Koszul calculus of preprojective algebras

Roland Berger (1) , Rachel Taillefer (2)

1 AGL - Algèbre, géométrie, logique
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

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 Poincar\'e Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.

Domaines: K-théorie et homologie [math.KT]; Anneaux et algèbres [math.RA]; Théorie des représentations [math.RT]

Roland Berger, Rachel Taillefer. Koszul calculus of preprojective algebras. Journal of the London Mathematical Society, 2020, Journal of the London Mathematical Society, 102 (3), pp.1241-1292. ⟨10.1112/jlms.12362⟩. ⟨hal-02132927v2⟩