Lifting spectral triples to noncommutative principal bundles
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.
scientific article