(n-1)-半单的n-李代数的几何描述

证明了实数域上(n-1)-半单的(n+1)维n-李代数A是n维欧氏空间的Lorentz群O(p,n-p)与n维Abel正规子群的半直积的n-李代数.且当p=0时,A是n维欧氏空间的等距变换群的n-李代数.并提出了关于(n-1)-半单的(n+1)维n-李代数的外导子的物理应用与几何应用问题.     

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.     

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.     

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.     

项链李代数的性质

研究项链李代数的性质,给出了其中心元的表示形式,证明了项链李代数非半单、非可解,通过构造项链李代数的可解非幂零子代数,证明了当箭图中有长度大于1的循环时,项链李代数非幂零.还给出了没有圈的箭图上项链李代数的分解.     

最简线状n-李代数

定义了一类特殊的幂零n-李代数,即最简线状n-李代数,它是最简线状李代数的推广.确定了m维最简线状n-李代数A的导子代数Der(A)和自同构群Aut(A),定义了n-李代数的全形h(A)=Der(A)( ) A,并证明了当A的基域F的特征P为零或P＞m-n时,Der(A)是不可解的完备李代数,而h(A)的一个子代数是可解的完备李代数,当F的特征为零时,Aut(A)是无中心的不可解群.     

单的n+1维n-李代数的表示

本文证明了不可约的L(A)-模是A-模的充要条件,给出了单的n 1-维n-李代数的有限维不可约表示的分类．     

（n-3）-Filiform李代数的极大环面

求出了（n-3）-filiform李代数的极大环面，并证明了（n-3）-filiform李代数是可完备化的．     

关于无限楔表示和孤立子方程的讲演(英文)

本文目的在于扼要介绍仿射李代数的重要概念及其基本表示的实现,然后把它们和孤立子方程的研究联系起来,用代数方法给出它们的解。 全文共分八讲,可议概括为两大部分。第一部分(前五讲)主要介绍了仿射李代数A_(n-1)~(1)以及A_∞的基本表示的两个不同的实现方法:一个称为Fock表示,另一个称为无限楔表示。证明了这两个表示的等价性,并且证明了在这两个表示空间里都可定义反变的、非退化的厄米特型。第二部分(后三讲)讨论了A_∞的上述表示和某一类偏微分方程(所谓的Hirota双线性方程组)的联系,用代数方法刻划了这一类偏微分方程以及它们的解。特别地将这些结果应用到最简单的仿射李代数A_1~(1)的基本表示上,可以得到KdV方程:的N-孤立子解。     

李代数SU(n)和阶化李代数SU(n/1)的不可约表示

本文对李代数SU(n)的生成元,引入了SU(n-1),SU(n-2),…SU(2)的不可约张量.并利用不可约张量的性质,求出了SU(n)的不可约表示,这些不可约表示的表达式比较简单.文中还给出了SU(n)的部分约化系数和Racah系数的代数表达式.在此基础上,求出了阶化李代数SU(n/1)的不可约表示.     

Analogs of the cartan criteria forn-Lie algebras

It is shown that Cartan's criteria for finite-dimensional Lie algebras to be semisimple and solvable are fully adaptable to n-Lie algebras, provided that ideals of an n-Lie algebra are understood to be solvable in the sense of Kuz'min. Specifically, we present a characterization of the Kuz'min radical in terms of a trace form associated with some representation ρ, which is analogous to the characterization which we have in the case of Lie algebras. One more analog of the Cartan theorem is proved for n-Lie algebras which are solvable in the sense of Filippov. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 274-287, May-June, 1995.     

Engel Subalgebras of n-Lie Algebras

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. 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. Cartan subalgebras are shown to have a property analogous to intravariance.     

紧 n -Lie 代数与不变双线性型

该文研究 n -Lie代数的非退化不变双线性型. 给出了实数域上紧 n -Lie 代数的概念, 并对其结构进行了研究.     

Exact and Semisimple Differential Graded Algebras

In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology.     

Automorphism groups of pointed Hopf algebras

The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.      

The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

A Frobenius–Schur Theorem for Hopf Algebras

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

关于BCI—代数伴随半群中的可逆元

给出了ＢＣＩ－代数的伴随半群中可逆元的刻划，说明ＢＣＩ－代数的ｐ－半单闭理想与其伴随半群的子群之间有一一对应关系，并证明最大的ｐ－半单闭理想是根理想。