CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Lie algebras determine connected Lie groups only locally

Statement

Assume countable choice. Connected real Lie groups with isomorphic Lie algebras have isomorphic identity neighborhoods as local Lie groups, but may differ globally through distinct discrete central quotients of the same simply connected integration.

Facts & Assumptions

Given: Countable choice and connected groups $G_{1}, G_{2}$ with isomorphic Lie algebras.

[L1]

Their simply connected integrations are isomorphic (Equivalence of simply connected Lie groups and real Lie algebras).

[L2]

Each connected group is a discrete central quotient of that integration (Connected Lie groups are central quotients of simply connected integrations).

Proof

technique · compare covering charts

1.1

Use [L1] to identify the two simply connected covers with one group $\tilde{G}$ . By [L2], write $G_{i} ≅ \tilde{G} / Γ_{i}$ for discrete central subgroups $Γ_{i}$ . Choose an identity neighborhood in $\tilde{G}$ meeting neither nonidentity kernel; both quotient maps restrict there to diffeomorphisms onto identity neighborhoods. Their composition is a local Lie-group isomorphism.

L1L2algebra

2.1

Nothing forces $Γ_{1} = Γ_{2}$ , so the global quotients need not be isomorphic; the line/circle and $S U (2) / S O (3)$ companion examples give exact witnesses. If both kernels are trivial, the groups are globally isomorphic. For the zero algebra all connected integrations are the one-point group. Countable choice is inherited exactly through [L1]–[L2].

L1L2step 1.1∎

Depends on

Used by

The circle and line have the same Lie algebra but different Lie groups Counterexample
SU(2) and SO(3): same local Lie theory, different groups Example

Dependency tree · two levels

8 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

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