Topics in Noncommutative Algebra

非对易代数是一个基本概念，它表示由几个非对易变量组成的方程组的解。非对易代数经常被用来编码网络、通信、量子计算、化学分子以及科学和工程中的许多其他对象的模型。通常不可能通过处理给定代数的元素来评估该代数，而代数的不变量捕获了该代数的许多关键特征。因此，理解不变量对于理解这些代数的不同方面变得非常有益。本课题主要研究非对易代数的新不变量，这些不变量来源于非对易射影几何、范畴理论、组合学以及无限维Hopf代数或量子群的研究。主要研究人员将包括研究生和博士后研究员。本项目涉及非对易代数领域的几个中心主题，与非对易几何、组合学、箭图的表示理论和范畴理论有关。主要研究人员计划研究非对易代数及其相关范畴的几个不变量，为新的研究方向奠定基础，并对描述非对易空间的代数对象进行分类。具体地说，该项目研究了非交换代数、箭图表示和张量三角范畴中的非交换判别式、Frobenius-Perron维、本原上同调以及其他有效的不变量和关键结构。主要的研究人员继续寻找解决非对易代数中自同构问题、同构问题以及不同版本的抵消问题的方法。最后，PI的最终目标之一是构造具有正特征的二维Gelfand-Kirillov新的Hopf代数域和有限表示型或驯服表示型箭图表示上的新张量三角结构。首席研究员将在与他的研究密切相关的领域对学生进行培训。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A noncommutative algebra is a fundamental concept that represents solutions to a system of equations of several non-commuting variables. Noncommutative algebras are frequently used to encode models for networks, communication, quantum computing, and chemical molecules, as well as many other objects in sciences and engineering. It is generally impossible to assess a given algebra by dealing with its elements, while invariants of an algebra capture many key features of the algebra. Then understanding invariants becomes extremely beneficial for understanding different aspects of these algebras. This project focuses on new invariants of noncommutative algebras that arise from several subjects such as noncommutative projective geometry, category theory, combinatorics, and the study of infinite dimensional Hopf algebras or quantum groups. The principal investigator will involve graduate students and postdoctoral fellows in this research.This project concerns several central topics in the field of noncommutative algebra with connections to noncommutative geometry, combinatorics, the representation theory of quivers and category theory. The principal investigator plans to study several invariants of noncommutative algebras and their associated categories, to develop foundations for new research directions, and to classify algebraic objects that describe noncommutative spaces. Specifically the project investigates noncommutative discriminants, Frobenius-Perron dimension, primitive cohomology, and other effective invariants and crucial structures in noncommutative algebras, quiver representations and tensor triangulated categories. The principal investigator continues to search for methods for the automorphism problem, the isomorphism problem, and different versions of the cancellation problem in noncommutative algebra. Finally one of PI's ultimate goals is to construct new Hopf algebra domains of Gelfand-Kirillov dimension two in positive characteristic and new tensor triangulated structures on the quiver representations of either finite or tame representation type. The principal investigator will train students in fields closely related to his research.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.