Classification of irreducible modules for Bershadsky-Polyakov algebra at certain levels

TL;DR

This paper classifies irreducible modules for the Bershadsky-Polyakov algebra at specific levels, revealing the structure of its Zhu algebra and identifying module types, including indecomposables at level zero.

Contribution

It provides a complete classification of modules in category O for certain levels and describes the Zhu algebra's structure, including indecomposable modules at level zero.

Findings

Zhu algebra is isomorphic to a quotient of the Smith algebra.

All modules in category O are classified for levels -5/3, -9/4, -1, 0.

The Zhu algebra at level zero admits 2-dimensional indecomposable modules.

Abstract

We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$. In particular, Zhu algebra of $\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category $\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ when $k=-5/3, -9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$ has $2$--dimensional indecomposable modules.