根系和图解演算。II。卡特图的二次型
Root systems and diagram calculus. II. Quadratic forms for the Carter diagrams
摘要：
对于任何含有4-圈的Cartan图$Γ$，我们引入了部分Cartan矩阵$B_L$，它类似于Dynkin图的Cartan矩阵.从$Γ$中得到一个连接图，通过将一个根与其键加在一起，使得得到的根的子集是线性无关的。在对偶部分Weyl群（与$B_L$相关）作用下连接的连接图构成连接系统，它类似于半单李代数表示论中出现的权系统.对于卡特图，$E_6(a_i)$和$E_6$（分别分别为$E_7(a_i)$和$E_7$。$D_n(a_i)$和$D_n$），连接系统分别具有2、1、1个组件，每个组件分别包含27、56个$2n$元件。数27、56和$2n$是半单李代数的最小基本表示的公知维数，分别是$E_6$、$E_7$和$D_n$。8-细胞纺锤形连锁子系统（称为loctet）在描述连锁系统中起着至关重要的作用。这说明重量系统也可以用八隅体来描述。
英文摘要：
For any Carter diagram $Γ$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent. The linkage diagrams connected under the action of dual partial Weyl group (associated with $B_L$) constitute the linkage system, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp. $E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has, respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56, $2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the smallest fundamental representations of semisimple Lie algebras, respectively, for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called loctets play the essential role in describing the linkage systems. It turns that weight systems also can be described by means of loctets.
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