Jordan triple systems with completely reducible derivation or structure algebras
Results that relate the derivations of an algebra to the structure of the algebra are extended in this article from Jordan algebras to Jordan triple systems. Specifically, let V be a finite dimensional Jordan triple system over a field k of characteristic zero. Then the structure algebra of V is completely reducible if and only if V is the direct sum of a semisimple and a trivial ideal. V is semisimple if and only if its derivation algebra is completely reducible and every derivation has trace zero. As a consequence, the author characterizes all real Jordan triple systems that have a compact automorphism group. The preceding theorems all follow from the result that, when k is algebraically closed, the derivation algebra of V is completely reducible if and only if V is the direct sum of two ideals such that one ideal is semisimple and the other is either trivial or a direct sum of ideals of the form \(X\oplus M\), where X and M are nonzero finite dimensional vector spaces over k, (\(\cdot,\cdot)\) is a nondegenerate symmetric bilinear form on X, and \(X\oplus M\) has quadratic representation \(P(x\oplus m)(y\oplus n)=(2(x,y)x-(x,x)y)\oplus ((x,x)n+2(x,y)m)\) for x,y\(\in X\) and m,n\(\in M\).
scientific article; zbMATH DE number 3889753