Infinitesimal deformations of the model $\mathbb{Z}_3$-filiform Lie algebra

In this work it is considered the vector space composed by the infinitesimal deformations of the model $\mathbb{Z}_3$-filiform Lie algebra $L^{n,m,p}$. By using these deformations all the $\mathbb{Z}_3$-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work together to those obtained in [Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 61(2011)1797-1808] and [Corrigendum to Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 62(2012)1571], leads to compute the total dimension of the mentioned space of deformations.


Introduction
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 can be obtained by a deformation of the model filiform algebra L n . In the same way as filiform Lie algebras, all filiform Lie superalgebras can be obtained by infinitesimal deformations of the model Lie superalgebra L n,m [1], [4], [8] and [9].
Continuing with the work of Vergne we have generalized the concept and the propierties of the filiform Lie algebras into the theory of color Lie superalgebras. Thus, filiform G-color Lie superalgebras and the model filiform G-color Lie superalgebra were obtained in [10].
In the present paper the focus of interest are color Lie superalgebras with a Z 3 -grading vector space, i.e. G = Z 3 , due to its physical applications [3], [6], [7], [13], [16] and [17]. Due to the fact that the one admissible commutation factor for Z 3 is exactly β(g, h) = 1 ∀g, h, Z 3 -color Lie superalgebras are indeed Z 3 -color Lie algebras or Z 3 -graded Lie algebras. Thus, we have studied the infinitesimal deformations of the model Z 3 -color Lie superalgebra, i.e. the model Z 3 -filiform Lie algebra L n,m,p . By means of these deformations all Z 3 -filiform Lie algebras can be obtained, hence the importance of these deformations.
We do assume that the reader is familiar with the standard theory of Lie algebras. 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.

Preliminaries
The vector space V is said to be Z n −graded if it admits a decomposition in direct sum, V = V 0 ⊕ V 1 ⊕ · · · V n−1 . An element X of V is called homogeneous of degree γ (deg(X) = d(X) = γ), γ ∈ Z n , if it is an element of V γ .
The mapping f is called a homomorphism of the Z n −graded vector space V into the Z n −graded vector space W if f is homogeneous of degree 0. Now it is evident how we define an isomorphism or an automorphism of Z n −graded vector spaces.
A superalgebra g is just a Z 2 −graded algebra g = g 0 ⊕ g 1 . That is, if we denote by [ , ] the bracket product of g, we have [g α , g β ] ⊂ g α+β(mod2) for all α, β ∈ Z 2 . Definition 2.1. [14] Let g = g 0 ⊕ g 1 be a superalgebra whose multiplication is denoted by the bracket product [ , ]. We call g a Lie superalgebra if the multiplication satisfies the following identities: for all X ∈ g α , Y ∈ g β , Z ∈ g γ with α, β, γ ∈ Z 2 . Identity 2 is called the graded Jacobi identity and it will be denoted by J g (X, Y, Z).
We observe that if g = g 0 ⊕ g 1 is a Lie superalgebra, we have that g 0 is a Lie algebra and g 1 has the structure of a g 0 −module.
Color Lie (super)algebras can be seen as a direct generalization of Lie (super)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 general for arbitrary color 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 2.3. Let G be an abelian group and β a commutation factor. The (complex or real) G−graded algebra Definition 2.4. A representation of a (G, β)-color Lie superalgebra is a mapping ρ : We observe that for all g, h ∈ G we have ρ(L g )V h ⊆ V g+h , which implies that any V g has the structure of a L 0 -module. In particular considering the adjoint representation ad L we have that every L g has the structure of a L 0 -module. 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 g ) = M g for any g ∈ G and also ϕ([x, y]) = [ϕ(x), ϕ(y)] for any x, y ∈ L.
Let L = g∈G Lg be a (G, β)-color Lie superalgebra. The descending central sequence of L is defined by If C k (L) = {0} for some k, the (G, β)-color Lie superalgebra is called nilpotent. The smallest integer k such as C k (L) = {0} is called the nilindex of L.
Also, we are going to define some new descending sequences of ideals, see [10]. Let L = g∈G Lg be a (G, β)-color Lie superalgebra. Then, we define the new descending sequences of ideals C k (L 0 ) (where 0 denotes the identity element of G) and C k (L g ) with g ∈ G \ {0}, as follows: 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 .
[10] Let L = g∈G Lg be a (G, β)-color Lie superalgebra. Then L is a filiform color Lie superalgebra if the following conditions hold: (1) L 0 is a filiform Lie algebra where 0 denotes the identity element of G.
(2) L g has structure of L 0 -filiform module, for all g ∈ G \ {0} Definition 2.9. Let L = g∈G Lg be a G-graded Lie algebra. Then L is a Gfiliform Lie algebra if the following conditions hold: (1) L 0 is a filiform Lie algebra where 0 denotes the identity element of G.
(2) L g has structure of L 0 -filiform module, for all g ∈ G \ {0} It is not difficult to see that for G = Z 3 , there is only one possibility for the commutation factor β, i. e.
From now on we will consider this commutation factor and we will write "Z 3color" instead of "(Z 3 , β)-color". We will note by L n,m,p the variety of all Z 3 -color Lie superalgebras L = L 0 ⊕ L 1 ⊕ L 2 with dim(L 0 ) = n + 1, dim(L 1 ) = m and dim(L 2 ) = p. N n,m,p will be the variety of all nilpotent Z 3 -color Lie superalgebras and F n,m,p is the subset of N n,m,p composed of all filiform color Lie superalgebras.
In the particular case of G = Z 3 the theorem of adapted basis rest as follows for L = L 0 ⊕ L 1 ⊕ L 2 ∈ F n,m,p : The model Z 3 -filiform Lie algebra, L n,m,p , is the simplest Z 3 -filiform Lie algebra and it is defined in an adapted basis {X 0 , X 1 , . . . , X n , Y 1 , . . . , Y m , Z 1 , . . . , Z p } by the following non-null bracket products

cocycles and infinitesimal deformations
Recall color Lie superalgebra cohomology is defined in the following well-known way (see e.g. [15]): in particular, the superspace of q-dimensional cocycles of the Z 3 -color Lie superalgebra L = L 0 ⊕ L 1 ⊕ L 2 with coefficients in the L-module V = V 0 ⊕ V 1 ⊕ V 2 will be given by The coboundary operator δ q : C q (L; V ) −→ C q+1 (L; V ), with δ q+1 • δ q = 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 where g ∈ C q (L; V ) of degree γ, and A 0 , A 1 , . . . , A q ∈ L are homogeneous with degrees α 0 , α 1 , . . . , α q respectively. The signˆindicates that the element below must be omitted and empty sums (like α 0 + · · · + α r−1 for r = 0 and α r+1 + · · · + α s−1 for s = r + 1) are set equal to zero. In particular, for q = 2 we obtain Let Z q (L; V ) denote the kernel of δ q and let B q (L; V ) denote the image of δ q−1 , then we have that B q (L; V ) ⊂ Z q (L; V ). The elements of Z q (L; V ) are called qcocycles, the elements of B q (L; V ) are the q-coboundaries. Thus, we can constuct the so-called cohomology groups We will focus our study in the 2-cocycles Z 2 0 (L n,m,p ; L n,m,p ) with L n,m,p the model filiform Z 3 -color Lie superalgebra. Thus G = Z 3 and the only admissible commutation factor is exactly β(g, h) = 1. Under all these restrictions the condition that have to verify ψ ∈ C 2 0 (L n,m,p ; L n,m,p ) to be a 2-cocycle rests for all A 0 , A 1 , A 2 ∈ L n,m,p . We observe that L n,m,p has the structure of a L n,m,pmodule via the adjoint representation.
Under these conditions we have the following lemma.
Thus, any Z 3 -filiform Lie algebra (filiform Z 3 -color Lie superalgebra) will be a linear deformation of the model Z 3 -filiform Lie algebra (the model Z 3 -color Lie superalgebra), i.e. L n,m,p is the model Z 3 -filiform Lie algebra an another arbitrary Z 3 -filiform Lie algebra will be equal to L n,m,p + ϕ, with ϕ an infinitesimal deformation of L n,m,p . Hence the importance of these deformations. So, in order to determine all the Z 3 -filiform Lie algebras it is only necessary to compute the infinitesimal deformations or so called 2-cocycles of degree 0, that vanish on the characteristic vector X 0 . Thanks to the following lemma these infinitesimal deformations will can be decomposed into 6 subspaces. Lemma 3.3. [11], [12] Let Z 2 (L; L) be the 2-cocycles Z 2 0 (L n,m,p ; L n,m,p ) that vanish on the characteristic vector X 0 . Then, Z 2 (L; L) can be divided into six subspaces, i.e. if L n,m,p = L = L 0 ⊕ L 1 ⊕ L 2 we will have that In order to obtain the dimension of A, B and C we are going to adapt the sl 2 (C)-module method that we have already used for Lie superalgebras [1], [4], [8] and for color Lie superalgebras [11], [12]. Next, we will do it explicitly for A = Z 2 (L; L) ∩ Hom(L 0 ∧ L 0 , L 0 ).

Dimension of
In general, any cocycle a ∈ Z 2 (L; L) ∩ Hom(L 0 ∧ L 0 , L 0 ) will be any skewsymmetric bilinear map from L 0 ∧ L 0 to L 0 such that: with a(X 0 , X) = 0 ∀X ∈ L. As X 0 / ∈ Im a and taking into account the bracket products of L then the equation (1) can be rewritten as follows 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].
Recall the following well-known facts about the Lie algebra sl(2, C) and its finitedimensional modules, see e.g. [2], [5]: sl(2, C) =< X − , H, X + > with the following commutation relations: Let V be a n-dimensional sl(2, C)-module, V =< e 1 , . . . , e n >. Then, up to isomorphism there exists a unique structure of an irreducible sl(2, C)-module in V given in a basis {e 1 , . . . , e n } as follows [2]: It is easy to see that e n is the maximal vector of V and its weight, called the highest weight of V , is equal to n − 1.
Let W 0 , W 1 , . . . , W k be sl(2, C)-modules, then the space Hom(⊗ k i=1 W i , W 0 ) is a sl(2, C)-module in the following natural manner: with ξ ∈ sl(2, C) and ϕ ∈ Hom(⊗ k i=1 W i , W 0 ). In particular, if k = 2 and W 0 = We are going to consider the structure of irreducible sl(2, C)-module in V 0 =< X 1 , . . . , X n >= L 0 /CX 0 , thus in particular: Next, we identify the multiplication of X + and X i in the sl(2, C)-module V 0 =< X 1 , . . . , X n >, with the bracket [X 0 , X i ] in L 0 and thanks to these identifications, the expressions (4.1) and (4.2) are equivalent. Thus we have the following result:  We use the fact that each irreducible module contains either a unique (up to scalar multiples) vector of weight 0 (in case the dimension of the irreducible module is odd) or a unique (up to scalar multiples) vector of weight 1 (in case the dimension of the irreducible module is even). We therefore have 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 A.
We consider a natural basis B of Hom(V 0 ∧ V 0 , V 0 ) consisting of the following maps: in all other cases where 1 ≤ i, j, k, l, s ≤ n, with i = j and ϕ s i,j = −ϕ s j,i .
Thanks to Corollary 5.1.2 it will be enough to find the basis vectors ϕ s i,j with weight 0 or 1. The weight of an element ϕ s i,j (with respect to H) is λ(ϕ s i,j ) = λ(X s ) − λ(X i ) − λ(X j ) = n + 2(s − i − j) + 1. In fact, We observe that 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 ϕ s i,j with weight 1 and if n is odd it will be sufficient to find those of them with weight 0.
We can consider the three sequences that correspond with the weights of V =< X 1 , X 2 , . . . , X n−1 , X n > 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. and we have to count the number of all possibilities to obtain 1 (if n is even) or 0 (if n is odd). Remember that λ(ϕ s i,j ) = λ(X s ) − λ(X i ) − λ(X j ), where λ(X s ) belongs to the last sequence, and λ(X i ), λ(X j ) 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 count the possibilities to sum the same quantity between the two first sequences. Taking into account the skew-symmetry of ϕ s i,j , that is ϕ s i,j = −ϕ s j,i and i = j, and repeating the above reasoning for all the elements of the last sequence we obtain the following theorem: Theorem 1. Let Z 2 (L; L) be the 2-cocycles Z 2 0 (L n,m,p ; L n,m,p ) that vanish on the characteristic vector X 0 . Then, if A = Z 2 (L; L) ∩ Hom(L 0 ∧ L 0 , L 0 ) we have that if n is even Proof. It is convenient to distinguish the following four cases where the reasoning for each case is not hard: (1). n ≡ 0 (mod 4).

Dimension of
In general, any cocycle b ∈ Z 2 (L; L) ∩ Hom(L 0 ∧ L 1 , L 1 ) will be any skewsymmetric bilinear map from L 0 ∧ L 1 to L 1 such that: with b(X 0 , X) = 0 ∀X ∈ L. This condition reduces to In order to obtain the dimension of the space of cocycles B we apply an adaptation of the sl(2, C)-module Method that we have already used in the precedent section.
In this case we are going to consider the structure of irreducible sl(2, C)-module in V 0 =< X 1 , . . . , X n >= L 0 /CX 0 and in V 1 =< Y 1 , . . . , Y n >= L 1 , thus in particular: We identify the multiplication of X + and X i in the sl(2, C)-module V 0 =< X 1 , . . . , X n >, with the bracket product [X 0 , X i ] in L 0 . Analogously with X + · Y j and [X 0 , Y j ]. 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 : V 0 ∧ V 1 −→ V 1 will be an element of B if and only if ϕ is a maximal vector of the sl(2, C)-module Hom Corollary 5.1.1. As each sl(2, C)-module has (up to nonzero scalar multiples) a unique maximal vector, then the dimension of B is equal to the number of summands of any decomposition of Hom (V 0 ∧ V 1 , V 1 ) into direct sum of irreducible sl(2, C)modules.
As each irreducible module contains either a unique (up to scalar multiples) vector of weight 0 or a unique vector of weight 1, then we have the following corollary.
Corollary 5.1.2. The dimension of B is equal to the dimension of the subspace of Hom (V 0 ∧ V 1 , V 1 ) spanned by the vectors of weight 0 or 1.
Next, we consider a natural basis of Hom(V 0 ∧ V 1 , V 1 ) consisting of the following maps where 1 ≤ s, j, l ≤ m and 1 ≤ i, k ≤ n: in all other cases Thanks to Corollary 5.1.2 it will be enough to find the basis vectors ϕ s i,j with weight 0 or 1. It is not difficult to see that the weight of an element ϕ s i,j (with respect to H) is Thus, 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 ϕ s i,j with weight 1 and if n is odd it will be sufficient to find those with weight 0. To do that we consider the three sequences that correspond with the weights of V 0 =< X 1 , . . . , X n >, 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(ϕ s where λ(Y s ) belongs to the last sequence, and λ(X i ), λ(Y j ) belong to the first and second sequences respectively. Thus, we obtain the following theorem.
Theorem 2. Let Z 2 (L; L) be the 2-cocycles Z 2 0 (L n,m,p ; L n,m,p ) that vanish on the characteristic vector X 0 . Then, if B = Z 2 (L; L) ∩ Hom(L 0 ∧ L 1 , L 1 ) we have that Proof. It is convenient to distinguish the following four cases where the reasoning for each case is not hard: (1). n ≡ 0 (mod 4).
Similarly to the previous section we can obtain the equivalent result for C.

conclusions
The Theorems 1, 2 and 3 together to those obtained in [11] and [12], leads to obtain the total dimension of the infinitesimal deformations of the model Z 3 -filiform Lie algebra L n,m,p . Thus, we have the following theorem.