Categorification of quasi-split i-quantum groups and related topics in representation theory

This is a project in representation theory which, roughly speaking, is the idea of understanding symmetry in the broadest sense by studying the different ways in which symmetries can be realized in terms of matrices. There are many applications, including to number theory, combinatorics, low-dimensional topology, theoretical physics and chemistry. Nearly forty years ago, quantum groups were discovered and shown to possess some remarkably rigid canonical bases. This had a dramatic impact on the study of the classical Lie groups which are the most fundamental symmetries in nature. In fact, quantum groups and their canonical bases are shadows of some even more remarkable higher structures, Kac-Moody 2-categories, which are often referred to as the categorifications of quantum groups. In classical mathematics, Lie groups go hand in hand with the symmetric spaces on which they act. Symmetric spaces admit quantizations, namely the i-quantum groups appearing in the title of the project, which were first introduced in 1998 and rapidly developed into a rich theory. This project will also provide research training activities for graduate students. The main goal of this project is to take the next step by categorifying all quasi-split i-quantum groups, building on the recent discovery by the PI and collaborators of a 2-category which categorifies the simplest rank one i-quantum group. This theory, which although algebraic in nature, has many connections to the geometry of the underlying Lie groups via the theory of singular Soergel bimodules. In addition, the PI will study more classical topics in representation theory by applying the deeper understanding of quantum and i-quantum groups that appear as hidden symmetries in more classical settings. 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.