Lifting spectral triples to noncommutative principal bundles

TL;DR

This paper develops a systematic method to lift spectral triples from fixed point algebras to the entire algebra under a compact Lie group action, enhancing the understanding of noncommutative principal bundles.

Contribution

It introduces a new construction for spectral triples on algebras with group actions, extending geometric insights from fixed point subalgebras to the full algebra.

Findings

Construction aligns with known examples

Provides a framework for noncommutative principal bundles

Enhances geometric interpretation in noncommutative geometry

Abstract

Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.