# Introduction

Lie algebras of finite dimension are well understood, and numerous procedures for performing calculations with them are described in Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie). 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 [[Kac, 1990](../../references.md#cite-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 $i \neq j$, 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 $\frak{g}(A)$. This Lie algebra is generated by $3n$ elements $e_i, f_i, h_i$ ($i = 1, \ldots, n$) satisfying the following defining relations:

$$
[h_i, h_j] = 0, [e_i, f_i] = h_i, [e_i, f_j] = 0 {\rm\ if\ } i \neq j,
$$

$$
[h_i, e_j] = a_{ij} e_j, [h_i, f_j] = -a_{ij} f_j,
$$

$$
({\rm ad} e_i)^{1-a_{ij}}e_j = 0, ({\rm ad} f_i)^{1-a_{ij}}f_j = 0  {\rm\ if\ } i \neq j.
$$

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

**(a)**
There is a vector $\theta$ of positive integers such $A\theta$ is a positive vector. In this case the Lie algebra $\frak{g}(A)$ is finite-dimensional and reductive.

**(b)**
There is a vector $\delta$ of positive integers such that $A\delta = 0$. In this case $\frak{g}(A)$ is infinite-dimensional, but is of polynomial growth. These Lie algebras are called *affine Lie algebras*.

**(c)**
There is a vector $\alpha$ of positive integers such that $A\alpha$ is negative. In this case $\frak{g}(A)$ is infinite-dimensional and of exponential growth.

The procedures for finite-dimensional Lie algebras are described in Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie). The affine Lie algebras are described in Section [Affine Kac–Moody Lie Algebras](affine.md#sectalgkacaff). The Kac–Moody Lie algebras of type (c) are not yet available.