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

Authors: Dražen Adamović, Ana Kontrec

Journal of Algebra and its applications (2020), https://doi.org/10.1142/S0219498821501024

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,

W

-algebras, Bershadsky–Polyakov algebra, Zhu’s algebra

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