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

TL;DR

This paper constructs an equivariant spectral triple for the quantum group U_q(2) with complex deformation parameters, computes its K-theory, and demonstrates the nontriviality of its K-homology class, advancing noncommutative geometry of quantum groups.

Contribution

It introduces a new spectral triple for U_q(2) with complex deformation, analyzing its properties and K-theory, which was not previously established.

Findings

K-theory of C(U_q(2)) computed for certain parameters

Constructed an even, 4+-summable spectral triple

Proved the nontriviality of the associated K-homology class

Abstract

Let $q=|q|e^{i\pi\theta},\,\theta\in(-1,1],$ be a nonzero complex number such that $|q|\neq 1$ and consider the compact quantum group $U_q(2)$. For $\theta\notin\mathbb{Q}\setminus\{0,1\}$, we obtain the $K$-theory of the $C^*$-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.