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Register a new userFusion subcategories of representation categories of twisted quantum doubles of finite groups
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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.
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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.
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 compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $mathfrak{u}_q(mathfrak{g})$, where $q$ is a root of unity.
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.
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.
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