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

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 $\textbf{U}_n(\mathfrak{L})$.