2-representation theory and categorification

2-表示理论和分类

• 批准号: EP/S017216/1
• 负责人: Vanessa Miemietz
• 金额: $ 41.99万
• 依托单位: University of East Anglia
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FS017216%2F1
• 关键词: representation; theory; categorification

While in mathematics it is very often helpful to break complicated problems down into less complicated ones by simplifying and forgetting data, the converse has often proved useful. The term categorification refers to the process of finding more complicated structures which, upon forgetting some information, reproduce the original problem that one wants to study. The more complicated structures often allow us to deduce useful information that was previously inaccessible. For example, an integer number solving a certain equation might a priori be anything, but if we then discover that this certain number, in fact, describes the number of elements in a set (e.g. cats in a household), it cannot be negative.Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.

尽管在数学方面，通过简化和忘记数据将复杂的问题分解为不太复杂的问题通常很有帮助，但相反的问题通常被证明是有用的。术语分类是指找到更复杂的结构的过程，这些结构在忘记一些信息后就重现了一个人想要研究的原始问题。更复杂的结构通常使我们能够推断出以前无法访问的有用信息。例如，一个整数求解某个方程式可能是先验的，但是如果我们发现该数字实际上描述了一组中的元素数量（例如，家庭中的猫），代表理论中的分类性不可能是负面的。通常以某些编码其他单词或其他单词为2个单词或其他单词的典型结构形式。为了开发两类的两种分量的理论（从有趣的示例中提取了某些不错的属性），并将它们应用于启发其定义的一些原始问题，这是该建议的目的。

项目成果

Basic Hopf algebras and symmetric bimodules

基本 Hopf 代数和对称双模

• DOI: 10.1016/j.jpaa.2023.107328
• 发表时间: 2023
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Hristova K

Kostant's problem for fully commutative permutations

完全交换排列的 Kostant 问题

• DOI: 10.4171/rmi/1428
• 发表时间: 2023
• 期刊: Revista Matemática Iberoamericana
• 作者: Mackaay M

Finitary birepresentations of finitary bicategories

有限二范畴的有限双表示

• DOI: 10.1515/forum-2021-0021
• 发表时间: 2021
• 期刊: Forum Mathematicum
• 影响因子: 0.8
• 作者: Mackaay M

Evaluation birepresentations of affine type A Soergel bimodules

仿射 A 型 Soergel 双模的评估双表示

• DOI: 10.1016/j.aim.2023.109401
• 发表时间: 2024
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Mackaay M

Pretriangulated 2-representations via dg algebra 1-morphisms

通过 dg 代数 1-态射进行预三角化 2-表示

• DOI: 10.48550/arxiv.2205.09999
• 发表时间: 2022
• 作者: Laugwitz R