Closed-subgroup theorem
[1][2][3] One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan,[4] who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.
Now let H be an arbitrary closed subgroup of G. It is necessary to show that H is a smooth embedded submanifold of G. The first step is to identify something that could be the Lie algebra of H, that is, the tangent space of H at the identity.
The challenge is that H is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space.
must be big enough to capture some interesting information about H. If, for example, H were some large subgroup of G but
In these coordinates, the lemma says that X corresponds to a point in H precisely if X belongs to
Thus, we have exhibited a "slice coordinate system" in which H ⊂ G looks locally like Rk ⊂ Rn, which is the condition for an embedded submanifold.
[9] It is worth noting that Rossmann shows that for any subgroup H of G (not necessarily closed), the Lie algebra
[10] Rossmann then goes on to introduce coordinates[11] on H that make the identity component of H into a Lie group.
[12] In the relative topology, a small open subset of H is composed of infinitely many almost parallel line segments on the surface of the torus.
In the group topology, the small open sets are single line segments on the surface of the torus and H is locally path connected.
The example shows that for some groups H one can find points in an arbitrarily small neighborhood U in the relative topology τr of the identity that are exponentials of elements of h, yet they cannot be connected to the identity with a path staying in U.
While the map exp : h → (H, τr) is an analytic bijection, its inverse is not continuous.
That is, if U ⊂ h corresponds to a small open interval −ε < θ < ε, there is no open V ⊂ (H, τr) with log(V) ⊂ U due to the appearance of the sets V. However, with the group topology τg, (H, τg) is a Lie group.
There are also examples of groups H for which one can find points in an arbitrarily small neighborhood (in the relative topology) of the identity that are not exponentials of elements of h.[14] For closed subgroups this is not the case as the proof below of the theorem shows.
Because of the conclusion of the theorem, some authors chose to define linear Lie groups or matrix Lie groups as closed subgroups of GL(n, R) or GL(n, C).
It follows every closed subgroup is an embedded submanifold of GL(n, C)[16] The homogeneous space construction theorem — If H ⊂ G is a closed Lie subgroup, then G/H, the left coset space, has a unique real-analytic manifold structure such that the quotient map π:G → G/H is an analytic submersion.
The closed subgroup theorem now simplifies the hypotheses considerably, a priori widening the class of homogeneous spaces.
A few sufficient conditions for H ⊂ G being closed, hence an embedded Lie group, are given below.
[24] The proof is given for matrix groups with G = GL(n, R) 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.
[5][6] The proof for general G is formally identical,[25] except that elements of the Lie algebra are left invariant vector fields on G and the exponential mapping is the time one flow of the vector field.
We begin by establishing the key lemma stated in the "overview" section above.
Let s = {S ∈ g | (S, T) = 0 ∀T ∈ h}, the orthogonal complement of h. Then g decomposes as the direct sum g = s ⊕ h, so each X ∈ g is uniquely expressed as X = S + T with S ∈ s, T ∈ h. Define a map Φ : g → GL(n, R) by (S, T) ↦ eSeT.
and the pushforward or differential at 0, Φ∗(S, T) = d/dtΦ(tS, tT)|t = 0 is seen to be S + T, i.e. Φ∗ = Id, the identity.
The hypothesis of the inverse function theorem is satisfied with Φ analytic, and thus there are open sets U1 ⊂ g, V1 ⊂ GL(n, R) with 0 ∈ U1 and I ∈ V1 such that Φ is a real-analytic bijection from U1 to V1 with analytic inverse.
It remains to show that U1 and V1 contain open sets U and V such that the conclusion of the theorem holds.
Consider a countable neighborhood basis Β at 0 ∈ g, linearly ordered by reverse inclusion with B1 ⊂ U1.
[a] Suppose for the purpose of obtaining a contradiction that for all i, Φ(Bi) ∩ H contains an element hi that is not on the form hi = eTi, Ti ∈ h. Then, since Φ is a bijection on the Bi, there is a unique sequence Xi = Si + Ti, with 0 ≠ Si ∈ s and Ti ∈ h such that Xi ∈ Bi converging to 0 because Β is a neighborhood basis, with eSieTi = hi.
For j ≥ i, the image in H of Bj under Φ form a neighborhood basis at I.
Furthermore, if h ∈ H, then φ1(h) ∈ h. By fixing a basis for g = h ⊕ s and identifying g with Rn, then in these coordinates φ1(h) = (x1(h), ..., xm(h), 0, ..., 0), where m is the dimension of h. This shows that (eU, φ1) is a slice chart.
