无限维李代数的权表示与非权表示

近年来随着在众多数学与物理分支对李代数这一有力工具的需求，李代数理论得到了很好的发展。当今李代数的重点研究集中在一些有很好背景的无限维李代数上。本项目将侧重于无限维李代数中具有较强物理背景的李代数(包括量子环面代数,广义Cartan型李代数,广义Virasoro代数，n-点Virasoro代数)的结构与表示的研究。一方面我们将继续研究这些代数的不可约权表示（已经有一些很好的研究结果），另一方面我们将开始研究这些代数的除Whittaker表示以外的不可约非权表示。另外，我们将用李代数方法继续探索Jacobi猜想,研究李代数理论在线性代数、可除代数中的应用。从而为利用代数方法解决物理及相关的数学问题提供新的方法与途径。

Nowadays as Lie theory has been widely used in many branches of mathematics and physics, it has been developed greatly. The focus of the study of Lie algebra theory is now on some infinite-dimensional Lie algebras with good background. The project of this application will study structure and representations oftheory of such Lie algebras including quantum torus Lie algebras, generalized Virasoro algebras, Cartan type Lie algebras, n-point Virasoro algebras. We will continue to study irreducible weight modules (we already have a lot of very good results on this aspect), at the same time we will start studying irreducible nonweight modules besides Whittaker modules. We also explore applications of Lie algebra theory to Jacobi conjecture, division algebra and linear algebra, try to find applications to other related mathematical and physical problems.