CHAPTER III - Lie groupoids and Lie algebroids

Summary

The philosophy behind this chapter is that Lie groupoids and Lie algebroids are much like Lie groups and Lie algebras, even with respect to those phenomena – such as connection theory – which have no parallel in the case of Lie groups and Lie algebras.

We begin therefore with an introductory section, §1, which treats the differentiable versions of the theory of topological groupoids, as developed in Chapter II, §§1-6. Note that a Lie groupoid is a differentiable groupoid which is locally trivial. Most care has to be paid to the question of the submanifold structure on the transitivity components, and on subgroupoids.

§2 introduces Lie algebroids, as briefly as is possible preparatory to the construction in §3 of the Lie algebroid of a differentiable groupoid. The construction given in §3 is presented so as to emphasize that it is a natural generalization of the construction of the Lie algebra of a Lie group. One difference that might appear arbitrary is that we use right-invariant vector fields to define the Lie algebroid bracket, rather than the left-invariant fields which are standard in Lie group theory. This is for compatibility with principal bundle theory, where it is universal to take the group action to be a right action.

In §4 we construct the exponential map of a differentiable groupoid, and give the groupoid version of the standard formulas relating the adjoint maps and the exponential.