有限张量范畴等变化的纤维函子

中文摘要 AI 辅助

设有限群\(G\)作用于有限张量范畴\(\mathcal{C}\)。我们根据\(\mathcal{C}\)上的等变正合模范畴对\(\mathcal{C}^G\)上的纤维函子进行分类，这些模范畴由\(G\)的子群索引。数据是子群\(H\subseteq G\)和一个\(H\) - 等变\(\mathcal{C}\) - 模范畴\(\mathcal{M}\)，其基础\(\mathcal{C}\) - 模范畴是不可分解、正合且半单的；当\(H\)对\(\mathcal{M}\)的单对象可迁作用且一个（从而每个）单对象的稳定子上循环非退化时，它们给出一个纤维函子。通过坦纳卡 - 克莱因重构，这描述了\(\mathcal{C}^G\)作为有限维霍普夫代数表示范畴的实现，对\(\mathcal{C}\)无半单性假设。作为应用，对于奇素数\(p\)，我们确定了\(\mathrm{Rep}(H_p)\)上的纤维函子，其中\(H_p\)表示维度为\(4p^2\)的尼克希奇半单霍普夫代数：当\(p\equiv 3\pmod 4\)时有一个等价类，当\(p\equiv 1\pmod 4\)时有两个。我们还利用分类进行规范，以确定格林和尼克希奇小维度列表中的哪些非点项是半单可因式分解霍普夫代数的表示范畴。

英文摘要

Let $G$ be a finite group acting on a finite tensor category $\mathcal{C}$. We classify fiber functors on the equivariantization $\mathcal{C}^G$ in terms of equivariant exact module categories over $\mathcal{C}$, indexed by subgroups of $G$. The data are a subgroup $H\subseteq G$ and an $H$-equivariant $\mathcal{C}$-module category $\mathcal{M}$ whose underlying $\mathcal{C}$-module category is indecomposable, exact, and semisimple; they give a fiber functor precisely when $H$ acts transitively on the simple objects of $\mathcal{M}$ and the stabilizer cocycle of one, hence every, simple object is non-degenerate. Through Tannaka-Krein reconstruction this describes realizations of $\mathcal{C}^G$ as the representation category of a finite-dimensional Hopf algebra, with no semisimplicity hypothesis on $\mathcal{C}$. As applications, for odd primes $p$ we determine the fiber functors on $\mathrm{Rep}(H_p)$, where $H_p$ denotes Nikshych's semisimple Hopf algebra of dimension $4p^2$: there is one equivalence class if $p\equiv 3\pmod 4$ and two if $p\equiv 1\pmod 4$. We also use the classification for gaugings to determine which non-pointed entries in the small-dimensional list of Green and Nikshych are representation categories of semisimple factorizable Hopf algebras.

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34 pages