Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven

TL;DR

This paper classifies filiform Lie algebras of dimension up to seven into isotopism and isomorphism classes, introducing new invariants and explicitly determining their distributions over finite fields.

Contribution

It introduces new isotopism invariants and explicitly classifies filiform Lie algebras up to dimension seven into isotopism and isomorphism classes.

Findings

Explicit classification of n-dimensional filiform Lie algebras for n ≤ 7.

Introduction of new isotopism invariants for Lie algebras.

Confirmation of known isomorphism class counts and new results over finite fields.

Abstract

Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of $n$-dimensional filiform Lie algebras, for $n\leq 7$. We also deal with the distribution of such algebras into isomorphism classes, for which we confirm some known results and we prove that there exist $p+8$ isomorphism classes of seven-dimensional filiform Lie algebras over the finite field $\mathbb{F}_p$ if $p\neq 2$.