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.

这个提议围绕着表示论的两个主要方向，并提出发展同伦方法。有限群的表示理论在很大程度上是由Alperin和Brou 'e在过去二十年中的著作推动的。该提案将为该地区带来新的拓扑方向和新的地理位置。其基本思想是用拓扑空间上的作用代替向量空间上的作用。高等表示论，研究给定的么半群范畴对范畴的作用，是一个最近的研究领域。它主要是与Kac-Moody代数有关的。最近的一个构造是张量结构。我们的建议将把同伦方法引入到理论中。它的目的还在于带来更高的代表性理论在该地区的geometers工作计数不变量和拓扑学家工作不变量的四流形，在延续霍瓦诺夫。该项目将带来新的拓扑洞察力代数，特别是在代表性理论，研究对称性。一个主要的问题是从局部对称性理解全局对称性。我们将在一个更广泛的拓扑环境中处理和重铸这一点。该项目还将产生在代数中构建对象的方法，作为几何空间作为家庭或“模空间”的经典结构的对应物。这将导致通过代数理解三维和四维几何的某些性质。