What question did this study set out to answer?

To show that no rigid complex filiform Lie algebras exist in dimensions 9 to 11.

March 29, 2026Open Access

There are No Rigid Filiform Lie Algebras of Low Dimension

Key Points

To show that no rigid complex filiform Lie algebras exist in dimensions 9 to 11.
Analyzed filiform Lie algebras up to dimension 11.
Constructed non-trivial linear deformations in a dense Zariski open set.
Examined Euclidean neighborhoods of filiform Lie brackets.
Proved that any nearby filiform Lie bracket is non-isomorphic.
Established the existence of linear deformations in specific dimensions.
Confirmed the known nature of lower-dimensional cases.

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.)

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