Degenerations to filiform Lie algebras of dimension 9

TL;DR

This paper investigates the degenerations of 9-dimensional filiform Lie algebras, showing that most can be obtained as degenerations of other non-isomorphic algebras, with a focus on characteristically nilpotent cases.

Contribution

It extends the understanding of degenerations to 9-dimensional filiform Lie algebras, especially characteristically nilpotent ones, which were previously less understood.

Findings

Most 9-dimensional filiform Lie algebras are degenerations of other non-isomorphic algebras.

Characteristically nilpotent Lie algebras are the primary focus due to known results for rank ≥ 1.

The study advances classification by identifying degeneration relationships among these algebras.

Abstract

For most complex 9-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 $\geq 1$, only the characteristically nilpotent ones should be considered.