Tohoku Mathematical Journal

Crossed actions of matched pairs of groups on tensor categories

Sonia Natale

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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.

Article information

Tohoku Math. J. (2), Volume 68, Number 3 (2016), 377-405.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Secondary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]

Tensor category exact sequence matched pair crossed action braided tensor category crossed braiding


Natale, Sonia. Crossed actions of matched pairs of groups on tensor categories. Tohoku Math. J. (2) 68 (2016), no. 3, 377--405. doi:10.2748/tmj/1474652265.

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