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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.09583 |
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| _version_ | 1866909030813794304 |
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| author | Towers, David A. Zuleta, Yesneri Gutierrez, Ismael |
| author_facet | Towers, David A. Zuleta, Yesneri Gutierrez, Ismael |
| contents | In this paper, we introduce the comaximal graph $Γ(L)$ 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 $μ$-algebras. We classify $Γ(L)$ 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 $Γ(L)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09583 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The comaximal graph of a finite-dimensional Lie algebra Towers, David A. Zuleta, Yesneri Gutierrez, Ismael Rings and Algebras Combinatorics 17B30, 17B45, 05C40, 05C25 In this paper, we introduce the comaximal graph $Γ(L)$ 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 $μ$-algebras. We classify $Γ(L)$ 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 $Γ(L)$ 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. |
| title | The comaximal graph of a finite-dimensional Lie algebra |
| topic | Rings and Algebras Combinatorics 17B30, 17B45, 05C40, 05C25 |
| url | https://arxiv.org/abs/2605.09583 |