User:Maximilian Janisch/latexlist/Algebraic Groups/Compact group

This page is a copy of the article Compact group in order to test automatic LaTeXification. This article is not my work.

A topological group that is compact as a topological space. For example, every finite group (in the discrete topology) is a compact group. An algebraic group, even though it is a compact topological space (with respect to the Zariski topology), is not a topological group with respect to this topology and therefore is not a compact group.

The following groups are two important classes of compact groups.

1) Locally connected compact groups. Examples of such compact groups are the group $U ( n , C )$ of all unitary complex matrices of order $12$, the group $O ( n , R )$ of all orthogonal real matrices of order $12$ (with the topology induced by the topology determined by the ordinary norm of the fields $m$ and $R$, respectively) and, more generally, any compact real Lie group.

2) Totally-disconnected compact groups. Of this type is the group $GL ( n , Z _ { p } )$ of invertible matrices of order $12$ with coefficients in the ring $Z _ { p }$ of $D$-adic integers (with the topology induced by that determined by the $D$-adic norm of $Z _ { p }$; see Totally-disconnected space).

Any totally-disconnected compact group is a profinite group, and conversely, every profinite group is a totally-disconnected compact group. The totally-disconnected compact Hausdorff groups can be characterized as the compact groups of topological dimension zero. If $k$ is locally connected and finite-dimensional, then $k$ is a real Lie group [1]. The structure of a compact group of general type is to a certain degree determined by the structure of these two types of compact groups. In an arbitrary finite-dimensional compact group $k$ there is zero-dimensional subgroup $M$ (lying in the centre of $k$) such that $G / N$ is a real Lie group and, furthermore, some neighbourhood of the identity in $k$ is the direct product of the group $M$ and a real local Lie group (cf. Lie group, local). Every connected finite-dimensional compact group has the form $( P \times C ) / Z$, where $P$ is a simply-connected compact semi-simple real Lie group, $C$ is a finite-dimensional connected commutative compact group and $7$ is a finite central normal subgroup for which only the identity lies in $C$. The study of the structure of connected compact real Lie groups has led to a complete classification of them (see Lie group, compact); the structure of commutative compact groups is elucidated in the theory of Pontryagin duality. Any compact group (not necessarily finite-dimensional) is the projective limit of compact real Lie groups [2]. The topological structure of the above two types of compact groups is as follows: Every locally connected finite-dimensional compact group is a topological manifold, while every infinite zero-dimensional compact group with a countable base is homeomorphic to the perfect Cantor set.

The study of structure of compact groups is based on the fact that every compact group $k$ has a sufficient system of finite-dimensional linear representations, that is, for any element $g \in G$ there exists a continuous finite-dimensional linear representation $0$ for which $g \notin \operatorname { Ker } \rho$. This fact is one of the important results of the well-developed general theory of linear representations of compact groups. This theory makes essential use of the fact that every compact group has a two-sided invariant measure $\mu ( g )$ (a Haar measure), which enables one to define invariant integration on $k$. The most important facts of this theory are as follows. Every continuous representation of a compact group $k$ in a pre-Hilbert space is equivalent to a unitary representation. Let $L _ { 2 } ( G )$ be the Hilbert space of square-integrable complex-valued functions on $k$ with respect to the invariant measure $\mu ( g )$. The action of the group $k$ on the functions by left and right translations determines on $L _ { 2 } ( G )$ the structure of a left and a right $k$-module. The corresponding representations are respectively called the left and right regular representation of $k$; they are unitary and unitarily equivalent. Let $\{ R ^ { \alpha } : \alpha \in I \}$ be the family of all possible pairwise inequivalent finite-dimensional irreducible unitary representations of the compact group $k$ and let $m _ { i j } ^ { \alpha } ( g )$, $j = 1 , \ldots , n _ { \alpha } = \operatorname { dim } R ^ { \alpha }$, be the set of matrix elements of the representation $R ^ { \alpha }$ in some orthonormal basis. Then the functions $m _ { i j } ^ { \alpha } ( g )$ lie in $L _ { 2 } ( G )$ and form in it a complete orthogonal system, the norm of the function $m _ { i j } ^ { \alpha } ( g )$ being $n _ { \alpha } - 1 / 2$. Any continuous complex-valued function on $k$ can, to any desired degree of accuracy, be uniformly approximated by finite linear combinations of the functions $m _ { i j } ^ { \alpha } ( g )$ (the Peter–Weyl theorem). The characters of the irreducible unitary finite-dimensional representations are pairwise orthogonal and have norm 1. Continuous finite-dimensional unitary representations are equivalent if and only if their characters are equal. A continuous finite-dimensional unitary representation is irreducible if and only if the norm of its character (which lies in $L _ { 2 } ( G )$) is equal to 1. Every irreducible continuous unitary representation of the group $k$ in a Hilbert space is finite-dimensional. Every continuous unitary representation of the group in a Hilbert space is an orthogonal direct sum of unitary representations that are multiples of finite-dimensional irreducible representations. In particular, the multiplicity of the imbedding of the representation $R ^ { \alpha }$ in the right regular representation is equal to $n _ { \alpha } = \operatorname { dim } R ^ { \alpha }$; furthermore, the sum of all $k$-submodules in the $k$-module $L _ { 2 } ( G )$ that are isomorphic to $R ^ { \alpha }$ is precisely the linear span of all the $m _ { i j } ^ { \alpha } ( g )$.

References

[1]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[2]	A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940) MR0005741
[3]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[4]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013

References

[a1]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1970) MR0262773 Zbl 0213.40103
[a2]	D. Montgomery, L. Zippin, "Topological transformation groups" , Interscience (1955) MR0073104 Zbl 0068.01904
[a3]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002