Tensor categories, dynamical R-matrices and double Hecke algebras

本计画主要研究三个主题：1）张量范畴与霍普夫代数; 2）动态R-矩阵与特殊函数; 3）契雷德尼克代数、卡洛格罗-莫泽系统。在第一个领域，PI计划构建有限张量范畴的新例子，包括半单和非半单，并在给定的张量范畴上对模范畴进行分类。在第二个领域，PI计划使用量子仿射代数和动态R-矩阵的表示理论来研究Felder和Varchenko的椭圆超几何函数，并证明关于量子Weyl群的De Concini猜想和关于椭圆超几何函数性质的Felder-Varchenko猜想。在第三个领域，PI计划研究辛反射代数的有限维表示，它们的大$N$极限（双Yangians），双曲类似物（Gan-Ginzburg代数）和“拓扑类似物”（各种辫子群的Hecke代数）。他计划证明代数PBW定理的变形组代数类似于双仿射赫克代数。他还计划研究这一主题与箭壶表现形式的联系。目前的项目是在几个数学领域的十字路口-表示论，数学物理，可积系统理论，特殊功能。Cherednik代数和Calogero-Moser空间的理论起源于一个著名的多元正交多项式族，称为Macdonald多项式。这些多项式是在氢原子研究中产生的经典勒让德多项式的非常广泛的推广，并且它们具有令人难以置信的丰富结构。与经典情形类似，麦克唐纳多项式是量子哈密顿量（麦克唐纳算符）的本征态，它定义了一个可积的量子力学系统，在这个意义上，人们可以精确地求解它。表象论允许我们使用对称性来理解麦克唐纳多项式的结构。类似于氢原子的旋转对称性如何有助于其量子力学性质的研究，麦克唐纳多项式的结构是使用量子群和切雷德尼克代数对称性揭示的。张量范畴理论是这个提议的另一个主题，它也有助于理解量子系统，因为表达许多这样的系统的对称性的表示论结构采取张量范畴的形式。

The current project proposes research on three subjects: 1) tensor categories and Hopf algebras; 2) dynamical R-matrices and special functions, and3) Cherednik algebras, Calogero-Moser systems. In the first area, the PI plans to construct new examples of finite tensor categories, both semisimple and nonsemisimple, and classify module categories over given tensor categories. In the second area, the PI plans to study elliptic hypergeometric functions of Felder and Varchenko using representation theory of quantum affine algebras and dynamical R-matrices, and to prove the De Concini conjecture about quantum Weyl groups and the Felder-Varchenko conjectureson the properties of elliptic hypergeometric functions. In the third area, the PI plans to study finite dimensional representations of symplectic reflection algebras, their large $N$ limits (double Yangians),hyperbolic analogs (Gan-Ginzburg algebras), and ``topologicalanalogs'' (Hecke algebras of various braid groups). He plans to prove algebraic PBW theorems for deformations of group algebras which are similar to double affine Hecke algebras. He also plans to study the connections of this subject with representations of quivers. The current project is at the crossroads of several mathematical areas -- representation theory, mathematical physics, theory of integrable systems, special functions. The theory of Cherednik algebras and Calogero-Moser spacesoriginates from a remarkable family of orthogonal polynomials of several variables called Macdonald polynomials. These polynomials are a very broad generalization of the classical Legendre polynomials arising in the study of the hydrogen atom, and they have an incredibly rich structure. Similarly to this classical case, Macdonald polynomials are eigenstates of a quantum Hamiltonian (Macdonald operator), which defines an integrable quantum-mechanical system, in the sense that one can exactly solve it. Representation theory allows us to use symmetry to understand the structure of Macdonald polynomials. Similarly to how the rotational symmetry of the hydrogen atom is instrumental in the study of its quantum-mechanical properties, the structure of Macdonald polynomials is uncovered using the quantum group and Cherednik algebra symmetry. The theory of tensor categories, which is another subject of this proposal, is also instrumental in understanding quantum systems, since the representation-theoretical structure expressing the symmetry of many such systems takes the shape of a tensor category.