Minimal inversion complete sets and maximal abelian ideals

TL;DR

This paper explores the relationship between maximal abelian ideals in a Borel subalgebra of a simple Lie algebra and minimal inversion complete sets in its Weyl group, revealing new structural insights.

Contribution

It establishes a novel connection between maximal abelian ideals and minimal inversion complete sets, advancing understanding of Lie algebra and Weyl group structures.

Findings

Identifies a correspondence between maximal abelian ideals and minimal inversion complete sets.

Provides new structural insights into the relationship between Lie algebra ideals and Weyl group elements.

Enhances the theoretical framework linking algebraic and combinatorial properties of Lie algebras.

Abstract

Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)