nLab Malcev completion




In a 1949 paper devoted to the study of the coset spaces of nilpotent Lie groups, Anatoly Malcev 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.


(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.


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


  • A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13, (1949) 9–32, MR28842.

    Original reference: А. И. Мальцев, Об одном классе однородных пространств, Известия Академии наук СССР. Серия математическая 13:1 (1949), 9–32. PDF.

  • Nicolas Bourbaki, Lie groups and Lie algebras

  • Jaume Amorós, On the Malcev completion of Kähler groups, Commentarii Mathematici Helvetici 71, n. 1, 192-212, doi alg-geom/9410013

  • Robert B. Warfield, The Malcev correspondence, Ch. 12 in Nilpotent Groups, Springer Lec. Notes in Math. 513, 1976, 104-111, doi

  • Bohumil Cenkl, Richard Porter, Mal’cev’s completion of a group and differential forms, MR628342, euclid

  • Benoit Fresse, Operads & Grothendieck-Teichmüller groups - draft document, pdf

Last revised on September 5, 2023 at 10:08:56. See the history of this page for a list of all contributions to it.