Nonlinear Lie triple derivations on parabolic subalgebras of finite-dimensional simple Lie algebras

Abstract

A map ϕ on a Lie algebra is called a nonlinear Lie triple derivation if ϕ([x, [y, z]]) = [ϕ(x), [y, z]] + [x, [ϕ(y), z]] + [x, [y, ϕ(z)]] for all , where ϕ may not be linear. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a standard parabolic subalgebra of L. In this article, we prove that a map ϕ on P is a nonlinear Lie triple derivation if and only if ϕ is a sum of an inner derivation and an additive quasi-derivation.