陈正新,郭丽玲.严格上三角矩阵李代数上的积零导子[J].数学研究及应用,2013,33(5):528~542 |
严格上三角矩阵李代数上的积零导子 |
Product Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras |
投稿时间:2012-05-30 修订日期:2012-08-15 |
DOI:10.3770/j.issn:2095-2651.2013.05.002 |
中文关键词: 积零导子 严格上三角矩阵李代数 李代数的导子. |
英文关键词:product zero derivations strictly upper triangular matrix Lie algebras derivations of Lie algebras. |
基金项目:国家自然科学基金(Grant No.11101084),福建省自然科学基金(Grant No.2013J01005). |
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中文摘要: |
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. |
英文摘要: |
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 ${\bf N}(n,\mathbb{F})$. |
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