Lie’s third theorem

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\Bbb R$ (or $\Bbb C$).

Theorem (existence and uniqueness up to simply connected cover)

There exists a connected, simply connected Lie group $G$ such that

as Lie algebras (via a Lie algebra isomorphism ).

Moreover, if $G_1$ and $G_2$ are connected simply connected Lie groups with Lie algebras isomorphic to $\mathfrak g$, then $G_1$ and $G_2$ are isomorphic as Lie groups . In other words, the simply connected integration of $\mathfrak g$ is unique up to Lie group isomorphism.

Context

Combined with the fact that every connected Lie group has a universal covering group which is again a Lie group (see existence of universal covering groups ), Lie’s third theorem explains why Lie algebras control the local—and, up to discrete central quotients, the global—structure of connected Lie groups (compare simply connected groups are determined by their Lie algebra ).

It also underlies the Lie correspondence : Lie subalgebras integrate to connected (immersed) Lie subgroups inside a given Lie group.