Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a reference for what the dimension actually is for each type. I'm also interested in knowing the dimension of $e\_i \Pi(Q)$, where $e\_i$ is the idempotent corresponding to vertex $i$ (i.e. the dimension of the space of all paths starting at vertex $i$).


I've already calculated by hand that the dimension of $\Pi(\mathbb{A}\_n)$ is given by ${n+2 \choose 3}$, and that the dimension of $e\_i \Pi(\mathbb{A}\_n)$ is given by the $i$th entry of the $n$th diagonal in the multiplication table for positive integers. However, the other Dynkin types seem to be more difficult.



Dimension of preprojective algebra of Dynkin type
It is something well-known but I do not know where it is written. q-deform the Cartan matrix into
$$C(q)=I+qB+q^2I.$$
$B$ has zeroes on the main diagonal; substituting $q=1$ gives you Cartan. Invert the thing:
$$C(q)^{-1}=I+qA\_1+\ldots q^nA\_n+\ldots.$$
Let $k$ be the smallest integer such that $A\_k$ has negative coefficients. Then
$$\sum\_{n=0}^{k-1}q^nA\_n$$
is the Poincare series of the preprojective algebra, i.e., all dimensions of all graded components of all $e\_i\Pi e\_j$ are written down.


