Extended affine Lie algebras, groups and representation theory

自19世纪以来，群的数学概念一直是数学家用来描述在许多不同的化身中出现的对称性的最基本的抽象结构之一，例如在科学和工程中。数学家们为群发展了一套复杂的理论，有着越来越深刻的应用。数学内部和外部的群体应用一直是一个引人注目的成功故事。最重要的群类型起源于挪威数学家索菲斯·李(1842--1899)的基础工作。这些所谓的李群和它们的代数类似物代数群的区别在于，人们可以将另一种数学结构--李代数--与它们联系起来。粗略地说，李代数是相应群的一阶近似。我的研究涉及某些类型的李代数和一些相关的结构。 20世纪上半叶，Cartan、Weyl、Jacobson和Chvalley创立了关于有限维半单李代数的宏伟理论。他们的理论后来被Kac和Moody推广到某些无限维李代数。特别是，所谓的仿射Kac-Moody代数一直是李代数研究的中心课题，有着许多深刻的应用。 受量子规范理论和奇点理论应用的启发，仿射Kac-Moody代数已被推广到所谓的扩展仿射李代数。这类新的李代数提供了令人兴奋的新可能性。它为我们提供了比以前所知的更多的结构和因此可能的应用。本研究的目的是发展扩展仿射李代数和其他相关李代数的结构和表示理论。我还将研究与这些李代数和另一种代数结构有关的群，即所谓的Jordan代数。

Since the 19th century the mathematical concept of a group has been one of the most basic abstract structures used by mathematicians to describe symmetry arising in many different incarnations, e.g. in science and engineering. Mathematicians have developed a sophisticated theory for groups with more and more profound applications. The applications of groups within and outside of mathematics have been a remarkable success story. The most important type of groups originates from the fundamental work of the Norwegian mathematician Sophus Lie (1842--1899). These so-called Lie groups and their algebraic analogues, the algebraic groups, are distinguished by the fact that one can associate another mathematical structure, a Lie algebra, to them. Roughly speaking, Lie algebras are first order approximations of the corresponding groups. My research is concerned with certain types of Lie algebras and some related structures. In the first part of the 20th century, Cartan, Weyl, Jacobson and Chevalley have created a magnificent theory of finite-dimensional semisimple Lie algebras. Their theory was later extended to certain infinite-dimensional Lie algebras by Kac and Moody. In particular, the so-called affine Kac-Moody algebras have been a central topic of research in Lie algebras with many profound applications. Motivated by applications in quantum gauge theory and singularity theory, affine Kac-Moody algebras have been generalized to so-called extended affine Lie algebras. This new class of Lie algebras offers exciting new possibilities. It provides us with many more structures and hence possible applications than what was previously known. The goal of my research is to develop the structure and representation theory of extended affine Lie algebras and some other related Lie algebras. I will also study the groups associated to these Lie algebras and another algebraic structure, so-called Jordan algebras.