几类无限维(量子)李(共形、双)代数的结构和表示

In this project, we shall study the structures and represen-.tations of some infinite-dimensional (quantized) Lie (conformal, bi) algebras .related to Schr?dinger-Virasoro type Lie algebras and W-algebras..The classification of Harish-Chandra modules is an important problem in the representation theory of Lie algebras. We shall study the irreducible Harish-Chandra module and the indecomposable modules of intermediate series over the twisted deformative Schr?dinger-Virasoro algebras, and investigate unitary Harish-Chandra modules over some particular Schr?dinger-Virasoro type Lie algebras..The notion of conformal algebras encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. The singular part of the operator product expansion encodes the commutation relations of fields, which leads to the notion of Lie conformal algebras. Simple finite Lie conformal algebras have been classified and all finite simple irreducible representations of simple finite Lie conformal algebras have been constructed. However, the structure theory, representations and cohomology theory of simple infinite Lie conformal algebras is far from being well developed. Therefore, it is very natural to first study some important examples. We shall construct some Lie conformal algebras related to Schr?dinger-Virasoro type Lie algebras and W-algebra W(a,b), and discuss the Low-dimensional conformal modules, cohomology groups and automorphism groups of these Lie conformal algebras and their related problems..The term"quantum group"first appeared in the theory of quantum integrable systems, which was then formalized by Drinfel'd and Jimbo as a particular class of Hopf algebra. In Drinfeld's approach, quantum groups arise as Hopf algebras and become universal enveloping algebras of a certain Lie algebra. Constructing quantization of Lie bialgebras is an important approach to producing new quantum groups. Actrually, investigating Lie bialgebra structures and quantization of a special Lie algebra is a complicated problem. We shall study the Lie bialgebra structures and quantizations of the generalized Schr?dinger-Virasoro algebra and construct some new quantum groups related to the Schr?dinger-Virasoro type Lie algebras and W-algebra W(a,b), and investigate their structures and representations..Block introduced a class of infinite-dimensional simple Lie algebras(usually referred to as Lie algebras of Block type). Partially due to their close relations to the Virasoro algebra and Cartan type Lie algebras, these algebras have attracted some attention in the literature. The structure theory of these algebras has been developed, however, their representation theory is far from being well developed, except for quasifinite representations of some particular Block type Lie algebras. On the basis of our previous research works, we shall study continually the simple modules over some particular Block type algebras.

本项目研究与Schr?dinger-Virasoro型李代数和W代数有关的几类无限维(量子)李(共形, 双)代数的结构和表示. 主要研究扭形变Schr?dinger-Virasoro李代数的不可约Harish-Chandra模以及不可分解的中间序列模,确定某些特殊Schr?dinger- Virasoro型李代数的酉模; 研究与Schr?dinger-Virasoro型以及W(a,b)李代数相对应的李共形代数, 对这些代数的低秩共形模进行讨论.同时研究这些李共形代数的同调群,自同构群等结构方面的性质; 研究广义Schr?dinger-Virasoro李代数的双代数结构并对其进行量子化. 同时构造与Schr?dinger-Virasoro型和W(a,b)李代数有关的新的量子群, 并对这些特殊的Hopf代数所涉及的相关性质进行研究. 在已有工作的基础上,对某些Block型单李代数的模进行研究.

无限维李代数的结构和表示是李理论和理论物理研究的重要内容。李共形代数在某种程度上是李代数的一般化，其公理化定义由Kac给出。它为研究满足某种局部性质的无限维李代数提供了一种有力的工具。量子群是由Drinfeld和Jimbo在研究Yang-Baxter方程时各自独立发现的。它是一类特殊的Hopf代数，是李代数的普遍包络代数的形变。当前，对量子群的研究是李理论研究的热点之一。Block型李代数是R. Block在1958年首先引入的，这类李代数与Virasoro代数以及Cartan型李代数密切相关。.本项目主要得到了以下结果。.1. 研究了一类Block型李代数, 给出了它的导子代数和自同构群。特别得到了这个李代数的外导子空间是1维的，内自同构群是平凡的；.2. 研究了一类Block型李代数的拟有限表示，得到了这个李代数上的不可约拟有限模是一个最高权模、一个最低权模、或者是一个中间序列模；.3. 研究了一类李代数的李双代数结构，这类李代数是在研究某种扩张仿射李代数时引入的。通过讨论得到了这个李代数上的李双代数都是三角的；.4. 得到了形变Schrodinger-Virasoro李代数的不可分解的中间序列模；.5. 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。通过计算，得到了这类李代数的导子代数和自同构群；.6. 构造了一类Schrodinger-Virasoro型李共形代数。这类李共形代数是非单的，可以看成Virasoro共形代数的某种扩张。通过计算得到了这个李共形代数的共形导子、二上同调群、以及秩为1的共形模。