The beauty of a physical object, such as a snowflake, is often a reflection of its regularity or symmetry. In mathematics such a regularity is described by the notion of groups or algebras. This research project seeks to study a certain deformation symmetry known as i-quantum groups, which exhibit a crystal-like discrete structure, not unlike what one sees in a snowflake. The PI plans to uncover higher symmetries (in categorical or geometric approaches) behind the i-quantum groups. This project provides research training opportunities for graduate students.
The i-quantum groups arising from quantum symmetric pairs are a vast generalization of quantum groups. The PI proposes a Hall algebra construction of i-quantum groups in greater generality, and constructs braid group symmetries of i-quantum groups along the way. The PI also proposes to provide a Drinfeld type construction of i-quantum groups of affine type, which will pave the way to their finite-dimensional representations and a geometric realization via classical flag varieties and more generally quiver varieties. In addition, a theory of cells for i-quantum groups will be developed. Finally, a Khovanov-Lauda-Rouquier type categorification of i-quantum groups is proposed, and it will have applications to modular representations of algebraic groups and quantum groups at roots of unity.
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.
