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Octonion Algebras over Rings Are Not Determined by their Norms

Published online by Cambridge University Press:  20 November 2018

Philippe Gille
Philippe Gille*
Affiliation:
UMR 8552 du CNRS, DMA, Ecole Normale Supérieure, F-75005 Paris, France e-mail: [email protected]
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Abstract

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Answering a question of H. Petersson, we provide a class of examples of a pair of octonion algebras over a ring having isometric norms.

Keywords

14L2420G41Octonion algebrastorsorsdescent
Type
Award Winners
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 303 - 309
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Bix, R., Isomorphism theorems for octonion planes over local rings. Trans. Amer. Math. Soc. 266 (1981, no. 2, 423–439. http://dx.doi.org/10.1090/S0002-9947-1981-0617543-0 CrossRefGoogle Scholar
[2] Colliot–Théléne, J.-L. and Sansuc, J.-J., Fibrés quadratiques et composantes connexes réelles. Math. Ann. 244 (1979, 105–134.http://dx.doi.org/10.1007/BF01420486 Google Scholar
[3] Demazure, M. and Gabriel, P., Groupes algébriques. North-Holland, Amsterdam, 1970.Google Scholar
[4] Giraud, J., Cohomologie non-abélienne. Die Grundlehren der mathematischenWissenschaften, 179, Springer-Verlag, Berlin-New York, 1971.Google Scholar
[5] Igusa, J.-I., A classification of spinors up to dimension twelve. Amer. J. Math. 92 (1970, 997–1028.http://dx.doi.org/10.2307/2373406 Google Scholar
[6] Knus, M.-A., Quadratic and hermitian forms over rings. Grundlehren der mathematischen Wissenschaften, 294, Springer-Verlag, Berlin, 1991.Google Scholar
[7] Knus, M.-A., Ojanguren, M., and Sridharan, R., Quadratic forms and Azumaya algebras. J. Reine Angew. Math. 303/304 (1978, 231–248.Google Scholar
[8] Loos, O., Petersson, H. P, and Racine, M. L., Inner derivations of alternative algebras over commutative rings. Algebra Number Theory 2 (2008, no. 8, 927–968.http://dx.doi.org/10.2140/ant.2008.2.927 Google Scholar
[9] Mimura, M., Homotopy theory of Lie groups. In: Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 951–991.CrossRefGoogle Scholar
[10] Petersson, H. P., Composition algebras over algebraic curves of genus zero. Trans. Amer. Math. Soc. 337 (1993, no. 1, 473–493.http://dx.doi.org/10.1090/S0002-9947-1993-1108613-X Google Scholar
[11] Séminaire de Géométrie algébrique de l’I.H.E.S., 1963-1964, schémas en groupes. dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, 151–153. Springer-Verlag, Berlin, 1970.Google Scholar
[12] Springer, T. A. and Veldkamp, F. D., Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.Google Scholar
[13] Steinmetz-Zikesch, A., Algèbres de Lie de dimension infinie et théorie de la descente. Mém. Soc. Math. Fr., to appear.Google Scholar
[14] van der Blij, F. and Springer, T. A., The arithmetics of octaves and of the group G2. Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959, 406–418.Google Scholar