On divisible weighted Dynkin diagrams and reachable elements

TL;DR

This paper classifies friendly pairs of nilpotent orbits with divisible weighted Dynkin diagrams in complex simple Lie algebras, explores properties of A2-pairs derived from subalgebras isomorphic to sl(3), and investigates the structure of their centralisers, especially focusing on reachable nilpotent elements.

Contribution

It introduces the classification of friendly pairs of nilpotent orbits with divisible weighted Dynkin diagrams and analyzes properties of A2-pairs and their centralisers in complex simple Lie algebras.

Findings

Classification of friendly pairs of nilpotent orbits.

A2-pairs are determined by subalgebras isomorphic to sl(3).

Centralisers of A2-pairs have a specific graded structure.

Abstract

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra $\g$. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in $\g$ determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.