Applications 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: 9343-2006
• 负责人: Pianzola, Arturo
• 金额: $ 1.75万
• 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: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363492
• 关键词: Applications; Galois; cohomology; infinite; dimensional

My work is in Lie Theory, so-named after the Norwegian mathematician Sophus Lie (pronounced Lee) who, late in the last century, began the study of this particular field of mathematics. At the heart of Lie Theory is the idea of symmetry, both discrete and continuous. Nature provides us with ample evidence of these symmetries. Snowflakes, molecules, and crystals are good examples of objects with a finite number of symmetries. Their existence is not only of pure mathematical interest, but reflects deeply on the physical characteristics of the objects themselves (how molecules bound for example). Continuous symmetries, which were at the centre of Lie's ideas,  have always played an important role into the insight and development of modern Physics. Here Lie theory arises as signs of the postulated symmetry in the laws of nature; for instance rotations and translations in space for classical mechanics, the Heisenberg group for quantum mechanics, and the Lorentz group in the case of special relativity. There is an important common philosophical thread weaving through these examples: the laws of nature tend to be as simple as possible once they are circumscribed by a given group of symmetries. In the case of Physics the belief in this principle is so pervasive, that the laws of some of the current unifying theories (like superstrings) are in fact obtained  this way. Outside the hard sciences we are also surrounded by symmetry as Escher's drawings, Bach's fugues, and Da Vinci's studies clearly show. It is thus not an exageration to say that symmetry (in a Platonic sense), is part of our very nature, and that Lie theory affords us a crucial tool for furthering our understanding of this relationship.

我的工作是在李理论，所谓的命名后，挪威数学家索菲斯李（发音李），谁，在上个世纪后期，开始研究这一特定领域的数学。李理论的核心是对称性的概念，包括离散和连续。大自然为我们提供了这些对称性的充分证据。雪花、分子和晶体都是具有有限对称性的物体的好例子。它们的存在不仅是纯粹的数学兴趣，而且深刻地反映了物体本身的物理特性（例如分子如何结合）。连续对称性，这是在李的思想中心，一直发挥了重要作用，到洞察力和现代物理学的发展。在这里，李理论作为自然定律中假设的对称性的标志而出现;例如经典力学中的空间旋转和平移，量子力学中的海森堡群，狭义相对论中的洛伦兹群。在这些例子中，有一条重要的共同哲学线索：一旦被给定的对称性所限制，自然定律往往会尽可能简单。在物理学中，对这一原理的信仰是如此普遍，以至于目前的一些统一理论（如超弦）的定律实际上是这样获得的。在硬科学之外，我们也被对称性所包围，正如埃舍尔的绘画、巴赫的赋格曲和达芬奇的研究清楚地表明的那样。因此，说对称性（在柏拉图的意义上）是我们本性的一部分，说李学理论为我们提供了一个重要工具来进一步理解这种关系，这并不是一种夸大。