Nonlinear maps satisfying derivability on the parabolic subalgebras of the general linear Lie algebras

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Let n be a positive integer greater than 2, 𝔽 be a field of characteristic other than 2 and contain at least n different elements, gl(n, 𝔽) the general linear Lie algebra over 𝔽, P a parabolic subalgebra of gl(n, 𝔽). A map ϕ on P is said to satisfy derivability if ϕ([x, y]) = [ϕ(x), y] + [x, ϕ(y)] for all x, y ∈ P, where ϕ may not be linear. In this article, we prove that a map ϕ on P satisfies derivability if and only if ϕ is a sum of an inner derivation, an additive quasi-derivation and a central quasi-derivation on P.