# the connection between Lie groups and Lie algebras

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

###### 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 $G/\Gamma$ by a discrete central subgroup $\Gamma$.

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

###### Theorem 4

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

Essentially, what these theorems tell us is the correspondence $\mathfrak{g}\mapsto G$ 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 TheConnectionBetweenLieGroupsAndLieAlgebras 2013-03-22 13:20:56 2013-03-22 13:20:56 bwebste (988) bwebste (988) 10 bwebste (988) Definition msc 22E60