Ad-nilpotent Ideals and Equivalence Relations

TL;DR

This paper explores the structure of ad-nilpotent ideals in complex simple Lie algebras, establishing their relations with affine Weyl groups, nilpotent orbits, and defining an equivalence relation compatible with existing algebraic structures.

Contribution

It introduces a new left equivalence relation for ad-nilpotent ideals and proves its compatibility with affine Weyl group structures and Lusztig's star operator in type A_{n-1}.

Findings

Defined a left equivalence relation for ad-nilpotent ideals.

Proved compatibility with affine Weyl group left cell structure.

Established connection with Lusztig's star operator in type A_{n-1}.

Abstract

In 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 $\tilde A_{n-1}$.