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the connection between Lie groups and Lie algebras
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(Definition)
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Given a finite dimensional Lie group $G$ , it has an associated Lie algebra $\fr g=\Lie(G)$ . 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 $\fr g$ be a finite dimensional Lie algebra over $\R$ or $\C$ . Then there exists a finite dimensional real or complex Lie group $G$ with $\Lie(G)=\fr g$ .
Even more important, is the fact that the correspondence $G\mapsto\fr g$ is functorial: given a homomorphism $\vp:G\to H$ of Lie groups, there is natural homomorphism defined on Lie algebras $\vp_*:\fr g\to\fr h$ , which just the derivative of the map $\vp$ 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 $\psi:\fr g\to\fr h$ be a homomorphism of Lie algebras. Then if $G$ is the unique connected, simply-connected group with Lie algebra $\fr g$ , and $H$ is any Lie group with Lie algebra $\fr h$ , there exists a homomorphism of Lie groups $\vp:G\to H$ with $\vp_*=\psi$ .
Theorem 4 (Uniqueness) Let $G$ be connected Lie group and $H$ an arbitrary Lie group. Then if two maps $\vp,\vp':G\to H$ induce the same maps on Lie algebras, then they are equal.
Essentially, what these theorems tell us is the correspondence $\fr g\mapsto G$ from Lie algebras to simply-connected Lie groups is functorial, and right adjoint to the functor $H\mapsto\Lie(H)$ from Lie groups to Lie algebras.
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"the connection between Lie groups and Lie algebras" is owned by bwebste. [ full author list (2) ]
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Cross-references: functor, right, induce, group, theorems, tangent space, identity, map, derivative, natural homomorphism, homomorphism, even, subgroup, discrete, quotient, finite-dimensional, connected, complex, real, information, Lie algebra, Lie group, finite dimensional
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This is version 7 of the connection between Lie groups and Lie algebras, born on 2003-01-02, modified 2004-11-03.
Object id is 3867, canonical name is ConnectionBetweenLieGroupsAndLieAlgebras.
Accessed 3235 times total.
Classification:
| AMS MSC: | 22E60 (Topological groups, Lie groups :: Lie groups :: Lie algebras of Lie groups) |
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Pending Errata and Addenda
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