Geodesic
A Survey on Invariant Cones inInfinite Dimensional Lie Algebras
Karl-Hermann Neeb1
1Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
Journal of Lie Theory, Tome 30 (2020) no. 2, pp. 513-564

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 22E65, 22E45
Mots-clés : Infinite-dimensional Lie group, infinite-dimensional Lie algebra, invariant cone, fixed points, double extensions, twisted loop algebras

Karl-Hermann Neeb  1

1 Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany 
Karl-Hermann Neeb. A Survey on Invariant Cones inInfinite Dimensional Lie Algebras. Journal of Lie Theory, Tome 30 (2020) no. 2, pp. 513-564. http://geodesic.mathdoc.fr/item/JOLT_2020_30_2_a12/
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     author = {Karl-Hermann Neeb},
     title = {A {Survey} on {Invariant} {Cones} {inInfinite} {Dimensional} {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {513--564},
     year = {2020},
     volume = {30},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2020_30_2_a12/}
}
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