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.