The Topology, Geometry and Algebra of Projective Linear Groups

射影线性群的拓扑、几何和代数

• 批准号: RGPIN-2016-03780
• 负责人: Williams, Thomas
• 金额: $ 1.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708544
• 关键词: Topology; Geometry; Algebra; Projective; Linear

An Azumaya algebra is a twisted form of a matrix algebra. Since matrices themselves are ubiquitous, these objects exist and are noteworthy in different mathematical contexts. In the context of algebras over a field, an Azumaya algebra is simply an algebra that becomes isomorphic to a matrix algebra upon extension to some (separable) algebraic extension. The prototypical example here is Hamilton's Quaternions over the real field, which becomes isomorphic to 2x2 complex matrices upon extension to the complex field. Azumaya algebras over fields are the Central Simple Algebras of classical importance. In the topological context, an Azumaya algebra over a space X is a family of algebras parametrized by X that is locally isomorphic to the trivial family of nxn complex matrices. Intermediate between the two contexts above is that of Azumaya algebras over varieties, since varieties straddle the worlds of topological spaces and of algebras over fields. This project aims to bring topological tools applicable to Azumaya algebras over topological spaces to bear in specific cases that relate to arithmetic or algebraic questions, and consequently to furnish a stream of examples and counterexamples. These examples will help to refine the study of Azumaya algebras in the algebraic context by either disproving or supporting existing conjectures, and by providing a guide to the behaviour one expects over high-dimensional varieties and their function fields, which can be hard to address directly using algebraic techniques. It also aims to transpose the calculations made in the topological context to an algebraic context, via etale cohomology and A1 or motivic homotopy theories, and to establish positive results in the algebraic theory by this method.

Azumaya代数是基质代数的扭曲形式。由于矩阵本身无处不在，因此存在这些对象，并且在不同的数学环境中值得注意。 在一个字段上的代数的背景下，Azumaya代数只是一个代数，在扩展到某些（可分离的）代数扩展时，它与矩阵代数同构。这里的原型示例是汉密尔顿在真实场上的四季度，它在扩展到复杂场后成为2x2复合矩阵的同构。田地上的Azumaya代数是经典重要性的中心简单代数。 在拓扑环境中，空间X上的Azumaya代数是由X参数术的代数家族，它是NXN复杂矩阵琐碎家族的局部同构。 上述两种情况之间的中间位置是品种上的Azumaya代数，因为品种跨越了拓扑空间和代数的世界。 该项目旨在将适用于Azumaya代数的拓扑工具在与算术或代数问题相关的特定情况下置于拓扑空间上，因此提供了一系列示例和反例。这些示例将有助于通过反对或支持现有猜想来完善对代数环境中Azumaya代数的研究，并通过为人们期望对高维品种及其功能领域的行为提供指南，这可能很难直接使用代数技术来解决。 它还旨在通过EtaLe Cromology和A1或动机同型理论将在拓扑环境中的计算转换为代数环境，并通过这种方法在代数理论中建立积极的结果。