Classification of non-compact real simple generalized Jordan triple systems of the second kind
A generalized Jordan triple system (GJTS) is a triple system satisfying the identity \((uv(xyz))=((uvx(yz))-(x(vuy)z)+(xy(uvz))\). Kantor constructed a \(\mathbb{Z}\)-graded Lie algebra from every GJTS; the GJTS is said to be of the \(n\)-th kind if all the subspaces with absolute values of indices greater than \(n\) are zero (and \(n\) is minimal). GJTS of the first kind are well known as Jordan triple systems. GJTS of the second kind have been discussed by Kantor and there is also a connection to the structurable algebras investigated by Allison and others. This article is about finite dimensional GJTS of the second kind over the real numbers. Such a GJTS is called nondegenerate resp. compact if Yamaguti's associative trace form is nondegenerate resp. positive definite. The ``classical'' compact simple GJTS of the second kind were classified by Kaneyuki and the author in an earlier paper. The main result of this paper states that a nondegenerate GJTS of the second kind can be obtained from a compact one by ``modification''. (Modification by an involutory automorphism is carried out by applying this automorphism to the middle element of the triple product.) It is then shown that a noncompact simple GJTS of the second kind is either modification of a simple compact one or modification of the direct sum of two copies of a simple compact GJTS by the exchange involution. Taken together, this yields a classification.
scientific article; zbMATH DE number 62997