the connection between Lie groups and Lie algebras

Given a finite dimensional Lie groupMathworldPlanetmath G, it has an associated Lie algebra 𝔤=Lie⁢(G). The Lie algebra encodes a great deal of information about the Lie group. I’ve collected a few results on this topic:

Theorem 1

(Existence) Let g be a finite dimensional Lie algebra over R or C. Then there exists a finite dimensional real or complex Lie group G with Lie⁢(G)=g.

Theorem 2

(Uniqueness) There is a unique connected simply-connected Lie group G with any given finite-dimensional Lie algebra. Every connected Lie group with this Lie algebra is a quotientPlanetmathPlanetmath G/Γ by a discrete central subgroup Γ.

Even more important, is the fact that the correspondence G↦𝔤 is functorial: given a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath φ:G→H of Lie groups, there is natural homomorphismMathworldPlanetmath defined on Lie algebras φ*:𝔤→𝔥, which just the derivative of the map φ at the identityPlanetmathPlanetmathPlanetmathPlanetmath (since the Lie algebra is canonically identified with the tangent space at the identity).

There are analogous existence and uniqueness theorems for maps:

Theorem 3

(Existence) Let ψ:g→h be a homomorphism of Lie algebras. Then if G is the unique connected, simply-connected group with Lie algebra g, and H is any Lie group with Lie algebra h, there exists a homomorphism of Lie groups φ:G→H with φ*=ψ.

Theorem 4

(Uniqueness) Let G be connected Lie group and H an arbitrary Lie group. Then if two maps φ,φ′:G→H induce the same maps on Lie algebras, then they are equal.

Essentially, what these theorems tell us is the correspondence 𝔤↦G from Lie algebras to simply-connected Lie groups is functorial, and right adjoint ( to the functor H↦Lie⁢(H) from Lie groups to Lie algebras.

Title the connection between Lie groups and Lie algebras
Canonical name TheConnectionBetweenLieGroupsAndLieAlgebras
Date of creation 2013-03-22 13:20:56
Last modified on 2013-03-22 13:20:56
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 10
Author bwebste (988)
Entry type Definition
Classification msc 22E60