Auslander-Reiten Components for Lie Algebras of Re

AUSLANDER-REITENCOMPONENTSFORLIEALGEBRASOFREDUCTIVEGROUPSRolfFarnsteiner0.IntroductionInthispaperweshallstudynonperiodicconnectedcomponentsofthestableAuslander-Reitenquiverofareducedenvelopingalgebrau(L;)associatedtoarestrictedLiealgebra(L;[p]).Accordingtoearlierresults(cf.[13,15])thetreeclassesoftheseAR-componentsareeitherEuclideandiagrams,ortheinniteDynkindiagramsA1,D1,A11.InthecontextofnitegroupsK.Erdmannhasrecentlyshownthatthelattertwotreescannotoccurforwildblocks(cf.[12]).Asdemonstratedin[13,18],asimilarresultholdsfortherestrictedenvelopingalgebrau(L):=u(L;0)incasetheunderlyingLiealgebraLisnilpotentandofcharacteristicp3.HereneithertameblocksnorcomponentsoftreeclassA11orD1exist.Forp=2therestrictedenvelopingalgebraofthe2-unipotentHeisenbergalgebrapossessescomponentsoftreeclassA11.Presently,noLiealgebraadmittingcomponentsoftreeclassD1isknown,andrecentwork[19]suggeststhatsuchcomponentswillberatherexceptional.Indefaultofageneralblocktheoryforenvelopingalgebrasoneisledtoeitheremploygeometrictechniques(cf.[19]),ortofocusonthosecases,wheretheblockstructureofu(L;)isgovernedbywell-understood\linkageprinciples(cf.[13,17,18]).TheformeraremosteectiveforLiealgebrasofalgebraicgroups,whilethelatterforinstanceplayar^oleintheAR-theoryofsupersolvableLiealgebras.Ourpapercanroughlybedividedintotwoparts.Thepurposeofsections1through3istofurnishbasicpropertiesofcomponentsofinniteDynkintype.ByusingamodicationofarecognitioncriterionduetoK.Erdmann[12],weprovideinsection2generalresultspertainingtotheaforementionedcomponents.ComponentsoftreeclassA1arediscussedinsection3.Inparticular,weshowthatsuchacomponentcontainsatmostonemoduleoflength2.01991MathematicsSubjectClassication:16G70,17B50SupportedbyN.S.A.GrantMDA904-96-1-0040andtheD.F.G.1ThesecondpartcombinestheseresultswithgeometrictechniquesincaseL=Lie(G)istheLiealgebraofanalgebraicgroupG.InthissettingtheadjointrepresentationAd:G!GL(L)inducesforeverylinearform2LanactionofitsstabilizerGonthesetofstableAuslander-Reitencomponentsofu(L;).Followingsometechnicalprepara-tions,weturninsection5tothestudyofAR-componentsofrestrictedenvelopingalgebrasforLiealgebrasofreductivegroups.Wefocusoninvariantcomponents,i.e.,thosethatarexedbytheaboveaction.Inviewof[20,(2.2)]thesearepreciselythosecomponentswhoseverticesareAd-stableinthesensethattwistingofthemodulebytheadjointrepresentationyieldsanisomorphicmodule.ComponentsofEuclideantreeclass,andthosecontainingindecomposableconstituentsofrationalG-modulesbelongtothisclass.Ourmainresults,Theorems5.2and5.5,showthatinvariantcomponentsareoftypeZ[A1]orZ[~A12],withthelatteroccurringonlyifSL(2)orPSL(2)isadirectfactorofG.Inparticular,Z[~A12]istheonlycomponentofEuclideantreeclass,aresultthatcontrastswiththeabsenceofsuchcomponentsforgroupalgebrasofnitegroupsofcharacteristicp3(cf.[40]).ThesamemethodsaordthedeterminationofthoseblocksoftherestrictedenvelopingalgebrasoftheLiealgebrasofreductivegroupsthatpossessasimplemoduleofcomplexity2(cf.(5.2)).Theseturnouttobefullmatrixalgebrasoverblocksofu(s‘(2)),therebyprovidinginthiscontextarenementofVoigt’swork(cf.[51])ontameenvelopingalgebras.InviewoftheMoritaequivalencegivenin[26],ourresultscontinuetoholdforreducedenvelopingalge-braswhosedeninglinearformsbelongtotheZariskidensesubsetofsemisimpleelements.However,fortheso-callednilpotentlinearformsdierentphenomenaareknowntooccur(cf.[17,44]).InthenalsectionweillustratehowrecentresultsbySuslin,FriedlanderandBendel[49]maybeemployedtostudyblocksofhigherFrobeniuskernelsofreductivegroups.Althoughsomeimportantfeaturesofthetheorynolongerholdinthiscontext,onecanstilldeterminetheMoritaequivalenceclassesofthetameblocks(cf.(7.1)).Workonthispaperwascompletedduringtheauthor’svisitattheSonderforschungsbe-reichoftheUniversityofBielefeld.IwouldliketotakethisopportunitytothankthemembersoftheMathematicsDepartmentingeneralandProfessorsRingelandVoigtinparticularfortheirhospitality.IamalsogratefultoAlexanderPremetforshowingmeconstructionsofnonsplitextensionsofsimplemodulesoveralgebraicgroups.1.PreliminariesThepurposeofthispreparatorysectionistomodifyvariousideasof[12]toobtaininforma-tionconcerningmodulesbelongingtocomponentsoftreeclassesA11;D1and~Dn.LetbeaFrobeniusalgebra,denedoveraeldF,withNakayamaautomorphism:!.Givena-moduleMandanautomorphism2Aut(),weletM()bethemodulewithunderlyingvectorspaceMandactionam:=1(a)m8a2;m2M:Fori2Z,weputM(i):=M(i).RecallthatthestableAuslander-Reitenquivers()hastheisomorphismtypes[M]ofnonprojectiveindecomposable-modulesasitsvertices.By2generaltheory(cf.[3,p.138])theAuslander-Reitentranslationofs()isgivenby([M])=2([M(1)])8[M]2s():HeredenotestheHelleroperator.Barringpossibleambiguitiesweshallwrite:=.Thereaderisreferredto[2]concerningbasicpropertiesofAR-quivers.Aninjectivehomomorphismg:M!Nbetweentwoindecomposable-modulesiscalledleftproperlyirreducibleifitisirreducible,butnotleftalmostsplit.Dually,asurjectivehomomorphismg:M!Nisreferredtoasrightproperlyirreducibleifitisirreducible,butnotrightalmostsplit.Wesaythatavert