Exponential Periods, Bispectrality and Affine Quantum Groups

对称性研究是科学中的一个基本范式，对结晶学、粒子物理学、广义相对论、量子计算和信号处理都有基本的影响。对给定的化学、物理或数学系统的对称性(双边、平移、旋转、标度和其他)的研究往往是理解它和最终解决它的关键，因为它们的存在极大地限制了系统的存在。量子群是自然界最基本对称性的变形。它们在20世纪80年代被发现，是一维和二维统计力学模型的对称性，例如，描述薄冰层的模型。量子群最近作为描述夸克等基本粒子相互作用的四维规范理论的对称性重新出现。量子群形成的层次结构取决于形变的强度和它们对光谱参数的依赖：常量、有理、三角或椭圆。随着层级的提高，一个有趣的取舍发生了：一些结构变得更加复杂，而另一些则从根本上简化了。该项目将通过在不同层次的成员之间建立概念桥梁来进一步揭示它们之间存在的关系。这些概念上的桥梁将对它们链接的量子群产生根本性的影响。一方面，它们将从根本上更简单地描述一个量子群的微分方程式或差分方程式的多值性，依据它的等级优势。另一方面，他们将从前者的角度对后者进行完备的描述，这除了澄清其结构外，还可能具有深远的算术后果。更准确地说，本项目将探索仿射量子群之间的关系。这个项目的一个主要主题是用量子Weyl群算子来描述可对称Kac-Moody代数的有理Casimir联络的单调性，推广到形变参数的数值，以及这种联络的差值模拟。该项目的第二个主要主题是利用延安和量子循环代数以及量子循环代数和椭圆量子群的有限维表示的亚纯张量等价，计算三角Casimir连接以及有理和三角qKZ方程的单调性，从而证明Frenkel和Reshetikin猜想的Drinfeld-Kohno定理的差异模拟。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The study of symmetry is a basic paradigm in science, with fundamental consequences for crystallography, particle physics, general relativity, quantum computing and signal processing. The study of the (bilateral, translation, rotational, scaling and other) symmetries of a given chemical, physical or mathematical system is often key to its understanding, and ultimate solution, since their presence greatly constrains the system. Quantum groups are deformations of the most basic symmetries of Nature. They were discovered in the 1980s as symmetries of one and two-dimensional statistical mechanical models that describe, for example, thin layers of ice. Quantum groups have re-emerged very recently as the symmetries of four-dimensional gauge theories that describe the interaction of elementary particles such as quarks. Quantum groups form a hierarchy that depends on the strength of the deformation and their dependence on a spectral parameter: constant, rational, trigonometric or elliptic. As the hierarchical level rises, an interesting trade-off occurs: some of the structure becomes far more intricate, while some simplifies radically. This project will further uncover relations that exist between different members of the hierarchy by building conceptual bridges between them. These conceptual bridges will have fundamental consequences for the quantum groups they link. On the one hand, they will give a radically simpler description of the multivaluedness of the differential or difference equations for one quantum group in terms of its hierarchical superior. On the other, they will give a self-contained description of the latter in terms of the former which, in addition to clarifying its structure, has potentially far reaching arithmetic consequences. More precisely, the present project will explore the relations between affine quantum groups. One main theme of the project is to extend the description of the monodromy of the rational Casimir connection of a symmetrisable Kac-Moody algebra, in terms of quantum Weyl group operators, to numerical values of the deformation parameter, and to the difference analogue of the connection. The second main theme of the project is to use the meromorphic tensor equivalence of finite dimensional representations of Yangians and quantum loop algebras, and of quantum loop algebras and elliptic quantum groups, to compute the monodromy of the trigonomeric Casimir connection and of the rational and trigonometric qKZ equations, thus proving the difference analog of the Drinfeld-Kohno theorem conjectured by Frenkel and Reshetikhin.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.