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Tensorproductsofq-superalgebrarepresentationsandq-seriesidentities.WonSangChungDepartmentofPhysics,GyeongsangNationalUniversity,Jinju,660-701,Korea.E.G.KalninsDepartmentofMathematicsandStatistics,UniversityofWaikato,Hamilton,NewZealand.andW.Miller,Jr. SchoolofMathematics,UniversityofMinnesota,Minneapolis,Minnesota,55455,U.S.A.December13,2001AbstractWeworkoutexamplesoftensorproductsfordistinctq-generalizationsofeu-clidean,oscillatorands‘(2)typesuperalgebras,incaseswherethemethodofhighestweightvectorswillnotapply.Inparticular,weusethethree-termrecurrencerelationsforAskey-Wilsonpolynomialstodecomposethetensorproductofrepresentationsfromthepositivediscreteseriesandrepresenta-tionsfromthenegativediscreteseries.Weshowthatvariousq-analogsoftheexponentialfunctioncanbeusedtomimictheexponentialmappingfromaLiealgebratoitsLiegroupandwecomputethecorrespondingmatrixel-ementsofthe\groupoperatorsontheserepresentationspaces.Weshow WorksupportedinpartbytheNationalScienceFoundationundergrantDMS94{005331991MathematicsSubjectClassi cation:33D55,33D45,17B37,81R501thatthematrixelementsthemselvestransformirreduciblyundertheactionofthequantumsuperalgebra.Themostimportantq-seriesidentitiesderivedhereareinterpretedastheexpansionofthematrixelementsofa\groupoperator(viatheexponentialmapping)inatensorproductbasisintermsofthematrixelementsinareducedbasis.Theyinvolveq-hypergeometricserieswithbase q;0q1.1IntroductionZhedanovandothershaveintroducedaproductofgeneralizeds‘q(2)algebrasthatallowsonetotaketensorproductsofrepresentationscorrespondingtotwodistinctalgebras,[28,8,7].Theirgeneralizationisanalgebra(v;u)withgeneratorsH,E+,E whichobeythecommutationrelations[H;E+]=E+;[H;E ]= E ;[E+;E ]= uq H vqH:(1)Here,uandvarerealnumbersand0q1.Foruv6=0thisalgebraisisomorphictooneofthetrues‘q(2)typealgebras,foruv=0;u2+v20itisisomorphictoaspecialrealizationoftheq-oscillatoralgebra,andforu=v=0itisisomorphictotheEuclideanLiealgebram(2),[12].ThisalgebrahasaninvariantelementC=E+E +vqH uq1 H1 q:(2)AspointedoutbyZhedanovandothers[28,8,7],thefamilyofalgebrasadmitsamultiplication(v;u) ( u;t) =(v;t),de nedbyF+= (E+)=E+ q12H+q 12H E+;F = (E )=E q12H+q 12H E ;(3)L= (H)=H I+I H:TheoperatorsF ,Lsatisfythecommutationrelations(1).Using(3)wecaneasilyde nethetensorproduct ofarepresentation of(v;u)andtherepresentation of( u;t),therebyobtainingarepresentationof(v;t).Thisconstructionyieldsaconvenientgeneralizationofthetensorproduct2computationsin,forexample,[13,14,15].Wefollowthisideatostudygeneralizationsoftheospq(1=2)algebra,[3,21,23].Inthispapertheq-superalgebra[v;u]isde nedbythegeneratorsH;V andrelations[H;V ]= 12V ;fV+;V g= uq 2H vq2H;(4)where[A;B]=AB BAandfA;Bg=AB+BA.Here,u;vareparameters.The\CasimiroperatorCfor[v;u]isC=V+V +v1+qq2H+u1+q 1q 2H(5)andsatis estherelations[H;C]=0;fV ;Cg=0:(6)NotethatC2isaninvariantoperator,i.e.,C2commuteswithallelementsofthe[v;u]algebra.Wede nethecoproductL= (H)=H I+I H;F = (V )=V qH+q H V ;(7)where(A B)(C D)=( 1)p(B)p(C)AC BD:(8)Herep(A)istheparityoftheoperatorA.InthiscasethebosonicvariablesH;qHhaveparity0andthefermionicvariablesV haveparity1,[2,21].Itfollowsthatthisalgebraadmitsamultiplicationoftheform[v;u] [ u;t]=[v;t];(9)wherethe rstfactorcorrespondstothealgebra[v;u]andthesecondfactortoalgebra[ u;t].Indeedtherelations[L;F ]= 12F ;fF+;F g= tq 2L vq2L;(10)aresatis ed.TheonlynontrivialpartoftheproofisfF ;F+g=3( uq 2H vq2H) q2H+q 2H ( tq 2H+uq2H)= tq 2H q 2H vq2H q2H= tq 2L vq2L:Inx2andx3westudyirreduciblerepresentationsofthe\Euclideanalgebra[0;0]andworkoutthe(nonunique)tensorproductdecompositionfor[0;0] [0;0] =[0;0]:WecomputetheClebsch-Gordancoe cientsfortheexpansionandusethemtoderiveq-seriesidentitiesforthespecialfunctionsthatappearnaturallyinthetheory.Theyareinterpretedhereasexpansionsofthematrixelementsofa\groupoperatorinatensorproductbasisintermsofthematrixelementsinareducedbasis.Theq-seriesarebase q,asfollowsnaturallyfromexpres-sions(7).Inx4wecarryouttheanalogousconstructionsforpositivediscreteseriesrepresentationsofthegeneralalgebra[v;u]forv2+u20tensoredwithnegativediscreteseriesrepresentations.Hereourmethodsleadnatu-rallytoathree-termrecurrencerelationfortheClebsch-Gordancoe cientsthatcanbesolvedthroughcomparisonwiththerecurrencerelationwithAskey-Wilsonpolynomials.Thisyieldsthemeasure(notnecessarilyposi-tive)determiningthedecompositionofthetensorproductintoirreduciblecomponents.Thenotationusedforq-seriesandq-integralsinthispaperfollowsthatofGasperandRahman[6].2Euclideanq-superalgebrarepresentationsThethreedimensionalq-supersymmetricLiealgebra[0;0]isdeterminedbyitsgeneratorsH,V+,V whichobeytherelations[H;V ]= V ;fV+;V g=0:(11)Weconsiderananalogy(!)ofthein nitedimensionalirreduciblerepresen-tationsoftheEuclideanLiealgebra,characterizedbythenonzerocomplexnumber!ThespectrumofHcorrespondingto(!)isthesetS=fn=2:n2Zgandthecomplexrepresentationspacehasbasisvectorsfm,m2S,suchthatV+fm=!fm+1;V fm=( 1)m!fm 1;Hfm=m2fm;(12)4whereC V+V isthe\Casimiroperator,withCfm=( 1)m!2fm.Notethattherepresentations(!)and( !)areequivalent.Asimplereali