Kac-Moody Groups

Abstract

This chapter is devoted to the construction of the “maximal” Kac—Moody group Ç associated to a Kac—Moody Lie algebra g = g(A), for A any£x£GCM, due to Tits. There are other versions of groups associated to g(A), e.g., we will discuss the “minimal” Kac—Moody group g^mindefined by Kac—Peterson in Section 7.4. We first construct certain groupsB,Nand {Pi}1<i<e, and then G is constructed as the amalgamated product of these groups. We now give an outline of the construction of these groups. Let n C g be the positive part of g, i.e., the direct sum of positive root spaces of g. Consider the completion û of n got by taking the directproductof the positive root spaces. Then ñ is canonically a pro-nilpotent pro-Lie algebra. LetUbe the pro-unipotent pro-group with Liel f =ñ (guaranteed by Theorem 4.4.19). Similarly, let U_i, 1 < i <Q, be the pro-unipotent pro-group with Lie algebra Lie U_i= û_i, where û_;C ñ is the direct product of all the positive root spaces except the one corresponding to the simple roota _i .