Vertex Operator Algebras and Associative Algebras

LetVbe a vertex operator algebra. We construct a sequence of associative algebrasAn(V) (n = 0, 1, 2,…) such thatAn(V) is a quotient ofAn + 1(V) and a pair of functors between the category ofAn(V)-modules which are notAn − 1(V)-modules and the category of admissibleV-modules. These functors exhibit a bijection between the simple modules in each category. We also show thatVis rational if and only if allAn(V) are finite-dimensional semisimple algebras.