Compact group

This article is about mathematics. For astronomy of galaxies, see galaxy group.

The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.

In mathematics, a compact (topological) group is a topological group whose topology is compact. 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

Lie groups form a very nice 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 groups O(n), the special orthogonal group SO(n) and its covering spin group Spin(n),
• the unitary group U(n) and the special unitary group SU(n),
• the symplectic group Sp(n),
• 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 (which already includes some redundancies).

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(G) which must be finite since G is compact. We therefore have a finite extension

$1\to G_{0}\to G\to \pi _{0}(G)\to 1.\,$

Now every compact, connected Lie group G₀ has a finite covering

$1\to A\to {\tilde {G}}_{0}\to G_{0}\to 1\,$

where $A\subset Z({\tilde {G}}_{0})$ is a finite abelian group and ${\tilde {G_{0}}}$ is a product of a torus and a compact, connected, simply-connected Lie group K:

${\tilde {G}}_{0}\cong \mathbb {T} ^{m}\times K.$

Finally, every compact, connected, simply-connected Lie group K is a product of compact, connected, simply-connected simple Lie groups K_i 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, 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 (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.

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. (See below.)

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.

Representation theory of a connected compact Lie group

The irreducible finite-dimensional representations of a connected compact group K are classified by a theorem of the highest weight, which is closely related to the analogous theorem classifying representations of a semisimple Lie algebra. Let T be a maximal torus of K and let t be the Lie algebra of T. One can construct a root system R inside t, which has all the usual properties of root system, except that the roots may not span t.

Given a representation $\Sigma$ of K, we can form the associated representation $\sigma$ of the Lie algebra k of K. One can then define the weights of $\Sigma$ to be the weights of $\sigma$. The theorem of the highest weight for representations of K is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an integral element is different. The weights $\lambda$ of a representation $\Sigma$ are analytically integral, meaning that if H in t satisfies $\mathrm {exp} (2\pi H)=e$, then $\langle \lambda ,H\rangle$ must be an integer. Every analytically integral element is integral in the Lie algebra sense, but not the other way around. (This phenomenon reflects that, in general, not every representation of the Lie algebra k comes from a representation of the group K.) On the other hand, if K is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.

In Weyl's analysis of the representations of K, the hardest part of the theorem, showing that every dominant, analytically integral element is actually the highest weight of some representation, is proved in totally different way from the usual Lie algebra construction using Verma modules. In Weyl's approach, the construction is based on the Peter–Weyl theorem and an analytic proof of the Weyl character formula. Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K.

Duality

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.

From compact to non-compact groups

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.

See also

• locally compact group
• p-compact group
• Protorus

Bibliography

• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 0-387-40122-9
• Hofmann, Karl H.; Morris, Sidney A. (1998), The structure of compact groups, Berlin: de Gruyter, ISBN 3-11-015268-1