Moduli spaces and higher representation theory

模空间和更高表示理论

• 批准号: EP/F065787/1
• 负责人: Raphael Rouquier
• 金额: $ 51.14万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FF065787%2F1
• 关键词: Moduli; spaces; higher; representation; theory

Representation theory is the study of symmetries via linear actions. Higher representation theory introduces a new paradigm, wherein spaces are replaced by higher structures (abelian or triangulated categories, or higher categorical structures). Such approaches have been advocated over the last twenty years, in particular by physicists working on quantum gravity, but very little has been achieved so far. It has appeared more and more important to study functors between categories, say to compare a category we are interested in with other categories we understand better. Our claim is that we should study the relations between these functors, and by doing so, we will discover some new symmetries of a fundamental type, analogous to classical symmetries for vector spaces. This would provide concrete (algebraic, numerical) information, while the current study of categories and functors is completed at an abstract level, and concrete data can be obtained only at the expense of a great loss of information. Such a study goes partly in line with usual representation theory: one defines interesting structures (classically one would, for example, consider symmetric groups, simple Lie algebras) and investigates the possible objects they can be symmetries of (classically one tries, for example, to classify simple representations, which are the building bricks for general representations). An important new feature is that, whilst vector spaces are fairly elementary structures, categories (abelian or triangulated) are not. An important consequence would be a better understanding of various categories of algebraic or geometric origin via the study of their higher symmetries. A crucial aspect of the proposal is to provide constructions of categories from other categories. Constructions of moduli spaces should be bypassed and the associated categorical structures should be constructed directly. Developing an algebraic substitute for moduli constructions is the main inspiration for the project. The aim of this project is to develop a new approach to counting invariants in (commutative and non-commutative) geometry, based on the PI's programme of higher representation theory.

表象理论是通过线性作用来研究对称性的。更高表示理论引入了一种新的范式，其中空间被更高的结构(阿贝尔或三角范畴，或更高的范畴结构)所取代。在过去的二十年里，这种方法一直被提倡，尤其是在量子引力方面工作的物理学家，但到目前为止，取得的成果很少。研究范畴之间的函子显得越来越重要，比如将我们感兴趣的范畴与我们更好地理解的其他范畴进行比较。我们的主张是，我们应该研究这些函子之间的关系，通过这样做，我们将发现一些新的基本类型的对称，类似于向量空间的经典对称。这将提供具体的(代数的，数值的)信息，而目前对范畴和函子的研究是在抽象的水平上完成的，只有以大量信息损失为代价才能获得具体的数据。这样的研究部分符合通常的表示理论：人们定义了有趣的结构(经典地，人们会考虑对称群、单李代数)，并研究它们可能是其对称性的对象(经典地，例如，人们试图对简单表示进行分类，简单表示是一般表示的基石)。一个重要的新特征是，虽然向量空间是相当基本的结构，但范畴(阿贝尔或三角化)不是。一个重要的结果将是通过研究它们的更高对称性来更好地理解各种代数或几何起源范畴。该提案的一个关键方面是提供来自其他类别的类别结构。应该绕过模空间的构造，而直接构造相关的范畴结构。开发模结构的代数替代品是该项目的主要灵感。这个项目的目的是开发一种新的方法来计算(交换和非交换)几何中的不变量，基于PI的更高表示理论的程序。