Would you please give some references concerning the number of indecomposable modules over preprojective algebras of type $A_n$?
More precisely, I need references about the following claim: The number of indecomposable modules over the preprojective algebra of type $A_n$ is:
1) only one for $n=1$,
2) four for $n=2$,
3) $12$ for $n=3$,
4) $40$ for $n=4$, and
5) an infinite number of indecomposable modules for $n\geq5$.
This is stated in Rigid modules over preprojective algebras by Geiss, Leclerc and Schröer, Sections 8.1-8.3. As noted in my comment a proof of the $n=5$ statement can be found in The module theoretic approach to quasi-hereditary algebras. (These are of course not the original references, I doubt that there is one original reference) Note also that you can split up the $n\geq 5$ case in $n=5$, which is the tame case, i.e. there is a classification of the infinitely many indecomposable modules, and the $n>5$ case, where a classification is not possible.
EDIT: There is a proof in Semicanonical bases and preprojective algebras by Geiss, Leclerc, Schröer, Section 9, via the theory of Galois coverings. The Auslander-Reiten quiver of these algebras can also be found there. The statement was long before and is probably part of the folklore in the representation theory of quivers.
