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<p>Let <i>A</i> be a given <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( 2\times 2 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> non-singular matrix over a general field <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( \mathbb {F} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation>, and let <i>m</i> be an arbitrary positive integer. In this article, we completely solve the generalized Yang-Baxter matrix equation, namely the equation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( A(XA)^{m}=X(AX)^{m} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mo>=</mo> <mi>X</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for the unknown non-singular matrix <i>X</i>.</p>

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THE GENERALIZED YANG-BAXTER MATRIX EQUATION \( A(XA)^{m}=X(AX)^{m} \) OVER \( \textrm{GL}(2,\mathbb {F}) \)

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月18日

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  • Boaz Cohen

摘要

Let A be a given \( 2\times 2 \) 2 × 2 non-singular matrix over a general field \( \mathbb {F} \) F , and let m be an arbitrary positive integer. In this article, we completely solve the generalized Yang-Baxter matrix equation, namely the equation \( A(XA)^{m}=X(AX)^{m} \) A ( X A ) m = X ( A X ) m for the unknown non-singular matrix X.