Invertible linear maps on simple Lie algebras preserving commutativity

Abstract:

Let $\mathfrak {g}$ be a finite-dimensional simple Lie algebra of rank $l$ over an algebraically closed field of characteristic zero. An invertible linear map $\varphi$ on $\mathfrak {g}$ is called preserving commutativity in both directions if, for any $x, y\in \mathfrak {g}$, $[x,y]=0$ $\Leftrightarrow$ $[\varphi (x),\varphi (y)]=0$. The group of all such maps on $\mathfrak {g}$ is denoted by $Pzp (\mathfrak {g})$. It is shown in this paper that, if $l=1$, then $Pzp(\mathfrak {g})=GL(\mathfrak {g})$; otherwise, $Pzp(\mathfrak {g})=Aut (\mathfrak {g})\times F^*I_{\mathfrak {g}}$, where $F^*I_{\mathfrak {g}}$ denotes the group of all non-zero scalar multiplication maps on $\mathfrak {g}$.