Lie algebra of a subgroup lemma

A Lie subgroup has Lie algebra equal to its tangent space at the identity, viewed as a Lie subalgebra.

Lie algebra of a subgroup lemma

Let $G$ be a Lie group and let $H\subseteq G$ be a Lie subgroup with inclusion map $\iota:H\hookrightarrow G$ .

Lemma

The differential at the identity

(d\iota)_e : \operatorname{Lie}(H)\to \operatorname{Lie}(G)

is injective, and its image identifies $\operatorname{Lie}(H)$ with the subspace $T_eH\subseteq T_eG$ . Under this identification, $\operatorname{Lie}(H)$ is a Lie subalgebra of $\operatorname{Lie}(G)$ .

Equivalently: the Lie algebra of a Lie subgroup is its tangent space at the identity, with the bracket inherited from $\operatorname{Lie}(G)$ .

Context

When $H$ is a closed subgroup, the closed subgroup theorem guarantees that $H$ is an embedded Lie subgroup, so $T_eH$ is literally a subspace of $T_eG$ in the usual embedded-submanifold sense. In that case, this lemma is the key step behind the subgroup–subalgebra bridge used in the Lie correspondence .