# Local and 2-local derivations and automorphisms on simple Leibniz algebras

## TL;DR

This paper investigates local and 2-local derivations and automorphisms on simple Leibniz algebras, proving that they are generally derivations or automorphisms, with exceptions in nilpotent cases.

## Contribution

It establishes that all local and 2-local derivations and automorphisms on simple Leibniz algebras are actual derivations and automorphisms, respectively, extending understanding of their structure.

## Key findings

- All local and 2-local derivations on simple Leibniz algebras are derivations.
- Every 2-local automorphism on simple Leibniz algebras is an automorphism.
- Nilpotent Leibniz algebras admit non-automorphism 2-local automorphisms.

## Abstract

The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show 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. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show 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.