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

Abstract

We study the Leibniz $n$-algebra ${\text{U}}_n(\mathfrak{L})$, 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 ${\text{U}}_n(\mathfrak{L})$ is simple if and only if $\mathfrak{L}$ is a simple Lie algebra. An analogue of Levi's theorem for Leibniz algebras in ${\text{U}}_n(\text{Lb})$ is established and it is proven that the Leibniz $n$-kernel of ${\text{U}}_n(\mathfrak{L})$ for any semisimple Leibniz algebra $\mathfrak{L}$ is the $n$-algebra ${\text{U}}_n(\mathfrak{L})$.