False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Isomorphic Lie algebras determine isomorphic connected Lie groups

Statement refuted

Assume countable choice. Connected real Lie groups with isomorphic Lie algebras are isomorphic as Lie groups.

Facts & Assumptions

Given: ${A C}_{ω}$ and the usual Lie-group structures on the line and circle.

[L1]

Every connected integration is a discrete central quotient of the simply connected integration (Connected Lie groups are central quotients of simply connected integrations).

[L2]

Countable choice is the declared weak-choice assumption (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Counterexample

technique · line versus circle

1.1

The groups $(R, +)$ and $S^{1}$ are connected one-dimensional Lie groups. Each Lie algebra is one-dimensional with zero bracket, so their Lie algebras are isomorphic.

givenalgebra

2.1

The circle is compact, whereas $R$ is not. A Lie-group isomorphism is a homeomorphism and would preserve compactness, so the groups are not isomorphic. Equivalently, they are the quotients $R / 0$ and $R / Z$ from the classification [L1]. This also displays the distinct discrete central kernels. Countable choice is the assumption [L2] used only through [L1]; the compactness witness itself is choice-free.

L1L2step 1.1∎

Depends on

Used by

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Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, §3.8 (standard reference, not scraped)