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Representations of quivers Michel Brion Lectures given at the summer school “Geometric methods in representation theory” (Grenoble, June 16 – July 4, 2008) Introduction Quivers are very simple mathematical objects: finite directed graphs. A representation of a quiver assigns a vector space to each vertex, and a linear map to each arrow. Quiver representations were originally introduced to treat problems of linear algebra, for exam- ple, the classification of tuples of subspaces of a prescribed vector space. But it soon turned out that quivers and their representations play an important role in representa- tion theory of finite-dimensional algebras; they also occur in less expected domains of mathematics including Kac-Moody Lie algebras, quantum groups, Coxeter groups, and geometric invariant theory. These notes present some fundamental results and examples of quiver representations, in its algebraic and geometric aspects. Our main goal is to give an account of a theo- rem of Gabriel characterizing quivers of finite representation type, that is, having only finitely many isomorphism classes of representations in any prescribed dimensions: such quivers are exactly the disjoint unions of Dynkin diagrams of types An, Dn, E6, E7, E8, equipped with arbitrary orientations. Moreover, the isomorphism classes of indecompos- able representations correspond bijectively to the positive roots of the associated root system. This beautiful result has many applications to problems of linear algebra.

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- unique arrow
- concerning quivers
- finite-dimensional representations
- representation theory
- zq0
- quiver representations
- has identity
- ext groups
- result has many

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Representations of

Michel Brion

Lectures given at the summer school “Geometric (Grenoble, June 16 – July 4, 2008)

Introduction

quivers

methods in representation

theory”

Quivers are very simple mathematical objects: nite directed graphs. A representation of a quiver assigns a vector space to each vertex, and a linear map to each arrow. Quiver

representations were originally introduced to treat problems of linear algebra, for exam-ple, the classication of tuples of subspaces of a prescribed vector space. But it soon turned out that quivers and their representations play an important role in representa-tion theory of nite-dimensional algebras; they also occur in less expected domains of mathematics including Kac-Moody Lie algebras, quantum groups, Coxeter groups, and geometric invariant theory. These notes present some fundamental results and examples of quiver representations, in its algebraic and geometric aspects. Our main goal is to give an account of a theo-remofGabrielcharacterizingquiversofniterepresentationtype,thatis,havingonly nitelymanyisomorphismclassesofrepresentationsinanyprescribeddimensions:such quivers are exactly the disjoint unions of Dynkin diagrams of typesAn,Dn,E6,E7,E8, equipped with arbitrary orientations. Moreover, the isomorphism classes of indecompos-able representations correspond bijectively to the positive roots of the associated root system. This beautiful result has many applications to problems of linear algebra. For example, when applied to an appropriate quiver of typeD4 of, it yields a classication triples of subspacesofaprescribedvectorspace,bynitelymanycombinatorialinvariants.The corresponding classication for quadruples of subspaces involves one-parameter families (the so-called tame case); forr-tuples withr5, one obtains families depending on an arbitrary number of parameters (the wild case).

2000heattimaMalCtissuSsccejbnoacit 16G20; Secondary 14L30, 16G60.. Primary

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Gabriel’stheoremholdsoveranarbitraryeld;inthesenotes,weonlyconsideral-gebraically closed elds, in order to keep the prerequisites at a minimum. Section 1 is devoted to the algebraic aspects of quiver representations; it requires very little back-ground. The geometric aspects are considered in Section 2, where familiarity with some anealgebraicgeometryisassumed.Section3,onrepresentationsofnitelygener-atedalgebras,isabitmoreadvanced,asituses(andillustrates)basicnotionsofane schemes. The reader will nd more detailed outlines, prerequisites, and suggestions for further reading, at the beginning of each section. Many important developments of quiver representations fall beyond the limited scope of these notes; among them, we mention Kac’s far-reaching generalization of Gabriel’s theorem (exposed in [10]), and the construction and study of moduli spaces (surveyed in the notes of Ginzburg, see also [16]).

Conventionsnotes, we consider vector spaces, linear maps, algebras, these . Throughout overaxedeldk All, assumed to be algebraically closed. algebras are assumed to be associative, with unit; modules are understood to be left modules, unless otherwise stated.

1 Quiver representations: the algebraic approach

In this section, we present fundamental notions and results on representations of quivers and of nite-dimensional algebras. Basic denitions concerning quivers and their representations are formulated in Sub-section1.1,andillustratedonthreeclassesofexamples.Inparticular,wedenequivers of nite representation type, and state their characterization in terms of Dynkin diagrams (Gabriel’s theorem). InSubsection1.2,wedenethequiveralgebra,andidentifyitsrepresentationswith those of the quiver. We also brie
y consider quivers with relations. The classes of simple, indecomposable, and projective representations are discussed in Subsection 1.3, in the general setting of representations of algebras. We illustrate these notions with results and examples from quiver algebras. Subsection 1.4 is devoted to the standard resolutions of quiver representations, with applications to extensions and to the Euler and Tits forms. The prerequisites are quite modest: basic material on rings and modules in Subsec-tions 1.1-1.3; some homological algebra (projective resolutions, Ext groups, extensions) in Subsection 1.4. We generally provide complete proofs, with the exception of some classical results for whichwereferto[3].Thereby,wemakeonlytherststepsintherepresentationtheoryof quiversandnite-dimensionalalgebras.Thereaderwillndmorecompleteexpositionsin

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the books [1, 2, 3] and in the notes [4]; the article [5] gives a nice overview of the subject.

1.1 Basic denitions and examples

Definition1.1.1.Aquiveris a nite directed graph, possibly with multiple arrows and loops. More specically , a quiver is a quadruple

Q= (Q0 Q1 s t) whereQ0,Q1 (the set of setsare nitevertices, resp.arrows) and

s t:Q1 →Q0 are maps assigning to each arrow itssource, resp.target.

We shall denote the vertices by lettersi j . . . arrow with source. Aniand targetj will be denoted by:i→j, or byi →jwhen depicting the quiver. For example, the quiver with verticesi jand arrows:i→jand1 2:j→jis depicted as follows: 1

ij 2

Definition1.1.2.ArepresentationMof a quiverQconsists of a family of vector spacesViindexed by the verticesi∈Q0, together with a family of linear mapsf:Vs()→ Vt()indexed by the arrows∈Q1. For example, a representation of the preceding quiver is just a diagram g1

V

f

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whereV,Ware vector spaces, andf g1 g2are linear maps. Definition1.1.3.Given two representationsM=(Vi)i∈Q0(f)∈Q1,N= (Wi g) of a quiverQ, amorphismu:M→Nis a family of linear maps (ui:Vi→Wi)i∈Q0such that the diagram Vs() f→Vt() us()yut() y g Ws() →Wt() commutes for any∈Q1.

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For any two morphismsu:M→Nandv:N→P, the family of compositions (viui)i∈Q0is a morphismvu:M→P. This denes the composition of morphisms, which is clearly associative and has identity elements idM:= (idVi)i∈Q0 we may consider the. So category of representations ofQ, that we denote by Rep(Q). Given two representationsM,Nas above, the set of all morphisms (of representa-tions) fromMtoNis a subspace ofQi∈Q0Hom(Vi Wi); we denote that subspace by HomQ(M N). IfM=N, then

EndQ(M) := HomQ(M N) is a subalgebra of the product algebraQi∈Q0End(Vi). Clearly,thecompositionofmorphismsisbilinear;also,wemaydenedirectsumsand exact sequences of representations in an obvious way. In fact, one may check that Rep(Q) is ak-linear abelian category; this will also follow from the equivalence of Rep(Q) with the category of modules over the quiver algebrakQ, see Proposition 1.2.2 below.

Definition1.1.4.A representationM= (Vi f) ofQisnite-dimensionalif so are all the vector spacesVi. Under that assumption, the family M:= (dimVi)i∈Q0 is thedimension vectorofM; it lies in the additive groupZQ0consisting of all tuples of integers = (ni)i∈Q0. We denote by (εi)i∈Q0the canonical basis ofZQ0, so that =Pi∈Q0niεi. Notethateveryexactsequenceofnite-dimensionalrepresentations 0 →M0 →M →M00 →0

satises

M M0+M00. =

Also,anytwoisomorphicnite-dimensionalrepresentationshavethesamedimension vector. A central problem of quiver theory isto describe the isomorphism classes of nite-dimensionalrepresentationsofaprescribedquiver,havingaprescribeddimension vector.

Examples1.1.5.1) Theloopis the quiverLhaving a unique vertexiand a unique arrow(thens() =t() =i). Thus, a representation ofLis a pair (V f), whereVis a vector space andfan endomorphism ofV; the dimension vector is just the dimension ofV.

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