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AbstractThe Schur Lie -multiplier of Leibniz algebras is the Schur multiplier of Leibniz algebras defined relative to the Liezation functor. In this paper, we study upper bounds for the dimension of the Schur Lie -multiplier of Lie -filiform Leibniz n-algebras and the Schur Lie -multiplier of its Lie -central factor. The upper bound obtained is associated to both the sequences of central binomial coefficients and the sum of the numbers located in the rhombus part of Pascal’s triangle. Also, the pattern of counting the number of Lie -brackets of a particular Leibniz n-algebra leads us to a new property of Pascal’s triangle. Moreover, we discuss some results which improve the existing upper bound for m-dimensional Lie -nilpotent Leibniz n-algebras with d-dimensional Lie -commutator. In particular, it is shown that if q is an m-dimensional Lie -nilpotent Leibniz 2-algebra with one-dimensional Lie -commutator, then dimMLie(q)≤12m(m−1)−1. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of isoclinism on Lie -central extensions of Leibniz n-algebras. We provide several equivalent conditions under which Lie -central extensions of Leibniz n-algebras are Lie -isoclinic. Also, we show that each Lie -isoclinic class of Lie -central extensions of Leibniz n-algebras contains a Lie -stem extension. Finally, we study the concept of Schur Lie -multiplier on Leibniz n-algebras using their Lie -central extensions, and analyze some connection with the concepts of Lie -isoclinism and Lie -stem cover. [ABSTRACT FROM AUTHOR]

In this article, we study the notion of central extensions of Leibniz n-algebras relative to n-Lie algebras to study properties of Schur Lie -multiplier and Lie -covers on Leibniz n-algebras. We provide a characterization of Lie -perfect Leibniz n-algebras by means of universal Lie -central extensions. It is also provided some inequalities on the dimension of the Schur Lie -multiplier of Leibniz n-algebras. Analogue to Wiegold and Green results on groups or Moneyhun results on Lie algebras, we provide upper bounds for the dimension of the Lie -commutator of a Leibniz n-algebra with finite dimensional Lie -central factor, and also for the dimension of the Schur Lie -multiplier of a finite dimensional Leibniz n-algebra. [ABSTRACT FROM AUTHOR]

We study the Leibniz n -algebra U n (L) , whose multiplication is defined via the bracket of a Leibniz algebra L as [ x 1 , ... , x n ] = [ x 1 , [ ... , [ x n - 2 , [ x n - 1 , x n ] ] ... ] ]. We show that U n (L) is simple if and only if L is a simple Lie algebra. An analog of Levi's theorem for Leibniz algebras in U n (Lb) is established and it is proven that the Leibniz n -kernel of U n (L) for any semisimple Leibniz algebra L is the n -algebra U n (L). [ABSTRACT FROM AUTHOR]

We analyze behaviors of generalized forgetful and Daletskii-Takhtajan's functors on perfect objects and crossed modules of Leibniz n -algebras. Then we give applications to homology and universal central extensions of Leibniz n -algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Zinbiel n-algebra and study some of its properties. Also, we explain the relation between Leibniz n-algebras and Zinbiel n-algebras. Finally, we classify some nilpotent Zinbiel n-algebras. [ABSTRACT FROM AUTHOR]

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras. [ABSTRACT FROM AUTHOR]

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established. [ABSTRACT FROM AUTHOR]

The concepts of solvable and nilpotent Leibniz n -algebra are introduced, and classical results of solvable and nilpotent Lie algebras theory are extended to Leibniz n -algebras category. A homological criterion similar to Stallings Theorem for Lie algebras is obtained in Leibniz n -algebras category by means of the homology with trivial coefficients of Leibniz n -algebras. [ABSTRACT FROM AUTHOR]

We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras ${}_nHL_1({\cal K})\to {}_nHL_1({\cal L}) \to {\rm M} \to {}_nHL_0 ({\cal K})\to {}_nHL_0({\cal L})\to 0$ associated to a central extension of Leibniz n-algebras $0\to {\rm M}\to {\cal K} \to {\cal L}\to 0$ by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra. [ABSTRACT FROM AUTHOR]

We introduce homology K-vector spaces with trivial coefficients for Leibniz n-algebras and we obtain exact sequences relating them. As a consequence we get a Hopf formula and several results on central extensions of Leibniz n-algebras. [ABSTRACT FROM AUTHOR]