Baaj, Saad 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 C. R. Acad. Sci., Paris, Sér. I 314, No. 13, 1021-1026 (1992). 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)]. Reviewer: N.Andruskiewitsch (Bonn) Cited in 1 ReviewCited in 8 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 46L30 States of selfadjoint operator algebras Keywords:Hopf \(C^*\)-algebra; quantum group \(E_ \mu(2)\); Haar measure; regular representation; Pontryagin dual Citations:Zbl 0743.46080 PDFBibTeX XMLCite \textit{S. Baaj}, C. R. Acad. Sci., Paris, Sér. I 314, No. 13, 1021--1026 (1992; Zbl 0771.17011)