华东师范大学学报(自然科学版) ›› 2011, Vol. 2011 ›› Issue (5): 93-102.

• 应用数学与基础数学 • 上一篇    下一篇

李代数的张量积所确定的Leibniz代数

颜倩倩   

  1. 华东师范大学 数学系, 上海 200241
  • 收稿日期:2011-02-01 修回日期:2011-05-01 出版日期:2011-09-25 发布日期:2011-11-22

Leibniz algebras defined by tensor product of Lie algebras

YAN Qian-qian   

  1. Department of Mathematics, East China Normal University, Shanghai 200241, China
  • Received:2011-02-01 Revised:2011-05-01 Online:2011-09-25 Published:2011-11-22

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摘要： 讨论了李代数\,$\mathcal{G}$\,以及由这个李代数诱导的\$\mathrm{Leibniz}$\,代数\,$\mathcal{G}\otimes\mathcal{G}$\,的一些性质, 主要从不变双线性型和导子看这两个代数之间的差异, 证明了在特定条件下两者的不变双线性型维数是一致的. 为进一步确定李代数\,$\mathcal{G}$\,和\,$\mathcal{G}\otimes\mathcal{G}$\的差异, 讨论了由\,$\mathcal{G}\otimes\mathcal{G}$\,诱导的一类重要的李代数\,$\mathcal{G}\boxtimes\mathcal{G}$; 最后证明了, 如果\,$\mathcal{G}$\,是有限维半单李代数, $\mathcal{G}$\,和\,$\mathcal{G}\boxtimes\mathcal{G}$\,是同构的.

关键词: Leibniz代数, 不变对称双线性型, 张量积, 导子, 边染色, 最大度, 第一类图

Abstract: By the definition of $\mathrm{Leibniz}$ algebra, we showed that \ $\mathcal{G}\otimes\mathcal{G}$\ was a $\mathrm{Leibniz}$\ algebra when \ $\mathcal{G}$\ was a $ \mathrm{Lie}$ algebra. We also proved that $\mathcal{G}\otimes\mathcal{G}$\ and $\mathcal{G}$\ have the same dimension of invariant symmetric bilinear forms in a special case, and the dimension of the derivation algebra of\ $\mathcal{G}$\ is always less than that of $\mathcal{G}\otimes\mathcal{G}$. $\mathcal{G}\boxtimes\mathcal{G}$\ is one of the important \ $\mathrm{Lie}$\ algebra induced by $\mathcal{G}\otimes\mathcal{G}$, and $\mathcal{G}\boxtimes\mathcal{G}$\ is isomorphic to $\mathcal{G}$\ when $\mathcal{G}$\ is a finite dimensional semi-simple\ $\mathrm{Lie}$\ algebra.

Key words: Leibniz algebra, invariant symmetric bilinear form, tensor product, derivation

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