The solvable Graph of a finite-dimensional Lie Algebra

TL;DR

We introduce and study the solvable graph $\Gamma_{\mathfrak S}(L)$ of a finite-dimensional Lie algebra $L$, encoding when two elements generate a solvable subalgebra via adjacency. The core tool is the solvabilizer $\mathrm{sol}_L(x)$, whose structure—especially a disjoint coset decomposition $\mathrm{sol}_L(x)=\bigsqcup_{a\in\mathcal{R}_x}(a+Fx)$—drives divisibility results, functoriality under quotients, and $S$-Lie algebra considerations; in characteristic zero, simple $S$-Lie algebras are shown to be abelian of dimension one. The paper computes and analyzes graphs for small and large matrix Lie algebras (notably $\mathfrak{sl}_2(\mathbb{F}_3)$, $\mathfrak{gl}_2(\mathbb{F}_q)$, and $\mathfrak{sl}_2(\mathbb{F}_q)$), revealing non-connectivity phenomena tied to spectral types of matrices and providing explicit degree sequences. An accompanying GAP/SageMath framework enables practical generation of these graphs, highlighting both parallels and novel differences with nilpotent-graph theory and suggesting that solvable graphs encode structural invariants in a genuinely new way for Lie theory.

Abstract

We introduce and investigate the solvable graph $Γ_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they generate a solvable subalgebra. After developing the basic properties of solvabilizers and $S$-Lie algebras, we establish divisibility conditions, coset decompositions, and degree constraints for solvable graphs. Explicit examples, such as $\mathfrak{sl}_2(\mathbb{F}_3)$, illustrate that solvable graphs may be non-connected, in sharp contrast with the group-theoretic setting. We further determine the degree sequences of $Γ_\mathfrak{S}(\mathfrak{gl}_2(\F_q))$ and $Γ_\mathfrak{S}(\mathfrak{sl}_2(\F_q))$, highlighting how spectral types of matrices dictate combinatorial patterns. An algorithmic framework based on GAP and SageMath is also provided for practical computations. Our results reveal both analogies and differences with the nilpotent graph of Lie algebras, and suggest that solvable graphs encode structural invariants in a genuinely new way. This work opens the door to a broader graphical approach to solvability in Lie theory.

Figures (2)

• Figure 1: The solvable graph $\Gamma_\mathfrak{S}(\mathfrak{sl}_2(\mathbb{F}_3))$
• Figure 2: The non-solvable graph $\Gamma_\mathfrak{S}(\mathfrak{sl}_2(\mathbb{F}_3))$

Theorems & Definitions (54)

• Definition 3.1
• Example 3.1
• Remark 3.2
• Theorem 3.3
• Example 3.4
• Definition 3.2
• Theorem 3.5
• proof
• Corollary 3.6
• Lemma 3.7