Infinite dimensional lie algebras and their coordinate algebras

有限维简单李代数是由索夫斯·李一百多年前提出的，作为对称的度量，它在几何和物理中具有重要的基础意义。大约30年前，Victor Kac和Robert Moody分别介绍了有限维简单李代数的无限维类似物，现在称为仿射Kac-Moody代数。从那时起，一个丰富的仿射Kac-Moody李代数的数学理论得到了发展。此外，这些代数已在物理学中用于研究发生在模拟基本粒子相互作用的场论中的对称性。1990年，两位数学物理学家Høegh-Krohn和Torresani引入了一类新的李代数，称为扩展仿射李代数。这类新的李代数包括所有有限维简单李代数和所有仿射Kac-Moody李代数。在过去十年中，对这些代数结构的研究取得了显著的进展，但仍有许多有趣的问题有待研究。在计划的研究中，我将继续研究扩展仿射李代数的结构以及一些相关的代数类。我的方法之一将是研究这些李代数的实现。实现允许人们对代数进行明确的计算。一种类型的实现是矩阵的坐标来自一个称为坐标代数的代数。我的部分研究将用于研究这些坐标代数。

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.