TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Kernels are closed embedded normal Lie subgroups

Statement

Assume ${A C}_{ω}$ . If $F : G \to H$ is a smooth Lie-group homomorphism, then $K = \ker F$ is a closed embedded normal Lie subgroup and

$Lie (K) = \ker (d F_{e}) .$

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth Lie-group homomorphism $F : G \to H$ .

[A1]

Closed subgroups have unique embedded Lie-subgroup structures under countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Cartan closed subgroup theorem.

[F1]

Lie-group homomorphisms have constant rank, and the constant-rank theorem gives the local form $(u, v) \mapsto (u, 0)$ . Lie-group homomorphisms have constant rank, The constant-rank theorem for manifolds.

Proof

technique · closed subgroup plus the constant-rank normal form

1.1

The identity singleton in $H$ is closed, so $K = F^{- 1} (e_{H})$ is closed. The homomorphism law gives $g K g^{- 1} \subseteq K$ for every $g \in G$ , and applying it to $g^{- 1}$ gives equality. Thus $K$ is a closed normal subgroup, and [A1] gives its unique embedded Lie-subgroup structure.

A1givenalgebra

2.1

Let $r = rank d F_{e}$ . By [F1], the rank is constant. Choose constant-rank charts at $e$ and $e_{H}$ that send these points to zero and in which $F$ is $(u, v) \mapsto (u, 0)$ . In the source chart, the fibre $F^{- 1} (e_{H}) = K$ is locally the slice $u = 0$ , whose tangent space at the origin is exactly the kernel of the displayed linear map. Because [A1] gives $K$ the embedded subspace structure, this slice tangent is $T_{e} K$ . Hence $T_{e} K = \ker d F_{e}$ .

A1F1step 1.1

3.1

By definition $Lie (K) = T_{e} K$ , so step 2.1 proves the formula. The trivial kernel, the constant map, disconnected groups, and ranks zero or full are included. Countable choice is used only through [A1].

A1step 2.1∎

Depends on

Used by

SL(n) as a closed Lie subgroup of GL(n) Example
First-isomorphism factorization for Lie group homomorphisms Proposition
Images are immersed Lie subgroups Theorem

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)