Exponential Periods, Bispectrality and Affine Quantum Groups

对称性的研究是科学中的一个基本范式，对晶体学、粒子物理学、广义相对论、量子计算和信号处理都有重要影响。研究一个特定的化学、物理或数学系统的（双边、平移、旋转、缩放和其他）对称性通常是理解和最终解决方案的关键，因为它们的存在极大地限制了系统。量子群是自然界最基本对称性的变形。它们是在20世纪80年代被发现的，作为描述例如薄冰层的一维和二维统计力学模型的对称性。量子群最近重新出现，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。量子群形成了一个等级，它取决于变形的强度和它们对谱参数的依赖性：常数，有理，三角或椭圆。随着层级的上升，一个有趣的权衡发生了：有些结构变得更加复杂，而有些结构则从根本上简化了。这个项目将进一步揭示存在于层次结构的不同成员之间的关系，通过在它们之间建立概念桥梁。这些概念桥梁将对它们所连接的量子群产生根本性的影响。一方面，他们将根据一个量子群的等级上级，对微分或差分方程的多值性给出一个极其简单的描述。另一方面，他们将根据前者对后者进行自成一体的描述，除了澄清其结构外，还可能产生深远的算术后果。更确切地说，本项目将探索仿射量子群之间的关系。该项目的一个主要主题是扩展描述的单值的合理卡西米尔连接的对称性卡茨-穆迪代数，在量子Weyl群运营商，数值的变形参数，和差异模拟的连接。该项目的第二个主题是使用Yangians和量子圈代数以及量子圈代数和椭圆量子群的有限维表示的亚纯张量等价，来计算三角卡西米尔连接的单值性以及有理和三角qKZ方程，因此证明了德林费尔德的差分类似物该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

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.