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A classification of some 3-Calabi-Yau algebras

Presented by: 
Paul Smith University of Washington
Wednesday 29th March 2017 -
10:00 to 11:00
INI Seminar Room 1
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^{\otimes 3} 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. 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons