Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras

Since its debut in the mathematical literature a little more than a decade ago, the theory of vertex operators has flourished. In addition to yielding all the finite-dimensional irreducible representations of the simple Lie algebras, vertex operators have provided a very natural setting for studying affine Lie algebras and their representations. Perhaps the major contribution to date of the theory is the construction of the Monster and the Moonshine module. Vertex operator theory also plays a fundamental role in string theory. In the present work, the author utilizes the vertex operator representations of the affine Lie algebras to give two equivalent descriptions of an integral basis for the affine Lie algebras and their associated universal enveloping algebras. The first basis exhibited is the vertex operator algebra version of the explicit Z-basis given by Garland and Mitzman. Next, the author examines the vertex algebra approach developed by Borcherds to give a (nonexplicit) description of an integral form for the enveloping algebras for the simply-laced affines. integral basis for the remaining unequal root length affine Lie algebras and their enveloping algebras.