Cartan Subalgebras of Leibniz n-Algebras

Abstract

The present paper is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.

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

“…For example, the Leibniz n-kernel can coincide with the n-algebra which is possible for n = 2 if and only if the Leibniz algebra is abelian. Also, there are examples of Leibniz n-algebras with invertible right multiplication operators and Cartan subalgebras of different dimensions [2], which are not in accordance with the corresponding results in Lie, Leibniz and n-Lie algebras.…”

Arbitrary triple systems admitting a multiplicative basis

“…Then, for a certain ∅ = I I ⊂ I, we can write I = j∈I J Fe j . Fix some i 0 ∈ I I being then (2) 0 = e i0 ∈ I.…”

Some radicals, Frattini and Cartan subalgebras of Leibnizn-algebras

“…Any right multiplication operator is a derivation and the space R(L) of all right multiplication operators forms a Lie ideal of Der(L) [2].…”

“…[2]). Let C be a nilpotent subalgebra of a Leibniz n-algebra L. Then C is a Cartan subalgebra if and only if it coincides with L 0 in the Fitting decomposition of the algebra L with respect to R(C).…”