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arXiv:0802.2993v1[math.OA]21Feb2008LiegroupextensionsassociatedtoprojectivemodulesofcontinuousinversealgebrasK.-H.NeebFebruary21,2008AbstractWecallaunitallocallyconvexalgebraAacontinuousinversealgebraifitsunitgroupA×isopenandinversionisacontinuousmap.Foranysmoothactionofa,possiblyinfinite-dimensional,connectedLiegroupGonacontinuousinversealgebraAbyautomorphismsandanyfinitelygeneratedprojectiverightA-moduleE,weconstructaLiegroupextensionbGofGbythegroupGLA(E)ofautomorphismsoftheA-moduleE.ThisLiegroupextensionisa“non-commutative”versionofthegroupAut(V)ofautomorphismofavectorbundleoveracompactmanifoldM,whicharisesforG=Diff(M),A=C∞(M,C)andE=ΓV.WealsoidentifytheLiealgebrabgofbGandexplainhowitisrelatedtoconnectionsoftheA-moduleE.AMCClassification:22E65,58B34Keywords:Continuousinversealgebra,infinitedimensionalLiegroup,vectorbundle,projectivemodule,semilinearautomorphism,covariantderivative,connectionIntroductionIn[ACM89]itisshownthatforafinite-dimensionalK-principalbundlePoveracompactmanifoldM,thegroupAut(P)ofallbundleautomorphismscarriesanaturalLiegroupstructurewhoseLiealgebraistheFr´echet–LiealgebraofV(P)KofK-invariantsmoothvectorfieldsonM.ThisappliesinparticulartothegroupAut(V)ofautomorphismsofafinite-dimensionalvectorbundlewithfiberVbecausethisgroupcanbeidentifiedwiththe1automorphismsgroupofthecorrespondingframebundleP=FrVwhichisaGL(V)-principalbundle.Inthispaper,weturntovariantsoftheLiegroupsAut(V)arisinginnon-commutativegeometry.Inviewof[Ko76],thegroupAut(V)canbeidentifiedwiththegroupofsemilinearautomorphismsoftheC∞(M,R)-moduleΓ(V)ofsmoothsectionsofV,which,accordingtoSwan’sTheorem,isafinitelygeneratedprojectivemodule.HerethegaugegroupGau(V)correspondstothegroupofC∞(M,R)-linearmoduleisomorphisms.Thissuggeststhefollowingsetup:Consideraunitallocallyconvexal-gebraAandafinitelygeneratedprojectiverightA-moduleE.WhencanweturngroupsofsemilinearautomorphismsofEintoLiegroups?Firstofall,wehavetorestrictourattentiontoanaturalclassofalgebraswhoseunitgroupsA×carrynaturalLiegroupstructures,whichisthecaseifA×isanopensubsetofAandtheinversionmapiscontinuous.Suchalge-brasarecalledcontinuousinversealgebras,CIAs,forshort.TheFr´echetalgebraC∞(M,R)isaCIAifandonlyifMiscompact.Thenitsautomor-phismgroupAut(C∞(M,R))∼=Diff(M)carriesanaturalLiegroupstructurewithLiealgebraV(M),theLiealgebraofsmoothvectorfieldsonM.An-otherimportantclassofCIAswhoseautomorphismgroupsareLiegroupsaresmooth2-dimensionalquantumtoriwithgenericdiophantineproperties(cf.[El86],[BEGJ89]).Unfortunately,ingeneral,automorphismgroupsofCIAsdonotalwayscarryanaturalLiegroupstructure,sothatitismuchmorenaturaltoconsidertriples(A,G,μA),whereAisaCIA,Gapossiblyinfinite-dimensionalLiegroup,andμA:G→Aut(A)agrouphomomorphismdefiningasmoothactionofGonA.Foranysuchtriple(A,G,μA)andanyfinitelygeneratedprojectiveA-moduleE,thesubgroupGEofallelementsofGforwhichμA(g)liftstoasemilinearautomorphismofEisanopensubgroup.Oneofourmainresults(Theorem3.3)isthatwethusobtainaLiegroupextension1→GLA(E)=AutA(E)→bGE→GE→1,wherebGEisaLiegroupactingsmoothlyonEbysemilinearautomorphisms.ForthespecialcasewhereMisacompactmanifold,A=C∞(M,R),E=Γ(V)forasmoothvectorbundleV,andG=Diff(M),theLiegroupbGisisomorphictothegroupAut(V)ofautomorphismsofthevectorbundleV,butourconstructioncontainsavarietyofotherinterestingsettings.Fromadifferentperspective,theLiegroupstructureonbGalsotellsusaboutpossible2smoothactionsofLiegroupsonfinitelygeneratedprojectiveA-modulesbysemilinearmapswhicharecompatiblewithasmoothactiononthealgebraA.AstartingpointofourconstructionistheobservationthattheconnectedcomponentsofthesetIdem(A)ofidempotentsofaCIAcoincidewiththeorbitsoftheidentitycomponentA×0ofA×undertheconjugationaction.UsingthenaturalmanifoldstructureonIdem(A)(cf.[Gram84]),theactionofA×onIdem(A)evenisasmoothLiegroupaction.OntheLiealgebraside,thesemilinearautomorphismsofEcorrespondtotheLiealgebraDEnd(E)ofderivativeendomorphisms,i.e.,thoseendomor-phismsϕ∈EndK(E)forwhichthereisacontinuousderivationD∈der(A)withϕ(s.a)=ϕ(s).a+s.(D.a)fors∈Eanda∈A.Theset\DEnd(E)ofallpairs(ϕ,D)∈EndK(E)×der(A)satisfyingthisconditionisaLiealgebraandweobtainaLiealgebraextension0→EndA(E)=glA(E)֒→\DEnd(E)→→der(A)→0.PullingthisextensionbackviatheLiealgebrahomomorphismg→der(A)inducedbytheactionofGonAyieldstheLiealgebrabgofthegroupbGfromabove(Proposition4.7).InSection5webrieflydiscusstherelationbetweenlinearsplittingsoftheLiealgebraextensionbgandcovariantderivativesinthecontextofnon-commutativegeometry(cf.[Co94],[MMM95],[DKM90]).Thanks:WethankHendrikGrundlingforreadingerlierversionsofthispaperandfornumerousremarkswhichleadtoseveralimprovementsofthepresentation.PreliminariesandnotationThroughoutthispaperwewriteI:=[0,1]fortheunitintervalinRandKeitherdenotesRorC.AlocallyconvexspaceEissaidtobeMackeycompleteifeachsmoothcurveγ:I→Ehasa(weak)integralinE.ForamoredetaileddiscussionofMackeycompletenessandequivalentconditions,wereferto[KM97,Th.2.14].ALiegroupGisagroupequippedwithasmoothmanifoldstructuremodeledonalocallyconvexspaceforwhichthegroupmultiplicationandtheinversionaresmoothmaps.Wewrite1∈Gforthe