Lie and Jordan algebras, and related groups

李代数和乔丹代数以及相关群

• 批准号: RGPIN-2016-04183
• 负责人: Neher, Erhard
• 金额: $ 1.31万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=660768
• 关键词: Lie; Jordan; algebras; related; groups

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.********

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发现对称性。它在我们的生活中无处不在。一个人只需要考虑音乐（巴赫）或绘画（巴赫）。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对称性的最基本的抽象结构之一。数学家们已经发展出了一种复杂的群理论，其应用越来越广泛。数学内外的群体应用是一个了不起的成功故事。** 最重要的一类群起源于19世纪挪威数学家Sophus Lie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。在本质上，李代数是相应群的一阶近似。我的研究涉及李代数，代数群和它们之间的相互作用。* 任何李代数理论的主要目标都是理解它们的内部结构和表示。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。这些李代数对当今的李代数理论很重要。* 例如，扩展仿射李代数推广了有限维半单和仿射Kac-Moody李代数。这两个例子的一个基本结果是它们的Cartan子代数的共轭性（Chevalley和Peterson-Kac的著名结果）。我的研究目标是将共轭性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影响。在群体领域，我将研究两种类型的群体，斯坦伯格群体和特殊群体。前者已经在许多不同类型中进行了研究。我与斯坦伯格卢什共同开发的斯坦伯格群理论将统一所有这些理论，从而使理论得到实质性简化。它通过系统地使用根系和Jordan对来引入新方法。例外群是所有代数群中最迷人的，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，它们大多是在基本场上研究的。我的目标是描述环上的特殊群，使用非结合代数，群方案和下降理论的高级方法的组合。从基域到基环的扩展将为这些群的构造提供新的线索。