Chapter 21Lie Color and Hom-Lie Algebras of Witt Typeand Their Central Extensions

Gunnar Sigurdsson and Sergei Silvestrov

Abstract In this article, two classes of Γ -graded Witt-type algebras, Lie color andhom-Lie algebras of Witt type, are considered. These algebras can be seen as gen-eralizations of Lie algebras of Witt type. One-dimensional central extensions of Liecolor and hom-Lie algebras of Witt type are investigated.

21.1 Introduction

Since 1980s, when research on quantum deformations (or q-deformations) of Liealgebras began a period of rapid expansion in connection with the introduction ofquantum groups motivated by applications to the Yang–Baxter equation and quan-tum inverse scattering methods, several other versions of (q-) deformed Lie alge-bras have appeared, especially in physical contexts such as string theory. The mainobjects for these deformations were infinite-dimensional algebras, primarily theHeisenberg algebras (oscillator algebras) and the Virasoro algebra, see [1–4] andthe references therein.

An important question about these algebras is whether they obey some deformed(twisted) versions of skew-symmetry or Jacobi identity. At the same time, in awell-known direct generalization of Lie algebras and Lie superalgebras to generalcommutative grading groups, the class of Lie color algebras, generalized skew-symmetry and Jacobi type identities, graded by a commutative group and twistedby a scalar bicharacter, hold (see [8]).

G. SigurdssonDepartment of Theoretical Physics, School of Engineering Sciences, Royal Institute of Technology,AlbaNova University Center, 106 91 Stockholm, Swedene-mail: gunnsi@kth.se

S. SilvestrovCentre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Swedene-mail: ssilvest@maths.lth.se

S. Silvestrov et al. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 247c© Springer-Verlag Berlin Heidelberg 2009

248 G. Sigurdsson and S. Silvestrov

Throughout we let F be a field of characteristic zero, F∗ = F \ {0} and (Γ ,+)

an abelian group. Recall that a Lie color algebra or Γ -graded ε-Lie algebra (see[10, 11]) is a Γ -graded linear space L with a bilinear multiplication 〈·, ·〉 satisfying:

• 〈x,y〉=−ε(γx,γy)〈y,x〉• ε(γz,γx)〈x,〈y,z〉〉+ ε(γx,γy)〈y,〈z,x〉〉+ ε(γy,γz)〈z,〈x,y〉〉= 0

for x ∈ Lγx ,y ∈ Lγy and z ∈ Lγz where L =⊕

γ∈Γ Lγ . Here γx, γy and γz are the gradeddegrees of x, y and z respectively . The map ε : Γ ×Γ → F

∗, called a commutationfactor, is a bicharacter on Γ with a symmetry property ε(γx,γy)ε(γy,γx) = 1. Noticethat Lie color algebras include both Γ -graded Lie algebras (with ε(γx,γy) = 1 for allγx,γy ∈ Γ ) and Lie superalgebras (with Γ = Z2 and ε(γx,γy) = (−1)γxγy). Lie coloralgebras are examples of Γ -graded quasi-Lie algebras (see [7]).

A remarkable and not yet fully exploited and understood feature of Lie superal-gebras and Lie color algebras is that they often appear simultaneously with usualLie algebras in various deformation families of algebras for initial, final or otherimportant special values of the deformation parameters.

Let s : Γ → F be a function defined over Γ and consider the Γ -graded vectorspace L =

⊕

g∈Γ Feg over F with a basis {eg |g ∈ Γ }. Assume that ε : Γ ×Γ → F∗

is a commutation factor on Γ . The linear space L endowed with the bracket product

[eg,eh] = (s(h)− ε(g,h)s(g))eg+h (21.1)

can be shown to be a Lie color algebra L(Γ ,ε,s) under certain conditions on thegrading group Γ and the mappings ε and s. L(Γ ,ε,s) is then said to be a Lie coloralgebra of Witt type. For further details, we refer to [13]. The special case given byε(g,h) = 1 for all g,h ∈ Γ and with s satisfying the condition

(s(g)− s(h))(s(g+h)− s(g)− s(h)+ s(0)) = 0 for all g,h ∈ Γ ,

is aΓ -graded Lie algebra and called a Lie algebra of Witt type (see [12] for instance).The main purpose of this article is to study Lie color algebra central extensions

of Witt-type Lie color algebras by a one-dimensional central part Fc and centralextensions of Γ -graded hom-Lie algebras of Witt type.

21.2 Central Extensions of Witt-Type Lie Color Algebras

Throughout this section, we assume that the additively written gradation group(Γ ,+) is 2-torsion free. Moreover, the commutation factor ε is supposed to be non-degenerate, which means that the set {g ∈ Γ | ε(g,h) = 1 for all h ∈ Γ }= {0}. No-tice that the mapping ε(·, ·) : Γ → F

∗, g �→ ε(g,g) is a homomorphism of the groupΓ into the multiplicative group {−1,1}. A group element g ∈ Γ is called even orodd if ε(g,g) is equal to 1 or −1, respectively. For a given commutation factor ε onΓ , it is common to define the subsets

21 Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions 249

Γ+ = {g ∈ Γ | ε(g,g) = 1}, Γ− = {g ∈ Γ | ε(g,g) =−1}.

Clearly there are two possibilities, either one has Γ+ = Γ , or Γ+ is a subgroup ofindex 2 in Γ . In the following, �g,h,k will be used to concisely denote summationover the cyclic permutation on g,h,k.

On the graded linear space L =⊕

g∈Γ Feg with one-dimensional homogeneouscomponents Feg, we introduce a bracket product given by (21.1). Putting S(g,h) =(s(h)− ε(g,h)s(g)), we have

[eg,eh] = (s(h)− ε(g,h)s(g))eg+h = S(g,h)eg+h.

Under the given restrictions on Γ and ε , it can be proved (see [13]) that L is a Liecolor algebra if and only if the function s : Γ → F satisfies the following conditions

s(g) = 0 or s(g) = s(0), (21.2)(s(h)− ε(g,h)s(g))(s(g+h)− s(g)− s(h)+ s(0)) = 0, (21.3)

for all g,h ∈ Γ . We then obtain a Lie color algebra of Witt type denoted asL(Γ ,ε,s). Note that if s(0) �= 0 (the non-abelian case) then by (21.2) the imageIm(s) = {0,s(0)} of the map s : Γ → F consists of two elements only. Hence s canbe expressed as s(g) = s(0)χsupp(s)(g) for any g ∈ Γ , the support of s being definedas supp(s) = {g ∈Γ | s(g) = s(0)} and where χA denotes the characteristic functionof a subset A⊆ Γ .

The following simple lemma is easy to prove by methods of basic group theory.

Lemma 21.1. Let G be a subset of an abelian group (Γ ,+). Then G is a subgroupof Γ if and only if:

(1) The identity element 0 ∈ G(2) g ∈ G, h /∈ G⇒ g+h /∈ G

By the conditions (21.2) and (21.3) together with Lemma 21.1, it immediatelyfollows that either supp(s) is empty (in case s(0) = 0) or supp(s) is a subgroup ofthe gradation group Γ . For more details on these matters we refer to [13].

We want to investigate Lie color algebra extensions L(Γ ,ε,s) of L(Γ ,ε,s) bya one-dimensional center Fc. The center can be considered as the one-dimensionaltrivial Lie color algebra with singleton basis {c} (the charge). We say that a Lie coloralgebra L(Γ ,ε,s) is a one-dimensional central extension of L(Γ ,ε,s) if L(Γ ,ε,s)can be written as a direct sum L(Γ ,ε,s)⊕Fc of L(Γ ,ε,s) and Fc as vector spaces,where the Lie color bracket 〈·, ·〉 in L(Γ ,ε,s) is given by

〈eg,eh〉= [eg,eh]+ϕ(eg,eh)c, (21.4)〈eg,c〉= 0 (21.5)

for all g,h ∈ Γ . Here [·, ·] is the Lie color bracket (21.1) and ϕ : L× L → F is abilinear form (the 2-cocycle) on L(Γ ,ε,s) such that the bracket (21.4) satisfies theaxioms of a Lie color bracket. The charge c is assumed to be homogeneous of degreezero. For further reference see [5, 6, 9].

250 G. Sigurdsson and S. Silvestrov

Proposition 21.1. Let L(Γ ,ε,s) be a Lie color algebra of Witt type over a field F

of characteristic zero. Assume that the gradation group Γ is 2-torsion free, and thecommutation factor ε is non-degenerate. Then L(Γ ,ε,s) admits a one-dimensionalcentral extension L(Γ ,ε,s)=L(Γ ,ε,s)⊕Fc defined by (21.4), (21.5) if and only if ϕis given by

ϕ(eg,eh) = δg,−hκ(h), g,h ∈ Γ ,

where κ is an F-valued function defined on Γ = Γ+∪Γ− such that:

(1) κ(−g) =−κ(g) if g ∈ Γ+, κ(−g) = κ(g) if g ∈ Γ−(2) κ(g) = ε(g,h)κ(h) for all g,h /∈ supp(s) such that g+h ∈ supp(s)(3) �g,h,k ε(k,g)(1− ε(h,k))κ(g) = 0 for all g,h,k ∈ supp(s) with g+h+ k = 0.

Proof. Given a basis {eg}g∈Γ of the linear space L, there is a well-defined mapb : Γ ×Γ → F, if we set b(g,h) := ϕ(eg,eh) for all g,h ∈ Γ . Then we have

〈eg,eh〉= [eg,eh]+b(g,h)c = S(g,h)eg+h +b(g,h)c. (21.6)

Applying (21.6), the double bracket 〈eg,〈eh,ek〉〉 can be expressed as

〈eg,〈eh,ek〉〉= 〈eg,S(h,k)eh+k +b(h,k)c〉= S(h,k)S(g,h+ k)eg+h+k +S(h,k)b(g,h+ k)c (21.7)

for all g,h,k ∈ Γ . Since L(Γ ,ε,s) is a Lie color algebra, the ε-Jacobi identity issatisfied by the bracket [·, ·] and hence by (21.7)

�g,h,k ε(k,g)S(h,k)S(g,h+ k)eg+h+k = 0.

Furthermore, by the ε-Jacobi identity for 〈·, ·〉, �g,h,k ε(k,g)〈eg,〈eh,ek〉〉 = 0, thisyields

�g,h,k ε(k,g)S(h,k)b(g,h+ k)c = 0,

or equivalently

�g,h,k ε(k,g)(s(k)− ε(h,k)s(h))b(g,h+ k) = 0. (21.8)

Writeb(g,h) := δg,−hκ(h). (21.9)

This means that〈eg,e−g〉= S(g,−g)e0 +κ(−g)c

for all g ∈ Γ so the grading condition is satisfied.Note that the ε-skew-symmetry condition implies

ε(g,h)b(h,g) = ε(g,h)δh,−gκ(g) =−δg,−hκ(h).

For g+h �= 0 this is trivially satisfied. Taking h =−g yields κ(−g)=−ε(g,−g)κ(g).Recall that ε(g,g)ε(g,−g) = ε(g,0) = 1. In the case where Γ+ = Γ then also

21 Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions 251

ε(g,−g) = 1 and the mapping κ is an odd function on Γ , κ(−g) = −κ(g) for allg ∈ Γ . On the other hand, if Γ− �= ∅ it follows that κ(−g) = κ(g) for all g ∈ Γ−,while we still have κ(−g) = −κ(g) on Γ+. Since ε(0,0) = 1, it is always true thatκ(0) = 1, if we assume that the characteristic of the field F is different from 2.

Going back to (21.8), this can be rewritten as

�g,h,k ε(k,g)(s(k)− ε(h,k)s(h))δg,−(h+k)κ(h+ k) = 0. (21.10)

The identity is trivially satisfied if g+h+k �= 0. For all g,h,k ∈ Γ such that g+h+k = 0 we obtain

�g,h,k ε(k,−h− k)(s(k)− ε(h,k)s(h))κ(h+ k) = 0.

Recalling that ε is a bihomomorphism on Γ ×Γ , we have

�g,h,k ε(k,−h)ε(k,−k)(s(k)− ε(h,k)s(h))κ(h+ k) = 0. (21.11)

Let s(0) �= 0 so that supp(s) is non-empty. For all g,h,k ∈Γ such that g,h /∈ supp(s),k ∈ supp(s) and g+h+ k = 0 it follows that (21.11) reduces to

ε(k,−k)ε(k,−h)κ(h+ k)− ε(g,−g)ε(g,−k)ε(k,g)κ(k +g) = 0. (21.12)

This equality holds if and only if

κ(−g)− ε(g,h)κ(−h) = 0,

for all g,h /∈ supp(s) such that g+h ∈ supp(s). We notice that ε(g,h) = ε(−g,−h)for all g,h∈Γ . Moreover, g,h /∈ supp(s) with g+h∈ supp(s) if and only if−g,−h /∈supp(s) and −(g+h) ∈ supp(s). The conclusion is that (21.12) holds iff

κ(g)− ε(g,h)κ(h) = 0,

for all g,h /∈ supp(s) such that g + h ∈ supp(s), i.e. condition (2). Note that in thiscase the factor (s(h)− ε(g,h)s(g)) in (21.3) vanishes.

In the next step, we consider the case where more than one of the elements g,h,kare contained in supp(s). Recall that we assume they satisfy the constraint g + h +k = 0. Since supp(s) is a subgroup of Γ , this means in particular that if two ofthe elements, say h and k, are inside supp(s) then also the third, g = −h− k mustbelong to supp(s). So it remains to consider the case where g,h,k ∈ supp(s), i.e.s(g) = s(h) = s(k) = s(0) �= 0. Then obviously also s(g+h) = s(0) so (21.3) is trulysatisfied in this case. The identity (21.10) now yields

�g,h,k ε(k,g)(1− ε(h,k))κ(−g) = 0.

Now by the symmetry property, ε(g,h) = ε(−g,−h) for all g,h∈Γ , this shows that�g,h,k ε(k,g)(1− ε(h,k))κ(g) = 0 for all g,h,k ∈ Γ such that g+h+ k = 0.

252 G. Sigurdsson and S. Silvestrov

Remark 21.1. By condition (2) of Proposition 21.1 we have κ(g) = ε(g,h)κ(h) forall g,h /∈ supp(s) such that g + h ∈ supp(s). Now if g /∈ supp(s), then also −g /∈supp(s) and g+(−g) = 0 ∈ supp(s), since supp(s) is a subgroup of Γ . This meansthat κ(g) = ε(g,−g)κ(−g). As ε(g,g)ε(g,−g) = 1, we have κ(−g) = ε(g,g)κ(g).If g is even, then ε(g,g) = 1 and κ(g) is odd by condition (1) implying that κ(g) = 0for all g �∈ supp(s), i.e. supp(κ)⊆ supp(s).

21.3 Central Extensions of Γ -Graded Hom-Lie Algebrasof Witt Type

Definition 21.1. A Γ -graded hom-Lie algebra is a triple (L,〈·, ·〉L,α) where

• L =⊕

γ∈Γ Lγ is a Γ -graded linear space over F

• 〈·, ·〉L : L×L→ L is a bilinear map called a product or bracket in L• α : L→ L is a linear map mapping ∪γ∈Γ Lγ to ∪γ∈Γ Lγ

such that the following conditions hold:

• (Γ -Grading axiom) 〈Lγ1 ,Lγ2〉L ⊆ Lγ1+γ2 , for all γ1,γ2 ∈ Γ• (Skew-symmetry) 〈x,y〉L =−〈y,x〉L, for all (x,y) ∈ ∪γ∈Γ Lγ ×∪γ∈Γ Lγ• (α-Twisted Jacobi identity) �x,y,z 〈α(x),〈y,z〉L〉L = 0, whenever x∈ Lγx ,y∈ Lγy

and z ∈ Lγz where γx,γy,γz ∈ Γ .

We form the Γ -graded F-vector space W =⊕

γ∈Γ Feγ , where F is our groundfield and {eγ |γ ∈ Γ } is a basis of W . Let s : Γ → F be a function on Γ and equip Wwith the bracket product

[eg,eh] = (s(h)− s(g))eg+h = S(g,h)eg+h. (21.13)

Without loss of generality, we can set s(0) = 0. When (W, [·, ·],α) is a hom-Liealgebra, then we call it a hom-Lie algebra of Witt type. Such a Γ -graded hom-Lie algebra will be denoted as W (Γ ,α,s). Obviously, the skew-symmetry condition[eg,eh] =−[eh,eg] is automatically satisfied since by construction S(g,h) =−S(h,g)for all g,h ∈ Γ . The α-twisted Jacobi identity yields

�g,h,k [α(eg), [eh,ek]] =�g,h,k S(h,k)[α(eg),eh+k] = 0 (21.14)

for all g,h,k ∈ Γ .We know that each homogeneous subspace Wγ = Feγ is one-dimensional. Given

that α : W →W is a linear map, mapping ∪γ∈ΓWγ to ∪γ∈ΓWγ , it follows that thereexists a function p :Γ →Γ such that α(eγ) = α p(γ)

γ ep(γ), where α p(γ)γ ∈ F, for every

γ ∈ Γ . By (21.14) we thus have

�g,h,k αp(g)g S(h,k)S(p(g),h+ k)ep(g)+h+k = 0 (21.15)

for all g,h,k ∈ Γ .

21 Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions 253

A special case of interest is when the linear map α is homogeneous of degree t,where t is a fixed element of the group Γ . Then we have p(γ) = γ+t for every γ ∈Γand the identity (21.15) can be written

�g,h,k αg+tg S(h,k)S(g+ t,h+ k)eg+t+h+k = 0.

This holds if and only if

�g,h,k αg+tg (s(k)− s(g))(s(h+ k)− s(g+ t)) = 0 (21.16)

for all g,h,k ∈ Γ .We look for hom-Lie algebra extensions W (Γ ,α,s) of W (Γ ,α,s) by a one-

dimensional center Fc. A hom-Lie algebra W (Γ ,α,s) is said to be a one-dimensionalcentral extension of W (Γ ,α,s) if W (Γ ,α,s) can be written as a direct sum ofW (Γ ,α,s) and Fc as vector spaces, W (Γ ,α,s)⊕Fc, where the hom-Lie bracket〈·, ·〉 in W (Γ ,α,s) is given by

〈eg,eh〉= [eg,eh]+ψ(eg,eh)c, (21.17)〈eg,c〉= 0 (21.18)

for all g,h ∈ Γ . By [·, ·] we mean the hom-Lie bracket (21.13) and ψ : L×L → F

denotes a bilinear form (the 2-cocycle) on W (Γ ,α,s) such that the bracket (21.17)satisfies the axioms of a hom-Lie bracket. The charge c is assumed to be homoge-neous of degree zero.

Let {eg}g∈Γ be a basis of the linear space W and define a map d : Γ ×Γ → F, bysetting d(g,h) := ψ(eg,eh) for all g,h ∈ Γ . Then we have by (21.17) and (21.13)

〈eg,eh〉= [eg,eh]+d(g,h)c = S(g,h)eg+h +d(g,h)c. (21.19)

Applying (21.19), the double bracket 〈ep(g),〈eh,ek〉〉 can be expressed as

〈ep(g),〈eh,ek〉〉= 〈ep(g),S(h,k)eh+k +d(h,k)c〉= S(h,k)S(p(g),h+ k)ep(g)+h+k +S(h,k)d(p(g),h+ k)c (21.20)

for all g,h,k ∈ Γ .Knowing that W (Γ ,α,s) is a Γ -graded hom-Lie algebra, the α-twisted Jacobi

identity is satisfied by the bracket [·, ·] and hence, for all g,h,k ∈ Γ ,

�g,h,k αp(g)g S(h,k)S(p(g),h+ k)ep(g)+h+k = 0.

By the α-twisted Jacobi identity for 〈·, ·〉, �g,h,k αp(g)g 〈ep(g),〈eh,ek〉〉 = 0, we thus

obtain�g,h,k α

p(g)g S(h,k)d(p(g),h+ k)c = 0,

or expressed differently

�g,h,k αp(g)g (s(k)− s(h))d(p(g),h+ k) = 0. (21.21)

254 G. Sigurdsson and S. Silvestrov

We writed(g,h) := δg,−hω(h).

This means that〈eg,e−g〉= S(g,−g)e0 +ω(−g)c

for all g ∈ Γ , so clearly the grading condition is fulfilled. We notice that the skew-symmetry condition 〈eg,eh〉=−〈eh,eg〉 implies d(h,g) =−d(g,h), i.e.

δh,−gω(g) =−δg,−hω(h).

For g+h �= 0 this is trivially satisfied. Taking h =−g this yields ω(g) =−ω(−g) soω :Γ → F has to be an odd function on Γ . Going back to (21.21), it can be rewrittenas

�g,h,k αp(g)g (s(k)− s(h))δp(g),−(h+k)ω(h+ k) = 0.

The identity is trivially satisfied if p(g) + h + k �= 0. For all g,h,k ∈ Γ such thatp(g)+h+ k = 0 we thus obtain

�g,h,k αp(g)g (s(−p(g)−h)− s(h))ω(p(g)) = 0.

We summarize our results as the following proposition.

Proposition 21.2. Let W (Γ ,α,s) be a Γ -graded hom-Lie algebra of Witt type overa field F of zero characteristic. Then W (Γ ,α,s) admits a one-dimensional centralextension W (Γ ,α,s) = W (Γ ,α,s)⊕Fc defined by (21.17), (21.18) if and only if ψis given by

ψ(eg,eh) = δg,−hω(h), g,h ∈ Γ ,

where ω is an F-valued function on Γ satisfying

(1) ω(−g) =−ω(g) for all g ∈ Γ(2) �g,h,k α

p(g)g (s(k)− s(h))ω(p(g)) = 0 for all g,h,k ∈ Γ with p(g)+h+ k = 0,

where p : Γ → Γ is a map such that α(eγ) = α p(γ)γ ep(γ) for all γ ∈ Γ .

Acknowledgements This work was supported by The Crafoord Foundation, The Royal SwedishAcademy of Sciences, The Royal Physiographic Society in Lund and The Swedish Foundation forInternational Cooperation in Research and Higher Education (STINT). The first author is gratefulto The Centre for Mathematical Sciences, Lund University for hospitality and support during hisvisits in Lund.

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