Higher Nakayama algebras I: Construction

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 ${\mathbb{Z}\mathbb{A}}_{\mathrm{\infty }}$ and the tubes.