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∞A∞-composition. To prove these results, we construct and study the minimal model of an A∞A∞-envelope of the derived category Db(Q)Db(Q) whose higher compositions encode the triangulated structure of Db(Q)Db(Q).