THE PROJECTIVE GEOMETRY OF A GROUP

WOLFGANG BERTRAM

Abstract We show that the pair (P(Ω) Gras(Ω)) given by the power set P =P(Ω) and by the ldquoGrassmannianrdquo Gras(Ω) of all subgroups of an arbitrary groupΩ behaves very much like a projective space P(W ) and its dual projective spaceP(W lowast) of a vector space W More precisely we generalize several results from thecase of the abelian group Ω = (W +) (cf [BeKi10a]) to the case of a general groupΩ Most notably pairs of subgroups (a b) of Ω parametrize torsor and semitorsorstructures on P The role of associative algebras and -pairs from [BeKi10a] is nowtaken by analogs of near-rings

1 Introduction and statement of main results

11 Projective geometry of an abelian group Before explaining our generalresults let us briefly recall the classical case of projective geometry of a vectorspace W let X = P(W ) be the projective space of W and X prime = P(W lowast) be itsdual projective space (space of hyperplanes) The ldquodualityrdquo between X and X prime isencoded on two levels

(1) on the level of incidence structures an element x = [v] isin PW is incidentwith an element a = [α] isin PW lowast if ldquox lies on ardquo ie if α(v) = 0 otherwisewe say that they are remote or transversal and we then write xgta

(2) on the level of (linear) algebra the set agt of elements x isin X that aretransversal to a is in a completely natural way an affine space

In [BeKi10a] the second point has been generalized for any pair (a b) isin X prime times X primethe intersection Uab = agtcapbgt of two ldquoaffine cellsrdquo carries a natural torsor structureRecall that ldquotorsors are for groups what affine spaces are for vector spacesrdquo1

Definition 11 A semitorsor is a set G together with a map G3 rarr G (x y z) 7rarr(xyz) such that the following identity called the para-associative law holds

(T1) (xy(zuv)) = (x(uzy)v) = ((xyz)uv)

A torsor is a semitorsor in which moreover the following idempotent law holds

(T2) (xxy) = y = (yxx)

Fixing the middle element y in a torsor G we get a group law xz = (xyz) withneutral element y and every group is obtained in this way Similarly semitorsorsgive rise to semigroups but the converse is more complicated The torsors Ua = Uaa

1991 Mathematics Subject Classification 08A02 20N10 16W10 16Y30 20A05 51N30Key words and phrases torsor (heap groud principal homogeneous space) semitorsor rela-

tions projective space Grassmannian near-ring generalized lattice1The concept used here goes back to J Certaine [Cer43] there are several equivalent versions

known under various other names such as groud heap or principal homogeneous space1

2 WOLFGANG BERTRAM

are the underlying torsors of the affine space agt hence are abelian whereas fora 6= b the torsors Uab are in general non-commutative Thus in a sense the torsorsUab are deformations of the abelian torsor Ua More generally in [BeKi10a] all thisis done for a pair (X X prime) of dual Grassmannians not only for projective spaces

12 Projective geometry of a general group In the present work the com-mutative group (W+) will be replaced by an arbitrary group Ω (however in orderto keep formulas easily readable we will still use an additive notation for the grouplaw of Ω) It turns out then that the role of X is taken by the power set P(Ω)of all subsets of Ω and the one of X prime by the ldquoGrassmannianrdquo of all subgroups ofΩ We call Ω the ldquobackground grouprdquo or just the background Its subsets will bedenoted by small latin letters a b x y and if possible elements of such sets bycorresponding greek letters α isin a ξ isin x and so on As said above ldquoprojectivegeometry on Ωrdquo in our sense has two ingredients which we are going to explain now

(1) a (fairly weak) incidence (or rather non-incidence) structure and(2) a much more relevant algebraic structure consisting of a collection of torsors

and semi-torsors

Definition 12 The projective geometry of a group (Ω+) is its power set P =P(Ω) We say that a pair (x y) isin P2 is left transversal if every ω isin Ω admits aunique decomposition ω = ξ+ η with ξ isin x and η isin y We then write xgty We saythat the pair (x y) is right transversal if ygtx and we let

xgt = y isin P | xgty gtx = y isin P | ygtx

The ldquo(non-) incidence structurerdquo thus defined is not very interesting in its ownright however in combination with the algebraic torsor structures it becomes quitepowerful There are two in a certain sense ldquopurerdquo special cases to consider thegeneral case is a sort of mixture of these two let a b be two subgroups of Ω

(A) the transversal case agtb then agt cap gtb is a torsor of ldquobijection typerdquo(B) the singular case a = b it corresponds to ldquopointwise torsorsrdquo gta and bgt

Protoypes for (A) are torsors of the type G = Bij(X Y ) (set of bijections f X rarr Ybetween two setsX and Y ) with torsor structure (fgh) = fgminus1h and prototypesfor (B) are torsors of the type G = Map(XA) (set of maps from X to A) whereA is a torsor and X a set together with their natural ldquopointwise productrdquo

Case (A) arises if when agtb we identify Ω with the cartesian product a times bthen elements z isin gtb can be identified with ldquoleft graphsrdquo (αZα) | α isin a ofmaps Z a rarr b The map Z is bijective iff this graph belongs to agt ThereforeG = gtbcap agt carries a natural torsor structure of ldquobijection typerdquo as a torsor it isisomorphic to Bij(a b) It may be empty if it is non-empty then it is isomorphicto (the underlying torsor of) the group Bij(a a) Note that the structure of thisgroup does not involve the one of Ω indeed the group structure of Ω enters hereonly implicitly via the identification of Ω with atimes b

On the other hand in the ldquosingular caserdquo (B) the set gta is naturally identifiedwith the set of sections of the canonical projection Ωrarr Ωa and this set is a torsorof pointwise type modelled on the ldquopointwise grouprdquo of all maps f Ωa rarr a It

THE PROJECTIVE GEOMETRY OF A GROUP 3

is abelian iff so is a Indeed such torsors correspond precisely to the ldquoaffine cellsrdquofrom usual projective geometry

13 The ldquobalancedrdquo torsors Uab and the ldquounbalancedrdquo torsors Ua Follow-ing the ideas developed in [BeKi10a] we consider the general torsors Uab as a sortof ldquodeformation of the pure case (B) in direction of (A)rdquo However for treatingthe case of a non-commutative group Ω we need several important modifications ofthe setting from [BeKi10a] first of all the projective geometry P and its ldquodualrdquoGras(Ω) are no longer the same objects (the subset Gras(Ω) sub P is no longer sta-ble under the various torsor laws) next for a = b we have to distinguish betweenseveral versions of torsor laws those that one can deform easily called balancedand those which seem to be more rigid and which we call unbalanced The mostconceptual way to present this is via the following algebraic structure maps

Definition 13 The structure maps of (Ω+) are the maps Γ P5 rarr P andΣ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z ξ = ω + β η = α + ω + β ζ = α + ω

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

For a fixed pair (a b) isin P2 resp a fixed element b isin P we let

(xyz)ab = Γ(x a y b z) (xyz)b = Σ(b x y z)

The following is a main result of the present work (Theorems 51 and 72)

Theorem 14 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a torsor structure on the set gtb Wedenote this torsor by Ub It is isomorphic to the torsor of sections of theprojection Ωrarr Ωb

(2) The map (x y z) 7rarr (xyz)ab defines a torsor structure on the set agt cap gtbWe denote this torsor by Uab If moreover agtb then it is isomorphic tothe group of bijections of a

Just as the torsor structures considered in [BeKi10a] these torsor laws extend tosemitorsor laws onto the whole projective geometry in the same way as the grouplaw of Bij(X) for a set X extends to a semigroup structure on Map(XX)

Theorem 15 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a semitorsor structure on P(2) The map (x y z) 7rarr (xyz)ab defines a semitorsor structure on P

We call the torsors Uab balanced and the torsors Ub unbalanced If Ω is non-abelianthe torsor Ubb is different from Ub ndash the latter are in general not members of atwo-parameter family This is due to the fact that the system of three equationsdefining Γ called the structure equations

(11)

ζ = α + ωη = α + ω + βξ = ω + β

4 WOLFGANG BERTRAM

is of a more symmetric nature than the one defining Σ We come back to this itembelow (Subsection 16)

14 Affine picture As usual in projective geometry a ldquoprojective statementrdquomay be translated into an ldquoaffine statementrdquo by choosing some ldquoaffinizationrdquo of P Thus one can rewrite the torsor law of Uab by an ldquoaffine formulardquo (Theorem 81)Here is a quite instructive special case consider two arbitrary groups (V+) and(W+) and fix a group homomorphism A W rarr V Let G = Map(VW ) be theset of all maps from V to W Then it is an easy exercise to show that

(12) X middotA Y = Y +X (idV + A Y )

defines an associative product on G (where + is the pointwise ldquosumrdquo of maps) withneutral element the ldquozero maprdquo 0 and which gives a group law on the set

(13) GA = X isin G | idV + A X is bijective The parameter A is a sort of ldquodeformation parameterrdquo if A = 0 we get pointwiseaddition if A is an isomorphism then GA is in fact isomorphic to the usual groupBij(V ) If V = W = Rn then one may do the same construction using continuousor smooth maps and thus gets a deformation of the abelian (additive) group Gof vector fields to the highly non-commutative group GA of diffeomorphisms of RnIf VW are linear spaces and XZ linear maps then (12) gives us back the lawX +XAY + Y considered in [BeKi10a]

15 Distributive laws and near-rings The reason why we are also interestedin the unbalanced torsors is that they are natural (being spaces of sections of aprincipal bundle over a homogeneous space) and interact nicely with the balancedstructures there is a base-point free version of a right distributive law which makesthe whole object a torsor-analog of a near-ring or a ldquogeneralized ringrdquo (cf [Pi77])

Definition 16 A (right) near-ring is a set N together with two binary operationsdenoted by + and middot such that

(1) (N+) is a group (not necessarily abelian)(2) (N middot) is a semigroup(3) we have the right distributive law (x+ y) middot z = x middot z + y middot z

A typical example is the set N of self-maps of a group (G+) where middot is compositionand + pointwise ldquoadditionrdquo In our context Γ takes the role of the product middot andΣ takes the one of the ldquoadditionrdquo + (cf Theorem 83)

Theorem 17 Let (a b) be a pair of subgroups of Ω Then we have the followingleft distributive law relating the unbalanced and the balanced torsor structures forall x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

Essentially this means that gtb looks like a ternary version of a near-ring whoseldquomultiplicativerdquo structure now depends on an additional parameter y As usual fornear-rings there is just one distributive law the other distributive law does nothold Compare with (12) which is ldquoaffinerdquo in X but not in Y

THE PROJECTIVE GEOMETRY OF A GROUP 5

16 Symmetry The preceding theorem makes it obvious that the definition of Γinvolves some arbitrary choices there is no reason why left distributivity shouldbe preferred to right distributivity Indeed if instead of Γ we looked at the mapΓ obtained by using everywhere the opposite group law of (Ω+) then we wouldget ldquorightrdquo instead of ldquoleft distributivityrdquo Thus Γ and Γ are in a certain senseldquoequivalentrdquo In the same way there is no reason to prefer the groups a or b totheir opposite groups in the structure equations we might replace α or β by theirnegatives without changing the whole theory Thus we are led to consider severalversions of the fundamental equations as ldquoessentially equivalentrdquo We investigatethis item in Section 9 there are in fact 24 signed (ie essentially equivalent) versionsof the structure equations on which a certain subgroup V of the permutation groupS6 permuting the six variables of (11) acts simply transitively (Theorem 96) Wecall V the Big Klein Group2 since it plays exactly the same role for the structureequations as the usual Klein Group V does for a single torsor structure (cf Lemma92) The group V is isomorphic to S4 sitting inside S6 as the subgroup preservingthe partition of six letters in three subsets ξ ζ α β and η ω (Lemma 94)Permutations from V leave invariant the general shape of (11) and introduce justcertain sign changes for some of the variables If we are willing to neglect such signchanges ndash like for instance in the ldquoprojectiverdquo framework of [BeKi10a] where onecan rescale by any invertible scalar ndash then the whole theory becomes invariant underthese permutations3 This explains partially why the associative geometries from[BeKi10a] (and their Jordan theoretic analogs) have such a high degree of symmetry(cf the ldquosymmetryrdquo and ldquoduality principlesrdquo for Jordan theory [Lo75]) If we agreeto neglect sign changes only with respect to α and β (which is reasonable since inTheorem 14 we assume that a and b are subgroups hence α isin a iff minusα isin a andsame for b) then we obtain as invariance group again a usual Klein Group V andthe orbit under V has 244 = 6 elements This in turn is completely analogous tothe behavior of the classical cross-ratio under S4 which is invariant under V andtakes generically 6 different values under permutations

17 Further topics Because of its generality the approach presented in this workis likely to interact with many other mathematical theories In the last section wemention some questions arising naturally in this context and we refer to Section 4of [BeKi10a] for some more remarks of a similar kind

Notation Throughout this paper Ω is a (possibly non-commutative) group calledthe background whose group law will be written additively Its neutral element willbe denoted by o We denote by P = P(Ω) its power set by Po = (Ω) the set ofsubsets of Ω containing the neutral element o and by Gras(Ω) the Grassmannianof Ω (the set of all subgroups of Ω) Transversality as defined in Definition 12above is denoted by xgty

2translated from the German Grosse Klein Gruppe3A side remark the author cannot help feeling being reminded by this situation to CPT-

invariance in physics where a very similar phenomenon occurs

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

2 WOLFGANG BERTRAM

are the underlying torsors of the affine space agt hence are abelian whereas fora 6= b the torsors Uab are in general non-commutative Thus in a sense the torsorsUab are deformations of the abelian torsor Ua More generally in [BeKi10a] all thisis done for a pair (X X prime) of dual Grassmannians not only for projective spaces

12 Projective geometry of a general group In the present work the com-mutative group (W+) will be replaced by an arbitrary group Ω (however in orderto keep formulas easily readable we will still use an additive notation for the grouplaw of Ω) It turns out then that the role of X is taken by the power set P(Ω)of all subsets of Ω and the one of X prime by the ldquoGrassmannianrdquo of all subgroups ofΩ We call Ω the ldquobackground grouprdquo or just the background Its subsets will bedenoted by small latin letters a b x y and if possible elements of such sets bycorresponding greek letters α isin a ξ isin x and so on As said above ldquoprojectivegeometry on Ωrdquo in our sense has two ingredients which we are going to explain now

(1) a (fairly weak) incidence (or rather non-incidence) structure and(2) a much more relevant algebraic structure consisting of a collection of torsors

and semi-torsors

Definition 12 The projective geometry of a group (Ω+) is its power set P =P(Ω) We say that a pair (x y) isin P2 is left transversal if every ω isin Ω admits aunique decomposition ω = ξ+ η with ξ isin x and η isin y We then write xgty We saythat the pair (x y) is right transversal if ygtx and we let

xgt = y isin P | xgty gtx = y isin P | ygtx

The ldquo(non-) incidence structurerdquo thus defined is not very interesting in its ownright however in combination with the algebraic torsor structures it becomes quitepowerful There are two in a certain sense ldquopurerdquo special cases to consider thegeneral case is a sort of mixture of these two let a b be two subgroups of Ω

(A) the transversal case agtb then agt cap gtb is a torsor of ldquobijection typerdquo(B) the singular case a = b it corresponds to ldquopointwise torsorsrdquo gta and bgt

Protoypes for (A) are torsors of the type G = Bij(X Y ) (set of bijections f X rarr Ybetween two setsX and Y ) with torsor structure (fgh) = fgminus1h and prototypesfor (B) are torsors of the type G = Map(XA) (set of maps from X to A) whereA is a torsor and X a set together with their natural ldquopointwise productrdquo

Case (A) arises if when agtb we identify Ω with the cartesian product a times bthen elements z isin gtb can be identified with ldquoleft graphsrdquo (αZα) | α isin a ofmaps Z a rarr b The map Z is bijective iff this graph belongs to agt ThereforeG = gtbcap agt carries a natural torsor structure of ldquobijection typerdquo as a torsor it isisomorphic to Bij(a b) It may be empty if it is non-empty then it is isomorphicto (the underlying torsor of) the group Bij(a a) Note that the structure of thisgroup does not involve the one of Ω indeed the group structure of Ω enters hereonly implicitly via the identification of Ω with atimes b

On the other hand in the ldquosingular caserdquo (B) the set gta is naturally identifiedwith the set of sections of the canonical projection Ωrarr Ωa and this set is a torsorof pointwise type modelled on the ldquopointwise grouprdquo of all maps f Ωa rarr a It

THE PROJECTIVE GEOMETRY OF A GROUP 3

is abelian iff so is a Indeed such torsors correspond precisely to the ldquoaffine cellsrdquofrom usual projective geometry

13 The ldquobalancedrdquo torsors Uab and the ldquounbalancedrdquo torsors Ua Follow-ing the ideas developed in [BeKi10a] we consider the general torsors Uab as a sortof ldquodeformation of the pure case (B) in direction of (A)rdquo However for treatingthe case of a non-commutative group Ω we need several important modifications ofthe setting from [BeKi10a] first of all the projective geometry P and its ldquodualrdquoGras(Ω) are no longer the same objects (the subset Gras(Ω) sub P is no longer sta-ble under the various torsor laws) next for a = b we have to distinguish betweenseveral versions of torsor laws those that one can deform easily called balancedand those which seem to be more rigid and which we call unbalanced The mostconceptual way to present this is via the following algebraic structure maps

Definition 13 The structure maps of (Ω+) are the maps Γ P5 rarr P andΣ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z ξ = ω + β η = α + ω + β ζ = α + ω

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

For a fixed pair (a b) isin P2 resp a fixed element b isin P we let

(xyz)ab = Γ(x a y b z) (xyz)b = Σ(b x y z)

The following is a main result of the present work (Theorems 51 and 72)

Theorem 14 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a torsor structure on the set gtb Wedenote this torsor by Ub It is isomorphic to the torsor of sections of theprojection Ωrarr Ωb

(2) The map (x y z) 7rarr (xyz)ab defines a torsor structure on the set agt cap gtbWe denote this torsor by Uab If moreover agtb then it is isomorphic tothe group of bijections of a

Just as the torsor structures considered in [BeKi10a] these torsor laws extend tosemitorsor laws onto the whole projective geometry in the same way as the grouplaw of Bij(X) for a set X extends to a semigroup structure on Map(XX)

Theorem 15 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a semitorsor structure on P(2) The map (x y z) 7rarr (xyz)ab defines a semitorsor structure on P

We call the torsors Uab balanced and the torsors Ub unbalanced If Ω is non-abelianthe torsor Ubb is different from Ub ndash the latter are in general not members of atwo-parameter family This is due to the fact that the system of three equationsdefining Γ called the structure equations

(11)

ζ = α + ωη = α + ω + βξ = ω + β

4 WOLFGANG BERTRAM

is of a more symmetric nature than the one defining Σ We come back to this itembelow (Subsection 16)

14 Affine picture As usual in projective geometry a ldquoprojective statementrdquomay be translated into an ldquoaffine statementrdquo by choosing some ldquoaffinizationrdquo of P Thus one can rewrite the torsor law of Uab by an ldquoaffine formulardquo (Theorem 81)Here is a quite instructive special case consider two arbitrary groups (V+) and(W+) and fix a group homomorphism A W rarr V Let G = Map(VW ) be theset of all maps from V to W Then it is an easy exercise to show that

(12) X middotA Y = Y +X (idV + A Y )

defines an associative product on G (where + is the pointwise ldquosumrdquo of maps) withneutral element the ldquozero maprdquo 0 and which gives a group law on the set

(13) GA = X isin G | idV + A X is bijective The parameter A is a sort of ldquodeformation parameterrdquo if A = 0 we get pointwiseaddition if A is an isomorphism then GA is in fact isomorphic to the usual groupBij(V ) If V = W = Rn then one may do the same construction using continuousor smooth maps and thus gets a deformation of the abelian (additive) group Gof vector fields to the highly non-commutative group GA of diffeomorphisms of RnIf VW are linear spaces and XZ linear maps then (12) gives us back the lawX +XAY + Y considered in [BeKi10a]

15 Distributive laws and near-rings The reason why we are also interestedin the unbalanced torsors is that they are natural (being spaces of sections of aprincipal bundle over a homogeneous space) and interact nicely with the balancedstructures there is a base-point free version of a right distributive law which makesthe whole object a torsor-analog of a near-ring or a ldquogeneralized ringrdquo (cf [Pi77])

Definition 16 A (right) near-ring is a set N together with two binary operationsdenoted by + and middot such that

(1) (N+) is a group (not necessarily abelian)(2) (N middot) is a semigroup(3) we have the right distributive law (x+ y) middot z = x middot z + y middot z

A typical example is the set N of self-maps of a group (G+) where middot is compositionand + pointwise ldquoadditionrdquo In our context Γ takes the role of the product middot andΣ takes the one of the ldquoadditionrdquo + (cf Theorem 83)

Theorem 17 Let (a b) be a pair of subgroups of Ω Then we have the followingleft distributive law relating the unbalanced and the balanced torsor structures forall x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

Essentially this means that gtb looks like a ternary version of a near-ring whoseldquomultiplicativerdquo structure now depends on an additional parameter y As usual fornear-rings there is just one distributive law the other distributive law does nothold Compare with (12) which is ldquoaffinerdquo in X but not in Y

THE PROJECTIVE GEOMETRY OF A GROUP 5

16 Symmetry The preceding theorem makes it obvious that the definition of Γinvolves some arbitrary choices there is no reason why left distributivity shouldbe preferred to right distributivity Indeed if instead of Γ we looked at the mapΓ obtained by using everywhere the opposite group law of (Ω+) then we wouldget ldquorightrdquo instead of ldquoleft distributivityrdquo Thus Γ and Γ are in a certain senseldquoequivalentrdquo In the same way there is no reason to prefer the groups a or b totheir opposite groups in the structure equations we might replace α or β by theirnegatives without changing the whole theory Thus we are led to consider severalversions of the fundamental equations as ldquoessentially equivalentrdquo We investigatethis item in Section 9 there are in fact 24 signed (ie essentially equivalent) versionsof the structure equations on which a certain subgroup V of the permutation groupS6 permuting the six variables of (11) acts simply transitively (Theorem 96) Wecall V the Big Klein Group2 since it plays exactly the same role for the structureequations as the usual Klein Group V does for a single torsor structure (cf Lemma92) The group V is isomorphic to S4 sitting inside S6 as the subgroup preservingthe partition of six letters in three subsets ξ ζ α β and η ω (Lemma 94)Permutations from V leave invariant the general shape of (11) and introduce justcertain sign changes for some of the variables If we are willing to neglect such signchanges ndash like for instance in the ldquoprojectiverdquo framework of [BeKi10a] where onecan rescale by any invertible scalar ndash then the whole theory becomes invariant underthese permutations3 This explains partially why the associative geometries from[BeKi10a] (and their Jordan theoretic analogs) have such a high degree of symmetry(cf the ldquosymmetryrdquo and ldquoduality principlesrdquo for Jordan theory [Lo75]) If we agreeto neglect sign changes only with respect to α and β (which is reasonable since inTheorem 14 we assume that a and b are subgroups hence α isin a iff minusα isin a andsame for b) then we obtain as invariance group again a usual Klein Group V andthe orbit under V has 244 = 6 elements This in turn is completely analogous tothe behavior of the classical cross-ratio under S4 which is invariant under V andtakes generically 6 different values under permutations

17 Further topics Because of its generality the approach presented in this workis likely to interact with many other mathematical theories In the last section wemention some questions arising naturally in this context and we refer to Section 4of [BeKi10a] for some more remarks of a similar kind

Notation Throughout this paper Ω is a (possibly non-commutative) group calledthe background whose group law will be written additively Its neutral element willbe denoted by o We denote by P = P(Ω) its power set by Po = (Ω) the set ofsubsets of Ω containing the neutral element o and by Gras(Ω) the Grassmannianof Ω (the set of all subgroups of Ω) Transversality as defined in Definition 12above is denoted by xgty

2translated from the German Grosse Klein Gruppe3A side remark the author cannot help feeling being reminded by this situation to CPT-

invariance in physics where a very similar phenomenon occurs

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 3

is abelian iff so is a Indeed such torsors correspond precisely to the ldquoaffine cellsrdquofrom usual projective geometry

13 The ldquobalancedrdquo torsors Uab and the ldquounbalancedrdquo torsors Ua Follow-ing the ideas developed in [BeKi10a] we consider the general torsors Uab as a sortof ldquodeformation of the pure case (B) in direction of (A)rdquo However for treatingthe case of a non-commutative group Ω we need several important modifications ofthe setting from [BeKi10a] first of all the projective geometry P and its ldquodualrdquoGras(Ω) are no longer the same objects (the subset Gras(Ω) sub P is no longer sta-ble under the various torsor laws) next for a = b we have to distinguish betweenseveral versions of torsor laws those that one can deform easily called balancedand those which seem to be more rigid and which we call unbalanced The mostconceptual way to present this is via the following algebraic structure maps

Definition 13 The structure maps of (Ω+) are the maps Γ P5 rarr P andΣ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z ξ = ω + β η = α + ω + β ζ = α + ω

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

For a fixed pair (a b) isin P2 resp a fixed element b isin P we let

(xyz)ab = Γ(x a y b z) (xyz)b = Σ(b x y z)

The following is a main result of the present work (Theorems 51 and 72)

Theorem 14 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a torsor structure on the set gtb Wedenote this torsor by Ub It is isomorphic to the torsor of sections of theprojection Ωrarr Ωb

(2) The map (x y z) 7rarr (xyz)ab defines a torsor structure on the set agt cap gtbWe denote this torsor by Uab If moreover agtb then it is isomorphic tothe group of bijections of a

Just as the torsor structures considered in [BeKi10a] these torsor laws extend tosemitorsor laws onto the whole projective geometry in the same way as the grouplaw of Bij(X) for a set X extends to a semigroup structure on Map(XX)

Theorem 15 Assume (a b) is a pair of subgroups of Ω Then the following holds

(1) The map (x y z) 7rarr (xyz)b defines a semitorsor structure on P(2) The map (x y z) 7rarr (xyz)ab defines a semitorsor structure on P

We call the torsors Uab balanced and the torsors Ub unbalanced If Ω is non-abelianthe torsor Ubb is different from Ub ndash the latter are in general not members of atwo-parameter family This is due to the fact that the system of three equationsdefining Γ called the structure equations

(11)

ζ = α + ωη = α + ω + βξ = ω + β

4 WOLFGANG BERTRAM

is of a more symmetric nature than the one defining Σ We come back to this itembelow (Subsection 16)

14 Affine picture As usual in projective geometry a ldquoprojective statementrdquomay be translated into an ldquoaffine statementrdquo by choosing some ldquoaffinizationrdquo of P Thus one can rewrite the torsor law of Uab by an ldquoaffine formulardquo (Theorem 81)Here is a quite instructive special case consider two arbitrary groups (V+) and(W+) and fix a group homomorphism A W rarr V Let G = Map(VW ) be theset of all maps from V to W Then it is an easy exercise to show that

(12) X middotA Y = Y +X (idV + A Y )

defines an associative product on G (where + is the pointwise ldquosumrdquo of maps) withneutral element the ldquozero maprdquo 0 and which gives a group law on the set

(13) GA = X isin G | idV + A X is bijective The parameter A is a sort of ldquodeformation parameterrdquo if A = 0 we get pointwiseaddition if A is an isomorphism then GA is in fact isomorphic to the usual groupBij(V ) If V = W = Rn then one may do the same construction using continuousor smooth maps and thus gets a deformation of the abelian (additive) group Gof vector fields to the highly non-commutative group GA of diffeomorphisms of RnIf VW are linear spaces and XZ linear maps then (12) gives us back the lawX +XAY + Y considered in [BeKi10a]

15 Distributive laws and near-rings The reason why we are also interestedin the unbalanced torsors is that they are natural (being spaces of sections of aprincipal bundle over a homogeneous space) and interact nicely with the balancedstructures there is a base-point free version of a right distributive law which makesthe whole object a torsor-analog of a near-ring or a ldquogeneralized ringrdquo (cf [Pi77])

Definition 16 A (right) near-ring is a set N together with two binary operationsdenoted by + and middot such that

(1) (N+) is a group (not necessarily abelian)(2) (N middot) is a semigroup(3) we have the right distributive law (x+ y) middot z = x middot z + y middot z

A typical example is the set N of self-maps of a group (G+) where middot is compositionand + pointwise ldquoadditionrdquo In our context Γ takes the role of the product middot andΣ takes the one of the ldquoadditionrdquo + (cf Theorem 83)

Theorem 17 Let (a b) be a pair of subgroups of Ω Then we have the followingleft distributive law relating the unbalanced and the balanced torsor structures forall x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

Essentially this means that gtb looks like a ternary version of a near-ring whoseldquomultiplicativerdquo structure now depends on an additional parameter y As usual fornear-rings there is just one distributive law the other distributive law does nothold Compare with (12) which is ldquoaffinerdquo in X but not in Y

THE PROJECTIVE GEOMETRY OF A GROUP 5

16 Symmetry The preceding theorem makes it obvious that the definition of Γinvolves some arbitrary choices there is no reason why left distributivity shouldbe preferred to right distributivity Indeed if instead of Γ we looked at the mapΓ obtained by using everywhere the opposite group law of (Ω+) then we wouldget ldquorightrdquo instead of ldquoleft distributivityrdquo Thus Γ and Γ are in a certain senseldquoequivalentrdquo In the same way there is no reason to prefer the groups a or b totheir opposite groups in the structure equations we might replace α or β by theirnegatives without changing the whole theory Thus we are led to consider severalversions of the fundamental equations as ldquoessentially equivalentrdquo We investigatethis item in Section 9 there are in fact 24 signed (ie essentially equivalent) versionsof the structure equations on which a certain subgroup V of the permutation groupS6 permuting the six variables of (11) acts simply transitively (Theorem 96) Wecall V the Big Klein Group2 since it plays exactly the same role for the structureequations as the usual Klein Group V does for a single torsor structure (cf Lemma92) The group V is isomorphic to S4 sitting inside S6 as the subgroup preservingthe partition of six letters in three subsets ξ ζ α β and η ω (Lemma 94)Permutations from V leave invariant the general shape of (11) and introduce justcertain sign changes for some of the variables If we are willing to neglect such signchanges ndash like for instance in the ldquoprojectiverdquo framework of [BeKi10a] where onecan rescale by any invertible scalar ndash then the whole theory becomes invariant underthese permutations3 This explains partially why the associative geometries from[BeKi10a] (and their Jordan theoretic analogs) have such a high degree of symmetry(cf the ldquosymmetryrdquo and ldquoduality principlesrdquo for Jordan theory [Lo75]) If we agreeto neglect sign changes only with respect to α and β (which is reasonable since inTheorem 14 we assume that a and b are subgroups hence α isin a iff minusα isin a andsame for b) then we obtain as invariance group again a usual Klein Group V andthe orbit under V has 244 = 6 elements This in turn is completely analogous tothe behavior of the classical cross-ratio under S4 which is invariant under V andtakes generically 6 different values under permutations

17 Further topics Because of its generality the approach presented in this workis likely to interact with many other mathematical theories In the last section wemention some questions arising naturally in this context and we refer to Section 4of [BeKi10a] for some more remarks of a similar kind

Notation Throughout this paper Ω is a (possibly non-commutative) group calledthe background whose group law will be written additively Its neutral element willbe denoted by o We denote by P = P(Ω) its power set by Po = (Ω) the set ofsubsets of Ω containing the neutral element o and by Gras(Ω) the Grassmannianof Ω (the set of all subgroups of Ω) Transversality as defined in Definition 12above is denoted by xgty

2translated from the German Grosse Klein Gruppe3A side remark the author cannot help feeling being reminded by this situation to CPT-

invariance in physics where a very similar phenomenon occurs

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

4 WOLFGANG BERTRAM

is of a more symmetric nature than the one defining Σ We come back to this itembelow (Subsection 16)

14 Affine picture As usual in projective geometry a ldquoprojective statementrdquomay be translated into an ldquoaffine statementrdquo by choosing some ldquoaffinizationrdquo of P Thus one can rewrite the torsor law of Uab by an ldquoaffine formulardquo (Theorem 81)Here is a quite instructive special case consider two arbitrary groups (V+) and(W+) and fix a group homomorphism A W rarr V Let G = Map(VW ) be theset of all maps from V to W Then it is an easy exercise to show that

(12) X middotA Y = Y +X (idV + A Y )

defines an associative product on G (where + is the pointwise ldquosumrdquo of maps) withneutral element the ldquozero maprdquo 0 and which gives a group law on the set

(13) GA = X isin G | idV + A X is bijective The parameter A is a sort of ldquodeformation parameterrdquo if A = 0 we get pointwiseaddition if A is an isomorphism then GA is in fact isomorphic to the usual groupBij(V ) If V = W = Rn then one may do the same construction using continuousor smooth maps and thus gets a deformation of the abelian (additive) group Gof vector fields to the highly non-commutative group GA of diffeomorphisms of RnIf VW are linear spaces and XZ linear maps then (12) gives us back the lawX +XAY + Y considered in [BeKi10a]

15 Distributive laws and near-rings The reason why we are also interestedin the unbalanced torsors is that they are natural (being spaces of sections of aprincipal bundle over a homogeneous space) and interact nicely with the balancedstructures there is a base-point free version of a right distributive law which makesthe whole object a torsor-analog of a near-ring or a ldquogeneralized ringrdquo (cf [Pi77])

Definition 16 A (right) near-ring is a set N together with two binary operationsdenoted by + and middot such that

(1) (N+) is a group (not necessarily abelian)(2) (N middot) is a semigroup(3) we have the right distributive law (x+ y) middot z = x middot z + y middot z

A typical example is the set N of self-maps of a group (G+) where middot is compositionand + pointwise ldquoadditionrdquo In our context Γ takes the role of the product middot andΣ takes the one of the ldquoadditionrdquo + (cf Theorem 83)

Theorem 17 Let (a b) be a pair of subgroups of Ω Then we have the followingleft distributive law relating the unbalanced and the balanced torsor structures forall x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

Essentially this means that gtb looks like a ternary version of a near-ring whoseldquomultiplicativerdquo structure now depends on an additional parameter y As usual fornear-rings there is just one distributive law the other distributive law does nothold Compare with (12) which is ldquoaffinerdquo in X but not in Y

THE PROJECTIVE GEOMETRY OF A GROUP 5

16 Symmetry The preceding theorem makes it obvious that the definition of Γinvolves some arbitrary choices there is no reason why left distributivity shouldbe preferred to right distributivity Indeed if instead of Γ we looked at the mapΓ obtained by using everywhere the opposite group law of (Ω+) then we wouldget ldquorightrdquo instead of ldquoleft distributivityrdquo Thus Γ and Γ are in a certain senseldquoequivalentrdquo In the same way there is no reason to prefer the groups a or b totheir opposite groups in the structure equations we might replace α or β by theirnegatives without changing the whole theory Thus we are led to consider severalversions of the fundamental equations as ldquoessentially equivalentrdquo We investigatethis item in Section 9 there are in fact 24 signed (ie essentially equivalent) versionsof the structure equations on which a certain subgroup V of the permutation groupS6 permuting the six variables of (11) acts simply transitively (Theorem 96) Wecall V the Big Klein Group2 since it plays exactly the same role for the structureequations as the usual Klein Group V does for a single torsor structure (cf Lemma92) The group V is isomorphic to S4 sitting inside S6 as the subgroup preservingthe partition of six letters in three subsets ξ ζ α β and η ω (Lemma 94)Permutations from V leave invariant the general shape of (11) and introduce justcertain sign changes for some of the variables If we are willing to neglect such signchanges ndash like for instance in the ldquoprojectiverdquo framework of [BeKi10a] where onecan rescale by any invertible scalar ndash then the whole theory becomes invariant underthese permutations3 This explains partially why the associative geometries from[BeKi10a] (and their Jordan theoretic analogs) have such a high degree of symmetry(cf the ldquosymmetryrdquo and ldquoduality principlesrdquo for Jordan theory [Lo75]) If we agreeto neglect sign changes only with respect to α and β (which is reasonable since inTheorem 14 we assume that a and b are subgroups hence α isin a iff minusα isin a andsame for b) then we obtain as invariance group again a usual Klein Group V andthe orbit under V has 244 = 6 elements This in turn is completely analogous tothe behavior of the classical cross-ratio under S4 which is invariant under V andtakes generically 6 different values under permutations

17 Further topics Because of its generality the approach presented in this workis likely to interact with many other mathematical theories In the last section wemention some questions arising naturally in this context and we refer to Section 4of [BeKi10a] for some more remarks of a similar kind

Notation Throughout this paper Ω is a (possibly non-commutative) group calledthe background whose group law will be written additively Its neutral element willbe denoted by o We denote by P = P(Ω) its power set by Po = (Ω) the set ofsubsets of Ω containing the neutral element o and by Gras(Ω) the Grassmannianof Ω (the set of all subgroups of Ω) Transversality as defined in Definition 12above is denoted by xgty

2translated from the German Grosse Klein Gruppe3A side remark the author cannot help feeling being reminded by this situation to CPT-

invariance in physics where a very similar phenomenon occurs

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 5

16 Symmetry The preceding theorem makes it obvious that the definition of Γinvolves some arbitrary choices there is no reason why left distributivity shouldbe preferred to right distributivity Indeed if instead of Γ we looked at the mapΓ obtained by using everywhere the opposite group law of (Ω+) then we wouldget ldquorightrdquo instead of ldquoleft distributivityrdquo Thus Γ and Γ are in a certain senseldquoequivalentrdquo In the same way there is no reason to prefer the groups a or b totheir opposite groups in the structure equations we might replace α or β by theirnegatives without changing the whole theory Thus we are led to consider severalversions of the fundamental equations as ldquoessentially equivalentrdquo We investigatethis item in Section 9 there are in fact 24 signed (ie essentially equivalent) versionsof the structure equations on which a certain subgroup V of the permutation groupS6 permuting the six variables of (11) acts simply transitively (Theorem 96) Wecall V the Big Klein Group2 since it plays exactly the same role for the structureequations as the usual Klein Group V does for a single torsor structure (cf Lemma92) The group V is isomorphic to S4 sitting inside S6 as the subgroup preservingthe partition of six letters in three subsets ξ ζ α β and η ω (Lemma 94)Permutations from V leave invariant the general shape of (11) and introduce justcertain sign changes for some of the variables If we are willing to neglect such signchanges ndash like for instance in the ldquoprojectiverdquo framework of [BeKi10a] where onecan rescale by any invertible scalar ndash then the whole theory becomes invariant underthese permutations3 This explains partially why the associative geometries from[BeKi10a] (and their Jordan theoretic analogs) have such a high degree of symmetry(cf the ldquosymmetryrdquo and ldquoduality principlesrdquo for Jordan theory [Lo75]) If we agreeto neglect sign changes only with respect to α and β (which is reasonable since inTheorem 14 we assume that a and b are subgroups hence α isin a iff minusα isin a andsame for b) then we obtain as invariance group again a usual Klein Group V andthe orbit under V has 244 = 6 elements This in turn is completely analogous tothe behavior of the classical cross-ratio under S4 which is invariant under V andtakes generically 6 different values under permutations

17 Further topics Because of its generality the approach presented in this workis likely to interact with many other mathematical theories In the last section wemention some questions arising naturally in this context and we refer to Section 4of [BeKi10a] for some more remarks of a similar kind

Notation Throughout this paper Ω is a (possibly non-commutative) group calledthe background whose group law will be written additively Its neutral element willbe denoted by o We denote by P = P(Ω) its power set by Po = (Ω) the set ofsubsets of Ω containing the neutral element o and by Gras(Ω) the Grassmannianof Ω (the set of all subgroups of Ω) Transversality as defined in Definition 12above is denoted by xgty

2translated from the German Grosse Klein Gruppe3A side remark the author cannot help feeling being reminded by this situation to CPT-

invariance in physics where a very similar phenomenon occurs

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

6 WOLFGANG BERTRAM

2 Structure maps and structure space

Definition 21 The structure maps of a group (Ω+) are the maps Γ P5 rarr PΓ P5 rarr P Σ P4 rarr P and Σ P4 rarr P defined for x a y b z isin P by

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin b existζ isin z η = α + ω + β ζ = α + ω ξ = ω + β

(21)

Γ(x a y b z) =

ω isin Ω

∣∣∣ existξ isin xexistα isin aexistη isin yexistβ isin bexistζ isin z η = β + ω + α ζ = ω + α ξ = β + ω

(22)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = ω + β η = ω + βprime + β ζ = ω + βprime

(23)

Σ(a x y z) =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ξ = β + ω η = β + βprime + ω ζ = βprime + ω

(24)

It is obvious that the set Po(Ω) of subsets containing o is stable under each ofthese maps and the corresponding restrictions of the four maps will also be calledstructure maps

Note that Γ resp Σ is obtained from Γ resp Σ simply by replacing the group lawin Ω by the opposite group law Hence if Ω is abelian we have Γ = Γ and Σ = ΣMoreover if Ω is abelian we obviously have

(25) Γ(x a y a z) = Σ(a x y z) = Σ(a z y x)

For general Ω the defining equations immediately imply the symmetry relation

(26) Γ(z b y a x) = Γ(x a y b z)

Definition 22 The system (11) of three equations for six variables in Ω is calledthe structure equations We say that another system of equations is equivalent tothe structure equations if it has the same set of solutions called the structure spaceof the group (Ω+)

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = α + ω + β ζ = α + ω ξ = ω + β

By definition the opposite structure space is the structure space of Ωopp

Γ =

(ξ ζα β η ω) isin Ω6∣∣∣ η = β + ω + α ζ = ω + α ξ = β + ω

The sets Σ sub Ω5 and Σ sub Ω5 can be defined similarly

Lemma 23 The following systems are all equivalent to the structure equations

(27)

α = η minus ξω = ξ minus η + ζβ = minusζ + η

(28)

η = α + ω + βη = α + ξη = ζ + β

η = ζ minus ω + ξη = α + ξη = ζ + β

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 7

(29)

ω = ξ minus η + ζω = ξ minus βω = minusα + ζ

ω = minusα + η minus βω = ξ minus βω = minusα + ζ

(210)

α = ζ + β minus ξα = η minus ξα = ζ minus ω

α = η minus β minus ωα = η minus ξα = ζ minus ω

(211)

β = minusζ + α + ξβ = minusω + ξβ = minusζ + η

β = minusω minus α + ηβ = minusω + ξβ = minusζ + η

(212)

ξ = minusα + ζ + βξ = minusα + ηξ = ω + β

ξ = ω minus ζ + ηξ = minusα + ηξ = ω + β

(213)

ζ = η minus ξ + ωζ = η minus βζ = α + ω

ζ = α + ξ minus βζ = η minus βζ = α + ω

(214)

η = α + ξβ = minusω + ξζ = α + ω

α = η minus ξζ = η minus βω = ξ minus β

α = ζ minus ωξ = ω + βη = ζ + β

The proof is by completely elementary computations Obviously the structurespace has certain symmetry properties with respect to permutations This will beinvestigated in more detail in Section 9 Note also that if Ω is abelian the structureequations are Z-linear and hence can be written in matrix form1 1 0

1 1 10 1 1

αωβ

=

ζηξ

Equations (27) then correspond to the inverse of this matrix

3 The semitorsor laws

Theorem 31 Assume that a and b are two subgroups of a group (Ω+) Then thepower set P and its subset Po become semitorsors under the ternary compositions

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)ab = Γ(x a y b z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

P3 rarr P (x y z) 7rarr (xyz)b = Σ(b x y z)

We denote these semitorsors by Pab Pab Pb Pb respectively

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

8 WOLFGANG BERTRAM

Proof We prove for x y z isin P(Ω) the identity

Γ(x a u bΓ(y a v b z)

)= Γ

(x aΓ(v a y b u) b z

)= Γ

(Γ(x a u b y) a v b z

)

ie the semitorsor law for (xyz)ab For the proof note that the definition ofΓ(x a y b z) can be written somewhat shorter as follows

Γ(x a y b z) =

ω isin Ω

∣∣∣ existα isin aexistβ isin b α + ω + β isin y α + ω isin z ω + β isin x

(31)

and similarly for Γ We refer to this description as (a b)-description Using thiswe have on the one hand

Γ(x a u bΓ(y a v b z)

)=

=

ω isin Ω

∣∣∣ existα isin a existβ isin b α + ω isin Γ(y a v b z) α + ω + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin aexistβ isin b existαprime isin aexistβprime isin b α + ω + β isin u ω + β isin x αprime + α + ω isin z

αprime + α + ω + βprime isin v α + ω + βprime isin y

On the other hand

Γ(x aΓ(v a y b u) b z

)=

=

ω isin Ω

∣∣∣ existαprimeprime isin aexistβprimeprime isin b αprimeprime + ω isin z αprimeprime + ω + βprimeprime isin Γ(v a y b u) ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin aexistβprimeprime isin bexistαprimeprimeprime isin aexistβprimeprimeprime isin b

αprimeprime + ω isin z ω + βprimeprime isin x αprimeprimeprime + αprimeprime + ω + βprimeprime isin uαprimeprime + ω + βprimeprime + βprimeprimeprime isin v αprimeprimeprime + αprimeprime + ω + βprimeprime + βprimeprimeprime isin y

Via the change of variables αprimeprime = αprime + α αprimeprimeprime = αprime βprimeprime = β βprimeprimeprime = minusβ + βprime we seethat these two subsets of Ω are the same (Here we use that a and b are groups)This proves the first defining equality of a semitorsor for Γ Since Ωopp is again agroup it holds also for Γ The second equality now follows from the first one usingthe symmetry relation (26)

Now consider the product (xyz)b Similarly as above we have

Σ(b x y z) =

ω isin Ω

∣∣∣ existβ βprime isin b ω + β isin x ω + βprime + β isin y ω + βprime isin z

(32)

Using (32) we have on the one hand

(x u (y v z)b)b =

=

ω isin Ω

∣∣∣ existα isin bexistβ isin b ω + α isin (y v z)b ω + α + β isin u ω + β isin x

=

ω isin Ω∣∣∣ existα isin bexistβ isin bexistαprime isin bexistβprime isin b ω + α + β isin u ω + β isin x ω + α + αprime isin z

ω + α + αprime + βprime isin v ω + α + βprime isin y

On the other hand

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 9(x (v y u)bz

)b

=

=

ω isin Ω

∣∣∣ existαprimeprime isin bexistβprimeprime isin b ω + αprimeprime isin z ω + αprimeprime + βprimeprime isin (v y u)b ω + βprimeprime isin x

=

ω isin Ω∣∣∣ existαprimeprime isin bexistβprimeprime isin bexistαprimeprimeprime isin bexistβprimeprimeprime isin b

ω + αprimeprime isin z ω + βprimeprime isin x ω + αprimeprime + βprimeprime + αprimeprimeprime isin uω + αprimeprime + βprimeprime + βprimeprimeprime isin v ω + αprimeprime + βprimeprime + αprimeprimeprime + βprimeprimeprime isin y

Via the change of variables β = βprimeprime βprime = βprimeprime + βprimeprimeprime α = αprimeprime + βprimeprime + αprimeprimeprime minus βprimeprimeprimeαprime = βprimeprime minus αprimeprimeprime minus βprimeprime we see that these two subsets of Ω are the same This provesthe first defining equality of a semitorsor for (xyz)b The proof of the other definingequality as well as for the semitorsor structure (xyz)b are similar

Definition 32 We call the semitorsors Pab Pab balanced and Pb Pb unbalanced

By the symmetry relation (26) Pba is the opposite semitorsor of Pab (where forany semitorsor (xyz) the opposite law is just (zyx)) whereas Pb is not the op-posite semitorsor of Pb Thus given a subgroup b sub Ω we have in general sixdifferent semitorsor laws on P Pbb Pb Pb along with their opposite laws If Ω iscommutative then of course these six semitorsor laws coincide More generally

Theorem 33 Assume that a and b are central subgroups of Ω Then

(1) Pab = Pab = Poppba and Pb = Pb = Popp = Poppb (2) Gras(Ω) is stable under all ternary laws from Theorem 31

Proof The first statement follows immediately from the definitions and the sec-ond by writing the structure equations for ω + ωprime resp for minusω with ω ωprime isinΓ(x a y b z) and using that variables from a and b commute with the others

Note that our condition is sufficient but not necessary with respect to item (2)for instance if Ω is a direct product of a and b (as a group) then the result of thenext section implies that Gras(Ω) is a subsemitorsor of Pab For general subgroupsa b this is no longer true the subsemitorsor generated by Gras(Ω) will be strictlybigger

4 The transversal case composition of relations in groups

Recall the definition of (left) transversality (Definition 12) denoted by agtb For afixed transversal pair we may identify Ω as a set with atimesb via (α β) 7rarr α+β Then(by definition) the power set P(Ω) is identified with the set Rel(a b) of relationsbetween a and b

Theorem 41 Let (a b) be a pair of left transversal subgroups of Ω Then theternary composition z yminus1 x of relations x y z isin P = Rel(a b) is given by

z yminus1 x = Γ(x a y b z)

If a and b commute then Gras(Ω) is stable under this ternary law

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

10 WOLFGANG BERTRAM

Proof The computation is the same as in [BeKi10a] Lemma 21 by respecting thepossible non-commutativity of Ω Recall first that if ABC are any sets wecan compose relations for x isin Rel(AB) y isin Rel(BC)

y x = yx = (uw) isin Atimes C | existv isin B (u v) isin x (v w) isin y Composition is associative both (z y) x and z (y x) are equal to

(41) z y x = (uw) isin AtimesD | exist(v1 v2) isin y (u v1) isin x (v2 w) isin z The reverse relation of x is

xminus1 = (w v) isin B times A | (v w) isin xFor x y z isin Rel(AB) we get another relation between A and B by zyminus1x (Thisternary composition satisfies the para-associative law and hence relations betweensets A and B form a semitorsor no structure on the sets A or B is needed here)Coming back to Ω = atimes b and switching to an additive notation we get

z yminus1 x =

ω = (αprime βprime) isin Ω

∣∣∣ existη = (αprimeprime βprimeprime) isin y (αprime βprimeprime) isin x (αprimeprime βprime) isin z

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin a existβprime βprimeprime isin b existη isin yexistξ isin xexistζ isin z ω = (αprime βprime) η = (αprimeprime βprimeprime) ξ = (αprime βprimeprime) ζ = (αprimeprime βprime)

=

ω isin Ω

∣∣∣ existαprime αprimeprime isin aexistβprime βprimeprime isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = αprimeprime + βprimeprime ξ = αprime + βprimeprime ζ = αprimeprime + βprime

Now use that a and b are transversal subgroups of Ω then the description of zyminus1xcan be rewritten by introducing the new variables α = αprime minus αprimeprime β = minusβprimeprime + βprime

(which belong again to a resp to b since these are subgroups)

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin bexistη isin yexistξ isin xexistζ isin z ω = αprime + βprime η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

Since agtb the first condition (existαprime isin a βprime isin b ω = αprime + βprime) in the precedingdescription is always satisfied and can hence be omitted in the description of zyminus1xThus

zyminus1x =

ω isin Ω

∣∣∣ existαprime α isin aexistβprime β isin b existη isin yexistξ isin x existζ isin z η = minusα + ω minus β ξ = ω minus β ζ = minusα + ω

= Γ(x a y b z)

Finally if a and b commute then the bijection Ω sim= a times b is also a group homo-morphism Since subgroups in a direct product of groups form a monoid undercomposition of relations it follows that Gras(Ω) is stable under the ternary com-position map

Recall that maps give rise to relations via their graphs In our setting

Definition 42 Assume (x y) is a left-transversal pair of subsets of Ω xgty andlet F xrarr y be a map The (left) graph of F is the subset

GF = ξ + F (ξ)| ξ isin x sub Ω

and if ygtx we define the right graph of F xrarr y to be

GF = F (ξ) + ξ| ξ isin x sub Ω

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 11

Lemma 43 Let b be a subgroup of Ω and y a subset such that ygtb Then thereare natural bijections between the following sets

(1) the set gtb(2) the set of sections σ Ωbrarr Ω of the canonical projection π Ωrarr Ωb(3) the set Map(y b) of maps from y to b

More precisely the bijection between (1) and (2) is given by the correspondencebetween σ and the image of σ and the one between (1) and (3) by Map(y b)rarr gtbF 7rarr GF If moreover y is a subgroup then we have

(B) the map F is bijective iff ygtGF

Proof Consider the equivalence relation given on Ω by ω sim ωprime iff minusω + ωprime isin bThen xgtb if and only if x is a set of representatives for this equivalence relationAs for any equivalence relation it follows therefore that gtb is in bijection with theset of sections of the canonical projection Ω rarr Ω sim Now let y = σ(Ωb) andx = σprime(Ωb) for two sections σ σprime Then f = minusσ+σprime is a map Ωbrarr b Converselygiven f Ωbrarr b σprime = σ + f is another section whose image is precisely the leftgraph of the map F = f π|y y rarr b

For the last statement assume that y is a subgroup and that F is bijective Ifω = η + β with β = F (ηprime) we have the decomposition ω = ηminus ηprime + ηprime + F (ηprime) withη minus ηprime isin y and ηprime + F (ηprime) isin GF which is unique since F is injective Hence ygtGF The converse is proved similarly

In a similar way if bgty every element of bgt is of the form GF with a unique mapF y rarr b

Theorem 44 Let (a b) be a pair of left-transversal subgroups of a group Ω Thenagt cap gtb is a subsemitorsor of Pab and it is actually a torsor denoted by Uab andnaturally isomorphic to the torsor of bijections F ararr b with its usual torsor law

(XY Z) = Z Y minus1 XIn other words if one fixes a bijection Y ararr b in order to identify a and b thenUab is the torsor corresponding to the group of all bijections of a 4

Proof By the lemma a relation r isin P belongs to agt cap gtb if and only if it is theleft graph of a bijection F ararr b Since composition of maps corresponds preciselyto the composition of their graphs the claim now follows from Theorem 41

Definition 45 A transversal triple of subgroups is a triple of subgroups (a b c)of Ω such that a and b commute and agtb bgtc and cgta

Theorem 46 Let (a b c) be a triple of subgroups such that agtb and a b commuteThen (a b c) is a transversal triple if and only if Ω sim= atimes a with a the first b thesecond factor and c the diagonal The subset U primeab = UabcapGras(Ω) of Uab is a torsorwith base point c isomorphic to the group Aut(a) of group automorphisms of a

Proof The assumption implies that Ω sim= atimes b as a group Let c isin Uab be the graphof the bijective map F b rarr a Then c sub Ω is a subgroup if and only if F is agroup morphism and the claim follows from the preceding results

4To be precise we get the opposite of the ldquousualrdquo composition This convention used alreadyin [BeKi10a] is in keeping with certain formulas from Jordan theory

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

12 WOLFGANG BERTRAM

5 The singular case pointwise torsors

Now we turn to the ldquosingularrdquo case a = b In this case Γ has to be replaced byΣ and torsors of bijections by ldquopointwise torsorsrdquo

Theorem 51 Assume b is a subgroup of Ω and let y isin P such that ygtb Thenthere are natural isomorphisms between the following torsors

(1) the set gtb which is a subsemitorsor of Pb and which becomes a torsordenoted by Ub with the induced law

(2) the torsor of all (images) of sections σ Ωbrarr Ω of the canonical projectionΩrarr Ωb with pointwise torsor structure (σσprimeσprimeprime)(u) = σ(u)minusσprime(u) +σprimeprime(u)

(3) the torsor Map(y b) of maps from y to b with its pointwise torsor structure

Similar statements hold for Ub = bgt which can be identified with sections of theprojection Ωrarr bΩ together with pointwise torsor structure

Proof On the level of sets these bijections have been established in Lemma 43 Itis immediately checked that the set of sections of the projection is stable under thepointwise torsor structure (σσprimeσprimeprime)(u) = σ(u) minus σprime(u) + σprimeprime(u) (as well as under itsopposite torsor structure) and it is clear then that the bijection between (2) and(3) becomes an isomorphism of torsors with pointwise torsor structures

In order to show that these torsor structures agree with the law described inTheorem 31 note that by a change of variables the unbalanced semitorsor lawcan also be written

(xyz)b =

ω isin Ω

∣∣∣ existξ isin xexistη isin yexistζ isin zexistβ βprime isin b ω = ξ minus η + ζ ζ = η + β ξ = η + βprime

(51)

Given three sections σ σprime σprimeprime we let ξ = σ(u) η = σprime(u) ζ = σprimeprime(u) so thatω = (σσprimeσprimeprime)(u) = σ(u)minus σprime(u) + σprimeprime(u) = ξ minus η + ζ which is the first condition in(51) The other two conditions just say that ξ η and ζ belong to the same cosetη+ b Thus the ternary structures from (1) (2) and (3) agree since (2) and (3) aretorsors the semitorsor law from (1) actually defines a torsor structure on gtb

The same arguments apply to sections of Ω rarr bΩ defining the two torsorstructures on bgt

6 An operator calculus on groups

In this section we generalize the various ldquoprojection operatorsrdquo used in [BeKi10a]to the case of general groups If Ω is abelian then all operators are Z-linear orZ-affine maps in general however they will not be endomorphisms of Ω In thefollowing all sums F +G and differences F minusG of maps FG Ωrarr Ω are pointwisesums resp differences and hence one has to respect orders in such expressions

Definition 61 Assume a and x are subsets of a group Ω such that agtx The leftresp right projection operators are defined by

P ax Ωrarr Ω ω = α + ξ 7rarr ξ

P xa Ωrarr Ω ω = α + ξ 7rarr α

where α isin a ξ isin x

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 13

Remark By dropping the transversality conditions some results of this section stillhold if one replaces projectors by generalized projectors ie by relations P x

a sub Ω2

of the form (η ω) isin Ω2 | existα isin a ξ isin x η = α + ξ ω = ξ for arbitrary x a isin P(see [BeKi10b] for the case of abelian Ω) This will be taken up elsewhere

Lemma 62 Let a b x y isin P such that agtx y and a bgtx Then

i) P xa + P a

x = idΩ that is P xa = idΩ minus P a

x and P ax = minusP x

a + idΩii) P a

x P bx = P b

x in particular P ax is idempotent (P a

x )2 = P ax

iii) if a is a subgroup then P ax P a

y = P ax

Proof i) If ω = α + ξ with α isin a ξ isin x then ξ = P ax (ω) and α = P x

a (ω) whencethe claim (Note in general P a

x + P xa will be different from the identity map)

ii) is obvious and iii) is proved by decomposing ω = α + ξ = αprime + η thenP ax P a

y (ω) = P ax (η) = ξ = P a

x (ω) since η = minusαprime + α + ξ

61 The ldquounbalanced operatorsrdquo

Definition 63 Let a b x y z isin P such that x ygtb and agtx y We define twotypes of transvection operators (from y to x along b reps a) by

T bxy = idminus P xb + P y

b = P bx + P y

b = P bx minus P b

y + id

T axy = P ya minus P x

a + id = P ya + P a

x = idminus P ay + P a

x

Notation is chosen such that for any operator A Ω rarr Ω which can be writtenas a ldquowordrdquo (iterated sum) in projection operators A denotes the correspondingoperator obtained by replacing Ω by Ωopp

Lemma 64 Assume a b are subgroups x ygtb and agtx y Then for all z isin P

Σ(b x y z) = T bxy(z)

Σ(a x y z) = T axy(z)

In particular it follows that if also ugtb

T bxx = idΩ T bxu T buy = T bxy

Thus T bxy | x y isin gtb is a group isomorphic to gtb as a torsor Similar remarks

apply to T bxy with respect to bgt

Proof (P bx minus P b

y + id)(z) is the set of ω isin Ω that can be written ω = ξ minus η + ζ

with ζ isin z ξ = P bx(ζ) η = P b

y (ζ) The last two conditions mean that there areβ βprime isin b with ζ = ξ + β and ζ = η + βprime Since b is a subgroup we see that ωsatisfies precisely the three conditions from (51) whence the first claim From thetorsor property we get now T bxu(T

buy(z)) = T bxy(z) Since this holds in particular

for singletons z = ζ the identity T bxu T buy = T bxy for operators on Ω followsThe remaining claims are clear

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

14 WOLFGANG BERTRAM

62 The ldquobalanced operatorsrdquo

Definition 65 Let a b x y z isin P If agtx and zgtb we define the middle multi-plication operators by

Mxabz = P ax minus id + P b

z

= P ax minus P z

b

= minusP xa + idminus P z

b

= minusP xa + P b

z

If agtx and ygtb we define the left multiplication operators by

Lxayb = minusP xa P b

y + id

and if (minusa)gty and zgtb we define the right multiplication operators by

Raybz = idminus P zb Pminusay

Lemma 66 Let a b x y z isin P

i) If agtx and zgtb then Γ(x a y b z) = Mxabz(y)ii) If agtx and ygtb then Γ(x a y b z) = Lxayb(z)

iii) If zgtb and (minusa)gty then Γ(x a y b z) = Raybz(x)

Proof i) (P ax minus id + P b

z )(y) is the set of all ω isin Ω that can be written in the formω = ξ minus η + ζ with ξ = P a

x (η) and ζ = P bz (η) for some η isin y This means in turn

that there is α isin a and ζ isin z such that η = α + ξ and η = ζ + β Summing up ωsatisfies exactly the three conditions from (27) hence they describe Γ(x a y b z)whence the equality of both sets

ii) (minusP xa P b

y + id)(z) is the set of all ω isin Ω that can be written as ω = minusα + ζ

with α = P xa P b

y (ζ) and ζ isin z This means that there is ξ isin x such that η = α+ ξ

for η = P by (ζ) And this means that there is η isin y and β isin b such that ζ = η minus β

Thus we end up with the three conditions

ω = minusα + ζ η = α + ξ ζ = η minus βwhich are equivalent to the structure equations

iii) (id minus P zb Pminusay )(x) is the set of all ω isin Ω that can be written as ω = ξ minus β

with ξ isin x and β = P zb (Pminusay (ξ)) This means that there is β isin b and ζ isin z such

that η = Pminusay (ξ) = ζ + β And this means that there is α isin a and η isin y such thatξ = minusα + η Again the three conditions thus obtained

ω = ξ minus β η = ζ + β ξ = minusα + η

are equivalent to the structure equations (Note it is not assumed in this lemmathat a or b are groups)

Lemma 67 Let a b x y z isin P Then

i) Assume b contains the origin of Ω If (minusa)gty and ygtb then Rayby = id henceΓ(x a y b y) = x for all x isin P

ii) If xgtb and agtx then Lxaxb = id hence Γ(x a x b z) = z for all z isin P

Proof i) We have P yb (y) = o since o isin y hence Rayby = idminus P y

b Pminusay = id

ii) Lxaxb = minusP xa P

bx + id = minus(idminus P a

x ) P bx + id = minus(P b

x minus P bx) + id = id

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 15

63 The ldquocanonical kernelrdquo

Definition 68 The canonical kernel is the family of maps defined by

Kaxy = P a

x |y y rarr x η 7rarr P ax (η)

Kbxy = P b

x|y y rarr x η 7rarr P bx(η)

where x y isin P a b isin Gras(Ω) with agtx and xgtb We let

Baxby = Ka

yx Kbxy = P a

y P bx|y y rarr y η 7rarr P a

y P bx(η)

Lemma 69 Let x y isin P a b isin Gras(Ω) with agtx and ygtb Then Kaxy y rarr x

is bijective iff agty and if this holds then for all η isin y

Kaxy(η) = T axy(η) = Lxayb(η)

Similarly Kbyx xrarr y is bijective iff xgtb and if this holds then for ξ isin x

Kbyx(ξ) = T byx(ξ) = Raxby(ξ)

It follows that agtx if and only if Bbxay y rarr y is bijective and if this is the case

Bbxay = T ayx T bxy|y y rarr y

(Bbxay

)minus1= T byx T axy y rarr y

Proof It is clear that y is another set of representatives for aΩ iff the projectionfrom y to x is a bijection If this holds then

T axy(η) = (P ya + P a

x )(η) = P ax (η)

Lxayb(η) = minusP xa P

by (η) + η = (minusP x

a + id)η = P ax (η)

The second claim is proved in the same way and the last statement follows

If x y a b are vector lines in R2 then Baxby is the linear map obtained by first

projecting y along b onto x and than back onto y along a Here is a quite generaldescription that applies to the case of vector spaces where one sees the close relationwith the so-called Bergman operators known in Jordan theory

Lemma 610 Assume Ω is a direct product of its subgroups y and b and let a bea subgroup such that ygta and x isin P such that xgtb Realize x = GX as a graph ofa map X y rarr b and a as a graph of a group homomorphism A brarr y Then

Baxby = idy minus A X y rarr y

In particular xgta if and only if idy minus A X is bijective

Proof We have to show that for all η isin y P ay (P b

xη) = η minus AXη Indeed since

η = η + Xη minus Xη is a decomposition according to xgtb we have P bxη = η + Xη

Next we decompose η + Xη = η minus AXη + AXη + Xη with η minus AXη isin y andAXη +Xη isin GA = a wence P a

y (P xb η) = P a

y (η +Xη) = η minus AXη

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

16 WOLFGANG BERTRAM

7 The balanced torsors Uab

Theorem 71 Fix a pair (a b) of subgroups of Ω Then the maps

Π+ab agt times gtbtimes agt rarr agt (x y z) 7rarr Γ(x a y b z)

Πminusab gtbtimes agt times gtbrarr gtb (x y z) 7rarr Γ(x a y b z)

are well-defined

Proof Assume agtx and use the bijection Ω sim= atimes x in order to write other subsetsas graphs let agtz and write z = GZ with a map Z xrarr a and let ygtb We haveto show that Lxayb(z) belongs again to agt that is that it can be written as is agraph In order to prove this let ζ = Zξ + ξ isin z where ξ isin x Then

Lxayb(Zξ + ξ) = (minusP xa P b

y + id)(Zξ + ξ) = minusP xa P b

y (Zξ + ξ) + Zξ + ξ

Define the map

F xrarr a ξ 7rarr minusP xa P b

y (Zξ + ξ)

so that Lxayb(ζ) = F (ξ) + Z(ξ) + ξ hence Lxaybz is the graph of F + Z x rarr aand it follows that Π+

ab is well-defined

Next assume that xgtb zgtb and agty and identify Ω sim= z times b Write x = GX asgraph of a map X z rarr b We have to show that Raybzx belongs again to gtb Letξ = ζ +Xζ isin x so

Raybz(ζ +Xζ) = (idminus P zb P a

y )(ζ +Xζ) = ζ +Xζ minus P zb (P a

y (ζ +Xζ))

and as above we see that Raybzx is the graph of a map X+F z rarr b hence belongsto gtb Thus Πminusab is well-defined

Remark See [BeKi10a] Theorem 18 for the case of abelian Ω and linear maps inthis case one can give a more explicit form for F in terms of block matrices

Theorem 72 Let Ω be a group and (a b) a pair of subgroups of Ω Then

(1) the set agt cap gtb = x isin P | agtx xgtb is a subsemitorsor of Pab and withrespect to the induced law it is a torsor which we will denote by Uab

(2) bgt cap gta is a subsemitorsor of Pab and with the induced law it becomes atorsor denoted by Uab

(3) the torsor Uba is the opposite torsor of Uab Uba = U oppab

Proof (1) The fact that agt cap gtb sub Pab is a subsemitotorsor follows directly fromthe preceding theorem The idempotent laws are satisfied by Lemma 67 Thus Uabis a torsor Now (2) and (3) follow by the symmetry relation (26) Note that theunderlying sets of Uba and Uab obviously agree

Definition 73 The tautological bundle of P(Ω) is the set

P = P(Ω) =

(y η) | y isin P(Ω) η isin y

and the map π P rarr P (y η) 7rarr y is called the canonical projection

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 17

Theorem 74 Let Ω be a group (a b) a pair of subgroups of Ω and fix y isin Uabconsidered as neutral element of the group (Uab y) defined by the preceding theoremThen there are natural left actions

Uab times P rarr P (x z) 7rarr xz = Γ(x a y b z)

Uab times P rarr P (x (z ζ)) 7rarr x(z ζ) =(xz Lxayb(ζ)

)

by ldquobundle mapsrdquo ie we have π(x(z ζ)) = x(π(z ζ)) Over agt this action canbe trivialized it is given in terms of the canonical kernel by (whith η isin y)

x(y η) = (xyKaxy(η))

Similarly we have natural right actions of Uab on P and on P which commute withthe left actions In particular Uab acts by conjugation on the fiber over the neutralelement and this action is given by the explicit formula

Uab times y rarr y η 7rarr xηxminus1 = Lxayb Raxby(η) = Baxby (η)

Proof Concerning the left action everything amounts to proving the following iden-tity for operators on Ω with x xprime isin Uab(71) Lxayb Lxprimeayb = LΓ(xaybxprime)ayb

Note that the operator on the right hand side is well-defined since Γ(x a y b xprime) isinUab by the preceding theorem Now para-associativity (Theorem 31 combinedwith Lemma 66) shows that applied to any subset z sub ω both operators give thesame result Taking for z singletons it follows that the operators coincide Theproof for the right action is similar and the fact that both actions commute againamounts to an operator identity

(72) Lxayb Raxprimeby = Raxprimeby Lxaybwhich is proved by the same arguments as (71) For x = xprime we use the definitionof the canonical kernel (Definition 68) and get the action by conjugation

Theorem 75 Assume (a b) is a pair of central subgroups Then the set Grasab =Gras(Ω) cap Uab is a subtorsor of Uab which acts from the left and from the right onthe Grassmannian Gras(Ω) and on the Grassmann tautological bundle

Gras(Ω) = (x ξ) | x isin Gras(Ω) ξ isin xProof This follows by combining the preceding result with Theorem 33

8 Distributive law and ldquoaffine picturerdquo

The following fairly explicit description of the group law of Uab is the analog ofthe ldquoaffine picturerdquo from the abelian and linear case given in Section 1 of [BeKi10a]

Theorem 81 Let (a b) be a pair of subgroups of Ω and x y z isin P such thatx y zgtb and agtx y Write x and z as left graphs with respect to the decompositionΩ sim= y times b ie x = GX z = GZ with maps XZ y rarr b Then

Γ(GX a y b GZ) = GX+ZBaxby

ie Γ(x a y b z) is the graph of the map X+ZBaxby y rarr b where Baxb

y y rarr yis the canonical kernel (Definition 68)

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

18 WOLFGANG BERTRAM

This ldquoaffine formulardquo may be written by identifying x with X and z with Z

(81) X middotayb Z = X + Z BaXby

Here y = o+ b = ominus are fixed ldquobasepointsrdquo and a is the ldquodeformation parameterrdquo

Proof Recall from Lemma 66 that

Γ(x a y b z) = (P ax minus id + P b

z )(y) = P ax (η)minus η + P b

z (η) | η isin y

Let η isin y Since P ax (η) isin x = GX and P b

z (η) isin z = GZ there exist unique ηprime isin yand ηprimeprime isin y such that

P ax (η) = ηprime +Xηprime P b

z (η) = ηprimeprime + Zηprimeprime

We determine ηprime and ηprimeprime as functions of η since ηprime isin y and Xηprime isin b we have bydefinition of the projection

ηprime = P by (P a

x (η)) = Bminus1(η)

where B = Baxby y rarr y is the canonical kernel (Definition 68) and in the same

way using Lemma 62 we get

ηprimeprime = P by P

bz (η) = P b

yη = η

whence P bz (η) = η+Zη Since the operator B y rarr y is bijective (Lemma 69) we

can make a change of variables ηprime = Bminus1η η = Bηprime and we get

(P ax minus id + P b

z )(η) = ηprime +Xηprime minus η + η + Zη= ηprime +Xηprime + ZBηprime

= ηprime + (X + Z B)ηprime

and hence invoking Lemma 66 Γ(x a y b z) is equal to

(P ax minus id + P b

z )(y) =ηprime + (X + Z Baxb

y )ηprime | ηprime isin y

that is to the (left) graph of the map X + Z Baxby y rarr b

Remark One may turn everything also the other way round assume (b+) is agroup y a set and let G = Map(y b) Assume given a map B G rarr Map(y y)X 7rarr BX such that B0 = idy and define a binary law on G by

X middot Z = X middotB Z = X + Z BX

where + denotes ldquopointwise additionrdquo in G It is straightforward to show that thislaw is associative iff B becomes a homomorphism in the sense that

BX+ZBX

= BX BZ

(cf [Pi77] p 243 for a similar construction in the context of near-fields) Theneutral element is the zero map 0 and an element X in G is invertible iff BX y rarr yis bijective and then its inverse is the ldquoquasi-inverserdquo

Xminus1 = minusX (BX)minus1

As a special case all this works if y and b are groups A brarr y a group homomor-phism and BX = idy + A X namely this is the affine picture of the following

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 19

Corollary 82 Assume that y is a subgroup commuting with b and write a = GA

with a group homomorphism A brarr y Then Formula (81) reads

Γ(GX GA y b GZ) = GX+Z(idyminusAX)

and Uab rarr Bij(y)op GX 7rarr idy minus AX is a group homomorphism

Proof Write (81) using that by Lemma 610 BaXby = idy minus A X The homo-

morphism property follows from Theorem 745

Theorem 83 Let (a b) be a pair of subgroups of Ω Then we have the followingldquoleft distributive lawrdquo for all x y isin Uab and u v w isin Ub

(xy(uvw)b)ab =((xyu)ab(xyv)ab(xyw)ab)b

In other words left multiplications Lxayb from Uab are automorphisms of the torsorUb Similarly right multiplications from Uba are automorphisms of the torsor Ua

Proof Let u v wgtb and denote by uppercase letters the corresponding maps y rarrb Then the law of the ldquopointwise torsorrdquo Ub is simply described by the pointwisetorsor structure U minus V + W of maps from y to b (Theorem 51) The claim nowfollows from Theorem 81

X + (U minus V +W ) Bx = X + U Bx minus V Bx +W Bx

= (X + U Bx)minus (X + V Bx) + (X +W Bx)

The ldquodualrdquo statement follows by replacing Ω by Ωopp

In general the law of Uab is not right distributive the laws xz = Γ(x a y b z)and x+ z = Σ(b x y z) define a near-ring and not a ring (cf Definition 16)

9 Permutation symmetries

We have already mentioned (remark after Lemma 23) that the structure map Γand the structure space Γ have certain invariance or ldquocovariancerdquo properties withrespect to permutations We start by a simple remark on torsors

Definition 91 The torsor graph of a torsor (G ( )) is

T = T(G) =

(ξ η ζ ω) isin G4 | ω = (ξηζ)

Using an additive notation the torsor graph of a group (Ω+) is thus given by

T = T(Ω) =

(ξ η ζ ω) isin Ω4 | ω = ξ minus η + ζ

Lemma 92 The torsor graph of a torsor is invariant under the Klein four-groupgenerated by the two double-transpositions (12)(34) and (13)(24)

Proof In additive notation this follows immediately from the fact that the torsorequation ω = ξ minus η + ζ is equivalent to η = ζ minus ω + ξ and to ζ = η minus ξ + ω

For an intrinsic proof without fixing a base point note that symmetry under(13)(24) is equivalent so saying that the middle multiplication operators Mxz(y) =(xyz) are invertible with inverse Mzx and symmetry under (12)(34) is equivalent

5 We have to use the opposite group structure on Bij(y) in order to be in keeping with ourconvention on Uab cf footnote 4

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

20 WOLFGANG BERTRAM

so saying that the left multiplication operators Lxy(z) = (xyz) are invertible withinverse Lyx (cf Appendix A of [BeKi10a])

Recall the definition of the structure space Γ(Ω) (Definition 22) and the equiv-alent versions of the structure equations (Lemma 23) Note that in System (29)the ldquotorsor equationrdquo appears hence the ldquo(ξ η ζ ω)-projectionrdquo

Γ(Ω)rarr T(Ω)(ξ ζα β η ω

)7rarr (ξ η ζ ω)

is well-defined Concerning other variables the ldquotorsor equationrdquo also appearsmodulo certain sign changes The relevant symmetry group here is a subgroup Vof the permutation group S6 playing a similar role as the Klein four-group V sub S4

in the preceding lemma

Definition 93 The Big Klein group is the subgroup V of permutations σ isin A6acting on six letters α β ξ ζ η ω and preserving the partition

α β ξ ζ η ω = A1 cup A2 cup A3 A1 = α β A2 = ξ ζ A3 = η ωie for i = 1 2 3 there is iprime isin 1 2 3 with σ(Ai) = Aiprime

Lemma 94 The Big Klein group V is isomorphic to S4 and its action on sixletters is equivalent to the natural action of S4 on the set K of all two-elementsubsets of 1 2 3 4

Proof We fix the following correspondence between our six letters and K

α = 1 2 β = 3 4 ξ = 1 3 ζ = 2 4 η = 1 4 ω = 2 3The natural action of S4 induces a homomorphism S4 rarr S6 letting act S4 on the sixletters α ω This homomorphism is obviously injective and its image belongsto V (note that each transposition from S4 acts by a double-transposition of thesesix letters hence the image belongs to A6) Let us prove that the homomorphism issurjective from the very definition of V we get a homomorphism Vrarr S3 sendingσ to the permutation i 7rarr iprime The kernel of this homomorphism is a Klein four-group and one easily constructs a section S3 rarr V so that |V| = 24 = |S4| whencethe claim

Definition 95 A vector s = (s1 s6) with si isin plusmn1 will be called a signvector Given a sign vector s the subspace

Γs =

(ξ ζα β η ω) isin Ω6∣∣∣ (s1ξ s2ζ s3α s4β s5η s6ω

)isin Γ

is called a signed version of the structure space

Since Ω rarr Ω v 7rarr minusv is an antiautomorphism of Ω we see that the oppositestructure space is a signed version of Γ namely

Γ = Γ(minus1minus1minus1minus1minus1minus1)

Theorem 96 The Big Klein group transforms Γ into signed versions of Γ foreach σ isin V there exists a sign vector s = s(σ) such that σΓ = Γs(σ) More preciselywe have the following table of elements σ isin V (given together with their correspond-ing element in S4 under the isomorphism from Lemma 94) and corresponding signvectors s(σ)

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 21

element of V sub S4 corresponding element σ isin V corresponding sign vector s(σ)id id (1 1 1 1 1 1)(12)(34) (ξζ)(ηω) (1 1minus1minus1 1 1)(13)(24) (αβ)(ηω) (minus1minus1 1 1minus1minus1)(14)(23) (αβ)(ξζ) (minus1minus1minus1minus1minus1minus1)

element of A4 V corresponding element σ isin V corresponding sign vector s(σ)(123) (αωξ)(βηζ) (1minus1minus1 1minus1minus1)(132) (αξω)(βζη) (minus1 1minus1minus1minus1 1)(124) (αζη)(βξω) (minus1 1 1 1 1minus1)(142) (αηζ)(βωξ) (minus1 1 1minus1 1 1)(134) (αωζ)(βηξ) (1minus1minus1 1 1 1)(143) (αζω)(βξη) (1minus1 1 1 1minus1)(234) (αξη)(βζω) (1minus1minus1minus1minus1 1)(243) (αηξ)(βωζ) (minus1 1 1minus1minus1minus1)

transposition in S4 corresponding element σ isin V corresponding sign vector s(σ)(12) (ξω)(ζη) (1 1 1minus1 1 1)(13) (αω)(βη) (1minus1 1minus1 1minus1)(14) (αζ)(βξ) (1 1 1 1 1minus1)(23) (αξ)(βζ) (minus1minus1minus1minus1minus1 1)(24) (αη)(βω) (minus1 1 1minus1 1minus1)(34) (ξη)(ζω) (1 1minus1 1 1 1)

elt of order 4 in S4 corresponding element σ isin V corresponding sign vector s(σ)(1234) (αωβη)(ξζ) (1minus1minus1 1minus1 1)(1243) (αζβξ)(ηω) (minus1minus1 1 1 1minus1)(1324) (αβ)(ξωζη) (minus1minus1 1minus1minus1minus1)(1342) (αξβζ)(ηω) (1 1minus1minus1minus1 1)(1423) (αβ)(ξηζω) (minus1minus1minus1 1minus1minus1)(1432) (αηβω)(ξζ) (minus1 1 1minus1minus1 1)

Proof The proof is by direct computation take any of the systems from (28) ndash(213) in Lemma 23 replace variables according to σ the system thus obtainedagrees up to sign changes with some other among the systems from (28) ndash (213)and by comparing one can read off the sign vector

A glance at the tables shows that elements from A4 induce an even number ofsign changes and elements of S4 A4 an odd number of sign changes The signvectors given in the tables form an S4-torsor under the induced action of V If Ωis non-commutative then different sign vectors s give rise to different spaces Γs ifΩ is commutative then Γ = Γ but (except for some degenerate examples) this isthe only case in which two signed versions of the structure space coincide If in theabelian case we work only with linear subspaces (subgroups) then we may ignoreall sign changes and hence all 24 permutations can be considered as equivalent Inthe general case the following statements also follow by direct inspection from thetables

Definition 97 A subset x in a group (Ω+) is a symmetric subset if minusx = x

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

22 WOLFGANG BERTRAM

Theorem 98 For any group (Ω+) and (x a y b z) isin P5 we have

(1) (behavior of Γ under the Klein group id (14) (14)(23) (23) sub S4)Γ(b z y x a) = minusΓ(x a y b z)Γ(z b y a x) = Γ(x a y b z)Γ(a x y z b) = minusΓ(x a y b z)and if x a y b z are symmetric subsets then Γ(x a y b z) = Γ(a x y z b)

(2) If we consider sign changes in α or β as negligible (and the the correspondingsigned versions as ldquoequivalentrdquo) then the equivalence class of Γ is invariantunder the Klein group id (12)(34) (12)(34) sub S4

Remark Following [BeKi10a] Theorem 211 and Remark 212 the second item maybe reformulated in another way invariance under (12) that is under (ξω)(ζη)amounts to the fact that the inverse of the relation lxayb sub Ω2 (which as in[BeKi10a] generalizes the operator Lxayb for arbitrary (x a y b) isin P4) is given bythe relation lyaxb Similarly invariance under (34) amounts to the analog for righttranslations and invariance under (12)(34) to the analog for middle multiplicationoperators

(lxayb)minus1 = lyaxb (raybz)

minus1 = razby (mxabz)minus1 = mzabx

Note that this is the exact analog of Lemma 92 however since a b need not betransversal subgroups these relations now apply to semitorsors as well In a certainsense this means that our semitorsors have the same ldquosymmetry type as a torsorrdquondash a property which certainly distinguishes them from ldquoarbitraryrdquo semitorsors

10 Generalized lattice structures

For abelian groups Theorem 24 of [BeKi10a] establishes a close link betweenthe structure map Γ and the lattice of subgroups For non-abelian groups Ω thesubgroups form no longer a lattice Nevertheless the two set-theoretic operations

x and y = x cap y x+ y = ω isin Ω | existξ isin x existη isin y ω = ξ + ηbehave very much like ldquomeetrdquo (and) and ldquojoinrdquo (or) as shows the following analog ofTheorem 24 of [BeKi10a]

Theorem 101 Let x a y b z be subgroups of Ω Then we have

(1 ) values of Γ on the ldquodiagonal x = yrdquo

Γ(x a x b z) =((x and a

)+ z)and (x+ b)

= (x and a) +(z and (x+ b)

)

(2 ) values of Γ on the ldquodiagonal a = zrdquo

Γ(x a y b a) =((x and (a+ y)

)+ b)and a

=(x+

((y + a) and b

))and a

(3 ) values of Γ on the ldquodiagonal b = zrdquo

Γ(x a y b b) =((a+ (y and b)

)and x)

+ b

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 23

=(a and

(x+ (y and b)

))+ b

This implies in particular that for all x a y b isin Gras(Ω)

Γ(x a x b x) = xΓ(a a y b b) = a+ bΓ(b a y b a) = a and b

Proof The technical details of the the proofs are exactly as in [BeKi10a] loc citby respecting the possible non-commutativity of Ω (and hence of the operation +)so let us here only prove the first equality from item (1) As in loc cit it is alwaysunderstood that α isin a ξ isin x β isin b η isin y ζ isin z We use System (29)

Let ω isin Γ(x a x b z) then ω = ξminusβ hence ω isin (x+ b) and ω = ξminus η+ ζ withv = ω minus ζ = ξ minus η isin x (since x = y) On the other hand v = ω minus ζ = minusα isin awhence ω = v+ ζ with v isin (xand a) proving one inclusion (Note that we have usedthat x and a are subgroups and that b is symmetric)

Conversely let ω isin ((xanda)+z)and(x+b) Then ω = ξ+β = α+ζ with α isin (xanda)Let η = minusα+ ξ Then η isin x and ω = ξ+ β = α+ η+ β hence ω isin Γ(x a x b z)

The remaining proofs are similar Note that since the systems in (28)ndash(214)come in pairs there are in fact two different expressions for one diagonal value Forthe final conclusion one uses the ldquoabsorption lawsrdquo in the following form

Lemma 102 Recall that Po = x isin P(Ω) | o isin x(1) If y isin Po then x and (x+ y) = x(2) If y isin Po and x isin Gras(Ω) then x+ (x and y) = x = (x and y) + x

Remark It is remarkable that for subgroups x a z a ldquonon-commutative modularlawrdquo is still satisfied (cf Remark 25 of [BeKi10a]) by letting b = x in (1) we get

((x and a) + z) and x = (x and a) + (z and x)

and letting a = x we get the ldquodual modular lawrdquo

x+ (z and (x+ b)) = (x+ z) and (x+ b)

More generally one has the impression that the formulas express some sort ofldquodualityrdquo between the operations and and + This would be rather mysterious sinceat a first glance the definitions of and and + look quite ldquonon-symmetricrdquo

11 Final remarks

Since groups and projective spaces are foundational concepts in mathematicsthe present approach is likely to interact with many areas of mathematics Theauthorrsquos original motivation came from non-associative algebra and in particularJordan theory ndash this domain does not really belong to todayrsquos mathematical main-stream and thus the shape of the approach presented here may seem quite unusualfor a ldquonormal user of group theoryrdquo The reader may find more motivation and gen-eral remarks in the introductory and concluding sections of preceding work eg[BeKi10a BeKi10b BeNe05 BeL08 Be02]

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

24 WOLFGANG BERTRAM

In the following let us add some short comments on aspects for which the gen-eralization of the framework to general groups and general torsors of mappingsmay be relevant Most importantly one needs to study categorial aspects moresystematically we hope to come back to such items in subsequent work

111 Morphisms Projective spaces or general Grassmannians can be turned intocategories in two essentially different ways (cf [Be02 BeKi10a]) The same is truein the present context One may conjecture that an analog of the ldquofundamentaltheorem of projective geometryrdquo holds every automorphism of (P(Ω)Γ) is inducedby an automorphism of Ω (if Ω is not too small) However the main difference is thatnow in the non-abelian case ldquointerior mapsrdquo (left- right- and middle multiplicationoperators) are no longer morphisms in either of the two categories One may wonderwhether they are morphisms in yet another sense

112 Antiautomorphisms involutions For the case of commutative Ω see[BeKi10b] If Ω is non-commutative the situation seems to change drasticallyfirst of all in principle each permutation of the 5 arguments of Γ gives rise toits own notion of ldquoanti-homomorphismrdquo The simplest case corresponds to anti-homomorphisms on the level of Ω they correspond to ldquousualrdquo morphisms betweenΓ and Γ For instance the inversion map of Ω induces an ldquoinvolutionrdquo of thiskind It corresponds to the permutation (ξζ)(αβ) which belongs to V On theother hand looking at permutations not belonging to V one may ask whetheranti-homomorphisms corresponding to the permutation (ξζ) exist these would gen-eralize the involutions considered in [BeKi10b] In the commutative case they aregiven by orthocomplementation maps In particular they induce lattice antiauto-morphisms As mentioned above our formulas suggest that some kind of dualityin this sense exists on the other hand there seems to be no hope to generalizeorthocomplementation maps in some obvious way

113 Subobjects A subspace of P(Ω) is a subset Y sub P(Ω) stable under Γ Suchsets may be defined by algebraic conditions (cf Theorem 33) or by topological ordifferential conditions eg if Ω is a topological or Lie group (and in particularfor Ω = R2n) we may consider spaces Y of closed subsets or of smooth or algebraicsubmanifolds One expects that Γ will have ldquosingularitiesrdquo so one possibly has toexclude some ldquosingular setsrdquo outside such sets we expect Γ to be fairly regular (cfrelated results in [BeNe05]) The analogy with near-rings suggests also to look atsubspaces corresponding to near-fields

114 Ideals inner ideals intrinsic subspaces As in rings or near-rings or inJordan algebraic structures one may define notions of certain subobjects playingthe role of various kinds of ldquoidealsrdquo and which will be of importance for a systematicldquostructure theoryrdquo Such ideals are defined as subobjects Y defining conditions likethe ldquoinner ideal conditionrdquo Γ(Y P Y P Y) sub Y In a Jordan theoretic contextsuch sets have been characterized in [BeL08] as ldquointrinsic subspacesrdquo

115 Products Our construction is compatible with direct products For instanceP(Ω1) times P2(Ω2) is a subspace of P(Ω1 times Ω2) Here the case Ω1 = Ω2 = Ω isparticularly important since elements of P(Ωtimes Ω) are nothing but relations on Ω

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

THE PROJECTIVE GEOMETRY OF A GROUP 25

and hence may play the role of endomorphisms as discussed above This situation ischaracterized by the existence of transversal triples (Theorem 46) and of projectionsthat are endomorphisms

116 Axiomatic approach base points and equivalence of categories Fol-lowing [BeKi10a] one may consider as base point a fixed pair (o+ ominus) of transversalsubgroups and then look at the ldquopair of ternary near-ringsrdquo (π+ πminus) defined byTheorem 71 as a sort of ldquotangent objectrdquo In which sense can one say then that thetheory of such objects is equivalent to the theory of P(Ω) ndash is there an equivalenceof categories between ldquotangent objectsrdquo and ldquogeometries with base pointrdquo

117 Symmetric spaces and Jordan theory Unlike the three ldquodiagonalsrdquomentioned in Theorem 101 the ldquodiagonal x = zrdquo is not related to lattice the-ory but rather to symmetric spaces in any torsor the ternary composition givesrise to a binary map micro(x y) = (xyx) which (in the case of a Lie group) is pre-cisely the underlying ldquosymmetric space structurerdquo (in the sense of [Lo69]) Thusautomatically our torsors Uab give rise to families of symmetric spaces Other sym-metric spaces can be constructed from these in presence of an involution Thereforesymmetric spaces are a main geometric ingredient for the theory corresponding tothe ldquodiagonal x = yrdquo which in turn would be a kind of ldquonon-commutative Jordantheoryrdquo Many of the surprising features of Jordan theory are due to the fact thatbecause of the symmetry (αζ)(ξβ) (Theorem 98) this theory essentially is the sameas the theory of the ldquodiagonal a = brdquo ie of the family of torsors Uaa thus thereshould be some kind of duality between certain families of symmetric spaces andcertain families of torsors

118 Flag geometries For Jordan theory the projective geometry of a Lie algebradefined in [BeNe04] gives a useful ldquouniversal geometric modelrdquo The construction issimilar in spirit to the present work however it is not clear at all how to carry it outon the level of groups Essentially one needs a definition of an analog of our mapsΓ and Σ for (short) flags that is for pairs of subsets (x xprime) with o isin x sub xprime sub Ωinstead of single subsets Some results by J Chenal ([Ch09]) point in the directionthat such constructions should be possible for general spaces of finite flags

119 Reductive groups finite groups For a reductive Lie group Ω how is theprojective geometry of Ω related to well-known structure theory Is there some linkwith the notion of building Is the projective geometry of the Weyl group relatedin some definite way to the projective geometry of Ω

1110 Supersymmetry We have the impression that Section 9 on ldquosymmetryrdquois closely related to the topic of supersymmetry indeed the behavior of the ldquosignedversionsrdquo under permutations reminds the ldquosign rulerdquo of supersymmetry On a veryfundamental level our map Γ takes account of the principle that there is no reasonto prefer a group to its opposite group in principle both should play symmetricroles However in presence of additional structure (two or more operations) thissymmetry may be broken eg there may be ldquoleft-distributivityrdquo but not ldquoright-distributivityrdquo As the tables in Theorem 96 show a complete book-keeping ofsuch situations is not entirely trivial The next level in such a book-keeping should

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr

26 WOLFGANG BERTRAM

be reached when we are dealing with P(Ωtimes Ω) namely subsets of Ωtimes Ω are justrelations on Ω and here again there is no reason to prefer ldquousualrdquo compositionto its opposite in other words we have to fix choices and conventions for thestructure maps ΓΩtimesΩ of the group Ω times Ω and similarly for iterated products ΩkSupersymmetry might turn out to be one of the sign-rules that are used for suchbook-keeping of iterated products

References

[Be02] W Bertram Generalized projective geometries general theory and equivalence withJordan structures Adv Geom 2 (2002) 329ndash369 (electronic version preprint 90 athttpmollefernuni-hagende~loosjordanindexhtml)

[Be04] W Bertram From linear algebra via affine algebra to projective algebra Linear Algebraand its Applications 378 (2004) 109ndash134

[Be08b] W Bertram Homotopes and conformal deformations of symmetric spaces J Lie Theory18 (2008) 301ndash333 arXiv mathRA0606449

[Be08] W Bertram On the Hermitian projective line as a home for the geometry of quantumtheory In Proceedings XXVII Workshop on Geometrical Methods in Physics Bia lowieza2008 arXiv math-ph08090561

[BeKi10a] W Bertram and M Kinyon Associative Geometries I Torsors Linear Relations andGrassmannians Journal of Lie Theory 20 (2) (2010) 215-252 arXiv mathRA09035441

[BeKi10b] W Bertram and M Kinyon Associative Geometries II Involutions the classicaltorsors and their homotopes Journal of Lie Theory 20 (2) (2010) 253-282 arXiv09094438

[BeL08] W Bertram and H Loewe Inner ideals and intrinsic subspaces Adv in Geometry 8(2008) 53ndash85 arXiv mathRA0606448

[BeNe04] W Bertram and K-H Neeb Projective completions of Jordan pairs Part I The gen-eralized projective geometry of a Lie algebra J of Algebra 227 2 (2004) 474-519arXiv mathRA0306272

[BeNe05] W Bertram and K-H Neeb Projective completions of Jordan pairs II Manifoldstructures and symmetric spaces Geom Dedicata 112 (2005) 73 ndash 113 arXivmathGR0401236

[Cer43] J Certaine The ternary operation (abc) = abminus1c of a group Bull Amer Math Soc49 (1943) 869ndash877

[Ch09] J Chenal Generalized flag geometries and manifolds associated to short Z-graded Liealgebras in arbitrary dimension C R Acad Sci Paris Ser I 347 (2009) 21ndash25 cfarXiv10074076v1[mathRA]

[Lo69] O Loos Symmetric Spaces I Benjamin New York 1969[Lo75] O Loos Jordan Pairs Lecture Notes in Math 460 Springer New York 1975[Pi77] G Pilz Near-Rings North-Holland Amsterdam 1977

Institut Elie Cartan Nancy Lorraine Universite CNRS INRIA Boulevard desAiguillettes BP 239 F-54506 Vandœuvre-les-Nancy France

E-mail address bertramiecnu-nancyfr