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

Hui Chen, Jian He, Yu-Zhe Liu

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.

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 . As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types and are described. Moreover, the representation types of the above Auslander algebras are also given exactly.

Paper Structure

This paper contains 10 sections, 11 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. String algebras
  4. Generalized diamond-stacked algebras
  5. Characterization of generalized diamond-stacked algebras
  6. The Auslander algebras of string algebras
  7. Examples and applications
  8. An example with non-empty gluing index and non-empty supplement
  9. The Auslander algebra of $\overrightarrow{\pmb{A}}_n$
  10. The Auslander algebra of $\overrightarrow{\pmb{D}}_n$

Key Result

Theorem 1.2

(Proposition prop:Aus-is-GDS and Theorem thm:main) The Auslander algebra $A^{\mathrm{Aus}}$ of arbitrary representation-finite string algebra $A$ is a GDS algebra. Moreover, $A^{\mathrm{Aus}}$ is a quotient of a gluing algebra of $\overrightarrow{\pmb{A}}_n^e = \overrightarrow{\pmb{A}}_n\otimes_{\m

Figures (3)

  • Figure 3.1: The underlying graph of $A_m \otimes_{\mathds{k}} A_n$ and the quiver of $\overrightarrow{\pmb{A}}_m\otimes_{\mathds{k}}\overrightarrow{\pmb{A}}_n$
  • Figure 5.1: The quivers of $\overrightarrow{\pmb{A}}_4^e$ and $A^{\mathrm{Aus}}$.
  • Figure 5.2: The quiver containing subquiver of Euclid type $\widetilde{\mathbb{D}}_4$

Theorems & Definitions (36)

  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: The irreducible morphisms over string algebras BR1987
  • Example 2.4
  • Definition 2.5: Diamond-stacked quivers
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 26 more