Applications of Galois cohomology to infinite dimensional Lie theory

我的工作是《谎言理论》，以挪威数学家索菲斯·李(发音为Lee)命名，他在上个世纪末开始了对这一特定数学领域的研究。李氏理论的核心是对称性，既有离散的，也有连续的。大自然为我们提供了充分的证据来证明这些对称性。雪花、分子和晶体是具有有限对称性的物体的很好的例子。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征(例如，分子如何结合)。连续对称性是李思想的核心，它在现代物理学的洞察和发展中一直发挥着重要作用。在这里，谎言理论作为自然规律中假设的对称性的标志而出现；例如，经典力学在空间中的旋转和平移，量子力学的海森堡群，以及狭义相对论中的洛伦兹群。有一条重要的共同哲学线索贯穿于这些例子中：一旦自然规律被一组给定的对称性所限定，它们往往会尽可能简单。就物理学而言，对这一原理的信仰是如此普遍，以至于当前的一些统一理论(如超弦)的定律实际上是通过这种方式获得的。正如埃舍尔的画作、巴赫的赋格和达芬奇的研究清楚地表明的那样，在硬科学之外，我们也被对称性所包围。因此，说对称性(在柏拉图意义上)是我们本性的一部分，并不是一种夸张，而谎言理论为我们进一步理解这种关系提供了一个至关重要的工具。

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.