On some algebras related to simple Lie triple systems (Q1306837)

Let \(T\) be a finite-dimensional simple Lie triple system over a field \(F\) of characteristic \(0\) and \(L = S\oplus T\) its standard embedding. The paper under review is devoted to determine the set of the homomorphisms of \(S\)-modules \(\text{Hom}_{S}(T\otimes _{F} T, T).\) Firstly the authors determine \(\text{Hom}_{S}(T\otimes _{F} T, T)\) under the assumption of \(F\) being an algebraically closed field. Here a fundamental role is played by the Dynkin diagrams associated with the Lie triple system \(T\) by \textit{J. R. Faulkner} [J. Algebra 62, 384-392 (1980; Zbl 0425.17007)]. In the last section this additional hypothesis is dropped. In the real case this provides a new proof for the determination by H. T. Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces.