从Leibniz超代数到Lie-Yamaguti超代数（英文）

本文研究了Lie-Yamaguti超代数的构造.利用左Leibniz超代数,先给出左Leibniz超代数的构造方法,再给出用左Leibniz超代数构造Lie-Yamaguti超代数的方法,获得了Lie-Yamaguti超代数的构造方法.将Leibniz代数和Lie-Yamaguti代数的构造推广到超代数的情形.     

n-李代数的结构

主要研究特征为2的代数闭域上(n+1)维n-李代数的结构,给出了(n+1)维n-李代数的分类,描述了其可解性与幂零性,刻画了(n+1)维n-李代数的导子代数与内导子代数的结构.     

n-李代数的结构

主要研究特征为2的代数闭域上(n+1)维n-李代数的结构,给出了(n+1)维n-李代数的分类,描述了其可解性与幂零性,刻画了(n+1)维n-李代数的导子代数与内导子代数的结构.     

n-李代数的中心扩张

对n-李代数的中心扩张问题进行了研究,提出了Heisenberg n-李代数的概念,并对任意一个线性空间V,给出了构造Heisenberg n-李代数H(V)的一种方法且研究了一类特殊类型Heisenberberg n-李代数的导子代数的结构.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

本原李超代数和李超代数的主因子

本文首先引入了本原李超代数,研究了三种类型本原李超代数及其相关的结构性质.接着引入了李超代数主因子,利用第三种类型的本原李超代数性质给出了李超代数的主因子之间存在的L-连接关系.最后,介绍了李超代数的CAP-子代数,证明了若李超代数L的所有极大阶化子代数都是CAP-子代数,那么L是可解的.     

一些无限维完备李代数

本文讨论了无限维完备李代数的一些性质,由Virasoro代数,Kac-Moody代数构造了几类无限维完备李代数.同时给出了Kac-Moody代数及其广义抛物子代数的导子代数的刻划.证明了完备李代数的Loop扩张仍为完备李代数.     

一些无限维完备李代数

本文讨论了无限继完备李代数的一些性质,由Virasoro代数,Kac－Moody代数构造了几类无限维完备李代数．同时给出了Kac－Moody代数及其广义抛物子代数的导子代数的刻划．证明了完备李代数的Loop扩张仍为完备李代数。     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的m-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数足特征单李代数.     

R0代数的Fuzzy子代数与Fuzzy关联MP滤子

引入了R0代数的Fuzzy 子代数、Fuzzy关联MP滤子的概念,给出了R0代数的Fuzzy集是Fuzzy子代数的几个等价刻画,讨论了R0代数的Fuzzy关联MP滤子的若干性质,证明了Fuzzy子代数(Fuzzy关联MP滤子)在R0代数同态(同构)下的不变性.     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

李Rinehart代数的子代数若干性质

本文主要把李代数的c-可补、E-代数的性质以及Frattini理论推广到更为广泛的李Rinehart代数,得到它们的若干性质,给出了可解李Rinehart代数的一个必要条件.同时,分别获得判断c-可补李Rinehart代数和E-李Rinehart代数的一个充分必要条件.     

Conjugacy of Cartan Subalgebras in Solvable Leibniz Algebras and Real Leibniz Algebras

We extend conjugacy results from Lie algebras to their Leibniz algebra generalizations. The proofs in the Lie case depend on anti-commutativity. Thus it is necessary to find other paths in the Leibniz case. Some of these results involve Cartan subalgebras. Our results can be used to extend other results on Cartan subalgebras. We show an example here and others will be shown in future work.     

Criteria for solvability and supersolvability in Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. Recently, analyzing the structure of subalgebras, David Towers gave some criteria for the solvability and supersolvability of Lie algebras. In this paper we define analogues concepts for Leibniz algebras and extend some of these results on solvability and supersolvability to that of Leibniz algebras.     

The growth of varieties of Leibniz algebras with nilpotent commutator subalgebra

It is proved that the exponents of certain varieties of Leibniz algebras with nilpotent commutator subalgebras exist and are integer.     

Classifying Several Classes of Leibniz Algebras

We extend results related to maximal subalgebras and ideals from Lie to Leibniz algebras. In particular, we classify minimal non-elementary Leibniz algebras and Leibniz algebras with a unique maximal ideal. In both cases, there are types of these algebras with no Lie algebra analogue. We also give a classification of E-Leibniz algebras which is very similiar to its Lie algebra counterpart. Note that a classification of elementary Leibniz algebras has been shown in Batten Ray et al. (2011).     

Triangulable Leibniz Algebras

A converse to Lie's theorem for Leibniz algebras is found and generalized. The result is used to find cases in which the generalized property, called triangulable, is 2-recognizable; that is, if all 2-generated subalgebras are triangulable, then the algebra is also. Triangulability joins solvability, supersolvability, strong solvability, and nilpotentcy as a 2-recognizable property for classes of Leibniz algebras.     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.