Advertisement

Letters in Mathematical Physics

, Volume 104, Issue 11, pp 1445–1468 | Cite as

The Magic Square of Lie Groups: The 2 × 2 Case

  • Tevian DrayEmail author
  • John Huerta
  • Joshua Kincaid
Article

Abstract

A unified treatment of the 2 × 2 analog of the Freudenthal–Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.

Mathematics Subject Classification

22E46 17A35 15A66 

Keywords

division algebras magic squares orthogonal groups Clifford algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Freudenthal H.: Lie groups in the foundations of geometry. Adv. Math. 1, 145–190 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Tits J.: Algèbres Alternatives, Algèbres de Jordan et algèbres de Lie Exceptionnelles. Indag. Math. 28, 223–237 (1966)MathSciNetGoogle Scholar
  3. 3.
    Vinberg E.B.: A construction of exceptional Lie groups (Russian). Tr. Semin. Vektorn. Tensorn. Anal. 13, 7–9 (1966)Google Scholar
  4. 4.
    Barton C.H., Sudbery A.: Magic squares and matrix models of Lie algebras. Adv. Math. 180, 596–647 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fairlie D.B., Manogue C.A.: Lorentz invariance and the composite string. Phys. Rev. D 34, 1832–1834 (1986)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Schray J.: The general classical solution of the superparticle. Class. Quant. Grav. 13, 27–38 (1996)MathSciNetCrossRefzbMATHADSGoogle Scholar
  7. 7.
    Baez, J., Huerta, J.: Division algebras and supersymmetry I. In: Doran, R.S., Friedman, G., Rosenberg, J. (eds) Superstrings, geometry, topology, and C*-algebras, pp. 65–80. American Mathematical Society, Providence (2010). arXiv:0909.0551
  8. 8.
    Manogue C.A., Schray J.: Finite Lorentz transformations, automorphisms, and division algebras. J. Math. Phys. 34, 3746–3767 (1993)MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    Manogue C.A., Dray T.: Dimensional reduction. Mod. Phys. Lett. A 14, 99–103 (1999)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Dray, T., Manogue, C.A.: Quaternionic spin. In: Rafał, A., Bertfried, F. (eds) Clifford algebras and mathematical physics, pp. 21–37. Birkhäuser, Boston (2000). arXiv:hep-th/9910010
  11. 11.
    Dray T., Manogue C.A.: Octonions and the structure of E 6. Comment. Math. Univ. Carolin. 51, 193–207 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Manogue C.A., Dray T.: Octonions, E 6, and particle physics. J. Phys. Conf. Ser. 254, 012005 (2010)CrossRefADSGoogle Scholar
  13. 13.
    Wangberg, A., Dray T.: E 6, the group: the structure of \({{\rm SL}(3,{\mathbb{O}})}\). J. Algebra Appl. (to appear). arXiv:1212.3182
  14. 14.
    Wangberg A., Dray T.: Discovering real Lie subalgebras of \({\mathfrak{e}_6}\) using Cartan decompositions. J. Math. Phys. 54, 081703 (2013)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Wangberg, A.: The structure of E 6. PhD thesis, Oregon State University (2007). arXiv:0711.3447
  16. 16.
    Dray T., Manogue C.A., Wilson R.A.: A symplectic representation of E 7. Comment. Math. Univ. Carolin 55, 387–399 (2014). arXiv:1311.0341
  17. 17.
    Wilson R.A.: A quaternionic construction of E 7. Proc. Am. Math. Soc. 142, 867–880 (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kincaid, J.J.: Division algebra representations of SO(4, 2). Master’s thesis, Oregon State University (2012). Available at http://ir.library.oregonstate.edu/xmlui/handle/1957/30682
  19. 19.
    Kincaid, J., Dray, T.: Division algebra representations of SO(4, 2). Mod. Phys. Lett. A 29, 1450128 (2014). arXiv:1312.7391
  20. 20.
    Albert A.A.: Quadratic forms permitting composition. Ann. Math. (2) 43, 161–177 (1942)CrossRefGoogle Scholar
  21. 21.
    Hurwitz, A.: Über die Komposition der quadratischen Formen. Math. Ann. 88, 1–25 (1923). Available at http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=GDZPPN002269074
  22. 22.
    Schafer, R.D.: An introduction to nonassociative algebras. Academic Press, New York (1966) (reprinted by Dover Publications 1995)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA
  2. 2.CAMGSD, Instituto Superior TécnicoLisboaPortugal
  3. 3.Department of PhysicsOregon State UniversityCorvallisUSA

Personalised recommendations