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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

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