Infinite dimensional lie algebras and their coordinate algebras

有限维单李代数，由Sophus Lie在一百多年前提出，作为对称性的度量，在几何和物理学中具有根本的重要性。 大约30年前，维克托卡茨和罗伯特穆迪独立地介绍了有限维单李代数的无限维类似，现在称为仿射卡茨-穆迪代数。 从那时起，仿射Kac-Moody李代数的丰富数学理论得到了发展。 此外，这些代数在物理学中被用来研究在模拟基本粒子相互作用的场论中出现的对称性。在1990年，一类新的李代数，称为扩展仿射李代数，由两位数学物理学家Høegh-Krohn和Torresani引入。 这个新的类包括所有有限维单李代数和所有仿射Kac-Moody李代数。 在过去的十年中，这些代数的结构研究取得了显著的进展，但仍有许多有趣的问题有待研究。 在本文的研究中，我将继续研究扩张仿射李代数的结构，沿着研究一些相关的代数类。我的方法之一将是研究这些李代数的实现。实现允许一个明确的计算涉及代数。一种实现方式是矩阵的坐标来自一个叫做坐标代数的代数. 我的部分研究将致力于这些坐标代数的研究。

Finite dimensional simple Lie algebras, introduced by Sophus Lie over a hundred years ago, are of fundamental importance in geometry and physics as a measure of symmetry.  About 30 years ago, Victor Kac and Robert Moody independently introduced some infinite dimensional analogs, now called affine Kac-Moody algebras, of finite dimensional simple Lie algebras.  A rich mathematical theory of affine Kac-Moody Lie algebras has been developed since that time.  Furthermore, these algebras have been used in physics to study the symmetries that occur in field theories that model interactions of elementary particles. In 1990 a new class of Lie algebras, called extended affine Lie algebras, was introduced by two mathematical physicists Høegh-Krohn and Torresani.  This new class includes all finite dimensional simple Lie algebras and all affine Kac-Moody Lie algebras.  Remarkable progress has been made in the last decade in the study of the structure of these algebras, but many interesting questions remain to be investigated.  In the proposed research, I will continue my study of the structure of extended affine Lie algebras along with some related classes of algebras. One of my approaches will be to investigate realizations of these Lie algebras. Realizations allow one to make explicit calculations involving the algebras. One type of realization is as matrices with coordinates from an algebra called a coordinate algebra.  Part of my investigation will be devoted to the study of these coordinate algebras.