Local and 2-Local Derivations and Automorphisms on Simple Leibniz Algebras

Abstract

This paper is devoted to study local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We show that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We prove that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further, we consider automorphisms of simple Leibniz algebras. It is established that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and that nilpotent Leibniz algebras admit 2-local automorphisms which are not automorphisms. A similar problem concerning local automorphism on simple Leibniz algebras is reduced to the case of simple Lie algebras.