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\begin{document}

%-------------------------------------------------------------------


\section*{local Lie group}

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

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

[[!include group theory - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \textbf{local Lie group} (rarely also called \emph{Lie groupuscule}) is a local / [[stalk]]wise version of a [[Lie group]], containing information about the product operation in arbitrary small [[open neighborhoods]] of the unit element; the multiplication is defined only when the factors are sufficiently close to the unit element, and if the consecutive products of triples in both ways are defined they are associative. To every local Lie group one functorially associates its Lie algebra.

Every real [[Lie algebra]] is a Lie algebra of some local Lie group. Or in more modern and precise phrasing, the category of real local Lie groups is equivalent to the category of finite-dimensional real Lie algebras. This has been proved by [[Sophus Lie]] as his famous [[Lie's three theorems|third theorem]]. The extension to the global Lie theory has been possible only after works of [[Élie Cartan]], who extended the equivalence to the equivalence between the category of real Lie algebras and connected simply connected Lie groups.

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

[[!include infinitesimal and local - table]]

[[!redirects local Lie groups]] [[!redirects Lie groupuscule]]

[[!redirects local group]] [[!redirects local groups]]



\end{document}