Representation theory and homotopical algebra

表示论和同伦代数

• 批准号: 1161999
• 负责人: Raphael Rouquier
• 金额: $ 63.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161999&HistoricalAwards=false
• 关键词: Representation; theory; homotopical; algebra

This proposal centers around two main directions in representation theory, and proposes to develop homotopical methods. The representation theory of finite groups has been driven in a large part by the conjectures of Alperin and Brou'e in the last twenty years. The proposal will bring new topological directions in the area, and new conjectures. The basic idea is to replace actions on vector spaces by actions on topological spaces. Higher representation theory, the study of actions of a given monoidal category on categories, is a much more recent area of research. It has been developed mostly in relation with Kac-Moody algebras. A recent construction is that of tensor structures. Our proposal will bring homotopical methods in the theory. It aims also at bringing higher representation theory in the area of geometers working on counting invariants and topologists working on invariants of four-manifolds, in the continuation of Khovanov.The project will bring new topological insight in algebra, and in particular in representation theory, the study of symmetries. A major problem is to understand global symmetries from local symmetries. We will approach and recast this within a wider topological setting. The project will also produce methods to build objects in algebra, as a counterpart of the classical constructions of spaces in geometry as families or "moduli spaces". This should lead to an understanding through algebra of some properties of three and four-dimensional geometry.

该提议围绕表示理论的两个主要方向，并提出开发同位方法。在过去的二十年中，有限群体的代表理论是由阿尔珀林和布鲁伊的猜想在很大程度上驱动的。该提案将在该地区带来新的拓扑指示以及新的猜想。基本思想是通过拓扑空间的动作代替矢量空间上的动作。更高的代表理论是对给定单体类别对类别的行为的研究，是一个较新的研究领域。它主要是与Kac-Moody代数相关的。最近的结构是张量结构。我们的提议将带来该理论中的同位方法。它还旨在在Khovanov的延续中引起更高的代表理论，以计算从事四个manifolds的不变性的几何学领域和拓扑师。一个主要问题是了解本地对称性的全球对称性。我们将在更广泛的拓扑设置中对其进行处理并重新铸造。该项目还将产生在代数中构建对象的方法，这是几何形状中的经典构建体的对应物，例如家族或“模量空间”。这应该通过代数对三维几何形状的某些特性进行理解。