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Invariant convexity in infinite dimensional Lie algebras

无限维李代数中的不变凸性

基本信息

批准号：
320351428
负责人：
Professor Dr. Karl-Hermann Neeb
金额：
--
依托单位：
Department of Mathematics and Statistics
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/320351428?language=en
关键词：
Invariant convexity infinite dimensional Lie

项目摘要

Infinite dimensional Lie groups 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 systematic understanding of convexity properties of infinite dimensional Lie algebras. More precisely, we are aiming at a classification of open convex cones in an infinite dimensional Lie algebra that are invariant under the adjoint action. In the dual of the Lie algebra we would like to determine those invariant convex subsets which are semi-equicontinuous, which means that their support functional is bounded in the neighborhood of some point. A key point of this project is to understand closed convex hulls of projections of adjoint and coadjoint orbits to subalgebras; results of this type are called convexity theorems. Classically convexity theorems mostly concern orbit projections onto abelian subalgebras, where they are often convex hulls of Weyl group orbits. The convexity theorems of Schur-Horn, Kostant, Atiyah-Pressley and Kac-Peterson are of this type. We are aiming at a systematic extension of these results to larger classes of Lie algebras and to projections onto more general subalgebras. This project is motivated to a large extent by its applications to unitary representations, where knowledge on open invariant cones is crucial to determine spectral bounds of operators from the derived representation. The set of all elements represented by operators bounded from below is an invariant convex cone. That it has interior points means that the representation is semibounded. Semiboundedness is a stable version of the positive energy condition which characterizes many representations arising in quantum mechanics. Typical Lie algebras we plan to study in this context are direct limits of finite dimensional Lie algebras and their completions, hermitian Lie algebras (corresponding to automorphism groups of symmetric Hilbert domains) and so-called double extensions of Hilbert-Lie algebras (close infinite dimensional relatives of compact Lie algebras) and of twisted loop algebras with infinite dimensional target groups. The latter lead to infinite rank generalizations of affine Kac-Moody Lie algebras. The focus of the present project lies on combining structural properties on infinite dimensional Lie algebras with functional analytic and geometric methods to obtain a concrete description of invariant convex cones and semi-equicontinuous coadjoint orbits.

无限维李群出现在数学和其他科学的所有领域，只要对称性依赖于无穷多个参数。这个项目的目标是发展一个系统的了解凸性质的无限维李代数。更准确地说，我们的目标是在一个无限维李代数的开凸锥的分类下，伴随作用下不变。在李代数的对偶中，我们希望确定那些半等度连续的不变凸子集，这意味着它们的支撑泛函在某个点的邻域内是有界的。这个项目的一个关键点是理解伴随轨道和余伴随轨道到子代数的投影的闭凸包;这种类型的结果被称为凸性定理。经典的凸性定理主要涉及到阿贝尔子代数上的轨道投影，它们通常是Weyl群轨道的凸包。Schur-Horn、Kostant、Atiyah-Pressley和Kac-Peterson的凸性定理就是这种类型。我们的目标是系统地扩展这些结果更大的类李代数和投影到更一般的子代数。这个项目的动机在很大程度上是由它的应用程序酉表示，其中知识的开放不变锥是至关重要的，以确定从派生的表示算子的谱界。由下有界算子表示的所有元素的集合是不变凸锥。它有内点意味着表示是半有界的。半有界性是正能量条件的稳定版本，它表征了量子力学中出现的许多表示。在这方面，我们计划研究的典型李代数是有限维李代数及其完备化的直接极限、厄米特李代数（对应于对称希尔伯特域的自同构群）和希尔伯特-李代数的所谓双扩展（紧李代数的密切无限维亲属）以及具有无限维目标群的扭环代数。后者导致仿射Kac-Moody李代数的无限秩推广。本项目的重点在于将无穷维李代数的结构性质与泛函分析和几何方法相结合，以获得不变凸锥和半等度连续余伴随轨道的具体描述。