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) $ .

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