Skip to main content
SpringerLink
Log in

Inner ideals and structurable algebras: Moufang sets, triangles and hexagons

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We construct Moufang sets, Moufang triangles and Moufang hexagons using inner ideals of Lie algebras obtained from structurable algebras via the Tits–Kantor–Koecher construction. The three different types of structurable algebras we use are, respectively: (1) structurable division algebras, (2) algebras D ⊕ D for some alternative division algebra D, equipped with the exchange involution, (3) matrix structurable algebras M (J, 1) for some cubic Jordan division algebra J. In each case, we also determine the root groups directly in terms of the structurable algebra.

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. B. N. Allison and J. R. Faulkner, A Cayley–Dickson process for a class of structurable algebras, Transactions of the American Mathematical Society 283 (1984), 185–210.

    Article  MathSciNet  Google Scholar 

  2. B. N. Allison and J. R. Faulkner, Elementary groups and invertibility for Kantor pairs, Communications in Algebra 27 (1999), 519–556.

    Article  MathSciNet  Google Scholar 

  3. B. N. Allison and W. Hein, Isotopes of some nonassociative algebras with involution, Journal of Algebra 69 (1981), 120–142.

    Article  MathSciNet  Google Scholar 

  4. B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Mathematische Annalen 237 (1978), 133–156.

    Article  MathSciNet  Google Scholar 

  5. B. N. Allison, Models of isotropic simple Lie algebras, Communications in Algebra 7 (1979), 1835–1875.

    Article  MathSciNet  Google Scholar 

  6. L. Boelaert and T. De Medts, Exceptional Moufang quadrangles and structurable algebras, Proceedings of the London Mathematical Society 107 (2013), 590–626.

    Article  MathSciNet  Google Scholar 

  7. L. Boelaert and T. De Medts, A new construction of Moufang quadrangles of type E6, E7and E8, Transactions of the American Mathematical Society 367 (2015), 3447–3480.

    Article  MathSciNet  Google Scholar 

  8. L. Boelaert, T. De Medts and A. Stavrova, Moufang sets and structurable division algebras, Memoirs of the American Mathematical Society 259 (2019).

  9. G. M. Benkart, Inner ideals and the structure of Lie algebras, Ph. D. Thesis, Yale University, New Haven, CN, 1974.

    Google Scholar 

  10. G. Benkart, On inner ideals and ad-nilpotent elements of Lie algebras, Transactions of the American Mathematical Society 232 (1977), 61–81.

    Article  MathSciNet  Google Scholar 

  11. A. M. Cohen and G. Ivanyos, Root filtration spaces from Lie algebras and abstract root groups, Journal of Algebra 300 (2006), 433–454.

    Article  MathSciNet  Google Scholar 

  12. A. M. Cohen and G. Ivanyos, Root shadow spaces, European Journal of Combinatorics 28 (2007), 1419–1441.

    Article  MathSciNet  Google Scholar 

  13. H. Cuypers and J. Meulewaeter, Extremal elements in Lie algebras, buildings and structurable algebras, Journal of Algebra 580 (2021), 1–42.

    Article  MathSciNet  Google Scholar 

  14. A. M. Cohen, Inner ideals in Lie algebras and spherical buildings, Indagationes Mathematicae 32 (2021), 1115–1138.

    Article  MathSciNet  Google Scholar 

  15. C. Draper, A. Fernández López, E. García and M. Gómez Lozano, The socle of a nondegenerate Lie algebra, Journal of Algebra 319 (2008), 2372–2394.

    Article  MathSciNet  Google Scholar 

  16. C. Draper, A. Fernández López, E. García and M. Gómez Lozano, The inner ideals of the simple finite dimensional Lie algebras, Journal of Lie Theory 22 (2012), 907–929.

    MathSciNet  Google Scholar 

  17. T. De Medts, Structurable algebras of skew-dimension one and hermitian cubic norm structures, Communications in Algebra 47 (2019), 154–172.

    Article  MathSciNet  Google Scholar 

  18. T. De Medts, Known examples of Moufang sets, https://algebra.ugent.be/~tdemedts/notes/moufsetexamples.pdf.

  19. T. De Medts and Y. Segev, Identities in Moufang sets, Transactions of the American Mathematical Society 360 (2008), 5831–5852.

    Article  MathSciNet  Google Scholar 

  20. T. De Medts and Y. Segev, A course on Moufang sets, Innovations in Incidence Geometry 9 (2009), 79–122.

    Article  MathSciNet  Google Scholar 

  21. T. De Medts and R. M. Weiss, Moufang sets and Jordan division algebras, Mathematische Annalen 335 (2006), 415–433.

    Article  MathSciNet  Google Scholar 

  22. J. R. Faulkner, On the geometry of inner ideals, Journal of Algebra 26 (1973), 1–9.

    Article  MathSciNet  Google Scholar 

  23. J. R. Faulkner, Groups with Steinberg relations and coordinatization of polygonal geometries, Memoirs of the American Mathematical Society 10 (1977).

  24. A. Fernández López, Jordan Structures in Lie Algebras, Mathematical Surveys and Monographs, Vol. 240, American Mathematical Society, Providence, RI, 2019.

    Google Scholar 

  25. R. S. Garibaldi, Structurable algebras and groups of type E6and E7, Journal of Algebra 236 (2001), 651–691.

    Article  MathSciNet  Google Scholar 

  26. E. García, M. Gómez Lozano and E. Neher, Nondegeneracy for Lie triple systems and Kantor pairs, Canadian Mathematical Bulletin 54 (2011), 442–455.

    Article  MathSciNet  Google Scholar 

  27. M. Grüninger, On the identities defining a quadratic Jordan division algebra, Archiv der Mathematik 104 (2015), 11–24.

    Article  MathSciNet  Google Scholar 

  28. M.-A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications, Vol. 44, American Mathematical Society, Providence, RI, 1998.

    Google Scholar 

  29. A. F. López, E. García, M. G. Lozano and E. Neher, A construction of gradings of Lie algebras, International Mathematics Research Notices 2007 (2007), Article no. 51.

  30. O. Loos, Jordan Pairs, Lecture Notes in Mathematics, Vol. 460, Springer, Berlin–New York, 1975.

    Google Scholar 

  31. K. McCrimmon, Inner ideals in quadratic Jordan algebras, Transactions of the American Mathematical Society 159 (1971), 445–468.

    Article  MathSciNet  Google Scholar 

  32. K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer, New York, 2004.

    Google Scholar 

  33. R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York–London, 1966.

    Google Scholar 

  34. A. Stavrova, 5-graded simple Lie algebras, structurable algebras, and Kantor pairs, Journal of Algebra 611 (2022), 561–588.

    Article  MathSciNet  Google Scholar 

  35. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer, Berlin–New York, 1974.

    Google Scholar 

  36. J. Tits and R. M. Weiss, Moufang Polygons, Springer Monographs in Mathematics, Springer, Berlin, 2002.

    Google Scholar 

  37. E. I. Zelmanov, Absolute zero-divisors in Jordan pairs and Lie algebras, Matematicheskiĭ Sbornik. Novaya Seriya 112 (1980), 611–629.

    MathSciNet  Google Scholar 

Download references

Acknowledgment

The idea to use inner ideals of Lie algebras has been inspired by a talk of Arjeh Cohen during the workshop “Buildings and Symmetry” at the University of Western Australia in September 2017. We thank Alice Devillers, Bernhard Mühlherr, James Parkinson and Hendrik Van Maldeghem for organizing this workshop. We are grateful to Arjeh Cohen for sharing his preprint [Coh21] with us. We thank Antonio Fernández López for his useful comments on an earlier version of this paper. In particular, he provided us with the proof of Proposition 3.15. Last but not least, we thank the referee for carefully reading our paper and providing insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics: Algebra and Geometry, Ghent University, Krijgslaan 281-S25, 9000, Gent, Belgium

    Tom De Medts & Jeroen Meulewaeter

Authors
  1. Tom De Medts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeroen Meulewaeter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom De Medts.

Additional information

The second author is a PhD Fellow of the Research Foundation Flanders (Belgium) (F.W.O.-Vlaanderen), 166032/1128720N.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Medts, T., Meulewaeter, J. Inner ideals and structurable algebras: Moufang sets, triangles and hexagons. Isr. J. Math. 259, 33–88 (2024). https://doi.org/10.1007/s11856-023-2491-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-023-2491-y

Search

Navigation