Crossed actions of matched pairs of groups on tensor categories

Abstract

We introduce the notion of $(G, Γ)$-crossed action on a tensor category, where $(G, Γ)$ is a matched pair of finite groups. A tensor category is called a $(G, Γ)$-crossed tensor category if it is endowed with a $(G, Γ)$-crossed action. We show that every $(G, Γ)$-crossed tensor category $\mathcal C$ gives rise to a tensor category $\mathcal C^{(G, Γ)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep G} \to \mathcal C^{(G, Γ)} \to \mathcal C$. We also define the notion of a $(G, Γ)$-braiding in a $(G, Γ)$-crossed tensor category, which is connected with certain set-theoretical solutions of the QYBE. This extends the notion of $G$-crossed braided tensor category due to Turaev. We show that if $\mathcal C$ is a $(G, Γ)$-crossed tensor category equipped with a $(G, Γ)$-braiding, then the tensor category $\mathcal C^{(G, Γ)}$ is a braided tensor category in a canonical way.