Replicated algebras derived equivalent to higher Auslander algebras of type A

TL;DR

This paper demonstrates that certain higher Auslander algebras of type A are derived equivalent to replicated algebras with specific properties, introducing new classes and methods for establishing derived equivalences.

Contribution

It establishes derived equivalences between higher Auslander algebras and replicated algebras, introduces 2-subhomogeneous m-representation finite algebras, and provides a method to connect fractionally Calabi-Yau and 2-subhomogeneous algebras.

Findings

Higher Auslander algebras of type A are derived equivalent to replicated algebras.

Replicated algebra B has global dimension nd and an nd-cluster tilting subcategory.

A new class of 2-subhomogeneous m-representation finite algebras is characterized.

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.