Minimal Model of Ginzburg Algebras

We compute the minimal model for Ginzburg algebras associated to acyclic quivers $Q$. In particular, we prove that there is a natural grading on the Ginzburg algebra making it formal and quasi-isomorphic to the preprojective algebra in non-Dynkin type, and in Dynkin type is quasi-isomorphic to a twisted polynomial algebra over the preprojective with a unique higher $A_\infty$-composition. To prove these results, we construct and study the minimal model of an $A_\infty$-envelope of the derived category $\mathcal{D}^b(Q)$ whose higher compositions encode the triangulated structure of $\mathcal{D}^b(Q)$.