Lie's third theorem

In mathematics represent the Lie's sets forth the relationship between Lie groups and Lie algebras.

Lie groups and Lie algebras

A Lie group is a differentiable manifold, which also has the structure of a group so that the group join and the inversion are infinitely differentiable.

The Lie algebra of a Lie group is the vector space of left - invariant vector fields with the commutator as Lie bracket. The Lie algebra can be identified in a canonical way with the tangent space at the neutral element of the Lie group:

Lie'sche sets

Set (third Lie'scher set, also set of Lie - Cartan ): For any finite- dimensional real Lie algebra, there is a simply connected Lie group whose Lie algebra is.

Sentence (second sentence Lie'scher ): Be Lie groups with Lie algebras and is simply connected. Then, for every Lie algebra homomorphism with a unique Lie group homomorphism.

Historical and Notes

The first Lie'sche set is a purely local statement, which describes the action of a Lie group on itself in local coordinates as the solution of certain differential equations with analytic coefficients.

The third Lie'sche set was originally proved by Sophus Lie in only a local version quoted here global shape goes back to Elie Cartan.

In the third Lie's theorem one another (simple non-contiguous ) receives, in addition to the simply connected Lie group Lie groups with Lie algebra as a factor group, a discrete subgroup of the center of being.