Lifting spectral triples to noncommutative principal bundles
type: Item
instance of: scholarly article
title: Lifting spectral triples to noncommutative principal bundles
zbMATH Open document ID: 1486.58002
author: Kay Schwieger
author: Stefan Wagner
published in: Advances in Mathematics
publication date: 2022-01-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/2012.15364
review text: In the noncommutative geometry, the notion of a free action of a quantum group on a \(C^{\ast}\)-algebra provides a natural framework for noncommutative principal bundles. Noncommutative principal bundles have various applications in geometry and mathematical physics (see references in the paper). In this paper, the authors restrict their study to Lie group actions. Their purpose is to give a methodical construction of spectral triples on noncommutative principal bundles. More precisely, let \(A\) be a unital \(C^{\ast}\)-algebra and \(G\) a compact Lie groups which aciting freely on \(A\). The authors construct a spectral triple on \(A\) systematically by means of the geometry of the fixed point algebra \(A^{G}\) and \(G\). Its spectral triple also can be constructed on non-trivial noncommutative principale bundles with non-Abelian structure groups of higher dimension. In the last 3 sections (section 6,7,8), the authors compare their construction with examples for crossed products, quantum 4-tori and homogeneous spaces.
reviewed by: Tatsuki Seto
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 7466428
zbMATH Keywords: noncommutative principal bundle
zbMATH Keywords: factor system
zbMATH Keywords: spectral triple
MaRDI profile type: Publication
OpenAlex ID: W3114649403
arXiv ID: 2012.15364
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DOI: 10.1016/J.AIM.2021.108160
description: scientific article