Représentation régulière du groupe quantique \(E_{\mu}(2)\) de Woronowicz. (Regular representation of the Woronowicz quantum group \(E_{\mu}(2)\)). (French. Abridged English version) Zbl 0771.17011

The aim of this paper is to provide a formula for the Haar measure on the quantum group \(E_ \mu(2)\) introduced by S. L. Woronowicz [Commun. Math. Phys. 136, 399-432 (1991; Zbl 0743.46080)]. The Haar measure \(\Phi\) is expressed as a sum of positive forms on the Hopf \(C^*\)-algebra of all “continuous functions on \(E_ \mu(2)\) vanishing at infinity”. It allows to deduce the regular representation of the Hopf \(C^*\)-algebra of all “continuous functions on the Pontryagin dual of \(E_ \mu(2)\) vanishing at infinity”. The Haar measure on this last quantum group, as well as in the quantum double of \(E_ \mu(2)\) are also computed. It is also shown that the modular theory of the Haar measures on the quantum groups \(E_ \mu(2)\) and its Pontryagin dual satisfy the conjecture in [G. Skandalis, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 997-1009 (1991)].


17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L30 States of selfadjoint operator algebras


Zbl 0743.46080