Cartan's theorem - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, three results in Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

theory are called Cartan's theorem, named after Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

1. The theorem that for a Lie group G, any closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

is a Lie subgroup

Lie subgroup

In mathematics, a Lie subgroup H of a Lie group G is a Lie group that is a subset of G and such that the inclusion map from H to G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie...

2. A theorem on highest weight vectors in the representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

of a semisimple Lie group.

3. The equivalence between the category of connected real Lie groups and finite dimensional real Lie algebras is called usually (in the literature of the second half of 20th century) Cartan's or Cartan-Lie theorem as it is proved by Élie Cartan whereas S. Lie has proved earlier just the infinitesimal version (local solvability of Maurer-Cartan equations (see Maurer-Cartan form

Maurer-Cartan form

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...

) or the equivalence between the finite dimensional Lie algebras and the category of local Lie groups). Lie listed his results as 3 direct and 3 converse theorems, the infinitesimal variant of Cartan's theorem was essentially his 3rd converse theorem, hence Serre has called it in an influential book, the "third Lie theorem", the name which is historically somewhat misleading, but more often used in the recent decade in the connection to many generalizations.