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4.1 Definitions

An n × n matrix A is called a “generalized Cartan matrix” (or just “Cartan matrix” for short) if it satisfies the following conditions9:
Aii = 2 ∀i = 1,⋅⋅⋅ ,n, (4.1 ) Aij ∈ ℤ− (i ⁄= j), (4.2 ) Aij = 0 ⇒ Aji = 0, (4.3 )
where ℤ − 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 i = 1,⋅⋅⋅ ,n, one associates a node in the diagram.
  2. One draws a line between the node i and the node j if A ⁄= 0 ij; if A = 0 ij (= A ji), one draws no line between i and j.
  3. One writes the pair (Aij,Aji) over the line joining i to j. When the products Aij ⋅ Aji are all ≤ 4 (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 i and j equal to max (|Aij|,|Aji|);
    2. one draws an arrow from j to i if |A | > |A |. ij ji
View Image

Figure 16: The Dynkin diagrams corresponding to the finite Lie algebras A2,B2 and G2 and to the affine Kac–Moody algebras A (22) and A+1.

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

( ) A [A ] = 2 − 1 , (4.4 ) 2 − 1 2
( ) A [B ] = 2 − 2 , (4.5 ) 2 − 1 2
( ) 2 − 3 A [G2] = − 1 2 , (4.6 )
(2) ( 2 − 4) A[A 2 ] = , (4.7 ) − 1 2
( ) + 2 − 2 A [A 1 ] = − 2 2 , (4.8 )
respectively. If the Dynkin diagram is connected, the matrix A 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 det A ⁄= 0; the second one is that A is symmetrizable. The restriction detA ⁄= 0 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 det A = 0 is then slightly more involved.

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

A = DS. (4.9 )
The matrix S is called a symmetrization of A and is unique up to an overall positive factor because A is indecomposable. To prove this, choose the first (diagonal) element ε > 0 1 of D arbitrarily. Since A is indecomposable, there exists a nonempty set J1 of indices j such that A1j ⁄= 0. One has A1j = ε1S1j and Aj1 = εjSj1. This fixes the εj’s > 0 in terms of ε1 since S1j = Sj1. If not all the elements εj are determined at this first step, we pursue the same construction with the elements Ajk = εjSjk and Akj = εkSkj = εkSkj with j ∈ J1 and, more generally, at step p, with j ∈ J ∩ J ⋅⋅⋅ ∩ J 1 2 p. As the matrix A is assumed to be indecomposable, all the elements εi of D and Sij of S can be obtained, depending only on the choice of ε1. One gets no contradicting values for the εj’s because the matrix A 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 A is of finite type if S is of Euclidean signature, and that it is of Lorentzian type if S is of Lorentzian signature.

Given a Cartan matrix A (with det A ⁄= 0), one defines the corresponding Kac–Moody algebra 𝔤 = 𝔤 (A ) as the algebra generated by 3n generators hi,ei,fi subject to the following “Chevalley–Serre” relations (in addition to the Jacobi identity and anti-symmetry of the Lie bracket),

[hi,hj ] = 0, [hi,ej] = Aijej (no summation on j), [hi,fj] = − Aijfj (no summation on j), (4.10 ) [ei,fj] = δijhj (no summation on j),
1− A 1− Aij adei ij(ej) = 0, ad fi (fj) = 0, i ⁄= j. (4.11 )
The relations (4.11View Equation), called Serre relations, read explicitly
[e◟i,[ei,[ei,⋅⋅⋅◝◜,[ei,ej]]⋅⋅⋅◞]= 0 (4.12 ) 1−A commutators ij
(and likewise for the fk’s).

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

𝔤 = 𝔫− ⊕ 𝔥 ⊕ 𝔫+, (4.13 )
where 𝔫− is the subalgebra involving the multicommutators [fi ,[fi ,⋅⋅⋅ ,[fi ,fi ]⋅⋅⋅] 1 2 k−1 k, 𝔫+ is the subalgebra involving the multicommutators [e ,[e ,⋅⋅⋅ ,[e ,e ]⋅⋅⋅] i1 i2 ik−1 ik and 𝔥 is the Abelian subalgebra containing the hi’s. This is called the Cartan subalgebra and its dimension n is the rank of the Kac–Moody algebra 𝔤. It should be stressed that the direct sum Equation (4.13View Equation) is a direct sum of 𝔫−, 𝔥 and 𝔫+ as vector spaces, not as subalgebras (since these subalgebras do not commute).

A priori, the numbers of the multicommutators

[fi1,[fi2,⋅⋅⋅ ,[fik−1,fik]⋅⋅⋅] and [ei1,[ei2,⋅⋅⋅ ,[eik−1,eik]⋅⋅⋅]

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 [116Jump To The Next Citation Point] that the Kac–Moody algebra is finite-dimensional if and only if the symmetrization S of A 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 A is of Lorentzian signature the Kac–Moody algebra 𝔤(A ), constructed from A 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.


Dynkin diagram







C n




















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