NONCOMMUTATIVE DEFORMATIONS OF KLEINIAN SINGULARITIES

WILLIAM CRAWLEY-BOEVEY AND MARTIN P. HOLLAND

Abstract. In this paper we study a family of noncommutative deformations of the coordinate ring of the Kleinian singularity K^{2}/Γ, where Γ is a finite subgroup of SL_{2}(K) and K is an algebraically closed field of characteristic zero. The family is defined by 𝒪^{λ} = e(K⟨x, y⟩ ∗ Γ)/(xy − yx − λ)e, for λ ∈ Z(KΓ), where K⟨x, y⟩ ∗ Γ is the skew group algebra for the action of Γ on the ring of noncommuting polynomials, and e is the average of the group elements. If λ has trace zero on KΓ then 𝒪^{λ} is commutative, and when K = ℂ we show that as complex analytic spaces the Spec 𝒪^{λ} are exactly the fibres of the semiuniversal deformation of ℂ^{2}/Γ. On the other hand, if λ has nonzero trace on KΓ we show that 𝒪^{λ} has only finitely many finite-dimensional simple modules. Using this we compute the ring-theoretic and homological properties of 𝒪^{λ}. They are controlled by the incidence of λ with hyperplanes in Z(KΓ) perpendicular to elements of the affine root system corresponding to the McKay graph of Γ. The main idea is to relate 𝒪^{λ} to a family of deformations of the preprojective algebra, and to study these using reflection functors.

Let K be an algebraically closed field of characteristic zero and let Γ be a nontrivial finite subgroup of SL_{2}(K). Deformations of the Kleinian singularity K^{2}/Γ have been studied for some time, especially the semiuniversal deformation (see, for example, Slodowy [34]). In this paper we define and study a family of deformations of its coordinate ring K[x, y]^{Γ}. Our rings, denoted 𝒪^{λ}, for λ ∈ Z(KΓ), are in general noncommutative, even simple rings. For example, 𝒪^{1} is the fixed ring of Γ on the Weyl algebra K⟨x, y | xy − yx = 1⟩. However, if the trace of λ on the regular representation KΓ is zero then 𝒪^{λ} is commutative, and (in case K = ℂ) the complex analytic spaces arising from the schemes Spec 𝒪^{λ} are precisely the fibres of the semiuniversal deformation.

In the special case when Γ is cyclic, a family of noncommutative deformations has been studied by Hodges [15] (see also [18, 36, 2]). These rings are defined by generators and relations and were originally studied as generalisations of the primitive factors of the enveloping algebra of 𝔰𝔩_{2}. Hodges noted that families of deformations should exist for general Γ but that it would be impractical to find them using his methods.

In retrospect, it is easy to write down deformations of K[x, y]^{Γ} for any Γ. The group acts naturally on the ring K⟨x, y⟩ of noncommuting polynomials, so one can form the skew group algebra K⟨x, y⟩ ∗ Γ. For λ ∈ Z(KΓ), let 𝒮^{λ} be the quotient (K⟨x, y⟩∗Γ)/(xy −yx−λ). We then define 𝒪^{λ} = e𝒮^{λ}e where e ∈ KΓ is the average of the group elements.

The ring 𝒪^{λ} is naturally filtered, and it is easy to see that the associated graded algebra is isomorphic to K[x, y]^{Γ}. It follows that 𝒪^{λ} is a noetherian domain, finitely generated as a K-algebra, of Gelfand-Kirillov dimension 2, is a maximal order in its quotient ring, is Auslander-Gorenstein and Cohen-Macaulay. In order to investigate how 𝒪^{λ} depends on λ we need the McKay graph [28] and a family of deformations of the preprojective algebra, both of which we now define.

1991 Mathematics Subject Classification. Primary 16S30, 32S30; Secondary 16G20.

Let N_{0}, …, N_{n} be the irreducible representations of Γ, with N_{0} trivial, and let V be the natural 2-dimensional representation of Γ. The McKay graph of Γ is the graph with vertex set I = {0, 1, …, n}, and with the number of edges between i and j being the multiplicity of N_{i} in V ⊗N_{j}. (Since V is self dual this is the same as the multiplicity of N_{j} in V ⊗ N_{i}.) McKay observed that this graph is an extended Dynkin diagram. Let Q be the quiver obtained from the McKay graph by choosing any orientation of the edges, and let δ ∈ ℕ^{I} be the vector with δ_{i} = dim N_{i}.

Henceforth we always identify Z(KΓ) = K^{I}, with λ ∈ Z(KΓ) corresponding to the vector whose i-th component λ_{i} is the trace of λ on N_{i}. Thus, for example, the dot product λ · δ = ∑_{i} λ_{i}δ_{i} is the same as the trace of λ ∈ Z(KΓ) on the regular representation KΓ.

We write \overline{Q} for the double of Q, by which we mean the quiver obtained by adding a reverse arrow a^{∗} : j → i for each arrow a : i → j in Q, and we write K\overline{Q} for its path algebra. This has basis the paths in \overline{Q}, including a trivial path e_{i} for each vertex i. If λ = (λ_{i}) ∈ K^{I} then the deformed preprojective algebra of weight λ is

Π^{λ} = K\overline{Q}/(c − λ),

where c is the sum of all commutators [a, a^{∗}], with a an arrow in Q, and λ is identified with the element ∑_{i} λ_{i}e_{i} of K\overline{Q}.

Theorem 1. 𝒪^{λ} ≅ e_{0}Π^{λ}e_{0} and 𝒮^{λ} is Morita equivalent to Π^{λ}.

The case when λ = 0, that K[x, y] ∗ Γ is Morita equivalent to Π^{0}, is already known, established during the proof of [33, Proposition 2.13]. The key calculation is [33, Proposition 2.4].

Representations of the quiver \overline{Q} of dimension vector δ are given by elements of the vector space

Rep(\overline{Q}, δ) = ⨁_{a:i→j} Hom(K^{δ_{i}}, K^{δ_{j}}) ⊕ Hom(K^{δ_{j}}, K^{δ_{i}})

where the sum is over the arrows in Q; the isomorphism classes correspond to orbits under the action of the group GL(δ) = ∏_{i∈I} GL(δ_{i}, K). Now, the assignment x = (x_{a}, x_{a^{∗}}) ↦ ∑_{a}[x_{a}, x_{a^{∗}}] defines a map μ : Rep(\overline{Q}, δ) → End(δ)_{0} where

End(δ)_{0} = {(θ_{i}) ∈ ∏_{i} End(K^{δ_{i}}) | ∑_{i} tr(θ_{i}) = 0}.

If λ ∈ K^{I} satisfies λ·δ = 0 then λ can be identified with an element in the centre of the Lie algebra End(δ)_{0}, and Rep(Π^{λ}, δ) = μ^{−1}(λ) is the space of representations of Π^{λ} of dimension vector δ. It is naturally a scheme, using the scheme-theoretic fibre of μ, and we write K[Rep(Π^{λ}, δ)] for its coordinate ring.

Theorem 2. If λ · δ = 0 then 𝒪^{λ} ≅ K[Rep(Π^{λ}, δ)]^{GL(δ)}.

In case K = ℂ we can relate this to work of Kronheimer [23]. By the discussion at the end of Section 8, if 𝔥 = {λ ∈ K^{I} | λ·δ = 0} then the map μ^{−1}(𝔥) // GL(δ) → 𝔥 is isomorphic in the category of complex analytic spaces to a lift through the Weyl group of the semiuniversal deformation of ℂ^{2}/Γ. Here the quotient μ^{−1}(𝔥) // GL(δ) is the usual one for the action of a reductive group on an affine scheme. Taking fibres, it follows that the quotients Rep(Π^{λ}, δ) // GL(δ) with λ·δ = 0 are isomorphic as complex analytic spaces to the fibres of the semiuniversal deformation. (Kronheimer also used hyperKähler quotients to construct the simultaneous resolution of the semiuniversal deformation. It would be interesting to have a noncommutative analogue of this.)

Using a notion of dominance we study 𝒪^{λ} when λ · δ ≠ 0. Its properties depend on the affine root system associated to the McKay graph. This is an infinite subset of ℤ^{I}, consisting of real roots, which are the images of the coordinate vectors under sequences of suitable reflections, and imaginary roots, which are the non-zero integer multiples of δ. By a Dynkin root we mean a root α with α_{0} = 0 (so α is a root for the Dynkin diagram obtained by deleting vertex 0).

Theorem 3. If λ · δ ≠ 0 then 𝒪^{λ} has only finitely many finite-dimensional simple modules up to isomorphism, and they are in 1–1 correspondence with the minimal positive roots α with λ · α = 0 which are non-Dynkin.

Finally, we can determine the homological and ring-theoretic properties of the 𝒪^{λ}. The commutativity condition follows immediately from the last two theorems.

Theorem 4. 𝒪^{λ} has the following properties:

1. It is commutative if and only if λ · δ = 0.

2. It is Morita equivalent to 𝒮^{λ} if and only if λ · α ≠ 0 for all Dynkin roots α.

3. It is a simple ring if and only if λ · α ≠ 0 for all non-Dynkin roots α.

4. It has global dimension

gl. dim 𝒪^{λ} = {1 if λ · α ≠ 0 for all roots α, 2 if λ · α = 0 for some root α, but λ · α ≠ 0 for all Dynkin roots α, ∞ if λ · α = 0 for some Dynkin root α.

5. It has Krull dimension

Kdim 𝒪^{λ} = \{{1 if λ · δ ≠ 0, \atop 2 if λ · δ = 0.}

In a final section we compute the K-theory of 𝒪^{λ}. We also obtain numerous results for the rings 𝒮^{λ} and for deformed preprojective algebras associated to other quivers Q (especially when Q is Dynkin).

1. The algebras 𝒮^{λ} and 𝒪^{λ}

Let K be an algebraically closed field of characteristic zero and let Γ ⊂ SL_{2}(K) be a non-trivial finite subgroup. The algebras 𝒮^{λ} and 𝒪^{λ} have been defined in the introduction. We shall also need to consider global versions of these algebras, which we now define. Let R be a commutative K-algebra and let λ ∈ Z(RG). The action of Γ extends to R⟨x, y⟩, with the action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) on R being trivial, gx = ax + cy and gy = bx + dy. We define

𝒮^{R,λ} = (R⟨x, y⟩ ∗ Γ)/(xy − yx − λ) and 𝒪^{R,λ} = e𝒮^{R,λ}e

where e = \frac{1}{|Γ|} ∑_{g∈Γ} g.

We filter 𝒮^{R,λ} with x and y in degree 1 and all elements of R and Γ in degree 0. This restricts to a filtration of 𝒪^{R,λ}.

Lemma 1.1. gr 𝒮^{R,λ} ≅ R[x, y] ∗ Γ and gr 𝒪^{R,λ} ≅ R[x, y]^{Γ}

Proof. 𝒮^{R,λ} is the free R-module with basis the elements x^{i}y^{j}g with i, j ≥ 0 and g ∈ Γ, for the relation xy − yx − λ enables any monomial in x and y to be rewritten with all xs preceeding all ys. The assertion for 𝒮^{R,λ} follows. The result for 𝒪^{R,λ} follows since e(R[x, y] ∗ Γ)e ≅ R[x, y]^{Γ}.

Lemma 1.2. If R is a domain then 𝒮^{R,λ} is a prime ring and 𝒪^{R,λ} is a domain. If R is noetherian then so are 𝒮^{R,λ} and 𝒪^{λ}. If R is finitely generated over K of dimension d then 𝒮^{λ} and 𝒪^{R,λ} are finitely generated over K of Gelfand-Kirillov dimension d + 2.

Proof. If F is the quotient field of R[x, y] then R[x, y] ∗ Γ is an order in F ∗ Γ. This is a simple ring by [27, Proposition 7.8.12], and hence R[x, y] ∗ Γ is prime. Of course R[x, y]^{Γ} is a domain. Now, the assertions for 𝒮^{R,λ} and 𝒪^{R,λ} follow by [27, Proposition 1.6.6]. Similarly the noetherianness follows from [27, Theorem 1.6.9]. Finally, to show that 𝒪^{R,λ} is finitely generated over K use [29, Proposition 1].

Lemma 1.3. If R is a noetherian integrally closed domain then 𝒪^{R,λ} and 𝒮^{R,λ} are maximal orders in their quotient rings.

Proof. For 𝒪^{R,λ} it suffices by [27, Theorem 5.1.6] to prove that its associated graded ring gr 𝒪^{R,λ} ≅ R[x, y]^{Γ} is integrally closed in its quotient field. This holds since R[x, y] is integrally closed in its quotient field.

For 𝒮^{R,λ} it suffices by the main result of [37] to prove that its associated graded ring gr 𝒮^{R,λ} ≅ R[x, y] ∗ Γ is a maximal order. This is well-known; see, for example, [26, Theorem 4.6].

Lemma 1.4. e𝒮^{λ} is a finitely generated reflexive 𝒪^{λ}-module and End_{𝒪^{λ}} (e𝒮^{λ}) ≅ 𝒮^{λ}.

Proof. It follows that e𝒮^{λ} is finitely generated from [29, Lemma 1].

For the first assertion, there are natural inclusions

𝒮^{λ} ⊆ End_{𝒪^{λ}} (e𝒮^{λ}) ⊆ End_{𝒮^{λ}}(𝒮^{λ}e𝒮^{λ}).

The point is that both endomorphism rings in the display can be identified with subrings of the simple artinian quotient ring Q of 𝒮^{λ}; do this, then it is trivial that

𝒮^{λ} ⊆ {q ∈ Q | e𝒮^{λ}q ⊆ e𝒮^{λ}} ⊆ {q ∈ Q | 𝒮^{λ}e𝒮^{λ}q ⊆ 𝒮^{λ}e𝒮^{λ}}.

But now it is an immediate consequence of the definition of a maximal order that the rightmost set here is equal to 𝒮^{λ}.

The second assertion now follows. For, e𝒮^{λ} is a torsionfree, faithful 𝒪^{λ}-module of finite uniform dimension and is a generator (𝒪^{λ} is a summand). Thus, we can apply [7].

Recall that a K-algebra R is said to be Auslander-Gorenstein if it has finite injective dimension and if for every finitely generated R-module M and every submodule N ⊆ Ext_{R}^{j}(M, R) one has Ext_{R}^{i}(N, R) = 0, for i < j. An Auslander-Gorenstein ring of finite global dimension is called Auslander-regular. Finally, an Auslander-Gorenstein ring R is said to be Cohen-Macaulay if j(M) + GK M = GK R, for all non-zero, finitely generated R-modules M. Here j(M) is the grade of M, that is inf{i : Ext^{i}(M, R) ≠ 0}, and GK M is the Gelfand-Kirillov dimension of M.

Theorem 1.5. The algebra 𝒮^{λ} is Auslander-regular and Cohen-Macaulay of GK dimension two. 𝒮^{λ} has global dimension one if and only if it has no non-zero finite-dimensional modules. It has global dimension two otherwise.

Proof. Once one knows that gr 𝒮^{λ} = K[x, y] ∗ Γ, this is a fairly standard argument. First, note that K[x, y] ∗ Γ is Auslander-regular of global dimension two and satisfies the Cohen-Macaulay property. This is well known; see, for example, [39, Proposition 3.9(ii)]. (One slightly subtle point is that the latter reference works with the Krull dimension version of the Cohen-Macaulay property. However, over a Noetherian, finitely generated, PI algebra, GK M = Kdim M, for a finitely generated module M. This follows from [27, Theorem 6.4.8, Proposition 6.4.12] and [21, Corollary 10.16, Lemma 10.18].)

In particular, by [27, Corollary 7.6.18], one knows that 𝒮^{λ} has global dimension at most two. Further, by [5, Theorem 4.1], 𝒮^{λ} is Auslander-Gorenstein (and hence Auslander-regular). Finally, by [27, Proposition 8.6.5] and [5, Theorem 4.3], it satisfies GK M + j(M) = 2, for all finitely generated modules M.

Recall that a Noetherian ring R of finite global dimension has global dimension equal to the common value of the injective dimension of R as a left (or right) module. Thus, we must compute the latter value, μ say. Note that μ is the maximal value of j(M), for a non-zero, finitely generated module M (conventionally j(0) = ∞). For, Ext^{μ}(N, 𝒮^{λ}) ≠ 0, for some right 𝒮^{λ}-module N. Since Ext commutes with direct limits in the first factor, we may assume that N is finitely generated. Now, the Auslander condition says that if we take M to be the finitely generated left module Ext^{μ}(N, 𝒮^{λ}) then j(M) ≥ μ.

Thus, 𝒮^{λ} has injective dimension equal to the maximal value of 2 − GK M, amongst non-zero finitely generated M. It follows quickly that 𝒮^{λ} has injective dimension one (two) in the case when it has no non-zero finite-dimensional module (has a non-zero finite-dimensional module). (Of course, 𝒮^{λ} is never semisimple artinian.)

Theorem 1.6. The algebra 𝒪^{λ} is Auslander-Gorenstein and Cohen-Macaulay. It has global dimension one, two or infinity. If it has finite global dimension then it has global dimension one exactly when it has no non-zero finite-dimensional modules.

Proof. This is a similar argument to the one for 𝒮^{λ}. Again, gr 𝒪^{λ} = 𝒪^{0} = K[x, y]^{Γ}. But the latter is a commutative Gorenstein K-algebra and so is certainly a Cohen-Macaulay, Auslander-Gorenstein ring. Now one can run through the same argument as for 𝒮^{λ}, reaching the same conclusions. The only difference being that 𝒪^{λ} may have infinite global dimension.

Remark 1.7. Later we will determine exactly when the possibilities of the last two theorems occur; see Theorems 7.7, 9.5 and Corollary 9.7 below.

2. Deformed preprojective algebras

In this section K is an arbitrary field and Q is an arbitrary (finite) quiver with vertex set I. We write a ∈ Q to mean that a is an arrow. Its head is at the vertex h(a) and its tail is at t(a). Thus, a : i → j means that h(a) = j and t(a) = i. The Ringel form for Q is the bilinear form on ℤ^{I} with

⟨α, β⟩ = ∑_{i∈I} α_{i}β_{i} − ∑_{a∈Q} α_{t(a)}β_{h(a)}.

for α, β ∈ ℤ^{I}, and the symmetric bilinear form is (α, β) = ⟨α, β⟩ + ⟨β, α⟩. The Dynkin quivers A_{n} (n ≥ 1), D_{n} (n ≥ 4), E_{6}, E_{7}, E_{8} are the connected quivers for which the corresponding quadratic form is positive definite. The extended Dynkin quivers Ã_{n} (n ≥ 0), D̃_{n} (n ≥ 4), Ẽ_{6}, Ẽ_{7}, Ẽ_{8} are the connected quivers for which the quadratic form is positive semi-definite (where Ã_{0} is the quiver with one vertex and one loop, and Ã_{1} is the quiver with two vertices joined by two arrows, in either direction). In this case we denote by δ the minimal vector in ℕ^{I} in the radical of (,). Any vertex i with δ_{i} = 1 is called an extending vertex, and deleting i one obtains the corresponding Dynkin quiver. If Q is extended Dynkin then we always number the vertices I = {0, 1, …, n} with 0 an extending vertex. This is the situation for an orientation of the McKay graph for a Kleinian singularity.

The double \overline{Q} of Q is the quiver obtained by adding a reverse arrow a^{∗} : j → i for each arrow a : i → j in Q. We write KQ and K\overline{Q} for the path algebras of Q and \overline{Q}, using the convention that if a : i → j and b : j → k are arrows then the composite path is ba : i → k. The trivial path for vertex i is denoted e_{i}.

Elements of K^{I} are called weights and if λ is a weight then we define the deformed preprojective algebra to be

Π^{λ}(Q) = K\overline{Q}/(c − λ),

where c is the commutator element of K\overline{Q}

c = ∑_{a∈Q}[a, a^{∗}]

(here, [a, a^{∗}] = aa^{∗} − a^{∗}a), and λ is identified with the element ∑_{i} λ_{i}e_{i} of K\overline{Q}. If the quiver is understood we just write Π^{λ}.

The algebra Π(Q) = Π^{0}(Q) is known as the preprojective algebra. If Q has no oriented cycles then Π(Q) is the preprojective algebra associated to the finitedimensional hereditary algebra KQ by Baer, Geigle and Lenzing [1]. (The preprojective algebra of Dlab and Ringel [9] is defined with different signs, essentially because the Auslander-Reiten translation differs from the Coxeter functor by a change-of-sign functor, see [12, Prop. 5.4]. However, if Q is bipartite the two types of preprojective algebra are isomorphic).

For technical reasons we also need to consider a global version of the deformed preprojective algebra, so if R is a commutative ring and λ ∈ R^{I}, we define Π^{R,λ}(Q) to be R\overline{Q}/(c − λ).

Lemma 2.1. The ideal (c − λ) is the same as the ideal with a generator

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} aa^{∗} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} a^{∗}a − λ_{i}e_{i}

for each vertex i.

Proof. These elements are of the form e_{i}(c−λ)e_{i} so they belong to the ideal (c−λ), but also c − λ is the sum of them, so the ideal they generate contains c − λ.

Lemma 2.2. Π^{R,λ}(Q) does not depend on the orientation of Q, and it is unchanged up to isomorphism if you multiply λ by an invertible element of R.

Proof. If Q′ is obtained from Q by reversing the arrow a, then there is an isomorphism Π^{R,λ}(Q) → Π^{R,λ}(Q′) which sends a to a^{∗} and a^{∗} to −a. The second assertion is clear.

There is a natural grading of K\overline{Q}, with the trivial paths e_{i} in degree zero and the arrows a, a^{∗} in degree one. This induces a grading of Π. It also gives rise to a filtration of K\overline{Q} which induces one of Π^{λ}.

Lemma 2.3. There is a natural surjective homomorphism Π → gr Π^{λ}.

Proof. Give the ideal (c − λ) the induced filtration. By [27, 7.6.13], there is an exact sequence 0 ⟶ gr(c − λ) ⟶ gr K\overline{Q} ⟶ gr(Π^{λ}) ⟶ 0. Clearly, we can identify gr K\overline{Q} = K\overline{Q} and then c ∈ gr(c − λ).

For Dynkin quivers the preprojective algebra is finite dimensional, so we obtain:

Lemma 2.4. If Q is Dynkin (or a disjoint union of Dynkin quivers) then Π^{λ}(Q) is finite dimensional.

3. Morita equivalence between Π^{λ} and 𝒮^{λ}

Let K be an algebraically closed field of characteristic zero and Γ a non-trivial finite subgoup of SL_{2}(K). In this section we prove Theorem 1, that 𝒮^{λ} is Morita equivalent to Π^{λ}(Q) and 𝒪^{λ} ≅ e_{0}Π^{λ}(Q)e_{0} where Q is any orientation of the McKay graph of Γ. For technical reasons we need to prove it for the global versions 𝒮^{R,λ} and Π^{R,λ}, so let R be a commutative K-algebra. As in the introduction, V is the natural 2-dimensional representation of Γ, the irreducible KΓ-modules are N_{0}, …, N_{n}, with N_{0} the trivial module, and I = {0, 1, …, n}.

If η_{i} (i ∈ I) are the central idempotents in KΓ corresponding to the irreducible representations N_{i} then the map

R^{I} → Z(RΓ), λ ↦ ∑_{i∈I} \frac{λ_{i}}{δ_{i}} η_{i}

is a bijection, and we use this to identify R^{I} and Z(RΓ). Note that in case R = K this agrees with the identification used in the introduction.

Let e ∈ KΓ be the average of the group elements, and for each i ∈ I let f_{i} be an idempotent in KΓ with KΓf_{i} ≅ N_{i}. The f_{i} are orthogonal, so f = f_{0} + ⋯ + f_{n} is idempotent. Also f_{0} = e, so e = fef.

If x, y is the standard basis of V then the symplectic form ⟨, ⟩ with

⟨x, y⟩ = 1, ⟨y, x⟩ = −1, ⟨x, x⟩ = ⟨y, y⟩ = 0.

is Γ-invariant so defines a homomorphism ⟨, ⟩ : V ⊗ V → K. There is also a homomorphism

ζ : K → V ⊗ V, ζ(1) = y ⊗ x − x ⊗ y.

The first lemma is elementary.

Lemma 3.1. If L and M are KΓ-modules then there are inverse bijections

Hom(L, V ⊗ M) {-→ \atop ←-}_{ϕ↦ϕ^{♭}}^{θ↦θ^{♯}} Hom(V ⊗ L, M)

where θ^{♯} = (⟨, ⟩ ⊗ 1)(1 ⊗ θ) and ϕ^{♭} = (1 ⊗ ϕ)(ζ ⊗ 1).

Our key lemma is as follows.

Lemma 3.2. To each arrow a : i → j in Q one can associate homomorphisms

θ_{a} : N_{i} → V ⊗ N_{j} and ϕ_{a} : N_{j} → V ⊗ N_{i}

in such a way that, for each vertex i,

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}}(1 ⊗ θ_{a})ϕ_{a} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}}(1 ⊗ ϕ_{a})θ_{a} = δ_{i}(ζ ⊗ 1)

as maps N_{i} → V ⊗ V ⊗ N_{i}; and the θ_{a}, ϕ_{a} (a ∈ Q) combine to give a basis for each of the spaces Hom(N_{i}, V ⊗ N_{j}).

Proof. First suppose that Q is of type Ã_{n}. Thus, V has a basis u, v with respect to which the action of Γ is diagonal. Moreover, by rescaling u one can ensure that v ⊗ u − u ⊗ v = y ⊗ x − x ⊗ y. It follows that Γ is cyclic of order n + 1, say with generator g acting via gu = ϵu and gv = ϵ^{−1}v, where ϵ is a primitive (n + 1)-th root of unity. The irreducible representations N_{i} are 1-dimensional, with g acting as multiplication by ϵ^{i}, Now the arrows a_{i} in Q connect vertices i and i+1 (identifying n + 1 with 0). According to the orientation, if a_{i} : i → i + 1 let θ_{a_{i}}(z) = v ⊗ z and ϕ_{a_{i}}(z) = −u ⊗ z, and if a_{i} : i + 1 → i let θ_{a_{i}}(z) = u ⊗ z and ϕ_{a_{i}}(z) = v ⊗ z. These maps clearly satisfy the lemma.

Now suppose that Q is of type D̃_{n} or Ẽ_{n}. Observe that in this case Q is a tree. For each arrow a ∈ Q choose a non-zero homomorphism θ_{a} : N_{t(a)} → V ⊗ N_{h(a)} and a non-zero homomorphism ϕ_{a} : N_{h(a)} → V ⊗ N_{t(a)}. Since all V ⊗ N_{i} are multiplicity-free, ϕ_{a}^{♯}θ_{a} is a non-zero endomorphism of N_{t(a)} and θ_{a}^{♯}ϕ_{a} is a non-zero endomorphism of N_{h(a)}. Now, by rescaling θ_{a}, one can simultaneously rescale these two endomorphisms. Then since Q is a tree one can arrange that there are non-zero scalars m_{i} for each vertex i, with

ϕ_{a}^{♯}θ_{a} = m_{h(a)}1_{N_{t(a)}} and θ_{a}^{♯}ϕ_{a} = −m_{t(a)}1_{N_{h(a)}}

for every arrow a. One proceeds as follows. First fix the value of m_{i} for one vertex. Then if the m_{i} have been defined for a connected piece of Q and a is an arrow connecting a vertex i in the piece to a vertex j outside, one of the two equations fixes θ_{a}, ϕ_{a} up to scalar multiples, and the other equation fixes m_{j}.

Thus, θ_{a}ϕ_{a}^{♯} is m_{h(a)} times the projection of V ⊗ N_{h(a)} onto N_{t(a)}, and ϕ_{a}θ_{a}^{♯} is −m_{t(a)} times the projection of V ⊗ N_{t(a)} onto N_{h(a)}. Thus, for any vertex i,

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} θ_{a}ϕ_{a}^{♯} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} ϕ_{a}θ_{a}^{♯} = m_{i} 1_{V ⊗N_{i}}.

If this map is ψ, then considering (1 ⊗ ψ)(ζ ⊗ 1) : N_{i} → V ⊗ V ⊗ N_{i} and using Lemma 3.1 one obtains

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}}(1 ⊗ θ_{a})ϕ_{a} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}}(1 ⊗ ϕ_{a})θ_{a} = m_{i}(ζ ⊗ 1).

Now, composing this with the map ⟨, ⟩ ⊗ 1 : V ⊗ V ⊗ N_{i} → N_{i}, the resulting endomorphism of N_{i} is multiplication by

− ∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} m_{t(a)} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} m_{h(a)} = −2m_{i}.

Extending the symmetric bilinear form (,) to K and considering the vector m ∈ K^{I} with components m_{i} one deduces that (m, ϵ_{i}) = 0 for all i, and this forces m to be a multiple of δ. The lemma now follows after rescaling all θ_{a} suitably.

Let C = K⟨x, y⟩ ∗ Γ, graded with x, y in degree 1 and elements of Γ in degree 0. Clearly C is the tensor algebra over KΓ of the bimodule V ⊗ KΓ, where the action of KΓ on the left is diagonal, but on the right it only acts on the second term in the tensor product.

Since the idempotent f induces a Morita equivalence, it follows that fCf is the tensor algebra over fC_{0}f of the bimodule fC_{1}f. Now, fC_{0}f ≅ K^{I} via the idempotents f_{i}, and we have the identification

f_{i}C_{1}f_{j} ≅ Hom(KΓf_{i}, C_{r}f_{j}) ≅ Hom(KΓf_{i}, V ⊗ KΓf_{j}) ≅ Hom(N_{i}, V ⊗ N_{j}).

Now we can restate the lemma.

Lemma 3.3. To each arrow a : i → j in Q one can associate elements

θ_{a} ∈ f_{j}C_{1}f_{i} and ϕ_{a} ∈ f_{i}C_{1}f_{j}

in such a way that, for each vertex i,

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} θ_{a}ϕ_{a} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} ϕ_{a}θ_{a} = δ_{i}f_{i}(yx − xy)

and the θ_{a}, ϕ_{a} (a ∈ Q) combine to give a basis for each of the spaces f_{i}C_{1}f_{j}.

Theorem 3.4. There is an isomorphism f𝒮^{R,λ}f ≅ Π^{R,λ}(Q). Under this isomorphism e corresponds to the trivial path e_{0}.

Proof. By the discussion above there is an isomorphism K\overline{Q} → fCf sending the trivial path e_{i} to f_{i}, sending an arrow a : i → j of Q to ϕ_{a} ∈ f_{j}Cf_{i} and sending a^{∗} to θ_{a}. Writing C^{R} for the induced R-algebra C ⊗ R we have an isomorphism RQ → C^{R}.

Observe that a(xy − yx − λ) = (xy − yx − λ)a for any a ∈ KΓ since x ⊗ y − y ⊗ x belongs to a trivial submodule of V ⊗ V and λ ∈ Z(RΓ) so it commutes with a. In particular f(xy − yx − λ) = (xy − yx − λ)f.

Now, if I is the ideal in C^{R} generated by xy −yx−λ then I ∩fC^{R}f is generated as an ideal in fC^{R}f by f(xy − yx − λ), for f corresponds to a Morita equivalence so there are a_{i}, b_{i} ∈ KΓ with ∑_{i} a_{i}fb_{i} = 1, and then the ideal in C^{R} generated by f(xy − yx − λ) contains

∑_{i} a_{i}((xy − yx − λ)f)b_{i} = ∑_{i}(xy − yx − λ)a_{i}fb_{i} = xy − yx − λ.

Equally, one can see that I ∩ fC^{R}f is generated by the elements f_{i}(xy − yx − λ). Identifying λ with an element of R^{I} this element becomes f_{i}(xy − yx − λ_{i}/δ_{i}), and then by the lemma this is equal to

\frac{1}{δ_{i}} (− ∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} θ_{a}ϕ_{a} + ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} ϕ_{a}θ_{a} − λ_{i}).

These are precisely the defining relations for Π^{R,λ}.

This has an immediate corollary, which has Theorem 1 as a special case.

Corollary 3.5. 𝒮^{R,λ} is Morita equivalent to Π^{R,λ}(Q) and 𝒪^{R,λ} ≅ e_{0}Π^{R,λ}(Q)e_{0}.

There is also the following consequence for deformed preprojective algebras.

Corollary 3.6. If Q is an extended Dynkin quiver with vertex set I and λ ∈ K^{I} then the natural map Π(Q) → gr Π^{λ}(Q) is an isomorphism and Π^{λ}(Q) is a prime noetherian ring of GK dimension 2.

Proof. Every extended Dynkin quiver arises by orienting the McKay graph of some Kleinian singularity (except the quiver consisting of one vertex and one loop, but for this quiver the assertion is elementary). Now, Π^{λ}(Q) is Morita equivalent to 𝒮^{λ} so is a prime noetherian ring of GK dimension 2. Also, the map Π(Q) → gr Π^{λ}(Q) is surjective, so it must also be injective since every proper quotient of Π(Q) has GK dimension strictly less than 2.

Remark 3.7. The corollary actually holds for arbitrary base fields. For arbitrary fields of characteristic zero it follows from the corollary, using that Π^{L,λ} ≅ Π^{λ} ⊗ L, where L is an algebraic closure of K. For fields of arbitrary characteristic we have checked the corollary by writing down, for each Q, a representation ρ : Π^{λ} → M_{n}(A) where A = K⟨s, t | st − ts = λ · δ⟩, and using a computer to check that ρ(Π^{λ}) has GK dimension at least 2. Now, Π(Q) is a prime noetherian ring of GK dimension 2, by [1], so it follows that Π(Q) → gr Π^{λ}(Q) must be an isomorphism and then that Π^{λ} is a prime noetherian ring of GK dimension 2.

4. Representations of deformed preprojective algebras

Again K is an arbitrary field and Q is an arbitrary quiver with vertex set I. Representations of Q of dimension vector α are given by elements of the vector space

Rep(Q, α) = ⨁_{a∈Q} Hom(K^{α_{t(a)}}, K^{α_{h(a)}}),

and representation of \overline{Q} of dimension vector α are given by elements of

Rep(\overline{Q}, α) = Rep(Q, α) ⊕ Rep(Q^{op}, α)

where Q^{op} is the opposite quiver to Q, with an arrow a^{∗} : j → i for each arrow a : i → j in Q. There is an action of the group

GL(α) = ∏_{i∈I} GL(α_{i}, K)

on both of these, and its orbits correspond to isomorphism classes of representations. This group has Lie algebra

End(α) = ⨁_{i∈I} End(K^{α_{i}}),

but more relevant is its subalgebra

End(α)_{0} = {(θ_{i}) ∈ End(α) | ∑_{i} tr(θ_{i}) = 0}.

Now, the assignment x = (x_{a}, x_{a^{∗}}) ↦ ∑_{a∈Q}[x_{a}, x_{a^{∗}}] defines a map

μ : Rep(\overline{Q}, α) → End(α)_{0}

where, in computing the commutator [x_{a}, x_{a^{∗}}] = x_{a}x_{a^{∗}} − x_{a^{∗}}x_{a}, one needs to send each term into the appropriate component of End(α). The map has image contained in End(α)_{0} since any commutator has trace zero.

Now, suppose that λ ∈ K^{I}. By Lemma 2.1, the category of Π^{λ}-modules is equivalent to the category of representations V of \overline{Q}, say with vector space V_{i} at vertex i and linear map V_{a} : V_{i} → V_{j} for each arrow a : i → j in \overline{Q}, which satisfy

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} V_{a}V_{a^{∗}} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} V_{a^{∗}}V_{a} = λ_{i}1_{V_{i}}

for each vertex i. Now, if λ · α = 0 then λ can be identified with the element (λ_{i}1_{K^{α_{i}}}) in the centre of End(α)_{0}, and

Rep(Π^{λ}, α) = μ^{−1}(λ)

is the scheme which classifies representations of Π^{λ} of dimension vector α. On the other hand the condition on traces implies:

Lemma 4.1. If λ · α ≠ 0 there is no representation of Π^{λ} of dimension vector α.

There is a partial converse which we now derive.

Lemma 4.2. If x = (x_{a})_{a∈Q} ∈ Rep(Q, α) and X is the corresponding KQ-module then there is an exact sequence

0 → D Ext^{1}(X, X) → Rep(Q^{op}, α) →^{c} End(α) → D End(X) → 0

where c sends (y_{a^{∗}}) ∈ Rep(Q^{op}, α) to ∑_{a∈Q}[x_{a}, y_{a^{∗}}] and D is duality with the field.

Proof. There is an exact sequence

0 → End(X) → End(α) →^{f} Rep(Q, α) → Ext^{1}(X, X) → 0

where f sends (θ_{i})_{i} to (θ_{h(a)}x_{a} − x_{a}θ_{t(a)})_{a}, for clearly the kernel of f is End(X), and by the Ringel formula

dim End(X) − dim Ext^{1}(X, X) = ⟨α, α⟩ = dim End(α) − dim Rep(Q, α)

so the cokernel has the same dimension as Ext^{1}(X, X). (In fact, using the standard projective resolution of X one can show that the cokernel is naturally isomorphic to Ext^{1}(X, X).) Now, the required exact sequence is the dual of this one, using the trace pairing

Hom(K^{a}, K^{b}) × Hom(K^{b}, K^{a}) → K, (θ, ϕ) ↦ tr(θϕ)

to identify Rep(Q^{op}, α) ≅ D Rep(Q, α) and End(α) ≅ D End(α).

Theorem 4.3. If λ · α = 0 and there is a representation of Q of dimension vector α and endomorphism algebra K, then there is a representation of Π^{λ} of dimension vector α.

Proof. If x ∈ Rep(Q, α) has endomorphism algebra K then the corresponding map c : Rep(Q^{op}, α) → End(α) has one-dimensional cokernel. Thus, c has image End(α)_{0}, for in any case its image is contained in this subspace. Now, if λ is identified with a central element of the ring End(α), then the condition λ · α = 0 ensures that λ ∈ End(α)_{0}. Thus, c^{−1}(λ) is non-empty. Now, any element of this fibre extends x to a representation of Π^{λ}.

In particular, this applies for extended Dynkin quivers and the dimension vector δ. In fact, one knows the following. (See, for example, [8].)

Lemma 4.4. If Q is an extended Dynkin quiver and δ is the minimal positive vector with (δ, δ) = 0 then there is a representation of Q of dimension vector δ whose endomorphism algebra is K. In addition, if K is algebraically closed then the representations with endomorphism algebra K form a dense open subset in Rep(Q, δ), and its complement is the union of only finitely many GL(δ)-orbits.

5. Reflections

In this section K is an arbitrary field and Q is an arbitrary quiver with vertex set I. We say that a vertex i is loopfree if there are no arrows a : i → i, and if so, the simple reflection s_{i} : ℤ^{I} → ℤ^{I} is defined by

s_{i}(α) = α − (α, ϵ_{i})ϵ_{i},

where ϵ_{i} is the coordinate vector at i. There is a dual reflection r_{i} : K^{I} → K^{I} with the property that r_{i}λ · α = λ · s_{i}α for all λ ∈ K^{I} and all α ∈ ℤ^{I}. (Explicitly, (r_{i}λ)_{j} = λ_{j} − (ϵ_{i}, ϵ_{j})λ_{i}.) The Weyl group W is the group of automorphisms of ℤ^{I} generated by the simple reflections for loopfree vertices. It acts on K^{I}, with

(wλ) · α = λ · (w^{−1}α)

for all w ∈ W, λ ∈ K^{I} and α ∈ ℤ^{I}.

The key tools for studying deformed preprojective algebras are certain reflection functors which are defined in the next theorem. It is interesting that these functors exist for deformed preprojective algebras but not for ordinary preprojective algebras. Note that our reflection functors are unlike those of Bernstein, Gelfand and Ponomarev [3] in that they are equivalences of categories and do not depend on the orientation of Q. We remark that Nakajima has mentioned the existence of reflection functors in an analogous situation, see [31, §9].

Theorem 5.1. If i is a loopfree vertex and λ_{i} ≠ 0 then there is an equivalence

E_{i} : Π^{λ}(Q)-modules ⟶ Π^{r_{i}λ}(Q)-modules

which acts as the simple reflection s_{i} on dimension vectors.

Proof. Since Π^{λ}(Q) does not depend on the orientation of Q we can assume that no arrows have tail at i.

We consider a Π^{λ}(Q)-module V as a representation of \overline{Q} with vector space V_{j} at vertex j and linear map V_{a} for each arrow a ∈ \overline{Q}, satisfying the relation

∑_{\begin{matrix}a∈Q \\ h(a)=j \end{matrix}} V_{a}V_{a^{∗}} − ∑_{\begin{matrix}a∈Q \\ t(a)=j \end{matrix}} V_{a^{∗}}V_{a} − λ_{j}1_{V_{j}}

for each vertex j. Given such a representation, let

V_{⊕} = ⨁_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} V_{t(a)}

and write μ_{a} and π_{a} for the inclusion and projection between V_{t(a)} and V_{⊕}. Define maps μ : V_{i} → V_{⊕} and π : V_{⊕} → V_{i} by

π = \frac{1}{λ_{i}} ∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} V_{a}π_{a} μ = ∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} μ_{a}V_{a^{∗}}

The relations ensure that πμ = 1_{V_{i}}, so μπ is an idempotent endomorphism of V_{⊕}.

We consider the representation V′ of \overline{Q} with V′_{i} = Im(1 − μπ) = Ker π and V′_{j} = V_{j} for j ≠ i, and with linear maps V′_{a} = V_{a} and V′_{a^{∗}} = V_{a^{∗}} if h(a) ≠ i, while if h(a) = i,

V′_{a} = −λ_{i}(1 − μπ)μ_{a} : V′_{t(a)} → V′_{i} V′_{a^{∗}} = π_{a}|_{V′_{i}} : V′_{i} → V′_{t(a)}.

We now show that V′ is a module for Π^{r_{i}λ}. For the vertex i,

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} V′_{a}V′_{a^{∗}} = ∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} −λ_{i}(1 − μπ)μ_{a}π_{a}|_{V′_{i}} = −λ_{i}(1 − μπ)|_{V′_{i}} = −λ_{i}1_{V′_{i}}

and (r_{i}λ)_{i} = −λ_{i}. Now, if a : j → i is an arrow in Q then

V′_{a^{∗}}V′_{a} = π_{a}(−λ_{i}(1 − μπ)μ_{a}) = −λ_{i}π_{a}μ_{a} + λ_{i}π_{a}μπμ_{a} = −λ_{i}1_{V_{t(a)}} + V_{a^{∗}}V_{a},

so if j ≠ i then

∑_{\begin{matrix}a∈Q \\ h(a)=j \end{matrix}} V′_{a}V′_{a^{∗}} − ∑_{\begin{matrix}a∈Q \\ t(a)=j \end{matrix}} V′_{a^{∗}}V′_{a} = ∑_{\begin{matrix}a∈Q \\ h(a)=j \end{matrix}} V_{a}V_{a^{∗}} − ∑_{\begin{matrix}a∈Q \\ t(a)=j \end{matrix}} V_{a^{∗}}V_{a} − (ϵ_{j}, ϵ_{i})λ_{i}1_{V_{j}}

since there are −(ϵ_{j}, ϵ_{i}) arrows from j to i. Then this reduces to

λ_{j}1_{V_{j}} − (ϵ_{j}, ϵ_{i})λ_{i}1_{V_{j}} = (r_{i}λ)_{j}1_{V_{j}}.

Now, the assignment V ↦ V′ clearly extends to a functor E_{i}, and the exact sequence

0 → V′_{i} → V_{⊕} →^{π} V_{i} → 0

shows that it acts as s_{i} on dimension vectors. Finally, the reflection also defines a functor

E′_{i} : Π^{r_{i}λ}(Q)-modules ⟶ Π^{λ}(Q)-modules

and by direct calculation there is a natural isomorphism V → E′_{i}(E_{i}(V)). It follows that E_{i} is an equivalence.

Corollary 5.2. Π^{λ} is Morita equivalent to Π^{μ} for any μ ∈ W λ. In fact if μ = wλ and w has minimal length with this property, then there is an equivalence

Π^{λ}-modules ⟶ Π^{μ}-modules

which acts as w on dimension vectors.

Proof. We prove this by induction on the length of w, and hence reduce to the case when w is a simple reflection s_{i}. Now, λ_{i} ≠ 0, for otherwise wλ = r_{i}λ = λ, contradicting the minimality of w. Thus, the reflection functor exists.

6. Roots

In this section K is an arbitrary field and Q is an arbitrary quiver with vertex set I. The simple roots for Q are the coordinate vectors of loopfree vertices. An element α ∈ ℤ^{I} is a real root if it is the image of a simple root under the Weyl group (as defined in the previous section). It is standard that any real root is either positive or negative, i.e. either α or −α belongs to ℕ^{I}. The fundamental region is

F = {α ∈ ℕ^{I} | α ≠ 0, support of α connected, and (α, ϵ_{i}) ≤ 0 for all i}

(in fact (α, ϵ_{i}) ≤ 0 is automatic if i is not loopfree), and an imaginary root is an element of ℤ^{I} of the form wβ or −wβ with w ∈ W and β ∈ F. A root is a real or imaginary root. The elements wβ all belong to ℕ^{I}, so these are the positive imaginary roots.

We say that λ ∈ K^{I} is on a wall if λ · α = 0 for some real root α

Lemma 6.1. Π^{λ} has a representation of dimension vector ϵ_{i} if and only if λ_{i} = 0. It is unique if i is loopfree.

Proof. Clear.

Theorem 6.2. Suppose that λ ∈ K^{I}.

1. If λ is on a wall then Π^{λ} has a simple representation whose dimension vector is a real root.

2. If λ is not on a wall then the dimension vector of any indecomposable finitedimensional representation of Π^{λ} is an imaginary root.

Proof. 1. Since λ is on a wall there is w ∈ W such that the weight μ = wλ has component μ_{i} = 0 for some loopfree vertex i. Now, Π^{μ} has a simple representation of dimension vector ϵ_{i} and by Corollary 5.2 there is a simple Π^{λ}-module of dimension vector w′ϵ_{i} for some w′ ∈ W.

2. Since λ is not on a wall one can apply any sequence of reflection functors. If you have an indecomposable representation then its support is connected, so if its dimension vector α is not in the fundamental chamber then (α, ϵ_{i}) > 0 for some loopfree vertex i. Applying the reflection at i one obtains an indecomposable representation (of a new deformed preprojective algebra) of dimension vector s_{i}α = α−(α, ϵ_{i})ϵ_{i}, which is strictly smaller than α. This can’t go on for ever, so eventually the dimension vector must be in the fundamental region. Thus, α is an imaginary root.

As a consequence we have

Corollary 6.3. If Q is Dynkin (or is a disjoint union of Dynkin quivers) then Π^{λ}(Q) ≠ 0 if and only if λ is on a wall.

Proof. If Q is a Dynkin quiver then Π^{λ}(Q) is finite-dimensional. Now, if λ is not on a wall, then since Dynkin quivers have no imaginary roots, Π^{λ}(Q) has no finitedimensional indecomposable representations. Thus, it must be zero.

Note that this is in complete contrast to extended Dynkin case, when Π^{λ}(Q) is always non-zero; see Corollary 3.6. For later use we have the following consequence.

Corollary 6.4. Let Q be an arbitrary quiver, λ ∈ K^{I} and J ⊂ I. Suppose that the Q′, the full subquiver of Q on J, is a disjoint union of Dynkin quivers. If e = ∑_{i∉J} e_{i} then the projective module Π^{λ}(Q)e is a generator if and only if the restriction λ′ of λ to J is not on a wall for Q′. In this case Π^{λ}(Q) is Morita equivalent to eΠ^{λ}(Q)e.

Proof. This follows from the isomorphism Π^{λ}(Q)/Π^{λ}(Q)eΠ^{λ}(Q) ≅ Π^{λ′}(Q′).

7. Dominance

In this section K is a field of characteristic zero, and we fix a total ordering ≺ on K satisfying

1. if a ≺ b then a + c ≺ b + c,

2. on integers ≺ coincides with the usual order, and

3. for any a ∈ K there is m ∈ ℤ with a ≺ m. For example, if B is a ℚ-basis of K which includes the element 1, choose a total ordering of B with the element 1 minimal, and use the lexicographic order, so that ∑ α_{b}b ≺ ∑ β_{b}b if and only if α_{a} < β_{a} for some a ∈ B, and α_{b} = β_{b} for all b < a. If K = ℂ one can instead use the order with z ≺ z′ if and only if either ℜz < ℜz′ or ℜz = ℜz′ and ℑz < ℑz′.

We say that a weight λ ∈ K^{I} is dominant if λ_{i} ⪰ 0, for all i ∈ I. If λ is dominant, or more generally a non-zero scalar multiple of a dominant weight, we define I_{λ} = {i ∈ I | λ_{i} = 0} and write Q_{λ} for the full subquiver of Q with vertex set I_{λ}. If i ∈ I_{λ} is a loopfree vertex we denote by S_{i} the simple module for Π^{λ} with dimension vector ϵ_{i}, as in Lemma 6.1.

Lemma 7.1. If λ is dominant then any finite-dimensional Π^{λ}-module has support contained in I_{λ}, so is a module for the algebra Π(Q_{λ}). In particular:

1. If Q is a disjoint union of Dynkin quivers then Π^{λ} ≅ Π(Q_{λ})

2. If Q_{λ} is a disjoint union of Dynkin quivers then the S_{i}, with i ∈ I_{λ}, are the finite-dimensional simple Π^{λ}-modules.

Proof. By Lemma 4.1, the dimension vector α of a finite-dimensional representation satisfies ∑_{i} α_{i}λ_{i} = 0. Now all λ ⪰ 0 so the sum is also ⪰ 0. This is impossible unless all terms in the sum vanish, which means that α_{i} = 0 whenever λ_{i} ≠ 0.

Now, if Q is a disjoint union of Dynkin quivers then Π^{λ}(Q) is finite-dimensional, so all modules are locally finite-dimensional, while if Q_{λ} is a disjoint union of Dynkin quivers then Π(Q_{λ}) is finite-dimensional so its simple modules are the S_{i}.

We now use reflections to transfer results known for dominant weights to general weights λ ∈ K^{I}. This is possible if either Q is a Dynkin quiver, or Q is an extended Dynkin quiver and λ · δ ≠ 0. In this case we write R_{λ} for the set of roots α with λ · α = 0, a finite set. If α ∈ R_{λ}, let s_{α} denote the automorphism of ℤ^{I} given by β ↦ β − (α, β)α, and let W_{λ} be the subgroup of W generated by these s_{α}. Let Σ_{λ} be the set of minimal positive elements of R_{λ}. Our next result is not explicitly in the literature. However, it can be fairly easily deduced from the results of [19] (see also, [17]) and so we leave its verification to the reader.

Lemma 7.2. Let Q be either Dynkin or extended Dynkin and let λ ∈ K^{I}. In the latter case suppose that λ · δ ≠ 0.

1. If Q is Dynkin then λ is conjugate under the Weyl group to a unique dominant weight.

2. If Q is extended Dynkin then λ/λ · δ is conjugate under the Weyl group to a unique dominant weight λ^{+}. Furthermore, there exists a unique element w^{+} ∈ W of minimal length with w^{+}(λ/λ · δ) = λ^{+}.

3. R_{λ} is the set of roots for a reduced root system (in the vector space it generates). W_{λ} acts faithfully on R_{λ} and identifies with the Weyl group for this root system.

4. Σ_{λ} is the unique basis of R_{λ} consisting of positive roots. Further, w^{+}Σ_{λ} = Σ_{λ^{+}}.

5. W_{λ} is the stabiliser of λ in W.

6. If λ is dominant then Σ_{λ} = {ϵ_{i} | i ∈ I_{λ}} and Q_{λ} is a disjoint union of Dynkin quivers.

Lemma 7.1 and Corollary 5.2 imply for Dynkin quivers:

Theorem 7.3. If Q is a disjoint union of Dynkin quivers and λ ∈ K^{I} then Π^{λ} is Morita equivalent to Π(Q′) for some full subquiver Q′ of Q.

For example, if Q is the quiver D_{4} with a central vertex 1 connected to three other vertices 2,3,4, then Π(Q) has dimension 28. We show that there is no λ such that Π^{λ}(Q) is semisimple of dimension 28. Namely, if its simple modules have dimension d_{j} then ∑_{j} d_{j}^{2} = 28, so there must be at least 4 simple modules. On the other hand if Π(Q′) is semisimple then Q′ has no arrows, so it has at most 3 vertices. (This result requires K to have characteristic zero. For example, if K has characteristic two and λ = (0, 1, 1, 1) then there are easily seen to be simple Π^{λ}- modules of dimensions (1, 0, 0, 0), (1, 1, 1, 0), (1, 1, 0, 1) and (1, 0, 1, 1). This implies that there is a surjective map Π^{λ} → K × M_{3}(K) × M_{3}(K) × M_{3}(K) and then this is an isomorphism by dimensions.)

For extended Dynkin quivers the same argument gives a description of the simple modules.

Theorem 7.4. If Q is extended Dynkin and λ·δ ≠ 0 then the isomorphism classes of finite-dimensional simple Π^{λ}-modules are in 1–1 correspondence with the set Σ_{λ}, under the assignment sending a simple module to its dimension vector.

Proof. This holds in case λ is dominant, for then Σ_{λ} can be identified with I_{λ}, and Q_{λ} is a disjoint union of Dynkin diagrams, so Lemma 7.1 applies.

For general λ with λ · δ ≠ 0 we may by rescaling suppose that λ · δ = 1. Then w^{+}λ = λ^{+}, by Lemma 7.2, so by Corollary 5.2, there is an equivalence from Π^{λ}- modules to Π^{λ^{+}}-modules which acts as w^{+} on dimension vectors. The assertion follows since w^{+}Σ_{λ} = Σ_{λ^{+}}.

Henceforth in this situation, if α ∈ Σ_{λ} we write S_{α} for the unique simple Π^{λ}- module with dimension vector α, and we write 𝒥(α) for its annihilator in Π^{λ}.

Corollary 7.5. If Q is extended Dynkin and λ · δ ≠ 0 then

Spec Π^{λ} = {𝒥(α) | α ∈ Σ_{λ}} ∪ {0}.

Proof. Π^{λ} is prime of GK dimension two, by Corollary 3.6. Suppose that J is a non-zero prime ideal. If J has finite codimension then J = 𝒥(α), for some α ∈ Σ_{λ}. Otherwise, Π^{λ}/J has GK dimension one. But this implies that Π^{λ}/J is PI and so has many finite-dimensional modules, contradicting Theorem 7.4.

If λ · δ ≠ 0 and S is a subset of Σ_{λ} we write 𝒥(S) = ∩_{α∈S}𝒥(α).

Corollary 7.6. Suppose that Q is extended Dynkin and λ · δ ≠ 0.

1. If S ⊆ Σ_{λ} then there is a unique minimal ideal of Π^{λ} with radical 𝒥(S), denoted by 𝒦(S). Moreover Π^{λ}/𝒦(S) is Morita equivalent to Π(Q′) for some full subquiver Q′ of Q.

2. 𝒦_{min} = 𝒦(Σ_{λ}) is the unique minimal non-zero ideal in Π^{λ}.

Proof. By Corollary 5.2 we may suppose that λ is dominant. Then S can be identified with proper subset of I. Define 𝒦(S) = ∑_{j∈I\S} Π^{λ}e_{j}Π^{λ}. Thus Π^{λ}/𝒦(S) is the preprojective algebra on the full subquiver Q′ of Q with vertex set S. Since this is finite dimensional it follows that the radical of 𝒦(S) is 𝒥(S). Now, if J is any ideal with radical 𝒥(S) then for r sufficiently large, J ⊇ 𝒥(S)^{r} ⊇ 𝒦(S)^{r} = 𝒦(S).

We can now characterise when Π^{λ} is a simple ring.

Theorem 7.7. If Q is extended Dynkin then Π^{λ} is simple if and only if it has no non-zero finite-dimensional modules if and only if λ · α ≠ 0 for all roots α.

Proof. If λ is on a wall then Theorem 6.2 says that Π^{λ} has a non-zero finitedimensional representation, while if λ · δ = 0 then Theorem 4.3 and Lemma 4.4 imply that Π^{λ} has a non-zero finite-dimensional representation. Thus, in both cases, Π^{λ} is not simple.

On the other hand, if λ is not on a wall then, by Theorem 6.2, the only possibility for an indecomposable finite-dimensional representation of Π^{λ} is one with dimension vector an imaginary root, so a multiple of δ, but this is impossible by the trace condition. Now the result follows from the general fact that a finitely generated, prime, Noetherian K-algebra R of GK dimension two and with no non-zero, finitedimensional modules must be simple. To see this note that if J is a non-zero ideal then R/J has GK dimension at most one. Thus, by [35], R/J is PI. In particular, either R/J = 0 or R has non-zero finite-dimensional modules.

Now suppose that Q is extended Dynkin with extending vertex 0, and let W^{∘} be the subgroup of W generated by the simple reflections s_{i} with i ≠ 0. It can be identified with the Weyl group of the Dynkin quiver obtained by deleting 0. The first part of Lemma 7.2 implies:

Lemma 7.8. Any λ ∈ K^{I} is conjugate under W^{∘} to a weight μ = (μ_{0}, …, μ_{n}) such that (μ_{1}, …, μ_{n}) is a dominant weight for the Dynkin part of Q.

This can be used to study weights λ with λ · δ = 0, and also to study the ring e_{0}Π^{λ}e_{0}, for we have:

Lemma 7.9. If w ∈ W^{∘} then e_{0}Π^{λ}e_{0} is Morita equivalent to e_{0}Π^{wλ}e_{0}

Proof. This follows from Corollary 5.2 since the reflection functor at a vertex i ≠ 0 sends modules M with e_{0}M = 0 to modules of the same form.

8. The case when λ · δ = 0

Henceforth K is an algebraically closed field of characteristic zero and Γ is a non-trivial finite subgroup of SL_{2}(K). Notation is as in the introduction, so μ is the map Rep(\overline{Q}, δ) → End(δ)_{0} and 𝔥 is the hyperplane \{λ ∈ K^{I} | λ · δ = 0\}. Recall that if λ ∈ 𝔥 then λ can be identified with the element (λ_{i}1_{K^{δ_{i}}}) in the centre of End(δ)_{0}, and Rep(Π^{λ}, δ) = μ^{−1}(λ), with its natural scheme structure. Now the coordinate ring of Rep(\overline{Q}, δ) is the polynomial ring

K[s_{apq} | a ∈ \overline{Q}, 1 ≤ p ≤ δ_{h(a)}, 1 ≤ q ≤ δ_{t(a)}]

where the indeterminate s_{apq} picks out the p, q entry of the matrix of x_{a}, corresponding to x ∈ Rep(\overline{Q}, δ). The next lemma is elementary.

Lemma 8.1. The scheme Rep(Π^{λ}, δ) has coordinate ring K[s_{apq}]/J_{λ} where J_{λ} is generated by the elements

∑_{\begin{matrix}a∈Q \\ h(a)=i \end{matrix}} ∑_{r=1}^{δ_{t(a)}} s_{apr}s_{a^{∗}rq} − ∑_{\begin{matrix}a∈Q \\ t(a)=i \end{matrix}} ∑_{r=1}^{δ_{h(a)}} s_{a^{∗}pr}s_{arq} − δ_{pq}λ_{i}

for each vertex i and for 1 ≤ p, q ≤ δ_{i} (where δ_{pq} is the Kronecker delta function).

Letting ℓ = ∑_{i} δ_{i}, the rows and columns of the matrix ring M_{ℓ}(K[s_{apq}]) can be indexed by pairs i, p where i is a vertex and 1 ≤ p ≤ δ_{i}. Then there is a natural ring homomorphism K\overline{Q} → M_{ℓ}(K[s_{apq}]) sending an arrow a to the matrix whose entries are the relevant s_{apq}. By the lemma this homomorphism descends to a map Π^{λ} → M_{ℓ}(K[Rep(Π^{λ}, δ)]). Now δ_{0} = 1, so this restricts to a homomorphism e_{0}Π^{λ}e_{0} → K[Rep(Π^{λ}, δ)]. Moreover, elements in the image of this map are invariant under the action of GL(δ), so it gives a map ϕ_{λ} : 𝒪^{λ} → A^{λ} where in this section we identify 𝒪^{λ} = e_{0}Π^{λ}e_{0} and we define

A^{λ} = K[Rep(Π^{λ}, δ)]^{GL(δ)} = (K[s_{apq}]/J)^{GL(δ)}.

We prove Theorem 2 by showing that ϕ_{λ} is an isomorphism. Our strategy is to first prove it for generic elements of 𝔥, explicitly for elements of the set

𝔥′ = {λ ∈ 𝔥 | λ · α ≠ 0 for all real roots α},

and then to extend to all elements of 𝔥 using the flatness of μ.

Now 𝒪^{λ} is a subring of Π^{λ}, so inherits its filtration. Also A^{λ} inherits a filtration from K[Rep(Π^{λ}, δ)] ≅ K[s_{apq}]/J, in which the s_{apq} have degree 1.

Lemma 8.2. The map ϕ_{λ} : 𝒪^{λ} → A^{λ} preserves the filtrations, and there is a commutative square

𝒪^{0} ⟶^{ϕ_{0}} A^{0}
↓  ↓
gr 𝒪^{λ} ⟶^{gr ϕ_{λ}} gr A^{λ}

where the left hand vertical map is an isomorphism and the right hand vertical map is surjective.

Proof. By Corollary 3.6 there is a natural isomorphism Π → gr Π^{λ}, and this induces an isomorphism 𝒪^{0} → gr 𝒪^{λ}. It is clear from Lemma 8.1 that there is a surjective ring homomorphism K[Rep(Π, δ)] → gr K[Rep(Π^{λ}, δ)], and since GL(δ) is linearly reductive this induces a surjection A^{0} → gr A^{λ}. Finally it is easy to check that ϕ_{λ} preserves the filtrations and that the square commutes.

Lemma 8.3. The map μ : Rep(\overline{Q}, δ) → End(δ)_{0} is surjective and every irreducible component of every fibre has dimension 1 + ∑_{i} δ_{i}^{2}. Thus, μ is flat.

Proof. By Lemma 4.4, and the argument of Theorem 4.3, the map μ is surjective. Thus, every irreducible component of every fibre has dimension at least

dim Rep(\overline{Q}, δ) − dim End(δ)_{0} = 1 + ∑_{i} δ_{i}^{2}.

For any θ ∈ End(δ)_{0} we analyse the projection

π : μ^{−1}(θ) → Rep(Q, δ)

further. By Lemma 4.2, the fibre of π over any point x is either empty, or of dimension dim Ext^{1}(X, X) where X is the representation of Q corresponding to x. Since ⟨δ, δ⟩ = 0 this dimension is the same as dim End(X). Now, Rep(Q, δ) is the union of a dense open subset Y consisting of the elements for which the representation has endomorphism algebra K, and a finite number of orbits O_{1}, …, O_{k}. The fibres of the map π^{−1}(Y) → Y all have dimension 1, so

dim π^{−1}(Y) = 1 + dim Y = 1 + dim Rep(Q, δ) = 1 + ∑_{a:i→j} δ_{i}δ_{j} = 1 + ∑_{i} δ_{i}^{2}

since ⟨δ, δ⟩ = 0. On the other hand, if x_{i} ∈ O_{i} and X_{i} is the corresponding representation, then π^{−1}(O_{i}) is either empty or of dimension

dim O_{i} + dim End(X_{i}) = dim GL(δ) = ∑_{i} δ_{i}^{2}

since End(X_{i}) has the same dimension as the automorphism group of X_{i}, and this is the stabiliser of x_{i} in GL(δ). Thus, dim μ^{−1}(θ) = 1 + ∑_{i} δ_{i}^{2}. It follows that μ is flat; see, for example, [11, Theorem 18.16].

In case K = ℂ the flatness of μ was already known—Kronheimer [22] reduced it to a conjecture about the dimension of a certain null cone, and this conjecture was subsequently proved by Lusztig [25, Theorem 12.3].

Lemma 8.4. Rep(Π^{λ}, δ) // GL(δ) has dimension at least 2 for λ ∈ 𝔥′.

Proof. Since λ ∈ 𝔥′ any representation of Π^{λ} of dimension vector δ is simple, for since δ_{0} = 1, one of the composition factors has dimension α with α_{0} = 0. It is then a representation for the deformed preprojective algebra Π^{λ′}(Q′) where λ′ and Q′ are obtained by deleting the vertex 0. But this algebra is zero.

By the representation theory of Q, cf [8], there is a one-dimensional subvariety C of Rep(Q, δ) consisting of non-isomorphic representations with endomorphism algebra K. Now, if π is the map used in the previous lemma, then π^{−1}(C) is a 2-dimensional subscheme of Rep(Π^{λ}, δ) consisting of pairwise non-isomorphic representations.

Finally, recall that Rep(Π^{λ}, δ) // GL(δ) parametrizes the isomorphism classes of semisimple Π^{λ}-modules of dimension δ, see for example [20, II.2.7 Satz 3 and II.3.2 Bemerkung 1]. Thus, it has dimension at least 2.

Lemma 8.5. ϕ_{λ} is an isomorphism for λ ∈ 𝔥′.

Proof. If x = (x_{a})_{a∈\overline{Q}} ∈ Rep(\overline{Q}, δ) and p = a_{1} … a_{m} is a path in K\overline{Q} which starts and ends at the same vertex then the function tr_{p} given by

tr_{p}(x) = tr(x_{a_{1}} … x_{a_{m}})

is invariant under the action of GL(δ), and, by [24], the algebra K[Rep(\overline{Q}, δ)]^{GL(δ)} is generated by the trace functions tr_{p}. Since GL(δ) is a reductive group and the field has characteristic zero, it follows that K[Rep(Π^{λ}, δ)]^{GL(δ)} is also generated by the trace functions.

Since λ ∈ 𝔥′, one has Π^{λ}e_{0}Π^{λ} = Π^{λ}, so any element of e_{i}Π^{λ}e_{i} can be written as a sum of paths p which pass through vertex 0. Furthermore, it is clear that a cyclic permutation of a path gives rise to the same trace function. Thus, K[Rep(Π^{λ}, δ)]^{GL(δ)} is generated by the trace functions for paths starting and ending at the vertex 0. But this is exactly Im(ϕ_{λ}), so ϕ_{λ} must be surjective.

Now, ϕ_{λ} is an isomorphism, for since 𝒪^{λ} is a domain of GK dimension 2, any proper quotient has GK dimension at most 1, but, by the previous lemma, the ring of invariants has dimension at least 2.

We now consider a global situation. Let R be a commutative ring and let λ ∈ R^{I} be an element with λ · δ = 0. Consider the fibre product

Y_{R,λ} ⟶ Spec R
↓  ↓λ̂
Rep(\overline{Q}, δ) ⟶^{μ} End(δ)_{0}

where λ̂ : Spec R → 𝔥 is the map corresponding to λ. Now K[Y_{R,λ}] = R[s_{apq}]/J_{λ}, where J_{λ} is the ideal generated by elements as in Lemma 8.1, except that the λ_{i} belong to R. Observe that GL(δ) acts naturally on Y_{R,λ}, in such a way that all maps in the fibre product are equivariant (where the action on Spec R is trivial). We define A^{R,λ} = K[Y_{R,λ}]^{GL(δ)} and we identify 𝒪^{R,λ} = e_{0}Π^{R,λ}e_{0}. This time there is a natural ring homomorphism R\overline{Q} → M_{ℓ}(R[s_{apq}]), which first descends to a homomorphism Π^{R,λ} → M_{ℓ}(K[Y_{R,λ}]), and then restricts to a homomorphism ϕ_{R,λ} : 𝒪^{R,λ} → A^{R,λ}. Observe that if f : R → S is a ring homomorphism then

𝒪^{S,f(λ)} ≅ 𝒪^{R,λ} ⊗_{R} S, A^{S,f(λ)} ≅ A^{R,λ} ⊗_{R} S, and ϕ_{S,f(λ)} = ϕ_{R,λ} ⊗ 1.

Lemma 8.6. A^{R,λ} is flat over R. (Equivalently Y_{R,λ} // GL(δ) → Spec R is flat.)

Proof. Y_{R,λ} → Spec R is flat since it is the pullback of μ, which is flat, by Lemma 8.3. Then A^{R,λ} is flat over R, for, by the Reynolds operator [30], it is a direct summand of K[Y_{R,λ}].

Lemma 8.7. If R is a domain with quotient field F, and λ·α ≠ 0 for all real roots α, then

ϕ_{R,λ} ⊗ 1 : 𝒪^{R,λ} ⊗ F → A^{R,λ} ⊗ F

is an isomorphism. In particular, A^{R,λ} is a domain.

Proof. It suffices to show that the map is an isomorphism on tensoring with an algebraic closure \overline{F} of F, but now, Lemma 8.5 applies with the base field K replaced by \overline{F}. Finally, by flatness, A^{R,λ} embeds in A^{R,λ} ⊗F, and this is isomorphic to 𝒪^{F,λ}, so is a domain.

Lemma 8.8. Rep(Π^{λ}, δ) // GL(δ) has dimension 2 for any λ ∈ 𝔥.

Proof. Take Spec R = 𝔥 and let λ be the canonical element. By the previous lemma, the map Y_{R,λ} // GL(δ) → 𝔥 is a morphism of irreducible varieties, and it is flat and surjective. Thus, all fibres have the same dimension. Since GL(δ) is linearly reductive the fibre over λ is Rep(Π^{λ}, δ) // GL(δ). Now, a general fibre has dimension 2, by Lemma 8.5.

Lemma 8.9. Suppose R is a Dedekind domain, 𝔪 is a maximal ideal and C is a torsion R-module, not necessarily finitely generated. If Tor_{1}^{R}(C, R/𝔪) = 0 then C ⊗ R/𝔪 = 0.

Proof. Localizing at 𝔪 we can assume that R is a discrete valuation ring. Now, if C ≠ 0 then since it is torsion it has a submodule isomorphic to R/𝔪. Moreover, since R is hereditary, the map Tor_{1}^{R}(R/𝔪, R/𝔪) → Tor_{1}^{R}(C, R/𝔪) is injective. But Tor_{1}^{R}(R/𝔪, R/𝔪) ≠ 0.

Theorem 8.10. The map ϕ_{0} : 𝒪^{0} → A^{0} is an isomorphism.

Proof. We show first that ϕ_{0} is injective, and for this it suffices to prove that the homomorphism F : Π → M_{ℓ}(K[Rep(Π, δ)]) is injective. Now Rep(Π, δ) // GL(δ) has dimension 2, so the semisimple representations of Π of dimension δ depend on 2 parameters. Since these representations all arise by specializing F to points of Rep(Π, δ), it follows that Π/ Ker(F) also has semisimple representations of dimension δ depending on 2 parameters. Now all but one of the composition factors of such a representation has dimension vector vanishing at the extending vertex 0, so is a simple representation for the preprojective algebra of the corresponding Dynkin quiver. But this algebra has only finitely many simple representations. We deduce that Π/ Ker(F) has simple representations (of some dimension) depending on 2 parameters, and it follows that Π/ Ker(F) has dimension at least 2, forcing Ker(F) to be zero.

Now fix some ν ∈ 𝔥′, let R = K[t] and let λ = νt ∈ R^{I}. Since 𝒪^{R,λ} is flat over R and ϕ_{R,λ} gives an isomorphism on tensoring with the quotient field of R, one has an exact sequence

0 → 𝒪^{R,λ} → A^{R,λ} → C → 0.

and C is a torsion R-module. Tensoring this sequence with the module K on which t acts as zero, the first map becomes ϕ_{0}, so remains injective. Thus, Tor_{1}^{R}(C, K) = 0, since A^{R,λ} is flat over R. By the previous lemma, C ⊗ K = 0, so ϕ_{0} is an isomorphism.

In terms of schemes the theorem says that K^{2}/Γ ≅ Rep(Π^{0}, δ) // GL(δ). In case K = ℂ this was known from work of Kronheimer. In his thesis, published as [23], he proved that these two spaces are isometric, and then in [22] he showed that there is an algebraic map.

Corollary 8.11. For any λ ∈ 𝔥 the map ϕ_{λ} : 𝒪^{λ} → A^{λ} is an isomorphism.

Proof. By Lemma 8.2, the map gr ϕ_{λ} : gr 𝒪^{λ} → gr A^{λ} is surjective. Now, A^{λ} has dimension 2 while gr 𝒪^{λ} ≅ 𝒪^{0} is a domain of dimension 2. Thus, gr ϕ_{λ} is an isomorphism, and hence so is ϕ_{λ}.

Corollary 8.12. If R is an integrally closed domain, finitely generated over K, and λ ∈ R^{I} satisfies λ·δ = 0, then the map ϕ_{R,λ} : 𝒪^{R,λ} → A^{R,λ} is an isomorphism.

Proof. By the previous result 𝒪^{F,λ} ≅ A^{F,λ}, where F is the quotient field of R. By flatness 𝒪^{R,λ} is a subring of 𝒪^{F,λ}, so it is commutative, and A^{R,λ} is a subring of A^{F,λ}, so it is a domain. Thus ϕ_{R,λ} induces a birational morphism of irreducible varieties

Spec A^{R,λ} → Spec 𝒪^{R,λ}.

This morphism is a bijection on closed points, for both varieties have a structure map to Spec R, and by the previous result it is an isomorphism on fibres over any maximal ideal of R. Now since 𝒪^{R,λ} is integrally closed by Lemma 1.3, a result of Richardson [20, §II.3.4] implies that the morphism is an isomorphism.

Under the hypotheses of the last corollary it follows that A^{R,λ} is an integrally closed domain, so the quotient scheme Y_{R,λ} // GL(δ) is normal. In particular, taking R to be the coordinate ring of 𝔥 and λ the canonical element, it follows that μ^{−1}(𝔥) // GL(δ) is normal. In case K = ℂ this variety is isomorphic to the variety Y of [23, §4], so Y is normal. (Actually Y was already known to be normal, as P. Slodowy has shown us another argument based only on [23] and the flatness of μ.) Thus there is no need to take the normalization in [23, Lemma 4.3], so the very last remark in Kronheimer’s paper says that (in the category of complex analytic spaces) the map μ^{−1}(𝔥) // GL(δ) → 𝔥 is obtained from the semi-universal deformation of ℂ^{2}/Γ by lifting through the Weyl group.

9. Properties of 𝒪^{λ}

In this section we prove Theorems 3 and 4. For simplicity of notation we fix an isomorphism between 𝒪^{λ} and e_{0}Π^{λ}e_{0}, and henceforth identify 𝒪^{λ} = e_{0}Π^{λ}e_{0}. We also obtain a number of results for 𝒮^{λ} (and hence, by Morita equivalence, for Π^{λ}).

Theorem 9.1. If λ · δ = 0 then 𝒪^{λ} is commutative and 𝒮^{λ} is a polynomial identity ring. If λ · δ ≠ 0 then neither is a polynomial identity ring.

Proof. By Lemma 1.4 and [27, Corollary 13.4.9], if 𝒪^{λ} is a PI ring then so is 𝒮^{λ}. Now, if λ · δ = 0 then we know that 𝒪^{λ} is commutative, while if λ · δ ≠ 0 then we know from Theorem 7.4 that 𝒮^{λ} has only finitely many finite-dimensional simple modules and so is not a PI ring.

We now compute the finite-dimensional simple modules for 𝒪^{λ} in case λ · δ ≠ 0. Note that we cannot reduce to the case when λ is dominant, since the Morita equivalence between Π^{λ} and Π^{wλ} does not, in general, induce a Morita equivalence between 𝒪^{λ} and 𝒪^{wλ}. Define Υ_{λ} = {α ∈ Σ_{λ} | α_{0} ≠ 0}.

Theorem 9.2. If λ · δ ≠ 0 then 𝒪^{λ} has a unique minimal non-zero ideal, and this ideal has finite codimension. There is a 1–1 correspondence between the finitedimensional simple 𝒪^{λ}-modules and Υ_{λ}, with α corresponding to the module e_{0}S_{α}.

Proof. Let ℐ_{min} = e_{0}𝒦_{min}e_{0} where 𝒦_{min} is as in Corollary 7.6. Then 𝒪^{λ}/ℐ_{min} ≅ e_{0} (Π^{λ}/𝒦_{min}) e_{0} is finite dimensional. Thus ℐ_{min} is non-zero, and it follows that it is the unique minimal non-zero ideal of 𝒪^{λ}. Also 𝒪^{λ}/ℐ_{min} has semisimple top e_{0}(Π^{λ}/𝒥(Σ_{λ}))e_{0}, and this implies that there is a bijection from

{S | S is a simple Π^{λ}/𝒥(Σ_{λ})-module and e_{0}S ≠ 0}

to the set of simple modules for e_{0}(Π^{λ}/𝒥(Σ_{λ}))e_{0} given by S ↦ e_{0}S.

Corollary 9.3. Spec 𝒪^{λ} = {Ann(e_{0}S_{α}) | α ∈ Υ_{λ}} ∪ {0}.

Proof. Clear.

Corollary 9.4. 𝒪^{λ} is simple if and only if λ · α ≠ 0 for all non-Dynkin roots α.

Proof. Clearly, if λ · α ≠ 0 for all non-Dynkin roots then Υ_{λ} = ∅. On the other hand, we can use Lemma 7.2 to see that if λ · α = 0, for some root α, then either α or −α is a non-negative integer combination of β ∈ Σ_{λ}. Thus, if Υ_{λ} is empty then α is a Dynkin root.

Theorem 9.5. 𝒪^{λ} has infinite global dimension if and only if λ · α = 0 for some Dynkin root α.

Proof. Note first that if λ doesn’t kill any Dynkin root then 𝒪^{λ} is Morita equivalent to 𝒮^{λ}, by Corollary 6.4. In particular, by Theorem 1.5 , it has finite global dimension.

On the other hand, suppose that 𝒪^{λ} has finite global dimension. We know already that this forces 𝒪^{λ} to have global dimension one or two. But, by Lemma 1.4, e𝒮^{λ} is a reflexive 𝒪^{λ}-module and so it will be projective. This module also has 𝒪^{λ} as a summand and it follows that e𝒮^{λ} is a progenerator with endomorphism ring 𝒮^{λ}. By standard theory, this means that e𝒮^{λ} is also a progenerator considered as a right module over 𝒮^{λ}. But we know that this condition forces that λ doesn’t annihilate any Dynkin root.

Corollary 9.6. 𝒪^{λ} is Morita equivalent to 𝒮^{λ} if and only if λ·α ≠ 0 for all Dynkin roots α.

Proof. We know from Corollary 6.4 that the condition ‘λ · α ≠ 0 for all Dynkin roots α’ is equivalent to the 𝒮^{λ}-module e𝒮^{λ} being a progenerator.

On the other hand, any Morita equivalence between 𝒪^{λ} and 𝒮^{λ} ensures that 𝒪^{λ} has global dimension one or two, by virture of Theorem 1.5. Now we can argue exactly as in the second paragraph of the proof of the preceding result.

Corollary 9.7. 𝒪^{λ} is hereditary if and only if λ · α ≠ 0 for all roots α.

Proof. Suppose that 𝒪^{λ} has finite global dimension. By Theorem 1.6, 𝒪^{λ} has global dimension two (or one) and that these possibilities occur exactly when 𝒪^{λ} does (does not) have a non-zero finite-dimensional module. The result follows from the previous theorem and Theorem 9.5.

Our next result computes the Krull dimension, in the sense of Gabriel and Rentschler, of 𝒮^{λ} and 𝒪^{λ}.

Theorem 9.8. If λ · δ = 0 then the rings 𝒮^{λ} and 𝒪^{λ} have Krull dimension two. If λ · δ ≠ 0 then the rings 𝒮^{λ} and 𝒪^{λ} have Krull dimension one.

Proof. If λ · δ = 0 then these are PI rings and so their Krull dimension coincides with their GK dimension (see [27, Proposition 13.10.6]).

Now, suppose that λ · δ ≠ 0. Note that as 𝒪^{λ} = e𝒮^{λ}e, and 𝒪^{λ} is not artinian, the result for 𝒪^{λ} is an easy consequence of the result for 𝒮^{λ}. Let E be an essential right ideal of 𝒮^{λ} and write M = 𝒮^{λ}/E. It is enough, by virtue of [27, Proposition 6.3.10], to show that such an M is artinian. Now, GK M ≤ 1, by [27, Corollary 8.3.6]. Suppose that M_{0} ⊃ M_{1} ⊃ ⋯ is a descending chain of submodules of M. If any M_{i} is finite dimensional then obviously the chain is finite, so we assume that all the M_{i} are infinite dimensional. Passing to associated graded modules we have gr M_{0} ⊃ gr M_{1} ⊃ ⋯ and each gr M_{i} is a finitely generated, graded module over the polynomial ring K[x, y]. As such it has a Hilbert polynomial. This gives rise to a multiplicity, e(gr M_{i}), say. Further, recall that multiplicty is additive on short exact sequences all of whose terms have fixed GK dimension. But all the M_{i} are infinite dimensional; which means that all the gr M_{i} have GK dimension one. It follows immediately that all but finitely many of the M_{i}/M_{i+1} are finite dimensional. Thus, in showing that the chain M_{0} ⊃ M_{1} ⊃ ⋯ is finite, we may assume that each M_{i}/M_{i+1} is finite dimensional. This means that 𝒦M_{i} ⊆ M_{i+1}, where 𝒦 is the unique minimal ideal of 𝒮^{λ}. Since 𝒦 is idempotent this forces that 𝒦M_{0} ⊆ ∩M_{i}. It follows that M_{0}/ ∩ M_{i} is finite dimensional, and so the chain is finite.

10. K-theory

Let us make some remarks about the Grothendieck groups of 𝒮^{λ} and 𝒪^{λ}. For a Noetherian ring R we denote by G_{i}(R) the ith Quillen K-group for the category of finitely generated R-modules and by K_{i}(R) the ith Quillen K-group for the category of finitely generated, projective R-modules.

Lemma 10.1. K_{i}(𝒮^{λ}) = G_{i}(𝒮^{λ}) ≅ K_{i}(K)^{I}. In particular, if λ · α ≠ 0 for all Dynkin roots α then K_{i}(𝒪^{λ}) = G_{i}(𝒪^{λ}) ≅ K_{i}(K)^{I}.

Proof. Since gr 𝒮^{λ} = K[x, y] ∗ Γ, one can apply Quillen’s result [32, §6, Theorem 7] to discover that G_{i}(𝒮^{λ}) = G_{i}((K[x, y] ∗ Γ)_{0}), where _{0} denotes the degree zero part. But this is the semisimple algebra KΓ. Now, if λ · α ≠ 0 for all Dynkin roots α then 𝒮^{λ} and 𝒪^{λ} are Morita equivalent, by Corollary 6.4.

So, in considering the Grothendieck groups of 𝒪^{λ}, the interesting case is when λ is such that λ·α = 0, for some Dynkin root α. Write μ = (λ_{1}, …, λ_{n}). By Lemma 7.9, we may suppose that that μ is dominant. Quillen’s localisation sequence [32, §5, Theorem 5] gives us a short exact sequence

G_{0}(Π^{μ}) → G_{0}(Π^{λ}) → G_{0}(𝒪^{λ}) → 0.

Let t_{i} denote the class of Π^{λ}e_{i} in G_{0}(Π^{λ}). Thus, G_{0}(Π^{λ}) is the free abelian group with basis t_{0}, …, t_{n}. If ϵ_{i} ∈ Σ_{μ} then the simple Π^{μ}-module S_{i} has a resolution over Π^{λ} given by

0 → Π^{λ}e_{i} → ⨁_{a : j → i in \overline{Q}} Π^{λ}e_{j} → Π^{λ}e_{i} → S_{i} → 0.

(To prove this note that one does have such a filtered complex. But now the associated graded complex is easily seen to be exact.) Thus, we see that the image of S_{i} in G_{0}(Π^{λ}) is 2t_{i} − ∑_{a:j→i} t_{j}.

Theorem 10.2. Suppose that λ = (λ_{0}, …, λ_{n}) is such that λ · α = 0, for some Dynkin root α. Suppose (without loss of generality) that μ = (λ_{1}, …, λ_{n}) is dominant. Then

G_{0}(𝒪^{λ}) = (ℤt_{0} ⊕ ⋯ ⊕ ℤt_{n}) / (2t_{i} − ∑_{a : j → i in \overline{Q}} t_{j} | i ≠ 0, λ_{i} = 0).

In particular, G_{0}(𝒪^{λ}) has rank n + 1 − |Σ_{μ}|.

Proof. The only thing left to prove is the statement about the rank. Note that is is equivalent to saying that the elements ξ_{1}, …, ξ_{n}, are linearly independent, where ξ_{i} = 2t_{i} −∑_{a:j→i} t_{j}. This is probably obvious combinatorially, but can also be seen from the fact that G_{0}(K[x, y]^{Γ}) has rank one (see [4]).

Now, we consider K_{0}(𝒪^{λ}) in the case when λ · α = 0 for some Dynkin root α.

Theorem 10.3. Suppose that λ = (λ_{0}, …, λ_{n}) is such that λ · α = 0 for some Dynkin root α. Suppose (without loss of generality) that μ = (λ_{1}, …, λ_{n}) is dominant. Then K_{0}(𝒪^{λ}) is free of rank n + 1 − |Σ_{μ}|. Further, the Cartan map c : K_{0}(𝒪^{λ}) → G_{0}(𝒪^{λ}) is a rational isomorphism.

Proof. We begin by showing that K_{0}(𝒪^{λ}) is free of rank ≤ n + 1 − |Σ_{μ}|. Recall (see [16, §2.3]) that there is a natural map Φ : K_{0}(𝒪^{λ}) → K_{0}(Π^{λ}), defined by [P] ↦ [Π^{λ}e_{0} ⊗_{𝒪^{λ}} P]. Further, Im Φ ⊆ Ker K_{0}(π), where π : Π^{λ} → Π^{λ}/Π^{λ}e_{0}Π^{λ} is the natural map. Now, Π^{λ}/Π^{λ}e_{0}Π^{λ} = Π^{μ} ≅ Π(Q_{μ}) where Q_{μ} is the full subquiver of Q with vertex set Σ_{μ}. It follows immediately that K_{0}(π) is surjective. Thus, Im Φ has rank at most n + 1 − |Σ_{μ}|.

On the other hand, by the main result of [16], if K_{1}(π) is surjective then Φ is injective. But, writing S for the subalgebra of Π^{λ} spanned by the trivial paths we have that S → Π^{λ} induces an isomorphism in K-theory. On the other hand, we claim that the composed map S → Π^{λ} → Π^{μ} = Π(Q_{μ}) is surjective on K_{1}. Accepting this for a moment, we certainly get that K_{1}(π) is surjective. It is enough, then, to show that K_{1}(Π(Q_{μ})) → K_{1}(S′) is an isomorphism (where S′ is the semisimple top of Π(Q_{μ})). This will certainly follow, if K_{1}(Π(Q_{μ}), J(Π(Q_{μ}))) = 0. But, by Goodwillie’s theorem (see [14], [38, Theorem A.1]), this relative K-group identifies with the shifted relative cyclic homology group HC_{0}(Π(Σ_{μ}), J(Π(Σ_{μ}))). This latter group is a homomorphic image of the corresponding group for the preprojective algebra of the Dynkin part of Q. Finally, this algebra is Morita equivalent to K[x, y] ∗ Γ/⟨e⟩ and so we are done, by (the proof of) [16, Theorem 3.8].

Now, let us consider the Cartan map. The vertex i = 0 or any vertex i ≠ 0 with λ_{i} ≠ 0, satisfies e_{i} = 0 in Π^{λ}/Π^{λ}e_{0}Π^{λ}, and so Π^{λ}e_{i}Π^{λ} ⊆ Π^{λ}e_{0}Π^{λ}. Because of this, e_{0}Π^{λ} ⊗_{Π^{λ}} Π^{λ}e_{i} is a projective 𝒪^{λ}-module (see, for example, [6, Theorem 1.6]). Now, c factors as

K_{0}(𝒪^{λ}) ⟶^{Π^{λ}e_{0}⊗_{𝒪^{λ}} −} K_{0}(Π^{λ}) ⟶^{e_{0}Π^{λ}⊗_{Π^{λ}} −} G_{0}(𝒪^{λ}),

so Im c contains \overline{t_{i}}. Now, it is clear from Theorem 10.2 that these elements generate G_{0}(𝒪^{λ}) modulo torsion.

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Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK

E-mail address: w.crawley-boevey@leeds.ac.uk

School of Mathematics, Pure Mathematics Section, University of Sheffield, Sheffield S3 7RH, UK

E-mail address: m.holland@shef.ac.uk   http://www.shef.ac.uk/˜ ms/staff/holland