%0 Journal Article
%T Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram
%A Dmitri I. Panyushev
%J Mathematics
%D 2010
%I arXiv
%X 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.
%U http://arxiv.org/abs/1008.1850v1