A classification of nilpotent compatible Lie algebras

TL;DR

This paper classifies nilpotent compatible Lie algebras up to dimension 4 over various fields, extending existing methods and identifying all isomorphism classes and parameter families.

Contribution

It extends the Skjelbred-Sund method to compatible Lie algebras and provides a complete classification up to dimension 4.

Findings

3 isomorphism classes and 1 one-parameter family in dimension 3

12 isomorphism classes, 6 one-parameter, and 2 two-parameter families in dimension 4

classification over cubically closed fields

Abstract

Working over an arbitrary field of characteristic different from $2$, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension $4$. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension $3$, and $12$ isomorphism classes, $6$ one-parameter families and two $2$-parameter families in dimension $4$