Local automorphisms of finite dimensional simple Lie algebras

Abstract

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over an algebraically closed field $K$ of characteristic $0$. A linear map $\phi :\mathfrak{g}\to \mathfrak{g}$ is called a local automorphism if for every $x$ in $\mathfrak{g}$ there is an automorphism ${\phi }_x$ of $\mathfrak{g}$ such that $\phi (x)={\phi }_x(x)$. We prove that a linear map $\phi :\mathfrak{g}\to \mathfrak{g}$ is local automorphism if and only if it is an automorphism or an anti-automorphism.

14 pages