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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

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.

Dijkgraaf–Witten theory, quantum double, transgression
Mathematical Subject Classification 2000
Primary: 57R56
Secondary: 16W30, 18B40
Received: 19 December 2006
Accepted: 10 July 2008
Published: 3 September 2008
Simon Willerton
Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield, S3 7RH