3-李代数的广义导子
Generalized Derivations of 3-Lie Algebras
Received: February 28, 2015  Revised: January 04, 2017
DOI：10.16205/j.cnki.cama.2017.0036

 Author Name Affiliation E-mail BAI Ruipu School of Mathematical Science, Huaiyin Normal University, Huai'an 223001, Jiangsu, China. bairuipu@hbu.edu.cn LI Qiyong School of Mathematical Science, Huaiyin Normal University, Huai'an 223001, Jiangsu, China. liqiyong2012@163.com ZHANG Kai School of Mathematical Science, Huaiyin Normal University, Huai'an 223001, Jiangsu, China. zhangkai@hebau.edu.cn
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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.
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.