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Pseudo-composition algebras. (English) Zbl 0788.17003

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.

MSC:

17A75 Composition algebras
17A65 Radical theory (nonassociative rings and algebras)
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References:

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