Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noetherianity of twisted Zhu algebra and bimodules

Jianqi Liu

Abstract

In this paper we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu algebras and bimodules (and their $g$-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

Noetherianity of twisted Zhu algebra and bimodules

Abstract

In this paper we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu algebras and bimodules (and their -twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

Paper Structure

This paper contains 21 sections, 25 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $g$-twisted modules over vertex operator algebras
  4. The $g$-twisted Zhu's algebra $A_{g}(V)$ and it bimodule $A_{g}(M)$
  5. Definition of $A_g(V)$
  6. Definition of $A_g(M)$
  7. Level filtration on $A_g(V)$ and $A_g(M)$
  8. The cofinite conditions of a VOA
  9. Noetherianity of twisted Zhu's algebra and its bimodule
  10. Noetherianity of $A_g(V)$ for $C_1$-cofinite VOA $V$
  11. Noetherianity of $A_g(M)$ for (weakly) $C_1$-cofinite $V$-module $M$
  12. Example of a non-$C_1$-cofinite CFT-type VOA whose Zhu's algebra is non-Noetherian
  13. The non-$C_1$-cofinite VOA $V_M$
  14. Non-Noetherianity of the Zhu's algebra of $V_M$
  15. A spanning set of $O(V_M)$
  16. ...and 6 more sections

Key Result

Theorem A

Let $V$ be a CFT-type VOA that is $C_1$-cofinite, and let $g\in \mathrm{Aut}(V)$ be a finite order automorphism. Then $A_g(V)$ is left and right Noetherian as an associative algebra.

Figures (1)

  • Figure 1:

Theorems & Definitions (51)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • ...and 41 more