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Quasi-finite Algebras Graded by Hamiltonian and Vertex Operator Algebras

Published online by Cambridge University Press:  06 July 2010

James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
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Summary

Abstract

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 C2-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 C2-finiteness (or C2-cofiniteness), saying that a certain quotient space V/C2(V) is finite-dimensional. We will call this condition Zhu's finiteness condition in the rest of the paper.

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Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 282 - 329
Publisher: Cambridge University Press
Print publication year: 2010

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