Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">F</mi> </math></EquationSource> </InlineEquation> be a free Leibniz algebra generated by a set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{x_{1},\ldots ,x_{n}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> over a field K of characteristic 0. In this study, we prove that for a homogeneous element u, an arbitrary endomorphism <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathfrak {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">F</mi> </math></EquationSource> </InlineEquation> is an automorphism if and only if u belongs to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi ( \mathfrak {F})^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">F</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. As a result, we obtain an algorithm for determining the rank of a homogeneous element and a particular automorphic image of this element that realizes the rank.

On the rank of an element in free Leibniz algebras

期刊论文
发表日期：

2025年12月20日

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Zeynep Özkurt

摘要

Let $\mathfrak {F}$ $F$ be a free Leibniz algebra generated by a set $\{x_{1},\ldots ,x_{n}\}$ ${x_{1}, \dots, x_{n}}$ over a field K of characteristic 0. In this study, we prove that for a homogeneous element u, an arbitrary endomorphism $\varphi $ $φ$ of $\mathfrak {F}$ $F$ is an automorphism if and only if u belongs to $\varphi ( \mathfrak {F})^m$ $φ {(F)}^{m}$ . As a result, we obtain an algorithm for determining the rank of a homogeneous element and a particular automorphic image of this element that realizes the rank.