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Nakayama-type phenomena in higher Auslander--Reiten theory

Gustavo Jasso, Julian Külshammer

Abstract

This paper surveys recent contructions in higher Auslander--Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type $\mathbb{A}$ quivers. These include higher dimensional analogues of Nakayama algebras, of the mesh category of type $\mathbb{Z}\mathbb{A}_\infty$ and the tubes, and of the triangulated category generated by an $m$-spherical object. For $m=2$, the latter category can be regarded as the higher cluster category of type $\mathbb{A}_\infty$ whose cluster-tilting combinatorics are controlled by the triangulations of the cylic apeirotope.

Nakayama-type phenomena in higher Auslander--Reiten theory

Abstract

This paper surveys recent contructions in higher Auslander--Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type quivers. These include higher dimensional analogues of Nakayama algebras, of the mesh category of type and the tubes, and of the triangulated category generated by an -spherical object. For , the latter category can be regarded as the higher cluster category of type whose cluster-tilting combinatorics are controlled by the triangulations of the cylic apeirotope.
Paper Structure (15 sections, 18 theorems, 32 equations)

This paper contains 15 sections, 18 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Auslander--Iyama correspondence
  4. Higher Auslander--Reiten theory
  5. Higher Nakayama algebras
  6. The universal $d$-Nakayama category
  7. Higher Nakayama algebras
  8. Obstructions to an alternative definition of higher Nakayama algebras
  9. Simple modules and the $\mathop{\mathrm{Ext}}\nolimits^d$-quiver of the higher Nakayama algebras
  10. Global dimension of the higher Nakayama algebras
  11. Dominant dimension of the higher Nakayama algebras
  12. Cluster categories of type $\mathbb{A}_n$ and $\mathbb{A}_\infty$
  13. Categorical construction
  14. Tilting modules for $A_n^{(d)}$
  15. Triangulations of cyclic polytopes and apeirotopes

Key Result

Theorem 2.1

There is a one-to-one correspondence between Morita equivalence classes of representation-finite algebras and Morita equivalence classes of Auslander algebras. The correspondence associates to a re-pre-sen-ta-tion-finite algebra $\Lambda$ the algebra $\mathop{\mathrm{End}}\nolimits_\Lambda(M)^{\math

Theorems & Definitions (35)

  • Theorem 2.1: Auslander correspondence
  • Theorem 2.2: Morita--Tachikawa correspondence
  • Definition 2.3: Iya07Iya11IO11
  • Theorem 2.4: $d$-dimensional Auslander--Iyama correspondence
  • Definition 2.5: Iya07, Iya07b, Jas16
  • Definition 2.6
  • Theorem 2.7: Iya11
  • Theorem 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 25 more