Topics in Noncommutative Algebra

非交换代数主题

• 批准号: 2001015
• 负责人: Jian Zhang
• 金额: $ 33万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001015&HistoricalAwards=false
• 关键词: Topics; Noncommutative; Algebra

A noncommutative algebra is a fundamental concept that represents solutions to a system of equations of several non-commuting variables. Noncommutative algebras are frequently used to encode models for networks, communication, quantum computing, and chemical molecules, as well as many other objects in sciences and engineering. It is generally impossible to assess a given algebra by dealing with its elements, while invariants of an algebra capture many key features of the algebra. Then understanding invariants becomes extremely beneficial for understanding different aspects of these algebras. This project focuses on new invariants of noncommutative algebras that arise from several subjects such as noncommutative projective geometry, category theory, combinatorics, and the study of infinite dimensional Hopf algebras or quantum groups. The principal investigator will involve graduate students and postdoctoral fellows in this research.This project concerns several central topics in the field of noncommutative algebra with connections to noncommutative geometry, combinatorics, the representation theory of quivers and category theory. The principal investigator plans to study several invariants of noncommutative algebras and their associated categories, to develop foundations for new research directions, and to classify algebraic objects that describe noncommutative spaces. Specifically the project investigates noncommutative discriminants, Frobenius-Perron dimension, primitive cohomology, and other effective invariants and crucial structures in noncommutative algebras, quiver representations and tensor triangulated categories. The principal investigator continues to search for methods for the automorphism problem, the isomorphism problem, and different versions of the cancellation problem in noncommutative algebra. Finally one of PI's ultimate goals is to construct new Hopf algebra domains of Gelfand-Kirillov dimension two in positive characteristic and new tensor triangulated structures on the quiver representations of either finite or tame representation type. The principal investigator will train students in fields closely related to his research.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.

非交换代数是一个基本概念，表示多个非交换变量的方程组的解。非交换代数经常用于对网络、通信、量子计算和化学分子以及科学和工程中的许多其他对象的模型进行编码。通常不可能通过处理给定代数的元素来评估它，而代数的不变量捕获了代数的许多关键特征。那么理解不变量对于理解这些代数的不同方面就变得非常有益。 该项目重点研究非交换代数的新不变量，这些不变量源自非交换射影几何、范畴论、组合学以及无限维霍普夫代数或量子群的研究等多个学科。首席研究员将让研究生和博士后参与这项研究。该项目涉及非交换代数领域的几个中心主题，与非交换几何、组合数学、箭袋表示论和范畴论有关。首席研究员计划研究非交换代数的几个不变量及其相关类别，为新的研究方向奠定基础，并对描述非交换空间的代数对象进行分类。具体来说，该项目研究非交换判别式、Frobenius-Perron 维数、原上同调以及非交换代数、箭袋表示和张量三角范畴中的其他有效不变量和关键结构。首席研究员继续寻找非交换代数中自同构问题、同构问题和不同版本的消去问题的方法。最后，PI 的最终目标之一是在正特征中构造新的 Gelfand-Kirillov 维二的 Hopf 代数域，以及在有限或驯服表示类型的箭袋表示上构造新的张量三角结构。 首席研究员将在与其研究密切相关的领域对学生进行培训。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

项目成果

Truncation of unitary operads

酉运算的截断

• DOI: 10.1016/j.aim.2020.107290
• 发表时间: 2020-10
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Bao, Yan;Ye, Yu;Zhang, James J.
• 通讯作者: Zhang, James J.

Reflexive hull discriminants and applications

自反船体判别式及应用

• DOI: 10.1007/s00029-021-00755-x
• 发表时间: 2022-04
• 期刊: Selecta Mathematica
• 作者: Chan, Kenneth;Gaddis, Jason;Won, Robert;Zhang, James J.
• 通讯作者: Zhang, James J.

Degree bounds for Hopf actions on Artin–Schelter regular algebras

ArtinäSchelter 正则代数上 Hopf 动作的度界

• DOI: 10.1016/j.aim.2022.108197
• 发表时间: 2022-03
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Kirkman, E.;Won, R.;Zhang, J.J.
• 通讯作者: Zhang, J.J.

Frobenius–Perron theory of representations of quivers

弗罗贝尼乌斯佩隆的箭袋表征理论

• DOI: 10.1007/s00209-021-02888-3
• 发表时间: 2022-03
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Zhang, J. J.;Zhou, J.
• 通讯作者: Zhou, J.

Iterated Hopf Ore extensions in positive characteristic

积极特征的迭代 Hopf 矿石扩展

• DOI: 10.4171/jncg/453
• 发表时间: 2020-04-30
• 期刊: Journal of Noncommutative Geometry
• 影响因子: 0.9
• 作者: K. Brown;James J. Zhang
• 通讯作者: James J. Zhang