Local automorphisms on finite-dimensional Lie and Leibniz algebras

TL;DR

This paper characterizes local automorphisms on finite-dimensional Lie and Leibniz algebras, showing they are mostly automorphisms or anti-automorphisms, with some exceptions in nilpotent Lie algebras.

Contribution

It provides a complete characterization of local automorphisms on simple Lie and Leibniz algebras, identifying when they coincide with automorphisms.

Findings

Local automorphisms on sl_n are automorphisms or anti-automorphisms.

On simple Leibniz algebras of a specific form, local automorphisms are automorphisms.

Existence of local automorphisms that are not automorphisms in certain nilpotent Lie algebras.

Abstract

We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form \(\mathfrak{sl}_n\dot +\mathcal{I}\) is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras \(\mathcal{L}\) with \(\dim \mathcal{L} \geq 3\) which admit local automorphisms which are not automorphisms.