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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741923
• 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.

该项目研究代数和拓扑之间的接口。我们研究同伦理论，即代数结构的对称群即使在连续变形后也不会改变的性质。该提案分为两部分。 第一种情况是代数对象由域 k 上的向量空间组成，没有附加结构。在这种情况下，该结构对由nxn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有效方法。在这个于 20 世纪 90 年代末首次建立的理论中，人们考虑那些可以由多项式函数定义的变形。在经典同伦理论中，有关空间 X 的许多信息都编码在其同伦群中：pi_n(X)，它记录了从球体到 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都很难计算，而A1-同伦群则更难确定。 GLn 和相关空间的 A1 同伦群以 k 的代数 K 理论的形式编码有关基础域 k 的微妙而神秘的信息，并且该提案将计算这些同伦群，以便提取和理解该信息信息。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。我们还将了解有关球体本身的同伦群的更多信息，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。 A 是代数，是数学中普遍存在的结构。对称性 G 受到 A 乘法的限制，并且比不存在乘法的情况更难理解。有一些与 (G,A) 的数据相关的特定几何空间：对 A 中的元素 r 元组进行参数化的空间，足以生成 A 的整个代数结构。迄今为止，对这些空间的研究很少，但因为它们是根据代数定义，我们可以使用代数技术来检验它们的普通同伦理论，从而促进许多显式计算。通过这种方式，我们将了解对称群 G 以及与 A 相关的代数结构。该项目将利用同伦理论来加深我们对几种不同类型广泛使用的代数结构的基础知识：代数、对合代数、向量代数对象和域上的丛（通过 K 理论）。它还将告诉我们更多有关球体之间地图拓扑的信息，球体是最基本的拓扑对象，但许多问题仍未得到解答。