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Universitéd’Abomey-Calavi(UAC), Bénin Institutde Mathématiques etde Sciences Physiques(IMSP), PortoNovo A dissertationsubmittedinpartial fulfillmentoftherequirementforthedegreeof DOCTORes-Sciences to the Universitéd’AbomeyCalavi 4 by 1 0 AncilleNGENDAKUMANA 2 n a J GroupTheoretical Construction ofPlanarNoncommutative 1 2 Systems ] h p - h t Jury a m President : Prof. Norbert HOUNKONNOU,CIPMA/UAC, Bénin [ Rapporters : Prof. Peter HORVATHY,UniversitédeTours,France 1 v : Prof. ParthaGUHA,S. N.Bose NationalCenterforBasicSciences, Inde 3 1 2 : Prof. JoachimNZOTUNGICIMPAYE,UniversityofRwanda, Rwanda 5 . : Prof. Leonard TODJIHOUNDE,IMSP/UAC, Bénin 1 0 4 Examiner : Prof. JoelTOSSA, IMSP/UAC, Bénin 1 : Directors : Prof. JoachimNZOTUNGICIMPAYE,UniversityofRwanda, Rwanda v i X : Prof. Leonard TODJIHOUNDE,IMSP/UAC, Bénin r a December2013 DEDICATION To Cyprien HORUGAVYE, DonDorel MUHINTAHE, Cyriel HORUGAVYE, DélissaNIYERA, Josh-MickyIHEZAGIRE. ACKNOWLEDGEMENTS IfirstandforemostexpressmyheartfeltthankstomyadvisorsProfessorJoachimNZO- TUNGICIMPAYEandProfessorLeonardTODJIHOUNDEwhoseguidance,unlimited patience and constant encouragement made this thesis possible. Their insight, passion forMathematicsand theirquestforperfection inspiredmeall alongourcollaboration. Furthermore,theircriticalfeedbackandinputallcontributedveryfundamentallytothis accomplishment. Iamthankfultomythesisreporters: ProfessorPeterHORVATHY,ProfessorPartha GUHA,ProfessorJoachimNZOTUNGICIMPAYEandProfessorLeonardTODJIHOU- NDE for their careful work in reading this thesis. Their remarks and suggestionshave considerably contributed to the improvement of this work. To my thesis committee membersProfessorNorbertHOUNKONNOUandProfessorJoelTOSSA,Iwishtoex- pressmy gratefulness. Bybeingatthe“InstitutdeMathématiquesetdeSciencesPhysiques”(IMSP),Imet a huge number of researchers, Physicists and Mathematicians. All interesting interac- tionsandexchangesthatwehadhadhelpedmetobecomeaMathematician. Ithankall the IMSP group of professors for contributing to the very positivework environment I found. Thankyousoverymuchforyourconstantencouragement. Iwouldalsoliketo thankMrsGisèleFOHOUNHEDOBANKOLEforalwaysbeinghelpfulwithadminis- trativeissues. Ialsowishtoexpressmygratefulnesstomycurrentofficemates:Joseph SalomonMBATAKOU,JonasDOUMATE,OumarSOWand Van Borhen NKOU with whomIhad interestingconversationsabout Mathematicsand much more.To DrFrank DJIDEME, Dr Stephane TCHUIAGA, Joel KPLE and Toussaint OKE for your help in LaTeX,Ithankyouallverymuch. IwouldalsoliketothankallresearchstudentsImet at IMSPformakingmystayin Benin enjoyable. God blessyouall. I would also liketo thank people who haveparticipated mostdirectly in my formation and initiated me into the fascinating world of Mathematics. I think about Profes- sor Joachim NZOTUNGICIMPAYE, Professor Jean NDIMUBANDI, Professor Jean- Bosco KAYOYA, Professor Gaspard BANGEREZAKO and Professor Isidore MA- HARA. I am also grateful to the Government of BURUNDI for its constant financial sup- port. I also wish to express my gratefulness to Dr Gaspard NTAHONKIRIYE and his familyforsupportingmyregularstaysinKigali(Rwanda)duringmy formation. I am thankful also to Burundian students in Bénin for their constant encourage- ments. In Particular I would like to thank Dr Donatien GAPARAYI, Charles GAT- URAGIand Jean-Berchmans BIZIMANA formakingmy stayinBénin so enjoyable. Last but not least, I thank my family who has endured my absence for years. They have always shown a lot of patience and devotion. Thank you so much dear mother, husband,children,sisters,sistersandbrothers-in-law. ABSTRACT In this thesis, we construct and classify planar noncommutative phase spaces by the coadjoint orbit method on the anisotropic and absolute time kinematical groups. We showthat noncommutativesymplecticstructures can be generated in the framework of centrally extended anisotropic kinematical groups as well as in the framework of non- centrallyabelian extendedabsolutetimekinematicalgroups. However,noncommutativephasespacesrealizedwithnoncentralabelianextensions ofthekinematicalgroupsarealgebraicallymoregeneralthanthoseconstructedontheir central extensions. As the coadjoint orbit construction has not been carried through someoftheseplanarkinematicalgroupsbefore,physicalinterpretationsofnewgenera- torsofthoseextendedstructuresaregiven. Furthermore,inallthecasesdiscussedhere, the noncommutativity is measured by naturally introduced fields, each corresponding toaminimalcoupling. This approach allows to not only construct directly a dynamical system when of course the symmetry group is known but also permits to eliminate the non minimal couplings in that system. Hence, we show also that the planar noncommutativephase spaces arise naturally by introducing minimal coupling. We introduce here new kinds of couplings. A coupling of position with a dual potential and a mixing model (that is minimal coupling of the momentum with a magnetic potential and of position with a dualpotential). Finally we show that this group theoretical discussion can be recovered by a linear deformation of the Poisson bracket. The reason why linear deformation of Poisson bracket is required here is that the noncommutative parameters (which are fields) are constant(theyarecomingfromcentralandnoncentralabelianextensionsofkinematical groups). Contents INTRODUCTION 1 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 PLANARKINEMATICALGROUPS,THEIRCENTRALANDNONCEN- TRAL EXTENSIONS 8 1.1 Possiblekinematicalgroups . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Bacry and Lévy-Leblondapproach . . . . . . . . . . . . . . . . 11 1.1.2 Planar anisotropickinematicalalgebras . . . . . . . . . . . . . 18 1.2 Extensionsofplanaranisotropickinematicalalgebras . . . . . . . . . . 19 1.2.1 mathematicalpreliminaries:abelian extensionsofLiealgebras 20 1.2.2 Central extensionsofplanaranisotropickinematicalalgebras . . 24 1.2.3 Noncentralabelianextensionsoftheabsolutetimeplanarkine- maticalalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 NONCOMMUTATIVEPHASESPACESBY MINIMALCOUPLINGS 32 2.1 NoncommutativePhaseSpaces . . . . . . . . . . . . . . . . . . . . . . 34 2.1.1 Symplecticmechanics onLiegroupsincommutativecoordinates 34 2.1.2 Lie-Poissonstructure . . . . . . . . . . . . . . . . . . . . . . 36 2.1.3 Noncommutativecoordinates . . . . . . . . . . . . . . . . . . 37 2.2 CouplingsinPlanar Mechanics . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 CouplingofmomentumwithamagneticField . . . . . . . . . 39 2.2.2 Couplingofpositionwithadualfield . . . . . . . . . . . . . . 41 2.2.3 Couplingwithamagneticfield and withadual magneticfield . 44 3 NONCOMMUTATIVEPHASESPACESCONSTRUCTEDGROUPTHE- ORETICALLY 46 3.1 Coadjointorbitand symplecticrealizationmethods . . . . . . . . . . . 48 3.1.1 coadjointorbitmethod . . . . . . . . . . . . . . . . . . . . . . 48 3.1.2 Symplecticrealizations . . . . . . . . . . . . . . . . . . . . . . 51 CONTENTS 3.2 NoncommutativephasespaceonAristotlegroup . . . . . . . . . . . . 51 3.2.1 Aristotlegroup . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Central extension of the Aristotle group and its maximal coad- jointorbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.3 Noncentrallyabelianextendedgroupanditsmaximalcoadjoint orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Noncommutative phase spaces constructed on anisotropic kinematical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 Newton-Hookenoncommutativephasespace . . . . . . . . . . 60 3.3.2 Galileannoncommutativephasespace . . . . . . . . . . . . . 74 3.3.3 Para-Galilean noncommutativephasespaces . . . . . . . . . . 77 3.3.4 Staticnoncommutativephasespaces . . . . . . . . . . . . . . 80 3.3.5 Carroll noncommutativephasespaces . . . . . . . . . . . . . . 82 3.4 Noncommutativephasespaces constructedonabsolutetimegroups . . 84 3.4.1 Galileinoncommutativephasespaces . . . . . . . . . . . . . . 84 3.4.2 Para-Galilei noncommutativephasespaces . . . . . . . . . . . 89 3.4.3 Newton-Hookenoncommutativephasespaces . . . . . . . . . . 96 3.4.4 Staticnoncommutativephasespacein theabsolutetimecase . . 96 3.5 Conclusionsand Classification . . . . . . . . . . . . . . . . . . . . . . 101 4 NONCOMMUTATIVE PHASE SPACES BY LINEAR DEFORMATION OFPOISSON BRACKETS 103 4.1 Poissonmanifoldsassociatedtotheplanaranisotropickinematicalgroups104 4.1.1 Poisson-Liestructureassociated toCarroll group . . . . . . . . 104 4.1.2 Poisson-Liestructuresassociatedtocentrallyextendedabsolute timeanisotropickinematicalgroups . . . . . . . . . . . . . . . 105 4.2 Poisson-Lie structures associated to noncentrally abelian extended ab- solutetimekinematicalgroups . . . . . . . . . . . . . . . . . . . . . . 108 4.3 Comparative analysis of the centrally and the noncentrally abelian extended results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4 Possible four-dimensional noncommutative phase spaces by linear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5 2n-dimensionalpossiblenoncommutativephasespacesbylineardefor- mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 INTRODUCTION 0.1 Introduction This thesis takes place within the framework of the titled domain noncommutative geometry. Noncommutativity appeared in nonrelativistic mechanics first in the work of Peierls [1] on the diamagnetism of conduction electrons. In relativistic quantum mechanics, the suggestions to use noncommutative coordinates goes back to Heisenberg andwasfirstlyformalizedin1947bySnyder[2]atsmalllengthscales. Sometimelater, Von Neumann [3] introduced the term noncommutativegeometry to refer in general to a geometry in which an algebra of functions is replaced by a more general associative algebra called noncommutative algebra. For him, operator algebra theory was a noncommutative outgrowth of measure theory. As in the quantization of classical phase space, coordinatesarereplaced bygenerators ofthealgebra. Thecorrespondencebetween“spaces”and“commutativealgebras”isfamiliarinMath- ematicsandintheoreticalPhysics. Thiscorrespondenceallowsanalgebraictranslation ofvariousgeometricalconceptsonspacesinappropriatealgebrasoffunctionsonthese spaces. Replacingthesecommutativealgebrasbynoncommutativealgebras,i.eforget- tingcommutativity,leads thentononcommutativegeneralizationsofgeometrieswhere notionsof“spaces ofpoints”are notinvolved. Interest in Snyder’s idea was revived much later when Mathematicians, notably Connes [4, 5]and Woronowicz[6], succeeded in generalizingthe notionof differential structure to noncommutative geometry. Such a noncommutative generalization was a need in Physics for the formulation of quantum theory and the understanding of its re- lations with classical Physics. Indeed, as the role of symplectic geometry and hence symplecticstructures has increased its importance in both Mathematics and Physics to constitutenowadaysanessentialtechniqueofdescribingandmodelingnaturalphenom- ena,thennoncommutativesymplecticstructuresofferanovelandpromisingframework for the construction of physical theories. Particularly, noncommutative phase spaces Introduction 2 providemathematicalbackgroundsforthestudyofmagneticfieldsin Physics. As applied to Physics, noncommutative geometry is understood mainly in two ap- proaches. The first one is the spectral triple of A. Connes [7] with the Dirac operator playing a central role in unifying, through the universal action principle, gravitation withstandard modelof fundamental interactions. Thesecond oneis thequantum field theoryonnoncommutativespaces [8]withMoyalproductas mainingredient. Besides these, a propositionby severalauthors [9, 10] corresponds to space coordinates that no longer commute. This was implemented by an extension of the Poisson structure on thecotangentspacesuchthatthebracketssatisfy{xk,xl} =6 0. Uponquantization,the correspondingoperatorsshouldthenalso benoncommutative. One motivation for this work is to demonstrate that this extension of the Poisson structure is achieved when we consider a Lie group G. Indeed, models associated withagivensymmetrygroupcan beconvenientlyconstructedusingthecoadjointorbit method also called Souriau’s method. His theorem says in fact that when a symmetry group G acts transitively on a phase space, then the latter is a coadjoint orbit of G equipped withits canonical symplecticform [11, 12, 13]. In otherwords, theclassical phase spaces of elementary systems correspond to coadjoint orbits of their symmetry groups. Thus, by considering a Lie group G, the problem is to find a symplecticman- ifold X whose symmetry group is G. Under some assumptions, this problem has a regularsolutionaccording totheSouriau’sapproach. The first applications that Souriau presented in his book [11] concern both the Poincaré and the Galilei groups for which coadjoint orbits represent elementary particles characterized by the invariants m (mass) and s (spin). Souriau himselfgoes one step further as he considers massless particles with spin, m = 0, s 6= 0 identified as relativisticand nonrelativisticspinrespectively. Souriau’s ideas were laterextendedto largergroups. TakingG = Poincare×H whereH is aninternal symmetrygroup 0 0 (e.g SU(2),SU(3),...)yieldsrelativisticparticleswithinternalstructure(formorede- tailssee[14,15]). The nonrelativistic kinematical groups admit nontrivial central extensions by one- dimensional algebra in dimension d ≥ 3. But in the plane, they admit an exotic [16] two-parameter central extension. The one-parameter central extension of the spatial Galilei group has been considered by Souriau in his book [11], the two-parameter central extension of the planar Galilei group was studied in [9, 16]. Furthermore, the coadjoint orbit method has recently also been applied to smaller spacetime symmetry groups. For example, in [17] a classical “photon ”model was constructed, based en- Introduction 3 tirely on the Euclidean group E(3), a subgroup of both the Poincaré and the Galilei groups. An other application of the Souriau’s method is found in [18] where the most general dynamical systems on which the nonrelativistic conformal groups act transi- tivelyassymmetriesareconstructed. A related motivation comes from a curiosity about a general solution to the above problem, that is, to find more general symplectic manifolds whose symmetry groups are the planar kinematical groups. This takes place in a well known fact that if a free particleiscoupledwithanexternalfield,thisreducesthesymmetrygrouptoasubgroup (of the Galilei or Poincaré group) and conversely, this reduced symmetry is consistent withthesymmetrysubgroup[11]. Thisthesisisdevotedtorealizeclassicaldynamicalsystemsassociatedwithanisotropic kinematicalgroups(kinematicalgroupswithoutrotationparameters[19])byuseofthe reciprocal schema. Precisely, we start with a model withanisotropickinematical symmetry and by applying Souriau’s method, we obtain models with additional terms (in thesymplectic2-form)interpretedasfields. Thelatterarelinkedtothefreeparticleso as to preservetheanisotropickinematicalsymmetries. More specifically,we studythe maximal coadjoint orbit (all invariants are nonvanishing) of the all planar anisotropic kinematicalgroups(oscillatingandexpandingNewton-HookeLiegroups,Galilei,Para- Galilei,Carroll and StaticLiegroups)according to theclassificationin [20]. Furthermore, equivalently to Souriau’s theorem, the dual G∗ of the Lie algebra G of G has a natural Poisson structure whose symplectic leaves are the coadjoint orbits. Depending on the Lie group, these orbits may provide noncommutative phase spaces. Notethatonitsgeneralform,anoncommutativephasespaceallowsfornonzero commutator among the coordinates and among the momenta. Thus, by applying the Souriau’s method, we construct and classify noncommutativephase spaces (which are effectively generalized or modified symplectic structures) associated to the extended kinematicalgroups. Asalreadyarguedweconsiderthecasewherethesymmetrygroups are the kinematical groups according to the classification in [20] and realize noncommutative phase spaces on their maximal coadjoints orbits by using central extensions of all anisotropic kinematical algebras (by relaxing the isotropy condition or dropping therotationgenerators[19]),thelatterbeingtheLiealgebrasforthekinematicalgroups. Note that for the one-parameter centrally extended kinematical algebras, the non- trivialLie bracket which contains theonly central extension parameter m (i.e the mass