the connection between Lie groups and Lie algebras
Given a finite dimensional Lie group , it has an associated Lie algebra . The Lie algebra encodes a great deal of information about the Lie group. I’ve collected a few results on this topic:
(Existence) Let be a finite dimensional Lie algebra over or . Then there exists a finite dimensional real or complex Lie group with .
Even more important, is the fact that the correspondence is functorial: given a homomorphism of Lie groups, there is natural homomorphism defined on Lie algebras , which just the derivative of the map 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:
(Existence) Let be a homomorphism of Lie algebras. Then if is the unique connected, simply-connected group with Lie algebra , and is any Lie group with Lie algebra , there exists a homomorphism of Lie groups with .
(Uniqueness) Let be connected Lie group and an arbitrary Lie group. Then if two maps induce the same maps on Lie algebras, then they are equal.
Essentially, what these theorems tell us is the correspondence from Lie algebras to simply-connected Lie groups is functorial, and right adjoint (http://planetmath.org/AdjointFunctor) to the functor from Lie groups to Lie algebras.
|Title||the connection between Lie groups and Lie algebras|
|Date of creation||2013-03-22 13:20:56|
|Last modified on||2013-03-22 13:20:56|
|Last modified by||bwebste (988)|