On the rigidity of k-step nilpotent graph Lie algebras

TL;DR

This paper investigates the rigidity of k-step nilpotent Lie algebras derived from simple graphs, revealing that only specific small graphs and the complete graph exhibit rigidity within this class.

Contribution

It identifies the precise conditions under which k-step nilpotent Lie algebras associated with graphs are rigid, introducing a construction for non-trivial deformations.

Findings

Only the complete graph yields rigid algebras for any k.

For k=2, graphs with at most 4 vertices produce rigid algebras.

A general deformation construction demonstrates non-rigidity in many cases.

Abstract

We thoroughly explore the class of k-step nilpotent Lie algebras associated with a simple graph looking for k-step nilpotent Lie algebras which are rigid in the variety of at most k-step nilpotent Lie algebras. We find out that, besides the complete graph, the only examples arise for k=2 and graphs of at most 4 vertices. A key tool to prove non-k-rigidity in this context, is a general construction of non-trivial deformations for naturally graded nilpotent Lie algebras.