We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group U¯qH(𝔤) of a simple Lie algebra 𝔤 at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the 𝔤 = 𝔰𝔩2 case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra WQ(r) of Feigin and Tipunin and the BQ(r) algebras.

中文翻译：

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Uprolling 展开的量子群

我们在展开的受限量子群的权模块类别中构造交换（超）代数对象族ü¯qH(𝔤)一个简单的李代数𝔤在统一的根源，并研究其本地模块的类别。我们确定它们的简单模块并推导出这些类别的条件是有限的、非退化的和带状的。受到众多例子的启发𝔤 = 𝔰𝔩2在这种情况下，我们希望其中一些类别能够很好地与顶点算子代数的模块类别进行比较。我们特别关注预期对应于较高等级三元组顶点代数的示例W问(r)费金和蒂普宁以及乙问(r)代数。