Borel子代数的Abel理想与Dynkin图的子集
Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram
摘要：
设$g$是单李代数，$Ab(g)$是$g$的Borel子代数的Abel理想集.本文讨论了$Ab(g)$和$g$的Dynkin图的子集之间的一个有趣的联系。我们注意到具有$k$生成元的交换理想的数目等于具有$k$连通分支的Dynkin图的子集的数目。对于类型为$A_n$或$C_n$的$g$，我们通过构造一个合适的双射来提供这种重合的组合解释。基于Peterson和Kostant的理论，我们还构造了$Ab(g)$与Dynkin图的子集之间的另一个一般双射.
英文摘要：
Let $g$ be a simple Lie algebra and $Ab(g)$ the set of Abelian ideals of a Borel subalgebra of $g$. In this note, an interesting connection between $Ab(g)$ and the subsets of the Dynkin diagram of $g$ is discussed. We notice that the number of abelian ideals with $k$ generators equals the number of subsets of the Dynkin diagram with $k$ connected components. For $g$ of type $A_n$ or $C_n$, we provide a combinatorial explanation of this coincidence by constructing a suitable bijection. We also construct another general bijection between $Ab(g)$ and the subsets of the Dynkin diagram, which is based on the theory developed by Peterson and Kostant.
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