# (Non-)formality for ADE preprojective algebras

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?

• The preprojective algebra is Koszul for not ADE, but only almost Koszul in the Dynkin cases, see Corollary 4.3 of Brenner, Butler, King: Periodic algebras which are almost Koszul. Since Koszulity is equivalent to formality of the Ext-algebra of the simples, this is related. – Julian Kuelshammer Dec 24 '17 at 12:00

This is a case in which various heuristics about the 'niceness' of ADE quivers can interfere with each other. One result in this area that might conform to your expectations is that the algebra $$\mathrm{H}^0(\Gamma_Q)$$ is finite-dimensional if and only if $$Q$$ is a Dynkin quiver. (If $$Q$$ is extended Dynkin, then it is infinite dimensional but Noetherian, and for other quivers even Noetherianity fails.) From a certain point of view, this makes the Dynkin case better behaved.
However, it also removes any possibility of $$\Gamma_Q$$ being formal. As you point out, $$\Gamma_Q$$ is a $$2$$-Calabi–Yau dg-algebra, so when it is formal, the ordinary algebra $$\mathrm{H}^0(\Gamma_Q)$$ is also $$2$$-Calabi–Yau. But finite-dimensional algebras (at least ungraded ones) cannot have this property: see this answer.
I am not sure which geometrical situations you have in mind, but sometimes one should replace a Dynkin diagram by its affine counterpart when passing from geometry to representation theory. For example, if $$R$$ is a Kleinian singularity corresponding to a Dynkin diagram $$\Delta$$, its category of Cohen–Macaulay modules is equivalent to projective modules over $$\mathrm{H}^0(\Gamma_Q)$$ for $$Q$$ any orientation of the extended Dynkin diagram $$\tilde{\Delta}$$ (and this preprojective algebra is a non-commutative crepant resolution of $$R$$.)