Noncommutative Algebra

代数是一个基本的数学概念，它对数学和其他学科中对象的信息进行编码。理解代数的结构本身以及数学的其他领域都是一项重要的任务。代数的不变量被认为能够捕捉到其所代表对象的显著特征，不变量之间的恒等式或方程揭示了这些对象背后的见解。本文主要研究非交换射影几何、非交换不变量理论和无限维Hopf代数的非交换代数。PI将调查几个长期项目；他的最终目标之一是发现表征非交换代数的强大不变量。该项目将包括研究生参与这项研究。本课题涉及高维Artin-Schelter正则代数、非交换不变量理论与非交换McKay对应、非交换代数的Zariski消去问题等三个非交换代数领域，它们之间有着密切的联系。PI继续为新的研究方向奠定基础，寻找非交换代数的不同不变量和不变量之间的恒等式，并研究该领域的重要开放性问题。这项工作涉及两个令人兴奋的思想，即非交换代数的判别式和同调恒等式，它们有许多令人惊讶的应用。不变量，如同调恒等式中的判别式不变量，触及了非交换代数结构的核心。该项目研究了大量有趣的问题，这些问题将刺激非交换代数和相关领域的研究。该项目的结果有望在本学科内外产生重大影响。

An algebra is a fundamental mathematical concept that encodes information about objects in mathematics and other disciplines. Understanding the structure of algebras is an important task on its own as well as for other fields in mathematics. Invariants of an algebra are supposed to capture distinguished features of its representing objects, and identities or equations between the invariants reveal the insights behind these objects. This proposal focuses on the research of noncommutative algebras that arise from several subjects such as noncommutative projective geometry, noncommutative invariant theory, and the study of infinite dimensional Hopf algebras. The PI will investigate several long-term projects; one of his ultimate goals is to discover powerful invariants that characterize noncommutative algebras. The PI will involve graduate students in this research. This research project concerns three topics in noncommutative algebra: higher dimensional Artin-Schelter regular algebras, noncommutative invariant theory and noncommutative McKay correspondence, and the Zariski cancellation problem for noncommutative algebras, all of which are intimately connected with each other. The PI continues to develop the foundations for new research directions, to search for distinct invariants of noncommutative algebras and identities between the invariants, and to work on important open questions in the field. The work involves two exciting ideas, discriminant of noncommutative algebras and homological identities, which have many surprising applications. Invariants such as the discriminant and those involved in homological identities touch the core of the structure of noncommutative algebras. This project investigates a large number of interesting questions that will stimulate research in noncommutative algebra and related areas. Results of the project have promise for significant impact in the subject and beyond.