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<p>We investigate the incidence algebras arising from one-branch extensions of “rectangles”. There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them as endomorphism algebra of tilting modules or tilting complexes over a Nakayama algebra. Meanwhile, an unexpected derived equivalence between Nakayama algebras <i>N</i>(2<i>r</i> − 1, <i>r</i>) and <i>N</i>(2<i>r</i> − 1, <i>r</i> + 1) has been found. As an application, we obtain the explicit formulas of the Coxeter polynomials for a large family of Nakayama algebras, i.e., the Nakayama algebras <i>N</i>(<i>n, r</i>) with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3287_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\({n \over 2} &lt; r &lt; n\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>&lt;</mo> <mi>r</mi> <mo>&lt;</mo> <mi>n</mi> </math></EquationSource> </InlineEquation>.</p>

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Derived Equivalences Between One-branch Extensions Algebras of “Rectangles”

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Qiang Dong,
  • Yanan Lin,
  • Shiquan Ruan

摘要

We investigate the incidence algebras arising from one-branch extensions of “rectangles”. There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them as endomorphism algebra of tilting modules or tilting complexes over a Nakayama algebra. Meanwhile, an unexpected derived equivalence between Nakayama algebras N(2r − 1, r) and N(2r − 1, r + 1) has been found. As an application, we obtain the explicit formulas of the Coxeter polynomials for a large family of Nakayama algebras, i.e., the Nakayama algebras N(n, r) with \({n \over 2} < r < n\) n 2 < r < n .