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

In this article, we study homogeneous spaces U<inf>q</inf>(2) / <inf>ϕ</inf>T and U<inf>q</inf>(2) / <inf>ψ</inf>T of the compact quantum group Uq(2),q∈C\{0}. The homogeneous space U<inf>q</inf>(2) / <inf>ϕ</inf>T is shown to be the braided quantum group SU<inf>q</inf>(2). The homogeneous space U<inf>q</inf>(2) / <inf>ψ</inf>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<inf>q</inf>(2) -equivariant and the other is T²-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 K¹(C(U<inf>q</inf>(2) / <inf>ψ</inf>T)).