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On the rigidity of k-step nilpotent graph Lie algebras

Josefina Barrionuevo, Paulo Tirao

TL;DR

The paper investigates rigidity of k-step nilpotent Lie algebras arising from simple graphs. It develops deformation- and cohomology-based criteria to detect non-rigidity and proves a general non-rigidity result for k≥3, showing that only complete graphs (yielding free k-step algebras) can be k-rigid; for k=2, rigidity persists only for the complete graph or a small finite set of four-vertex graphs. The results provide a near-complete classification of rigidity within this graph-based family and illustrate the effectiveness of explicit deformations and Hall-type bases in analyzing rigidity questions for nilpotent Lie algebras. The findings underscore the rarity of rigidity in graph-associated k-step nilpotent algebras beyond the well-known free and Heisenberg families, and they connect to broader questions about Vergne-type conjectures in nilpotent Lie theory.

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.

On the rigidity of k-step nilpotent graph Lie algebras

TL;DR

The paper investigates rigidity of k-step nilpotent Lie algebras arising from simple graphs. It develops deformation- and cohomology-based criteria to detect non-rigidity and proves a general non-rigidity result for k≥3, showing that only complete graphs (yielding free k-step algebras) can be k-rigid; for k=2, rigidity persists only for the complete graph or a small finite set of four-vertex graphs. The results provide a near-complete classification of rigidity within this graph-based family and illustrate the effectiveness of explicit deformations and Hall-type bases in analyzing rigidity questions for nilpotent Lie algebras. The findings underscore the rarity of rigidity in graph-associated k-step nilpotent algebras beyond the well-known free and Heisenberg families, and they connect to broader questions about Vergne-type conjectures in nilpotent Lie theory.

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.

Paper Structure

This paper contains 9 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $k$-step nilpotent graph Lie algebras
  3. A Hall type basis of $\mathfrak{g}(k,G)$
  4. On the rigidity of nilpotent Lie algebras
  5. Some known results
  6. Linear deformations
  7. Criteria for 2-step nilpotent Lie algebras
  8. The rigidity problem for $k$-step nilpotent graph Lie algebras, for $k\ge 3$
  9. The rigidity problem for $2$-step nilpotent graph Lie algebras

Key Result

Theorem 2.2

Let $\mathfrak{g}$ be a nilpotent Lie algebra with bracket $\mu$. Let $\mathfrak{h}$ be a codimension 2 subalgebra of $\mathfrak{g}$ and let $a_1,a_2$ be such that $\mathfrak{g}= \langle a_1,a_2\rangle \oplus \mathfrak{h}$. Then for every $y$ in the centralizer of $\mathfrak{h}$, $\mu_t=\mu+t \sigma

Theorems & Definitions (19)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Theorem 2.2: BTS
  • Theorem 2.3: BTS
  • Lemma 2.4: A
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 9 more