Basic definitions

Abstract

The central object of our study is a certain class of infinite-dimensional Lie algebras alternatively known as contragredient Lie algebras, generalized Cartan matrix Lie algebras or Kac-Moody algebras. Their definition is a rather straightforward “infinite-dimensional” generalization of the definition of semisimple Lie algebras via the Cartan matrix and Chevalley generators. The slight technical difficulty that occurs in the case det A = 0 is handled by introducing the “realization” in the “Cartan subalgebra” h. The Lie algebra o(A) is then a quotient of the Lie algebra õ(A) with generators e i , f i and h, and defining relations (1.2.1), by the maximal ideal intersecting h trivially. Some of the advantages of this definition as compared to the one given in the introduction, as we will see, are as follows: the definition of roots and weights is natural; the Weyl group acts on a nice convex cone; the characters have a nice region of convergence.KeywordsCartan MatrixFormal TopologyChevalley GeneratorGeneralize Cartan MatrixRoot Space DecompositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.