nLab
topological group

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Group Theoryc

Contents

Idea

A topological group is a topological space with a continuous group structure: a group object in the category Top.

Definition

Definition

A topological group is

  1. a group, hence

    1. a set GG,

    2. a neutral element eGe \in G,

    3. a associative unitality function

    4. ()():G×GG(-)\cdot (-) \;\colon\; G \times G \to G,

    5. a function () 1:GG(-)^{-1} \;\colon\; G \to G such that gg 1=e=g 1gg \cdot g^{-1} = e = g^{-1} \cdot g for all gGg \in G;

  2. a topology τ GP(G)\tau_G \subset P(G) giving GG the structure of a topological space

such that the operations () 1(-)^{-1} and ()()(-)\cdot (-) are continuous functions (the latter with respect to the product topology).

Properties

General

Lemma

(open subgroups of topological groups are closed)

Every open subgroup HGH \subset G of a topological group is closed.

(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)

Proof

The set of HH-cosets is a cover of GG by disjoint open subsets. One of these cosets is HH itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.

Proposition

(connected locally compact topological groups are sigma-compact)

Every connected locally compact topological group is sigma-compact.

Every locally compact topological group is paracompact.

(e.g. Arhangel’skii-Tkachenko 08, cor. 3.1.4, cor. 3.1.5)

Proof

By assumption of local compactness, there exists a compact neighbourhood C eGC_e \subset G of the neutral element. We may assume without restriction of generality that with gC eg \in C_e any element, then also the inverse element g 1C eg^{-1} \in C_e.

For if this is not the case, then we may enlarge C eC_e by including its inverse elements, and the result is still a compact neighbourhood of the neutral element: Since taking inverse elements () 1:GG(-)^{-1} \colon G \to G is a continuous function, and since continuous images of compact spaces are compact, it follows that also the set of inverse elements to elements in C eC_e is compact, and the union of two compact subspaces is still compact (obviously, otherwise see this prop).

Now for nn \in \mathbb{N}, write C e nGC_e^n \subset G for the image of k{1,n}C ek{1,,n}G\underset{k \in \{1, \cdots n\}}{\prod} C_e \subset \underset{k \in \{1, \cdots, n\}}{\prod} G under the iterated group product operation k{1,,n}GG\underset{k \in \{1, \cdots, n\}}{\prod} G \longrightarrow G.

Then

HnC e nG H \coloneqq \underset{n \in \mathbb{N}}{\cup} C_e^n \;\subset\; G

is clearly a topological subgroup of GG.

Observe that each C e nC_e^n is compact. This is because k{1,,n}C e\underset{k \in \{1, \cdots, n\}}{\prod}C_e is compact by the Tychonoff theorem, and since continuous images of compact spaces are compact. Thus

H=nC e n H = \underset{n \in \mathbb{N}}{\cup} C_e^n

is a countable union of compact subspaces, making it sigma-compact. Since locally compact and sigma-compact spaces are paracompact, this implies that HH is paracompact.

Observe also that the subgroup HH is open, because it contains with the interior of C eC_e a non-empty open subset Int(C e)HInt(C_e) \subset H and we may hence write HH as a union of open subsets

H=hHInt(C e)h. H = \underset{h \in H}{\cup} Int(C_e) \cdot h \,.

Finally, as indicated in the proof of Lemma 1, the cosets of the open subgroup HH are all open and partition GG as a disjoint union space (coproduct in Top) of these open cosets. From this we may draw the following conclusions.

  • In the particular case where GG is connected, then there is just one such coset, namely HH itself. The argument above thus shows that a connected locally compact topological group is σ\sigma-compact and (by local compactness) also paracompact.

  • In the general case, all the cosets are homeomorphic to HH which we have just shown to be a paracompact group. Thus GG is a disjoint union space of paracompact spaces. This is again paracompact (by this prop.).

Uniform structure

A topological group GG carries two canonical uniformities: a right and left uniformity. The right uniformity consists of entourages l,U\sim_{l, U} where x l,Uyx \sim_{l, U} y if xy 1Ux y^{-1} \in U; here UU ranges over neighborhoods of the identity that are symmetric: gUg 1Ug \in U \Leftrightarrow g^{-1} \in U. The left uniformity similarly consists of entourages r,U\sim_{r, U} where x r,Uyx \sim_{r, U} y if x 1yUx^{-1} y \in U. The uniform topology for either coincides with the topology of GG.

Obviously when GG is commutative, the left and right uniformities coincide. They also coincide if GG is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.

Let GG, HH be topological groups, and equip each with their left uniformities. Let f:GHf: G \to H be a group homomorphism.

Proposition

The following are equivalent:

  • The map ff is continuous at a point of GG;

  • The map ff is continuous;

  • The map ff is uniformly continuous.

Proof

Suppose ff is continuous at gGg \in G. Since neighborhoods of a point xx are xx-translates of neighborhoods of the identity ee, continuity at gg means that for all neighborhoods VV of eHe \in H, there exists a neighborhood UU of eGe \in G such that

f(gU)f(g)Vf(g U) \subseteq f(g) V

Since ff is a homomorphism, it follows immediately from cancellation that f(U)Vf(U) \subseteq V. Therefore, for every neighborhood VV of eHe \in H, there exists a neighborhood UU of eGe \in G such that

xy 1Uf(x)f(y) 1=f(xy 1)Vx y^{-1} \in U \Rightarrow f(x) f(y)^{-1} = f(x y^{-1}) \in V

in other words such that x Uyf(x) Vf(y)x \sim_U y \Rightarrow f(x) \sim_V f(y). Hence ff is uniformly continuous with respect to the right uniformity. By similar reasoning, ff is uniformly continuous with respect to the right uniformity.

Unitary representation on Hilbert spaces

Definition

A unitary representation RR of a topological group GG in a Hilbert space \mathcal{H} is a continuous group homomorphism

R:G𝒰() R \colon G \to \mathcal{U}(\mathcal{H})

where 𝒰()\mathcal{U}(\mathcal{H}) is the group of unitary operators on \mathcal{H} with respect to the strong topology.

Remark

Here 𝒰()\mathcal{U}(\mathcal{H}) is a complete, metrizable topological group in the strong topology, see (Schottenloher, prop. 3.11).

Remark

In physics, when a classical mechanical system is symmetric, i.e. invariant in a proper sense, with respect to the action of a topological group GG, then an unitary representation of GG is sometimes called a quantization of GG. See at geometric quantization and orbit method for more on this.

Why the strong topology is used

The reason that in the definition of a unitary representation, the strong operator topology on 𝒰()\mathcal{U}(\mathcal{H}) is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.

Example: let GG be a compact Lie group and L 2(G)L^2(G) be the Hilbert space of square integrable measurable functions with respect to its Haar measure. The right regular representation of GG on L 2(G)L^2(G) is defined as

R:G𝒰(L 2(G)) R: G \to \mathcal{U}(L^2(G))
g(R g:f(x)f(xg)) g \mapsto (R_g: f(x) \mapsto f(x g))

and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.

Which topological groups admit Lie group structure?

Protomodularity

Proposition

The category TopGrp of topological groups and continuous group homomorphisms between them is a protomodular category.

A proof is spelled out by Todd Trimble here on MO.

Examples

References

  • Alexander Arhangel’skii, Mikhail Tkachenko, Topological Groups and Related Structures, Atlantis Press 2008

The following monograph is not particulary about group representations, but some content of this page is based on it:

Revised on May 22, 2017 12:56:46 by Urs Schreiber (92.218.150.85)