The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers

In this paper, we compute the cup product structure of the preprojective algebra Dynkin quivers of type D and E over a field of characteristic zero. This is a continuation of the work done in [P. Etingof, C. Eu, Hochschild and cyclic homology of preprojective algebras of ADE quivers, arXiv: math.AG/0609006] where the additive structure of the Hochschild cohomology (together with its grading) was computed. Together with the results in [K. Erdmann, N. Snashall, On Hochschild cohomology of preprojective algebras. I, J. Algebra 205 (2) (1998) 391–412, II, J. Algebra 205 (2) (1998) 413–434] (where the A-case was studied), this yields a complete description of the product in the Hochschild cohomology of ADE quivers over a field of characteristic zero.