Superpotentials, Calabi-Yau algebras, and PBW deformations

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 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] \rtimes G. We show there are no nontrivial PBW deformations when G is a small subgroup.

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, Φ n = c a1...an a 1 . . . a n , in the path algebra of Q satisfying the n superpotential condition: c aq = (−1) n−1 c qa for all arrows a and paths q. From such a superpotential Φ n and a non-negative integer k we construct an algebra D(Φ n , k) := CQ 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 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, N -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 N -Koszul, CY, superpotential algebra D(Φ n , k) are of the form D(Φ ′ , k) for a inhomogenous superpotential Φ ′ = Φ n + φ n−1 + · · · + φ k . We then specialise to the case of 2-Koszul, n-CY, D(Φ n , n − 2), and consider PBW deformations of the form D(Φ n + φ n−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 D(Φ 2n + φ 2n−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 D(Φ 2 , 0) Morita equivalent to C[C 2 ] ⋊ G for G a finite subgroup of GL 2 (C). In particular we show there are no nontrivial PBW deformations when G is a small subgroup not contained in SL 2 (C).
1.4. Acknowledgments. The author is an EPSRC funded student at the university of Edinburgh, 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.

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

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 = (Q 0 , Q 1 ), with Q 0 the set of vertices and Q 1 the set of arrows. The set of arrows is equipped with head and tail maps h, t : Q 1 → Q 0 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 = a r . . . a 2 a 1 with a i ∈ Q 1 satisfying h(a i ) = t(a i+1 ) for 1 ≤ i ≤ r − 1.
a 2 a r We will reuse the notation of the head and tail maps for the head and tail of a path, defining h(p) = h(a r ) and t(p) = t(a 1 ) when p = a r . . . a 2 a 1 . We also define a trivial path e v for each vertex v ∈ Q 0 , 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 = a r . . . a 1 , where each a i 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 be a field. We define the path algebra of the quiver Q, Q, as follows: Q as a -vector space has a basis given by the paths in the quiver; an associative multiplication is defined by concatenation of paths.
p.q = pq if h(q) = t(p) 0 otherwise We define S to be the subalgebra of this generated by the trivial paths, and V to be the vector subspace of Q spanned by the arrows, a ∈ Q 1 . Then S is a semisimple algebra, with one simple module for each vertex.
For an arrow a, we have a = e h(a) .a.e t(a) , and so V has the structure of a left S e := S ⊗ S op module and Q can be identified with the tensor algebra T S (V ) = S ⊕ V ⊕ (V ⊗ S V ) ⊕ . . . , equating the path a 1 . . . a r with a 1 ⊗ S · · · ⊗ S a r .
The algebra T S (V ) = Q is equipped with a grading and filtration by pathlength, the graded part in degree n is T S (V ) n := V ⊗Sn , and the filtered part is F n (T S (V )) := V ⊗Sn ⊕ · · · ⊕ V ⊕ S.
Given R ⊂ T S (V ) we define I(R) to be the two sided ideal in T S (V ) generated by R. We then define Q R := Q I(R) and refer to it as the path algebra with relations R.
We define < R > to be the S e submodule of T S (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.
where the sum is taken over all paths p of pathlength n with coefficients c p in . Our convention will be that c 0 = 0. For example if t(a) = h(b) then ab = 0 and hence c ab = c 0 = 0. We will say Φ n satisfies the n superpotential condition if c aq = (−1) n−1 c qa for all a ∈ Q 1 and paths q, and if Φ n satisfies the n superpotential condition we will call it a homogeneous superpotential of degree n. In particular this requires c p = 0 if p is not a closed path.
We will say Φ n satisfies the twisted n superpotential condition if c σ(a)q = (−1) n−1 c qa for a ∈ Q 1 and paths q. We will say Φ n is a σ-twisted homogeneous superpotential of degree n if it satisfies the twisted n superpotential condition.
Definition 2.1.4. Consider an element Φ ∈ F n (T S (V )) written Φ = Σ m≤n Σ |p|=m c p p as above. We will say Φ is an inhomogeneous superpotential of degree n if it satisfies the conditions c aq = (−1) n−1 c qa for any path q and arrow a, and for some p of length n the coefficient c p is non-zero. An inhomogeneous superpotential Φ can be written in homogeneous parts Φ = φ n + φ n−1 + · · · + φ 0 where each of the φ m satisfies the nsuperpotential condition and φ n is non-zero. Note that if n is even, m odd, and char = 2 then φ m = 0, and also that any φ 0 ∈ S satisfies the n superpotential condition.
We will say this is an σ-twisted inhomogeneous superpotential of degree n if it satisfies c σ(a)q = (−1) n−1 c qa for any path q and arrow a. A twisted inhomogeneous superpotential Φ can be written in homogeneous parts Φ = φ n + φ n−1 + · · · + φ 0 where each of the φ m satisfies the n-twisted superpotential condition.
Differentiation. We define differentiation by paths.
Definition 2.1.5. Let p be a path in Q, and a ∈ Q 1 . We then define left/right derivative of p by a as; We extend this so that if q = a 1 . . . a n is a path of length n then δ q = δ an . . . δ a1 , and δ ′ q = δ ′ an . . . δ ′ a1 . We will call the operation δ q left differentiation by the path q, and δ ′ q right differentiation by the path q.
Note that for Φ a degree n inhomogeneous superpotential, b, c ∈ Q 1 , and p any path Definition 2.1.6. Given an inhomogeneous superpotential Φ of degree n we can consider the S e -modules W n−k ⊂ Q given by: We then define the algebra D(Φ, k) to be We will call this the superpotential algebra D(Φ, k).
Example 2.1.7. Consider the quiver, Q, with one vertex, •, and two arrows x and y. Then Q is the free algebra on two elements, < x, y > • x y We consider some superpotentials on Q. 1) Φ 2 = xy − yx is a degree 2 homogeneous superpotential.
2) Φ = xy − yx − e • is an inhomogeneous superpotential of degree 2 3) Φ 3 = xyx + xxy + yxx is a degree 3 homogeneous superpotential. These give us the algebras the first Weyl algebra, is the quiver with relations given by 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 A e -module resolution of length n implies an algebra is n-Calabi-Yau.
Let A be an associative algebra, set A e = A ⊗ A op and D(A e ) to be the unbounded derived category of left A e modules. We note that there are two A e -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 A e with the outer structure for any A e module, M , we can make Hom A e (M, A e ) an A e -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 A e -modules, and In the paper [9] self-duality of a complex is used. We give the definition here. Denote Definition 2.2.2. Define a complex of A e modules, C • , of length n to be self dual, written if there exist A e -module isomorphisms α i such that the following diagram commutes: By construction the existence of a length n self dual projective A e -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 -algebra, where 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 where * denotes the standard 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.

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], [4], [5], and [6]. It generalises the concept of a Koszul algebra, which is defined here to be a 2-Koszul algebra.
Now we can define an N -complex (In an N -complex d N = 0 rather than d 2 = 0) To define the differentials let µ be multiplication, and d l , d r : and hence can consider the restriction of d l , d r to K i (A). Now choose q ∈ a primitive N th root of unity, which we need to assume exists in , and define d : As d l and d r commute with d N l = 0 and d N r = 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 We say that A is N -Koszul if this complex is exact, and hence gives an A e -module resolution of A.
We call 2-Koszul algebras Koszul. In this case the N -complex is in fact a complex.

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 Higer Order Derivations. Here we state several results from the paper [9] concerning superpotential algebras.
Let Q be a quiver, with path algebra CQ. Let Φ n be a homogeneous superpotential of degree n, and A = D(Φ n , n − 2). For i = 0, . . . n we have the S e -modules W i as in Definition 2.1.6 above, and we can define a complex W • by for p any path of length n − i.
The differential d i is defined as This is a complex as d l , d r commute and have square 0. We show this for d l , the other case being similar. Writing Φ n = |t|=n c t t, we have where a, b ∈ Q 1 , 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.
The complex (1) is the relevant complex for D(Φ 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 e -modules W j .
for q a primitive N th root of unity, which we assume exists in . This can be contracted into by composing the differentials in W • , as in Definition 2.3.1, where m is the largest integer such that m ≤ n/N .
R 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 D(Φ n , k) where Φ n is an 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 [V ] ⋊ G which is the semi direct product with the action of G given by that of GL(V ). The multiplication of (f 1 , For such a group G and representation V we can construct the McKay quiver. We now assume our field is 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, There is a recipe to compute a superpotential Φ n such that C[V ]⋊G is Morita equivalent to the superpotential algebra D(Φ n , n − 2) attached to the McKay quiver (G, V ). The recipe is given in [ 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 in the case A an N -Koszul S e -module, relative to S a semisimple algebra, and will make use of the setup and results of [5].
Let S be a semisimple algebra, V a left S e -module, and T S (V ) the tensor algebra of V over S. Consider the (Z ≥0 ) grading on this with degree n part T S (V ) n = V ⊗S n , and filtered parts F n = V ⊗S n ⊕ · · · ⊕ V ⊕ S. For R an S e -submodule of V ⊗SN define I(R) = i,j≥0 V ⊗i RV ⊗j to be the two sided ideal in T S (V ) generated by R, which is , and as R is homogeneous this is a graded algebra with degree n part A n := V ⊗ S n I(R)n . Now define the projection map π N : F N → V ⊗S N , and let P be an S e -submodule of F N such that π(P ) = R. Let I(P ) be the 2 sided ideal in T S (V ) generated by P . This is not graded but is filtered by I(P ) n = I(P ) ∩ F n , and hence A = TS (V ) I(P ) is also a filtered algebra with A n = F n I(P ) n . We can construct the associated graded algebra gr(A), which is graded with gr(A) n = A n A n−1 . Identify gr(A) n with the S e -module F n I(P ) n +F n−1 by noting I(P ) n ∩ F n−1 = I(P ) n−1 , and consider the maps φ n : V ⊗n → gr(A) n defined as the composition V ⊗n ֒→ F n → F n I(P ) n +F n−1 . This allows us to define a surjective algebra morphism φ = n≥0 φ n : 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 ⊂ F N to be of PBW type it must be the case that P ∩ F N −1 = {0}. Hence any PBW type P can be given as P = {r − θ(r) : r ∈ R} for an S emodule map θ : R → F N −1 . Such a map can be written in homogeneous components as Consider the map ψ(θ j ) defined for each θ j as The S e -module P is of PBW type if and only if the following conditions are satisfied 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 Gaitgory, [10]. They also show any PBW deformation gives a graded deformation. With the notation as above we define a graded deformation of A, A t , to be a graded [t] algebra with t in degree 1. That is A t is free as a module over [t] and there is an isomorphism A t /tA t → 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(Φ n , n − 2) Koszul and S a field there is also the following theorem of Wu and Zhu, [16], 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 Φ n be a homogeneous superpotential on a single vertex quiver. Let A = D(Φ n , n − 2). If A is n-CY Noetherian and Koszul and A is a PBW deformation given by θ 0 , θ 1 then A is n-CY if and only if Proof. See [16,Theorem 3.1]. This proves such a PBW deformation is n-CY if and only if

Main results
We state and prove our main results concerning the PBW deformations of an (k + 2)-CY, (n − k)-Koszul, superpotential algebra A := D(Φ n , k), where Φ n is a homogeneous superpotential. We will prove a condition for a PBW deformation to be of the form D(Φ ′ , k) for an inhomogeneous superpotential Φ ′ = Φ n + φ n−1 + · · · + φ k , and prove that inhomogeneous superpotentials with degree n part Φ n 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 Φ n ∈ CQ to be a homogeneous superpotential of degree n on some quiver Q. This will have inhomogenous superpotentials associated to it, denoted Φ ′ = Φ n + φ n−1 + · · · + φ k .
We introduce two new definitions we make use of in the proof: Definition 3.0.5. We will call Φ ′ , an inhomogeneous superpotential, k-coherent if for any 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 where ψ and θ are defined as in Section 2.5. We will say P is of zeroPBW type if A is a zeroPBW deformation of A.
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 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.
R be a path algebra with relations which is (n − k)-Koszul and (k + 2)-CY. Then A = D(Φ n , k), for Φ n a homogeneous superpotential of degree n, and the zeroPBW deformations of A correspond exactly to the algebras D(Φ ′ , k) defined by k-coherent inhomogeneous superpotentials of the form Φ ′ = Φ n + φ n−1 + · · · + φ k .
Proof. As A is (k + 2)-CY and (n − k)-Koszul by Theorem 2.4.3 A = D(Φ n , k) for Φ n a degree n homogeneous superpotential.
We define inverse maps between P of zeroPBW type and coherent superpotentials Φ ′ 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.
Firstly F (P ) is not k-coherent precisely when there exist coefficients λ p ∈ C such that |p|=k λ p δ p Φ n = 0 with ω = |p|=k λ p δ p Φ ′ (θ) = 0. But this only occurs when there exists ω ∈ P ∩ F n−1 which is non zero. But as P is of zeroPBW type PBW1) holds, and P ∩ F n−1 = {0}. Hence F (P ) is k-coherent. Now we show that F (P ) is a superpotential. As P is of zeroPBW type by Lemma 3.0.7 Im ψ(θ j ) = {0} for j = 0, . . . , n − k − 1 . We evaluate ψ(θ j ) on elements of the form δ q Φ n where |q| = k − 1, as in Example 2.5.3, and use θ j (δ p Φ n ) = −δ p φ k+j (θ j ) to deduce that F (P ) is a superpotential. Hence each φ j+k satisfies the n superpotential condition, and F (P ) is a inhomogenous superpotential of degree n.
3.2. CY property of deformations. We now consider when zeroPBW deformations are CY. We consider the case where A = D(Φ n , n − 2) is n-CY and 2-Koszul. We currently prove a result only in the case of a zeroPBW deformation with θ 1 = 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 [16, 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 precisely the 3-CY PBW deformations. Proof. By Theorem 3.1.1 such a deformation is given by a superpotential Φ ′ = Φ n +φ n−2 = t c t t and A = D(Φ ′ , 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 A e -module resolution of A. We then define a complex W The S e -modules W k , and differential d, are defined as in Section 2.4, i.e. W j =< {δ p Φ n : |p| = n − j} >, and 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 d j−1 • d j = 0 for j = 2, . . . , n and that µ • d 1 = 0. In the calculations |p| = n − j, and the sums are taken over all a, b ∈ Q 1 and paths q of length j − 2. We now construct isomorphisms α k between the complex W • A and its dual 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 S e -module let F T be the A e -module A ⊗ S T ⊗ S A, and let T * denote the C dual of T . Then gives an isomorphism of A e -modules. Moreover, as S is semisimple, tensoring A ⊗ S (−) ⊗ S A is flat, hence constructing an isomorphism of S e -modules W j → W * n−j will give us an isomorphism of A e -modules F Wj → F W * n−j , and composing with the above isomorphism an will give us an isomorphism F Wj → F ∨ Wn−j .
For |p| = n−j define ∂ p ∈ W * n−j by ∂ p (δ q Φ n ) = c qp , where Φ n = |t|=n c t t and |q| = j. Then there are S e -module homomorphisms for γ j arbitrary nonzero constants.
There are C vector space isomorphisms between eW j f and (f W n−j e) * for e, f ∈ Q 0 arising from the pairings, As eW j f and f W n−j e are finite dimensional C vector spaces the injectivity of η j implies this pairing is perfect, thus η j is an isomorphism. Now we use this to define isomorphisms α j i.e. the following diagram commutes; If |p| = n − j and |q| = j − 1 then where the sums are taken over all a ∈ Q 1 . Thus for these to be equal we require −γ j ǫ n−j+1 c qap = γ j−1 ǫ j (−1) j+n−1 c qap and −γ j ǫ n−j+1 (−1) j c aqp = γ j−1 ǫ j c qpa = (−1) n−1 γ j−1 ǫ j c aqp for all a ∈ Q 1 . These follow if (−1) n γ j−1 ǫ j = γ j ǫ n−j+1 (−1) j for j = 1, . . . , n. As the γ j were arbitrary non-zero scalars, we can choose the γ j so that this is satisfied, and the proof is completed.
Let Q be a single vertex quiver and A = D(Φ n , n − 2) be a Noetherian, Koszul, n-CY algebra. We consider a PBW deformation, A, given by θ 1 , θ 0 , and set φ n−1 = p pθ 1 (δ p Φ n ) = c q q. If the deformation is a zeroPBW deformation φ n−1 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 Proof. 1. Let A be a PBW deformation of A, defined by a map θ. Then define Φ ′ (θ) = Φ n + φ n−1 + φ n−2 by Write φ n−1 = t c t t. We will show that A is n-CY if and only if for any q = q 1 . . . q n−1 , with q j ∈ Q 1 . In particular this shows any zeroPBW deformation is n-CY, as then Φ ′ (θ) is a superpotential and when n is even φ n−1 = c t t = 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 Considering the coefficients of the paths, which are linearly independent, and calculating the coefficient of q 1 . . . q n−1 we find the condition on φ n−1 to be for all q 1 . . . q n−1 . 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 ⊗ θ 1 − θ 1 ⊗ id = ψ(θ 1 ) = {0} which is the zeroPBW condition. When n = 2 it gives the condition θ 1 = 0 which is the zeroPBW condition for even n.
Moreover when n = 3 any superpotential, Φ ′ , 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 ∈ Q 1 and the relations δ a Φ 3 . Hence as the arrows are linearly independent so are the relations, and a∈Q1 λ a δ a Φ 3 = 0 ⇒ λ a = 0 ⇒ λ a δ a Φ ′ = 0, so Φ ′ is 1-coherent. When n = 2 the superpotential is only differentiated by paths of length 0, so it is clearly 0-coherent. This result in the 3-CY case is is already known in the context of a general quiver due to the following result of Berger and Taillefer.

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 ω and G a symplectic reflection group which acts faithfully on V preserving the symplectic form. We say We consider the skew group algebra C[V ]⋊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].
where κ t,c (x, y) = tω V * (x, y) − Σ s ω s (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 ω V * on V * is induced from ω on V , and ω s is defined as ω V * restricted to (id − s)(V * ).
In particular they fall into the even dimensional case considered in Section 3 above, with the PBW parameter θ 1 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 ω((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 ] ⋊ G are the Rational Cherednik Algebras. • Wreath Product Algebras. Let K be a finite subgroup of SL 2 (C), and S n the symmetric group of order n. Then the wreath product group G = S n ≀ K is a symplectic reflection group acting on V = (C 2 ) n .

Symplectic reflection algebras as superpotential algebras.
Here we show that symplectic reflection algebras are Morita equivalent to superpotential algebras. for some homogeneous superpotential Φ 2n , and is 2n-CY and Koszul.

Any H t,c is Morita equivalent to
inhomogeneous superpotential which is (2n − 2)-coherent. 3. Any (2n − 2)-coherent superpotential of the form Φ ′ = Φ 2n + φ 2n−2 , gives an algebra A = D(Φ ′ , 2n − 2) that is Morita equivalent to a symplectic reflection algebra H t,c  Here we consider the symplectic reflection algebra corresponding to the group S 3 acting on h ⊕ h * in the manner of a rational Cherednik algebra, where the representation h is given by with ε 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, Φ 4 +4bB LL − 4bBLL + 4LbB L + 4bBLL − 2bB bB + bBbB + bB bB +cyclic permutations We now wish to look at the zeroPBW deformations, which by Theorem 4.1.1 correspond to 2-coherent superpotentials Φ ′ = Φ 4 +φ 2 . Writing φ 2 = c xy xy the PBW deformations are parametrised by the c xy such that Φ ′ is a 2-coherent superpotential. We see that Φ ′ is a superpotential if c xy = −c yx for all arrows x, y. A superpotential Φ ′ is 2-coherent if the δ xy Φ ′ = δ xy Φ 4 + c xy e h(x) e t(y) satisfy the same linear relations as the δ xy Φ 4 . For instance as δ Aa Φ = 0 and δ Aa Φ = −δ Aa we require that c Aa = 0 and c Aa = −c Aa .
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 So we have 2 degrees of freedom in our parameters, exactly as we do for the t, c in H t,c for S 3 acting on h ⊕ h * . a a By choosing a G-equivariant basis we calculate the superpotential We now calculate the zeroPBW deformations of A := D(Φ 4 , 2). We write φ 2 = c xy xy, and by Theorem 4.1.1 the zeroPBW deformations of A are parameterised by the c xy such that Φ ′ = Φ 4 + φ 2 is a 2-coherent superpotential.
In particular we require c xy = −c yx and for the c xy to satisfy the same linear relations as the δ xy Φ 4 . Making these calculations in this example we see the only non-zero c xy , and dependency relations among them, are; Hence to obtain a 2-coherent superpotential we require a φ 2 of the form 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 D 8 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[C 2 ] ⋊ G for G is a finite subgroup of SL 2 (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 a 1 , ..., a k , and the arrows in the opposite direction a * 1 , . . . a * k . Then C[C 2 ] ⋊ G is Morita equivalent to A = D(Φ 2 , 0) for the homogeneous superpotential, Φ 2 = [a, a * ]. This is the preprojective algebra Now we consider the PBW deformations of A. We apply Theorem 4.1.1, and deduce PBW deformations correspond to 0-coherent inhomogeneous superpotentials, Φ 2 +φ 0 . We note that φ 0 := − i∈Q0 λ i e i ∈ 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 Working with homogeneous superpotentials, Φ n = c p p, we have c p = 0 for any p that is not a closed path. This is no longer the case for twisted homogeneous superpotentials, here we find c p = 0 unless h(p) = σ(t(p)). Since the twist for C[W ] ⋊ G is given by tensoring by det W , c p is non zero only for paths from W i to det W ⊗ C W i , where the W i are the irreducible representations corresponding to vertices in the McKay quiver.
There are two different cases of finite subgroups of GL n (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 GL 2 (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 W i and head det W ⊗ C W i . Theorem 5.0.6. Let G be a small finite subgroup of GL 2 (C), which is not contained in SL 2 (C). Then C[C 2 ] ⋊ G has no nontrivial (relative to CG) PBW deformations.
Proof. The algebra C[C 2 ] ⋊ G can be written as T CG (C 2 * ⊗ C CG)

<[x,y]⊗1>
and is Morita equivalent to a path algebra with relations CQ/R = TS (V ) R = D(Φ 2 , 0) for some twisted homogeneous superpotential Φ 2 , where we use notation as in 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[C 2 ] ⋊ 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(Φ 2 , 0) has no nontrivial PBW deformations.
There can only possibly exist PBW deformations if there exists some non zero θ 0 , θ 1 as in Section 2.5. But θ 1 ∈ Hom S e (R, V ) and θ 0 ∈ Hom S e (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 the Appendix, Lemma A.0.1, that the tail and head of any relation are vertices which are distance greater than one apart. Hence, as S e module maps preserve heads and tails, the sets Hom S e (R, V ) and Hom S e (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 ε m denote a primitive m th root of unity.  Then PBW deformations of CQ/R are classified by θ 1 , θ 0 : R → A which are S e -module maps; θ 1 ∈ Hom S e (R, V ) and θ 0 ∈ Hom S e (R, S).
As S e -module maps preserve heads and tails we see that θ 1 must be zero, and θ 0 must be zero on all relations but the central one aA + dD − bB − cC. At this relation θ 0 (aA + dD − bB − cC) = λe 4 for some λ ∈ C. In this case (R ⊗ S V ) ∩ (V ⊗ S R) = 0, and so any such θ 0 gives us a PBW deformation. Hence there is a one parameter collection of PBW deformations.
Appendix A. McKay quivers for finite small subgroups of GL 2 (C) We use the classification of McKay quivers for small finite subgroups of GL 2 (C), [2], to prove the following: Lemma A.0.1. Let G < GL 2 (C) be a small finite subgroup, given by a representation W ∼ = C 2 . Let W i be an irreducible representation of G. Then the shortest path from W i to det W ⊗ W i 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 GL 2 (C) up to conjugacy, as in [2, Section 2].
Let Z n = g = ε 0 0 ε , and ε be a primitive n th root of unity. Any finite small subgroup of GL 2 (C) is, up to conjugacy, one of the following: 1. Z a cyclic subgroup, Z = g = ε 0 0 ε q for 1 ≤ q < n.
2. Z n D = {zd | z ∈ Z n , d ∈ D} for D a finite, non cyclic, subgroup of SL 2 (C).
3.H. To defineH let D < SL 2 (C) be a binary dihedral group, with A a cyclic subgroup of index 2, and define H < Z 2n × D to be ThenH is the image of H under the map H → GL 2 (C).

4.K.
To defineK let D < SL 2 (C) be the binary tetrahedral group, with A a normal binary dihedral subgroup of index 3, let n ≥ 3, and define K < Z 3n × D to be ThenK is the image of K under the map K → GL 2 (C).
We note that if n = 1, 2 thenK 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 distance two apart.
We first look at cyclic subgroups. Let Z be as above. Such a representation is in SL 2 (C) only when q + 1 = n. We suppose q + 1 = n.
Such a group has n irreducible one dimensional representations, which we label W 0 , . . . W n−1 , where W i is given by g → ε i . The defining representation is reducible as C 2 = W 1 ⊕ W q , and its determinant is the representation W 1+q . Hence the McKay quiver has n vertices corresponding to the W i 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 W i have head W i+q+1 module n. We see that the two vertices W i and W i+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 Z n × D and then taking the image of this under the map to GL 2 (C). If we calculate the McKay quiver for the subgroup then the image in GL 2 (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 Z n × D.
We first do this for the case Z n D. We note that this is contained in SL 2 (C) for n = 1, 2, hence we assume n > 2. In this case we consider the McKay quiver of Z n × D. Let the irreducible representations of D be labeled D 0 , . . . , D r−1 where D 0 is trivial, and D 1 is the given 2 dimensional representation. Let R i , for i = 0, . . . n − 1, be the n one dimensional irreducible representations of Z n with R i given by g → ε i . Then Z n × D has nr irreducible representations given by R i ⊗ D j for 0 ≤ i < n and 0 ≤ j < r. Then we consider the McKay quiver for the defining representation R 1 ⊗ D 1 , as this corresponds to the defining representation in GL 2 (C). In particular the McKay quiver has n groups of r representations labeled by representation of Z n , with group i corresponding to the set of representations {R i ⊗ D j | 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 R 1 ⊗ D 1 the determinant representation is R 2 ⊗ D 0 , 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 casesH andK we take the McKay quiver for Z n ×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 D 1 is the representation in SL 2 (C) and D 0 is the trivial representation. Now we label the representations of Z 2n as R i for i = 0 . . . 2n − 1 as above, and Z 2n × D has 2n(r + 2) irreducible representations given by their tensor products. The defining representation in GL 2 (C) is given as R 1 ⊗ D 1 , and hence the determinant representation is R 2 ⊗ D 0 . Now all representations of Z 2n × D are still irreducible for H but some are identified, [2,Proposition 7 (d)]. The representations which are identified are R i ⊗ D j with R n+i ⊗ D j for j = 1 . . . r − 1, modulo 2n, and R i ⊗ D j with R n+i ⊗ D ′ j modulo 2n for j = 0, r. The McKay quiver for H is the McKay quiver for Z 2n × D with these identifications made. In this case we group the irreducible representations into n groups labeled by the representation of Z 2n 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 SL 2 (C).
The final case is to consider K < Z 3n × D. Again we label the representations of Z 3n by R i for i = 0 . . . 3n − 1, and we label the representations of D by where D 0 is the trivial representation, and D 1 the defining representation.
The representation defining the group in GL 2 (C) is R 1 ⊗ D 1 , and the determinant representation is R 2 ⊗ D 0 . Now all the irreducible representations for Z 3n × D remain so for K, however some are identified [2, Proposition 7 (f)]. This time triples are identified: j modulo 3n, for j = 0, 1 and R i ⊗ D 2 ∼ = R i+n ⊗ D 2 ∼ = R i+2n ⊗ D 2 modulo 3n, as representations of K.
Once again we note that this splits the quiver into n groups, labeled by the representation of Z 3n 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 GL 2 (C) not contained in SL 2 (C) the determinant in the McKay quiver maps between vertices which are distance greater than 1 apart.