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Introduction

Lie algebras of finite dimension are well understood, and numerous procedures for performing calculations with them are described in Chapter LIE ALGEBRAS. An important class of infinite dimensional Lie algebras is that of Kac-Moody Lie algebras. The principal text on this subject is a book by Kac [Kac90]. Let us briefly introduce these Lie algebras.

A generalized Cartan matrix is an integral matrix A = (a_ij)_{i, j=1}ⁿ such that a_ii = 2, a_ij < 0 for i != j, and a_ij = 0 implies a_ji = 0. (Note that in particular, a Cartan matrix in the usual sense is a generalized Cartan matrix.)

To a generalized Cartan matrix we associate a Kac-Moody Lie algebra g(A). This Lie algebra is generated by 3n elements e_i, f_i, h_i (i = 1, ..., n) satisfying the following defining relations:

[h_i, h_j] = 0, [e_i, f_i] = h_i, [e_i, f_j] = 0 (if ) i != j,

[h_i, e_j] = a_ij e_j, [h_i, f_j] = - a_ij f_j,

((ad) e_i)^{1 - a_ij}e_j = 0, ((ad) f_i)^{1 - a_ij}f_j = 0 (if ) i != j.

The class of Kac-Moody Lie algebras breaks up into three subclasses:

(a)

There is a vector θ of positive integers such Aθ is a positive vector. In this case the Lie algebra g(A) is finite-dimensional and reductive.

(b)

There is a vector δ of positive integers such that Aδ = 0. In this case g(A) is infinite-dimensional, but is of polynomial growth. These Lie algebras are called affine Lie algebras.

(c)

There is a vector α of positive integers such that Aα is negative. In this case g(A) is infinite-dimensional and of exponential growth.

The procedures for finite-dimensional Lie algebras are described in Chapter LIE ALGEBRAS. The affine Lie algebras are described in Section Affine Kac-Moody Lie Algebras. The Kac-Moody Lie algebras of type (c) are not yet available.

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