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\begin{document}

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\section*{topological group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{group_theoryc}{}\paragraph*{{Group Theoryc}}\label{group_theoryc}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{uniform_structure}{Uniform structure}\dotfill \pageref*{uniform_structure} \linebreak
\noindent\hyperlink{unitary_representation_on_hilbert_spaces}{Unitary representation on Hilbert spaces}\dotfill \pageref*{unitary_representation_on_hilbert_spaces} \linebreak
\noindent\hyperlink{why_the_strong_topology_is_used}{Why the strong topology is used}\dotfill \pageref*{why_the_strong_topology_is_used} \linebreak
\noindent\hyperlink{which_topological_groups_admit_lie_group_structure}{Which topological groups admit Lie group structure?}\dotfill \pageref*{which_topological_groups_admit_lie_group_structure} \linebreak
\noindent\hyperlink{Protomodularity}{Protomodularity}\dotfill \pageref*{Protomodularity} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{topological group} is a [[topological space]] with a continuous [[group]] structure: a [[group object]] in the [[category]] [[Top]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \emph{topological group} is

\begin{enumerate}%
\item a [[group]], hence

\begin{enumerate}%
\item a [[set]] $G$,


\item a [[neutral element]] $e \in G$,


\item a [[associativity|associative]] [[unitality|unitality]] [[function]]


\item $(-)\cdot (-) \;\colon\; G \times G \to G$,


\item a function $(-)^{-1} \;\colon\; G \to G$ such that $g \cdot g^{-1} = e = g^{-1} \cdot g$ for all $g \in G$;



\end{enumerate}

\item a topology $\tau_G \subset P(G)$ giving $G$ the structure of a [[topological space]]



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

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{lemma}
\label{OpenSubgroupOfTopologicalGroupIsClosed}\hypertarget{OpenSubgroupOfTopologicalGroupIsClosed}{}
\textbf{(open subgroups of topological groups are closed)}

Every [[open subset|open]] [[subgroup]] $H \subset G$ of a topological group is [[closed subset|closed]].

\end{lemma}
(e.g \hyperlink{ArhangelskiiTkachenko08}{Arhangel'skii-Tkachenko 08, theorem 1.3.5})

\begin{proof}
The set of $H$-[[cosets]] is a [[cover]] of $G$ by [[disjoint subsets|disjoint]] [[open subsets]]. One of these cosets is $H$ itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.

\end{proof}
\begin{prop}
\label{ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact}\hypertarget{ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact}{}
\textbf{([[connected space|connected]] [[locally compact topological space|locally compact]] [[topological groups]] are [[sigma-compact topological space|sigma-compact]])}

Every [[connected topological space|connected]] [[locally compact topological space|locally compact]] topological group is [[sigma-compact topological space|sigma-compact]].

Every [[locally compact topological space|locally compact]] topological group is [[paracompact topological space|paracompact]].

\end{prop}
(e.g. \hyperlink{ArhangelskiiTkachenko08}{Arhangel'skii-Tkachenko 08, cor. 3.1.4, cor. 3.1.5})

\begin{proof}
By assumption of local compactness, there exists a [[compact topological space|compact]] [[neighbourhood]] $C_e \subset G$ of the [[neutral element]]. We may assume without restriction of generality that with $g \in C_e$ any element, then also the [[inverse element]] $g^{-1} \in C_e$.

For if this is not the case, then we may enlarge $C_e$ by including its inverse elements, and the result is still a compact neighbourhood of the neutral element: Since taking [[inverse elements]] $(-)^{-1} \colon G \to G$ 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 $C_e$ is compact, and the union of two compact subspaces is still compact (obviously, otherwise see \href{compact+space#UnionsAndIntersectionOfCompactSubspaces}{this prop}).

Now for $n \in \mathbb{N}$, write $C_e^n  \subset G$ for the [[image]] of $\underset{k \in \{1, \cdots n\}}{\prod} C_e \subset \underset{k \in \{1, \cdots, n\}}{\prod} G$ under the iterated group product operation $\underset{k \in \{1, \cdots, n\}}{\prod} G \longrightarrow G$.

Then

\begin{displaymath}
H \coloneqq \underset{n \in \mathbb{N}}{\cup} C_e^n
    \;\subset\; 
  G
\end{displaymath}
is clearly a topological subgroup of $G$.

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

\begin{displaymath}
H = \underset{n \in \mathbb{N}}{\cup} C_e^n
\end{displaymath}
is a countable union of compact subspaces, making it [[sigma-compact]]. Since [[locally compact and sigma-compact spaces are paracompact]], this implies that $H$ is paracompact.

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

\begin{displaymath}
H = \underset{h \in H}{\cup} Int(C_e) \cdot h
  \,.
\end{displaymath}
Finally, as indicated in the proof of Lemma \ref{OpenSubgroupOfTopologicalGroupIsClosed}, the cosets of the open subgroup $H$ are all open and partition $G$ as a [[disjoint union space]] ([[coproduct]] in [[Top]]) of these open cosets. From this we may draw the following conclusions.

\begin{itemize}%
\item In the particular case where $G$ is connected, then there is just one such coset, namely $H$ itself. The argument above thus shows that a connected locally compact topological group is $\sigma$-compact and (by local compactness) also paracompact.


\item In the general case, all the cosets are homeomorphic to $H$ which we have just shown to be a paracompact group. Thus $G$ is a [[disjoint union space]] of paracompact spaces. This is again paracompact (by \href{paracompact+topological+space#ParacompactnessPreservedByDisjointUnion}{this prop.}).



\end{itemize}
\end{proof}
\hypertarget{uniform_structure}{}\subsubsection*{{Uniform structure}}\label{uniform_structure}

A topological group $G$ carries two canonical [[uniform space|uniformities]]: a right and left uniformity. The \textbf{right 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: $g \in U \Leftrightarrow g^{-1} \in U$. The \textbf{left 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.

\begin{prop}
\label{}\hypertarget{}{}
The following are equivalent:

\begin{itemize}%
\item The map $f$ is continuous at a point of $G$;


\item The map $f$ is continuous;


\item The map $f$ is uniformly continuous.



\end{itemize}
\end{prop}
\begin{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

\begin{displaymath}
f(g U) \subseteq f(g) V
\end{displaymath}
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

\begin{displaymath}
x y^{-1} \in U \Rightarrow f(x) f(y)^{-1} = f(x y^{-1}) \in V
\end{displaymath}
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 right uniformity. By similar reasoning, $f$ is uniformly continuous with respect to the right uniformity.

\end{proof}
\hypertarget{unitary_representation_on_hilbert_spaces}{}\subsubsection*{{Unitary representation on Hilbert spaces}}\label{unitary_representation_on_hilbert_spaces}

\begin{defn}
\label{}\hypertarget{}{}
A unitary [[representation]] $R$ of a topological group $G$ in a [[Hilbert space]] $\mathcal{H}$ is a continuous [[group homomorphism]]

\begin{displaymath}
R \colon G \to \mathcal{U}(\mathcal{H})
\end{displaymath}
where $\mathcal{U}(\mathcal{H})$ is the group of [[unitary operator]]s on $\mathcal{H}$ with respect to the [[strong topology]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Here $\mathcal{U}(\mathcal{H})$ is a complete, metrizable topological group in the [[strong topology]], see (\hyperlink{Schottenloher}{Schottenloher, prop. 3.11}).

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
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 \textbf{quantization} of $G$. See at \emph{[[geometric quantization]]} and \emph{[[orbit method]]} for more on this.

\end{remark}
\hypertarget{why_the_strong_topology_is_used}{}\paragraph*{{Why the strong topology is used}}\label{why_the_strong_topology_is_used}

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

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

\begin{displaymath}
R: G \to \mathcal{U}(L^2(G))
\end{displaymath}
\begin{displaymath}
g \mapsto (R_g: f(x) \mapsto f(x g))
\end{displaymath}
and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.

\hypertarget{which_topological_groups_admit_lie_group_structure}{}\subsubsection*{{Which topological groups admit Lie group structure?}}\label{which_topological_groups_admit_lie_group_structure}

\begin{itemize}%
\item \emph{[[Hilbert's fifth problem]]}

\end{itemize}
\hypertarget{Protomodularity}{}\subsubsection*{{Protomodularity}}\label{Protomodularity}

\begin{prop}
\label{}\hypertarget{}{}
The [[category]] [[TopGrp]] of topological groups and [[continuous function|continuous]] [[group homomorphisms]] between them is a [[protomodular category]].

\end{prop}
A proof is spelled out by [[Todd Trimble]] \href{http://mathoverflow.net/questions/133110/why-is-topgrp-the-category-of-topological-groups-and-continous-homomorphisms-prot/133601#133601}{here on MO}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The [[classical Lie groups]] are in particular topological groups, such as the [[general linear group]] and its subgroups.


\item \ldots{}



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[group]]


\item [[discrete group]], [[finite group]]


\item [[topological monoid]]


\item \textbf{topological group},

\begin{itemize}%
\item [[locally compact topological group]]


\item [[compact topological group]]


\item [[amenable topological group]]


\item [[maximal compact subgroup]]


\item [[loop group]]



\end{itemize}

\item [[Lie group]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Alexander Arhangel'skii, Mikhail Tkachenko, \emph{Topological Groups and Related Structures}, Atlantis Press 2008

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

\begin{itemize}%
\item [[Martin Schottenloher]], \emph{A mathematical introduction to conformal field theory.} Springer, 2nd edition 2008 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1161.17014&format=complete}{ZMATH entry})

\end{itemize}
[[!redirects topological groups]]



\end{document}