Abstract
In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.
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References
Albert, A. A. Structure of algebras. Colloq. Publ., Vol. 24, Amer. Math. Soc., Providence, R. I., 1939
— A construction of exceptional Jordan division algebras. Ann. of Math. (2)67 (1958), 1–28
— On exceptional Jordan division algebras. Pacific J. Math.15 (1965), 377–404
Ferrar, J. C. Generic splitting fields of composition algebras. Trans. Amer. Math. Soc.128 (1967), 506–514
Jacobson, N. Structure and representations of Jordan algebras. Colloq. Publ., Vol. 39, Amer. Math. Soc., Providence, R. I., 1968
Lang, S. Algebra. Addison-Wesley Publishing Company, Reading, MS, 1965
Loos, O. Jordan pairs. Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-Heidelberg-New York, 1975
McCrimmon, K. The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras. Trans. Amer. Math. Soc.139 (1969), 495–510
— The Freudenthal-Springer-Tits constructions revisited. Trans. Amer. Math. Soc.148 (1970), 293–314
Petersson, H. P. Composition algebras over a field with a discrete valuation. J. of Algebra29 (1974), 414–426
Petersson, H. P. and M. L. Racine. Exceptional Jordan division algebras. Contemporary Mathematics13 (1982), 307–316
- Jordan algebras of degree 3 and the Tits process. To appear
- Pure and blende Tits constructions of exceptional Jordan division algebras. To appear
Springer, T. A. Sur les formes quadratiques d'indice zéro. C. R. Acad. Sci. Paris234 (1952), 1517–1519
— Oktaven, Jordan-Algebren und Ausnahmegruppen. University of Göttingen Lecture Notes, Göttingen, 1963
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The research of the second author was supported in part by an NSERC grant and by a von Humboldt Fellowship at the University of Münster
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Petersson, H.P., Racine, M.L. Springer forms and the first Tits construction of exceptional Jordan division algebras. Manuscripta Math 45, 249–272 (1984). https://doi.org/10.1007/BF01158039
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DOI: https://doi.org/10.1007/BF01158039