Closed-subgroup theorem


In mathematics, the closed-subgroup theorem is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group, then is an embedded Lie group with the smooth structure 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.

Overview

Let be a Lie group with Lie algebra. Now let be an arbitrary closed subgroup of. Our goal is to show that is a smooth embedded submanifold of. Our 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, we 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 we might hope could be the tangent space of at the identity. For this idea to work, however, we need to know that is 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 to us.
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, we need to show that the following critical lemma holds:
Once this has been established, one can use exponential coordinates on, that is, writing each 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.
It is worth noting that Rossmann shows that for any subgroup of , the Lie algebra of is a Lie subalgebra of. Rossmann then goes on to introduce coordinates on that make the identity component of into a Lie group. It is important to note, however, that the topology on coming from these coordinates is not the subset topology. That it so say, the identity component of is an immersed submanifold of but not an embedded submanifold.
In particular, the lemma stated above does not hold if is not closed.

Example of a non-closed subgroup

For an example of a subgroup that is not an embedded Lie subgroup, consider the torus and an "irrational winding of the torus".
and its subgroup
with irrational. Then is dense in and hence not closed. In the relative topology, a small open subset of is composed of infinitely many almost parallel line segments on the surface of the torus. This means that is not locally path connected. In the group topology, the small open sets are single line segments on the surface of the torus and is locally path connected.
The example shows that for some groups one can find points in an arbitrarily small neighborhood in the relative topology of the identity that are exponentials of elements of, yet they cannot be connected to the identity with a path staying in. The group is not a Lie group. While the map is an analytic bijection, its inverse is not continuous. That is, if corresponds to a small open interval, there is no open with due to the appearance of the sets. However, with the group topology, is a Lie group. With this topology the injection is an analytic injective immersion, but not a homeomorphism, hence not an embedding. There are also examples of groups for which one can find points in an arbitrarily small neighborhood of the identity that are not exponentials of elements of. For closed subgroups this is not the case as the proof below of the theorem shows.

Applications

Because of the conclusion of the theorem, some authors chose to define linear Lie groups or matrix Lie groups as closed subgroups of or. In this setting, one proves that every element of the group sufficiently close to the identity is the exponential of an element of the Lie algebra. It follows every closed subgroup is an embedded submanifold of
The homogeneous space construction theorem states
The closed subgroup theorem now simplifies the hypotheses considerably, a priori widening the class of homogeneous spaces. Every closed subgroup yields a homogeneous space.
In a similar way, the closed subgroup theorem simplifies the hypothesis in the following theorem.

Conditions for being closed

A few sufficient conditions for being closed, hence an embedded Lie group, is given below.
An embedded Lie subgroup is closed so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, is an embedded Lie subgroup if and only if its group topology equals its relative topology.

Proof

The proof is given for matrix groups with for concreteness and relative simplicity, since matrices and their exponential mapping are easier concepts than in the general case. Historically, this case was proven first, by John von Neumann in 1929, and inspired Cartan to prove the full closed subgroup theorem in 1930. The proof for general is formally identical, except that elements of the Lie algebra are left invariant vector fields on and the exponential mapping is the time one flow of the vector field. If with closed in, then is closed in, so the specialization to instead of arbitrary matters little.

Proof of the key lemma

We begin by establishing the key lemma stated in the "overview" section above.
Endow with an inner product, and let be the Lie algebra of defined as. Let, the orthogonal complement of. Then decomposes as the direct sum, so each is uniquely expressed as with.
Define a map by. Expand the exponentials,
and the pushforward or differential at, is seen to be, i.e., the identity. The hypothesis of the inverse function theorem is satisfied with analytic, and thus there are open sets with and such that is a real-analytic bijection from to with analytic inverse. It remains to show that and contain open sets and such that the conclusion of the theorem holds.
Consider a countable neighborhood basis at, linearly ordered by reverse inclusion with. Suppose for the purpose of obtaining a contradiction that for all, contains an element that is not on the form. Then, since is a bijection on the, there is a unique sequence, with and such that converging to because is a neighborhood basis, with. Since and, as well.
Normalize the sequence in,. It takes its values in the unit sphere in and since it is compact, there is a convergent subsequence converging to. The index henceforth refers to this subsequence. It will be shown that. Fix and choose a sequence of integers such that as. For example, such that will do, as → 0. Then
Since is a group, the left hand side is in for all. Since is closed,, hence. This is a contradiction. Hence, for some the sets and satisfy and the exponential restricted to the open set is in analytic bijection with the open set. This proves the lemma.

Proof of the theorem

For, the image in of under form a neighborhood basis at. This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in is analytic, the left and right translates of this neighborhood basis by a group element gives a neighborhood basis at. These bases restricted to gives neighborhood bases at all. The topology generated by these bases is the relative topology. The conclusion is that the relative topology is the same as the group topology.
Next, construct coordinate charts on. First define. This is an analytic bijection with analytic inverse. Furthermore, if, then. By fixing a basis for and identifying with, then in these coordinates, where is the dimension of. This shows that is a slice chart. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in. This shows that is an embedded submanifold of.
Moreover, multiplication, and inversion in are analytic since these operations are analytic in and restriction to a submanifold with the relative topology again yield analytic operations and. But since is embedded, and are analytic as well.