Topics in noncommutative algebra 2022: homological regularities

2022 年非交换代数专题：同调正则

• 批准号: 2302087
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302087&HistoricalAwards=false
• 关键词: Topics; noncommutative; algebra; 2022; homological

Many phenomena in sciences, specially in mathematics and physics, can be described by noncommuting variables, that is, the product XY of two variables X and Y may not equal to the product YX in the opposite order. A noncommutative algebra is a mathematical concept that encodes such a phenomenon, and the subject of noncommutative algebra naturally has many applications in quantum mechanics, quantum field theory, string theory, and so on. The study of noncommutativity has become more and more common in modern science and technology. For example, a quantum group is equivalent to a Hopf algebra whose underlying algebraic structure is noncommutative and/or whose underlying coalgebraic structure is noncocommutative. The study of noncommutative algebras is both important and challenging. In many cases, noncommutative algebras arise as noncommutative analogues and generalizations of classical objects coming from other fields such as commutative algebra, algebraic geometry, and Lie theory. To understand the structure of noncommutative algebras, the PI will investigate invariants that are defined by homological means. These invariants capture hidden structures and symmetries of noncommutative algebras. This research project combines ideas and methodology from several areas of mathematics such as algebraic geometry, commutative algebra, linear algebra, homological algebra, and combinatorics. The proposed research activities will make contributions to teaching, undergraduate and graduate student training, and outreach. A major part of the project concerns homological invariants such as Castelnuovo-Mumford regularity, Tor-regularity, and Artin-Schelter regularity for connected graded algebras. A weighted version of homological regularities with an extra parameter will also be introduced that incorporates different classical homological concepts. This is a new research direction in noncommutative algebra with connections to representation theory, noncommutative algebraic geometry, and noncommutative invariant theory. The PI will investigate and prove homological identities involving these invariants for connected graded algebras and their associated categories, and thus expand the foundations for this theory. One particular sub-project is the classification of non-Artin-Schelter regular algebras that are close to being Artin-Schelter regular. In addition this project concerns other active research topics in noncommutative algebra: the automorphism problem, the cancellation problem, operad theory, and noncommutative discriminants.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.

科学中的许多现象，特别是数学和物理中的现象，都可以用非交换变量来描述，即两个变量 X 和 Y 的乘积 XY 可能不等于相反顺序的乘积 YX。非交换代数是编码这种现象的数学概念，非交换代数学科自然在量子力学、量子场论、弦论等方面有很多应用。非交换性的研究在现代科学技术中变得越来越普遍。例如，量子群等价于其底层代数结构是非交换性的和/或其底层煤代数结构是非共交换的 Hopf 代数。非交换代数的研究既重要又具有挑战性。在许多情况下，非交换代数是作为来自其他领域（例如交换代数、代数几何和李理论）的经典对象的非交换类比和概括而出现的。为了理解非交换代数的结构，PI 将研究通过同调方法定义的不变量。这些不变量捕获了非交换代数的隐藏结构和对称性。该研究项目结合了代数几何、交换代数、线性代数、同调代数和组合数学等多个数学领域的思想和方法。拟议的研究活动将为教学、本科生和研究生培训以及推广做出贡献。该项目的主要部分涉及同调不变量，例如连通分级代数的 Castelnuovo-Mumford 正则、Tor 正则和 Artin-Schelter 正则。还将引入具有额外参数的同调规律的加权版本，其中结合了不同的经典同调概念。这是非交换代数的一个新研究方向，与表示论、非交换代数几何和非交换不变量理论相关。 PI 将研究并证明涉及连通分级代数及其相关类别的这些不变量的同调恒等式，从而扩展该理论的基础。一个特定的子项目是接近 Artin-Schelter 正则代数的非 Artin-Schelter 正则代数的分类。此外，该项目还涉及非交换代数中的其他活跃研究主题：自同构问题、取消问题、运算理论和非交换判别式。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评估进行评估，认为值得支持。影响审查标准。