Ideals in Parabolic Subalgebras of Simple Lie Algebras

TL;DR

This paper characterizes ad-nilpotent ideals in parabolic subalgebras of simple Lie algebras using antichains in root systems, providing necessary and sufficient conditions and explicit classifications for classical types.

Contribution

It offers a complete characterization of ad-nilpotent ideals via antichains and recovers known results for classical Lie algebras, advancing understanding of their structure.

Findings

Provides necessary and sufficient conditions for antichains to determine ad-nilpotent ideals.

Classifies all such antichains for classical simple Lie algebras.

Identifies the unique irreducible ideal in a parabolic subalgebra as a module.

Abstract

We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by the corresponding subset of roots.