CROSSED ACTIONS OF MATCHED PAIRS OF GROUPS ON TENSOR CATEGORIES

Abstract

We introduce the notion of $(G, \varGamma)$-crossed action on a tensor category, where $(G, \varGamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \varGamma)$-crossed tensor category if it is endowed with a $(G, \varGamma)$-crossed action. We show that every $(G, \varGamma)$-crossed tensor category $\mathcal{C}$ gives rise to a tensor category $\mathcal{C}^{(G, \varGamma)}$ that fits into an exact sequence of tensor categories Rep $G \longrightarrow\mathcal{C}^{(G, \varGamma)} \longrightarrow \mathcal{C}$. We also define the notion of a $(G, \varGamma)$-braiding in a $(G, \varGamma)$-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 $(G, \varGamma)$-crossed tensor category equipped with a $(G, \varGamma)$-braiding, then the tensor category $\mathcal{C}^{(G, \varGamma)}$ is a braided tensor category in a canonical way.