Regularity properties of infinite-dimensional Lie groups, and exponential laws

无限维李群的正则性质和指数定律

• 批准号: 384439538
• 负责人: Professor Dr. Helge Glöckner
• 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: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/384439538?language=en
• 关键词: Regularity; properties; infinite; dimensional; Lie

Exponential laws enable functions with values in a function space to be interpreted as ordinary functions of two variables, and thus make them easier to handle. They are central tools in infinite-dimensional differential calculus and used, for example, to establish smoothness of the group operations for prominent examples of infinite-dimensional Lie groups. Frequently, exponential laws are also the key for the proof of regularity of such groups, i.e., the existence and smooth parameter-dependence of solutions to relevant differential equations on G. One goal of the project is to provide new exponential laws. The main goal is to develop further the theory of regular infinite-dimensional Lie groups, notably the theory of measurable regularity. Recent research showed that integral curves for left-invariant vector fields with (merely) measurable dependence on time are of particular usefulness; for example, the Trotter product formula and the commutator formula for one-parameter groups (which are otherwise difficult to prove) automatically hold in measurably regular Lie groups (in which existence and smooth parameter-dependence is available for the Lie group-valued evolutions to measurable Lie algebra-valued curves). Using suitable exponential laws or alternative strategies, measurable regularity shall be established for further important classes of infinite-dimensional Lie groups.

指数定律使函数空间中具有值的函数能够被解释为两个变量的普通函数，从而使它们更容易处理。它们是无限维微分微积分中的核心工具，例如，用于为无限维李群的突出例子建立群运算的平滑性。通常，指数定律也是证明此类群正则性的关键，即 G 上相关微分方程解的存在性和平滑参数依赖性。该项目的目标之一是提供新的指数定律。主要目标是进一步发展正则无限维李群理论，特别是可测正则性理论。最近的研究表明，（仅）可测量地依赖于时间的左不变向量场的积分曲线特别有用。例如，单参数群的 Trotter 乘积公式和交换子公式（否则很难证明）在可测正则李群中自动成立（其中存在性和平滑的参数依赖性可用于李群值演化可测量的李代数值曲线）。使用合适的指数定律或替代策略，应为无限维李群的进一步重要类别建立可测量的规律性。

项目成果

Differentiability along one-parameter subgroups compared to differentiability on Lie groups as manifolds

沿单参数子群的可微性与作为流形的李群的可微性相比

• DOI: 10.4064/bc113-0-17
• 发表时间: 2017
• 期刊: Banach Center Publications
• 作者: Nikitin