Product Zero Derivations of the Parabolic Subalgebras of Simple Lie Algebras

D. Wang ; W. Zhang ; Z. Chen

Journal of Lie theory, Tome 20 (2010) no. 1, pp. 167-174

Résumé

\def\b{{\frak b}} \def\g{{\frak g}} \def\p{{\frak p}} Let $\g$ be a simple Lie algebra of rank $l$ over an algebraic closed field of characteristic zero, $\b$ a Borel subalgebra of $\g$, $\p$ a parabolic subalgebra of $\g$ containing $\b$. A linear map $\varphi$ on $\p$ is called a product zero derivation if, for $x, y\in \p$, $[x,y]=0$ implies $[\varphi(x), y]+[x,\varphi(y)]=0$. It is shown in this paper that a product zero derivation $\varphi$ on $\p$ is just a sum of an inner derivation and a scalar multiplication map in case that $l\geq 2$.

Détail

Classification : 17B20, 17B30, 17B40
Mots-clés : Simple Lie algebras, parabolic subalgebras, derivations of Lie algebras