In mathematics, there are two basic results in Lie group theory that go by the name Cartan's theorem. They are both named for Élie Cartan. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a 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 subgroup is a Lie subgroup.

2. A theorem on highest weight vectors in the representation theory of a semisimple Lie group.

See also Cartan's theorems A and B, results of Henri Cartan. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is 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 is a group operation... In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ...