The twisted Drinfeld double of a finite group via gerbes and finite groupoids

Willerton, Simon

doi:10.2140/agt.2008.8.1419

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The twisted Drinfeld double of a finite group via gerbes and finite groupoids

Simon Willerton

Algebraic & Geometric Topology 8 (2008) 1419–1457

DOI: 10.2140/agt.2008.8.1419

Abstract

The twisted Drinfeld double (or quasi-quantum double) of a finite group with a $3$ –cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the $3$ –cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters.

This is all motivated by gerbes and $3$ –dimensional quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the “space of sections” associated to a transgressed gerbe over the loop groupoid.

Keywords

Dijkgraaf–Witten theory, quantum double, transgression

Mathematical Subject Classification 2000

Primary: 57R56

Secondary: 16W30, 18B40

References

Publication

Received: 19 December 2006

Accepted: 10 July 2008

Published: 3 September 2008

Authors

Simon Willerton
Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield, S3 7RH UK

Volume 8, issue 3 (2008)