Geometric Insights in Noncommutative Algebra

交换代数研究乘法顺序无关的代数系统，而非交换代数允许两个乘积XY和YX可能不同。历史上，非对易代数出现在对称性的代数研究中，出现在线性代数中，也许最深刻的出现在量子力学中。自从代数几何问世以来，交换代数就被几何思想彻底改变了。同样，非对易几何在过去几十年的发展激发了非对易代数的许多新思想和新观点，但到目前为止，它带来的挑战比它所解决的问题更多。例如，许多非交换代数由于其结构的几何性质而被期望表现良好，但我们缺乏代数工具来证明这一点。此外，非对易几何在很大程度上仍然是一个松散相关的框架的集合，这些框架彼此之间不直接兼容，这可能会使这个领域对新手来说特别困难。这个项目将直接解决这些基本问题，通过创造新的技术来推导几何构造的非对易代数的良好代数性质，并提供新的工具来表示与非对易代数相对应的几何结构。这个项目将包括研究生的研究和培训机会。这个项目的第一部分集中在非对易代数几何中的同调问题。PI和合作者将使用Koszul对偶的方法和有限维代数的表示来攻击一个长期存在的猜想，即Artin-Schelter(AS)正则代数是域。相关的技巧将被用来理解广义为不一定相连的正则代数，如分次Calabi-Yau代数是素环的情况。这个项目的第二部分集中在非对易几何中的谱问题。PI将使用各种方法来理解非对易离散空间，并利用它们来构造环和C*-代数的非对易谱函子。将研究的主题包括非对易空间的结构层，对偶余代数的刻画方法，C*-代数的离散化，以及环的射影表示理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Commutative algebra studies algebraic systems for which the order of multiplication is irrelevant, while noncommutative algebra allows for the possibility that the two products XY and YX may not be the same. Noncommutative algebra historically arose in the algebraic study of symmetry, in linear algebra, and perhaps most profoundly in quantum mechanics. Since the advent of algebraic geometry, commutative algebra has been revolutionized by geometric ideas. In a similar way, the development of noncommutative geometry over the past several decades has stimulated many new ideas and perspectives in noncommutative algebra, but to date it has posed more challenges than it has solved. For instance, many noncommutative algebras are expected to be well-behaved due to the geometric nature of their construction, but we lack the algebraic tools to prove that this is the case. Furthermore, noncommutative geometry largely remains a collection of loosely related frameworks that are not directly compatible with one another, which can make the field especially difficult for newcomers. This project will directly address these fundamental problems by creating new techniques to deduce good algebraic properties for geometrically constructed noncommutative algebras and providing new tools to represent geometric structures corresponding to noncommutative algebras. The project will involve research and training opportunities for graduate students.The first part of this project focuses on homological problems in noncommutative algebraic geometry. The PI and collaborators will use methods of Koszul duality and the representations of finite-dimensional algebras to attack a longstanding conjecture that Artin-Schelter (AS) regular algebras are domains. Related techniques will be used to understand when generalized AS regular algebras that are not necessarily connected, such as graded Calabi-Yau algebras, are prime rings. The second part of this project is focused on spectral problems in noncommutative geometry. The PI will use a variety of approaches to understand noncommutative discrete spaces and utilize them to construct noncommutative spectrum functors for both rings and C*-algebras. Topics to be investigated include structure sheaves for noncommutative spaces, methods to characterize dual coalgebras, discretization of C*-algebras, and a projective representation theory for rings.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.