lectures on representations of quivers

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1、LecturesonRepresentationsofQuiversbyWilliamCrawley-BoeveyContents§1.Pathalgebras......................3§2.Bricks.........................9§3.Thevarietyofrepresentations.............11§4.DynkinandEuclideandiagrams..............15§5.Finiterepresentation

2、type...............19§6.Morehomologicalalgebra................21§7.Euclideancase.Preprojectivesandpreinjectives....25§8.Euclideancase.Regularmodules.............28§9.Euclideancase.Regularsimplesandroots........32§10.Furthertopics.....................3

3、61A'quiver'isadirectedgraph,andarepresentationisdefinedbyavectorspaceforeachvertexandalinearmapforeacharrow.Thetheoryofrepresentationsofquiverstoucheslinearalgebra,invarianttheory,finitedimensionalalgebras,freeidealrings,Kac-MoodyLiealgebras,andmanyot

4、herfields.ThesearethenotesforacourseofeightlecturesgiveninOxfordinspring1992.Myaimwastheclassificationoftherepresentationsforthe~~~~~EuclideandiagramsA,D,E,E,E.Itseemedambitiousforeightnn678lectures,butturnedouttobeeasierthanIexpected.TheDynkincaseisa

5、nalysedusinganargumentofJ.Tits,P.GabrielandC.M.Ringel,whichinvolvesactionsofalgebraicgroups,astudyofrootsystems,andsomecleverhomologicalalgebra.TheEuclideancaseistreatedusingthesametools,andinadditiontheAuslander-Reitentranslations-,,andthenotionofa

6、'regularuniserialmodule'.Ihaveavoidedtheuseofreflectionfunctors,Auslander-Reitensequences,andcase-by-caseanalyses.Theprerequisitesforthiscoursearequitemodest,consistingofthebasic1notionsaboutringsandmodules;alittlehomologicalalgebra,uptoExtandlongexac

7、tsequences;theZariskitopologyon
n;andmaybesomeideasfromcategorytheory.InthelastsectionIhavelistedsometopicswhicharetheobjectofcurrentresearch.Ihopetheselecturesareausefulpreparationforreadingthepaperslistedthere.WilliamCrawley-Boevey,MathematicalInsti

8、tute,OxfordUniversity24-29St.Giles,OxfordOX13LB,England2§1.PathalgebrasOnceandforall,wefixanalgebraicallyclosedfieldk.DEFINITIONS.(1)AquiverQ=(Q,Q,s,t:Q
011Q0)isgivenbyasetQofvertices,whichforuswillbe{1,2,...,n},and0asetQofarrows,whi