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

Abstract: We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}k(sl_3,f{\theta})$. 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.