Product Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras

DOI：10.3770/j.issn:2095-2651.2013.05.002

 作者 单位 陈正新 福建师范大学数学与计算机科学学院, 福建 福州 350007 郭丽玲 福建师范大学数学与计算机科学学院, 福建 福州 350007

设$F$ 为域, $n\geq 3$, $\bf{N}$$(n,\mathbb{F}) 为域\mathbb{F} 上所有n\times n 阶严格上三角矩阵构成的严格上三角矩阵李代数, 其李运算为[x,y]=xy-yx. \bf{N}$$(n, \mathbb{F})$ 上一线性映射$\varphi$ 称为积零导子,如果由$[x,y]=0, x,y\in \bf{N}$$(n,\mathbb{F}),总可推出 [\varphi(x), y]+[x,\varphi(y)]=0. 本文证明 \bf{N}$$(n,\mathbb{F})$上一线性映射 $\varphi$ 为积零导子当且仅当 $\varphi$ 为$\bf{N}$$(n,\mathbb{F})$ 上内导子, 对角线导子, 极端导子, 中心导子和标量乘法的和.

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 ${\bf N}(n,\mathbb{F})$.