Filiform Lie algebras of dimension 8 as degenerations

TL;DR

This paper investigates degenerations among 8-dimensional filiform Lie algebras, identifying specific degenerations to each algebra, with a focus on characteristically nilpotent cases, advancing understanding of their structural relationships.

Contribution

It provides a comprehensive classification of degenerations among 8-dimensional filiform Lie algebras, especially highlighting the role of characteristically nilpotent cases.

Findings

Each 8-dimensional filiform Lie algebra degenerates from a non-isomorphic algebra.

Degenerations are explicitly constructed for all such algebras.

Focus on characteristically nilpotent algebras due to prior knowledge on nilpotent Lie algebras of rank ≥ 1.

Abstract

For each complex 8-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank $\ge 1$, only the caracteristically nilpotent ones should be considered.