We introduce and investigate the solvable graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma _\mathfrak {S}(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi mathvariant="fraktur">S</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a finite-dimensional Lie algebra L over a field F. The vertices are the elements outside the solvabilizer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{sol}(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>sol</mtext> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathfrak {sl}_2(\mathbb {F}_3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">sl</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, illustrate that solvable graphs may be non-connected, in sharp contrast with the group-theoretic setting. We further determine the degree sequences of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma _\mathfrak {S}(\mathfrak {gl}_2(\mathbb {F}_q))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi mathvariant="fraktur">S</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="fraktur">gl</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma _\mathfrak {S}(\mathfrak {sl}_2(\mathbb {F}_q))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi mathvariant="fraktur">S</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="fraktur">sl</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, 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.

The solvable graph of a finite-dimensional Lie Algebra

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2026年4月2日

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David Towers,
Ismael Gutierrez,
Luis Fernandez

摘要

We introduce and investigate the solvable graph $\Gamma _\mathfrak {S}(L)$ $Γ_{S} (L)$ of a finite-dimensional Lie algebra L over a field F. The vertices are the elements outside the solvabilizer $\textrm{sol}(L)$ $sol (L)$ , 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 $\mathfrak {sl}_2(\mathbb {F}_3)$ ${sl}_{2} (F_{3})$ , illustrate that solvable graphs may be non-connected, in sharp contrast with the group-theoretic setting. We further determine the degree sequences of $\Gamma _\mathfrak {S}(\mathfrak {gl}_2(\mathbb {F}_q))$ $Γ_{S} ({gl}_{2} (F_{q}))$ and $\Gamma _\mathfrak {S}(\mathfrak {sl}_2(\mathbb {F}_q))$ $Γ_{S} ({sl}_{2} (F_{q}))$ , 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.