Compact group

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In mathematics, compact groups are topological groups whose topology is compact . Compact groups generalize finite groups with the discrete topology and many properties can be transferred. There is a well-understood representation theory for compact groups .

The unit circle in the complex plane is a Lie group with the complex multiplication

In the following we assume that all topological groups are Hausdorff spaces .

Compact Lie groups

The Lie groups form a class of topological groups and there is a particularly well-developed theory for compact Lie groups. Basic examples of compact Lie groups are:

  • the circle group and the torus groups
  • the orthogonal groups and the special orthogonal groups as well as their overlays and the spin groups
  • the unitary groups and the special unitary groups
  • the symplectic groups
  • The compact forms of the exceptional Lie groups , , , and .

The classification theorem for compact Lie groups says that this list is complete except for finite extensions and overlay groups (and already contains redundancies). This classification is described in more detail in the following paragraph.

classification

For a given compact Lie group, let the connected component be one , which is a connected normal divisor . The quotient group is the group of the components that has to be finite because of the compactness of . We thus have a finite extension:

For connected, compact Lie groups we have the following result:

Theorem: Every connected, compact Lie group is (except for isomorphism) the quotient of the product of a connected, simply connected , compact Lie group and a torus group after a finite central subgroup .

In this way, the classification of the connected Lie groups can in principle be traced back to the knowledge of the compact, connected, simply connected Lie groups and their centers. (For more information on the Center, see the Fundamental Group and Center section below.)

Every simply connected, compact Lie group is the product of simply connected, compact, and simple groups, each of which is isomorphic to exactly one of the following:

  • the compact, symplectic group
  • the special, unitary group
  • the spin group
  • one of the exceptional groups , , , or .

The conditions on were set to exclude isomorphisms between the groups on the list. The center is known for each of these groups. The classification takes place via assigned root systems (for a fixed maximum torus, see below), which in turn can be classified using Dynkin diagrams .

The classification of the simply connected, compact Lie groups is the same as the classification of the complex, semi-simple Lie algebras . If, in fact, is a simply connected, compact Lie group, then the complexification of the corresponding Lie algebra is semi-simple. Conversely, every complex semi-simple Lie algebra has a real form that is isomorphic to a simply connected, compact Lie group.

Maximum tori and root systems

The main idea in the study of a contiguous, compact Lie group is the concept of the maximal torus , a subgroup that is isomorphic to the product of multiple copies of and is not included in any other subgroup of this type. A typical example is the subgroup of diagonal matrices in . The so-called theorem of the maximal torus is a fundamental result, according to which every element from lies in a maximal torus and every two maximal tori are conjugated to each other .

A maximal torus in a coherent, compact Lie group plays a role similar to that of the Cartan subalgebra of a complex, semi-simple Lie algebra. In particular, after choosing a maximal torus, one can define a root system and a Weyl group , much like in the theory of semi-simple Lie algebras . These structures then play an essential role in the classification of the connected, compact Lie groups (as described above) and in their representation theory (see below).

The following root systems appear in the classification of simply connected, compact Lie groups:

  • The special unitary group belongs to the root system .
  • The odd spin group belongs to the root system .
  • The compact, symplectic group belongs to the root system .
  • The straight spin group belongs to the root system .
  • The exceptional compact Lie groups are among the five exceptional root systems , , , or .

Fundamental group and center

It is important to know of a connected, compact Lie group whether it is simply connected, and if not, to determine its fundamental group . There are two basic approaches to this for compact Lie groups. The first is the classical, compact groups , , and and used induction by . The second approach uses the root systems and works for all coherent, compact Lie groups.

In the above classification, it is also important to know the center of the connected, compact Lie group. The centers of the classical groups can easily be calculated “by hand”, in most cases they are simply suitable multiples of the identity. The group is an exception here because as an Abelian group it coincides with its center, which therefore contains elements that are not multiples of the identity. For example, the center of consists of the -th roots of unity times identity, which is a cyclic group of order .

In general, the center can be described using the root system and the kernel of the exponential map of the maximum torus. This general method shows, for example, that the simply connected, compact group to the exceptional root system has a trivial center. This makes the compact -group one of the few examples of compact Lie groups that are at the same time simply connected and have a trivial center, the others being and .

Further examples

The compact groups, which are not Lie groups and therefore do not have the structure of a manifold , include the solenoid and the additive group of p -adic integers and groups constructed from them. Indeed, every pro-finite group is compact. This means that Galois groups are compact, a fundamental result of the theory of algebraic extensions of finite degree .

Pontryagin duality provides a rich arsenal of commutative, compact groups. These are the dual groups of discrete groups.

The hairy measure

Compact groups have a hair-like measure that is invariant with regard to both left and right translations , because the modular function maps the group to a compact subgroup of , so it must be constant equal to 1. In other words, compact groups are unimodular . Therefore, the Haar measure can easily be normalized to a probability measure such as on the circle group.

Such a hair-cut measure can easily be calculated in many cases. The Haar measure on the orthogonal groups was already known to Adolf Hurwitz , and in the case of Lie groups it can be described as an invariant differential form. In pro-finite groups there are many subgroups with finite indexes , so that the hairy measure of a minor class is equal to the reciprocal value of the index. Therefore, integrals with regard to Haar's measure can often be calculated directly, which is often used in number theory .

If there is a compact group with a Haar's measure , then the Peter-Weyl theorem provides a decomposition of the Hilbert space as an orthogonal sum of finite-dimensional subspaces on which the group operates as an irreducible matrix representation .

Representation theory

The representation theory of a compact group (not necessarily a Lie group and also not necessarily connected) was established by the Peter-Weyl theorem . Hermann Weyl developed this into a detailed character theory based on the maximum tori . The resulting Weylian character formula was an influential result for the mathematics of the twentieth century. The combination of Peter-Weyl's theorem and Weyl's character formula led Weyl to a complete classification of the representations of connected Lie groups, which is described in the following section.

Weyl's work and Cartan's theorem on Lie groups provide an overview of the representation theory of compact groups . According to Peter-Weyl's theorem, the images of the irreducible, unitary representations of lie in the unitary groups (finite dimension) and are closed subgroups of the unitary group because of their compactness. According to Cartan's theorem, the image must be a Lie subgroup of the unitary group. If there is no Lie group itself, it must have a non-trivial core. In this way one can construct an inverse system of finite-dimensional unitary representations with ever smaller kernels, so that finally compact Lie groups are identified with an inverse limit . The fact that you get a faithful representation of in the Limes is a further consequence of Peter-Weyl's theorem.

The unknown part of representation theory of compact groups is, roughly speaking, reduced to the complex representation theory of finite groups . This theory is very extensive, but qualitatively well understood.

Representation theory of a coherent, compact Lie group

Some simple cases of representation theory of compact Lie groups can be computed by hand, such as the representations of the rotation group or the special unitary groups and . See also the parallel representation theory of Lie algebras .

In this section we consider a fixed, connected, compact Lie group and a fixed maximal torus.

Representation theory by T

Since is commutative, we know from Schur's lemma that an irreducible representation of must be one-dimensional:

.

Since it is also compact, it even has to be mapped.

For a concrete description of these representations, let the Lie algebra of and we write points as

Regarding such coordinates, the form takes

a linear functional on . Since the exponential map is not injective, not every linear functional defines a map in this way . Namely, be the core of the exponential mapping

,

where the neutral element of is (we have scaled the exponential mapping here with the factor in order to avoid it in other places). Then , in order to produce a well-defined mapping using the above formula , the condition

satisfy, where is the set of integers. A linear functional that fulfills this condition is called an analytically integer element. This integer condition corresponds, albeit not in all details, to analog integer conditions from the theory of semi-simple Lie algebras.

Let's look at the simplest case , the set of complex numbers of magnitude 1. The associated Lie algebra is the set of purely imaginary numbers and the core of the (scaled) exponential mapping consists of the numbers . A linear functional takes on integer values ​​for all these numbers if and only if it is of the form for an integer . The associated irreducible representations are in this case

Representation theory by K

Now be a finite-dimensional, irreducible representation of (over ). Then we consider the constraint of on . Although this is not irreducible (unless it is one-dimensional), it breaks down into a direct sum of irreducible representations of . (Note that irreducible representations can occur several times, one says with multiplicity). Now every irreducible representation of the above is described by a linear functional . Every such thing that occurs at least once in the decomposition of the restriction of to is called a weight of . The strategy pursued in representation theory by is now the classification of the irreducible representations by means of their weights.

We now briefly describe the structures needed to formulate the sentence. We need the notion of the root system of (relative to the chosen torus ). The construction of this root system is similar to the construction in the theory of complex, semi-simple Lie algebras. More precisely, the weights of are the weights of the adjoint group action of on the complexified Lie algebra of . The root system has the usual properties of a root system with the exception that the elements of are not quite spanning. We then choose a base of and say an integer element is dominant if for all . Finally, we say that one weight is greater than another if the difference can be expressed as a linear combination of elements from with non-negative coefficients.

The irreducible, finite-dimensional representations of are then classified in analogy to the theory of representations of semi-simple Lie algebras by the theorem of highest weight . This says:

(1) Every irreducible representation has the highest weight.
(2) The highest weight is always a dominant, analytically integer element.
(3) Two irreducible representations with the same highest weight are isomorphic.
(4) Each dominant, analytically integer element appears as the highest weight of an irreducible representation.

This theorem of highest weight for representations of is almost the same as for semi-simple Lie algebras, with one important exception: the concepts of the integer element are different. The weights of a representation are analytically integral in the sense described above. Every analytically integral element is also integral in the Lie algebra sense, but not the other way around. This phenomenon reflects the fact that not every representation of the associated Lie algebra comes from a group representation of . On the other hand , if it is simply connected, then the set of the possible highest weights in the group sense is the same as the set of the possible highest weights in the sense of Lie algebras.

Weyl's character formula

If there is a representation of , then the character is through

defined function, where is the lane mapping . It's easy to see that the character is a class function , that is, it applies

   for everyone    .

Therefore it is already determined by its restriction on .

The study of characters is an important part of compact group representation theory. A decisive result, which is a corollary to the Peter-Weyl theorem, says that the characters form an orthonormal basis of the square-integrable class functions . A second key result is Weyl's character formula, which provides an explicit formula for the characters, more precisely for the limitations of the characters , by means of the highest weight of the representation.

In the closely related representation theory of semi-simple Lie algebras, Weyl's character formula is an additional result that is proven according to the classification theorem. In Weyl's analysis of compact groups, on the other hand, the character formula is an essential part of the classification itself. In particular, the most difficult part of the proof, namely that every dominant, analytically integer weight comes from a representation, is proven very differently than the usual Lie algebra construction using Verma modules . Weyl's approach is based on Peter-Weyl's theorem and an analytical proof of the character formula. Finally, the irreducible representations from in the space of continuous functions are realized.

The case of the SU (2)

To clarify what has been said so far, let us consider the case of the unitary group . The representations are usually viewed from the standpoint of Lie algebra, but here we take the group view. We choose the set of matrices as the maximum torus

.

As discussed above in the section on representation theory of T , the analytically integer elements are also represented here by integers, so that the dominant, analytically integer elements are the non-negative integers . The general theory therefore gives us an irreducible representation of the highest weight for each .

The character, which codes information for representation, is based on Weyl's character formula

given. We can also write this as the sum of exponential functions as follows:

(If one applies the formula for the finite geometric series to this expression, one can again derive the first-mentioned formula from it.) From this last expression and the Weylian character formula one can read off that the weights of this representation

are, each with multiplicity 1. The weights are the integers that appear in the exponents of the above sum, their multiplicities are the coefficients of the associated exponential terms. Since we have all in all weights with multiplicity 1, the dimension of the representation is the same . This shows how one can obtain information about the representations that are usually obtained from calculations in Lie algebras.

duality

The subject of how to win back a group from its representation theory is dealt with in what is known as the Tannaka-Kerin duality .

From compact to non-compact groups

The influence of the theory of compact groups on non-compact groups was formulated by Weyl in his so-called unitary trick. Within a semisimple Lie group sits a maximal, compact subgroup, and the representation theory of the semisimple Lie groups, as it was largely developed by Harish-Chandra , makes extensive use of the restriction of a representation to such a subgroup, where then the Weyl's character theory can be applied.

See also

Individual evidence

  1. Hall 2015, Section 1.2
  2. Bröcker 1985, Chapter V, Sections 7 and 8
  3. Hall 2015, Chapter 11
  4. Hall 2015, Section 7.7
  5. Hall 2015, Section 13.8
  6. ^ André Weil: L'intégration dans les groupes topologiques et ses applications , Actualités Scientifiques et Industrielles, Volume 869, Hermann-Verlag 1941
  7. ^ F. Peter, H. Weyl: The completeness of the primitive representations of a closed continuous group , Math. Ann. (1927), Volume 97, pages 737-755
  8. Hall 2015, Part III
  9. Hall 2015, sentence 12.9
  10. Hall 2015, Section 12.2
  11. Hall 2015, Section 11.7
  12. Hall 2015, Section 12
  13. Hall 2015, Section 12.2
  14. ^ Hall 2015, Corollary 13.20
  15. Hall 2015, Sections 12.4 and 12.5

literature

  • Theodor Bröcker, Tammo tom Dieck: Representations of Compact Lie Groups , Graduate Texts in Mathematics 98, Springer-Verlag (1985), ISBN 978-3-642-05725-0
  • Brian C. Hall: Lie Groups, Lie Algebras, and Representations An Elementary Introduction , Graduate Texts in Mathematics, 222 (2nd ed.), Springer-Verlag (2015), ISBN 978-3-319-13466-6
  • Karl A. Hofmann, Sidney A. Morris: The structure of compact groups , de Gruyter (1998), ISBN 3-11-015268-1