On generalized Jordan triple systems and their psi-modifications

Kantor定义了一个广义Jordan三系(GJTS)，并由此构造了一个分次李代数(GLA)。在GLA是经典的情况下，S.Kaneyuki-H.Asano给出了二阶实简单紧GJTS的分类。H.Asano试图通过一个使用*-修饰的过程将二阶非紧实简单GJTS分类。他成功地对它们进行了分类，以防它们的GLA是古典的。另一方面，K.Yamaguti定义了统一GJTS和Freudenthal三系(FTS)的U(Epsilon)-代数(epsilon=*1)。我们这项研究的主要目的是在相关的GLA例外的情况下对GJTS进行分类。为此，我们将Yamaguti的U(Epsilon)-代数推广到epsilon是三元系的自同构的情况。在这项研究中，我们给出了一个与Asano理论平行的一般理论，这是进行U(Epsilon)-代数(epsilon=*1)分类的基础。在续集中，GJTS的分类和FTS的分类同时完成。

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.