Cubic and symmetric compositions over rings (Q2465206)

On the space of trace zero elements of a degree \(3\) separable alternative algebra \(A\) over a field \(k\) of characteristic \(\neq 3\), \textit{S. Okubo} [Hadronic J. 1, No. 4, 1250--1278 (1978; Zbl 0417.17011)] and \textit{J. R. Faulkner} [Proc. Am. Math. Soc. 104, No. 4, 1027--1030 (1988; Zbl 0698.17004)] found a multiplication \(\star\) which has the composition and associativity property: \(q(x\star y)=q(x)q(y)\), \(b(x\star y,z)=b(x,y\star z)\), where \(q\) is \(-1/3\) times the quadratic trace of \(A\) and \(b\) is the polar form of \(q\). This was used by \textit{A. Elduque} and \textit{H. C. Myung} [Commun. Algebra 21, No. 7, 2481--2505 (1993; Zbl 0781.17002)] to establish an equivalence between symmetric composition algebras of dimension \(\geq 2\) and separable alternative algebras of degree \(3\) (in case the ground field contains the cubic roots of \(1\), otherwise one has to consider separable alternative algebras with an involution of the second kind over the field obtained by adjoining to \(k\) these cubic roots). By results of \textit{R. D. Schafer} [J. Math. Mech. 12, No. 5, 777--792 (1963; Zbl 0115.25802)] these last algebras are the same as unital algebras of dimension \(\geq 3\) with nondegenerate multiplicative cubic form. In remark (34.26) of \textit{The book of involutions} by \textit{M. A. Knus} et al. [Am. Math. Soc., Providence, RI (1998; Zbl 0955.16001)], the question was posed of finding a direct proof of the equivalence between the symmetric compositions and the multiplicative cubic forms. The paper under review gives a direct proof of this correspondence, and it does with the extra important features of not making any non-degeneracy assumption and of working over arbitrary unital commutative base rings. The main result of the paper asserts that, assuming \(3\) is invertible in the base ring \(k\) and that \(k\) contains a primitive sixth root of unity, then the categories of unital cubic compositions and of symmetric compositions over \(k\) (both defined without non-degeneracy conditions) are equivalent. The assumption on the sixth root of unity is dropped by considering the category of unital cubic compositions of the second kind, which involve working with algebras over \(K=k[t]/(t^2-t+1)\).