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\begin{document}

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\section*{Mal&#39;cev completion}

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

In a 1949 paper devoted to the study of the coset spaces of nilpotent Lie groups, Mal'cev exhibited an equivalence between the category of torsion free radicable nilpotent finite rank groups and the category of finite dimensional, nilpotent rational Lie algebras. This involves a completion construction which is used also in the general formulation of Hausdorff series (cf. Bourbaki) which takes values in the Mal'cev completion of the universal enveloping algebra on two generators.

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

(There are several variants of the definition.)

Mal'cev completion is a left [[adjoint functor]] to the embedding of the category of uniquely divisible nilpotent groups into the category of nilpotent groups.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

Hausdorff series, [[rational homotopy theory]], surgery, combinatorial group theory

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

\begin{itemize}%
\item A. Mal'cev, \emph{On a class of homogeneous spaces}, Izvestiya Akad. Nauk. SSSR. Ser. Mat. \textbf{13}, (1949) 9--32, \href{http://www.ams.org/mathscinet-getitem?mr=28842}{MR28842}


\item [[Nicolas Bourbaki]], \emph{Lie groups and Lie algebras}


\item Jaume Amor\'o{}s, \emph{On the Malcev completion of K\"a{}hler groups}, Commentarii Mathematici Helvetici \textbf{71}, n. 1, 192-212, \href{http://dx.doi.org/10.1007/BF02566416}{doi} \href{http://arxiv.org/abs/alg-geom/9410013}{alg-geom/9410013}


\item Robert B. Warfield, \emph{The Malcev correspondence}, Ch. 12 in \textbf{Nilpotent Groups}, Springer Lec. Notes in Math. \textbf{513}, 1976, 104-111, \href{http://dx.doi.org/10.1007/BFb0080164}{doi}


\item Bohumil Cenkl, Richard Porter, \emph{Mal'cev's completion of a group and differential forms}, \href{http://www.ams.org/mathscinet-getitem?mr=628342}{MR628342}, \href{http://projecteuclid.org/euclid.jdg/1214435841}{euclid}


\item [[Benoit Fresse]], \emph{Operads \& Grothendieck-Teichm\"u{}ller groups - draft document}, \href{https://hal.archives-ouvertes.fr/hal-00656333}{pdf} [[!redirects Malcev completion]]



\end{itemize}


\end{document}