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Twisted bimodules and associative algebras associated to VOAs

Shun Xu

TL;DR

The paper provides a unified, streamlined proof that twisted Zhu-type algebras $A_{g,n}(V)$ and their bimodules $A_{g,n,m}(V)$ are isomorphic to subquotients of the universal enveloping algebra $U(V[g])$ associated with a finite-order automorphism $g$. By constructing the map $\varphi_{n,m}$ and proving its algebra and bimodule properties, it establishes the isomorphisms $A_{g,n}(V) \cong U(V[g])_0/U(V[g])_0^{-n-1/T}$ and $A_{g,n,m}(V) \cong U(V[g])_{n-m}/U(V[g])_{n-m}^{-m-1/T}$, unifying prior results by Han–Xiao and Dong–Jiang. The approach relies on a careful analysis of the universal enveloping algebra $U(V[g])$, the action on twisted modules, and reductions of monomials, yielding a robust link between twisted representation theory and subquotients of $U(V[g])$. This work clarifies how associative algebras and bimodules arising in twisted VOA theory can be realized concretely inside the universal enveloping framework, with potential applications to classification and construction of twisted modules.

Abstract

Let $V$ be a vertex operator algebra, $g$ be an automorphism of $V$ of order $T$, and $m, n \in (1/T)\mathbb{N}$. In~\cite{HX2} and~\cite{HXX1}, it was shown respectively that the associative algebra $A_{g,n}(V)$ constructed by Dong, Li, and Mason~\cite{DLM3}, and the $A_{g,n}(V)\!-\!A_{g,m}(V)$-bimodule $A_{g,n,m}(V)$ constructed by Dong and Jiang~\cite{DJ2}, are both isomorphic to certain subquotients of $U(V[g])$, where $U(V[g])$ denotes the universal enveloping algebra of $V$ with respect to $g$. In this paper, we give a unified and concise proof of these isomorphisms.

Twisted bimodules and associative algebras associated to VOAs

TL;DR

The paper provides a unified, streamlined proof that twisted Zhu-type algebras and their bimodules are isomorphic to subquotients of the universal enveloping algebra associated with a finite-order automorphism . By constructing the map and proving its algebra and bimodule properties, it establishes the isomorphisms and , unifying prior results by Han–Xiao and Dong–Jiang. The approach relies on a careful analysis of the universal enveloping algebra , the action on twisted modules, and reductions of monomials, yielding a robust link between twisted representation theory and subquotients of . This work clarifies how associative algebras and bimodules arising in twisted VOA theory can be realized concretely inside the universal enveloping framework, with potential applications to classification and construction of twisted modules.

Abstract

Let be a vertex operator algebra, be an automorphism of of order , and . In~\cite{HX2} and~\cite{HXX1}, it was shown respectively that the associative algebra constructed by Dong, Li, and Mason~\cite{DLM3}, and the -bimodule constructed by Dong and Jiang~\cite{DJ2}, are both isomorphic to certain subquotients of , where denotes the universal enveloping algebra of with respect to . In this paper, we give a unified and concise proof of these isomorphisms.

Paper Structure

This paper contains 5 sections, 5 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. $A_{g,n}(V)\!-\!A_{g,m}(V)$-Bimodules $A_{g,n,m}(V)$
  4. Universal enveloping algebra $U(V[g])$ of $V$ with respect to $g$
  5. The proof of Theorem \ref{['thm1.1']}

Key Result

Theorem 1.1

For any $n,m \in (1/T)\mathbb{N}$, define a linear map by sending $u + O_{n, m}(V)$ to $J_{m-n}(u) + U(V[g])_{n-m}^{-m - 1/T}$. Then $\varphi_{n, n}$ is an algebra isomorphism, and more generally, $\varphi_{n, m}$ is an $A_{g,n}(V)$--$A_{g,m}(V)$-bimodule isomorphism.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Remark 4.2
  • ...and 4 more