q.Inthiscase,star
structuresdi erentfrom(3.10)havebeenproposed,ofthetypexD
becomesastaralgebra,theyleadtostarstructuresonUi=±xi[8,4].WhileCthenthesecondtype(3.5),q
qof
whichwehavediscardedabove.However,wewillseebelowthatthereisastaronthesemidirectproductalgebraUq
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
situationforq∈Riscompletelyanalogous.Atrootsofunityhowever,thestructureofpolynomialschangescompletely,andwewillseethattheanaloguesoftheclassicalscalar eldscaninfactbewrittenaspolynomialsintheti.
Toseethis,wehavetostudyCDqinmoredetail.ConsiderthesetofhomogeneouspolynomialsMqk CDqwithdegreekinthexi,whichformsasubmoduleofthek–foldtensorproductrepresentationV k
xDofUqres.Mqkisnotirreducible,becausethemetricprojectormay
benontrivial.Clearly1...x1=(x1)k∈Mqkisahighestweightvectorwithweightkλ1.Itgeneratesthehighestweightmodule
F(k):=Uqres·(x1)k Mqk.(4.1)
Ifqisgeneric,thenF(k)isairreduciblerepresentationwithhighestweightkλ1correspondingtoatotally(q–)symmetrictracelesstensor,andMqk=F(k)⊕x2
Ifqisarootofunity,thenMqkisnotcompletelyreducibleanyF(more,k 2)which⊕...asisclassically.atypicalphenomenonatrootsofunity.ThefullstructureofMk
discussedelsewhere.Hereweonlyconsiderthosemodesqisquitecomplicated,andwillbe
whichwillbeneededfortheHilbert
spaceofscalar eldsonAdSq.
First,weidentifythepolynomialsF(k)inCDqwhichhaveessentiallythesamestructureasclassically,whichmeansthecharacterofLres(kλ1)ing(2.24),Theorem2.1impliesthatχ(Lres(kλ1))=χ(kλ1)fork≤
indeed,anypositiverootαcanthenbewrittenasα=
3)d1 MSniα i(withD n3)i,whichis≤M≤(assuming=ai,whereDai≥are4;
theCoxeterlabels.Hence(kλ1+ρ,α)≤(k+Dd1M1providedk≤M1
However, (Dthis 3).boundForDcan=3,betheimprovedboundisusingk≤MS
the strong1/2).linkageprinciple,seee.g.[38].
Wecanassumethatk<MS;thenitimpliesthatthecharacterofLres(kλ1)=Lfin(kλ1)isthesameasclassically,unlessitcontainsadominantintegralweightµwhichisintheorbitofkλ1undertheWeylgroupactingwithcenterMSλ1 ρ(sinceMsλ1isaspecialweight).Since(λ1
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
(ForoddD,thereisasimilarphenomenon).Thisisrelatedtothescalarsingleton eldontheAdSspace,aswewillseeinthenextsection.Onecancheckthate.g.theCasimirv(2.13)indeedbecomesdegeneratefortheseweights.kTosummarize,thestructureofMqisthesameasclassicallyforsmallk:
Theorem4.1Forn0≤kSandrootsofunityoftheform(2.24),
F(n0)～=Lres(n0λ1)=Lfin(n0λ1)
n0hasthesamecharacterχ(n0λ1)asclassically.Moreover,Mqisthedirectsum
n0Mq(4.5)=
0≤k≤n0/2 (x2)kF(n0 2k).(4.6)
TheproofiscompletedinAppendixC.2
NextconsiderF(kMS),whichisofcentralimportancetous.ItshighestweightkMSλ1isaspecialweight(2.33).Using(2.6)andthecommutationrelations(2.20),onehas
X1·(x1)n=qS (n 1)/2
(n 1)/2
(n 1)/2x2(x1)n 1+qS (n 3)/2x1x2(x1)n 2+...+qS)x2(x1)n 1(n 1)/2(x1)n 1x2(4.7)=qS2(1+qS+...+qS2(n 1)
=qS[n]qSx2(x1)n 1.
Since[kMS]qS=0,itfollowsthatXi ·(x1)kMS=0foralli,hence(x1)kMSisaone–
findimensionalrepresentationofUq.Asdiscussed inSection2.1below(2.35),thisimplies
thatallweightsofF(kMS)havetheformλz=ziMiΛi,andF(kMS)isarepresentation ±andH i.Moreover,itis )withgeneratorsXoftheclassicaluniversalenvelopingalgebraU(giahighestweightmodulewithhighestweightvector(x1)kMS.Bytheclassicalrepresentationtheory,itfollowsthatitisirreducible,hence
F(kMS)～=Lres(kMSλ1),(4.8)
(exceptforD=3,whereitisL(kΛ1)).whichisessentiallyL(kλ1)ofg
Nowconsidermoregenerallyn=n0+kMSwith0≤n0<kSandk∈N.ThenbothF(n)andF(n0) F(kMS)arehighestweightmoduleswithhighestweightnλ1.ClearlyF(n) F(n0)F(kMS),by(4.1)and(2.6).Ontheotherhand,F(n0)F(kMS) F(n0) F(kMS),whichisisomorphictoLres(nλ1)byTheorem2.3.Hencewehaveshownthat
14
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
Theorem4.2Forn=n0+kMSwith0≤n0<kSandk∈N,
F(n)=F(n0)F(kMS)～=Lres(nλ1).=F(n0) F(kMS)～(4.9)
Inparticular,itisessentiallythetensorproductofF(n0)withtheirreduciblerepresentation
.L(kλ1)(orL(kΛ1)forD=3)oftheclassicalalgebrag
5
5.1RealformsandHilbertspacerepresentationsofthequantumspacesD 1D 1HilbertspacesforSqandAdSq
Nowwearereadytodiscusstherealitystructureatrootsofunity.Aswaspointedoutbefore,(3.10)and(3.11)arenotconsistentwiththealgebraforqaphase.To ndthecorrectde nition,we rstconstructtheHilbertspaces,andthensimplycalculatetheadjointoftheoperatorsofinterest.Wewanttoemphasizethattheinnerproductsonirreduciblerepresentationsaredetermineduniquelyby(3.2).Indeedonanyhighestweightmodule,thereexistsaunique(uptonormalization)invariantinnerproductforagivenstarstructureoftheform(3.4).Thisisbecauseoncetheinnerproductisde nedonthehighestweightvector,itcanbecalculatedforalldescendantvectorsusing(3.2);ingeneral,itisnotunitary.Theresultinginvariantinnerproductisnon–degenerateiftherepresentationisirreducible.