Jordan algebraic and differential geometric study of homogeneous complex manifolds

对于任何复单李代数，Freudenthal簇被定义为对应的伴随簇关于复接触型阶化的第一阶阶化子空间的线性部分.在这项研究中[KY]，同质性的弗赖登塔尔品种证明了先验的方式。并利用0阶分次李子代数所对应的伴随群的李群对第一阶分次子空间的轨道结构给出了统一的描述。然后仅从一阶分次子空间所具有的辛三系公理出发，得到了Fredenthal簇作为复射影簇的若干性质。此外，还为一阶分次子空间享受辛三系公理提供了一种简捷的交替证明。这些结果是基于K.Yamaguti-H.Asano（1975）由单复辛三系构造出秩不小于2的所有复单李代数，以及H. Asano的varsa构造出秩不小于2的所有复单李代数。（1975年）。注意，这些结果避免了根据所有复单李代数的分类进行个案论证，并且这些结果是通过考虑辛三系的子结构而得到的。在代数上，H.Asano研究了辛三系的结构理论（未发表）。这些结果为辛三系的代数结构理论提供了一个几何的线索。在本研究[Y]中，我们找到了两个具有一维中心的非同构约化李群使得它们的李代数同构的典型例子。在文献中，没有证据表明这些例子。这一结果首先通过考虑B型和D型复单李代数的0阶李子代数所对应的伴随群的李群对1阶分次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 Algebraically, a structure theory of symplectic triple systems is studied by H.Asano (unpublished).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.