Polynomial super-gl(n) algebras 

导读:3002ebF512v902103/0ht-pe:hviXraNovember2002UTAS-PHYS-02-03Polynomialsuper-gl(n)algebrasPDJarvis?andGRudolph??,SchoolofMathematicsandPhysics,UniversityofTasmaniaGPOBox252-21,HobartT

Polynomial super-gl(n) algebras 

3002 ebF 51 2v902103/0ht-pe:hviXraNovember2002UTAS-PHYS-02-03

Polynomialsuper-gl(n)algebras

PDJarvis?andGRudolph??,

SchoolofMathematicsandPhysics,UniversityofTasmania

GPOBox252-21,HobartTas7001,Australia

Weintroduceaclassof?nitedimensionalnonlinearsuperalgebrasL=LvidinggradingsofL

1

pro-

Qb,and

evengeneratorsEab,a,b=1,...,n,athreeparameterfamilyofquadraticsuper-gl(n)algebras(deformationsofsl(n/1))isde?ned.Ingeneral,additionalcovariantSerre-typeconditionsareimposed,inorderthattheJacobiidentitiesbeful?lled.Forthesequadraticsuper-gl(n)algebras,theconstructionofKacmodules,andconditionsforatypicality,arebrie?yconsidered.Applicationsinquantum?eldtheory,includingHamiltonianlatticeQCDandspace-timesupersymmetry,arediscussed.

§1Introduction

Theinterplaybetweentheapplicationofsymmetryprinciplestomodelsofphysicalsys-tems,andstudyoftheclassi?cation,propertiesandrepresentationtheoryofunderlyingalgebraicstructures,haslongbeenamajorthemeinmathematicalphysics.Abroadspectrumofgeneralisedsymmetryalgebrasisunderactivestudy,includingin?nitedi-mensionalalgebrasandsuperalgebras,deformationsofuniversalenvelopingalgebras,andvariousternaryandothernon-associativealgebras.Aparticularclassistheso-calledW-algebrasandsuperalgebras,arisingfromHamiltonianreductionofsystemswith?rstclassconstraintsde?nedonLie-Poissonmanifolds.Althoughthegeneralstudyofnon-linearLie(super-)algebrasbelongstoabstractdeformationtheory,inspeci?ccontextsenoughstructureexiststoallowprogressonclassi?cationandrepresentationtheory.Forexample,theW-(super)algebras,althoughnotLie(super)algebras,arerigidlyconstrainedbytheiroriginsinHamiltonianreduction.

Inthisspiritwestudyinthispaperaclassof?nitedimensionalnonlinearsuperal-gebras,withattentiontotheircovariantclosurerelations,andwhicharede?nedalge-braically,withoutreferencetoadditionalstructure.Namely,weconsidersuperalgebras(Z2-gradedalgebras)L=L1withevensubalgebraL1,withde?ningrelationsoftheform:

[L

0]0

?L

0,L1,

{L

1}

?U(L

algebras’possessoddgeneratorsL

0),

thatis,onpolynomials(ofquadraticorhigher

degree)intheevengeneratorsL

0=gl(n)?sl(n)+gl(1).

Studyoftheclassi?cationofsuchsuperalgebrasdevolvestoexaminationofpossibleL

1,andadmissiblestructureconstants(1)whichareconsistentwiththeJacobi

identities.Thisistakenupin§2below,wherewediscussthestructureofcandidatepolynomialsuper-gl(n)algebrasof‘typeI’:thatis,whereL

0-module{λ}togetherwithitscontragredientrepresentation

{λ})(wherekisthemaximaldegreewithinU(L

1,L

Qbinthede?ningn-dimensional

representationofgl(n),anditscontragredient.Athree-parameterfamilyofquadratical-gebrasgl2(n/{1}+{

super-gl(n)algebra,namelythesimpleLiesuperalgebrasl(n/1)≡gl1(n/{1}+{

Qb=q

3})aregiven,allowing(indecomposable)modules

tobeidenti?edinagl(n)basis,bothonthefermionicFockspace,andviatheadjointac-tionontheassociatedCli?ordalgebra(seetables1and2forthen=1andn=2cases).Finally(§A.4),forthecasen=4therelationbetweenthealgebrasgl2(4/{13}+{

1})a,α,β(§4)isstudied.

§2Polynomialsuper-gl(n)algebrasglk(n/{λ}+{

withrespectto(the

adjointactionof)theevensubalgebra,and{L1}transformsunderadL

1?L0-moduleL

0-module

containscommonirreduciblesubmodules,withthebranching

multiplicityofthelatterdeterminingtheirnumberandtype.

Studyoftheclassi?cationofsuchsuperalgebrasdevolvestoexaminationofpossibleL1,andadmissiblestructureconstants(1).Similarquestionsariseinthe

0

0,and{Qα}abasisforL

andtheanticommutator{Qα,Qβ}transformsinthetensorproductoftherelevantrepresentationsoftheevensubalgebra.

0,

studyofsimpleLiesuperalgebras[1](whereL0)in(1)above).Ananalo-goussituationisaddressedintheWittconstruction[2],whereoneconsidersLiealgebrasassociatedwithagivenLiealgebraL0-module(atrivialexten-sionbeingthesemi-directproduct,withthemodulegiventhestructureofanabelianalgebra)2.Theseconsiderationsaremoretractableifweturntothe‘typeI’super-gl(n)algebras:theL1isthesumofasingleirreduciblerepresentationanditscon-tragredient.TheZ2-gradingisthusinheritedfromaZ-gradingofL,associatedwiththespectrumoftheadjointactionoftheabeliansummandofgl(n)?sl(n)+gl(1).ThusL=L?1+L0+L1,withL1=L?1+L1.Thisentails{L±1,L±1}=0,and{L+1,L?1}?U(L0).WithoutlossofgeneralitywemayassumethatL+1isanirreduciblerepresentationofthesemisimplepartsl(n),withL?1thecorrespondingcontragredient.NowforsemisimpleLiealgebras,Joseph’stheorem[4]statesthattheenvelopingalgebraU(L0)isisomorphicasanL0-module,tothesumoveralldominantintegralweights,ofthetensorproductofthecorresponding(?nite-dimensional)highestweightmodule,withitscontragredient.Thus,itispossibletoinvestigatewhethertheanticommutators{L+1,L?1}canbeassociatedwithauniqueelementofU(L0).However,asJoseph’stheoremdoesnotmandateanyrelationbetweenthepolynomialdegreewithinU(L0)ofelementsofagiventensorproductcontributingtothesum,weproceedmoregenerally.Let,then,{λ}denoteadominantintegralweightofgl(n)and{

1}·{1}(correspondingtothetensorproductofthen-dimensionalde?ningrepresentationwithitscontragredient)withirreducibleparts{

λ}·{λ}.Let

?betheweightof{λ}asapartition,andkbethepolynomialdegreeofnonlinearityintheenvelopingalgebraofgl(n)characterisingthealgebra.Thenwehave,

{

({

μ;ν};μ;ν}.

(2)

Themultiplicitiesnkμν,nλμνarede?nedin§A.1intermsofthestandardLittlewood-Richardsoncoe?cients.In§A.1itisshownthatnkμν≥nλμνprovidedk≥?,and

ρ;σ}correspondstoanirreducibletensorofcontravariantsymmetry

typeρ,andcovariantsymmetrytypeσ,andtracelesswithrespecttocontractionsbetweencontravariantandcovariantindices.See§A.1,andalso[6].

moreoverif?>k,thennkμν=0forμ,ν?k+1,...,?.Inthelattercase,generalisedstructureconstantsforthecorrespondingsymmetrytypes{λ}·{λ}donotexist.

Tocompletethediscussionofcouplingsinspeci?ccases,itisnecessarytoadoptanexplicitnotation.Recallthewell-knownpresentationofgl(n)viatheGel’fandgeneratorsEab,1≤a,b≤n(seeforexamplethe?rstlineof(10)below).De?nethematrixpowers(Ek+1)ab=(Ek)acEcbintheobviousway,andtheirtraces??Ek??=(Ek)cc(thestandardCasimirinvariants,see§A.2).BearinginmindJoseph’stheorem[4],thefollowinggeneralisedpermanents[7]provideasuitablespanningsetforU(gl(n)):for{λ}??,and{μ}an?-partpartition(ofweightk,withnonzeropartsaugmentedbyzeroesasnecessary),de?ne

[{λ};{μ}]a1a2...a?b1b2...b?=

1

λ}·{λ},and{μ}thedistributionoftensor

contractions,forpolynomialsinthegeneratorsEabbelongingtotheenvelopingalgebraU(gl(n)).Inthisnotationthematrixpowersareofcourse(Ek)ab≡[{1};{k}]ab,whiletheCasimiroperatorsaresimplyfoundbycontractionofany[{λ};{μ}]with

?λa1a2...a?b1b2...b?=[{λ};{0}]a1a2...a?b1b2...b?

Forthecasesconsideredinthesequel,elementarytensornotationsu?cesforexplicitconstructions.Asanexampleletusanalyseindetailthecasek=?=3and{λ}={2,1}.Explicitly,wehave(see§A.1)

{({

2,1;2,1}+{3;3}+{

1,1;1,1}+4{

1;1}+{0},1,1,1;1,1,1}+2{

2,1})has10typesofstructureconstant(or9freeparameters

fortheassociatedreducedmatrixelements,uptooverallnormalisation),correspondingtothemaximummultiplicitiesofthecommonirreduciblecomponentsoftheabovetwodecompositions(excludingadditionalstructureconstantsarisingfromlowerdegree).Tocompletetheconstructionofcouplingsforthiscase,de?nethefollowingobjectsintheenvelopingalgebra,

{F·F′·F′′}abcpqr≡(FapF′bq+FbpF′aq)F′′cr

?(FcpF′bq+FapF′cq)F′′ar+...,

(5)

beingofmixedsymmetrytype[{2,1};{μ1,μ2,μ3}]withrespecttocontravariantandcovariantindices(where‘...’representsthreeadditionalquartetsoftermsestablishingmixedsymmetrywithrespecttothepqrlabelpermutations(thetermsshownexplicitly

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