辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

二次李三系的分解定理

有限维李三系称为二次的,如果它容许一个非退化的不变对称双线性型.无论是李三系分解为不可分解理想的直和,还是二次李三系分解为不可分解非退化理想的正交直和,证明了这两类分解在同构意义下都是唯一的.     

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, $\{I_{[\alpha]},T,I_{[\beta]}\}=\{I_{[\alpha]},I_{[\beta]},T\}=\{T,I_{[\alpha]},I_{[\beta]}\}=0$.     

三次可解型非退化李代数及其导子李代数的结构

本文给出了非退化可解李代数的两个类型:三次可解型非退化李代数和扩充的 Heisenberg李代数,并确定三次可解型非退化李代数及其导子李代数的结构．     

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.     

李color三系的Frattini子系的性质

给出了李color三系的Frattini子系的定义,得到了李color三系的Frattini子系的一些性质·特别的,证明了李color三系T有分解T=T_1⊕T_2⊕…⊕T_m,则φ(T)有分解φ(T)=φ(T_1)⊕φ(T_2)⊕…⊕φ(T_m).     

一类带有非退化对称不变双线性型的李代数

本文给出了一类带有非退化对称不变双线性型的李代数的特征性质、结构及 实现．     

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).     

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 378–397, September, 2008.     

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.     

SUPERSYMMETRIC INVARIANCE AND UNIVERSAL CENTRAL EXTENSIONS OF LIE SUPERTRIPLE SYSTEMS

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland＇s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

Supersymmetric invariance and universal central extensions of lie supertriple systems

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland's theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

一类带有非退化不变对称双线性函数的幂零李代数

本文证明了一类带有非退化不变对称双线性函数、幂零指数为Ｎ的李代数 满足：定理１．（ｉ）ｃ（ｇ）＝ｇ~Ｎ; （ｉｉ）ｄｉｍｃ（ｇ）＝ｌ．ｌ是ｇ的生成元个数．定理３给出了这类李代数结构的充分且必要条件．     

李三系的Frattini子系

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.针对李三系的Frattini子系和基本李三系的问题进行了研究,给出了Frattini子系和基本李三系的一些性质,并证明了李三系的非嵌入定理,同时得到了幂零李三系是基本李三系的一个充要条件.     

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。     

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.     

Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.     

$T^∗$-Extension of Lie Supertriple Systems

In this article, we study the Lie supertriple system (LSTS) $T$ over a field$\mathbb{K}$ admitting a nondegenerate invariant supersymmetric bilinear form (call such a $T$ metrisable). We give the definition of $T^∗_ω$-extension of an LSTS $T$, prove a necessary and sufficient condition for a metrised LSTS ($T$, $ϕ$) to be isometric to a $T^∗$-extension of some LSTS, and determine when two $T^∗$-extensions of an LSTS are "same", i.e., they are equivalent or isometrically equivalent.     

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.     

具拟周期摄动的近可积Hamiltonian系统的KAM型定理

In this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition, then the majority of invariant tori is persistent under the perturbation.