Double Loop Groups and Algebras, Central Extensions, and their Representations

双环群和代数、中心推广及其表示

• 批准号: 1313894
• 负责人: Xinwen Zhu
• 金额: $ 0.57万
• 依托单位: Northwestern University
• 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: 2012-12-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313894&HistoricalAwards=false
• 关键词: Double; Loop; Groups; Algebras; Central

The proposed research is to study representation theory of double loop groups and algebras. The main idea of this project is that the "right" objects to study are not the groups nor algebras themselves, but their categorical central-extensions. The PI and his collaborator have already constructed these central-extensions and their representations. The PI will continue to study the finer structures on these extensions and their representations, and their interaction with algebraic geometry, K-theory, and topology. The PI also proposed to study various aspects of the Geometrical Langlands Duality. The proposal plans to understand the geometry of moduli space of local systems (the Langlands parameter), in particular the wild ramified local systems.Representation theory is a branch of mathematics that studies symmetries via linear algebra (or linear vector spaces), which is so far proved to be a very brilliant idea and a very powerful tool in the study of many other disciplines, such as physics and number theory. However, in recent years, there are many clues that there are symmetries that are complicated enough so that one should study them not via linear vector spaces, but via linear categories. The PI's research provides (probably the first) examples of such complicated symmetry which is better studied by categories rather than vector spaces. It is also hoped that studying this symmetry would shade lights on two-dimensional Langlands duality and four-dimensional quantum field theory.

拟议的研究是研究双环群和代数的表示论。该项目的主要思想是，“正确”的研究对象不是群或代数本身，而是它们的分类中心扩展。 PI 和他的合作者已经构建了这些中央扩展及其表示。 PI 将继续研究这些扩展的更精细结构及其表示形式，以及它们与代数几何、K 理论和拓扑的相互作用。 PI 还提议研究几何朗兰兹对偶性的各个方面。该提案计划理解局域系统的模空间几何（朗兰兹参数），特别是狂野的分支局域系统。表示论是通过线性代数（或线性向量空间）研究对称性的数学分支，迄今为止已被证明是一个非常聪明的想法，也是许多其他学科（例如物理和数论）研究中非常强大的工具。然而，近年来，有许多线索表明存在足够复杂的对称性，因此人们不应该通过线性向量空间而是通过线性类别来研究它们。 PI 的研究提供了（可能是第一个）这种复杂对称性的例子，通过类别而不是向量空间可以更好地研究这种对称性。人们还希望研究这种对称性能够为二维朗兰兹对偶性和四维量子场论带来光明。