Semibounded unitary representations of infinite dimensional Lie groups

无限维李群和它们的表示出现在数学和其他科学的所有领域，在那里依赖于无穷多个参数的对称性出现。这个项目的目标是发展一个几何方法的重要类的半有界酉表示的无限维李群。在这种背景下出现的典型群是希尔伯特李群的双重扩展，包括阿贝尔情形下的振子群、基于无限维目标的循环群的仿射Kac Moody群以及大量其李代数是Z-分次的群。酉表示的半有界性是“正能量”条件的稳定版本，该条件表征了数学物理中出现的许多表示，场理论对于李群的酉表示，这意味着来自导出表示的自伴算子在李代数的某个开子集上一致有界。我们的目标是理解分解理论和这类不可约表示。本项目的重点在于结合理论的代数，几何和分析方面，如全纯丛中的实现和与算子谱性质相关的动量映射的凸性性质，以获得分类结果。

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.