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:
  • 金额:
    $ 33.18万
  • 依托单位:
  • 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

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交换变量的数学设置。然而,从量子力学开始,出现了大量的模型,这些模型导致了涉及不再交换的变量的重要数学问题。非交换代数是研究这些结构的主要数学领域之一。这个项目解决了非交换对象的性质和对称性的关键问题,以及相应的非交换代数的表示。表示论的后一个领域通过所有可能的方式来研究非交换代数,以矩阵的形式来表示它们。三个主要的方法被使用:(1)泊松几何-几何所产生的变形的交换对象的非交换的,(2)集群代数-一种组合方法的基础上复杂的内部变换的对象,所谓的集群突变和(3)monoidal三角范畴-一般抽象代数结构所产生的考虑所有表示的代数同时。这些研究活动将被用来作为基础的研究生和本科生的培训和数学博士后的指导。 更详细地说,在这个项目中,PI将研究量子对称空间的结构,Nichols代数,monoidal三角范畴,以及有限维代数的支持理论,更一般地说,有限张量范畴。本论文的主要研究方向有三个:(1)单位根量子簇代数的研究分为两个方面:判别理想的描述和不可约表示的分类。这将基于泊松序和凯莱-汉密尔顿代数的理论。(2)量子簇代数结构将在量子旗簇、量子Bott-Samelson簇和量子对称空间上构造。它们的单位根量子对应物的不可约表示和判别理想将通过第1部分中开发的技术来研究。利用星星积和泊松序,我们将发展量子对称对和量子超群在单位根上的表示理论。(3)将发展有限张量范畴的稳定范畴的非对易巴耳末谱的分类方法。他们将被用于描述有限维Hopf代数的上同调支撑,更一般地说,有限张量categories.This奖反映了NSF的法定使命,并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Root of unity quantum cluster algebras and Cayley–Hamilton algebras
单位根量子簇代数和凯莱汉密尔顿代数
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Milen Yakimov其他文献

A Deodhar-type stratification on the double flag variety
  • DOI:
    10.1007/s00031-007-0061-8
  • 发表时间:
    2007-11-27
  • 期刊:
  • 影响因子:
    0.400
  • 作者:
    Ben Webster;Milen Yakimov
  • 通讯作者:
    Milen Yakimov
Poisson geometry and Azumaya loci of cluster algebras
簇代数的泊松几何与阿祖梅亚轨迹
  • DOI:
    10.1016/j.aim.2024.109822
  • 发表时间:
    2024-09-01
  • 期刊:
  • 影响因子:
    1.500
  • 作者:
    Greg Muller;Bach Nguyen;Kurt Trampel;Milen Yakimov
  • 通讯作者:
    Milen Yakimov
Partitions of the wonderful group compactification
  • DOI:
    10.1007/s00031-007-0062-7
  • 发表时间:
    2007-11-27
  • 期刊:
  • 影响因子:
    0.400
  • 作者:
    Jiang-Hua Lu;Milen Yakimov
  • 通讯作者:
    Milen Yakimov

Milen Yakimov的其他文献

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{{ truncateString('Milen Yakimov', 18)}}的其他基金

Noncommutative Algebras and Related Categorical Structures
非交换代数和相关分类结构
  • 批准号:
    2131243
  • 财政年份:
    2021
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Continuing Grant
Noncommutative Algebras and Related Categorical Structures
非交换代数和相关分类结构
  • 批准号:
    1901830
  • 财政年份:
    2019
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Continuing Grant
International Conference on Representation Theory, Mathematical Physics and Integrable Systems
表示论、数学物理和可积系统国际会议
  • 批准号:
    1803265
  • 财政年份:
    2018
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Standard Grant
Research in Noncommutative Algebra
非交换代数研究
  • 批准号:
    1601862
  • 财政年份:
    2016
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Continuing Grant
Quantum Groups and Quantum Cluster Algebras
量子群和量子簇代数
  • 批准号:
    1303038
  • 财政年份:
    2013
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Standard Grant
Quantum Groups, Poisson Lie Groups, and Combinatorics
量子群、泊松李群和组合学
  • 批准号:
    1001632
  • 财政年份:
    2010
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Standard Grant
Poisson Lie groups, representation theory, combinatorics, and integrable systems
泊松李群、表示论、组合学和可积系统
  • 批准号:
    0701107
  • 财政年份:
    2007
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Standard Grant
Poisson Lie groups, integrable systems, and representation theory
泊松李群、可积系统和表示论
  • 批准号:
    0406057
  • 财政年份:
    2004
  • 资助金额:
    $ 33.18万
  • 项目类别:
    Standard Grant

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量子群、W 代数和布劳尔-考夫曼范畴
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