Closed-subgroup theorem

In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations. Let be a Lie group with Lie algebra . Now let be an arbitrary closed subgroup of . It is necessary to show that is a smooth embedded submanifold of . The first step is to identify something that could be the Lie algebra of , that is, the tangent space of at the identity. The challenge is that is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra" of by the formula It is not difficult to show that is a Lie subalgebra of . In particular, is a subspace of , which one might hope to be the tangent space of at the identity. For this idea to work, however, must be big enough to capture some interesting information about . If, for example, were some large subgroup of but turned out to be zero, would not be helpful. The key step, then, is to show that actually captures all the elements of that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma: Once this has been established, one can use exponential coordinates on , that is, writing each (not necessarily in ) as for . In these coordinates, the lemma says that corresponds to a point in precisely if belongs to . That is to say, in exponential coordinates near the identity, looks like . Since is just a subspace of , this means that is just like , with and . Thus, we have exhibited a "slice coordinate system" in which looks locally like , which is the condition for an embedded submanifold.