Jordan algebraic and differential geometric study of homogeneous complex manifolds

对于任何复简单李代数，Freudenthal变量被定义为对应的伴随变量的线性截面，由第1阶子空间相对于李代数的复接触型级积。本研究[KY]采用先验方法证明了Freudenthal品种的同质性。用第0阶李子代数对应的伴随群的李子群，给出了第一阶李子空间轨道结构的统一描述。然后，仅从1阶子空间所具有的辛三重系统的公理出发，导出了Freidenthal变异体作为复投影变异体的若干性质。此外，它还为1- 1阶子空间享受辛三重系统公理提供了一个捷径交替证明。这些结果是基于K.Yamaguti-H在简单复辛三重系统上构造了秩不小于2的所有复单李代数。浅野（1975），反之亦然（1975）。请注意，这些结果避免了根据所有复简单李代数的分类逐个讨论，并且这些结果是通过考虑辛三重系统的子结构来驱动的。在代数上，H.Asano（未发表）研究了辛三重系统的结构理论。这些结果对辛三元系统的代数结构理论提供了一个几何启示。在本研究[Y]中，找到了两个具有一维中心的非同构约李群的典型例子，使得它们的李代数同构。在文献中，没有关于这些例子的证据。该结果首次通过考虑B和D型复简单李代数的第0阶李子代数对应的伴随群的李子群对第1阶子空间的轨道分解的结果得到。

For any complex simple Lie algebra, the Freudenthal variety is defined as the linear section of the corresponding adjoint variety by the 1-st graded subspace with respect to the complex contact type gradation of the Lie algebra. In this research [KY], the homogeneity of the Freudenthal varieties is proved in a priori way. And a unified description is given for the orbit structure of the first graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra. Then several properties of the Freidenthal varieties as complex projective varieties are derived only from the axioms of symplectic triple systems which are enjoyed by the 1-st graded subspace. Moreover, it provides a short cutting alternating proof for the 1-st graded subspace to enjoy the axioms of symplectic triple systems. These results are based on the construction of all complex simple Lie algebra of rank non less than two from simple complex symplectic triple systems by K.Yamaguti-H.Asano (1975), and vice varsa by H.Asano.(1975). Note that these results avoid case-by-case arguments according to the classification of all complex simple Lie algebras, and that these results are drived by considering substructure of symplectic triple systems. Algebraically, a structure theory of symplectic triple systems is studied by H.Asano (unpublished). These results give a geometric light on this algebraic structure theory of symplectic triple systems.In this research [Y], typical examples are found on two non-isomorphic reductive Lie groups with one dimensional center such that their Lie algebras are isomorphic. In the literature, no proof was published on these examples. This result is firstly observed by considering the result of the orbit decomposition of the 1-st graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra of complex simple Lie algebras of type B and D.