In this final chapter we will provide a brief introduction to the theory of Lie algebroids.
Lie algebroids arise naturally as the infinitesimal parts of Lie groupoids, in complete analogy to the way that Lie algebras arise as the in-finitesimal part of Lie groups. Once isolated, the concept of a Lie algebroid turns out to be a very natural one, which unifies various different types of infinitesimal structure. For example, foliated manifolds, Poisson manifolds, infinitesimal actions of Lie algebras on manifolds, and many other structures can be naturally viewed as Lie algebroids. In this way, Lie algebroids connect various themes of this book: Lie groupoids and foliations provide examples of Lie algebroids, while conversely, we will see that the basic theory of foliations which has been developed in earlier chapters can be applied to prove some of the basic structure theorems about Lie algebroids.
The plan of this chapter is as follows. In the first section, we will isolate the infinitesimal part of a given Lie groupoid, as an important way of constructing Lie algebroids. In the next section, we will introduce the abstract notion of a Lie algebroid, and present some basic examples.
The rest of this chapter is devoted to the Lie theory for Lie groupoids and Lie algebroids. The classical correspondence between (finite dimensional) Lie groups and Lie algebras is described by three ‘Lie theorems’. These theorems assert that any connected Lie group can be covered by a simply connected Lie group, that maps from a simply connected Lie group into an arbitrary Lie group correspond exactly to maps between their Lie algebras, and, finally, that any Lie algebra is the Lie algebra of a Lie group.