Geometric Methods in the Representation Theory of Affine Hecke Algebras and Quantum Groups

仿射Hecke代数和量子群表示论中的几何方法

基本信息

批准号：
9732805
负责人：
George Lusztig
金额：
$ 56.85万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1998
资助国家：
美国
起止时间：
1998-07-01 至 2004-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732805&HistoricalAwards=false
关键词：
Geometric Methods Representation Theory Affine

项目摘要

9732805 Lusztig This award supports the continued study of periodic W-graphs attached to affine Hecke algebras and the investigation of their possible connections with unrestricted representations of semisimple Lie algebras in positive characteristic. The principal investigator will also study canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. He will continue the study of unipotent representations of semisimple p-adic groups, as well as character sheaves on (possibly disconnected) reductive groups. Representation theory of Lie algebras and Lie groups, initially developed at the turn of the century, studies the structure of symmetries and their realizations. This theory encompasses many areas in mathematics and has fundamental applications to theoretical physics. In the past twenty years, the study of representation theory of a special class of affine Lie algebras and Lie groups led to many new unexpected discoveries in mathematics and theoretical physics, as well as to a synthesis of many established areas in both disciplines. In particular, it yielded a pure mathematical description of a simplest quantum field theory -- a prototype theory of fundamental interactions. It is expected that representation theory of new classes of infinite-dimensional Lie algebras and groups might substantially deepen our understanding of mathematics and theoretical physics during the next decade.

9732805 Lusztig该奖项支持对仿射Hecke代数附加的周期性W指法的持续研究，并研究了其可能的连接，并与半神经代数的不受限制表示，具有积极特征。首席研究者还将从变形带轮的角度研究量化包围代数的规范基础。他将继续研究半神经p-adic群体的一能表现，以及（可能是断开连接）还原群的特征束。谎言代数和谎言群体的代表理论最初在世纪之交，研究对称性及其实现的结构。该理论涵盖了数学的许多领域，并在理论物理学上具有基本应用。在过去的二十年中，对特殊类别的代数和谎言组的表示理论的研究导致了许多新的数学和理论物理学发现，以及在这两个学科中许多既定领域的综合。特别是，它对最简单的量子场理论产生了纯粹的数学描述 - 基本相互作用的原型理论。可以预期，在未来十年中，新的无限二维谎言代数和群体的代表理论可能会大大加深我们对数学和理论物理学的理解。