Proof — Lie's third theorem (Lie algebra → Lie group)
Statement
Every finite-dimensional Lie algebra over the real numbers is the Lie algebra of some Lie group. Equivalently (modern global form), for each such Lie algebra 𝔤 there exists a simply connected Lie group G whose Lie algebra is isomorphic to 𝔤. Lie established the local integration of a Lie algebra to a local Lie group; the global simply connected statement is the standard package attributed to Lie’s program and later rigorization (Cartan, Ado, van Est, etc.). Proofs use ordinary differential equations for the Maurer–Cartan form and monodromy of structure constants; formalizable in analysis and differential geometry (ZFC).
Formula
∀ finite-dimensional real Lie algebra 𝔤 ∃ simply connected Lie group G Lie(G) ≅ 𝔤
Proof sketch
Classical route: construct a local group from structure constants via Maurer–Cartan equations; extend to a global simply connected group using covering-space / monodromy arguments in the modern treatment. See standard Lie theory texts (e.g. Warner, Helgason, Hall).
Source
Lie, M. S. Theorie der Transformationsgruppen (with Engel), Teubner, Leipzig, 1888–1893. Modern expositions: Hall, Lie Groups, Lie Algebras, and Representations; Warner, Foundations of Differentiable Manifolds and Lie Groups.
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mathematics
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- Theorem: Lie's third theorem (Lie algebra → Lie group)
- Proof: Lie's third theorem (Lie algebra → Lie group)