5.2Productspaces
Fromtheabovediscussion,itwouldseemmuchmorenaturaltoconsiderallpolynomials
fininthetiinsteadofjustcertainUq·(t1)n.Forsimplicity,wewillrestrictourselvestothe
respolynomialsoftheformF(n)=Uq·(t1)nforallkMS<n<kMS+kSandallk∈N,and
1=λ1forD>3,andλ 1=λ1/2=Λ1forD=3.Thenstudytheir eldcontent.Letλ
F(n)=F(n0)F(kMS)using(4.9),wherethesecondfactorisessentiallytherepresentation 1)oftheclassicalsymmetryalgebrag ,whichconnectsthevariouscomponentswithL(kλ
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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
di erentλz′.Hence
⊕F(n)= F(n0) k∈N0≤n0<kS 1),L(kλ(5.9)
wherecertainmodeswereomittedasinSection5.1forsimplicity.Forthemomentweignoretherealitystructure. 1)areveryparticularrepresentationsoftheclassical ObservethattheL(kλg,whichhaveaniceinterpretation.Consider rstD=2r.Thenthedualalgebraisso (2r)=so(2r)as 1)canbeviewedasafunctionontheclassicalsphereSD 1.Henceshownin[37],andL(kλ 1)～⊕k∈NL(kλ=Fun(SD 1),whichisthespaceofpolynomialfunctionsontheclassicalsphereSD 1. 1=Λ1,andF(kMS)isNext,considerD=2r+1,whichislessobvious.Thenλ
thehighest–weightrepresentationL(kΛ1)ofso (2r+1)=sp(2r)withhighestweightkΛ1.Observethatthe2r–dimensionalrepresentationL(Λ1)isnotreal,inthesensethatthe2rvariablesziarenecessarilycomplex,andcanbeconsideredas4rrealvariablesxi.Thecompactform USp(2r)actsbymultiplyingtheziwithaunitarymatrix.Thereforethe
2radiusx=izi