On derivations of parabolic Lie algebras

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed, characteristic zero field or over $\mathbb{R}$. Let $\mathfrak{q}$ be a parabolic subalgebra of $\mathfrak{g}$. We characterize the derivations of $\mathfrak{q}$ by decomposing the derivation algebra as the direct sum of two ideals: one of which being the image of the adjoint representation and the other consisting of all linear transformations on $\mathfrak{q}$ that map into the center of $\mathfrak{q}$ and map the derived algebra of $\mathfrak{q}$ to $0$.