Some radicals, Frattini and Cartan subalgebras of Leibnizn-algebras

Abstract

In the present work we introduce notions such as k-solvability, s-and K 1 -nilpotency and the corresponding radicals. We prove that these radicals are invariant under derivations of Leibniz n-algebras. The Frattini and Cartan subalgebras of Leibniz n-algebras are studied. In particular, we construct examples that show that a classical result on conjugacy of Cartan subalgebras of Lie algebras, which also holds in Leibniz algebras and Lie n-algebras, is not true for Leibniz n-algebras.

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

“…Definition 1.12. [13] An ideal H of a Leibniz n-algebra L is said to be k-solvable with index of k-solvability equal to m if there exists m ∈ N such that H (m) k = 0 and H (m−1) k = 0. An ideal H is called solvable if it is n-solvable.…”

“…In [13,Theorem 4.5] the invariance of a k-radical under a derivation of a Leibniz n-algebra is proven.…”

“…While the structure theory of Leibniz and n-Lie algebras has been well-developed, the same cannot be said about n-Leibniz algebras. Although a notion of a solvable radical and some other classical properties of the radical of Leibniz n-algebras are obtained in [13], Levi decomposition for Leibniz n-algebras has not yet been established, while for n-Lie and Leibniz algebras they are proven by Ling [18] and Barnes [6], respectively. Even in small dimensions, there is only a partial classification of two-dimensional Leibniz n-algebras over complex numbers in [1] and [8].…”

Some necessary and sufficient conditions for nilpotent n-Lie superalgebras

“…n-Lie superalgebras are generalizations of n-Lie algebras and Lie superalgebras. As the structural properties of n-Lie superalgebras mostly remain unexplored and motivated by the investigation on Engel's theorem and nilpotency of n-Lie algebras [4], [8], [9], [13], [15] and Leibniz n-algebras [1], [5], [7], [12], it is natural to ask about the extension of these properties to the n-Lie superalgebras category. As is well known, for n-Lie algebras and Leibniz n-algebras, Engel's theorem and nilpotency play a predominant role in Lie theory.…”