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Computing the index of Lie algebras/Lie algebrate indeksi arvutamine.

1. INTRODUCTION

The index theory of Lie algebras was intensively studied by Elashvili (see [5-8]), in particular the case of semi-simple Lie algebras and Frobenius Lie algebras. He classified all the algebraic Frobenius algebras up to dimension 6. In [3], the authors connect the computation of the index to combinatorial theory of meanders and evaluate the index of a Lie algebra of seaweed type, which is equal to the number of cycles in an associated permutation. The index of semi-simple Lie algebras was also studied in [21]. The authors of that paper consider a semi-simple Lie algebra G with a Cartan subalgebra h, R its corresponding root system, [pi] a base of R, and S, T subsets of [pi]. They provide an upper bound for the index of [G.sub.S,T], the direct sum of h, and the sum of the root spaces for the positive roots in the space spanned by S and the sum of the root spaces for the negative roots in the space spanned by T. They then verify that this inequality is actually an equality in a number of special cases and conjecture that equality holds in all cases. See also [20], where the index of a Borel subalgebra of a semi-simple Lie algebra is determined.

The aim of this paper is to compute the index of Lie algebras in low dimensions and in general for some special cases. In Section 2 we summarize the index theory of Lie algebras. Then, in Section 3, we recall the classification of n-dimensional Lie algebras for n < 5 and compute the indexes for all these Lie algebras. Section 4 is dedicated to nilpotent Lie algebras and specially to filiform Lie algebras. We consider the generalized Heisenberg Lie algebras and the two graded filiform Lie algebras [L.sub.n] and [Q.sub.n]. Notice that [L.sub.n] plays an important role in the study of filiform and nilpotent Lie algebras. It is known that any n-dimensional filiform Lie algebra may be obtained by deformation of the one of the filiform Lie algebras [L.sub.n]. In the last Section we study the evolution by deformation of the index of a Lie algebra. We prove that the index of a Lie algebra decreases by deformation.

2. INDEX OF LIE ALGEBRAS

Throughout this paper K is an algebraically closed field of characteristic 0. In this Section we summarize the index theory of Lie algebras.

Definition 1. A Lie algebra G over K is a pair consisting of a vector space V = G and a skew-symmetric bilinear map [,] : G x G [right arrow] G (x, y) [right arrow] [x, y] satisfying the Jacobi identity

[x, [y,z]] + [y, [z,x]] + [z, [x,y]] = 0 [for all]x,y,z [member of] G.

Let x [member of] G. We denote by adx the endomorphism of G defined by adx (y) = [x, y] [for all]y [member of] G.

Let V be a finite-dimensional vector space over K provided with the Zariski topology, G be a Lie algebra and [G.sup.*] its dual. Then G acts on [G.sup.*] as follows:

G x [G.sup.*] [right arrow] [G.sup.*],

(x, f) [??] x x f, where [for all]y[member of] G : (x x f )(y) = f ([x, y]).

Let f [member of] [G.sup.*] and [[PHI].sub.f] be a skew-symmetric bilinear form defined by

[[PHI].sub.f] : G x G [right arrow] K, (x,y) [??] [[PHI].sub.f](x,y)= f ([x,y]).

We denote the kernel of the map [[PHI].sub.f] by [G.sup.f]:

[G.sup.f] = {x [member of] G : f ([x,y]) = 0 [for all]y[member of] G}.

Definition 2. The index of Lie algebra G is the integer [X.sub.G] = inf {dim [G.sup.f]; f [member of] [G.sup.*]} . A linear functional f [member of] [G.sup.*] is called regular if dim [G.sup.f] = [X.sub.G] ? The set of all regular linear functionals is denoted by [[G.sup.*.sub.r].

Remark 3. The set [[G.sup.*.sub.r] of all regular linear functionals is a nonempty Zariski open set.

Let {[x.sub.1],....,[x.sub.n]} be a basis of G. We can express the index using the matrix [([[x.sub.i], [x.sub.j]).sub.1[less than or equal to]i[less than or equal to]j[less than or equal to]n] as a matrix over the ring S(G), (see [4]). We have the following proposition:

Proposition 4. The index of an n-dimensional Lie algebra G is the integer

[X.sub.G] = n - [Rank.sub.R(G)] [([[x.sub.i], [x.sub.j]).sub.1[less than or equal to]i[less than or equal to]j[less than or equal to]n],

where R(G) is the quotient field of the symmetric algebra S(G).

Remark 5. The index of an n-dimensional Abelian Lie algebra is n.

Definition 6. A Lie algebra G over an algebraically closed field of characteristic 0 is said to be Frobenius if there exists a linear form f [member of] [G.sup.*] such that the bilinear form [[PHI].sub.f] f on G is nondegenerate.

In [7] the author described all the Frobenius algebraic Lie algebras G = R+N whose nilpotent radical N is Abelian in the following two cases: the reductive Levi subalgebra R acts on N irreducibly; R is simple. He classified all the algebraic Frobenius algebras up to dimension 6. See also [16-18] for further computations.

3. LIE ALGEBRAS OF DIMENSION n < 5

In this section we compute the index of n-dimensional Lie algebras with n < 5. Let G be an n-dimensional Lie algebra and {[x.sub.1], [x.sub.2],...,[x.sub.n]} be a fixed basis of V = G.

Any n-dimensional Lie algebra with n < 5 is isomorphic to one of the following Lie algebras.

Dimension 2

[G.sup.1.sub.2] : [[x.sub.1],[x.sub.2]] = [x.sub.2].

Dimension 3

[G.sup.1.sub.3] : [[x.sub.1], [x.sub.2]] = [x.sub.3].

[G.sup.2.sub.3] : [[x.sub.1],[x.sub.2]] = [x.sub.2], [[x.sub.1],[x.sub.3]] = [alpha][x.sub.3], [alpha] [not equal to] 0.

[G.sup.3.sub.3] : [[x.sub.1], [x.sub.2]] = [x.sub.2] + [x.sub.3], [[x.sub.1], [x.sub.3]] = [x.sub.3].

[G.sup.3.sub.4] : [[x.sub.1],[x.sub.3]] = -2[x.sub.2], [[x.sub.1],[x.sub.3]] = -2[x.sub.3].

Dimension 4

[G.sup.1.sub.4] : [[x.sub.1],[x.sub.2]] = [x.sub.2], [[x.sub.1],[x.sub.3]] = [alpha] [x.sub.3], [[x.sub.1],[x.sub.4]] = (1 + [alpha])[x.sub.4], [[x.sub.2],[x.sub.3]] = [x.sub.4].

[G.sup.2.sub.4] : [[x.sub.1],[x.sub.2]] = [x.sub.2] + [x.sub.3], [[x.sub.1],[x.sub.3]] = [x.sub.3], [[x.sub.1],[x.sub.4]] = 2[x.sub.4], [[x.sub.2],[x.sub.3]] = [x.sub.4].

[G.sup.3.sub.4] : [[x.sub.1],[x.sub.3]] = [x.sub.3], [[x.sub.1],[x.sub.4]] = [x.sub.4], [[x.sub.2],[x.sub.3]] = [x.sub.4].

[G.sup.4.sub.4] : [[x.sub.1],[x.sub.2]] = [x.sub.2], [[x.sub.1],[x.sub.3]] = a[x.sub.3], [[x.sub.1],[x.sub.4]] = [beta][x.sub.3].

[G.sup.5.sub.4] : [[x.sub.1],[x.sub.2]] = [alpha][x.sub.2], [[x.sub.1],[x.sub.3]] = [x.sub.3] + [x.sub.4], [[x.sub.1],[x.sub.4]] = [x.sub.4].

[G.sup.6.sub.4] : [[x.sub.1],[x.sub.2]] = [x.sub.2] + [x.sub.3], [[x.sub.1],[x.sub.3]] = [x.sub.3] + [x.sub.4], [[x.sub.1],= [x.sub.4].

[G.sup.7.sub.4] : [[x.sub.1], [x.sub.2]] = [x.sub.3], [[x.sub.1], [x.sub.4]] = [x.sub.4].

[G.sup.8.sub.4] : [[x.sub.1],[x.sub.2]] = [x.sub.3], [[x.sub.1],[x.sub.3]] = [x.sub.4].

[G.sup.9.sub.4] : [[x.sub.1], [x.sub.2]] = 2[x.sub.2], [[x.sub.1], [x.sub.3]] = -2[x.sub.3].

The computations of the index using Proposition 4 lead to the following result.

Proposition 7. The index of n-dimensional Lie algebras with n < 5 is

X (G.sup.1.sub.2]) = 0,

X (G.sup.i.sub.3]) = 1 for i = 1,2,3,4,

X (G.sup.1.sub.4]) = 0 if [alpha] [not equal to] -1 and [chi] ([g.sup.1.sub.4]) = 2 if [alpha] = -1,

X (G.sup.i.sub.4]) = 0 for i = 2,3, [chi]([G.sup.i.sub.4]) = 2 for i = 4,...,9.

Proof. By direct computations we obtain:

Index of the 2-dimensional Lie algebra: The corresponding matrix of [G.sup.1.sub.2] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since its rank is 2, [chi] [G.sup.1.sub.2] = 0.

Index of 3-dimensional Lie algebras:

We make the computation for [G.sup.1.sub.3]. The corresponding matrix is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is of rank 2, then [chi]([G.sup.1.sub.3]) = 1 .

The corresponding matrices of Lie algebras [G.sup.2.sub.3], [G.sup.3.sub.3], [G.sup.4.sub.3] are of rank 2, so the index is equal to 1.

Index of 4-dimensional Lie algebras: We make the computation for [G.sup.1.sub.4]. The corresponding matrix of [G.sup.1.sub.4] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The determinant of this matrix is [(1 + [alpha]).sup.2][x.sup.2.sub.4]. Then it is of rank 4 if [alpha] = [not equal to] -1. When [alpha] [not equal to] -1, the matrix is of rank 2. Thus, [chi]([G.sup.1.sub.4]) = 0 if [alpha] = [not equal to] -1 and [chi] ([G.sup.1.sub.4]) = 2 if [alpha] = - 1.

In a similar way we find that the corresponding matrices for the Lie algebras [G.sup.2.sub.4], [G.sup.3.sub.4] are of rank 4, so their index is equal to 0, and the corresponding matrices for the Lie algebras [G.sup.4.sub.4],..., [G.sup.9.sub.4] are of rank 2, so their index is equal to 2. Details of calculations can be found in [1].

4. INDEX OF NILPOTENT AND FILIFORM LIE ALGEBRAS

Let G be a Lie algebra. We set [C.sup.0]G = G and [C.sup.k]G = [[C.sup.k-1]G, G], for k > 0. A Lie algebra G is said to be nilpotent if there exists an integer p such that [C.sup.p]G = 0. The smallest p such that [C.sup.p]G = 0 is called the nilindex of G. Then a nilpotent Lie algebra has a natural filtration given by the central descending sequence: G = [C.sup.0]G [contains or equal to] [C.sup.1]G [contains or equal to] ... [C.sup.p-1]G [contains or equal to] [C.sup.p]G = 0.

We have the following characterization of nilpotent Lie algebras (Engel's theorem).

Theorem 8. A Lie algebra G is nilpotent if and only if the operator adx is nilpotent for all x in G.

Example 9. We consider the generalized Heisenberg algebra, which is a (2n + 1)-dimensional Lie algebra G given, with respect to a basis {[x.sub.1], [x.sub.2],...,[x.sub.2n+1]}, by the following nontrivial brackets:

[[x.sub.2i+1], [x.sub.2i+2] = [x.sub.2n+1]; i = 0, ..., n - 1.

The associated matrix of G is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This matrix is of rank 2n, then the index of G is [chi](G) = 1. The regular vectors are of the form f = [[summation].sup.2k.sub.i=1] [g.sub.i][x.sup.*.sub.i] + [x.sup.*.sub.2k+1].

In the study of nilpotent Lie algebras the filiform Lie algebras play an important role. This class was introduced by Vergne [22]. An n-dimensional nilpotent Lie algebra is called filiform if its nilindex p = n [right arrow] 1. The filiform Lie algebras are the nilpotent algebras with the largest nilindex. If G is an n-dimensional filiform Lie algebra, we have dim [C.sup.i]G = n - i for 2 [less than or equal to] i [less than or equal to] n.

Another characterization of filiform Lie algebras uses characteristic sequences c(G) = sup{c(x) : x [member of] G \ [G,G]}, where c(x) is the sequence, in decreasing order, of dimensions of characteristic subspaces of the nilpotent operator adx. Thus an n-dimensional nilpotent Lie algebra is filiform if its characteristic sequence is of the form c (G) = (n - 1, 1).

The classification of filiform Lie algebras was given by Vergne ([22]) until dimension 6 and was extended to dimension 11 by several authors (see [2,13,14,19]).

Throughout the classification of n-dimensional Lie algebra n < 5, there are only two isomorphic classes of filiform Lie algebras, that is [G.sup.1.sub.3] and [G.sup.8.sub.4], and their indexes are [chi]([G.sup.1.sub.3]) = 1, [chi]([G.sup.8.sub.4]) = 2.

The 5-dimensional filiform Lie algebras are isomorphic to one of the following Lie algebras:

[G.sup.1.sub.5] : [[x.sub.1],[x.sub.i]] = [x.sub.i+1], for i = 2,3,4, [G.sup.2.sub.5] : [[x.sub.1],[x.sub.i]] = [x.sub.i+1], for i = 2,3,4 and [[x.sub.2],[x.sub.3]] = [x.sub.5].

Their indexes are [chi] ([G.sup.1.sub.5]) = 3, [chi] ([G.sup.2.sub.5]) = 1. The regular vectors of [G.sup.1.sub.5] are of the form f = [g.sub.1][x.sup.*.sub.1] + [g.sub.2][x.sup.*.sub.2] + g([x.sup.*.sub.3] + [x.sup.*.sub.4] + [x.sup.*.sub.5]) with g [not equal to] 0 and the regular vectors of [G.sup.2.sub.5] are of the form f = ([[summation].sup.4.sub.i=1][g.sub.i][x.sup.*.sub.i]) + [x.sup.*.sub.5].

In the general case there are two classes [L.sub.n] and [Q.sub.n] of filiform Lie algebras which play an important role in the study of the algebraic varieties of filiform and more generally nilpotent Lie algebras.

Let {[x.sub.1],...,[x.sub.n]} be a basis of the K vector space [L.sub.n]. The Lie algebra structure of [L.sub.n] is defined by the following nontrivial brackets:

[[x.sub.1], [x.sub.i]]= [x.sub.i+1], i = 2,...,n - 1. (1)

Let {[x.sub.1],...,[x.sub.n=2k]} be a basis of the K vector space [Q.sub.n]. The Lie algebra structure of [Q.sub.n] is defined by the following nontrivial brackets:

[[x.sub.1], [x.sub.i]]= [x.sub.i+1], i = 2,..., n - 1,

[[x.sub.i],[x.sub.n-i+1] =[(-1).sup.i+1][x.sub.n], i=2,...,k, where n=2k. (2)

The classification of n-dimensional graded filiform Lie algebras yields two isomorphic classes [L.sub.n] and [Q.sub.n] when n is odd and only the Lie algebra [L.sub.n] when n is even.

It turns out that any filiform Lie algebra is isomorphic to a Lie algebra obtained as a deformation of a Lie algebra [L.sub.n].

We aim to compute the indexes of [L.sub.n] and [Q.sub.n] and regular vectors.

Let {[x.sub.1],[x.sub.2],...,[x.sub.n]} be a fixed basis of the vector space V = [L.sub.n] (resp. V = [Q.sub.n]) and {[x.sup.*.sub.1],..., [x.sup.*.sub.n]} be a basis of the dual space. Define the Lie algebra [L.sub.n] (resp. [Q.sub.n]) with respect to the basis by the brackets (1) (resp. (2)). Set f = [[summation].sub.i[less than or equal to]0][g.sub.i][x.sup.*.sub.i] [member of] [V.sup.*].

Proposition 10. For n [greater than or equal to] 3, the index of the n-dimensional filiform Lie algebra [L.sub.n] is [chi] ([L.sub.n]) = n - 2. The regular vectors of [L.sub.n] are of the form f = [[summation].sup.n.sub.i=1][g.sub.i][x.sup.*.sub.i] with one of [g.sub.i] [not equal to] 0 where i [member of] {3,...,n}.

Proof. Since the corresponding matrix to the Lie algebra [L.sub.n] is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and its rank is 2, [chi]([L.sub.n]) = n - 2. The second assertion is obtained by a direct calculation.

Proposition 11. For n = 2k and k [greater than or equal to] 2, the index of the n-dimensional filiform Lie algebra [Q.sub.n] is [chi] ([Q.sub.n]) = 2.

The regular vectors of [Q.sub.n] are of the form f = [[summation].sup.n.sub.i=1][g.sub.i][x.sup.*.sub.i] with [g.sub.n] [not equal to] 0.

Proof. Since the corresponding matrix to the Lie algebra [Q.sub.n] is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and its rank is n - 2, [chi]([Q.sub.n]) = 2. The second assertion is obtained by a direct calculation.

5. INDEX AND DEFORMATIONS

We study now the evolution by deformation of the index of a Lie algebra. About deformation theory we refer to [9-12] and [15]. Let V be a K-vector space and [G.sub.0] = (V, [[,].sub.0]) be a Lie algebra. Let K[[t]] be the power series ring in one variable t and coefficients in K and V[[t]] be the set of formal power series whose coefficients are elements of V. A formal Lie deformation of [G.sub.0] is given by the K[[t]]-bilinear map [[,].sub.t] : V[[t]] x V[[t]] [right arrow] V[[t]] of the form [[,].sub.t] = [[summation].sub.i[greater than or equal to]0] [[,].sub.i][t.sup.i], where each [[,].sub.i] is a K-bilinear map [[,].sub.i] : V x V [right arrow] V, satisfying the skew-symmetry and the Jacobi identity.

Proposition 12. The index of a Lie algebra decreases by deformation.

Proof. The rank of the matrix [([X.sub.i],[X.sub.j].sub.ij] increases by deformation, consequently the index decreases.

Corollary 13. The index of a filiform Lie algebra is less than or equal ton - 2.

Proof. Any filiform Lie algebra N is obtained as a deformation of the Lie algebra [L.sub.n]. Since [chi] ([L.sub.n]) = n - 2 using the previous lemma, one has [chi] (N) < n - 2.

doi: 10.3176/proc.2010.4.03

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Hadjer Adimi (a) and Abdenacer Makhlouf (b) *

(a) Centre universitaire de Bordj Bou Arreridj, Departement de Mathematiques, Bordj Bou Arreridj, Algeria

(b) Universite de Haute Alsace, Laboratoire de Mathematiques, Informatique et Applications, 4, rue des Freres Lumiere F-68093 Mulhouse, France

Received 17 April 2009, accepted 21 October 2009

* Corresponding author, Abdenacer.Makhlouf@uha.fr
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Title Annotation:MATHEMATICS
Author:Adimi, Hadjer; Makhlouf, Abdenacer
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