李超代数上的不变双线性型

本文通过定义李超代数上的形心和零次形心来考察其性质.证明了二次李超代数(G,B)上的不变数积的集合和其形心中的可逆B-超对称元素的集合之间存在一一对应.而对实单李超代数分为两种不同的类型或者是一个忽略了复结构的复李超代数或者是一个复单李超代数的实形式.     

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.     

Hom-李超代数的结构

类比于单李超代数的结构性质,证明了单Hom-李超代数没有任何非平凡的左(右)理想、理想.通过给出保积Hom-李超代数的若干性质,建立了保积Hom-李超代数与李超代数之间的关系.特别地,证明了正则Hom-李超代数是可解(幂零)的充要条件是其容许李超代数是可解(幂零)的,并给出了正则Hom-李超代数是单的必要条件为其容许李超代数是单的.     

李超代数的弱C-理想及其相关性质

本文首先引入了李超代数的弱c-理想、弱c-单李超代数、弱c-理想可补的概念,然后研究了特征不为2,3的基域上李超代数与弱c-理想相关的一些结构性质,给出一个李超代数是弱c-单李超代数的充要条件,并利用Frattini理想,给出了李超代数的一个弱c-理想是其理想的充分条件,同时给出其商代数的子代数有子理想补的充要条件;最...     

有限维模李超代数(-W)

本文构造了有限维模李超代数(-W).利用已有的模李超代数W(m,n,t_),证明了(-W)是单李超代数,获得了(-W)是限制李超代数的一个充分必要条件,进而证明了-W没有非退化结合型.     

关于单李超三系的分类(英文)

本文证明了特征零代数闭域上的单李超三系可分为两部分:一是将单李超代数视为李超三系;二是单李超代数的对合自同构的(-1)-特征子空间.这些推广了Lister和Meyberg给出的关于单李三系分类的相应结果.     

論实半簡单李代数的正則定理

<正> 曾經証明了复牛筒单李代数[算]的最大非半简单子代数都包含了[g]的一个Cartan子代数.本文的目的是証明这个定理对实牛簡单李代数来耕也是正确的. 以下我們以g代表实牛簡单李代数,M代表g的最大非牛簡单子代数.以代表某一个实李代数L的复化.     

无限维L-型模李超代数

研究素特征域上无限维L-型李超代数.证明了它的单性,利用ad-幂零元,证明了L-型李超代数的自然滤过在其自同构下是不变的.由此可得到L-型李超代数的一个特性.     

一类Weyi型单李超代数

本文研究了单李超代数的构造理论.借助于张量积方法,定义了一类Weyl型结合超代数和一类Weyl型李超代数,并且证明了这类Weyl型结合超代数和Weyl型李超代数是单的充分必要条件.     

扩张Hamilton李超代数的Borel子代数

设为特征零的代数闭域上秩为5的有限维Z-分次Hamilton单李超代数H通过添加次数导子得到的扩张李超代数.本文通过对正则元的分类,证明了关于典范环面共有160个正根系,从而得到160个Borel子代数;通过单根以及连接的定义,确定了每一个正根系的单根系,进而刻画了任意两个Borel子代数的连接关系;最后证明了共有48个Borel子代数是极大可解子代数.本文所得结果可用于进一步研究Cartan型单李超代数的结构与表示.     

Super-biderivations of classical simple Lie superalgebras

In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field \(\mathbb {C}\). Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras.     

奇Contact李超代数偶部的导子

在特征p>2的情况下,利用奇Contact李超代数偶部的生成元集,通过计算导子在其生成元集上的作用的方法,确定了奇Contact李超代数偶部的-1次数的导子.     

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type Ⅰ.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

反李三系的不变双线性型

本文考察了反李三系和它的标准嵌入李超代数的Killing型之间的关系,并且证明了反李三系的反对称不变双线性型可以被唯一地扩张到它的标准嵌入李超代数。作为扩张定理的一个应用,得到了二次李和反李三系的唯一分解定理。     

On complex lie supergroups and split homogeneous supermanifolds

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, $ \mathcal{O} $ _G ) is a complex Lie supergroup and H ? G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, $ \mathcal{O} $ _G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, $ \mathcal{O} $ _G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold. It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [OS1]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.     

Uniqueness of the decomposition of lie superalgebras and quadratic lie superalgebras

A Lie superalgebra endowed with a non-degenerate super-symmetric and invariant bilinear form is called a quadratic Lie superalgebra. ln this paper, we consider the decomposition of a Lie superalgebra and the first main result is that the decomposition of a Lie superalgebra into indecomposable graded ideals is unique up to an isomorphism. Next, we obtain the uniqueness of the decomposition of an arbitrary quadratic Lie superalgebra into irreducible graded ideals up to an isometry.     

2-Local Automorphisms on Basic Classical Lie Superalgebras

Let G be a basic classical Lie superalgebra except A(n, n) and D(2, 1, α) over the complex number field C. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on G is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra spl(3, 3).     

ON SOLVABLE COMPLETE LIE SUPERALGEBRAS

The authors discuss the properties of solvable complete Lie superalgebra, proving that solvable Lie superalgebras of maximal rank are complete.