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.

理解对称性，以及它如何以及为什么在自然界中出现，在数学和物理学中都很重要。 测量对称性的数学对象称为群。李群，以挪威人命名 数学家索菲斯·李在世纪发现并首次研究了这些天体， 自然是物理学中许多理论的基本对称性。 世纪中期见证了代数群的诞生，这些对象抓住了李群的精神， 更加普遍。当代数学中许多最引人注目的结果都利用了代数 组它们的起源可以追溯到韦伊、Chevalley、Borel、Serre、Grothendieck、Demazure的基本工作，他们在20世纪40年代和50年代系统地发展了Lie和Cartan在代数几何中的思想。在随后的几十年中，代数群理论被用来统一处理代数的几个关键思想，包括二次和厄米特形式，中心单代数，对合代数和非关联代数。 该研究项目的重点是了解代数群本身的本质及其在数学和物理的多个领域的应用。给定一个代数群，人们可以把不同的几何对象联系起来。特别感兴趣的是torsors和射影齐次簇。最近的结果在代数表明，未来的进展，在这一领域的数学取决于如何深入了解他们的几何性质。我的课题是研究这些物体的性质。作为一个应用，我们计划将我们的结果应用到物理学中出现的无限维李代数的分类。