On $$\mathsf {Lie}$$ Lie -central extensions of Leibniz algebras

Abstract

Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called $$\mathsf {Lie}$$ -central extensions. We obtain a six-term exact homology sequence associated to a $$\mathsf {Lie}$$ -central extension. This sequence, together with the relative commutators, allows us to characterize several classes of $$\mathsf {Lie}$$ -central extensions, such as $$\mathsf {Lie}$$ -trivial extensions, $$\mathsf {Lie}$$ -stem extensions and $$\mathsf {Lie}$$ -stem covers, and to introduce and characterize $$\mathsf {Lie}$$ -unicentral, $$\mathsf {Lie}$$ -capable, $$\mathsf {Lie}$$ -solvable and $$\mathsf {Lie}$$ -nilpotent Leibniz algebras.