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Replicated algebras derived equivalent to higher Auslander algebras of type A

Wei Xing

TL;DR

The paper constructs explicit derived equivalences between higher Auslander algebras of type $A$ and replicated algebras by using tilting complexes built from Auslander translates of projectives. For $ ext{gcd}(n,d)=1$, it shows that the endomorphism algebra $B= ext{End}_{ ext{D}^b(A)}(T)$ of a carefully chosen tilting object $T$ is a $2$-subhomogeneous $nd$-representation finite algebra, with $B$ naturally realized as the $(n+d)$-fold replicated algebra of a base algebra $B_0$ and with the $(nd+1)$-preprojective algebra given by a graded trivial extension of $B_0$. The work introduces lattice-path and rational Dyck-path frameworks to model indecomposable modules and to guide tilting constructions, yielding a concrete combinatorial description of the derived equivalence, the endomorphism algebras, and Calabi–Yau properties, thereby linking higher Auslander algebras, replication, and fractionally Calabi–Yau phenomena with potential applications to symplectic geometry and related categories.

Abstract

We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster tilting subcategory consisting of all direct sums of projective modules and injective modules. We introduce a class of algebras called $2$-subhomogeneous $m$-representation finite to characterize the homological properties of $B$ and give a method to obtain derived equivalences between fractionally Calabi-Yau algebras and $2$-subhomogeneous algebras using certain tilting complexes.

Replicated algebras derived equivalent to higher Auslander algebras of type A

TL;DR

The paper constructs explicit derived equivalences between higher Auslander algebras of type and replicated algebras by using tilting complexes built from Auslander translates of projectives. For , it shows that the endomorphism algebra of a carefully chosen tilting object is a -subhomogeneous -representation finite algebra, with naturally realized as the -fold replicated algebra of a base algebra and with the -preprojective algebra given by a graded trivial extension of . The work introduces lattice-path and rational Dyck-path frameworks to model indecomposable modules and to guide tilting constructions, yielding a concrete combinatorial description of the derived equivalence, the endomorphism algebras, and Calabi–Yau properties, thereby linking higher Auslander algebras, replication, and fractionally Calabi–Yau phenomena with potential applications to symplectic geometry and related categories.

Abstract

We show that every higher Auslander algebra of type such that is derived equivalent to a certain replicated algebra . Moreover and admits an -cluster tilting subcategory consisting of all direct sums of projective modules and injective modules. We introduce a class of algebras called -subhomogeneous -representation finite to characterize the homological properties of and give a method to obtain derived equivalences between fractionally Calabi-Yau algebras and -subhomogeneous algebras using certain tilting complexes.

Paper Structure

This paper contains 15 sections, 29 theorems, 81 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and notations
  4. $d$-cluster tilting subcategories
  5. Higher Auslander algebras and higher preprojective algebras
  6. Tilting complexes and derived equivalences
  7. $2$-subhomogeneous $d$-representation finite algebras
  8. Definitions and properties
  9. Construction
  10. A derived equivalence for higher Auslander algebras of type A
  11. Higher Auslander algebras of type A
  12. Lattice paths
  13. Rational Dyck paths
  14. The tilting complex
  15. Endomorphism algebras

Key Result

Theorem 1.2

(Theorem thm_construction_minimal_dRF_alg) Let $A$ be a finite dimensional $k$-algebra such that ${\rm{gldim}} A < \infty$ and $A$ is $\frac{d}{a+1}$-Calabi-Yau. Suppose $X\in \mathcal{D}^b(A)$ such that $T=\bigoplus_{i=0}^{a-1}\nu^iX$ is a tilting complex. Denote by $B={\rm{End}}_{D^b(A)}(T)$ the e

Figures (8)

  • Figure 1: $(d\times n)$-rectangle
  • Figure 2: $\ell_1<\ell_2<\ell_3$
  • Figure 3: $\delta_{(x,y)}$ and $R_{(x,y)}$
  • Figure 4: the region $S_D$ for $D\in \Delta$
  • Figure 5: $\ell$ and $r^i(\ell)$
  • ...and 3 more figures

Theorems & Definitions (78)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Iya11IY08IJ17
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 68 more