| 白瑞蒲,李奇勇,张凯.3-李代数的广义导子[J].数学年刊A辑,2017,38(4):447~460 |
| 3-李代数的广义导子 |
| Generalized Derivations of 3-Lie Algebras |
| 投稿时间：2015-02-28 修订日期：2017-01-04 |
| DOI：10.16205/j.cnki.cama.2017.0036 |
| 中文关键词: 3-Lie algebra, Derivation, Generalized derivation, Quasiderivation, Quasicentroid |
| 英文关键词:3-Lie algebra, Derivation, Generalized derivation, Quasiderivation, Quasicentroid |
| 基金项目:本文受到国家自然科学基金(No.11371245)和河北省自然科学基金(No.A2014201006)的资助. |
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| 摘要点击次数: 887 |
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| 中文摘要: |
| 给出了$3${-}李代数的广义导子、拟导子、拟型心的定义, 研究了他们之间的结构关系,
并对具有极大对角环面的$3${-}李代数的拟导子和拟型心结构进行了系统的研究.
证明了 (1) 广义导代数$G{\rm Der}(A)$ 可以分解成拟导子代数$Q{\rm Der}(A)$ 和拟型心
$Q\Gamma(A)$ 的直和; (2) $3${-}李代数$A$的拟导子可以扩张成一个具有较大维数的
$3${-}李代数的导子; (3) 拟导子代数$Q{\rm Der}(A)$ 包含在拟型心的正规化子中, 表示为
$[Q{\rm Der}(A), Q\Gamma(A)]\subseteq Q\Gamma(A)$; (4) 如果$A$ 包含极大对角环面$T$,
那么$Q{\rm Der}(A)$和$Q\Gamma(A)$是$T$的对角模, 也就是$(T, T)$半单地作用在$Q{\rm Der}(A)$和$Q\Gamma(A)$上. |
| 英文摘要: |
| The authors introduce generalized derivations, quasiderivations and
quasicentroid of 3-Lie algebras, and studied their relations. They also
investigate the structure of quasiderivations and quasicentroid of
$3$-Lie algebras that contains a maximal diagonalized torus. It is
proved that (1) the generalized derivation algebra $G{\rm Der}(A)$ of a 3-Lie algebra $A$
is the direct sum of quasiderivation algebra $Q{\rm Der}(A)$ and quasicentroid
$Q\Gamma(A)$; (2) quasiderivations of $A$ can be embedded as derivations in a larger algebra;
(3) quasiderivation algebra $Q{\rm Der}(A)$ normalizes quasicentroid, that is,
$[Q{\rm Der}(A), Q\Gamma(A)]\subseteq Q\Gamma(A)$; (4) if $A$ contains a maximal diagonalized torus $T$,
then $Q{\rm Der}(A)$ and $Q\Gamma(A)$ are diagonalized $T$-modules, that is, as $T$-modules, $(T, T)$
semi-simplely acts on $Q{\rm Der}(A)$ and $Q\Gamma(A)$, respectively. |
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