Noncommutative Algebra

代数是一个基本的数学概念，它编码了数学和其他学科中对象的信息。理解代数的结构本身就是一项重要的任务，对于数学的其他领域也是如此。代数的不变量应该捕捉其表示对象的显著特征，并且不变量之间的恒等式或等式揭示了这些对象背后的见解。这个建议的重点是研究非交换代数，产生于几个学科，如非交换射影几何，非交换不变理论，研究无限维Hopf代数。PI将研究几个长期项目;他的最终目标之一是发现表征非交换代数的强大不变量。 PI将让研究生参与这项研究。 本研究计画涉及非交换代数中的三个主题：高维Artin Schelter正则代数、非交换不变量理论与非交换McKay对应、以及非交换代数的Zerkiki消去问题，所有这些主题彼此密切相关。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身份，并致力于该领域的重要开放问题。工作涉及两个令人兴奋的想法，非交换代数的判别式和同调恒等式，其中有许多令人惊讶的应用。不变量如判别式和同调恒等式中的不变量触及非交换代数结构的核心。这个项目调查了大量有趣的问题，将刺激非交换代数和相关领域的研究。该项目的结果有希望在该主题及其他领域产生重大影响。

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.