Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram

Details Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram Abelian ideals of a Borel subalgebra and subsets of the Dynkin diagram

2010-08-11

arxiv.org/abs/1008.1850v1

Mathematics

Combinatorics

8 pages

Scientific paper

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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