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
schmuedgen@math.uni-leipzig.de
Elmar Wagner
Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, 28045 Colima, Mexico
Elmar.Wagner@math.uni-leipzig.de
[Abstract-pdf]
\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
[ Fulltext-pdf (368 KB)] for subscribers only.