Categorical centers, cactus actions, and diagram algebras

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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.