Monoidal Triangular Categories: Representation Theory, Cohomology, and Geometry

幺半群三角范畴：表示论、上同调和几何

基本信息

批准号：
2101941
负责人：
Daniel Nakano
金额：
$ 23.24万
依托单位：
University of Georgia Research Foundation Inc
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101941&HistoricalAwards=false
关键词：
Monoidal Triangular Categories Representation Theory

项目摘要

Representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology and number theory. In addition, it has many applications to chemistry and physics. Representations are mappings of complicated algebraic objects (such as groups, rings, Lie algebras, and Lie superalgebras) to an array of numbers (matrices). These realizations by matrices encode important data that can yield deep insights into these complicated algebraic objects. In recent years, a useful approach has been to understand the entire collection of representations of an object. Representations for a certain algebraic object often form a tensor triangulated category. Techniques from homological algebra can be used to build bridges between tensor triangulated categories and geometric objects. Uncovering this hidden geometry often leads to new insights about the algebraic object and its representations. This project includes research and training opportunities for graduate students and postdoctoral fellows in algebra and representation theory. In this project the PI will develop new methods to understand tensor structures in monoidal tensor categories. This will entail the development of monoidal triangular geometry and explicit computations of Balmer spectra. The PI will also introduce new geometric and topological methods to provide concrete calculations of cohomology for algebraic/finite groups, Lie superalgebras, quantum groups, and Frobenius kernels. The development of the cohomology theory was an important tool to resolving 30 year old problems that deal with tilting modules in connection with filtrations of representations of algebraic groups. The PI will study representations of Lie superalgebras via the application of super geometry to construct representations and to compute higher sheaf cohomology. The PI will develop a new Lie theory that entails the use of detecting and parabolic subalgebras, in addition to, a nilpotent cone to formulate a geometric setting in order to compute characters for irreducible representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

表示论已成为现代数学的核心领域，与组合数学、代数几何、拓扑和数论相关。此外，它在化学和物理方面也有许多应用。表示是复杂代数对象（例如群、环、李代数和李超代数）到数字数组（矩阵）的映射。这些矩阵的实现编码了重要的数据，可以深入了解这些复杂的代数对象。近年来，一种有用的方法是理解对象的整个表示集合。某个代数对象的表示通常形成张量三角范畴。同调代数的技术可用于在张量三角类别和几何对象之间建立桥梁。揭示这种隐藏的几何图形通常会带来关于代数对象及其表示的新见解。该项目包括为代数和表示论方面的研究生和博士后研究员提供研究和培训机会。在这个项目中，PI 将开发新方法来理解幺半群张量类别中的张量结构。这将需要开发幺半群三角几何和巴尔默谱的显式计算。 PI 还将引入新的几何和拓扑方法，为代数/有限群、李超代数、量子群和 Frobenius 核提供上同调的具体计算。上同调理论的发展是解决 30 年历史问题的重要工具，这些问题涉及与代数群表示过滤相关的倾斜模。 PI 将通过应用超几何来构建表示并计算更高层上同调来研究李超代数的表示。 PI 将开发一种新的李理论，该理论需要使用检测和抛物线子代数，此外，还需要使用幂零锥来制定几何设置，以便计算不可约表示的字符。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。