On polynomial identities of Jordan pairs of rectangular matrices
If \(k\) is a field of characteristic different from 2 and 3, a pair \(V=(V^+,V^-)\) of \(k\)-vector spaces is called a Jordan pair if there is a trilinear mapping \(V^{\sigma}\times V^{-\sigma}\times V^{\sigma}\to V^{\sigma}\), \(\sigma=\pm\), denoted by \(\{xay\}\), \(x,y\in V^{\sigma}\), \(a\in V^{-\sigma}\), such that \(\{xay\}=\{yax\}\) and \(\{xa\{ybz\}\}-\{yb\{xaz\}\}=\{\{xay\}bz\}-\{y\{axb\}z\}\) for all \(x,y,z\in V^{\sigma}\), \(a,b\in V^{-\sigma}\), \(\sigma=\pm\). This definition has a natural generalization when \(k\) is any associative commutative ring and \(V^+,V^-\) are \(k\)-modules. In the paper under review the author studies the polynomial identities of the Jordan pair \(P(m,n)=(M_{m,n},M_{n,m})\), where \(M_{i,j}\) is the set of all \(i\times j\) matrices over \(k\) and the operation is given by \(\{xyz\}=xyz+zyx\). The first result is that over any field the polynomial identities of the pair \(P(m',n')\) are consequences of these of \(P(m,n)\) if and only if \(m\leq m'\) and \(n\leq n'\). Then the author gives a minimal basis of the identities of the pair \(P(1,n)\), \(n=1,2,\ldots,\infty\), over a field of characteristic 0. As a byproduct of his approach the author establishes that central simple (not necessarily associative) algebras of different dimension can be distinguished by means of Capelli identities.
scientific article; zbMATH DE number 1040497
scientific article