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Journal of Lie Theory 30 (2020), No. 2, 513--564
Copyright Heldermann Verlag 2020

A Survey on Invariant Cones inInfinite Dimensional Lie Algebras

Karl-Hermann Neeb
Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany

For the Lie algebra g of a connected infinite-dimensional Lie group G, there is a natural duality between so-called semi-equicontinuous 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, Kac-Moody algebras and twisted loop algebras with infinite-dimensional range spaces. We also formulate various open problems.

Keywords: Infinite-dimensional Lie group, infinite-dimensional Lie algebra, invariant cone, fixed points, double extensions, twisted loop algebras.

MSC: 22E65, 22E45.

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