A decomposition of the Brauer–Picard group of the representation category of a finite group

Abstract

Abstract. We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra H and thus the Brauer-Picard group of H-mod. We consider two natural subgroups and a subset as candidates for generators. In this article H is the group algebra of a finite group G. As our main result we prove that any element of the Brauer-Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups G our decomposition reduces to the Bruhat decomposition of the Brauer-Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in 3d-TQFT and group extensions of fusion categories.

Low-dimensional topology, low-dimensional field theory and representation theory

“…Since the Brauer-Picard group of the monoidal category H-mod is isomorphic to the group of braided equivalences of the Drinfeld center Z(H-mod) [11], it is invariant under partial dualization. On the other hand it is not hard to give conditions [22] that ensure the existence of a Hopf isomorphism f : r A (H) This result has an interesting application in three-dimensional topological field theories of Turaev-Viro type. A Turaev-Viro theory is a fully extended oriented three-dimensional topological field theory, which assigns a spherical fusion category A to the point.…”

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

“…The modular tensor category corresponding to the Dijkgraaf-Witten theory with gauge group D and topological action ω ∈ Z 3 (BD; U (1)) is the category of finite-dimensional modules over the ω-twisted Drinfeld double of the group algebra C[D] defined in [DPR90]. The corresponding Brauer-Picard group for ω = 0 is studied in detail in [LP17a]. The more general case of the representation category of Hopf algebras which includes the case of non-trivial ω is studied in [LP17b].…”

On Monoidal Autoequivalences of the Category of Yetter-Drinfeld Modules Over a Group: The Lazy Case

“…In this article we address the case H = kG the group algebra of a finite group G and restrict ourselves to so-called lazy 2-cocycles. We do not provide a complete decomposition, but we achieve partial results, that are however sufficient for our calculation of the corresponding subgroup of the Brauer-Picard group of kG-mod in [LP15].…”

“…These maps together with the following Lemmas are partial result that are needed to provide the full decomposition and are in addition necessary for our application in [LP15].…”