Deformations and rigidity in varieties of Lie algebras

TL;DR

This paper introduces a new method for constructing linear deformations of Lie algebras, demonstrating that many classes are not rigid and analyzing the rigidity properties of specific algebra families.

Contribution

It presents a novel construction of linear deformations and applies it to prove non-rigidity of various Lie algebra classes, including those with abelian factors and nilpotent structures.

Findings

Lie algebras with abelian factors are generally not rigid.

k-step free nilpotent Lie algebras are k-rigid but not (k+1)-rigid.

Heisenberg Lie algebras are 2-rigid but not 3-rigid.

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. We consider the family of Lie algebras with an abelian factor showing that, in general, they are not rigid even for the case of a 1-dimensional abelian factor. We also address the problem of k-rigidity for k-step nilpotent Lie algebras. Using our construction and recent results in [BCC] we prove the that the k-step free nilpotent Lie algebras are k-rigid but not (k + 1)-rigid and that Heisenberg Lie algebras are 2-rigid but not 3-rigid.