On the Schur $\mathsf{Lie}$-multiplier and $\mathsf{Lie}$-covers of Leibniz $n$-algebras

TL;DR

This paper investigates the properties of the Schur Lie-multiplier and Lie-covers in Leibniz n-algebras, providing characterizations, inequalities, and bounds that extend classical results from group and Lie algebra theory.

Contribution

It introduces a characterization of Lie-perfect Leibniz n-algebras via universal Lie-central extensions and establishes new bounds on the dimensions of related algebraic structures.

Findings

Characterization of Lie-perfect Leibniz n-algebras using universal Lie-central extensions

Inequalities on the dimension of the Schur Lie-multiplier

Upper bounds for the dimension of the Lie-commutator and Schur Lie-multiplier

Abstract

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.