Invariant convexity in infinite dimensional Lie algebras

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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.