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是通过群的概念来描述的。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。许多重要的对称性都是通过称为李群的连续群来描述的，以数学家 Sophus Lie 的名字命名；这些群是由相关的李代数生成的。 量子群是这种对称性的变形，也是更高结构的影子。该研究项目旨在通过所谓的 i 量子群进一步开发一种新方法来表示李代数和李超代数。这种新方法旨在揭示 i 量子群的基本几何结构和更高的分类结构。 预计结果也可应用于结理论。由于最近发现了与旗簇几何、规范基和分类的联系，i-量子群（Drinfeld-Jimbo 量子群的共理想子代数）已被证明在量子群理论中发挥着越来越重要的作用以及李代数和李超代数的表示。在这个项目中，研究者计划在 Kac-Moody 设置中开发 i 量子群及其模块的规范基和分类行为理论。特别是，仿射量子群的分类将被制定并应用于研究统一根的经典类型量子（超）群和素性特征的经典代数群的模表示论。特殊李超代数的 Bernstein-Gelfand-Gelfand 类别中的特征公式也将被制定。

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