Қырылымды топологиялық топтар

Topological group with compact topology
mathematics

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). 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.

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 (The group SO(2) is an exception—the center is the whole group, even though most elements are not roots of the identity.) 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. (The others are and .)

Amongst groups that are not Lie groups, and so do not carry the structure of a 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 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.

Compact groups all carry a Haar measure, which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (R+, ×), and so 1). 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 .

The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) 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 (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.

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(3), the special unitary group SU(2), and the special unitary group SU(3). 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.

The topic of recovering a compact group from its representation theory is the subject of the Tannaka–Krein duality, now often recast in terms of Tannakian category theory.

The influence of the compact group theory on non compact groups was formulated by Weyl in his unitarian trick. 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.