Canonical Bases, Categorification, and Modular Representations

规范基础、分类和模块化表示

• 批准号: 1702254
• 负责人: Weiqiang Wang
• 金额: $ 31.81万
• 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: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702254&HistoricalAwards=false
• 关键词: Canonical; Bases; Categorification; Modular; Representations

The symmetries of a snowflake or a baseball are described by the notion of a group. Groups also describe more abstract symmetries, including supersymmetry in theoretical particle physics. Many important symmetries are described by continuous groups known as Lie groups, after mathematician Sophus Lie; these groups are generated by associated Lie algebras. Quantum groups, which are deformations of such symmetries, also serve as a shadow of higher structures. This research project aims to further develop a new approach, via so-called i-quantum groups, to representations of Lie algebras and Lie superalgebras. This new approach aims to uncover the underlying geometric and higher categorical structures of i-quantum groups. Results are expected also to have applications to knot theory. Because of recently discovered connections to geometry of flag varieties, canonical bases, and categorification, i-quantum groups, which are co-ideal subalgebras of Drinfeld-Jimbo quantum groups, have been shown to play an increasingly important role in the theory of quantum groups and representations of Lie algebras and Lie superalgebras. In this project the investigator plans to develop the theory of canonical bases and categorical actions of i-quantum groups and their modules in the Kac-Moody setting. In particular, a categorification of affine -quantum groups will be formulated and applied to study the modular representation theory of quantum (super)groups of classical type at roots of unity and classical algebraic groups in prime characteristic. Character formulae in Bernstein-Gelfand-Gelfand category for exceptional Lie superalgebras will also be formulated.

雪花或棒球的对称性用群的概念来描述。群还描述了更抽象的对称性，包括理论粒子物理学中的超对称性。许多重要的对称被称为李群的连续群来描述，这是数学家Sophus Lie的名字；这些群由相关的李代数生成。量子群是这种对称性的变形，也是更高结构的影子。这个研究项目旨在进一步发展一种新的方法，通过所谓的I-量子群来表示李代数和李超代数。这一新方法旨在揭示i-量子群的基本几何结构和更高范畴结构。这一结果也有望应用于纽结理论。由于最近发现了与标志簇、标准基和范畴几何的联系，I-量子群作为Drinfeld-Jimbo量子群的上理想子代数，在量子群理论和李代数和李超代数的表示中发挥着越来越重要的作用。在这个项目中，研究者计划在Kac-Moody环境下发展I-量子群及其模的典范基和范畴作用的理论。特别地，建立了仿射量子群的范畴，并将其应用于研究单位根处的经典类型量子(超)群和素数特征的经典代数群的模表示理论。还将建立特殊李超代数的Bernstein-Gelfand-Gelfand范畴的特征标公式。

项目成果

Odd Singular Vector Formula for General Linear Lie Superalgebras

一般线性李超代数的奇奇异向量公式

• DOI: 10.21915/bimas.2019401
• 发表时间: 2019
• 期刊: Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES
• 影响因子: 0.2
• 作者: Liu, Jie;Wang, Li Luo
• 通讯作者: Wang, Li Luo

Quantum supergroups VI: roots of 1

量子超群 VI：1 的根

• DOI: 10.1007/s11005-019-01209-4
• 发表时间: 2019
• 期刊: Letters in Mathematical Physics
• 影响因子: 1.2
• 作者: Chung, Christopher;Sale, Thomas;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang

Spin nilHecke algebras of classical type

经典类型的自旋 nilHecke 代数

• DOI: 10.1016/j.jalgebra.2017.10.024
• 发表时间: 2018
• 期刊: Journal of algebra
• 影响因子: 0.9
• 作者: Johnson, Ian;Wang, Weiqiang
• 通讯作者: Wang, Weiqiang