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?



Classification of indecomposable modules in tame hereditary algebras
In the Dynkin case, [Gabriel's theorem](https://en.wikipedia.org/wiki/Gabriel%27s_theorem) states that the indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram. You can read about it for instance in chapter VII of


*Ibrahim Assem, Daniel Simson, and Andrzej Skowronski*, MR 2197389 [**Elements of the representation theory of associative algebras. Vol. 1**](http://dx.doi.org/10.1017/CBO9780511614309), ISBN: 978-0-521-58423-4; 978-0-521-58631-3; 0-521-58631-3.


The Euclidean case is treated in chapter XIII of *Elements of the representation theory of associative algebras. Vol. 2* by the same authors (can't find it with the citation helper).


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> 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.
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