A note on the $m$-extended module categories of Nakayama algebras

TL;DR

This paper investigates the structure of extended module categories of Nakayama algebras, providing classifications, algorithms, and explicit AR-quivers, thereby advancing understanding of their representation theory.

Contribution

It introduces new results on the existence of postprojective components and classifies finite type cases for extended module categories of Nakayama algebras.

Findings

Existence of postprojective components in certain extended module categories

Classification of Nakayama algebras of finite type in this context

Explicit AR-quivers for specific Nakayama algebras

Abstract

We study the extended module category $m\operatorname{-mod}\Lambda$ of an algebra $\Lambda$, recently introduced by Gupta and Zhou. The existence of a postprojective component of $m\operatorname{-mod}\Lambda$ is shown for certain algebras $\Lambda$, and a knitting algorithm through cohomological dimension vectors is provided. In particular, the extended module category of Nakayama algebras with homogeneous relations are investigated, and it is classified when they are of finite type. The paper give fully calculated AR-quivers for several Nakayama algebras.