### 4.1 Definitions

An matrix A is called a “generalized Cartan matrix” (or just “Cartan matrix” for short) if it satisfies the following conditions:
where denotes the non-positive integers. One can encode the Cartan matrix in terms of a Dynkin diagram, which is obtained as follows:

1. For each , one associates a node in the diagram.
2. One draws a line between the node and the node if ; if (), one draws no line between and .
3. One writes the pair over the line joining to . When the products are all (which is the only situation we shall meet in practice), this third rule can be replaced by the following rules:
1. one draws a number of lines between and equal to ;
2. one draws an arrow from to if

So, for instance, the Dynkin diagrams in Figure 16 correspond to the Cartan matrices

respectively. If the Dynkin diagram is connected, the matrix is indecomposable. This is what shall be assumed in the following.

Although this is not necessary for developing the general theory, we shall impose two restrictions on the Cartan matrix. The first one is that ; the second one is that is symmetrizable. The restriction excludes the important class of affine algebras and will be lifted below. We impose it at first because the technical definition of the Kac–Moody algebra when is then slightly more involved.

The second restriction imposes that there exists an invertible diagonal matrix with positive elements and a symmetric matrix such that

The matrix is called a symmetrization of and is unique up to an overall positive factor because is indecomposable. To prove this, choose the first (diagonal) element of arbitrarily. Since is indecomposable, there exists a nonempty set of indices such that . One has and . This fixes the ’s in terms of since . If not all the elements are determined at this first step, we pursue the same construction with the elements and with and, more generally, at step , with . As the matrix is assumed to be indecomposable, all the elements of and of can be obtained, depending only on the choice of . One gets no contradicting values for the ’s because the matrix is assumed to be symmetrizable.

In the symmetrizable case, one can characterize the Cartan matrix according to the signature of (any of) its symmetrization(s). One says that is of finite type if is of Euclidean signature, and that it is of Lorentzian type if is of Lorentzian signature.

Given a Cartan matrix (with ), one defines the corresponding Kac–Moody algebra as the algebra generated by generators subject to the following “Chevalley–Serre” relations (in addition to the Jacobi identity and anti-symmetry of the Lie bracket),

The relations (4.11), called Serre relations, read explicitly
(and likewise for the ’s).

Any multicommutator can be reduced, using the Jacobi identity and the above relations, to a multicommutator involving only the ’s, or only the ’s. Hence, the Kac–Moody algebra splits as a direct sum (“triangular decomposition”)

where is the subalgebra involving the multicommutators , is the subalgebra involving the multicommutators and is the Abelian subalgebra containing the ’s. This is called the Cartan subalgebra and its dimension is the rank of the Kac–Moody algebra . It should be stressed that the direct sum Equation (4.13) is a direct sum of , and as vector spaces, not as subalgebras (since these subalgebras do not commute).

A priori, the numbers of the multicommutators

are infinite, even after one has taken into account the Jacobi identity. However, the Serre relations impose non-trivial relations among them, which, in some cases, make the Kac–Moody algebra finite-dimensional. Three worked examples are given in Section 4.4 to illustrate the use of the Serre relations. In fact, one can show [116] that the Kac–Moody algebra is finite-dimensional if and only if the symmetrization of is positive definite. In that case, the algebra is one of the finite-dimensional simple Lie algebras given by the Cartan classification. The list is given in Table 10.

When the Cartan matrix is of Lorentzian signature the Kac–Moody algebra , constructed from using the Chevalley–Serre relations, is called a Lorentzian Kac–Moody algebra. This is the case of main interest for the remainder of this paper.

 Table 10: Finite Lie algebras.
 Name Dynkin diagram