6

S. PREVOST

algebras, as in [G] and [M].

In Sections 4.3 - 4.7 we first present a version of Borcherds' use of vertex algebras to

give a (nonexplicit) description of an integral form of the universal enveloping algebras

for the simply-laced affine algebras. The relationship between Schur polynomials and

a basis for the symmetric algebra 5(h.z) pl a v a n important role in verifying Borcherds'

assertion that this description is an integral form. One difference between Borcherds'

approach and our approach is that one considers the affine Lie algebra as a sub quotient

of the vertex algebra instead of looking at the Lie algebra generated by the components

of the vertex operators associated with the weight one elements of a vertex operator

algebra. Secondly, we show that the integral form of the universal enveloping algebra

obtained using the vertex algebra approach is (essentially) (cf. (4.6.3)) the same as

the integral form obtained from the subalgebra generated by the Z-basis explicitly

described in Theorem 4.2.6. Finally, in Section 4.7 we extend the Borcherds-type

vertex algebra description of a Z-form of the universal enveloping algebra to include

the remaining unequal root length affines.

Acknowledgements. This paper is a revised version of the author's Ph.D. dissertation

at Rutgers University, 1989. I would like to thank my advisor Professor Robert Wilson

and Professor James Lepowsky for all of their invaluable help throughout my stay at

Rutgers University, and especially for their time spent reading this work.