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

Abstract

We study the representation theory of the Bershadsky–Polyakov algebra [Formula: see text]. In particular, Zhu algebra of [Formula: see text] is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category [Formula: see text] for the Bershadsky–Polyakov algebra [Formula: see text] for [Formula: see text]. In the case [Formula: see text], we show that the Zhu algebra [Formula: see text] has two-dimensional indecomposable modules.