# Compact group

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 $U ( n ,\ \mathbf C )$ of all unitary complex matrices of order $n$ , the group $\textrm{ O }( n ,\ \mathbf R )$ of all orthogonal real matrices of order $n$ (with the topology induced by the topology determined by the ordinary norm of the fields $\mathbf C$ and $\mathbf R$ , respectively) and, more generally, any compact real Lie group.

2) Totally-disconnected compact groups. Of this type is the group $\mathop{\rm GL}\nolimits ( n ,\ \mathbf Z _{p} )$ of invertible matrices of order $n$ with coefficients in the ring $\mathbf Z _{p}$ of $p$ -adic integers (with the topology induced by that determined by the $p$ -adic norm of $\mathbf Z _{p}$ ; 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 $G$ is locally connected and finite-dimensional, then $G$ 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 $G$ there is zero-dimensional subgroup $N$ (lying in the centre of $G$ ) such that $G / N$ is a real Lie group and, furthermore, some neighbourhood of the identity in $G$ is the direct product of the group $N$ 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 $Z$ 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 $G$ 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 $\rho$ for which $g \notin \mathop{\rm Ker}\nolimits \ \rho$ . 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 $G$ . The most important facts of this theory are as follows. Every continuous representation of a compact group $G$ 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 $G$ with respect to the invariant measure $\mu (g)$ . The action of the group $G$ on the functions by left and right translations determines on $L _{2} (G)$ the structure of a left and a right $G$ -module. The corresponding representations are respectively called the left and right regular representation of $G$ ; 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 $G$ and let $m _{ij} ^ \alpha (g)$ , $i ,\ j = 1 \dots n _ \alpha = \mathop{\rm dim}\nolimits \ R ^ \alpha$ , be the set of matrix elements of the representation $R ^ \alpha$ in some orthonormal basis. Then the functions $m _{ij} ^ \alpha (g)$ lie in $L _{2} (G)$ and form in it a complete orthogonal system, the norm of the function $m _{ij} ^ \alpha (g)$ being $n _ \alpha ^{-1}/2$ . Any continuous complex-valued function on $G$ can, to any desired degree of accuracy, be uniformly approximated by finite linear combinations of the functions $m _{ij} ^ \alpha (g)$ (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 $G$ 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 $n _ \alpha = \mathop{\rm dim}\nolimits \ R ^ \alpha$ ; furthermore, the sum of all $G$ -submodules in the $G$ -module $L _{2} (G)$ that are isomorphic to $R ^ \alpha$ is precisely the linear span of all the $m _{ij} ^ \alpha (g)$ .

#### 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