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<p>M.&#xa0;Goncharov introduced and studied a Rota–Baxter operator on a cocommutative Hopf algebra. In the present paper we define relative Rota–Baxter operators on an arbitrary Hopf algebra. A particular case of this definition is Goncharov’s operator. On a Hopf algebra with a relative Rota–Baxter operator we define a new associative operation and construct a new Hopf algebra and Hopf brace. Further, we construct Rota–Baxter operators of integer weights on some groups. The question on a possibility to define an operator of zero weight on groups was formulated by X. Gao, L. Guo, Y. Liu, and Z.-C. Zhu. In the last section we construct a family of two generated Hopf algebras. This family includes some known Hopf algebras, in particular, the 4-dimensional Sweedler algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2025_976_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

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Relative Rota–Baxter Operators on Groups and Hopf Algebras

  • 期刊论文

  • 发表日期：

    2025年6月16日

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  • Valeriy G. Bardakov,
  • Igor M. Nikonov

摘要

M. Goncharov introduced and studied a Rota–Baxter operator on a cocommutative Hopf algebra. In the present paper we define relative Rota–Baxter operators on an arbitrary Hopf algebra. A particular case of this definition is Goncharov’s operator. On a Hopf algebra with a relative Rota–Baxter operator we define a new associative operation and construct a new Hopf algebra and Hopf brace. Further, we construct Rota–Baxter operators of integer weights on some groups. The question on a possibility to define an operator of zero weight on groups was formulated by X. Gao, L. Guo, Y. Liu, and Z.-C. Zhu. In the last section we construct a family of two generated Hopf algebras. This family includes some known Hopf algebras, in particular, the 4-dimensional Sweedler algebra \(H_4\) H 4 .