nLab
topological group

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 $G$ carries two canonical uniformities: a right and left uniformity. The left uniformity consists of entourages ${\sim }_{l,U}$ where $x{\sim }_{l,U}y$ if $x{y}^{-1}\in U$; here $U$ ranges over neighborhoods of the identity that are symmetric: $g\in U\iff g^{-1}\in U$. The right uniformity similarly consists of entourages ${\sim }_{r,U}$ where $x{\sim }_{r,U}y$ if $x^{-1}y\in U$. The uniform topology for either coincides with the topology of $G$.

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

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

Proposition

The following are equivalent:

• The map $f$ is continuous at a point of $G$;

• The map $f$ is continuous;

• The map $f$ is uniformly continuous.

Proof

Suppose $f$ is continuous at $g\in G$. Since neighborhoods of a point $x$ are $x$-translates of neighborhoods of the identity $e$, continuity at $g$ means that for all neighborhoods $V$ of $e\in H$, there exists a neighborhood $U$ of $e\in G$ such that

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

Since $f$ is a homomorphism, it follows immediately from cancellation that $f(U)\subseteq V$. Therefore, for every neighborhood $V$ of $e\in H$, there exists a neighborhood $U$ of $e\in G$ such that

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

in other words such that $x{\sim }_{U}y\Rightarrow f(x){\sim }_{V}f(y)$. Hence $f$ is uniformly continuous with respect to the left uniformities. By similar reasoning, $f$ is uniformly continuous with respect to the right uniformities.

Unitary representation on Hilbert spaces

Definition

A unitary representation $R$ of a topological group $G$ in a Hilbert space $\mathscr{H}$ is a continuous group homomorphism

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

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

Remark

Here $𝒰(\mathscr{H})$ 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 $G$, then an unitary representation of $G$ is sometimes called a quantization of $G$. 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 $𝒰(\mathscr{H})$ is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.

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

$$R:G\to 𝒰(L^2(G))$$R: G \to \mathcal{U}(L^2(G))

$$g\mapsto (R_g:f(x)\mapsto f(xg))$$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?

• Hilbert's fifth problem

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.

Related concepts

• group

• discrete group, finite group

• topological group,

  • locally compact topological group

  • compact topological group

  • maximal compact subgroup

  • loop group

• Lie group

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)