Noncommutative Algebra

非交换代数与其他学科的相互作用是现代非交换代数研究的主题之一。非交换代数的范畴和同调方法已经非常成功。主要研究非交换代数的结构及其相关的非交换方案，特别是这些对象的同调方面。本文的主要课题之一是四维Artin-Schelter正则代数的分类。解决这个问题的一个新方法是使用A无穷代数。许多物理现象都可以用非交换方程来解释。利用非交换Hopf代数发现了新的拓扑不变量。代数几何中的一些去奇点化问题被解释为非交换代数中由Morita上下文引起的派生等价。弦理论中的许多问题都是用非交换代数来表述的。人们也可以使用非交换代数来编码网络、量子计算和化学分子的模型。进一步深入地了解非交换代数将对数学、物理和其他学科的许多领域大有裨益。

The interaction between noncommutative algebra and other subjects is one of the main themes in modern noncommutative algebra. The categorical and homological approaches to noncommutative algebras have been very successful. The Principal Investigator is studying the structure of noncommutative algebras and associated noncommutative schemes, in particular, the homological aspects of those objects. One of the main projects of this proposal is the classification of Artin-Schelter regular algebras of dimension four. A new approach to this question is to use A-infinity algebras.Many physical phenomena have been explained by noncommutative equations. New topological invariants have been discovered by using noncommutative Hopf algebras. Some de-singularization problems in algebraic geometry have been interpreted as derived equivalences induced by a Morita context in noncommutative algebra. Many questions in string theory have been formulated by using noncommutative algebras. One may also use noncommutative algebra to encode models for networks, quantum computing and chemical molecules. Further and deeper understanding of noncommutative algebras will be greatly beneficial to many areas in mathematics, physics and other disciplines.