Invertible linear maps on simple Lie algebras preserving commutativity

Abstract

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