Some Theorems on Leibniz $n$-Algebras from the Category ${\mathbf{U}}_n(\mathbf{Lb})$

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