Solvable Infinite Filiform Lie Algebras

TL;DR

This paper investigates solvable infinite filiform Lie algebras, proving that their derived algebra is abelian and clarifying their classification over the complex numbers with additional computations.

Contribution

It completes the classification of solvable infinite filiform Lie algebras by providing necessary computations and clarifying previous incomplete results.

Findings

If L is solvable, then [L,L] is abelian.

The isomorphism classes are characterized over the complex numbers.

Provides additional computations to complete previous classifications.

Abstract

An infinite filiform Lie algebra L is residually nilpotent and its graded associated with respect to the lower central series has smallest possible dimension in each degree but is still infinite. This means that gr(L) is of dimension two in degree one and of dimension one in all higher degrees. We prove that if L is solvable, then already [L,L] is abelian. The isomorphism classes in this case are given in a paper by Bratzlavsky, but the proof there is incomplete. We make the necessary additional computations and restate Bratzlavskys result when the ground field is the complex numbers.