Infinite-dimensional Lie algebras and their associated groups

纯数学研究的主要目的是研究结构。由于这种抽象的方法，数学结果可以应用于不同的领域，比如金融、物理或人类基因组。我感兴趣的结构是代数的一部分。它源于对方程解的研究。这些解通常具有一定的对称性，从而形成数学家称之为群的结构。它满足一些很好的内在规律，就像日常生活中的一群朋友。我所研究的群体不仅有无限的成员，而且距离很近的成员看起来也差不多。19世纪晚期，挪威数学家索菲斯·李（发音为李）发现，人们可以用一种更好的结构来研究这些群的许多性质，这种结构是一种近似的近似，现在被称为李代数。李代数的研究一直是20世纪数学的中心。李代数在数学内外都有许多应用，例如在粒子物理学中。上世纪上半叶的大多数研究都是关于有限维李代数的，即那些只允许有限数量的自由的李代数，近几十年来，兴趣已经转移到无限维李代数。提出的研究致力于一类特殊的无限维李代数，即物理学家发明的所谓扩展仿射李代数。我将确定这些李代数的精确结构。该建议的第二部分处理可以与这些李代数相关联的群。重点是运用第一部分中获得的知识来组织小组。

The principal goal of research in pure mathematics is to study structures. Because of the abstract approach the mathematical results can then be aplied in different areas, like finance or physics or the human genome. The structure I am interested in is part of algebra. It arises from the study of solutions of equations. These solutions often have some symmetry, leading to a structure that mathematicians call a group. It satisfies some nice internal laws, like a group of friends in ordinary life. The groups I am studying not only have infinitely members, but also members which are close-by look more or less the same. In the late 19th century the Norwegian mathematician Sophus Lie (pronouced Lee) discovered that one can study lots of properties of these groups with the means of an even nicer structure, a sort of approximationby approximation, which nowadays is called a Lie algebra. The study of Lie algebras has been at the centre of 20th century mathematics. Lie algebras have found many applications within and outside of mathematics, for example in particle physics. While most of the studies in the first part of the last century were concerned with finite-dimensional Lie algebras, i.e., those allowing only a finite number of freedoms, in recent decades the interest has shifted to infinite-dimensional Lie algebras. The proposed research is devoted to a special class of infinite-dimensional Lie algebras, the so-called extended affine Lie algebras which have been invented by physicists. I will determine the precise structure of these Lie algebras. The second part of the proposal deals with groups one can associate to these Lie algebras. The main point will be in constructing the groups, using the knowledge obtained in the first part.