Given a quiver $Q$, I can associate to $Q$ a certain 2-Calabi-Yau (dg-)algebra $\Gamma_Q$ by a 2-dimensional version of the "Ginzburg dga" construction: *i.e.*, start by doubling $Q$, and then impose the "preprojective" relation $\sum_{x\in Q_1}[x,x^*]=0.$ (See, *e.g.*, this article of Van Den Bergh for more details.) Then $H^0(\Gamma_Q)$ is what is normally referred to as the "preprojective algebra" associated to $Q$.

I am interested the formality of $\Gamma_Q.$ According to this article, we have the following result, assuming the quiver $Q$ is acyclic:

**Theorem** [Hermes]: The algebra $\Gamma_Q$ is formal if and only if the quiver $Q$ is *not* of ADE type.

This result is quite surprising to me. First, the heuristic I have always had in mind for ADE quivers is that they are strictly better-behaved than non-ADE quivers, whereas in this case it is the ADE type quivers which seem more complicated. Second, the algebras $\Gamma_Q$ show up in geometrical situations where there is a natural bigrading, and there doesn't seem to be any natural difference in the ADE & non-ADE geometries, so I might hope that I might be able to make a formality argument there which works uniformly across quiver types--but clearly such an argument is bound to fail in type ADE.

The proof in *loc. cit.* is a calculation, which I don't find particularly enlightening or at least in which I can't find answers to the above questions which satisfy me. So my question is a request for another explanation of this phenomenon. In other words:

Question: What is special about preprojective algebras in type ADE? Why should I have expected a formality argument to fail in this case where it succeeded for all other quivers?