Thisfollowsfromthestronglinkageprinciple,whichwas rstshownin[2];foramoreelementaryapproach,see[38].Moreover,Lfin(λ)=Lres(λ)fortheseweightsλ,sincethe±(M)Xiiacttrivially.
Forgeneralλ,thestructureofLres(λ)isdi culttoanalyze.Howeverforthe“specialweights” λz=MiziΛiforzi∈Z(2.33)itcanbeunderstoodeasily,andthiswillbethekeyformuchofthefollowing.Therelation
res(2.5)togetherwith(2.25)impliesthatforanyhighestweightmoduleUq·wλzwithhighest
weightλz,
(2.34) i:=Xi ·wλz
isahighestweightvector(possiblyzero)foranyi,i.e.Xi+· j=0foralli,j.BecauseLres(λz)isirreduciblebyde nition,itfollowsthat
Xi ·wλz=0foralli(2.35)
fin(so(D))areone–inLres(λz).Inparticular,theirreduciblerepresentationsLfin(λz)ofUq±(M)resdimensional.Howeverthe“large”generatorsXii∈UqdoactnontriviallyonLres(λz),
aswewillseenext.
resUsingthecommutationrelations,anyelementofUq(so(D))canbewrittenasasum
fin1.ItfollowsthatallweightsofLres(λz)haveoftermsoftheformX...XikkUqi1 theformλz′=λz iniMiαiwithni∈N.Inotherwords,Lres(λz)isadirectsumof
fin,sinceallλz′arespecialpoints.Infact,one–dimensionalrepresentationsLfin(λz′)ofUqtheweightsλzhavethestructureofaweightlatticewith“fundamentalweights”MiΛi.This ij=Aji ,withCartanmatrixAturnsouttobetherescaledlatticeofadualLiealgebrag
=so(D)ifDiseven,andg =sp(D 1)ifprovided(2.24)holds[37].Inthepresentcase,g
res“contains”acorrespondingclassicalLiealgebraasaquotient.ThisisDisodd.Infact,UqtheessenceofaremarkableresultofLusztig[23],andcanbemadeexplicitasfollows([37],
Theorem4.2):
Letai∈{0,1}suchthatai+aj=1ifAij=0andi=j;thisisalwayspossible.De ne i=qdiMiHi,andK (Mi) (Mi)
+=X+(Mi)K ai,Xiii
2 =X (Mi)K 1 aiqMi,Xiiii i=[X +,X ]Hii
8(2.36)
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
Thenonecanshowthefollowing:
Theorem2.2Forallspecialweightsλz,Lres(λz)isanirreduciblehighest–weightrepresen- ±andH i.If tationoftheclassicaluniversalenvelopingalgebraU(g),withgeneratorsXi ′ i·vz′=z′vz′.vz′∈Lres(λz)hasweightjzjMjΛj,thenHi
Inparticularifλzisadominantweight,thenthecharacterofLres(λz)isinvariantundertheWeylgroup,andcanbeobtainedfromWeyl’ingthis,thestructureofLres(λ)with“large”dominantintegralλcanbedescribedasfollows[5]:∨Theorem2.3Letλzasin(2.33)andλ0beanintegralweightswith0≤(λ0,αi)<Mifor
alli.Then
Lres(λ0+λz)=Lres(λ0) Lres(λz).(2.37)
Thisisnothardtoprove,see[5]or[37].Moreover,thegenerators(2.36)essentiallyacton
finthesecondfactorin(2.37)andUqonthe rst,butwithacertain“twisting”[37].
3Realitystructuresandsymmetryalgebra
Sincetheproperchoiceoftherealitystructureiscrucialinthefollowing,we rstdiscusstherelationoftherealstructuresonthespaceswiththeirsymmetryalgebras.
ThealgebraoffunctionsonbothclassicalandquantumD–dimensionalcomplexEu-clideanspaceisgeneratedbycoordinatefunctionsxi,whichtransforminthevectorrepre-sentationVDofUq(so(D,C)).Thetensorproductof2suchrepresentationscontainsauniquetrivialrepresentation;inotherwords,thereisaninvariantbilinearform<,>:VD×VD→C.InHopf–algebralanguage,invariancemeans<f,g>=<u(1)·f,u(2)·g>forf,g∈VD,where (u)=u 1+1 u=u(1) u(2)denotesthecoproductofu∈U(so(D,C))orUq.Thisextendsimmediatelytopolynomialfunctions.
Intheclassicalcase,thealgebraofcomplexfunctionsonrealEuclideanspace(oranyrealmanifold)isequippedwitha –structure,i.e.anantilinearmapwhosesquareistheidentity,de nedbythecomplexconjugation.Theaboveinvariantbilinearformtheninducesahermitianinnerproductby
(f,g)=<f ,g>,(3.1)
whichsatis es(f,g) =(g,f).Thesignature(p,D p)ofthisinnerproductisthesignatureofthe(pseudo)Euclideanrealspace,anditispositivede niteonlyintheEuclideancase.
9
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
forallu∈U(so(D,C)).Thisde nestherealformU(so(p,D p)),andonecanthenstudyitsunitaryrepresentations,whicharein nite–dimensionalexceptintheeuclideancase.Intheq–deformedcase,weusethisconnectionbetweentherealformofafunctionspaceanditssymmetryalgebraintheotherdirection:therewillbeaclearchoiceoftherealformofUq,whichthendeterminestherealformofthefunctionalgebras.Arealformor –structureofUqisanantilinearanti–algebramap onUqwhosesquareistheidentity.AninvariantinnerproductonarepresentationofUqisahermitianinnerproductwhichsatis es(3.2)forallu∈Uq;inotherwords,thestaronUqisimplementedbytheadjoint,whichiswell–de nedfornondegenerateinnerproducts.ThisisparticularlyintuitiveforelementsofUqoftheformg=exp(itu)withu =u,sincetheng =g 1.
ArepresentationofUqisunitarywithrespecttoarealformofUqifithasapositivede niteinvariantinnerproduct.Then