Nilpotent Lie algebras with two centralizer dimensions over a finite field

TL;DR

This paper classifies all 3-step nilpotent Lie algebras over finite fields that have exactly two distinct centralizer dimensions, extending previous results on their structure and properties.

Contribution

It provides a complete classification of 3-step nilpotent Lie algebras with two centralizer dimensions over finite fields, building on earlier work by Barnea and Isaacs.

Findings

Classification of all such Lie algebras achieved

Identification of structural properties specific to 3-step nilpotency

Extension of known results to a broader class of Lie algebras

Abstract

A result of Barnea and Isaacs states that if $L$ is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of $L$ is either $2$ or $3$. In this article, we classify all such finite dimensional $3$-step nilpotent Lie algebras over a finite field.