Morita equivalences for Zhu's algebra

TL;DR

This work develops a Morita-theoretic framework linking Zhu's algebra $\mathsf A$ to higher-degree data via mode transition algebras $\mathfrak A_d$ and zig-zag algebras ${\mathcal Z}_d$. By introducing strong identity elements and treating $\mathfrak A$ as a Peirce algebra, the authors prove equivalences between $\mathfrak A_d$-modules and ${\mathsf Z}_d$-modules, and, under suitable hypotheses, between $\mathfrak A_d$-modules and $\mathsf A$-modules. This leads to explicit presentations for higher Zhu algebras $\mathsf A_d$, especially for rational VOAs, where $\mathsf A_d(V)\cong\prod_{j=0}^d\prod_{i=1}^m\mathrm{Mat}_{\dim(S_j^i)}(\mathbb{C})$, and to detailed computations for examples such as rank-$n$ Heisenberg VOAs, Virasoro, and lattice VOAs. The results unify and extend previous level-$d$ Zhu algebra descriptions, illuminate when higher-degree data reduce to lower-degree information, and provide tools for studying rationality, induced modules, and contragredients in VOA theory.

Abstract

Through the introduction of new ideals, and with the assistance of the $d$-th mode transition algebras $\mathfrak{A}_d$, for $d\in \mathbb{N}$, we show how Zhu's associative algebra $\mathsf{A}$, conventionally valued for tracking information about the degree $0$ part of an $\mathbb{N}$-graded module over a vertex operator algebra $V$, also contains information about components of higher degree. As an application, equivalent conditions are given for rationality of $V$, and explicit presentations for higher-level Zhu algebras are given, including for a large class of non-rational VOAs.

Theorems & Definitions (63)

• Remark 1.1.1
• Definition 1.2.1
• Lemma 2.0.1
• proof
• Lemma 2.0.2
• proof
• Lemma 2.0.3
• proof
• Theorem 2.0.4
• proof