Algebras of acyclic cluster type: tree type and type $\widetilde{A}$

TL;DR

This paper classifies algebras of global dimension at most 2 related to acyclic quivers of tree and type Ã, providing explicit descriptions and derived equivalence classifications.

Contribution

It establishes derived equivalence classifications for algebras associated with acyclic quivers of tree and type Ã, including explicit descriptions for type A_n.

Findings

Algebras cluster equivalent to tree quivers are derived equivalent to their path algebras.

Explicit classification of algebras of cluster type A_n for all orientations.

Complete classification of algebras cluster equivalent to tame acyclic quivers.

Abstract

In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type $\A_n$ for each possible orientation of $\A_n$. We give an explicit way to read off in which derived equivalence class such an algebra lies, and describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.