Cyclic compositions and trisotopies
\textit{T. A. Springer} introduced the notion of cyclic composition in a work published at Universität Göttingen, mimeographed notes (1963) entitled ``Oktaven, Jordan-Algebren und Ausnahmegruppen'' [see the revised English version ``Octonions, Jordan algebras and exceptional groups.'' Berlin: Springer (2000; Zbl 1087.17001)]. The original name of the notion ``twisted composition'' appears also in ``The Book of Involutions'' by \textit{M.-A. Knus, A. S. Merkurjev, M. Rost} and \textit{J.-P. Tignol} (1998; Zbl 0955.16001). A quintuple \(S=(E,\rho,M,Q,*)\) is a cyclic composition if (1) \((E,\rho)\) is a cyclic cubic étale \(k\)-algebra, (2) \((M,Q)\) is a quadratic space over \(E\), so \(Q: M\to E\) is non-singular, (3) \((M,*)\) is a nonassociative algebra, (4) \(Q\) permits twisted composition: \(Q(x*y)=\rho(Q(x))\rho^2(Q(y))\), and (5) \(Q\) is twisted associative: \(Q(x*y,z)=\rho(Q(y*z,x))\). The concept was a tool for the study of Albert algebras (i.e. exceptional simple Jordan algebras). It was \textit{A. A. Albert} himself who in a work in Ann. Math. (2) 67, 1--28 (1958; Zbl 0079.04701) had introduced a different tool called ``cyclic trisotopies'' in the terminology of the paper under review. These ``cyclic trisotopies'' are a kind of composition algebras over étale cubic algebras. To be more precise a quintuple \(A=(E,\rho,C,g,b)\) is called a cyclic trisotopy when (1) \((E,\rho)\) is a cyclic cubic étale \(k\)-algebra, (2) \(C\) is a composition algebra over \(E\), (3) \(b\) is an invertible element of \(C\), (4) \(g: C\to C\) is a \(\rho\)-semilinear map satisfying the identity \(g(xy)=[g(x)b][b^{-1}g(y)]\) and (5) \(g\) stabilizes the line through \(b\) and \(g^3(x)=bxb^{-1}\). The work under review studies both concepts relating them by means of a quadrupling of composition algebras. This allows the author to enumerate cyclic trisotopies and compositions in a rational way (without scalar extensions of the ground field). The author relates cyclic trisotopies to simple associative algebras of degree \(3\) with involution, and to the Tits process of cubic Jordan algebras.
scientific article; zbMATH DE number 5118805
scientific article