Pseudo-composition algebras
Let \(F\) be an algebraically closed field of characteristic not 2 or 3, and let \(A\) be a commutative algebra over \(F\). Then \(A\) is called a pseudo-composition algebra if there exists a symmetric bilinear form \(\varphi\neq 0\) such that \(x^ 3= \varphi(x,x)x\), for all \(x\in A\). The pseudo-composition algebra \(A\) with identity element \(e\) is of quadratic type if every element \(x\in A\) satisfies an equation of the form \(x^ 2+\beta(x)x+ \gamma(x,x)e=0\). In this paper it is shown that in any pseudo-composition algebra \(A\) the radical \(\text{Rad }\varphi\) is an ideal, and that either (i) \(A\) is of quadratic type, (ii) \(A/\text{Rad }\varphi\) is of quadratic type, or (iii) \(A\) may be constructed from an alternative quadratic algebra.
scientific article; zbMATH DE number 561286
scientific article