Connections on Lie algebroids and on derivation-based noncommutative geometry
arXiv:1003.6106 · doi:10.1016/j.geomphys.2011.11.002
Abstract
In this paper we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a suitable differential calculus. Two examples allow this comparison: on the one hand, the Atiyah Lie algebroid of a principal fiber bundle and, on the other hand, the space of derivations of the algebra of endomorphisms of a -vector bundle. Gauge transformations are also considered in this comparison.
v3 : version to be published in Journal of Geometry and Physics
References in corpus (6)
- Gravity and the standard model with neutrino mixing
- Lectures on Integrability of Lie Brackets
- Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
- Differential calculus on a Lie algebroid and Poisson manifolds
- An informal introduction to the ideas and concepts of noncommutative geometry
- Noncommutative generalization of SU(n)-principal fiber bundles: a review
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