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.

非公共代数是一个基本概念，它代表了几个非交通变量方程系统的解决方案。非共同代数通常用于编码网络，通信，量子计算和化学分子以及科学和工程中的许多其他物体的模型。通常不可能通过处理其元素来评估给定代数，而代数的不变符捕获代数的许多关键特征。然后，理解不变的人对理解这些代数的不同方面非常有益。 该项目着重于非共同代数的新不变式，这些代数来自多个主题，例如非交通性投影几何，类别理论，组合学和无限维度Hopf代数或量子群的研究。这项研究中的主要研究人员将涉及研究生和博士后研究员。该项目涉及非交通代数领域的几个中心主题，并与非共同几何学，组合学，Quivers和类别理论的代表理论有关。首席研究者计划研究非共同代数及其相关类别的几个不变性，以开发新的研究方向的基础，并对描述非共同空间的代数对象进行分类。该项目具体研究了非交通性代数，颤动表示和张量三角调节类别中的非交通歧视因素，Frobenius-Perron维度，原始共同体以及其他有效的不变结构和关键结构。主要研究者继续寻找无态性问题，同构问题和不同版本的取消代数中的不同版本的方法。最终，PI的最终目标之一是在有限或驯服表示类型的Quiver表示上构建Gelfand-Kirillov Dimension二维的新的Hopf代数域。 首席调查员将培训与他的研究密切相关的领域学生。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影响的评论标准来评估值得支持的。

项目成果

Iterated Hopf Ore extensions in positive characteristic

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

Frobenius-Perron theory for projective schemes

• DOI: 10.1090/tran/8624
• 发表时间: 2019-07
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: J. Chen;Z. Gao;E. Wicks;J. J. Zhang-J.;X-.H. Zhang;H. Zhu

Cancellation of Morita and skew types

• DOI: 10.1007/s11856-021-2199-9
• 发表时间: 2021-09
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: Xin Tang;James J. Zhang;Xiangui Zhao

Frobenius–Perron theory of representations of quivers

• DOI: 10.1007/s00209-021-02888-3
• 发表时间: 2020-04
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: J. J. Zhang-J.;J.-H. Zhou

Truncation of unitary operads

酉运算的截断

• DOI: 10.1016/j.aim.2020.107290
• 发表时间: 2018-05
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Yan-Hong Bao;Yu Ye;James J. Zhang
• 通讯作者: James J. Zhang