Applications of torsors in algebra and Lie theory

扭转量在代数和李理论中的应用

• 批准号: 298447-2012
• 负责人: Chernousov, Vladimir
• 金额: $ 2.19万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571496
• 关键词: Applications; torsors; algebra; Lie; theory

Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. The mathematical objects that measure symmetry are called Groups. Lie groups, named after the Norwegian mathematician Sophus Lie who discovered and first studied these objects late in the 19th century, arise naturally as the underlying symmetry of many theories in Physics. The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of Lie groups yet are much more universal. Many of the most striking results in contemporary Mathematics make use of algebraic groups. Their origins go back the fundamental work of Weil, Chevalley, Borel, Serre, Grothendieck, Demazure, who in the 1940s and 50s systematically developed the ideas of Lie and Cartan in the context of algebraic geometry. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key ideas of algebra, including the theories of quadratic and hermitian forms, central simple algebras, algebras with involutions and non-associated algebras. The research project centers on understanding the very nature of algebraic groups themselves, and their applications to several areas of Mathematics, as well as Physics. Given an algebraic group one can associate different geometric objects. Of special interest are torsors and projective homogeneous varieties. Recent results in algebra show that future progress in this area of mathematics depends how deeply we can understand their geometric properties. My project concerns on studying properties of these objects. As an application we plan to apply our results to classification of infinite dimensional Lie algebras which appear in Physics.

了解对称性，以及它在自然界中的出现以及为什么在数学和物理学中都很重要。 测量对称性的数学对象称为组。以挪威人的名字命名 数学家Sophus谎言发现并在19世纪末首次研究了这些物体 自然是许多物理学理论的基础对称性。 20世纪中叶看到了代数群体的诞生，捕捉谎言群体精神的物体是 更普遍。当代数学中，许多最引人注目的结果利用了代数 组。他们的起源返回了Weil，Chevalley，Borel，Serre，Grothendieck，Amazure的基本工作。在随后几十年的过程中，代数群体的理论已被用来对代数的几个关键思想进行统一处理，包括二次和遗传形式的理论，中央简单代数，代数，具有相关性和非相关代数。 研究项目集中在理解代数群体本身的本质以及它们在数学领域以及物理学领域的应用。给定代数组，可以将不同的几何对象关联。特别感兴趣的是Torsors和投射性均匀品种。代数的最新结果表明，这一数学领域的未来进展取决于我们能够深入了解它们的几何特性。我的项目涉及研究这些对象的属性。作为应用程序，我们计划将结果应用于物理学中出现的无限尺寸谎言代数的分类。