Chapter 1
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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