Connections on Lie algebroids and on derivation-based noncommutative geometry
Copyright policy )Connections on Lie algebroids and on derivation-based noncommutative geometry
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 $SL(n, \mathbb{C})$-vector bundle. Gauge transformations are also considered in this comparison.
v3 : version to be published in Journal of Geometry and Physics
Mathematics - Differential Geometry, Noncommutative geometry (à la Connes), connection, FOS: Physical sciences, Mathematical Physics (math-ph), Yang-Mills and other gauge theories in quantum field theory, Differential Geometry (math.DG), Pseudogroups and differentiable groupoids, differential algebra, FOS: Mathematics, Noncommutative differential geometry, differential geometry, noncommutative geometry, 57Rxx, 58Axx, 53C05 (Primary) 46L87, 81T13 (Secondary), Lie algebroid, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematical Physics, Connections (general theory)
Mathematics - Differential Geometry, Noncommutative geometry (à la Connes), connection, FOS: Physical sciences, Mathematical Physics (math-ph), Yang-Mills and other gauge theories in quantum field theory, Differential Geometry (math.DG), Pseudogroups and differentiable groupoids, differential algebra, FOS: Mathematics, Noncommutative differential geometry, differential geometry, noncommutative geometry, 57Rxx, 58Axx, 53C05 (Primary) 46L87, 81T13 (Secondary), Lie algebroid, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematical Physics, Connections (general theory)
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