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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 𝔤=F 1𝔤F 2𝔤\mathfrak{g}=F_1 \mathfrak{g}\supset F_2 \mathfrak{g}\supset\cdots such that

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

  • the filtered object 𝔤\mathfrak{g} is complete: the canonical map 𝔤lim n𝔤/F n𝔤\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.