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.

一个非交换代数是一个基本概念，表示解决方案的方程组的几个非交换变量。非交换代数经常被用于网络、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编码模型。通常不可能通过处理其元素来评估给定的代数，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是非常有益的。 这个项目的重点是新的不变量的非交换代数产生的几个科目，如非交换射影几何，范畴论，组合数学，研究无限维的霍普夫代数或量子群。主要研究者将包括研究生和博士后研究员在这项研究中。这个项目涉及几个中心议题，在非交换代数领域的连接到非交换几何，组合数学，表示理论的箭图和范畴理论。首席研究员计划研究几个不变量的非交换代数及其相关的类别，发展新的研究方向的基础，并分类代数对象描述非交换空间。具体来说，该项目研究非交换判别式、Frobenius-Perron维度、本原上同调以及非交换代数、箭图表示和张量三角化范畴中的其他有效不变量和关键结构。主要研究人员继续寻找方法的自同构问题，同构问题，和不同版本的取消问题的非交换代数。最后，PI的最终目标之一是构建新的Hopf代数域的Gelfand-Kirillov维度2的积极特征和新的张量三角结构的有限或驯服的表示类型的表示。 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

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

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

Degree bounds for Hopf actions on Artin–Schelter regular algebras

• DOI: 10.1016/j.aim.2022.108197
• 发表时间: 2020-08
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: E. Kirkman;R. Won;James J. Zhang