In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a maximal compact subgroup ''K'' of a
topological group
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 st ...
''G'' is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''K'' that is a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, in the
subspace topology, and
maximal amongst such subgroups.
Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are ''not'' in general unique, but are unique up to
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change ...
– they are
essentially unique.
Example
An example would be the subgroup O(2), the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, inside the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(2, R). A related example is the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
SO(2) inside
SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as any
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 ...
has an associated orthogonal group, and the essential uniqueness corresponds to the essential uniqueness of the inner product.
Definition
A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a ''maximal (compact subgroup)'' – rather than being (alternate possible reading) a
maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...
that happens to be compact; which would probably be called a ''compact (maximal subgroup)'', but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact).
Existence and uniqueness
The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected
locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a
semisimple Lie group uniqueness is a consequence of the
Cartan fixed point theorem, which asserts that if a compact group acts by isometries on a complete simply connected
negatively curved Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
then it has a fixed point.
Maximal compact subgroups of connected Lie groups are usually ''not'' unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups ''K'' and ''L'', there is an element ''g'' ∈ ''G'' such that
[Note that this element ''g'' is not unique – any element in the same coset ''gK'' would do as well.] ''gKg''
−1 = ''L''. Hence a maximal compact subgroup is
essentially unique, and people often speak of "the" maximal compact subgroup.
For the example of the general linear group GL(''n'', R), this corresponds to the fact that ''any''
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 ...
on R
''n'' defines a (compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O(''n'', R).
Proofs
For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in and . and discuss the extension to connected Lie groups and connected locally compact groups.
For semisimple groups, existence is a consequence of the existence of a compact
real form of the noncompact semisimple Lie group and the corresponding
Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding
Riemannian symmetric space ''G''/''K'' has
negative curvature and
Cartan's fixed point theorem. showed that the derivative of the exponential map at any point of ''G''/''K'' satisfies , d exp ''X'', ≥ , X, . This implies that ''G''/''K'' is a
Hadamard space, i.e. a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the
Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.
Proof of uniqueness for semisimple groups
also related the general problem for semisimple groups to the case of GL(''n'', R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in .
Let
be a real semisimple Lie algebra with
Cartan involution σ. Thus the
fixed point subgroup of σ is the maximal compact subgroup ''K'' and there is an eigenspace decomposition
:
where
, the Lie algebra of ''K'', is the +1 eigenspace. The Cartan decomposition gives
:
If ''B'' is the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
on
given by ''B''(''X'',''Y'') = Tr (ad X)(ad Y), then
:
is a real inner product on
. Under the adjoint representation, ''K'' is the subgroup of ''G'' that preserves this inner product.
If ''H'' is another compact subgroup of ''G'', then averaging the inner product over ''H'' with respect to the Haar measure gives an inner product invariant under ''H''. The operators Ad ''p'' with ''p'' in ''P'' are positive symmetric operators. This new inner produst can be written as
:
where ''S'' is a positive symmetric operator on
such that
Ad(''h'')
''t''''S'' Ad ''h'' = ''S'' for ''h'' in ''H'' (with the transposes computed with respect to the inner product). Moreover, for ''x'' in ''G'',
:
So for ''h'' in ''H'',
:
For ''X'' in
define
:
If ''e''
''i'' is an orthonormal basis of eigenvectors for ''S'' with ''Se''
''i'' = λ
''i'' ''e''
''i'', then
:
so that ''f'' is strictly positive and tends to ∞ as , ''X'', tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad ''X'' and each non-zero eigenvalue occurs with its negative, since i ad ''X'' is a ''skew-adjoint operator'' on the compact real form
.
So ''f'' has a global minimum at ''Y'' say. This minimum is unique, because if ''Z'' were another then
:
where ''X'' in
is defined by the Cartan decomposition
:
If ''f''
''i'' is an orthonormal basis of eigenvectors of ad ''X'' with corresponding real eigenvalues μ
''i'', then
:
Since the right hand side is a positive combination of exponentials, the real-valued function ''g'' is
strictly convex if ''X'' ≠ 0, so has a unique minimum. On the other hand, it has local minima at ''t'' = 0 and ''t'' = 1, hence ''X'' = 0 and ''p'' = exp ''Y'' is the unique global minimum. By construction
''f''(''x'') = ''f''(σ(''h'')''xh''
−1) for ''h'' in ''H'', so that ''p'' = σ(''h'')''ph''
−1 for ''h'' in ''H''. Hence σ(''h'')= ''php''
−1. Consequently, if ''g'' = exp ''Y''/2, ''gHg''
−1 is fixed by σ and therefore lies in ''K''.
Applications
Representation theory
Maximal compact subgroups play a basic role in the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
when ''G'' is not compact. In that case a maximal compact subgroup ''K'' is a
compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.
The operations relating the representation theories of ''G'' and ''K'' are
restricting representations from ''G'' to ''K'', and
inducing representations from ''K'' to ''G'', and these are quite well understood; their theory includes that of
spherical functions.
Topology
The
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
of the Lie groups is also largely carried by a maximal compact subgroup ''K''. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact ''K'' and a Euclidean space – ''G'' = ''K'' × R
''d'' – thus in particular ''K'' is a
deformation retract of ''G,'' and is
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, and thus they have the same
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
. Indeed, the inclusion
and the deformation retraction
are
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
s.
For the general linear group, this decomposition is the
QR decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decomp ...
, and the deformation retraction is the
Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the
Iwasawa decomposition of ''G'' as ''G'' = ''KAN'' in which ''K'' occurs in a product with a
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
subgroup ''AN''.
See also
*
Hyperspecial subgroup In the theory of reductive groups over local fields, a hyperspecial subgroup of a reductive group ''G'' is a certain type of compact subgroup of ''G''.
In particular, let ''F'' be a nonarchimedean local field, ''O'' its ring of integers, ''k'' its ...
*
Complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
Notes
References
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* {{citation, first=K., last=Iwasawa, title=On some types of topological groups, journal= Ann. of Math., volume=50, year=1949, pages= 507–558, doi=10.2307/1969548
Topological groups
Lie groups