Double Affine Hecke Algebras

双仿射赫克代数

• 批准号: 1363138
• 负责人: Ivan Cherednik
• 金额: $ 52.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363138&HistoricalAwards=false
• 关键词: Double; Affine; Hecke; Algebras

This project sits at the interface of the mathematical subfields of harmonic analysis, representation theory, and combinatorics. The project is expected to have applications to the classification of knots in three-dimensional space. It is also expected to have connections to theoretical physics. At the center of the project is a family of algebras, introduced by the PI in the middle 1990's, that have two actions by a symmetry group. These algebras are known as double affine Hecke algebras and they have found uses in several areas of mathematics, starting with their use in the proof of a conjecture of Ian Macdonald about the properties of certain orthogonal polynomials that arise from the study of symmetry.The major themes of the proposed work are expected to be the following: (1) the theory of invariants of torus knots using the structure of double affine Hecke algebras (abbreviated DAHA), (2) a new theory of Rogers-Ramanujan identities based on nil-DAHA, (3) a surprising formula for the minimal number of creation operators in terms of non-symmetric Macdonald polynomials, and (4) the action of the absolute Galois group on the ramified covers of elliptic curves associated with perfect DAHA modules.

该项目位于调和分析，表示论和组合数学的数学子领域的接口。 该项目预计将在三维空间中的节点分类的应用。 它也有望与理论物理学有联系。 该项目的中心是PI在20世纪90年代中期引入的一族代数，它们具有对称群的两个作用。 这些代数被称为双仿射Hecke代数，他们已经发现在数学的几个领域的用途，开始与他们的使用证明的猜想的伊恩麦克唐纳关于性质的某些正交多项式的研究所产生的对称性。主要主题的拟议工作预计将如下：（1）利用双仿射Hecke代数的结构研究环面纽结的不变量（2）基于nil-DAHA的Rogers-Ramanujan恒等式的新理论，（3）关于非对称Macdonald多项式的最小生成算子数的一个令人惊讶的公式;（4）绝对Galois群在与完全DAHA模相关的椭圆曲线的分歧覆盖上的作用。