de la PE NA Universidad Nacional Aut57524onoma de M57524exico M57524exico Proyecciones Vol 20 N 3 pp 323337 December 2001 Universidad Cat57524olica del Norte Antofagasta Chile Abstract Let kQI be a basic and connected 64257nite dimension algebra ov ID: 67385
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C.NovoaandJ.A.delaPe~naisarepresentationin¯nitetiltedalgebraoftype),withacompletesliceinthepreinjectivecomponent,and(respectively,)isanindecomposableregular-moduleofregularlength2(respectively,regularlength1)lyinginatubein¡having2rays,and+1(2)isthenumberofobjectsininN;t]whicharenotin2)Everyproperconvexsubcategoryofisofpolynomialgrowth.Inparticular,wesaythatthealgebrais2-tubularififM],whereM2indgDnisregularindecomposableoflength2lyinginatubein¡having2rays.Proposition2.1(in[P1]).Apg-criticalalgebraisderived-equivalenttoanalgebragivenbythefollowingquiver:Figure1.Withthecommutativerelations,indicatedbydottededges.bea¯nite-dimensional-algebra,andHom;kdenotethestandarddualityon.Therepetitivealgebra(see[HW])ofistheself-injective,locally¯nite-dimensionalalgebrawithoutidentity,de¯nedby:DAADAA OntherepresentationtypeofcertaintrivialextensionsFigure2.withcommutativityrelationsgivenby:x;v®y;xx®x;x;¯xyx;xWeconsiderednow,thefollowingclannishalgebra,givenbythefollowingquiver:Figure3. OntherepresentationtypeofcertaintrivialextensionsWeconsiderthealgebra,givenbythequiverinthefollowingFigure4.wherethedottedlinesiszerorelation.WehavethatthatM]wherethemoduleisde¯nedby:10:=00thatisliesinthemouththetubeofrank1,thealgebraisatiltedalgebraoftypethenthereexistafunctor¡!D),thatgiveatriangularequivalencesuchthat)lyiesinthetubeofrank,andisaregularoflenghtWeconsidernowthealgebrade¯nedbythefollowingquiver:Figure5. C.NovoaandJ.A.delaPe~naHerethemoduleisthesameasbefore.Then,byBarot-Lenzing(see[BL]),wehavethat:that:½(M)]),wherewhere½(M)]isaone-pointextensionbymodule)ofregularlength2,thenthen½(M)]).Usingthesimilarconstructionasthelemma31withclannishalgebrasisnotdi¯culttoseethatisequivalenttoaclannishalgebragivenbythefollowingquiver:Figure6.isspecialloop.ByGeissandDelaPe~na(see44in[GEP])thetrivialextension)istame,thus)istame.Since),therefore T(D)»=mod ),thenby[Kr]wehavethat)istame.Beforetostatethenextresult,weconsiderkQ=Ibeanalge-bra,whereisaquiverwithoutorientedcycles.Let¡!¡!bethequadraticformsde¯nedby:)=Ext;S)=i;j=(;:::;v),i;j)=I=IJJI,andistheidealgeneratedbyarrowsofthequiver.ThequadraticformcalledtheEulerformofthealgebra,andiscalledtheTitsform.ItsknowthatifgldimA2,then