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.


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