There are no rigid filiform Lie algebras of low dimension

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.)