## Tohoku Mathematical Journal 2016 September SECOND SERIES VOL. 68, NO. 3

 Tohoku Math. J. 68 (2016), 377-405

Title CROSSED ACTIONS OF MATCHED PAIRS OF GROUPS ON TENSOR CATEGORIES

Author Sonia Natale

(Received June 9, 2014, revised October 29, 2014)
Abstract. We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category, where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G, \Gamma)$-crossed action. We show that every $(G,\Gamma)$-crossed tensor category $\mathcal{C}$ gives rise to a tensor category $\mathcal{C}^{(G, \Gamma)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep} G \toto \mathcal{C}^{(G, \Gamma)} \toto \mathcal{C}$. We also define the notion of a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-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 $(G, \Gamma)$-crossed tensor category equipped with a $(G, \Gamma)$-braiding, then the tensor category $\mathcal{C}^{(G, \Gamma)}$ is a braided tensor category in a canonical way.

Mathematics Subject Classification. Primary 18D10; Secondary 16T05.

Key words and phrases. Tensor category, exact sequence, matched pair, crossed action, braided tensor category, crossed braiding.