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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] A. A. Albert,Quadratic forms permitting composition, Annals of Mathematics (2)43 (1942), 161–177.
[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.
[D] L. E. Dickson,Linear Algebras with associativity not assumed, Duke Mathematical Journal1 (1935), 113–125.
[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.
[Kn] M. Kneser,Composition of binary quadratic forms, Journal of Number Theory15 (1982), 406–413.
[Knu] M.-A. Knus,Quadratic and Hermitian Forms over Rings, Springer-Verlag, New York-Heidelberg-Berlin, 1991.
[L] H. J. Lee,Maximal orders in split nonassociative quaternion algebras, Journal of Algebra146 (1992), 427–440.
[L-W] H. J. Lee, W. C. Waterhouse,Maximal orders in nonassociative quaternion algebras, Journal of Algebra146 (1992), 441–453.
[Mc] K. McCrimmon,Nonassociative algebras with scalar involutions, Pacific Journal of Mathematics116 (1985), 85–109.
[P] H. P. Petersson,Composition algebras over algebraic curves of genus 0, Transactions of the American Mathematical Society337 (1993), 473–491.
[Pf] A. Pfister,Quadratic lattices in function fields of genus 0, Proceedings of the London Mathematical Society66 (1993), 257–278.
[Pu1] S. Pumplün,Composition algebras over rings of genus zero, Transactions of the American Mathematical Society351 (1999), 1277–1292.
[Pu2] S. Pumplün, Composition algebras over\(k[t,\sqrt {at^2 + b]} \), Indagationes Mathematicae. New Series9 (1998), 417–429.
[Pu3] S. Pumplün,Composition algebras over a ring of fractions, Journal of Algebra187 (1997), 474–492.
[W] W. C. Waterhouse,Nonassociative quaternion algebras, Algebras, Groups and Geometries4 (1987), 365–378.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pumplün, S., Astier, V. Nonassociative quaternion algebras over rings. Isr. J. Math. 155, 125–147 (2006). https://doi.org/10.1007/BF02773952
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02773952