Coadjoint representation of Virasoro type Lie algebras and differential operators

Valentin Ovsienko

Description

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


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• adler-gelfand-dickey poisson


• moyal-weil star


• geometrical picture


• kirillov has


• gelfand-fuchs cocycle

Sujets

• Gelfand
• Has Been

Informations

• Publié par: mijec
• Nombre de lectures: 54
• Langue: English

Extrait

CoadjointrepresentationofVirasoro-type Liealgebrasanddifferentialoperators ontensor-densities ValentinYu.Ovsienko CentredePhysiqueThe´orique C.N.R.S. Luminy–Case907 F-13288MarseilleCedex9 France email:[e-mail] TomyteacherAlexanderAlexandrovichKirillov Abstract WediscussthegeometricalnatureofthecoadjointrepresentationoftheVira- soroalgebraandsomeofitsgeneralizations.Theisomorphismofthecoadjoint representationoftheVirasorogrouptotheDiff(S 1 )-actiononthespaceofSturm- LiouvilleoperatorswasdiscoveredbyA.A.KirillovandG.Segal.Thisdeepand fruitfulresultrelatesthistopictotheclassicalproblemsofprojectivedifferen- tialgeometry(lineardifferentialoperators,projectivestructuresonS 1 etc.)The purposeofthistalkistogiveadetailedexplanationoftheA.A.Kirillovmethod [14]forthegeometricrealizationofthecoadjointrepresentationintermsoflinear differentialoperators.Kirillov’smethodisbasedonLiesuperalgebrasgeneralizing theVirasoroalgebra.OneobtainstheSturm-Liouvilleoperatorsdirectlyfromthe coadjointrepresentationoftheseLiesuperalgebras.Wewillshowthatthismethod isuniversal.Wewillconsiderafewexamplesofinfinite-dimensionalLiealgebras andshowthattheKirillovmethodcanbeappliedtothem.Thistalkispurely expository:alltheresultsareknown. MathematicalSubjectClassification(2000)Primary:17B68. Secondary:17B65,81R10. 1

TableofContents Introduction 1CoadjointrepresentationofVirasorogroupandSturm-Liouvilleoperators; Schwarzianderivativeasa1-cocycle 2ProjectivelyinvariantversionoftheGelfand-Fuchscocycleandofthe Schwarzianderivative 3Kirillov’smethodofLiesuperalgebras 4InvariantsofcoadjointrepresentationoftheVirasorogroup 5ExtensionoftheLiealgebraoffirstorderlineardifferentialoperatorsonS 1 andmatrixanalogueoftheSturm-Liouvilleoperator 6GeometricaldefinitionoftheGelfand-Dickeybracketandtherelationtothe Moyal-Weilstar-product References Introduction Thecoadjointrepresentationofinfinite-dimensionalLiegroupsandLiealgebrasis oneofthemostinterestingsubjectsofKirillov’sorbitmethod.Geometricalprob- lemsrelatedtothissubjectlinktogethersuchfundamentaldomainsas:symplectic andKa¨hlergeometry,harmonicanalysis,integrablesystemsandmanyothers. Themainpurposeofthistalkistodescribea“geometricalpicture”dueto Kirillov,forthecoadjointrepresentationoftheVirasorogroupandtheVirasoro algebra.Wewillalsoconsidersomeoftheirgeneralizations. 1.TheVirasorogroupistheunique(moduloequivalence)nontrivialcentralexten- sionofthegroupofdiffeomorphismsofthecircle.ThecorrespondingLiealgebra, calledtheVirasoroalgebra,isdefinedastheunique(moduloequivalence)non- trivialcentralextensionoftheLiealgebraofvectorfieldsonS 1 .Thecoadjoint representationoftheVirasorogroupandtheVirasoroalgebrawasstudiedinpio- neeringworksbyA.A.Kirillov[13]andG.Segal[27].Theirresultisasfollows. ThedualspacetotheVirasoroalgebracanberealizedasthespaceofSturm- Liouvilleoperators: 2 d L=c 2 +u(x)(1) xd whereu(x+2π)=u(x)isaperiodicfunction,c∈R(orC)isaconstant.The coadjointrepresentationoftheVirasorogroupcoincideswiththenaturalaction ofthegroupofdiffeomorphismsofS 1 onthespaceofoperators(1). Thisrealizationgivesageometricinterpretationofthecoadjointrepresen- tationoftheVirasorogroup(andtheVirasoroalgebra).Itrelatesthecoadjoint representationoftheVirasorogrouptoclassicalworksondifferentialoperators 2