Invariant convex cone

In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under the inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.

For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to gave have a Hermitian structure, i.e. the maximal compact subgroup has a center isomorphic to the circle group. The invariant convex cone generated by a generator of the center is closed and is the minimal invariant convex cone. The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely dtermined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersections are invariant under the Weyl group of the maximal torus and the orbit of every point in the interior of the cone intersects the interior of the Weyl group invariant cone.

For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones.

Invariant convex cones arise in the analysis of the holomorphic semigroups introduced by Olshanskii. These are naturally associated with Hermitian symmetric spaces and the associated holomorphic discrete series.

References

• Hilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D. (1989), Lie groups, convex cones, and semigroups, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853569-4
• Hilgert, Joachim; Neeb, Karl-Hermann (1993), Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552, Springer-Verlag, ISBN 3540569545
• Kumaresan, S.; Ranjan, A. (1982), "On invariant convex cones in simple Lie algebras", Proc. Indian Acad. Sci. Math. Sci., 91: 167–182
• Paneitz, Stephen M. (1981), "Invariant convex cones and causality in semisimple Lie algebras and groups", J. Funct. Anal., 313–359
• Paneitz, Stephen M. (1983), "Determination of invariant convex cones in simple Lie algebras", Ark. Mat., 21: 217–228
• Olshanskii, G. I. (1981), "Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series", Funct. Anal. Appl., 15: 275–285
• Vinberg, E. B. (1980), "Invariant convex cones and orderings in Lie groups", Funct. Anal. Appl., 14: 1–10