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The Schur $\mathsf{Lie}$-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 $\mathsf{Lie}$-multiplier of $\mathsf{Lie}$-filiform Leibniz $n$-algebras and the Schur $\mathsf{Lie}$-multiplier of its $\mathsf{Lie}$-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 $\mathsf{Lie}$-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 published in [23] for $m$-dimensional $\mathsf{Lie}$-nilpotent Leibniz $n$-algebras with $d$-dimensional $\mathsf{Lie}$-commutator. In particular, it is shown that if $\mathfrak{q}$ is an $m$-dimensional $\mathsf{Lie}$-nilpotent Leibniz $2$-algebra with one-dimensional $\mathsf{Lie}$-commutator, then $\dim\mathcal{M}_{\mathsf{Lie}}(\mathfrak{q})\leq \frac{1}{2}m(m-1)-1.$

In this paper, we define on one hand, the notions of characteristics as well as central characteristics ideals of a given Leibniz algebra g and provide a necessary condition under which for two given subalgebras J and K of g such that, J IS a a non empty subset of K. J is a central characteristics two-sided ideal of K. On the other hand, we introduce the class of quasi-Noetherian Leibniz algebras. This generalizes both the class of Noetherian Leibniz algebras and that of quasi-Noetherian Lie algebras introduced by Falih and Stewart. We provide a necessary condition for a Leibniz algebra to be quasi-Noetherian. As in the case of Lie algebras, quasi-Noetherian Leibniz algebras are shown to be closed under quotients, but not under extensions. Finally, we leverage the maximal condition of abelian ideals to provide a characterization of Noetherian Leibniz algebras., Comment: Research paper with 14 pages

Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem.The theorem classically asserts that there is a cochain equivalence between the usual singular cochain complex of a simplicial complex and the relative Hochschild complex of its incidence algebra, and a quasi-isomorphism with the standard Hochschild complex. Here, we will be mostly interested in its application to arbitrary posets (or, equivalently, finite topologies) and their incidence algebras. We construct various structures, classical and new, on the above two complexes: cosimplicial, differential graded algebra, operadic and brace algebra structures and show that the comparison theorem preserves all of them. These results provide non standard insights on links between the theory of posets, incidence algebras, endomorphism operads and finite and combinatorial topology. By non standard, we refer here to the use of relative versions of Hochschild complexes and operads., Comment: arXiv admin note: text overlap with arXiv:math/0112177

In these notes, we define a new simplicial structure on a connected multiplicative operad and call it connected multiplicative simplicial operad (for short; simplicial operad). Next we introduce on this simplicial operad a brace algebra structure analogous to that of Gerstenhaber-Voronov that we call right brace algebra structure. This permits us to obtain on the operad with the above mentioned properties a bicomplex structure one of whose two differential operators is a coboundary and the other one is a boundary. Moreover we define on one hand on the above simplicial operad together with its right brace algebra structure, two distinct products up to a sign respectively called dot-product and odot-product. Then we show that the coboundary and the boundary together with the odot-product provide to this simplicial operad two distinct differential graded algebra structures. On the other hand we obtain through the Alexander-Withney map, a differential graded coalgebra structure on a simplicial operad. We end by illustrating our constructions with some examples, Comment: Le papier s'\'etend sur 24 pages. Le papier poss\`ede 8 figures

In this paper, we study the category of trigroups as a generalization of the notion of digroup [4] and analyze their relationship with 3-racks [1] and Leibniz 3-algebras [6]. Trigroups are essentially associative trioids in which there are bar-units and bar-inverses. We prove that 3-racks can be constructed by conjugating trigroups. We also prove that trigroups equipped with a smooth manifold structure produce Leibniz 3-algebras via their associated Lie 3-racks., Comment: 13 pages