A twisted spectral triple for quantum SU(2)

TL;DR

This paper develops a twisted spectral triple framework for quantum SU(2), revealing novel properties such as bounded twisted commutators and a meromorphic zeta function, advancing noncommutative geometry of quantum groups.

Contribution

It introduces a new twisted spectral triple for quantum SU(2) with unique boundedness and compactness properties, and analyzes its spectral and cohomological features.

Findings

Twisted commutators are bounded, unlike in classical spectral triples.

The resolvent is compact with respect to a trace on a semifinite von Neumann algebra.

The zeta function admits a meromorphic continuation to the entire complex plane.

Abstract

We initiate the study of a q-deformed geometry for quantum SU(2). In contrast with the usual properties of a spectral triple, we get that only twisted commutators between algebra elements and our Dirac operator are bounded. Furthermore, the resolvent only becomes compact when measured with respect to a trace on a semifinite von Neumann algebra which does not contain the quantum group. We show that the zeta function at the identity has a meromorphic continuation to the whole complex plane and that a large family of local Hochschild cocycles associated with our twisted spectral triple are twisted coboundaries.