Lie correspondence

Let $G$ be a Lie group with Lie algebra $\mathfrak g=\operatorname{Lie}(G)$.

Theorem (subgroup–subalgebra correspondence)

1. If $H\subseteq G$ is a connected Lie subgroup , then $\operatorname{Lie}(H)$ is a Lie subalgebra of $\mathfrak g$ (by the Lie algebra of a subgroup lemma ).

2. Conversely, for every Lie subalgebra $\mathfrak h\subseteq \mathfrak g$ there exists a unique connected immersed Lie subgroup $H\subseteq G$ such that

   $$\operatorname{Lie}(H)=\mathfrak h.$$

   This subgroup is often denoted $H=\langle \exp(\mathfrak h)\rangle$, emphasizing that it is generated by exponentials of elements of $\mathfrak h$ via the exponential map .

3. The immersed subgroup $H$ is embedded if and only if it is closed in $G$; this is where closed subgroups are Lie subgroups becomes decisive.

Motivation

This correspondence packages the idea that “connected subgroups are determined infinitesimally.” One often uses it in the form: a Lie subalgebra $\mathfrak h$ determines a unique connected subgroup, and computations can be done in $\mathfrak h$ using the bracket before passing back to $H$ (compare connected subgroups are determined by their Lie algebra ).

It is also a key input in global existence results such as Lie’s third theorem , which asserts that every finite-dimensional Lie algebra integrates to a Lie group.