Quantum Group Covariance and the Braided Structure of Deform

1、quantum group covariance and the braided structure of deform the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroup

2、s and their repre 2 002 lju 81 aq.tham 1v1517020/tham:vixraquantumgroupcovariancedeformedandoscillators thebraidedstructureof a.yildiz fezagurseyinstitute,p.o.box6,81220,cengelkoy,istanbul,turkey1. abstract theconnectionbetweenbraidedhopfalgebrastructureandquantumgroupcovarianceofthe deformedoscilla

3、torsisconstructedexplicitly.inthiscontextweprovidedeformationsofthehopfalgebraoffunctionsonsu(1,1).quantumsubgroupsandtheirrepresentationsarealsodiscussed. 1introduction thecovarianceoftheoscillatoralgebrasattractedalotofattentionandisdiscussedindi erentcontexts1.thecovarianceofanalgebraundertheacti

4、onofanoncommutativealgebradeformsthenotionofde ningidenticalcopiesinthetransformedalgebraandthisleadstothedeformationoftheusualtensorproductnamelybraidedtensorproduct.thehopfalgebraaxiomsarereplacedbythebraidedhopfalgebraaxioms2.hencebraidedgrouptheory(selfcontainedreviewscanbefoundinref.3)uni esthe

5、notionsofsymmetryandstatistics.recently,wefoundthegeneralbraidedhopfalgebrasolutionsofthegeneralizedoscillators4.inthisworkweshowthatsomeofthesesolutionsareconnectedwiththequantumgroupcovarianceandwe ndther-matricescontrollingthebraidingstructureandthequantumgroup.wealsodiscusstherepresentationsofqu

6、antumsubgroups. 2 thegeneralizedoscillator,itscovarianceandbraidedhopfstructure supposethatthegeneralizedoscillatoralgebra aa q1a a=q2n aqn=na(1) qna =qa qn iscovariantunderthetransformation (a)=ak1+qnk2+a k3 (a )=a k1 +qnk2 +ak3 (2) (qn)=al1+qnl2+a l 1 the connection between braided hopf algebra st

7、ructure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre wherethedeformationparametersq1andqarepositiveandwiththe -structure(a ) =aand (qn) =qn.th

8、eelementsk1,k1,k2,k2,k3,k3,l1,l 1andl2(l2=l2)generatesomealgebra.ouraimisto ndthatalgebraifthetransformationisaquantumgrouptransformation.wewritetheabovetransformationasacovectortransformation x=xt where x=(aa qn), thematrixtisaquantummatrixsatisfying k1k3l1 l andt= k3k11 . k2k2l2 (3) (4)(5) rt1t2=t

9、2t1r andrsatis esqybe r12r13r23=r23r13r12. (6) to ndther-matrixandhencethequantumgroup,wewritetheoscillatoralgebraasacovector algebra x1x2=x2x1r(7)where x1x2=x2x1= aa 2 aa aq n aa(a) 2 aq n qaqaaq n nn n 2n2n (8)(9) andthegeneralformofther-matrix 2 aaqaaa n (a) 2 qa n aq r= 10000a10a600a400a20a70000

10、000000a5000000a30a8000000000a1500a11000000a16100000a90a13000a12000a100a1400000a17 theconstants(ai)appearinginther-matrixistobedeterminedfromtheconsistencyof(7)with theoscillatorrelations(1)andfrom(6).thecovarianceofacovectoralgebraundertheactionofaquantumgroupinducesabraidedhopfalgebrastructurewhose

11、axiomsarecollectivelygivenby m (id m)=m (m id)m (id )=m ( id)=id(id ) =( id) . (10) the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su

12、(1,1). quantum subgroups and their repre ( id) m (id s) (m id) (id m)(id ) ( id) ms m s m ( id) (id ) ( id)=(id ) =id m (s id) = (id m) ( id) (id )(m id) (id ) ( id)( id) (id ) ( id)(id )( id) (id ) (m m)(id id) ( )m (s s)(s s) (id ) ( id) (id ). (11) the -structureforabraidedalgebrabisdi erentfromt

13、henon-braidedonesuchthat = ( ) s = s (a b) =b a , a,bb. (12) thebraidedcovectoralgebrahasabraidedhopfalgebrastructure (x)=x 1+1 x, (x)=0,s(x)= x withthebraidingrelations (x1 x2)=x2 x1r, i.e., ab (xi xj)=xb xarij. (13) (14) thematrixrwhichcontrolsthebraidingrelationsshouldsatisfythefollowingcondition

14、s r12r13r23 r12r13r23r12r13r23 (pr+1)(pr 1) r21r = r23r13r12r23r13r12r23r13r120r21r (15) wherepisthepermutationmatrix.hencetheproblemof ndingthequantumgroupleavingthegeneralizedoscillatoralgebracovariantandthebraidingsinducedbythequantumgroupisreducedto ndingthematricesrandr. the connection between

15、braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre thegeneralformofthematrixrcanbewrittenas r= c1000000000c20c70000c1

16、200c5000c11000c30c80000c130000c1000000000c100c6000c6000c100000000c110c500c40c90000c14 (16) whichgivesthegeneralformofthebraidingrelations. forthethreedeformationparametersq1andqfree,itturnsoutthatthereisauniquesolutionforthematricesrandrnamely 10 0q21 0000000 00000 (q2 q1) 0000q2 q00(q2 q1) 0q1 00 0

17、0001010 000 000q2 00000 r= q 00 01 00 0q1 q10000 0q102 q q1 00q 000q21 00 00q2 0000 0000 0000 0000 00 q1 q1 00 00 0q q2 q1 q21 00 q2 the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide de

18、formations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre 1 similartothermatrixofsuq(2)thematrixrisproportionaltor(r=q2q 1r).theentriesofthequantummatrix(4)generatethealgebra k1k1 =k1 k1+q2q 12l 1l1+q 2q1(q2 q1)k3 k3 k1k2 = 11k2k1 k1k2 =q 1q1k2 k1+ 11l2l1+q 2q1(q2 q1)

19、k3 k2 k1k3 =q2q 12 k3k1 k1k3 =q1k3 k1+l21k1l1=ql1k1 k1l 1= 11l 1k1+q 1(q2 q1)l1k3k1l2 =l2k1+q 1(q2 q1)l1k2 k2k2 =q1k2 k2+q 2q21k3 k3 k1 k1+l2 2 k2k3 = 11k3k2 k2k3 =q 1q21k3 k2+l2l1k2l1=q1l1k2 k2l 1=l 1k2+q 1(q2 q1)l2k3 k2l2 =ql2k2 11l 1k1+q 1 q1l1k3) k3k3 =q 2q31k3 k3+l 1l1k3l1=q 1q21l1k3k3l 1=ql 1k

20、3k3l2=q1l2k3 l1l 1=q2q 12l 1l1 l1l2 = 11l2l1. thehopfalgebrastructureisgivenbythegrouphopfalgebra (t)=t t, (t)=1,s(t)=t 1 wheretheinversematrixisgivenby l2k 1l k q 12l2k3 +q 1q 11l1k2 12l 1k3 q 1l1k t 1= q211 121l2k3+1l 1k2l2k1 q 1q1l1k2 11 q q21l1k3 ql 1k1 q2q 11k3k2 qk2k1 q 2q1k3 k2 q 1k2 k1 k 1 k

21、1 q 2q21k3 k3 theelementwhichisde nedtobe 1 . l2k1 k1 q 2q21l2k3 k3+l1k3k2 +l 1k3 k2 11l 1k2 k1 q 1q1l1k2k1 hasgrouplikehopfalgebrastructure ()= , ()=1, s()= 1 andsatis es (19) (20) (21) (22) (23) the connection between braided hopf algebra structure and the quantum group covariance of deformed osci

22、llators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre 24 2 k1=k1,k2=q 1q21k2,k3=1k3,l1=1l1,l2=l2 (24) andtheir*-conjugateswith =. thebraidedhopfalgebrastructureofthegeneralizedoscillator(1)impliedbyth

23、equantumgroupcovarianceisgivenbythecoproducts (qn)=qn 1+1 qn, (a)=a 1+1 a, (a )=a 1+1 a , thecounits (qn)= (a)= (a )=0, theantipodes s(qn)= qn,s(a)= a,s(a )= a andthebraidingsimpliedby(18) (qn qn)(qn a)(a qn)(qn a )(a qn)(a a)(a a )(a a )(a a) = 1nn q2q 1q q, 1n 1a q, 1n 1q a, 12n qa qn+q 1(q q1)q a

24、, 12n n a+q 1(q q1)a q, 1 q2q 1a a, 1 q2q 1a a,12 nnq 1(q q1)a a+q1a a+q q, 2nn2 2 q2q 1q q+1a a. (25) (26) (27) (28) incontrasttothethreeparameterdeformedcasewherethereisauniquesolutionforthebraidings, thetwoparameterdeformedcaseq1=q2hasthreemoresolutionsapartfromthesolutionobtainedbysubstitutingq1

25、=q2into(28).thesesolutionsaresol1: c1=1,c2=q2,c3=0,c4=0,c5=q,c6=0,c7=0, c8=q 2,c9=0,c10=q 1,c11=0,c12=0,c13=0,c14= 1 sol2: c1=1,c2=q2,c3=0,c4=2,c5=q,c6=0,c7=0, c8=q 2,c9=0,c10=q 1,c11=0,c12=0,c13=0,c14=1 sol3: c1=1,c2=q2,c3=0,c4=0,c5=q,c6=0,c7=0, c8=q 2,c9= 2q 2,c10=q 1,c11=0,c12=0,c13=0,c14=1. (29)

26、 (30) (31) the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre theq1=q2caseisspecialnotonlybeca

27、usetherearethreemoresolutionsforthebraidings, 12 butalsother-matrixistriangular(r 12=r21)ands=idissatis edforthequantumgroup.wealsonotethatinthegeneralbraidedhopfalgebrasolutionsgiveninref.4,onlythesolutionswegiveinthissectionarerelatedwiththequantumgroupcovariance. weshouldalsonotethatwhenl1=l 1=k2

28、=k2=0andl2=1thetransformationmatrixisanelementofsu(1,1)intheq=q1=1limit.hencethegroupwede necanbeinterpretedasdeformationsofsu(1,1). 3subgroupsandrepresentations inthegeneralformofthetransformationofthegeneralizedoscillator,theinvariancequantumgroupisanine-parameterquantumgroupwiththreedeformationpa

29、rameters.thisquantumgrouphassevenand veparametersubgroupswhichwearegoingtodiscuss.a:thesevenparametersubgroupcanbeobtainedbysettingl1=l 1=0in(19).thentheconsistencyoftherelationsrequiresq1=q2,i.e.,fortheoscillator aa q2a a=q2n aqn=naqna =qa qn thetransformation (aa qn)=(aa qn k1k30 0 ) k3k1 k2k2l2 (

30、32) leavesthealgebracovariantwheretheentriesofthequantummatrixsatisfy k1k1k1k2 k1k2k1k3 k1k3k1l2 k2k2k2k3 k2k3k2l2 k3k3k3l2 (33) = k1k1,q 1k2k1, qk2k1,q 2k3k1, k1,q2k3 l2k1, q2k2k2+q2k3k3 k1k1+l22,q 1k3k2, k2,q3k3 ql2k2, q4k3k3,q2l2k3. (34) thehopfalgebrastructureisgivenbythegrouphopfalgebra,i.e., (

31、t)=t t, (t)=1,s(t)=t 1 (35) the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre andthematrixinv

32、erseis t 1 = l2k1 q4l2k3 k3k2 qk2k1 wheretheelement q0 1 l2k10 k3k2 q 1k2k1k1k1 q2k3k3 4 l2k3 (36) l2(k1k1 q2k3k3) (37)(38) isgrouplikeandsatis es ()= , ()=1,s()= 1 k1=k1,k2=q3k2,k3=q6k3,l2=l2, =. (39) toconstructtherepresentation,wetakethegeneratorsofthisalgebraasoperatorsactingon somespace.we rst

33、ndthesimultaneouslydiagonalizibleoperators:theoperatorsl2,k1and k1commuteamongthemselvesandtakingintoaccountthatl 2=l2wecantaketheseoperatorsasdiagonaloperators.wetaketheeigenvalueofthehermitianoperatoras l2|n =aqn|n whereaisarealconstant.therelationsofthealgebrasuggestthat k2|n |n 1 , k3|n |n 2 (41

34、)(40) andhencewetaketheactionsofthegeneratorsas k1|n =k1,n|n , k1|n =k1,n|n , k2|n =k2,n|n 1 k3|n =k3,n|n 2 k2|n =k2k3|n =k3,n+1|n+1 ,n+2|n+2 . (42) substitutingtheseinto(34)weobtain k1,n=bqn, k2,n=cqn, k3,n=dqn (43) whereb,canddarecomplexconstantssatisfying |b|2=a2+q2|d|2. (44) wenotethatthereprese

35、ntationisin nitedimensional. b:the veparametersubgroupcanbeobtainedbysettingl1=l 1=k3=k3=0inthetransformation(2),i.e.,forthealgebra aa q1a a=q2n aqn=naqna =qa qn thetransformation (45) the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is c

36、onstructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre (aa qn)=(aa qn leavesthealgebracovariantwheretheentriesofthequantummatrixsatisfy k1k1k1k2 k1k2k1l2 k2k2k2l2 k100 0 ) 0k1 k2k2l2 (46) = k1k1, 1 1k2k1, q 1q1k2k

37、1,l2k1 q1k2k2 k1k1+l22ql2k2. (47) thehopfalgebrastructureisgivenbythegrouphopfalgebra (t)=t t, theinversematrixis t where 1 (t)=1, s(t)=t 1. (48) = l2k1 qk2k1 00 1 l2k10 q 1k2k1k1k1 (49) l2k1k1 (50) s()= 1 l2=l2. (51)(52) isgrouplike ()= , andsatis es k1=k1, ()=1, k2=q 1q21k2, similartotheconstructi

38、onoftherepresentationofthesevenparametersubgroup,theelements l2andk1canbetakenasdiagonaloperatorsandk2andk2canbetakenasloweringandraisingoperatorsrespectively.wetaketheeigenvalueofthehermitianoperatoras l2|n =aqn|n andfortheotheroperatorswetake k1|n =k1,n|n ,k2|n =k2,n|n 1 , k1|n =k1,n|n ,k2|n =k2,n

39、+1|n+1 . substitutingtheseinto(47)weobtain (53) (54) k1,n=b q1 q1 q2 |b|2 qn1 q1 q1q1 the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on

40、su(1,1). quantum subgroups and their repre whereaisrealandbiscomplex.thequadraticcasimirofthealgebrawhichisfoundtobe c=k1k1+(q 2 1)k2k2+q 2l22 (56)(57) hastheeigenvalue c|n =(a2+q 2|b|2)|n . inthealgebra(47)whenweidentify l2qh,k1=k1q h,k2(q q 1)1/2x ,k2(q q 1)1/2x+,q1=1 (58) thealgebraturnsouttobe q

41、h x=q1 xqh , 2hx+x x x+= q q 2h the connection between braided hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. in this context we provide deformations of the hopf algebra of functions on su(1,1). quantum subgroups and their repre connections

42、betweenquantumgroupsandnonextensivestatisticalmechanics6.inthisworkweuserealdeformationparameters,however,thefractionalsupersymmetricstructuresrequireatleastoneofthedeformationparameterstobearootofunity7asthegeneralizationofthe(-1)factorinthefermioniccase.onemorethingwhichdeservesaseparatestudyisthe

43、decouplingoftheoscillatorsortheunbraidingtransformations8. thebraidedcovectoralgebrascovariantunderquantumgroupsarealsocovariantunderthebraidedgroupsobtainedfromquantumgroupsbyatransmutationprocess3.themainingredientofthisconstructionisther-matrix.hencether-matrixwefoundde nesanewbraidedgroupwhichwedonotconsiderhere. the conne