Journal of Lie Theory Volume 7 (1997) 101–120 ◯^{C}1997 Heldermann Verlag

The Quantum Double of a (locally) Compact Group

T. H. Koornwinder and N. M. Muller

Communicated by B. Ørsted

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, IR).

Department of Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands thk@wins.uva.nl Institute for Theoretical Physics University of Amsterdam Valckenierstraat 65 1018 XE Amsterdam The Netherlands nmuller@phys.uva.nl

Received June 26, 1996

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag