MODELS OF q-ALGEBRA REPRESENTATIONS MATRIX ELEMENT

MODELSOFq-ALGEBRAREPRESENTATIONS:MATRIXELEMENTSOFTHEq-OSCILLATORALGEBRAE.G.KALNINSy,WILLARDMILLER,Jr.zANDSANCHITAMUKHERJEE*Abstract.Thispapercontinuesastudyoffunctionspacemodelsofirreduciblerepresentationsofq-analogsofLieenvelopingalgebras,motivatedbyrecurrencere-lationssatisedbyq-hypergeometricfunctions.Hereweconsideraq-analogoftheoscillatoralgebra(notaquantumalgebra).Weshowthatvariousq-analogsoftheexponentialfunctioncanbeusedtomimictheexponentialmappingfromaLieal-gebratoitsLiegroupandwecomputethecorrespondingmatrixelementsofthe\groupoperatorsontheserepresentationspaces.This\localapproachappliestomoregeneralfamiliesofspecialfunctions,e.g.,withcomplexargumentsandparam-eters,thandoesthequantumgroupapproach.Weshowthatthematrixelementsthemselvestransformirreduciblyundertheactionofthealgebra.Wendq-analogsofaformulafortheproductoftwohypergeometricfunctions1F1andtheproductofa1F1andaBesselfunction.Theyareinterpretedhereasexpansionsofthematrixelementsofa\groupoperator(viatheexponentialmapping)inatensorproductbasis(forthetensorproductoftwoirreducibleoscillatoralgebrarepresentations)intermsofthematrixelementsinareducedbasis.Asabyproductofthisanalysiswendaninterestingneworthonormalbasisforaq-analogoftheBargmann-SegalHilbertspaceofentirefunctions.PACS:02.20.+b,03.65.Fd1.Introduction.Thispapercontinuesthestudyoffunctionspacemodelsofirreduciblerepresentationsofq-algebras[1,2,3].Thesealgebrasandmodelsaremotivatedbyrecurrencerelationssatisedbyq-hypergeometricfunctions[4]andourtreatmentisanalternativetothetheoryofquantumgroups.Here,weconsidertheirreduciblerepresentationsofaq-analogoftheoscillatoralgebra(notaquantumalgebra).WereplacetheusualexponentialfunctionmappingfromtheLiealgebratotheLiegroupbytheq-exponentialmappingsEqandeq.Inplaceoftheusualmatrixelementsonthegroup(arisingfromanirreduciblerepresentation)whichareexpressibleintermsofLaguerrepolynomialsandfunctions,wendseventypesofmatrixelementsexpressibleintermsofq-hypergeometricseries.Theseq-matrixelementsdonotsatisfygrouphomomorphismproperties,sotheydonotleadtoyDepartmentofMathematicsandStatistics,UniversityofWaikato,Hamilton,NewZealandzSchoolofMathematicsandInstituteforMathematicsanditsApplications,UniversityofMinnesota,Minneapolis,Minnesota55455.WorksupportedinpartbytheNationalScienceFoundationundergrantDMS91{100324*InstituteforMathematicsanditsApplications,UniversityofMinnesota,Minneapolis,Minnesota55455.WorksupportedbytheStudyAbroadfellowshipoftheGovernmentofIndia.TypesetbyAMS-TEX12E.G.KALNINSy,WILLARDMILLER,JR.zANDSANCHITAMUKHERJEE*additiontheoremsintheusualsense.However,theydosatisfyorthogonalityrela-tions.Furthermore,inanalogywithtruegrouprepresentationtheorywecanshowthateachofthesevenfamiliesofmatrixelementsdeterminesatwo-variablemodelforirreduciblerepresentationsoftheq-oscillatoralgebra.Inx3weshowhowthistwo-variablemodelleadstoorthogonalityrelationsforthematrixelements.Inx4wendaq-analogofaformulafortheproductoftwohypergeometricfunctions1F1.Thisisinterpretedhereasanexpansionofthematrixelementsofa\groupoperator(viatheexponentialmapping)inatensorproductbasis(forthetensorproductoftwoirreducibleoscillatoralgebrarepresentations)intermsofthematrixelementsinareducedbasis.Inx5wendaq-analogofaformulafortheproductofa1F1andaBesselfunction.Thisisinterpretedhereasanexpansionofthematrixelementsofthe\groupoperatorinatensorproductbasis(forthetensorproductofanirreducibleoscillatoralgebrarepresentationandanirreduciblerepresentationofthequantummotiongroup)intermsofthematrixelementsinareducedbasis.Asabyproductofthisanalysiswendaninterestingneworthonormalbasisforaq-analogoftheBargmann-SegalHilbertspaceofentirefunctions.Ourapproachtothederivationandunderstandingofq-seriesidentitiesisbasedonthestudyofq-algebrasasq-analogsofLiealgebras,[5,6].Weareattemptingtondq-analogsofthetheoryrelatingLiealgebraandlocalLietransformationgroups[7,8].AsimilarapproachhasbeenadoptedbyFloreaniniandVinet[9-12].Thisisanalternativetotheelegantpapers[13-21]whicharebasedprimarilyonthetheoryofquantumgroups.Themainjusticationofthe\localapproachisthatitismoregeneral;itappliestomoregeneralfamiliesofspecialfunctionsthandoesthequantumgroupapproach.Thenotationusedforq-seriesinthispaperfollowsthatofGasperandRahman[22].2.Modelsofoscillatoralgebrarepresentations.In[1]aq-analogoftheoscillatoralgebrawasintroduced.ThisistheassociativealgebrageneratedbythefourelementsH,E+,E,Ethatobeythecommutationrelations[H;E+]=E+;[H;E]=E;[E+;E]=qHE;[E;E]=[E;H]=0:(2.1)Itadmitsaclassofalgebraicallyirreduciblerepresentations‘;where‘;arecomplexnumbersand‘6=0.Thesearedenedonavectorspacewithbasisfen:n=0;1;2;g,suchthatE+en=‘sqn111qen+1Een=‘sqn11qen1(2.2)Hen=(+n)en;Een=‘2q1en:Ifand‘arerealwith‘0(aswewillassumeinthispaper)then‘;isdenedontheHilbertspaceK0withorthonormalbasisfengandonthisspaceMODELSOFq-ALGEBRAREPRESENTATIONS:MATRIXELEMENTSOFTHEq-OSCILLATORALGEBRA3wehaveE+=(E),H=HandE=E.AsecondconvenientbasisforK0isffn:n=0;1;gwhereE+fn=‘q(n+1)=2fn+1Efn=‘qn=21qn1qfn1(2.3)Hfn=(+n)fnEfn=‘2q1fn:Here