Nilpotent Lie Algebras of breadth type $(0,3)$

TL;DR

This paper classifies finite-dimensional nilpotent Lie algebras of breadth type (0,3) over various fields, providing a complete classification over finite fields of odd characteristic and partial results over other fields.

Contribution

It offers a complete classification of nilpotent Lie algebras of breadth type (0,3) over finite fields of odd characteristic and extends partial classifications to other fields.

Findings

Complete classification over finite fields of odd characteristic.

Partial classification over finite fields of even characteristic, ields of \u00c4, ields of ields.

Discussion of 2-step nilpotent Camina Lie algebras.

Abstract

For a natural number $m$, a Lie algebra $L$ over a field $k$ is said to be of breadth type $(0, m)$ if the co-dimension of the centralizer of every non-central element is of dimension $m$. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type $(0, 3)$ over $\mathbb F_q$ of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, $\mathbb C$ and $\mathbb R$. We also discuss $2$-step nilpotent Camina Lie algebras.