Advertisement

Communications in Mathematical Physics

, Volume 283, Issue 3, pp 663–674 | Cite as

A Geometric Construction of the Exceptional Lie Algebras F 4 and E 8

  • José Figueroa-O’FarrillEmail author
Article

Abstract

We present a geometric construction of the exceptional Lie algebras F 4 and E 8 starting from the round spheres S 8 and S 15, respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background.

Keywords

Clifford Algebra Jacobi Identity Geometric Construction Spinor Module Killing Spinor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, J.F.: Lectures on exceptional Lie groups, Chicago, IL: The University of Chicago Press, 1996 Z. Mahmud, M. Mimura, (eds.)Google Scholar
  2. 2.
    Green M., Schwarz J., Witten E.: Superstring Theory. 2 vols.. Cambridge University Press, Cambridge (1987)Google Scholar
  3. 3.
    Figueroa-O’Farrill J.M., Meessen P., Philip S.: Supersymmetry and homogeneity of M-theory backgrounds. Class Quant. Grav. 22, 207–226 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Figueroa-O’Farrill J.M., Hackett-Jones E., Moutsopoulos G.: The Killing superalgebra of ten-dimensional supergravity backgrounds. Class. Quant. Grav. 24, 3291–3308 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Atiyah M.F., Bott R., Shapiro A.: Clifford modules. Topology 3, 3–38 (1964)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Lawson H., Michelsohn M.: Spin geometry. Princeton University Press, Princeton, NJ (1989)zbMATHGoogle Scholar
  7. 7.
    Harvey F.: Spinors and calibrations. Academic Press, London-New York (1990)zbMATHGoogle Scholar
  8. 8.
    Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing spinors on riemannian manifolds. No.108 in Seminarberichte. Berlin, Humboldt-Universität, 1990Google Scholar
  9. 9.
    Bär C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)zbMATHCrossRefADSGoogle Scholar
  10. 10.
    Gallot S.: Equations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. 12, 235–267 (1979)MathSciNetGoogle Scholar
  11. 11.
    Wang M.: Parallel spinors and parallel forms. Ann Global Anal. Geom. 7(1), 59–68 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Figueroa-O’Farrill, J.M., Leitner, F., Simón, J.: “Supersymmetric Freund–Rubin backgrounds,” In preparationGoogle Scholar
  13. 13.
    Kosmann Y.: Dérivées de Lie des spineurs. Annali di Mat. Pura Appl. (IV) 91, 317–395 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Figueroa-O’Farrill J.M.: On the supersymmetries of Anti-de Sitter vacua. Class. Quant. Grav. 16, 2043–2055 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    “248-dimension maths puzzle solved.” BBC News, March, 2007 available at http://news.bbc.co.uk/1/hi/sci/tech/6466129.stm, 2007
  16. 16.
    van Leeuwen M.A.A.: LiE, a software package for Lie group computations. Euromath Bull. 1(2), 83–94 (1994)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Maxwell Institute and School of MathematicsUniversity of EdinburghEdinburghUK

Personalised recommendations