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 $A_{\mathrm{\infty }}$-composition. To prove these results, we construct and study the minimal model of an $A_{\mathrm{\infty }}$-envelope of the derived category ${\mathcal{D}}^b(Q)$ whose higher compositions encode the triangulated structure of ${\mathcal{D}}^b(Q)$.