
Journal of Lie Theory 20 (2010), No. 1, 167174 Copyright Heldermann Verlag 2010 Product Zero Derivations of the Parabolic Subalgebras of Simple Lie Algebras Dengyin Wang Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, P. R. China wdengyin@126.com Wei Zhang Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, P. R. China Zhengxin Chen School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, P. R. China chenzx@fjnu.edu.cn [Abstractpdf] \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 $l\geq 2$. Keywords: Simple Lie algebras, parabolic subalgebras, derivations of Lie algebras. MSC: 17B20, 17B30, 17B40 [ Fulltextpdf (153 KB)] for subscribers only. 