Closed-subgroup theorem

Template:Short description 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 Template:Math is a closed subgroup of a Lie group Template:Math, then Template:Math is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.Template:SfnTemplate:SfnTemplate:Sfn One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan,Template:Sfn who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.Template:SfnTemplate:Sfn

Overview

Let Template:Math be a Lie group with Lie algebra $𝔤$. Now let Template:Math be an arbitrary closed subgroup of Template:Math. It is necessary to show that Template:Math is a smooth embedded submanifold of Template:Math. The first step is to identify something that could be the Lie algebra of Template:Math, that is, the tangent space of Template:Math at the identity. The challenge is that Template:Math 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" $𝔥$ of Template:Math by the formula $$𝔥=\{X\mid e^{tX}\in H,\mathrm{\forall }t\in 𝐑\}.$$

It is not difficult to show that $𝔥$ is a Lie subalgebra of $𝔤$.Template:Sfn In particular, $𝔥$ is a subspace of $𝔤$, which one might hope to be the tangent space of Template:Math at the identity. For this idea to work, however, $𝔥$ must be big enough to capture some interesting information about Template:Math. If, for example, Template:Math were some large subgroup of Template:Math but $𝔥$ turned out to be zero, $𝔥$ would not be helpful.

The key step, then, is to show that $𝔥$ actually captures all the elements of Template:Math that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma: Template:Math theorem

Once this has been established, one can use exponential coordinates on Template:Math, that is, writing each Template:Math (not necessarily in Template:Math) as Template:Math for Template:Math. In these coordinates, the lemma says that Template:Math corresponds to a point in Template:Math precisely if Template:Math belongs to $𝔥\subset 𝔤$. That is to say, in exponential coordinates near the identity, Template:Math looks like $𝔥\subset 𝔤$. Since $𝔥$ is just a subspace of $𝔤$, this means that $𝔥\subset 𝔤$ is just like Template:Math, with $k=\dim (𝔥)$ and $n=\dim (𝔤)$. Thus, we have exhibited a "slice coordinate system" in which Template:Math looks locally like Template:Math, which is the condition for an embedded submanifold.Template:Sfn

It is worth noting that Rossmann shows that for any subgroup Template:Math of Template:Math (not necessarily closed), the Lie algebra $𝔥$ of Template:Math is a Lie subalgebra of $𝔤$.Template:Sfn Rossmann then goes on to introduce coordinatesTemplate:Sfn on Template:Math that make the identity component of Template:Math into a Lie group. It is important to note, however, that the topology on Template:Math coming from these coordinates is not the subset topology. That it so say, the identity component of Template:Math is an immersed submanifold of Template:Math but not an embedded submanifold.

In particular, the lemma stated above does not hold if Template:Math is not closed.

Example of a non-closed subgroup

Error creating thumbnail:

The torus Template:Math. Imagine a bent helix laid out on the surface picturing Template:Math. If Template:Math in lowest terms, the helix will close up on itself at Template:Math after Template:Math rotations in Template:Math and q rotations in Template:Math. If Template:Math 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". $$G={𝕋}^2=\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi i\varphi }\end{array}\right)|\theta ,\varphi \in 𝐑\},$$ and its subgroup $$H=\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi ia\theta }\end{array}\right)|\theta \in 𝐑\}\text{with Lie algebra }𝔥=\{\left(\begin{array}{cc}i\theta & 0\\ 0& ia\theta \end{array}\right)|\theta \in 𝐑\},$$ with Template:Math irrational. Then Template:Math is dense in Template:Math and hence not closed.Template:Sfn In the relative topology, a small open subset of Template:Math is composed of infinitely many almost parallel line segments on the surface of the torus. This means that Template:Math is not locally path connected. In the group topology, the small open sets are single line segments on the surface of the torus and Template:Math is locally path connected.

The example shows that for some groups Template:Math one can find points in an arbitrarily small neighborhood Template:Math in the relative topology Template:Math of the identity that are exponentials of elements of Template:Math, yet they cannot be connected to the identity with a path staying in Template:Math.Template:Sfn The group Template:Math is not a Lie group. While the map Template:Math is an analytic bijection, its inverse is not continuous. That is, if Template:Math corresponds to a small open interval Template:Math, there is no open Template:Math with Template:Math due to the appearance of the sets Template:Math. However, with the group topology Template:Math, Template:Math is a Lie group. With this topology the injection Template:Math is an analytic injective immersion, but not a homeomorphism, hence not an embedding. There are also examples of groups Template:Math 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 Template:Math.Template:Sfn For closed subgroups this is not the case as the proof below of the theorem shows.

Applications

Template:Lie groups Because of the conclusion of the theorem, some authors chose to define linear Lie groups or matrix Lie groups as closed subgroups of Template:Math or Template:Math. 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.Template:Sfn (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 Template:MathTemplate:Sfn

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 Template:Math is a set with transitive group action and the isotropy group or stabilizer of a point Template:Math is a closed Lie subgroup, then Template:Math has a unique smooth manifold structure such that the action is smooth.

Conditions for being closed

A few sufficient conditions for Template:Math being closed, hence an embedded Lie group, are given below.

• All classical groups are closed in Template:Math, where Template:Math is Template:Math, Template:Math, or Template:Math, the quaternions.
• A subgroup that is locally closed is closed.Template:Sfn A subgroup is locally closed if every point has a neighborhood in Template:Math such that Template:Math is closed in Template:Math.
• If Template:Math, where Template:Math is a compact group and Template:Math is a closed set, then Template:Math is closed.Template:Sfn
• If Template:Math is a Lie subalgebra such that for no Template:Math, then Template:Math, the group generated by Template:Math, is closed in Template:Math.Template:Sfn
• If Template:Math, then the one-parameter subgroup generated by Template:Math is not closed if and only if Template:Math is similar over Template:Math to a diagonal matrix with two entries of irrational ratio.Template:Sfn
• Let Template:Math be a Lie subalgebra. If there is a simply connected compact group Template:Math with Template:Math isomorphic to Template:Math, then Template:Math is closed in Template:Math. Template:Sfn
• If G is simply connected and Template:Math is an ideal, then the connected Lie subgroup with Lie algebra Template:Math is closed. Template:Sfn

Converse

An embedded Lie subgroup Template:Math is closedTemplate:Sfn so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, Template:Math is an embedded Lie subgroup if and only if its group topology equals its relative topology.Template:Sfn

Proof

File:JohnvonNeumann-LosAlamos.gif

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 Template:Math 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.Template:SfnTemplate:Sfn The proof for general Template:Math is formally identical, except that elements of the Lie algebra are left invariant vector fields on Template:Math and the exponential mapping is the time one flow of the vector field. If Template:Math with Template:Math closed in Template:Math, then Template:Math is closed in Template:Math, so the specialization to Template:Math instead of arbitrary Template:Math matters little.

Proof of the key lemma

We begin by establishing the key lemma stated in the "overview" section above.

Endow Template:Math with an inner product (e.g., the Hilbert–Schmidt inner product), and let Template:Math be the Lie algebra of Template:Math defined as Template:Math. Let Template:Math, the orthogonal complement of Template:Math. Then Template:Math decomposes as the direct sum Template:Math, so each Template:Math is uniquely expressed as Template:Math with Template:Math.

Define a map Template:Math by Template:Math. Expand the exponentials, $$\mathrm{\Phi }(S,T)=e^{tS}{e}^{tT}=I+tS+tT+O(t^2),$$ and the pushforward or differential at Template:Math, Template:Math is seen to be Template:Math, i.e. Template:Math, the identity. The hypothesis of the inverse function theorem is satisfied with Template:Math analytic, and thus there are open sets Template:Math with Template:Math and Template:Math such that Template:Math is a real-analytic bijection from Template:Math to Template:Math with analytic inverse. It remains to show that Template:Math and Template:Math contain open sets Template:Math and Template:Math such that the conclusion of the theorem holds.

Consider a countable neighborhood basis Template:Math at Template:Math, linearly ordered by reverse inclusion with Template:Math.Template:Efn Suppose for the purpose of obtaining a contradiction that for all Template:Math, Template:Math contains an element Template:Math that is not on the form Template:Math. Then, since Template:Math is a bijection on the Template:Math, there is a unique sequence Template:Math, with Template:Math and Template:Math such that Template:Math converging to Template:Math because Template:Math is a neighborhood basis, with Template:Math. Since Template:Math and Template:Math, Template:Math as well.

Normalize the sequence in Template:Math, Template:Math. It takes its values in the unit sphere in Template:Math and since it is compact, there is a convergent subsequence converging to Template:Math.Template:Sfn The index Template:Math henceforth refers to this subsequence. It will be shown that Template:Math. Fix Template:Math and choose a sequence Template:Math of integers such that Template:Math as Template:Math. For example, Template:Math such that Template:Math will do, as Template:Math. Then $$(e^{S_i}{)}^{m_i}=e^{m_i{S}_i}=e^{m_i\Vert S_i\Vert Y_i}\to e^{tY}.$$

Since Template:Math is a group, the left hand side is in Template:Math for all Template:Math. Since Template:Math is closed, Template:Math,Template:Sfn hence Template:Math. This is a contradiction. Hence, for some Template:Math the sets Template:Math and Template:Math satisfy Template:Math and the exponential restricted to the open set Template:Math is in analytic bijection with the open set Template:Math. This proves the lemma.

Proof of the theorem

For Template:Math, the image in Template:Math of Template:Math under Template:Math form a neighborhood basis at Template:Math. This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in Template:Math is analytic, the left and right translates of this neighborhood basis by a group element Template:Math gives a neighborhood basis at Template:Math. These bases restricted to Template:Math gives neighborhood bases at all Template:Math. 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 Template:Math. First define Template:Math. This is an analytic bijection with analytic inverse. Furthermore, if Template:Math, then Template:Math. By fixing a basis for Template:Math and identifying Template:Math with Template:Math, then in these coordinates Template:Math, where Template:Math is the dimension of Template:Math. This shows that Template:Math is a slice chart. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in Template:Math. This shows that Template:Math is an embedded submanifold of Template:Math.

Moreover, multiplication Template:Math, and inversion Template:Math in Template:Math are analytic since these operations are analytic in Template:Math and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations Template:Math and Template:Math.Template:Sfn But since Template:Math is embedded, Template:Math and Template:Math are analytic as well.Template:Sfn

See also

• Inverse function theorem
• Lie correspondence

References

• Template:Citation. See in particular p. 441.
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation