Nonlinear Strong Commutativity Preserving Maps and Nonlinear Strong Product Zero Derivations on Simple Lie Algebras

Let L be a finite-dimensional simple Lie algebra over an algebrically closed field F of characteristic zero. A nonlinear map f: L → L is called a strong commutativity preserving map if f is invertible and for any x,y ∈ L,[f( x) ,f( y) ] = [x,y]. It shows that a strong commutativity preserving map over L is just an identical mapping or negative identical mapping. A nonlinear map δ: L → L is called a nonlinear strong product zero derivation if for any x,y ∈ L,[δ( x) ,y]+[x,δ( y) ] = 0 . It is shown that a strong product zero derivation is just a zero map.