Higher Auslander algebras of finite representation type

TL;DR

This paper characterizes when higher Auslander algebras of finite representation type are representation-finite based on the finiteness of certain indecomposable modules, and classifies specific linearly oriented type A cases.

Contribution

It provides a characterization of representation-finiteness for higher Auslander algebras and classifies those of linearly oriented type A.

Findings

A higher Auslander algebra is representation-finite iff finitely many indecomposable modules have projective dimension n+1.

Complete classification of representation-finite higher Auslander algebras of linearly oriented type A.

Explicit calculation of the number of indecomposable modules over these algebras.

Abstract

Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension $n+1$ is finite. As an application, we classify the representation-finite higher Auslander algebras of linearly oriented type $\mathbb{A}$ in the sense of Iyama and calculate the number of non-isomorphic indecomposable modules over these algebras.