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The Lie algebra of a Lie subgroup is a Lie subalgebra

Statement

Assume ${A C}_{ω}$ . If $i : H \to G$ is a Lie-subgroup inclusion, then

$d i_{e} : Lie (H) = T_{e} H ⟶ Lie (G) = T_{e} G$

is injective and identifies $Lie (H)$ with the Lie subalgebra $d i_{e} (T_{e} H)$ of $Lie (G)$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , a finite-dimensional real Lie group $G$ , and a Lie subgroup $i : H \to G$ .

[F1]

A Lie-subgroup inclusion is an injective immersion and a smooth Lie-group homomorphism. Immersed, embedded, and closed Lie subgroups.

[F2]

The differential of an immersion is injective at every point. Immersed, embedded, and closed Lie subgroups.

[F3]

Under ${A C}_{ω}$ , the identity differential of a smooth Lie-group homomorphism is linear and bracket preserving. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Differential of a Lie-group homomorphism is a Lie-algebra homomorphism.

[F4]

A Lie subalgebra is a linear subspace closed under the ambient bracket. Lie subalgebras and ideals.

Proof

technique · identify the tangent algebra with the image of the identity differential

1.1F1F2algebra

Since $i$ is an immersion by [F1], its differential $d i_{e} : T_{e} H \to T_{e} G$ is injective by [F2]. It is therefore a linear isomorphism from $T_{e} H$ onto the linear subspace $h : = d i_{e} (T_{e} H) \subseteq T_{e} G$ .

2.1F1F3step 1.1

Because $i$ is also a smooth Lie-group homomorphism, [F3] gives $d i_{e} ([X, Y]_{H}) = [d i_{e} X, d i_{e} Y]_{G}$ for all $X, Y \in T_{e} H$ . Hence the bracket of any two vectors in $h$ again lies in $h$ .

3.1F3F4step 1.1step 2.1∎

Thus $h$ is a bracket-closed linear subspace of $T_{e} G$ and so is a Lie subalgebra by [F4]. Step 1.1 identifies $Lie (H)$ with it, and step 2.1 shows that this identification respects Lie brackets. This includes the zero-dimensional and full-dimensional cases. The only choice used is the stated ${A C}_{ω}$ inherited by [F3].

The Lie algebra of a Lie subgroup is a Lie subalgebra

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources