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

We study the Leibniz n-algebra Un (L), whose multiplication is defined via the bracket of a Leibniz algebra L as [ x1, ..., xn]= [x1, [...,[ xn-2, [ x n-1, xn] ] ... ] ]. We show that Un (L) is simple if and only if L is a simple Lie algebra. An analog of Levi's theorem for Leibniz algebras in Un (Lb) is established and it is proven that the Leibniz n-kernel of Un(L) for any semisimple Leibniz algebra L is the n-algebra Un (L).

Automated summary

The study focused on the Leibniz n-algebra Un(L) and its simplicity condition, as well as an analog of Levi's theorem for Leibniz algebras in Un(Lb). It was proven that the Leibniz n-kernel of Un(L) for any semisimple Leibniz algebra L is the n-algebra Un(L).