ON DEFORMATIONS OF THE FILIFORM LIE SUPERALGEBRA Ln,m

Many work was done for filiform Lie algebras defined by M. Vergne [8]. An interesting fact is that this algebras are obtained by deformations of the filiform Lie algebra Ln,m. This was used for classifications in [4]. Like filiform Lie algebras, filiform Lie superalgebras are obtained by nilpotent deformations of the Lie superalgebra Ln,m. In this paper, we recall this fact and we study even cocycles of the superalgebra Ln,m which give this nilpotent deformations. A family of independent bilinear maps will help us to describe this cocycles. At the end an evaluation of the dimension of the space Z 0 (Ln,m, Ln,m) is established. The description of this cocycles can help us to get some classifications which was done in [2, 3]. 1. Deformation of Lie superalgebras 1.1. Nilpotent Lie superalgebras. Definition 1.1. A Z2-graded vector space G = G0 ⊕ G1 over an algebraic closed field is a Lie superalgebra if there exists a bilinear product [, ] over G such that [Gα,Gβ ] ⊂ Gα+β mod 2, [gα, gβ ] = (−1) α.β [gβ, gα] for all gα ∈ Gα and gβ ∈ Gβ and satisfying Jacobi identity: (−1)[A, [B,C]] + (−1)[B, [C,A]] + (−1) [C, [A,B]] = 0 for all A ∈ Gα, B ∈ Gβ and C ∈ Gγ . For such a Lie superalgebra we define the lower central series { C(G) = G, C(G) = [G, C(G)]. Definition 1.2. A Lie superalgebra G is nilpotent if there exist an integer n such that C(G) = {0}. We define for a Lie superalgebra G = G0 ⊕ G1 two sequences : C(G0) = G0, C (G0) = [G0, C (G0)] and C(G1) = G1, C (G1) = [G0, C (G1)] Theorem 1.1. Let G = G0 ⊕G1 be a Lie superalgebras. Then G is nilpotent if and only if there exist (p, q) ∈ N such that C(G0) = {0} and C(G1) = {0}. 1 2 M. GILG Proof. If the Lie superalgebra G = G0 ⊕ G1 is nilpotent the existence of (p, q) such that C(G0) = {0} and C(G1) = {0} is obvious. For the converse, assume that there exist (p, q) such that C(G0) = {0} and C(G1) = {0}, then every operator ad(X) with X ∈ G0 is nilpotent. Let Y ∈ G1, as ad(Y ) ◦ ad(Y ) = 1 2 ad([Y, Y ]) [Y, Y ] is an element of G0, then ad([Y, Y ]) is nilpotent. This implies that ad(Y ) is nilpotent for every Y ∈ G1. By Engel’s theorem for Lie superalgebras [6], this implies that G is nilpotent Lie superalgebra. Definition 1.3. Let G be a nilpotent Lie superalgebra, the super-nilindex of G is the pair (p, q) such that : C(G0) = {0}, C(G0) 6= {0} and C(G1) = {0}, C(G1) 6= {0}. It is and invariant up to isomorphism. 1.2. Cohomology. We recall some definition from [1]. By definition, the superspace of q-dimensional cocycles of the Lie superalgebra G = G0 ⊕ G1 with coefficient in the G-module A = A0 ⊕A1 is given by C(G;A) = ⊕