Geometric Insights in Noncommutative Algebra

非交换代数中的几何见解

• 批准号: 2201273
• 负责人: Manuel Reyes
• 金额: $ 20万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201273&HistoricalAwards=false
• 关键词: Geometric; Insights; Noncommutative; Algebra

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.

换数的代数研究代数系统无关紧要，而非共同代数允许两种产品XY和YX可能不一样的可能性。历史上，非共同代数在对称的代数研究中出现，线性代数，也许是量子力学中最深刻的。自代数几何形状出现以来，几何思想就彻底改变了交换代数。以类似的方式，在过去的几十年中，非交通性几何形状的发展激发了非交通性代数的许多新思想和观点，但迄今为止，它提出的挑战超出了其所解决的挑战。例如，由于其构建的几何特性，预计许多非交通性代数将得到良好的行为，但是我们缺乏代数工具来证明这种情况是这种情况。此外，非共同的几何形状在很大程度上仍然是与彼此不直接兼容的松散相关框架的集合，这对于新移民而言可能使该领域特别困难。该项目将通过创建新技术来直接解决这些基本问题，从而推断出良好的代数属性，用于几何构造的非交通性代数，并提供新工具来表示与非交通性代数相对应的几何结构。该项目将涉及研究生的研究和培训机会。该项目的第一部分着重于非共同代数几何学的同源问题。 PI和合作者将使用Koszul二元性方法以及有限维代数的表示，以攻击长期以来的猜想，即Artin-Schelter（AS）常规代数是域。相关技术将用于理解当不一定连接的常规代数时，例如渐变的卡拉比（Calabi-Yau）代数是主要环。该项目的第二部分集中在非交通性几何形状中的光谱问题上。 PI将使用多种方法来理解非交换空间，并利用它们为环和C*-Algebras构造非交通频谱函数。要研究的主题包括针对非交通空间的结构束线，表征双卵石的方法，C* - algebras的离散化以及戒指的投射代表理论。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和广泛的范围来评估的。