A comparison of deformations and geometric study o

arXiv:math/0601766v1[math.RA]31Jan2006AcomparisonofdeformationsandgeometricstudyofassociativealgebrasvarietiesAbdenacerMAKHLOUFFebruary2,2008LaboratoiredeMath´ematiquesetApplications,Universit´edeHauteAlsace,Mulhouse,FranceEmail:[e-mail]iveanoverviewandtocomparethedifferentdefor-mationtheoriesofalgebraicstructures.Wedescribeineachcasethecorrespondingnotionsofdegenerationandrigidity.Weillustratethesenotionswithexamplesandgivesomegeneralproperties.Thelastpartofthisworkshowshowthesenotionshelpinthestudyofassociativealgebrasvarieties.ThefirstandpopulardeformationapproachwasintroducedbyM.Gerstenhaberforringsandalgebrasusingformalpowerseries.AnoncommutativeversionwasgivenbyPinczonandgeneralizedbyF.Nadaud.Amoregeneralapproachcalledglobaldeformationfollowsfromagen-eraltheoryofSchlessingerandwasdevelopedbyA.FialowskiinordertodeforminfinitedimensionalnilpotentLiealgebras.InaNonstandardframework,M.GozeintroducedthenotionofperturbationforstudyingtherigidityoffinitedimensionalcomplexLiealgebras.Alltheseapproacheshasincommonthefactthatwemakean”extension”ofthefield.Thesetheoriesmaybeappliedtoanymultilinearstructure.Weshallbeconcernedinthispaperwiththecategoryofassociativealgebras.AMSClassification2000:16-xx,13-xx,14-Rxx,14-Fxx,81-xx.1IntroductionThroughoutthispaperKwillbeanalgebraicallyclosedfield,andAdenotesanassociativeK-algebra.Mostexampleswillbeinthefinitedimensionalcase,letVbetheunderlyingn-dimensionalvectorspaceofAoverKand(e1,···,en)beabasisofV.ThebilinearmapμdenotesthemultiplicationofAonV,ande1istheunity.Bylinearitythiscanbedonebyspecifyingthen3structureconstantsCkij∈Kwhereμ(ei,ej)=Pnk=1Ckijek.Theassociativityconditionlimitsthesetsofstructureconstants,CkijtoasubvarietyofKn3whichwedenotebyAlgn.ItisgeneratedbythepolynomialrelationsnXl=1ClijCslk−CsilCljk=0andCj1i=Cji1=δij1≤i,j,k,s≤n.1Thisvarietyisquadratic,nonregularandingeneralnonreduced.ThenaturalactionofthegroupGL(n,K)correspondstothechangeofbasis:twoalgebrasμ1andμ2overVareisomorphicifthereexistsfinGL(n,K)suchthat:∀X,Y∈Vμ2(X,Y)=(f·μ1)(X,Y)=f−1(μ1(f(X),f(Y)))TheorbitofanalgebraAwithmultiplicationμ0,denotedbyϑ(μ0),isthesetofallitsisomorphicalgebras.Thedeformationtechniquesareusedtodothegeometricstudyofthesevarieties.Thedeformationattemptstounderstandwhichalgebrawecangetfromtheoriginalonebydeforming.Atthesametimeitgivesmoreinformationsaboutthestructureofthealgebra,forexamplewecanlookforwhichpropertiesarestableunderdeformation.Thedeformationofmathematicalobjectsisoneoftheoldesttechniquesusedbymathematicians.Thedifferentareaswherethenotionofdeformationappearsaregeometry,complexmanifolds(KodairaandSpencer1958,Kuranishi1962),algebraicmanifolds(Artin,Schlessinger1968),Liealgebras(Nijenhuis,Richardson1967)andringsandassociativealgebras(Gerstenhaber1964).Thedualnotionofdeformation(insomesense)isthatofdegenerationwhichappearsfirstinphysicsliterature(Segal1951,InonuandWigner1953).Degenerationisalsocalledspecialisationorcontraction.Thequantummechanicsandquantumgroupsgaveimpulsetothetheoryofde-formation.ThetheoryofquantizationwasintroducedbyBayen,Flato,Fronsdal,LichnerowiczandSternheimer[3]todescribethequantummechanicsasadefor-mationofclassicalmechanics.ItassociatestoaPoissonmanifoldastar-product,whichisaoneparameterfamilyofassociativealgebras.Inanotherhandthequan-tumgroupsareobtainedbydeformingtheHopfstructureofanalgebra,inparticularenvelopingalgebras.In[4],weshowtheexistenceofassociativedeformationofanenvelopingalgebrausingthelinearPoissonstructureoftheLiealgebra.Inthefollowing,wewillrecallfirstthedifferentnotionsofdeformation.ThemostusedoneistheformaldeformationintroducedbyGerstenhaberforringsandalgebras[13],itusesaformalseriesandconnectsthetheoryofdeformationtoHochschildcohomology.Anoncommutativeversion,wheretheparameternolongercommuteswiththeelementofalgebra,wasintroducedbyPinzson[37]andgeneralizedbyNadaud[33].TheydescribethecorrespondingcohomologyandshowthattheWeylalgebra,whichisrigidforformaldeformation,isnonrigidinthenoncommutativecase.InaNonstandardframework,GozeintroducedthenotionofperturbationforstudyingtherigidityofLiealgebras[17][1],itwasalsousedtodescribethe6-dimensionalrigidassociativealgebras[18][28].Theperturbationneedstheconceptofinfinitelysmallelements,theseelementsareobtainedhereinanalgebraicway.Weconstructanextensionofrealorcomplexnumberssetscontainingtheseelements.AmoregeneralnotioncalledglobaldeformationwasintroducedbyFialowski,followingSchlessinger,forLiealgebras[7].Alltheseapproachesarecomparedinsection3.Thesection4isdevotedtouniversalandversaldeformation.InSection5,wedescribethecorrespondingnotionsofdegenerationwithsomegeneralpropertiesandexamples.TheSection6isdevotedtothestudyofrigidityofalgebraineachframework.ThelastsectionconcernsthegeometricstudyofthealgebraicvarietiesAlgnusingthedeformationtools.2InformaldeformationthepropertiesaredescribedusingtheHochschildcoho-mologygroups.Theglobaldeformationseemstobethegoodframeworktosolvetheproblemofuniversalorversaldeformations,thedeformationswhichgeneratetheothers.Whereas,theperturb