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.

该项目位于谐波分析，表示理论和组合学数学子场的界面。 预计该项目将在三维空间中的结分类中应用。 它也有望与理论物理有联系。 该项目的中心是由PI在1990年代中期引入的代数家族，其对称群体有两种动作。 这些代数被称为双仿射hecke代数，他们在数学的几个领域中发现了用途，首先要证明伊恩·麦克唐纳（Ian MacDonald）关于某些正交多项式的特性的猜想，这些概念是由对称的研究产生的。 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.