Crossed pointed categories and their equivariantizations

Abstract

We propose a notion, quasiabelian third cohomology of crossed modules, which generalizes Eilenberg and Mac Lane's abelian and Ospel's quasiabelian cohomology. We 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. We give a criterion for these categories to be modular. We describe the quasitriangular quasi-Hopf algebras underlying these categories. IntroductionTuraev's notion [2000; 2008] of a crossed category (short for braided group-crossed category) has attracted much attention recently [Drinfeld et al. 2010;Kirillov 2001a;2001b;Müger 2004;. Roughly, a crossed category consists of a group G, a G-graded tensor category Ꮿ, an action g → T g of G on Ꮿ by tensor autoequivalences, and G-braidings c(X, Y ) : X ⊗ Y ∼ − → T g (Y ) ⊗ X for X, Y ∈ Ꮿ, satisfying certain compatibility conditions. Crossed categories are known to arise in various contexts; for instance, Müger [2004] showed that Galois extensions of braided tensor categories have a natural structure of crossed categories. In [2005], Müger established a connection between 1-dimensional quantum field theories and crossed categories. Kirillov [2001b] showed that crossed categories arise in the theory of vertex operator algebras.A fusion category is said to be pointed if all its simple objects are invertible. One of the goals of this paper is to classify all crossed pointed categories. From [Joyal and Street 1993], it is known that braided pointed categories are classified by Eilenberg and Mac Lane's abelian cohomology H 3 ab (A, ‫ދ‬ × ), where A is a finite abelian group. On the other hand, certain crossed pointed categories in which the group action is strict were described by Turaev [2000;2008] in terms of Ospel's quasiabelian cohomology H 3 qa (G, ‫ދ‬ × ), where G is a (not necessarily abelian) finite MSC2000: primary 18D10; secondary 16W30.

On the Equivalence of Module Categories over a Group-Theoretical Fusion Category

“…The equivariantization C G with respect to this action is equivalent to the category of finite-dimensional representations of the twisted quantum double D ω G (see [5,Lemma 6.3…”

On Gauging Symmetry of Modular Categories

“…By [25,Theorem 5.3] every braided group-theoretical fusion category B can be obtained as a gauging of a pointed G-crossed braided fusion category C. The pair (G, Inv(C)) is an ordinary crossed module, where the G-action on X is induced by the G-action on C and the morphism ∂ : X → G is defined by the G-grading.…”

Modular categories from finite crossed modules

“…The category M(X) associated to a crossed module is known to be modular, iff the boundary map ∂ is an isomorphism [13,Proposition 5.6]. In this case, it is equivalent to the representation category of the Drinfeld double of a finite group.…”

“…For a proof of this assertion that does not directly use Bruigières' criterion, we refer to [13,Proposition 5.6]. …”

“…An important example of a symmetric special Frobenius algebra in the symmetric tensor category of k[G]-modules is the algebra of functions k(G) on G, the regular representation. (13) in which the regular representation ρ R : G(X) → Aut(G(X)) is given by ρ R (χ , c)(χ ,c) := (χ˜cχ , cc). (14) It is convenient to perform a partial Fourier transform with respect to K to introduce also the basis…”