# Classification of indecomposable modules in tame hereditary algebras

An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite representation type if the number of distinct isomorphism classes of indecomposable $A$-modules is finite. Thus if an algebra is tame (or finite), there is considered to be at least some hope of classifying the indecomposable $A$-modules.

Let $Q$ be a quiver (without orientated cycles), $K$ an algebraically closed field and $KQ$ be the path algebra of $Q$ over $K$.

The tame hereditary algebras (over $K$) are known to be precisely those Morita equivalent to a path algebra $KQ$, where $Q$ is a simply laced Dynkin quiver (in the finite case) or Euclidean quiver (in the infinite case).

The precise classification of indecomposable $KQ$-modules is known for the quivers of type $A_n$ and $\widetilde{A}_n$. These are string (specifically gentle) algebras, and thus are given by string and band modules. What about $D_n$, $\widetilde{D}_n$, $E_6$, $E_7$, $E_8$, $\widetilde{E}_6$, $\widetilde{E}_7$ and $\widetilde{E}_8$? String, tree and band modules are not sufficient in each of these cases.

Would someone be able to point me towards a reference (or references) which classify the indecomposable modules for each of these cases?

The aim of this chapter is to present a classification of indecomposable $A$-modules and a detailed description of the Auslander-Reiten quiver $\Gamma (\operatorname {mod} A)$ of any hereditary path algebra $$A=KQ$$ of a finite acyclic quiver $Q$ whose underlying graph $\overline Q$ is Euclidean.