The twisted Drinfeld double of a finite group via gerbes and finite groupoids
type: Item
instance of: scholarly article
title: The twisted Drinfeld double of a finite group via gerbes and finite groupoids
zbMATH Open document ID: 1154.57029
author: Simon Willerton
published in: Algebraic \& Geometric Topology
publication date: 2008-09-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0503266
review text: This goals of this paper are to understand the twisted Drinfeld double of a finite group, and to understand a finite `toy model' for Freed-Hopkins-Teleman's deep result identifying the Grothendieck ring of the category of highest weight representations of a loop group with the twisted equivariant \(K\)-theory of \(G\). It does this by interpreting both in the context of the categorical description of 3-dimensional quantum field theory (in fact Dijkgraaf-Witten theory) in terms of gerbes and groupoids. The twisted Drinfeld double of a group corresponds to a quantum group for the quantum field theory of a finite group twisted by a 3-cocycle. The formal algebraic definition of the twisted Drinfeld double looks mysterious, but by geometrically representing the 3-cocycle as a gerbe it turns out to have a natural interpretation one step up the categorical ladder. First (and this is perhaps the key insight), the paper identifies an analogue of a `loop group' for a finite group \(G\), or rather for its categorification. Let \(\overline{\mathbb{Z}}\) and \(\overline{G}\) (the author considers also more general groupoids \(\mathcal{G}\)) be categories with one object whose sets of morphisms are isomorphic to the integers \(\mathbb{Z}\) and to the group \(G\) correspondingly. Define \(\Lambda \overline{G}\) to be \(Hom(\overline{Z},\overline{G})\). By the Parmesan Theorem (wonderful name!) \(\Lambda \overline{G}\) indeed behaves as one would expect a loop group for \(G\) to behave, namely \(B\Lambda\overline{G}\equiv \mathcal{L}B\overline{G}\) where \(\mathcal{L}\) means to take the loop space. The Drinfeld double of \(G\) twisted by a 3-cocycle \(\omega\) can now be naturally understood as a twisting of \(\mathbb{C}(\Lambda\overline{G})\) by the 2-cocycle \(\tau(\omega)\), where \(\tau\) is the `transgression map' taking \(\omega\) to a 2-cocycle on \(\Lambda\overline{G}\). The paper points out that indeed, this can be taken as the definition of the twisted Drinfeld double. I…
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 18B40
zbMATH DE Number: 5344773
zbMATH Keywords: twisted Drinfeld double
zbMATH Keywords: equivariant K-theory
zbMATH Keywords: gerbe
zbMATH Keywords: groupoid
zbMATH Keywords: loop groupoid
zbMATH Keywords: representation category
zbMATH Keywords: transgression
zbMATH Keywords: Dijkgraaf-Witten theory
zbMATH Keywords: quantum double
MaRDI profile type: Publication
OpenAlex ID: W3098501141
arXiv ID: math/0503266
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DOI: 10.2140/AGT.2008.8.1419
description: scientific article