Some theorems on Leibniz $n$-algebras from the category $\textbf{U}_n(\textbf{Lb})$

TL;DR

This paper investigates Leibniz n-algebras derived from Leibniz algebras, establishing conditions for simplicity, an analogue of Levi's theorem, and characterizing the Leibniz n-kernel for semisimple cases.

Contribution

It introduces new structural results for Leibniz n-algebras, including simplicity criteria and Levi-type decompositions, expanding understanding of their algebraic properties.

Findings

Leibniz n-algebra is simple iff the underlying Leibniz algebra is a simple Lie algebra.

An analogue of Levi's theorem is established for Leibniz n-algebras.

The Leibniz n-kernel of U_n(L) equals U_n(L) for any semisimple Leibniz algebra L.

Abstract

We study the Leibniz $n$-algebra $\textbf{U}_n(\mathfrak{L})$, whose multiplication is defined via the bracket of a Leibniz algebra $\mathfrak{L}$ as $[x_1,\dots,x_n]=[x_1,[\dots, [x_{n-2},[x_{n-1},x_n]]\dots]]$. We show that $\textbf{U}_n(\mathfrak{L})$ is simple if and only if $\mathfrak{L}$ is a simple Lie algebra. An analogue of Levi's theorem for Leibniz algebras in $\textbf{U}_n(\textbf{Lb})$ is established and it is proven that the Leibniz $n$-kernel of $\textbf{U}_n(\mathfrak{L})$ for any semisimple Leibniz algebra $\mathfrak{L}$ is the $n$-algebra $\textbf{U}_n(\mathfrak{L})$.