Superpotentials and higher order derivations

We consider algebras defined from quivers with relations that are kkth order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GLn for all nn, and four-dimensional Sklyanin algebras. More generally, we show that any NN-Koszul, (twisted) Calabi–Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between NN-Koszul twisted Calabi–Yau algebras AA and algebras defined by a superpotential ωω such that an associated complex is a bimodule resolution of AA. Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of an extension of SL2(Z/4).