Infinite Dimensional Lie Algebras, Quantum Groups, & their Applications

9401163 列舍季欣 本项目主要研究量子化泛包络代数及其表示，以及它们在低维拓扑和1+1维量子场论模型中的应用。 主要研究量子化泛包络代数上的模范畴的结构与相应的仿射李代数之间的关系。 他将进一步研究3-流形的不变量，以更好地理解拓扑场论的组合方法与微分几何方法之间的关系。 他将继续研究量子化的通用包络代数的无限维李代数及其表示理论，特别是当参数的变形是根1。 量子群是数学家和物理学家研究的一个新领域。 在数学方面，它结合了三个最古老的领域的“纯”数学，代数，分析和几何，但它是非常感兴趣的物理学家工作的共形量子场论。

9401163 Reshetikhin This project is concerned with the study of quantized universal enveloping algebras and their representations together with their applications to low-dimensional topology and to 1+1-dimension models of quantum field theory. The principal investigator will investigate the relations between the structure of the category of modules over quantized universal enveloping algebras and corresponding affine Lie algebras. He will further study invariants of 3-manifolds to better understand the relation between the combinatorial approach to topological field theory and the differential-geometric one. He will continue studying the quantized universal enveloping algebras of infinite-dimensional Lie algebras and their representation theory and, in particular, when the parameter of deformations is a root of 1. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.