Algebras and Representation Theory

, Volume 15, Issue 6, pp 1163–1203 | Cite as

A New Invariant of Quadratic Lie Algebras

  • Minh Thanh Duong
  • Georges Pinczon
  • Rosane Ushirobira
Article

Abstract

We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in \(\mathfrak{o}(n)\).

Keywords

Quadratic Lie algebras Invariants Double extensions Adjoint orbits Solvable Lie algebras 

Mathematics Subject Classifications (2010)

17B05 17B20 17B30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bajo, I., Benayadi, S.: Lie algebras admitting a unique quadratic structure. Commun. Algebra 25(9), 2795–2805 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bajo, I., Benayadi, S.: Lie algebras with quadratic dimension equal to 2. J. Pure Appl. Algebra 209(3), 725–737 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Benayadi, S.: Socle and some invariants of quadratic Lie superalgebras. J. Algebra 261(2), 245–291 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bourbaki, N.: Eléments de Mathématiques. Algèbre, Algèbre Multilinéaire, vol. Fasc. VII, Livre II. Hermann, Paris (1958)Google Scholar
  5. 5.
    Bourbaki, N.: Eléments de Mathématiques. Algèbre, Formes sesquilinéaires et formes quadratiques, vol. Fasc. XXIV, Livre II. Hermann, Paris (1959)Google Scholar
  6. 6.
    Bourbaki, N.: Eléments de Mathématiques. Groupes et Algèbres de Lie, Chapitre I, Algèbres de Lie. Hermann, Paris (1971)Google Scholar
  7. 7.
    Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie. Algebras, p. 186. Van Nostrand Reihnhold Mathematics Series, New York (1993)Google Scholar
  8. 8.
    Dixmier, J.: Algèbres Enveloppantes, p. 349. Cahiers scientifiques, fasc.37, Gauthier-Villars, Paris (1974)Google Scholar
  9. 9.
    Favre, G., Santharoubane, L.J.: Symmetric, invariant, non-degenerate bilinear form on a Lie algebra. J. Algebra 105, 451–464 (1987)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kac, V.: Infinite-Dimensional Lie Algebras, xvii + 280 pp. Cambridge University Press, New York (1985)Google Scholar
  11. 11.
    Magnin, L.: Determination of 7-dimensional indecomposable Lie algebras by adjoining a derivation to 6-dimensional Lie algebras. Algebr. Represent. Theory 13, 723–753 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Medina, A., Revoy, Ph.: Algèbres de Lie et produit scalaire invariant. Ann. Sci. École Norm. Sup. 4, 553–561 (1985)MathSciNetGoogle Scholar
  13. 13.
    Ooms, A.: Computing invariants and semi-invariants by means of Frobenius Lie algebras. J. Algebra 4, 1293–1312 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pinczon, G., Ushirobira, R.: New applications of graded Lie algebras to Lie algebras, generalized Lie algebras, and cohomology. J. Lie Theory 17(3), 633–668 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Samelson, H.: Notes on Lie Algebras. Universitext. Springer (1980)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Minh Thanh Duong
    • 1
  • Georges Pinczon
    • 1
  • Rosane Ushirobira
    • 1
  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijon CedexFrance

Personalised recommendations