Upper bounds on the dimension of the Schur Lie-multiplier of Lie-nilpotent Leibniz n-algebras

The 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 published in 23 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 ₋₈₄ (q) 12m (m-1) -1.