Reduced elements in Jordan triple systems
Reduced elements in Jordan triple systems J over k (i.e. elements \(c\in J\) with P(c)J\(\subset kc)\) have proven useful in several instances. They appear in E. Zel'manov's work, and also in U. Hirzebruch's work on Riemannian symmetric spaces of rank 1, in the construction of generic reducing fields by N. Jacobson and H. Petersson and in the study of \(JBW^*\)-triples (Y. Friedman, B. Russo and others). Recently, \textit{J. Faulkner} [J. Algebra 89, 318-322 (1984; Zbl 0536.17001)] used reduced elements to construct an invariant symmetric bilinear form for a wide class of Jordan and Lie triple systems over fields of characteristic \(\neq 2.\) The author studies \(\rho (J)=\{c\in J\); c reduced\(\}\) in general: \(\rho\) (J) always spans an ideal R(J) in J. If k is a domain acting torsion- freely on \(\rho\) (J), R(J) carries an invariant symmetric bilinear form f, the Faulkner form, which is characterized by \(P(c)x=f(c,x)c\), \(c\in \rho (J)\), \(x\in R(J)\). The Faulkner form coincides with Loos's generic trace form if J is reduced \((J=R(J))\) and finite-dimensional separable over a field, and it coincides with Albert's trace form if J is a reduced Jordan algebra. The author gives a new characterization of nondegeneracy: J is nondegenerate (as a triple system) iff the Faulkner form f is ''quadratically'' nondegenerate, i.e. \[ Q Rad f=\{z\in R(J);\quad f(z,R(J))=0=f(P(z)J,R(J))\}=0. \] In this case R(J) has a Dieudonné decomposition into a direct sum of finitely many simple ideals as soon as R(J) has minimum condition on ideals.
scientific article; zbMATH DE number 3918612