### 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 A _{n} (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 A _{n−1} (V) is isomorphic to a direct summand of A _{n} (V) affects the types of indecomposable V-modules which can be constructed by inducing from an A _{n} (V)-module, and in particular whether there are V-modules induced from A _{n} (V)-modules that were not already induced by A _{0} (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 A _{1} (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 A _{1} (V) in relationship to A _{0} (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.

Original language | English (US) |
---|---|

Pages (from-to) | 3295-3317 |

Number of pages | 23 |

Journal | Journal of Pure and Applied Algebra |

Volume | 223 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2019 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Pure and Applied Algebra*,

*223*(8), 3295-3317. https://doi.org/10.1016/j.jpaa.2018.11.002

**Higher level Zhu algebras and modules for vertex operator algebras.** / Barron, Katrina; Vander Werf, Nathan; Yang, Jinwei.

Research output: Contribution to journal › Article

*Journal of Pure and Applied Algebra*, vol. 223, no. 8, pp. 3295-3317. https://doi.org/10.1016/j.jpaa.2018.11.002

}

TY - JOUR

T1 - Higher level Zhu algebras and modules for vertex operator algebras

AU - Barron, Katrina

AU - Vander Werf, Nathan

AU - Yang, Jinwei

PY - 2019/8

Y1 - 2019/8

N2 - 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 A n (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 A n−1 (V) is isomorphic to a direct summand of A n (V) affects the types of indecomposable V-modules which can be constructed by inducing from an A n (V)-module, and in particular whether there are V-modules induced from A n (V)-modules that were not already induced by A 0 (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 A 1 (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 A 1 (V) in relationship to A 0 (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.

AB - 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 A n (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 A n−1 (V) is isomorphic to a direct summand of A n (V) affects the types of indecomposable V-modules which can be constructed by inducing from an A n (V)-module, and in particular whether there are V-modules induced from A n (V)-modules that were not already induced by A 0 (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 A 1 (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 A 1 (V) in relationship to A 0 (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.

UR - http://www.scopus.com/inward/record.url?scp=85056287436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056287436&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2018.11.002

DO - 10.1016/j.jpaa.2018.11.002

M3 - Article

AN - SCOPUS:85056287436

VL - 223

SP - 3295

EP - 3317

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 8

ER -