Representations of cross product algebras of Podles quantum spheres

Abstract

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.