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