Inmathematics, theclosed-subgroup theorem (sometimes referred to asCartan's theorem) is atheorem in the theory ofLie groups. It states that ifH is aclosed subgroup of aLie groupG, thenH is anembeddedLie group with thesmooth structure (and hence thegroup topology) agreeing with the embedding.^[1][2][3]One of several results known asCartan's theorem, it was first published in 1930 byÉlie Cartan,^[4] who was inspired byJohn von Neumann's 1929 proof of a special case for groups oflinear transformations.^[5][6]

LetG be a Lie group with Lie algebra${\displaystyle \mathfrak{g}}$. Now letH be an arbitrary closed subgroup ofG. It is necessary to show thatH is a smooth embedded submanifold ofG. The first step is to identify something that could be the Lie algebra ofH, that is, the tangent space ofH at the identity. The challenge is thatH 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"${\displaystyle \mathfrak{h}}$ ofH by the formula$${\displaystyle \mathfrak{h}=\left\{X\mid e^{tX}\in H,\mathrm{\forall }t\in \mathbf{R}\right\}.}$$

It is not difficult to show that${\displaystyle \mathfrak{h}}$ is a Lie subalgebra of${\displaystyle \mathfrak{g}}$.^[7] In particular,${\displaystyle \mathfrak{h}}$ is a subspace of${\displaystyle \mathfrak{g}}$, which one might hope to be the tangent space ofH at the identity. For this idea to work, however,${\displaystyle \mathfrak{h}}$ must be big enough to capture some interesting information aboutH. If, for example,H were some large subgroup ofG but${\displaystyle \mathfrak{h}}$ turned out to be zero,${\displaystyle \mathfrak{h}}$ would not be helpful.

The key step, then, is to show that${\displaystyle \mathfrak{h}}$ actually captures all the elements ofH that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma:

Once this has been established, one can useexponential coordinates onW, that is, writing eachg ∈W (not necessarily inH) asg =e^X forX = log(g). In these coordinates, the lemma says thatX corresponds to a point inH precisely ifX belongs to${\displaystyle \mathfrak{h}\subset \mathfrak{g}}$. That is to say, in exponential coordinates near the identity,H looks like${\displaystyle \mathfrak{h}\subset \mathfrak{g}}$. Since${\displaystyle \mathfrak{h}}$ is just a subspace of${\displaystyle \mathfrak{g}}$, this means that${\displaystyle \mathfrak{h}\subset \mathfrak{g}}$ is just likeR^k ⊂Rⁿ, with${\displaystyle k=\dim (\mathfrak{h})}$ and${\displaystyle n=\dim (\mathfrak{g})}$. Thus, we have exhibited a "slice coordinate system" in whichH ⊂G looks locally likeR^k ⊂Rⁿ, which is the condition for an embedded submanifold.^[9]

It is worth noting that Rossmann shows that forany subgroupH ofG (not necessarily closed), the Lie algebra${\displaystyle \mathfrak{h}}$ ofH is a Lie subalgebra of${\displaystyle \mathfrak{g}}$.^[10] Rossmann then goes on to introduce coordinates^[11] onH that make the identity component ofH into a Lie group. It is important to note, however, that the topology onH coming from these coordinates is not the subset topology. That it so say, the identity component ofH is an immersed submanifold ofG but not an embedded submanifold.

In particular, the lemma stated above does not hold ifH is not closed.

For an example of a subgroup that is not an embedded Lie subgroup, consider thetorus and an "irrational winding of the torus".and its subgroupwitha irrational. ThenH isdense inG and hence not closed.^[12] In therelative topology, a small open subset ofH is composed of infinitely many almost parallel line segments on the surface of the torus. This means thatH is notlocally path connected. In the group topology, the small open sets aresingle line segments on the surface of the torus andHis locally path connected.

The example shows that for some groupsH one can find points in an arbitrarily small neighborhoodU in the relative topologyτ_r of the identity that are exponentials of elements ofh, yet they cannot be connected to the identity with a path staying inU.^[13] The group(H,τ_r) is not a Lie group. While the mapexp :h → (H,τ_r) is an analytic bijection, its inverse is not continuous. That is, ifU ⊂h corresponds to a small open interval−ε <θ <ε, there is no openV ⊂ (H,τ_r) withlog(V) ⊂U due to the appearance of the setsV. However, with the group topologyτ_g,(H,τ_g) is a Lie group. With this topology the injectionι : (H,τ_g) →G is an analyticinjective immersion, but not ahomeomorphism, hence not an embedding. There are also examples of groupsH for which one can find points in an arbitrarily small neighborhood (in the relative topology) of the identity that arenot exponentials of elements ofh.^[14] For closed subgroups this is not the case as the proof below of the theorem shows.

Lie groups andLie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups inphysics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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Because of the conclusion of the theorem, some authors chose todefinelinear Lie groups ormatrix Lie groups as closed subgroups ofGL(n,R) orGL(n,C).^[15] 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.^[8] (The proof is practically identical to the proof of the closed subgroup theorem presented below.) It follows every closed subgroup is anembedded submanifold ofGL(n,C)^[16]

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.

IfX is a set withtransitive group action and theisotropy group orstabilizer of a pointx ∈X is a closed Lie subgroup, thenX has a unique smooth manifold structure such that the action is smooth.

A few sufficient conditions forH ⊂G being closed, hence an embedded Lie group, are given below.

• Allclassical groups are closed inGL(F,n), whereF isR,C, orH, thequaternions.
• A subgroup that islocally closed is closed.^[17] A subgroup is locally closed if every point has a neighborhood inU ⊂G such thatH ∩U is closed inU.
• IfH =AB = {ab |a ∈A,b ∈B}, whereA is a compact group andB is a closed set, thenH is closed.^[18]
• Ifh ⊂g is a Lie subalgebra such that for noX ∈g ∖h, [X,h] ∈h, thenΓ(h), the group generated bye^h, is closed inG.^[19]
• IfX ∈g, then theone-parameter subgroup generated byX isnot closed if and only ifX is similar overC to a diagonal matrix with two entries of irrational ratio.^[20]
• Leth ⊂g be a Lie subalgebra. If there is asimply connected compact groupk withk isomorphic toh, thenΓ(h) is closed inG.^[21]
• IfG is simply connected andh ⊂g is anideal, then the connected Lie subgroup with Lie algebrah is closed.^[22]

An embedded Lie subgroupH ⊂G is closed^[23] so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently,H is an embedded Lie subgroup if and only if its group topology equals its relative topology.^[24]

The proof is given formatrix groups withG = 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 generalG is formally identical,^[25] except that elements of the Lie algebra areleft invariantvector fields onG and the exponential mapping is the time oneflow of the vector field. IfH ⊂G withG closed inGL(n,R), thenH is closed inGL(n,R), so the specialization toGL(n,R) instead of arbitraryG ⊂ GL(n,R) matters little.

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

Endowg with aninner product (e.g., theHilbert–Schmidt inner product), and leth be the Lie algebra ofH defined ash = {X ∈ M_n(R) =g |e^tX ∈H ∀t ∈R}. Lets = {S ∈g | (S,T) = 0 ∀T ∈h}, theorthogonal complement ofh. Theng decomposes as thedirect sumg =s ⊕h, so eachX ∈g is uniquely expressed asX =S +T withS ∈s,T ∈h.

Define a mapΦ :g → GL(n,R) by(S,T) ↦e^Se^T. Expand the exponentials,$${\displaystyle \mathrm{\Phi }(S,T)=e^{tS}{e}^{tT}=I+tS+tT+O(t^2),}$$and thepushforward ordifferential at0,Φ_∗(S,T) =⁠d/dt⁠Φ(tS,tT)|_{t = 0} is seen to beS +T, i.e.Φ_∗ = Id, the identity. The hypothesis of theinverse function theorem is satisfied withΦ analytic, and thus there are open setsU₁ ⊂g,V₁ ⊂ GL(n,R) with0 ∈U₁ andI ∈V₁ such thatΦ is areal-analytic bijection fromU₁ toV₁ with analytic inverse. It remains to show thatU₁ andV₁ contain open setsU andV such that the conclusion of the theorem holds.

Consider acountableneighborhood basisΒ at0 ∈g, linearly ordered by reverse inclusion withB₁ ⊂U₁.^[a] Suppose for the purpose of obtaining a contradiction that for alli,Φ(B_i) ∩H contains an elementh_i that isnot on the formh_i =e^T_i,T_i ∈h. Then, sinceΦ is a bijection on theB_i, there is a unique sequenceX_i =S_i +T_i, with0 ≠S_i ∈s andT_i ∈h such thatX_i ∈B_i converging to0 becauseΒ is a neighborhood basis, withe^S_ie^T_i =h_i. Sincee^T_i ∈H andh_i ∈H,e^S_i ∈H as well.

Normalize the sequence ins,Y_i =⁠S_i/||S_i||⁠. It takes its values in the unit sphere ins and since it iscompact, there is a convergent subsequence converging toY ∈s.^[26] The indexi henceforth refers to this subsequence. It will be shown thate^tY ∈H, ∀t ∈R. Fixt and choose a sequencem_i of integers such thatm_i ||S_i|| →t asi → ∞. For example,m_i such thatm_i ||S_i|| ≤t ≤ (m_i + 1) ||S_i|| will do, asS_i → 0. Then$${\displaystyle (e^{S_i}{)}^{m_i}=e^{m_i{S}_i}=e^{m_i\Vert S_i\Vert Y_i}\to e^{tY}.}$$

SinceH is a group, the left hand side is inH for alli. SinceH is closed,e^tY ∈H, ∀t,^[27] henceY ∈h. This is a contradiction. Hence, for somei the setsU = Β_i andV = Φ(Β_i) satisfye^U∩h =H ∩V and the exponential restricted to the open set(U ∩h) ⊂h is in analytic bijection with the open setΦ(U) ∩H ⊂H. This proves the lemma.

Forj ≥i, the image inH ofB_j underΦ form a neighborhood basis atI. This is, by the way it is constructed, a neighborhood basis both in the group topology and therelative topology. Since multiplication inG is analytic, the left and right translates of this neighborhood basis by a group elementg ∈G gives a neighborhood basis atg. These bases restricted toH gives neighborhood bases at allh ∈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 onH. First defineφ₁ :e^(U) ⊂G →g,g ↦ log(g). This is an analytic bijection with analytic inverse. Furthermore, ifh ∈H, thenφ₁(h) ∈h. By fixing a basis forg =h ⊕s and identifyingg withRⁿ, then in these coordinatesφ₁(h) = (x₁(h), ...,x_m(h), 0, ..., 0), wherem is the dimension ofh. This shows that(e^U,φ₁) is aslice chart. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point inH. This shows thatH is an embedded submanifold ofG.

Moreover, multiplicationm, and inversioni inH are analytic since these operations are analytic inG and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operationsm :H ×H →G andi :H ×H →G.^[28] But sinceH is embedded,m :H ×H →H andi :H ×H →H are analytic as well.^[29]

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