i (π(u)ji)=πj(u)Butinanycase,itde nesa –structureonthealgebraU(so(D,C)),i.e.anantilinearanti–algebramap,by(f,u·g)=(u ·f,g)(3.2)(3.3)
foranyu∈Uq,inanorthonormalbasis.
Ourguidingprincipleto ndtheappropriateq–deformedrealspacesisthatthestarstructureonUqshouldadmitasu cientlylargeclassofunitaryrepresentationsofthequan-tumAdSgroupinordertodescribeelementaryparticles,inthespiritofWigner.TheAdSgroupisparticularlywellsuitedforsuchanapproach,becauseithasunitaryrepresenta-tionsforanyhalf–integerspincorrespondingtomassiveaswellasmasslessparticleswithpositiveenergy,inanydimension.Infact,onecanchoosetheCartansubalgebrasuchthattheenergyisoneofitsgenerators,andtheunitaryrepresentationsarethenlowest–weightrepresentationswithpositiveenergyanddiscretespectrum.TheunitaryrepresentationsofthePoincar´egrouparerecoveredinthe atlimit.Wewanttomaintainthesefeaturesintheq–deformedcase.Thiswilluniquelyselecttherealstructure.
3.1StarstructuresonUq
Xi±=siXi , Thereareessentially2typesofstarstructures3onUq[37],the rstoftheformHi =Hi,(3.4)
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
andthesecondoftheform
Xi±=siXi±, Hi = Hi,(3.5)
withsi=±1.Theyde neconsistentstaralgebrasforbothrealqandqarootofunity.Thecompatibilityconditionswiththecoproductaredi erentforrealqandqaphase,whichwillshowupindi erentrealitystructuresoftheassociatedquantumspaces.
Itisknownthatthereexist nite–dimensionalunitaryrepresentationsofthe rsttype(3.4)ifqisarootofunity[38,37],whichhavethedesiredpropertiesincludingahigh–energycuto intheAdScase.Forq∈R,unitaryrepresentationsofnoncompactformsalsoexist,buttheyhavenocuto .Therearealsocertainunitaryrepresentationsofthesecondtypeforqaphase,e.g.forUq(so(2,1))[32],buttheyarenothighest–weightrepresentations;inparticular,theCartansubalgebracannotbediagonalized,andtheenergyintheAdScaseisnotpositivede nite(noticethattheCartansubalgebraisdistinguishedforq=1,unlikeintheclassicalcase).Finite–dimensionalunitaryrepresentationsofthesecondtypecannotexist,sincethenHicouldbediagonalizedwithpurelyimaginaryspectrum,whichisincontradictionwiththecommutationrelation(2.4);ThereforeweonlyconsiderstarstructuresonUqofthe rsttype(3.4)fromnowon,withqarootofunity.
ConsiderthevectorrepresentationVDofUq,withbasisxiandweightsλifori=1,...,D.VDisunitarywithrespecttothecompactform
Xi±=Xi , Hi =Hi,(3.6)
whichde nesUq(so(D))forbothrealqandqaphase.Ingeneral,thereisauniqueinvariantinnerproductonhighestweightmodulessatisfying(3.2)forstarstructuresofthetype(3.4).Theweightvectorsxiarethenorthogonalfordi erentweights,andtheycande nedtobeorthonormal,i.e.
(xi,xj)=δij.(3.7)
Thisisthestandardconventionintheliterature.
Nowitiseasyto ndthede nitionofUq(so(2,D 2)).LetEbetheelementoftheCartansubalgebrawhichisdualtoλ1/dS,sothat<E,λ>=(λ1,λ)/dS.ThentheeigenvaluesofEonVDareEi:=(λ1,λi)/dS=(1,0,...,0, 1);Ewillturnouttobetheenergy.Weclaimthatthestarstructurede ningUq(so(2,D 2))is
Xi±=( 1)Eθ(Xi±)( 1)E=siXi , Hi =Hiwithsi=( 1)<E,αi>,(3.8)wheresi=( 1,1...,1)forD=4,andsi=( 1, 1)forD=4.Thisisastaralgebraofthe rsttype(3.4)whichcanbeconsideredforbothq∈Randqarootofunity,andthereexistunitaryrepresentationsinbothcases.ThemaximalcompactsubalgebraisUq(so(D 2))×±±Uq(so(2))whereUq(so(D 2))generatedbyX2,...,Xr,H2,...,Hr(forD=4),andUq(so(2))
11
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
byE;thesesubalgebrascommute.Theunique(uptonormalization)correspondinginvariantinnerproductonVDwhichsatis es(3.2)is
(xi,xj)=( 1)Eiδij(3.9)
intheabovebasis,andhasthecorrectsignaturefortheAdScase.Wewillalso ndthedesiredunitaryrepresentationsofU
de nethe“physical”quantumAnti–deqfin(so(2,D 2)),provided(2.24)holds.Thereforewe
Sittergroupatrootsofunitytobetherealform
(3.8)ofUqfin(so(2,D 2)).
3.2QuantumEuclideanandAnti–deSitterspaceforrealqForrealq,thereisastandardstarstructureonthealgebraCDq.ThealgebraRDqontherealquantumEuclideanspaceisde nedby[8]
x i=xjgji.(3.10)
Thisisaconsistentstaralgebra,whichindeedleadstotheinvariantinnerproduct(xi,xj)=δij,andby(3.1)correspondstotherealform(3.6)onUq.Itisalsoconsistentwiththeconstraintxixjgij=1,therebyde ningthequantumEuclideansphereSD 1forrealq.ThealgebraAdSqD 1onquantumAnti–deSitterspaceforrealqissimilarlyqde nedby
t i= ( 1)Eitjgji,(3.11)
togetherwith
t2=titjgij=1.(3.12)
Thisisconsistentwith(2.17)becauseEisintheCartansubalgebra,andhasthecorrectclassicallimit.By(3.1),thisleadstotheinvariantinnerproduct(ti,tj)=( 1)Eiδij.Ifqisaphase,(3.10)doesnotextendasastaronthealgebraCD