The Auslander-Reiten components seen as Quasi-hereditary Categories

TL;DR

This paper introduces quasi-hereditary categories to analyze Auslander-Reiten components, linking representation theory of algebras with functor categories to provide new insights into their structure.

Contribution

It defines quasi-hereditary categories and applies this concept to Auslander-Reiten components, offering a novel perspective in the study of finite-dimensional algebras.

Findings

Introduces the concept of quasi-hereditary categories.

Establishes a connection between Auslander-Reiten components and quasi-hereditary categories.

Provides applications to functor categories related to Auslander-Reiten quivers.

Abstract

Quasi-hereditary were introduced by L. Scott \cite{Scott, CPS1,CPS2} in order to deal highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups, and they have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander [A], [AQM] and used in his proof of the first Brauer-Thrall conjecture [A2] and later on used systematically in his joint work with I. Reiten on stable equivalence [AR], [AR2] and many other applications. Recently, functor categories were used in [MVS3] to study the Auslander-Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category, and we can think of the components of the Auslander-Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category $\mathrm{Mod}(\mathcal{C} )$, with $\mathcal C$ a component of the Auslander-Reiten quiver.