On derivations of parabolic Lie algebras

TL;DR

This paper characterizes the derivations of parabolic subalgebras in reductive Lie algebras over algebraically closed fields or the real numbers, decomposing the derivation algebra into specific ideal components.

Contribution

It provides a detailed decomposition of derivation algebras of parabolic subalgebras, revealing their structure in terms of adjoint images and central mappings.

Findings

Derivation algebra decomposes into two ideals.

One ideal is the image of the adjoint representation.

The other consists of linear maps into the center that vanish on the derived algebra.

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 $0$.