Koszul calculus of preprojective algebras

Abstract

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A 1 and A 2 , vanishes in any (co)homological degree p > 2 . Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees p and 2 − p , and we prove a generalised version of the 2-Calabi–Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi–Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi–Yau property and we say that they are Koszul complex Calabi–Yau (Kc–Calabi–Yau) of dimension 2. For Kc–Calabi–Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.