Representations of cross product algebras of Podles quantum spheres
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Hilbert space representations of the cross product *-algebras of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qr}) of Podles spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf *-algebra O(SU_q(2)) of the quantum group SU_q(2) decomposes into an orthogonal sum of projective Hopf modules corresponding to irreducible integrable *-representations of the cross product algebras and that each irreducible integrable *-representation appears with multiplicity one. The projections of these projective modules are computed. The decompositions of tensor products of irreducible integrable *-representations with spin l representations of U_q(su_2) are given. The invariant state h on O(S^2_{qr}) is studied in detail. By passing to function algebras over the quantum spheres 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.
50 pages, references updated, typos corrected
50 pages, references updated, typos corrected