On Compact Generalized Jordan Triple Systems of the Second Kind
On Compact Generalized Jordan Triple Systems of the Second Kind
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第二类紧广义乔丹三系统
DOI：
10.3836/tjm/1270134265
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发表时间：
1988
影响因子：
0.6
通讯作者：
S. Kaneyuki
中科院分区：
文献类型：
--
作者：
H. Asano;S. Kaneyuki
$(uv(xyz))=((uvx)yz)-(x(vuy)z)+(xy(uvz))$ 适用于$u,$$v,$$x,$$y,$$z\in U$。此外，如果关系$(xyz)=(zyx)$对$x,$$y,$$z\in U$成立，则称$B$是一个Jordan三重系统。Koecher[5]和Meyberg[7]研究了非退化迹型Jordan三重系统与对称李代数之间的有趣关系$(\mathcal{G}, \tau)$；其中$\mathcal{G}$是具有$\mathcal{G}_{0}=[\mathcal{G}_{-1}, \mathcal{G}_{1}]$的第一类半简单分级李代数，$\tau$是具有$\mathcal{G}$的分级反转对合。我们主要关心的是将这种联系推广到广义Jordan三重系统的情况。已知（Kantor[3]），对于$U$上的广义Jordan三重系统$B$，有一个与$U_{-1}=U$对应的分级李代数$-\mathscr{G}(B)=\sum U_{i}$。如果分级李代数$\mathscr{L}(B)$是第v类，则称三重系统$B$为第v类。对于$B$（参见\S 1），在一定条件(a)下，$\mathscr{L}(B)$允许级数反转对合$\tau_{B}$。对$(\mathscr{L}(B), \tau_{B})$被认为是对应于Jordan三重系统的对称李代数的推广。另一方面，K. Yamaguti[8]为一类更广泛的三重系统引入了双线性形式$\gamma_{B}$。对于广义Jordan三重系统$B$，形式$\gamma_{B}$是对称的，并且，正如本文所看到的，它与Jordan三重系统的迹形起着相同的作用。现在假设$B$属于第二种。本文的第一个目的是证明下列含义（命题2.4、2.5、2.10和定理2.8）：
$(uv(xyz))=((uvx)yz)-(x(vuy)z)+(xy(uvz))$ is valid for $u,$ $v,$ $x,$ $y,$ $z\in U$. If, in addition, the relation $(xyz)=(zyx)$ holds for $x,$ $y,$ $z\in U$, then $B$ is said to be a Jordan triple system. Koecher [5] and Meyberg [7] studied interesting relationship between Jordan triple systems with nondegenerate trace forms and symmetric Lie algebras $(\mathcal{G}, \tau)$ ; here $\mathcal{G}$ is a semisimple graded Lie algebra of the 1st kind with $\mathcal{G}_{0}=[\mathcal{G}_{-1}, \mathcal{G}_{1}]$ , and $\tau$ is a gradereversing involution of $\mathcal{G}$. Our main concern is to generalize this connection to the case of generalized Jordan triple systems. It is known (Kantor [3]) that to a generalized Jordan triple system $B$ on $U$ there corresponds a graded Lie algebra $-\mathscr{G}(B)=\sum U_{i}$ with $U_{-1}=U$. The triple system $B$ is called of the v-th kind, if the graded Lie algebra $\mathscr{L}(B)$ is of the v-th kind. Under a certain condition (A) for $B$ (cf. \S 1), $\mathscr{L}(B)$ admits a grade-reversing involution $\tau_{B}$ . The pair $(\mathscr{L}(B), \tau_{B})$ is considered to be a generalization of the symmetric Lie algebra corresponding to a Jordan triple system. On the other hand, K. Yamaguti [8] introduced the bilinear forms $\gamma_{B}$ for a wider class of triple systems. For a generalized Jordan triple system $B$ , the form $\gamma_{B}$ is symmetric, and, as is seen in the present paper, it plays the same role as the trace form for a Jordan triple system does. Now suppose $B$ is of the 2nd kind. The first aim of this paper is to prove the following implications (Propositions 2.4, 2.5, 2.10 and Theorem 2.8):
Freudenthal-Kantor三重系统的结构理论
DOI：
--
发表时间：
2010
期刊：
Bull.Aust.Math.
影响因子：
--
作者：
S.Fukasawa;H.Kaji;楫元;神谷徳昭
通讯作者：
神谷徳昭