CONICET | Buscador de Institutos y Recursos Humanos

We introduce the notion of $(G, \Gamma)$-crossed action on atensor category, where $(G, \Gamma)$ is a matched pair of finite groups. Atensor category is called a $(G, \Gamma)$-crossed tensor category if it isendowed with a $(G, \Gamma)$-crossed action. We show that every $(G,\Gamma)$-crossed tensor category $\C$ gives rise to a tensorcategory $\C^{(G, \Gamma)}$ that fits into an exact sequence of tensorcategories $\Rep G \toto \C^{(G, \Gamma)} \toto \C$. We also define the notionof a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensor category, whichis connected with certain set-theoretical solutions of the QYBE. This extendsthe notionof $G$-crossed braided tensor category due to Turaev. We show that if $\C$ is a$(G, \Gamma)$-crossed tensor category equipped with a $(G, \Gamma)$-braiding,then the tensor category $\C^{(G, \Gamma)}$ is a braided tensor category in acanonical way.