INFINITESIMAL DEFORMATIONS OF THE MODEL Z -FILIFORM LIE ALGEBRA 3 3 1 R.M.NAVARRO 0 2 Abstract. Inthisworkitisconsideredthevectorspacecomposedbythein- n finitesimaldeformationsofthemodelZ3-filiformLiealgebraLn,m,p. Byusing a thesedeformationsalltheZ3-filiformLiealgebrascanbeobtained, hencethe J importanceofthesedeformations. Theresultsobtainedinthisworktogether 7 tothoseobtainedin[11]and[12],leadstocomputethetotaldimensionofthe 1 mentioned spaceofdeformations. ] T 2000 MSC: 17B30; 17B70; 17B75; 17B56 R Key-Words: graded Lie algebras, cohomology, deformation, nilpotent, filiform. . h t 1. Introduction a m The concept of filiform Lie algebras was firstly introduced in [18] by Vergne. [ This type of nilpotent Lie algebra has important properties; in particular, every filiform Lie algebra canbe obtainedby a deformationof the model filiform algebra 1 v Ln. In the same way as filiform Lie algebras, all filiform Lie superalgebras can be 7 obtained by infinitesimal deformations of the model Lie superalgebra Ln,m [1], [4], 4 [8] and [9]. 0 4 Continuing with the work of Vergne we have generalized the concept and the . 1 propierties of the filiform Lie algebras into the theory of color Lie superalgebras. 0 Thus, filiform G-color Lie superalgebras and the model filiform G-color Lie super- 3 1 algebra were obtained in [10]. : v In the present paper the focus of interest are color Lie superalgebras with a i X Z -grading vector space, i.e. G = Z , due to its physical applications [3],[6],[7], 3 3 r [13], [16] and [17]. Due to the fact that the one admissible commutation factor a for Z is exactly β(g,h) = 1 ∀g,h, Z -color Lie superalgebras are indeed Z -color 3 3 3 Lie algebras or Z -graded Lie algebras. Thus, we have studied the infinitesimal 3 deformationsof the model Z -colorLie superalgebra,i.e. the model Z -filiform Lie 3 3 algebra Ln,m,p. By means of these deformations all Z -filiform Lie algebras can be 3 obtained, hence the importance of these deformations. In [11] and [12], the authors decomposed the space of these infinitesimal defor- mations, noted by Z2(L;L), into six subspaces of deformations: Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕ 0 0 0 0 1 1 Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕ 0 2 2 1 1 2 Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L ) 1 2 0 2 2 1 =A⊕B⊕C⊕D⊕E⊕F 1 2 R.M.NAVARRO In the present paper it is given a method that will allow to determine the di- mension of the subspaces A, B and C, giving explicitly the total dimension of all of them (Theorems 1, 2 and 3). This result together to those obtained in [11] and [12], leads to obtain the total dimension of the infinitesimal deformations of the model Z -filiform Lie algebra Ln,m,p (Main Theorem). 3 WedoassumethatthereaderisfamiliarwiththestandardtheoryofLiealgebras. All the vector spaces that appear in this paper (and thus, all the algebras) are assumed to be F-vector spaces (F=C or R) with finite dimension. 2. Preliminaries The vector space V is said to be Z −graded if it admits a decomposition in n direct sum, V = V0⊕V1⊕···Vn−1. An element X of V is called homogeneous of degree γ (deg(X)=d(X)=γ), γ ∈Z , if it is an element of V . n γ Let V =V0⊕V1⊕···Vn−1 and W =W0⊕W1⊕···Wn−1 be two gradedvector spaces. A linear mapping f : V −→ W is said to be homogeneous of degree γ (deg(f)=d(f)=γ), γ ∈Z , iff(V )⊂W forall α∈Z . The mapping n α α+γ(mod n) n f is calleda homomorphismofthe Z −gradedvectorspace V into the Z −graded n n vector space W if f is homogeneous of degree 0. Now it is evident how we define an isomorphism or an automorphism of Z −graded vector spaces. n A superalgebrag is just a Z −gradedalgebra g=g ⊕g . That is, if we denote 2 0 1 by [ , ] the bracket product of g, we have [g ,g ]⊂g for all α,β ∈Z . α β α+β(mod2) 2 Definition 2.1. [14]Let g=g ⊕g be a superalgebrawhose multiplicationis de- 0 1 notedbythebracketproduct[,]. WecallgaLiesuperalgebraifthemultiplication satisfies the following identities: 1. [X,Y]=−(−1)α·β[Y,X], ∀X ∈g ,∀Y ∈g . α β 2. (−1)γ·α[X,[Y,Z]]+(−1)α·β[Y,[Z,X]]+(−1)β·γ[Z,[X,Y]]=0 for all X ∈g ,Y ∈g ,Z ∈g with α,β,γ ∈Z . α β γ 2 Identity2iscalledthegradedJacobiidentityanditwillbedenotedbyJ (X,Y,Z). g We observe that if g = g ⊕g is a Lie superalgebra, we have that g is a Lie 0 1 0 algebra and g has the structure of a g −module. 1 0 Color Lie (super)algebras can be seen as a direct generalization of Lie (su- per)algebras. Indeed, the latter are defined through antisymmetric (commutator) or symmetric (anticommutator) products, although for the former the product is neither symmetric nor antisymmetric and is defined by means of a commutation factor. This commutation factor is equal to ± 1 for (super)Lie algebras and more generalfor arbitrarycolor Lie (super)algebras. As happened for Lie superalgebras, the basic tool to define color Lie (super)algebras is a grading determined by an abelian group. Definition 2.2. Let G be an abelian group . A commutation factor β is a map β : G×G−→F\{0}, (F=C or R), satisfying the following constraints: (1) β(g,h)β(h,g)=1 for all g,h∈G (2) β(g,h+k)=β(g,h)β(g,k) for all g,h,k∈G INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 3 (3) β(g+h,k)=β(g,k)β(h,k) for all g,h,k∈G The definition above implies, in particular, the following relations: β(0,g)=β(g,0)=1, β(g,h)=β(−h,g), β(g,g)=±1 ∀g,h∈G where 0 denotes the identity element of G. In particular, fixing g one element of G, the induced mapping β :G−→F\{0} defines a homomorphism of groups. g Definition 2.3. Let G be an abelian group and β a commutation factor. The (complex or real) G−graded algebra L= Lg g∈G M with bracket product [ , ], is called a (G,β)-color Lie superalgebra if for any X ∈ L , Y ∈L , and Z ∈L we have g h (1) [X,Y]=−β(g,h)[Y,X] (anticommutative identity) (2) [[X,Y],Z]=[X,[Y,Z]]−β(g,h)[Y,[X,Z]] (Jacobi identity) Corollary 2.3.1. Let L = Lg be a (G,β)-color Lie superalgebra. Then we g∈G have L (1) L is a (complex or real) Lie algebra where 0 denotes the identity element 0 of G. (2) For all g ∈ G\{0}, L is a representation of L . If X ∈ L and Y ∈ L , g 0 0 g then [X,Y] denotes the action of X on Y. Examples. For the particular case G = {0}, L = L reduces to a Lie algebra. 0 If G = Z = {0,1} and β(1,1) = −1 we have ordinary Lie superalgebras, i.e. a 2 Lie superalgebra is a (Z ,β)-color Lie superalgebra where β(i,j) = (−1)ij for all 2 i,j ∈Z . 2 Definition 2.4. A representation of a (G,β)-color Lie superalgebra is a mapping ρ:L−→End(V), where V = V is a graded vector space such that g∈G g [ρ(X),ρ(Y)]L=ρ(X)ρ(Y)−β(g,h)ρ(Y)ρ(X) for all X ∈L , Y ∈L g h We observe that for all g,h ∈ G we have ρ(L )V ⊆ V , which implies that g h g+h any V has the structure of a L -module. In particular considering the adjoint g 0 representationad we have that every L has the structure of a L -module. L g 0 Two (G,β)-color Lie superalgebras L and M are called isomorphic if there is a linear isomorphism ϕ : L −→ M such that ϕ(L ) = M for any g ∈ G and also g g ϕ([x,y])=[ϕ(x),ϕ(y)] for any x,y ∈L. LetL= Lg bea(G,β)-colorLiesuperalgebra. Thedescending central g∈G sequence of L is defined by L C0(L)=L, Ck+1(L)=[Ck(L),L] ∀k ≥0 If Ck(L) = {0} for some k, the (G,β)-color Lie superalgebra is called nilpotent. The smallest integer k such as Ck(L)={0} is called the nilindex of L. 4 R.M.NAVARRO Also, we are going to define some new descending sequences of ideals, see [10]. Let L = Lg be a (G,β)-color Lie superalgebra. Then, we define the g∈G new descending sequences of ideals Ck(L ) (where 0 denotes the identity element L 0 of G) and Ck(L ) with g ∈G\{0}, as follows: g C0(L )=L , Ck+1(L )=[L ,Ck(L )], k ≥0 0 0 0 0 0 and C0(L )=L , Ck+1(L )=[L ,Ck(L )], k ≥0, g ∈G\{0} g g g 0 g Using the descending sequences of ideals defined above we give an invariant of color Lie superalgebras called color-nilindex. We are going to particularize this definition for G=Z . 3 Definition 2.5. [11] If L = L ⊕L ⊕L is a nilpotent (Z ,β)-color Lie superal- 0 1 2 3 gebra, then L has color-nilindex (p ,p ,p ), if the following conditions hold: 0 1 2 (Cp0−1(L ))(Cp1−1(L ))(Cp2−1(L ))6=0 0 1 2 and Cp0(L )=Cp1(L )=Cp2(L )=0 0 1 2 Definition 2.6. [10] Let L = Lg be a (G,β)-color Lie superalgebra. L is g∈G g called a L -filiform module if there exists a decreasing subsequence of vectorial 0 L subspaces in its underlying vectorial space V, V = V ⊃ ··· ⊃ V ⊃ V , with m 1 0 dimensions m,m−1,...0, respectively, m>0, and such that [L ,V ]=V . 0 i+1 i Remark 2.7. ThedefinitionoffiliformmoduleisalsovalidforG-gradedLiealgebras. Definition 2.8. [10] Let L = Lg be a (G,β)-color Lie superalgebra. Then g∈G L is a filiform color Lie superalgebra if the following conditions hold: L (1) L is a filiform Lie algebra where 0 denotes the identity element of G. 0 (2) L has structure of L -filiform module, for all g ∈G\{0} g 0 Definition 2.9. Let L = Lg be a G-graded Lie algebra. Then L is a G- g∈G filiform Lie algebra if the following conditions hold: L (1) L is a filiform Lie algebra where 0 denotes the identity element of G. 0 (2) L has structure of L -filiform module, for all g ∈G\{0} g 0 It is not difficult to see that for G = Z , there is only one possibility for the 3 commutation factor β, i. e. β(g,h)=1 ∀ g, h ∈Z ={0,1,2} 3 From now on we will consider this commutation factor and we will write “Z - 3 color”instead of “(Z ,β)-color”. We will note by Ln,m,p the variety of all Z -color 3 3 Lie superalgebras L = L ⊕L ⊕L with dim(L ) = n+1, dim(L ) = m and 0 1 2 0 1 dim(L )=p. Nn,m,p will be the variety ofall nilpotent Z -colorLie superalgebras 2 3 andFn,m,p isthe subsetofNn,m,p composedofallfiliformcolorLiesuperalgebras. Remark 2.10. If G = Z then β(g,h) = 1 ∀ g, h. Thus, Z -color Lie super- 3 3 algebras are effectively Z -graded Lie algebras and filiform Z -color Lie 3 3 superalgebras are Z -filiform Lie algebras. 3 INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 5 In the particular case ofG=Z the theoremof adaptedbasis restas follows for 3 L=L ⊕L ⊕L ∈Fn,m,p: 0 1 2 [X ,X ]=X , 1≤i≤n−1, 0 i i+1 [X ,X ]=0,  0 n  [[XX00,,YYmj]]==Y0j+1, 1≤j ≤m−1, [X ,Z ]=Z , 1≤k ≤p−1, 0 k k+1 with {X0,X1,..., Xn}a[bXa0s,isZpo]f=L00,.{Y1,...,Ym} a basis of L1 and {Z1,...,Zp} abasis ofL . The model Z -filiformLie algebra,Ln,m,p, is the simplestZ -filiform 2 3 3 Lie algebra and it is defined in an adapted basis {X ,X ,...,X , Y ,...,Y , 0 1 n 1 m Z ,...,Z } by the following non-null bracket products 1 p [X ,X ]=X , 1≤i≤n−1 0 i i+1 Ln,m,p = [X ,Y ]=Y , 1≤j ≤m−1  0 j j+1  [X0,Zk]=Zk+1 1≤k ≤p−1 3. cocycles and infinitesimal deformations Recall that a module V = V ⊕V ⊕V of the Z -color Lie superalgebra L is a 0 1 2 3 bilinear map of degree 0, L×V →V satisfying ∀ X ∈L , Y ∈L v ∈V : X(Yv)−Y(Xv)=[X,Y]v g h color Lie superalgebra cohomology is defined in the following well-known way (see e.g. [15]): inparticular,thesuperspaceofq-dimensional cocyclesoftheZ -colorLie 3 superalgebraL=L ⊕L ⊕L with coefficients in the L-module V =V ⊕V ⊕V 0 1 2 0 1 2 will be given by Cq(L;V)= Hom(∧q0L ⊗∧q1L ⊗∧q2L ,V) 0 1 2 q0+qM1+q2=q This space is graded by Cq(L;V)=Cq(L;V)⊕Cq(L;V)⊕Cq(L;V) with 0 1 2 Cq(L;V)= Hom(∧q0L ⊗∧q1L ⊗∧q2L ,V ) p 0 1 2 r q0+q1M+q2=q q1+2q2+p≡rmod3 The coboundary operator δq : Cq(L;V) −→ Cq+1(L;V), with δq+1 ◦δq = 0 is defined in general, with L an arbitrary (G,β)-color Lie superalgebra and V an L-module, by the following formula for q ≥1 (δqg) A ,A ,...,A = 0 1 q q (cid:0) (cid:1) (−1)rβ(γ+α0+···+αr−1,αr)Ar ·g A0,...,Aˆr,...,Aq r=0 + (−1)sβ(αr+1+X···+αs−1,αs)g A0,...,Ar−1,[Ar,As],A(cid:0)r+1,...,Aˆs,...Aq (cid:1), r<s X (cid:0) (cid:1) where g ∈ Cq(L;V) of degree γ, and A ,A ,...,A ∈ L are homogeneous with 0 1 q degreesα ,α ,...,α respectively. Thesignˆindicatesthattheelementbelowmust 0 1 q 6 R.M.NAVARRO be omitted and empty sums (like α0+···+αr−1 for r =0 and αr+1+···+αs−1 for s=r+1) are set equal to zero. In particular, for q =2 we obtain (δ2g) A ,A ,A = β(γ,α )A ·g A ,A −β(γ+α ,α )A ·g A ,A + 0 1 2 0 0 1 2 0 1 1 0 2 β(γ+α +α ,α )A ·g A ,A 0 1 2 2 0 1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) −g([A ,A ],A )+β(α ,α )g([A ,A ],A )+g(A ,[A ,A ]). 0 1 2 1 2 0 2 1 0 1 2 (cid:0) (cid:1) Let Zq(L;V) denote the kernelofδq and letBq(L;V)denote the imageof δq−1, then we have that Bq(L;V) ⊂ Zq(L;V). The elements of Zq(L;V) are called q- cocycles, the elements of Bq(L;V) are the q-coboundaries. Thus, we can constuct the so-called cohomology groups Hq(L;V)=Zq(L;V)/Bq(L;V) Hq(L;V)=Zq(L;V) Bq(L;V),if G=Z then p=0,1,2 p p p 3 Two elements of Zq(L;V) are(cid:14)said to be cohomologous if their residue classes modulo Bq(L;V) coincide, i.e., if their difference lies in Bq(L;V). We will focus our study in the 2-cocycles Z2(Ln,m,p;Ln,m,p) with Ln,m,p the 0 model filiform Z -color Lie superalgebra. Thus G = Z and the only admissible 3 3 commutationfactorisexactlyβ(g,h)=1. Underalltheserestrictionsthecondition that have to verify ψ ∈C2(Ln,m,p;Ln,m,p) to be a 2-cocycle rests 0 (δ2ψ) A ,A ,A = [A ,ψ A ,A ]−[A ,ψ A ,A ]+ 0 1 2 0 1 2 1 0 2 [A ,ψ A ,A ]−ψ([A ,A ],A )+ 2 0 1 0 1 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) ψ([A ,A ],A )+ψ(A ,[A ,A ])=0 0 2 1 0 1 2 (cid:0) (cid:1) for all A ,A ,A ∈Ln,m,p. We observe that Ln,m,p has the structure of a Ln,m,p- 0 1 2 module via the adjoint representation. We consider an homogeneous basis of Ln,m,p = L ⊕L ⊕L , in particular an 0 1 2 adapted basis {X ,X ,..., X ,Y ,...,Y ,Z ,...,Z } with {X ,X ,..., X } a 0 1 n 1 m 1 p 0 1 n basis of L , {Y ,...,Y } a basis of L and {Z ,...,Z } a basis of L . 0 1 m 1 1 p 2 Under these conditions we have the following lemma. Lemma 3.1. [11], [12] Let ψ be such that ψ ∈ C2(Ln,m,p;Ln,m,p), then ψ is a 2- 0 cocycle, ψ ∈Z2(Ln,m,p;Ln,m,p),iffthe10conditionsbelowholdforallX ,X ,X ∈ 0 i j k L , Y ,Y ,Y ∈L and Z ,Z ,Z ∈L 0 i j k 1 i j k 2 (1) [X ,ψ(X ,X )]−[X ,ψ(X ,X )]+[X ,ψ(X ,X )]−ψ([X ,X ],X )+ i j k j i k k i j i j k ψ([X ,X ],X )+ψ(X ,[X ,X ])=0 i k j i j k (2) [X ,ψ(X ,Y )]−[X ,ψ(X ,Y )]+[Y ,ψ(X ,X )]−ψ([X ,X ],Y )+ i j k j i k k i j i j k ψ([X ,Y ],X )+ψ(X ,[X ,Y ])=0 i k j i j k (3) [X ,ψ(X ,Z )]−[X ,ψ(X ,Z )]+[Z ,ψ(X ,X )]−ψ([X ,X ],Z )+ i j k j i k k i j i j k ψ([X ,Z ],X )+ψ(X ,[X ,Z ])=0 i k j i j k (4) [X ,ψ(Y ,Y )]−[Y ,ψ(X ,Y )]+[Y ,ψ(X ,Y )]−ψ([X ,Y ],Y )+ i j k j i k k i j i j k ψ([X ,Y ],Y )+ψ(X ,[Y ,Y ])=0 i k j i j k (5) [X ,ψ(Y ,Z )]−[Y ,ψ(X ,Z )]+[Z ,ψ(X ,Y )]−ψ([X ,Y ],Z )+ i j k j i k k i j i j k ψ([X ,Z ],Y )+ψ(X ,[Y ,Z ])=0 i k j i j k (6) [X ,ψ(Z ,Z )]−[Z ,ψ(X ,Z )]+[Z ,ψ(X ,Z )]−ψ([X ,Z ],Z )+ i j k j i k k i j i j k ψ([X ,Z ],Z )+ψ(X ,[Z ,Z ])=0 i k j i j k INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 7 (7) [Y ,ψ(Y ,Y )]−[Y ,ψ(Y ,Y )]+[Y ,ψ(Y ,Y )]−ψ([Y ,Y ],Y )+ i j k j i k k i j i j k ψ([Y ,Y ],Y )+ψ(Y ,[Y ,Y ])=0 i k j i j k (8) [Y ,ψ(Y ,Z )]−[Y ,ψ(Y ,Z )]+[Z ,ψ(Y ,Y )]−ψ([Y ,Y ],Z )+ i j k j i k k i j i j k ψ([Y ,Z ],Y )+ψ(Y ,[Y ,Z ])=0 i k j i j k (9) [Y ,ψ(Z ,Z )]−[Z ,ψ(Y ,Z )]+[Z ,ψ(Y ,Z )]−ψ([Y ,Z ],Z )+ i j k j i k k i j i j k ψ([Y ,Z ],Z )+ψ(Y ,[Z ,Z ])=0 i k j i j k (10) [Z ,ψ(Z ,Z )]−[Z ,ψ(Z ,Z )]+[Z ,ψ(Z ,Z )]−ψ([Z ,Z ],Z )+ i j k j i k k i j i j k ψ([Z ,Z ],Z )+ψ(Z ,[Z ,Z ])=0 i k j i j k Proposition 3.2. [11] ψ is an infinitesimal deformation of Ln,m,p iff ψ is a 2- cocycle of degree 0, ψ ∈Z2(Ln,m,p;Ln,m,p). 0 Theorem 3.2.1. [10] (1) Any filiform (G,β)-color Lie superalgebra law µ is iso- morphic to µ +ϕ where µ is the law of the model filiform (G,β)-color Lie super- 0 0 algebra and ϕ is an infinitesimal deformation of µ verifying that ϕ(X ,X)=0 for 0 0 all X ∈L, with X the characteristic vector of model one. 0 (2) Conversely, if ϕ is an infinitesimal deformation of a model filiform (G,β)- color Lie superalgebra law µ with ϕ(X ,X)=0 for all X ∈L, then the law µ +ϕ 0 0 0 is a filiform (G,β)-color Lie superalgebra law iff ϕ◦ϕ=0. Thus, any Z -filiform Lie algebra (filiform Z -color Lie superalgebra) will be a 3 3 linear deformation of the model Z -filiform Lie algebra (the model Z -color Lie 3 3 superalgebra), i.e. Ln,m,p is the model Z -filiform Lie algebra an another arbi- 3 trary Z -filiform Lie algebra will be equal to Ln,m,p+ϕ, with ϕ an infinitesimal 3 deformation of Ln,m,p. Hence the importance of these deformations. So, in or- der to determine all the Z -filiform Lie algebras it is only necessary to compute 3 the infinitesimal deformations or so called 2-cocycles of degree 0, that vanish on the characteristic vector X . Thanks to the following lemma these infinitesimal 0 deformations will can be decomposed into 6 subspaces. Lemma3.3. [11],[12]LetZ2(L;L)bethe2-cocyclesZ2(Ln,m,p;Ln,m,p)thatvanish 0 on the characteristic vector X . Then, Z2(L;L) can be divided into six subspaces, 0 i.e. if Ln,m,p =L=L ⊕L ⊕L we will have that 0 1 2 Z2(L;L)= Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕ 0 0 0 0 1 1 Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕ 0 2 2 1 1 2 Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L ) 1 2 0 2 2 1 = A⊕B⊕C⊕D⊕E⊕F In order to obtain the dimension of A, B and C we are going to adapt the sl (C)-module method that we have already used for Lie superalgebras [1], [4], [8] 2 and for color Lie superalgebras [11], [12]. Next, we will do it explicitly for A = Z2(L;L)∩Hom(L ∧L ,L ). 0 0 0 4. Dimension of A=Z2(L;L)∩Hom(L ∧L ,L ) 0 0 0 In general, any cocycle a ∈ Z2(L;L) ∩ Hom(L ∧ L ,L ) will be any skew- 0 0 0 symmetric bilinear map from L ∧L to L such that: 0 0 0 (1) [X ,a(X ,X )]−[X ,a(X ,X )]+[X ,a(X ,X )]−a([X ,X ],X )+ i j k j i k k i j i j k a([X ,X ],X )+a(X ,[X ,X ])=0 ∀ X ,X ,X ∈L i k j i j k i j k 0 8 R.M.NAVARRO with a(X ,X) = 0 ∀X ∈ L. As X ∈/ Im a and taking into account the bracket 0 0 products of L then the equation (1) can be rewritten as follows (4.1) [X ,a(X ,X )]−a([X ,X ],X )−a(X ,[X ,X ])=0, 1≤j <k ≤n 0 j k 0 j k j 0 k In order to obtain the dimension of the space of cocycles for A we apply an adaptation of the sl(2,C)-module Method that we used in [11]. Recallthefollowingwell-knownfactsabouttheLiealgebrasl(2,C)anditsfinite- dimensional modules, see e.g. [2], [5]: sl(2,C)=<X−,H,X+ > with the following commutation relations: [X+,X−]=H [H,X ]=2X ,  + +  [H,X−]=−2X−. LetV bean-dimensionalsl(2,C)-module,V =<e1,...,en >. Then,uptoisomor- phism there exists a unique structure of an irreducible sl(2,C)-module in V given in a basis {e ,...,e } as follows [2]: 1 n X ·e =e , 1≤i≤n−1, + i i+1 X ·e =0,  + n  H ·ei =(−n+2i−1)ei, 1≤i≤n. It is easy to seethat en is the maximal vector of V and its weight, called the highest weight of V, is equal to n−1. Let W ,W ,...,W be sl(2,C)-modules, then the space Hom(⊗k W ,W ) is a 0 1 k i=1 i 0 sl(2,C)-module in the following natural manner: k (ξ·ϕ)(x ,...,x )=ξ·ϕ(x ,...,x )− ϕ(x ,...,ξ·x ,x ,...,x ) 1 k 1 k 1 i i+1 n i=1 X with ξ ∈ sl(2,C) and ϕ ∈ Hom(⊗k W ,W ). In particular, if k = 2 and W = i=1 i 0 0 W =W =V , then 1 2 0 (ξ·ϕ)(x ,x )=ξ·ϕ(x ,x )−ϕ(ξ·x ,x )−ϕ(x ,ξ·x ). 1 2 1 2 1 2 1 2 An element ϕ∈Hom(V ⊗V ,V ) is said to be invariant if X ·ϕ=0, that is 0 0 0 + (4.2) X ·ϕ(x ,x )−ϕ(X ·x ,x )−ϕ(x ,X ·x )=0, ∀x ,x ∈V. + 1 2 + 1 2 1 + 2 1 2 Note thatϕ∈Hom(V ⊗V ,V )isinvariantifandonlyifϕisamaximalvector. 0 0 0 We are going to consider the structure of irreducible sl(2,C)-module in V =< 0 X ,...,X >=L /CX , thus in particular: 1 n 0 0 X ·X =X , 1≤i≤n−1, + i i+1 ( X+·Xn =0, Next, we identify the multiplicationofX andX inthe sl(2,C)-module V =< + i 0 X ,...,X >, with the bracket [X ,X ] in L and thanks to these identifications, 1 n 0 i 0 the expressions (4.1) and (4.2) are equivalent. Thus we have the following result: INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 9 Proposition 4.1. Any skew-symmetric bilinear map ϕ, ϕ:V ∧V −→V will be 0 0 0 an element of the space of cocycles A if and only if ϕ is a maximal vector of the sl(2,C)-module Hom(V ∧V ,V ), with V =hX ,...,X i. 0 0 0 0 1 n Corollary 4.1.1. As each irreducible sl(2,C)-module has (up to nonzero scalar multiples) a unique maximal vector, then the dimension of the space of cocycles A is equal to the number of summands of any decomposition of Hom(V ∧V ,V ) into 0 0 0 the direct sum of irreducible sl(2,C)-modules. We use the fact that each irreducible module contains either a unique (up to scalarmultiples)vectorofweight0(incasethedimensionoftheirreduciblemodule isodd)oraunique(uptoscalarmultiples)vectorofweight1(incasethedimension of the irreducible module is even). We therefore have Corollary 4.1.2. The dimension of the space of cocycles A is equal to the dimen- sion of the subspace of Hom (V ∧V ,V ) spanned by the vectors of weight 0 or 0 0 0 1. At this point, we aregoingto apply the sl(2,C)-module method aforementioned in order to obtain the dimension of the space of cocycles A. We consider a natural basis B of Hom(V ∧V ,V ) consisting of the following 0 0 0 maps: X if (i,j)=(k,l) ϕs (X ,X )= s i,j k l 0 in all other cases (cid:26) where 1≤i,j,k,l,s≤n, with i6=j and ϕs =−ϕs . i,j j,i Thanks to Corollary 5.1.2 it will be enough to find the basis vectors ϕs with i,j weight 0 or 1. The weight of an element ϕs (with respect to H) is i,j λ(ϕs )=λ(X )−λ(X )−λ(X )=n+2(s−i−j)+1. i,j s i j In fact, (H ·ϕs )(X ,X ) =H ·ϕs (X ,X )−ϕs (H ·X ,X )−ϕs (X ,H ·X ) i,j i j i,j i j i,j i j i,j i j =H ·X −ϕs ((−n−1+2i)X ,X )−ϕs (X ,(−n−1+2j)X ) s i,j i j i,j i j =(−n−1+2s)X −(−n−1+2i)X −(−n−1+2j)X s s s =[n+2(s−i−j)+1]X s We observethat if n is even then λ(ϕ) is odd, and if n is odd then λ(ϕ) is even. So, if n is even it will be sufficient to find the elements ϕs with weight 1 and if n i,j is odd it will be sufficient to find those of them with weight 0. We can consider the three sequences that correspondwith the weights of V =< X1,X2,...,Xn−1,Xn > in order to find the elements with weight 0 or 1: −n+1,−n+3,...,n−3,n−1; −n+1,−n+3,...,n−3,n−1; −n+1,−n+3,...,n−3,n−1. andwehavetocountthenumberofallpossibilitiestoobtain1(ifniseven)or0(if n is odd). Remember that λ(ϕs )=λ(X )−λ(X )−λ(X ), where λ(X ) belongs i,j s i j s 10 R.M.NAVARRO to the last sequence, and λ(X ), λ(X ) belong to the first and second sequences i j respectively. Forexample,ifnisodd,wehavetoobtain0,sowecanfixanelement (a weight) of the last sequence and then count the possibilities to sum the same quantity between the two first sequences. Taking into account the skew-symmetry ofϕs ,thatisϕs =−ϕs andi6=j, andrepeatingthe abovereasoningforallthe i,j i,j j,i elements of the last sequence we obtain the following theorem: Theorem 1. Let Z2(L;L)be the 2-cocyclesZ2(Ln,m,p;Ln,m,p)thatvanishonthe 0 characteristic vector X . Then, if A=Z2(L;L)∩Hom(L ∧L ,L ) we have that 0 0 0 0 n(3n−2) if n is even 8 dim A=  3n2−4n+1 +⌊n+1⌋ if n is odd 8 4 Proof. It is convenient todistinguish the following four cases where the reasoning for each case is not hard: (1). n≡0 (mod 4). (2). n≡1 (mod 4). (3). n≡2 (mod 4). (4). n≡3 (mod 4). (cid:3) 5. Dimension of B =Z2(L;L)∩Hom(L ∧L ,L ) 0 1 1 In general, any cocycle b ∈ Z2(L;L) ∩ Hom(L ∧ L ,L ) will be any skew- 0 1 1 symmetric bilinear map from L ∧L to L such that: 0 1 1 (2) [X ,b(X ,Y )]−[X ,b(X ,Y )]−b([X ,X ],Y )+ i j k j i k i j k b([X ,Y ],X )+b(X ,[X ,Y ])=0 ∀ X ,X ∈L , Y ∈L i k j i j k i j 0 k 1 with b(X ,X)=0 ∀X ∈L. This condition reduces to 0 (5.1) [X ,b(X ,Y )]−b([X ,X ],Y )−b(X ,[X ,Y ])=0, 1≤j ≤n, 1≤k ≤m 0 j k 0 j k j 0 k In order to obtain the dimension of the space of cocycles B we apply an adap- tation of the sl(2,C)-module Method that we have already used in the precedent section. RecallthatifW ,W ,...,W aresl(2,C)-modules,thenthespaceHom(⊗k W ,W ) 0 1 k i=1 i 0 will be a sl(2,C)-module in the following natural manner: k (ξ·ϕ)(x ,...,x )=ξ·ϕ(x ,...,x )− ϕ(x ,...,ξ·x ,x ,...,x ) 1 k 1 k 1 i i+1 n i=1 X