The Commuting Graphs of Certain Solvable Lie Algebras

TL;DR

The paper analyzes commuting graphs Γ(ℒ) of solvable Lie algebras, focusing on dimensions up to 4. It proves that under specific conditions on the derived algebra and center, Γ(ℒ) is disconnected with a precise partition into complete components, and it determines exact component counts over finite fields. Using known classifications, it explicitly describes Γ(ℒ) for all 3D noncommutative solvable algebras and all 4D cases, revealing that components are typically complete, windmill, or their unions. The results yield a complete catalog of low-dimensional solvable Lie-algebra commuting graphs and clarify how algebraic structure governs graph connectivity.

Abstract

Let $Z(\cal L)$ be the center of a Lie algebra $\cal L$ with Lie bracket $[\cdot, \cdot]$. %We then define The commuting graph of $\cal L$ is then defined by the simple undirected graph $Γ({\cal L})=(V_{\cal L},E_{\cal L})$ in which the vertex set is $V_{\mathcal L}=\mathcal L \setminus Z(\mathcal L)$ and the set of edges $E_{\cal L}=\left\{ \{x,y\} \mid [x,y] =0 \right\}$. The main purpose of this paper is to accurately describe the connected components of the commuting graph of solvable Lie algebras of dimension at most 4.

Theorems & Definitions (19)

• Definition 1
• Lemma 2
• Corollary 3
• Example 4
• Theorem 5
• proof
• Corollary 6
• proof
• Lemma 7
• proof