Superpotentials, Calabi–Yau algebras, and PBW deformations

J. Karmazyn

Research output: Contribution to journalArticlepeer-review

Abstract

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 C[V]⋊G, for V a finite dimensional C vector space and G a finite subgroup of GL(V), is Morita equivalent to a path algebra with relations given by a superpotential, and is Calabi–Yau for G<SL(V). 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 GL2(C) not contained in SL2(C), we consider PBW deformations of a path algebra with relations which is Morita equivalent to C[x,y]⋊G. We show there are no non-trivial PBW deformations when G is a small subgroup.
Original languageEnglish
Pages (from-to)100-134
JournalJournal of Algebra
Volume413
DOIs
Publication statusPublished - 1 Sep 2014

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