Compact topological group

A compact topological group or compact group is a topological Groupe G such as the topological Espace subjacent is compact. The compact groups are groups unimodulaires, whose compactness simplifies the study. These groups include/understand in particular the finished groups, and the groups of Dregs compact. Any compact topological group is limiting projective compact groups of Dregs.

Examples of compact groups

Group finished

See also: Group finished

Any finished group, provided with discrete topology, is a compact topological group. Indeed:

• Any finished unit provided with discrete topology is separate and admits a finished number of open; it is thus compact.
• Any application between spaces provided with discrete topology is continuous; the multiplication and the inversion are thus automatically continuous.

The study itself of the finished groups does not call upon the topological concepts. However:

• Of the properties concerning the compact groups is always checked for the finished groups.
• the discrete sub-groups of a compact group are finished groups.

Construction of compact groups

Compact groups can be built by using the general methods of Construction of the topological groups:

• the produces direct compact groups is a compact group.
• a Sous-groupe closed of a compact group is a compact group.
• the quotient of a compact group by a normal Sous-groupe closed is a compact group.
• the core of a clean continuous morphism between topological groups $G\; \backslash \; rightarrow\; H$ is a sub-group compact of G .
• the projective Limite of a family of compact topological groups is a compact group. In particular, the groups profinis are examples of compact groups.

Group of Dregs compact

See also: compact Group of Dregs

A compact group of Dregs is a Groupe of Dregs (real or complex) whose subjacent topological space is compact. Among them, one can quote the compact groups of traditional Dregs; of which here the list:

• the orthogonal Group $O\_n\; (\backslash \; mathbf\; \{R\})$, of dimension $n\; (n-1)\; /2$;
• the orthogonal special Group $SO\_n\; (\backslash \; mathbf\; \{R\})$, which is its Composante neutral;
• the unit Group $U\_n\; (\backslash \; mathbf\; \{C\})$, of dimension $n^2$;
• the unit special Group $SU\_n\; (\backslash \; mathbf\; \{C\})$, of dimension $n^2-1$;
• the compact Group symplectic $SP\; (N)$, of dimension $n\; (2n+1)$.

It is to be noticed that a compact group of Dregs complex is necessarily commutative.

Integration on the compact groups

On any topological group Locally compact and Separable G , there exists a invariant Mesure borélienne by the translations on the left, called Mesure of Haar, single except for multiplicative coefficient. It is finished on the compact parts of G . If G is itself compact, any measurement of Haar is finished, and it is possible to standardize it so that its mass is worth 1. One thus lays out on any compact topological group G of single a Mesure of probability which is invariant on the left by translations, that one names by abuse language the measurement of Haar of G , noted $\backslash \; lambda$ in the continuation of the article.

In practice, the measurement of Haar makes it possible to realize the objects on G to obtain invariants objects.

Unimodulaire

A topological group locally compact and separable is known as Unimodulaire if one (and thus all) measurement of Haar is invariant on the right. Any compact topological group is unimodulaire.

In general, translété on the right of a measurement of Haar $\backslash \; lambda$ by G is a measurement of Haar, therefore is written $\backslash \; Delta\; (G).\; \backslash \; lambda$. The real coefficient $\backslash \; Delta\; (G)$ is independent of the choice of $\backslash \; lambda$ and $\backslash \; Delta:\; G\; \backslash \; rightarrow\; \backslash \; mathbf\; \{R\}\; \_+^*$ is a continuous morphism of topological groups. If G is compact, the image of $\backslash \; Delta$ is a sub-group compact of $\backslash \; mathbf\; \{R\}\; \_+^*$: a fortiori, it is commonplace. The application $\backslash \; Delta$ is in this case constant equal to 1. Any measurement of Haar is invariant on the right by the translations.

Theorem of the point fixes of Kakutani

See also: Théorème of the point fixes of Kakutani

The following theorem was shown by Shizuo Kakutani in 1941:

• Théorème of the point fixes of Kakutani: Is V a topological vector Space locally convex separate, G a compact topological group, and $G\; \backslash \; times\; V\; \backslash \; rightarrow\; V$ a linear action continuous of G on V . Let us suppose data a convex part compacts not vacuum of V overall stable by the action of G . Then there exists a point C in C fixed by all the elements G of G ( G . C = C ).

The theorem of Kakutani gives an elementary proof of the existence of the measurement of Haar on a compact group G . Indeed, the space of finished real measurements $\backslash \; mathcal\; \{M\}\; (G)$ is by the Théorème of Riesz the topological Dual of the Espace of Banach of the real continuous functions $C\; (G,\; \backslash \; mathbf\; \{R\})$, provided with the uniform Norme of convergence. Boréliennes measurements of probability form a convex subset and *faiblement compact. The action of G on itself by translations on the left induced a linear action of G on $\backslash \; mathcal\; \{M\}\; (G)$, which is continuous for topology *faible. Convex the C is stable by G : in fact, the theorem of Kakutani applies and gives the existence of a fixed point of the action of G in C , in other words of a borélienne measurement of invariant probability on G by translation on the left. The existence of the measurement of Haar is thus established. (However, compactness does not make it possible to simplify the proof of unicity.)

Representation of the compact groups

Reductibility supplements

A representation of a topological group G is a linear continuous action of G on a real or complex topological vector space V . The representation is known as of finished size if V is of finished size: in this case, the representation can be defined by a continuous morphism $G\; \backslash \; rightarrow\; GL\; (V)$. It is known as unit if V is an Euclidean or square space and if $\backslash \; rho$ is with values in the orthogonal Groupe O (V) or unit U (V).

• Toute real or complex of size finished of a compact topological group '' G '' is equivalent to a unit representation. '''

Indeed, E a real or complex vector space of finished size and $\backslash \; rho\; are$: G \ rightarrow GL (E) a continuous morphism. Let us take an Euclidean or square structure unspecified on E . Let us pose for all vectors v and W :

. It is noted that $(.\; |.)$ defines a Forme sesquilinéaire on E and definite positive (thus an Euclidean or square structure). Let us check that it is invariant under the action of G . For H in G , and v and W in E , it comes: . The last equality comes from the invariance on the left of the measurement of Haar $\backslash \; lambda$.

A representation of a topological group G in a space V is known as Irréductible if there does not exist any vectorial subspace closed in V overall invariant by the linear action of G . It is known as completely reducible if space V is the direct sum of a family of closed subspaces invariants, whose induced Représentation on each one is irreducible.

• Any representation of a compact group of finished size is completely reducible.

Any unit representation of finished size is completely reducible (to read unit Représentation of a topological group). However, any representation of finished size equivalent to a completely reducible representation is itself completely reducible. From where the result by applying the preceding property.

Theorem of Peter-Weyl

See also: Theorem of Peter-Weyl

The theorem of Peter-Weyl was shown by Hermann Weyl (1885 - 1955) . The collection of the irreducible representations of finished size with equivalence close to a compact topological group G given forms a unit. Once fixed a measurement of Haar on G , one lays out of a natural representation of the group $G\; \backslash \; times\; G$ in the Algèbre hilbertienne $L^2\; (G)$ given by:

$(H,\; K)\; .f:\; G\; \backslash \; mapsto\; F\; (h^\; \{-\; 1\}\; gk)$ This representation is unit. The theorem of Peter-Weyl affirms that this representation is equivalent to the orthogonal sum of the tensorial products of the unit representations has G by their dual representations has ^*, where the unit representations are identified except for equivalence.

This theorem has noncommonplace consequences. It makes it possible for example to show that any compact topological group is limiting of a projective system of compact groups of Dregs.

Character

See also: Character of a compact topological group

Being given a complex representation of finished size of a compact topological group G , the associated character is defined by:

$\backslash \; chi\_\; \{\backslash \; rho\}\; (G)\; =Tr\; (\backslash \; rho\; (G))$ The characters of two equivalent representations are equal. One speaks about character and irreducible nature to indicate the characters respectively associated with a complex representation and an irreducible complex representation.

A central Fonction is a function (measurable, continues,…) $G\; \backslash \; rightarrow\; \backslash \; mathbf\; \{C\}$ constant on the classes of conjugation of G . The characters are examples of central functions. The collection of all the irreducible natures of a compact topological group trains a orthonormée family of $L^2\; (G,\; \backslash \; lambda)$. More exactly, it is a Base hilbertienne subspace of Hilbert of the central functions of class $L^2$.

An important example is the determination of the characters of the Tore and the classification of its representations.

Representation of the finished groups

See also: Representations of a group finished

The properties of the representations of the compact groups specialize for the finished groups. The study is simplified, in particular because the number of characters is finished. In particular, the irreducible characters of a group finished G form a base of the vector space of size finished of the central functions. The dimension of this space is thus the number of classes of conjugation of G .

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