Equivariant spectral triple for the quantum group $U_q(2)$ for complex deformation parameters

Abstract

Let $q=|q|e^{i\pi \theta },\theta \in (-1,1],$ be a nonzero complex number such that $|q|\ne 1$ and consider the compact quantum group $U_q(2)$. For $\theta \notin \mathbb{Q}\setminus \{0,1\}$, we obtain the $K$-theory of the $C^{\ast }$-algebra $C(U_q(2))$. We construct a spectral triple on $U_q(2)$ 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 $L^2$-space of $U_q(2)$. The $K$-homology class of the associated Fredholm module is shown to be nontrivial.

Title shortened, To appear in Journal of Geometry and Physics