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Quiver Representations 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 f1; 2; : : : ; ng. 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 = f1; 2; 3g, Q1 = fα; β; γ; λ, µg, 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 2 Q0 and a linear map Xρ : Xs(ρ) ! Xt(ρ) for each ρ 2 Q1.A morphism of representations 0 0 0 θ : X ! X is given by linear maps θi : Xi ! Xi for each i 2 Q0 such that Xρθs(ρ) = θt(ρ)Xρ for each ρ 2 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) i2Q0 ρ2Q1 defined by (φx)x2Q0 7! (φt(ρ)Xρ − Yρφs(ρ))ρ2Q1 , 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 2 Q0. 1 ρ Example 2.4. Let Q be the quiver 2 σ 3 . Consider the following represen- τ 4 tations of Q: 2 3 1 4 5 2 3 k 0 1 k 4 5 1 1 X : k2 k ;Y : k 1 k : 2 3 1 k 0 k 4 5 1 ∼ 2 In this case we have that HomQ(X; Y ) = k . Definition 2.5. Let X; Y 2 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 2 Q0, ρ 2 Q1. We say that a representation is trivial if Xi = 0 for all i 2 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 2 3 Ir 0 4 5 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.
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