A ] 2 1 Ju l 2 00 3 Representations of cross product algebras of Podleś quantum spheres

Hilbert space representations of the cross product ∗-algebras of the Hopf ∗-algebra Uq(su2) and its module ∗-algebras O(Sqr) of Podleś 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(SUq(2)) of the quantum group SUq(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 Uq(su2) are given. The invariant state h on O(Sqr) is studied in detail. By passing to function algebras over the quantum spheres Sqr, we give chart descriptions of quantum line bundles and describe the representations from the first approach by means of the second approach.