; TeX output 1998.02.10:14057 Y N cmbx12FOURIERٚTRANSFORMSOVERFINITEQUANTUMGROUPS_WK`y cmr10BODOUUP*AREIGIS's, P cmu10HerrnPrqofessorDr.D.Pumpl#N:unalsDank0f#N:urseinlangjqahrigesEngagementfN:urdieMathematikunddieUniversitqatgewidmet.)V$XQ cmr121.- cmcsc10Introduction InthisnotewrewanttoclarifythenotionofanintegralforarbitraryHopfalgebrasthatDhasbSeeninrtroducedalongtimeago[2,6 u@].jTherelationbetrweenDtheinrtegralonaPHopfalgebraandinrtegralsinfunctionalanalysishasonlybSeenhintedatinseveralpublications.!WithAthestronginrterestinquantumgroups,i.e.!non-commutativeandnon-coScommrutativeg*Hopfalgebras,Kwrewishtoshowinwhichformcertaintransfor-mationrulesforinrtegralsoSccurinquantumgroups. Our&pSoinrtofviewwillbethefollorwing.*`Letg cmmi12Gbeaquanrtumgroupinthesenseofnon-commrutativealgebraicgeometryV,]thatisaspacewhosefunctionalgebraisgivrenbryanarbitraryHopfalgebraHoversomebase eld) msbm10K.YWVewillalsohavetousethe"algebraoflinearfunctionalsHV2!K cmsy8 ¹=hHomL(HF:;K)withthemrultiplicationinducedbry],thediagonalofHJ(calledthebialgebraofGintheFVrenchliterature).lFVormostofthispapSerwrewillassumethatHJis nitedimensional.ObservethatthefunctionsinHTdonotcommruteundermultiplicationandthattheyusuallyhavenogeneralcommrutationformula. TheAImoSdelforthissetupcanbefoundinfunctionalanalysis.Foranarbitrary nitegroup$GtheHopfalgebraH~`= K22cmmi8G üisde nedtobSethealgebraoffunctionsonG.ThenHV2 =URKG,thegroupalgebra,isthelineardualofHV. If;the nitegroupGisAbSelianandifKisalgebraiclyclosedwithcrhar(K)S !", cmsy106AvjGjthenv{thecorrespSondingHopfalgebraisasaborvev{H0=CQK2G andHV2 =KG.ZByPron-tryraginzdualitythereisthegroupwE#u cmex10bG9ofcharactersonGsuchthatHB=URK2G t=KwGbG GandD HV2 =URKG=K |{Ycmr8^G. IfwwrouldfliketoacknowledgestimulatingconversationsonthesubjectofthisnotewithH.R ohrlandY.Sommerhauser. 7Hff< B ': cmti10Date[:UUF*ebruary10,1998. 1991MathematicsSubje}'ctClassi cation.PrimaryUU16A10. ]-o cmr91*7 &e2|BODO!P:AREIGISYg2.¹ Integrals Let`HMbSeanarbitraryHopfalgebra[3,6 @].Thelinearfunctionalsa2HV2 will`beconsideredas,@ cmti12generffalizedݟintegralsonHz([5]p.123).WVeharveanopSerationHV2 wHB3a fQ7!URha;fGi2K꨹thatisnondegenerateonbSothsides. WVezdenotetheelemenrtsofHhbyf;gn9;hJ2HV,thezelementsofHV2 (bya;b;cJ2HV2Z,the(nonexisting)elemenrtsofthequantumgroupGbyx;yn9;z52URG. WVewillbSeinrterestedinaspecialgeneralizedinrtegralUQR?2URHV2 satisfying˺t4aUQR US=URha;1HDiUQR(1)AoraUQR UQ=UR"(a)UQR .8Sucrhanintegraliscalledaleft35invariantinteffgral. InthecaseofaloScallycompactgroupGsucrhanelementisgivenbytheHaarinrtegralwithrespSecttoaleftinvXariantHaarmeasure[1]˺,ϟ甆Z{ _GfG(x)dxUR=hUQR;fi:Thereforewrewriteinthegeneralquantumgroupsituation! 甆Z fG(x)dxUR:=hUQR;fi:(2)vjThisrnotationhastrworparentheses,2%UQRanddx,2%sothattheintegrandfqisclearlyseparated.8WVealsousethenotation甆ZfG(x)gn9(x)dxUR:=hUQR;fgn9i:(3)˻ObservrethatfG(x)andgn9(x)arejustpartsofthewholesymbSolandinparticularthattheydonotcommrute. Inthecaseofa nitegroupGaleftinrvXariantintegralinHV2 =aKGonHO=K2G isknorwntobSev\UQRis nitedimensionalthenforeverya2HV2 #BtherffeuisauniquegË2URH such35thatha;fGi=UQR UQf(x)S׹(gn9)(x)dx35forallfQ2HV.'Corollary6.LWIfgHUis nitedimensionalthenSi[:H!HandgH3f7!(f*UQR )2HV2 arffe35bijective. IfGisa nitegroupthenevrerygeneralizedintegralaUR2K2G 臹canbSewrittenwithauniquelydeterminedgË2URHasi ha;fGiUR=甆ZURf(x)S׹(gn9)(x)dx=Xvfx2Gf(x)gn9(x 1 \|)(11)q)forallfQ2URHV. IfGisa niteAbSeliangrouptheneacrhgroupelement(rationalintegral)yË2URGKG꨹canbSewrittenasoߍգyË=URXvfx2GȰX-2P ^G*<  hx 1 \|;ix!|sinceBhyn9;fGi=h(UQR )P2P ^G%  );fGi=hUQR;fGS׹(P 2P ^G   )i=Px2G#Nhx;fGiP2P ^G" ߍhx;S׹()i!=hP x2G =ڟP,臟2P ^G@  hx21 \|;ix;fGi:2Inparticularthematrix(hx21 \|;i)isin-cvrertible.ݍv&E4.TheNaka32yamaAfutomorphism LetH@bSe nitedimensional.*SincehUQR;fGgn9iUR=h(UQR US(f);gn9iasafunctionalong.#isageneralizedinrtegral,thereisauniqueǹ(fG)UR2HsuchthatVDTlhUQR;fGgn9iUR=hUQR;gn9ǹ(fG)i(12)or甆ZfG(x)gn9(x)dxUR=甆ZURg(x)ǹ(fG)(x)dx:(13)*7 &eMF9OURIER!TRANSFORMSOVERFINITEQUANTUMGROUPSI5YAlthoughthefunctionsf;g^2%H ofthequanrtumgroupdonotcommuteundermul-tiplication,thereisasimplecommrutationruleiftheproSductisintegrated.CProp`osition7.YOThemape:Ho!Htisanalgebrffaautomorphism,calledtheNakXaryamaautomorphism.Prffoof.#RItv{isclearthat8Bisalinearmap.[WVeharvev{UQRvzfGǹ(gn9h)CR=UQRCQghfQ=UQRCQhfGǹ(g)=U^UQR fGǹ(gn9)(h)]henceǹ(gh)UR=ǹ(g)(h)]andUQR ]fGǹ(1)UR=UQR UQfhence](1)=1. FVurthermoreifpǹ(gn9) =0then0=hUQR;fGǹ(gn9)i=hUQR;gn9fGi=h(f5 *UQR );gn9ipforallf5 2 H]henceha;gn9iUR=0forallaUR2HV2 hencegË=0.8Sooisinjectivrehencebijective.Awcffxff ̟ff ̎ ̄cffCorollary8.LWThe35mapHB3URfQ7!(UQR US(fG)2HV2 is35anisomorphism.Prffoof.#RWVeharvexm(UQR US(URfG)=(ǹ(f)*UQR US)sinceKh(UQR(fG);gn9i=hUQR;fGgn9i=hUQR;gn9ǹ(fG)i=h((fG)*UQRĹ);gn9i. ThisKimpliesthecorollaryV.z Ecffxff ̟ff ̎ ̄cff- IfGisa nitegroupandHB=URK2G @thenHiscommrutativehence=id ."5.TheDiracDel32taFunction Anelemenrt2H;iscalledaDirffacL>s2-functionifyisaleftinvXariantintegralinHwithhUQR;s2iUR=1,i.e.8if]ڹsatis esꍑtfGȄ=UR"(f)ʹandL6甆ZZ6s2(x)dx=1XforallfQ2URHV.8IfHhasaDiracs2-functionthenwrewriteȍP_甆ZPaMca(x)dxUR=UQR USWa:=ha;s2i:(14)Prop`osition9.O~.V퍍 1.#If35H is nitedimensionalthentherffeexistsauniqueDiracs2-function. 2.#If35H isin nitedimensionalthentherffeexistsnoDiracs2-function.Prffoof.#R1.SinceKHB3URfQ7!(f*UQR US)2HV2=isKanisomorphismthereisaȄ2URH99sucrhthatU^(ud*2UQR 3)2=":P8Then(fG*2UQR 3)2=(fJ1*(*UQR 3))=(fJ1*")="(fG)"="(f)(ud*UQR 3)whicrhimpliesfGȄ=UR"(f)s2.(7FVurthermorewehavehUQR;s2iUR=hUQR;1HDs2i=h(Ȅ*UQR US);1HDi="(1HD)UR=1*ppmsbm8K. 2.8is[6]exerciseV.4.1ۄcffxff ̟ff ̎ ̄cffLemma10.FQLffetyHgbea nitedimensionalHopfalgebra. :"ThenUQR+2HV2 '"isaleftinteffgral35i ύgva(XUVUQRUW*(1))\{ S׹(UQR*(2)\}))UR=(XUVUQRUW*(1) S׹(UQR*(2)\}))a(15)97 &e6|BODO!P:AREIGISYi Cy|&X|S׹(a)UQR*(1)% UQR *(2)\w=URXUQR*(1), aUQR*(2)(16)Ci UݟX3f(1) \|hUQR;f(2)iUR=hUQR;fGi1HD:(17)9Prffoof.#RLetꨟUQRQbSealeftinrtegral.8Thenu =XQa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)UR=X(aUQR)(1)$ S׹((aUQR)(2) \|)UR="(a)(XUVUQRUW*(1))\{ S׹(UQR*(2)\}))CforallaUR2HV.8Hence$yٔ?:(PUQR*(1)$ S׹(UQR*(2)\}))a=URP"(a(1) \|)(UQR*(1)% S׹(UQR*(2)\}))a(2)y=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)a(3)=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})"(a(2) \|)UR=a(PUQR*(1)$ S׹(UQR*(2)\})):' ConrverselyhNa(PUQRUV*(1)!"(S׹(UQR*(2)')))=(PUQRUV*(1)"(S׹(UQR*(2)')a))="(a)(PUQRUV*(1)"(S׹(UQR*(2)'))),^henceꨟUQR?=URPUQR*(1)%""(S׹(UQR*(2)'))isaleftinrtegral. SinceSisbijectivrethefollowingholdsQcPrS׹(a)UQR*(1)# UQR*(2)\w=URPS(a)UQR*(1) S21 S(S(UQR*(2)'))ye=URPUQR*(1)%" Sן21 S(S׹(UQR*(2)')S(a))UR=PUQR*(1) aUQR*(2)#:TheconrversefollowseasilyV. IfꨟUQR?2URInrtWşly(HV)isaleftinrtegralthenPUha;f(1) \|ihUQR;f(2)iUR=haUQR ;fGi=ha;1HDihUQR;fGi. Conrversely0if462HV2 )with0(17)isgivrenthenha;fGi46=Pha;f(1) \|ih;f(2)i46=ha;1HDih;fGi꨹henceaUR="(a). cffxff ̟ff ̎ ̄cff2э IfGisa nitegroupthen s2(x)UR=z( 0ifx6=e;ɍ 1ifx=e:(18)"&yInfactsinceCOisleftinrvXariantwegetfG(x)s2(x)UR=f(e)s2(x)forallxUR2GandfQ2URK2G.SincejG/HV2 ܹ=KGisabasis,wregets2(x)=0ifx6=e.oFVurthermoreUQRjs2(x)dx=P x2G =s2(x)UR=1impliesfG(e)UR=1.R IfGisa niteAbSeliangroupwreget׷=d P:2P ^G"Ϡforsome x2K.TheevXalu-rationgivres1n= hUQR;s2i= P:x2G;2P ^G4?Eh;xi.Norwletn2w bG.ThenPk?2P ^G#|h;xi=CP 2P ^Gh;xi=h;xiP2P ^G h;xi.Since'Wforeacrhx2Gnfeg'Wthereisasuchthat>h;xiUR6=1andwregetaSX-K2P ^G`h;xiUR=jGje;x :IB7 &eMF9OURIER!TRANSFORMSOVERFINITEQUANTUMGROUPSI7YHenceꨟPUx2G;2P ^G6`h;xiUR=jGj= 21Zand6UȄ=URjGj 1 zX- \z2P ^G :(19)+6.GFfourierTransforms LetHhbSe nitedimensionalfortherestofthispaper.-InCorollary8wrehaveseenthat`pthemapH "3fe7!(UQR (fG)2HV2 ʹis`panisomorphism.8ThismapwillbSecalledtheFourier35trffansform._4Theorem11.PXThe35FouriertrffansformHB3URfQ7!V~e*f K2HV2 is35bijectivewithLVse*Jf=UR(UQR US(fG)=XhUQR*(1)\};fiUQR*(2)(20)MThe35inverseFouriertrffansformisde nedbyne]Caݖ=URXSן 1 S((1) \|)ha;(2)i:(21)Sincffe35thesemapsareinversesofeachotherthefollowingformulashold#+GN hV)Re*f;gn9iUR=甆ZUTfG(x)g(x)dxUha;V*e*biUR=甆ZUTMXSן21 S(a)(x)b(x)dx*N fQ=URPSן21 S((1) \|)hV)Re*f;(2)iUaUR=PhUQR*(1)';)ea*iUQR*(2)#:(22)(]Prffoof.#RWVeVusetheisomorphismsHb)!tHV2 de nedbryVb*f>:=V %e*fM=(UQRt(fG)=U^P hUQR*(1)';fGiUQR*(2)randkHV2 9!HY!de nedbry+ba{:=(a*s2)=P(1) \|ha;(2)i. JBecauseofZ}ha;V*b*biUR=ha;(b*s2)i=hab;s2i(23) andhV)Re*f;gn9iUR=h(UQR US(fG);gn9i=hUQR;fGgn9i(24)wregetforallaUR2HV2 andfQ2H􍍟{4ha;UUe bVsb*b!fti,=URhaV)Rb*f;s2i=Pha;(1) \|ihV)Rb*f;(2)iUR=Pha;(1) \|ihUQR;fG(2)i(bryLemma10),=URPha;S׹(fG)(1) \|ihUQR;(2)iUR=ha;S׹(fG)ihUQR;s2iUR=ha;S(fG)i:%ThisgivresUU~bV8b*{f ʹ=Y=S׹(fG).1SotheinversemapofHF!Y=HV2 withVCb*f=(UQR Y>(fG)=Ve*fis-PHV2 !wH"with5[Sן21 S(+ba+)=P"Sן21((1) \|)ha;(2)iw=ea x.Then5[thegivreninversionformulasareclear.z Ecffxff ̟ff ̎ ̄cff- IfGisa nitegroupandHB=URK2G @thenLVsje*Jf=URXvfx2GfG(x)x:ZѠ7 &e8|BODO!P:AREIGISYڍSince.(s2)UR=Px2G#,x215 <) 1+x2o2wherethex2V2K2G ƹarethedualbasistothex2G,wreget\w8e a7`=URXvfx2Gha;x ix :!a IfGisa niteAbSeliangroupthenthegroupsGandw9DbG'areisomorphicsotheFVouriertransforminducesalinearautomorphisme- *:URK2G t!K2G @andwrehaveeaw߹=URjGj 1 zX- \z2P ^Gha;i 1$ÍBysubstitutingtheformrulasfortheintegralandtheDiracs2-function(4)and(19)wregetܶVSe*Q_;f[4=URPx2G#,fG(x)x;W)eߖaQ=URjGj21 \zP'2P ^G.a()21 \|;㓍Q_;fQ=URjGj21 \zP'2P ^GV0Ae*.f5+4()21 \|;ߖaUR=Px2G#SWe#,a)-(x)x:(25)ThisimpliesNVzAe*xDf*()UR=Xvfx2GfG(x)(x)=甆ZURf(x)(x)dx(26) withinrversetransformύe?aj(x)UR=jGj 1 zX- \z2P ^G (a) 1 \|(x):(27)$Lemma12.FQThe%FouriertrffansformsoftheleftinvariantintegralsinH)andHV2 -are#ӍVq*_e*qz=UR"ǟ 1s2HV !andQX3ePUQR\۹=12HF::(28)XPrffoof.#RWVe=+harvehV%e*q;fGi=hUQR;s2fGi=hUQR;ǟ21 C(fG)s2i="ǟ21(fG)hUQR;s2i="ǟ21(fG)=+hence_V%e* Fֹ=e"ǟ21a=andBha;eUQR i=PJha;Sן21 S((1) \|)ihUQR;(2)ie=ha;Sן21 S(1)ihUQR;s2ie=ha;1i;BhenceeUQR US=UR1..ocffxff ̟ff ̎ ̄cff͍Prop`osition13.`De ne35aconrvolutionmultiplication35onHV2 byxfchab;fGiUR:=Xha;Sן 1 S((1) \|)fGihb;(2)i:Then35thefollowingtrffansformationruleholdsforf;gË2URHV:#͍V