Compact group
Encyclopedia
In mathematics
, a compact (topological, often understood) group is a topological group
whose topology
is compact
. Compact groups are a natural generalisation of finite group
s with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group action
s and representation theory
.
In the following we will assume all groups are Hausdorff
.
s form the nicest class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
The classification theorem
of compact Lie groups states that up to finite extensions
and finite covers this exhausts the list of examples (which already includes some redundancies).
G0, which is connected
. The quotient group
G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension
Now every compact, connected Lie group G0 has a finite covering
where
is a finite abelian group
and
is a product of a torus and a compact, connected, simply-connected Lie group K:
Finally, every compact, connected, simply-connected Lie group K is a product of compact, connected, simply-connected simple Lie group
s Ki each of which is isomorphic to exactly one of
, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois group
s are compact groups, a basic fact for the theory of algebraic extension
s in the case of infinite degree.
Pontryagin duality
provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete group
s.
, which will be invariant by both left and right translation (the modulus function
must be a continuous homomorphism
to the positive multiplicative reals, and so 1). In other words these groups are unimodular. Haar measure is easily normalised to be a probability measure
, analogous to dθ/2π on the circle.
Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Hurwitz
, and in the Lie group cases can always be given by an invariant differential form
. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore integrals are often computable quite directly, a fact applied constantly in number theory
.
went on to give the detailed character theory
of the compact connected Lie groups, based on maximal torus
theory. The resulting Weyl character formula
was one of the influential results of twentieth century mathematics.
A combination of Weyl's work and Cartan's theorem
gives a survey of the whole representation theory of compact groups G . That is, by the Peter-Weyl theorem the irreducible unitary representation
s ρ of G are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If G is not itself a Lie group, there must be a kernel to ρ. Further one can form an inverse system
, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies G as an inverse limit
of compact Lie groups. Here the fact that in the limit a faithful representation
of G is found is another consequence of the Peter-Weyl theorem,
The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the complex representations of finite groups. This theory is rather rich in detail, but is qualitatively well understood.
, now often recast in term of tannakian category
theory.
. Inside a general semisimple Lie group there is a maximal compact subgroup
, and the representation theory of such groups, developed largely by Harish-Chandra
, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a compact (topological, often understood) group is a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
whose topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Compact groups are a natural generalisation of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s and representation theory
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 studiesmodules over these abstract algebraic structures...
.
In the following we will assume all groups are Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
Compact Lie groups
Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s form the nicest class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
- the circle group T and the torus groups Tn,
- the orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
s O(n), the special orthogonal group SO(n) and its covering spin group Spin(n), - the unitary groupUnitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
U(n) and the special unitary groupSpecial unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
SU(n), - the symplectic groupSymplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
Sp(n), - the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and E8E8 (mathematics)In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...
,
The classification theorem
Classification theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class....
of compact Lie groups states that up to finite extensions
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
and finite covers this exhausts the list of examples (which already includes some redundancies).
Classification
Given any compact Lie group G one can take its identity componentIdentity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
G0, which is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. The quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension
Now every compact, connected Lie group G0 has a finite covering
where
is a finite abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
and
is a product of a torus and a compact, connected, simply-connected Lie group K:Finally, every compact, connected, simply-connected Lie group K is a product of compact, connected, simply-connected simple Lie group
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
s Ki each of which is isomorphic to exactly one of
- Sp(n), n ≥ 1
- SU(n), n ≥ 3
- Spin(n), n ≥ 7
- G2, F4, E6, E7, or E8
Further examples
Amongst groups that are not Lie groups, and so do not carry the structure of a manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s are compact groups, a basic fact for the theory of algebraic extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
s in the case of infinite degree.
Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
s.
Haar measure
Compact groups all carry a Haar measureHaar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
, which will be invariant by both left and right translation (the modulus function
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
must be a continuous homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
to the positive multiplicative reals, and so 1). In other words these groups are unimodular. Haar measure is easily normalised to be a probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
, analogous to dθ/2π on the circle.
Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Hurwitz
Hurwitz
Hurwitz is a surname and may refer to:*Aaron Hurwitz, musician, see Live on Breeze Hill*Adolf Hurwitz , German mathematician**Hurwitz polynomial**Hurwitz matrix**Hurwitz quaternion**Hurwitz's automorphisms theorem**Hurwitz zeta function...
, and in the Lie group cases can always be given by an invariant differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore integrals are often computable quite directly, a fact applied constantly in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Representation theory
The representation theory of compact groups was founded by the Peter-Weyl theorem. Hermann WeylHermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
went on to give the detailed character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of the compact connected Lie groups, based on maximal torus
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
theory. The resulting Weyl character formula
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
was one of the influential results of twentieth century mathematics.
A combination of Weyl's work and Cartan's theorem
Cartan's theorem
In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....
gives a survey of the whole representation theory of compact groups G . That is, by the Peter-Weyl theorem the irreducible unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
s ρ of G are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If G is not itself a Lie group, there must be a kernel to ρ. Further one can form an inverse system
Inverse system
In mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inverse system is sometimes called a pro-object in C. The dual concept is a direct system.-The category of inverse systems:...
, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies G as an inverse limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
of compact Lie groups. Here the fact that in the limit a faithful representation
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
of G is found is another consequence of the Peter-Weyl theorem,
The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the complex representations of finite groups. This theory is rather rich in detail, but is qualitatively well understood.
Duality
The topic of recovering a compact group from its representation theory is the subject of the Tannaka-Krein dualityTannaka-Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. Its natural extension to the non-Abelian case is the Grothendieck duality theory....
, now often recast in term of tannakian category
Tannakian category
In mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K...
theory.
From compact to non-compact groups
The influence of the compact group theory on non-compact groups was formulated by Weyl in his unitarian trickUnitarian trick
In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some group G is in a qualitative way controlled by that...
. Inside a general semisimple Lie group there is a maximal compact subgroup
Maximal compact subgroup
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups....
, and the representation theory of such groups, developed largely by Harish-Chandra
Harish-Chandra
Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory.