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 代数是矩阵代数的扭曲形式。由于矩阵本身无处不在，因此这些对象在不同的​​数学环境中都存在并且值得注意。 在域上的代数背景下，Azumaya 代数只是在扩展到某个（可分离的）代数扩展时与矩阵代数同构的代数。这里的典型例子是实数域上的汉密尔顿四元数，它在扩展到复数域后变得与 2x2 复数矩阵同构。域上的 Azumaya 代数是具有经典重要性的中心简单代数。 在拓扑背景下，空间 X 上的东屋代数是由 X 参数化的代数族，该代数族与 nxn 复矩阵的平凡族局部同构。 介于上述两个上下文之间的是 Azumaya 代数的簇，因为簇跨越了拓扑空间和域上的代数的世界。 该项目旨在将适用于 Azumaya 代数的拓扑工具引入拓扑空间，以应用于与算术或代数问题相关的特定情况，从而提供一系列示例和反例。这些例子将有助于在代数背景下完善对 Azumaya 代数的研究，反驳或支持现有的猜想，并为人们在高维簇及其函数域上预期的行为提供指导，而这些行为很难直接解决使用代数技术。 它还旨在通过 etale 上同调和 A1 或动机同伦理论将拓扑环境中进行的计算转置到代数环境中，并通过这种方法在代数理论中建立积极的结果。