Regularity of Lie Groups and Lie's Third Theorem

李群的正则性和李第三定理

• 批准号: 405865003
• 负责人: Dr. Maximilian Hanusch
• 金额: --
• 依托单位: Fachgebiet Unendlich-dimensionale Analysis und Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/405865003?language=en
• 关键词: Regularity; Lie; Groups; Third; Theorem

Besides their relevance in mathematics, infinite-dimensional Lie groups play an important role in physics, where they occur as phase spaces and symmetry groups. Although particular classes like diffeomorphism-, gauge-, and operator groups are well-understood, the general theory is still in the initial stage. This is essentially due to the absence of a general theory of ODE's in the (Hausdorff) locally convex case. Besides the question under which circumstances a given Lie algebra is enlargible (Lie's third Theorem), regularity properties of Lie groups play a central role in this context. Regularity is concerned with the domain-, continuity-, and smoothness properties of the product integral -- a notion that naturally generalizes the concept of the Riemann integral to Lie groups (Lie algebra-valued curves are thus integrated to Lie group elements). Due to the preparatory work done in prior to, and within the scope of my project, essentially the domain problem as well as parts of the continuity problem are left open in this context. A further issue to be investigated in this project is the question of higher differentiability (not smoothness) of the product integral in the semiregular context. Semiregularity means that all Lie algebra-valued curves of a fixed differentiability class are contained in the domain of the product integral (are integrable). One of my recent results states that the product integral is automatically differentiable in the semiregular case, and the guess is that this statement extends to higher orders of differentiability. Related to this is another aim of this project, namely to generalize Milnor's integrability result for Lie algebra homomorphisms (that originally motivated him to introduce the notion of regularity in the context of infinite-dimensional Lie groups) from the regular to the semiregular case. A further focus of my project is on the enlargibility problem for Lie algebras (Lie's third theorem) in the infinite-dimensional asymptotic estimate and sequentially complete context. The enlargibility problem is concerned with the question under which circumstances a given infinite-dimensional Lie algebra arises as the Lie algebra of a (simply connected) Lie group.

除了在数学中的相关性之外，无限维李群在物理学中也发挥着重要作用，它们以相空间和对称群的形式出现。尽管像微分同胚、规范群和算子群这样的特殊类已经被很好地理解，但一般理论仍处于初始阶段。这本质上是由于在（豪斯多夫）局部凸情况下缺乏常微分方程的一般理论。除了给定李代数在何种情况下可放大的问题（李第三定理）之外，李群的正则性质在这种情况下也发挥着核心作用。正则性与乘积积分的域、连续性和平滑性特性有关——这一概念自然地将黎曼积分的概念推广到李群（李代数值曲线因此被积分到李群元素）。由于在我的项目之前和项目范围内完成的准备工作，本质上领域问题以及部分连续性问题在这种情况下仍然悬而未决。该项目要研究的另一个问题是半正则环境中乘积积分的更高可微性（而不是平滑度）的问题。半正则性意味着固定可微分类的所有李代数值曲线都包含在乘积积分的域中（可积）。我最近的结果之一指出，乘积积分在半正则情况下是自动可微分的，并且猜测该陈述扩展到更高阶的可微分性。与此相关的是该项目的另一个目标，即将米尔诺的李代数同态可积性结果（最初促使他在无限维李群的背景下引入正则性概念）从正则情况推广到半正则情况。 我的项目的另一个重点是无限维渐近估计和顺序完整上下文中李代数（李第三定理）的可扩展性问题。可扩展性问题涉及在什么情况下给定的无限维李代数作为（单连通）李群的李代数出现的问题。