The Schur multiplier and stem covers of Leibniz $n$-algebras

Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

“…For that the homomorphism θ * (e) in exact sequence (1) plays a central role. When a Leibniz algebra is regarded as a Leibniz crossed module in the two usual ways (Example 2.3 (i)), then the subsequent results recover the corresponding ones for stem extensions and stem cover of Leibniz algebras in [10,13]. Also, if (a, b, σ) ∼ = M(n, q, δ), then the stem extension (e) is called a stem cover or covering of (n, q, δ).…”

“…This structure is not only important by algebraic reasons, but also for its applications in other branches such as Geometry or Physics (see for instance [11,18,20,25] An active research line consists in the extension of properties from Lie algebras to Leibniz algebras. As an example of these generalizations, stem covers and stem extensions of a Leibniz algebras where studied in [10]; in [19] was extended to Leibniz algebras the notion of non-abelian tensor product of Lie algebras introduced by Ellis in [14]; in [12], authors investigated the interplay between the non-abelian tensor and exterior product of Leibniz algebras with the low dimensional Leibniz homology of Leibniz algebras.…”

The second relative homology of Leibniz algebras

“…The converse of theorem follows from Propositions 3.1(ii) and 4.3(i). One observes that the relative stem cover δ : m −→ g of the pair (g, g) is the usual stem cover of g, which was introduced by Casas and Ladra [9]. In this case,…”

“…The homology theory of Leibniz algebras was flourished at the beginning of the development of these algebras in a paper by Loday and Pirashvili [27], and immediately several concepts related to the theory of abelian extensions tied to low dimensional homologies, see [9]. From the practical standpoint to the theory of Baer-invariants, the relative Lie-central extension, Lie-cover and Liemultiplier of Leibniz algebras with respect to the subctegory of Lie algebras were studied in a series of papers, see for instance [11,8].…”

On the universal <inline-formula><tex-math id="M1">$ \alpha $</tex-math></inline-formula>-central extensions of the semi-direct product of Hom-preLie algebras

“…In recent year, the universal central extension of a perfect Leibniz algebra was studied in several articles [2,6,3,1,8,9,10]. In [4,5,7], authors study universal (α)-central extension.…”