Categorical centers, cactus actions, and diagram algebras

该项目深入研究表征理论中的几个研究方向，表征理论是通过物体的对称性和保存它们的操作来研究物体的数学框架。这样的操作可以携带一个经典的，甚至更有趣的量子代数结构。最初出现在统计力学和量子可积系统的物理模型中，量子群和围绕它们的理论现在是发现新的数学原理的一个蓬勃发展的来源。该项目将通过为量子群和相关图代数的研究结合代数、组合和高结构分类技术建立一个共同的基础，从而加深对这一理论的理解。这将导致更统一的方法，并提供几个数学领域之间的联系，以及潜在的物理应用。该项目将涉及本科生的参与，并为早期职业研究人员之间的讨论和合作创造机会。具体而言，该项目将从范畴和组合的角度回答有关量子群及其相关代数的表示理论的几个问题。更详细地说，量子群的中心是理解其表示结构的关键，编织和仙人掌群捕获了进一步的关键对称性。量子群的行为与Hecke和Temperley-Lieb代数的行为交织在一起，相互告知对方的行为和它们的表示结构。该项目的主要目标是首先构造和对角化分类量子群中心的元素，并使用所得的幂等元集合来研究分类表示的分解。随后，PI将开发新的工具，建立编织和仙人掌群体行为之间的联系，在经典和更高结构的分类水平上，并在此过程中发展仙人掌群体的表征理论。最后，PI将研究一组密切相关的图代数——两界Temperley-Lieb代数——并将其标准模块的三个径向视点统一起来，这将使研究它们的分解和不可约性成为可能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project delves into several research directions within representation theory, which is the mathematical framework for studying objects through their symmetries and the operations which preserve them. Such operations can carry a classical, or even more intriguingly a quantum algebraic structure. Originally appearing in physical models within statistical mechanics and quantum integrable systems, quantum groups and the theory surrounding them are now a thriving source of uncovering new mathematical principles. This project will develop a richer understanding of this theory by building a common ground for combining algebraic, combinatorial, and higher-structural categorical techniques for the study of quantum groups and associated diagram algebras. This will lead to a more unified approach and provide connections between several areas of mathematics, as well as potential physical applications. The project will involve the participation of undergraduate students and create opportunities for discussion and collaboration among early-career researchers. Specifically, the project will answer several questions about the representation theory of quantum groups and their related algebras from a categorical and a combinatorial perspective. In more detail, the centers of quantum groups are key to understanding the structure of their representations, which have further key symmetries captured by braid and cactus groups. The actions of quantum groups are intertwined with actions of Hecke and Temperley-Lieb algebras, informing each other's behavior and the structures of their representations. The main goals of the project are to first construct and diagonalize elements of the centers of categorified quantum groups, and use the resulting collection of idempotents to study the decomposition of categorical representations. Subsequently, the PI will develop new tools and build connections between braid and cactus group actions, both on the classical and higher-structure, categorical level, and in the process develop the representation theory of the cactus group. Finally, the PI will investigate a family of closely related diagram algebras—the two-boundary Temperley-Lieb algebras—and unite three radial viewpoints of their standard modules, which will enable the study of their decomposition and irreducibility.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.