A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Abstract

Abstract. Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. Nilpotent Lie algebras and invertible derivationsWhich information about the structure of a Lie algebra is contained in its group of automorphisms or in its Lie algebra of derivations? Several partial answers to this question, in terms of sufficient conditions for the nilpotency or solvability of the Lie algebra, are known -see for example the theorems by Borel, Serre and Mostow [BS, BM]. We will mainly be interested in the following well-known theorem by Jacobson on the invertibility of Lie algebra derivations. Once again, it was asked whether the converse were true; but counterexamples were found: nilpotent Lie algebras admitting only nilpotent pre-derivations. Such Lie algebras are called strongly nilpotent, [Bu]. By stating Jacobson's theorem even more generally, in terms of

A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

“…Moreover, Moens proved the following Lemma (actually, he proved it only for the right Leibniz-derivations, but one can easily that this lemma is true for the left Leibniz-derivations and Leibniz-derivations): Lemma 5. [34] If s, t ∈ N and s|t then LDer l(s+1) (A) ⊆ LDer l(t+1) (A), LDer s+1 (A) ⊆ LDer t+1 (A).…”

“…We will need the following proposition by Moens: Proposition 6. [34] For n ≥ 2 the subsets LDer(A), LDer n (A) and LDer l(n) (A) are in fact subalgebras of the Lie algebra of endomorphisms of A and we have the chains…”

“…In paper [34] a generalization of derivations and pre-derivations of Lie algebras is defined as a Leibnizderivation of order k. Moens proved that a finite-dimensional Lie algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz-derivation. After that, Fialowski, Khudoyberdiyev and Omirov [13] showed that with the definition of Leibniz-derivations from [34] the similar result for non-Lie Leibniz algebras is not true. Namely, they gave an example of non-nilpotent Leibniz algebra which admits an invertible Leibniz-derivation.…”

On some structures of Leibniz algebras

“…Dixmier and Lister [16] constructed an example to show the converse does not hold directly. Allowing a weaker form of operator than derivation, Moens (see [25]) developed a converse. A similar process has been obtained for Leibniz algebras in [17].…”

Alternative algebras admitting derivations with invertible values and invertible derivations

“…The definition of an invertible derivation as an invertible mapping first arose in [7], where the nilpotency of a Lie algebra admitting an invertible derivation was proved. The research on that topic was then continued in [8,9].…”