$2$-local automorphisms on finite-dimensional Lie algebras

TL;DR

This paper proves that all 2-local automorphisms on finite-dimensional semi-simple Lie algebras are automorphisms, but provides examples of non-automorphic 2-local automorphisms on certain nilpotent Lie algebras.

Contribution

It establishes the equivalence of 2-local automorphisms and automorphisms for semi-simple Lie algebras and shows existence of non-automorphic 2-local automorphisms in nilpotent cases.

Findings

All 2-local automorphisms on semi-simple Lie algebras are automorphisms.

Existence of non-automorphic 2-local automorphisms in nilpotent Lie algebras.

Characterization of 2-local automorphisms depends on Lie algebra structure.

Abstract

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an automorphism.