Pseudo-Riemannian Spectral Triples for $\mathrm{SU}(1,1)$

TL;DR

This paper constructs and analyzes a pseudo-Riemannian spectral triple for the group SU(1,1) using harmonic analysis, extending the framework of indefinite spectral triples to this non-compact setting.

Contribution

It introduces a new pseudo-Riemannian spectral triple for SU(1,1) based on Kostant's cubic Dirac operator, expanding the class of indefinite spectral triples.

Findings

The triple satisfies the conditions of a pseudo-Riemannian spectral triple.

It also qualifies as an indefinite spectral triple in the sense of Van den Dungen and Rennie.

The construction leverages harmonic analysis on SU(1,1).

Abstract

We use the harmonic analysis of $\mathrm{SU}(1,1)$ to show that the triple $(\mathcal{A},\mathcal{H},D)$, with $D$ (the closure of) Kostant's cubic Dirac operator acting on the Hilbert space $\mathcal{H}=L^2(\mathrm{SU}(1,1))\otimes\mathbb{C}^2$, and with $*$-algebra $\mathcal{A}=C^\infty_c(\mathrm{SU}(1,1))\otimes 1$, forms both a pseudo-Riemannian spectral triple in the sense of Van den Dungen, Paschke and Rennie, and an indefinite spectral triple in the sense of Van den Dungen and Rennie.