Classical and A1-homotopy theory of linear algebraic groups

线性代数群的经典和A1-同伦论

• 批准号: RGPIN-2021-02603
• 负责人: Williams, Thomas
• 金额: $ 1.53万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758602
• 关键词: Classical; A1; homotopy; theory; linear

This project studies the interface between algebra and topology. We study the homotopy theory, i.e., the properties that do not change even after continuous deformations, of the symmetry groups of algebraic structures. The proposal is in two parts. The first is when the algebraic objects consist of vector spaces over a field k with no additional structure. In this case, the structure groups the general linear groups GLn, which are comprised of nxn invertible matrices. We study the A1-homotopy theory of GLn. A1-homotopy is a powerful way to define a homotopy theory of algebraically-defined objects. In this theory, first established in the late 1990s, one considers those deformations that may be defined by polynomial functions. In classical homotopy theory, much information about a space X is encoded in its homotopy groups: pi_n(X), which record the different homotopy-classes of continuous functions from spheres to X. In A1-homotopy, one may analogously define homotopy groups, but now the sense of homotopy is the A1-homotopy. The ordinary homotopy groups are difficult to calculate in most cases, and the A1-homotopy groups are even more difficult to determine. The A1-homotopy groups of GLn and related spaces encode subtle and mysterious information about the underlying field k, in the guise of the algebraic K-theory of k, and this proposal will calculate these homotopy groups in order to extract and make sense of that information. We will gain insight into the way in which the theory of vector bundles over an algebraic-geometric object X relates to the algebraic K-theory of X. We will also learn more about the homotopy groups of the spheres themselves, since the group GLn is a symmetry group of the A1-homotopy sphere A^n-0. The second part of the proposal examines what happens when the vector space A has an additional structure, such as multiplication. A is then an algebra, a prevalent structure in mathematics. The symmetries G are restricted by the multiplication of A and are harder to understand than in the case where the multiplication was absent. There are particular geometric spaces associated to the data of (G,A): spaces parametrizing r-tuples of elements in A that are sufficient to generate the entire algebraic structure of A. These spaces are little-studied to date, but because they are algebraically defined, we may use algebraic techniques to examine their ordinary homotopy theory, facilitating a number of explicit calculations. In this way, we will cast light on the symmetry group G and on algebraic structures related to A. The project will use homotopy theory to deepen our fundamental knowledge about several different kinds of widely-used algebraic structures: algebras, algebras with involution, vector bundles on algebraic objects, and fields (through the K-theory). It will also tell us more about the topology of maps between spheres, which are the most fundamental topological objects but about which many questions remain unanswered.

这个项目研究代数和拓扑之间的接口。我们学习同伦理论，即，代数结构的对称群即使在连续变形后也不改变的性质。该建议分为两部分。 第一种是当代数对象由域k上的向量空间组成时，没有额外的结构。在这种情况下，该结构对由n × n可逆矩阵组成的一般线性群GLn进行分组。我们研究了GLN的A1-同伦理论。A1-同伦是定义代数定义对象的同伦理论的一种强有力的方法。在这个理论中，首次建立在20世纪90年代末，人们认为这些变形可以由多项式函数定义。在经典同伦理论中，空间X的许多信息都被编码在它的同伦群pi_n（X）中，它记录了从球面到X的连续函数的不同同伦类。在A1-同伦中，可以类似地定义同伦群，但现在同伦的意义是A1-同伦。普通同伦群在大多数情况下是很难计算的，而A1-同伦群更是难以确定。GLn和相关空间的A1-同伦群以k的代数K-理论为幌子，编码了关于基础域k的微妙而神秘的信息，本提案将计算这些同伦群，以便提取和理解这些信息。我们将深入了解代数几何对象X上的向量丛理论与X的代数K理论的关系。我们还将学习更多关于球面自身的同伦群，因为群GLn是A1-同伦球面A^n-0的对称群。该提案的第二部分研究了当向量空间A具有附加结构（例如乘法）时会发生什么。那么A就是一个代数，一个数学中普遍存在的结构。对称性G受到A的乘法的限制，并且比没有乘法的情况更难理解。存在与（G，A）的数据相关联的特定几何空间：参数化A中元素的r元组的空间，这些空间足以生成A的整个代数结构。这些空间很少被研究，但因为它们是代数定义的，我们可以使用代数技巧来检查它们的普通同伦理论，从而方便了一些显式计算。这样，我们就能阐明对称群G和与A有关的代数结构。该项目将使用同伦理论来加深我们对几种不同类型的广泛使用的代数结构的基础知识：代数，代数与对合，代数对象上的向量丛，和字段（通过K理论）。它还将告诉我们更多关于球面之间映射的拓扑，球面是最基本的拓扑对象，但关于球面的许多问题仍然没有答案。