关于可除加权Dynkin图与可达元
On divisible weighted Dynkin diagrams and reachable elements
摘要：
设D（e）表示复单李代数$\g$中幂零元$e$的加权Dynkin图.我们说D（e）是可除的，如果D（e）/2又是一个加权Dynkin图。(That是，整除的必要条件是$e$是偶数。）相应的一对幂零轨道被认为是友好的。在这篇笔记中，我们对友好对进行了分类，并描述了它们的一些性质。我们还观察到$\g$中的任何子代数sl（3）决定一个友好对。这样的对被称为A2-对。结果表明，A2-对中低轨道的中心化子具有一些显著的性质。设$Gx$是这样的轨道，而$h$是$x$的特征。然后$h$确定扶正器$z=z(x)$的Z分级。证明了$z$是由Levi子代数$z(0)$和$z(1)$中的两个元素生成的.特别地，（1）$z$的幂零自由基由$z(1)$和（2）$x\in [z,z]$生成。具有最后一个属性的幂零元素称为可达的。
英文摘要：
Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra $\g$. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in $\g$ determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.
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