# the connection between Lie groups and Lie algebras

Given a finite dimensional Lie group^{} $G$, it has an associated Lie algebra $\U0001d524=\mathrm{Lie}(G)$. 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

Even more important, is the fact that the correspondence $G\mapsto \U0001d524$ is functorial: given a homomorphism^{} $\phi :G\to H$ of Lie groups, there is natural
homomorphism^{} defined on Lie algebras ${\phi}_{*}:\U0001d524\to \U0001d525$, 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 $\U0001d524\mapsto G$ 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 |