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\begin{document}

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\section*{universal central extension}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

[[!include group theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_discrete_groups}{For discrete groups}\dotfill \pageref*{for_discrete_groups} \linebreak
\noindent\hyperlink{for_lie_groups}{For Lie groups}\dotfill \pageref*{for_lie_groups} \linebreak
\noindent\hyperlink{for_super_lie_algebras}{For super Lie algebras}\dotfill \pageref*{for_super_lie_algebras} \linebreak


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

For any [[algebra|algebraic]] object, such as an [[associative algebra]], [[group]], [[Lie algebra]], etc. such that there is a notion of [[central extension]], the \emph{universal} central extension is, if it exists, the [[initial object]] in the [[category]] of central extensions of that object.

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

For $G$ a [[discrete group]] which is [[perfect group|perfect]], the \emph{universal [[central extension]]} of $G$ is the essentially unique [[group]] $K$ that is a [[Schur covering group]] of $G$.

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

\begin{itemize}%
\item The universal central extension of a perfect group is also perfect.


\item The universal central extension of a perfect group is a Schur-trivial group, and hence a superperfect group (superperfect means that it's perfect and Schur-trivial).


\item The universal central extension operator is idempotent, i.e., the universal central extension of the universal central extension is the universal central extension. This follows directly from the universal central extension being a Schur-trivial group.



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

\begin{itemize}%
\item [[maximal central extension]]


\item [[universal higher central extension]]



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

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

Discussion in [[semi-abelian categories]]:

\begin{itemize}%
\item Jose Casas, Tim Van der Linden, \emph{A relative theory of universal central extensions} (\href{https://arxiv.org/abs/0908.3762}{arXiv:0908.3762})

\end{itemize}
\hypertarget{for_discrete_groups}{}\subsubsection*{{For discrete groups}}\label{for_discrete_groups}

\begin{itemize}%
\item groupprops wiki \emph{\href{http://groupprops.subwiki.org/wiki/Universal_central_extension}{Universal central extension}}

\end{itemize}
\hypertarget{for_lie_groups}{}\subsubsection*{{For Lie groups}}\label{for_lie_groups}

\begin{itemize}%
\item [[Karl-Hermann Neeb]], \emph{Universal Central Extensions of Lie Groups}, Acta Applicandae Mathematicae (2002) 73: 175 ()

\end{itemize}
\hypertarget{for_super_lie_algebras}{}\subsubsection*{{For super Lie algebras}}\label{for_super_lie_algebras}

For [[super Lie algebras]]

\begin{itemize}%
\item Erhard Neher, \emph{An introduction to universal central extensions of Lie superalgebras} ([[NeherUniversalCentralExtension.pdf:file]])

\end{itemize}
[[!redirects universal central extension]]



\end{document}