VERTEX OPERATORS AND INTEGRAL BASES 3

algebras. This played a role in a generalization of the definition of a Chevalley group

[Stein] which utilized representations other than the adjoint representation of a Lie

algebra (cf. [H]).

To describe a natural integral form for the universal enveloping algebra of an affine

Lie algebra, one has a choice of two methods: the explicit construction of a Z-basis of

the enveloping algebra via a Chevalley-type basis of the affine Lie algebra, as in [G]

or [M], or a (nonexplicit) description in which some set of conditions is used to define

a Z-form of the enveloping algebra. The second method is used in [T] and [B]. Tits in

[T] used a certain set of generators to obtain an integral form. For the simply-laced

affine Lie algebras, Borcherds in [B] employed the theory of vertex algebras to define

a new family of irreducible integrable modules for these affine algebras and to find

integral forms for the modules. He then defined the integral form of the universal

enveloping algebra to consist of the elements of the enveloping algebra which preserve

the integral forms of the modules in the aforementioned family.

In the present work we give a new proof for the identities which allow one to

straighten the Z-basis elements in the integral form of the universal enveloping al-

gebras. The proof offered here differs in two respects from the previous proof. We

first understand how to prove the identities in the enveloping algebras associated to

the simply-laced affines. Then we observe that the identities for the remaining affines

follow as corollaries to the identities in the equal root length affines. Secondly, the

proof we present is constructive in nature: the earlier proofs merely "checked" the

validity, but didn't give much insight into the discovery of the identities.

Next we consider the vertex operator representations of the affine Lie algebras.

It was discovered in [L-Wl], [K-K-L-W], [F-K] and [Seg] that affine Lie algebras

could be constructed explicitly using vertex operator representations. See also the

introduction in [F-L-M] for a discussion of the contributions to this phenomenom

made by physicists.

We also give a detailed version of the method employed by Borcherds in his an-