The solvable Graph of a finite-dimensional Lie Algebra

TL;DR

This paper introduces the solvable graph of a finite-dimensional Lie algebra, exploring its properties, examples, and computational methods, revealing new insights into the algebra's solvability structure through graph theory.

Contribution

It defines and studies the solvable graph of Lie algebras, establishing properties, examples, and computational tools, and highlighting differences from group-theoretic graphs.

Findings

Solvable graphs can be non-connected, unlike group-theoretic counterparts.

Degree sequences of specific Lie algebra graphs are characterized.

An algorithmic framework for computing solvable graphs is developed.

Abstract

We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ 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 $\Gamma_\mathfrak{S}(\mathfrak{gl}_2(\F_q))$ and $\Gamma_\mathfrak{S}(\mathfrak{sl}_2(\F_q))$, 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.