Ad-nilpotent Ideals of Minimal Dimension

TL;DR

This paper explores ad-nilpotent ideals in complex simple Lie algebras, establishing their relations with affine Weyl groups and nilpotent orbits, and introduces a new equivalence relation compatible with existing algebraic structures.

Contribution

It defines a novel left equivalence relation for ad-nilpotent ideals and proves its compatibility with affine Weyl group structures and Lusztig's star operator in type A.

Findings

Established a left equivalence relation for ad-nilpotent ideals.

Proved compatibility with affine Weyl group left cell structure.

Connected the ideals' structure with Lusztig's star operator.

Abstract

n this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and generators, and prove that the equivalence relation is compatible with the left cell structure of affine Weyl group of $\ccg$ and Lusztig's star operator for type $\tilde A_{n-1}$.