Algebraic constructions in the category of lie algebroids

Abstract

The aim of this paper is to give the first general and abstract treatment of the algebraic properties of Lie algebroids. The concept of Lie algebroid was introduced in 1967 by Pradines [lS], as the basic infinitesimal invariant of a differentiable groupoid; the construction of the Lie algebroid of a differentiable groupoid is a direct generalization of the construction of the Lie algebra of a Lie group, and [ 183 described a full Lie theory for differentiable groupoids and Lie algebroids, encompassing many phenomena in the foundations of differential geometry. (A more detailed account and further references are given in [13].) However, at this stage the algebraic properties of Lie algebroids were not pursued. In [ 13, III, Section 2, IV, Section 11, one of us gave a fairly detailed account of the abstract algebra of Lie algebroids over a fixed base: Lie algebroids are vector bundles with several additional structures, and a category of Lie algebroids on a given base, and morphisms which preserve that base, has properties similar to those of the category of Lie algebras. (Using this algebra, it was demonstrated in [13] that the Lie theory of a category of locally trivial differentiable ( = Lie) groupoids over a fixed base, and the corresponding category of (transitive) Lie algebroids over that base, is coextensive with the standard theory of connections in principal bundles.) We now propose to show that the Lie functor from the category of all differentiable groupoids (over arbitrary bases) and arbitrary smooth morphisms, to the category of all Lie algebroids, preserves the basic algebraic constructions known to be possible in the category of (differentiable) groupoids; the main work is to give abstract formulations of the corresponding Lie algebroid concepts. The groupoid results with which we