Orbits of the automorphism group of the exceptional Jordan algebra.

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 $\Im$.