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.
Highlights
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 [9] a complex, W, is defined which depends only on the superpotential, and a path algebra with relations is N -Koszul and CY if and only if it is of the form D(Φn, k) for a superpotential Φn and W is a resolution
We define an inhomogeneous superpotential of degree n to be an element of the path algebra Φ′ := Φn +φn−1 +· · ·+φk, such that each φj := cpp is a sum of elements of the path algebra of length j, and each φj satisfies the n superpotential condition
Summary
We set up the definitions and give a summary of results which we use later. A is 2-Koszul and n-CY if and only if A is of the form D(Φn, n−2) for some homogeneous superpotential Φn of degree n and the attached complex W is a resolution of A In this case the resolution equals the Koszul complex of Definition 2.3.1 with each Wi = Ki. The complex (1) is the relevant complex for D(Φn, n − 2), where the relations are obtained by differentiation by paths of length n − 2. In the case with D(Φn, n − 2) Koszul and S a field there is the following theorem of Wu and Zhu, [16], which classifies when PBW deformations of a Noetherian n-CY algebra are n-CY In our language this applies when there is only a single vertex in the quiver.
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