Skip to main content
SpringerLink
Log in

Subalgebras of the Split Octonions

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

We classify the subalgebras of the split octonions, paying particular attention to the null subalgebras and their extensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barton, C.H., Sudbery, A.: Magic squares and matrix models of Lie Algebras. Adv. Math. 180, 596–647 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bentz, L.: Subalgebras of the Split Octonions, MS paper, Oregon State University (2017). http://ir.library.oregonstate.edu/xmlui/handle/1957/61755. Accessed 9 Aug 2017

  3. Borsten, L., Marrani, A.: A kind of magic (2017). arXiv:1707.02072

  4. Chung, K.W., Sudbery, A.: Octonions and the Lorentz and conformal groups of ten-dimensional space-time. Phys. Lett. B 198, 161 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  5. Dickson, L.E.: On quaternions and their generalization and the history of the eight square theorem. Ann. Math. 20, 155–171 (1919)

  6. Dray, T., Manogue, C.A.: Octonionic Cayley Spinors and \(E_6\). Comment. Math. Univ. Carolin. 51, 193–207 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Dray, T., Manogue, C.A.: The Geometry of the Octonions. World Scientific, Singapore (2015)

    Book  MATH  Google Scholar 

  8. Dray, T., Manogue, C.A.: Quaternionic spin. In: Ablamowicz, R., Fauser, B. (eds.) Clifford Algebras and their Applications in Mathematical Physics. Birkhäuser, Boston, pp. 29–46 (2000)

  9. Dray, T., Manogue, C.A., Wilson, R.A.: A symplectic representation of \(E_7\). Comment. Math. Univ. Carolin. 55, 387–399 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Fairlie, D.B., Manogue, C.A.: Lorentz invariance and the composite string. Phys. Rev. D 34, 1832–1834 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  11. Fairlie, D.B., Manogue, C.A.: A parametrization of the covariant superstring. Phys. Rev. D 36, 475–479 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  12. Freudenthal, H.: Lie groups in the foundations of geometry. Adv. Math. 1, 145–190 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grishkov, A.N., Giuliani, M.D.L.M., Zavarnitsine, A.V.: Classification of subalgebras of the Cayley algebra over a finite field. J. Algebra Appl. 9, 791–808 (2010)

  14. Günaydin, M., Gürsey, F.: Quark statistics and octonions. Phys. Rev. D 9, 3387–3391 (1974)

    Article  ADS  Google Scholar 

  15. Gürsey, F., Tze, C.-H.: On the Role of Division, Jordan, and Related Algebras in Particle Physics. World Scientific, Singapore (1996)

    Book  MATH  Google Scholar 

  16. Gürsey, F., Ramond, P., Sikivie, P.: A universal gauge theory model based on \(E_6\). Phys. Lett. B 60, 177–180 (1976)

    Article  ADS  Google Scholar 

  17. Hurwitz, A.: Über die Komposition der quadratischen Formen. Math. Ann. 88, 1–25 (1923)

    Article  MATH  Google Scholar 

  18. Manogue, C.A., Dray, T.: Dimensional reduction. Mod. Phys. Lett. A 14, 93–97 (1999)

  19. Manogue, C.A., Dray, T.: Octonions, \(E_6\), and particle physics. J. Phys. Conf. Ser. 254, 012005 (2010)

  20. Manogue, C.A., Sudbery, A.: General solutions of covariant superstring equations of motion. Phys. Rev. D 40, 4073–4077 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  21. Manogue, C.A., Schray, J.: Finite Lorentz transformations, automorphisms, and division algebras. J. Math. Phys. 34, 3746–3767 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Okubo, S.: Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  23. Racine, M.L.: On maximal subalgebras. J. Algebra 30, 155–180 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wangberg, A., Dray, T.: \(E_6\), the Group: the structure of \(SL(3,\mathbb{O})\). J. Algebra Appl. 14, 1550091 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Oregon State University, Corvallis, OR , 97331, USA

    Lida Bentz & Tevian Dray

Authors
  1. Lida Bentz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tevian Dray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tevian Dray.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bentz, L., Dray, T. Subalgebras of the Split Octonions. Adv. Appl. Clifford Algebras 28, 40 (2018). https://doi.org/10.1007/s00006-018-0859-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-018-0859-6

Keywords

Search

Navigation