Invariant convexity in infinite dimensional Lie algebras

无限维李群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对无限维李代数的凸性性质的系统理解。更准确地说，我们的目标是在伴随作用下不变的无限维李代数中的开凸锥的分类。在李代数的对偶中，我们想要确定那些半等连续的不变凸子集，这意味着它们的支持泛函在某一点的邻域内有界。本课题的一个重点是理解伴随轨道和伴随轨道对子代数的投影的闭合凸包；这类结果称为凸性定理。经典的凸性定理主要涉及到阿贝尔子代数上的轨道投影，这些子代数通常是Weyl群轨道的凸壳。Schur-Horn、Kostant、Atiyah-Pressley和Kac-Peterson的凸性定理都属于这一类。我们的目标是将这些结果系统地推广到更大的李代数类和更一般的子代数上的投影。该项目在很大程度上是由其在酉表示中的应用所激发的，其中关于开不变锥的知识对于从派生表示中确定算子的谱界至关重要。由以下有界的算子表示的所有元素的集合是一个不变凸锥。它有内点意味着这个表示是半有界的。半有界性是正能量条件的一个稳定版本，它是量子力学中许多表征的特征。在这种情况下，我们计划研究的典型李代数是有限维李代数及其补全的直接极限，埃尔米特李代数（对应于对称希尔伯特域的自同构群）和希尔伯特-李代数（紧李代数的密切无限维亲戚）和具有无限维目标群的扭曲环代数的所谓双重扩展。后者导致仿射Kac-Moody李代数的无限阶推广。本课题的重点是将无限维李代数的结构性质与泛函解析和几何方法相结合，得到不变凸锥和半等连续共轨的具体描述。

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.