Connections on Lie algebroids and on derivation-based noncommutative geometry

Serge Lazzarini (1, 2, 3) , Thierry Masson (2, 3, 1)

1 CPT - E2 Géométrie, Physique et Symétries
2 CPT - Centre de Physique Théorique - UMR 6207
3 CPT - Centre de Physique Théorique - UMR 7332

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.

Domaines: Géométrie différentielle [math.DG]

Serge Lazzarini, Thierry Masson. Connections on Lie algebroids and on derivation-based noncommutative geometry. Journal of Geometry and Physics, 2012, 62, pp.387-402. ⟨10.1016/j.geomphys.2011.11.002⟩. ⟨hal-00468913⟩