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\begin{document}

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\section*{maximal compact subgroup}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

[[!include topology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ExamplesForLieGroups}{For Lie groups}\dotfill \pageref*{ExamplesForLieGroups} \linebreak
\noindent\hyperlink{counterexamples}{Counterexamples}\dotfill \pageref*{counterexamples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $G$ a [[topological group]] a \emph{compact subgroup} is a topological [[subgroup]] $K \subset G$ which is a [[compact group]].

\begin{defn}
\label{}\hypertarget{}{}
A [[compact topological space|compact]] [[subgroup]] $K \hookrightarrow G$ is called \textbf{maximal compact} if every compact subgroup of $G$ is [[conjugation|conjugate]] to a subgroup of $K$.

\end{defn}
If it exists then, by definition, it is unique up to conjugation.

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

\begin{defn}
\label{AlmostConnected}\hypertarget{AlmostConnected}{}
A [[locally compact topological group]] $G$ is called \textbf{[[almost connected topological group|almost connected]]} if the [[quotient]] [[topological space]] $G/G_0$ (of $G$ by the [[connected component]] of the neutral element) is [[compact topological space|compact]].

\end{defn}
See for instance (\hyperlink{HofmannMorris}{Hofmann-Morris, def. 4.24}).

\begin{defn}
\label{}\hypertarget{}{}
Every [[compact topological space|compact]] and every [[connected topological space|connected]] [[topological group]] is almost connected.

Also every [[quotient]] of an almost connected group is almost connected.

\end{defn}
\begin{theorem}
\label{MalcevIwasawa}\hypertarget{MalcevIwasawa}{}
Let $G$ be a [[locally compact topological space|locally compact]] \hyperlink{AlmostConnected}{almost connected} [[topological group]].

Then

\begin{itemize}%
\item $G$ has a maximal compact subgroup $K$;


\item the [[coset]] space $G/K$ is [[homeomorphism|homeomorphic]] to a [[Euclidean space]].



\end{itemize}
\end{theorem}
This is due to (\hyperlink{Malcev}{Malcev}) and (\hyperlink{Iwasawa}{Iwasawa}). See for instance (\hyperlink{Stroppel}{Stroppel, theorem 32.5}).

\begin{theorem}
\label{}\hypertarget{}{}
Let $G$ be a [[locally compact topological space|locally compact]] \hyperlink{AlmostConnected}{almost connected} [[topological group]].

Then a [[compact topological space|compact]] subgroup $K \hookrightarrow G$ is maximal compact precisely if the [[coset]] space $G/K$ is [[contractible space|contractible]]

(in which case, due to theorem \ref{MalcevIwasawa}, it is necessarily [[homeomorphism|homeomorphic]] to a [[Euclidean space]]).

\end{theorem}
This is (\hyperlink{Antonyan}{Antonyan, theorem 1.2}).

\begin{remark}
\label{}\hypertarget{}{}
In particular, in the above situation the [[subgroup]] inclusion

\begin{displaymath}
K \hookrightarrow G
\end{displaymath}
is a [[homotopy equivalence]] of [[topological spaces]].

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

\hypertarget{ExamplesForLieGroups}{}\subsubsection*{{For Lie groups}}\label{ExamplesForLieGroups}

The following table lists some [[Lie groups]] and their maximal compact Lie subgroups (e.g. \hyperlink{Conrad}{Conrad}). See also \emph{[[compact Lie group]]}.

\newline | Lorentz / AdS [[pin group]] $Pin(q,p)$ | $Pin(q) \times Pin(q) / \{(1,1), (-1,-1)\}$ |

The following table lists specifically the maximal compact subgroups of the ``$E$-series'' of Lie groups culminating in the [[exceptional Lie groups]] $E_n$.

\begin{tabular}{l|l|l|l|l}
$n$&[[real form]] $E_{n(n)}$&maximal compact subgroup $H_n$&$dim(E_{n(n)})$&$dim(E_{n(n)}/H_n )$\\
\hline 
2&$SL(2, \mathbb{R}) \times \mathbb{R}$&$SO(2)$&4&3\\
3&$SL(3,\mathbb{R}) \times SL(2,\mathbb{R})$&$SO(3) \times SO(2)$&11&7\\
4&$SL(5, \mathbb{R})$&$SO(5)$&24&14\\
5&$Spin(5,5)$&$(Sp(2) \times Sp(2))/\mathbb{Z}_2$&45&25\\
6&[[E6&E6(6)]]&$Sp(4)/\mathbb{Z}_2$&78\\
7&[[E7&E7(7)]]&$SU(8)/\mathbb{Z}_2$&133\\
8&[[E8&E8(8)]]&$Spin(16)/\mathbb{Z}_2$&248\\
\end{tabular}

\hypertarget{counterexamples}{}\subsubsection*{{Counterexamples}}\label{counterexamples}

A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the [[Prüfer group]] $\mathbb{Z}[1/p]/\mathbb{Z}$ endowed with the discrete ($0$-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.

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

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


\item [[compact Lie group]]


\item [[maximal subgroup]]


\item [[maximal torus]]


\item [[reduction and lift of structure groups]]



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

Textbooks with relevant material include

\begin{itemize}%
\item M. Stroppel, \emph{Locally compact groups}, European Math. Soc., (2006)


\item Karl Hofmann, Sidney Morris, \emph{The Lie theory of connected pro-Lie groups}, Tracts in Mathematics 2, European Mathematical Society, (2000)



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Maximal_compact_subgroup}{Maximal compact subgroup}}

\end{itemize}
Original articles include

\begin{itemize}%
\item A. Malcev, \emph{On the theory of the Lie groups in the large}, Mat.Sbornik N.S. vol. 16 (1945) pp. 163-189


\item K. Iwasawa, \emph{On some types of topological groups}, Ann. of Math. vol.50 (1949) pp. 507-558.


\item M. Peyrovian, \emph{Maximal compact normal subgroups}, Proceedings of the American Mathematical Society, Vol. 99, No. 2, (1987) (\href{http://www.jstor.org/pss/2046647}{jstor})


\item Sergey A. Antonyan, \emph{Characterizing maximal compact subgroups} (\href{http://arxiv.org/abs/1104.1820v1}{arXiv:1104.1820v1})



\end{itemize}
The maximal compact subgroups inside the (indefinite) rotation groups

\begin{itemize}%
\item [[Brian Conrad]], \emph{Examples of maximal compact subgroups} (\href{http://virtualmath1.stanford.edu/~conrad/210CPage/handouts/maxcompact.pdf}{pdf})

\end{itemize}
[[!redirects maximal compact subgroups]]



\end{document}