On triagonal elements and centralizers in Jordan triple systems
Let V be a Jordan triple system over a field of characteristic \(\neq 2\). Necessary and sufficient conditions are given for an element x of V to be triagonal, that is, a linear combination of orthogonal tripotents. More generally, necessary and sufficient conditions are given for x to be separable, that is, for the linear span of the odd powers of x to be finite dimensional and separable. (A subsystem of V is called separable if it remains semisimple under arbitrary field extensions.) Let X be a finite dimensional separable subsystem of V. The centralizer \(C_ V(X)\) of X in V consists of all \(y\in V\) such that \(\{xyz\}=\{yxz\}\) for all \(x\in X\) and \(z\in V\). Triagonal and separable elements are used to prove that \(C_ V(X)\) is a subsystem of V and that \(C_ W(X)=W\cap C_ V(X)\) if W is a subsystem of V containing X. Moreover, if V is finite dimensional semisimple and k has characteristic 0, then \(C_ V(X)\) is semisimple. If k is perfect and V is finite dimensional, each \(x\in V\) has a unique Jordan-Chevalley decomposition as the sum of a separable and a nilpotent element that commute with each other.
scientific article; zbMATH DE number 3871596