A complete description of all the infinitesimal deformations of the Lie superalgebra Ln,m

Journal of Geometry and Physics 60 (2010) 131–141

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A complete description of all the infinitesimal deformations of the Lie superalgebra Ln,m Yu. Khakimdjanov a , R.M. Navarro b,∗ a

Laboratoire de mathématiques et applications, Université de Haute Alsace, Mulhouse, France

b

Dpto. de Matemáticas, Universidad de Extremadura, Cáceres 10071, Spain

article

info

Article history: Received 2 June 2009 Received in revised form 30 July 2009 Accepted 20 September 2009 Available online 25 September 2009 MSC: 17B30 17B56 Keywords: Lie algebras Lie superalgebras Cohomology Deformation Nilpotent Filiform

abstract In this paper, we find the dimension and a method to obtain a basis of some infinitesimal deformations of the model Lie superalgebra Ln,m . These deformations lie in Hom(Ln ∧V1 , V1 ) being Ln and V1 the even and odd parts of the Lie superalgebra Ln,m respectively. As Ln corresponds to the model filiform Lie algebra, then these deformations can also be identified with the space of the infinitesimal deformations of the filiform Ln -module V1 . Combining with Bordemann et al. (2007) [2], Gómez et al. (2008) [3] and Khakimdjanov and Navarro [4] we therefore obtain a complete classification of all the infinitesimal deformations of the model Lie superalgebra Ln,m . © 2009 Elsevier B.V. All rights reserved.

1. Introduction The concept of filiform Lie algebras was firstly introduced in [1] by Vergne. This type of nilpotent Lie algebra has important properties; in particular, every filiform Lie algebra can be obtained by a deformation of the model filiform algebra Ln . The present work is about filiform Lie superalgebras. In the same way as filiform Lie algebras, all filiform Lie superalgebras can be obtained by infinitesimal deformations of the model Lie superalgebra Ln,m thus being the analogue of the filiform Lie algebra Ln in the theory of Lie superalgebras. We shall therefore consider infinitesimal deformations of Ln,m which are defined by even 2-cocycles in Z 2 (Ln,m , Ln,m ). n,m This latter space has an obvious direct decomposition into three subspaces according to the even part L0 and the odd part n,m n,m n,m n ,m n,m n ,m 2 n ,m L1 of L . The first and the last component (which lie in Hom(L0 ∧ L0 , L0 ) or in Hom(S L1 , L0 ) have already been n ,m n ,m n,m dealt with in [2–4]). The aim of this article is a detailed study of the second space Z 2 (Ln,m , Ln,m ) ∩ Hom(L0 ∧ L1 , L1 ). As n,m L0 corresponds to the model filiform Lie algebra Ln , then these deformations can also be identified with the space of the n ,m n ,m infinitesimal deformations of the filiform Ln -module L1 . To simplify we shall denote by V1 the subspace L1 . Representation theory of sl(2, C) will allow us to determine the dimension of this space of deformations (Theorems 1 and 2), for a similar sl(2, C)-type computation of the dimension see [5, p. 197], [2–4]. Furthermore, we shall develop a method for calculating an expression for the basis of the above space of deformations (Propositions 6.1–6.3), giving explicitly a basis for some concrete dimensions of n and generic m (Propositions 6.4–6.7).

∗

Corresponding author. Tel.: +34 927257213; fax: +34 927257203. E-mail addresses: [email protected] (Yu. Khakimdjanov), [email protected] (R.M. Navarro).

0393-0440/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2009.09.002

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Combining with [2–4] we therefore obtain a complete classification of all the infinitesimal deformations of the model Lie superalgebra Ln,m (Theorem of dimension). All the vector spaces that appear in this paper (and thus, all the algebras) are assumed to be C-vector spaces of finite dimension. Moreover, we shall use the well-known convention that for the definition of a (super) Lie bracket in terms of a basis only the nonvanishing brackets in some ordering of the base are explicitly mentioned. 2. Preliminaries Recall that a superspace is a vector space with a Z2 -grading: V = V0 ⊕ V1 . Usually, elements of the space V0 are called even, and elements of the space V1 , odd; the indices 0 and 1 are modulo 2. A linear map φ : V → W between two super vector spaces is called even iff φ(V0 ) ⊂ W0 and φ(V1 ) ⊂ W1 and is called odd iff φ(V0 ) ⊂ W1 and φ(V1 ) ⊂ W0 . Clearly, Hom(V , W ) = Hom(V , W )0 ⊕ Hom(V , W )1 where the first summand comprises all the even and the second summand all the odd linear maps. Tensor products V ⊗ W are Z2 graded by means of (V ⊗ W )0 := (V0 ⊗ W0 ) ⊕ (V1 ⊗ W1 ) and (V ⊗ W )1 := (V0 ⊗ W1 ) ⊕ (V1 ⊗ W0 ). A Lie superalgebra (see [6,7]) is a superspace g = g0 ⊕ g1 , with an even bilinear commutation operation (or ‘‘supercommutation’’) [ , ], which satisfies the conditions: 1. [X , Y ] = −(−1)α·β [Y , X ] ∀X ∈ gα , ∀Y ∈ gβ . 2. (−1)γ ·α [X , [Y , Z ]] + (−1)α·β [Y , [Z , X ]] + (−1)β·γ [Z , [X , Y ]] = 0 for all X ∈ gα , Y ∈ gβ , Z ∈ gγ with α, β, γ ∈ Z2 (Graded Jacobi identity). Thus, g0 is an ordinary Lie algebra, and g1 is a module over g0 ; the Lie superalgebra structure also contains the symmetric pairing S 2 g1 −→ g0 , which is a g0 -homomorphism and satisfies the graded Jacobi identity applied to three elements of the space g1 . The descending central sequence of a Lie superalgebra (as Lie algebras) g = g0 ⊕ g1 is defined by C 0 (g) = g, C k+1 (g) = k [C (g), g] for all k ≥ 0. If C k (g) = {0} for some k, the Lie superalgebra is called nilpotent. The smallest integer k such as C k (g) = {0} is called the nilindex of g. We define two new descending sequences, C k (g0 ) and C k (g1 ), as follows: C 0 (gi ) = gi , C k+1 (gi ) = [g0 , C k (gi )], k ≥ 0, i ∈ {0, 1}. If g = g0 ⊕ g1 is a nilpotent Lie superalgebra, then g has super-nilindex or s-nilindex (p, q), if the following conditions hold:

(C p−1 (g0 )) 6= 0

(C q−1 (g1 )) 6= 0,

C p (g0 ) = C q (g1 ) = 0.

Recall that a module A = A0 ⊕ A1 over the Lie superalgebra g is an even bilinear map g × A → A satisfying

∀X ∈ gα , Y ∈ gβ , a ∈ A : X (Ya) − (−1)αβ Y (Xa) = [X , Y ]a. Lie superalgebra cohomology is defined in the following well-known way (see e.g. [6,8]): the superspace of q-dimensional cocycles of the Lie superalgebra g = g0 ⊕ g1 with coefficients in the g-module A = A0 ⊕ A1 is given by C q (g; A) =

M

Hom ∧q0 g0 ⊗ S q1 g1 , A .




q0 +q1 =q q

q

This space is graded by C q (g; A) = C0 (g; A) ⊕ C1 (g; A) with Cpq (g; A) =

M

Hom ∧q0 g0 ⊗ S q1 g1 , Ar .




cq0 +q1 =q q1 +r ≡p mod 2

The differential d : C q (g; A) −→ C q+1 (g; A) is defined by the formula

X


(dc ) g1 , . . . , gq0 , h1 , . . . , hq1 =

(−1)s+t −1 c [gs , gt ], g1 , . . . , gˆs , . . . , gˆt , . . . , gq0 , h1 , . . . , hq1




1≤s
+

q0 X q1   X (−1)s−1 c g1 , . . . , gˆs , . . . , gq0 , [gs , ht ], h1 , . . . , hˆ t , . . . , hq1 s=1 t =1

+

X



c [hs , ht ], g1 , . . . , gq0 , h1 , . . . , hˆ s , . . . , hˆ t , . . . , hq1

1≤s
+

q0 X


(−1)s gs c (g1 , . . . , gˆs , . . . , gq0 , h1 , . . . , hq1 )

s=1

+ (−1)q0 −1+p(c )

q1 X s=1





hs c (g1 , . . . , gq0 , h1 , . . . , hˆ s , . . . , hq1 )



Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 60 (2010) 131–141

133 q

q +1

where c ∈ C q (g; A), g1 , . . . , gq0 ∈ g0 and h1 , . . . , hq1 ∈ g1 . Obviously, d ◦ d = 0, and d(Cp (g; A)) ⊂ Cp q = 0, 1, 2, . . . and p = 0, 1. Then we have the cohomology groups

(g; A) for

Hpq (g; A) = Zpq (g; A)/Bqp (g; A) q

q

where the elements of Z0 (g; A) and Z1 (g; A) are called even q-cocycles and odd q-cocycles respectively. Analogously, the q q elements of B0 (g; A) and B1 (g; A) will be even q-coboundaries and odd q-coboundaries respectively. Two elements of Z q (g; A) are said to be cohomologous if their residue classes modulo Bq (g; A) coincide, i.e., if their difference lies in Bq (g; A). 3. Deformations of L n,m Recall that N n+1 is the variety of (n + 1)-dimensional Lie algebras. If we denote by N n+1,m the variety of nilpotent Lie superalgebras g = g0 ⊕ g1 with dim g0 = n + 1 and dim g1 = m, we will have the following definition: Definition 3.1 ([9]). Any nilpotent Lie superalgebra g = g0 ⊕ g1 ∈ N n+1,m with s-nilindex (n, m) is called filiform. We denote by F n+1,m the subset of N n+1,m consisting of all the filiform Lie superalgebras. Before we study this family of Lie superalgebras, it is convenient to solve the problem of finding a suitable basis, a so-called adapted basis. Theorem 3.1 ([9]). If g = g0 ⊕ g1 ∈ F n+1,m , then there exists an adapted basis of g, namely {X0 , X1 , . . . , Xn , Y1 , . . . , Ym }, with {X0 , X1 , . . . , Xn } a basis of g0 and {Y1 , . . . , Ym } a basis of g1 , such that:

 [X , X ] = Xi+1 ,   0 i [X0 , Xn ] = 0,  [X0 , Yj ] = Yj+1 , [X0 , Ym ] = 0.

1 ≤ i ≤ n − 1, 1 ≤ j ≤ m − 1,

X0 is called the characteristic vector. From the preceding theorem it can be observed that the simplest filiform Lie superalgebra, denoted by Ln,m , will be defined by the following brackets: L

n ,m

 :

[X0 , Xi ] = Xi+1 , [X0 , Yj ] = Yj+1 ,

1 ≤ i ≤ n − 1, 1 ≤ j ≤ m − 1,

with {X0 , X1 , . . . , Xn , Y1 , . . . , Ym } a basis of Ln,m . We shall frequently write V0 := hX1 , . . . , Xn i,

V1 := hY1 , . . . , Ym i,

n ,m L0

n ,m

whence = hX0 i ⊕ V0 and L1 = V1 . Recall that the simplest filiform Lie algebra is Ln : Ln : [X0 , Xi ] = Xi+1 ,



1≤i≤n−1

with {X0 , X1 , . . . , Xn } a basis of Ln . This algebra is very important because of every filiform Lie algebra can be obtained by a deformation of it [1]. In complete analogy to Lie algebras, Ln,m will be the most important filiform Lie superalgebra since all the other filiform Lie superalgebras can be obtained from it by deformations. So, we are going to consider its infinitesimal deformations that will be given by the even 2-cocycles, Z02 (Ln,m , Ln,m ). An infinitesimal deformation of Ln,m will thus be an element of the following space Z02 (Ln,m , Ln,m ) = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g0 , g0 ) ⊕ Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 )

⊕ Z 2 (Ln,m , Ln,m ) ∩ Hom(S 2 g1 , g0 ) n,m

n ,m

where g0 = L0 and g1 = L1 . The third component has been determined in [2,3], the first component has been determined in [4]. n ,m About the second component, we can consider in L1 = V1 := hY1 , . . . , Ym i the structure of Ln -module by the product: X0 · Yi = Yi+1 ,

i = 1, . . . , m − 1. n,m

Then, the Lie superalgebra Ln,m can be viewed as the semidirect product of the Lie algebra Ln by L1 that is the Lie algebra defined in the basis {X0 , X1 , . . . , Xn , Y1 , . . . , Ym } by the brackets:



[X0 , Xi ] = Xi+1 , [X0 , Yj ] = Yj+1 ,

= V1 := hY1 , . . . , Ym i,

1 ≤ i ≤ n − 1, 1 ≤ j ≤ m − 1.

Thus, the second space Z 2 (Ln,m , Ln,m )∩ Hom(g0 ∧ g1 , g1 ) can be identified with the space of the infinitesimal deformations of the filiform Ln -module V1 .

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4. Deformations of the filiform Ln -module V1 In order to determine the infinitesimal deformations of the filiform Ln -module V1 we are going to consider the Z-graduation of Ln,m that follows:

(Ln,m )i = hXi−1 i, 1 ≤ i ≤ n + 1 (Ln,m )i = hYm+i i, −m + 1 ≤ i ≤ 0 (Ln,m )i = 0, with i ≤ −m or i ≥ n + 2. As this graduation is compatible with the coboundary operator d : C q (g; A) −→ C q+1 (g; A), we have the following Z-graduation of B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) B=

M

(B)i

i∈Z

(B)i = {Ψ ∈ B : Ψ ((Ln,m )k , (Ln,m )t ) ⊂ (Ln,m )k+t +i }. Thus, it will be sufficient to find a basis of B constituted by homogeneous cocycles Ψ , that is Ψ (Xi , Yj ) = aYs and if

Ψ (Xi , Yj ) = aYs , a 6= 0 Ψ (Xr , Yt ) = bYq , b = 6 0 then s − i − j = q − r − t. Furthermore, we distinguish different subspaces of deformations in B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ). Thus, we will consider: Case 1. Cocycles Ψ such that,

Ψ : hX1 , . . . , Xn i ∧ hY1 , . . . , Ym i −→ hY1 , . . . , Ym i. This subspace of B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) will be denoted by B1 . Case 2. B2 will be the subspace of B generated by the homogeneous cocycles Ψ defined by a skew-symmetric map: hX0 i ∧ hYk i −→ hYt i with 1 ≤ k, t ≤ m. Then, we will have that B = B1 ⊕ B2 , so the problems of calculating the dimension and finding a basis are reduced to solving them in each Bi . 5. Dimension of the deformations of the filiform Ln -module V1 5.1. Dimension of B1 . In general, any cocycle Ψ ∈ Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) = B1 ⊕ B2 will be any skew-symmetric bilinear map from g0 ∧ g1 to g1 such that dΨ is equivalent to 0, that is:

Ψ ([Xi , Xj ], Yk ) + Ψ ([Yk , Xi ], Xj ) + Ψ ([Xj , Yk ], Xi ) − [Xi , Ψ (Xj , Yk )] − [Xj , Ψ (Yk , Xi )] = 0 ∀ Xi , Xj ∈ g0 , Yk ∈ g1 . In our case B1 , that is Ψ : hX1 , . . . , Xn i ∧ hY1 , . . . , Ym i −→ hY1 , . . . , Ym i, this condition reduces to

[X0 , Ψ (Xj , Yk )] − Ψ ([X0 , Xj ], Yk ) − Ψ (Xj , [X0 , Yk ]) = 0,

1 ≤ j ≤ n, 1 ≤ k ≤ m .

(5.1)

In order to obtain the dimension of the space of cocycles for the Case 1, we apply an adaptation of the sl(2, C)-module method that we used in [2–4]. Recall the following well-known facts about the Lie algebra sl(2, C) and its finite-dimensional modules; see e.g. [10,11]: sl(2, C) = hX− , H , X+ i with the following commutation relations:

( [X+ , X− ] = H [H , X+ ] = 2X+ , [H , X− ] = −2X− . Let V be an n-dimensional sl(2, C)-module, V = he1 , . . . , en i. Then, up to isomorphism there exists a unique structure of an irreducible sl(2, C)-module in V given in a basis e1 , . . . , en as follows [10]: X+ · ei = ei+1 , X+ · en = 0, H · ei = (−n + 2i − 1)ei ,

(

1 ≤ i ≤ n − 1, 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 W0 , W1 , . . . , Wk be sl(2, C)-modules, then the space Hom(⊗ki=1 Wi , W0 ) is a sl(2, C)-module in the following natural manner:

(ξ · ϕ)(x1 , . . . , xk ) = ξ · ϕ(x1 , . . . , xk ) −

i =k X i =1

ϕ(x1 , . . . , ξ · xi , xi+1 , . . . , xn )

Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 60 (2010) 131–141

135

with ξ ∈ sl(2, C) and ϕ ∈ Hom(⊗ki=1 Wi , W0 ). In particular, if k = 2 and V0 = W1 , V1 = W2 = W0 , then

(ξ · ϕ)(x1 , x2 ) = ξ · ϕ(x1 , x2 ) − ϕ(ξ · x1 , x2 ) − ϕ(x1 , ξ · x2 ). An element ϕ ∈ Hom(V0 ⊗ V1 , V1 ) is said to be invariant if X+ · ϕ = 0, that is X+ · ϕ(x1 , x2 ) − ϕ(X+ · x1 , x2 ) − ϕ(x1 , X+ · x2 ) = 0,

∀x1 ∈ V0 , ∀x2 ∈ V1 .

(5.2)

Note that ϕ ∈ Hom(V0 ⊗ V1 , V1 ) is invariant if and only if ϕ is a maximal vector. n,m We are going to consider the structure of irreducible sl(2, C)-module in V0 = hX1 , . . . , Xn i = L0 − {X0 } and in n ,m V1 = hY1 , . . . , Ym i = L1 , thus in particular:

 X · X = Xi+1 ,   + i X+ · Xn = 0  X+ · Yj = Yj+1 , X+ · Ym = 0.

1≤i≤n−1 1≤j≤m−1

We identify the multiplication of X+ and Xi in the sl(2, C)-module V0 = hX1 , . . . , Xn i, with the bracket product [X0 , Xi ] n,m in L0 . Analogously with X+ · Yj and [X0 , Yj ]. Thanks to these identifications, the expressions (5.1) and (5.2) are equivalent, so we have the following result: Proposition 5.1. Any skew-symmetric bilinear map ϕ , ϕ : V0 ∧ V1 −→ V1 will be an element of B1 if and only if ϕ is a maximal n,m vector of the sl(2, C)-module Hom(V0 ∧ V1 , V1 ), with V0 = hX1 , . . . , Xn i and V1 = L1 . Corollary 5.1. As each sl(2, C)-module has (up to nonzero scalar multiples) a unique maximal vector, then the dimension of B1 is equal to the number of summands of any decomposition of Hom(V0 ∧ V1 , V1 ) into direct sum of irreducible sl(2, C)-modules. But instead of looking at the maximal vectors, we can equally well use the fact that each irreducible module contains either a unique (up to scalar multiples) vector of weight 0 (in the case where the dimension of the irreducible module is odd) or a unique (up to scalar multiples) vector of weight 1 (in the case where the dimension of the irreducible module is even). We therefore have the following: Corollary 5.2. The dimension of B1 is equal to the dimension of the subspace of Hom(V0 ∧ V1 , V1 ) spanned by the vectors of weight 0 or 1. At this point, we are going to apply the sl(2, C)-module method aforementioned in order to obtain the dimension of the space of cocycles B1 . Firstly, we consider a natural basis of Hom (V0 ∧ V1 , V1 ) consisting of the following maps where 1 ≤ s, j, l ≤ m and 1 ≤ i, k ≤ n:

ϕis,j (Xk , Yl ) =



Ys 0

if (i, j) = (k, l) in all other cases.

Thanks to Corollary 5.2 it will be enough to find the basis vectors ϕis,j with weight 0 or 1. The weight of an element ϕis,j (with respect to H) is

λ(ϕis,j ) = λ(Ys ) − λ(Xi ) − λ(Yj ) = (−m + 2s − 1) − (−n + 2i − 1) − (−m + 2j − 1) = n + 2(s − i − j) + 1. In fact,

(H · ϕis,j )(Xi , Yj ) = H · ϕis,j (Xi , Yj ) − ϕis,j (H · Xi , Yj ) − ϕis,j (Xi , H · Yj ) = H · Ys − ϕis,j ((−n + 2i − 1)Xi , Yj ) − ϕis,j (Xi , (−m + 2j − 1)Yj ) = (−m + 2s − 1)Ys − (−n + 2i − 1)Ys − (−m + 2j − 1)Ys = [n + 2(s − i − j) + 1]Ys . We are going to introduce a simpler weight, it corresponds to the action of the diagonalizable derivation d, d ∈ Der Ln , defined by: d(X0 ) = X0 ,

d(Xi ) = iXi ;

1 ≤ i ≤ n.

This weight will be denoted by p(ϕ). We have that p(ϕis,j ) = s − i − j. We have the following relationships between the two weights:

λ(ϕ) = 2p(ϕ) + n + 1, p(ϕ) =

1 2

(λ(ϕ) − n − 1).

Remark 5.1. 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 ϕis,j with weight 1 and if n is odd it will be sufficient to find those with weight 0.

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In order to find the elements with weight 0 or 1, we can consider the three sequences that correspond to the weights of V0 = hX1 , . . . , Xn i, V1 = hY1 , Y2 , . . . , Ym i and V1 = hY1 , Y2 , . . . , Ym i:

−n + 1, −n + 3, . . . , n − 3, n − 1; −m + 1, −m + 3, . . . , m − 3, m − 1; −m + 1, −m + 3, . . . , m − 3, m − 1. We shall have to count the number of all possibilities to obtain 1 (if n is even) or 0 (if n is odd). Remember that

λ(ϕis,j ) = λ(Ys ) − λ(Xi ) − λ(Yj ), where λ(Ys ) belongs to the last sequence, and λ(Xi ), λ(Yj ) belong to the first and second sequences respectively. For example, if n is odd we have to obtain 0, so we can fix an element (a weight) of the last sequence and then to count the possibilities to sum the same quantity between the two first sequences. Repeating the above reasoning for all the elements of the last sequence we obtain the following theorem. Theorem 1. If B1 is the subspace of B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) spanned by the cocycles Ψ such that,

Ψ : hX1 , . . . , Xn i ∧ hY1 , . . . , Ym i −→ hY1 , . . . , Ym i. Then, we have the following values for the dimension of B1

 4nm − n2 + 1     4 dim B1 = 4nm − n2    4  2

m

if n is odd, if n is even,

n < 2m + 1 n < 2m + 1

if n ≥ 2m + 1.

Proof. It is convenient to distinguish the following four cases where the reasoning for each case is not hard: (1) (2) (3) (4)

n n n n

≡ 0 (mod 4). ≡ 1 (mod 4). ≡ 2 (mod 4). ≡ 3 (mod 4).



5.2. Dimension and a basis of B2 In this section we are going to consider B2 that is the subspace of B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) generated by the homogeneous cocycles Ψ defined by a skew-symmetric map: hX0 i ∧ hYk i −→ hYt i with 1 ≤ k, t ≤ m. It is easy to see the following results. Proposition 5.2. The following maps j

Ψ0,i (X0 , Yl ) =



Yj 0

if i = l in all other cases

where 1 ≤ i, j, l ≤ m, form a basis of B2 . Corollary 5.3. dim B2 = m2 . 5.3. Total dimension of B Thanks to the preceding subsections we can obtain the total dimension of the space of deformations B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ). Theorem 2. B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then

 4nm − n2 + 1  2   m + 4 4nm − n2 dim B = 2  m +   4  2m2

if n is odd, if n is even,

n < 2m + 1 n < 2m + 1

if n ≥ 2m + 1.

6. Computing a basis of B In this section we are going to develop a method that permits us to calculate a basis of B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ) = B1 ⊕ B2 . As we have yet seen, thanks to Proposition 5.2 we have a basis of the subspace B2 . Thus, it only rests to compute a basis for B1 .

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Let ϕ be an element of B1 , with weight λ(ϕ). As ϕ is a maximal vector of the sl(2, C)-module Hom (V0 ∧ V1 , V1 ), its weight λ(ϕ) is always a nonnegative integer, λ(ϕ) ≥ 0. On the other hand, p(ϕ) is always less than or equal to m − 2, p(ϕ) ≤ m − 2. In fact, ϕ1m,1 is an element with maximal weight p(ϕ), p(ϕ1m,1 ) = m − 2. So, we have the following estimates for p(ϕ):

−n − 1

≤ p(ϕ) ≤ m − 2. (6.1) 2 In order to get a basis of B1 it is enough to obtain the basis for each subspace B1 (p) of B1 , spanned by all the elements with weight p such that p satisfies (6.1). Let Ψks,1 be an element of B1 with weight p, p(Ψks,1 ) = s − k − 1, and defined by Ψks,1

(Xi , Y1 ) =



if i = k in all other cases

Ys 0

with 1 ≤ k, i ≤ n, 1 ≤ s ≤ m and satisfying the equations

[X0 , Ψks,1 (Xi , Yj )] − Ψks,1 ([X0 , Xi ], Yj ) − Ψks,1 (Xi , [X0 , Yj ]) = 0, 1 ≤ i ≤ n, 1 ≤ j ≤ m − 1. (6.2) Thanks to Eqs. (5.1) we observe that Ψks,1 is not always a cocycle of B1 . In particular, Ψks,1 will be a cocycle of B1 if and only

if it satisfies the equations

[X0 , Ψks,1 (Xi , Ym )] − Ψks,1 (Xi+1 , Ym ) = 0, with 1 ≤ i ≤ n. We observe that if i = n then Xi+1 vanishes. The following formula for Ψks,1 can be proved by induction: Ψks,1 (Xi , Yj ) = −Ψks,1 (Yj , Xi ) = (−1)k−i Cjk−−1i Yi+j+s−k−1 q

with 1 ≤ s ≤ m, 1 ≤ i ≤ k ≤ n and 1 ≤ i + j + s − k − 1 ≤ m. We suppose that Ct = 0 if q < 0 or t < 0 or q > t, and q C00 = Ctt = Ct0 = 1 with t > 0. In the remaining cases we have Ct = q!(tt−! q)! . Proposition 6.1. The 2-skew symmetric bilinear map Ψks,1 defined by the formula

Ψks,1 (Xi , Yj ) = (−1)k−i Cjk−−1i Yi+j+s−k−1 ,

1 ≤ i ≤ k, 1 ≤ j ≤ m

is a cocycle of B1 iff p(Ψks,1 ) = s − k − 1 ≥ −1. Proof. We only have to check whether Ψks,1 satisfies or not the equations

[X0 , Ψks,1 (Xi , Ym )] − Ψks,1 (Xi+1 , Ym ) = 0,

with 1 ≤ i ≤ n.

If p(Ψks,1 ) = s − k − 1 = −1, then

Ψks,1 (X1 , Ym ) = (−1)k−1 Cmk−−11 Ym and Ψks,1 (X2 , Ym ) = · · · = Ψks,1 (Xn , Ym ) = 0 which clearly satisfies the above equations. If p(Ψks,1 ) > −1, then Ψks,1 (X1 , Ym ) = · · · = Ψks,1 (Xn , Ym ) = 0 and also satisfies the above equations. If p(Ψks,1 ) < −1, then

Ψks,1 (X1 , Ym ) = (−1)k−1 Cmk−−11 Yt with t < m. If we apply the cocycle equations we have

[X0 , Ψks,1 (X1 , Ym )] = Ψks,1 (X2 , Ym ) = (−1)k−2 Cmk−−21 Xt +1 , but

[X0 , Ψks,1 (X1 , Ym )] = [X0 , (−1)k−1 Cmk−−11 Xt ] = (−1)k−1 Cmk−−11 Xt +1 and then k −2 k−1 Cm −1 = −Cm−1 ,

which is a contradiction.



Proposition 6.2. Let Ψ ∈ B1 be a cocycle with weight p = p(Ψ ) ≤ −2. Then

Ψ =

X

ak Ψks,1

s−k−1=p

for some numbers ak .

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Proof. Let Ψ ∈ B1 be a cocycle with weight p. Then Ψ (Xi , Y1 ) = ai Yi+1+p . We are going to consider the difference

X

ϕ=Ψ −

ak Ψks,1 .

s−k−1=p

It is easy to check that ϕ is a 2-skew symmetric bilinear map such that

ϕ(X1 , Y1 ) = ϕ(X2 , Y1 ) = · · · = ϕ(Xn , Y1 ) = 0. As

Ψks,1

satisfies the equations

[X0 , Ψks,1 (Xi , Yj )] − Ψks,1 ([X0 , Xi ], Yj ) − Ψks,1 (Xi , [X0 , Yj ]) = 0,

1 ≤ i ≤ n, 1 ≤ j ≤ m − 1

then ϕ satisfies them too, which implies that ϕ vanishes. In fact, if we apply ad X0 to ϕ(Xi , Y1 ) with 1 ≤ i ≤ n we obtain that

ϕ(X1 , Y2 ) = ϕ(X2 , Y2 ) = · · · = ϕ(Xn , Y2 ) = 0. Repeating the process for ϕ(Xi , Y2 ) with 1 ≤ i ≤ n it can be obtained that

ϕ(X1 , Y3 ) = ϕ(X2 , Y3 ) = · · · = ϕ(Xn , Y3 ) = 0. And thus, successively it can be obtained that ϕ vanishes.



Proposition 6.3. Let Ψ a linear combination such that

Ψ =

X

ak Ψks,1

s−k−1=p

with

≤ p ≤ −2. Then Ψ is a cocycle of B1 iff

−n−1 2

(ad X0 )−p−1 (Ψ (X1 , Ym )) = (ad X0 )−p−2 (Ψ (X2 , Ym )) = · · · = (ad X0 )(Ψ (X−p−1 , Ym )) = Ψ (X−p , Ym ). Proof. As each Ψks,1 verifies Eq. (6.2), Ψ satisfies them too. Thus, we have that Ψ will be a cocycle of B1 iff Ψ satisfies the equations

[X0 , Ψ (Xi , Ym )] = Ψ (Xi+1 , Ym ), which proves the result.

with 1 ≤ i ≤ n



6.1. Basis for concrete dimensions of n and generic m In Proposition 5.2 we have yet obtained a basis of B2 . For B1 as Proposition 6.1 gives us the description of the cocycles with p ≥ −1, it remains to describe a basis of the cocycles of B1 such that:

−n − 1 2

≤ p ≤ −2.

If we fix p satisfying −n2−1 ≤ p ≤ −2, then all the mappings Ψks,1 with weight p will be p+k+1

Ψk,1

with 1 ≤ k ≤ n, 1 ≤ p + k + 1 ≤ m.

Also we have that max(1, −p) ≤ k ≤ min(n, m − p − 1). As 2 ≤ −p, then −p ≤ k ≤ min(n, m − p − 1). Thus, all the mappings Ψks,1 with weight p will be p+t +1

Ψ−1 p,1 , Ψ−2 p+1,1 , . . . , Ψt ,1

with t = min(n, m − p − 1). Let Ψ be

p+t +1

Ψ = a−p Ψ−1 p,1 + a−p+1 Ψ−2 p+1,1 + · · · + at Ψt ,1

.

Proposition 6.3 gives us −p − 1 linear equations in a−p , . . . , at :

(ad X0 )i (Ψ (X−p−i , Ym )) = Ψ (X−p , Ym ),

1 ≤ i ≤ −p − 1.

The solutions of these systems will give us a basis of the space of cocycles. The main problem is that the matrices of the resulting systems are not reduced as happened in [2] or [3], so it will be necessary to fix concrete dimensions for n to solve the problem. Proposition 6.4. If B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then a basis of B for n = 2 and arbitrary m, m ≥ 2, is constituted by the following m2 + 2m − 1 cocycles:

n

o  j Ψ0,i , 1 ≤ i, j ≤ m ∪ Ψ11,1 , Ψ12,1 , . . . , Ψ1m,1 , Ψ22,1 , . . . , Ψ2m,1 .

Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 60 (2010) 131–141

139

Proof. In Proposition 5.2 we have the following maps j Ψ0 ,i

(X0 , Yl ) =



Yj 0

if i = l in all other cases j

where 1 ≤ i, j, l ≤ m, as a basis of B2 . Hence we have the basis cocycles Ψ0,i , with 1 ≤ i, j ≤ m. For B1 as p verifies that

−n−1 2

≤ p ≤ m−2 and n = 2 we have the following restriction for the values of p: −1 ≤ p ≤ m−2. p+2

Thus, thanks to Proposition 6.1 we obtain the basis cocycles of the form Ψks,1 . In particular we obtain the cocycles Ψ1,1 and p+3

Ψ2,1 for each p with −1 ≤ p ≤ m−3 and the cocycle Ψ1m,1 for p = m−2. Thus, we have {Ψ11,1 , Ψ12,1 , . . . , Ψ1m,1 , Ψ22,1 , . . . , Ψ2m,1 }. So, we have obtained m2 + 2m − 1 linearly independent cocycles. Thanks to Theorem 2 we conclude the proof.  Proposition 6.5. If B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then a basis of B for n = 3 and arbitrary m, m ≥ 3, is constituted by the following m2 + 3m − 2 cocycles:

n

o n o 2 j Ψ0,i , 1 ≤ i, j ≤ m ∪ Ψ11,1 , Ψ12,1 , . . . , Ψ1m,1 , Ψ22,1 , . . . , Ψ2m,1 , Ψ33,1 , . . . , Ψ3m,1 , Ψ 3,1

2

with Ψ 3,1 =

m−1 Ψ21,1 2

+ Ψ32,1 . j

Proof. Thanks to Proposition 5.2 we have the maps Ψ0,i with 1 ≤ i, j ≤ m as a basis of B2 . −n −1

For B1 as p verifies that 2 ≤ p ≤ m − 2, then as n = 3 we have the following restriction for the values of p, that is −2 ≤ p ≤ m − 2. For p ≥ −1 thanks to Proposition 6.1 we obtain the basis cocycles of the form Ψks,1 . In particular we p+2

p+3

p+4

obtain the cocycles Ψ1,1 , Ψ2,1 and Ψ3,1 for each p such that −1 ≤ p ≤ m − 4; we obtain the cocycles Ψ1m,1−1 and Ψ2m,1 for p = m − 3, and the cocycle Ψ1m,1 for p = m − 2. Finally for p = −2 the mappings Ψks,1 with weight p are Ψ21,1 and Ψ32,1 , but in this case they are not cocycles. We have to consider a linear combination of them called Ψ :

Ψ = a2 Ψ21,1 + a3 Ψ32,1 . As Proposition 6.3 gives us the linear equation:

−a2 Cm1 −1 + a3 Cm2 −1 = a2 Cm0 −1 − a3 Cm1 −1 2

1 1 giving to a3 the value of 1 we obtain a basis of solutions: a2 = m− and a3 = 1, leading to Ψ 3,1 = m− Ψ21,1 + Ψ32,1 . 2 2 2 So, we have obtained m + 3m − 2 linearly independent cocycles. Thanks to Theorem 2 we conclude the proof.



Remark 6.1. It is easy to see that for the following values of the dimension pair (n, m): (3, 2) and (4, 2), it can be obtained that the basis cocycles are constituted by the following 8 cocycles

n

o n o 2 j Ψ0,i , 1 ≤ i, j ≤ 2 ∪ Ψ11,1 , Ψ12,1 , Ψ22,1 , Ψ 3,1

2

with Ψ 3,1 =

1 Ψ1 2 2 ,1

+ Ψ32,1 , and being each cocycle the respective expression according to the dimension.

Proposition 6.6. If B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then a basis of B for n = 4 and arbitrary m, m ≥ 4, is constituted by the following m2 + 4m − 4 cocycles:

n

o n o 2 3 j Ψ0,i , 1 ≤ i, j ≤ m ∪ Ψ11,1 , Ψ12,1 , . . . , Ψ1m,1 , Ψ22,1 , . . . , Ψ2m,1 , Ψ33,1 , . . . , Ψ3m,1 , Ψ44,1 , . . . , Ψ4m,1 , Ψ 3,1 , Ψ 4,1

2

with Ψ 3,1 =

m−1 Ψ21,1 2

3

+ Ψ32,1 and Ψ 4,1 =

−(m−1)(m−2) 6

Ψ21,1 + Ψ43,1 . j

Proof. Thanks to Proposition 5.2 we have the maps Ψ0,i with 1 ≤ i, j ≤ m as a basis of B2 .

For B1 as p verifies that −n2−1 ≤ p ≤ m − 2, then as n = 4 we have the following restriction for the values of p, that is −2 ≤ p ≤ m − 2. For p ≥ −1 thanks to Proposition 6.1 we obtain the basis cocycles of the form Ψks,1 . In particular we obtain p+5 p+2 p+3 p+4 the cocycles Ψ1,1 , Ψ2,1 , Ψ3,1 and Ψ4,1 for each p such that −1 ≤ p ≤ m − 5; we obtain the cocycles Ψ1m,1−2 , Ψ2m,1−1 and m−1 m Ψ3,1 for p = m − 4, the cocycles Ψ1,1 and Ψ2m,1 for p = m − 3, and the cocycle Ψ1m,1 for p = m − 2. Finally for p = −2 the mappings Ψks,1 with weight p are Ψ21,1 , Ψ32,1 and Ψ43,1 but in this case they are not cocycles. We have to consider a linear combination of them called Ψ :

Ψ = a2 Ψ21,1 + a3 Ψ32,1 + a4 Ψ43,1 . As Proposition 6.3 gives us the linear equation:

−a2 Cm1 −1 + a3 Cm2 −1 − a4 Cm3 −1 = a2 Cm0 −1 − a3 Cm1 −1 + a4 Cm2 −1 .

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Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 60 (2010) 131–141

Two linearly independent solutions correspond to the following possibilities for the pair (a3 , a4 ): (1, 0) and (0, 1). If

(a3 , a4 ) = (1, 0) then a1 =

−(m−1)(m−2)

2

3

obtaining Ψ 3,1 , and if (a3 , a4 ) = (0, 1) then a1 = obtaining in this case Ψ 4,1 . 6 So, we have obtained m + 4m − 4 linearly independent cocycles. Thanks to Theorem 2 we conclude the proof.  2

m−1 2

Remark 6.2. It is easy to see that for the concrete dimension n = 4 and m = 3 it can be obtained that the basis cocycles are constituted by the following 17 cocycles

n

o o n 2 3 j Ψ0,i , 1 ≤ i, j ≤ 3 ∪ Ψ11,1 , Ψ12,1 , Ψ13,1 , Ψ22,1 , Ψ23,1 , Ψ33,1 , Ψ 3,1 , Ψ 4,1

2 3 with Ψ 3,1 = Ψ21,1 + Ψ32,1 and Ψ 4,1 = −31 Ψ21,1 + Ψ43,1 .

Proposition 6.7. If B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then a basis of B for n = 5 and arbitrary m, m ≥ 5, is constituted by the following m2 + 5m − 6 cocycles:

n

o n o 2 3 4 3 j Ψ0,i , 1 ≤ i, j ≤ m ∪ Ψ 3,1 , Ψ 4,1 , Ψ 5,1 , Ψ 5,1  ∪ Ψ11,1 , . . . , Ψ1m,1 , Ψ22,1 , . . . , Ψ2m,1 , Ψ33,1 , . . . , Ψ3m,1 , Ψ44,1 , . . . , Ψ4m,1 , Ψ55,1 , . . . , Ψ5m,1

with 2

m−1

4

2 (m − 1)(m − 2)(m − 3)

Ψ 3,1 = Ψ 5,1 =

Ψ21,1 + Ψ32,1 , 24

3

Ψ 4 ,1 =

−(m − 1)(m − 2) 6

Ψ21,1 + Ψ43,1 ,

3

Ψ21,1 + Ψ54,1 ,

Ψ 5,1 = a3 Ψ31,1 + a4 Ψ42,1 + Ψ53,1

being a4 = a3 =

3 2 4 2 2 m(Cm −1 − Cm−1 ) + (Cm−1 + Cm−1 )(Cm−1 − 1)

−m(Cm3 −1 − Cm1 −1 ) + (Cm2 −1 + Cm1 −1 )(Cm2 −1 − 1) 3 2 4 1 Cm −1 − Cm−1 + (Cm−1 − Cm−1 )a4 2 Cm −1 − 1

,

. j

Proof. Thanks to Proposition 5.2 we have the maps Ψ0,i with 1 ≤ i, j ≤ m as a basis of B2 . In the case of B1 we have the following restriction for the values of p: −3 ≤ p ≤ m − 2. For p ≥ −1 thanks to Proposition 6.1 we obtain the basis cocycles of the form Ψks,1 . Finally when p = −2 or p = −3 we consider the mappings Ψks,1 with weight p. After we take a linear combination of 2

3

4

them and with Proposition 6.3 we obtain a linear system whose basis solutions are Ψ 3,1 , Ψ 4,1 , and Ψ 5,1 for p = −2 and 3 Ψ 5,1 for p = −3. So, we have obtained m2 + 4m − 4 linearly independent cocycles. Thanks to Theorem 2 we conclude the proof.  Remark 6.3. If B = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 ), then a basis of B for the following values of the pair of dimensions (n, m) is 2

2

• For (5, 2) the cocycles basis is {Ψ0j,i , 1 ≤ i, j ≤ 2} ∪ {Ψ11,1 , Ψ12,1 , Ψ22,1 , Ψ 3,1 } with Ψ 3,1 = 21 Ψ21,1 + Ψ32,1 . 2 3 3 2 • For (5, 3) the cocycles basis is {Ψ0j,i , 1 ≤ i, j ≤ 3} ∪ {Ψ11,1 , Ψ12,1 , Ψ13,1 , Ψ22,1 } ∪{Ψ23,1 , Ψ33,1 , Ψ 3,1 , Ψ 4,1 , Ψ 5,1 } with Ψ 3,1 = 3

Ψ21,1 + Ψ32,1 , Ψ 4,1 =

−1 3

3

Ψ21,1 + Ψ43,1 , Ψ 5,1 = 61 Ψ31,1 + 12 Ψ42,1 + Ψ53,1 .

2

3

• For (5, 4) the cocycles basis is {Ψ0j,i , 1 ≤ i, j ≤ 4} ∪ {Ψ11,1 , Ψ12,1 , Ψ13,1 , Ψ14,1 , Ψ22,1 } ∪ {Ψ23,1 , Ψ24,1 , Ψ33,1 , Ψ34,1 , Ψ44,1 , Ψ 3,1 , Ψ 4,1 , 4 Ψ 5,1

} with

2 Ψ 3 ,1

= 32 Ψ21,1 +

3 Ψ32,1 , Ψ 4,1

= −Ψ21,1 +

4 Ψ43,1 , Ψ 5,1

= 14 Ψ21,1 + Ψ54,1 .

Remark 6.4. As we have done in the precedent propositions it can be done for any value of n and arbitrary m. It will be necessary to do apart the particular cases: m = 2, m = 3, . . . , m = n + d −n2−1 e + 1; and then we would give the general expression of the basis cocycles for the pair of dimensions (n, m) with m ≥ n + d −n2−1 e + 2. 7. Complete description of the deformations Z02 (L n,m , L n,m ) Theorem of dimension. If we consider the space of the infinitesimal deformations of the Lie superalgebra Ln,m , Z02 (Ln,m , Ln,m ), then the dimension of this space of deformations will be given by dim(Z02 (Ln,m , Ln,m )) = dim A + dim B + dim C

Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 60 (2010) 131–141

141

with

dim A =

 n(3n − 2)  + n2  2

8

if n is even



3n − 4n + 1 n+1   + + n2 8

 4nm − n2 + 1  2  m +  4 4nm − n2 dim B = 2  m +   4  2

2m

dim C =



if n is odd

4

if n is odd, if n is even,

n < 2m + 1 n < 2m + 1

if n ≥ 2m + 1

 m(m + 1)    2    1   (4mn − n2 + 2n + 3)    8 1   (4mn − n2 + 2n − 1)   8       1  (4mn − n2 + 2n) 8

n ,m

Proof. If we write g0 and g1 in place of L0

if n ≥ 2m − 1 if n < 2m − 1,

n ≡ 1 (mod 4) and m odd, or n ≡ 3 (mod 4) and m even

if n < 2m − 1,

n ≡ 3 (mod 4) and m odd, or n ≡ 1 (mod 4) and m even

if n < 2m − 1

and n even.

n ,m

and L1 , then

Z02 (Ln,m , Ln,m ) = Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g0 , g0 ) ⊕ Z 2 (Ln,m , Ln,m ) ∩ Hom(g0 ∧ g1 , g1 )

⊕ Z 2 (Ln,m , Ln,m ) ∩ Hom(S 2 g1 , g0 ) =: A ⊕ B ⊕ C . So, dim(Z02 (Ln,m , Ln,m )) = dim A + dim B + dim C . dim C has been determined in [2] (Theorem 1), dim A has been determined in [4] (Theorem 2) and the expression of dim B is in Theorem 2 of present paper.  Basis of Z02 (Ln,m , Ln,m ). In order to obtain a basis of Z02 (Ln,m , Ln,m ) = A ⊕ B ⊕ C it is only necessary to obtain a basis of each subspace. In this paper we have given a method to obtain a basis of B (Propositions 6.1–6.3), giving explicitly a basis for some concrete dimensions of n and generic m (Propositions 6.4–6.7). A similar result for C can be found in [2,3] and for A in [4]. So, combining these results it can be obtained a complete description of all the infinitesimal deformations Z02 (Ln,m , Ln,m ). References [1] M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes, Bull. Soc. Math. France 98 (1970) 81–116. [2] M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras, J. Geom. Phys. 57 (2007) 1391–1403. [3] J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Infinitesimal deformations of the Lie superalgebra Ln,m , J. Geom. Phys. 58 (2008) 849–859. [4] Yu. Khakimdjanov, R.M. Navarro, Deformations of filiform Lie algebras and superalgebras, J. Geom. Phys. (submitted for publication). [5] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. LXVI (2) (1997) 151–201. [6] D.B. Fuks, Cohomology of Infinite-dimensional Lie Algebras, Plenum Publishing Copr., 1986. [7] M. Scheunert, The Theory of Lie Superalgebras, in: Lecture Notes in Math., vol. 716, 1979. [8] M. Scheunert, R.B. Zhang, Cohomology of Lie superalgebras and of their generalizations, J. Math. Phys. 39 (1998) 5024–5061. [9] J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some problems concerning to nilpotent Lie superalgebras, J. Geom. Phys. 51 (2004) 473–486. [10] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1975, (Chapter 7–8). [11] James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1987.