# Superpotentials, Calabi–Yau algebras, and PBW deformationsAcademic research paper on "Mathematics"

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## Abstract of research paper on Mathematics, author of scientific article — J. Karmazyn

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 &lt; 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 GL 2 ( C ) not contained in SL 2 ( 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.

## Academic research paper on topic "Superpotentials, Calabi–Yau algebras, and PBW deformations"

﻿ELSEVIER

Superpotentials, Calabi-Yau algebras, and PBW deformations

J. Karmazyn

a r t i c l e i n f o a b s t r a c t

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] x 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] x G. We show there are no non-trivial PBW deformations when g is a small subgroup.

Contents

1. Introduction...................................................... 101

http://dx.doi.org/10.1016/j-jalgebra.2014.05.007

Contents lists available at ScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

I CrossMark

Article history: Received 21 June 2013 Available online 2 June 2014 Communicated by Michel Van den Bergh

Keywords: Path algebra Superpotential Calabi—Yau algebras Koszul algebras PBW deformations

1.1. Introduction ..................................................................................................101

1.2. Main results ..................................................................................................102

1.3. Contents........................................................................................................103

2. Preliminaries............................................................................................................103

2.1. Quivers and superpotentials..............................................................................103

2.2. Calabi—Yau algebras........................................................................................107

2.3. Koszul algebras..............................................................................................108

2.4. Superpotentials and higher order derivations......................................................109

2.5. PBW deformations..........................................................................................111

3. Main results ............................................................................................................113

3.1. Deformations of superpotential algebras ............................................................114

3.2. CY property of deformations............................................................................117

4. Application: symplectic reflection algebras....................................................................122

4.1. Symplectic reflection algebras as superpotential algebras......................................123

4.1.1. Examples..........................................................................................124

5. Application: PBW deformations of skew group rings for GL2..........................................127

Acknowledgments..............................................................................................................130

Appendix A. McKay quivers for finite small subgroups of GL2(C)..........................................130

References........................................................................................................................134

1. Introduction

1.1. Introduction

In this paper we consider path algebras of quivers with certain relations, in particular studying relations produced from a superpotential. Given a quiver Q, a homogeneous superpotential of degree n is an element, <Pn = cai...ana1 ... an, in the path algebra of Q satisfying the n superpotential condition: caq = (—1)n-1 cqa for all arrows a and paths q. From such a superpotential <Pn and a non-negative integer k we construct an algebra T>(<&n, k) := R as a path algebra with relations R. These relations are constructed by the process of differentiation, where we define the left derivative of a path p by a path q, denoted Sqp, to be t if p = qt and 0 otherwise, and the relations are given by

R = {{SP^n : |p| = k».

Algebras of this form are considered in [9], where they are related to Calabi-Yau (CY), W-Koszul algebras. In [9] a complex, W*, is defined which depends only on the superpotential, and a path algebra with relations is W-Koszul and CY if and only if it is of the form T>(&n, k) for a superpotential <Pn and W* is a resolution.

Skew group algebras, C[V] xG, for G a finite subgroup of GL(^), are Morita equivalent to path algebras of this form. These are 2-Koszul, and CY when G < SL(^), and hence their relations can be given by a superpotential. An explicit way to calculate this superpotential is given in [9, Theorem 3.2].

We prove two results concerning the PBW deformations of (n — k)-Koszul, (k + 2)-CY algebras of the form T>(\$n, k). We define an inhomogeneous superpotential of degree n to be an element of the path algebra <P' := <Pn + + ... + , such that each := J2cpp is a sum of elements of the path algebra of length j, and each satisfies the n superpotential condition. Such a superpotential defines relations P = ({Sp<P' : |p| = k}},

and we define V(<&', k) := -jr. Theorem 3.1.1 classifies which PBW deformations are of this form. This is known for the case of PBW deformations of D(<Pn, 1) [7, Theorem 3.1, 3.2], but here we extend this to include higher differentials, k > 1.

We then prove that certain classes of PBW deformations of a 2-Koszul, n-CY T>(<£n,n — 2) are n-CY in Theorem 3.2.1. This is already known in the 3-CY case due to [7] which proves that any PBW deformation of an W-Koszul, 3-CY T>(<£n, 1) given by inhomogeneous superpotential is 3-CY [7, Theorem 3.6], and in the case of a one vertex quiver there is a result [17, Theorem 3.1], which finds a necessary and sufficient condition for a PBW deformation of a Noetherian, 2-Koszul, n-CY algebra to be n-CY.

Next we give an application of these results to symplectic reflection algebras. Sym-plectic reflection algebras are defined in [12] as PBW deformations of certain skew group algebras C[V] x G, and hence we consider the Morita equivalent path algebra with relations. Applying the previous results, and a result of [12], we deduce that these path algebras with relations are of the form T>(@2n + \$2n-2, 2n — 2) and are 2n-CY.

We go on to consider PBW deformations of the path algebras with relations Morita equivalent to C[C2] x G when G is a finite subgroup of GL2(C) not contained is SL2(C). We show that if G does not contain pseudo-reflections there are no PBW deformations.

1.2. Main results

1) A classification of the PBW deformations of a (k + 2)-CY, (n — k)-Koszul, superpotential algebra D(<Pn, k), whose relations are given by inhomogeneous superpotentials. These are proved to be the PBW deformations satisfying one additional property, which we call the zeroPBW condition, Definition 3.0.6, and the corresponding superpotentials are shown to be k-coherent superpotentials, Definition 3.0.5.

Theorem (Theorem 3.1.1). Let A = T>(@n, k), for <Pn a homogeneous superpotential of degree n, be (n — k)-Koszul and (k + 2)-CY. Then the zeroPBW deformations of A correspond exactly to the algebras T>(<i>', k) defined by k-coherent inhomogeneous superpotentials of the form <P' = <Pn + 0n-1 + ... + 4*k.

2) A proof that these PBW deformations are CY in certain cases. We consider A = T>(&n, n — 2) which is n-CY and 2-Koszul, and a zeroPBW deformation, A, which by the previous results is of the form A := T>(\$>', n — 2) for <P' = <Pn + \$n-1 + 0n-2.

Theorem (Theorem 3.2.1). Suppose 4>n-1 = 0, then A is n-CY.

3) An application to symplectic reflection algebras. Let H be a path algebra with relations Morita equivalent to an undeformed symplectic reflection algebra. So H is 2-Koszul and 2n-CY and H = T>(&2n, 2n — 2) for some homogeneous degree 2n superpotential &2n.

Theorem (Theorem 4.1.1). Any PBW deformation of H is a zeroPBW deformation and of the form T>(<i>', n — 2) for = <p2n + 02n-2 an inhomogeneous superpotential. Hence any PBW deformation of H is 2n-CY, and all symplectic reflection algebras are Morita equivalent to 2n-CY algebras of the form D(<i>', 2n — 2).

1.3. Contents

We outline the structure of the paper.

Section 2: We discuss preliminaries, listing definitions and results fundamental to the rest of the paper. These concern quivers, superpotentials, CY algebras, W-Koszul algebras, and PBW deformations. In particular we recall a Theorem classifying PBW deformations from [5], and several Theorems from [9] concerning path algebras with relations defined by superpotentials and CY algebras.

Section 3: The main results of the paper are stated and proved here. We begin by making two definitions required to state our theorems, k-coherent superpotentials and zeroPBW PBW deformations. We then prove Theorem 3.1.1 classifying which PBW deformations of an W-Koszul, CY, superpotential algebra D(<Pn, k) are of the form T>(<P', k) for an inhomogeneous superpotential <P' = <Pn + \$n-1 + ... + . We then specialise to the case of 2-Koszul, n-CY, T>(<£n, n — 2), and consider PBW deformations of the form + 0n-2, n — 2), proving they are n-CY in Theorem 3.2.1. We note known related results.

Section 4: We apply our results to symplectic reflection algebras. We recall the definition of symplectic reflection algebras, and their classification as PBW deformations by [12]. We consider the Morita equivalent path algebras with relations, which are of the form T>(@2n + 02n-2, 2n — 2) and 2n-CY by the previous results. We calculate several examples, including the case of preprojective algebras.

Section 5: Finally we give an analysis of PBW deformations of algebras , 0) Morita equivalent to C[C2] x G for G a finite subgroup of GL2(C). In particular we show there are no nontrivial PBW deformations when G is a small subgroup not contained in SL2(C).

2. Preliminaries

In this section we set up the definitions and give a summary of results which we use later.

2.1. Quivers and superpotentials

Here we give the definition of a quiver, its path algebra, and its path algebra with relations. We define certain elements of the path algebra to be superpotentials, and give a construction to create relations on the quiver from the superpotential. We are following the set up of [9].

Quivers

Definitions 2.1.1. A quiver is a directed multigraph. We will denote a quiver Q by Q = (Qo,Qi), with Qo the set of vertices and Qi the set of arrows. The set of arrows is equipped with head and tail maps h, t : Q1 ^ Q0 which take an arrow to the vertices that are its head and tail respectively.

A non-trivial path in the quiver is defined to be a sequence of arrows

p = ar . .. a2a1 with ai G Q1 satisfying h(ai) = t(ai+1) for 1 < i < r — 1.

We will reuse the notation of the head and tail maps for the head and tail of a path, defining h(p) = h(ar) and t(p) = t(a1) when p = ar ... a2a1. We also define a trivial path ev for each vertex v G Q0, which has both head and tail equal to v. A path p is called closed if h(p) = t(p). The pathlength of a nontrivial path p = ar . ..a]^, where each ai is an arrow, is defined to be r. A trivial path is defined to have pathlength 0. We will denote the pathlength of a path p by |p|.

Definition 2.1.2. Let k be a field. We define the path algebra of the quiver Q, kQ, as follows: kQ as a k-vector space has a basis given by the paths in the quiver; an associative multiplication is defined by concatenation of paths.

\ pq if h(q) = t(p)

I 0 otherwise

We define S to be the subalgebra of this generated by the trivial paths, and V to be the k vector subspace of kQ spanned by the arrows, a G Q1. Then S is a semisimple algebra, with one simple module for each vertex.

For an arrow a, we have a = eh(a).a.et(a), and so V has the structure of a left Se := S Sop module and kQ can be identified with the tensor algebra TS(V) = S © V © (V ®S V) © ..., equating the path a1 ...ar with a1 ®S ... ®S ar.

The algebra TS(V) = kQ is equipped with a grading and filtration by pathlength, the graded part in degree n is TS(V)n := V®Sn, and the filtered part is Fn(TS(V)) := V n © ...© V © S.

Given R C TS(V) we define I(R) to be the two sided ideal in TS(V) generated by R. We then define

R ' I(R)

and refer to it as the path algebra with relations R.

We define (R) to be the Se submodule of Ts(V) generated by R.

Superpotentials We define superpotentials and twisted superpotentials in the homogeneous and inhomogeneous cases. In the majority of this text we will be working with non-twisted superpotentials, but in Section 5 we will consider the twisted case.

Definition 2.1.3. Let <&n G TS(V)n, written <&n = Spcpp, where the sum is taken over all paths p of pathlength n with coefficients cp in k. Our convention will be that c0 = 0. For example if t(a) = h(b) then ab = 0 and hence cab = c0 = 0.

We will say <&n satisfies the n superpotential condition if caq = (—1)n-1cqa for all a G Q1 and paths q, and if <Pn satisfies the n superpotential condition we will call it a homogeneous superpotential of degree n. In particular this requires cp = 0 if p is not a closed path.

Let a G Autk(kQ) be a graded automorphism, so that a(TS(V)n) = TS(V)n for all n > 0, and assume that a(h(p)) = h(a(p)), a(t(p)) = t(a(p)) for any path p.

We will say <Pn satisfies the twisted n superpotential condition if ca(a)q = (—1)n-1cqa for a G Q1 and paths q. We will say <Pn is a a-twisted homogeneous superpotential of degree n if it satisfies the twisted n superpotential condition.

Definition 2.1.4. Consider an element <P G Fn(TS(V)) written <P = Em<nE\p\=mcpp as above. We will say is an inhomogeneous superpotential of degree n if it satisfies the conditions caq = (—1)n-1cqa for any path q and arrow a, and for some p of length n the coefficient cp is non-zero. An inhomogeneous superpotential <P can be written in homogeneous parts <P = 0n + 0n-1 + ... + 0o where each of the satisfies the n-superpotential condition and 0n is non-zero. Note that if n is even, m odd, and char k = 2 then = 0, and also that any 0o G S satisfies the n superpotential condition.

We will say this is a a-twisted inhomogeneous superpotential of degree n if it satisfies ca(a)q = (—1)n-1 cqa for any path q and arrow a. A twisted inhomogeneous superpotential <P can be written in homogeneous parts <P = 0n + 0n-1 + ... + 0o where each of the satisfies the n-twisted superpotential condition.

Differentiation We define differentiation by paths.

Definition 2.1.5. Let p be a path in kQ, and a G Q1. We then define left/right derivative of p by a as;

s p = J q, if p = aq ps' = J q, if p = qa

a 0, otherwise a 0, otherwise

We extend this so that if q = a1. ..an is a path of length n then Sq = San . ..Sai, and sq = 5'an . ..5'ai. We will call the operation Sq left differentiation by the path q, and S'q right differentiation by the path q.

Note that for & a degree n inhomogeneous superpotential, b, c G Q1, and p any path

5b\$ =( — 1)n-1<P5'b, 5b<P5'c = (—1)n-15b5c& = (—1)n-1Scb&,

Sp& = Y^ aSaSp& = SP&S'aa.

aGQi aGQi

Definition 2.1.6. Given an inhomogeneous superpotential & of degree n we can consider the Se-modules Wn-k C kQ given by:

Wn-k = {Sp& : |p| = h) We then define the algebra D(&, h) to be

h) = kW-J

We will call this the superpotential algebra D(&, k).

Example 2.1.7. Consider the quiver, Q, with one vertex, •, and two arrows x and y. Then kQ is the free algebra on two elements, k{x, y)

We consider some superpotentials on Q.

1) = xy — yx is a degree 2 homogeneous superpotential.

2) <P = xy — yx — e, is an inhomogeneous superpotential of degree 2.

3) = xyx + xxy + yxx is a degree 3 homogeneous superpotential.

These give us the algebras

1) D(^2, 0) = k[x, y], as W2 = (xy — yx) gives relations R = {xy — yx = 0}

2) V(<P, 0) = A1(k), the first Weyl algebra, is the quiver with relations given by R = {xy — yx = e,}

3) , 1) is the quiver with relations given by R = {yx + xy = 0, xx = 0}

2.2. Calabi-Yau algebras

We define Calabi-Yau algebras following the definitions of [13] and [1]. We recall the definition of self-dual used in [9], and note that the existence of a self-dual finite projective Ae-module resolution of length n implies an algebra is n-Calabi-Yau.

Let A be an associative k algebra, set Ae = A < Aop and D(Ae) to be the unbounded derived category of left Ae modules. We note that there are two Ae-module structures on A < A, the inner structure given by (a < b)(x < y) = (xb < ay), and the outer structure given by (a < b)(x < y) = (ax < yb). By considering Ae with the outer structure for any Ae module, M, we can make Hom^e (M, Ae) an Ae-module using the inner structure.

Definition 2.2.1. Let n > 2. Then A is n bimodule Calabi-Yau (n-CY) if A has a finite length resolution by finitely generated projective Ae-modules, and

RHomAe (A, Ae) [n] ^ A in D(Ae)

In the paper [9] self-duality of a complex is used. We give the definition here. Denote (-)v := HomAe(-, Ae) : Ae-Mod ^ Ae-Mod.

Definition 2.2.2. Define a complex of Ae modules, of length n to be self-dual, written

HomAe (C •,Ae) = C n-\

if there exist Ae-module isomorphisms a such that the following diagram commutes:

dn ^ 1 dn di Cn -Cn-1 -' ' ' -Ci -*■ Co

r<V C0

By construction the existence of a length n self-dual projective Ae-module resolution of A implies that A is n-CY, and we will use this later in Section 3 to show algebras are n-CY.

We note some properties of Calabi-Yau algebras Lemma 2.2.3. Let A be n-CY k-algebra, where k is algebraically closed. Then

1. If there exists a non-zero finite dimensional A-module, then A has global dimension n.

2. For X, Y € D(A) with finite dimensional total homology

HomD{A)(X,Y) = HomD(A)(Y,X[n])*

where * denotes the standard k dual.

Proof. These are some of the standard properties of CY algebras, see for example [1, Proposition 2.4] and[7, Section 2] for proofs. □

2.3. Koszul algebras

Here we give the definitions of N-Koszul algebras.

The concept of an N-Koszul algebra was introduced by Berger, and is defined and studied in the papers [3-6,14]. It generalises the concept of a Koszul algebra, which is defined here to be a 2-Koszul algebra.

Definition 2.3.1. Let S be a semisimple ring, V a left Se-module, and TS(V) the tensor algebra. An algebra is N-homogeneous if it is given in the form A = TffV for R an Se-submodule of V03N.

We now define the Koszul N-complex. For N-homogeneous A = TfiRV) define

Ki = ] (V0j ®s R ®s V0k) i > N

j+N+k=i

Ki = V03i 0 < i < N Ko = S

and let

Ki(A) = A ®S Ki ®S A

Now we can define an N-complex (In an N-complex dN = 0 rather than d2 = 0.)

... —b Kn(A) —b ... —b K0(A) —b A b 0

To define the differentials let ¡i be multiplication, and di, dr : A <3S V0Si ®S A b A 0S V03(i-1) 0S A be defined by

dl(a <g) v1 v2 . ..vi ® ¡3) = av1 ® v2 ...vi ® ¡3 dr (a ® v1 ... vi-1vi ® 3) = a <g) v1 ... vi-1 ® vi3

As R C V0SN we see Ki C V0Si and hence can consider the restriction of dl, dr to Ki(A). Now choose q G k a primitive Nth root of unity, which we need to assume exists in k, and define d : Ki(A) b Ki-1(A) as d = dl Ki^) — (q)i-1dr Ki^).

As dl and dr commute with dN = 0 and dN = 0, d defines an N differential. To such an N-complex we can associate a complex by contracting several terms together. This is done by splitting the N-complex into sections of N consecutive differentials from the right. Each one of these is collapsed to a term with two differentials

by keeping the right most differential, and composing the other N — 1. This defines a complex of the form

... u kn +1(A) —U KN ———U K0(A) —U A U 0

which exists regardless of the field k, with d = (di — dr)\Kj(a) and dN-1 = (dN-1 + dN 2dr + ... + dN-2)\Kj(A).

We say that A is N-Koszul if this complex is exact, and hence gives an Ae-module resolution of A.

We call 2-Koszul algebras Koszul. In this case the N-complex is in fact a complex.

2.4. Superpotentials and higher order derivations

We summarise some results from the paper [9]. Later we will be particularly interested the applications to skew group algebras so we recall their definition and the relevant results from [9].

Superpotentials and higher order derivations Here we state several results from the paper [9] concerning superpotential algebras.

Let Q be a quiver, with path algebra CQ. Let <Pn be a homogeneous superpotential of degree n, and A = T>(&n, n — 2). For i = 0,...,n we have the Se-modules Wi as in Definition 2.1.6 above, and we can define a complex W• by

0 U A <S Wn <S A ——U A <S Wn-1 <S A -——U ... —U A <S Wo <S A U 0. (1) To define the differential first define di, dr : A <S Wi <S A U A <S Wi-1 <S A by

di(a < 5p\$n < 3) = aa < Sa^p^n < 3

dr (a < 5p\$n < 3) = a < SP&nS'a < a/

for p any path of length n — i. The differential di is defined as

(l ^ ) u i(—1)i(n-i) if i< (n + 1)/2

di = e^di + (—1)idr) where ei otherwise

This is a complex as dl, dr commute and have square 0. We show this for dl, the other case being similar. Writing <Pn = ^\t\=n ctt, we have

dj-1 o dj (1 0 Sp<Pn 0 ah ® SabSv\$n 0 1

Y ab 0 Cpabqq 0 1

= J2(-1)(n-1)(j-2)Cqpabab 0 q 0 1 a,b,q

= E(-1)(n-1)j5qp®n 0 q 0 1 = 0 q

where a, b G Q1, p is a path of length (n — j), q a path of length j — 2, and the sums are taken over all such a, b and q.

Theorem 2.4.1. (See [9, Section 6].) Let A = T>(<£n, n — 2), then W' is a self-dual complex of projective Ae-modules.

Theorem 2.4.2. (See [9, Theorem 6.2].) Let A = CQ be a path algebra with relations. Then A is 2-Koszul and n-CY if and only if A is of the form T>(@n, n — 2) for some homogeneous superpotential <Pn 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 T>(&n, n — 2), where the relations are obtained by differentiation by paths of length n — 2. More generally, differentiating by paths of length k, we need another complex. Let N = n — k, and in this case we define an N-complex W', again making use of the S^-modules Wj.

0 ^ A 0s Wn 0s A — A 0s Wn-1 0s A ... — A 0s Wo 0s A b 0 (2)

with differential di : A 0S Wi 0S A b A 0S Wi-1 0S A defined by di = di + (q)idr, for q a primitive Nth root of unity, which we assume exists in k. This can be contracted into a 2-complex, W'

0 b A 0S WmN +1 0S A —b A 0S WmN 0S A —b A 0S W(m-1)N+1 0S A —b ... —b A 0S Wo 0S A b 0

by composing the differentials in W', as in Definition 2.3.1, where m is the largest integer such that m < n/N.

Theorem 2.4.3. (See [9, Theorem 6.8].) Let A = CQ be a path algebra with relations. Then A is (n — k)-Koszul and (k + 2)-Calabi-Yau if and only if it is of the form T>(@n, k) where <Pn is a homogeneous superpotential of degree n and W' is a resolution of A. In this case W' equals the Koszul (n — k)-complex as in Definition 2.3.1.

Skew group algebras Let G be a finite subgroup of GL(V). We can form the smash product k[V] x G which is the semi direct product with the action of G given by that of GL(V). The multiplication of (f1, g1), (f2, g2) € k[V] x G is given by

(fi ,gi).(f2,g2) = (fif2S1 ,gig2)

This is a graded algebra with G in degree 0 and V* in degree 1, where k[V] = Sym(V*).

For such a group G and representation V we can construct the McKay quiver. We now assume our field has characteristic not dividing the order of G and is algebraically closed. The McKay quiver for (G, V) has a vertex, i, for each irreducible representation, Vi, of G over k, and dimk Hom(V V, Vj) arrows from i to j.

Theorem 2.4.4. (See [9, Theorem 3.2, Lemma 6.1].) Let V be an n-dimensional C-vector space and G be a finite subgroup of GL(V). Then C[V] x G is Morita equivalent to T>(@n, n — 2) for some homogeneous superpotential <Pn of degree n attached to the McKay quiver (G, V). Moreover T>(@n, n — 2) is n-CY and Koszul for G < SL(V).

There is a recipe to compute a superpotential <Pn such that C[V] x G is Morita equivalent to the superpotential algebra T>(<£n, n — 2) attached to the McKay quiver (G, V). The recipe is given in [9, Theorem 3.2]. When G is abelian the superpotential algebra T>(\$n, n — 2) is in fact isomorphic to C[V] x G.

2.5. PBW deformations

In this subsection we will recall the definition of the PBW deformations of a graded algebra A. We will be considering PBW deformations of A an N-Koszul algebra relative to S a semisimple algebra, and we will make use of the setup and results of [5].

Let S be a semisimple algebra, V a left Se-module, and TS(V) the tensor algebra of V over S. Consider the (Z>0) grading on this with degree n part TS(V)n = V®Sn, and filtered parts Fn = V®Sn 0 ... 0 V 0 S. For R an Se-submodule of V®SN define I (R) = Ei,j>0 V0iRVto be the two sided ideal in TS(V) generated by R, which is graded by I(R)n = I(R) fl V®Sn. Define A := tsrV) := ^jr), and as R is homogeneous this is a graded algebra with degree n part An := J(RS .

Now define the projection map nN : FN U V®SN, and let P be an Se-submodule of FN such that n(P) = R. Let I(P) be the 2 sided ideal in TS(V) generated by P. This is not graded but is filtered by I(P)n = I(P) f Fn, and hence A = ts(p) is also a filtered algebra with An = jpn.

We can construct the associated graded algebra gr (A), which is graded with gr (A)n = A—. Identify gr(A)n with the Se-module j(P)f+fn-1 by noting I(P)n f Fn-1 = I(P)n-1, and consider the maps 4>n : V0n U gr(A)n defined as the composition V®n u Fn U j(p),F+fn-1. This allows us to define a surjective algebra morphism

0 = 0n>0 0n : Ts(V) b gr(A). Now 0n(R) = 0 as P + FN-1 = R © FN-1, and hence this defines a surjective morphism of Z>0-graded S e-modules

p : A b gr (A)

Definition 2.5.1. We say that A = Tspf) is a PBW deformation of A = Ts(V) if p is an isomorphism. We say that P is of PBW type if A is a PBW deformation of A.

The papers [7] and [5] prove a collection of conditions on P equivalent to it being PBW type which we state here.

First note that for P C FN to be of PBW type it must be the case that P n FN-1 = {0}. Hence any PBW type P can be given as P = {r — 0(r) : r G R} for an Se-module map 0 : R b FN-1. Such a map can be written in homogeneous components as 0 = 6N-1 + ... + 00, with 0j : R b V0Sj.

Consider the map ^(0j) defined for each 0j as

^(0j) := id 0 0j — 0j 0 id : (V 0S R) n (R 0S V) b V0S(j+1) where (V 0S R) n (R 0S V) C V0s(N +1)

Theorem 2.5.2. (See [5, Section 3].) The Se-module P is of PBW type if and only if the following conditions are satisfied

PBW1) P n FN-1 = {0} PBW2) Im ^(0N_1) C R

PBW3) Im(0j(4>(0n_1)) + ^(0j_1)) = {0} for all 1 < j < N — 1 PBW4) Im00(V>(0n-1)) = {0}

Example 2.5.3. Let A = D(<Pn, k) be an N-Koszul (k + 2)-CY superpotential algebra, where N = n — k. Then Theorem 2.4.3 tells us that the Koszul N-complex equals the complex W*, and Wi = Ki, of Definition 2.3.1. Hence (V0S R) n (R0S V) = WN+1, and R = WN. We can also use the expression Sp<Pn = J2aeq1 a0 Sa5p<Pn = J2aeq1 Sp^nS'a 0 a to give an explicit form for ^(0j) with j = 0, ..., N — 1:

V>(0j) : Wn+1 b V0S(j+1)

5p& b a 0 0j (Sa5p&) — 0j(Sp&5'a) 0 a where \p\ = k — 1

We note that in the case of S being a field, and A being 2-Koszul, Theorem 2.5.2 was proved by Braverman and Gaitsgory [10]. They also show any PBW deformation gives a graded deformation. With the notation as above we define a graded deformation of A, At, to be a graded k[i] algebra with t in degree 1. That is At is free as a module over k[t] and there is an isomorphism At/tAt b A. It is shown that for a PBW deformation

A there is a graded deformation of A with fibre at t =1 canonically isomorphic to A [10, Theorem 4.1].

In the case with D(<Pn, n — 2) Koszul and S a field there is also the following theorem of Wu and Zhu [17], which classifies when PBW deformations of a Noetherian n-CY algebra are also n-CY. In our language this applies when there is only a single vertex in the quiver.

Theorem 2.5.4. Let <&n be a homogeneous superpotential on a single vertex quiver. Let A = T>(@n, n — 2). If A is n-CY Noetherian and Koszul and A is a PBW deformation given by do, 0i then A is n-CY if and only if

YJ(—1)iid0 di 0 id®(n-2-i)(^n) = 0 i=0

Proof. See [17, Theorem 3.1]. This proves such a PBW deformation is n-CY if and only if

J2(—1)iid0i 0 d1 0 id®(n-2-i) : p| V0i ® R 0 V®(n-2-i) ^ V®n-1 i=o i

is 0. Note that f|i V0i 0 R 0 V®(n-2-i) = Wn = (@n) in this case, giving the result. □ 3. Main results

We state and prove our main results concerning the PBW deformations of a (k + 2)-CY, (n — k)-Koszul, superpotential algebra A := D(<Pn, k), where <&n is a homogeneous superpotential. We will prove a condition for a PBW deformation to be of the form T>(<P', k) for an inhomogeneous superpotential = <Pn + 0n-1 + ... + and prove that inhomogeneous superpotentials with degree n part <Pn define PBW deformations of A. In the Koszul case we will show that certain PBW deformations of such an n-CY algebra are also n-CY.

Throughout this section consider <Pn G CQ to be a homogeneous superpotential of degree n on some quiver Q. This will have inhomogeneous superpotentials associated to it, denoted <P' = &n + 0n-1 + ... + .

Throughout this section we work over C in order to apply the results of [9], in particular Theorem 2.4.3. We note that this is the only way in which the assumption k = C is used in the proofs of Theorems 3.1.1 and 3.2.1, and we could instead assume the conclusions of Theorem 2.4.3 as additional conditions on A; that A = D(<Pn, k) and that the Koszul resolution of A equals the resolution W*.

We introduce two new definitions we make use of in the proof:

Definition 3.0.5. We will call &, an inhomogeneous superpotential, k-coherent if for any Xp e C

XPSP\$n = 0 e CQ ^ Xp5p& = 0 e CQ

\p\=k |p| = k

Definition 3.0.6. Let A = and A = be as in Definition 2.5.1. We will say

that A is a zeroPBW deformation of A if it is a PBW deformation which also satisfies the zeroPBW condition

Im( 4>(6n-1)) = {0},

where ^ and 0 are defined as in Section 2.5.

We will say P is of zeroPBW type if A is a zeroPBW deformation of A.

Lemma 3.0.7. An Se module P is of zeroPBW type if and only if

PBW1) P n FN-1 = {0}

ZPBW) Im(^(éj )) = {0} for j = 0 ...N — 1.

Proof. By definition P is of zeroPBW type if and only if it is of PBW type and also satisfies the zeroPBW condition. It is of PBW type if and only if it satisfies conditions PBW1, 2, 3, 4) of Theorem 2.5.2. Satisfying the zeroPBW condition is equivalent to reducing the conditions PBW2, 3, 4) to the condition ZPBW). □

3.1. Deformations of superpotential algebras

We state and prove our results relating superpotentials and PBW deformations.

Theorem 3.1.1. Let A = CQ be a path algebra with relations which is (n — k)-Koszul and (k + 2)-CY. Then A = D(&n, k), for <Pn a homogeneous superpotential of degree n, and the zeroPBW deformations of A correspond exactly to the algebras T>(<i>', k) defined by k-coherent inhomogeneous superpotentials of the form & = <Pn + 0n_i + ... + 0k.

Proof. As A is (k + 2)-CY and (n — k)-Koszul by Theorem 2.4.3 A = V(@n, k) for @n a degree n homogeneous superpotential.

We define inverse maps between P of zeroPBW type and coherent superpotentials which will give us the correspondence. We note that zeroPBW deformations exactly correspond to P of zeroPBW type.

We first define a map, F, taking P of zeroPBW type to k-coherent superpotentials. We define F(P) as an element of CQ, then show it is k-coherent and a superpotential.

Any zeroPBW deformation defined by P is given by a map 6 = 6n-k-1 + ••• + 60 with 6j : R b V®Sj such that P and 6 satisfy the conditions PBW1) and ZPBW) of Lemma 3.0.7. We construct F(P) := <Pn + 0n-1(6n-k-1) + ••• + (60) by defining

0n-j(6n-k-j) := p6n-k-j(Sp&n) =: ^ CsS

\p\ = k |s|=n-j

and show that this is a fc-coherent superpotential. We note that by this definition Sp^n-j(6n-k-j) = -6n-k-j(Sp\$n) for any \p\ = fc.

Firstly F(P) is not fc-coherent precisely when there exist coefficients Xp G C such that J2\p\=k \pSp<Pn = 0 with w = \p\=k (6) = 0. But this only occurs when

there exists w G P H Fn-1 which is non-zero. But as P is of zeroPBW type PBW1) holds, and P H Fn-1 = {0}. Hence F(P) is fc-coherent.

Now we show that F(P) is a superpotential. As P is of zeroPBW type by Lemma 3.0.7 Im ^(6j) = {0} for j = 0, • ••, n — fc — 1. We evaluate ^(6j) on elements of the form 5q<&n where \q\ = fc — 1, as in Example 2.5.3, and use 6j(5p&n) = —5p4>k+j (6j) to deduce that F(P) is a superpotential.

0 = V(6j )(Sq \$n) = ^ (°6j (^n) + ( —1)n6j (Saq \$n)a)

= —^2 (adqa^k+j (6j ) + (—1)n^aq0k+j (6j )a)

= —Y (Cqapap + (—1)nCaqpPa) a,p

= — ^ ] ((cqapi...pj + ( —1) cpj qapi...pj-i) ap)

with the sums over all a G Q1 and p = p1 • • • pj of length j.

Hence considering the coefficient of a path ap we see 0k+j = s css satisfies the n superpotential condition

cqapi ...pj = (—1) cpj qapi...pj-i

Hence each 0j+k satisfies the n superpotential condition, and F(P) is an inhomoge-neous superpotential of degree n.

We note that the zeroPBW deformation defined by P is, by construction, D(F(P), fc) as P = {{r — 6(r) : r G R}) = {{SpF(P) : \p\ = fc}).

Now we define a map, G, sending fc-coherent superpotentials = <&n + 0n-1 + • • • + &n-k to P of zeroPBW type. We define Gas an Se-module, show that Gsatisfies the PBW1) condition, and then define a map 6® as in Lemma 3.0.7. We then show that G(&) is of zeroPBW type.

The map G is defined by G(\$') := {{&p\$r : \p\ = fc}). We first check that G(\$') satisfies PBW1). Indeed as is fc-coherent there are no w = ^2\p\=k= 0, with Xp G C,

satisfying J2\p\=k ^pdp&n = 0. This implies G(&'n) n Fn 1 = {0}, hence G(&'n) satisfies PBW1).

Then, as G«) satisfies PBW1), = {{r - 0®' (r) : r G R}} where 0®' are the

maps defined by 0® := 0 1 0j!j+k with each oj!j+k defined on elements Sp<fin by

n ^ \$p0j + k

Q9i + k . R ^ y®Sj

We next show that G(&') is of zeroPBW type. By Lemma 3.0.7, to show that G(&') is of zeroPBW type it is enough to show that G(&') and 0® satisfy the conditions PBW1) and ZPBW). We have already shown PBW1) is satisfied, so need only check ZPBW); Im ^(0fj+k) = {0} for j = 0, ..., n - k - 1. As in Example 2.5.3 (V ®S R) n (R ®S V) = Wn- k+i = {Sp@n : \p\ = k—1}, hence we calculate ^(0jj+k )(Sq<&n), for j =0, ..., n—k—1, where \q\ = k — 1.

)(Sq\$n ) = Y,(a0tj+k (Sqa^n) + (-1)n0f+k (Saq \$n)a)

= (aSqa^j+h + (-1)nSaq\$j + ha)

= - ^2 (Cqapap + (-1)nCaqppa) a,p

= - ^2 dCqapi---pj + (-1) Cpj qa-pi-.-pj-l) ap)

where we write 0j = \t\=j ctt, and the sums are over all a G Q1 and p = p1. ..pj of length j. Hence as 0j satisfies the n superpotential condition

cqapi ...pr = ( —1) cPr qapi...pr-i

then ^(ef+k)(Sq\$n) = 0 for all q and j. Therefore Im^(6?k+i ) = {0} for all j, and G(^') is of zeroPBW type.

We note that as G(^') = : |p| = k}} by construction the zeroPBW deformation

defined by G(\$') is indeed V(&', k).

Now the maps F and G are inverses by construction, and provide a bijection between zeroPBW deformations and k-coherent superpotentials of the form <Pn + 0n _ 1 + ... + Moreover the zeroPBW deformation defined by P equals D(F(P), k), and the zeroPBW deformation defined by Gequals k). Hence the theorem is proved. □

Remark 3.1.2. Let A = V(\$n, n - 2) be a 2-Koszul n-CY algebra, and A = V(&, n - 2), as in Theorem 3.1.1, be a zeroPBW deformation of defined by 61, 60. Then if n is even 01 = 0.

Proof. Note that if n is even then any 0n-1 satisfying the n superpotential condition is 0, as observed after Definition 2.1.4. In particular a zeroPBW deformation defined by 60, 61 corresponds to a superpotential <&n + 0n-1 + 0n-2 by the bijection of Theorem 3.1.1, and superpotentials with 0n-1 = 0 correspond to zeroPBW deformations with 61 = 0. Hence for even n we have 0n-1 = 0 and so 61 = 0. □

3.2. CY property of deformations

We now consider when zeroPBW deformations are CY. We consider the case where A = D(<Pn, n — 2) is n-CY and 2-Koszul. We currently prove a result only in the case of a zeroPBW deformation with 61 =0, which — by Remark 3.1.2 — covers all even dimensional cases.

We then briefly mention two results concerning PBW deformations of CY algebras. Our results are weaker, but in a more general setting. One is the result of [17, Theorem 3.1], which is quoted as Theorem 2.5.4 above, giving a complete characterisation of CY PBW deformations of a Noetherian CY algebra over a field. The other is the paper [7] which proves that the zeroPBW deformations of a 3-CY superpotential algebra are 3-CY PBW deformations.

Theorem 3.2.1. Let A be a zeroPBW deformation of A with 61 = 0. Then A is n-CY.

Proof. By Theorem 3.1.1 such a deformation is given by a superpotential = <&n + 0n-2 = ctt and A = T>(<P', n — 2). We construct a resolution for A and show it is self-dual. We recall the complex W*, defined as (1) in Section 2.4, and, by Theorem 2.4.1, that W* is an Ae-module resolution of A. We then define a complex W*

0 bA®s Wn ®s A——+A®s Wn-1 ®S ••• —bA^s Wo ®S A—bAb0.

The Se-modules Wk, and differential d, are defined as in Section 2.4, i.e. Wj = {{5p&n : \p\ = n — j}), and di = €i(d\ + ( — 1)d) where

_ i ( —1)i(n-i) if i< (n + 1)/2 i 1 1 otherwise

and di, dr : A <S>s Wi ®s A bA<S>s Wi-1 <S>s A are defined by

di(a ® 5p<Pn ® /3) = Y aa ® ¿aSp&n ® ¡3

di (a ® 5p\$n ® 3) = a ® Sp^nK ® a/3

for p any path of length n — i. We will show this is a self-dual resolution of A.

We first check this is a complex, checking dj_1 o dj = 0 for j = 2,.n and that 1 o d1 =0. In the calculations \p\ = n — j, and the sums are taken over all a, b G Q1 and paths q of length j — 2.

dj-1 o dj (1 0 5p<Pn 0 1) = ejtj-1^2(ab 0 dpab&n 0 1 — 1 0 ¿p&nS'ab 0 ab)

= ejej_1 ^ (cpabqab 0 q 0 1 — 1 0 q 0 Opqabab)

= ejej_1 ^ ((—1)(n_1)jCqpabab 0 q 0 1 — 1 0 q 0 Cpqabab)

= ejej_1 (( —1)(n 1)jSqp^n 0 q 0 1 — 1 0 q 0 ¿pq#n)

= —ejej_1 ((—1)(n_1)jSqp\$n_2 0 q 0 1 — 1 0 q 0 ¿pq0n_2) q

= — ejej_1^2((—1)(n_1)jCqp 1 0 q 0 1 — 1 0 q 0 1cpq)

= —ejej_1 ^(cpq 1 0 q 0 1 — 1 0 q 0 1cpq) = 0 q

¡i o d1(1 0 Sp\$n 0 1) = enf a 0 Spa&n 0 1 — 1 0 ¿p®n5'a 0 a

e1 y^ (cpaa — (—1)n 1cap^ = e1 ^ (cpaa — Cpaa) = 0

Since A is a PBW deformation of A we have grA = A and so grW* = W*, where we use the product filtration, Fn(A 0s Wj 0S A) := Ei+j+fe<n Fi (A) 0s Wj 0s Fk(A). Since W* is exact, so is W* [16, Lemma 1], and since W* consists of finitely generated projectives so does W* [15, Lemmas 6.11, 6.16]. Hence W* gives a resolution of A by finitely generated projectives. Now we need only check that this is still self-dual, implying A is n-CY.

We now construct isomorphisms ak between the complex W* and its dual

A 0s Wn 0S A —^ A 0S Wn_1 0S A ■

d.2 d\ -* A 0s W1 0s A->- A 0s Wo 0s A

(A 0s Wo 0s A)v —i (A 0s W1 0s A)x

-dg _dn-1

(A 0S Wn_1 0S A)v -"(A 0S Wn 0S A)v

such that all the squares commute.

We will use the notation and result of [8, Section 4], working over C. For T a finite dimensional Se-module let FT be the Ae-module A <s T <s A, and let T* denote the C dual of T. Then

Ft * —^ FT

(1 ® 0 ® 1) — ((1 ® p < 1) — 0(epf )e <g>c f ^ e,fEQo

gives an isomorphism of Ae-modules. Moreover, as S is semisimple, tensoring A <s (—) <S A is flat, hence constructing an isomorphism of Se-modules Wj — W*-j will give us an isomorphism of Ae-modules Fwj — Fw**_ ., and composing with the above isomorphism a will give us an isomorphism Fw3- — FW _..

For \p\ = n — j define dp e W*- by dp(Sq= cqp, where &n = E|t|=n ctt and \q\ = j. Then there are Se-module homomorphisms

nj : Wj — W*-j

n — Yj dp

for Yj arbitrary nonzero constants.

To see nj is injective suppose nj(E XpSp\$n) = 0. Then J2p ^pdp(Sq\$n) = J2p ^qp = 0 for all q. Hence ]Tp \Sp\$n = Ep,q Xpcpqq = Eq(Ep Xpcpq)q = 0, so the map is injective.

There are C vector space isomorphisms between eWj f and (f Wn-je)* for e, f e Q0 arising from the pairings,

eWj f <c f Wn-je — C

eSp^nf < fSq&ne — Yj cqp = nj (e&p\$nf )(fSq^e)

= (—1)(n-1)|q|^ nn-j (fSq \$ne)(eSp\$nf)

where we note that dp(Sq\$n) = cqp = (—1)(n-1)|q|cpq = ( —1)(n-1)|q|dq(Sp\$n), and to be non-zero we require h(p) = t(q) and t(p) = h(q).

As eWjf and f Wn-je are finite dimensional C vector spaces the injectivity of nj and nn-j implies this pairing is perfect, thus nj is an isomorphism. Now we use this to define isomorphisms aj

aj : A <s Wj <s A — (A <s Wn-j <s A)v

(a1 < Sp\$n < a2) — ((61 < Sq\$n < ¿2) — (M2 < nj(Sp\$n)(Sq\$n) < a^)) = ((¿1 < Sq<Pn < ¿2) — (Yjcqpb1h(p)a2 <C a1 t(p)b2))

where we note that for this to be non-zero h(p) = t(q) = h(a2) = t(b1) and h(q) = t(p) = h(b2) = t(a1).

It remains to check a.j-i o dj = —o aj, i.e. the following diagram commutes;

A 0 s W7- 0 s A ■

A0s W,-_i 0S A

(A0s Wn-j 0 s A)

If \p\ = n - j and \q\ = j - 1 then

(aj-i o dj(1 0 Sp& 0 1))(1 0 Sq& 0 1)

(A0s wn-j+i 0s a)x

= aj-i( a ® Spa & 0 1 + (—1)j+(n-1)1 0 Sap\$ 0 a) (1 0 Sq & 0 1)

= Yj-iej X] Cqpat(q) 0 a + 7j-i£j ( —1)j+n-^ Cqapa 0 h(g)

whilst

(-'dn-j+i o aj (1 0 Sp& 0 1))(1 0 Sq & 0 1) = aj (1 0 Sp& 0 1)( -tn-j+i^2 a ® Sqa& 0 1 + (-1)j 1 0 Saq & 0 a J

= — Yj6n-j + ^ Cqapa 0 t(p) - Yj€n-j+i(-1)j ^ Caqp0 a aa

where the sums are taken over all a G Qi. Thus for these to be equal we require

—Yj tn-j+iCqap = Yj-iej (-1)j+n iC

-Yjen-j+i(-1)j Caqp = Yj-i£jcqpa = (-1)n Yj-i^

3^-aqp

for all a G Qi. These follow if (-1)nYj-i£j = Yj£n-j+i(- 1)j for j = 1, ■ ■■, n. As the Yj were arbitrary non-zero scalars, we can choose the Yj so that this is satisfied, and the proof is completed. □

Let Q be a single vertex quiver and A = D(&n, n - 2) be a Noetherian, Koszul, n-CY algebra. We consider a PBW deformation, A, given by Qi, Q0, and set 0n-i = EpP^i(Sp&n) = Ecqq. If the deformation is a zeroPBW deformation 0n-i has the n-superpotential property.

Theorem 3.2.2. Keeping the above notation and assumptions

1. Any zeroPBW deformation of A is n-CY

2. Let n = 2 or 3. Then the zeroPBW deformations of A are exactly the n-CY PBW deformations of A, and moreover any superpotential = <Pn + 0n-1 + 0n-2 is (n — 2)-coherent.

Proof. 1. Let A be a PBW deformation of A, defined by a map Q. Then define &'(Q) =

\$n + 0 n — 1 + 0 2 by

0n-2+j := — Y PQj (dp&n)-|p|=n-2

Write 0n-1 = Et ctt. We will show that A is n-CY if and only if

0 = —1) cqi+2---qn-iqi---qi+i i=0

for any q = q1 ... qn-1, with qj G Q1. In particular this shows any zeroPBW deformation is n-CY, as then ^'(Q) is a superpotential and when n is even 0n-1 = E ctt = 0 and when n is odd the terms cancel in pairs by the superpotential property. Referring to Theorem 2.5.4 A is n-CY if and only if

0 = 1)iid® Q1 ® id®(n-2-i)(^n)

= ^( —1)i XI P1 ...PiQ^ 5Pi-Pi ^nSPi+Lpn-^ Pi+1 . . . Pn-2 i P=P1 ■■■Pn-2

= E(-l)i+(n-1)(n-2-i) ^ P1 . . .PiQ1(SPi+i...Pn-2Pl.P &n)Pi+1 . . . Pn-2

i using Q1(Jq\$n) = ^q0n-1 and writing 0n -1 = E crr J

^ H=n-1 '

= E(-1)in X P1 . . . PicPi+i ■■■Pn-2Pi ■■■PiaaPi+1 . . .Pn-2 i P,a

= X(-1)in X cqi+2■■■qn—lql ■■■qiqi+i q1 . . . qiqi+1qi+2 . . . qn-1 iq

Considering the coefficients of the paths, which are linearly independent, and calculating the coefficient of q1 ... qn-1 we find the condition on 0n-1 to be

0 = ^ y(-1) Cqi+2---qn-iqi---qi+i

for all qi ■■■ qn-i.

2. By part 1/Theorem 2.5.4 we have a condition for A to be n-CY. In the n = 3 case the condition gives that Im id 0 9i - 9i 0 id = ^(0i) = {0} which is the zeroPBW condition. When n = 2 it gives the condition Qi = 0 which is the zeroPBW condition for even n.

Moreover when n = 3 any superpotential, &', is 1-coherent. In particular the fact that A is 3-CY gives a duality in W• between the 1st and 2nd terms — the arrows a G Qi and the relations Sa&3. Hence as the arrows are linearly independent so are the relations, and EaeQi ^aSa&3 = 0 ^ \a = 0 ^ ^ ^aSa&' = 0, so &' is 1-coherent. When n = 2 the superpotential is only differentiated by paths of length 0, so it is clearly 0-coherent. □

A similar result in the 3-CY case is already known in the context of a general quiver due to the following result of Berger and Taillefer. We note that their result is for potentials rather than superpotentials, where a potential is defined to be an element W G CQ/[CQ, CQ], and there is a map into CQ defined by c : an ■■■ai ^ ai ■ ■■ aian ■ ■ ■ ai+i, see [7, Section 2].

Theorem 3.2.3. (See [7, Section 3].) Let A = D(c(WN +i), 1) be N Koszul and 3-CY, with Wn+i a potential on some quiver Q. Then a PBW deformation A of A is 3-CY if it is a zeroPBW deformation. Moreover the zeroPBW deformations correspond to D(c(W'), 1) for W' = Wn+i + Wn + ■■■ + Wi an inhomogeneous potential with each Wi in grade i.

We also note that Theorem 3.2.3 makes use of the results of [8], in particular using the bimodule resolution of a graded 3-CY algebra given in [8, Section 4.2], whereas our results use the bimodule resolution of a Koszul n-CY algebra given in [9, Theorem 6.8].

4. Application: symplectic reflection algebras

In this section we recall the definition of symplectic reflection algebras and deduce they are Morita equivalent to CY superpotential algebras by applying Theorems 2.4.4, 3.1.1, and 3.2.1. We go on to calculate some examples, and consider the interpretation of the parameters of a symplectic reflection algebra in the superpotential setting.

Let V be a 2n dimensional space, equipped with symplectic form w and G a symplectic reflection group which acts faithfully on V preserving the symplectic form. We say (G, V) is indecomposable if there is no G-stable splitting V = Vi © V2 with w(Vi, V2) = 0. We consider the skew group algebra C[V] x G and the symplectic reflection algebras are defined to be the PBW deformations of this relative to CG, and were classified by Etingof and Ginzburg [12].

Theorem 4.0.4. (See [12, Theorem 1.3].) Any PBW deformation of such an indecomposable C[V] x G is of the form

H _ TCG(V *)

{[x,y] - Ktc(x,y) : x,y G V*}

where Kt,c(x, y) = twv* (x, y) - Ssws(x, y)c(s)s with the sum taken over the symplectic reflections s, and c a complex valued class function on symplectic reflections. The symplectic form wv* on V* is induced from w on V, and ws is defined as wv* restricted to (id - s)(V*).

In particular they fall into the even dimensional case considered in Section 3 above, with the PBW parameter Qi equal to zero, and they are PBW deformations relative to CG, which under Morita equivalence with a quiver of relations correspond to PBW deformations relative to S.

Two infinite classes of symplectic reflection algebras are the rational Cherednik and wreath product algebras.

• Rational Cherednik algebras. Let h be a finite dimensional vector space and G a finite subgroup of GL(h). Then V = h © h* has the natural symplectic form w((x, f), (y, g)) = f (y) - g(x), and an action of G as a symplectic reflection group. Then the symplectic reflection algebras given by PBW deformations of C[V] x G are the Rational Cherednik Algebras.

• Wreath product algebras. Let K be a finite subgroup of SL2(C), and Sn the symmetric group of order n. Then the wreath product group G = Sn I K is a symplectic reflection group acting on V = (C2 )n.

4.1. Symplectic reflection algebras as superpotential algebras

Here we show that symplectic reflection algebras are Morita equivalent to superpotential algebras.

Theorem 4.1.1. Let Htc be as in Theorem 4.0.4.

1. Ho,o is Morita equivalent to A = D(&2n, 2n-2) for some homogeneous superpotential &2n, and is 2n-CY and Koszul.

2. Any Ht c is Morita equivalent to A = D(&', 2n - 2) for &' = &2n + 02n-2 an inho-mogeneous superpotential which is (2n - 2)-coherent.

3. Any (2n - 2)-coherent superpotential of the form &' = &2n + 02n-2, gives an algebra A = D(&', 2n - 2) that is Morita equivalent to a symplectic reflection algebra Ht c.

4. All Ht c are 2n-CY algebras.

Proof. 1: H0j0 is just C[V] x G. Hence Theorem 2.4.4 applies, and H0 0 is Morita equivalent to a 2n-CY, Koszul algebra A := T>(@2n, 2n — 2) for the McKay quiver for (G, V).

2 and 3: The Morita equivalence between C[V] x G and A switches CG with S, respects the gradings, and respects the Koszul resolutions. Hence any zeroPBW deformation of C[V] x G corresponds to a zeroPBW deformation of A, and 2 and 3 follow from Theorem 3.1.1 once we note by Theorem 4.0.4 all PBW deformations of C[V] x G are zeroPBW.

4: From 2 and 3 we know any symplectic reflection algebra is Morita equivalent to some A := T>(@2n + 02n-2, 2n — 2). By Theorem 3.2.1 A is 2n-CY: A has a finite length resolution by finitely generated projective modules and RHom(A, Ae)[n] = A. Then the Morita equivalent symplectic reflection algebra, Htc, also has a finite length resolution by finitely generated projective modules, as these properties are preserved under Morita equivalence. The isomorphism in the derived category RHom(A, Ae)[n] = A transfers to the isomorphism RHom(Htc, Htec)[n] = Htc under the Morita equivalence, hence Htc is 2n-CY.

4.1.1. Examples

We give three examples.

Example 4.1.2. Here we consider the symplectic reflection algebra corresponding to the group S3 acting on h © h* in the manner of a rational Cherednik algebra, where the representation h is given by

S3 = V = 0 £§ ^

with £3 a primitive third root of unity. The McKay quiver is;

Now following the calculation in BSW [9, Theorem 3.2] we can calculate a superpotential to accompany this quiver,

= — AaAa + 2AaAa + 4ALLa — 4ALLa — 4A6Ba + 2AbBa + 2AbBa — AaAa — 4ALLa + 4ALLa + 2AbBa + 2AbBa — 4AbBa — 8LLLL + 4bBLL — 4bBLL + 4LbBL + 4bBLL — 2bBbB + bBbB + bBbB + cyclic permutations

We now wish to look at the zeroPBW deformations, which by Theorem 4.1.1 correspond to 2-coherent superpotentials & = &4 + 02. Writing 02 = E cxyxy the PBW deformations are parametrised by the cxy such that & is a 2-coherent superpotential. We see that & is a superpotential if cxy = -cyx for all arrows x, y. A superpotential & is 2-coherent if the Sxy& = Sxy&4 + cxyeh(x)et(y) satisfy the same linear relations as the Sxy&4. For instance as Sao.& = 0 and Sa&& = -¿Aa we require that cAa = 0 and

cAa = -cAa.

Making these calculations in this example we see the only non-zero c, and dependency relations among them, are:

Hence we have a 2-coherent superpotential for any

02 = caA(aA — Aa + Aa — aA) + cbB (bB — Bb + Bb — bB) + (caA + cbB )(LL — LL)

So we have 2 degrees of freedom in our parameters, exactly as we do for the t, c in Ht,c for S3 acting on h © h * .

Example 4.1.3. Here we consider the symplectic reflection algebra corresponding to the dihedral group of order 8, D8, acting on h © h* in the manner of a rational Cherednik algebra. The representation h is given as

caA = -cAa = cAa = -caA

cbB = -csb = cb6 = -c6B

caA + cbB = cll = -cLL

where £4 is a primitive fourth root of unity. This has McKay quiver

By choosing a G-equivariant basis we calculate the superpotential

\$ = -AaAa + 2AaAa - 4AdDa + 2AdDa + 2AdDa + 2AbBa - 2AbBa + 2AcCa

- 2AcCa - DdDd + 2DdDd + 2DbBd - 2DbBd + 2DcCd - 2DcCd - DdDd

+ 2DbBd - 2DbBd + 2DcCd - 2DcCd - BbBb + 2BbBb - 4BcCb + 2BcCb

+ 2BcCb - 4BcCb - BbBb + 2BcCb + 2BcCb - CcCc + 2CcCc - CcCc

+ cyclic permutations

We now calculate the zeroPBW deformations of A := D(\$4, 2). We write 02 = J2cxyxy, and by Theorem 4.1.1 the zeroPBW deformations of A are parameterised by the cxy such that \$ = + 02 is a 2-coherent superpotential.

In particular we require cxy = -cyx and for the cxy to satisfy the same linear relations as the 5xy \$4. Making these calculations in this example we see the only non-zero cxy, and dependency relations among them, are;

caA = - cAa = cAa = -caA

cbB = - c_Bb = cb6 = -c6B ccC = - cCc = cCc = -ccC cdD = -cDd = cDd = -cdD caA + cdD = - cbB - ccC = cbB + ccc

Hence to obtain a 2-coherent superpotential we require a 02 of the form

02 = caA(aA - Aa + Aa - aA) + cbB(bB - Bb + Bb - bB)

+ ccC(cC - Cc + Cc - cC) + (cbB + ccC - caA)(dD - Dd + Dd - dD)

So we see here there are 3 degrees of freedom in our parameters exactly as for the parameters t and c in the symplectic reflection algebra for D8 acting on h © h*.

Example 4.1.4. A special case of path algebras Morita equivalent to symplectic reflection algebras are the deformed preprojective algebras of [11]. These can be given as superpotential algebras in the n = 2, differentiation by paths of length 0, case.

We consider a skew group algebras C[C2] x G for G is a finite subgroup of SL2(C). We construct the McKay quiver and label the arrows in a particular way; between any two vertices we choose a direction, label the arrows in this direction ai, ■■■, ak, and the arrows in the opposite direction al^^^a^,. Then C[C2] x G is Morita equivalent to A = D(\$2, 0) for the homogeneous superpotential, = a*]. This is the preprojective algebra

A = D(\$2,0)= CQ

E[a,a*]

Now we consider the PBW deformations of A. We apply Theorem 4.1.1, and deduce PBW deformations correspond to 0-coherent inhomogeneous superpotentials, &2 + 0o. We note that 0o := - E^Qo ^ S can in fact be arbitrary as any element of S satisfies the superpotential property, and the 0-coherent property is always satisfied. Hence we recover that PBW deformations of the preprojective algebra are the deformed preprojective algebras

A = D(& + 0o, 0) CQ

E[a,a1 - Exiei'

which are parameterised by a scalar, Aj, for each vertex, i. By Theorem 4.1.1 these are 2-CY.

5. Application: PBW deformations of skew group rings for GL2

So far we have been considering subgroups of SL(W) where W is a finite dimensional vector space. This corresponds to non-twisted superpotentials. Here we consider algebras Morita equivalent to C[W] x G for G a finite subgroup of GL(W), and the existence of PBW deformations for C[W] x G. We recall

Theorem 5.0.5. (See [9, 3.2, 6.1 and 6.8].) Let W = Cn, and G be a finite subgroup of GL(W). Then C[W] x G is Morita equivalent to D(&n, n - 2) for the McKay quiver (G, W), &n a twisted homogeneous superpotential, and the twist automorphism given by (-) ®C det W. There is a recipe to construct the twisted superpotential.

Working with homogeneous superpotentials, &n = E cpp, we have cp = 0 for any p that is not a closed path. This is no longer the case for twisted homogeneous superpotentials, here we find cp = 0 unless h(p) = a(t(p)). Since the twist for C[W] x G is given by tensoring by det W, cp is non-zero only for paths from Wi to det W Wi, where the Wi are the irreducible representations corresponding to vertices in the McKay quiver.

There are two different cases of finite subgroups of GLn(C) we consider, those that contain pseudo-reflections, and those that do not. Those that do not are known as small subgroups.

As a particular case we will consider GL2(C), where differentiation is by paths of length 0, and so our relations are given by the superpotential, and any relations are a sum of paths with tail Wi and head det W Wj,.

Theorem 5.0.6. Let G be a small finite subgroup of GL2(C), which is not contained in SL2(C). Then C[C2] x G has no nontrivial (relative to CG) PBW deformations.

Proof. The algebra C[C2] x G can be written as ^"[x^^lC^ and is Morita equivalent to a path algebra with relations CQ/R = TsR^ = D(&2, 0) for some twisted homogeneous

superpotential \$2, where we use notation as in Section 2.1. In particular the Morita equivalence switches CG with S, and respects the gradings and Koszul resolutions. Hence considering PBW deformations as in Section 2.5 we see that under the Morita equivalence any PBW deformation of C[C2] x G would give a PBW deformation of CQ/R, noting that in one case considering PBW deformations relative to CG, and in the other relative to S.

Hence it is enough to show that the Morita equivalent twisted superpotential algebra A := D(\$2, 0) has no nontrivial PBW deformations.

There can only possibly exist PBW deformations if there exists some non-zero Q0, 0i as in Section 2.5. But Qi G Hom.se (R, V) and Q0 G Hom.se (R, S), so if both these sets are {0} there are no nontrivial PBW deformations.

Define the distance between two vertices in the quiver to be the minimal length of a path from one to the other. It is shown in Appendix A, Lemma A.0.1, that the tail and head of any relation are vertices which are distance greater than one apart. Hence, as Se module maps preserve heads and tails, the sets Hom.se (R, V) and Hom.se (R, S) are both {0} and there are no nontrivial PBW deformations.

We look at examples of a small and non-small subgroup, using the calculations from [9]. We let em denote a primitive mth root of unity.

Example 5.0.7. We first consider a small subgroup D5 2 with representation as

/(£4 0 \ f 0 eA ( 0 £6 \ \ \lv 0 e-y , y£4 0 J ^£6 0 J/

This is given as an example in [9, Example 5.4].

It has McKay quiver

where the first column of vertices equals the final column of vertices. Tensoring by the determinant representation maps from a vertex to the next vertex in line to the right, as indicated by the dashed arrows, wrapping around from the right side of the diagram to the left.

The relations are

Consider the head and tail of any element of R. These vertices are related by the twist, and we see the shortest path between the tail and head is always length 2. Then Horns« (R, V) and Hom^e (R, S) are both {0}, hence there can be no nontrivial PBW deformations.

Example 5.0.8. We now consider D8 as the non-small subgroup with representation

biai = 0, dici = 0, fiei = 0 for i = 1, 2,3,4

J2 cj bj = 0, ej dj = 0, aj fj =0

j=i j=i j=i

and McKay quiver and relations, Q and R.

Da = 0 Cb = 0

Ad = 0 Bc = 0

aA + dD - bB - cC = 0

Then PBW deformations of CQ/R are classified by Q\, Q0 : R b A which are Se-module maps; Q\ G Homs« (R, V) and Q0 G Homs« (R, S).

As Se-module maps preserve heads and tails we see that Qi must be zero, and Q0 must be zero on all relations but the central one aA + dD — bB — cC. At this relation 60(aA + dD — bB — cC) = Ae4 for some A G C. In this case (RV) n (VR) = 0, and so any such Q0 gives us a PBW deformation. Hence there is a one parameter collection of PBW deformations.

Acknowledgments

The author is a student at the University of Edinburgh funded via a doctoral training grant from the Engineering and Physical Sciences Research Council [grant number EP/J500410/1], and this material will form part of his PhD thesis. The author would like to express his thanks to his supervisors, Prof. Iain Gordon and Dr. Michael Wemyss, for much guidance and patience, and also to the EPSRC.

Appendix A. McKay quivers for finite small subgroups of GL2(C)

We use the classification of McKay quivers for small finite subgroups of GL2 (C) [2], to prove the following:

Lemma A.0.1. Let G < GL2(C) be a small finite subgroup, given by a representation W = C2. Let Wi be an irreducible representation of G. Then the shortest path from Wi to det W 0 Wi has length > 2.

Proof. We outline this case by case by examining the McKay quivers, showing there are no length 0 or 1 paths between a vertex in the quiver and the vertex related by tensoring by the determinant.

We list the small finite subgroups of GL2(C) up to conjugacy, as in [2, Section 2]. Let Zn = (g = (0°)), and e be a primitive nth root of unity. Any finite small subgroup of GL2(C) is, up to conjugacy, one of the following:

1. Z a cyclic subgroup, Z = (g = (0 ^ )) for 1 < q < n.

2. ZnD = {zd | z G Zn, d G D} for D a finite, non-cyclic, subgroup of SL2(C).

3. H. To define Hi let D < SL2(C) be a binary dihedral group, with A a cyclic subgroup of index 2, and define H < Z2n x D to be

H = {(z, d) G Z x D | d + A = z + Zm in Z2 = D/A = Z2n/Znj.

Then H is the image of H under the map H ^ GL2(C).

4. K. To define K let D < SL2(C) be the binary tetrahedral group, with A a normal binary dihedral subgroup of index 3, let n > 3, and define K < Z3n x D to be

K = {(z, d) G Z x D | d + A = z + Zn in Z3 = D/A = Z3M/Zn }.

Then K is the image of K under the map K ^ GL2(C).

We note that if n =1, 2 then K is the binary tetrahedral group with defining representation containing pseudo reflections, so is not small.

The McKay quivers for these groups are described in [2, Proposition 7], and we look at the determinant representation in each case and show tensoring by it relates vertices at least distance two apart.

We first look at cyclic subgroups. Let Z be as above. Such a representation is in SL2(C) only when q +1 = n. We suppose q + 1 = n.

Such a group has n irreducible one dimensional representations, which we label W0,..., Wn-1, where Wi is given by g b sj. The defining representation is reducible as C2 = W1 © Wq, and its determinant is the representation W1+q. Hence the McKay quiver has n vertices corresponding to the Wi and at vertex i has two arrows to vertices i + 1 and i + q modulo n. The relations on the McKay quiver have head and tail related by tensoring by the determinant, hence any relations with tail Wi have head Wi+q+1 module n. We see that the two vertices Wi and Wi+q+1 are distance 2 apart; they are not distance 0 as i = i + 1+ q module n, and they are not distance 1 as the only arrows from i are to i + 1 or i + q, and neither of these equals i + 1 + q modulo n.

All the remaining groups are constructed by taking a subgroup of Zn x D and then taking the image of this under the map to GL2(C). If we calculate the McKay quiver for the subgroup then the image in GL2(C) has McKay quiver which is a subquiver. Hence for our purposes it is enough to calculate the McKay quivers for the various subgroups of Zn x D.

We first do this for the case ZnD. We note that this is contained in SL2(C) for n = 1, 2, hence we assume n > 2. In this case we consider the McKay quiver of Zn x D. Let the irreducible representations of D be labelled D0,..., Dr-1 where D0 is trivial, and D1 is the given 2 dimensional representation. Let Ri, for i = 0,...,n — 1, be the n one dimensional irreducible representations of Zn with Ri given by g b £i. Then Zn x D has nr irreducible representations given by Ri ® Dj for 0 < i < n and 0 < j < r. Then we consider the McKay quiver for the defining representation R1 ® D1, as this corresponds to the defining representation in GL2(C). In particular the McKay quiver has n groups of r representations labelled by representation of Zn, with group i corresponding to the set of representations {Ri ® Dj | j = 0,...,r — 1}. By definition any arrows in the quiver go from group i to group i + 1 modulo n. As the defining representation is R1 ® D1 the determinant representation is R2 ® D0, and the determinant maps from group i to group i + 2 modulo n. In particular, as n > 2, any two vertices related by this are not distance zero or one apart.

For the cases H and K we take the McKay quiver for Zn x D, make some identifications to account for certain irreducible representations being identified for the subgroups H, K.

We first consider H. In this case we label the representations of the binary dihedral group as

where Di is the representation in SL2(C) and D0 is the trivial representation.

Now we label the representations of Z2n as Ri for i = 0 ... 2n— 1as above, and Z2n x D has 2n(r + 2) irreducible representations given by their tensor products. The defining representation in GL2(C) is given as Ri 0 Di, and hence the determinant representation is R2 0 D0. Now all representations of Z2n x D are still irreducible for H but some are identified [2, Proposition 7(d)]. The representations which are identified are Ri 0 Dj with Rn+i 0 Dj for j = 1 . ..r — 1, modulo 2n, and Ri 0 Dj with Rn+i 0 Dj modulo 2n for j = 0, r. The McKay quiver for H is the McKay quiver for Z2n x D with these identifications made. In this case we group the irreducible representations into n groups labelled by the representation of Z2n modulo n, so arrows go from group i to i + 1 and determinant from group i to i + 2 module n. Hence, for n > 2, the head and tail of two vertices related by the determinant are not distance zero or one apart. When n = 1, 2 then H = D and then the representation is contained in SL2 (C).

The final case is to consider K < Z3n x D. Again we label the representations of Z3n by Ri for i = 0 ... 3n — 1, and we label the representations of D by

Do Di D2 D'i D'o

where D0 is the trivial representation, and D1 the defining representation.

The representation defining the group in GL2(C) is R1 0 D1, and the determinant representation is R2 0 D0.

Now all the irreducible representations for Z3n x D remain so for K, however some are identified [2, Proposition 7 (f)]. This time triples are identified: Ri 0 Dj = Ri+n 0 Dj = Ri+2n 0 Dj modulo 3n, for j = 0, 1 and Ri 0 D2 = Ri+n 0 D2 = Ri+2n 0 D2 modulo 3n, as representations of K.

Once again we note that this splits the quiver into n groups, labelled by the representation of Z3n module n, with arrows from group i to i + 1 and determinant from i to

i + 2 modulo n. Hence, as n > 2, the determinant maps between vertices which are not distance 0 or 1 apart.

Hence for any small finite subgroup of GL2(C) not contained in SL2(C) the determinant in the McKay quiver maps between vertices which are distance greater than 1 apart. □

We also note that, while we do not know a complete proof without the use of the classification, given Wi = det W ® Wi there is an elementary character theoretic proof of Lemma A.0.1.

Lemma A.0.2. Let G < GL2(C) be a small finite subgroup, given by a representation W = C2. Let Wi be an irreducible representation of G. Then, assuming Wi = det W<&Wi, there are no arrows from Wi to det W<g>Wi hence the shortest path from Wi to det W<g>Wi has length > 2.

Proof. We suppose that G is a small finite subgroup of GL2(C), and that Wi = det W ® Wi. Hence there are no length 0 paths from Wi to det W ® Wi. We show that there are no length one paths Wi to det W ® Wi

We let (—, —) denote the standard inner product for characters of G, and suppose that the determinant representation of W has order d. We note that for any two characters A, B

(ch(W)A,B) = (A, ch(W) ch(detW)d-1B)

as if g G G has W(g) with eigenvalues e1, e2 then

tr W(g-1) = (e-1 + e-1) = (e^)-^ + £2) = tr(det W(g))-1 tr(W(g))

Suppose there were an arrow from Wi to det W ® Wi, then

1 < a = (ch(Wi)ch(W), ch(det W)ch(Wi)) = (ch(Wi), ch(W) ch(Wi))

By considering dimensions we see a =1, and W ® Wi = Wi © det(W) ® Wi.

Suppose g G G, W(g) has eigenvalues e1, e2, and Wi(g) has eigenvalues ¡i1,...,^r. Then

(£1 + £2)(M1 + ... + Vr) = ch(W ® Wi) = ch(Wi © det(W) ® Wi) = (1 + £1£2)(V1 + ... + Vr)

As Wi is an irreducible representation, and G is nontrivial, we can choose g G G nontrivial such that ch Wi(g) = v1 + ... + Vr = 0. The eigenvalues of such a g satisfy 1 + e1e2 = e1 + e2. Hence 1 is an eigenvalue of W(g), so g is a pseudo reflection, which is a contradiction as G is small.

Hence there are no arrows from Wi to Wi ® det W. □

References

C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen—Macaulay modules and cluster categories. arXiv:1104.3658v1.

M. Auslander, I. Reiten, McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1) (1986) 293-301.

R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2) (2001) 705-734.

R. Berger, M. Dubois-Violette, M. Wambst, Homogeneous algebras, J. Algebra 261 (1) (2003)

172-185.

R. Berger, V. Ginzburg, Higher symplectic reflection algebras and non-homogeneous N-Koszul property, J. Algebra 304 (1) (2006) 577-601.

R. Berger, N. Marconnet, Koszul and Gorenstein properties for homogeneous algebras, Algebr. Represent. Theory 9 (2006) 67-97, http://dx.doi.org/10.1007/s10468-005-9002-1. R. Berger, R. Taillefer, Poincare-Birkhoff-Witt deformations of Calabi-Yau algebras, J. Noncom-mut. Geom. 1 (2) (2007) 241-270.

R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (1) (2008) 14-32.

R. Bocklandt, T. Schedler, M. Wemyss, Superpotentials and higher order derivations, J. Pure Appl. Algebra 214 (9) (2010) 1501-1522.

A. Braverman, D. Gaitsgory, Poincare-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (2) (1996) 315-328.

W. Crawley-Boevey, M.P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (3) (1998) 605-635.

P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2) (2002) 243-348. V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139v3.

E.L. Green, E.N. Marcos, R. Martinez-Villa, P. Zhang, C-Koszul algebras, J. Pure Appl. Algebra 193 (1-3) (2004) 141-162.

J.C. McConnell, J.C. Robson, L.W. Small, Noncommutative Noetherian Rings, Grad. Stud. Math., American Mathematical Society, 2001.

G. Sjodin, On filtered modules and their associated graded modules, Math. Scand. 33 (1974) (1973) 229-249.

Q. Wu, C. Zhu, Poincare-Birkhoff-Witt deformation of Koszul Calabi-Yau algebras, Algebr. Represent. Theory 16 (2) (2013) 405-420, http://dx.doi.org/10.1007/s10468-011-9312-4.