Product Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras
Received:May 30, 2012  Revised:August 15, 2012
Key Word: product zero derivations   strictly upper triangular matrix Lie algebras   derivations of Lie algebras.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11101084) and the Natural Science Foundation of Fujian Province (Grant No.2013J01005).
 Author Name Affiliation Zhengxi CHEN School of Mathematics and Computer Science, Fujian Normal University, Fujian 350007, P. R. China Liling GUO School of Mathematics and Computer Science, Fujian Normal University, Fujian 350007, P. R. China
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Let $\mathbb{F}$ be a field, $n\geq 3$, ${\bf N}(n,\mathbb{F})$ the strictly upper triangular matrix Lie algebra consisting of the $n\times n$ strictly upper triangular matrices and with the bracket operation $[x,y]=xy-yx$. A linear map $\varphi$ on ${\bf N}(n, \mathbb{F})$ is said to be a product zero derivation if $[\varphi(x), y]+[x, \varphi(y)]=0$ whenever $[x,y]=0, x,y\in {\bf N}(n,\mathbb{F})$. In this paper, we prove that a linear map on ${\bf N}(n,\mathbb{F})$ is a product zero derivation if and only if $\varphi$ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on ${\bf N}(n,\mathbb{F})$.