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2016 Crossed actions of matched pairs of groups on tensor categories
Sonia Natale
Tohoku Math. J. (2) 68(3): 377-405 (2016). DOI: 10.2748/tmj/1474652265


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


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Sonia Natale. "Crossed actions of matched pairs of groups on tensor categories." Tohoku Math. J. (2) 68 (3) 377 - 405, 2016.


Received: 9 June 2014; Revised: 29 October 2014; Published: 2016
First available in Project Euclid: 23 September 2016

zbMATH: 1354.18007
MathSciNet: MR3550925
Digital Object Identifier: 10.2748/tmj/1474652265

Primary: 18D10
Secondary: 16T05

Keywords: braided tensor category , crossed action , crossed braiding , exact sequence , matched pair , tensor category

Rights: Copyright © 2016 Tohoku University


Vol.68 • No. 3 • 2016
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