OnLie-isoclinic Leibniz algebras

$$\mathsf {Lie}$$Lie-Isoclinism of Pairs of Leibniz Algebras

“…Remark 2.3. When m i = q i , i = 1, 2, then we recover the notion of Lie-isoclinism of Leibniz algebras given in [1].…”

“…In the papers [1,5] was initiated a study of properties of Leibniz algebras relative to the Liezation functor, which assigns to a Leibniz algebra q the Lie algebra q Lie = q/ {[x, x] : x ∈ q} , as opposed to the absolute ones, the corresponding to the abelianization functor. The origin of this point of view comes from the general theory of central extensions relative to a chosen subcategory of a base category introduced in [10] and considered in the context of semi-abelian categories relative to a Birkhoff subcategory in [11].…”

On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras

“…The category of Lie algebras forms a Birkhoff subcategory of the category of Leibniz algebras. This means that we can study relative commutator theory o Leibniz algebras respect to Lie, giving rise to interesting developments in the comprehension of both algebraic structures [2,3,5,7,33]. Moreover, the interplay between these two categories can be somehow tricky, since it is also possible to find Leibniz algebras as a subcategory of a certain type of Lie algebras [25], although many interesting categorical properties are not preserved [15,16,17,18].…”

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].…”