Representations of Quantum Groups, Special Functions, and Geometry

量子群、特殊函数和几何的表示

• 批准号: 9610201
• 负责人: George Lusztig
• 金额: $ 6.35万
• 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: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9610201&HistoricalAwards=false
• 关键词: Representations; Quantum; Groups; Special; Functions

KIRILLOV This research is concerned with the representation theory of quantum groups and its relation with combinatorics and mathematical physics. The principal investigator will work on two problems. The first problem concerns a geometric realization of Lusztig's canonical basis, using the identification of modules over a quantum group with the homology of a certain local system on the configuration space, which is due to Schechtman and Varchenko. Such a realization would reveal a new deep relation between combinatorics and the geometric structures of mathematical physics. The second continues the study of special functions appearing in representation theory and related combinatorial identities. In particular, the principal investigator will work on the generalization of the inner product identities for Macdonald's polynomials to affine root systems, and on on further properties of this inner product, such as positivity and its relation with the inner product on the space of conformal blocks in conformal field theory. This study is in the general area of representation theory of Lie algebras and related objects, and is important both for mathematics and physics.

本研究涉及量子群的表示理论及其与组合学和数学物理的关系。首席研究员将研究两个问题。第一个问题涉及到Lusztig正则基的几何实现，利用具有组态空间上某个局部系统同调的量子群上的模的识别，这是由于Schechtman和Varchenko。这种认识将揭示组合学与数学物理的几何结构之间的一种新的深刻关系。第二部分继续研究在表示理论和相关组合恒等式中出现的特殊函数。特别地，主要研究者将研究Macdonald多项式内积恒等式到仿射根的推广，以及该内积的进一步性质，如正性及其与共形场论中共形块空间内积的关系。本研究属于李代数及其相关对象的表示理论的一般领域，具有重要的数学和物理意义。