限制李三系的Frattinip-子系

主要把群的Frattini理论发展到限制李三系,得到限制李三系的Frattini-子系、Frattini p-子系的性质,给出Frattini p-子系为零时限制李三系的分解.     

限制李三系的半单元

研究了限制李三系的半单元的一些重要性质,给出了若干个限制李三系是可换的条件,得到了限制李三系的有环面元基的几个条件,刻划了限制李三系的Frattini p-子系的一些性质.同时,研究了中心为零的所有元素是半单元的限制李三系的一些重要性质.     

李三系的中心扩张

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.对李三系的中心扩张问题进行了研究,提出了Heisenberg李三系的概念,并对任意线性空间给出了构造Heisenberg李三系的一种方法.     

李三系的根基与次理想

本文讨论李三系的可解根基和Hopkins幂零的某些性质及导子作用下的不变性，讨论了李三系次理想的某些性质，证明了李三系为幂零的当且仅当每个子系都是次理想．     

具有导子的李三系的中心扩张与形变

考虑具有导子的李三系.由李三系和一个导子称为LietsDer对.定义系数在表示中的LietsDer对的上同调理论.研究LietsDer对的中心扩张.接下来,将形变理论推广到由李三系和导子构成LietsDer对上,它由带有系数的LietsDer对的上同调所支配.     

辛三代数的Frattini子代数

本文研究了辛三代数的Frattini子代数和基本辛三代数的问题.利用Frattini子代数和基本辛三代数的性质,得到了辛三代数的非嵌入定理,从而推广了李三系中关于Frattini子系的结果.     

分裂的$\delta$-Jordan 李三系

本文研究了任意分裂的$\delta$-Jordan李三系的结构,其为分裂的李三系的结构的推广. 利用这种三系的根连通, 得到了带有对称根系的分裂的 $\delta$-Jordan 李三系可以表示成 $T=U+\sum_{[\alpha]\in \Lambda^{1}/\sim} I_{[\alpha]}$,其中$U$是0根空间$T_{0}$的子空间,任意$I_{[\alpha]}$为$T$的理想, 并且满足 当$[\alpha]\neq [\beta]$时, $\{I_{[\alpha]},T,I_{[\beta]}\}=\{I_{[\alpha]},I_{[\beta]},T\}=\{T,I_{[\alpha]},I_{[\beta]}\}=0$.     

可度量化李三系

研究域K上可定义非退化不变对称双线性形的李三系■(称这样的李三系■为可度量化的).对一个李三系■,我们定义了该李三系的T_ω~*-扩张,给出了一个度量化李三系(■,φ)同构于某李三系的T~*-扩张的充分必要条件,并且讨论了什么时候两个T~*-扩张是等价或度量等价的.最后证明当基域的特征为零时,偶数维幂零的度量化李三系与某李三系的T~*-扩张同构,而奇数维幂零的度量化李三系则同构于某李三系的T~*-扩张的余维数为1的非退化理想.     

关于李三系的分类

首先基于复单李代数的所有对合自同构的分类,研究了复单李三系的分类.然后研究了实单李三系的分类,得到了实单李二三系或者同构于一个复单李三系的实形式,或者同构于复单李三系的实化的结论.同时还给出了关于实李三系复化和复李三系实化的部分结果.     

李三系分解的唯一性

李三系是从黎曼对称空间产生的三元运算的代数体系，近来得到许多数学家的重视．但是至今为止，李三系的研究集中在半单与单李三系上．本文主要讨论中心为零的李三系的一些基本问题，特别是分解唯一性问题．我们的结果包含了半单李三系的分解唯一性．     

Elementary and Φ-Free Lie Triple Systems

We introduce elementary and Φ-free Lie triple systems and study the properties of these systems. In particular, structures of subsystems of an elementary Lie triple system and a class of Φ-free Lie triple systems are investigated.     

李三系与Laurent多项式代数F[t,t~(-1)]

本文通过讨论Laurent多项式代数及其导子代数的对合自同构确定了一类具体的无限维单李三系, 并且提供了一种利用Novikov代数上自然的李代数结构来构造李三系的方法.     

Systems of Quotients of Lie Triple Systems

In this article, we introduce the notion of system of quotients of Lie triple systems and investigate some properties which can be lifted from a Lie triple system to its systems of quotients. We relate the notion of Lie triple system of Martindale-like quotients with respect to a filter of ideals and the notion of system of quotients, and prove that the system of quotients of a Lie triple system is equivalent to the algebra of quotients of a Lie algebra in some sense, and these allow us to construct the maximal system of quotients for nondegenerate Lie triple systems.     

On the Representation Theory of Lie Triple Systems

In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems.

     

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

Dialgebras and related triple systems

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz-Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz-Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an (ε, δ)-Freudenthal-Kantor triple system on a dialgebra.