Representations of Crossed Product Algebras of Podles Quantum Spheres

K. Schm�dgen ; E. Wagner

Journal of Lie theory, Tome 17 (2007) no. 4, pp. 751-79.

Résumé

\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.

Détail

Classification : 17B37, 81R50,46L87
Mots-clés : Quantum groups, unbounded representations