nLab
topological group

Context

Topology

Group Theory

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 an internal group object in the category of topological spaces.

More explicitly, it is a group equipped with a topology such that the multiplication and inversion maps are continuous.

Properties

Uniform structure

A topological group GG carries two canonical uniformities: a right and left uniformity. The left 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 right 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 left uniformities. By similar reasoning, ff is uniformly continuous with respect to the right uniformities.

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.

References

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

  • Martin Schottenloher: A mathematical introduction to conformal field theory. Springer, 2nd edition 2008 (ZMATH entry)

Revised on August 14, 2013 03:20:40 by Urs Schreiber (65.196.126.11)