Consider the preprojective algebra of type $A\_n$. It is well known that this algebra is of finite representation type when $n<5$, of tame representation type when $n=5$, and of wild representation type when $n>5$. In particular, this means that there exists a family of indecomposable representations of the preprojective algebra of type $A\_5$ which depends on a continuous parameter. I was wondering if anyone knew of an explicit example of such a family. Any examples of two-parameter families in type $A\_n$ for $n>5$ would be also helpful.



One-Parameter Families of Indecomposable Representations of the Preprojective Algebra of type A5
I only know the answer for type A5.
The smallest dimension-vector for a one-parameter family of indecomposable representations is (1, 2, 2, 2, 1). All the representations in the family have socle and head both equal to S\_2 \oplus S\_4. (Here S\_i is the one-dimensional representation attached to vertex i, with the standard labelling of the vertices.) With these informations, you will easily construct the desired representations.
The paper *Semicanonical bases and preprojective algebras* by Geiss, Leclerc and Schröer (Ann. Sci. Ecole Norm. Sup. 38 (2005), 193-253) contains a precise description of the representations of the preprojective algebra of type A5.


