Topology of Kac-Moody groups
Abstract (summary)
The class of complex semi-simple Lie-algebras can be extended to include infinite dimensional Lie algebras known as Kac-Moody Lie algebras. The correspondence between complex semi-simple Lie algebras and complex, connected, simply-connected Lie groups has been extended by Kac-Peterson to a correspondence between Kac-Moody Lie algebras and certain simply-connected topological groups known as Kac-Moody groups. These groups behave like Lie groups in that they have a maximal torus of finite rank and a Weyl group. One can also construct flag varieties that are projective varieties and admit a Bruhat decomposition.
Kac-Moody groups fall into three general types: The finite type, the affine type and the indefinite type. The groups of finite type are the usual simply-connected Lie groups. The groups of affine type are closely related to loop-groups and have been extensively studied in topology and physics. The groups of indefinite type constitute the majority among Kac-Moody groups and very little is known about them.
In this thesis, we explore the topology of Kac-Moody groups. Chapter 1 gives a general overview of the theory of Kac-Moody groups and is a condensed version of (6). Chapter 2 contains some results about the Hopf-algebra structure of the cohomology of Kac-Moody groups. Chapter 3 uses the results of chapter 2 to compute the cohomology of all the Kac-Moody groups of rank 2 which are not of finite type. In this chapter we also compute the cohomology of the classifying spaces of these groups. In chapter 4 we show that the classifying space of a Kac-Moody group which is not of finite type can be realized as a certain homotopy-colimit of the classifying space of its proper parabolics. Using this we show that the classifying space of the Kac-Moody group is approximated by the classifying space of the normalizer of its maximal torus at all primes which do not appear in the torsion of the Weyl group. This extends a classical result to the class of all Kac-Moody groups. Finally, in chapter 5 we construct a fibration using rank 2 Kac-Moody groups and end with a conjecture relating this fibration to certain well known fibrations known as Anick fibrations (1). (Copies available exclusively from MIT Libraries Rm. 14-0551, Cambridge, MA 02139-4307 Ph. 617-253-5668; Fax 617-253-1690.)