nLab Malcev group

Definition

A Malcev group is a filtered group $G=F_1 G\supset F_2 G\supset\cdots$ such that

• the associated graded Lie algebra $gr(G)$ over integers is uniquely divisible, hence a rational Lie algebra;

• the graded rational Lie algebra $gr(G)$ is generated as a Lie algebra by its degree 1 component;

• the filtered object $G$ is complete: the canonical map $G\to lim_n G/F_n G$ is an isomorphism.

Morphisms of Malcev groups are morphisms of filtered groups?.

Properties

The category of Malcev groups is equivalent to the category of rational complete Hopf algebras. See there for more information.

Related concepts

• Malcev Lie algebra

• complete Hopf algebra

References

• А. И. Мальцев, Нильпотентные группы без кручения, Изв. АН СССР. Сер. матем. 13:3 (1949), 201–212. PDF.

• Daniel Quillen, Rational homotopy theory, Annals of Mathematics 90:2 (1969), 205. doi.