The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers

The Hochschild cohomology ring of any associative algebra, together with the Hochschild homology, forms a structure of calculus. This was proved in Daletski et al. (1990) [1]. In this paper, we compute the calculus structure for the preprojective algebras of Dynkin quivers over a field of characteristic zero, using the Batalin–Vilkovisky structure of the Hochschild cohomology. Together with the results of Crawley-Boevey et al. [2], where the Batalin–Vilkovisky structure is computed for non-ADE quivers (and the calculus can be easily computed from that), this work gives us a complete description of the calculus for any quiver.