On generalized Jordan triple systems and their psi-modifications

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I.L. Kantor defined a generalized Jordan triple system (GJTS), and he constructed a graded Lie algebra (GLA) from it. The classification of real simple compact GJTS's of the 2nd order was given by S.Kaneyuki-H.Asano in case that the associated GLA's were classical. H.Asano tried to clasify non-compact real simple GJTS's of the 2nd order by a procedure to use the *-modification. He succeeded in classifying them in case that their GLA's were classical. On the other hand, K.Yamaguti defined a U(epsilon)-algebra (epsilon = *1)unifying a GJTS and a Freudenthal triple system (FTS). Our main purpose of this research was to classify GJTS's in case that the associated GLA's were exceptional. For this purpose, we extended Yamaguti's U(epsilon)-algebra to the case that epsilon was an automorphism of the triple system. In this research, we gave a general theory parallel to Asano's one, which is basic for carring out the classification of U(epsilon)-algebras (epsilon=*1). In the sequel, the classification of GJTS's and that of FTS's were completed simultaneously.