Cyclic compositions and trisotopies

Abstract

Cyclic compositions in the sense of Springer [T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Universität Göttingen, 1963] and Knus, Merkurjev, Rost and Tignol [M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. With a preface in French by J. Tits] are investigated by means of cyclic trisotopies, a concept originally due to Albert [A.A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math.(2) 67 (1958) 1-28]. Using the quadrupling of composition algebras, we enumerate cyclic trisotopies and compositions in a rational manner, i.e., without extending the base field. We relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the Tits process of cubic Jordan algebras.

The cyclicity problem for Albert algebras

“…IX). The aforementioned question, which derives its importance not only from Albert's own work in ( [1], [2]) on the subject, but also from Springer's theory of cyclic compositions (see [6], [11], [18], [19]), has been answered affirmatively by Petersson-Racine ( [12], Thm. 4) in the presence of the cube roots of unity, and by Petersson ([8] in characteristic 3.…”