On some twisted Kac-Moody groups

Abstract

This thesis consists of two distinct parts. The first part comprises the first three chapters and is largely of an expository nature. The second part comprises the last three chapters all of which are, to the best of our knowledge, original. In the first part we cover the background material which we shall require in the sequel. Thus Chapter 1 deals with the theory of Kac-Moody algebras and is drawn from two main sources, namely [Kac90] and [BdK90]. Two enlightening examples are given at the end of this chapter. Chapter 2 introduces the notion of the Kac-Moody group functor. This material is drawn largely, but not exclusively, from an extensive body of work on the topic by J. Tits. We give a presentation for Kac-Moody groups over fields and describe some of their properties. In Chapter 3 we give an overview of some results on Kac-Moody groups. First we describe the work of J-Y. Hee generalizing the notion of twisted Chevalley groups to the Kac-Moody situation. We then give an exposition of the work of R.W. Carter and Y. Chen on the automorphisms of complex simply-connected affine Kac-Moody groups arising from extended Cartan matrices and we describe the classification of such automorphisms. In particular, we note that the family of diagonal automorphisms of such groups behave in a manner which has no analogy in the classical theory. We conclude the Chapter with an example demonstrating the limitation of Hee’s results with regards to this type of automorphism. Chapter 4 makes use of the results on Kac-Moody algebras described in §1.5 to extend the results of Hee. Suppose A is a simply-laced extended Cartan matrix and let β(K) be a Kac-Moody group associated to A. In Chapter 4 we extend the results of Hee to the fixed point subgroup, β(K) say, of β(K) under a particular graph-by-diagonal automorphism. We then establish an isomorphism between the subgroup β(K) so obtained and a Kac-Moody group associated to an affine Cartan matrix of type II or III. Thus Chapter 4 contains our main contributions for two reasons. Firstly, it provides a realization of Kac-Moody groups of types II and III in terms of those arising from extended Cartan matrices. More precisely, Propositions 4.4.3, 4.5.6, and 4.6.4 prove the following result