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

本提案涉及量子群理论的各个方面。该提案的目标可以简短地表述如下。1.完成PI与D·Kazhdan合作开始的Lie双代数的普适量子化理论，并回答Drinfeld关于量子化的大部分尚未解决的问题。特别地，研究了结合元的量子化的依赖性，并证明了Kac-Moody代数的普适量子化与它们通常的量子化是一致的。研究Poisson齐次空间的量子化。研究正特征中的量子化理论。研究经典动力学Yang-Baxter方程及其Poisson群胚的解的量子化。2.继续发展由费尔德于1994年提出的动力学量子群理论。特别地，将交换结构(在PI与Varchenko的联合工作中引入)推广到仿射李代数的情况。利用这种构造建立延安、量子仿射代数和椭圆代数的适当表示范畴的等价性。研究通过对单李代数的广义Belavin-Drinfeld三元组进行量子化得到的动力学量子群的结构。3.继续发展广义麦克唐纳函数理论，开始于国际和平组织与瓦尔琴科、基里洛夫和斯蒂尔卡的联合工作。特别地，利用表示理论将Feld和Varchenko关于椭圆超几何函数的最新结果从sl(2)推广到任意单李代数。利用量子仿射代数的表示理论，发展了(A类)仿射根系的Macdonald理论。发展了量子群的扭迹函数理论，并推导出量子群的扭迹函数的差分方程组，其中包含由Belavin-Drinfeld三元组量子化得到的R-矩阵。4.研究有限维Hopf代数理论。寻找半单Hopf代数的新例子。研究余三角Hopf代数表示的张量积。5.继续研究量子杨-巴克斯特方程的集合论解。量子群论是20世纪80年代中期S在群论和量子理论的结合点上兴起的一个相对较新的数学领域。群论创立于19世纪，是一种用来精确描述对称现象的数学理论，它是现代粒子物理学的基础。对称性是自然界的基本属性。从发明车轮开始，它也在人类的所有技术成就中发挥了至关重要的作用：车轮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果，奠定了本世纪技术革命的基础。它是一种物理理论，可以描述非常小的物体的行为，比如原子和电子。在八十年代后期，人们意识到在用量子理论描述的某些系统中，存在着一种新的对称性，这是群论无法描述的。这种新的、非常奇特的对称性被称为量子对称性。为了从数学上描述这种对称性，数学家们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，并探讨其与数学和物理的其他领域的相互作用。

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.