Connections on Lie algebroids and on derivation-based noncommutative geometry

TL;DR

This paper explores the relationship between connections on Lie algebroids and derivation-based noncommutative geometry, providing algebraic comparisons through examples like Atiyah algebroids and derivations of endomorphism algebras.

Contribution

It establishes a formal connection between Lie algebroid connections and noncommutative geometric connections using algebraic methods and differential calculus.

Findings

Comparison of connections on Lie algebroids and noncommutative geometry frameworks

Analysis of gauge transformations in both settings

Illustrative examples with Atiyah algebroids and derivation algebras

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 $SL(n, \mathbb{C})$-vector bundle. Gauge transformations are also considered in this comparison.