Topics in noncommutative algebra 2022: homological regularities

科学中的许多现象，特别是数学和物理中的许多现象，都可以用不可交换变量来描述，即两个变量X和Y的乘积XY不一定等于相反顺序的乘积YX。非交换代数是对这种现象进行编码的数学概念，而非交换代数的主题自然在量子力学、量子场论、弦理论等方面有许多应用。在现代科学技术中，对非交换性的研究越来越普遍。例如，一个量子群等价于一个Hopf代数，其基础代数结构是非交换的和/或其基础共代数结构是非交换的。非交换代数的研究既重要又具有挑战性。在许多情况下，非交换代数是作为经典对象的非交换类似物和推广而出现的，这些对象来自其他领域，如交换代数、代数几何和李论。为了理解非交换代数的结构，PI将研究由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。这个研究项目结合了几个数学领域的思想和方法，如代数几何、交换代数、线性代数、同调代数和组合学。建议的研究活动将有助于教学，本科生和研究生的培训和推广。该项目的主要部分是关于连通梯度代数的同调不变量，如Castelnuovo-Mumford正则、tor正则和Artin-Schelter正则。还将引入一个带有额外参数的同调规律的加权版本，它包含了不同的经典同调概念。这是非交换代数中与表示理论、非交换代数几何和非交换不变量理论相联系的一个新的研究方向。PI将研究和证明涉及这些不变量的连通梯度代数及其相关范畴的同调恒等式，从而扩展该理论的基础。一个特别的子课题是对接近于Artin-Schelter正则的非Artin-Schelter正则代数的分类。此外，本项目还涉及非交换代数中其他活跃的研究课题：自同构问题、消去问题、运算理论和非交换判别式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.