Semibounded unitary representations of infinite dimensional Lie groups

无限维李群的半有界酉表示

• 批准号: 122817625
• 负责人: Professor Dr. Karl-Hermann Neeb
• 金额: --
• 依托单位: Lehrstuhl für Mathematik (Prof. Dr. Karl-Hermann Neeb)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/122817625?language=en
• 关键词: Semibounded; unitary; representations; infinite; dimensional

Infinite dimensional Lie groups and their representations show up in all areas of mathematics and other sciences, wherever symmetries depending on infinitely many parameters arise. The goal of this project is to develop a geometric approach to the important class of semibounded unitary representations of infinite dimensional Lie groups. Typical groups arising in this context are double extensions of Hilbert Lie groups, which include oscillator groups in the abelian case, afine Kac Moody groups based on loop groups with infinite dimensional targets and a large number of groups whose Lie algebras are Z-graded. Semiboundedness of a unitary representation is a stable version of the „positive energy" condition which characterizes many representations arising in mathematical physics, resp., field theories. For a unitary representation of a Lie group it means that the selfadjoint operators from the derived representation are uniformly bounded below on some open subset of the Lie algebra. Our goal is to understand the decomposition theory and the irreducible representations in this class.The focus of the present project lies on combining algebraic, geometric and analytic aspects of the theory, such as realizations in holomorphic bundles and convexity properties of momentum maps related to spectral properties of operators to obtain classification results.

无限维李群及其表示出现在数学和其他科学的所有领域，只要存在依赖无限多个参数的对称性。本计画的目标是发展一种几何方法来处理无限维李群的半有界酉表示的重要类别。在这种情况下产生的典型群是Hilbert Lie群的双重扩展，其中包括在阿贝尔情况下的振子群，基于无限维目标环群的优良Kac Moody群，以及大量李代数为z级的群。酉表示的半有界性是“正能量”条件的稳定版本，它表征了数学物理中出现的许多表示，如：，场论。对于李群的酉表示，它意味着该表示的自伴随算子在李代数的某个开子集上是一致有界的。我们的目标是在这门课上理解分解理论和不可约表示。本项目的重点在于结合理论的代数、几何和解析方面，如全纯束的实现和与算子谱性质相关的动量映射的凹凸性，从而获得分类结果。