Abstract

This is a report on joint work with Izuru Mori and work of Mori and Ueyama.
A graded algebra A is Calabi-Yau of dimension n if the homological shift A[n] is a dualizing object in the appropriate derived category. For example, polynomial rings are Calabi-Yau algebras. Although many examples are known, there are few if any classification results. Bocklandt proved that connected graded Calabi-Yau algebras are of the form TV/(dw) where TV denotes the tensor algebra on a vector space V and (dw) is the ideal generated by the cyclic partial derivatives of an element w in TV. However, it is not known exactly which w give rise to a Calabi-Yau algebra. We present a classification of those w for which TV/(dw) is Calabi-Yau in two cases: when dim(V)=3 and w is in V ^{and when dim(V)=2 and w is in V}{\otimes 4}.  We also describe the structure of TV/(dw)  in these two cases and show that (most) of them are deformation quantizations of the polynomial ring on three variables.