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.

该项目研究代数与拓扑之间的界面。我们研究了代数结构的对称基团，即使在连续变形后也不会改变的属性理论，即即使在连续变形之后也不会改变。该提议分为两个部分。 首先是代数对象由没有其他结构的场k上的矢量空间组成。在这种情况下，结构分组的一般线性基团GLN由NXN可逆矩阵组成。我们研究了GLN的A1-homotopicy理论。 A1-HOMOTOPY是定义代数定义对象的同质理论的有力方法。在这一理论中，最初是在1990年代后期建立的，人们考虑了可能由多项式函数定义的变形。在经典同质理论中，有关空间X的许多信息在其同型组中进行了编码：PI_N（X），它记录了从球到X的连续函数的不同同质类别的类别。在大多数情况下，普通同型组很难计算，并且A1-HOMOTOPY组更难确定。 GLN和相关空间的A1-HOMOTOPY组用代数K理论的幌子编码了有关基础域K的微妙而神秘的信息，该建议将计算这些同型组，以提取并了解该信息。我们将深入了解代数几何对象上的矢量捆绑理论与X的代数K理论有关。我们还将了解球体本身的同型群体群体，因为GLN组是A1-HOMOTOPY范围的对称组a^n-0。该提案的第二部分研究了矢量空间A具有附加结构（例如乘法）时会发生什么。然后，A是一个代数，这是数学中普遍的结构。与不存在乘法的情况下，对称g受A的乘法限制，并且难以理解。与（g，a）的数据相关的特定几何空间：在A中的参数为r-tubles的空间足以生成A. A的整个代数结构。这些空间迄今为止尚未研究，但是由于它们是代数定义的，我们可能会使用代数技术来研究其普通同型理论，并计算出一个数字。通过这种方式，我们将阐明与A的对称性组和代数结构有关。该项目将使用同型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本知识：代数，代数，参与度，参与，矢量套件，代数捆绑在代数的对象，以及ktheory（通过K-theory）。它还将告诉我们更多有关球之间地图拓扑的拓扑，这是最基本的拓扑对象，但许多问题仍然没有解决。