Equivariant Spectral Triples for Homogeneous Spaces of the Compact Quantum Group Uq(2)

In this article, we study homogeneous spaces U-q(2)/(phi) T and U-q(2)/(psi) T of the compact quantum group U-q (2), q is an element of C \ {0}. The homogeneous space U-q(2)/(phi) T is shown to be the braided quantum group SUq (2). The homogeneous space U-q(2)/(psi) T is established as a universal C*-algebra given by a finite set of generators and relations. Its K-groups are computed and two families of finitely summable odd spectral triples, one is U-q (2)-equivariant and the other is T-2-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the K-homology group K-1 (C (U-q(2)/(psi) T)).