Quiver Varieties

Andy Sanders Universit¨atHeidelberg

SoSe 2019 Contents

1 Chapter 1

Introduction

As a motivation, let us consider a compact K¨ahlermanifold M and π = π1(M) its fundamental group. If G denotes a complex reductive Lie group, then the character variety Hom (π, G)/G exhibits various interesting structures such as the structure of a symplectic space, or even the structure of a hyper-K¨ahlerspace. As a matter of fact for now, quiver varieties present the same structure as character varieties is a slightly richer manner. Therefore, anyone interested in the study of character varieties should be curious about quiver varieties. One way to introduce quiver varieties consists in starting from groups representation. Let G ! End (V ) be a representation of some group G in a vector space V . For g ∈ G, write Ag : V ! V for the associated linear transformation. The group G can be drawn as a graph with a single vertex (representing G) and an arrow for every element of the group. The same cartoon can be drawn to illustrate the representation A• : G ! End (V ).

What makes the study of groups with linear representation very hard is the non- linearity arising with the constraints

−1 Ag−1 = (Ag) ,Agh = Ag ◦ Ah, ∀g, h ∈ G.

If the constraints are dropped, the resulting object is a graph (on a single vertex) where each arrow is freely chosen to be any linear transformation of the vector space. Such an object, is called the representation of a quiver. This survey is an introduction to quivers and the study of their representations.

2 Ah−1 Ag−1 V Ag Ah h−1 g−1 G g h

Figure 1.1: Visualization of the linear representation A• : G ! End (V ) as a graph. Observe that the graph does not transcribe the inherent group laws.

3 Chapter 2

Quiver varieties and representations

We start with the deﬁnition of a quiver.

Definition 1 (Quiver). A quiver Q is a finite directed graph (Q0,Q1, s, t) where Q0 is the set of vertices, Q1 is the set of edges and s, t: Q1 ! Q0 are the two functions indicating the source and the target of each edge. Note that multiple edges between the same pair of vertices is allowed in the definition of a quiver. Consider the following first example of a quiver.

Example 2. Let the quiver Q be given by Q0 = {0, 1}, Q1 = {α, β, γ} and s, t being deﬁned as follows: s(α) = 1, t(α) = 2 and s(β) = t(β) = s(γ) = t(γ) = 2. The quiver Q can be drawn as:

α 1 2

Figure 2.1: Illustration of the graph corresponding to the quiver Q.

Another source of examples of quivers are finite groups that give rise to quivers via the graph introduced in Figure ??. The next crucial concept is the definition of a representation of quiver. Let k denote a field. Definition 3 (Representation of a quiver). A k-representation of a quiver Q is a collection of k-vector spaces {Vi}i∈Q0 together with a collection of k-linear maps {fα : Vs(α) !

Vt(α)}α∈Q1 .

4 It was mentioned in introduction that representation of quivers should be thought of group representations where constraints corresponding to the underlying group structures have been dropped. In particular, we emphasize that no conditions of commuta- tivity have been assumed in the deﬁnition of a representation of a quiver. As with any new mathematical object, a notion of morphism shall be deﬁned.

Deﬁnition 4 (Morphism of representation). A morphism between two representations M = ({Vi}, {fα}) and N = ({Wi}, {gα}) of the same quiver Q = (Q0,Q1, s, t) is a collection {ui}i∈Q0 of k-linear maps such that the following diagram commutes for every α in Q1.

fα Vs(α) Vt(α)

us(α) ut(α)

gα Ws(α) Wt(α)

Such a representation is denoted u: M ! N. Immediate properties of morphisms of quiver representations can be derived from the deﬁnition. Let M = ({Vi}, {fα}),N,P be representations of a quiver Q = (Q0,Q1, s, t).

• Given two morphisms u: M ! N and v : N ! P , then there is a well-deﬁned composed morphism u ◦ v : M ! P given by the compositions of the underlying linear maps: (u ◦ v)i = ui ◦ vi for every i ∈ Q0. • The composition of quiver representations morphisms is associative because composition of linear maps is associative.

• There is an obvious identity morphism id: M ! M. These three properties deﬁne a category Rep (Q) whose objects are representations of the quiver Q. Observe furthermore that

• two morphisms u: M ! N and v : M ! N give rise to a sum morphism u + v : M ! N deﬁned by (u + v)i = ui + vi for every i ∈ Q0.,

• and that given a morphism u: M ! N, then Ker (u) deﬁnes a new object of

Rep (Q) whose associated vector spaces are the kernels {Ker (ui)}i∈Q0 and asso-

ciated linear maps {fα Ker (us(α))}α∈Q1 are the restrictions to the kernels of the linear maps associated to the representations M.

Cokernel representation can be deﬁned in the same way. Therefore, the category Rep (Q) is an abelian category. Note that the zero object in Rep (Q) is given by the representation of Q that associates the vector space {0} to every vertex. Observe, that

5 Rep (Q), as an abelian category, is thus equivalent to a full subcategory of LeftR-mod for some ring R. From now on, we will be studying representations of quivers in which every vector space has ﬁnite dimension. Let Q = (Q0 = {1,..., |Q0|},Q1, s, t) be a quiver and M = ({Vi}, {fα}) be a representation of Q such that dim k(Vi) < ∞ for all i ∈ Q0. Deﬁnition 5 (Dimension vector). The dimension vector of M is

dim (M) := (n1, . . . , n|Q0|), ni = dim k(Vi), i ∈ Q0.

For a choice of nonegative integers n1, . . . , n|Q |, denote by Rep (n ,...,n )(Q) the sub- 0 1 |Q0| category of Rep (Q) of representations with dimension vector (n1, . . . , n|Q0|). One of the major goals in the study of quiver varieties is to classify the representations with a ﬁxed dimension vector.

Example 6. Consider the following quiver Q with one vertex and one edge:

1 α

A representation M of Q is simply a pair (f, V ) of a k-vector space V and a linear map f : V ! V . Further, an isomorphism between two representations (f, V ) and (g, W ) is simply a linear isomorphism u: V ! W such that g = ufu−1. In particular, if Mn×n(k) denotes the n × n-matrices with entries in k, then to every n matrix A ∈ Mn×n(k) we can associate the representation (A, k ) of Q. More precisely, if we introduce the category Mn×n(k) induced by the action by conjugation of GLn(k) on Mn×n(k), i.e. the category whose objects are the matrices Mn×n(k) and morphisms −1 between A, B ∈ Mn×n(k) are the elements g ∈ GLn(k) such that B = gAg , then we get a functor F : Mn×n(k) ! Rep n(Q). This functor is faithful. Moreover, it is essentially surjective, as for any repre- ∼ n sentation (f, V ) of Q with dim k(V ) = n, by choosing an isomorphism V = K and conjugating f by the corresponding change of basis, one gets an isomorphic represen- n tation (Af , k ) in the image of F . Nevertheless, the functor F is not necessarily full as the following pair of morphisms needs not be invertible.

kn A kn u u kn A kn

6 Hence, F is not an equivalence of category. However, there is a bijection at the level of isomorphisms induced by F :

Isom (Rep n(Q)) ↔ Isom (Mn×n(Q)).

Observe that isomorphism classes in Mn×n(Q) are nothing but conjugacy classes for the action of GLn(k) on Mn×n(k). Thus Isom (Mn×n(Q)) = Mn×n(k)/GLn(k). Note furthermore that if k is algebraically closed, then such classes are enumerated by Jordan canonical forms. As a last remark, observe that the action of GLn(k) on Mn×n(k) is not proper and therefore the quotient Mn×n(k)/GLn(k) is not Hausdorﬀ and is not an algebraic variety in any reasonable way.

As a moral consequence of Example ??, we emphasize the omnipresence of geometric invariant theory in the study of quiver varieties. Let us consider some more examples.

Example 7. Consider the following quiver Q with one vertex and two edges:

1 α β

Similar to Example ??, a representation M of Q is a k-vector space V and two linear maps fα, fβ : V ! V . Further, an isomorphism between two representations (fα, fβ,V ) −1 and (gα, gβ,W ) is simply a linear isomorphism u: V ! W satisfying gα = ufαu and −1 gβ = ufβu . As in Example ??, we can associate to a pair (A, B) of matrices in n Mn×n(k) a representation (A, B, k ) of Q and obtain a faithful functor

2 F : Mn×n(k) ! Rep n(Q) and a bijection at the level of isomorphism classes

2 Mn×n(k) /GLn(k) ↔ Isom (Rep n(Q)).

Example 8. We can extend Example ?? to a quiver Q with one vertex and r edges:

7 1 α1 α2 ... αr

As in Example ??, we obtain a faithful functor

r F : Mn×n(k) ! Rep n(Q) and a bijection at the level of isomorphism classes

r Mn×n(k) /GLn(k) ↔ Isom (Rep n(Q)).

Example 9. Consider the following quiver Q with three vertices and two edges:

2 1 3 α β

A representation of Q consists of three vectorspaces V,W2,W3 and linear maps fα : W2 ! n V and fβ : W3 ! V . Let V have dimension n and ﬁx an identiﬁcation ϕ: V ! K . Then we have the following diagram:

fα W2 / V o W3 fβ ϕ n _ ϕ ◦ fα(W2) / k o ? W3

n Fix linear subspaces X,Y ⊆ k with dimk(X) = dim(W2) and dimk(Y ) = dim(W3). Then GL(n, k) acts from the left on triples (kn,X,Y ), and we can map a triple (kn,X,Y ) to (X,! kn -Y ) (the maps denoting the inclusions). The latter is a representation of Q. It is an exercise to check that this gives a bijection on isomorphism classes.

Definition 10 (Finite orbit type). A quiver Q has finite orbit type if there are finitely many isomorphism classes of representations for any dimension vector.

8 Our ﬁrst goal is to classify all quivers of ﬁnite orbit type.

Theorem 11 (Gabriel’s theorem). Suppose Q has ﬁnite orbit type. Then the underlying graph of Q is a symply laced Dynkin diagram, i.e. of type A, D or E.

These types of Dynkin diagrams are: A E6

D E7

Dynkin diagrams are useful for classifying complex simple Lie algebras. For instance, the Dynkin Diagram of type A is associated with sl(2, C) and D is associated with so(2n, C). Reminder. Let A, B be categories and F,G: A ! B two functors. A natural transformation from A to B is a collection of maps {ϕA}A∈A such that for all A, B ∈ A the diagram

F (A) / F (B)

ϕA ϕB G(A) / G(B) commutes. F is an equivalence if it has a quasi-inverse, i.e. there exists a functor H : B ! A such that F ◦ H is naturally isomorphic to idB and H ◦ F is naturally isomorphic to idA. Reminder. Let G be a ﬁnite group. Then there is an equivalence of categeories ( )

X representations ρ: G ! GLn(K) ↔ Left modules overK[G] = λigi λi ∈ K, gi ∈ G . i If g is a Lie algebra over K, then there is an equivalence

representations ϕ: g ! EndK (V ) ↔ Left modules overUg

9 The fact that representations for groups and Lie algebras can be identiﬁed with left modules brings us to the question if there is also such an identiﬁcation for representations of quivers. Indeed, if Q is a quiver, there is an equivalence

Rep Q ↔ Left modules over the quiver algebra KQ.

We will establish this equivalence in the next section, where we start with introduction the quiver algebra. Before, we remind the following:

Reminder. A k-algebra A is a k-vectorspace equipped with a k-linear map

A ⊗k A ! A

a ⊗k b 7! a · b.

Giving a left A-module M is the same as giving a morphism A ! End k(M).

2.1 Quiver algebras

Let Q = (Q0,Q1, s, t) be a quiver and let M = {{Vi}i∈Q0 , {fα}α∈Q1 } ∈ Rep Q be a representation of Q. Set r := |Q0|. Deﬁne the vectorspace

r M V = Vi i=1 and for all i ∈ Q0 and α ∈ Q1 the maps

ei : V ! V, (2.1)

(v1, . . . , vr) 7! (0, . . . , vi,..., 0), (2.2) and

α: V ! V, (2.3)

(v1, . . . , vs(α), . . . , vr) 7! (0, . . . , fα(vs(α)),..., 0). (2.4)

It is immediate that these maps satisfy the following relations:

2 1. ei = ei for all i ∈ Qo (i.e. the ei are idempotent).

2. eiej = 0 for i 6= j.

3. et(α)α = α = αes(α) for all α ∈ Q1.

It is an exercise to check that this is a complete set of relations among the set {α, ei}i∈Q0,α∈Q1 , i.e. any other relation can be deduced from those.

10 Deﬁnition 12 (Quiver algebra). The quiver algebra kQ is the associative unital algebra freely generated by the symbols {ei}i∈Q0 and {α}α∈Q1 subject to the relations ??, ?? and ??.

We will now introduce another algebra that is sometimes easier to work with in practice and then see that is isomorphic to the quiver algebra. To motivate it, consider the following graph with three vertices and two oriented edges:

α β 1 2 3

Interpret the oriented edges α, β as paths and in addition, consider the constant paths ei ≡ i for i = 1, 2, 3. We can concatenate these paths, as long as the terminal point of the ﬁrst path equals the inital point of the second one. E.g. we can look at the path βα and αe1 = α. For two paths that we cannot concatenat, we set their concatenation to be zero, e.g. α2 =. This construction gives us another algebra associated with Q:

Deﬁnition 13 (Path algebra). The path algebra associated with Q is the free algebra on the symbols {ei}i∈Q0 and {α}α∈Q1 , subject to the multiplication rule given by concatenation of paths when possible and 0 otherwise.

Proposition 14. The quiver algebra KQ and the path algebra are isomorphic.

We do not prove the proposition, but the idea of the proof is to derive the relations for the path algebra from the relations (??), (??) and (??). For instance, in the example considered above,

( 2 2 (??) α if s(α) = t(α) by (??) α = α · α = α · es(α) · et(α) · α = 0 if s(α) 6= t(α) by (??).

With Proposition ?? we can deduce how the quiver algebra looks like for some quivers we have already seen.

Example 15. For the quiver Q with one vertex and one edge as in Example ??, we have

kQ ' pathQ ' K[α].

Here, we have that e1 = idV . Example 16. For the quiver Q with one vertex and two edges as in Example ??, we have

kQ ' pathQ ' K < α, β > where K < α, β > are the non-commutative polynomials in α and β.

11 Theorem 17. There exists a functor F from Rep Q into kQ-modules and it is an equivalence.

Proof. We deﬁne the functor F as follows: Map a representation M = {{Vi}i∈Q0 , {fα}α∈Q1 } ∈ r Rep Q to the left kQ-module V = ⊕i=1Vi where the morphism kQ ! End k(V ) is given by ei, α as in (??) and (??). To check that this is a functor, let N =

{{Wi}i∈Q0 , {gα}α∈Q1 } ∈ Rep Q be another representation of Q and let u: M ! N be a morphism. The diagram

fα Vs(α / Vt(α)

us(α ut(α)

gα Ws(α / Wt(α) commutes at the level of k-vectorspaces. For u: V ! W , deﬁned using ui on every part, we still have to check that for every α ∈ Q1, v ∈ V , u(α · v) = α · u(v) and the same for ei for every i ∈ Q0. We have u (v1, . . . , vr) / (u1(v1), . . . , ur(vr)) _ _ α α

u (0, . . . , fα(vs(α),..., 0) / (0, . . . , gα(us(α)(vs(α))),..., 0)

(0, . . . , ut(α)(fα(vs(α))),..., 0) so indeed, u(αv¨) = α · u(v). The computation for ei works similarly. To show that is is an equivalence, we need to construct a quasi-inverse G: Left kQ − modules ! Rep Q.

For a left kQ-module V deﬁne ({Vi}i∈Q0 , {fα}α∈Q1 ) by Vi := ei · V and fα : Vs(α) ! Vt(α) by fα(v) := α·v. To check that we really land in the right vectorspace, let v = es(α) ·w ∈ Vi = ei · V with w ∈ W . Then

fα(v) = α · v = α · es(α) · w = et(α) · α · w ∈ Vt(α) = et(α) · V. Finally, we have to check that this is quasi-inverse’ to the construction above. What happens when we compose the two constructions? G F left kQ−modules −! Rep Q −! left kQ−modules r 0 M V, ei, α 7−!{{ei · V }i∈Q0 , {fα}α∈Q1 } 7−!V = ei · V, ei, α i=1 What is the relation between V and V 0? r 0 M V !˜ V = ei · V i=1 r r ! X X v 7! ei · v = ei v = 1 · v i=1 i=1

12 Note that in the second last equality, we are cheating in notation, because in fact, the left-hand side is an external direct sum, not an internal. Thus, the map is not the identity itself, but is naturally isomorphic to the identity. To ﬁnish the proof, we also have to check what happens when we compose the constructions in the opposite order:

F G Rep Q −! left kQ−modules −! Rep Q r M {{Vi}i∈Q0 , {fα}α∈Q1 } 7−!V = Vi, ei, α 7−!{{ei · V }i∈Q0 , {fα}α∈Q1 } i=1 In this case, the composition is the identity, so we are done.