Coadjoint representation of Virasoro type Lie algebras and differential operators

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Niveau: Supérieur, Doctorat, Bac+8
ar X iv :m at h- ph /0 60 20 09 v1 3 F eb 2 00 6 Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities Valentin Yu. Ovsienko Centre de Physique Theorique C.N.R.S. Luminy – Case 907 F-13288 Marseille Cedex 9 France email: To my teacher Alexander Alexandrovich Kirillov Abstract We discuss the geometrical nature of the coadjoint representation of the Vira- soro algebra and some of its generalizations. The isomorphism of the coadjoint representation of the Virasoro group to the Diff(S1)-action on the space of Sturm- Liouville operators was discovered by A.A. Kirillov and G. Segal. This deep and fruitful result relates this topic to the classical problems of projective differen- tial geometry (linear differential operators, projective structures on S1 etc.) The purpose of this talk is to give a detailed explanation of the A.A. Kirillov method [14] for the geometric realization of the coadjoint representation in terms of linear differential operators. Kirillov's method is based on Lie superalgebras generalizing the Virasoro algebra. One obtains the Sturm-Liouville operators directly from the coadjoint representation of these Lie superalgebras. We will show that this method is universal.

  • dimensional lie

  • product references

  • has been

  • adler-gelfand-dickey poisson

  • moyal-weil star

  • geometrical picture

  • kirillov has

  • gelfand-fuchs cocycle


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CoadjointrepresentationofVirasoro-typeLiealgebrasanddifferentialoperatorsontensor-densitiesValentinYu.OvsienkoCentredePhysiqueThe´oriqueC.N.R.S.Luminy–Case907F-13288MarseilleCedex9Franceemail:Valentin.Ovsienko@cpt.univ-mrs.frTomyteacherAlexanderAlexandrovichKirillovAbstractWediscussthegeometricalnatureofthecoadjointrepresentationoftheVira-soroalgebraandsomeofitsgeneralizations.TheisomorphismofthecoadjointrepresentationoftheVirasorogrouptotheDiff(S1)-actiononthespaceofSturm-LiouvilleoperatorswasdiscoveredbyA.A.KirillovandG.Segal.Thisdeepandfruitfulresultrelatesthistopictotheclassicalproblemsofprojectivedifferen-tialgeometry(lineardifferentialoperators,projectivestructuresonS1etc.)ThepurposeofthistalkistogiveadetailedexplanationoftheA.A.Kirillovmethod[14]forthegeometricrealizationofthecoadjointrepresentationintermsoflineardifferentialoperators.Kirillov’smethodisbasedonLiesuperalgebrasgeneralizingtheVirasoroalgebra.OneobtainstheSturm-LiouvilleoperatorsdirectlyfromthecoadjointrepresentationoftheseLiesuperalgebras.Wewillshowthatthismethodisuniversal.Wewillconsiderafewexamplesofinfinite-dimensionalLiealgebrasandshowthattheKirillovmethodcanbeappliedtothem.Thistalkispurelyexpository:alltheresultsareknown.MathematicalSubjectClassification(2000)Primary:17B68.Secondary:17B65,81R10.1
TableofContentsIntroduction1CoadjointrepresentationofVirasorogroupandSturm-Liouvilleoperators;Schwarzianderivativeasa1-cocycle2ProjectivelyinvariantversionoftheGelfand-FuchscocycleandoftheSchwarzianderivative3Kirillov’smethodofLiesuperalgebras4InvariantsofcoadjointrepresentationoftheVirasorogroup5ExtensionoftheLiealgebraoffirstorderlineardifferentialoperatorsonS1andmatrixanalogueoftheSturm-Liouvilleoperator6GeometricaldefinitionoftheGelfand-DickeybracketandtherelationtotheMoyal-Weilstar-productReferencesIntroductionThecoadjointrepresentationofinfinite-dimensionalLiegroupsandLiealgebrasisoneofthemostinterestingsubjectsofKirillov’sorbitmethod.Geometricalprob-lemsrelatedtothissubjectlinktogethersuchfundamentaldomainsas:symplecticandKa¨hlergeometry,harmonicanalysis,integrablesystemsandmanyothers.Themainpurposeofthistalkistodescribea“geometricalpicture”duetoKirillov,forthecoadjointrepresentationoftheVirasorogroupandtheVirasoroalgebra.Wewillalsoconsidersomeoftheirgeneralizations.1.TheVirasorogroupistheunique(moduloequivalence)nontrivialcentralexten-sionofthegroupofdiffeomorphismsofthecircle.ThecorrespondingLiealgebra,calledtheVirasoroalgebra,isdefinedastheunique(moduloequivalence)non-trivialcentralextensionoftheLiealgebraofvectorfieldsonS1.ThecoadjointrepresentationoftheVirasorogroupandtheVirasoroalgebrawasstudiedinpio-neeringworksbyA.A.Kirillov[13]andG.Segal[27].Theirresultisasfollows.ThedualspacetotheVirasoroalgebracanberealizedasthespaceofSturm-Liouvilleoperators:2dL=c2+u(x)(1)xdwhereu(x+2π)=u(x)isaperiodicfunction,cR(orC)isaconstant.ThecoadjointrepresentationoftheVirasorogroupcoincideswiththenaturalactionofthegroupofdiffeomorphismsofS1onthespaceofoperators(1).Thisrealizationgivesageometricinterpretationofthecoadjointrepresen-tationoftheVirasorogroup(andtheVirasoroalgebra).ItrelatesthecoadjointrepresentationoftheVirasorogrouptoclassicalworksondifferentialoperators2