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

In this article, we study homogeneous spaces \(U_q(2)/_\phi \mathbb {T}\) and \(U_q(2)/_\psi \mathbb {T}\) of the compact quantum group \(U_q(2),\,q\in {\mathbb {C}}\setminus \{0\}\). The homogeneous space \(U_q(2)/_\phi \mathbb {T}\) is shown to be the braided quantum group \(SU_q(2)\). The homogeneous space \(U_q(2)/_\psi \mathbb {T}\) is established as a universal \(C^*\)-algebra given by a finite set of generators and relations. Its \({\mathcal {K}}\)-groups are computed and two families of finitely summable odd spectral triples, one is \(U_q(2)\)-equivariant and the other is \(\mathbb {T}^2\)-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 \({\mathcal {K}}\)-homology group \(K^1(C(U_q(2)/_\psi \mathbb {T}))\).