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.

Related concepts

• Malcev group

• Malcev completion

• 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.