Quantum Symmetric Pairs, Categorification, and Geometry

物理物体的美，如雪花，往往是其规律性或对称性的反映。在数学中，这种正则性是用群或代数的概念来描述的。这项研究项目试图研究一种被称为I-量子群的变形对称性，它展示了一种类似晶体的离散结构，与人们在雪花中看到的情况没有什么不同。PI计划发现I-量子群背后的更高对称性(在分类或几何方法中)。该项目为研究生提供了科研培训机会。由量子对称对产生的I-量子群是量子群的广泛推广。PI提出了更一般的I-量子群的Hall代数构造，并在此过程中构造了I-量子群的辫子群对称。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旗簇和更一般的箭形簇的几何实现铺平道路。此外，还将发展I-量子群的胞元理论。最后，提出了I-量子群的Khovanov-Lauda-Rouquier型分类，它将应用于代数群和单位根处的量子群的模表示。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.