Groups associated with some types of infinite dimensional Lie algebras

Abstract

In this note, we shall discuss a Hopf algebraic approach to the construction of ind-affine groups (afline groups of Shafarevich types) associated with some types of infinite dimensional Lie algebras. It is well known that a complex Lie algebra determines a Lie group up to local isomorphisms, and therefore determines a unique connected simply connected Lie group. As for the algebraic groups, G. Hochschild has shown that for a finite dimensional Lie algebra L over an algebraically closed field F of characteristic 0 whose radical is nilpotent, there exists a connected simply connected algebraic group G such that the Lie algebra of G is isomorphic to L. In particular, he has shown that if L = [L, L], the dual Hopf algebra U(L)’ of the universal enveloping algebra U(L) of L is the coordinate ring of the algebraic group G (cf. G. Hochschild [3]). For a semi-simple Lie algebra, there are some other methods to construct such an algebraic group attached to it. One method is to use linear representation of the given Lie algebra (C. Chevalley); the other is to determine the group by generators and relations (R. Steinberg). These methods have been generalized to construct groups attached to Kac-Moody Lie algebras and have been investigated by many authors (G. V. Kac [7,8], G. V. Kac and D. H. Peterson [9], J. Tits [12, 133). G. V. Kac [9] has shown that the group has a structure of an infinite dimensional algebraic group in the sense of Shafarevich (cf. 3.3 and I. R. Shafarevich [lo]). In this paper, generalizing Hochschild’s theory, we shall give a method to construct indafline groups (a generalization of groups introduced by Shafarevich) directly from the given Lie algebra of some types called integrable. Here the definition of integrable Lie algebra is somewhat different from that given by