Monatshefte für Mathematik

, Volume 100, Issue 3, pp 183–210 | Cite as

Lorentzian cones in real Lie algebras

  • Joachim Hilgert
  • Karl H. Hofmann
Article

Abstract

A Lorentzian coneW in a finite dimensional real Lie algebraL is the convex closed cone bounded by one half of the zero-set of a Lorentzian formq onL with the additional property, that for all sufficiently smallx, yW the Campbell-Hausdorff productx*y=x+y+1/2[x,y]+..., is also inW. We characterize Lorentzian cones completely; in particular, with the exception of one class of “almost abelian” solvable algebras, the Lorentzian formq is invariant, i.e., satisfiesq([x, y], z)=q (x,[y, z]).

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References

  1. [Gr 83]Graham, G.: Differentiable semigroups. Lect. Notes Math.998, 57–127. Berlin-Heidelberg-New York: Springer. 1983.Google Scholar
  2. [Gu 77]Guts, A. K.: Invariant orders of three-dimensional Lie groups. Siber. math. J.17, 731–736 (1977).Google Scholar
  3. [GL 84]Guts, A. K., Levichev, A. V. K osnovanyam teorii otnositel'nosti. Dokladii AN USSR277, 1299–1302 (1984). (On the foundations of the theory of relativity).Google Scholar
  4. [HH 84a]Hilgert, J., Hofmann, K. H.: Old and new on S1 (2). Manuscripta mathematica. To appear.Google Scholar
  5. [HH 84b]Hilgert, J., Hofmann, K. H.: On Sophus Lie's Fundamental theorem. J. Funct. Analysis. To appear.Google Scholar
  6. [HH 84c]Hilgert, J., Hofmann, K. H.: The invariance of cones and wedges under flows, Preprint No. 796. TH Darmstadt (1983). Submitted.Google Scholar
  7. [HH 85a]Hilgert, J., Hofmánn, K. H.: Semigroups in Lie groups, Lie semialgebras in Lie algebras. Trans. Amer. Math. Soc.288, 481–504 (1985).Google Scholar
  8. [HH 85b]Hilgert, J., Hofmann, K. H.: Lie Semialgebras are real phenomena. Math. Ann.270, 97–103 (1985).Google Scholar
  9. [HL 81]Hofmann, K. H., Lawson, J. D.: The local theory of semigroups in nilpotent Lie groups. Semigroup Forum23, 343–357 (1981).Google Scholar
  10. [HK 85]Hofmann, K. H., Keith, V. S.: Invariant quadratic forms on finite dimensional Lie algebras, Bull. Austr. Math. Soc. To appear.Google Scholar
  11. [HL 83a]Hofmann, K. H., Lawson, J. D.: Foundations of Lie semigroups. Lect. Notes Math.998, 128–201. Berlin-Heidelberg-New York: Springer. 1983.Google Scholar
  12. [HL 83b]Hofmann, K. H., Lawson, J. D.: Divisible subsemigroups of Lie groups. J. London Math. Soc. (2),27, 427–434 (1983).Google Scholar
  13. [Ke 84]Keith, V.: On bilinear forms on finite dimensional Lie algebras, Dissertation. New Orleans: Tulane University. 1984.Google Scholar
  14. [Le 84]Levichev, A.: On elliptic semialgebras. Letters April 29, 1984 and June 23, 1984.Google Scholar
  15. [Le 84b]Levichev, A. V.: Some applications of Lie semigroups theory in relativity. Workshop on the Lie Theory of semigroups. Darmstadt: Technological Institute (THD). 1984.Google Scholar
  16. [M 82]Medina, A.: Groupes de Lie munis de pseudométriques de Riemann biinvariantes, Séminaire de Géométrie Diff., 1981–82, Exp. No 6, 41 pp., Montpellier.Google Scholar
  17. [MR 83]Medina, A., Revoy, P.: Sur une géométrie Lorentzienne du groupe oscillateur, Séminaire de Géométrie Diff., 1982–83, Montpellier.Google Scholar
  18. [MR 84a]Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant, Séminaire de Géométrie Diff., 1983–84, 63–93, Montpellier.Google Scholar
  19. [MR 84b]Medina, A., Revoy, P.: Caractérisation des groupes de Lie ayant une pseudométrique bi-invariante. Applications. Travaux en cours, Séminaire Sud-Rhodanien de Géométrie III. Paris: Hermann. 1984.Google Scholar
  20. [Ol 81]Ol'shanksii, G. I.: Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series. Functional Anal. Appl.15, 275–285 (1981).Google Scholar
  21. [Ol 82]Ol'shanskii, G. I.: Convex cones in symmetric Lie algebras, Lie semigroups and invariant causal (order) structures on pseudo Riemannian symmetric spaces. Dokl. Soviet. Math.26, 97–101 (1982).Google Scholar
  22. [Pa 81]Paneitz, S. M.: Invariant convex cones and causality in semisimple Lie algebras and groups, J. Funct. Anal.43, 1313–359 (1981).Google Scholar
  23. [Pa 83]Paneitz, S. M.: Classification of invariant convex cones in simple Lie algebras. Arkiv f. Mat.21, 217–228 (1984).Google Scholar
  24. [Ro 80]Rothkrantz, L. J. M.: Transformatiehalfgroepen van niet-compacte hermitische symmetrische Ruimten. Dissertation. University of Amsterdam. 1980.Google Scholar
  25. [St 35]Straszewicz, S.: Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math.24, 139–143 (1935).Google Scholar
  26. [Vi 80]Vinberg, E. B.: Invariant cones and orderings in Lie groups. Functional Anal. Appl.14, 1–13 (1980).Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Joachim Hilgert
    • 1
  • Karl H. Hofmann
    • 1
  1. 1.Fachbereich MathematikTHDDarmstadtGermany

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