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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 $n\geq5$.

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up vote 5 down vote accepted

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.

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