Advertisement

Mathematische Annalen

, Volume 351, Issue 1, pp 109–148 | Cite as

Valuations on algebras with involution

  • J.-P. Tignol
  • A. R. WadsworthEmail author
Open Access
Article

Abstract

Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ(x)x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.

Mathematics Subject Classification (2000)

16W10 16K20 16W60 11E39 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Bourbaki, N.: Elements of Mathematics, Commutative Algebra. Addison-Wesley, Reading (1972). English trans. of Éléments de Mathématique, Algèbre CommutativeGoogle Scholar
  2. 2.
    Dejaiffe I.: Somme orthogonale d’algèbres à involution et algèbre de Clifford. Comm. Algebra 26, 1589–1612 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dejaiffe I., Lewis D.W., Tignol J.-P.: Witt equivalence of central simple algebras with involution. Rend. Circ. Mat. Palermo 49(2), 325–342 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dixmier, J.: Les C *-algèbres et leurs représentations. Gauthier-Villars & Cie, Éditeur, Paris (1964). English trans.: C *-Algebras. North-Holland, Amsterdam (1977)Google Scholar
  5. 5.
    Ershov, Yu.L.: Valued division rings. In: Fifth All Union Symposium, Theory of Rings, Algebras, and Modules, pp. 53–55. Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk (1982, in Russian)Google Scholar
  6. 6.
    Ershov Yu.L.: Henselian valuations of division rings and the group SK 1. Math. USSR Sbornik 45, 63–71 (1983)zbMATHCrossRefGoogle Scholar
  7. 7.
    Ershov Yu.L.: Multi-Valued Fields. Kluwer, New York (2001)CrossRefGoogle Scholar
  8. 8.
    Endler O.: Valuation Theory. Springer, New York (1972)zbMATHGoogle Scholar
  9. 9.
    Engler A.J., Prestel A.: Valued Fields. Springer, Berlin (2005)zbMATHGoogle Scholar
  10. 10.
    Hwang Y.-S., Wadsworth A.R.: Algebraic extensions of graded and valued fields. Comm. Algebra 27, 821–840 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hwang Y.-S., Wadsworth A.R.: Correspondences between valued division algebras and graded division algebras. J. Algebra 220, 73–114 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Knus, M.-A., Merkurjev, A.S., Rost, M., Tignol, J.-P.: The Book of Involutions. Coll. Pub. 44. Amer. Math. Soc., Providence (1998)Google Scholar
  13. 13.
    Larmour D.W.: A Springer Theorem for Hermitian forms. Math. Z. 252, 459–472 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Morandi P.: The Henselization of a valued division algebra. J. Algebra 122, 232–243 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pierce R.S.: Associative Algebras. Springer, New York (1982)zbMATHGoogle Scholar
  16. 16.
    Renard J.-F., Tignol J.-P., Wadsworth A.R.: Graded Hermitian forms and Springer’s theorem. Indag. Math., N.S. 18, 97–134 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schilling O.F.G.: The Theory of Valuations. Amer. Math. Soc., New York (1950)zbMATHGoogle Scholar
  18. 18.
    Tignol J.-P., Wadsworth A.R.: Value functions and associated graded rings for semisimple algebras. Trans. Am. Math. Soc. 362, 687–726 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Wadsworth A.R.: Extending valuations to finite-dimensional division algebras. Proc. Am. Math. Soc. 98, 20–22 (1986)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Weil A.: Algebras with involutions and the classical groups. J. Indian Math. Soc. (N.S.) 24, 589–623 (1960)MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institute for Information and Communication Technologies, Electronics and Applied MathematicsUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations