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

Abstract

We study the representation theory of the Bershadsky-Polyakov algebra ${\mathcal{W}}_k={\mathcal{W}}_k(s{l}_3,f_{\theta })$. 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.

Latex, 33 pages