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Minimal identities for Jordan algebras of degree 2. (English) Zbl 0579.17014

The purpose of the paper under review is the investigation of the polynomial identities satisfied by all Jordan algebras of degree 2 (i.e. for all algebras J(Q,1) obtained from a quadratic form Q). The main result claims that over a field K of characteristic zero all identities of minimal degree are consequences of \[ k=x^ 2\quad S_ 3(V(x_ 1),V(x_ 2),V(x_ 3))-x\quad S_ 3(V(x_ 1),V(x_ 2),V(x_ 3))\circ x, \] where \(S_ 3\) is the standard identity and \(zV(x)=z\circ x\). It is shown that the Jordan element of \(K<x,y,z>\lambda =[[x,y]^ 2,[x,z]]\) is an identity of J(Q,1) but \(\lambda =0\) does not follow from \(k=0\). The author states the conjecture that the polynomials k and \(\lambda\) form a basis for the T-ideal of J(Q,1). Some identities for the Jordan algebras of degree 3 are obtained as well.
Reviewer: V.Drensky

MSC:

17C05 Identities and free Jordan structures
16Rxx Rings with polynomial identity
Full Text: DOI

References:

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