Homotopical Approaches to Algebraic Structures

代数结构的同伦方法

• 批准号: 1105766
• 负责人: Julia Bergner
• 金额: $ 11.51万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105766&HistoricalAwards=false
• 关键词: Homotopical; Approaches; Algebraic; Structures

In this proposal, the PI seeks to extend previous research on the homotopy theory of homotopy theories in several ways. A major topic of interest is the continuation of work in progress on higher-categorical generalizations of these structures. While a few models have been introduced for doing so, in part motivated by work in algebraic geometry and topological quantum field theory, it is important to establish that they are all equivalent to one another analogous to the PI's previous work. In related work, analogues of these models in the world of operads are being developed, and again it will be important to see which known results can be carried over and how comparisons can be made.In another direction, the PI seeks to use homotopy-theoretic tools to explore further the relationship between Hall algebras and quantum groups. A longtime open question in representation theory is to find a derived version of the Hall algebra from which all of the corresponding quantum group can be recovered. It is anticipated that newly-developed techniques in homotopy theory will provide new insights into solving this conjecture.

在这个提议中，PI试图以几种方式扩展以前对同伦理论的同伦理论的研究。 感兴趣的一个主要议题是继续进行中的工作更高类别的概括这些结构。 虽然已经引入了一些模型来这样做，部分是出于代数几何和拓扑量子场论的工作，但重要的是要建立它们都是等价的，类似于PI以前的工作。 在相关工作中，正在开发这些模型在运算世界中的类似物，同样重要的是要看看哪些已知结果可以继承以及如何进行比较。在另一个方向上，PI寻求使用同伦理论工具来进一步探索霍尔代数和量子群之间的关系。 在表示论中一个长期悬而未决的问题是找到一个霍尔代数的衍生版本，从这个版本中可以恢复所有相应的量子群。 同伦理论中新发展的技术将为解决这一猜想提供新的见解。