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.

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.