Higher Nakayama algebras I: Construction

TL;DR

This paper introduces higher dimensional analogues of Nakayama algebras using Iyama's higher Auslander--Reiten theory, constructing new algebras with $d$-cluster-tilting modules and higher analogues of classical categories.

Contribution

It constructs finite dimensional algebras $A^{(d)}$ with $d$-cluster-tilting modules, extending Nakayama algebras into higher dimensions.

Findings

Constructed higher dimensional Nakayama algebras $A^{(d)}$.

Developed higher analogues of mesh categories and tubes.

Established properties of the $d$-cluster-tilting modules.

Abstract

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{ZA}_\infty$ and the tubes.