Computing the structure of the Hochschild cohomology of an algebra can be hard work, but Etingof and Eu (2006) showed that it can be done surprisingly easily for preprojective algebras associated to ADE Dynkin diagrams, at least if you only want to know the graded vector space structure of each Hochschild cohomology group. Their method has since been used by Evans and Pugh (2012) on higher preprojective algebras that arise from “higher” ADE Dynkin diagrams, although the type A case was only recently completed by Morigi in his thesis (2022) up to assuming the truth of a conjectured formula for the determinant of the graded Cartan matrix of such an algebra.

In this talk, we present a generalization of the method used by Etingof and Eu (jt. with Jon W. Anundsen) obtained in part through a theory of projective resolutions of almost T-Koszul algebras (jt. with Johanne Haugland). In many cases, this generalization is easier to use, and we present applications to computations of Hochschild cohomology for other classes of higher preprojective algebras and how we can recover Morigi's results without the dependence on the conjectured formula mentioned above.

This work is in part motivated by potential applications of Hochschild cohomology to the periodicity conjecture and the conjectured acyclicity of the quivers of d-hereditary algebras.

The talk is based on joint work with Jon Wallem Anundsen and joint work with Johanne Haugland.