Journal of Lie Theory 20 (2010), No. 1, 167--174
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
[Abstract-pdf]
\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
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