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<p>The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1930_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak {g}(\!(x)\!) \times \mathfrak {g}[x]/x^m \mathfrak {g}[x]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">g</mi> <mrow> <mo stretchy="false">(</mo> <mspace width="-0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mi mathvariant="fraktur">g</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mi>x</mi> <mi>m</mi> </msup> <mi mathvariant="fraktur">g</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.</p>

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Generalized classical Yang-Baxter equation and regular decompositions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月13日

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  • R. Abedin,
  • S. Maximov,
  • A. Stolin

摘要

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of \(\mathfrak {g}(\!(x)\!) \times \mathfrak {g}[x]/x^m \mathfrak {g}[x]\) g ( ( x ) ) × g [ x ] / x m g [ x ] . The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.