The Auslander-Reiten Components Seen as Quasi-Hereditary Categories

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 , with a component of the Auslander-Reiten quiver.