Enumeration of ad-nilpotent ideals of parabolic subalgebras for exceptional types

TL;DR

This paper uses computational methods to count ad-nilpotent and abelian ideals in parabolic subalgebras of exceptional Lie algebras, providing explicit enumeration for types E, F, and G.

Contribution

It introduces a computational approach to enumerate ad-nilpotent and abelian ideals in exceptional Lie algebra parabolic subalgebras, filling a gap in explicit counts.

Findings

Number of ad-nilpotent ideals for each exceptional type

Number of abelian ideals for each exceptional type

Implementation of GAP4 for algebraic enumeration

Abstract

Using GAP4, we determine the number of ad-nilpotent and abelian ideals of a parabolic subalgebra of a simple Lie algebra of exceptional types E, F or G.