Normalisers of abelian ideals of a Borel subalgebra and $\mathbb Z$-gradings of a simple Lie algebra

TL;DR

This paper characterizes the normalisers of abelian ideals in a Borel subalgebra of a simple Lie algebra and explores their connection to $ ext{Z}$-gradings, providing explicit descriptions for a broad class of ideals.

Contribution

It offers an explicit description of normalisers for a class of abelian ideals, including all maximal ones, and investigates their link to $ ext{Z}$-gradings of the Lie algebra.

Findings

Normalisers of certain abelian ideals are explicitly described.

A relationship between abelian ideals and $ ext{Z}$-gradings is established.

Includes all maximal abelian ideals in the description.

Abstract

Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of all abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. If $\mathfrak a\in\mathfrak{Ab}$, then the normaliser of $\mathfrak a$ is a standard parabolic subalgebra of $\mathfrak g$. We give an explicit description of the normaliser for a class of abelian ideals that includes all maximal abelian ideals. We also elaborate on a relationship between abelian ideals and $\mathbb Z$-gradings of $\mathfrak g$ associated with their normalisers.