$\Gamma$ of §\[sect3.1\]. We require $Q$ to be acyclic, that is, $Q$ has no oriented cycle. Let $\overline{Q}$ denote the double quiver obtained from $Q$ by adjoining to every arrow $a\colon i\to j$ an opposite arrow $a^*\colon j \to i$. Consider the element $$\rho = \sum (aa^* - a^*a)$$ of the path algebra $\C\overline{Q}$ of $\overline{Q}$, where the sum is over all arrows $a$ of $Q$. Following [@GP; @Ri], we define the *preprojective algebra* $\L$ as the quotient of $\C\overline{Q}$ by the two-sided ideal generated by $\rho$. It is well-known that $\L$ is independent of the choice of orientation $Q$ of $\Gamma$. Moreover, $\L$ is finite-dimensional if and only if the Kac-Moody algebra $\g$ is finite-dimensional, that is, if and only if $\Gamma$ is a Dynkin diagram of type $A, D, E$.

We say that a finite-dimensional $\L$-module is *nilpotent* if all its composition factors are one-dimensional. Let $\nil(\L)$ denote the category of nilpotent $\L$-modules. This is an abelian category with infinitely many isomorphism classes of indecomposable objects, except if $\g$ has type $A_n$ with $n\le 4$. It is remarkable that these few exceptional cases coincide precisely with the cases when the cluster algebra $\C[N]$ has finitely many cluster variables. 