Crossed actions of matched pairs of groups on tensor categories

TL;DR

This paper introduces a new framework for $(G, \Gamma)$-crossed actions on tensor categories, extending existing concepts and connecting to solutions of the quantum Yang-Baxter equation, with implications for braided tensor categories.

Contribution

It defines $(G, \Gamma)$-crossed actions and braidings on tensor categories, generalizing $G$-crossed categories and linking to set-theoretical solutions of QYBE.

Findings

Every $(G, \\Gamma)$-crossed tensor category yields an associated tensor category with an exact sequence.

Introduces $(G, \\Gamma)$-braiding linked to solutions of the QYBE.

Establishes that a $(G, \\Gamma)$-crossed tensor category with a braiding produces a braided tensor category.

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 $\operatorname{Rep G} \to \mathcal C^{(G, \Gamma)} \to \mathcal C$. 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.