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由一群人的概念描述。组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。在数学家Sophus谎言之后，连续的群体将许多重要的对称性描述。这些组是由相关的谎言代数产生的。 量子组是这种对称性的变形，也是较高结构的阴影。该研究项目旨在通过所谓的i-Quantum群体进一步开发一种新方法，以代表代数和谎言级别的级别代表。这种新方法旨在揭示I-量子组的潜在几何形状和更高的分类结构。 结果预计还将对结理论应用。由于最近发现了与国旗品种，规范碱基和分类的几何形状的联系，因此，I-Quantum群体是Drinfeld-Jimbo量子群的共核亚代词，已被证明在lie elgebras和Superalgebras的量子群和代表中起着越来越重要的作用。在这个项目中，研究人员计划在Kac-Moody环境中发展I-Quantum群体及其模块的规范基础和分类行动的理论。特别是，将制定仿射量子组的分类，并应用于研究主要特征的统一和经典代数群的经典类型的量子（超级）基团的模块化表示理论。伯恩斯坦 - 吉尔夫甘夫甘富夫（Bernstein-Gelfand-Gelfand）类别中的特征公式也将制定为超级甲壳虫的特殊谎言。

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

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

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