1αijdiHi djHj+res resR=q1 1+Uq Uq(2.12)
whereαij=(αi,αj).Inthispaper,weconsiderg=so(2r+1)=Brandg=so(2r)=Dr.Aswasshownin[7],thefollowingremarkableelement
ρv=(SR2)R1q 2 (2.13)
isinthecenterofUq,andwillbecalledDrinfeld–Casimir.Hereρ isdualtotheWeylvectorρ=1
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
ConsiderthematrixRij
kl=πik
S=qfor πjl(R)whereπikistheD–dimensional(“vector”)
sentationofUq,andletqso(2r),and2qS=q2forso(2r+1).ThenR repre-
ij=Rji
decomposesasR ij=(qSP+ q 1P +q1 DP0)ij
kl,whereP+,P andP0klkl
aretheinvariant
projectorsonthekltracelesssymmetric,StheSantisymmetric,andthesingletcomponentinthetensorproductof2vectorrepresentations,respectively.Theinvarianttensorgijisgiven(P0)ijq2by 1
kl=S
1+qDqρi′x2,i<i′(2.20)
S 2
Bothalgebrasarecovariantundertherightcoactionxi→xj AjiofFunq(SO(D,C)),whichisequivalenttoaleftaction
xi→u·xi=xjπji(u)(2.21)
foru∈Uq(so(D,C)).Wewillusuallyworkwiththelatter,whichismorefamiliarfromtheclassicalLiealgebras;thenπij(uu′)=πik(u)πkj(u′).Wecanusethisforaquickcheckofthe rstrelationin(2.20):
X+1/2
1·(x1x2 qSx2x1)=qSx1x1 qSq 1/2
Sx1x1=0(2.22)
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
using(2.6)andAppendixA,asitmustbe.Theotherrelationscanbecheckedsimilarly.ThealgebraoffunctionsAdSqD 1de ningthequantumAnti–deSitterspacewillbede nedasarealformofthiscomplexquantumsphere,witha(co)actionofthequantumAnti-deSittergroup.Thereforeasanalgebra,itisagainde nedby(P )ij2Onecouldintroduceaphysicalscalebysettingkltitj=0,t=1.
yi:=tiR(2.23)
foraconstantR>0,sothaty2=R2.Wewillsimplyusetheunitsde nedbyR=1;physicallyspeaking,thescalewillbesetbythe“radius”ofAdSspace.
Sofar,allthesespacesarecomplex.Thecrucialissueisto ndthecorrectde nitionofthecorrespondingrealquantumspaces.Thisisnotobviousespeciallyifqisaphase,andinfactdi erentpossibilitieshavebeenproposedintheliterature[4,13].Thiswillbediscussedindetailinlatersections.
2.1RootsofUnityandRepresentations
Sinceweareprimarilyinterestedinthecasewhereqisarootofunity,wewillconsideramorepowerfulversionoftheabove,theso–called“restrictedspecialization”Uqres:=Uqres(so(D,C))
[22]withgeneratorsX±(k)(X±
i)k
i=
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
Wewillonlyconsider nite–dimensionalrepresentationsinthispaper.TheCartangen-eratorscanthenbediagonalized,witheigenvalues
<Hi,λ>=
∨satisfy(Λi,αj)=δi,j,therefore(α,α)(αi,λ)isthecorootofα.ThefundamentalweightsΛi
(2.27)<Hi,Λj>=δij,
andspanthelatticeofintegralweights.Theirreduciblehighest–weightrepresentationswithhighestweightλwillbedenotedbyLres(λ).
ThevectorrepresentationVDofUq(so(D,C))istherepresentationLres(λ1)withbasisxi(orti)fori=1,...,D.TheirweightsλiaregivenexplicitlyinAppendixA.IfD>4,thenthehighestweightλ1isequaltoΛ1.Wealsode ne
dS=(λ1,λ1),MS=M/dS,(2.28)
sothatqS=qdSwhichwasusedabove.
Itiswell–known[31]thatforgenericq,therepresentationtheoryisessentiallythesameasintheclassicalcase.Inparticular,all nite–dimensionalrepresentations(=modules)ofresUqaredirectsumsofsomeLres(λ).Theircharacter
resλχ(L(λ))=edimLres(λ)ηe η=:χ(λ)(2.29)
η>0
isgivenbyWeylsformula.HereLres(λ)ηistheweightspaceofLres(λ)withweightλ η.
finTheirreduciblehighestweightrepresentationsofUqwillbedenotedbyLfin(λ).
ThevalueoftheDrinfeld–Casimirv(2.13)onLres(λ)(andonanyhighest–weightmodulewithhighestweightλ)was rstdeterminedin[30]:
v·w=q cλwforw∈Lres(λ),(2.30)
wherecλ=(λ,λ+2ρ)isthevalueoftheclassicalquadraticCasimironL(λ).Inparticularforhighestweightsoftheformλ=kλ1,theclassicalCasimirforso(2r+1)is
ckλ1=2k2+2k(D 2),
andforso(2r)itis
ckλ1=k2+k(D 2).(2.31)(2.32)
Finally,wequoteafewimportantfactsaboutirreduciblerepresentationsatrootsofunity.The rstone[2,5]statesthatthestructureofLreswith“small”highestweightλisthesameasclassically:
7
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
Theorem2.1Assumethatλisadominantintegralweightwith(λ+ρ,α)≤Mforallpositiverootsα.ThenthehighestweightrepresentationLres(λ)hasthesamecharacterχasintheclassicalcase,givenbyWeyl’scharacterformula.