There are no rigid filiform Lie algebras of low dimension

TL;DR

This paper demonstrates that in dimensions up to 11, complex filiform Lie algebras are not rigid, as they admit non-trivial deformations, expanding understanding of their structural flexibility.

Contribution

It proves the non-existence of rigid complex filiform Lie algebras in dimensions ≤11 by constructing explicit deformations.

Findings

No rigid filiform Lie algebras in dimensions ≤11

Existence of non-trivial deformations in dimensions 9, 10, 11

Deformations form a Zariski open dense set in the variety

Abstract

We prove that there are no rigid complex filiform Lie algebras in the variety of (filiform) Lie algebras of dimension less than or equal to 11. More precisely we show that in any Euclidean neighborhood of a filiform Lie bracket (of low dimension), there is a non-isomorphic filiform Lie bracket. This follows by constructing non trivial linear deformations in a Zariski open dense set of the variety of filiform Lie algebras of dimension 9, 10 and 11. (In lower dimensions this is well known.)