Higher level Zhu algebras and modules for vertex operator algebras

Abstract

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted An(V), for n∈N, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not An−1(V) is isomorphic to a direct summand of An(V) affects the types of indecomposable V-modules which can be constructed by inducing from an An(V)-module, and in particular whether there are V-modules induced from An(V)-modules that were not already induced by A0(V). We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of A1(V): when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of A1(V) in relationship to A0(V) determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.