Nonlinear maps satisfying derivability on standard parabolic subalgebras of finite-dimensional simple Lie algebras

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. 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 be not linear. Note that maps satisfying derivability on P may be not derivations on P. In this article, we prove that a map ϕ (without linearity assumption) on P satisfies derivability if and only if ϕ is a sum of an inner derivation and an additive quasi-derivation on P. Moreover, we obtain a corollary that any derivation of P is an inner derivation.

Перевод пока недоступен