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.

表示理论是通过线性动作对对称性的研究。较高的表示理论引入了一个新的范式，其中空间被更高的结构（Abelian或三角形类别或更高的分类结构）取代。在过去的二十年中，尤其是从致力于量子重力的物理学家提倡这种方法，但到目前为止几乎没有实现。在类别之间研究函子的研究似乎越来越重要，可以比较我们对其他类别感兴趣的类别进行比较。我们的说法是，我们应该研究这些函子之间的关系，通过这样做，我们将发现一些基本类型的新对称性，类似于向量空间的经典对称性。这将提供具体的（代数，数值）信息，而当前的类别和函子的研究以抽象的水平完成，并且只能以极大的信息损失而获得具体数据。这样的研究部分符合通常的表示理论：一个人定义了有趣的结构（例如，一个经典的结构会考虑对称群体，简单的谎言代数），并研究了可能的对象，它们可以是对称的对称性（例如，经典的一个尝试，例如对简单的表示，这是对一般表示形式的建筑砖块）。一个重要的新功能是，尽管向量空间是相当基本的结构，但类别（Abelian或Trianguald）却不是。一个重要的结果是通过研究其较高的对称性来更好地理解代数或几何起源的各种类别。该提案的一个关键方面是提供其他类别的类别结构。应绕过模量空间的结构，并应直接构建相关的分类结构。开发代数替代模量结构是该项目的主要灵感。该项目的目的是基于PI的“高度代表理论”计划，开发一种新的方法来计算（交换性和非共同）几何形状（交换性和非共同）几何形状。