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

TL;DR

This paper explores the relationship between Abelian ideals of Borel subalgebras in simple Lie algebras and subsets of their Dynkin diagrams, revealing combinatorial correspondences and constructing explicit bijections.

Contribution

It establishes a connection between Abelian ideals and Dynkin diagram subsets, providing combinatorial explanations and explicit bijections for types A and C.

Findings

Number of Abelian ideals with k generators equals subsets with k connected components

Constructed bijections between Abelian ideals and Dynkin diagram subsets for types A and C

Linked the theory of Peterson and Kostant to the combinatorial structure of Abelian ideals

Abstract

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.