Affine Chevalley algebras

Abstract

Chevalley algebras associated with finite-dimensional simple Lie algebras L over the complex field were defined in 151, and their ideal structure was worked out there in terms of the ideal structure of the underlying commutative ring R with identity (See also ]lO].) Recently, Garland [2] has shown the existence of a Chevalley basis for infinite-dimensional Kac-Moody Lie algebras [6, 7 ] of affine type [8 J over the complex field. This makes possible the construction of Chevalley algebras by transfer of the scalars to a commutative ring R with identity, just as in the iinitedimensional case. We study here the ideal structure of these algebras, which we call aflne Chevalley algebras. In Section 1, we establish the notation we need to state results on the ideal structure of iinitc dimensional Chevalley algebras. In Section 2, we describe Kac-Moody Lie algebras and Garland’s Chevalley basis theorem for such algebras associated with afline generalized Cartan matrices. In Section 3, we study an important exact sequence for atline Chevalley algebras, extending a result of Kac and Moody [6,8] on the structure of Euclidean Kac-Moody Lie algebras. In Section 4, the ideal structure of the alline Chevalley algebras over Noctherian integral domains is studied, using the results of Section 3 and the known structure of finite-dimensional Chevalley algebras over Laurent polynomial rings. Our Main Theorem 4.7 takes the form of a sandwich result reminiscent of the situation in classical Chevalley algebras [ 5, Theorems 3.4-3.6; 10, Theorem 3.11. Our result constitutes a partial generalization of a theorem of Moody [7, Theorem 4] on the ideal structure of Euclidean Lie algebras over a field of characteristic zero. (see