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PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS

PAUL-ANDI NAGY

Abstract. We study the skew-symmetric prolongation of a Lie subalgebra g⊆ so(n), in otherwords the intersection Λ3 ∩ (Λ1 ⊗ g). We compute this space in full generality. Applicationsinclude uniqueness results for connections with skew-symmetric torsion and also the proof ofa conjecture posed in [19] concerning a class of Plucker-type embeddings. We also derive aclassification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finitedimension. Next we study the space of invariant 4-forms of a given metric representation andapply these considerations to classify the holonomy representation of metric connections withvectorial torsion, that is with torsion contained in Λ1 ⊗ Λ2.

Contents

1. Introduction 12. Metric connections with torsion 42.1. Prolongations of Lie algebras 62.2. Algebraic curvature tensors and Berger algebras 73. Structure results and flat holonomy algebras 93.1. Reducible representations 134. Uniqueness of connections with totally skew-symmetric torsion 155. A class of Plucker type embeddings and k-Lie algebras 155.1. Metric n-Lie algebras 176. The holonomy of connections with vectorial torsion 176.1. Invariant forms and Casimir operators 186.2. Non degeneracy properties and the classification 197. Some facts peculiar to dimension 8 238. Appendix: The skew-symmetric prolongation of an adjoint representation 24References 26

1. Introduction

Let (Mn, g) be an oriented Riemannian manifold of dimension n. A G-structureon M consists in a reduction of the frame bundle of (Mn, g) to a Lie subgroupG ⊆ SO(n). While this is an interesting object by itself, it appears naturally in thecontext of many partial differential equations where it is often additionally equippedwith a connection preserving the G-structure, generally with non-vanishing torsion.Without further assumptions this is known to be fairly unrestrictive [23]. Sometimesthe reduction of the frame bundle of (Mn, g) is realised by means of the isotropygroup of a tensor field on M of specific algebraic type, as required for instance by

Date: March 28, 2019.2000 Mathematics Subject Classification. 53C12, 53C24, 53C55.Key words and phrases. skew-symmetric prolongation, connection with skew symmetric, vectorial torsion .

1

2 P.-A.NAGY

models in string theory [33]. When combining the classification of G-structureswith the study of the torsion tensor of a given G-connection (see [9, 21, 10] forinstance) it is possible to distinguish several types of geometries. An approach tothe classification of each type could be through the study of the holonomy of theG-connection involved.

In the case of connections of Levi-Civita or torsion-free type powerful classifica-tion results (see [7, 8, 29]) are available and each of the resulting geometries e.g.Kahler, Calabi-Yau, quaternion-Kahler, Joyce manifolds is in the mainstream ofcurrent research. The theory of connections with non-vanishing torsion is less wellestablished and many classical results from Riemannian holonomy theory fail tohold, such as the deRham splitting teorem.

In this paper we shall mainly consider metric connections with skew-symmetric orvectorial torsion on (Mn, g), meaning that their torsion is assumed to be in Λ3M orΛ1M ⊆ Λ1M ⊗ Λ2M , respectively. These classes are labelled W1 and W3 and shallbe given a more detailed description later on. There are situations when the classW1 emerges naturally from a specific class of G-manifolds such as nearly-Kahler[20, 30] or Spin(7)-structures [25], for instance. Concerning the class W1 we wishto address the question of uniqueness when the holonomy representation is fixed,that is if the requirement on a connection D in the class W1 to have holonomycontained in a fixed Lie subgroup G ⊂ SO(n) implies its uniqueness. Note thatsuch a connection might not always exist, and that the uniqueness problem hasbeen solved for those representations arising as Riemannian, non-locally symmetricholonomies in [25, 16, 17], as well as for the irreducible SO(3) representation indimension 5 [6]. To treat the general case we will consider, for a given faithfulrepresentation (g, V ) of a Lie algebra on an Euclidean vector its skew-symmetricprolongation

Λ3V ∩ (Λ1V ⊗ g) = T ∈ Λ3V : XyT ∈ g, for all X ∈ V .

We are able to compute explicitly this space as follows.

Theorem 1.1. Let (g, V ) be a faithful and irreducible representation of a Lie algebraonto an Euclidean space. If g is proper then either:

(i) Λ3V ∩ (Λ1V ⊗ g) = 0;or

(ii) (g, V ) is the adjoint representation of a compact, simple Lie algebra, whenΛ3V ∩ (Λ1V ⊗ g) is 1-dimensional, generated by the Cartan form of g.

We also show how this can be used to compute the skew-symmetric prolongationof an arbitrary orthogonal representation. As a direct application of Theorem 1.1we obtain that:

Theorem 1.2. Let (Mn, g) be a connected Riemannian manifold admitting a metricconnection D such that:

(i) D has skew-symmetric torsion;(ii) Hol(D) ⊆ G ⊂ O(n) for some Lie subgroup G of O(n).

If the representation (G, V ) is irreducible, then D is unique with these properties,exception made of the case when (G, V ) is the adjoint representation of a compact,simple Lie group.

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 3

Theorem 1.1 has a few other applications we shall briefly describe now. The firstconsists in the explicit description of the holonomy of connections with constanttorsion in flat space, or equivalently of holonomy algebras generated by a 3-form, thestudy of which has been initiated in [2]. The second is related to a class of Pluckertype relations [19], having their origins in the study of maximally supersymmetricsolutions of ten and eleven dimensional supergravity theories [18]. The last is toobtain a classification of the so-called n-Lie (or Filipov) algebras, starting from theobservation [19] that those are in 1 : 1 correspondence with the Plucker relationsmentioned above.

In the second part of the paper we classify the holonomy of metric connectionswith vectorial torsion. More precisely we look at Riemannian manifolds (Mn, g)equipped with a metric connection D of the form

DX = ∇X +X ∧ θ

for all X in TM , where ∇ is the Levi-Civita connection of the metric g and θ isa 1-form on M . The triple (Mn, g, D) will be called closed if θ is closed, that isdθ = 0. The main class of examples in this direction is obtained when (Mn, g)has a locally parallel structure (l.c.p. for short). That is we require g not to belocally conformally symmetric and moreover it must be locally conformal to a metricadmitting a Riemannian holonomy reduction ([32] for details). Note that such ametric needs not be globally conformally parallel. For results and examples in thisdirection we refer the reader to [26, 14, 31] and references therein. We prove:

Theorem 1.3. Let (Mn, g) be a connected and oriented Riemannian manifold andlet D be a metric connection with vectorial torsion. If the holonomy representation(G, V ) of D is irreducible with G proper in SO(n) then:

(i) (Mn, g, D) is closed. Moreover, either:(ii) g is conformal to a non-flat locally symmetric Riemannian metric on M ,(iii) (Mn, g) is locally conformally flat,or

(iv) g admits a l.c.p. structure.

The paper is organised as follows. In Section 2 we start by presenting some wellknown facts related to the classification of connections with torsion and also tosome of their basic properties including the Bianchi identity and the symmetries ofthe non-Riemannian curvature tensors. Next, and by following [10] essentially, werecall the results on Berger algebras and algebraic curvature tensors we will need inwhat follows,

Section 3 is devoted to the proof of Theorem 1.1 which is build on the observa-tion that the second symmetric tensor power of the skew-symmetric prolongationspace maps into the space of algebraic curvature tensors. This is initially done forirreducible representations and we also apply the results to compute the holonomyof metric connections with constant 3-form torsion in flat space. Finally we extendTheorem 1.1 to the case of an arbitrary representation, by using a decompositionalgorithm from [10].

In Section 4 we present the proof of Theorem 1.3 and in the section after we provethe other applications of Theorem 1.1 mentioned previously. Section 6 contains the

4 P.-A.NAGY

necessary ingredients for the proof of Theorem 1.3. We use the construction ofinvariant 4-forms for an orthogonal metric representation given in [27] (see also [3])and show that in most cases those 4-forms are multisymplectic on 2-forms, in a senseto be precised. This enables us to use results from [1] and [10] in order to obtaina classification.The case when the holonomy admits no non-zero invariant 4-formsis given a separate treatement by relating the spectral properties of a Casimir typeoperator to the geometry of the situation.

Finally, in Section 7 we look at a class of representation which occur naturallywhen one considers the existence problem for metric connections with skew-torsion.Those representations were classified in [17] under the requirement that their Liealgebra has the involution property. Through arguments similar to those used in theproof of Theorem 1.1, we show that the involutivity assumption can be removed.The paper ends by an Appendix where, for self-containedness reasons, the skew-symmetric prolongation of the adjoint representation of a simple orthogonal Liealgebra is computed in detail.

2. Metric connections with torsion

Let (Mn, g) be an oriented Riemannian manifold. To the metric g one attachesits Levi-Civita connection ∇ which leaves g invariant and moreover it is torsionfree. More generally, a metric connection D on (Mn, g) is a linear connection onTM preserving the Riemannian metric, that is Dg = 0. Any such connection D canbe written as D = ∇ + η where the tensor η belongs to Λ1M ⊗ Λ2M . The torsiontensor Tor in Λ2M ⊗ Λ1M of the connection D is given by

Tor(X, Y ) =DXY −DYX − [X, Y ]

=ηXY − ηYX

for all X, Y in TM . Moreover, the connection D is extended to Λ⋆M by defining

DXϕ = ∇Xϕ+ [ηX , ϕ]

for all X in TM and whenever ϕ belongs to TM . Here we have defined the com-mutator

[α, γ] =

n∑

i=1

(eiyα) ∧ (eiyγ)

for all α in Λ2M and for all γ in Λ⋆M , where ei, 1 ≤ i ≤ n is some local or-thonormal basis in TM . In what follows we shall identify Λ2M and so(TM), bywriting any 2-form α as α = g(F ·, ·) for some skew-symmetric endomorphism ofTM . Also, we shall systematically use the metric g to identify Λ1M to TM . Forfurther computations, it is also useful to note that in some local orthonormal basisei, 1 ≤ i ≤ n of TM , one has

g([F,G]·, ·) =n

∑

i=1

(eiyα) ∧ (eiyβ)

for all F,G in so(TM) with dual 2-forms α = g(F ·, ·) and β = g(G·, ·), where thebracket in l.h.s. is the usual Lie bracket of so(TM).

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 5

Consider now the Cartan decomposition

Λ1M ⊗ Λ2M = W1 ⊕W2 ⊕W3

into irreducible components under the action of SO(n). Explicitly, one has

W1 =Λ3M

W2 =(Λ1M ⊗ Λ2M) ∩Kera ∩Kert

W3 =Λ1M.

Here, for any p ≥ 0 the total alternation map a : Λ1M ⊗ ΛpM → Λp+1M and thetrace map t : Λ1M ⊗ ΛpM → Λp−1M are given, respectively, by

a(η) =

n∑

i=1

ei ∧ ηei

t(η) =n

∑

i=1

eiyηei

for all η in Λ1M ⊗ ΛpM , where ei, 1 ≤ i ≤ n is some local orthonormal frameon M . We also note that the embedding of Λ1M into Λ1M ⊗ Λ2M is given byX 7→ X ∧ ·. Since the torsion tensor of any metric connection on (Mn, g) lives inΛ1M ⊗Λ2M it follows [9] that there are nine main classes to be considered. In thispaper we shall be mainly interested in the class W1, when the torsion is given by a3-form onM and in the classW3 when the torsion tensor is determined by a 1-form.

Explicitly, a metric connection D belongs to the class W1 if and only if it is ofthe form

DX = ∇X +1

2XyT

for all X in TM , where T belongs to Λ3M . In this case one has Tor = T . If theconnection D is in W3 it can be written as

DX = ∇X +X ∧ θ

for all X in TM , where θ is in Λ1M .Given any metric connection D its curvature tensor RD in Λ2M⊗Λ2M is defined

by RD(X, Y ) = −D2X,Y +D2

Y,X −DTor(X,Y ) for all X, Y in TM . In the case of theLevi-Civita connection it will be simply denoted by R. When D = ∇+ η for someη in Λ1M ⊗ Λ2M one obtains the following comparison formula

(2.1) RD(X, Y ) = R(X, Y )−

[

(DXη)Y − (DY η)X

]

+ [ηX , ηY ]− ηTor(X,Y )

for all X, Y in TM . Let us recall now that the Bianchi operator b1 : Λ2M⊗Λ2M →

Λ1M ⊗ Λ3M is defined by

(b1Q)X =

n∑

i=1

ei ∧Q(ei, X)

for all (X,Q) in TM × (Λ2M ⊗Λ2M), where ei, 1 ≤ i ≤ n is some local orthonor-mal frame on M . The following Proposition provides first order information on thebehaviour of a connection in the class W3.

6 P.-A.NAGY

Proposition 2.1. Let D belong to the class W3, that is ηX = X ∧ θ for all X inTM and for some θ in Λ1M . The following hold:

(i) [ηX , ηY ]− ηTor(X,Y ) = |θ|2X ∧ Y whenever X, Y are in TM ;(ii) (b1R

D)X = X ∧ dθ for all X in TM ;(iii) RD(X, Y, Z, U) − RD(Z, U,X, Y ) = −〈[X ∧ Y, dθ], Z ∧ U〉 for all X, Y, Z, U

in TM .

The proof is a straightforward manipulation involving mainly the difference for-mula (2.1), and it is therefore left to the reader.

2.1. Prolongations of Lie algebras. Let (V n, g) be a Euclidean vector space.Given a Lie algebra g we shall say that (g, V ) is an orthogonal representation ifthere exists a Lie algebra morphism g → so(V ). In case this is injective (g, V )is called faithful and the Lie algebra g can be identified with a Lie subalgebra ofso(V ).Unless otherwise specified we shall deal in what follows only with faithfulorthogonal representations.

For any Lie sub-algebra g ⊆ so(V ) we define its first skew-symmetric prolongationby

(2.2) Λ3V ∩ (Λ1V ⊗ g) = T ∈ Λ3V : xyT ∈ g for all x ∈ V

When g = 0 or so(V ), the skew-symmetric prolongation space equals 0 or Λ3V

respectively, therefore we shall assume in what follows that g is proper.The skew-symmetric prolongation enters naturally in the sequence

0 → Λ3V ∩ (Λ1V ⊗ g) → Λ3Vε⊥

→ Λ1V ⊗ g⊥,

where the map ε⊥ is obtained by considering the splitting

id = ε+ ε⊥

along Λ1V ⊗ Λ2V = (Λ1V ⊗ g)⊕ (Λ1V ⊗ g⊥). Explicitly,

ε⊥(T )X = (XyT )g⊥

for all X in V and whenever T is in Λ3V . By means of an orthogonality argumentit is easy to see that moreover

(2.3) Λ1V ⊗ g⊥ = Imε⊥ ⊕

[

(Λ1V ⊗ g⊥) ∩Ker(a)

]

.

The projection map ε⊥ plays an important role in the theory of metric connectionsin the class W1. For, given a G-manifold (Mn, g, G) such a G-connection exists iffthe intrinsic torsion tensor η belongs to Imε⊥ [16, 17].

Remark 2.1. Following [17] we note that those G-structures such that ε⊥ is anisomorphism always admit a (unique) connection with totally skew-symmetric tor-sion. Therefore it is of interest to characterise this maximal case. When G is a Liegroup having the involution property it has been showed in [17] that the only suchrepresentation is (Spin(7),R8).

In fact, ε⊥ is an isomorphism if and only if the g-modules

Λ3V ∩ (Λ1V ⊗ g), (Λ1V ∩ g⊥) ∩Ker(a)

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 7

vanish identically. We shall use this observation later on to describe, in full gen-erality, the case when ε⊥ is an isomorphism. An important case when the skew-symmetric prolongation space is a priori understood is described below.

Theorem 2.1. [2] Let (g, V ) be an orthogonal representation and suppose that thereexists a spinor ψ 6= 0 such that gψ = 0. Then Λ3V ∩ (Λ1V ∩ g) = 0.

2.2. Algebraic curvature tensors and Berger algebras. In order to study theg-module Λ3V ∩ (Λ1V ⊗ g) in the case when the defining representation (g, V ) isirreducible, we need to recall a number of facts related to formal curvature tensorsand to the Berger algebra of (g, V ). To begin with, the space of algebraic curvaturetensors on V is given by

K(so(V )) = R ∈ Λ2V ⊗ Λ2V : b1R = 0,

where b1 : Λ2V ⊗ Λ2V → Λ1V ⊗ Λ3V is the Bianchi map defined in Section 2.Alternatively, one has

K(so(V )) = S2(Λ2V ) ∩Ker(a)

where a : S2(Λ2V ) → Λ4V is the total alternation map.If Q belongs to Λ2V ⊗Λ2V ,its Ricci tensor Ric(Q) in ⊗2V is defined by

Ric(Q)(X, Y ) =

n∑

i=1

Q(X, ei, Y, ei)

for all X, Y in V , and where ei, 1 ≤ i ≤ n is some orthonormal basis in V .Note that Ric : Λ2V ⊗ Λ2V → ⊗2V preserves tensor type, w.r.t the splittingΛ2V ⊗ Λ2V = S2(Λ2V )⊕ Λ2(Λ2V ).

For any Lie sub-algebra g ⊆ so(V ) the space of g-valued curvature tensors of Vis given by

K(g) = K(so(n)) ∩ (Λ2V ⊗ g).

Definition 2.1. The Berger algebra g of the the metric representation (g, V ) is thesmallest subspace p ⊆ g such that K(p) = K(g).

Directly from this definition it follows that if h ⊆ g satisfies K(h) = K(g), theng ⊆ h and also that

g = spanR(X, Y ) : X, Y ∈ V,R ∈ K(g).

The case when g = g is of special interest and then (g, V ) is called a Riemannian ho-lonomy representation. Now we recall that the Berger list of irreducible, orthogonalrepresentations is

(2.4)

g Vso(n) Rn

u(m) R2m

su(m) R2m

sp(m)⊕ sp(1) R4m

sp(m) R4m

spin(7) R8

g2 R7

8 P.-A.NAGY

For irreducible orthogonal representations a precise description of the space of for-mal curvature tensors is given below (see also [29] for the non-metric case).

Theorem 2.2. [10] Let (g, V ) be an irreducible, orthogonal representation. If (g, V )is faithful then either:

(i) K(g) = 0;(ii) (g, V ) belongs to the Berger list;(iii) K(g) ∼= R;

or(iv) (g, V ) = (sp(m)⊕ u(1), [[EL]]).

When K(g) ∼= R the Lie algebra g is the isotropy algebra of a symmetric spaceof compact type and the isomorphism is realised by the Ricci contraction map.Also note that instances in (ii) and (iii) correspond to the case when (g, V ) is aRiemannian holonomy representation, that is g = g. For a proof of these facts andmore detail on the representations in the Berger list we refer the reader to [7, 8, 10].We equally need to recall that the main result in [10, Thm.4.6] also implies:

Proposition 2.2. Let (G, V ) be an irreducible orthogonal representation of a Liegroup G, with Lie algebra g. If the representation (g, V ) is not irreducible thenK(g, V ) = 0.

We end this section with a few facts, the first of which is well known [24], concern-ing the Ricci tensors of elements in K(g), when in presence of additional invariantobjects.The second fact below has been proved in [10] for irreducible representationsand here we give it a direct proof.

Proposition 2.3. Let (g, V ) be a metric representation. The following hold:

(i) if there exists a spinor x 6= 0 such that gx = 0 then Ric(R) = 0 for all R inK(g);

(ii) the same conclusion holds if (g, V ) is faithful and there exists a non-degenerate,g-invariant element, in Λ3V ∩ (Λ1V ⊗ g⊥).

Proof. (i) Let R belong to K(g), so that R(X, Y )x = 0 for all X, Y in V . Since the

Clifford contractionn∑

i=1

eiR(ei, X)x = 0 for all X in V , after using that b1R = 0 we

arrive at (RicX)x = 0 for all X in V and the claim follows.(ii) Let T in Λ3V ∩ (Λ1V ⊗ g⊥) be g-invariant and non-degenerate. If R belongs toK(g) we have [R(X, Y ), T ] = 0 for all X, Y in V . In particular,

n∑

i=1

eiy[R(ei, X), T ] = 0

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 9

for all X in V , where ei, 1 ≤ i ≤ n is some orthonormal basis. But

n∑

i=1

eiy[R(ei, X), T ] =∑

i,k

eiy(R(ei, X)ek ∧ ekyT )

=−RicXyT −∑

i,k

R(ei, X)ek ∧ eiyekyT

=−RicXyT +1

2

∑

i,k

R(ek, ei)X ∧ eiyekyT

for all X in V , after using the Bianchi identity for R. Now the last sum equalsn∑

p=1

R(epyT )X ∧ ep and since Y yT ∈ g⊥ for all Y in V , it vanishes. We have shown

that RicXyT = 0 for all X in V , therefore the non-degeneracy of R yields thevanishing of the Ricci contraction of R.

3. Structure results and flat holonomy algebras

We shall compute in this section the first skew-symmetric prolongation of a properLie sub-algebra g of so(V ), i.e the intersection Λ3V ∩(Λ1V ⊗g), under the additionalassumption that the representation (g, V ) is irreducible.

For any T in Λ3V and for any X in V let TX in so(V ) be the dual of XyT w.r.t.to the metric g, that is XyT = g(TX ·, ·). If γ belongs to Λ⋆V , let Lγ : Λ⋆V → Λ⋆V

be the exterior multiplication with γ, and let L⋆γ be the adjoint of Lγ . Note that if

α = g(F, ·, ·) is a 2-form then one has

L⋆αϕ =

1

2

n∑

i=1

eiyFeiyϕ

whenever ϕ belongs to Λ⋆V . Our starting point is the following simple observation.

Lemma 3.1. Let T belong to Λ3V ∩ (Λ1V ⊗ g). The tensor RT in Λ2V ⊗ g definedby

RT (X, Y ) = [TX , TY ] + 2TTXY

for all X, Y in V satisfies the following:

(i) RT belongs to K(g);(ii) Ric(RT )(X, Y ) = 3〈XyT, Y yT 〉 for all X, Y in V ;(iii) RT = 0 if and only if T = 0.

Proof. (i) Since TX belongs to g for all X in V we have that RT (X, Y ) ∈ g for allX, Y in V . That b1R

T = 0 and also the proof of (ii) are direct easy computationswhich are left to the reader. Now (iii) follows directly from (ii) by a positivityargument.

Remark 3.1. (i) The result in (i) of the Lemma above still holds, with unchangedproof, if the metric g has arbitrary signature.(ii) If g has indefinite signature, there are examples [13] of 3-forms T 6= 0 such thatTXTY = 0 for all X, Y in V , and hence RT = 0.

10 P.-A.NAGY

Using the structure results on the space of K(g) in Theorem 2.2 we proceed nowto the computation of the first skew-symmetric prolongation Λ3V ∩ (Λ1V ⊗ g), inseveral steps.

Lemma 3.2. Let (g, V ) be irreducible with K(g) ∼= R. If T 6= 0 is in Λ3V ∩(Λ1V ⊗g)then:

(i) the 4-form ΩT = 12

n∑

i=1

Tei ∧ Tei belongs to Λ4V ∩ (Λ2V ⊗ g);

(ii)

[TTXY , TZ ] + [TTZX , TY ] + [TTY Z , TX ] = −TΩT (X,Y,Z)

for all X, Y, Z in V ;(iii) [ΩT (X, Y ), T ] = −2TXY yΩ

T for all X, Y in V ;(iv) X 7→ L⋆

TXΩT belongs to Λ3V ∩ (Λ1V ⊗ g).

Proof. (i) follows directly from

(3.1) ΩT (X, Y ) = −[TX , TY ] + TTXY

for all X, Y in V and the fact that T is in Λ3V ∩ (Λ1V ⊗ g).(ii) since K(g) is 1-dimensional any of its elements is g-invariant. Hence RT isg-invariant and since TX ∈ g for all X in V we get

RT (TXY, Z) +RT (Y, TXZ) + [RT (Y, Z), TX ] = 0

for all X, Y, Z in V . After expanding RT following its definition, this yields

[TTXY , TZ ] + [TTZX , TY ] + 2[TTY Z , TX ]− 2TTZTXY+TY TZX + [[TY , TZ ], TX ] = 0

for all X, Y, Z in V . The claim follows now by taking the cyclic sum above whileusing the Jacobi identity combined with (3.1).(iii) when rewritten by using ΩT under the form given in (3.1) to absorb doublecommutators, (ii) yields

σX,Y,ZΩT (TXY, Z) = 2TΩT (X,Y,Z)

for all X, Y, Z in V .Since ΩT belongs to S2(Λ2V ) it follows

σX,Y,ZΩT (U1, U2, TXY, Z) = 2ΩT (X, Y, Z, TU1

U2) = −2ΩT (TU1U2, X, Y, Z)

for all X, Y, Z, U1, U2 in V and the conclusion is now straightforward.(iv) Using (iii) we obtain

n∑

i=1

eiy[ΩT (ei, X), T ] = −2

n∑

i=1

eiyTeiXyΩT = −4L⋆TX

ΩT

for all X in V . Butn

∑

i=1

eiy[ΩT (ei, X), T ] =−

∑

i,k

ΩT (ei, X)ek ∧ Tekei = −∑

k,p

ΩT (Tekep, ek)X ∧ ep

=−Xy

∑

p,k

ΩT (Tekep, ek) ∧ ep +

n∑

k=1

ΩT (TekX, ek).

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 11

Now some elementary considerations lead to L⋆TX

ΩT = 13Xy(

n∑

p=1

ep ∧ L⋆Tep

ΩT ) and

the claim follows by also using that ΩT ∈ Λ2V ⊗ g.

Another preliminary result we need is:

Lemma 3.3. Let (g, V ) be irreducible with K(g) ∼= R. If T 6= 0 is in Λ3V ∩(Λ1V ⊗g)and there exists ϕ in Λ3V ∩ (Λ1V ⊗ g) such that

[Xyϕ, T ] = 0

for all X in V , then either ϕ = 0 or (g, V ) is the adjoint representation of a simpleLie algebra.

Proof. By a contraction argument, one also has [TX , ϕ] = 0 for all X in V , hence[RT (X, Y ), ϕ] = 0 whenever X, Y are in V by using the Jacobi identity. Since K(g)is one dimensional we have that

g = g = spanRT (X, Y ) : X, Y ∈ V

hence [g, ϕ] = 0. Now, if (g, V ) is not the adjoint representation of a simple Liealgebra, any g-invariant element in Λ3V ∩ (Λ1V ⊗ g) vanishes (see [10]) and theproof is complete.

Proposition 3.1. Let (g, V ) be irreducible and proper. The following hold:

(i) if K(g) = 0 we must have Λ3V ∩ (Λ1V ⊗ g) = 0;(ii) if K(g) is 1-dimensional then either Λ3V ∩ (Λ1V ⊗ g) = 0 or (g, V ) is the

adjoint representation of a compact, simple Lie algebra. In the latter case,Λ3V ∩ (Λ1V ⊗ g) is 1-dimensional, generated by the Cartan 3-form of g.

Proof. (i) follows directly from Lemma 3.1.(ii) Let us suppose first that (g, V ) is not the adjoint representation of a compact,simple Lie algebra. Let T 6= 0 belong to Λ3V ∩ (Λ1V ⊗ g). Since RT is g-invariantit follows that its Ricci contraction is proportional to 1V . Therefore, Lemma 3.1,(ii) yields 〈XyT, Y yT 〉 = λ〈X, Y 〉 whenever X, Y are in V , where λ 6= 0 is some

real number. Equivalently, we have −n∑

i=1

T 2ei= λ1V , where ei, 1 ≤ i ≤ n is some

orthonormal basis in V . Now

[L⋆TX

ΩT , T ] =1

2

n∑

i=1

[ΩT (ei, TXei), T ] =n

∑

i=1

T 2eiXyΩT = −λXyΩT = λ[TX , T ]

for all X in V , after making use of (iii) in Lemma 3.2. Because X 7→ L⋆TX

ΩT is inΛ3V ∩ (Λ1V ⊗ g) by (iv) of Lemma 3.2, we apply Lemma 3.3 to find that

L⋆TX

ΩT = λTX

for all X in V . Keeping this in mind we compute

〈[TTXY , TZ ], TU〉 =〈TTXY , [TZ , TU ]〉 = −〈TTXY ,ΩT (Z, U) + TTZU〉

=− 〈ΩT (TTXY )Z, U〉 − λ〈TXY, TZU〉 = −λ〈TTXY Z, U〉 − λ〈TXY, TZU〉

=2λ〈TZTXY, U〉

12 P.-A.NAGY

for all X, Y, Z, U in V . It follows that

〈[TTXY , TZ ] + [TTZX , TY ] + [TTY Z , TX ], TU〉 = 2λΩT (X, Y, Z, U).

Comparison with (ii) in Lemma 3.2 leads to 2λΩT (X, Y, Z, U) = −λΩT (X, Y, Z, U)for all X, Y, Z, U in V hence ΩT = 0. It follows that RT (X, Y ) = 3TTXY for all X, Yin V . In other words RT ∈ h, where the subalgebra h ⊆ g is given by

h = TX : X ∈ V .

Since K(h) ⊆ K(g) and the latter is 1-dimensional we conclude that K(h) = K(g)hence g ⊆ h. Because g = g we must have h = g, in other words (g, V ) is the adjointrepresentation of a compact, simple Lie algebra, a contradiction. We have showedthat Λ3V ∩ (Λ1V ⊗ g) = 0 in this case.

If (g, V ) is the adjoint representation of a compact, simple Lie algebra, the com-putation of the space Λ3V ∩ (Λ1V ⊗ g) is contained in the Appendix, Proposition8.1.

Remark 3.2. From the proof of (i) above we see that Λ3V ∩ (Λ1V ⊗ g) vanisheswhen K(g) does, regardless of the irreducibility of the representation involved.

We are now ready to prove our main result in this section.

Theorem 3.1. Let g ⊆ so(V n) be proper such that (g, V ) is irreducible. Thefollowing hold:

(i) Λ3V ∩ (Λ1V ⊗ g) = 0 unless (g, V ) is the adjoint representation of a com-pact, simple Lie algebra;

(ii) ΛpV ∩ (Λp−2V ⊗ g) = 0 for 4 ≤ p ≤ n.

Proof. (i) By Proposition 3.1 it is enough to study only the representations en-tering in the Berger list. When g = su(n), sp(m), sp(m) ⊕ u(1), spin(7), g2 theskew-symmetric prolongation vanishes, because in these cases the algebra g fixes aspinor and Theorem 2.1 applies. When g = u(m), sp(m)⊕ sp(1) it is an easy directexercise to verify that the skew-symmetric prolongation vanishes.(ii) the proof consists in the examination of the various instances for Λ3V ∩(Λ1V ⊗g),as indicated in (i).

(1) Λ3V ∩ (Λ1V ⊗ g) = 0;IfXk, 1 ≤ k ≤ p−3 are fixed, thenX1y . . .Xp−3yT belongs to Λ3V ∩(Λ1V ⊗g)and hence it vanishes, and we conclude that T = 0.

(2) Λ3V ∩ (Λ1V ⊗ g) 6= 0;In analogy with the inductive argument above it is enough to see that (Λ2V ⊗g) ∩ Λ4V = 0. Since in this case (g, V ) is the adjoint representation of asimple Lie algebra, let ϕ in Λ3V ∩ (Λ1V ⊗ g) be its Cartan form. Then anyelement β in (Λ2V ⊗ g) ∩ Λ4V satisfies

Xyβ = 〈X, ζ〉ϕ

for all X in V , where ζ belongs to V . If ζ 6= 0 we get that ζyϕ = 0, acontradiction since the Cartan form of a simple Lie algebra is non-degenerate.Therefore ζ = 0, hence β = 0, and we have showed that (Λ2V ⊗ g) ∩ Λ4V =0 in this case as well.

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 13

Corollary 3.1. Let (g, V n), n ≥ 3 be an irreducible and faithful orthogonal repre-

sentation. If g is proper we have dimRg ≤ n2−13

.

Proof. By making use of Theorem 3.1, (i) it is clear that we need to treat only thecase when Λ3V ∩ (Λ1V ⊗ g) = 0. Then ε⊥ : Λ3V → Λ1V ⊗ g⊥ is injective, inparticular dimRΛ

3V ≤ dimR(Λ1V ⊗ g⊥) and the claim follows easily.

Theorem (3.1) now be directly applied to obtain a classification of the holonomyalgebras generated by a 3-form or equivalently of the holonomy algebras of connec-tions with constant torsion in flat space. Indeed, given a linear 3-form T 6= 0 inΛ3V one forms the connection

∇TX = ∇X +

1

2XyT

for all X in V . Here V is equipped with its flat metric and ∇ is the associatedLevi-Civita connection. The connection ∇T is metric and its torsion is totallyskew-symmetric, given by the form T . The holonomy algebra h⋆T is computed ash⋆T = [g⋆T , g

⋆T ] where the Lie subalgebra g⋆T of so(V ) is defined by

g⋆T = LieXyT : X ∈ V .

It is known [2] that g⋆T is semisimple hence h⋆T = g⋆T . It has also been shown in [2]that for any T there is an orthogonal splitting of representations

(g⋆T , V ) = V0 ⊕d

⊕

i=1

(g⋆Ti, Vi)

where V0 is trivial factor and the representations (g⋆Ti, Vi) where Ti is in Λ3Vi are

irreducible for 1 ≤ i ≤ d.Therefore the study of the representation (g⋆T , V ) reduces to the case when it is

irreducible. Since by construction we have that T belongs to Λ3V ∩ (Λ1V ⊗ g⋆T )applying Theorem 3.1 yields immediately:

Theorem 3.2. Let T 6= 0 in Λ3V be such that the representation (g⋆T , V ) is ir-reducible. Then either g⋆T = so(V ), or the representation (g⋆T , V ) is the adjointrepresentation of a simple Lie group g on its Lie algebra, with T proportional to theCartan 3-form. In the latter case we have that g⋆T = g.

3.1. Reducible representations. To compute Λ3V ∩(Λ1V ∩g) in full generality itis necessary to examine the case when the representation (g, V ) is reducible. Indeed,we consider the orthogonal splitting

V = V0 ⊕d

⊕

k=1

Vk

where g acts trivially on V0 and each of the representations (g, Vk), 1 ≤ k ≤ d areirreducible. Moreover we define

Vi =

d⊕

k 6=i

k=1

Vk

14 P.-A.NAGY

and let πk : g → so(Vk) the restriction of our original representation to Vk. Finally,we consider the ideal

gk := (Kerπk)

of g given by for all 1 ≤ k ≤ d. Then each of the orthogonal representations (gk, Vk)is faithful.

Proposition 3.2. We have that

Λ3V ∩ (Λ1V ⊗ g) =d

⊕

k=1

Λ3Vk ∩ (Λ1Vk ⊗ gk)

Proof. Let us pick T in Λ3V ∩ (Λ1⊗ g), that is T in Λ3V such that Tv belongs to g,for all v in V . First of all, since gV0 = 0 we get that TvV0 = 0 for all v in V henceTv0 = 0 for all v0 in V0. Now since gVk ⊆ Vk it follows that TvVk ⊆ Vk for all v in

V and for all 1 ≤ k ≤ d, in other words T is ind⊕

k=1

Λ3Vk. Let us write T =d∑

k=1

Tk

for the corresponding decomposition of T . Then for any v in Vk, 1 ≤ k ≤ d we haveTvVk = 0 and since Tv belongs to g it follows that Tk belongs to Λ3Vk ∩ (Λ1Vk ⊗ gk)for all 1 ≤ k ≤ d and the claim follows easily.

The decomposition algorithm used above has been first presented and used toobtain a similar result for the space K(g) in [10]. Note however that the represen-tations (gi, Vi) may be still reducible. To circumvent this difficulty, it is helpful tomake the following straightforward:

Lemma 3.4. Let (g, V ) be a faithful orthogonal representation admitting a g-invariant and orthogonal splitting V = V1 ⊕ V2. If l is the ideal of g given byl = (g1 ⊕ g2)

⊥ then:

(i) the representations (gk ⊕ l, Vk), k = 1, 2 are faithful;(ii) if g acts irreducibly on V1 then (g1 ⊕ l, V1) is an irreducible representation.

Whereas this does not iterate well, it can be used for the computation of a g-module splitting as the skew-symmetric prolongation does.

Theorem 3.3. Let (g, V ) be a faithful orthogonal representation. Then (g, V ) splitsas a direct sum of representations of the following type:

(i) the trivial representation;(ii) (so(W ),W ) where W is some Euclidean vector space;(iii) adjoint representations of compact, simple Lie algebras;(iv) representations with vanishing skew-symmetric prolongation.

Proof. If Λ3V ∩(Λ1V⊗g) = 0 the representation is of type (iv) and there is nothingto prove. Otherwise, and if the representation is not trivial, by using Proposition3.2 we find that there exists 1 ≤ p ≤ d such that Λ3Vp ∩ (Λ1Vp ⊗ gp) 6= 0.

We split orthogonally g = gp ⊕ gp ⊕ l where gp = so(Vp) ∩ g and l = (gp ⊕ gp)⊥

and note that (gp ⊕ l, Vp) is faithful and irreducible by Lemma 3.4. MoreoverΛ3Vp∩(Λ1Vp⊗(gp⊕ l)) 6= 0 since it contains the non-vanishing Λ3Vp∩(Λ1Vp⊗ gp).By applying Theorem 3.1 when taking into account that gp 6= 0 we find that theideal l vanishes and hence (gp, Vp) splits out as a direct factor of type (ii) or (iii)

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 15

with the exception of the case when Vp is 4-dimensional and gp = Λ2+Vp

∼= so(3).Butthis case is not eligible since then (gp, Vp) is irreducible hence its skew prolongationvanishes, again by Theorem 3.1, a contradiction.

4. Uniqueness of connections with totally skew-symmetric torsion

Let (Mn, g) be a Riemannian manifold equipped with a metric connection D inthe class W1 That is, D = ∇ + 1

2T where T belongs to Λ3M . Let G = Hol(D) ⊆

O(n) be the holonomy group of the connection D and we also denote by g theholonomy algebra of the Lie group G, at some point of the manifold M . We shallinvestigate now the uniqueness of such a connection. Namely, let us assume that

D′ = ∇+1

2T ′

belongs to the class W1 and is such that Hol(D′) ⊆ G. Since the holonomy sub-bundle g ⊆ Λ2M is preserved by D and D′ it follows that

[Xy(T − T ′), g] ⊆ g

for all X in TM . This can be easily seen to lead to the difference T − T ′ beingcontained in Λ3V ∩(Λ1V ⊗g) where we have defined the extension g of the holonomyalgebra g by

g = g⊕ F ∈ g⊥ : [F, g] = 0.

We can now make:

Theorem 4.1. Let (Mn, g) be a Riemannian manifold admitting a connection D

in the class W1 with holonomy Hol(D) ⊆ G ⊂ O(n). If the representation (G, V )is irreducible, then D is unique with these properties, exception made of the casewhen (G, V ) is the adjoint representation of a simple Lie group.

Proof. Let us consider the Lie subgroup of O(n) given by G = g ∈ O(n) : g⋆g ⊆ g.The Lie algebra of G clearly equals g and moreover (G, V ) is irreducible since

G ⊆ G. As indicated above, we need to examine the skew-symmetric prolongationspace Λ3V ∩ (Λ1V ⊗ g). If (g, V ) does not act irreducibly then K(g) = 0 byProposition 2.2 and then the skew-symmetric prolongation space vanishes as well(see also Remark 3.2). When (g, V ) is irreducible, using Theorem (3.1) yields thatΛ3V ∩ (Λ1V ⊗ g) = 0 unless (g, V ) is the adjoint representation of a compact,simple, Lie algebra. If this is the case, g = g, and the claim follows.

5. A class of Plucker type embeddings and k-Lie algebras

Using the computation of the skew-symmetric prolongation of a Lie algebra ofcompact type we have obtained previously, we shall present here a class of Pluckertype relations, which have been first considered and treated in dimensions up to 8in [19]. Our present context will be that of a Euclidean vector space (V n, g). Weare interested here in the class of forms T in ΛpV, p ≥ 3 satisfying

(5.1) [L⋆ζT, T ] = 0

for all ζ in Λp−2V , where we recall that for any ζ in Λp−2V the map L⋆ζ is the

adjoint of the exterior multiplication by ζ . When p = 3 every form T subject to

16 P.-A.NAGY

(5.1) induces a Lie algebra structure on V which is given by [X, Y ] = TXY for allX, Y in V (see also [34] for a related question).

Note that the classical Plucker relations (see [22, 15]) for a p-form T state thatT is decomposable if and only if it satisfies

(5.2) L⋆ζT ∧ T = 0

whenever ζ belongs to Λp−1V . Here a p-form is called decomposable if it can bewritten as the sum of mutually orthogonal simple p-forms.

Let us first observe the impact of having (5.1) satisfied for some form T on itsisotropy algebra. Recall that the latter is defined by

gT = α ∈ Λ2V : [α, T ] = 0

and also that the representation (gT , V ) is metric. At the same time let us considerthe Lie algebra

rT := LieL⋆ζT : ζ ∈ Λp−2V .

It is clear that T satisfies the Plucker relations (5.1) if and only if one has rT ⊆ gT .Both gT and rT have good splitting properties and in order to progress in thatdirection we make:

Definition 5.1. Let T belong to ΛpV . We call T irreducible iff the orthogonalrepresentation (rT , V ) is irreducible.

Proposition 5.1. Let T in ΛpV satisfy (5.1). Then V admits an orthogonal, directsum decomposition

V = V0 ⊕d

⊕

k=1

Vk

such that T vanishes on V0 and T =d∑

k=1

Tk where:

(i) Tk belong to ΛpVk and satisfy (5.1) for 1 ≤ k ≤ d;(ii) the forms Tk, 1 ≤ k ≤ d are irreducible.

Proof. (i) Since (rT , V ) is a metric representation it splits as

V = V0 ⊕d

⊕

k=1

Vk

on orthogonal direct sum, where rT acts trivially on V and (rT , Vk), 1 ≤ k ≤ d areirreducible. It follows that 〈viyvjyT, ζ〉 = 0 for all vi in Vi, vj in Vj and for all ζ inΛp−2V . It is now easy to conclude that T splits as indicated and (i) holds.To prove (ii) we notice that since (5.1) holds for Tk we have that rTk

⊆ gTkfor all

1 ≤ k ≤ d, hence the latter acts irreducibility on Vk as the former does.

Theorem 5.1. Let T in ΛpV, p ≥ 4 satisfy (5.1). If T 6= 0 is irreducible then wemust have p = dimRV , therefore T is proportional to a volume form.

Proof. The defining equation (5.1) actually says that T belongs to

(Λp−2V ⊗ gT ) ∩ ΛpV.

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 17

Since T 6= 0, by using Theorem 3.1, (ii) it follows that gT = so(V ). In this case Tmust be invariant under so(V ), hence proportional to a volume form. This forcesthe dimension of V to equal p.

In the case of indefinite signature, examples where Theorem 5.1 fails to hold fora class of 4-forms have been constructed in [12].

5.1. Metric n-Lie algebras. The Plucker relations (5.1) are related, as it has beenobserved in [19] to the class of the so-called n-Lie algebras. More precisely,

Definition 5.2. Let V be a vector space over R (not necessarily of finite dimension).Then a n-Lie algebra structure on V consists in a map

[·, . . . , ·] : ΛnV → V

such that the generalised Jacobi identity

[X1, . . . , Xn−1, [Y1, . . . , Yn]] =

n∑

i=1

[Y1, . . . , [X1, . . . , Xn−1, Yi], . . . , Yn]

holds, whenever Xk, Yk, 1 ≤ k ≤ n− 1 are in V .An n-Lie algebra is called metric if there is a non-degenerate, symmetric bilinearform 〈·, ·〉 on V such that

〈[X1, . . . , Xn], Xn+1〉 = −〈[X1, . . . , Xn+1], Xn〉

Finally, let us call a metric n-Lie algebra Euclidean iff its associated bilinear formis positive definite. To see that one can classify finite dimensional, Euclidean n-Liealgebra we need to recall first

Proposition 5.2. [19] Let (V,< ·, · >, [·, . . . , ·]) be a metric n-Lie algebra. Thenthe n+ 1-form T in Λn+1V defined by

T (X1, . . . , Xn, Xn+1) = 〈[X1, . . . , Xn], Xn+1〉

whenever Xk, 1 ≤ k ≤ n+ 1 belong to V , satisfies the Plucker type relations (5.1).

This is very similar to the construction of the Cartan 3-form of a metric Liealgebra. By using Theorem 5.1 we immediately obtain the following.

Theorem 5.2. Let V be a finite dimensional n-algebra. If the form T is irreduciblethen either V is a Lie algebra or n = dimRV − 1, in which case T is proportionalto a volume form.

6. The holonomy of connections with vectorial torsion

This section is devoted to the study of the holonomy representation of metricconnections with torsion in Λ1 ⊗ Λ2. Particular attention will be paid to the prop-erties of invariant 4-forms, for those shall be used to obtain geometric informationin our situation.

18 P.-A.NAGY

6.1. Invariant forms and Casimir operators. Let us consider again a Euclideanvector space (V n, g), n ≥ 5 equipped with a faithful action of some Lie algebra g,therefore considered as a subalgebra of so(V ). The existence problem of g-invariantforms on V has been considered in [27] by means of the following construction.

Definition 6.1. The characteristic form of the representation (g, V ) is given by

T g = a(1g)

where a : S2g → Λ4V is the alternation map.

Clearly T g in Λ4V is g-invariant and if the representation (g, V ) is induced bythat of a Lie group G then T g is invariant under G as well.To consider the casewhen the characteristic form vanishes we also need:

Definition 6.2. Let (g, V ) be a faithful metric representation. It is called minimalif one has T g = 0.

On a given representation minimality turns out to be a strong requirement as thefollowing shows.

Proposition 6.1. Let (g, V ) be faithful and metric. Then:

(i) (g, V ) is minimal iff 1g belongs to K(g, V );(ii) if there exists a spinor x 6= 0 such that gx = 0 then T g 6= 0.

Proof. (i) follows immediately from the definition of T g, since 1g belongs to S2(g).(ii) supposing that T g = 0, it follows by (i) that 1g ∈ K(g) hence Ric(1g) = 0 byProposition 2.3, since g preserves a spinor. But

Ric(1g)(X, Y ) =

n∑

i=1

〈(ei ∧X)g, ei ∧ Y 〉

=∑

Fk∈g

〈FkX,FkY 〉

for all X, Y in V , where Fk is an orthonormal basis in g. It follows that Ric(1g)vanishes if and only if (g, V ) is the trivial representation, a contradiction.

All entries in the Berger list (2.4) but the first and u(m), sp(m)⊕sp(1) fix a spinorand therefore cannot be minimal.Moreover a direct verification using this time (i) inthe Proposition above shows that u(m) and sp(m)⊕ sp(1) are not minimal neither,making that only so(n), n 6= 4 is. On the contrary, the adjoint representation of asemisimple Lie algebra is minimal as well as any irreducible representation of a Liealgebra on a 5-dimensional Euclidean space. In fact, the list of minimal irreduciblerepresentations has been given in [27] in terms of their associated symmetric spaces.

For our purposes, the vanishing of the characteristic form is best described interms of the Casimir operators

Ck : ΛkV → ΛkV

of the representation (g, V ), defined by Ck(ϕ) = −∑

xk∈g

[xk, [xk, ϕ]] for some or-

thonormal basis xk in g and for all ϕ in ΛkV, 0 ≤ k ≤ n. We have that Ck ≥ 0 forall k ≥ 0 and when k = 1 the operator C1 gives actually an element in S2V , which

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 19

is invariant under g, that is [g, C1] = 0. A technical observation we shall need isthat a straightforward verification yields

(6.1) L⋆α1(α2 ∧ α3) = g(F3F1F2 + F2F1F3, ·) +

1

2〈F1, F2〉α3 +

1

2〈F3, F1〉α2

for all αi = g(Fi·, ·), 1 ≤ i ≤ 3 in Λ2V .

Proposition 6.2. The following hold:

(i) T g = 0 if and only if C2 = C1, · − 1g;(ii) if an element B in g⊥ satisfies [B, g⊥] ⊆ g then

C2B = (n− 2)B.

Proof. (i) It is enough to note that after using (6.1), one has

L⋆FT

g = Fg + 2∑

k

xkFxk = 1g(F ) + C2F − C1, F

for all F in Λ2V .(ii) let yk be an orthonormal basis in g⊥. We compute, for a given F in g⊥

|[yk, F ]|2 = |[yk, F ]g|

2 + |[yk, F ]g⊥|2.

But∑

k

|[yk, F ]g|2 =

∑

k,i

〈[yk, F ], xi〉2 = 〈C2F, F 〉

after using that [g, g⊥] ⊆ g⊥, and where xi is an orthonormal basis in g. Thereforewe end up with

(6.2)∑

k

|[yk, F ]g⊥|2 = 〈(C − 2C2)F, F 〉

where C denotes the Casimir operator of so(n) which, as well known, is given byC = 2(n− 2). Our claim follows now immediately by polarising (6.2).

6.2. Non degeneracy properties and the classification. We shall now inves-tigate up to what extent T g is a multisymplectic 4-form. As we shall see later onthis is directly related to the geometry of G-structures of vectorial type.

Proposition 6.3. Let us (G, V ) be an irreducible orthogonal representation of someLie group G, with Lie algebra denoted by g. If dimRV 6= 8 and T g 6= 0, the mapL⊕ L⋆ : Λ2V → Λ6V ⊕ Λ2V given by

(L⊕ L⋆)α = (α ∧ T g, L⋆αT

g)

for all α in Λ2V is injective.

Proof. Let α = 〈F ·, ·〉 belong to Ker(L⊕ L⋆), that is α ∧ T g = 0 and L⋆αT

g = 0. Ifei, 1 ≤ i ≤ n denotes an orthonormal basis in V we have

0 = L⋆α(α ∧ T g) =

1

2Feiyeiy(α ∧ T g) =

1

2Feiy(Fei ∧ T

g + α ∧ eiyTg)

=1

2|Fei|

2T g+ F 2ei ∧ eiyTg+

1

2α ∧ (FeiyeiyT

g).

20 P.-A.NAGY

Because L⋆αT

g = 0 it follows after summation that

1

2|F |2T g +

n∑

i=1

F 2ei ∧ eiyTg = 0.

The irreducibility of (G, V ) implies after using the G-invariance of T g that

〈xyT g, yyT g〉 = c〈x, y〉

for all x, y in V , for some c in R⋆. It is computed by nc = 4|T g|2. Therefore bytaking the scalar product with T g in our last equation we arrive at

1

2|F |2|T g|2 = c|F |2 =

4

n|F |2|T g|2

and the claim follows since n 6= 8 and T g is non-vanishing.

Remark 6.1. When n = 8, let us equip with V an orientation, given by a volumeform ν in Λ8V and split T g = T+ + T− along the decomposition Λ4 = Λ4

+V ⊕Λ4−V .

Then any 2-form α in Ker(L⊕ L⋆) satisfies

α ∧ T± = L⋆αT

± = 0.

It is not true in general that α must vanishes. Examples of self-dual four forms withnon-trivial annihilator in the sense above have been constructed in [4].

Another auxiliary result we shall need in order to treat the case when the holo-nomy representation of a connection with vectorial torsion has vanishing character-istic form is the following.

Lemma 6.1. Let (g, V n), n > 4 be a faithful and irreducible orthogonal representa-tion such that T g = 0. If F in Λ2V satisfies

[F, g⊥] ⊆ g

then F = 0.

Proof. We split F = A +B along Λ2V = g⊕ g⊥ and notice that

[C2F, g⊥] = [C2A+ C2B, g⊥] ⊆ g

because of [g, g⊥] ⊆ g⊥ and after use of the Jacobi identity. Since our representationis irreducible, we have C1 = µ1V where

(6.3) µ =dimRg

n.

Because T g = 0, from Proposition 6.2, (i) we get

[(2µ− 1)A+ 2µB, g⊥] ⊆ g

which combined with our original assumption yields [A, g⊥] ⊆ g. Hence [A, g⊥] = 0by using again that [g, g⊥] ⊆ g⊥. It follows that CA − C2A = 0 in other words

(2µ − 1 − 2(n − 2))A = 0. If A 6= 0 by using (6.3) we get dimRg = n(2n−3)2

and

since dimRg ≤ n(n−1)2

it follows that n ≤ 2, a contradiction, so A = 0. We are left

with [B, g⊥] ⊆ g and let us assume that B 6= 0. Using (ii) in Proposition 6.2 yields

then C2B = (n − 2)B whence dimR g = n(n−2)2

again by (i) in Proposition 6.2 and

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 21

(6.3).The dimension bound in Corollary (3.1) implies then n ≤ 5 hence n ≤ 4 sincen 6= 5 for arithmetic reasons. We get a contradiction with the assumption thatB 6= 0 and the claim is proved.

The last ingredient we need in order to state and prove our main result in thissection is the following Lemma.

Lemma 6.2. [11] Let M be a connected manifold and let θ in Λ1M be a closed1-form, that is dθ = 0. If λ is a smooth, non-identically zero, function on M suchthat

dλ+ λθ = 0

then λ is nowhere vanishing and θ = −dln|λ|.

We can now make the following.

Theorem 6.1. Let (Mn, g), n 6= 4, 8 be a connected and oriented Riemannian mani-fold and let D be a metric connection with vectorial torsion θ in Λ1M ⊆ Λ1M⊗Λ2M .If the holonomy representation (G, V ) of D is irreducible and proper then:

(i) dθ = 0;(ii) if (G, V ) is not on the Berger list then either:

(a) the connection D is flat,or

(b) the metric g is conformal to a non-flat,locally symmetric Riemannianmetric;

(iii) if (G, V ) is on the Berger list then (M, g) has a l.c.p. structure.

Proof. Let us denote by g the Lie algebra of the connected Lie group G. At agiven point of the manifold we consider the G-invariant form T g which can be thenextended over M , by using parallel transport, to a parallel form w.r.t to D.

(i) We subdivide the proof into several cases.

(1) T g 6= 0.Since DT g = 0 we have [1]

dT g = 4θ ∧ T g, d⋆T g = −(n− 4)θyT g.

Applying d respectively d⋆ to these equations it follows easily that

(L⊕ L⋆)dθ = 0.

provided that n 6= 4. By using Proposition 6.3 we conclude that θ is closed,that is dθ = 0, provided that n 6= 8, too.

(2) T g = 0.The curvature tensor RD of the connection D lies in Λ2M ⊗ g hence byevaluation of (iii) in Proposition 6.1 on g⊥ ⊗ g⊥ we find that [dθ, g⊥] ⊆ g.Moreover by (i) in Proposition 6.1 we must have K(g) 6= 0 so (g, V ) must beirreducible too, by Proposition 2.2. Applying now Lemma 6.1 we get thatdθ = 0.

We have shown that θ is a closed 1-form in both cases, hence (i) is completelyproved.(ii) By (i) the 1-form θ is closed hence the curvature tensor RD belongs to K(g)at any point of M , after also using (ii) in Proposition 2.1. If K(g) = 0 then the

22 P.-A.NAGY

connection D must be flat. Otherwise (g, V ) is again irreducible by Proposition 2.2and thus K(g) is 1-dimensional, so that

(6.4) RD = λRg

for some smooth function λ on M , which moreover turns out to be non identicallyzero. Indeed the vanishing of λ implies that of RD and hence the vanishing ofg by using the Ambrose-Singer theorem, a contradiction with the irreducibility of(g, V ).Here Rg denotes the parallel transport over M of the (G-invariant) generatorof K(g, V ) overM . Now, the differential Bianchi identity for the connection D reads(see [5]):

σX,Y,Z(DXRD)(Y, Z) +RD(Tor(X, Y ), Z) = 0

for all X, Y, Z in TM , where σ denotes the cyclic sum. Because the torsion tensorof D is given by Tor(X, Y ) = θ(X)Y − θ(Y )X for all X, Y in TM we obtain

σX,Y,ZRD(Tor(X, Y ), Z) = 2σX,Y,Zθ(X)RD(Y, Z)

whenever X, Y, Z in TM . By also using (6.4) and the fact that DRg = 0 we arriveat

σX,Y,Z(dλ+ 2λθ)X · Rg(Y, Z) = 0

for all X, Y, Z in TM , which is easily seen to imply that dλ + 2λθ = 0. UsingLemma 6.2, it follows that λ is nowhere zero and θ = −dln

√

|λ|. Let us consider

now the Riemannian metric g = ε2g, where ε =√

|λ|. We will show that (M, g)is a locally symmetric space by examination of the conformal transformation rulesfor the relevant objects. First of all, using the Koszul formula for the Levi-Civitaconnections ∇ and ∇g one easily arrives at

∇gXY = DXY − θ(X)Y

for any smooth vector fields onM . It follows directly that Rg(X, Y )Z = RD(X, Y )Zwhenever X, Y, Z belong to TM . Now, a short computation using only the definitionof the covariant derivative gives

(∇gXR

g)(Y, Z, U) = (DXRD)(Y, Z, U) + 2θ(X)RD(Y, Z)U

for all X, Y, Z, U in TM . On the other hand side, since DRg = 0 we have by using(6.4):

(DXRD)(Y, Z, U) = (Xλ)Rg(Y, Z)U = 2(Xlnε)RD(Y, Z)U = −2θ(X)RD(Y, Z)U

for all X, Y, Z, U in TM . This shows that ∇gRg = 0, in other words (Mn, g) is alocally symmetric space.(iii) is an immediate consequence of the fact that θ is closed and of the behaviourof the holonomy algebra of D, after operating a conformal change in the metric, asthe calculations in (ii) show.

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 23

7. Some facts peculiar to dimension 8

Let (g, V ) be a faithful metric representation. We seek here to obtain a descriptionof representations having the property that the map

(7.1) ε⊥ : Λ3V → Λ1V ⊗ g⊥ is surjective.

Under the additional assumption that g has the involution property, which includesthe case when g is a simple Lie algebra it was shown in [17] that (7.1) forces theequality (g, V ) = (spin(7),R8). We shall treat in this section, from a differentperspective the general case. We observe first that:

Lemma 7.1. Any representation (g, V ) satisfying (7.1) must be irreducible.

Proof. As observed before, if (7.1) holds then

(Λ1V ⊗ g⊥) ∩Ker(a) = 0.

In particular, it follows that the map

a : S2g⊥ → Λ4V

is injective. Let us suppose now that we have an orthogonal splitting V = V1 ⊕ V2which is also g-invariant. Then V1 ∧ V2 ⊆ g⊥ and since

a((v1 ∧ v2)⊗ (v1 ∧ v2)) = 0

for all vk in Vk, k = 1, 2 we obtain a contradiction.

We also need to establish the following general fact.

Proposition 7.1. Let (g, V ) be metric and faithful such that Λ4V ∩(Λ2V ⊗g) = 0.The map

I : (K(g))⊥ ∩Ker(Ric) ⊆ S2(g) → S2(g⊥)

given byI(S)α = L⋆

αa(S)

for all α in g⊥ is well defined and injective.

Proof. Let us show first that I is well defined. If S is in S2(g) a short computationusing (6.1) shows that

L⋆αa(S) = Sα− ric(S), α −

∑

xk∈g

[xk, [Sxk, α]]

for all α in g. If moreover ric(S) = 0, it follows that L⋆αa(S) belongs to g for all α

in g and one sees that I(S) belongs to S2(g⊥) by using an orthogonality argument.Let now S in S2g ∩ ker(Ric) be such that I(S) = 0. Then L⋆

αa(S) = 0 for allα in g⊥, and since L⋆

αa(S) belongs to g for all α in g it follows that a(S) is inΛ4V ∩ (Λ2V ⊗ g), hence a(S) = 0. But then S must belong to K(g) and so itvanishes.

We can now prove the following.

Theorem 7.1. Let (g, V ) be an orthogonal and faithful representation such that(7.1) holds. Then (g, V ) = (spin(7),R8).

24 P.-A.NAGY

Proof. Since our representation must be irreducible, we shall treat the various oc-currences for K(g) as Theorem 2.2 indicates. Also note that the dimension of theLie algebra g is given by

(7.2) dimR g⊥ =

1

6(n− 1)(n− 2).

since ε⊥ is an isomorphism. If K(g) = 0 or if it is 1-dimensional, then we musthave Λ4V ∩ (Λ2V ⊗ g) = 0 for n ≥ 4, as it follows from Theorem 3.1, (ii). Nowby Proposition 7.1, the map I : S2g ∩Ker(Ric) → S2(g⊥) is injective, hence

dimRS2g ≤ dimRS

2g⊥ + dimR S2V ≤ dimRS

2(g⊥ ⊕ V ).

It follows that dimRg ≤ dimRg⊥ + n, which leads to n2 ≤ 4 + 3n after using

(7.2). Thus n ≤ 4 and the equality case can be excluded because then g would be5-dimensional and

dimRS2g = 15 dimRS

2g⊥ + dimR S2V = 1 + 10 = 11.

It remains to treat the entries present in the Berger list (2.4). It is an easy exerciseto see that all entries but the 6-th can be excluded for dimensional reasons. Henceonly the case (g, V ) = (spin(7),R8) is eligible, when moreover (7.1) is known tohold [25, 17], and the proof is completed.

Remark 7.1. Another interesting g-module is (Λ1V ⊗g)∩Kera ⊆ Λ1V ⊗g. It ap-pears naturally in the study of holonomy groups of Lorentzian metrics in connectionwith the so-called weak Berger algebras [28]. It seems not unlikely that techniquessimilar to those in this paper could lead to a characterisation of the instances when(Λ1V ⊗ g) ∩Kera vanishes.

8. Appendix: The skew-symmetric prolongation of an adjoint

representation

Let (g, β) be an orthogonal Lie algebra, that is a Lie algebra g equipped with ascalar product β : g× g → R, which is moreover g-invariant in the sense that

β([x, y], z) = −β(y, [x, z])

for all x, y, z in g. The adjoint representation of g shall be denoted by (g, V ) andin what follows we will compute its first skew-symmetric prolongation, under theadditional assumption that the Lie algebra g is simple. Recall that the Cartan3-form in Λ3V is defined by

(x, y, z) 7→ β([x, y], z)

for all x, y, z in V = g and moreover that it is g-invariant.

Proposition 8.1. If g is simple, the space Λ3V ∩ (Λ1V ⊗ g) is 1-dimensional,generated by the Cartan form of (g, β).

Proof. If T belongs to Λ3V ∩ (Λ1V ⊗ g), it can be written as TX = adQX for all Xin V = g where Q belongs to EndR(g). In other words, T (X, Y, Z) = β([QX, Y ], Z)for all X, Y, Z in g, and since T is a 3-form it follows that

(8.1) [QX, Y ] = [X,QY ]

PROLONGATIONS OF LIE ALGEBRAS AND APPLICATIONS 25

for all X, Y in g. If C denotes the Casimir operator of g we must have C = λ1g forsome non-zero λ in R since g is simple. We deduce that

λQX = −∑

i

[ei, [Qei, X ]]

for all X in g, where ei is some orthonormal basis in g. It is easy to see that thedual of the above equation reads

λQ⋆X = −∑

i

[ei, [Q⋆ei, X ]]

for all X in g, where Q⋆ the adjoint of Q w.r.t. β. Therefore, the skew-symmetricendomorphism F of g, defined by F = Q−Q⋆ satisfies

λFX = −∑

i

[ei, [Fei, X ]]

for all X in g. From the Jacobi identity it follows that

[ei, [Fei, X ]] + [Fei, [X, ei]] + [X, [ei, F ei]] = 0

hence after summation we get 2∑

i

[ei, [Fei, X ]] = −[X, ζ ] for all X in g, since F is

skew-symmetric. Here the element ζ of g is defined by ζ =∑

i

[ei, F ei]. In particular,

one has

λFX =1

2[X, ζ ]

for all X in g. Using the definition of ζ , this leads to

λζ =1

2

∑

i

[ei, [ei, ζ ]] = −1

2Cζ = −

λ

2ζ

whence ζ = 0. In other words, Q is a symmetric endomorphism of g, and let

g =

q⊕

k=1

gk

be its spectral decomposition, with corresponding eigenvalues λi, 1 ≤ i ≤ q, whichwe assume pairwise distinct. From (8.1) we get that [gi, gj ] = 0 whenever 1 ≤ i 6=j ≤ q. Because β is g-invariant it follows that gi is orthogonal (w.r.t. β) to

⊕

i 6=j

gj

hence [gi, gi] ⊆ gi for all 1 ≤ i ≤ q. Therefore, each of the subspaces gi, 1 ≤ i ≤ q isan ideal in g hence at most one can be non-vanishing because g is simple. We haveshowed that Q is proportional to the identity thus T is proportional to the Cartanform of (g, β) and the proof is finished.

Acknowledgements: This research has been partly supported by the Royal Soci-ety of New Zealand, Marsden Grant no. 06-UOA-029 and the University of Hamburgwhere the final part of this work has been carried out. It is a pleasure to thankR.Gover, Th.Leistner, P.Nurowski and especially V.Cortes for useful discussions.

26 P.-A.NAGY

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(P.-A. Nagy) Department of Mathematics, University of Auckland, Private Bag 92019, Auckland,

New Zealand

E-mail address: nagy@math.auckland.ac.nz