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