Quantum double of a (locally) compact group

TL;DR

This paper extends the quantum double construction to function algebras on (locally) compact groups, classifies irreducible *-representations, and explores explicit examples like SU(2) and SL(2,R).

Contribution

It generalizes the quantum double construction to (Hopf) algebras of functions on (locally) compact groups and classifies their irreducible *-representations.

Findings

Classified irreducible *-representations using transformation group algebras.

Compared the results for finite groups with existing literature.

Explicitly analyzed examples of SU(2) and SL(2,R).

Abstract

We generalise the quantum double construction of Drinfel'd to the case of the (Hopf) algebra of suitable functions on a compact or locally compact group. We will concentrate on the *-algebra structure of the quantum double. If the conjugacy classes in the group are countably separated, then we classify the irreducible *-representations by using the connection with so-called transformation group algebras. For finite groups, we will compare our description to the result of Dijkgraaf, Pasquier and Roche. Finally we will work out the explicit examples of SU(2) and SL(2,R).