Compact group
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In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. Such groups G have a well-understood theory, in relation to group actions and representation theory, at least when G is also assumed to be Hausdorff. This doesn't cover all interesting cases, since affine algebraic groups carry the Zariski topology which is not Hausdorff, unless G happens to be finite. But it is the compact Hausdorff groups, natural generalisations of finite groups with their discrete topology, that have the properties that carry over in significant fashion. In the following the Hausdorff property will be assumed.
Examples of compact groups include the unit circle, which is the first in both the families of unitary groups U(n) and special orthogonal groups SO(n); and also the related special unitary groups SU(n) and orthogonal groups O(n) of definite quadratic forms. There are further compact Lie groups known, which now play an important role in theoretical physics as well as mathematics. Also every torus is a compact group.
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.
Compact groups all carry a Haar measure, which will be invariant by both left and right translation (since the modulus function must be a continuous homomorphism to the positive multiplicative reals, and so 1). It is easily normalised to be a probability measure. 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 be given by a 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.
The representation theory of compact groups was founded by the Peter-Weyl theorem. Weyl went on to give the detailed character theory of the compact connected Lie groups, based on maximal torus theory; a combination of these results and Cartan's theorem shows that the whole representation theory is thereby, roughly speaking, thrown back onto the complex representations of finite groups. The latter theory is rather rich in detail, but qualitatively well understood.
The topic of recovering a compact group from its representation theory is the subject of the Tannaka-Krein duality, now often recast in term 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 restriction to such a subgroup, and also the model of Weyl's character theory.