# nLab Malcev Lie algebra

## Definition

We work over a field of characteristic 0, such as the rationals.

A Malcev Lie algebra is a filtered Lie algebra $\mathfrak{g}=F_1 \mathfrak{g}\supset F_2 \mathfrak{g}\supset\cdots$ such that

• the graded Lie algebra $gr(\mathfrak{g})$ is generated as a Lie algebra by its degree 1 component;

• the filtered object $\mathfrak{g}$ is complete: the canonical map $\mathfrak{g}\to lim_n \mathfrak{g}/F_n \mathfrak{g}$ is an isomorphism.

Morphisms of Malcev Lie algebras are morphisms of filtered Lie algebras?.

## Properties

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

## 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.

Last revised on March 28, 2021 at 14:22:27. See the history of this page for a list of all contributions to it.