Compact group

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 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). This classification is described in more detail in the next subsection.
Classification.
Given any compact Lie group "G" one can take its identity component "G"0, which is connected. The quotient group "G"/"G"0 is the group of components π0("G") 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. (For information about the center, see the section below on fundamental group and center.)
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 the following:
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 (for a fixed maximal torus), 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 formula_5 and that is not contained in any larger subgroup of this type. A basic example is the case formula_6, in which case we may take formula_7 to be the group of diagonal elements in formula_8. A basic result is the "torus theorem" which states that every element of formula_8 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 formula_10 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 (described above) and in the representation theory of a fixed such group (described below).
The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:
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 formula_11, formula_20, formula_21, and formula_15 and proceeds by induction on formula_23. 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 formula_24 can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in formula_24. (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 formula_11 consists of "n"th roots of unity times the identity, a cyclic group of order formula_23.
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 formula_28 has trivial center. Thus, the compact formula_28 group is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are formula_30 and formula_31.)
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.
If formula_8 is a compact group and formula_33 is the associated Haar measure, the Peter–Weyl theorem provides a decomposition of formula_34 as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of formula_8.
Representation theory.
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.
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 , 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".
Representation theory of "T".
Since "T" is commutative, Schur's lemma tells us that each irreducible representation formula_36 of "T" is one-dimensional:
Since, also, "T" is compact, formula_36 must actually map into formula_39.
To describe these representations concretely, we let formula_40 be the Lie algebra of "T" and we write points formula_41 as
In such coordinates, formula_36 will have the form
for some linear functional formula_45 on formula_40. 
Now, since the exponential map formula_47 is not injective, not every such linear functional formula_45 gives rise to a well-defined map of "T" into formula_5. Rather, let formula_50 denote the kernel of the exponential map:
where formula_52 is the identity element of "T". (We scale the exponential map here by a factor of formula_53 in order to avoid such factors elsewhere.) 
Then for formula_45 to give a well-defined map formula_36, formula_45 must satisfy
where formula_58 is the set of integers. A linear functional formula_45 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 formula_5 of complex numbers formula_61 of absolute value 1. The Lie algebra is the set of purely imaginary numbers, formula_62 and the kernel of the (scaled) exponential map is the set of numbers of the form formula_63 where formula_23 is an integer. A linear functional formula_45 takes integer values on all such numbers if and only if it is of the form formula_66 for some integer formula_67. The irreducible representations of "T" in this case are one-dimensional and of the form
Representation theory of "K".
We now let formula_69 denote a finite-dimensional irreducible representation of "K" (over formula_70). We then consider the restriction of formula_69 to "T". This restriction is not irreducible unless formula_69 is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of "T". (Note that a given irreducible representation of "T" may occur more than once.) Now, each irreducible representation of "T" is described by a linear functional formula_45 as in the preceding subsection. If a given formula_45 occurs at least once in the decomposition of the restriction of formula_69 to "T", we call formula_45 a weight of formula_69. The strategy of the representation theory of "K" is to classify the irreducible representations in terms of their weights.
We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on weights in representation theory. We need the notion of a root system for "K" (relative to a given maximal torus "T"). The construction of this root system formula_78 is very similar to the construction for complex semisimple Lie algebras. Specifically, the weights are the nonzero weights for the adjoint action of "T" on the complexified Lie algebra of "K". The root system "R" has all the usual properties of a root system, except that the elements of "R" may not span formula_40. We then choose a base formula_80 for "R" and we say that an integral element formula_45 is dominant if formula_82 for all formula_83. Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of formula_80 with non-negative coefficients.
The irreducible finite-dimensional representations of "K" are then classified by a theorem of the highest weight, which is closely related to the analogous theorem classifying representations of a semisimple Lie algebra. The result says that: 
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 formula_45 of a representation formula_69 are analytically integral in the sense described in the previous subsection. 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 formula_87 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.
The Weyl character formula.
If formula_88 is representation of "K", we define the character of formula_89 to be the function formula_90 given by
This function is easily seen to be a class function, i.e., formula_92 for all formula_93 and formula_94 in "K". Thus, formula_95 is determined by its restriction to "T".
The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the Peter–Weyl theorem, is that the characters form an orthonormal basis for the set of square-integrable class functions in "K". A second key result is the Weyl character formula, which gives an explicit formula for the character—or, rather, the restriction of the character to "T"—in terms of the highest weight of the representation. 
In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established "after" the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, 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 a 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".
The SU(2) case.
We now consider the case of the compact group SU(2). The representations are often considered from the Lie algebra point of view, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form
According to the example discussed above in the section on representations of "T", the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers formula_33. The general theory then tells us that for each formula_33, there is a unique irreducible representation of SU(2) with highest weight formula_33. 
Much information about the representation corresponding to a given formula_33 is encoded in its character. Now, the Weyl character formula says, in this case, that the character is given by
We can also write the character as sum of exponentials as follows:
From this last expression and the standard formula for the character in terms of the weights of the representation, we can read off that the weights of the representation are 
each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are formula_104 weights, each with multiplicity 1, the dimension of the representation is formula_104. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.
An outline of the proof.
We now outline the proof of the theorem of the highest weight, following the original argument of Hermann Weyl. We continue to let formula_8 be a connected compact Lie group and formula_7 a fixed maximal torus in formula_8. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.
The tools for the proof are the following:
With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the Weyl character formula. The formula states that if formula_89 is an irreducible representation with highest weight formula_45, then the character formula_95 of formula_89 satisfies:
for all formula_115 in the Lie algebra of formula_7. Here formula_36 is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of formula_118 in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the formula_119 norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.
Next, we let formula_120 denote the function on the right-hand side of the character formula. We show that "even if formula_45 is not known to be the highest weight of a representation", formula_120 is a well-defined, Weyl-invariant function on formula_7, which therefore extends to a class function on formula_8. Then using the Weyl integral formula, one can show that as formula_45 ranges over the set of dominant, analytically integral elements, the functions formula_120 form an orthonormal family of class functions. We emphasize that we do not currently know that every such formula_45 is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions formula_120, and these functions are orthonormal.
Now comes the conclusion. The set of all formula_120—with formula_45 ranging over the dominant, analytically integral elements—forms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the formula_120's. And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some formula_45 that is not the highest weight of a representation, then the corresponding formula_120 would not be the character of a representation. Thus, the characters would be a "proper" subset of the set of formula_120's. But then we have an impossible situation: an orthonormal "basis" (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of formula_120's). Thus, every formula_45 must actually be the highest weight of a representation.
Duality.
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.
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.