In
mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
in the theory of
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
. It states that if is a
closed subgroup
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...
of a
Lie group , then is an
embedded Lie group with the
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...
(and hence the
group topology) agreeing with the embedding.
One of several results known as
Cartan's theorem, it was first published in 1930 by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
, who was inspired by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
's 1929 proof of a special case for groups of
linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
.
[; .]
Overview
Let
be a Lie group with Lie algebra
. Now let
be an arbitrary closed subgroup of
. It is necessary to show that
is a smooth embedded submanifold of
. The 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, define the "Lie algebra"
of
by the formula
It is not difficult to show that
is a Lie subalgebra of
. In particular,
is a subspace of
, which one might hope to be the tangent space of
at the identity. For this idea to work, however,
must be 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.
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, it is necessary to prove the following critical lemma:
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.
It is worth noting that Rossmann shows that for ''any'' subgroup
of
(not necessarily closed), the Lie algebra
of
is a Lie subalgebra of
. Rossmann then goes on to introduce coordinates 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.
In particular, the lemma stated above does not hold if
is not closed.
Example of a non-closed subgroup
For an example of a subgroup that is not an embedded Lie subgroup, consider the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
and an "
irrational winding of the torus".
and its subgroup
with irrational. Then is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in and hence not closed. In the
relative topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced ...
, a small open subset of is composed of infinitely many almost parallel line segments on the surface of the torus. This means that is not
locally path connected. In the group topology, the small open sets are ''single'' line segments on the surface of the torus and ''is'' locally path connected.
The example shows that for some groups one can find points in an arbitrarily small neighborhood in the relative topology of the identity that are exponentials of elements of , yet they cannot be connected to the identity with a path staying in .
[ See comment to Corollary 5, Section 2.2.] The group is not a Lie group. While the map is an analytic bijection, its inverse is not continuous. That is, if corresponds to a small open interval , there is no open with due to the appearance of the sets . However, with the group topology , is a Lie group. With this topology the injection is an analytic
injective immersion, but not a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
, hence not an embedding. There are also examples of groups 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 .
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 or . 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. (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
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 is a set with
transitive group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
and the
isotropy group
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
or stabilizer of a point is a closed Lie subgroup, then has a unique smooth manifold structure such that the action is smooth.
Conditions for being closed
A few sufficient conditions for being closed, hence an embedded Lie group, are given below.
*All
classical group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...
s are closed in , where is
,
, or
, the
quaternions.
*A subgroup that is ''locally closed'' is closed. A subgroup is locally closed if every point has a neighborhood in such that is closed in .
*If , where is a compact group and is a closed set, then is closed.
*If is a Lie subalgebra such that for no , then , the group generated by , is closed in .
*If , then the
one-parameter subgroup generated by is ''not closed'' if and only if is similar over
to a diagonal matrix with two entries of irrational ratio.
*Let be a Lie subalgebra. If there is a
simply connected compact group
''K'' with isomorphic to , then is closed in .
*If ''G'' is simply connected and is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
, then the connected Lie subgroup with Lie algebra is closed.
Converse
An embedded Lie subgroup is closed so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, is an embedded Lie subgroup if and only if its group topology equals its relative topology.
Proof
The proof is given for
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
s with 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.
The proof for general is formally identical, except that elements of the Lie algebra are left invariant
vector fields on and the exponential mapping is the time one flow of the vector field. If with closed in , then is closed in , so the specialization to instead of arbitrary matters little.
Proof of the key lemma
We begin by establishing the key lemma stated in the "overview" section above.
Endow with an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(e.g., the
Hilbert–Schmidt inner product In mathematics, Hilbert–Schmidt may refer to
* a Hilbert–Schmidt operator;
** a Hilbert–Schmidt integral operator In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open ...
), and let be the Lie algebra of defined as . Let , the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of . Then decomposes as the
direct sum , so each is uniquely expressed as with .
Define a map by . Expand the exponentials,
and the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
or differential at , is seen to be , i.e. , the identity. The hypothesis of the
inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
is satisfied with analytic, and thus there are open sets with and such that is a
real-analytic bijection from to with analytic inverse. It remains to show that and contain open sets and such that the conclusion of the theorem holds.
Consider a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
neighborhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
at , linearly ordered by reverse inclusion with . Suppose for the purpose of obtaining a contradiction that for all , contains an element that is ''not'' on the form . Then, since is a bijection on the , there is a unique sequence , with and such that converging to because is a neighborhood basis, with . Since and , as well.
Normalize the sequence in , . It takes its values in the unit sphere in and since it is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, there is a convergent subsequence converging to . The index henceforth refers to this subsequence. It will be shown that . Fix and choose a sequence of integers such that as . For example, such that will do, as → 0. Then
Since is a group, the left hand side is in for all . Since is closed, , hence . This is a contradiction. Hence, for some the sets and satisfy and the exponential restricted to the open set is in analytic bijection with the open set . This proves the lemma.
Proof of the theorem
For , the image in of under form a neighborhood basis at . This is, by the way it is constructed, a neighborhood basis both in the group topology and the
relative topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced ...
. Since multiplication in is analytic, the left and right translates of this neighborhood basis by a group element gives a neighborhood basis at . These bases restricted to gives neighborhood bases at all . 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 . First define . This is an analytic bijection with analytic inverse. Furthermore, if , then . By fixing a basis for and identifying with
, then in these coordinates , where is the dimension of . This shows that is a
slice chart. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in . This shows that is an embedded submanifold of .
Moreover, multiplication , and inversion in are analytic since these operations are analytic in and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations and . But since is embedded, and are analytic as well.
[ Corollary 8.25.]
See also
*
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
*
Lie correspondence
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be int ...
Notes
References
*. See in particula
p. 441
*
*
*
*
*
*{{Citation, last=Willard, first=Stephen, title=General Topology, publisher=Dover Publications, year=1970, isbn=0-486-43479-6
Lie groups
Theorems in group theory