Compact group

In mathematics, a compact 'group' is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.
In the following we will assume all groups are Hausdorff spaces.

Compact Lie groups

s form a 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 group O, the special orthogonal group SO and its covering spin group Spin,
the compact symplectic group USp,
the unitary group U and the special unitary group SU,
the compact forms of the exceptional Lie groups: G₂, F₄, E₆, E₇, and E₈.

The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples. This classification is described in more detail in the next subsection.

Classification

Given any compact Lie group G one can take its identity component G₀, which is connected. The quotient group G/''G₀ is the group of components π₀ which must be finite since G'' is compact. We therefore have a finite extension
Meanwhile, for connected compact Lie groups, we have the following result:
Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers.
Finally, every compact, connected, simply-connected Lie group K is a product of finitely many compact, connected, simply-connected simple Lie groups K_i each of which is isomorphic to exactly one of the following:

The compact symplectic group
The special unitary group
The spin group

or one of the five exceptional groups G₂, F₄, E₆, E₇, and E₈. The restrictions on n are to avoid special isomorphisms among the various families for small values of n. For each of these groups, the center is known explicitly. The classification is through the associated root system, which in turn are classified by their Dynkin diagrams.
The classification of compact, simply connected Lie groups is the same as the classification of complex semisimple Lie algebras. Indeed, if K is a simply connected compact Lie group, then the complexification of the Lie algebra of K is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group.

Maximal tori and root systems

A key idea in the study of a connected compact Lie group K is the concept of a maximal torus, that is a subgroup T of K that is isomorphic to a product of several copies of and that is not contained in any larger subgroup of this type. A basic example is the case, in which case we may take to be the group of diagonal elements in. A basic result is the torus theorem which states that every element of belongs to a maximal torus and that all maximal tori are conjugate.
The maximal torus in a compact group plays a role analogous to that of the Cartan subalgebra in a complex semisimple Lie algebra. In particular, once a maximal torus has been chosen, one can define a root system and a Weyl group similar to what one has for semisimple Lie algebras. These structures then play an essential role both in the classification of connected compact groups and in the representation theory of a fixed such group.
The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:

The special unitary groups correspond to the root system
The odd spin groups correspond to the root system
The compact symplectic groups correspond to the root system
The even spin groups correspond to the root system
The exceptional compact Lie groups correspond to the five exceptional root systems G₂, F₄, E₆, E₇, or E₈
Fundamental group and center

It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its fundamental group. For compact Lie groups, there are two basic approaches to computing the fundamental group. The first approach applies to the classical compact groups,,, and and proceeds by induction on. The second approach uses the root system and applies to all connected compact Lie groups.
It is also important to know the center of a connected compact Lie group. The center of a classical group can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in. Thus, for example, the center of consists of nth roots of unity times the identity, a cyclic group of order.
In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus. The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system has trivial center. Thus, the compact group is one of very few simple compact groups that are simultaneously simply connected and center free. | and E8

Further examples

Amongst groups that are not Lie groups, and so do not carry the structure of a manifold, examples are the additive group Z_p of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions 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 groups.

Haar measure

Compact groups all carry a Haar measure, which will be invariant by both left and right translation. In other words, these groups are unimodular. Haar measure is easily normalized 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 Adolf 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.
If is a compact group and is the associated Haar measure, the Peter–Weyl theorem provides a decomposition of as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of.

Representation theory

The representation theory of compact groups was founded by the Peter–Weyl theorem. Hermann Weyl 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. The combination of the Peter–Weyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section.
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 representations ρ of G are into a unitary group and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im 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.

Representation theory of a connected compact Lie group

Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the rotation group SO, the special unitary group SU, and the special unitary group SU. We focus here on the general theory. See also the parallel theory of representations of a semisimple Lie algebra.
Throughout this section, we fix a connected compact Lie group K and a maximal torus T in K.

Representation theory of ''T''

Since T is commutative, Schur's lemma tells us that each irreducible representation of T is one-dimensional:
Since, also, T is compact, must actually map into.
To describe these representations concretely, we let be the Lie algebra of T and we write points as
In such coordinates, will have the form
for some linear functional on.
Now, since the exponential map is not injective, not every such linear functional gives rise to a well-defined map of T into. Rather, let denote the kernel of the exponential map:
where is the identity element of T.
Then for to give a well-defined map, must satisfy
where is the set of integers. A linear functional satisfying this condition is called an analytically integral element. This integrality condition is related to, but not identical to, the notion of integral element in the setting of semisimple Lie algebras.
Suppose, for example, T is just the group of complex numbers of absolute value 1. The Lie algebra is the set of purely imaginary numbers, and the kernel of the exponential map is the set of numbers of the form where is an integer. A linear functional takes integer values on all such numbers if and only if it is of the form for some integer. The irreducible representations of T in this case are one-dimensional and of the form