Chapter 1

Introduction

The vertex operator made its debut in the mathematics literature a little more than

a decade ago. Since then, the theory of vertex operators has nourished. Vertex oper-

ator representations have been shown to yield all of the finite-dimensional irreducible

representations of the simple Lie algebras. Also vertex operators have provided a very

natural setting in which to study affine Lie algebras and their representations. And

perhaps the most important contribution, to date, of vertex operator theory is the

construction of the Monster and the Moonshine module V* [F-L-M], [B]. It should

also be noted that vertex operator theory plays a fundamental role in string theory.

The reader may refer to the introduction in [F-L-M] for a discussion of contributions

to vertex operator theory by physicists.

This paper utilizes the vertex operator representations of the affine Lie algebras to

give two different, yet equivalent, descriptions of integral bases for the affine Lie alge-

bras and their associated universal enveloping algebras. First we present a new proof

for the straightening arguments used by Garland in [G] and later by Mitzman in [M],

to exhibit an explicit Z-form for the universal enveloping algebras associated to the

type 1, and type 2 and type 3 Lie algebras. We remark that the identities which allow

one to "straighten" the Z-basis elements are the most crucial part of these proofs.

Secondly, we shift our attention to the vertex operator descriptions of an integral

(or Z-) form of the universal enveloping algebra of an affine Lie algebra. We first

use vertex operator algebra theory to rewrite the explicit Z-bases for the enveloping

algebras of the affine Lie algebras exhibited by both Garland [G] and Mitzman [M].

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