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 C gives rise to a tensor category C (G,Γ) that fits into an exact sequence of tensor categories Rep G −→ C(G,Γ) −→ 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 C is a (G, Γ)-crossed tensor category equipped with a (G, Γ)- braiding, then the tensor category C (G,Γ) is a braided tensor category in a canonical way.