Higher level Zhu algebras and modules for vertex operator algebras |
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Authors: | Katrina Barron Nathan Vander Werf Jinwei Yang |
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Affiliation: | 1. Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, IN 46656, United States of America;2. Department of Mathematics and Statistics, Warner Hall 2036, University of Nebraska, 2504 9th Ave., Kearny, NE 68849, United States of America;3. Department of Mathematics, University of Alberta, 632 Central Academic Building, Edmonton, AB T6G 2G1, Canada |
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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 , for , 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 is isomorphic to a direct summand of affects the types of indecomposable V-modules which can be constructed by inducing from an -module, and in particular whether there are V-modules induced from -modules that were not already induced by . 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 : 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 in relationship to 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. |
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Keywords: | 17B68 17B69 17B81 81R10 81T40 81T60 |
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