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Examples of (non-)braided tensor categories. (English) Zbl 1436.18016
Let \(C_2\) be the group with two elements and let \(\mathcal{C} = \mathsf{Comod}(H)\) be the category of finite-dimensional \(H\)-comodules over a supergroup algebra \(H\) (a particular kind of finite-dimensional Hopf algebra). A \(C_2\)-crossed product extension of \(\mathcal{C}\) in the sense of [C. Galindo, J. Algebra 337, No. 1, 233–252 (2011; Zbl 1239.18007)] is a \(C_2\)-grading \(\mathcal{D}=\mathcal{D}_e \oplus \mathcal{D}_g\) on a finite tensor category \(\mathcal{D}\) such that \(\mathcal{D}_e \cong \mathcal{C}\) and each one of \(\mathcal{D}_e\) and \(\mathcal{D}_g\) has at least one invertible object of \(\mathcal{D}\).
In [A. Mejía Castaño and M. Mombelli, Int. J. Math. 26, No. 9, Article ID 1550067, 26 p. (2015; Zbl 1328.18011)], eight examples of non-equivalent \(C_2\)-crossed products which are extensions of \(\mathcal{C}\) were presented and their monoidal structure is described.
The paper under review is devoted to investigate the possible braidings over them and the conclusion is that two out these eight categories can be braided with the braiding coming from \(\mathcal{C}\) and the other six do not admit a braiding at all.
18M05 Monoidal categories, symmetric monoidal categories
18M15 Braided monoidal categories and ribbon categories
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