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<p>We consider actions of Taft algebras on noetherian graded down-up algebras. We classify all such actions and determine properties of the corresponding invariant rings <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="31_2025_9921_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(A^T\)</EquationSource> </InlineEquation>. We identify precisely when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="31_2025_9921_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(A^T\)</EquationSource> </InlineEquation> is commutative, when it is Artin–Schelter regular, and give sufficient conditions for it to be Artin–Schelter Gorenstein. Our results show that many results and conjectures in the literature concerning actions of semisimple Hopf algebras on Artin–Schelter regular algebras can fail when the semisimple hypothesis is omitted.</p>

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Actions of Taft Algebras on Noetherian Down-Up Algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月3日

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  • Simon Crawford,
  • Jason Gaddis,
  • Robert Won

摘要

We consider actions of Taft algebras on noetherian graded down-up algebras. We classify all such actions and determine properties of the corresponding invariant rings \(A^T\) . We identify precisely when \(A^T\) is commutative, when it is Artin–Schelter regular, and give sufficient conditions for it to be Artin–Schelter Gorenstein. Our results show that many results and conjectures in the literature concerning actions of semisimple Hopf algebras on Artin–Schelter regular algebras can fail when the semisimple hypothesis is omitted.