In mathematics, a compact (topological) group is a topological group whose topology is compact (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.
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.
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
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 finitely many compact, connected, simply-connected simple Lie groups Ki 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.
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 (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:
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.
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.
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. (We scale the exponential map here by a factor of in order to avoid such factors elsewhere.) 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 (scaled) 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
We now let denote a finite-dimensional irreducible representation of K (over ). We then consider the restriction of to T. This restriction is not irreducible unless 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 as in the preceding subsection. If a given occurs at least once in the decomposition of the restriction of to T, we call a weight of . 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 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 . We then choose a base for R and we say that an integral element is dominant if for all . Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of 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 of a representation 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 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.
If is representation of K, we define the character of to be the function given by
This function is easily seen to be a class function, i.e., for all and in K. Thus, 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.
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 . The general theory then tells us that for each , there is a unique irreducible representation of SU(2) with highest weight .
Much information about the representation corresponding to a given 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:
(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)
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 weights, each with multiplicity 1, the dimension of the representation is . Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.
We now outline the proof of the theorem of the highest weight, following the original argument of Hermann Weyl. We continue to let be a connected compact Lie group and a fixed maximal torus in . 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 is an irreducible representation with highest weight , then the character of satisfies:
for all in the Lie algebra of . Here is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the 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 denote the function on the right-hand side of the character formula. We show that even if is not known to be the highest weight of a representation, is a well-defined, Weyl-invariant function on , which therefore extends to a class function on . Then using the Weyl integral formula, one can show that as ranges over the set of dominant, analytically integral elements, the functions form an orthonormal family of class functions. We emphasize that we do not currently know that every such 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 , and these functions are orthonormal.
Now comes the conclusion. The set of all —with 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 '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 that is not the highest weight of a representation, then the corresponding would not be the character of a representation. Thus, the characters would be a proper subset of the set of '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 's). Thus, every must actually be the highest weight of a representation.
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.