On the Schur Lie-multiplier and Lie-covers of Leibniz n-algebras

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.

Automated summary

This article studies central extensions of Leibniz n-algebras relative to n-Lie algebras to analyze the properties of Schur Lie-multipliers and Lie-covers. The authors provide a characterization of Lie-perfect Leibniz n-algebras using universal Lie-central extensions and present some inequalities regarding the dimension of the Schur Lie-multipliers of Leibniz n-algebras. Additionally, upper bounds are given for the dimension of the Lie-commutator and the dimension of the Schur Lie-multiplier of finite dimensional Leibniz n-algebras, similar to the results on groups and Lie algebras by Wiegold, Green, and Moneyhun.