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 particular they show that the algebra , for V a finite dimensional vector space and G a finite subgroup of , is Morita equivalent to a path algebra with relations given by a superpotential, and is Calabi–Yau for

. 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 not contained in , we consider PBW deformations of a path algebra with relations which is Morita equivalent to . We show there are no non-trivial PBW deformations when G is a small subgroup.