Jordan structures and non associative geometry

Wolfgang Bertram

Description

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

Sujets

• Bertram
• Cartan
• Álgebra
• Lie algebra

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 12
• Langue: English

Extrait

Jordanstructuresandnon-associative geometry WolfgangBertram InstitutElieCartan,Universite´NancyI,Faculte´desSciences,B.P.239,54506 Vandœuvre-le`s-Nancy,Cedex,France [e-mail] Abstract. WegiveanoverviewofconstructionsofgeometriesassociatedtoJor- danstructures(algebras,triplesystemsandpairs),featuringanalogsofthesecon- structionswiththeLiefunctorontheonehandandwiththeapproachofnon- commutativegeometryontheotherhand. Keywords: Jordanpair,Lietriplesystem,gradedLiealgebra,filteredLiealge- bra,generalizedprojectivegeometry,flaggeometries,(non-)associativealgebraand geometry AMSsubjectclassification: 17C37,17B70,53C15 Introduction Letuscomparetwoaspectsofthevastmathematicaltopic“linksbetweenge- ometryandalgebra”:ontheonehand,theLiefunctorestablishesacloserela- tionbetweenLiegroups(geometricside)andLiealgebras(algebraicside);this isgeneralizedbyacorrespondencebetweensymmetricspacesandLietriple systems(see[Lo69]).Ontheotherhand,thephilosophyofNon-Commutative Geometrygeneralizestherelationbetweenusual,geometricpoint-spacesM (e.g.,manifolds)andthecommutativeandassociativealgebraReg(M, K )of “regular”(e.g.,bounded,smooth,algebraic,...,accordingtothecontext) K - valuedfunctionsonM,where K = R or C ,byreplacingthealgebraReg(M, K ) bymoregeneral,possiblynon-commutativealgebras A .Theinteractionbe- tweenthesetwoaspectsseemstoberatherweak;indeed,insomesensethey are“orthogonal”toeachother:first,theclassicalsettingofLie’sthirdthe- oremisfinite-dimensional,whereasthealgebrasAofcommutativeornon- commutativegeometryaretypicallyinfinitedimensional;second,andmore importantly,takingacommutativeandassociativealgebraasinput,weob- tainasLiebracket[x,y]=xy−yx=0,andhence(ifweforgettheassociative structureandretainonlytheLiebracket)weareleftwithconstructingaLie

2WolfgangBertram groupwithzeroLiebracket.Butthisisnotveryinteresting:itdoesnoteven capturethespecificinformationencodedin“commutativegeometries”. 1 Thisremarksuggeststhat,ifonelooksforalinkbetweenthetwoaspects justmentioned,itshouldbeinterestingtoaskforananalogoftheLiefunc- torforaclassofalgebrasthatcontainsfaithfullytheclassofcommutative associativealgebras,butrathersacrificesassociativitythancommutativity. AtthispointletusnotethatthealgebrasAofnon-commutativegeometry areofcoursealwayssupposedtobeassociative,sothattheterm“associative geometry”mightbemoreappropriatethan“non-commutativegeometry”. Indeed,behindthegardenofassociativealgebrasstartstherealmofgeneral algebras,genericallyneitherassociativenorcommutative,and,forthetime being,nobodyhasanideaofwhattheir“geometricinterpretation”might be.Fortunately,twooffspringsofassociativealgebrasgrownottoofaraway behindthegardenwalls:Liealgebrasrightontheotherside(thebranches reachoverthewallsoabundantlythatsomepeopleevenconsiderthemas stillbelongingtothegarden),andJordanalgebrasabitfurther.Letusjust recallthattheformeraretypicallyobtainedbyskew-symmetrizinganasso- ciativealgebra,[x,y]=xy−yx,whereasthelatteraretypicallyobtainedby symmetrizingthem,x•y= 2 1 (xy+yx)(thefactor 2 1 beingconventional,in ordertoobtainthesamepowersx k asintheassociativealgebra). SoletuslookatJordanalgebras–theiradvantagebeingthattheclassof commutativeassociativealgebrasisfaithfullyembedded.Inthissurveypaper wewillexplaintheirgeometricinterpretationviacertaingeneralizedprojec- tivegeometries,emphasizingthatthisinterpretationreallycombinesboth aspectsmentionedabove.Ofcourse,itisthenimportanttoincludethecase ofinfinite-dimensionalalgebras,andtotreattheminessentiallythesameway asfinite-dimensionalones.Thisisbestdoneinapurelyalgebraicframework, leavingasideallquestionsoftopologizingouralgebrasandgeometricspaces. Ofcourse,suchquestionsformaninterestingtopicforthefurtherdevelopment ofthetheory:inaverygeneralsetting(topologicalalgebraicstructuresover generaltopologicalfieldsorrings)basicresultsaregivenin[BeNe05],andcer- tainlymanyresultsfromthemorespecificsettingofJordanoperatoralgebras (Banach-Jordanstructuresover K = R or C ;see[HOS84])admitinteresting geometricinterpretationsinourframework.BeyondtheBanach-setting,it shouldbeinterestingtodevelopatheoryoflocallyconvextopologicalJordan structuresandtheirgeometries,takinguptheassociativetheory(cf.[Bil04]). ThegeneralconstructionforJordanalgebras(andforothermembersofthe familyofJordanalgebraicstructures,namelyJordanpairsandJordantriple systems,whichinsomesenseareeasiertounderstandthanJordanalgebras; cf.Section1)beingexplainedinthemaintext(Section2andSection3),letus 1 Ofcourse,thisdoesnotexcludethat other waysofassociatingLiegroupsto functionalgebrasareinteresting,forexamplebylookingattheirderivationsand automorphisms;butourpointispreciselythatsuchconstructionsneedmore inputthanthetrivialLiebracketonthealgebraoffunctions.