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つづき
A systematic study of Kac-Moody algebras was started independently by V.G. Kac [Ka] and R.V. Moody [Mo], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac-Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. Casimir element), which can be constructed provided that the matrix A
is symmetrizable, i.e. A=DB
for some invertible diagonal matrix D
and symmetric matrix B
[Ka2]. In the non-symmetrizable case more sophisticated geometric methods are required [Ku], [Ma].
One of the most important ingredients of the theory of Kac-Moody algebras are integrable highest-weight representations (cf. also Representation with a highest weight vector).
The numerous applications of Kac-Moody algebras are mainly related to the fact that the Kac-Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.)
These Kac-Moody algebras are called affine algebras.
This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [PrSe]).
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