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Crossed actions of matched pairs of groups on tensor categories. (English) Zbl 1354.18007

Let \((G,\Gamma)\) be a matched pair of finite groups. In this paper, the author introduce the notion of \((G, \Gamma)\)-crossed action on a tensor category. A tensor category \({\mathcal C}\) is called a \((G, \Gamma)\)-crossed tensor category if it is endowed with a \((G, \Gamma)\)-crossed action. Using the notion of Hopf monad, introduced by A. Bruguières and A. Virelizier [Adv. Math. 215, No. 2, 679–733 (2007; Zbl 1168.18002)] (see also the paper of A. Bruguières et al. [Adv. Math. 227, No. 2, 745–800 (2011; Zbl 1233.18002)]), in Theorem 5., from a \((G, \Gamma)\)-crossed tensor category \({\mathcal C}\), we can find a way to produce a new tensor category denoted \({\mathcal C}^{(G, \Gamma)}\). The category, \({\mathcal C}^{(G, \Gamma)}\) is a finite tensor category if and only if the neutral homogeneous component \({\mathcal D} = {\mathcal C}_e\) of the associated \(\Gamma\)-grading is a finite tensor category. On the other hand, \( {\mathcal C}^{(G, \Gamma)}\) is a fusion category if and only if \({\mathcal D}\) is a fusion category and the characteristic of \(k\) does not divide the order of \(G\) (see Proposition 6.2). Moreover, like in the case of an equivariantization under a group action by tensor autoequivalences, the category \({\mathcal C}^{(G, \Gamma)}\) fits into an exact sequence (see Theorem 6.1) \[ \text{Rep} \;G \rightarrow \; {\mathcal C}^{(G, \Gamma)} \;\rightarrow \;{\mathcal C}, \] in the sense of the definition given in joint paper of the author and A. Bruguières [J. Math. Soc. Japan 66, No. 1, 257–287 (2014; Zbl 1311.18009)].
In Section 7 the author introduces the 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 new notion is an extension of the definition of \(G\)-crossed braided tensor category due to V. Turaev [Homotopy quantum field theory. With appendices by Michael Müger and Alexis Virelizier. Zürich: European Mathematical Society (EMS) (2010; Zbl 1243.81016)]. She also proves (see Theorem 7.5) that a \((G, \Gamma)\)-crossed tensor category equipped with a \((G, \Gamma)\)-crossed braiding gives rise to a braided tensor category. Finally, in the last section of the paper, we can find interesting examples of the general theory linked to abelian extensions of Hopf algebras.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16T05 Hopf algebras and their applications
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