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

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.