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Invariant quadratic forms on finite dimensional lie algebras

Published online by Cambridge University Press:  17 April 2009

Karl H. Hofmann  and
Verena S. Keith
Karl H. Hofmann
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, Germany (FRG). and Department of Mathematics, Tulane University, New Orleans, La. 70118.
Verena S. Keith
Affiliation:
Center for Naval Analyses, P.O. Box 16268, Alexandria, VA 22302.
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Abstract

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Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 1 , February 1986 , pp. 21 - 36
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Bourbaki, N., Groupes et algèbres de Lie, Chap. 1, (Hermann, Paris, 1960).Google Scholar
[2]Hilgert, J. and Hofmann, K.H., “Lorentzian cones in Lie algebras”, Monatshefte für mathematik (to appear) (Preprint nr. 855 Oktober 1984, FB Mathematik, Technische Hochschule Darmstadt, 24 pp.)Google Scholar
[3]Guts, A.K. and Levichev, A.V., “On the foundations of relativity theory”, Soviet Math. Dokl. 30 (1984), 253–257.Google Scholar
[4]Keith, V.S., “On invariant bilinear forms on finits dimensional Lie algebras”, Dissertation, Tulane University, New orleans, 1984, 93 pp., University Microfilm International, P.O. Box 1764, An Arbor, Michigan 48106.Google Scholar
[5]Koszul, J.L., “Homologie et cohomologie des algèbres de Lie”, Bull. Soc. Math. France 78 (1950), 65–127.CrossRefGoogle Scholar
[6]Medina, A., “Groupes de Lie munis de pseudométriques de Riemann biinvariantes”, Séminaire de Géometrie Diff., Exp No. 6 M.R. 84c:53063 (Montpellier, 1981–1982).Google Scholar
[7]Medina, A. et Revoy, Ph., “Algèbre de Lie et produit scalaire invariant”, Séminaire de Géometrie Diff., (Montpellier, 1983–1984).Google Scholar
[8]Medina, A. et Revoy, Ph.. “Sur une géometrie Lorentzienne du groupe oszillateur”, Séminaire de Géometrie Diff., (Montpellier, 1982–1983).Google Scholar
[9]medina et Ph. Revoy, A., “Caractérisation des groupes de Lie ayant une pseudométrique biinvariante, Applications, Travaux en cours”, Séminaire Sud-Rhodanien de Géometrie III, (Hermann, paris 1984).Google Scholar
[10]Milnor, J., “Curvatures of left invariant metrics on Lie groups”, Advances in Math. 21 (1976), 283–329.CrossRefGoogle Scholar