Quasi-finite Algebras Graded by Hamiltonian and Vertex Operator Algebras

A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced and various module categories over quasi-finite algebras are described. When applied to the current algebras (universal enveloping algebras) of vertex operator algebras satisfying Zhu's C 2 -finiteness condition, our general consideration derives important consequences on representation theory of such vertex operator algebras. In particular, the category of modules over such a vertex operator algebra is shown to be equivalent to the category of modules over a finite-dimensional associative algebra. Introduction In order to construct conformal field theories on Riemann surfaces associated with a vertex operator algebra V and to obtain their properties such as the finite-dimensionality of the space of conformal blocks, factorization of the blocks along the boundaries of the moduli space of Riemann surfaces and the fusion functors or the tensor product of V -modules, we need first to impose an appropriate finiteness condition on V and second to study the structure of the abelian category of V -modules to some extent. One of the candidates of such a finiteness condition is the one introduced by Y.-C. Zhu ([Zhu]), usually called the C 2 -finiteness (or C 2 -cofiniteness), saying that a certain quotient space V /C 2 ( V ) is finite-dimensional. We will call this condition Zhu's finiteness condition in the rest of the paper.