On generalized Jordan triple systems and their psi-modifications

Kantor定义了广义Jordan三重系统（GJTS），并以此构造了一个梯度李代数（GLA）。S.Kaneyuki-H给出了二阶实简单紧化GJTS的分类。如果相关的GLA是经典的。h.a asano尝试用*-修正的方法对二阶非紧凑型实简单GJTS进行分类。他成功地对它们进行了分类，以防它们的GLA是经典的。另一方面，k.a yamaguti定义了一个统一GJTS和Freudenthal三重系统（FTS）的U(epsilon)-代数（epsilon = *1）。我们这项研究的主要目的是在相关GLA异常的情况下对GJTS进行分类。为此，我们将Yamaguti的U()-代数推广到是三重系统的自同构的情况。在本研究中，我们给出了一个与Asano的理论平行的一般理论，这是进行U(epsilon)-代数（epsilon=*1）分类的基础。在后续研究中，同时完成了GJTS和FTS的分类。

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.