Vertex operator algebras and associative algebras

Let V be a vertex operator algebra. We construct a sequence of associative algebras A(n)(V) (n = 0, 1, 2,...) such that A(n)(V) is a quotient of A(n+1)(V) and a pair of functors between the category of A(n)(V)-modules which are not A(n-1)(V)-modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is rational if and only if all A,(V) are finite-dimensional semisimple algebras. (C) 1998 Academic Press.