单李代数

Let g be a finite dimensional complex semisimple Lie algebra with the Cartan subalgebra h.

令g为一个有限维的复半单李代数，而h是g的Cartan子李代数。

We prove that any irreduciblesymmetric self-dual Lie algebra 〓 whose Cartan subalgebra consists of semi-simpleelements is in one of the following classes:(1)〓 is simple;(2)〓 is 1-dimensional;(3)〓is solvable.

第五节讨论Cartan子代数由半单元构成的对称自对偶李代数的结构，证明了不可约对称自对偶李代数g如果它的Cartan子代数由半单元构成，那么它只能是下列情形之一：（1）g为单李代数；（2）g是1维的；（3）g是可解的。

Libermann.The early researcheson this kind of manifolds were closely related to Physics and Mechanics.But since1991,S.Kaneyuki published his result on the algebraic condition for the existence ofinvariant〓structures on a coset space,Lie theory has played the most impor-tant role in the study of this kind of manifolds.In particular,dipolarizations in a Liealgebra are closely related to the homogeneous〓manifolds.Dipolarizationsin semisimple Lie algebras and the homogeneous〓manifolds associated withthese dipolarizations have been studied by S.Kaneyuki,Z.X.Hou and S.Q.Deng.Inthe partⅡ of this thesis we study the dipolarizations in some quadratic Lie algebrasand the homogeneous parakahler manifolds associated with these dipolarizations.

Libermann给出的，早期的有关类流形的研究与物理和力学密切相关，自从1991年金行壮二发表了陪集空间上存在不变仿凯勒结构的代数化结果后，李群及李代数理论在这类流形的研究中起着主要作用，特别地，李代数的双极化与这类流形密切相关，半单李代数的双极化的相关几何，金行壮二，候自新和邓少强等人已作了研究，二次李代数是比半单李代数更广且带有非退化不变双线性型的李代数，本文主要研究了二次代数的双极化及相关几何。

In this paper, we mainly use complex semisimple algebras knowledge. First, we give the structure formula of C. Second, we show that any n x n symmetric matrix A is congruent under the symplectic group to the direct sum of a identity matrix and a tridiagonal matrix.

记正交代数和辛代数的交集构成的李代数为L，本文主要是运用复半单李代数的知识，首先给出了L的结构公式，然后给出了任一n×n对称矩阵A辛合同于一个单位矩阵和一个三对角矩阵的直和。

In recent years, the theory of infinite dimensional simple Lie algebra has become an important branch in Lie theory.

近年来无限维单李代数的结构理论以及表示理论已经成为李代数研究中的重要分支，无限维单李代数的构造，自同构群的确定，同构分类，二上同调群的计。。。

The solvable Lie algebra is corresponding to a cascade decomposition of the system and the semisimple Lie algebra is corresponding to a qasi-parallel decomposition such that the system has a parallel form of a cascade decomposition and a qasi-parallel decomposition.

任一李代数都可分解为一可解李代数与一半单李代数的半直和，可解李代数对应于系统的级联分解，半单李代数对应的是系统的准平行分解，将二者合并起来，就得到一般李群下的非线性系统的结构分解，这是一级联形式与一准平行形式的并联形式分解。

This result is a generalization of the Chevalley basis in case a simply-laced complex simple Lie algebra. Meanwhile, it generalizes also the corresponding Chevalley forms in case affine Lie algebras and 2- extended affine Lie algebras.

该结果是simply-laced复半单李代数情形下Chevalley基的推广，同时也是仿射李代数和2-扩大仿射李代数的相应Chevalley结构形式的推广。

Throughout the present paper, the author mainly discuss the complexification of n-Lie algebras and the classification of real simple n-Lie algebras.

就李代数而言，单李代数的复化和复李代数的实形是一个非常重要的问题，同时其分类也是人们关注的焦点，本文将这两个结论放在n-李代数中进行深入的研究。

In the fifth chapter,we study dipolarizations in some quadratic Lie algebras.Inthe first section,we obtain some results on the classification of dipolarizations in gen-eral quadratic Lie algebras,and prove that there exist dipolarizations in the solvablequadratic Lie algebras whose Cartan subalgebras consist of semisimple elements.

第五章讨论了某些二次李代数的双极化，在第一节中，我们给出了二次李代数的双极化的一些分类结果；特别证明Cartan子代数是由半单元组成的二次李代数上存在双极化，第二节确定了四维扩张Heisenberg代数的所有双极化，在第三节中，我们构造了2n+2维扩张Heisenberg代数的六类双极化，我们发现两个不同于半单李代数情形的有趣事实：（1）在扩张Heisenberg代数上同时存在对称和非对称双极化；（2）对应于扩张Heisenberg代数的双极化的特征元有的是半单的有的是幂零的。

Using the concept of regular elements,we obtain some results on the classificationof dipolarizations in general symmetric self-dual Lie algebras.

在第六节中，我们把半单李代数的特征元的概念推广到任意带有双极化的对称自对偶李代数上〓用正则元的概念，得到了一般对称自对偶李代数上双极化的一些分类结果。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。