Deformations and rigidity in varieties of Lie algebras

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