INTEGRAL BASES FOR AFFINE LIE ALGEBRAS

AND THEIR UNIVERSAL ENVELOPING ALGEBRAS

David Mitzman

1

1. INTRODUCTION

During the past ten years the field of Kac-Moody Lie algebras has

experienced enormous growth and development. Interesting accounts of

this development have recently been given by two protagonists in the

field (see [19], [20] and [15]). In addition, some recent general

references on Kac-Moody Lie algebras are [6], [9], [17], [23]. To

place the present work in its context we briefly review the relevant

background in the theories of semisimple and Kac-Moody Lie algebras.

Let lF denote a field of characteristic 0.

If

A is an algebra

(or Lie algebra) over lF, then an integral form for A is an algebra

(or Lie algebra) All. over

71.

(in the obvious sense) such that

A71. @ll.lF

=

A.

Integral forms for finite-dimensional semisimple Lie algebras were

first constructed and studied by Chevalley [3]. They provide the

1980 Mathematics Subject Classification. 17B65, 17B35, 17B20, 17B05.

1This

work is a revised version of the author's Ph.D. thesis

written under the supervision of

J.

Lepowsky at Rutgers University, 1983.

The author was partially supported by a Graduate Research Fellowship

from

Rut~ers

University.

1

http://dx.doi.org/10.1090/conm/040