Skip to main content
SpringerLink
Log in

Bilinear and Quadratic Algebra

  • Chapter
  • First Online:
Automorphic Forms and Even Unimodular Lattices

  • 1115 Accesses

Abstract

This chapter essentially recalls classical material. First we introduce the definitions and results from the theory of symmetric bilinear forms, quadratic forms, alternating forms, and their associated (classical) groups, that are used in this book. Next we explain the relation between root systems and even unimodular lattices. We recall in particular the remarkable method, introduced by Venkov, for retrieving the classification, due to Niemeier, of even unimodular lattices of dimension 24.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Barge, J. Lannes, Suites de Sturm, indice de Maslov et périodicité de Bott, Progress in Math., vol. 267 (Birkhäuser Verlag, Basel, 2008).

    Google Scholar 

  2. J. Barge, J. Lannes, F. Latour, P. Vogel, Λ-sphères, Ann. Sci. Éc. Norm. Sup. 7 (1974), pp. 463–505.

    Article  Google Scholar 

  3. N. Bourbaki, Éléments de mathématique, Algèbre, Chapitre 9 (Masson, Paris, 1981).

    MATH  Google Scholar 

  4. N. Bourbaki, Éléments de mathématique, Groupes et algèbres de Lie, Chapitres 4, 5 et 6 (Masson, Paris, 1981).

    MATH  Google Scholar 

  5. C. Chevalley, The algebraic theory of spinors and Clifford algebras, oeuvres complètes vol. 2 (Springer-Verlag, Berlin, 1997).

    MATH  Google Scholar 

  6. B. Conrad, Reductive group schemes, Notes of the Summer School SGA3 (Luminy, 2011), available at http://math.stanford.edu/~conrad/papers/.

  7. J. H. Conway, A characterisation of Leech’s lattice, Invent. Math. 7 (1969), pp. 137–142.

    Article  MathSciNet  Google Scholar 

  8. J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren math. Wiss., vol. 290 (Springer-Verlag, New York, 1999).

    Google Scholar 

  9. P. Deligne, N. Katz, Groupes de monodromie en géométrie algébrique, SGA7, Tome II, Séminaire de géométrie algébrique du Bois Marie 1967–69, Lecture Notes in Math., vol. 340 (Springer Verlag, 1973).

    Google Scholar 

  10. M. Demazure, P. Gabriel, Groupes algébriques, Tome 1, (Masson, Paris, 1970).

    MATH  Google Scholar 

  11. J. Dieudonné, Pseudo-discriminant and Dickson invariant, Pacific. J. Math. 5 (1955), pp. 907–910.

    Article  MathSciNet  Google Scholar 

  12. J. Dieudonné, Sur les groupes classiques (Hermann, Paris, 1997).

    MATH  Google Scholar 

  13. J. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., vol. 29 (Cambridge Univ. Press, 1990).

    Google Scholar 

  14. J. Jantzen, Representations of algebraic groups, 2nd edn., Math. Survey Monogr., vol. 107 (Amer. Math. Soc., 2007).

    Google Scholar 

  15. J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1964), pp. 657–682.

    Article  MathSciNet  Google Scholar 

  16. J. Milnor, D. Husemoller, Symmetric bilinear forms, Ergebn. Math. Grenzgeb., vol. 73 (Springer Verlag, 1973).

    Google Scholar 

  17. H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number Theory 5 (1973), pp. 142–178.

    Article  MathSciNet  Google Scholar 

  18. V. Platonov, A. Rapinchuk, Algebraic groups and number theory, Pure Appl. Math., vol 139 (1994).

    Google Scholar 

  19. V. Pless, On the uniqueness of the Golay codes, J. Combin. Theory 5 (1968), pp. 215–228.

    Article  MathSciNet  Google Scholar 

  20. V. Pless, N. J. A. Sloane, Binary self-dual codes of length 24, Bull. Amer. Math. Soc. 80 (1974), pp. 1173–1178.

    Article  MathSciNet  Google Scholar 

  21. J.-P. Serre, Cours d’arithmétique (Publ. Univ. France, Paris, 1970).

    MATH  Google Scholar 

  22. J. R. Stembridge, The partial order of dominant weights, Adv. in Math. 136 (1998), pp. 340–364.

    Article  MathSciNet  Google Scholar 

  23. B. Venkov, On the classification of integral even unimodular 24-dimensional quadratic forms, Chapter 18 in J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren math. Wiss., vol. 290 (Springer-Verlag, New York, 1999).

    Google Scholar 

  24. W. Wu, Classes caractéristiques et i-carrés d’une variété, C. R. Acad. Sci. (Paris, 1950), pp. 508–511.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, Institut de Mathématique d’Orsay, Université Paris-Sud, Orsay, France

    Gaëtan Chenevier

  2. Institut de Mathématiques de Jussieu, Université Paris Diderot, Paris, France

    Jean Lannes

Authors
  1. Gaëtan Chenevier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean Lannes
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chenevier, G., Lannes, J. (2019). Bilinear and Quadratic Algebra. In: Automorphic Forms and Even Unimodular Lattices. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-95891-0_2

Download citation

Publish with us

Policies and ethics

Search

Navigation