具有三角分解李代数的广义限制单模

通过广义限制李代数的定义,得到了所有具有三角分解的李代数的广义限制单模,并且作为一个例子,计算了李代数V3G的所有广义限制单模以及它们的维数.     

阶化Cartan型李代数的上同调

本文利用广义限制李代数的概念和应用Frobenius代数的一些性质来研究广义限制李代数的广义限制完备上同调,并利用广义限制上同调与通常上同调的关系尝试着给出一种计算系数为不可约模的阶化Cartan型李代数上同调的方法.     

阶化Cartan型李代数的上同调

本文利用广义限制李代数的概念和应用Frobenius代数的一些性质来研究广义限制李代数的广义限制完备上同调,并利用广义限制上同调与通常上同调的关系尝试着给出一种计算系数为不可约模的阶化Cartan型李代数上同调的方法.     

限制pre-李代数

研究限制pre-李代数的结构,给出了限制pre-李代数和可限制pre-李代数的定义,得到了限制pre-李代数的p-映射性质和可限制pre-李代数的性质,还讨论了带有半单元的限制pre-李代数及拟环面限制pre-李代数和分解唯一性定理.     

广义Baby-TKK李代数的一类顶点表示

利用广义 Virasoro- Toroidal李代数的顶点表示理论研究了广义 Baby- TKK李代数的一类顶点表示 .     

一些无限维完备李代数

本文讨论了无限维完备李代数的一些性质,由Virasoro代数,Kac-Moody代数构造了几类无限维完备李代数.同时给出了Kac-Moody代数及其广义抛物子代数的导子代数的刻划.证明了完备李代数的Loop扩张仍为完备李代数.     

一类W-代数的导子和自同构

讨论了一类W-代数,这类李代数包含无中心的广义Virasoro子代数.本文确定了这类李代数的导子和自同构.     

广义仿射李代数上的非交换Poisson代数结构

非交换的Poisson代数同时具有(未必交换的)结合代数和李代数两种代数结构,且结合代数和李代数之间满足所谓的Leibniz法则.本文确定了一般广义仿射李代数上所有的Poisson代数结构.     

与广义Cartan K型李代数有关的一类无限维李代数

本文主要引入了一类与广义Cartan K型李代数有关的无限维李代数,并讨论了 它的单性及任两个这类李代数的同构问题.     

具有三角分解可解李代数的表示

本文研究具有三角分解可解李代数和它的表示,探讨了具有三角分解可解李代数是广义限制李代数的条件,对于某些s∈Map(B,F),在uφ2(L,S)-模的范畴里,确定了不可约模和主不可分解模,并对upuφ2,L,S)的块进行了描述.     

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.     

A duality theorem for hope module algebras over dedekind rings

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖_M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&;(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].     

素特征的“李定理”

设L为代数闭域F上有限维李代数,著名的李定理说:若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=P>0,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的; (3)若char F=P>7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.     

Contact代数K(m,n)的不可约模的结构

利用广义限制李代数的概念和性质,研究Contact代数K(m,n)的不可约表示,给出了特征标高度大于等于2且小于p-3的不可约K(m,n)-模的结构.     

Contact代数K(m,n)的不可约模的结构

利用广义限制李代数的概念和性质,研究Contact代数K(m,n)的不可约表示,给出了特征标高度大于等于2且小于P-3的不可约K(m,n)-模的结构.     

与一类unit型相联系的有限维单李代数的结构

Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

Application of a semi-direct sum of Lie algebras to the TD hierarchy

We construct a Lie algebra G by using a semi-direct sum of Lie algebra G₁ with Lie algebra G₂. A direct application to the TD hierarchy leads to a novel hierarchy of integrable couplings of the TD hierarchy. Furthermore, the generalized variational identity is applied to Lie algebra G to obtain quasi-Hamiltonian structures of the associated integrable couplings.     

Restricted Lie algebras all whose elements are semisimple

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra.     

Formality in generalized Kähler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kähler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kähler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.