Yoneda algebras of the triplet vertex operator algebra

TL;DR

This paper investigates the algebraic structures associated with the triplet vertex operator algebra, comparing categories and their Ext quivers, Morita equivalences, and Yoneda algebras, revealing infinite global dimension and Koszul properties.

Contribution

It provides a detailed analysis of the abelian categories related to the triplet VOA, including their Ext quivers, Morita equivalences, and Yoneda algebras, especially in the non-rational case.

Findings

The categories have infinite global dimension.

The Zhu algebra and associated graded algebra are isomorphic and also have infinite global dimension.

The module categories exhibit specific Koszul properties.

Abstract

Given a vertex operator algebra $V$, one can construct two associative algebras, the Zhu algebra $A(V)$ and the $C_2$-algebra $R(V)$. This gives rise to two abelian categories $A(V)-\text{Mod}$ and $R(V)-\text{Mod}$, in addition to the category of admissible modules of $V$. In case $V$ is rational and $C_2$-cofinite, the category of admissible $V$-modules and the category of all $A(V)$-modules are equivalent. However, when $V$ is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra $\mathcal{W}(p)$, as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra $ \text{End} (\oplus_{L\in \text{Irr}} P_L)^{op}$, and the Yoneda algebra $\text{Ext}^{*}(\oplus_{L\in \text{Irr}}L, \oplus_{L\in \text{Irr}}L)$. As a consequence, the category of admissible log-modules for the triplet VOA $ \mathcal W(p)$ has infinite global dimension, as do the Zhu algebra $A(\mathcal W(p))$, and the associated graded algebra $\text{gr} \ A(\mathcal W(p))$ which is isomorphic to $R(\mathcal W(p))$. We also describe the Koszul properties of the module categories of $ \mathcal W(p)$, $A(\mathcal W(p))$ and $\text{gr} \ A(\mathcal W(p))$.