Extensions of tensor categories by finite group fusion categories

Abstract

We study exact sequences of finite tensor categories of the form $\Rep G \to \C \to \D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $\Gamma$ and mutual actions by permutations $\rhd: \Gamma \times G \to G$ and $\lhd: \Gamma \times G \to \Gamma$ that make $(G, \Gamma)$ into matched pair of groups endowed with a natural crossed action on $\D$ such that $\C$ is equivalent to a certain associated crossed extension $\D^{(G, \Gamma)}$ of $\D$. Dually, we show that an exact sequence of finite tensor categories $\vect_G \to \C \to \D$ induces an $\Aut(G)$-grading on $\C$ whose neutral homogeneous component is a $(Z(G), \Gamma)$-crossed extension of a tensor subcategory of $\D$. As an application we prove that such extensions $\C$ of $\D$ are weakly group-theoretical fusion categories if and only if $\D$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.