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Lectures given at the summer school “Geometric methods in representation theory” Grenoble June July

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Niveau: Supérieur, Doctorat, Bac+8
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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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 classi cation 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-remofGabrielcharacterizingquiversof niterepresentationtype,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 classi cation triples of subspacesofaprescribedvectorspace,by nitelymanycombinatorialinvariants.The corresponding classi cation 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).
2000heattimaMalCt issuSsccejbnoacit 16G20; Secondary 14L30, 16G60.. Primary
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Gabrielstheoremholdsoveranarbitrary eld;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,onrepresentationsof nitelygener-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 overa xed eldk 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 de nitions concerning quivers and their representations are formulated in Sub-section1.1,andillustratedonthreeclassesofexamples.Inparticular,wede nequivers of nite representation type, and state their characterization in terms of Dynkin diagrams (Gabriel’s theorem). InSubsection1.2,wede nethequiveralgebra,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,wemakeonlythe rststepsintherepresentationtheoryof quiversand nite-dimensionalalgebras.Thereaderwill ndmorecompleteexpositionsin
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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 de nitions and examples
Definition1.1.1.Aquiveris a nite directed graph, possibly with multiple arrows and loops. More speci cally , 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 :ij, or byi  jwhen depicting the quiver. For example, the quiver with verticesi jand arrows :ijand 12:jjis depicted as follows: 1
i j 2
Definition1.1.2.ArepresentationMof a quiverQconsists of a family of vector spacesViindexed by the verticesiQ0, 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
2
whereV,Ware vector spaces, andf g1 g2are linear maps. Definition1.1.3.Given two representationsM=(Vi)iQ0(f ) Q1,N= (Wi g ) of a quiverQ, amorphismu:MNis a family of linear maps (ui:ViWi)iQ0such that the diagram Vs( )  f Vt( ) us( )yut( )y g Ws( ) →Wt( ) commutes for any Q1.
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For any two morphismsu:MNandv:NP, the family of compositions (viui)iQ0is a morphismvu:MP. This de nes the composition of morphisms, which is clearly associative and has identity elements idM:= (idVi)iQ0 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 ofQiQ0Hom(Vi Wi); we denote that subspace by HomQ(M N). IfM=N, then
EndQ(M) := HomQ(M N) is a subalgebra of the product algebraQiQ0End(Vi). Clearly,thecompositionofmorphismsisbilinear;also,wemayde nedirectsumsand 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 ) ofQis nite-dimensionalif so are all the vector spacesVi. Under that assumption, the family M:= (dimVi)iQ0 is thedimension vectorofM; it lies in the additive groupZQ0consisting of all tuples of integers = (ni)iQ0. We denote by (εi)iQ0the canonical basis ofZQ0, so that =PiQ0niεi. Notethateveryexactsequenceof nite-dimensionalrepresentations 0 →M0 →M →M00 →0
satis es
M M0+M00. =
Also,anytwoisomorphic nite-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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