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

摘要

A map φ on a Lie algebra g 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 x, y, z ∈ g 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.