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 Lie-central extensions. We obtain a six-term exact homology sequence associated to a Lie-central extension. This sequence, together with the relative commutators, allows us to characterize several classes of Lie-central extensions, such as Lie-trivial, Lie-stem and Lie-stem cover, to introduce and characterize Lie-unicentral, Lie-capable, Lie-solvable and Lie-nilpotent Leibniz algebras.

A study of n-Lie-isoclinic Leibniz algebras

“…Let m, n be two-sided ideals of a Leibniz algebra g. The following notions come from [7], which were derived from [8].…”

“…Definition 2.1 [7] Let n be a two-sided ideal of a Leibniz algebra g. The lower Lie-central series of g relative to n is the sequence…”

“…Definition 2.3 [7] The upper Lie-central series of a Leibniz algebra g is the sequence of two-sided ideals, called i-Lie centers, i=0, 1, 2, . .…”

“…This philosophy comes from the categorical theory of central extensions relative to a chosen Birkhoff subcategory of a semiabelian category. We refer to [3,7,8] and references given therein for a detailed explanation.…”

The c-nilpotent Schur Lie-multiplier of Leibniz algebras

“…In the recent papers [9,11,12] authors approached the relative theory of Leibniz algebras with respect to the Liezation functor, yielding to the introduction of new notions of central extensions, capability, nilpotency, stem cover, isoclinism and Schur multiplier relative to the Liezation functor, the so-called Lie-central extensions, Lie-capability, Lie-nilpotency, Lie-stem cover, Lie-isoclinism and Schur Lie-multiplier.…”