Applications of Galois cohomology to infinite dimensional Lie theory

我的工作是李论，李论是以挪威数学家索夫斯·李（发音为李）的名字命名的，他在上世纪末开始研究这一特殊的数学领域。李氏理论的核心思想是对称，离散的和连续的。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数量对称性的物体的好例子。它们的存在不仅具有纯粹的数学意义，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）。连续的对称性，作为李博士思想的核心，一直在现代物理学的洞察力和发展中扮演着重要的角色。在这里，李论是作为自然法则中假定的对称性的标志而出现的；例如经典力学中的旋转和平移，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：一旦自然法则被一组给定的对称性所限制，它们就会变得尽可能简单。在物理学中，对这一原理的信仰是如此普遍，以至于一些目前的统一理论（如超弦）的定律实际上就是以这种方式获得的。在硬科学之外，我们也被对称所包围，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，毫不夸张地说，对称（在柏拉图的意义上）是我们本性的一部分，而李氏理论为我们进一步理解这种关系提供了一个重要的工具。

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.