## 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.

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 language | English |
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Pages (from-to) | 100-134 |

Journal | Journal of Algebra |

Volume | 413 |

DOIs | |

Publication status | Published - 1 Sep 2014 |