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.

尺寸和对称性等不变量是数学和其他学科中的有用工具。理解不同不变量之间的联系是现代数学中的一项艰巨任务。本研究项目旨在利用代数、组合、几何和其他不变量来研究非交换代数（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研究领域之间架起一座桥梁。 PI 将研究几个重要代数族的结构，并研究该学科的核心问题。由于非交换代数已被广泛使用，该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。该提案的中心主题是非交换麦凯对应关系，这是一个受经典麦凯对应关系启发的概念最近扩展到几个新领域。具体主题包括 Artin-Schelter 正则代数上 Hopf 代数作用的非交换商奇点；偏态 Calabi-Yau 性质和 Artin-Schelter Gorenstein 代数的 Nakayama 自同构；以及在其中心上模有限的代数的非交换判别式。 PI 引入了许多不变量，在自同构群和非交换代数的局部幂零推导以及非交换 Zariski 消除问题的研究中具有富有成效的应用。 PI 将继续寻找非交换代数的独特不变量，为新的研究方向奠定基础，并致力于解决该领域的核心开放问题。非交换麦凯对应关系是非交换代数、霍普夫代数和量子群理论、非交换不变量理论和非交换代数几何之间相互作用的重要指南。

项目成果

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: 2021-04-22
• 期刊: Selecta Mathematica
• 作者: K. Chan;Jason Gaddis;R. Won;James J. Zhang
• 通讯作者: James J. Zhang

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: 2019-06
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Wicks; Elizabeth
• 通讯作者: Elizabeth

Cancellation of Morita and skew types

取消森田和倾斜类型

• DOI: 10.1007/s11856-021-2199-9
• 发表时间: 2020-01-18
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: Xin Tang;James J. Zhang;Xiangui Zhao
• 通讯作者: Xiangui Zhao

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

NOETHERIAN 分级自下向上群作用的 AUSLANDER 定理

• DOI: 10.1007/s00031-020-09565-5
• 发表时间: 2018-01-27
• 期刊: Transformation Groups
• 影响因子: 0.7
• 作者: Jianmin Chen;E. Kirkman;J. J. Zhang
• 通讯作者: J. J. Zhang

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: 2020-04-20
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: J. J. Zhang;J.
• 通讯作者: J.