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<p>The aim of this article is to study the Auslander algebra of any representation-finite string algebra. More precisely, we introduce the notion of gluing algebras and show that the Auslander algebra of a representation-finite string algebra is a quotient of a gluing algebra of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\overrightarrow{\boldsymbol{A}}_{n}^{e}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mover> <mi mathvariant="bold-italic">A</mi> <mo>→</mo> </mover> <mrow> <mi>n</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{A}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{D}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> </math></EquationSource> </InlineEquation> are described. Moreover, the representation types of the above Auslander algebras are also given exactly.</p>

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The Tensorial Description of the Auslander Algebras of Representation-finite String Algebras

  • 期刊论文

  • 发表日期：

    2026年3月20日

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  • Hui Chen,
  • Jian He,
  • Yuzhe Liu

摘要

The aim of this article is to study the Auslander algebra of any representation-finite string algebra. More precisely, we introduce the notion of gluing algebras and show that the Auslander algebra of a representation-finite string algebra is a quotient of a gluing algebra of \(\overrightarrow{\boldsymbol{A}}_{n}^{e}\) A → n e . As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types \(\mathbb{A}\) A and \(\mathbb{D}\) D are described. Moreover, the representation types of the above Auslander algebras are also given exactly.