Computing the index of Lie algebras

The aim of this paper is to compute and discuss the index of Lie algebras. We consider the n-dimensional Lie algebras for n < 5 and the case of filiform Lie algebras which form a special class of nilpotent Lie algebras. We compute the index of generalized Heisenberg algebras and graded filiform Lie algebras Ln and Qn. We also discuss the evolution of the Lie algebra index by deformation.


INTRODUCTION
The index theory of Lie algebras was intensively studied by Elashvili (see [5][6][7][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, π a base of R, and S, T subsets of π.They provide an upper bound for the index of G 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 n and Q n .Notice that L 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 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.

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 [ , ] : Let x ∈ G .We denote by adx the endomorphism of G defined by adx (y) = [x, y] ∀y ∈ G .
Let V be a finite-dimensional vector space over K provided with the Zariski topology, G be a Lie algebra and G * its dual.Then G acts on G * as follows: where ∀y ∈ G : Let f ∈ G * and Φ f be a skew-symmetric bilinear form defined by We denote the kernel of the map Φ f by G f : The set of all regular linear functionals is denoted by G * r .
Remark 3. The set G * r of all regular linear functionals is a nonempty Zariski open set.Let {x 1 , . . ., x n } be a basis of G .We can express the index using the matrix ([x i , x j ]) 1≤i< j≤n as a matrix over the ring S(G ), (see [4]).We have the following proposition: 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 ∈ G * such that the bilinear form Φ 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][17][18] for further computations.

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 1 , x 2 , . . ., x 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.
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 Proof.By direct computations we obtain: Since its rank is 2, χ G 1 2 = 0. Index of 3-dimensional Lie algebras: We make the computation for G 1 3 .The corresponding matrix is It is of rank 2, then χ G 1 3 = 1.The corresponding matrices of Lie algebras G 2  3 , G 3 3 , G 4 3 are of rank 2, so the index is equal to 1.
Index of 4-dimensional Lie algebras: We make the computation for G 1  4 .The corresponding matrix of The determinant of this matrix is In a similar way we find that the corresponding matrices for the Lie algebras G 2  4 , G 3 4 are of rank 4, so their index is equal to 0, and the corresponding matrices for the Lie algebras G 4  4 , . . ., G 9 4 are of rank 2, so their index is equal to 2. Details of calculations can be found in [1].

INDEX OF NILPOTENT AND FILIFORM LIE ALGEBRAS
Let G be a Lie algebra.We set A Lie algebra G is said to be nilpotent if there exists an integer p such that C p G = 0.The smallest p such that C p G = 0 is called the nilindex of G .Then a nilpotent Lie algebra has a natural filtration given by the central descending sequence: 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 1 , x 2 , . . ., x 2n+1 }, by the following nontrivial brackets: The associated matrix of G is of the form This matrix is of rank 2n, then the index of G is χ (G ) = 1.The regular vectors are of the form 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 − 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 Another characterization of filiform Lie algebras uses characteristic sequences c(G ) = sup{c(x) : x ∈ 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) .
Throughout the classification of n-dimensional Lie algebra n < 5, there are only two isomorphic classes of filiform Lie algebras, that is G 1  3 and G 8  4 , and their indexes are The 5-dimensional filiform Lie algebras are isomorphic to one of the following Lie algebras: ) with g = 0 and the regular vectors of G 2 5 are of the form f = (∑ 4 i=1 g i x * i ) + x * 5 .In the general case there are two classes L n and Q 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 1 , . . ., x n } be a basis of the K vector space L n .The Lie algebra structure of L n is defined by the following nontrivial brackets: Let {x 1 , . . ., x n=2k } be a basis of the K vector space Q n .The Lie algebra structure of Q n is defined by the following nontrivial brackets: The classification of n-dimensional graded filiform Lie algebras yields two isomorphic classes L n and Q n when n is odd and only the Lie algebra L 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 n .
We aim to compute the indexes of L n and Q n and regular vectors.Let {x 1 , x 2 , . . ., x n } be a fixed basis of the vector space V = L n (resp.V = Q n ) and {x * 1 , . . ., x * n } be a basis of the dual space.Define the Lie algebra L n (resp.Q n ) with respect to the basis by the brackets (1) (resp.( 2)).Set f = ∑ i≥0 g i x * i ∈ V * .Proposition 10.For n ≥ 3, the index of the n-dimensional filiform Lie algebra L n is χ (L n ) = n − 2. The regular vectors of L n are of the form f = ∑ n i=1 g i x * i with one of g i = 0 where i ∈ {3, . . ., n}.Proof.Since the corresponding matrix to the Lie algebra L n is of the form The second assertion is obtained by a direct calculation.
Proposition 11.For n = 2k and k ≥ 2, the index of the n-dimensional filiform Lie algebra The regular vectors of Q n are of the form f = ∑ n i=1 g i x * i with g n = 0. Proof.Since the corresponding matrix to the Lie algebra The second assertion is obtained by a direct calculation.

INDEX AND DEFORMATIONS
We study now the evolution by deformation of the index of a Lie algebra.About deformation theory we refer to [9][10][11][12] and [15].Let V be a K-vector space and G 0 = (V, [ , ] 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 0 is given by the K[ ] of the form [ , ] t = ∑ i≥0 [ , ] i t i , where each [ , ] i is a K-bilinear map [ , ] i : V × V → 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 i , X j ]) i j increases by deformation, consequently the index decreases.
Corollary 13.The index of a filiform Lie algebra is less than or equal to n − 2.
Proof.Any filiform Lie algebra N is obtained as a deformation of the Lie algebra L n .Since χ (L n ) = n − 2 using the previous lemma, one has χ (N ) ≤ n − 2.