Editing Lie algebroid (section)

{{Short description|Infinitesimal version of Lie groupoid}}
In [[mathematics]], a Lie algebroid is a [[vector bundle]] <math>A \rightarrow M</math> together with a [[Lie bracket]] on its space of [[Section (fiber bundle)|sections]] <math>\Gamma(A)</math> and a vector bundle morphism <math>\rho: A \rightarrow TM</math>, satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a [[Lie algebra]].

Lie algebroids play a similar same role in the theory of [[Lie groupoid]]s that Lie algebras play in the theory of [[Lie groups]]: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid.

Lie algebroids were introduced in 1967 by Jean Pradines.<ref name=":0">{{Cite journal|last=Pradines|first=Jean|date=1967|title=Théorie de Lie pour les groupoïdes dif́férentiables. Calcul différentiel dans la caté́gorie des groupoïdes infinitésimaux|url=https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f259.image|journal=C. R. Acad. Sci. Paris|language=fr|volume=264|pages=245–248}}</ref>