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Register a new userLagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
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We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfelds characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.
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We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories.
We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $cal B$ (consisting of invertible central $cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.
Let $W$ be a finite dimensional purely odd supervector space over $mathbb{C}$, and let $sRep(W)$ be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup $W$. We show that the set of equivalence classes of finite non-degenerate braided tensor categories $C$ containing $sRep(W)$ as a Lagrangian subcategory is a torsor over the cyclic group $mathbb{Z}/16mathbb{Z}$. In particular, we obtain that there are $8$ non-equivalent such braided tensor categories $C$ which are integral and $8$ which are non-integral.
Given a pair of finite groups $F, G$ and a normalized 3-cocycle $omega$ of $G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension $Bbbk^G_omega#_c,Bbbk F$ where $c$ denotes some suitable cohomological data. When $Frightarrow overline{F}:=F/A$ is a quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type $Bbbk^G_omega#_c, Bbbk Frightarrow Bbbk^G_omega#_{overline{c}} , Bbbk overline{F}$. Our construction is particularly natural when $F=G$ acts on $G$ by conjugation, and $Bbbk^G_omega#_c Bbbk G$ is a twisted quantum double $D^{omega}(G)$. In this case, we give necessary and sufficient conditions that Rep($Bbbk^G_omega#_{overline{c}} , Bbbk overline{G}$) is a modular tensor category.
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.
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