
Journal of Lie Theory 30 (2020), No. 2, 513564 Copyright Heldermann Verlag 2020 A Survey on Invariant Cones inInfinite Dimensional Lie Algebras KarlHermann Neeb Department Mathematik, FriedrichAlexanderUniversität ErlangenNürnberg, 91058 Erlangen, Germany neeb@math.fau.de For the Lie algebra g of a connected infinitedimensional Lie group G, there is a natural duality between socalled semiequicontinuous weak*closed convex Ad*(G)invariant subsets of the dual space g' and Ad(G)invariant lower semicontinuous positively homogeneous convex functions on open convex cones in g. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of infinite Lie algebras, such as completions of locally finite Lie algebras, KacMoody algebras and twisted loop algebras with infinitedimensional range spaces. We also formulate various open problems. Keywords: Infinitedimensional Lie group, infinitedimensional Lie algebra, invariant cone, fixed points, double extensions, twisted loop algebras. MSC: 22E65, 22E45. [ Fulltextpdf (346 KB)] for subscribers only. 