Quantization, Quantum Groups, the Yang-Baxter Equation, Integrable Systems, and Special Functions

量化、量子群、杨-巴克斯特方程、可积系统和特殊函数

• 批准号: 9988796
• 负责人: Pavel Etingof
• 金额: $ 37.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988796&HistoricalAwards=false
• 关键词: Quantization; Quantum; Groups; Yang; Baxter

The present proposal is concerned with various aspects of the theory of quantum groups. The goals of the proposal could be briefly formulated as follows. 1. To complete the theory of universal quantization of Lie bialgebras, begun in the PI's joint work with D.Kazhdan, and answer most of the remaining open questions of Drinfeld about quantization. In particular, to investigate the dependence of the quantization of an associator, and to show that the universal quantization of Kac-Moody algebras coincides with their usual quantization. To study quantization of Poisson homogeneous spaces. To study quantization theory in positive characteristic. To study quantization of solutions of the classical dynamical Yang-Baxter equations, and associated Poisson groupoids. 2. To continue to develop the theory of dynamical quantum groups, originated by Felder in 1994. In particular, to generalize the exchange construction (introduced in the PI's joint work with Varchenko) to the case of affine Lie algebras. To use this construction to establish the equivalence of suitable categories of representations for Yangians, quantum affine algebras, and elliptic algebras. To study the structure of dynamical quantum groups obtained by quantizing generalized Belavin-Drinfeld triples for simple Lie algebras. 3. To continue to develop the theory of generalized Macdonald functions, begun in the PI's joint work with Varchenko, Kirillov, and Styrkas. In particular, to use representation theory to generalize the recent results of Felder and Varchenko on elliptic hypergeometric functions from sl(2) to any simple Lie algebra. To develop Macdonald's theory for affine root systems (of type A) using the representation theory of quantum affine algebras. To develop a theory of twisted trace functions for quantum groups, and deduce difference equations for them which involve R-matrices obtained by quantizing Belavin-Drinfeld triples. 4. To work on the theory of finite dimensional Hopf algebras. To look for new examples of semisimple Hopf algebras. To study tensor products of representations of cotriangular Hopf algebras. 5. To continue to study set-theoretical solutions of the quantum Yang-Baxter equation. The theory of quantum groups is a relatively new area of mathematics which arose in mid 1980-s at the junction of the theory of groups and quantum theory. The theory of groups, which was created in 19-th century, is a mathematical theory which is designed to describe precisely the phenomenon of symmetry, and which is fundamental in modern particle physics. Symmetry is a basic property of nature. It also played a crucial role in all technological achievements of mankind, starting from the invention of a wheel: the ability of a wheel to roll comes from its symmetry with respect to rotations. Quantum theory was the most important achievement of theoretical physics of the 20-th century, which lies at the foundation of this century's technological revolution. It is a physical theory which allows to describe the behavior of very small objects, like atoms and electrons. In the late eighties, it was realized that in certain systems described by quantum theory, there is a new kind of symmetry which the theory of groups fails to describe. This new, very peculiar kind of symmetry, is called quantum symmetry. In order to describe this symmetry mathematically, mathematicians introduced new mathematical objects called quantum groups, which are generalizations of ordinary groups. The present proposal seeks to further develop this theory and to explore its interactions with other areas of mathematics and physics.

本提案涉及量子群理论的各个方面。建议的目标可以简单地表述如下。1. 完成李双代数的普遍量子化理论，开始于PI与D.Kazhdan的联合工作，并回答德林菲尔德关于量子化的大部分剩余开放问题。特别地，研究了关联子量子化的依赖性，并证明了Kac-Moody代数的普遍量子化与它们的通常量子化是一致的。研究泊松齐次空间的量子化问题。研究量化理论的积极特征。研究经典动力学Yang-Baxter方程及其相关泊松群解的量子化问题。2. 继续发展费尔德于1994年提出的动态量子群理论。特别地，将交换构造（在PI与Varchenko的联合工作中引入）推广到仿射李代数的情况。利用这一构造建立了杨算子、量子仿射代数和椭圆代数的合适表示范畴的等价性。研究由简单李代数的广义Belavin-Drinfeld三元组量子化得到的动态量子群的结构。3. 继续发展广义麦克唐纳函数理论，开始于PI与Varchenko， Kirillov和Styrkas的联合工作。特别地，利用表征理论将Felder和Varchenko关于椭圆型超几何函数的最新结果从sl(2)推广到任何简单李代数。利用量子仿射代数的表示理论发展麦克唐纳的仿射根系（A型）理论。建立了量子群的扭曲迹函数理论，并推导了量子群的差分方程，其中包含了通过量子化Belavin-Drinfeld三元组得到的r -矩阵。4. 研究有限维霍普夫代数的理论。寻找半简单Hopf代数的新例子。研究共三角Hopf代数表示的张量积。5. 继续研究量子Yang-Baxter方程的集理论解。量子群理论是20世纪80年代中期在群理论和量子论的交汇处兴起的一个相对较新的数学领域。群理论创建于19世纪，是一种旨在精确描述对称现象的数学理论，是现代粒子物理学的基础。对称是自然的基本属性。它还在人类所有的技术成就中发挥了至关重要的作用，从轮子的发明开始：轮子滚动的能力来自于它相对于旋转的对称性。量子理论是20世纪理论物理学最重要的成就，是本世纪技术革命的基础。它是一种物理理论，允许描述非常小的物体，如原子和电子的行为。在八十年代后期，人们认识到在量子理论所描述的某些系统中，存在一种群理论无法描述的新的对称性。这种新的、非常奇特的对称被称为量子对称。为了在数学上描述这种对称性，数学家引入了新的数学对象，称为量子群，它是普通群的推广。本提案旨在进一步发展这一理论，并探索其与其他数学和物理领域的相互作用。