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<p>In this paper, we introduce the notion of an extended Rota–Baxter operator that generalizes the Rota–Baxter operator and the modified Rota–Baxter operator on pre-Lie algebras, and give examples and general properties of extended Rota–Baxter operators. Later, we define a representation of extended Rota–Baxter pre-Lie algebras to define a suitable cohomology theory. We define a cohomology theory for extended Rota–Baxter pre-Lie algebras generalizing the cohomology of pre-Lie algebras. Next, we introduce the notion of 2-term extended Rota–Baxter pre-Lie<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2411_Article_IEq1.gif" Format="GIF" Height="8" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>∞</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebras and show that skeletal 2-term extended Rota–Baxter pre-Lie<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2411_Article_IEq1.gif" Format="GIF" Height="8" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>∞</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebras are classified by 3-cocycles of extended Rota–Baxter pre-Lie algebras. Later, we introduce crossed modules of extended Rota–Baxter pre-Lie algebras and characterize strict 2-term extended Rota–Baxter pre-Lie<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2411_Article_IEq1.gif" Format="GIF" Height="8" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>∞</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebras. Furthermore, we study non-abelian extensions of an extended Rota–Baxter pre-Lie algebra by another extended Rota–Baxter pre-Lie algebra. Finally, we study inducibility of pair of extended Rota–Baxter pre-Lie algebra automorphisms and the fundamental exact sequence of Wells.</p>

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2-Term Extended Rota–Baxter Pre-Lie\(_{\infty }\)-Algebra and Non-Abelian Extensions of Extended Rota–Baxter Pre-Lie Algebras

  • 期刊论文

  • 发表日期：

    2025年5月2日

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  • Shuangjian Guo,
  • Yizheng Li,
  • Dingguo Wang

摘要

In this paper, we introduce the notion of an extended Rota–Baxter operator that generalizes the Rota–Baxter operator and the modified Rota–Baxter operator on pre-Lie algebras, and give examples and general properties of extended Rota–Baxter operators. Later, we define a representation of extended Rota–Baxter pre-Lie algebras to define a suitable cohomology theory. We define a cohomology theory for extended Rota–Baxter pre-Lie algebras generalizing the cohomology of pre-Lie algebras. Next, we introduce the notion of 2-term extended Rota–Baxter pre-Lie \(_{\infty }\) ∞ -algebras and show that skeletal 2-term extended Rota–Baxter pre-Lie \(_{\infty }\) ∞ -algebras are classified by 3-cocycles of extended Rota–Baxter pre-Lie algebras. Later, we introduce crossed modules of extended Rota–Baxter pre-Lie algebras and characterize strict 2-term extended Rota–Baxter pre-Lie \(_{\infty }\) ∞ -algebras. Furthermore, we study non-abelian extensions of an extended Rota–Baxter pre-Lie algebra by another extended Rota–Baxter pre-Lie algebra. Finally, we study inducibility of pair of extended Rota–Baxter pre-Lie algebra automorphisms and the fundamental exact sequence of Wells.