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 very nice class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include^[1]

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). This classification is described in more detail in the next subsection.

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.\,

Meanwhile, for connected compact Lie groups, we have the following result:^[2]

Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.

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.

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:

The compact symplectic group Sp(n), n ≥ 1
The special unitary group SU(n), n ≥ 3
The spin group Spin(n), n ≥ 7

or one of the five exceptional groups G₂, F₄, E₆, E₇, and E₈. For each of these groups, the center is known explicitly.

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,^[3] 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 (not necessarily Lie groups and not necessarily connected) was founded by the Peter–Weyl theorem.^[4] Hermann Weyl went on to give the detailed character theory of the compact connected Lie groups, based on maximal torus theory.^[5] The resulting Weyl character formula was one of the influential results of twentieth century mathematics. (This theory is described in detail 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 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.

Representation theory of T

Since T is commutative, Schur's lemma tells us that each irreducible representation $\rho$ of T is one-dimensional:

\rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}

.

Since, also, T is compact, $\rho$ must actually map into $S^{1}\subset \mathbb {C}$ .

To describe these representations concretely, we let ${\mathfrak {t}}$ be the Lie algebra of T and we write points $h\in T$ as

h=e^{H},\quad H\in {\mathfrak {t}}

.

In such coordinates, $\rho$ will have the form

\rho (e^{H})=e^{i\lambda (H)}

for some linear functional $\lambda$ on ${\mathfrak {t}}$ .

Now, since the exponential map $H\mapsto e^{H}$ is not injective, not every such linear functional $\lambda$ gives rise to a well-defined map of T into $S^{1}$ . Rather, let $\Gamma$ denote the kernel of the exponential map:

\Gamma =\{H\in {\mathfrak {t}}|e^{2\pi H}=\mathrm {Id} \}

,

where $\mathrm {Id}$ is the identity element of T. (We scale the exponential map here by a factor of $2\pi$ in order to avoid such factors elsewhere.) Then for $\lambda$ to give a well-defined map $\rho$ , $\lambda$ must satisfy

\lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma

,

where $\mathbb {Z}$ is the set of integers.^[6] A linear functional $\lambda$ 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.^[7]

Representation theory of K

Example of the weights of a representation of the group SU(3)

We now let $\Sigma$ denote a finite-dimensional irreducible representation of K (over $\mathbb {C}$ ). We then consider the restriction of $\Sigma$ to T. This restriction not irreducible unless $\Sigma$ 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 $\lambda$ as in the preceding subsection. If a given $\lambda$ occurs in the decomposition of the restriction of $\Sigma$ to T, we call $\lambda$ a weight of $\Sigma$ . 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 $R\subset {\mathfrak {t}}$ is very similar to the construction for complex semisimple Lie algebras. The root system R has all the usual properties of a root system, except that the elements of R may not span ${\mathfrak {t}}$ .^[8] We then choose a base $\Delta$ for R and we say that an integral element $\lambda$ is dominant if $\lambda (\alpha )\geq 0$ for all $\lambda \in \Delta$ . Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of $\Delta$ with non-negative coefficients.

The irreducible finite-dimensional representations of K are then classified by a theorem of the highest weight,^[9] which is closely related to the analogous theorem classifying representations of a semisimple Lie algebra. The result says that:

(1) every irreducible representation has highest weight,

(2) the highest weight is always a dominant, analytically integral element,

(3) two irreducible representations with the same highest weight are irreducible, and

(4) every dominant, analytically integral element arises as the highest weight of an irreducible representation.

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 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.^[10] (This phenomenon reflects that, in general, not every representation of the Lie algebra ${\mathfrak {k}}$ 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.^[11]

The Weyl character formula

In the closely related representation theory of semisimple Lie algebras, an important result about the representations is the Weyl character formula. In the Lie algebra setting, however, the 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 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.^[12] 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.

References

^ Hall 2015 Section 1.2
^ Bröcker 1985 Chapter V, Sections 7 and 8
^ Weil, André (1940), L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, vol. 869, Paris: Hermann
^ Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann., 97: 737–755, doi:10.1007/BF01447892.
^ Hall 2015 Part III
^ Hall 2015 Proposition 12.9
^ Hall 2015 Section 12.2
^ Hall 2015 Section 11.7
^ Hall 2015 Chapter 12
^ Hall 2015 Section 12.2
^ Hall 2015 Corollary 13.20
^ Hall 2015 Sections 12.4 and 12.5

Bibliography

Bröcker, Theodor; tom Dieck, Tammo (1985), Representations of Compact Lie Groups, Graduate Texts in Mathematics, vol. 98, Springer
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

[1] Hall 2015 Section 1.2

[2] Bröcker 1985 Chapter V, Sections 7 and 8

[3] Weil, André (1940), L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, vol. 869, Paris: Hermann

[4] Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann., 97: 737–755, doi:10.1007/BF01447892.

[5] Hall 2015 Part III

[6] Hall 2015 Proposition 12.9

[7] Hall 2015 Section 12.2

[8] Hall 2015 Section 11.7

[9] Hall 2015 Chapter 12

[10] Hall 2015 Section 12.2

[11] Hall 2015 Corollary 13.20

[12] Hall 2015 Sections 12.4 and 12.5

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