Quantum Groups, W-algebras, and Brauer-Kauffmann Categories

Symmetries are patterns that repeat or stay the same when certain changes are made, like rotating a shape or reflecting it in a mirror. They are everywhere in nature, from the spirals of a seashell to the orbits of planets around the sun. They also hide behind mathematical objects and the laws of physics. Quantum groups and Lie algebras are tools mathematicians use to study these symmetries. This project is a deep dive into understanding the underlying structure of these patterns, even when they're slightly changed or twisted, and how they influence the behavior of everything around us. The project will also provide research training opportunities for graduate students. In more detail, the PI will develop emerging directions in i-quantum groups arising from quantum symmetric pairs as well as develop applications in various settings of classical types beyond type A. The topics include braid group actions for i-quantum groups; Drinfeld presentations for affine i-quantum groups and twisted Yangians, and applications to W-algebras; character formulas in parabolic categories of modules for finite W-algebras; and categorification of i-quantum groups, and applications to Hecke, Brauer and Schur categories. 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.