Research in Noncommutative Algebra: Hopf Algebra Actions on Noetherian Artin-Schelter Regular Algebras and Noncommutative McKay Correspondence

非交换代数研究：Noetherian Artin-Schelter 正则代数上的 Hopf 代数作用和非交换麦凯对应

• 批准号: 1700825
• 负责人: Jian Zhang
• 金额: $ 16.9万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-15 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700825&HistoricalAwards=false
• 关键词: Research; Noncommutative; Algebra; Hopf; Actions

Invariants such as dimensions and symmetries are useful tools in mathematics and other disciplines. Understanding links between different invariants is a demanding task in modern mathematics. This research project is to study noncommutative algebras (mathematical structures in which xy does not necessarily equal yx) by using algebraic, combinatorial, geometric, and other invariants, and to build a bridge between the subject of noncommutative algebra and other active research areas. The PI will investigate the structure of several important families of algebras and work on central questions in the subject. Since noncommutative algebras have been used extensively, this project will deepen the understanding of other mathematical areas including noncommutative algebraic geometry, commutative algebra and mathematical physics.A central theme of the proposal is the noncommutative McKay correspondence, a concept motivated by the classical McKay correspondence that has recently been extended to several new areas. Specific topics include noncommutative quotient singularities of Hopf algebra actions on Artin-Schelter regular algebras; skew Calabi-Yau property and the Nakayama automorphism of Artin-Schelter Gorenstein algebras; and the noncommutative discriminant of algebras which are module-finite over their center. The PI has introduced a number of invariants with fruitful applications in the study of automorphism groups and locally nilpotent derivations of noncommutative algebras, as well as the noncommutative Zariski cancellation problem. The PI will continue to search for distinct invariants of noncommutative algebras, to develop foundations for new research directions, and to work on central open questions in the field. The noncommutative McKay correspondence is one essential guideline for the interplay between noncommutative algebra, Hopf algebra and theory of quantum groups, noncommutative invariant theory, and noncommutative algebraic geometry.

不变量，如维度和对称性，在数学和其他学科中是有用的工具。在现代数学中，理解不同不变量之间的联系是一项艰巨的任务。本研究项目旨在利用代数、组合、几何等不变量来研究非交换代数(其中xy不一定等于yx的数学结构)，并在非交换代数学科和其他活跃的研究领域之间架起一座桥梁。PI将研究几个重要的代数族的结构，并致力于这门学科的中心问题。由于非交换代数已经得到了广泛的应用，这个项目将加深对其他数学领域的理解，包括非交换代数几何、交换代数和数学物理。该提议的一个中心主题是非交换McKay对应，这一概念源于经典的McKay对应，最近被扩展到几个新的领域。具体的主题包括：Artin-Schelter正则代数上Hopf代数作用的非对易商奇性；Artin-Schelter Gorenstein代数的斜Calabi-Yau性质和Nakayama自同构；以及中心模有限的代数的非对易判别式。PI引入了一些不变量，在研究非交换代数的自同构群和局部幂零导子以及非交换Zariski消去问题方面有很好的应用。PI将继续寻找非交换代数的独特不变量，为新的研究方向发展基础，并致力于该领域的中心开放问题。非对易McKay对应是非对易代数、Hopf代数与量子群理论、非对易不变理论、非对易代数几何之间相互作用的一个基本准则。

项目成果

Frobenius–Perron theory of modified ADE bound quiver algebras

修正的 ADE 边界箭袋代数的 Frobenius 佩隆理论

• DOI: 10.1016/j.jpaa.2018.09.013
• 发表时间: 2019
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Wicks, Elizabeth

AUSLANDER’S THEOREM FOR GROUP COACTIONS ON NOETHERIAN GRADED DOWN-UP ALGEBRAS

• DOI: 10.1007/s00031-020-09565-5
• 发表时间: 2018-01
• 期刊: Transformation Groups
• 影响因子: 0.7
• 作者: Jianmin Chen;E. Kirkman;J. J. Zhang-J.

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

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

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