李代数表示及其应用

李代数及其表示被广泛应用于物理及其他数学分支。 近年来有限维非半单李代数、无限维李代数的表示理论、内在联系及其应用被大量数学家及物理学家所关注，不断涌现出新的重要成果。本课题将主要研究Schrodinger代数、conformal Galilei代数、Heisenberg代数、Kac-Moody代数、Virasoro相关代数、量子环面李代数、广义Cartan W, S型李代数等李代数的不可约Harish-Chandra模分类、权空间无限维的权模构造、非权模的构造及刻画等问题；系统总结一般方法并探索其在其他数学及物理分支上的应用。

Lie algebra and its representation theory have been widely used in in many other mathematical branches and physics areas. New important results on finite dimensional non-semisimple Lie algebras and infinite dimensional Lie algebras have well developed in recent few years. The represenatation theory, inherent connection and application of those algebras have attracted extensive attention of many mathematicians and physicists. The main research topics of this project is to classify irreducible Harish-Chandra modules, constuct and characterize the weight modules and non-weight modules over various Lie algebras, including Schrodinger algebras, conformal Galilei algebras, Heisenberg algebras, Kac-Moody algebras, Virasoro related algebras, quantum Lie algebras, generalized Cartan W,S Lie algebras. And we will establish the general methods and try to use them in other mathematical branches and physics areas.