Toroidal李代数的量子化、Hall代数实现以及仿射Nash群的结构

This project is on the background of researching the relations between the representation theory of algebras and Lie theory. The objects we focus on are toroidal Lie algebras, which are very important generalizations of Kac-Moody algebras. It had been proved that 2-toroidal Lie algebras can be presented by generators of generalized Heisenberg algebras and Chevalley generators of simple Lie algebras, subject to certain amalgamation relations. On the basis of this new presentations of toroidal Lie algebras, we plan to discuss the quantized toroidal algebras defined by quantized these amalgamation relations. Furthermore, we will try to construct a realization of quantized toroidal algebras via Hall algebras of suitable algebras in representation theory. This work will contribute to find the profound relations of the representation theory of algebras and quamtum groups by the Hall approach..Another goal of this project is to develop the structure theory of affine Nash groups. Parallel to the structure theory of algebraic groups, we plan to generalize these results to the affine Nash groups. Because some class of Nash groups, such as almost linear Nash groups, whose structures are simpler than that of general Lie groups but their infinite dimensional representations are more flexible than linear algebraic groups. So this work is significant to the research of infinite dimensional representations of Lie groups.

本项目是以研究代数表示论与李理论的交叉联系为背景，以Kac-Moody李代数的重要推广形式toroidal李代数为研究对象，从把toroidal李代数通过粘合关系表示成复单李代数与广义Heisenberg李代数的粘合代数这一新的表现形式出发，利用复单李代数、Heisenberg李代数的量子化结果，尝试通过把粘合关系进行量子化的新方法，对toroidal李代数的量子化问题进行研究，并研究toroidal李代数及其量子化由代数表示论中合适代数的Hall代数来实现的问题。这对通过Hall代数理论揭示代数表示论与李代数和量子群之间的深刻联系，都有着重要的学术意义。.同时，本项目还着力于发展仿射Nash群的结构理论。以在李群上赋予半代数结构的Nash群为研究对象，从代数群的结构理论出发，把代数群的结构理论和重要定理推广到仿射Nash群的情形。这对于研究李群的无限维表示是非常有益的。

本项目发展了仿射Nash群的结构理论。以在李群上赋予半代数结构的Nash群为研究对象，从代数群的结构理论出发，把代数群的结构理论和重要定理推广到仿射Nash群的情形。这对于研究李群的无限维表示是非常有益的。与此同时，我们通过建立局部Godement-Jacquet L-函数的极点与有奇异支撑的矩阵空间的分布之间的关系，证明了任意局部域上的一般线性群GLn(F)的泛不可约表示的全θ提升是不可约的。证明过程中我们使用傅里叶变换，用表示论刻画Godement-Jacquet L-函数在1/2处极点的存在性问题。因为一般线性群GLn(F)的泛不可约表示也出现在尖自守型表示的局部分支中，所以该结果在表示理论和自守形式中有非常重要的应用。