Application of Galois cohomology to infinite dimensional Lie theory

伽罗瓦上同调在无限维李理论中的应用

• 批准号: RGPIN-2016-04651
• 负责人: Pianzola, Arturo
• 金额: $ 1.97万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=662104
• 关键词: Application; Galois; cohomology; infinite; dimensional

Lie theory, which owes its origins to the famous Norwegian mathematician Sophus Lie at the end of the 19th century, has evolved into a major mathematical subject, with far reaching tentacles in many other mathematical areas. A by-product is two central intrinsically related objects, viz., Lie groups and Lie algebras.***Groups (and algebras) are present in all of mathematics. One of their most important applications is to detect/describe symmetry and invariance of objects (the latter being the permanence of a property under certain symmetries). The importance of symmetry to many areas of science is paramount. The quintessential examples of this can be found in Einstein's theory of special relativity. The famous formula e=mc2 follows from a simple algebraic manipulation if one assumes that the basic laws of physics have to be invariant under translations and rotations. The Lie groups in question are examples of affine and orthogonal groups. This line of thought is extremely fertile. For example, some of the current exotic theories in particle physics (e.g., any of the superstring theories) derive formulas by assuming a priori that the theory must respect certain symmetries. The Lie algebras encode much of the information of the groups but in an easier language to manipulate. The representations of the Lie algebras, for examples, correspond to elementary particles in certain physical theories.***In the area of Geometry, Lie groups and algebras are typically regarded as “continuously varying” objects. For mathematical reasons, based on applications to number theory, starting in the 1950's, new “algebraic” versions of geometry and Lie theory were developed. This was the birth of algebraic geometry and algebraic groups, which went on to amalgamate with the theory of schemes and reductive group schemes, as developed by A. Grothendieck and his collaborators in the late 60's. Some of the deepest mathematical results established over the last three decades (including A. Wiles's proof of Fermat's last theorem) could not have been possible without Grothendiecks's revolutionary “language” of geometry.***My work centers in trying to discover connections between Grothendieck's creations and infinite dimensional Lie theory. The “infinite dimensional” part appears naturally in string theory. It is also important from a strictly mathematical point of view. This has furnished powerful new machinery to the study of Lie theory and that led to fruition a number of fundamental results. Recently discovered new connections to other areas of mathematics indicate that the proposed research approach will continue to produce results of the highest scientific standards.**************

李理论起源于 19 世纪末挪威著名数学家索弗斯·李 (Sophus Lie)，现已发展成为一门重要的数学学科，在许多其他数学领域具有深远的触角，其副产品是两个本质上相关的核心对象。 ，即李群和李代数。***群（和代数）存在于所有数学中，它们最重要的应用之一是检测/描述对称性和不变性。对称性对于许多科学领域来说至关重要，这一点的典型例子可以在爱因斯坦的狭义相对论中找到。如果假设物理基本定律在平移和旋转下保持不变，则这是一种简单的代数运算。所讨论的李群是仿射群和正交群的例子。例如，当前粒子物理学中的一些奇异理论（例如任何超弦理论）通过先验地假设该理论必须尊重某些对称性来推导公式。例如，李代数的表示对应于某些物理理论中的基本粒子。***在几何领域，李群和代数通常被视为出于数学原因，从 20 世纪 50 年代开始，基于数论的应用，新的“代数”版本的几何和李理论被开发出来，这就是代数几何和代数群的诞生。与 A. Grothendieck 及其合作者在 60 年代末发展的图式理论和约简群图式相结合，这是过去三十年中建立的一些最深刻的数学结果。 （包括 A. 怀尔斯对费马大定理的证明）如果没有格洛腾迪克革命性的几何“语言”，这一切都是不可能实现的。***我的工作重点是试图发现格洛腾迪克的创造与无限维李理论之间的联系。从严格的数学角度来看，这部分自然地出现在李理论中，这也为李理论的研究提供了强大的新机制，并导致了最近发现的许多基本结果。与其他数学领域的新联系表明，所提出的研究方法将继续产生最高科学标准的结果。****************