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Register a new userCrossed pointed categories and their equivariantizations
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Authors
Deepak Naidu
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We propose the notion of quasi-abelian third cohomology of crossed modules, generalizing Eilenberg and MacLanes abelian cohomology and Ospels quasi-abelian cohomology, and classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. A criterion for these categories to be modular is given. We also describe the quasi-triangular quasi-Hopf algebras underlying these categories.
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We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of certain metric data, generalizing the well-known result of Joyal and Street for fusion categories. We study symmetric centers and ribbon structures of pointed braided tensor categories and examine their Drinfeld centers.
We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical finite groups. As an application, we construct new examples of weakly group-theoretical fusion categories.
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.
We develop a method for generating the complete set of basic data under the torsorial actions of $H^2_{[rho]}(G,mathcal{A})$ and $H^3(G,U(1))$ on a $G$-crossed braided tensor category $mathcal{C}_G^times$, where $mathcal{A}$ is the set of invertible simple objects in the braided tensor category $mathcal{C}$. When $mathcal{C}$ is a modular tensor category, the $H^2_{[rho]}(G,mathcal{A})$ and $H^3(G,U(1))$ torsorial action gives a complete generation of possible $G$-crossed extensions, and hence provides a classification. This torsorial classification can be (partially) collapsed by relabeling equivalences that appear when computing the set of $G$-crossed braided extensions of $mathcal{C}$. The torsor method presented here reduces these redundancies by systematizing relabelings by $mathcal{A}$-valued $1$-cochains.
We study actions of pointed Hopf algebras on matrix algebras. Our approach is based on known facts about group gradings of matrix algebras.
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