The tensorial description of the Auslander algebras of representation-finite string algebras

TL;DR

This paper investigates the structure of Auslander algebras associated with representation-finite string algebras, introducing gluing algebras and describing their tensorial properties, with specific focus on Dynkin type A and D cases.

Contribution

It introduces the concept of gluing algebras and characterizes Auslander algebras of representation-finite string algebras as quotients of these, providing explicit descriptions for Dynkin types A and D.

Findings

Auslander algebra of a string algebra is a quotient of a gluing algebra.

Explicit descriptions of Auslander algebras for Dynkin types A and D.

Determination of the representation types of these Auslander 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 $\vec{A}^e_n $. As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types $A$ and $D$ are described. Moreover, the representation types of the above Auslander algebras are also given exactly.