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Invariant convex sets and functions in Lie algebras

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Abstract

We say that an invariant convex coneW in a Lie algebras\(\mathfrak{g}\) is elliptic if its interior consists of elliptic elements of\(\mathfrak{g}\). If such a cone exists, then\(\mathfrak{g}\) has a compactly embedded Cartan subalgebra. The first main result, of this paper is a characterization of those Lie algebras, which contain elliptic invariant cones. If\(D \subseteq W\) is an invariant domain in such a cone, then we characterize the invariant locally convex functions onD by their restrictions to\(D \cap \mathfrak{t}\) where\(\mathfrak{t}\) is a compactly embedded Cartan subalgebra.

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References

  1. Azad, H., and J. J. Loeb,Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N. S.3(4) (1992), 365–375.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourbaki, N.,Groupes et algèbres de Lie, Chapitres 7 et 8, Masson, Paris, 1990.

    Google Scholar 

  3. Helgason, S., “Groups and geometric analysis,” Acad. Press, London, 1984.

    MATH  Google Scholar 

  4. Hilgert, J., K. H., Hofmann, and J. D. Lawson, “Lie Groups, Convex Cones, and Semigroups,” Oxford University Press, 1989.

  5. Hilgert, J., and K.-H. Neeb, “Lie-Gruppen und Lie-Algebren,” Vieweg Verlag, Wiesbaden, 1991.

    MATH  Google Scholar 

  6. Hilgert, J., “Lie semigroups and their applications,” Lecture Notes in Math.1552, Springer, 1993.

  7. Hilgert, J., K.-H. Neeb, and W. Plank,Symplectic convexity theorems and coadjoint orbits, Comp. Math.94 (1994), 129–180.

    MATH  MathSciNet  Google Scholar 

  8. Humphreys, J. E., “Reflection Groups and Coxeter Groups,” Cambridge studies in advanced mathematics29, Cambridge University Press, 1992.

  9. Neeb, K.-H.,Invariant subsemigroups of Lie groups, Memoirs of the Amer. Math. Soc.499, 1993.

  10. —,Holomorphic representation theory II, Acta math.173:1 (1994), 103–133.

    Article  MATH  MathSciNet  Google Scholar 

  11. Neeb, K.-H.,Convexity properties of the coadjoint action of non-compact Lie groups, submitted.

  12. —,On closedness and simple connectedness of adjoint and coadjoint orbits, Manuscripta Math.82 (1994), 51–65.

    MATH  MathSciNet  Google Scholar 

  13. —,A convexity theorem for semisimple symmetric spaces, Pacific Journal of Math.162:2 (1994), 305–349.

    MATH  MathSciNet  Google Scholar 

  14. Neeb, K.-H.,On the complex and convex geometry of Ol'shanskiî semigroups, submitted.

  15. Neeb, K.-H., “Holomorphy and Convexity in Lie Theory,” book in preparation.

  16. Rockafellar, R. T., “Convex analysis,” Princeton, New Jersey, Princeton University Press 1970.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054, Erlangen, Germany

    Karl-Hermann Neeb

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  1. Karl-Hermann Neeb
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Communicated by Karl H. Hofmann

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Neeb, KH. Invariant convex sets and functions in Lie algebras. Semigroup Forum 53, 230–261 (1996). https://doi.org/10.1007/BF02574139

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  • DOI: https://doi.org/10.1007/BF02574139

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