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Journal of Lie Theory 17 (2007), No. 4, 751--790
Copyright Heldermann Verlag 2007

Representations of Crossed Product Algebras of Podles Quantum Spheres

Konrad Schmüdgen
Mathematisches Institut, Universität Leipzig, Johannisgasse 26, 04103 Leipzig, Germany

Elmar Wagner
Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, 28045 Colima, Mexico


\def\cU{{\cal{U}}} \def\cO{{\cal{O}}} Hilbert space representations of the crossed product $\ast$-algebras of the Hopf $\ast$-algebra $\cU_q(\rm{su}_2)$ and its module $\ast$-algebras $\cO(\rm{S}^2_{qr})$ of Podle\'s spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf $\ast$-algebra $\cO(\rm{SU}_q(2))$ of the quantum group $\rm{SU}_q(2)$ decomposes into an orthogonal sum of projective Hopf modules corresponding to irreducible integrable $\ast$-representations of the crossed product algebras and that each irreducible integrable $\ast$-representation appears with multiplicity one. The projections of these projective modules are computed. The decompositions of tensor products of irreducible integrable $\ast$-representations with spin $l$ representations of $\cU_q(\rm{su}_2)$ are given. The invariant state $h$ on $\cO(\rm{S}^2_{qr})$ is studied in detail. By passing to function algebras over the quantum spheres $\rm{S}^2_{qr}$, we give chart descriptions of quantum line bundles and describe the representations from the first approach by means of the second approach.

Keywords: Quantum groups, unbounded representations.

MSC: 17B37, 81R50,46L87

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