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Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.
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Let $mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majids automorphism braided group of $mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $mathcal{C}$. We show that the center of $mathcal{C}$ is isomorphic to the category of left $B$-comodules in $mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$operatorname*{Comod}_{mathcal{C}}$ into a direct sum of indecomposable $mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $mathcal{C}$ is graded over a group $Gamma$, a torus-valued 3-cocycle on $Gamma$ can be used to deform the associator of $mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Gamma$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
We study exact sequences of finite tensor categories of the form $Rep G to C to D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $Gamma$ and mutual actions by permutations $rhd: Gamma times G to G$ and $lhd: Gamma times G to Gamma$ that make $(G, Gamma)$ into matched pair of groups endowed with a natural crossed action on $D$ such that $C$ is equivalent to a certain associated crossed extension $D^{(G, Gamma)}$ of $D$. Dually, we show that an exact sequence of finite tensor categories $vect_G to C to D$ induces an $Aut(G)$-grading on $C$ whose neutral homogeneous component is a $(Z(G), Gamma)$-crossed extension of a tensor subcategory of $D$. As an application we prove that such extensions $C$ of $D$ are weakly group-theoretical fusion categories if and only if $D$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.
We show that braidings on a fusion category $mathcal{C}$ correspond to certain fusion subcategories of the center of $mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.
We introduce and study the new notion of an {em exact factorization} $mathcal{B}=mathcal{A}bullet mathcal{C}$ of a fusion category $mathcal{B}$ into a product of two fusion subcategories $mathcal{A},mathcal{C}subseteq mathcal{B}$ of $mathcal{B}$. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied by P. Etingof and the author in cite{EG}. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.
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