無窮維李代數(shù)和頂點算子代數(shù).pdf

1、上海交通大學(xué)博士后學(xué)位論文無窮維李代數(shù)和頂點算子代數(shù)姓名：裴玉峰申請學(xué)位級別：博士后專業(yè)：基礎(chǔ)數(shù)學(xué)指導(dǎo)教師：姜翠波20090421第，’章引言11研究背景第一章引言Vertexoperatoralgebras，developedin【9】【26】and【27】areanewfundamentalclassofalgebraicstructures，whichcanberegardedascomplexanaloguesofbothLie

2、algebrasandcommutativeassociativealgebrasThenotionofvertex(operator)algebraispreciselyanalgebraiccounterpartofthenotionofwhatisnowusuallycalledchiralalgebraintwodimensionalconformalquantumfieldtheory，formulatedinsuchwork

3、sas【6】ManyexamplesofvertexoperatoralgebraswereconstructedbyusinginfinitedimensionalLiealgebrasandtheirhighestweightmodulesAmongtheimportantexamples，arethoseassociatedwithaffineLiealgebras(includinginfinitedimensionalHeis

4、enbergalgebras)，theVirasoroalgebl。a【28】andtheLiealgebraofdifferentialoperatorsonthecircle【24]Asisknown，higherspinextensionsoftheVirasoroalgebra，whicharecommonlycalledWalgebrasthatemergeinthestudyofbosonicextendedsymmetry

5、alsoplayaveryimportantroleinconformMfieldtheoryAmongthesealgebras，thealgebraWlooanditssubalgebraWooarethesimplestinstructureandhaveappearedinmathematicalphysicsinverydifferentproblemsBytakinglargeNlimitofZamolodchikov’SW

6、lvalgebra【53]，IBakas【3，4】obtainedaninfinitedimensionalLiealgebraWl∞，whichisgivenintermsofgenerators％，rn∈Z，i=1，0，1withtheLiebracket【L‰，磁】=(D1)m一(i十1)n)Lm禮AsasubalgebraofWloo，thealgebraWoohasabasisL‰，m∈Z，i=0，1ThealgebraWoo

7、hasacleargeometricalmeaning：itcanbeviewedasthealgebraofareapreservingsymplecticdiffeomorphismsofatwodimensionalcylinder【2，32，43]Anotherremarkableapt)earmlceofWlooanditssubalgebras，asasymmetryofthec=1string，werediscovered

8、in[34]and【50]andhavebeenstudiedextensivelyinconnectiontotwodimensionalstringtheoryandquantumgravity(See[29】andreferencesthereinforreviewandmoredetails)Moreover，thealgebrasWlooandWooalsoappearedintheTodatheoryIintegrablem