Vertex algebras and integral bases for the enveloping algebras of affine Lie algebras

The explicit construction of integral bases for affine Kac-Moody algebras of type 1 and their associated universal enveloping algebras was first given by Garland (G). Later Mitzman extended these results and gave a description of the integral bases for the enveloping algebras of the type 2 and 3 affine Lie algebras (M). In this paper we will again describe integral bases of the above structures, but from the viewpoint of vertex operator representations of the affine Kac-Moody algebras on a vertex operator algebra {dollar}Vsb L, L{dollar} a suitable root lattice. The vertex operator constructions give us more natural objects and simplify the proofs found in (M). For example, the commutator identities needed for the straightening arguments are consequences of the commutator formula (B) or the "Jacobi identity" (F-L-M) for vertex operator algebras. In addition to the above integral bases, we give a description of a Z-basis for the vertex operator algebra {dollar}Vsb L{dollar} and also for a slightly more general structure called a vertex algebra. The results rely heavily on the work of (G), (M), (F-L-M) and (B).; Also we give a detailed version of Borcherds' treatment of integral forms for vertex algebras. Here we show that the integral forms for the simply-laced affines, derived in Borcherds' work, are (essentially) the same integral forms generated by the Z-basis elements described in (M). Then we extend Borcherds' methods to obtain a description of the integral forms associated to the unequal root length affines.