The Schur Lie-multiplier of Leibniz algebras

Leibniz algebras with absolute maximal Lie subalgebras

“…These notions of Lie-center and Lie-centralizer were studied in [4,6,8,9] as a result of approaching the relative theory of Leibniz algebras with respect to the Liezation functor. Note that Z Lie (g…”

The c-nilpotent Schur Lie-multiplier of Leibniz algebras

“…Since Leib is a category of interest (see [10]), hence is a category of Ω-groups, and following Proposition 4.3.2 in [19] we can conclude that the collection of all nilpotent objects of class ≤ c in Leib form a variety. Now following [15,Proposition 7.8], it can be showed that M Lie (q) is the Schur Lie-multiplier of a Leibniz algebra q (see [9,12]). [12]).…”

“…Now following [15,Proposition 7.8], it can be showed that M Lie (q) is the Schur Lie-multiplier of a Leibniz algebra q (see [9,12]). [12]). Then the surjective homomorphism of Leibniz algebras f : g ։ q defined by f (a 1 ) = e 2 , f (a 2 ) = e 1 , f (a 3 ) = 0 and f (a 4 ) = 0 is a 2-Lie-central extension.…”

“…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.…”

Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

“…The approached properties are closely related to the relative notions of central extension in a semi-abelian category with respect to a Birkhoff subcategory (see [11] and [14]). A recent research line deals with the development of absolute properties of Leibniz algebras (absolute are the usual properties and it means relative to the abelianization functor) in the relative setting (with respect to the Liezation functor); in general, absolute properties have the corresponding relative ones, but not all absolute properties immediately hold in the relative case, so new requirements are needed as it can be seen in the papers [3]- [5], [8], [10] and [19].…”