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Register a new userSome properties of group-theoretical categories
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We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category.
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We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs).
We show any slightly degenerate weakly group-theoretical fusion category admits a minimal extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $mathcal{C}$, assume that $text{FPdim}(mathcal{C})=nd$ and $(n,d)=1$. If $(text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $mathcal{C}$, then we show that $mathcal{C}$ contains a non-degenerate fusion subcategory $mathcal{C}(mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $mathcal{C}subseteq text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.
The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum--Weber, 2015) and skew categories of partitions (more general; due to Maassen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph.
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${mathcal C}$ to be equivalent. This concludes the classification of such module categories.
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