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 an SL(n, C)-vector bundle. Gauge transformations are also considered in this comparison. (C) 2011 Elsevier B.V. All rights reserved.