几类无限维李代数与超共形代数的研究

无限维李代数与超共形代数被广泛应用于物理及其他数学分支，特别是Heisenberg代数、Kac-Moody代数、Virasoro代数、广义Virasoro代数、Weyl代数等李代数以及N =2 超共形代数等超共形代数被大量数学家及物理学家所关注。同时无限维李代数与超共形代数的研究联系密切。本课题主要研究几类无限维李代数与超共形代数的结构与表示理论。我们计划研究几类无限维李代数与超共形代数的基本代数性质，包括自同构群、上同调群、李双代数结构等；研究这些代数的权模理论，包括这些代数的不可约Harish-Chandra模分类、不可分解模的结构及权模的构造等；研究这些代数的非权模理论，包括Virasoro等代数上非分次模的构造，Whittaker模、几乎分次模等的构造与分类问题；研究相关代数上的顶点算子代数的结构与表示，并通过顶点代数的表示理论刻画相关李（超）代数的表示理论。

The infinite dimensional Lie algebra and superconformal algebra theory has been widely used in many physics areas and other mathematical branches.Various infinite dimensional Lie algebras and superconformal algebras, for example, the Heisenberg algebra, Kac-Moody algebra, the Virasoro algebra and generalized Virasoro algerbas,the Weyl algebra, the N=2 superconformal algebras, have attracted extensive attention of many mathematicians and physicists. Meanwhile,there are many connections between the studies on infinite dimensional Lie algebras and superconformal algebras. The main research topics of this project is to develop the structure and representation theory of some infinite dimensional Lie algebras and superconformal algebras. On the structure theory, we will focus on their algebraic propertes: automorphism groups, cohomology groups and Lie bialgebra structures，etc. On the representation theory, we shall study the weight module theory and classify all irreducible Harish-Chandra modules, describe the structure of the highest weight modules, and constuct the realizations of operator type over these Lie (super)algebras. Furthermore we plan to study their non-weight module theory, we will concern the constuction and partial classification of non-weight modules over the Virasoro algebra, which incule the Whittaker modules and the almost-graded modules. Finally, we will study structure and representation theory over some vertex operator algebras related to these algebras. In terms of these constructions, we shall try to characterize the representation theory of the related Lie (super)algebras.