Clifford theory for graded fusion categories

Abstract

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant Ce-module categories and extensions of Ce-module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of this modules categories. We associate a G-crossed product fusion category to each G-invariant Cemodule category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.1991 Mathematics Subject Classification. 16W30. 1 PreliminariesThroughout the paper we work over an algebraically closed field k of characteristic 0. All categories considered in this paper are finite, abelian, semisimple, and k-linear. All functors and bifunctors considered in this paper are additive and k-linear.

Morita equivalence methods in classification of fusion categories

“…This grading is universal in the sense that any faithful grading (16) [13]. Namely, G-extensions of a fusion category A correspond to homomorphisms G → BrPic(A), where BrPic(A) is the Brauer-Picard group of A consisting of invertible A-module categories, and certain cohomological data.…”

“…By construction, B is categorically Morita equivalent to A. On the other hand, there is a classification of module categories over a given graded fusion category in terms of module categories over its trivial component [16,24]. So, in principle, weakly group-theoretical fusion categories can be described in terms of finite groups and their cohomology.…”

Generalized and Quasi-Localizations of Braid Group Representations

“…Let M be an indecomposable C-module category, then by the complete unitarity of C e and Theorem 5.12, M is equivalent to a C-module * -category. Moreover, if M and N are Cmodule * -categories equivalent as C-module categories, by [Ga2,Proposition 4.6], Remark 5.14 and [Ga2, Theorem 1.3], M and N are equivalent as C-module *categories and every C-module equivalence is equivalent to a C-module * -functor equivalence.…”

Tensor functors between Morita duals of fusion categories

“…[16, Corollary 6.4] Let G be a finite group and C = x∈G C x be a G-graded fusion category. Let M be an indecomposable C-module category which remains indecomposable as a C e -module category.…”

“…Let C = T Y(Z/pZ, χ, τ ) be a non group-theoretical Tambara-Yamagami category, that is, χ(1, 1) = e 2πk pwhere k ∈ Z/pZ is a quadratic non-residue. It follows by[16, Proposition 5.7] that the only indecomposable C-module category is C itself. Thus, T (C) = {T Y(Z/pZ, χ, τ )} and hence BrPic(T Y(Z/pZ, χ, τ )) = Out ⊗ (T Y(Z/pZχ, τ )).…”