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

Abstract

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{\mathit{A}}}_n^e

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. As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types

\mathbb{A}

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and

\mathbb{D}

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are described. Moreover, the representation types of the above Auslander algebras are also given exactly.

Keywords

String algebra / Auslander algebra / enveloping algebra / 16G10 / 16G60 / 16G70

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