Derivations, Automorphisms And Local Derivations of Null-Filiform Leibniz Algebras

Abstract

The paper is devoted to the study of derivations, automorphisms, local derivations and local automorphisms of the null-filiform Leibniz algebra NFn, which is the unique n-dimensional complex nilpotent Leibniz algebra of maximal nilindex. We give explicit descriptions of the derivation algebra and the automorphism group and show that the derivation algebra is an n-dimensional solvable Lie algebra with an abelian ideal of codimension one. The main result states that a linear map on NFn is a local derivation (respectively, a local automorphism) if and only if its matrix in the natural basis is lower triangular (respectively, lower triangular and invertible). Consequently, for every n ≥ 2 the algebra NFn admits n(n − 1)/2 linearly independent local derivations which are not derivations.