Higher Nakayama algebras I: Construction

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.

v5: 50 pages, further minor corrections following referee report. With an appendix by the second named author and Chrysostomos Psaroudakis and an appendix by Sondre Kvamme

References in corpus (10)

• Calabi-Yau algebras
• Auslander correspondence
• $n$-angulated categories
• $n$-abelian and $n$-exact categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• Selfinjective quivers with potential and 2-representation-finite algebras
• d-Representation-finite self-injective algebras
• Simplicial structures in higher Auslander-Reiten theory
• Higher Auslander correspondence for dualizing $R$-varieties
• ${\tau }^2$-stable tilting complexes over weighted projective lines

Cited by in corpus (14)

• d-Representation-finite self-injective algebras
• Simplicial structures in higher Auslander-Reiten theory
• $n$-exangulated categories
• Higher Auslander correspondence for dualizing $R$-varieties
• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• Maximal ${\tau }_d$-rigid pairs
• Gluing of $n$-cluster tilting subcategories for representation-directed algebras
• Fabric idempotents and homological dimensions
• $d\mathbb{Z}$-cluster tilting subcategories of singularity categories
• A characterisation of higher torsion classes
• Nakayama-type phenomena in higher Auslander--Reiten theory
• Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $S$-constructions
• Classification of the d-representation-finite symmetric k-algebras of finite representation type
• $n$-cluster tilting subcategories from gluing systems of representation-directed algebras