“Closed-subgroup theorem”的意思、由来-开放百科全书

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 {{math|H}} is a closed subgroup of a Lie group {{math|G}}, then {{math|H}} is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.^[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.^[5]

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"

by the formula

It is not difficult to show that

is a Lie subalgebra of

.^[6] 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

(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.^[8]

It is worth noting that Rossmann shows that for any subgroup

(not necessarily closed), the Lie algebra

is a Lie subalgebra of

.^[9] Rossmann then goes on to introduce coordinates^[10] 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.

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".

with {{math|a}} irrational. Then {{math|H}} is dense in {{math|G}} and hence not closed.^[11] In the relative topology, a small open subset of {{math|H}} is composed of infinitely many almost parallel line segments on the surface of the torus. This means that {{math|H}} is not locally path connected. In the group topology, the small open sets are single line segments on the surface of the torus and {{math|H}} is locally path connected.

The example shows that for some groups {{math|H}} one can find points in an arbitrarily small neighborhood {{math|U}} in the relative topology {{math|τ_r}} of the identity that are exponentials of elements of {{math|h}}, yet they cannot be connected to the identity with a path staying in {{math|U}}.^[12] The group {{math|(H, τ_r)}} is not a Lie group. While the map {{math|exp:h → (H, τ_r)}} is an analytic bijection, its inverse is not continuous. That is, if {{math|U ⊂ h}} corresponds to a small open interval {{math|−ε < θ < ε}}, there is no open {{math|V ⊂ (H, τ_r)}} with {{math|log(V) ⊂ U}} due to the appearance of the sets {{math|V}}. However, with the group topology {{math|τ_g}}, {{math|(H, τ_g)}} is a Lie group. With this topology the injection {{math|ι:(H, τ_g) → G}} is an analytic injective immersion, but not a homeomorphism, hence not an embedding. There are also examples of groups {{math|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 {{math|h}}.^[12] 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 {{math|GL(n, ℝ)}} or {{math|GL(n, ℂ)}}.^[13] 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.^[14] (The proof is practically identical to the proof of the closed subgroup theorem presented below.) It follows every closed subgroup is an embedded submanifold of {{math|GL(n, ℂ)}}^[15]

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 {{math|H ⊂ G}} being closed, hence an embedded Lie group, is given below.

Converse

An embedded Lie subgroup {{math|H ⊂ G}} is closed^[22] so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, {{math|H}} is an embedded Lie subgroup if and only if its group topology equals its relative topology.^[23]

Proof

The proof is given for matrix groups with {{math|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] The proof for general {{math|G}} is formally identical,^[24] except that elements of the Lie algebra are left invariant vector fields on {{math|G}} and the exponential mapping is the time one flow of the vector field. If {{math|H ⊂ G}} with {{math|G}} closed in {{math|GL(n,ℝ)}}, then {{math|H}} is closed in {{math|GL(n, ℝ)}}, so the specialization to {{math|GL(n, ℝ)}} instead of arbitrary {{math|G ⊂ GL(n, ℝ)}} matters little.

Proof of the key lemma

Endow {{math|g}} with the Hilbert–Schmidt inner product, {{math|(X, Y) → Tr(X^tY)}}, and let {{math|h}} be the Lie algebra of {{math|H}} defined as {{math|h {{=}} {H ∈ M_n(ℝ) {{=}} g{{!}}e^tH ∈ H ∀t ∈ ℝ}}}. Let {{math|s {{=}} {S ∈ g{{!}} (S, H) {{=}} 0 ∀H ∈ h}}}, the orthogonal complement of {{math|h}}. Then {{math|g}} decomposes as the direct sum {{math|g {{=}} s ⊕ h}}, so each {{math|X ∈ g}} is uniquely expressed as {{math|X {{=}} S + H}} with {{math|S ∈ s, H ∈ h}}.

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

and the pushforward or differential at {{math|0}}, {{math|Φ_∗(S, H) {{=}} {{frac|d|dt}}Φ(tS, tH){{!}}_{t {{=}} 0}}} is seen to be {{math|S + H}}, i.e. {{math|Φ_∗ {{=}} Id}}, the identity. The hypothesis of the inverse function theorem is satisfied with {{math|Φ}} analytic, and thus there are open sets {{math|U₁ ⊂ g, V₁ ⊂ GL(n, ℝ)}} with {{math|0 ∈ U₁}} and {{math|I ∈ V₁}} such that {{math|Φ}} is an analytic bijection from {{math|U₁}} to {{math|V₁}} with analytic inverse. It remains to show that {{math|U₁}} and {{math|V₁}} contain open sets {{math|U}} and {{math|V}} such that the conclusion of the theorem holds.

Consider a countable neighborhood basis {{math|Β}} at {{math|0 ∈ g}}, linearly ordered by reverse inclusion with {{math|B₁ ⊂ U₁}}.^[25] Suppose for the purpose of obtaining a contradiction that for all {{math|i}}, {{math|e^Φ(B_i) ∩ H}} contains an element {{math|h_i}} that is not on the form {{math|h_i {{=}} e^H_i,H_i ∈ h}}. Then, since {{math|Φ}} is a bijection on the {{math|B_i}}, there is a unique sequence {{math|X_i {{=}} S_i + H_i}}, with {{math|S_i ∈ s}} and {{math|H_i ∈ h}} such that {{math|X_i ∈ B_i}} converging to {{math|0}} because {{math|Β}} is a neighborhood basis, with {{math|e^S_ie^H_i {{=}} h_i}}. Since {{math|e^H_i ∈ H}} and {{math|h_i ∈ H}}, {{math|e^S_i ∈ H}} as well.

Normalize the sequence in {{math|s}}, {{math|Y_i {{=}} {{fraction|S_i|{{!}}{{!}}S_i{{!}}{{!}}}}}}. It takes its values in the unit sphere in {{math|s}} and since it is compact, there is a convergent subsequence converging to {{math|Y ∈ s}}.^[26] The index {{math|i}} henceforth refers to this subsequence. It will be shown that {{math|e^tY ∈ H, ∀t ∈ ℝ}}. Fix {{math|t {{=}} τ}} and fix a sequence {{math|(m_i(τ))}} of real numbers such that {{math|m_i(τ){{!}}{{!}}Y_i{{!}}{{!}} → τ}}. For this, {{math|m_i(τ)}} such that {{math|m_i(τ){{!}}{{!}}Y_i{{!}}{{!}} ≤ τ ≤ (m_i(τ) + 1){{!}}{{!}}Y_i{{!}}{{!}}}} will do. Then

Since {{math|H}} is a group, the left hand side is in {{math|H}} for all {{math|i}}. Since {{math|H}} is closed, the limit as {{math|i}} goes to infinity, {{math|e^tY ∈ H, ∀t}},^[27] hence {{math|Y ∈ h}}. This is a contradiction. For some {{math|i}} the sets {{math|U {{=}} Β_i}} and {{math|V {{=}} Φ(Β_i)}} satisfy {{math|e^{(U ∩ h)} {{=}} H ∩ V}} and the exponential restricted to the open set {{math|(U ∩ h) ⊂ h}} is in analytic bijection with the open set {{math|Φ(U) ∩ H ⊂ H}}. This proves the lemma.

Proof of the theorem

For {{math|j ≥ i}}, the image in {{math|H}} of {{math|B_j}} under {{math|Φ}} form a neighborhood basis at {{math|I}}. This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in {{math|G}} is analytic, the left and right translates of this neighborhood basis by a group element {{math|g ∈ G}} gives a neighborhood basis at {{math|g}}. These bases restricted to {{math|H}} gives neighborhood bases at all {{math|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 {{math|H}}. First define {{math|φ₁:e^(U) ⊂ G → g, g ↦ log(g)}}. This is an analytic bijection with analytic inverse. Furthermore, if {{math|h ∈ H}}, then {{math|φ₁(h) ∈ h}}. By fixing a basis for {{math|g {{=}} h ⊕ s}} and identifying {{math|g}} with {{math|ℝⁿ}}, then in these coordinates {{math|φ₁(h) {{=}} (x₁(h), …, x_m(h), 0, …, 0)}}, where {{math|m}} is the dimension of {{math|h}}. This shows that {{math|(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 {{math|H}}. This shows that {{math|H}} is an embedded submanifold of {{math|G}}.

Moreover, multiplication {{math|m}}, and inversion {{math|i}} in {{math|H}} are analytic since these operations are analytic in {{math|G}} and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations {{math|m:H × H → G}} and {{math|i:H × H → G}}.^[28] But since {{math|H}} is embedded, {{math|m:H × H → H}} and {{math|i:H × H → H}} are analytic as well.^[29]