标题： 根单位的Hecke-Clifford代数和共形嵌入 显示英文标题

标题： Hecke-Clifford algebras at roots of unity and conformal embeddings

摘要： 在本文中，我们给出了由第一作者和Snyder最近引入的范畴$\mathcal{E}_q$和$\overline{\mathcal{SE}_N}$的柯西完备的组合描述。 这进而给出了范畴$\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}_{A}$的组合描述，其中$A$是对应于共形嵌入$\mathfrak{sl}_N$ level$N$到$\mathfrak{so}_{N^2-1}$level 1 的平展代数对象。 特别是，我们给出了这些范畴的简单对象的分类，它们的量子维度的公式，以及与定义对象张量乘法的融合规则。 我们获得这些结果的方法是研究某些同态代数在$\mathcal{E}_q$和$\mathcal{SE}_N$中的表示理论，这些代数已知是 Hecke-Clifford 代数的子代数。我们基于现有文献来研究单位根处的 Hecke-Clifford 代数的表示理论。 显示英文摘要

摘要： In this paper we give a combinatorial description of the Cauchy completion of the categories $\mathcal{E}_q$ and $\overline{\mathcal{SE}_N}$ recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}_{A}$ where $A$ is the \`etale algebra object corresponding to the conformal embedding $\mathfrak{sl}_N$ level $N$ into $\mathfrak{so}_{N^2-1}$ level 1. In particular we give a classification of the simple objects of these categories, a formula for their quantum dimensions, and fusion rules for tensoring with the defining object. Our method of obtaining these results is the Schur-Weyl approach of studying the representation theory of certain endomorphism algebras in $\mathcal{E}_q$ and $\mathcal{SE}_N$, which are known to be subalgebras of Hecke-Clifford algebras. We build on existing literature to study the representation theory of the Hecke-Clifford algebras at roots of unity.