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

Authors: Dražen Adamović, Ana Kontrec
Journal of Algebra and its applications (2020),
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 for 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.

Keywords: Vertex algebra, WW-algebras, Bershadsky–Polyakov algebra, Zhu’s algebra