Higher Auslander Algebras arising from Dynkin Quivers and n-Representation Finite Algebras

TL;DR

This paper constructs higher Auslander algebras from Dynkin quivers and n-representation finite algebras, extending the higher Auslander correspondence and exploring their module categories and cluster-tilting objects.

Contribution

It introduces a canonical construction of higher Auslander algebras from Dynkin quivers and n-representation finite algebras, generalizing existing theories.

Findings

Constructed higher Auslander algebras with specified global dimensions.

Described module categories with higher cluster-tilting objects.

Connected these algebras to cluster categories and their subcategories.

Abstract

In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend this construction for higher analogues of representation finite and hereditary algebras. Specifically, if $M$ is an n-cluster tilting object in the bounded derived category of n-representation finite and n-hereditary algebra, then we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $(n+2)k+n+1$ for any $k\geq 1$. As an application, we revisit the higher Auslander correspondence. Firstly, we describe the corresponding module categories that have higher cluster-tilting objects, and then we discuss their relationship with certain full subcategories of the derived category. Consequently, we obtain a vast list of n-representation finite n-hereditary algebras whose n-cluster tilting objects are always minimal generator-cogenerator. Moreover, resulting algebras can be realized as endomorphism algebras of certain full subcategories of (higher) cluster categories.