Some Theorems on Leibniz n-Algebras from the Category Un (Lb)

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