K -theory and equivariant spectral triple for the quantum group Uq(2) for complex deformation parameters

Let q=|q|ei??,??(?1,1], be a nonzero complex number such that |q|?1 and consider the compact quantum group Uq(2). For ??Q?{0,1}, we obtain the K-theory of the C?-algebra C(Uq(2)). Then, we produce a spectral triple on Uq(2) which is equivariant under its own comultiplication action. The spectral triple obtained here is even, 4+-summable, non-degenerate, and the Dirac operator acts on two copies of the L2-space of Uq(2). The Chern character of the associated Fredholm module is shown to be nontrivial. At the end, we compute the spectral dimension of Uq(2).