Superpotentials, Calabi-Yau algebras, and PBW deformations

The paper [9] by Bocklandt, Schedler and Wemyss considers path algebras with relations given by the higher derivations of a superpotential, giving a condition for such an algebra to be Calabi-Yau. In this paper we extend these results, giving a condition for a PBW deformation of a Calabi-Yau, Koszul path algebra with relations given by a superpotential to have relations given by a superpotential, and proving these are Calabi-Yau in certain cases. We apply our methods to symplectic reflection algebras, where we show that every symplectic reflection algebra is Morita equivalent to a path algebra whose relations are given by the higher derivations of an inhomogeneous superpotential. In particular we show these are Calabi-Yau regardless of the deformation parameter. Also, for G a finite subgroup of GL_2(C) not contained in SL_2(C), we consider PBW deformations of a path algebra with relations which is Morita equivalent to C[x,y] \rtimes G. We show there are no nontrivial PBW deformations when G is a small subgroup.