In this article, we study the notion of central extension of Leibniz $n$-algebras relative to $n$-Lie algebras to study properties of Schur $\mathsf{ Lie}$-multiplier and $\mathsf{ Lie}$-covers on Leibniz $n$-algebras. We provide a characterization of $\mathsf{ Lie}$-perfect Leibniz $n$-algebras by means of universal $\mathsf{ Lie}$-central extensions. It is also provided some inequalities on the dimension of the Schur $\mathsf{ Lie}$-multiplier of Leibniz $n$-algebras. Analogue to Wiegold [38] and Green [17] results on groups or Moneyhun [26] result on Lie algebras, we provide upper bounds for the dimension of the $\mathsf{ Lie}$-commutator of a Leibniz $n$-algebra with finite dimensional $\mathsf{ Lie}$-central factor, and also for the dimension of the Schur $\mathsf{ Lie}$-multiplier of a finite dimensional Leibniz $n$-algebra.
Categories: math.RA