We study parafermion vertex algebras N−3/2(𝔰𝔩(2)) and N−3/2(𝔰𝔩(3)). Using the isomorphism between N−3/2(𝔰𝔩(3)) and the logarithmic vertex algebra 𝒲0(2) A2 from [D. Adamović, A realization of certain modules for the N = 4 superconformal algebra and the affine Lie algebra A2(1), Transform. Groups 21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra 𝒲(2, 3) of central charge c = −10, and that N−3/2(𝔰𝔩(3)) is a direct sum of irreducible N−3/2(𝔰𝔩(2))-modules. As a byproduct, we prove certain conjectures about the vertex algebra 𝒲0(p) A2. We also obtain a vertex-algebraic proof of the irreducibility of a family of 𝒲(2, 3)c modules at c = −10.

中文翻译：

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关于 𝔰𝔩(2) 和 𝔰𝔩(3) 在级别 −3 2 的准费米子顶点代数

我们研究 parafermion 顶点代数ñ-3/2(𝔰𝔩(2))和ñ-3/2(𝔰𝔩(3)). 使用之间的同构ñ-3/2(𝔰𝔩(3))和对数顶点代数𝒲0(2) 一种2来自 [D. Adamović，实现某些模块的ñ = 4超共形代数和仿射李代数一种2(1),转变。团体 21(2) (2016) 299-327]，我们证明这些对角顶点代数是 Zamolodchikov 代数的不可约模的无限直和𝒲(2, 3)中央收费C = -10， 然后ñ-3/2(𝔰𝔩(3))是不可约的直接和ñ-3/2(𝔰𝔩(2))-模块。作为副产品，我们证明了关于顶点代数的某些猜想𝒲0(p) 一种2. 我们还获得了一个族的不可约性的顶点代数证明𝒲(2, 3)C模块在C = -10.