李代数的张量积所确定的Leibniz代数

颜倩倩

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

详细信息

中图分类号: O152
计量
- 文章访问数: 2595
- HTML全文浏览量: 16
- PDF下载量: 191
- 被引次数: 0
出版历程
- 收稿日期: 2011-02-01
- 修回日期: 2011-05-01
- 刊出日期: 2011-09-25

Leibniz algebras defined by tensor product of Lie algebras

YAN Qian-qian

1.
Department of Mathematics, East China Normal University, Shanghai 200241, China

摘要

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.
- 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.
- Leibniz algebra /
- invariant symmetric bilinear form /
- tensor product /
- derivation /

HTML全文

参考文献(1)

[1]

{1}BLOCH~A. On a generalization of Lie Algebra [J]. Math in USSR Doklady, 1965, 165(3): 471-473.
{2}LODAY~J~L. Cyclic Homology [M]. Berlin: Springer-Verlag, 1992.
{3}LODAY~J~L, Pirashvili~T. Universal enveloping algebras of Leibniz algebras and (co)homology [J]. Mathematische Annalen, 1993, 296(1): 139-158.
{4}LODAY~J~L. K\"{u}nneth-style formula for the homology of Leibniz algebras [J]. Mathematics Zeitschrift, 1996, 221(1): 41-47.
{5}PIRASHVILI~T. On Leibniz homology [J]. Annales De L'Institut Fourier, 1994, 44(2): 401-411.
{6}AYUPOV~A~SH, OMIROV~B~A. On Leibniz algebra [C]. Algebra and Operator Theory Proceeding of the Colloquium in Tashkent, 1997: 1-12.
{7}AYUPOV~A~SH, OMIROV~B~A. On some classes of nilpotent Leibniz algebra [J]. Siberian Mathematical, 2001, 42(1): 15-24.
{8}ALBEVERIO~S, AYUPOV~A~SH, OMIROV~B~A. On nilpotent and simple Leibniz algebra [J]. Communications in Algebra, 2005, 33(1): 159-172.
{9}KURDIANI~R, PIRASHVILI~T. A Leibniz Algebra Structure on the Second Tensor Power[J]. Journal of Lie Theory, 2002: 583-596.
{10}LIU D, LIN L. On the toroidal Leibniz algebras[J]. Acta Mathematica Sinica, English Series, 2008, 24(2): 227-240.
{11}LIU D, HU NH. Leibniz Algebras Graded by Finite Root Systems [J]. Algebra Colloquium, 2010, 17(3): 431-446.
{12}蒋启芬. 三维\ $\mathrm{Leibniz}$\ 代数的分类 [J]. 数学研究与评论, 2007, 27(4): 677-686.\\

JIANG Q F. Classification of 3-Dimensional Leibniz Algebras [J]. Journal of Mathematical Research and Exposition, 2007, 27(4): 677-686.
{13}DATTOLI~G, LEVI~D, WINTERNITZ~P. Heisenberg algebra umbral calculus and orthogonal polynomials [J]. Journal of Mathematical Physics, 2008, 49(5)：9-19.

施引文献

资源附件(0)

访问统计