The comaximal graph of a finite-dimensional Lie algebra

TL;DR

This paper introduces and analyzes the comaximal graph of finite-dimensional Lie algebras, classifying it for small dimensions and exploring properties for $ ext{sl}_2( ext{F}_q)$, revealing diverse structural behaviors.

Contribution

It defines the comaximal graph for Lie algebras, characterizes its properties, classifies it for low dimensions, and computes invariants for $ ext{sl}_2( ext{F}_q)$, advancing understanding of Lie algebra substructure interactions.

Findings

Classified comaximal graphs for Lie algebras of dimension up to three.

Determined graph invariants for $ ext{sl}_2( ext{F}_q)$, including degree sequence and chromatic number.

Established structural properties relating to isolated vertices and completeness of the graph.

Abstract

In this paper, we introduce the comaximal graph $\Gamma(L)$ of a finite-dimensional Lie algebra $L$, whose vertices are the nontrivial proper Lie subalgebras of $L$ over a field $\mathbb{F}$, and two vertices $A$ and $B$ are adjacent if and only if $\langle A, B\rangle =L$. We establish general structural properties, including a characterization of isolated vertices via the Frattini subalgebra and a criterion for completeness in terms of $\mu$-algebras. We classify $\Gamma(L)$ for all Lie algebras of dimension at most three over a finite field $\mathbb{F}_q$, providing an explicit description in each case. The resulting graphs exhibit a rich range of behaviors, depending on the structure of the derived algebra and the action of $\operatorname{ad}x$. For $L\cong \mathfrak{sl}_2(\mathbb{F}_q)$, we determine several graph invariants, including the degree sequence, clique number, chromatic number, domination number, diameter, and radius, and show that $\Gamma(L)$ is connected and non-planar. The graph contains a large clique formed by the nonsplit semisimple lines together with the Borel subalgebras, while the nilpotent and split semisimple lines have a more restricted adjacency structure governed by their containment in Borel subalgebras.