Noncommutative Geometry Approach to Principal and Associated Bundles

Details Noncommutative Geometry Approach to Principal and Associated Bundles Noncommutative Geometry Approach to Principal and Associated Bundles

2006-12-31

arxiv.org/abs/math/0701033v2

Mathematics

Differential Geometry

45 pages, 6 figures

Scientific paper

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of C*-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over S^2.

Affiliated with

Also associated with

No associations

Rating

Noncommutative Geometry Approach to Principal and Associated Bundles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Noncommutative Geometry Approach to Principal and Associated Bundles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Noncommutative Geometry Approach to Principal and Associated Bundles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-294097