It is well-known that there are finitely many indecomposable module over the preprojective algebra associated to a quiver $Q$ if and only if $Q=A\_2,A\_3,A\_4$ and tame type for $A\_5$ and wild for others.
Now let say $Q$ is a ADE Dynkin diagram, and $V$ be an indecomposable preprojective algebra. What can we say about the dimension vector $|V|$ about $V$, does $|V| \in \mathbb{N}\_0^I$ have an explicit bound?


I think for $Q=A\_5$, for any indecomposable module the dimension vector of which is bounded by $(1,2,2,2,1)$.
What about other types?



dimension vector of indecomposable module over preprojective algebra
For every finite dimensional algebra that does not have finite representation type, there is no bound on the dimension of indecomposable modules. This is known as the first Brauer-Thrall conjecture (now a theorem).


