On the Brauer–Picard groups of fusion categories

Abstract

Abstract. We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.

Equivalences of graded fusion categories

“…For any fusion category C, there is an important invariant BrPic(C), the Brauer-Picard 3category of C. This invariant is defined as the 3-group of invertible bimodules, bimodule equivalences, and bimodule functor natural isomorphisms. The 3-category BrPic(C) has many important applications to various areas of mathematics, and thus there exists significant literature dedicated to computing BrPic(C) for various examples [4,2,1,12,10].…”

Homomorphisms and Rigid Isomorphisms of Twisted Group Doubles

“…Nikshych and Riepel [24] gave a procedure that can, in principle, allow the computation of Aut br (Rep(D(G))) for a given group G, but this remains an ad hoc procedure. More recently, and in a similar vein, Marshall and Nikshych [20] introduced several other methods for studying the Brauer-Picard group of Vec ω G , and applied their methods to completely describe the Brauer-Picard groups associated to certain classes of fusion categories of prime power dimension. The case with G abelian and trivial 3-cocycle was considered in [9], and the special case of the (unique) non-abelian group of order p 3 and exponent p was handled by Riepel [25].…”

The Brauer–Picard groups of fusion categories coming from the ADE subfactors

“…Given a fusion category C the results of [8] allow us to classify G-graded extensions of C. This main ingredient of such a classification is BrPic(C), the group of Morita auto-equivalences of C. As well as being useful in classification problems this group also appears in the study of subfactors. If C is unitary then BrPic(C) classifies all subfactors whose even and dual even parts are both C. The process of computing Brauer-Picard groups of fusion categories is currently receiving attention in the literature by both researchers interested in subfactors [12,13], and fusion categories [31,25,4].…”