Orbits of the automorphism group of the exceptional Jordan algebra.
HTML articles powered by AMS MathViewer
- by John R. Faulkner PDF
- Trans. Amer. Math. Soc. 151 (1970), 433-441 Request permission
Abstract:
Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra $\Im$ to be conjugate under the automorphism group $\mathrm {Aut} \Im$ of $\Im$ are obtained. It was known previously that if $\Im$ is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under $\mathrm {Aut} \Im$ if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of $\Im$.References
- N. Jacobson, A coordinatization theorem for Jordan algebras, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1154–1160. MR 140553, DOI 10.1073/pnas.48.7.1154 —, Structure and representations of Jordan algebras. Chapter IX, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R. I., 1969.
- Kevin McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495–510. MR 238916, DOI 10.1090/S0002-9947-1969-0238916-9
- T. A. Springer and F. D. Veldkamp, Elliptic and hyperbolic octave planes. I, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 413–428. MR 0155227
- J. Williamson, The Equivalence of Non-Singular Pencils of Hermitian Matrices in an Arbitrary Field, Amer. J. Math. 57 (1935), no. 3, 475–490. MR 1507088, DOI 10.2307/2371179
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 151 (1970), 433-441
- MSC: Primary 17.40
- DOI: https://doi.org/10.1090/S0002-9947-1970-0263886-5
- MathSciNet review: 0263886