Double Affine Hecke Algebras

双仿射赫克代数

• 批准号: 0456445
• 负责人: Ivan Cherednik
• 金额: $ 14.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456445&HistoricalAwards=false
• 关键词: Double; Affine; Hecke; Algebras

Double Affine Hecke AlgebrasIvan CherednikThe aim of the project is to study double affine Heckealgebras (DAHA) and their representations, especially thosewith applications in harmonic analysis. The focus is thestructure of the polynomial representation of DAHA, itsdecomposition and special quotients for generic q, at rootsof unity and for unimodular q. The main examples are finitedimensional quotients of the polynomial representationgeneralizing the Verlinde algebras from conformal fieldtheory; infinite dimensional and non-semisimple quotientsare a natural next step. Deep connections to the classicaltheory of affine Hecke algebras, the theory of Kac-Moodyalgebras, and modern mathematical physics are expected.Lie groups and Lie algebras formalize the concept of symmetryin the theory of special functions, physics, geometry, andcombinatorics. In a similar way, DAHA are candidates for aformalization of the notion of the Fourier transform. In thesimplest one-dimensional case, they are directly related tothe celebrated sl(2). There are indications that they servethe multidimensional theory of Fourier transform better thanLie and Kac-Moody algebras. DAHA are also a source of new unitaryinfinite dimensional theories, that, presumably, will findfundamental applications in mathematics and physics, thetheory of special functions and combinatorics. There are otherimportant directions of the DAHA theory, mainly of algebraicnature, but unitary representaions and Fourier analysis remainof high priority.

双仿射Hecke代数项目的目的是研究双仿射Hecke代数（DAHA）及其表示，特别是那些在谐波分析中的应用。重点讨论了DAHA的多项式表示的结构、一般q、单位根和非模q的分解和特殊商。主要的例子是多项式表示的有限维商，推广了共形场论中的Verlinde代数；无限维和非半单商自然是下一步。期望与经典仿射Hecke代数理论，Kac-Moodyalgebras理论和现代数学物理有深刻的联系。李群和李代数形式化了特殊函数理论、物理、几何和组合学中对称的概念。类似地，DAHA是傅里叶变换概念形式化的候选项。在最简单的一维情况下，它们与著名的sl(2)直接相关。有迹象表明，它们比lie代数和Kac-Moody代数更好地服务于傅里叶变换的多维理论。DAHA也是新的酉无限维理论的来源，这些理论可能会在数学和物理、特殊函数理论和组合学中找到基本的应用。DAHA理论还有其他重要的方向，主要是代数性质的，但酉表示和傅立叶分析仍然是高度优先的。