Right n-Nakayama algebras and their representations

TL;DR

This paper investigates right n-Nakayama algebras, establishing their connection to representation-finite algebras, classifying hereditary cases, and explicitly describing modules and quivers for right 2-Nakayama algebras.

Contribution

It provides a classification of right n-Nakayama algebras, characterizes their relation to representation-finite algebras, and explicitly describes modules and quivers for the case n=2.

Findings

An artin algebra is representation-finite iff it is right n-Nakayama for some n.

Hereditary right n-Nakayama algebras are classified.

Finite dimensional right 2-Nakayama algebras are classified via quivers with relations.

Abstract

In this paper we study right $n$-Nakayama algebras. Right $n$-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra $\Lambda$ is representation-finite if and only if $\Lambda$ is right $n$-Nakayama for some positive integer $n$. We classify hereditary right $n$-Nakayama algebras. We also define right $n$-coNakayama algebras and show that an artin algebra $\Lambda$ is right $n$-coNakayama if and only if $\Lambda$ is left $n$-Nakayama. We then study right $2$-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right $2$-Nakayama algebra. We end by classifying finite dimensional right $2$-Nakayama algebras in terms of their quivers with relations.