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.

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