The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

Abstract

Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories 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, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martinez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think 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.