Generalized standard Auslander-Reiten components

Abstract

A$ the category of finitely generated right $A$ -modules, $rad^{\infty}(mod A)$ the infinite radical of $mod A$ , and $\Gamma_{A}$ the Auslander- Reiten quiver of $A$ .It is well-known (see [6]) that $\Gamma_{A}$ describes the quotient category $mod A/rad^{\infty}(mod A)$ .We are interested in the behaviour of the con- nected components of $\Gamma_{A}$ in the category $mod A$ .In the rePresentation theory of finite dimensional algebras over an alge- braically closed field $k$ , an important role is played by the standard Auslander- Reiten components.Recall that following [12], [36], a connected component $C$ of the Auslander-Reiten quiver $\Gamma_{A}$ of a finite dimensional $k$ -algebra $\Lambda$ is called standard if the full subcategory of $mod \Lambda$ formed by all modules from $C$ is equivalent to the mesh-category $k(C)$ of $C$ .If $\Lambda$ is representation-finite (basic, connected), then $\Gamma_{A}$ is standard if and only if $\Lambda$ admits a simply connected Galois covering [12], [13].Moreover, if $k$ is of characteristic 2, then there