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Journal of Generalized Lie Theory and Applications
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Infinitesimal Deformations of the Model Z3-Filiform Lie Algebra

Rosa Maria Navarro

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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Abstract

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.

1 Introduction

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 Ln. 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 [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 Image -grading vector space (i.e., G = Image, due to its physical applications) [3,7,6,13,16,17]. Due to the fact that the one admissible commutation factor for Image is exactly β(g,h) = 1 ∀g,h, Image -color Lie superalgebras are indeed Image-color Lie algebras or Image-graded Lie algebras. Thus, we have studied the infinitesimal deformations of the model Image-color Lie superalgebra (i.e., the model Image-filiform Lie algebra Ln,m,p). By means of these deformations, all Image-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 Z2(L;L), into six subspaces of deformations:

Image

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 Image-filiform Lie algebra Ln,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 Image-vector spaces (Image= Image or Image) with finite dimension.

2 Preliminaries

The vector space V is said to be Image-graded if it admits a decomposition in direct sum, V = V0 ⊕V1⊕· · ·Vn−1. An element X of V is called homogeneous of degree γ (deg(X) = d(X) = γ), γ ∈ Zn, if it is an element of Vγ.

Let V = V0 ⊕V1⊕· · ·Vn−1 and W = W0 ⊕W1⊕· · ·Wn−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 ∈ Image, if f(Vα) ⊂ Wα+γ (modn) for all α ∈ Image. The mapping f is called a homomorphism of the Image-graded vector space V into the Image-graded vector space W, if f is homogeneous of degree 0. Now it is evident how we define an isomorphism or an automorphism of Image-graded vector spaces.

A superalgebra g is just a Z2-graded algebra g = g0 ⊕g1. That is, if we denote by [ , ] the bracket product of g, we have [gα,gβ] ⊂ gα+β (mod2) for all α,β ∈ Z2.

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:

Image

Identity (2) is called the graded Jacobi identity, and it will be denoted by Jg(X,Y,Z).

We observe that if g = g0⊕g1 is a Lie superalgebra, we have that g0 is a Lie algebra and g1 has the structure of a g0-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. 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, fiXing 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

Image

with bracket product [ , ], is called a (G,β)-color Lie superalgebra if for any X ∈ Lg, Y ∈ Lh, 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 Image be a (G,β)-color Lie superalgebra. Then we have:

(1) L0 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 L0. If X ∈ L0 and Y ∈ Lg, then [X,Y ] denotes the action of X on Y .

Examples. For the particular case G = {0}, L = L0 reduces to a Lie algebra. If G = Z2 = {0,1} and β(1,1) = −1, we have ordinary Lie superalgebras; that is, a Lie superalgebra is a (Z2,β)-color Lie superalgebra where β(i, j) = (−1)ij for all i, j ∈ Z2.

Definition 5. A representation of a (G,β)-color Lie superalgebra is a mapping ρ : L → End(V ), where V =Image 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 ∈ Lg, Y ∈ Lh.

We observe that for all g,h ∈ G we have ρ(Lg)Vh ⊆ Vg+h, which implies that any Vg has the structure of a L0-module. In particular considering the adjoint representation adL we have that every Lg has the structure of a L0-module.

Two (G,β)-color Lie superalgebras L and M are called isomorphic if there is a linear isomorphism Image such thatImage for any g ∈ G and also Image for any x,y ∈ L.

Let Image be a (G,β)-color Lie superalgebra. The descending central sequence of L is defined by

Image

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.

Also, we are going to define some new descending sequences of ideals, see [10]. Let Image be a (G,β)- color Lie superalgebra. Then, we define the new descending sequences of ideals Ck(L0) (where 0 denotes the identity element of G) and Ck(Lg) with g ∈ G\{0}, as follows:

Image

and

Image

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 = Z3.

Definition 6 (see [11]). If L = L0⊕L1⊕L2 is a nilpotent (Z3,β)-color Lie superalgebra, then L has color-nilindex (p0,p1,p2), if the following conditions hold:

Image

and

Image

Definition 7 (see [10]). Let Image be a (G,β)-color Lie superalgebra. Lg is called a L0-filiform module if there eXists a decreasing subsequence of vectorial subspaces in its underlying vectorial space V , V = Vm ⊃ · · · ⊃ V1 ⊃ V0, with dimensions m,m−1, . . . 0, respectively,m>0, and such that [L0,Vi+1] = Vi.

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

Definition 9 (see [10]). Let Image be a (G,β)-color Lie superalgebra. Then L is a filiform color Lie superalgebra if the following conditions hold:

(1) L0 is a filiform Lie algebra where 0 denotes the identity element of G.

(2) Lg has structure of L0-filiform module, for all g ∈ G\{0}

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

(1) L0 is a filiform Lie algebra where 0 denotes the identity element of G.

(2) Lg has structure of L0-filiform module, for all g ∈ G\{0}

It is not difficult to see that for G = Z3, there is only one possibility for the commutation factor β, that is:

β(g,h) = 1 ∀g,h ∈ Z3 = {0,1,2}.

From now on, we will consider this commutation factor, and we will write “Z3-color” instead of “(Z3,β)-color”. We will note by Ln,m,p, the variety of all Z3-color Lie superalgebras L = L0 ⊕L1 ⊕L2 with dim(L0) = n+1, dim(L1) = m and dim(L2) = p. Nn,m,p will be the variety of all nilpotent Z3-color Lie superalgebras, and Fn,m,p is the subset of Nn,m,p composed of all filiform color Lie superalgebras.

Remark 11. If G = Z3, then β(g,h) = 1 ∀g,h. Thus, Z3-color Lie superalgebras are effectively Z3-graded Lie algebras, and filiform Z3-color Lie superalgebras are Z3-filiform Lie algebras.

In the particular case of G = Z3, the theorem of adapted basis rests as follows for Image

Image

with {X0,X1,...,Xn} a basis of L0, {Y1, . . . , Ym} a basis of L1, and {Z1,...,Zp} a basis of L2. The model Z3-filiform Lie algebra, Ln,m,p, is the simplest Z3-filiform Lie algebra; and it is defined in an adapted basis {X0,X1,...,Xn,Y1, . . . , Ym,Z1,...,Zp} by the following non-null bracket products:

Image

3 Cocycles and infinitesimal deformations

Recall that a module V = V0⊕V1⊕V2 of the Z3-color Lie superalgebra L is a bilinear map of degree 0, L×V →V satisfying:

Image

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 Z3-color Lie superalgebra L = L0 ⊕L1 ⊕L2 with coefficients in the L-module V = V0⊕V1⊕V2 will be given by:

Image

This space is graded by Image with

Image

The coboundary operator Image, with Image 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:

Image

where Image of degree γ, and A0,A1,...,Aq ∈ 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:

Image

Let Zq(L;V ) denote the kernel of δq and let Bq(L;V ) denote the image of 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 construct the so-called cohomology groups:

Image

Two elements of Zq(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 (Ln,m,p;Ln,m,p) with Ln,m,p, the model filiform Z3-color Lie superalgebra. Thus, G = Z3 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 (Ln,m,p;Ln,m,p) to be a 2-cocycle rests

Image

for all A0,A1,A2 ∈ Ln,m,p. We observe that Ln,m,p has the structure of a Ln,m,p-module via the adjoint representation.

We consider a homogeneous basis of Ln,m,p = L0 ⊕L1 ⊕L2, in particular an adapted basis {X0,X1,...,Xn, Y1, . . . , Ym,Z1,...,Zp} with {X0,X1,...,Xn} a basis of L0, {Y1, . . . , Ym} a basis of L1 and {Z1,...,Zp} a basis of L2.

Under these conditions, we have the following lemma.

Lemma 12 (see [11,12]). Let ψ be such that Image then ψ is a 2-cocycle, Imageiff the 10 conditions below hold for all Xi,Xj,Xk ∈ L0, Yi,Yj,Yk ∈ L1 and Zi,Zj,Zk ∈ L2

Image

Proposition 13 (see [11]). ψ is an infinitesimal deformation of Ln,m,p iff ψ is a 2-cocycle of degree 0, Image

Theorem 14 (see [10]). (1) Any filiform (G,β)-color Lie superalgebra law μ is isomorphic to Image where μ0 is the law of the model filiform (G,β)-color Lie superalgebra, and Image is an infinitesimal deformation of μ0 verifying that Image for all X ∈ L, with X0 the characteristic vector of model one.

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

Thus, any Z3-filiform Lie algebra (filiform Z3-color Lie superalgebra) will be a linear deformation of the model Z3-filiform Lie algebra (the model Z3-color Lie superalgebra); that is, Ln,m,p is the model Z3-filiform Lie algebra, and another arbitrary Z3-filiform Lie algebra will be equal to Image, with Image an infinitesimal deformation of Ln,m,p, hence the importance of these deformations. So, in order to determine all the Z3-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 X0. 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 Image that vanish on the characteristic vector X0. Then Z2(L;L) can be divided into six subspaces; that is, if Ln,m,p = L = L0⊕L1⊕L2, we will have:

Image

In order to obtain the dimension of A, B, and C, we are going to adapt the sL2(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(L0 ∧L0,L0).

4 Dimension of A = Z2(L;L)∩Hom(L0 ∧L0,L0)

In general, any cocycle a ∈ Z2(L;L)∩Hom(L0 ∧L0,L0) will be any skew-sYmmetric bilinear map from L0 ∧L0 to L0 such that:

Image (4.1)

with a(X0,X) = 0 ∀X ∈ L. As X0 ∉ Ima and taking into account the bracket products of L, then (4.1) can be rewritten as follows:

Image (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]:

Image with the following commutation relations:

Image

Let V be a n-dimensional sl(2,C)-module, Image . 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 [2]:

Image

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 Image is a sl(2,C)-module in the following natural manner:

Image

with Ξ ∈ sl(2,C) and Image. In particular, if k = 2 and W0 =W1 =W2 = V0, then:

Image

An element Image is said to be invariant ifImage that is:

Image (4.3)

Note that Image is invariant if and only if Image is a maXimal vector.

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

Image

Next, we identify the multiplication of X+ and Xi in the sl(2,C)-module Image with the bracket [X0,Xi] in L0 and thanks to these identifications, the expressions (4.2) and (4.3) are equivalent. Thus, we have the following result:

Proposition 16. Any skew-sYmmetric bilinear map Image will be an element of the space of cocycles A if and only if Imageis a maXimal vector of the sl(2,C)-module Image withImage.

Corollary 17. 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 Image 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 Image 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 Image consisting of the following maps:

Image

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

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

Image.

In fact,

Image

We observe that if n is even, then Image is odd; and if n is odd, then Image is even. So, if n is even, it will be sufficient to find the elements Image 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 Image in order to find the elements with weight 0 or 1:

Image.

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 Image where λ(Xs) belongs to the last sequence, and λ(Xi), 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-sYmmetry of Image , that isImage 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 Image that vanish on the characteristic vector X0. Then, if Image, we have that

Image

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).

5 Dimension of Image

In general, any cocycle Image will be any skew-sYmmetric bilinear map from L0 ∧L1 to L1 such that:

Image(5.1)

with b(X0,X) = 0 ∀X ∈ L. This condition reduces to

Image.

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 ifW0,W1,...,Wk are sl(2,C)-modules, then the space Image will be a sl(2,C)-module in the following natural manner:

Image

with Ξ ∈ sl(2,C) and Image. In particular, if k = 2 and V0 =W1, V1 =W2 =W0, then:

Image.

An element Image is said to be invariant if that is:

Image. (5.3)

Note that Image is invariant if and only if Image is a maXimal vector.

In this case, we are going to consider the structure of irreducible sl(2,C)-module in Image L0/CX0 and inImage. Thus, in particular:

Image

We identify the multiplication of X+ and Xi in the sl(2,C)-module Image, with the bracket product [X0,Xi] in L0. Analogously withX+ ·Yj and [X0,Yj ]. Thanks to these identifications, the expressions (5.2) and (5.3) are equivalent, so we have the following result:

Proposition 20. Any skew-sYmmetric bilinear map Image will be an element of B if and only if Image is a maXimal vector of the sl(2,C)-module Hom(V0 ∧V1,V1), with Image and V1 = L1.

Corollary 21. 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(V0 ∧V1,V1) 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(V0 ∧V1,V1) spanned by the vectors of weight 0 or 1.

Next, 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:

Image

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

Image.

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

Image

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 Image, where λ(Ys) belongs to the last sequence, and λ(Xi), 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 Image that vanish on the characteristic vector X0. Then, if B = Z2(L;L)∩Hom(L0 ∧L1,L1), we have that:

Image

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(L0 ∧L2,L2)

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

Theorem 24. Let Z2(L;L) be the 2-cocycles Image that vanish on the characteristic vector X0. Then, if C = Z2(L;L)∩Hom(L0 ∧L2,L2), we have that:

Image

7 Conclusions

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 Z3-filiform Lie algebra Ln,m,p. Thus, we have the following theorem.

Main theorem. The dimension of the space of infinitesimal deformations of the model Z3-filiform Lie algebra Ln,m,p that vanish on the characteristic vector X0, is exactly A+B+C +D+E+F where

Image

Image

(1) If m+p−n is even, then

Image

(2) If m+p−n is odd, then

Image

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