Minimal model of Ginzburg algebras

Abstract

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

Koszul duality patterns in Floer theory

“…Theorem 8 (Hermes [40], and also Cor. (27)) (1) Suppose Γ is non-Dynkin, then H * (G Γ ) = Π Γ is supported in degree 0 and is quasi-isomorphic to G Γ .…”

“…Its cohomology has locally finite grading. Indeed, for an (algebraically closed) field with characteristic zero, it was computed in [40] that…”

“…For a non-Dynkin tree Γ, working over K of characteristic 0, Hermes [40] proved that G Γ is formal, that is, G Γ is quasi-isomorphic to its homology Π Γ (see also Cor. (27) for another proof that works over any field).…”

“…It was first proven by Hermes [40] under the assumption that K is algebraically closed and characteristic 0. In Cor.…”