Quantum Symmetric Pairs, Categorification, and Geometry

量子对称对、分类和几何

• 批准号: 2001351
• 负责人: Weiqiang Wang
• 金额: $ 34.5万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001351&HistoricalAwards=false
• 关键词: Quantum; Symmetric; Pairs; Categorification; Geometry

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.

一个物理对象的美，比如雪花，通常是它的规则性或对称性的反映。在数学中，这样的正则性用群或代数的概念来描述。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，它表现出类似晶体的离散结构，与人们在雪花中看到的没有什么不同。PI计划揭示i-量子群背后的更高对称性（分类或几何方法）。该项目为研究生提供了研究培训机会。由量子对称偶产生的i-量子群是量子群的一个广泛推广。PI在更一般的情况下提出了i-量子群的Hall代数构造，并在此基础上沿着构造了i-量子群的辫群对称。PI还提议提供仿射型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标志簇和更一般的标志簇的几何实现铺平道路。此外，理论的细胞的i-量子群将被开发。最后，Khovanov-Lauda-Rouquier型分类的i-量子群的建议，它将有应用程序的代数群和量子群的根unity.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

Braid group action and quasi-split affine ?quantum groups I

辫群作用和准分裂仿射？量子群 I

• DOI: 10.1090/ert/657
• 发表时间: 2023
• 期刊: Representation Theory of the American Mathematical Society
• 作者: Lu, Ming;Wang, Weiqiang;Zhang, Weinan
• 通讯作者: Zhang, Weinan

Serre–Lusztig relations for $$\imath $$quantum groups II

$$imath $$量子群 II 的 Serre‐Lusztig 关系

• DOI: 10.1007/s11005-021-01497-9
• 发表时间: 2022
• 期刊: Letters in Mathematical Physics
• 影响因子: 1.2
• 作者: Chen, Xinhong;Letzter, Gail;Lu, Ming;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang

Serre-Lusztig relations for ıquantum groups III

ä±量子群 III 的 Serre-Lusztig 关系

• DOI: 10.1016/j.jpaa.2022.107253
• 发表时间: 2023
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Chen, Xinhong;Lu, Ming;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang

Formulae of ı-divided powers in Uq(sl2) , III

Uq(sl2) , III 中的 ä± 幂公式

• DOI: 10.1016/j.jalgebra.2022.12.001
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Chen, Xinhong;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang

?Hall algebra of the projective line and ?-Onsager algebra

?射影线的霍尔代数和?-Onsager 代数

• DOI: 10.1090/tran/8798
• 发表时间: 2023
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Lu, Ming;Ruan, Shiquan;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang