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Simple noncommutative Jordan algebras satisfying \(([x,y],y,y)=0\). (English) Zbl 0812.17001

Noncommutative Jordan rings satisfying the identity \(([ x,y], y,y) =0\) (where \((x,y,z)= (xy)z- x(yz)\) and \([ x,y]= xy- yx\) are, respectively, the associator and commutator) were studied by I. P. Shestakov [Algebra Logic 10, 252-280 (1973); translation from Algebra Logika 10, 407-448 (1971; Zbl 0259.17001)].
The aim of the paper under review is to completely characterize the finite dimensional central simple noncommutative Jordan algebras over fields of characteristic not two satisfying this identity. Thus, the main result is that these algebras are either central simple commutative Jordan algebras, central simple alternative algebras or forms of a specific seven dimensional algebra. This result is obtained by considering a mutation of the original algebra. This satisfies an identity verified by the algebra of color introduced by G. Domokos and S. Kövesi-Domokos [J. Math. Phys. 19, 1477-1481 (1978; Zbl 0384.17001)]. Then the author applies previous results by himself [J. Algebra 160, 93-129 (1993; Zbl 0788.17001)]. The multiplication table of the forms of the seven dimensional algebra considered is obtained by relating them to the forms of the color algebra, which were studied by the reviewer.

MSC:

17A15 Noncommutative Jordan algebras
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