bibkey: heib2026structural authors: T. Heib; D. E. Bruschi year: 2026 title: “On the structural properties of Lie algebras via associated labeled directed graphs” doi: 10.48550/arXiv.2601.16161 url: https://arxiv.org/abs/2601.16161v1 claim: “Conjecture 83 states that a minimal-graph-admissible Lie algebra with trivial center either has no proper non-empty vertex subset with the ideal-graph-property in its minimal graph, or is a direct sum of centerless components each of whose minimal graphs has no such subset.” strata_touched:
- D5/S3/Quantum/Algebra/HeibBruschiCenterlessIdealGraphRefutation license: citation-only triage: anchor
Heib–Bruschi, Lie algebras via labeled directed graphs
T. Heib and D. E. Bruschi, On the structural properties of Lie algebras via associated labeled directed graphs, arXiv:2601.16161v1 (22 January 2026; math-ph, cross-listed quant-ph). Numbers below are those of the v1 PDF.
Verified locator
DOI: 10.48550/arXiv.2601.16161.
Primary version: https://arxiv.org/abs/2601.16161v1 (the only version).
The TeX source pr09_arXiv_01.tex of v1 and the v1 PDF supply §I (conventions),
Definition 16 and Eq. (7) in §II.A, Algorithm 1 and the minimal-graph definition in §II,
and Definition 75 and Conjecture 83 (p. 48) in §IV.D.
Source statements
Conventions (§I): “We denote any field with the symbol (\mathbb{F}) … we write (x\propto y) if and only if there exists a constant (\kappa\in\mathbb{F}^*) such that (x=\kappa y)”.
Definition 16 (§II.A, Graph-admissible Lie algebra): “Let (\mathfrak g) be an (n)-dimensional Lie algebra. We say that (\mathfrak g) is minimal-graph-admissible if it admits a basis ({x_j}{j=1}^n) such that the Lie bracket satisfies: ([x_j,x_k]=\alpha{jk} x_{\delta(j,k)}) for all (j,k\in\mathcal{N}:={1,\ldots,n}), (7) where (\boldsymbol{\alpha}\in\mathbb{F}^{n\times n}) is an antisymmetric matrix and (\delta:\mathcal{N}\times\mathcal{N}\to\mathcal{N}) is symmetric function … To ensure consistency and for later convenience, we define (\delta(j,k):=0) whenever (\alpha_{jk}=0) …, and set (x_0:=0)”.
Algorithm 1 (§II): “draw a vertex (v_j) for every basis element (x_j\in\mathcal{B}) … If there exists an element (x_\ell\in\mathcal{B}) such that ([x_j,x_k]\propto x_\ell), one draws a directed edge from (v_j) to (v_\ell), labeled by (v_k).” A graph associated with (\mathfrak g) “is a minimal graph if (|V|=\dim(\mathfrak g))”.
Definition 75 (§IV.D): “A subset (W\subseteq V) is said to satisfy the ideal-graph-property if and only if no edge (e\in E) points from a vertex (w\in W) to a vertex (v\in V\setminus W).”
Conjecture 83 (§IV.D, p. 48): “Let (\mathfrak g) be a minimal-graph-admissible Lie algebra associated with the minimal graph (G(V,E)). Suppose the center of (\mathfrak g) is trivial, i.e., (\mathcal{Z}(\mathfrak g)={0}). Then one of the following conditions must hold: (i) The vertex set (V) contains no proper non-empty subset (W\subsetneq V) that satisfies the ideal-graph-property or (ii) The Lie algebra (\mathfrak g) admits a decomposition as a direct sum (\mathfrak g=\bigoplus_{j\in\mathcal{J}}\mathfrak g_j), such that each component satisfies (\mathcal{Z}(\mathfrak g_j)={0}), and every minimal graph (G(V_j,E_j)) associated with each (\mathfrak g_j) contains no proper non-empty subset (W_j\subsetneq V_j) that satisfies the ideal-graph-property.”
Scope
The paper motivates Conjecture 83 with (\mathfrak{su}(2)\oplus\mathfrak{su}(2)), whose two summands have the ideal-graph-property. It names the two-dimensional affine algebra (\mathfrak{aff}(\mathbb F)) (§I) as a solvable non-nilpotent example. arXiv lists only v1.