<p>This paper introduces and studies the <i>relative commutativity degree</i> of a subalgebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> for a finite-dimensional Lie algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation> over a finite field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d(H, L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This notion generalizes the known commutativity degree <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a Lie algebra. We establish several foundational properties, bounds, and structural theorems for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d(H, L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, we provide upper bounds depending on whether <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> is abelian or non-abelian and characterize the subalgebras that attain these upper bounds for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d(H, L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the relative commutativity degree of subalgebra of a finite-dimensional Lie algebra

期刊论文
发表日期：

2026年7月20日

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Akram Chareh Khah Moghaddam,
Ahmad Erfanian,
Afsaneh Shamsaki

摘要

This paper introduces and studies the relative commutativity degree of a subalgebra $H$ $H$ for a finite-dimensional Lie algebra $L$ $L$ over a finite field $\mathbb {F}_q$ $F_{q}$ , denoted by $d(H, L)$ $d (H, L)$ . This notion generalizes the known commutativity degree $d(L)$ $d (L)$ of a Lie algebra. We establish several foundational properties, bounds, and structural theorems for $d(H, L)$ $d (H, L)$ . In particular, we provide upper bounds depending on whether $H$ $H$ is abelian or non-abelian and characterize the subalgebras that attain these upper bounds for $d(H, L)$ $d (H, L)$ .