Noncommutative Algebras and Monoidal Triangulated Categories

非交换代数和幺半群三角范畴

基本信息

批准号：
2200762
负责人：
Milen Yakimov
金额：
$ 33.18万
依托单位：
Northeastern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200762&HistoricalAwards=false
关键词：
Noncommutative Algebras Monoidal Triangulated Categories

项目摘要

Many models in the sciences and engineering are based on mathematical settings that involve commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to important mathematical problems involving variables that no longer commute. Noncommutative Algebra is one of the major areas of mathematics that studies those structures. This project addresses key problems about the properties and symmetries of noncommutative objects, as well as the representations of the corresponding noncommutative algebras. The latter area of representations theory investigates noncommutative algebras through all possible ways to present them in terms of matrices. Three major approaches are used: (1) Poisson geometry--geometry arising from the deformation of commutative objects to noncommutative ones, (2) cluster algebras--a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations and (3) monoidal triangulated categories--general abstract algebraic structures arising from considering all representations of an algebra simultaneously. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring of mathematics postdocs. In more detail, in this project the PI will investigate the structure of quantum symmetric spaces, Nichols algebras, monoidal triangulated categories, as well as support theories for finite dimensional algebras and, more generally, finite tensor categories. The following three broad directions will be pursued: (1) Root of unity quantum cluster algebras will be investigated in two interrelated plans: description of their discriminant ideals and classification of irreducible representations. This will be based on the theory of Poisson orders and Cayley-Hamilton algebras. (2) Quantum cluster algebra structures will be constructed on quantum flag varieties, quantum Bott-Samelson varieties and quantum symmetric spaces. The irreducible representations and discriminant ideals of their root of unity quantum counterparts will be studied through the techniques developed in part 1. The representation theory of quantum symmetric pairs and quantum supergroups at roots of unity will be developed using star products and Poisson orders. (3) Methods for the classification of the noncommutative Balmer spectra of the stable categories of finite tensor categories will be developed. They will be used for the descriptions of the cohomological supports of finite dimensional Hopf algebras, and more generally, finite tensor categories.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.

科学和工程中的许多模型都是基于涉及通勤变量的数学设置。但是，从量子力学开始，出现了许多模型，这些模型导致了重要的数学问题，涉及不再通勤的变量。非共同代数是研究这些结构的数学的主要领域之一。该项目解决了有关非共同对象的属性和对称性的关键问题，以及相应的非共同代数的表示。代表理论的后一个领域通过所有可能的方式来调查非共同代数，以用矩阵呈现它们。 Three major approaches are used: (1) Poisson geometry--geometry arising from the deformation of commutative objects to noncommutative ones, (2) cluster algebras--a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations and (3) monoidal triangulated categories--general abstract algebraic structures arising from considering all representations of an algebra simultaneously.这些研究活动将用作培训研究生和本科生的基础，并指导数学博士后。更详细地，在此项目中，PI将研究量子对称空间，尼科尔代数，单三角类别的结构，并支持有限维代数的理论以及更一般而言的有限张量类别。将提出以下三个广泛的方向：（1）将在两个相互关联的计划中调查统一量子群集代数的根源：对其判别理想的描述和对不可约表示的分类。这将基于Poisson命令和Cayley-Hamilton代数的理论。（2）将在量子标志品种，量子bott-samelson品种和量子对称空间上构建量子群集代数结构。将通过第1部分中开发的技术研究其统一量子量子根的不可约说明和判别理想。统一根部的量子对称对和量子超级组的表示理论将使用星级产品和Poisson Orders开发。（3）将开发有限张量类别类别的稳定类别的非交通性Balmer光谱的分类方法。它们将用于描述有限维数代数的共同体支持，更普遍地是有限的张量类别。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。