Hochschild (co)homology and cyclic homology via a graded Euler characteristic with applications to higher preprojective algebras

Abstract

Computing the structure of the Hochschild (co)homology and the cyclic homology of an algebra can be hard work, but Etingof and Eu showed that it can be done surprisingly easily for preprojective algebras of ADE Dynkin type, at least if one only wants to know the graded vector space structure of each Hochschild cohomology group. Their method is based on exploiting strong structural features of such a preprojective algebra via a graded Euler characteristic that can computed using the algebra's graded Cartan matrix. In this paper, we present a generalization of the method used by Etingof and Eu to higher preprojective algebras. We also apply our generalization to the higher preprojective algebras of the 2-representation finite algebras that arise as tensor products of representation finite hereditary algebras of type A. For this, it turns out to be enough to know the graded vector space structure of the center and the zeroth Hochschild homology to be able to deduce the graded vector space structure of the Hochschild (co)homology and the cyclic homology in all other degrees.