Classification of indecomposable modules in tame hereditary algebras

Question

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?

Answer 1

In the Dynkin case, Gabriel's 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, 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).

Quote:

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.