Canonical Bases, Categorification, and Modular Representations

雪花或棒球的对称性是用群的概念来描述的。群还描述了更抽象的对称性，包括理论粒子物理中的超对称性。许多重要的对称性是由连续群描述的，称为李群，以数学家索夫斯·李命名；这些群是由相关李代数生成的。量子群是这种对称性的变形，也可以作为更高结构的影子。该研究项目旨在通过所谓的i-量子群进一步开发一种新的方法来表示李代数和李超代数。这种新方法旨在揭示i-量子群的潜在几何和更高的分类结构。研究结果也有望应用于结理论。由于最近发现了旗簇、正则基和范畴的几何联系，i-量子群作为Drinfeld-Jimbo量子群的共理想子代数，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-量子群及其模块的规范基和范畴作用理论。特别地，我们将提出仿射量子群的一个范畴，并应用它来研究在单位根处的经典型量子（超）群和素数特征处的经典代数群的模表示理论。并给出了例外李超代数的Bernstein-Gelfand-Gelfand范畴的特征公式。

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.