Noncommutative Algebraic Geometry in Representation Theory

表示论中的非交换代数几何

• 批准号: 0602007
• 负责人: Kobi Kremnizer
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602007&HistoricalAwards=false
• 关键词: Noncommutative; Algebraic; Geometry; Representation; Theory

This project, involving the work of Kobi Kremnizer, is devoted to the application of methods from noncommutative algebraic geometry, to the study of representation theory. The first aim is the study of the representation theory of the real quantum group. The quantization of the theory of real (algebraic) reductive groups is a new and important subject. The project proposes using quantum D-modules to study quantum Harish-Chandra modules thus making the classification an accessible task. The classical theory of Harish-Chandra modules might benefit from this study of its deformations. The second aim is to give a geometric proof of Lusztig`s conjectures concerning characters of nonrestricted representations of reductive Lie algebras in positive characteristic. The proof will use quantum differential operators on the quantum flag variety at a root of unity. Lusztig`s conjectures are only proven for the restricted case by a very computational method. The project proposes to give a proof for the general case of these conjectures using the geometry of the Springer resolution. The project also suggests showing the equivalence of the category of representations of the nonrestricted quantum group at a root of unity and the category of representations of the affine Lie algebra at the critical level by connecting both categories to the Springer resolution. The third aim is to give a geometric construction of the double affine Hecke algebra and to prove a quantum version of the twisted Harish-Chandra homomorphism. This should help study the representation theory of this algebra at a root of unity and connect it to the representation theory of the trigonometric Cherednik algebra in positive characteristic. This project lies in the meeting point of representation theory, noncommutative algebraic geometry, geometry in positive characteristic, infinite dimensional geometry and conformal field theory. It will hopefully help clarify the close relations between loop space geometry and noncommutative geometry that is observed in string theory and will open new pathways for research. The methods it offers to use should be useful in other cases as well, making noncommutative geometry an important tool in representation theory and in algebraic geometry.

该项目涉及Kobi Kremnizer的工作，致力于将非交换代数几何方法应用于表征理论的研究。第一个目的是研究实量子群的表示理论。实（代数）约化群理论的量化是一个新的重要课题。该项目建议使用量子d模块来研究量子哈利希-钱德拉模块，从而使分类成为一项可访问的任务。哈里什-钱德拉模组的经典理论可能会受益于对其变形的研究。第二个目的是给出鲁兹提格关于约化李代数的非限制表示的正特征的猜想的几何证明。证明将使用量子微分算子在单位根处的量子标志变异上。Lusztig的猜想只能通过一种非常计算的方法来证明。本课题提出利用施普林格分辨率的几何来证明这些猜想的一般情况。该项目还建议通过将非受限量子群的表示范畴与施普林格分辨率联系起来，在单位根上显示出非受限量子群的表示范畴与仿射李代数的表示范畴在临界水平上的等价性。第三个目的是给出双仿射Hecke代数的几何构造，并证明扭曲Harish-Chandra同态的量子版本。这有助于研究该代数在单位根上的表示理论，并将其与三角Cherednik代数的正特征表示理论联系起来。本课题是表征论、非交换代数几何、正特征几何、无限维几何和共形场论的交汇点。这将有助于澄清环空间几何和弦理论中观察到的非交换几何之间的密切关系，并为研究开辟新的途径。它提供的方法在其他情况下也应该有用，使非交换几何成为表示理论和代数几何中的重要工具。