arxiv: 2605.09583 · v1 · submitted 2026-05-10 · 🧮 math.RA · math.CO

Recognition: no theorem link

The comaximal graph of a finite-dimensional Lie algebra

David A. Towers, Ismael Gutierrez, Yesneri Zuleta

Pith reviewed 2026-05-12 04:42 UTC · model grok-4.3

classification 🧮 math.RA math.CO

keywords comaximal graphLie algebrafinite fieldsl_2graph invariantssubalgebra generationconnectivity

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The pith

The comaximal graph of sl_2(F_q) is connected and non-planar, with a large clique from nonsplit semisimple lines and Borel subalgebras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines the comaximal graph of a finite-dimensional Lie algebra L, whose vertices are the nontrivial proper subalgebras and whose edges join pairs that generate all of L. General properties are proved, including a characterization of isolated vertices via the Frattini subalgebra and a criterion for the graph to be complete. All Lie algebras of dimension at most three over a finite field F_q are classified by the structure of their comaximal graphs, which depend on the derived algebra and the action of ad x. For L isomorphic to sl_2(F_q) the graph is shown to be connected and non-planar, and explicit values are obtained for its degree sequence, clique number, chromatic number, domination number, diameter and radius.

Core claim

The comaximal graph Γ(L) for L ≅ sl_2(F_q) is connected and non-planar. It contains a large clique formed by the nonsplit semisimple lines together with the Borel subalgebras, while the nilpotent and split semisimple lines have restricted adjacency governed by their containment in Borel subalgebras. Explicit values are given for the degree sequence, clique number, chromatic number, domination number, diameter, and radius.

What carries the argument

The comaximal graph Γ(L), whose vertices are the nontrivial proper Lie subalgebras of L and with adjacency between A and B precisely when the subalgebra generated by A and B equals L.

If this is right

For every Lie algebra of dimension at most three over F_q the structure of Γ(L) is determined by the derived algebra and the action of ad x.
In the case L ≅ sl_2(F_q) the graph contains a clique whose size is fixed by the number of nonsplit semisimple lines together with the Borel subalgebras.
The diameter and radius of Γ(L) for sl_2(F_q) are finite and can be read off from the containment relations among nilpotent and split lines.
Nilpotent and split semisimple lines are adjacent only when they lie in a common Borel subalgebra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same adjacency relation could be applied to subalgebras of other algebraic structures to obtain analogous connectivity and planarity results.
Non-planarity for sl_2 suggests that subalgebra-generation graphs of simple Lie algebras will generally require crossings when drawn in the plane.
Higher-dimensional simple Lie algebras may admit similar clique descriptions built from parabolic subalgebras and root-space lines.

Load-bearing premise

The classification for dimensions at most three depends on a complete enumeration of all Lie algebra structures over F_q in those dimensions together with the definition of adjacency via generated subalgebra equaling L.

What would settle it

An explicit listing of all proper subalgebras of one concrete three-dimensional Lie algebra over a small finite field such as F_2, followed by a check that the pairs generating the whole algebra do not produce the graph claimed in the classification.

Figures

Figures reproduced from arXiv: 2605.09583 by David A. Towers, Ismael Gutierrez, Yesneri Zuleta.

**Figure 1.** Figure 1: The comaximal graph of a 3-dimensional abelian Lie algebra over F2. Case 2A. dim L ′ = 1 - The nonabelian nilpotent algebra (The Heisenberg algebra). Let B = (e, f, h) be a basis for L, with brackets [e, f] = h, and [e, h] = [f, h] = 0. Let Z := Z(L) = L ′ = ⟨h⟩ be its center and derived algebra. Note that in this case, the derived algebra is central, so the bracket may produce a new independent direction,… view at source ↗

**Figure 2.** Figure 2: The comaximal graph Γ(H3(F2)). Blue vertices are noncentral lines, red vertices are planes, and the isolated black vertex is the central line Z = ⟨h⟩. Corollary 4.4. The graph Γ(L) ∗ is (q 2 + q)-regular of order (q + 1)2 . Proof. By Proposition 4.3, every vertex in V \ {Z} has degree q 2 + q. The order of V \ {Z} is (q 2 + q + 1) − 1 + (q + 1) = (q + 1)2 . □ Case 2B. dim L ′ = 1 - A solvable non-nilpotent… view at source ↗

**Figure 3.** Figure 3: The comaximal graphs for q = 2 in cases 1 and 2 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: The comaximal graphs for q = 2 in cases 3 and 4 Case 4. dim L ′ = 3 - the perfect case. We would expect that the following result is well known, but we were unable to find a reference for it. Theorem 4.9. Let L be a perfect three-dimensional Lie algebra over a finite field Fq where q ̸= 2. Then L ∼= sl2(Fq) (the split case) or L ∼= su2(Fq). In [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: The comaximal graph of sl2(F3). Blue vertices are split semisimple lines, red vertices are Borels, green vertices are nilpotent lines, and black vertices are nonsplit semisimple lines. The following result is well-known. Lemma 4.14. Each Borel subalgebra of sl2(Fq) contains exactly one nilpotent line and exactly q split semisimple lines [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new comaximal graph on Lie subalgebras, classifies it for all cases in dimension at most three over finite fields, and computes invariants for sl_2, with the enumeration step as the main point to check.

read the letter

The paper introduces the comaximal graph of a Lie algebra, where vertices are the nontrivial proper subalgebras and edges connect pairs that generate the entire algebra. It classifies this graph for all Lie algebras of dimension at most three over finite fields and works out several invariants for the special linear algebra sl_2. What stands out is the new definition and the explicit work in small dimensions. They establish some general facts first, including when vertices are isolated using the Frattini subalgebra and when the graph is complete. Then they go through each possible Lie algebra structure in dimensions one, two, and three over F_q and describe the resulting graph. For sl_2(F_q) they show the graph is connected and non-planar, identify a large clique consisting of nonsplit semisimple lines and Borel subalgebras, and give the degree sequence, clique number, chromatic number, domination number, diameter, and radius. The sl_2 part looks well-supported by the standard subalgebra theory of that algebra. The general properties also seem straightforward from the definitions. The main soft spot is the completeness of the low-dimensional classification. Classifying all Lie algebras up to isomorphism in dimension three over F_q is not trivial, as the number of classes increases with q and varies by characteristic. The paper claims to do this for every such algebra, but without an external reference or computational check mentioned in the abstract, there is a risk that some structures were overlooked. If the list is incomplete, then the stated graph properties would miss some cases. This is the only real concern I see; everything else follows directly once the algebras are listed. Readers who work on graph invariants of algebraic structures or on the subalgebra structure of Lie algebras will get the most from this. It supplies concrete examples and numbers that could serve as test cases for broader conjectures. I would send this to peer review. The new construction and the detailed small-case results are worth expert scrutiny, particularly on the enumeration step.

Referee Report

1 major / 2 minor

Summary. The paper defines the comaximal graph Γ(L) of a finite-dimensional Lie algebra L over a field F, with vertices the nontrivial proper subalgebras of L and an edge between A and B precisely when the subalgebra they generate equals L. General structural results are proved, including a characterization of isolated vertices in terms of the Frattini subalgebra and a criterion for the graph to be complete when L is a μ-algebra. The manuscript then classifies Γ(L) completely for every Lie algebra of dimension at most 3 over a finite field F_q, describing the resulting graphs case-by-case according to the structure of the derived algebra and the action of ad x. For L ≅ sl_2(F_q) the paper computes the degree sequence, clique number, chromatic number, domination number, diameter and radius, and proves that Γ(L) is connected and non-planar, with a distinguished clique consisting of the nonsplit semisimple lines together with the Borel subalgebras.

Significance. If the low-dimensional classification is exhaustive, the work supplies a new combinatorial invariant that directly encodes the generation properties of subalgebras. The explicit invariants obtained for sl_2(F_q) illustrate how Lie-algebraic features (semisimplicity, nilpotency, Borel containment) translate into concrete graph-theoretic quantities, and the absence of fitted parameters or self-referential constructions strengthens the internal coherence of the approach. The results could serve as a foundation for further study of subalgebra lattices via graph theory, provided the enumeration of isomorphism classes is verified against the literature.

major comments (1)

[Classification for dimension at most three] The classification of Γ(L) for all Lie algebras of dimension ≤3 over F_q (the central claim of the paper) rests on an asserted exhaustive list of all isomorphism classes of such algebras. No reference is supplied to standard tables of low-dimensional Lie algebras over finite fields, nor is any verification method (computational or otherwise) described that would confirm all non-split extensions and characteristic-p phenomena have been included. Any omitted structure would render the corresponding graph description incomplete.

minor comments (2)

[Abstract] Notation for the base field alternates between F and F_q in the abstract and early sections; a single consistent symbol should be adopted from the outset.
The explicit formulas or tables for the degree sequence, clique number, etc., of Γ(sl_2(F_q)) would benefit from a compact summary table that also records the dependence on q and characteristic.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point about the presentation of the low-dimensional classification. We address the major comment below and commit to revisions that will strengthen the exposition.

read point-by-point responses

Referee: The classification of Γ(L) for all Lie algebras of dimension ≤3 over F_q (the central claim of the paper) rests on an asserted exhaustive list of all isomorphism classes of such algebras. No reference is supplied to standard tables of low-dimensional Lie algebras over finite fields, nor is any verification method (computational or otherwise) described that would confirm all non-split extensions and characteristic-p phenomena have been included. Any omitted structure would render the corresponding graph description incomplete.

Authors: We agree that the absence of an explicit reference and a brief verification outline is a presentational shortcoming. In the revised manuscript we will insert a short subsection (or paragraph in the introduction to the classification) that cites the standard literature on the classification of Lie algebras of dimension at most three over finite fields and outlines the exhaustive case division we employed: the possible dimensions of the derived algebra L' (0, 1, 2 or 3) together with the possible actions of ad x for x ∉ L', including all split and non-split extensions and the characteristic-dependent phenomena that arise in the Jordan canonical forms or nilpotency indices. Because the dimension is small, these cases are completely determined by the structure theory of Lie algebras; the added reference and outline will make the completeness of the list transparent to the reader. revision: yes

Circularity Check

0 steps flagged

No circularity: direct definition and case-by-case classification

full rationale

The paper defines Γ(L) explicitly via subalgebras and the generation condition <A,B>=L with no parameters, no fitted quantities, and no equations that reduce outputs to inputs by construction. Classification for dim≤3 enumerates isomorphism classes of Lie algebras (a standard external enumeration) and computes adjacency and invariants directly for each; this is falsifiable against independent lists of low-dimensional Lie algebras over F_q and does not rely on self-citation chains or ansatzes smuggled from prior work. No load-bearing step collapses to a self-definition or renamed fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard definition and axioms of Lie algebras together with the newly introduced graph construction; no numerical parameters are fitted and no new physical or algebraic entities beyond the graph itself are postulated.

axioms (1)

standard math A Lie algebra is a vector space equipped with a bilinear, skew-symmetric bracket satisfying the Jacobi identity
Invoked throughout as the ambient structure whose subalgebras form the vertices.

invented entities (1)

Comaximal graph Γ(L) no independent evidence
purpose: To encode adjacency of subalgebras whose join is the whole algebra
Newly defined object whose properties are then studied; no independent falsifiable prediction outside the paper is supplied.

pith-pipeline@v0.9.0 · 5541 in / 1644 out tokens · 45483 ms · 2026-05-12T04:42:18.961355+00:00 · methodology

discussion (0)

Reference graph

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