Closed subgroup theorem

In mathematics, the closed subgroup 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 relative topology being the same as the group topology.^[1][2] One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan,^[3] who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.^[4]

Informal description

Let G be a Lie group, whose associated Lie algebra under the Lie correspondence is g. If H ⊂ G is any subgroup of G, not necessarily a closed subgroup, then h = {X ∈ g | ∀t ∈ ℝ, exp(tX) ∈ H} is a Lie subalgebra of g; here exp(x) is the exponential map from the Lie algebra to the Lie group. As a consequence of the Lie correspondence, the connected component of H is the closed subgroup of G generated by exp(h).^[5] An open set U ⊂ h that contains the origin is said to be "small enough" when the exponential map has an inverse function log on exp(U) satisfying the following two conditions:

1. For every X in U, log(exp(X)) = X, and
2. For every x in exp(U), exp(log(x)) = x.

Then the sets of the form h exp(U) for group elements h in H and for small enough open sets U ⊂ h form a basis for a topology on H, called the group topology.^[6] Also, the function φ₁ that maps each element h of exp(U) to the element log(h) of h provides coordinates in small enough neighborhoods of the identity of H, and more generally the function φ_g that maps h in g exp(U) to log(g⁻¹h) in h provides smooth coordinates around any element g of H.^[7] These are called exponential coordinates. Their values belong to the Lie algebra h rather than being real numbers, but this is immaterial because h is a vector space so (given a basis for this space) its elements can be expanded into tuples of real-number coordinates. In exponential coordinates the group H with the group topology is a Lie group, i.e. an analytic manifold with analytic group operations with respect to the topology.

Embedded submanifolds are characterized by so-called slice charts on the embedding manifold. If (U, φ) is a slice chart in M on N ⊂ M, then φ(p) = (φ₁(p), …, φ_n(p), 0, …, 0) ∀p ∈ N, where n is the dimension of N. If each p ∈ N is in the domain of a slice chart on N in M, then N is an embedded submanifold.^[8] Embedded submanifolds always have the subspace topology. It is not necessarily true that the subspace topology is the same as the group topology. The group topology is often finer than the relative topology. If this is the case, then H is not an embedded submanifold of G, but may instead be an immersed submanifold.

The closed subgroup theorem provides a sufficient condition for a subgroup to be an embedded Lie subgroup.

Example of a non-closed subgroup

The torus G. Imagine a bent helix laid out on the surface picturing H. If a =  ^p⁄_q in lowest terms, the helix will close up on itself at (1, 1) after p rotations in φ and q rotations in θ. If a is irrational, the helix winds indefinitely.

For an example of a subgroup that is not an embedded Lie subgroup, consider the torus and an "irrational winding of the torus".

$G={\mathbb {T}}^{2}=\left\{\left.\left({\begin{matrix}e^{{2\pi i\theta }}&0\\0&e^{{2\pi i\phi }}\end{matrix}}\right)\right|\theta ,\phi \in {\mathbb {R}}\right\},$

and its subgroup

$H=\left\{\left.\left({\begin{matrix}e^{{2\pi i\theta }}&0\\0&e^{{2\pi ia\theta }}\end{matrix}}\right)\right|\theta \in {\mathbb {R}}\right\}{\text{with Lie algebra }}{\mathfrak {h}}=\left\{\left.\left({\begin{matrix}i\theta &0\\0&ia\theta \end{matrix}}\right)\right|\theta \in {\mathbb {R}}\right\},$

with a irrational. Then H is dense in G and hence not closed.^[9] 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. This means that H is not locally path connected.^[10] 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 τ_r of the identity that are exponentials of elements of h, yet they cannot be connected to the identity with a path staying in U.^[11] The group (H, τ_r) is not a Lie group. While the map exp:h → (H, τ_r) 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, τ_r) with log(V) ⊂ U due to the appearance of the sets V. However, with the group topology τ_g, (H, τ_g) is a Lie group. With this topology the injection ι:(H, τ_g) → G is an analytic injective immersion, but not a homeomorphism, hence not an embedding. 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.^[11] For closed subgroups this is not the case as the proof below of the theorem shows.

Applications

The Homogeneous space construction theorem states

If H ⊂ G is a closed Lie subgroup, then G/H, the left coset space, has a unique analytic manifold structure such that the quotient map π:G → G/H is an analytic submersion. The left action given by g₁ ⋅ (g₂H) = (g₁g₂)H turns G/H into a homogeneous G-space.

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.

If X is a set with transitive group action and the isotropy group or stabilizer of a point x ∈ X is a closed Lie subgroup, then X has a unique smooth manifold structure such that the action is smooth.

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).^[12] In this setting, there is no closed subgroup theorem. There is, on the other hand, a theorem that says that (in the subspace topology) there is an analytical bijection between small enough neighborhoods of the identity in the group and the origin in the Lie algebra respectively.^[13] Its proof is practically identical to the proof of the closed subgroup theorem presented below.

A few sufficient conditions for H ⊂ G being closed, hence an embedded Lie group, is given below.

• All classical groups are closed in GL(F, n), where F = ℝ, ℂ, or ℍ, the quaternions.
• A subgroup that is locally closed is closed.^[14] A subgroup is locally closed if every point has a neighborhood in U ⊂G such that H ∩ U is closed in U.
• If H = AB = {ab| a ∈ A, b ∈ B}, where A is a compact group and B is a closed set, then H is closed.^[15]
• If h ⊂ g is a Lie subalgebra such that for no X ∈ g\h, [X, h] ∈ h, then Γ(h), the group generated by e^h, is closed in G.^[16]
• If X ∈ g, then the one-parameter subgroup generated by X is not closed if and only if X is similar over ℂ to a diagonal matrix with two entries of irrational ratio.^[17]
• Let h ⊂ g be a Lie subalgebra. If there is a simply connected compact group K with k isomorphic to h, then Γ(h) is closed in G.^[18]

Converse

An embedded Lie subgroup H ⊂ G is closed^[19] so a subgroup is an embedded Lie subgroup if and only of it is closed. Equivalently, H is an embedded Lie subgroup if and only if its group topology equals its relative topology.^[20]

Proof

John von Neumann in 1929 proved the theorem in the case of matrix groups as given here. He was prominent in many areas, including quantum mechanics, set theory and the foundations of mathematics.

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.^[4] The proof for general G is formally identical,^[21] 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. If H ⊂ G with G closed in GL(n,ℝ), then H is closed in GL(n, ℝ), so the specialization to GL(n, ℝ) instead of arbitrary G ⊂ GL(n, ℝ) matters little.

The theorem relies on the following all-important lemma based on the inverse function theorem:

Let H be a closed subgroup of GL(n, ℝ). There exist an open neighborhood U of 0 in g = M_n(ℝ) and an open neighborhood V of I in GL(n, ℝ) such that e^{(U ∩ h)} = H ∩ V and exp:U → exp(U) is an analytic bijection onto the open neighborhood H ∩ V of I in H with analytic inverse.^[22]

Endow g with the Hilbert-Schmidt inner product, (X, Y) → Tr(XY), and let h be the Lie algebra of H defined as h = {H ∈ M_n(ℝ) = g|e^tH ∈ H ∀t ∈ ℝ}. Let s = {S ∈ g| (S, H) = 0 ∀H ∈ 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 + H with S ∈ s, H ∈ h.

Define a map Φ:g → GL(n, ℝ) by (S, H) ↦ e^Se^H. Expand the exponentials,

$\Phi(S,H) = e^{tS}e^{tH} = (I + tS + O(t^2))(I + tH + O(t^2)) = I + tS + tH + O(t^2),$

and the pushforward or differential at 0, Φ_∗(S, H) =  ^d⁄_dtΦ(tS, tH)|_{t = 0} is seen to be S + H, 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 an 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.^[23] Suppose for the purpose of obtaining a contradiction that for all i, e^(Φ(Bi)) ∩ H contains an element h_i that is not on the form h_i = e^H_i,H_i ∈ h. Then, since Φ is a bijection on the B_i, there is a unique sequence X_i = S_i + H_i, with S_i ∈ s and H_i ∈ h such that X_i ∈ B_i converging to 0 because Β is a neighborhood basis, with e^S_ie^H_i = h_i. Since e^H_i ∈ H and h_i ∈ H, e^S_i ∈ H as well.

Normalize the sequence in s, Y_i =  ^S_i⁄_||S||. It takes its values in the unit sphere in s and since it is compact, there is a convergent subsequence converging to Y ∈ s.^[24] The index i henceforth refers to this subsequence. It will be shown that e^tY ∈ H, ∀t. Fix t = τ and fix a sequence (m_i(τ)) such that m_i(τ)||Y_i|| → τ. For this, m_i(τ) such that m_i(τ)||Y_i|| ≤ τ ≤ (m_i(τ) + 1)||Y_i|| will do. Then

$(e^{{S_{i}}})^{{m_{i}}}=e^{{m_{i}S_{i}}}=e^{{m_{i}\|S_{i}\|{\frac {S_{i}}{\|S_{i}\|}}}}\rightarrow e^{{\tau Y}}.$

Since H is a group, the left hand side is in H for all i. Since H is closed, the limit as i goes to infinity, e^tY ∈ H, ∀t,^[25] hence Y ∈ h. This is a contradiction. For some i the sets U = Β_i and V = Φ(Β_i) satisfy e^{(U ∩ h)} = H ∩ V and the exponential restricted to the open set (U ∩ h) ⊂ h is in analytic bijection with the open set Φ(U) ∩ H ⊂ H. This proves the lemma.

For j ≥ i, the image in H of B_j under Φ form a neighborhood basis at I. This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in G is analytic, the left and right translates of this neighborhood basis by a group element g ∈ G gives a neighborhood basis at g. These bases restricted to H gives neighborhood bases at all h ∈ H. 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 H. First define φ₁:e^(U) ⊂ G → g, g ↦ log(g). This is an analytic bijection with analytic inverse. 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), …, x_m(h), 0, …, 0), where m is the dimension of h. This shows that (e^U, φ₁) is a slice chart. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in H. This shows that H is an embedded submanifold of G.

Moreover, multiplication m, and inversion i in H are analytic since these operations are analytic in G and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations m:H × H → G and i:H × H → G.^[26] But since H is embedded, m:H × H → H and i:H × H → H are analytic as well.^[27]

See also

• Inverse function theorem
• Lie correspondence

Notes

1. Lee 2003 Theorem 20.10. Lee states and proves this theorem in all generality.
2. Rossmann 2002 Theorem 1, Section 2.7 Rossmann states the theorem for linear groups. The statement is that there is an open subset U ⊂ g such that U × H → G, (X, H) → e^XH is an analytic bijection onto an open neighborhood of H in G.
3. Cartan 1930 See § 26.
4. von Neumann (1929); Bochner (1958).
5. Rossmann 2002 Section 2.6.
6. Willard 1970 At each fixed h, the sets hU form a neighborhood basis. Willard's theorem 5.4 then guarantees that hU as both h and U varies form a basis for some topology.
7. Rossmann 2002 Section 4.1.
8. Lee 2003 Embedded manifolds are defined in Chapter 8.
9. Lee 2003 Example 7.3
10. Hall 2003 Hall (Chapter 3) mentions this as being a common phenomenon.
11. Rossmann 2002 See comment to Corollary 5, Section 2.2.
12. E.g. Hall 2003. See definition in Chapter 1.
13. Hall 2003. Corollary 2.29
14. Rossmann 2002 Problem 1. Section 2.7
15. Rossmann 2002 Problem 3. Section 2.7
16. Rossmann 2002 Problem 4. Section 2.7
17. Rossmann 2002 Problem 5. Section 2.7
18. Hall 2003 Problem 18, Chapter 3.
19. Lee 2003 Proposition 8.30.
20. Rossmann 2002 Problem 2. Section 2.7.
21. See for instance Lee 2002 Chapter 21
22. Hall 2003 Theorem 2.27.
23. For this one can choose open balls, Β = {B_k| diam(B_k) =  ¹⁄_{(k + m}, k ∈ ℕ} for some large enough m such that B₁ ⊂ U₁. Here the metric obnained from the Hilbert-Schmidt inner product is used.
24. Willard 1970 By problem 17G, s is sequentially compact, meaning every sequence has a convergent subsequence.
25. Willard 1979 Corollary 10.5.
26. Lee 2003 Proposition 8.22.
27. Lee 2003 Corollary 8.25.

References

• Bochner, S. (1958), "John von Neumann 1903–1957" (PDF), Biographical Memoirs of the National Academy of Sciences: 438–456. See in particular p. 441.
• Cartan, Élie (1930), "La théorie des groupes finis et continus et l'Analysis Situs", Mémorial Sc. Math., XLII, pp. 1–61
• Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Springer, ISBN 0-387-40122-9
• Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, 218, ISBN 0-387-95448-1
• von Neumann, John (1929), "Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen", Mathematische Zeitschrift (in German), 30 (1): 3–42, doi:10.1007/BF01187749
• Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0 19 859683 9
• Willard, Stephen (1970), General Topology, Dover Publications, ISBN 0-486-43479-6