Freudenthal algebra

In algebra, Freudenthal algebras are certain Jordan algebras constructed from composition algebras.

Definition

Suppose that C is a composition algebra over a field F and a is a diagonal matrix in GL_n(F). A reduced Freudenthal algebra is defined to be a Jordan algebra equal to the set of 3 by 3 matrices X over C such that X^Ta=aX. A Freudenthal algebra is any twisted form of a reduced Freudental algebra.

References

Freudenthal, Hans (1985) [1951], "Oktaven, Ausnahmegruppen und Oktavengeometrie", Geom. Dedicata, 19 (1): 7–63, doi:10.1007/BF00233101, MR 0797151, S2CID 121496094
Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, vol. 44, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, Zbl 0955.16001

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