the connection between Lie groups and Lie algebras

Given a finite dimensional Lie group $G$, it has an associated Lie algebra $𝔤=\mathrm{Lie}(G)$. 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 $\mathrm{g}$ be a finite dimensional Lie algebra over $\mathrm{R}$ or $\mathrm{C}$. Then there exists a finite dimensional real or complex Lie group $G$ with $\mathrm{Lie}\mathit{}\mathrm{(}G\mathrm{)}\mathrm{=}\mathrm{g}$.

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 quotient $G\mathrm{/}\mathrm{\Gamma }$ by a discrete central subgroup $\mathrm{\Gamma }$.

Even more important, is the fact that the correspondence $G\mapsto 𝔤$ is functorial: given a homomorphism $\phi :G\to H$ of Lie groups, there is natural homomorphism defined on Lie algebras ${\phi }_{*}:𝔤\to 𝔥$, which just the derivative of the map $\phi$ at the identity (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 $\psi \mathrm{:}\mathrm{g}\mathrm{\to }\mathrm{h}$ be a homomorphism of Lie algebras. Then if $G$ is the unique connected, simply-connected group with Lie algebra $\mathrm{g}$, and $H$ is any Lie group with Lie algebra $\mathrm{h}$, there exists a homomorphism of Lie groups $\phi \mathrm{:}G\mathrm{\to }H$ with ${\phi }_{\mathrm{*}}\mathrm{=}\psi$.

Theorem 4

(Uniqueness) Let $G$ be connected Lie group and $H$ an arbitrary Lie group. Then if two maps $\phi \mathrm{,}{\phi }^{\mathrm{\prime }}\mathrm{:}G\mathrm{\to }H$ induce the same maps on Lie algebras, then they are equal.

Essentially, what these theorems tell us is the correspondence $𝔤\mapsto G$ from Lie algebras to simply-connected Lie groups is functorial, and right adjoint (http://planetmath.org/AdjointFunctor) to the functor $H\mapsto \mathrm{Lie}(H)$ 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