Un Theoreme de Poincaré–Birkhoff–Witt pour les Algebres de Leibniz

Abstract

Poincaré–Birkhoff–Witt's theorem for Leibniz algebras. The dialgebras have been recently introduced by J. L. Loday. For any Leibniz algebra is defined an enveloping dialgebra which has a good universal property. In this work we give, in the category of dialgebras, a Poincaré–Birkhoff–Witt's theorem for Leibniz algebras: Let g be a Leibniz algebra, g Lie the associated universal Lie algebra and S(g Lie ) the symmetric algebra on the K-module g Lie . Let grUd(g) be the associated graded dialgebra of the enveloping dialgebra Ud(g) of g. Then, if g is free as K-module, there is an isomorphism of graded dialgebras S(g Lie ) ⊗ g = grUd(g).