We study the representation theory of the Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3,f𝜃). In particular, Zhu algebra of 𝒲k is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category 𝒪 for the Bershadsky–Polyakov algebra 𝒲k for k = −5/3,−9/4,−1, 0. In the case k = 0, we show that the Zhu algebra A(𝒲k) has two-dimensional indecomposable modules.

中文翻译：

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Bershadsky-Polyakov代数在一定水平上的不可约模分类

我们研究了 Bershadsky-Polyakov 代数的表示论𝒲ķ = 𝒲ķ(sl3,F𝜃). 特别是朱代数𝒲ķ在改变 Virasoro 向量后，与 Smith 代数的某个商同构。我们对类别中的所有模块进行分类𝒪Bershadsky-Polyakov 代数𝒲ķ为了ķ = -5/3,-9/4,-1, 0. 在这种情况下ķ = 0，我们证明了朱代数一种(𝒲ķ)具有二维不可分解的模块。