Quantum Symmetric Pairs, Categorification, and Geometry

物体的美，如雪花，往往是其规律性或对称性的反映。在数学中，这种规律性是用群或代数的概念来描述的。这个研究项目旨在研究一种被称为i-量子群的变形对称性，它表现出一种晶体状的离散结构，就像人们在雪花中看到的那样。PI计划揭示i-量子群背后更高的对称性（以分类或几何方法）。本项目为研究生提供研究训练机会。由量子对称对产生的i-量子群是量子群的广泛推广。本文提出了广义i-量子群的霍尔代数构造，并在此基础上构造了i-量子群的编织群对称性。PI还建议提供仿射型i-量子群的Drinfeld型构造，这将为它们的有限维表示和通过经典的旗子变体和更普遍的颤振变体的几何实现铺平道路。此外，还将发展i-量子群的细胞理论。最后，提出了i-量子群的Khovanov-Lauda-Rouquier型分类，并将其应用于代数群和量子群在单位根处的模表示。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。

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.