关于半单李代数的抛物子代数中的一类双极化

本文给出了一种构造复半单李代数抛物子代数中双极化的方法,并给出了其实形式.一般情况下,构造的双极化是非对称的.这种构造方法给出了一大类非可解李代数中极化的例子.后者在表示理论和物理,特别是力学中有重要应用.     

偶交叉双积Hopf代数B_T_RH的构造

本文给出了DoiY．构造的偶交叉积BT■H的代数结构与ReshetikhimN．构造的双代数B■RH的余代数结构在张量空间B■H上构成双代数（记为Bτ■RH）的充要条件,利用此结论具体构造了一个有趣的例子B4■KZ2;证明了当B,H均为Hopf．代数时Bτ■RH也为Hopf代数,最后给出这类双代数的映射刻划。     

偶交叉双积Hopf代数B_T_RH的构造

本文给出了DoiY．构造的偶交叉积BT■H的代数结构与ReshetikhimN．构造的双代数B■RH的余代数结构在张量空间B■H上构成双代数（记为Bτ■RH）的充要条件,利用此结论具体构造了一个有趣的例子B4■KZ2;证明了当B,H均为Hopf．代数时Bτ■RH也为Hopf代数,最后给出这类双代数的映射刻划。     

Hom-扭曲积和Hom-扭曲余积

首先给出了对偶Hom-双代数和余对偶Hom-双代数的概念,其次构造了Hom-扭曲积和Hom-扭曲余积并讨论了他们的一些性质．     

偶交叉双积Hopf代数B RH的构造

本文给出了Doi Y.构造的偶交叉积B H的代数结构与Reshetikhim～N.构造的双代数BRH的余代数结构在张量空间B H上构成双代数(记为BRH)的充要条件,利用此结论具体构造了一个有趣的例子H4 KZ2,证明了当B,H均为Hopf.代数时BRH也为Hopf代数,最后给出这类双代数的映射刻划.     

扭Smash双积

设H是双代数 ,A是H 双模代数 ,且为左H 余模余代数 .本文构造一种新的代数—扭Smash双积A×H ,推广了扭Smash积和Smash双积 .我们给出了扭Smash双积A×H作成双代数的充要条件 ,证明了当A ,H都是Hopf代数时 ,A×H 也是Hopf代数 ,利用映射系统刻画了双代数A×H的结构 ,并考虑了它的对偶情况 .     

乘子Hopf代数的 L-R Smash余积

设 $A$ 是正则乘子Hopf代数, $R$ 是 $A$-\!\!双余模代数, 首先定义了 $A$-\!\!双余模双代数, 并利用它构造了 L-R Smash 积的对偶形式, 即 $R\otimes A$ 上一种非平凡的乘子Hopf代数结构, 称之为 L-R Smash余积. 然后给出了 L-R Smash余积上的积分和$*$-\!\!结构.     

Lie余模和Lie双代数的构造

首先给出Lie余模的直和分解, 然后根据Lie余模理论由Lie余代数构造某些(三角)Lie双代数.     

树型偏序集的双扭指标代数及其Ringel对偶

本文给出了有限树型偏序集的双扭指标代数的典范模及特征模的构造，描述了其Ｒｉｎｇｅｌ对偶的通常箭图的形状．     

广义弱群像代数

本文首先介绍了广义弱群像代数的概念, 并构造了一系列具体的例子, 通过例子可以看出有限群胚代数和群像代数都可以看作是广义弱群像代数; 接着本文证明了广义弱群像代数也可以看作是一类广义弱双Frobenius代数, 并探讨了广义弱双Frobenius代数成为有限群胚代数的条件; 最后本文给出了低维广义弱群像代数的具体结构.     

Lie代数双极化与齐性抗仿凯勒流形的新进展

本文介绍Lie代数双极化与齐性仿凯勒流形的若干新进展,并提出了若干相关问题,指出该领域可能发展的若干方向。     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

The solution to the embedding problem of a (differential) Lie algebra into its Wronskian envelope

Any commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

Classification of complex simply connected homogeneous spaces of dimension not greater than 2

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.     

Twisted Hamiltonian Lie algebras and their multiplicity-free representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.     

Tanaka prolongation of free Lie algebras

With the exception of the three step real free Lie algebra on two generators, all real free Lie algebras of step at least three are shown to have trivial Tanaka prolongation. This result, together with the known results concerning the step two real free Lie algebras and the step three real free Lie algebra on two generators, gives a complete list of Tanaka prolongations for real free Lie algebras.