University of Seville
Degree Final Dissertation Double Degree in Physics and Mathematics
Quiver Representations
Javier Linares
Tutor: Prof. Fernando Muro
June 2019 Abstract A quiver representation is a collection of vector spaces and linear maps indexed by a directed graph: the quiver. In this paper we intend to give an introduction to the theory of quiver representations developing basic techniques that allow us to give a proof of Gabriel’s Theorem [Gab72]: A connected quiver is of finite type if and only if its underlying graph is a Dynkin diagram. This surprising theorem has a second part which links isomorphism classes of indecomposable representations of a given quiver with the root system of the graph underlying such a quiver. Throughout this project, we study the category of quiver representations from an algebraic approach, viewing representations as modules and using tools from homological algebra, but also from a geometrical point of view, defining quiver varieties and studying the corresponding root systems.
Contents
1. Introduction 1
2. First definitions and examples 2
3. Path algebras 6
4. The standard resolution 9
5. Bricks 14
6. The variety of representations RepQ(α) 16
7. Dynkin and Euclidean diagrams: classification 19
8. Gabriel’s Theorem 23
9. Conclusions 24 1 INTRODUCTION 1
1. Introduction
Broadly speaking, representation theory is the branch of mathematics which studies symmetry in linear spaces. Its beginning corresponds to the early twentieth century with the work of German mathematician F.G. Frobenius. During the 30s, Emmy Noether gave a more modern view interpreting representations as modules. Since then, algebraic objects such as groups, associative algebras or Lie algebras have been studied within the framework of this theory, where the idea is representing elements of a certain algebraic structure as linear transformations in a vector space, giving rise to an infinite number of applications in number theory, geometry, probability theory, quantum mechanics or quantum field theory, among others. Each finite dimensional associative algebra over an algebraically closed field k corre- sponds to a geometric structure, the quiver, which is a directed graph1. Conversely, to each quiver we can attach an associative k-algebra, which has identity element and finite dimension under suitable conditions. Using the quiver associated with an algebra A, we can visualize A-modules of finite dimension as a collection of k-vector spaces connected by linear maps, the so-called quiver representations. The idea of this graphic representation appeared in the latest 40s with Thrall [Thr47], Grothendieck [Gro57] and then Gabriel [Gab60], but they began to spread in the early 70s with [Gab72] where Peter Gabriel ex- plicitly defined the notion of quiver representation in addition to proving the theorem that constitutes the main goal of this work. It is considered the starting point of the modern theory of representations of associative algebras. Nowadays, quiver representation theory is one of the most active lines of research inside this field [CBKK11]; there are many basic open problems about quiver representations, some of them on the construction of indecomposible representations in general. One of the main reasons why we decided to choose this subject for a final degree dissertation is the natural way in which the theory of quiver representations appears in certain problems of linear algebra, such as the simultaneous diagonalization of two matrices of the same size (Example 2.8) or the problem of determining the possible configurations of n subspaces of a given vector space (Example 2.10), that can be stated at undergraduate levels. The theory developed in this project will allow us to study them in a more general setting with a sophisticated language. Some prerequisites are needed before getting into this project (which undergraduate students as me may not have):
Basic notions in linear algebra and non-commutative rings and modules.
Some tools from homological algebra: the use of functors Hom and Ext as well as a good handling of long exact sequences. In our case the main reference is [Rot08].
Some ideas from category theory: quiver representation introductory books as [Sch14], [DW17] and [Bar15] alternate their principal content with a presentation of categor- ical language.
Algebraic Geometry: Zariski topology of affine space, some dimension reasonings and algebraic group actions.
In Section 2, we give some definitions and present a collection of problems which motivate the study of quiver representations, in addition to stating Gabriel’s Theorem. In
1The distinction between these two terms is due to the difference in the motivation of quiver theory and classical graph theory. 2 FIRST DEFINITIONS AND EXAMPLES 2
Section 3, we associate each quiver with an associative algebra, the path algebra, so that the representations of a quiver correspond precisely to the modules over these algebras. In Section 4 and 5 we study properties of the category of modules over path algebras, establishing relations between the dimension of endomorphisms spaces and of extensions of indecomposable modules under certain conditions. In Section 6 we give a geometric approach to quiver representations, defining the so-called variety of representations so that by studying the action of a certain algebraic group we can deduce information from the dimensions of the corresponding orbits. In Section 7 we classify connected graphs into three types: Dynkin diagrams, Euclidean diagrams and wild graphs, and we study the root systems in the first two cases. Finally, in 8 we combine the development made in the previous sections to prove Gabriel’s Theorem. This theorem is true for fields in general [Die99, Sec 4.7.], but for simplicity we will work with an algebraically closed field, which will be denoted by k.
2. First definitions and examples
Definition 2.1. A quiver Q = (Q0,Q1, s, t) is given by
A finite set of vertices, Q0, which for us is {1, 2, . . . , n}.
A set of arrows, Q1, which is also finite.
Maps s, t : Q1 → Q0. Given an arrow ρ, we will say that ρ starts at s(ρ) and terminates at t(ρ). It will be ρ indicated as s(ρ) −→ t(ρ).
Example 2.2. The following quiver is given by: Q0 = {1, 2, 3}, Q1 = {α, β, γ, λ, µ}, s(α) = 3, s(β) = 2, s(γ) = 3, s(λ) = 1, s(µ) = 1, t(α) = 2, t(β) = 1, t(γ) = 3, t(λ) = 3, t(µ) = 3: 2
β α
λ 1 3 γ µ
Definition 2.3. A representation X of Q is given by a k-vectorial space Xi for each i ∈ Q0 and a linear map Xρ : Xs(ρ) → Xt(ρ) for each ρ ∈ Q1.A morphism of representations 0 0 0 θ : X → X is given by linear maps θi : Xi → Xi for each i ∈ Q0 such that Xρθs(ρ) = θt(ρ)Xρ for each ρ ∈ Q1, i.e., the diagram
Xρ Xs(ρ) Xt(ρ)
θs(ρ) θt(ρ) X0 0 ρ 0 Xs(ρ) Xt(ρ) commutes. We define the composition of morphisms θ : X → X0, φ : X0 → X00 as (φ ◦ θ)i = φi ◦ θi. We will say that X is of finite dimension if each Xi is. The set of 0 0 morphisms between two representations X and X will be denoted by HomQ(X,X ). The category whose objects are representations of Q and whose morphisms are morphisms of representations will be called Rep(Q). 2 FIRST DEFINITIONS AND EXAMPLES 3
Note that the spaces HomQ(X,Y ) have a k-vector space structure. In fact, if we consider the following homomorphism between vector spaces M M d : Homk(Xi,Yi) −→ Homk(Xs(ρ),Yt(ρ)) (2.1) i∈Q0 ρ∈Q1 defined by (φx)x∈Q0 7→ (φt(ρ)Xρ − Yρφs(ρ))ρ∈Q1 , we have that HomQ(X,Y ) = Ker(d).A isomorphism in Rep(Q) is a morphism θ such that θi is an isomorphism of vector spaces, i ∈ Q0.
1 ρ Example 2.4. Let Q be the quiver 2 σ 3 . Consider the following represen- τ 4 tations of Q: 1 k 0 1 k 1 1 X : k2 k ,Y : k 1 k .
1 k 0 k 1
∼ 2 In this case we have that HomQ(X,Y ) = k . Definition 2.5. Let X,Y ∈ Rep(Q). We define the direct sum X ⊕ Y by Xρ 0 (X ⊕ Y )i = Xi ⊕ Yi, (X ⊕ Y )ρ = , 0 Yρ for all i ∈ Q0, ρ ∈ Q1. We say that a representation is trivial if Xi = 0 for all i ∈ Q0. If X is isomorphic to V ⊕ W with V , W nontrivial, we say that X is decomposable. Otherwise, we say that X es indecomposable. This terminology will be justified in 3.6 when proving the equivalence between the category Rep(Q) and the category of left modules over a certain k-algebra. We will also study that a finite dimensional representation has a unique decomposition as a direct sum of indecomposible ones up to isomorphism (5.2). Therefore, the problem of classifying the representations of a quiver is reduced to classifying the indecomposible ones. Example 2.6. A representation of the quiver ρ 1 −→ 2 is a couple of vector spaces X1, X2 together with a linear map Xρ : X1 → X2. We know from linear algebra that we can choose basis in X1 and X2 such that the associated matrix of Xρ is given by I 0 r , 0 0 where r is the range of Xρ and Ir is the identity matrix of size r. Hence, thanks to a change of basis, every representation X is isomorphic to Ir 0 0 0 X1 X2 , 2 FIRST DEFINITIONS AND EXAMPLES 4
so two representations X and Y are isomorphic if and only if dim X1 = dim Y1, dim X2 = dim Y2 and the range of Xρ and Yρ are the same. For this quiver, there are 3 indecompos- able representations
A : k −→ 0,B : 0 −→ k, C : k −→1 k, because every representation X is isomorphic to
X =∼ Ad1−r ⊕ Bd2−r ⊕ Cr, where d1 = dim X1, d2 = dim X2 and r is the range of Xρ.
Example 2.7. Let Q be the quiver
ρ
1 a representation of Q is just a vector space X1 together with an endomorphism Xρ.A morphism of representations X → Y is a linear map θ : X1 → Y1 such that θXρ = Yρθ. In particular, endomorphisms of X are exactly endomorphisms of X1 that commutes with Xρ. Since k is algebraically closed, we can take a basis in X1 such that the matrix of Xρ is of the form J(n1, λ1) 0 ··· 0 0 J(n2, λ2) 0 . . . , . .. . 0 0 ··· J(nr, λr) where J(n, λ) denotes the Jordan block n×n with eigenvalue λ ∈ k. Since Jordan matrices are unique up to block permutations, two representations X and Y are isomorphic if and only if Xρ and Yρ has the same normal form. Furthermore, we can see that indecomposible representations are in correspondence with Jordan blocks, so we conclude there is a infinite number of those, parameterized by a continuous parameter λ ∈ k and another discrete, n.
Example 2.8. The problem of classifying indecomposable representations of the double loop quiver
ρ 1 σ is equivalent to classifying couples of matrices up to simultaneous conjugation. This is a classical problem in linear algebra which has not been satisfactory solved. Although there exists some algorithms that compute a set of invariant properties which allow us to distinguish two nonisomorphic representations of a given dimension d (see [Fri83] y [Bel00]), the number of steps in those algorithms and of invariants to compute grows with d, so there is no uniform description for the solution.
Example 2.9. Generalizing example 2.6, we can consider the Kronecker quiver with r arrows, which has two vertices and r arrows ρ1, . . . , ρr : 1 → 2. The classification of indecomposible representations for this quiver is equivalent to classifying r-tuples of 0 0 matrices of the same size under the relation (A1,...,Ar) ∼ (A1,...,Ar) if there exists 0 −1 invertible matrices U and T with Ai = TAiU for i = 1, . . . , r. 2 FIRST DEFINITIONS AND EXAMPLES 5
For r = 2, the classification was given by Kronecker itself. The list of indecomposable is the following [Bar15, Prop. 1.6.]:
In T h i z z In Im J(m,0) kn kn+1 , km km , km km , kn+1 kn . h i T J(m,λ) Im z In z In where n ≥ 0, m ≥ 1 are natural numbers, λ ∈ k y z ∈ kn denotes the zero vector. When r = 3, the problem gets complicated and we can find a family of indecomposables given by two continuous parameters. This representations Xλ,µ are given in [Bar15, Prop. 1.8.]: 2 1 0 λ (Xλ,µ) = k, (Xλ,µ) = k , (Xλ,µ) = , (Xλ,µ) = y (Xλ,µ) = . 1 2 ρ1 0 ρ2 1 ρ3 µ
Example 2.10. Generalizing again example 2.6, we consider the n-subspace quiver, given by n + 1 vertices and n arrows ρi : i → n + 1 (1 ≤ i ≤ n). A representation X where all linear maps are injective can be viewed as a configuration of n subspaces of Xn+1. For the two subspace problem we have a finite list of indecomposables [Bar15, Prop. 1.5.] 0 −→ k ←− 0 k −→ 0 ←− 0 0 −→ 0 ←− k k −→1 k ←− 0 0 −→ k ←−1 k k −→1 k ←−1 k.
In the case n = 5, again we find a two parameter family of indecomposable representations:
k 1 1 0 0 k 1 1 k
k2 1 1 λ µ
k k
There exists other families of indecomposable representations where the number of pa- rameter can be arbitrary large. We have observed different behaviors in the classification of indecomposable represen- tation for some quivers. In examples 2.6 and 2.10 for n = 2 we obtained a finite number of indecomposibles. This kind of quiver is called of finite type. However, if there is a infi- nite number of indecomposable representations but they can be parameterized by a single continuous parameter, they are called of tame type. In situations similar to examples 2.9 for r = 3 or 2.10 with n = 5, the quiver is of wild type. For a precise definition of tame and wild, see for example [KJ16, Cap. 7]. It is well known that every quiver is finite, tame or wild, and only one of them. This trichotomy is also true for every finite dimensional algebra over an algebraically closed field [CB88]. Forgetting about the orientation in arrows of a quiver, we obtain its underlying graph. The following surprising theorem was proved by Gabriel in [Gab72] and constitutes the fundamental goal of this project. 3 PATH ALGEBRAS 6
Teorema 2.11 (Gabriel, part I). A connected quiver is of finite type if and only if its underlying graph is a Dynkin diagram of type ADE, which are the followings (n ≥ 1, m ≥ 4): •
• • • ··· • • • ··· • An : Dm : •
• • E6 : E7 : • • • • • • • • • • •
• E8 : • • • • • • •
Subscripts in Dynkin diagrams indicate the number of vertices of the graph. These diagrams play an important role in the classification of simple Lie algebras, in crystallo- graphic root systems and Coxeter groups, and other objects of “finite type”. Therefore, Gabriel’s Theorem allows us to distinguish between finite and infinite quivers (tame and wild). The proof we present here follows the steps taken in the notes of William Crawley- Boevey [CB92], where the main argument does not involve the use of reflection functors as in [ASS06] or [DW17].
3. Path algebras
Given a quiver Q, we can define an associative k-algebra, the path algebra, which we will be denoted by kQ. This concept is key because after proving the equivalence of categories Rep(Q) ' kQ-Mod, we will be able to apply powerful tools from module theory to quiver representations. In addition, path algebras are essential in the theory of representations of associative algebras since for all k-algebra A of finite dimension, A is 2 Morita equivalent to a k-algebra of type kQA/I (see [ASS06, I.6.10. and II.3.7.]), where 3 QA is a quiver associated with A and I is an admissible ideal . Therefore if we have a property of a k-algebra of finite dimension on an algebraically closed field that canbe detected in the category A-Mod, such as being tame or wild, it is sufficient to study it for path algebras with relations, that is, of type kQA/I. Consequently, we can say that the role played by path algebras in the theory of finite dimensional associative algebras is similar to that played by polynomial rings in commutative algebras.
Definition 3.1. A nontrivial path of length m in Q is a sequence ρ1 . . . ρm (m ≥ 1) of ρ1 ρ2 ρm arrows satisfying t(ρi+1) = s(ρi) for 1 ≤ i ≤ m. A diagram such • ←− • ←− · · · ←− • represents a path which starts at s(ρm) and terminates at t(ρ1). For each vertex i ∈ Q0, we will denote by ei the trivial path which starts and ends at i. We will denote s(x) and t(x) the starting and terminating vertex of x, respectively. A oriented cycle is a nontrivial path x such that s(x) = t(x). 2Two k-algebras A and B are Morita equivalent if the categories A-Mod and B-Mod are equivalent. Informally we can say that, from the point of view of representation theory, A and B are equal. 3 m 2 It means that RQ ⊆ I ⊆ RQ for a certain m ≥ 2, where RQ is the ideal (on both sides) generated by the arrows. Informally, these conditions imply that paths with length greater or equal to m are zero and that we are not allowed to “cut the arrows”. 3 PATH ALGEBRAS 7
Definition 3.2. The path algebra kQ is the k-algebra whose basis are paths in Q and where the product of two paths x, y is given by concatenation xy when t(y) = s(x) or zero otherwise. ρ σ Example 3.3. Consider the quiver 1 −→ 2 −→ 3. Then kQ has as basis the paths e1, e2, e3, ρ, σ and σρ. The product of paths σ and ρ is given by σρ, while ρσ = 0. Other products are ρρ = 0, e1ρ = 0, e2ρ = ρ, e3(σρ) = σρ, e1e1 = e1, e1e2 = 0, etc. Note that throughout this work path composition is written as usual application com- position, giving priority to left kQ-modules. In some reference, such as [Sch14] or [DW17], theory is developed composing on the right and dealing with right kQ-modules. Example 3.4. We now consider a couple of examples of k-algebras that can be built has path algebras:
t
1. If Q is the quiver 1 then kQ =∼ k[t]. If Q has 1 vertex and r loops, then kQ is the free associative k-algebra in r letters. 2. If Q is such that at most there is a path between two vertices, then kQ is isomorphic to the subalgebra
{C ∈ Mn(k) | Cij = 0 if there are no paths fromj to i }
of Mn(k), by mapping ei 7→ Eii, ρ 7→ Eji, where s(ρ) = i, t(ρ) = j for ρ ∈ Q1, i ∈ Q0 and Eji denotes a matrix with all its entries zero except (Eji)ji = 1. In particular, lower triangular matrices can be identified with the path algebra of An : 1 −→ 2 −→ · · · −→ n. Let Q be a quiver and A = kQ. The following observations describe properties of A and can be summarized with the sentence “the path algebra of a quiver is a basic k-algebra and {e1, . . . , en} is a complete set of primitive orthogonal idempotents”. We will denote the set of homomorphisms of left A-modules from X to Y as HomA(X,Y ) and EndA(X) = HomA(X,X). Recall that an A-module is projective if for all surjective homomorphism f : X → Y and all morphism g : P → Y , there exists a morphism h : P → X such that g = fh. Free modules are projective and if P = P1 ⊕ · · · ⊕ Pn, P is projective if and only if each Pi (1 ≤ i ≤ n) is.
2 1. The ei’s are orthogonal idempotents, i.e., eiej = 0 if i 6= j and ei = ei. Pn 2. A has an identity element given by 1 = i=1 ei. That is, the ei’s are a complete set of orthogonal idempotents.
3. The k-vector spaces Aei, ejA, ejAei has as basis the paths starting at i, terminating at j and those which start at i and terminate at j, respectively. Ln 4. We have that A = i=1 Aei (as a left A-module). Therefore, the Aei’s are projective A-modules. ∼ 5. If X is a left A-module, then HomA(Aei,X) = eiX (as vector spaces) by the iso- morphism ϕ ∈ HomA(Aei,X) 7→ ϕ(ei) ∈ eiX.
6. If 0 6= f ∈ Aei and 0 6= g ∈ eiA, then fg 6= 0: note that if x and y are paths of maximal length in the expression of f y g respectively, the coefficient of xy in fg is nonzero. 3 PATH ALGEBRAS 8
7. The ei’s are primitive idempotents, that is, Aei is a indecomposable A-module. ∼ 2 Indeed, if EndA(Aei) = eiAei contains an idempotent f, we have f = f = fei, but then f(ei − f) = 0 and now we use 6.
8. If ei ∈ AejA, then i = j, since AejA has as basis the paths going thought the vertex j.
9. A is a basic k-algebra, i.e., Aei Aej for i 6= j: if an isomorphism φ : Aei → Aej −1 −1 exists, we have φ(ei) ∈ eiAej and φ (ej) ∈ ejAei, hence ei = φ (φ(ei)ej) = −1 φ(ei)φ (ej) ∈ AejA, which contradicts 8.
In the next proposition, we list some other properties of path algebras, but proofs will not be included since they will not be used henceforth.
Proposition 3.5. 1. A is finite dimensional if and only if Q has no oriented cycles.
2. A is prime, i.e., IJ 6= 0 for nontrivial ideals I, J if and only if for all i, j ∈ Q0, there exists a path from i to j.
3. A is left (right) noetherian if and only if for every oriented cycle that passes through i, there exists a unique arrow that starts (termiantes) at i.
4. A basis of the Jacobson radical of A are paths x such that there is no path from t(x) to s(x).
5. The center of A is given by k × k × · · · × k × k[t] × k[t] × · · · × k[t], with one factor for each connected component C of Q so that a factor is k[t] if and only if C is a cycle.
Next we prove the equivalence between the categories Rep(Q) and kQ-Mod which was announced previously. We will only do the construction of the quasi-inverse functors, since the verification that the corresponding compositions are naturally equivalent tothe identity functors is tedious and they are not interesting in the context of this work.
Lemma 3.6. The category Rep(Q) is equivalent to kQ-Mod.
Proof. First define F : kQ-Mod→ Rep(Q) so that to each kQ-module M we define the representation given by Xi = eiM for each i ∈ Q0 and Xρ : es(ρ)M → et(ρ)M, m 7→ ρm for ρ ∈ Q1. If φ : M → N is a morphism of kQ-modules, then define F(φ): F(M) → F(N) given by F(φ)i = F(φ) . eiM On the other hand, we define G : Rep(Q) → kQ-Mod the following way. Given a rep- ε π X Q G(X) = L X X →i G(X) →i X resentation of , define i∈Q0 i. Let i i the canonical mor- phisms. Then define the action of kQ on G(X) as ρ1 ··· ρmx = εt(ρ1)Xρ1 ...Xρm πs(ρm)(x), e x = ε π (x) θ : X → X0 G(θ): L X → i i i . If is a morphism of representations, define i∈Q0 i L X0 θ i∈Q0 i such as the direct sum of the i’s. Example 3.7. Let Q be a quiver and A = kQ. Let us explicitly write the representation corresponding to the projective A-modules Aei, i ∈ Q0. If we call X = F(Aei), then the representation X is given by Xj = ejAei, j ∈ Q0 and Xρ : es(ρ)Aei → et(ρ)Aei, a 7→ ρa, ρ ∈ Q1. In particular, if Q is the quiver
1 3 4
2 4 THE STANDARD RESOLUTION 9
then we have that Ae1 corresponds with the representation
" # 0 1 0 1 0 1 k k2 k2
1 . 1 k 0
Thanks to this equivalence, we can translate the language of module categories to that of quiver representations, so that we can talk about subrepresentations, kernels, images, exact sequences, etc. From now on, we will use the same notation to refer to both modules and representations.
Definition 3.8. Let Q be a quiver with n vertices. The dimension vector of a finite n dimensional representation X is the vector dimX ∈ N given by (dimX)i = dim Xi = dim HomA(Aei,X) (1 ≤ i ≤ n). Here N = {0, 1,...} denotes the set of nonnegative n integers. We define the Euler form as the bilinear form in Z given by n X X hα, βi = αiβi − αs(ρ)βt(ρ) i=1 ρ∈Q1
n for α, β ∈ Z . We define the Tits form q(α) = hα, αi, which is a quadratic form in n Z . Finally, we define the symmetric bilinear form or Cartan form as (α, β) = hα, βi + hβ, αi.
Example 3.9. For the quiver 1 −→ 2 −→ 3 3 the matrices of the Euler form and Cartan form with respect to the canonical basis of Z are, respectively,
1 −1 0 2 −1 0 E = 0 1 −1 ,C = −1 2 −1 . 0 0 1 0 −1 2
4. The standard resolution
In this section we prove that path algebras are hereditary, that is, submodules of projective modules are projective4. For this, we will see that in the category kQ-Mod there always are projective resolutions that end after the second step, which we will call standard resolutions. In fact, for an ideal I of any kQ, the k-algebra kQ/I is hereditary if and only if I = 0 [GR97, Sec. 8.2.].
4A necessary remark about the definition of k-hereditary algebra. We should have defined what a left hereditary k-algebra is, since the property of submodules of projective module being projective is a condition imposed in the category of left kQ-modules which, in general, does not imply that the category of right kQ-modules, Mod-kQ, satisfies also that property. However, for path algebras we have the equivalence of categories Mod-kQ ' kQop-Mod, where Qop is the quiver obtained by reversing the direction of the arrows in Q. Therefore Mod-kQ is also hereditary. 4 THE STANDARD RESOLUTION 10
Let us recall some notions of homological algebra: given a ring R, a projective resolution of a R-module M is a exact sequence
δn δ1 δ0 · · · −→ Pn −→ Pn−1 −→ · · · −→ P1 −→ P0 −→ M −→ 0 i where each Pi is projective. Throughout this section, we will use functors ExtR(−,M), i which are defined as the right derived functors of HomR(−,M), that is, ExtR(N,M) is the i-th cohomology group (in our case k-vector space) of the cochain complex obtained when applying HomR(−,M) to some projective resolution of an R-module N. The abelian 1 group ExtR(M,N) can be also constructed by considering extensions of M by N, that is, exact sequences
0 −→ N −→a E −→b M −→ 0 a b a0 b0 under the equivalence relacion 0 → N → E → M → 0 ∼ 0 → N → E0 → M → 0 if there exists an isomorphism ζ such that a0 = ζa and b = b0ζ. Summation in this group is given by the so-called “Baer sum” (see [Rot08, Sec. 6.2 y 7.2] for the general case or [Bar15, Sec. 5.2.] for R a k-álgebra), while the neutral element in such a group is the f g extension 0 → N → N ⊕ M → M → 0. Furthermore, if 0 → X → Y → Z → 0 is a long exact sequence, when applying the functor HomR(−,M) yields a long exact sequence 0 (recall that HomR(−,M) being left exact implies ExtR(−,M) = HomR(−,M)):
g∗ f ∗ 0 HomR(Z,M) HomR(Y,M) HomR(X,M)
∆1 ∗ ∗ 1 g1 1 f1 1 ExtR(Z,M) ExtR(Y,M) ExtR(X,M)
∆2 ∗ ∗ 2 g2 2 f2 2 ∆3 ExtR(Z,M) ExtR(Y,M) ExtR(X,M) ···
1 ∗ 1 let us see how the homomorphisms ∆1 : HomR(X,M) → ExtR(Z,M) and g1 : ExtR(Z,M) → 1 ExtR(Y,M) work. If φ ∈ HomR(X,M), then ∆1(φ) is the extension obtained by consid- ering the pushout of f and φ:
f g 0 X Y Z 0 φ
∆1(φ): 0 M E Z 0
a b ∗ Analogously, if ξ : 0 → M → E → Z → 0 is an extension of Z by M, g1(ξ) is obtained as the pullback of g and b:
∗ 0 g1(ξ): 0 M E Y 0 g
0 M a E b Z 0 Let Q be a quiver, A = kQ and X be a left A-module. Let us start by describing the projective A-modules and homomorphisms that will appear in the standard resolution of X. Define M M P0 = Aei ⊗k eiX,P1 = Aet(ρ) ⊗k es(ρ)X i∈Q0 ρ∈Q1 4 THE STANDARD RESOLUTION 11
and define g : P0 → X and f : P1 → P0 as
g(a ⊗ x) = ax for each a ∈ Aei, x ∈ eiX, f(a ⊗ x) = aρ ⊗ x − a ⊗ ρx for each a ∈ Aet(ρ), x ∈ es(ρ)X.
Note that both P0 and P1 only depend on Q and the dimension vector of X, while f and g do depend on the linear applications corresponding to the arrows of the quiver. We also have that for all ξ ∈ P0, we can express ξ uniquely as X X ξ = a ⊗ xa,
i∈Q0 a∈Ai
5 where Ai = {Paths a with s(a) = i} and with xa ∈ eiX allmost all nonzero . Define the degree of ξ, Deg(ξ), as the maximum of lengths of paths a with xa 6= 0. The following lemma will be necessary for proving the main theorem of this section: ξ = P P a ⊗ x ∈ P ξ + (f) = {ξ + η | η ∈ (f)} Lemma 4.1. Let i∈Q0 a∈Ai a 0. Then Im Im has an element of degree 0.
0 Proof. Let a ∈ Ai be a nontrivial path such that xa 6= 0, then we can express a = a ρ 0 0 with ρ ∈ Q1 such that s(ρ) = i and with a another path satisfying s(a ) = t(ρ). Viewing 0 a ⊗ xa as an element of the ρ-th component of P1, we have
0 0 f(a ⊗ xa) = a ⊗ xa − a ⊗ ρxa. Suppose that ξ has degree d > 0, then calling
d 0 0 Ai = Paths a = a ρ of length d with ρ ∈ Q1 such that s(ρ) = i, t(ρ) = s(a ) , we have that X X 0 ξ − f a ⊗ xa i∈Q0 d a∈Ai has degree strictly lower than d, so we can conclude by induction.
Teorema 4.2. Let Q be a quiver and A = kQ. If X is a left A-module, then we have an exact sequence f g 0 −→ P1 −→ P0 −→ X −→ 0. It is called the standard resolution of X.
Proof. First note that for all a ⊗ x ∈ Aet(ρ) ⊗ es(ρ)X, we have
f(g(a ⊗ x)) = f(aρ ⊗ x − a ⊗ ρx) = aρx − aρx = 0 hence Im(f) ⊆ Ker(g). For the other inclusion, let ξ ∈ Ker(g) and due to Lemma 4.1 we can take ξ0 ∈ ξ + Im(f) of degree 0. Then
n ! n 0 X 0 X 0 0 = g(ξ) = g(ξ ) = g ei ⊗ xei = xei , i=1 i=1
5 Recall that if Q has oriented cycles then the spaces Aei may have infinite dimension. 4 THE STANDARD RESOLUTION 12
0 Ln 0 0 with xei ∈ eiX. Since X = i=1 eiX, each xei must be zero and therefore ξ = 0 and ξ ∈ Im(f). Thus g is surjective since if we are given x ∈ X, we have X g ei ⊗ eix = (e1 + ··· + en)x = 1x = x. i∈Q0
Finally, let us prove Ker(f) = 0. Let ξ ∈ Ker(f), then we can write it as X X ξ = a ⊗ xρ,a
ρ∈Q1 a∈At(ρ) where xρ,a ∈ es(ρ)X allmost all zero. If ξ 6= 0, take a˜ a path of maximal length xρ,˜ a˜ 6= 0 for some ρ˜ ∈ Q1. However, X X X X f(ξ) = aρ ⊗ xρ,a − a ⊗ ρxρ,a
ρ∈Q1 a∈Aρ ρ∈Q1 a∈Aρ satisfies that the coefficient of a˜ρ˜ ⊗ xρ,˜ a˜ 6= 0, which is a contradiction.
f g Since the standard resolution 0 −→ P1 −→ P0 −→ X −→ 0 terminates in the second 1 ∗ ∗ step, we conclude that ExtA(X,Y ) = Coker(f ), with f : HomA(P0,Y ) → HomA(P1,Y ) defined as f ∗(φ) = φ ◦ f.
σ 4 ρ Example 4.3. Consider the following representation X of D4 = 1 2 , τ 3
h i " # 0 1 k 1 0 X : k k2
h i 1 1 k and denote by {x} , {y1, y2} , {v} , {w} the basis taken in X1,X2,X3,X4, respectively. The f g sequence P1 ,→ P0 X is written in terms of quiver representations as: 4 THE STANDARD RESOLUTION 13
1 0 he1 ⊗ xi hxi
1 1 0 0 0
1 −1 1 1 0 0 0 0 1 hρ⊗x,e2⊗y1, he2 ⊗ xi hy1, y2i e2⊗y2i
1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0
hτ⊗x,e3⊗y1, hσ⊗x,e4⊗y1, hτρ⊗x,τ⊗y1, hσρ⊗x,σ⊗y1, hvi hwi e3⊗y2i e4⊗y2i τ⊗y2,e3⊗vi σ⊗y2,e4⊗wi
0 0 1 1 1 1 1 1 1 0 0 1 0 0 −1 1 0 −1 1 0 0 0 1 0 0 1 0 1 1 0 1 1
Standard resolutions allow us to prove that path algebras are hereditary and fur- 1 thermore we can relate the dimensions of the spaces HomA(X,Y ) and ExtA(X,Y ) with dimension vectors of X and Y using the Euler form.
i Corollary 4.4. 1. If X is a left A-module, then ExtA(X,Y ) = 0 for all A-module Y , i ≥ 2.
2. A is hereditary, that is, if X is a submodule of a projective module P , then X is projective.
1 3. If X,Y are finite dimensional, we have that dim HomA(X,Y ) − dim ExtA(X,Y ) = hdimX, dimY i.
1 4. If X is finite dimensional, then dim EndA(X) − dim ExtA(X,X) = q(dimX).
Proof. 1. The A-modules Aei ⊗ V are isomorphic to the direct sum of dim V (also understand the case in which dim V is infinite) copies of Aei, which are projective for being direct summands of A, which is projective for being free. Hence the standard resolution is a projective resolution of X. The claim follows from the definition of i ExtA(X,Y ), i ≥ 2.
1 2 f g 2. For all A-module Y , ExtA(X,Y ) = ExtA(P/X, Y ) = 0, since if 0 → P1 −→ P0 −→ X → 0 is the standard resolution of X, then the exact sequence
f g˜ π 0 −→ P1 −→ P0 −→ P −→ P/X −→ 0,
g where g˜ : P0 −→ X,→ P and π is the canonical projection, is a projective resolution 1 ∗ 2 of P/X, so that ExtA(X,Y ) = Coker(f ) = ExtA(P/X, Y ) = 0. We conclude 1 observing that an A-module X is projective if and only if ExtA(X,Y ) = 0 for all A-module Y . 5 BRICKS 14
3. Note that given a exact sequence of finite dimensional vector spaces 0 → V1 → V2 → V3 → V4 → 0, we have that dim V1 + dim V3 = dim V2 + dim V4. With that in mind, apply the functor HomA(−,Y ) to the standard resolution to obtain a long 1 1 exact sequence (recall that ExtA(P0,Y ) = ExtA(P1,Y ) = 0 for P0 and P1 being projective):
1 0 −→ HomA(X,Y ) −→ HomA (P0,Y ) −→ HomA (P1,Y ) −→ ExtA(X,Y ) −→ 0.
Pn 6 Nevertheless, dim HomA (P0,Y ) = i=1 dim Xi dim Yi since M ∼ M HomA (P0,Y ) = HomA( Aei ⊗ eiX,Y ) = HomA (Aei ⊗ eiX,Y )
i∈Q0 i∈Q0 ∼ M ∼ M = Homk (eiX, HomA (Aei,Y )) = Homk(Xi,Yi)
i∈Q0 i∈Q0 Hom (P ,Y ) ∼ L Hom (X ,Y ) dim Hom (P ,Y ) = and analogously 1 = ρ∈Q1 k s(ρ) t(ρ) thus A 1 P dim X dim Y h X, Y i ρ∈Q1 s(ρ) t(ρ). The result follows from the definition of dim dim . 4. Apply 3 to X = Y .
5. Bricks
In this section we consider finite dimensional A-modules with A a hereditary k-algebra i or equivalently ExtA(X,Y ) = 0 for all A-module X,Y and i ≥ 2. In particular, results will be true when A is the path algebra of a quiver. Our goal is studying the relationship 1 between spaces ExtA(X,X) and EndA(X) when X is an indecomposable A-module. Let us first recall a couple of general results, sometimes known in the literature asFitting Lemma and Krull-Schmidt Theorem (see, for example, [ASS06, I.4.8, I.4.10]): Lemma 5.1 (Fitting Lemma). Let A be k-algebra and X a left A-module a izquierda.
1. If the k-algebra EndA(X) is local, then X is indecomposable.
2. If X is indecomposable and finite dimensional, then EndA(X) is local and each element is nilpotent or invertible. Teorema 5.2 (Krull-Schmidt Theorem). Let A be k-algebra and X a left A-module. If X is of finite dimension, then X is the direct sum of indecomposable that, up to permutation, are uniquely determined up to isomorphism.
Definition 5.3. We will say that an A-module X is a brick if EndA(X) = k. By 5.1, we have that every block is indecomposable. Next we prove a couple of lemmas that will be of great importance in the proof of Gabriel’s Theorem: 1 Lemma 5.4. Suppose that X,Y are indecomposable. If ExtA(Y,X) = 0, then every nonzero morphism θ : X → Y is injective or surjective.
6 Here we are using that the functor HomA(−,Y ) transforms sums in products and in particular com- mutes with finite direct sums and then the adjoint isomorphism described in[Rot08, Th. 2.76.]: Let R and S be rings. Let A be a left R-module, B a S-R-bimodule and C a left S-module. Then we have
HomS (B ⊗R A, C) =∼ HomR (A, HomS (B,C)) . 5 BRICKS 15
Proof. Consider the exact sequences
i π ι θ˜ ξ : 0 −→ Im(θ) −→ Y −→ Coker(θ) −→ 0, η : 0 −→ Ker(θ) −→ X −→ Im(θ) −→ 0. Where i, π e ι are canonical inclusions and projections and θ˜ is defined as θ. Apply HomA(Coker(θ), −) to the sequence η to obtain a long exact sequence
˜ 1 ι∗1 1 θ∗1 1 · · · −→ ExtA(Coker(θ), Ker(θ)) −→ ExtA(Coker(θ),X) −→ ExtA(Coker(θ), Im(θ)) −→ 0. i where we are using that Ext (X,Y ) = 0 for i ≥ 2 since A is hereditary, so that θ˜∗1 is ˜ 1 surjective and ξ = θ∗1(ζ) for some ζ ∈ ExtA(Coker(θ),X). Suppose that ζ is in corre- a b spondence with the exact sequence 0 → X → Z → Coker(θ) → 0, so that there exists an homomorphism β : X → Im(θ) bringing the following commutative diagram
ζ : 0 X a Z b Coker(θ) 0 β γ
ξ : 0 Im(θ) i Y π Coker(θ) 0
However, the sequence a h i β γ −i 0 X Z ⊕ Im(θ) Y 0
1 ∼ is exact and it splits since ExtA(X,Y ) = 0 by hypothesis and hence X ⊕ Y = Z ⊕ Im(θ). If Im(θ) 6= 0, X or Y is a summand of Im(θ) by Krull-Schmidt Theorem. But if θ is not injective or surjective, then dim Im(θ) < dim X, dim Y, which is a contradiction.
1 Corollary 5.5. If X is indecomposable with no self-extensions, that is, ExtA(X,X) = 0, then X is a brick.
Proof. Every θ ∈ EndA(X) is inyecitve ir surjective and hence an isomorphism. In that case EndA(X) is a finite dimensional division algebra over k, which is algebraically closed, ∼ therefore EndA(X) = k.
Lemma 5.6. If X is not indecomposable and not a brick, then X has a submodule which 1 is a brick with self-extensions, that is, ExtA(X,X) 6= 0. Proof. It is sufficient to prove that if X is indecomposable and not a brick, then there is a proper submodule U ⊂ X such that it is indecomposable and with self-extensions, since if U is not a brick we can argue by induction in the dimension of X. Choose θ ∈ EndA(X) with I = Im(θ) of nonzero minimal dimension. Since X is in- 2 decomposable and not a brick, EndA(X) 6= k is local, then θ is nilpotent and θ = 0 by Lr minimality. We have that I ⊆ Ker(θ). Let Ker(θ) = i=1 Ki with each Ki indecom- posable, and take j such that the composition α : I,→ Ker(θ) Kj is nonzero. We α claim that we can take U = Kj. Indeed, since the map X I → Kj ,→ X has image Im(α) 6= 0, we deduce that α must be injective by minimality. On the other hand, we 1 have ExtA(I,Kj) 6= 0. If this were not the case, consider the pushout
0 Ker(θ) i X I 0
π
0 Kj Y I 0 6 THE VARIETY OF REPRESENTATIONS REPQ(α) 16
where i : Ker(θ) → X and π : Ker(θ) → Kj are the canonical inclusion and projection, ∼ respectively. Here the second row splits, with Y = X, hence Kj would be a summand of X, which is a contradiction. α Finally, Kj must have self-extensions since the exact sequence 0 → I → Kj → X → 0 1 1 induces a surjective map ExtA(Kj,Kj) ExtA(I,Kj) 6= 0.
6. The variety of representations RepQ(α)
In this section let Q be a quiver with n vertices and A = kQ. We will define the variety n of representations of Q with dimension vector α ∈ N and describe some of its elementary properties. We will use some dimension arguments of algebraic geometry. To go in depth into quiver varieties, see [KJ16] and for an introduction to the theory of the actions of algebraic groups, we have consulted [Bri10]. r We will denote the r-dimensional affine space with the Zariski topology as A and we r will consider locally closed subspaces U ⊆ A , that is, open in its closure U. Recall that a subspace U is said to be irreducible if all nonempty open sets in U are dense in U. Recall that the dimension of a subspace U is
sup {n | ∃ Z0 ⊂ Z1 ⊂ · · · ⊂ Zn irreducible closed sets in U} .
We have that dim U = dim U, and if W = U ∪ V , dim W = max {dim U, dim V } and that r the space A has dimension r. An algebraic group is a locally closed set which has group structure, so that multi- r plication and taking inverses are regular maps. If an algebraic group G acts on A , then r r for each x ∈ A , the orbits Ox = {g · x ∈ A | g ∈ G} have the following properties ([KJ16, Cap. 2]):
1. Each orbit Ox is locally closed.
2. Ox is a union of orbits. Furthermore, Ox − Ox is a union of orbits of strictly smaller dimension than dim Ox.
r 3. For each x ∈ A , the stabilizer subgroup Gx = {g ∈ G | g · x = x} is a closed alge- braic subgroup of G. Moreover, we have a canonical isomorphism ∼ Ox = G/Gx.
In particular, dim G = dim Ox + dim Gx. n Definition 6.1. Let Q be a quiver and α ∈ N , where n is the number of vertices in Q. We define the variety of representations of Q with dimension vector α as
Y αs(ρ) αt(ρ) RepQ(α) = Homk(k , k ). ρ∈Q1 (α) r r = P α α We have that RepQ is isomorphic to A , where ρ∈Q1 t(ρ) s(ρ). Each element x ∈ RepQ(α) gives us a representation in Q, which will be denoted by R(x) and defined α by R(x)i = k i for i ∈ Q0 and R(x)ρ is the linear map whose corresponding matrix with respect to the canonical basis is xρ, for ρ ∈ Q1. n Definition 6.2. Let α ∈ N (n ≥ 1). We define n Y GL(α) = GL(αi, k). i=1 6 THE VARIETY OF REPRESENTATIONS REPQ(α) 17
s Pn 2 It is an algebraic group which is open in A , where s = i=1 αi . If Q is a quiver with n vertices, GL(α) acts on RepQ(α) by conjugation, that is,
−1 (g · x)ρ = gt(ρ)xρgs(ρ) for g ∈ GL(α) and x ∈ RepQ(α). Note that stabilizers
n −1 o GL(α)x = g ∈ GL(α) | gs(ρ)xρgt(ρ) = xρ, ρ ∈ Q1 are exactly the invertible endomorphisms of R(x), which will be denoted by AutA(R(x)). Given a representation X ∈ Rep(Q) with dimX = α, we have that X =∼ R(x) for some x ∈ RepQ(α) by choosing a basis. Furthermore, two elements x, y ∈ RepQ(α) define ∼ isomorphic representations R(x) = R(y) if and only if y ∈ Ox. In other words, given a n quiver Q with n vertices and α ∈ N , we have a bijection n o Isomorphism classes in Rep(Q) ←→ (α) . of dimension vector α Orbits in RepQ ∼ For X ∈ Rep(Q) denote OX = Ox, with x ∈ RepQ(α) taken such that X = R(x). 3 ∼ 2 Example 6.3. Let Q = A3 and α = (1, 1, 1) ∈ N . Then RepQ(α) = A , GL(α) = 3 0 0 (A − {0}) and two points (λ, µ), (λ , µ ) ∈ RepQ(α) are in the same orbit if and only if there exist nonzero a, b, c ∈ k such that the following diagram is commutative
µ k λ k k
a b c 0 µ0 k λ k k that is, λ = 0 if and only if λ0 = 0 and µ = 0 if and only if µ0 = 0. Therefore 2 2 there are 4 orbits, {(0, 0)}, (λ, µ) ∈ A | λ = 0, µ 6= 0 , (λ, µ) ∈ A | µ = 0, λ 6= 0 y 2 (λ, µ) ∈ A | λ, µ 6= 0 . The corresponding representations are, respectively
k −→0 k −→0 k, k −→1 k −→0 k, k −→0 k −→1 k, k −→1 k −→1 k,
We will soon see that the fact that the orbit with largest dimension corresponds to the unique indecomposable representation is not a coincidence. Lemma 6.4. Let X ∈ Rep(Q) with dimX = α. We have that
1 dim RepQ(α) − dim OX = dim ExtA(X,X). ∼ Proof. Suppose that X = R(x) for certain x ∈ RepQ(α). Note that GL(α) is a nonempty r r = P α2 dim (α) = s (α) = open of A , with i∈Q0 i , so it is dence, thus GL . Analogously, GL x AutA(R(x)) is also a nonempty open of the linear variety EndA(R(x)), then also dense, so dim AutA(R(x)) = dim EndA(R(x)) = dim EndA(X). Therefore
dim OX = dim GL(α) − dim GL(α)x = r − dim EndA(X).
q(α) s = P α α considering the expression of , calling ρ∈Q1 s(ρ) t(ρ) and using 4.4, we finally get
1 dim RepQ(α)−dim OX = r−s−dim EndA(X) = q(α)−dim EndA(X) = dim ExtA(X,X). 6 THE VARIETY OF REPRESENTATIONS REPQ(α) 18
Corollary 6.5. 1. If α 6= 0 y q(α) ≤ 0, there are infinitely many orbits in RepQ(α).
2. OX is open if and only if X has no self-extensions.
3. Up to isomorphism, there is at most one module with dimension vector α without self-extensions.
Proof. 1. If X is a module with dimension vector α, then EndA(X) is nonzero so
1 dim RepQ(α) − dim OX = dim ExtA(X,X) = dim EndA(X) − q(α) > 0. and by dimension arguments there must be infinitely many orbits.
1 2. If ExtA(X,X) = 0 then dim OX = dim RepQ(α) if and only if dim OX = dim RepQ(α) and in that case OX = RepQ(α), because a subset of an irreducible space which is proper and closed has strictly smaller dimension. Since OX is locally closed, it is open. Conversely, if OX is open in RepQ(α), then OX = RepQ(α) since RepQ(α) is 1 irreducible and therefore their dimensions are equal and thus dim ExtA(X,X) = 0.
3. If X, Y are such modules, OX , OY ⊆ RepQ(α) are open and therefore dense, so their intersection is nonempty and then OX = OY .
Lemma 6.6. If ξ : 0 −→ U −→ X −→ V −→ 0 is a non-splitting exact sequence, then OU⊕V ⊆ OX and furthermore OU⊕V 6= OX . ∼ Proof. Let us take u ∈ RepQ(dimU) and v ∈ RepQ(dimV ) such that U = R(u) and ∼ V = R(v). For each vertex i ∈ Q0, we can identify Ui with a subspace of Xi. Choosing basis in Ui and extending them to basis in Xi, we can select x ∈ OX such that xρ is of the form uρ wρ xρ = . 0 vρ λIdUi 0 For nonzero λ ∈ k, we define gλ ∈ GL(α) by (gλ)i = for each i ∈ Q0. Then 0 IdVi we have that uρ λwρ (gλx)ρ = . 0 vρ uρ 0 Hence OX contains the point with matrices , which corresponds to U ⊕V . Finally, 0 vρ let us see that X is not isomorphic to U ⊕V : applying functor HomA(−,U) to the sequence ξ we obtain a long exact sequence
∆1 1 0 −→ HomA(V,U) −→ HomA(X,U) −→ HomA(U, U) −→ ExtA(V,U) −→ · · · so that
dim HomA(V,U) − dim HomA(X,U) + dim HomA(U, U) − dim Im(∆1)
= dim HomA(V ⊕ U, U) − dim HomA(X,U) − dim Im(∆1) = 0 but ∆1(IdU ) = ξ, which is a nontrivial extension, so dim Im(∆1) 6= 0 and then
dim HomA(U ⊕ V,U) 6= dim HomA(X,U), from where we deduce that X U ⊕ V . 7 DYNKIN AND EUCLIDEAN DIAGRAMS: CLASSIFICATION 19
Corollary 6.7. If OX is an orbit in RepQ(α) of maximal dimension and X = U ⊕ V , 1 then ExtA(U, V ) = 0.
Proof. If there were a non-splitting sequence 0 −→ U −→ E −→ V −→ 0, then OX ⊆ OE − OE, so that dim OX < dim OE, which is a contradiction.
7. Dynkin and Euclidean diagrams: classification
In this section we classify connected graphs into 3 types: Dynkin diagrams, Euclidean diagrams and the rest, sometimes called wild graphs. Given a connected graph Γ with n n vertices, we will define a quadratic form q in Z associated to it in a similar way to what we have done for quivers, so that Γ will be Dynkin, Euclidean or wild according to q being positive definite, positive semi-defined or indefinite, respectively. In the first twocaseswe study the corresponding root systems, concluding preparations for Gabriel’s Theorem.
Definition 7.1. Let Γ be a finite graph with vertices {1, 2, . . . , n}, where multiple edges and loops are allowed, so that Γ is given by the set of natural numbers
{nij = nji number of edges between i and j} .
n Let h−, −i be the bilinear form in Z defined by
n n X X hα, βi = αiβi − nijαiβj i=1 i≤j and let q(α) = hα, αi, (α, β) = hα, βi + hβ, αi. Note that knowing Γ, q or (−, −) we 1 can determine the others since q(α) = 2 (α, α), (α, β) = q(α + β) − q(α) − q(β) and n (εi, εj) = −nij (i 6= j), where εi denotes the i-th vector of the canonical basis in Z . Given a quiver Q and Γ its underlying graph, the definitions of q and (−, −) coincides with those given in 3.8. However, h−, −i depends on the orientation of arrows and both definitions are equal if the vertices are listed so that for every ρ ∈ Q1, s(ρ) < t(ρ).
n Definition 7.2. Let q a quadratic form in Z . We say that q is positive definite if n q(α) > 0 for all 0 6= α ∈ Z . We say that q is positive semi-definite if q(α) ≥ 0 for all n α ∈ Z . The radical of q is the set
n rad(q) = {α ∈ Z | (α, −) = 0} .
n We say that α ∈ Z is sincere if each component α is nonzero. We establish a partial n n ordering in Z given by α ≤ β if β − α ∈ N . Lemma 7.3. If Γ is a connected graph and there exists nonzero radical vector β ≥ 0, then n β is sincere and q is positive semi-definite. Furthermore, we have that for α ∈ Z
q(α) = 0 ⇐⇒ α ∈ Qβ ⇐⇒ α ∈ rad(q). where Qβ = {qβ | q ∈ Q}. Proof. By assumption we have that, for each i = 1, . . . , n, X 0 = (εi, β) = (2 − 2nii)βi − nijβj, (7.1) i6=j 7 DYNKIN AND EUCLIDEAN DIAGRAMS: CLASSIFICATION 20
P If βi = 0, then i6=j nijβj = 0 since every term is greater or equal to 0, we have that βj = 0 if there is an edge from i to j. Using that Γ is connected, it follows that β = 0, which is a contradiction. Therefore β is sincere. Nonetheless, using 7.1, n n X 2 X X 1 2 X q(α) = αi − nijαiαj = (2 − 2nii)βi αi − nijαiαi 2βi i=1 i≤j i=1 i X βj 2 X = nij αi − nijαiαj 2βi i6=j i It follows that q is positive semi-definite. If q(α) = 0, then αi/βi = αj/βj if there is an edge from i to j, and again, since Γ is connected, we have that α ∈ Qβ. If α ∈ Qβ, then α ∈ rad(q), since β ∈ rad(q) by assumption. Finally if α ∈ rad(q) we clearly have that q(α) = 0. Definition 7.4. 1. The following graphs are Dynkin diagrams (n ≥ 0, m ≥ 4): • • • • ··· • • • ··· • An : Dm : • • • E6 : E7 : • • • • • • • • • • • • E8 : • • • • • • • 2. The following graphs are Euclidean diagrams (n ≥ 0, m ≥ 4): • ··· • • • ˜ ˜ An : • • Dm : • ··· • • ··· • • • • • ˜ ˜ E6 : • E7 : • • • • • • • • • • • • • ˜ E8 : • • • • • • • • 7 DYNKIN AND EUCLIDEAN DIAGRAMS: CLASSIFICATION 21 ˜ ˜ Note that A0 has a vertex with a loop and A1 has two vertices joined by two edges. In Dynkin diagrams, subscripts indicate the number of vertices while in the Euclidean it corresponds to the number of vertices minus one. In Lie theory, other families of Dynkin and Euclidean diagrams appear. More precisely, we have defined here Dynkin and Euclidean diagrams of the ADE type. Now weprove a combinatorial lemma before giving the theorem that will allow us to classify connected graphs according to q. Lemma 7.5. Let Γ a connected graph. If Γ is not Dynkin or Euclidean, then Γ contains an Euclidean subgraph. ˜ Proof. If Γ does not contain any An, then it has no cycles and hence it is a tree. If it does ˜ not contain any Dn, it cannot have vertices of degree ≥ 4 and neither it is possible more that two vertices of degree 3. If every vertex is of degree 2 then Γ = An. Let v of degree 3 and let d1 ≤ d2 ≤ d3 be the distances from v to the degree-one vertices. If d1 ≥ 2, then ˜ Γ contains a E6. Suppose then that d1 = 1. If d2 = 1, then Γ = Dn for some n and if ˜ d2 ≥ 3, then Γ contains a E7. Suppose then that d2 = 2. If d3 = 3 o 4, then Γ = E7 or ˜ Γ = E8, respectively. Finally, if d3 ≥ 5, then Γ contains a E8. Definition 7.6. Let Γ an Euclidean diagram. We call radical vectors, and are denoted by δ, those given by: 1 ··· 1 1 1 ˜ ˜ An : 1 1 Dm : 2 ··· 2 1 ··· 1 1 1 1 2 ˜ ˜ E6 : 2 E7 : 1 2 3 4 3 2 1 1 2 3 2 1 3 ˜ E8 : 2 4 6 5 4 3 2 1 vertices i with δi = 1 are denominated extending vertices. Note that if we remove an extending vertex of an Euclidean diagram we obtain its corresponding Dynkin diagram. Teorema 7.7. Let Γ be a connected graph. 1. If Γ is an Euclidean diagram, q is positive semi-definte and rad(q) = Zδ, where δ is the corresponding radical vector. 2. If Γ is a Dynkin diagram, then q is positive definite. n 3. In another case, it exists α ∈ Z such that q(α) < 0. Proof. 7 DYNKIN AND EUCLIDEAN DIAGRAMS: CLASSIFICATION 22 ˜ 1. Checking that each δ is radical is left to the reader. For example, for E6, (δ, α) = 2α1 + 4α2 + 6α3 + 4α4 + 2α5 + 4α6 + 2α7 − 2α1 − α2 − 2α3 − 3α4 − 2α5 − 3α6 − 2α7 − 3α2 − 2α3 − α4 − α6 − 2α3 = 0. δ 2δ = P δ The key step is verify that for each i, i j | nij 6=0 j is satisfied, since the α (δ, α) 2δ − P δ q coefficient of i in is precisely i j | nij 6=0 j. Therefore is positive semi- definite by Lemma 7.3. Since δi = 1 for some i in all cases, n rad(q) = Qδ ∩ Z = Zδ. 2. We can extend Γ to its corresponding Euclidean diagram, Γ˜. Let q˜ be its quadratic n form and δ its radical vector. Suppose that there exists nonzero α ∈ Z such that q(α) ≤ 0 and let α˜ defined as α at vertices of Γ and zero at the extending vertex. We have that q˜(˜α) = q(α) ≤ 0 and hence q˜(˜α) = 0, since q˜ is positive semi-definite. It follows from Lemma 7.3 that α˜ ∈ Qδ, but it is impossible since α˜ is zero at the extending vertex. 3. Thanks to Lemma 7.5, there exists an Euclidean subgraph Γ0 and let δ be its radical vector. If every vertex in Γ is in Γ0, then we can take α = δ. If i is a vertex which is not in Γ0 such that there exists an edge connecting i with Γ0, then we can take α = 2δ + εi, since q(α) = q(2δ) + q(εi) + (2δ, εi) < 0, because q(2δ) = 4q(δ) = 0, q(εi) = 1 − nii ≤ 1 and (2δ, εi) = 2(δ, εi) ≤ −2. Definition 7.8. Let Γ be a connected graph. We define the set of roots of Γ as n ∆ = {α ∈ Z | α 6= 0, q(α) ≤ 1} . We say that a root α is real if q(α) = 1 and that it is imaginary is q(α) = 0. We say that α is positive if α ≥ 0 and that it is negative if α ≤ 0. n Each εi ∈ Z is always a root. Furthermore, for all α ∈ ∆ ∪ {0} and β ∈ rad(q), −α, α + β ∈ ∆ ∪ {0}, since q(α ± β) = q(β) + q(α) ± (β, α) = q(α). If Γ is a Dynkin diagram, all roots are real. In the case of Γ being Euclidean, imaginary roots are exactly rad(q) − {0}. Lemma 7.9. Let Γ be a Dynkin or Euclidean diagram. Every root α ∈ ∆ is positive or negative. Proof. Suppose that α = α+ − α−, with nonzero α+, α− ≥ 0 and disjoint support, i.e., + − for every vertex i, αi > 0 if and only if αi = 0 and vice versa. Clearly we have that (α+, α−) ≤ 0, so that 1 ≥ q(α) = q(α+) + q(α−) − (α+, α−) ≥ q(α+) + q(α−). Hence α+ o α− must be an imaginary root and as a consequence a sincere vector, so the other one must be zero, contradiction. 8 GABRIEL’S THEOREM 23 Proposition 7.10. If Γ is Euclidean, then (∆ ∪ {0}) / ∼ is finite, where α ∼ β if and only if α − β ∈ Zδ. Proof. Let i be a extending vertex. If α is a root with αi = 0, then δ − α y δ + α are roots with positive i-th component, thus by 7.9 we have that δ − α, δ + α ≥ 0. It follows that n {α ∈ ∆ ∪ {0} | αi = 0} ⊆ {α ∈ Z | − δ ≤ α ≤ δ} , which is finite. Finally, if β ∈ ∆ ∪ {0}, then β − βiδ ∈ {α ∈ ∆ ∪ {0} | αi = 0}. Corollary 7.11. If Γ is Dynkin, then ∆ is finite. Proof. Let Γ˜ be the Euclidean graph obtained by extending Γ through i. Then we can see every root α of Γ as a root of Γ˜ with αi = 0, so that we conclude using 7.10. 8. Gabriel’s Theorem In this section we combine almost everything we have done in order to prove Gabriel’s Theorem. The crucial step is observing that if Q is a quiver and its underlying graph is Dynkin, then every indecomposable module is a brick without self-extensions. Teorema 8.1 (Gabriel, part II). Suppose that Q is a quiver such that its underlying graph is Dynkin, and let q be its Tits form. The mapping X 7→ dimX induces a bijection between isomorphism classes of idecomposable representations of Q and positive roots of q. Proof. Firstly, let us prove that if X is indecomposable, dimX is a root of q. We claim that X is a brick. Otherwise by 5.6 there exists a proper submodule U ⊂ X which is brick with self-extensions and since q is positive definite 1 1 0 < q(dimU) = dim EndA(U) − dim ExtA(U, U) = 1 − dim ExtA(U, U) ≤ 0, which is a contradiction. It follows from 1 1 0 < q(dimX) = dim EndA(X) − dim ExtA(X,X) = 1 − dim ExtA(X,X) that X cannot have self-extensions and furthermore dimX is a positive root. On the other hand, let α be a positive root. Let us prove that there exists an in- decomposable module X with dimX = α. To do this, consider an orbit OX of maximal dimension in RepQ(α) and let us prove that X is indecomposable. Otherwise, suppose that 1 1 X = U ⊕ V , for some nonzero U, V . By 6.7, we have that ExtA(U, V ) = ExtA(V,U) = 0, then 1 = q(α) = q(dimU) + q(dimV ) + hdimU, dimV i + hdimV, dimUi = q(dimU) + q(dimV ) + dim HomA(U, V ) + dim HomA(V,U) ≥ 2, which is a contradiction, so X must be indecomposable. Finally, if X, X0 are indecom- posable modules with the same dimension vector, then X =∼ X0 by 6.5(3). Teorema 8.2 (Gabriel, part I). If Q is a connected quiver with underlying graph Γ, then there is a finite number of indecomposable representations if and onlyif Γ is Dynkin. 9 CONCLUSIONS 24 Proof. If Γ is Dynkin then the indecomposable modules correspond to the positive roots, which by 7.11 are a finite number. Conversely, suppose that there is a finite number of indecomposable representations. Every module is a direct sum of indecomposables, so up to isomorphism, there is a finite n number of modules with vector dimensions 0 6= α ∈ N . Therefore there is a finite number of orbits in RepQ(α). By 6.5 (1) we have that q(α) > 0. We conclude by the classification of graphs given in Theorem 7.7. If we compute the roots of Dynkin diagrams, we can see that the number of isomorphic classes of representations of a quiver Q of finite type isASS06 ([ , VII.5.10.]): An Dm E6 E7 E8 1 2 2 n(n + 1) m − m 36 63 120 For quivers of tame type, there is a similar description in terms of Euclidean diagrams [DF73]: Teorema 8.3. A quiver Q which is not of finite type is tame if and only if its underlying graph is a union of Dynkin and Euclidean diagrams. One of the consequences of Gabriel’s Theorem is that the dimension vectors of the indecomposable modules of a quiver Q with an underlying Dynkin graph do not depend on the orientation of the arrows. The following result, proved by Kac in 1980, states that this property is true for an arbitrary quiver [Kac80]: Teorema 8.4. For an arbitrary quiver Q the set of dimension vectors of indecomposable representations of Q does not depend on the orientation of the arrows. The dimension vectors of indecomposable representations correspond to the positive roots of the associated Tits form. 9. Conclusions One of the main ideas which may conclude this work is that quiver representation theory techniques give us a convenient way to visualize finite dimensional algebras and their modules. However, we have not dealt with procedures that attack the problem of computing indecomposable modules and homomorphisms between them. Therefore, the content presented here can be a good introduction to start different lines of continuation of studies. On the one hand, we have the Auslander-Reiten Theory, originally introduced by M. Auslander in [Aus74], where powerful tools from Representation Theory are defined such as the Auslander Reiten sequences, which are exact sequences that in a certain sense are minimal among all the non-splitting exact sequences, or the quiver of Auslander, whose vertices are the indecomposable modules and arrows are the so-called irreducible morphisms. This theory allows us to know, thanks to a combinatorial algorithm called “knitting”, certain parts (sometimes even all) of the category of modules over an algebra. On the other hand, we have the Theory of tilting [HR82], where the basic idea is, in broad terms, to conveniently replace an algebra A whose representation theory is complicated to study with another algebra B, so that the categories A-Mod and B-Mod are similar but the problem can be addressed. REFERENCES 25 It is also important to remark that this project has been a first contact with math- ematics, perhaps not first line of research, but beyond undergraduate studies. However, there is an obvious relationship between what we have studied here and courses included in the mathematics degree program at University of Seville, among which I would highlight “Linear Algebra and Geometry I”, “Algebraic Structures”, “Commutative Algebra and Algebraic Geometry” and others beyond the Department of Algebra, such as “Simplicial Homology”, where a different use of homological algebra has helped me understand thever- satility of this mathematical theory. Undoubtedly, this work has given me the opportunity to extend my knowledge and have a more global vision of what I have learned these years. In particular, I would like to mention notions learned from category theory (although they are not used decisively in this project). I have found that the way in which certain typical objects from different mathematical theories share universal properties that can be stated in terms of composition of arrows is very interesting. From my point of view I would have liked being introduced to this language throughout my undergraduate studies. Finally, it is well known that Representation Theory plays an important role in Quan- tum Mechanics and especially in Particle Physics, fact that relates the content of this work with my studies in Physics. Quiver theory is also used in this context, as shown in this series of articles where it is applied in different fields of theoretical physics: [Ura01] (Particle Physics), [FM00] (Supersymmetry) and [DM96] (String Theory). References [ASS06] Ibrahim Assem, Andrzej Skowronski, and Daniel Simson. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Rep- resentation Theory, volume 65. Cambridge University Press, 2006. 6, 14, 24 [Aus74] Maurice Auslander. Representation theory of artin algebras ii. Communications in algebra, 1(4):269–310, 1974. 24 [Bar15] Michael Barot. Introduction to the representation theory of algebras. 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