Advertisement

Israel Journal of Mathematics

, Volume 155, Issue 1, pp 125–147 | Cite as

Nonassociative quaternion algebras over rings

  • S. Pumplün
  • V. Astier
Article

Abstract

Non-split nonassociative quaternion algebras over fields were first discovered over the real numbers independently by Dickson and Albert. They were later classified over arbitrary fields by Waterhouse. These algebras naturally appeared as the most interesting case in the classification of the four-dimensional nonassociative algebras which contain a separable field extension of the base field in their nucleus. We investigate algebras of constant rank 4 over an arbitrary ringR which contain a quadratic étale subalgebraS overR in their nucleus. A generalized Cayley-Dickson doubling process is introduced to construct a special class of these algebras.

Keywords

Zero Divisor Quaternion Algebra Constant Rank Composition Algebra Principal Ideal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A] A. A. Albert,Quadratic forms permitting composition, Annals of Mathematics (2)43 (1942), 161–177.CrossRefMathSciNetGoogle Scholar
  2. [A-H-K] C. Althoen, K. D. Hansen and L. D. Kugler, ℂAssociative algebras of dimension 4 over ℝ, Algebras, Groups and Geometries3 (1986), 329–360.zbMATHMathSciNetGoogle Scholar
  3. [D] L. E. Dickson,Linear Algebras with associativity not assumed, Duke Mathematical Journal1 (1935), 113–125.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [K] M. Knebusch,Grothendiek- und Wittringe von nicht-ausgearteten symmetrischen Bilinearformen, Sitzungsberichte der Heidelberg Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, Springer-Verlag, New York-Heidelberg-Berlin, 1970.Google Scholar
  5. [Kn] M. Kneser,Composition of binary quadratic forms, Journal of Number Theory15 (1982), 406–413.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Knu] M.-A. Knus,Quadratic and Hermitian Forms over Rings, Springer-Verlag, New York-Heidelberg-Berlin, 1991.zbMATHGoogle Scholar
  7. [L] H. J. Lee,Maximal orders in split nonassociative quaternion algebras, Journal of Algebra146 (1992), 427–440.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [L-W] H. J. Lee, W. C. Waterhouse,Maximal orders in nonassociative quaternion algebras, Journal of Algebra146 (1992), 441–453.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Mc] K. McCrimmon,Nonassociative algebras with scalar involutions, Pacific Journal of Mathematics116 (1985), 85–109.zbMATHMathSciNetGoogle Scholar
  10. [P] H. P. Petersson,Composition algebras over algebraic curves of genus 0, Transactions of the American Mathematical Society337 (1993), 473–491.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Pf] A. Pfister,Quadratic lattices in function fields of genus 0, Proceedings of the London Mathematical Society66 (1993), 257–278.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Pu1] S. Pumplün,Composition algebras over rings of genus zero, Transactions of the American Mathematical Society351 (1999), 1277–1292.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Pu2] S. Pumplün, Composition algebras over\(k[t,\sqrt {at^2 + b]} \), Indagationes Mathematicae. New Series9 (1998), 417–429.zbMATHCrossRefGoogle Scholar
  14. [Pu3] S. Pumplün,Composition algebras over a ring of fractions, Journal of Algebra187 (1997), 474–492.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [W] W. C. Waterhouse,Nonassociative quaternion algebras, Algebras, Groups and Geometries4 (1987), 365–378.zbMATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2006

Authors and Affiliations

  • S. Pumplün
    • 1
  • V. Astier
    • 1
  1. 1.School of MathematicsUniversity of NottinghamNottinghamUK

Personalised recommendations