PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exponential map is natural for Lie-group homomorphisms

Statement

Assume ${A C}_{ω}$ . If $F : G \to H$ is a homomorphism of finite-dimensional real Lie groups, then for every $X \in g = T_{e} G$ ,

$F (\exp_{G} X) = \exp_{H} (d F_{e} X) .$

The countable-choice assumption is used exactly through the supplied one-parameter-subgroup/exponential characterization.

Facts & Assumptions

Given: ${A C}_{ω}$ , a Lie-group homomorphism $F : G \to H$ , and $X \in g = T_{e} G$ .

[F1]

${A C}_{ω}$ is countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F2]

A one-parameter subgroup with initial velocity $Y$ is uniquely the curve $t \mapsto \exp (t Y)$ . One-parameter subgroups are exactly exponentials.

[F3]

The map $F$ is smooth, preserves identities, and satisfies $F (g h) = F (g) F (h)$ . Lie-group homomorphism, isomorphism, and automorphism.

[F4]

Differentials of smooth maps obey the chain rule. The chain rule for differentials of smooth maps.

Proof

technique · direct

1.1F2F3

By [F2], $a_{X} (t) = \exp_{G} (t X)$ is a one-parameter subgroup. Since [F3] makes $F$ a smooth group homomorphism, the composite $c (t) = F (a_{X} (t))$ is a one-parameter subgroup of $H$ .

2.1F2F3F4step 1.1

The same characterization [F2] gives $a_{X}^{'} (0) = X$ . Hence the chain rule [F4] yields $c^{'} (0) = d F_{e} (a_{X}^{'} (0)) = d F_{e} X .$

3.1F2step 1.1step 2.1

Apply [F2] in $H$ : the unique one-parameter subgroup with initial velocity $d F_{e} X$ is $t \mapsto \exp_{H} (t d F_{e} X)$ . Steps 1.1--2.1 identify $c$ with this curve. Evaluating at $t = 1$ gives $F (\exp_{G} X) = \exp_{H} (d F_{e} X) .$

4.1F1F2F3F4step 1.1step 2.1step 3.1∎

Both Lie groups are nonempty and boundaryless. Zero-dimensional source or target Lie algebras and dimension one require no change, including $X = 0$ . The one-parameter curves are global, so no endpoint issue occurs. No metric or nondegeneracy condition occurs. The only choice use is the stated ${A C}_{ω}$ , inherited through [F2]; composition with one supplied homomorphism and evaluation at one add no choice. No biconditional is asserted.

Depends on

Used by

Every compact Lie group is a closed matrix group Corollary
Roots of a compact connected Lie group Definition
Peter–Weyl decomposition of L2(SU(2)) Example
Global cartan and iwasawa decompositions hold for every nonlinear cover without modified k False statement
Algebraic root subgroups from root exponentials Lemma
Bruhat double cosets from rank-one multiplication Lemma
Minimal parabolic from one negative simple root Lemma
Projective orbit constructions for G/B and G/Pₐlpha Lemma
Rank-one SL2 homomorphism and Weyl representative Lemma
Rational highest-weight modules from adjoint Plücker vectors Lemma
Adjoint intertwines the exponential map Proposition
Central quotients and intermediate character lattices Proposition
Differentiation and integration of highest weights Proposition
Root and weight lattice sandwich Proposition
A homomorphism from a connected Lie group is determined by its differential at the identity Theorem
Analytic and root-system Weyl groups agree Theorem
Compact connected Lie groups are classified by root data Theorem
Compact roots form a reduced crystallographic root system Theorem
Complete reducibility and the Reynolds operator for a complex reductive group Theorem
Conjugacy of maximal tori Theorem
Continuous homomorphisms between Lie groups are smooth Theorem
Global Cartan decomposition for a connected finite center semisimple Lie group Theorem
Highest weights for compact connected groups Theorem
Reduction in stages for free proper regular actions Theorem
Semisimple compact groups up to isogeny Theorem
The compact Weyl group is finite Theorem
Weyl integration formula Theorem

Dependency tree · two levels

16 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

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)