Medical, Pharma, Engineering, Science, Technology and Business

Departamento de Matem´aticas, Universidad de Extremadura, 10071 C´aceres, Spain Address correspondence to Rosa Mar´ıa Navarro, [email protected]

**Received Date:** 22 January 2013; **Accepted Date:** 30 January 2013

**Copyright:** © 2013 Rosa Mar´ıa Navarro. This is an open access article distributed under the terms of the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of Generalized Lie Theory and Applications

In this work, it is considered that the vector space is composed by the infinitesimal deformations of the model Z3-filiform Lie algebra Ln,m,p. By using these deformations, all the Z3-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work, together with those obtained by Khakimdjanov and Navarro (J. Geom. Phys. 2011 and 2012), lead to compute the total dimension of the mentioned space of deformations.

The concept of filiform Lie algebras was firstly introduced by Vergne [18]. 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} [1,4,8,9].

Continuing with the work of Vergne, we have generalized the concept and the properties 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 a previous study [10].

In the present, paper the focus of interest are color Lie superalgebras with a -grading vector space (i.e., G = , due to its physical applications) [3,7,6,13,16,17]. Due to the fact that the one admissible commutation
factor for is exactly β(g,h) = 1 ∀g,h, -color Lie superalgebras are indeed -color Lie algebras or -graded
Lie algebras. Thus, we have studied the infinitesimal deformations of the model -color Lie superalgebra (i.e.,
the model -filiform Lie algebra L^{n,m,p}). By means of these deformations, all -filiform Lie algebras can be
obtained, hence the importance of these deformations.

Khakimdjanov and Navarro [11,12] decomposed the space of these infinitesimal deformations, noted by
Z^{2}(L;L), into six subspaces of deformations:

In the present paper, a method is given that will allow to determine the dimension of the subspaces A, B, and
C, giving explicitly the total dimension of all of them (Theorems 19, 23, and 24). This result, together with those
obtained by Khakimdjanov and Navarro [11,12], leads to obtain the total dimension of the infinitesimal deformations
of the model -filiform Lie algebra L^{n,m,p} (Main theorem).

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 -vector spaces (= or ) with finite dimension.

The vector space V is said to be -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}.

Let V = V_{0} ⊕V_{1}⊕· · ·V_{n−1} and W = W_{0} ⊕W_{1}⊕· · ·W_{n−1} be two graded vector spaces. A linear mapping
f : V →W is said to be homogeneous of degree γ (deg(f) = d(f) = γ), γ ∈ , if f(V_{α}) ⊂ W_{α+γ} (modn) for all
α ∈ . The mapping f is called a homomorphism of the -graded vector space V into the -graded vector
space W, if f is homogeneous of degree 0. Now it is evident how we define an isomorphism or an automorphism of
-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}] ⊂ gα+β (mod2) for all α,β ∈ Z_{2}.

**Definition 1** (see [14]). Let g = g0 ⊕g1 be a superalgebra whose multiplication is denoted by the bracket product
[ , ]. We call g a Lie superalgebra if the multiplication satisfies the following identities:

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 antisY_{m}metric (commutator) or sY_{m}metric (anticommutator) products, although for the former, the
product is neither sY_{m}metric nor antisY_{m}metric 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.** 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 ∀g,h ∈ G

(2) β(g,h+k) = β(g,h)β(g,k) ∀g,h,k ∈ G

(3) β(g+h,k) = β(g,k)β(h,k) ∀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,

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, fiX_{i}ng g one element of G, the induced mapping βg : G→
F\{0} defines a homomorphism of groups.

**Definition 3.** Let G be an abelian group and β a commutation factor. The (complex or real) G-graded algebra

with bracket product [ , ], is called a (G,β)-color Lie superalgebra if for any X ∈ L_{g}, Y ∈ L_{h}, and Z ∈ L, we have:

(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 4.** Let be a (G,β)-color Lie superalgebra. Then we have:

(1) L_{0} is a (complex or real) Lie algebra where 0 denotes the identity element of G.

(2) For all g ∈ G\{0}, Lg is a representation of L_{0}. If X ∈ L_{0} and Y ∈ Lg, then [X,Y ] denotes the action of X on
Y .

**Examples.** For the particular case G = {0}, L = L_{0} reduces to a Lie algebra. If G = Z_{2} = {0,1} and β(1,1) = −1,
we have ordinary Lie superalgebras; that is, a Lie superalgebra is a (Z_{2},β)-color Lie superalgebra where β(i, j) =
(−1)^{ij} for all i, j ∈ Z_{2}.

**Definition 5.** A representation of a (G,β)-color Lie superalgebra is a mapping ρ : L → End(V ), where V = is a graded vector space such that:

[ρ(X),ρ(Y )] = ρ(X)ρ(Y )−β(g,h)ρ(Y )ρ(X)

for all X ∈ L_{g}, Y ∈ L_{h}.

We observe that for all g,h ∈ G we have ρ(L_{g})V_{h} ⊆ V_{g+h}, which implies that any Vg has the structure of a
L_{0}-module. In particular considering the adjoint representation adL we have that every Lg 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 such that for any g ∈ G and also for any x,y ∈ L.

Let 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 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}(Lg) with g ∈ G\{0}, as follows:

and

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 6** (see [11]). If L = L_{0}⊕L_{1}⊕L_{2} is a nilpotent (Z_{3},β)-color Lie superalgebra, then L has color-nilindex
(p_{0},p_{1},p_{2}), if the following conditions hold:

and

**Definition 7** (see [10]). Let be a (G,β)-color Lie superalgebra. Lg is called a L_{0}-filiform module if
there eX_{i}sts a decreasing subsequence of vectorial subspaces in its underlying vectorial space V , V = Vm ⊃ · · · ⊃
V_{1} ⊃ V_{0}, with dimensions m,m−1, . . . 0, respectively,m>0, and such that [L_{0},Vi+1] = Vi.

Remark 8. The definition of filiform module is also valid for G-graded Lie algebras.

**Definition 9** (see [10]). Let 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) Lg has structure of L_{0}-filiform module, for all g ∈ G\{0}

**Definition 10.** Let be a G-graded Lie algebra. Then L is a G-filiform Lie algebra if the following
conditions hold:

(1) L_{0} is a filiform Lie algebra where 0 denotes the identity element of G.

(2) Lg 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, that is:

β(g,h) = 1 ∀g,h ∈ Z_{3} = {0,1,2}.

From now on, we will consider this commutation factor, and we will write “Z_{3}-color” 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.

Remark 11. If G = Z_{3}, then β(g,h) = 1 ∀g,h. Thus, Z_{3}-color Lie superalgebras are effectively Z_{3}-graded Lie
algebras, and filiform Z_{3}-color Lie superalgebras are Z_{3}-filiform Lie algebras.

In the particular case of G = Z_{3}, the theorem of adapted basis rests as follows for

with {X_{0},X_{1},...,X_{n}} a basis of L_{0}, {Y_{1}, . . . , Y_{m}} a basis of L_{1}, and {Z_{1},...,Z_{p}} a basis of L_{2}. 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:

Recall that a module V = V_{0}⊕V_{1}⊕V2 of the Z_{3}-color Lie superalgebra L is a bilinear map of degree 0, L×V →V
satisfying:

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}⊕V2 will be given by:

This space is graded by with

The coboundary operator , with 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 of degree γ, and A_{0},A_{1},...,A_{q} ∈ L are homogeneous with degrees α0,α1, . . . , αq, respectively.
The sign ˆ indicates that the element below must 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:

Let Z^{q}(L;V ) denote the kernel of δ^{q} and let Bq(L;V ) denote the image of δ^{q−1}, then we have that Bq(L;V ) ⊂
Z^{q}(L;V ). The elements of Z^{q}(L;V ) are called q-cocycles; the elements of Bq(L;V ) are the q-coboundaries. Thus,
we can construct the so-called cohomology groups:

Two elements of Z^{q}(L;V ) are said to be cohomologous if their residue classes modulo Bq(L;V ) coincide, that
is, if their difference lies in Bq(L;V ).

We will focus our study in the 2-cocycles Z2
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 ∈ C2
0 (L^{n,m,p};L^{n,m,p}) to be a 2-cocycle rests

for all A_{0},A_{1},A2 ∈ L^{n,m,p}. We observe that L^{n,m,p} has the structure of a L^{n,m,p}-module via the adjoint representation.

We consider a homogeneous basis of L^{n,m,p} = L_{0} ⊕L_{1} ⊕L_{2}, in particular an adapted basis {X_{0},X_{1},...,X_{n},
Y_{1}, . . . , Y_{m},Z_{1},...,Z_{p}} with {X_{0},X_{1},...,X_{n}} a basis of L_{0}, {Y_{1}, . . . , Y_{m}} a basis of L_{1} and {Z_{1},...,Z_{p}} a basis of
L_{2}.

Under these conditions, we have the following lemma.

**Lemma 12** (see [11,12]). Let ψ be such that then ψ is a 2-cocycle, iff the 10 conditions below hold for all X_{i},Xj,Xk ∈ L_{0}, Yi,Yj,Yk ∈ L_{1} and Zi,Zj,Zk ∈ L_{2}

**Proposition 13** (see [11]). ψ is an infinitesimal deformation of L^{n,m,p} iff ψ is a 2-cocycle of degree 0,

**Theorem 14** (see [10]). (1) Any filiform (G,β)-color Lie superalgebra law μ is isomorphic to where μ0 is
the law of the model filiform (G,β)-color Lie superalgebra, and is an infinitesimal deformation of μ0 verifying
that for all X ∈ L, with X_{0} the characteristic vector of model one.

(2) Conversely, if is an infinitesimal deformation of a model filiform (G,β)-color Lie superalgebra law μ0 with for all X ∈ L, then the law is a filiform (G,β)-color Lie superalgebra law iff .

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); that is, L^{n,m,p} is the model Z_{3}-filiform Lie algebra,
and another arbitrary Z_{3}-filiform Lie algebra will be equal to , with 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 can be decomposed into
six subspaces.

**Lemma 15** (see [11,12]). Let Z2(L;L) be the 2-cocycles that vanish on the characteristic
vector X_{0}. Then Z2(L;L) can be divided into six subspaces; that is, if L^{n,m,p} = L = L_{0}⊕L_{1}⊕L_{2}, we will have:

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 = Z2(L;L)∩Hom(L_{0} ∧L_{0},L_{0}).

**4 Dimension of** A = Z2(L;L)∩Hom(L_{0} ∧L_{0},L_{0})

In general, any cocycle a ∈ Z2(L;L)∩Hom(L_{0} ∧L_{0},L_{0}) will be any skew-sY_{m}metric bilinear map from L_{0} ∧L_{0} to
L_{0} such that:

(4.1)

with a(X_{0},X) = 0 ∀X ∈ L. As X_{0} ∉ Ima and taking into account the bracket products of L, then (4.1) can be
rewritten as follows:

(4.2)

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 a previous study [11].

Recall the following well-known facts about the Lie algebra sl(2,C) and its finite-dimensional modules, see, for example, [2,5]:

with the following commutation relations:

Let V be a n-dimensional sl(2,C)-module, . Then, up to isomorphism, there eX_{i}sts 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 en is the maX_{i}mal vector of V ; and its weight, called the highest weight of V , is equal to
n−1.

Let W_{0},W_{1},...,Wk be sl(2,C)-modules, then the space is a sl(2,C)-module in the following
natural manner:

with &X_{i}; ∈ sl(2,C) and . In particular, if k = 2 and W_{0} =W_{1} =W_{2} = V_{0}, then:

An element is said to be invariant if that is:

(4.3)

Note that is invariant if and only if is a maX_{i}mal vector.

We are going to consider the structure of irreducible sl(2,C)-module in thus in particular:

Next, we identify the multiplication of X+ and X_{i} in the sl(2,C)-module with the bracket
[X_{0},X_{i}] in L_{0} and thanks to these identifications, the expressions (4.2) and (4.3) are equivalent. Thus, we have the
following result:

**Proposition 16.** Any skew-sY_{m}metric bilinear map will be an element of the space of cocycles
A if and only if is a maX_{i}mal vector of the sl(2,C)-module with.

**Corollary 17.** As each irreducible sl(2,C)-module has (up to nonzero scalar multiples) a unique maX_{i}mal vector,
then the dimension of the space of cocycles A is equal to the number of summands of any decomposition of into the direct sum of irreducible sl(2,C)-modules.

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:

**Corollary 18.** The dimension of the space of cocycles A is equal to the dimension of the subspace of 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 A.

We consider a natural basis B of consisting of the following maps:

where 1 ≤ i, j,k, l,s ≤ n, with i ≠ j and .

Thanks to Corollary 18, it will be enough to find the basis vectors with weight 0 or 1. The weight of an element (with respect to H) is:

.

In fact,

We observe that if n is even, then is odd; and if n is odd, then is even. So, if n is even, it will be sufficient to find the elements 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 in order to find the elements with weight 0 or 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 where λ(Xs) belongs to the last sequence, and λ(X_{i}), λ(Xj) 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-sY_{m}metry of , that is and i ≠ j, and repeating the above reasoning for all the
elements of the last sequence, we obtain the following theorem:

**Theorem 19.** Let Z2(L;L) be the 2-cocycles that vanish on the characteristic vector X_{0}. Then, if , 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 (mod4).

(2) n ≡ 1 (mod4).

(3) n ≡ 2 (mod4).

(4) n ≡ 3 (mod4).

In general, any cocycle will be any skew-sY_{m}metric bilinear map from L_{0} ∧L_{1} to
L_{1} such that:

(5.1)

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.

Recall that ifW_{0},W_{1},...,Wk are sl(2,C)-modules, then the space will be a sl(2,C)-module in the following natural manner:

with &X_{i}; ∈ sl(2,C) and . In particular, if k = 2 and V_{0} =W_{1}, V_{1} =W_{2} =W_{0}, then:

.

An element is said to be invariant if that is:

. (5.3)

Note that is invariant if and only if is a maX_{i}mal vector.

In this case, we are going to consider the structure of irreducible sl(2,C)-module in L_{0}/CX_{0} and in. Thus, in particular:

We identify the multiplication of X+ and X_{i} in the sl(2,C)-module , with the bracket product
[X_{0},X_{i}] in L_{0}. Analogously withX+ ·Yj and [X_{0},Yj ]. Thanks to these identifications, the expressions (5.2) and (5.3)
are equivalent, so we have the following result:

**Proposition 20.** Any skew-sY_{m}metric bilinear map will be an element of B if and only if is a maX_{i}mal vector of the sl(2,C)-module Hom(V_{0} ∧V_{1},V_{1}), with and V_{1} = L_{1}.

**Corollary 21.** As each sl(2,C)-module has (up to nonzero scalar multiples) a unique maX_{i}mal 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 22.** 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:

Thanks to Corollary 22, it will be enough to find the basis vectors with weight 0 or 1. It is not difficult to see that the weight of an element (with respect to H) is:

.

Thus, if n is even, then is odd; and if n is odd, then is even. So, if n is even, it will be sufficient to find the elements 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 and.

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 , where λ(Ys) belongs to the last sequence, and λ(X_{i}), λ(Yj) belong to the first
and second sequences respectively. Thus, we obtain the following theorem.

**Theorem 23.** Let Z2(L;L) be the 2-cocycles that vanish on the characteristic vector X_{0}. Then,
if B = Z2(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 (mod4).

(2) n ≡ 1 (mod4).

(3) n ≡ 2 (mod4).

(4) n ≡ 3 (mod4).

**6 Dimension of** C = Z2(L;L)∩Hom(L_{0} ∧L_{2},L_{2})

Similarly to the previous section, we can obtain the equivalent result for C.

**Theorem 24.** Let Z2(L;L) be the 2-cocycles that vanish on the characteristic vector X_{0}. Then,
if C = Z2(L;L)∩Hom(L_{0} ∧L_{2},L_{2}), we have that:

Theorems 1, 2, and 3, together with those obtained by Khakimdjanov and Navarro [11,12], lead 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.

**Main theorem.** The dimension of the space of infinitesimal deformations of the model Z_{3}-filiform Lie algebra
L^{n,m,p} that vanish on the characteristic vector X_{0}, is exactly A+B+C +D+E+F where

(1) If m+p−n is even, then

(2) If m+p−n is odd, then

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