
Journal of Lie Theory 23 (2013), No. 1, 229250 Copyright Heldermann Verlag 2013 Induced *Representations and C^{*}Envelopes of Some Quantum *Algebras Philip A. Dowerk MPI für Mathematik, Inselstrasse 22, 04103 Leipzig, Germany dowerk@mis.mpg.de Yurii Savchuk Universität Leipzig, Mathematisches Institut, Johannisgasse 26, 04103 Leipzig, Germany savchuk@math.unileipzig.de [Abstractpdf] We consider three quantum algebras: the $q$oscillator algebra, the Podle\'s sphere and the $q$deformed enveloping algebra of $su(2)$. To each of these $*$algebras we associate a certain partial dynamical system and perform the ``Mackey analysis'' of $*$representations developed by Yu. Savchuk and K. Schm\"udgen [``Unbounded induced representations of $*$algebras'', Algebr. Represent. Theory, DOI: 10.1007/s1046801193106]. As a result we get the description of ``standard'' irreducible $*$representations. Further, for each of these examples we show the existence of a ``$C^*$envelope'' which is canonically isomorphic to the covariance $C^*$algebra of the partial dynamical system. Finally, for the $q$oscillator algebra and the $q$deformed ${\cal U}(su(2))$ we show the existence of ``bad'' representations. Keywords: Induced representations, group graded algebras, wellbehaved representations, partial action of a group, Mackey analysis, C*envelope, qdeformed enveloping algebra, Podles sphere, qoscillator. MSC: 20G42, 47L60, 17B37; 16G99, 22D30, 16W50, 47L65 [ Fulltextpdf (598 KB)] for subscribers only. 