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Jordan structures and non associative geometry

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21 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
Jordan structures and non-associative geometry Wolfgang Bertram Institut Elie Cartan, Universite Nancy I, Faculte des Sciences, B.P. 239, 54506 Vandœuvre-les-Nancy, Cedex, France Abstract. We give an overview of constructions of geometries associated to Jor- dan structures (algebras, triple systems and pairs), featuring analogs of these con- structions with the Lie functor on the one hand and with the approach of non- commutative geometry on the other hand. Keywords: Jordan pair, Lie triple system, graded Lie algebra, filtered Lie alge- bra, generalized projective geometry, flag geometries, (non-) associative algebra and geometry AMS subject classification: 17C37, 17B70, 53C15 Introduction Let us compare two aspects of the vast mathematical topic “links between ge- ometry and algebra”: on the one hand, the Lie functor establishes a close rela- tion between Lie groups (geometric side) and Lie algebras (algebraic side); this is generalized by a correspondence between symmetric spaces and Lie triple systems (see [Lo69]). On the other hand, the philosophy of Non-Commutative Geometry generalizes the relation between usual, geometric point-spaces M (e.g., manifolds) and the commutative and associative algebra Reg(M,K) of “regular” (e.

  • reg

  • algebra

  • associative geometry

  • geometric space

  • banach-jordan

  • graded lie

  • commutative geometry

  • purely geometric

  • jordan pairs

  • lie algebras


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Jordanstructuresandnon-associativegeometryWolfgangBertramInstitutElieCartan,Universite´NancyI,Faculte´desSciences,B.P.239,54506Vandœuvre-le`s-Nancy,Cedex,Francebertram@iecn.u-nancy.frAbstract.WegiveanoverviewofconstructionsofgeometriesassociatedtoJor-danstructures(algebras,triplesystemsandpairs),featuringanalogsofthesecon-structionswiththeLiefunctorontheonehandandwiththeapproachofnon-commutativegeometryontheotherhand.Keywords:Jordanpair,Lietriplesystem,gradedLiealgebra,filteredLiealge-bra,generalizedprojectivegeometry,flaggeometries,(non-)associativealgebraandgeometryAMSsubjectclassification:17C37,17B70,53C15IntroductionLetuscomparetwoaspectsofthevastmathematicaltopic“linksbetweenge-ometryandalgebra”:ontheonehand,theLiefunctorestablishesacloserela-tionbetweenLiegroups(geometricside)andLiealgebras(algebraicside);thisisgeneralizedbyacorrespondencebetweensymmetricspacesandLietriplesystems(see[Lo69]).Ontheotherhand,thephilosophyofNon-CommutativeGeometrygeneralizestherelationbetweenusual,geometricpoint-spacesM(e.g.,manifolds)andthecommutativeandassociativealgebraReg(M,K)of“regular”(e.g.,bounded,smooth,algebraic,...,accordingtothecontext)K-valuedfunctionsonM,whereK=RorC,byreplacingthealgebraReg(M,K)bymoregeneral,possiblynon-commutativealgebrasA.Theinteractionbe-tweenthesetwoaspectsseemstoberatherweak;indeed,insomesensetheyare“orthogonal”toeachother:first,theclassicalsettingofLie’sthirdthe-oremisfinite-dimensional,whereasthealgebrasAofcommutativeornon-commutativegeometryaretypicallyinfinitedimensional;second,andmoreimportantly,takingacommutativeandassociativealgebraasinput,weob-tainasLiebracket[x,y]=xyyx=0,andhence(ifweforgettheassociativestructureandretainonlytheLiebracket)weareleftwithconstructingaLie
2WolfgangBertramgroupwithzeroLiebracket.Butthisisnotveryinteresting:itdoesnotevencapturethespecificinformationencodedin“commutativegeometries”.1Thisremarksuggeststhat,ifonelooksforalinkbetweenthetwoaspectsjustmentioned,itshouldbeinterestingtoaskforananalogoftheLiefunc-torforaclassofalgebrasthatcontainsfaithfullytheclassofcommutativeassociativealgebras,butrathersacrificesassociativitythancommutativity.AtthispointletusnotethatthealgebrasAofnon-commutativegeometryareofcoursealwayssupposedtobeassociative,sothattheterm“associativegeometry”mightbemoreappropriatethan“non-commutativegeometry”.Indeed,behindthegardenofassociativealgebrasstartstherealmofgeneralalgebras,genericallyneitherassociativenorcommutative,and,forthetimebeing,nobodyhasanideaofwhattheir“geometricinterpretation”mightbe.Fortunately,twooffspringsofassociativealgebrasgrownottoofarawaybehindthegardenwalls:Liealgebrasrightontheotherside(thebranchesreachoverthewallsoabundantlythatsomepeopleevenconsiderthemasstillbelongingtothegarden),andJordanalgebrasabitfurther.Letusjustrecallthattheformeraretypicallyobtainedbyskew-symmetrizinganasso-ciativealgebra,[x,y]=xyyx,whereasthelatteraretypicallyobtainedbysymmetrizingthem,xy=21(xy+yx)(thefactor21beingconventional,inordertoobtainthesamepowersxkasintheassociativealgebra).SoletuslookatJordanalgebras–theiradvantagebeingthattheclassofcommutativeassociativealgebrasisfaithfullyembedded.Inthissurveypaperwewillexplaintheirgeometricinterpretationviacertaingeneralizedprojec-tivegeometries,emphasizingthatthisinterpretationreallycombinesbothaspectsmentionedabove.Ofcourse,itisthenimportanttoincludethecaseofinfinite-dimensionalalgebras,andtotreattheminessentiallythesamewayasfinite-dimensionalones.Thisisbestdoneinapurelyalgebraicframework,leavingasideallquestionsoftopologizingouralgebrasandgeometricspaces.Ofcourse,suchquestionsformaninterestingtopicforthefurtherdevelopmentofthetheory:inaverygeneralsetting(topologicalalgebraicstructuresovergeneraltopologicalfieldsorrings)basicresultsaregivenin[BeNe05],andcer-tainlymanyresultsfromthemorespecificsettingofJordanoperatoralgebras(Banach-JordanstructuresoverK=RorC;see[HOS84])admitinterestinggeometricinterpretationsinourframework.BeyondtheBanach-setting,itshouldbeinterestingtodevelopatheoryoflocallyconvextopologicalJordanstructuresandtheirgeometries,takinguptheassociativetheory(cf.[Bil04]).ThegeneralconstructionforJordanalgebras(andforothermembersofthefamilyofJordanalgebraicstructures,namelyJordanpairsandJordantriplesystems,whichinsomesenseareeasiertounderstandthanJordanalgebras;cf.Section1)beingexplainedinthemaintext(Section2andSection3),letus1Ofcourse,thisdoesnotexcludethatotherwaysofassociatingLiegroupstofunctionalgebrasareinteresting,forexamplebylookingattheirderivationsandautomorphisms;butourpointispreciselythatsuchconstructionsneedmoreinputthanthetrivialLiebracketonthealgebraoffunctions.