有限维Cartan型模李超代数的保积Hom-结构

在特征大于3的代数闭域上,利用Cartan型模李超代数的自然滤过不变性,通过计算Cartan型模李超代数的Hom-映射在其-1分支上的作用,本文证明了有限维Cartan型模李超代数的保积Hom-结构是平凡的.     

有限维单Cartan型模李超代数HO

本文构造了一族有限维Cartan型模李超代数-奇Hamilton模李超代数HO, 并证明了其单性.通过建立Cartan型模李超代数W,H,K,S和HO的维数公式,讨 论了奇Hamilton模李超代数HO与Cartan型模李超代数W,S,H,K的同构关系.     

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.     

李超代数■(2)到奇Hamilton超代数HO的低维上同调

本文在特征p> 3的域上,采用将典型李超代数■(2)嵌入到奇Hamilton超代数HO的零阶化分支的方法,使得在伴随表示的意义下,HO成为■(2)-模.通过对HO进行子模分解和权空间分解,得到了■(2)到HO的低维上同调.     

素特征域上广义Witt李超代数的自同构群

设W是素特征域上无限维或有限维广义Witt李超代数.本文利用W的自然滤过不变性和W的底代数的不变维数性质,证明了W的自同构群AutW同构于W的底代数的容许自同构群,还证明了在此群同构之下,AutW的标准正规列恰好对应W的底代数的容许自同构群的标准正规列,并给出AutW若干较为细致的性质.     

无限维模特殊奇Hamiltonian超代数的自然滤过

用研究幂零元的方法证明了无限维模特殊奇Hamiltonian超代数的自然滤过在自同构下是不变的,进而得到了此类李超代数的几个重要性质.     

模李超代数研究的若干进展

近几年，模李超代数（即素特征域上的李超代数）的研究取得了一些进展。本文分析以下6个方面介绍近几年在模李超代数研究中取得的一些新结果。1，限制李超代数，2，Cartan型模李超代数；3，深度1的Z－阶化李超代数；4，深度1的滤过李超代数。5；Cartan型模李超代数的某些内蕴性质；6，公开的问题。     

非单W和S型李超代数的滤过和分类

设F是特征p3的域,g是F上外代数与有限维广义Witt李代数或特殊李代数的张量积所构成的有限维非单李超代数.本文通过确定ad幂零元的方法证明了g的标准滤过是g的自同构群的不变量,进而证明了定义这类李超代数的参数组是内蕴的,从而给出了这两类李超代数在同构意义下的分类.     

限制李超代数的环面秩

本文将模李代数中环面与环面秩的理论推广到模李超代数中.应用模李超代数的限制包络得到了环面秩的若干重要性质.作为应用,计算了典型李超代数slm|n与限制Cartan型李超代数W(m,n,1),S(m,n,1)的绝对环面秩及S(m,n,1)在W(m,n,1)中的环面秩.     

δ-Jordan李超代数的交换扩张

通过δ-Jordan李超代数T的表示和上同调理论,构造δ-Jordan李超代数T■V.证明了δ-Jordan李超代数的等价交换扩张给出相同的表示.通过δ-Jordan李超代数的表示和其交换扩张得到2-上圈.     

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.     

Finite-dimensional simple modular Lie superalgebra ℳ

A new family of finite-dimensional simple modular Lie superalgebra ? is constructed based on results of Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601–3619]. The simplicity and generators of ? are discussed and the derivation superalgebra of ? is characterized. Furthermore, the invariance of the nonnatural filtration of ? is determined by the method of minimal dimension of image spaces.     

Finite-dimensional simple modular Lie superalgebra M

A new family of finite-dimensional simple modular Lie superalgebra M is constructed based on results of Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601–3619]. The simplicity and generators of M are discussed and the derivation superalgebra of M is characterized. Furthermore, the invariance of the nonnatural filtration of M is determined by the method of minimal dimension of image spaces.     

Modular Lie Superalgebras of H-type

In this article the Z-graded transitive modular Lie superalgebra -1 Li,whose repre-sentation of L_0 in L_(-1) is isomorphic to the natural representation of osp(L_(-1)),is determined.     

Classification of four-dimensional transitive local Lie groups of transformations of R 4 and their two-point invariants

In the theory of physical structures the classification of metric functions (both on a single set and on two ones) plays an important role. A metric function represents a two-point invariant of a certain local Lie transformation group. Moreover, one can uniquely restore this group with the help of the invariance condition. According to this theorem, in order to find all metric functions, it suffices to construct the complete classification of local Lie transformation groups. In this paper we classify Lie algebras of simply transitive local Lie groups of local transformations of a four-dimensional space, and then we define metric functions. The obtained results admit application in physics, in particular, in thermodynamics.     

On the Reduction of Some Dispersionless Integrable Systems

We study the invariance, reduction, exact solutions and conservation laws of the dispersionless Kadomtsev-Petviashivili and the heavenly equation. The existence of nontrivial conservation laws lead to repeated reductions paving the way for determining exact solutions. A variety of such solutions exist dependent on the subalgebra of Lie point symmetries that is chosen in the reduction procedure.     

Integrating Klein-Gordon-Fock equations in an external electromagnetic field on Lie groups

We investigate the structure of the Klein-Gordon-Fock equation symmetry algebra on pseudo-Riemannian manifolds with motions in the presence of an external electromagnetic field. We show that in the case of an invariant electromagnetic field tensor, this algebra is a one-dimensional central extension of the Lie algebra of the group of motions. Based on the coadjoint orbit method and harmonic analysis on Lie groups, we propose a method for integrating the Klein-Gordon-Fock equation in an external field on manifolds with simply transitive group actions. We consider a nontrivial example on the four-dimensional group E(2)×? in detail.     

Modular classes of Loday algebroids

We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive. To cite this article: M. Stiénon, P. Xu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS

This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primary decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.