%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Cartan3.dvi %%Pages: 38 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Cartan3.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.02.11:1843 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1100 600 600 (Cartan3.dvi) @start %DVIPSBitmapFont: Fa cmtt8 8.8 18 /Fa 18 123 df<007FB512F8B612FCA46C14F81E067C9827>45 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A728927>I64 D<3803FFC0000F13F8487F4813FF819038807FC0141F6C486C7E00041307C7FCEB03FF13 3F48B5FC1207121F481307EA7FE01380EAFE005AA46C130F007F131FEBC0FF6CB612806C 15C07E000313F1C69038807F80221F7C9E27>97 D99 DII104 D<131E133F497EA46DC7FC131E90C8FCA7EA7FFFB57EA37EEA001FB3A300 7FB512C0B612E0A36C14C01B2E7AAD27>I107 D<3A7F07E01F803AFFBFF0FFC0D9FFF97F91B57E7E3A 0FFC7FF1F89038F03FC001E0138001C01300A2EB803EB03A7FF0FFC3FF486C01E3138001 F913E701F813E36C4801C31300291F819E27>109 D<397FE03FC039FFF1FFF001F77F90 B57E6C80000313E0EC007F497F5B5BA25BAF3A7FFF83FFF8B500C713FCA36C018313F826 1F809E27>II<397FFC01FE9039FE07FF 8000FF011F13C0007F017F13E091B5FCD8007F130F14FC9138F007C09138E001004AC7FC 5CA291C8FCA2137EAC007FB57EA2B67E6C5CA2231F7F9E27>114 D<9038FFF3800007EBFFC0121F5A5AEB803F38FC000F5AA2EC078000FE90C7FCEA7FE0EB FF806C13F8000F13FE0003EBFF80C66C13C0010113E0EB000F0078EB03F000FC1301A27E 14036CEB07E0EBE01F90B512C01580150000FB13FC38707FF01C1F7B9E27>I<133C137E A8007FB512F0B612F8A36C14F0D8007EC7FCAD1518157EA415FE6D13FC1483ECFFF86D13 F06D13E0010313C0010013001F287EA727>I<397FE007FE486C487EA3007F7F0003EB00 3FB15DA25DEBFC076CB612F816FC7E013F13BF90390FFC1FF8261F809E27>I<001FB512 FE4814FFA490380001FEEC03FCEC0FF8EC1FF0001EEB3FE0C7EA7F80ECFF00495A495A49 5A495A495A495A495A49C7FC4848131ED807FC133F485A485AEA3FC0485AB7FCA46C14FE 201F7E9E27>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmmi6 6.6 1 /Fb 1 111 df<390F801FC0391FE07FF03939F1E0F83930F3807C3870FF00EA60FE5BEA E1F812C15B000314FC5D5BA2000713015DD9C003136015E0000F15E0020713C0018013C1 1680001F14C3913803C70090380001FE000EEB0078231A7C982A>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmbx8 8.8 16 /Fc 16 85 df48 D<1478EB01F81307133FEA07FFB5FCA213DFEAF81F1200 B3B1B612FEA41F3079AF2D>III< 151F5D5DA25D5C5C5C5C5C5C143D147914F11301EB03E1EB07C1EB0F811401131E133C13 7C13F8EA01F0EA03E013C0EA0780EA0F00121E123E5A5AB8FCA4C700031300A80107B6FC A428307EAF2D>I<000E140FD80FF013FF90B6FC5D5D5D5D5D158092C7FC14FC14C00180 C8FCA6EB87FF01BF13E090B512F89038FC07FC9038E003FE90388001FF90C71380000E15 C0C8EA7FE0A316F0A2121F487E487E12FF7FA216E05B15FF6C4814C0267E00011380123E 261F80071300390FF01FFE6CB512F86C5CC61480D91FFCC7FC24317CAF2D>II<123C123F90B612FCA44815F816F016E016C016801600007CC7127E00785C4A5A00F8 5C4813034A5A4A5AC7485A4AC7FCA2147E14FE5C130113035CA21307A2130FA25C131FA4 133FA96D5AA2EB078026327BB02D>III65 D68 D77 D80 D<913803FFC0023F13FC49B67E0107010013E0D91FF8EB1FF849486D7ED9FFC0EB03FF48 496D13804890C813C04817E049157F4848ED3FF0001F17F8A24848ED1FFCA3007F17FE49 150FA300FF17FFAB007F17FEA26D151F003F17FCA36C6CED3FF8A2000F02FE14F03C07FE 01FF807FE0D9FF07EBC0FF6CDA83E013C06C90268F00F113806C01CE017B1300D97FEEEB 7FFED91FFFEB3FF80107903880FFE0010190B5EA80026D6C15070203EBDFC091C7381FF0 1F94B5FC82A218FE82A27013FC18F8827013F0EF7FE0EF3F8038417BB243>I<003FB812 F8A4D9F003EB801FD87F80ED03FC01001501007E1600007C177CA20078173CA448171EA4 C71600B3A9011FB612F0A437317DB03E>84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmti8 8.8 59 /Fd 59 123 df11 D<923807FF80031F13E092387E00F8 DA01F8133CEC03E04A48137C17FC4A5AA217704AC8FCA45C143EA4147E147C013FB612E0 A217C0903900FC00074A130F1780A30101141F4A1400A35E163E13035C167E167CA21307 4AEBFC1CEEF83C1738A2130F4A1478EEF07017F0EEF8E0011F147991C7EA3FC0EE0F0093 C7FCA2133EA3EA383CEAFC7CA21378485A12F0EAF1E0EA7FC0001FCAFC2E4381B32C>I< DB07F8EB07FEDB3FFE90393FFF8FC0923BFC0F80FC03EF912901F00781F0011380913C03 E00FC3E003FF913B07C01FC7C0071B00DA0F80EB8F80DC070FEB01DFDC001FC7123F4AC7 153EA34D147E4A023E147C143EA21AFC057E5C027E147C147C013FB9FC62A2D9007CC738 F8000102FC16034A5FA216014D13070101605CA2190F04035D01035D5C191F97C7FC1607 01075D4AEE3F07F13E0F1A0E160F5F010F181E4AEE3C1CA2041FEC3E3894C7EA1E7849C7 6F5AF103C04C91C8FC163E133EA25E38383C0E26FC7C3F137816F80178495A3AF8F83E03 E03AF0F03C07C03AF1E03E0F80277FC00FFECBFC391F0003F84A4381B346>15 D<127812F87EA2127EA27E7EEA0F80A2EA07C01203EA01E0A2EA00800B0F6AB228>18 D<130F131F133F137F13FEEA01FCEA03F0EA07E0EA0FC0EA1F80EA3E005A5A5A100E67B2 28>I39 D<157015E0EC01C0EC0380EC0700141E5C143814785C495A495A 13075C49C7FC5B131E133E133C5B13F8A2485AA25B12035B1207A25B120FA290C8FC5AA3 121E123EA3123C127CA41278A512F8A412701278A71238123CA2121CA2121E120E7EA26C 7E6C7EA21C4A75B620>I<1430A2143880A280140F80A21580A2140315C0A71401EC03E0 A415C0A51407A41580140FA4EC1F00A3141E143EA2143C147CA2147814F85C1301A2495A A25C495A130F91C7FC5B131E5B137C13785B485A485A5B120748C8FC121C5A5A5A5A1B4A 7EB620>I44 DI<15C0140114031407EC0F80 141F143F147F903801FF00133FEB7FBFEB7E3FEB003EA2147EA2147CA214FCA25CA21301 A25CA21303A25CA21307A25CA2130FA25CA2131FA291C7FCA25BA2133E137E387FFFFEA2 B5FC1A3178B028>49 D<1370EA01F8EA03FCA313F813F0EA01E0C7FCAF121C127F5AA45A 12380E1F779E18>58 D<161C163C163E167EA216FE1501A21503A2ED077E150F150E151C 167F4B7EA2157015F015E0EC01C0A2EC038014071500140E17804A131FA25C147814705C 91B6FC5BA290390380001F130791C7FC010EEC0FC0A25B133C13385B13F0A2485A1203D8 0FF0EC1FE0D8FFFE0103B5FCA330357BB43A>65 D<010FB67E4915F017FC903A007E0001 FEEE007F027C801880171F14FC4A15C0A3010116805C173F180001035D4A14FE4C5A4C5A 01074A5A4AEB1FC0047FC7FC91B512FC4980913980003F80EE0FC0707E011F6E7E91C77F 1601A24981133EA3017E5D017C1403A24C5A13FC494A5A4C5A4C5A00014BC7FC49EB01FE 0003EC0FFCB712F016C04BC8FC32327AB137>II<010FB67E4915F017FC903A007F0003FEEE007F027EEC3F80EF0FC0A202FE EC07E04A140318F0A2010115014A15F8A313035CA3010715035CA3010F16F04A1407A301 1F16E04A140F18C0A2013FED1F8091C8FCEF3F00A249157E017E5DA24C5A01FE4A5A494A 5A4C5A4C5A0001037EC7FC49495A0003EC0FF0B75A93C8FC15F835327AB13B>I<010FB7 12C05BA2D9007FC7123F170F027E1407A2170302FE15805CA313015C1638EE7807010302 7013004A91C7FCA216F001075CECE001150791B5FC5B5EECC007A2011F6D5A1480170E03 07131E013FEC001C0200143C173892C7FC491578017E157017F04C5A13FE4914034C5A16 0F00014B5A49143F0003DA03FFC7FCB8FCA25E32327AB135>I<010FB712805BA2D9007F C7127F171F027E140FA2170702FE15005CA313015C1638EE780F0103EC700E4A91C7FCA2 16F001075CECE001150791B5FC5B5EECC007A2011F6D5A1480A21507013F91C8FC1400A2 92C9FC5B137EA313FE5BA312015B1203B512F8A331327AB133>II<90270FFFFE01B512C0495CA2D9007FC738 0FE000A2027E5DA2171F14FE4A5DA2173F13014A92C7FCA25F13034A147EA217FE13074A 5C91B6FCA25B9139C00001F8A21603131F4A5CA21607133F91C75BA2160F5B017E5DA216 1F13FE495DA2163F12014992C8FC00035DB539E01FFFFCA33A327AB13A>I<010FB5FCA3 9038003F80A21500A35C147EA314FE5CA313015CA313035CA313075CA3130F5CA3131F5C A3133F91C7FCA35B137EA313FE5B1201387FFFF0B5FCA220327BB11E>I<90260FFFFE90 380FFFE05BA2D9007FC73803FE0018F0027E4A5AEF0F804DC7FC02FE143C4A5C4C5A4C5A 0101EC07804A49C8FC163E5E010314F04A485AED07C04B5A0107131FECE03FEDFFC014E1 90380FE3CF9138C78FE0ECDF0702FE7FEB1FF8ECF00302E07FECC001D93F807F91C7FC82 A249147E017E147F828313FE49141F83160F0001825B000382B500E090B57EA295C7FC3B 327AB13C>75 D<010FB57E5B93C7FCD9007FC8FCA2147EA314FE5CA313015CA313035CA3 13075CA3130F5CA3131F5CEE0180EE03C0013F158091C7FC16071700495C137E161EA201 FE143E49143C167C5E000114034913070003EC3FF0B7FCA25E2A327AB131>I<90260FFF 80ED7FFC4917FF60D9007FEEFE006F140302775EF0077CF00EFC14F702E74B5A18381839 902601E3E0147102C35E18E1EF01C31303028391380383E0A2EF07070107150EDA01F05D 171CEF380F130F010E03705BA2EFE01F011EEC01C090261C00F892C7FCEE03804C485A13 3C0138020E133EA24C137E01785C0170017C147C5E18FC01F05C49D97DC05BA200019138 7F8001160000035F486C013E1303B592B512C0153C153846327AB146>III<010FB6FC4915E017F8903A007F0003FCEE00FE027E 147F8302FE1580171F5CA21301173F5CA21303EF7F005C17FE01075D4C5A4A495A4C5A01 0FEC1FC004FFC7FC91B512FC16F0D91FC0C9FCA25CA2133FA291CAFCA25BA2137EA213FE A25BA21201A25B1203B512E0A331327AB135>II<010FB512FC49ECFF8017E0903A00 7F000FF8EE01FC027E6D7E177E177F02FE805CA301015D5CA2177E010315FE4A5C4C5A4C 5A01074A5A4A495A043FC7FCED01FC49B512F016809138C003E06F7E011F6D7E4A7F167C 167E133F91C7FCA216FE5B137EA24B5A13FE5BEF01C01703000117805B00031607B5D8E0 001400170EEE7E1CC9EA1FF8EE07E032347AB139>I<913901FC01C091380FFF81023F13 C391397E03E780903901F800FF4948137F495A4948EB3F005C49C77EA2013E141EA3137E 161CA393C7FC137F806D7E14F86DB47E15F06D13FC010313FF010080141F02017FEC003F 151F6F7E1507A3120EA2001E5D121CA2150F003C5DA2003E4AC7FC5D007F143E5D6D5B39 7BC003F039F1F80FE000F0B5128026E03FFEC8FC38C007F82A367AB32C>I<0007B8FCA2 5A9039F001FC00018049133ED81F00151E001E1303A2485CA200380107141CA200785C12 70020F143C00F01638485CC71500141FA25DA2143FA292C8FCA25CA2147EA214FEA25CA2 1301A25CA21303A25CA21307A25C497E001FB512F0A3303173B038>I<267FFFC090380F FFE0B5FCA2D803F8C73801FC00496E5A5F4C5A5F6D4A5A0001150794C7FC160EA25E163C 16385E7F00005D15015E4B5A150793C8FC150E7F017E5B153C15385DA25D14015D90387F 0380133F4AC9FC140EA25C143C14385C14F06D5AA25C5CA291CAFCA2131E333471B13A> 86 D89 D97 D<137EEA1FFE5BA21200A25BA312015BA312035BA312075B EBC3F0EBCFFC380FFE1EEBF80F01F01380EBE007D81FC013C013801300A24814E0123EA2 EC0FC0127E127CA2141F00FC14805AA2EC3F00A2147EA200785B5C495A6C485A495A6C48 5AD807FEC7FCEA01F01B3478B224>I<14FC903807FF8090381F03C090387E01E0EBF800 3801F00300031307EA07E0EA0FC015C0391F80010091C7FC48C8FCA25A127EA312FE5AA4 1560007C14F0A2EC01E06CEB07C0EC0F806CEB3E00380F81F83803FFE0C690C7FC1C2178 9F24>III<153EEDFF80913801E3C0 EC03C3913807C7E0158F140F16C0ED83804AC7FCA45C143EA4147E147CA290383FFFFCA2 5DD900F8C7FCA413015CA513035CA413075CA4130F5CA4131F91C8FCA35B133EA3133CEA 387C12FC1378A2485A12F0EAF1E0EA7F80001FC9FC234381B318>I<143FECFFC0903803 E0E7903907C07F80EB0F8090383F003F133E49140001FC7F48485BA24848133EA2157E12 0749137CA215FC120F495BA21401A200075C14031407140F6C6C485A0001133F3800F07F EB7FE790381F87C0EB0007140FA25DA2141F92C7FC00385B007E133E147E00FE5B38FC01 F0387807E0383FFF80D80FFCC8FC212F7C9F24>III<153815FCA315F815701500AB143EECFF80903801C3C0EB0783EC03E0130E13 1E131C133CEB38071378017013C01300140FA21580A2141FA21500A25CA2143EA2147EA2 147CA214FCA25CA21301A25CA213035CEA3807007E5B495AD8FE1FC7FCEAFC3EEA787CEA 3FF0EA0FC01E4083B018>II<133FEA07FFEA0FFEA2EA007EA2137CA213FCA213F8A21201A213F0 A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2127CA2 EAFC1C133CEAF838A21378137012F013F0EAF8E01279EA3FC0EA0F00103479B214>I<27 07C003F0133F3C0FF01FFE01FFE03C1C783C1F03C1F0003C903AF00F8F00F826387DE013 9E90277FC007BC13780078018001F8137C007001005B017E5CD8F0FE010F14FC48484A5B A2C6485C0001021F130160491400A200034A13036049133E17070007027EECC1E0050F13 C049017C1481A2000F02FC148394381F038049491407050F1300001F0101150E181E0100 49EB07F8000E6D48EB01F03B217A9F40>I<3903C003F0390FF01FFE391C783C1F003C90 38F00F8038387DE090387FC0070078018013C000701300137ED8F0FE130F48481480A2C6 5A0001141F16005BA200035C153E5B157E0007EC7C1EEDFC1C4913F8A2000F153C913801 F0380180147802001370001F15E015F190C7EA7F80000EEC1F0027217A9F2C>I<14FE90 3807FF8090381F83C090383E01E09038F800F0484813F812034848137C485AA2485AA248 C7FC15FC5A127EA2140100FE14F85AA2EC03F0A2EC07E0007C14C0140FEC1F806CEB3F00 147E6C13F8380F83F03803FFC0C648C7FC1E21789F28>I<011E137E90397F81FF809039 E3C7C3C000019038CF01E0D9C3FE13F0ECFC0000034913F8EB83F05CEA0787010714FC5C 1200010FEB01F8A21480A2011F130316F01400A249EB07E0A2013EEB0FC0A2017EEB1F80 017F1400153E6E5A01FF5B9038FDC1F09038F8FFC0023EC7FC000190C8FCA25BA21203A2 5BA21207A2EA7FFFB5FCA2262E7F9F28>I<903803F00C90380FFC1C90383E0E3C90387C 077801F813F83801F003EA03E0D807C013F0000F13011380001F1303003F14E01300A248 1307007E14C0A300FE130F481480A3141F15005C127C5C14FEEA3C01EA1E03381F0F7E38 07FE7CEA01F0C7FC14FC5CA313015CA21303A290B51280A31E2E789F24>I<3903C00FC0 390FF03FF8391E78F03C383C79E039387FC07EEC80FE00781300D8707E13FC1538D8F0FC 130012E05B12001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1F 217A9F21>II<1307EB0F80 A2131FA21400A25BA2133EA2137EA2137CA2387FFFFCB5FC14F83800F8001201A25BA212 03A25BA21207A25BA2120FA25BA2001F13781470130014F014E01301381E03C01480381F 0700EA0F1EEA07FCEA01F0162F7AAD1A>IIII<011F137E90397FC1FF803A01E1F3C3C03903C0F7833A07807F 07E09038007E0F120E001E017C13C0001CEC038002FCC7FC003C5B1238C7FC13015CA313 035CA30107EB0780160014C000385CD87C0F130E00FC141E011F131C5D39F03FE0703978 79E1E0393FE0FFC0260F803FC7FC23217C9F24>II<903807800E90381FE01E90387FF01C9038FFF83CEC FC783901F07FF03903E00FE09038C001C0EB80039038000780C7EA0F00141E5C5C5CEB03 E0EB078049C7FC131E5B49133049137848481370485A484813F0390F0001E0381FF80390 38FE07C0393C1FFF80487E397007FE0038F003FC38E001F01F217C9F20>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmbx7 7.7 1 /Fe 1 114 df<90390FF803C090387FFE0748B5128F3907FE07CF390FF801FF48487E48 48137F127F49133FA2485AA86C7EA2157F6C7E6C6C13FF380FF8033807FC0F6CB512BFC6 EBFE3FEB1FF090C7FCA8913803FFFCA426287D9B2C>113 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmr5 5.5 3 /Ff 3 51 df<1470B1B712F8A3C70070C7FCB125257C9D2F>43 D<13E01201120F12FF12 F11201B3A4487EB512C0A2121E7A9D1F>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmsy5 5.5 1 /Fg 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmex7 7.7 2 /Fh 2 102 df<143014FCEB03FF010F13C0013F13F09038FF03FC3903FC00FFD80FE0EB 1FC0003FC7EA03F000FCEC00FC00E0151C260B7FAE29>98 D<013F14082601FFC0131C00 0701F01378489038FC01E0D81E00B512C00078013F138000E090380FFE000040EB03F026 087FAD29>101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi msam10 11 2 /Fi 2 87 df<007FB912E0BA12F0A300F0CBFCB3B3B2BAFCA36C18E03C3E7BBD47>3 D86 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsl10 11 56 /Fj 56 123 df<9339FFC003F8030F9038F01FFE923A3FC07C7E0F923BFE001FF81F80DA 03F890383FF07F4A48D9FFE013C0EC1FE04A4848EBC0FF03804A1380DA7F00157F4A9238 003E004A6D91C7FC8301015D4A5CA4160113034A5CA416030007B812FCA3290007F00003 F8C8FCA21607130F4A5CA4160F131F4A5CA4161F133F4A5CA4163F137F91C75BA4167F5B 4992C9FCA31201486C49487EB5D8F83F13FF5DA242407EBF35>11 DI<1518153815F0EC 01E0EC03C0EC0780EC0F00141E5C5C14F8495A495AA2495A495AA249C7FC5B133E137E13 7C13FC5B12015B1203A2485AA25B120FA3485AA348C8FCA45A127EA512FE5AAB5AA37EA4 127CA57EA3121E121FA27E7F1207A26C7EA26C7E6C7E13707F133C131C130C1D5A74C324 >40 D<497E806D7E1470147880141C141E80A2EC0780A215C0140315E0A2140115F0A315 F81400A515FCAEEC01F8A6140315F0A4140715E0A3EC0FC0A31580141FA215005CA2147E A2147C14FC5C13015C495AA2495A495AA249C7FC133E133C137C5B485A5B485A485A48C8 FC121C5A12F05A1E5A7FC324>I44 D<007FB5FCA2B512FEA418067C961E>I<121EEA3F80EA7FC012FFA41380EA7F00123C0A 0A788919>I48 D<157015F014011407143F903803FFE090B5FC14CFEBF80F1300141F15 C0A5143F1580A5147F1500A55C5CA513015CA513035CA513075CA5130F5CA3131FEB3FF8 B612FCA215F81E3D77BC2E>III<1638167816F8A21501ED03F01507A2150F151F153BED77E01567 15E7EC01C7EC0387EC07079138060FC0140E141C1438147014609138E01F80EB01C0EB03 80EB07001306130E49EB3F005B5B136013E0485A4848137E48C7FC1206120E5A5A485C12 60B712FEA3C73803FC00A25DA41407A25DA3140F4A7E010FB512F05BA2273E7ABD2E>I< 010C1403010F141F9138F001FE91B512FC4914F816F016E01680EDFE0015F8013C13C001 38C8FCA513781370A513F09038E03FE0ECFFF89038E7E07E9038EF001F01FCEB0F80D801 F014C049EB07E05B16F0C8120316F8A61507A21206EA3F80487EA2150F00FF15F0A24914 E090C7121F00FC15C000F0143F00701580ED7F0000785C00385C003C495A6CEB07F06C49 5A3907C07FC06CB5C7FCC613F8EB3FC0283F79BC2E>I55 DI<13F0EA01FC1203EA07 FEA313FCA2EA03F8EA01E0C7FCB3121EEA3F80EA7FC012FFA41380EA7F00123C0F2778A6 19>58 D<133C137F5B481380A31400A26C5A137890C7FCB3EA0780EA0FE0121F123FA512 1FEA0F601200A213E05BA212015B120390C7FC5A1206120E5A5A123012705A5A11397AA6 19>I<17E016011603831607A2160FA2161F83163FA2167F167716F7EEE7FCED01E316C3 150316831507EE03FEED0F01150E151E151C153C03387FED7800157015F05D4A4880177F 4A5AA24AC7FCA2020E81173F5C021FB6FC5CA20270C7EA3FE0171F5CA2495AA249488117 0F49C8FCA2130EA24982013C1507A2137CD801FE4B7E2607FF80EC3FFEB500F00107B512 FC19F85E3E417DC044>65 D<013FB7FC18E018F8903B007FE00007FE6E48EB01FF711380 4BEC7FC0A2F03FE0A219F0147F92C8FCA54A16E04A157F19C0A2F0FF804D130001014B5A 4A4A5A4D5A4D5AEF7FC0DC03FFC7FC49B612F8EFFF8002F8C7EA3FE0EF0FF0EF07FC717E 010715014A81711380A319C0130F5CA5011F4B13805C19005F601707013F4B5A4A4A5A4D 5A4D5A017F913801FF8001FF020F90C7FCB812FC17F094C8FC3C3E7CBD41>II<013FB7FC18F018FC903B007FF00007FF6E48 01007FF03FC04B6E7E727E727E727E1801027F824B6E7EA21A80197FA202FF17C092C9FC A54918E05CA41AC013034A16FFA41A8013074A5DA21A00A24E5A130F4A5E180761180F61 011F4C5A5C4E5A4E5A4EC7FC4D5A013F4B5A4A4A5AEF1FE0EF7FC0017F4A485A01FFDA0F FEC8FCB812F817C004FCC9FC433E7DBD46>I<013FB812E0A3903A007FF000016E48EB00 1F180F4B140718031801A3147F4B1400A514FF92C71270A3190017F0495D5C1601160316 07161F49B65AA39138FC003F160F160701075D4A1303A5010F4AC8FC5C93C9FCA4131F5C A5133F5CA3137FEBFFF0B612F8A33B3E7DBD3C>70 D<013FB5D8F807B6FC04F015FEA290 26007FF0C7380FFE006E486E5AA24B5DA4180F147F4B5DA4181F14FF92C85BA4183F5B4A 5EA491B8FC5B6102FCC8127FA318FF13074A93C7FCA45F130F4A5DA41703131F4A5DA417 07133F4A5DA3017F150F496C4A7EB6D8E01FB512FC6115C0483E7DBD44>72 D<011FB512FC5BA29039003FF8006E5AA25DA5143F5DA5147F5DA514FF92C7FCA55B5CA5 13035CA513075CA5130F5CA5131F5CA3133F497E007FB512F0A2B6FC263E7EBD21>I<01 3FB512FEA25E9026007FF8C8FCEC3FE0A25DA5147F5DA514FF92C9FCA55B5CA513035CA5 13075CA21838A21870130F5CA218E0A3011F15014A15C01703A21707EF0F80013F151F4A 143F177FEFFF00017F140701FF143FB9FC5FA2353E7DBD39>76 D<90263FFFF093380FFF FC5013F8629026007FF8EFF800023F4D5AA2023BEFEFE0A2DA39FCED01CFA2F1039F1479 02719338071FC0DA70FE150EA2191C1A3FDAF07F153802E0601970A219E06F6C157F0101 EE01C002C0DB038090C7FCA2F007006F7E060E5B130302804B5B6F7E60A2F07001010716 E04A6C6C5DEF01C0A2EF0380190349903903F80700010E60170EA25F6F6C481307131E01 1C4B5CA26F6C5AA27048130F133C01384B5C013C147F017C92C7FC01FE027E141F2607FF 804C7EB500FC017C013FB512F0163C4A01385E563E7CBD54>I<90263FFFE0023FB5FC6F 16FEA29026003FF8020313C0021F030013004A6C157C023B163C6F153814398102381678 02787FDA707F157082153F82031F15F002F07FDAE00F5D8215078203031401010180DAC0 015D82811780047F1303010315C04A013F5C17E0161F17F0040F1307010715F891C70007 91C7FC17FC160317FE04015B4915FF010E6E130E188E177F18CEEF3FDE011E16FE011C6F 5AA2170FA21707133C01386F5A133C017C150113FE2607FF801400B512FC18705C483E7D BD44>I<0007B912F0A33C0FFE000FF8003F01F0160F01C04A13034848160190C7FC121E F000E048141F5E1238A212781270153F5E5AA3C81600157F5EA515FF93C9FCA55C5DA514 035DA514075DA5140F5DA3141FEC7FFC0003B7FCA33C3D76BC42>84 D87 D89 D97 DIIIII<17FC91390FF003FE91397FFE0F1F903A01FC1F9C3F903903F007F890 390FC003E0011F14F0903A3F8001F80C020014005B01FE80A315031201495CA300004A5A 5E150F017E495A013E5C013F017EC7FC90387FC1FC9038E3FFF03901C07F8091C9FC485A A47FA27F90B512F815FF6C15C016F06C6C8048B6FC3A07C0000FFCD81F80130148C87E00 7E157E127C00FC153E5AA2167E167CA25E6C1401007C4A5A6C4A5A6CEC1F80D80FC0017F C7FC3903F803FCC6B512E0D90FFEC8FC303D7EA82E>I<147FEB3FFFA313017FA25CA513 015CA513035CA4ED07F80107EB1FFF9139F0781FC09138F1E00F9139F38007E0ECF70002 FE14F0495A5CA25CA24A130F131F4A14E0A4161F133F4A14C0A4163F137F91C71380A416 7F5B491500A300015D486C491380B5D8F87F13FCA32E3F7DBE33>I<1478EB01FE130314 FFA25B14FE130314FCEB00F01400ACEB03F8EA01FF14F0A2EA001F130FA314E0A5131F14 C0A5133F1480A5137F1400A55B5BA4EA03FF007F13F0A2B5FC183E7DBD1A>II<147FEB3FFFA313017FA25CA513015CA513035CA501070103B5FC02F014FEA26F13F0 6F1380EEFE00010F14F84A485AED03C04B5A031FC7FC153E011F1378ECC1F0ECC3E0ECC7 F0ECCFF814FF497F14F1ECE1FE14C04A7E4A7E4980017E133F82151F82150F01FE804913 0782A2000181486C49B4FCB5D8F03F13F04B13E0A2303F7EBE30>I<143FEB1FFF5BA213 017FA214FEA5130114FCA5130314F8A5130714F0A5130F14E0A5131F14C0A5133F1480A5 137F1400A55B5BA4EA03FF007F13F8A2B5FC183F7DBE1A>I<902707F007F8EB03FCD803 FFD91FFF90380FFF80913CE0781FC03C0FE09126E1E00FEBF0073E001FE38007E1C003F0 90260FE700EBE38002EEDAF70013F802FC14FE02D85C14F84A5CA24A5C011F020F14074A 4A14F0A5013F021F140F4A4A14E0A5017F023F141F91C74914C0A549027F143F4992C713 80A300014B147F486C496DEBFFC0B5D8F87FD9FC3F13FEA347287DA74C>I<903907F007 F8D803FFEB1FFF9139E0781FC09138E1E00F3B001FE38007E090380FE70002EE14F014FC 14D814F85CA24A130F131F4A14E0A4161F133F4A14C0A4163F137F91C71380A4167F5B49 1500A300015D486C491380B5D8F87F13FCA32E287DA733>II<91387F01FE903A7FFF0FFFC09139FE3E03F09238F801F8903A03FF E000FE6D49137F4B7F92C713804A15C04A141FA218E0A20103150F5C18F0A3171F010716 E05CA3173F18C0130F4A147F1880A2EFFF004C5A011F5D16034C5A6E495AEE1FC06E495A D93FDC017EC7FC91388F01F8913883FFE0028090C8FC92C9FC137FA291CAFCA45BA25BA3 1201487EB512F8A3343A81A733>I<91390FE003C0DAFFFC1380903903F81E0790390FE0 070F90391F80038F90397F0001DF13FE4848903800FF00485A1207485A8248485C123F49 5CA2485AA2150112FF90C75BA41503A25EA37E15077F003F4A5A151F6C6C133F6C6C137F 000714FF3903F003CF3A00FC0F8FE090383FFE0FEB0FF090C7FC151F5EA5153F5EA4157F 4B7E023F13FEA32A3A7AA730>I<903907F01F80D803FFEB7FE09138E1E1F09138E387F8 39001FE707EB0FE614EE02FC13F002D813E09138F801804AC7FCA25C131FA25CA4133F5C A5137F91C8FCA55B5BA31201487EB512FEA325287EA724>I<9138FF81C0010713E39038 1F807F90397C003F8049131F4848130F5B00031407A248481400A27FA27F6D90C7FCEBFF 8014FC6C13FF6C14C015F06C6C7F011F7F13079038007FFE1403140100381300157EA212 3C153E157E007C147CA2007E147815F8007F495A4A5A486C485A26F9E01FC7FC38E0FFFC 38C01FE0222A7DA824>II<01FE147F00FFEC7FFF4914FEA20007140300031401A34914FCA4150312074914F8A4 1507120F4914F0A4150F121F4914E0A2151FA3153F4914C0157F15FFEC01DF3A0FC003BF E09138073FFF3803F01E3801FFF826003FE01380282977A733>III<48B539C07FFFC0A33C000FFE003FF8006D48EB1FE0010315800101023EC7FC 6E133C01005C027F5B6F5A91383F81C0EDC380DA1FC7C8FC15EFEC0FFE6E5A5D140381A2 4A7E140FEC1E7F023C7FEC383F02707FECE01F010180903803C00F49486C7ED90F007F49 1303017E80D801FE80D807FF497EB5D8803F13F8A332277FA630>I<90B539E007FFF05E 18E0902707FE000313006D48EB01FC705A5F01014A5A5F16036E5C0100140794C7FC160E 805E805E1678ED8070023F13F05EED81C015C191381FC38015C793C8FC15EF15EEEC0FFC A25DA26E5AA25DA26E5A5DA24AC9FC5C140E141E141C5C121C003F5B5A485B495A130300 FE5B4848CAFCEA701EEA783CEA3FF0EA0FC0343A80A630>I<90B612FEA2EC80019039FC 0003FC4848EB07F849EB0FF049EB1FE049EB3FC0ED7F8000031500495B4A5A4A5A484848 5A4A5AC7485A4A5A5D147F4AC7FC495A495A49481370494813E0495A5C133F90387F8001 D9FF0013C0485A48481303485A4848EB078049130F4848131F003F143F397F8001FFB712 00A227277DA629>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmmi5 5.5 6 /Fk 6 107 df<90B612FCA290390F8000F81638131FA291C7121816385BED6030133E16 00017E13E0140190387FFFC0A2EBFC01A201F85BA2120191C8FC5BA21203A25BA21207B5 FCA2261F7A9E28>70 D77 D96 D<13F0121FA2EA01E0A21203A25BA21207A2EB87E0 EB9FF8380FB83C13E0EBC01CEB801EEA1F005C121EA2123E5C123C5C007C143015603878 01E015C012F8903800E38048EBFF000060133C1C217A9F26>104 D<1318133C137CA213381300A5EA03C0EA0FE0EA1CF0EA307013F01260A2EAC1E01201A2 EA03C0A2EA0780A21383EA0F06A2130C120EEA0F38EA07F0EA03C010207C9E18>I<1470 14F8A214F014E01400A5131FEB3F80EB61C0EBC1E0EA0181EA0301A2380603C01200A3EB 0780A4EB0F00A4131EA45B127012F85B485AEAE1E0EA7FC06CC7FC15297E9E1B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmex10 11 25 /Fl 25 122 df<140E141E143C147814F8EB01F0EB03E0EB07C0A2EB0F80131F1400133E 137E137C13FCA2485AA2485AA3485AA3485AA3121F5BA2123FA390C7FCA25AA7127E12FE B3A3127E127FA77EA27FA3121FA27F120FA36C7EA36C7EA36C7EA26C7EA2137C137E133E 7F1480130FEB07C0A2EB03E0EB01F0EB00F81478143C141E140E176D72832A>0 D<12E07E12787E123E7E6C7E6C7EA26C7E7F12016C7E7F137C137EA27FA2EB1F80A3EB0F C0A3EB07E0A314F01303A214F8A31301A214FCA7130014FEB3A314FC1301A714F8A21303 A314F0A2130714E0A3EB0FC0A3EB1F80A3EB3F00A2137EA2137C13FC5B485A12035B485A A2485A48C7FC123E123C5A5A5A176D7C832A>I16 D<12F07E127C7E7E6C7E6C7E6C7E7F6C7E6C7E137C137E7F6D7EA26D7E6D7EA26D7EA26D 7E6D7EA2147EA2147F8081141FA26E7EA2811407A281140381A21401A281A2140081A315 7E157FA4811680A5151F16C0A9ED0FE0B3A8ED1FC0A91680153FA516005DA4157E15FEA3 5D1401A25DA21403A25D14075DA2140F5DA24A5AA2143F92C7FC5C147EA25CA2495A495A A2495AA2495A495AA249C8FC137E137C13FC485A485A5B485A485A48C9FC123E5A5A5A23 A47D8337>I<17F01601EE03E0EE07C0EE0F80EE1F00163E5E16FC4B5A4B5A5E15074B5A 4B5A4BC7FCA2157E5DA24A5A4A5AA24A5A140F5D141F5D143F4AC8FCA214FEA2495AA249 5AA25C1307A2495AA2495AA3495AA349C9FCA313FEA312015BA21203A25BA21207A25BA2 120FA35BA2121FA45BA2123FA65B127FAA90CAFC5AB3AD7E7FAA123F7FA6121FA27FA412 0FA27FA31207A27FA21203A27FA21201A27F1200A3137FA36D7EA36D7EA36D7EA26D7EA2 130380A26D7EA26D7EA2147FA26E7E141F81140F8114076E7EA26E7E6E7EA2157E81A26F 7E6F7E6F7E1503826F7E6F7E167C8282EE0F80EE07C0EE03E0EE01F016002CDB6D8343> I<12F07E127C7E7E6C7E6C7E6C7E7F6C7E6C7E137C137E7F6D7E6D7EA26D7E6D7EA26D7E 6D7EA2147E147F8081141F816E7EA26E7EA26E7EA26E7EA2140081A2157FA26F7EA36F7E A36F7EA36F7EA3821503A282A21501A282A21500A282A382A21780A4163FA217C0A6161F 17E0AA160F17F0B3AD17E0161FAA17C0163FA61780A2167FA41700A25EA35EA21501A25E A21503A25EA215075EA34B5AA34B5AA34B5AA34BC7FCA215FEA25D1401A24A5AA24A5AA2 4A5AA24A5A5D143F92C8FC5C147E5CA2495A495AA2495A495AA2495A49C9FC137E137C13 FC485A485A5B485A485A48CAFC123E5A5A5A2CDB7D8343>III[<173E177E17FCEE01F8160317F0EE07E0EE0FC0161FEE3F801700 167E16FE4B5A5E15034B5A5E150F4B5AA24B5AA24BC7FC15FEA24A5AA24A5AA24A5AA24A 5AA2141F5D143F5D147F92C8FC5CA25C13015C1303A25C1307A2495AA3495AA3133F5CA3 495AA313FF91C9FCA35A5BA31203A25BA31207A25BA3120FA45BA2121FA55BA2123FA75B A2127FAD485AB3B3A56C7EAD123FA27FA7121FA27FA5120FA27FA41207A37FA21203A37F A21201A37F7EA380137FA36D7EA380131FA36D7EA36D7EA2130380A2130180130080A280 81143F81141F81140FA26E7EA26E7EA26E7EA26E7EA2157F6F7EA26F7EA26F7E1507826F 7E1501826F7E167E821780EE1FC0160FEE07E0EE03F017F81601EE00FC177E173E>47 273 107 131 72 32 D[<12F87E127E7E7F121F6C7E6C7E7F6C7E12016C7E7F137F7F80 6D7E130F806D7EA26D7EA26D7E6D7EA2147FA26E7EA26E7EA26E7EA28114078114038114 0181A21400818182A2153F82A26F7EA36F7EA3821507A36F7EA3821501A38281A31780A2 167FA317C0A2163FA317E0A4161FA217F0A5160FA217F8A71607A217FCADEE03FEB3B3A5 EE07FCAD17F8A2160FA717F0A2161FA517E0A2163FA417C0A3167FA21780A316FFA21700 A35D5EA315035EA34B5AA3150F5EA34B5AA34B5AA25E157FA293C7FC5D5D1401A25D1403 5D14075D140F5DA24A5AA24A5AA24AC8FCA214FEA2495A495AA2495AA2495A5C131F495A 91C9FC5B13FE5B485A1203485A5B485A485A123F90CAFC127E5A5A>47 273 125 131 72 I48 D<12F87E127EA27E6C7E6C 7EA26C7E6C7E7F12017F6C7E137E137F6D7EA26D7E6D7EA26D7EA26D7EA26D7E80130080 8081143F81141F81A26E7EA26E7EA36E7EA26E7EA28180A282157FA282153F82A2151F82 A2150F82A21507A282A2150382A36F7EA46F7FA4707EA5707EA583161FA483A2160FA483 A31607A483A51603A383A882A21880AF82A218C0B3A732A57D8250>I<12F0B3B3B3A304 3964803D>63 DIIII80 D88 DI<1560EC01F84A7EEC0FFF023F13C091B512F049 EB0FF8903907FC03FE903A1FF000FF80D97FC0EB3FE04848C7EA0FF8D803FCEC03FCD80F F0EC00FFD83F80ED1FC000FEC9EA07F00078EE01E00060EE0060341181C333>98 D101 D104 DI<143CB3B100C0150300F0 150F00FC153FB415FFD83FC0EB03FCD80FE0EB07F0D803F8EB1FC0D800FCEB3F00017E13 7E011F13F890380FBDF06DB45A6D5B6D5B6D90C7FC147E143CA214182836767F3D>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm eufm10 11 5 /Fm 5 118 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fn msbm7 7.7 2 /Fn 2 125 df<001FB612F8A23A187E00E03001E0EBC070261BC0011360D81F00EB80E0 003E90380381C0003C9038070180EC06030038010E1300EC0C070030EB1C0EEC380CEC30 1CC7EA7038ECE030ECC07001011360EC80E090380381C09038070180EB0603010E90C7FC EB0C07EB1C0EEB380CEB301CD970181306EB6038EBE0702601C060130EEB80E0260381C0 130C26070180131CEA0603000E90C7123CD80C07147CD81C0E14ECD8380CEB01CCD8301C EB039CD87018EB0F18D86038133CB712F8A2272C7EAB3A>90 D124 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsy7 7.7 11 /Fo 11 89 df<007FB71280B812C0A26C16802A04799139>0 D<123C127E12FFA4127E12 3C0808799316>I<0060EC018000F0EC03C06C14076C140F007EEC1F806CEC3F006C6C13 7E6C6C5B6C6C485A6C6C485A6C6C485A6C6C485A90387E1F806D48C7FCEB1FFE6D5A6D5A 6D5AA2497E497E497EEB3F3F90387E1F80496C7E48486C7E48486C7E48486C7E48486C7E 4848137E48C77E007EEC1F8048EC0FC04814074814030060EC0180222475A139>I<1318 133CA500E0130700F0130F38FC183F00FE137F383F99FC380FDBF03803FFC038007E00A2 3803FFC0380FDBF0383F99FC38FE187F00FC133F38F03C0F00E0130700001300A5131818 1C7A9D25>I10 D20 D<126012F812FEEA7F80EA3FE0EA0FF8EA 03FEC66C7EEB3FE0EB0FF8EB03FCEB00FFEC3FC0EC0FF0EC07FCEC01FF9138007FC0ED1F F0ED07FCED01FF9238007F80EE1FC0A2EE7F80923801FF00ED07FCED1FF0ED7FC04A48C7 FCEC07FCEC0FF0EC3FC002FFC8FCEB03FCEB0FF8EB3FE0EBFF80D803FEC9FCEA0FF8EA3F E0EA7F8000FECAFC12F81260CBFCAC007FB71280B812C0A26C16802A3C79AB39>I<01FE 15082603FF80141C4813E04813F8487F4801FF143C01017F277C007FE013780078D91FF8 13F848903907FE03F06EB5FC48010014E06F13C0031F1380030713000040EC01FCCBFCAE 007FB712F8B812FCA26C16F82E227B9F39>39 D<133813FEA31201A213FCA2EA03F8A313 F01207A213E0120F13C0A3EA1F80A21300A25A123EA2127E127CA2127812F8A25A0F217D A315>48 D<91B51280010714C0131F491480D9FF80C7FCD801FCC8FCEA03F0485A485A5B 48C9FC5A123E5AA21278A212F8A25AB7128016C0A2168000F0C9FC7EA21278A2127CA27E 123F7E6C7E7F6C7E6C7EEA01FC6CB47E013FB512806D14C0130701001480222C79A531> 50 D88 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi7 7.7 50 /Fp 50 123 df11 D<133FD9FFC01330000301F01370486D13604815E0261F80 7C13C0393E003C010038011E138048EB0E030207130000605C00E0EB03060040140EC713 8CEC019C159815B815B0A215F05DA35DA25DA21403A44AC7FCA35CA2140EA35CA3242A7E 9C26>13 DI<1460A214E0A21460EC7FE015F0EB01FF903803DFE049C7 FC131E5B5B5BA2485A485A485AA248C8FCA2121EA25AA3127C1278A312F8A6127C127EA2 6C7E13E0EA1FFC6CB47E6C13F000017F38003FFCEB07FE1300143E141EA4EB601CEB783C EB1FF0EB07E01C397CAB21>16 D<39078007E0391FE03FF83938F0783E3830F9E039707B 801F38607F0013FE12E0485A5B00C1143F120149133EA20003147EA249137CA2000714FC A24913F8A2000F1301A2018013F0A2001F1303A2010013E0120EC71207A215C0A2140FA2 1580A2141FA21500140E202A7D9C26>I<147C49B4FC903807C78090380F03C0131E013E 13E0137CEBF801A2D801F013F0120313E01207EBC003120FA2EA1F80A2EC07E0123F1300 90B5FCA25A15C0387E000F141F15805AA2EC3F00A2143E48137E147C14FC5C387801F0A2 387C03E0495A003C5B49C7FCEA1E3EEA0FF8EA03E01C2E7CAC23>I<131C013EEB038001 7E1307150F137CA201FC131F16005BA200015C153E5BA20003147E157C5BA2000714FCED F8185BA2000F010113381630EC03F0D9E0071370001F90380EF8609039F03C78E090397F F03FC090391FC00F0048C9FCA2123EA2127EA2127CA212FCA25A1270252A7D9C2C>22 D<0107B512E0011F14F0137F90B612E03A03F81F80003907E00FC0380FC007138048486C 7E5A123E127E127C140700FC5C5AA2140F5DA24AC7FC143EA200785B007C5B383C01E038 1F07C0D807FFC8FCEA01F8241D7C9B2A>27 D<90B6FC00031580120F481500261E00E0C7 FC00385BEA700112E012C0A2C6485AA31307A349C8FCA35BA2131E133EA45BA21338211D 7D9B22>I<156015E0A25DA21401A25DA21403A292C7FCA25CA2EC7FC0903803FFF89038 0FCE3E90393E0E0F809039F80C07C0D801F0EB03E03903E01C01D807C014F0390F801800 D81F0014F8481338123E007E1330127CEC700112FC48016013F0150302E013E0150702C0 13C00078EC0F80D87C01EB1F00003C143E003E495A001F5C390F8383E03903E39F802600 FFFEC7FCEB1FF00107C8FCA21306A2130EA2130CA2131CA21318A225397CAB2C>30 DI<150615 0E150CA3151C1518A315381530A315701560D803E01407D807F8EC1F80D81E3C13E00018 14C0EA383E0070150F006001011307D8E07E1380D8C07C140313FCEBF803000002001300 120113F04A5B00030106130601E0140E160C020E131C0007010C131801C0143800035DD9 E01C5B91381801C001F0495A00014AC7FC3900F8381E90383F30F890380FFFE0010190C8 FCEB00701460A314E05CA313015CA329397DAB30>I<123C127E12FFA4127E123C080879 8716>58 D<123C127EB4FC1380A3127F123D1201A212031300A25A1206120E5A5A5A1260 0914798716>III<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB 03FCEB00FFEC3FC0EC0FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007F80 EE1FC0A2EE7F80923801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC0FF0EC3FC002FF C8FCEB03FCEB0FF8EB3FE0EBFF80D803FEC9FCEA0FF8EA3FE0EA7F8000FECAFC12F81260 2A2C79A539>I<013FB512FEEEFFC017F0903A00FC0007F8EE01FCEE007E0101814AEC1F 80A2EF0FC013034A140718E0A213075CA3010F150F5CA3011F16C04A141FA21880013F15 3F91C8FC1800177E5B017E5D4C5AA201FE4A5A494A5A4C5AEE3F800001037EC7FC49495A 0003EC1FF0B712C093C8FC15F8332C7BAB3C>68 D<013FB7FCA3D900FCC7FC171E170E13 015C1706A213035CA2EEE00E0107150C4AEBC000A21501010F13034A485A91B5FCA25BDA 800FC7FC81A2013F1518EC0006173817304990C71270017E156017E0A201FEEC01C05BEE 038016070001ED0F0049143F00034AB4FCB75AA25E302C7BAB36>I<013FB7FCA3D900FC C7FC171E170E13015C1706A213035CA2EE700E0107150C4AEB6000A216E0010F13014A48 5A91B5FCA25B91388007801503A2133F020090C7FCA34990C8FC137EA313FE5BA312015B 1203B512F8A25C302C7CAB2E>I<903B3FFFF80FFFFEA3D900FEC7EA3F804A15005F1301 4A147EA217FE13034A5CA2160113074A5CA21603130F4A5C91B6FCA25B9139800007E0A2 160F133F91C75BA2161F5B017E5DA2163F13FE4992C7FCA25E120149147E000315FEB539 E03FFFF8A3372C7BAB3C>72 D<90383FFFF8A3903800FE005CA213015CA313035CA31307 5CA3130F5CA3131F5CA3133F91C7FCA35B137EA313FE5BA312015B1203B512E0A31D2C7C AB20>I77 D I<013FB512FCEEFF8017E0903A00FC000FF0EE03F80101EC01FC16004A14FEA21303A25C A21307EE01FC5CEE03F8010F15F0EE07E04AEB0FC0EE3F80011F903801FE0091B512F816 E00280C8FC133FA291C9FCA25BA2137EA213FEA25BA21201A25B1203B512E0A32F2C7BAB 2E>80 D<013FB512F016FE707E903A00FC001FE0EE07F0707E010114014A80A313035CA2 5F010714034A5C4C5A4C5A010F4A5A4A017EC7FCED03F891B512E04991C8FC9138800FC0 ED03E06F7E013F801400150015015B137EA2150313FE5BA2EF0380000117005B00035EB5 39E001FC0E173C9238007FF8C9EA0FC0312D7BAB37>82 D<140FEC1FC0147814F0A2EB01 E0EB03C0A2EB0780130F1401011F1380A2EB3E0315005CEB7C06140E140CEBFC1C495A14 30000113705CEBF1C0EBF38001F7C7FC13F6EA03EE13FC5B5B5BA21207120F121F123912 7100E114C0000113013900F00780EC0E00EB787CEB3FF0EB0F801A2E7FAC1F>96 DI<13F8123FA25B120112 03A25BA21207A25BA2120FA2EB87E0EB9FF8381FF83CEBE03E497E138048487E1580123E A2007E131FA2127CA2143F12FC481400A2147EA35C5C495AEA7803495A383C0F806C48C7 FCEA0FFCEA03F0192D7CAB20>II<151FEC07FFA215FE EC003E157EA2157CA215FCA215F8A21401A2903807E1F0EB1FF1EB7C3FEBF80FD803F013 E03807E00713C0EA0F80001F14C0EA3F00140FA2007E1480A2141F12FE481400A25C1506 48133EA2EC7E0E0078140C007C13FC0101131C393E03BE18391E0F1E38390FFC0FF03903 F003C0202D7CAB26>II<15F8EC03FEEC078791380F0780EC1F1F141E143E1600150E92C7FC5CA514FC5C90 387FFFF8A3D901F8C7FC5CA513035CA413075CA5130F5CA4131F91C8FCA4133EA21238EA 7C3C12FC5BA2EAF0F0EA71E0EA3FC06CC9FC213A7BAC24>I<14FCEB03FE90380F879890 383F01FC137CEBFC00485A5B000314F8485A1401120F01C013F0A21403121F018013E0A2 1407A215C0A2000F130F141F0007EB3F80EBC07F3803E1FF3800FF9F90383E1F0013005C A2143EA20038137E00FC137C14FC495A495A38F00FC0D87FFFC7FCEA1FF81E2A7D9C24> I<131FEA07FFA25BEA003E137EA2137CA313FC5BA31201EBF03F9038F1FFC09038F3C1F0 3803FF019038FC00F85B5B12075B5B1401000F5C1380A21403001F5C130014075D4815C0 003E130F15801581007E90381F0180007C1403ED0700EC0F0600FC141E48EB07F80070EB 01E0222D7CAB2B>I<1307EB0F80131FA3EB0E0090C7FCA8EA03E0EA07F8EA1E3C1218EA 383E12701260EAE07EEAC07C13FC5B120012015B1203A25B12075B14C0120F1380EA1F81 EB01801303EB0700EA0F06131EEA07F8EA01F0122C7DAA1A>I<15E0EC03F0A315E0EC01 C091C7FCA814F8EB03FEEB070F010E1380EB1C071338EB700F136013E013C0141F130015 00A25CA2143EA2147EA2147CA214FCA25CA21301A25CA21303A25CA2EA380700FC5B495A 49C7FC133EEAF07CEA7FF0EA1F801C3880AA1E>I<131FEA07FFA25BEA003E137EA2137C A313FC5BA312014913F0EC03FCEC0F0E0003EB1C1E9038E0387E147014E03907E1C07C90 38C3803801C7C7FC13DEEA0FF85B13FEEBFFC0381F87E0EB03F06D7E130048140C123EA2 151C007E1418007CEBF038ECF830157000FCEB78E048EB3FC00070EB0F001F2D7CAB27> I<27078007F0137E3C1FE01FFC03FF803C38F0783E0783E03B30F8E01F1E033C7079C00F B801F026607B8013F0D9FF0013E0EAE0FE484814C01680D8C1F8011F130300015F491400 A200034A13076049133E170F0007027E5CF0818049017C131F1801000F02FCEB3F03053E 130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB03E0391E7D9C41> 109 D<39078007E0391FE03FF83938F0783E3830F9E039707B801F38607F0013FE12E048 5A5B00C15C0001143E5BA20003147E157C5B15FC00075C1618EBC00115F0000F01031338 EDE0300180147016E0001F010113C015E390C7EAFF00000E143E251E7D9C2D>II<90387801F83901FE07FE39038F1E0FD90FB8 13803A0707F007C0000613E090380FC003000E15E0000C1380A2011F130712001400A249 130FA2013E14C0A2017EEB1F80A2137CED3F0001FC133E5D15FC6D485A0001EB03E09038 FF87C0D9F1FFC7FCEBF0FC000390C8FCA25BA21207A25BA2EA7FFE12FF5B2329819C26> I<903807E03090381FF07090387C3CF0EBF80D3903F00FE03807E00713C0EA0F80001F14 C0EA3F00A2140F007E1480A300FE131F481400A35C48133EA2147E1278007C5B1301EA3E 03EA1E0F380FFCF8EA03F0C7FC13015CA313035C130790B5FCA25C1C297C9C21>I<3907 801F80391FE07FE03938F0E0703930FBC07839707F01F8EA607E13FED8E0FC13F039C0F8 00E0150012C112015BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1D1E7D 9C22>II<13075B5BA35BA2133EA213 7EA2137C387FFFFCA2B5FC3800F800A21201A25BA21203A25BA21207A25BA2120F141813 80143814301470EB00E0EB01C0EB838038078700EA03FEEA00F816297EA71C>III<013F137C9038FFC1FF3A03C1E383803A0701F703C039 0E00FE0F000C13FC121C48158000309038F8070092C7FC130100005BA313035CA31307EC C003A21238D87C0F5B00FC140E150C011F131C39F03BE038D8707113F0393FE0FFC0260F 803FC7FC221E7D9C29>120 D<011F131890387F80389038FFC03048EBE070ECF0E03903 C0FFC09038801F8039070007000006130EC75A5C5C5C495AEB078049C7FC131C5B491320 491360485A484813E039070001C0380FE003391FFC0F80383C7FFFD8701F130038600FFE 38E007F838C003E01D1E7C9C23>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmsy8 8.8 4 /Fq 4 84 df<130FA28091C7FCA30020144000F8EB01F000FC130300FE130738FF8F1F39 3FC63FC0390FF6FF003801FFF838007FE0EB1F80EB7FE03801FFF8380FF6FF393FC63FC0 39FF8F1FF038FE0F0700FC130300F813010020EB004000001400A38091C7FCA21C1F7BA1 27>3 D65 D<030E1820033F18601CE04B17011B031B0782F30FC01B1F03FF173F 70167F1BFF501380A2912601EFE05D6203CFEE0FBFF21F3F912603C7F0033F13001A7E03 87167C505A91260783F8EC01F0F103E00303ED07C070020F137EDA0F01ED1F80020EEE3F 00077E13FE91261E00FE147C021C5E4E5A4A6D495A70495A0278030F5C02706E485A043F 49C7FC4ADAC07E130160010191381FE1F84AECE3F093380FF7E04948ECFFC0705B010793 C7FC49C76C5A00385ED87E1E6E5AD87FFE4B16184991C9EBFE784848F1FFF01CE0491AC0 6C487213006C48197C000FCFFC55387EB35C>77 D83 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr7 7.7 19 /Fr 19 127 df0 D<010FB512C0A39026001FE0C7FC6E5AA500FEED01 FCD8FF801407D81FC0EC0FE0000F16C0A201E0141F00071680A7D803F0EC3F00A3D801F8 147EA2D800FC5C017C5C017E13C190391F8FC7E090390FEFDFC0902601FFFEC7FC903800 3FF0EC0FC0A64A7E010FB512C0A32E2C7BAB38>9 D<13031307130E131C1338137013E0 1201EA03C0A2EA0780A2EA0F00A2121EA2123E123CA35AA512F8A25AAA7EA21278A57EA3 123E121EA27EA2EA0780A2EA03C0A2EA01E0120013701338131C130E13071303104078AF 1D>40 D<12C07E12707E7E7E7E1380EA03C0A2EA01E0A2EA00F0A21378A2137C133CA313 1EA5131FA2130FAA131FA2131EA5133CA3137C1378A213F0A2EA01E0A2EA03C0A2EA0780 1300120E5A5A5A5A5A10407BAF1D>I<14034A7EB3A3007FB712F8B812FCA26C16F8C7D8 0780C7FCB3A36EC8FC2E307BA738>43 D48 D<131C133C13FC120312FF137C12FC1200B3AD13FEB512FEA3172A79A924>III<140E141EA2143E147E14FEA21301EB03BE143E 1306130E131C13181330137013E013C0EA01801203EA07001206120E5A12185A12705AB6 12FCA3C7EA3E00A8147F90381FFFFCA31E2B7DAA24>I<000CEB0180380FC01F90B51200 5C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EBFFF8380FE07EEB001F000EEB0F80000C14 C0C7120715E0140315F0A31230127C12FCA3EC07E012F0006014C00070130F6CEB1F8000 1EEB3F00380F80FE3807FFF800015B38007F801C2B7CA924>II<1230123C003FB512F8A34814F015E00070C7FC0060EB01C0EC03 8048EB07001406140EC75A5C143014705C495AA2495A130791C7FC5BA2131EA3133EA213 3C137CA413FCA813781D2C7BAA24>III<007FB712F8B812FC A26C16F8CBFCAC007FB712F8B812FCA26C16F82E147B9938>61 D<15F8141FA314011400 AAEB0FE0EB7FF83801F81E3803E007380FC003381F80011300481300127EA2127C12FCA8 127C127EA2003E13017EEB8003000F497E3A07C00EFFC03801F03C38007FF090391FC0F8 00222D7DAB28>100 D105 D<3807C008381FF01C383FFFF84813F038E03F E038400F80160679AA24>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmmi10 11 72 /Fs 72 125 df11 DIII<15E014015DA5EDDFF8913800FFFC913801F01C913807FFFC91380E1FF0023C C7FC5C5C495A495A495A49C8FC131E5B133813785B485AA2485AA2485A120F90C9FC5A12 1EA25AA3127C1278A312F8A25AA47EA47EA27E127E127F7FEA3FE0EA1FF813FF6C13E000 0313FC6CEBFF806C6C13E0011F7F01037FEB007FEC0FFC14011400157CA3153CA2157C15 78010C5B130E90380783E0903801FF80D9007EC7FC26527CBE28>16 D<01F8EB0FF0D803FEEB3FFC3A078F80F03E3A0F0F83C01F3B0E07C7800F80001CEBCF00 02FE14C0003C5B00385B495A127800705BA200F049131F011F158000005BA2163F133F91 C71300A25E5B017E147EA216FE13FE495CA215011201495CA215031203495CA215071207 495CEA01C0C8120FA25EA2151FA25EA2153FA293C7FCA25DA2157EA3157C15382A3C7EA7 2D>I<15FCEC03FF91380F87C091383E03E0EC7C0102F813F01301903903F000F8495A01 0F14FC5C495A133F91C7FC4914FE13FEA212015B12034913011207A25B000F15FC150312 1F5BA21507003F15F890B6FCA33A7FC0000FF05BA2151F16E048C7FCA2ED3FC0A2481580 157F1600A215FEA24A5AA24A5A007E5C14075D4A5A003E5C141F4AC7FC6C137E5C380F81 F03807C3E03801FF80D8007EC8FC27417DBF2B>I<1338137C13FCA312015BA312035BA3 12075BA3120F5BA2121F5BA2123F90C7FCA248130E127EA200FE131E48131C143C147848 137014F0EB01E0EB03C0EB0F00EA7C3EEA3FF8EA0FC017297BA720>I22 D24 D<011FB612FE017F15FF48B8FC5A4816FE3B0FC03801C000EA1F 00003E1403003C01785B4813705AECF0075AC712E0010191C7FCA25DEB03C0A313071480 A2010F5BA2EB1F0082A2133EA2137E825B150F0001815B120315075BC648EB038030287D A634>I<020FB512FE027F14FF49B7FC1307011F15FE903A3FE03FE00090387F000F01FE 6D7E4848130348488048481301485A5B121F5B123F90C7FC5A127EA2150300FE5D5AA24B 5AA2150F5E4B5AA2007C4AC7FC157E157C6C5C001E495A001FEB07E0390F800F802603E0 7EC8FC3800FFF8EB3FC030287DA634>27 D<011FB612C0017F15E048B7FC5A4816C0260F C007C8FC48C65A123E003C130E48131E5AA25AC75AA3147CA2147814F8A4495AA31303A2 5CA21307A3495AA3131FA25C6DC9FC2B287DA628>I<16F0A25EA21501A25EA21503A25E A21507A293C7FCA25DA2150EA2151EA2151C4AB47E020F13F091387F3CFC903901F8381F D907E0EB0F80903A0F807807C0D93F00EB03E0017E90387001F04915F84848EBF0004848 15FC48485B4848157C1401EA1F805DEA3F00020314FC5A007E5CA20207130100FE16F848 140016034A14F01607020E14E0007CED0FC0141E007EED1F80003E011CEB3F00167E6C01 3C5B0180495A000F90383803E0D807E0EB0FC02701F0783FC7FC3900FC79FC90381FFFE0 D903FEC8FCEB00F0A25CA21301A25CA21303A25CA21307A291C9FCA25BA22E527BBE36> 30 D<13FE2603FF80157026078FE015F0260F07F01401000E6D15E00103ED03C0000C6D EC0780D80001ED0F006E141E01005D5F027F5C4C5A91383F80035F4C5A6E6C48C7FC161E 5E6E6C5A5EEDE1E0913807E3C015F75E6EB4C8FC5D5D5D6E7EA2140314074A7EA2141EEC 3C7F147814F049486C7EEB03C0EB078049486C7EA2131E496D7E5B498048481307485A48 486D7E48C7FC48EDFC03001E0201EB07804803FE1300486E6C5A48ED7F1E0060ED1FFCC9 EA03F0343B7EA739>I I<0120ED01C00178ED07F001F8150F000117F85B485A5B0007160749150348C9EA01F0A2 121E1700121C003C023814E0003814FCA217010078010115C000705CA217034B148000F0 130317074B14005F5D0207141E6F133E6C010F5C4A7E6C013F5C007E9038FFF8033B7F87 FDFF0FF0D9FFF8EBFFE06C495C6C496C5B4A6C90C7FC00079038001FFCD801F8EB03F035 297FA739>I39 D<13F8EA03FC120FEA1FF8EA3F80EA 7E00127C5AA25AA47EA2127C127EEA3F80EA1FF8EA0FFC1203EA00F80E167BAA19>44 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919>58 D<121EEA7F8012FF13C0A2 13E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C 798919>I<183818FC1703EF0FF8EF3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03 FEC7FCED0FF8ED3FE0EDFF80DA03FEC8FCEC0FF8EC3FE0ECFF80D903FEC9FCEB0FF8EB3F E0EBFF80D803FECAFCEA0FF8EA3FE0EA7F8000FECBFCA2EA7F80EA3FE0EA0FF8EA03FEC6 6C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8 ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FC1700183836 3678B147>I<16C0ED01E01503A216C01507A21680150FA216005DA2151E153EA25DA215 7815F8A25D1401A25D1403A25D1407A24A5AA292C7FC5CA2141E143EA2143C147CA21478 14F8A2495AA25C1303A25C1307A25C130FA291C8FC5BA2131E133EA25BA2137813F8A25B 1201A25B1203A25B1207A2485AA290C9FC5AA2121E123EA2123C127CA2127812F8A25A12 60235B7BC32E>I<126012F8B4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FC EB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0 EE1FF0EE07FCEE01FF9338007FC0EF1FF0EF07F8EF01FCA2EF07F8EF1FF0EF7FC0933801 FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7F C04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC12703636 78B147>I<17075F84171FA2173F177FA217FFA25E5EA24C6C7EA2EE0E3F161E161C1638 A21670A216E0ED01C084ED0380171FED07005D150E5DA25D157815705D844A5A170F4A5A 4AC7FC92B6FC5CA2021CC7120F143C14384A81A24A140713015C495AA249C8FC5B130E13 1E4982137E13FED807FFED1FFEB500F00107B512FCA219F83E417DC044>65 D<49B712F818FF19C090260001FEC7EA3FE0F00FF84B1407F003FCF001FE14034B15FFA2 8414074B5CA3020F16FE4B140319FCA2021FED07F84BEC0FF0181FF03FE0023FED7F804B ECFF00EF03FEEF0FF8027FEC7FE092B6C7FC18E092C7EA07F84AEC01FE4A6E7EF07F80F0 3FC013014A16E0181FA213034A16F0A3010717E04A153FA219C0010F167F4A168018FF4D 1300011F5E4A14034D5A013FED1FF04D5A4AECFFC0017F020790C7FCB812FC17F094C8FC 403E7CBD45>II<49B712FCF0FF 8019E0D9000190C7EA3FF8F007FC4BEC01FE727EF17F800203EE3FC04B151FF10FE0A202 07EE07F05DA21AF8140F4B1503A21AFC141F5DA21907143F5DA21AF8147F4B150FA302FF 17F092C9121FA21AE049173F5C1AC0197F010318805CF1FF0061010716014A4B5A611807 010F4C5A4A5E4E5A4E5A011F4CC7FC4AEC01FEEF03F8013FED0FF0EF3FE04AECFF80017F DA07FEC8FCB812F817C004FCC9FC463E7DBD4C>I<49B912C0A3D9000190C71201F0003F 4B151F190F1A80020316075DA314075D1A00A2140F4B1307A24D5B021F020E130E4B92C7 FC171EA2023F5C5D177CEE01FC4AB55AA3ED800302FF6D5A92C7FCA3495D5C1938040114 7801034B13704A16F093C85AA2010716014A5E180361010F16074A4BC7FCA260011F163E 4A157E60013F15014D5A4A140F017F15FFB95AA260423E7DBD43>I<49B9FCA3D9000190 C7120718004B157F193F191F14035D190EA214075DA3140F5D17074D131E021F020E131C 4B1500A2171E023F141C4B133C177C17FC027FEB03F892B5FCA39139FF8003F0ED000116 00A2495D5CA2160101035D5CA293C9FC13075CA3130F5CA3131F5CA2133FA25C497EB612 F8A3403E7DBD3B>II<49B6D8C03FB512F81BF01780D900010180C7383FF00093C85B4B5EA2197F14 034B5EA219FF14074B93C7FCA260140F4B5DA21803141F4B5DA21807143F4B5DA2180F4A B7FC61A20380C7121F14FF92C85BA2183F5B4A5EA2187F13034A5EA218FF13074A93C8FC A25F130F4A5DA21703131F4A5DA2013F1507A24A5D496C4A7EB6D8E01FB512FCA2614D3E 7DBD4C>I<49B612C05BA2D90001EB800093C7FC5DA314035DA314075DA3140F5DA3141F 5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F5CA213 3FA25CEBFFE0B612E0A32A3E7DBD28>I<92B612E0A39239003FF000161F5FA2163F5FA3 167F5FA316FF94C7FCA35D5EA315035EA315075EA3150F5EA3151FA25EA2153FA25EA215 7FA25EA2D80F8013FFEA3FC0486C91C8FCA25CD8FFC05B140301805B49485A00FC5C0070 495A0078495A0038495A001E017EC9FC380F81FC3803FFE0C690CAFC33407ABD33>I<49 B612F0A3D900010180C7FC93C8FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3 147F5DA314FF92C9FCA35B5C180C181E0103161C5C183C183813074A1578187018F0130F 4AEC01E0A21703011FED07C04A140F171F013FED3F8017FF4A1303017F021F1300B9FCA2 5F373E7DBD3E>76 D<49B56C93387FFFE098B5FC5014C0D90001F1C000704B5B03DF95C7 FC507EF20EFF1403039F4C5A1A381A399126078FE01571030F5F1AE1F101C3140F020E93 380383F8F107036F6C1507021E160E021C041C5BA2F1380F143C023804705B19E06F6C15 1F0278ED01C00270DB03805BA2953807003F14F002E0030E5C606F6C157F01015E02C04B 91C8FCA24E5B130302804A485B4D5ADB00FE14010107ED070091C7010E5CA24D1303495D 010E60EE7F701907011E5D011C4B5CA2013C4B130F017C92C7FC01FE60D803FF023E4A7E B500FC033FB512F0163C16385B3E7CBD59>I<49B56C49B512F81BF0A290C76D9039000F FE004AEE03F0705D735A03DF150302037F038F5E82190791380787FC030793C7FC150370 5C140F91260E01FF140EA26F151E021E80021C017F141C83193C023C6D7E02381638161F 711378147802706D6C1370A2040714F002F0804A01035C8318010101EC01FF4A5E821883 13034A91387FC380A2EF3FC7010716E791C8001F90C8FC18F718FF4981010E5E1707A213 1E011C6F5AA2013C1501137C01FE6F5AEA03FFB512FC187818704D3E7DBD49>I<49B712 F018FF19C0D9000190C76C7EF00FF84BEC03FC1801020382727E5DA214071A805DA2140F 4E13005DA2021F5E18034B5D1807023F5E4E5A4B4A5A4E5A027F4B5A06FEC7FC4BEB03FC EF3FF091B712C005FCC8FC92CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2131F A25CA2133FA25C497EB612E0A3413E7DBD3B>80 D<49B77E18F818FFD90001D900017F94 38003FE04BEC0FF0727E727E14034B6E7EA30207825DA3020F4B5A5DA24E5A141F4B4A5A 614E5A023F4B5A4B4A5A06FEC7FCEF03FC027FEC0FF04BEBFF8092B500FCC8FC5F9139FF 8001FE92C7EA7F80EF1FC084496F7E4A1407A28413035CA2170F13075C60171F130F5CA3 011F033F5B4AEE038018E0013F17071A004A021F5B496C160EB600E090380FF01E05075B 716C5ACBEAFFE0F03F8041407DBD45>82 DI<48B912FC A25A913A0003FE000F01F84A1301D807E0EE00F8491307491778000F5D90C7FC001E140F A2001C4B1470123C0038141FA200785D1270033F15F000F018E0485DC81600157FA25EA2 15FFA293C9FCA25CA25DA21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2 147FA214FF497F001FB612FCA25E3E3D7FBC35>I86 DI<027FB5D88007B512C091B6FCA202 0101F8C7EBF8009126007FE0EC7F804C92C7FC033F157C701478616F6C495A4E5A6F6C49 5A4EC8FC180E6F6C5B606F6C5B6017016F6C485A4D5A6F018FC9FC179E17BCEE7FF85F70 5AA3707EA283163F167FEEF7FCED01E7EEC3FEED0383ED070392380E01FF151E4B6C7F5D 5D4A486D7E4A5A4A486D7E92C7FC140E4A6E7E5C4A6E7E14F0495A49486E7E1307D91F80 6E7ED97FC014072603FFE0EC1FFF007F01FC49B512FEB55CA24A3E7EBD4C>I<151EED7F 80913801F1C0EC03C1EC07C0ED80E0EC0F005C141E91383E01C0147CA214F81503D901F0 1380A21303ECE007010714005D90380FC00EA2151E90381F801C153C5D133F4A5A5D1401 49485A017E5B14074AC7FCEBFE1E13FC5C5C5C3801F9E0EBFBC0A2EBFF8091C8FC5B5B5B 5BA212031207120F121F123D127800F0140300E0EC0780C66CEB0F000178131E157C6D13 F04A5A90381E0F80D90FFEC7FCEB03F823417FBF26>96 D<143FECFFC0903903E0F0E090 390FC07BF090381F803B90387F001F01FE5C4848130F5B120348485C120F5B001F141F5E 485AA2153F007F92C7FC5BA25D00FF147E90C7FCA215FE9238FC03805AA20201130703F8 1300007E1303A20207130E6C130F021F131E6C013C131C260F8070133C3A07C1E07C783A 01FF801FF03A007E0007C029297CA730>IIIII<16 3EEEFFC0923803E1E0923807C0F0ED0F811687ED1F8F160F153FA217E092387E038093C7 FCA45DA514015DA30103B512FCA390260003F0C7FCA314075DA4140F5DA5141F5DA4143F 92C8FCA45C147EA414FE5CA413015CA4495AA35CEA1E07127F5C12FF495AA200FE90C9FC EAF81EEA703EEA7878EA1FF0EA07C02C537BBF2D>III<143C14FEA21301A314FCEB00701400AD137E38 01FF803803C7C0EA0703000F13E0120E121C13071238A2EA780F007013C0A2EAF01F1480 1200133F14005B137EA213FE5BA212015B0003130E13F0A20007131EEBE01CA2143CEBC0 381478147014E013C13803E3C03801FF00EA007C173E7EBC1F>IIII<01F8D907F0EB07F8D803 FED93FFEEB1FFE28078F80F81FEB781F3E0F0F81C00F81E00F803E0E07C78007C3C007C0 001CD9CF00EBC78002FEDAEF007F003C4914FE0038495C49485C12780070495CA200F049 4948130F011F600000495CA2041F141F013F6091C75B193F043F92C7FC5B017E92C75A19 7E5E01FE9438FE01C049027E14FCA204FE01011303000106F81380495CF20700030115F0 0003190E494A151E1A1C03035E0007943800F8F0494AEC7FE0D801C0D900E0EC1F804A29 7EA750>I<01F8EB0FF0D803FEEB3FFC3A078F80F03E3A0F0F83C01F3B0E07C7800F8000 1CEBCF0002FE80003C5B00385B495A127800705BA200F049131F011F5D00005BA2163F01 3F92C7FC91C7FC5E167E5B017E14FE5EA201FE0101EB03804914F8A203031307000103F0 13005B170E16E000035E49153C17385F0007913801F1E0496DB45AD801C0023FC7FC3129 7EA737>III<023F13189138FFC038903903E0 F0F890390FC078F090381F803990387F001F13FE4848EB0FE05B1203485A000F15C05B12 1F151F48481480A3007F143F491400A300FF5C90C7127EA315FE485CA31401007E495AA2 14076C130F4A5A6C133F380F80733807C1E73901FF87E038007E071300140F5DA3141F5D A3143F92C7FCA25C5C013F13FE5B5D253A7CA729>III<14 7014FC1301A25CA21303A25CA21307A25CA2130FA25CA2007FB512F0B6FC15E039001F80 00133FA291C7FCA25BA2137EA213FEA25BA21201A25BA21203A25BA21207EC01C013E014 03000F1480A2EBC0071500140E141E5C000713385C3803E1E03801FF80D8003EC7FC1C3A 7EB821>I<137C48B4EC03802603C7C0EB0FC0EA0703000F7F000E151F121C0107158012 38163FEA780F0070491400A2D8F01F5C5C0000157E133F91C712FEA2495C137E150113FE 495CA215030001161C4914F0A21507173CEEE038150F031F1378000016706D133F017C01 7313F0017E01E313E0903A3F03C1F1C0903A0FFF007F80D901FCEB1F002E297EA734>I< 013E1438D9FF8013FE3901C3E001D8038114FF380701F0120E121ED81C037F003C157F00 38153EEA780700705B161EEAF00F4A131C1200131F4A133C1638133F91C7FC1678491470 137E16F016E013FE49130116C01503168015071600150E137C017E5B013E5B6D5B90380F 81E0903807FF80D900FEC7FC28297EA72C>I<013EEE0380D9FF800107EB0FE02601C3E0 90381F801FD8038117F0380701F0000E153F001E1600D81C03160F003C170700384BEB03 E0D87807147E00705B1801D8F00F14FE4A4914C01200131FDA800114034C1480133F1400 03031407494A1400137EA26001FE0107140E495C60A360150F017C5E017E011F14F0705B 6D0139495A6D903970F8038090280FC0E07C0FC7FC903A03FFC01FFC903A007F0007F03C 297EA741>II<137C48B4EC03802603C7C0EB0FC0EA0703000F7F000E15 1F001C168013071238163FD8780F150000705BA2D8F01F5C4A137E1200133F91C712FE5E 5B137E150113FE495CA2150300015D5BA215075EA2150F151F00005D6D133F017C137F01 7E13FF90393F03DF8090380FFF1FEB01FC90C7123F93C7FCA25DD80380137ED80FE013FE 001F5C4A5AA24848485A4A5A6CC6485A001C495A001E49C8FC000E137C380781F03803FF C0C648C9FC2A3B7EA72D>I<02F0131CD903FC133CD90FFE1338496C137849EB80F049EB C1E090397C07E3C09038F801FF9039F0003F8049EB0F000001141E6C485B90C75A5D4A5A 4A5A4A5A4AC7FC141E5C5C5C495A495A495A011FC8FC013C14E05B5B4913010001EC03C0 48481307D807F8EB0F8048B4131F3A1F0FE07F00391E07FFFE486C5B486C5BD870005B00 F0EB7FC048011FC7FC26297CA72A>I124 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmti10 11 43 /Ft 43 122 df14 D<387FFFFCA3B5FCA216 05799521>45 D<120FEA3FC0127FA212FFA31380EA7F00123C0A0A77891C>I<15FE9138 07FFC091381F03E091387C00F04A13F84948137C4948133C4948133E495A91C7FC5B013E 143FA2137E5BA212015B167F12035B0007157EA24914FE120FA34848EB01FCA44848EB03 F8A448C7EA07F0A316E0007E140F12FE16C0151FA2481580153F1600A2157EA2157C15FC 5D4A5A007C13035D4A5A6C495A001E49C7FC001F137C380FC1F83803FFE0C690C8FC283F 76BC2F>48 D<15031507150F151F151E153E15FE1401EC03FC140F141FEB01FF90381FF3 F814C3EB1E07130015F0A2140FA215E0A2141FA215C0A2143FA21580A2147FA21500A25C A25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CEB7FE0B612F0A3203D 76BC2F>I<157F913803FFC091380F81F091381E00F80278137C4A7F4948133F4A7F4948 148049C7FC5BD90E0C14C0131EEB1C0EEB3C06133813781370020E133FD9F00C148013E0 141C0218137F00011600EBC0384A13FEEC600102E05B3A00E3C003F89039FF0007F0013C 5C90C7485A4B5A4BC7FC157EEC01F84A5AEC07C0EC1F80023EC8FC14F8EB03F0EB07C049 5A011EC9FC49140C49141E49141C485A4848143C485A48C85A120E001E5DD81FF0130148 B4495A393C3FF80FD87807B55AD870035CD8F00091C7FC486D5AEC1FF8EC07E02A3F79BC 2F>II<161C163F 167F167EA316FE16FCA3150116F8A2150316F0A2ED07E0A3ED0FC0A2ED1F80A216005D15 3E157E157C15FC5D14015D14035D14075D4A5AA24AC7FC91383E0380ED0FC0147CECF81F 02F01380EB01E01303903807C03FD90F8013001400131E495B49137E5B485A484813FE48 B46C5A001F13F04813FE267C007F130400F090380FFFFE00601301C714F0913803F8005D A314075DA3140F5DA3141F5DA3020EC7FC284F7DBC2F>I<131EEB3F80137FEBFFC05AA2 14806C13005B133C90C7FCB3120FEA3FC0127FA212FFA35B6CC7FC123C122777A61C>58 D<171C173C177CA217FCA216011603A21607A24C7EA2161DA216391679167116E1A2ED01 C1A2ED038115071601150EA2031C7FA24B7EA25D15F05D4A5AA24A5AA24AC7FC5C140E5C 021FB6FC4A81A20270C7127FA25C13015C495AA249C8FCA2130E131E131C133C5B01F882 487ED807FEEC01FFB500E0017FEBFF80A25C39417BC044>65 D<9339FF8001C0030F13E0 033F9038F803809239FF807E07913A03FC001F0FDA0FF0EB071FDA1FC0ECBF00DA7F806D B4FC4AC77E495AD903F86E5A495A130F4948157E4948157C495A13FF91C9FC4848167812 035B1207491670120FA2485A95C7FC485AA3127F5BA312FF5BA490CCFCA2170FA2170EA2 171E171C173C173817786C16706D15F04C5A003F5E6D1403001F4B5A6D4AC8FC000F151E 6C6C5C6C6C14F86C6C495A6C6CEB07C090397FC03F8090261FFFFEC9FC010713F0010013 803A4272BF41>67 D<49B812F85BA290260003FEC7121F18074B14031801F000F014075D A3140F5D19E0A2141F5D1738EF7801023F027013C04B91C7FCA217F0027F5CED8001A216 0302FFEB0FC092B5FCA3499038001F804A130F1607A2010392C8FC5CA25E0107140E5CA2 93C9FC130F5CA3131F5CA3133F5CA2137FA25C497EB612E0A33D3E7BBD3C>70 DI<4AB61280A2180091C713C0167F5FA216FF94 C7FCA35D5EA315035EA315075EA3150F5EA3151F5EA3153F5EA3157FA25EA215FFA293C8 FCA25CA25DA2380F8003EA3FC0D87FE05BA21407D8FFC05B140F01805B49485A12FC0070 495A4A5A6C01FEC9FC383C01FC380F07F03807FFC0C648CAFC314079BD30>74 D<49B612C0A25FD9000390C8FC5D5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA3 14FF92C9FCA35B5CA313035C18C0EF01E0010716C05C17031880130F4A140718005F131F 4A141EA2173E013F5D4A14FC1601017F4A5A16074A131F01FFECFFF0B8FCA25F333E7BBD 39>76 D<902601FFFE020FB5FC496D5CA2D900016D010013C04AEE3F00193E70141C193C EC07BFDB3FE01438151F1978020F7FDA0E0F15708219F0EC1E07021C6D5CA20303140102 3C7FDA38015DA2701303EC7800027002805BA2047F130702F014C04A013F91C7FCA2715A 0101141F4AECF00EA2040F131E010315F84A151C1607EFFC3C0107140391C7143817FE04 0113784915FF010E16708218F0131E011C6F5AA2173F133C01385E171F137813F8486C6F 5AEA07FEB500F01407A295C8FC483E7BBD44>78 D<49B77E18F018FC903B0003FE0003FE EF00FF4BEC7F80F03FC00207151F19E05DA2020F16F0A25DA2141FF03FE05DA2023F16C0 187F4B1580A2027FEDFF00604B495A4D5A02FF4A5A4D5A92C7EA3FC04CB4C7FC4990B512 FC17E04ACAFCA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA25C 497EB67EA33C3E7BBD3E>80 D<49B612FE49EDFFC018F0903B0003FE000FFCEF01FE4BEB 007FF03F8019C00207ED1FE05DA219F0140F5DA3021FED3FE05DA2F07FC0143F4B1580F0 FF004D5A027F5D4B495A4D5AEF1FC002FF027FC7FC92380003FC92B512E0178049903800 07E04AEB03F8707E707E0103814A80A28413075CA217FF130F5C95C7FC5E131F5CA3013F 4A14204A16705F017F17F019E04A01011301496C16C0B6D8800013039438FF078094383F 0F00CAEA1FFEEF03F83C407BBD43>82 D<92391FE00380ED7FFC913A01FFFE0700913907 F01F8F91390FC007DF4AC66CB4FC023E6D5A4A130014FC495A4948147CA2495AA2010F15 785CA3011F1570A46E91C7FCA2808014FE90380FFFE015FC6DEBFF8016E06D806D806D6C 7F141F02037FEC003FED07FF1501A281A282A212075A167E120EA2001E15FE5EA25E003E 14015E003F14034B5A486C5C150F6D495A6D49C8FCD8F9F0137C39F8FE01F839F03FFFF0 D8E00F13C026C001FEC9FC314279BF33>I<48B9FCA25A903AFE001FF00101F89138E000 7FD807E0163E49013F141E5B48C75BA2001E147FA2001C4B131C123C003814FFA2007892 C7FC12704A153C00F01738485CC716001403A25DA21407A25DA2140FA25DA2141FA25DA2 143FA25DA2147FA25DA214FFA292C9FCA25BA25CA21303A25CEB0FFE003FB67E5AA2383D 71BC41>I<143F903801FFC0903907E1E1C090390F80F7E090383F0077017E133F4914C0 4848131F12035B48481480120FA24848133F1600485AA25D007F147E90C7FCA215FE485C 5AA21401EDF81C5AA21403EDF03C02071338127C020F1378007E011F1370003E133F6C01 7913F002F113E03A0F83C0F1C03A03FF007F80D800FCEB1F00262977A72F>97 DIIII<167C4BB4FC923807C78092380F83C0ED1F87161FED3F3FA2157EA2 1780EE0E004BC7FCA414015DA414035DA30103B512F8A390260007E0C7FCA3140F5DA514 1F5DA4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495AA4495A121E127F5C12FF 49C9FCA2EAFE1EEAF83C1270EA7878EA3FE0EA0F802A5383BF1C>III<1478EB01FCA21303A314F8EB00E01400 AD137C48B4FC38038F80EA0707000E13C0121E121CEA3C0F1238A2EA781F00701380A2EA F03F140012005B137E13FE5BA212015BA212035B1438120713E0000F1378EBC070A214F0 EB80E0A2EB81C01383148038078700EA03FEEA00F8163E79BC1C>I<1507ED1FC0A2153F A31680ED0E0092C7FCADEC07C0EC3FF0EC78F8ECE07CEB01C01303EC807EEB0700A2010E 13FE5D131E131CEB3C01A201005BA21403A25DA21407A25DA2140FA25DA2141FA25DA214 3FA292C7FCA25CA2147EA214FEA25CA213015CA2121C387F03F012FF495A5C495A4848C8 FCEAF83EEA707CEA3FF0EA0FC0225083BC1C>IIIIIII114 DII<13 7C48B4141C26038F80137EEA0707000E7F001E15FE121CD83C0F5C12381501EA781F0070 01805BA2D8F03F1303140000005D5B017E1307A201FE5C5B150F1201495CA2151F0003ED C1C0491481A2153F1683EE0380A2ED7F07000102FF13005C01F8EBDF0F00009038079F0E 90397C0F0F1C90391FFC07F8903907F001F02A2979A731>I<017CEB01C048B4EB07F038 038F80EA0707000E01C013F8121E001C1403EA3C0F0038EC01F0A2D8781F130000705BA2 EAF03F91C712E012005B017E130116C013FE5B1503000115805BA2ED07001203495B150E A25DA25D1578000114706D5B0000495A6D485AD97E0FC7FCEB1FFEEB03F0252979A72A> I<017C163848B4023813FC26038F8001FC13FED807071501000E7F001E0201EB00FF001C 4B137FEA3C0F0038173E1503D8781F4A131E00701380A2D8F03F1307020049131C12005B 017E010F143C4C133813FE5B031F147800014B13705BA218F00003023F14E04991C7FC17 0118C0A2EF03805D0001EE07004B5B6DEC800E0000D901CF5BD97C036D5A903A3F0787E0 F0903A0FFE01FFE0902703F8003FC7FC382979A73D>I<137C48B4143826038F8013FCEA 0707000E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F130702 0013E012005B017E130F16C013FE5B151F1201491480A2153F000315005BA25D157EA315 FE5D00011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA21407000E5CEA3F 80007F495AA24A5AD8FF0090C7FC143E007C137E00705B387801F0383803E0381E0FC06C B4C8FCEA03F8263B79A72C>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu msbm10 11 5 /Fu 5 125 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmsy10 11 31 /Fv 31 121 df<007FB812F8B912FCA26C17F83604789847>0 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A799B19>I<0060166000F816F06C1501007E15036CED07E06C6C EC0FC06C6CEC1F806C6CEC3F006C6C147E6C6C5C6C6C495A017E495A6D495A6D6C485A6D 6C485A6D6C48C7FC903803F07E6D6C5A903800FDF8EC7FF06E5A6E5AA24A7E4A7EECFDF8 903801F8FC903803F07E49487E49486C7E49486C7E49486C7E017E6D7E496D7E48486D7E 4848147E4848804848EC1F804848EC0FC048C8EA07E0007EED03F0481501481500006016 602C2C73AC47>I8 D10 D14 D<007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E0CDFCAE007FB912E0 BA12F0A26C18E03C287BAA47>17 D<0207B612F8023F15FC49B7FC4916F8D90FFCC9FCEB 1FE0017FCAFC13FEEA01F8485A485A5B485A121F90CBFC123EA25AA21278A212F8A25AA8 7EA21278A2127CA27EA27E7F120F6C7E7F6C7E6C7EEA00FE137FEB1FE0EB0FFC0103B712 F86D16FCEB003F020715F891CAFCAE001FB812F84817FCA26C17F8364878B947>I<1818 187CEF01FCEF07F8EF1FF0EF7FC0933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED 1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA 07FCEA1FF0EA7FC048CBFC5AEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE 903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0 EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FC170018381800AE007FB812F8B912FCA2 6C17F8364878B947>20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB 0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800 FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FC1701EF07F8EF1FF0EF7FC0 933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1F F0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC12 70CCFCAE007FB812F8B912FCA26C17F8364878B947>I24 D<0207B612F8023F15FC49B7 FC4916F8D90FFCC9FCEB1FE0017FCAFC13FEEA01F8485A485A5B485A121F90CBFC123EA2 5AA21278A212F8A25AA87EA21278A2127CA27EA27E7F120F6C7E7F6C7E6C7EEA00FE137F EB1FE0EB0FFC0103B712F86D16FCEB003F020715F8363678B147>26 D<19301978A2197C193CA2193E191EA2191F737EA2737E737EA2737E737E1A7C1A7EF21F 80F20FC0F207F0007FBB12FCBDFCA26C1AFCCDEA07F0F20FC0F21F80F27E001A7C624F5A 4F5AA24F5A4F5AA24FC7FC191EA2193E193CA2197C1978A2193050307BAE5B>33 D39 DII<0207B512E0023F14F049B6FC4915E0D90FFCC8FCEB1FE0017FC9FC13FEEA01F8485A 485A5B485A121F90CAFC123EA25AA21278A212F8A25AA2B812E017F0A217E000F0CAFCA2 7EA21278A2127CA27EA27E7F120F6C7E7F6C7E6C7EEA00FE137FEB1FE0EB0FFC0103B612 E06D15F0EB003F020714E02C3678B13D>50 D<176017F01601A2EE03E0A2EE07C0A2EE0F 80A2EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA2 4A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2495AA249 C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA248CAFCA2123EA25AA25AA25A1260 2C5473C000>54 D<126012F0AE12FC12FEA212FC12F0AE126007227BA700>I<0060EE01 8000F0EE03C06C1607A200781780007C160FA2003C1700003E5EA26C163EA26C163C6D15 7CA2000716786D15F8A26C6C4A5AA200015E6D140390B7FC6C5EA3017CC7EA0F80A2013C 92C7FC013E5CA2011E141E011F143EA26D6C5BA2010714786E13F8A26D6C485AA201015C ECF003A201005CECF807A291387C0F80A2023C90C8FCEC3E1FA2EC1E1EEC1F3EA2EC0FFC A26E5AA36E5AA36E5A6E5A324180BE33>I<4AB512FC023FECFFE049B712FC0107EEFF80 011F8390277FE1FC0114F0D9FC01D9000F7FD803F003017FD807C09238003FFE260F8003 6F7ED81F001607487113804883007E4A6E13C012FE48187F00F019E000C00107163FC7FC 5D191FA3140F5DA21AC0A24A5AA2F13F80A24A5A1A0061197E4AC9FC61A2027E4B5A02FE 5E18034A4B5A01015F4E5A4A4BC7FC0103163E604A5D0107ED03F04AEC07C0EF1F80010F 037EC8FC4A495A011FEC0FF04AEB7FC0DB0FFFC9FC49B512FC90B612E04892CAFC4814F8 4891CBFC433E7EBD46>68 D83 D88 DI92 D<157EEC03FEEC0FC0EC3F0014FE495A5C495AA2495AB3AA130F5C131F495A91C7FC13FE EA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE133F806D7E130F801307B3AA6D7EA26D7E80 6D7E143FEC0FC0EC03FEEC007E1F5B79C32E>102 D<12FCEAFFC0EA07F0EA01FCEA007E 6D7E131F6D7EA26D7EB3AA8013038013016D7E147EEC1F80EC07FCEC01FEEC07FCEC1F80 EC7E005C495A13035C13075CB3AA495AA2495A133F017EC7FC485AEA07F0EAFFC000FCC8 FC1F5B79C32E>I<1430147814F8A214F01301A2EB03E0A214C01307A2EB0F80A214005B A2133EA2133C137CA25BA25B1201A25B1203A2485AA25B120FA248C7FCA2121E123EA25A A2127812F8A41278127CA27EA2121E121FA26C7EA212077FA26C7EA212017FA212007FA2 137CA2133C133EA27FA27F1480A2EB07C0A2130314E0A2EB01F0A2130014F8A214781430 155A77C324>I<126012F07EA21278127CA27EA2121E121FA26C7EA212077FA26C7EA212 017FA26C7EA21378137CA2133C133EA27FA27F1480A2EB07C0A2130314E0A2EB01F0A213 0014F8A414F01301A2EB03E0A214C01307A2EB0F80A214005BA2133EA2133C137CA21378 13F8A2485AA25B1203A2485AA25B120FA248C7FCA2121E123EA25AA2127812F8A25A1260 155A7BC324>I<126012F0B3B3B3B3B11260045B76C319>I120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr6 6.6 4 /Fw 4 53 df48 D<13FE3807FFC0381F03F0383800F84813FC0060137E00FC133E6C 133FA3141F127C0010133FC7FC143E147E147C14FC14F8EB01F0EB03E0EB07C0EB0F80EB 1F00133C5BEBF003EA01E03803C006EA0700120E381FFFFE5A4813FCB5FCA218247CA321 >50 DI<14381478A214F813 011303A21306130E131C131813301370136013C01201EA038013001206120E5A12185A12 705AB612C0A2C7EAF800A7497E90383FFFC0A21A257DA421>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmmi8 8.8 7 /Fx 7 114 df<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E120C 121C5A5A12600916798716>59 D71 D<010FB56C48B5FCA219FE9026003FC0C7EA7FE019004B147C60EF01E0 027FEC07C092C7485A051EC7FC177C4A5C4AEB01E0EE07C04C5A0101021EC8FC4A137C5E 4B5A010313074A487E151F157F0107EBFBFCECF1E39138F7C1FEECFF814948C67E5C4A6D 7E14E0011F6E7E5C707EA2013F6E7E5C707EA2017F6E7E91C7FC707EA249140049810001 4B7FB500FC011F13FCA25C40327DB141>75 D97 D108 D<013E133F90397F80FFC09039E3E3C1F03A01C3E780F80003 9038FE00FCD983FC137C0007157EEB03F85C1307120F000E5B1200010F14FEA25CA2011F 130116FC1480A2013FEB03F8A291380007F016E049130F16C09138801F80ED3F009038FF C07EECE0F89038FC7FE0EC1F80000190C8FCA25BA21203A25BA21207A2387FFF80B5FCA2 272E829F27>112 D<9138FC0180903807FF0790381F878F90397E01DF0001FC13FF4848 7E485A4848137E120F5B001F14FE495B123FA2007F130101005BA3481303485CA314075D 127E140F141F003E495A003F137F6C13FF380F83DF3903FF1F80EA00FC1300143F92C7FC A35C147EA214FEA290383FFFF0A3212E7E9F23>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmr8 8.8 76 /Fy 76 128 df12 D<137813FCA212011203EA07F8EA0FE0EA1FC0EA3F80130012 7C5A5A5A0E0E70B227>19 D22 D<14C01301EB0380EB0F00130E 5B133C5B5BA2485A12035B12075B120F90C7FC5AA3123EA45AA512FCA25AAC7EA2127CA5 7EA47EA37E7F12077F12037F12016C7EA213787F131C7F130FEB0380EB01C01300124A79 B61E>40 D<12C07E1270123C121C7E120F6C7E6C7EA26C7E7F12007F1378137C133C133E A37FA4EB0F80A514C0A21307AC130FA21480A5EB1F00A4133EA3133C137C137813F85B12 015B485AA2485A48C7FC120E5A123C12705A5A124A7CB61E>I<123C127EB4FCA21380A2 127F123D1201A312031300A25A1206120E120C121C5A5A12600916798716>44 DI<123C127E12FFA4127E123C0808798716>I48 D<13075B133F13FF120FB5FC133F12F0 1200B3B3497E007FB512C0A31A3179B027>IIII<001C1460391FC007E0 90B512C0158015005C14F814C0D81C1CC7FC90C8FCA9EB0FE0EB7FF8EBF03E391FC01F80 9038000FC0001EEB07E0A2001CEB03F0C7FC15F81401A215FCA31218127E12FEA415F800 F81303007014F01278EC07E06C130F6CEB1FC0001FEB3F80390FC0FF003803FFFCC613F0 EB3F801E337CB027>I<14FF010313C090380F81F090383E00784913384913FC3801F001 EA03E0EA07C0120F90388000F8001F1400A248C8FCA25AA2127EEB07F838FE1FFE903838 0F8090387007C09038E003E039FFC001F0018013F8EC00FC90C7FC157EA2157F5AA4127E A3127F7E157EA26C14FE6D13FC000F130101C013F83907E003F03903F007E03901F81FC0 6CB5128090383FFE00EB07F020337DB027>I<1238123E003FB61280A3481500A25D0078 C7123C0070143815785D48495A5D1403C7485A4AC7FC140E141E5C143814785CA2495AA2 13035C1307A2495AA3131FA349C8FCA55BA9131C21337CB027>III<123C127E12FFA4127E123C1200 AF123C127E12FFA4127E123C081F799E16>I<123C127E12FFA4127E123C1200AF123C12 7E12FE12FFA3127F123F1203A412071206A2120E120C121C1238123012701260082D799E 16>I<15E04A7EA34A7EA34A7EA24A7E140EA2EC1EFFEC1C7FA2023C7FEC383FA24A6C7E A34A6C7EA2010180ECC007A2010380EC8003A201076D7E1400A2010E6D7EA2010FB5FC49 80A2013CC76C7E0138143FA20178810170141FA2496E7EA300016F7E1203000782D81FF8 4A7EB549B512E0A333357DB43A>65 DIIIIIIII<90383FFFFEA39038003FE0EC1FC0B3B0123CB4FCA4EC3F80 5A007814005C6C13FE6C5B380F83F03803FFC0C690C7FC1F347CB128>IIIII IIII<90381FE0039038FFFC0748EBFF0F3907F01F CF390F8003FF48487E003E7F814880815A81A3817EA27E007F91C7FC7F7FEA3FF8EBFFC0 6C13FC6CEBFF806C14E00001806C6C7F010F7F01007FEC07FF1400ED7F80153FED1FC015 0F12E0A21507A37EA26C1580150F7E6CEC1F006C143E7F01E013FC39F1FC03F800F0B512 E0D8E01F138026C003FEC7FC22367CB32B>I<007FB8FCA390398007F000D87E00153F00 7C82007882007082A200F01780A2481603A6C792C7FCB3AA4A7E011FB512FCA331317DB0 38>IIII<267FFFF890B512C0A3000101C090383FF800 6C49EB0FE0017F15806D6C91C7FC6D6C131E161C6D6C133C6D6C5B6D6C137016F06D6C48 5AD900FF5B4B5AEC7F87DA3FCFC8FCEC1FCE15FE6E5A6E5AA26E7E6E7EA24A7E4A7F153F 020F7F91381E1FE04A6C7E143891387807F84A6C7E14E001016D7E49486C7E49487F91C7 7F496E7E011E6E7E131C013C6E7E017C6E7ED801FC81D807FE4A7EB500C090387FFFF8A3 35327EB13A>II91 D93 D97 DII<157EEC0FFEA31400157EAEEB07F0EB3FFCEB FC0F3903F003FE3807E001380FC000485A157E48C7FC5AA2127E12FEA9127E127FA27E7F 001F14FE380FC0010007497E2603F00713F03900F81F7FEB3FFC90390FE07E0024347DB2 2B>IIIIIIIII<2703E01FE013FF00FF9026 7FF80313C0903BE1E07E0F03F0903AE3803E1C012807E7003F387FD803EE147001FC6D48 6C7E495CA3495CB2486C496C487EB53BC7FFFE3FFFF0A33C207D9F41>I<3907C03F8000 FFEBFFE09038C3C1F89038C700FCEA0FCED807DC137C01F8137E5BA35BB2486C13FFB500 0F13F0A324207C9F2B>II<3807E0FF D8FFE313C09038EF81F09038FE00FCD80FF8137E6C487F168049EB1FC0A2ED0FE0A3ED07 F0A9ED0FE0A3ED1FC016806D133F6DEB7F00157E01FE5B9038EF03F09038E3FFC0D9E0FE C7FC91C8FCA9487EB5FCA3242E7D9F2B>I<903807F00E90383FFC1EEBFC0F3903F007BE 3907E003FE000F1301381FC0005B003F147E48C7FCA312FEA9127FA36C7E15FE6C7E000F 13013807E0033803F0073900F81E7EEB3FFCEB0FE090C7FCA915FF020F13F0A3242E7D9F 29>I<3807C0FC38FFC3FEEBC71F9038CE3F80EA0FDCEA07D813F89038F01F0091C7FCA3 5BB1487EB512C0A319207E9F1E>II<1370A513F0A41201A212031207120F381FFFFEB5FCA23803F000AF 1407A83801F80EA23800FC1CEB7E3CEB3FF8EB07E0182E7FAC1E>IIII< 3A7FFF01FFF8A32603FC0013800001ECFC0000005C017E5B6D485AEC83C090381FC78001 0F90C7FCEB07EFEB03FE5C1301130080497E497F9038079FC0EB0F0F90381E07E06E7E01 3C7F496C7EEBF800D803F0137ED80FF8EBFF80B46C4813FCA3261F7F9E29>II<003FB512F8A290388007F0383E000F003C14E00038EB1FC00078EB3F 80147F0070EBFF005C495A495AEA0007495A5C495A495A017F1338EBFF005B485A485A00 071478485A491370484813F0485A007F1303EB000FB6FCA21D1F7E9E23>II126 D<00381370387E01F800FE13FCA3007E13F838380070160778B127> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmcsc8 8.8 43 /Fz 43 128 df<137813FCA212011203EA07F8EA0FE0EA1FC0EA3F801300127C5A5A5A0E 0E6EB22A>19 D<146014E0EB03C0EB0780EB0F00131E5BA25B5B12015B12035B12075B12 0F90C7FC5AA2121E123EA35AA512FCA25AAC7EA2127CA57EA3121E121FA27E7F12077F12 037F12017F120013787FA27F7FEB0780EB03C0EB00E01460134A77B621>40 D<12C07E12787E7E7E6C7EA26C7E6C7E7F12007F1378137C133C133E131E131FA27F1480 A3EB07C0A514E0A21303AC1307A214C0A5EB0F80A314005BA2131E133E133C137C137813 F85B12015B485A485AA248C7FC121E5A5A12E05A134A7BB621>I 45 D<123C127E12FFA4127E123C0808788718>I48 D51 D<000C1438390FE003F890B512F015E015C0150014FC14F0D80E0E C7FC90C8FCA9EB07F0EB3FFE9038F81F80390FC007C090388003E090380001F0000E14F8 000CEB00FCC8FC15FE157EA2157FA31218127E12FEA415FE12F8007014FC007813010038 14F8003CEB03F06C1307390F800FC03907E03F806CB51200C613FCEB1FE020337BB02A> 53 D56 DI<123C127E12FFA4127E123C1200AF123C127E12FFA4127E123C081F789E18>I<15 38157CA315FEA24A7EA3913803BF80A202077F151FA291380E0FE0A2021E7FEC1C07A24A 6C7EA202787FEC7001A202F07FECE000A24948137FA20103814A133FA249C76C7E91B6FC 4981A2010EC7120F496E7EA2013C8101381403A2496E7EA201F081491400A2000182486C 81D80FF8EDFF80B5021F13FEA337357CB43F>65 D<4AB4EB0180021FEBE00391B5EAF807 0103EB007ED907F8EB0F0FD91FE0EB07DFD93F80EB01FF49C8FCD801FE157F5B4848153F 4848151F120F49150F121F491507123F5B007F1603A390C9FC4893C7FCAA7E6DED0380A3 123F7F001F16076D1600120F6D5D0007160E6C6C151E6C6C5D6D5DD8007F15F86D6C495A D91FE0495AD907F8EB0F806DB4017FC7FC0100EBFFFC021F13F00201138031367AB33D> 67 D70 D<4AB4EB0180021FEBE00391B5EAF8070103EB007ED907F8EB0F0FD91FE0EB07DFD9 3F80EB01FF49C8FCD801FE157F5B4848153F4848151F120F49150F121F491507123F5B00 7F1603A390C9FC4893C7FCA993B512FE7E7F9338007FC0EF3F80123F7F121F7F120F7F12 076C7E6C7E6D157FEA007F6D6C14FFD91FE0EB01EFD907F8EB07C7903A03FF803F830100 9038FFFE01021F01F8C7FC0201138037367AB342>I II<011FB5FCA39038001F F0EC0FE0B3B0123CB4FCA4EC1FC05A00781480143F6CEB7F006C13FE380F81F83803FFE0 C66CC7FC20347BB12B>I77 DI<90391FF0018090387FFE03 48B512873907F00FC7390FC001FF48C7FC003E147F153F48141F150F5A1507A46C1403A2 7E007F91C7FC7F7FEA3FFC381FFFC014FE6CEBFFC0000314F06C806C6C7F010F7F903800 7FFF02071380EC007FED3FC0151FED0FE0A200E01407A21503A37EA26C15C015077E6CEC 0F806CEC1F006D5B01E0137E39F1FC01F839F07FFFF0D8E01F13C026C003FEC7FC23367A B32F>83 D<007FB812F0A3903AC001FC001FD87E001503007C160100781600A200701770 00F01778A2481738A6C71600B3AA4A7E0107B6FCA335317CB03D>II<1438147CA314FEA2497EA39038039F80A39038070FC0A2010F7FEB0E07A2496C 7EA2013C7FEB3801A290387800FC1370A290B57EA23901E0007F497FA20003EC1F805B12 07ED0FC0120FD83FC0EB1FE0D8FFF0EBFFFEA227267DA52E>97 DI<02FF13300107EBE07090391F80F8F090387E003DD801F8130F48481307485A48 481303001F14015B003F140090C8FC5A1670127E12FE1600A8007E1570127FA27E6D14F0 001F15E06D1301000F15C06C6C13036C6CEB07806C6CEB0F00D8007E131E90381F80F890 3807FFE00100138024277CA52D>III<02FF13300107EBE07090391F80F8F090387E003D D801F8130F48481307485A48481303001F14015B003F140090C8FC5A1670127E12FE1600 A74AB5FC127E007F90380007F8ED03F07E7F121F7F120F6C7E6C7E6C6C1307D8007E130F 90391F807E70903907FFF8300100EBC00028277CA531>103 DI< B5FCA2EA0FF0EA07E0B3ADEA0FF0B5FCA210257DA417>I107 D109 DI<49B4FC010F13E090383F01F89038FC007ED801F0131F4848EB 0F804848EB07C0000F15E04848EB03F0A248C7EA01F8A24815FC007E1400A200FE15FEA9 007E15FC007F1401A26C15F86D1303001F15F0A26C6CEB07E06C6CEB0FC06C6CEB1F806C 6CEB3F006C6C137E90383F01F890380FFFE0010190C7FC27277CA530>I114 DI<007FB612F8A2397E00FC010078EC0078A20070153800F0153CA248151CA4 C71400B3A3497E90387FFFF8A226247DA32D>III<3DFFFE03FFF803FFC0A2291FF0003FC00013006C486D48137C1207701378 7F000317704B7E6C6C5EA24B7E6C6C0173495AA2EDE1F8017E4B5AA2913801C0FC013F4B C7FCA2913803807ED91F83140EA2913887003FD90FC75CA202CEEB1FBCD907EE14B8A202 FCEB0FF801035DA24A130701015DA24A130301005DA23A267EA43F>I121 D<0038137000FC13FCEAFE01A3EAFC000038137016 0776B12A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA cmr10 11 88 /FA 88 128 df0 D<16E04B7EA24B7EA24B7EA24B7EA2 ED1DFFA203387FA29238787FC015709238F03FE015E002016D7E15C002036D7E15800207 6D7E15004A6D7E140E021E6D7E141C023C6D7F143802786E7E147002F06E7E5C01016F7E 5C01036F7E5C01076F7E91C8FC496F7E130E011E6F7E131C013C6F7F13380178707E1370 01F0707E5B0001717E5B0003717E5B0007717E90CAFC48717E120E001E717E001FBAFC48 1980A24819C0A2BB12E0A243417CC04C>I<017FED0FE02603FFC0EC3FFC4801F0ECFFFE 486D497F486D491480486D4914C06E5B486E4814E03C7E007FC03FE007007C013FECC003 00F0903B1FE07F8000F0020F91C7FC480107017E14706E6C5AA2C7000149140015F90200 5BA3ED7FE0A56F5AB3B04B7E4B7E0103B612FCA33C407BBF47>7 D<010FB612E0A3D900030180C7FC6E90C8FC6E5AA7913807FFC0027F13FC903A03FCFE7F 80D90FE0EB0FE0D93F80EB03F8D9FE00EB00FE4848157F4848ED3F804848ED1FC0000F17 E04848ED0FF0003F17F8A24848ED07FCA200FF17FEA8007F17FCA26C6CED0FF8A2001F17 F06C6CED1FE0000717C06C6CED3F806C6CED7F006C6C15FED93F80EB03F8D90FE0EB0FE0 D903FCEB7F809027007FFFFCC7FC020713C0DA00FEC8FCA74A7E4A7F010FB612E0A3373E 7BBD42>I<49B612FCA390C7D87FF0C8FC6F5A6F5AA7B4EF0FF001C0163FD81FE0EE7F80 6C6CEEFF006C6C4B5A00035FA26D150300015FAB12006D4B5AA4017F4B5AA26D5E028014 1FD91FC05D010F153F02E04AC7FCD907F0147ED903F85CD900FCEBC3F8027FEBC7E09139 1FDFFF80912607FFFEC8FC9138007FF0ED1FC0A74B7E4B7E49B612FCA33C3E7BBD47>I< 4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383FF03FD907F090397FE07F80 494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01FE6D91C7FC82ADB97EA3C6 48C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35>11 DIIIII<133E133F137F13FF5A13FEEA03FCEA07F8EA0FF013E0EA1F80EA3F00127E12 7C5A12E0124010116DBE2E>19 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AC121EAB120C C7FCA8121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A4179C019>33 D<001EEB0780397F801FE000FF133F01C013F0A201E013F8A3007F131F391E6007980000 EB0018A401E01338491330A200011470491360A2000314E090C712C0481301000EEB0380 48EB0700001813060078131E0030130C1D1C7DBE2E>I<4B6C130C4B6C131EA20307143E A24C133CA2030F147CA293C71278A24B14F8A2031E5CA2033E1301A2033C5CA3037C1303 A203785CA203F81307A24B5CA20201140F007FBAFCBB1280A26C1900C72707C0003EC8FC 4B133CA3020F147CA292C71278A24A14F8A2021E5CA3023E1301007FBAFCBB1280A26C19 00C727F80007C0C8FC4A5CA20101140FA24A91C9FCA301035CA24A131EA20107143EA24A 133CA2010F147CA291C71278A34914F8A2011E5CA2013E1301A2013C5CA201186D5A4151 7BBE4C>I<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A31201138012 0313005A120E5A1218123812300B1C79BE19>39 D<14181438147014E0EB01C0EB038013 07EB0F00131E133E133C5B13F85B12015B1203A2485AA2120F5BA2121F90C7FCA25AA212 3EA2127EA5127C12FCB2127C127EA5123EA2123FA27EA27F120FA27F1207A26C7EA21201 7F12007F13787F133E131E7FEB07801303EB01C0EB00E0147014381418155A77C324>I< 12C07E12707E7E7E120F6C7E6C7E7F12016C7E7F1378137C133C133EA27FA21480130FA2 14C01307A214E0A21303A214F0A5130114F8B214F01303A514E0A21307A214C0A2130F14 80A2131F1400A2133EA2133C137C137813F85B485A12035B485A48C7FC120E5A5A5A5A5A 155A7BC324>I<1506150FB3A9007FB912E0BA12F0A26C18E0C8000FC9FCB3A915063C3C 7BB447>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380 120313005A120E5A1218123812300B1C798919>II<121EEA7F80 A2EAFFC0A4EA7F80A2EA1E000A0A798919>I48 D<14E013011303130F133FEA03FFB5FC13CFEAFC0F1200B3B3ACEB3FF8007FB512FCA31E 3D78BC2E>III<151C153CA2157C15FCA214011403A214071406140E141C141814381470A214 E0EB01C0A2EB038014005B130EA25B5BA25B136013E0485A5B120348C7FCA2120E5AA25A 123012705AB712FEA3C73801FC00AC4A7E0103B512FEA3273E7DBD2E>I<000C1403D80F 80131F9038F801FE90B5FC5D5D15E05D92C7FC14FC380E7FE090C9FCACEB03FC90380FFF 8090383C07E09038F001F8390FC000FC49137E90C7127F000E801680C8EA1FC0A216E015 0FA316F0A3120C127F7F12FFA416E090C7121F12FC007015C00078143F00381580003CEC 7F007E15FE6C6C485A6C6C485A3903F01FE0C6B51280D93FFEC7FCEB0FF0243F7BBC2E> II<1238123C123F90 B612FEA316FC5A16F816F00078C812E000701401ED03C01680150748EC0F00151E151C15 3CC85A157015F04A5A5D14034A5A92C7FC5C5C141E143EA25CA214FCA25C1301A21303A3 13075CA3130FA6131FAA6D5AEB038027407BBD2E>III<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A2779A619>I<121EEA7F80A2EAFFC0A4EA7F80A2 EA1E00C7FCB3121E127FEAFF80A213C0A4127F121E1200A412011380A3120313005A1206 120E120C121C5A1230A20A3979A619>I<007FB912E0BA12F0A26C18E0CDFCAE007FB912 E0BA12F0A26C18E03C167BA147>61 D<15074B7EA34B7EA34B7EA34B7EA34B7E15E7A291 3801C7FC15C3A291380381FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D7EA34A6D 7EA349486D7E91B6FCA249819138800001A249C87EA24982010E157FA2011E82011C153F A2013C820138151FA2017882170F13FCD803FE4B7ED80FFF4B7EB500F0010FB512F8A33D 417DC044>65 DIIIIIIII<011FB512FCA3D900071300 6E5A1401B3B3A6123FEA7F80EAFFC0A44A5A1380D87F005B007C130700385C003C495A6C 495A6C495A2603E07EC7FC3800FFF8EB3FC026407CBD2F>IIIII< ED7FE0913807FFFE91391FC03F8091397E0007E04948EB03F8D907F0EB00FE4948147F49 486E7E49486E7E49C86C7E01FE6F7E00018349150300038348486F7EA248486F7EA2001F 188049167F003F18C0A3007F18E049163FA300FF18F0AC007F18E06D167FA4003F18C0A2 6C6CEEFF80A36C6C4B1300A26C6C4B5A00035F6D150700015F6C6C4B5A6D5E6D6C4A5A6D 6C4A5A6D6C4AC7FC6D6C14FED901FCEB03F8D9007FEB0FE091391FC03F80912607FFFEC8 FC9138007FE03C427BBF47>II 82 DI<003FB91280A3903AF0007FE0 01018090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00E0A5C8 1600B3B14B7E4B7E0107B612FEA33B3D7DBC42>II87 D89 D91 D93 D97 DI<49B47E010713F090381F80FE90 387E000F49EB1F80D803F8133F4848EB7FC05B120F485AA24848EB3F80ED0E00007F91C7 FCA290C9FC5AAB6C7EA3003FEC01C07F001F14036C6C148015076C6C14006C6C5B6C6C13 1ED8007E5B90383F81F090380FFFC0010190C7FC222A7DA829>III I<167C903907F803FF903A3FFF078F809039FC0FDE1F3901F003F83903E001F0000714F8 3B0FC000FC0600001F6EC7FC49137E003F147FA8001F147E6D13FE000F5C6C6C485A0003 5C6D485A3907FC0FC0260E3FFFC8FCEB07F8000CCAFC121CA2121EA2121FA26C7E90B512 E015FC6CECFF806C816C15F05A3A0F80001FF848C7EA03FC003E1401481400167E5A163E A46C157E007C157C007E15FC6C4A5A6C6C495A6C6C495AD803F0EB1F80D800FE01FEC7FC 90383FFFF801031380293D7DA82E>III<1478EB01FEA2EB03FF A4EB01FEA2EB00781400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14FEA2 14FCEB81F8EA7F01387C03F0381E07C0380FFF803801FC00185185BD1C>III<2701F801FE14FF00FF902707FFC00313 E0913B1E07E00F03F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F0001 01805C01FBD900FF147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F87FD9 FC3F13FEA347287DA74C>I<3901F801FE00FF903807FFC091381E07E091387803F00007 9038E001F82603F9C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8F87F 13FCA32E287DA733>I<49B4FC010F13E090383F01F890387C007C497F48487F4848EB0F 804848EB07C0000F15E04848EB03F0A2003F15F8A248C7EA01FCA44815FEA96C15FCA36D 1303003F15F8A2001F15F06C6CEB07E0A26C6CEB0FC06C6CEB1F806C6CEB3F00D8007C13 7C90383F01F890380FFFE0010190C7FC272A7DA82E>I<3901FC03FC00FF90381FFF8091 387C0FE09039FDE003F03A07FFC001FC6C496C7E6C90C7127F49EC3F805BEE1FC017E0A2 EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F89039 FDE007F09039FC780FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>I<02 FF131C0107EBC03C90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC150748 481303121F485A1501485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F6C7E 6C6C133DD8007E137990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8A32D 3A7DA730>I<3901F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB1401EC 00FC01FF1330491300A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FFF8E0 3807C03F381F000F003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA7FF8 EBFFC06C13F86C13FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800E013 03A214017E1400A27E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FFFC38 C01FE01D2A7DA824>I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6FCA2 D801FCC7FCB31570AA000014F06D13E0A290387F01C0133F90381F8380903807FF00EB01 FC1C397EB724>III< B53BC3FFFE03FFF8A3290FFE003FF00013C06C48D91FC0EB3F806C4817006D010F141E00 016F131C15076D163C00004A6C1338A2017F5E4B7E151DD93F805DED3DFC1538D91FC04A 5AED78FE9238707E03D90FE0017F5BEDE03F02F0140701070387C7FC9138F1C01F02F914 8F010315CE9138FB800F02FF14DE6D15FCED00076D5DA24A1303027E5CA2027C1301023C 5C023813003D287EA642>III<003FB61280A2 EBE00090C71300003E495A003C495A12384A5A0078495A141F00705C4A5A147F5D4AC7FC C6485AA2495A495A130F5C495A90393FC00380A2EB7F80EBFF005A5B4848130712074914 00485A48485BA248485B4848137F00FF495A90B6FCA221277DA629>II<01F01310D803FC1338486C1370390FFF80E0391E3FE3C039380FFF80D8700313 00486C5A384000781D0978BC2E>126 D<001E130F007FEB1FC0EB803F00FF14E0A3007F 14C0EB001F001EEB0F001B0977BD2E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FB cmcsc10 11 26 /FB 26 128 df<130FEB1F80133F137F13FF4813005B485A485A485AEA1FC05B48C7FC12 7C5A5A126011116CBE32>19 D45 D65 D72 D<0107B6FCA3D9000113C06E 1380157FB3B3A6123F487E487EA4EDFF005B6CC75A007C13010038495A003C5C6C495A6C 6C485A3903E03F802600FFFEC7FCEB1FF028407BBD34>74 D78 D 82 DI97 D99 DIIIIII107 DI110 DI114 D<90383FC00C9038FFF81C0003EBFE3C390FE03FFC381F8007EB0003003E130148130015 7C5A153CA36C141CA27E6C14006C7E13E013FE383FFFE06C13FE6CEBFF806C14E0000114 F06C6C13F8010F13FC1300EC07FE14011400157F153F12E0151FA37EA2151E6C143E6C14 3C6C147C6C14F89038C001F039FBF807E000F1B512C0D8E07F130038C007FC20317BAF2A >I<007FB712F8A39039801FF0073A7E000FE00000781678A20070163800F0163CA34816 1CA5C71500B3A8EC3FF8011FB512F0A32E2E7CAD36>II119 D<003C130F007FEB3F80 4814C0EB807FA3EB003F6C1480003CEB0F001A0975BD32>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmbx10 11 53 /FC 53 123 df12 D44 DII49 D<903803FFC0013F13FC90B6FC000315C0488126 0FF80F7F261FE0017F3A3F80007FFC486C6D7E6D6D7E487E6D6D1380A26F13C0A46C5A6C 5A6C5A0007C7FCC8FC4B1380A217005D5E4B5AA24B5A4B5A5E4A13804A90C7FC4A5A4A5A EC1FF04A5AEC7F809139FF0007C0495A495AD907F0EB0F80495A495A495A017EC7121F5B 48B7FC4816005A5A5A5A5AB8FC5EA42A3C7BBB35>II<16FC15011503A21507150F151F153F157FA215FF5C5C5C15DFEC0F9F 141FEC3F1F147E147C14F81301EB03F0EB07E014C0EB0F80131FEB3F00137E5B5B120148 5A485A485A5B48C7FC5A127E5AB812FCA5C8383FFC00AA91B612FCA52E3C7DBB35>I<00 07151C01E014FC01FF131F91B5FC5E5E5E5E5E93C7FC15FC5D15E092C8FC01C0C9FCA9EC FFE001C713FC01DFEBFF8090B67E028013F09039FC003FF801F06D7E13C06F7E4980C8FC 6F1380A317C0A21207EA1FC0487E487E12FF7FA21780A2495B17006C5A0180495A6CC7FC 01C0495A6C6CEBFFF8260FFC035B6CB612C06C5DC64AC7FC013F13F8010713802A3D7BBB 35>II<121F 7F13F890B712F8A45A17F017E017C0178017005E485D007EC8FC4B5A007C4A5A4B5A4B5A 4B5A484AC7FC153E157EC85A4A5A4A5AA24A5A140F5D141F143F5D147FA24AC8FCA25BA2 5BA25B5CA3130FA4131FAA6D5A6D5A6D5A2D3F7ABD35>II<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC 7F5D167F020180EDF03F02038015E0820207814B7E020F81158082021F814B7E4A81023E 7FA2027E81027C7F02FC814A147FA249B77EA24982A3D907E0C7001F7F4A80010F834A80 A2011F8391C87E4983013E81A2017E83017C81B500FC91B612FCA546407CBF4F>65 D<922607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE01039138803F FB499039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F484915075C481703 48491501485B18004890CAFC197E5A5B193E127FA349170012FFAD127F7F193EA2123FA2 7F6C187E197C6C7F19FC6C6D16F86C6D15016C18F06E15036C6DED07E0D97FFEED0FC06D 6CED3F80010F01C0ECFF006D01F8EB03FE6D9039FF801FFC010091B55A023F15E0020715 80020002FCC7FC030713C03F417ABF4C>67 DIII<92 2607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE01039138803FFB49 9039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F484915075C4817034849 1501485B18004890CAFC197E5A5B193E127FA34994C7FC12FFAC0407B612FC127F7FA300 3F92C7383FFE00A27F7E807EA26C7F6C7F6C7FA26C13FC6D6C157F6D6C7E010F01E014FF 6D01F813036D9038FFC01F010091B512F3023F15C00207ED803E02009138FE000E030701 E090C7FC46417ABF53>III76 DII80 DI<903A03FFC001C0011FEBF803017FEBFE0748B6128F4815DF4801 0013FFD80FF8130F48481303497F4848EB007FA24848143F161FA200FF150FA36D1407A2 7F7F6D91C7FC13FE387FFFE014FF15F06C14FF16C06C15F06C816C816C816C816C168001 3F15C0010F15E01301D9001F14F01400150F030113F881167F163F0078151F12F8A2160F A27EA217F07E161F6C16E06D143F01E015C001F8EC7F8001FEEB01FF9026FFE007130048 90B55A486C14F8D8F81F5CD8F00314C027E0003FFEC7FC2D417ABF3A>83 D<003FB912FCA5903BFE003FFE003FD87FF0EE0FFE01C0160349160190C71500197E127E A2007C183EA400FC183F48181FA5C81600B3B0010FB712F8A5403E7CBD49>I<903807FF C0013F13F890B512FE00036E7E2607FE037F260FF8007F6D6D7E486C6D7E151F82A26F7E A26C5A6C5A6C5AC8FCA215FF91B5FC130F137F3901FFFC0F000713E0481380381FFE0048 5A5B485A12FF5BA4151F7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFFE1 00031480C6EC003FD91FF890C7FC2F2C7DAA33>97 D<13FFB5FCA512077EAEEDFFE00207 13FC021F13FF027F14C0DAFF817F9139FC003FF802F06D7E4A130F4A6D7E4A80701380A3 7013C0A318E0AB18C0A25E1880A218006E5B4C5A6E5C6E495A02FCEB7FF0903AFCFF01FF E0D9F83FB55A496C91C7FCD9E00713FCC813C033407DBE3A>IIIII<903A03FFC003F8011F9038F81FFC017F9038FE7FFE48B8FC48 018113FC3A07FE007FE148486D7E49011F13FE001FEDF87C49010F1300003F81A8001F5D 6D131F000F5D6D133F6C6C495A3A03FF81FFC091B55A484AC7FC019F13F8D80F8313C090 CAFC7FA37F13F090B67E16F816FE6C816C16C017E06C16F01207001F16F8393FF00001D8 7FC0EB001F49EC0FFC00FF150790C81203A46D1407007F16F86D140F003F16F0D81FF0EC 3FE0D80FFCECFFC02707FFC00F13806C90B61200C615FC011F14E0010149C7FC303E7DA9 35>I<13FFB5FCA512077EAEED1FF8EDFFFE02036D7E4A80DA0FE07F91381F007F023E80 147802706D7E14F05CA25CA25CB3A5B5D8FE0FB512E0A5333F7CBE3A>III<13FFB5FCA51207 7EAF92380FFFFEA5DB01FEC7FC4B5A4B5AED0FE04B5AED7F804BC8FC4A5A4A5AEC07F014 0F143F4A7E4A7E818114F702E37F02C37F02017F806F7E6F7E826F7E150F6F7E6F7F8381 6F7FB5D8FC07EBFFC0A5323F7DBE37>I<13FFB5FCA512077EB3B3AFB512FCA5163F7CBE 1D>I<01FFD91FF8ECFFC0B590B5010713F84ADAC00F13FE0207DAE03F7FDA0FE09026F0 7F077F91261F803FEBFC010007D93E00D9F9F0806C49ECFBE002786DB4486C7F4A5D4A92 C7FCA24A5CA34A5CB3A4B5D8FE07B5D8F03FEBFF80A5512A7CA958>I<01FFEB1FF8B5EB FFFE02036D7E4A80DA0FE07F91381F007F0007013E806C137802706D7E14F05CA25CA25C B3A5B5D8FE0FB512E0A5332A7CA93A>II<01FFEBFFE0B5000713FC021F13FF027F14C0DAFF817F9139FC007FF8000701F0 6D7E6C49131F4A6D7E4A80701380A28218C0A37013E0AB4C13C0A318805E18006E5B5F6E 495A6E495A02FCEBFFF0DAFF035B02BFB55A029F91C7FC028713FC028013C092C9FCACB5 12FEA5333C7DA93A>II<3901FE 01FE00FF903807FF804A13E04A13F0EC3F1F91387C3FF8000713F8000313F0EBFFE0A291 38C01FF0ED0FE091388007C092C7FCA391C8FCB3A3B6FCA5252A7DA92B>I<90381FFC1E 48B512BE000714FE5A381FF00F383F800190C7FC48147E127E00FE143EA27EA201C090C7 FC13F013FF6C13FCECFF806C14E0816C806C800003806C80D8003F14801303D9001F13C0 140100787F00F8143FA26C141FA27E16807E153F6D140001E013FEEBFC0390B55A00FC14 F0486C13C026E00FFEC7FC222C7DAA29>IIIIIII<003FB612FCA4D9F8 0013F801E014F0EB8001494813E04A13C0007E15805C4A1300007C495A5D147F4A5A5DC6 5A495B495B5D5B49EB003E495A5C137F4948137E4A137C5A485B484913FC5C4814014813 004848130349130F007FEC7FF8B7FCA427297DA82F>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 593 438 a FC(Finite)36 b(Quan)m(tum)f(Groups)g(and)h(Cartan)g (Matrices)394 823 y FB(Nicol)658 815 y(\023)655 823 y(as)e(Andr)n (uskiewitsch)f(and)g(Hans-J)2125 815 y(\177)2122 823 y(ur)n(gen)g(Schneider)1307 1035 y FA(F)-8 b(ebruary)29 b(2,)h(2000)219 1285 y Fz(Abstra)n(ct.)42 b Fy(W)-6 b(e)19 b(consider)h(a)g(class)g(of)g(Hopf)g(algebras)g(ha)n(ving)h(as)e(an)i (in)n(v)l(arian)n(t)g(a)f(generalized)219 1376 y(Cartan)j(matrix.)34 b(F)-6 b(or)22 b(a)g(Hopf)h(algebra)f(of)g(this)i(kind,)f(w)n(e)f(pro)n (v)n(e)g(that)h(it)h(is)f(\014nite)g(dimensional)219 1467 y(if)i(and)g(only)g(if)g(its)g(generalized)g(Cartan)g(matrix)g(is) g(actually)g(a)g(\014nite)g(Cartan)g(matrix)g(\(under)219 1559 y(some)37 b(mild)i(h)n(yp)r(othesis\).)70 b(These)37 b(results)h(allo)n(w)h(to)f(classify)g(all)h(the)e(\014nite)i (dimensional)219 1650 y(coradically)29 b(graded)f(p)r(oin)n(ted)g(Hopf) g(algebras)g(whose)f(coradical)h(has)g(o)r(dd)g(prime)g(dimension)219 1741 y Fx(p)p Fy(.)34 b(W)-6 b(e)25 b(also)h(c)n(haracterize)e (coradically)i(graded)g(p)r(oin)n(ted)g(Hopf)g(algebras)f(of)g(order)h Fx(p)2780 1716 y Fw(4)2818 1741 y Fy(.)0 2160 y Fv(x)p FC(1.)46 b(In)m(tro)s(duction.)110 2271 y FA(W)-8 b(e)30 b(shall)f(w)m(ork)h(o)m(v)m(er)h(an)f(algebraically)25 b(closed)k(\014eld)g Fu(|)15 b FA(of)30 b(c)m(haracteristic)c(0.)40 b(In)31 b(this)e(ar-)0 2381 y(ticle,)c(w)m(e)i(are)e(concerned)e(with)j (the)f(classi\014cation)d(problem)i(of)i(\014nite)f(dimensional)d(p)s (oin)m(ted)0 2490 y(Hopf)38 b(algebras)e(o)m(v)m(er)i Fu(|)-9 b FA(.)59 b(Our)37 b(references)f(for)i(the)f(theory)f(of)i (Hopf)g(algebras)e(are)i([Mo1],)0 2600 y([Sw].)110 2770 y Ft(1.1)h(The)h(gener)-5 b(al)41 b(metho)-5 b(d.)46 b FA(Let)38 b(us)g(recall)e(that)h(a)h(Hopf)h(algebra)d(is)i Ft(p)-5 b(ointe)g(d)39 b FA(if)f(all)f(its)0 2879 y(simple)25 b(sub)s(coalgebras)e(are)j(one-dimensional,)e(that)h(is)h(spanned)f(b)m (y)j(a)e(group-lik)m(e)f(elemen)m(t.)0 2989 y(The)d(class)e(of)i(p)s (oin)m(ted)e(Hopf)i(algebras)e(includes)f(man)m(y)k(imp)s(ortan)m(t)c (examples)h(suc)m(h)i(as)g(group)0 3099 y(algebras,)i(en)m(v)m(eloping) f(algebras,)h(quan)m(tized)f(en)m(v)m(eloping)g(algebras)g(and)h(F)-8 b(rob)s(enius-Lusztig)0 3208 y(k)m(ernels.)85 b(In)45 b(fact,)k(an)m(y)c(Hopf)h(algebra)d(generated)f(b)m(y)k(group-lik)m(e)e (and)h(sk)m(ew-primitiv)m(e)0 3318 y(elemen)m(ts)28 b(is)i(p)s(oin)m (ted.)110 3429 y(Finite)36 b(dimensional)g(co)s(comm)m(utativ)m(e)g (Hopf)k(algebras)c(o)m(v)m(er)k Fu(|)24 b FA(are)39 b(necessarily)d (group)0 3539 y(algebras;)43 b(it)c(is)g(also)g(kno)m(wn)g(that)g(p)s (oin)m(ted)e(semisimple)h(Hopf)h(algebras)f(are)h(just)g(group)0 3648 y(algebras.)106 b(The)53 b(main)f(kno)m(wn)h(examples)e(of)i (\014nite)e(dimensional)e(p)s(oin)m(ted)i(and)h(non-)0 3758 y(co)s(comm)m(utativ)m(e)27 b(Hopf)j(algebras)e(are)i(the)f(F)-8 b(rob)s(enius-Lusztig)26 b(k)m(ernels)j([L1],)j([L2].)110 3869 y(Recen)m(tly)-8 b(,)37 b(the)e(classi\014cation)d(problem)i(of)i (\014nite)f(dimensional)e(Hopf)j(algebras,)g(or)g(ac-)0 3979 y(cording)28 b(to)h(Drinfeld,)f(\014nite)g(quan)m(tum)i(groups,)f (attracted)e(the)i(in)m(terest)f(of)i(man)m(y)g(sp)s(ecial-)0 4089 y(ists.)63 b(Whereas)37 b(there)f(w)m(ere)i(sev)m(eral)f(imp)s (ortan)m(t)e(con)m(tributions)f(in)j(the)g(semisimple)f(case)0 4198 y(\(see)42 b(the)f(surv)m(ey)i([Mo2]\),)j(only)c(few)g(results)f (are)h(kno)m(wn)h(in)f(the)g(non-semisimple)d(case.)0 4308 y(P)m(oin)m(ted)25 b(non-semisimple)e(Hopf)j(algebras)e(of)i (dimension)d Fs(p)2112 4275 y Fr(3)2180 4308 y FA(are)i(classi\014ed)e (in)j([AS2],)i([CD],)0 4417 y([SvO];)41 b(in\014nite)c(families)g(of)i (p)s(oin)m(ted)e(non-semisimple)g(Hopf)i(algebras)e(of)j(dimension)d Fs(p)3247 4384 y Fr(4)0 4527 y FA(w)m(ere)f(constructed)d(in)i([AS2],)k ([BDG],)e([G].)g(The)f(full)f(classi\014cation)d(w)m(as)37 b(w)m(ork)m(ed)f(out)f(for)0 4637 y(some)d(small)f(dimensions{)e(see)i ([Mo2];)j(also,)e(p)s(oin)m(ted)d(Hopf)k(algebras)c(of)j(order)f(16)h (and)f(32)0 4746 y(w)m(ere)h(resp)s(ectiv)m(ely)d(classi\014ed)g(in)j ([CDR],)i([Gr1].)46 b(Still)30 b(less)i(is)g(kno)m(wn)g(in)f(the)h (non-p)s(oin)m(ted)0 4856 y(non-semisimple)27 b(case,)j(see)g(ho)m(w)m (ev)m(er)g([AS1].)p 0 4972 548 4 v 110 5059 a Fy(This)e(w)n(ork)g(w)n (as)f(partially)i(supp)r(orted)f(b)n(y)g(CONICET,)g(CONICOR,)h(D)n (AAD,)f(the)f(Graduiertenk)n(olleg)0 5151 y(of)e(the)h(Math.)34 b(Institut)27 b(\(Univ)n(ersit\177)-39 b(at)53 b(M)r(\177)-41 b(unc)n(hen\),)25 b(TW)-9 b(AS)26 b(\(T)-6 b(rieste\))26 b(and)g(Secyt)f(\(UNC\).)2535 5370 y(T)n(yp)r(eset)g(b)n(y)h Fq(A)2988 5386 y(M)3070 5370 y(S)p Fy(-T)3186 5386 y(E)3230 5370 y(X)1624 5589 y(1)p eop %%Page: 2 2 2 1 bop 0 -128 a Fy(2)696 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g(H.{J.)e (SCHNEIDER)110 91 y FA(Our)k(approac)m(h)f(to)h(the)f(classi\014cation) d(problem)j(is)h(in)g(the)g(spirit)e(of)i(Lie)g(theory)f(and)h(is)0 201 y(based)f(on)h(our)f(previous)f(pap)s(er)g([AS2],)37 b(whic)m(h)c(can)g(b)s(e)g(view)m(ed)g(as)h(a)g(con)m(tin)m(uation)c (of)k(the)0 311 y(w)m(ork)c(of)g(Nic)m(hols)g([N].)110 420 y(Let)j Fs(A)h FA(b)s(e)f(a)h(p)s(oin)m(ted)e(Hopf)i(algebra,)e (let)h(\000)h(b)s(e)f(the)g(group)g(of)h(its)f(group-lik)m(e)f(elemen)m (ts)0 530 y(and)27 b(let)g Fs(A)373 544 y Fr(0)440 530 y FA(=)f Fu(|)-10 b FA(\000)20 b Fv(\022)25 b Fs(A)834 544 y Fr(1)901 530 y Fv(\022)g Fs(:)15 b(:)g(:)45 b FA(b)s(e)28 b(its)f(coradical)e(\014ltration.)36 b(Then)28 b(the)f(asso)s(ciated)e (graded)0 639 y(coalgebra)i(gr)15 b Fs(A)26 b FA(=)f Fv(\010)764 653 y Fp(n)p Fo(\025)p Fr(0)922 639 y FA(gr)15 b Fs(A)p FA(\()p Fs(n)p FA(\))30 b(is)f(a)h(p)s(oin)m(ted)e(graded)g (Hopf)i(algebra;)f Fu(|)-10 b FA(\000)20 b Fv(')25 b FA(gr)15 b Fs(A)p FA(\(0\))29 b(is)h(a)0 749 y(Hopf)35 b(subalgebra)c(and)j(the)g(pro)5 b(jection)31 b Fs(\031)36 b FA(:)d(gr)15 b Fs(A)33 b Fv(!)g FA(gr)15 b Fs(A)p FA(\(0\))34 b(with)f(k)m(ernel)h Fv(\010)2813 763 y Fp(n>)p Fr(0)2971 749 y FA(gr)15 b Fs(A)p FA(\()p Fs(n)p FA(\),)0 858 y(is)34 b(a)h(Hopf)g(algebra)d(map)j(and)f(a)g(retraction)d(of)k(the)f (inclusion.)51 b(Let)34 b Fs(R)i FA(b)s(e)e(the)g(algebra)e(of)0 968 y(coin)m(v)-5 b(arian)m(ts)28 b(of)i Fs(\031)s FA(.)110 1078 y(W)-8 b(e)35 b(denote)f(b)m(y)704 1044 y Fn(|)-17 b Fr(\000)704 1103 y Fn(|)g Fr(\000)790 1078 y Fv(Y)8 b(D)38 b FA(the)d(braided)e(category)h(of)h(Y)-8 b(etter-Drinfeld)33 b(mo)s(dules)g(o)m(v)m(er)j Fu(|)-9 b FA(\000,)0 1187 y(see)32 b([Mo1].)48 b(Y)-8 b(etter-Drinfeld)30 b(mo)s(dules)h(o)m(v)m (er)i(\000)g(w)m(ere)f(already)f(considered)f(b)m(y)j(Whitehead)0 1297 y(under)c(the)g(name)h(of)g("crossed)e(\000-mo)s(dules".)110 1406 y(Then)i Fs(R)j FA(is)e(a)g(braided)e(Hopf)i(algebra)e(in)1605 1373 y Fn(|)-18 b Fr(\000)1605 1432 y Fn(|)g Fr(\000)1690 1406 y Fv(Y)8 b(D)34 b FA(and)c(gr)15 b Fs(A)32 b FA(can)f(b)s(e)f (reconstructed)d(from)0 1516 y Fs(R)k FA(and)f Fu(|)-9 b FA(\000)24 b(as)30 b(a)h(bipro)s(duct)26 b([Ra]:)1361 1732 y(gr)14 b Fs(A)26 b Fv(')g Fs(R)q FA(#)p Fu(|)-9 b FA(\000)p Fs(:)0 1947 y FA(This)35 b(reconstruction)c(is)36 b(called)d(b)s(osonization,)g(see)i([Mj1].)58 b(The)35 b(passage)g(from)g Fs(A)i FA(to)e(the)0 2057 y(bipro)s(duct)d Fs(R)q FA(#)p Fu(|)-9 b FA(\000)29 b(is)35 b(a)g(substitute)d(of)j(the) f(w)m(ell-kno)m(wn)h(Cartier-Kostan)m(t-Milnor-Mo)s(o)o(re)0 2166 y(theorem)h(for)h(co)s(comm)m(utativ)m(e)e(Hopf)i(algebras.)61 b(Notice)36 b(that)h(in)g(the)f(\014nite)g(dimensional)0 2276 y(case,)30 b(dim)14 b Fs(A)26 b FA(=)f(dim)15 b(gr)f Fs(A)26 b FA(=)g(dim)14 b Fs(R)i FA(ord)f(\000.)110 2386 y(The)29 b("in\014nitesimal)d(part")j Fs(R)i FA(satis\014es)e(the)g (follo)m(wing)f(conditions:)86 2522 y(\(1.1\))54 b Fs(R)26 b FA(=)g Fv(\010)592 2536 y Fp(n)p Fo(\025)p Fr(0)735 2522 y Fs(R)q FA(\()p Fs(n)p FA(\))j(is)h(a)g(graded)f(braided)f(Hopf)i (algebra.)86 2632 y(\(1.2\))54 b Fs(R)q FA(\(0\))24 b(=)h Fu(|)-9 b FA(1)24 b(\(hence)k(the)i(coradical)c(is)k(trivial,)e Ft(cf.)41 b FA([Sw,)31 b(Chapter)d(11]\).)86 2741 y(\(1.3\))54 b Fs(R)q FA(\(1\))24 b(=)h Fs(P)13 b FA(\()p Fs(R)q FA(\))29 b(\(the)g(space)g(of)h(primitiv)m(e)e(elemen)m(ts)g(of)i Fs(R)q FA(\).)110 2878 y(A)i(\014rst)f(rough)g(in)m(v)-5 b(arian)m(t)30 b(of)h(suc)m(h)h Fs(R)h FA(is)f(the)e(dimension)g(of)h Fs(P)13 b FA(\()p Fs(R)q FA(\))31 b(and)g(w)m(e)h(call)e(this)h(the)0 2987 y Ft(r)-5 b(ank)31 b FA(of)f Fs(R)q FA(.)43 b(It)30 b(is)g(in)g(general)f(not)g(true)h(that)f(a)i(braided)d(Hopf)j(algebra) d Fs(R)k FA(satisfying)d(\(1.1\),)0 3097 y(\(1.2\))g(and)h(\(1.3\))f (also)g(satis\014es)86 3234 y(\(1.4\))54 b Fs(R)31 b FA(is)f(generated)d(as)j(an)g(algebra)e(o)m(v)m(er)j Fu(|)15 b FA(b)m(y)31 b Fs(R)q FA(\(1\).)110 3370 y(The)k(subalgebra)f Fs(R)837 3337 y Fo(0)899 3370 y FA(of)i Fs(R)h FA(generated)c(b)m(y)k Fs(R)q FA(\(1\),)g(a)f(Hopf)g(subalgebra)e(of)i Fs(R)h FA(satisfying)0 3480 y(\(1.1\),)28 b(\(1.2\),)h(\(1.3\))f(and)g (\(1.4\),)h(is)f(an)h(in)m(v)-5 b(arian)m(t)27 b(of)h(our)h(initial)c (Hopf)k(algebra)e Fs(A)p FA(.)41 b(W)-8 b(e)29 b(call)e(a)0 3589 y(braided)h(Hopf)j(algebra)e(satisfying)f(\(1.1\),)i(\(1.2\),)g (\(1.3\))g(and)g(\(1.4\))f(a)i Ft(Nichols)i(algebr)-5 b(a)30 b FA(since)0 3699 y(the)21 b(notion)g(app)s(ears)f(for)i(the)f (\014rst)g(time)h(in)f(the)g(article)f([N].)k(It)e(can)f(b)s(e)h(sho)m (wn)g(that)f(a)h(Nic)m(hols)0 3809 y(algebra)29 b(with)h(prescrib)s(ed) e(space)i(of)h(primitiv)m(e)d(elemen)m(ts)i(is)h(unique)e(up)i(to)g (isomorphisms;)0 3918 y(that)21 b(is,)j(the)e(notions)e(of)j("Nic)m (hols)e(algebra")f(and)i("Y)-8 b(etter-Drinfeld)19 b(mo)s(dule")h(are)i (naturally)0 4028 y(equiv)-5 b(alen)m(t.)38 b(W)-8 b(e)30 b(shall)e(denote)g(b)m(y)j Fm(B)p FA(\()p Fs(V)19 b FA(\))30 b(the)f(Nic)m(hols)f(algebra)g(whose)h(space)g(of)g(primitiv)m(e)0 4137 y(elemen)m(ts)f(is)i(isomorphic)e(to)h(the)h(Y)-8 b(etter-Drinfeld)27 b(mo)s(dule)h Fs(V)20 b FA(.)110 4247 y(This)39 b(notion,)h(or)g(the)e(related)g(one)h(of)g("quan)m(tum) g(sh)m(u\017e)h(algebras")d(w)m(as)j(considered)0 4357 y(b)m(y)34 b(sev)m(eral)e(authors)g(under)g(v)-5 b(arious)32 b(presen)m(tations;)g(see)g([L3],)k([M)s(\177)-48 b(u],)33 b([Ro2],)i([Ro3],)g([CR],)0 4466 y([Rz],)j([BD],)d([Sbg],)i([W].)g(W)-8 b(e)35 b(refer)f(to)g(the)h(surv)m(ey)g(article)d([A)m(G])37 b(for)e(details)d(and)j(further)0 4576 y(references.)110 4685 y(No)m(w)29 b(w)m(e)g(can)f(describ)s(e)e(our)i(metho)s(d)f(for)h (classifying)e(\014nite)h(dimensional)f(p)s(oin)m(ted)g(Hopf)0 4795 y(algebras)i(o)m(v)m(er)i Fu(|)-9 b FA(.)35 b(W)-8 b(e)30 b(\014x)h(a)f(\014nite)e(group)h(\000.)41 b(Then)157 4932 y(\(a\))54 b(Determine)38 b(all)h(\014nite)g(dimensional)e(Y)-8 b(etter-Drinfeld)37 b(mo)s(dules)i Fs(V)61 b FA(o)m(v)m(er)41 b(\000)g(suc)m(h)329 5041 y(that)29 b Fm(B)p FA(\()p Fs(V)19 b FA(\))30 b(is)g(\014nite)f(dimensional.)152 5151 y(\(b\))54 b(F)-8 b(or)30 b Fs(V)51 b FA(as)30 b(in)f(step)h (\(a\),)f(compute)f(the)i(braided)e(Hopf)i(algebra)e Fm(B)p FA(\()p Fs(V)20 b FA(\).)p eop %%Page: 3 3 3 2 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)554 b(3)162 91 y FA(\(c\))54 b(F)-8 b(or)37 b Fm(B)p FA(\()p Fs(V)20 b FA(\))37 b(as)g(in)g(step)g (\(b\),)i(compute)c(all)h(Hopf)i(algebras)d Fs(A)j FA(suc)m(h)g(that)e (gr)15 b Fs(A)38 b Fv(')329 201 y Fm(B)p FA(\()p Fs(V)20 b FA(\)#)p Fu(|)-10 b FA(\000)24 b(\("lifting"\))i(up)k(to)g (isomorphism.)152 311 y(\(d\))54 b(In)m(v)m(estigate)21 b(whether)g(an)m(y)j(\014nite)d(dimensional)f(braided)h(Hopf)j(algebra) d Fs(R)j FA(in)3061 277 y Fn(|)-18 b Fr(\000)3061 336 y Fn(|)g Fr(\000)3146 311 y Fv(Y)8 b(D)329 420 y FA(satisfying)25 b(\(1.1\),)j(\(1.2\))f(and)g(\(1.3\))g(is)g(generated)e(b)m(y)k(its)e (primitiv)m(e)e(elemen)m(ts,)i(that)329 530 y(is)j(also)f(ful\014lls)f (\(1.4\).)110 664 y(As)f(to)f(step)f(\(a\),)i(it)f(is)g(in)g(general)e (v)m(ery)j(di\016cult)d(to)i(decide)e(when)i Fm(B)p FA(\()p Fs(V)20 b FA(\))26 b(is)g(\014nite)f(dimen-)0 774 y(sional)j(since)h (an)g(e\016cien)m(t)f(description)f(of)j(the)f(de\014ning)e(relations)g (of)j Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b(is)f(not)h(kno)m(wn.)0 884 y(Step)d(\(c\))f(should)g(b)s(e)h(p)s(ossible)e(once)i(the)f(answ)m (er)h(to)h(step)e(\(b\))h(is)g(kno)m(wn,)i(as)f(the)e(exp)s(erience)0 993 y(seems)32 b(to)f(indicate)e([AS2].)48 b(Condition)29 b(\(d\))i(means)h(that)f(an)m(y)h(p)s(oin)m(ted)e(\014nite)g (dimensional)0 1103 y(Hopf)j(algebra)d(with)h(coradical)f Fu(|)-9 b FA(\000)26 b(is)33 b(generated)c(b)m(y)34 b(group-lik)m(e)d (and)h(sk)m(ew-primitiv)m(e)f(ele-)0 1212 y(men)m(ts.)110 1322 y(In)f(this)f(pap)s(er)f(w)m(e)j(mainly)e(con)m(tribute)e(to)i (\(a\))g(and)h(\(d\).)110 1533 y Ft(1.2)f(The)g(main)h(r)-5 b(esults.)48 b FA(Let)26 b(us)h(assume)g(from)f(no)m(w)h(on)g(that)f (\000)h(is)f(a)h(\014xed)g(\014nite)e(ab)s(elian)0 1642 y(group.)49 b(A)34 b(\014nite)d(dimensional)f(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)i Fs(V)54 b FA(o)m(v)m(er)33 b Fu(|)-9 b FA(\000)28 b(admits)k(a)h(basis)f Fs(x)3221 1656 y Fr(1)3262 1642 y FA(,)0 1758 y Fs(:)15 b(:)g(:)32 b FA(,)i Fs(x)248 1772 y Fp(\022)322 1758 y FA(suc)m(h)f(that,)f(for)g(some)h(elemen)m (ts)e Fs(g)s FA(\(1\),)g Fs(:)15 b(:)g(:)32 b FA(,)i Fs(g)s FA(\()p Fs(\022)s FA(\))28 b Fv(2)h FA(\000,)34 b Fs(\037)p FA(\(1\),)f Fs(:)15 b(:)g(:)31 b FA(,)j Fs(\037)p FA(\()p Fs(\022)s FA(\))28 b Fv(2)3048 1735 y Fl(b)3045 1758 y FA(\000,)34 b(the)0 1868 y(action)28 b(and)i(coaction)d(of)j (\000)g(are)g(giv)m(en)f(b)m(y)0 2177 y(\(1.5\))306 b Fs(h:x)625 2191 y Fp(j)690 2177 y FA(=)25 b Fs(\037)p FA(\()p Fs(j)5 b FA(\)\()p Fs(h)p FA(\))p Fs(x)1135 2191 y Fp(j)1171 2177 y Fs(;)198 b(\016)s FA(\()p Fs(x)1526 2191 y Fp(j)1565 2177 y FA(\))25 b(=)g Fs(g)s FA(\()p Fs(j)5 b FA(\))19 b Fv(\012)h Fs(x)2046 2191 y Fp(j)2085 2177 y Fs(;)198 b(j)30 b FA(=)c(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(\022)s(:) 0 2377 y FA(Since)k(w)m(e)i(are)f(in)m(terested)e(in)i(\014nite)f (dimensional)e(braided)h(Hopf)j(algebras)d Fs(R)q FA(,)24 b(w)m(e)d(can)f(assume)0 2486 y(that)0 2686 y(\(1.6\))1148 b Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)24 b(6)p FA(=)h(1)p Fs(;)0 2886 y Ft(cf.)41 b FA([AS2,)30 b(Lemma)g(4.1].)41 b(The)29 b(braiding)f Fs(c)h FA(is)h(giv)m(en)f(with)g(resp)s(ect)e(to)i(the)g (basis)g Fs(x)2929 2900 y Fp(i)2980 2886 y Fv(\012)19 b Fs(x)3122 2900 y Fp(j)3191 2886 y FA(b)m(y)0 3085 y(\(1.7\))982 b Fs(c)p FA(\()p Fs(x)1299 3099 y Fp(i)1349 3085 y Fv(\012)20 b Fs(x)1492 3099 y Fp(j)1531 3085 y FA(\))25 b(=)g Fs(b)1727 3099 y Fp(ij)1807 3085 y Fs(x)1859 3099 y Fp(j)1918 3085 y Fv(\012)20 b Fs(x)2061 3099 y Fp(i)2092 3085 y Fs(;)0 3285 y FA(where)29 b(\()p Fs(b)339 3299 y Fp(ij)403 3285 y FA(\))439 3299 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)737 3285 y FA(=)d(\()p Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)p FA(\))1340 3299 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)1610 3285 y FA(.)0 3445 y FC(De\014nition.)53 b FA(W)-8 b(e)26 b(shall)e(sa)m(y)j(that)d(a)i(braiding)d(giv)m(en)i(b) m(y)h(a)g(matrix)f FC(b)g FA(=)h(\()p Fs(b)2650 3459 y Fp(ij)2714 3445 y FA(\))2750 3459 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)3049 3445 y FA(whose)0 3555 y(en)m(tries)35 b(are)g(ro)s(ots)g(of)i(unit)m(y)e(is)h Ft(of)i(Cartan)h(typ)-5 b(e)37 b FA(if)f(for)g(all)f Fs(i;)15 b(j)5 b FA(,)40 b Fs(b)2361 3569 y Fp(ii)2453 3555 y Fv(6)p FA(=)c(1)g(and)g(there)f(exists)0 3664 y Fs(a)48 3678 y Fp(ij)138 3664 y Fv(2)25 b Fu(Z)i FA(suc)m(h)j(that)0 3973 y(\(1.8\))1207 b Fs(b)1435 3987 y Fp(ij)1499 3973 y Fs(b)1538 3987 y Fp(j)t(i)1628 3973 y FA(=)26 b Fs(b)1764 3924 y Fp(a)1804 3933 y Fk(ij)1764 3999 y Fp(ii)1866 3973 y Fs(:)110 4173 y FA(The)j(in)m(tegers)f Fs(a)683 4187 y Fp(ij)778 4173 y FA(are)i(determined)d(once)i(they)h(are)f(c)m (hosen)g(in)h(the)f(follo)m(wing)f(w)m(a)m(y:)86 4308 y(\(1.9\))54 b(If)30 b Fs(i)c FA(=)f Fs(j)36 b FA(w)m(e)30 b(tak)m(e)g Fs(a)1027 4322 y Fp(ii)1109 4308 y FA(=)25 b(2;)41 4417 y(\(1.10\))53 b(if)29 b Fs(i)d Fv(6)p FA(=)g Fs(j)5 b FA(,)30 b(w)m(e)h(select)d(the)h(unique)g Fs(a)1550 4431 y Fp(ij)1645 4417 y FA(suc)m(h)h(that)f Fv(\000)15 b FA(ord)g Fs(b)2322 4431 y Fp(ii)2403 4417 y Fs(<)26 b(a)2548 4431 y Fp(ij)2638 4417 y Fv(\024)f FA(0.)110 4552 y(Then)h(\()p Fs(a)429 4566 y Fp(ij)493 4552 y FA(\))g(is)g(a)h (generalized)c(Cartan)i(matrix)g([K].)j(W)-8 b(e)27 b(transfer)e(the)h (terminology)d(from)0 4662 y(generalized)f(Cartan)j(matrices)f(and)h (Dynkin)g(diagrams)g(to)g(braidings)f(of)i(Cartan)e(t)m(yp)s(e.)39 b(W)-8 b(e)0 4771 y(shall)31 b(sa)m(y)j(that)e(a)h(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)i Fs(V)54 b FA(is)33 b Ft(of)i(Cartan)g(typ)-5 b(e)34 b FA(\(resp.,)f(connected,)e(...\))0 4881 y(if)f(the)f(matrix)g(\()p Fs(b)613 4895 y Fp(ij)677 4881 y FA(\))h(as)g(ab)s(o)m(v)m(e)g(is)g(of)g(Cartan)e(t)m(yp)s(e)i (\(resp.,)g(connected,)d(...\).)110 5041 y(The)50 b(main)h(kno)m(wn)g (examples)e(of)i(braidings)e(of)i(Cartan)f(t)m(yp)s(e)g(are)h(giv)m(en) f(b)m(y)i FC(b)61 b FA(=)0 5151 y(\()p Fs(q)80 5118 y Fp(i:j)166 5151 y FA(\))202 5165 y Fp(i;j)t Fo(2)p Fp(I)374 5151 y FA(,)29 b(where)f Fs(q)k FA(is)c(a)g(ro)s(ot)f(of)i(unit)m(y)e (and)h(\()p Fs(I)7 b(;)15 b(:)p FA(\))30 b(is)e(a)g(Cartan)f(datum)h (as)h(in)f([L3,)h(Ch.)40 b(1].)p eop %%Page: 4 4 4 3 bop 0 -128 a Fy(4)696 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g(H.{J.)e (SCHNEIDER)0 91 y FC(De\014nition.)53 b FA(Let)39 b FC(b)g FA(b)s(e)g(a)g(braiding)d(of)j(Cartan)f(t)m(yp)s(e)h(with)f(asso)s (ciated)d(Cartan)j(matrix)0 201 y(\()p Fs(a)84 215 y Fp(ij)148 201 y FA(\))27 b(as)h(in)f(\(1.9\),)g(\(1.10\).)39 b(W)-8 b(e)28 b(sa)m(y)g(that)e FC(b)i FA(is)f(of)h Ft(FL-typ)-5 b(e)28 b FA(if)f(there)f(exists)h(p)s(ositiv)m(e)e(in)m(tegers)0 311 y Fs(d)48 325 y Fr(1)89 311 y Fs(;)15 b(:)g(:)g(:)i(;)e(d)339 325 y Fp(\022)411 311 y FA(suc)m(h)30 b(that)f(for)g(all)g Fs(i;)15 b(j)5 b FA(,)41 452 y(\(1.11\))53 b Fs(d)377 466 y Fp(i)407 452 y Fs(a)455 466 y Fp(ij)545 452 y FA(=)25 b Fs(d)689 466 y Fp(j)727 452 y Fs(a)775 466 y Fp(j)t(i)870 452 y FA(\(hence)j(\()p Fs(a)1243 466 y Fp(ij)1307 452 y FA(\))i(is)g(symmetrizable\).)41 564 y(\(1.12\))53 b(There)29 b(exists)g Fs(q)f Fv(2)e Fu(|)15 b FA(suc)m(h)30 b(that)f Fs(b)1525 578 y Fp(ij)1615 564 y FA(=)c Fs(q)1755 531 y Fp(d)1793 540 y Fk(i)1822 531 y Fp(a)1862 540 y Fk(ij)1925 564 y FA(.)110 706 y(F)-8 b(urthermore,)26 b(w)m(e)i(shall)f(sa)m(y)i(that)d(a)i(braiding)e FC(b)j FA(is)e Ft(lo)-5 b(c)g(al)5 b(ly)32 b(of)f(FL-typ)-5 b(e)28 b FA(if)g(an)m(y)g(principal)0 815 y(2)13 b Fv(\002)g FA(2)26 b(submatrix)f(of)h FC(b)i FA(giv)m(es)e(a)g(braiding)e(of)i (FL-t)m(yp)s(e.)39 b(Recall)25 b(that)g(a)h(principal)e(submatrix)0 925 y(of)30 b(\()p Fs(b)179 939 y Fp(ij)243 925 y FA(\))g(is)g(a)g (matrix)f(\()p Fs(b)849 939 y Fp(ij)913 925 y FA(\))949 939 y Fp(i;j)t Fo(2)p Fp(I)1152 925 y FA(for)g(some)h(subset)f Fs(I)k Fv(\032)25 b(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(\022)s Fv(g)p FA(.)110 1102 y(Let)22 b(\()p Fs(b)341 1116 y Fp(ij)405 1102 y FA(\))g(b)s(e)g(a)h(braiding)d(of)j(Cartan)e(t)m(yp)s (e.)38 b(Let)22 b Fv(X)37 b FA(b)s(e)22 b(the)g(set)g(of)g(connected)e (comp)s(onen)m(ts)0 1211 y(of)30 b(the)g(Dynkin)g(diagram)e(corresp)s (onding)e(to)k(it.)41 b(F)-8 b(or)30 b(eac)m(h)g Fs(I)j Fv(2)26 b(X)13 b FA(,)31 b(w)m(e)g(let)e Fm(g)2745 1225 y Fp(I)2818 1211 y FA(b)s(e)g(the)h(Kac-)0 1321 y(Mo)s(o)s(dy)i(Lie)h (algebra)f(corresp)s(onding)d(to)34 b(the)f(generalized)c(Cartan)k (matrix)f(\()p Fs(a)2834 1335 y Fp(ij)2898 1321 y FA(\))2934 1335 y Fp(i;j)t Fo(2)p Fp(I)3141 1321 y FA(and)0 1430 y Fm(n)48 1444 y Fp(I)120 1430 y FA(b)s(e)e(the)f(Lie)g(subalgebra)f (of)i Fm(g)1163 1444 y Fp(I)1235 1430 y FA(spanned)f(b)m(y)i(all)e(its) g(p)s(ositiv)m(e)f(ro)s(ots.)0 1607 y FC(Theorem)52 b(1.1.)g Fj(Let)44 b FC(b)51 b FA(=)f(\()p Fs(b)1162 1621 y Fp(ij)1226 1607 y FA(\))45 b Fj(b)s(e)f(a)h(braiding)d(of)j(Cartan)e(t)m(yp)s(e,) 49 b(corresp)s(onding)40 b(to)0 1717 y Fs(c)25 b FA(:)h Fs(V)40 b Fv(\012)21 b Fs(V)46 b Fv(!)25 b Fs(V)41 b Fv(\012)21 b Fs(V)f Fj(.)41 b(W)-8 b(e)30 b(also)g(assume)f(that)g Fs(b)1734 1731 y Fp(ij)1829 1717 y Fj(has)h(o)s(dd)f(order)g(for)h(all) f Fs(i;)15 b(j)5 b Fj(.)110 1831 y(\(i\).)37 b(If)25 b FC(b)g Fj(is)g(of)f(\014nite)f(Cartan)h(t)m(yp)s(e,)h(then)f Fm(B)p FA(\()p Fs(V)19 b FA(\))25 b Fj(is)f(\014nite)f(dimensional,)g (and)h(if)g(moreo)m(v)m(er)0 1940 y(3)29 b(do)s(es)g(not)f(divide)g (the)g(order)h(of)g Fs(b)1250 1954 y Fp(ii)1336 1940 y Fj(for)g(all)f Fs(i)i Fj(in)f(a)g(connected)e(comp)s(onen)m(t)g(of)i (the)g(Dynkin)0 2050 y(diagram)f(of)i(t)m(yp)s(e)g Fs(G)737 2064 y Fr(2)778 2050 y Fj(,)g(then)1119 2308 y FA(dim)15 b Fm(B)p FA(\()p Fs(V)20 b FA(\))25 b(=)1650 2221 y Fl(Y)1634 2417 y Fp(I)5 b Fo(2X)1797 2308 y Fs(N)1880 2262 y FA(dim)15 b Fm(n)2096 2276 y Fp(I)1870 2334 y(I)2143 2308 y Fs(;)0 2630 y Fj(where)29 b Fs(N)337 2644 y Fp(I)405 2630 y FA(=)c Fs(or)s(d)p FA(\()p Fs(b)712 2644 y Fp(ii)768 2630 y FA(\))30 b Fj(for)f(all)g Fs(i)d Fv(2)f Fs(I)38 b Fj(and)30 b Fs(I)i Fv(2)26 b(X)13 b Fj(.)110 2744 y(\(ii\).)51 b(Assume)35 b(that)e FC(b)j Fj(is)e(lo)s(cally)d(of)k(FL-t)m(yp)s(e)f (and)g(that,)g(for)h(all)e Fs(i)p Fj(,)k(the)c(order)h(of)g Fs(b)3135 2758 y Fp(ii)3226 2744 y Fj(is)0 2854 y(relativ)m(ely)d (prime)i(to)h(3)g(whenev)m(er)e Fs(a)1313 2868 y Fp(ij)1410 2854 y FA(=)g Fv(\000)p FA(3)h Fj(for)h(some)g Fs(j)5 b Fj(,)35 b(and)f(is)f(di\013eren)m(t)f(from)i(3,)h(5,)h(7,)0 2963 y(11,)30 b(13,)g(17.)41 b(If)30 b Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b Fj(b)s(e)f(\014nite)f(dimensional,)f(then)j FC(b)g Fj(is)g(of)g(\014nite)f(t)m(yp)s(e.)110 3140 y FA(Giv)m(en)f(a)h(\014nite)e(dimensional)f(Y)-8 b(etter-Drinfeld)26 b(mo)s(dule)h Fs(V)50 b FA(o)m(v)m(er)30 b(\000,)g(it)e(is)h(easy)g(to) f(decide)0 3250 y(whether)23 b(it)h(is)g(of)h(Cartan)f(t)m(yp)s(e)g (and)g(lo)s(cally)e(of)j(FL-t)m(yp)s(e.)38 b(Let)24 b Fs(p)h FA(b)s(e)f(an)h(o)s(dd)e(prime)h(n)m(um)m(b)s(er.)0 3359 y(Assume)31 b(that)e FC(b)e FA(=)f(\()p Fs(b)794 3373 y Fp(ij)858 3359 y FA(\))k(satis\014es)f Fs(b)1300 3373 y Fp(ii)1383 3359 y Fv(6)p FA(=)d(1)31 b(and)f(the)g(order)f(of)i Fs(b)2276 3373 y Fp(ij)2371 3359 y FA(is)f(either)f Fs(p)i FA(or)f(1)h(for)f(all)f Fs(i)0 3469 y FA(and)g Fs(j)5 b FA(.)41 b(Then)30 b FC(b)g FA(is)g(of)g(Cartan)f(t)m(yp)s(e.)40 b(If)30 b Fs(b)1490 3483 y Fp(ij)1580 3469 y FA(=)25 b Fs(b)1715 3483 y Fp(j)t(i)1810 3469 y FA(for)30 b(all)e Fs(i;)15 b(j)5 b FA(,)32 b(then)d(it)g(is)h(also)f(not)g(di\016cult)0 3579 y(to)38 b(see)h(that)f FC(b)h FA(is)g(lo)s(cally)d(of)j(FL-t)m(yp) s(e,)i(see)d(Lemma)h(4.3,)j(using)37 b(\(1.10\))h(for)h Fs(p)h FA(=)g(3.)67 b(W)-8 b(e)0 3688 y(conclude)29 b(from)i(Theorem)e (1.1)j(and)e(the)h(reduction)d(to)j(the)f(symmetric)g(connected)e(case) j(in)0 3798 y(Section)d(4.2:)0 3975 y FC(Corollary)37 b(1.2.)52 b Fj(Let)31 b Fs(p)g Fj(b)s(e)g(an)g(o)s(dd)f(prime)h(n)m(um) m(b)s(er,)g FA(\000)h Fj(a)f(\014nite)f(direct)f(sum)j(of)f(copies)f (of)0 4084 y Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))f Fj(and)k Fs(V)55 b Fj(a)34 b(\014nite)e(dimensional)f(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)h(o)m(v)m(er)i FA(\000)h Fj(with)e(braiding)e FC(b)p Fj(.)0 4194 y(W)-8 b(e)30 b(assume)g(that)f Fs(b)710 4208 y Fp(ii)792 4194 y Fv(6)p FA(=)c(1)30 b Fj(for)g(all)f Fs(i)p Fj(.)41 b(Then)30 b FC(b)g Fj(is)g(of)g(Cartan)f(t)m(yp)s(e)g (and)110 4308 y(\(i\).)41 b(If)31 b FC(b)h Fj(is)e(of)h(\014nite)e(t)m (yp)s(e,)i(then)f Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b Fj(is)h(\014nite)e(dimensional,)f(and)i FA(dim)15 b Fm(B)p FA(\()p Fs(V)20 b FA(\))26 b(=)g Fs(p)3181 4274 y Fp(M)3262 4308 y Fj(,)0 4417 y(where)j Fs(M)35 b FA(=)484 4349 y Fl(P)581 4445 y Fp(I)5 b Fo(2X)749 4417 y FA(dim)14 b Fm(n)964 4431 y Fp(I)1037 4417 y Fj(is)30 b(the)f(n)m(um)m(b)s(er)g (of)h(p)s(ositiv)m(e)e(ro)s(ots)h(of)h(the)f(ro)s(ot)g(system)h(of)g FC(b)p Fj(.)110 4531 y(\(ii\).)38 b(If)31 b Fm(B)p FA(\()p Fs(V)19 b FA(\))30 b Fj(is)g(\014nite)f(dimensional)d(and)k Fs(p)25 b(>)h FA(17)p Fj(,)k(then)f FC(b)i Fj(is)e(of)h(\014nite)f(t)m (yp)s(e.)110 4708 y FA(W)-8 b(e)31 b(remark)h(ho)m(w)m(ev)m(er)f(that)g (there)f(are)h(examples)f(of)i(\014nite)e(dimensional)e(braided)i(Hopf) 0 4818 y(algebras)g Fm(B)p FA(\()p Fs(V)20 b FA(\))33 b(of)f(rank)g(2)h(o)m(v)m(er)g(cyclic)d(groups)i(of)g(ev)m(en)h(order)e (whic)m(h)h(are)g Ft(not)h FA(of)f(Cartan)0 4927 y(t)m(yp)s(e,)e(see)f ([N.)j(pp.)41 b(1540)29 b(\013.].)110 5041 y(W)-8 b(e)33 b(w)m(an)m(t)g(to)g(apply)f(Theorem)g(1)i(to)e(solv)m(e)h(step)g(\(a\)) f(of)h(the)g(general)e(metho)s(d;)i(w)m(e)h(ha)m(v)m(e)0 5151 y(to)c(determine)e(whic)m(h)h(braidings)f FC(b)j FA(of)f(\014nite)f(Cartan)g(t)m(yp)s(e)h(actually)d(app)s(ear)i(o)m(v)m (er)i(\000.)42 b(This)p eop %%Page: 5 5 5 4 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)554 b(5)0 91 y FA(reduces,)31 b(for)g(eac)m(h)f(\014xed)h(\014nite)f(Cartan)g(matrix)h(\()p Fs(a)1851 105 y Fp(ij)1915 91 y FA(\))c Fv(2)g Fu(Z)2131 58 y Fp(\022)r Fo(\002)p Fp(\022)2262 91 y FA(,)33 b(to)e(\014nd)g(all) f(the)h(sequences)0 216 y Fs(g)s FA(\(1\),)e Fs(:)15 b(:)g(:)32 b FA(,)f Fs(g)s FA(\()p Fs(\022)s FA(\))24 b Fv(2)h FA(\000,)31 b Fs(\037)p FA(\(1\),)f Fs(:)15 b(:)g(:)32 b FA(,)e Fs(\037)p FA(\()p Fs(\022)s FA(\))24 b Fv(2)1511 193 y Fl(b)1508 216 y FA(\000)30 b(suc)m(h)g(that)41 344 y(\(1.13\))53 b Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)24 b(6)p FA(=)h(1,)31 b(for)e(all)g Fs(i)i FA(and)41 454 y(\(1.14\))53 b Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(ih)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)g(g)s FA(\()p Fs(j)5 b FA(\))p Fv(i)21 b FA(=)k Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)1736 421 y Fp(a)1776 430 y Fk(ij)1838 454 y FA(,)30 b(for)g(all)f Fs(i;)15 b(j)5 b FA(.)110 582 y(This)29 b(can)g(b)s(e)g(in)m(terpreted)e(as)j(a)f (problem)g(of)g(computational)d(n)m(um)m(b)s(er)j(theory:)39 b(to)30 b(com-)0 692 y(pute)e(all)h(the)f(solutions)f(of)i(a)h(system)f (of)h(quadratic)d(congruences.)37 b(See)29 b(Section)e(8.)40 b(In)30 b(fact,)0 801 y(w)m(e)35 b(can)e(sho)m(w)i(that)e(the)h(n)m(um) m(b)s(er)g(of)g(Y)-8 b(etter-Drinfeld)31 b(mo)s(dules)i Fs(V)55 b FA(\(up)34 b(to)g(isomorphism\))0 911 y(of)c(\014nite)e (Cartan)h(t)m(yp)s(e)h(o)m(v)m(er)g(our)g(\014xed)g(group)f(\000)h(is)g (\014nite.)110 1020 y(W)-8 b(e)27 b(illustrate)d(this)i(in)h(the)f (case)h(\000)e Fv(')g Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\),)f(where)i Fs(p)h FA(is)g(a)h(prime)e(n)m(um)m(b)s(er.)39 b(W)-8 b(e)28 b(assume)0 1130 y(that)h Fs(p)h FA(is)g(o)s(dd;)g(the)f(case)g Fs(p)c FA(=)h(2)k(is)g(co)m(v)m(ered)f(b)m(y)i([N,)h(Th.)40 b(4.2.1].)110 1240 y(Before)24 b(stating)f(the)h(next)h(Theorem,)h(w)m (e)f(recall)f(that)g(a)h(graded)f(coalgebra)f(is)i Ft(c)-5 b(or)g(adic)g(al)5 b(ly)0 1349 y(gr)-5 b(ade)g(d)22 b FA(if)g(its)f(coradical)f(\014ltration)f(coincides)g(with)i(the)h (\014ltration)d(coming)h(from)j(the)e(grading)0 1459 y([CM].)0 1606 y FC(Theorem)48 b(1.3.)k Fj(Let)40 b FA(\000)k Fv(')g Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))p Fj(,)c(where)g Fs(p)h Fj(is)g(an)g(o)s(dd)f(prime)h(n)m(um)m(b)s(er.)73 b(The)41 b(follo)m(w-)0 1716 y(ing)33 b(list)f(con)m(tains)g(all)g(p)s (ossible)f(Nic)m(hols)i(algebras)e(of)i(\014nite)f(dimension)g(o)m(v)m (er)i FA(\000)g Fj(and)f(\014nite)0 1825 y(dimensional)27 b(coradically)f(graded)j(Hopf)h(algebras)e(with)h(coradical)e (isomorphic)g(to)j Fs(k)s FA(\000)p Fj(.)110 1935 y(\(i\).)50 b(The)34 b(quan)m(tum)f(lines)g(and)g(planes)f(discussed)h(in)g FA([AS2])p Fj(.)53 b(By)34 b(b)s(osonization)c(w)m(e)k(get)0 2044 y(resp)s(ectiv)m(ely)27 b(T)-8 b(aft)30 b(algebras)e(and)h(b)s(o)s (ok)g(Hopf)h(algebras)e Ft(cf.)41 b FA([AS1])p Fj(.)110 2154 y(\(ii\).)d(There)27 b(exists)g(a)h(Nic)m(hols)f(algebra)f(with)h (Dynkin)g(diagram)f Fs(A)2467 2168 y Fr(2)2537 2154 y Fj(if)i(and)f(only)h(if)f Fs(p)f FA(=)f(3)0 2264 y Fj(or)e Fs(p)7 b Fv(\000)g FA(1)23 b Fj(is)g(divisible)d(b)m(y)k(3.)38 b(F)-8 b(or)23 b Fs(p)j FA(=)f(3)p Fj(,)g(w)m(e)f(obtain)d(b)m(y)j(b)s (osonization)19 b(from)k(Nic)m(hols)f(algebras)0 2373 y(of)36 b(dimension)d FA(27)j Fj(exactly)e(2)i(non-isomorphic)d(p)s (oin)m(ted)g(Hopf)j(algebras)e(of)i(dimension)d FA(81)0 2483 y Fj(with)e(coradical)f(of)i(dimension)e FA(3)p Fj(.)48 b(F)-8 b(or)32 b Fs(p)e Fv(\021)f FA(1)61 b(mo)s(d)29 b(3)p Fj(,)34 b(w)m(e)e(obtain)f(b)m(y)i(b)s(osonization)28 b(from)0 2592 y(Nic)m(hols)21 b(algebras)g(of)h(dimension)e Fs(p)1227 2559 y Fr(3)1291 2592 y Fj(exactly)g Fs(p)5 b Fv(\000)g FA(1)22 b Fj(non-isomorphic)e(p)s(oin)m(ted)g(Hopf)i (algebras)0 2702 y(of)30 b(dimension)e Fs(p)582 2669 y Fr(4)653 2702 y Fj(with)h(coradical)e(of)j(dimension)e Fs(p)p Fj(.)110 2812 y(\(iii\).)36 b(There)25 b(exists)h(a)g(Nic)m (hols)f(algebra)f(with)h(Dynkin)g(diagram)f Fs(B)2477 2826 y Fr(2)2545 2812 y Fj(if)i(and)f(only)h(if)f Fs(p)h Fv(\021)f FA(1)0 2921 y(mo)s(d)k(4)p Fj(.)56 b(F)-8 b(or)34 b(eac)m(h)h(suc)m(h)g(prime,)g(w)m(e)g(obtain)e(b)m(y)j(b)s (osonization)31 b(from)j(Nic)m(hols)g(algebras)f(of)0 3031 y(dimension)d Fs(p)480 2998 y Fr(4)553 3031 y Fj(exactly)g FA(2\()p Fs(p)21 b Fv(\000)h FA(1\))31 b Fj(non-isomorphic)e(p)s(oin)m (ted)h(Hopf)i(algebras)d(of)j(dimension)0 3140 y Fs(p)46 3107 y Fr(5)117 3140 y Fj(with)d(coradical)e(of)j(dimension)e Fs(p)p Fj(.)110 3250 y(\(iv\).)38 b(There)25 b(exists)g(a)h(Nic)m(hols) f(algebra)f(with)h(Dynkin)g(diagram)g Fs(G)2478 3264 y Fr(2)2545 3250 y Fj(if)h(and)f(only)h(if)f Fs(p)h Fv(\021)f FA(1)0 3360 y(mo)s(d)k(3)p Fj(.)56 b(F)-8 b(or)34 b(eac)m(h)h(suc)m(h)g (prime,)g(w)m(e)g(obtain)e(b)m(y)j(b)s(osonization)31 b(from)j(Nic)m(hols)g(algebras)f(of)0 3469 y(dimension)d Fs(p)480 3436 y Fr(6)553 3469 y Fj(exactly)g FA(2\()p Fs(p)21 b Fv(\000)h FA(1\))31 b Fj(non-isomorphic)e(p)s(oin)m(ted)h (Hopf)i(algebras)d(of)j(dimension)0 3579 y Fs(p)46 3546 y Fr(7)117 3579 y Fj(with)d(coradical)e(of)j(dimension)e Fs(p)p Fj(.)110 3688 y(\(v\).)73 b(There)41 b(exist)f(Nic)m(hols)g (algebras)g(with)g(\014nite)f(Dynkin)i(diagram)f(of)h(rank)g Fv(\025)j FA(3)d Fj(if)0 3798 y(and)33 b(only)g(if)g Fs(p)e FA(=)g(3)i Fj(and)g(the)g(corresp)s(onding)c(Dynkin)k(diagram)f (is)h Fs(A)2511 3812 y Fr(2)2575 3798 y Fv(\002)23 b Fs(A)2737 3812 y Fr(1)2812 3798 y Fj(or)33 b Fs(A)2995 3812 y Fr(2)3059 3798 y Fv(\002)23 b Fs(A)3221 3812 y Fr(2)3262 3798 y Fj(.)0 3907 y(W)-8 b(e)38 b(obtain)e(resp)s(ectiv)m (ely)f(braided)h(Hopf)i(algebras)d(of)j(dimension)e FA(3)2493 3874 y Fr(4)2572 3907 y Fj(and)h(braided)f(Hopf)0 4017 y(algebras)42 b(of)i(dimension)e FA(3)980 3984 y Fr(6)1066 4017 y Fj(o)m(v)m(er)i Fu(Z)p Fs(=)p FA(\(3\))p Fj(;)j(hence,)f(w)m(e)f (get)f(exactly)e(4)j(Hopf)f(algebras)e(of)0 4127 y(dimension)f FA(3)491 4094 y Fr(5)574 4127 y Fj(and)i(2)g(Hopf)g(algebras)e(of)h (dimension)f FA(3)2066 4094 y Fr(7)2150 4127 y Fj(with)h(coradical)e (of)i(dimension)0 4236 y(3.)91 b Fi(\003)110 4384 y FA(These)25 b(Hopf)h(algebras)e(are)h(new;)j(but)d(the)g(Hopf)h(algebras)e(of)i (order)f(81)h(\()p Fs(p)f FA(=)g(3,)i(t)m(yp)s(e)f Fs(A)3211 4398 y Fr(2)3252 4384 y FA(\))0 4493 y(w)m(ere)i(kno)m(wn)g([N].)i(An)e (application)c(of)k(Theorem)f(1.3)h(is)f(the)g(determination)d(of)k (all)f(p)s(ossible)0 4603 y(\014nite)36 b(dimensional)e(p)s(oin)m(ted)h Ft(c)-5 b(or)g(adic)g(al)5 b(ly)39 b(gr)-5 b(ade)g(d)37 b FA(Hopf)h(algebras)d(of)i(dimension)e Fs(p)3027 4570 y Fr(4)3068 4603 y FA(.)63 b(See)0 4712 y(Theorem)29 b(7.1.)110 4822 y(Theorem)37 b(1.3)i(sa)m(ys)h(in)e(particular)e(that)h (an)m(y)j(\014nite)d(dimensional)e(coradically)h(graded)0 4932 y(Hopf)27 b(algebra)d(with)i(coradical)d(of)k(o)s(dd)e(prime)h (dimension)e(is)i(generated)e(b)m(y)k(sk)m(ew-primitiv)m(e)0 5041 y(and)i(group-lik)m(e)e(elemen)m(ts.)110 5151 y(In)i(general,)e(w) m(e)i(conjecture)p eop %%Page: 6 6 6 5 bop 0 -128 a Fy(6)696 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g(H.{J.)e (SCHNEIDER)0 91 y FC(Conjecture)f(1.4.)52 b Fj(Let)25 b Fs(A)g Fj(b)s(e)f(a)h(\014nite)e(dimensional)f(p)s(oin)m(ted)h(Hopf)i (algebra)d(o)m(v)m(er)k Fu(|)-10 b Fj(.)34 b(Then)0 201 y Fs(A)d Fj(is)f(generated)d(as)j(an)g(algebra)e(b)m(y)j(group-lik)m(e) d(and)i(sk)m(ew-primitiv)m(e)e(elemen)m(ts.)110 366 y FA(Finally)h(in)i(the)f(last)h(Section)e(w)m(e)j(giv)m(e)f(some)g(b)s (ounds)f(on)h(the)g(n)m(um)m(b)s(er)g(of)g(isomorphism)0 476 y(classes)h(of)h(Y)-8 b(etter-Drinfeld)30 b(mo)s(dules)i(of)h (\014nite)f(Cartan)g(t)m(yp)s(e)g(o)m(v)m(er)i(\014nite)e(ab)s(elian)e (groups)0 585 y(of)g(o)s(dd)f(order.)110 806 y(The)d(pap)s(er)f(is)h (organized)d(as)k(follo)m(ws:)37 b(W)-8 b(e)27 b(discuss)e(the)h (necessary)f(facts)h(concerning)d(the)0 915 y(t)m(wisting)h(op)s (eration)e(\(resp.,)27 b(F)-8 b(rob)s(enius-Lusztig)22 b(k)m(ernels\))i(in)i(Section)d(2)j(\(resp.,)h(Section)c(3\);)0 1025 y(w)m(e)34 b(pro)m(v)m(e)g(Theorem)e(1.1)i(in)f(Section)e(4.)51 b(Section)31 b(5)j(is)f(dev)m(oted)f(to)h(Nic)m(hols)g(algebras)f(o)m (v)m(er)0 1135 y Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))18 b(and)k(the)g(existence)f(part)h(of)h(Theorem)f(1.3.)38 b(W)-8 b(e)23 b(discuss)f(isomorphisms)f(b)s(et)m(w)m(een)g(dif-)0 1244 y(feren)m(t)28 b(b)s(osonizations)c(in)29 b(Section)d(6;)k(this)e (concludes)e(the)i(pro)s(of)g(of)h(Theorem)e(1.3.)40 b(Section)0 1354 y(7)26 b(is)f(dev)m(oted)g(to)g(p)s(oin)m(ted)e(Hopf)j (algebras)e(of)h(order)g Fs(p)1878 1321 y Fr(4)1919 1354 y FA(.)39 b(In)26 b(Section)d(8,)k(w)m(e)f(pro)m(v)m(e)g(b)s(ounds)e (on)0 1463 y(the)34 b(dimension)f(of)j(Y)-8 b(etter-Drinfeld)32 b(mo)s(dules)h(of)i(\014nite)f(Cartan)g(t)m(yp)s(e.)55 b(Our)35 b(con)m(v)m(en)m(tions)0 1573 y(are)29 b(mostly)h(standard)e (and)i(w)m(ere)f(already)g(used)g(in)h([AS2].)110 1738 y Ft(A)-5 b(dde)g(d)30 b(in)i(January)e(2000:.)46 b FA(Using)28 b(the)f(results)g(and)h(metho)s(ds)e(of)i(this)g(pap)s(er)e(w)m(e)j (mean-)0 1848 y(while)f(completely)g(classi\014ed)f (\014nite-dimensional)e(p)s(oin)m(ted)j(Hopf)i(algebras)e(with)h (coradical)0 1957 y Fu(|)-9 b FA(\000,)25 b(where)k(\000)h(is)g(a)g (\014nite)e(direct)g(sum)i(of)g(copies)e(of)i Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))25 b(and)30 b Fs(p)25 b(>)g FA(17)30 b(is)g(prime,)f(or)h(where)0 2067 y(\000)35 b(is)g(cyclic)e(of)h(prime)g(order)g Fs(>)f FA(7)i([AS3].)56 b(In)35 b(particular,)d(Conjecture)h(1.4)h(is)h(true)f(in)g(this)0 2176 y(case.)110 2286 y(In)c(an)g(earlier)e(v)m(ersion)i(of)h(this)e (pap)s(er)g(w)m(e)i(conjectured)c(that)j(up)g(to)g(isomorphism)e(there) 0 2396 y(are)j(only)f(\014nitely)f(man)m(y)j(\014nite-dimensional)26 b(Nic)m(hols)31 b(algebras)e(o)m(v)m(er)i(a)h(\014xed)f(\014nite)e (group)0 2506 y(\(of)22 b(o)s(dd)g(order\).)37 b(This)22 b(conjecture)f(is)h(sho)m(wn)h(to)g(b)s(e)f(true)g(b)m(y)i(Gra)s(~)-48 b(na)20 b([Gr2])k(as)f(a)g(consequence)0 2615 y(of)41 b(Theorem)e(3.1)i(\(2\))e(in)i(the)e(presen)m(t)h(pap)s(er)f(whic)m(h)h (follo)m(ws)f(from)i([L3].)73 b(Ov)m(er)41 b(ab)s(elian)0 2725 y(groups)33 b(of)h(ev)m(en)h(order)e(there)g(are)g(in\014nitely)f (man)m(y)i(\014nite-dimensional)c(Nic)m(hols)j(algebras)0 2834 y(b)m(y)e([N].)110 2999 y Ft(A)-5 b(cknow)5 b(le)-5 b(dgement.)47 b FA(W)-8 b(e)22 b(are)g(grateful)e(to)i(Eric)f(M)s(\177) -48 b(uller)20 b(for)i(crucial)e(con)m(v)m(ersations)g(ab)s(out)0 3109 y(F)-8 b(rob)s(enius-Lusztig)35 b(k)m(ernels.)67 b(W)-8 b(e)40 b(also)e(thank)h(Mat)-10 b(\023)-36 b(\020as)37 b(Gra)s(~)-48 b(na)37 b(for)i(man)m(y)h(con)m(v)m(ersations)0 3219 y(ab)s(out)33 b(Nic)m(hols)g(algebras;)i(Jorge)f(and)g(Juan)g (Guccione)e(for)i(enligh)m(tening)d(remarks)j(ab)s(out)0 3328 y(t)m(wisting,)28 b(and)i(Sorin)f(Dascalescu)e(for)j(in)m (teresting)d(con)m(v)m(ersations.)0 3549 y Fv(x)p FC(2.)46 b(Twisting.)110 3659 y FA(The)c(idea)f(of)i(t)m(wisting)d(w)m(as)j(in)m (tro)s(duced)c(b)m(y)44 b(Drinfeld)c([Dr2],)46 b(see)c(also)g([Dr1],)k ([Dr3].)0 3768 y(General)23 b(facts)i(are)g(discussed)f(in)h([Mj2],)i ([Mj3];)h(for)d(applications)c(to)26 b(quan)m(tized)d(en)m(v)m(eloping) 0 3878 y(algebras)j(see)i([Re],)i(for)d(applications)d(to)k(semisimple) e(Hopf)i(algebras)e(see)i([Nk].)42 b(W)-8 b(e)28 b(brie\015y)0 3987 y(discuss)36 b(the)g(general)e(idea)i(referring)e(to)i(the)g (literature)d(for)k(more)f(details)e(and)j(w)m(ork)g(out)0 4097 y(accurately)27 b(the)i(case)h(of)g(our)f(in)m(terest.)110 4317 y(Let)k Fs(A)h FA(b)s(e)f(a)h(Hopf)f(algebra.)49 b(Giv)m(en)33 b(an)g(in)m(v)m(ertible)e Fs(F)44 b Fv(2)31 b Fs(A)23 b Fv(\012)g Fs(A)p FA(,)36 b(w)m(e)e(can)e(consider)g(the)0 4427 y("t)m(wisted")c(com)m(ultiplication)c(\001)1157 4441 y Fp(F)1248 4427 y FA(giv)m(en)30 b(b)m(y)1220 4647 y(\001)1296 4661 y Fp(F)1357 4647 y FA(\()p Fs(x)p FA(\))25 b(=)g Fs(F)13 b FA(\001\()p Fs(x)p FA(\))p Fs(F)1946 4610 y Fo(\000)p Fr(1)2042 4647 y Fs(:)110 4822 y FA(Under)20 b(certain)e(conditions)f(on)k Fs(F)13 b FA(,)23 b Fs(A)1405 4836 y Fp(F)1487 4822 y FA(\(the)c(same)h(algebra)f Fs(A)i FA(but)f(with)g(com)m(ultiplication)0 4932 y(\001)76 4946 y Fp(F)137 4932 y FA(\))41 b(is)g(again)e(a)j(Hopf)f(algebra.)71 b(W)-8 b(e)42 b(are)e(in)m(terested)f(in)h(the)h(particular)c(case)k (when)f Fs(F)0 5041 y FA(b)s(elongs)28 b(indeed)h(to)h Fs(H)e Fv(\012)21 b Fs(H)7 b FA(,)32 b(where)e Fs(H)38 b FA(is)30 b(a)h(Hopf)g(subalgebra)d(of)i Fs(A)p FA(.)44 b(W)-8 b(e)30 b(further)g(assume)0 5151 y(that)j(the)h(inclusion)d Fs(H)40 b(,)-15 b Fv(!)33 b Fs(A)j FA(has)e(a)g(Hopf)h(algebra)d (retraction)f Fs(\031)36 b FA(:)d Fs(A)g Fv(!)g Fs(H)7 b FA(;)38 b(so)c Fs(A)i FA(is)e(the)p eop %%Page: 7 7 7 6 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)554 b(7)0 91 y FA(b)s(osonization)30 b(of)k Fs(R)i FA(b)m(y)f Fs(H)42 b FA(where)33 b Fs(R)j FA(is)e(a)g(braided)f(Hopf)h(algebra)e(in)i(the)g(category)3077 58 y Fp(H)3077 116 y(H)3146 91 y Fv(Y)8 b(D)0 201 y FA(of)30 b(Y)-8 b(etter-Drinfeld)27 b(mo)s(dules)i(o)m(v)m(er)h Fs(H)7 b FA(.)42 b(Then)30 b Fs(H)1759 215 y Fp(F)1850 201 y FA(is)g(a)g(Hopf)g(subalgebra)e(of)i Fs(A)2871 215 y Fp(F)2963 201 y FA(and)f Fs(\031)34 b FA(is)0 311 y(again)c(a)h(Hopf)g(algebra)e(map.)43 b(If)31 b Fs(H)1276 325 y Fp(F)1364 311 y Fv(')c Fs(H)39 b FA(\(this)29 b(happ)s(ens)g(for) i(instance)e(if)i Fs(H)38 b FA(is)31 b(ab)s(elian\),)0 420 y(then)e Fs(F)42 b FA(induces)28 b(an)i(auto)s(equiv)-5 b(alence)24 b Fs(V)46 b Fv(7!)26 b Fs(V)1691 434 y Fp(F)1782 420 y FA(of)k(the)f(braided)f(category)2740 387 y Fp(H)2740 445 y(H)2810 420 y Fv(Y)8 b(D)32 b FA(and)d Fs(A)3226 434 y Fp(F)0 530 y FA(is)h(in)f(fact)g(the)h(b)s(osonization)25 b(of)30 b Fs(R)1239 544 y Fp(F)1331 530 y FA(b)m(y)h Fs(H)1534 544 y Fp(F)1594 530 y FA(.)41 b(See)30 b([Mj2].)110 759 y(W)-8 b(e)29 b(no)m(w)f(assume)h(that)e Fs(H)33 b FA(=)25 b Fu(|)-9 b FA(\000)23 b(where)28 b(\000)h(is)g(a)g(\014nite) e(ab)s(elian)e(group.)39 b(W)-8 b(e)29 b(\014x)g(non-zero)0 868 y(elemen)m(ts)f Fs(y)s FA(\(1\))p Fs(;)15 b(:)g(:)g(:)i(;)e(y)s FA(\()p Fs(M)10 b FA(\))29 b(in)h(\000)g(suc)m(h)g(that)f(ord)14 b Fs(y)s FA(\()p Fs(`)p FA(\))25 b(=)g Fs(E)2080 882 y Fp(`)2116 868 y FA(,)0 1205 y(\(2.1\))147 b(\000)25 b(=)p Fs(<)g(y)s FA(\(1\))f Fs(>)i Fv(\010)15 b(\001)g(\001)g(\001)i (\010)25 b Fs(<)h(y)s FA(\()p Fs(M)10 b FA(\))24 b Fs(>;)107 b FA(and)90 b Fs(E)2024 1219 y Fp(`)2060 1205 y Fv(j)p Fs(E)2152 1219 y Fp(`)p Fr(+1)2310 1205 y FA(for)30 b Fs(`)c FA(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(M)31 b Fv(\000)20 b FA(1)p Fs(:)110 1432 y FA(Let)29 b Fs(q)34 b FA(b)s(e)c(a)g(ro)s(ot)f (of)h(unit)m(y)g(of)g(order)f Fs(E)1498 1446 y Fp(M)1580 1432 y FA(.)41 b(F)-8 b(or)30 b(all)f Fs(i)i FA(de\014ne)e(the)h(in)m (teger)e Fs(D)2799 1446 y Fp(i)2860 1432 y FA(b)m(y)j Fs(D)3063 1446 y Fp(i)3093 1432 y Fs(E)3160 1446 y Fp(i)3217 1432 y FA(=)0 1542 y Fs(E)67 1556 y Fp(M)148 1542 y FA(.)41 b(Then)30 b Fs(q)494 1556 y Fp(i)549 1542 y FA(=)c Fs(q)690 1509 y Fp(D)750 1518 y Fk(i)814 1542 y FA(is)j(a)h(ro)s(ot)f(of)h(unit) m(y)g(of)g(order)e Fs(E)1929 1556 y Fp(i)1960 1542 y FA(.)110 1782 y(Let)i Fs(\015)5 b FA(\()p Fs(i)p FA(\))26 b Fv(2)546 1759 y Fl(b)543 1782 y FA(\000)31 b(b)s(e)f(the)g(unique)f (c)m(haracter)g(suc)m(h)h(that)g Fv(h)p Fs(\015)5 b FA(\()p Fs(i)p FA(\))p Fs(;)15 b(y)s FA(\()p Fs(j)5 b FA(\))p Fv(i)25 b FA(=)h Fs(q)2612 1732 y Fp(\016)2645 1741 y Fk(ij)2609 1807 y Fp(i)2709 1782 y FA(,)31 b(for)g(1)26 b Fv(\024)g Fs(i;)15 b(j)33 b Fv(\024)0 1891 y Fs(M)10 b FA(.)41 b(Then)29 b(ord)14 b Fs(\015)5 b FA(\()p Fs(`)p FA(\))25 b(=)h Fs(E)901 1905 y Fp(`)967 1891 y FA(and)992 2095 y Fl(b)989 2119 y FA(\000)f(=)p Fs(<)g(\015)5 b FA(\(1\))25 b Fs(>)g Fv(\010)15 b(\001)g(\001)g(\001)i(\010)26 b Fs(<)f(\015)5 b FA(\()p Fs(M)10 b FA(\))24 b Fs(>)i(:)110 2311 y FA(Giv)m(en)j Fs(g)f Fv(2)e FA(\000,)k Fs(\037)c Fv(2)817 2288 y Fl(b)814 2311 y FA(\000)31 b(w)m(e)f(shall)f(use)h(the) f(notation)e Fs(g)1974 2325 y Fp(i)2004 2311 y FA(,)k Fs(\037)2117 2325 y Fp(j)2186 2311 y FA(meaning)d(that)0 2648 y(\(2.2\))218 b Fs(g)28 b FA(=)d Fs(g)619 2662 y Fr(1)660 2648 y Fs(y)s FA(\(1\))19 b(+)h Fv(\001)15 b(\001)g(\001)22 b FA(+)e Fs(g)1198 2662 y Fp(M)1279 2648 y Fs(y)s FA(\()p Fs(M)10 b FA(\))p Fs(;)197 b(\037)25 b FA(=)h Fs(\037)1956 2662 y Fr(1)1997 2648 y Fs(\015)5 b FA(\(1\))19 b(+)h Fv(\001)15 b(\001)g(\001)22 b FA(+)f Fs(\037)2553 2662 y Fp(M)2634 2648 y Fs(\015)5 b FA(\()p Fs(M)10 b FA(\))p Fs(:)110 2893 y FA(W)-8 b(e)29 b(iden)m(tify)e Fs(H)36 b FA(with)28 b(the)g(Hopf)h(algebra)e Fu(|)1672 2844 y Fh(b)1670 2860 y Fr(\000)1743 2893 y FA(of)i(functions)d(on)j(the)f (group)2781 2870 y Fl(b)2778 2893 y FA(\000)h(iden)m(tifying)0 3009 y(an)36 b(elemen)m(t)f(in)593 2986 y Fl(b)590 3009 y FA(\000)i(with)e(its)h(ev)-5 b(aluation;)36 b(w)m(e)h(denote)e(b)m(y) i Fs(\016)2121 3023 y Fp(\034)2202 3009 y Fv(2)f Fs(H)44 b FA(the)35 b(function)f(giv)m(en)i(b)m(y)0 3125 y Fs(\016)41 3139 y Fp(\034)86 3125 y FA(\()p Fs(\020)7 b FA(\))24 b(=)i Fs(\016)367 3139 y Fp(\034)s(;\020)467 3125 y FA(,)31 b Fs(\034)5 b(;)15 b(\020)33 b Fv(2)770 3102 y Fl(b)767 3125 y FA(\000.)41 b(Then)1214 3372 y Fs(\016)1255 3386 y Fp(\034)1326 3372 y FA(=)1464 3310 y(1)p 1433 3351 108 4 v 1433 3435 a Fv(j)p FA(\000)p Fv(j)1567 3285 y Fl(X)1567 3481 y Fp(g)r Fo(2)p Fr(\000)1700 3372 y Fv(h)p Fs(\034)5 b(;)15 b(g)1868 3334 y Fo(\000)p Fr(1)1966 3372 y Fv(i)p Fs(g)s(:)110 3652 y FA(Hence)29 b Fs(\020)7 b FA(\()p Fs(\016)505 3666 y Fp(\034)549 3652 y FA(\))25 b(=)h Fs(\016)748 3666 y Fp(\034)s(;\020)848 3652 y Fs(:)110 3881 y FA(W)-8 b(e)24 b(shall)e(describ)s(e)g(the)h(datum)g(for)h(t)m (wisting)e Fs(F)36 b FA(in)24 b(terms)f(of)h(2-co)s(cycles,)f(follo)m (wing)f([Nk],)0 3991 y([Mv].)41 b(The)28 b(theory)g(of)g(the)g(co)s (cycles)e(w)m(e)j(are)g(in)m(terested)c(in)k(go)s(es)e(bac)m(k)i(to)f (Sc)m(h)m(ur)g(and)g(there)0 4100 y(is)i(an)g(extensiv)m(e)e (literature)f(on)j(it;)f(w)m(e)i(refer)e(to)g([BT].)110 4223 y(W)-8 b(e)30 b(recall)e(that)h Fs(!)f FA(:)848 4200 y Fl(b)845 4223 y FA(\000)20 b Fv(\002)1017 4200 y Fl(b)1013 4223 y FA(\000)26 b Fv(!)g Fu(|)1269 4190 y Fo(\002)1355 4223 y FA(is)k(a)g(2-co)s(cycle)e(if)i Fs(!)s FA(\()p Fs(\034)5 b(;)15 b FA(1\))25 b(=)g Fs(!)s FA(\(1)p Fs(;)15 b(\034)10 b FA(\))25 b(=)g(1)31 b(and)0 4560 y(\(2.3\))457 b Fs(!)s FA(\()p Fs(\034)5 b(;)15 b(\020)7 b FA(\))p Fs(!)s FA(\()p Fs(\034)j(\020)d(;)15 b(\021)s FA(\))26 b(=)f Fs(!)s FA(\()p Fs(\034)5 b(;)15 b(\020)7 b(\021)s FA(\))p Fs(!)s FA(\()p Fs(\020)g(;)15 b(\021)s FA(\))31 b(for)f(all)e Fs(\034)5 b(;)15 b(\020)7 b(;)15 b(\021)31 b Fv(2)2562 4537 y Fl(b)2559 4560 y FA(\000)p Fs(:)110 4787 y FA(It)f(is)f(not)h(di\016cult)e(to)h(deduce)g (that)0 5012 y(\(2.4\))356 b Fs(!)s FA(\()p Fs(\034)10 b(\020)d(;)15 b(\037)p FA(\))p Fs(!)s FA(\()p Fs(\037;)g(\034)10 b(\020)d FA(\))1197 4975 y Fo(\000)p Fr(1)1295 5012 y Fs(!)s FA(\()p Fs(\020)g(;)15 b(\037)p FA(\))1571 4975 y Fo(\000)p Fr(1)1669 5012 y Fs(!)s FA(\()p Fs(\037;)g(\020)7 b FA(\))24 b(=)i Fs(!)s FA(\()p Fs(\034)5 b(;)15 b(\037)p FA(\))p Fs(!)s FA(\()p Fs(\037;)g(\034)10 b FA(\))2619 4975 y Fo(\000)p Fr(1)2717 5012 y Fs(:)p eop %%Page: 8 8 8 7 bop 0 -128 a Fy(8)696 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g(H.{J.)e (SCHNEIDER)0 91 y FC(Lemma)42 b(2.1.)52 b Fj(Let)36 b Fs(c)806 105 y Fp(ij)869 91 y Fj(,)j FA(1)d Fv(\024)g Fs(i)g(<)g(j)41 b Fv(\024)36 b Fs(M)10 b Fj(,)38 b(b)s(e)e(in)m(tegers) f(suc)m(h)h(that)f FA(0)h Fv(\024)g Fs(c)2765 105 y Fp(ij)2865 91 y Fs(<)g(E)3039 105 y Fp(i)3069 91 y Fj(.)60 b(Let)0 216 y Fs(!)28 b FA(:)139 193 y Fl(b)136 216 y FA(\000)21 b Fv(\002)308 193 y Fl(b)305 216 y FA(\000)k Fv(!)h Fu(|)561 182 y Fo(\002)647 216 y Fj(b)s(e)j(the)g(map)h(de\014ned)f(b)m(y)0 673 y FA(\(2.5\))453 b Fs(!)s FA(\()p Fs(\034)5 b(;)15 b(\037)p FA(\))25 b(=)1151 587 y Fl(Y)1038 783 y Fr(1)p Fo(\024)p Fp(i)25 b Fv(\000)p Fs(a)673 4460 y Fp(ij)738 4446 y FA(,)31 b(where)e Fs(N)1131 4460 y Fp(i)1192 4446 y FA(is)h(the)f(order)g(of)h Fs(b)1823 4460 y Fp(ii)1880 4446 y FA(,)g(for)g(all)f Fs(i;)15 b(j)5 b FA(.)110 4603 y(No)m(w)28 b FC(b)h FA(is)e(of)h (Cartan)e(t)m(yp)s(e)i(and)f(b)m(y)h(\(3.1\),)g(it)f(is)h(of)f(FL-t)m (yp)s(e)h(and)f(its)g(asso)s(ciated)d(Cartan)0 4712 y(matrix)34 b(is)g(\()p Fs(a)482 4726 y Fp(ij)546 4712 y FA(\),)j(with)c(the)h Fs(d)1066 4726 y Fp(i)1096 4712 y FA('s)i(as)e(ab)s(o)m(v)m(e.)55 b(Con)m(v)m(ersely)-8 b(,)36 b(an)m(y)f(braiding)d(of)j(FL-t)m(yp)s(e)f (suc)m(h)0 4822 y(that)29 b Fs(b)237 4836 y Fp(ij)332 4822 y FA(has)h(o)s(dd)f(order)g(for)h(all)e Fs(i;)15 b(j)37 b FA(arises)29 b(in)g(this)g(w)m(a)m(y)-8 b(.)110 4932 y(Our)25 b(aim)g(is)g(to)f(sk)m(etc)m(h)i(the)e(main)g(ideas)g(of) i(the)e(pro)s(of)g(of)h(the)g(follo)m(wing)d(Theorem,)k(whic)m(h)0 5041 y(follo)m(ws)k(from)i(deep)e(results)g(of)i(Lusztig)e(and)h(in)g (addition)d(from)k(the)f(w)m(ork)h(of)f(M)s(\177)-48 b(uller)28 b([M)s(\177)-48 b(u])0 5151 y(and)30 b(Rosso)g([Ro1].)p eop %%Page: 12 12 12 11 bop 0 -128 a Fy(12)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Theorem)35 b(3.1.)157 225 y FA(\(1\))54 b Fj(Assume)33 b(that)f FA(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))34 b Fj(is)f(of)g(\014nite)e(t)m(yp)s(e.)49 b(Then)32 b Fm(B)p FA(\()p Fs(V)20 b FA(\))33 b Fj(is)g (\014nite-dimensional.)45 b(As-)329 334 y(sume)40 b(moreo)m(v)m(er)g (that)f FA(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))41 b Fj(is)f(connected)d(and)j(if)g(the)f(corresp)s(onding)e(Dynkin)329 444 y(diagram)32 b(has)i(t)m(yp)s(e)f Fs(G)1136 458 y Fr(2)1211 444 y Fj(that)g(3)h(do)s(es)e(not)h(divide)g(the)g(order)g (of)g Fs(b)2694 458 y Fp(ii)2785 444 y Fj(for)h(all)e Fs(i)g Fv(2)g Fs(I)7 b Fj(.)329 554 y(Then)42 b Fm(B)p FA(\()p Fs(V)20 b FA(\))44 b Fj(is)f(isomorphic)d(to)j(the)g(p)s (ositiv)m(e)e(part)h(of)i(the)e(F)-8 b(rob)s(enius-Lusztig)329 663 y(k)m(ernel)29 b Fm(u)p Fj(,)i Ft(cf.)41 b FA([L2,)30 b(8.2])p Fj(,)h(and)1292 854 y FA(dim)14 b Fm(B)p FA(\()p Fs(V)20 b FA(\))25 b(=)g Fs(N)1889 817 y Fp(M)1971 854 y Fs(;)329 1042 y Fj(where)i Fs(M)38 b Fj(is)28 b(the)f(n)m(um)m(b)s (er)h(of)g(p)s(ositiv)m(e)e(ro)s(ots)h(of)h(the)f(ro)s(ot)g(system)h (of)g FA(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))29 b Fj(and)e Fs(N)329 1152 y Fj(is)j(the)f(order)g(of)h Fs(b)960 1166 y Fp(ii)1047 1152 y Fj(for)g(all)e Fs(i)e Fv(2)f Fs(I)7 b Fj(.)157 1261 y FA(\(2\))54 b Fj(Assume)29 b(that)e FA(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))29 b Fj(is)f(not)g(of)g (\014nite)f(t)m(yp)s(e.)40 b(Then)28 b Fm(B)p FA(\()p Fs(V)19 b FA(\))29 b Fj(is)f(in\014nite)e(dimensional.)0 1419 y Ft(Pr)-5 b(o)g(of.)46 b FA(\(1\))21 b(The)g(\014niteness)e(of)j (the)e(dimension)f(of)j Fm(B)p FA(\()p Fs(V)e FA(\))h(follo)m(ws)f (from)i(the)e(c)m(haracterization)0 1528 y(in)30 b([M)s(\177)-48 b(u,)31 b(Section)e(2])i(or)g([Ro1];)h(in)f(the)f(terminology)e(of)j(M) s(\177)-48 b(uller,)28 b Fs(q)35 b FA(\014ts)30 b(to)2663 1542 y Fp(k)2708 1528 y Fs(f)41 b FA(b)m(y)31 b(\(3.1\))g(and)0 1638 y([M)s(\177)-48 b(u,)30 b(2.9].)110 1755 y(If)d(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))29 b(is)e(connected,)f(then)g(there)g (are)h(in)m(tegers)f Fs(d)i FA(and)f(1)e Fv(\024)2306 1731 y FA(~)2290 1755 y Fs(d)2338 1769 y Fp(i)2394 1755 y Fv(\024)g FA(3)j(suc)m(h)f(that)f Fs(d)3010 1769 y Fp(i)3066 1755 y FA(=)f Fs(d)3226 1731 y FA(~)3210 1755 y Fs(d)3258 1769 y Fp(i)0 1880 y FA(for)i(all)e Fs(i)p FA(.)41 b(De\014ne)25 b Fs(l)j FA(=)d Fs(q)831 1847 y Fp(d)874 1880 y FA(.)40 b(Then)26 b(for)g(all)g Fs(i;)15 b(j)5 b FA(,)29 b Fs(b)1641 1894 y Fp(ij)1731 1880 y FA(=)d Fs(q)1872 1847 y Fp(d)1910 1856 y Fk(i)1939 1847 y Fp(a)1979 1856 y Fk(ij)2067 1880 y FA(=)f Fs(l)2207 1830 y Fr(~)2192 1847 y Fp(d)2230 1856 y Fk(i)2259 1847 y Fp(a)2299 1856 y Fk(ij)2362 1880 y FA(,)j(and)e Fs(b)2628 1894 y Fp(ii)2685 1880 y FA(,)i Fs(b)2777 1894 y Fp(j)t(j)2877 1880 y FA(and)e Fs(l)k FA(ha)m(v)m(e)0 1989 y(the)e(same)g(order)g(b)m (y)i(our)e(assumption)f(ab)s(out)g Fs(G)1722 2003 y Fr(2)1762 1989 y FA(.)41 b(Hence)28 b(the)g(dimension)e(form)m(ula)i(follo)m(ws)0 2099 y(from)i([L2,)h(8.3,)f(8.4].)110 2208 y(W)-8 b(e)34 b(sk)m(etc)m(h)f(no)m(w)h(the)f(pro)s(of)g(of)g(P)m(art)h(\(2\).)51 b(W)-8 b(e)34 b(k)m(eep)f(the)g(notation)e(from)j([L3].)52 b(It)34 b(is)g(in)0 2318 y(principle)26 b(not)j(alw)m(a)m(ys)h(true)e (that)g(the)h(p)s(ositiv)m(e)e(part)i Fm(u)1957 2285 y Fr(+)2047 2318 y FA(of)h(the)f(F)-8 b(rob)s(enius-Lusztig)25 b(k)m(ernel)0 2428 y(is)30 b(isomorphic)d(to)j Fm(B)p FA(\()p Fs(V)20 b FA(\).)40 b(W)-8 b(e)30 b(ha)m(v)m(e)h(ho)m(w)m(ev)m (er)f(an)g(epimorphism)e Fm(u)2393 2395 y Fr(+)2479 2428 y Fv(!)e Fm(B)p FA(\()p Fs(V)20 b FA(\).)110 2585 y(Since)g(\()p Fs(I)7 b(;)15 b Fv(\001)p FA(\))24 b(is)e(not)g(of)g(\014nite)f(t)m(yp) s(e,)j(its)e(W)-8 b(eyl)22 b(group)f(is)i(not)e(\014nite.)37 b(Hence,)24 b(for)e(an)m(y)h Fs(M)35 b(>)25 b FA(0)0 2695 y(there)39 b(exists)h(an)g(elemen)m(t)f Fs(w)46 b Fv(2)c Fs(W)54 b FA(of)41 b(length)d Fs(M)10 b FA(,)44 b(sa)m(y)d(with)f(reduced)e(expression)h Fs(w)46 b FA(=)0 2804 y Fs(s)43 2818 y Fp(i)69 2827 y Ff(1)124 2804 y Fs(:)15 b(:)g(:)i(s)289 2818 y Fp(i)315 2827 y Fk(M)387 2804 y FA(.)55 b(W)-8 b(e)35 b(shall)e(sho)m(w)i(that)e Fm(B)p FA(\()p Fs(V)20 b FA(\))35 b(con)m(tains)d(2)1949 2771 y Fp(M)2065 2804 y FA(linearly)g(indep)s(enden)m(t)f(elemen)m(ts;) 0 2914 y(this)e(concludes)f(the)h(pro)s(of)g(of)h(\(2\).)110 3023 y(W)-8 b(e)33 b(w)m(ork)h(\014rst)e(in)h(the)g(transcenden)m(tal)c (case.)49 b(W)-8 b(e)34 b(use)f(no)m(w)g(sp)s(eci\014cally)c(the)k (notation)0 3133 y(from)43 b([L3,)j(38.2.2].)79 b(The)43 b(sequence)e FC(h)47 b FA(=)f(\()p Fs(i)1672 3147 y Fr(1)1713 3133 y Fs(;)15 b(:)g(:)g(:)i(;)e(i)1946 3147 y Fp(M)2029 3133 y FA(\))42 b(is)h(admissible)d(b)s(ecause)g(of)j([L3,)0 3243 y(Prop)s(osition)36 b(40.2.1)k(\(a\)].)69 b(It)40 b(is)f(clear)f(that)h(the)g(elemen)m(t)f Fs(x)k FA(=)f(1)f(is)g (adapted)d(to)j(\()p FC(h)p Fs(;)15 b FA(0\).)0 3352 y(Let)33 b Fs(c)p FA(,)g Fs(c)305 3319 y Fo(0)364 3352 y FA(b)s(e)f(as)h([L3,)i(Prop)s(osition)29 b(38.2.3])k(and)g(supp)s (ose)e(that)h(their)f(en)m(tries)h(are)g(either)f(0)0 3462 y(or)j(1.)54 b(Then)34 b Fs(L)p FA(\()p FC(h)p Fs(;)15 b(c;)g FA(0)p Fs(;)g FA(1\))35 b(is)f(orthogonal)d(\(resp,)k(not)f (orthogonal\))c(to)k Fs(L)p FA(\()p FC(h)p Fs(;)15 b(c)2795 3429 y Fo(0)2821 3462 y Fs(;)g FA(0)p Fs(;)g FA(1\))35 b(if)f Fs(c)g FA(is)0 3571 y(di\013eren)m(t)c(\(resp.,)i(equal\))e(to)h Fs(c)1075 3538 y Fo(0)1100 3571 y FA(.)45 b(But)31 b(w)m(e)h(can)f (pass)g(from)g(the)g(transcenden)m(tal)d(case)i(to)i(our)0 3681 y(case)j(\()p Fs(q)j FA(a)d(ro)s(ot)f(of)h(unit)m(y\))g(thanks)f (to)h(the)g(results)e(in)i([M)s(\177)-48 b(u,)36 b(Section)d(2],)38 b(notably)33 b(Lemma)0 3791 y(2.2,)e(Prop)s(osition)26 b(2.3)k(and)g(Theorem)f(2.11)g(\(a\).)91 b Fi(\003)0 3996 y Fv(x)p FC(4.)46 b(Pro)s(of)34 b(of)h(Theorem)g(1.1.)110 4153 y Ft(4.1)d(Sketch)h(of)f(the)h(pr)-5 b(o)g(of.)110 4263 y FA(Let)25 b(us)g(\014rst)g(outline)e(the)i(pro)s(of)f(of)i (Theorem)e(1.1.)39 b(Let)26 b Fs(V)46 b FA(b)s(e)25 b(a)g(Y)-8 b(etter-Drinfeld)23 b(mo)s(dule)0 4372 y(o)m(v)m(er)32 b(our)e(\014xed)h(group)f(\000)h(as)g(in)g(\(1.5\))f(and)g(let)g Fs(R)e FA(=)f Fm(B)p FA(\()p Fs(V)20 b FA(\).)43 b(W)-8 b(e)32 b(b)s(egin)d(b)m(y)j(passing)d(from)i(a)0 4482 y(general)e(braiding)g(\(1.7\))h(to)h(a)h(symmetric)d(one.)44 b(It)31 b(is)h(natural)d(to)i(try)g(to)g(apply)f(a)h(Drinfeld)0 4591 y(t)m(wist)i(to)g(the)g(Hopf)h(algebra)e Fu(|)-9 b FA(\000#)p Fs(R)29 b FA(\(compare)j(with)g([Ro2]\).)52 b(Unfortunately)31 b(this)i(is)h(not)0 4701 y(alw)m(a)m(ys)29 b(p)s(ossible.)37 b(It)29 b(fails)e(already)h(for)g(quan)m(tum)g (linear)f(planes)g(o)m(v)m(er)i(cyclic)e(groups,)h(since)0 4811 y(no)20 b(t)m(wist)g(is)g(p)s(ossible)e(in)i(the)g(cyclic)e(group) i(case.)37 b(Ho)m(w)m(ev)m(er,)23 b(w)m(e)e(can)f(o)m(v)m(ercome)g (this)g(di\016cult)m(y)0 4920 y(b)s(ecause)26 b(the)h(algebra)f(and)i (coalgebra)d(structures)h(of)i Fs(R)i FA(do)e(not)f(dep)s(end)g(on)h (\000)g(but)g(only)f(on)0 5030 y(the)g(braiding)f(\(1.7\).)39 b(That)27 b(is,)i(let)e(\007)h(b)s(e)f(another)f(\014nite)h(ab)s(elian) e(group)i(and)g(let)g Fs(h)p FA(\(1\),)g Fs(:)15 b(:)g(:)32 b FA(,)0 5151 y Fs(h)p FA(\()p Fs(\022)s FA(\))23 b Fv(2)j FA(\007,)31 b Fs(\021)s FA(\(1\),)f Fs(:)15 b(:)g(:)32 b FA(,)e Fs(\021)s FA(\()p Fs(\022)s FA(\))25 b Fv(2)1108 5128 y Fl(b)1098 5151 y FA(\007)31 b(b)s(e)e(sequences)f(satisfying)p eop %%Page: 13 13 13 12 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(13)993 221 y Fv(h)p Fs(\021)s FA(\()p Fs(i)p FA(\))p Fs(;)15 b(h)p FA(\()p Fs(j)5 b FA(\))p Fv(i)24 b FA(=)h Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(j)5 b FA(\))p Fv(i)p Fs(;)105 b Fv(8)p Fs(i;)15 b(j:)110 422 y FA(Let)34 b Fs(T)48 b FA(b)s(e)34 b(the)g(Nic)m(hols)g(algebra)f(o)m(v)m(er)i (\007)h(suc)m(h)e(that)g Fs(T)13 b FA(\(1\))32 b(=)i Fs(V)55 b FA(with)34 b(Y)-8 b(etter-Drinfeld)0 532 y(mo)s(dule)38 b(structure)g(giv)m(en)h(b)m(y)i(\(1.5\))e(but)g(with)g Fs(h)p FA(\()p Fs(i)p FA(\)'s)h(and)f Fs(\021)s FA(\()p Fs(j)5 b FA(\)'s)41 b(instead)d(of)h Fs(g)s FA(\()p Fs(i)p FA(\)'s)i(and)0 641 y Fs(\037)p FA(\()p Fs(j)5 b FA(\)'s.)43 b(Then)30 b Fs(T)45 b FA(is)31 b(isomorphic)d(to)i Fs(R)j FA(b)s(oth)c(as)i(an)g(algebra)e(and)h(coalgebra.)41 b(This)30 b(follo)m(ws)0 751 y(at)g(once)f(from)h(the)f(description)d (of)k Fs(T)44 b FA(and)29 b Fs(R)j FA(in)e(terms)f(of)h("quan)m(tum)f (an)m(tisymmetrizers";)0 860 y(see)g(for)g(instance)e([Sbg],)j([Ro2],)g ([A)m(G].)i(So)d(w)m(e)h(c)m(hange)e(the)g(group,)h(t)m(wist)g(and)f (assume)h(that)0 970 y(the)g(braiding)f(\(1.7\))h(is)h(symmetric.)110 1155 y(W)-8 b(e)21 b(then)g(discuss)f(brie\015y)h(the)f(relations)f(b)s (et)m(w)m(een)h(the)h(notions)f(of)h(Cartan)f(t)m(yp)s(e,)j(FL-t)m(yp)s (e)0 1264 y(and)35 b(lo)s(cally)d(of)j(FL-t)m(yp)s(e.)55 b(W)-8 b(e)36 b(state)d(the)i(last)f(one)g(in)h(arithmetical)c(terms)j (and)h(describ)s(e)0 1374 y(sev)m(eral)29 b(instances)f(where)h(it)g (holds.)40 b(One)29 b(of)h(them)g(is)g(the)f(follo)m(wing:)110 1490 y(Let)36 b FC(b)g FA(=)h(\()p Fs(b)557 1504 y Fp(ij)621 1490 y FA(\))f(b)s(e)g(a)h(connected)d(braiding)g(of)i(Cartan)g(t)m(yp) s(e)g(whose)g(en)m(tries)f(ha)m(v)m(e)i(o)s(dd)0 1600 y(order.)g(W)-8 b(e)23 b(sa)m(y)g(that)f FC(b)h FA(satis\014es)e(the)h Ft(r)-5 b(elative)26 b(primeness)g(c)-5 b(ondition)23 b FA(if)f Fs(a)2574 1614 y Fp(ij)2662 1600 y FA(is)g(0)h(or)g(relativ)m (ely)0 1709 y(prime)29 b(to)h(the)f(order)g(of)h Fs(b)910 1723 y Fp(ii)997 1709 y FA(for)g(all)e Fs(i;)15 b(j)5 b FA(.)110 1826 y(W)-8 b(e)31 b(pro)m(v)m(e)g(then)f(that)f(a)i (braiding)d(of)j(symmetrizable)d(Cartan)h(t)m(yp)s(e)i(satisfying)d (the)j(rel-)0 1935 y(ativ)m(e)e(primeness)g(condition)d(is)k(of)g(FL-t) m(yp)s(e.)40 b(By)31 b(the)e(results)f(ev)m(ok)m(ed)j(in)e(Section)f (3,)i(this)f(is)0 2045 y(enough)g(for)g(part)g(\(i\))g(of)h(Theorem)f (1.1.)110 2161 y(F)-8 b(or)38 b(the)g(con)m(v)m(erse)g(part)g(\(ii\))e (of)j(Theorem)f(1.1,)j(w)m(e)e(pro)m(v)m(e)g(that)e(if)h Fm(B)p FA(\()p Fs(V)20 b FA(\))39 b(is)f(\014nite)f(di-)0 2271 y(mensional)31 b(and)j FC(b)g FA(is)g(lo)s(cally)c(of)k(FL-t)m(yp) s(e)f(then)g(it)g(is)g(symmetrizable.)49 b(By)34 b(the)f(results)f(of)0 2380 y(Section)c(3)i(again,)f(this)g(concludes)e(the)i(pro)s(of)g(of)h (the)f(theorem.)110 2634 y Ft(4.2)j(R)-5 b(e)g(duction)33 b(to)f(the)h(symmetric)g(c)-5 b(onne)g(cte)g(d)34 b(c)-5 b(ase.)110 2750 y FA(Let)30 b(\()p Fs(b)349 2764 y Fp(ij)413 2750 y FA(\))449 2764 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)753 2750 y FA(b)s(e)g(a)h(matrix)f(with)f(en)m(tries)h(in)g Fu(|)1919 2717 y Fo(\002)1974 2750 y FA(.)44 b(Let)30 b Fs(V)52 b FA(b)s(e)30 b(a)h(v)m(ector)e(space)h(with)g(a)0 2860 y(basis)f Fs(x)276 2874 y Fr(1)317 2860 y Fs(;)15 b(:)g(:)g(:)j(;)d(x)572 2874 y Fp(\022)644 2860 y FA(and)30 b(let)f Fs(c)c FA(:)g Fs(V)41 b Fv(\012)21 b Fs(V)46 b Fv(!)25 b Fs(V)41 b Fv(\012)20 b Fs(V)51 b FA(b)s(e)30 b(the)f(linear)f(map)i(giv)m(en)f(b)m(y)0 3215 y(\(4.1\))699 b Fs(c)p FA(\()p Fs(x)1016 3229 y Fp(i)1066 3215 y Fv(\012)20 b Fs(x)1209 3229 y Fp(j)1248 3215 y FA(\))k(=)i Fs(b)1444 3229 y Fp(ij)1508 3215 y Fs(x)1560 3229 y Fp(j)1619 3215 y Fv(\012)21 b Fs(x)1763 3229 y Fp(i)1793 3215 y Fs(;)107 b FA(1)25 b Fv(\024)h Fs(i;)15 b(j)31 b Fv(\024)25 b Fs(\022)s(:)110 3462 y FA(It)30 b(is)f(easy)h(to)g(v)m(erify)g(that)f Fs(c)h FA(satis\014es)e(the)h(Braid)g(relation:)714 3708 y(\()p Fs(c)19 b Fv(\012)h FA(id)o(\)\(id)14 b Fv(\012)p Fs(c)p FA(\)\()p Fs(c)k Fv(\012)i FA(id)o(\))25 b(=)h(\(id)14 b Fv(\012)p Fs(c)p FA(\)\()p Fs(c)k Fv(\012)i FA(id)o(\)\(id)14 b Fv(\012)p Fs(c)p FA(\))p Fs(:)110 3909 y FA(Therefore,)22 b(it)g(induces)e(represen)m(tations)f(of)j(the)g(Braid)f(group)g Fu(B)2367 3923 y Fp(n)2446 3909 y FA(on)h Fs(V)2639 3876 y Fo(\012)p Fp(n)2768 3909 y FA(for)h(an)m(y)f Fs(n)k Fv(\025)f FA(2)0 4018 y(and)i(this)f(in)h(turn)f(allo)m(ws)g(to)h (de\014ne)f("braided)e(an)m(tisymmetrizers")g(whic)m(h)j(are)f(linear)f (maps)0 4128 y Fm(S)76 4142 y Fp(n)165 4128 y FA(:)40 b Fs(V)304 4095 y Fo(\012)p Fp(n)450 4128 y Fv(!)g Fs(V)655 4095 y Fo(\012)p Fp(n)762 4128 y FA(.)66 b(Let)39 b Fs(R)h FA(=)g Fv(\010)1317 4142 y Fp(n)p Fo(\025)p Fr(0)1460 4128 y Fs(V)1534 4095 y Fo(\012)p Fp(n)1641 4128 y Fs(=)15 b FA(k)m(er)g Fm(S)1915 4142 y Fp(n)1964 4128 y FA(;)44 b(it)38 b(is)g(a)h(graded)e(algebra.)64 b(See)38 b(for)0 4238 y(instance)26 b([Sbg],)k([A)m(G].)h(Using)d(the)g(metho)s(ds)f(of) i([Mj3,)h(Th.)40 b(10.2.1])29 b(it)f(is)g(p)s(ossible)e(to)j(pro)m(v)m (e)0 4347 y(that)39 b Fs(R)i FA(is)f(a)g(braided)e(Hopf)i(algebra.)69 b(In)40 b(our)g(case)f(this)g(will)f(follo)m(w)h(from)h(Lemma)g(4.1)0 4457 y(b)s(elo)m(w.)110 4710 y(If)32 b(\000)h(is)f(an)g(ab)s(elian)e (group,)i Fs(g)s FA(\(1\),)g Fs(:)15 b(:)g(:)32 b FA(,)h Fs(g)s FA(\()p Fs(\022)s FA(\))28 b Fv(2)h FA(\000,)34 b Fs(\037)p FA(\(1\),)d Fs(:)15 b(:)g(:)32 b FA(,)i Fs(\037)p FA(\()p Fs(\022)s FA(\))28 b Fv(2)2662 4687 y Fl(b)2659 4710 y FA(\000)33 b(are)f(sequences)0 4819 y(suc)m(h)g(that)e Fs(b)446 4833 y Fp(ij)539 4819 y FA(=)e Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)30 b FA(and)h(w)m(e)h(consider)e Fs(V)52 b FA(as)32 b(a)g(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)g(o)m(v)m(er)k (\000)0 4932 y(via)25 b(\(1.5\),)i(then)d Fs(c)i FA(is)g(the)f("comm)m (utativit)m(y)e(isomorphism")g(in)i(the)g(braided)f(category)3071 4898 y Fr(\000)3071 4957 y(\000)3121 4932 y Fv(Y)8 b(D)r FA(.)0 5041 y(Moreo)m(v)m(er,)26 b Fs(R)h FA(has)e(a)g(braided)e(Hopf)j (algebra)d(structure)g(and)h(is)h(in)g(fact)f(the)h(Nic)m(hols)f (algebra)0 5151 y(asso)s(ciated)j(to)i Fs(V)21 b FA(.)p eop %%Page: 14 14 14 13 bop 0 -128 a Fy(14)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Lemma)36 b(4.1.)51 b Fj(Let)31 b FA(\()p Fs(b)829 105 y Fp(ij)893 91 y FA(\))g Fj(b)s(e)f(a)h (braiding)e(of)i(Cartan)f(t)m(yp)s(e)g(and)h(let)f(us)h(assume)f(that)g Fs(b)3130 105 y Fp(ij)3226 91 y Fj(is)0 201 y(a)37 b(ro)s(ot)e(of)i(1)g (of)f(o)s(dd)g(order)g(for)g(an)m(y)h Fs(i;)15 b(j)5 b Fj(.)62 b(Then)37 b(there)e(exists)h(a)h(\014nite)e(ab)s(elian)f (group)h FA(\000)p Fj(,)0 317 y(sequences)g Fs(g)s FA(\(1\))p Fj(,)h Fs(:)15 b(:)g(:)32 b Fj(,)39 b Fs(g)s FA(\()p Fs(\022)s FA(\))34 b Fv(2)i FA(\000)p Fj(,)j Fs(\037)p FA(\(1\))p Fj(,)d Fs(:)15 b(:)g(:)32 b Fj(,)39 b Fs(\037)p FA(\()p Fs(\022)s FA(\))c Fv(2)2012 294 y Fl(b)2009 317 y FA(\000)i Fj(and)f Fs(F)48 b Fv(2)36 b Fu(|)-9 b FA(\000)18 b Fv(\012)25 b Fu(|)-9 b FA(\000)31 b Fj(giv)m(en)36 b(b)m(y)h(a)0 438 y(2-co)s(cycle)28 b(of)498 415 y Fl(b)495 438 y FA(\000)i Fj(as)g(in)g(Lemma)g(2.2)g(suc)m(h)g(that)178 575 y FA(\(i\))53 b Fj(for)30 b FA(1)25 b Fv(\024)g Fs(i;)15 b(j)31 b Fv(\024)26 b Fs(\022)s Fj(,)k(w)m(e)g(ha)m(v)m(e)0 907 y FA(\(4.2\))1114 b Fs(b)1342 921 y Fp(ij)1431 907 y FA(=)26 b Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)p FA(;)152 1128 y(\(ii\))53 b Fj(the)35 b(braiding)f(\(1.7\))h(corresp)s(onding)d (to)k(the)f(Nic)m(hols)g(algebra)f Fs(R)2678 1142 y Fp(F)2776 1128 y Fj(is)h(symmetric)329 1238 y(and)28 b(of)h(Cartan)f(t)m(yp)s(e,) h(with)f(the)g(same)h(asso)s(ciated)d(Cartan)h(matrix)h(and)h(diagonal) 329 1347 y(elemen)m(ts.)110 1485 y(Here)j Fs(R)i Fj(is)e(the)g(Nic)m (hols)f(algebra)g(asso)s(ciated)e(to)j(the)g(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)i Fs(V)53 b Fj(with)0 1594 y(basis)29 b Fs(x)276 1608 y Fr(1)317 1594 y Fs(;)15 b(:)g(:)g(:)j(;)d(x)572 1608 y Fp(\022)644 1594 y Fj(and)30 b(structure)e(de\014ned)g(b)m(y)j (\(1.5\).)0 1761 y Ft(Pr)-5 b(o)g(of.)46 b FA(\(i\):)k(W)-8 b(e)35 b(de\014ne)f(\000)g(:=)p Fs(<)g(y)s FA(\(1\))f Fs(>)h Fv(\010)15 b(\001)g(\001)g(\001)i(\010)34 b Fs(<)g(y)s FA(\()p Fs(\022)s FA(\))e Fs(>)p FA(,)37 b(where)e(w)m(e)g(imp)s(ose)f (that)g(the)0 1870 y(order)e Fs(E)309 1884 y Fp(j)382 1870 y FA(of)i Fs(y)s(j)39 b FA(is)33 b(the)g(least)f(common)g(m)m (ultiple)g(of)h(the)g(orders)f(of)i Fs(b)2511 1884 y Fp(i`)2572 1870 y FA(,)h(for)e(all)g Fs(i)p FA(,)i(and)e(for)0 1980 y Fs(`)41 b Fv(\024)g Fs(j)5 b FA(.)68 b(Then)39 b Fs(E)642 1994 y Fp(j)681 1980 y Fv(j)p Fs(E)773 1994 y Fp(j)t Fr(+1)905 1980 y FA(:)59 b(\(2.1\))38 b(holds.)68 b(Let)39 b Fs(q)k Fv(2)e Fs(k)h FA(b)s(e)d(a)g(ro)s(ot)f(of)h(1)h(of)f (order)f Fs(E)3019 1994 y Fp(\022)3061 1980 y FA(.)68 b(Let)0 2139 y Fs(D)76 2153 y Fp(i)144 2139 y FA(=)264 2078 y Fs(E)331 2092 y Fp(\022)p 264 2118 109 4 v 269 2202 a Fs(E)336 2216 y Fp(i)384 2139 y FA(,)40 b(1)e Fv(\024)g Fs(i)g Fv(\024)g Fs(\022)j FA(and)c(let)g Fs(q)1270 2153 y Fp(i)1338 2139 y FA(=)h Fs(q)1491 2106 y Fp(D)1551 2115 y Fk(i)1584 2139 y FA(.)64 b(Then)37 b Fs(q)1960 2153 y Fp(i)2028 2139 y FA(is)h(a)f(ro)s(ot)f(of)i(1)g(of)f(order)g Fs(E)3035 2153 y Fp(i)3066 2139 y FA(.)63 b(Let)0 2319 y Fs(\015)5 b FA(\()p Fs(i)p FA(\))37 b Fv(2)292 2296 y Fl(b)289 2319 y FA(\000)h(b)s(e)e(the)g(unique)f(c)m(haracter)g(suc)m (h)i(that)e Fv(h)p Fs(\015)5 b FA(\()p Fs(i)p FA(\))p Fs(;)15 b(y)s FA(\()p Fs(j)5 b FA(\))p Fv(i)36 b FA(=)g Fs(q)2422 2269 y Fp(\016)2455 2278 y Fk(ij)2419 2344 y Fp(i)2519 2319 y FA(,)j(for)d(1)h Fv(\024)g Fs(i;)15 b(j)42 b Fv(\024)37 b Fs(N)10 b FA(.)0 2442 y(Then)32 b(ord)15 b Fs(\015)5 b FA(\()p Fs(`)p FA(\))29 b(=)i Fs(E)749 2456 y Fp(`)817 2442 y FA(and)1001 2419 y Fl(b)998 2442 y FA(\000)f(=)p Fs(<)g(\015)5 b FA(\(1\))30 b Fs(>)g Fv(\010)15 b(\001)g(\001)g(\001)i(\010)30 b Fs(<)g(\015)5 b FA(\()p Fs(M)10 b FA(\))30 b Fs(>)p FA(.)49 b(W)-8 b(e)33 b(c)m(ho)s(ose)f Fs(\037)p FA(\()p Fs(j)5 b FA(\))29 b(=)h Fs(\015)5 b FA(\()p Fs(j)g FA(\))0 2552 y(for)35 b(all)g Fs(j)41 b FA(and)35 b(de\014ne)f Fs(g)s FA(\()p Fs(i)p FA(\))h(b)m(y)i(\(4.2\);)h(it)c(mak)m(es)i(sense)f(b)s(ecause)f (the)h(order)f(of)i Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)0 2661 y FA(divides)28 b Fs(E)374 2675 y Fp(j)444 2661 y FA(for)i(all)e Fs(i;)15 b(j)5 b FA(.)110 2828 y(\(ii\):)49 b(Let)35 b Fs(\013)535 2842 y Fp(ij)636 2828 y FA(b)s(e)g(giv)m(en)f(b)m(y)j (\(2.9\).)56 b(W)-8 b(e)35 b(are)g(lo)s(oking)e(for)i(in)m(tegers)f Fs(c)2586 2842 y Fp(ij)2686 2828 y FA(suc)m(h)h(that,)h(if)f Fs(F)0 2940 y FA(is)k(asso)s(ciated)d(to)j(the)f(co)s(cycle)f Fs(!)43 b FA(giv)m(en)c(b)m(y)h(\(2.5\),)h(then)d(the)g(matrix)h Fs(\013)2643 2907 y Fp(F)2643 2964 y(ij)2747 2940 y FA(is)g(symmetric.) 0 3054 y(Assume)30 b(that)f Fs(i)d(<)g(j)5 b FA(.)40 b(By)31 b(\(2.10\))d(and)i(our)g(preceding)d(c)m(hoices,)i(w)m(e)h(ha)m (v)m(e)1251 3275 y Fs(\013)1309 3289 y Fp(ij)1399 3275 y FA(=)c Fs(D)1572 3289 y Fp(i)1602 3275 y Fs(g)s FA(\()p Fs(j)5 b FA(\))1764 3289 y Fp(i)1793 3275 y Fs(;)1251 3429 y(\013)1309 3392 y Fp(F)1309 3452 y(ij)1399 3429 y FA(=)26 b Fv(\000)p Fs(D)1643 3443 y Fp(i)1673 3429 y Fs(c)1713 3443 y Fp(ij)1797 3429 y FA(+)20 b Fs(\013)1946 3443 y Fp(ij)2011 3429 y Fs(;)1251 3584 y(\013)1309 3546 y Fp(F)1309 3606 y(j)t(i)1399 3584 y FA(=)26 b Fs(D)1572 3598 y Fp(i)1602 3584 y Fs(c)1642 3598 y Fp(ij)1726 3584 y FA(+)20 b Fs(\013)1875 3598 y Fp(j)t(i)1940 3584 y FA(;)0 3818 y(hence)29 b Fs(\013)312 3785 y Fp(F)312 3842 y(ij)402 3818 y Fv(\021)c Fs(\013)556 3785 y Fp(F)556 3842 y(j)t(i)682 3818 y FA(mo)s(d)k Fs(E)954 3832 y Fp(\022)1026 3818 y FA(if)h(and)g(only)f(if)0 4043 y(\(4.3\))456 b(2)p Fs(D)767 4057 y Fp(i)797 4043 y Fs(c)837 4057 y Fp(ij)926 4043 y Fv(\021)25 b Fs(\013)1080 4057 y Fp(ij)1165 4043 y Fv(\000)c Fs(\013)1315 4057 y Fp(j)t(i)1405 4043 y Fv(\021)k Fs(D)1577 4057 y Fp(i)1607 4043 y Fs(g)s FA(\()p Fs(j)5 b FA(\))1769 4057 y Fp(i)1819 4043 y Fv(\000)20 b Fs(D)1986 4057 y Fp(j)2024 4043 y Fs(g)s FA(\()p Fs(i)p FA(\))2174 4057 y Fp(j)2303 4043 y FA(mo)s(d)30 b Fs(E)2576 4057 y Fp(\022)2617 4043 y Fs(:)110 4265 y FA(W)-8 b(e)36 b(solv)m(e)f(\(4.3\).)58 b(The)35 b(factor)g(2)g(can)h(b)s(e)f(ignored) f(b)s(ecause)g Fs(E)2362 4279 y Fp(\022)2440 4265 y FA(is)h(o)s(dd.)57 b(That)36 b(is,)h(it)e(is)0 4374 y(enough)29 b(to)g(solv)m(e)0 4706 y(\(4.4\))718 b Fs(D)983 4720 y Fp(i)1013 4706 y Fl(e)-51 b Fs(c)1053 4720 y Fp(ij)1142 4706 y Fv(\021)26 b Fs(D)1315 4720 y Fp(i)1345 4706 y Fs(g)s FA(\()p Fs(j)5 b FA(\))1507 4720 y Fp(i)1556 4706 y Fv(\000)21 b Fs(D)1724 4720 y Fp(j)1762 4706 y Fs(g)s FA(\()p Fs(i)p FA(\))1912 4720 y Fp(j)2041 4706 y FA(mo)s(d)29 b Fs(E)2313 4720 y Fp(\022)2355 4706 y Fs(:)110 4928 y FA(No)m(w,)i(since)e Fs(g)s FA(\()p Fs(i)p FA(\))24 b(=)838 4859 y Fl(P)934 4955 y Fp(t)982 4928 y Fs(g)s FA(\()p Fs(i)p FA(\))1132 4942 y Fp(t)1163 4928 y Fs(y)s FA(\()p Fs(t)p FA(\))30 b(in)f(\000,)i(w)m(e)f(ha)m(v)m(e)930 5149 y Fs(b)969 5163 y Fp(ij)1059 5149 y FA(=)c Fv(h)p Fs(\037)p FA(\()p Fs(j)5 b FA(\))p Fs(;)15 b(g)s FA(\()p Fs(i)p FA(\))p Fv(i)23 b FA(=)i Fs(q)1753 5100 y Fp(g)r Fr(\()p Fp(i)p Fr(\))1875 5109 y Fk(j)1750 5175 y Fp(j)1938 5149 y FA(=)h Fs(q)2079 5112 y Fp(D)2139 5121 y Fk(j)2172 5112 y Fp(g)r Fr(\()p Fp(i)p Fr(\))2294 5121 y Fk(j)2332 5149 y Fs(:)p eop %%Page: 15 15 15 14 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(15)110 91 y FA(As)31 b(the)e(braiding)e(\()p Fs(b)837 105 y Fp(ij)901 91 y FA(\))j(is)g(of)g(Cartan)f(t)m(yp)s(e,)h(w)m(e)g(see)g(that)869 343 y Fs(q)913 305 y Fp(D)973 314 y Fk(i)1002 305 y Fp(g)r Fr(\()p Fp(j)t Fr(\))1132 314 y Fk(i)1160 305 y Fr(+)p Fp(D)1276 314 y Fk(j)1310 305 y Fp(g)r Fr(\()p Fp(i)p Fr(\))1432 314 y Fk(j)1494 343 y FA(=)c Fs(b)1630 357 y Fp(ij)1694 343 y Fs(b)1733 357 y Fp(j)t(i)1823 343 y FA(=)g Fs(b)1959 293 y Fp(a)1999 302 y Fk(ij)1959 368 y Fp(ii)2087 343 y FA(=)f Fs(q)2227 305 y Fp(D)2287 314 y Fk(i)2316 305 y Fp(a)2356 314 y Fk(ij)0 583 y FA(and)30 b(therefore)903 830 y Fs(D)979 844 y Fp(j)1017 830 y Fs(g)s FA(\()p Fs(i)p FA(\))1167 844 y Fp(j)1230 830 y Fv(\021)c Fs(D)1403 844 y Fp(i)1433 830 y Fs(a)1481 844 y Fp(ij)1565 830 y Fv(\000)21 b Fs(D)1733 844 y Fp(i)1763 830 y Fs(g)s FA(\()p Fs(j)5 b FA(\))1925 844 y Fp(i)2045 830 y FA(mo)s(d)30 b Fs(E)2318 844 y Fp(\022)2359 830 y Fs(:)110 1031 y FA(Th)m(us)k Fl(e)-50 b Fs(c)386 1045 y Fp(ij)483 1031 y FA(=)34 b(2)p Fs(g)s FA(\()p Fs(j)5 b FA(\))796 1045 y Fp(i)849 1031 y Fv(\000)23 b Fs(a)991 1045 y Fp(ij)1091 1031 y FA(is)35 b(a)g(solution)e(of)i(\(4.4\).)54 b(The)35 b(last)f(statemen)m(t)f(follo)m(ws)h(from)0 1141 y(Remark)c(2.4)91 b Fi(\003)110 1326 y FA(In)30 b(the)g(rest)f(of)h(this)g(section,)e(w)m(e)j(shall)e(consider)f (matrices)h FC(b)d FA(=)g(\()p Fs(b)2501 1340 y Fp(ij)2565 1326 y FA(\))2601 1340 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)2904 1326 y FA(of)k(Cartan)0 1436 y(t)m(yp)s(e,)h(suc)m(h)h (that)e Fs(b)678 1450 y Fp(ij)774 1436 y FA(is)h(a)g(ro)s(ot)f(of)h(1)h (of)f(o)s(dd)f(order)h(and)f Fs(b)2068 1450 y Fp(ij)2160 1436 y FA(=)e Fs(b)2298 1450 y Fp(ij)2394 1436 y FA(for)j(all)f Fs(i;)15 b(j)5 b FA(.)46 b(By)31 b(Lemma)0 1545 y(4.1,)41 b(w)m(e)f(can)e(consider)e(its)j(asso)s(ciated)c(Nic)m(hols)j(algebra)f Fs(R)q FA(;)44 b(the)38 b(com)m(ultiplication)33 b(is)38 b(an)0 1655 y(algebra)28 b(map)i(where)f(the)g(pro)s(duct)f(in)h Fs(R)22 b Fv(\012)e Fs(R)32 b FA(is)d(determined)f(b)m(y)923 1895 y(\()p Fs(x)1011 1909 y Fp(r)1071 1895 y Fv(\012)21 b Fs(x)1215 1909 y Fp(i)1245 1895 y FA(\)\()p Fs(x)1369 1909 y Fp(j)1427 1895 y Fv(\012)g Fs(x)1571 1909 y Fp(s)1610 1895 y FA(\))k(=)g Fs(b)1806 1909 y Fp(ij)1871 1895 y Fs(x)1923 1909 y Fp(r)1963 1895 y Fs(x)2015 1909 y Fp(j)2074 1895 y Fv(\012)c Fs(x)2218 1909 y Fp(i)2248 1895 y Fs(x)2300 1909 y Fp(s)2339 1895 y Fs(:)110 2142 y FA(If)34 b Fs(I)39 b Fv(\032)31 b(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(\022)s Fv(g)34 b FA(then)f(w)m(e)h(denote)e(b)m(y)j Fs(R)q FA(\()p Fs(I)7 b FA(\))33 b(the)g(Nic)m(hols)g(algebra)f(corresp)s(onding)d(to) 0 2252 y(the)i(braiding)e(\()p Fs(b)596 2266 y Fp(ij)660 2252 y FA(\))696 2266 y Fp(i;j)t Fo(2)p Fp(I)868 2252 y FA(.)45 b(Clearly)-8 b(,)31 b(there)f(is)i(an)f(injectiv)m(e)e(map)j (of)f(braided)e(Hopf)j(algebras)0 2361 y Fs(R)q FA(\()p Fs(I)7 b FA(\))25 b Fv(!)g Fs(R)q FA(.)0 2547 y FC(Lemma)34 b(4.2.)52 b Fj(\(i\).)40 b(If)30 b Fs(b)881 2561 y Fp(ij)971 2547 y FA(=)25 b(1)30 b Fj(then)f Fs(x)1403 2561 y Fp(i)1434 2547 y Fs(x)1486 2561 y Fp(j)1550 2547 y FA(=)c Fs(x)1698 2561 y Fp(j)1737 2547 y Fs(x)1789 2561 y Fp(i)1820 2547 y Fj(.)110 2663 y(\(ii\).)58 b(Assume)37 b(there)f(is)g Fs(I)44 b Fv(\032)36 b(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(\022)s Fv(g)37 b Fj(suc)m(h)f(that)g Fs(b)2071 2677 y Fp(ij)2172 2663 y FA(=)g(1)h Fj(for)g(all)e Fs(i)i Fv(2)f Fs(I)7 b Fj(,)40 b Fs(j)h Fv(2)36 b Fs(J)45 b FA(:=)0 2773 y Fv(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(\022)s Fv(g)20 b(\000)g Fs(I)7 b Fj(.)41 b(Then)30 b Fs(R)c Fv(')f Fs(R)q FA(\()p Fs(I)7 b FA(\))20 b Fv(\012)g Fs(R)q FA(\()p Fs(J)9 b FA(\))p Fj(.)0 2958 y Ft(Pr)-5 b(o)g(of.)46 b FA(By)f(a)g(direct)e(computation,)i(w)m(e)h(see)e(that)g Fs(x)1963 2972 y Fp(i)1993 2958 y Fs(x)2045 2972 y Fp(j)2114 2958 y Fv(\000)30 b Fs(x)2267 2972 y Fp(j)2306 2958 y Fs(x)2358 2972 y Fp(i)2434 2958 y FA(is)44 b(primitiv)m(e;)51 b(but)44 b(the)0 3068 y(primitiv)m(e)32 b(elemen)m(ts)g(are)h(concen)m (trated)d(in)k(degree)e(one,)i(so)g(it)f(should)g(b)s(e)g(0.)52 b(This)33 b(pro)m(v)m(es)0 3177 y(\(i\).)39 b(Let)30 b Fs(S)h FA(=)25 b Fs(R)q FA(\()p Fs(I)7 b FA(\))20 b Fv(\012)g Fs(R)q FA(\()p Fs(J)9 b FA(\);)30 b(it)f(is)h(a)g(graded)f (algebra)f(and)h(coalgebra)e(with)i(resp)s(ect)f(to)i(the)0 3287 y(grading)20 b Fs(S)5 b FA(\()p Fs(h)p FA(\))25 b(=)g Fv(\010)700 3301 y Fp(`)735 3287 y Fs(R)q FA(\()p Fs(I)7 b FA(\)\()p Fs(`)p FA(\))e Fv(\012)g Fs(R)q FA(\()p Fs(J)k FA(\)\()p Fs(h)c Fv(\000)g Fs(`)p FA(\))18 b(and)k(the)g(usual)f (tensor)g(pro)s(duct)f(m)m(ultiplication)0 3396 y(and)31 b(com)m(ultiplication.)38 b(Then)30 b Fs(S)37 b FA(is)31 b(a)g(strictly)e(graded)h(coalgebra,)f Ft(i.e.)44 b FA(\(1.1\),)31 b(\(1.2\),)g(\(1.3\))0 3506 y(hold,)36 b Ft(cf.)57 b FA([Sw,)38 b(p.)57 b(240].)g(On)36 b(the)f(other)f(hand,)i Fs(S)42 b FA(is)35 b(a)h(Y)-8 b(etter-Drinfeld)32 b(mo)s(dule)i(and)h (it)0 3615 y(is)f(easy)f(to)g(conclude)e(from)j(the)f(h)m(yp)s(othesis) f(and)h(\(i\))f(that)h(it)g(is)g(a)h(braided)e(Hopf)i(algebra.)0 3725 y(Clearly)-8 b(,)32 b Fs(S)38 b FA(is)33 b(generated)c(as)k (algebra)d(b)m(y)k Fs(S)5 b FA(\(1\))28 b Fv(')i Fs(V)20 b FA(,)34 b Ft(i.e.)48 b FA(it)32 b(satis\014es)f(\(1.4\).)47 b(Th)m(us)33 b Fs(S)38 b FA(is)32 b(a)0 3835 y(Nic)m(hols)f(algebra,)f (hence)h(isomorphic)e(to)i Fs(R)j FA(b)s(ecause)29 b(their)h(spaces)h (of)h(primitiv)m(e)e(elemen)m(ts)0 3944 y(coincide.)88 b Fi(\003)110 4198 y FA(W)-8 b(e)26 b(consider)e(the)i(follo)m(wing)e (equiv)-5 b(alence)23 b(relation)h(on)i Fv(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)h(;)f(\022)s Fv(g)p FA(:)39 b Fs(i)26 b Fv(\030)f Fs(j)32 b FA(if)26 b(there)f(exists)0 4308 y(a)35 b(sequence)f(of)h(elemen)m(ts)f Fs(i)g FA(=)g Fs(h)1176 4322 y Fr(0)1217 4308 y Fs(;)15 b(:)g(:)g(:)i(;)e(h)1472 4322 y Fp(P)1567 4308 y FA(=)34 b Fs(j)41 b FA(in)35 b Fv(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)h(;)f(\022)s Fv(g)36 b FA(suc)m(h)f(that)f Fs(h)2751 4322 y Fp(`)2820 4308 y Fv(6)p FA(=)g Fs(h)2978 4322 y Fp(`)p Fr(+1)3141 4308 y FA(and)0 4417 y Fs(b)39 4431 y Fp(h)82 4441 y Fk(`)114 4431 y Fp(h)157 4441 y Fk(`)p Ff(+1)304 4417 y Fv(6)p FA(=)f(1.)55 b(Note)35 b(that)e Fs(P)46 b FA(=)33 b(0)i(is)f(allo)m(w)m (ed)g(and)g(giv)m(es)h Fs(i)e Fv(\030)g Fs(i)p FA(.)56 b(A)36 b(class)d(of)i(this)f(relation)0 4527 y(is)40 b(called)e(a)j(connected)c(comp)s(onen)m(t.)70 b(This)40 b(is)g(equiv)-5 b(alen)m(t)38 b(to)i(the)g(notion)f(of)h(connected)0 4637 y(Dynkin)29 b(diagram)f(or)i(indecomp)s(osable)25 b(matrix)k(as)h(in)g([K,)h(1.7.1].)41 b(The)29 b(preceding)e(Lemma)0 4746 y(4.2)k(sho)m(ws)g(that)e(the)h(Nic)m(hols)g(algebra)f Fs(R)j FA(is)e(the)g(tensor)f(pro)s(duct)g(of)h(the)g(Nic)m(hols)g (algebras)0 4856 y(corresp)s(onding)d(to)k(its)f(connected)e(comp)s (onen)m(ts.)42 b(Therefore,)30 b(to)g(pro)m(v)m(e)i(Theorem)e(1.1)h(it) f(is)0 4965 y(enough)f(to)g(assume)h(that)f FC(b)h FA(is)g Ft(c)-5 b(onne)g(cte)g(d)p FA(.)110 5151 y Ft(4.3)32 b(Conditions)h(for)f(FL-typ)-5 b(e.)p eop %%Page: 16 16 16 15 bop 0 -128 a Fy(16)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Lemma)36 b(4.3.)52 b Fj(Let)31 b FC(b)d FA(=)f(\()p Fs(b)1014 105 y Fp(ij)1078 91 y FA(\))1114 105 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)1419 91 y Fj(b)s(e)k(a)g(braiding)e(of)i(Cartan)f(t)m(yp)s(e)h (suc)m(h)g(that)g Fs(b)3053 105 y Fp(ij)3149 91 y Fj(is)g(a)0 201 y(ro)s(ot)e(of)h(1)g(of)g(o)s(dd)f(order)g(and)g Fs(b)1115 215 y Fp(ij)1205 201 y FA(=)d Fs(b)1341 215 y Fp(j)t(i)1436 201 y Fj(for)j(all)g Fs(i;)15 b(j)5 b Fj(.)110 311 y(If)34 b FC(b)h Fj(is)e(symmetrizable)f(and)h (satis\014es)f(the)i(relativ)m(e)e(primeness)g(condition)f(\()p Fs(a)2932 325 y Fp(ij)3030 311 y Fj(is)j(0)g(or)0 420 y(relativ)m(ely)28 b(prime)h(to)g(the)g(order)g(of)h Fs(b)1308 434 y Fp(ii)1395 420 y Fj(for)g(all)f Fs(i;)15 b(j)5 b Fj(\))31 b(then)e(it)g(is)h(of)g(FL-t)m(yp)s(e.)0 582 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)30 b(can)g(easily)f(reduce)f (to)i(the)f(case)g(when)h FC(b)g FA(is)g(connected.)110 692 y(W)-8 b(e)29 b(claim)e(\014rst)h(that)g(the)g(order)f(of)i Fs(b)1427 706 y Fp(k)r(k)1542 692 y FA(is)f(the)g(same)h(for)f(all)g Fs(k)s FA(.)40 b(Indeed,)28 b(let)g(us)h(\014x)g Fs(i)c Fv(6)p FA(=)h Fs(j)0 802 y FA(suc)m(h)40 b(that)f Fs(a)472 816 y Fp(ij)578 802 y Fv(6)p FA(=)j(0)e(\(if)f(no)g(suc)m(h)h(pair)f Fs(i;)15 b(j)46 b FA(exists,)c(then)d(b)m(y)h(connectedness)d Fs(\022)44 b FA(=)e(1)e(and)0 914 y(there)34 b(is)i(nothing)e(to)h(pro) m(v)m(e\).)58 b(Then)35 b Fs(b)1413 864 y Fp(a)1453 873 y Fk(ij)1413 939 y Fp(ii)1551 914 y FA(=)g Fs(b)1696 864 y Fp(a)1736 873 y Fk(j)s(i)1696 939 y Fp(j)t(j)1835 914 y FA(and)g(ord)15 b Fs(b)2205 928 y Fp(ii)2296 914 y FA(=)35 b(ord)15 b Fs(b)2589 928 y Fp(j)t(j)2697 914 y FA(b)m(y)37 b(the)e(relativ)m(e)0 1024 y(primeness)28 b(condition.)38 b(The)29 b(claim)g(follo)m(ws)g(b)m(y)i(connectedness.) 37 b(W)-8 b(e)30 b(call)e Fs(N)36 b FA(=)26 b(ord)14 b Fs(b)3058 1038 y Fp(ii)3114 1024 y FA(.)110 1134 y(Let)33 b Fs(d)325 1148 y Fp(k)404 1134 y FA(b)s(e)g(in)m(tegers)f(as)i(in)g (\(1.11\).)51 b(W)-8 b(e)34 b(can)f(assume)h(that)f(they)g(are)g (relativ)m(ely)f(prime,)0 1243 y(that)e(is,)h(\()p Fs(d)401 1257 y Fr(1)441 1243 y Fs(;)15 b(:)g(:)g(:)i(;)e(d)691 1257 y Fp(\022)732 1243 y FA(\))27 b(=)f(1.)43 b(Then)30 b(it)g(is)h(easy)f(to)g(see)h(that)e Fs(N)42 b FA(and)30 b Fs(d)2431 1257 y Fp(k)2507 1243 y FA(are)g(relativ)m(ely)e(prime)0 1353 y(for)c(all)e Fs(k)s FA(.)39 b(Indeed,)24 b(if)f Fs(t)i FA(divides)d Fs(N)35 b FA(and)23 b Fs(d)1449 1367 y Fp(i)1503 1353 y FA(then)g Fs(t)h FA(divides)f Fs(d)2112 1367 y Fp(j)2150 1353 y FA(,)j(since)c Fs(t)i FA(divides)f Fs(d)2825 1367 y Fp(i)2855 1353 y Fs(a)2903 1367 y Fp(ij)2993 1353 y FA(=)i Fs(d)3137 1367 y Fp(j)3175 1353 y Fs(a)3223 1367 y Fp(j)t(i)0 1463 y FA(and)30 b(\()p Fs(N)5 b(;)15 b(a)380 1477 y Fp(j)t(i)445 1463 y FA(\))24 b(=)i(1.)40 b(Again,)30 b(the)f(claim)g(follo)m(ws)g(b)m(y)i(connectedness.)110 1575 y(In)h(particular,)f(there)g(is)i(a)g(unique)e(ro)s(ot)h(of)g (unit)m(y)g Fs(q)37 b FA(of)32 b(order)g Fs(N)44 b FA(suc)m(h)32 b(that)g Fs(b)2923 1589 y Fp(ii)3009 1575 y FA(=)e Fs(q)3154 1542 y Fr(2)p Fp(d)3228 1551 y Fk(i)3262 1575 y FA(.)0 1685 y(W)-8 b(e)37 b(claim)f(\014nally)g(that)f(this)i Fs(q)j FA(satis\014es)c(\(1.12\).)60 b(Indeed,)38 b Fs(b)2204 1636 y Fp(a)2244 1645 y Fk(ij)2204 1710 y Fp(ii)2344 1685 y FA(=)f Fs(q)2496 1652 y Fr(2)p Fp(d)2570 1661 y Fk(i)2600 1652 y Fp(a)2640 1661 y Fk(ij)2740 1685 y FA(=)g Fs(q)2892 1652 y Fr(2)p Fp(d)2966 1661 y Fk(j)3000 1652 y Fp(a)3040 1661 y Fk(j)s(i)3141 1685 y FA(and)0 1797 y Fs(b)39 1748 y Fp(a)79 1757 y Fk(ij)39 1823 y Fp(ii)191 1797 y FA(=)50 b Fs(b)351 1764 y Fr(2)351 1821 y Fp(ij)465 1797 y FA(=)g Fs(b)625 1748 y Fp(a)665 1757 y Fk(j)s(i)625 1823 y Fp(j)t(j)728 1797 y FA(.)84 b(Th)m(us,)49 b Fs(b)1151 1811 y Fp(ij)1265 1797 y FA(=)h Fs(q)1430 1764 y Fp(d)1468 1773 y Fk(i)1497 1764 y Fp(a)1537 1773 y Fk(ij)1644 1797 y FA(\(since)43 b Fs(N)55 b FA(is)45 b(o)s(dd\))e(and)h Fs(b)2614 1811 y Fp(j)t(j)2736 1797 y FA(=)50 b Fs(q)2901 1764 y Fr(2)p Fp(d)2975 1773 y Fk(j)3059 1797 y FA(\(since)0 1913 y(\()p Fs(N)5 b(;)15 b(a)202 1927 y Fp(j)t(i)267 1913 y FA(\))25 b(=)g(1\).)40 b(Once)30 b(more,)g(the)f(claim)f(follo)m(ws)h(b)m(y)i(connectedness.) 88 b Fi(\003)110 2128 y FA(W)-8 b(e)31 b(no)m(w)h(in)m(v)m(estigate)d (conditions)g(for)i(FL-t)m(yp)s(e)g(in)g(rank)g(2.)45 b(Let)32 b FC(b)c FA(=)g(\()p Fs(b)2712 2142 y Fp(ij)2776 2128 y FA(\))2812 2142 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fr(2)3116 2128 y FA(b)s(e)j(a)0 2237 y(braiding)17 b(of)j(connected)d (Cartan)i(t)m(yp)s(e)g(whose)h(en)m(tries)e(ha)m(v)m(e)j(o)s(dd)e (order,)i(suc)m(h)f(that)e Fs(b)2946 2251 y Fr(12)3049 2237 y FA(=)26 b Fs(b)3185 2251 y Fr(21)3262 2237 y FA(.)0 2347 y(Then)37 b(it)g(is)g(automatically)c(symmetrizable)h(since)j(the) f(rank)h(is)h(2.)62 b("Connected")35 b(means)0 2456 y(that)30 b Fs(a)247 2470 y Fr(12)353 2456 y Fv(6)p FA(=)e(0.)46 b(Let)31 b Fs(d)782 2470 y Fr(1)823 2456 y FA(,)h Fs(d)928 2470 y Fr(2)1001 2456 y FA(b)s(e)f(relativ)m(ely)e(prime)i(in)m(tegers) e(suc)m(h)j(that)f Fs(d)2583 2470 y Fr(1)2623 2456 y Fs(a)2671 2470 y Fr(12)2777 2456 y FA(=)d Fs(d)2924 2470 y Fr(2)2964 2456 y Fs(a)3012 2470 y Fr(21)3090 2456 y FA(.)46 b(W)-8 b(e)0 2566 y(denote)28 b(b)m(y)j Fs(N)494 2580 y Fp(i)555 2566 y FA(the)f(order)f(of)h Fs(b)1095 2580 y Fp(ii)1151 2566 y FA(.)0 2728 y FC(Lemma)k(4.4.)52 b Fj(The)30 b(follo)m(wing)e(are)h(equiv)-5 b(alen)m(t:)178 2864 y FA(\(i\))53 b FC(b)31 b Fj(is)e(of)h(FL-t)m(yp)s(e.)152 2974 y FA(\(ii\))53 b Fj(There)29 b(exists)g Fs(u)d Fv(2)f Fu(|)15 b Fj(of)30 b(o)s(dd)f(order)g(suc)m(h)h(that)f Fs(u)2069 2941 y Fr(2)p Fp(d)2143 2950 y Fk(i)2203 2974 y FA(=)c Fs(b)2338 2988 y Fp(ii)2394 2974 y Fj(,)31 b Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2)p Fj(.)127 3083 y FA(\(iii\))52 b Fj(There)29 b(exists)g Fs(v)g Fv(2)c Fu(|)15 b Fj(suc)m(h)30 b(that)f Fs(v)1536 3050 y Fp(d)1574 3059 y Fk(i)1634 3083 y FA(=)c Fs(b)1769 3097 y Fp(ii)1825 3083 y Fj(,)31 b Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2)p Fj(.)0 3272 y Ft(Pr)-5 b(o)g(of.)46 b FA(\(iii\))61 b(=)-15 b Fv(\))63 b FA(\(ii\):)45 b(Note)32 b(that)g(the)g(order)g(of)h Fs(v)j FA(divides)c Fs(d)2274 3286 y Fr(1)2314 3272 y Fs(N)2387 3286 y Fr(1)2462 3272 y FA(and)g Fs(d)2690 3286 y Fr(2)2731 3272 y Fs(N)2804 3286 y Fr(2)2845 3272 y FA(.)50 b(Since)31 b(the)0 3382 y Fs(N)73 3396 y Fp(i)104 3382 y FA('s)i(are)e(o)s(dd)g(and)g Fs(d)761 3396 y Fr(1)802 3382 y FA(,)i Fs(d)908 3396 y Fr(2)981 3382 y FA(are)e(relativ)m(ely)f (prime,)i(the)f(order)g(of)h Fs(v)k FA(is)31 b(o)s(dd.)46 b(Hence)31 b Fs(v)36 b FA(has)c(a)0 3491 y(square)d(ro)s(ot)g Fs(u)i FA(of)f(o)s(dd)f(order.)110 3601 y(\(ii\))55 b(=)-15 b Fv(\))58 b FA(\(i\):)41 b(W)-8 b(e)32 b(ha)m(v)m(e)f Fs(b)1067 3568 y Fr(2)1067 3624 y(12)1171 3601 y FA(=)c Fs(b)1308 3560 y Fp(a)1348 3569 y Ff(12)1308 3625 y Fr(11)1446 3601 y FA(=)g Fs(u)1596 3568 y Fr(2)p Fp(d)1670 3577 y Ff(1)1706 3568 y Fp(a)1746 3577 y Ff(12)1817 3601 y FA(.)43 b(Hence)30 b Fs(b)2196 3615 y Fr(12)2301 3601 y FA(=)c Fs(u)2450 3568 y Fp(d)2488 3577 y Ff(1)2524 3568 y Fp(a)2564 3577 y Ff(12)2667 3601 y FA(since)j(b)s(oth)h(ha)m(v)m (e)0 3711 y(o)s(dd)f(order.)110 3820 y(\(i\))54 b(=)-15 b Fv(\))56 b FA(\(iii\):)38 b(If)30 b Fs(q)k FA(satis\014es)28 b(\(1.12\),)h(tak)m(e)h Fs(v)f FA(=)c Fs(q)1880 3787 y Fr(2)1921 3820 y FA(.)92 b Fi(\003)110 3982 y FA(W)-8 b(e)31 b(w)m(an)m(t)g(to)f(giv)m(e)g(a)h(criterion)d(for)i(the)g (condition)e(\(iii\))g(in)i(the)g(preceding)e(Lemma.)42 b(Let)0 4092 y Fs(e)43 4106 y Fp(i)103 4092 y FA(,)31 b Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2)31 b(b)s(e)e(non-zero)g(in)m (tegers)f(suc)m(h)i(that)1188 4302 y Fs(e)1231 4316 y Fr(1)1272 4302 y Fs(d)1320 4316 y Fr(1)1360 4302 y Fs(N)1433 4316 y Fr(1)1500 4302 y FA(=)c Fs(e)1640 4316 y Fr(2)1680 4302 y Fs(d)1728 4316 y Fr(2)1769 4302 y Fs(N)1842 4316 y Fr(2)1908 4302 y FA(=:)g Fs(r)s FA(;)0 4512 y(for)21 b(instance)e(w)m(e)j(could)d(tak)m(e)j Fs(r)i FA(the)d(lo)m(w)m(est)f (common)h(m)m(ultiple)e(of)i Fs(d)2356 4526 y Fr(1)2397 4512 y Fs(N)2470 4526 y Fr(1)2533 4512 y FA(and)f Fs(d)2749 4526 y Fr(2)2790 4512 y Fs(N)2863 4526 y Fr(2)2904 4512 y FA(.)38 b(Observ)m(e)0 4621 y(that)29 b(there)g(exists)g Fs(s)c Fv(2)g Fu(Z)h FA(suc)m(h)k(that)f Fs(r)f FA(=)e Fs(d)1547 4635 y Fr(1)1587 4621 y Fs(d)1635 4635 y Fr(2)1676 4621 y Fs(s)p FA(.)110 4731 y(Let)i(no)m(w)i Fs(\030)f Fv(2)d Fu(|)14 b FA(b)s(e)28 b(a)i(primitiv)m(e)d Fs(r)s FA(-th)i(ro)s(ot)f(of)h(1)g(and)g(c)m(ho)s(ose)f(in)m(tegers)f Fs(k)2701 4745 y Fr(1)2742 4731 y FA(,)j Fs(k)2845 4745 y Fr(2)2915 4731 y FA(suc)m(h)f(that)1399 4941 y Fs(b)1438 4955 y Fp(ii)1520 4941 y FA(=)c Fs(\030)1660 4903 y Fp(e)1695 4912 y Fk(i)1724 4903 y Fp(d)1762 4912 y Fk(i)1791 4903 y Fp(k)1830 4912 y Fk(i)1863 4941 y FA(;)0 5151 y(this)k(is)h(p)s (ossible)d(b)s(ecause)h Fs(\030)995 5118 y Fp(e)1030 5127 y Fk(i)1059 5118 y Fp(d)1097 5127 y Fk(i)1161 5151 y FA(has)i(order)f Fs(N)1636 5165 y Fp(i)1667 5151 y FA(.)p eop %%Page: 17 17 17 16 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(17)0 91 y FC(Lemma)34 b(4.5.)52 b Fj(Condition)27 b(\(iii\))h(in)i(Lemma)g(4.4)g(is)g(equiv) -5 b(alen)m(t)27 b(to)86 222 y FA(\(4.5\))54 b Fs(e)372 236 y Fr(1)412 222 y Fs(k)460 236 y Fr(1)526 222 y Fv(\021)26 b Fs(e)666 236 y Fr(2)706 222 y Fs(k)754 236 y Fr(2)855 222 y FA(mo)s(d)k Fs(s:)0 395 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)30 b(claim)f(\014rst)g(that)g(\(iii\))f(is)i(equiv)-5 b(alen)m(t)28 b(to)h(the)g(follo)m(wing)f(statemen)m(t:)129 526 y(\(iv\))54 b(There)29 b(exists)g Fs(t)878 540 y Fp(i)934 526 y Fv(2)c Fu(Z)p FA(,)i Fs(i)f FA(=)f(1)p Fs(;)15 b FA(2)30 b(suc)m(h)g(that)1000 689 y Fs(e)1043 703 y Fp(i)1073 689 y Fs(d)1121 703 y Fp(i)1151 689 y Fs(k)1199 703 y Fp(i)1255 689 y Fv(\021)25 b Fs(e)1394 703 y Fp(i)1424 689 y Fs(d)1472 703 y Fp(i)1502 689 y Fs(t)1535 703 y Fp(i)1626 689 y FA(mo)s(d)30 b Fs(r)m(;)107 b(i)26 b FA(=)f(1)p Fs(;)15 b FA(2)-2287 b(\(4.6\))1072 831 y Fs(e)1115 845 y Fr(1)1155 831 y Fs(t)1188 845 y Fr(1)1255 831 y Fv(\021)25 b Fs(e)1394 845 y Fr(2)1434 831 y Fs(t)1467 845 y Fr(2)1569 831 y FA(mo)s(d)30 b Fs(r)m(:)-1838 b FA(\(4.7\))0 1029 y(Indeed,)30 b(if)g(\(iii\))e(holds) h(then)h Fs(q)1077 996 y Fp(d)1115 1005 y Fk(i)1144 996 y Fp(N)1202 1005 y Fk(i)1261 1029 y FA(=)c(1;)32 b(as)e Fs(\030)1617 996 y Fp(e)1652 1005 y Fk(i)1716 1029 y FA(has)h(order)e Fs(d)2167 1043 y Fp(i)2197 1029 y Fs(N)2270 1043 y Fp(i)2301 1029 y FA(,)i(there)e(exists)h Fs(t)2877 1043 y Fp(i)2933 1029 y Fv(2)c Fu(Z)h FA(suc)m(h)0 1139 y(that)33 b Fs(q)i FA(=)e Fs(\030)426 1106 y Fp(e)461 1115 y Fk(i)489 1106 y Fp(t)517 1115 y Fk(i)550 1139 y FA(.)54 b(Then)33 b(\(4.6\),)i(\(4.7\))e(follo)m(w)g(no)m(w)i (without)d(di\016cult)m(y)-8 b(.)52 b(Con)m(v)m(ersely)-8 b(,)35 b(if)f(\(iv\))0 1248 y(holds,)k(tak)m(e)e Fs(q)k FA(=)d Fs(\030)710 1215 y Fp(e)745 1224 y Ff(1)780 1215 y Fp(t)808 1224 y Ff(1)885 1248 y FA(=)f Fs(\030)1036 1215 y Fp(e)1071 1224 y Ff(2)1107 1215 y Fp(t)1135 1224 y Ff(2)1174 1248 y FA(,)k(b)m(y)d(\(4.7\).)60 b(Then)36 b(\(iii\))f(is)h(true)g(b)m(y)i(\(4.6\).)60 b(The)36 b(claim)f(is)0 1358 y(pro)m(v)m(ed.)110 1468 y(No)m(w)c(\(4.6\))e(is)h (equiv)-5 b(alen)m(t)27 b(to)j Fs(k)1221 1482 y Fp(i)1276 1468 y Fv(\021)c Fs(t)1406 1482 y Fp(i)1497 1468 y FA(mo)s(d)j Fs(N)1775 1482 y Fp(i)1806 1468 y FA(,)i Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2.)110 1577 y(Assume)31 b(\(iv\).)43 b(Then)31 b(there)f(exist)g Fs(x)1407 1591 y Fp(i)1465 1577 y Fv(2)d Fu(Z)h FA(suc)m(h)j(that)f Fs(t)2085 1591 y Fp(i)2143 1577 y FA(=)d Fs(k)2289 1591 y Fp(i)2340 1577 y FA(+)21 b Fs(N)2505 1591 y Fp(i)2536 1577 y Fs(x)2588 1591 y Fp(i)2619 1577 y FA(,)32 b Fs(i)c FA(=)f(1)p Fs(;)15 b FA(2.)45 b(No)m(w)32 b Fs(r)0 1687 y FA(\(and)d Ft(a)j(fortiori)e Fs(s)p FA(\))f(divides)727 1850 y Fs(e)770 1864 y Fr(1)810 1850 y Fs(t)843 1864 y Fr(1)904 1850 y Fv(\000)21 b Fs(e)1039 1864 y Fr(2)1079 1850 y Fs(t)1112 1864 y Fr(2)1178 1850 y FA(=)26 b Fs(e)1318 1864 y Fr(1)1358 1850 y Fs(k)1406 1864 y Fr(1)1467 1850 y Fv(\000)20 b Fs(e)1601 1864 y Fr(2)1642 1850 y Fs(k)1690 1864 y Fr(2)1751 1850 y FA(+)g Fs(e)1885 1864 y Fr(1)1926 1850 y Fs(N)1999 1864 y Fr(1)2040 1850 y Fs(x)2092 1864 y Fr(1)2153 1850 y Fv(\000)h Fs(e)2288 1864 y Fr(2)2328 1850 y Fs(N)2401 1864 y Fr(2)2443 1850 y Fs(x)2495 1864 y Fr(2)2536 1850 y FA(;)0 2014 y(but)29 b Fs(e)210 2028 y Fr(1)251 2014 y Fs(N)324 2028 y Fr(1)391 2014 y FA(=)c Fs(d)535 2028 y Fr(2)576 2014 y Fs(s)30 b FA(and)f Fs(e)869 2028 y Fr(2)910 2014 y Fs(N)983 2028 y Fr(2)1050 2014 y FA(=)c Fs(d)1194 2028 y Fr(1)1234 2014 y Fs(s)p FA(.)41 b(So)30 b(\(4.5\))f(holds.)110 2123 y(If)d(\(4.5\))g(holds,)g(let)f Fs(y)k Fv(2)c Fu(Z)e FA(suc)m(h)j(that)f Fs(e)1488 2137 y Fr(1)1529 2123 y Fs(k)1577 2137 y Fr(1)1630 2123 y Fv(\000)13 b Fs(e)1757 2137 y Fr(2)1798 2123 y Fs(k)1846 2137 y Fr(2)1912 2123 y FA(=)25 b Fv(\000)p Fs(y)s(s)p FA(.)40 b(As)27 b Fs(d)2414 2137 y Fr(1)2482 2123 y FA(and)e Fs(d)2703 2137 y Fr(2)2771 2123 y FA(are)h(relativ)m(ely)0 2233 y(prime,)h(there)f(exists)g Fs(x)814 2247 y Fr(1)855 2233 y Fs(;)15 b(x)947 2247 y Fr(2)1016 2233 y FA(suc)m(h)27 b(that)f Fs(y)i FA(=)e Fs(d)1632 2247 y Fr(2)1672 2233 y Fs(x)1724 2247 y Fr(1)1780 2233 y Fv(\000)14 b Fs(d)1913 2247 y Fr(1)1953 2233 y Fs(x)2005 2247 y Fr(2)2047 2233 y FA(.)39 b(If)28 b(no)m(w)f(w)m(e)g (tak)m(e)g Fs(t)2746 2247 y Fp(i)2802 2233 y FA(=)e Fs(k)2946 2247 y Fp(i)2991 2233 y FA(+)14 b Fs(N)3149 2247 y Fp(i)3180 2233 y Fs(x)3232 2247 y Fp(i)3262 2233 y FA(,)0 2342 y Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2,)31 b(then)e(\(4.6\))g(holds)g(b) m(y)i(de\014nition;)c(and)18 2506 y Fs(e)61 2520 y Fr(1)102 2506 y Fs(t)135 2520 y Fr(1)196 2506 y Fv(\000)20 b Fs(e)330 2520 y Fr(2)371 2506 y Fs(t)404 2520 y Fr(2)470 2506 y FA(=)25 b Fs(e)609 2520 y Fr(1)650 2506 y Fs(k)698 2520 y Fr(1)759 2506 y Fv(\000)20 b Fs(e)893 2520 y Fr(2)933 2506 y Fs(k)981 2520 y Fr(2)1042 2506 y FA(+)h Fs(e)1177 2520 y Fr(1)1217 2506 y Fs(N)1290 2520 y Fr(1)1331 2506 y Fs(x)1383 2520 y Fr(1)1445 2506 y Fv(\000)f Fs(e)1579 2520 y Fr(2)1620 2506 y Fs(N)1693 2520 y Fr(2)1734 2506 y Fs(x)1786 2520 y Fr(2)1853 2506 y FA(=)25 b Fs(e)1992 2520 y Fr(1)2032 2506 y Fs(k)2080 2520 y Fr(1)2141 2506 y Fv(\000)c Fs(e)2276 2520 y Fr(2)2316 2506 y Fs(k)2364 2520 y Fr(2)2425 2506 y FA(+)f Fs(d)2564 2520 y Fr(2)2605 2506 y Fs(sx)2700 2520 y Fr(1)2761 2506 y Fv(\000)g Fs(d)2900 2520 y Fr(1)2941 2506 y Fs(sx)3036 2520 y Fr(2)3102 2506 y FA(=)26 b(0)p Fs(:)0 2669 y FA(So,)k(\(4.7\))f(follo)m(ws.)90 b Fi(\003)110 2821 y FA(Lemma)28 b(4.5)g(describ)s(es)d(an)j(easy)g (algorithm)d(to)j(decide)e(whether)h(a)h(giv)m(en)f FC(b)i FA(is)f(lo)s(cally)d(of)0 2931 y(FL-t)m(yp)s(e.)40 b(As)31 b(an)f(example)f(w)m(e)h(note:)0 3083 y FC(Corollary)37 b(4.6.)52 b Fj(There)31 b(are)h(braidings)d(of)j(symmetrizable)d (Cartan)i(t)m(yp)s(e)g(whic)m(h)g(are)h(not)0 3192 y(of)e(FL-t)m(yp)s (e.)0 3344 y Ft(Pr)-5 b(o)g(of.)46 b FA(Let)34 b Fs(p)i FA(b)s(e)d(an)i(o)s(dd)f(prime)f(n)m(um)m(b)s(er)i(and)f(tak)m(e)g Fs(N)2040 3358 y Fr(1)2115 3344 y FA(=)f Fs(N)2292 3358 y Fr(2)2367 3344 y FA(=)g Fs(p)2517 3311 y Fr(2)2558 3344 y FA(,)j Fs(a)2667 3358 y Fr(12)2778 3344 y FA(=)d Fs(a)2930 3358 y Fr(12)3041 3344 y FA(=)g Fv(\000)p Fs(p)p FA(.)0 3454 y(Let)d Fs(k)212 3468 y Fr(1)252 3454 y FA(,)h Fs(k)356 3468 y Fr(2)427 3454 y FA(b)s(e)e(t)m(w)m(o)i(elemen)m(ts)d (not)h(divisible)f(b)m(y)j Fs(p)f FA(suc)m(h)g(that)858 3617 y Fs(k)906 3631 y Fr(1)971 3617 y Fv(6\021)c Fs(k)1116 3631 y Fr(2)1248 3617 y FA(mo)s(d)j Fs(p)1499 3580 y Fr(2)1540 3617 y Fs(;)198 b(k)1811 3631 y Fr(1)1877 3617 y Fv(\021)26 b Fs(k)2022 3631 y Fr(2)2153 3617 y FA(mo)s(d)k Fs(p:)0 3781 y FA(Let)g Fs(q)j FA(b)s(e)d(a)g(ro)s(ot)e(of)j(unit)m(y)e (of)h(order)f Fs(p)1366 3748 y Fr(2)1438 3781 y FA(and)g(let)g Fs(b)1786 3795 y Fp(ii)1868 3781 y FA(=)d Fs(q)2009 3748 y Fp(k)2048 3757 y Fk(i)2081 3781 y FA(,)31 b Fs(i)26 b FA(=)f(1)p Fs(;)15 b FA(2,)31 b Fs(b)2517 3795 y Fr(12)2620 3781 y FA(=)26 b Fs(b)2756 3795 y Fr(21)2864 3781 y FA(the)j(unique)0 3890 y(ro)s(ot)35 b(of)h(unit)m(y)g(of)g(o)s(dd)f(order)g(suc)m(h)h (that)f Fs(b)1553 3857 y Fr(2)1553 3913 y(12)1667 3890 y FA(=)g Fs(b)1812 3846 y Fp(p)1812 3915 y Fr(11)1890 3890 y FA(.)59 b(Then)36 b Fs(b)2258 3857 y Fr(2)2258 3913 y(12)2371 3890 y FA(=)g Fs(b)2517 3846 y Fp(p)2517 3915 y Fr(22)2595 3890 y FA(,)i(but)e Fs(s)f FA(=)h Fs(p)3063 3857 y Fr(2)3141 3890 y FA(and)0 4000 y Fs(e)43 4014 y Fr(1)109 4000 y FA(=)25 b Fs(e)248 4014 y Fr(2)314 4000 y FA(=)h(1,)k(so)g(\(4.5\))f(do)s(es)g(not)h(hold.)90 b Fi(\003)0 4152 y FC(Example)44 b(4.7.)51 b Fj(There)36 b(are)h(braidings)d(of)j(Cartan)f(t)m(yp)s(e)g(whic)m(h)g(are)h(lo)s (cally)d(of)i(FL-t)m(yp)s(e)0 4261 y(but)29 b(not)h(symmetrizable.)0 4413 y Ft(Pr)-5 b(o)g(of.)46 b FA(T)-8 b(ak)m(e)33 b FC(b)c FA(=)f(\()p Fs(q)776 4380 y Fp(d)814 4389 y Fk(i)843 4380 y Fp(a)883 4389 y Fk(ij)945 4413 y FA(\),)33 b(where)e Fs(q)k FA(is)d(a)g(ro)s(ot)e(of)i(1)g(of)g(order)f(5,)i Fs(d)2433 4427 y Fr(1)2502 4413 y FA(=)28 b(1,)33 b Fs(d)2753 4427 y Fr(2)2822 4413 y FA(=)c(2,)j Fs(d)3073 4427 y Fr(3)3142 4413 y FA(=)d(3)0 4523 y(and)1109 4719 y(\()p Fs(a)1193 4733 y Fp(ij)1257 4719 y FA(\))c(=)1415 4536 y Fl(0)1415 4700 y(@)1545 4610 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(3)1510 4719 y Fv(\000)p FA(4)126 b(2)h Fv(\000)p FA(3)1510 4829 y Fv(\000)p FA(1)91 b Fv(\000)p FA(2)126 b(2)2058 4536 y Fl(1)2058 4700 y(A)2153 4719 y Fs(:)0 4973 y FA(Then)31 b FC(b)h FA(is)f(of)g(Cartan)f(t)m(yp)s (e)h(since)f Fs(d)1319 4987 y Fp(i)1349 4973 y Fs(a)1397 4987 y Fp(ij)1489 4973 y FA(=)d Fs(d)1635 4987 y Fp(j)1673 4973 y Fs(a)1721 4987 y Fp(j)t(i)1847 4973 y FA(mo)s(d)i(5)j(for)f(all) f Fs(i;)15 b(j)5 b FA(.)45 b(By)32 b(Lemma)f(4.3,)h FC(b)0 5082 y FA(is)e(lo)s(cally)d(of)j(FL-t)m(yp)s(e.)40 b(But)30 b(\()p Fs(a)1139 5096 y Fp(ij)1203 5082 y FA(\))g(is)f(not)h (symmetrizable.)88 b Fi(\003)p eop %%Page: 18 18 18 17 bop 0 -128 a Fy(18)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Lemma)f(4.8.)52 b Fj(Let)24 b FC(b)i Fj(b)s(e)e(a)h(braiding)e(of)i(Cartan)e(t)m(yp)s(e)i(and)f (rank)h(2)g(and)g(assume)f Fs(b)2946 105 y Fr(12)3049 91 y FA(=)i Fs(b)3185 105 y Fr(21)3262 91 y Fj(.)0 201 y(If)k Fs(a)139 215 y Fr(12)242 201 y FA(=)c Fv(\000)p FA(1)k Fj(and)f Fs(a)711 215 y Fr(21)820 201 y Fj(is)g(o)s(dd,)h(then)f FC(b)i Fj(is)e(of)h(FL-t)m(yp)s(e.)0 353 y Ft(Pr)-5 b(o)g(of.)46 b FA(Let)34 b Fs(n)e FA(=)g Fv(\000)p Fs(a)764 367 y Fr(21)842 353 y FA(.)53 b(Let)38 b Fl(e)-55 b Fs(q)37 b FA(b)s(e)d(a)g(ro)s(ot)f(of)h(1)g(suc)m(h)g(that)k Fl(e)-56 b Fs(q)2219 319 y Fp(n)2301 353 y FA(=)32 b Fs(b)2443 314 y Fo(\000)p Fr(1)2443 377 y(12)2542 353 y FA(;)37 b(then)c Fs(b)2855 319 y Fp(n)2855 375 y Fr(22)2965 353 y FA(=)f Fs(b)3107 367 y Fr(11)3217 353 y FA(=)0 466 y Fs(b)39 427 y Fo(\000)p Fr(2)39 491 y(12)166 466 y FA(=)h Fl(e)-55 b Fs(q)310 433 y Fr(2)p Fp(n)396 466 y FA(.)48 b(Hence)31 b(there)g(exists)h(an)g Fs(n)p FA(-th)g(ro)s(ot)e (of)j(1)f Fs(!)k FA(suc)m(h)c(that)f Fs(b)2488 480 y Fr(22)2594 466 y FA(=)e Fs(!)2754 433 y Fr(2)2800 466 y Fl(e)-56 b Fs(q)2839 433 y Fr(2)2913 466 y FA(since)31 b Fs(n)h FA(is)0 576 y(o)s(dd.)40 b(W)-8 b(e)30 b(c)m(ho)s(ose)f(then)g Fs(q)f FA(=)d Fs(!)8 b Fl(e)-56 b Fs(q)t FA(.)91 b Fi(\003)110 728 y Ft(4.4)32 b(Pr)-5 b(o)g(of)32 b(of)g(The)-5 b(or)g(em)32 b(1.1.)110 837 y FA(By)g(Lemma)g(4.1)g(w)m(e)h(can)f(assume)f(that)g FC(b)i FA(is)f(symmetric,)g(that)f(is)h Fs(b)2547 851 y Fp(ij)2640 837 y FA(=)d Fs(b)2779 851 y Fp(j)t(i)2877 837 y FA(for)i(all)g Fs(i;)15 b(j)5 b FA(.)0 947 y(By)30 b(Lemma)g(4.2)g(w)m(e)h(can)e(assume)h(that)f FC(b)i FA(is)e(connected.)110 1099 y(P)m(art)i(\(i\).)45 b(By)32 b(Lemma)g(4.3)g(\(if)f(the)g(t)m(yp)s(e)g(is)h(di\013eren)m(t)e(from)i Fs(G)2341 1113 y Fr(2)2382 1099 y FA(\))f(or)h(Lemma)g(4.8)g(\(if)f (the)0 1208 y(t)m(yp)s(e)e(is)h Fs(G)369 1222 y Fr(2)410 1208 y FA(\),)g(the)g(braiding)d FC(b)k FA(is)f(of)g(FL-t)m(yp)s(e.)40 b(W)-8 b(e)30 b(no)m(w)g(apply)g(Theorem)e(3.1.)110 1360 y(P)m(art)20 b(\(ii\).)36 b(F)-8 b(or)21 b(an)m(y)g(\014xed)g(pair)f Fs(i;)15 b(j)27 b FA(in)20 b Fv(f)p FA(1)p Fs(;)15 b(:)g(:)g(:)i(;)e (\022)s Fv(g)p FA(,)23 b(the)d(braiding)f(of)h(rank)h(2)g(corresp)s (onding)0 1469 y(to)36 b(the)f(submatrix)f(supp)s(orted)g(b)m(y)j Fs(i;)15 b(j)42 b FA(is)36 b(of)g(FL-t)m(yp)s(e)f(and)h Fm(B)p FA(\()p Fs(V)20 b FA(\))36 b(is)g(\014nite)e(dimensional.)0 1579 y(By)e(Theorem)e(3.1,)i(w)m(e)g(conclude)d(that)h Fs(a)1466 1593 y Fp(ij)1531 1579 y Fs(a)1579 1593 y Fp(j)t(i)1671 1579 y FA(=)e(0)p Fs(;)15 b FA(1)p Fs(;)g FA(2)32 b(or)f(3.)45 b(Since)29 b(3)j(is)f(relativ)m(ely)e(prime)0 1689 y(to)h(the)g(orders) f(of)h(the)g Fs(b)846 1703 y Fp(ii)933 1689 y FA(whenev)m(er)g Fs(a)1383 1703 y Fp(ij)1474 1689 y FA(=)c Fv(\000)p FA(3)k(for)g(some)g Fs(j)36 b FA(b)m(y)31 b(h)m(yp)s(othesis,)e(w)m(e)i(can)f(apply)0 1798 y(Lemma)e(4.3)h(and)f(conclude)e(again)h(from)h(Theorem)f(3.1)h (that)g FC(b)h FA(is)f(of)g(\014nite)f(t)m(yp)s(e)h(whenev)m(er)0 1908 y(it)h(is)h(symmetrizable.)110 2060 y(So,)g(it)f(only)h(remains)e (to)i(sho)m(w)g(that:)40 b FC(b)30 b FA(is)g(necessarily)d (symmetrizable.)110 2169 y(It)32 b(is)h(kno)m(wn)g(that)f(a)h(matrix)f (whic)m(h)g(is)h(either)e(simply-laced)f(or)j(has)g(no)f(cycles)g(is)h (sym-)0 2279 y(metrizable,)e(cf.)49 b([K,)34 b(ex.)49 b(2.1].)g(W)-8 b(e)33 b(are)g(reduced)e(to)h(pro)m(v)m(e)h(that)f(no)h (cycle)e(with)h(a)h(double)0 2388 y(or)d(triple)e(arro)m(w)h(can)h (arise.)39 b(W)-8 b(e)30 b(pro)s(ceed)e(b)m(y)j(induction)c(on)j Fs(\022)s FA(.)110 2498 y(Assume)38 b(\014rst)g(that)f FC(b)h FA(is)g(a)h(cycle)d(with)i(a)g(triple)e(arro)m(w.)64 b(If)38 b Fs(\022)k Fv(\025)d FA(4,)h(then)d(w)m(e)i(remo)m(v)m(e)0 2608 y(a)33 b(suitable)d(v)m(ertex)j(and)f(get)g(a)h(sub)s(diagram)e (with)h(no)g(cycles)g(whic)m(h)g(is)h(not)f(of)h(\014nite)e(t)m(yp)s (e.)0 2717 y(This)f(is)f(a)h(con)m(tradiction.)110 2827 y(So,)35 b(assume)f Fs(\022)h FA(=)e(3.)53 b(Then)34 b(there)f(are)h(sev)m(eral)f(p)s(ossibilities)d(of)k(cycles)f(with)h (\(at)f(least\))0 2936 y(one)24 b(triple)e(arro)m(w.)39 b(W)-8 b(e)24 b(can)g(discard)f(t)m(w)m(o)i(of)g(them)f(whic)m(h)f(are) h(symmetrizable.)36 b(W)-8 b(e)25 b(discard)0 3046 y(the)35 b(rest)h(b)s(ecause)d(of)j(the)g(restrictions)c(on)k(the)f(orders)g(of) h Fs(b)2170 3060 y Fp(ii)2227 3046 y FA(.)58 b(Let)36 b(us)g(pro)s(ceed)e(with)h(the)0 3155 y(details.)j(Consider)28 b(a)j(Cartan)d(matrix)1111 3461 y(\()p Fs(a)1195 3475 y Fp(ij)1260 3461 y FA(\))c(=)1417 3278 y Fl(0)1417 3442 y(@)1548 3352 y FA(2)128 b Fv(\000)p FA(3)91 b Fv(\000)p Fs(b)1513 3461 y Fv(\000)p FA(1)127 b(2)g Fv(\000)p Fs(c)1512 3571 y Fv(\000)p Fs(d)92 b Fv(\000)p Fs(e)125 b FA(2)2056 3278 y Fl(1)2056 3442 y(A)2151 3461 y Fs(:)0 3714 y FA(Here)30 b Fs(b;)15 b(c;)g(d;)g(e)31 b FA(are)e(p)s(ositiv)m(e)f(in)m(tegers)g (suc)m(h)i(that)f Fs(bd;)15 b(ce)25 b FA(=)g(1)p Fs(;)15 b FA(2)31 b(or)f(3.)40 b(This)30 b(means)f(that)967 3876 y Fs(b)1006 3839 y Fr(3)1006 3899 y(11)1109 3876 y FA(=)c Fs(b)1244 3890 y Fr(22)1322 3876 y Fs(;)107 b(b)1493 3839 y Fp(c)1493 3899 y Fr(22)1596 3876 y FA(=)25 b Fs(b)1731 3839 y Fp(e)1731 3899 y Fr(33)1809 3876 y Fs(;)107 b(b)1980 3839 y Fp(b)1980 3899 y Fr(11)2082 3876 y FA(=)26 b Fs(b)2218 3839 y Fp(d)2218 3899 y Fr(33)2295 3876 y Fs(:)0 4038 y FA(Hence,)1210 4148 y Fs(b)1249 4110 y Fp(b)1249 4171 y Fr(11)1352 4148 y FA(=)g Fs(b)1488 4110 y Fp(d)1488 4171 y Fr(33)1565 4148 y Fs(;)107 b(b)1736 4110 y Fr(3)p Fp(c)1736 4171 y Fr(11)1839 4148 y FA(=)25 b Fs(b)1974 4110 y Fp(e)1974 4171 y Fr(33)2052 4148 y Fs(:)0 4302 y FA(W)-8 b(e)34 b(kno)m(w)f(that)f Fs(b)f FA(=)g(1)i(or)g Fs(d)e FA(=)g(1.)50 b(Assume)34 b Fs(b)c FA(=)h(1.)51 b(Then)32 b Fs(b)2186 4263 y Fr(3)p Fp(cd)p Fo(\000)p Fp(e)2186 4326 y Fr(33)2421 4302 y FA(=)e(1.)51 b(W)-8 b(e)33 b(ha)m(v)m(e)h(sev)m(eral)0 4411 y(sub)s(cases)28 b(according)g(to)h(the)g(v)-5 b(alues)29 b(of)h Fs(c;)15 b(d;)g(e)p FA(.)178 4555 y(\(i\))53 b(If)30 b Fs(e)25 b FA(=)g(3,)31 b(then)e Fs(c)c FA(=)g(1)30 b(and)g Fs(b)1348 4507 y Fr(3\()p Fp(d)p Fo(\000)p Fr(1\))1348 4579 y(33)1603 4555 y FA(=)25 b(1.)41 b(There)29 b(are)g(three)g(p)s(ossibilities:)110 4685 y(If)k Fs(d)c FA(=)h(1)i(the)g(matrix)g(is)h(symmetrizable)c(and)j (w)m(e)i(can)e(apply)f(Theorem)h(3.1.)49 b(If)33 b Fs(d)c FA(=)h(2,)0 4795 y Fs(b)39 4762 y Fr(3)39 4818 y(33)145 4795 y FA(=)e(1.)46 b(If)32 b Fs(d)27 b FA(=)i(3,)j Fs(b)771 4762 y Fr(6)771 4818 y(33)877 4795 y FA(=)c(1.)46 b(As)33 b(w)m(e)f(assume)f(that)g(the)g(orders)f(are)h(o)s(dd,)h(w)m(e)g (should)f(ha)m(v)m(e)0 4905 y Fs(b)39 4871 y Fr(3)39 4927 y(33)154 4905 y FA(=)37 b(1.)61 b(These)36 b(t)m(w)m(o)i(p)s (ossibilites)32 b(are)37 b(not)f(p)s(ossible)f(b)m(y)i(our)g(c)m(hoice) e(follo)m(wing)g(the)h(rule)0 5014 y(\(1.10\).)152 5151 y(\(ii\))53 b(If)30 b Fs(e)25 b FA(=)g(2,)31 b(then)e Fs(c)c FA(=)g(1)30 b(and)g Fs(b)1348 5112 y Fr(3)p Fp(d)p Fo(\000)p Fr(2)1348 5175 y(33)1546 5151 y FA(=)25 b(1.)41 b(There)29 b(are)g(three)g(p)s(ossibilities:)p eop %%Page: 19 19 19 18 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(19)110 91 y FA(If)26 b Fs(d)f FA(=)g(1)i Fs(b)478 105 y Fr(33)581 91 y FA(=)e(1,)j(a)e(p)s (ossibilit)m(y)d(that)i(w)m(e)i(excluded.)37 b(If)27 b Fs(d)e FA(=)g(2,)i Fs(b)2410 58 y Fr(4)2410 114 y(33)2513 91 y FA(=)f(1,)h(but)f(w)m(e)h(assume)0 201 y(that)i(the)g(orders)g (are)h(o)s(dd.)39 b(If)31 b Fs(d)24 b FA(=)i(3,)k Fs(b)1399 168 y Fr(7)1399 224 y(33)1502 201 y FA(=)c(1.)40 b(F)-8 b(or)30 b(this)f(reason)g(w)m(e)i(exclude)d(the)h(order)g(7.)127 342 y(\(iii\))52 b(If)30 b Fs(e)25 b FA(=)g(1,)31 b(then)e Fs(b)933 303 y Fr(3)p Fp(cd)p Fo(\000)p Fr(1)933 366 y(33)1164 342 y FA(=)c(1.)41 b(There)29 b(are)g(v)-5 b(arious)29 b(p)s(ossibilities:)110 477 y(As)36 b Fs(cd)e FA(=)g(1)p Fs(;)15 b FA(2)p Fs(;)g FA(3)p Fs(;)g FA(4)p Fs(;)g FA(6)p Fs(;)g FA(9,)39 b(3)p Fs(cd)23 b Fv(\000)h FA(1)35 b(could)f(tak)m(e)i(the)f(v)-5 b(alues)34 b(2)p Fs(;)15 b FA(5)p Fs(;)g FA(8)p Fs(;)g FA(11)p Fs(;)g FA(17)p Fs(;)g FA(26.)58 b(Since)34 b(w)m(e)0 587 y(assume)c(that)e (the)i(orders)f(are)g(o)s(dd,)h(w)m(e)g(ha)m(v)m(e)h(to)e(exclude)g (the)g(orders)g(5,)h(11,)g(13,)h(17.)110 697 y(Let)e(us)h(no)m(w)h (assume)e(that)g Fs(d)c FA(=)g(1.)41 b(Then)29 b Fs(b)1652 658 y Fr(3)p Fp(c)p Fo(\000)p Fp(be)1652 721 y Fr(11)1875 697 y FA(=)d(1.)129 840 y(\(iv\))54 b(If)30 b Fs(c)25 b FA(=)g(3,)31 b(then)e Fs(e)c FA(=)g(1)30 b(and)g Fs(b)1348 800 y Fr(9)p Fo(\000)p Fp(b)1348 864 y Fr(11)1504 840 y FA(=)25 b(1.)41 b(There)29 b(are)g(three)g(p)s(ossibilities:)110 975 y(If)24 b Fs(b)i FA(=)f(1,)h Fs(b)492 942 y Fr(8)492 997 y(11)595 975 y FA(=)f(1;)i(but)c(the)g(orders)h(are)f(o)s(dd.)38 b(If)24 b Fs(b)i FA(=)f(2,)h Fs(b)2113 942 y Fr(7)2113 997 y(11)2216 975 y FA(=)f(1.)39 b(This)24 b(is)f(a)i(new)e(instance)0 1084 y(where)32 b(7)i(should)d(b)s(e)i(excluded.)48 b(If)33 b Fs(b)e FA(=)g(3,)j Fs(b)1600 1051 y Fr(6)1600 1107 y(11)1709 1084 y FA(=)c(1.)50 b(As)35 b(w)m(e)e(assume)g(that)f(the)g (orders)h(are)0 1194 y(o)s(dd,)c(w)m(e)i(should)e(ha)m(v)m(e)h Fs(b)879 1161 y Fr(3)879 1217 y(11)982 1194 y FA(=)c(1.)40 b(This)30 b(con)m(tradicts)d(the)i(rule)g(\(1.10\).)155 1335 y(\(v\))54 b(If)30 b Fs(c)25 b FA(=)g(2,)31 b(then)e Fs(e)c FA(=)g(1)30 b(and)g Fs(b)1348 1296 y Fr(6)p Fo(\000)p Fp(b)1348 1359 y Fr(11)1504 1335 y FA(=)25 b(1.)41 b(There)29 b(are)g(three)g(p)s(ossibilities:)110 1470 y(If)j Fs(b)d FA(=)h(1,)j Fs(b)515 1484 y Fr(11)622 1470 y FA(=)c(5.)47 b(This)32 b(is)g(a)g(new)g(instance)e(where)i(5)g(should)f(b)s(e)g (excluded.)46 b(If)32 b Fs(b)d FA(=)h(2,)0 1580 y Fs(b)39 1547 y Fr(4)39 1602 y(11)142 1580 y FA(=)25 b(1,)k(but)f(w)m(e)h (assume)f(that)f(the)g(orders)h(are)f(o)s(dd.)39 b(If)29 b Fs(b)c FA(=)h(3,)j Fs(b)2331 1547 y Fr(3)2331 1602 y(11)2434 1580 y FA(=)c(1.)40 b(This)28 b(con)m(tradicts)0 1689 y(the)h(rule)g(\(1.10\).)129 1830 y(\(vi\))54 b(If)30 b Fs(c)25 b FA(=)g(1,)31 b(then)e Fs(b)930 1791 y Fr(3)p Fo(\000)p Fp(be)930 1855 y Fr(11)1120 1830 y FA(=)d(1)f(=)g Fs(b)1423 1791 y Fr(3)p Fo(\000)p Fp(be)1423 1855 y Fr(33)1588 1830 y FA(.)41 b(There)29 b(are)h(v)-5 b(arious)28 b(p)s(ossibilities:) 110 1966 y(As)h Fs(be)c FA(=)h(1)p Fs(;)15 b FA(2)p Fs(;)g FA(3)p Fs(;)g FA(4)p Fs(;)g FA(6)p Fs(;)g FA(9,)30 b(3)17 b Fv(\000)g Fs(be)29 b FA(could)e(tak)m(e)h(the)g(v)-5 b(alues)27 b(2)p Fs(;)15 b FA(1)p Fs(;)g FA(0)p Fs(;)g Fv(\000)p FA(1)p Fs(;)g Fv(\000)p FA(3)p Fs(;)g Fv(\000)p FA(6.)42 b(The)28 b(v)-5 b(alues)0 2075 y(2)p Fs(;)15 b FA(1)p Fs(;)g Fv(\000)p FA(1)41 b(are)g(excluded)d(b)m(y)k(h)m(yp)s (othesis;)j(the)40 b(v)-5 b(alues)39 b Fv(\000)p FA(3)p Fs(;)15 b Fv(\000)p FA(6)42 b(are)e(excluded)f(b)m(y)i(the)f(rule)0 2185 y(\(1.10\).)56 b(The)35 b(v)-5 b(alue)34 b(0)i(arises)e(only)h (when)g FC(b)h FA(is)g(symmetrizable,)e(but)h(then)f(w)m(e)i(can)f (apply)0 2294 y(Theorem)29 b(3.1.)110 2404 y(W)-8 b(e)27 b(next)g(assume)h(that)e FC(b)i FA(is)g(a)f(cycle)g(with)f(no)i(triple) d(arro)m(w)i(but)g(at)h(least)e(a)h(double)f(one.)0 2514 y(If)36 b Fs(\022)h Fv(\025)e FA(6,)j(again)c(w)m(e)j(ha)m(v)m(e)f(a)g (con)m(tradiction)31 b(b)m(y)37 b(remo)m(ving)e(a)g(suitable)f(v)m (ertex.)57 b(If)36 b Fs(\022)i FA(=)d(5,)0 2623 y(w)m(e)k(remo)m(v)m(e) g(systematically)c(di\013eren)m(t)i(v)m(ertices.)65 b(Since)37 b(the)h(resulting)e(sub)s(diagrams)g(are)0 2733 y(symmetrizable)30 b(and)i(hence)f(of)h(\014nite)f(t)m(yp)s(e,)i(w)m(e)g(see)f(that)g FC(b)h FA(has)f(only)g(one)g(double)f(arro)m(w.)0 2842 y(W)-8 b(e)30 b(can)g(easily)e(get)i(a)g(con)m(tradiction.)110 2952 y(No)m(w)h(w)m(e)f(assume)g Fs(\022)d FA(=)f(4,)k(with)f(a)h (Cartan)f(matrix)1001 3305 y(\()p Fs(a)1085 3319 y Fp(ij)1149 3305 y FA(\))c(=)1306 3094 y Fl(0)1306 3255 y(B)1306 3313 y(@)1441 3141 y FA(2)132 b Fv(\000)p FA(2)c(0)h Fv(\000)p Fs(b)1406 3250 y Fv(\000)p FA(1)j(2)f Fv(\000)p Fs(c)d FA(0)1441 3360 y(0)j Fv(\000)p Fs(d)c FA(2)g Fv(\000)p Fs(e)1401 3470 y Fv(\000)p Fs(f)138 b FA(0)127 b Fv(\000)p Fs(g)h FA(2)2166 3094 y Fl(1)2166 3255 y(C)2166 3313 y(A)2261 3305 y Fs(:)0 3659 y FA(Here)30 b Fs(b;)15 b(c;)g(d;)g(e;)g(f) 5 b(;)15 b(g)35 b FA(are)c(p)s(ositiv)m(e)d(in)m(tegers)h(suc)m(h)h (that)g Fs(bf)5 b(;)15 b(cd;)g(eg)28 b FA(=)f(1)p Fs(;)15 b FA(2.)42 b(By)31 b(remo)m(ving)f(the)0 3768 y(third,)38 b(resp)s(ectiv)m(ely)d(the)h(fourth,)j(v)m(ertex,)g(w)m(e)f(conclude)d (that)h Fs(bf)48 b FA(=)38 b Fs(cd)e FA(=)i(1.)63 b(There)37 b(are)0 3878 y(three)j(p)s(ossibilities:)59 b(\()p Fs(e;)15 b(g)s FA(\))44 b(=)g(\(1)p Fs(;)15 b FA(1\))p Fs(;)g FA(\(1)p Fs(;)g FA(2\))41 b(or)g(\(2)p Fs(;)15 b FA(1\).)75 b(The)41 b(\014rst)g(is)h(not)f(p)s(ossible)e(b)m(y)j(a)0 3987 y(similar)28 b(argumen)m(t)h(as)h(for)g Fs(\022)f FA(=)c(5.)41 b(The)30 b(second)f(corresp)s(onds)f(to)i(a)g (symmetrizable)d(matrix)0 4097 y(and)j(the)f(third)f(is)i(the)g (\014rst)f(instance)f(forcing)g(to)h(exclude)g(the)g(order)g(3.)110 4207 y(W)-8 b(e)36 b(\014nally)f(assume)h(that)f Fs(\022)j FA(=)e(3.)60 b(W)-8 b(e)37 b(ha)m(v)m(e)f(a)h(3-cycle)e(with)g(no)i (triple)d(arro)m(w)i(but)f(at)0 4316 y(least)h(a)i(double)d(one.)63 b(Arguing)36 b(as)i(in)f(the)f(case)h(of)h(triple)d(arro)m(ws,)k(w)m(e) f(eliminate)d(all)h(the)0 4426 y(p)s(ossibilities)25 b(except)k(t)m(w)m(o;)i(for)e(those)g(w)m(e)i(exclude)d(the)h(orders)g (3)h(and)g(7.)41 b(F)-8 b(or)30 b(instance,)1109 4733 y(\()p Fs(a)1193 4747 y Fp(ij)1257 4733 y FA(\))25 b(=)1415 4550 y Fl(0)1415 4714 y(@)1545 4624 y FA(2)127 b Fv(\000)p FA(2)91 b Fv(\000)p FA(1)1510 4733 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(2)1510 4843 y Fv(\000)p FA(2)91 b Fv(\000)p FA(1)126 b(2)2058 4550 y Fl(1)2058 4714 y(A)2153 4733 y Fs(;)0 5041 y FA(then)967 5151 y Fs(b)1006 5113 y Fr(2)1006 5173 y(11)1109 5151 y FA(=)25 b Fs(b)1244 5165 y Fr(22)1322 5151 y Fs(;)107 b(b)1493 5113 y Fr(2)1493 5173 y(22)1596 5151 y FA(=)25 b Fs(b)1731 5165 y Fr(33)1809 5151 y Fs(;)107 b(b)1980 5165 y Fr(11)2082 5151 y FA(=)26 b Fs(b)2218 5113 y Fr(2)2218 5173 y(33)2295 5151 y Fs(:)p eop %%Page: 20 20 20 19 bop 0 -128 a Fy(20)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FA(Hence)f Fs(b)310 105 y Fr(11)413 91 y FA(=)d Fs(b)549 58 y Fr(8)549 114 y(11)626 91 y FA(,)31 b(whic)m(h)e(is)h(not)g(p)s(ossible)d(b)s(ecause)h(the)h(order) g(of)h Fs(b)2418 105 y Fr(11)2526 91 y FA(is)g(not)f(7.)0 257 y Ft(R)-5 b(emark.)46 b FA(W)-8 b(e)31 b(observ)m(e)f(that)f(the)h (preceding)e(pro)s(of)i(of)h(Theorem)e(1.1)i(P)m(art)g(\(2\))f(only)g (needs)0 366 y(the)f(h)m(yp)s(othesis)f("lo)s(cally)f(of)j(FL-t)m(yp)s (e")f(to)h(conclude)e(that)86 503 y(\(4.8\))54 b(A)29 b(\014nite)e(dimensional)f(Nic)m(hols)h(algebra)g Fm(B)p FA(\()p Fs(V)20 b FA(\))28 b(of)h(rank)f(2)h(is)f(necessarily)e(of)j (\014nite)329 613 y(t)m(yp)s(e.)110 750 y(If)h(\000)h(is)e(an)m(y)i (\014nite)d(ab)s(elian)f(group)j(suc)m(h)g(that)e(\(4.8\))i(holds)f (for)g Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b(in)2726 717 y Fr(\000)2726 775 y(\000)2776 750 y Fv(Y)8 b Fs(D)32 b FA(then)d(the)0 861 y(conclusion)e(of)j(Theorem)f(1.1)h(P)m(art)g (\(2\))f(is)h(v)-5 b(alid)29 b(for)g(all)g Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b(in)2374 828 y Fr(\000)2374 887 y(\000)2424 861 y Fv(Y)8 b Fs(D)r FA(.)110 1082 y(F)-8 b(or)33 b(further)f(use,)j (w)m(e)f(list)e(the)h(Cartan)f(matrices,)h(up)g(to)g(n)m(um)m(b)s (ering,)h(causing)d(trouble)0 1192 y(for)f(small)f(v)-5 b(alues)29 b(of)h Fs(p)p FA(:)86 1329 y(\(4.9\))54 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(3,)31 b(the)e(matrix)1109 1658 y(\()p Fs(a)1193 1672 y Fp(ij)1257 1658 y FA(\))c(=)1415 1475 y Fl(0)1415 1639 y(@)1545 1549 y FA(2)127 b Fv(\000)p FA(2)91 b Fv(\000)p FA(1)1510 1658 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(1)1510 1768 y Fv(\000)p FA(2)91 b Fv(\000)p FA(1)126 b(2)2058 1475 y Fl(1)2058 1639 y(A)2153 1658 y Fs(:)41 1988 y FA(\(4.10\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(3,)31 b(the)e(matrix)1005 2363 y(\()p Fs(a)1089 2377 y Fp(ij)1153 2363 y FA(\))c(=)1311 2152 y Fl(0)1311 2313 y(B)1311 2371 y(@)1441 2199 y FA(2)127 b Fv(\000)p FA(2)f(0)h Fv(\000)p FA(1)1406 2308 y Fv(\000)p FA(1)f(2)h Fv(\000)p FA(1)f(0)1441 2418 y(0)h Fv(\000)p FA(1)f(2)h Fv(\000)p FA(2)1406 2528 y Fv(\000)p FA(1)f(0)h Fv(\000)p FA(1)f(2)2162 2152 y Fl(1)2162 2313 y(C)2162 2371 y(A)2257 2363 y Fs(:)41 2738 y FA(\(4.11\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(5,)31 b(the)e(matrix)1109 3068 y(\()p Fs(a)1193 3082 y Fp(ij)1257 3068 y FA(\))c(=)1415 2885 y Fl(0)1415 3049 y(@)1545 2958 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 3068 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(1)1510 3178 y Fv(\000)p FA(2)91 b Fv(\000)p FA(1)126 b(2)2058 2885 y Fl(1)2058 3049 y(A)2153 3068 y Fs(:)41 3398 y FA(\(4.12\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(5,)31 b(the)e(matrix)1109 3727 y(\()p Fs(a)1193 3741 y Fp(ij)1257 3727 y FA(\))c(=)1415 3544 y Fl(0)1415 3708 y(@)1545 3618 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 3727 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(2)1510 3837 y Fv(\000)p FA(1)91 b Fv(\000)p FA(1)126 b(2)2058 3544 y Fl(1)2058 3708 y(A)2153 3727 y Fs(:)41 4057 y FA(\(4.13\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(7,)31 b(the)e(matrix)1109 4386 y(\()p Fs(a)1193 4400 y Fp(ij)1257 4386 y FA(\))c(=)1415 4203 y Fl(0)1415 4367 y(@)1545 4277 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 4386 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(1)1510 4496 y Fv(\000)p FA(3)91 b Fv(\000)p FA(2)126 b(2)2058 4203 y Fl(1)2058 4367 y(A)2153 4386 y Fs(:)41 4716 y FA(\(4.14\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(7,)31 b(the)e(matrix)1109 5046 y(\()p Fs(a)1193 5060 y Fp(ij)1257 5046 y FA(\))c(=)1415 4862 y Fl(0)1415 5026 y(@)1545 4936 y FA(2)127 b Fv(\000)p FA(2)91 b Fv(\000)p FA(1)1510 5046 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(2)1510 5155 y Fv(\000)p FA(2)91 b Fv(\000)p FA(1)126 b(2)2058 4862 y Fl(1)2058 5026 y(A)2153 5046 y Fs(:)p eop %%Page: 21 21 21 20 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(21)41 91 y FA(\(4.15\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(11,)30 b(the)g(matrix)1109 434 y(\()p Fs(a)1193 448 y Fp(ij)1257 434 y FA(\))25 b(=)1415 250 y Fl(0)1415 415 y(@)1545 324 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 434 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(2)1510 543 y Fv(\000)p FA(2)91 b Fv(\000)p FA(1)126 b(2)2058 250 y Fl(1)2058 415 y(A)2153 434 y Fs(:)41 776 y FA(\(4.16\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(13,)30 b(the)g(matrix)1109 1119 y(\()p Fs(a)1193 1133 y Fp(ij)1257 1119 y FA(\))25 b(=)1415 935 y Fl(0)1415 1100 y(@)1545 1009 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 1119 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(3)1510 1228 y Fv(\000)p FA(3)91 b Fv(\000)p FA(1)126 b(2)2058 935 y Fl(1)2058 1100 y(A)2153 1119 y Fs(:)41 1461 y FA(\(4.17\))53 b(F)-8 b(or)30 b Fs(p)25 b FA(=)g(17,)30 b(the)g(matrix)1109 1804 y(\()p Fs(a)1193 1818 y Fp(ij)1257 1804 y FA(\))25 b(=)1415 1620 y Fl(0)1415 1785 y(@)1545 1694 y FA(2)127 b Fv(\000)p FA(3)91 b Fv(\000)p FA(1)1510 1804 y Fv(\000)p FA(1)126 b(2)h Fv(\000)p FA(2)1510 1913 y Fv(\000)p FA(3)91 b Fv(\000)p FA(1)126 b(2)2058 1620 y Fl(1)2058 1785 y(A)2153 1804 y Fs(:)0 2274 y Fv(x)p FC(5.)46 b(Nic)m(hols)36 b(algebras)f(o)m(v)m(er)h Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))p FC(.)110 2388 y FA(In)g(this)f(section,)h(\000)h(will)e(denote) f Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\),)f Fs(p)k FA(an)f(o)s(dd)f(prime.) 59 b(W)-8 b(e)36 b(\014rst)g(discuss)f(Nic)m(hols)0 2498 y(algebras)24 b(of)j(rank)f(2.)39 b(If)27 b Fs(V)47 b FA(is)27 b(a)f(Y)-8 b(etter-Drinfeld)24 b(mo)s(dule)g(of)j(dimension)d (2)i(satisfying)f(\(1.6\),)0 2608 y(then)k(there)g(exists)g(a)h (generator)d Fs(u)k FA(of)f(\000,)h Fs(q)d Fv(2)e Fs(k)1683 2574 y Fo(\002)1774 2608 y FA(of)k(order)f Fs(p)h FA(and)g(in)m(tegers) e Fs(b)p FA(,)j Fs(d)f FA(suc)m(h)g(that)0 2955 y(\(5.1\))364 b Fs(g)s FA(\(1\))23 b(=)j Fs(u;)107 b(g)s FA(\(2\))24 b(=)h Fs(u)1359 2917 y Fp(b)1396 2955 y Fs(;)107 b Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)s(;)107 b Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fs(q)2667 2917 y Fp(d)2710 2955 y Fs(:)110 3197 y FA(So)k Fs(b)275 3211 y Fr(11)378 3197 y FA(=)d Fs(q)s(;)107 b(b)690 3211 y Fr(22)793 3197 y FA(=)25 b Fs(q)933 3159 y Fp(bd)1008 3197 y FA(.)0 3375 y FC(Dynkin)36 b(diagram)f(of)g(t)m(yp) s(e)h Fs(A)1206 3389 y Fr(2)1247 3375 y FC(.)53 b FA(W)-8 b(e)30 b(should)f(ha)m(v)m(e)923 3612 y Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(g)s FA(\(2\))p Fv(ih)p Fs(\037)p FA(\(2\))p Fs(;)g(g)s FA(\(1\))p Fv(i)k FA(=)26 b Fs(b)1982 3573 y Fo(\000)p Fr(1)1982 3637 y(11)2105 3612 y FA(=)g Fs(b)2241 3573 y Fo(\000)p Fr(1)2241 3637 y(22)2339 3612 y Fs(:)110 3804 y FA(This)35 b(means)h Fs(b)24 b FA(+)h Fs(d)35 b Fv(\021)h(\000)p FA(1)f Fv(\021)h(\000)p Fs(bd)60 b FA(mo)s(d)30 b Fs(p)p FA(.)59 b(It)36 b(follo)m(ws)f(readily)f(that)h Fs(b)2734 3771 y Fr(2)2800 3804 y FA(+)24 b Fs(b)h FA(+)f(1)35 b Fv(\021)h FA(0)0 3914 y(mo)s(d)29 b Fs(p)p FA(.)41 b(Hence)28 b Fs(p)d FA(=)h(3)i(and)h Fs(b)c Fv(\021)h FA(1)60 b(mo)s(d)30 b Fs(p)p FA(,)f(or)g Fs(b)c Fv(6\021)h FA(1)60 b(mo)s(d)30 b Fs(p)f FA(is)f(a)h(cubic)e(ro)s(ot)g(of)i(1.)40 b(In)29 b(the)0 4023 y(last)g(case,)g Fs(b)i FA(exists)e(if)g(and)h (only)f(if)g Fs(p)20 b Fv(\000)g FA(1)30 b(is)g(divisible)d(b)m(y)j(3.) 41 b(The)29 b(corresp)s(onding)d(matrices)0 4133 y(are)j(not)h (symmetric)f(if)g Fs(p)d Fv(6)p FA(=)f(3.)0 4375 y FC(Dynkin)36 b(diagram)f(of)g(t)m(yp)s(e)h Fs(B)1207 4389 y Fr(2)1248 4375 y FC(.)52 b FA(W)-8 b(e)31 b(should)d(ha)m(v)m(e)923 4612 y Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(g)s FA(\(2\))p Fv(ih)p Fs(\037)p FA(\(2\))p Fs(;)g(g)s FA(\(1\))p Fv(i)k FA(=)26 b Fs(b)1982 4573 y Fo(\000)p Fr(1)1982 4637 y(11)2105 4612 y FA(=)g Fs(b)2241 4573 y Fo(\000)p Fr(2)2241 4637 y(22)2339 4612 y Fs(:)110 4804 y FA(This)38 b(means)g Fs(b)26 b FA(+)g Fs(d)40 b Fv(\021)f(\000)p FA(1)h Fv(\021)g(\000)p FA(2)p Fs(bd)60 b FA(mo)s(d)29 b Fs(p)p FA(.)67 b(Th)m(us)39 b(2)p Fs(b)2175 4771 y Fr(2)2242 4804 y FA(+)26 b(2)p Fs(b)g FA(+)g(1)40 b Fv(\021)f FA(0)61 b(mo)s(d)30 b Fs(p)39 b FA(and)0 4914 y(lo)s(oking)23 b(at)j(the)f(discriminan)m(t)d (of)k(this)f(equation)e(w)m(e)j(see)g(that)e(it)h(has)h(a)g(solution)d (if)i(and)h(only)0 5023 y(if)k Fv(\000)p FA(1)g(is)g(a)g(square,)g Ft(i.e.)41 b FA(exactly)29 b(when)g Fs(p)d Fv(\021)g FA(1)60 b(mo)s(d)30 b(4.)41 b(The)29 b(corresp)s(onding)d(matrices)j (are)0 5133 y(not)g(symmetric.)p eop %%Page: 22 22 22 21 bop 0 -128 a Fy(22)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Dynkin)36 b(diagram)f(of)g(t)m(yp)s(e)h Fs(G)1210 105 y Fr(2)1250 91 y FC(.)53 b FA(By)30 b(\(1.10\),)g(w)m(e)g (ha)m(v)m(e)h Fs(p)25 b(>)g FA(3.)41 b(W)-8 b(e)30 b(should)f(ha)m(v)m (e)923 277 y Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(g)s FA(\(2\))p Fv(ih)p Fs(\037)p FA(\(2\))p Fs(;)g(g)s FA(\(1\))p Fv(i)k FA(=)26 b Fs(b)1982 238 y Fo(\000)p Fr(1)1982 302 y(11)2105 277 y FA(=)g Fs(b)2241 238 y Fo(\000)p Fr(3)2241 302 y(22)2339 277 y Fs(:)110 461 y FA(This)38 b(means)g Fs(b)26 b FA(+)g Fs(d)40 b Fv(\021)f(\000)p FA(1)h Fv(\021)g(\000)p FA(3)p Fs(bd)60 b FA(mo)s(d)29 b Fs(p)p FA(.)67 b(Th)m(us)39 b(3)p Fs(b)2175 428 y Fr(2)2242 461 y FA(+)26 b(3)p Fs(b)g FA(+)g(1)40 b Fv(\021)f FA(0)61 b(mo)s(d)30 b Fs(p)39 b FA(and)0 570 y(lo)s(oking)23 b(at)j(the)f(discriminan)m(t)d(of)k(this)f(equation)e(w)m(e)j(see)g (that)e(it)h(has)h(a)g(solution)d(if)i(and)h(only)0 680 y(if)33 b Fv(\000)p FA(3)g(is)g(a)h(square)93 b(mo)s(d)30 b Fs(p)p FA(.)50 b(By)34 b(the)e(quadratic)f(recipro)s(cit)m(y)f(la)m (w,)k(this)e(happ)s(ens)g(exactly)0 789 y(when)d Fs(p)d Fv(\021)f FA(1)61 b(mo)s(d)29 b(3.)41 b(The)30 b(matrices)e(are)h(not)g (symmetric.)110 993 y(W)-8 b(e)30 b(no)m(w)g(conclude)e(the)h(pro)s(of) g(of)h(the)f(existence)e(part)j(of)g(Theorem)e(1.3.)0 1149 y FC(Prop)s(osition)38 b(5.1.)51 b Fj(There)32 b(is)g(no)g(Nic)m (hols)g(algebra)e(of)j(rank)f Fv(\025)d FA(3)k Fj(o)m(v)m(er)g Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))27 b Fj(with)32 b(\014nite)0 1259 y(Cartan)21 b(matrix)h(except)f(when)h Fs(p)k FA(=)f(3)e Fj(and)f(the)g(corresp)s(onding)c(Dynkin)k(diagram)f(is)i Fs(A)3055 1273 y Fr(2)3102 1259 y Fv(\002)5 b Fs(A)3246 1273 y Fr(1)0 1368 y Fj(or)30 b Fs(A)180 1382 y Fr(2)242 1368 y Fv(\002)20 b Fs(A)401 1382 y Fr(2)442 1368 y Fj(.)0 1525 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)30 b(\014rst)g(consider)e (Dynkin)h(diagrams)g(of)h(rank)g(3.)40 b(W)-8 b(e)30 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(cases:)383 1845 y(\()p Fs(a)p FA(\))15 b Fs(A)586 1859 y Fr(2)647 1845 y Fv(\002)21 b Fs(A)807 1859 y Fr(1)848 1845 y Fs(;)186 b FA(\()p Fs(b)p FA(\))15 b Fs(B)1254 1859 y Fr(2)1315 1845 y Fv(\002)21 b Fs(A)1475 1859 y Fr(1)1516 1845 y Fs(;)198 b FA(\()p Fs(c)p FA(\))14 b Fs(G)1937 1859 y Fr(2)1998 1845 y Fv(\002)20 b Fs(A)2157 1859 y Fr(1)2199 1845 y Fs(;)384 1987 y FA(\()p Fs(d)p FA(\))14 b Fs(A)586 2001 y Fr(3)627 1987 y Fs(;)404 b FA(\()p Fs(e)p FA(\))14 b Fs(B)1254 2001 y Fr(3)1295 1987 y Fs(;)404 b FA(\()p Fs(f)10 b FA(\))k Fs(C)1930 2001 y Fr(3)1971 1987 y Fs(;)199 b FA(\()p Fs(g)s FA(\))15 b Fs(A)2397 2001 y Fr(1)2457 1987 y Fv(\002)21 b Fs(A)2617 2001 y Fr(1)2678 1987 y Fv(\002)g Fs(A)2838 2001 y Fr(1)2879 1987 y Fs(:)110 2175 y FA(The)31 b(case)g(\(g\))g(do)s(es)f(not)h (arise,)h(as)g(sho)m(wn)f(in)g([AS2,)j(Section)29 b(4].)46 b(Therefore)30 b(w)m(e)i(can)f(\014x)0 2284 y(a)f(n)m(umeration)e(of)j (the)e(v)m(ertices)g(of)h(the)g(Dynkin)f(diagram)g(suc)m(h)h(that)f (the)h(v)m(ertices)f(1)h(and)g(2)0 2394 y(\(resp.,)e(1)h(and)e(3\))h (are)g(connected)d(\(resp.,)j(not)g(connected\).)36 b(W)-8 b(e)29 b(also)e(assume)h(that)f(there)f(is)0 2503 y(only)j(one)h(arro)m (w)g(b)s(et)m(w)m(een)e(1)i(and)g(2)g(in)f(cases)h(\(e\))f(an)h(\(f)7 b(\).)40 b(That)29 b(is,)502 2824 y(\()p Fs(a)p FA(\))35 b Fv(\016)662 2880 y Fr(1)p 734 2803 92 4 v 857 2824 a Fv(\016)861 2880 y Fr(2)1105 2824 y Fv(\016)1110 2880 y Fr(3)1151 2824 y Fs(;)186 b FA(\()p Fs(b)p FA(\))15 b Fv(\016)1492 2880 y Fr(1)1559 2824 y Fv(\))25 b(\016)1680 2880 y Fr(2)1904 2824 y Fv(\016)1908 2880 y Fr(3)1949 2824 y Fs(;)198 b FA(\()p Fs(c)p FA(\))14 b Fv(\016)2303 2880 y Fr(1)2369 2824 y Fi(V)26 b Fv(\016)2491 2880 y Fr(2)2714 2824 y Fv(\016)2719 2880 y Fr(3)2760 2824 y Fs(;)503 2994 y FA(\()p Fs(d)p FA(\))34 b Fv(\016)662 3051 y Fr(1)p 734 2973 V 857 2994 a Fv(\016)861 3051 y Fr(2)p 934 2973 V 1036 2994 a Fv(\016)1041 3051 y Fr(3)1082 2994 y Fs(;)252 b FA(\()p Fs(e)p FA(\))34 b Fv(\016)1513 3051 y Fr(1)p 1585 2973 V 1687 2994 a Fv(\016)1692 3051 y Fr(2)1758 2994 y Fv(\))26 b(\016)1880 3051 y Fr(3)1921 2994 y Fs(;)211 b FA(\()p Fs(f)10 b FA(\))35 b Fv(\016)2323 3051 y Fr(1)p 2395 2973 V 2498 2994 a Fv(\016)2502 3051 y Fr(2)2569 2994 y Fv(\()25 b(\016)2690 3051 y Fr(3)2731 2994 y Fs(:)110 3211 y FA(If)34 b Fs(V)56 b FA(is)34 b(a)h(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)i(of)i(dimension)d(3)j (satisfying)d(\(1.6\))i(and)g(of)g(Cartan)0 3321 y(t)m(yp)s(e)c(\(a\),) f(\(b\),)h Fs(:)15 b(:)g(:)47 b FA(or)30 b(\(f)7 b(\),)30 b(then)f(there)g(exist)g(a)h(generator)e Fs(u)i FA(of)g(\000,)h Fs(q)e Fv(2)c Fs(k)2629 3288 y Fo(\002)2721 3321 y FA(of)30 b(order)f Fs(p)i FA(and)0 3430 y(in)m(tegers)d Fs(b;)15 b(d;)g(e;)g(f)41 b FA(\(none)29 b(of)h(them)f(divisible)e(b)m(y)k Fs(p)p FA(\))f(suc)m(h)g(that)792 3751 y Fs(g)s FA(\(1\))24 b(=)h Fs(u;)358 b(g)s FA(\(2\))23 b(=)j Fs(u)1849 3713 y Fp(b)1886 3751 y Fs(;)357 b(g)s FA(\(3\))24 b(=)h Fs(u)2605 3713 y Fp(e)2645 3751 y Fs(;)618 3906 y Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fs(q)s(;)191 b Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)1841 3868 y Fp(d)1883 3906 y Fs(;)186 b Fv(h)p Fs(\037)p FA(\(3\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fs(q)2597 3868 y Fp(f)2645 3906 y Fs(:)110 4103 y FA(So)k Fs(b)275 4117 y Fr(11)378 4103 y FA(=)d Fs(q)s FA(,)31 b Fs(b)614 4117 y Fr(22)717 4103 y FA(=)25 b Fs(q)857 4070 y Fp(bd)932 4103 y FA(,)30 b Fs(b)1026 4117 y Fr(33)1129 4103 y FA(=)c Fs(q)1270 4070 y Fp(ef)1352 4103 y FA(.)41 b(Also,)30 b Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(g)s FA(\(3\))p Fv(ih)p Fs(\037)p FA(\(3\))p Fs(;)g(g)s FA(\(1\))p Fv(i)20 b FA(=)25 b(1)30 b(means)0 4396 y(\(5.2\))1083 b Fs(e)20 b FA(+)g Fs(f)35 b Fv(\021)25 b FA(0)91 b(mo)s(d)30 b Fs(p:)110 4579 y FA(No)m(w,)d(considering)21 b(the)i(sub)s(diagram)f (supp)s(orted)g(b)m(y)j(the)f(v)m(ertices)f(1)i(and)f(2,)i(w)m(e)f (conclude)0 4689 y(from)30 b(the)f(argumen)m(ts)g(ab)s(o)m(v)m(e)h(for) f(the)h(rank)f(2)i(case)e(that)86 4822 y(\(5.3\))54 b(In)36 b(cases)f(\(a\),)i(\(d\),)g(\(e\))e(and)h(\(f)7 b(\),)37 b Fs(b)1589 4789 y Fr(2)1655 4822 y FA(+)24 b Fs(b)h FA(+)f(1)35 b Fv(\021)g FA(0)61 b(mo)s(d)29 b Fs(p)p FA(,)39 b Fs(bd)c Fv(\021)g FA(1)61 b(mo)s(d)29 b Fs(p)37 b FA(and)329 4932 y Fs(p)25 b FA(=)g(3)31 b(or)e Fs(p)d Fv(\021)f FA(1)61 b(mo)s(d)29 b(3.)86 5041 y(\(5.4\))54 b(In)30 b(case)f(\(b\),)h(2)p Fs(b)899 5008 y Fr(2)960 5041 y FA(+)20 b(2)p Fs(b)g FA(+)h(1)k Fv(\021)g FA(0)61 b(mo)s(d)29 b Fs(p)p FA(,)i(2)p Fs(bd)25 b Fv(\021)g FA(1)61 b(mo)s(d)29 b Fs(p)h FA(and)g Fs(p)25 b Fv(\021)h FA(1)60 b(mo)s(d)30 b(4.)86 5151 y(\(5.5\))54 b(In)30 b(case)f(\(c\),)h(3)p Fs(b)889 5118 y Fr(2)950 5151 y FA(+)20 b(3)p Fs(b)g FA(+)h(1)k Fv(\021)g FA(0)61 b(mo)s(d)29 b Fs(p)p FA(,)i(3)p Fs(bd)25 b Fv(\021)g FA(1)60 b(mo)s(d)30 b Fs(p)g FA(and)g Fs(p)25 b Fv(\021)h FA(1)60 b(mo)s(d)30 b(3.)p eop %%Page: 23 23 23 22 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(23)110 91 y FA(On)26 b(the)g(other)g(hand,)g(in)h(cases)e(\(a\),)i(\(b\))f(and)g(\(c\))g(w)m (e)h(ha)m(v)m(e)g Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(g)s FA(\(3\))p Fv(ih)p Fs(\037)p FA(\(3\))p Fs(;)g(g)s FA(\(2\))p Fv(i)k FA(=)26 b(1,)0 201 y Ft(i.e.)0 505 y FA(\(5.6\))1039 b Fs(ed)20 b FA(+)g Fs(bf)35 b Fv(\021)26 b FA(0)91 b(mo)s(d)29 b Fs(p:)110 699 y FA(No)m(w,)i(com)m(bining)d (\(5.6\))h(with)g(\(5.2\))g(w)m(e)i(conclude)c(that)i Fs(b)d Fv(\021)f Fs(d)60 b FA(mo)s(d)30 b Fs(p)p FA(.)41 b(Hence:)110 858 y(In)30 b(case)f(\(a\),)h Fs(b)629 825 y Fr(2)695 858 y Fv(\021)25 b FA(1)61 b(mo)s(d)29 b Fs(p)p FA(.)41 b(But)30 b(also)f Fs(b)1619 825 y Fr(3)1685 858 y Fv(\021)d FA(1)60 b(mo)s(d)30 b Fs(p)p FA(,)h(so)f Fs(b)25 b Fv(\021)h FA(1)60 b(mo)s(d)30 b Fs(p)g FA(and)g Fs(p)25 b FA(=)g(3.)110 1016 y(In)43 b(case)f(\(b\),)k(2)p Fs(b)722 983 y Fr(2)810 1016 y Fv(\021)h FA(1)60 b(mo)s(d)30 b Fs(p)43 b FA(and)g(th)m(us)g Fs(b)k Fv(\021)g(\000)p FA(1)61 b(mo)s(d)29 b Fs(p)44 b FA(and)e(1)47 b Fv(\021)g FA(0)61 b(mo)s(d)29 b Fs(p)p FA(,)47 b(a)0 1126 y(con)m(tradiction.)110 1285 y(In)38 b(case)f(\(c\),)j(3)p Fs(b)696 1252 y Fr(2)775 1285 y Fv(\021)f FA(1)61 b(mo)s(d)29 b Fs(p)p FA(.)65 b(Plugging)36 b(this)h(in)m(to)h(the)f(\014rst)g(equation)f(of)i (\(5.5\))g(w)m(e)0 1394 y(easily)29 b(get)g(a)h(con)m(tradiction.)110 1553 y(No)m(w)h(w)m(e)f(turn)f(to)h(the)f(remaining)f(cases.)39 b(In)31 b(case)e(\(d\),)g(w)m(e)i(ha)m(v)m(e)923 1750 y Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(g)s FA(\(3\))p Fv(ih)p Fs(\037)p FA(\(3\))p Fs(;)g(g)s FA(\(2\))p Fv(i)k FA(=)26 b Fs(b)1982 1711 y Fo(\000)p Fr(1)1982 1774 y(22)2105 1750 y FA(=)g Fs(b)2241 1711 y Fo(\000)p Fr(1)2241 1774 y(33)2339 1750 y FA(;)0 1944 y(using)j(\(5.2\),)g(this)h(implies)1070 2163 y Fs(f)10 b FA(\()p Fs(b)19 b Fv(\000)i Fs(d)p FA(\))j Fv(\021)i(\000)p FA(1)f Fv(\021)g Fs(f)1809 2126 y Fr(2)1941 2163 y FA(mo)s(d)k Fs(p:)110 2326 y FA(Th)m(us)e Fs(b)14 b Fv(\000)g Fs(d)25 b Fv(\021)g Fs(f)71 b FA(mo)s(d)29 b Fs(p)e FA(and)g(\()p Fs(b)14 b Fv(\000)g Fs(d)p FA(\))1472 2293 y Fr(2)1537 2326 y Fv(\021)25 b Fs(b)1672 2293 y Fr(2)1727 2326 y FA(+)14 b Fs(b)g Fv(\000)g FA(2)25 b Fv(\021)h(\000)p FA(1)60 b(mo)s(d)30 b Fs(p)p FA(;)e(this)e(last)g (equation)0 2438 y(con)m(tradicts)j(the)j(\014rst)g(of)g(\(5.3\).)47 b(In)33 b(case)e(\(e\),)i(w)m(e)g(ha)m(v)m(e)f Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(g)s FA(\(3\))p Fv(ih)p Fs(\037)p FA(\(3\))p Fs(;)g(g)s FA(\(2\))p Fv(i)24 b FA(=)29 b Fs(b)3089 2399 y Fo(\000)p Fr(1)3089 2462 y(22)3217 2438 y FA(=)0 2552 y Fs(b)39 2513 y Fo(\000)p Fr(2)39 2576 y(33)137 2552 y FA(;)i(using)e(\(5.2\),)h(this)f(implies)1047 2771 y Fs(f)10 b FA(\()p Fs(b)20 b Fv(\000)g Fs(d)p FA(\))k Fv(\021)i(\000)p FA(1)f Fv(\021)g FA(2)p Fs(f)1832 2733 y Fr(2)1964 2771 y FA(mo)s(d)k Fs(p:)110 2933 y FA(Th)m(us)h Fs(b)20 b Fv(\000)g Fs(d)25 b Fv(\021)g FA(2)p Fs(f)70 b FA(mo)s(d)30 b Fs(p)g FA(and)f(\()p Fs(b)20 b Fv(\000)g Fs(d)p FA(\))1550 2900 y Fr(2)1615 2933 y Fv(\021)25 b Fs(b)1750 2900 y Fr(2)1811 2933 y FA(+)20 b Fs(b)g Fv(\000)g FA(2)25 b Fv(\021)h FA(4)p Fs(f)2321 2900 y Fr(2)2386 2933 y Fv(\021)g(\000)p FA(2)60 b(mo)s(d)30 b Fs(p)p FA(;)g(this)f(last)0 3043 y(equation)21 b(con)m(tradicts)g (the)h(\014rst)h(of)g(\(5.3\).)38 b(In)23 b(case)g(\(f)7 b(\),)25 b(w)m(e)e(ha)m(v)m(e)h Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(g)s FA(\(3\))p Fv(ih)p Fs(\037)p FA(\(3\))p Fs(;)g(g)s FA(\(2\))p Fv(i)20 b FA(=)0 3155 y Fs(b)39 3116 y Fo(\000)p Fr(2)39 3180 y(22)163 3155 y FA(=)25 b Fs(b)298 3116 y Fo(\000)p Fr(1)298 3180 y(33)396 3155 y FA(;)31 b(using)e(\(5.2\),)h(this)f(implies)1070 3374 y Fs(f)10 b FA(\()p Fs(b)19 b Fv(\000)i Fs(d)p FA(\))j Fv(\021)i(\000)p FA(2)f Fv(\021)g Fs(f)1809 3337 y Fr(2)1941 3374 y FA(mo)s(d)k Fs(p:)110 3537 y FA(Th)m(us)34 b Fs(b)22 b Fv(\000)h Fs(d)31 b Fv(\021)g Fs(f)71 b FA(mo)s(d)29 b Fs(p)34 b FA(and)f(\()p Fs(b)23 b Fv(\000)g Fs(d)p FA(\))1539 3504 y Fr(2)1610 3537 y Fv(\021)31 b Fs(b)1751 3504 y Fr(2)1815 3537 y FA(+)23 b Fs(b)g Fv(\000)g FA(2)31 b Fv(\021)g Fs(f)2299 3504 y Fr(2)2371 3537 y Fv(\021)g(\000)p FA(2)61 b(mo)s(d)29 b Fs(p)p FA(;)36 b(this)d(last)0 3646 y(equation)28 b(con)m(tradicts)f(the)i(\014rst)h(of)g(\(5.3\).)110 3805 y(W)-8 b(e)30 b(ha)m(v)m(e)g(sho)m(wn)g(that)e(the)h(only)g(p)s (ossibilit)m(y)e(in)i(rank)h(3)g(is)f(case)g(\(a\))g(with)g Fs(p)c FA(=)h(3.)40 b(Then,)0 3915 y(if)30 b Fs(V)52 b FA(is)31 b(a)f(Y)-8 b(etter-Drinfeld)28 b(mo)s(dule)h(of)i(Cartan)e (t)m(yp)s(e)h(satisfying)f(\(1.6\))h(with)f(\014nite)g(Cartan)0 4024 y(matrix)i(and)h(rank)g Fv(\025)d FA(3,)k(then)e Fs(p)d FA(=)h(3,)k(the)f(corresp)s(onding)c(Dynkin)j(diagram)g(should)g (ha)m(v)m(e)0 4134 y(at)g(most)f(2)i(connected)c(comp)s(onen)m(ts{)g Ft(cf.)44 b FA(case)30 b(\(g\){)g(and)h(eac)m(h)f(comp)s(onen)m(t)f(is) i(of)g(t)m(yp)s(e)f Fs(A)3246 4148 y Fr(1)0 4244 y FA(or)g Fs(A)180 4258 y Fr(2)221 4244 y FA(.)42 b(On)30 b(the)g(other)e(hand,)i (let)f Fs(u)i FA(b)s(e)f(a)g(generator)d(of)k Fu(Z)p Fs(=)p FA(\(3\))24 b(and)30 b Fs(q)k FA(a)c(ro)s(ot)f(of)h(1)g(of)g (order)0 4353 y(3.)41 b(It)30 b(is)f(easy)h(to)g(see)g(that)e(the)i (sequences)0 4707 y(\(5.7\))749 4625 y Fs(g)s FA(\(1\))23 b(=)j Fs(u)f FA(=)h Fs(g)s FA(\(2\))p Fs(;)105 b(g)s FA(\(3\))24 b(=)i Fs(u)1841 4587 y Fp(e)1905 4625 y FA(=)g Fs(g)s FA(\(4\))p Fs(;)574 4775 y Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)i FA(=)e Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)p Fs(;)105 b Fv(h)p Fs(\037)p FA(\(3\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)2173 4738 y Fo(\000)p Fp(e)2294 4775 y FA(=)g Fv(h)p Fs(\037)p FA(\(4\))p Fs(;)15 b(u)p Fv(i)0 4974 y FA(de\014ne)29 b(a)i(Y)-8 b(etter-Drinfeld)27 b(mo)s(dule)h(of)j(rank)f(4)g(o)m(v)m (er)h Fu(Z)p Fs(=)p FA(\(3\),)26 b(whic)m(h)j(is)h(of)h(Cartan)e(t)m (yp)s(e)h(and)0 5084 y(has)g(Dynkin)f(diagram)g Fs(A)910 5098 y Fr(2)971 5084 y Fv(\002)21 b Fs(A)1131 5098 y Fr(2)1172 5084 y FA(.)92 b Fi(\003)p eop %%Page: 24 24 24 23 bop 0 -128 a Fy(24)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Lemma)35 b(5.2.)52 b Fj(Let)31 b Fm(B)p FA(\()p Fs(V)19 b FA(\))31 b Fj(b)s(e)f(a)h(\014nite)e (dimensional)f(Nic)m(hols)i(algebra)e(o)m(v)m(er)k Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))p Fj(.)38 b(Then)0 201 y Fs(V)51 b Fj(is)30 b(of)g(\014nite)e(Cartan)h(t)m(yp)s(e.)0 365 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)35 b(kno)m(w)g(that)e(the)h (Lemma)h(is)f(true)g(whenev)m(er)g Fs(p)h FA(is)f(di\013eren)m(t)f (from)h(3,)j(5,)f(7,)g(11,)0 475 y(13,)e(17.)49 b(Assume)33 b(that)f Fs(p)h FA(is)g(one)f(of)h(these)e(small)h(primes.)48 b(By)34 b(the)e(pro)s(of)f(of)i(Theorem)f(1.1)0 584 y(\(ii\),)h(it)h(w) m(ould)f(b)s(e)g(enough)f(to)i(sho)m(w)g(that)f(no)h(Cartan)e(matrix)h (as)h(in)g(\(4.9\),)f Fs(:)15 b(:)g(:)32 b FA(,)k(\(4.17\))c(is)0 694 y(p)s(ossible)22 b(o)m(v)m(er)j Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\).)34 b(Note)24 b(that)g(the)g(relativ)m(e)e(primeness)h (condition)f(is)i(satis\014ed)f(for)h Fs(p)i FA(=)f(3)0 804 y(since)33 b(no)g(sub)s(diagram)f(of)i(t)m(yp)s(e)f Fs(G)1247 818 y Fr(2)1322 804 y FA(is)h(p)s(ossible)e(o)m(v)m(er)i Fu(Z)p Fs(=)p FA(\(3\).)47 b(All)34 b(these)f(matrices)f(ha)m(v)m(e)i (a)0 913 y(sub)s(diagram)c(of)j(rank)f(2)h(of)g(t)m(yp)s(e)f Fs(B)1281 927 y Fr(2)1355 913 y FA(or)h Fs(G)1542 927 y Fr(2)1582 913 y FA(.)49 b(This)32 b(eliminates)e(all)h(the)h(primes)g (except)f(for)0 1023 y(13.)59 b(W)-8 b(e)36 b(sho)m(w)h(then)e(that)g (no)h(matrix)f(lik)m(e)h(\(4.16\))f(exists)g(o)m(v)m(er)i Fu(Z)p Fs(=)p FA(\(13\).)54 b(Supp)s(ose)34 b(in)i(the)0 1132 y(con)m(trary)g(that)g(it)h(exists.)62 b(Then)37 b(there)f(exists)g(a)i(generator)c Fs(u)k FA(of)g(\000,)i Fs(q)h Fv(2)c Fs(k)2785 1099 y Fo(\002)2884 1132 y FA(of)g(order)g Fs(p)0 1242 y FA(and)31 b(in)m(tegers)e Fs(b;)15 b(c;)g(d;)g(e)32 b FA(\(none)e(of)h(them)f(divisible)f(b)m(y)j(13\))e(suc)m(h)h(that)f Fs(g)s FA(\(1\))c(=)h Fs(u)p FA(,)32 b Fs(g)s FA(\(2\))26 b(=)h Fs(u)3225 1209 y Fp(b)3262 1242 y FA(,)0 1352 y Fs(g)s FA(\(3\))d(=)h Fs(u)337 1318 y Fp(c)375 1352 y FA(,)30 b Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)s FA(,)k Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)25 b FA(=)g Fs(q)1493 1318 y Fp(d)1536 1352 y FA(,)30 b Fv(h)p Fs(\037)p FA(\(3\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)2095 1318 y Fp(e)2134 1352 y FA(;)31 b(and)e(the)g(follo)m(wing)e(equations)0 1461 y(hold)1176 1790 y Fs(b)21 b FA(+)f Fs(d)25 b Fv(\021)g(\000)p FA(3)g Fv(\021)h(\000)p Fs(bd)60 b FA(mo)s(d)30 b(13)p Fs(;)-2276 b FA(\(5.8\))1140 1932 y Fs(c)20 b FA(+)g Fs(e)25 b Fv(\021)g(\000)p FA(1)g Fv(\021)h(\000)p FA(3)p Fs(ec)59 b FA(mo)s(d)30 b(13)p Fs(;)-2276 b FA(\(5.9\))1012 2073 y Fs(be)20 b FA(+)g Fs(dc)25 b Fv(\021)g(\000)p Fs(ec)g Fv(\021)g(\000)p FA(3)p Fs(bd)60 b FA(mo)s(d)30 b(13)p Fs(:)-2276 b FA(\(5.10\))0 2297 y(No)m(w)33 b(\(5.8\))e(implies)f(\()p Fs(b;)15 b(d)p FA(\))27 b(=)i(\(2)p Fs(;)15 b FA(8\))31 b(or)h(\(8)p Fs(;)15 b FA(2\);)32 b(and)f(\(5.9\))g(implies)f(\()p Fs(c;)15 b(e)p FA(\))27 b(=)i(\(5)p Fs(;)15 b FA(7\))31 b(or)g(\(7)p Fs(;)15 b FA(5\).)0 2406 y(W)-8 b(e)32 b(ha)m(v)m(e)g (four)g(p)s(ossibilities)26 b(for)32 b Fs(be)21 b FA(+)h Fs(dc)31 b FA(but)g(none)g(of)h(them)f(giv)m(es)g Fv(\000)p FA(9)61 b(mo)s(d)30 b(13;)i(that)f(is,)0 2516 y(\(5.10\))e(do)s(es)g (not)g(hold.)90 b Fi(\003)0 2680 y FC(Lemma)44 b(5.3.)52 b Fj(Let)38 b Fs(R)i Fj(b)s(e)e(a)g(\014nite)f(dimensional)e(braided)i (Hopf)h(algebra)f(in)2863 2647 y Fr(\000)2863 2706 y(\000)2912 2680 y Fv(Y)8 b(D)s Fj(,)41 b FA(\000)f(=)0 2790 y Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))p Fj(.)35 b(Then)30 b FA(dim)14 b Fs(P)f FA(\()p Fs(R)q FA(\))24 b Fv(\024)i FA(2)k Fj(for)f Fs(p)d(>)f FA(3)p Fj(;)31 b(and)e FA(dim)15 b Fs(P)e FA(\()p Fs(R)q FA(\))23 b Fv(\024)j FA(4)k Fj(for)g Fs(p)25 b FA(=)g(3)p Fj(.)0 2954 y Ft(Pr)-5 b(o)g(of.)46 b FA(Consider)29 b(\014rst)h(the)f(coradical)e(\014ltration)h(of)i Fs(R)i FA(and)e(the)f(subalgebra)f Fs(S)36 b FA(of)31 b(the)e(cor-)0 3064 y(resp)s(onding)f(graded)h(coalgebra)f(generated)g(b)m(y)j Fs(P)13 b FA(\()p Fs(R)q FA(\).)42 b(Then)30 b Fs(S)37 b FA(is)30 b(a)h(Nic)m(hols)f(algebra)f(and)0 3174 y(the)g(claim)g (follo)m(ws)g(from)h(Prop)s(osition)c(5.1)k(and)g(Lemma)g(5.2.)91 b Fi(\003)0 3338 y FC(Prop)s(osition)46 b(5.4.)52 b Fj(Let)39 b Fs(R)i Fj(b)s(e)e(a)h(\014nite)e(dimensional)e(braided)i(Hopf)h (algebra)f(in)3071 3305 y Fr(\000)3071 3363 y(\000)3121 3338 y Fv(Y)8 b(D)r Fj(,)0 3448 y FA(\000)25 b(=)h Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\))p Fj(.)35 b(Assume)c(that)e(\(1.1\),)g (\(1.2\))g(and)h(\(1.3\))f(hold.)39 b(Then)30 b Fs(R)h Fj(is)f(a)g(Nic)m(hols)g(algebra.)0 3612 y Ft(Pr)-5 b(o)g(of.)46 b FA(Let)34 b Fs(S)39 b FA(=)33 b Fs(R)724 3579 y Fo(\003)766 3612 y FA(;)38 b(it)c(is)h(a)f(braided)f(Hopf)i(algebra)e(in)2101 3579 y Fr(\000)2101 3637 y(\000)2151 3612 y Fv(Y)8 b(D)37 b FA(and)d(satis\014es)g(\(1.1\),)h(\(1.2\))0 3722 y(and)d(\(1.4\);)g (that)f(is,)i Fs(S)38 b FA(is)32 b(graded,)g Fs(S)1318 3736 y Fr(0)1388 3722 y FA(=)d Fu(|)-10 b FA(1)26 b(and)32 b(is)g(generated)e(in)h(degree)g(1.)47 b(W)-8 b(e)32 b(w)m(an)m(t)g(to)0 3831 y(sho)m(w)e(that)f(\(1.3\))g(holds)g(for)h Fs(S)5 b FA(;)31 b(hence)e(\(1.4\))g(holds)g(for)h Fs(R)q FA(.)41 b(See)29 b(Lemma)h(5.5)g(b)s(elo)m(w.)110 3941 y(W)-8 b(e)39 b(kno)m(w)g(that)f Fs(V)60 b FA(:=)41 b Fs(S)5 b FA(\(1\))39 b Fv(\022)h Fs(P)13 b FA(\()p Fs(S)5 b FA(\))38 b(and)g Fm(B)p FA(\()p Fs(V)20 b FA(\))39 b(is)f(an)h(epimorphic)d(image)i(of)g Fs(S)45 b FA(as)0 4050 y(braided)e(Hopf)i(algebras.)83 b(W)-8 b(e)45 b(w)m(an)m(t)g(to)f (pro)m(v)m(e)i Fs(V)70 b FA(=)50 b Fs(P)13 b FA(\()p Fs(S)5 b FA(\),)48 b(that)c(is)h Fs(S)55 b FA(=)50 b Fm(B)p FA(\()p Fs(V)20 b FA(\).)85 b(If)0 4160 y(dim)14 b Fs(V)49 b FA(=)29 b(1,)k(w)m(e)f(are)f(done)g(b)m(y)i([AS2,)g(Th.)47 b(3.2].)f(If)32 b(dim)15 b Fs(V)49 b Fv(\025)28 b FA(2)k(and)g Fs(p)c(>)h FA(3,)j(or)g(dim)15 b Fs(V)49 b Fv(\025)28 b FA(4)0 4269 y(and)e Fs(p)g FA(=)f(3,)j(then)e(w)m(e)h(are)g(done)f(b) m(y)h(Lemma)g(5.3)g(applied)e(to)h Fs(S)5 b FA(.)40 b(Th)m(us)27 b(the)g(remaining)d(cases)0 4379 y(are)29 b(dim)15 b Fs(V)46 b FA(=)25 b(2)30 b(or)g(3)g(and)g Fs(p)25 b FA(=)h(3.)110 4489 y(W)-8 b(e)33 b(assume)g(\014rst)g(that)f Fs(V)54 b FA(is)33 b(2-dimensional)d(with)j(basis)f Fs(x)2257 4503 y Fr(1)2298 4489 y Fs(;)15 b(x)2390 4503 y Fr(2)2432 4489 y FA(.)51 b(Then)33 b(there)f(exists)g(a)0 4598 y(generator)27 b Fs(u)k FA(of)f Fu(Z)p Fs(=)p FA(\(3\),)25 b Fs(q)k Fv(2)c Fs(k)1079 4565 y Fo(\002)1170 4598 y FA(of)30 b(order)f(3)h(and)g(in)m(tegers)e Fs(b)p FA(,)j Fs(d)f FA(suc)m(h)g(that)0 4927 y(\(5.1\))364 b Fs(g)s FA(\(1\))23 b(=)j Fs(u;)107 b(g)s FA(\(2\))24 b(=)h Fs(u)1359 4889 y Fp(b)1396 4927 y Fs(;)107 b Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)i Fs(q)s(;)107 b Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fs(q)2667 4889 y Fp(d)2710 4927 y Fs(:)110 5151 y FA(So)k Fs(b)275 5165 y Fr(11)378 5151 y FA(=)d Fs(q)s(;)107 b(b)690 5165 y Fr(22)793 5151 y FA(=)25 b Fs(q)933 5113 y Fp(bd)1008 5151 y Fs(;)107 b(b)1179 5165 y Fr(12)1282 5151 y FA(=)25 b Fs(q)1422 5113 y Fp(d)1465 5151 y Fs(;)107 b(b)1636 5165 y Fr(21)1739 5151 y FA(=)25 b Fs(q)1879 5113 y Fp(b)1916 5151 y FA(.)p eop %%Page: 25 25 25 24 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(25)110 91 y FA(Let)29 b Fs(z)h FA(:=)c(ad)563 105 y Fp(c)600 91 y FA(\()p Fs(x)688 105 y Fr(1)729 91 y FA(\)\()p Fs(x)853 105 y Fr(2)893 91 y FA(\))f(=)h Fs(x)1103 105 y Fr(1)1144 91 y Fs(x)1196 105 y Fr(2)1257 91 y Fv(\000)21 b Fs(b)1388 105 y Fr(12)1465 91 y Fs(x)1517 105 y Fr(2)1558 91 y Fs(x)1610 105 y Fr(1)1682 91 y FA(b)s(e)29 b(the)g(braided)f(comm)m(utator.)0 201 y(There)h(are)h(t)m(w)m(o)g(cases:)40 b(a)30 b(quan)m(tum)g(plane)e(or) i Fs(A)1739 215 y Fr(2)1780 201 y FA(.)110 311 y(In)g(the)f(\014rst)h (case)f(w)m(e)h(ha)m(v)m(e)h(to)e(sho)m(w)i Fs(z)f FA(=)25 b(0)30 b(and)g Fs(x)1920 277 y Fr(3)1920 333 y(1)1986 311 y FA(=)c(0)p Fs(;)15 b(x)2221 277 y Fr(3)2221 333 y(2)2287 311 y FA(=)26 b(0.)110 420 y(But)j Fs(z)35 b FA(is)30 b(primitiv)m(e)e(in)h(this)g(case)h(and)972 632 y Fs(\016)s FA(\()p Fs(z)t FA(\))25 b(=)h Fs(u)1308 594 y Fr(1+)p Fp(b)1457 632 y Fv(\012)21 b Fs(z)t(;)198 b(u:z)31 b FA(=)25 b Fs(q)2108 594 y Fr(1+)p Fp(d)2243 632 y Fs(z)t(:)0 847 y FA(No)m(w)30 b Fs(u)259 814 y Fr(1+)p Fp(b)388 847 y Fs(:z)h FA(=)25 b Fs(q)626 814 y Fr(\(1+)p Fp(b)p Fr(\)\(1+)p Fp(d)p Fr(\))1000 847 y Fs(z)t FA(.)41 b(Since)27 b Fs(b)f FA(=)f Fv(\000)p Fs(d)p FA(,)30 b(the)e(exp)s(onen)m(t)f(\(1)17 b(+)i Fs(b)p FA(\)\(1)e(+)h Fs(d)p FA(\))24 b(=)i(1)18 b Fv(\000)g Fs(b)3081 814 y Fr(2)3151 847 y FA(is)29 b(0)0 957 y(mo)s(d)g(3.)41 b(This)30 b(is)f(a)i(con)m(tradiction)25 b(b)m(y)31 b([AS2,)g(Lemma)f (3.1])h(unless)e Fs(z)h FA(=)25 b(0.)110 1066 y(Since)42 b(the)i(third)f(p)s(o)m(w)m(ers)h(of)g(the)f(primitiv)m(e)g(elemen)m (ts)f Fs(x)2228 1080 y Fr(1)2269 1066 y Fs(;)15 b(x)2361 1080 y Fr(2)2448 1066 y FA(are)44 b(primitiv)m(e)e(b)m(y)j(the)0 1176 y(quan)m(tum)24 b(binomial)e(form)m(ula)i(and)g(ha)m(v)m(e)i (trivial)c(coaction,)h(w)m(e)i(see)g(that)e Fs(x)2621 1143 y Fr(3)2621 1199 y(1)2688 1176 y FA(=)i(0)g(and)f Fs(x)3079 1143 y Fr(3)3079 1199 y(2)3146 1176 y FA(=)h(0)0 1286 y(again)k(b)m(y)h([AS2,)h(Lemma)f(3.1].)110 1395 y(In)g(the)f(second)g(case)g(w)m(e)i(ha)m(v)m(e)f(to)g(sho)m(w)g(the)f (Serre)g(relations)651 1607 y Fs(s)694 1621 y Fr(1)760 1607 y FA(:=)d(\(ad)1014 1621 y Fp(c)1051 1607 y FA(\()p Fs(x)1139 1621 y Fr(1)1180 1607 y FA(\)\))1252 1569 y Fr(2)1292 1607 y FA(\()p Fs(x)1380 1621 y Fr(2)1421 1607 y FA(\))e(=)i(0)p Fs(;)15 b(s)1707 1621 y Fr(2)1773 1607 y FA(:=)26 b(\(ad)2027 1621 y Fp(c)2064 1607 y FA(\()p Fs(x)2152 1621 y Fr(2)2193 1607 y FA(\)\))2265 1569 y Fr(2)2305 1607 y FA(\()p Fs(x)2393 1621 y Fr(1)2434 1607 y FA(\))e(=)i(0)0 1819 y(and)i(also)f(the)g(ro)s(ot)g(v)m(ector)g (relations)e Fs(x)1399 1786 y Fr(3)1399 1841 y(1)1466 1819 y FA(=)g(0)p Fs(;)15 b(x)1700 1786 y Fr(3)1700 1841 y(2)1767 1819 y FA(=)25 b(0)j(and)g Fs(z)2159 1786 y Fr(3)2226 1819 y FA(=)d(0,)k(since)e(it)g(is)h(not)f(di\016cult)0 1928 y(to)i(see)g(that)g(these)f(are)h(the)g(de\014ning)e(relations)g (of)i Fm(B)p FA(\()p Fs(V)20 b FA(\))30 b(when)f(the)g(t)m(yp)s(e)f(is) i Fs(A)2831 1942 y Fr(2)2902 1928 y FA(\(cf.)40 b([AS4,)0 2038 y(part)29 b(3)i(of)f(the)g(pro)s(of)f(of)h(Theorem)f(3.6]\).)41 b(In)31 b(this)e(situation,)f Fs(b)e FA(=)g Fs(d)f FA(=)h(1.)42 b(But)29 b(for)h Fs(i)d FA(=)e(1)p Fs(;)15 b FA(2,)0 2147 y Fs(\016)s FA(\()p Fs(s)123 2161 y Fp(i)153 2147 y FA(\))29 b(=)h Fs(u)371 2114 y Fr(2+1)527 2147 y Fv(\012)22 b Fs(s)663 2161 y Fp(i)694 2147 y FA(.)49 b(Again,)33 b(this)f(is)h(a)g(con)m(tradiction)28 b(unless)k Fs(s)2292 2161 y Fp(i)2352 2147 y FA(=)e(0,)k(since)e(the)g(elemen)m(ts)0 2257 y Fs(s)43 2271 y Fp(i)104 2257 y FA(are)d(primitiv)m(e)f(\(cf.)40 b(Lemma)30 b(A.1\).)110 2367 y(By)d(the)g(same)h(argumen)m(t)e(as)i(b)s (efore,)e Fs(x)1497 2334 y Fr(3)1497 2389 y(1)1564 2367 y FA(=)f(0)p Fs(;)15 b(x)1798 2334 y Fr(3)1798 2389 y(2)1865 2367 y FA(=)25 b(0.)40 b(Then)27 b(it)g(is)h(easy)f(to)g(see)h(that)e (also)0 2476 y(the)j(third)g(ro)s(ot)f(v)m(ector)i Fs(z)35 b FA(is)29 b(primitiv)m(e.)39 b(Hence)29 b(again)g(w)m(e)h(conclude)e Fs(z)2522 2443 y Fr(3)2589 2476 y FA(=)d(0.)110 2586 y(W)-8 b(e)25 b(assume)f(\014nally)f(that)h(dim)14 b Fs(S)5 b FA(\(1\))25 b(is)f(3.)39 b(In)25 b(this)f(case,)h Fs(S)5 b FA(\(1\))25 b(is)f(the)g(direct)f(sum)i Fs(V)3036 2600 y Fr(1)3087 2586 y Fv(\010)10 b Fs(V)3221 2600 y Fr(2)3262 2586 y FA(,)0 2695 y(where)29 b(sa)m(y)i Fs(V)475 2709 y Fr(1)546 2695 y FA(is)f(of)g(t)m(yp)s(e)g Fs(A)1016 2709 y Fr(2)1088 2695 y FA(and)f Fs(V)1318 2709 y Fr(2)1390 2695 y FA(is)h(of)g(t)m(yp)s(e)f Fs(A)1859 2709 y Fr(1)1901 2695 y FA(,)h(cf.)41 b(Prop)s(osition)26 b(5.1.)41 b(Applying)29 b(the)0 2805 y(considerations)24 b(for)k(case)g(2,)h(w)m(e)g(see)f (that)f Fs(S)34 b FA(is)28 b(the)g(t)m(wisted)e(tensor)h(pro)s(duct)f (of)j Fm(B)p FA(\()p Fs(V)3036 2819 y Fr(1)3076 2805 y FA(\))g(and)0 2915 y Fm(B)p FA(\()p Fs(V)170 2929 y Fr(2)210 2915 y FA(\).)41 b(It)30 b(follo)m(ws)f(that)g Fs(S)35 b FA(is)30 b Fm(B)p FA(\()p Fs(V)20 b FA(\),)30 b(as)g(claimed.)90 b Fi(\003)110 3077 y FA(Then)34 b(next)g(Lemma)g(is) g(implicit)e(in)i([A)m(G,)i(Example)e(3.2.8].)54 b(It)35 b(w)m(as)g(also)f(observ)m(ed)f(b)m(y)0 3187 y(the)23 b(referee)e(of)i([AS2].)40 b(W)-8 b(e)23 b(\014rst)g(recall)e(the)h (basic)g(notions)g(o)m(v)m(er)i(arbitrary)d(Hopf)i(algebras)e Fs(H)0 3297 y FA(with)30 b(bijectiv)m(e)e(an)m(tip)s(o)s(de)g Fs(S)36 b FA(\(see)30 b(for)h(example)e([A)m(G,)j(Section)d(2]\).)42 b(If)31 b Fs(V)47 b Fv(2)2693 3263 y Fp(H)2693 3322 y(H)2789 3297 y Fv(Y)8 b(D)34 b FA(is)c(\014nite-)0 3408 y(dimensional,)36 b(the)g(dual)g Fs(V)985 3375 y Fo(\003)1064 3408 y FA(=)h(Hom\()p Fs(V)5 b(;)15 b Fu(|)-9 b FA(\))31 b(is)37 b(in)1833 3375 y Fp(H)1833 3433 y(H)1902 3408 y Fv(Y)8 b(D)40 b FA(with)c(the)g(usual)g(structure.)59 b(Let)0 3517 y Fs(f)35 b Fv(2)25 b Fs(V)240 3484 y Fo(\003)282 3517 y FA(.)40 b(Then)26 b(\()p Fs(h)14 b Fv(\001)g Fs(f)c FA(\)\()p Fs(v)s FA(\))24 b(=)h Fs(f)10 b FA(\()p Fs(S)5 b FA(\()p Fs(v)s FA(\)\))26 b(for)h(all)f Fs(h)f Fv(2)g Fs(H)r(;)15 b(v)30 b Fv(2)25 b Fs(V)5 b(;)28 b FA(de\014nes)e(the)g (action)f(of)i Fs(H)7 b FA(;)29 b(the)0 3627 y(coaction)e Fs(\016)s FA(\()p Fs(f)10 b FA(\))24 b(=)h Fs(f)696 3643 y Fr(\()p Fo(\000)p Fr(1\))871 3627 y Fv(\012)20 b Fs(f)1007 3643 y Fr(\(0\))1135 3627 y FA(is)30 b(giv)m(en)f(b)m(y)i(the)e (equation)e Fs(f)2166 3643 y Fr(\()p Fo(\000)p Fr(1\))2321 3627 y Fs(f)2366 3643 y Fr(\(0\))2464 3627 y FA(\()p Fs(v)s FA(\))e(=)g Fs(S)2765 3594 y Fo(\000)p Fr(1)2864 3627 y FA(\()p Fs(v)2944 3641 y Fo(\000)p Fr(1)3041 3627 y FA(\))p Fs(f)10 b FA(\()p Fs(v)3212 3641 y Fr(0)3252 3627 y FA(\))0 3737 y(for)30 b(all)f Fs(v)g Fv(2)c Fs(V)20 b FA(,)31 b(where)e Fs(\016)s FA(\()p Fs(v)s FA(\))c(=)h Fs(v)1148 3751 y Fo(\000)p Fr(1)1266 3737 y Fv(\012)21 b Fs(v)1402 3751 y Fr(0)1473 3737 y FA(denotes)29 b(the)g(coaction)e (of)j Fs(v)s FA(.)110 3847 y(Let)j Fs(R)g FA(=)g Fv(\010)554 3861 y Fp(n)p Fo(\025)p Fr(0)697 3847 y Fs(R)q FA(\()p Fs(n)p FA(\))g(b)s(e)h(a)g(\014nite-dimensional)29 b(graded)k(Hopf)h (algebra)e(in)2841 3813 y Fp(H)2841 3872 y(H)2911 3847 y Fv(Y)8 b(D)r FA(.)53 b(The)0 3958 y(dual)31 b Fs(S)j FA(=)28 b Fs(R)464 3925 y Fo(\003)535 3958 y FA(=)h Fv(\010)706 3972 y Fp(n)p Fo(\025)p Fr(0)849 3958 y Fs(R)q FA(\()p Fs(n)p FA(\))1046 3925 y Fo(\003)1119 3958 y FA(is)j(in)1321 3925 y Fp(H)1321 3983 y(H)1391 3958 y Fv(Y)8 b(D)34 b FA(as)e(just)g(explained.)44 b(T)-8 b(o)32 b(de\014ne)f(S)h(as)g(a)g (braided)0 4070 y(Hopf)24 b(algebra)e(w)m(e)i(in)m(tro)s(duce)d(for)i (an)m(y)i(\014nite-dimensional)19 b Fs(V)5 b(;)15 b(W)39 b Fv(2)2373 4036 y Fp(H)2373 4095 y(H)2467 4070 y Fv(Y)8 b(D)27 b FA(the)c(isomorphism)1119 4281 y(\010)i(:)h Fs(W)1360 4244 y Fo(\003)1422 4281 y Fv(\012)21 b Fs(V)1587 4244 y Fo(\003)1655 4281 y Fv(!)k FA(\()p Fs(V)41 b Fv(\012)20 b Fs(W)13 b FA(\))2127 4244 y Fo(\003)0 4493 y FA(with)29 b(\010\()p Fs(')20 b Fv(\012)g Fs( )s FA(\)\()p Fs(v)j Fv(\012)e Fs(w)r FA(\))26 b(=)f Fs( )s FA(\()p Fs(v)s FA(\))p Fs(')p FA(\()p Fs(w)r FA(\))30 b(for)g(all)f Fs(')c Fv(2)g Fs(W)1946 4460 y Fo(\003)1988 4493 y Fs(;)15 b( )30 b Fv(2)25 b Fs(V)2277 4460 y Fo(\003)2319 4493 y Fs(;)15 b(v)29 b Fv(2)c Fs(V)5 b(;)15 b(w)30 b Fv(2)25 b Fs(W)13 b FA(.)110 4603 y(Then)43 b Fs(S)50 b FA(is)45 b(a)f(graded)f(Hopf)i(algebra)d(in)1672 4570 y Fp(H)1672 4628 y(H)1741 4603 y Fv(Y)8 b(D)47 b FA(with)d(m)m(ultiplication)39 b(\001)2826 4570 y Fo(\003)2868 4603 y FA(\010)45 b(and)f(co-)0 4712 y(m)m(ultiplication)35 b(\010)662 4679 y Fo(\000)p Fr(1)760 4712 y Fs(\026)815 4679 y Fo(\003)857 4712 y FA(,)44 b(where)c(\001)h(and)f Fs(\026)h FA(denote)e(the)g(com)m (ultiplication)c(and)40 b(m)m(ultipli-)0 4822 y(cation)j(of)h Fs(R)q FA(.)85 b(\(This)43 b(is)i Fs(R)1022 4789 y Fo(\003)p Fp(bop)1215 4822 y FA(in)f(the)g(notation)e(of)i([A)m(G,)j(Section)42 b(2])j(and)f(other)f(v)m(er-)0 4932 y(sions)32 b(of)h(the)f(braided)e (dual)i(w)m(ould)g(also)g(w)m(ork\).)48 b(If)34 b Fs(R)g FA(is)e(not)g(\014nite-dimensional)c(but)k(has)0 5041 y(\014nite-dimensional)20 b(homogeneous)i(comp)s(onen)m(ts,)j(then)e (in)i(the)f(same)h(w)m(a)m(y)h(the)e(graded)f(dual)0 5151 y Fs(S)31 b FA(=)25 b Fs(R)253 5118 y Fo(\003)320 5151 y FA(=)h Fv(\010)488 5165 y Fp(n)p Fo(\025)p Fr(0)631 5151 y Fs(R)q FA(\()p Fs(n)p FA(\))828 5118 y Fo(\003)899 5151 y FA(is)k(a)g(graded)f(Hopf)h(algebra)e(in)2017 5118 y Fp(H)2017 5176 y(H)2086 5151 y Fv(Y)8 b(D)r FA(.)p eop %%Page: 26 26 26 25 bop 0 -128 a Fy(26)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Lemma)f(5.5.)52 b Fj(Let)25 b Fs(H)34 b Fj(b)s(e)24 b(a)i(Hopf)g(algebra)d(with)i(bijectiv)m(e)e (an)m(tip)s(o)s(de)g(and)i Fs(R)h FA(=)g Fv(\010)2949 105 y Fp(n)p Fo(\025)p Fr(0)3092 91 y Fs(R)q FA(\()p Fs(n)p FA(\))0 203 y Fj(a)35 b(graded)f(Hopf)h(algebra)e(in)1051 170 y Fp(H)1051 228 y(H)1120 203 y Fv(Y)8 b(D)38 b Fj(with)c (\014nite-dimensional)d(homogeneous)h(comp)s(onen)m(ts)0 312 y Fs(R)q FA(\()p Fs(n)p FA(\))p Fs(;)15 b(n)27 b Fv(\025)h FA(0)p Fj(,)33 b(and)e Fs(R)q FA(\(0\))26 b(=)i Fu(|)-9 b FA(1)p Fj(.)39 b(Let)32 b Fs(S)h FA(=)28 b Fv(\010)1606 326 y Fp(n)p Fo(\025)p Fr(0)1749 312 y Fs(R)q FA(\()p Fs(n)p FA(\))1946 279 y Fo(\003)2019 312 y Fj(b)s(e)j(the)g (graded)f(dual)g(of)i Fs(R)q Fj(.)45 b Fs(S)38 b Fj(is)31 b(a)0 424 y(graded)f(Hopf)i(algebra)d(in)955 391 y Fp(H)955 449 y(H)1024 424 y Fv(Y)8 b(D)35 b Fj(with)30 b Fs(S)5 b FA(\(0\))27 b(=)g Fu(|)-9 b FA(1)p Fj(.)39 b(Then)31 b Fs(R)q FA(\(1\))26 b(=)i Fs(P)13 b FA(\()p Fs(R)q FA(\))30 b Fj(if)h(and)g(only)g(if)g Fs(S)0 534 y Fj(is)f(generated)d(as)j(an)g (algebra)e(b)m(y)j Fs(S)5 b FA(\(1\))p Fs(:)0 690 y Ft(Pr)-5 b(o)g(of.)46 b FA(Let)31 b(\001)528 657 y Fp(k)602 690 y FA(:)d Fs(R)h Fv(!)g Fs(R)943 657 y Fo(\012)p Fr(\()p Fp(k)r Fr(+1\))1222 690 y FA(=)f Fs(R)23 b Fv(\012)e(\001)15 b(\001)g(\001)23 b(\012)e Fs(R)34 b FA(\()p Fs(k)23 b FA(+)e(1)32 b(times\),)f Fs(k)f Fv(\025)f FA(0,)j(b)s(e)f(the)g (iterated)0 799 y(com)m(ultiplication)36 b(de\014ned)41 b(b)m(y)i(\001)1226 766 y Fr(0)1312 799 y FA(=)i(id)o Fs(;)15 b FA(\001)1620 766 y Fr(1)1707 799 y FA(=)46 b(\001)p Fs(;)15 b FA(\001)2016 766 y Fp(k)2107 799 y FA(=)45 b(\(id)27 b Fv(\012)h FA(\001\)\001)2650 766 y Fp(k)r Fo(\000)p Fr(1)2788 799 y Fs(;)15 b(k)49 b Fv(\025)c FA(2)p Fs(:)d FA(Let)0 909 y Fs(\031)29 b FA(:)d Fs(R)g Fv(!)g Fs(R)q FA(\(1\))j(b)s(e)g(the)h(pro)5 b(jection.)38 b(It)30 b(is)f(easy)h(to)g(see)g(that)f(the)g(Lemma)h(follo)m(ws)f (from)h(the)0 1019 y(equiv)-5 b(alence)27 b(of)j(the)f(follo)m(wing)f (statemen)m(ts:)110 1128 y(\(1\))h(R\(1\))g(=)i(P\(R\).)110 1238 y(\(2\))e(F)-8 b(or)30 b(all)e Fs(n)e Fv(\025)f FA(2,)30 b Fs(\036)879 1252 y Fp(n)955 1238 y FA(:=)c Fs(\031)1132 1205 y Fo(\012)p Fp(n)1239 1238 y FA(\001)1315 1205 y Fp(n)p Fo(\000)p Fr(1)1484 1238 y FA(:)f Fs(R)i Fv(!)e Fs(R)q FA(\(1\))1934 1205 y Fo(\012)p Fp(n)2070 1238 y FA(is)30 b(injectiv)m(e)e(on)i Fs(R)q FA(\()p Fs(n)p FA(\).)110 1347 y(Assume)36 b(\(2\).)56 b(Then)35 b(the)g(only)g(primitiv)m(e)f(elemen)m(ts)g(are)h(in)g(degree)f(one.)57 b(This)35 b(pro)m(v)m(es)0 1457 y(\(1\),)30 b(since)e(in)i(general)e Fs(R)q FA(\(1\))c Fv(\032)h Fs(P)13 b FA(\()p Fs(R)q FA(\))29 b(b)m(y)i(the)e(assumption)f Fs(R)q FA(\(0\))c(=)i Fu(|)-10 b FA(1.)110 1566 y(Assume)36 b(\(1\))g(and)g(let)f Fs(x)h Fv(2)g Fs(R)q FA(\()p Fs(n)p FA(\))p Fs(;)15 b(n)35 b Fv(\025)i FA(2)p Fs(;)f FA(with)g Fs(\036)1926 1580 y Fp(n)1976 1566 y FA(\()p Fs(x)p FA(\))f(=)h(0)p Fs(:)h FA(W)-8 b(rite)35 b(\001\()p Fs(x)p FA(\))h(=)2953 1498 y Fl(P)3049 1521 y Fp(n)3049 1594 y(i)p Fr(=0)3187 1566 y Fs(y)3232 1580 y Fp(i)3262 1566 y Fs(;)0 1676 y FA(where)e Fs(y)314 1690 y Fp(i)377 1676 y Fv(2)f Fs(R)q FA(\()p Fs(i)p FA(\))23 b Fv(\012)g Fs(R)q FA(\()p Fs(n)g Fv(\000)h Fs(i)p FA(\),)36 b(0)d Fv(\024)g Fs(i)h Fv(\024)f Fs(n:)i FA(By)g(coasso)s(ciativit)m(y)-8 b(,)32 b(for)j(all)e(1)g Fv(\024)g Fs(i)h Fv(\024)f Fs(n)23 b Fv(\000)h FA(1,)0 1786 y Fs(\036)54 1800 y Fp(n)129 1786 y FA(=)i(\()p Fs(\036)316 1800 y Fp(i)355 1786 y Fv(\012)8 b Fs(\036)488 1800 y Fp(n)p Fo(\000)p Fp(i)621 1786 y FA(\)\001)p Fs(:)24 b FA(F)-8 b(or)24 b(all)f(0)i Fv(\024)h Fs(j)k Fv(\024)25 b Fs(n;)15 b(i)27 b Fv(6)p FA(=)e Fs(j;)g FA(\001)1806 1752 y Fp(i)p Fo(\000)p Fr(1)1930 1786 y FA(\()p Fs(R)q FA(\()p Fs(j)5 b FA(\)\))22 b(has)i(no)g(non-zero)e(comp)s(onen)m(t)0 1895 y(in)30 b Fs(R)q FA(\(1\))295 1862 y Fo(\012)p Fp(i)381 1895 y FA(.)41 b(Hence)721 2114 y Fs(\036)775 2128 y Fp(n)825 2114 y FA(\()p Fs(x)p FA(\))25 b(=)g(\()p Fs(\036)1160 2128 y Fp(i)1211 2114 y Fv(\012)20 b Fs(\036)1356 2128 y Fp(n)p Fo(\000)p Fp(i)1489 2114 y FA(\)\()p Fs(y)1606 2128 y Fp(i)1635 2114 y FA(\))p Fs(;)46 b FA(for)30 b(all)f(1)c Fv(\024)g Fs(i)h Fv(\024)f Fs(n)20 b Fv(\000)h FA(1)p Fs(:)0 2265 y FA(Th)m(us)39 b(w)m(e)g(see)g(b)m(y)g(induction)d(that)i Fs(y)1342 2279 y Fp(i)1412 2265 y FA(=)i(0)f(for)f(all)g(1)i Fv(\024)g Fs(i)g Fv(\024)g Fs(n)26 b Fv(\000)g FA(1.)67 b(Hence)38 b Fs(x)i FA(m)m(ust)f(b)s(e)0 2374 y(primitiv)m(e.)f(By)31 b(\(1\),)e(w)m(e)i(conclude)c Fs(x)f Fv(2)f Fs(R)q FA(\(1\))k(and)h (then)f Fs(x)c FA(=)h(0,)k(since)f Fs(n)c Fv(\025)g FA(2.)92 b Fi(\003)110 2527 y FA(Finally)28 b(w)m(e)j(men)m(tion)e(t)m(w)m(o)h (easy)h(coun)m(terexamples)c(to)j(Conjecture)e(1.4)i(when)g(the)g(Hopf) 0 2637 y(algebra)e(is)i(not)f(\014nite-dimensional)c(or)30 b(the)f(c)m(haracteristic)d(is)k(not)f(0:)110 2789 y(\(5.11\).)48 b(Let)33 b Fs(F)46 b FA(b)s(e)33 b(a)g(\014eld)f(of)h(p)s(ositiv)m(e)e (c)m(haracteristic)e Fs(p)p FA(.)50 b(Let)33 b Fs(S)39 b FA(b)s(e)32 b(the)h(\(usual\))e(Hopf)0 2911 y(algebra)e Fs(F)13 b FA([)p Fs(x)p FA(])p Fs(=)p Fv(h)p Fs(x)629 2878 y Fp(p)667 2851 y Ff(2)706 2911 y Fv(i)31 b FA(with)e Fs(x)e Fv(2)f Fs(P)13 b FA(\()p Fs(S)5 b FA(\).)42 b(Then)30 b Fs(x)1710 2878 y Fp(p)1779 2911 y Fv(2)c Fs(P)13 b FA(\()p Fs(S)5 b FA(\).)41 b(Hence)30 b Fs(R)e FA(=)e Fs(S)2664 2878 y Fo(\003)2738 2911 y FA(satis\014es)j(\(1.1\),)0 3021 y(\(1.2\))g(and)h(\(1.3\))f(but)g(not)h(\(1.4\).)110 3174 y(\(5.12\).)42 b(Let)31 b Fs(S)i FA(=)27 b Fu(|)-9 b FA([)p Fs(X)7 b FA(])21 b(=)28 b Fv(\010)1145 3188 y Fp(n)p Fo(\025)p Fr(0)1288 3174 y Fs(S)5 b FA(\()p Fs(n)p FA(\))30 b(b)s(e)h(a)g(p)s(olynomial)d(algebra)h(in)i(one)f(v)-5 b(ariable.)42 b(W)-8 b(e)0 3285 y(consider)29 b Fs(S)38 b FA(as)31 b(a)h(braided)e(Hopf)i(algebra)d(in)1628 3252 y Fp(H)1628 3310 y(H)1697 3285 y Fv(Y)8 b(D)s FA(,)33 b(where)d Fs(H)36 b FA(=)28 b Fu(|)-9 b FA(\000,)26 b(\000)32 b(an)g(in\014nite)d(cyclic)0 3395 y(group)g(with)g(generator)e Fs(g)s FA(,)k(with)e(action,)f(coaction)g(and)h(com)m(ultiplication)24 b(giv)m(en)30 b(b)m(y)476 3562 y Fs(\016)s FA(\()p Fs(X)639 3524 y Fp(n)688 3562 y FA(\))25 b(=)g Fs(g)892 3524 y Fp(n)962 3562 y Fv(\012)20 b Fs(X)1136 3524 y Fp(n)1186 3562 y Fs(;)107 b(g)s(:X)32 b FA(=)25 b Fs(q)s(X)r(;)107 b FA(\001\()p Fs(X)7 b FA(\))24 b(=)i Fs(X)h Fv(\012)20 b FA(1)g(+)h(1)f Fv(\012)g Fs(X)r(:)0 3729 y FA(Here)31 b Fs(q)f Fv(2)c Fu(|)16 b FA(is)31 b(a)g(ro)s(ot)e(of)i(1)g(of)g(order) f Fs(N)10 b FA(.)43 b(That)31 b(is,)g Fs(S)36 b FA(is)31 b(a)g(so-called)e(quan)m(tum)h(line.)42 b(Then)0 3838 y Fs(S)32 b FA(satis\014es)25 b(\(1.1\),)h(\(1.2\))g(and)g(\(1.4\))f (but)h(not)g(\(1.3\))f(since)g Fs(X)2104 3805 y Fp(N)2200 3838 y FA(is)h(also)g(primitiv)m(e.)37 b(Hence)26 b(the)0 3948 y(graded)j(dual)h Fs(R)e FA(=)f Fs(S)760 3915 y Fp(d)830 3948 y FA(=)f Fv(\010)998 3962 y Fp(n)p Fo(\025)p Fr(0)1142 3948 y Fs(S)5 b FA(\()p Fs(n)p FA(\))1330 3915 y Fo(\003)1402 3948 y FA(is)31 b(a)g(braided)e(Hopf)i(algebra)e (satisfying)f(\(1.1\),)j(\(1.2\))0 4057 y(and)e(\(1.3\))g(but)f(not)h (\(1.4\).)40 b(Let)29 b(us)g(presen)m(t)f(explicitly)f Fs(R)q FA(.)40 b(It)30 b(is)f(a)g(v)m(ector)g(space)f(with)h(basis)0 4167 y Fs(y)45 4181 y Fp(n)94 4167 y FA(,)i Fs(n)25 b Fv(\025)h FA(0;)k(the)g(structure)d(is)j(giv)m(en)g(b)m(y)g Fs(y)1479 4181 y Fr(0)1546 4167 y FA(=)25 b(1,)596 4411 y Fs(y)641 4425 y Fp(n)690 4411 y Fs(y)735 4425 y Fp(m)830 4411 y FA(=)926 4282 y Fl(\024)989 4356 y Fs(m)c FA(+)f Fs(n)1085 4466 y(n)1251 4282 y Fl(\025)1299 4502 y Fp(q)1355 4411 y Fs(y)1400 4425 y Fp(n)p Fr(+)p Fp(m)1570 4411 y Fs(;)107 b FA(\001\()p Fs(y)1859 4425 y Fp(n)1908 4411 y FA(\))25 b(=)2111 4324 y Fl(X)2066 4520 y Fr(0)p Fo(\024)p Fp(i)p Fo(\024)p Fp(n)2302 4411 y Fs(y)2347 4425 y Fp(i)2398 4411 y Fv(\012)20 b Fs(y)2534 4425 y Fp(n)p Fo(\000)p Fp(i)2667 4411 y Fs(;)953 4655 y(\016)s FA(\()p Fs(y)1078 4669 y Fp(n)1127 4655 y FA(\))25 b(=)g Fs(g)1331 4617 y Fo(\000)p Fp(n)1458 4655 y Fv(\012)20 b Fs(y)1594 4669 y Fp(n)1644 4655 y Fs(;)107 b(g)s(:y)1893 4669 y Fp(n)1968 4655 y FA(=)25 b Fs(q)2108 4617 y Fo(\000)p Fp(n)2215 4655 y Fs(y)2260 4669 y Fp(n)2310 4655 y Fs(:)110 4822 y FA(W)-8 b(e)37 b(do)g(not)g(kno)m(w)h(an)m(y)g(\014nite)e (dimensional)e(coun)m(terexample)g(in)j(c)m(haracteristic)c(zero.)0 4932 y(Note)21 b(that)f(in)h(b)s(oth)f(examples)g(ab)s(o)m(v)m(e)h(a)h (F)-8 b(rob)s(enius)19 b(homomorphism)g(is)j(giv)m(en)f(implicitly)-8 b(.)34 b(In)0 5041 y(c)m(haracteristic)17 b(zero,)23 b(Lusztig)d(said)h(that)g("there)f(are)h(no)g(p)s(o)m(w)m(ers)h(of)f (the)g(quan)m(tum)h(F)-8 b(rob)s(enius)0 5151 y(map")30 b([L4,)h(8.5,)f(p.)41 b(58];)30 b(this)f(could)g(b)s(e)g(related)f(to)h (our)h(lac)m(k)g(of)g(examples.)p eop %%Page: 27 27 27 26 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(27)0 91 y Fv(x)p FC(6.)46 b(Isomorphisms)35 b(b)s(et)m(w)m(een)h(b)s(osonizations.)110 201 y FA(W)-8 b(e)24 b(b)s(egin)f(with)g(a)i(general)d(Lemma)i(whose)g (pro)s(of)f(is)h(straigh)m(tforw)m(ard.)36 b(W)-8 b(e)25 b(shall)e(denote)0 311 y(the)31 b(com)m(ultiplication)25 b(of)32 b Fs(R)h FA(b)m(y)f(\001)1244 325 y Fp(R)1304 311 y FA(\()p Fs(r)s FA(\))27 b(=)1546 242 y Fl(P)1657 311 y Fs(r)1701 277 y Fr(\(1\))1820 311 y Fv(\012)21 b Fs(r)1956 277 y Fr(\(2\))2054 311 y FA(;)33 b(w)m(e)e(omit)g(most)g (of)h(the)e(time)h(the)0 420 y(summation)d(sign.)0 583 y FC(Lemma)44 b(6.1.)52 b Fj(Let)37 b Fs(H)46 b Fj(b)s(e)38 b(a)g(Hopf)g(algebra,)h Fs( )j FA(:)e Fs(H)46 b Fv(!)39 b Fs(H)46 b Fj(an)38 b(automorphism)e(of)i(Hopf)0 692 y(algebras,)28 b Fs(V)5 b(;)15 b(W)45 b Fj(Y)-8 b(etter-Drinfeld)27 b(mo)s(dules)h(o)m(v)m(er)i Fs(H)7 b Fj(.)157 828 y FA(\(1\))54 b Fj(Let)29 b Fs(V)566 795 y Fp( )652 828 y Fj(b)s(e)g(the)g(same)h (space)f(underlying)e Fs(V)51 b Fj(but)30 b(with)f(action)f(and)h (coaction)0 1044 y FA(\(6.1\))304 b Fs(h:)571 1058 y Fp( )626 1044 y Fs(v)29 b FA(=)d Fs( )s FA(\()p Fs(h)p FA(\))p Fs(:v)s(;)107 b(\016)1231 1007 y Fp( )1286 1044 y FA(\()p Fs(v)s FA(\))25 b(=)1526 970 y Fl(\000)1568 1044 y Fs( )1630 1007 y Fo(\000)p Fr(1)1749 1044 y Fv(\012)c FA(id)1917 970 y Fl(\001)1974 1044 y Fs(\016)s FA(\()p Fs(v)s FA(\))p Fs(;)106 b(h)25 b Fv(2)h Fs(H)r(;)15 b(v)29 b Fv(2)d Fs(V)5 b(:)438 1265 y Fj(Then)34 b Fs(V)754 1232 y Fp( )844 1265 y Fj(is)g(also)f(a)h(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)h(o)m(v)m(er)i Fs(H)7 b Fj(.)54 b(If)34 b Fs(T)45 b FA(:)32 b Fs(V)53 b Fv(!)32 b Fs(W)47 b Fj(is)34 b(a)329 1374 y(morphism)i(in)876 1341 y Fp(H)876 1400 y(H)945 1374 y Fv(Y)8 b Fs(D)r Fj(,)39 b(then)d Fs(T)50 b FA(:)37 b Fs(V)1609 1341 y Fp( )1701 1374 y Fv(!)g Fs(W)1928 1341 y Fp( )2021 1374 y Fj(also)f(is.)60 b(Moreo)m(v)m(er,)39 b(the)d(braiding)329 1486 y Fs(c)25 b FA(:)g Fs(V)518 1453 y Fp( )586 1486 y Fv(\012)13 b Fs(W)769 1453 y Fp( )849 1486 y Fv(!)25 b Fs(W)1064 1453 y Fp( )1132 1486 y Fv(\012)13 b Fs(V)1289 1453 y Fp( )1371 1486 y Fj(coincides)23 b(with)i(the)h (braiding)d Fs(c)i FA(:)h Fs(V)33 b Fv(\012)13 b Fs(W)38 b Fv(!)26 b Fs(W)f Fv(\012)13 b Fs(V)20 b Fj(.)157 1649 y FA(\(2\))54 b Fj(If)37 b Fs(R)h Fj(is)f(an)g(algebra)e(\(resp.,)j(a)f (coalgebra,)f(a)h(Hopf)g(algebra\))e(in)2694 1616 y Fp(H)2694 1674 y(H)2763 1649 y Fv(Y)8 b Fs(D)40 b Fj(then)35 b Fs(R)3232 1616 y Fp( )329 1758 y Fj(also)29 b(is,)h(with)f(the)g(same)h (structural)d(maps.)157 1921 y FA(\(3\))54 b Fj(Let)38 b Fs(R)i Fj(b)s(e)d(a)i(Hopf)g(algebra)d(in)1502 1888 y Fp(H)1502 1946 y(H)1571 1921 y Fv(Y)8 b(D)s Fj(.)66 b(Then)38 b(the)g(map)g FA(\011)i(:)f Fs(R)2673 1888 y Fp( )2729 1921 y FA(#)p Fs(H)46 b Fv(!)40 b Fs(R)q FA(#)p Fs(H)329 2031 y Fj(giv)m(en)29 b(b)m(y)i FA(\011\()p Fs(r)s FA(#)p Fs(h)p FA(\))24 b(=)h Fs(r)s FA(#)p Fs( )s FA(\()p Fs(h)p FA(\))30 b Fj(is)g(an)g(isomorphism)e(of)i(Hopf)g (algebras.)0 2220 y Ft(Pr)-5 b(o)g(of.)46 b FA(The)34 b(pro)s(of)f(of)h(\(1\))f(is)h(a)g(direct)e(computation)e(that)j(w)m(e) i(lea)m(v)m(e)e(to)h(the)f(reader.)51 b(The)0 2329 y(statemen)m(t)28 b(\(2\))h(follo)m(ws)g(from)h(\(1\).)40 b(Let)29 b(us)h(c)m(hec)m(k)h (\(3\).)39 b(If)31 b Fs(r)m(;)15 b(s)26 b Fv(2)f Fs(R)32 b FA(and)d Fs(h;)15 b(g)29 b Fv(2)c Fs(H)38 b FA(then)348 2650 y(\011)15 b(\(\()p Fs(r)s FA(#)p Fs(h)p FA(\)\()p Fs(s)p FA(#)p Fs(g)s FA(\)\))22 b(=)j(\011)1193 2577 y Fl(\000)1235 2650 y Fs(r)s(h)1332 2667 y Fr(\(1\))1430 2650 y Fs(:)1455 2664 y Fp( )1510 2650 y Fs(s)p FA(#)p Fs(h)1682 2667 y Fr(\(2\))1780 2650 y Fs(g)1827 2577 y Fl(\001)1894 2650 y FA(=)g Fs(r)s( )s FA(\()p Fs(h)2185 2667 y Fr(\(1\))2283 2650 y FA(\))p Fs(:s)p FA(#)p Fs( )s FA(\()p Fs(h)2614 2667 y Fr(\(2\))2712 2650 y Fs(g)s FA(\);)363 2792 y(\011\()p Fs(r)s FA(#)p Fs(h)p FA(\)\011\()p Fs(s)p FA(#)p Fs(g)s FA(\))e(=)i(\()p Fs(r)s FA(#)p Fs( )s FA(\()p Fs(h)p FA(\)\)\()p Fs(s)p FA(#)p Fs( )s FA(\()p Fs(g)s FA(\)\))d(=)k Fs(r)s( )s FA(\()p Fs(h)p FA(\))2208 2809 y Fr(\(1\))2305 2792 y Fs(:s)p FA(#)p Fs( )s FA(\()p Fs(h)p FA(\))2636 2809 y Fr(\(2\))2734 2792 y Fs( )s FA(\()p Fs(g)s FA(\))p Fs(:)110 3003 y FA(That)j(is,)h(\011)h(is)e(an)h (algebra)e(map.)41 b(On)30 b(the)f(other)g(hand,)388 3324 y(\001\011\()p Fs(r)s FA(#)p Fs(h)p FA(\))24 b(=)h(\001\()p Fs(r)s FA(#)p Fs( )s FA(\()p Fs(h)p FA(\)\))f(=)h Fs(r)1519 3287 y Fr(\(1\))1617 3324 y FA(#\()p Fs(r)1773 3287 y Fr(\(2\))1870 3324 y FA(\))1906 3341 y Fr(\()p Fo(\000)p Fr(1\))2061 3324 y Fs( )s FA(\()p Fs(h)p FA(\))2248 3341 y Fr(\(1\))2366 3324 y Fv(\012)20 b FA(\()p Fs(r)2537 3287 y Fr(\(2\))2634 3324 y FA(\))2670 3341 y Fr(\(0\))2768 3324 y FA(#)p Fs( )s FA(\()p Fs(h)p FA(\))3031 3341 y Fr(\(2\))3128 3324 y FA(;)134 3503 y(\(\011)g Fv(\012)g FA(\011\)\001\()p Fs(r)s FA(#)p Fs(h)p FA(\))k(=)h(\(\011)20 b Fv(\012)g FA(\011\))1240 3401 y Fl(\020)1294 3503 y Fs(r)1338 3465 y Fr(\(1\))1436 3503 y FA(#)p Fs( )1574 3465 y Fo(\000)p Fr(1)1688 3401 y Fl(\020)1743 3503 y FA(\()p Fs(r)1823 3465 y Fr(\(2\))1920 3503 y FA(\))1956 3519 y Fr(\()p Fo(\000)p Fr(1\))2111 3401 y Fl(\021)2181 3503 y Fs(h)2234 3519 y Fr(\(1\))2352 3503 y Fv(\012)g FA(\()p Fs(r)2523 3465 y Fr(\(2\))2620 3503 y FA(\))2656 3519 y Fr(\(0\))2754 3503 y FA(#)p Fs(h)2883 3519 y Fr(\(2\))2980 3401 y Fl(\021)804 3684 y FA(=)25 b Fs(r)944 3646 y Fr(\(1\))1042 3684 y FA(#\()p Fs(r)1198 3646 y Fr(\(2\))1295 3684 y FA(\))1331 3701 y Fr(\()p Fo(\000)p Fr(1\))1486 3684 y Fs( )s FA(\()p Fs(h)1637 3701 y Fr(\(1\))1735 3684 y FA(\))20 b Fv(\012)g FA(\()p Fs(r)1962 3646 y Fr(\(2\))2059 3684 y FA(\))2095 3701 y Fr(\(0\))2193 3684 y FA(#)p Fs( )s FA(\()p Fs(h)2420 3701 y Fr(\(2\))2518 3684 y FA(\))p Fs(:)110 3896 y FA(Th)m(us)30 b(\011)g(is)g(a)g(bialgebra)d (map)j(and)g Ft(a)i(fortiori)d FA(a)h(Hopf)h(algebra)d(map.)91 b Fi(\003)110 4111 y FA(Let)38 b Fs(H)47 b FA(b)s(e)39 b(a)g(cosemisimple)d(Hopf)k(algebra.)66 b(Let)39 b Fs(A)i FA(=)g Fv(\010)2263 4125 y Fp(n)p Fo(\025)p Fr(0)2406 4111 y Fs(A)p FA(\()p Fs(n)p FA(\))e(b)s(e)f(a)i(coradically)0 4221 y(graded)29 b(Hopf)i(algebra)e(with)g(coradical)f Fs(A)1508 4235 y Fr(0)1575 4221 y FA(=)f Fs(A)p FA(\(0\))j(isomorphic)e (to)i Fs(H)7 b FA(.)43 b(Let)31 b Fs(\023)26 b FA(:)h Fs(A)p FA(\(0\))e Fv(!)i Fs(A)0 4330 y FA(denote)c(the)h(inclusion.)36 b(Let)24 b Fs(\031)29 b FA(:)c Fs(A)h Fv(!)g Fs(A)p FA(\(0\))e(b)s(e)g (the)g(unique)g(graded)f(pro)5 b(jection)22 b(with)h(image)0 4440 y Fs(A)p FA(\(0\))32 b(and)g(let)f Fs(R)f FA(=)f Fv(f)p Fs(a)g Fv(2)g Fs(A)h FA(:)f(\(id)14 b Fv(\012)p Fs(\031)s FA(\)\001\()p Fs(a)p FA(\))28 b(=)i Fs(a)22 b Fv(\012)f FA(1)p Fv(g)p FA(.)47 b(Let)32 b Fs(B)38 b FA(b)s(e)31 b(another)g(Hopf)h(algebra)0 4550 y(with)d(coradical)e Fs(B)663 4564 y Fr(0)735 4550 y FA(isomorphic)g(to)j Fs(H)7 b FA(.)0 4712 y FC(Lemma)63 b(6.2.)51 b Fj(Let)k FA(\011)66 b(:)g Fs(A)h Fv(!)g Fs(B)60 b Fj(b)s(e)53 b(an)i(isomorphism)d(of)j(Hopf)g(algebras.)111 b(Let)0 4822 y Fs(B)5 b FA(\()p Fs(n)p FA(\))45 b(:=)i(\011)15 b(\()p Fs(A)p FA(\()p Fs(n)p FA(\)\))o Fj(;)50 b(then)42 b Fs(B)48 b Fj(is)42 b(also)g(coradically)d(graded)j(with)f(resp)s(ect) g(to)h(the)g(grad-)0 4932 y(ing)f Fs(B)49 b FA(=)44 b Fv(\010)468 4946 y Fp(n)p Fo(\025)p Fr(0)611 4932 y Fs(B)5 b FA(\()p Fs(n)p FA(\))p Fj(.)74 b(Let)41 b Fs( )48 b FA(:)d Fs(H)52 b Fv(!)44 b Fs(H)50 b Fj(denote)39 b(the)i(isomorphism)e (induced)h(b)m(y)i(the)0 5041 y(restriction)31 b(of)k FA(\011)p Fj(,)i(with)c(resp)s(ect)g(to)h(\014xed)h(iden)m (ti\014cations)30 b Fs(A)2215 5055 y Fr(0)2290 5041 y Fv(')j Fs(H)7 b Fj(,)37 b Fs(B)2608 5055 y Fr(0)2683 5041 y Fv(')c Fs(H)7 b Fj(.)55 b(Let)35 b Fs(|)p Fj(,)i Fs(\020)7 b Fj(,)0 5151 y Fs(S)38 b Fj(denote)31 b(the)h(same)g(ob)5 b(jects)31 b(as)i Fs(\023)p Fj(,)h Fs(\031)s Fj(,)h Fs(R)f Fj(but)e(with)f(resp)s(ect)g(to)h(the)g(grading)e(in)i Fs(B)5 b Fj(.)49 b(Then)p eop %%Page: 28 28 28 27 bop 0 -128 a Fy(28)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FA(\011\()p Fs(R)q FA(\))h(=)g Fs(S)39 b Fj(and)34 b(the)f(restriction)d Fs(\036)h FA(:)h Fs(R)h Fv(!)f Fs(S)39 b Fj(of)34 b FA(\011)g Fj(is)f(a)h(morphism)f Fs(R)f Fv(!)g Fs(S)2791 58 y Fp( )2880 91 y Fj(of)i(braided)0 201 y(Hopf)c(algebras)e(in)686 168 y Fp(H)686 226 y(H)755 201 y Fv(Y)8 b(D)r Fj(.)41 b(Moreo)m(v)m(er,)30 b(the)f(follo)m(wing)f (diagram)g(comm)m(utes)1317 444 y Fs(A)1620 372 y Fr(\011)1511 444 y Fv(\000)-30 b(\000)-20 b(\000)f(\000)-30 b(!)119 b Fs(B)1263 620 y Fo(')1320 522 y Fl(?)1320 577 y(?)1320 631 y(y)1854 620 y Fo(')1911 522 y Fl(?)1911 577 y(?)1911 631 y(y)1236 843 y Fs(R)q FA(#)p Fs(H)1572 767 y Fp(\036)p Fo(\012)p Fp( )1511 843 y Fv(\000)-30 b(\000)-20 b(\000)f(\000)-30 b(!)46 b Fs(S)5 b FA(#)p Fs(H)110 991 y Fj(where)29 b(the)g(v)m (ertical)f(isomorphisms)g(are)h(the)h(in)m(v)m(erses)f(of)h(the)g(b)s (osonization)25 b(maps.)110 1199 y FA(Note)32 b(that)g Fs(R)q FA(#)p Fs(H)825 1123 y Fp(\036)p Fo(\012)p Fp( )794 1199 y Fv(\000)-35 b(\000)-20 b(\000)-35 b(!)31 b Fs(S)5 b FA(#)p Fs(H)41 b FA(factorizes)30 b(as)j(the)g(comp)s(osition)c(of)k (t)m(w)m(o)h(Hopf)f(algebra)0 1355 y(maps:)40 b Fs(R)q FA(#)p Fs(H)561 1279 y Fp(\036)p Fo(\012)p Fr(id)529 1355 y Fv(\000)-30 b(\000)-20 b(\000)-30 b(!)26 b Fs(S)840 1322 y Fp( )896 1355 y FA(#)p Fs(H)1111 1279 y Fr(id)13 b Fo(\012)p Fp( )1080 1355 y Fv(\000)-20 b(\000)g(\000)g(!)25 b Fs(S)5 b FA(#)p Fs(H)i FA(.)0 1517 y Ft(Pr)-5 b(o)g(of.)46 b FA(It)d(is)g(clear)f(that)g Fs(B)52 b FA(=)47 b Fv(\010)1257 1531 y Fp(n)p Fo(\025)p Fr(0)1400 1517 y Fs(B)5 b FA(\()p Fs(n)p FA(\))42 b(is)h(a)h(grading)d(of)i(Hopf)g(algebras.)78 b(Since)42 b(\011)0 1626 y(preserv)m(es)29 b(the)g(terms)h(of)g(the)f (coradical)e(\014ltration,)g(w)m(e)k(ha)m(v)m(e)627 1845 y Fs(B)696 1859 y Fp(m)791 1845 y FA(=)26 b(\011\()p Fs(A)1063 1859 y Fp(m)1132 1845 y FA(\))f(=)h(\011)15 b(\()o Fv(\010)1482 1859 y Fr(0)p Fo(\024)p Fp(n)p Fo(\024)p Fp(m)1747 1845 y Fs(A)p FA(\()p Fs(n)p FA(\)\))25 b(=)g Fv(\010)2170 1859 y Fr(0)p Fo(\024)p Fp(n)p Fo(\024)p Fp(m)2435 1845 y Fs(B)5 b FA(\()p Fs(n)p FA(\);)0 2013 y(that)29 b(is,)h Fs(B)35 b FA(is)30 b(coradically)d(graded.)39 b(It)30 b(follo)m(ws)e(at)i(once)f(that)0 2218 y(\(6.2\))1028 b Fs(|)p FA(\011)25 b(=)h(\011)p Fs(\023;)107 b FA(\011)p Fs(\031)29 b FA(=)c Fs(\020)7 b FA(\011)p Fs(:)110 2423 y FA(Hence)33 b(\011\()p Fs(R)q FA(\))f(=)h Fs(S)5 b FA(.)54 b(W)-8 b(e)35 b(c)m(hec)m(k)f(that)g Fs(\036)h FA(is)f(a)h(morphism)e(from)h Fs(R)i FA(to)e Fs(S)2688 2390 y Fp( )2743 2423 y FA(.)55 b(It)34 b(preserv)m(es)0 2532 y(the)28 b(m)m(ultiplication)23 b(b)s(ecause)k(is)h(the)g (restriction)c(of)29 b(the)f(algebra)e(map)i(\011.)41 b(It)28 b(preserv)m(es)g(the)0 2642 y(com)m(ultiplication)20 b(b)s(ecause)k(\011)i(is)g(a)g(coalgebra)d(map)j(and)f(b)m(y)i (\(6.2\).)38 b(It)26 b(preserv)m(es)f(the)g(action)0 2752 y(and)30 b(the)f(coaction:)427 3066 y Fs(\036)p FA(\()p Fs(h:r)s FA(\))c(=)g(\011)882 2992 y Fl(\000)924 3066 y Fs(h)977 3083 y Fr(\(1\))1075 3066 y Fs(r)18 b Fv(S)7 b FA(\()p Fs(h)1285 3083 y Fr(\(2\))1382 3066 y FA(\))1418 2992 y Fl(\001)1485 3066 y FA(=)25 b Fs( )1659 2992 y Fl(\000)1701 3066 y Fs(h)1754 3083 y Fr(\(1\))1852 2992 y Fl(\001)1909 3066 y Fs(\036)p FA(\()p Fs(r)s FA(\))p Fs( )2156 2992 y Fl(\000)2198 3066 y Fv(S)7 b FA(\()p Fs(h)2349 3083 y Fr(\(2\))2446 3066 y FA(\))2482 2992 y Fl(\001)700 3217 y FA(=)25 b(\()p Fs( )s FA(\()p Fs(h)p FA(\)\))1055 3234 y Fr(\(1\))1152 3217 y Fs(\036)p FA(\()p Fs(r)s FA(\))15 b Fv(S)1413 3143 y Fl(\000)1455 3217 y FA(\()p Fs( )s FA(\()p Fs(h)p FA(\)\))1714 3234 y Fr(\(2\))1811 3143 y Fl(\001)1878 3217 y FA(=)25 b Fs( )s FA(\()p Fs(h)p FA(\))p Fs(:\036)p FA(\()p Fs(r)s FA(\))g(=)g Fs(h:)2555 3231 y Fp( )2610 3217 y Fs(\036)p FA(\()p Fs(r)s FA(\);)406 3372 y Fs(\016)450 3335 y Fp( )505 3372 y Fs(\036)p FA(\()p Fs(r)s FA(\))g(=)g(\()p Fs( )894 3335 y Fo(\000)p Fr(1)1013 3372 y Fv(\012)20 b FA(id)o(\)\()p Fs(\020)26 b Fv(\012)21 b FA(id)o(\)\(\011)e Fv(\012)h FA(\011\)\001\()p Fs(r)s FA(\))k(=)i(\()p Fs( )2257 3335 y Fo(\000)p Fr(1)2355 3372 y Fs(\020)7 b FA(\011)20 b Fv(\012)g FA(\011\)\001\()p Fs(r)s FA(\))700 3514 y(=)25 b(\()p Fs(\031)e Fv(\012)e FA(\011\)\001\()p Fs(r)s FA(\))i(=)j(\(id)14 b Fv(\012)p Fs(\036)p FA(\))p Fs(\016)s FA(\()p Fs(r)s FA(\))p Fs(:)110 3723 y FA(Here)35 b(again)g(w)m(e)h(used)g(\(6.2\).)58 b(The)35 b(last)g(statemen)m(t)f(follo)m(ws)h(b)s(ecause)f(\011)j(is)e (an)h(algebra)0 3833 y(map.)91 b Fi(\003)110 3994 y FA(Let)36 b(no)m(w)i Fs(H)44 b FA(=)38 b Fs(k)s FA(\000,)h(where)d(\000)i(is)f(a) g(\014nite)e(ab)s(elian)g(group.)61 b(Let)36 b Fs(g)s FA(\(1\),)h Fs(:)15 b(:)g(:)32 b FA(,)40 b Fs(g)s FA(\()p Fs(\022)s FA(\))35 b Fv(2)i FA(\000,)0 4115 y Fs(\037)p FA(\(1\),)32 b Fs(:)15 b(:)g(:)32 b FA(,)h Fs(\037)p FA(\()p Fs(\022)s FA(\))27 b Fv(2)722 4092 y Fl(b)719 4115 y FA(\000)32 b(and)g(let)f Fs(V)53 b FA(b)s(e)31 b(the)h(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)h(with)h(structure)f(giv)m (en)0 4236 y(b)m(y)c(\(1.5\).)39 b(Let)25 b(also)f Fs(h)p FA(\(1\),)h Fs(:)15 b(:)g(:)32 b FA(,)27 b Fs(h)p FA(\()p Fs(\022)s FA(\))d Fv(2)h FA(\000,)i Fs(\021)s FA(\(1\),)f Fs(:)15 b(:)g(:)32 b FA(,)27 b Fs(\021)s FA(\()p Fs(\022)s FA(\))d Fv(2)2198 4213 y Fl(b)2195 4236 y FA(\000)i(and)f(let)f (analogously)f Fs(W)39 b FA(b)s(e)0 4346 y(the)31 b(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)i(with)g(structure)f(giv)m(en)h(b)m(y)i (\(1.5\))e(with)g Fs(g)s FA(\()p Fs(i)p FA(\)'s,)j(resp.)46 b Fs(\037)p FA(\()p Fs(j)5 b FA(\)'s,)0 4456 y(replaced)34 b(b)m(y)k Fs(h)p FA(\()p Fs(i)p FA(\)'s,)h(resp.)61 b Fs(\021)s FA(\()p Fs(j)5 b FA(\)'s.)61 b(Let)37 b Fs(R)q FA(,)i Fs(S)k FA(b)s(e)36 b(the)g(corresp)s(onding)d(Nic)m(hols)j (algebras)0 4565 y(and)30 b Fs(A)25 b FA(=)h Fs(R)q FA(#)p Fs(H)7 b FA(,)31 b Fs(B)f FA(=)c Fs(S)5 b FA(#)p Fs(H)i FA(.)0 4726 y FC(Prop)s(osition)28 b(6.3.)52 b Fj(Assume)24 b(that)f Fs(R)j Fj(and)d Fs(S)30 b Fj(are)23 b(\014nite)g(dimensional.) 35 b(The)24 b(Hopf)g(algebras)0 4836 y Fs(A)29 b Fj(and)f Fs(B)33 b Fj(are)28 b(isomorphic)d(if)j(and)g(only)f(if)h(there)f (exist)g Fs(')e Fv(2)g FA(Aut\(\000\))j Fj(and)f Fs(\033)i Fv(2)d Fu(S)2852 4850 y Fp(\022)2916 4836 y Fj(suc)m(h)i(that)0 5151 y FA(\(6.3\))363 b Fs(g)s FA(\()p Fs(j)5 b FA(\))24 b(=)i Fs(')895 5113 y Fo(\000)p Fr(1)1008 5151 y FA(\()o Fs(h)p FA(\()p Fs(\033)s(j)5 b FA(\)\))14 b Fs(;)198 b(\037)p FA(\()p Fs(j)5 b FA(\))24 b(=)i Fs(\021)s FA(\()p Fs(\033)s(j)5 b FA(\))p Fs(';)198 b FA(1)25 b Fv(\024)g Fs(j)30 b Fv(\024)c Fs(\022)s(:)p eop %%Page: 29 29 29 28 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(29)0 91 y Ft(Pr)-5 b(o)g(of.)46 b FA(Assume)27 b(that)e(\011)h(:)f Fs(A)h Fv(!)g Fs(B)32 b FA(is)26 b(an)g(isomorphism)e(of)j(Hopf)f(algebras.)38 b(W)-8 b(e)26 b(claim)f(that)0 201 y(\011\()p Fs(A)p FA(\()p Fs(n)p FA(\)\))f(=)h Fs(B)5 b FA(\()p Fs(n)p FA(\))21 b(for)h(all)f Fs(n)26 b Fv(\025)f FA(0.)38 b(This)22 b(is)g(clear)e(for)i Fs(n)j FA(=)h(0)c(since)f(\011)h(preserv)m(es)g (the)f(coradical)0 311 y(\014ltration.)40 b(F)-8 b(or)31 b Fs(n)26 b FA(=)h(1)k(w)m(e)g(argue)f(as)h(in)f([AS2,)j(Lemma)d(5.4].) 44 b(Indeed,)30 b(let)g(us)h(consider)e(the)0 420 y(adjoin)m(t)e (action)g(of)h(\000)h(on)g Fs(A)p FA(.)40 b(The)29 b(decomp)s(osition) 24 b Fs(A)1883 434 y Fr(1)1950 420 y FA(=)h Fs(A)p FA(\(0\))17 b Fv(\010)g Fs(A)p FA(\(1\))28 b(is)h(a)f(decomp)s(osition)0 530 y(of)c(\000-mo)s(dules.)37 b(By)24 b([AS2,)j(Lemma)c(3.1])i Fs(A)p FA(\(0\))e(is)h(the)f(isot)m(ypical)e(comp)s(onen)m(t)h(of)i (trivial)e(t)m(yp)s(e)0 639 y(of)33 b(the)f(\000-mo)s(dule)g Fs(A)748 653 y Fr(1)822 639 y FA(\(here)g(w)m(e)h(use)g(that)f Fs(A)i FA(is)e(\014nite)g(dimensional\).)45 b(Hence)33 b Fs(A)p FA(\(1\))f(is)h(the)0 749 y(sum)22 b(of)g(all)f(the)g(other)g (isot)m(ypical)e(comp)s(onen)m(ts.)35 b(This)22 b(implies)e(that)g (\011\()p Fs(A)p FA(\(1\)\))k(=)i Fs(B)5 b FA(\(1\),)22 b(since)0 858 y(\011)29 b(in)m(tert)m(wines)e(the)h(resp)s(ectiv)m(e)f (adjoin)m(t)g(actions)g(of)i(\000.)41 b(It)29 b(is)g(not)f(di\016cult)f (to)i(see)f(that)g Fs(A)i FA(is)0 968 y(generated)f(b)m(y)k Fs(A)615 982 y Fr(1)688 968 y FA(as)f(an)g(algebra,)f(b)m(y)h (de\014nition)d(of)j(Nic)m(hols)f(algebra)e(and)j([AS2,)h(Lemma)0 1078 y(2.4)c(\(iii\)].)38 b(Hence)28 b(\011\()p Fs(A)p FA(\()p Fs(n)p FA(\)\))c(=)h Fs(B)5 b FA(\()p Fs(n)p FA(\))28 b(for)g(all)g Fs(n)d Fv(\025)g FA(0.)40 b(W)-8 b(e)29 b(can)f(apply)g(no)m(w)h(\(the)f(second)f(half)0 1187 y(of)7 b(\))28 b(Lemma)g(6.2.)40 b(The)28 b(automorphism)d Fs( )k Fv(2)d FA(Aut)o(\()p Fs(H)7 b FA(\))28 b(is)g(determined)d(b)m (y)k Fs(')c Fv(2)h FA(Aut)o(\(\000\).)40 b(W)-8 b(e)0 1297 y(also)25 b(kno)m(w)i(that)e Fs(R)h Fv(')g Fs(S)861 1264 y Fp( )943 1297 y FA(as)g(braided)e(Hopf)i(algebras;)g(but)f(this) h(is)g(equiv)-5 b(alen)m(t)24 b(to)h Fs(V)46 b Fv(')26 b Fs(W)3233 1264 y Fp( )0 1406 y FA(in)102 1373 y Fp(H)102 1432 y(H)172 1406 y Fv(Y)8 b(D)r FA(.)40 b(This)25 b(is)h(p)s(ossible)d (if)i(and)h(only)f(if)g(\(6.3\))g(holds.)39 b(If)26 b(\(6.3\))f(holds,) h(w)m(e)g(do)f(not)h(need)e(to)0 1518 y(assume)31 b(that)g Fs(R)i FA(and)f Fs(S)37 b FA(are)32 b(\014nite)e(dimensional.)43 b(By)32 b(what)f(w)m(e)h(ha)m(v)m(e)h(just)e(said,)h Fs(R)d Fv(')g Fs(S)3207 1485 y Fp( )3262 1518 y FA(.)0 1628 y(Hence)g Fs(R)q FA(#)p Fs(H)k Fv(')26 b Fs(S)684 1595 y Fp( )739 1628 y FA(#)p Fs(H)7 b FA(.)42 b(By)30 b(Lemma)g(6.1,)g(w)m(e)h(infer)e(that)f Fs(A)e Fv(')g Fs(B)5 b FA(.)91 b Fi(\003)110 1880 y FA(W)-8 b(e)35 b(no)m(w)h(\014nish)e(the)h(pro)s(of)f(of)h(Theorem)f(1.3.)57 b(W)-8 b(e)35 b(shall)f(apply)h(Prop)s(osition)d(6.3)j(with)0 1996 y(\000)25 b(=)h Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\).)34 b(Let)24 b(us)h(\014rst)g(assume)f Fs(\022)k FA(=)d(2.)39 b(Let)25 b Fs(g)s FA(\(1\))p Fs(;)15 b(g)s FA(\(2\))23 b Fv(2)i FA(\000)g(and)g Fs(\037)p FA(\(1\))p Fs(;)15 b(\037)p FA(\(2\))24 b Fv(2)2904 1973 y Fl(b)2901 1996 y FA(\000)h(de\014ne)f(a)0 2105 y(Nic)m(hols)j(algebra)e(of)i(\014nite) f(Cartan)g(t)m(yp)s(e.)40 b(Let)27 b Fs(u)e Fv(2)h FA(\000)15 b Fv(\000)g FA(1,)29 b Fs(q)f Fv(2)d Fs(k)2304 2072 y Fo(\002)2393 2105 y FA(of)j(order)e Fs(p)i FA(and)f(in)m(tegers)0 2215 y Fs(b)p FA(,)35 b Fs(d)f FA(suc)m(h)g(that)e(the)h Fs(g)s FA(\()p Fs(i)p FA(\)'s)h(and)f Fs(\037)p FA(\()p Fs(j)5 b FA(\)'s)33 b(are)g(giv)m(en)g(b)m(y)i(\(5.1\).)50 b(Note)34 b(that)e Fs(d)i FA(is)f(determined)0 2325 y(b)m(y)40 b Fs(b)p FA(.)69 b(W)-8 b(e)40 b(denote)e(b)m(y)i Fs(R)q FA(\()p Fs(q)s(;)15 b(b)p FA(\))40 b(the)e(Nic)m(hols)h(algebra)e (corresp)s(onding)e(to)k(the)g Fs(g)s FA(\()p Fs(i)p FA(\)'s)h(and)0 2446 y Fs(\037)p FA(\()p Fs(j)5 b FA(\)'s.)59 b(Let)36 b(no)m(w)h Fs(h)p FA(\(1\))p Fs(;)15 b(h)p FA(\(2\))33 b Fv(2)j FA(\000,)j Fs(\021)s FA(\(1\))p Fs(;)15 b(\021)s FA(\(2\))35 b Fv(2)1824 2422 y Fl(b)1820 2446 y FA(\000)q(\))h (de\014ne)f(another)f(Nic)m(hols)i(algebra)e(of)0 2555 y(\014nite)28 b(Cartan)h(t)m(yp)s(e.)40 b(Let)30 b(also)f Fs(v)g Fv(2)c FA(\000)20 b Fv(\000)h FA(1,)34 b Fl(e)-55 b Fs(q)29 b Fv(2)c Fs(k)1767 2522 y Fo(\002)1858 2555 y FA(of)30 b(order)f Fs(p)p FA(,)i(in)m(tegers)d Fs(s)p FA(,)j Fs(t)f FA(and)569 2801 y Fs(h)p FA(\(1\))23 b(=)j Fs(v)s(;)107 b(h)p FA(\(2\))24 b(=)h Fs(v)1377 2763 y Fp(s)1417 2801 y Fs(;)107 b Fv(h)p Fs(\021)s FA(\(1\))p Fs(;)15 b(v)s Fv(i)25 b FA(=)k Fl(e)-55 b Fs(q)s(;)107 b Fv(h)p Fs(\021)s FA(\(2\))p Fs(;)15 b(v)s Fv(i)25 b FA(=)30 b Fl(e)-56 b Fs(q)2661 2763 y Fp(t)2694 2801 y Fs(:)110 3000 y FA(W)-8 b(e)44 b(can)f(assume,)k(b)m(y)d(Prop)s (osition)d(6.3,)47 b(that)c Fs(u)48 b FA(=)h Fs(v)s FA(.)82 b(Then)43 b(the)g(Nic)m(hols)g(algebra)0 3110 y(corresp)s(onding)26 b(to)k(the)f Fs(h)p FA(\()p Fs(i)p FA(\)'s)h(and)f Fs(\021)s FA(\()p Fs(j)5 b FA(\)'s)31 b(is)f Fs(R)q FA(\()t Fl(e)-55 b Fs(q)r(;)15 b(s)p FA(\).)0 3294 y FC(Lemma)29 b(6.4.)52 b Fj(The)24 b(Hopf)i(algebras)d Fs(A)j FA(=)f Fs(R)q FA(\()p Fs(q)s(;)15 b(b)p FA(\)#)p Fs(k)s FA(\000)25 b Fj(and)g Fs(B)30 b FA(=)25 b Fs(R)q FA(\()t Fl(e)-55 b Fs(q)s(;)15 b(s)p FA(\)#)p Fs(k)s FA(\000)25 b Fj(are)g(isomor-)0 3404 y(phic)k(if)h(and)f(only)h(if)178 3547 y FA(\(i\))53 b Fs(q)28 b FA(=)i Fl(e)-55 b Fs(q)33 b Fj(and)d Fs(b)25 b Fv(\021)h Fs(s)60 b FA(mo)s(d)30 b Fs(p)g Fj(when)g(the)f(t)m(yp)s(e) g(is)h Fs(B)2055 3561 y Fr(2)2127 3547 y Fj(or)g Fs(G)2311 3561 y Fr(2)2352 3547 y Fj(,)152 3657 y FA(\(ii\))53 b Fs(q)28 b FA(=)i Fl(e)-55 b Fs(q)33 b Fj(and)d Fs(b)25 b Fv(\021)h Fs(s)k Fj(or)g Fv(\021)25 b(\000)p FA(1)20 b Fv(\000)g Fs(s)61 b FA(mo)s(d)30 b Fs(p)g Fj(when)f(the)h(t)m(yp)s(e) f(is)h Fs(A)2564 3671 y Fr(2)2636 3657 y Fj(.)0 3875 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)38 b(assume)g(that)e Fs(A)j FA(and)e Fs(B)43 b FA(are)38 b(isomorphic;)h(it)e(is)h(clear)e (that)h(they)g(are)g(of)h(the)0 3984 y(same)g(t)m(yp)s(e.)65 b(Let)38 b Fs(')p FA(,)j Fs(\033)h FA(b)s(e)c(as)g(in)g(Prop)s(osition) d(6.3,)41 b Ft(i.e.)65 b FA(\(6.3\))38 b(holds.)64 b(If)39 b Fs(\033)k FA(=)c(id)o(,)i(then)0 4094 y Fs(')p FA(\()p Fs(u)p FA(\))24 b(=)i Fs(u)31 b FA(and)e(hence)g Fs(q)f FA(=)i Fl(e)-55 b Fs(q)s FA(,)30 b Fs(b)c FA(=)f Fs(s)p FA(.)41 b(If)30 b Fs(\033)f Fv(6)p FA(=)d(id)o(,)31 b(then)e Fs(')p FA(\()p Fs(u)p FA(\))24 b(=)h Fs(u)2319 4061 y Fp(s)2359 4094 y FA(,)31 b Fs(')p FA(\()p Fs(u)2563 4061 y Fp(b)2599 4094 y FA(\))25 b(=)g Fs(u)p FA(,)286 4449 y Fs(q)j FA(=)e Fv(h)p Fs(\037)p FA(\(1\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fv(h)p Fs(\021)s FA(\(2\))p Fs(;)15 b(u)1205 4411 y Fp(s)1244 4449 y Fv(i)25 b FA(=)30 b Fl(e)-56 b Fs(q)1445 4411 y Fp(st)1512 4449 y Fs(;)198 b(q)1779 4411 y Fp(d)1847 4449 y FA(=)26 b Fv(h)p Fs(\037)p FA(\(2\))p Fs(;)15 b(u)p Fv(i)24 b FA(=)h Fv(h)p Fs(\021)s FA(\(1\))p Fs(;)15 b(u)2697 4411 y Fp(s)2736 4449 y Fv(i)25 b FA(=)k Fl(e)-55 b Fs(q)2937 4411 y Fp(s)2976 4449 y Fs(:)110 4695 y FA(These)45 b(equalities)d(imply)j Fs(bs)51 b Fv(\021)h FA(1)60 b(mo)s(d)30 b Fs(p)p FA(,)50 b Fs(dt)h Fv(\021)g FA(1)61 b(mo)s(d)29 b Fs(p)p FA(.)88 b(Hence)45 b Fs(st)51 b Fv(\021)h FA(\()p Fs(bd)p FA(\))3191 4661 y Fo(\000)p Fr(1)0 4804 y FA(mo)s(d)29 b Fs(p)p FA(.)65 b(W)-8 b(e)38 b(kno)m(w)g(that)f(\()p Fs(bd)p FA(\))1117 4771 y Fo(\000)p Fr(1)1252 4804 y Fv(\021)i FA(\()p Fs(st)p FA(\))1510 4771 y Fo(\000)p Fr(1)1645 4804 y Fv(\021)g FA(1)p Fs(;)15 b FA(2)38 b(or)f(3)61 b(mo)s(d)29 b Fs(p)39 b FA(if)e(the)g(t)m(yp)s(e)h(is)f(resp)s(ec-)0 4914 y(tiv)m(ely)h Fs(A)328 4928 y Fr(2)370 4914 y FA(,)k Fs(B)506 4928 y Fr(2)586 4914 y FA(or)d Fs(G)779 4928 y Fr(2)820 4914 y FA(.)68 b(This)38 b(leads)g(immediately)e(to)j(a)g(con)m(tradiction) 34 b(unless)k(the)h(t)m(yp)s(e)0 5023 y(is)e Fs(A)167 5037 y Fr(2)209 5023 y FA(.)64 b(But)37 b(in)g(this)f(last)h(case,)i Fs(s)f Fv(\021)g Fs(d)60 b FA(mo)s(d)30 b Fs(p)38 b FA(and)f Fs(t)h Fv(\021)g Fs(b)61 b FA(mo)s(d)30 b Fs(p)p FA(.)63 b(Hence)37 b Fs(q)k FA(=)i Fl(e)-56 b Fs(q)42 b FA(and)0 5133 y Fs(b)26 b Fv(\021)f(\000)p FA(1)20 b Fv(\000)g Fs(s)61 b FA(mo)s(d)29 b Fs(p)p FA(.)92 b Fi(\003)p eop %%Page: 30 30 30 29 bop 0 -128 a Fy(30)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)110 91 y FA(W)-8 b(e)24 b(no)m(w)h(pass)f(to)g(the)f (cases)h Fs(\022)j FA(=)f(3)e(and)g Fs(\022)k FA(=)d(4.)39 b(W)-8 b(e)24 b(denote)f(b)m(y)i Fs(R)2436 105 y Fr(4)2477 91 y FA(\()p Fs(q)s(;)15 b(e)p FA(\),)26 b(resp.)38 b Fs(R)3023 105 y Fr(3)3064 91 y FA(\()p Fs(q)s(;)15 b(e)p FA(\),)0 201 y(the)34 b(Nic)m(hols)g(algebra)f(corresp)s(onding)d(to)35 b(the)f(sequence)f(in)h(\(5.7\),)i(resp.)54 b(the)34 b(sequence)f(of)0 311 y(the)40 b(\014rst)h(three)e(terms)i(in)f (\(5.7\).)72 b(Here)41 b Fs(q)k FA(is)40 b(a)h(third)f(ro)s(ot)f(of)i (1)g(and)g Fs(e)i FA(=)h(1)d(or)f(2.)74 b(By)0 420 y(Prop)s(osition)27 b(6.3,)j(arguing)e(as)i(for)g Fs(\022)e FA(=)d(2,)31 b(w)m(e)f(are)g(reduced)e(to)h(pro)m(v)m(e:)0 581 y FC(Lemma)34 b(6.5.)52 b Fj(Let)30 b Fs(q)s(;)20 b Fl(e)-56 b Fs(q)34 b Fj(b)s(e)29 b(third)g(ro)s(ots)f(of)j(1)f(and)f Fs(e;)16 b Fl(e)-52 b Fs(e)25 b FA(=)h(1)k Fj(or)g(2.)110 690 y(\(i\).)46 b(The)32 b(Hopf)h(algebras)d Fs(R)1122 704 y Fr(4)1163 690 y FA(\()p Fs(q)s(;)15 b(e)p FA(\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))26 b Fj(and)32 b Fs(R)1993 704 y Fr(4)2034 690 y FA(\()t Fl(e)-55 b Fs(q)s(;)16 b Fl(e)-52 b Fs(e)p FA(\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))26 b Fj(are)32 b(isomorphic)d(if)0 800 y(and)h(only)f(if)h Fs(e)25 b FA(=)h Fl(e)-52 b Fs(e)p Fj(.)110 909 y(\(ii\).)38 b(The)30 b(Hopf)g(algebras)e Fs(R)1133 923 y Fr(3)1175 909 y FA(\()p Fs(q)s(;)15 b(e)p FA(\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\()o(3\))24 b Fj(and)30 b Fs(R)2000 923 y Fr(3)2041 909 y FA(\()t Fl(e)-55 b Fs(q)s(;)16 b Fl(e)-52 b Fs(e)p FA(\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3)o(\))24 b Fj(are)30 b(isomorphic)d(if)0 1019 y(and)j(only)f(if)h Fs(e)25 b FA(=)h Fl(e)-52 b Fs(e)30 b Fj(and)g Fs(q)e FA(=)i Fl(e)-56 b Fs(q)s Fj(.)0 1180 y Ft(Pr)-5 b(o)g(of.)46 b FA(All)31 b(the)g(elemen)m(ts)f Fs(g)s FA(\()p Fs(i)p FA(\))i(are)f(equal)f(for)i Fs(e)27 b FA(=)h(1)k(but)f Fs(g)s FA(\(1\))26 b Fv(6)p FA(=)i Fs(g)s FA(\(3\))j(for)g Fs(e)d FA(=)g(2.)45 b(Hence,)0 1289 y(b)m(y)36 b(Prop)s(osion)e(6.3,)j Fs(e)d FA(=)i Fl(e)-52 b Fs(e)36 b FA(if)f(the)g(Hopf)h(algebras)d(are) i(isomorphic.)55 b(By)36 b(Prop)s(osition)c(6.3)0 1399 y(again,)k Fs(\036)f FA(=)g(id)h(resp.)57 b Fs(\036)p FA(\()p Fs(u)p FA(\))35 b(=)g Fs(u)1201 1366 y Fr(2)1278 1399 y FA(and)g Fs(\033)s FA(\(1\))g(=)f(3)p Fs(;)15 b(\033)s FA(\(2\))35 b(=)g(4)p Fs(;)15 b(\033)s FA(\(3\))34 b(=)h(1)p Fs(;)15 b(\033)s FA(\(4\))34 b(=)h(2)h(de\014ne)0 1508 y(an)g(isomorphism)e Fs(R)742 1522 y Fr(4)783 1508 y FA(\()p Fs(q)s(;)15 b FA(1\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))1370 1484 y Fv(\030)1370 1513 y FA(=)1476 1508 y Fs(R)1545 1522 y Fr(4)1587 1508 y FA(\()p Fs(q)1667 1475 y Fo(\000)p Fr(1)1764 1508 y Fs(;)g FA(1\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))30 b(resp.)59 b Fs(R)2589 1522 y Fr(4)2630 1508 y FA(\()p Fs(q)s(;)15 b FA(2\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))3217 1484 y Fv(\030)3217 1513 y FA(=)0 1618 y Fs(R)69 1632 y Fr(4)110 1618 y FA(\()p Fs(q)190 1585 y Fo(\000)p Fr(1)288 1618 y Fs(;)g FA(2\)#)p Fs(k)s Fu(Z)p Fs(=)p FA(\(3\))o(.)33 b(This)24 b(pro)m(v)m(es)g(\(i\).) 38 b(In)24 b(case)f(\(ii\))f(it)i(follo)m(ws)f(from)g(Prop)s(osition)e (6.3)j(that)0 1728 y(an)30 b(isomorphism)e(is)i(only)f(p)s(ossible)f (if)h Fs(q)g FA(=)g Fl(e)-55 b Fs(q)s FA(.)92 b Fi(\003)0 1939 y Fv(x)p FC(7.)46 b(P)m(oin)m(ted)35 b(Hopf)g(algebras)h(of)f (order)g Fs(p)1673 1906 y Fr(4)1714 1939 y FC(.)110 2049 y FA(Let)26 b Fs(p)i FA(b)s(e)f(an)g(o)s(dd)f(prime)g(n)m(um)m(b)s(er.) 39 b(W)-8 b(e)28 b(no)m(w)f(c)m(haracterize)d(coradically)g(graded)h(p) s(oin)m(ted)0 2158 y(Hopf)30 b(algebras)e(of)i(order)f Fs(p)968 2125 y Fr(4)1009 2158 y FA(.)0 2319 y FC(Theorem)41 b(7.1.)52 b Fj(Let)34 b Fs(G)i Fj(b)s(e)e(a)h(coradically)d(graded)h(p) s(oin)m(ted)g(Hopf)j(algebras)d(of)i(order)f Fs(p)3247 2286 y Fr(4)0 2428 y Fj(with)29 b(coradical)e Fu(|)-9 b FA(\000)p Fj(.)35 b(Then)29 b Fs(G)c Fv(')h Fs(R)q FA(#)p Fu(|)-9 b FA(\000)p Fj(,)24 b(where)86 2564 y FA(\(7.2\))54 b Fs(R)31 b Fj(is)f(a)g(quan)m(tum)g(line)e(if)i FA(\000)h Fj(has)e(order)g Fs(p)1774 2530 y Fr(3)1815 2564 y Fj(;)86 2673 y FA(\(7.3\))54 b Fs(R)31 b Fj(is)f(a)g(quan)m(tum) g(line)e(or)i(plane)f(if)g FA(\000)i Fj(has)f(order)f Fs(p)2130 2640 y Fr(2)2171 2673 y Fj(;)86 2783 y FA(\(7.4\))54 b Fs(R)26 b FA(=)g Fm(B)p FA(\()p Fs(V)19 b FA(\))p Fj(,)30 b(where)e Fs(V)49 b Fj(has)29 b(Dynkin)f(diagram)f Fs(A)2072 2797 y Fr(2)2143 2783 y Fj(if)h FA(\000)h Fj(has)g(order)f Fs(p)p Fj(,)h(where)f Fs(p)e FA(=)f(3)329 2892 y Fj(or)30 b Fs(p)20 b Fv(\000)g FA(1)30 b Fj(is)g(divisible)d(b)m(y)k(3.)0 3078 y Ft(Pr)-5 b(o)g(of.)46 b FA(The)25 b(case)g(\(7.2\))f(follo)m(ws) g(from)i(Prop)s(osition)c(7.5)j(b)s(elo)m(w;)h(w)m(e)g(can)e(not)h (apply)g(directly)0 3188 y([AS2])i(since)d(there)g(are)h(non-ab)s (elian)d(groups)j(of)g(order)g Fs(p)1999 3155 y Fr(3)2040 3188 y FA(.)39 b(The)26 b(case)e(\(7.3\))h(follo)m(ws)g(directly)0 3298 y(from)30 b([AS2,)h(Lemma)f(5.6].)42 b(If)30 b(\000)h(has)f(order) f Fs(p)p FA(,)i(then)e(w)m(e)i(see)e(from)h(Theorem)f(1.3)i(that)e (only)0 3407 y(the)g(case)h(\(ii\),)e(that)h(is)h(\(7.4\),)f(is)h(p)s (ossible.)89 b Fi(\003)0 3568 y FC(Prop)s(osition)48 b(7.5.)j Fj(Let)41 b FA(\000)h Fj(b)s(e)e(a)h(\014nite)f (\(non-necessarily)d(ab)s(elian\))h(group.)73 b(Let)40 b Fs(R)j Fj(b)s(e)0 3677 y(a)c(braided)d(Hopf)j(algebra)e(in)1099 3644 y Fr(\000)1099 3702 y(\000)1149 3677 y Fv(Y)8 b Fs(D)41 b Fj(of)d(prime)g(order)g Fs(p)p Fj(,)j(where)d Fs(p)h Fj(is)f(the)g(smallest)f(prime)0 3787 y(dividing)28 b(the)h(order)g(of)h FA(\000)p Fj(.)41 b(Assume)30 b(that)f Fs(R)1581 3801 y Fr(0)1648 3787 y FA(=)c Fu(|)-9 b FA(1)p Fj(.)35 b(Then)29 b Fs(R)j Fj(is)d(a)h(quan)m(tum)g(line.)0 3947 y Ft(Pr)-5 b(o)g(of.)46 b FA(As)27 b Fs(P)13 b FA(\()p Fs(R)q FA(\))24 b(is)i(a)g(Y)-8 b(etter-Drinfeld)22 b(submo)s(dule)i (of)h Fs(R)i FA(and)f(it)f(is)g(non-zero)g(b)m(y)h(h)m(yp)s(othe-)0 4057 y(sis,)k(there)e(exists)h(1)c Fv(6)p FA(=)h Fs(g)i Fv(2)d FA(\000)31 b(suc)m(h)e(that)g Fs(P)13 b FA(\()p Fs(R)q FA(\))1669 4071 y Fp(g)1735 4057 y Fv(6)p FA(=)26 b(0.)40 b(But)29 b(the)h(Y)-8 b(etter-Drinfeld)26 b(condition)0 4167 y(sho)m(ws)36 b(that)e Fs(P)13 b FA(\()p Fs(R)q FA(\))685 4181 y Fp(g)761 4167 y FA(is)36 b(stable)d(under)i(the)f (action)g(of)h Fs(g)s FA(.)57 b(Hence,)36 b(there)e(exists)g Fs(y)k Fv(2)c Fs(P)13 b FA(\()p Fs(R)q FA(\),)0 4276 y Fs(y)28 b Fv(6)p FA(=)e(0,)k(suc)m(h)g(that)1152 4386 y Fs(g)s(:y)f FA(=)c Fs(q)s(y)s(;)107 b(\016)s FA(\()p Fs(y)s FA(\))25 b(=)g Fs(g)e Fv(\012)e Fs(y)s(:)0 4552 y FA(Let)36 b Fs(N)48 b FA(b)s(e)36 b(the)g(order)g(of)h Fs(q)s FA(.)62 b(By)37 b([AS2,)i(Lemma)e(3.1])g Fs(N)48 b FA(is)37 b(not)f(1.)61 b(No)m(w)38 b Fs(N)48 b FA(divides)35 b(the)0 4661 y(order)24 b(of)h(\000,)i(hence)d Fs(N)35 b Fv(\025)26 b Fs(p)p FA(.)39 b(Since)23 b(the)i(dimension)e(of)i Fs(k)s FA([)p Fs(y)s FA(])g(is)g Fs(N)36 b FA(b)m(y)26 b(the)e(quan)m(tum)h(binomial)0 4771 y(form)m(ula)k(\(cf.)40 b([AS2,)31 b(Lemma)f(3.3]\),)h(w)m(e)f(conclude)d(that)i Fs(N)36 b FA(=)25 b Fs(p:)92 b Fi(\003)0 4932 y Ft(R)-5 b(emark.)46 b FA(Prop)s(osition)20 b(7.5)k(allo)m(ws)e(the)h (classi\014cation)c(of)24 b(\014nite)e(dimensional)e(p)s(oin)m(ted)i (Hopf)0 5041 y(algebras)34 b(of)h(dimension)f Fs(p)955 5008 y Fp(n)1040 5041 y FA(whose)h(coradical)e(has)i(index)g Fs(p)p FA(.)58 b(This)35 b(last)g(result)f(w)m(as)i(inde-)0 5151 y(p)s(endien)m(tly)27 b(giv)m(en)i(in)h([D])h(b)m(y)g(a)f (di\013eren)m(t)e(metho)s(d.)p eop %%Page: 31 31 31 30 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(31)0 91 y Fv(x)p FC(8.)46 b(A)35 b(b)s(ound)h(on)f(the)g(dimension.)110 201 y FA(Let)30 b(\000)g(b)s(e)g(again)f(our)h(\014xed)g(\014nite)e(ab)s(elian)g(group) h(of)i(o)s(dd)e(order.)40 b(Theorem)30 b(1.1)g(giv)m(es)g(a)0 311 y(metho)s(d)i(to)h(obtain)f(\014nite)g(dimensional)e(Nic)m(hols)j (algebras)f(o)m(v)m(er)i(\000:)48 b(pic)m(k)34 b(a)f(\014nite)f(Cartan) 0 420 y(matrix)25 b(\()p Fs(a)377 434 y Fp(ij)441 420 y FA(\))h(and)f(a)h(Y)-8 b(etter-Drinfeld)23 b(mo)s(dule)h Fs(V)47 b FA(o)m(v)m(er)26 b(\000)h(of)f(Cartan)e(t)m(yp)s(e)h(with)g (asso)s(ciated)0 530 y(matrix)34 b(\()p Fs(a)386 544 y Fp(ij)450 530 y FA(\);)k(then)c Fm(B)p FA(\()p Fs(V)20 b FA(\))35 b(is)g(\014nite)e(dimensional)f(and)j(w)m(e)g(kno)m(w)h(in)e (fact)h(its)f(dimension.)0 639 y(W)-8 b(e)24 b(w)m(an)m(t)g(no)m(w)f (to)h(describ)s(e)c(explicitly)h(a)i(w)m(a)m(y)i(to)e(construct)e(Y)-8 b(etter-Drinfeld)21 b(mo)s(dules)h(o)m(v)m(er)0 749 y(\000)31 b(of)f(Cartan)e(t)m(yp)s(e)i(with)f(asso)s(ciated)e(matrix)i(\()p Fs(a)1731 763 y Fp(ij)1795 749 y FA(\).)110 858 y(W)-8 b(e)23 b(\014x)g(elemen)m(ts)e Fs(y)793 872 y Fr(1)834 858 y FA(,)i Fs(:)15 b(:)g(:)32 b FA(,)25 b Fs(y)1114 872 y Fp(M)1219 858 y FA(of)d(\000)i(suc)m(h)e(that)g(ord)15 b Fs(y)1978 872 y Fp(`)2038 858 y FA(=)25 b Fs(E)2201 872 y Fp(`)2237 858 y FA(,)g(\000)g(=)h Fv(h)p Fs(y)s FA(\(1\))p Fv(i)6 b(\010)g(\001)15 b(\001)g(\001)5 b(\010)h(h)p Fs(y)s FA(\()p Fs(M)k FA(\))p Fv(i)p FA(,)0 968 y Fs(E)67 982 y Fp(`)134 968 y FA(divides)28 b Fs(E)508 982 y Fp(`)p Fr(+1)667 968 y FA(and)j(set)f Fs(E)i FA(=)26 b Fs(E)1252 982 y Fp(M)1333 968 y FA(.)43 b(Then)30 b(\000)c(=)g Fv(\010)1891 982 y Fr(1)p Fo(\024)p Fp(`)p Fo(\024)p Fp(M)2153 968 y Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))p Fs(y)s FA(\()p Fs(`)p FA(\))26 b(is)k(a)h(direct)d(sum)j(of)0 1078 y(cyclic)h Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\)-mo)s(dul)o(es)28 b Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))p Fs(y)s FA(\()p Fs(`)p FA(\))28 b(of)33 b(order)g Fs(E)1755 1092 y Fp(`)1790 1078 y FA(.)51 b(Let)33 b Fs(q)k FA(b)s(e)c(a)g(primitiv)m(e)f Fs(E)5 b FA(-th)34 b(ro)s(ot)e(of)h(1)0 1187 y(in)d Fu(|)-10 b FA(.)110 1297 y(A)34 b(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)h Fs(V)55 b FA(o)m(v)m(er)34 b(\000)g(of)g(Cartan)e(t)m(yp)s(e)i(\()p Fs(a)2291 1311 y Fp(ij)2355 1297 y FA(\))2391 1311 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)2698 1297 y FA(is)f(giv)m(en)g(\(up)g(to)0 1421 y(p)s(erm)m(utation\))k(b)m(y)42 b(sequences)e Fs(g)s FA(\(1\))p Fs(;)15 b(:)g(:)g(:)h(;)f(g)s FA(\()p Fs(\022)s FA(\))42 b Fv(2)i FA(\000,)h Fs(\037)p FA(\(1\))p Fs(;)15 b(:)g(:)g(:)h(;)f(\037)p FA(\()p Fs(\022)s FA(\))43 b Fv(2)2640 1398 y Fl(b)2636 1421 y FA(\000)f(suc)m(h)f(that)f (the)0 1531 y(elemen)m(ts)28 b Fs(b)412 1545 y Fp(j)t(i)502 1531 y FA(=)e Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(j)5 b FA(\))p Fv(i)28 b FA(whic)m(h)h(w)m(e)i (write)d(as)j Fs(b)1842 1545 y Fp(j)t(i)1931 1531 y FA(=)26 b Fs(q)2072 1498 y Fp(z)2107 1507 y Fk(ij)2170 1531 y FA(,)31 b Fs(z)2268 1545 y Fp(ij)2358 1531 y Fv(2)25 b Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\),)26 b(satisfy)86 1662 y(\(8.1\))54 b Fs(z)371 1676 y Fp(ij)456 1662 y FA(+)20 b Fs(z)589 1676 y Fp(j)t(i)679 1662 y FA(=)26 b Fs(z)818 1676 y Fp(ii)874 1662 y Fs(a)922 1676 y Fp(ij)987 1662 y FA(,)31 b(for)f(all)e Fs(i)p FA(,)k Fs(j)5 b FA(;)86 1771 y(\(8.2\))54 b Fs(z)371 1785 y Fp(ii)453 1771 y Fv(6)p FA(=)25 b(0,)31 b(for)e(all)g Fs(i)p FA(.)0 1902 y(Clearly)-8 b(,)29 b(\(8.1\))g(is)h(equiv)-5 b(alen)m(t)28 b(to)i Fs(z)1233 1916 y Fp(ij)1318 1902 y FA(+)20 b Fs(z)1451 1916 y Fp(j)t(i)1541 1902 y FA(=)25 b Fs(z)1679 1916 y Fp(ii)1736 1902 y Fs(a)1784 1916 y Fp(ij)1879 1902 y FA(and)30 b Fs(z)2099 1916 y Fp(ij)2189 1902 y FA(=)25 b Fs(z)2327 1916 y Fp(j)t(i)2392 1902 y FA(,)31 b(for)f(all)f Fs(i)c(<)h(j)5 b FA(.)110 2012 y(Supp)s(ose)36 b(w)m(e)j(ha)m(v)m(e)h (solv)m(ed)e(the)g(system)g(of)h(linear)d(equations)h(\(8.1\))h(o)m(v)m (er)h Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))34 b(suc)m(h)0 2122 y(that)28 b(\(8.2\))g(holds.)39 b(T)-8 b(o)30 b(realize)c(\()p Fs(b)1180 2136 y Fp(ij)1244 2122 y FA(\))j(o)m(v)m(er)h(\000)f(w)m(e)h (ha)m(v)m(e)f(to)g(\014nd)f(elemen)m(ts)g Fs(x)2650 2136 y Fp(i`)2711 2122 y FA(,)i Fs(w)2831 2136 y Fp(`j)2926 2122 y Fv(2)25 b Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\),)0 2231 y(1)33 b Fv(\024)f Fs(i;)15 b(j)39 b Fv(\024)33 b Fs(\022)s FA(,)j(1)d Fv(\024)g Fs(`)g Fv(\024)f Fs(M)10 b FA(,)36 b(with)e Fs(E)1338 2245 y Fp(`)1374 2231 y Fs(x)1426 2245 y Fp(i`)1520 2231 y FA(=)f(0,)j(suc)m(h)e(that)g Fs(b)2183 2245 y Fp(j)t(i)2280 2231 y FA(=)f Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(j)5 b FA(\))p Fv(i)33 b FA(for)h(all)g Fs(i)p FA(,)i Fs(j)5 b FA(,)0 2341 y(where)1074 2507 y Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(y)s FA(\()p Fs(`)p FA(\))p Fv(i)24 b FA(=)h Fs(q)1668 2469 y Fp(x)1709 2479 y Fk(i`)1342 2676 y Fs(g)s FA(\()p Fs(j)5 b FA(\))24 b(=)1687 2589 y Fl(X)1624 2785 y Fr(1)p Fo(\024)p Fp(`)p Fo(\024)p Fp(M)1897 2676 y Fs(w)1962 2690 y Fp(`j)2031 2676 y Fs(y)s FA(\()p Fs(`)p FA(\))p Fs(;)0 2935 y FA(1)k Fv(\024)h Fs(i;)15 b(j)34 b Fv(\024)29 b Fs(\022)s FA(,)k(1)28 b Fv(\024)g Fs(`)h Fv(\024)g Fs(M)10 b FA(.)46 b(W)-8 b(e)32 b(ha)m(v)m(e)h(then)e(to)g(solv)m(e)h(the)f(follo)m(wing)f (system)h(of)h(quadratic)0 3045 y(equations)c(o)m(v)m(er)j Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))25 b(in)30 b(the)f(2)p Fs(\022)s(M)39 b FA(unkno)m(wns)30 b Fs(x)1848 3059 y Fp(i`)1909 3045 y FA(,)h Fs(w)2030 3059 y Fp(`j)2099 3045 y FA(:)0 3320 y(\(8.3\))1020 3234 y Fl(X)957 3430 y Fr(1)p Fo(\024)p Fp(`)p Fo(\024)p Fp(M)1230 3320 y Fs(x)1282 3334 y Fp(i`)1343 3320 y Fs(w)1408 3334 y Fp(`j)1503 3320 y FA(=)25 b Fs(z)1641 3334 y Fp(ij)1706 3320 y Fs(;)198 b FA(for)30 b(all)f Fs(i;)15 b(j:)0 3580 y FA(The)35 b(analysis)g(of)g(this)g(system,)i(and)f(in)f(particular)d(the)j (existence)e(of)j(solutions,)f(in)m(v)m(olv)m(es)0 3689 y(arithmetic)27 b(to)s(ols,)h(namely)h(the)h(quadratic)d(recipro)s(cit) m(y)g(la)m(w.)110 3799 y(Let)33 b Fs(F)47 b FA(b)s(e)33 b(the)g(free)g Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\)-mo)s(dule)28 b(with)33 b(basis)g Fs(e)p FA(\(1\))p Fs(;)15 b(:)g(:)g(:)h(;)f(e)p FA(\()p Fs(\022)s FA(\).)51 b(De\014ne)32 b Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\)-linear)0 3909 y(maps)44 b Fs(x)49 b FA(:)h(\000)f Fv(!)g Fs(F)13 b FA(,)48 b Fs(w)k FA(:)e Fs(F)61 b Fv(!)50 b FA(\000)44 b(and)g Fs(z)54 b FA(:)49 b(\000)g Fv(!)h FA(\000)44 b(b)m(y)h Fs(x)p FA(\()p Fs(y)s FA(\()p Fs(`)p FA(\)\))j(=)2675 3840 y Fl(P)2771 3936 y Fr(1)p Fo(\024)p Fp(i)p Fo(\024)p Fp(\022)3004 3909 y Fs(x)3056 3923 y Fp(i`)3117 3909 y Fs(e)p FA(\()p Fs(i)p FA(\),)0 4026 y Fs(w)r FA(\()p Fs(e)p FA(\()p Fs(j)5 b FA(\)\))24 b(=)418 3957 y Fl(P)514 4053 y Fr(1)p Fo(\024)p Fp(`)p Fo(\024)p Fp(M)791 4026 y Fs(w)856 4040 y Fp(`j)926 4026 y Fs(y)s FA(\()p Fs(`)p FA(\))29 b(and)h Fs(z)t FA(\()p Fs(e)p FA(\()p Fs(j)5 b FA(\)\))24 b(=)1687 3957 y Fl(P)1784 4053 y Fr(1)p Fo(\024)p Fp(i)p Fo(\024)p Fp(\022)2017 4026 y Fs(z)2059 4040 y Fp(ij)2124 4026 y Fs(e)p FA(\()p Fs(i)p FA(\))29 b(for)h(all)f Fs(`)p FA(,)i Fs(j)5 b FA(.)41 b(Then)30 b(\(8.3\))f(is)0 4137 y(equiv)-5 b(alen)m(t)28 b(to)0 4303 y(\(8.4\))1308 b Fs(xy)29 b FA(=)c Fs(z)t(:)0 4480 y FA(Th)m(us)d(w)m(e)g(see)g(that)e (giving)h Fs(g)s FA(\(1\))p Fs(;)15 b(:)g(:)g(:)h(;)f(g)s FA(\()p Fs(\022)s FA(\))24 b Fv(2)h FA(\000,)f Fs(\037)p FA(\(1\))p Fs(;)15 b(:)g(:)g(:)i(;)e(\037)p FA(\()p Fs(\022)s FA(\))24 b Fv(2)2355 4457 y Fl(b)2351 4480 y FA(\000)f(with)d Fs(b)2669 4494 y Fp(j)t(i)2759 4480 y FA(=)26 b Fv(h)p Fs(\037)p FA(\()p Fs(i)p FA(\))p Fs(;)15 b(g)s FA(\()p Fs(j)5 b FA(\))p Fv(i)0 4590 y FA(for)22 b(all)e Fs(i)p FA(,)25 b Fs(j)i FA(is)22 b(equiv)-5 b(alen)m(t)19 b(to)j(factorizing)c Fs(z)26 b FA(o)m(v)m(er)d(\000.)38 b(Suc)m(h)22 b(a)g(factorization)17 b(exists)k(if)g(and)h(only)0 4700 y(if)g(the)g(ro)m(w)h(space)e(of)i (\()p Fs(z)805 4714 y Fp(ij)869 4700 y FA(\))g(is)f(an)g(epimorphic)e (image)h(of)i(a)f(subgroup)f(of)i(\000,)i(or)d(equiv)-5 b(alen)m(tly)20 b(of)0 4809 y(\000)29 b(\(since)e(b)m(y)i(dualit)m(y)-8 b(,)28 b(an)m(y)h(subgroup)e(of)h(\000)h(is)g(an)f(epimorphic)e(image)h (of)h(\000\).)41 b(In)28 b(particular,)0 4932 y(if)23 b(\()p Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\)\))399 4886 y Fp(\022)464 4932 y FA(itself)22 b(is)i(an)f(an)h(epimorphic)c(image)j (of)h(\000,)h(then)e(for)h(an)m(y)g(solution)d(\()p Fs(z)2853 4946 y Fp(ij)2917 4932 y FA(\))j(of)g(\(8.1\),)0 5041 y(\(8.2\))34 b(there)g(is)h(a)g(Y)-8 b(etter-Drinfeld)32 b(mo)s(dule)h(o)m(v)m(er)j(\000)f(with)f(braiding)f Fs(b)2519 5055 y Fp(j)t(i)2617 5041 y FA(=)h Fs(q)2766 5008 y Fp(z)2801 5017 y Fk(ij)2899 5041 y FA(of)h(Cartan)0 5151 y(t)m(yp)s(e)29 b(\()p Fs(a)289 5165 y Fp(ij)354 5151 y FA(\))390 5165 y Fr(1)p Fo(\024)p Fp(i;j)t Fo(\024)p Fp(\022)662 5151 y FA(.)p eop %%Page: 32 32 32 31 bop 0 -128 a Fy(32)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y FC(Prop)s(osition)42 b(8.1.)52 b Fj(If)37 b FA(\000)f Fj(is)h(a)f(\014nite)f(ab)s(elian)e(group)i(of)i (o)s(dd)e(order)h(and)f Fs(V)58 b Fj(is)36 b(a)g(Y)-8 b(etter-)0 201 y(Drinfeld)28 b(mo)s(dule)g(o)m(v)m(er)i FA(\000)h Fj(of)f(\014nite)e(Cartan)h(t)m(yp)s(e,)h(then)0 412 y FA(\(8.5\))1080 b(dim)14 b Fs(V)46 b Fv(\024)25 b FA(2\(ord)14 b(\000\))1953 374 y Fr(2)1994 412 y Fs(:)0 676 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)30 b(describ)s(e)c Fs(V)50 b FA(as)30 b(ab)s(o)m(v)m(e.)40 b(By)29 b(\(8.4\),)g(the)f (image)g(of)h Fs(z)35 b FA(is)29 b(con)m(tained)d(in)j(the)f(image)0 786 y(of)38 b Fs(x)p FA(;)43 b(hence)37 b(ord)o(\(Im)15 b Fs(z)t FA(\))39 b Fv(\024)g FA(ord)15 b(\000.)65 b(W)-8 b(e)39 b(no)m(w)f(apply)f(the)h(dualit)m(y)e(functor)h Fs(M)48 b Fv(7!)40 b Fs(M)3111 753 y Fo(\003)3191 786 y FA(:=)0 895 y(Hom)190 912 y Fn(Z)-9 b Fp(=)p Fr(\()p Fp(E)s Fr(\))389 895 y FA(\()p Fs(M)5 b(;)15 b Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\)\))20 b(to)25 b(\(8.4\))f(and)h(conclude)e (that)h(also)g(ord\(Im)14 b Fs(z)2443 862 y Fo(\003)2486 895 y FA(\))25 b Fv(\024)g FA(ord)14 b(\000)26 b(since)e(\000)3149 862 y Fo(\003)3217 895 y Fv(')0 1009 y FA(\000.)45 b(Th)m(us)31 b(w)m(e)h(obtain)d(from)i(\(8.1\))g(that)f(the)g(ro)m(w)i(space)e(of)h (\()p Fs(z)2179 1023 y Fp(ij)2244 1009 y FA(\))20 b(+)2371 976 y Fp(t)2425 1009 y FA(\()p Fs(z)2503 1023 y Fp(ij)2567 1009 y FA(\))27 b(=)g(\()p Fs(z)2806 1023 y Fp(ii)2863 1009 y Fs(a)2911 1023 y Fp(ij)2975 1009 y FA(\))k(has)g(at)0 1119 y(most)f(ord)o(\(Im)15 b Fs(z)t FA(\))g(ord)o(\(Im)g Fs(z)951 1086 y Fo(\003)993 1119 y FA(\))25 b Fv(\024)h FA(\(ord)14 b(\000\))1427 1086 y Fr(2)1498 1119 y FA(elemen)m(ts.)110 1229 y(W)-8 b(e)30 b(compute)e(no)m(w)i(a)h(lo)m(w)m(er)e(b)s(ound)g (for)h(the)f(n)m(um)m(b)s(er)h(of)g(elemen)m(ts)e(in)h(the)h(ro)m(w)g (space)f(of)0 1338 y(\()p Fs(z)78 1352 y Fp(ii)134 1338 y Fs(a)182 1352 y Fp(ij)247 1338 y FA(\))c(o)m(v)m(er)h Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\).)34 b(W)-8 b(e)26 b(can)e(assume)h(that)f(\()p Fs(a)1716 1352 y Fp(ij)1780 1338 y FA(\))h(is)g(a)g(diagonal)e(of)i(blo)s(c)m(ks)f(corresp)s (onding)0 1448 y(to)g(matrices)e(of)j(connected)c(\014nite)i(Cartan)g (t)m(yp)s(e.)38 b(It)24 b(is)g(easy)g(to)g(c)m(hec)m(k)h(the)e(follo)m (wing)f(\(rather)0 1557 y(w)m(eak\))37 b(claim)f(ab)s(out)g(the)h(ro)m (w)g(space)g(of)g(a)h(connected)d(\014nite)g(Cartan)i(matrix)f Fs(C)45 b FA(mo)s(dulo)0 1667 y Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\),)19 b(not)j(of)f(t)m(yp)s(e)g Fs(A)814 1681 y Fr(2)856 1667 y FA(:)37 b(Let)21 b Fs(\013;)15 b(\014)28 b FA(b)s(e)21 b(non-zero)f(elemen)m(ts)h(in)g Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))17 b(and)22 b Fs(a)p FA(,)i Fs(b)e FA(t)m(w)m(o)h(di\013eren)m(t)0 1777 y(ro)m(ws)30 b(in)g Fs(C)7 b FA(.)41 b(Then)29 b Fs(\013a)d Fv(6)p FA(=)g(0,)k Fs(\014)5 b(b)25 b Fv(6)p FA(=)h(0)k(and)f Fs(\013a)e Fv(6)p FA(=)e Fs(\014)5 b(b)p FA(.)110 1886 y(\(If)32 b Fs(C)40 b FA(is)32 b(of)h(t)m(yp)s(e)f Fs(A)821 1900 y Fr(2)862 1886 y FA(,)i Fs(a)c FA(=)f(\(2)p Fs(;)15 b Fv(\000)p FA(1\))32 b(and)g Fs(b)e FA(=)g(\()p Fv(\000)p FA(1)p Fs(;)15 b FA(2\))31 b(then)h Fs(\013a)f FA(=)e Fs(\014)5 b(b)33 b FA(whenev)m(er)e Fs(\013)g FA(=)e Fs(\014)0 1996 y FA(and)h(3)p Fs(\013)25 b FA(=)h(0\).)110 2105 y(Let)21 b Fs(s)g FA(b)s(e)g(the)g(n)m(um)m(b)s(er)g(of)h (connected)c(comp)s(onen)m(ts)h(of)j(\()p Fs(a)2105 2119 y Fp(ij)2169 2105 y FA(\))g(of)f(t)m(yp)s(e)g Fs(A)2587 2119 y Fr(2)2628 2105 y FA(.)39 b(Then)20 b Fs(\022)28 b FA(=)e(2)p Fs(s)s FA(+)s Fs(t)0 2215 y FA(for)k(some)g Fs(t)25 b Fv(\025)g FA(0)30 b(and)g(the)f(ro)m(w)i(space)e(of)h(\()p Fs(z)1536 2229 y Fp(ii)1592 2215 y Fs(a)1640 2229 y Fp(ij)1704 2215 y FA(\))g(has)g(at)g(least)e Fs(s)21 b FA(+)f Fs(t)30 b FA(elemen)m(ts.)39 b(Therefore)849 2431 y Fs(\022)28 b FA(=)d(2)p Fs(s)20 b FA(+)h Fs(t)k Fv(\024)g FA(2\()p Fs(s)20 b FA(+)g Fs(t)p FA(\))25 b Fv(\024)g FA(2\(ord)14 b(\000\))2119 2393 y Fr(2)2159 2431 y Fs(:)183 b Fi(\003)0 2695 y FC(Corollary)33 b(8.2.)52 b Fj(Let)27 b FA(\000)h Fj(b)s(e)f(a)h(\014xed)g(\014nite)e(ab)s(elian)f(group)i(of)g(o)s(dd)g (order.)39 b(Then)27 b(there)g(are)0 2804 y(\(up)35 b(to)g (isomorphisms\))f(only)h(\014nitely)f(man)m(y)i(Y)-8 b(etter-Drinfeld)33 b(mo)s(dules)g(o)m(v)m(er)k FA(\000)f Fj(of)g(\014nite)0 2914 y(Cartan)29 b(t)m(yp)s(e.)0 3077 y Ft(Pr)-5 b(o)g(of.)46 b FA(This)29 b(follo)m(ws)f(from)h(Prop)s (osition)d(8.1)k(since)e(there)g(are)g(only)h(\014nitely)e(man)m(y)j(Y) -8 b(etter-)0 3186 y(Drinfeld)28 b(mo)s(dules)g(o)m(v)m(er)i(\000)h(of) f(a)g(giv)m(en)g(dimension)d(\(up)j(to)f(isomorphisms\).)89 b Fi(\003)110 3349 y FA(It)36 b(is)h(easy)f(to)h(impro)m(v)m(e)f(the)g (b)s(ound)f(in)i(\(8.5\))f(under)f(additional)e(assumptions.)59 b(If)37 b(the)0 3458 y(Cartan)29 b(matrix)g(of)h Fs(V)51 b FA(is)30 b(in)m(v)m(ertible)d(o)m(v)m(er)k Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\),)26 b(then)1258 3671 y Fs(f)1303 3685 y Fr(1)1359 3671 y Fs(:)15 b(:)g(:)h(f)1525 3685 y Fp(\022)1592 3671 y Fv(\024)25 b FA(\(ord)14 b(\000\))1964 3633 y Fr(2)2005 3671 y Fs(;)0 3882 y FA(where)36 b(1)h Fv(6)p FA(=)h Fs(f)508 3896 y Fp(i)575 3882 y FA(:=)g(ord)o(\()p Fu(Z)p Fs(=)p FA(\()p Fs(E)5 b FA(\))p Fs(z)1174 3896 y Fp(ii)1226 3882 y FA(\))37 b(is)g(a)g(divisor)f(of)h Fs(E)5 b FA(.)64 b(This)36 b(follo)m(ws)h(from)g(the)f(pro)s(of)g(of)0 3992 y(Prop)s(osition)27 b(8.1.)42 b(Let)29 b Fs(f)41 b FA(the)29 b(least)g(divisor)g(of)h Fs(E)37 b FA(greater)28 b(than)i(one;)g(then)f Fs(f)2782 3958 y Fp(\022)2849 3992 y Fv(\024)d FA(\(ord)14 b(\000\))3222 3958 y Fr(2)3262 3992 y FA(,)0 4101 y(so)1297 4211 y Fs(\022)28 b Fv(\024)d FA(2)15 b(log)1643 4233 y Fp(f)1690 4211 y FA(\(ord)f(\000\))p Fs(:)0 4384 y FA(In)30 b(particular)c(w)m(e)k(get)f(a)h(b)s(etter)d(b)s (ound)h(for)i(quan)m(tum)f(linear)e(spaces)i(\(all)f(connected)f(com-)0 4493 y(p)s(onen)m(ts)33 b(are)i(of)g(t)m(yp)s(e)f Fs(A)890 4507 y Fr(1)932 4493 y FA(\).)55 b(On)36 b(the)e(other)g(hand,)i(if)e (\000)i(is)f(a)g(pro)s(duct)e(of)i Fs(r)j FA(cyclic)33 b(groups)0 4603 y(\(not)e(in)g(the)g(canonical)e(form)j(ab)s(o)m(v)m (e,)g(so)g Fs(r)j FA(could)c(b)s(e)g(m)m(uc)m(h)h(greater)e(than)h Fs(M)10 b FA(\))32 b(then)e(there)0 4712 y(exists)36 b(at)g(least)g(a)g(quan)m(tum)h(linear)d(space)i(o)m(v)m(er)h(\000)h (of)e(dimension)f(2)p Fs(r)s FA(,)j(see)f([AS2,)i(Lemma)0 4822 y(4.1].)110 4932 y(Finally)-8 b(,)35 b(w)m(e)h(consider)d(the)i (sp)s(ecial)e(case)i(when)g(\000)g Fv(')f FA(\()p Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\)\))2347 4886 y Fp(r)2387 4932 y FA(,)k Fs(p)e FA(an)f(o)s(dd)g(prime.)56 b(W)-8 b(e)0 5041 y(ha)m(v)m(e)27 b(sho)m(wn)f(that)g(there)f(is)h(alw)m(a)m(ys)g(a) h(Y)-8 b(etter-Drinfeld)23 b(mo)s(dule)i(o)m(v)m(er)i(\000)g(of)f(an)m (y)h(giv)m(en)f(\014nite)0 5151 y(Cartan)j(t)m(yp)s(e)g(of)h(rank)g Fs(\022)j FA(if)d Fs(\022)d Fv(\024)f Fs(r)s FA(.)40 b(In)31 b(general)c(w)m(e)k(ha)m(v)m(e)f(the)g(follo)m(wing)d(b)s (ound.)p eop %%Page: 33 33 33 32 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(33)0 91 y FC(Prop)s(osition)41 b(8.3.)51 b Fj(Let)35 b Fs(p)f(>)f FA(3)i Fj(b)s(e)g(a)g(prime,)g Fs(r)i Fv(\025)c FA(1)i Fj(a)g(natural)e(n)m(um)m(b)s(er,)j FA(\000)e Fv(')g FA(\()o Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\)\))3247 45 y Fp(r)0 201 y Fj(and)c Fs(V)51 b Fj(a)30 b(Y)-8 b(etter-Drinfeld)27 b(mo)s(dule)h(o)m(v)m(er)i FA(\000)h Fj(of)f(\014nite)e(Cartan)h(t)m(yp)s(e.)40 b(Then)0 448 y FA(\(8.6\))1103 b(dim)15 b Fs(V)46 b Fv(\024)25 b FA(2)p Fs(r)1755 386 y(p)c Fv(\000)f FA(1)p 1755 427 204 4 v 1755 510 a Fs(p)h Fv(\000)f FA(2)1970 448 y Fs(:)0 752 y Ft(Pr)-5 b(o)g(of.)46 b FA(W)-8 b(e)36 b(w)m(ork)f(as)h(in)e(the) h(pro)s(of)f(of)h(Prop)s(osition)c(9.1,)37 b(but)e(with)f Fs(E)40 b FA(=)34 b Fs(p)h FA(and)g(o)m(v)m(er)h(the)0 862 y(\014eld)22 b Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\).)34 b(F)-8 b(rom)23 b(\(8.1\))g(and)g(\(8.4\))f(w)m(e)i(obtain)d(an)j(upp)s (er)e(b)s(ound)g(for)h(the)g(rank)g(o)m(v)m(er)h Fu(Z)p Fs(=)p FA(\()p Fs(p)p FA(\):)1123 1071 y(rk\()p Fs(z)1285 1085 y Fp(ii)1341 1071 y Fs(a)1389 1085 y Fp(ij)1454 1071 y FA(\))g Fv(\024)i FA(2)15 b(rk\()p Fs(z)1834 1085 y Fp(ij)1898 1071 y FA(\))25 b Fv(\024)g FA(2)p Fs(r)m(:)0 1280 y FA(If)30 b(\()p Fs(a)175 1294 y Fp(ij)239 1280 y FA(\))g(is)g(in)m(v)m(ertible)e(mo)s(d)h Fs(p)i FA(w)m(e)f(conclude)e (that)1505 1489 y Fs(\022)g Fv(\024)e FA(2)p Fs(r)m(:)0 1697 y FA(The)31 b(determinan)m(ts)e(of)i(the)g(connected)e(Cartan)h (matrices)f(of)j(\014nite)e(t)m(yp)s(e)h(are)g Fs(`)21 b FA(+)g(1)32 b(for)f Fs(A)3252 1711 y Fp(`)0 1807 y FA(and)25 b(1)p Fs(;)15 b FA(2)p Fs(;)g FA(3)27 b(or)f(4)g(for)f(the)g (other)g(t)m(yp)s(es.)38 b(Let)26 b Fs(C)1621 1821 y Fp(i)1652 1807 y FA(,)h(1)e Fv(\024)h Fs(i)f Fv(\024)h Fs(s)p FA(,)h(b)s(e)e(the)g(connected)e(comp)s(onen)m(ts)0 1917 y(of)29 b(\()p Fs(a)187 1931 y Fp(ij)251 1917 y FA(\))h(of)f(t)m(yp)s(e)g Fs(A)693 1931 y Fp(`)724 1940 y Fk(i)787 1917 y FA(suc)m(h)g(that)f Fs(p)p Fv(j)p FA(\()p Fs(`)1334 1931 y Fp(i)1384 1917 y FA(+)18 b(1\);)30 b(hence)e Fs(`)1901 1931 y Fp(i)1957 1917 y Fv(\025)d Fs(p)19 b Fv(\000)g FA(1.)40 b(W)-8 b(rite)28 b Fs(\022)g FA(=)2740 1848 y Fl(P)2836 1944 y Fr(1)p Fo(\024)p Fp(i)p Fo(\024)p Fp(s)3067 1917 y Fs(`)3105 1931 y Fp(i)3154 1917 y FA(+)19 b Fs(r)0 2034 y FA(for)30 b(some)g Fs(r)e Fv(\024)d FA(0.)41 b(Then)29 b Fs(\022)f Fv(\025)d Fs(s)p FA(\()p Fs(p)20 b Fv(\000)g FA(1\))30 b(and)0 2290 y(\(8.7\))894 b Fs(\022)23 b Fv(\000)d Fs(s)25 b Fv(\025)h Fs(\022)c Fv(\000)1652 2228 y Fs(\022)p 1573 2269 V 1573 2353 a(p)e Fv(\000)g FA(1)1812 2290 y(=)1920 2228 y Fs(p)g Fv(\000)g FA(2)p 1920 2269 V 1920 2353 a Fs(p)g Fv(\000)g FA(1)2134 2290 y Fs(\022)s(:)0 2547 y FA(W)-8 b(e)28 b(no)m(w)h(replace)c(in)j(\()p Fs(a)840 2561 y Fp(ij)904 2547 y FA(\))g(eac)m(h)g(diagonal)d(blo)s(c)m (k)i Fs(C)1839 2561 y Fp(i)1898 2547 y FA(b)m(y)i(the)e(smaller)g(blo)s (c)m(k)g(of)h(t)m(yp)s(e)f Fs(A)3104 2561 y Fp(`)3135 2570 y Fk(i)3164 2561 y Fo(\000)p Fr(1)3262 2547 y FA(.)0 2657 y(The)34 b(new)g(Cartan)g(matrix)f(is)i(in)m(v)m(ertible)d(mo)s(d) i Fs(p)h FA(of)f(rank)g Fs(\022)26 b Fv(\000)d Fs(s)35 b FA(since)e Fs(p)i FA(do)s(es)f(not)g(divide)0 2766 y Fs(`)38 2780 y Fp(i)89 2766 y FA(+)20 b(1.)41 b(Hence)29 b(w)m(e)i(kno)m(w)0 2975 y(\(8.8\))1239 b Fs(\022)23 b Fv(\000)d Fs(s)25 b Fv(\024)h FA(2)p Fs(r)m(:)0 3184 y FA(No)m(w)31 b(\(8.6\))e(follo)m(ws)g(from)h(\(8.7\))f(and)h (\(8.8\).)90 b Fi(\003)110 3346 y FA(Note)25 b(that)g(Prop)s(osition)e (8.3)j(giv)m(es)g(an)g(alternativ)m(e)d(pro)s(of)i(of)h(Prop)s(osition) d(5.1)j(for)g Fs(p)f(>)h FA(3.)0 3561 y FC(App)s(endix.)49 b(Primitiv)m(e)36 b(elemen)m(ts)g(from)e(quan)m(tum)i(Serre)g (relations.)110 3723 y Ft(Gaussian)c(binomial)h(c)-5 b(o)g(e\016cients.)110 3833 y FA(In)29 b(the)g(p)s(olynomial)d(algebra) h Fu(Z)p FA([)p FC(q)p FA(],)g(w)m(e)j(consider)d(the)h(Gaussian,)h(or) g FC(q)p FA(-binomial,)f(co)s(e\016-)0 3942 y(cien)m(ts)1181 3966 y Fl(\022)1249 4033 y Fs(n)1260 4158 y(i)1303 3966 y Fl(\023)1371 4187 y Fe(q)1444 4095 y FA(=)1711 4020 y(\()p Fs(n)p FA(\)!)1862 4047 y Fe(q)p 1551 4074 519 4 v 1551 4158 a FA(\()p Fs(n)20 b Fv(\000)g Fs(i)p FA(\)!)1846 4185 y Fe(q)1894 4158 y FA(\()p Fs(i)p FA(\)!)2022 4185 y Fe(q)2081 4095 y Fs(;)0 4342 y FA(where)k(\()p Fs(n)p FA(\)!)410 4369 y Fe(q)484 4342 y FA(=)h(\()p Fs(n)p FA(\))706 4369 y Fe(q)769 4342 y Fs(:)15 b(:)g(:)i FA(\(2\))1008 4369 y Fe(q)1056 4342 y FA(\(1\))1173 4369 y Fe(q)1221 4342 y FA(,)26 b(and)f(\()p Fs(n)p FA(\))1570 4369 y Fe(q)1644 4342 y FA(=)g(1)10 b(+)g FC(q)g FA(+)g Fv(\001)15 b(\001)g(\001)c FA(+)f FC(q)2275 4309 y Fp(n)p Fo(\000)p Fr(1)2418 4342 y Fs(;)25 b FA(for)g Fs(n)g Fv(2)g Fu(N)7 b FA(,)33 b(0)25 b Fv(\024)g Fs(i)h Fv(\024)f Fs(n)p FA(.)0 4454 y(W)-8 b(e)30 b(ha)m(v)m(e)h(the)e(iden)m(tit)m(y)0 4713 y(\(A.1\))46 b FC(q)312 4675 y Fp(h)360 4584 y Fl(\022)427 4651 y Fs(n)428 4775 y(h)482 4584 y Fl(\023)549 4804 y Fe(q)617 4713 y FA(+)709 4584 y Fl(\022)854 4651 y Fs(n)776 4775 y(h)20 b Fv(\000)g FA(1)986 4584 y Fl(\023)1053 4804 y Fe(q)1127 4713 y FA(=)1223 4584 y Fl(\022)1290 4651 y Fs(n)1291 4775 y(h)1345 4584 y Fl(\023)1412 4804 y Fe(q)1481 4713 y FA(+)g FC(q)1627 4675 y Fp(n)p Fr(+1)p Fo(\000)p Fp(h)1870 4584 y Fl(\022)2014 4651 y Fs(n)1937 4775 y(h)g Fv(\000)g FA(1)2147 4584 y Fl(\023)2214 4804 y Fe(q)2288 4713 y FA(=)2384 4584 y Fl(\022)2451 4651 y Fs(n)g FA(+)h(1)2531 4775 y Fs(h)2663 4584 y Fl(\023)2731 4804 y Fe(q)2779 4713 y Fs(;)15 b FA(1)25 b Fv(\024)h Fs(h)f Fv(\024)g Fs(n)p FA(;)0 5003 y(This)38 b(implies)e(that)741 4929 y Fl(\000)783 4962 y Fp(n)793 5034 y(i)828 4929 y Fl(\001)870 5039 y Fe(q)958 5003 y Fv(2)j Fu(Z)p FA([)p FC(q)p FA(].)63 b(If)38 b Fs(A)i FA(is)e(an)g(asso)s(ciativ)m(e)d (algebra)i(o)m(v)m(er)h Fs(k)k FA(and)37 b Fs(q)43 b Fv(2)c Fs(k)s FA(,)0 5138 y(then)207 5064 y Fl(\000)249 5098 y Fp(n)259 5170 y(i)294 5064 y Fl(\001)336 5175 y Fp(q)406 5138 y FA(denotes)28 b(the)g(sp)s(ecialization)c(of)1561 5064 y Fl(\000)1603 5098 y Fp(n)1612 5170 y(i)1648 5064 y Fl(\001)1690 5175 y Fe(q)1768 5138 y FA(in)k Fs(q)s FA(.)41 b(If)30 b Fs(x;)15 b(y)29 b Fv(2)c Fs(A)30 b FA(are)f(t)m(w)m(o)h(elemen)m(ts)d(that)p eop %%Page: 34 34 34 33 bop 0 -128 a Fy(34)657 b(N.)30 b(ANDR)n(USKIEWITSCH)i(AND)g (H.{J.)e(SCHNEIDER)0 91 y Fs(q)s FA(-comm)m(ute,)43 b(i.e.)71 b Fs(xy)46 b FA(=)d Fs(q)s(y)s(x)p FA(,)h(then)39 b(the)h(quan)m(tum)g (binomial)e(form)m(ula)h(holds)h(for)g(ev)m(ery)0 201 y Fs(n)25 b Fv(2)g Fu(N)7 b FA(:)0 478 y(\(A.2\))873 b(\()p Fs(x)20 b FA(+)g Fs(y)s FA(\))1367 440 y Fp(n)1442 478 y FA(=)1582 364 y Fp(n)1538 391 y Fl(X)1545 586 y Fp(i)p Fr(=0)1685 349 y Fl(\022)1753 416 y Fs(n)1764 541 y(i)1808 349 y Fl(\023)1875 569 y Fp(q)1915 478 y Fs(y)1963 440 y Fp(i)1994 478 y Fs(x)2046 440 y Fp(n)p Fo(\000)p Fp(i)2178 478 y Fs(:)0 781 y FA(Let)30 b(us)g(also)f(record)0 1004 y(\(A.3\))934 b(\()p Fs(r)s FA(\))1260 1032 y Fe(q)1329 1004 y FA(+)20 b FC(q)1475 967 y Fp(r)1516 1004 y FA(\()p Fs(s)p FA(\))1630 1032 y Fe(q)1703 1004 y FA(=)26 b(\()p Fs(r)c FA(+)e Fs(s)p FA(\))2069 1032 y Fe(q)2117 1004 y Fs(:)110 1232 y FA(There)26 b(is)h(another)e(v)m(ersion)i(of)g(")p FC(q)p FA(-binomial)e(co)s(e\016cien)m(ts":)36 b(in)27 b(the)f(Lauren)m(t)g(p)s(olynomial)0 1341 y(algebra)i Fu(Z)p FA([)p FC(q)p Fs(;)15 b FC(q)560 1308 y Fo(\000)p Fr(1)656 1341 y FA(],)31 b(w)m(e)f(consider)74 1502 y Fl(\024)137 1576 y Fs(n)149 1686 y(i)207 1502 y Fl(\025)256 1722 y Fe(q)329 1631 y FA(=)586 1556 y([)p Fs(n)p FA(]!)717 1583 y Fe(q)p 437 1610 478 4 v 437 1694 a FA([)p Fs(n)20 b Fv(\000)g Fs(i)p FA(]!)711 1721 y Fe(q)759 1694 y FA([)p Fs(i)p FA(]!)866 1721 y Fe(q)926 1631 y Fs(;)106 b FA(where)30 b([)p Fs(n)p FA(]!)1452 1658 y Fe(q)1526 1631 y FA(=)25 b([)p Fs(n)p FA(])1728 1658 y Fe(q)1791 1631 y Fs(:)15 b(:)g(:)i FA([2])2010 1658 y Fe(q)2058 1631 y FA([1])2154 1658 y Fe(q)2202 1631 y Fs(;)107 b FA(and)30 b([)p Fs(n)p FA(])2617 1658 y Fe(q)2691 1631 y FA(=)2798 1569 y FC(q)2853 1536 y Fp(n)2924 1569 y Fv(\000)20 b FC(q)3070 1536 y Fo(\000)p Fp(n)p 2798 1610 379 4 v 2828 1694 a FC(q)g Fv(\000)h FC(q)3050 1667 y Fo(\000)p Fr(1)3188 1631 y Fs(:)0 1921 y FA(Clearly)-8 b(,)1289 2035 y(\()p Fs(n)p FA(\))1415 2062 y Fe(q)1459 2044 y Ff(2)1524 2035 y FA(=)26 b FC(q)1676 1997 y Fp(n)p Fo(\000)p Fr(1)1819 2035 y FA([)p Fs(n)p FA(])1925 2062 y Fe(q)1973 2035 y Fs(;)0 2215 y FA(and)k(hence)1183 2232 y Fl(\022)1251 2299 y Fs(n)1262 2423 y(i)1305 2232 y Fl(\023)1373 2452 y Fe(q)1417 2434 y Ff(2)1482 2361 y FA(=)25 b FC(q)1633 2323 y Fp(i)p Fr(\()p Fp(n)p Fo(\000)p Fp(i)p Fr(\))1849 2232 y Fl(\024)1913 2306 y Fs(n)1924 2415 y(i)1983 2232 y Fl(\025)2031 2452 y Fe(q)2079 2361 y Fs(:)0 2610 y FA(In)30 b(particular,)d(for)j Fs(n)25 b Fv(\025)g FA(1,)31 b(the)e(follo)m(wing)f(equalit)m(y-)h(see) g([L3,)i(1.3.4)g(\(a\)]-)0 2995 y(\(A.4\))1173 2880 y Fp(n)1129 2908 y Fl(X)1136 3102 y Fp(i)p Fr(=0)1261 2995 y FA(\()p Fv(\000)p FA(1\))1450 2957 y Fp(i)1479 2995 y FC(q)1534 2957 y Fp(i)p Fr(\(1)p Fo(\000)p Fp(n)p Fr(\))1761 2866 y Fl(\024)1824 2940 y Fs(n)1836 3049 y(i)1894 2866 y Fl(\025)1943 3086 y Fe(q)2016 2995 y FA(=)26 b(0)0 3298 y(translates)h(in)m(to)0 3571 y(\(A.5\))1066 3457 y Fp(n)1023 3484 y Fl(X)1029 3679 y Fp(i)p Fr(=0)1154 3571 y FA(\()p Fv(\000)p FA(1\))1343 3534 y Fp(i)1373 3571 y FC(q)1428 3534 y Fr(\()p Fp(i)1483 3506 y Ff(2)1518 3534 y Fr(+)p Fp(i)p Fr(\))p Fp(=)p Fr(2)p Fo(\000)p Fp(ni)1835 3442 y Fl(\022)1902 3509 y Fs(n)1914 3634 y(i)1957 3442 y Fl(\023)2024 3663 y Fe(q)2098 3571 y FA(=)e(0)p Fs(:)0 3874 y FA(This)30 b(last)f(equalit)m(y)f(is)i(also)f (equiv)-5 b(alen)m(t,)28 b(b)m(y)j(a)f(c)m(hange)f(of)h(the)f (summation)g(index,)g(to)0 4166 y(\(A.6\))1104 4052 y Fp(n)1061 4079 y Fl(X)1059 4276 y Fp(h)p Fr(=0)1194 4166 y FA(\()p Fv(\000)p FA(1\))1383 4128 y Fp(h)1429 4166 y FC(q)1484 4128 y Fr(\()p Fp(h)1556 4100 y Ff(2)1592 4128 y Fo(\000)p Fp(h)p Fr(\))p Fp(=)p Fr(2)1798 4037 y Fl(\022)1866 4104 y Fs(n)1867 4228 y(h)1920 4037 y Fl(\023)1988 4257 y Fe(q)2061 4166 y FA(=)d(0)p Fs(:)110 4472 y Ft(Primitive)32 b(elements.)110 4583 y FA(Let)g Fs(S)38 b FA(b)s(e)31 b(a)i(braided)d(Hopf)j(algebra)d(o)m(v)m(er)j (our)f(an)g(ab)s(elian)d(group)j(\000.)48 b(Let)32 b Fs(x;)15 b(y)33 b Fv(2)c Fs(P)13 b FA(\()p Fs(S)5 b FA(\))0 4693 y(suc)m(h)30 b(that)286 4916 y Fs(\016)s FA(\()p Fs(x)p FA(\))25 b(=)g Fs(g)f Fv(\012)c Fs(x;)107 b(h:x)26 b FA(=)f Fs(\037)p FA(\()p Fs(h)p FA(\))p Fs(x)90 b(;)15 b(\016)s FA(\()p Fs(y)s FA(\))25 b(=)h Fs(t)20 b Fv(\012)g Fs(y)s(;)107 b(h:y)29 b FA(=)c Fs(\021)s FA(\()p Fs(h)p FA(\))p Fs(y)94 b Fv(8)p Fs(h)25 b Fv(2)g FA(\000)p Fs(;)0 5151 y FA(for)30 b(some)g Fs(g)s(;)15 b(t)25 b Fv(2)h FA(\000,)31 b Fs(\037;)15 b(\021)29 b Fv(2)975 5128 y Fl(b)971 5151 y FA(\000)q(.)41 b(W)-8 b(e)30 b(assume)f(that)g(the)h (orders)e(of)j Fs(\037)p FA(\()p Fs(g)s FA(\),)e Fs(\021)s FA(\()p Fs(t)p FA(\))h(are)g(p)s(ositiv)m(e.)p eop %%Page: 35 35 35 34 bop 593 -128 a Fy(FINITE)31 b(QUANTUM)h(GR)n(OUPS)f(AND)h(CAR)-6 b(T)g(AN)31 b(MA)-6 b(TRICES)515 b(35)0 91 y FC(De\014nition.)53 b FA(The)27 b Ft(br)-5 b(aide)g(d)30 b(adjoint)g(r)-5 b(epr)g(esentation)35 b FA(is)27 b(the)g(linear)e(map)j(ad)2731 105 y Fp(c)2794 91 y FA(:)d Fs(S)31 b Fv(!)26 b FA(End)14 b Fs(S)0 201 y FA(giv)m(en)29 b(b)m(y)997 318 y(ad)1093 332 y Fp(c)1131 318 y FA(\()p Fs(u)p FA(\)\()p Fs(v)s FA(\))24 b(=)h Fs(\026)p FA(\()p Fs(u)20 b Fv(\012)h Fs(v)i Fv(\000)e Fs(c)p FA(\()p Fs(u)e Fv(\012)i Fs(v)s FA(\)\);)0 498 y(here)29 b Fs(\026)i FA(is)e(the)h(m)m(ultiplication)25 b(map)30 b(of)g Fs(S)35 b FA(and)30 b Fs(c)g FA(is)g(the)f(comm)m (utativit)m(y)f(constrain)m(t.)110 670 y(The)h(follo)m(wing)f(prop)s (ert)m(y)g(holds:)0 896 y(\(A.7\))430 b(ad)737 910 y Fp(c)775 896 y FA(\()p Fs(u)p FA(\)\()p Fs(v)s(w)r FA(\))25 b(=)g(ad)1302 910 y Fp(c)1340 896 y FA(\()p Fs(u)p FA(\)\()p Fs(v)s FA(\))p Fs(w)d FA(+)e(\()p Fs(u)1849 913 y Fr(\()p Fo(\000)p Fr(1\))2004 896 y Fs(:v)s FA(\))15 b(ad)2224 910 y Fp(c)2261 896 y FA(\()p Fs(u)2349 913 y Fr(\(0\))2447 896 y FA(\)\()p Fs(w)r FA(\))p Fs(:)110 1126 y FA(In)37 b(our)f(situation,)g(w)m(e)i(de\014ne)d(inductiv)m(ely)f Fs(z)1745 1140 y Fr(1)1824 1126 y FA(=)j(ad)2028 1140 y Fp(c)2065 1126 y FA(\()p Fs(x)p FA(\)\()p Fs(y)s FA(\))e(=)i Fs(xy)29 b Fv(\000)24 b Fs(\021)s FA(\()p Fs(g)s FA(\))p Fs(y)s(x)p FA(;)41 b Fs(z)3048 1140 y Fp(j)t Fr(+1)3217 1126 y FA(=)0 1235 y(ad)96 1249 y Fp(c)134 1235 y FA(\()p Fs(x)p FA(\)\()p Fs(z)336 1249 y Fp(j)373 1235 y FA(\).)g(F)-8 b(or)30 b(instance,)291 1466 y Fs(z)333 1480 y Fr(2)400 1466 y FA(=)25 b Fs(x)548 1428 y Fr(2)589 1466 y Fs(y)f Fv(\000)c Fs(\021)s FA(\()p Fs(g)s FA(\))15 b(\()o(1)20 b(+)h Fs(\037)p FA(\()p Fs(g)s FA(\)\))13 b Fs(xy)s(x)21 b FA(+)f Fs(\021)s FA(\()p Fs(g)s FA(\))1780 1428 y Fr(2)1821 1466 y Fs(\037)p FA(\()p Fs(g)s FA(\))p Fs(y)s(x)2097 1428 y Fr(2)2137 1466 y Fs(;)291 1618 y(z)333 1632 y Fr(3)400 1618 y FA(=)25 b Fs(x)548 1580 y Fr(3)589 1618 y Fs(y)f Fv(\000)c Fs(\021)s FA(\()p Fs(g)s FA(\)\(3\))1034 1635 y Fp(\037)p Fr(\()p Fp(g)r Fr(\))1178 1618 y Fs(x)1230 1580 y Fr(2)1271 1618 y Fs(y)s(x)g FA(+)h Fs(\021)s FA(\()p Fs(g)s FA(\))1650 1580 y Fr(2)1690 1618 y Fs(\037)p FA(\()p Fs(g)s FA(\)\(3\))1984 1635 y Fp(\037)p Fr(\()p Fp(g)r Fr(\))2127 1618 y Fs(xy)s(x)2279 1580 y Fr(2)2341 1618 y Fv(\000)f Fs(\021)s FA(\()p Fs(g)s FA(\))2599 1580 y Fr(3)2639 1618 y Fs(\037)p FA(\()p Fs(g)s FA(\))2815 1580 y Fr(3)2855 1618 y Fs(y)s(x)2955 1580 y Fr(3)0 1844 y FA(and)30 b(in)f(general)0 2155 y(\(A.8\))446 b Fs(z)699 2169 y Fp(N)794 2155 y FA(=)924 2041 y Fp(N)890 2068 y Fl(X)897 2263 y Fp(i)p Fr(=0)1022 2155 y FA(\()p Fv(\000)p FA(1\))1211 2117 y Fp(i)1240 2026 y Fl(\022)1307 2093 y Fs(N)1333 2218 y(i)1391 2026 y Fl(\023)1458 2246 y Fp(\037)p Fr(\()p Fp(g)r Fr(\))1603 2155 y Fs(\037)p FA(\()p Fs(g)s FA(\))1779 2117 y Fp(i)p Fr(\()p Fp(i)p Fo(\000)p Fr(1\))p Fp(=)p Fr(2)2059 2155 y Fs(\021)s FA(\()p Fs(g)s FA(\))2226 2117 y Fp(i)2271 2155 y Fs(x)2323 2117 y Fp(N)7 b Fo(\000)p Fp(i)2475 2155 y Fs(y)s(x)2575 2117 y Fp(i)2606 2155 y Fs(:)0 2466 y FA(The)30 b(pro)s(of)e(of)i (\(A.8\))g(is)g(b)m(y)h(induction)c(using)i(\(A.7\).)110 2578 y(The)22 b(follo)m(wing)f(Lemma)i(is)g(probably)e(w)m(ell-kno)m (wn.)37 b(F)-8 b(or)23 b(completeness,)f(w)m(e)h(giv)m(e)g(a)g(direct)0 2688 y(pro)s(of.)0 2860 y FC(Lemma)34 b(A.1.)53 b Fj(Let)30 b Fs(r)e Fv(\025)d FA(1)30 b Fj(b)s(e)g(a)g(natural)e(n)m(um)m(b)s(er.) 40 b(Assume)30 b(that)0 3086 y FA(\(A.9\))1031 b Fs(\021)s FA(\()p Fs(g)s FA(\))p Fs(\037)p FA(\()p Fs(t)p FA(\))p Fs(\037)p FA(\()p Fs(g)s FA(\))1747 3048 y Fp(r)r Fo(\000)p Fr(1)1904 3086 y FA(=)25 b(1)0 3312 y Fj(and)30 b Fs(r)23 b Fv(\000)d FA(1)25 b Fs(<)g FA(ord\()p Fs(\037)p FA(\()p Fs(g)s FA(\)\))p Fj(.)39 b(Then)29 b FA(\(ad)1315 3326 y Fp(c)1367 3312 y Fs(x)p FA(\))1455 3279 y Fp(r)1495 3312 y Fs(y)34 b Fj(is)c(primitiv)m(e.)0 3484 y Ft(Pr)-5 b(o)g(of.)46 b FA(Let)39 b Fs(z)44 b FA(=)657 3415 y Fl(P)754 3438 y Fp(r)754 3511 y(i)p Fr(=0)892 3484 y Fs(\013)950 3498 y Fp(i)981 3484 y Fs(x)1033 3451 y Fp(i)1063 3484 y Fs(y)s(x)1163 3451 y Fp(r)r Fo(\000)p Fp(i)1287 3484 y FA(,)e(where)c Fs(\013)1685 3498 y Fp(i)1755 3484 y Fv(2)i Fs(k)s FA(.)67 b(Then)38 b(b)m(y)i(the)e(quan)m(tum)h (binomial)0 3593 y(form)m(ula,)0 3903 y(\001\()p Fs(z)t FA(\))25 b(=)363 3788 y Fp(r)315 3816 y Fl(X)322 4010 y Fp(i)p Fr(=0)516 3788 y Fp(i)463 3816 y Fl(X)467 4012 y Fp(`)p Fr(=0)618 3788 y Fp(r)r Fo(\000)p Fp(i)611 3816 y Fl(X)610 4012 y Fp(h)p Fr(=0)760 3903 y Fs(\013)818 3917 y Fp(i)849 3774 y Fl(\022)919 3841 y Fs(i)916 3965 y(`)954 3774 y Fl(\023)1021 3994 y Fp(\037)p Fr(\()p Fp(g)r Fr(\))1167 3774 y Fl(\022)1234 3841 y Fs(r)e Fv(\000)d Fs(i)1301 3965 y(h)1421 3774 y Fl(\023)1488 3994 y Fp(\037)p Fr(\()p Fp(g)r Fr(\))1648 3801 y Fl(h)1691 3903 y Fs(\021)s FA(\()p Fs(g)s FA(\))1858 3865 y Fp(i)p Fo(\000)p Fp(`)1976 3903 y Fs(\037)p FA(\()p Fs(g)s FA(\))2152 3865 y Fp(h)p Fr(\()p Fp(i)p Fo(\000)p Fp(`)p Fr(\))2369 3903 y Fs(x)2421 3865 y Fp(`)2456 3903 y Fs(y)s(x)2556 3865 y Fp(h)2624 3903 y Fv(\012)g Fs(x)2767 3865 y Fp(r)r Fo(\000)p Fp(`)p Fo(\000)p Fp(h)214 4163 y FA(+)31 b Fs(\037)p FA(\()p Fs(t)p FA(\))478 4125 y Fp(h)525 4163 y Fs(\037)p FA(\()p Fs(g)s FA(\))701 4125 y Fp(h)p Fr(\()p Fp(i)p Fo(\000)p Fp(`)p Fr(\))918 4163 y Fs(x)970 4125 y Fp(`)p Fr(+)p Fp(h)1124 4163 y Fv(\012)20 b Fs(x)1267 4125 y Fp(i)p Fo(\000)p Fp(`)1386 4163 y Fs(y)s(x)1486 4125 y Fp(r)r Fo(\000)p Fp(i)p Fo(\000)p Fp(h)1709 4062 y Fl(i)219 4323 y FA(=)25 b Fs(z)g Fv(\012)20 b FA(1)g(+)h(1)f Fv(\012)g Fs(z)214 4583 y FA(+)380 4497 y Fl(X)305 4693 y Fp(`)p Fo(\025)p Fr(0)p Fp(;h)p Fo(\025)p Fr(0)335 4759 y Fp(`)p Fr(+)p Fp(h