Lie and Jordan algebras, and related groups

对称性在自然界中无处不在，也是我们所有人享受的源泉。但我们不仅在自然界中找到了对称性。它在我们的生活中无处不在。人们只需考虑音乐(巴赫)或绘画(埃舍尔)。更重要的是，对称性是我们理解物理世界的基础。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化身中出现的对称性的最基本的抽象结构之一。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19世纪挪威数学家索菲斯·李的基本工作。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。本质上，李代数是相应群的一阶近似。我的研究涉及李代数、代数群以及它们之间的相互作用。*任何李代数理论的主要目的都是了解它们的内部结构及其表示。这些也是我计划研究的李代数的长期目标：扩展仿射李代数、根分次李代数和等变映射李代数。这些李代数对于当今的李代数理论是重要的。*例如，扩展仿射李代数推广了有限维半单李代数和仿射Kac-Moody李代数。推广的仿射李代数的这两个例子的一个基本结果是它们的Cartan子代数的共轭(Chvalley和Peterson-Kac的著名结果)。我的研究目的是将共轭推广到所有的扩展仿射李代数。这将对他们的结构理论产生重要的影响。*在群领域，我将研究两种类型的群，斯坦伯格群和例外群。对前者的研究有许多不同的类型。我和奥特马尔·卢斯共同开发的斯坦伯格群理论将统一所有这些理论，从而允许理论的实质性简化。它通过系统地利用根系和Jordan对引入了新的方法。*例外群是所有代数群中最吸引人的，但也是最复杂的。它们在数学和物理学的许多领域都有应用。到目前为止，对它们的研究大多是在基场上进行的。我的目标是使用非结合代数、群方案和下降理论的高级方法的组合来描述环上的例外群。从基场到基环的延伸将给这些群的构造带来新的曙光。

Symmetry is omnipresent in nature and the source of much enjoyment for all of us. But it is not only in nature that we find symmetry. It is everywhere in our lives. One just has to think about music (Bach) or paintings (Escher). Even more, symmetry is basic for our understanding of the physical world.******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. 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 19th-century Norwegian mathematician Sophus Lie. 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. In essence, Lie algebras are first order approximations of the corresponding groups. My research is concerned with Lie algebras, algebraic groups and the interplay between them. ***The main goals in any theory of Lie algebras are to understand their internal structure and their representations. These are also my long-term goals for the Lie algebras I plan to study: extended affine Lie algebras, root-graded Lie algebras and equivariant map Lie algebras. These Lie algebras are important for present-day Lie algebra theory. ******For example, extended affine Lie algebras generalize finite-dimensional semisimple and affine Kac-Moody Lie algebras. A fundamental result for these two examples of extended affine Lie algebras is conjugacy of their Cartan subalgebras (celebrated results of Chevalley and Peterson-Kac). My research aims to extend conjugacy to all extended affine Lie algebras. This will have important consequences for their structure theory.******In the area of groups, I will be working on two types of groups, Steinberg groups and exceptional groups. The former have been studied in many different types. The theory of Steinberg groups that I have developed jointly with Ottmar Loos will unify all of them, thus allowing substantial simplifications of the theory. It introduces new methods by systematically employing root systems and Jordan pairs.******Exceptional groups are the most fascinating, but also most complicated of all algebraic groups. They have found applications in many areas of mathematics and in physics. So far, they have mostly been studied over base fields. My goal is to describe exceptional groups over rings, using a combination of advanced methods from nonassociative algebras, group schemes and descent theory. The extension from base fields to base rings will shed new light on the construction of these groups.********