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

维数和对称性等不变量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。本研究项目是通过使用代数、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），并在非交换代数的主题和其他活跃的研究领域之间建立一座桥梁。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，本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对易麦凯对应，这是一个由经典麦凯对应所激发的概念，最近被扩展到几个新的领域。具体的主题包括Artin-Schelter正则代数上的Hopf代数作用的非交换商奇点；Artin-Schelter Gorenstein代数的歪斜Calabi-Yau性质和中山自同构以及在中心上模有限的代数的非交换判别式。PI在非交换代数的自同构群、局部幂零导数以及非交换Zariski消去问题的研究中引入了许多不变量，并取得了丰硕的成果。PI将继续寻找非交换代数的不同不变量，为新的研究方向奠定基础，并致力于该领域的核心开放性问题。非交换McKay对应是研究非交换代数、Hopf代数与量子群理论、非交换不变理论和非交换代数几何之间相互作用的重要准则。

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.