Deformations and rigidity in varieties of Lie algebras

Abstract

We present a novel construction of linear deformations for Lie algebras and use it to prove the non-rigidity of several classes of Lie algebras in different varieties. In particular, we address the problem of k-rigidity for k-step nilpotent Lie algebras and k-solvable Lie algebras. We show that Lie algebras with an abelian factor are not rigid, even for the case of a 1-dimensional abelian factor. This holds in the more restricted case of k-rigidity. We also prove that the k-step free nilpotent Lie algebras are not (k+1)-rigid, but however they are k-rigid.