Annals of Global Analysis and Geometry

, Volume 43, Issue 2, pp 107–121 | Cite as

Invariant four-forms and symmetric pairs

  • Andrei MoroianuEmail author
  • Uwe Semmelmann


We give criteria for real, complex, and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.


S-Representations Exceptional Lie algebras Irreducible representations Representation of Lie type 

Mathematics Subject Classification (2000)

Primary 22E46 20C35 15A66 17B25 53C35 57T15 


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  1. 1.
    Adams, J.: Lectures on Exceptional Lie Groups. Mahmud, Z., Mimira, M. (eds.) University of Chicago Press, Chicago (1996)Google Scholar
  2. 2.
    Baez J.: The Octonions. Bull. Amer. Math. Soc. 39(2), 145–205 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Besse, A.: Einstein Manifolds [Ergebnisse der Mathematik und ihrer Grenzgebiete (3)], vol. 10. Springer, Berlin (1987)Google Scholar
  4. 4.
    Dynkin E.B.: Maximal subgroups of the classical groups. Amer. Math. Soc. Transl. Ser. 2 6, 245–378 (1957)zbMATHGoogle Scholar
  5. 5.
    Eschenburg J.-H., Heintze E.: Polar representations and symmetric spaces. J. Reine Angew. Math. 507, 93–106 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Figueroa-O’Farrill J.: A geometric construction of the exceptional Lie algebras F4 and E8. Commun. Math. Phys. 283(3), 663–674 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Green M., Schwartz J., Witten E.: Superstring Theory, vol. 1. Cambridge University Press, Cambridge (1987)Google Scholar
  8. 8.
    Heintze E., Ziller W.: Isotropy irreducible spaces and s-representations. Differential Geom. Appl. 6(2), 181–188 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kostant B.: On invariant skew-tensors. Proc. Natl. Acad. Sci. USA 42, 148–151 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kostant B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100(3), 447–501 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lawson B., Michelson M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  12. 12.
    Landsberg J.M., Manivel L.: Construction and classification of complex simple Lie algebras via projective geometry. Selecta Math. (N.S.) 8(1), 137–159 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
  14. 14.
    Moroianu, A., Pilca, M.: Higher rank homogeneous Clifford structures. arXiv:1110.4260Google Scholar
  15. 15.
    Moroianu A., Semmelmann U.: Clifford structure on Riemannian manifolds. Adv. Math. 228(2), 940–967 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Wang, M., Ziller, W.: On the isotropy representations of a symmetric space. Rend. Sem. Mat. Univ. Politec. Torino, Special Issue 253–261 (1984)Google Scholar
  17. 17.
    Wang M., Ziller W.: Symmetric spaces and strongly isotropy irreducible spaces. Math. Ann. 296, 285–326 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.CMLS, École PolytechniquePalaiseauFrance
  2. 2.Institut für Geometrie und Topologie, Fachbereich MathematikUniversität StuttgartStuttgartGermany

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