Skip to main content
SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Advances in Applied Clifford Algebras
  3. Article

The E 8 Geometry from a Clifford Perspective

  • Open Access
  • Published: 28 April 2016
  • volume 27, pages 397–421 (2017)
Download PDF

You have full access to this open access article

Advances in Applied Clifford Algebras Aims and scope Submit manuscript
The E 8 Geometry from a Clifford Perspective
Download PDF
  • Pierre-Philippe Dechant  ORCID: orcid.org/0000-0002-4694-40101 
  • 630 Accesses

  • 7 Citations

  • 5 Altmetric

  • Explore all metrics

  • Cite this article

Abstract

This paper considers the geometry of E 8 from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system H 3 gives rise to the largest (and therefore exceptional) non-crystallographic root system H 4. Arnold’s trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and E 8. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the 120 elements of the icosahedral group H 3 are doubly covered by 240 8-component objects, which endowed with a ‘reduced inner product’ are exactly the E 8 root system. It was previously known that E 8 splits into H 4-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, D 6 as well as E 8, whose Coxeter versor factorises as \({W = \exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)}\) . This explicitly describes 30-fold rotations in 4 orthogonal planes with the correct exponents \({\{1, 7, 11, 13, 17, 19, 23, 29\}}\) arising completely algebraically from the factorisation.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Arnold, V.I.: Symplectization, complexification and mathematical trinities. In: Bierstone E, Khesin B, Khovanskii A, Marsden JE (eds.) The Arnoldfest, pp. 23–37. American Mathematical Society, Providence, RI (1999)

  2. Arnold, V.I.: Mathematics: Frontiers and Perspectives. American Mathematical Society, Providence (2000)

  3. Bourbaki N.: Groupes et algèbres de Lie, chapitres 4, 5 et 6. Masson, Paris (1981)

    MATH  Google Scholar 

  4. Dechant P.-P., Boehm C., Twarock R.: Novel Kac–Moody-type affine extensions of non-crystallographic Coxeter groups. J. Phys. A: Math. Theor. 45(28), 285202 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dechant P.-P., Boehm C., Twarock R.: Affine extensions of non-crystallographic Coxeter groups induced by projection. J. Math. Phys. 54(9), 093508 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Dechant, P.-P: Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups. Adv. Appl. Clifford Algebras 23, 301–321 (2013). doi:10.1007/s00006-012-0371-3

  7. Dechant P.-P.: Platonic solids generate their four-dimensional analogues. Acta Crystallogr. Sect. A: Found. Crystallogr. 69(6), 592–602 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dechant P.-P.: A Clifford algebraic framework for Coxeter group theoretic computations. Adv. Appl. Clifford Algebras 24(1), 89–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dechant P.-P.: Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction. J. Phys. Conf. Ser. 597(1), 012027 (2015)

    Article  Google Scholar 

  10. Dechant, P.-P.: The birth of E 8 out of the spinors of the icosahedron. Proc. R. Soc. A, vol. 472, pp. 20150504. The Royal Society (2016)

  11. Doran C., Hestenes D., Sommen F., Acker N.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Hestenes, D.: Point Groups and Space Groups in Geometric Algebra, pp. 3–34. Birkhäuser, Boston (2002)

  13. Hestenes D., Holt J.W.: The crystallographic space groups in geometric algebra. J. Math. Phys. 48, 023514 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Hitzer, E., Helmstetter, J., Abłamowicz, R.: Square roots of −1 in real Clifford algebras. In: Quaternion and Clifford Fourier Transforms and Wavelets, pp. 123–153. Springer, Berlin (2013)

  15. Hitzer, E., Perwass, C.: Interactive 3D space group visualization with CLUCalc and the Clifford Geometric Algebra description of space groups. Adv. Appl. Clifford Algebras 20, 631–658 (2010). doi:10.1007/s00006-010-0214-z

  16. Humphreys J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  17. Koca, M., Al-Barwani, M., Koç, R.: Quaternionic root systems and subgroups of the Aut(F4). J. Math. Phys. 47(4), 043507 (2006)

  18. Koca M., Koç R., Al-Barwani M.: Quaternionic roots of SO(8), SO(9), F 4 and the related Weyl groups. J. Math. Phys. 44, 3123–3140 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Lusztig G.: Some examples of square integrable representations of semisimple p-adic groups. Trans. Am. Math. Soc. 277(2), 623–653 (1983)

    MathSciNet  MATH  Google Scholar 

  20. McKay, J.: Graphs, singularities, and finite groups. In: Proceedings of Symposia in Pure Mathematics, vol. 37, pp. 183–186 (1980)

  21. Moody R.V., Patera J.: Quasicrystals and icosians. J. Phys. A: Math. Gen. 26(12), 2829 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Shcherbak O.P.: Wavefronts and reflection groups. Russ. Math. Surv. 43(3), 149 (1988)

    Article  MATH  Google Scholar 

  23. Sekiguchi J., Yano T.: A note on the Coxeter group of type H 3. Sci. Rep. Saitama Univ. Ser. A 9, 33–44 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Wilson, R.A.: The geometry of the Hall–Janko group as a quaternionic reflection group. Geometriae Dedicata 20, 157–173 (1986). doi:10.1007/BF00164397

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Biology, York Centre for Complex Systems Analysis, University of York, Heslington, York, YO10 5GE, UK

    Pierre-Philippe Dechant

Authors
  1. Pierre-Philippe Dechant
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre-Philippe Dechant.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dechant, PP. The E 8 Geometry from a Clifford Perspective. Adv. Appl. Clifford Algebras 27, 397–421 (2017). https://doi.org/10.1007/s00006-016-0675-9

Download citation

  • Received: 18 February 2016

  • Accepted: 01 April 2016

  • Published: 28 April 2016

  • Issue Date: March 2017

  • DOI: https://doi.org/10.1007/s00006-016-0675-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification

  • 52B10
  • 52B12
  • 52B15
  • 15A66
  • 20F55
  • 17B22
  • 14E16

Keywords

  • E8
  • Exceptional phenomena
  • Clifford algebras
  • Icosahedral symmetry
  • Coxeter groups
  • Root systems
  • Spinors
  • Coxeter plane
  • Lie algebras
  • Lie groups
  • Representation theory
  • Quantum algebras
  • Trinities
  • McKay correspondence

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature