%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: Integralewww.dvi %%CreationDate: Wed Sep 30 16:06:18 1998 %%Pages: 60 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips Integralewww.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.09.30:1605 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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 1000 300 300 (/tmp_mnt/home/math/sommerh/P/EigArb/Integrale/Integralewww.dvi) @start /Fa 1 86 df85 D E /Fb 3 51 df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c 3 116 df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d 2 53 df<3E00418080C0C0C000C000C001 8003000400084030407F80FF800A0D7E8C0E>50 D<0300070007000B00130023006300C3 00FFC00300030003001FC00A0D7E8C0E>52 D E /Fe 3 73 df<207040F8410882088208 841864303FE01F800C000C00180018000D0D7E8813>39 D<001800003800003800005800 009800008C00010C00020C00060C0007FE00080600100600300600F81F80110E7E8D16> 65 D<1FC3F80600C00600C00600C00C01800C01800FFF800C0180180300180300180300 180300300600FC1F80150E7E8D17>72 D E /Ff 2 64 df<8000F000FC00FF00FFE0FFFE FFFEFFE0FF00FC00F00080000F0C67852A>45 D63 D E /Fg 2 18 df3 D<04040000000408000000080800000018100000001020 0000006040000000FFFFFFFFF0FFFFFFFFF0604000000010200000001810000000080800 000004080000000404000000240E7D902A>17 D E /Fh 9 106 df20 DI<00001C00003C0000F80001E00003C0000780000F00000E00001E00003C 00003C00003C000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 0000780000780000780000780000F00000F00000F00001E00001E00003C0000380000700 000E00001C0000780000E00000E000007800001C00000E000007000003800003C00001E0 0001E00000F00000F00000F0000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 00007800007800007800007800007800007800003C00003C00003C00001E00000E00000F 000007800003C00001E00000F800003C00001C167C7B8121>40 D<0000FF8000000007FF F00000001FFFFC0000007F007F000000F8000F800001E00003C00003C00001E000078000 00F0000FC00001F8001FE00003FC001CF000079C003878000F0E00383C001E0E00701E00 3C0700700F00780700700780F007007003C1E00700E001E3C00380E000F7800380E0007F 000380E0003E000380E0003E000380E0007F000380E000F7800380E001E3C003807003C1 E00700700780F00700700F00780700701E003C0700383C001E0E003878000F0E001CF000 079C001FE00003FC000FC00001F80007800000F00003C00001E00001E00003C00000F800 0F8000007F007F0000001FFFFC00000007FFF000000000FF800000292A7E7F2E>78 D80 D88 DI104 DI E /Fi 4 69 df0 D<0C000C00EDC07F801E007F80EDC00C000C 000A097E890F>3 D<03F00C301030202060404000C000C000C000C000E000602070C03F 000C0E7E8D10>67 D<1FFC00230F004303808301C00300C00300C00200C0060080060180 0601000402000C0C000870001F8000120E7E8D16>I E /Fj 8 69 df0 D<040004000400C460E4E03F800E003F80E4E0C46004 00040004000B0D7E8D11>3 D<01F8000606000801001000802801402402404402204204 208108108090108060108060108090108108104204204402202402402801401000800801 0006060001F80014167E911A>10 D<0E001F00318060C060C0C060C060C060C06060C060 C031801F000E000B0E7E8D11>14 D<040E0E1C1C1C38383070706060C0C0070F7F8F0A> 48 D<03FC0FFC1C003000600060006000C000C000FFFCFFFCC000C00060006000600030 001C000FFC03FC0E147D9016>50 D<003F00FF030704060806180C3008300060006000E0 00E000E000E000F000F002780C7C103FE01F801014809312>67 D<03FFC00FFFF030E1FC 60C07CC0C01E81C01E01C00E01C00E01C00E01800C01800C03801C030018030010070020 0600400601800C0E000FF8001FC00017147F931A>I E /Fk 25 121 df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l 13 94 df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m 30 114 df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n 31 122 df<003FC00001F0300003C0380007C0 7C000F807C000F807C000F8038000F8000000F8000000F8000000F800000FFFFFC00FFFF FC000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C000F80 7C000F807C000F807C000F807C000F807C000F807C007FE1FF807FE1FF80191D809C1B> 12 D<78FCFCFCFC7806067D850D>46 D<03F8000F1E001C07003C07803803807803C078 03C07803C0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F8 03E0F803E07803C07803C03803803C07801C07000F1E0003F800131B7E9A18>48 D<00600001E0000FE000FFE000F3E00003E00003E00003E00003E00003E00003E00003E0 0003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0 0003E0007FFF807FFF80111B7D9A18>I<07F8001FFE00383F80780FC0FC07C0FC07E0FC 03E0FC03E07803E00007E00007C00007C0000F80001F00001E0000380000700000E00001 80600300600600600800E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131B7E9A18>I<03F800 1FFE003C1F003C0F807C07C07E07C07C07C03807C0000F80000F80001E00003C0003F800 001E00000F800007C00007C00007E03007E07807E0FC07E0FC07E0FC07C0780F80781F00 1FFE0007F800131B7E9A18>I<000180000380000780000F80001F80003F80006F8000CF 80008F80018F80030F80060F800C0F80180F80300F80600F80C00F80FFFFF8FFFFF8000F 80000F80000F80000F80000F80000F8001FFF801FFF8151B7F9A18>I<1801801FFF001F FE001FFC001FF8001FC00018000018000018000018000019F8001E0E00180F8010078000 07C00007E00007E00007E07807E0F807E0F807E0F807C0F007C0600F80381F001FFE0007 F000131B7E9A18>I<007E0003FF000781800F03C01E07C03C07C03C0380780000780000 F80000F8F800FB0E00FA0780FC0380FC03C0F803E0F803E0F803E0F803E07803E07803E0 7803C03C03C03C07801E0F0007FE0003F800131B7E9A18>I<6000007FFFE07FFFE07FFF C07FFF807FFF80E00300C00600C00C00C0180000300000300000600000E00000E00001E0 0001C00003C00003C00003C00003C00007C00007C00007C00007C00007C00007C0000380 00131C7D9B18>I<03F8000FFE001E0F803807803803C07803C07803C07E03C07F83807F C7003FFE001FFC000FFE0007FF801DFF80387FC0781FE0F007E0F003E0F001E0F001E0F0 01E07801C07803803E07801FFE0003F800131B7E9A18>I<03F8000FFE001E0F003C0780 7807807803C0F803C0F803C0F803E0F803E0F803E0F803E07807E03807E03C0BE00E1BE0 03E3E00003E00003C00003C03807C07C07807C0700780F00383C001FF8000FE000131B7E 9A18>I<001FE02000FFF8E003F80FE007C003E00F8001E01F0000E03E0000E03E000060 7E0000607C000060FC000000FC000000FC000000FC000000FC000000FC000000FC000000 FC0000007C0000607E0000603E0000603E0000C01F0000C00F80018007C0030003F80E00 00FFFC00001FE0001B1C7D9B22>67 DI76 D80 D<7FFFFFE07FFFFFE0781F81E070 1F80E0601F8060E01F8070C01F8030C01F8030C01F8030C01F8030001F8000001F800000 1F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800000 1F8000001F8000001F8000001F8000001F800007FFFE0007FFFE001C1C7E9B21>84 D<0FF8001C1E003E0F803E07803E07C01C07C00007C0007FC007E7C01F07C03C07C07C07 C0F807C0F807C0F807C0780BC03E13F80FE1F815127F9117>97 D<01FC000F07001C0380 3C01C07801C07801E0F801E0F801E0FFFFE0F80000F80000F800007800007C00603C0060 1E00C00F038001FC0013127F9116>101 D<007F0001E38003C7C00787C00F87C00F8380 0F80000F80000F80000F80000F8000FFF800FFF8000F80000F80000F80000F80000F8000 0F80000F80000F80000F80000F80000F80000F80000F80000F80007FF8007FF800121D80 9C0F>I104 D<1E003F003F003F003F001E00000000000000000000000000FF00FF001F001F001F001F 001F001F001F001F001F001F001F001F001F001F00FFE0FFE00B1E7F9D0E>I108 DII<01FC000F07801C01C03C01E07800F07800F0F800F8F800F8F800F8F800F8F8 00F8F800F87800F07800F03C01E01E03C00F078001FC0015127F9118>II114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007CC01CC01CE01CE018F830CF C00E127E9113>I<0300030003000300070007000F000F003FFCFFFC1F001F001F001F00 1F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F9913>I121 D E /Fo 69 122 df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p 85 128 df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q 38 123 df<3C007F00FF80FF80FFC0FFC0FFC07FC03EC000C000C0018001800180 0300030006000E001C00380030000A157BA913>39 D45 D<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE007FFFFE7FFFFE7FFFFE17277BA622>49 D<00FF800007FFF0000FFFFC001E 03FE003800FF807C003F80FE003FC0FF001FC0FF001FE0FF000FE0FF000FE07E000FE03C 001FE000001FE000001FC000001FC000003F8000003F0000007E000000FC000000F80000 01F0000003E00000078000000F0000001E0000003C00E0007000E000E000E001C001C003 8001C0060001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF801B 277DA622>I<007F800003FFF00007FFFC000F80FE001F007F003F807F003F803F803F80 3F803F803F801F803F801F003F8000007F0000007F0000007E000000FC000001F8000007 F00000FFC00000FFC0000001F80000007E0000003F0000003F8000001FC000001FC00000 1FE000001FE03C001FE07E001FE0FF001FE0FF001FE0FF001FC0FF003FC0FE003F807C00 7F003F00FE001FFFFC0007FFF00000FF80001B277DA622>I<00000E0000001E0000003E 0000007E000000FE000000FE000001FE000003FE0000077E00000E7E00000E7E00001C7E 0000387E0000707E0000E07E0000E07E0001C07E0003807E0007007E000E007E000E007E 001C007E0038007E0070007E00E0007E00FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE 000000FE000000FE000000FE000000FE000000FE000000FE00007FFFF8007FFFF8007FFF F81D277EA622>I<180003001F801F001FFFFE001FFFFC001FFFF8001FFFF0001FFFC000 1FFF00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C7FC000 1DFFF8001F80FC001E003F0008003F0000001F8000001FC000001FC000001FE000001FE0 18001FE07C001FE0FE001FE0FE001FE0FE001FE0FE001FC0FC001FC078003F8078003F80 3C007F001F01FE000FFFFC0003FFF00000FF80001B277DA622>I<0007F800003FFE0000 FFFF0001FC078003F00FC007C01FC00F801FC01F801FC01F001FC03F000F803F0000007E 0000007E0000007E000000FE020000FE1FF000FE3FFC00FE603E00FE801F00FF801F80FF 000FC0FF000FC0FE000FE0FE000FE0FE000FE0FE000FE07E000FE07E000FE07E000FE07E 000FE03E000FE03F000FC01F000FC01F001F800F801F0007E07E0003FFFC0001FFF80000 3FC0001B277DA622>I<00000780000000000780000000000FC0000000000FC000000000 0FC0000000001FE0000000001FE0000000003FF0000000003FF0000000003FF000000000 77F80000000077F800000000F7FC00000000E3FC00000000E3FC00000001C1FE00000001 C1FE00000003C1FF0000000380FF0000000380FF00000007007F80000007007F8000000F 007FC000000E003FC000000E003FC000001C001FE000001C001FE000003FFFFFF000003F FFFFF000003FFFFFF00000700007F80000700007F80000F00007FC0000E00003FC0000E0 0003FC0001C00001FE0001C00001FE0003C00001FF00FFFE003FFFFCFFFE003FFFFCFFFE 003FFFFC2E297EA833>65 D68 D70 D73 D76 D78 D82 D<7FFFFFFFFF807FFFFFFFFF807FFFFFFFFF807F807F807F807C007F800F8078 007F80078078007F80078070007F800380F0007F8003C0F0007F8003C0E0007F8001C0E0 007F8001C0E0007F8001C0E0007F8001C0E0007F8001C000007F80000000007F80000000 007F80000000007F80000000007F80000000007F80000000007F80000000007F80000000 007F80000000007F80000000007F80000000007F80000000007F80000000007F80000000 007F80000000007F80000000007F80000000007F80000000007F80000000007F80000000 007F80000000007F80000000FFFFFFC00000FFFFFFC00000FFFFFFC0002A287EA72F>84 D89 D<03FF80000FFFF0001F01FC003F80FE003F807F003F803F003F803F801F003F8000 003F8000003F8000003F8000003F80003FFF8001FC3F800FE03F801F803F803F003F807E 003F80FC003F80FC003F80FC003F80FC003F80FC005F807E00DF803F839FFC1FFE0FFC03 F803FC1E1B7E9A21>97 DI<003FF00001FFFC0003F03E000FC07F001F807F003F007F003F007F007F00 3E007E0000007E000000FE000000FE000000FE000000FE000000FE000000FE000000FE00 00007E0000007E0000007F0000003F0003803F8003801F8007000FE00E0003F83C0001FF F800003FC000191B7E9A1E>I<00007FF000007FF000007FF0000007F0000007F0000007 F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 F0003F87F001FFF7F007F03FF00FC00FF01F8007F03F0007F03F0007F07E0007F07E0007 F07E0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007 F07E0007F07E0007F03F0007F03F0007F01F800FF00FC01FF007E07FFF01FFE7FF007F87 FF202A7EA925>I<003FC00001FFF00003E07C000F803E001F801F001F001F003F000F80 7E000F807E000FC07E000FC0FE0007C0FE0007C0FFFFFFC0FFFFFFC0FE000000FE000000 FE0000007E0000007E0000007F0000003F0001C01F0001C00F80038007C0070003F01E00 00FFFC00003FE0001A1B7E9A1F>I<0007F8003FFC007E3E01FC7F03F87F03F07F07F07F 07F03E07F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FFFFC007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F0007FFF807FFF807FFF80182A7E A915>I<007F80F001FFE3F807C0FE1C0F807C7C1F003E7C1F003E103F003F003F003F00 3F003F003F003F003F003F003F003F001F003E001F003E000F807C0007C0F80005FFE000 0C7F8000180000001C0000001C0000001E0000001FFFF8001FFFFF000FFFFFC007FFFFE0 03FFFFF00FFFFFF03E0007F07C0001F8F80000F8F80000F8F80000F8F80000F87C0001F0 7C0001F03F0007E00FC01F8007FFFF00007FF0001E287E9A22>II<07000F801FC03FE03FE03FE01FC0 0F8007000000000000000000000000000000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE 0F2B7EAA12>I107 DIII<003FE00001FFFC0003F07E000FC01F801F800FC03F0007E03F00 07E07E0003F07E0003F07E0003F0FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE00 03F8FE0003F8FE0003F87E0003F07E0003F03F0007E03F0007E01F800FC00FC01F8007F0 7F0001FFFC00003FE0001D1B7E9A22>II< FFC3E0FFC7F8FFCC7C0FD8FE0FD0FE0FD0FE0FF0FE0FE07C0FE0000FE0000FE0000FE000 0FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE000 FFFF00FFFF00FFFF00171B7E9A1B>114 D<03FE300FFFF03E03F07800F07000F0F00070 F00070F80070FE0000FFE0007FFF007FFFC03FFFE01FFFF007FFF800FFF80007FC0000FC E0007CE0003CF0003CF00038F80038FC0070FF01E0E7FFC0C1FF00161B7E9A1B>I<0070 0000700000700000700000F00000F00000F00001F00003F00003F00007F0001FFFE0FFFF E0FFFFE007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F0 0007F00007F00007F07007F07007F07007F07007F07007F07007F07003F0E001F8C000FF C0003F0014267FA51A>II 121 D<3FFFFF3FFFFF3F00FE3C01FE3803FC7803F87807F0700FF0700FE0701FC0003FC0 003F80007F0000FF0000FE0001FC0703FC0703F80707F0070FF00F0FE00F1FC00E3FC01E 7F803E7F00FEFFFFFEFFFFFE181B7E9A1E>I E /Fr 27 122 df<60F0F8680808081010 20C0050B7D990B>39 D45 D<60F0F06004047D830B>I68 D72 D87 D89 D<3F8070C070E020700070007007 F01C7030707070E070E071E071E0F171FB1E3C10107E8F13>97 DI<07F80C1C381C30087000E000E000E000E000E000E0007000300438080C1807E0 0E107F8F11>I<007E00000E00000E00000E00000E00000E00000E00000E00000E00000E 0003CE000C3E00380E00300E00700E00E00E00E00E00E00E00E00E00E00E00E00E00600E 00700E00381E001C2E0007CFC0121A7F9915>I<07C01C3030187018600CE00CFFFCE000 E000E000E0006000300438080C1807E00E107F8F11>I<01F0031807380E100E000E000E 000E000E000E00FFC00E000E000E000E000E000E000E000E000E000E000E000E000E000E 007FE00D1A80990C>I<0FCE187330307038703870387038303018602FC0200060007000 3FF03FFC1FFE600FC003C003C003C0036006381C07E010187F8F13>II<18003C003C001800000000000000000000000000FC001C001C001C001C00 1C001C001C001C001C001C001C001C001C001C00FF80091A80990A>I108 DII< 07E01C38300C700E6006E007E007E007E007E007E0076006700E381C1C3807E010107F8F 13>II114 D<1F2060E04020C020C020F0007F003FC01FE000F080708030 C030C020F0408F800C107F8F0F>I<0400040004000C000C001C003C00FFC01C001C001C 001C001C001C001C001C001C201C201C201C201C200E4003800B177F960F>II119 D121 D E /Fs 7 117 df<00030000000780000007800000078000000FC000000FC000001BE000001BE000001B E0000031F0000031F0000060F8000060F80000E0FC0000C07C0000C07C0001803E0001FF FE0003FFFF0003001F0003001F0006000F8006000F800E000FC0FFC07FFCFFC07FFC1E1A 7F9921>65 D<0FF0001C3C003E1E003E0E003E0F001C0F00000F0000FF000FCF003E0F00 7C0F00F80F00F80F00F80F00F817007C27E01FC3E013117F9015>97 DI<03FC000F0E001C1F003C1F00781F00780E00F80000F8 0000F80000F80000F800007800007800003C01801C03000F060003FC0011117F9014>I< FC78FC9C1D3E1D3E1E3E1E1C1E001E001E001E001E001E001E001E001E00FFC0FFC00F11 7F9012>114 D<1FB020704030C030C030F000FF807FE03FF807F8003CC00CC00CE00CE0 08F830CFE00E117F9011>I<06000600060006000E000E001E003FF0FFF01E001E001E00 1E001E001E001E001E001E181E181E181E181E180F3003E00D187F9711>I E /Ft 13 128 df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u 22 118 df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end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 415 436 a Fu(Ribb)r(on)20 b(transformations,)h(In)n(tegrals,)h (and)587 527 y(T)-6 b(riangular)21 b(Decomp)r(ositions)748 682 y Ft(Y)l(orc)o(k)16 b(Sommerh\177)-24 b(auser)889 953 y Fs(Abstract)364 1034 y Fr(W)m(e)17 b(study)g(the)g(theory)g(of)g (in)o(tegrals)h(in)g(Y)m(etter-Drinfel'd)f(Hopf)g(algebras)h(and)361 1080 y(use)12 b(the)f(results)i(to)e(determine)i(the)e(in)o(tegrals)i (of)e(Hopf)g(algebras)i(with)f(triangular)361 1126 y(decomp)q(osition.) 257 1297 y Fq(1)67 b(In)n(tro)r(duction)257 1422 y Fp(The)16 b(question)g(whether)g(it)f(is)h(p)q(ossible)f(to)g(understand)i(the)f (algebra)f(structure)i(of)e(de-)257 1472 y(formed)d(en)o(v)o(eloping)g (algebras)h(in)f(terms)g(of)h(the)g(algebra)f(structure)j(of)d(the)i (three)g(subal-)257 1522 y(gebras)f(app)q(earing)g(in)f(the)h (triangular)e(decomp)q(osition)g(w)o(as)i(addressed)h(in)e([31)o(])g (\(cf.)g(also)257 1572 y([32)o(]\).)j(F)m(or)g(this)g(purp)q(ose,)h (one)f(can)h(use)g(t)o(w)o(o)f(Hopf-algebraic)f(constructions,)i(the)g (\014rst)257 1622 y(one)d(of)g(these)h(t)o(w)o(o)e(b)q(eing)h(a)g(v)o (ery)g(general)g(one)g(whic)o(h)g(migh)o(t)e(describ)q(e)k(the)e (structure)i(of)257 1672 y(other)i(Hopf)e(algebras,)h(to)q(o.)f(In)h (these)h(constructions,)g(t)o(w)o(o)f(of)f(the)i(three)g(comp)q(onen)o (ts)257 1721 y(are)12 b(not)e(ordinary)g(Hopf)h(algebras)f(but)h(Hopf)g (algebras)f(in)h(a)f(certain)h(t)o(wisted)g(sense,)h(that)257 1771 y(is,)i(Hopf)g(algebras)g(in)g(the)h(category)g(of)f(Y)m (etter-Drinfel'd)g(mo)q(dules.)f(It)h(is)g(the)h(purp)q(ose)257 1821 y(of)c(this)h(pap)q(er)g(to)f(determine)h(the)g(in)o(tegrals)f(of) g(Hopf)g(algebras)g(that)h(admit)d(a)i(triangular)257 1871 y(decomp)q(osition)h(whic)o(h)g(is)h(similar)d(to)j(the)g (triangular)f(decomp)q(osition)f(of)h(deformed)g(en-)257 1921 y(v)o(eloping)g(algebras.)g(F)m(or)g(this)h(purp)q(ose,)h(w)o(e)f (carry)g(out)f(a)h(comprehensiv)o(e)g(in)o(v)o(estigation)257 1970 y(of)h(the)g(prop)q(erties)h(of)f(in)o(tegrals)f(of)h(Y)m (etter-Drinfel'd)f(Hopf)h(algebras.)257 2055 y(The)c(article)g(is)f (organized)h(as)f(follo)o(ws:)f(In)h(section)i(2,)d(w)o(e)i(recall)f (some)g(results)i(of)e(D.)f(Fisc)o(h-)257 2105 y(man,)k(S.)h(Mon)o (tgomery)f(and)h(H.-J.)g(Sc)o(hneider)h(\(cf.)g([5)o(]\))f(concerning)h (the)g(existence)i(and)257 2154 y(uniqueness)c(of)d(in)o(tegrals)g(in)h (Y)m(etter-Drinfel'd)f(Hopf)h(algebras,)f(as)h(w)o(ell)f(as)h(the)h (de\014nition)257 2204 y(of)h(the)h(in)o(tegral)f(c)o(haracter)i(and)e (the)h(in)o(tegral)f(group)g(elemen)o(t.)g(It)g(m)o(ust)g(b)q(e)h (emphasized)257 2254 y(that)g(these)g(authors)g(ha)o(v)o(e)f (established)h(these)h(results)g(ev)o(en)f(under)g(m)o(uc)o(h)e(more)g (general)257 2304 y(h)o(yp)q(otheses.)22 b(In)e(con)o(trast)h(to)f (their)h(metho)q(ds,)e(the)i(pro)q(ofs)f(in)g(section)h(2)f(are)h (direct)257 2354 y(analogues)12 b(to)g(the)h(pro)q(ofs)f(for)g (ordinary)f(Hopf)h(algebras.)g(These)h(pro)q(ofs)f(also)g(carry)g(o)o (v)o(er)257 2403 y(directly)k(to)f(more)g(general)g(quasisymmetric)f (categories)i(as)g(already)f(observ)o(ed)h(earlier)257 2453 y(b)o(y)11 b(V.)f(Lyubashenk)o(o)g(\(cf.)h([16)o(],[17)n(]\).)f (As)g(H.-J.)g(Sc)o(hneider)i(has)f(p)q(oin)o(ted)f(out,)g(these)i (facts)257 2503 y(are)i(also)f(recalled)i(in)e(a)g(recen)o(t)i(preprin) o(t)g(of)e(Y.)g(Doi)f(\(cf.)i([4)o(]\).)f(It)h(should)f(b)q(e)h(noted)g (that)963 2628 y(1)p eop %%Page: 2 2 2 1 bop 257 262 a Fp(imp)q(ortan)o(t)16 b(parts)h(of)g(these)i(results) f(ha)o(v)o(e)f(already)g(b)q(een)h(sho)o(wn)f(m)o(uc)o(h)f(earlier)h(b) o(y)g(D.)257 311 y(Radford)12 b(\(cf.)h([23)o(]\).)f(After)i(this)f (rep)q(etition,)g(w)o(e)g(study)g(the)h(prop)q(erties)g(of)f(the)g(mo)q (dular)257 361 y(functions)k(and)f(elemen)o(ts)h(as)f(w)o(ell)g(as)h (the)g(prop)q(erties)h(of)e(in)o(tegral)g(c)o(haracter)h(and)g(the)257 411 y(in)o(tegral)e(group)h(elemen)o(t.)f(In)h(particular,)f(w)o(e)h (sho)o(w)f(that)h(these)h(are)f(cen)o(tral)h(elemen)o(ts)257 461 y(in)d(a)f(certain)i(sense.)257 546 y(In)f(section)h(3,)e(w)o(e)h (la)o(y)f(the)h(abstract)h(foundations)e(that)h(mak)o(e)e(it)i(easier)g (to)g(understand)257 596 y(the)d(nature)f(of)f(certain)h(maps)e(that)i (arise)f(naturally)g(when)h(considering)g(Y)m(etter-Drinfel'd)257 646 y(Hopf)k(algebras.)g(W)m(e)f(recall)h(the)h(notion)e(of)h(a)g (monoidal)d(transformation)h(and)i(con)o(trast)257 696 y(it)20 b(with)g(the)h(related)g(notion)f(of)g(a)g(ribb)q(on)g (transformation.)e(W)m(e)i(exhibit)g(examples)257 746 y(for)g(monoidal)d(and)j(ribb)q(on)g(transformations)e(in)h(the)i (category)f(of)g(Y)m(etter-Drinfel'd)257 795 y(mo)q(dules,)11 b(and)i(explain)f(the)h(e\013ect)h(of)e(the)h(action)f(of)g(suc)o(h)h (transformations)e(on)i(Y)m(etter-)257 845 y(Drinfel'd)i(Hopf)h (algebras.)f(Tw)o(o)h(kinds)g(of)f(monoidal)f(transformations,)g(the)i (mo)q(dular)257 895 y(transformations)e(and)h(the)g(in)o(tegral)g (transformation,)d(pla)o(y)j(the)g(essen)o(tial)h(role)f(in)f(sec-)257 945 y(tion)g(4)f(where)j(w)o(e)e(in)o(tro)q(duce)h(the)f(t)o(wisted)h (Nak)n(a)o(y)o(ama)c(automorphism)o(s)h(and)i(pro)o(v)o(e)g(an)257 995 y(analogue)i(of)f(Radford's)g(form)o(ula)f(for)h(the)i(fourth)f(p)q (o)o(w)o(er)g(of)g(the)h(an)o(tip)q(o)q(de)f(of)f(a)h(Hopf)257 1044 y(algebra)e(in)f(the)i(Y)m(etter-Drinfel'd)e(case.)257 1130 y(In)18 b(section)h(5,)e(w)o(e)h(determine)g(the)g(in)o(tegrals)f (of)h(Hopf)f(algebras)h(with)f(triangular)g(de-)257 1180 y(comp)q(osition)g(that)h(arise)h(from)d(the)j(ab)q(o)o(v)o(e)f(men)o (tioned)g(constructions.)h(It)f(turns)i(out)257 1230 y(to)f(b)q(e)f(di\016cult)g(to)g(determine)g(the)h(in)o(tegrals)f(for)g (the)h(\014rst)g(construction,)g(while)f(the)257 1279 y(in)o(tegrals)f(of)g(the)h(second)h(construction)f(can)g(b)q(e)g (easily)f(obtained)g(afterw)o(ards.)g(These)257 1329 y(results)h(con)o(tain)f(as)g(a)g(sp)q(ecial)g(case)h(the)g(form)o(ula) c(for)j(the)g(in)o(tegrals)g(of)f(the)i(Drinfel'd)257 1379 y(double)c(construction)h(obtained)f(b)o(y)g(D.)f(Radford)g(\(cf.) h([24)o(]\).)f(In)h(the)h(last)e(section)i(6,)e(w)o(e)257 1429 y(illustrate)f(the)h(theory)f(b)o(y)g(considering)g(the)h(example) e(of)g(the)i(F)m(rob)q(enius-Lusztig)f(k)o(ernel)257 1479 y(of)18 b Fo(sl)q Fp(\(2\).)g(The)g(F)m(rob)q(enius-Lusztig)g(k)o (ernels)h(of)e(deformed)h(en)o(v)o(eloping)f(algebras)h(w)o(ere)257 1529 y(de\014ned)i(b)o(y)f(G.)f(Lusztig)h(\(cf.)f([14)o(],)g([15)o (]\).)g(Their)h(presen)o(t)i(name)c(w)o(as)i(in)o(tro)q(duced)h(b)o(y) 257 1578 y(N.)14 b(Andruskiewitsc)o(h)h(and)f(H.)f(J.)h(Sc)o(hneider)h (\(cf.)f([1)o(]\).)257 1664 y(In)j(this)f(article,)f(all)g(v)o(ector)i (spaces)h(o)q(ccurring)f(are)f(de\014ned)h(o)o(v)o(er)g(a)e(base)i (\014eld)f Fo(K)s Fp(.)g(An)257 1714 y(excellen)o(t)f(general)f (reference)i(for)e(all)f(topics)h(discussed)h(here)g(is)f([19)o(].)257 1887 y Fq(2)67 b(In)n(tegrals)23 b(and)g(Y)-6 b(etter-Drinfel'd)24 b(mo)r(dules)257 2013 y Fn(2.1)48 b Fp(In)13 b(this)g(preliminary)e (section)i(w)o(e)g(giv)o(e)g(a)f(self-con)o(tained)h(exp)q(osition)g (of)f(some)g(re-)257 2063 y(sults)g(of)e(D.)g(Fisc)o(hman,)f(S.)h(Mon)o (tgomery)f(and)i(H.-J.)f(Sc)o(hneider)i(\(cf.)e([5)o(]\).)g(Some)g(of)g (these)257 2113 y(results)18 b(w)o(ere)f(pro)o(v)o(ed)g(earlier)g(b)o (y)f(D.)f(Radford)h(\(cf.)g([23)o(]\).)g(W)m(e)g(w)o(ork)g(in)g(the)h (follo)o(wing)257 2163 y(situation:)g Fo(H)k Fp(denotes)e(a)e(Hopf)h (algebra)f(with)h(bijectiv)o(e)f(an)o(tip)q(o)q(de)h(de\014ned)h(o)o(v) o(er)f(the)257 2213 y(\014eld)12 b Fo(K)j Fp(ha)o(ving)10 b(the)j(com)o(ultiplicati)o(on)c(\001)934 2219 y Fm(H)976 2213 y Fp(and)j(the)g(counit)g Fo(\017)1266 2219 y Fm(H)1297 2213 y Fp(.)f(W)m(e)g(use)i(the)f(follo)o(wing)257 2262 y(Heyneman-Sw)o(eedler)j(sigma)c(notation:)817 2354 y(\001)852 2360 y Fm(H)883 2354 y Fp(\()p Fo(h)p Fp(\))h(=)f Fo(h)1018 2360 y Fl(1)1046 2354 y Fk(\012)f Fo(h)1112 2360 y Fl(2)257 2445 y Fp(Recall)k(the)i(notion)e(of)g(a)g(left)g(Y)m(etter-Drinfel'd)h (mo)q(dule)e(\(cf.)h([38],)f([19)o(],)h(Def.)g(10.6.10\):)257 2495 y(This)c(is)g(a)f(left)h Fo(H)s Fp(-como)q(dule)e Fo(V)19 b Fp(whic)o(h)10 b(is)f(also)g(a)h(left)f Fo(H)s Fp(-mo)q(dule)f(suc)o(h)j(that)f(the)g(follo)o(wing)963 2628 y(2)p eop %%Page: 3 3 3 2 bop 257 262 a Fp(compatibilit)o(y)11 b(condition)i(is)h (satis\014ed:)563 341 y Fo(h)587 347 y Fl(1)605 341 y Fo(v)626 324 y Fl(1)655 341 y Fk(\012)9 b Fp(\()p Fo(h)736 347 y Fl(2)767 341 y Fk(!)i Fo(v)841 324 y Fl(2)860 341 y Fp(\))h(=)f(\()p Fo(h)971 347 y Fl(1)1002 341 y Fk(!)g Fo(v)q Fp(\))1092 324 y Fl(1)1111 341 y Fo(h)1135 347 y Fl(2)1163 341 y Fk(\012)e Fp(\()p Fo(h)1244 347 y Fl(1)1275 341 y Fk(!)i Fo(v)q Fp(\))1365 324 y Fl(2)257 420 y Fp(for)17 b(all)e Fo(h)h Fk(2)g Fo(H)k Fp(and)c Fo(v)i Fk(2)e Fo(V)9 b Fp(.)16 b(Here)i(w)o(e)f(ha)o(v)o(e)g(used)g(the)h(follo)o(wing)c(Sw) o(eedler)j(notation)257 470 y(for)g(the)g(coaction:)f Fo(\016)r Fp(\()p Fo(v)q Fp(\))h(=)g Fo(v)736 455 y Fl(1)766 470 y Fk(\012)11 b Fo(v)830 455 y Fl(2)866 470 y Fk(2)16 b Fo(H)e Fk(\012)d Fo(V)f Fp(.)16 b(The)h(arro)o(w)f Fk(!)h Fp(denotes)h(the)f(mo)q(dule)257 520 y(action.)12 b(W)m(e)h(also)f(de\014ne)h(righ)o(t)g(Y)m(etter-Drinfel'd)f(mo)q (dules,)f(whic)o(h)i(are)g(the)g(left)g(Y)m(etter-)257 570 y(Drinfel'd)d(mo)q(dules)f(o)o(v)o(er)h(the)i(opp)q(osite)e(and)h (co)q(opp)q(osite)g(Hopf)f(algebra.)f(They)i(are)g(righ)o(t)257 619 y(como)q(dules)i(and)h(righ)o(t)g(mo)q(dules)e(that)i(satisfy:)563 699 y(\()p Fo(v)600 681 y Fl(1)631 699 y Fk( )d Fo(h)708 705 y Fl(1)726 699 y Fp(\))f Fk(\012)f Fo(v)814 681 y Fl(2)834 699 y Fo(h)858 705 y Fl(2)888 699 y Fp(=)i(\()p Fo(v)j Fk( )d Fo(h)1058 705 y Fl(2)1076 699 y Fp(\))1092 681 y Fl(1)1120 699 y Fk(\012)f Fo(h)1186 705 y Fl(1)1204 699 y Fp(\()p Fo(v)k Fk( )d Fo(h)1331 705 y Fl(2)1349 699 y Fp(\))1365 681 y Fl(2)257 778 y Fp(Of)j(course)i(one)e(can)h (also)e(de\014ne)i(left-righ)o(t)f(and)g(righ)o(t-left)f(Y)m (etter-Drinfel'd)h(mo)q(dules,)257 828 y(whic)o(h)i(are)g(the)h(left)e (Y)m(etter-Drinfel'd)h(mo)q(dules)f(o)o(v)o(er)h(the)g(opp)q(osite)g (resp.)h(co)q(opp)q(osite)257 878 y(Hopf)d(algebra,)f(but)h(they)g(are) h(not)e(used)i(in)f(this)g(article.)257 1008 y Fn(2.2)48 b Fp(The)22 b(tensor)h(pro)q(duct)g(of)e(t)o(w)o(o)h(Y)m (etter-Drinfel'd)f(mo)q(dules)g(b)q(ecomes)h(again)f(a)257 1057 y(Y)m(etter-Drinfel'd)12 b(mo)q(dule)e(if)h(it)h(is)g(endo)o(w)o (ed)g(with)g(the)g(diagonal)e(mo)q(dule)h(and)g(the)i(co)q(di-)257 1107 y(agonal)c(como)q(dule)h(structure)i(\(cf.)f([19)o(],)e(Example)g (10.6.14\).)f(The)j(base)g(\014eld)g Fo(K)i Fp(b)q(ecomes)257 1157 y(a)f(Y)m(etter-Drinfel'd)g(mo)q(dule)e(via)h(the)i(trivial)d(mo)q (dule)h(structure)j Fo(h)d Fk(!)g Fo(\030)j Fp(:=)d Fo(\017)1503 1163 y Fm(H)1534 1157 y Fp(\()p Fo(h)p Fp(\))p Fo(\030)k Fp(and)257 1207 y(the)j(trivial)d(como)q(dule)g(structure)k Fo(\016)845 1213 y Fm(K)877 1207 y Fp(\()p Fo(\030)r Fp(\))e(:=)f(1)11 b Fk(\012)g Fo(\030)r Fp(.)16 b(The)h(left)g(Y)m (etter-Drinfel'd)f(mo)q(d-)257 1257 y(ules,)11 b(and)g(also)f(the)i (righ)o(t)e(ones,)h(therefore)i(constitute)f(a)e(monoidal)e(category)m (.)j(But)g(these)257 1307 y(categories)k(also)e(p)q(ossess)j (braidings,)d(whic)o(h)h(are)g(in)f(the)i(left)f(case)g(giv)o(en)g(b)o (y:)736 1386 y Fo(\033)760 1392 y Fm(V)r(;W)840 1386 y Fp(:)d Fo(V)19 b Fk(\012)9 b Fo(W)18 b Fk(\000)-7 b(!)11 b Fo(W)k Fk(\012)10 b Fo(V)760 1453 y(v)h Fk(\012)e Fo(w)k Fk(7!)e Fp(\()p Fo(v)965 1436 y Fl(1)996 1453 y Fk(!)g Fo(w)q Fp(\))e Fk(\012)h Fo(v)1168 1436 y Fl(2)257 1533 y Fp(The)15 b(corresp)q(onding)g(form)o(ula)c(in)i(the)i(righ)o(t)e (case)i(reads:)692 1612 y Fo(\033)716 1618 y Fm(V)r(;W)785 1612 y Fp(\()p Fo(v)c Fk(\012)e Fo(w)q Fp(\))j(=)f Fo(w)1006 1595 y Fl(1)1034 1612 y Fk(\012)f Fp(\()p Fo(v)j Fk( )e Fo(w)1209 1595 y Fl(2)1227 1612 y Fp(\))p Fo(:)257 1691 y Fp(These)16 b(mappings)c(are)i(bijectiv)o(e)g(since)g Fo(H)j Fp(has)d(a)g(bijectiv)o(e)g(an)o(tip)q(o)q(de.)257 1771 y(Since)h(w)o(e)f(ha)o(v)o(e)g(the)h(notion)e(of)g(a)h(Hopf)g (algebra)f(inside)h(a)g(braided)g(monoidal)d(category)257 1821 y(\(cf.)j([28)o(]\),)f(w)o(e)h(can)g(sp)q(eak)h(of)e(Y)m (etter-Drinfel'd)h(Hopf)f(algebras.)257 1951 y Fn(2.3)48 b Fp(There)21 b(are)f(sev)o(eral)g(elemen)o(tary)e(op)q(erations)i (with)f(Y)m(etter-Drinfel'd)h(mo)q(dules)257 2001 y(whic)o(h)14 b(will)f(b)q(e)h(needed)h(in)f(the)g(sequel.)257 2094 y Fn(Lemma)36 b Fp(If)20 b Fo(V)29 b Fp(is)20 b(a)f(\014nite)h (dimensional)d(left)j(Y)m(etter-Drinfel'd)g(mo)q(dule,)d(then)k(the)257 2143 y(dual)13 b(space)i Fo(V)493 2128 y Fj(\003)526 2143 y Fp(is)e(in)g(a)h(unique)f(w)o(a)o(y)g(a)g(righ)o(t)h(Y)m (etter-Drinfel'd)f(mo)q(dule)f(suc)o(h)i(that)g(the)257 2193 y(natural)g(pairing)671 2243 y Fk(h\001)p Fo(;)7 b Fk(\001i)k Fp(:)g Fo(V)18 b Fk(\002)10 b Fo(V)897 2226 y Fj(\003)928 2243 y Fk(!)h Fo(K)q(;)c Fp(\()p Fo(v)q(;)g(f)t Fp(\))12 b Fk(7!)f Fo(f)t Fp(\()p Fo(v)q Fp(\))257 2311 y(is)j(a)g(Y)m(etter-Drinfel'd)f(form)f(\(cf.)i([32)o(],)f(subsection)i (2)p Fo(:)p Fp(4\),)e(i.)g(e.)g(that)h(w)o(e)g(ha)o(v)o(e)308 2415 y(1.)20 b Fk(h)p Fo(h)12 b Fk(!)f Fo(v)q(;)c(f)t Fk(i)12 b Fp(=)g Fk(h)p Fo(v)q(;)7 b(f)17 b Fk( )11 b Fo(h)p Fk(i)308 2493 y Fp(2.)20 b Fk(h)p Fo(v)q(;)7 b(f)441 2478 y Fl(1)461 2493 y Fk(i)p Fo(f)501 2478 y Fl(2)532 2493 y Fp(=)12 b Fo(v)597 2478 y Fl(1)616 2493 y Fk(h)p Fo(v)653 2478 y Fl(2)672 2493 y Fo(;)7 b(f)t Fk(i)963 2628 y Fp(3)p eop %%Page: 4 4 4 3 bop 257 262 a Fn(Pro)q(of.)36 b Fp(An)13 b(elemen)o(t)f(of)g Fo(H)j Fp(acts)e(on)g Fo(V)912 246 y Fj(\003)944 262 y Fp(via)e(the)i(transp)q(ose)h(of)e(the)h(action)f(on)h Fo(V)c Fp(.)j(The)257 311 y(como)q(dule-structure)j(is)f(giv)o(en)f(b)o (y)h(the)g(form)o(ula:)680 413 y Fo(\016)698 419 y Fm(V)725 411 y Fi(\003)744 413 y Fp(\()p Fo(f)t Fp(\))f(=)877 361 y Fm(n)857 374 y Fh(X)860 462 y Fm(i)p Fl(=1)924 413 y Fo(v)945 396 y Fl(\()p Fm(i)p Fl(\))p Fj(\003)1011 413 y Fk(\012)d Fo(f)t Fp(\()p Fo(v)1113 420 y Fl(\()p Fm(i)p Fl(\))1154 396 y(2)1172 413 y Fp(\))p Fo(v)1208 420 y Fl(\()p Fm(i)p Fl(\))1248 396 y(1)257 528 y Fp(where)15 b Fo(v)397 535 y Fl(\(1\))442 528 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(v)562 535 y Fl(\()p Fm(n)p Fl(\))624 528 y Fp(is)13 b(a)h(basis)g(of)f Fo(V)24 b Fp(with)13 b(dual)g(basis)h Fo(v)1206 513 y Fl(\(1\))p Fj(\003)1268 528 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(v)1389 513 y Fl(\()p Fm(n)p Fl(\))p Fj(\003)1455 528 y Fp(.)13 b Fg(\003)257 632 y Fp(The)24 b(transp)q(ose)g(of)e(an)g Fo(H)s Fp(-linear)g(and)h(colinear)f(map)g (b)q(et)o(w)o(een)i(\014nite-dimensional)257 682 y(Y)m(etter-Drinfel'd) k(mo)q(dules)f(is)g(linear)h(and)g(colinear.)f(If)g Fo(V)1288 688 y Fl(1)1335 682 y Fp(and)h Fo(V)1454 688 y Fl(2)1500 682 y Fp(are)g(\014nite-)257 732 y(dimensional)21 b(left)i(Y)m (etter-Drinfel'd)f(mo)q(dules,)g(then)h Fo(V)1197 738 y Fl(1)1216 714 y Fj(\003)1250 732 y Fk(\012)16 b Fo(V)1322 738 y Fl(2)1341 714 y Fj(\003)1383 732 y Fp(is)22 b(isomorphic)g(to)257 782 y(\()p Fo(V)297 788 y Fl(1)326 782 y Fk(\012)9 b Fo(V)391 788 y Fl(2)410 782 y Fp(\))426 767 y Fj(\003)459 782 y Fp(as)14 b(a)g(righ)o(t)f(Y)m(etter-Drinfel'd)h(mo)q(dule)e(via)h (the)i(isomorphism)432 853 y Fo(V)456 859 y Fl(1)474 834 y Fj(\003)503 853 y Fk(\012)9 b Fo(V)568 859 y Fl(2)587 834 y Fj(\003)618 853 y Fk(!)i Fp(\()p Fo(V)711 859 y Fl(1)739 853 y Fk(\012)e Fo(V)804 859 y Fl(2)823 853 y Fp(\))839 835 y Fj(\003)859 853 y Fo(;)e(f)898 859 y Fl(1)925 853 y Fk(\012)j Fo(f)987 859 y Fl(2)1017 853 y Fk(7!)h Fp(\()p Fo(v)1106 859 y Fl(1)1134 853 y Fk(\012)f Fo(v)1196 859 y Fl(2)1226 853 y Fk(7!)h Fo(f)1299 859 y Fl(1)1318 853 y Fp(\()p Fo(v)1354 859 y Fl(1)1373 853 y Fp(\))p Fo(f)1409 859 y Fl(2)1428 853 y Fp(\()p Fo(v)1464 859 y Fl(2)1483 853 y Fp(\)\))257 923 y(Up)f(to)g(this)f(isomorphism,)e (the)j(braiding)e(on)i Fo(V)999 929 y Fl(1)1018 905 y Fj(\003)1038 923 y Fk(\012)q Fo(V)1095 929 y Fl(2)1114 905 y Fj(\003)1142 923 y Fp(is)g(the)g(transp)q(ose)h(of)e(the)h (braiding)257 973 y(on)k Fo(V)339 979 y Fl(1)366 973 y Fk(\012)9 b Fo(V)431 979 y Fl(2)450 973 y Fp(.)k(One)h(can)g(express) h(these)g(facts)f(b)o(y)g(sa)o(ying)f(that)g(taking)g(the)h(dual)f (space)i(is)257 1023 y(a)f(\(non-strict\))g(braided)g(monoidal)c (functor)k(from)e(the)i(category)g(of)f(\014nite-dimensional)257 1073 y(left)h(Y)m(etter-Drinfel'd)f(mo)q(dules)g(to)g(the)h(category)g (of)f(\014nite-dimensional)e(righ)o(t)j(Y)m(etter-)257 1123 y(Drinfel'd)k(mo)q(dules)f(\(cf.)h([9)o(],)g(De\014nition)f (2.3\).)h(As)g(a)h(consequence,)h(if)e Fo(A)g Fp(is)g(a)g(\014nite-)257 1172 y(dimensional)e(left)i(Y)m(etter-Drinfel'd)g(Hopf)g(algebra,)f (then)h(the)h(dual)f(space)h Fo(B)i Fp(:=)d Fo(A)1670 1157 y Fj(\003)257 1222 y Fp(is)e(in)e(a)i(unique)f(w)o(a)o(y)g(a)g (righ)o(t)g(Y)m(etter-Drinfel'd)g(Hopf)g(algebra)f(suc)o(h)j(that)e (the)h(natural)257 1272 y(pairing)h(describ)q(ed)i(ab)q(o)o(v)o(e)f(is) f(a)h(bialgebra)f(form)e(\(cf.)j([32)o(],)f(subsection)i(2.6\),)d(i.)h (e.)g(w)o(e)257 1322 y(ha)o(v)o(e:)308 1420 y(1.)j Fk(h)p Fo(a)9 b Fk(\012)h Fo(a)472 1405 y Fj(0)484 1420 y Fo(;)d Fp(\001)538 1426 y Fm(B)565 1420 y Fp(\()p Fo(b)p Fp(\))p Fk(i)12 b Fp(=)g Fk(h)p Fo(aa)747 1405 y Fj(0)758 1420 y Fo(;)7 b(b)p Fk(i)308 1495 y Fp(2.)20 b Fk(h)p Fo(a;)7 b(bb)454 1480 y Fj(0)465 1495 y Fk(i)12 b Fp(=)f Fk(h)p Fp(\001)587 1501 y Fm(A)614 1495 y Fp(\()p Fo(a)p Fp(\))p Fo(;)c(b)i Fk(\012)g Fo(b)773 1480 y Fj(0)785 1495 y Fk(i)308 1569 y Fp(3.)20 b Fk(h)p Fp(1)p Fo(;)7 b(b)p Fk(i)k Fp(=)h Fo(\017)523 1575 y Fm(B)551 1569 y Fp(\()p Fo(b)p Fp(\))p Fo(;)i Fk(h)p Fo(a;)7 b Fp(1)p Fk(i)k Fp(=)g Fo(\017)792 1575 y Fm(A)819 1569 y Fp(\()p Fo(a)p Fp(\))257 1696 y Fn(2.4)48 b Fp(If)15 b Fo(H)j Fp(is)e (\014nite-dimensional,)c(the)k(pro)q(cess)i(of)d(dualization)f(can)h (also)g(b)q(e)h(applied)257 1746 y(to)e Fo(H)s Fp(:)257 1833 y Fn(Lemma)36 b Fp(Supp)q(ose)18 b(that)e Fo(H)j Fp(is)d(\014nite)g(dimensional.)e(If)h Fo(V)26 b Fp(is)16 b(a)g(left)g(Y)m(etter-Drinfel'd)257 1883 y(mo)q(dule)c(o)o(v)o(er)h Fo(H)s Fp(,)f(then)i Fo(V)23 b Fp(b)q(ecomes)13 b(a)g(righ)o(t)f(Y)m (etter-Drinfel'd)h(mo)q(dule)f(o)o(v)o(er)h Fo(H)1563 1868 y Fj(\003)1595 1883 y Fp(using)257 1933 y(the)i(mo)q(dule)d (structure)816 1983 y Fo(v)h Fk( )e Fo(f)16 b Fp(:=)c Fo(f)t Fp(\()p Fo(v)1055 1965 y Fl(1)1075 1983 y Fp(\))p Fo(v)1112 1965 y Fl(2)257 2045 y Fp(for)i Fo(v)f Fk(2)e Fo(V)24 b Fp(and)13 b Fo(f)k Fk(2)11 b Fo(V)630 2030 y Fj(\003)649 2045 y Fp(,)j(and)f(the)i(como)q(dule)e(structure)706 2147 y Fo(\016)724 2153 y Fj(\003)744 2147 y Fp(\()p Fo(v)q Fp(\))f(:=)885 2095 y Fm(n)865 2107 y Fh(X)868 2196 y Fm(i)p Fl(=1)925 2147 y Fp(\()p Fo(h)965 2154 y Fl(\()p Fm(i)p Fl(\))1016 2147 y Fk(!)f Fo(v)q Fp(\))f Fk(\012)g Fo(h)1182 2130 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)257 2262 y Fp(where)15 b Fo(h)401 2269 y Fl(\(1\))446 2262 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(h)570 2269 y Fl(\()p Fm(n)p Fl(\))631 2262 y Fp(is)14 b(a)g(basis)g(of)f Fo(H)k Fp(with)c(dual)g(basis)h Fo(h)1221 2247 y Fl(\(1\))p Fj(\003)1283 2262 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(h)1407 2247 y Fl(\()p Fm(n)p Fl(\))p Fj(\003)1471 2262 y Fp(.)257 2349 y Fn(Pro)q(of.)36 b Fp(This)14 b(rests)h(on)f(direct)h (computation)d(\(cf.)i([19)o(],)f(Lemma)e(1.6.4\).)h Fg(\003)257 2453 y Fp(If)j Fo(H)j Fp(is)e(\014nite)f(dimensional,)e(a)i (mapping)e(b)q(et)o(w)o(een)j(t)o(w)o(o)f(Y)m(etter-Drinfel'd)g(mo)q (dules)g(is)257 2503 y(linear)e(and)f(colinear)g(with)h(resp)q(ect)h (to)f Fo(H)i Fp(if)d(and)h(only)e(if)h(it)g(is)h(linear)f(and)g (colinear)h(with)963 2628 y(4)p eop %%Page: 5 5 5 4 bop 257 262 a Fp(resp)q(ect)14 b(to)e Fo(H)484 246 y Fj(\003)503 262 y Fp(.)f(The)h(dualization)e(pro)q(cess)k(describ)q (ed)f(in)e(this)h(Lemma)d(comm)o(utes)h(with)257 311 y(taking)j(the)g(tensor)h(pro)q(duct)g(of)f(t)o(w)o(o)g(Y)m (etter-Drinfel'd)f(mo)q(dules,)g(and)h(the)h(braiding)e(on)257 361 y(the)19 b(tensor)f(pro)q(duct)h(is)e(the)h(same)f(b)q(efore)h(and) g(after)g(the)g(dualization.)e(This)h(can)h(b)q(e)257 411 y(expressed)j(b)o(y)d(sa)o(ying)g(that)g(dualization)g(with)g(resp) q(ect)i(to)f Fo(H)i Fp(giv)o(es)d(rise)h(to)f(a)h(strict)257 461 y(braided)h(monoidal)d(functor)j(from)e(the)i(category)g(of)f(left) g(Y)m(etter-Drinfel'd)h(mo)q(dules)257 511 y(o)o(v)o(er)13 b Fo(H)j Fp(to)d(righ)o(t)g(Y)m(etter-Drinfel'd)g(mo)q(dules)f(o)o(v)o (er)h Fo(H)1134 496 y Fj(\003)1153 511 y Fp(.)f(Therefore,)i(a)f(Hopf)f (algebra)h(in)257 560 y(the)i(former)e(category)h(remains)f(a)g(Hopf)h (algebra)f(in)h(the)g(latter)g(category)m(.)257 691 y Fn(2.5)48 b Fp(Since)14 b(left)f(Y)m(etter-Drinfel'd)g(mo)q(dules)g (are)h(the)g(same)e(as)i(righ)o(t)f(Y)m(etter-Drinfel'd)257 740 y(mo)q(dules)g(o)o(v)o(er)h(the)h(opp)q(osite)f(and)f(co)q(opp)q (osite)i(Hopf)e(algebra,)g(w)o(e)h(ha)o(v)o(e:)257 824 y Fn(Lemma)308 874 y Fp(1.)20 b(If)13 b Fo(A)h Fp(is)f(a)g(left)h(Y)m (etter-Drinfel'd)f(Hopf)g(algebra)g(o)o(v)o(er)h Fo(H)s Fp(,)f(then)h(the)g(opp)q(osite)g(and)361 923 y(co)q(opp)q(osite)j (Hopf)f(algebra)g Fo(A)852 908 y Fm(op)f(cop)966 923 y Fp(is)i(a)f(righ)o(t)g(Y)m(etter-Drinfel'd)g(Hopf)g(algebra)361 973 y(o)o(v)o(er)e Fo(H)488 958 y Fm(op)g(cop)586 973 y Fp(.)308 1051 y(2.)20 b(If)13 b Fo(B)i Fp(is)e(a)g(righ)o(t)g(Y)m (etter-Drinfel'd)f(Hopf)h(algebra)g(o)o(v)o(er)g Fo(H)s Fp(,)f(then)i Fo(B)1445 1036 y Fm(op)h(cop)1556 1051 y Fp(is)e(a)g(left)361 1101 y(Y)m(etter-Drinfel'd)h(Hopf)f(algebra)h(o) o(v)o(er)g Fo(H)1036 1086 y Fm(op)g(cop)1133 1101 y Fp(.)257 1211 y(The)h(pro)q(of)e(is)h(omitted.)257 1341 y Fn(2.6)48 b Fp(W)m(e)13 b(no)o(w)f(pro)q(ceed)j(to)e(explain)f(a)h(sligh)o(tly)f (more)g(in)o(teresting)h(op)q(eration)g(that)g(can)257 1391 y(b)q(e)g(carried)g(out)f(with)g(Y)m(etter-Drinfel'd)f(Hopf)h (algebras.)f(In)i(this)f(article,)f(a)h(c)o(haracter)i(is)257 1441 y(an)h(algebra)f(map)f(to)h(the)h(base)g(\014eld.)f(A)h(c)o (haracter)h(is)e(the)h(same)f(as)h(an)f(augmen)o(tation.)257 1491 y(A)i(c)o(haracter)h(in)e(the)h(sense)i(used)e(here)h(is)e(a)h (one-dimensional)d(c)o(haracter)k(in)e(the)h(more)257 1541 y(general)e(sense)i(of)d(the)i(w)o(ord)f(used)g(in)g(represen)o (tation)h(theory)m(.)257 1633 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)18 b(that)f Fo(A)h Fp(is)f(a)f(left)h(Y)m (etter-Drinfel'd)g(Hopf)g(algebra)f(o)o(v)o(er)i Fo(H)257 1683 y Fp(and)c(that)g Fo(!)f Fp(:)e Fo(A)h Fk(!)f Fo(K)17 b Fp(is)c(an)h Fo(H)s Fp(-linear)f(and)h(colinear)g(c)o(haracter.)g(W)m (e)g(ha)o(v)o(e:)308 1787 y(1.)20 b Fo(A)14 b Fp(can)h(b)q(e)f (regarded)i(in)d(another)i(w)o(a)o(y)e(as)i(a)e(left)h(Y)m (etter-Drinfel'd)g(Hopf)g(algebra)361 1837 y(if)f(the)i(com)o (ultiplicatio)o(n)c(is)j(c)o(hanged)g(to)818 1916 y(\001)853 1899 y Fm(!)853 1926 y(A)880 1916 y Fp(\()p Fo(a)p Fp(\))d(:=)h Fo(a)1023 1922 y Fl(1)1050 1916 y Fk(\012)e Fo(!)q Fp(\()p Fo(a)1157 1922 y Fl(2)1176 1916 y Fp(\))p Fo(a)1214 1922 y Fl(3)361 1995 y Fp(and)k(counit)g(and)f(an)o(tip)q(o)q(de)h(are)h(c)o (hanged)f(to)529 2075 y Fo(\017)546 2058 y Fm(!)546 2085 y(A)573 2075 y Fp(\()p Fo(a)p Fp(\))e(:=)f Fo(!)721 2058 y Fj(\000)p Fl(1)766 2075 y Fp(\()p Fo(a)p Fp(\))p Fo(;)90 b(S)949 2058 y Fm(!)947 2085 y(A)974 2075 y Fp(\()p Fo(a)p Fp(\))12 b(:=)f Fo(!)1122 2058 y Fj(\000)p Fl(1)1167 2075 y Fp(\()p Fo(a)1205 2081 y Fl(1)1224 2075 y Fp(\))p Fo(S)1265 2081 y Fm(A)1292 2075 y Fp(\()p Fo(a)1330 2081 y Fl(2)1349 2075 y Fp(\))p Fo(!)1392 2058 y Fj(\000)p Fl(1)1437 2075 y Fp(\()p Fo(a)1475 2081 y Fl(3)1494 2075 y Fp(\))p Fo(:)361 2154 y Fp(Here)16 b(w)o(e)e(ha)o(v)o(e)g(adopted)g (the)h(notation)e Fo(!)1041 2139 y Fj(\000)p Fl(1)1086 2154 y Fp(\()p Fo(a)p Fp(\))g(:=)e Fo(!)q Fp(\()p Fo(S)1276 2160 y Fm(A)1304 2154 y Fp(\()p Fo(a)p Fp(\)\).)j(W)m(e)g(shall)g (denote)361 2204 y Fo(A)g Fp(b)o(y)g Fo(A)495 2189 y Fm(!)533 2204 y Fp(if)f(it)g(is)h(endo)o(w)o(ed)g(with)g(this)g (structure.)308 2282 y(2.)20 b(The)14 b(mappings)799 2362 y Fo(\036)e Fp(:)f Fo(A)g Fk(!)g Fo(A)985 2344 y Fm(!)1009 2362 y Fo(;)c(a)k Fk(7!)g Fo(!)q Fp(\()p Fo(a)1179 2368 y Fl(1)1198 2362 y Fp(\))p Fo(a)1236 2368 y Fl(2)795 2424 y Fo( )j Fp(:)d Fo(A)g Fk(!)g Fo(A)985 2407 y Fm(!)1009 2424 y Fo(;)c(a)k Fk(7!)g Fo(a)1136 2430 y Fl(1)1155 2424 y Fo(!)q Fp(\()p Fo(a)1220 2430 y Fl(2)1239 2424 y Fp(\))361 2503 y(are)j(isomorphisms)e(of)h(Y)m(etter-Drinfel'd)g (Hopf)h(algebras.)963 2628 y(5)p eop %%Page: 6 6 6 5 bop 257 262 a Fn(Pro)q(of.)36 b Fp(It)12 b(is)f(of)g(course)i (su\016cien)o(t)f(to)f(pro)o(v)o(e)h(the)g(second)h(assertion,)e(b)q (ecause)j(then)e(the)257 311 y(structure)k(in)e(1.)e(will)h(app)q(ear)h (as)g(the)g(structure)i(of)d Fo(A)h Fp(carried)g(o)o(v)o(er)g(to)g (somewhere)g(else)257 361 y(via)h(a)g(bijectiv)o(e)h(map.)d(Since)j(up) f(to)h(the)g(iden)o(ti\014cation)f(of)f Fo(K)g Fk(\012)d Fo(A)k Fp(resp.)h Fo(A)11 b Fk(\012)f Fo(K)19 b Fp(with)257 411 y Fo(A)p Fp(,)14 b(whic)o(h)g(is)f Fo(H)s Fp(-linear)h(and)f (colinear,)g(w)o(e)i(ha)o(v)o(e:)570 500 y Fo(\036)c Fp(=)h(\()p Fo(!)f Fk(\012)e Fo(id)780 506 y Fm(A)807 500 y Fp(\))h Fk(\016)f Fp(\001)898 506 y Fm(A)1007 500 y Fo( )k Fp(=)f(\()p Fo(id)1143 506 y Fm(A)1179 500 y Fk(\012)e Fo(!)q Fp(\))g Fk(\016)f Fp(\001)1339 506 y Fm(A)1365 500 y Fo(;)257 589 y Fp(w)o(e)i(see)h(that)e Fo(\036)g Fp(and)h Fo( )h Fp(are)f(linear)f(and)g(colinear)g(as)h(comp) q(ositions)e(of)g(linear)h(and)h(colinear)257 639 y(maps.)i(They)h(are) g(ob)o(viously)f(bijectiv)o(e)h(with)f(in)o(v)o(erses:)566 729 y Fo(\036)591 711 y Fj(\000)p Fl(1)635 729 y Fp(\()p Fo(a)p Fp(\))f(=)g Fo(!)772 711 y Fj(\000)p Fl(1)817 729 y Fp(\()p Fo(a)855 735 y Fl(1)873 729 y Fp(\))p Fo(a)911 735 y Fl(2)1013 729 y Fo( )1041 711 y Fj(\000)p Fl(1)1086 729 y Fp(\()p Fo(a)p Fp(\))g(=)g Fo(a)1218 735 y Fl(1)1236 729 y Fo(!)1263 711 y Fj(\000)p Fl(1)1308 729 y Fp(\()p Fo(a)1346 735 y Fl(2)1365 729 y Fp(\))257 818 y Fo(\036)i Fp(is)g(an)f(algebra)h(map)e(since)j Fo(!)g Fp(is)f(linear:)636 907 y Fo(\036)p Fp(\()p Fo(aa)721 890 y Fj(0)733 907 y Fp(\))d(=)h Fo(!)q Fp(\()p Fo(a)869 913 y Fl(1)888 907 y Fp(\()p Fo(a)926 913 y Fl(2)945 890 y(1)975 907 y Fk(!)f Fo(a)1050 890 y Fj(0)1050 917 y Fl(1)1069 907 y Fp(\)\))p Fo(a)1123 913 y Fl(2)1142 890 y(2)1160 907 y Fo(a)1182 890 y Fj(0)1182 917 y Fl(2)760 969 y Fp(=)h Fo(!)q Fp(\()p Fo(a)869 975 y Fl(1)888 969 y Fp(\))p Fo(!)q Fp(\()p Fo(a)969 952 y Fj(0)969 980 y Fl(1)988 969 y Fp(\))p Fo(a)1026 975 y Fl(2)1045 969 y Fo(a)1067 952 y Fj(0)1067 980 y Fl(2)1097 969 y Fp(=)g Fo(\036)p Fp(\()p Fo(a)p Fp(\))p Fo(\036)p Fp(\()p Fo(a)1283 952 y Fj(0)1294 969 y Fp(\))257 1059 y Fo( )k Fp(is)d(an)h(algebra)g(map)e (since)i Fo(!)i Fp(is)d(colinear:)631 1148 y Fo( )q Fp(\()p Fo(aa)719 1131 y Fj(0)731 1148 y Fp(\))e(=)h Fo(a)824 1154 y Fl(1)843 1148 y Fp(\()p Fo(a)881 1154 y Fl(2)900 1131 y(1)930 1148 y Fk(!)f Fo(a)1005 1131 y Fj(0)1005 1158 y Fl(1)1023 1148 y Fp(\))p Fo(!)q Fp(\()p Fo(a)1104 1154 y Fl(2)1123 1131 y(2)1142 1148 y Fo(a)1164 1131 y Fj(0)1164 1158 y Fl(2)1183 1148 y Fp(\))758 1210 y(=)h Fo(a)824 1216 y Fl(1)843 1210 y Fo(a)865 1193 y Fj(0)865 1220 y Fl(1)883 1210 y Fo(!)q Fp(\()p Fo(a)948 1216 y Fl(2)967 1210 y Fp(\))p Fo(!)q Fp(\()p Fo(a)1048 1193 y Fj(0)1048 1220 y Fl(2)1068 1210 y Fp(\))f(=)h Fo( )q Fp(\()p Fo(a)p Fp(\))p Fo( )q Fp(\()p Fo(a)1287 1193 y Fj(0)1300 1210 y Fp(\))257 1299 y Fo(\036)j Fp(and)h Fo( )g Fp(ob)o(viously)f(preserv)o(e)i(the)f(unit)f(and)g(the)h (counit.)f(They)h(are)g(coalgebra)f(maps)257 1349 y(b)q(ecause)h(w)o(e) e(ha)o(v)o(e:)520 1438 y(\001)555 1421 y Fm(!)555 1449 y(A)581 1438 y Fp(\()p Fo(\036)p Fp(\()p Fo(a)p Fp(\)\))e(=)g Fo(!)q Fp(\()p Fo(a)813 1444 y Fl(1)832 1438 y Fp(\))p Fo(a)870 1444 y Fl(2)898 1438 y Fk(\012)e Fo(!)q Fp(\()p Fo(a)1005 1444 y Fl(3)1024 1438 y Fp(\))p Fo(a)1062 1444 y Fl(4)1092 1438 y Fp(=)i(\()p Fo(\036)d Fk(\012)g Fo(\036)p Fp(\))g Fk(\016)g Fp(\001)1342 1444 y Fm(A)1369 1438 y Fp(\()p Fo(a)p Fp(\))516 1501 y(\001)551 1484 y Fm(!)551 1511 y(A)578 1501 y Fp(\()p Fo( )q Fp(\()p Fo(a)p Fp(\)\))j(=)g Fo(a)770 1507 y Fl(1)789 1501 y Fo(!)q Fp(\()p Fo(a)854 1507 y Fl(2)873 1501 y Fp(\))d Fk(\012)h Fo(a)962 1507 y Fl(3)980 1501 y Fo(!)q Fp(\()p Fo(a)1045 1507 y Fl(4)1064 1501 y Fp(\))i(=)g(\()p Fo( )f Fk(\012)e Fo( )q Fp(\))h Fk(\016)f Fp(\001)1350 1507 y Fm(A)1377 1501 y Fp(\()p Fo(a)p Fp(\))257 1590 y(Finally)m(,)j(w)o(e)i(observ)o(e)h(that)f Fo(\036)f Fp(and)h Fo( )h Fp(comm)o(ute)d(with)i(the)g(an)o(tip)q(o)q (de:)606 1679 y Fo(\036)p Fp(\()p Fo(S)672 1685 y Fm(A)700 1679 y Fp(\()p Fo(a)p Fp(\)\))e(=)f Fo(!)q Fp(\()p Fo(S)893 1685 y Fm(A)921 1679 y Fp(\()p Fo(a)959 1685 y Fl(1)978 1662 y(1)1008 1679 y Fk(!)g Fo(a)1083 1685 y Fl(2)1102 1679 y Fp(\)\))p Fo(S)1159 1685 y Fm(A)1187 1679 y Fp(\()p Fo(a)1225 1685 y Fl(1)1243 1662 y(2)1262 1679 y Fp(\))782 1741 y(=)g Fo(!)q Fp(\()p Fo(S)893 1747 y Fm(A)921 1741 y Fp(\()p Fo(a)959 1747 y Fl(2)978 1741 y Fp(\)\))p Fo(S)1035 1747 y Fm(A)1063 1741 y Fp(\()p Fo(a)1101 1747 y Fl(1)1120 1741 y Fp(\))782 1809 y(=)g Fo(!)q Fp(\()p Fo(a)890 1815 y Fl(1)910 1809 y Fp(\))p Fo(!)953 1792 y Fj(\000)p Fl(1)998 1809 y Fp(\()p Fo(a)1036 1815 y Fl(2)1054 1809 y Fp(\))p Fo(S)1095 1815 y Fm(A)1123 1809 y Fp(\()p Fo(a)1161 1815 y Fl(3)1180 1809 y Fp(\))p Fo(!)1223 1792 y Fj(\000)p Fl(1)1268 1809 y Fp(\()p Fo(a)1306 1815 y Fl(4)1324 1809 y Fp(\))782 1871 y(=)g Fo(S)852 1854 y Fm(!)850 1881 y(A)878 1871 y Fp(\()p Fo(\036)p Fp(\()p Fo(a)p Fp(\)\))257 1960 y(b)o(y)j(the)h(linearit)o(y)d(of)i Fo(!)h Fp(and)604 2050 y Fo( )q Fp(\()p Fo(S)673 2056 y Fm(A)702 2050 y Fp(\()p Fo(a)p Fp(\)\))d(=)f Fo(S)852 2056 y Fm(A)880 2050 y Fp(\()p Fo(a)918 2056 y Fl(1)937 2033 y(1)967 2050 y Fk(!)g Fo(a)1042 2056 y Fl(2)1060 2050 y Fp(\))p Fo(!)q Fp(\()p Fo(S)1144 2056 y Fm(A)1172 2050 y Fp(\()p Fo(a)1210 2056 y Fl(1)1229 2033 y(2)1248 2050 y Fp(\)\))784 2117 y(=)g Fo(S)852 2123 y Fm(A)880 2117 y Fp(\()p Fo(a)918 2123 y Fl(2)937 2117 y Fp(\))p Fo(!)980 2100 y Fj(\000)p Fl(1)1025 2117 y Fp(\()p Fo(a)1063 2123 y Fl(1)1081 2117 y Fp(\))784 2185 y(=)g Fo(!)854 2167 y Fj(\000)p Fl(1)899 2185 y Fp(\()p Fo(a)937 2191 y Fl(1)956 2185 y Fp(\))p Fo(S)997 2191 y Fm(A)1025 2185 y Fp(\()p Fo(a)1063 2191 y Fl(2)1081 2185 y Fp(\))p Fo(!)1124 2167 y Fj(\000)p Fl(1)1169 2185 y Fp(\()p Fo(a)1207 2191 y Fl(3)1226 2185 y Fp(\))p Fo(!)q Fp(\()p Fo(a)1307 2191 y Fl(4)1326 2185 y Fp(\))784 2247 y(=)g Fo(S)854 2230 y Fm(!)852 2257 y(A)880 2247 y Fp(\()p Fo( )q Fp(\()p Fo(a)p Fp(\)\))257 2336 y(b)o(y)j(the)h(colinearit)o(y)e(of)g Fo(!)q Fp(.)h Fg(\003)257 2453 y Fp(This)i(Prop)q(osition)e(of)h(course)i(also)d (holds)h(if)g Fo(H)j Fp(is)d(equal)g(to)g(the)h(base)g(\014eld)g Fo(K)s Fp(;)f(in)f(this)257 2503 y(case)19 b(Y)m(etter-Drinfel'd)f (Hopf)g(algebras)f(are)i(ordinary)e(Hopf)h(algebras.)f(W)m(e)h (therefore)963 2628 y(6)p eop %%Page: 7 7 7 6 bop 257 262 a Fp(see)20 b(that)f(there)h(is)f(no)f(essen)o(tial)i (di\013erence)g(b)q(et)o(w)o(een)g(the)g(counit)e(and)h(an)g(arbitrary) 257 311 y(c)o(haracter:)c(If)f(one)g(c)o(haracter)i(of)d(an)h(algebra)g (arises)h(as)f(the)h(counit)f(of)f(a)h(suitable)g(Hopf)257 361 y(algebra)g(structure,)h(then)g(ev)o(ery)g(c)o(haracter)g(arises)f (in)g(this)g(w)o(a)o(y)m(.)257 447 y(It)e(should)f(b)q(e)h(observ)o(ed) g(that)g(in)e(pro)o(ving)h(that)g Fo(\036)g Fp(is)g(an)h(algebra)e(map) g(only)h(the)g(linearit)o(y)257 497 y(of)j Fo(!)h Fp(w)o(as)e(needed,)i (whereas)g(the)g(fact)f(that)g Fo( )h Fp(is)f(an)f(algebra)g(map)g (follo)o(ws)f(solely)h(from)257 546 y(the)i(colinearit)o(y)e(of)g Fo(!)q Fp(.)h(This)g(fact)g(will)e(b)q(e)i(needed)i(in)d(the)i(sequel.) 257 682 y Fn(2.7)48 b Fp(Recall)16 b(that)h(a)g(F)m(rob)q(enius)g (algebra)g(is)g(a)f(\014nite-dimensional)f(algebra)i Fo(A)g Fp(whic)o(h)257 732 y(admits)c(a)g(nondegenerate)i(bilinear)e (form)f Fk(h\001)p Fo(;)7 b Fk(\001i)k Fp(:)g Fo(A)e Fk(\012)g Fo(A)j Fk(!)f Fo(K)17 b Fp(whic)o(h)d(is)f(asso)q(ciativ)o(e) h(in)257 781 y(the)f(sense)h(that)f(w)o(e)f(ha)o(v)o(e)g Fk(h)p Fo(aa)734 766 y Fj(0)746 781 y Fo(;)7 b(a)787 766 y Fj(0)o(0)808 781 y Fk(i)k Fp(=)h Fk(h)p Fo(a;)7 b(a)958 766 y Fj(0)969 781 y Fo(a)991 766 y Fj(00)1013 781 y Fk(i)12 b Fp(for)g(all)f Fo(a;)c(a)1222 766 y Fj(0)1245 781 y Fp(and)13 b Fo(a)1347 766 y Fj(0)o(0)1379 781 y Fk(2)e Fo(A)p Fp(.)h(Suc)o(h)h(a)f(form)257 831 y(ob)o(viously)h(can)h (b)q(e)h(written)f(as)836 922 y Fk(h)p Fo(a;)7 b(a)915 905 y Fj(0)926 922 y Fk(i)12 b Fp(=)g Fo(f)t Fp(\()p Fo(aa)1082 905 y Fj(0)1094 922 y Fp(\))257 1014 y(for)k(some)f(linear)g (form)f Fo(f)19 b Fp(:)c Fo(A)f Fk(!)h Fo(K)j Fp(whic)o(h)e(is)g (determined)g(b)o(y)f(the)h(bilinear)f(form)f(via)257 1064 y Fo(f)t Fp(\()p Fo(a)p Fp(\))g(=)f Fk(h)p Fo(a;)7 b Fp(1)p Fk(i)12 b Fp(=)h Fk(h)p Fp(1)p Fo(;)7 b(a)p Fk(i)p Fp(.)13 b(This)h(linear)g(form)f(is)h(called)h(the)g(F)m(rob)q (enius)g(homom)o(orphism)o(.)257 1113 y(W)m(e)e(note)h(that)g(this)f (notion)g(is)g(not)h(related)g(to)f(the)h(same)f(term)g(used)h(in)f (Galois)f(theory)m(.)257 1199 y(The)j(follo)o(wing)c(Prop)q(osition)i (w)o(as)h(explained)g(to)g(me)e(b)o(y)i(H.-J.)f(Sc)o(hneider:)257 1299 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)18 b(that)e Fo(A)h Fp(is)g(a)f(F)m(rob)q(enius)h(algebra)f(whic)o(h)g(is)h(augmen)o (ted)f(b)o(y)257 1348 y Fo(\017)h Fp(:)g Fo(A)h Fk(!)e Fo(K)s Fp(.)h(Then)h(the)g(space)h(of)d(left)h(in)o(tegrals)g Fk(f)p Fo(x)g Fk(2)g Fo(A)12 b Fk(j)f(8)p Fo(a)17 b Fk(2)g Fo(A)g Fp(:)g Fo(ax)g Fp(=)g Fo(\017)p Fp(\()p Fo(a)p Fp(\))p Fo(x)p Fk(g)p Fp(,)257 1398 y(and)h(also)f(the)i(space)f(of)g (righ)o(t)f(in)o(tegrals)g Fk(f)p Fo(x)h Fk(2)g Fo(A)12 b Fk(j)f(8)p Fo(a)18 b Fk(2)g Fo(A)g Fp(:)g Fo(xa)g Fp(=)g Fo(\017)p Fp(\()p Fo(a)p Fp(\))p Fo(x)p Fk(g)p Fp(,)f(is)g(one-)257 1448 y(dimensional.)257 1548 y Fn(Pro)q(of.)36 b Fp(Consider)14 b(the)h(left)e(coregular)h(action)f(of)h Fo(A)f Fp(on)h Fo(A)1225 1533 y Fj(\003)1244 1548 y Fp(:)f(An)h(elemen)o(t)f Fo(a)h Fp(of)f Fo(A)h Fp(acts)257 1597 y(on)j(an)h(elemen)o(t)e Fo(g)j Fp(of)e Fo(A)657 1582 y Fj(\003)693 1597 y Fp(yielding)f(the)i (elemen)o(t)f Fo(a)g Fk(!)g Fo(g)h Fp(of)f Fo(A)1302 1582 y Fj(\003)1339 1597 y Fp(whic)o(h)g(is)g(de\014ned)h(as)257 1647 y(\()p Fo(a)c Fk(!)f Fo(g)q Fp(\)\()p Fo(a)439 1632 y Fj(0)451 1647 y Fp(\))g(=)h Fo(g)q Fp(\()p Fo(a)585 1632 y Fj(0)597 1647 y Fo(a)p Fp(\).)g(One)i(can)f(restate)i(the)e (de\014nition)g(of)f(a)h(F)m(rob)q(enius)g(algebra)g(b)o(y)257 1697 y(sa)o(ying)e(that)h(the)h(mapping)776 1788 y Fo(A)d Fk(!)f Fo(A)903 1771 y Fj(\003)922 1788 y Fo(;)c(a)k Fk(7!)g Fp(\()p Fo(a)h Fk(!)f Fo(f)t Fp(\))257 1880 y(is)18 b(bijectiv)o(e,)f(that)h(is,)f Fo(A)665 1865 y Fj(\003)702 1880 y Fp(is)g(a)h(free)g(cyclic)g Fo(A)p Fp(-mo)q(dule)e(generated)j (b)o(y)f(the)g(F)m(rob)q(enius)257 1930 y(homomorphism)11 b Fo(f)t Fp(.)k(Our)h(assertion)g(will)e(b)q(e)h(pro)o(v)o(ed)h(if)e(w) o(e)h(can)h(sho)o(w)f(that)g Fo(x)g Fp(is)g(a)g(left)257 1979 y(in)o(tegral)f(if)f(and)g(only)g(if)g Fo(x)f Fk(!)f Fo(f)18 b Fp(is)c(a)g(m)o(ultiple)d(of)j Fo(\017)p Fp(.)f(But)h(observ) o(e)h(that:)311 2071 y Fk(9)p Fo(\025)c Fk(2)h Fo(K)i Fp(:)e Fo(x)f Fk(!)g Fo(f)16 b Fp(=)c Fo(\025\017)g Fk(,)f(9)p Fo(\025)g Fk(2)h Fo(K)g Fk(8)p Fo(a)g Fk(2)f Fo(A)g Fp(:)h Fo(f)t Fp(\()p Fo(ax)p Fp(\))g(=)g Fo(\025\017)p Fp(\()p Fo(a)p Fp(\))703 2133 y Fk(,)f(8)p Fo(a)g Fk(2)g Fo(A)h Fp(:)f Fo(f)t Fp(\()p Fo(ax)p Fp(\))h(=)g Fo(f)t Fp(\()p Fo(x)p Fp(\))p Fo(\017)p Fp(\()p Fo(a)p Fp(\))703 2195 y Fk(,)f(8)p Fo(a;)c(a)842 2178 y Fj(0)864 2195 y Fk(2)k Fo(A)h Fp(:)f Fo(f)t Fp(\()p Fo(aa)1053 2178 y Fj(0)1066 2195 y Fo(x)p Fp(\))g(=)h Fo(f)t Fp(\()p Fo(x)p Fp(\))p Fo(\017)p Fp(\()p Fo(aa)1318 2178 y Fj(0)1330 2195 y Fp(\))g(=)g Fo(f)t Fp(\()p Fo(x)p Fp(\))p Fo(\017)p Fp(\()p Fo(a)p Fp(\))p Fo(\017)p Fp(\()p Fo(a)1608 2178 y Fj(0)1620 2195 y Fp(\))703 2258 y Fk(,)f(8)p Fo(a;)c(a)842 2240 y Fj(0)864 2258 y Fk(2)k Fo(A)h Fp(:)f Fo(f)t Fp(\()p Fo(aa)1053 2240 y Fj(0)1066 2258 y Fo(x)p Fp(\))g(=)h Fo(f)t Fp(\()p Fo(a\017)p Fp(\()p Fo(a)1278 2240 y Fj(0)1290 2258 y Fp(\))p Fo(x)p Fp(\))703 2320 y Fk(,)f(8)p Fo(a)801 2303 y Fj(0)824 2320 y Fk(2)g Fo(A)h Fp(:)f Fo(a)951 2303 y Fj(0)962 2320 y Fo(x)h Fp(=)f Fo(\017)p Fp(\()p Fo(a)1096 2303 y Fj(0)1108 2320 y Fp(\))p Fo(x)257 2411 y Fp(The)j(assertion)f(on)g(righ)o(t)f(in)o(tegrals)g(follo)o(ws)g(b)o (y)g(considering)h(the)g(opp)q(osite)g(algebra)g Fo(A)1654 2396 y Fm(op)257 2461 y Fp(instead)i(of)e Fo(A)p Fp(.)g Fg(\003)963 2628 y Fp(7)p eop %%Page: 8 8 8 7 bop 257 262 a Fp(Since)17 b(the)h(form)c Fk(h\001)p Fo(;)7 b Fk(\001i)16 b Fp(is)g(nondegenerate,)i(w)o(e)e(can)h(c)o(ho)q (ose)g(dual)f(bases)i Fo(x)1474 269 y Fl(\(1\))1518 262 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(x)1642 269 y Fl(\()p Fm(n)p Fl(\))257 311 y Fp(and)14 b Fo(y)358 318 y Fl(\(1\))403 311 y Fo(;)7 b(:)g(:)g(:)k(;)c(y)522 318 y Fl(\()p Fm(n)p Fl(\))584 311 y Fp(satisfying)13 b Fk(h)p Fo(y)805 318 y Fl(\()p Fm(i)p Fl(\))845 311 y Fo(;)7 b(x)888 318 y Fl(\()p Fm(j)r Fl(\))931 311 y Fk(i)12 b Fp(=)f Fo(\016)1020 317 y Fm(ij)1050 311 y Fp(.)i(F)m(rom)f(linear)h(algebra)g(w)o(e)h(kno) o(w)f(that)257 386 y(w)o(e)i(ha)o(v)o(e)f Fo(a)e Fp(=)510 347 y Fm(n)498 355 y Fh(P)493 423 y Fm(i)p Fl(=1)547 386 y Fk(h)p Fo(a;)7 b(x)628 393 y Fl(\()p Fm(i)p Fl(\))667 386 y Fk(i)p Fo(y)703 393 y Fl(\()p Fm(i)p Fl(\))755 386 y Fp(=)816 347 y Fm(n)804 355 y Fh(P)799 423 y Fm(i)p Fl(=1)853 386 y Fk(h)p Fo(y)889 393 y Fl(\()p Fm(i)p Fl(\))929 386 y Fo(;)g(a)p Fk(i)p Fo(x)1010 393 y Fl(\()p Fm(i)p Fl(\))1064 386 y Fp(for)13 b(all)g Fo(a)f Fk(2)g Fo(A)p Fp(.)i(This)g(implies)e(that)i(w)o(e)257 460 y(ha)o(v)o(e:)702 481 y Fm(n)683 493 y Fh(X)686 582 y Fm(i)p Fl(=1)749 533 y Fo(ax)795 540 y Fl(\()p Fm(i)p Fl(\))844 533 y Fk(\012)c Fo(y)906 540 y Fl(\()p Fm(i)p Fl(\))957 533 y Fp(=)1021 481 y Fm(n)1001 493 y Fh(X)1004 582 y Fm(i)p Fl(=1)1068 533 y Fo(x)1092 540 y Fl(\()p Fm(i)p Fl(\))1141 533 y Fk(\012)f Fo(y)1202 540 y Fl(\()p Fm(i)p Fl(\))1242 533 y Fo(a)257 671 y Fp(The)15 b(elemen)o(t)513 632 y Fm(n)501 640 y Fh(P)496 708 y Fm(i)p Fl(=1)557 671 y Fo(x)581 678 y Fl(\()p Fm(i)p Fl(\))630 671 y Fk(\012)10 b Fo(y)692 678 y Fl(\()p Fm(i)p Fl(\))746 671 y Fp(is)k(therefore)h (called)f(the)h(Casimir)d(elemen)o(t)h(of)h Fo(A)p Fp(.)g(It)g(do)q(es) 257 745 y(not)f(dep)q(end)h(on)f(the)h(c)o(hoice)f(of)g(the)g(dual)f (bases,)i(but)f(of)f(course)j(it)d(do)q(es)i(dep)q(end)g(on)f(the)257 794 y(bilinear)g(form.)257 930 y Fn(2.8)48 b Fp(W)m(e)13 b(no)o(w)g(pro)q(ceed)j(to)d(pro)o(v)o(e)h(the)g(existence)h(a)f (non-zero)g(in)o(tegral)f(in)g(ev)o(ery)h(\014nite)257 980 y(dimensional)c(Y)m(etter-Drinfel'd)i(Hopf)g(algebra)f(strictly)i (along)d(the)j(lines)f(of)g(the)g(original)257 1029 y(argumen)o(t)19 b(of)g(R.)g(Larson)g(and)h(M.)f(Sw)o(eedler)i(\(cf.)e([12)o(],)g(see)i (also)e([19)o(]\).)g(It)g(m)o(ust)g(b)q(e)257 1079 y(emphasized)i(that) f(this)h(result)g(is)f(not)h(new,)f(but)h(is)f(a)h(sp)q(ecial)f(case)i (of)e(a)g(Theorem)257 1129 y(of)c(D.)g(Fisc)o(hman,)e(S.)i(Mon)o (tgomery)f(and)h(H.-J.)g(Sc)o(hneider)i(on)e(the)h(one)f(hand)h(and)f (V.)257 1179 y(Lyubashenk)o(o)g(on)g(the)g(other)g(hand)g(\(cf.)g([5)o (],)f([16)o(],)f([17)o(]\).)h(W)m(e)h(include)f(the)i(result)f(here)257 1229 y(for)i(the)h(sak)o(e)f(of)f(completeness)i(and)f(b)q(ecause)i (the)e(argumen)o(ts)g(of)f(the)i(pro)q(of)e(will)g(b)q(e)257 1279 y(needed)f(later)e(on.)257 1364 y(The)k(basic)g(ingredien)o(t)f (of)g(the)h(pro)q(of)f(is)g(the)h(structure)i(theorem)d(for)g(Hopf)g (mo)q(dules.)257 1414 y(Recall)12 b(the)h(notion)e(of)h(a)g(Hopf)g(mo)q (dule)f(in)h(the)h(category)f(of)g(Y)m(etter-Drinfel'd)g(mo)q(dules:) 257 1514 y Fn(De\014nition)33 b Fp(Supp)q(ose)16 b(that)f Fo(A)g Fp(is)g(a)f(left)h(Y)m(etter-Drinfel'd)f(Hopf)h(algebra)f(o)o(v) o(er)h Fo(H)s Fp(.)f(A)257 1563 y(righ)o(t)i(Y)m(etter-Drinfel'd)g (Hopf)g(mo)q(dule)f(is)h(a)g(left)g(Y)m(etter-Drinfel'd)g(mo)q(dule)f Fo(M)21 b Fp(whic)o(h)257 1613 y(is)14 b(a)g(righ)o(t)f Fo(A)p Fp(-mo)q(dule)g(via:)807 1663 y Fo(\036)832 1669 y Fm(M)880 1663 y Fp(:)f Fo(M)i Fk(\012)9 b Fo(A)j Fk(!)f Fo(M)257 1738 y Fp(and)j(a)g(righ)o(t)f Fo(A)p Fp(-como)q(dule)g(via:) 810 1829 y Fo(\016)828 1835 y Fm(M)877 1829 y Fp(:)e Fo(M)17 b Fk(!)11 b Fo(M)j Fk(\012)9 b Fo(A)257 1920 y Fp(suc)o(h)16 b(that)f Fo(\036)468 1926 y Fm(M)520 1920 y Fp(and)f Fo(\016)619 1926 y Fm(M)672 1920 y Fp(are)h Fo(H)s Fp(-linear)f(and)h Fo(H)s Fp(-colinear)g(and)f(the)i(follo)o (wing)c(compati-)257 1970 y(bilit)o(y)h(condition)g(is)h(satis\014ed:) 418 2061 y(\()p Fo(m)e Fg(\021)f Fo(a)p Fp(\))573 2044 y Fl(\(1\))627 2061 y Fk(\012)e Fp(\()p Fo(m)j Fg(\021)f Fo(a)p Fp(\))823 2044 y Fl(\(2\))880 2061 y Fp(=)g(\()p Fo(m)975 2044 y Fl(\(1\))1032 2061 y Fg(\021)g Fp(\()p Fo(m)1137 2044 y Fl(\(2\))r(1)1212 2061 y Fk(!)g Fo(a)1287 2067 y Fl(1)1306 2061 y Fp(\)\))f Fk(\012)f Fo(m)1425 2044 y Fl(\(2\))r(2)1489 2061 y Fo(a)1511 2067 y Fl(2)257 2153 y Fp(for)14 b(all)f Fo(m)f Fk(2)f Fo(M)18 b Fp(and)c Fo(a)e Fk(2)f Fo(A)p Fp(,)i(where)i(w)o(e)f(ha)o(v)o(e)g(used)h(the)f (notation:)545 2244 y Fo(\036)570 2250 y Fm(M)607 2244 y Fp(\()p Fo(m)c Fk(\012)f Fo(a)p Fp(\))j(=)g Fo(m)g Fg(\021)f Fo(a)83 b(\016)1028 2250 y Fm(M)1065 2244 y Fp(\()p Fo(m)p Fp(\))12 b(=)g Fo(m)1225 2227 y Fl(\(1\))1279 2244 y Fk(\012)e Fo(m)1357 2227 y Fl(\(2\))257 2371 y Fp(The)17 b(coaction)g Fo(\016)530 2377 y Fm(M)583 2371 y Fp(should)f(b)q(e)h(distinguished)g(from)d(the)k(coaction)e Fo(\016)i Fp(:)d Fo(M)20 b Fk(!)c Fo(H)d Fk(\012)f Fo(M)257 2421 y Fp(whic)o(h)j(comes)g(from)e(the)i(Y)m(etter-Drinfel'd)g (structure.)i(Examples)c(of)i(Y)m(etter-Drinfel'd)963 2628 y(8)p eop %%Page: 9 9 9 8 bop 257 262 a Fp(Hopf)19 b(mo)q(dules)f(can)h(b)q(e)g(easily)f (obtained)h(in)f(the)i(follo)o(wing)c(w)o(a)o(y:)i(If)g Fo(A)h Fp(is)g(a)g(Y)m(etter-)257 311 y(Drinfel'd)12 b(Hopf)g(algebra)g(and)h Fo(V)22 b Fp(is)12 b(an)o(y)h(Y)m (etter-Drinfel'd)f(mo)q(dule,)f(then)i Fo(M)k Fp(:=)11 b Fo(V)16 b Fk(\012)7 b Fo(A)257 361 y Fp(b)q(ecomes)12 b(a)g(Y)m(etter-Drinfel'd)g(Hopf)f(mo)q(dule)f(with)i(resp)q(ect)i(to)e (the)g(structure)i(elemen)o(ts:)532 452 y Fo(\016)550 458 y Fm(M)599 452 y Fp(:)d Fo(V)19 b Fk(\012)9 b Fo(A)j Fk(!)f Fp(\()p Fo(V)19 b Fk(\012)9 b Fo(A)p Fp(\))h Fk(\012)f Fo(A;)e(v)k Fk(\012)e Fo(a)j Fk(7!)f Fo(v)g Fk(\012)e Fo(a)1303 458 y Fl(1)1331 452 y Fk(\012)g Fo(a)1394 458 y Fl(2)526 515 y Fo(\036)551 521 y Fm(M)599 515 y Fp(:)i(\()p Fo(V)19 b Fk(\012)9 b Fo(A)p Fp(\))h Fk(\012)f Fo(A)j Fk(!)f Fo(V)19 b Fk(\012)9 b Fo(A;)e(v)k Fk(\012)e Fo(a)g Fk(\012)h Fo(a)1217 498 y Fj(0)1240 515 y Fk(7!)h Fo(v)g Fk(\012)e Fo(aa)1409 498 y Fj(0)257 606 y Fp(The)18 b(structure)i (theorem)d(for)g(Hopf)g(mo)q(dules)f(no)o(w)h(asserts)i(that)e(this)h (is)f(already)g(the)257 656 y(general)d(form)f(of)g(a)g(Y)m (etter-Drinfel'd)h(Hopf)g(mo)q(dule:)257 756 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)17 b(that)g Fo(A)f Fp(is)g(a)g(Y)m(etter-Drinfel'd)g (Hopf)g(algebra)g(o)o(v)o(er)g Fo(H)k Fp(and)257 805 y(that)g Fo(M)j Fp(is)c(a)g(righ)o(t)g(Y)m(etter-Drinfel'd)g(Hopf)g(mo) q(dule)e(o)o(v)o(er)i Fo(A)p Fp(.)g(De\014ne)h(the)g(space)g(of)257 855 y(coin)o(v)n(arian)o(ts:)596 905 y Fo(V)h Fp(:=)12 b Fo(M)742 888 y Fm(coA)811 905 y Fp(=)g Fk(f)p Fo(m)g Fk(2)f Fo(M)16 b Fk(j)11 b Fo(\016)1060 911 y Fm(M)1098 905 y Fp(\()p Fo(m)p Fp(\))h(=)g Fo(m)d Fk(\012)h Fp(1)p Fk(g)257 980 y Fp(Then)15 b Fo(V)23 b Fp(is)14 b(a)f(Y)m (etter-Drinfel'd)h(submo)q(dule)f(of)g Fo(M)19 b Fp(and)646 1071 y Fo(p)12 b Fp(:)f Fo(M)16 b Fk(!)11 b Fo(V)r(;)c(m)12 b Fk(7!)f Fp(\()p Fo(m)1009 1054 y Fl(\(1\))1066 1071 y Fg(\021)g Fo(S)1144 1077 y Fm(A)1171 1071 y Fp(\()p Fo(m)1223 1054 y Fl(\(2\))1268 1071 y Fp(\)\))257 1162 y(is)j(a)g(pro)r(jection)g(on)o(to)f Fo(V)d Fp(.)j(F)m(urthermore,)g (the)i(mappings)639 1254 y Fo(f)h Fp(:)11 b Fo(V)19 b Fk(\012)9 b Fo(A)j Fk(!)f Fo(M)5 b(;)i(v)j Fk(\012)g Fo(a)h Fk(7!)g Fp(\()p Fo(v)i Fg(\021)e Fo(a)p Fp(\))642 1324 y Fo(g)i Fp(:)e Fo(M)16 b Fk(!)11 b Fo(V)19 b Fk(\012)10 b Fo(A;)d(m)k Fk(7!)g Fo(p)p Fp(\()p Fo(m)1115 1307 y Fl(\(1\))1160 1324 y Fp(\))f Fk(\012)f Fo(m)1263 1307 y Fl(\(2\))257 1415 y Fp(are)18 b(linear)f(and)g(colinear)g(with)g (resp)q(ect)j(to)d Fo(A)g Fp(and)g Fo(H)j Fp(and)d(are)h(m)o(utually)d (in)o(v)o(erse)j(to)257 1465 y(eac)o(h)d(other.)257 1565 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(\014rst)g(pro)o(v)o(e)g(that)g Fo(p)g Fp(really)f(maps)g(to)h(the)g(coin)o(v)n(arian)o(ts:)340 1656 y Fo(\016)358 1662 y Fm(M)395 1656 y Fp(\()p Fo(p)p Fp(\()p Fo(a)p Fp(\)\))e(=)g Fo(\016)576 1662 y Fm(M)613 1656 y Fp(\()p Fo(m)665 1639 y Fl(\(1\))722 1656 y Fg(\021)f Fo(S)800 1662 y Fm(A)827 1656 y Fp(\()p Fo(m)879 1639 y Fl(\(2\))925 1656 y Fp(\)\))514 1727 y(=)h(\()p Fo(m)610 1710 y Fl(\(1\))667 1727 y Fg(\021)f Fp(\()p Fo(m)772 1710 y Fl(\(2\))r(1)847 1727 y Fk(!)g Fo(S)925 1733 y Fm(A)952 1727 y Fp(\()p Fo(m)1004 1710 y Fl(\(3\))1050 1727 y Fp(\))1066 1733 y Fl(1)1084 1727 y Fp(\)\))f Fk(\012)f Fo(m)1203 1710 y Fl(\(2\))r(2)1267 1727 y Fo(S)1292 1733 y Fm(A)1320 1727 y Fp(\()p Fo(m)1372 1710 y Fl(\(3\))1417 1727 y Fp(\))1433 1733 y Fl(2)514 1797 y Fp(=)j(\()p Fo(m)610 1780 y Fl(\(1\))667 1797 y Fg(\021)f Fp(\()p Fo(S)761 1803 y Fm(A)788 1797 y Fp(\()p Fo(m)840 1780 y Fl(\(2\))r(1)904 1797 y Fo(m)940 1780 y Fl(\(3\))r(1)1015 1797 y Fk(!)g Fo(m)1104 1780 y Fl(\(4\))1149 1797 y Fp(\)\)\))f Fk(\012)f Fo(m)1284 1780 y Fl(\(2\))r(2)1348 1797 y Fo(S)1373 1803 y Fm(A)1401 1797 y Fp(\()p Fo(m)1453 1780 y Fl(\(3\))r(2)1516 1797 y Fp(\))514 1868 y(=)j(\()p Fo(m)610 1851 y Fl(\(1\))667 1868 y Fg(\021)f Fp(\()p Fo(S)761 1874 y Fm(A)788 1868 y Fp(\()p Fo(m)840 1851 y Fl(\(2\))885 1874 y(1)904 1851 y(1)923 1868 y Fo(m)959 1851 y Fl(\(2\))1004 1874 y(2)1022 1851 y(1)1052 1868 y Fk(!)h Fo(m)1142 1851 y Fl(\(3\))1187 1868 y Fp(\)\)\))d Fk(\012)h Fo(m)1322 1851 y Fl(\(2\))1367 1874 y(1)1385 1851 y(2)1404 1868 y Fo(S)1429 1874 y Fm(A)1456 1868 y Fp(\()p Fo(m)1508 1851 y Fl(\(2\))1554 1874 y(2)1572 1851 y(2)1591 1868 y Fp(\))514 1938 y(=)i(\()p Fo(m)610 1921 y Fl(\(1\))667 1938 y Fg(\021)f Fp(\()p Fo(S)761 1944 y Fm(A)788 1938 y Fp(\()p Fo(m)840 1921 y Fl(\(2\))r(1)916 1938 y Fk(!)g Fo(m)1005 1921 y Fl(\(3\))1050 1938 y Fp(\)\)\))e Fk(\012)h Fo(m)1185 1921 y Fl(\(2\))r(2)1248 1944 y(1)1267 1938 y Fo(S)1292 1944 y Fm(A)1320 1938 y Fp(\()p Fo(m)1372 1921 y Fl(\(2\))r(2)1435 1944 y(2)1454 1938 y Fp(\))514 2009 y(=)i(\()p Fo(m)610 1991 y Fl(\(1\))667 2009 y Fg(\021)f Fp(\()p Fo(S)761 2015 y Fm(A)788 2009 y Fp(\()p Fo(m)840 1991 y Fl(\(2\))r(1)916 2009 y Fk(!)g Fo(m)1005 1991 y Fl(\(3\))1050 2009 y Fp(\)\)\))e Fk(\012)h Fo(\017)1166 2015 y Fm(A)1193 2009 y Fp(\()p Fo(m)1245 1991 y Fl(\(2\))r(2)1308 2009 y Fp(\)1)514 2079 y(=)i(\()p Fo(m)610 2062 y Fl(\(1\))667 2079 y Fg(\021)f Fo(S)745 2085 y Fm(A)772 2079 y Fp(\()p Fo(m)824 2062 y Fl(\(2\))869 2079 y Fp(\)\))f Fk(\012)f Fp(1)257 2170 y(W)m(e)i(no)o(w)g(pro)o(v)o(e)g(that)h Fo(V)20 b Fp(is)11 b(an)g Fo(H)s Fp(-sub)q(como)q(dule)g(of)f Fo(M)5 b Fp(.)11 b(Since)g(w)o(e)h(ha)o(v)o(e)f Fo(\016)1431 2176 y Fm(M)1468 2170 y Fp(\()p Fo(m)p Fp(\))h(=)g Fo(m)t Fk(\012)t Fp(1)257 2220 y(for)i Fo(m)e Fk(2)f Fo(V)f Fp(,)j(w)o(e)h(can)g(compute:)428 2312 y Fo(m)464 2294 y Fl(1)492 2312 y Fk(\012)9 b Fo(\016)551 2318 y Fm(M)589 2312 y Fp(\()p Fo(m)641 2294 y Fl(2)660 2312 y Fp(\))j(=)f Fo(m)767 2294 y Fl(\(1\))r(1)831 2312 y Fo(m)867 2294 y Fl(\(2\))r(1)940 2312 y Fk(\012)e Fo(m)1017 2294 y Fl(\(1\))r(2)1090 2312 y Fk(\012)h Fo(m)1168 2294 y Fl(\(2\))r(2)1243 2312 y Fp(=)i Fo(m)1323 2294 y Fl(1)1351 2312 y Fk(\012)e Fo(m)1429 2294 y Fl(2)1457 2312 y Fk(\012)f Fp(1)257 2403 y(W)m(riting)i Fo(m)444 2388 y Fl(1)470 2403 y Fk(\012)6 b Fo(m)544 2388 y Fl(2)575 2403 y Fp(=)619 2372 y Fh(P)663 2415 y Fm(b)p Fj(2)p Fm(B)735 2403 y Fo(b)g Fk(\012)g Fo(x)821 2409 y Fm(b)851 2403 y Fp(for)12 b(some)g(basis)g Fo(B)j Fp(of)d Fo(H)s Fp(,)g(w)o(e)h(see)g(that)g Fo(x)1508 2409 y Fm(b)1536 2403 y Fk(2)e Fo(M)1620 2388 y Fm(coA)1678 2403 y Fp(.)257 2453 y(In)h(a)g(similar)d(fashion,)i(it)g(is)h(p)q (ossible)g(to)g(pro)o(v)o(e)g(that)f Fo(V)22 b Fp(is)11 b(an)h Fo(H)s Fp(-submo)q(dule)f(of)g Fo(M)5 b Fp(.)11 b(The)257 2503 y(Y)m(etter-Drinfel'd)k(condition)f(therefore)i(holds)e (for)h Fo(V)24 b Fp(since)15 b(it)g(holds)f(for)g Fo(M)5 b Fp(.)14 b(Next,)h(w)o(e)963 2628 y(9)p eop %%Page: 10 10 10 9 bop 257 262 a Fp(observ)o(e)15 b(that)e(the)h(mappings)d Fo(p)p Fp(,)i Fo(f)k Fp(and)d Fo(g)g Fp(are)g(linear)e(and)h(colinear)g (with)g(resp)q(ect)j(to)d Fo(H)257 311 y Fp(b)q(ecause)h(they)f(are)f (comp)q(ositions)e(of)i Fo(H)s Fp(-linear)f(and)h(colinear)f(maps.)g(W) m(e)g(pro)o(v)o(e)h(that)g(w)o(e)257 361 y(ha)o(v)o(e)i Fo(p)p Fp(\()p Fo(m)e Fg(\021)f Fo(a)p Fp(\))h(=)g Fo(\017)602 367 y Fm(A)628 361 y Fp(\()p Fo(a)p Fp(\))p Fo(p)p Fp(\()p Fo(m)p Fp(\))j(for)f Fo(m)e Fk(2)f Fo(M)5 b Fp(,)13 b Fo(a)e Fk(2)h Fo(A)p Fp(:)429 452 y Fo(p)p Fp(\()p Fo(m)g Fg(\021)f Fo(a)p Fp(\))h(=)g(\()p Fo(m)g Fg(\021)f Fo(a)p Fp(\))816 435 y Fl(\(1\))872 452 y Fg(\021)g Fo(S)950 458 y Fm(A)978 452 y Fp(\(\()p Fo(m)h Fg(\021)f Fo(a)p Fp(\))1149 435 y Fl(\(2\))1194 452 y Fp(\))617 523 y(=)h(\()p Fo(m)713 506 y Fl(\(1\))770 523 y Fg(\021)f Fp(\()p Fo(m)875 506 y Fl(\(2\))r(1)950 523 y Fk(!)g Fo(a)1025 529 y Fl(1)1044 523 y Fp(\)\))g Fg(\021)g Fo(S)1165 529 y Fm(A)1193 523 y Fp(\()p Fo(m)1245 506 y Fl(\(2\))r(2)1309 523 y Fo(a)1331 529 y Fl(2)1349 523 y Fp(\))617 593 y(=)h Fo(m)697 576 y Fl(\(1\))753 593 y Fg(\021)f Fp(\()p Fo(m)858 576 y Fl(\(2\))s(1)934 593 y Fk(!)g Fo(a)1009 599 y Fl(1)1027 593 y Fp(\))p Fo(S)1068 599 y Fm(A)1096 593 y Fp(\()p Fo(m)1148 576 y Fl(\(2\))r(2)1223 593 y Fk(!)g Fo(a)1298 599 y Fl(2)1317 593 y Fp(\))p Fo(S)1358 599 y Fm(A)1386 593 y Fp(\()p Fo(m)1438 576 y Fl(\(2\))r(3)1501 593 y Fp(\))617 664 y(=)h Fo(m)697 647 y Fl(\(1\))753 664 y Fg(\021)f Fp(\()p Fo(m)858 647 y Fl(\(2\))s(1)934 664 y Fk(!)g Fp(\()p Fo(a)1025 670 y Fl(1)1044 664 y Fo(S)1069 670 y Fm(A)1096 664 y Fp(\()p Fo(a)1134 670 y Fl(2)1153 664 y Fp(\)\)\))p Fo(S)1226 670 y Fm(A)1254 664 y Fp(\()p Fo(m)1306 647 y Fl(\(2\))r(2)1369 664 y Fp(\))617 734 y(=)h Fo(\017)678 740 y Fm(A)705 734 y Fp(\()p Fo(a)p Fp(\))p Fo(m)795 717 y Fl(\(1\))852 734 y Fg(\021)f Fo(S)930 740 y Fm(A)957 734 y Fp(\()p Fo(m)1009 717 y Fl(\(2\))1054 734 y Fp(\))257 826 y(W)m(e)j(no)o(w)g(pro)o(v)o(e)h(that)f Fo(f)19 b Fp(and)c Fo(g)g Fp(are)g(m)o(utually)d(in)o(v)o(erse.)i(It)h (is)f(ob)o(vious)g(that)g Fo(f)g Fk(\016)c Fo(g)j Fp(=)g Fo(id)p Fp(.)257 876 y(On)i(the)f(other)h(hand,)e(w)o(e)h(ha)o(v)o(e)g (b)o(y)f(the)i(preceding)g(calculation:)268 967 y(\()p Fo(g)c Fk(\016)e Fo(f)t Fp(\)\()p Fo(v)j Fk(\012)d Fo(a)p Fp(\))j(=)f Fo(p)p Fp(\(\()p Fo(v)j Fg(\021)d Fo(a)p Fp(\))745 950 y Fl(\(1\))789 967 y Fp(\))f Fk(\012)f Fp(\()p Fo(v)14 b Fg(\021)d Fo(a)p Fp(\))997 950 y Fl(\(2\))1053 967 y Fp(=)h Fo(p)p Fp(\()p Fo(v)1155 950 y Fl(\(1\))1211 967 y Fg(\021)f Fp(\()p Fo(v)1301 950 y Fl(\(2\))s(1)1377 967 y Fk(!)g Fo(a)1452 973 y Fl(1)1471 967 y Fp(\)\))e Fk(\012)h Fo(v)1575 950 y Fl(\(2\))r(2)1638 967 y Fo(a)1660 973 y Fl(2)801 1037 y Fp(=)i Fo(p)p Fp(\()p Fo(v)903 1020 y Fl(\(1\))948 1037 y Fp(\))p Fo(\017)981 1043 y Fm(A)1008 1037 y Fp(\()p Fo(v)1045 1020 y Fl(\(2\))r(1)1121 1037 y Fk(!)f Fo(a)1196 1043 y Fl(1)1214 1037 y Fp(\))e Fk(\012)h Fo(v)1302 1020 y Fl(\(2\))r(2)1366 1037 y Fo(a)1388 1043 y Fl(2)801 1108 y Fp(=)i Fo(p)p Fp(\()p Fo(v)903 1091 y Fl(\(1\))948 1108 y Fp(\))d Fk(\012)h Fo(v)1036 1091 y Fl(\(2\))1081 1108 y Fo(a)i Fp(=)f Fo(p)p Fp(\()p Fo(v)q Fp(\))f Fk(\012)g Fo(a)257 1199 y Fp(It)20 b(is)g(ob)o(vious)f (that)h Fo(f)25 b Fp(is)20 b Fo(A)p Fp(-linear,)f(and)g(therefore)j (its)e(in)o(v)o(erse)g Fo(g)h Fp(is)f(also)f Fo(A)p Fp(-linear.)257 1249 y(Similarly)m(,)10 b(its)k(ob)o(vious)f(that)h Fo(g)h Fp(is)f Fo(A)p Fp(-colinear,)f(and)g(therefore)j Fo(f)i Fp(is)c(also)f Fo(A)p Fp(-colinear.)g Fg(\003)257 1384 y Fn(2.9)48 b Fp(W)m(e)11 b(shall)g(no)o(w)g(see)i(that)e(the)h(dual)f (v)o(ector)h(space)h(of)e(a)g(\014nite)g(dimensional)f(Y)m(etter-)257 1434 y(Drinfel'd)j(Hopf)g(algebra)h(is)g(a)f(Y)m(etter-Drinfel'd)h (Hopf)f(mo)q(dule:)257 1534 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)e Fo(A)h Fp(is)f(a)g(\014nite)h(dimensional)d (Y)m(etter-Drinfel'd)i(Hopf)g(al-)257 1584 y(gebra.)19 b(Cho)q(ose)g(a)f(basis)h Fo(a)704 1591 y Fl(\(1\))749 1584 y Fo(;)7 b(:)g(:)g(:)k(;)c(a)870 1591 y Fl(\()p Fm(n)p Fl(\))937 1584 y Fp(of)18 b Fo(A)h Fp(and)g(denote)g(the)g (corresp)q(onding)h(dual)257 1639 y(basis)14 b(b)o(y)g Fo(a)439 1624 y Fl(\(1\))q Fj(\003)503 1639 y Fo(;)7 b(:)g(:)g(:)k(;)c(a)624 1624 y Fl(\()p Fm(n)p Fl(\))q Fj(\003)691 1639 y Fp(.)308 1758 y(1.)20 b(The)15 b(dual)f(v)o(ector)h (space)g Fo(A)807 1743 y Fj(\003)841 1758 y Fp(b)q(ecomes)f(a)g(left)g (Y)m(etter-Drinfel'd)g(mo)q(dule)f(o)o(v)o(er)i Fo(H)361 1808 y Fp(b)o(y)f(the)g(co)q(op)q(eration:)685 1930 y Fo(\016)705 1913 y Fm(H)703 1940 y(A)728 1932 y Fi(\003)748 1930 y Fp(\()p Fo(f)t Fp(\))e(=)880 1878 y Fm(n)860 1891 y Fh(X)863 1979 y Fm(i)p Fl(=1)927 1930 y Fo(S)952 1936 y Fm(H)984 1930 y Fp(\()p Fo(a)1022 1937 y Fl(\()p Fm(i)p Fl(\))1062 1913 y(1)1081 1930 y Fp(\))d Fk(\012)g Fo(f)t Fp(\()p Fo(a)1209 1937 y Fl(\()p Fm(i)p Fl(\))1250 1913 y(2)1269 1930 y Fp(\))p Fo(a)1307 1913 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)361 2058 y Fp(and)14 b(the)g(op)q(eration:)762 2149 y(\()p Fo(h)d Fk(!)g Fo(f)t Fp(\)\()p Fo(a)p Fp(\))i(=)f Fo(f)t Fp(\()p Fo(S)1084 2131 y Fj(\000)p Fl(1)1082 2161 y Fm(H)1130 2149 y Fp(\()p Fo(h)p Fp(\))g Fk(!)f Fo(a)p Fp(\))361 2240 y(for)j Fo(f)i Fk(2)11 b Fo(A)531 2225 y Fj(\003)550 2240 y Fp(,)j Fo(a)d Fk(2)g Fo(A)j Fp(and)g Fo(h)e Fk(2)f Fo(H)s Fp(.)308 2324 y(2.)20 b Fo(A)392 2308 y Fj(\003)425 2324 y Fp(is)14 b(a)g(righ)o(t)f(Y)m (etter-Drinfel'd)h(Hopf)f(mo)q(dule)f(with)i(the)h(action:)410 2446 y Fo(\036)435 2452 y Fm(A)460 2444 y Fi(\003)491 2446 y Fp(:)c Fo(A)545 2429 y Fj(\003)574 2446 y Fk(\012)e Fo(A)j Fk(!)f Fo(A)742 2429 y Fj(\003)761 2446 y Fo(;)c(f)13 b Fk(\012)d Fo(a)h Fk(7!)h Fp(\()p Fo(f)k Fg(\021)11 b Fo(a)p Fp(\))h(=)1160 2394 y Fm(n)1140 2406 y Fh(X)1144 2495 y Fm(i)p Fl(=1)1207 2446 y Fo(a)1229 2429 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1288 2446 y Fo(f)1312 2429 y Fl(2)1332 2446 y Fp(\()p Fo(a)1370 2453 y Fl(\()p Fm(i)p Fl(\))1409 2446 y Fo(S)1434 2452 y Fm(A)1462 2446 y Fp(\()p Fo(f)1502 2429 y Fl(1)1533 2446 y Fk(!)f Fo(a)p Fp(\)\))953 2628 y(10)p eop %%Page: 11 11 11 10 bop 361 262 a Fp(and)14 b(the)g(coaction:)469 370 y Fo(\016)489 353 y Fm(A)487 380 y(A)512 372 y Fi(\003)543 370 y Fp(:)d Fo(A)597 353 y Fj(\003)628 370 y Fk(!)g Fo(A)712 353 y Fj(\003)740 370 y Fk(\012)f Fo(A;)d(f)16 b Fk(7!)11 b Fo(f)945 353 y Fl(\(1\))999 370 y Fk(\012)f Fo(f)1065 353 y Fl(\(2\))1122 370 y Fp(=)1185 318 y Fm(n)1166 331 y Fh(X)1169 419 y Fm(i)p Fl(=1)1233 370 y Fo(a)1255 353 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)1323 370 y Fk(\012)f Fo(a)1386 377 y Fl(\()p Fm(i)p Fl(\))1426 376 y(1)1444 370 y Fo(f)t Fp(\()p Fo(a)1506 377 y Fl(\()p Fm(i)p Fl(\))1547 376 y(2)1566 370 y Fp(\))257 502 y Fn(Pro)q(of.)36 b Fp(That)18 b Fo(A)562 487 y Fj(\003)600 502 y Fp(b)q(ecomes)g(a)g(left) g(Y)m(etter-Drinfel'd)g(mo)q(dule)f(in)g(this)i(w)o(a)o(y)e(follo)o(ws) 257 552 y(from)e(subsections)j(2.3)e(and)g(2.5,)f(b)q(ecause)j(the)f (dual)f(v)o(ector)h(space)h(is)e(a)h(righ)o(t)f(Y)m(etter-)257 602 y(Drinfel'd)d(mo)q(dule)g(o)o(v)o(er)h Fo(H)j Fp(and)e(therefore)g (also)f(a)g(left)g(Y)m(etter-Drinfel'd)g(mo)q(dule)e(o)o(v)o(er)257 652 y Fo(H)295 637 y Fm(op)j(cop)393 652 y Fp(.)c(The)h(sp)q(eci\014ed) i(structure)g(no)o(w)d(arises)h(from)e(applying)h(the)h(isomorphism)c Fo(S)1634 658 y Fm(H)1678 652 y Fp(:)257 702 y Fo(H)295 687 y Fm(op)15 b(cop)405 702 y Fk(!)c Fo(H)s Fp(.)i(T)m(o)g(pro)o(v)o (e)h(the)g(second)h(statemen)o(t,)f(w)o(e)g(note)g(\014rst)h(that)f (the)g(mappings)619 787 y Fo(ev)f Fp(:)f Fo(A)d Fk(\012)g Fo(A)807 770 y Fj(\003)838 787 y Fk(!)i Fo(K)s(;)c(a)i Fk(\012)h Fo(f)16 b Fk(7!)11 b Fo(f)t Fp(\()p Fo(a)p Fp(\))621 887 y Fo(db)f Fp(:)i Fo(K)i Fk(!)d Fo(A)828 870 y Fj(\003)857 887 y Fk(\012)e Fo(A;)e(\025)12 b Fk(7!)f Fo(\025)1087 836 y Fm(n)1068 848 y Fh(X)1071 936 y Fm(i)p Fl(=1)1135 887 y Fo(a)1157 870 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)1225 887 y Fk(\012)e Fo(a)1288 894 y Fl(\()p Fm(i)p Fl(\))257 1009 y Fp(are)20 b(linear)e(and)g(colinear)h(o)o(v)o (er)g Fo(H)s Fp(,)f(as)h(one)g(can)g(v)o(erify)f(b)o(y)h(direct)g (calculation.)f(This)257 1059 y(implies)12 b(that)h Fo(\036)512 1065 y Fm(A)537 1057 y Fi(\003)569 1059 y Fp(and)h Fo(\016)670 1044 y Fm(A)668 1071 y(A)693 1062 y Fi(\003)726 1059 y Fp(are)g(also)e(linear)h(and)g(colinear)g(o)o(v)o(er)h Fo(H)s Fp(,)e(since)i(they)g(can)g(b)q(e)257 1109 y(written)h(as)f(the) g(comp)q(osition)e(of)i Fo(H)s Fp(-linear)f(and)g(colinear)h(maps:)373 1195 y Fo(\036)398 1201 y Fm(A)423 1192 y Fi(\003)453 1195 y Fp(=)e(\()p Fo(id)549 1201 y Fm(A)574 1192 y Fi(\003)603 1195 y Fk(\012)d Fo(ev)q Fp(\))i Fk(\016)e Fp(\()p Fo(id)793 1201 y Fm(A)818 1192 y Fi(\003)846 1195 y Fk(\012)h Fo(\026)913 1201 y Fm(A)949 1195 y Fk(\012)g Fo(id)1027 1201 y Fm(A)1052 1192 y Fi(\003)1071 1195 y Fp(\))f Fk(\016)g Fp(\()p Fo(db)g Fk(\012)g Fo(S)1257 1201 y Fm(A)1294 1195 y Fk(\012)h Fo(id)1372 1201 y Fm(A)1397 1192 y Fi(\003)1416 1195 y Fp(\))f Fk(\016)g Fo(\033)1495 1201 y Fm(A)1520 1192 y Fi(\003)1537 1201 y Fm(;A)379 1263 y Fo(\016)399 1246 y Fm(A)397 1273 y(A)422 1265 y Fi(\003)453 1263 y Fp(=)j(\()p Fo(id)549 1269 y Fm(A)574 1261 y Fi(\003)603 1263 y Fk(\012)d Fo(id)680 1269 y Fm(A)716 1263 y Fk(\012)h Fo(ev)q Fp(\))g Fk(\016)f Fp(\()p Fo(id)906 1269 y Fm(A)931 1261 y Fi(\003)960 1263 y Fk(\012)g Fp(\001)1036 1269 y Fm(A)1072 1263 y Fk(\012)h Fo(id)1150 1269 y Fm(A)1177 1263 y Fp(\))f Fk(\016)g Fp(\()p Fo(db)g Fk(\012)g Fo(id)1374 1269 y Fm(A)1399 1261 y Fi(\003)1419 1263 y Fp(\))257 1349 y(No)o(w)14 b Fo(A)383 1334 y Fj(\003)416 1349 y Fp(is)g(a)f(como)q(dule)g(o)o(v)o (er)h Fo(A)g Fp(b)q(ecause)h(w)o(e)g(ha)o(v)o(e:)418 1457 y Fo(\016)438 1440 y Fm(A)436 1467 y(A)461 1459 y Fi(\003)481 1457 y Fp(\()p Fo(f)521 1440 y Fl(\(1\))566 1457 y Fp(\))10 b Fk(\012)f Fo(f)657 1440 y Fl(\(2\))714 1457 y Fp(=)787 1405 y Fm(n)768 1418 y Fh(X)758 1506 y Fm(i;j)r Fl(=1)844 1457 y Fo(a)866 1440 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)934 1457 y Fk(\012)h Fo(a)998 1464 y Fl(\()p Fm(i)p Fl(\))1037 1463 y(1)1056 1457 y Fo(a)1078 1440 y Fl(\()p Fm(j)r Fl(\))q Fj(\003)1140 1457 y Fp(\()p Fo(a)1178 1464 y Fl(\()p Fm(i)p Fl(\))1218 1463 y(2)1237 1457 y Fp(\))f Fk(\012)h Fo(a)1326 1464 y Fl(\()p Fm(j)r Fl(\))1369 1463 y(1)1388 1457 y Fo(f)t Fp(\()p Fo(a)1450 1464 y Fl(\()p Fm(j)r Fl(\))1494 1463 y(2)1513 1457 y Fp(\))714 1601 y(=)778 1549 y Fm(n)758 1562 y Fh(X)761 1650 y Fm(i)p Fl(=1)825 1601 y Fo(a)847 1584 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)915 1601 y Fk(\012)f Fo(a)978 1608 y Fl(\()p Fm(i)p Fl(\))1018 1607 y(1)1046 1601 y Fk(\012)h Fo(a)1110 1608 y Fl(\()p Fm(i)p Fl(\))1149 1607 y(2)1168 1601 y Fo(f)t Fp(\()p Fo(a)1230 1608 y Fl(\()p Fm(i)p Fl(\))1270 1607 y(3)1289 1601 y Fp(\))714 1710 y(=)i Fo(f)782 1693 y Fl(\(1\))837 1710 y Fk(\012)d Fp(\001)913 1716 y Fm(A)940 1710 y Fp(\()p Fo(f)980 1693 y Fl(\(2\))1025 1710 y Fp(\))257 1795 y(T)m(o)k(pro)o(v)o(e)h(that)g Fo(A)551 1780 y Fj(\003)584 1795 y Fp(is)g(a)g(righ)o(t)f Fo(A)p Fp(-mo)q(dule,)f(w)o(e)i(calculate:)332 1912 y(\()p Fo(f)i Fg(\021)11 b Fo(a)p Fp(\))h Fg(\021)f Fo(a)562 1895 y Fj(0)585 1912 y Fp(=)648 1860 y Fm(n)629 1873 y Fh(X)632 1961 y Fm(i)p Fl(=1)696 1912 y Fo(a)718 1895 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)776 1912 y Fp(\()p Fo(f)17 b Fg(\021)11 b Fo(a)p Fp(\))920 1895 y Fl(2)939 1912 y Fp(\()p Fo(a)977 1919 y Fl(\()p Fm(i)p Fl(\))1016 1912 y Fo(S)1041 1918 y Fm(A)1069 1912 y Fp(\(\()p Fo(f)17 b Fg(\021)11 b Fo(a)p Fp(\))1229 1895 y Fl(1)1259 1912 y Fk(!)g Fo(a)1334 1895 y Fj(0)1345 1912 y Fp(\)\))585 2050 y(=)648 1998 y Fm(n)629 2011 y Fh(X)632 2099 y Fm(i)p Fl(=1)696 2050 y Fo(a)718 2033 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)776 2050 y Fp(\()p Fo(f)816 2033 y Fl(2)847 2050 y Fg(\021)h Fo(a)923 2033 y Fl(2)941 2050 y Fp(\)\()p Fo(a)995 2057 y Fl(\()p Fm(i)p Fl(\))1035 2050 y Fo(S)1060 2056 y Fm(A)1088 2050 y Fp(\()p Fo(f)1128 2033 y Fl(1)1147 2050 y Fo(a)1169 2033 y Fl(1)1199 2050 y Fk(!)f Fo(a)1274 2033 y Fj(0)1286 2050 y Fp(\)\))585 2189 y(=)658 2137 y Fm(n)638 2149 y Fh(X)629 2238 y Fm(i;j)r Fl(=1)715 2189 y Fo(a)737 2171 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)795 2189 y Fo(a)817 2171 y Fl(\()p Fm(j)r Fl(\))r Fj(\003)880 2189 y Fp(\()p Fo(a)918 2196 y Fl(\()p Fm(i)p Fl(\))958 2189 y Fo(S)983 2195 y Fm(A)1010 2189 y Fp(\()p Fo(f)1050 2171 y Fl(1)1070 2189 y Fo(a)1092 2171 y Fl(1)1122 2189 y Fk(!)g Fo(a)1197 2171 y Fj(0)1209 2189 y Fp(\)\))p Fo(f)1265 2171 y Fl(3)1284 2189 y Fp(\()p Fo(a)1322 2196 y Fl(\()p Fm(j)r Fl(\))1366 2189 y Fo(S)1391 2195 y Fm(A)1418 2189 y Fp(\()p Fo(f)1458 2171 y Fl(2)1489 2189 y Fk(!)g Fo(a)1564 2171 y Fl(2)1583 2189 y Fp(\)\))585 2332 y(=)648 2281 y Fm(n)629 2293 y Fh(X)632 2381 y Fm(i)p Fl(=1)696 2332 y Fo(a)718 2315 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)776 2332 y Fo(f)800 2315 y Fl(3)820 2332 y Fp(\()p Fo(a)858 2339 y Fl(\()p Fm(i)p Fl(\))898 2332 y Fo(S)923 2338 y Fm(A)950 2332 y Fp(\()p Fo(f)990 2315 y Fl(1)1010 2332 y Fo(a)1032 2315 y Fl(1)1062 2332 y Fk(!)g Fo(a)1137 2315 y Fj(0)1148 2332 y Fp(\))p Fo(S)1189 2338 y Fm(A)1217 2332 y Fp(\()p Fo(f)1257 2315 y Fl(2)1288 2332 y Fk(!)g Fo(a)1363 2315 y Fl(2)1382 2332 y Fp(\)\))585 2471 y(=)648 2419 y Fm(n)629 2431 y Fh(X)632 2520 y Fm(i)p Fl(=1)696 2471 y Fo(a)718 2454 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)776 2471 y Fo(f)800 2454 y Fl(2)820 2471 y Fp(\()p Fo(a)858 2478 y Fl(\()p Fm(i)p Fl(\))898 2471 y Fp([)p Fo(f)934 2454 y Fl(1)964 2471 y Fk(!)g Fp(\()p Fo(S)1058 2477 y Fm(A)1086 2471 y Fp(\()p Fo(a)1124 2454 y Fl(1)1154 2471 y Fk(!)g Fo(a)1229 2454 y Fj(0)1241 2471 y Fp(\))p Fo(S)1282 2477 y Fm(A)1309 2471 y Fp(\()p Fo(a)1347 2454 y Fl(2)1366 2471 y Fp(\)\)]\))953 2628 y(11)p eop %%Page: 12 12 12 11 bop 585 289 a Fp(=)648 237 y Fm(n)629 249 y Fh(X)632 338 y Fm(i)p Fl(=1)696 289 y Fo(a)718 271 y Fl(\()p Fm(i)p Fl(\))q Fj(\003)776 289 y Fo(f)800 271 y Fl(2)820 289 y Fp(\()p Fo(a)858 296 y Fl(\()p Fm(i)p Fl(\))898 289 y Fp(\()p Fo(f)938 271 y Fl(1)969 289 y Fk(!)11 b Fo(S)1047 295 y Fm(A)1074 289 y Fp(\()p Fo(aa)1134 271 y Fj(0)1146 289 y Fp(\)\)\))h(=)g Fo(f)k Fg(\021)11 b Fp(\()p Fo(aa)1399 271 y Fj(0)1411 289 y Fp(\))275 413 y(Finally)m(,)16 b(w)o(e)i(ha)o(v)o(e)g(to)g(v)o(erify)f(the)i(Y)m(etter-Drinfel'd)f (Hopf)g(mo)q(dule)e(condition.)h(Note)257 462 y(\014rst)e(that)f(this)g (condition,)f(if)g(written)h(without)g(Sw)o(eedler)h(notation,)d(reads) j(as)f(follo)o(ws:)434 550 y Fo(\016)454 533 y Fm(A)452 560 y(A)477 552 y Fi(\003)506 550 y Fk(\016)9 b Fo(\036)561 556 y Fm(A)586 548 y Fi(\003)617 550 y Fp(=)i(\()p Fo(\036)701 556 y Fm(A)726 548 y Fi(\003)755 550 y Fk(\012)e Fo(\026)821 556 y Fm(A)848 550 y Fp(\))h Fk(\016)f Fp(\()p Fo(id)956 556 y Fm(A)981 548 y Fi(\003)1009 550 y Fk(\012)h Fo(\033)1075 556 y Fm(A;A)1146 550 y Fk(\012)f Fo(id)1223 556 y Fm(A)1250 550 y Fp(\))h Fk(\016)f Fp(\()p Fo(\016)1342 533 y Fm(A)1340 560 y(A)1365 552 y Fi(\003)1394 550 y Fk(\012)g Fp(\001)1470 556 y Fm(A)1497 550 y Fp(\))257 638 y(where)j Fo(\026)399 644 y Fm(A)436 638 y Fp(denotes)g(the)f(m)o(ultiplicati)o(on)c(of)j Fo(A)p Fp(.)g(Since)h(w)o(e)g(ha)o(v)o(e)f(for)g Fo(a;)d(b)j Fk(2)i Fo(A)e Fp(and)g Fo(f)17 b Fk(2)11 b Fo(A)1659 623 y Fj(\003)1678 638 y Fp(,)257 687 y(using)j(the)g(Y)m (etter-Drinfel'd)g(condition)f(and)h(the)g(colinearit)o(y)g(of)f Fo(ev)q Fp(,)h(that:)353 775 y(\()p Fo(aS)416 781 y Fm(A)444 775 y Fp(\()p Fo(f)484 758 y Fl(1)515 775 y Fk(!)d Fo(b)586 781 y Fl(1)604 775 y Fp(\)\))636 781 y Fl(1)655 775 y Fo(b)673 781 y Fl(2)691 775 y Fo(f)715 758 y Fl(2)735 775 y Fp(\(\()p Fo(aS)814 781 y Fm(A)842 775 y Fp(\()p Fo(f)882 758 y Fl(1)913 775 y Fk(!)g Fo(b)984 781 y Fl(1)1002 775 y Fp(\)\))1034 781 y Fl(2)1053 775 y Fp(\))471 843 y(=)h Fo(a)537 849 y Fl(1)556 843 y Fp(\()p Fo(a)594 849 y Fl(2)612 825 y(1)642 843 y Fk(!)f Fp(\()p Fo(S)736 849 y Fm(A)764 843 y Fp(\()p Fo(f)804 825 y Fl(1)835 843 y Fk(!)g Fo(b)906 849 y Fl(1)925 843 y Fp(\)\))957 849 y Fl(1)976 843 y Fp(\))p Fo(b)1010 849 y Fl(2)1040 843 y Fo(f)1064 825 y Fl(2)1083 843 y Fp(\()p Fo(a)1121 849 y Fl(2)1140 825 y(2)1158 843 y Fo(S)1183 849 y Fm(A)1211 843 y Fp(\()p Fo(f)1251 825 y Fl(1)1282 843 y Fk(!)g Fo(b)1353 849 y Fl(1)1371 843 y Fp(\))1387 849 y Fl(2)1406 843 y Fp(\))471 910 y(=)h Fo(a)537 916 y Fl(1)556 910 y Fp(\()p Fo(a)594 916 y Fl(2)612 893 y(1)631 910 y Fo(f)655 893 y Fl(1)686 910 y Fk(!)f Fo(S)764 916 y Fm(A)791 910 y Fp(\()p Fo(b)825 916 y Fl(1)844 893 y(1)874 910 y Fk(!)g Fo(b)945 916 y Fl(2)964 910 y Fp(\)\))p Fo(b)1014 916 y Fl(3)1044 910 y Fo(f)1068 893 y Fl(3)1087 910 y Fp(\()p Fo(a)1125 916 y Fl(2)1144 893 y(2)1163 910 y Fp(\()p Fo(f)1203 893 y Fl(2)1234 910 y Fk(!)g Fo(S)1312 916 y Fm(A)1339 910 y Fp(\()p Fo(b)1373 916 y Fl(1)1392 893 y(2)1410 910 y Fp(\)\)\))471 977 y(=)h Fo(a)537 983 y Fl(1)556 977 y Fo(S)581 983 y Fm(A)608 977 y Fp(\()p Fo(a)646 983 y Fl(2)665 960 y(1)683 977 y Fo(f)707 960 y Fl(1)727 977 y Fo(b)745 983 y Fl(1)763 960 y(1)793 977 y Fk(!)f Fo(b)864 983 y Fl(2)883 977 y Fp(\))p Fo(b)917 983 y Fl(3)947 977 y Fo(ev)q Fp(\()p Fo(a)1025 983 y Fl(2)1045 960 y(2)1063 977 y Fo(S)1088 983 y Fm(A)1116 977 y Fp(\()p Fo(f)1156 960 y Fl(2)1187 977 y Fk(!)g Fo(b)1258 983 y Fl(1)1276 960 y(2)1295 977 y Fp(\))e Fk(\012)h Fo(f)1386 960 y Fl(3)1405 977 y Fp(\))471 1045 y(=)i Fo(a)537 1051 y Fl(1)556 1045 y Fo(S)581 1051 y Fm(A)608 1045 y Fp(\()p Fo(a)646 1051 y Fl(2)665 1028 y(1)683 1045 y Fp(\()p Fo(f)723 1028 y Fl(1)754 1045 y Fk(!)g Fo(b)826 1051 y Fl(1)844 1045 y Fp(\))860 1028 y Fl(1)879 1045 y Fo(f)903 1028 y Fl(2)934 1045 y Fk(!)f Fo(b)1005 1051 y Fl(2)1023 1045 y Fp(\))p Fo(b)1057 1051 y Fl(3)1087 1045 y Fo(ev)q Fp(\()p Fo(a)1165 1051 y Fl(2)1185 1028 y(2)1204 1045 y Fo(S)1229 1051 y Fm(A)1256 1045 y Fp(\(\()p Fo(f)1312 1028 y Fl(1)1343 1045 y Fk(!)g Fo(b)1414 1051 y Fl(1)1433 1045 y Fp(\))1449 1028 y Fl(2)1467 1045 y Fp(\))f Fk(\012)f Fo(f)1558 1028 y Fl(3)1578 1045 y Fp(\))471 1112 y(=)j Fo(a)537 1118 y Fl(1)556 1112 y Fo(S)581 1118 y Fm(A)608 1112 y Fp(\()p Fo(b)642 1118 y Fl(2)661 1112 y Fp(\))p Fo(b)695 1118 y Fl(3)725 1112 y Fo(ev)q Fp(\()p Fo(a)803 1118 y Fl(2)822 1112 y Fo(S)847 1118 y Fm(A)875 1112 y Fp(\()p Fo(f)915 1095 y Fl(1)946 1112 y Fk(!)f Fo(b)1017 1118 y Fl(1)1035 1112 y Fp(\))f Fk(\012)f Fo(f)1126 1095 y Fl(2)1146 1112 y Fp(\))471 1180 y(=)j Fo(a)537 1186 y Fl(1)567 1180 y Fo(f)591 1162 y Fl(2)611 1180 y Fp(\()p Fo(a)649 1186 y Fl(2)667 1180 y Fo(S)692 1186 y Fm(A)720 1180 y Fp(\()p Fo(f)760 1162 y Fl(1)791 1180 y Fk(!)f Fo(b)p Fp(\)\))257 1267 y(w)o(e)j(can)h (expand)f(the)g(righ)o(t)g(hand)f(side)i(as)f(follo)o(ws:)422 1355 y(\()p Fo(\036)463 1361 y Fm(A)488 1353 y Fi(\003)517 1355 y Fk(\012)9 b Fo(\026)583 1361 y Fm(A)610 1355 y Fp(\))g Fk(\016)g Fp(\()p Fo(id)717 1361 y Fm(A)742 1353 y Fi(\003)771 1355 y Fk(\012)h Fo(\033)837 1361 y Fm(A;A)907 1355 y Fk(\012)g Fo(id)985 1361 y Fm(A)1012 1355 y Fp(\))f Fk(\016)g Fp(\()p Fo(\016)1103 1338 y Fm(A)1101 1365 y(A)1126 1357 y Fi(\003)1155 1355 y Fk(\012)h Fp(\001)1232 1361 y Fm(A)1259 1355 y Fp(\)\()p Fo(f)k Fk(\012)9 b Fo(a)p Fp(\))474 1425 y(=)j Fo(\036)543 1431 y Fm(A)568 1423 y Fi(\003)587 1425 y Fp(\()p Fo(f)627 1408 y Fl(\(1\))682 1425 y Fk(\012)e Fp(\()p Fo(f)764 1408 y Fl(\(2\))r(1)839 1425 y Fk(!)h Fo(a)914 1431 y Fl(1)933 1425 y Fp(\)\))e Fk(\012)h Fo(f)1040 1408 y Fl(\(2\))r(2)1104 1425 y Fo(a)1126 1431 y Fl(2)474 1525 y Fp(=)538 1474 y Fm(n)518 1486 y Fh(X)521 1574 y Fm(i)p Fl(=1)585 1525 y Fo(a)607 1508 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)666 1525 y Fo(f)690 1508 y Fl(\(1\))r(2)754 1525 y Fp(\()p Fo(a)792 1532 y Fl(\()p Fm(i)p Fl(\))832 1525 y Fo(S)857 1531 y Fm(A)884 1525 y Fp(\()p Fo(f)924 1508 y Fl(\(1\))s(1)988 1525 y Fo(f)1012 1508 y Fl(\(2\))s(1)1088 1525 y Fk(!)h Fo(a)1163 1531 y Fl(1)1181 1525 y Fp(\)\))f Fk(\012)g Fo(f)1289 1508 y Fl(\(2\))r(2)1353 1525 y Fo(a)1375 1531 y Fl(2)474 1664 y Fp(=)538 1612 y Fm(n)518 1624 y Fh(X)521 1713 y Fm(i)p Fl(=1)585 1664 y Fo(a)607 1647 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)666 1664 y Fo(f)690 1647 y Fl(2)r(\(1\))754 1664 y Fp(\()p Fo(a)792 1671 y Fl(\()p Fm(i)p Fl(\))832 1664 y Fo(S)857 1670 y Fm(A)884 1664 y Fp(\()p Fo(f)924 1647 y Fl(1)955 1664 y Fk(!)h Fo(a)1030 1670 y Fl(1)1049 1664 y Fp(\)\))e Fk(\012)h Fo(f)1156 1647 y Fl(2)r(\(2\))1220 1664 y Fo(a)1242 1670 y Fl(2)474 1802 y Fp(=)548 1750 y Fm(n)528 1763 y Fh(X)518 1851 y Fm(i;j)r Fl(=1)604 1802 y Fo(a)626 1785 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)685 1802 y Fo(a)707 1785 y Fl(\()p Fm(j)r Fl(\))q Fj(\003)769 1802 y Fp(\()p Fo(a)807 1808 y Fm(i)821 1802 y Fo(S)846 1808 y Fm(A)874 1802 y Fp(\()p Fo(f)914 1785 y Fl(1)945 1802 y Fk(!)h Fo(a)1020 1808 y Fl(1)1039 1802 y Fp(\)\))e Fk(\012)h Fo(a)1144 1809 y Fl(\()p Fm(j)r Fl(\))1187 1808 y(1)1206 1802 y Fo(f)1230 1785 y Fl(2)1249 1802 y Fp(\()p Fo(a)1287 1809 y Fl(\()p Fm(j)r Fl(\))1331 1808 y(2)1349 1802 y Fp(\))p Fo(a)1387 1808 y Fl(2)474 1946 y Fp(=)538 1894 y Fm(n)518 1907 y Fh(X)521 1995 y Fm(i)p Fl(=1)585 1946 y Fo(a)607 1929 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)675 1946 y Fk(\012)g Fp(\()p Fo(a)755 1953 y Fl(\()p Fm(i)p Fl(\))795 1946 y Fo(S)820 1952 y Fm(A)847 1946 y Fp(\()p Fo(f)887 1929 y Fl(1)918 1946 y Fk(!)h Fo(a)993 1952 y Fl(1)1012 1946 y Fp(\)\))1044 1952 y Fl(1)1063 1946 y Fo(a)1085 1952 y Fl(2)1103 1946 y Fo(f)1127 1929 y Fl(2)1147 1946 y Fp(\(\()p Fo(a)1201 1953 y Fl(\()p Fm(i)p Fl(\))1241 1946 y Fo(S)1266 1952 y Fm(A)1293 1946 y Fp(\()p Fo(f)1333 1929 y Fl(1)1364 1946 y Fk(!)g Fo(a)1439 1952 y Fl(1)1458 1946 y Fp(\)\))1490 1952 y Fl(2)1509 1946 y Fp(\))474 2084 y(=)538 2032 y Fm(n)518 2045 y Fh(X)521 2133 y Fm(i)p Fl(=1)585 2084 y Fo(a)607 2067 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)675 2084 y Fk(\012)f Fo(a)739 2091 y Fl(\()p Fm(i)p Fl(\))778 2090 y(1)809 2084 y Fo(f)833 2067 y Fl(2)852 2084 y Fp(\()p Fo(a)890 2091 y Fl(\()p Fm(i)p Fl(\))930 2090 y(2)948 2084 y Fo(S)973 2090 y Fm(A)1001 2084 y Fp(\()p Fo(f)1041 2067 y Fl(1)1072 2084 y Fk(!)h Fo(a)p Fp(\)\))271 2208 y(On)j(the)h(other)f(hand,)g(w)o(e)g(ha)o(v)o(e:)411 2327 y Fo(\016)431 2310 y Fm(A)429 2337 y(A)454 2329 y Fi(\003)474 2327 y Fp(\()p Fo(f)j Fg(\021)11 b Fo(a)p Fp(\))g(=)693 2275 y Fm(n)673 2287 y Fh(X)676 2376 y Fm(i)p Fl(=1)740 2327 y Fo(a)762 2310 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)830 2327 y Fk(\012)f Fo(a)894 2334 y Fl(\()p Fm(i)p Fl(\))933 2333 y(1)952 2327 y Fp(\()p Fo(f)16 b Fg(\021)11 b Fo(a)p Fp(\)\()p Fo(a)1133 2334 y Fl(\()p Fm(i)p Fl(\))1173 2333 y(2)1192 2327 y Fp(\))629 2465 y(=)702 2413 y Fm(n)683 2426 y Fh(X)673 2514 y Fm(i;j)r Fl(=1)759 2465 y Fo(a)781 2448 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)849 2465 y Fk(\012)f Fo(a)913 2472 y Fl(\()p Fm(i)p Fl(\))952 2471 y(1)971 2465 y Fo(a)993 2448 y Fl(\()p Fm(j)r Fl(\))q Fj(\003)1055 2465 y Fp(\()p Fo(a)1093 2472 y Fl(\()p Fm(i)p Fl(\))1133 2471 y(2)1152 2465 y Fp(\))i Fo(f)1204 2448 y Fl(2)1223 2465 y Fp(\()p Fo(a)1261 2472 y Fl(\()p Fm(j)r Fl(\))1305 2465 y Fo(S)1330 2471 y Fm(A)1357 2465 y Fp(\()p Fo(f)1397 2448 y Fl(1)1428 2465 y Fk(!)f Fo(a)p Fp(\)\))953 2628 y(12)p eop %%Page: 13 13 13 12 bop 629 289 a Fp(=)693 237 y Fm(n)673 249 y Fh(X)676 338 y Fm(i)p Fl(=1)740 289 y Fo(a)762 271 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)830 289 y Fk(\012)10 b Fo(a)894 296 y Fl(\()p Fm(i)p Fl(\))933 295 y(1)963 289 y Fo(f)987 271 y Fl(2)1007 289 y Fp(\()p Fo(a)1045 296 y Fl(\()p Fm(i)p Fl(\))1085 295 y(2)1103 289 y Fo(S)1128 295 y Fm(A)1156 289 y Fp(\()p Fo(f)1196 271 y Fl(1)1227 289 y Fk(!)h Fo(a)p Fp(\)\))257 416 y(and)j(therefore)h(b)q(oth)f(sides)h(are)f (equal.)f Fg(\003)257 552 y Fn(2.10)48 b Fp(W)m(e)16 b(no)o(w)g(reac)o(h)h(our)f(\014rst)h(goal:)e(T)m(o)g(pro)o(v)o(e)h (the)h(existence)h(and)f(uniqueness)g(of)257 601 y(in)o(tegrals)j(in)g (\014nite)h(dimensional)d(Y)m(etter-Drinfel'd)i(Hopf)g(algebras.)g (Note)g(that)h(the)257 651 y(follo)o(wing)13 b(Prop)q(osition)i(is)g (not)g(new,)g(it)g(is)g(\(a)g(v)n(arian)o(t)f(of)s(\))h([5)o(],)f (Corollary)g(5.8)g(\(cf.)h(also)257 701 y([23)o(],)e(section)i(2,)e (Prop.)h(3)f(and)h(4\).)257 801 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)13 b(that)f Fo(A)g Fp(is)g(a)f(\014nite-dimensional)f (left)h(Y)m(etter-Drinfel'd)h(Hopf)257 850 y(algebra)i(o)o(v)o(er)g Fo(H)s Fp(.)f(Then)h(w)o(e)g(ha)o(v)o(e:)308 969 y(1.)20 b(There)e(is)e(a)h(nonzero)g(left)g(in)o(tegral)f(\003)982 975 y Fm(A)1025 969 y Fp(of)g Fo(A)p Fp(,)g(a)h(nonzero)g(righ)o(t)g (in)o(tegral)e(\000)1612 975 y Fm(A)1656 969 y Fp(of)361 1019 y Fo(A)p Fp(,)e(a)g(nonzero)h(left)f(in)o(tegral)f Fo(\025)851 1025 y Fm(A)892 1019 y Fp(of)g Fo(A)969 1004 y Fj(\003)1001 1019 y Fp(and)i(a)e(nonzero)i(righ)o(t)f(in)o(tegral)g Fo(\032)1541 1025 y Fm(A)1581 1019 y Fp(of)g Fo(A)1659 1004 y Fj(\003)1678 1019 y Fp(.)361 1069 y(Suc)o(h)h(elemen)o(ts)g (satisfy:)770 1160 y Fo(\025)794 1166 y Fm(A)821 1160 y Fp(\(\003)866 1166 y Fm(A)893 1160 y Fp(\))e Fk(6)p Fp(=)f(0)83 b Fo(\025)1092 1166 y Fm(A)1120 1160 y Fp(\(\000)1162 1166 y Fm(A)1189 1160 y Fp(\))11 b Fk(6)p Fp(=)h(0)772 1222 y Fo(\032)793 1228 y Fm(A)821 1222 y Fp(\(\003)866 1228 y Fm(A)893 1222 y Fp(\))g Fk(6)p Fp(=)f(0)83 b Fo(\032)1089 1228 y Fm(A)1117 1222 y Fp(\(\000)1159 1228 y Fm(A)1186 1222 y Fp(\))12 b Fk(6)p Fp(=)f(0)308 1330 y(2.)20 b(There)c(is)f(a)f (c)o(haracter)i Fo(\023)757 1336 y Fm(A)797 1330 y Fp(:)d Fo(H)j Fk(!)c Fo(K)s Fp(,)j(called)f(the)i(in)o(tegral)e(c)o(haracter)i (of)e Fo(A)p Fp(,)h(and)361 1380 y(a)f(grouplik)o(e)f(elemen)o(t)h Fo(g)750 1386 y Fm(A)789 1380 y Fk(2)e Fo(H)s Fp(,)h(called)h(the)h(in) o(tegral)f(group)g(elemen)o(t)f(of)h Fo(A)p Fp(,)g(suc)o(h)361 1430 y(that)g(w)o(e)g(ha)o(v)o(e:)621 1521 y Fo(h)d Fk(!)g Fp(\003)738 1527 y Fm(A)777 1521 y Fp(=)h Fo(\023)836 1527 y Fm(A)862 1521 y Fp(\()p Fo(h)p Fp(\)\003)947 1527 y Fm(A)1057 1521 y Fo(\016)r Fp(\(\003)1122 1527 y Fm(A)1149 1521 y Fp(\))g(=)g Fo(g)1241 1527 y Fm(A)1277 1521 y Fk(\012)e Fp(\003)1348 1527 y Fm(A)627 1584 y Fo(h)h Fk(!)g Fp(\000)741 1590 y Fm(A)780 1584 y Fp(=)g Fo(\023)838 1590 y Fm(A)865 1584 y Fp(\()p Fo(h)p Fp(\)\000)947 1590 y Fm(A)1057 1584 y Fo(\016)r Fp(\(\000)1119 1590 y Fm(A)1147 1584 y Fp(\))g(=)h Fo(g)1238 1590 y Fm(A)1274 1584 y Fk(\012)e Fp(\000)1342 1590 y Fm(A)630 1651 y Fo(\025)654 1657 y Fm(A)693 1651 y Fk( )h Fo(h)g Fp(=)h Fo(\023)840 1657 y Fm(A)867 1651 y Fp(\()p Fo(h)p Fp(\))p Fo(\025)947 1657 y Fm(A)1057 1651 y Fo(a)1079 1634 y Fl(1)1098 1651 y Fo(\025)1122 1657 y Fm(A)1149 1651 y Fp(\()p Fo(a)1187 1634 y Fl(2)1206 1651 y Fp(\))g(=)f Fo(\025)1301 1657 y Fm(A)1329 1651 y Fp(\()p Fo(a)p Fp(\))p Fo(g)1403 1657 y Fm(A)636 1718 y Fo(\032)657 1724 y Fm(A)696 1718 y Fk( )g Fo(h)g Fp(=)h Fo(\023)843 1724 y Fm(A)870 1718 y Fp(\()p Fo(h)p Fp(\))p Fo(\032)947 1724 y Fm(A)1057 1718 y Fo(a)1079 1701 y Fl(1)1098 1718 y Fo(\032)1119 1724 y Fm(A)1146 1718 y Fp(\()p Fo(a)1184 1701 y Fl(2)1203 1718 y Fp(\))g(=)g Fo(\032)1296 1724 y Fm(A)1323 1718 y Fp(\()p Fo(a)p Fp(\))p Fo(g)1397 1724 y Fm(A)361 1810 y Fp(for)18 b(ev)o(ery)h(left)e(in)o(tegral)h(\003)804 1816 y Fm(A)848 1810 y Fp(and)g(righ)o(t)g(in)o(tegral)f(\000)1219 1816 y Fm(A)1264 1810 y Fp(of)g Fo(A)h Fp(and)g(for)g(ev)o(ery)h(left) 361 1860 y(in)o(tegral)13 b Fo(\025)536 1866 y Fm(A)577 1860 y Fp(and)h(righ)o(t)g(in)o(tegral)f Fo(\032)931 1866 y Fm(A)972 1860 y Fp(of)g Fo(A)1050 1844 y Fj(\003)1070 1860 y Fp(.)308 1943 y(3.)20 b Fo(A)14 b Fp(is)f(a)g(F)m(rob)q(enius)h (algebra)f(with)g(a)g(nonzero)h(righ)o(t)f(in)o(tegral)g Fo(\032)1371 1949 y Fm(A)1411 1943 y Fp(of)g Fo(A)1489 1927 y Fj(\003)1522 1943 y Fp(as)g(F)m(rob)q(e-)361 1992 y(nius)k(homomo)o(rphism)o(.)c(The)k(Casimir)e(elemen)o(t)h(for)g(this) h(F)m(rob)q(enius)g(homomo)o(r-)361 2042 y(phism)c(is)781 2092 y(\003)810 2098 y Fm(A)r Fl(1)865 2092 y Fk(\012)d Fo(\023)922 2074 y Fj(\000)p Fl(1)922 2104 y Fm(A)966 2092 y Fp(\(\003)1011 2098 y Fm(A)r Fl(2)1057 2074 y(1)1075 2092 y Fp(\))p Fo(S)1116 2098 y Fm(A)1144 2092 y Fp(\(\003)1189 2098 y Fm(A)r Fl(2)1234 2074 y(2)1253 2092 y Fp(\))361 2167 y(where)k(\003)509 2173 y Fm(A)548 2167 y Fp(is)e(a)g(left)h(in)o (tegral)e(of)h Fo(A)h Fp(satisfying)e Fo(\032)1136 2173 y Fm(A)1164 2167 y Fp(\(\003)1209 2173 y Fm(A)1236 2167 y Fp(\))g(=)h(1)g(and)h Fo(\023)1435 2149 y Fj(\000)p Fl(1)1435 2179 y Fm(A)1491 2167 y Fp(:=)e Fo(\023)1561 2173 y Fm(A)1594 2167 y Fk(\016)6 b Fo(S)1646 2173 y Fm(H)1678 2167 y Fp(.)308 2250 y(4.)20 b(The)14 b(an)o(tip)q(o)q(de)g (of)g Fo(A)g Fp(is)f(bijectiv)o(e.)257 2349 y Fn(Pro)q(of.)36 b Fp(This)14 b(will)e(b)q(e)j(pro)o(v)o(ed)f(in)f(sev)o(eral)i(steps:) 953 2628 y(13)p eop %%Page: 14 14 14 13 bop 257 262 a Fp(\(1\))21 b(Using)9 b(the)h(righ)o(t)f(Y)m (etter-Drinfel'd)g(Hopf)g(mo)q(dule)f(structure)j(from)d(Prop)q (osition)g(2.9,)257 311 y(w)o(e)13 b(see)h(that)f(the)g(coin)o(v)n (arian)o(ts)f(are)h(nothing)f(but)h(the)g(left)g(in)o(tegrals,)f(since) h(w)o(e)g(ha)o(v)o(e)g(for)257 361 y Fo(f)k Fk(2)11 b Fo(A)364 346 y Fj(\003)383 361 y Fp(:)426 448 y Fo(\016)446 431 y Fm(A)444 459 y(A)469 450 y Fi(\003)489 448 y Fp(\()p Fo(f)t Fp(\))h(=)g Fo(f)i Fk(\012)c Fp(1)22 b Fk(,)h(8)p Fo(a)11 b Fk(2)h Fo(A)f Fp(:)23 b Fo(f)982 431 y Fl(\(1\))1027 448 y Fp(\()p Fo(a)p Fp(\))p Fo(f)1105 431 y Fl(\(2\))1162 448 y Fp(=)12 b Fo(f)t Fp(\()p Fo(a)p Fp(\)1)720 548 y Fk(,)f(8)p Fo(a)h Fk(2)f Fo(A)h Fp(:)966 496 y Fm(n)946 509 y Fh(X)949 597 y Fm(i)p Fl(=1)1013 548 y Fo(a)1035 531 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1094 548 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1170 555 y Fl(\()p Fm(i)p Fl(\))1210 554 y(1)1229 548 y Fo(f)t Fp(\()p Fo(a)1291 555 y Fl(\()p Fm(i)p Fl(\))1331 554 y(2)1350 548 y Fp(\))g(=)f Fo(f)t Fp(\()p Fo(a)p Fp(\)1)720 653 y Fk(,)g(8)p Fo(a)h Fk(2)f Fo(A)h Fp(:)22 b Fo(a)968 659 y Fl(1)987 653 y Fo(f)t Fp(\()p Fo(a)1049 659 y Fl(2)1068 653 y Fp(\))12 b(=)g Fo(f)t Fp(\()p Fo(a)p Fp(\)1)257 746 y(where)19 b Fo(a)403 753 y Fl(\(1\))447 746 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(a)569 753 y Fl(\()p Fm(n)p Fl(\))634 746 y Fp(is)17 b(a)g(basis)h(of)f Fo(A)g Fp(with)g(dual)g(basis)g Fo(a)1243 731 y Fl(\(1\))p Fj(\003)1305 746 y Fo(;)7 b(:)g(:)g(:)k(;)c(a)1426 731 y Fl(\()p Fm(n)p Fl(\))p Fj(\003)1491 746 y Fp(.)17 b(Since)h(the)257 796 y(mapping)660 846 y Fo(A)691 829 y Fj(\003)p Fm(coA)775 846 y Fk(\012)10 b Fo(A)i Fk(!)f Fo(A)944 829 y Fj(\003)963 846 y Fo(;)c(f)13 b Fk(\012)d Fo(a)h Fk(7!)g Fp(\()p Fo(f)17 b Fg(\021)11 b Fo(a)p Fp(\))257 918 y(is)17 b(a)e(bijection)h(b)o(y)g(Prop)q(osition)g(2.8,)f (the)i(dimensions)e(of)h(b)q(oth)g(sides)h(m)o(ust)e(b)q(e)i(equal:)257 968 y Fo(dim)p Fp(\()p Fo(A)376 953 y Fj(\003)p Fm(coA)461 968 y Fk(\012)10 b Fo(A)p Fp(\))i(=)g Fo(dimA)709 953 y Fj(\003)728 968 y Fp(,)i(and)f(therefore)j Fo(dimA)1112 953 y Fj(\003)p Fm(coA)1199 968 y Fp(=)c(1.)257 1049 y(\(2\))21 b(If)c Fo(\025)400 1055 y Fm(A)443 1049 y Fk(2)f Fo(A)518 1034 y Fj(\003)554 1049 y Fp(is)h(a)f(left)h(in)o (tegral,)e(then)j(for)e(all)f Fo(h)i Fk(2)f Fo(H)j Fp(w)o(e)e(ha)o(v)o (e)g(that)g Fo(\025)1520 1055 y Fm(A)1563 1049 y Fk( )f Fo(h)h Fp(is)257 1099 y(again)c(a)h(left)f(in)o(tegral:)442 1186 y Fo(a)464 1192 y Fl(1)482 1186 y Fp(\()p Fo(\025)522 1192 y Fm(A)561 1186 y Fk( )e Fo(h)p Fp(\)\()p Fo(a)692 1192 y Fl(2)711 1186 y Fp(\))h(=)f Fo(S)807 1192 y Fm(H)839 1186 y Fp(\()p Fo(h)879 1192 y Fl(1)898 1186 y Fp(\))p Fo(h)938 1192 y Fl(2)968 1186 y Fk(!)g Fo(a)1043 1192 y Fl(1)1062 1186 y Fo(\025)1086 1192 y Fm(A)1113 1186 y Fp(\()p Fo(h)1153 1192 y Fl(3)1183 1186 y Fk(!)g Fo(a)1258 1192 y Fl(2)1277 1186 y Fp(\))739 1248 y(=)g Fo(S)807 1254 y Fm(H)839 1248 y Fp(\()p Fo(h)879 1254 y Fl(1)898 1248 y Fp(\))h Fk(!)f Fp(1)g Fo(\025)1035 1254 y Fm(A)1062 1248 y Fp(\()p Fo(h)1102 1254 y Fl(2)1133 1248 y Fk(!)g Fo(a)p Fp(\))g(=)h(1)f Fo(\025)1335 1254 y Fm(A)1363 1248 y Fp(\()p Fo(h)g Fk(!)g Fo(a)p Fp(\))257 1335 y(Since)k(the)g (space)g(of)e(in)o(tegrals)h(is)g(one)g(dimensional,)e(w)o(e)i(ha)o(v)o (e)g Fo(\025)1304 1341 y Fm(A)1343 1335 y Fk( )e Fo(h)f Fp(=)i Fo(\023)1492 1341 y Fm(A)1518 1335 y Fp(\()p Fo(h)p Fp(\))p Fo(\025)1598 1341 y Fm(A)1640 1335 y Fp(for)257 1385 y(some)g(n)o(um)o(b)q(er)h Fo(\023)528 1391 y Fm(A)554 1385 y Fp(\()p Fo(h)p Fp(\).)g(It)g(is)g(ob)o(vious)f(that)h Fo(\023)978 1391 y Fm(A)1018 1385 y Fp(is)g(an)g(algebra)f(homomorphi)o (sm)o(.)257 1451 y(No)o(w)h(write:)648 1531 y Fo(\016)668 1514 y Fm(H)666 1541 y(A)691 1533 y Fi(\003)710 1531 y Fp(\()p Fo(\025)750 1537 y Fm(A)778 1531 y Fp(\))e(=)874 1479 y Fm(l)849 1492 y Fh(X)851 1580 y Fm(j)r Fl(=1)916 1531 y Fo(h)940 1538 y Fl(\()p Fm(j)r Fl(\))993 1531 y Fk(\012)d Fo(f)1054 1538 y Fl(\()p Fm(j)r Fl(\))1110 1531 y Fk(2)i Fo(H)h Fk(\012)d Fo(A)1268 1514 y Fj(\003)1288 1531 y Fo(;)257 1645 y Fp(where)18 b Fo(h)404 1652 y Fl(\(1\))449 1645 y Fo(;)7 b(:)g(:)g(:)k(;)c(h)572 1652 y Fl(\()p Fm(l)p Fl(\))627 1645 y Fp(as)17 b(w)o(ell)f(as)h Fo(f)842 1652 y Fl(\(1\))886 1645 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(f)1006 1652 y Fl(\()p Fm(l)p Fl(\))1061 1645 y Fp(are)17 b(linearly)f(indep)q(enden)o(t.)i(W)m(e)e(then)257 1695 y(ha)o(v)o(e:)455 1776 y Fo(\016)475 1758 y Fm(H)473 1786 y(A)498 1777 y Fi(\003)517 1776 y Fp(\()p Fo(\025)557 1782 y Fm(A)585 1776 y Fp(\))c(=)681 1724 y Fm(l)656 1736 y Fh(X)658 1825 y Fm(j)r Fl(=1)723 1776 y Fo(h)747 1783 y Fl(\()p Fm(j)r Fl(\))800 1776 y Fk(\012)d Fo(f)861 1783 y Fl(\()p Fm(j)r Fl(\))917 1776 y Fp(=)980 1724 y Fm(n)960 1736 y Fh(X)964 1825 y Fm(i)p Fl(=1)1027 1776 y Fo(S)1052 1782 y Fm(H)1084 1776 y Fp(\()p Fo(a)1122 1783 y Fl(\()p Fm(i)p Fl(\))1162 1758 y(1)1181 1776 y Fp(\))g Fk(\012)h Fo(\025)1272 1782 y Fm(A)1299 1776 y Fp(\()p Fo(a)1337 1783 y Fl(\()p Fm(i)p Fl(\))1377 1758 y(2)1395 1776 y Fp(\))p Fo(a)1433 1758 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)257 1890 y Fp(Because)16 b(w)o(e)e(ha)o(v)o(e:) 450 1977 y Fo(S)475 1983 y Fm(H)507 1977 y Fp(\()p Fo(a)545 1983 y Fl(2)564 1960 y(1)582 1977 y Fp(\))c Fk(\012)f Fo(a)671 1983 y Fl(1)690 1977 y Fo(\025)714 1983 y Fm(A)741 1977 y Fp(\()p Fo(a)779 1983 y Fl(2)798 1960 y(2)817 1977 y Fp(\))i(=)h Fo(S)913 1983 y Fm(H)945 1977 y Fp(\()p Fo(a)983 1983 y Fl(2)1002 1960 y(1)1020 1977 y Fp(\))p Fo(S)1061 1983 y Fm(H)1093 1977 y Fp(\()p Fo(a)1131 1983 y Fl(1)1150 1960 y(1)1169 1977 y Fp(\))p Fo(a)1207 1983 y Fl(1)1225 1960 y(2)1253 1977 y Fk(\012)e Fo(a)1317 1983 y Fl(1)1335 1960 y(3)1354 1977 y Fo(\025)1378 1983 y Fm(A)1405 1977 y Fp(\()p Fo(a)1443 1983 y Fl(2)1462 1960 y(2)1481 1977 y Fp(\))844 2044 y(=)i Fo(S)913 2050 y Fm(H)945 2044 y Fp(\()p Fo(a)983 2050 y Fl(1)1002 2027 y(1)1020 2044 y Fo(a)1042 2050 y Fl(2)1061 2027 y(1)1079 2044 y Fp(\))p Fo(a)1117 2050 y Fl(1)1136 2027 y(2)1164 2044 y Fk(\012)e Fo(a)1228 2050 y Fl(1)1246 2027 y(3)1265 2044 y Fo(\025)1289 2050 y Fm(A)1316 2044 y Fp(\()p Fo(a)1354 2050 y Fl(2)1373 2027 y(2)1391 2044 y Fp(\))844 2112 y(=)i Fo(S)913 2118 y Fm(H)945 2112 y Fp(\()p Fo(a)983 2094 y Fl(1)1002 2112 y Fp(\))p Fo(a)1040 2094 y Fl(2)1058 2118 y(1)1077 2094 y(1)1105 2112 y Fk(\012)d Fo(a)1168 2094 y Fl(2)1187 2118 y(1)1206 2094 y(2)1224 2112 y Fo(\025)1248 2118 y Fm(A)1275 2112 y Fp(\()p Fo(a)1313 2094 y Fl(2)1332 2118 y(2)1351 2112 y Fp(\))844 2179 y(=)j Fo(S)913 2185 y Fm(H)945 2179 y Fp(\()p Fo(a)983 2162 y Fl(1)1002 2179 y Fp(\))d Fk(\012)h Fp(1)h Fo(\025)1125 2185 y Fm(A)1152 2179 y Fp(\()p Fo(a)1190 2162 y Fl(2)1209 2179 y Fp(\))257 2266 y(this)j(implies:)467 2340 y Fm(l)442 2352 y Fh(X)443 2441 y Fm(j)r Fl(=1)509 2392 y Fo(h)533 2399 y Fl(\()p Fm(j)r Fl(\))586 2392 y Fk(\012)9 b Fo(a)649 2398 y Fl(1)668 2392 y Fo(f)688 2399 y Fl(\()p Fm(j)r Fl(\))731 2392 y Fp(\()p Fo(a)769 2398 y Fl(2)788 2392 y Fp(\))j(=)879 2340 y Fm(n)860 2352 y Fh(X)863 2441 y Fm(i)p Fl(=1)926 2392 y Fo(S)951 2398 y Fm(H)983 2392 y Fp(\()p Fo(a)1021 2399 y Fl(\()p Fm(i)p Fl(\))1061 2375 y(1)1080 2392 y Fp(\))d Fk(\012)h Fo(a)1169 2398 y Fl(1)1187 2392 y Fo(\025)1211 2398 y Fm(A)1239 2392 y Fp(\()p Fo(a)1277 2399 y Fl(\()p Fm(i)p Fl(\))1316 2375 y(2)1335 2392 y Fp(\))p Fo(a)1373 2375 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1432 2392 y Fp(\()p Fo(a)1470 2398 y Fl(2)1489 2392 y Fp(\))816 2503 y(=)i Fo(S)885 2509 y Fm(H)916 2503 y Fp(\()p Fo(a)954 2509 y Fl(2)973 2486 y(1)992 2503 y Fp(\))d Fk(\012)h Fo(a)1081 2509 y Fl(1)1099 2503 y Fo(\025)1123 2509 y Fm(A)1151 2503 y Fp(\()p Fo(a)1189 2509 y Fl(2)1207 2486 y(2)1226 2503 y Fp(\))953 2628 y(14)p eop %%Page: 15 15 15 14 bop 446 289 a Fp(=)12 b Fo(S)515 295 y Fm(H)547 289 y Fp(\()p Fo(a)585 271 y Fl(1)604 289 y Fp(\))d Fk(\012)h Fp(1)h Fo(\025)727 295 y Fm(A)754 289 y Fp(\()p Fo(a)792 271 y Fl(2)811 289 y Fp(\))g(=)902 237 y Fm(n)882 249 y Fh(X)885 338 y Fm(i)p Fl(=1)949 289 y Fo(S)974 295 y Fm(H)1006 289 y Fp(\()p Fo(a)1044 296 y Fl(\()p Fm(i)p Fl(\))1084 271 y(1)1103 289 y Fp(\))e Fk(\012)g Fp(1)j Fo(\025)1226 295 y Fm(A)1253 289 y Fp(\()p Fo(a)1291 296 y Fl(\()p Fm(i)p Fl(\))1331 271 y(2)1349 289 y Fp(\))p Fo(a)1387 271 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1446 289 y Fp(\()p Fo(a)p Fp(\))838 435 y(=)907 383 y Fm(l)882 395 y Fh(X)884 483 y Fm(j)r Fl(=1)949 435 y Fo(h)973 442 y Fl(\()p Fm(j)r Fl(\))1026 435 y Fk(\012)d Fp(1)j Fo(f)1120 442 y Fl(\()p Fm(j)r Fl(\))1163 435 y Fp(\()p Fo(a)p Fp(\))257 554 y(Since)19 b Fo(h)394 561 y Fl(\(1\))438 554 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(h)562 561 y Fl(\()p Fm(l)p Fl(\))617 554 y Fp(are)19 b(linearly)d(indep)q(enden)o(t,)j(w)o (e)f(can)g(compare)f(co)q(e\016cien)o(ts)i(and)257 604 y(conclude)14 b(that)e Fo(a)537 610 y Fl(1)556 604 y Fo(f)576 611 y Fl(\()p Fm(j)r Fl(\))620 604 y Fp(\()p Fo(a)658 610 y Fl(2)676 604 y Fp(\))g(=)g(1)p Fo(f)789 611 y Fl(\()p Fm(j)r Fl(\))832 604 y Fp(\()p Fo(a)p Fp(\).)h(This)f (pro)o(v)o(es)h(that)g Fo(f)1240 611 y Fl(\()p Fm(j)r Fl(\))1296 604 y Fp(is)g(itself)f(a)g(left)h(in)o(tegral,)257 654 y(and)h(therefore)g(m)o(ust)f(b)q(e)h(a)f(m)o(ultiple)e(a)i(of)g Fo(\025)971 660 y Fm(A)998 654 y Fp(,)g(that)g(is)h Fo(f)1174 661 y Fl(\()p Fm(j)r Fl(\))1229 654 y Fp(=)e Fo(\026)1298 660 y Fm(j)1315 654 y Fo(\025)1339 660 y Fm(A)1380 654 y Fp(for)h(some)g Fo(\026)1572 660 y Fm(j)1601 654 y Fk(2)e Fo(K)s Fp(.)257 713 y(If)k(w)o(e)g(de\014ne:)j(~)-23 b Fo(g)515 719 y Fm(A)556 713 y Fp(=)601 682 y Fh(P)645 693 y Fm(l)645 726 y(j)r Fl(=1)712 713 y Fo(\026)737 719 y Fm(j)754 713 y Fo(h)778 720 y Fl(\()p Fm(j)r Fl(\))822 713 y Fp(,)14 b(w)o(e)h(see)i(that)e Fo(\016)1090 698 y Fm(H)1088 725 y(A)1113 717 y Fi(\003)1133 713 y Fp(\()p Fo(\025)1173 719 y Fm(A)1200 713 y Fp(\))f(=)h(~)-22 b Fo(g)1296 719 y Fm(A)1333 713 y Fk(\012)10 b Fo(\025)1399 719 y Fm(A)1426 713 y Fp(.)15 b(Therefore)h(w)o(e)257 763 y(ha)o(v)o(e:)449 836 y(~)-23 b Fo(g)467 842 y Fm(A)494 836 y Fo(\025)518 842 y Fm(A)545 836 y Fp(\()p Fo(a)p Fp(\))12 b(=)675 784 y Fm(n)655 797 y Fh(X)658 885 y Fm(i)p Fl(=1)722 836 y Fo(S)747 842 y Fm(H)779 836 y Fp(\()p Fo(a)817 843 y Fl(\()p Fm(i)p Fl(\))857 819 y(1)875 836 y Fp(\))p Fo(\025)915 842 y Fm(A)943 836 y Fp(\()p Fo(a)981 843 y Fl(\()p Fm(i)p Fl(\))1021 819 y(2)1039 836 y Fp(\))p Fo(a)1077 819 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1136 836 y Fp(\()p Fo(a)p Fp(\))g(=)g Fo(S)1271 842 y Fm(H)1303 836 y Fp(\()p Fo(a)1341 819 y Fl(1)1359 836 y Fp(\))p Fo(\025)1399 842 y Fm(A)1427 836 y Fp(\()p Fo(a)1465 819 y Fl(2)1483 836 y Fp(\))257 939 y(The)17 b(como)q(dule)e(axiom)e(no)o(w)j(implies)e(that)k(~)-23 b Fo(g)1002 945 y Fm(A)1045 939 y Fp(is)16 b(a)g(grouplik)o(e)f(elemen) o(t.)g(No)o(w)h(de\014ne:)257 988 y Fo(g)277 994 y Fm(A)316 988 y Fp(:=)d(~)-23 b Fo(g)392 971 y Fj(\000)p Fl(1)391 1001 y Fm(A)437 988 y Fp(.)257 1066 y(\(3\))21 b(W)m(e)10 b(kno)o(w)g(from)f(the)i(structure)h(theorem)e(for)g(Y)m (etter-Drinfel'd)g(Hopf)g(mo)q(dules)f(that)257 1116 y(for)14 b(a)f(nonzero)i(left)f(in)o(tegral)f Fo(\025)758 1122 y Fm(A)785 1116 y Fp(,)h(the)g(map)763 1193 y Fo(A)e Fk(!)f Fo(A)890 1176 y Fj(\003)909 1193 y Fo(;)c(a)k Fk(7!)g Fp(\()p Fo(\025)1054 1199 y Fm(A)1093 1193 y Fg(\021)g Fo(a)p Fp(\))257 1271 y(is)j(bijectiv)o(e.)g(No)o(w)f(w)o(e)h (ha)o(v)o(e:)453 1379 y(\()p Fo(\025)493 1385 y Fm(A)532 1379 y Fg(\021)d Fo(a)p Fp(\)\()p Fo(a)661 1362 y Fj(0)673 1379 y Fp(\))h(=)764 1327 y Fm(n)744 1340 y Fh(X)748 1428 y Fm(i)p Fl(=1)811 1379 y Fo(a)833 1362 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)892 1379 y Fp(\()p Fo(a)930 1362 y Fj(0)942 1379 y Fp(\))p Fo(\025)982 1362 y Fl(2)982 1389 y Fm(A)1009 1379 y Fp(\()p Fo(a)1047 1386 y Fl(\()p Fm(i)p Fl(\))1087 1379 y Fo(S)1112 1385 y Fm(A)1140 1379 y Fp(\()p Fo(\025)1180 1362 y Fl(1)1180 1389 y Fm(A)1218 1379 y Fk(!)g Fo(a)p Fp(\)\))701 1485 y(=)f Fo(\025)768 1468 y Fl(2)768 1495 y Fm(A)796 1485 y Fp(\()p Fo(a)834 1468 y Fj(0)845 1485 y Fo(S)870 1491 y Fm(A)898 1485 y Fp(\()p Fo(\025)938 1468 y Fl(1)938 1495 y Fm(A)977 1485 y Fk(!)g Fo(a)p Fp(\)\))h(=)f Fo(\025)1163 1491 y Fm(A)1191 1485 y Fp(\()p Fo(a)1229 1468 y Fj(0)1240 1485 y Fo(S)1265 1491 y Fm(A)1293 1485 y Fp(\()p Fo(g)1330 1467 y Fj(\000)p Fl(1)1329 1497 y Fm(A)1386 1485 y Fk(!)g Fo(a)p Fp(\)\))257 1563 y(Therefore,)k(if)e Fo(S)520 1569 y Fm(A)548 1563 y Fp(\()p Fo(a)p Fp(\))e(=)h(0,)h(then)i(w)o(e)f (ha)o(v)o(e)g Fo(\025)979 1569 y Fm(A)1017 1563 y Fg(\021)e Fo(a)f Fp(=)h(0,)h(and)h(therefore)h Fo(a)c Fp(=)h(0.)257 1640 y(\(4\))21 b(The)15 b(fact)f(that)g(the)h(map)e Fo(A)f Fk(!)g Fo(A)882 1625 y Fj(\003)901 1640 y Fo(;)7 b(a)12 b Fk(7!)f Fp(\()p Fo(\025)1047 1646 y Fm(A)1087 1640 y Fg(\021)h Fo(a)p Fp(\))i(is)g(bijectiv)o(e)g(implies)f(that)h (the)257 1690 y(bilinear)f(form)452 1767 y Fo(A)c Fk(\002)h Fo(A)h Fk(!)g Fo(A;)c Fp(\()p Fo(a;)g(a)758 1750 y Fj(0)769 1767 y Fp(\))12 b Fk(7!)f Fp(\()p Fo(\025)890 1773 y Fm(A)929 1767 y Fg(\021)g Fo(a)p Fp(\)\()p Fo(a)1058 1750 y Fj(0)1069 1767 y Fp(\))h(=)g Fo(\025)1165 1773 y Fm(A)1192 1767 y Fp(\()p Fo(a)1230 1750 y Fj(0)1242 1767 y Fo(S)1267 1773 y Fm(A)1294 1767 y Fp(\()p Fo(g)1331 1750 y Fj(\000)p Fl(1)1330 1780 y Fm(A)1388 1767 y Fk(!)f Fo(a)p Fp(\)\))257 1845 y(is)j(nondegenerate.)h(This)f(implies)e(that)i (also)f(the)i(bilinear)e(form)705 1922 y Fo(A)c Fk(\002)h Fo(A)h Fk(!)g Fo(A;)c Fp(\()p Fo(a;)g(a)1011 1905 y Fj(0)1022 1922 y Fp(\))12 b Fk(7!)f Fo(\025)1127 1928 y Fm(A)1154 1922 y Fp(\()p Fo(aa)1214 1905 y Fj(0)1226 1922 y Fp(\))257 2000 y(is)h(nondegenerate,)g(that)f(is,)g Fo(A)h Fp(is)f(a)g(F)m(rob)q (enius)h(algebra)e(with)h(resp)q(ect)j(to)d(the)h(F)m(rob)q(enius)257 2049 y(homomorphism)d Fo(\025)574 2055 y Fm(A)602 2049 y Fp(.)k(W)m(e)g(therefore)j(conclude)e(from)e(Prop)q(osition)h(2.7)g (that)h(there)h(is)f(a)257 2099 y(nonzero)h(left)f(in)o(tegral)g(\003) 666 2105 y Fm(A)707 2099 y Fp(and)g(a)g(nonzero)h(righ)o(t)f(in)o (tegral)f(\000)1256 2105 y Fm(A)1297 2099 y Fp(that)h(are)h(unique)f (up)h(to)257 2149 y(scalar)f(m)o(ultiples.)e(W)m(e)h(kno)o(w)h(the)g (pro)q(of)g(of)f(that)h(Prop)q(osition)f(that)h(w)o(e)g(ha)o(v)o(e:)718 2226 y Fo(\025)742 2232 y Fm(A)769 2226 y Fp(\(\003)814 2232 y Fm(A)841 2226 y Fp(\))e Fk(6)p Fp(=)g(0)82 b Fo(\025)1040 2232 y Fm(A)1068 2226 y Fp(\(\000)1110 2232 y Fm(A)1137 2226 y Fp(\))11 b Fk(6)p Fp(=)h(0)257 2304 y(Since)e(w)o(e)g(kno)o(w)f (from)f(subsection)i(2.3)f(that)g Fo(A)991 2289 y Fj(\003)1020 2304 y Fp(is)g(righ)o(t)g(Y)m(etter-Drinfel'd)g(Hopf)g(algebra,)257 2354 y(and)16 b(therefore)i(from)c(subsection)j(2.5)e(that)i Fo(A)1013 2339 y Fj(\003)p Fm(op)d(cop)1144 2354 y Fp(is)h(a)h(left)g (Y)m(etter-Drinfel'd)g(Hopf)257 2403 y(algebra)d(o)o(v)o(er)g Fo(H)528 2388 y Fm(op)i(cop)626 2403 y Fp(,)e(w)o(e)g(conclude)h(that)g Fo(A)1002 2388 y Fj(\003)p Fm(op)g(cop)1130 2403 y Fp(con)o(tains)f(a)g (nonzero)h(left)f(in)o(tegral)257 2453 y Fo(\032)278 2459 y Fm(A)323 2453 y Fp(whic)o(h)j(ob)o(viously)g(is)g(a)h(righ)o(t)f (in)o(tegral)g(in)g Fo(A)1052 2438 y Fj(\003)1088 2453 y Fp(that)h(also)f(do)q(es)i(not)e(v)n(anish)g(on)h(the)257 2503 y(in)o(tegrals)d(of)f Fo(A)p Fp(.)h(Therefore,)g(the)h(\014rst)g (assertion)f(in)g(the)g(Prop)q(osition)f(is)h(no)o(w)g(pro)o(v)o(ed.) 953 2628 y(15)p eop %%Page: 16 16 16 15 bop 257 262 a Fp(\(5\))21 b(W)m(e)14 b(no)o(w)f(pro)q(ceed)j(to)d (pro)o(v)o(e)h(the)h(second)g(assertion.)f(W)m(e)f(ha)o(v)o(e:)511 353 y Fo(\025)535 359 y Fm(A)562 353 y Fp(\()p Fo(a)p Fp(\()p Fo(h)f Fk(!)f Fp(\003)734 359 y Fm(A)761 353 y Fp(\)\))h(=)f Fo(\025)872 359 y Fm(A)900 353 y Fp(\(\()p Fo(h)956 359 y Fl(2)975 353 y Fo(S)1002 335 y Fj(\000)p Fl(1)1000 365 y Fm(H)1047 353 y Fp(\()p Fo(h)1087 359 y Fl(1)1106 353 y Fp(\))g Fk(!)g Fo(a)p Fp(\)\()p Fo(h)1264 359 y Fl(3)1295 353 y Fk(!)g Fp(\003)1377 359 y Fm(A)1404 353 y Fp(\)\))805 421 y(=)g(\()p Fo(\025)888 427 y Fm(A)927 421 y Fk( )g Fo(h)1004 427 y Fl(2)1023 421 y Fp(\)\(\()p Fo(S)1098 403 y Fj(\000)p Fl(1)1096 433 y Fm(H)1144 421 y Fp(\()p Fo(h)1184 427 y Fl(1)1203 421 y Fp(\))g Fk(!)g Fo(a)p Fp(\)\003)1350 427 y Fm(A)1377 421 y Fp(\))805 489 y(=)g Fo(\023)863 495 y Fm(A)890 489 y Fp(\()p Fo(h)930 495 y Fl(2)949 489 y Fp(\))p Fo(\017)982 495 y Fm(A)1009 489 y Fp(\()p Fo(S)1052 471 y Fj(\000)p Fl(1)1050 501 y Fm(H)1097 489 y Fp(\()p Fo(h)1137 495 y Fl(1)1156 489 y Fp(\))h Fk(!)f Fo(a)p Fp(\))p Fo(\025)1299 495 y Fm(A)1326 489 y Fp(\(\003)1371 495 y Fm(A)1398 489 y Fp(\))805 551 y(=)g Fo(\023)863 557 y Fm(A)890 551 y Fp(\()p Fo(h)p Fp(\))p Fo(\025)970 557 y Fm(A)998 551 y Fp(\()p Fo(a)p Fp(\003)1065 557 y Fm(A)1092 551 y Fp(\))257 642 y(This)k(implies)d Fo(h)h Fk(!)f Fp(\003)614 648 y Fm(A)653 642 y Fp(=)h Fo(\023)713 648 y Fm(A)740 642 y Fp(\()p Fo(h)p Fp(\)\003)825 648 y Fm(A)852 642 y Fp(,)h(since)h(the)g(bilinear)f(form)e(considered) k(ab)q(o)o(v)o(e)f(w)o(as)257 692 y(nondegenerate.)h(By)f(a)f(similar)e (calculation,)h(w)o(e)i(can)g(sho)o(w)g(that)f Fo(h)f Fk(!)f Fp(\000)1454 698 y Fm(A)1494 692 y Fp(=)h Fo(\023)1554 698 y Fm(A)1580 692 y Fp(\()p Fo(h)p Fp(\)\000)1662 698 y Fm(A)257 742 y Fp(and)h(in)g(turn)g(that)g Fo(\032)590 748 y Fm(A)629 742 y Fk( )d Fo(h)g Fp(=)h Fo(\023)776 748 y Fm(A)803 742 y Fp(\()p Fo(h)p Fp(\))p Fo(\032)880 748 y Fm(A)908 742 y Fp(.)257 809 y(W)m(e)i(ha)o(v)o(e)g(already)f (seen)i(the)g(equalit)o(y)e Fo(a)913 793 y Fl(1)931 809 y Fo(\025)955 815 y Fm(A)983 809 y Fp(\()p Fo(a)1021 793 y Fl(2)1039 809 y Fp(\))f(=)g Fo(g)1131 815 y Fm(A)1158 809 y Fo(\025)1182 815 y Fm(A)1209 809 y Fp(\()p Fo(a)p Fp(\).)h(This)h(implies:)428 900 y(\003)457 883 y Fl(1)457 910 y Fm(A)484 900 y Fo(\025)508 906 y Fm(A)535 900 y Fp(\()p Fo(a)p Fp(\003)602 883 y Fl(2)602 910 y Fm(A)629 900 y Fp(\))e(=)g Fo(S)726 906 y Fm(H)758 900 y Fp(\()p Fo(a)796 883 y Fl(1)814 900 y Fp(\))p Fo(a)852 883 y Fl(2)871 900 y Fp(\003)900 883 y Fl(1)900 910 y Fm(A)927 900 y Fo(\025)951 906 y Fm(A)978 900 y Fp(\()p Fo(a)1016 883 y Fl(3)1035 900 y Fp(\003)1064 883 y Fl(2)1064 910 y Fm(A)1091 900 y Fp(\))f(=)h Fo(S)1187 906 y Fm(H)1219 900 y Fp(\()p Fo(a)1257 883 y Fl(1)1276 900 y Fp(\))p Fo(g)1312 906 y Fm(A)1339 900 y Fo(\025)1363 906 y Fm(A)1390 900 y Fp(\()p Fo(a)1428 883 y Fl(2)1447 900 y Fp(\003)1476 906 y Fm(A)1503 900 y Fp(\))657 962 y(=)g Fo(\017)718 968 y Fm(A)745 962 y Fp(\()p Fo(a)p Fp(\))p Fo(g)819 968 y Fm(A)846 962 y Fo(\025)870 968 y Fm(A)897 962 y Fp(\(\003)942 968 y Fm(A)969 962 y Fp(\))g(=)f Fo(g)1060 968 y Fm(A)1087 962 y Fo(\025)1111 968 y Fm(A)1138 962 y Fp(\()p Fo(a)p Fp(\003)1205 968 y Fm(A)1232 962 y Fp(\))257 1053 y(This)20 b(implies)e(that)i Fo(\016)r Fp(\(\003)666 1059 y Fm(A)693 1053 y Fp(\))h(=)h Fo(g)804 1059 y Fm(A)844 1053 y Fk(\012)14 b Fp(\003)919 1059 y Fm(A)946 1053 y Fp(.)19 b(Similarly)l(,)e(w)o(e)j(can)g(pro)o(v)o(e)g(that)f Fo(\016)r Fp(\(\000)1592 1059 y Fm(A)1620 1053 y Fp(\))i(=)257 1103 y Fo(g)277 1109 y Fm(A)315 1103 y Fk(\012)12 b Fp(\000)385 1109 y Fm(A)412 1103 y Fp(.)k(The)h(fact)f(that)h Fo(a)727 1088 y Fl(1)745 1103 y Fo(\032)766 1109 y Fm(A)794 1103 y Fp(\()p Fo(a)832 1088 y Fl(2)851 1103 y Fp(\))f(=)g Fo(g)951 1109 y Fm(A)978 1103 y Fo(\032)999 1109 y Fm(A)1026 1103 y Fp(\()p Fo(a)p Fp(\))h(can)g(b)q(e)g(deduced)h(b)o(y)f(applying) e(the)257 1153 y(preceding)j(argumen)o(ts)e(to)h Fo(A)737 1138 y Fj(\003)8 b Fm(op)g(cop)853 1153 y Fp(,)17 b(whic)o(h)f(is)h(a)g (left)f(Y)m(etter-Drinfel'd)h(mo)q(dule)e(o)o(v)o(er)257 1203 y Fo(H)295 1188 y Fm(op)8 b(cop)401 1203 y Fp(b)o(y)13 b(Lemma)e(2.3)i(and)h(Lemma)d(2.5.)257 1286 y(\(6\))21 b(W)m(e)13 b(shall)g(no)o(w)g(pro)o(v)o(e)g(part)h(\(3\))f(of)g(the)h (Prop)q(osition.)f(Applying)f(the)i(com)o(ultiplica-)257 1336 y(tion)g(to)f(the)i(equation)e Fo(a)p Fp(\003)685 1342 y Fm(A)724 1336 y Fp(=)f Fo(\017)785 1342 y Fm(A)811 1336 y Fp(\()p Fo(a)p Fp(\)\003)894 1342 y Fm(A)935 1336 y Fp(yields:)563 1427 y Fo(a)585 1433 y Fl(1)604 1427 y Fp(\()p Fo(a)642 1433 y Fl(2)661 1410 y(1)691 1427 y Fk(!)f Fp(\003)773 1433 y Fm(A)r Fl(1)818 1427 y Fp(\))f Fk(\012)f Fo(a)907 1433 y Fl(2)926 1410 y(2)944 1427 y Fp(\003)973 1433 y Fm(A)r Fl(2)1030 1427 y Fp(=)j Fo(\017)1091 1433 y Fm(A)1118 1427 y Fp(\()p Fo(a)p Fp(\)\003)1201 1433 y Fm(A)r Fl(1)1256 1427 y Fk(\012)e Fp(\003)1327 1433 y Fm(A)q Fl(2)1372 1427 y Fo(:)257 1518 y Fp(W)m(e)k(therefore)h (ha)o(v)o(e:)380 1610 y Fo(\017)397 1616 y Fm(A)424 1610 y Fp(\()p Fo(a)p Fp(\)\003)507 1616 y Fm(A)r Fl(1)562 1610 y Fk(\012)9 b Fo(S)628 1616 y Fm(A)656 1610 y Fp(\(\003)701 1616 y Fm(A)r Fl(2)747 1610 y Fp(\))i(=)h Fo(a)840 1616 y Fl(1)859 1610 y Fp(\()p Fo(a)897 1616 y Fl(2)915 1593 y(1)945 1610 y Fk(!)f Fp(\003)1027 1616 y Fm(A)r Fl(1)1073 1610 y Fp(\))e Fk(\012)h Fo(S)1165 1616 y Fm(A)1192 1610 y Fp(\()p Fo(a)1230 1616 y Fl(2)1249 1593 y(2)1268 1610 y Fp(\003)1297 1616 y Fm(A)r Fl(2)1342 1610 y Fp(\))774 1677 y(=)i Fo(a)840 1683 y Fl(1)859 1677 y Fp(\()p Fo(a)897 1683 y Fl(2)915 1660 y(1)945 1677 y Fk(!)f Fp(\003)1027 1683 y Fm(A)r Fl(1)1073 1677 y Fp(\))e Fk(\012)h Fo(S)1165 1683 y Fm(A)1192 1677 y Fp(\()p Fo(a)1230 1683 y Fl(2)1249 1660 y(2)1279 1677 y Fk(!)h Fp(\003)1361 1683 y Fm(A)r Fl(2)1407 1677 y Fp(\))p Fo(S)1448 1683 y Fm(A)1475 1677 y Fp(\()p Fo(a)1513 1683 y Fl(2)1532 1660 y(3)1551 1677 y Fp(\))774 1745 y(=)h Fo(a)840 1751 y Fl(1)859 1745 y Fp(\003)888 1751 y Fm(A)q Fl(1)942 1745 y Fk(\012)e Fo(S)1009 1751 y Fm(A)1036 1745 y Fp(\(\003)1081 1751 y Fm(A)r Fl(2)1127 1745 y Fp(\))p Fo(S)1168 1751 y Fm(A)1196 1745 y Fp(\()p Fo(\023)1227 1751 y Fm(A)1253 1745 y Fp(\()p Fo(a)1291 1751 y Fl(2)1310 1727 y(1)1329 1745 y Fp(\))p Fo(a)1367 1751 y Fl(2)1385 1727 y(2)1404 1745 y Fp(\))257 1836 y(Con)o(v)o(olution)i(in)o(v)o(ersion)i(yields:)488 1927 y Fo(S)513 1933 y Fm(A)540 1927 y Fp(\()p Fo(a)p Fp(\)\003)623 1933 y Fm(A)s Fl(1)678 1927 y Fk(\012)c Fo(S)745 1933 y Fm(A)772 1927 y Fp(\(\003)817 1933 y Fm(A)r Fl(2)863 1927 y Fp(\))i(=)g(\003)964 1933 y Fm(A)q Fl(1)1018 1927 y Fk(\012)e Fo(S)1085 1933 y Fm(A)1112 1927 y Fp(\(\003)1157 1933 y Fm(A)r Fl(2)1203 1927 y Fp(\))p Fo(S)1244 1933 y Fm(A)1271 1927 y Fp(\()p Fo(\023)1302 1933 y Fm(A)1329 1927 y Fp(\()p Fo(a)1367 1910 y Fl(1)1386 1927 y Fp(\))p Fo(a)1424 1910 y Fl(2)1443 1927 y Fp(\))257 2019 y(If)g(w)o(e)h(apply)e(the)i(mapping)d Fo(a)j Fk(7!)g Fo(\023)799 2001 y Fj(\000)p Fl(1)799 2031 y Fm(A)844 2019 y Fp(\()p Fo(a)882 2003 y Fl(1)900 2019 y Fp(\))p Fo(a)938 2003 y Fl(2)967 2019 y Fp(to)f(the)h(second)g(tensorand)g(in)f (this)g(equation,)257 2068 y(w)o(e)k(get:)351 2160 y Fo(S)376 2166 y Fm(A)404 2160 y Fp(\()p Fo(a)p Fp(\)\003)487 2166 y Fm(A)r Fl(1)542 2160 y Fk(\012)9 b Fo(\023)598 2142 y Fj(\000)p Fl(1)598 2172 y Fm(A)642 2160 y Fp(\(\003)687 2166 y Fm(A)r Fl(2)733 2142 y(1)752 2160 y Fp(\))p Fo(S)793 2166 y Fm(A)820 2160 y Fp(\(\003)865 2166 y Fm(A)r Fl(2)911 2142 y(2)930 2160 y Fp(\))i(=)h(\003)1030 2166 y Fm(A)r Fl(1)1085 2160 y Fk(\012)d Fo(\023)1141 2142 y Fj(\000)p Fl(1)1141 2172 y Fm(A)1186 2160 y Fp(\(\003)1231 2166 y Fm(A)r Fl(2)1276 2142 y(1)1295 2160 y Fp(\))p Fo(S)1336 2166 y Fm(A)1363 2160 y Fp(\(\003)1408 2166 y Fm(A)r Fl(2)1454 2142 y(2)1473 2160 y Fp(\))p Fo(S)1514 2166 y Fm(A)1541 2160 y Fp(\()p Fo(a)p Fp(\))257 2251 y(Since)14 b(the)f(an)o(tip)q(o)q(de)f(of)g Fo(A)h Fp(is)g(bijectiv)o(e,)f(w)o(e)h (can)g(replace)g Fo(S)1210 2257 y Fm(A)1238 2251 y Fp(\()p Fo(a)p Fp(\))f(b)o(y)h Fo(a)f Fp(in)h(this)f(equation.)257 2301 y(W)m(e)i(no)o(w)f(ha)o(v)o(e:)367 2392 y Fo(\032)388 2398 y Fm(A)416 2392 y Fp(\()p Fo(a)p Fp(\003)483 2398 y Fm(A)r Fl(1)528 2392 y Fp(\))p Fo(\023)559 2374 y Fj(\000)p Fl(1)559 2404 y Fm(A)604 2392 y Fp(\(\003)649 2398 y Fm(A)r Fl(2)694 2374 y(1)713 2392 y Fp(\))p Fo(S)754 2398 y Fm(A)782 2392 y Fp(\(\003)827 2398 y Fm(A)r Fl(2)872 2374 y(2)891 2392 y Fp(\))f(=)f Fo(\032)983 2398 y Fm(A)1011 2392 y Fp(\(\003)1056 2398 y Fm(A)r Fl(1)1102 2392 y Fp(\))p Fo(\023)1133 2374 y Fj(\000)p Fl(1)1133 2404 y Fm(A)1177 2392 y Fp(\(\003)1222 2398 y Fm(A)r Fl(2)1268 2374 y(1)1286 2392 y Fp(\))p Fo(S)1327 2398 y Fm(A)1355 2392 y Fp(\(\003)1400 2398 y Fm(A)r Fl(2)1446 2374 y(2)1464 2392 y Fp(\))p Fo(a)h Fp(=)g Fo(a)257 2483 y Fp(This)i(pro)o(v)o(es)g (the)h(assertion.)f Fg(\003)953 2628 y Fp(16)p eop %%Page: 17 17 17 16 bop 257 262 a Fn(2.11)48 b Fp(The)12 b(in)o(tegral)e(c)o (haracter)j(and)e(the)h(in)o(tegral)f(group)g(elemen)o(t)g(are)g(not)h (totally)e(un-)257 311 y(related:)i(Conjugation)e(b)o(y)i(the)g(in)o (tegral)f(group)g(elemen)o(t)g(is)h(the)g(same)e(as)i(co)q(conjugation) 257 361 y(b)o(y)g(the)h(in)o(tegral)f(c)o(haracter.)h(Neither,)f(they)h (can)f(b)q(e)h(arbitrary)f(c)o(haracters)i(or)e(grouplik)o(e)257 411 y(elemen)o(ts,)h(they)h(ha)o(v)o(e)g(to)f(b)q(e)h(cen)o(tral)g (among)e(the)i(other)g(grouplik)o(es.)f(Recall)g(from)e([19)o(],)257 461 y(Example)16 b(9.1.4)f(that)i(the)h(c)o(haracters)h(are)e (precisely)h(the)g(grouplik)o(e)e(elemen)o(ts)h(in)g(the)257 511 y(\014nite)d(dual)g Fo(H)494 496 y Fj(\016)526 511 y Fp(of)g Fo(H)s Fp(.)257 610 y Fn(Theorem)36 b Fp(Supp)q(ose)13 b(that)e Fo(\023)744 616 y Fm(A)783 610 y Fp(and)h Fo(g)882 616 y Fm(A)920 610 y Fp(are)h(the)f(in)o(tegral)f(c)o(haracter)i(resp.) g(the)f(in)o(tegral)257 660 y(group)j(elemen)o(t)f(of)h(a)f(\014nite)h (dimensional)e(left)h(Y)m(etter-Drinfel'd)h(Hopf)f(algebra)g Fo(A)p Fp(.)h(W)m(e)257 710 y(ha)o(v)o(e:)308 829 y(1.)20 b Fk(8)9 b Fo(h)j Fk(2)f Fo(H)j Fp(:)e Fo(g)561 835 y Fm(A)587 829 y Fo(hg)632 811 y Fj(\000)p Fl(1)631 841 y Fm(A)689 829 y Fp(=)g Fo(\023)748 811 y Fj(\000)p Fl(1)748 841 y Fm(A)792 829 y Fp(\()p Fo(h)832 835 y Fl(1)850 829 y Fp(\))p Fo(h)890 835 y Fl(2)909 829 y Fo(\023)924 835 y Fm(A)951 829 y Fp(\()p Fo(h)991 835 y Fl(3)1010 829 y Fp(\))308 912 y(2.)20 b Fo(g)381 918 y Fm(A)420 912 y Fk(2)11 b Fo(Z)s Fp(\()p Fo(G)p Fp(\()p Fo(H)s Fp(\)\),)i(the)i(cen)o(ter)g(of)f Fo(G)p Fp(\()p Fo(H)s Fp(\).)308 995 y(3.)20 b Fo(\023)376 1001 y Fm(A)414 995 y Fk(2)12 b Fo(Z)s Fp(\()p Fo(G)p Fp(\()p Fo(H)588 980 y Fj(\016)607 995 y Fp(\)\))257 1094 y Fn(Pro)q(of.)36 b Fp(Select)15 b(a)f(righ)o(t)f(in)o(tegral)g Fo(\032)850 1100 y Fm(A)890 1094 y Fk(2)e Fo(A)960 1079 y Fj(\003)993 1094 y Fp(and)j(a)f(left)h(in)o(tegral)f(\003)1361 1100 y Fm(A)1400 1094 y Fk(2)e Fo(A)j Fp(that)g(satisfy)257 1144 y Fo(\032)278 1150 y Fm(A)306 1144 y Fp(\(\003)351 1150 y Fm(A)378 1144 y Fp(\))e(=)f(1.)i(The)i(Y)m(etter-Drinfel'd)e (condition)h(implies:)480 1235 y(\()p Fo(h)520 1241 y Fl(1)551 1235 y Fk(!)d Fp(\003)633 1241 y Fm(A)660 1235 y Fp(\))676 1218 y Fl(1)694 1235 y Fo(h)718 1241 y Fl(2)737 1235 y Fo(\032)758 1241 y Fm(A)785 1235 y Fp(\(\()p Fo(h)841 1241 y Fl(1)872 1235 y Fk(!)g Fp(\003)954 1241 y Fm(A)981 1235 y Fp(\))997 1218 y Fl(2)1015 1235 y Fp(\))h(=)g Fo(h)1111 1241 y Fl(1)1129 1235 y Fp(\003)1158 1241 y Fm(A)1185 1218 y Fl(1)1204 1235 y Fo(\032)1225 1241 y Fm(A)1252 1235 y Fp(\()p Fo(h)1292 1241 y Fl(2)1323 1235 y Fk(!)f Fp(\003)1405 1241 y Fm(A)1432 1218 y Fl(2)1450 1235 y Fp(\))257 1327 y(No)o(w)g(the)h(left)e(hand)h(side)h(is)e Fo(\023)722 1333 y Fm(A)749 1327 y Fp(\()p Fo(h)789 1333 y Fl(1)808 1327 y Fp(\)\003)853 1333 y Fm(A)880 1312 y Fl(1)898 1327 y Fo(h)922 1333 y Fl(2)941 1327 y Fo(\032)962 1333 y Fm(A)990 1327 y Fp(\(\003)1035 1333 y Fm(A)1062 1312 y Fl(2)1080 1327 y Fp(\))i(=)g Fo(\023)1167 1333 y Fm(A)1193 1327 y Fp(\()p Fo(h)1233 1333 y Fl(1)1252 1327 y Fp(\))p Fo(g)1288 1333 y Fm(A)1315 1327 y Fo(h)1339 1333 y Fl(2)1358 1327 y Fp(,)e(whereas)i(the)g(righ)o(t)257 1377 y(hand)k(side)h(is)f Fo(h)517 1383 y Fl(1)535 1377 y Fo(g)555 1383 y Fm(A)582 1377 y Fo(\032)603 1383 y Fm(A)631 1377 y Fp(\()p Fo(h)671 1383 y Fl(2)704 1377 y Fk(!)f Fp(\003)790 1383 y Fm(A)817 1377 y Fp(\))g(=)h Fo(h)920 1383 y Fl(1)938 1377 y Fo(g)958 1383 y Fm(A)985 1377 y Fo(\023)1000 1383 y Fm(A)1027 1377 y Fp(\()p Fo(h)1067 1383 y Fl(2)1086 1377 y Fp(\).)f(This)h(pro)o(v)o(es)h(the)f(\014rst)h (assertion.)257 1426 y(T)m(o)f(pro)o(v)o(e)h(the)g(second)h(assertion,) f(assume)f(that)g Fo(g)i Fk(2)e Fo(G)p Fp(\()p Fo(H)s Fp(\))g(is)g(a)h(grouplik)o(e)e(elemen)o(t.)257 1476 y(Then)g(w)o(e)f(ha)o(v)o(e:)726 1526 y Fo(g)746 1532 y Fm(A)773 1526 y Fo(g)q(g)815 1508 y Fj(\000)p Fl(1)814 1538 y Fm(A)871 1526 y Fp(=)e Fo(\023)930 1508 y Fj(\000)p Fl(1)930 1538 y Fm(A)974 1526 y Fp(\()p Fo(g)q Fp(\))p Fo(g)q(\023)1063 1532 y Fm(A)1091 1526 y Fp(\()p Fo(g)q Fp(\))g(=)g Fo(g)257 1601 y Fp(This)20 b(pro)o(v)o(es)h(the)f(second)h (statemen)o(t.)e(The)i(third)f(statemen)o(t)f(follo)o(ws)g(from)f (similar)257 1651 y(considerations.)c Fg(\003)257 1786 y Fn(2.12)48 b Fp(The)16 b(fact)f(that)h(the)g(space)g(of)f(in)o (tegrals)g(is)g(one-dimensional)e(leads)j(to)f(the)h(ex-)257 1836 y(istence)21 b(of)d(certain)i(elemen)o(ts)f(whic)o(h)g(w)o(e)g (study)h(no)o(w.)e(Supp)q(ose)i(that)f Fo(A)h Fp(is)f(a)f(\014nite-)257 1886 y(dimensional)e(left)j(Y)m(etter-Drinfel'd)f(Hopf)g(algebra)g(o)o (v)o(er)g Fo(H)j Fp(and)e(that)f(\003)1487 1892 y Fm(A)1532 1886 y Fp(resp.)i(\000)1663 1892 y Fm(A)257 1935 y Fp(are)e(nonzero)h (left)e(resp.)h(righ)o(t)f(in)o(tegrals)g(of)g Fo(A)p Fp(.)g(Since)h(the)h(spaces)g(of)d(left)i(resp.)g(righ)o(t)257 1985 y(in)o(tegrals)13 b(are)g(one-dimensional,)d(and)j(since)h(\003) 1017 1991 y Fm(A)1043 1985 y Fo(a)f Fp(resp.)h Fo(a)p Fp(\000)1225 1991 y Fm(A)1264 1985 y Fp(is)f(ob)o(viously)f(again)f(a)i (left)257 2035 y(resp.)i(righ)o(t)e(in)o(tegral,)g(w)o(e)h(ha)o(v)o(e:) 638 2126 y(\003)667 2132 y Fm(A)694 2126 y Fo(a)d Fp(=)h Fo(\013)798 2109 y Fm(L)798 2137 y(A)825 2126 y Fp(\()p Fo(a)p Fp(\)\003)908 2132 y Fm(A)1018 2126 y Fo(a)p Fp(\000)1066 2132 y Fm(A)1104 2126 y Fp(=)g Fo(\013)1175 2109 y Fm(R)1175 2137 y(A)1202 2126 y Fp(\()p Fo(a)p Fp(\)\000)1282 2132 y Fm(A)257 2218 y Fp(for)j(some)f(n)o(um)o(b)q(ers)h Fo(\013)623 2203 y Fm(L)623 2229 y(A)650 2218 y Fp(\()p Fo(a)p Fp(\))g(and)g Fo(\013)828 2203 y Fm(R)828 2229 y(A)855 2218 y Fp(\()p Fo(a)p Fp(\).)g Fo(\013)963 2203 y Fm(L)963 2229 y(A)1005 2218 y Fp(and)g Fo(\013)1114 2203 y Fm(R)1114 2229 y(A)1156 2218 y Fp(are)g(ob)o(viously)f(c)o (haracters)j(of)d Fo(A)257 2267 y Fp(whic)o(h)k(are)f(indep)q(enden)o (t)i(of)e(the)g(c)o(hoice)h(of)f(the)h(in)o(tegrals.)e(They)i(are)g (called)f(the)h(left)257 2317 y(resp.)g(the)f(righ)o(t)f(mo)q(dular)f (function)h(of)g Fo(A)p Fp(.)h(Dually)m(,)d(if)i Fo(\025)1190 2323 y Fm(A)1234 2317 y Fp(resp.)h Fo(\032)1357 2323 y Fm(A)1401 2317 y Fp(are)g(nonzero)h(left)257 2367 y(resp.)d(righ)o(t) e(in)o(tegrals)h(of)f Fo(A)703 2352 y Fj(\003)722 2367 y Fp(,)h(w)o(e)g(ha)o(v)o(e)g(for)f(all)g Fo(a)e Fk(2)g Fo(A)p Fp(:)560 2458 y Fo(\025)584 2464 y Fm(A)611 2458 y Fp(\()p Fo(a)649 2464 y Fl(1)668 2458 y Fp(\))p Fo(a)706 2464 y Fl(2)736 2458 y Fp(=)h Fo(\025)804 2464 y Fm(A)831 2458 y Fp(\()p Fo(a)p Fp(\))p Fo(a)907 2441 y Fm(L)907 2469 y(A)1018 2458 y Fo(a)1040 2464 y Fl(1)1058 2458 y Fo(\032)1079 2464 y Fm(A)1107 2458 y Fp(\()p Fo(a)1145 2464 y Fl(2)1163 2458 y Fp(\))g(=)g Fo(\032)1256 2464 y Fm(A)1283 2458 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1359 2441 y Fm(R)1359 2469 y(A)953 2628 y Fp(17)p eop %%Page: 18 18 18 17 bop 257 262 a Fp(for)16 b(t)o(w)o(o)f(grouplik)o(e)f(elemen)o(ts) i Fo(a)778 246 y Fm(L)778 273 y(A)820 262 y Fp(and)f Fo(a)924 246 y Fm(R)924 273 y(A)967 262 y Fp(of)g Fo(A)p Fp(.)g(These)i(elemen)o(ts)e(are)h(called)g(the)g(left)257 311 y(resp.)h(righ)o(t)f(mo)q(dular)e(elemen)o(ts)i(of)g Fo(A)p Fp(.)g(Ob)o(viously)m(,)e(the)j(mo)q(dular)d(functions)i(of)g Fo(A)g Fp(are)257 361 y(the)g(mo)q(dular)d(elemen)o(ts)h(of)g Fo(A)745 346 y Fj(\003)765 361 y Fp(,)g(whereas)i(the)f(mo)q(dular)e (elemen)o(ts)i(of)f Fo(A)h Fp(represen)o(t)i(the)257 411 y(mo)q(dular)12 b(functions)i(of)g Fo(A)680 396 y Fj(\003)713 411 y Fp(in)f(the)i(bidual)d(space.)257 511 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)13 b(that)f Fo(A)g Fp(is)g(a)f(\014nite-dimensional)f(left)h(Y)m(etter-Drinfel'd)h(Hopf) 257 560 y(algebra)i(o)o(v)o(er)g Fo(H)s Fp(.)308 687 y(1.)20 b(The)13 b(an)o(tip)q(o)q(de)e(of)h Fo(A)g Fp(maps)e(left)i(in) o(tegrals)g(to)f(righ)o(t)h(in)o(tegrals)g(and)f(righ)o(t)h(in)o (tegrals)361 737 y(to)i(left)g(in)o(tegrals.)308 820 y(2.)20 b Fo(\013)388 805 y Fm(L)388 832 y(A)429 820 y Fp(and)13 b Fo(\013)536 805 y Fm(R)536 832 y(A)577 820 y Fp(are)h(H-linear)g(and)f(colinear.)308 903 y(3.)20 b(W)m(e)14 b(ha)o(v)o(e:)672 995 y Fo(h)e Fk(!)f Fo(a)783 978 y Fm(L)783 1005 y(A)821 995 y Fp(=)h Fo(\017)882 1001 y Fm(H)913 995 y Fp(\()p Fo(h)p Fp(\))p Fo(a)991 978 y Fm(L)991 1005 y(A)1101 995 y Fo(\016)r Fp(\()p Fo(a)1159 978 y Fm(L)1159 1005 y(A)1187 995 y Fp(\))f(=)h(1)d Fk(\012)h Fo(a)1352 978 y Fm(L)1352 1005 y(A)672 1063 y Fo(h)h Fk(!)g Fo(a)782 1046 y Fm(R)782 1073 y(A)821 1063 y Fp(=)h Fo(\017)882 1069 y Fm(H)913 1063 y Fp(\()p Fo(h)p Fp(\))p Fo(a)991 1046 y Fm(R)991 1073 y(A)1101 1063 y Fo(\016)r Fp(\()p Fo(a)1159 1046 y Fm(R)1159 1073 y(A)1187 1063 y Fp(\))f(=)h(1)d Fk(\012)h Fo(a)1352 1046 y Fm(R)1352 1073 y(A)361 1159 y Fp(that)20 b(is,)g(the)g(mapping)e Fo(K)25 b Fk(!)c Fo(A;)7 b(\030)24 b Fk(7!)d Fo(\030)r(a)1091 1144 y Fm(L)1091 1170 y(A)1138 1159 y Fp(is)f Fo(H)s Fp(-linear)f(and)h(colinear)f(\(and)361 1209 y(similarly)11 b(for)j Fo(a)617 1193 y Fm(R)617 1220 y(A)644 1209 y Fp(\).)308 1292 y(4.)20 b Fo(a)383 1276 y Fm(L)383 1303 y(A)422 1292 y Fp(=)11 b(\()p Fo(a)503 1276 y Fm(R)503 1303 y(A)531 1292 y Fp(\))547 1276 y Fj(\000)p Fl(1)605 1292 y Fp(and)j Fo(\013)713 1276 y Fm(L)713 1303 y(A)751 1292 y Fp(=)e(\()p Fo(\013)838 1276 y Fm(R)838 1303 y(A)865 1292 y Fp(\))881 1276 y Fj(\000)p Fl(1)926 1292 y Fp(,)h(where)i(\()p Fo(\013)1114 1276 y Fm(R)1114 1303 y(A)1141 1292 y Fp(\))1157 1276 y Fj(\000)p Fl(1)1214 1292 y Fp(=)c Fo(\013)1284 1276 y Fm(R)1284 1303 y(A)1320 1292 y Fk(\016)e Fo(S)1375 1298 y Fm(A)1403 1292 y Fp(.)257 1391 y Fn(Pro)q(of.)36 b Fp(T)m(o)12 b(pro)o(v)o(e)h(the)g(\014rst)g(assertion,)g(observ)o(e)h (that)e(for)h(a)f(left)g(in)o(tegral)g(\003)1513 1397 y Fm(A)1553 1391 y Fp(of)g Fo(A)g Fp(w)o(e)257 1441 y(ha)o(v)o(e:)636 1532 y Fo(\017)653 1538 y Fm(A)680 1532 y Fp(\()p Fo(a)p Fp(\))p Fo(S)759 1538 y Fm(A)787 1532 y Fp(\(\003)832 1538 y Fm(A)859 1532 y Fp(\))f(=)h Fo(S)955 1538 y Fm(A)983 1532 y Fp(\()p Fo(a)p Fp(\003)1050 1538 y Fm(A)1077 1532 y Fp(\))886 1600 y(=)g Fo(S)955 1606 y Fm(A)983 1600 y Fp(\()p Fo(a)1021 1583 y Fl(1)1051 1600 y Fk(!)f Fp(\003)1133 1606 y Fm(A)1160 1600 y Fp(\))p Fo(S)1201 1606 y Fm(A)1228 1600 y Fp(\()p Fo(a)1266 1583 y Fl(2)1285 1600 y Fp(\))886 1667 y(=)h Fo(S)955 1673 y Fm(A)983 1667 y Fp(\(\003)1028 1673 y Fm(A)1055 1667 y Fp(\))p Fo(S)1096 1673 y Fm(A)1123 1667 y Fp(\()p Fo(\023)1154 1673 y Fm(A)1181 1667 y Fp(\()p Fo(a)1219 1650 y Fl(1)1238 1667 y Fp(\))p Fo(a)1276 1650 y Fl(2)1295 1667 y Fp(\))257 1764 y(where)20 b Fo(\023)397 1770 y Fm(A)442 1764 y Fp(is)e(the)g(in)o(tegral)g(c)o(haracter)h(from) e(Prop)q(osition)h(2.10.)e(Inserting)j Fo(\023)1532 1746 y Fj(\000)p Fl(1)1532 1776 y Fm(A)1576 1764 y Fp(\()p Fo(a)1614 1749 y Fl(1)1633 1764 y Fp(\))p Fo(a)1671 1749 y Fl(2)257 1814 y Fp(instead)i(of)e Fo(a)h Fp(and)f(using)h(the)g (colinearit)o(y)f(of)h Fo(\017)1060 1820 y Fm(A)1087 1814 y Fp(,)f(w)o(e)h(see)h(that)f Fo(S)1379 1820 y Fm(A)1407 1814 y Fp(\(\003)1452 1820 y Fm(A)1479 1814 y Fp(\))g(is)f(a)h(righ)o (t)257 1864 y(in)o(tegral.)14 b(A)h(similar)e(calculation)h(sho)o(ws)h (that)g(the)h(an)o(tip)q(o)q(de)f(maps)e(righ)o(t)i(in)o(tegrals)g(to) 257 1914 y(left)f(in)o(tegrals.)257 1999 y(F)m(or)g(the)g(second)h (assertion,)f(w)o(e)h(pro)o(v)o(e)e(that)h Fo(\013)1012 1984 y Fm(R)1012 2011 y(A)1053 1999 y Fp(is)g Fo(H)s Fp(-linear:)576 2091 y Fo(\013)603 2073 y Fm(R)603 2101 y(A)630 2091 y Fp(\()p Fo(h)670 2097 y Fl(1)701 2091 y Fk(!)d Fo(a)p Fp(\))p Fo(\023)807 2097 y Fm(A)833 2091 y Fp(\()p Fo(h)873 2097 y Fl(2)892 2091 y Fp(\)\000)934 2097 y Fm(A)973 2091 y Fp(=)h(\()p Fo(h)1057 2097 y Fl(1)1087 2091 y Fk(!)f Fo(a)p Fp(\)\()p Fo(h)1218 2097 y Fl(2)1248 2091 y Fk(!)g Fp(\000)1327 2097 y Fm(A)1354 2091 y Fp(\))973 2153 y(=)h Fo(h)f Fk(!)g Fp(\()p Fo(a)p Fp(\000)1169 2159 y Fm(A)1196 2153 y Fp(\))973 2221 y(=)h Fo(\013)1044 2204 y Fm(R)1044 2232 y(A)1071 2221 y Fp(\()p Fo(a)p Fp(\))p Fo(h)f Fk(!)g Fp(\000)1239 2227 y Fm(A)973 2290 y Fp(=)h Fo(\013)1044 2273 y Fm(R)1044 2300 y(A)1071 2290 y Fp(\()p Fo(a)p Fp(\))p Fo(\023)1140 2296 y Fm(A)1167 2290 y Fp(\()p Fo(h)p Fp(\)\000)1249 2296 y Fm(A)257 2385 y Fp(and)21 b(therefore)h Fo(\013)553 2370 y Fm(R)553 2397 y(A)580 2385 y Fp(\()p Fo(h)620 2391 y Fl(1)661 2385 y Fk(!)h Fo(a)p Fp(\))p Fo(\023)779 2391 y Fm(A)805 2385 y Fp(\()p Fo(h)845 2391 y Fl(2)864 2385 y Fp(\))g(=)g Fo(\013)985 2370 y Fm(R)985 2397 y(A)1012 2385 y Fp(\()p Fo(a)p Fp(\))p Fo(\023)1081 2391 y Fm(A)1108 2385 y Fp(\()p Fo(h)p Fp(\).)d(Con)o(v)o(olution)f(m)o(ultiplicatio)o(n)257 2435 y(with)f Fo(\023)371 2417 y Fj(\000)p Fl(1)371 2447 y Fm(A)434 2435 y Fp(on)g(b)q(oth)g(sides)h(yields)f(the)h Fo(H)s Fp(-linearit)o(y)e(of)g Fo(\013)1197 2420 y Fm(R)1197 2447 y(A)1224 2435 y Fp(.)h(One)h(can)f(sho)o(w)g(similarly)257 2485 y(that)c Fo(\013)374 2470 y Fm(R)374 2497 y(A)414 2485 y Fp(is)f(also)g(colinear,)g(and)g(the)h(same)f(is)g(true)i(for)e Fo(\013)1178 2470 y Fm(L)1178 2497 y(A)1204 2485 y Fp(.)g(The)h(third)f (assertion)i(is)e(the)953 2628 y(18)p eop %%Page: 19 19 19 18 bop 257 262 a Fp(dualization)17 b(of)g(the)h(second)h(assertion)g (to)e Fo(A)1003 246 y Fj(\003)s Fm(op)8 b(cop)1132 262 y Fp(o)o(v)o(er)18 b Fo(H)1263 246 y Fm(op)8 b(cop)1372 262 y Fp(using)17 b(Lemma)e(2.3)257 311 y(and)f(Lemma)d(2.5.)257 397 y(It)i(remains)f(to)g(sho)o(w)h(the)h(fourth)e(assertion)i(in)e (the)h(Prop)q(osition.)f(Select)i(a)e(nonzero)i(left)257 447 y(in)o(tegral)k(\003)442 453 y Fm(A)488 447 y Fp(in)g Fo(A)p Fp(.)g(Using)g(the)h(grouplik)o(e)f(elemen)o(t)g Fo(g)1162 453 y Fm(A)1207 447 y Fp(from)f(Prop)q(osition)h(2.10,)f(w)o (e)257 497 y(calculate:)599 588 y Fo(\013)626 571 y Fm(L)626 598 y(A)653 588 y Fp(\()p Fo(a)p Fp(\))p Fo(S)732 594 y Fm(A)759 588 y Fp(\(\003)804 594 y Fm(A)831 588 y Fp(\))12 b(=)g Fo(S)928 594 y Fm(A)955 588 y Fp(\(\003)1000 594 y Fm(A)1027 588 y Fo(a)p Fp(\))859 656 y(=)g Fo(S)928 662 y Fm(A)955 656 y Fp(\(\003)1000 662 y Fm(A)1027 638 y Fl(1)1057 656 y Fk(!)g Fo(a)p Fp(\))p Fo(S)1174 662 y Fm(A)1201 656 y Fp(\(\003)1246 662 y Fm(A)1273 638 y Fl(2)1292 656 y Fp(\))859 718 y(=)g Fo(S)928 724 y Fm(A)955 718 y Fp(\()p Fo(g)991 724 y Fm(A)1030 718 y Fk(!)f Fo(a)p Fp(\))p Fo(S)1146 724 y Fm(A)1173 718 y Fp(\(\003)1218 724 y Fm(A)1245 718 y Fp(\))859 787 y(=)h Fo(\013)930 770 y Fm(R)930 797 y(A)957 787 y Fp(\()p Fo(S)998 793 y Fm(A)1025 787 y Fp(\()p Fo(g)1061 793 y Fm(A)1100 787 y Fk(!)f Fo(a)p Fp(\)\))p Fo(S)1232 793 y Fm(A)1260 787 y Fp(\(\003)1305 793 y Fm(A)1332 787 y Fp(\))859 856 y(=)h Fo(\013)930 838 y Fm(R)930 866 y(A)957 856 y Fp(\()p Fo(S)998 862 y Fm(A)1025 856 y Fp(\()p Fo(a)p Fp(\)\))p Fo(S)1120 862 y Fm(A)1148 856 y Fp(\(\003)1193 862 y Fm(A)1220 856 y Fp(\))257 947 y(The)17 b(assertion)f(ab)q(out)g(the)g(mo)q(dular)e(elemen)o(ts)i(is,) f(as)h(ab)q(o)o(v)o(e,)g(the)g(dualization)f(of)g(the)257 997 y(statemen)o(t)f(ab)q(out)g(the)g(mo)q(dular)e(functions.)i Fg(\003)257 1132 y Fn(2.13)48 b Fp(If)12 b Fo(H)j Fp(is)d (\014nite-dimensional,)e(there)k(are)f(of)e(course)j(also)e(the)h(mo)q (dular)d(functions)257 1182 y Fo(\013)284 1167 y Fm(L)284 1193 y(H)331 1182 y Fp(and)16 b Fo(\013)441 1167 y Fm(R)441 1193 y(A)484 1182 y Fp(as)g(w)o(ell)f(as)h(the)h(mo)q(dular)d(elemen)o (ts)h Fo(a)1110 1167 y Fm(L)1110 1193 y(H)1158 1182 y Fp(and)h Fo(a)1263 1167 y Fm(R)1263 1193 y(H)1310 1182 y Fp(for)f Fo(H)s Fp(.)h(The)g(follo)o(wing)257 1232 y(simple)h(iden)o(tit)o(y)h(relates)h(them)f(to)g(the)h(in)o(tegral)f (c)o(haracter)i(and)e(the)h(in)o(tegral)f(group)257 1282 y(elemen)o(t:)257 1381 y Fn(Prop)q(osition)33 b Fp(W)m(e)14 b(ha)o(v)o(e:)f Fo(\013)742 1366 y Fm(R)742 1393 y(H)773 1381 y Fp(\()p Fo(g)809 1387 y Fm(A)836 1381 y Fp(\))p Fo(\023)867 1387 y Fm(A)894 1381 y Fp(\()p Fo(a)932 1366 y Fm(R)932 1393 y(H)963 1381 y Fp(\))f(=)g(1)257 1481 y Fn(Pro)q(of.)36 b Fp(Select)16 b(righ)o(t)e(in)o(tegrals)g(\000)839 1487 y Fm(H)884 1481 y Fk(2)e Fo(H)17 b Fp(and)e Fo(\032)1079 1487 y Fm(H)1124 1481 y Fk(2)d Fo(H)1202 1466 y Fj(\003)1236 1481 y Fp(that)i(satisfy)h Fo(\032)1478 1487 y Fm(H)1510 1481 y Fp(\(\000)1552 1487 y Fm(H)1583 1481 y Fp(\))e(=)g(1.)257 1531 y(W)m(e)h(kno)o(w)f(from)f(Theorem)i(2.11)e(that:)673 1622 y Fo(g)693 1628 y Fm(A)720 1622 y Fo(\023)735 1628 y Fm(A)761 1622 y Fp(\(\000)803 1628 y Fm(H)5 b Fl(1)854 1622 y Fp(\)\000)896 1628 y Fm(H)t Fl(2)957 1622 y Fp(=)12 b(\000)1027 1628 y Fm(H)5 b Fl(1)1077 1622 y Fo(\023)1092 1628 y Fm(A)1119 1622 y Fp(\(\000)1161 1628 y Fm(H)t Fl(2)1211 1622 y Fp(\))p Fo(g)1247 1628 y Fm(A)257 1713 y Fp(If)14 b(w)o(e)g(apply)f Fo(\032)495 1719 y Fm(H)541 1713 y Fp(to)h(this)g(equation,)f(the)h(left)g(hand)g(side)g (simpli\014es)e(to:)472 1805 y Fo(\023)487 1811 y Fm(A)514 1805 y Fp(\(\000)556 1811 y Fm(H)5 b Fl(1)606 1805 y Fp(\))p Fo(\032)643 1811 y Fm(H)675 1805 y Fp(\()p Fo(g)711 1811 y Fm(A)738 1805 y Fp(\000)764 1811 y Fm(H)g Fl(2)815 1805 y Fp(\))11 b(=)h Fo(\023)901 1811 y Fm(A)928 1805 y Fp(\()p Fo(g)965 1787 y Fj(\000)p Fl(1)964 1817 y Fm(A)1010 1805 y Fp(\))p Fo(\023)1041 1811 y Fm(A)1068 1805 y Fp(\()p Fo(g)1104 1811 y Fm(A)1131 1805 y Fp(\000)1157 1811 y Fm(H)t Fl(1)1207 1805 y Fp(\))p Fo(\032)1244 1811 y Fm(H)1276 1805 y Fp(\()p Fo(g)1312 1811 y Fm(A)1339 1805 y Fp(\000)1365 1811 y Fm(H)t Fl(2)1415 1805 y Fp(\))842 1873 y(=)g Fo(\023)901 1879 y Fm(A)928 1873 y Fp(\()p Fo(g)965 1855 y Fj(\000)p Fl(1)964 1885 y Fm(A)1010 1873 y Fp(\))p Fo(\013)1053 1856 y Fm(R)1053 1883 y(H)1084 1873 y Fp(\()p Fo(g)1120 1879 y Fm(A)1147 1873 y Fp(\))p Fo(\023)1178 1879 y Fm(A)1205 1873 y Fp(\(\000)1247 1879 y Fm(H)t Fl(1)1297 1873 y Fp(\))p Fo(\032)1334 1879 y Fm(H)1366 1873 y Fp(\(\000)1408 1879 y Fm(H)5 b Fl(2)1458 1873 y Fp(\))842 1942 y(=)12 b Fo(\023)901 1948 y Fm(A)928 1942 y Fp(\()p Fo(g)965 1924 y Fj(\000)p Fl(1)964 1954 y Fm(A)1010 1942 y Fp(\))p Fo(\013)1053 1925 y Fm(R)1053 1952 y(H)1084 1942 y Fp(\()p Fo(g)1120 1948 y Fm(A)1147 1942 y Fp(\))p Fo(\023)1178 1948 y Fm(A)1205 1942 y Fp(\()p Fo(a)1243 1925 y Fm(R)1243 1952 y(H)1274 1942 y Fp(\))257 2033 y(whereas)k(the)e(righ)o(t)f(hand)h (side)g(yields:)494 2124 y Fo(\032)515 2130 y Fm(H)547 2124 y Fp(\(\000)589 2130 y Fm(H)5 b Fl(1)639 2124 y Fo(g)659 2130 y Fm(A)686 2124 y Fp(\))p Fo(\023)717 2130 y Fm(A)744 2124 y Fp(\(\000)786 2130 y Fm(H)g Fl(2)836 2124 y Fp(\))12 b(=)g Fo(\023)923 2130 y Fm(A)949 2124 y Fp(\()p Fo(g)986 2107 y Fj(\000)p Fl(1)985 2136 y Fm(A)1031 2124 y Fp(\))p Fo(\032)1068 2130 y Fm(H)1101 2124 y Fp(\(\000)1143 2130 y Fm(H)t Fl(1)1193 2124 y Fo(g)1213 2130 y Fm(A)1240 2124 y Fp(\))p Fo(\023)1271 2130 y Fm(A)1297 2124 y Fp(\(\000)1339 2130 y Fm(H)5 b Fl(2)1390 2124 y Fo(g)1410 2130 y Fm(A)1437 2124 y Fp(\))864 2192 y(=)12 b Fo(\023)923 2198 y Fm(A)949 2192 y Fp(\()p Fo(g)986 2175 y Fj(\000)p Fl(1)985 2205 y Fm(A)1031 2192 y Fp(\))p Fo(\032)1068 2198 y Fm(H)1101 2192 y Fp(\(\000)1143 2198 y Fm(H)t Fl(1)1193 2192 y Fp(\))p Fo(\023)1224 2198 y Fm(A)1251 2192 y Fp(\(\000)1293 2198 y Fm(H)t Fl(2)1343 2192 y Fp(\))864 2260 y(=)g Fo(\023)923 2266 y Fm(A)949 2260 y Fp(\()p Fo(g)986 2243 y Fj(\000)p Fl(1)985 2273 y Fm(A)1031 2260 y Fp(\))257 2352 y(Multiplication)g(b)o (y)i Fo(\023)601 2358 y Fm(A)628 2352 y Fp(\()p Fo(g)664 2358 y Fm(A)691 2352 y Fp(\))g(yields)f(the)i(assertion.)f Fg(\003)953 2628 y Fp(19)p eop %%Page: 20 20 20 19 bop 257 262 a Fn(2.14)48 b Fp(As)16 b(for)f(ordinary)g(Hopf)g (algebras,)g(there)i(is)e(a)g(v)o(ersion)h(of)f(Masc)o(hk)o(e's)h (theorem)257 311 y(for)e(Y)m(etter-Drinfel'd)g(Hopf)f(algebras.)h(This) g(w)o(as)f(already)h(observ)o(ed)h(b)o(y)f(D.)f(Fisc)o(hman,)257 361 y(S.)19 b(Mon)o(tgomery)e(and)i(H.)g(J.)g(Sc)o(hneider)h(\(cf.)f ([5)o(],)f(Cor.)g(5.8\),)g(in)h(fact,)f(they)i(pro)o(v)o(e)f(a)257 411 y(more)c(general)g(theorem)g(that)g(holds)g(for)g(the)h(subalgebra) g(of)e(coin)o(v)n(arian)o(ts)g(that)i(arises)257 461 y(from)j(a)h(cleft)g(Hopf)g(algebra)f(surjection.)i(Their)f(pro)q(of)g (rests)h(on)f(a)g(nice)h(Lemma)c(of)257 511 y(M.)h(Koppinen)g(\(cf.)f ([11)o(],)g(Prop.)h(5.2,)e(p.)i(442\).)f(W)m(e)g(also)g(note)h(that)g (it)g(is)g(p)q(ossible)g(to)257 560 y(generalize)f(this)g(theorem)f(to) g(Hopf)g(algebras)g(in)g(fairly)e(general)j(categories,)g(ho)o(w)o(ev)o (er,)257 610 y(it)e(seems)g(that)f(a)h(pro)q(of)f(of)g(this)h(result)g (has)g(not)f(y)o(et)h(b)q(een)h(published.)f(F)m(or)f(the)h(sak)o(e)g (of)257 660 y(completeness,)i(w)o(e)f(include)h(a)f(\(v)o(ery)h(sligh)o (tly)e(more)g(general\))i(v)n(arian)o(t)e(of)h(their)h(result)257 710 y(here,)c(and)f(giv)o(e)g(a)g(simple)f(pro)q(of)h(that)g(sta)o(ys)h (completely)e(within)h(the)g(framew)o(ork)f(of)h(the)257 760 y(metho)q(ds)i(dev)o(elop)q(ed)g(in)g(this)g(section.)257 859 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)13 b(that)f Fo(A)g Fp(is)g(a)f(\014nite-dimensional)f(left)h(Y)m(etter-Drinfel'd)h (Hopf)257 909 y(algebra.)g(Pic)o(k)h(a)f(nonzero)h(left)g(in)o(tegral)f (\003)942 915 y Fm(A)980 909 y Fk(2)f Fo(A)p Fp(.)i(Then)g(the)g(follo) o(wing)d(statemen)o(ts)j(are)257 959 y(equiv)n(alen)o(t:)308 1078 y(1.)20 b Fo(A)14 b Fp(is)g(a)f(semisimple)f(algebra.)308 1161 y(2.)20 b Fo(\017)378 1167 y Fm(A)405 1161 y Fp(\(\003)450 1167 y Fm(A)477 1161 y Fp(\))12 b Fk(6)p Fp(=)g(0.)257 1279 y(In)i(this)g(case,)h(w)o(e)f(ha)o(v)o(e)f Fo(\023)661 1285 y Fm(A)700 1279 y Fp(=)e Fo(\017)760 1285 y Fm(H)792 1279 y Fp(,)i Fo(g)837 1285 y Fm(A)876 1279 y Fp(=)e(1)940 1285 y Fm(H)985 1279 y Fp(and)j Fo(\013)1093 1264 y Fm(L)1093 1291 y(A)1131 1279 y Fp(=)e Fo(\013)1202 1264 y Fm(R)1202 1291 y(A)1241 1279 y Fp(=)f Fo(\017)1301 1285 y Fm(A)1328 1279 y Fp(.)257 1379 y Fn(Pro)q(of.)36 b Fp(If)16 b Fo(A)h Fp(is)f(semisimple,)d(then)k(the)g Fo(A)p Fp(-linear)f(map)e Fo(\017)1228 1385 y Fm(A)1271 1379 y Fp(:)h Fo(A)h Fk(!)f Fo(K)k Fp(from)c(the)i(left)257 1429 y(regular)i Fo(A)p Fp(-mo)q(dule)e Fo(A)i Fp(to)g(the)g(trivial)e Fo(A)p Fp(-mo)q(dule)g Fo(K)22 b Fp(has)d(a)f(righ)o(t)h(in)o(v)o(erse)g Fo(K)k Fk(!)c Fo(A)p Fp(.)257 1479 y(This)d(righ)o(t)g(in)o(v)o(erse)g (maps)f(the)h(unit)g(elemen)o(t)f(of)g Fo(K)k Fp(to)d(a)g(left)f(in)o (tegral)g(\003)1474 1485 y Fm(A)1517 1479 y Fp(of)g Fo(A)h Fp(that)257 1529 y(ob)o(viously)g(satis\014es)i Fo(\017)617 1535 y Fm(A)644 1529 y Fp(\(\003)689 1535 y Fm(A)716 1529 y Fp(\))e Fk(6)p Fp(=)h(0.)f(F)m(or)h(the)g(con)o(v)o(erse,)h (supp)q(ose)g(that)f(w)o(e)g(ha)o(v)o(e)g(a)g(left)257 1578 y(in)o(tegral)d(\003)438 1584 y Fm(A)478 1578 y Fp(that)g(satis\014es)h Fo(\017)738 1584 y Fm(A)765 1578 y Fp(\(\003)810 1584 y Fm(A)837 1578 y Fp(\))d(=)f(1.)j(Then)g(w)o(e)g (ha)o(v)o(e:)575 1670 y Fo(\023)590 1676 y Fm(A)616 1670 y Fp(\()p Fo(h)p Fp(\))p Fo(\017)689 1676 y Fm(A)716 1670 y Fp(\(\003)761 1676 y Fm(A)788 1670 y Fp(\))e(=)g Fo(\017)877 1676 y Fm(A)904 1670 y Fp(\()p Fo(h)f Fk(!)g Fp(\003)1037 1676 y Fm(A)1064 1670 y Fp(\))h(=)g Fo(\017)1153 1676 y Fm(H)1184 1670 y Fp(\()p Fo(h)p Fp(\))p Fo(\017)1257 1676 y Fm(A)1284 1670 y Fp(\(\003)1329 1676 y Fm(A)1356 1670 y Fp(\))626 1737 y Fo(g)646 1743 y Fm(A)672 1737 y Fo(\017)689 1743 y Fm(A)716 1737 y Fp(\(\003)761 1743 y Fm(A)788 1737 y Fp(\))g(=)g(\003)889 1720 y Fl(1)889 1747 y Fm(A)916 1737 y Fo(\017)933 1743 y Fm(A)960 1737 y Fp(\(\003)1005 1720 y Fl(2)1005 1747 y Fm(A)1032 1737 y Fp(\))f(=)h(1)p Fo(\017)1141 1743 y Fm(A)1168 1737 y Fp(\(\003)1213 1743 y Fm(A)1240 1737 y Fp(\))257 1833 y(In)i(addition,)f(w)o(e)h(ha)o(v)o(e)g Fo(\013)669 1818 y Fm(L)669 1844 y(A)695 1833 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)766 1839 y Fm(A)794 1833 y Fp(\(\003)839 1839 y Fm(A)866 1833 y Fp(\)\003)911 1839 y Fm(A)949 1833 y Fp(=)e(\003)1022 1839 y Fm(A)1049 1833 y Fo(a)p Fp(\003)1100 1839 y Fm(A)1139 1833 y Fp(=)g Fo(\017)1200 1839 y Fm(A)1227 1833 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1298 1839 y Fm(A)1325 1833 y Fp(\(\003)1370 1839 y Fm(A)1397 1833 y Fp(\)\003)1442 1839 y Fm(A)1469 1833 y Fp(.)h(In)h(particu-)257 1882 y(lar,)c(since)i Fo(\023)442 1888 y Fm(A)481 1882 y Fp(=)f Fo(\017)541 1888 y Fm(H)573 1882 y Fp(,)f(w)o(e)i(ha)o(v)o(e)e(b)o(y)h (Prop)q(osition)g(2.10)f(that)h(\003)1223 1888 y Fm(A)r Fl(1)1272 1882 y Fk(\012)t Fo(S)1333 1888 y Fm(A)1360 1882 y Fp(\(\003)1405 1888 y Fm(A)r Fl(2)1451 1882 y Fp(\))g(is)g(a)f(Casimir)257 1932 y(elemen)o(t.)i(Therefore,)i(if)e Fo(M)17 b Fp(is)c(an)f Fo(A)p Fp(-mo)q(dule)g(with)g(submo)q(dule)g Fo(N)5 b Fp(,)12 b(w)o(e)h(can)g(build)f(from)257 1982 y(a)i Fo(K)s Fp(-linear)f(pro)r(jection)i Fo(\031)d Fp(:)f Fo(M)17 b Fk(!)11 b Fo(N)19 b Fp(a)13 b(map)650 2073 y(~)-23 b Fo(\031)12 b Fp(:)f Fo(M)17 b Fk(!)11 b Fo(N)r(;)c(m)12 b Fk(7!)f Fp(\003)1001 2079 y Fm(A)r Fl(1)1046 2073 y Fo(\031)q Fp(\()p Fo(S)1112 2079 y Fm(A)1140 2073 y Fp(\(\003)1185 2079 y Fm(A)r Fl(2)1231 2073 y Fp(\))p Fo(m)p Fp(\))257 2165 y(whic)o(h)j(is)g(ob)o(viously)e(still)h(a)h(pro)r(jection)g(but)g (in)g(addition)f(is)g Fo(A)p Fp(-linear:)427 2256 y(~)-23 b Fo(\031)q Fp(\()p Fo(am)p Fp(\))12 b(=)g(\003)625 2262 y Fm(A)r Fl(1)671 2256 y Fo(\031)q Fp(\()p Fo(S)737 2262 y Fm(A)764 2256 y Fp(\(\003)809 2262 y Fm(A)r Fl(2)855 2256 y Fp(\))p Fo(am)p Fp(\))g(=)g Fo(a)p Fp(\003)1052 2262 y Fm(A)r Fl(1)1098 2256 y Fo(\031)q Fp(\()p Fo(S)1164 2262 y Fm(A)1191 2256 y Fp(\(\003)1236 2262 y Fm(A)r Fl(2)1282 2256 y Fp(\))p Fo(m)p Fp(\))g(=)g Fo(a)r Fp(~)-23 b Fo(\031)q Fp(\()p Fo(m)p Fp(\))257 2347 y(Therefore,)15 b Fo(A)f Fp(is)g(semisimple.)d Fg(\003)257 2466 y Fp(Using)j(Lemma)d (2.3)i(and)h(Lemma)d(2.5,)h(w)o(e)j(can)f(easily)f(deriv)o(e)h(a)g (dualization:)953 2628 y(20)p eop %%Page: 21 21 21 20 bop 257 262 a Fn(Corollary)35 b Fp(Supp)q(ose)19 b(that)f Fo(A)f Fp(is)h(a)f(\014nite-dimensional)e(left)j(Y)m (etter-Drinfel'd)f(Hopf)257 311 y(algebra.)e(Pic)o(k)g(a)f(nonzero)i (righ)o(t)f(in)o(tegral)f Fo(\032)976 317 y Fm(A)1017 311 y Fk(2)g Fo(A)1090 296 y Fj(\003)1109 311 y Fp(.)g(Then)i(the)g (follo)o(wing)c(statemen)o(ts)257 361 y(are)j(equiv)n(alen)o(t:)308 457 y(1.)20 b Fo(A)14 b Fp(is)g(a)f(cosemisimple)f(coalgebra.)308 533 y(2.)20 b Fo(\032)382 539 y Fm(A)410 533 y Fp(\(1)447 539 y Fm(A)474 533 y Fp(\))11 b Fk(6)p Fp(=)h(0.)257 628 y(In)i(this)g(case,)h(w)o(e)f(ha)o(v)o(e)f Fo(\023)661 634 y Fm(A)700 628 y Fp(=)e Fo(\017)760 634 y Fm(H)792 628 y Fp(,)i Fo(g)837 634 y Fm(A)876 628 y Fp(=)e(1)940 634 y Fm(H)985 628 y Fp(and)j Fo(a)1088 613 y Fm(L)1088 640 y(A)1127 628 y Fp(=)e Fo(a)1193 613 y Fm(R)1193 640 y(A)1231 628 y Fp(=)g(1)1296 634 y Fm(A)1323 628 y Fp(.)257 790 y Fq(3)67 b(Ribb)r(on)28 b(transformations)d(and)i(monoidal)g (trans-)358 865 y(formations)257 983 y Fn(3.1)48 b Fp(In)18 b(the)g(further)h(analysis)e(of)h(the)g(prop)q(erties)h(of)f(in)o (tegrals)f(in)h(Y)m(etter-Drinfel'd)257 1033 y(Hopf)d(algebras)g(w)o(e) h(shall)e(encoun)o(ter)j(sev)o(eral)f(natural)e(transformations)g(b)q (et)o(w)o(een)j(cer-)257 1083 y(tain)12 b(quasisymmetric)d(monoidal)g (functors)k(in)e(the)i(category)f(of)f(Y)m(etter-Drinfel'd)h(mo)q(d-) 257 1133 y(ules.)e(Since)h(their)f(prop)q(erties)h(are)g(b)q(etter)g (understo)q(o)q(d)h(from)c(a)h(more)g(general)i(viewp)q(oin)o(t,)257 1183 y(w)o(e)j(include)f(here)h(some)f(basic)g(remarks)g(on)g(monoidal) d(categories.)k(Our)f(basic)h(example)257 1233 y(con)o(tin)o(ues)i(to)f (b)q(e)g(the)g(category)h(of)e(left)h(Y)m(etter-Drinfel'd)f(mo)q(dules) g(considered)i(in)f(the)257 1282 y(preceding)e(section,)e(and)h (therefore)h(w)o(e)e(shall)g(also)g(in)f(this)i(section)g(denote)g(b)o (y)g Fo(H)i Fp(a)d(Hopf)257 1332 y(algebra)j(with)f(bijectiv)o(e)h(an)o (tip)q(o)q(de.)257 1459 y Fn(3.2)48 b Fp(Supp)q(ose)16 b(that)f Fk(C)i Fp(and)e Fk(D)h Fp(are)g(quasisymmetric)d(monoidal)f (categories)k(and)e(that)257 1509 y Fo(F)32 b Fp(:)25 b Fk(C)j(!)e(D)d Fp(and)g Fo(G)i Fp(:)h Fk(C)i(!)d(D)f Fp(are)f(t)o(w)o(o)f(quasisymmetric)e(monoidal)g(functors)257 1559 y(\(cf.)g([9)o(],p.)f(25\).)g(Suc)o(h)h(functors)g(come,)f(b)o(y)g (de\014nition,)g(along)g(with)h(certain)g(natural)257 1609 y(transformations)502 1682 y Fo(\036)527 1688 y Fl(2)p Fm(;A;B)628 1682 y Fp(:)11 b Fo(F)6 b(A)j Fk(\012)g Fo(F)d(B)14 b Fk(!)d Fo(F)6 b Fp(\()p Fo(A)j Fk(\012)g Fo(B)r Fp(\))84 b Fo(\036)1184 1688 y Fl(0)1214 1682 y Fp(:)11 b Fo(E)1268 1688 y Fj(D)1308 1682 y Fk(!)g Fo(F)6 b(E)1425 1688 y Fj(C)499 1744 y Fo( )526 1750 y Fl(2)p Fm(;A;B)627 1744 y Fp(:)11 b Fo(GA)e Fk(\012)h Fo(GB)k Fk(!)d Fo(G)p Fp(\()p Fo(A)e Fk(\012)g Fo(B)r Fp(\))84 b Fo( )1186 1750 y Fl(0)1216 1744 y Fp(:)11 b Fo(E)1270 1750 y Fj(D)1310 1744 y Fk(!)g Fo(GE)1427 1750 y Fj(C)257 1817 y Fp(that)f(satisfy)g(their)g(de\014ning)g (conditions)f(\(cf.)h([9)o(]\).)f(Here)i Fo(E)1186 1823 y Fj(C)1217 1817 y Fp(and)f Fo(E)1325 1823 y Fj(D)1363 1817 y Fp(denote)h(the)f(neutral)257 1866 y(ob)r(jects)17 b(of)e(the)h(monoidal)d(categories)j Fk(C)i Fp(and)e Fk(D)q Fp(.)f(W)m(e)g(consider)h(t)o(w)o(o)f(kinds)h(of)f(natural)257 1916 y(transformations)e Fo(\022)g Fp(:)e Fo(F)17 b Fk(!)11 b Fo(G)p Fp(:)257 2005 y Fn(De\014nition)33 b Fp(Supp)q(ose)21 b(that)f Fo(\022)k Fp(:)e Fo(F)27 b Fk(!)21 b Fo(G)f Fp(is)g(a)g(natural)g(transformation)e(b)q(et)o(w)o(een)257 2054 y(quasisymmetric)12 b(monoidal)f(functors.)308 2150 y(1.)20 b Fo(\022)e Fp(is)d(called)h(a)g(monoidal)c(transformation)i (\(cf.)i([9)o(],)f(p.)h(38\))f(if)g(the)i(follo)o(wing)c(t)o(w)o(o)361 2200 y(diagrams)f(comm)o(ute:)668 2473 y Fo(GA)d Fk(\012)g Fo(GB)403 b(G)p Fp(\()p Fo(A)9 b Fk(\012)g Fo(B)r Fp(\))p 861 2463 376 2 v 1195 2462 a Ff(-)1026 2499 y Fo( )1053 2505 y Fl(2)668 2307 y Fo(F)d(A)j Fk(\012)g Fo(F)d(B)403 b(F)6 b Fp(\()p Fo(A)j Fk(\012)g Fo(B)r Fp(\))p 861 2297 V 1195 2296 a Ff(-)1027 2280 y Fo(\036)1052 2286 y Fl(2)p 757 2433 2 108 v 758 2433 a Ff(?)596 2391 y Fo(\022)615 2397 y Fm(A)652 2391 y Fk(\012)g Fo(\022)712 2397 y Fm(B)p 1338 2433 V 1339 2433 a Ff(?)1356 2389 y Fo(\022)1375 2395 y Fm(A)p Fj(\012)p Fm(B)953 2628 y Fp(21)p eop %%Page: 22 22 22 21 bop 854 450 a Fo(E)885 456 y Fj(D)1132 450 y Fo(GE)1196 456 y Fj(C)p 926 440 194 2 v 1078 439 a Ff(-)1000 476 y Fo( )1027 482 y Fl(0)854 284 y Fo(E)885 290 y Fj(D)1132 284 y Fo(F)6 b(E)1196 290 y Fj(C)p 926 274 V 1078 273 a Ff(-)1001 257 y Fo(\036)1026 263 y Fl(0)p 883 410 2 108 v 884 410 a Ff(?)795 364 y Fp(1)816 370 y Fm(E)840 374 y Fi(D)p 1173 410 V 1174 410 a Ff(?)1191 365 y Fo(\022)1210 371 y Fm(E)1234 375 y Fi(C)308 582 y Fp(2.)20 b Fo(\022)13 b Fp(is)d(called)h(a)f(ribb)q(on)h(transformation)e(if)h(the)i(follo)o (wing)c(t)o(w)o(o)i(diagrams)f(comm)o(ute:)458 719 y Fo(F)d(A)j Fk(\012)g Fo(F)d(B)278 b(F)6 b Fp(\()p Fo(A)j Fk(\012)h Fo(B)r Fp(\))p 651 710 252 2 v 861 709 a Ff(-)755 693 y Fo(\036)780 699 y Fl(2)1371 719 y Fo(F)c Fp(\()p Fo(A)j Fk(\012)g Fo(B)r Fp(\))p 1107 710 V 1317 709 a Ff(-)1094 689 y Fo(F)d Fp(\()p Fo(\033)1167 695 y Fm(B)q(;A)1239 689 y Fk(\016)j Fo(\033)1293 695 y Fm(A;B)1356 689 y Fp(\))p 547 929 2 191 v 548 929 a Ff(?)386 845 y Fo(\022)405 851 y Fm(A)442 845 y Fk(\012)h Fo(\022)503 851 y Fm(B)p 1460 929 V 1461 929 a Ff(?)1478 843 y Fo(\022)1497 849 y Fm(A)p Fj(\012)p Fm(B)458 968 y Fo(GA)f Fk(\012)g Fo(GB)735 b(G)p Fp(\()p Fo(A)9 b Fk(\012)g Fo(B)r Fp(\))p 651 959 708 2 v 1317 958 a Ff(-)982 995 y Fo( )1009 1001 y Fl(2)854 1331 y Fo(E)885 1337 y Fj(D)1132 1331 y Fo(GE)1196 1337 y Fj(C)p 926 1321 194 2 v 1078 1320 a Ff(-)1000 1357 y Fo( )1027 1363 y Fl(0)854 1166 y Fo(E)885 1172 y Fj(D)1132 1166 y Fo(F)d(E)1196 1172 y Fj(C)p 926 1155 V 1078 1154 a Ff(-)1001 1138 y Fo(\036)1026 1144 y Fl(0)p 883 1291 2 108 v 884 1291 a Ff(?)795 1246 y Fp(1)816 1252 y Fm(E)840 1256 y Fi(D)p 1173 1291 V 1174 1291 a Ff(?)1191 1247 y Fo(\022)1210 1253 y Fm(E)1234 1257 y Fi(C)257 1474 y Fp(This)14 b(comm)o(utativi)o(t)o(y)d(is)j(of)f(course)i(required)g (for)f(all)e(ob)r(jects)j Fo(A;)7 b(B)14 b Fk(2)d(C)r Fp(.)257 1589 y(The)j(notion)e(of)g(a)g(ribb)q(on)h(transformation)e (is)i(closely)f(related)i(to)e(the)i(notion)e(of)g(a)g(t)o(wist)257 1639 y(\(cf.)17 b([9)o(],)f(Def.)h(6.1,)e(p.)i(65\):)f(If)h Fo(F)22 b Fp(and)17 b Fo(G)f Fp(are)i(the)f(iden)o(tit)o(y)g(functors,) g(then)h(in)o(v)o(ertible)257 1689 y(ribb)q(on)13 b(transformations)e (are)i(t)o(wists)g(in)f(the)h(category)g Fk(C)1180 1674 y Fj(0)1204 1689 y Fp(where)h Fo(\033)g Fp(is)e(replaced)h(b)o(y)g Fo(\033)1645 1674 y Fj(\000)p Fl(1)257 1739 y Fp(\(cf.)h([9)o(],)f (Def.)g(2.1,)g(p.)g(33\).)257 1872 y Fn(3.3)48 b Fp(In)11 b(our)h(con)o(text,)f(monoidal)d(functors)k(arise)g(in)f(the)h(follo)o (wing)c(w)o(a)o(y:)i(Supp)q(ose)i(that)257 1922 y Fo(')g Fp(:)f Fo(H)k Fk(!)c Fo(H)i Fp(is)e(a)g(Hopf)f(algebra)h(automorphism)d (and)i(that)h Fo(V)21 b Fp(is)10 b(a)h(left)g(Y)m(etter-Drinfel'd)257 1972 y(mo)q(dule)j(o)o(v)o(er)i Fo(H)s Fp(.)f(Then)g(w)o(e)h(can)g(use) g(the)g(isomorphism)c(to)j(in)o(tro)q(duce)i(a)e(new)h(Y)m(etter-)257 2022 y(Drinfel'd)e(mo)q(dule)e(structure)17 b(on)d Fo(V)24 b Fp(in)14 b(the)h(follo)o(wing)d(w)o(a)o(y:)h(The)i(new)g(action)f(is) g(giv)o(en)257 2072 y(as:)798 2121 y Fo(h)d(,)-7 b Fk(!)11 b Fo(v)i Fp(:=)e Fo(')p Fp(\()p Fo(h)p Fp(\))h Fk(!)f Fo(v)257 2193 y Fp(and)j(the)h(new)f(coaction)g(is)f(giv)o(en)h(as:)774 2280 y Fo(\016)792 2286 y Fm(')817 2280 y Fp(\()p Fo(v)q Fp(\))e(:=)f Fo(')964 2263 y Fj(\000)p Fl(1)1009 2280 y Fp(\()p Fo(v)1046 2263 y Fl(1)1065 2280 y Fp(\))f Fk(\012)f Fo(v)1153 2263 y Fl(2)257 2367 y Fp(If)14 b Fo(V)23 b Fp(is)14 b(regarded)h(as)e(a)h(Y)m(etter-Drinfel'd)g(mo)q(dule)e(in)h (this)h(w)o(a)o(y)m(,)e(w)o(e)i(shall)g(denote)g(it)g(b)o(y)257 2416 y Fo(V)281 2422 y Fm(')305 2416 y Fp(.)g(What)f(w)o(e)h(ha)o(v)o (e)g(established)h(is)f(a)f(functor)711 2503 y Fo(F)738 2509 y Fm(')773 2503 y Fp(:)797 2486 y Fm(H)797 2513 y(H)828 2503 y Fo(Y)c(D)k Fk(!)961 2486 y Fm(H)961 2513 y(H)993 2503 y Fo(Y)c(D)q(;)16 b(V)21 b Fk(7!)11 b Fo(V)1211 2509 y Fm(')953 2628 y Fp(22)p eop %%Page: 23 23 23 22 bop 257 262 a Fp(of)12 b(the)h(category)g(of)f(Y)m (etter-Drinfel'd)g(mo)q(dules)g(whic)o(h)g(is)g(the)h(iden)o(tit)o(y)f (on)g(morphisms.)257 311 y(It)k(is)f(easy)h(to)f(see)i(that)e Fo(F)679 317 y Fm(')718 311 y Fp(is)g(strictly)h(monoidal)c(and)k (strictly)f(quasisymmetric,)e(i.)i(e.)257 361 y(w)o(e)f(ha)o(v)o(e:)692 446 y Fo(F)719 452 y Fm(')743 446 y Fp(\()p Fo(V)19 b Fk(\012)9 b Fo(W)d Fp(\))12 b(=)g Fo(F)987 452 y Fm(')1010 446 y Fp(\()p Fo(V)e Fp(\))f Fk(\012)h Fo(F)1154 452 y Fm(')1177 446 y Fp(\()p Fo(W)c Fp(\))729 508 y Fo(F)756 514 y Fm(')779 508 y Fp(\()p Fo(\033)819 514 y Fm(V)r(;W)888 508 y Fp(\))12 b(=)g Fo(\033)984 515 y Fm(F)1005 519 y Fe(')1026 515 y Fl(\()p Fm(V)6 b Fl(\))p Fm(;F)1109 519 y Fe(')1130 515 y Fl(\()p Fm(W)t Fl(\))257 640 y Fn(3.4)48 b Fp(No)o(w,)11 b(w)o(e)h(lo)q(ok)f(at)h(a)g(sp)q(ecial)g (case.)g(Supp)q(ose)h(that)f Fo(g)h Fk(2)e Fo(H)k Fp(is)c(a)h(grouplik) o(e)f(elemen)o(t)257 690 y(and)j(that)g Fo(\015)g Fp(:)d Fo(H)k Fk(!)c Fo(K)17 b Fp(is)d(a)f(c)o(haracter.)i(W)m(e)f(consider)g (the)h(comp)q(osition)630 775 y Fo(')c Fp(:)g Fo(H)k Fk(!)c Fo(H)q(;)k(h)d Fk(7!)f Fo(\015)r Fp(\()p Fo(h)1009 781 y Fl(1)1028 775 y Fp(\))p Fo(g)q(h)1089 781 y Fl(2)1108 775 y Fo(g)1129 757 y Fj(\000)p Fl(1)1174 775 y Fo(\015)1197 757 y Fj(\000)p Fl(1)1242 775 y Fp(\()p Fo(h)1282 781 y Fl(3)1301 775 y Fp(\))257 859 y(of)k(the)h(conjugation)f(with)g Fo(g)i Fp(and)e(the)h(co)q(conjugation)f(with)g Fo(\015)1280 844 y Fj(\000)p Fl(1)1326 859 y Fp(.)g(It)g(is)h(easy)g(to)f(v)o(erify) 257 909 y(that)f Fo(')g Fp(is)g(a)g(Hopf)f(algebra)h(isomorphism)o(.) 257 991 y(No)o(w)f(supp)q(ose)g(that)g Fo(V)22 b Fp(is)12 b(a)g(left)h(Y)m(etter-Drinfel'd)f(mo)q(dule)f(o)o(v)o(er)i Fo(H)s Fp(.)e(W)m(e)i(in)o(tro)q(duce)g(the)257 1041 y(map:)686 1091 y Fo( )713 1097 y Fm(V)753 1091 y Fp(:)e Fo(V)21 b Fk(!)11 b Fo(V)898 1097 y Fm(')922 1091 y Fo(;)18 b(v)c Fk(7!)d Fo(\015)r Fp(\()p Fo(v)1099 1074 y Fl(1)1119 1091 y Fp(\))p Fo(g)i Fk(!)e Fo(v)1242 1074 y Fl(2)257 1162 y Fp(This)j(no)o(w)g(turns)g(out)g(to)g(b)q(e)g(an)g(example)f (for)g(our)h(abstract)h(notion:)257 1257 y Fn(Prop)q(osition)33 b Fo( )15 b Fp(is)d(a)h(monoidal)c(transformation)i(from)g(the)j(iden)o (tit)o(y)e(functor)h(on)g(the)257 1307 y(category)i(of)e(Y)m (etter-Drinfel'd)h(mo)q(dules)e(to)i(the)h(functor)f Fo(F)1229 1313 y Fm(')1252 1307 y Fp(.)257 1403 y Fn(Pro)q(of.)36 b Fp(W)m(e)15 b(ha)o(v)o(e)f(to)h(pro)o(v)o(e)g(that)g Fo( )875 1409 y Fm(V)918 1403 y Fp(is)g(a)g(morphism)d(inside)i(the)i (category)m(,)e(that)h(is,)257 1452 y(that)f Fo( )374 1458 y Fm(V)417 1452 y Fp(is)g Fo(H)s Fp(-linear)f(and)h(colinear:)415 1537 y Fo( )442 1543 y Fm(V)470 1537 y Fp(\()p Fo(h)e Fk(!)f Fo(v)q Fp(\))h(=)g Fo(')p Fp(\()p Fo(h)p Fp(\))g Fk(!)f Fo( )843 1543 y Fm(V)872 1537 y Fp(\()p Fo(v)q Fp(\))84 b Fo(\016)1027 1543 y Fm(V)1056 1537 y Fp(\()p Fo( )1099 1543 y Fm(V)1128 1537 y Fp(\()p Fo(v)q Fp(\)\))13 b(=)f Fo(')p Fp(\()p Fo(v)1318 1520 y Fl(1)1337 1537 y Fp(\))d Fk(\012)h Fo( )1431 1543 y Fm(V)1460 1537 y Fp(\()p Fo(v)1497 1520 y Fl(2)1516 1537 y Fp(\))257 1621 y(W)m(e)16 b(only)f(pro)o(v)o(e)h(the)g(\014rst)g(equalit)o(y)f(and)h (lea)o(v)o(e)f(the)i(second)g(one)f(as)f(an)h(easy)g(exercise.)257 1671 y(W)m(e)e(ha)o(v)o(e:)550 1756 y Fo( )577 1762 y Fm(V)606 1756 y Fp(\()p Fo(h)d Fk(!)g Fo(v)q Fp(\))h(=)g Fo(\015)r Fp(\(\()p Fo(h)h Fk(!)e Fo(v)q Fp(\))985 1738 y Fl(1)1004 1756 y Fp(\))p Fo(g)i Fk(!)e Fp(\()p Fo(h)h Fk(!)f Fo(v)q Fp(\))1248 1738 y Fl(2)759 1823 y Fp(=)h Fo(\015)r Fp(\()p Fo(h)866 1829 y Fl(1)886 1823 y Fo(v)907 1806 y Fl(1)926 1823 y Fo(S)951 1829 y Fm(H)983 1823 y Fp(\()p Fo(h)1023 1829 y Fl(3)1042 1823 y Fp(\)\))p Fo(g)q(h)1119 1829 y Fl(2)1149 1823 y Fk(!)f Fo(v)1223 1806 y Fl(2)759 1890 y Fp(=)h Fo(\015)r Fp(\()p Fo(h)866 1896 y Fl(1)886 1890 y Fp(\))p Fo(\015)r Fp(\()p Fo(v)962 1873 y Fl(1)982 1890 y Fp(\))p Fo(\015)1021 1873 y Fj(\000)p Fl(1)1067 1890 y Fp(\()p Fo(h)1107 1896 y Fl(3)1125 1890 y Fp(\))p Fo(g)q(h)1186 1896 y Fl(2)1205 1890 y Fo(g)1226 1873 y Fj(\000)p Fl(1)1271 1890 y Fo(g)h Fk(!)e Fo(v)1378 1873 y Fl(2)759 1953 y Fp(=)h Fo(')p Fp(\()p Fo(h)p Fp(\))g Fk(!)f Fo( )978 1959 y Fm(V)1007 1953 y Fp(\()p Fo(v)q Fp(\))257 2037 y(It)16 b(is)f(imm)o(ediate)e(that)i Fo( )i Fp(is)e(natural,)f(and)h(the)h(assertion)g(that)f Fo( )i Fp(is)e(monoidal)d(means)257 2087 y(that)i(w)o(e)g(ha)o(v)o(e)g Fo( )531 2093 y Fm(V)7 b Fj(\012)p Fm(W)633 2087 y Fp(=)12 b Fo( )704 2093 y Fm(V)742 2087 y Fk(\012)e Fo( )811 2093 y Fm(W)848 2087 y Fp(,)k(whic)o(h)g(is)f(also)h(easily)f(v)o (eri\014ed.)h Fg(\003)257 2219 y Fn(3.5)48 b Fp(W)m(e)19 b(kno)o(w)g(from)f([27)o(])g(that,)h(in)g(general,)g(there)i(is)e(no)h (t)o(wist)f(on)g(the)h(category)257 2269 y(of)15 b(Y)m(etter-Drinfel'd) g(mo)q(dules,)f(and)h(therefore)i(no)e(in)o(v)o(ertible)g(ribb)q(on)g (transformation)257 2319 y(from)h(the)i(iden)o(tit)o(y)f(functor)h(to)f (itself.)g(Ho)o(w)o(ev)o(er,)g(there)i(is)e(a)g(ribb)q(on)g (transformation)257 2369 y(b)q(et)o(w)o(een)12 b(the)f(iden)o(tit)o(y)f (functor)h(and)f(the)h(functor)g Fo(F)1086 2378 y Fm(S)1108 2368 y Fd(2)1106 2388 y Fe(H)1135 2369 y Fp(.)e(F)m(or)h(ev)o(ery)i (left)e(Y)m(etter-Drinfel'd)257 2419 y(mo)q(dule)j Fo(V)23 b Fp(o)o(v)o(er)14 b Fo(H)s Fp(,)f(w)o(e)h(de\014ne:)655 2503 y Fo(\022)674 2509 y Fm(V)715 2503 y Fp(:)d Fo(V)21 b Fk(!)11 b Fo(V)860 2512 y Fm(S)882 2502 y Fd(2)880 2522 y Fe(H)909 2503 y Fo(;)18 b(v)13 b Fk(7!)e Fp(\()p Fo(S)1066 2509 y Fm(H)1098 2503 y Fp(\()p Fo(v)1135 2486 y Fl(1)1155 2503 y Fp(\))h Fk(!)f Fo(v)1257 2486 y Fl(2)1276 2503 y Fp(\))953 2628 y(23)p eop %%Page: 24 24 24 23 bop 257 262 a Fp(It)14 b(is)g(easy)g(to)g(see)h(that)f Fo(\022)662 268 y Fm(V)705 262 y Fp(is)g(an)g(isomorphism)c(with)k(in)o (v)o(erse:)761 349 y Fo(\022)781 331 y Fj(\000)p Fl(1)780 361 y Fm(V)827 349 y Fp(\()p Fo(v)q Fp(\))e(=)g Fo(S)963 331 y Fj(\000)p Fl(2)961 361 y Fm(H)1008 349 y Fp(\()p Fo(v)1045 332 y Fl(1)1065 349 y Fp(\))f Fk(!)g Fo(v)1166 332 y Fl(2)257 446 y Fn(Prop)q(osition)33 b Fo(\022)16 b Fp(is)d(a)h(ribb)q(on)g(transformation)e(from)g(the)i(iden)o(tit)o(y) g(functor)g(to)g Fo(F)1608 455 y Fm(S)1630 445 y Fd(2)1628 465 y Fe(H)1657 446 y Fp(.)257 543 y Fn(Pro)q(of.)36 b Fp(As)14 b(ab)q(o)o(v)o(e,)e(w)o(e)h(ha)o(v)o(e)f(to)h(pro)o(v)o(e)g (\014rst)h(that)f Fo(\022)1120 549 y Fm(V)1162 543 y Fp(is)f Fo(H)s Fp(-linear)h(and)f(colinear,)g(that)257 593 y(is,)i(that)g(w)o(e)g(ha)o(v)o(e:)400 680 y Fo(\022)419 686 y Fm(V)448 680 y Fp(\()p Fo(h)e Fk(!)f Fo(v)q Fp(\))h(=)g Fo(S)673 663 y Fl(2)671 691 y Fm(H)703 680 y Fp(\()p Fo(h)p Fp(\))g Fk(!)f Fo(\022)843 686 y Fm(V)872 680 y Fp(\()p Fo(v)q Fp(\))84 b Fo(\016)1027 686 y Fm(V)1056 680 y Fp(\()p Fo(\022)1091 686 y Fm(V)1121 680 y Fp(\()p Fo(v)q Fp(\)\))12 b(=)g Fo(S)1273 663 y Fl(2)1271 691 y Fm(H)1303 680 y Fp(\()p Fo(v)1340 663 y Fl(1)1359 680 y Fp(\))e Fk(\012)f Fo(\022)1445 686 y Fm(V)1474 680 y Fp(\()p Fo(v)1511 663 y Fl(2)1531 680 y Fp(\))257 768 y(W)m(e)14 b(pro)o(v)o(e)g(the)g(\014rst)h(form)o(ula,)c(lea)o(ving)i (the)h(second)h(one)f(as)g(an)g(exercise:)576 855 y Fo(\022)595 861 y Fm(V)625 855 y Fp(\()p Fo(h)d Fk(!)g Fo(v)q Fp(\))i(=)e Fo(S)847 861 y Fm(H)879 855 y Fp(\(\()p Fo(h)h Fk(!)f Fo(v)q Fp(\))1037 838 y Fl(1)1057 855 y Fp(\))g Fk(!)g Fp(\()p Fo(h)h Fk(!)f Fo(v)q Fp(\))1279 838 y Fl(2)779 922 y Fp(=)g Fo(S)847 928 y Fm(H)879 922 y Fp(\()p Fo(h)919 928 y Fl(1)938 922 y Fo(v)959 905 y Fl(1)978 922 y Fo(S)1003 928 y Fm(H)1035 922 y Fp(\()p Fo(h)1075 928 y Fl(3)1094 922 y Fp(\)\))h Fk(!)f Fp(\()p Fo(h)1231 928 y Fl(2)1261 922 y Fk(!)g Fo(v)1335 905 y Fl(2)1354 922 y Fp(\))779 990 y(=)g Fo(S)849 973 y Fl(2)847 1000 y Fm(H)879 990 y Fp(\()p Fo(h)919 996 y Fl(3)938 990 y Fp(\))p Fo(S)979 996 y Fm(H)1011 990 y Fp(\()p Fo(v)1048 973 y Fl(1)1068 990 y Fp(\))p Fo(S)1109 996 y Fm(H)1141 990 y Fp(\()p Fo(h)1181 996 y Fl(1)1199 990 y Fp(\))p Fo(h)1239 996 y Fl(2)1270 990 y Fk(!)g Fo(v)1344 973 y Fl(2)779 1057 y Fp(=)g Fo(S)849 1040 y Fl(2)847 1067 y Fm(H)879 1057 y Fp(\()p Fo(h)p Fp(\))h Fk(!)f Fo(\022)1019 1063 y Fm(V)1049 1057 y Fp(\()p Fo(v)q Fp(\))257 1144 y(T)m(o)i(pro)o(v)o(e)h(that)g Fo(\022)i Fp(really)d(is)h(a)f(ribb)q(on)h(transformation,)e(that)i (is,)f(that)h(w)o(e)g(ha)o(v)o(e)681 1232 y Fo(\022)700 1238 y Fm(V)7 b Fj(\012)p Fm(W)800 1232 y Fk(\016)i Fo(\033)854 1238 y Fm(W)o(;V)932 1232 y Fk(\016)g Fo(\033)986 1238 y Fm(V)r(;W)1066 1232 y Fp(=)j Fo(\022)1129 1238 y Fm(V)1167 1232 y Fk(\012)e Fo(\022)1228 1238 y Fm(W)257 1319 y Fp(is)k(sligh)o(tly)f(more)g(cum)o(b)q(ersome:)305 1406 y(\()p Fo(\022)340 1412 y Fm(V)379 1406 y Fk(\012)c Fo(\022)439 1412 y Fm(W)478 1406 y Fp(\))g Fk(\016)g Fo(\033)558 1389 y Fj(\000)p Fl(1)557 1418 y Fm(V)r(;W)635 1406 y Fk(\016)g Fo(\033)690 1389 y Fj(\000)p Fl(1)689 1418 y Fm(W)o(;V)758 1406 y Fp(\()p Fo(v)i Fk(\012)e Fo(w)q Fp(\))j(=)f(\()p Fo(\022)983 1412 y Fm(V)1022 1406 y Fk(\012)f Fo(\022)1083 1412 y Fm(W)1121 1406 y Fp(\))f Fk(\016)g Fo(\033)1201 1389 y Fj(\000)p Fl(1)1200 1418 y Fm(V)r(;W)1269 1406 y Fp(\()p Fo(w)1316 1389 y Fl(2)1344 1406 y Fk(\012)g Fo(S)1412 1389 y Fj(\000)p Fl(1)1410 1418 y Fm(H)1458 1406 y Fp(\()p Fo(w)1505 1389 y Fl(1)1523 1406 y Fp(\))j Fk(!)f Fo(v)q Fp(\))489 1477 y(=)h(\()p Fo(\022)568 1483 y Fm(V)607 1477 y Fk(\012)d Fo(\022)667 1483 y Fm(W)705 1477 y Fp(\)\(\()p Fo(S)780 1459 y Fj(\000)p Fl(1)778 1489 y Fm(H)826 1477 y Fp(\()p Fo(w)873 1460 y Fl(1)892 1477 y Fp(\))i Fk(!)h Fo(v)q Fp(\))1010 1460 y Fl(2)1038 1477 y Fk(\012)e Fo(S)1107 1459 y Fj(\000)p Fl(1)1105 1489 y Fm(H)1152 1477 y Fp(\(\()p Fo(S)1211 1459 y Fj(\000)p Fl(1)1209 1489 y Fm(H)1257 1477 y Fp(\()p Fo(w)1304 1460 y Fl(1)1322 1477 y Fp(\))i Fk(!)f Fo(v)q Fp(\))1440 1460 y Fl(1)1459 1477 y Fp(\))h Fk(!)f Fo(w)1571 1460 y Fl(2)1590 1477 y Fp(\))489 1545 y(=)h(\()p Fo(\022)568 1551 y Fm(V)607 1545 y Fk(\012)d Fo(\022)667 1551 y Fm(W)705 1545 y Fp(\)\()p Fo(S)764 1528 y Fj(\000)p Fl(1)762 1557 y Fm(H)810 1545 y Fp(\()p Fo(w)857 1528 y Fl(2)876 1545 y Fp(\))i Fk(!)g Fo(v)977 1528 y Fl(2)1006 1545 y Fk(\012)e Fo(S)1074 1528 y Fj(\000)p Fl(1)1072 1557 y Fm(H)1120 1545 y Fp(\()p Fo(S)1163 1528 y Fj(\000)p Fl(1)1161 1557 y Fm(H)1208 1545 y Fp(\()p Fo(w)1255 1528 y Fl(3)1274 1545 y Fp(\))p Fo(v)1311 1528 y Fl(1)1330 1545 y Fo(w)1361 1528 y Fl(1)1380 1545 y Fp(\))i Fk(!)g Fo(w)1491 1528 y Fl(4)1510 1545 y Fp(\))489 1613 y(=)h(\()p Fo(\022)568 1619 y Fm(V)607 1613 y Fk(\012)d Fo(\022)667 1619 y Fm(W)705 1613 y Fp(\)\()p Fo(S)764 1596 y Fj(\000)p Fl(1)762 1625 y Fm(H)810 1613 y Fp(\()p Fo(w)857 1596 y Fl(2)876 1613 y Fp(\))i Fk(!)g Fo(v)977 1596 y Fl(2)1006 1613 y Fk(\012)e Fo(S)1074 1596 y Fj(\000)p Fl(1)1072 1625 y Fm(H)1120 1613 y Fp(\()p Fo(v)1157 1596 y Fl(1)1176 1613 y Fo(w)1207 1596 y Fl(1)1226 1613 y Fp(\))p Fo(S)1269 1596 y Fj(\000)p Fl(2)1267 1625 y Fm(H)1314 1613 y Fp(\()p Fo(w)1361 1596 y Fl(3)1380 1613 y Fp(\))i Fk(!)g Fo(w)1491 1596 y Fl(4)1510 1613 y Fp(\))489 1681 y(=)h(\()p Fo(\022)568 1687 y Fm(V)607 1681 y Fk(\012)d Fo(\022)667 1687 y Fm(W)705 1681 y Fp(\)\(\()p Fo(S)780 1664 y Fj(\000)p Fl(1)778 1694 y Fm(H)826 1681 y Fp(\()p Fo(w)873 1664 y Fl(2)892 1681 y Fp(\))i Fk(!)h Fo(v)994 1664 y Fl(2)1013 1681 y Fp(\))d Fk(\012)h Fp(\()p Fo(S)1123 1664 y Fj(\000)p Fl(1)1121 1694 y Fm(H)1168 1681 y Fp(\()p Fo(v)1205 1664 y Fl(1)1225 1681 y Fo(w)1256 1664 y Fl(1)1274 1681 y Fp(\))i Fk(!)f Fo(\022)1375 1664 y Fj(\000)p Fl(1)1374 1694 y Fm(W)1420 1681 y Fp(\()p Fo(w)1467 1664 y Fl(3)1486 1681 y Fp(\)\)\))489 1749 y(=)h(\()p Fo(S)574 1755 y Fm(H)606 1749 y Fp(\()p Fo(w)653 1732 y Fl(2)672 1749 y Fp(\))f Fk(!)g Fo(\022)771 1755 y Fm(V)801 1749 y Fp(\()p Fo(v)838 1732 y Fl(2)857 1749 y Fp(\)\))f Fk(\012)f Fp(\()p Fo(S)981 1755 y Fm(H)1013 1749 y Fp(\()p Fo(v)1050 1732 y Fl(1)1069 1749 y Fo(w)1100 1732 y Fl(1)1119 1749 y Fp(\))j Fk(!)f Fo(w)1231 1732 y Fl(3)1249 1749 y Fp(\))489 1816 y(=)h(\()p Fo(S)574 1822 y Fm(H)606 1816 y Fp(\()p Fo(v)643 1799 y Fl(2)662 1816 y Fo(w)693 1799 y Fl(2)712 1816 y Fp(\))g Fk(!)f Fo(v)814 1799 y Fl(3)833 1816 y Fp(\))e Fk(\012)h Fp(\()p Fo(S)941 1822 y Fm(H)973 1816 y Fp(\()p Fo(v)1010 1799 y Fl(1)1029 1816 y Fo(w)1060 1799 y Fl(1)1079 1816 y Fp(\))h Fk(!)g Fo(w)1190 1799 y Fl(3)1209 1816 y Fp(\))489 1878 y(=)h Fo(\022)552 1884 y Fm(V)7 b Fj(\012)p Fm(W)643 1878 y Fp(\()p Fo(v)k Fk(\012)e Fo(w)q Fp(\))257 1966 y(It)14 b(is)g(easy)g(to)g(see)h(that)f Fo(\022)i Fp(is)d(really)h (natural.)f Fg(\003)257 2099 y Fn(3.6)48 b Fp(Next,)15 b(w)o(e)f(w)o(an)o(t)g(to)g(understand)i(ho)o(w)e(the)h(monoidal)c (transformations)i(and)h(the)257 2149 y(ribb)q(on)j(transformations)e (in)o(teract.)i(Supp)q(ose)g(that)g Fo(g)h Fp(and)e Fo(g)1263 2134 y Fj(0)1291 2149 y Fp(are)h(t)o(w)o(o)g(grouplik)o(e)e(ele-)257 2199 y(men)o(ts)g(in)f Fo(H)s Fp(,)g(and)h(that)g Fo(\015)h Fp(:)c Fo(H)k Fk(!)d Fo(K)18 b Fp(and)d Fo(\015)991 2184 y Fj(0)1016 2199 y Fp(:)e Fo(H)j Fk(!)c Fo(K)18 b Fp(are)e(t)o(w)o(o)e (c)o(haracters,)i(that)f(is,)257 2249 y(t)o(w)o(o)g(grouplik)o(e)f (elemen)o(ts)h(of)g(the)h(\014nite)f(dual)f Fo(H)1049 2234 y Fj(\016)1068 2249 y Fp(.)h(Then)h(w)o(e)f(can)g(form)f(t)o(w)o (o)g(monoidal)257 2299 y(transformations)e Fo( )i Fp(and)f Fo( )705 2284 y Fj(0)730 2299 y Fp(that)g(are)h(giv)o(en)e(on)h(a)g (\014xed)g(left)g(Y)m(etter-Drinfel'd)g(mo)q(dule)257 2348 y Fo(V)24 b Fp(as:)671 2436 y Fo( )698 2442 y Fm(V)739 2436 y Fp(:)11 b Fo(V)21 b Fk(!)11 b Fo(V)e(;)e(v)13 b Fk(7!)e Fp(\()p Fo(\015)r Fp(\()p Fo(v)1074 2419 y Fl(1)1094 2436 y Fp(\))p Fo(g)i Fk(!)e Fo(v)1217 2419 y Fl(2)1236 2436 y Fp(\))671 2503 y Fo( )699 2486 y Fj(0)698 2513 y Fm(V)739 2503 y Fp(:)g Fo(V)21 b Fk(!)11 b Fo(V)e(;)e(v)13 b Fk(7!)e Fp(\()p Fo(\015)1037 2486 y Fj(0)1049 2503 y Fp(\()p Fo(v)1086 2486 y Fl(1)1106 2503 y Fp(\))p Fo(g)1143 2486 y Fj(0)1166 2503 y Fk(!)g Fo(v)1240 2486 y Fl(2)1260 2503 y Fp(\))953 2628 y(24)p eop %%Page: 25 25 25 24 bop 257 262 a Fp(If)14 b(w)o(e)g(consider)h Fo( )549 268 y Fm(V)591 262 y Fp(and)f Fo( )700 246 y Fj(0)699 273 y Fm(V)742 262 y Fp(only)f(as)h Fo(K)s Fp(-linear)f(maps,)g(then)h (w)o(e)g(see)h(that:)519 353 y Fo( )547 336 y Fj(0)546 363 y Fm(V)585 353 y Fk(\016)8 b Fo( )641 359 y Fm(V)670 353 y Fp(\()p Fo(v)q Fp(\))13 b(=)f Fo(\015)r Fp(\()p Fo(v)840 336 y Fl(1)860 353 y Fp(\))p Fo(\015)899 336 y Fj(0)911 353 y Fp(\(\()p Fo(g)h Fk(!)f Fo(v)1051 336 y Fl(2)1070 353 y Fp(\))1086 336 y Fl(1)1105 353 y Fp(\)\()p Fo(g)1158 336 y Fj(0)1181 353 y Fk(!)f Fp(\()p Fo(g)i Fk(!)e Fo(v)1357 336 y Fl(2)1377 353 y Fp(\))1393 336 y Fl(2)1411 353 y Fp(\))736 420 y(=)h Fo(\015)r Fp(\()p Fo(v)840 403 y Fl(1)860 420 y Fp(\))p Fo(\015)899 403 y Fj(0)911 420 y Fp(\()p Fo(g)q(v)969 403 y Fl(1)989 420 y Fo(g)1010 403 y Fj(\000)p Fl(1)1055 420 y Fp(\)\()p Fo(g)1108 403 y Fj(0)1120 420 y Fo(g)h Fk(!)e Fo(v)1227 403 y Fl(2)1246 420 y Fp(\))736 488 y(=)h(\()p Fo(\015)r(\015)842 471 y Fj(0)855 488 y Fp(\)\()p Fo(v)908 471 y Fl(1)928 488 y Fp(\)\()p Fo(g)981 471 y Fj(0)993 488 y Fo(g)h Fk(!)e Fo(v)1100 471 y Fl(2)1119 488 y Fp(\))257 579 y(Therefore,)k(w)o(e)f(ha)o(v)o(e)g(constructed)i(a)e(group)f (homomorphism)o(:)257 679 y Fn(Prop)q(osition)33 b Fo(G)p Fp(\()p Fo(H)623 664 y Fj(\016)642 679 y Fp(\))658 664 y Fm(op)704 679 y Fk(\002)11 b Fo(G)p Fp(\()p Fo(H)s Fp(\))k Fk(!)f Fo(GL)p Fp(\()p Fo(V)c Fp(\))p Fo(;)d Fp(\()p Fo(\015)r(;)g(g)q Fp(\))15 b Fk(7!)f Fo( )1260 685 y Fm(V)1305 679 y Fp(is)i(a)f(group)h(homomo)o(r-)257 728 y(phism.)257 847 y(In)10 b(addition,)e(w)o(e)i(ha)o(v)o(e)g(to)f (understand)i(the)f(relation)f(of)h(these)h(monoidal)6 b(transformation)257 897 y(with)14 b(the)g(ribb)q(on)g(transformation)e (as)i Fo(K)s Fp(-linear)g(maps:)257 997 y Fn(Lemma)36 b Fo( )474 1003 y Fm(V)512 997 y Fk(\016)9 b Fo(\022)561 1003 y Fm(V)602 997 y Fp(=)j Fo(\022)665 1003 y Fm(V)704 997 y Fk(\016)d Fo( )761 1003 y Fm(V)257 1096 y Fn(Pro)q(of.)36 b Fp(By)23 b(the)g Fo(H)s Fp(-linearit)o(y)e(and)h(colinearit)o(y)f (prop)q(erties)j(of)d Fo(\022)1379 1102 y Fm(V)1431 1096 y Fp(from)g(Prop)q(osi-)257 1146 y(tion)14 b(3.5,)e(w)o(e)i(ha)o(v)o (e:)277 1237 y Fo( )304 1243 y Fm(V)333 1237 y Fp(\()p Fo(\022)368 1243 y Fm(V)397 1237 y Fp(\()p Fo(v)q Fp(\)\))f(=)f Fo(\015)r Fp(\()p Fo(S)589 1220 y Fl(2)587 1248 y Fm(H)619 1237 y Fp(\()p Fo(v)656 1220 y Fl(1)676 1237 y Fp(\)\))p Fo(g)h Fk(!)e Fo(\022)813 1243 y Fm(V)842 1237 y Fp(\()p Fo(v)879 1220 y Fl(2)899 1237 y Fp(\))g(=)h Fo(\015)r Fp(\()p Fo(v)1030 1220 y Fl(1)1050 1237 y Fp(\))p Fo(\022)1085 1243 y Fm(V)1115 1237 y Fp(\()p Fo(S)1158 1220 y Fj(\000)p Fl(2)1156 1250 y Fm(H)1203 1237 y Fp(\()p Fo(g)q Fp(\))g Fk(!)f Fo(v)1342 1220 y Fl(2)1362 1237 y Fp(\))g(=)h Fo(\022)1452 1243 y Fm(V)1482 1237 y Fp(\()p Fo( )q Fp(\()p Fo(v)q Fp(\)\))43 b Fg(\003)257 1364 y Fp(These)16 b(results)f(will)d (b)q(e)i(used)h(throughout)f(in)g(section)g(4.)257 1500 y Fn(3.7)48 b Fp(If)15 b Fo(V)24 b Fp(is)15 b(\014nite-dimensional,)d (the)k(ribb)q(on)f(transformation)e Fo(\022)1357 1506 y Fm(V)1401 1500 y Fp(admits)g(a)i(simple)257 1550 y(in)o(terpretation) f(in)e(terms)h(of)f(dualit)o(y)m(.)f(W)m(e)i(ha)o(v)o(e)f(already)h (men)o(tioned)f(in)g(subsection)i(2.9)257 1599 y(that)i(w)o(e)f(can)h (regard)g(the)g(dual)e(v)o(ector)i(space)h Fo(V)1058 1584 y Fj(\003)1092 1599 y Fp(as)f(a)f(left)g(Y)m(etter-Drinfel'd)g(mo) q(dule)257 1649 y(o)o(v)o(er)f Fo(H)s Fp(,)g(and)f(w)o(e)h(also)g(said) f(there)i(that)f(the)h(mappings)608 1741 y Fo(ev)647 1747 y Fm(V)688 1741 y Fp(:)c Fo(V)19 b Fk(\012)9 b Fo(V)829 1723 y Fj(\003)859 1741 y Fk(!)i Fo(K)s(;)c(v)k Fk(\012)e Fo(f)17 b Fk(7!)11 b Fo(f)t Fp(\()p Fo(v)q Fp(\))608 1841 y Fo(db)648 1847 y Fm(V)688 1841 y Fp(:)g Fo(K)k Fk(!)c Fo(V)847 1824 y Fj(\003)876 1841 y Fk(\012)e Fo(V)g(;)e(\020)15 b Fk(7!)c Fo(\020)1103 1789 y Fm(n)1083 1801 y Fh(X)1086 1890 y Fm(i)p Fl(=1)1150 1841 y Fo(v)1171 1824 y Fl(\()p Fm(i)p Fl(\))p Fj(\003)1237 1841 y Fk(\012)f Fo(v)1299 1848 y Fl(\()p Fm(i)p Fl(\))257 1968 y Fp(are)17 b Fo(H)s Fp(-linear)f(and)h(colinear,)e(where)j Fo(v)894 1975 y Fl(\(1\))939 1968 y Fo(;)7 b(:)g(:)g(:)k(;)c(v)1058 1975 y Fl(\()p Fm(n)p Fl(\))1123 1968 y Fp(is)17 b(a)f(basis)g(of)g Fo(V)26 b Fp(with)16 b(dual)g(basis)257 2024 y Fo(v)278 2009 y Fl(\(1\))p Fj(\003)341 2024 y Fo(;)7 b(:)g(:)g(:)k(;)c(v)461 2009 y Fl(\()p Fm(n)p Fl(\))p Fj(\003)527 2024 y Fp(.)13 b(The)h(canonical)f(isomorphism)d Fo(\030)1079 2030 y Fm(V)1121 2024 y Fp(b)q(et)o(w)o(een)15 b Fo(V)23 b Fp(and)13 b(its)h(bidual)e(space)257 2074 y(b)q(ecomes)i Fo(H)s Fp(-linear)g(and)f(colinear)h(if)f(considered)i(as)f(a)g(map)822 2165 y Fo(\030)840 2171 y Fm(V)881 2165 y Fp(:)d Fo(V)928 2178 y Fm(S)950 2165 y Fi(\000)p Fd(2)948 2188 y Fe(H)1002 2165 y Fk(!)g Fo(V)1089 2148 y Fj(\003\003)257 2263 y Fp(Up)18 b(to)g(this)g(iden)o(ti\014cation,)e Fo(\022)753 2269 y Fm(V)800 2263 y Fp(can)i(b)q(e)h(built)e(up)h(from)e(the)i(ev)n (aluation)f Fo(ev)i Fp(and)f(the)257 2312 y(co)q(ev)n(aluation)13 b Fo(db)p Fp(:)257 2412 y Fn(Prop)q(osition)33 b Fo(\030)554 2418 y Fm(V)591 2412 y Fk(\016)8 b Fo(F)647 2425 y Fm(S)669 2412 y Fi(\000)p Fd(2)667 2435 y Fe(H)709 2412 y Fp(\()p Fo(\022)745 2394 y Fj(\000)p Fl(1)744 2424 y Fm(V)790 2412 y Fp(\))k(=)g(\()p Fo(id)914 2418 y Fm(V)941 2410 y Fi(\003\003)983 2412 y Fk(\012)c Fo(ev)1062 2418 y Fm(V)1091 2412 y Fp(\))g Fk(\016)g Fp(\()p Fo(\033)1184 2418 y Fm(V)r(;V)1242 2410 y Fi(\003\003)1284 2412 y Fk(\012)g Fo(id)1360 2418 y Fm(V)1387 2410 y Fi(\003)1407 2412 y Fp(\))f Fk(\016)h Fp(\()p Fo(id)1511 2418 y Fm(V)1548 2412 y Fk(\012)g Fo(db)1628 2418 y Fm(V)1654 2410 y Fi(\003)1673 2412 y Fp(\))953 2628 y(25)p eop %%Page: 26 26 26 25 bop 257 262 a Fn(Pro)q(of.)273 353 y Fp(\()p Fo(id)325 359 y Fm(V)352 351 y Fi(\003\003)396 353 y Fk(\012)9 b Fo(ev)476 359 y Fm(V)505 353 y Fp(\))h Fk(\016)f Fp(\()p Fo(\033)601 359 y Fm(V)r(;V)659 351 y Fi(\003\003)702 353 y Fk(\012)h Fo(id)780 359 y Fm(V)807 351 y Fi(\003)826 353 y Fp(\))f Fk(\016)g Fp(\()p Fo(id)933 359 y Fm(V)972 353 y Fk(\012)g Fo(db)1053 359 y Fm(V)1079 351 y Fi(\003)1099 353 y Fp(\)\()p Fo(v)q Fp(\))517 453 y(=)j(\()p Fo(id)613 459 y Fm(V)639 451 y Fi(\003\003)683 453 y Fk(\012)e Fo(ev)764 459 y Fm(V)793 453 y Fp(\))g Fk(\016)f Fp(\()p Fo(\033)889 459 y Fm(V)r(;V)946 451 y Fi(\003\003)990 453 y Fk(\012)h Fo(id)1068 459 y Fm(V)1095 451 y Fi(\003)1114 453 y Fp(\)\()1166 401 y Fm(n)1146 414 y Fh(X)1149 502 y Fm(i)p Fl(=1)1213 453 y Fo(v)h Fk(\012)e Fo(\030)1303 459 y Fm(V)1332 453 y Fp(\()p Fo(v)1368 460 y Fl(\()p Fm(i)p Fl(\))1408 453 y Fp(\))h Fk(\012)f Fo(v)1496 436 y Fl(\()p Fm(i)p Fl(\))p Fj(\003)1554 453 y Fp(\))517 591 y(=)j(\()p Fo(id)613 597 y Fm(V)639 589 y Fi(\003\003)683 591 y Fk(\012)e Fo(ev)764 597 y Fm(V)793 591 y Fp(\)\()845 539 y Fm(n)825 552 y Fh(X)828 640 y Fm(i)p Fl(=1)892 591 y Fo(v)913 574 y Fl(1)944 591 y Fk(!)h Fo(\030)1015 597 y Fm(V)1044 591 y Fp(\()p Fo(v)1080 598 y Fl(\()p Fm(i)p Fl(\))1120 591 y Fp(\))f Fk(\012)f Fo(v)1208 574 y Fl(2)1237 591 y Fk(\012)g Fo(v)1299 574 y Fl(\()p Fm(i)p Fl(\))p Fj(\003)1356 591 y Fp(\))517 697 y(=)j Fo(v)582 680 y Fl(1)612 697 y Fk(!)f Fo(\030)683 703 y Fm(V)712 697 y Fp(\()p Fo(v)749 680 y Fl(2)769 697 y Fp(\))h(=)f Fo(\030)858 703 y Fm(V)887 697 y Fp(\()p Fo(S)930 680 y Fj(\000)p Fl(2)928 710 y Fm(H)976 697 y Fp(\()p Fo(v)1013 680 y Fl(1)1032 697 y Fp(\))h Fk(!)f Fo(v)1134 680 y Fl(2)1153 697 y Fp(\))473 b Fg(\003)257 833 y Fn(3.8)48 b Fp(It)13 b(is)f(a)g(v)o(ery)h(strange)g(fact)g(that)f(it)g(is)h(p)q (ossible)f(to)h(calculate)f(the)h(p)q(o)o(w)o(ers)g(of)f Fo(\022)1613 839 y Fm(V)1655 833 y Fp(in)257 883 y(a)j(uni\014ed)g (form.)e(T)m(o)i(deriv)o(e)g(this)g(form)o(ula,)d(w)o(e)k(\014rst)f(in) o(tro)q(duce)h(a)f(strange)h(expression:)257 932 y(F)m(or)e Fo(h)d Fk(2)h Fo(H)s Fp(,)h(de\014ne:)726 1013 y Fo(P)753 1019 y Fm(n)775 1013 y Fp(\()p Fo(h)p Fp(\))f(:=)918 961 y Fm(n)902 974 y Fh(Y)898 1063 y Fm(k)q Fl(=1)965 1013 y Fo(S)992 992 y Fl(2\()p Fm(n)p Fj(\000)p Fm(k)q Fl(\)+1)990 1025 y Fm(H)1145 1013 y Fp(\()p Fo(h)1185 1019 y Fm(k)1205 1013 y Fp(\))257 1127 y(where)j(b)o(y)d(con)o(v)o(en)o (tion)h Fo(P)667 1133 y Fl(0)685 1127 y Fp(\()p Fo(h)p Fp(\))f(=)g Fo(\017)814 1133 y Fm(H)845 1127 y Fp(\()p Fo(h)p Fp(\)1)922 1133 y Fm(H)954 1127 y Fp(.)g(In)h(the)h(ab)q(o)o(v)o (e)f(form)o(ula,)d(the)k(index)f Fo(k)h Fp(in)e Fo(h)1669 1133 y Fm(k)257 1177 y Fp(denotes)i(a)f(Sw)o(eedler)g(index,)g(that)f (is,)g(the)i Fo(n)7 b Fk(\000)g Fp(1-times)k(iterated)i(copro)q(duct)h (is)f(denoted)257 1227 y(b)o(y)313 1196 y Fh(N)359 1206 y Fm(n)359 1240 y(k)q Fl(=1)428 1227 y Fo(h)452 1233 y Fm(k)473 1227 y Fp(.)e(Using)g(this,)h(w)o(e)g(no)o(w)f(can)h (calculate)g(the)g(p)q(o)o(w)o(ers)g(of)f Fo(\022)1357 1233 y Fm(V)1398 1227 y Fp(for)h(a)f(left)h(Y)m(etter-)257 1277 y(Drinfel'd)h(mo)q(dule)f Fo(V)24 b Fp(in)13 b(the)i(follo)o(wing) c(form:)257 1377 y Fn(Prop)q(osition)33 b Fo(\022)556 1361 y Fm(n)555 1388 y(V)585 1377 y Fp(\()p Fo(v)q Fp(\))12 b(=)g Fo(P)721 1383 y Fm(n)743 1377 y Fp(\()p Fo(v)780 1361 y Fl(1)799 1377 y Fp(\))g Fk(!)f Fo(v)901 1361 y Fl(2)257 1476 y Fn(Pro)q(of.)36 b Fp(W)m(e)15 b(pro)q(ceed)h(b)o(y)f (induction)g(on)f Fo(n)p Fp(,)h(the)g(case)h Fo(n)e Fp(=)f(0)i(b)q (eing)g(ob)o(vious)f(and)h(the)257 1526 y(case)k Fo(n)e Fp(=)h(1)f(to)q(o)g(b)q(ecause)i(w)o(e)f(ha)o(v)o(e)f Fo(P)903 1532 y Fl(1)922 1526 y Fp(\()p Fo(h)p Fp(\))g(=)h Fo(S)1070 1532 y Fm(H)1102 1526 y Fp(\()p Fo(h)p Fp(\).)f(It)h(follo)o (ws)d(directly)j(from)e(the)257 1576 y(de\014nition)e(that)g(w)o(e)g (ha)o(v)o(e)g Fo(P)716 1582 y Fm(n)p Fl(+1)780 1576 y Fp(\()p Fo(h)p Fp(\))e(=)f Fo(S)918 1561 y Fl(2)916 1587 y Fm(H)948 1576 y Fp(\()p Fo(P)991 1582 y Fm(n)1014 1576 y Fp(\()p Fo(h)1054 1582 y Fl(1)1072 1576 y Fp(\)\))p Fo(S)1129 1582 y Fm(H)1162 1576 y Fp(\()p Fo(h)1202 1582 y Fl(2)1220 1576 y Fp(\).)j(No)o(w,)f(w)o(e)h(calculate:)654 1667 y Fo(\022)674 1649 y Fm(n)p Fl(+1)673 1679 y Fm(V)739 1667 y Fp(\()p Fo(v)q Fp(\))e(=)g Fo(\022)867 1673 y Fm(V)896 1667 y Fp(\()p Fo(P)939 1673 y Fm(n)962 1667 y Fp(\()p Fo(v)999 1650 y Fl(1)1018 1667 y Fp(\))g Fk(!)f Fo(v)1120 1650 y Fl(2)1139 1667 y Fp(\))804 1735 y(=)h Fo(S)875 1717 y Fl(2)873 1745 y Fm(H)905 1735 y Fp(\()p Fo(P)948 1741 y Fm(n)970 1735 y Fp(\()p Fo(v)1007 1717 y Fl(1)1027 1735 y Fp(\)\))g Fk(!)f Fo(\022)1143 1741 y Fm(V)1172 1735 y Fp(\()p Fo(v)1209 1717 y Fl(2)1228 1735 y Fp(\))804 1802 y(=)h Fo(S)875 1785 y Fl(2)873 1812 y Fm(H)905 1802 y Fp(\()p Fo(P)948 1808 y Fm(n)970 1802 y Fp(\()p Fo(v)1007 1785 y Fl(1)1027 1802 y Fp(\)\))p Fo(S)1084 1808 y Fm(H)1116 1802 y Fp(\()p Fo(v)1153 1785 y Fl(2)1172 1802 y Fp(\))g Fk(!)f Fo(v)1274 1785 y Fl(3)804 1869 y Fp(=)h Fo(P)875 1875 y Fm(n)p Fl(+1)939 1869 y Fp(\()p Fo(v)976 1852 y Fl(1)996 1869 y Fp(\))f Fk(!)g Fo(v)1097 1852 y Fl(2)257 1961 y Fp(whic)o(h)j(establishes)h(the)g (inductiv)o(e)e(step.)i Fg(\003)257 2096 y Fn(3.9)48 b Fp(If)10 b Fo(H)k Fp(is)d(\014nite-dimensional,)d(the)j(natural)f (transformations)g(discussed)i(so)f(far)f(can)257 2146 y(also)j(b)q(e)h(understo)q(o)q(d)g(from)e(the)i(p)q(oin)o(t)f(of)f (view)h(of)g(the)h(Drinfel'd)e(double)h(construction,)257 2196 y(and)20 b(in)g(this)h(w)o(a)o(y)e(they)i(b)q(ecome)f(v)o(ery)h (simple.)d(First,)i(observ)o(e)i(that)e(for)g(an)o(y)g(Hopf)257 2246 y(algebra)14 b Fo(H)i Fp(and)e(an)o(y)g(grouplik)o(e)f(elemen)o(t) g Fo(g)g Fk(2)e Fo(H)17 b Fp(the)d(conjugation)f(with)h Fo(g)q Fp(:)759 2337 y Fo(')d Fp(:)h Fo(H)i Fk(!)d Fo(H)q(;)18 b(h)11 b Fk(7!)g Fo(g)q(hg)1143 2320 y Fj(\000)p Fl(1)257 2428 y Fp(is)i(ob)o(viously)e(a)h(Hopf)g(algebra)g(automorphism)o(,)d (and)k(if)e(w)o(e)i(de\014ne)g(for)f(ev)o(ery)h Fo(H)s Fp(-mo)q(dule)257 2478 y Fo(V)25 b Fp(a)16 b(new)g(mo)q(dule)e (structure)j Fo(V)781 2484 y Fm(')821 2478 y Fp(b)o(y)e(pullbac)o(k)g (via)g Fo(')g Fp(as)h(in)f(subsection)i(3.3,)d(then)i(w)o(e)953 2628 y(26)p eop %%Page: 27 27 27 26 bop 257 262 a Fp(obtain)13 b(a)h(monoidal)d(functor)j(from)e(the) j(category)f(of)f(all)g Fo(H)s Fp(-mo)q(dules)f(in)o(to)h(itself.)g(No) o(w,)257 311 y(if)g(w)o(e)i(de\014ne)f(for)g(all)f Fo(H)s Fp(-mo)q(dules)f Fo(V)23 b Fp(the)15 b(map)728 403 y Fo( )755 409 y Fm(V)796 403 y Fp(:)c Fo(V)21 b Fk(!)11 b Fo(V)941 409 y Fm(')965 403 y Fo(;)16 b(v)d Fk(7!)e Fp(\()p Fo(g)i Fk(!)e Fo(v)q Fp(\))257 494 y(then)17 b(this)f(is)g(ob)o(viously)e(a)i(monoidal)d(transformation)h(from)g (the)j(iden)o(tit)o(y)e(functor)i(on)257 544 y(the)e(category)f(of)f Fo(H)s Fp(-mo)q(dules)g(to)h(this)g(functor.)257 629 y(W)m(e)c(no)o(w)h(explain)e(ho)o(w)h(the)i(monoidal)7 b(transformations)i(constructed)k(in)d(subsection)h(3.3)257 679 y(can)17 b(b)q(e)g(understo)q(o)q(d)h(from)d(this)i(p)q(oin)o(t)f (of)g(view.)g(W)m(e)g(shall)g(use)h(the)g(Drinfel'd)f(double)257 729 y(construction)h(\(cf.)e([3)o(],)g Fk(x)p Fp(13,)g([19)o(],)f Fk(x)p Fp(10.3,)h([24)o(],)f(sec.)i(3\).)f(Let)h Fo(D)g Fp(=)f Fo(D)q Fp(\()p Fo(H)1437 714 y Fm(cop)1488 729 y Fp(\))1504 714 y Fm(cop)1570 729 y Fp(denote)257 779 y(the)i(Drinfel'd)e(double)g(of)h(the)g(co)q(opp)q(osite)h(Hopf)e (algebra)h Fo(H)1262 764 y Fm(cop)1328 779 y Fp(with)f(the)i(co)q(opp)q (osite)257 829 y(copro)q(duct.)12 b Fo(D)q Fp(,)e(whic)o(h)h(is)f (isomorphic)f(to)i Fo(H)965 814 y Fj(\003)987 829 y Fk(\012)s Fo(H)i Fp(as)e(a)f(v)o(ector)i(space,)f(is)f(a)h(Hopf)f(algebra)257 878 y(with)k(m)o(ultiplicatio)o(n:)547 970 y(\()p Fo(p)c Fk(\012)f Fo(h)p Fp(\)\()p Fo(p)712 953 y Fj(0)733 970 y Fk(\012)h Fo(h)799 953 y Fj(0)810 970 y Fp(\))i(=)g(\()p Fo(h)922 976 y Fl(3)952 970 y Fk(!)f Fo(p)1026 953 y Fj(0)1049 970 y Fk( )g Fo(S)1127 976 y Fm(H)1159 970 y Fp(\()p Fo(h)1199 976 y Fl(1)1218 970 y Fp(\)\))p Fo(p)e Fk(\012)g Fo(h)1345 976 y Fl(2)1364 970 y Fo(h)1388 953 y Fj(0)257 1061 y Fp(where)15 b(the)g(arro)o(ws)f(denote)h(the)f (coregular)g(actions,)g(com)o(ultiplicatio)o(n:)659 1152 y(\001)694 1158 y Fm(D)724 1152 y Fp(\()p Fo(p)9 b Fk(\012)h Fo(h)p Fp(\))h(=)h(\()p Fo(p)944 1158 y Fl(1)972 1152 y Fk(\012)d Fo(h)1037 1158 y Fl(1)1056 1152 y Fp(\))g Fk(\012)h Fp(\()p Fo(p)1160 1158 y Fl(2)1188 1152 y Fk(\012)f Fo(h)1253 1158 y Fl(2)1272 1152 y Fp(\))257 1244 y(unit)14 b(1)366 1250 y Fm(D)407 1244 y Fp(=)e Fo(\017)468 1250 y Fm(H)509 1244 y Fk(\012)d Fp(1)571 1250 y Fm(H)603 1244 y Fp(,)k(counit:)753 1335 y Fo(\017)770 1341 y Fm(D)800 1335 y Fp(\()p Fo(p)d Fk(\012)f Fo(h)p Fp(\))j(=)g Fo(p)p Fp(\(1)1042 1341 y Fm(H)1073 1335 y Fp(\))p Fo(\017)1106 1341 y Fm(H)1137 1335 y Fp(\()p Fo(h)p Fp(\))257 1426 y(and)i(an)o(tip)q(o)q(de:)575 1518 y Fo(S)600 1524 y Fm(D)631 1518 y Fp(\()p Fo(p)9 b Fk(\012)g Fo(h)p Fp(\))j(=)g(\()p Fo(\017)847 1524 y Fm(H)888 1518 y Fk(\012)d Fo(S)954 1524 y Fm(H)986 1518 y Fp(\()p Fo(h)p Fp(\)\))h Fk(\012)f Fp(\()p Fo(S)1152 1500 y Fj(\000)p Fl(1)1150 1530 y Fm(H)1179 1522 y Fi(\003)1199 1518 y Fp(\()p Fo(p)p Fp(\))h Fk(\012)f Fp(1)1324 1524 y Fm(H)1356 1518 y Fp(\))257 1609 y Fo(D)16 b Fp(is)d(also)h(quasitriangular)e(with)i Fo(R)p Fp(-matrix:)668 1731 y Fo(R)d Fp(=)775 1679 y Fm(n)755 1692 y Fh(X)758 1780 y Fm(i)p Fl(=1)815 1731 y Fp(\()p Fo(h)855 1714 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)923 1731 y Fk(\012)f Fp(1)986 1737 y Fm(H)1017 1731 y Fp(\))f Fk(\012)h Fp(\()p Fo(\017)1117 1737 y Fm(H)1158 1731 y Fk(\012)f Fo(h)1223 1738 y Fl(\()p Fm(i)p Fl(\))1263 1731 y Fp(\))257 1867 y(where)15 b Fo(h)401 1874 y Fl(\(1\))446 1867 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(h)570 1874 y Fl(\()p Fm(n)p Fl(\))631 1867 y Fp(is)14 b(a)g(basis)g(of)f Fo(H)k Fp(with)c(dual)g(basis)h Fo(h)1221 1852 y Fl(\(1\))r Fj(\003)1285 1867 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(h)1409 1852 y Fl(\()p Fm(n)p Fl(\))q Fj(\003)1476 1867 y Fp(.)257 1953 y(No)o(w,)16 b Fo(D)i Fp(is)e(related)h(to)f(our)h(con)o(text)g(as)f(follo)o(ws:)f(It)h(is)g (easy)h(to)f(see)i(that)e(left)h(Y)m(etter-)257 2002 y(Drinfel'd)g(mo)q(dules)g(are)i(the)g(same)e(as)h Fo(D)q Fp(-mo)q(dules:)f(If)h Fo(V)28 b Fp(is)18 b(a)g(left)g(Y)m (etter-Drinfel'd)257 2052 y(mo)q(dule,)12 b(then)j Fo(V)23 b Fp(can)14 b(b)q(e)h(turned)g(in)o(to)e(a)g Fo(D)q Fp(-mo)q(dule)g (via)g(the)i(action:)645 2143 y(\()p Fo(p)10 b Fk(\012)f Fo(h)p Fp(\))j Fk(!)f Fo(v)i Fp(:=)e(\()p Fo(h)h Fk(!)f Fo(v)q Fp(\))1068 2126 y Fl(2)1087 2143 y Fo(p)p Fp(\(\()p Fo(h)h Fk(!)f Fo(v)q Fp(\))1266 2126 y Fl(1)1285 2143 y Fp(\))257 2235 y(In)i(this)g(w)o(a)o(y)m(,)e(w)o(e)j(obtain)e(a)g (functor)i(from)d(the)i(category)h(of)e(left)g(Y)m(etter-Drinfel'd)h (mo)q(d-)257 2285 y(ules)22 b(in)o(to)e(the)i(category)f(of)g(left)g Fo(D)q Fp(-mo)q(dules)f(whic)o(h)h(is)g(easily)f(seen)j(to)d(b)q(e)i (strictly)257 2334 y(monoidal,)14 b(strictly)i(quasisymmetric)f(and)h (an)h(isomorphism)c(of)j(categories.)i(Supp)q(ose)257 2384 y(no)o(w)d(that)f(w)o(e)h(are)g(giv)o(en)g(a)f(grouplik)o(e)g (elemen)o(t)g Fo(g)g Fk(2)f Fo(H)k Fp(and)e(a)f(c)o(haracter)i Fo(\015)g Fp(:)d Fo(H)i Fk(!)e Fo(K)s Fp(.)257 2434 y(The)j(elemen)o(t) f Fo(g)519 2440 y Fm(D)562 2434 y Fp(:=)f(\()p Fo(\017)653 2440 y Fm(H)694 2434 y Fk(\012)d Fo(g)q Fp(\)\()p Fo(\015)i Fk(\012)d Fp(1)887 2440 y Fm(H)919 2434 y Fp(\))15 b(is)g(a)g(grouplik) o(e)f(elemen)o(t)h(of)f Fo(D)j Fp(b)q(ecause)g(it)e(is)257 2484 y(the)j(pro)q(duct)f(of)g(t)o(w)o(o)f(grouplik)o(e)g(elemen)o(ts,) g(and)h(w)o(e)g(ha)o(v)o(e)f Fo(g)1255 2490 y Fm(D)1301 2484 y Fk(!)g Fo(v)i Fp(=)f Fo(\015)r Fp(\()p Fo(v)1506 2469 y Fl(1)1526 2484 y Fp(\))p Fo(g)h Fk(!)e Fo(v)1659 2469 y Fl(2)1678 2484 y Fp(.)953 2628 y(27)p eop %%Page: 28 28 28 27 bop 257 262 a Fp(Therefore,)14 b(the)g(monoidal)c (transformations)i(considered)i(in)f(subsection)i(3.3)d(reduce)j(to)257 311 y(the)g(action)e(of)h(grouplik)o(e)f(elemen)o(ts)g(considered)j(ab) q(o)o(v)o(e.)257 391 y(In)h(a)g(similar)e(fashion,)g(the)j(ribb)q(on)f (transformation)e Fo(\022)k Fp(can)e(b)q(e)g(in)o(terpreted)i(in)e (terms)257 441 y(of)e Fo(D)q Fp(.)g(As)h(for)f(ev)o(ery)h (quasitriangular)e(Hopf)h(algebra,)g(w)o(e)g(get)h(from)e(the)i Fo(R)p Fp(-matrix)d(an)257 491 y(elemen)o(t:)342 592 y Fo(u)366 598 y Fm(D)408 592 y Fp(=)471 540 y Fm(n)451 552 y Fh(X)455 641 y Fm(i)p Fl(=1)518 592 y Fo(S)543 598 y Fm(D)574 592 y Fp(\()p Fo(\017)607 598 y Fm(H)648 592 y Fk(\012)c Fo(h)713 599 y Fl(\()p Fm(i)p Fl(\))753 592 y Fp(\)\()p Fo(h)809 574 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)877 592 y Fk(\012)h Fp(1)940 598 y Fm(H)971 592 y Fp(\))i(=)1062 540 y Fm(n)1042 552 y Fh(X)1045 641 y Fm(i)p Fl(=1)1102 592 y Fp(\()p Fo(\017)1135 598 y Fm(H)1176 592 y Fk(\012)e Fo(S)1243 598 y Fm(H)1274 592 y Fp(\()p Fo(h)1314 599 y Fl(\()p Fm(i)p Fl(\))1354 592 y Fp(\)\)\()p Fo(h)1426 574 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1495 592 y Fk(\012)f Fp(1)1557 598 y Fm(H)1588 592 y Fp(\))257 706 y(W)m(e)14 b(then)g(ha)o(v)o(e:)400 807 y Fo(u)424 813 y Fm(D)465 807 y Fk(!)d Fo(v)i Fp(=)615 755 y Fm(n)595 767 y Fh(X)598 856 y Fm(i)p Fl(=1)655 807 y Fp(\()p Fo(S)696 813 y Fm(H)728 807 y Fp(\()p Fo(h)768 814 y Fl(\()p Fm(i)p Fl(\))808 807 y Fp(\))e Fk(!)h Fo(v)910 789 y Fl(2)929 807 y Fp(\))p Fo(h)969 789 y Fl(\()p Fm(i)p Fl(\))r Fj(\003)1028 807 y Fp(\()p Fo(v)1065 789 y Fl(1)1084 807 y Fp(\))g(=)g Fo(S)1181 813 y Fm(H)1212 807 y Fp(\()p Fo(v)1249 789 y Fl(1)1269 807 y Fp(\))g Fk(!)f Fo(v)1371 789 y Fl(2)1401 807 y Fp(=)h Fo(\022)1464 813 y Fm(V)1493 807 y Fp(\()p Fo(v)q Fp(\))257 921 y(This)19 b(means)f(that)g(our)h (ribb)q(on)f(transformation)f(coincides)i(with)g(the)g(action)f(of)g Fo(u)1648 927 y Fm(D)1678 921 y Fp(.)257 971 y(The)f(fact)g(that)g Fo(\022)h Fp(is)e(a)h(ribb)q(on)f(transformation)f(no)o(w)h(turns)h (out)g(to)f(b)q(e)i(equiv)n(alen)o(t)e(to)257 1020 y(the)f(w)o(ell-kno) o(wn)d(form)o(ula)g(\(cf.)h([19)o(],)g(Theorem)h(10.1.13,)d(form)o(ula) g(\(*\),)j(p.)f(181\):)726 1098 y Fo(u)750 1104 y Fm(D)789 1098 y Fk(\012)c Fo(u)854 1104 y Fm(D)896 1098 y Fp(=)i(\001)974 1104 y Fm(D)1004 1098 y Fp(\()p Fo(u)1044 1104 y Fm(D)1074 1098 y Fp(\)\()p Fo(R)1138 1081 y Fl(21)1173 1098 y Fo(R)p Fp(\))257 1176 y(T)m(o)i(see)j(this,)d(w)o(e)h(write)g Fo(R)e Fp(=)733 1145 y Fh(P)776 1189 y Fm(i)797 1176 y Fo(a)819 1183 y Fl(\()p Fm(i)p Fl(\))868 1176 y Fk(\012)e Fo(b)928 1183 y Fl(\()p Fm(i)p Fl(\))967 1176 y Fp(.)k(W)m(e)f(then)i (ha)o(v)o(e:)341 1267 y Fo(\022)360 1273 y Fm(V)7 b Fj(\012)p Fm(W)460 1267 y Fk(\016)i Fo(\033)514 1273 y Fm(W)o(;V)592 1267 y Fk(\016)g Fo(\033)646 1273 y Fm(V)r(;W)715 1267 y Fp(\()p Fo(v)i Fk(\012)e Fo(w)q Fp(\))j(=)g Fo(\022)925 1273 y Fm(V)7 b Fj(\012)p Fm(W)1025 1267 y Fk(\016)i Fo(\033)1079 1273 y Fm(W)o(;V)1148 1267 y Fp(\()1164 1227 y Fh(X)1186 1316 y Fm(j)1231 1267 y Fo(b)1249 1274 y Fl(\()p Fm(j)r Fl(\))1303 1267 y Fk(!)i Fo(w)g Fk(\012)e Fo(a)1460 1274 y Fl(\()p Fm(j)r Fl(\))1515 1267 y Fk(!)i Fo(v)q Fp(\))862 1386 y(=)h Fo(u)930 1392 y Fm(D)971 1386 y Fk(!)f Fp(\()1040 1346 y Fh(X)1052 1435 y Fm(i;j)1107 1386 y Fo(b)1125 1393 y Fl(\()p Fm(i)p Fl(\))1165 1386 y Fo(a)1187 1393 y Fl(\()p Fm(j)r Fl(\))1242 1386 y Fk(!)g Fo(v)g Fk(\012)e Fo(a)1389 1393 y Fl(\()p Fm(i)p Fl(\))1429 1386 y Fo(b)1447 1393 y Fl(\()p Fm(j)r Fl(\))1502 1386 y Fk(!)i Fo(w)q Fp(\))862 1496 y(=)h Fo(u)930 1502 y Fm(D)971 1496 y Fk(!)f Fo(v)g Fk(\012)f Fo(u)1121 1502 y Fm(D)1162 1496 y Fk(!)h Fo(w)862 1558 y Fp(=)h(\()p Fo(\022)941 1564 y Fm(V)979 1558 y Fk(\012)e Fo(\022)1040 1564 y Fm(W)1078 1558 y Fp(\)\()p Fo(v)h Fk(\012)f Fo(w)q Fp(\))257 1688 y Fn(3.10)48 b Fp(Monoidal)11 b(transformations)f(and)i (ribb)q(on)g(transformations)f(ma)o(y)f(in)i(particular)257 1738 y(b)q(e)19 b(de\014ned)h(for)e(Y)m(etter-Drinfel'd)g(Hopf)g (algebras.)g(Here,)h(they)g(lead)f(to)g(algebra)g(au-)257 1787 y(tomorphisms)d(in)h(the)h(case)h(of)e(monoidal)e (transformations,)g(and)j(also)f(in)g(the)h(case)h(of)257 1837 y(ribb)q(on)c(transformations)e(if)h(comp)q(osed)h(with)g(the)g (square)h(of)e(the)h(an)o(tip)q(o)q(de:)257 1929 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)e Fo(g)g Fk(2)e Fo(H)17 b Fp(is)c(a)h (grouplik)o(e)e(elemen)o(t)h(and)h(that)f Fo(\015)i Fp(:)c Fo(H)j Fk(!)d Fo(K)257 1979 y Fp(is)j(a)g(c)o(haracter.)g(De\014ne:)630 2057 y Fo(')d Fp(:)g Fo(H)k Fk(!)c Fo(H)q(;)k(h)d Fk(7!)f Fo(\015)r Fp(\()p Fo(h)1009 2063 y Fl(1)1028 2057 y Fp(\))p Fo(g)q(h)1089 2063 y Fl(2)1108 2057 y Fo(g)1129 2039 y Fj(\000)p Fl(1)1174 2057 y Fo(\015)1197 2039 y Fj(\000)p Fl(1)1242 2057 y Fp(\()p Fo(h)1282 2063 y Fl(3)1301 2057 y Fp(\))257 2134 y(Supp)q(ose)k(that)f Fo(A)g Fp(is)g(a)f(left)h(Y)m (etter-Drinfel'd)g(Hopf)f(algebra.)308 2237 y(1.)20 b Fo(A)392 2243 y Fm(')429 2237 y Fp(is)13 b(again)f(a)g(Y)m (etter-Drinfel'd)h(Hopf)g(algebra)f(with)h(the)g(same)f(m)o (ultiplication,)361 2287 y(com)o(ultiplication,)e(unit,)j(counit)h(and) g(an)o(tip)q(o)q(de.)308 2364 y(2.)20 b(The)e(map)d Fo( )572 2370 y Fm(A)616 2364 y Fp(:)h Fo(A)h Fk(!)g Fo(A)782 2370 y Fm(')806 2364 y Fo(;)g(a)g Fk(7!)f Fp(\()p Fo(\015)r Fp(\()p Fo(a)1009 2349 y Fl(1)1029 2364 y Fp(\))p Fo(g)i Fk(!)f Fo(a)1164 2349 y Fl(2)1182 2364 y Fp(\))g(is)g(a)g(Y)m (etter-Drinfel'd)g(Hopf)361 2414 y(algebra)d(isomorphism)o(.)308 2492 y(3.)20 b Fo(S)388 2477 y Fl(2)386 2503 y Fm(A)423 2492 y Fk(\016)9 b Fo(\022)472 2498 y Fm(A)513 2492 y Fp(is)14 b(a)g(Y)m(etter-Drinfel'd)f(Hopf)h(algebra)f(endomorphism)e (of)j Fo(A)p Fp(.)953 2628 y(28)p eop %%Page: 29 29 29 28 bop 257 262 a Fn(Pro)q(of.)36 b Fp(The)14 b(\014rst)f(statemen)o (t)g(is)g(rather)h(ob)o(vious.)d(T)m(o)h(pro)o(v)o(e)h(the)h(second,)f (w)o(e)g(observ)o(e)257 311 y(\014rst)i(that)f Fo( )460 317 y Fm(A)501 311 y Fp(comm)o(utes)e(with)i(the)g(m)o(ultiplication)c Fo(\026)1153 317 y Fm(A)1192 311 y Fp(=)i Fo(\026)1261 317 y Fm(A)1286 321 y Fe(')1309 311 y Fp(:)606 388 y Fo( )633 394 y Fm(A)669 388 y Fk(\016)d Fo(\026)724 394 y Fm(A)763 388 y Fp(=)j Fo(\026)832 394 y Fm(A)868 388 y Fk(\016)d Fo( )925 394 y Fm(A)p Fj(\012)p Fm(A)1015 388 y Fp(=)i Fo(\026)1083 394 y Fm(A)1120 388 y Fk(\016)e Fp(\()p Fo( )1193 394 y Fm(A)1229 388 y Fk(\012)h Fo( )1298 394 y Fm(A)1325 388 y Fp(\))257 466 y(Here,)18 b(the)f(\014rst)g (equalit)o(y)f(holds)h(b)q(ecause)h Fo( )g Fp(is)e(natural,)g(and)h (the)g(second)h(b)q(ecause)g Fo( )257 515 y Fp(is)h(monoidal.)c(By)k(a) g(similar)d(calculation,)h(it)i(follo)o(ws)e(that)i Fo( )1287 521 y Fm(A)1332 515 y Fp(comm)o(utes)e(with)i(the)257 565 y(com)o(ultiplication.)8 b(It)k(is)g(clear)g(that)g(it)f(preserv)o (es)k(the)d(unit)g(and)f(the)i(counit.)e(Finally)m(,)e(w)o(e)257 615 y(ha)o(v)o(e)k Fo( )379 621 y Fm(A)414 615 y Fk(\016)8 b Fo(S)468 621 y Fm(A)507 615 y Fp(=)k Fo(S)576 621 y Fm(A)611 615 y Fk(\016)c Fo( )667 621 y Fm(A)707 615 y Fp(since)14 b Fo( )835 621 y Fm(A)876 615 y Fp(is)f(natural.)f(This)h (pro)o(v)o(es)h(the)g(second)g(statemen)o(t.)257 665 y(F)m(or)g(the)g(last)g(one,)g(observ)o(e)h(that)f(w)o(e)g(ha)o(v)o(e:) 559 742 y Fo(\022)578 748 y Fm(A)615 742 y Fk(\016)9 b Fo(\026)670 748 y Fm(A)708 742 y Fp(=)j Fo(\026)777 748 y Fm(A)813 742 y Fk(\016)d Fo(\022)862 748 y Fm(A)p Fj(\012)p Fm(A)952 742 y Fp(=)j Fo(\026)1021 748 y Fm(A)1057 742 y Fk(\016)d Fp(\()p Fo(\022)1122 748 y Fm(A)1159 742 y Fk(\012)h Fo(\022)1220 748 y Fm(A)1247 742 y Fp(\))g Fk(\016)e Fo(\033)1327 724 y Fj(\000)p Fl(2)1326 754 y Fm(A;A)257 819 y Fp(Again,)h(the)i(\014rst)h(equalit)o(y)d(holds)h(b) q(ecause)i Fo(\022)g Fp(is)f(natural,)e(and)h(the)h(second)h(holds)e(b) q(ecause)257 869 y Fo(\022)19 b Fp(is)d(a)g(ribb)q(on)h (transformation.)d(No)o(w,)i(it)g(is)g(w)o(ell)g(kno)o(wn)g(that)h(w)o (e)g(ha)o(v)o(e:)f Fo(S)1519 875 y Fm(A)1557 869 y Fk(\016)11 b Fo(\026)1614 875 y Fm(A)1657 869 y Fp(=)257 919 y Fo(\026)282 925 y Fm(A)319 919 y Fk(\016)e Fp(\()p Fo(S)390 925 y Fm(A)426 919 y Fk(\012)h Fo(S)493 925 y Fm(A)520 919 y Fp(\))g Fk(\016)f Fo(\033)600 925 y Fm(A;A)661 919 y Fp(,)14 b(and)g(therefore)h(w)o(e)f(ha)o(v)o(e:)314 996 y Fo(S)341 979 y Fl(2)339 1006 y Fm(A)376 996 y Fk(\016)9 b Fo(\022)425 1002 y Fm(A)462 996 y Fk(\016)g Fo(\026)517 1002 y Fm(A)555 996 y Fp(=)j Fo(S)626 979 y Fl(2)624 1006 y Fm(A)661 996 y Fk(\016)d Fo(\026)716 1002 y Fm(A)752 996 y Fk(\016)g Fp(\()p Fo(\022)817 1002 y Fm(A)854 996 y Fk(\012)g Fo(\022)914 1002 y Fm(A)942 996 y Fp(\))g Fk(\016)g Fo(\033)1022 978 y Fj(\000)p Fl(2)1021 1008 y Fm(A;A)1094 996 y Fp(=)j Fo(\026)1163 1002 y Fm(A)1199 996 y Fk(\016)d Fp(\()p Fo(S)1272 979 y Fl(2)1270 1006 y Fm(A)1307 996 y Fk(\012)h Fo(S)1376 979 y Fl(2)1374 1006 y Fm(A)1401 996 y Fp(\))g Fk(\016)f Fp(\()p Fo(\022)1492 1002 y Fm(A)1529 996 y Fk(\012)g Fo(\022)1589 1002 y Fm(A)1617 996 y Fp(\))257 1073 y(It)j(follo)o(ws)e(again)h(b)o(y)g (similar)f(calculations)h(that)h Fo(S)1070 1079 y Fm(A)1102 1073 y Fk(\016)5 b Fo(\022)1147 1079 y Fm(A)1186 1073 y Fp(comm)o(utes)10 b(with)h(the)i(com)o(ulti-)257 1123 y(plication)f(and)h(preserv)o(es)i(the)f(unit)e(and)h(the)h(counit.)e (It)h(comm)o(utes)e(with)i(the)h(an)o(tip)q(o)q(de)257 1173 y(since)h Fo(\022)h Fp(is)d(natural.)g Fg(\003)257 1302 y Fn(3.11)48 b Fp(The)13 b(morphism)c Fo(\022)685 1308 y Fm(A)725 1302 y Fp(also)j(o)q(ccurs)i(in)e(the)h(form)o(ula)d (for)i(the)h(squared)g(an)o(tip)q(o)q(de)f(of)257 1352 y(the)k(Radford)e(bipro)q(duct)i(\(cf.)f([23)o(],)f([19)o(],)p Fk(x)p Fp(10.6\).)f(Recall)i(that)g(the)h(Radford)e(bipro)q(duct)257 1402 y(construction,)i(whic)o(h)g(is)f(the)h(analogue)e(of)h(a)g (semidirect)h(pro)q(duct)g(for)f(Hopf)g(algebras,)257 1452 y(assigns)c(to)f(ev)o(ery)i(Y)m(etter-Drinfel'd)e(Hopf)g(algebra)g Fo(A)g Fp(an)g(ordinary)g(Hopf)h(algebra)e Fo(A)r Fk(\012)r Fo(H)257 1501 y Fp(with)14 b(m)o(ultiplicatio)o(n:)643 1578 y(\()p Fo(a)c Fk(\012)f Fo(h)p Fp(\)\()p Fo(a)810 1561 y Fj(0)831 1578 y Fk(\012)h Fo(h)897 1561 y Fj(0)908 1578 y Fp(\))i(=)g Fo(a)p Fp(\()p Fo(h)1042 1584 y Fl(1)1072 1578 y Fk(!)f Fo(a)1147 1561 y Fj(0)1159 1578 y Fp(\))e Fk(\012)g Fo(h)1249 1584 y Fl(2)1268 1578 y Fo(h)1292 1561 y Fj(0)257 1656 y Fp(com)o(ultiplication:)634 1733 y(\001\()p Fo(a)g Fk(\012)g Fo(h)p Fp(\))j(=)f(\()p Fo(a)890 1739 y Fl(1)918 1733 y Fk(\012)f Fo(a)982 1739 y Fl(2)1000 1716 y(1)1019 1733 y Fo(h)1043 1739 y Fl(1)1062 1733 y Fp(\))f Fk(\012)g Fp(\()p Fo(a)1166 1739 y Fl(2)1185 1716 y(2)1213 1733 y Fk(\012)h Fo(h)1279 1739 y Fl(2)1297 1733 y Fp(\))257 1810 y(unit)k(1)366 1816 y Fm(A)402 1810 y Fk(\012)c Fp(1)465 1816 y Fm(H)496 1810 y Fp(,)j(counit)h Fo(\017)665 1816 y Fm(A)701 1810 y Fk(\012)c Fo(\017)760 1816 y Fm(H)805 1810 y Fp(and)j(an)o(tip)q(o)q(de:)555 1887 y Fo(S)580 1893 y Fm(A)p Fj(\012)p Fm(H)663 1887 y Fp(\()p Fo(a)c Fk(\012)g Fo(h)p Fp(\))j(=)g(\(1)884 1893 y Fm(A)920 1887 y Fk(\012)e Fo(S)987 1893 y Fm(H)1019 1887 y Fp(\()p Fo(a)1057 1870 y Fl(1)1075 1887 y Fo(h)p Fp(\)\)\()p Fo(S)1172 1893 y Fm(A)1200 1887 y Fp(\()p Fo(a)1238 1870 y Fl(2)1257 1887 y Fp(\))f Fk(\012)h Fp(1)1345 1893 y Fm(H)1376 1887 y Fp(\))257 1964 y(W)m(e)k(ha)o(v)o(e)g(the)g (follo)o(wing)d(form)o(ula)h(for)h(the)i(square)f(of)g(the)g(an)o(tip)q (o)q(de)g(in)f Fo(A)d Fk(\012)f Fo(H)s Fp(:)257 2055 y Fn(Prop)q(osition)33 b Fo(S)563 2040 y Fl(2)561 2067 y Fm(A)p Fj(\012)p Fm(H)644 2055 y Fp(\()p Fo(a)9 b Fk(\012)h Fo(h)p Fp(\))h(=)h Fo(S)855 2040 y Fl(2)853 2067 y Fm(A)881 2055 y Fp(\()p Fo(\022)916 2061 y Fm(A)943 2055 y Fp(\()p Fo(a)p Fp(\)\))e Fk(\012)g Fo(S)1092 2040 y Fl(2)1090 2067 y Fm(H)1122 2055 y Fp(\()p Fo(h)p Fp(\))257 2147 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(ha)o(v)o(e:)425 2224 y Fo(S)452 2207 y Fl(2)450 2234 y Fm(A)p Fj(\012)p Fm(H)532 2224 y Fp(\()p Fo(a)c Fk(\012)f Fo(h)p Fp(\))j(=)g Fo(S)742 2230 y Fm(A)p Fj(\012)p Fm(H)824 2224 y Fp(\()p Fo(S)865 2230 y Fm(A)893 2224 y Fp(\()p Fo(a)931 2207 y Fl(2)950 2224 y Fp(\))d Fk(\012)h Fp(1)1038 2230 y Fm(H)1069 2224 y Fp(\))p Fo(S)1110 2230 y Fm(A)p Fj(\012)p Fm(H)1193 2224 y Fp(\(1)1230 2230 y Fm(A)1266 2224 y Fk(\012)g Fo(S)1333 2230 y Fm(H)1365 2224 y Fp(\()p Fo(a)1403 2207 y Fl(1)1421 2224 y Fo(h)p Fp(\)\))673 2291 y(=)i(\(1)754 2297 y Fm(A)790 2291 y Fk(\012)d Fo(S)856 2297 y Fm(H)888 2291 y Fp(\()p Fo(a)926 2274 y Fl(2)945 2291 y Fp(\)\)\()p Fo(S)1020 2274 y Fl(2)1018 2301 y Fm(A)1046 2291 y Fp(\()p Fo(a)1084 2274 y Fl(3)1103 2291 y Fp(\))g Fk(\012)g Fp(1)1190 2297 y Fm(H)1222 2291 y Fp(\)\(1)1275 2297 y Fm(A)1311 2291 y Fk(\012)g Fo(S)1379 2274 y Fl(2)1377 2301 y Fm(H)1409 2291 y Fp(\()p Fo(a)1447 2274 y Fl(1)1466 2291 y Fo(h)p Fp(\)\))673 2359 y(=)j Fo(S)744 2341 y Fl(2)742 2369 y Fm(A)769 2359 y Fp(\()p Fo(S)810 2365 y Fm(H)842 2359 y Fp(\()p Fo(a)880 2341 y Fl(3)899 2359 y Fp(\))g Fk(!)f Fo(a)1002 2341 y Fl(4)1020 2359 y Fp(\))f Fk(\012)f Fo(S)1112 2365 y Fm(H)1144 2359 y Fp(\()p Fo(a)1182 2341 y Fl(2)1201 2359 y Fp(\))p Fo(S)1244 2341 y Fl(2)1242 2369 y Fm(H)1274 2359 y Fp(\()p Fo(a)1312 2341 y Fl(1)1330 2359 y Fp(\))p Fo(S)1373 2341 y Fl(2)1371 2369 y Fm(H)1404 2359 y Fp(\()p Fo(h)p Fp(\))673 2426 y(=)j Fo(S)744 2409 y Fl(2)742 2436 y Fm(A)769 2426 y Fp(\()p Fo(\022)804 2432 y Fm(A)832 2426 y Fp(\()p Fo(a)p Fp(\)\))d Fk(\012)h Fo(S)980 2409 y Fl(2)978 2436 y Fm(H)1010 2426 y Fp(\()p Fo(h)p Fp(\))257 2503 y(since)15 b(the)g(an)o(tip)q(o)q(de)e(is)h(an)g(algebra)f(an)o (tihomom)o(orphism)o(.)d Fg(\003)953 2628 y Fp(29)p eop %%Page: 30 30 30 29 bop 257 262 a Fp(W)m(e)19 b(ha)o(v)o(e)f(to)h(emphasize)f(that)h (the)h(ab)q(o)o(v)o(e)e(form)o(ula)e(w)o(as)j(noted)g(indep)q(enden)o (tly)h(and)257 311 y(earlier)13 b(b)o(y)f(N.)g(Andruskiewitsc)o(h)i (and)f(H.-J.)e(Sc)o(hneider)j(\(cf.)e([2],)f(p.)h(7,)g(eq.)g(\(4.5\)\)) g(where)257 361 y(also)j(the)i(algebra)e(homom)o(orphism)d Fo(S)885 346 y Fl(2)883 373 y Fm(A)921 361 y Fk(\016)e Fo(\022)971 367 y Fm(A)1014 361 y Fp(is)15 b(considered)i(in)e(this)h (manner,)e(that)i(is,)257 411 y(b)o(y)e(restriction)h(from)d(the)i (Radford)f(bipro)q(duct.)257 545 y Fn(3.12)48 b Fp(Tw)o(o)16 b(monoidal)e(transformations)h(that)i(will)e(pla)o(y)h(an)g(imp)q (ortan)o(t)f(role)i(in)f(the)257 595 y(next)e(section)g(deserv)o(e)i (sp)q(ecial)d(names.)g(In)g(fact,)g(the)h(sole)f(purp)q(ose)i(of)e (this)g(section)i(w)o(as)257 645 y(to)f(describ)q(e)i(their)e(prop)q (erties)h(from)d(a)i(more)f(abstract)i(p)q(oin)o(t)e(of)g(view.)257 742 y Fn(De\014nition)33 b Fp(Supp)q(ose)15 b(that)f Fo(V)23 b Fp(is)14 b(a)f(left)h(Y)m(etter-Drinfel'd)g(mo)q(dule)e(o)o (v)o(er)i Fo(H)s Fp(.)308 857 y(1.)20 b(If)12 b Fo(H)j Fp(is)d(\014nite-dimensional)e(and)i Fo(\013)930 842 y Fm(R)930 869 y(H)973 857 y Fp(\(resp.)h Fo(a)1109 842 y Fm(R)1109 869 y(H)1141 857 y Fp(\))f(is)g(the)h(righ)o(t)e(mo)q (dular)g(function)361 907 y(\(resp.)k(mo)q(dular)d(elemen)o(t\),)h (de\014ne)i(the)f(righ)o(t)g(mo)q(dular)e(transformation)g(as:)673 995 y Fo(M)718 978 y Fm(R)713 1006 y(V)757 995 y Fp(:)f Fo(V)21 b Fk(!)11 b Fo(V)902 1004 y Fm(S)924 994 y Fd(4)922 1014 y Fe(H)951 995 y Fo(;)c(v)13 b Fk(7!)e Fo(\013)1083 978 y Fm(R)1083 1006 y(H)1114 995 y Fp(\()p Fo(v)1151 978 y Fl(1)1171 995 y Fp(\)\()p Fo(a)1225 978 y Fm(R)1225 1006 y(H)1268 995 y Fk(!)g Fo(v)1342 978 y Fl(2)1361 995 y Fp(\))361 1084 y(Similarly)l(,)g(de\014ne)k(the)f(left)g(mo)q (dular)e(transformation)g(as:)668 1172 y Fo(M)713 1155 y Fm(L)708 1182 y(V)749 1172 y Fp(:)f Fo(V)21 b Fk(!)11 b Fo(V)894 1185 y Fm(S)916 1171 y Fi(\000)p Fd(4)914 1195 y Fe(H)957 1172 y Fo(;)c(v)13 b Fk(7!)e Fo(\013)1089 1155 y Fm(L)1089 1182 y(H)1120 1172 y Fp(\()p Fo(v)1157 1155 y Fl(1)1176 1172 y Fp(\)\()p Fo(a)1230 1155 y Fm(L)1230 1182 y(H)1273 1172 y Fk(!)g Fo(v)1347 1155 y Fl(2)1367 1172 y Fp(\))361 1272 y(where)20 b Fo(\013)513 1257 y Fm(L)513 1284 y(H)563 1272 y Fp(\(resp.)f Fo(a)705 1257 y Fm(L)705 1284 y(H)737 1272 y Fp(\))g(denotes)h(the)f(left)f(mo)q (dular)f(function)i(\(resp.)g(mo)q(dular)361 1322 y(elemen)o(t\).)308 1404 y(2.)h(If)13 b Fo(A)g Fp(is)f(a)h(\014nite-dimensional)e(left)h(Y) m(etter-Drinfel'd)h(Hopf)g(algebra)f(o)o(v)o(er)h Fo(H)j Fp(with)361 1454 y(in)o(tegral)11 b(c)o(haracter)i Fo(\023)704 1460 y Fm(A)742 1454 y Fp(and)f(in)o(tegral)f(group)g(elemen)o(t)g Fo(g)1256 1460 y Fm(A)1283 1454 y Fp(,)g(w)o(e)h(de\014ne)h(the)f(in)o (tegral)361 1503 y(transformation)g(as:)721 1592 y Fo(I)739 1598 y Fm(V)779 1592 y Fp(:)f Fo(V)21 b Fk(!)11 b Fo(V)r(;)c(v)14 b Fk(7!)d Fo(\023)1047 1598 y Fm(A)1073 1592 y Fp(\()p Fo(v)1110 1574 y Fl(1)1130 1592 y Fp(\)\()p Fo(g)1182 1598 y Fm(A)1220 1592 y Fk(!)g Fo(v)1294 1574 y Fl(2)1314 1592 y Fp(\))257 1706 y(W)m(e)f(note)g(that)f(a)h(relation)f(of)g(the)h (use)h(of)e(the)h(term)f(`mo)q(dular)e(transformation')h(used)i(here) 257 1756 y(to)j(the)h(use)g(of)f(the)g(same)f(term)h(in)g([17)o(])f(is) h(not)g(in)o(tended.)h(Similarl)o(y)m(,)9 b(the)14 b(term)f(`in)o (tegral)257 1806 y(transform')g(is)g(not)h(in)o(tended)h(to)e(mean)g (something)g(lik)o(e)g(a)g(`F)m(ourier)h(transform'.)257 1940 y Fn(3.13)48 b Fp(As)14 b(w)o(e)g(ha)o(v)o(e)f(already)g (explained)h(in)f(subsection)i(3.10,)d(the)i(comp)q(osite)f Fo(S)1580 1925 y Fl(2)1578 1952 y Fm(A)1614 1940 y Fk(\016)8 b Fo(\022)1662 1946 y Fm(A)257 1990 y Fp(is)16 b(a)g(Y)m (etter-Drinfel'd)g(Hopf)g(algebra)g(morphism.)c(The)17 b(follo)o(wing)d(Prop)q(osition)h(refor-)257 2040 y(m)o(ulates)20 b(in)h(the)h(con)o(text)g(of)e(ribb)q(on)h(transformations)f(a)h(nice)g (form)o(ula)e(of)i(N.)f(An-)257 2090 y(druskiewitsc)o(h)15 b(and)f(H.)f(J.)h(Sc)o(hneider)h(for)f(its)g(trace:)257 2187 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(\032)810 2193 y Fm(A)849 2187 y Fk(2)d Fo(A)919 2172 y Fj(\003)952 2187 y Fp(and)j(\000)1059 2193 y Fm(A)1097 2187 y Fk(2)d Fo(A)j Fp(are)h(righ)o(t)e(in)o(tegrals)g(satisfying)257 2237 y Fo(\032)278 2243 y Fm(A)306 2237 y Fp(\(\000)348 2243 y Fm(A)375 2237 y Fp(\))f(=)f(1.)j(Then)g(w)o(e)g(ha)o(v)o(e:)f Fo(T)6 b(r)q Fp(\()p Fo(S)862 2222 y Fl(2)860 2249 y Fm(A)897 2237 y Fk(\016)j Fo(\022)946 2243 y Fm(A)974 2237 y Fp(\))i(=)h Fo(\032)1066 2243 y Fm(A)1094 2237 y Fp(\(1\))p Fo(\017)1164 2243 y Fm(A)1191 2237 y Fp(\(\000)1233 2243 y Fm(A)1260 2237 y Fp(\))257 2354 y(F)m(or)k(a)f(pro)q(of,)g(w)o (e)h(refer)g(to)g([2)o(],)f(Theorem)g(7.3.)f(Andruskiewitsc)o(h)j(and)e (Sc)o(hneider)i(also)257 2403 y(pro)o(v)o(e)h(that)g(the)h(ordinary)e (trace)i(of)e Fo(S)900 2388 y Fl(2)898 2415 y Fm(A)937 2403 y Fk(\016)12 b Fo(\022)989 2409 y Fm(A)1034 2403 y Fp(coincides)19 b(with)e(the)i(categorical)e(trace)257 2453 y(of)d Fo(S)332 2438 y Fl(2)330 2465 y Fm(A)357 2453 y Fp(.)g(Ho)o(w)o(ev)o(er,)g(this)g(result)h(dep)q(ends)g(on)f (their)g(de\014nition)g(of)f(the)i(categorical)f(trace:)257 2503 y(Their)d(de\014nition)f(agrees)i(with)e(the)h(de\014nition)f(of)g (P)m(.)f(J.)i(F)m(reyd)f(and)h(D.)e(N.)h(Y)m(etter)i(\(cf.)e([6)o(],) 953 2628 y(30)p eop %%Page: 31 31 31 30 bop 257 262 a Fp(Def.)10 b(1.5,)e(p.)h(160\),)g(but)h(not)g(with) f(the)i(re\014ned)g(de\014nition)e(of)h(V.)f(G.)g(T)m(uraev)h(\(cf.)f ([36],)f([37)o(],)257 311 y(sec.)16 b(I.1.5,)e(p.)h(21\).)f(Ho)o(w)o (ev)o(er,)h(since)h(T)m(uraev's)f(de\014nition)g(in)o(v)o(olv)o(es)f(a) h(t)o(wist,)g(it)g(is)g(only)257 361 y(de\014ned)c(for)e(ribb)q(on)h (categories,)g(and,)f(as)h(w)o(e)f(ha)o(v)o(e)h(already)f(noted)h(ab)q (o)o(v)o(e,)f(w)o(e)h(kno)o(w)f(from)257 411 y([27)o(])16 b(that)h(the)g(category)g(of)f(Y)m(etter-Drinfel'd)g(mo)q(dules)f(is,)h (in)g(general,)g(not)g(a)h(ribb)q(on)257 461 y(category)m(.)12 b(But)h(w)o(e)g(ha)o(v)o(e)f(already)g(explained)g(that)h(the)g(ribb)q (on)f(transformation)f Fo(\022)1586 467 y Fm(V)1627 461 y Fp(can)257 511 y(in)g(a)f(w)o(a)o(y)g(serv)o(e)i(as)f(a)g(substitute) h(for)e(this.)g(Therefore,)i(one)f(reasonable)g(de\014nition)g(of)f (the)257 560 y(categorical)j(trace)h(of)f(an)g Fo(H)s Fp(-linear)f(and)h(colinear)f(endomorphism)f Fo(f)16 b Fp(:)11 b Fo(V)21 b Fk(!)11 b Fo(V)23 b Fp(of)12 b(some)257 610 y(Y)m(etter-Drinfel'd)i(mo)q(dule)e Fo(V)24 b Fp(w)o(ould)13 b(b)q(e)h(the)h(image)d(of)h(the)h(unit)g(in)g(the)g(comp)q(osition)591 700 y Fo(ev)e Fk(\016)d Fp(\()p Fo(\022)708 682 y Fj(\000)p Fl(1)707 712 y Fm(V)762 700 y Fk(\012)h Fo(id)840 706 y Fm(V)866 698 y Fi(\003)886 700 y Fp(\))f Fk(\016)g Fo(\033)965 706 y Fm(V)992 698 y Fi(\003)1009 706 y Fm(;V)1057 700 y Fk(\016)g Fp(\()p Fo(id)1139 706 y Fm(V)1166 698 y Fi(\003)1194 700 y Fk(\012)h Fo(f)t Fp(\))g Fk(\016)f Fo(db)257 790 y Fp(from)j Fo(K)k Fp(to)d Fo(K)s Fp(,)g(where)h Fo(ev)h Fp(and)e Fo(db)f Fp(are)i(de\014ned)g(as)f(in)f(subsection)j (3.7.)c(This)i(is)g(justi\014ed)257 840 y(in)e(particular)g(b)q(ecause) i(a)f(simple)d(calculation)i(sho)o(ws)g(that)h(this)f(notion)g(of)g(a)g (categorical)257 890 y(trace)k(agrees)g(with)f(the)g(ordinary)g(trace.) 257 1062 y Fq(4)67 b(An)n(tip)r(o)r(des)23 b(and)g(Nak)l(a)n(y)n(ama)f (automorphisms)257 1188 y Fn(4.1)48 b Fp(In)13 b(this)g(section,)h(w)o (e)f(pro)o(v)o(e)g(an)g(analogue)f(of)g(a)h(form)o(ula)e(of)h(D.)g (Radford)h(\(cf.)f([22)o(],)257 1238 y(Prop.)19 b(6,)f(p.)h(347\))f (for)h(the)h(fourth)f(p)q(o)o(w)o(er)g(of)f(the)i(an)o(tip)q(o)q(de)f (in)g(the)g(case)h(of)f(Y)m(etter-)257 1288 y(Drinfel'd)e(Hopf)g (algebras.)h(In)f(con)o(trast)i(to)f(Radford's)e(result)j(on)f (ordinary)f(Hopf)g(al-)257 1338 y(gebras,)g(the)g(form)o(ula)d(in)o(v)o (olv)o(es)h(in)h(the)h(Y)m(etter-Drinfel'd)g(case)g(the)g(mo)q(dular)e (and)h(the)257 1387 y(in)o(tegral)10 b(transformations)f(as)i(w)o(ell)f (as)h(the)g(ribb)q(on)g(transformation)d Fo(\022)13 b Fp(considered)f(in)e(the)257 1437 y(previous)15 b(section.)f(The)g(pro) q(of)g(of)f(the)i(form)o(ula)c(follo)o(ws)h(a)i(pro)q(of)f(of)h(H.-J.)f (Sc)o(hneider)i(of)257 1487 y(Radford's)f(result)h(\(cf.)f([30)o(]\).)f (In)h(order)h(to)g(pro)o(v)o(e)f(this)g(form)o(ula,)d(w)o(e)k(in)o(tro) q(duce)g(t)o(wisted)257 1537 y(v)o(ersions)c(of)e(the)i(Nak)n(a)o(y)o (ama)c(automorphism)g(and)i(deriv)o(e)i(explicit)e(form)o(ulas)f(for)i (them)f(in)257 1587 y(terms)14 b(of)e(the)i(square)g(of)f(the)h(an)o (tip)q(o)q(de.)f(W)m(e)g(also)g(consider)h(the)g(in)o(terrelation)f(b)q (et)o(w)o(een)257 1637 y(the)19 b(t)o(wisted)f(Nak)n(a)o(y)o(ama)d (automorphisms,)g(the)j(mo)q(dular)e(and)i(in)o(tegral)f(transforma-) 257 1686 y(tions)d(and)f(the)i(ribb)q(on)e(transformation)f Fo(\022)j Fp(considered)g(in)e(the)i(last)e(section.)h(Along)f(the)257 1736 y(w)o(a)o(y)m(,)g(w)o(e)h(calculate)g(the)g(eigen)o(v)n(alue)g(of) f(the)h(squared)h(an)o(tip)q(o)q(de)f(on)f(an)h(in)o(tegral.)257 1821 y(In)i(the)g(whole)f(section,)h Fo(A)f Fp(denotes)i(a)e (\014nite-dimensional)e(left)j(Y)m(etter-Drinfel'd)f(Hopf)257 1871 y(algebra)f(o)o(v)o(er)g(a)g(Hopf)f(algebra)h Fo(H)s Fp(.)f(W)m(e)h(assume)f(that)h(the)h(an)o(tip)q(o)q(de)f(of)f Fo(H)k Fp(is)d(bijectiv)o(e.)257 1921 y(W)m(e)e(shall)e(use)j(the)f (notation)f(of)f(the)j(preceding)f(sections,)g(in)f(particular)h(for)f (the)h(in)o(tegral)257 1971 y(c)o(haracter)21 b Fo(\023)459 1977 y Fm(A)486 1971 y Fp(,)e(the)i(in)o(tegral)e(group)g(elemen)o(t)g Fo(g)1054 1977 y Fm(A)1081 1971 y Fp(,)g(and)h(the)g(mo)q(dular)e (elemen)o(ts)i(and)257 2020 y(functions)15 b Fo(a)459 2005 y Fm(L)459 2032 y(A)486 2020 y Fp(,)e Fo(a)533 2005 y Fm(R)533 2032 y(A)560 2020 y Fp(,)h Fo(\013)613 2005 y Fm(L)613 2032 y(A)653 2020 y Fp(and)g Fo(\013)761 2005 y Fm(R)761 2032 y(A)802 2020 y Fp(of)g Fo(A)p Fp(.)g Fo(\032)928 2026 y Fm(A)967 2020 y Fk(2)d Fo(A)1037 2005 y Fj(\003)1071 2020 y Fp(and)j(\000)1178 2026 y Fm(A)1216 2020 y Fk(2)e Fo(A)i Fp(denote)h(righ)o(t)f(in)o(tegrals)257 2070 y(that)f(satisfy)f Fo(\032)495 2076 y Fm(A)522 2070 y Fp(\(\000)564 2076 y Fm(A)591 2070 y Fp(\))g(=)g(1.)f Fo(M)752 2055 y Fm(R)747 2082 y(V)780 2070 y Fp(,)g Fo(M)848 2055 y Fm(L)843 2082 y(V)873 2070 y Fp(,)h Fo(I)915 2076 y Fm(V)956 2070 y Fp(and)g Fo(\022)1054 2076 y Fm(V)1096 2070 y Fp(denote)h(the)g(mo)q(dular,)d(in)o(tegral)i(and)257 2120 y(the)j(ribb)q(on)f(transformation)e(in)o(tro)q(duced)i(in)g (section)g(3.)257 2255 y Fn(4.2)48 b Fp(W)m(e)18 b(ha)o(v)o(e)f (already)h(calculated)g(in)g(Prop)q(osition)f(2.10)g(the)i(Casimir)d (elemen)o(t)h(of)257 2305 y Fo(A)f Fp(that)g(arises)h(from)d(a)i (nonzero)g(righ)o(t)g(in)o(tegral.)e(Ho)o(w)o(ev)o(er,)i(w)o(e)g(shall) g(need)g(t)o(w)o(o)g(other)257 2354 y(forms)d(of)g(this)h(elemen)o(t.) 257 2453 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(\032)810 2459 y Fm(A)849 2453 y Fk(2)d Fo(A)919 2438 y Fj(\003)952 2453 y Fp(and)j(\000)1059 2459 y Fm(A)1097 2453 y Fk(2)d Fo(A)j Fp(are)h(righ)o(t)e(in)o(tegrals)g(satisfying)257 2503 y Fo(\032)278 2509 y Fm(A)306 2503 y Fp(\(\000)348 2509 y Fm(A)375 2503 y Fp(\))f(=)f(1.)j(Then)g(the)g(Casimir)e(elemen)o (t)h(for)h(the)g(F)m(rob)q(enius)g(homomorphism)9 b Fo(\032)1620 2509 y Fm(A)1662 2503 y Fp(is)953 2628 y(31)p eop %%Page: 32 32 32 31 bop 257 262 a Fp(giv)o(en)14 b(b)o(y:)346 344 y Fo(S)371 350 y Fm(A)399 344 y Fp(\()p Fo(g)436 326 y Fj(\000)p Fl(1)435 356 y Fm(A)492 344 y Fk(!)d Fp(\000)571 350 y Fm(A)r Fl(1)617 344 y Fp(\))p Fo(a)655 327 y Fm(R)655 354 y(A)691 344 y Fk(\012)f Fp(\000)759 350 y Fm(A)r Fl(2)816 344 y Fp(=)i Fo(S)887 326 y Fj(\000)p Fl(1)885 356 y Fm(A)932 344 y Fp(\(\000)974 350 y Fm(A)r Fl(2)1020 327 y(3)1039 344 y Fp(\))d Fk(\012)g Fo(\023)1120 350 y Fm(A)1147 344 y Fp(\(\000)1189 350 y Fm(A)r Fl(2)1235 327 y(2)1254 344 y Fp(\))p Fo(S)1297 326 y Fj(\000)p Fl(1)1295 356 y Fm(H)1342 344 y Fp(\(\000)1384 350 y Fm(A)r Fl(2)1430 327 y(1)1448 344 y Fp(\))j Fk(!)f Fp(\000)1555 350 y Fm(A)r Fl(1)257 438 y Fn(Pro)q(of.)36 b Fp(The)15 b(\014rst)f(elemen)o(t)g(is)f(the)i(Casimir)d(elemen)o(t)h(of)g Fo(\032)1237 444 y Fm(A)1279 438 y Fp(b)q(ecause)i(w)o(e)f(ha)o(v)o(e:) 278 521 y Fo(S)303 527 y Fm(A)331 521 y Fp(\()p Fo(g)368 503 y Fj(\000)p Fl(1)367 533 y Fm(A)424 521 y Fk(!)d Fp(\000)503 527 y Fm(A)r Fl(1)549 521 y Fp(\))p Fo(a)587 504 y Fm(R)587 531 y(A)614 521 y Fo(\032)635 527 y Fm(A)663 521 y Fp(\(\000)705 527 y Fm(A)r Fl(2)750 521 y Fo(a)p Fp(\))h(=)g Fo(S)869 527 y Fm(A)896 521 y Fp(\()p Fo(S)937 527 y Fm(H)969 521 y Fp(\(\000)1011 527 y Fm(A)1038 504 y Fl(1)1057 521 y Fp(\))g Fk(!)f Fp(\000)1164 527 y Fm(A)1191 503 y Fl(2)1209 527 y(1)1228 521 y Fp(\))p Fo(a)1266 504 y Fm(R)1266 531 y(A)1293 521 y Fo(\032)1314 527 y Fm(A)1342 521 y Fp(\(\000)1384 527 y Fm(A)1411 503 y Fl(2)1430 527 y(2)1448 521 y Fo(a)p Fp(\))626 589 y(=)g Fo(S)694 595 y Fm(A)722 589 y Fp(\()p Fo(S)763 595 y Fm(H)795 589 y Fp(\(\000)837 595 y Fm(A)864 572 y Fl(2)883 589 y Fp(\))g Fk(!)g Fp(\000)989 595 y Fm(A)1016 571 y Fl(3)1035 595 y(1)1054 589 y Fp(\))p Fo(\017)1087 595 y Fm(H)1118 589 y Fp(\(\000)1160 595 y Fm(A)1187 572 y Fl(1)1206 589 y Fp(\))p Fo(a)1244 572 y Fm(R)1244 600 y(A)1271 589 y Fo(\032)1292 595 y Fm(A)1320 589 y Fp(\(\000)1362 595 y Fm(A)1389 571 y Fl(3)1407 595 y(2)1426 589 y Fo(a)p Fp(\))626 658 y(=)g Fo(S)694 664 y Fm(H)726 658 y Fp(\(\000)768 664 y Fm(A)795 641 y Fl(1)814 658 y Fp(\))h Fk(!)f Fp(\()p Fo(S)936 664 y Fm(A)963 658 y Fp(\(\000)1005 664 y Fm(A)1033 640 y Fl(2)1051 664 y(1)1070 658 y Fp(\))p Fo(a)1108 641 y Fm(R)1108 668 y(A)1135 658 y Fo(\032)1156 664 y Fm(A)1184 658 y Fp(\(\000)1226 664 y Fm(A)1253 640 y Fl(2)1271 664 y(2)1290 658 y Fo(a)p Fp(\)\))626 726 y(=)g Fo(S)694 732 y Fm(H)726 726 y Fp(\(\000)768 732 y Fm(A)795 709 y Fl(1)814 726 y Fp(\))h Fk(!)f Fp(\()p Fo(S)936 732 y Fm(A)963 726 y Fp(\(\000)1005 732 y Fm(A)1033 708 y Fl(2)1051 732 y(1)1070 726 y Fp(\)\000)1112 732 y Fm(A)1139 709 y Fl(2)1158 732 y(2)1176 726 y Fp(\(\000)1218 732 y Fm(A)1245 709 y Fl(2)1264 732 y(3)1283 709 y(1)1313 726 y Fk(!)g Fo(a)1388 732 y Fl(1)1406 726 y Fp(\)\))p Fo(\032)1459 732 y Fm(A)1487 726 y Fp(\(\000)1529 732 y Fm(A)1556 709 y Fl(2)1575 732 y(3)1593 709 y(2)1612 726 y Fo(a)1634 732 y Fl(2)1653 726 y Fp(\))626 794 y(=)g Fo(S)694 800 y Fm(H)726 794 y Fp(\(\000)768 800 y Fm(A)795 777 y Fl(1)814 794 y Fp(\))h Fk(!)f Fp(\(\000)937 800 y Fm(A)964 777 y Fl(2)994 794 y Fk(!)g Fo(a)1069 800 y Fl(1)1088 794 y Fp(\))p Fo(\032)1125 800 y Fm(A)1152 794 y Fp(\(\000)1194 800 y Fm(A)1221 777 y Fl(3)1240 794 y Fo(a)1262 800 y Fl(2)1281 794 y Fp(\))626 856 y(=)g Fo(a)691 862 y Fl(1)710 856 y Fo(\032)731 862 y Fm(A)758 856 y Fp(\(\000)800 862 y Fm(A)828 856 y Fo(a)850 862 y Fl(2)868 856 y Fp(\))h(=)g Fo(a)257 939 y Fp(T)m(o)20 b(sho)o(w)g(that)g(the)g(second)i(elemen)o(t)d(is)h(the)h(Casimir)d (elemen)o(t)h(of)h Fo(\032)1428 945 y Fm(A)1455 939 y Fp(,)g(w)o(e)g(use)h(the)257 988 y(sk)o(ew-an)o(tip)q(o)q(de)15 b(equation:)292 1071 y Fo(S)319 1053 y Fj(\000)p Fl(1)317 1083 y Fm(A)365 1071 y Fp(\(\000)407 1077 y Fm(A)r Fl(2)452 1054 y(3)471 1071 y Fp(\))p Fo(\023)502 1077 y Fm(A)529 1071 y Fp(\(\000)571 1077 y Fm(A)r Fl(2)617 1054 y(2)635 1071 y Fp(\))p Fo(\032)672 1077 y Fm(A)700 1071 y Fp(\(\()p Fo(S)759 1053 y Fj(\000)p Fl(1)757 1083 y Fm(H)805 1071 y Fp(\(\000)847 1077 y Fm(A)r Fl(2)892 1054 y(1)911 1071 y Fp(\))d Fk(!)f Fp(\000)1018 1077 y Fm(A)r Fl(1)1063 1071 y Fp(\))p Fo(a)p Fp(\))304 1139 y(=)h Fo(S)375 1121 y Fj(\000)p Fl(1)373 1151 y Fm(A)420 1139 y Fp(\(\000)462 1145 y Fm(A)r Fl(2)508 1122 y(3)526 1139 y Fp(\)\()p Fo(\032)579 1145 y Fm(A)619 1139 y Fk( )f Fp(\000)698 1145 y Fm(A)r Fl(2)743 1122 y(2)762 1139 y Fp(\)\(\()p Fo(S)837 1121 y Fj(\000)p Fl(1)835 1151 y Fm(H)883 1139 y Fp(\(\000)925 1145 y Fm(A)r Fl(2)970 1122 y(1)989 1139 y Fp(\))h Fk(!)f Fp(\000)1096 1145 y Fm(A)r Fl(1)1141 1139 y Fp(\))p Fo(a)p Fp(\))304 1207 y(=)h Fo(S)375 1189 y Fj(\000)p Fl(1)373 1219 y Fm(A)420 1207 y Fp(\(\000)462 1213 y Fm(A)r Fl(2)508 1190 y(2)526 1207 y Fp(\))p Fo(\032)563 1213 y Fm(A)591 1207 y Fp(\(\000)633 1213 y Fm(A)r Fl(1)679 1207 y Fp(\(\000)721 1213 y Fm(A)r Fl(2)766 1190 y(1)797 1207 y Fk(!)f Fo(a)p Fp(\)\))304 1275 y(=)h Fo(S)375 1257 y Fj(\000)p Fl(1)373 1287 y Fm(A)420 1275 y Fp(\(\000)462 1281 y Fm(A)r Fl(2)508 1258 y(3)526 1275 y Fp(\)\()p Fo(S)585 1257 y Fj(\000)p Fl(1)583 1287 y Fm(H)631 1275 y Fp(\(\000)673 1281 y Fm(A)r Fl(2)719 1258 y(2)737 1275 y Fp(\))g Fk(!)f Fo(\032)839 1281 y Fm(A)867 1275 y Fp(\(\000)909 1281 y Fm(A)r Fl(1)954 1275 y Fp(\(\000)996 1281 y Fm(A)r Fl(2)1042 1258 y(1)1072 1275 y Fk(!)g Fo(a)p Fp(\)\)1)1200 1281 y Fm(A)1227 1275 y Fp(\))304 1343 y(=)h Fo(S)375 1325 y Fj(\000)p Fl(1)373 1355 y Fm(A)420 1343 y Fp(\(\000)462 1349 y Fm(A)r Fl(3)508 1326 y(4)526 1343 y Fp(\)\()p Fo(S)585 1325 y Fj(\000)p Fl(1)583 1355 y Fm(H)631 1343 y Fp(\(\000)673 1349 y Fm(A)r Fl(3)719 1326 y(3)737 1343 y Fp(\))g Fk(!)f Fo(\032)839 1349 y Fm(A)867 1343 y Fp(\(\000)909 1349 y Fm(A)r Fl(1)954 1343 y Fp(\(\000)996 1349 y Fm(A)r Fl(2)1042 1326 y(1)1061 1343 y Fp(\000)1087 1349 y Fm(A)r Fl(3)1132 1326 y(1)1162 1343 y Fk(!)h Fo(a)1238 1349 y Fl(1)1256 1343 y Fp(\)\)\(\000)1330 1349 y Fm(A)s Fl(2)1376 1326 y(2)1395 1343 y Fp(\(\000)1437 1349 y Fm(A)r Fl(3)1482 1326 y(2)1513 1343 y Fk(!)f Fo(a)1588 1349 y Fl(2)1606 1343 y Fp(\)\)\))304 1411 y(=)h Fo(\032)369 1417 y Fm(A)396 1411 y Fp(\(\000)438 1417 y Fm(A)r Fl(1)484 1411 y Fp(\(\000)526 1417 y Fm(A)r Fl(2)572 1394 y(1)590 1411 y Fp(\000)616 1417 y Fm(A)r Fl(3)662 1394 y(1)692 1411 y Fk(!)f Fo(a)767 1417 y Fl(1)786 1411 y Fp(\)\))p Fo(S)845 1393 y Fj(\000)p Fl(1)843 1423 y Fm(A)890 1411 y Fp(\(\000)932 1417 y Fm(A)r Fl(3)978 1394 y(3)997 1411 y Fp(\)\()p Fo(S)1056 1393 y Fj(\000)p Fl(1)1054 1423 y Fm(H)1101 1411 y Fp(\(\000)1143 1417 y Fm(A)r Fl(3)1189 1394 y(2)1208 1411 y Fp(\))g Fk(!)g Fp(\000)1314 1417 y Fm(A)r Fl(2)1360 1394 y(2)1379 1411 y Fp(\))p Fo(a)1417 1417 y Fl(2)304 1479 y Fp(=)h Fo(\032)369 1485 y Fm(A)396 1479 y Fp(\(\000)438 1485 y Fm(A)r Fl(1)484 1479 y Fp(\(\000)526 1485 y Fm(A)r Fl(2)572 1462 y(1)602 1479 y Fk(!)f Fo(a)677 1485 y Fl(1)695 1479 y Fp(\)\))p Fo(S)754 1461 y Fj(\000)p Fl(1)752 1491 y Fm(A)800 1479 y Fp(\(\000)842 1485 y Fm(A)r Fl(2)888 1462 y(2)906 1485 y(2)925 1462 y(2)944 1479 y Fp(\)\()p Fo(S)1003 1461 y Fj(\000)p Fl(1)1001 1491 y Fm(H)1048 1479 y Fp(\(\000)1090 1485 y Fm(A)r Fl(2)1136 1462 y(2)1155 1485 y(2)1173 1462 y(1)1192 1479 y Fp(\))h Fk(!)f Fp(\000)1299 1485 y Fm(A)r Fl(2)1344 1462 y(2)1363 1485 y(1)1381 1479 y Fp(\))p Fo(a)1419 1485 y Fl(2)304 1546 y Fp(=)h Fo(\032)369 1552 y Fm(A)396 1546 y Fp(\(\000)438 1552 y Fm(A)r Fl(1)484 1546 y Fp(\(\000)526 1552 y Fm(A)r Fl(2)572 1529 y(1)602 1546 y Fk(!)f Fo(a)677 1552 y Fl(1)695 1546 y Fp(\)\))p Fo(\017)744 1552 y Fm(A)771 1546 y Fp(\(\000)813 1552 y Fm(A)s Fl(2)859 1529 y(2)878 1546 y Fp(\))p Fo(a)916 1552 y Fl(2)304 1609 y Fp(=)h Fo(\032)369 1615 y Fm(A)396 1609 y Fp(\(\000)438 1615 y Fm(A)465 1609 y Fo(a)487 1615 y Fl(1)506 1609 y Fp(\))p Fo(a)544 1615 y Fl(2)574 1609 y Fp(=)g Fo(a)257 1691 y Fp(where)j(w)o(e)g(ha)o(v)o(e)e(used)i (the)f(sk)o(ew-an)o(tip)q(o)q(de)h(equation)e(in)h(the)g(sev)o(en)o(th) h(equalit)o(y)m(.)d Fg(\003)257 1804 y Fp(Let)j(us)f(dra)o(w)f(one)h (ob)o(vious)f(conclusion.)h(W)m(e)f(ha)o(v)o(e)h(seen)h(in)e(Prop)q (osition)g(2.12)g(that)h(the)257 1853 y(an)o(tip)q(o)q(de)d(maps)f (left)h(in)o(tegrals)f(to)h(righ)o(t)g(in)o(tegrals,)f(and)h(similarly) d(maps)h(righ)o(t)i(in)o(tegrals)257 1903 y(to)h(left)f(in)o(tegrals.)f (Since)i(the)g(space)g(of)f(righ)o(t)g(in)o(tegrals)g(is)g (one-dimensional,)e(ev)o(ery)j(righ)o(t)257 1953 y(in)o(tegral)h(is)g (an)h(eigen)o(v)o(ector)g(for)f(the)h(squared)g(an)o(tip)q(o)q(de.)f (It)g(a)h(natural)f(question)g(to)g(ask)257 2003 y(for)h(the)g(eigen)o (v)n(alue:)257 2097 y Fn(Corollary)35 b Fp(If)13 b(\003)560 2103 y Fm(A)600 2097 y Fp(and)g(\000)706 2103 y Fm(A)746 2097 y Fp(are)g(a)g(left)f(and)h(a)g(righ)o(t)f(in)o(tegral)h(resp)q (ectiv)o(ely)m(,)g(then)h(they)257 2147 y(are)h(eigen)o(v)o(ectors)g (of)e Fo(S)633 2132 y Fl(2)631 2158 y Fm(A)672 2147 y Fp(corresp)q(onding)i(to)f(the)g(eigen)o(v)n(alue)g Fo(\023)1275 2153 y Fm(A)1302 2147 y Fp(\()p Fo(g)1338 2153 y Fm(A)1364 2147 y Fp(\))1380 2132 y Fl(2)1399 2147 y Fo(\013)1426 2132 y Fm(R)1426 2158 y(A)1453 2147 y Fp(\()p Fo(a)1491 2132 y Fm(R)1491 2158 y(A)1519 2147 y Fp(\))257 2241 y Fn(Pro)q(of.)36 b Fp(Apply)14 b Fo(id)584 2247 y Fm(A)620 2241 y Fk(\012)9 b Fo(\017)678 2247 y Fm(A)719 2241 y Fp(to)14 b(the)g(equalit)o(y)f(in)h(the)g(ab)q(o)o(v)o(e)g(Prop)q (osition.)f Fg(\003)257 2354 y Fp(Note)i(that)g(if)f(w)o(e)h(set)g Fo(H)h Fp(=)d Fo(K)s Fp(,)h(the)h(base)h(\014eld,)e(then)h(Y)m (etter-Drinfel'd)f(Hopf)h(algebras)257 2403 y(b)q(ecome)g(ordinary)g (Hopf)f(algebras.)h(In)f(this)h(case,)h(the)f(ab)q(o)o(v)o(e)g(Prop)q (osition)g(reduces)i(to)257 2453 y(the)e(form)o(ula:)632 2503 y Fo(S)657 2509 y Fm(A)685 2503 y Fp(\(\000)727 2509 y Fm(A)r Fl(1)773 2503 y Fp(\))p Fo(a)811 2486 y Fm(R)811 2513 y(A)847 2503 y Fk(\012)10 b Fp(\000)915 2509 y Fm(A)r Fl(2)972 2503 y Fp(=)i Fo(S)1043 2485 y Fj(\000)p Fl(1)1041 2515 y Fm(A)1088 2503 y Fp(\(\000)1130 2509 y Fm(A)r Fl(2)1176 2503 y Fp(\))d Fk(\012)h Fp(\000)1269 2509 y Fm(A)r Fl(1)953 2628 y Fp(32)p eop %%Page: 33 33 33 32 bop 257 262 a Fp(This)14 b(v)o(ersion)g(of)f(the)i(form)o(ula)c (w)o(as)j(pro)o(v)o(ed)g(earlier)g(b)o(y)g(D.)f(Radford)g(\(cf.)g([26)o (],)g(Thm.)f(3,)257 311 y(p.)f(595\),)e(while)h(the)i(form)o(ula)c(for) i(the)i(eigen)o(v)n(alue)e(of)g(the)h(squared)h(an)o(tip)q(o)q(de)e(on) h(in)o(tegrals)257 361 y(reduces)16 b(to)e(still)f(another)h(form)o (ula)e(of)h(D.)g(Radford)g(\(cf.)h([22)o(],)f(Cor.)g(5,)g(p.)g(345\).) 257 497 y Fn(4.3)48 b Fp(By)14 b(the)h(v)o(ery)f(de\014nition)f(of)h(a) f(F)m(rob)q(enius)h(algebra,)f(the)i(form)820 588 y(\()p Fo(a;)7 b(a)899 571 y Fj(0)910 588 y Fp(\))12 b Fk(7!)f Fo(\032)1012 594 y Fm(A)1039 588 y Fp(\()p Fo(aa)1099 571 y Fj(0)1111 588 y Fp(\))257 679 y(is)k(nondegenerate.)g(Since)g (for)g(a)f(\014xed)h Fo(a)d Fk(2)g Fo(A)j Fp(the)g(mapping)d Fo(a)1281 664 y Fj(0)1306 679 y Fk(7!)g Fo(\032)1381 685 y Fm(A)1408 679 y Fp(\()p Fo(aa)1468 664 y Fj(0)1480 679 y Fp(\))i(is)h(a)f(linear)257 729 y(form)e(on)i Fo(A)p Fp(,)g(there)h(m)o(ust)e(b)q(e)h(an)g(elemen)o(t)f Fo(\027)967 735 y Fm(A)994 729 y Fp(\()p Fo(a)p Fp(\))h(suc)o(h)g(that)g(w)o(e)h (ha)o(v)o(e:)770 820 y Fo(\032)791 826 y Fm(A)818 820 y Fp(\()p Fo(aa)878 803 y Fj(0)890 820 y Fp(\))c(=)h Fo(\032)982 826 y Fm(A)1010 820 y Fp(\()p Fo(a)1048 803 y Fj(0)1059 820 y Fo(\027)1080 826 y Fm(A)1107 820 y Fp(\()p Fo(a)p Fp(\)\))257 912 y(for)i(all)f Fo(a)401 897 y Fj(0)425 912 y Fk(2)f Fo(A)p Fp(.)i(This)h(determines)f(a)g (mapping)e Fo(\027)1055 918 y Fm(A)1096 912 y Fp(from)h Fo(A)h Fp(to)g Fo(A)h Fp(whic)o(h)f(is)g(easily)g(seen)257 961 y(to)g(b)q(e)f(an)h(algebra)e(automorphism.)e(It)k(is)f(called)g (the)h(Nak)n(a)o(y)o(ama)c(automorphism)g(of)j Fo(A)p Fp(.)257 1047 y(Ho)o(w)o(ev)o(er,)e(from)e(the)i(viewp)q(oin)o(t)f(of)g (braided)h(monoidal)d(categories,)j(the)g(ab)q(o)o(v)o(e)g(equation)257 1097 y(in)o(v)o(olv)o(es)i(a)g(t)o(wist)g(of)g Fo(a)h Fp(and)f Fo(a)734 1082 y Fj(0)759 1097 y Fp(whic)o(h)g(w)o(as)h(not)f (tak)o(en)h(to)f(b)q(e)h(the)g(t)o(wist)g(map)d(inside)j(the)257 1147 y(category)j(of)f(Y)m(etter-Drinfel'd)g(mo)q(dules.)f(It)i(m)o (ust)e(b)q(e)i(exp)q(ected)i(that)e(the)g(analogous)257 1196 y(mappings)i(in)o(v)o(olving)g(the)i(braiding)f(of)g(the)h (category)g(also)f(do)h(pla)o(y)f(an)g(imp)q(ortan)o(t)257 1246 y(role,)f(p)q(erhaps)h(ev)o(en)g(ha)o(v)o(e)f(nicer)h(prop)q (erties.)g(W)m(e)f(therefore)h(in)o(tro)q(duce)g(the)g(t)o(wisted)257 1296 y(Nak)n(a)o(y)o(ama)11 b(automorphisms,)g(whic)o(h)j(exist)g(b)o (y)g(a)f(similar)f(reasoning.)257 1396 y Fn(De\014nition)33 b Fp(Supp)q(ose)15 b(that)f Fo(\032)775 1402 y Fm(A)814 1396 y Fk(2)d Fo(A)884 1381 y Fj(\003)917 1396 y Fp(is)j(a)f(nonzero)i (righ)o(t)e(in)o(tegral.)308 1514 y(1.)20 b(W)m(e)13 b(de\014ne)i(the)f(p)q(ositiv)o(ely)f(t)o(wisted)h(Nak)n(a)o(y)o(ama)c (automorphism)h Fo(\027)1451 1520 y Fl(+)1489 1514 y Fp(:)g Fo(A)h Fk(!)f Fo(A)j Fp(to)361 1564 y(b)q(e)h(the)f(unique)g (map)e(satisfying)636 1656 y Fk(8)p Fo(a;)7 b(a)722 1638 y Fj(0)745 1656 y Fk(2)k Fo(A)h Fp(:)f Fo(\032)871 1662 y Fm(A)898 1656 y Fp(\()p Fo(a\027)957 1662 y Fl(+)985 1656 y Fp(\()p Fo(a)1023 1638 y Fj(0)1034 1656 y Fp(\)\))h(=)g Fo(\032)1143 1662 y Fm(A)1170 1656 y Fp(\(\()p Fo(a)1224 1638 y Fl(1)1255 1656 y Fk(!)f Fo(a)1330 1638 y Fj(0)1341 1656 y Fp(\))p Fo(a)1379 1638 y Fl(2)1398 1656 y Fp(\))308 1764 y(2.)20 b(W)m(e)12 b(de\014ne)h(the)g(negativ)o(ely)f(t)o(wisted)g (Nak)n(a)o(y)o(ama)d(automorphism)g Fo(\027)1451 1770 y Fj(\000)1491 1764 y Fp(:)i Fo(A)g Fk(!)g Fo(A)i Fp(to)361 1813 y(b)q(e)i(the)f(unique)g(map)e(satisfying)584 1905 y Fk(8)p Fo(a;)7 b(a)670 1888 y Fj(0)692 1905 y Fk(2)12 b Fo(A)f Fp(:)g Fo(\032)818 1911 y Fm(A)846 1905 y Fp(\()p Fo(a)884 1888 y Fj(0)896 1905 y Fo(\027)917 1911 y Fj(\000)944 1905 y Fp(\()p Fo(a)p Fp(\)\))h(=)g Fo(\032)1091 1911 y Fm(A)1118 1905 y Fp(\()p Fo(a)1156 1888 y Fl(2)1175 1905 y Fp(\()p Fo(S)1218 1887 y Fj(\000)p Fl(1)1216 1917 y Fm(H)1264 1905 y Fp(\()p Fo(a)1302 1888 y Fl(1)1320 1905 y Fp(\))g Fk(!)f Fo(a)1423 1888 y Fj(0)1435 1905 y Fp(\)\))257 2040 y Fn(4.4)48 b Fp(W)m(e)18 b(collect)g(some)f(of)h (the)g(elemen)o(tary)g(prop)q(erties)h(of)e(the)i(t)o(wisted)g(Nak)n(a) o(y)o(ama)257 2090 y(automorphisms.)9 b(Although)j(they)h(ha)o(v)o(e)f (-)g(in)g(comparison)f(to)h(the)h(ordinary)f(Nak)n(a)o(y)o(ama)257 2140 y(automorphism)g Fo(\027)554 2146 y Fm(A)594 2140 y Fp(-)i(the)h(adv)n(an)o(tage)f(of)f(b)q(eing)i Fo(H)s Fp(-linear)e(and)h(colinear,)g(they)h(are)g(not)257 2189 y(algebra)f(homomo)o(rphism)o(s)e(in)h(general,)h(but)g(satisfy)g(a)g (kind)g(of)f(doubly)g(t)o(wisted)i(m)o(ulti-)257 2239 y(plicativit)o(y)m(,)d(as)i(the)g(an)o(tip)q(o)q(de)g(or)g(the)g(ribb)q (on)g(transformation)e Fo(\022)q Fp(.)257 2330 y Fn(Prop)q(osition)308 2380 y Fp(1.)20 b(W)m(e)14 b(ha)o(v)o(e)f(for)h(all)f Fo(a;)7 b(a)713 2365 y Fj(0)735 2380 y Fk(2)k Fo(A)p Fp(:)471 2477 y Fo(\032)492 2483 y Fm(A)519 2477 y Fp(\()p Fo(a)557 2460 y Fj(0)r Fl(2)588 2477 y Fp(\()p Fo(\027)625 2483 y Fl(+)652 2477 y Fp(\()p Fo(S)695 2459 y Fj(\000)p Fl(1)693 2489 y Fm(H)740 2477 y Fp(\()p Fo(a)778 2460 y Fj(0)r Fl(1)809 2477 y Fp(\))g Fk(!)g Fo(a)p Fp(\)\)\))h(=)g Fo(\032)1036 2483 y Fm(A)1064 2477 y Fp(\()p Fo(aa)1124 2460 y Fj(0)1135 2477 y Fp(\))g(=)g Fo(\032)1228 2483 y Fm(A)1255 2477 y Fp(\(\()p Fo(a)1309 2460 y Fl(1)1340 2477 y Fk(!)f Fo(a)1415 2460 y Fj(0)1426 2477 y Fp(\))p Fo(\027)1463 2483 y Fj(\000)1491 2477 y Fp(\()p Fo(a)1529 2460 y Fl(2)1548 2477 y Fp(\)\))953 2628 y(33)p eop %%Page: 34 34 34 33 bop 308 262 a Fp(2.)20 b Fo(\027)382 268 y Fl(+)423 262 y Fp(and)14 b Fo(\027)525 268 y Fj(\000)566 262 y Fp(are)g Fo(H)s Fp(-linear)g(and)f(colinear.)308 343 y(3.)20 b(The)14 b(t)o(wisted)h(Nak)n(a)o(y)o(ama)c(automorphisms)g (satisfy:)730 430 y Fo(\027)751 436 y Fl(+)788 430 y Fk(\016)e Fo(\026)843 436 y Fm(A)881 430 y Fp(=)j Fo(\026)950 436 y Fm(A)986 430 y Fk(\016)d Fp(\()p Fo(\027)1053 436 y Fl(+)1090 430 y Fk(\012)g Fo(\027)1152 436 y Fl(+)1179 430 y Fp(\))h Fk(\016)f Fo(\033)1260 412 y Fj(\000)p Fl(2)1259 442 y Fm(A;A)730 500 y Fo(\027)751 506 y Fj(\000)787 500 y Fk(\016)g Fo(\026)842 506 y Fm(A)881 500 y Fp(=)j Fo(\026)950 506 y Fm(A)986 500 y Fk(\016)d Fp(\()p Fo(\027)1053 506 y Fj(\000)1090 500 y Fk(\012)g Fo(\027)1152 506 y Fj(\000)1180 500 y Fp(\))g Fk(\016)g Fo(\033)1260 483 y Fl(2)1259 511 y Fm(A;A)361 588 y Fp(where)19 b Fo(\033)h Fp(denotes)f(the)g(quasisymmetry)d(in)i(the)h(category)f(of)g(Y)m (etter-Drinfel'd)361 637 y(mo)q(dules)13 b(and)h Fo(\026)630 643 y Fm(A)671 637 y Fp(denotes)h(the)f(m)o(ultiplication)d(map)h(of)h Fo(A)p Fp(.)308 719 y(4.)20 b(W)m(e)14 b(ha)o(v)o(e)f(for)h(all)f Fo(a;)7 b(a)713 704 y Fj(0)735 719 y Fk(2)k Fo(A)p Fp(:)604 806 y Fo(\032)625 812 y Fm(A)653 806 y Fp(\()p Fo(\027)690 812 y Fl(+)717 806 y Fp(\()p Fo(a)p Fp(\))p Fo(\027)792 812 y Fj(\000)820 806 y Fp(\()p Fo(a)858 789 y Fj(0)870 806 y Fp(\)\))g(=)h Fo(\032)978 812 y Fm(A)1006 806 y Fp(\()p Fo(aa)1066 789 y Fj(0)1077 806 y Fp(\))g(=)g Fo(\032)1170 812 y Fm(A)1197 806 y Fp(\()p Fo(\027)1234 812 y Fj(\000)1262 806 y Fp(\()p Fo(a)p Fp(\))p Fo(\027)1337 812 y Fl(+)1364 806 y Fp(\()p Fo(a)1402 789 y Fj(0)1414 806 y Fp(\)\))257 903 y Fn(Pro)q(of.)36 b Fp(W)m(e)20 b(\014rst)g(pro)o(v)o(e)g(1.)f(The)h(\014rst)h(equalit)o(y)e(sa)o(ys,)g (written)i(in)e(terms)h(of)f(maps)257 953 y(without)14 b(using)g(elemen)o(ts,)f(that)h(w)o(e)g(ha)o(v)o(e:)627 1040 y Fo(\032)648 1046 y Fm(A)685 1040 y Fk(\016)9 b Fo(\026)740 1046 y Fm(A)776 1040 y Fk(\016)g Fp(\()p Fo(id)858 1046 y Fm(A)894 1040 y Fk(\012)h Fo(\027)957 1046 y Fl(+)984 1040 y Fp(\))f Fk(\016)g Fo(\033)1064 1022 y Fj(\000)p Fl(1)1063 1052 y Fm(A;A)1136 1040 y Fp(=)j Fo(\032)1201 1046 y Fm(A)1238 1040 y Fk(\016)d Fo(\026)1293 1046 y Fm(A)257 1127 y Fp(whereas)16 b(b)o(y)d (de\014nition)h(of)f Fo(\027)725 1133 y Fl(+)766 1127 y Fp(w)o(e)h(ha)o(v)o(e:)627 1214 y Fo(\032)648 1220 y Fm(A)685 1214 y Fk(\016)9 b Fo(\026)740 1220 y Fm(A)776 1214 y Fk(\016)g Fp(\()p Fo(id)858 1220 y Fm(A)894 1214 y Fk(\012)h Fo(\027)957 1220 y Fl(+)984 1214 y Fp(\))i(=)f Fo(\032)1076 1220 y Fm(A)1113 1214 y Fk(\016)e Fo(\026)1168 1220 y Fm(A)1204 1214 y Fk(\016)g Fo(\033)1258 1220 y Fm(A;A)257 1301 y Fp(The)17 b(second)g(equation)f(is)h(of)e(similar)f (di\016cult)o(y)m(.)g(W)m(e)i(shall)g(sho)o(w)g(no)o(w)g(that)g Fo(\027)1550 1307 y Fl(+)1594 1301 y Fp(is)g Fo(H)s Fp(-)257 1351 y(linear:)317 1438 y Fo(\032)338 1444 y Fm(A)366 1438 y Fp(\()p Fo(a)p Fp(\()p Fo(h)11 b Fk(!)g Fo(\027)529 1444 y Fl(+)556 1438 y Fp(\()p Fo(a)594 1421 y Fj(0)606 1438 y Fp(\)\)\))h(=)g Fo(\032)731 1444 y Fm(A)758 1438 y Fp(\(\()p Fo(h)814 1444 y Fl(2)833 1438 y Fo(S)860 1421 y Fj(\000)p Fl(1)858 1451 y Fm(H)906 1438 y Fp(\()p Fo(h)946 1444 y Fl(1)964 1438 y Fp(\))g Fk(!)f Fo(a)p Fp(\)\()p Fo(h)1123 1444 y Fl(3)1153 1438 y Fk(!)g Fo(\027)1227 1444 y Fl(+)1254 1438 y Fp(\()p Fo(a)1292 1421 y Fj(0)1304 1438 y Fp(\)\)\))666 1506 y(=)h(\()p Fo(\032)747 1512 y Fm(A)786 1506 y Fk( )f Fo(h)863 1512 y Fl(2)882 1506 y Fp(\)\(\()p Fo(S)957 1489 y Fj(\000)p Fl(1)955 1519 y Fm(H)1002 1506 y Fp(\()p Fo(h)1042 1512 y Fl(1)1061 1506 y Fp(\))h Fk(!)f Fo(a)p Fp(\))p Fo(\027)1201 1512 y Fl(+)1228 1506 y Fp(\()p Fo(a)1266 1489 y Fj(0)1278 1506 y Fp(\)\))666 1574 y(=)h Fo(\023)725 1580 y Fm(A)752 1574 y Fp(\()p Fo(h)792 1580 y Fl(2)810 1574 y Fp(\))p Fo(\032)847 1580 y Fm(A)875 1574 y Fp(\(\()p Fo(S)934 1557 y Fj(\000)p Fl(1)932 1587 y Fm(H)980 1574 y Fp(\()p Fo(h)1020 1580 y Fl(1)1038 1574 y Fp(\))g Fk(!)f Fo(a)p Fp(\))p Fo(\027)1178 1580 y Fl(+)1205 1574 y Fp(\()p Fo(a)1243 1557 y Fj(0)1255 1574 y Fp(\)\))666 1642 y(=)h(\()p Fo(\032)747 1648 y Fm(A)786 1642 y Fk( )f Fo(h)863 1648 y Fl(2)882 1642 y Fp(\)\(\(\()p Fo(S)973 1625 y Fj(\000)p Fl(1)971 1655 y Fm(H)1019 1642 y Fp(\()p Fo(h)1059 1648 y Fl(1)1077 1642 y Fp(\))h Fk(!)f Fo(a)p Fp(\))1196 1625 y Fl(1)1226 1642 y Fk(!)g Fo(a)1301 1625 y Fj(0)1313 1642 y Fp(\)\()p Fo(S)1372 1625 y Fj(\000)p Fl(1)1370 1655 y Fm(H)1418 1642 y Fp(\()p Fo(h)1458 1648 y Fl(1)1476 1642 y Fp(\))h Fk(!)f Fo(a)p Fp(\))1595 1625 y Fl(2)1614 1642 y Fp(\))666 1710 y(=)h(\()p Fo(\032)747 1716 y Fm(A)786 1710 y Fk( )f Fo(h)863 1716 y Fl(4)882 1710 y Fp(\)\(\()p Fo(S)957 1693 y Fj(\000)p Fl(1)955 1723 y Fm(H)1002 1710 y Fp(\()p Fo(h)1042 1716 y Fl(3)1061 1710 y Fp(\))p Fo(a)1099 1693 y Fl(1)1118 1710 y Fo(h)1142 1716 y Fl(1)1172 1710 y Fk(!)g Fo(a)1247 1693 y Fj(0)1259 1710 y Fp(\)\()p Fo(S)1318 1693 y Fj(\000)p Fl(1)1316 1723 y Fm(H)1363 1710 y Fp(\()p Fo(h)1403 1716 y Fl(2)1422 1710 y Fp(\))h Fk(!)f Fo(a)1525 1693 y Fl(2)1543 1710 y Fp(\)\))666 1778 y(=)h Fo(\032)731 1784 y Fm(A)758 1778 y Fp(\(\()p Fo(a)812 1761 y Fl(1)831 1778 y Fo(h)g Fk(!)f Fo(a)942 1761 y Fj(0)953 1778 y Fp(\))p Fo(a)991 1761 y Fl(2)1010 1778 y Fp(\))666 1840 y(=)h Fo(\032)731 1846 y Fm(A)758 1840 y Fp(\()p Fo(a\027)817 1846 y Fl(+)845 1840 y Fp(\()p Fo(h)f Fk(!)g Fo(a)971 1823 y Fj(0)983 1840 y Fp(\)\))257 1927 y(where)i(w)o(e)e(ha)o(v)o(e)g(used)h(a)f(v)n(arian)o(t)f(of)h (the)h(Y)m(etter-Drinfel'd)f(condition)f(in)h(the)h(\014fth)f(equal-) 257 1977 y(it)o(y)m(.)i(W)m(e)g(shall)g(sho)o(w)h(no)o(w)g(that)g Fo(\027)797 1983 y Fj(\000)838 1977 y Fp(is)g(colinear.)f(W)m(e)g(ha)o (v)o(e)h(b)o(y)g(the)g(\014rst)h(part:)317 2064 y Fo(a)339 2047 y Fl(1)358 2064 y Fo(a)380 2047 y Fj(0)q Fl(1)410 2064 y Fo(\032)431 2070 y Fm(A)458 2064 y Fp(\(\()p Fo(a)512 2047 y Fl(2)543 2064 y Fk(!)c Fo(a)618 2047 y Fj(0)q Fl(2)648 2064 y Fp(\))p Fo(\027)685 2070 y Fj(\000)713 2064 y Fp(\()p Fo(a)751 2047 y Fl(3)769 2064 y Fp(\)\))h(=)g Fo(a)879 2047 y Fl(1)897 2064 y Fo(a)919 2047 y Fj(0)r Fl(1)950 2064 y Fo(\032)971 2070 y Fm(A)998 2064 y Fp(\()p Fo(a)1036 2047 y Fl(2)1055 2064 y Fo(a)1077 2047 y Fj(0)q Fl(2)1107 2064 y Fp(\))813 2132 y(=)g Fo(g)877 2138 y Fm(A)904 2132 y Fo(\032)925 2138 y Fm(A)952 2132 y Fp(\()p Fo(aa)1012 2115 y Fj(0)1024 2132 y Fp(\))g(=)f Fo(g)1115 2138 y Fm(A)1142 2132 y Fo(\032)1163 2138 y Fm(A)1191 2132 y Fp(\(\()p Fo(a)1245 2115 y Fl(1)1275 2132 y Fk(!)g Fo(a)1350 2115 y Fj(0)1362 2132 y Fp(\))p Fo(\027)1399 2138 y Fj(\000)1426 2132 y Fp(\()p Fo(a)1464 2115 y Fl(2)1483 2132 y Fp(\)\))813 2199 y(=)h(\()p Fo(a)895 2182 y Fl(1)925 2199 y Fk(!)f Fo(a)1000 2182 y Fj(0)1012 2199 y Fp(\))1028 2182 y Fl(1)1047 2199 y Fo(\027)1068 2205 y Fj(\000)1095 2199 y Fp(\()p Fo(a)1133 2182 y Fl(2)1152 2199 y Fp(\))1168 2182 y Fl(1)1187 2199 y Fo(\032)1208 2205 y Fm(A)1235 2199 y Fp(\(\()p Fo(a)1289 2182 y Fl(1)1319 2199 y Fk(!)g Fo(a)1394 2182 y Fj(0)1406 2199 y Fp(\))1422 2182 y Fl(2)1441 2199 y Fo(\027)1462 2205 y Fj(\000)1489 2199 y Fp(\()p Fo(a)1527 2182 y Fl(2)1546 2199 y Fp(\))1562 2182 y Fl(2)1581 2199 y Fp(\))813 2267 y(=)h Fo(a)879 2249 y Fl(1)897 2267 y Fo(a)919 2249 y Fj(0)r Fl(1)950 2267 y Fo(S)975 2273 y Fm(H)1007 2267 y Fp(\()p Fo(a)1045 2249 y Fl(3)1063 2267 y Fp(\))p Fo(\027)1100 2273 y Fj(\000)1128 2267 y Fp(\()p Fo(a)1166 2249 y Fl(4)1185 2267 y Fp(\))1201 2249 y Fl(1)1219 2267 y Fo(\032)1240 2273 y Fm(A)1268 2267 y Fp(\(\()p Fo(a)1322 2249 y Fl(2)1352 2267 y Fk(!)f Fo(a)1427 2249 y Fj(0)r Fl(2)1458 2267 y Fp(\))p Fo(\027)1495 2273 y Fj(\000)1522 2267 y Fp(\()p Fo(a)1560 2249 y Fl(4)1579 2267 y Fp(\))1595 2249 y Fl(2)1614 2267 y Fp(\))257 2354 y(This)19 b(implies)d(1)523 2360 y Fm(H)554 2354 y Fo(\032)575 2360 y Fm(A)603 2354 y Fp(\(\()p Fo(a)657 2339 y Fl(1)695 2354 y Fk(!)i Fo(a)777 2339 y Fj(0)789 2354 y Fp(\))p Fo(\027)826 2360 y Fj(\000)854 2354 y Fp(\()p Fo(a)892 2339 y Fl(2)910 2354 y Fp(\)\))i(=)f Fo(S)1038 2360 y Fm(H)1070 2354 y Fp(\()p Fo(a)1108 2339 y Fl(2)1127 2354 y Fp(\))p Fo(\027)1164 2360 y Fj(\000)1191 2354 y Fp(\()p Fo(a)1229 2339 y Fl(3)1248 2354 y Fp(\))1264 2339 y Fl(1)1283 2354 y Fo(\032)1304 2360 y Fm(A)1331 2354 y Fp(\(\()p Fo(a)1385 2339 y Fl(1)1423 2354 y Fk(!)g Fo(a)1506 2339 y Fj(0)1517 2354 y Fp(\))p Fo(\027)1554 2360 y Fj(\000)1582 2354 y Fp(\()p Fo(a)1620 2339 y Fl(3)1639 2354 y Fp(\))1655 2339 y Fl(2)1673 2354 y Fp(\))257 2403 y(whic)o(h)12 b(in)f(turn)h(implies)e(1)670 2409 y Fm(H)701 2403 y Fo(\032)722 2409 y Fm(A)750 2403 y Fp(\()p Fo(a)788 2388 y Fj(0)799 2403 y Fo(\027)820 2409 y Fj(\000)848 2403 y Fp(\()p Fo(a)p Fp(\)\))i(=)g Fo(S)999 2409 y Fm(H)1031 2403 y Fp(\()p Fo(a)1069 2388 y Fl(1)1087 2403 y Fp(\))p Fo(\027)1124 2409 y Fj(\000)1152 2403 y Fp(\()p Fo(a)1190 2388 y Fl(2)1209 2403 y Fp(\))1225 2388 y Fl(1)1244 2403 y Fo(\032)1265 2409 y Fm(A)1292 2403 y Fp(\()p Fo(a)1330 2388 y Fj(0)1342 2403 y Fo(\027)1363 2409 y Fj(\000)1390 2403 y Fp(\()p Fo(a)1428 2388 y Fl(2)1447 2403 y Fp(\))1463 2388 y Fl(2)1482 2403 y Fp(\).)f(Hence)i(w)o(e)257 2453 y(\014nally)g(see:)629 2503 y Fo(a)651 2486 y Fl(1)670 2503 y Fo(\032)691 2509 y Fm(A)718 2503 y Fp(\()p Fo(a)756 2486 y Fj(0)768 2503 y Fo(\027)789 2509 y Fj(\000)816 2503 y Fp(\()p Fo(a)854 2486 y Fl(2)873 2503 y Fp(\)\))f(=)g Fo(\027)982 2509 y Fj(\000)1009 2503 y Fp(\()p Fo(a)p Fp(\))1063 2486 y Fl(1)1082 2503 y Fo(\032)1103 2509 y Fm(A)1131 2503 y Fp(\()p Fo(a)1169 2486 y Fj(0)1180 2503 y Fo(\027)1201 2509 y Fj(\000)1229 2503 y Fp(\()p Fo(a)p Fp(\))1283 2486 y Fl(2)1302 2503 y Fp(\))953 2628 y(34)p eop %%Page: 35 35 35 34 bop 257 262 a Fp(The)12 b(pro)q(of)f(that)g Fo(\027)554 268 y Fl(+)592 262 y Fp(is)g(colinear)g(and)g Fo(\027)882 268 y Fj(\000)920 262 y Fp(is)g(linear)g(rests)i(on)e(similar)d (calculations.)i(Next,)257 311 y(w)o(e)k(pro)o(v)o(e)g(3.)g(The)g (\014rst)h(equation)e(can)h(b)q(e)h(written)f(in)g(the)g(form:)616 403 y Fo(\027)637 409 y Fl(+)673 403 y Fk(\016)9 b Fo(\026)728 409 y Fm(A)765 403 y Fk(\016)g Fo(\033)819 409 y Fm(A;A)892 403 y Fp(=)j Fo(\026)961 409 y Fm(A)997 403 y Fk(\016)d Fp(\()p Fo(\027)1064 409 y Fl(+)1100 403 y Fk(\012)h Fo(\027)1163 409 y Fl(+)1190 403 y Fp(\))f Fk(\016)g Fo(\033)1270 385 y Fj(\000)p Fl(1)1269 415 y Fm(A;A)257 494 y Fp(Using)14 b(the)g(form)e(of)h(the)h(de\014nition)g(of)f Fo(\027)914 500 y Fl(+)954 494 y Fp(giv)o(en)g(during)h(the)g(pro)q(of) f(of)g(part)h(1,)f(w)o(e)h(no)o(w)257 544 y(carry)h(out)f(the)g(follo)o (wing)d(calculation:)282 635 y Fo(\032)303 641 y Fm(A)331 635 y Fk(\016)p Fo(\026)377 641 y Fm(A)413 635 y Fk(\016)e Fp(\()p Fo(id)495 641 y Fm(A)531 635 y Fk(\012)h Fo(\027)594 641 y Fl(+)621 635 y Fp(\))f Fk(\016)g Fp(\()p Fo(id)728 641 y Fm(A)764 635 y Fk(\012)h Fo(\026)831 641 y Fm(A)858 635 y Fp(\))f Fk(\016)g Fo(\033)938 617 y Fj(\000)p Fl(1)937 647 y Fm(A;A)p Fj(\012)p Fm(A)363 706 y Fp(=)j Fo(\032)428 712 y Fm(A)465 706 y Fk(\016)d Fo(\026)520 712 y Fm(A)556 706 y Fk(\016)g Fp(\()p Fo(id)638 712 y Fm(A)674 706 y Fk(\012)h Fo(\027)737 712 y Fl(+)764 706 y Fp(\))f Fk(\016)g Fo(\033)844 688 y Fj(\000)p Fl(1)843 718 y Fm(A;A)914 706 y Fk(\016)g Fp(\()p Fo(\026)985 712 y Fm(A)1021 706 y Fk(\012)h Fo(id)1099 712 y Fm(A)1126 706 y Fp(\))363 775 y(=)i Fo(\032)428 781 y Fm(A)465 775 y Fk(\016)d Fo(\026)520 781 y Fm(A)556 775 y Fk(\016)g Fp(\()p Fo(\026)627 781 y Fm(A)663 775 y Fk(\012)h Fo(id)741 781 y Fm(A)768 775 y Fp(\))363 838 y(=)i Fo(\032)428 844 y Fm(A)465 838 y Fk(\016)d Fo(\026)520 844 y Fm(A)556 838 y Fk(\016)g Fp(\()p Fo(id)638 844 y Fm(A)674 838 y Fk(\012)h Fo(\026)741 844 y Fm(A)768 838 y Fp(\))363 906 y(=)i Fo(\032)428 912 y Fm(A)465 906 y Fk(\016)d Fo(\026)520 912 y Fm(A)556 906 y Fk(\016)g Fp(\()p Fo(id)638 912 y Fm(A)674 906 y Fk(\012)h Fo(\027)737 912 y Fl(+)764 906 y Fp(\))f Fk(\016)g Fo(\033)844 888 y Fj(\000)p Fl(1)843 918 y Fm(A;A)914 906 y Fk(\016)g Fp(\()p Fo(id)996 912 y Fm(A)1032 906 y Fk(\012)h Fo(\026)1099 912 y Fm(A)1126 906 y Fp(\))363 977 y(=)i Fo(\032)428 983 y Fm(A)465 977 y Fk(\016)d Fo(\026)520 983 y Fm(A)556 977 y Fk(\016)g Fp(\()p Fo(id)638 983 y Fm(A)674 977 y Fk(\012)h Fo(\027)737 983 y Fl(+)764 977 y Fp(\))f Fk(\016)g Fp(\()p Fo(\026)860 983 y Fm(A)897 977 y Fk(\012)g Fo(id)974 983 y Fm(A)1001 977 y Fp(\))g Fk(\016)g Fo(\033)1081 959 y Fj(\000)p Fl(1)1080 989 y Fm(A)p Fj(\012)p Fm(A;A)363 1048 y Fp(=)j Fo(\032)428 1054 y Fm(A)465 1048 y Fk(\016)d Fo(\026)520 1054 y Fm(A)556 1048 y Fk(\016)g Fp(\()p Fo(id)638 1054 y Fm(A)674 1048 y Fk(\012)h Fo(\026)741 1054 y Fm(A)768 1048 y Fp(\))f Fk(\016)g Fp(\()p Fo(id)875 1054 y Fm(A)911 1048 y Fk(\012)h Fo(id)989 1054 y Fm(A)1025 1048 y Fk(\012)g Fo(\027)1088 1054 y Fl(+)1115 1048 y Fp(\))f Fk(\016)g Fo(\033)1195 1030 y Fj(\000)p Fl(1)1194 1060 y Fm(A)p Fj(\012)p Fm(A;A)363 1118 y Fp(=)j Fo(\032)428 1124 y Fm(A)465 1118 y Fk(\016)d Fo(\026)520 1124 y Fm(A)556 1118 y Fk(\016)g Fp(\()p Fo(id)638 1124 y Fm(A)674 1118 y Fk(\012)h Fo(\027)737 1124 y Fl(+)764 1118 y Fp(\))f Fk(\016)g Fo(\033)844 1101 y Fj(\000)p Fl(1)843 1131 y Fm(A;A)914 1118 y Fk(\016)g Fp(\()p Fo(id)996 1124 y Fm(A)1032 1118 y Fk(\012)h Fo(\026)1099 1124 y Fm(A)1126 1118 y Fp(\))f Fk(\016)g Fp(\()p Fo(id)1233 1124 y Fm(A)1269 1118 y Fk(\012)h Fo(id)1347 1124 y Fm(A)1383 1118 y Fk(\012)g Fo(\027)1446 1124 y Fl(+)1473 1118 y Fp(\))f Fk(\016)g Fo(\033)1553 1101 y Fj(\000)p Fl(1)1552 1131 y Fm(A)p Fj(\012)p Fm(A;A)363 1189 y Fp(=)j Fo(\032)428 1195 y Fm(A)465 1189 y Fk(\016)d Fo(\026)520 1195 y Fm(A)556 1189 y Fk(\016)g Fp(\()p Fo(\026)627 1195 y Fm(A)663 1189 y Fk(\012)h Fo(\027)726 1195 y Fl(+)753 1189 y Fp(\))f Fk(\016)g Fo(\033)833 1172 y Fj(\000)p Fl(1)832 1202 y Fm(A)p Fj(\012)p Fm(A;A)954 1189 y Fk(\016)g Fp(\()p Fo(id)1036 1195 y Fm(A)1072 1189 y Fk(\012)h Fo(id)1150 1195 y Fm(A)1186 1189 y Fk(\012)f Fo(\027)1248 1195 y Fl(+)1275 1189 y Fp(\))h Fk(\016)f Fo(\033)1356 1172 y Fj(\000)p Fl(1)1355 1202 y Fm(A)p Fj(\012)p Fm(A;A)363 1260 y Fp(=)j Fo(\032)428 1266 y Fm(A)465 1260 y Fk(\016)d Fo(\026)520 1266 y Fm(A)556 1260 y Fk(\016)g Fp(\()p Fo(\026)627 1266 y Fm(A)663 1260 y Fk(\012)h Fo(id)741 1266 y Fm(A)768 1260 y Fp(\))f Fk(\016)g Fp(\()p Fo(id)875 1266 y Fm(A)911 1260 y Fk(\012)h Fo(\027)974 1266 y Fl(+)1010 1260 y Fk(\012)g Fo(\027)1073 1266 y Fl(+)1100 1260 y Fp(\))f Fk(\016)g Fo(\033)1180 1243 y Fj(\000)p Fl(2)1179 1273 y Fm(A)p Fj(\012)p Fm(A;A)257 1352 y Fp(Using)14 b(the)h(Y)m(ang-Baxter)e(equation,)g(this)h(can)g(b)q(e)h(reduced)h (to:)257 1443 y Fo(\032)278 1449 y Fm(A)307 1443 y Fk(\016)q Fo(\026)354 1449 y Fm(A)382 1443 y Fk(\016)q Fp(\()p Fo(id)456 1449 y Fm(A)484 1443 y Fk(\012)q Fo(\027)538 1449 y Fl(+)565 1443 y Fp(\))q Fk(\016)q Fp(\()p Fo(id)656 1449 y Fm(A)684 1443 y Fk(\012)q Fo(\026)742 1449 y Fm(A)769 1443 y Fp(\))c(=)g Fo(\032)862 1449 y Fm(A)890 1443 y Fk(\016)q Fo(\026)937 1449 y Fm(A)965 1443 y Fk(\016)q Fp(\()p Fo(id)1039 1449 y Fm(A)1067 1443 y Fk(\012)q Fo(\026)1125 1449 y Fm(A)1153 1443 y Fp(\))q Fk(\016)q Fp(\()p Fo(id)1244 1449 y Fm(A)1272 1443 y Fk(\012)q Fo(\027)1326 1449 y Fl(+)1354 1443 y Fk(\012)q Fo(\027)1408 1449 y Fl(+)1435 1443 y Fp(\))q Fk(\016)q Fp(\()p Fo(id)1526 1449 y Fm(A)1554 1443 y Fk(\012)q Fo(\033)1612 1425 y Fj(\000)p Fl(2)1611 1455 y Fm(A;A)1673 1443 y Fp(\))257 1534 y(whic)o(h)g(yields)g(the)h(\014rst)g(equation)e(in)h(3.)f(b)o(y)h (nondegeneracy)m(.)h(The)f(second)i(equation)d(can)257 1584 y(b)q(e)k(sho)o(wn)f(similarly)l(.)d(The)j(last)g(part)g(follo)o (ws)e(b)o(y)i(inserting)g(the)g(de\014nitions.)g Fg(\003)257 1720 y Fn(4.5)48 b Fp(W)m(e)19 b(no)o(w)g(pro)q(ceed)i(to)f(deriv)o(e)g (explicit)f(form)o(ulas)e(for)i(the)i(t)o(wisted)f(Nak)n(a)o(y)o(ama) 257 1769 y(automorphisms.)c(W)m(e)i(shall)g(need)i(the)f(follo)o(wing)d (Lemma,)g(whic)o(h)j(tells)f(us)i(ho)o(w)e(the)257 1819 y(mappings)12 b(\()p Fo(\032)481 1825 y Fm(A)518 1819 y Fk(\016)d Fo(\026)573 1825 y Fm(A)600 1819 y Fp(\))h Fk(\012)f Fo(id)703 1825 y Fm(A)744 1819 y Fp(and)14 b Fo(id)861 1825 y Fm(A)897 1819 y Fk(\012)9 b Fp(\()p Fo(\032)975 1825 y Fm(A)1012 1819 y Fk(\016)g Fo(\026)1067 1825 y Fm(A)1094 1819 y Fp(\))14 b(are)h(in)o(terrelated:)257 1919 y Fn(Lemma)36 b Fp(W)m(e)14 b(ha)o(v)o(e)g(for)f(all)g Fo(a;)7 b(a)799 1904 y Fj(0)821 1919 y Fk(2)k Fo(A)p Fp(:)308 2038 y(1.)20 b Fo(\032)382 2044 y Fm(A)410 2038 y Fp(\()p Fo(a)448 2022 y Fl(2)466 2038 y Fo(a)488 2022 y Fj(0)488 2048 y Fl(2)507 2038 y Fp(\))p Fo(a)545 2022 y Fm(R)545 2049 y(A)572 2038 y Fo(S)599 2020 y Fj(\000)p Fl(1)597 2050 y Fm(A)645 2038 y Fp(\()p Fo(a)683 2022 y Fl(1)713 2038 y Fk(!)11 b Fo(a)788 2022 y Fj(0)788 2048 y Fl(1)806 2038 y Fp(\))h(=)g Fo(a)900 2044 y Fl(1)919 2038 y Fo(\032)940 2044 y Fm(A)967 2038 y Fp(\()p Fo(a)1005 2044 y Fl(2)1024 2038 y Fo(a)1046 2022 y Fj(0)1057 2038 y Fp(\))308 2121 y(2.)20 b Fo(\032)382 2127 y Fm(A)410 2121 y Fp(\()p Fo(aa)470 2105 y Fj(0)470 2131 y Fl(2)488 2105 y(2)507 2121 y Fp(\))p Fo(a)545 2105 y Fm(R)545 2132 y(A)572 2121 y Fo(S)599 2103 y Fj(\000)p Fl(1)597 2133 y Fm(A)645 2121 y Fp(\()p Fo(S)688 2103 y Fj(\000)p Fl(1)686 2133 y Fm(H)733 2121 y Fp(\()p Fo(a)771 2105 y Fj(0)771 2131 y Fl(2)790 2105 y(1)809 2121 y Fp(\))11 b Fk(!)g Fo(a)911 2105 y Fj(0)911 2131 y Fl(1)930 2121 y Fp(\))g(=)h Fo(g)1022 2103 y Fj(\000)p Fl(1)1021 2133 y Fm(A)1079 2121 y Fk(!)f Fo(a)1154 2127 y Fl(1)1172 2121 y Fo(\032)1193 2127 y Fm(A)1221 2121 y Fp(\()p Fo(a)1259 2127 y Fl(2)1278 2121 y Fo(a)1300 2105 y Fj(0)1311 2121 y Fp(\))953 2628 y(35)p eop %%Page: 36 36 36 35 bop 257 262 a Fn(Pro)q(of.)36 b Fp(The)15 b(\014rst)f(statemen)o (t)g(follo)o(ws)e(from)g(the)j(follo)o(wing)c(calculation:)280 351 y Fo(\032)301 357 y Fm(A)328 351 y Fp(\()p Fo(a)366 334 y Fl(2)385 351 y Fo(a)407 334 y Fj(0)407 361 y Fl(2)425 351 y Fp(\))p Fo(a)463 334 y Fm(R)463 361 y(A)491 351 y Fo(S)518 333 y Fj(\000)p Fl(1)516 363 y Fm(A)563 351 y Fp(\()p Fo(a)601 334 y Fl(1)631 351 y Fk(!)h Fo(a)707 334 y Fj(0)707 361 y Fl(1)725 351 y Fp(\))396 420 y(=)g Fo(\032)461 426 y Fm(A)489 420 y Fp(\()p Fo(a)527 402 y Fl(2)545 420 y Fo(a)567 402 y Fj(0)567 430 y Fl(2)586 420 y Fp(\))p Fo(a)624 402 y Fm(R)624 430 y(A)651 402 y Fl(2)670 420 y Fo(S)697 402 y Fj(\000)p Fl(1)695 432 y Fm(A)742 420 y Fp(\()p Fo(S)785 402 y Fj(\000)p Fl(1)783 432 y Fm(H)831 420 y Fp(\()p Fo(a)869 402 y Fm(R)869 430 y(A)896 402 y Fl(1)915 420 y Fp(\))p Fo(a)953 402 y Fl(1)983 420 y Fk(!)f Fo(a)1058 402 y Fj(0)1058 430 y Fl(1)1077 420 y Fp(\))396 488 y(=)h Fo(\032)461 494 y Fm(A)489 488 y Fp(\()p Fo(a)527 470 y Fl(2)545 494 y(2)564 470 y(2)583 488 y Fo(a)605 470 y Fj(0)605 498 y Fl(3)623 488 y Fp(\)\()p Fo(a)677 470 y Fl(2)696 494 y(1)715 488 y Fp(\()p Fo(a)753 470 y Fl(2)771 494 y(2)790 470 y(1)820 488 y Fk(!)f Fo(a)895 470 y Fj(0)895 498 y Fl(2)914 488 y Fp(\)\))946 470 y Fl(2)965 488 y Fo(S)992 470 y Fj(\000)p Fl(1)990 500 y Fm(A)1037 488 y Fp(\()p Fo(S)1080 470 y Fj(\000)p Fl(1)1078 500 y Fm(H)1126 488 y Fp(\(\()p Fo(a)1180 470 y Fl(2)1199 494 y(1)1217 488 y Fp(\()p Fo(a)1255 470 y Fl(2)1274 494 y(2)1293 470 y(1)1323 488 y Fk(!)g Fo(a)1398 470 y Fj(0)1398 498 y Fl(2)1416 488 y Fp(\)\))1448 470 y Fl(1)1467 488 y Fp(\))p Fo(a)1505 470 y Fl(1)1535 488 y Fk(!)g Fo(a)1610 470 y Fj(0)1610 498 y Fl(1)1629 488 y Fp(\))396 556 y(=)h Fo(\032)461 562 y Fm(A)489 556 y Fp(\()p Fo(a)527 562 y Fl(2)545 538 y(3)564 556 y Fo(a)586 538 y Fj(0)586 566 y Fl(3)605 556 y Fp(\)\()p Fo(a)659 562 y Fl(1)678 538 y(2)696 556 y Fp(\()p Fo(a)734 562 y Fl(2)753 538 y(2)783 556 y Fk(!)f Fo(a)858 538 y Fj(0)858 566 y Fl(2)877 556 y Fp(\)\))909 538 y Fl(2)928 556 y Fo(S)955 538 y Fj(\000)p Fl(1)953 568 y Fm(A)1000 556 y Fp(\()p Fo(S)1043 538 y Fj(\000)p Fl(1)1041 568 y Fm(H)1088 556 y Fp(\(\()p Fo(a)1142 562 y Fl(1)1161 538 y(2)1180 556 y Fp(\()p Fo(a)1218 562 y Fl(2)1237 538 y(2)1267 556 y Fk(!)g Fo(a)1342 538 y Fj(0)1342 566 y Fl(2)1360 556 y Fp(\)\))1392 538 y Fl(1)1411 556 y Fp(\))p Fo(a)1449 562 y Fl(1)1468 538 y(1)1487 556 y Fo(a)1509 562 y Fl(2)1527 538 y(1)1557 556 y Fk(!)g Fo(a)1632 538 y Fj(0)1632 566 y Fl(1)1651 556 y Fp(\))396 624 y(=)h Fo(\032)461 630 y Fm(A)489 624 y Fp(\()p Fo(a)527 630 y Fl(2)545 607 y(5)564 624 y Fo(a)586 607 y Fj(0)586 634 y Fl(3)605 624 y Fp(\))p Fo(a)643 630 y Fl(1)661 607 y(3)680 624 y Fp(\()p Fo(a)718 630 y Fl(2)737 607 y(3)767 624 y Fk(!)f Fo(a)842 607 y Fj(0)842 634 y Fl(2)860 607 y(2)879 624 y Fp(\))p Fo(S)922 606 y Fj(\000)p Fl(1)920 636 y Fm(A)968 624 y Fp(\()p Fo(S)1011 606 y Fj(\000)p Fl(1)1009 636 y Fm(H)1056 624 y Fp(\()p Fo(a)1094 630 y Fl(1)1113 607 y(2)1132 624 y Fo(a)1154 630 y Fl(2)1172 607 y(2)1191 624 y Fo(a)1213 607 y Fj(0)1213 634 y Fl(2)1231 607 y(1)1250 624 y Fo(S)1275 630 y Fm(H)1307 624 y Fp(\()p Fo(a)1345 630 y Fl(2)1364 607 y(4)1382 624 y Fp(\)\))p Fo(a)1436 630 y Fl(1)1455 607 y(1)1474 624 y Fo(a)1496 630 y Fl(2)1514 607 y(1)1544 624 y Fk(!)g Fo(a)1619 607 y Fj(0)1619 634 y Fl(1)1638 624 y Fp(\))396 692 y(=)h Fo(\032)461 698 y Fm(A)489 692 y Fp(\()p Fo(a)527 698 y Fl(2)545 675 y(3)564 692 y Fo(a)586 675 y Fj(0)586 702 y Fl(3)605 692 y Fp(\))p Fo(a)643 698 y Fl(1)661 692 y Fp(\()p Fo(a)699 698 y Fl(2)718 675 y(1)748 692 y Fk(!)f Fo(a)823 675 y Fj(0)823 702 y Fl(2)842 675 y(2)860 692 y Fp(\))p Fo(S)903 674 y Fj(\000)p Fl(1)901 704 y Fm(A)949 692 y Fp(\()p Fo(a)987 698 y Fl(2)1006 675 y(2)1024 692 y Fo(S)1051 674 y Fj(\000)p Fl(1)1049 704 y Fm(H)1097 692 y Fp(\()p Fo(a)1135 675 y Fj(0)1135 702 y Fl(2)1153 675 y(1)1172 692 y Fp(\))h Fk(!)f Fo(a)1275 675 y Fj(0)1275 702 y Fl(1)1293 692 y Fp(\))396 760 y(=)h Fo(\032)461 766 y Fm(A)489 760 y Fp(\()p Fo(a)527 766 y Fl(2)545 743 y(2)564 760 y Fo(a)586 743 y Fj(0)586 770 y Fl(3)605 760 y Fp(\))p Fo(a)643 766 y Fl(1)661 760 y Fp(\()p Fo(a)699 766 y Fl(2)718 743 y(1)748 760 y Fk(!)f Fp([)p Fo(a)835 743 y Fj(0)835 770 y Fl(2)853 743 y(2)872 760 y Fo(S)899 742 y Fj(\000)p Fl(1)897 772 y Fm(A)944 760 y Fp(\()p Fo(S)987 742 y Fj(\000)p Fl(1)985 772 y Fm(H)1033 760 y Fp(\()p Fo(a)1071 743 y Fj(0)1071 770 y Fl(2)1090 743 y(1)1108 760 y Fp(\))h Fk(!)f Fo(a)1211 743 y Fj(0)1211 770 y Fl(1)1230 760 y Fp(\)]\))396 827 y(=)h Fo(\032)461 833 y Fm(A)489 827 y Fp(\()p Fo(a)527 833 y Fl(2)545 810 y(2)564 827 y Fo(a)586 810 y Fj(0)586 837 y Fl(2)605 827 y Fp(\))p Fo(a)643 833 y Fl(1)661 827 y Fp(\()p Fo(a)699 833 y Fl(2)718 810 y(1)748 827 y Fk(!)f Fo(\017)818 833 y Fm(A)845 827 y Fp(\()p Fo(a)883 810 y Fj(0)883 837 y Fl(1)902 827 y Fp(\)1\))396 889 y(=)h Fo(a)462 895 y Fl(1)481 889 y Fo(\032)502 895 y Fm(A)529 889 y Fp(\()p Fo(a)567 895 y Fl(2)586 889 y Fo(a)608 872 y Fj(0)620 889 y Fp(\))257 979 y(T)m(o)h(pro)o(v)o(e)h(the)g(second)h(statemen)o(t,)e(observ)o(e)i (that)f(w)o(e)g(ha)o(v)o(e)g(b)o(y)f(Prop)q(osition)g(2.10)g(that:)351 1068 y Fo(a)373 1051 y Fl(2)391 1068 y Fo(a)413 1051 y Fj(0)413 1079 y Fl(2)432 1051 y(1)460 1068 y Fk(\012)c Fo(\032)522 1074 y Fm(A)550 1068 y Fp(\()p Fo(a)588 1051 y Fl(3)607 1068 y Fo(a)629 1051 y Fj(0)629 1079 y Fl(2)647 1051 y(2)666 1068 y Fp(\))p Fo(a)704 1051 y Fm(R)704 1079 y(A)731 1068 y Fo(S)758 1051 y Fj(\000)p Fl(1)756 1081 y Fm(A)804 1068 y Fp(\()p Fo(a)842 1051 y Fl(1)872 1068 y Fk(!)i Fo(a)947 1051 y Fj(0)947 1079 y Fl(1)965 1068 y Fp(\))h(=)g Fo(g)1057 1074 y Fm(A)1093 1068 y Fk(\012)d Fo(\032)1155 1074 y Fm(A)1183 1068 y Fp(\()p Fo(a)1221 1051 y Fl(2)1240 1068 y Fo(a)1262 1051 y Fj(0)1262 1079 y Fl(2)1280 1068 y Fp(\))p Fo(a)1318 1051 y Fm(R)1318 1079 y(A)1346 1068 y Fo(S)1373 1051 y Fj(\000)p Fl(1)1371 1081 y Fm(A)1418 1068 y Fp(\()p Fo(a)1456 1051 y Fl(1)1486 1068 y Fk(!)i Fo(a)1561 1051 y Fj(0)1561 1079 y Fl(1)1580 1068 y Fp(\))993 1131 y(=)h Fo(g)1057 1137 y Fm(A)1093 1131 y Fk(\012)d Fo(a)1156 1137 y Fl(1)1175 1131 y Fo(\032)1196 1137 y Fm(A)1224 1131 y Fp(\()p Fo(a)1262 1137 y Fl(2)1280 1131 y Fo(a)1302 1113 y Fj(0)1314 1131 y Fp(\))257 1220 y(and)14 b(therefore)h(w)o(e)g(conclude)f(from)e(Prop)q(osition)i(2.12) e(that:)268 1310 y Fo(g)289 1292 y Fj(\000)p Fl(1)288 1322 y Fm(A)345 1310 y Fk(!)f Fo(a)420 1316 y Fl(1)439 1310 y Fo(\032)460 1316 y Fm(A)487 1310 y Fp(\()p Fo(a)525 1316 y Fl(2)544 1310 y Fo(a)566 1292 y Fj(0)577 1310 y Fp(\))h(=)g Fo(S)676 1292 y Fj(\000)p Fl(1)674 1322 y Fm(H)721 1310 y Fp(\()p Fo(a)759 1292 y Fl(2)778 1310 y Fo(a)800 1292 y Fj(0)800 1320 y Fl(2)819 1292 y(1)837 1310 y Fp(\))g Fk(!)f Fp(\()p Fo(\032)955 1316 y Fm(A)982 1310 y Fp(\()p Fo(a)1020 1292 y Fl(3)1039 1310 y Fo(a)1061 1292 y Fj(0)1061 1320 y Fl(2)1080 1292 y(2)1098 1310 y Fp(\))p Fo(a)1136 1292 y Fm(R)1136 1320 y(A)1164 1310 y Fo(S)1191 1292 y Fj(\000)p Fl(1)1189 1322 y Fm(A)1236 1310 y Fp(\()p Fo(a)1274 1292 y Fl(1)1304 1310 y Fk(!)g Fo(a)1379 1292 y Fj(0)1379 1320 y Fl(1)1398 1310 y Fp(\)\))605 1378 y(=)h Fo(\032)670 1384 y Fm(A)697 1378 y Fp(\()p Fo(aa)757 1361 y Fj(0)757 1388 y Fl(2)776 1361 y(2)795 1378 y Fp(\))p Fo(a)833 1361 y Fm(R)833 1388 y(A)860 1378 y Fo(S)887 1360 y Fj(\000)p Fl(1)885 1390 y Fm(A)932 1378 y Fp(\()p Fo(S)975 1360 y Fj(\000)p Fl(1)973 1390 y Fm(H)1021 1378 y Fp(\()p Fo(a)1059 1361 y Fj(0)1059 1388 y Fl(2)1078 1361 y(1)1096 1378 y Fp(\))g Fk(!)f Fo(a)1199 1361 y Fj(0)1199 1388 y Fl(1)1218 1378 y Fp(\))413 b Fg(\003)257 1503 y Fp(The)15 b(second)g(form)o(ula)d(in)i(the)g(ab)q (o)o(v)o(e)g(Lemma)e(can)i(b)q(e)h(b)q(etter)h(understo)q(o)q(d)f(in)f (terms)g(of)257 1552 y(the)h(categorically)e(co)q(opp)q(osite)h(Hopf)f (algebra)g Fo(A)1052 1537 y Fm(copp)1133 1552 y Fp(\(cf.)h([32)o(],)f (subsection)i(4.3\).)d Fo(A)1622 1537 y Fm(copp)257 1602 y Fp(has)h(the)g(same)e(m)o(ultiplication,)e(unit)j(and)g(counit)g(as)h Fo(A)p Fp(,)f(but)h(the)g(com)o(ultiplicati)o(on)d(and)257 1652 y(the)15 b(an)o(tip)q(o)q(de)f(of)f Fo(A)579 1637 y Fm(copp)660 1652 y Fp(are)i(giv)o(en)e(b)o(y:)648 1742 y(\001)683 1722 y Fm(copp)683 1754 y(A)762 1742 y Fp(=)f Fo(\033)831 1724 y Fj(\000)p Fl(1)830 1754 y Fm(A;A)901 1742 y Fk(\016)d Fp(\001)966 1748 y Fm(A)1075 1742 y Fo(S)1102 1722 y Fm(copp)1100 1754 y(A)1182 1742 y Fp(=)j Fo(S)1253 1724 y Fj(\000)p Fl(1)1251 1754 y Fm(A)257 1831 y Fo(A)288 1816 y Fm(copp)372 1831 y Fp(is)j(not)g(a)h(Y)m (etter-Drinfel'd)f(Hopf)g(algebra,)g(but)h(rather)g(a)f(Hopf)h(algebra) f(in)g(the)257 1881 y(category)d(where)g(the)g(usual)f(quasisymmetry)e (is)i(replaced)i(b)o(y)e(its)g(in)o(v)o(erse)h(\(cf.)f([9)o(],)f(sec.)i (2,)257 1931 y(p.)k(33\).)f(If)h(w)o(e)g(temp)q(orarily)e(use)j(the)g (Sw)o(eedler)g(notation)e(\001)1253 1911 y Fm(copp)1253 1943 y(A)1320 1931 y Fp(\()p Fo(a)p Fp(\))h(=)f Fo(a)1459 1938 y Fl(\(1\))1514 1931 y Fk(\012)c Fo(a)1579 1938 y Fl(\(2\))1640 1931 y Fp(for)257 1980 y(the)k(co)q(opp)q(osite)f(com)o (ultiplication,)c(then)15 b(the)f(second)h(form)o(ula)d(tak)o(es)i(the) h(form:)588 2070 y Fo(\032)609 2076 y Fm(A)636 2070 y Fp(\()p Fo(aa)696 2053 y Fj(0)696 2081 y Fl(\(1\))741 2070 y Fp(\))p Fo(a)779 2053 y Fm(R)779 2080 y(A)806 2070 y Fo(S)833 2052 y Fj(\000)p Fl(1)831 2082 y Fm(A)879 2070 y Fp(\()p Fo(a)917 2053 y Fj(0)917 2081 y Fl(\(2\))961 2070 y Fp(\))d(=)g Fo(g)1054 2052 y Fj(\000)p Fl(1)1053 2082 y Fm(A)1110 2070 y Fk(!)f Fo(a)1185 2076 y Fl(1)1204 2070 y Fo(\032)1225 2076 y Fm(A)1252 2070 y Fp(\()p Fo(a)1290 2076 y Fl(2)1309 2070 y Fo(a)1331 2053 y Fj(0)1343 2070 y Fp(\))257 2194 y(W)m(e)j(no)o(w)f(giv)o(e)h(the)g(explicit)g(form)e (of)h(the)h(t)o(wisted)h(Nak)n(a)o(y)o(ama)c(automorphisms:)257 2293 y Fn(Theorem)36 b Fp(W)m(e)13 b(ha)o(v)o(e)h(for)g Fo(a)d Fk(2)g Fo(A)p Fp(:)308 2409 y(1.)20 b Fo(\027)382 2415 y Fl(+)409 2409 y Fp(\()p Fo(I)446 2392 y Fj(\000)p Fl(1)443 2422 y Fm(A)491 2409 y Fp(\()p Fo(a)p Fp(\)\))12 b(=)g Fo(\013)644 2394 y Fm(R)644 2421 y(A)671 2409 y Fp(\()p Fo(a)709 2415 y Fl(1)728 2409 y Fp(\))p Fo(S)771 2392 y Fj(\000)p Fl(2)769 2422 y Fm(A)816 2409 y Fp(\()p Fo(a)854 2415 y Fl(2)873 2409 y Fp(\))308 2492 y(2.)20 b Fo(\027)382 2498 y Fj(\000)410 2492 y Fp(\()p Fo(I)444 2498 y Fm(A)471 2492 y Fp(\()p Fo(a)p Fp(\)\))12 b(=)g Fo(a)619 2477 y Fm(L)619 2503 y(A)646 2492 y Fo(S)673 2477 y Fl(2)671 2503 y Fm(A)698 2492 y Fp(\()p Fo(a)736 2498 y Fl(1)755 2492 y Fp(\))p Fo(a)793 2477 y Fm(R)793 2503 y(A)820 2492 y Fo(\013)847 2477 y Fm(R)847 2503 y(A)874 2492 y Fp(\()p Fo(a)912 2498 y Fl(2)931 2492 y Fp(\))953 2628 y(36)p eop %%Page: 37 37 37 36 bop 257 262 a Fn(Pro)q(of.)36 b Fp(Using)19 b(the)h(forms)e(of)h (the)h(Casimir)d(elemen)o(t)i(in)g(Prop)q(osition)f(4.2)h(and)g(the)257 311 y Fo(H)s Fp(-linearit)o(y)13 b(of)g Fo(\027)541 317 y Fl(+)568 311 y Fp(,)h(w)o(e)g(calculate:)430 398 y Fo(\027)451 404 y Fl(+)478 398 y Fp(\()p Fo(a)p Fp(\))e(=)g Fo(S)615 380 y Fj(\000)p Fl(1)613 410 y Fm(A)660 398 y Fp(\(\000)702 404 y Fm(A)s Fl(2)748 381 y(3)767 398 y Fp(\))p Fo(\023)798 404 y Fm(A)825 398 y Fp(\(\000)867 404 y Fm(A)r Fl(2)912 381 y(2)931 398 y Fp(\))p Fo(\032)968 404 y Fm(A)996 398 y Fp(\(\()p Fo(S)1055 380 y Fj(\000)p Fl(1)1053 410 y Fm(H)1100 398 y Fp(\(\000)1142 404 y Fm(A)s Fl(2)1188 381 y(1)1207 398 y Fp(\))f Fk(!)g Fp(\000)1313 404 y Fm(A)r Fl(1)1359 398 y Fp(\))p Fo(\027)1396 404 y Fl(+)1423 398 y Fp(\()p Fo(a)p Fp(\)\))544 466 y(=)h Fo(S)615 448 y Fj(\000)p Fl(1)613 478 y Fm(A)660 466 y Fp(\(\000)702 472 y Fm(A)s Fl(2)748 449 y(3)767 466 y Fp(\)\()p Fo(\032)820 472 y Fm(A)859 466 y Fk( )f Fp(\000)938 472 y Fm(A)r Fl(2)984 449 y(2)1002 466 y Fp(\)\(\()p Fo(S)1077 448 y Fj(\000)p Fl(1)1075 478 y Fm(H)1123 466 y Fp(\(\000)1165 472 y Fm(A)r Fl(2)1211 449 y(1)1230 466 y Fp(\))g Fk(!)g Fp(\000)1336 472 y Fm(A)r Fl(1)1382 466 y Fp(\))p Fo(\027)1419 472 y Fl(+)1446 466 y Fp(\()p Fo(a)p Fp(\)\))544 534 y(=)h Fo(S)615 516 y Fj(\000)p Fl(1)613 546 y Fm(A)660 534 y Fp(\(\000)702 540 y Fm(A)s Fl(2)748 517 y(2)767 534 y Fp(\))p Fo(\032)804 540 y Fm(A)831 534 y Fp(\(\000)873 540 y Fm(A)s Fl(1)919 534 y Fo(\027)940 540 y Fl(+)967 534 y Fp(\(\000)1009 540 y Fm(A)r Fl(2)1055 517 y(1)1085 534 y Fk(!)f Fo(a)p Fp(\)\))544 602 y(=)h Fo(S)615 584 y Fj(\000)p Fl(1)613 614 y Fm(A)660 602 y Fp(\(\000)702 608 y Fm(A)s Fl(2)748 585 y(2)767 602 y Fp(\))p Fo(\032)804 608 y Fm(A)831 602 y Fp(\(\(\000)889 608 y Fm(A)s Fl(1)935 585 y(1)954 602 y Fp(\000)980 608 y Fm(A)r Fl(2)1026 585 y(1)1056 602 y Fk(!)f Fo(a)p Fp(\)\000)1173 608 y Fm(A)r Fl(1)1218 585 y(2)1237 602 y Fp(\))257 688 y(W)m(e)j(conclude:)258 775 y Fo(S)285 757 y Fl(2)283 785 y Fm(A)311 775 y Fp(\()p Fo(\027)348 781 y Fl(+)375 775 y Fp(\()p Fo(a)p Fp(\)\))e(=)g Fo(\032)522 781 y Fm(A)549 775 y Fp(\(\(\000)607 781 y Fm(A)634 757 y Fl(1)664 775 y Fk(!)f Fo(a)p Fp(\)\000)781 781 y Fm(A)809 757 y Fl(2)827 781 y(1)846 775 y Fp(\))p Fo(S)887 781 y Fm(A)914 775 y Fp(\(\000)956 781 y Fm(A)984 757 y Fl(2)1002 781 y(2)1021 775 y Fp(\))457 842 y(=)h Fo(\032)522 848 y Fm(A)549 842 y Fp(\(\(\000)607 848 y Fm(A)634 825 y Fl(1)664 842 y Fk(!)f Fo(a)739 848 y Fl(1)758 842 y Fp(\)[\(\000)828 848 y Fm(A)855 825 y Fl(2)885 842 y Fk(!)g Fo(a)960 848 y Fl(2)979 842 y Fp(\))995 825 y Fl(1)1025 842 y Fk(!)g Fp(\000)1104 848 y Fm(A)1131 825 y Fl(3)1150 848 y(1)1168 842 y Fp(]\)\(\000)1238 848 y Fm(A)1265 825 y Fl(2)1295 842 y Fk(!)g Fo(a)1370 848 y Fl(2)1389 842 y Fp(\))1405 825 y Fl(2)1424 842 y Fp(\000)1450 848 y Fm(A)1477 825 y Fl(3)1495 848 y(2)1514 842 y Fo(S)1539 848 y Fm(A)1566 842 y Fp(\(\000)1608 848 y Fm(A)1635 825 y Fl(3)1654 848 y(3)1673 842 y Fp(\))457 909 y(=)h Fo(\032)522 915 y Fm(A)549 909 y Fp(\(\(\000)607 915 y Fm(A)634 892 y Fl(1)664 909 y Fk(!)f Fo(a)739 915 y Fl(1)758 909 y Fp(\)[\(\000)828 915 y Fm(A)855 892 y Fl(2)885 909 y Fk(!)g Fo(a)960 915 y Fl(2)979 909 y Fp(\))995 892 y Fl(1)1025 909 y Fk(!)g Fp(\000)1104 915 y Fm(A)1131 892 y Fl(3)1150 909 y Fp(]\)\(\000)1220 915 y Fm(A)1246 892 y Fl(2)1277 909 y Fk(!)g Fo(a)1352 915 y Fl(2)1370 909 y Fp(\))1386 892 y Fl(2)257 996 y Fp(Using)j(Prop)q(osition)g(2.10,)e(this)i(reduces)i(to:)352 1082 y Fo(S)379 1065 y Fl(2)377 1092 y Fm(A)404 1082 y Fp(\()p Fo(\027)441 1088 y Fl(+)469 1082 y Fp(\()p Fo(a)p Fp(\)\))c(=)f Fo(\032)615 1088 y Fm(A)643 1082 y Fp(\(\()p Fo(g)695 1088 y Fm(A)733 1082 y Fk(!)h Fo(a)809 1088 y Fl(1)827 1082 y Fp(\)[\()p Fo(g)891 1088 y Fm(A)929 1082 y Fk(!)f Fo(a)1004 1088 y Fl(2)1023 1082 y Fp(\))1039 1065 y Fl(1)1069 1082 y Fk(!)g Fp(\000)1148 1088 y Fm(A)1175 1082 y Fp(]\)\()p Fo(g)1239 1088 y Fm(A)1277 1082 y Fk(!)g Fo(a)1352 1088 y Fl(2)1371 1082 y Fp(\))1387 1065 y Fl(2)551 1150 y Fp(=)g Fo(\023)609 1156 y Fm(A)636 1150 y Fp(\(\()p Fo(g)688 1156 y Fm(A)727 1150 y Fk(!)g Fo(a)802 1156 y Fl(2)820 1150 y Fp(\))836 1132 y Fl(1)855 1150 y Fp(\))p Fo(\032)892 1156 y Fm(A)920 1150 y Fp(\(\()p Fo(g)972 1156 y Fm(A)1010 1150 y Fk(!)g Fo(a)1085 1156 y Fl(1)1104 1150 y Fp(\)\000)1146 1156 y Fm(A)1173 1150 y Fp(\)\()p Fo(g)1225 1156 y Fm(A)1264 1150 y Fk(!)g Fo(a)1339 1156 y Fl(2)1357 1150 y Fp(\))1373 1132 y Fl(2)551 1218 y Fp(=)g Fo(\023)609 1224 y Fm(A)636 1218 y Fp(\(\()p Fo(g)688 1224 y Fm(A)727 1218 y Fk(!)g Fo(a)802 1224 y Fl(2)820 1218 y Fp(\))836 1201 y Fl(1)855 1218 y Fp(\))p Fo(\013)898 1201 y Fm(R)898 1228 y(A)925 1218 y Fp(\()p Fo(g)961 1224 y Fm(A)1000 1218 y Fk(!)g Fo(a)1075 1224 y Fl(1)1093 1218 y Fp(\)\()p Fo(g)1145 1224 y Fm(A)1184 1218 y Fk(!)g Fo(a)1259 1224 y Fl(2)1278 1218 y Fp(\))1294 1201 y Fl(2)551 1287 y Fp(=)g Fo(\023)609 1293 y Fm(A)636 1287 y Fp(\()p Fo(a)674 1293 y Fl(2)693 1269 y(1)711 1287 y Fp(\))p Fo(\013)754 1269 y Fm(R)754 1297 y(A)781 1287 y Fp(\()p Fo(a)819 1293 y Fl(1)838 1287 y Fp(\))p Fo(g)874 1293 y Fm(A)913 1287 y Fk(!)g Fo(a)988 1293 y Fl(2)1006 1269 y(2)551 1355 y Fp(=)g Fo(\013)621 1338 y Fm(R)621 1365 y(A)648 1355 y Fp(\()p Fo(a)686 1361 y Fl(1)705 1355 y Fp(\))p Fo(I)739 1361 y Fm(A)766 1355 y Fp(\()p Fo(a)804 1361 y Fl(2)823 1355 y Fp(\))h(=)g Fo(\013)922 1338 y Fm(R)922 1365 y(A)949 1355 y Fp(\()p Fo(I)983 1361 y Fm(A)1010 1355 y Fp(\()p Fo(a)1048 1361 y Fl(1)1067 1355 y Fp(\)\))p Fo(I)1117 1361 y Fm(A)1144 1355 y Fp(\()p Fo(a)1182 1361 y Fl(2)1201 1355 y Fp(\))g(=)f Fo(\013)1299 1338 y Fm(R)1299 1365 y(A)1326 1355 y Fp(\()p Fo(I)1360 1361 y Fm(A)1388 1355 y Fp(\()p Fo(a)p Fp(\))1442 1361 y Fl(1)1461 1355 y Fp(\))p Fo(I)1495 1361 y Fm(A)1522 1355 y Fp(\()p Fo(a)p Fp(\))1576 1361 y Fl(2)257 1442 y Fp(In)j(order)h(to)f(pro)o(v)o(e)g(the)g(second)h(form)o(ula,)c(w)o (e)j(apply)f(the)i(preceding)g(Lemma:)312 1528 y Fo(\032)333 1534 y Fm(A)361 1528 y Fp(\()p Fo(a)399 1511 y Fj(0)399 1538 y Fl(2)417 1528 y Fo(\027)438 1534 y Fj(\000)466 1528 y Fp(\()p Fo(a)p Fp(\)\))p Fo(a)558 1511 y Fm(R)558 1538 y(A)585 1528 y Fo(S)612 1510 y Fj(\000)p Fl(1)610 1540 y Fm(A)658 1528 y Fp(\()p Fo(a)696 1511 y Fj(0)696 1538 y Fl(1)714 1528 y Fp(\))d(=)g Fo(\032)807 1534 y Fm(A)834 1528 y Fp(\()p Fo(a)872 1511 y Fl(2)891 1528 y Fp(\()p Fo(S)934 1510 y Fj(\000)p Fl(1)932 1540 y Fm(H)980 1528 y Fp(\()p Fo(a)1018 1511 y Fl(1)1036 1528 y Fp(\))g Fk(!)f Fo(a)1139 1511 y Fj(0)1139 1538 y Fl(2)1158 1528 y Fp(\)\))p Fo(a)1212 1511 y Fm(R)1212 1538 y(A)1239 1528 y Fo(S)1266 1510 y Fj(\000)p Fl(1)1264 1540 y Fm(A)1312 1528 y Fp(\()p Fo(a)1350 1511 y Fj(0)1350 1538 y Fl(1)1368 1528 y Fp(\))742 1596 y(=)h Fo(\032)807 1602 y Fm(A)834 1596 y Fp(\()p Fo(a)872 1579 y Fl(4)891 1596 y Fp(\()p Fo(S)934 1579 y Fj(\000)p Fl(1)932 1609 y Fm(H)980 1596 y Fp(\()p Fo(a)1018 1579 y Fl(1)1036 1596 y Fp(\))g Fk(!)f Fo(a)1139 1579 y Fj(0)1139 1607 y Fl(2)1158 1596 y Fp(\)\))p Fo(a)1212 1579 y Fm(R)1212 1607 y(A)1239 1596 y Fo(S)1266 1579 y Fj(\000)p Fl(1)1264 1609 y Fm(A)1312 1596 y Fp(\()p Fo(a)1350 1579 y Fl(3)1368 1596 y Fo(S)1395 1579 y Fj(\000)p Fl(1)1393 1609 y Fm(H)1441 1596 y Fp(\()p Fo(a)1479 1579 y Fl(2)1497 1596 y Fp(\))h Fk(!)f Fo(a)1600 1579 y Fj(0)1600 1607 y Fl(1)1619 1596 y Fp(\))742 1664 y(=)h Fo(a)808 1647 y Fl(2)827 1670 y(1)845 1664 y Fo(\032)866 1670 y Fm(A)894 1664 y Fp(\()p Fo(a)932 1647 y Fl(2)950 1670 y(2)969 1664 y Fp(\()p Fo(S)1012 1647 y Fj(\000)p Fl(1)1010 1677 y Fm(H)1058 1664 y Fp(\()p Fo(a)1096 1647 y Fl(1)1114 1664 y Fp(\))g Fk(!)f Fo(a)1217 1647 y Fj(0)1229 1664 y Fp(\)\))257 1751 y(Inserting)k(\000)458 1757 y Fm(A)499 1751 y Fp(for)e Fo(a)584 1736 y Fj(0)596 1751 y Fp(,)g(w)o(e)h(get:)537 1837 y Fo(\023)552 1819 y Fj(\000)p Fl(1)552 1849 y Fm(A)596 1837 y Fp(\()p Fo(a)634 1820 y Fl(1)653 1837 y Fp(\))p Fo(a)691 1820 y Fl(2)710 1843 y(1)728 1837 y Fo(\013)755 1820 y Fm(R)755 1847 y(A)782 1837 y Fp(\()p Fo(a)820 1820 y Fl(2)839 1843 y(2)858 1837 y Fp(\))d(=)h Fo(a)951 1820 y Fm(R)951 1847 y(A)978 1837 y Fo(S)1005 1819 y Fj(\000)p Fl(1)1003 1849 y Fm(A)1051 1837 y Fp(\(\000)1093 1843 y Fm(A)r Fl(1)1138 1837 y Fp(\))p Fo(\032)1175 1843 y Fm(A)1203 1837 y Fp(\(\000)1245 1843 y Fm(A)r Fl(2)1291 1837 y Fo(\027)1312 1843 y Fj(\000)1339 1837 y Fp(\()p Fo(a)p Fp(\)\))257 1924 y(But)j(this)f(implies)e(b)o(y)h(Prop)q (osition)h(4.2)f(that:)321 2010 y Fo(\023)336 1992 y Fj(\000)p Fl(1)336 2022 y Fm(A)380 2010 y Fp(\()p Fo(a)418 1993 y Fl(1)437 2010 y Fp(\))p Fo(a)475 1993 y Fm(R)475 2020 y(A)502 1993 y Fj(\000)p Fl(1)547 2010 y Fo(S)574 1993 y Fl(2)572 2020 y Fm(A)599 2010 y Fp(\()p Fo(g)636 1992 y Fj(\000)p Fl(1)635 2022 y Fm(A)693 2010 y Fk(!)e Fo(a)768 1993 y Fl(2)786 2016 y(1)805 2010 y Fp(\))p Fo(a)843 1993 y Fm(R)843 2020 y(A)870 2010 y Fo(\013)897 1993 y Fm(R)897 2020 y(A)924 2010 y Fp(\()p Fo(a)962 1993 y Fl(2)981 2016 y(2)1000 2010 y Fp(\))833 2078 y(=)h Fo(S)902 2084 y Fm(A)929 2078 y Fp(\()p Fo(g)966 2061 y Fj(\000)p Fl(1)965 2091 y Fm(A)1023 2078 y Fk(!)f Fp(\000)1102 2084 y Fm(A)r Fl(1)1147 2078 y Fp(\))p Fo(a)1185 2061 y Fm(R)1185 2089 y(A)1213 2078 y Fo(\032)1234 2084 y Fm(A)1261 2078 y Fp(\(\000)1303 2084 y Fm(A)r Fl(2)1349 2078 y Fo(\027)1370 2084 y Fj(\000)1397 2078 y Fp(\()p Fo(a)p Fp(\)\))h(=)g Fo(\027)1544 2084 y Fj(\000)1572 2078 y Fp(\()p Fo(a)p Fp(\))257 2171 y(whic)o(h)i(sa)o(ys)g(that)g Fo(a)576 2156 y Fm(L)576 2182 y(A)603 2171 y Fo(S)630 2156 y Fl(2)628 2182 y Fm(A)656 2171 y Fp(\()p Fo(I)693 2153 y Fj(\000)p Fl(1)690 2183 y Fm(A)738 2171 y Fp(\()p Fo(a)p Fp(\))792 2177 y Fl(1)811 2171 y Fp(\))p Fo(a)849 2156 y Fm(R)849 2182 y(A)876 2171 y Fo(\013)903 2156 y Fm(R)903 2182 y(A)930 2171 y Fp(\()p Fo(I)967 2153 y Fj(\000)p Fl(1)964 2183 y Fm(A)1012 2171 y Fp(\()p Fo(a)p Fp(\))1066 2177 y Fl(2)1085 2171 y Fp(\))e(=)g Fo(\027)1178 2177 y Fj(\000)1205 2171 y Fp(\()p Fo(a)p Fp(\).)i Fg(\003)257 2304 y Fn(4.6)48 b Fp(It)11 b(is)f(natural)g(to)g (ask)g(whether)i(the)f(t)o(wisted)g(Nak)n(a)o(y)o(ama)c(automorphisms)h (and)i(the)257 2354 y(ordinary)16 b(Nak)n(a)o(y)o(ama)e(automorphism)f (are)k(in)o(terrelated.)g(W)m(e)g(can)f(deriv)o(e)i(an)e(explicit)257 2403 y(form)o(ula)10 b(for)i(the)g(ordinary)g(Nak)n(a)o(y)o(ama)d (automorphism)g(in)j(terms)g(of)f(the)i(negativ)o(e)f(one.)257 2453 y(It)20 b(will)d(b)q(ecome)i(clear)h(later)f(that)h(this)f(is)g (not)g(so)g(easy)h(to)f(deriv)o(e)h(a)f(corresp)q(onding)257 2503 y(form)o(ula)12 b(in)h(the)i(case)f(of)g(the)g(p)q(ositiv)o(e)g (one.)953 2628 y(37)p eop %%Page: 38 38 38 37 bop 257 262 a Fn(Prop)q(osition)33 b Fp(F)m(or)14 b(all)e Fo(a)g Fk(2)f Fo(A)p Fp(,)j(w)o(e)g(ha)o(v)o(e:)f Fo(\027)987 268 y Fm(A)1013 262 y Fp(\()p Fo(a)p Fp(\))f(=)g Fo(\023)1138 268 y Fm(A)1165 262 y Fp(\()p Fo(a)1203 246 y Fl(1)1221 262 y Fp(\))p Fo(\027)1258 268 y Fj(\000)1286 262 y Fp(\()p Fo(\022)1321 268 y Fm(A)1349 262 y Fp(\()p Fo(a)1387 246 y Fl(2)1405 262 y Fp(\)\))257 361 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(ha)o(v)o(e:)445 452 y Fo(\032)466 458 y Fm(A)493 452 y Fp(\()p Fo(a)531 435 y Fj(0)543 452 y Fo(\027)564 458 y Fj(\000)592 452 y Fp(\()p Fo(a)p Fp(\)\))e(=)f Fo(\032)738 458 y Fm(A)766 452 y Fp(\()p Fo(a)804 435 y Fl(2)823 452 y Fp(\()p Fo(S)866 435 y Fj(\000)p Fl(1)864 465 y Fm(H)911 452 y Fp(\()p Fo(a)949 435 y Fl(1)968 452 y Fp(\))h Fk(!)f Fo(a)1071 435 y Fj(0)1082 452 y Fp(\)\))674 521 y(=)g Fo(\032)738 527 y Fm(A)766 521 y Fp(\(\()p Fo(S)825 503 y Fj(\000)p Fl(1)823 533 y Fm(H)871 521 y Fp(\()p Fo(a)909 503 y Fl(2)927 521 y Fp(\))p Fo(S)970 503 y Fj(\000)p Fl(2)968 533 y Fm(H)1016 521 y Fp(\()p Fo(a)1054 503 y Fl(3)1073 521 y Fp(\))g Fk(!)g Fo(a)1175 503 y Fl(4)1194 521 y Fp(\)\()p Fo(S)1253 503 y Fj(\000)p Fl(1)1251 533 y Fm(H)1299 521 y Fp(\()p Fo(a)1337 503 y Fl(1)1355 521 y Fp(\))h Fk(!)f Fo(a)1458 503 y Fj(0)1470 521 y Fp(\)\))674 589 y(=)g(\()p Fo(\032)754 595 y Fm(A)794 589 y Fk( )g Fo(S)874 571 y Fj(\000)p Fl(1)872 601 y Fm(H)919 589 y Fp(\()p Fo(a)957 571 y Fl(1)976 589 y Fp(\)\)\(\()p Fo(S)1067 571 y Fj(\000)p Fl(2)1065 601 y Fm(H)1113 589 y Fp(\()p Fo(a)1151 571 y Fl(2)1169 589 y Fp(\))h Fk(!)f Fo(a)1272 571 y Fl(3)1291 589 y Fp(\))p Fo(a)1329 571 y Fj(0)1340 589 y Fp(\))674 657 y(=)g Fo(\023)732 639 y Fj(\000)p Fl(1)732 669 y Fm(A)777 657 y Fp(\()p Fo(a)815 639 y Fl(1)833 657 y Fp(\))p Fo(\032)870 663 y Fm(A)898 657 y Fp(\()p Fo(\022)934 639 y Fj(\000)p Fl(1)933 669 y Fm(A)979 657 y Fp(\()p Fo(a)1017 639 y Fl(2)1036 657 y Fp(\))p Fo(a)1074 639 y Fj(0)1086 657 y Fp(\))674 725 y(=)g Fo(\023)732 707 y Fj(\000)p Fl(1)732 737 y Fm(A)777 725 y Fp(\()p Fo(a)815 707 y Fl(1)833 725 y Fp(\))p Fo(\032)870 731 y Fm(A)898 725 y Fp(\()p Fo(a)936 707 y Fj(0)948 725 y Fo(\027)969 731 y Fm(A)995 725 y Fp(\()p Fo(\022)1031 707 y Fj(\000)p Fl(1)1030 737 y Fm(A)1077 725 y Fp(\()p Fo(a)1115 707 y Fl(2)1133 725 y Fp(\)\)\))257 822 y(By)k(nondegeneracy)m(,)g(this)g (implies)d Fo(\027)856 828 y Fj(\000)884 822 y Fp(\()p Fo(a)p Fp(\))g(=)h Fo(\023)1010 804 y Fj(\000)p Fl(1)1010 834 y Fm(A)1054 822 y Fp(\()p Fo(a)1092 807 y Fl(1)1111 822 y Fp(\))p Fo(\027)1148 828 y Fm(A)1175 822 y Fp(\()p Fo(\022)1211 804 y Fj(\000)p Fl(1)1210 834 y Fm(A)1256 822 y Fp(\()p Fo(a)1294 807 y Fl(2)1313 822 y Fp(\)\),)h(whic)o(h)g (implies)f(the)257 872 y(assertion.)h Fg(\003)257 990 y Fp(F)m(rom)g(the)i(Prop)q(osition,)f(w)o(e)h(see)h(that)f(the)g (negativ)o(ely)f(t)o(wisted)h(Nak)n(a)o(y)o(ama)c(automor-)257 1040 y(phism)i(comm)o(utes)f(with)i(the)h(ordinary)e(Nak)n(a)o(y)o(ama) e(automorphism,)f(b)q(ecause)17 b(it)e(com-)257 1090 y(m)o(utes)f(with)g(itself,)f Fo(\022)q Fp(,)h(and,)g(b)o(y)f (naturalit)o(y)m(,)g(with)g(the)i(monoidal)c(transformation)h Fo(a)g Fk(7!)257 1140 y Fo(\023)272 1146 y Fm(A)299 1140 y Fp(\()p Fo(a)337 1125 y Fl(1)356 1140 y Fp(\))p Fo(a)394 1125 y Fl(2)413 1140 y Fp(.)i(But)i(then,)g(the)g(p)q(ositiv)o(ely)e(t) o(wisted)i(Nak)n(a)o(y)o(ama)c(automorphism)g(also)j(com-)257 1190 y(m)o(utes)f(with)f(the)i(ordinary)e(one,)h(b)q(ecause)h(w)o(e)f (ha)o(v)o(e)g(b)o(y)g(Prop)q(osition)f(4.4:)443 1281 y Fo(\032)464 1287 y Fm(A)492 1281 y Fp(\()p Fo(a\027)551 1287 y Fl(+)578 1281 y Fp(\()p Fo(\027)615 1287 y Fm(A)642 1281 y Fp(\()p Fo(a)680 1264 y Fj(0)691 1281 y Fp(\)\)\))f(=)g Fo(\032)816 1287 y Fm(A)844 1281 y Fp(\()p Fo(\027)884 1263 y Fj(\000)p Fl(1)881 1291 y Fj(\000)928 1281 y Fp(\()p Fo(a)p Fp(\))p Fo(\027)1003 1287 y Fm(A)1029 1281 y Fp(\()p Fo(a)1067 1264 y Fj(0)1079 1281 y Fp(\)\))751 1349 y(=)g Fo(\032)816 1355 y Fm(A)844 1349 y Fp(\()p Fo(a)882 1332 y Fj(0)893 1349 y Fo(\027)917 1331 y Fj(\000)p Fl(1)914 1359 y Fj(\000)961 1349 y Fp(\()p Fo(a)p Fp(\)\))g(=)g Fo(\032)1108 1355 y Fm(A)1135 1349 y Fp(\()p Fo(\027)1175 1331 y Fj(\000)p Fl(1)1172 1361 y Fm(A)1219 1349 y Fp(\()p Fo(\027)1259 1331 y Fj(\000)p Fl(1)1256 1359 y Fj(\000)1303 1349 y Fp(\()p Fo(a)p Fp(\)\))p Fo(a)1395 1332 y Fj(0)1407 1349 y Fp(\))751 1417 y(=)g Fo(\032)816 1423 y Fm(A)844 1417 y Fp(\()p Fo(\027)884 1399 y Fj(\000)p Fl(1)881 1427 y Fj(\000)928 1417 y Fp(\()p Fo(\027)968 1399 y Fj(\000)p Fl(1)965 1429 y Fm(A)1011 1417 y Fp(\()p Fo(a)p Fp(\)\))p Fo(a)1103 1400 y Fj(0)1115 1417 y Fp(\))g(=)g Fo(\032)1208 1423 y Fm(A)1235 1417 y Fp(\()p Fo(\027)1275 1399 y Fj(\000)p Fl(1)1272 1429 y Fm(A)1319 1417 y Fp(\()p Fo(a)p Fp(\))p Fo(\027)1394 1423 y Fl(+)1422 1417 y Fp(\()p Fo(a)1460 1400 y Fj(0)1471 1417 y Fp(\)\))751 1479 y(=)g Fo(\032)816 1485 y Fm(A)844 1479 y Fp(\()p Fo(\027)881 1485 y Fl(+)908 1479 y Fp(\()p Fo(a)946 1462 y Fj(0)958 1479 y Fp(\))p Fo(a)p Fp(\))f(=)h Fo(\032)1088 1485 y Fm(A)1116 1479 y Fp(\()p Fo(a\027)1175 1485 y Fm(A)1201 1479 y Fp(\()p Fo(\027)1238 1485 y Fl(+)1266 1479 y Fp(\()p Fo(a)1304 1462 y Fj(0)1315 1479 y Fp(\)\)\))257 1571 y(whic)o(h)i(implies)e Fo(\027)538 1577 y Fl(+)565 1571 y Fp(\()p Fo(\027)602 1577 y Fm(A)629 1571 y Fp(\()p Fo(a)667 1555 y Fj(0)679 1571 y Fp(\)\))f(=)h Fo(\027)787 1577 y Fm(A)814 1571 y Fp(\()p Fo(\027)851 1577 y Fl(+)878 1571 y Fp(\()p Fo(a)916 1555 y Fj(0)928 1571 y Fp(\)\))i(b)o(y)f (nondegeneracy)m(.)257 1656 y(W)m(e)d(note)g(that)f(the)h(ab)q(o)o(v)o (e)g(Prop)q(osition)f(also)g(enables)h(us)g(to)g(calculate)f(the)i (compatibilit)o(y)257 1706 y(of)20 b(the)h(ordinary)f(Nak)n(a)o(y)o (ama)d(automorphism)h(with)i(the)h(action)f(and)g(the)h(coaction.)257 1756 y(Using)14 b(the)h(notation:)484 1847 y Fo(')c Fp(:)g Fo(H)k Fk(!)c Fo(H)q(;)k(h)d Fk(7!)f Fo(S)827 1830 y Fl(2)825 1857 y Fm(H)857 1847 y Fp(\()p Fo(g)894 1829 y Fj(\000)p Fl(1)893 1859 y Fm(A)939 1847 y Fo(hg)983 1853 y Fm(A)1009 1847 y Fp(\))h(=)g Fo(\023)1096 1853 y Fm(A)1123 1847 y Fp(\()p Fo(h)1163 1853 y Fl(1)1181 1847 y Fp(\))p Fo(S)1224 1830 y Fl(2)1222 1857 y Fm(H)1254 1847 y Fp(\()p Fo(h)1294 1853 y Fl(2)1313 1847 y Fp(\))p Fo(\023)1344 1829 y Fj(\000)p Fl(1)1344 1859 y Fm(A)1388 1847 y Fp(\()p Fo(h)1428 1853 y Fl(3)1447 1847 y Fp(\))257 1938 y(w)o(e)j(see)g(that)f Fo(\027)497 1944 y Fm(A)537 1938 y Fp(is)g(an)g Fo(H)s Fp(-linear)f(and)h(colinear)f(map)g(from)f Fo(A)i Fp(to)g Fo(A)1357 1944 y Fm(')1395 1938 y Fp(b)q(ecause)h(it)f (is)g(the)257 1988 y(comp)q(osition)e(of)i(the)g Fo(H)s Fp(-linear)f(and)h(colinear)g(mappings:)740 2092 y Fo(A)795 2064 y Fm(\027)812 2068 y Fi(\000)782 2092 y Fk(\000)-6 b(!)11 b Fo(A)917 2068 y Fm(\022)933 2072 y Fe(A)903 2092 y Fk(\000)-6 b(!)11 b Fo(A)1013 2101 y Fm(S)1035 2091 y Fd(2)1033 2111 y Fe(H)1084 2066 y Fm( )1106 2070 y Fe(A)1074 2092 y Fk(\000)-7 b(!)11 b Fo(A)1183 2098 y Fm(')257 2190 y Fp(where)k Fo( )h Fp(denotes)f(the)f(monoidal)d (transformation)h Fo( )1131 2196 y Fm(A)1158 2190 y Fp(\()p Fo(a)p Fp(\))g(=)g Fo(\023)1283 2196 y Fm(A)1310 2190 y Fp(\()p Fo(a)1348 2175 y Fl(1)1366 2190 y Fp(\))p Fo(a)1404 2175 y Fl(2)1423 2190 y Fp(.)257 2325 y Fn(4.7)48 b Fp(In)11 b(order)h(to)e(form)o(ulate)f(our)i(analogue)f(of)g(Radford's)g(form)o (ula,)e(w)o(e)j(m)o(ust)f(consider)257 2375 y(adjoin)o(ts)16 b(of)h(linear)f(maps)f(with)i(resp)q(ect)i(to)e(the)g(bilinear)f(form)f (whic)o(h)i(is)f(induced)i(b)o(y)257 2425 y(our)c(righ)o(t)g(in)o (tegral)f Fo(\032)604 2431 y Fm(A)631 2425 y Fp(:)953 2628 y(38)p eop %%Page: 39 39 39 38 bop 257 262 a Fn(De\014nition)33 b Fp(If)13 b Fo(f)j Fp(:)c Fo(A)f Fk(!)g Fo(A)j Fp(is)f(a)g Fo(K)s Fp(-linear)h(map,)d (denote)j(b)o(y)g Fo(f)1303 246 y Fj(\003)1334 262 y Fp(:)d Fo(A)h Fk(!)f Fo(A)i Fp(the)h(unique)257 311 y(linear)g(map)e (that)i(satis\014es)h(for)e(all)g Fo(a;)7 b(a)892 296 y Fj(0)915 311 y Fk(2)k Fo(A)p Fp(:)743 400 y Fo(\032)764 406 y Fm(A)791 400 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\))p Fo(a)907 383 y Fj(0)920 400 y Fp(\))h(=)g Fo(\032)1013 406 y Fm(A)1040 400 y Fp(\()p Fo(af)1102 383 y Fj(\003)1122 400 y Fp(\()p Fo(a)1160 383 y Fj(0)1172 400 y Fp(\)\))257 488 y Fo(f)281 473 y Fj(\003)320 488 y Fp(will)18 b(b)q(e)i(called)f (the)h(adjoin)o(t)e(mapping)f(of)h Fo(f)t Fp(,)i(more)e(precisely)i (the)g(righ)o(t)f(adjoin)o(t)257 538 y(mapping.)257 655 y(The)g(follo)o(wing)d(Prop)q(osition)i(summarizes)f(the)i(basic)g (prop)q(erties)h(of)e(the)h(adjunction)257 704 y(pro)q(cess:)257 802 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(f)i Fp(:)11 b Fo(A)h Fk(!)f Fo(A)j Fp(is)g(a)f Fo(K)s Fp(-linear)h(map.)308 917 y(1.)20 b(If)11 b Fo(f)16 b Fp(is)c(an)f(algebra)g(automorphism)d(whic)o(h)j(is)g(sim)o (ultaneously)f(a)h(coalgebra)g(auto-)361 967 y(morphism,)f(then)15 b Fo(\032)682 973 y Fm(A)723 967 y Fp(is)e(an)h(eigen)o(v)o(ector)g (for)f(the)i(transp)q(ose)g(of)e Fo(f)t Fp(,)g(that)h(is,)f(there)361 1017 y(is)h Fo(\020)h Fk(2)c Fo(K)17 b Fp(suc)o(h)e(that)e(w)o(e)i(ha)o (v)o(e)e(for)h(all)e Fo(a)g Fk(2)f Fo(A)p Fp(:)856 1106 y Fo(\032)877 1112 y Fm(A)904 1106 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\)\))i(=)f Fo(\020)s(\032)1113 1112 y Fm(A)1141 1106 y Fp(\()p Fo(a)p Fp(\))361 1194 y(The)i(adjoin)o(t)f(of)g Fo(f)19 b Fp(is:)13 b Fo(f)750 1179 y Fj(\003)782 1194 y Fp(=)f Fo(\020)s(f)871 1179 y Fj(\000)p Fl(1)308 1276 y Fp(2.)20 b(Supp)q(ose)d(that)f Fo(')f Fp(:)g Fo(H)j Fk(!)c Fo(H)19 b Fp(is)d(a)f(Hopf)h(algebra)g(automorphism)c(that)k (satis\014es)361 1326 y Fo(')p Fp(\()p Fo(g)424 1332 y Fm(A)451 1326 y Fp(\))c(=)g Fo(g)543 1332 y Fm(A)583 1326 y Fp(and)h Fo(\023)678 1332 y Fm(A)713 1326 y Fk(\016)8 b Fo(')j Fp(=)h Fo(\023)839 1332 y Fm(A)866 1326 y Fp(.)h(If)g Fo(f)j Fp(:)11 b Fo(A)h Fk(!)f Fo(A)1118 1332 y Fm(')1155 1326 y Fp(is)i Fo(H)s Fp(-linear)g(and)g(colinear,)g(then)361 1375 y Fo(f)385 1360 y Fj(\003)419 1375 y Fp(is)h(an)f Fo(H)s Fp(-linear)g(and)h(colinear)g(map)e(from)g Fo(A)1142 1381 y Fm(')1180 1375 y Fp(to)i Fo(A)p Fp(.)257 1473 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(\014rst)g(c)o(hec)o(k)h(that)f Fo(\032)803 1479 y Fm(A)840 1473 y Fk(\016)9 b Fo(f)19 b Fp(is)13 b(again)g(a)h(righ)o(t)f(in)o(tegral:)334 1562 y Fo(\032)355 1568 y Fm(A)382 1562 y Fp(\()p Fo(f)t Fp(\()p Fo(a)460 1568 y Fl(1)480 1562 y Fp(\)\))p Fo(a)534 1568 y Fl(2)564 1562 y Fp(=)f Fo(f)632 1545 y Fj(\000)p Fl(1)677 1562 y Fp(\()p Fo(\032)714 1568 y Fm(A)742 1562 y Fp(\()p Fo(f)t Fp(\()p Fo(a)820 1568 y Fl(1)840 1562 y Fp(\)\))p Fo(f)t Fp(\()p Fo(a)934 1568 y Fl(2)953 1562 y Fp(\)\))g(=)g Fo(f)1065 1545 y Fj(\000)p Fl(1)1110 1562 y Fp(\()p Fo(\032)1147 1568 y Fm(A)1175 1562 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\)\)1)1306 1568 y Fm(A)1334 1562 y Fp(\))g(=)g Fo(\032)1427 1568 y Fm(A)1454 1562 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\)\)1)1585 1568 y Fm(A)257 1650 y Fp(Since)k(the)h(space)f(of)f(righ)o(t)h(in)o(tegrals)f (is)g(one-dimensional,)e(there)k(is)f Fo(\020)h Fk(2)d Fo(K)19 b Fp(suc)o(h)d(that)257 1700 y Fo(\032)278 1706 y Fm(A)306 1700 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\)\))d(=)f Fo(\020)s(\032)515 1706 y Fm(A)542 1700 y Fp(\()p Fo(a)p Fp(\).)i(W)m(e)f(no)o(w)h(ha)o(v)o(e:)617 1788 y Fo(\032)638 1794 y Fm(A)666 1788 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\))p Fo(f)t Fp(\()p Fo(a)822 1771 y Fj(0)836 1788 y Fp(\)\))d(=)h Fo(\032)944 1794 y Fm(A)972 1788 y Fp(\()p Fo(f)t Fp(\()p Fo(aa)1072 1771 y Fj(0)1084 1788 y Fp(\)\))g(=)g Fo(\020)s(\032)1214 1794 y Fm(A)1242 1788 y Fp(\()p Fo(aa)1302 1771 y Fj(0)1313 1788 y Fp(\))257 1877 y(Inserting)17 b Fo(f)458 1862 y Fj(\000)p Fl(1)503 1877 y Fp(\()p Fo(a)541 1862 y Fj(0)553 1877 y Fp(\))f(instead)g(of)g Fo(a)802 1862 y Fj(0)829 1877 y Fp(yields)g(the)h(form)o(ula)c(for)j(the)g (adjoin)o(t)f(mapping.)e(T)m(o)257 1927 y(pro)o(v)o(e)h(the)h(second)g (statemen)o(t,)e(w)o(e)h(carry)h(out)f(the)g(follo)o(wing)d (calculation:)372 2015 y Fo(\032)393 2021 y Fm(A)421 2015 y Fp(\()p Fo(af)483 1998 y Fj(\003)502 2015 y Fp(\()p Fo(h)h Fk(!)f Fo(a)629 1998 y Fj(0)641 2015 y Fp(\)\))h(=)f Fo(\032)749 2021 y Fm(A)777 2015 y Fp(\()p Fo(f)t Fp(\()p Fo(a)p Fp(\)\()p Fo(h)i Fk(!)e Fo(a)999 1998 y Fj(0)1010 2015 y Fp(\)\))685 2083 y(=)g Fo(\032)749 2089 y Fm(A)777 2083 y Fp(\(\()p Fo(h)833 2089 y Fl(2)852 2083 y Fo(S)879 2065 y Fj(\000)p Fl(1)877 2095 y Fm(H)924 2083 y Fp(\()p Fo(h)964 2089 y Fl(1)983 2083 y Fp(\))g Fk(!)g Fo(f)t Fp(\()p Fo(a)p Fp(\)\)\()p Fo(h)1197 2089 y Fl(3)1229 2083 y Fk(!)g Fo(a)1304 2066 y Fj(0)1315 2083 y Fp(\)\))685 2151 y(=)g Fo(\023)743 2157 y Fm(A)770 2151 y Fp(\()p Fo(h)810 2157 y Fl(2)829 2151 y Fp(\))p Fo(\032)866 2157 y Fm(A)893 2151 y Fp(\()p Fo(f)t Fp(\()p Fo(')976 2134 y Fj(\000)p Fl(1)1022 2151 y Fp(\()p Fo(S)1065 2133 y Fj(\000)p Fl(1)1063 2163 y Fm(H)1111 2151 y Fp(\()p Fo(h)1151 2157 y Fl(1)1169 2151 y Fp(\)\))h Fk(!)f Fo(a)p Fp(\))p Fo(a)1326 2134 y Fj(0)1338 2151 y Fp(\))685 2219 y(=)g Fo(\023)743 2225 y Fm(A)770 2219 y Fp(\()p Fo(')813 2202 y Fj(\000)p Fl(1)858 2219 y Fp(\()p Fo(h)898 2225 y Fl(2)917 2219 y Fp(\)\))p Fo(\032)970 2225 y Fm(A)997 2219 y Fp(\(\()p Fo(')1056 2202 y Fj(\000)p Fl(1)1101 2219 y Fp(\()p Fo(S)1144 2201 y Fj(\000)p Fl(1)1142 2231 y Fm(H)1190 2219 y Fp(\()p Fo(h)1230 2225 y Fl(1)1249 2219 y Fp(\)\))g Fk(!)g Fo(a)p Fp(\))p Fo(f)1407 2202 y Fj(\003)1427 2219 y Fp(\()p Fo(a)1465 2202 y Fj(0)1477 2219 y Fp(\)\))685 2287 y(=)g Fo(\032)749 2293 y Fm(A)777 2287 y Fp(\(\()p Fo(')836 2270 y Fj(\000)p Fl(1)881 2287 y Fp(\()p Fo(h)921 2293 y Fl(2)940 2287 y Fo(S)967 2269 y Fj(\000)p Fl(1)965 2299 y Fm(H)1012 2287 y Fp(\()p Fo(h)1052 2293 y Fl(1)1071 2287 y Fp(\)\))g Fk(!)g Fo(a)p Fp(\)\()p Fo(')1248 2270 y Fj(\000)p Fl(1)1293 2287 y Fp(\()p Fo(h)1333 2293 y Fl(3)1352 2287 y Fp(\))h Fk(!)f Fo(f)1457 2270 y Fj(\003)1477 2287 y Fp(\()p Fo(a)1515 2270 y Fj(0)1526 2287 y Fp(\)\)\))685 2355 y(=)g Fo(\032)749 2361 y Fm(A)777 2355 y Fp(\()p Fo(a)p Fp(\()p Fo(')858 2337 y Fj(\000)p Fl(1)903 2355 y Fp(\()p Fo(h)p Fp(\))h Fk(!)f Fo(f)1048 2337 y Fj(\003)1067 2355 y Fp(\()p Fo(a)1105 2337 y Fj(0)1117 2355 y Fp(\)\)\))257 2443 y(By)g(the)g(nondegeneracy)h(of)d(our)i(bilinear)e(form,)f(w)o(e)j (obtain)f(the)g(equation)g Fo(f)1471 2428 y Fj(\003)1491 2443 y Fp(\()p Fo(h)i Fk(!)f Fo(a)1618 2428 y Fj(0)1630 2443 y Fp(\))g(=)257 2493 y Fo(')284 2478 y Fj(\000)p Fl(1)329 2493 y Fp(\()p Fo(h)p Fp(\))h Fk(!)f Fo(f)474 2478 y Fj(\003)494 2493 y Fp(\()p Fo(a)532 2478 y Fj(0)543 2493 y Fp(\).)j(W)m(e)f(lea)o(v)o(e)h(the)h(v)o(eri\014cation)e(of)g (the)i(colinearit)o(y)e(to)h(the)g(reader.)h Fg(\003)953 2628 y Fp(39)p eop %%Page: 40 40 40 39 bop 257 262 a Fn(4.8)48 b Fp(Using)12 b(Prop)q(osition)g(4.7,)f (it)h(is)g(no)o(w)g(easy)h(to)f(calculate)h(the)g(adjoin)o(ts)e(of)h (the)h(map-)257 311 y(pings)g(of)f(in)o(terest)i(here,)g(b)q(ecause)g (to)f(calculate)g(the)g(adjoin)o(t)f(of,)g(for)g(example,)f(the)j(in)o (te-)257 361 y(gral)h(transformation,)f(w)o(e)i(need)h(only)e(to)g (calculate)h(the)g(eigen)o(v)n(alue)g(of)f(the)h(transp)q(ose)257 411 y(on)e Fo(\032)336 417 y Fm(A)364 411 y Fp(,)f(whic)o(h)h(is)f (easy:)257 509 y Fn(Prop)q(osition)33 b Fp(W)m(e)14 b(ha)o(v)o(e:)308 624 y(1.)20 b(\()p Fo(M)422 609 y Fm(R)417 636 y(A)449 624 y Fp(\))465 609 y Fj(\003)496 624 y Fp(=)12 b Fo(M)585 609 y Fm(L)580 636 y(A)693 624 y Fp(\()p Fo(M)754 609 y Fm(L)749 636 y(A)778 624 y Fp(\))794 609 y Fj(\003)825 624 y Fp(=)g Fo(M)914 609 y Fm(R)909 636 y(A)1038 624 y Fp(\(if)h Fo(H)k Fp(is)c(\014nite-dimensional\))308 706 y(2.)20 b Fo(I)382 691 y Fj(\003)379 717 y Fm(A)418 706 y Fp(=)12 b Fo(\023)477 712 y Fm(A)504 706 y Fp(\()p Fo(g)540 712 y Fm(A)566 706 y Fp(\))582 691 y Fl(2)601 706 y Fo(I)622 688 y Fj(\000)p Fl(1)619 718 y Fm(A)308 788 y Fp(3.)20 b(\()p Fo(S)404 773 y Fl(2)402 799 y Fm(A)439 788 y Fk(\016)9 b Fo(\022)488 794 y Fm(A)516 788 y Fp(\))532 773 y Fj(\003)562 788 y Fp(=)j Fo(\023)621 794 y Fm(A)648 788 y Fp(\()p Fo(g)684 794 y Fm(A)711 788 y Fp(\))p Fo(\013)754 773 y Fm(R)754 799 y(A)781 788 y Fp(\()p Fo(a)819 773 y Fm(R)819 799 y(A)846 788 y Fp(\)\()p Fo(S)905 770 y Fj(\000)p Fl(2)903 800 y Fm(A)960 788 y Fk(\016)d Fo(\022)1010 770 y Fj(\000)p Fl(1)1009 800 y Fm(A)1055 788 y Fp(\))257 886 y Fn(Pro)q(of.)36 b Fp(Note)14 b(\014rst)g(that)g Fo(M)743 871 y Fm(L)738 897 y(A)781 886 y Fp(is)f(the)h(in)o(v)o(erse)g (of)f Fo(M)1122 871 y Fm(R)1117 897 y(A)1149 886 y Fp(.)g(Therefore,)h (the)h(\014rst)f(assertion)257 936 y(follo)o(ws)c(from)g(the)j (preceding)f(Prop)q(osition)f(if)g(w)o(e)h(can)g(sho)o(w)g(that)f(the)h (transp)q(ose)h(of)e Fo(M)1662 921 y Fm(R)1657 947 y(A)257 985 y Fp(tak)o(es)16 b(the)f(eigen)o(v)n(alue)g(1)g(on)f(the)i(righ)o (t)e(in)o(tegral)h Fo(\032)1079 991 y Fm(A)1106 985 y Fp(,)g(b)q(ecause)h Fo(M)1332 970 y Fm(R)1327 997 y(A)1374 985 y Fp(is)f(an)g(algebra)g(and)257 1035 y(a)f(coalgebra)g (automorphism)c(b)o(y)k(Prop)q(osition)f(3.10.)g(No)o(w)g(w)o(e)h(ha)o (v)o(e:)440 1124 y Fo(\032)461 1130 y Fm(A)488 1124 y Fp(\()p Fo(M)549 1107 y Fm(R)544 1134 y(A)576 1124 y Fp(\()p Fo(a)p Fp(\)\))e(=)g Fo(\013)729 1107 y Fm(R)729 1134 y(H)760 1124 y Fp(\()p Fo(a)798 1107 y Fl(1)817 1124 y Fp(\))p Fo(\032)854 1130 y Fm(A)881 1124 y Fp(\()p Fo(a)919 1107 y Fm(R)919 1134 y(H)962 1124 y Fk(!)g Fo(a)1038 1107 y Fl(2)1056 1124 y Fp(\))g(=)g Fo(\013)1155 1107 y Fm(R)1155 1134 y(H)1186 1124 y Fp(\()p Fo(a)1224 1107 y Fl(1)1242 1124 y Fp(\))p Fo(\023)1273 1130 y Fm(A)1300 1124 y Fp(\()p Fo(a)1338 1107 y Fm(R)1338 1134 y(H)1370 1124 y Fp(\))p Fo(\032)1407 1130 y Fm(A)1434 1124 y Fp(\()p Fo(a)1472 1107 y Fl(2)1491 1124 y Fp(\))1084 1192 y(=)g Fo(\013)1155 1175 y Fm(R)1155 1203 y(H)1186 1192 y Fp(\()p Fo(g)1222 1198 y Fm(A)1249 1192 y Fp(\))p Fo(\023)1280 1198 y Fm(A)1306 1192 y Fp(\()p Fo(a)1344 1175 y Fm(R)1344 1203 y(H)1376 1192 y Fp(\))p Fo(\032)1413 1198 y Fm(A)1441 1192 y Fp(\()p Fo(a)p Fp(\))257 1285 y(and)17 b(b)o(y)g(Prop)q(osition) g(2.13)f(w)o(e)h(ha)o(v)o(e)g Fo(\013)910 1270 y Fm(R)910 1296 y(H)941 1285 y Fp(\()p Fo(g)977 1291 y Fm(A)1004 1285 y Fp(\))p Fo(\023)1035 1291 y Fm(A)1062 1285 y Fp(\()p Fo(a)1100 1270 y Fm(R)1100 1296 y(H)1131 1285 y Fp(\))g(=)g(1.)g(The)g (second)h(part)f(of)g(the)257 1335 y(\014rst)c(statemen)o(t)f(as)h(w)o (ell)e(as)h(the)h(second)g(statemen)o(t)f(follo)o(w)e(b)o(y)i(a)g (similar)e(reasoning.)h(T)m(o)257 1385 y(pro)o(v)o(e)j(the)h(last)e (statemen)o(t,)h(w)o(e)g(observ)o(e)h(that:)394 1473 y Fo(\032)415 1479 y Fm(A)443 1473 y Fp(\()p Fo(\022)478 1479 y Fm(A)506 1473 y Fp(\()p Fo(a)p Fp(\)\))d(=)f Fo(\032)652 1479 y Fm(A)680 1473 y Fp(\()p Fo(S)721 1479 y Fm(H)753 1473 y Fp(\()p Fo(a)791 1456 y Fl(1)810 1473 y Fp(\))g Fk(!)g Fo(a)912 1456 y Fl(2)931 1473 y Fp(\))h(=)f Fo(\023)1017 1455 y Fj(\000)p Fl(1)1017 1485 y Fm(A)1062 1473 y Fp(\()p Fo(a)1100 1456 y Fl(1)1118 1473 y Fp(\))p Fo(\032)1155 1479 y Fm(A)1183 1473 y Fp(\()p Fo(a)1221 1456 y Fl(2)1240 1473 y Fp(\))g(=)h Fo(\023)1326 1455 y Fj(\000)p Fl(1)1326 1485 y Fm(A)1371 1473 y Fp(\()p Fo(g)1407 1479 y Fm(A)1433 1473 y Fp(\))p Fo(\032)1470 1479 y Fm(A)1498 1473 y Fp(\()p Fo(a)p Fp(\))257 1562 y(Since)h(it)e(follo)o(ws)g(from)f(Corollary)g (4.2)h(that)h(the)g(transp)q(ose)i(of)d Fo(S)1294 1547 y Fl(2)1292 1573 y Fm(A)1331 1562 y Fp(tak)o(es)i(the)f(eigen)o(v)n (alue)257 1611 y Fo(\023)272 1617 y Fm(A)299 1611 y Fp(\()p Fo(g)335 1617 y Fm(A)362 1611 y Fp(\))378 1596 y Fl(2)397 1611 y Fo(\013)424 1596 y Fm(R)424 1623 y(A)451 1611 y Fp(\()p Fo(a)489 1596 y Fm(R)489 1623 y(A)516 1611 y Fp(\))j(on)f Fo(\032)626 1617 y Fm(A)654 1611 y Fp(,)g(w)o(e)h(kno)o (w)f(that)h(the)h(transp)q(ose)g(of)e Fo(S)1277 1596 y Fl(2)1275 1623 y Fm(A)1312 1611 y Fk(\016)c Fo(\022)1362 1617 y Fm(A)1404 1611 y Fp(tak)o(es)15 b(the)g(eigen-)257 1661 y(v)n(alue)i Fo(\023)383 1667 y Fm(A)410 1661 y Fp(\()p Fo(g)446 1667 y Fm(A)473 1661 y Fp(\))p Fo(\013)516 1646 y Fm(R)516 1673 y(A)543 1661 y Fp(\()p Fo(a)581 1646 y Fm(R)581 1673 y(A)608 1661 y Fp(\))h(on)f Fo(\032)724 1667 y Fm(A)752 1661 y Fp(.)g(No)o(w)g(the)h(assertion)g(follo)o(ws)e (b)q(ecause)j(w)o(e)f(kno)o(w)f(from)257 1711 y(Prop)q(osition)d(3.10)e (that)i Fo(S)685 1696 y Fl(2)683 1722 y Fm(A)720 1711 y Fk(\016)9 b Fo(\022)769 1717 y Fm(A)810 1711 y Fp(is)14 b(an)g(algebra)f(and)h(a)g(coalgebra)f(automorphism.)e Fg(\003)257 1845 y Fn(4.9)48 b Fp(In)10 b(the)g(pro)q(of)f(of)g(the)h (analogue)e(of)h(Radford's)g(form)o(ula,)d(w)o(e)k(need)g(to)g(kno)o(w) f(ho)o(w)g(the)257 1895 y(the)j(t)o(wisted)f(Nak)n(a)o(y)o(ama)c (automorphisms)h(and)j(the)g(m)o(ultiplications)d(with)i(the)h(mo)q (dular)257 1945 y(elemen)o(ts)j(comm)o(ute:)257 2043 y Fn(Prop)q(osition)33 b Fp(W)m(e)14 b(ha)o(v)o(e)f(for)h(all)f Fo(a)e Fk(2)g Fo(A)p Fp(:)308 2158 y(1.)490 2208 y Fo(\027)511 2214 y Fl(+)538 2208 y Fp(\()p Fo(aa)598 2191 y Fm(R)598 2218 y(A)626 2208 y Fp(\))g(=)h Fo(\013)724 2191 y Fm(R)724 2218 y(A)751 2208 y Fp(\()p Fo(a)789 2191 y Fm(R)789 2218 y(A)816 2208 y Fp(\))p Fo(\027)853 2214 y Fl(+)881 2208 y Fp(\()p Fo(a)p Fp(\))p Fo(a)957 2191 y Fm(R)957 2218 y(A)1067 2208 y Fo(\027)1088 2214 y Fl(+)1115 2208 y Fp(\()p Fo(aa)1175 2191 y Fm(L)1175 2218 y(A)1202 2208 y Fp(\))g(=)g Fo(\013)1301 2191 y Fm(R)1301 2218 y(A)1328 2208 y Fp(\()p Fo(a)1366 2191 y Fm(L)1366 2218 y(A)1393 2208 y Fp(\))p Fo(\027)1430 2214 y Fl(+)1457 2208 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1533 2191 y Fm(L)1533 2218 y(A)490 2281 y Fo(\027)511 2287 y Fl(+)538 2281 y Fp(\()p Fo(a)576 2264 y Fm(R)576 2291 y(A)604 2281 y Fo(a)p Fp(\))f(=)h Fo(\013)724 2264 y Fm(R)724 2291 y(A)751 2281 y Fp(\()p Fo(a)789 2264 y Fm(R)789 2291 y(A)816 2281 y Fp(\))p Fo(a)854 2264 y Fm(R)854 2291 y(A)882 2281 y Fo(\027)903 2287 y Fl(+)930 2281 y Fp(\()p Fo(a)p Fp(\))83 b Fo(\027)1088 2287 y Fl(+)1115 2281 y Fp(\()p Fo(a)1153 2264 y Fm(L)1153 2291 y(A)1180 2281 y Fo(a)p Fp(\))12 b(=)g Fo(\013)1301 2264 y Fm(R)1301 2291 y(A)1328 2281 y Fp(\()p Fo(a)1366 2264 y Fm(L)1366 2291 y(A)1393 2281 y Fp(\))p Fo(a)1431 2264 y Fm(L)1431 2291 y(A)1458 2281 y Fo(\027)1479 2287 y Fl(+)1506 2281 y Fp(\()p Fo(a)p Fp(\))308 2370 y(2.)489 2420 y Fo(\027)510 2426 y Fj(\000)538 2420 y Fp(\()p Fo(aa)598 2403 y Fm(R)598 2430 y(A)625 2420 y Fp(\))g(=)g Fo(\013)724 2403 y Fm(R)724 2430 y(A)751 2420 y Fp(\()p Fo(a)789 2403 y Fm(R)789 2430 y(A)816 2420 y Fp(\))p Fo(\027)853 2426 y Fj(\000)881 2420 y Fp(\()p Fo(a)p Fp(\))p Fo(a)957 2403 y Fm(R)957 2430 y(A)1067 2420 y Fo(\027)1088 2426 y Fj(\000)1116 2420 y Fp(\()p Fo(aa)1176 2403 y Fm(L)1176 2430 y(A)1203 2420 y Fp(\))f(=)h Fo(\013)1301 2403 y Fm(R)1301 2430 y(A)1328 2420 y Fp(\()p Fo(a)1366 2403 y Fm(L)1366 2430 y(A)1393 2420 y Fp(\))p Fo(\027)1430 2426 y Fj(\000)1458 2420 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1534 2403 y Fm(L)1534 2430 y(A)489 2493 y Fo(\027)510 2499 y Fj(\000)538 2493 y Fp(\()p Fo(a)576 2476 y Fm(R)576 2503 y(A)603 2493 y Fo(a)p Fp(\))g(=)g Fo(\013)724 2476 y Fm(R)724 2503 y(A)751 2493 y Fp(\()p Fo(a)789 2476 y Fm(R)789 2503 y(A)816 2493 y Fp(\))p Fo(a)854 2476 y Fm(R)854 2503 y(A)881 2493 y Fo(\027)902 2499 y Fj(\000)930 2493 y Fp(\()p Fo(a)p Fp(\))83 b Fo(\027)1088 2499 y Fj(\000)1116 2493 y Fp(\()p Fo(a)1154 2476 y Fm(L)1154 2503 y(A)1181 2493 y Fo(a)p Fp(\))11 b(=)h Fo(\013)1301 2476 y Fm(R)1301 2503 y(A)1328 2493 y Fp(\()p Fo(a)1366 2476 y Fm(L)1366 2503 y(A)1393 2493 y Fp(\))p Fo(a)1431 2476 y Fm(L)1431 2503 y(A)1458 2493 y Fo(\027)1479 2499 y Fj(\000)1507 2493 y Fp(\()p Fo(a)p Fp(\))953 2628 y(40)p eop %%Page: 41 41 41 40 bop 257 262 a Fn(Pro)q(of.)36 b Fp(Observ)o(e)15 b(\014rst)f(that)f(Prop)q(osition)g(2.12)f(implies)f(that)j Fo(I)1314 268 y Fm(A)1341 262 y Fp(\()p Fo(a)1379 246 y Fm(R)1379 273 y(A)1406 262 y Fp(\))e(=)g Fo(a)1500 246 y Fm(R)1500 273 y(A)1540 262 y Fp(and)h(also)257 311 y(that)i Fo(\022)367 317 y Fm(A)395 311 y Fp(\()p Fo(a)433 296 y Fm(R)433 323 y(A)460 311 y Fp(\))e(=)g Fo(a)556 296 y Fm(R)556 323 y(A)583 311 y Fp(.)h(Since)h Fo(a)740 296 y Fm(R)740 323 y(A)782 311 y Fp(is)g(a)f(grouplik)o(e)g (elemen)o(t,)f(w)o(e)i(ha)o(v)o(e)g Fo(S)1392 296 y Fl(2)1390 323 y Fm(A)1417 311 y Fp(\()p Fo(a)1455 296 y Fm(R)1455 323 y(A)1483 311 y Fp(\))d(=)i Fo(a)1579 296 y Fm(R)1579 323 y(A)1606 311 y Fp(.)g(W)m(e)257 361 y(can)g(no)o(w)g(apply)f (Theorem)g(4.5)g(to)h(obtain:)475 447 y Fo(\027)496 453 y Fl(+)523 447 y Fp(\()p Fo(a)561 430 y Fm(R)561 457 y(A)588 447 y Fp(\))e(=)g Fo(\027)681 453 y Fl(+)708 447 y Fp(\()p Fo(I)745 429 y Fj(\000)p Fl(1)742 459 y Fm(A)790 447 y Fp(\()p Fo(a)828 430 y Fm(R)828 457 y(A)855 447 y Fp(\)\))g(=)g Fo(\013)970 430 y Fm(R)970 457 y(A)997 447 y Fp(\()p Fo(a)1035 430 y Fm(R)1035 457 y(A)1062 447 y Fp(\))p Fo(S)1105 429 y Fj(\000)p Fl(2)1103 459 y Fm(A)1151 447 y Fp(\()p Fo(a)1189 430 y Fm(R)1189 457 y(A)1216 447 y Fp(\))g(=)g Fo(\013)1315 430 y Fm(R)1315 457 y(A)1341 447 y Fp(\()p Fo(a)1379 430 y Fm(R)1379 457 y(A)1407 447 y Fp(\))p Fo(a)1445 430 y Fm(R)1445 457 y(A)257 532 y Fp(The)i(next)f(step)h(is)f(to)g(observ)o(e)h(from)d (Prop)q(osition)i(2.12)f(that)h(w)o(e)g(ha)o(v)o(e)g Fo(\033)1434 538 y Fm(A;A)1495 532 y Fp(\()p Fo(a)8 b Fk(\012)f Fo(a)1602 517 y Fm(R)1602 544 y(A)1630 532 y Fp(\))k(=)257 582 y Fo(a)279 567 y Fm(R)279 594 y(A)316 582 y Fk(\012)e Fo(a)14 b Fp(and)g Fo(\033)498 588 y Fm(A;A)559 582 y Fp(\()p Fo(a)597 567 y Fm(R)597 594 y(A)634 582 y Fk(\012)9 b Fo(a)p Fp(\))j(=)g Fo(a)d Fk(\012)h Fo(a)864 567 y Fm(R)864 594 y(A)891 582 y Fp(.)j(This)h(implies)e(b)o (y)h(Prop)q(osition)h(4.4)f(that:)492 668 y(\()p Fo(\027)529 674 y Fl(+)566 668 y Fk(\016)8 b Fo(\026)620 674 y Fm(A)648 668 y Fp(\)\()p Fo(a)h Fk(\012)h Fo(a)775 651 y Fm(R)775 678 y(A)802 668 y Fp(\))h(=)h(\()p Fo(\026)914 674 y Fm(A)951 668 y Fk(\016)d Fp(\()p Fo(\027)1018 674 y Fl(+)1054 668 y Fk(\012)h Fo(\027)1117 674 y Fl(+)1144 668 y Fp(\))f Fk(\016)g Fo(\033)1224 650 y Fj(\000)p Fl(2)1223 680 y Fm(A;A)1285 668 y Fp(\)\()p Fo(a)g Fk(\012)g Fo(a)1411 651 y Fm(R)1411 678 y(A)1439 668 y Fp(\))829 739 y(=)j(\()p Fo(\026)914 745 y Fm(A)951 739 y Fk(\016)d Fp(\()p Fo(\027)1018 745 y Fl(+)1054 739 y Fk(\012)h Fo(\027)1117 745 y Fl(+)1144 739 y Fp(\)\)\()p Fo(a)f Fk(\012)h Fo(a)1287 722 y Fm(R)1287 750 y(A)1314 739 y Fp(\))829 808 y(=)i Fo(\013)900 791 y Fm(R)900 818 y(A)927 808 y Fp(\()p Fo(a)965 791 y Fm(R)965 818 y(A)992 808 y Fp(\))p Fo(\026)1033 814 y Fm(A)1061 808 y Fp(\()p Fo(\027)1098 814 y Fl(+)1125 808 y Fp(\()p Fo(a)p Fp(\))d Fk(\012)h Fo(a)1252 791 y Fm(R)1252 818 y(A)1279 808 y Fp(\))257 894 y(This)j(pro)o(v)o(es)h(the)g(\014rst)g (equation.)e(The)h(other)h(form)o(ulas)d(can)j(b)q(e)f(obtained)g(b)o (y)g(a)g(similar)257 943 y(reasoning.)h Fg(\003)257 1058 y Fp(In)h(the)h(pro)q(of)e(of)h(the)g(analogue)f(of)h(Radford's)e(form) o(ula,)f(w)o(e)j(shall)g(need)h(another)f(com-)257 1108 y(m)o(utativit)o(y)h(result.)i(First,)g(w)o(e)g(in)o(tro)q(duce)h(an)e (abbreviation)h(for)f(the)i(righ)o(t)e(coregular)257 1158 y(action)d(of)f Fo(\013)456 1143 y Fm(L)456 1169 y(A)483 1158 y Fp(:)738 1208 y Fo(L)f Fp(:)f Fo(A)g Fk(!)h Fo(A;)j(a)d Fk(7!)f Fo(\013)1069 1190 y Fm(L)1069 1218 y(A)1095 1208 y Fp(\()p Fo(a)1133 1214 y Fl(1)1152 1208 y Fp(\))p Fo(a)1190 1214 y Fl(2)257 1279 y Fp(Observ)o(e)18 b(that)d(the)i(adjoin)o(t)d Fo(L)754 1264 y Fj(\003)789 1279 y Fp(of)h Fo(L)h Fp(is,)f(up)h(to)g(scalar)g(m)o(ultiple,)d(the)j (righ)o(t)f(coregular)257 1329 y(action)f(of)f Fo(\013)456 1314 y Fm(R)456 1340 y(A)483 1329 y Fp(.)g(T)m(o)h(see)h(this,)e(w)o(e) h(apply)f Fo(\013)931 1314 y Fm(L)931 1340 y(A)972 1329 y Fp(to)h(the)g(equation)g(in)f(Lemma)e(4.5)i(to)h(get:)603 1414 y Fo(\013)630 1397 y Fm(L)630 1425 y(A)656 1414 y Fp(\()p Fo(a)694 1420 y Fl(1)713 1414 y Fp(\))p Fo(\032)750 1420 y Fm(A)778 1414 y Fp(\()p Fo(a)816 1420 y Fl(2)834 1414 y Fo(a)856 1397 y Fj(0)868 1414 y Fp(\))e(=)f Fo(\013)966 1397 y Fm(L)966 1425 y(A)993 1414 y Fp(\()p Fo(a)1031 1397 y Fm(L)1031 1425 y(A)1058 1414 y Fp(\))p Fo(\013)1101 1397 y Fm(R)1101 1425 y(A)1128 1414 y Fp(\()p Fo(a)1166 1397 y Fj(0)1166 1425 y Fl(1)1185 1414 y Fp(\))p Fo(\032)1222 1420 y Fm(A)1249 1414 y Fp(\()p Fo(aa)1309 1397 y Fj(0)1309 1425 y Fl(2)1328 1414 y Fp(\))257 1500 y(where)18 b(w)o(e)e(ha)o(v)o(e) h(used)g(the)g Fo(H)s Fp(-linearit)o(y)e(of)h Fo(\013)1008 1485 y Fm(R)1008 1511 y(A)1051 1500 y Fp(from)e(Prop)q(osition)i(2.12.) f(This)h(means)257 1550 y(that)j Fo(\032)373 1556 y Fm(A)401 1550 y Fp(\()p Fo(aL)467 1535 y Fj(\003)486 1550 y Fp(\()p Fo(a)524 1535 y Fj(0)536 1550 y Fp(\)\))h(=)g Fo(\032)661 1556 y Fm(A)688 1550 y Fp(\()p Fo(L)p Fp(\()p Fo(a)p Fp(\))p Fo(a)808 1535 y Fj(0)821 1550 y Fp(\))f(=)h Fo(\013)935 1535 y Fm(L)935 1561 y(A)962 1550 y Fp(\()p Fo(a)1000 1535 y Fm(R)1000 1561 y(A)1027 1550 y Fp(\))p Fo(\013)1070 1535 y Fm(R)1070 1561 y(A)1097 1550 y Fp(\()p Fo(a)1135 1535 y Fj(0)1135 1560 y Fl(1)1154 1550 y Fp(\))p Fo(\032)1191 1556 y Fm(A)1219 1550 y Fp(\()p Fo(aa)1279 1535 y Fj(0)1279 1560 y Fl(2)1297 1550 y Fp(\),)f(whic)o(h)g(implies)d(that)257 1600 y Fo(L)285 1585 y Fj(\003)305 1600 y Fp(\()p Fo(a)343 1585 y Fj(0)354 1600 y Fp(\))c(=)g Fo(\013)453 1585 y Fm(L)453 1611 y(A)480 1600 y Fp(\()p Fo(a)518 1585 y Fm(R)518 1611 y(A)545 1600 y Fp(\))p Fo(\013)588 1585 y Fm(R)588 1611 y(A)615 1600 y Fp(\()p Fo(a)653 1585 y Fj(0)653 1610 y Fl(1)672 1600 y Fp(\))p Fo(a)710 1585 y Fj(0)710 1610 y Fl(2)742 1600 y Fp(b)o(y)i(the)g(nondegeneracy)i(of)d (our)h(bilinear)f(form.)257 1683 y(No)o(w)h(w)o(e)g(can)g(state)h(our)f (comm)o(utativi)o(t)o(y)d(result:)257 1779 y Fn(Lemma)36 b Fo(L)475 1764 y Fj(\003)495 1779 y Fp(\()p Fo(a)533 1764 y Fm(R)533 1790 y(A)569 1779 y Fo(a)p Fp(\))12 b(=)g Fo(\013)690 1764 y Fm(R)690 1790 y(A)717 1779 y Fp(\()p Fo(a)755 1764 y Fm(R)755 1790 y(A)782 1779 y Fp(\))d Fo(a)829 1764 y Fm(R)829 1790 y(A)857 1779 y Fo(L)885 1764 y Fj(\003)904 1779 y Fp(\()p Fo(a)p Fp(\))83 b Fo(L)1069 1764 y Fj(\003)1088 1779 y Fp(\()p Fo(aa)1148 1764 y Fm(L)1148 1790 y(A)1175 1779 y Fp(\))12 b(=)g Fo(\013)1274 1764 y Fm(R)1274 1790 y(A)1301 1779 y Fp(\()p Fo(a)1339 1764 y Fm(L)1339 1790 y(A)1366 1779 y Fp(\))d Fo(L)1419 1764 y Fj(\003)1439 1779 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1515 1764 y Fm(L)1515 1790 y(A)257 1875 y Fn(Pro)q(of.)36 b Fp(Since)15 b(a)f(scalar)g(factor)h(do)q(es)g(not)f(matter)f(in)h (the)h(ab)q(o)o(v)o(e)f(equations,)g(the)h(\014rst)257 1925 y(assertion)g(follo)o(ws)d(from)g(the)j(calculation:)319 2011 y Fo(\013)346 1994 y Fm(R)346 2021 y(A)373 2011 y Fp(\(\()p Fo(a)427 1994 y Fm(R)427 2021 y(A)463 2011 y Fo(a)p Fp(\))501 2017 y Fl(1)520 2011 y Fp(\)\()p Fo(a)574 1994 y Fm(R)574 2021 y(A)611 2011 y Fo(a)p Fp(\))649 2017 y Fl(2)679 2011 y Fp(=)d Fo(\013)750 1994 y Fm(R)750 2021 y(A)777 2011 y Fp(\()p Fo(a)815 1994 y Fm(R)815 2021 y(A)p Fl(1)858 2011 y Fp(\()p Fo(a)896 1994 y Fm(R)896 2021 y(A)p Fl(2)940 1994 y(1)970 2011 y Fk(!)f Fo(a)1045 2017 y Fl(1)1064 2011 y Fp(\)\))p Fo(a)1118 1994 y Fm(R)1118 2021 y(A)p Fl(2)1162 1994 y(2)1180 2011 y Fo(a)1202 2017 y Fl(2)1232 2011 y Fp(=)h Fo(\013)1303 1994 y Fm(R)1303 2021 y(A)1330 2011 y Fp(\()p Fo(a)1368 1994 y Fm(R)1368 2021 y(A)1395 2011 y Fp(\))p Fo(\013)1438 1994 y Fm(R)1438 2021 y(A)1466 2011 y Fp(\()p Fo(a)1504 2017 y Fl(1)1522 2011 y Fp(\))p Fo(a)1560 1994 y Fm(R)1560 2021 y(A)1588 2011 y Fo(a)1610 2017 y Fl(2)257 2096 y Fp(The)j(second)g(assertion)f (follo)o(ws)f(from)f(a)h(similar)f(calculation.)g Fg(\003)257 2229 y Fn(4.10)48 b Fp(W)m(e)13 b(no)o(w)g(pro)q(ceed)i(to)e(pro)o(v)o (e)g(the)h(main)d(result)j(of)f(this)g(section,)h(an)f(analogue)f(of) 257 2279 y(Radford's)h(form)o(ula.)e(W)m(e)j(shall)f(need)i(one)f (further)g(tec)o(hnical)h(preparation:)257 2366 y Fn(Lemma)308 2416 y Fp(1.)20 b Fo(\022)381 2401 y Fj(\003)380 2428 y Fm(A)422 2416 y Fp(comm)o(utes)12 b(with)h Fo(\022)730 2422 y Fm(A)772 2416 y Fp(and)g Fo(S)879 2401 y Fl(2)877 2428 y Fm(A)905 2416 y Fp(.)308 2497 y(2.)20 b Fo(L)389 2482 y Fj(\003)422 2497 y Fp(comm)o(utes)12 b(with)i Fo(I)730 2503 y Fm(A)757 2497 y Fp(.)953 2628 y(41)p eop %%Page: 42 42 42 41 bop 257 262 a Fn(Pro)q(of.)36 b Fp(T)m(o)13 b(pro)o(v)o(e)h(the)g (\014rst)g(statemen)o(t,)f(note)h(that)g(it)f(follo)o(ws)f(from)g(Prop) q(osition)h(4.7)257 311 y(that)h(w)o(e)g(ha)o(v)o(e:)401 403 y Fo(\022)421 386 y Fj(\003)420 413 y Fm(A)447 403 y Fp(\()p Fo(h)e Fk(!)f Fo(a)p Fp(\))h(=)f Fo(S)672 385 y Fj(\000)p Fl(2)670 415 y Fm(H)718 403 y Fp(\()p Fo(h)p Fp(\))g Fk(!)h Fo(\022)859 386 y Fj(\003)858 413 y Fm(A)885 403 y Fp(\()p Fo(a)p Fp(\))83 b Fo(\016)r Fp(\()p Fo(\022)1078 386 y Fj(\003)1077 413 y Fm(A)1105 403 y Fp(\()p Fo(a)p Fp(\)\))12 b(=)g Fo(S)1258 385 y Fj(\000)p Fl(2)1256 415 y Fm(H)1303 403 y Fp(\()p Fo(a)1341 386 y Fl(1)1360 403 y Fp(\))d Fk(\012)h Fo(\022)1447 386 y Fj(\003)1446 413 y Fm(A)1473 403 y Fp(\()p Fo(a)1511 386 y Fl(2)1530 403 y Fp(\))257 494 y(This)k(implies:)574 544 y Fo(\022)593 550 y Fm(A)620 544 y Fp(\()p Fo(\022)656 527 y Fj(\003)655 554 y Fm(A)683 544 y Fp(\()p Fo(a)p Fp(\)\))e(=)g Fo(S)836 526 y Fj(\000)p Fl(1)834 556 y Fm(H)881 544 y Fp(\()p Fo(a)919 527 y Fl(1)938 544 y Fp(\))g Fk(!)f Fo(\022)1039 527 y Fj(\003)1038 554 y Fm(A)1065 544 y Fp(\()p Fo(a)1103 527 y Fl(2)1122 544 y Fp(\))h(=)f Fo(\022)1213 527 y Fj(\003)1212 554 y Fm(A)1240 544 y Fp(\()p Fo(\022)1275 550 y Fm(A)1303 544 y Fp(\()p Fo(a)p Fp(\)\))257 619 y(T)m(o)e(pro)o(v)o(e)h(that)g Fo(\022)528 603 y Fj(\003)527 630 y Fm(A)564 619 y Fp(also)f(comm)o(utes)f(with)i Fo(S)952 603 y Fl(2)950 630 y Fm(A)977 619 y Fp(,)g(w)o(e)g(argue)g(as)f(follo)o (ws:)f(Since)i Fo(\022)i Fp(is)d(a)h(natural)257 668 y(transformation,)j Fo(\022)572 674 y Fm(A)614 668 y Fp(comm)o(utes)f(with)i Fo(S)931 653 y Fl(2)929 680 y Fm(A)971 668 y Fp(and)h(therefore)h(with)e Fo(S)1350 653 y Fl(2)1348 680 y Fm(A)1385 668 y Fk(\016)9 b Fo(\022)1434 674 y Fm(A)1462 668 y Fp(.)14 b(Adjoin)o(ts)g(of)257 718 y(comm)o(uting)i(maps)h(comm)o(ute,)e(and)k(therefore)h(w)o(e)e (see)i(b)o(y)e(Prop)q(osition)g(4.8)f(that)i Fo(\022)1663 703 y Fj(\003)1662 730 y Fm(A)257 768 y Fp(comm)o(utes)12 b(with)g Fo(S)572 750 y Fj(\000)p Fl(2)570 780 y Fm(A)625 768 y Fk(\016)c Fo(\022)674 750 y Fj(\000)p Fl(1)673 780 y Fm(A)719 768 y Fp(.)k(Since)i(w)o(e)f(already)g(kno)o(w)g(that)g Fo(\022)1273 774 y Fm(A)1313 768 y Fp(and)g Fo(\022)1413 753 y Fj(\003)1412 779 y Fm(A)1453 768 y Fp(comm)o(ute,)d(w)o(e)257 818 y(conclude)h(that)g Fo(\022)531 803 y Fj(\003)530 829 y Fm(A)567 818 y Fp(comm)o(utes)e(with)g Fo(S)876 803 y Fl(2)874 829 y Fm(A)902 818 y Fp(.)g(Finally)m(,)f(w)o(e)i(ha)o (v)o(e)g(to)g(sho)o(w)h(that)f Fo(L)1479 803 y Fj(\003)1508 818 y Fp(comm)o(utes)257 868 y(with)k Fo(I)370 874 y Fm(A)397 868 y Fp(.)g(F)m(rom)e(Prop)q(osition)h(2.12)g(w)o(e)h (conclude)h(that)e Fo(\013)1191 853 y Fm(L)1191 879 y(A)1227 868 y Fk(\016)c Fo(I)1275 874 y Fm(A)1314 868 y Fp(=)j Fo(\013)1385 853 y Fm(L)1385 879 y(A)1412 868 y Fp(:)427 959 y(\()p Fo(\013)470 942 y Fm(L)470 969 y(A)506 959 y Fk(\016)d Fo(I)554 965 y Fm(A)582 959 y Fp(\)\()p Fo(a)p Fp(\))j(=)f Fo(\023)722 965 y Fm(A)749 959 y Fp(\()p Fo(a)787 942 y Fl(1)806 959 y Fp(\))p Fo(\013)849 942 y Fm(L)849 969 y(A)876 959 y Fp(\()p Fo(g)912 965 y Fm(A)950 959 y Fk(!)g Fo(a)1025 942 y Fl(2)1044 959 y Fp(\))g(=)h Fo(\023)1130 965 y Fm(A)1157 959 y Fp(\()p Fo(a)1195 942 y Fl(1)1214 959 y Fp(\))p Fo(\013)1257 942 y Fm(L)1257 969 y(A)1283 959 y Fp(\()p Fo(a)1321 942 y Fl(2)1340 959 y Fp(\))g(=)g Fo(\013)1439 942 y Fm(L)1439 969 y(A)1465 959 y Fp(\()p Fo(a)p Fp(\))257 1050 y(This)g(implies)d Fo(L)p Fp(\()p Fo(I)550 1056 y Fm(A)578 1050 y Fp(\()p Fo(a)p Fp(\)\))j(=)f Fo(\013)730 1035 y Fm(L)730 1062 y(A)757 1050 y Fp(\()p Fo(I)791 1056 y Fm(A)818 1050 y Fp(\()p Fo(a)856 1056 y Fl(1)875 1050 y Fp(\)\))p Fo(I)925 1056 y Fm(A)953 1050 y Fp(\()p Fo(a)991 1056 y Fl(2)1009 1050 y Fp(\))h(=)g Fo(I)1099 1056 y Fm(A)1126 1050 y Fp(\()p Fo(L)p Fp(\()p Fo(a)p Fp(\)\).)g(Therefore)h Fo(L)1478 1035 y Fj(\003)1508 1050 y Fp(comm)o(utes)257 1100 y(with)h Fo(I)373 1085 y Fj(\003)370 1111 y Fm(A)409 1100 y Fp(=)e Fo(\023)468 1106 y Fm(A)494 1100 y Fp(\()p Fo(g)530 1106 y Fm(A)557 1100 y Fp(\))573 1085 y Fl(2)592 1100 y Fo(I)613 1082 y Fj(\000)p Fl(1)610 1112 y Fm(A)658 1100 y Fp(,)h(whic)o(h)h(implies)e(the)j(assertion.)f Fg(\003)257 1219 y Fp(After)h(all)e(these)i(preparations,)f(the)g (actual)g(pro)q(of)f(of)g(the)i(form)o(ula)c(is)j(easy:)257 1318 y Fn(Theorem)36 b Fp(F)m(or)13 b(all)g Fo(a)e Fk(2)h Fo(A)p Fp(,)h(w)o(e)h(ha)o(v)o(e:)486 1410 y(\()p Fo(S)529 1393 y Fl(4)527 1420 y Fm(A)564 1410 y Fk(\016)9 b Fo(\022)614 1393 y Fj(\003)613 1420 y Fm(A)649 1410 y Fk(\016)g Fo(\022)698 1416 y Fm(A)726 1410 y Fp(\)\()p Fo(a)p Fp(\))j(=)g Fo(\023)867 1416 y Fm(A)893 1410 y Fp(\()p Fo(g)930 1392 y Fj(\000)p Fl(1)929 1422 y Fm(A)975 1410 y Fp(\))p Fo(\013)1018 1393 y Fm(R)1018 1420 y(A)1045 1410 y Fp(\()p Fo(a)1083 1416 y Fl(1)1102 1410 y Fp(\))p Fo(a)1140 1393 y Fm(R)1140 1420 y(A)1167 1410 y Fo(I)1188 1393 y Fl(2)1185 1420 y Fm(A)1213 1410 y Fp(\()p Fo(a)1251 1416 y Fl(2)1269 1410 y Fp(\))p Fo(a)1307 1393 y Fm(L)1307 1420 y(A)1335 1410 y Fo(\013)1362 1393 y Fm(L)1362 1420 y(A)1388 1410 y Fp(\()p Fo(a)1426 1416 y Fl(3)1445 1410 y Fp(\))257 1509 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(ha)o(v)o(e)f(from)g(Theorem)g (4.5)g(the)h(follo)o(wing)e(form)o(ulas)f(for)j Fo(S)1437 1494 y Fl(2)1435 1521 y Fm(A)1462 1509 y Fp(:)371 1601 y Fo(S)398 1583 y Fj(\000)p Fl(2)396 1613 y Fm(A)443 1601 y Fp(\()p Fo(a)p Fp(\))e(=)g Fo(\013)580 1584 y Fm(L)580 1611 y(A)606 1601 y Fp(\()p Fo(a)644 1607 y Fl(1)663 1601 y Fp(\))p Fo(\027)700 1607 y Fl(+)727 1601 y Fp(\()p Fo(I)764 1583 y Fj(\000)p Fl(1)761 1613 y Fm(A)809 1601 y Fp(\()p Fo(a)847 1607 y Fl(2)866 1601 y Fp(\)\))83 b Fo(S)1008 1584 y Fl(2)1006 1611 y Fm(A)1034 1601 y Fp(\()p Fo(a)p Fp(\))12 b(=)f Fo(a)1165 1584 y Fm(R)1165 1611 y(A)1202 1601 y Fo(\027)1223 1607 y Fj(\000)1250 1601 y Fp(\()p Fo(I)1284 1607 y Fm(A)1312 1601 y Fp(\()p Fo(a)1350 1607 y Fl(1)1369 1601 y Fp(\)\))p Fo(a)1423 1584 y Fm(L)1423 1611 y(A)1450 1601 y Fo(\013)1477 1584 y Fm(L)1477 1611 y(A)1503 1601 y Fp(\()p Fo(a)1541 1607 y Fl(2)1560 1601 y Fp(\))257 1692 y(By)i(Prop)q(osition)f(4.9,)e(the)j (second)g(form)o(ula)d(can)i(also)g(b)q(e)g(written)h(in)f(the)h(form)d Fo(S)1566 1677 y Fl(2)1564 1703 y Fm(A)1591 1692 y Fp(\()p Fo(a)p Fp(\))i(=)257 1742 y Fo(\027)278 1748 y Fj(\000)306 1742 y Fp(\()p Fo(a)344 1727 y Fm(R)344 1753 y(A)371 1742 y Fo(I)389 1748 y Fm(A)416 1742 y Fp(\()p Fo(a)454 1748 y Fl(1)473 1742 y Fp(\))p Fo(a)511 1727 y Fm(L)511 1753 y(A)538 1742 y Fp(\))p Fo(\013)581 1727 y Fm(L)581 1753 y(A)608 1742 y Fp(\()p Fo(a)646 1748 y Fl(2)665 1742 y Fp(\).)h(This)h(implies)e(b)o(y)i(Prop)q(osition)f(4.4:)349 1833 y Fo(\032)370 1839 y Fm(A)398 1833 y Fp(\()p Fo(a)p Fp(\()p Fo(S)479 1816 y Fl(4)477 1843 y Fm(A)514 1833 y Fk(\016)c Fo(\022)564 1816 y Fj(\003)563 1843 y Fm(A)599 1833 y Fk(\016)g Fo(\022)648 1839 y Fm(A)676 1833 y Fp(\)\()p Fo(a)730 1816 y Fj(0)742 1833 y Fp(\)\))j(=)g Fo(\032)851 1839 y Fm(A)878 1833 y Fp(\()p Fo(a)p Fp(\()p Fo(\022)952 1816 y Fj(\003)951 1843 y Fm(A)988 1833 y Fk(\016)d Fo(S)1045 1816 y Fl(2)1043 1843 y Fm(A)1080 1833 y Fk(\016)g Fo(\022)1129 1839 y Fm(A)1165 1833 y Fk(\016)g Fo(S)1222 1816 y Fl(2)1220 1843 y Fm(A)1248 1833 y Fp(\)\()p Fo(a)1302 1816 y Fj(0)1314 1833 y Fp(\)\))447 1901 y(=)j Fo(\032)512 1907 y Fm(A)540 1901 y Fp(\()p Fo(\022)575 1907 y Fm(A)602 1901 y Fp(\()p Fo(a)p Fp(\)\()p Fo(S)699 1883 y Fl(2)697 1911 y Fm(A)735 1901 y Fk(\016)c Fo(\022)783 1907 y Fm(A)820 1901 y Fk(\016)h Fo(S)877 1883 y Fl(2)875 1911 y Fm(A)903 1901 y Fp(\)\()p Fo(a)957 1883 y Fj(0)969 1901 y Fp(\)\))447 1969 y(=)j Fo(\023)506 1975 y Fm(A)533 1969 y Fp(\()p Fo(g)569 1975 y Fm(A)596 1969 y Fp(\))p Fo(\013)639 1952 y Fm(R)639 1979 y(A)666 1969 y Fp(\()p Fo(a)704 1952 y Fm(R)704 1979 y(A)731 1969 y Fp(\))p Fo(\032)768 1975 y Fm(A)796 1969 y Fp(\()p Fo(S)839 1951 y Fj(\000)p Fl(2)837 1981 y Fm(A)885 1969 y Fp(\()p Fo(a)p Fp(\))p Fo(S)966 1952 y Fl(2)964 1979 y Fm(A)991 1969 y Fp(\()p Fo(a)1029 1952 y Fj(0)1041 1969 y Fp(\)\))447 2038 y(=)g Fo(\023)506 2044 y Fm(A)533 2038 y Fp(\()p Fo(g)569 2044 y Fm(A)596 2038 y Fp(\))p Fo(\013)639 2021 y Fm(R)639 2048 y(A)666 2038 y Fp(\()p Fo(a)704 2021 y Fm(R)704 2048 y(A)731 2038 y Fp(\))p Fo(\013)774 2021 y Fm(L)774 2048 y(A)801 2038 y Fp(\()p Fo(a)839 2044 y Fl(1)858 2038 y Fp(\))p Fo(\032)895 2044 y Fm(A)922 2038 y Fp(\()p Fo(\027)959 2044 y Fl(+)987 2038 y Fp(\()p Fo(I)1024 2020 y Fj(\000)p Fl(1)1021 2050 y Fm(A)1069 2038 y Fp(\()p Fo(a)1107 2044 y Fl(2)1126 2038 y Fp(\)\))p Fo(\027)1179 2044 y Fj(\000)1206 2038 y Fp(\()p Fo(a)1244 2021 y Fm(R)1244 2048 y(A)1272 2038 y Fo(I)1290 2044 y Fm(A)1317 2038 y Fp(\()p Fo(a)1355 2021 y Fj(0)1355 2048 y Fl(1)1374 2038 y Fp(\))p Fo(a)1412 2021 y Fm(L)1412 2048 y(A)1439 2038 y Fp(\)\))p Fo(\013)1498 2021 y Fm(L)1498 2048 y(A)1525 2038 y Fp(\()p Fo(a)1563 2021 y Fj(0)1563 2048 y Fl(2)1581 2038 y Fp(\))447 2106 y(=)g Fo(\023)506 2112 y Fm(A)533 2106 y Fp(\()p Fo(g)569 2112 y Fm(A)596 2106 y Fp(\))p Fo(\013)639 2089 y Fm(R)639 2116 y(A)666 2106 y Fp(\()p Fo(a)704 2089 y Fm(R)704 2116 y(A)731 2106 y Fp(\))p Fo(\013)774 2089 y Fm(L)774 2116 y(A)801 2106 y Fp(\()p Fo(a)839 2112 y Fl(1)858 2106 y Fp(\))p Fo(\032)895 2112 y Fm(A)922 2106 y Fp(\()p Fo(I)959 2088 y Fj(\000)p Fl(1)956 2118 y Fm(A)1005 2106 y Fp(\()p Fo(a)1043 2112 y Fl(2)1061 2106 y Fp(\))p Fo(a)1099 2089 y Fm(R)1099 2116 y(A)1127 2106 y Fo(I)1145 2112 y Fm(A)1172 2106 y Fp(\()p Fo(a)1210 2089 y Fj(0)1210 2116 y Fl(1)1229 2106 y Fp(\))p Fo(a)1267 2089 y Fm(L)1267 2116 y(A)1294 2106 y Fp(\))p Fo(\013)1337 2089 y Fm(L)1337 2116 y(A)1364 2106 y Fp(\()p Fo(a)1402 2089 y Fj(0)1402 2116 y Fl(2)1420 2106 y Fp(\))447 2175 y(=)g Fo(\023)506 2181 y Fm(A)533 2175 y Fp(\()p Fo(g)569 2181 y Fm(A)596 2175 y Fp(\))p Fo(\013)639 2158 y Fm(R)639 2185 y(A)666 2175 y Fp(\()p Fo(a)704 2158 y Fm(R)704 2185 y(A)731 2175 y Fp(\))p Fo(\032)768 2181 y Fm(A)796 2175 y Fp(\()p Fo(I)833 2157 y Fj(\000)p Fl(1)830 2187 y Fm(A)878 2175 y Fp(\()p Fo(L)p Fp(\()p Fo(a)p Fp(\)\))p Fo(a)1014 2158 y Fm(R)1014 2185 y(A)1042 2175 y Fo(I)1060 2181 y Fm(A)1087 2175 y Fp(\()p Fo(a)1125 2158 y Fj(0)1125 2185 y Fl(1)1144 2175 y Fp(\))p Fo(a)1182 2158 y Fm(L)1182 2185 y(A)1209 2175 y Fp(\))p Fo(\013)1252 2158 y Fm(L)1252 2185 y(A)1279 2175 y Fp(\()p Fo(a)1317 2158 y Fj(0)1317 2185 y Fl(2)1336 2175 y Fp(\))447 2243 y(=)g Fo(\023)506 2249 y Fm(A)533 2243 y Fp(\()p Fo(g)570 2226 y Fj(\000)p Fl(1)569 2256 y Fm(A)615 2243 y Fp(\))p Fo(\013)658 2226 y Fm(R)658 2254 y(A)685 2243 y Fp(\()p Fo(a)723 2226 y Fm(R)723 2254 y(A)750 2243 y Fp(\))p Fo(\032)787 2249 y Fm(A)815 2243 y Fp(\()p Fo(L)p Fp(\()p Fo(a)p Fp(\))p Fo(a)935 2226 y Fm(R)935 2254 y(A)963 2243 y Fo(I)984 2226 y Fl(2)981 2254 y Fm(A)1008 2243 y Fp(\()p Fo(a)1046 2226 y Fj(0)1046 2254 y Fl(1)1065 2243 y Fp(\))p Fo(a)1103 2226 y Fm(L)1103 2254 y(A)1130 2243 y Fp(\))p Fo(\013)1173 2226 y Fm(L)1173 2254 y(A)1200 2243 y Fp(\()p Fo(a)1238 2226 y Fj(0)1238 2254 y Fl(2)1256 2243 y Fp(\))447 2312 y(=)g Fo(\023)506 2318 y Fm(A)533 2312 y Fp(\()p Fo(g)570 2294 y Fj(\000)p Fl(1)569 2324 y Fm(A)615 2312 y Fp(\))p Fo(\013)658 2295 y Fm(R)658 2322 y(A)685 2312 y Fp(\()p Fo(a)723 2295 y Fj(0)723 2322 y Fl(1)742 2312 y Fp(\))p Fo(\032)779 2318 y Fm(A)806 2312 y Fp(\()p Fo(aa)866 2295 y Fm(R)866 2322 y(A)894 2312 y Fo(I)915 2295 y Fl(2)912 2322 y Fm(A)939 2312 y Fp(\()p Fo(a)977 2295 y Fj(0)977 2322 y Fl(2)996 2312 y Fp(\))p Fo(a)1034 2295 y Fm(L)1034 2322 y(A)1061 2312 y Fp(\))p Fo(\013)1104 2295 y Fm(L)1104 2322 y(A)1131 2312 y Fp(\()p Fo(a)1169 2295 y Fj(0)1169 2322 y Fl(3)1187 2312 y Fp(\))257 2403 y(Because)22 b(the)f(bilinear)f(form)e(\()p Fo(a;)7 b(a)840 2388 y Fj(0)851 2403 y Fp(\))23 b Fk(7!)e Fo(\032)974 2409 y Fm(A)1002 2403 y Fp(\()p Fo(aa)1062 2388 y Fj(0)1073 2403 y Fp(\))g(is)f(nondegenerate,)h(the)g(Theorem)257 2453 y(follo)o(ws.)12 b Fg(\003)953 2628 y Fp(42)p eop %%Page: 43 43 43 42 bop 257 262 a Fp(Note)14 b(that)g(if)f Fo(H)h Fp(=)e Fo(K)s Fp(,)h(the)h(base)h(\014eld,)e(then)h(Y)m(etter-Drinfel'd)f (Hopf)g(algebras)h(are)g(the)257 311 y(same)j(as)h(ordinary)f(Hopf)g (algebras.)g(In)g(this)h(case,)g Fo(\022)1135 317 y Fm(A)1162 311 y Fp(,)f Fo(\022)1211 296 y Fj(\003)1210 323 y Fm(A)1255 311 y Fp(and)g Fo(I)1357 317 y Fm(A)1402 311 y Fp(are)h(the)g(iden)o (tit)o(y)257 361 y(mappings,)12 b(and)i(the)g(ab)q(o)o(v)o(e)g(form)o (ula)d(reduces)16 b(to:)696 449 y Fo(S)723 432 y Fl(4)721 459 y Fm(A)749 449 y Fp(\()p Fo(a)p Fp(\))c(=)f Fo(\013)885 432 y Fm(R)885 459 y(A)912 449 y Fp(\()p Fo(a)950 455 y Fl(1)969 449 y Fp(\))p Fo(a)1007 432 y Fm(R)1007 459 y(A)1034 449 y Fo(a)1056 455 y Fl(2)1075 449 y Fo(a)1097 432 y Fm(L)1097 459 y(A)1124 449 y Fo(\013)1151 432 y Fm(L)1151 459 y(A)1178 449 y Fp(\()p Fo(a)1216 455 y Fl(3)1234 449 y Fp(\))257 536 y(whic)o(h)j(is)g(just)g(Radford's)f(w)o (ell)g(kno)o(wn)g(form)o(ula)f(\(cf.)h([22)o(],)g(Prop.)h(6,p.)f (347\).)257 670 y Fn(4.11)48 b Fp(There)16 b(is)e(one)h(p)q(oin)o(t)f (ab)q(out)g(the)h(ab)q(o)o(v)o(e)f(form)o(ula)e(whic)o(h)j(migh)o(t)d (app)q(ear)j(as)g(not)257 720 y(completely)d(satisfactory:)g(It)g(in)o (v)o(olv)o(es)g(the)h(adjoin)o(t)e(of)h Fo(\022)1169 726 y Fm(A)1196 720 y Fp(,)g(whic)o(h)g(is)h(not)f(written)h(do)o(wn) 257 769 y(explicitly)g(in)f(terms)h(of)g(the)h(other)g(structure)h (elemen)o(ts)e(and)g(ma)o(y)e(not)j(b)q(e)f(easily)g(acces-)257 819 y(sible.)f(W)m(e)g(no)o(w)g(pro)q(ceed)h(to)f(pro)o(v)o(e)h(a)f (second)h(v)o(ersion)f(of)g(Radford's)f(form)o(ula)f(under)j(the)257 869 y(additional)8 b(assumption)h(that)h(also)f Fo(H)j Fp(is)e(\014nite-dimensional.)d(Since)j(the)h(form)o(ula)c(w)o(e)j(are) 257 919 y(going)h(to)h(pro)o(v)o(e)g(in)o(v)o(olv)o(es)g(the)g(mo)q (dular)e(transformation,)g(whic)o(h)i(is)g(not)g(de\014ned)h(if)f Fo(H)j Fp(is)257 969 y(in\014nite-dimensional,)c(it)i(is)h(not)f(ev)o (en)i(p)q(ossible)e(to)h(write)g(do)o(wn)f(the)i(second)f(v)o(ersion)g (of)257 1019 y(the)i(form)o(ula)d(without)i(this)g(assumption.)e(Once)k (again,)c(w)o(e)j(need)g(some)f(preparations.)257 1068 y(Recall)j(the)h(Radford)f(bipro)q(duct)h(construction)h(from)d (subsection)j(3.11.)d(The)i(follo)o(w-)257 1118 y(ing)d(Prop)q(osition) h(expresses)i(the)e(mo)q(dular)e(elemen)o(ts)i(and)f(functions)h(of)f (the)h(Radford)257 1168 y(bipro)q(duct)e(in)e(terms)h(of)f(the)i (corresp)q(onding)f(elemen)o(ts)g(of)f Fo(A)h Fp(and)g Fo(H)s Fp(:)257 1265 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(H)j Fp(is)c(\014nite-dimensional.)308 1379 y(1.)20 b(The)14 b(mo)q(dular)e(functions)i(of)g(the)g(Radford)f(bipro) q(duct)i(are)f(giv)o(en)f(b)o(y:)361 1467 y Fo(\013)388 1450 y Fm(R)388 1477 y(A)p Fj(\012)p Fm(H)470 1467 y Fp(\()p Fo(a)q Fk(\012)q Fo(h)p Fp(\))g(=)e Fo(\013)665 1450 y Fm(R)665 1477 y(A)692 1467 y Fp(\()p Fo(a)p Fp(\))p Fo(\023)761 1473 y Fm(A)788 1467 y Fp(\()p Fo(h)828 1473 y Fl(1)847 1467 y Fp(\))p Fo(\013)890 1450 y Fm(R)890 1477 y(H)921 1467 y Fp(\()p Fo(h)961 1473 y Fl(2)980 1467 y Fp(\))41 b Fo(\013)1064 1450 y Fm(L)1064 1477 y(A)p Fj(\012)p Fm(H)1146 1467 y Fp(\()p Fo(a)q Fk(\012)q Fo(h)p Fp(\))13 b(=)f Fo(\013)1342 1450 y Fm(L)1342 1477 y(A)1368 1467 y Fp(\()p Fo(a)p Fp(\))p Fo(\013)1449 1450 y Fm(L)1449 1477 y(H)1481 1467 y Fp(\()p Fo(h)1521 1473 y Fl(1)1539 1467 y Fp(\))p Fo(\023)1570 1449 y Fj(\000)p Fl(1)1570 1479 y Fm(A)1615 1467 y Fp(\()p Fo(h)1655 1473 y Fl(2)1673 1467 y Fp(\))308 1570 y(2.)20 b(The)14 b(mo)q(dular)e (elemen)o(ts)i(of)f(the)i(Radford)e(bipro)q(duct)h(are)h(giv)o(en)e(b)o (y:)615 1658 y Fo(a)637 1641 y Fm(R)637 1668 y(A)p Fj(\012)p Fm(H)730 1658 y Fp(=)f Fo(a)796 1641 y Fm(R)796 1668 y(A)833 1658 y Fk(\012)d Fo(g)894 1664 y Fm(A)921 1658 y Fo(a)943 1641 y Fm(R)943 1668 y(H)1057 1658 y Fo(a)1079 1641 y Fm(L)1079 1668 y(A)p Fj(\012)p Fm(H)1173 1658 y Fp(=)j Fo(a)1239 1641 y Fm(L)1239 1668 y(A)1275 1658 y Fk(\012)e Fo(g)1338 1640 y Fj(\000)p Fl(1)1337 1670 y Fm(A)1383 1658 y Fo(a)1405 1641 y Fm(L)1405 1668 y(H)257 1755 y Fn(Pro)q(of.)36 b Fp(W)m(e)11 b(kno)o(w)f(from)g([23)o(],)g (Prop.)h(3,)f(p.)h(333)f(that)h(\003)1169 1761 y Fm(A)1199 1755 y Fk(\012)t Fp(\003)1264 1761 y Fm(H)t Fl(1)1314 1755 y Fo(\023)1329 1761 y Fm(A)1356 1755 y Fp(\(\003)1401 1761 y Fm(H)t Fl(2)1451 1755 y Fp(\))g(is)g(a)g(nonzero)257 1805 y(left)k(in)o(tegral)f(of)g Fo(A)c Fk(\012)g Fo(H)17 b Fp(if)d(\003)734 1811 y Fm(A)774 1805 y Fk(2)e Fo(A)j Fp(and)g(\003)971 1811 y Fm(H)1015 1805 y Fk(2)d Fo(H)18 b Fp(are)d(nonzero)g(left)g(in)o(tegrals.)f(Using)257 1855 y(Prop)q(osition)g(2.12,)e(w)o(e)i(no)o(w)g(calculate:)339 1942 y(\(\003)384 1948 y Fm(A)421 1942 y Fk(\012)9 b Fp(\003)491 1948 y Fm(H)t Fl(1)541 1942 y Fo(\023)556 1948 y Fm(A)583 1942 y Fp(\(\003)628 1948 y Fm(H)t Fl(2)678 1942 y Fp(\)\)\()p Fo(a)g Fk(\012)h Fo(h)p Fp(\))h(=)h(\003)923 1948 y Fm(A)950 1942 y Fp(\(\003)995 1948 y Fm(H)5 b Fl(1)1057 1942 y Fk(!)11 b Fo(a)p Fp(\))e Fk(\012)h Fp(\003)1228 1948 y Fm(H)t Fl(2)1278 1942 y Fo(h\023)1317 1948 y Fm(A)1343 1942 y Fp(\(\003)1388 1948 y Fm(H)5 b Fl(3)1438 1942 y Fp(\))850 2011 y(=)12 b Fo(\013)921 1994 y Fm(L)921 2021 y(A)948 2011 y Fp(\(\003)993 2017 y Fm(H)t Fl(1)1055 2011 y Fk(!)f Fo(a)p Fp(\)\003)1175 2017 y Fm(A)1211 2011 y Fk(\012)e Fp(\003)1281 2017 y Fm(H)c Fl(2)1331 2011 y Fo(h\023)1370 2017 y Fm(A)1397 2011 y Fp(\(\003)1442 2017 y Fm(H)t Fl(3)1492 2011 y Fp(\))850 2079 y(=)12 b Fo(\013)921 2062 y Fm(L)921 2090 y(A)948 2079 y Fp(\()p Fo(a)p Fp(\)\003)1031 2085 y Fm(A)1067 2079 y Fk(\012)e Fp(\003)1138 2085 y Fm(H)t Fl(1)1188 2079 y Fo(h\023)1227 2085 y Fm(A)1253 2079 y Fp(\(\003)1298 2085 y Fm(H)5 b Fl(2)1348 2079 y Fp(\))850 2148 y(=)12 b Fo(\013)921 2131 y Fm(L)921 2158 y(A)948 2148 y Fp(\()p Fo(a)p Fp(\)\003)1031 2154 y Fm(A)1067 2148 y Fk(\012)e Fp(\003)1138 2154 y Fm(H)t Fl(1)1188 2148 y Fo(h)1212 2154 y Fl(1)1230 2148 y Fo(\023)1245 2154 y Fm(A)1272 2148 y Fp(\(\003)1317 2154 y Fm(H)t Fl(2)1367 2148 y Fo(h)1391 2154 y Fl(2)1410 2148 y Fp(\))p Fo(\023)1441 2130 y Fj(\000)p Fl(1)1441 2160 y Fm(A)1485 2148 y Fp(\()p Fo(h)1525 2154 y Fl(3)1544 2148 y Fp(\))850 2216 y(=)i Fo(\013)921 2199 y Fm(L)921 2227 y(A)948 2216 y Fp(\()p Fo(a)p Fp(\)\003)1031 2222 y Fm(A)1067 2216 y Fk(\012)e Fp(\003)1138 2222 y Fm(H)t Fl(1)1188 2216 y Fo(\023)1203 2222 y Fm(A)1229 2216 y Fp(\(\003)1274 2222 y Fm(H)5 b Fl(2)1324 2216 y Fp(\))p Fo(\013)1367 2199 y Fm(L)1367 2227 y(H)1399 2216 y Fp(\()p Fo(h)1439 2222 y Fl(1)1457 2216 y Fp(\))p Fo(\023)1488 2199 y Fj(\000)p Fl(1)1488 2229 y Fm(A)1533 2216 y Fp(\()p Fo(h)1573 2222 y Fl(2)1591 2216 y Fp(\))257 2304 y(This)13 b(establishes)h(the)g(form)d(of)i(the)g(left)g(mo)q(dular)e(function.)h (The)i(righ)o(t)e(mo)q(dular)f(func-)257 2354 y(tion)k(ma)o(y)f(b)q(e)i (deriv)o(ed)h(b)o(y)e(a)g(similar)f(calculation)g(using)i(the)g(fact)f (that)h(\000)1472 2360 y Fm(A)1510 2354 y Fk(\012)10 b Fp(\000)1578 2360 y Fm(H)1625 2354 y Fp(is)16 b(a)257 2403 y(nonzero)j(righ)o(t)f(in)o(tegral)f(of)g(the)i(Radford)e(bipro)q (duct)i(if)e(\000)1231 2409 y Fm(A)1276 2403 y Fk(2)h Fo(A)g Fp(and)g(\000)1482 2409 y Fm(H)1532 2403 y Fk(2)g Fo(H)j Fp(are)257 2453 y(nonzero)f(righ)o(t)f(in)o(tegrals,)f(whic)o(h) h(w)o(e)g(kno)o(w)g(from)e([23)o(],)h(Prop.)h(3)f(or)h(b)o(y)g(straigh) o(tfor-)257 2503 y(w)o(ard)14 b(v)o(eri\014cation.)f(No)o(w)g(w)o(e)h (turn)g(to)g(the)g(left)f(mo)q(dular)f(elemen)o(t.)h(Select)h(nonzero)h (left)953 2628 y(43)p eop %%Page: 44 44 44 43 bop 257 262 a Fp(in)o(tegrals)18 b Fo(\025)453 268 y Fm(A)497 262 y Fk(2)f Fo(A)573 246 y Fj(\003)610 262 y Fp(and)g Fo(\025)718 268 y Fm(H)768 262 y Fk(2)g Fo(H)851 246 y Fj(\003)870 262 y Fp(.)g(Then)h(w)o(e)g(kno)o(w)f(from)e ([23)o(],)i(Prop.)g(4)g(that)h(the)257 311 y(function)696 361 y Fo(\025)720 367 y Fm(A)p Fj(\012)p Fm(H)803 361 y Fp(\()p Fo(a)9 b Fk(\012)g Fo(h)p Fp(\))j(=)g Fo(\025)1011 367 y Fm(A)1038 361 y Fp(\()p Fo(a)p Fp(\))p Fo(\025)1116 367 y Fm(H)1148 361 y Fp(\()p Fo(g)1184 367 y Fm(A)1211 361 y Fo(h)p Fp(\))257 436 y(is)i(a)g(nonzero)g(left)g(in)o(tegral)f (in)h(\()p Fo(A)9 b Fk(\012)h Fo(H)s Fp(\))913 421 y Fj(\003)932 436 y Fp(.)j(F)m(rom)f(this,)i(w)o(e)g(obtain:)460 527 y Fo(\025)484 533 y Fm(A)p Fj(\012)p Fm(H)567 527 y Fp(\(\()p Fo(a)9 b Fk(\012)g Fo(h)p Fp(\))711 533 y Fl(1)730 527 y Fp(\)\()p Fo(a)h Fk(\012)f Fo(h)p Fp(\))875 533 y Fl(2)905 527 y Fp(=)j Fo(\025)973 533 y Fm(A)1000 527 y Fp(\()p Fo(a)1038 533 y Fl(1)1057 527 y Fp(\))p Fo(\025)1097 533 y Fm(H)1129 527 y Fp(\()p Fo(g)1165 533 y Fm(A)1192 527 y Fo(a)1214 533 y Fl(2)1233 510 y(1)1251 527 y Fo(h)1275 533 y Fl(1)1294 527 y Fp(\))p Fo(a)1332 533 y Fl(2)1350 510 y(2)1378 527 y Fk(\012)e Fo(h)1444 533 y Fl(2)905 596 y Fp(=)i Fo(\025)973 602 y Fm(A)1000 596 y Fp(\()p Fo(a)p Fp(\))p Fo(\025)1078 602 y Fm(H)1110 596 y Fp(\()p Fo(g)1146 602 y Fm(A)1173 596 y Fo(h)1197 602 y Fl(1)1216 596 y Fp(\))p Fo(a)1254 579 y Fm(L)1254 606 y(A)1290 596 y Fk(\012)e Fo(h)1356 602 y Fl(2)905 664 y Fp(=)i Fo(\025)973 670 y Fm(A)1000 664 y Fp(\()p Fo(a)p Fp(\))p Fo(\025)1078 670 y Fm(H)1110 664 y Fp(\()p Fo(g)1146 670 y Fm(A)1173 664 y Fo(h)1197 670 y Fl(1)1216 664 y Fp(\))p Fo(a)1254 647 y Fm(L)1254 675 y(A)1290 664 y Fk(\012)e Fo(g)1353 647 y Fj(\000)p Fl(1)1352 676 y Fm(A)1398 664 y Fo(g)1418 670 y Fm(A)1444 664 y Fo(h)1468 670 y Fl(2)905 733 y Fp(=)i Fo(\025)973 739 y Fm(A)1000 733 y Fp(\()p Fo(a)p Fp(\))p Fo(\025)1078 739 y Fm(H)1110 733 y Fp(\()p Fo(g)1146 739 y Fm(A)1173 733 y Fo(h)p Fp(\))p Fo(a)1235 716 y Fm(L)1235 743 y(A)1272 733 y Fk(\012)d Fo(g)1334 715 y Fj(\000)p Fl(1)1333 745 y Fm(A)1379 733 y Fo(a)1401 716 y Fm(L)1401 743 y(H)257 824 y Fp(This)18 b(establishes)h(the)f(form)f(of)g(the)h(left)g(mo)q(dular)e(elemen)o (t.)h(The)h(form)e(of)h(the)i(righ)o(t)257 874 y(mo)q(dular)12 b(elemen)o(t)g(can)i(b)q(e)g(established)g(b)o(y)f(a)g(similar)d (calculation)i(using)h(the)h(fact)f(that)257 924 y Fo(\032)278 930 y Fm(A)317 924 y Fk(\012)e Fo(\032)381 930 y Fm(H)428 924 y Fk(2)k Fp(\()p Fo(A)10 b Fk(\012)h Fo(H)s Fp(\))625 909 y Fj(\003)661 924 y Fp(is)16 b(a)f(nonzero)i(righ)o(t)f(in)o (tegral)f(if)g Fo(\032)1215 930 y Fm(A)1258 924 y Fk(2)g Fo(A)1332 909 y Fj(\003)1367 924 y Fp(and)h Fo(\032)1471 930 y Fm(H)1518 924 y Fk(2)f Fo(H)1599 909 y Fj(\003)1634 924 y Fp(are)257 974 y(nonzero)g(righ)o(t)e(in)o(tegrals.)h Fg(\003)257 1092 y Fp(W)m(e)g(note)g(that)g(a)g(similar)d(form)o(ula)g (has)j(app)q(eared)h(earlier)f(in)g([5)o(],)f(Remark)f(5.9.)257 1178 y(W)m(e)i(no)o(w)f(deriv)o(e)i(the)f(\014nite-dimensional)e(form)g (of)h(the)i(form)o(ula:)257 1278 y Fn(Theorem)36 b Fp(W)m(e)13 b(ha)o(v)o(e)h(for)g(all)e Fo(a)g Fk(2)f Fo(A)p Fp(:)537 1369 y(\()p Fo(S)580 1352 y Fl(2)578 1379 y Fm(A)615 1369 y Fk(\016)e Fo(\022)664 1375 y Fm(A)692 1369 y Fp(\))708 1352 y Fl(2)726 1369 y Fp(\()p Fo(a)p Fp(\))j(=)g Fo(\013)863 1352 y Fm(R)863 1379 y(A)890 1369 y Fp(\()p Fo(a)928 1375 y Fl(1)947 1369 y Fp(\))p Fo(a)985 1352 y Fm(R)985 1379 y(A)1012 1369 y Fo(I)1030 1375 y Fm(A)1057 1369 y Fp(\()p Fo(M)1118 1352 y Fm(R)1113 1379 y(A)1145 1369 y Fp(\()p Fo(a)1183 1375 y Fl(2)1202 1369 y Fp(\)\))p Fo(a)1256 1352 y Fm(L)1256 1379 y(A)1283 1369 y Fo(\013)1310 1352 y Fm(L)1310 1379 y(A)1337 1369 y Fp(\()p Fo(a)1375 1375 y Fl(3)1394 1369 y Fp(\))257 1468 y Fn(Pro)q(of.)36 b Fp(By)14 b(Radford's)f(form)o(ula)f(for)h(ordinary)h(Hopf)f(algebras) h(w)o(e)g(ha)o(v)o(e:)367 1560 y Fo(S)394 1543 y Fl(4)392 1570 y Fm(A)p Fj(\012)p Fm(H)475 1560 y Fp(\()p Fo(a)9 b Fk(\012)h Fp(1\))h(=)h Fo(\013)683 1543 y Fm(R)683 1570 y(A)p Fj(\012)p Fm(H)765 1560 y Fp(\(\()p Fo(a)d Fk(\012)h Fp(1\))907 1566 y Fl(1)925 1560 y Fp(\))p Fo(a)963 1543 y Fm(R)963 1570 y(A)p Fj(\012)p Fm(H)1046 1560 y Fp(\()p Fo(a)f Fk(\012)h Fp(1\))1172 1566 y Fl(2)1190 1560 y Fo(a)1212 1543 y Fm(L)1212 1570 y(A)p Fj(\012)p Fm(H)1294 1560 y Fo(\013)1321 1543 y Fm(L)1321 1570 y(A)p Fj(\012)p Fm(H)1403 1560 y Fp(\(\()p Fo(a)g Fk(\012)f Fp(1\))1545 1566 y Fl(3)1564 1560 y Fp(\))257 1651 y(W)m(e)k(can)g (calculate)g(the)g(left)g(hand)g(side)g(b)o(y)g(Prop)q(osition)f(3.11)g (and)g(the)i(righ)o(t)e(hand)h(side)257 1701 y(b)o(y)h(the)h(preceding) f(Prop)q(osition.)f(Using)h(the)h(form)o(ula)471 1792 y(\001)506 1775 y Fl(2)506 1803 y Fm(A)p Fj(\012)p Fm(H)587 1792 y Fp(\()p Fo(a)10 b Fk(\012)f Fp(1\))j(=)g(\()p Fo(a)807 1798 y Fl(1)834 1792 y Fk(\012)e Fo(a)898 1798 y Fl(2)917 1775 y(1)935 1792 y Fo(a)957 1798 y Fl(3)976 1775 y(1)994 1792 y Fp(\))g Fk(\012)f Fp(\()p Fo(a)1099 1798 y Fl(2)1118 1775 y(2)1146 1792 y Fk(\012)g Fo(a)1209 1798 y Fl(3)1228 1775 y(2)1247 1792 y Fp(\))g Fk(\012)g Fp(\()p Fo(a)1351 1798 y Fl(3)1370 1775 y(3)1398 1792 y Fk(\012)g Fp(1\))257 1884 y(for)14 b(the)g(t)o(w)o(ofold)f(copro)q (duct,)h(w)o(e)g(arriv)o(e)g(at:)267 1975 y Fo(S)294 1958 y Fl(4)292 1985 y Fm(A)319 1975 y Fp(\()p Fo(\022)355 1958 y Fl(2)354 1985 y Fm(A)382 1975 y Fp(\()p Fo(a)p Fp(\)\))9 b Fk(\012)h Fp(1)278 2043 y(=)i Fo(\013)349 2026 y Fm(R)349 2054 y(A)376 2043 y Fp(\()p Fo(a)414 2049 y Fl(1)433 2043 y Fp(\))p Fo(\023)464 2049 y Fm(A)491 2043 y Fp(\()p Fo(a)529 2049 y Fl(2)547 2026 y(1)566 2043 y Fo(a)588 2049 y Fl(3)606 2026 y(1)625 2043 y Fp(\))p Fo(\013)668 2026 y Fm(R)668 2054 y(H)699 2043 y Fp(\()p Fo(a)737 2049 y Fl(2)756 2026 y(2)775 2043 y Fo(a)797 2049 y Fl(3)815 2026 y(2)834 2043 y Fp(\)\()p Fo(a)888 2026 y Fm(R)888 2054 y(A)924 2043 y Fk(\012)e Fo(g)986 2049 y Fm(A)1013 2043 y Fo(a)1035 2026 y Fm(R)1035 2054 y(H)1066 2043 y Fp(\)\()p Fo(a)1120 2049 y Fl(2)1139 2026 y(3)1167 2043 y Fk(\012)f Fo(a)1230 2049 y Fl(3)1249 2026 y(3)1268 2043 y Fp(\)\()p Fo(a)1322 2026 y Fm(L)1322 2054 y(A)1358 2043 y Fk(\012)h Fo(g)1421 2026 y Fj(\000)p Fl(1)1420 2056 y Fm(A)1466 2043 y Fo(a)1488 2026 y Fm(L)1488 2054 y(H)1519 2043 y Fp(\))p Fo(\013)1562 2026 y Fm(L)1562 2054 y(A)1589 2043 y Fp(\()p Fo(a)1627 2049 y Fl(3)1645 2026 y(4)1664 2043 y Fp(\))278 2112 y(=)i Fo(\013)349 2095 y Fm(R)349 2122 y(A)376 2112 y Fp(\()p Fo(a)414 2118 y Fl(1)433 2112 y Fp(\))p Fo(\023)464 2118 y Fm(A)491 2112 y Fp(\()p Fo(a)529 2118 y Fl(2)547 2095 y(1)566 2112 y Fp(\))p Fo(\013)609 2095 y Fm(R)609 2122 y(H)640 2112 y Fp(\()p Fo(a)678 2118 y Fl(2)697 2095 y(2)715 2112 y Fp(\)\()p Fo(a)769 2095 y Fm(R)769 2122 y(A)806 2112 y Fk(\012)e Fo(g)868 2118 y Fm(A)894 2112 y Fo(a)916 2095 y Fm(R)916 2122 y(H)948 2112 y Fp(\)\()p Fo(a)1002 2118 y Fl(2)1021 2095 y(3)1049 2112 y Fk(\012)f Fp(1\)\()p Fo(a)1165 2095 y Fm(L)1165 2122 y(A)1201 2112 y Fk(\012)h Fo(g)1264 2094 y Fj(\000)p Fl(1)1263 2124 y Fm(A)1309 2112 y Fo(a)1331 2095 y Fm(L)1331 2122 y(H)1362 2112 y Fp(\))p Fo(\013)1405 2095 y Fm(L)1405 2122 y(A)1432 2112 y Fp(\()p Fo(a)1470 2118 y Fl(3)1489 2112 y Fp(\))278 2181 y(=)i Fo(\013)349 2163 y Fm(R)349 2191 y(A)376 2181 y Fp(\()p Fo(a)414 2187 y Fl(1)433 2181 y Fp(\))p Fo(\023)464 2187 y Fm(A)491 2181 y Fp(\()p Fo(a)529 2187 y Fl(2)547 2163 y(1)566 2181 y Fp(\))p Fo(\013)609 2163 y Fm(R)609 2191 y(H)640 2181 y Fp(\()p Fo(a)678 2187 y Fl(2)697 2163 y(2)715 2181 y Fp(\)\()p Fo(a)769 2163 y Fm(R)769 2191 y(A)797 2181 y Fp(\()p Fo(g)833 2187 y Fm(A)860 2181 y Fo(a)882 2163 y Fm(R)882 2191 y(H)925 2181 y Fk(!)f Fo(a)1000 2187 y Fl(2)1018 2163 y(3)1037 2181 y Fp(\))p Fo(a)1075 2163 y Fm(L)1075 2191 y(A)1111 2181 y Fk(\012)f Fp(1\))p Fo(\013)1217 2163 y Fm(L)1217 2191 y(A)1243 2181 y Fp(\()p Fo(a)1281 2187 y Fl(3)1300 2181 y Fp(\))278 2249 y(=)i Fo(\013)349 2232 y Fm(R)349 2259 y(A)376 2249 y Fp(\()p Fo(a)414 2255 y Fl(1)433 2249 y Fp(\)\()p Fo(a)487 2232 y Fm(R)487 2259 y(A)514 2249 y Fo(I)532 2255 y Fm(A)559 2249 y Fp(\()p Fo(M)620 2232 y Fm(R)615 2259 y(A)648 2249 y Fp(\()p Fo(a)686 2255 y Fl(2)704 2249 y Fp(\)\))p Fo(a)758 2232 y Fm(L)758 2259 y(A)795 2249 y Fk(\012)d Fp(1\))p Fo(\013)900 2232 y Fm(L)900 2259 y(A)927 2249 y Fp(\()p Fo(a)965 2255 y Fl(3)984 2249 y Fp(\))257 2340 y(The)15 b(consideration)f(of)f(the)h Fo(A)p Fp(-comp)q(onen)o(t)g (yields)f(the)i(assertion.)f Fg(\003)953 2628 y Fp(44)p eop %%Page: 45 45 45 44 bop 257 262 a Fn(4.12)48 b Fp(A)12 b(comparison)f(b)q(et)o(w)o (een)i(b)q(oth)f(v)n(arian)o(ts)g(of)f(Radford's)g(form)o(ulas)f(that)i (w)o(e)h(ha)o(v)o(e)257 311 y(pro)o(v)o(ed)k(so)f(far)g(yields)f(in)o (teresting)i(in)o(terrelations)f(b)q(et)o(w)o(een)i(the)e(natural)g (transforma-)257 361 y(tions)e(that)g(w)o(e)g(are)g(considering.)257 452 y Fn(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(H)j Fp(is)c(\014nite-dimensional.)f(Then)i(w)o(e)g(ha)o(v)o(e:)308 553 y(1.)20 b Fo(\032)382 559 y Fm(A)410 553 y Fp(\()p Fo(\022)445 559 y Fm(A)472 553 y Fp(\()p Fo(M)533 538 y Fm(L)528 564 y(A)558 553 y Fp(\()p Fo(a)p Fp(\)\))p Fo(a)650 538 y Fj(0)662 553 y Fp(\))12 b(=)g Fo(\023)749 559 y Fm(A)775 553 y Fp(\()p Fo(g)812 535 y Fj(\000)p Fl(1)811 565 y Fm(A)857 553 y Fp(\))p Fo(\032)894 559 y Fm(A)922 553 y Fp(\()p Fo(a\022)979 559 y Fm(A)1007 553 y Fp(\()p Fo(I)1041 559 y Fm(A)1068 553 y Fp(\()p Fo(a)1106 538 y Fj(0)1118 553 y Fp(\)\)\))308 630 y(2.)20 b Fo(\022)381 615 y Fj(\003)380 641 y Fm(A)419 630 y Fp(=)12 b Fo(\023)478 636 y Fm(A)505 630 y Fp(\()p Fo(g)542 612 y Fj(\000)p Fl(1)541 642 y Fm(A)587 630 y Fp(\))p Fo(\027)624 636 y Fl(+)660 630 y Fk(\016)d Fo(\022)710 612 y Fj(\000)p Fl(1)709 642 y Fm(A)765 630 y Fk(\016)g Fo(\027)819 612 y Fj(\000)p Fl(1)816 640 y Fj(\000)874 630 y Fp(=)j Fo(\023)933 636 y Fm(A)959 630 y Fp(\()p Fo(g)996 612 y Fj(\000)p Fl(1)995 642 y Fm(A)1041 630 y Fp(\))p Fo(\022)1076 636 y Fm(A)1113 630 y Fk(\016)d Fo(M)1188 615 y Fm(L)1183 641 y(A)1222 630 y Fk(\016)g Fo(I)1270 636 y Fm(A)257 721 y Fn(Pro)q(of.)36 b Fp(By)19 b(comparing)e(the)i(t)o(w)o(o)f(form)o(ulas)e(for)j Fo(S)1129 706 y Fl(4)1127 732 y Fm(A)1173 721 y Fp(in)f(Theorem)g(4.10)f(and)h (Theo-)257 771 y(rem)c(4.11,)e(w)o(e)i(see)h(that:)423 847 y(\()p Fo(\022)459 830 y Fj(\003)458 858 y Fm(A)495 847 y Fk(\016)9 b Fo(\022)545 830 y Fj(\000)p Fl(1)544 860 y Fm(A)590 847 y Fp(\)\()p Fo(\013)649 830 y Fm(R)649 858 y(A)676 847 y Fp(\()p Fo(a)714 853 y Fl(1)733 847 y Fp(\))p Fo(a)771 830 y Fm(R)771 858 y(A)798 847 y Fo(I)816 853 y Fm(A)843 847 y Fp(\()p Fo(M)904 830 y Fm(R)899 858 y(A)931 847 y Fp(\()p Fo(a)969 853 y Fl(2)988 847 y Fp(\)\))p Fo(a)1042 830 y Fm(L)1042 858 y(A)1069 847 y Fp(\))p Fo(\013)1112 830 y Fm(L)1112 858 y(A)1139 847 y Fp(\()p Fo(a)1177 853 y Fl(3)1196 847 y Fp(\))871 916 y(=)j Fo(\023)930 922 y Fm(A)957 916 y Fp(\()p Fo(g)994 898 y Fj(\000)p Fl(1)993 928 y Fm(A)1039 916 y Fp(\))p Fo(\013)1082 899 y Fm(R)1082 926 y(A)1109 916 y Fp(\()p Fo(a)1147 922 y Fl(1)1165 916 y Fp(\))p Fo(a)1203 899 y Fm(R)1203 926 y(A)1231 916 y Fo(I)1252 899 y Fl(2)1249 926 y Fm(A)1276 916 y Fp(\()p Fo(a)1314 922 y Fl(2)1333 916 y Fp(\))p Fo(a)1371 899 y Fm(L)1371 926 y(A)1398 916 y Fo(\013)1425 899 y Fm(L)1425 926 y(A)1451 916 y Fp(\()p Fo(a)1489 922 y Fl(3)1508 916 y Fp(\))257 997 y(Inserting)j Fo(\013)459 982 y Fm(L)459 1008 y(A)485 997 y Fp(\()p Fo(a)523 1003 y Fl(1)542 997 y Fp(\))p Fo(a)580 1003 y Fl(2)599 997 y Fo(\013)626 982 y Fm(R)626 1008 y(A)653 997 y Fp(\()p Fo(a)691 1003 y Fl(3)709 997 y Fp(\))f(in)g(place)g(of)f Fo(a)p Fp(,)h(w)o(e)g(get:)518 1074 y(\()p Fo(\022)554 1056 y Fj(\003)553 1084 y Fm(A)590 1074 y Fk(\016)9 b Fo(\022)640 1056 y Fj(\000)p Fl(1)639 1086 y Fm(A)685 1074 y Fp(\)\()p Fo(a)739 1056 y Fm(R)739 1084 y(A)767 1074 y Fo(I)785 1080 y Fm(A)812 1074 y Fp(\()p Fo(M)873 1056 y Fm(R)868 1084 y(A)900 1074 y Fp(\()p Fo(a)p Fp(\)\))p Fo(a)992 1056 y Fm(L)992 1084 y(A)1020 1074 y Fp(\))i(=)h Fo(\023)1106 1080 y Fm(A)1133 1074 y Fp(\()p Fo(g)1170 1056 y Fj(\000)p Fl(1)1169 1086 y Fm(A)1215 1074 y Fp(\))p Fo(a)1253 1056 y Fm(R)1253 1084 y(A)1280 1074 y Fo(I)1301 1056 y Fl(2)1298 1084 y Fm(A)1325 1074 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1401 1056 y Fm(L)1401 1084 y(A)257 1150 y Fp(Since)17 b Fo(I)386 1156 y Fm(A)430 1150 y Fp(and)g Fo(M)559 1135 y Fm(R)554 1162 y(A)602 1150 y Fp(are)g(algebra)f(homomorphism)o(s)e(b)o(y)i(Prop)q(osition)g (3.10,)f(it)h(follo)o(ws)257 1200 y(easily)10 b(from)d(Prop)q(osition)j (2.12)e(that)i(w)o(e)g(ha)o(v)o(e)f(for)g(example)g Fo(I)1234 1206 y Fm(A)1261 1200 y Fp(\()p Fo(a)1299 1185 y Fm(L)1299 1212 y(A)1326 1200 y Fo(a)p Fp(\))j(=)g Fo(I)1438 1206 y Fm(A)1465 1200 y Fp(\()p Fo(a)1503 1185 y Fm(L)1503 1212 y(A)1530 1200 y Fp(\))p Fo(I)1564 1206 y Fm(A)1591 1200 y Fp(\()p Fo(a)p Fp(\))g(=)257 1250 y Fo(a)279 1235 y Fm(L)279 1261 y(A)306 1250 y Fo(I)324 1256 y Fm(A)352 1250 y Fp(\()p Fo(a)p Fp(\).)18 b(Therefore,)h(inserting)f Fo(a)837 1235 y Fm(L)837 1261 y(A)864 1250 y Fo(aa)908 1235 y Fm(R)908 1261 y(A)954 1250 y Fp(for)g Fo(a)p Fp(,)g(w)o(e)g(get) h(\()p Fo(\022)1249 1235 y Fj(\003)1248 1261 y Fm(A)1288 1250 y Fk(\016)12 b Fo(\022)1341 1232 y Fj(\000)p Fl(1)1340 1262 y Fm(A)1386 1250 y Fp(\)\()p Fo(I)1436 1256 y Fm(A)1464 1250 y Fp(\()p Fo(M)1525 1235 y Fm(R)1520 1261 y(A)1552 1250 y Fp(\()p Fo(a)p Fp(\)\)\))19 b(=)257 1300 y Fo(\023)272 1306 y Fm(A)299 1300 y Fp(\()p Fo(g)336 1282 y Fj(\000)p Fl(1)335 1312 y Fm(A)381 1300 y Fp(\))p Fo(I)418 1285 y Fl(2)415 1311 y Fm(A)443 1300 y Fp(\()p Fo(a)p Fp(\).)12 b(It)i(is)f(easy)g(to)h(deduce)g(from)e(Theorem)g(2.11)g(and)h(Prop)q (osition)g(3.6)f(that)257 1350 y Fo(I)275 1356 y Fm(A)320 1350 y Fp(comm)o(utes)j(with)i Fo(M)661 1335 y Fm(R)656 1361 y(A)705 1350 y Fp(and,)f(since)i Fo(\022)924 1356 y Fm(A)968 1350 y Fp(comm)o(utes)e(with)g Fo(\022)1284 1335 y Fj(\003)1283 1361 y Fm(A)1328 1350 y Fp(b)o(y)h(Lemma)d(4.10,)h (w)o(e)257 1399 y(get:)f Fo(\022)358 1384 y Fj(\003)357 1411 y Fm(A)394 1399 y Fk(\016)9 b Fo(M)469 1384 y Fm(R)464 1411 y(A)507 1399 y Fp(=)j Fo(\023)566 1405 y Fm(A)593 1399 y Fp(\()p Fo(g)630 1382 y Fj(\000)p Fl(1)629 1412 y Fm(A)675 1399 y Fp(\))p Fo(\022)710 1405 y Fm(A)747 1399 y Fk(\016)d Fo(I)795 1405 y Fm(A)822 1399 y Fp(.)k(This)h (implies:)570 1476 y Fo(\032)591 1482 y Fm(A)618 1476 y Fp(\()p Fo(\022)653 1482 y Fm(A)681 1476 y Fp(\()p Fo(a)p Fp(\))p Fo(M)780 1459 y Fm(R)775 1486 y(A)807 1476 y Fp(\()p Fo(a)845 1459 y Fj(0)857 1476 y Fp(\)\))e(=)g Fo(\023)960 1482 y Fm(A)986 1476 y Fp(\()p Fo(g)1023 1458 y Fj(\000)p Fl(1)1022 1488 y Fm(A)1068 1476 y Fp(\))p Fo(\032)1105 1482 y Fm(A)1133 1476 y Fp(\()p Fo(a\022)1190 1482 y Fm(A)1218 1476 y Fp(\()p Fo(I)1252 1482 y Fm(A)1279 1476 y Fp(\()p Fo(a)1317 1459 y Fj(0)1329 1476 y Fp(\)\)\))257 1553 y(whic)o(h)i(implies)e(the)j(\014rst)f(assertion)h(b)o(y)e(Prop)q (osition)h(4.8.)257 1632 y(T)m(o)g(pro)o(v)o(e)g(the)g(second)h (assertion,)f(w)o(e)h(observ)o(e)g(that,)e(since)i Fo(\022)h Fp(is)e(a)f(ribb)q(on)h(transforma-)257 1682 y(tion,)f(w)o(e)h(ha)o(v)o (e)g(as)g(in)f(subsection)j(3.10:)559 1759 y Fo(\022)578 1765 y Fm(A)615 1759 y Fk(\016)9 b Fo(\026)670 1765 y Fm(A)708 1759 y Fp(=)j Fo(\026)777 1765 y Fm(A)813 1759 y Fk(\016)d Fo(\022)862 1765 y Fm(A)p Fj(\012)p Fm(A)952 1759 y Fp(=)j Fo(\026)1021 1765 y Fm(A)1057 1759 y Fk(\016)d Fp(\()p Fo(\022)1122 1765 y Fm(A)1159 1759 y Fk(\012)h Fo(\022)1220 1765 y Fm(A)1247 1759 y Fp(\))g Fk(\016)e Fo(\033)1327 1741 y Fj(\000)p Fl(2)1326 1771 y Fm(A;A)257 1835 y Fp(where)14 b Fo(\026)401 1841 y Fm(A)439 1835 y Fp(:)d Fo(A)6 b Fk(\012)g Fo(A)13 b Fk(!)e Fo(A)h Fp(denotes)h(the)g (m)o(ultiplication)c(map.)h(Since)j(w)o(e)g(kno)o(w)e(from)g(the)257 1885 y(pro)q(of)j(of)f(Prop)q(osition)h(4.8)e(that)i(w)o(e)h(ha)o(v)o (e)e Fo(\032)971 1891 y Fm(A)1008 1885 y Fk(\016)c Fo(\022)1057 1891 y Fm(A)1096 1885 y Fp(=)j Fo(\023)1155 1891 y Fm(A)1182 1885 y Fp(\()p Fo(g)1219 1867 y Fj(\000)p Fl(1)1218 1897 y Fm(A)1264 1885 y Fp(\))p Fo(\032)1301 1891 y Fm(A)1328 1885 y Fp(,)i(w)o(e)g(ha)o(v)o(e:)504 1962 y Fo(\032)525 1968 y Fm(A)561 1962 y Fk(\016)9 b Fo(\026)616 1968 y Fm(A)653 1962 y Fk(\016)g Fo(\033)708 1944 y Fj(\000)p Fl(1)707 1974 y Fm(A;A)777 1962 y Fk(\016)g Fp(\()p Fo(\022)842 1968 y Fm(A)879 1962 y Fk(\012)h Fo(\022)940 1968 y Fm(A)967 1962 y Fp(\))i(=)g Fo(\023)1054 1968 y Fm(A)1081 1962 y Fp(\()p Fo(g)1118 1944 y Fj(\000)p Fl(1)1117 1974 y Fm(A)1163 1962 y Fp(\))p Fo(\032)1200 1968 y Fm(A)1236 1962 y Fk(\016)d Fo(\026)1291 1968 y Fm(A)1328 1962 y Fk(\016)g Fo(\033)1382 1968 y Fm(A;A)257 2039 y Fp(But)k(this)f (implies)f(b)o(y)h(the)h(de\014nition)e(of)h(the)h(t)o(wisted)g(Nak)n (a)o(y)o(ama)c(automorphisms)g(that)257 2089 y(w)o(e)14 b(ha)o(v)o(e:)395 2165 y Fo(\032)416 2171 y Fm(A)453 2165 y Fk(\016)9 b Fo(\026)508 2171 y Fm(A)544 2165 y Fk(\016)g Fp(\()p Fo(id)626 2171 y Fm(A)662 2165 y Fk(\012)h Fo(\027)725 2171 y Fj(\000)752 2165 y Fp(\))g Fk(\016)f Fp(\()p Fo(\022)843 2171 y Fm(A)880 2165 y Fk(\012)g Fo(\022)940 2171 y Fm(A)968 2165 y Fp(\))i(=)h Fo(\023)1054 2171 y Fm(A)1081 2165 y Fp(\()p Fo(g)1118 2147 y Fj(\000)p Fl(1)1117 2177 y Fm(A)1163 2165 y Fp(\))p Fo(\032)1200 2171 y Fm(A)1237 2165 y Fk(\016)d Fo(\026)1292 2171 y Fm(A)1328 2165 y Fk(\016)g Fp(\()p Fo(id)1410 2171 y Fm(A)1446 2165 y Fk(\012)h Fo(\027)1509 2171 y Fl(+)1536 2165 y Fp(\))257 2242 y(Since)k(the)g(left)f(hand)g(side)g(is)g(also)g Fo(\032)838 2248 y Fm(A)873 2242 y Fk(\016)7 b Fo(\026)926 2248 y Fm(A)961 2242 y Fk(\016)h Fp(\()p Fo(id)1042 2248 y Fm(A)1076 2242 y Fk(\012)g Fo(\022)1136 2227 y Fj(\003)1135 2253 y Fm(A)1163 2242 y Fp(\))g Fk(\016)f Fp(\()p Fo(id)1267 2248 y Fm(A)1302 2242 y Fk(\012)h Fo(\027)1363 2248 y Fj(\000)1391 2242 y Fp(\))g Fk(\016)f Fp(\()p Fo(id)1495 2248 y Fm(A)1530 2242 y Fk(\012)h Fo(\022)1589 2248 y Fm(A)1616 2242 y Fp(\))14 b(b)o(y)257 2292 y(the)h(de\014nition)e(of)h (adjoin)o(t)e(maps,)g(w)o(e)j(get)f(b)o(y)f(nondegeneracy)j(that:)742 2368 y Fo(\023)757 2374 y Fm(A)784 2368 y Fp(\()p Fo(g)821 2351 y Fj(\000)p Fl(1)820 2381 y Fm(A)866 2368 y Fp(\))p Fo(\027)903 2374 y Fl(+)941 2368 y Fp(=)c Fo(\022)1005 2351 y Fj(\003)1004 2379 y Fm(A)1041 2368 y Fk(\016)d Fo(\027)1092 2374 y Fj(\000)1129 2368 y Fk(\016)g Fo(\022)1178 2374 y Fm(A)257 2445 y Fp(This)20 b(implies)d(the)j(\014rst)g(equalit)o (y)m(.)e(W)m(e)h(ha)o(v)o(e)g(already)g(seen)i(ab)q(o)o(v)o(e)e(a)g(v)n (arian)o(t)f(of)h(the)257 2495 y(second)c(equalit)o(y)m(.)d Fg(\003)953 2628 y Fp(45)p eop %%Page: 46 46 46 45 bop 257 262 a Fq(5)67 b(In)n(tegrals)23 b(and)g(triangular)h (decomp)r(ositions)257 388 y Fn(5.1)48 b Fp(In)18 b(this)g(section,)g Fo(A)g Fp(\(resp.)h Fo(B)r Fp(\))f(is)g(a)g(\014xed)g(left)f(\(resp.)i (righ)o(t\))f(Y)m(etter-Drinfel'd)257 438 y(bialgebra)i(o)o(v)o(er)g(a) f(bialgebra)h Fo(H)s Fp(.)f(\001)870 444 y Fm(A)917 438 y Fp(\(resp.)h(\001)1073 444 y Fm(B)1102 438 y Fp(\))g(and)g Fo(\017)1242 444 y Fm(A)1289 438 y Fp(\(resp.)g Fo(\017)1427 444 y Fm(B)1456 438 y Fp(\))g(denote)h(the)257 488 y(com)o (ultiplication)11 b(and)j(the)g(counit.)g(Recall)f(the)h(\014rst)h (construction)g(of)e([31)o(]:)257 581 y Fn(De\014nition)257 631 y Fp(A)g(pair)e(\()p Fo(A;)c(B)r Fp(\))13 b(consisting)f(of)g(a)g (left)g(and)g(a)g(righ)o(t)f(Y)m(etter-Drinfel'd)h(bialgebra)f (together)257 681 y(with)16 b(linear)f(mappings)f Fo(*)p Fp(:)g Fo(B)f Fk(\012)d Fo(A)15 b Fk(!)f Fo(A)p Fp(,)h Fo(\()p Fp(:)f Fo(B)f Fk(\012)e Fo(A)k Fk(!)f Fo(B)k Fp(and)e Fo(])e Fp(:)g Fo(B)f Fk(\012)e Fo(A)k Fk(!)f Fo(H)19 b Fp(is)257 730 y(called)14 b(a)g(Y)m(etter-Drinfel'd)f (bialgebra)g(pair)h(if:)308 849 y(1.)20 b Fo(A)14 b Fp(is)g(a)f(left)h Fo(B)r Fp(-mo)q(dule)f(via)g Fo(*)p Fp(.)308 932 y(2.)20 b Fo(B)d Fp(is)c(a)h(righ)o(t)f Fo(A)p Fp(-mo)q(dule)g(via)g Fo(\()p Fp(.)257 1051 y(and)h(the)h(follo)o(wing)c(compatibilit)o(y)g (conditions)i(are)h(satis\014ed:)308 1178 y(1.)20 b(\001)396 1184 y Fm(A)423 1178 y Fp(\()p Fo(b)11 b(*)g(a)p Fp(\))h(=)g(\()p Fo(b)649 1184 y Fl(1)667 1159 y(1)697 1178 y Fo(*)g(a)773 1184 y Fl(1)791 1178 y Fp(\))d Fk(\012)h Fp(\()p Fo(b)892 1184 y Fl(1)911 1159 y(2)941 1178 y Fk(!)h Fp(\()p Fo(b)1028 1184 y Fl(2)1058 1178 y Fo(*)g(a)1133 1184 y Fl(2)1151 1178 y Fp(\)\))361 1244 y(\001)396 1250 y Fm(B)424 1244 y Fp(\()p Fo(b)h(\()f(a)p Fp(\))g(=)h(\(\()p Fo(b)666 1250 y Fl(1)697 1244 y Fo(\()f(a)772 1250 y Fl(1)790 1244 y Fp(\))h Fk( )f Fo(a)893 1250 y Fl(2)911 1229 y(1)930 1244 y Fp(\))e Fk(\012)h Fp(\()p Fo(b)1031 1250 y Fl(2)1061 1244 y Fo(\()h(a)1136 1250 y Fl(2)1155 1229 y(2)1173 1244 y Fp(\))308 1327 y(2.)20 b(\001)396 1333 y Fm(H)427 1327 y Fp(\()p Fo(b]a)p Fp(\))12 b(=)g(\()p Fo(b)605 1333 y Fl(1)623 1309 y(1)642 1327 y Fo(]a)680 1333 y Fl(1)699 1327 y Fp(\))p Fo(a)737 1333 y Fl(2)755 1312 y(1)783 1327 y Fk(\012)e Fo(b)843 1333 y Fl(1)861 1309 y(2)880 1327 y Fp(\()p Fo(b)914 1333 y Fl(2)932 1327 y Fo(]a)970 1333 y Fl(2)989 1312 y(2)1008 1327 y Fp(\))308 1411 y(3.)20 b Fo(b)11 b(*)h Fp(\()p Fo(aa)504 1395 y Fj(0)515 1411 y Fp(\))g(=)g(\()p Fo(b)621 1417 y Fl(1)639 1392 y(1)669 1411 y Fo(*)f(a)744 1417 y Fl(1)763 1411 y Fp(\)\()p Fo(b)813 1417 y Fl(1)832 1392 y(2)850 1411 y Fp(\()p Fo(b)884 1417 y Fl(2)903 1411 y Fo(]a)941 1417 y Fl(2)960 1411 y Fp(\))p Fo(a)998 1417 y Fl(3)1016 1395 y(1)1047 1411 y Fk(!)g Fp([\()p Fo(b)1146 1417 y Fl(3)1175 1411 y Fo(\()g(a)1250 1417 y Fl(3)1269 1395 y(2)1287 1411 y Fp(\))h Fo(*)f(a)1390 1395 y Fj(0)1402 1411 y Fp(]\))361 1477 y(\()p Fo(bb)413 1462 y Fj(0)425 1477 y Fp(\))g Fo(\()g(a)h Fp(=)g(\([)p Fo(b)f(\()g Fp(\()p Fo(b)727 1462 y Fj(0)727 1487 y Fl(1)745 1462 y(1)775 1477 y Fo(*)g(a)850 1483 y Fl(1)869 1477 y Fp(\)])g Fk( )g Fo(b)979 1462 y Fj(0)979 1487 y Fl(1)998 1462 y(2)1016 1477 y Fp(\()p Fo(b)1050 1462 y Fj(0)1050 1487 y Fl(2)1069 1477 y Fo(]a)1107 1483 y Fl(2)1126 1477 y Fp(\))p Fo(a)1164 1483 y Fl(3)1182 1462 y(1)1201 1477 y Fp(\)\()p Fo(b)1251 1462 y Fj(0)1251 1487 y Fl(3)1281 1477 y Fo(\()g(a)1356 1483 y Fl(3)1375 1462 y(2)1393 1477 y Fp(\))308 1560 y(4.)20 b Fo(b])p Fp(\()p Fo(aa)455 1545 y Fj(0)467 1560 y Fp(\))11 b(=)h(\()p Fo(b)572 1566 y Fl(1)591 1560 y Fo(]a)629 1566 y Fl(1)648 1560 y Fp(\))p Fo(a)686 1566 y Fl(2)704 1545 y(1)723 1560 y Fp(\(\()p Fo(b)773 1566 y Fl(2)803 1560 y Fo(\()f(a)878 1566 y Fl(2)897 1545 y(2)915 1560 y Fp(\))p Fo(]a)969 1545 y Fj(0)981 1560 y Fp(\))361 1626 y(\()p Fo(bb)413 1611 y Fj(0)425 1626 y Fp(\))p Fo(]a)g Fp(=)h(\()p Fo(b])p Fp(\()p Fo(b)618 1611 y Fj(0)618 1637 y Fl(1)637 1611 y(1)667 1626 y Fo(*)f(a)742 1632 y Fl(1)761 1626 y Fp(\)\))p Fo(b)811 1611 y Fj(0)811 1637 y Fl(1)829 1611 y(2)848 1626 y Fp(\()p Fo(b)882 1611 y Fj(0)882 1637 y Fl(2)901 1626 y Fo(]a)939 1632 y Fl(2)957 1626 y Fp(\))308 1709 y(5.)20 b Fo(\017)378 1715 y Fm(H)409 1709 y Fp(\()p Fo(b]a)p Fp(\))12 b(=)g Fo(\017)570 1715 y Fm(A)597 1709 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)668 1715 y Fm(B)697 1709 y Fp(\()p Fo(b)p Fp(\))308 1792 y(6.)20 b Fo(b)11 b(*)h Fp(1)f(=)h Fo(\017)537 1798 y Fm(B)565 1792 y Fp(\()p Fo(b)p Fp(\)1)p Fo(;)20 b Fp(1)11 b Fo(\()h(a)f Fp(=)h Fo(\017)848 1798 y Fm(A)875 1792 y Fp(\()p Fo(a)p Fp(\)1)308 1875 y(7.)20 b Fo(b])p Fp(1)11 b(=)h Fo(\017)488 1881 y Fm(B)517 1875 y Fp(\()p Fo(b)p Fp(\)1)p Fo(;)20 b Fp(1)p Fo(]a)11 b Fp(=)h Fo(\017)751 1881 y Fm(A)778 1875 y Fp(\()p Fo(a)p Fp(\)1)308 1958 y(8.)20 b(\()p Fo(b)395 1964 y Fl(1)414 1940 y(1)444 1958 y Fo(*)11 b(a)519 1964 y Fl(1)537 1958 y Fp(\))553 1943 y Fl(1)572 1958 y Fo(b)590 1964 y Fl(1)609 1940 y(2)627 1958 y Fp(\()p Fo(b)661 1964 y Fl(2)680 1958 y Fo(]a)718 1964 y Fl(2)737 1958 y Fp(\))e Fk(\012)g Fp(\()p Fo(b)837 1964 y Fl(1)856 1940 y(1)886 1958 y Fo(*)i(a)961 1964 y Fl(1)980 1958 y Fp(\))996 1943 y Fl(2)1026 1958 y Fp(=)h(\()p Fo(b)1104 1964 y Fl(1)1123 1940 y(1)1141 1958 y Fo(]a)1179 1964 y Fl(1)1198 1958 y Fp(\))p Fo(a)1236 1964 y Fl(2)1255 1943 y(1)1282 1958 y Fk(\012)e Fp(\()p Fo(b)1358 1964 y Fl(1)1376 1940 y(2)1407 1958 y Fk(!)h Fp(\()p Fo(b)1494 1964 y Fl(2)1524 1958 y Fo(*)g(a)1599 1964 y Fl(2)1617 1943 y(2)1636 1958 y Fp(\)\))361 2025 y(\()p Fo(b)395 2031 y Fl(2)425 2025 y Fo(\()g(a)500 2031 y Fl(2)519 2010 y(2)537 2025 y Fp(\))553 2010 y Fl(1)581 2025 y Fk(\012)f Fp(\()p Fo(b)657 2031 y Fl(1)676 2025 y Fo(]a)714 2031 y Fl(1)732 2025 y Fp(\))p Fo(a)770 2031 y Fl(2)789 2010 y(1)808 2025 y Fp(\()p Fo(b)842 2031 y Fl(2)872 2025 y Fo(\()h(a)947 2031 y Fl(2)965 2010 y(2)984 2025 y Fp(\))1000 2010 y Fl(2)1030 2025 y Fp(=)h(\(\()p Fo(b)1124 2031 y Fl(1)1143 2006 y(1)1173 2025 y Fo(\()f(a)1248 2031 y Fl(1)1267 2025 y Fp(\))g Fk( )g Fo(a)1369 2031 y Fl(2)1388 2010 y(1)1406 2025 y Fp(\))f Fk(\012)f Fo(b)1491 2031 y Fl(1)1510 2006 y(2)1528 2025 y Fp(\()p Fo(b)1562 2031 y Fl(2)1581 2025 y Fo(]a)1619 2031 y Fl(2)1638 2010 y(2)1656 2025 y Fp(\))308 2108 y(9.)20 b Fo(b)11 b(*)h Fp(\()p Fo(h)f Fk(!)g Fo(a)p Fp(\))h(=)g Fo(h)666 2114 y Fl(1)696 2108 y Fk(!)f Fp(\(\()p Fo(b)g Fk( )g Fo(h)887 2114 y Fl(2)906 2108 y Fp(\))h Fo(*)f(a)p Fp(\))361 2174 y(\()p Fo(b)h Fk( )f Fo(h)p Fp(\))g Fo(\()g(a)h Fp(=)g(\()p Fo(b)f(\()g Fp(\()p Fo(h)780 2180 y Fl(1)810 2174 y Fk(!)g Fo(a)p Fp(\)\))h Fk( )f Fo(h)1006 2180 y Fl(2)287 2257 y Fp(10.)20 b(\()p Fo(b])p Fp(\()p Fo(h)451 2263 y Fl(1)481 2257 y Fk(!)12 b Fo(a)p Fp(\)\))p Fo(h)613 2263 y Fl(2)643 2257 y Fp(=)g Fo(h)711 2263 y Fl(1)729 2257 y Fp(\(\()p Fo(b)g Fk( )f Fo(h)868 2263 y Fl(2)886 2257 y Fp(\))p Fo(]a)p Fp(\))287 2340 y(11.)20 b(\()p Fo(b)395 2346 y Fl(1)425 2340 y Fo(*)11 b(a)500 2346 y Fl(1)519 2340 y Fp(\))e Fk(\012)h Fp(\()p Fo(b)620 2346 y Fl(2)650 2340 y Fo(\()h(a)725 2346 y Fl(2)743 2340 y Fp(\))h(=)g(\()p Fo(b)849 2346 y Fl(1)868 2322 y(2)898 2340 y Fk(!)f Fp(\()p Fo(b)985 2346 y Fl(2)1015 2340 y Fo(*)g(a)1090 2346 y Fl(2)1108 2325 y(2)1127 2340 y Fp(\)\))f Fk(\012)f Fp(\(\()p Fo(b)1260 2346 y Fl(1)1279 2322 y(1)1309 2340 y Fo(\()i(a)1384 2346 y Fl(1)1403 2340 y Fp(\))g Fk( )g Fo(a)1505 2346 y Fl(2)1524 2325 y(1)1542 2340 y Fp(\))257 2467 y(These)16 b(conditions)d(are)h(of)g (course)h(required)g(for)e(all)g Fo(a;)i(a)1175 2452 y Fj(0)1199 2467 y Fk(2)c Fo(A)p Fp(,)i Fo(b;)j(b)1358 2452 y Fj(0)1381 2467 y Fk(2)11 b Fo(B)16 b Fp(and)e Fo(h)e Fk(2)f Fo(H)s Fp(.)953 2628 y(46)p eop %%Page: 47 47 47 46 bop 257 262 a Fp(In)14 b(this)g(situation,)f(w)o(e)h(can)g(carry) h(out)e(the)i(\014rst)g(construction:)257 354 y Fn(Theorem)52 b Fp(Giv)o(en)13 b(a)h(Y)m(etter-Drinfel'd)f(bialgebra)g(pair,)g Fo(A)c Fk(\012)h Fo(H)i Fk(\012)e Fo(B)16 b Fp(is)e(a)f(bialgebra)257 404 y(with)h(m)o(ultiplicatio)o(n)482 484 y Fo(\026)d Fp(:)g(\()p Fo(A)f Fk(\012)f Fo(H)k Fk(\012)c Fo(B)r Fp(\))p Fk(\012)q Fp(\()p Fo(A)h Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(\))i Fk(!)f Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)276 546 y Fp(\()p Fo(a)h Fk(\012)f Fo(h)g Fk(\012)h Fo(b)p Fp(\))f Fk(\012)h Fp(\()p Fo(a)563 529 y Fj(0)584 546 y Fk(\012)f Fo(h)649 529 y Fj(0)670 546 y Fk(\012)g Fo(b)729 529 y Fj(0)741 546 y Fp(\))j Fk(7!)541 619 y Fo(a)p Fp(\()p Fo(h)603 625 y Fl(1)633 619 y Fk(!)f Fp(\()p Fo(b)720 625 y Fl(1)738 601 y(1)769 619 y Fo(*)o(a)832 602 y Fj(0)832 629 y Fl(1)851 619 y Fp(\)\))e Fk(\012)h Fo(h)958 625 y Fl(2)976 619 y Fo(b)994 625 y Fl(1)1013 601 y(2)1031 619 y Fp(\()p Fo(b)1065 625 y Fl(2)1084 619 y Fo(]a)1122 602 y Fj(0)1122 629 y Fl(2)1140 619 y Fp(\))p Fo(a)1178 602 y Fj(0)1178 629 y Fl(3)1197 596 y(1)1216 619 y Fo(h)1240 602 y Fj(0)1240 629 y Fl(1)1268 619 y Fk(\012)f Fp(\(\()p Fo(b)1359 625 y Fl(3)1389 619 y Fo(\()i(a)1464 602 y Fj(0)1464 629 y Fl(3)1483 596 y(2)1502 619 y Fp(\))g Fk( )g Fo(h)1606 602 y Fj(0)1606 629 y Fl(2)1625 619 y Fp(\))p Fo(b)1659 602 y Fj(0)257 699 y Fp(unit)j(elemen)o(t)f(1)c Fk(\012)h Fp(1)f Fk(\012)g Fp(1,)k(com)o(ultiplication)543 778 y(\001)e(:)g Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)15 b Fk(!)c Fp(\()p Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(\))h Fk(\012)g Fp(\()p Fo(A)f Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(\))504 858 y Fo(a)g Fk(\012)h Fo(h)f Fk(\012)g Fo(b)j Fk(7!)f Fp(\()p Fo(a)772 864 y Fl(1)800 858 y Fk(\012)e Fo(a)863 864 y Fl(2)882 841 y(1)901 858 y Fo(h)925 864 y Fl(1)952 858 y Fk(\012)h Fo(b)1012 864 y Fl(1)1030 840 y(1)1049 858 y Fp(\))f Fk(\012)h Fp(\()p Fo(a)1154 864 y Fl(2)1172 841 y(2)1200 858 y Fk(\012)g Fo(h)1266 864 y Fl(2)1284 858 y Fo(b)1302 864 y Fl(1)1321 840 y(2)1349 858 y Fk(\012)f Fo(b)1408 864 y Fl(2)1427 858 y Fp(\))257 926 y(and)14 b(counit)794 976 y Fo(\017)e Fp(:)f Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)14 b Fk(!)d Fo(K)709 1043 y(a)f Fk(\012)f Fo(h)g Fk(\012)h Fo(b)h Fk(7!)g Fo(\017)956 1049 y Fm(A)983 1043 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1054 1049 y Fm(H)1086 1043 y Fp(\()p Fo(h)p Fp(\))p Fo(\017)1159 1049 y Fm(B)1187 1043 y Fp(\()p Fo(b)p Fp(\))257 1111 y(If)j Fo(A)p Fp(,)f Fo(H)k Fp(and)d Fo(B)i Fp(p)q(ossess)g(an)o(tip)q (o)q(des,)e(then)g Fo(A)c Fk(\012)f Fo(H)j Fk(\012)e Fo(B)16 b Fp(has)e(the)h(an)o(tip)q(o)q(de)351 1191 y Fo(S)r Fp(\()p Fo(a)10 b Fk(\012)g Fo(h)f Fk(\012)g Fo(b)p Fp(\))j(=)g(\(1)d Fk(\012)g Fp(1)g Fk(\012)h Fo(S)816 1197 y Fm(B)845 1191 y Fp(\()p Fo(b)879 1174 y Fl(1)897 1191 y Fp(\)\)\(1)g Fk(\012)f Fo(S)1042 1197 y Fm(H)1074 1191 y Fp(\()p Fo(a)1112 1174 y Fl(1)1131 1191 y Fo(hb)1173 1174 y Fl(2)1191 1191 y Fp(\))h Fk(\012)f Fp(1\)\()p Fo(S)1336 1197 y Fm(A)1364 1191 y Fp(\()p Fo(a)1402 1174 y Fl(2)1421 1191 y Fp(\))g Fk(\012)g Fp(1)g Fk(\012)h Fp(1\))257 1301 y(The)h(form)o(ulas)d(for)j(the)g(m)o(ultipli)o(cation) d(and)i(the)h(com)o(ultiplication)c(in)j(this)g(construction)257 1351 y(are)k(strictly)g(symmetric.)d(This)j(can)f(b)q(e)h(formalized)e (as)h(follo)o(ws:)f(Consider)i Fo(B)1513 1336 y Fm(op)8 b(cop)1618 1351 y Fp(as)14 b(a)257 1401 y(left)c(Y)m(etter-Drinfel'd)f (bialgebra)g(o)o(v)o(er)g Fo(H)920 1386 y Fm(op)f(cop)1021 1401 y Fp(and)i Fo(A)1129 1386 y Fm(op)e(cop)1230 1401 y Fp(as)h(a)h(righ)o(t)f(Y)m(etter-Drinfel'd)257 1451 y(bialgebra)17 b(o)o(v)o(er)h Fo(H)572 1435 y Fm(op)8 b(cop)681 1451 y Fp(as)17 b(in)g(subsection)i(2.5.)d(In)o(tro)q(duce)j (a)f(left)f Fo(A)1412 1435 y Fm(op)8 b(cop)1504 1451 y Fp(-action)17 b(on)257 1500 y Fo(B)290 1485 y Fm(op)9 b(cop)396 1500 y Fp(via)574 1550 y Fo(A)605 1533 y Fm(op)g(cop)706 1550 y Fk(\012)h Fo(B)781 1533 y Fm(op)e(cop)884 1550 y Fk(!)j Fo(B)970 1533 y Fm(op)e(cop)1062 1550 y Fo(;)e(a)i Fk(\012)g Fo(b)j Fk(7!)f Fp(\()p Fo(b)g(\()g(a)p Fp(\))257 1618 y(In)o(tro)q(duce)k(a)f(righ)o(t)f Fo(B)613 1603 y Fm(op)c(cop)705 1618 y Fp(-action)14 b(on)g Fo(A)933 1603 y Fm(op)8 b(cop)1038 1618 y Fp(via)828 1698 y Fo(a)h Fk(\012)g Fo(b)j Fk(7!)f Fp(\()p Fo(b)g(*)g(a)p Fp(\))257 1777 y(Similarly)m(,)f(w)o(e)k(can)g(in)o(tro)q(duce)h(a)e(mapping)613 1857 y Fo(A)644 1840 y Fm(op)8 b(cop)744 1857 y Fk(\012)i Fo(B)819 1840 y Fm(op)f(cop)922 1857 y Fk(!)j Fo(H)1014 1840 y Fm(op)7 b(cop)1105 1857 y Fo(;)g(a)h Fk(\012)i Fo(b)h Fk(7!)g Fo(b]a)257 1936 y Fp(Using)j(these)h(structure)h(elemen) o(ts,)e(w)o(e)g(can)g(form)e Fo(B)1108 1921 y Fm(op)d(cop)1209 1936 y Fk(\012)h Fo(H)1289 1921 y Fm(op)e(cop)1389 1936 y Fk(\012)i Fo(A)1462 1921 y Fm(op)e(cop)1553 1936 y Fp(.)257 2029 y Fn(Lemma)36 b Fp(The)15 b(map)538 2109 y Fo(B)571 2092 y Fm(op)9 b(cop)672 2109 y Fk(\012)h Fo(H)752 2092 y Fm(op)e(cop)852 2109 y Fk(\012)i Fo(A)925 2092 y Fm(op)e(cop)1028 2109 y Fk(!)j Fp(\()p Fo(A)f Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(\))1317 2088 y Fm(op)e(cop)851 2171 y Fo(b)h Fk(\012)h Fo(h)f Fk(\012)h Fo(a)h Fk(7!)g Fo(a)e Fk(\012)h Fo(h)f Fk(\012)g Fo(b)257 2251 y Fp(is)14 b(a)g(bialgebra)f(isomorphism.)257 2343 y Fn(Pro)q(of.)36 b Fp(This)12 b(is)g(clear)g(from)f(the)h(de\014nitions.)g(Observ)o(e)h (that)f(it)g(is)g(not)g(ev)o(en)g(necessary)257 2393 y(to)18 b(c)o(hec)o(k)h(the)f(compatibilit)o(y)d(conditions)i(ab)q(o)o (v)o(e)h(since)g(b)o(y)g([32)o(],)f(subsection)i(3.10)d(w)o(e)257 2443 y(kno)o(w)i(that)h(these)h(are)f(also)f(necessary)j(conditions)d (for)g Fo(B)1231 2428 y Fm(op)9 b(cop)1336 2443 y Fk(\012)k Fo(H)1419 2428 y Fm(op)7 b(cop)1522 2443 y Fk(\012)13 b Fo(A)1598 2428 y Fm(op)8 b(cop)257 2493 y Fp(b)q(eing)14 b(a)g(bialgebra,)e(whic)o(h)i(is)g(clear)g(since)h(it)e(is)h (isomorphic)e(to)i(\()p Fo(A)c Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(\))1537 2472 y Fm(op)e(cop)1629 2493 y Fp(.)13 b Fg(\003)953 2628 y Fp(47)p eop %%Page: 48 48 48 47 bop 257 262 a Fn(5.2)48 b Fp(W)m(e)15 b(shall)g(assume)g(for)h (the)g(rest)h(of)e(this)h(section)g(that)g Fo(A)p Fp(,)f Fo(H)k Fp(and)c Fo(B)j Fp(are)e(\014nite-)257 311 y(dimensional)e(and)i (do)f(ha)o(v)o(e)h(an)o(tip)q(o)q(des)g Fo(S)943 317 y Fm(A)971 311 y Fp(,)f Fo(S)1023 317 y Fm(H)1071 311 y Fp(and)h Fo(S)1179 317 y Fm(B)1224 311 y Fp(resp)q(ectiv)o(ely)m(.)g (The)g(purp)q(ose)257 361 y(of)e(this)g(subsection)h(is)e(to)h(pro)o(v) o(e)g(the)h(follo)o(wing)c(theorem:)257 451 y Fn(Theorem)308 501 y Fp(1.)20 b(Supp)q(ose)d(that)g(\003)648 507 y Fm(A)675 501 y Fp(,)f(\003)732 507 y Fm(H)779 501 y Fp(and)g(\003)891 507 y Fm(B)936 501 y Fp(are)h(nonzero)g(left)f(in)o(tegrals)g(of)g Fo(A)p Fp(,)g Fo(H)j Fp(and)d Fo(B)361 551 y Fp(resp)q(ectiv)o(ely)m(.) c(Supp)q(ose)h(that)e Fo(\032)862 557 y Fm(A)901 551 y Fp(is)h(a)f(righ)o(t)g(in)o(tegral)g(of)g Fo(A)1296 536 y Fj(\003)1326 551 y Fp(satisfying)g Fo(\032)1530 557 y Fm(A)1558 551 y Fp(\(\003)1603 557 y Fm(A)1630 551 y Fp(\))g(=)361 600 y(1.)i(Set:)637 650 y Fo(!)663 656 y Fm(B)703 650 y Fp(:)e Fo(B)j Fk(!)e Fo(K)q(;)7 b(b)j Fk(7!)h Fo(\032)982 656 y Fm(A)1010 650 y Fp(\()p Fo(S)1051 656 y Fm(B)1080 650 y Fp(\()p Fo(b)1114 633 y Fl(1)1133 650 y Fp(\))g Fo(*)g Fp(\003)1242 656 y Fm(A)1269 650 y Fp(\))p Fo(\023)1300 633 y Fj(\000)p Fl(1)1300 662 y Fm(A)1345 650 y Fp(\()p Fo(b)1379 633 y Fl(2)1397 650 y Fp(\))361 724 y(where)k Fo(\023)496 730 y Fm(A)536 724 y Fp(is)e(the)h(in)o(tegral)e(c)o(haracter)j(of)e Fo(A)p Fp(.)g(Then)h Fo(!)1216 730 y Fm(B)1257 724 y Fp(is)f(an)g Fo(H)s Fp(-linear)g(c)o(haracter)361 774 y(of)g Fo(B)k Fp(and)703 824 y(\003)732 830 y Fm(A)768 824 y Fk(\012)9 b Fp(\003)838 830 y Fm(H)c Fl(1)888 824 y Fo(\023)903 830 y Fm(A)930 824 y Fp(\(\003)975 830 y Fm(H)t Fl(2)1025 824 y Fp(\))k Fk(\012)h Fp(\003)1121 830 y Fm(B)s Fl(1)1168 824 y Fo(!)1195 806 y Fj(\000)p Fl(1)1194 836 y Fm(B)1240 824 y Fp(\(\003)1285 830 y Fm(B)s Fl(2)1332 824 y Fp(\))361 897 y(is)k(a)f(nonzero)i(left)f(in)o (tegral)f(of)g Fo(A)d Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(.)308 980 y(2.)20 b(Supp)q(ose)c(that)f(\000)642 986 y Fm(A)669 980 y Fp(,)g(\000)722 986 y Fm(H)768 980 y Fp(and)g(\000)876 986 y Fm(B)919 980 y Fp(are)h(nonzero)f(righ)o(t)g (in)o(tegrals)f(of)h Fo(A)p Fp(,)f Fo(H)k Fp(and)d Fo(B)361 1030 y Fp(resp)q(ectiv)o(ely)m(.)f(Supp)q(ose)g(that)f Fo(\025)870 1036 y Fm(B)912 1030 y Fp(is)g(a)g(left)g(in)o(tegral)f(of) h Fo(B)1289 1014 y Fj(\003)1322 1030 y Fp(satisfying)f Fo(\025)1530 1036 y Fm(B)1559 1030 y Fp(\(\000)1601 1036 y Fm(B)1630 1030 y Fp(\))f(=)361 1079 y(1.)i(Set:)632 1129 y Fo(!)658 1135 y Fm(A)697 1129 y Fp(:)e Fo(A)h Fk(!)f Fo(K)q(;)c(a)k Fk(7!)g Fo(\023)972 1111 y Fj(\000)p Fl(1)972 1141 y Fm(B)1016 1129 y Fp(\()p Fo(a)1054 1112 y Fl(1)1073 1129 y Fp(\))p Fo(\025)1113 1135 y Fm(B)1141 1129 y Fp(\(\000)1183 1135 y Fm(B)1224 1129 y Fo(\()g(S)1302 1135 y Fm(A)1329 1129 y Fp(\()p Fo(a)1367 1112 y Fl(2)1386 1129 y Fp(\)\))361 1203 y(where)j Fo(\023)495 1209 y Fm(B)537 1203 y Fp(is)f(the)h(in)o(tegral)e(c)o(haracter)i(of)f Fo(B)r Fp(.)g(Then)h Fo(!)1218 1209 y Fm(A)1258 1203 y Fp(is)f(an)g Fo(H)s Fp(-linear)f(c)o(haracter)361 1253 y(of)h Fo(A)h Fp(and)710 1302 y Fo(!)737 1285 y Fj(\000)p Fl(1)736 1315 y Fm(A)782 1302 y Fp(\(\000)824 1308 y Fm(A)r Fl(1)869 1302 y Fp(\)\000)911 1308 y Fm(A)s Fl(2)966 1302 y Fk(\012)c Fo(\023)1023 1308 y Fm(B)1051 1302 y Fp(\(\000)1093 1308 y Fm(H)5 b Fl(1)1143 1302 y Fp(\)\000)1185 1308 y Fm(H)g Fl(2)1245 1302 y Fk(\012)k Fp(\000)1312 1308 y Fm(B)361 1376 y Fp(is)14 b(a)f(nonzero)i(righ)o(t)f(in)o(tegral) f(of)g Fo(A)c Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(.)257 1475 y Fn(Pro)q(of.)36 b Fp(The)15 b(second)h(part)f(is)f(the)h (dualization)e(of)h(the)i(\014rst)f(part)g(via)e(Lemma)f(5.1.)h(It)257 1525 y(remains)g(to)h(sho)o(w)g(the)g(\014rst)h(part.)f(This)f(pro)q (ceeds)j(in)e(sev)o(eral)g(steps.)257 1641 y(\(1\))21 b(W)m(e)c(sho)o(w)f(\014rst)i(that)f Fo(!)719 1647 y Fm(B)764 1641 y Fp(is)g Fo(H)s Fp(-linear.)e(By)i(condition)g(\(9\))f (in)h(De\014nition)f(5.1,)f(w)o(e)257 1691 y(ha)o(v)o(e:)312 1781 y Fo(S)337 1787 y Fm(H)369 1781 y Fp(\()p Fo(h)409 1787 y Fl(1)428 1781 y Fp(\))c Fk(!)g Fp(\()p Fo(b)h(*)f Fp(\()p Fo(h)647 1787 y Fl(2)677 1781 y Fk(!)g Fo(a)p Fp(\)\))h(=)g Fo(S)865 1787 y Fm(H)897 1781 y Fp(\()p Fo(h)937 1787 y Fl(1)955 1781 y Fp(\))p Fo(h)995 1787 y Fl(2)1025 1781 y Fk(!)f Fp(\(\()p Fo(b)h Fk( )f Fo(h)1217 1787 y Fl(3)1236 1781 y Fp(\))g Fo(*)g(a)p Fp(\))h(=)g(\()p Fo(b)f Fk( )g Fo(h)p Fp(\))h Fo(*)f(a)257 1871 y Fp(W)m(e)j(no)o(w)f (calculate,)h(using)f(the)i(Y)m(etter-Drinfel'd)f(condition)f(and)g (Prop)q(osition)h(2.10:)314 1960 y Fo(!)340 1966 y Fm(B)368 1960 y Fp(\()p Fo(b)e Fk( )f Fo(h)p Fp(\))g(=)h Fo(\032)583 1966 y Fm(A)611 1960 y Fp(\()p Fo(S)652 1966 y Fm(B)681 1960 y Fp(\(\()p Fo(b)g Fk( )f Fo(h)p Fp(\))836 1939 y Fl(1)854 1960 y Fp(\))h Fo(*)f Fp(\003)964 1966 y Fm(A)991 1960 y Fp(\))p Fo(\023)1022 1942 y Fj(\000)p Fl(1)1022 1972 y Fm(A)1066 1960 y Fp(\(\()p Fo(b)h Fk( )f Fo(h)p Fp(\))1221 1939 y Fl(2)1240 1960 y Fp(\))518 2028 y(=)h Fo(\032)583 2034 y Fm(A)611 2028 y Fp(\()p Fo(S)652 2034 y Fm(B)681 2028 y Fp(\()p Fo(b)715 2011 y Fl(1)745 2028 y Fk( )f Fo(h)822 2034 y Fl(2)841 2028 y Fp(\))g Fo(*)g Fp(\003)950 2034 y Fm(A)977 2028 y Fp(\))p Fo(\023)1008 2010 y Fj(\000)p Fl(1)1008 2040 y Fm(A)1053 2028 y Fp(\()p Fo(S)1094 2034 y Fm(H)1126 2028 y Fp(\()p Fo(h)1166 2034 y Fl(1)1184 2028 y Fp(\))p Fo(b)1218 2011 y Fl(2)1237 2028 y Fo(h)1261 2034 y Fl(3)1280 2028 y Fp(\))518 2096 y(=)h Fo(\032)583 2102 y Fm(A)611 2096 y Fp(\()p Fo(S)652 2102 y Fm(H)684 2096 y Fp(\()p Fo(h)724 2102 y Fl(2)743 2096 y Fp(\))f Fk(!)g Fp(\()p Fo(S)864 2102 y Fm(B)893 2096 y Fp(\()p Fo(b)927 2079 y Fl(1)946 2096 y Fp(\))h Fo(*)f Fp(\()p Fo(h)1067 2102 y Fl(3)1097 2096 y Fk(!)g Fp(\003)1179 2102 y Fm(A)1206 2096 y Fp(\)\)\))p Fo(\023)1269 2102 y Fm(A)1296 2096 y Fp(\()p Fo(h)1336 2102 y Fl(1)1355 2096 y Fp(\))p Fo(\023)1386 2078 y Fj(\000)p Fl(1)1386 2108 y Fm(A)1430 2096 y Fp(\()p Fo(b)1464 2079 y Fl(2)1483 2096 y Fp(\))p Fo(\023)1514 2078 y Fj(\000)p Fl(1)1514 2108 y Fm(A)1558 2096 y Fp(\()p Fo(h)1598 2102 y Fl(4)1617 2096 y Fp(\))518 2164 y(=)h Fo(\023)577 2170 y Fm(A)604 2164 y Fp(\()p Fo(S)645 2170 y Fm(H)677 2164 y Fp(\()p Fo(h)717 2170 y Fl(2)736 2164 y Fp(\)\))p Fo(\032)789 2170 y Fm(A)817 2164 y Fp(\()p Fo(S)858 2170 y Fm(B)887 2164 y Fp(\()p Fo(b)921 2147 y Fl(1)939 2164 y Fp(\))g Fo(*)f(\023)1035 2170 y Fm(A)1062 2164 y Fp(\()p Fo(h)1102 2170 y Fl(3)1120 2164 y Fp(\)\003)1165 2170 y Fm(A)1192 2164 y Fp(\))p Fo(\023)1223 2170 y Fm(A)1250 2164 y Fp(\()p Fo(h)1290 2170 y Fl(1)1309 2164 y Fp(\))p Fo(\023)1340 2147 y Fj(\000)p Fl(1)1340 2176 y Fm(A)1384 2164 y Fp(\()p Fo(b)1418 2147 y Fl(2)1437 2164 y Fp(\))p Fo(\023)1468 2147 y Fj(\000)p Fl(1)1468 2176 y Fm(A)1512 2164 y Fp(\()p Fo(h)1552 2170 y Fl(4)1571 2164 y Fp(\))518 2227 y(=)h Fo(\017)579 2233 y Fm(H)611 2227 y Fp(\()p Fo(h)p Fp(\))p Fo(!)693 2233 y Fm(B)721 2227 y Fp(\()p Fo(b)p Fp(\))257 2332 y(\(2\))21 b(By)14 b([19)o(],)f(Theorem)g(2.1.3,)f(p.)h(18)g(w)o (e)h(kno)o(w)g(that)g Fo(A)9 b Fk(\012)g Fo(H)j Fk(\012)d Fo(B)17 b Fp(con)o(tains)c(a)h(nonzero)257 2382 y(in)o(tegral)g(\003.)f (W)m(rite:)756 2471 y(\003)e(=)861 2419 y Fm(k)840 2431 y Fh(X)843 2520 y Fm(i)p Fl(=1)907 2471 y Fo(a)929 2478 y Fl(\()p Fm(i)p Fl(\))978 2471 y Fk(\012)e Fo(h)1043 2478 y Fl(\()p Fm(i)p Fl(\))1092 2471 y Fk(\012)h Fo(b)1152 2478 y Fl(\()p Fm(i)p Fl(\))953 2628 y Fp(48)p eop %%Page: 49 49 49 48 bop 257 262 a Fp(where)21 b Fo(k)g Fp(is)e(minima)o(l)d(so)k (that)g Fo(a)815 269 y Fl(\(1\))873 262 y Fk(\012)13 b Fo(h)942 269 y Fl(\(1\))986 262 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(a)1108 269 y Fl(\()p Fm(k)q Fl(\))1167 262 y Fk(\012)13 b Fo(h)1236 269 y Fl(\()p Fm(k)q Fl(\))1302 262 y Fp(and)20 b Fo(b)1407 269 y Fl(\(1\))1451 262 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(b)1569 269 y Fl(\()p Fm(k)q Fl(\))1634 262 y Fp(are)257 311 y(linearly)13 b(indep)q(enden)o(t.)i(W) m(e)e(ha)o(v)o(e)h(for)g Fo(a)d Fk(2)g Fo(A)p Fp(,)j Fo(h)d Fk(2)h Fo(H)s Fp(:)421 386 y Fm(k)400 398 y Fh(X)403 486 y Fm(i)p Fl(=1)460 437 y Fp(\()p Fo(a)d Fk(\012)h Fo(h)p Fp(\)\()p Fo(a)627 444 y Fl(\()p Fm(i)p Fl(\))676 437 y Fk(\012)f Fo(h)741 444 y Fl(\()p Fm(i)p Fl(\))781 437 y Fp(\))g Fk(\012)h Fo(b)866 444 y Fl(\()p Fm(i)p Fl(\))917 437 y Fp(=)i(\()p Fo(a)d Fk(\012)h Fo(h)f Fk(\012)g Fp(1\)\003)i(=)h Fo(\017)1262 443 y Fm(A)1289 437 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1360 443 y Fm(H)1392 437 y Fp(\()p Fo(h)p Fp(\)\003)917 583 y(=)982 531 y Fm(k)961 544 y Fh(X)964 632 y Fm(i)p Fl(=1)1028 583 y Fo(\017)1045 589 y Fm(A)1072 583 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1143 589 y Fm(H)1174 583 y Fp(\()p Fo(h)p Fp(\)\()p Fo(a)1268 590 y Fl(\()p Fm(i)p Fl(\))1317 583 y Fk(\012)e Fo(h)1383 590 y Fl(\()p Fm(i)p Fl(\))1423 583 y Fp(\))f Fk(\012)g Fo(b)1507 590 y Fl(\()p Fm(i)p Fl(\))257 707 y Fp(and)17 b(therefore,)h(for)e(all)g Fo(i)h Fp(=)g(1)p Fo(;)7 b(:)g(:)g(:)k(;)c (k)q Fp(,)16 b Fo(a)930 714 y Fl(\()p Fm(i)p Fl(\))981 707 y Fk(\012)11 b Fo(h)1048 714 y Fl(\()p Fm(i)p Fl(\))1105 707 y Fp(is)16 b(a)h(left)f(in)o(tegral)h(in)f(the)h(Radford)257 757 y(bipro)q(duct)d Fo(A)8 b Fk(\012)g Fo(H)s Fp(.)13 b(W)m(e)g(kno)o(w)g(from)e([23)o(],Prop.)h(3,)h(p.)g(333)f(that)h(\003) 1341 763 y Fm(A)1376 757 y Fk(\012)c Fp(\003)1446 763 y Fm(H)t Fl(1)1495 757 y Fo(\023)1510 763 y Fm(A)1537 757 y Fp(\(\003)1582 763 y Fm(H)c Fl(2)1632 757 y Fp(\))14 b(is)257 807 y(a)g(left)g(in)o(tegral)f(of)g Fo(A)c Fk(\012)h Fo(H)17 b Fp(and)c(therefore)j(w)o(e)e(ha)o(v)o(e,)f(for)h(all)e Fo(i)g Fp(=)g(1)p Fo(;)7 b(:)g(:)g(:)k(;)c(k)q Fp(:)672 894 y Fo(a)694 901 y Fl(\()p Fm(i)p Fl(\))743 894 y Fk(\012)j Fo(h)809 901 y Fl(\()p Fm(i)p Fl(\))860 894 y Fp(=)i Fo(\020)922 900 y Fm(i)936 894 y Fp(\003)965 900 y Fm(A)1001 894 y Fk(\012)e Fp(\003)1072 900 y Fm(H)t Fl(1)1121 894 y Fo(\023)1136 900 y Fm(A)1163 894 y Fp(\(\003)1208 900 y Fm(H)5 b Fl(2)1258 894 y Fp(\))257 1015 y(for)14 b(some)f Fo(\020)443 1021 y Fm(i)468 1015 y Fk(2)f Fo(K)s Fp(.)h(If)h(w)o(e)g (set)h Fo(x)763 1021 y Fm(B)802 1015 y Fp(=)864 975 y Fm(k)851 983 y Fh(P)846 1051 y Fm(i)p Fl(=1)907 1015 y Fo(\020)925 1021 y Fm(i)939 1015 y Fo(b)957 1022 y Fl(\()p Fm(i)p Fl(\))997 1015 y Fp(,)e(w)o(e)h(see)h(that:)711 1128 y(\003)c(=)h(\003)824 1134 y Fm(A)860 1128 y Fk(\012)d Fp(\003)930 1134 y Fm(H)c Fl(1)980 1128 y Fo(\023)995 1134 y Fm(A)1022 1128 y Fp(\(\003)1067 1134 y Fm(H)t Fl(2)1117 1128 y Fp(\))k Fk(\012)h Fo(x)1208 1134 y Fm(B)257 1232 y Fp(\(3\))21 b(W)m(e)12 b(denote)g(b)o(y)g Fo(!)615 1216 y Fj(00)614 1243 y Fm(B)654 1232 y Fp(the)h(restriction)f(of)f (the)i(righ)o(t)e(mo)q(dular)f(function)i(of)f Fo(A)5 b Fk(\012)g Fo(H)k Fk(\012)c Fo(B)257 1281 y Fp(to)14 b Fo(B)r Fp(:)711 1331 y Fo(!)738 1314 y Fj(0)o(0)737 1341 y Fm(B)765 1331 y Fp(\()p Fo(b)p Fp(\))e(:=)f Fo(\013)909 1314 y Fm(R)909 1341 y(A)p Fj(\012)p Fm(H)r Fj(\012)p Fm(B)1043 1331 y Fp(\(1)e Fk(\012)h Fp(1)f Fk(\012)g Fo(b)p Fp(\))257 1404 y(W)m(e)14 b(ha)o(v)o(e:)304 1491 y(\003)333 1497 y Fm(A)370 1491 y Fk(\012)9 b Fp(\003)440 1497 y Fm(H)t Fl(1)490 1491 y Fo(\023)505 1497 y Fm(A)532 1491 y Fp(\(\003)577 1497 y Fm(H)t Fl(2)627 1491 y Fp(\))g Fk(\012)h Fo(x)718 1497 y Fm(B)746 1491 y Fo(b)h Fp(=)h(\003\(1)d Fk(\012)h Fp(1)f Fk(\012)g Fo(b)p Fp(\))j(=)f(\003)1125 1497 y Fm(A)1161 1491 y Fk(\012)f Fp(\003)1232 1497 y Fm(H)t Fl(1)1282 1491 y Fo(\023)1297 1497 y Fm(A)1324 1491 y Fp(\(\003)1369 1497 y Fm(H)t Fl(2)1419 1491 y Fp(\))f Fk(\012)h Fo(x)1510 1497 y Fm(B)1538 1491 y Fo(!)1565 1474 y Fj(00)1564 1501 y Fm(B)1592 1491 y Fp(\()p Fo(b)p Fp(\))257 1578 y(and)20 b(therefore)h(w)o(e)e(conclude)i(that)e Fo(x)886 1584 y Fm(B)914 1578 y Fo(b)i Fp(=)g Fo(!)1033 1563 y Fj(00)1032 1590 y Fm(B)1061 1578 y Fp(\()p Fo(b)p Fp(\))p Fo(x)1135 1584 y Fm(B)1163 1578 y Fp(.)e(So)g Fo(!)1284 1563 y Fj(00)1283 1590 y Fm(B)1331 1578 y Fp(is)g(an)h (augmen)o(tation)257 1628 y(and)14 b Fo(x)362 1634 y Fm(B)405 1628 y Fp(is)g(the)h(corresp)q(onding)g(in)o(tegral.)e (Dualizing)f(Prop)q(osition)i(2.10)f(with)h(the)g(help)257 1678 y(of)19 b(Lemma)d(2.5,)h(w)o(e)i(see)h(that)f Fo(B)i Fp(is)e(a)g(F)m(rob)q(enius)g(algebra)f(and)h(therefore)h(w)o(e)f(kno)o (w)257 1728 y(b)o(y)d(Prop)q(osition)f(2.7)g(that)h(the)g(space)h(of)e (suc)o(h)i(in)o(tegrals)e(is)h(one-dimensional.)d(By)j(the)257 1778 y(reasoning)e(w)o(e)g(ha)o(v)o(e)g(already)g(encoun)o(tered)h(sev) o(eral)g(times)e(w)o(e)h(conclude)h(that)f(w)o(e)g(ha)o(v)o(e)832 1865 y Fo(bx)874 1871 y Fm(B)914 1865 y Fp(=)e Fo(!)985 1848 y Fj(0)984 1875 y Fm(B)1012 1865 y Fp(\()p Fo(b)p Fp(\))p Fo(x)1086 1871 y Fm(B)257 1953 y Fp(for)i(some)f(c)o(haracter)i Fo(!)633 1938 y Fj(0)632 1964 y Fm(B)674 1953 y Fp(of)f Fo(B)r Fp(.)257 2034 y(\(4\))21 b(Let)15 b(us)f(no)o(w)f(do)h(a)g (little)f(calculation.)f(W)m(e)i(ha)o(v)o(e:)258 2122 y(\003)287 2128 y Fm(A)323 2122 y Fk(\012)9 b Fp(\003)393 2128 y Fm(H)c Fl(1)443 2122 y Fo(\023)458 2128 y Fm(A)485 2122 y Fp(\(\003)530 2128 y Fm(H)t Fl(2)580 2122 y Fp(\))k Fk(\012)h Fo(\017)664 2128 y Fm(B)692 2122 y Fp(\()p Fo(b)p Fp(\))p Fo(x)766 2128 y Fm(B)806 2122 y Fp(=)i(\(1)d Fk(\012)h Fp(1)f Fk(\012)g Fo(b)p Fp(\)\003)1072 2128 y Fm(A)1108 2122 y Fk(\012)h Fp(\003)1179 2128 y Fm(H)t Fl(1)1229 2122 y Fo(\023)1244 2128 y Fm(A)1270 2122 y Fp(\(\003)1315 2128 y Fm(H)5 b Fl(2)1365 2122 y Fp(\))10 b Fk(\012)f Fo(x)1456 2128 y Fm(B)269 2190 y Fp(=)j(\()p Fo(b)347 2196 y Fl(1)366 2172 y(1)396 2190 y Fo(*)f Fp(\003)478 2196 y Fm(A)r Fl(1)523 2190 y Fp(\))f Fk(\012)f Fo(b)608 2196 y Fl(1)627 2172 y(2)645 2190 y Fp(\()p Fo(b)679 2196 y Fl(2)698 2190 y Fo(])p Fp(\003)743 2196 y Fm(A)r Fl(2)788 2190 y Fp(\)\003)833 2196 y Fm(A)r Fl(3)879 2172 y(1)898 2190 y Fp(\003)927 2196 y Fm(H)t Fl(1)986 2190 y Fk(\012)g Fp(\(\()p Fo(b)1077 2196 y Fl(3)1108 2190 y Fo(\()i Fp(\003)1190 2196 y Fm(A)r Fl(3)1235 2172 y(2)1254 2190 y Fp(\))g Fk( )g Fp(\003)1363 2196 y Fm(H)5 b Fl(2)1413 2190 y Fp(\))p Fo(x)1453 2196 y Fm(B)1482 2190 y Fo(\023)1497 2196 y Fm(A)1524 2190 y Fp(\(\003)1569 2196 y Fm(H)t Fl(3)1619 2190 y Fp(\))269 2259 y(=)12 b(\()p Fo(b)347 2265 y Fl(1)366 2241 y(1)396 2259 y Fo(*)f Fp(\003)478 2265 y Fm(A)r Fl(1)523 2259 y Fp(\))f Fk(\012)f Fo(b)608 2265 y Fl(1)627 2241 y(2)645 2259 y Fp(\()p Fo(b)679 2265 y Fl(2)698 2259 y Fo(])p Fp(\003)743 2265 y Fm(A)r Fl(2)788 2259 y Fp(\)\003)833 2265 y Fm(A)r Fl(3)879 2241 y(1)898 2259 y Fp(\003)927 2265 y Fm(H)t Fl(1)986 2259 y Fk(\012)g Fo(!)1054 2242 y Fj(0)1053 2270 y Fm(B)1082 2259 y Fp(\(\()p Fo(b)1132 2265 y Fl(3)1162 2259 y Fo(\()i Fp(\003)1244 2265 y Fm(A)r Fl(3)1290 2241 y(2)1308 2259 y Fp(\))h Fk( )f Fp(\003)1418 2265 y Fm(H)t Fl(2)1468 2259 y Fp(\))p Fo(\023)1499 2265 y Fm(A)1526 2259 y Fp(\(\003)1571 2265 y Fm(H)t Fl(3)1621 2259 y Fp(\))p Fo(x)1661 2265 y Fm(B)257 2347 y Fp(and)j(can)g(conclude)h (that)337 2434 y(\003)366 2440 y Fm(A)402 2434 y Fk(\012)9 b Fp(\003)472 2440 y Fm(H)c Fl(1)522 2434 y Fo(\023)537 2440 y Fm(A)564 2434 y Fp(\(\003)609 2440 y Fm(H)g Fl(2)659 2434 y Fp(\))p Fo(\017)692 2440 y Fm(B)721 2434 y Fp(\()p Fo(b)p Fp(\))11 b(=)337 2503 y(\()p Fo(b)371 2509 y Fl(1)389 2485 y(1)420 2503 y Fo(*)g Fp(\003)502 2509 y Fm(A)q Fl(1)547 2503 y Fp(\))e Fk(\012)h Fo(b)632 2509 y Fl(1)650 2485 y(2)669 2503 y Fp(\()p Fo(b)703 2509 y Fl(2)722 2503 y Fo(])p Fp(\003)767 2509 y Fm(A)r Fl(2)812 2503 y Fp(\)\003)857 2509 y Fm(A)r Fl(3)903 2485 y(1)922 2503 y Fp(\003)951 2509 y Fm(H)t Fl(1)1000 2503 y Fo(!)1027 2486 y Fj(0)1026 2513 y Fm(B)1055 2503 y Fp(\(\()p Fo(b)1105 2509 y Fl(3)1135 2503 y Fo(\()h Fp(\003)1217 2509 y Fm(A)r Fl(3)1263 2485 y(2)1281 2503 y Fp(\))h Fk( )f Fp(\003)1391 2509 y Fm(H)t Fl(2)1441 2503 y Fp(\))p Fo(\023)1472 2509 y Fm(A)1499 2503 y Fp(\(\003)1544 2509 y Fm(H)t Fl(3)1594 2503 y Fp(\))953 2628 y(49)p eop %%Page: 50 50 50 49 bop 257 262 a Fp(This)14 b(in)g(turn)g(implies:)413 353 y Fo(S)438 359 y Fm(B)467 353 y Fp(\()p Fo(b)501 336 y Fl(1)519 353 y Fp(\))e Fo(*)f Fp(\003)629 359 y Fm(A)665 353 y Fk(\012)f Fo(b)725 336 y Fl(2)743 353 y Fp(\003)772 359 y Fm(H)t Fl(1)822 353 y Fo(\023)837 359 y Fm(A)864 353 y Fp(\(\003)909 359 y Fm(H)t Fl(2)959 353 y Fp(\))424 422 y(=)i Fo(S)493 428 y Fm(B)522 422 y Fp(\()p Fo(b)556 428 y Fl(1)575 403 y(1)593 422 y Fp(\))g Fo(*)f(\017)691 428 y Fm(B)719 422 y Fp(\()p Fo(b)753 428 y Fl(2)772 422 y Fp(\)\003)817 428 y Fm(A)853 422 y Fk(\012)f Fo(b)913 428 y Fl(1)931 403 y(2)950 422 y Fp(\003)979 428 y Fm(H)t Fl(1)1029 422 y Fo(\023)1044 428 y Fm(A)1070 422 y Fp(\(\003)1115 428 y Fm(H)5 b Fl(2)1166 422 y Fp(\))424 491 y(=)12 b Fo(S)493 497 y Fm(B)522 491 y Fp(\()p Fo(b)556 497 y Fl(1)575 472 y(1)593 491 y Fp(\))p Fo(b)627 497 y Fl(2)646 472 y(1)676 491 y Fo(*)f Fp(\003)758 497 y Fm(A)r Fl(1)813 491 y Fk(\012)477 559 y Fo(b)495 565 y Fl(1)514 541 y(2)532 559 y Fo(b)550 565 y Fl(2)569 541 y(2)587 559 y Fp(\()p Fo(b)621 565 y Fl(3)640 559 y Fo(])p Fp(\003)685 565 y Fm(A)r Fl(2)731 559 y Fp(\)\003)776 565 y Fm(A)r Fl(3)821 541 y(1)840 559 y Fp(\003)869 565 y Fm(H)t Fl(1)919 559 y Fo(\023)934 565 y Fm(A)960 559 y Fp(\(\003)1005 565 y Fm(H)5 b Fl(3)1056 559 y Fp(\))p Fo(!)1099 542 y Fj(0)1098 570 y Fm(B)1126 559 y Fp(\(\()p Fo(b)1176 565 y Fl(4)1206 559 y Fo(\()11 b Fp(\003)1288 565 y Fm(A)r Fl(3)1334 541 y(2)1353 559 y Fp(\))g Fk( )g Fp(\003)1462 565 y Fm(H)5 b Fl(2)1512 559 y Fp(\))424 628 y(=)12 b Fo(S)493 634 y Fm(B)522 628 y Fp(\()p Fo(b)556 634 y Fl(1)575 610 y(1)593 634 y(1)612 628 y Fp(\))p Fo(b)646 634 y Fl(1)664 610 y(1)683 634 y(2)713 628 y Fo(*)f Fp(\003)795 634 y Fm(A)r Fl(1)850 628 y Fk(\012)477 697 y Fo(b)495 703 y Fl(1)514 678 y(2)532 697 y Fp(\()p Fo(b)566 703 y Fl(2)585 697 y Fo(])p Fp(\003)630 703 y Fm(A)r Fl(2)676 697 y Fp(\)\003)721 703 y Fm(A)r Fl(3)766 679 y(1)785 697 y Fp(\003)814 703 y Fm(H)t Fl(1)864 697 y Fo(\023)879 703 y Fm(A)905 697 y Fp(\(\003)950 703 y Fm(H)5 b Fl(3)1001 697 y Fp(\))p Fo(!)1044 680 y Fj(0)1043 707 y Fm(B)1071 697 y Fp(\(\()p Fo(b)1121 703 y Fl(3)1151 697 y Fo(\()11 b Fp(\003)1233 703 y Fm(A)r Fl(3)1279 679 y(2)1297 697 y Fp(\))h Fk( )f Fp(\003)1407 703 y Fm(H)t Fl(2)1457 697 y Fp(\))424 765 y(=)h(\003)497 771 y Fm(A)r Fl(1)552 765 y Fk(\012)d Fp(\()p Fo(b)627 771 y Fl(1)646 765 y Fo(])p Fp(\003)691 771 y Fm(A)r Fl(2)736 765 y Fp(\)\003)781 771 y Fm(A)r Fl(3)827 747 y(1)846 765 y Fp(\003)875 771 y Fm(H)t Fl(1)925 765 y Fo(\023)940 771 y Fm(A)966 765 y Fp(\(\003)1011 771 y Fm(H)c Fl(3)1061 765 y Fp(\))p Fo(!)1104 748 y Fj(0)1103 776 y Fm(B)1132 765 y Fp(\(\()p Fo(b)1182 771 y Fl(2)1212 765 y Fo(\()11 b Fp(\003)1294 771 y Fm(A)r Fl(3)1340 747 y(2)1358 765 y Fp(\))h Fk( )f Fp(\003)1468 771 y Fm(H)t Fl(2)1518 765 y Fp(\))257 857 y(No)o(w)j(recall)g(that)g(b)o(y)f (Prop)q(osition)h(2.10)e(w)o(e)j(ha)o(v)o(e:)690 948 y Fo(\032)711 954 y Fm(A)738 948 y Fp(\()p Fo(a)p Fp(\003)805 954 y Fm(A)r Fl(1)851 948 y Fp(\)\003)896 954 y Fm(A)r Fl(2)953 948 y Fp(=)d Fo(\023)1012 954 y Fm(A)1039 948 y Fp(\()p Fo(a)1077 931 y Fl(1)1095 948 y Fp(\))p Fo(S)1138 930 y Fj(\000)p Fl(1)1136 960 y Fm(A)1184 948 y Fp(\()p Fo(a)1222 931 y Fl(2)1241 948 y Fp(\))257 1039 y(Th)o(us)j(w)o(e)f(see) h(that:)284 1131 y Fo(\032)305 1137 y Fm(A)332 1131 y Fp(\()p Fo(a)p Fp(\()p Fo(S)411 1137 y Fm(B)441 1131 y Fp(\()p Fo(b)475 1114 y Fl(1)493 1131 y Fp(\))d Fo(*)f Fp(\003)603 1137 y Fm(A)630 1131 y Fp(\)\))p Fo(b)680 1114 y Fl(2)699 1131 y Fp(\003)728 1137 y Fm(H)t Fl(1)777 1131 y Fo(\023)792 1137 y Fm(A)819 1131 y Fp(\(\003)864 1137 y Fm(H)5 b Fl(2)914 1131 y Fp(\))295 1199 y(=)12 b Fo(\032)360 1205 y Fm(A)388 1199 y Fp(\()p Fo(a)p Fp(\003)455 1205 y Fm(A)r Fl(1)500 1199 y Fp(\)\()p Fo(b)550 1205 y Fl(1)569 1199 y Fo(])p Fp(\003)614 1205 y Fm(A)r Fl(2)660 1199 y Fp(\)\003)705 1205 y Fm(A)r Fl(3)750 1181 y(1)769 1199 y Fp(\003)798 1205 y Fm(H)t Fl(1)848 1199 y Fo(\023)863 1205 y Fm(A)890 1199 y Fp(\(\003)935 1205 y Fm(H)t Fl(3)985 1199 y Fp(\))p Fo(!)1028 1182 y Fj(0)1027 1209 y Fm(B)1055 1199 y Fp(\(\()p Fo(b)1105 1205 y Fl(2)1135 1199 y Fo(\()g Fp(\003)1218 1205 y Fm(A)q Fl(3)1263 1181 y(2)1282 1199 y Fp(\))f Fk( )g Fp(\003)1391 1205 y Fm(H)5 b Fl(2)1441 1199 y Fp(\))295 1267 y(=)12 b Fo(\023)354 1273 y Fm(A)381 1267 y Fp(\()p Fo(a)419 1250 y Fl(1)438 1267 y Fp(\)\()p Fo(b)488 1273 y Fl(1)506 1267 y Fo(]S)549 1249 y Fj(\000)p Fl(1)547 1279 y Fm(A)595 1267 y Fp(\()p Fo(a)633 1250 y Fl(2)652 1267 y Fp(\))668 1273 y Fl(1)686 1267 y Fp(\))p Fo(S)729 1249 y Fj(\000)p Fl(1)727 1279 y Fm(A)775 1267 y Fp(\()p Fo(a)813 1250 y Fl(2)832 1267 y Fp(\))848 1273 y Fl(2)866 1250 y(1)885 1267 y Fp(\003)914 1273 y Fm(H)t Fl(1)964 1267 y Fo(\023)979 1273 y Fm(A)1006 1267 y Fp(\(\003)1051 1273 y Fm(H)t Fl(3)1101 1267 y Fp(\))p Fo(!)1144 1250 y Fj(0)1143 1277 y Fm(B)1171 1267 y Fp(\(\()p Fo(b)1221 1273 y Fl(2)1252 1267 y Fo(\()f(S)1332 1249 y Fj(\000)p Fl(1)1330 1279 y Fm(A)1377 1267 y Fp(\()p Fo(a)1415 1250 y Fl(2)1434 1267 y Fp(\))1450 1273 y Fl(2)1469 1250 y(2)1487 1267 y Fp(\))h Fk( )f Fp(\003)1597 1273 y Fm(H)t Fl(2)1647 1267 y Fp(\))257 1358 y(Prop)q(osition)f(2.10)g(implies)e(for)i(the)i (ordinary)e(Hopf)g(algebra)g Fo(H)j Fp(that)e(\003)1396 1364 y Fm(H)t Fl(1)1448 1358 y Fk(\012)r Fo(S)1507 1364 y Fm(H)1540 1358 y Fp(\(\003)1585 1364 y Fm(H)t Fl(2)1635 1358 y Fp(\))g(is)257 1408 y(a)f(Casimir)e(elemen)o(t.)g(Therefore)j(w) o(e)f(ha)o(v)o(e)g(that)g Fo(h)p Fp(\003)1071 1414 y Fm(H)t Fl(1)1121 1408 y Fo(\023)1136 1414 y Fm(A)1162 1408 y Fp(\(\003)1207 1414 y Fm(H)5 b Fl(2)1258 1408 y Fp(\))11 b(=)h(\003)1358 1414 y Fm(H)t Fl(1)1408 1408 y Fo(\023)1423 1414 y Fm(A)1450 1408 y Fp(\()p Fo(S)1493 1391 y Fj(\000)p Fl(1)1491 1420 y Fm(H)1538 1408 y Fp(\()p Fo(h)p Fp(\)\003)1623 1414 y Fm(H)5 b Fl(2)1673 1408 y Fp(\))257 1458 y(=)12 b Fo(\023)316 1440 y Fj(\000)p Fl(1)316 1470 y Fm(A)360 1458 y Fp(\()p Fo(h)p Fp(\)\003)445 1464 y Fm(H)5 b Fl(1)496 1458 y Fo(\023)511 1464 y Fm(A)537 1458 y Fp(\(\003)582 1464 y Fm(H)g Fl(2)632 1458 y Fp(\).)14 b(No)o(w,)f(inserting)h Fo(a)d Fp(=)h(1)i(in)o(to)f(the)i(ab)q(o)o(v)o (e)e(equation)h(yields:)364 1549 y Fo(\032)385 1555 y Fm(A)412 1549 y Fp(\()p Fo(S)453 1555 y Fm(B)483 1549 y Fp(\()p Fo(b)517 1532 y Fl(1)535 1549 y Fp(\))e Fo(*)f Fp(\003)645 1555 y Fm(A)672 1549 y Fp(\))p Fo(\023)703 1532 y Fj(\000)p Fl(1)703 1562 y Fm(A)747 1549 y Fp(\()p Fo(b)781 1532 y Fl(2)800 1549 y Fp(\)\003)845 1555 y Fm(H)t Fl(1)895 1549 y Fo(\023)910 1555 y Fm(A)937 1549 y Fp(\(\003)982 1555 y Fm(H)t Fl(2)1032 1549 y Fp(\))g(=)h(\003)1132 1555 y Fm(H)t Fl(1)1182 1549 y Fo(\023)1197 1555 y Fm(A)1224 1549 y Fp(\(\003)1269 1555 y Fm(H)t Fl(3)1319 1549 y Fp(\))p Fo(!)1362 1532 y Fj(0)1361 1560 y Fm(B)1389 1549 y Fp(\()p Fo(b)g Fk( )f Fp(\003)1517 1555 y Fm(H)t Fl(2)1567 1549 y Fp(\))257 1641 y(By)h(applying)e(Prop)q(osition)h(2.10)f(to)i Fo(H)880 1626 y Fm(cop)941 1641 y Fp(instead)g(of)f Fo(A)p Fp(,)g(w)o(e)g(see)i(that)f Fo(H)i Fp(is)d(a)g(F)m(rob)q(enius)257 1691 y(algebra)g(with)g(F)m(rob)q(enius)h(morphism)c Fo(\025)896 1697 y Fm(H)928 1691 y Fp(,)j(a)g(left)g(in)o(tegral)g(of)g Fo(H)1285 1675 y Fj(\003)1315 1691 y Fp(satisfying)f Fo(\025)1521 1697 y Fm(H)1553 1691 y Fp(\(\003)1598 1697 y Fm(H)1630 1691 y Fp(\))h(=)257 1740 y(1,)h(and)g(corresp)q(onding)h (Casimir)e(elemen)o(t)g(\003)980 1746 y Fm(H)5 b Fl(2)1036 1740 y Fk(\012)h Fo(S)1101 1723 y Fj(\000)p Fl(1)1099 1753 y Fm(H)1147 1740 y Fp(\(\003)1192 1746 y Fm(H)f Fl(1)1242 1740 y Fp(\).)12 b(As)h(ab)q(o)o(v)o(e,)e(w)o(e)i(therefore) 257 1790 y(ha)o(v)o(e)h(for)g(all)e Fo(h)g Fk(2)f Fo(H)s Fp(:)747 1840 y Fo(h)g Fp(=)h Fo(\025)850 1846 y Fm(H)882 1840 y Fp(\()p Fo(S)925 1822 y Fj(\000)p Fl(1)923 1852 y Fm(H)970 1840 y Fp(\(\003)1015 1846 y Fm(H)5 b Fl(1)1065 1840 y Fp(\))p Fo(h)p Fp(\)\003)1150 1846 y Fm(H)g Fl(2)257 1915 y Fp(Using)14 b(this)g(and)g(the)g(ab)q(o)o(v)o(e)g(equation,)f(w) o(e)h(see:)334 2006 y Fo(\032)355 2012 y Fm(A)382 2006 y Fp(\()p Fo(S)423 2012 y Fm(B)452 2006 y Fp(\()p Fo(b)486 1989 y Fl(1)505 2006 y Fp(\))d Fo(*)h Fp(\003)615 2012 y Fm(A)641 2006 y Fp(\))q Fo(\023)673 1988 y Fj(\000)p Fl(1)673 2018 y Fm(A)717 2006 y Fp(\()p Fo(b)751 1989 y Fl(2)769 2006 y Fp(\))p Fo(\023)800 2012 y Fm(A)827 2006 y Fp(\()p Fo(h)p Fp(\))669 2074 y(=)g Fo(\032)734 2080 y Fm(A)761 2074 y Fp(\()p Fo(S)802 2080 y Fm(B)832 2074 y Fp(\()p Fo(b)866 2057 y Fl(1)884 2074 y Fp(\))g Fo(*)f Fp(\003)994 2080 y Fm(A)1021 2074 y Fp(\))p Fo(\023)1052 2056 y Fj(\000)p Fl(1)1052 2086 y Fm(A)1096 2074 y Fp(\()p Fo(b)1130 2057 y Fl(2)1149 2074 y Fp(\))p Fo(\025)1189 2080 y Fm(H)1221 2074 y Fp(\()p Fo(S)1264 2056 y Fj(\000)p Fl(1)1262 2086 y Fm(H)1309 2074 y Fp(\(\003)1354 2080 y Fm(H)5 b Fl(1)1404 2074 y Fp(\))p Fo(h)p Fp(\))p Fo(\023)1475 2080 y Fm(A)1502 2074 y Fp(\(\003)1547 2080 y Fm(H)g Fl(2)1597 2074 y Fp(\))669 2142 y(=)12 b Fo(\025)737 2148 y Fm(H)769 2142 y Fp(\()p Fo(S)812 2124 y Fj(\000)p Fl(1)810 2154 y Fm(H)857 2142 y Fp(\(\003)902 2148 y Fm(H)5 b Fl(1)952 2142 y Fp(\))p Fo(h)p Fp(\))p Fo(\023)1023 2148 y Fm(A)1050 2142 y Fp(\(\003)1095 2148 y Fm(H)g Fl(3)1145 2142 y Fp(\))p Fo(!)1188 2125 y Fj(0)1187 2152 y Fm(B)1216 2142 y Fp(\()p Fo(b)11 b Fk( )g Fp(\003)1343 2148 y Fm(H)5 b Fl(2)1393 2142 y Fp(\))669 2204 y(=)12 b Fo(\023)728 2210 y Fm(A)755 2204 y Fp(\()p Fo(h)795 2210 y Fl(2)813 2204 y Fp(\))p Fo(!)856 2187 y Fj(0)855 2215 y Fm(B)884 2204 y Fp(\()p Fo(b)f Fk( )g Fo(h)1006 2210 y Fl(1)1025 2204 y Fp(\))257 2296 y(Inserting)k Fo(h)c Fp(=)h(1)i(in)o(to)f(this)h(equation)f(w)o(e)h(get:)668 2387 y Fo(!)695 2370 y Fj(0)694 2397 y Fm(B)722 2387 y Fp(\()p Fo(b)p Fp(\))e(=)f Fo(\032)848 2393 y Fm(A)876 2387 y Fp(\()p Fo(S)917 2393 y Fm(B)946 2387 y Fp(\()p Fo(b)980 2370 y Fl(1)999 2387 y Fp(\))g Fo(*)g Fp(\003)1108 2393 y Fm(A)1135 2387 y Fp(\))p Fo(\023)1166 2369 y Fj(\000)p Fl(1)1166 2399 y Fm(A)1211 2387 y Fp(\()p Fo(b)1245 2370 y Fl(2)1263 2387 y Fp(\))257 2478 y(That)j(is,)f(w)o(e)i(\014nally)d (conclude)j(that)f Fo(!)888 2484 y Fm(B)930 2478 y Fp(and)g Fo(!)1038 2463 y Fj(0)1037 2490 y Fm(B)1079 2478 y Fp(are)g(equal.)953 2628 y(50)p eop %%Page: 51 51 51 50 bop 257 262 a Fp(\(5\))21 b(Since)g(w)o(e)g(ha)o(v)o(e)g(already) f(seen)i(that)e Fo(!)989 268 y Fm(B)1038 262 y Fp(is)h Fo(H)s Fp(-linear,)e(w)o(e)i(kno)o(w)f(from)f(subsec-)257 311 y(tion)e(2.6)g(dualized)h(via)e(Lemma)f(2.5)i(that)h Fo(\036)f Fp(:)h Fo(B)i Fk(!)e Fo(B)r(;)7 b(b)17 b Fk(7!)h Fo(b)1321 317 y Fl(1)1339 311 y Fo(!)1366 294 y Fj(\000)p Fl(1)1365 324 y Fm(B)1411 311 y Fp(\()p Fo(b)1445 317 y Fl(2)1463 311 y Fp(\))g(is)g(an)f(alge-)257 361 y(bra)h(isomorphism.) c(Therefore,)k(w)o(e)g(ha)o(v)o(e)f Fo(\036)p Fp(\()p Fo(b)p Fp(\))p Fo(\036)p Fp(\(\003)1105 367 y Fm(B)1133 361 y Fp(\))h(=)g Fo(\017)1234 367 y Fm(B)1263 361 y Fp(\()p Fo(b)p Fp(\))p Fo(\036)p Fp(\(\003)1383 367 y Fm(B)1411 361 y Fp(\))g(and)f(therefore)257 411 y Fo(\036)p Fp(\(\003)327 417 y Fm(B)356 411 y Fp(\))c(=)g(\003)459 417 y Fm(B)s Fl(1)506 411 y Fo(!)533 393 y Fj(\000)p Fl(1)532 423 y Fm(B)578 411 y Fp(\(\003)623 417 y Fm(B)s Fl(2)670 411 y Fp(\))i(is)f(a)h(nonzero)g(left)g(in)o(tegral)f(with)g (resp)q(ect)j(to)e(the)g(c)o(haracter)257 461 y Fo(!)283 467 y Fm(B)312 461 y Fp(.)h(But)i(no)o(w)f(b)o(y)g(Prop)q(osition)g (2.7,)e(w)o(e)j(kno)o(w)e(that)h(the)h(space)g(of)f(suc)o(h)h(in)o (tegrals)f(is)257 511 y(one-dimensional,)10 b(and)i(therefore)h Fo(x)847 517 y Fm(B)887 511 y Fp(m)o(ust)e(b)q(e)i(prop)q(ortional)e (to)h(\003)1358 517 y Fm(B)s Fl(1)1405 511 y Fo(!)1432 493 y Fj(\000)p Fl(1)1431 523 y Fm(B)1477 511 y Fp(\(\003)1522 517 y Fm(B)s Fl(2)1569 511 y Fp(\).)g(This)257 560 y(\014nishes)j(the)g (pro)q(of)e(of)g(the)i(\014rst)g(part)f(of)f(the)h(Prop)q(osition.)f Fg(\003)257 676 y Fp(It)h(should)g(b)q(e)g(noted)g(that)g(the)g (assumption)e(that)i Fo(A)p Fp(,)g Fo(H)i Fp(and)e Fo(B)i Fp(is)d(necessary)m(,)i(since)g(b)o(y)257 726 y([34)o(],)c(Cor.)g(2.7,) f(p.)i(330)f(or)g([33)o(],)g(Chap.)g(V,)g(Ex.)g(4,)g(p.)h(108)e(an)i (in\014nite-dimensional)d(Hopf)257 775 y(algebra)14 b(do)q(es)g(not)g (con)o(tain)g(a)f(nonzero)i(in)o(tegral.)257 910 y Fn(5.3)48 b Fp(As)11 b(explained)f(in)g([31)o(],)f(the)i(construction)g (discussed)h(ab)q(o)o(v)o(e)f(con)o(tains)f(as)g(a)g(sp)q(ecial)257 959 y(case)19 b(the)f(so-called)f(double)g(crosspro)q(duct)j(\(cf.)d ([35)o(]\).)f(Here,)i Fo(A)g Fp(and)f Fo(B)j Fp(are)e(ordinary)257 1009 y(Hopf)c(algebras)g(acting)f(on)h(eac)o(h)g(other)h(via)649 1098 y Fo(*)p Fp(:)c Fo(B)h Fk(\012)d Fo(A)j Fk(!)f Fo(A)94 b(\()p Fp(:)11 b Fo(B)h Fk(\012)d Fo(A)j Fk(!)f Fo(B)257 1186 y Fp(suc)o(h)k(that)f(the)g(follo)o(wing)e(conditions)h(are)i (satis\014ed:)308 1302 y(1.)20 b(\001)396 1308 y Fm(A)423 1302 y Fp(\()p Fo(b)11 b(*)g(a)p Fp(\))h(=)g(\()p Fo(b)649 1308 y Fl(1)679 1302 y Fo(*)f(a)754 1308 y Fl(1)772 1302 y Fp(\))f Fk(\012)f Fp(\()p Fo(b)873 1308 y Fl(2)903 1302 y Fo(*)i(a)978 1308 y Fl(2)997 1302 y Fp(\))361 1364 y(\001)396 1370 y Fm(B)424 1364 y Fp(\()p Fo(b)h(\()f(a)p Fp(\))g(=)h(\()p Fo(b)650 1370 y Fl(1)680 1364 y Fo(\()f(a)755 1370 y Fl(1)774 1364 y Fp(\))e Fk(\012)h Fp(\()p Fo(b)875 1370 y Fl(2)905 1364 y Fo(\()h(a)980 1370 y Fl(2)999 1364 y Fp(\))308 1445 y(2.)20 b Fo(b)11 b(*)h Fp(\()p Fo(aa)504 1430 y Fj(0)515 1445 y Fp(\))g(=)g(\()p Fo(b)621 1451 y Fl(1)651 1445 y Fo(*)f(a)726 1451 y Fl(1)744 1445 y Fp(\)\(\()p Fo(b)810 1451 y Fl(2)841 1445 y Fo(\()g(a)916 1451 y Fl(2)934 1445 y Fp(\))h Fo(*)f(a)1037 1430 y Fj(0)1049 1445 y Fp(\))361 1507 y(\()p Fo(bb)413 1492 y Fj(0)425 1507 y Fp(\))g Fo(\()g(a)h Fp(=)g(\()p Fo(b)f(\()g Fp(\()p Fo(b)715 1492 y Fj(0)715 1517 y Fl(1)745 1507 y Fo(*)g(a)820 1513 y Fl(1)839 1507 y Fp(\)\)\()p Fo(b)905 1492 y Fj(0)905 1517 y Fl(2)935 1507 y Fo(\()g(a)1010 1513 y Fl(2)1029 1507 y Fp(\))308 1589 y(3.)20 b Fo(\017)378 1595 y Fm(A)405 1589 y Fp(\()p Fo(b)12 b(*)f(a)p Fp(\))g(=)h Fo(\017)614 1595 y Fm(B)642 1589 y Fp(\()p Fo(b)p Fp(\))p Fo(\017)709 1595 y Fm(A)736 1589 y Fp(\()p Fo(a)p Fp(\))g(=)g Fo(\017)863 1595 y Fm(B)891 1589 y Fp(\()p Fo(b)g(\()f(a)p Fp(\))308 1671 y(4.)20 b Fo(b)11 b(*)h Fp(1)f(=)h Fo(\017)537 1677 y Fm(B)565 1671 y Fp(\()p Fo(b)p Fp(\)1)p Fo(;)7 b Fp(1)k Fo(\()g(a)g Fp(=)h Fo(\017)834 1677 y Fm(A)861 1671 y Fp(\()p Fo(a)p Fp(\)1)308 1753 y(5.)20 b(\()p Fo(b)395 1759 y Fl(1)425 1753 y Fo(*)11 b(a)500 1759 y Fl(1)519 1753 y Fp(\))e Fk(\012)h Fp(\()p Fo(b)620 1759 y Fl(2)650 1753 y Fo(\()h(a)725 1759 y Fl(2)743 1753 y Fp(\))h(=)g(\()p Fo(b)849 1759 y Fl(2)879 1753 y Fo(*)f(a)954 1759 y Fl(2)973 1753 y Fp(\))e Fk(\012)h Fp(\()p Fo(b)1074 1759 y Fl(1)1104 1753 y Fo(\()h(a)1179 1759 y Fl(1)1197 1753 y Fp(\))257 1868 y(In)j(this)g(situation,)f Fo(A)c Fk(\012)h Fo(B)16 b Fp(is)e(a)g(Hopf)f(algebra)g(with)h(m)o(ultiplicatio)o(n)586 1957 y(\()p Fo(a)c Fk(\012)f Fo(b)p Fp(\)\()p Fo(a)747 1940 y Fj(0)768 1957 y Fk(\012)h Fo(b)828 1940 y Fj(0)839 1957 y Fp(\))i(=)g Fo(a)p Fp(\()p Fo(b)967 1963 y Fl(1)997 1957 y Fo(*)f(a)1072 1940 y Fj(0)1072 1967 y Fl(1)1090 1957 y Fp(\))f Fk(\012)f Fp(\()p Fo(b)1191 1963 y Fl(2)1221 1957 y Fo(\()i(a)1296 1940 y Fj(0)1296 1967 y Fl(2)1315 1957 y Fp(\))p Fo(b)1349 1940 y Fj(0)257 2046 y Fp(unit)j(elemen)o(t)f (1)c Fk(\012)h Fp(1,)j(com)o(ultiplication)682 2134 y(\001\()p Fo(a)c Fk(\012)g Fo(b)p Fp(\))j(=)f(\()p Fo(a)932 2140 y Fl(1)960 2134 y Fk(\012)f Fo(b)1020 2140 y Fl(1)1038 2134 y Fp(\))g Fk(\012)f Fp(\()p Fo(a)1143 2140 y Fl(2)1171 2134 y Fk(\012)h Fo(b)1231 2140 y Fl(2)1249 2134 y Fp(\))257 2223 y(counit)779 2273 y Fo(\017)p Fp(\()p Fo(a)f Fk(\012)h Fo(b)p Fp(\))h(=)h Fo(\017)991 2279 y Fm(A)1018 2273 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1089 2279 y Fm(B)1118 2273 y Fp(\()p Fo(b)p Fp(\))257 2346 y(and)i(an)o(tip)q(o)q(de)661 2395 y Fo(S)r Fp(\()p Fo(a)c Fk(\012)g Fo(b)p Fp(\))h(=)h(\(1)d Fk(\012)h Fo(S)980 2401 y Fm(B)1009 2395 y Fp(\()p Fo(b)p Fp(\)\)\()p Fo(S)1116 2401 y Fm(A)1144 2395 y Fp(\()p Fo(a)p Fp(\))f Fk(\012)h Fp(1\))257 2503 y(As)15 b(a)e(corollary)m(,)f (w)o(e)j(can)f(calculate)g(the)g(in)o(tegrals)g(of)f(the)h(double)g (crosspro)q(duct:)953 2628 y(51)p eop %%Page: 52 52 52 51 bop 257 249 a Fn(Corollary)257 299 y Fp(Supp)q(ose)15 b(that)f Fo(A)g Fp(and)g Fo(B)i Fp(are)f(\014nite-dimensional.)308 417 y(1.)20 b(Supp)q(ose)14 b(that)f(\003)641 423 y Fm(A)681 417 y Fp(resp.)h(\003)809 423 y Fm(B)850 417 y Fp(are)g(nonzero)g(left) f(in)o(tegrals)f(of)h Fo(A)g Fp(resp.)h Fo(B)r Fp(.)f(Cho)q(ose)361 467 y(a)h(righ)o(t)f(in)o(tegral)g Fo(\032)668 473 y Fm(A)710 467 y Fp(of)g Fo(A)788 452 y Fj(\003)821 467 y Fp(satisfying)g Fo(\032)1027 473 y Fm(A)1054 467 y Fp(\(\003)1099 473 y Fm(A)1126 467 y Fp(\))f(=)g(1)i(and)f(de\014ne)705 559 y Fo(!)731 565 y Fm(B)771 559 y Fp(:)e Fo(B)j Fk(!)d Fo(K)q(;)18 b(b)11 b Fk(7!)g Fo(\032)1061 565 y Fm(A)1089 559 y Fp(\()p Fo(S)1130 565 y Fm(B)1159 559 y Fp(\()p Fo(b)p Fp(\))h Fo(*)f Fp(\003)1303 565 y Fm(A)1330 559 y Fp(\))361 650 y(Then)18 b Fo(!)499 656 y Fm(B)546 650 y Fp(is)f(a)h(c)o(haracter)h(of)e Fo(B)k Fp(and)c(\003)1031 656 y Fm(A)1070 650 y Fk(\012)12 b Fp(\003)1143 656 y Fm(B)s Fl(1)1190 650 y Fo(!)1217 632 y Fj(\000)p Fl(1)1216 662 y Fm(B)1262 650 y Fp(\(\003)1307 656 y Fm(B)s Fl(2)1354 650 y Fp(\))18 b(is)g(a)f(nonzero)i(left)361 700 y(in)o(tegral)13 b(of)h(the)g(double)g(crosspro)q(duct)i Fo(A)9 b Fk(\012)h Fo(B)r Fp(.)308 783 y(2.)20 b(Supp)q(ose)12 b(that)g(\000)635 789 y Fm(A)673 783 y Fp(resp.)g(\000)796 789 y Fm(B)836 783 y Fp(are)g(nonzero)g(righ)o(t)f(in)o(tegrals)g(of)g Fo(A)g Fp(resp.)h Fo(B)r Fp(.)g(Cho)q(ose)361 833 y(a)i(left)f(in)o (tegral)h Fo(\025)644 839 y Fm(B)686 833 y Fp(of)f Fo(B)766 817 y Fj(\003)800 833 y Fp(satisfying)g Fo(\025)1009 839 y Fm(B)1038 833 y Fp(\(\000)1080 839 y Fm(B)1108 833 y Fp(\))f(=)g(1)h(and)h(de\014ne)702 924 y Fo(!)728 930 y Fm(A)766 924 y Fp(:)e Fo(A)f Fk(!)g Fo(K)q(;)18 b(a)12 b Fk(7!)f Fo(\025)1062 930 y Fm(B)1090 924 y Fp(\(\000)1132 930 y Fm(B)1173 924 y Fo(\()g(S)1251 930 y Fm(A)1278 924 y Fp(\()p Fo(a)p Fp(\)\))361 1015 y(Then)k Fo(!)496 1021 y Fm(A)536 1015 y Fp(is)f(a)g(c)o(haracter)h(of)e Fo(A)h Fp(and)849 1106 y Fo(!)876 1089 y Fj(\000)p Fl(1)875 1119 y Fm(A)921 1106 y Fp(\(\000)963 1112 y Fm(A)r Fl(1)1009 1106 y Fp(\)\000)1051 1112 y Fm(A)r Fl(2)1106 1106 y Fk(\012)9 b Fp(\000)1173 1112 y Fm(B)361 1198 y Fp(is)14 b(a)f(nonzero)i(righ)o(t)f(in)o(tegral)f(of)g(the)h(double)g(crosspro)q (duct)i Fo(A)10 b Fk(\012)f Fo(B)r Fp(.)257 1333 y Fn(5.4)48 b Fp(In)18 b(this)g(subsection)h(w)o(e)f(calculate)g(the)g(in)o (tegrals)g(of)f(the)i(second)f(construction)257 1383 y(giv)o(en)j(in)f([31)o(].)g(W)m(e)g(w)o(ork)h(in)f(the)i(follo)o(wing) c(situation:)i Fo(H)j Fp(is)e(a)g(\014nite-dimensional)257 1433 y(comm)o(utativ)o(e)14 b(and)h(co)q(comm)o(utativ)o(e)f(Hopf)i (algebra.)f(W)m(e)h(assume)f(that)h Fo(A)h Fp(is)f(a)f(\014nite-)257 1483 y(dimensional)c(left)h(Y)m(etter-Drinfel'd)g(Hopf)g(algebra)h(o)o (v)o(er)f Fo(H)s Fp(.)g(W)m(e)g(imp)q(ose)g(the)h(follo)o(wing)257 1532 y(main)f(assumption)h(on)g Fo(A)p Fp(:)569 1624 y Fk(8)p Fo(a;)7 b(a)655 1607 y Fj(0)677 1624 y Fk(2)12 b Fo(A)f Fp(:)g(\()p Fo(a)820 1607 y Fl(1)851 1624 y Fk(!)g Fo(a)926 1607 y Fj(0)937 1624 y Fp(\))f Fk(\012)f Fo(a)1026 1607 y Fl(2)1056 1624 y Fp(=)j Fo(a)1122 1607 y Fj(0)1134 1609 y Fl(2)1163 1624 y Fk(\012)e Fp(\()p Fo(a)1243 1607 y Fj(0)1255 1609 y Fl(1)1287 1624 y Fk(!)h Fo(a)p Fp(\))257 1715 y(W)m(e)j(in)o(tro)q(duce)g(a)f(righ)o(t)h(Y)m (etter-Drinfel'd)f(Hopf)g(algebra)h Fo(B)i Fp(in)d(the)h(follo)o(wing)d (w)o(a)o(y:)i(W)m(e)257 1765 y(set)i Fo(B)f Fp(=)e Fo(A)442 1750 y Fj(\003)474 1765 y Fp(as)h(an)h(algebra)e(and)i(as)f(an)g Fo(H)s Fp(-mo)q(dule)f(in)h(the)h(sense)h(of)d(subsection)j(2.3.)d(If) 257 1815 y Fo(\016)275 1821 y Fm(A)300 1813 y Fi(\003)335 1815 y Fp(and)j(\001)452 1821 y Fm(A)477 1813 y Fi(\003)511 1815 y Fp(denote)h(the)g(co)q(op)q(eration)f(and)g(com)o(ultiplication) d(from)h(subsection)j(2.3)257 1865 y(resp)q(ectiv)o(ely)m(,)f(w)o(e)f (de\014ne)h(the)f(co)q(op)q(eration)g(and)g(the)h(com)o(ultiplicatio)o (n)c(of)i Fo(B)k Fp(b)o(y:)541 1956 y Fo(\016)559 1962 y Fm(B)600 1956 y Fp(=)11 b(\()p Fo(id)695 1962 y Fm(A)720 1954 y Fi(\003)749 1956 y Fk(\012)f Fo(S)816 1962 y Fm(H)848 1956 y Fp(\))f Fk(\016)g Fo(\016)921 1962 y Fm(A)946 1954 y Fi(\003)1049 1956 y Fp(\001)1084 1962 y Fm(B)1123 1956 y Fp(=)j Fo(\033)1192 1938 y Fj(\000)p Fl(1)1191 1968 y Fm(A)1216 1960 y Fi(\003)1233 1968 y Fm(;A)1268 1960 y Fi(\003)1297 1956 y Fk(\016)d Fp(\001)1362 1962 y Fm(A)1387 1954 y Fi(\003)257 2047 y Fp(where)14 b Fo(\033)f Fp(is)f(the)h(braiding)e(in)g(the)i(category)g(of)e(righ)o(t)h(Y)m (etter-Drinfel'd)g(mo)q(dules.)e(When)257 2097 y(w)o(e)k(use)g(Sw)o (eedler's)g(notation,)e(w)o(e)h(refer)h(to)f(the)h(structure)h(of)e Fo(B)j Fp(and)d(not)g(to)g(the)g(struc-)257 2147 y(ture)i(of)e Fo(A)423 2132 y Fj(\003)443 2147 y Fp(.)257 2232 y(Let)f Fk(h\001)p Fo(;)7 b Fk(\001i)404 2238 y Fm(A)441 2232 y Fp(:)k Fo(A)s Fk(\012)s Fo(B)j Fk(!)e Fo(K)q(;)7 b(a)s Fk(\012)s Fo(b)j Fk(7!)h Fo(b)p Fp(\()p Fo(a)p Fp(\))g(denote)h(the)f (natural)f(pairing)g(b)q(et)o(w)o(een)i(a)f(v)o(ector)257 2282 y(space)g(and)e(its)g(dual.)f(W)m(e)h(also)g(use)h(the)g(form)e Fk(h\001)p Fo(;)f Fk(\001i)1041 2288 y Fm(B)1080 2282 y Fp(:)k Fo(A)p Fk(\012)p Fo(B)j Fk(!)e Fo(K)q(;)7 b(a)p Fk(\012)p Fo(b)k Fk(7!)g(h)p Fo(S)1499 2264 y Fj(\000)p Fl(1)1497 2294 y Fm(A)1544 2282 y Fp(\()p Fo(a)p Fp(\))p Fo(;)c(b)p Fk(i)1651 2288 y Fm(A)1678 2282 y Fp(.)257 2368 y(Recall)13 b(the)i(second)g(construction)g(from)d([31)o(]:)953 2628 y(52)p eop %%Page: 53 53 53 52 bop 257 249 a Fn(Prop)q(osition)257 299 y Fo(A)10 b Fk(\012)f Fo(H)j Fk(\012)e Fo(B)16 b Fp(is)e(a)g(Hopf)f(algebra)g (with)h(com)o(ultiplication:)543 383 y(\001)d(:)g Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)15 b Fk(!)c Fp(\()p Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(\))h Fk(\012)g Fp(\()p Fo(A)f Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(\))504 467 y Fo(a)g Fk(\012)h Fo(h)f Fk(\012)g Fo(b)j Fk(7!)f Fp(\()p Fo(a)772 473 y Fl(1)800 467 y Fk(\012)e Fo(a)863 473 y Fl(2)882 450 y(1)901 467 y Fo(h)925 473 y Fl(1)952 467 y Fk(\012)h Fo(b)1012 473 y Fl(1)1030 449 y(1)1049 467 y Fp(\))f Fk(\012)h Fp(\()p Fo(a)1154 473 y Fl(2)1172 450 y(2)1200 467 y Fk(\012)g Fo(h)1266 473 y Fl(2)1284 467 y Fo(b)1302 473 y Fl(1)1321 449 y(2)1349 467 y Fk(\012)f Fo(b)1408 473 y Fl(2)1427 467 y Fp(\))271 538 y(m)o(ultiplication:)263 622 y Fo(\026)i Fp(:)g(\()p Fo(A)f Fk(\012)f Fo(H)k Fk(\012)c Fo(B)s Fp(\))g Fk(\012)h Fp(\()p Fo(A)f Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(\))j Fk(!)f Fo(A)f Fk(\012)f Fo(H)j Fk(\012)e Fo(B)361 684 y Fp(\()p Fo(a)g Fk(\012)f Fo(h)g Fk(\012)h Fo(b)p Fp(\))f Fk(\012)h Fp(\()p Fo(a)648 667 y Fj(0)668 684 y Fk(\012)g Fo(h)734 667 y Fj(0)755 684 y Fk(\012)f Fo(b)814 667 y Fj(0)826 684 y Fp(\))i Fk(7!)543 757 y Fo(a)p Fp(\()p Fo(h)605 763 y Fl(1)635 757 y Fk(!)g Fp(\()p Fo(b)722 763 y Fl(1)740 739 y(1)771 757 y Fo(*)g(a)846 740 y Fj(0)846 768 y Fl(1)864 757 y Fp(\)\))f Fk(\012)f Fo(h)971 763 y Fl(2)990 757 y Fo(b)1008 763 y Fl(1)1026 739 y(2)1045 757 y Fp(\()p Fo(b)1079 763 y Fl(2)1097 757 y Fo(]a)1135 740 y Fj(0)1135 768 y Fl(2)1154 757 y Fp(\))p Fo(a)1192 740 y Fj(0)1192 768 y Fl(3)1211 734 y(1)1229 757 y Fo(h)1253 740 y Fj(0)1253 768 y Fl(1)1281 757 y Fk(\012)h Fp(\(\()p Fo(b)1373 763 y Fl(3)1403 757 y Fo(\()h(a)1478 740 y Fj(0)1478 768 y Fl(3)1497 734 y(2)1515 757 y Fp(\))h Fk( )f Fo(h)1620 740 y Fj(0)1620 768 y Fl(2)1638 757 y Fp(\))p Fo(b)1672 740 y Fj(0)257 842 y Fp(counit:)794 891 y Fo(\017)h Fp(:)f Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)14 b Fk(!)d Fo(K)709 962 y(a)f Fk(\012)f Fo(h)g Fk(\012)h Fo(b)h Fk(7!)g Fo(\017)956 968 y Fm(A)983 962 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)1054 968 y Fm(H)1086 962 y Fp(\()p Fo(h)p Fp(\))p Fo(\017)1159 968 y Fm(B)1187 962 y Fp(\()p Fo(b)p Fp(\))257 1032 y(unit)j(1)9 b Fk(\012)g Fp(1)g Fk(\012)h Fp(1)k(and)f(an)o(tip)q(o)q(de:)706 1116 y Fo(S)h Fp(:)d Fo(A)f Fk(\012)f Fo(H)j Fk(\012)e Fo(B)k Fk(!)d Fo(A)e Fk(\012)h Fo(H)i Fk(\012)d Fo(B)377 1201 y(a)g Fk(\012)g Fo(h)g Fk(\012)h Fo(b)h Fk(7!)g Fp(\(1)e Fk(\012)h Fp(1)f Fk(\012)g Fo(S)790 1207 y Fm(B)819 1201 y Fp(\()p Fo(b)853 1184 y Fl(1)872 1201 y Fp(\)\)\(1)g Fk(\012)h Fo(S)1017 1207 y Fm(H)1049 1201 y Fp(\()p Fo(a)1087 1184 y Fl(1)1106 1201 y Fo(hb)1148 1184 y Fl(2)1166 1201 y Fp(\))f Fk(\012)h Fp(1\)\()p Fo(S)1311 1207 y Fm(A)1338 1201 y Fp(\()p Fo(a)1376 1184 y Fl(2)1395 1201 y Fp(\))f Fk(\012)h Fp(1)f Fk(\012)g Fp(1\))257 1271 y(where)15 b Fo(*)p Fp(,)e Fo(\()h Fp(are)g(the)g(coadjoin)o(t)f(actions)610 1355 y Fo(b)e(*)g(a)h Fp(=)g Fk(h)p Fp(\()p Fo(S)827 1361 y Fm(H)859 1355 y Fp(\()p Fo(a)897 1361 y Fl(2)915 1338 y(1)934 1355 y Fp(\))g Fk(!)f Fo(a)1037 1361 y Fl(1)1055 1355 y Fp(\))p Fo(S)1096 1361 y Fm(A)1124 1355 y Fp(\()p Fo(a)1162 1361 y Fl(3)1181 1355 y Fp(\))p Fo(;)c(b)p Fk(i)1250 1361 y Fm(B)1278 1355 y Fo(a)1300 1361 y Fl(2)1318 1338 y(2)610 1424 y Fo(b)k(\()g(a)h Fp(=)g Fk(h)p Fo(a;)7 b(S)852 1430 y Fm(B)880 1424 y Fp(\()p Fo(b)914 1430 y Fl(1)933 1424 y Fp(\)\()p Fo(b)983 1430 y Fl(3)1013 1424 y Fk( )k Fo(S)1091 1430 y Fm(H)1123 1424 y Fp(\()p Fo(b)1157 1430 y Fl(2)1175 1406 y(2)1194 1424 y Fp(\)\))p Fk(i)1242 1430 y Fm(A)1270 1424 y Fo(b)1288 1430 y Fl(2)1306 1406 y(1)257 1508 y Fp(and)j Fo(])g Fp(is)g(de\014ned)h(as:)681 1593 y Fo(b]a)c Fp(:=)h Fk(h)p Fo(a)842 1599 y Fl(1)860 1593 y Fo(;)7 b(b)897 1599 y Fl(1)915 1574 y(1)934 1593 y Fk(i)950 1599 y Fm(B)978 1593 y Fo(b)996 1599 y Fl(1)1015 1574 y(2)1034 1593 y Fo(a)1056 1599 y Fl(2)1074 1575 y(1)1093 1593 y Fk(h)p Fo(a)1131 1599 y Fl(2)1149 1575 y(2)1168 1593 y Fo(;)g(b)1205 1599 y Fl(2)1223 1593 y Fk(i)1239 1599 y Fm(A)257 1709 y Fp(W)m(e)14 b(shall)f(need)i(the)f (follo)o(wing)e(Lemma.)257 1796 y Fn(Lemma)308 1846 y Fp(1.)20 b(Supp)q(ose)15 b(that)f(\003)643 1852 y Fm(A)684 1846 y Fp(and)f(\000)790 1852 y Fm(A)831 1846 y Fp(are)h(a)g(left)g (and)f(a)h(righ)o(t)f(in)o(tegral)h(of)f Fo(A)p Fp(.)g(Then)831 1930 y Fo(B)h Fk(!)e Fo(K)q(;)7 b(b)j Fk(7!)h(h)p Fp(\003)1111 1936 y Fm(A)1138 1930 y Fo(;)c(b)p Fk(i)1191 1936 y Fm(B)361 2014 y Fp(is)14 b(a)f(left)h(in)o(tegral)f(and)833 2064 y Fo(B)h Fk(!)d Fo(K)q(;)c(b)k Fk(7!)g(h)p Fp(\000)1110 2070 y Fm(A)1137 2064 y Fo(;)c(b)p Fk(i)1190 2070 y Fm(B)361 2134 y Fp(is)14 b(a)f(righ)o(t)h(in)o(tegral)f(of)g Fo(B)769 2119 y Fj(\003)308 2214 y Fp(2.)20 b(Supp)q(ose)15 b(that)f(\003)643 2220 y Fm(B)685 2214 y Fp(and)g(\000)792 2220 y Fm(B)834 2214 y Fp(are)h(a)e(left)h(and)f(a)h(righ)o(t)f(in)o(tegral)h(of)f Fo(B)r Fp(.)h(Then)828 2299 y Fo(A)e Fk(!)f Fo(K)q(;)c(a)k Fk(7!)g(h)p Fo(a;)c Fp(\003)1151 2305 y Fm(B)1179 2299 y Fk(i)1195 2305 y Fm(A)361 2383 y Fp(is)14 b(a)f(left)h(in)o(tegral)f (and)830 2433 y Fo(A)f Fk(!)f Fo(K)q(;)c(a)j Fk(7!)h(h)p Fo(a;)c Fp(\000)1149 2439 y Fm(B)1178 2433 y Fk(i)1194 2439 y Fm(A)361 2503 y Fp(is)14 b(a)f(righ)o(t)h(in)o(tegral)f(of)g Fo(A)767 2488 y Fj(\003)787 2503 y Fp(.)953 2628 y(53)p eop %%Page: 54 54 54 53 bop 308 262 a Fp(3.)20 b(If)e Fo(a)429 246 y Fm(L)429 273 y(A)475 262 y Fp(\(resp.)h Fo(a)617 246 y Fm(R)617 273 y(A)644 262 y Fp(\))g(is)f(the)h(left)f(\(resp.)i(righ)o(t\))e(mo)q (dular)f(elemen)o(t)h(of)f Fo(A)i Fp(and)f Fo(\013)1661 246 y Fm(L)1661 273 y(B)361 311 y Fp(\(resp.)d Fo(\013)504 296 y Fm(R)504 323 y(B)532 311 y Fp(\))f(is)g(the)g(left)g(\(resp.)g (righ)o(t\))g(mo)q(dular)e(function)i(of)f Fo(B)r Fp(,)h(w)o(e)g(ha)o (v)o(e:)679 403 y Fk(h)p Fo(a)717 386 y Fm(L)717 413 y(A)744 403 y Fo(;)7 b(b)p Fk(i)797 409 y Fm(A)835 403 y Fp(=)k Fo(\013)905 386 y Fm(L)905 413 y(B)934 403 y Fp(\()p Fo(b)p Fp(\))83 b Fk(h)p Fo(a)1105 386 y Fm(R)1105 413 y(A)1132 403 y Fo(;)7 b(b)p Fk(i)1185 409 y Fm(A)1223 403 y Fp(=)12 b Fo(\013)1294 386 y Fm(R)1294 413 y(B)1322 403 y Fp(\()p Fo(b)p Fp(\))308 511 y(4.)20 b(If)e Fo(a)429 496 y Fm(L)429 522 y(B)476 511 y Fp(\(resp.)h Fo(a)618 496 y Fm(R)618 522 y(B)646 511 y Fp(\))g(is)f(the)g(left)g(\(resp.)h (righ)o(t\))f(mo)q(dular)f(elemen)o(t)g(of)h Fo(B)j Fp(and)d Fo(\013)1663 496 y Fm(L)1663 522 y(A)361 560 y Fp(\(resp.)d Fo(\013)504 545 y Fm(R)504 572 y(A)531 560 y Fp(\))f(is)f(the)i(left)f (\(resp.)g(righ)o(t\))g(mo)q(dular)e(function)h(of)h Fo(A)p Fp(,)f(w)o(e)h(ha)o(v)o(e:)669 652 y Fk(h)p Fo(a;)7 b(a)748 635 y Fm(L)748 662 y(A)775 652 y Fk(i)791 658 y Fm(B)831 652 y Fp(=)12 b Fo(\013)902 635 y Fm(L)902 662 y(A)929 652 y Fp(\()p Fo(a)p Fp(\))83 b Fk(h)p Fo(a;)7 b(a)1145 635 y Fm(R)1145 662 y(B)1173 652 y Fk(i)1189 658 y Fm(B)1229 652 y Fp(=)12 b Fo(\013)1300 635 y Fm(R)1300 662 y(A)1327 652 y Fp(\()p Fo(a)p Fp(\))257 751 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(ha)o(v)o(e)f(b)o(y)h([32)o(],)f(Prop)q(osition)g(4.2:) 385 843 y Fk(h)p Fo(a;)7 b(b)460 849 y Fl(1)478 843 y Fk(i)494 849 y Fm(B)523 843 y Fk(h)p Fp(\003)568 849 y Fm(A)595 843 y Fo(;)g(b)632 849 y Fl(2)650 843 y Fk(i)666 849 y Fm(B)706 843 y Fp(=)12 b Fk(h)p Fo(a)p Fp(\003)817 849 y Fm(A)844 843 y Fo(;)7 b(b)p Fk(i)897 849 y Fm(B)936 843 y Fp(=)12 b Fo(\017)997 849 y Fm(A)1024 843 y Fp(\()p Fo(a)p Fp(\))p Fk(h)p Fp(\003)1123 849 y Fm(A)1150 843 y Fo(;)7 b(b)p Fk(i)1203 849 y Fm(B)1243 843 y Fp(=)12 b Fk(h)p Fo(a;)7 b Fp(1)p Fk(i)1381 849 y Fm(B)1409 843 y Fk(h)p Fp(\003)1454 849 y Fm(A)1481 843 y Fo(;)g(b)p Fk(i)1534 849 y Fm(B)257 934 y Fp(W)m(e)14 b(conclude:)g Fo(b)529 940 y Fl(1)547 934 y Fk(h)p Fp(\003)592 940 y Fm(A)619 934 y Fo(;)7 b(b)656 940 y Fl(2)674 934 y Fk(i)690 940 y Fm(B)731 934 y Fp(=)k(1)p Fk(h)p Fp(\003)840 940 y Fm(A)867 934 y Fo(;)c(b)p Fk(i)920 940 y Fm(B)962 934 y Fp(The)14 b(other)h(v)o(eri\014cations)f(are)g(similar.)d Fg(\003)257 1034 y Fn(Theorem)36 b Fp(Supp)q(ose)16 b(that)g(\003)766 1040 y Fm(A)808 1034 y Fp(and)g(\003)920 1040 y Fm(H)966 1034 y Fp(are)g(left)g(in)o(tegrals)f(of)g Fo(A)g Fp(resp.)i Fo(H)h Fp(and)d(that)257 1083 y(\000)283 1089 y Fm(B)326 1083 y Fp(is)e(a)h(righ)o(t)f(in)o(tegral)h(of)f Fo(B)r Fp(.)h(Then)752 1175 y(\003)781 1181 y Fm(A)817 1175 y Fk(\012)9 b Fp(\003)887 1181 y Fm(H)918 1183 y Fl(1)937 1175 y Fo(\023)952 1181 y Fm(A)979 1175 y Fp(\(\003)1024 1181 y Fm(H)1055 1183 y Fl(2)1074 1175 y Fp(\))g Fk(\012)h Fp(\000)1167 1181 y Fm(B)257 1266 y Fp(is)j(a)g(left)g(and)g(a)g(righ)o (t)g(in)o(tegral)g(of)f Fo(A)c Fk(\012)g Fo(H)j Fk(\012)d Fo(B)r Fp(,)14 b(where)g Fo(\023)1173 1272 y Fm(A)1213 1266 y Fp(is)f(the)h(in)o(tegral)e(c)o(haracter)j(of)257 1316 y Fo(A)p Fp(.)f(In)g(particular,)f Fo(A)c Fk(\012)h Fo(H)i Fk(\012)d Fo(B)17 b Fp(is)d(unimo)q(dular.)257 1416 y Fn(Pro)q(of.)36 b Fp(W)m(e)14 b(can)h(assume)f(that)h(the)g(in)o (tegrals)f(in)o(v)o(olv)o(ed)f(are)i(nonzero.)g(Select)h(a)e(righ)o(t) 257 1465 y(in)o(tegral)g Fo(\032)430 1471 y Fm(A)471 1465 y Fp(of)f Fo(A)549 1450 y Fj(\003)582 1465 y Fp(satisfying)g Fo(\032)788 1471 y Fm(A)816 1465 y Fp(\(\003)861 1471 y Fm(A)888 1465 y Fp(\))e(=)h(1)i(and)f(de\014ne:)586 1557 y Fo(!)612 1563 y Fm(B)651 1557 y Fp(:)f Fo(B)i Fk(!)d Fo(K)q(;)c(b)j Fk(7!)h Fo(\032)930 1563 y Fm(A)958 1557 y Fp(\()p Fo(S)999 1563 y Fm(B)1028 1557 y Fp(\()p Fo(b)1062 1540 y Fl(1)1081 1557 y Fp(\))g Fo(*)g Fp(\003)1190 1563 y Fm(A)1217 1557 y Fp(\))p Fo(\023)1248 1539 y Fj(\000)p Fl(1)1248 1569 y Fm(A)1293 1557 y Fp(\()p Fo(b)1327 1540 y Fl(2)1345 1557 y Fp(\))257 1648 y(Using)j(the)h(in)o(tegral)e(group)g (elemen)o(t)h Fo(g)888 1654 y Fm(A)928 1648 y Fp(of)g Fo(A)p Fp(,)f(w)o(e)h(ha)o(v)o(e:)385 1739 y Fo(!)411 1745 y Fm(B)439 1739 y Fp(\()p Fo(b)p Fp(\))e(=)g Fk(h)p Fp(\()p Fo(S)602 1745 y Fm(H)634 1739 y Fp(\(\003)679 1745 y Fm(A)r Fl(2)725 1721 y(1)743 1739 y Fp(\))g Fk(!)f Fp(\003)853 1745 y Fm(A)r Fl(1)899 1739 y Fp(\))p Fo(S)940 1745 y Fm(A)967 1739 y Fp(\(\003)1012 1745 y Fm(A)r Fl(3)1058 1739 y Fp(\))p Fo(;)c(S)1118 1745 y Fm(B)1146 1739 y Fp(\()p Fo(b)1180 1722 y Fl(1)1199 1739 y Fp(\))p Fk(i)1231 1745 y Fm(B)1260 1739 y Fo(\032)1281 1745 y Fm(A)1308 1739 y Fp(\(\003)1353 1745 y Fm(A)r Fl(2)1399 1721 y(2)1418 1739 y Fp(\))p Fo(\023)1449 1722 y Fj(\000)p Fl(1)1449 1752 y Fm(A)1493 1739 y Fp(\()p Fo(b)1527 1722 y Fl(2)1546 1739 y Fp(\))501 1807 y(=)12 b Fk(h)p Fp(\()p Fo(S)602 1813 y Fm(H)634 1807 y Fp(\()p Fo(g)670 1813 y Fm(A)697 1807 y Fp(\))g Fk(!)f Fp(\003)807 1813 y Fm(A)r Fl(1)852 1807 y Fp(\))p Fo(S)893 1813 y Fm(A)921 1807 y Fp(\(\003)966 1813 y Fm(A)r Fl(3)1012 1807 y Fp(\))p Fo(;)c(S)1072 1813 y Fm(B)1100 1807 y Fp(\()p Fo(b)1134 1790 y Fl(1)1153 1807 y Fp(\))p Fk(i)1185 1813 y Fm(B)1214 1807 y Fo(\032)1235 1813 y Fm(A)1262 1807 y Fp(\(\003)1307 1813 y Fm(A)r Fl(2)1353 1807 y Fp(\))p Fo(\023)1384 1790 y Fj(\000)p Fl(1)1384 1820 y Fm(A)1428 1807 y Fp(\()p Fo(b)1462 1790 y Fl(2)1481 1807 y Fp(\))501 1876 y(=)12 b Fk(h)p Fp(\()p Fo(S)602 1882 y Fm(H)634 1876 y Fp(\()p Fo(g)670 1882 y Fm(A)697 1876 y Fp(\))g Fk(!)f Fo(a)800 1859 y Fm(R)800 1886 y(A)827 1876 y Fp(\))p Fo(;)c(S)887 1882 y Fm(B)916 1876 y Fp(\()p Fo(b)950 1859 y Fl(1)968 1876 y Fp(\))p Fk(i)1000 1882 y Fm(B)1029 1876 y Fo(\032)1050 1882 y Fm(A)1078 1876 y Fp(\(\003)1123 1882 y Fm(A)1150 1876 y Fp(\))p Fo(\023)1181 1858 y Fj(\000)p Fl(1)1181 1888 y Fm(A)1225 1876 y Fp(\()p Fo(b)1259 1859 y Fl(2)1278 1876 y Fp(\))501 1944 y(=)12 b Fk(h)p Fo(a)583 1927 y Fm(R)583 1955 y(A)610 1944 y Fo(;)7 b(S)654 1950 y Fm(B)683 1944 y Fp(\()p Fo(b)717 1927 y Fl(1)735 1944 y Fp(\))p Fk(i)767 1950 y Fm(B)796 1944 y Fo(\023)811 1927 y Fj(\000)p Fl(1)811 1957 y Fm(A)855 1944 y Fp(\()p Fo(b)889 1927 y Fl(2)908 1944 y Fp(\))501 2013 y(=)12 b Fk(h)p Fo(a)583 1996 y Fm(L)583 2023 y(A)610 2013 y Fo(;)7 b(b)647 1996 y Fl(1)665 2013 y Fk(i)681 2019 y Fm(B)710 2013 y Fo(\023)725 1995 y Fj(\000)p Fl(1)725 2025 y Fm(A)769 2013 y Fp(\()p Fo(b)803 1996 y Fl(2)821 2013 y Fp(\))501 2082 y(=)12 b Fk(h)p Fo(a)583 2064 y Fm(R)583 2092 y(A)610 2082 y Fo(;)7 b(b)647 2064 y Fl(1)665 2082 y Fk(i)681 2088 y Fm(A)708 2082 y Fo(\023)723 2064 y Fj(\000)p Fl(1)723 2094 y Fm(A)768 2082 y Fp(\()p Fo(b)802 2064 y Fl(2)820 2082 y Fp(\))12 b(=)g Fo(\013)919 2064 y Fm(R)919 2092 y(B)947 2082 y Fp(\()p Fo(b)981 2064 y Fl(1)999 2082 y Fp(\))p Fo(\023)1030 2064 y Fj(\000)p Fl(1)1030 2094 y Fm(A)1075 2082 y Fp(\()p Fo(b)1109 2064 y Fl(2)1127 2082 y Fp(\))g(=)g Fo(\013)1226 2064 y Fm(R)1226 2092 y(B)1254 2082 y Fp(\()p Fo(b)p Fp(\))257 2173 y(W)m(e)i(therefore)h (see)h(that)e Fo(!)687 2179 y Fm(B)729 2173 y Fp(is)g(in)f(this)i(case) g(the)f(righ)o(t)g(mo)q(dular)e(function)i(of)f Fo(B)r Fp(.)h(Since)257 2223 y(left)k(in)o(tegrals)f(corresp)q(onding)h(to)f (this)h(c)o(haracter)h(are)f(precisely)g(the)g(righ)o(t)f(in)o (tegrals,)257 2272 y(w)o(e)g(see)h(b)o(y)f(Theorem)f(5.2)f(that)i(the)g (ab)q(o)o(v)o(e)g(elemen)o(t)f(is)h(a)f(left)g(in)o(tegral.)g(By)h (applying)257 2322 y(part)h(\(2\))g(of)f(Theorem)h(5.2,)e(w)o(e)i (deriv)o(e)g(b)o(y)g(a)g(similar)d(calculation)i(that)h(the)g(elemen)o (t)257 2372 y(\003)286 2378 y Fm(A)322 2372 y Fk(\012)10 b Fo(\023)379 2378 y Fm(A)406 2372 y Fp(\(\000)448 2378 y Fm(H)t Fl(1)498 2372 y Fp(\)\000)540 2378 y Fm(H)t Fl(2)599 2372 y Fk(\012)g Fp(\000)667 2378 y Fm(B)711 2372 y Fp(is)15 b(a)g(righ)o(t)g(in)o(tegral)g(of)g Fo(A)10 b Fk(\012)h Fo(H)i Fk(\012)d Fo(B)r Fp(,)16 b(where)h(\000)1477 2378 y Fm(H)1523 2372 y Fp(is)f(a)f(righ)o(t)257 2422 y(in)o(tegral)h(of)g Fo(H)s Fp(.)f(Since)i Fo(H)i Fp(is)d(comm)o (utativ)o(e)e(and)i(co)q(comm)o(utativ)o(e,)d Fo(H)19 b Fp(is)d(unimo)q(dular,)257 2472 y(and)e(left)g(and)f(righ)o(t)h(in)o (tegrals)f(of)h Fo(A)9 b Fk(\012)h Fo(H)i Fk(\012)d Fo(B)17 b Fp(coincide.)c Fg(\003)953 2628 y Fp(54)p eop %%Page: 55 55 55 54 bop 257 262 a Fn(5.5)48 b Fp(The)14 b(ab)q(o)o(v)o(e)f(Theorem)f (generalizes)i(a)f(result)h(of)f(D.)f(Radford)h(on)f(the)i(in)o (tegrals)f(of)257 311 y(the)k(Drinfel'd)e(double)h(construction)h (\(cf.)f([24)o(],)g(Thm.)e(4,)h(p.)h(303\).)f(W)m(e)h(note)h(that)f (the)257 361 y(same)f(result)h(w)o(as)f(pro)o(v)o(ed)g(b)o(y)g(M.)g(A.) g(Hennings)g(\(cf.)g([7]\))f(and)h(K.)g(Ho\013mann)f(\(unpub-)257 411 y(lished\).)f(Recall)g(from)f([32)o(],)g(subsection)j(4.18)d(ho)o (w)h(the)h(Drinfel'd)e(double)h(construction)257 461 y(arises)20 b(as)f(a)f(sp)q(ecial)h(case)h(of)e(the)h(second)h (construction:)f(First,)g(w)o(e)g(see)g(from)e(Theo-)257 511 y(rem)d(5.4)f(ab)q(o)o(v)o(e)h(that)g(if)g(w)o(e)g(set)h Fo(H)g Fp(=)e Fo(K)k Fp(in)d(the)h(second)g(construction,)g(then)g (\003)1557 517 y Fm(A)1593 511 y Fk(\012)10 b Fp(\000)1661 517 y Fm(B)257 560 y Fp(is)20 b(an)f(in)o(tegral)g(of)g Fo(A)13 b Fk(\012)h Fo(B)r Fp(,)19 b(where)i(\003)887 566 y Fm(A)935 560 y Fk(2)g Fo(A)e Fp(is)h(a)f(left)h(in)o(tegral)e (and)i(\000)1470 566 y Fm(B)1519 560 y Fk(2)h Fo(B)h Fp(is)e(a)257 610 y(righ)o(t)13 b(in)o(tegral.)f(Since)h Fo(A)8 b Fk(\012)f Fo(B)16 b Fp(is)d(unimo)q(dular,)d(this)j(remains)f (an)h(in)o(tegral)f(if)g(w)o(e)h(pass)h(to)257 660 y(\()p Fo(A)5 b Fk(\012)g Fo(B)r Fp(\))395 645 y Fm(op)10 b(cop)488 660 y Fp(.)h(F)m(ollo)o(wing)e([32)o(]\),)i(w)o(e)h(no)o(w)g(ha)o(v)o (e)f(to)h(in)o(terc)o(hange)g(the)h(tensorands)g Fo(A)f Fp(and)257 710 y Fo(B)17 b Fp(to)e(obtain)f(the)h(Drinfel'd)e(double)i (of)e Fo(A)946 695 y Fm(op)c(cop)1038 710 y Fp(.)14 b(Therefore,)h (\000)1290 716 y Fm(B)1329 710 y Fk(\012)10 b Fp(\003)1400 716 y Fm(A)1441 710 y Fp(is)k(a)h(t)o(w)o(o-sided)257 760 y(in)o(tegral)e(of)g(the)h(Drinfel'd)f(double)g(of)g Fo(A)911 745 y Fm(op)8 b(cop)1003 760 y Fp(,)13 b(whic)o(h)g(implies)f (that)i(an)f(in)o(tegral)g(of)g(the)257 809 y(Drinfel'd)h(double)h(of)f Fo(A)h Fp(is)g(of)f(the)i(form)d(\003)953 815 y Fm(B)991 809 y Fk(\012)d Fp(\000)1059 815 y Fm(A)1101 809 y Fp(for)15 b(a)f(left)h(in)o(tegral)f(\003)1456 815 y Fm(B)1498 809 y Fk(2)f Fo(B)k Fp(and)e(a)257 859 y(righ)o(t)f(in)o(tegral)f(\000) 535 865 y Fm(A)573 859 y Fk(2)f Fo(A)p Fp(.)257 1032 y Fq(6)67 b(The)22 b(F)-6 b(rob)r(enius-Lusztig)26 b(k)n(ernel)d(of)f Fc(U)1358 1041 y Fo(q)1380 1032 y Fb(\()p Fc(sl)q Fb(\(2\)\))257 1159 y Fn(6.1)48 b Fp(T)m(o)10 b(illustrate)g(the)g(tec)o(hniques)i (dev)o(elop)q(ed)f(in)f(the)h(previous)f(section,)h(w)o(e)f(calculate) 257 1209 y(in)19 b(this)g(section)h(the)g(in)o(tegral)f(for)g(a)g (\014nite-dimensional)e(quotien)o(t)i(of)f(the)i(deformed)257 1258 y(en)o(v)o(eloping)15 b(algebra)g Fo(U)638 1264 y Fm(q)657 1258 y Fp(\()p Fo(sl)q Fp(\(2\)\).)h(Supp)q(ose)g(that)g Fo(q)g Fp(is)g(a)f(primitiv)o(e)e(m-th)h(ro)q(ot)i(of)f(unit)o(y)257 1308 y(in)e(the)i(\014eld)e Fo(K)s Fp(.)h(W)m(e)f(assume)g(that)g Fo(m)h Fp(is)g(larger)f(than)h(2.)f(Note)h(that)f(the)h(existence)i(of) d(a)257 1358 y(primitiv)o(e)c(ro)q(ot)j(of)f(unit)o(y)f(implies)g(that) h(the)h(c)o(haracteristic)h(of)d(the)i(\014eld)g(do)q(es)g(not)f (divide)257 1408 y(its)j(order)h Fo(m)p Fp(.)f(W)m(e)f(in)o(tro)q(duce) i(the)f(n)o(um)o(b)q(er)f Fo(k)i Fp(b)o(y:)769 1534 y Fo(k)e Fp(=)847 1463 y Fh(\()881 1506 y Fo(m)45 b Fp(if)13 b Fo(m)h Fp(is)g(o)q(dd)886 1549 y Fm(m)p 886 1556 30 2 v 892 1580 a Fl(2)962 1566 y Fp(if)f Fo(m)h Fp(is)g(ev)o(en)257 1663 y(Let)d Fo(G)f Fp(b)q(e)h(the)f(cyclic)h(group)f(of)g(order)g Fo(k)i Fp(generated)f(b)o(y)f Fo(g)q Fp(.)g(W)m(e)g(set)h Fo(H)j Fp(=)e Fo(K)s Fp([)p Fo(G)p Fp(],)d(the)i(group)257 1713 y(ring)k(of)f Fo(G)p Fp(,)g(with)g(its)h(usual)f(Hopf)g(algebra)h (structure.)h(W)m(e)e(shall)g(turn)i(the)f(p)q(olynomial)257 1762 y(algebra)g Fo(K)s Fp([)p Fo(\022)q Fp(])h(o)o(v)o(er)f(the)h (indeterminate)f Fo(\022)i Fp(in)o(to)e(a)g(left)g(Y)m(etter-Drinfel'd) g(mo)q(dule)f(o)o(v)o(er)257 1812 y Fo(H)j Fp(via)c(the)i(mo)q(dule)d (op)q(eration)830 1903 y Fo(g)h Fk(!)e Fo(\022)936 1886 y Fm(n)971 1903 y Fp(=)g Fo(q)1034 1886 y Fl(2)p Fm(n)1074 1903 y Fo(\022)1094 1886 y Fm(n)257 1995 y Fp(and)j(the)h(como)q(dule)d (co)q(op)q(eration)796 2086 y Fo(\016)814 2093 y Fm(K)r Fl([)p Fm(\022)q Fl(])882 2086 y Fp(\()p Fo(\022)918 2069 y Fm(n)941 2086 y Fp(\))g(=)g Fo(g)1034 2069 y Fm(n)1066 2086 y Fk(\012)e Fo(\022)1128 2069 y Fm(n)257 2177 y Fp(It)15 b(can)g(b)q(e)g(v)o(eri\014ed)g(directly)g(that)g Fo(K)s Fp([)p Fo(\022)q Fp(])g(b)q(ecomes)g(an)f(algebra)g(in)g(the)i (category)f(of)f(left)257 2227 y(Y)m(etter-Drinfel'd)g(mo)q(dules)f(in) g(this)h(w)o(a)o(y)m(.)257 2312 y(Consider)20 b(the)f(tensor)h(pro)q (duct)f(algebra)g Fo(K)s Fp([)p Fo(\022)q Fp(])12 b Fk(\012)h Fo(K)s Fp([)p Fo(\022)q Fp(])18 b(formed)g(inside)h(the)g(category)257 2362 y(of)h(Y)m(etter-Drinfel'd)f(mo)q(dules.)f(W)m(e)i(in)o(tro)q (duce)g(a)g(coalgebra)f(structure)j(on)e Fo(K)s Fp([)p Fo(\022)q Fp(])g(b)o(y)257 2412 y(requiring)14 b(\001)469 2419 y Fm(K)r Fl([)p Fm(\022)q Fl(])550 2412 y Fp(to)g(b)q(e)g(the)h (unique)f(algebra)f(homomorphi)o(sm)d(satisfying)751 2503 y(\001)786 2510 y Fm(K)r Fl([)p Fm(\022)q Fl(])853 2503 y Fp(\()p Fo(\022)q Fp(\))i(=)g Fo(\022)f Fk(\012)e Fp(1)g(+)h(1)f Fk(\012)g Fo(\022)953 2628 y Fp(55)p eop %%Page: 56 56 56 55 bop 257 262 a Fp(W)m(e)13 b(de\014ne)g(the)g(counit)g Fo(\017)659 269 y Fm(K)r Fl([)p Fm(\022)q Fl(])738 262 y Fp(:)e Fo(K)s Fp([)p Fo(\022)q Fp(])g Fk(!)g Fo(K)16 b Fp(to)c(b)q(e)h(the)h(unique)e(algebra)g(homomorphism)257 311 y(annihilating)f Fo(\022)q Fp(,)i(and)f(de\014ne)i(the)g(an)o(tip)q (o)q(de)e Fo(S)996 318 y Fm(K)r Fl([)p Fm(\022)q Fl(])1076 311 y Fp(:)f Fo(K)s Fp([)p Fo(\022)q Fp(])g Fk(!)h Fo(K)s Fp([)p Fo(\022)q Fp(])1328 296 y Fm(opp)1393 311 y Fp(to)h(b)q(e)g(the) g(unique)257 361 y(algebra)j(homomorphism)c(satisfying)j Fo(S)913 368 y Fm(K)r Fl([)p Fm(\022)q Fl(])981 361 y Fp(\()p Fo(\022)q Fp(\))h(=)g Fk(\000)p Fo(\022)q Fp(.)g(Here)i Fo(K)s Fp([)p Fo(\022)q Fp(])1360 346 y Fm(opp)1429 361 y Fp(means)d(the)i(op-)257 411 y(p)q(osite)h(algebra)f(in)g(the)h (categorical)f(sense:)i(The)e(m)o(ultiplication)d Fo(\026)1374 391 y Fm(opp)1374 425 y(K)r Fl([)p Fm(\022)q Fl(])1459 411 y Fp(of)j Fo(K)s Fp([)p Fo(\022)q Fp(])1592 396 y Fm(opp)1662 411 y Fp(is)257 465 y(giv)o(en)d(b)o(y:)750 515 y Fo(\026)775 495 y Fm(opp)775 529 y(K)r Fl([)p Fm(\022)q Fl(])854 515 y Fp(=)e Fo(\026)923 522 y Fm(K)r Fl([)p Fm(\022)q Fl(])1000 515 y Fk(\016)d Fo(\033)1054 522 y Fm(K)r Fl([)p Fm(\022)q Fl(])p Fm(;K)r Fl([)p Fm(\022)q Fl(])257 597 y Fp(where)21 b Fo(\033)f Fp(is)f(the)h(braiding)e(as)h (in)g(subsection)i(2.2,)c(and)i(the)h(unit)f(of)g Fo(K)s Fp([)p Fo(\022)q Fp(])1511 581 y Fm(opp)1582 597 y Fp(is)h(not)257 646 y(c)o(hanged.)12 b(Note)h(that)f(this)g(opp)q(osite)g(m)o (ultiplicatio)o(n)d(is)j(di\013eren)o(t)h(from)d(the)j(one)f(used)h(in) 257 696 y(subsection)k(4.5,)d(where)j(w)o(e)e(used)i(the)f(in)o(v)o (erse)g(braiding)e(in)h(con)o(trast)i(to)e(the)h(ordinary)257 746 y(braiding)h(used)h(here.)g(These)h(morphisms)c(are)j Fo(H)s Fp(-linear)e(and)i(colinear,)e(and)i(one)f(can)257 796 y(v)o(erify)d(directly)g(that)g Fo(K)s Fp([)p Fo(\022)q Fp(])g(is)f(a)h(Y)m(etter-Drinfel'd)g(Hopf)f(algebra.)257 931 y Fn(6.2)48 b Fp(Consider)18 b(the)g(principal)f(ideal)g(\()p Fo(\022)934 916 y Fm(k)955 931 y Fp(\))h(generated)g(b)o(y)g Fo(\022)1263 916 y Fm(k)1284 931 y Fp(.)f(It)g(is)h(immedia)o(te)d (that)257 981 y(this)i(ideal)e(is)h(an)g Fo(H)s Fp(-submo)q(dule)f(and) i(a)f(sub)q(como)q(dule.)f(W)m(e)h(rep)q(eat)h(brie\015y)g(wh)o(y)f(it) g(is)257 1031 y(also)10 b(a)h(coideal.)f(It)h(follo)o(ws)e(from)g(the)i (Gaussian)f(binomial)e(form)o(ula)g(\(cf.)i([13)o(],)g(subsection)257 1080 y(1.3.5,)i(p.)h(10\))h(that)642 1205 y(\001)677 1212 y Fm(K)r Fl([)p Fm(\022)q Fl(])744 1205 y Fp(\()p Fo(\022)780 1188 y Fm(n)804 1205 y Fp(\))d(=)895 1153 y Fm(n)875 1165 y Fh(X)878 1254 y Fm(i)p Fl(=0)942 1205 y Fo(q)962 1188 y Fm(i)p Fl(\()p Fm(n)p Fj(\000)p Fm(i)p Fl(\))1060 1146 y Fh(\024)1087 1177 y Fo(n)1092 1233 y(i)1117 1146 y Fh(\025)1139 1205 y Fo(\022)1159 1188 y Fm(i)1182 1205 y Fk(\012)f Fo(\022)1244 1188 y Fm(n)p Fj(\000)p Fm(i)257 1332 y Fp(Here,)k(w)o(e)e(ha)o(v)o(e)h(used)g(the)g (so)g(called)f(Gaussian)g(binomial)d(co)q(e\016cien)o(ts)14 b(that)e(are)h(de\014ned)257 1382 y(recursiv)o(ely)i(as:)455 1437 y Fh(\024)482 1467 y Fo(n)484 1524 y Fp(0)512 1437 y Fh(\025)545 1495 y Fp(=)589 1437 y Fh(\024)616 1467 y Fo(n)616 1524 y(n)646 1437 y Fh(\025)679 1495 y Fp(=)d(1)827 1437 y Fh(\024)854 1467 y Fo(n)d Fp(+)g(1)895 1524 y Fo(i)955 1437 y Fh(\025)988 1495 y Fp(=)j Fo(q)1052 1478 y Fj(\000)p Fm(n)1101 1437 y Fh(\024)1128 1467 y Fo(n)1133 1524 y(i)1158 1437 y Fh(\025)1189 1495 y Fp(+)d Fo(q)1250 1478 y Fm(n)p Fj(\000)p Fm(i)p Fl(+1)1353 1437 y Fh(\024)1410 1467 y Fo(n)1379 1524 y(i)h Fk(\000)f Fp(1)1470 1437 y Fh(\025)257 1611 y Fp(Since)i(it)e(can)h(b)q(e)h(sho)o(wn)f(\(cf.)g ([13)o(],)e(Lemma)g(34.1.2,)f(p.)j(265\))f(that)h(the)g(Gaussian)g (binomial)257 1673 y(co)q(e\016cien)o(ts)466 1627 y Fh(h)491 1655 y Fm(k)494 1688 y(i)514 1627 y Fh(i)546 1673 y Fp(v)n(anish)h(for) h Fo(i)g Fp(=)g(1)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(k)f Fk(\000)g Fp(1,)12 b(w)o(e)h(ha)o(v)o(e)f(\001)1225 1680 y Fm(K)r Fl([)p Fm(\022)q Fl(])1292 1673 y Fp(\()p Fo(\022)1328 1658 y Fm(k)1349 1673 y Fp(\))g(=)g Fo(\022)1441 1658 y Fm(k)1468 1673 y Fk(\012)6 b Fp(1)g(+)g(1)g Fk(\012)g Fo(\022)1656 1658 y Fm(k)1678 1673 y Fp(,)257 1740 y(whic)o(h)11 b(pro)o(v)o(es)g(that)g(\()p Fo(\022)621 1725 y Fm(k)642 1740 y Fp(\))g(is)f(a)g(coideal.)g(Since)h(it)f(follo)o(ws)f(from)g (the)i(form)e(of)h(the)i(an)o(tip)q(o)q(de)257 1790 y(in)k(the)h (general)f(case)h(\(cf.)f([32)o(],)f(Prop.)h(5.1,)f(p.)g(56\))h(or)g (from)e(a)i(simple)f(induction)g(that)257 1839 y(w)o(e)f(ha)o(v)o(e)714 1889 y Fo(S)739 1896 y Fm(K)r Fl([)p Fm(\022)q Fl(])807 1889 y Fp(\()p Fo(\022)843 1872 y Fm(n)867 1889 y Fp(\))d(=)h(\()p Fk(\000)p Fp(1\))1023 1872 y Fm(n)1046 1889 y Fo(q)1066 1872 y Fm(n)p Fl(\()p Fm(n)p Fj(\000)p Fl(1\))1178 1889 y Fo(\022)1198 1872 y Fm(n)1221 1889 y Fo(;)257 1968 y Fp(the)j(ideal)e(\()p Fo(\022)464 1953 y Fm(k)485 1968 y Fp(\))h(is)g(also)f(in)o(v)n(arian)o(t)f(with)i(resp)q(ect)i(to)e (the)g(an)o(tip)q(o)q(de.)257 2054 y(The)19 b(factor)g(algebra)f Fo(A)h Fp(=)g Fo(K)s Fp([)p Fo(\022)q Fp(])p Fo(=)p Fp(\()p Fo(\022)861 2038 y Fm(k)882 2054 y Fp(\))g(is)f(therefore)i(again)d(a)i (Y)m(etter-Drinfel'd)f(Hopf)257 2103 y(algebra)12 b(o)o(v)o(er)f Fo(H)k Fp(via)c(the)h(induced)g(structures.)i(It)e(ob)o(viously)e(has)i (dimension)e Fo(k)j Fp(o)o(v)o(er)f Fo(K)s Fp(.)257 2153 y(W)m(e)h(denote)h(the)g(dual)e(basis)h(of)g(1)p Fo(;)7 b(\022)q(;)g(\022)870 2138 y Fl(2)888 2153 y Fo(;)g(:)g(:)g(:)12 b(;)7 b(\022)1008 2138 y Fm(k)q Fj(\000)p Fl(1)1084 2153 y Fp(b)o(y)13 b Fo(p)1162 2159 y Fl(0)1180 2153 y Fo(;)7 b(p)1220 2159 y Fl(1)1238 2153 y Fo(;)g(:)g(:)g(:)12 b(;)7 b(p)1359 2159 y Fm(k)q Fj(\000)p Fl(1)1421 2153 y Fp(,)13 b(thereb)o(y)h(using)257 2203 y(the)h(notation)e Fo(\022)i Fp(also)f(for)f(the)i(equiv)n(alence)f(class)g(of)f Fo(\022)j Fp(in)d Fo(A)p Fp(.)257 2288 y(Denote)19 b(b)o(y)f Fo(B)j Fp(the)e(dual)f(of)g Fo(A)g Fp(mo)q(di\014ed)f(as)i(describ)q (ed)h(in)e(subsection)h(5.4.)e(W)m(e)h(can)257 2338 y(no)o(w)d(carry)g (out)f(the)h(second)h(construction)f(and)g(obtain)f(a)g(Hopf)g(algebra) g(structure)j(on)257 2388 y Fo(A)10 b Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(.)k(W)m(e)f(in)o(tro)q(duce)i(the)f(follo)o(wing)e (elemen)o(ts:)471 2487 y Fo(E)h Fp(=)f Fo(\022)f Fk(\012)e Fp(1)g Fk(\012)h Fp(1)p Fo(;)47 b(K)15 b Fp(=)d(1)d Fk(\012)h Fo(g)g Fk(\012)g Fp(1)p Fo(;)47 b(F)17 b Fp(=)12 b(1)d Fk(\012)h Fp(1)f Fk(\012)1384 2459 y Fo(p)1405 2465 y Fl(1)p 1336 2478 136 2 v 1336 2516 a Fo(q)1356 2504 y Fj(\000)p Fl(1)1410 2516 y Fk(\000)g Fo(q)953 2628 y Fp(56)p eop %%Page: 57 57 57 56 bop 257 262 a Fp(It)13 b(is)g(ob)o(vious)f(that)h Fo(K)i Fp(is)e(in)o(v)o(ertible)f(with)h(in)o(v)o(erse)g(1)7 b Fk(\012)g Fo(g)1171 246 y Fj(\000)p Fl(1)1223 262 y Fk(\012)g Fp(1.)12 b(The)h(follo)o(wing)d(Prop)q(o-)257 311 y(sition)k(describ)q(es)i(precisely)e(ho)o(w)g Fo(A)9 b Fk(\012)h Fo(H)i Fk(\012)d Fo(B)17 b Fp(arises)d(as)g(a)g(quotien)o (t)g(of)f Fo(U)1460 317 y Fm(q)1479 311 y Fp(\()p Fo(sl)q Fp(\(2\)\):)257 400 y Fn(Prop)q(osition)33 b Fp(The)15 b(elemen)o(ts)e Fo(E)r(;)7 b(F)q(;)g(K)16 b Fp(and)e Fo(K)1060 385 y Fj(\000)p Fl(1)1118 400 y Fp(satisfy:)824 474 y Fo(K)s(K)900 456 y Fj(\000)p Fl(1)957 474 y Fp(=1)d(=)h Fo(K)1103 456 y Fj(\000)p Fl(1)1148 474 y Fo(K)670 541 y(K)s(E)h Fp(=)f Fo(q)816 524 y Fl(2)835 541 y Fo(E)r(K)86 b(K)s(F)17 b Fp(=)12 b Fo(q)1135 524 y Fj(\000)p Fl(2)1180 541 y Fo(F)6 b(K)807 634 y(E)r(F)15 b Fk(\000)10 b Fo(F)c(E)13 b Fp(=)1049 606 y Fo(K)g Fk(\000)c Fo(K)1176 591 y Fj(\000)p Fl(1)p 1049 625 172 2 v 1068 663 a Fo(q)h Fk(\000)f Fo(q)1158 651 y Fj(\000)p Fl(1)718 725 y Fo(K)756 708 y Fm(k)788 725 y Fp(=)j(1)83 b Fo(E)969 708 y Fm(k)1001 725 y Fp(=)11 b(0)83 b Fo(F)1181 708 y Fm(k)1213 725 y Fp(=)11 b(0)257 798 y(and)17 b(this)h(constitutes)g(a)f(presen)o(tation)h(of)e Fo(A)c Fk(\012)g Fo(H)i Fk(\012)e Fo(B)19 b Fp(in)e(terms)g(of)f (generators)j(and)257 848 y(relations.)257 937 y Fn(Pro)q(of.)36 b Fp(\(1\))21 b(W)m(e)14 b(sho)o(w)h(\014rst)g(that)g(the)g(elemen)o (ts)g Fo(E)r Fp(,)f Fo(F)6 b Fp(,)13 b Fo(K)18 b Fp(and)d Fo(K)1383 922 y Fj(\000)p Fl(1)1442 937 y Fp(in)f(fact)h(satisfy)257 987 y(the)j(ab)q(o)o(v)o(e)f(relations.)f(Observ)o(e)j(that)e Fo(p)913 993 y Fl(1)948 987 y Fp(is)g(primitiv)o(e.)d(The)k(relations)e (are)i(all)e(rather)257 1037 y(ob)o(vious)d(except)j(for)d(the)i (fourth)f(one.)f(W)m(e)h(ha)o(v)o(e:)603 1110 y Fo(p)624 1116 y Fl(1)654 1110 y Fo(*)d(\022)j Fp(=)d Fk(h)p Fo(\022)q(;)c(p)859 1116 y Fl(1)878 1110 y Fk(i)894 1116 y Fm(B)923 1110 y Fp(1)i(+)g Fk(h\000)p Fo(\022)q(;)e(p)1102 1116 y Fl(1)1121 1110 y Fk(i)1137 1116 y Fm(B)1166 1110 y Fp(1)k(=)h(0)603 1172 y Fo(p)624 1178 y Fl(1)654 1172 y Fo(\()f(\022)j Fp(=)d Fk(h)p Fo(\022)q(;)c Fk(\000)p Fo(p)891 1178 y Fl(1)910 1172 y Fk(i)926 1178 y Fm(A)954 1172 y Fp(1)i(+)g Fk(h)p Fo(\022)q(;)e(p)1101 1178 y Fl(1)1120 1172 y Fk(i)1136 1178 y Fm(A)1163 1172 y Fp(1)k(=)h(0)603 1240 y Fo(p)624 1246 y Fl(1)643 1240 y Fo(]\022)h Fp(=)f Fk(h)p Fo(\022)q(;)7 b(p)811 1246 y Fl(1)830 1240 y Fk(i)846 1246 y Fm(B)874 1240 y Fo(g)895 1223 y Fj(\000)p Fl(1)950 1240 y Fp(+)i Fo(g)q Fk(h)p Fo(\022)q(;)e(p)1088 1246 y Fl(1)1107 1240 y Fk(i)1123 1246 y Fm(A)1162 1240 y Fp(=)12 b Fo(g)e Fk(\000)g Fo(g)1299 1223 y Fj(\000)p Fl(1)257 1313 y Fp(W)m(e)k(therefore)h(can)f(calculate:)565 1387 y Fo(F)6 b(E)13 b Fp(=)f(\(1)d Fk(\012)g Fp(1)g Fk(\012)898 1358 y Fo(p)919 1364 y Fl(1)p 850 1377 136 2 v 850 1415 a Fo(q)870 1403 y Fj(\000)p Fl(1)924 1415 y Fk(\000)g Fo(q)990 1387 y Fp(\)\()p Fo(\022)i Fk(\012)f Fp(1)f Fk(\012)g Fp(1\))642 1486 y(=)j Fo(\022)f Fk(\012)e Fp(1)g Fk(\012)881 1458 y Fo(p)902 1464 y Fl(1)p 834 1476 V 834 1514 a Fo(q)854 1502 y Fj(\000)p Fl(1)907 1514 y Fk(\000)h Fo(q)983 1486 y Fp(+)g(1)f Fk(\012)g Fp(\()1165 1458 y Fo(p)1186 1464 y Fl(1)p 1117 1476 V 1117 1514 a Fo(q)1137 1502 y Fj(\000)p Fl(1)1191 1514 y Fk(\000)h Fo(q)1258 1486 y(]\022)q Fp(\))g Fk(\012)f Fp(1)642 1601 y(=)j Fo(E)r(F)i Fp(+)807 1573 y Fo(K)e Fk(\000)e Fo(K)934 1558 y Fj(\000)p Fl(1)p 807 1591 172 2 v 825 1629 a Fo(q)845 1617 y Fj(\000)p Fl(1)899 1629 y Fk(\000)f Fo(q)257 1709 y Fp(\(2\))21 b(No)o(w)16 b(supp)q(ose)i(that)e Fo(X)k Fp(is)c(the)h(algebra)f(generated)i(b)o(y) e(sym)o(b)q(ols)f Fo(E)1445 1694 y Fj(0)1457 1709 y Fp(,)g Fo(F)1517 1694 y Fj(0)1529 1709 y Fp(,)g Fo(K)1594 1694 y Fj(0)1623 1709 y Fp(and)257 1759 y Fo(K)295 1744 y Fj(0)r(\000)p Fl(1)368 1759 y Fp(sub)r(ject)j(to)f(the)g(ab)q(o)o(v)o (e)f(relations.)g(It)h(is)g(clear)g(that)f(w)o(e)h(ha)o(v)o(e)g(an)f (algebra)g(mor-)257 1809 y(phism)f Fo(\031)e Fp(:)e Fo(X)k Fk(!)c Fo(A)f Fk(\012)f Fo(H)j Fk(\012)e Fo(B)r Fp(.)16 b(It)h(is)f(ob)o(vious)g(from)f(the)i(relations)f(that)h(the)g(elemen)o (ts)257 1859 y Fo(E)290 1844 y Fj(0)r Fm(i)316 1859 y Fo(K)354 1844 y Fj(0)r Fm(n)388 1859 y Fo(F)421 1844 y Fj(0)q Fm(j)466 1859 y Fp(for)f(0)f Fk(\024)h Fo(i;)7 b(j;)g(n)14 b Fk(\024)i Fo(k)c Fk(\000)f Fp(1)k(generate)j Fo(X)i Fp(as)c(a)g(v)o(ector)h(space.)g(On)f(the)h(other)257 1908 y(hand,)e(the)g(elemen)o(ts)g Fo(E)650 1893 y Fm(i)664 1908 y Fo(K)702 1893 y Fm(n)725 1908 y Fo(F)758 1893 y Fm(j)790 1908 y Fp(for)f(0)f Fk(\024)h Fo(i;)7 b(j;)g(n)12 b Fk(\024)i Fo(k)d Fk(\000)f Fp(1)15 b(form)e(a)h(basis)h(of)g Fo(A)10 b Fk(\012)g Fo(H)j Fk(\012)d Fo(B)257 1958 y Fp(b)o(y)k(construction.)h(Since)g(a)f(non)o(trivial)f(linear)g (relation)h(for)g(the)h(primed)e(system)h(w)o(ould)257 2008 y(map)f(to)h(a)g(non)o(trivial)f(linear)h(relation)g(for)g(the)h (unprimed)e(system,)h(b)q(oth)h(m)o(ust)e(form)g(a)257 2058 y(basis.)h(Therefore)h Fo(\031)g Fp(is)f(an)f(isomorphism.)e Fg(\003)257 2186 y Fn(6.3)48 b Fp(In)12 b(order)h(to)f(calculate)h(the) g(in)o(tegral)e(of)h Fo(A)6 b Fk(\012)g Fo(H)j Fk(\012)d Fo(B)r Fp(,)13 b(w)o(e)f(m)o(ust)g(\014nd)g(the)h(in)o(tegrals)257 2235 y(of)h(the)g(comp)q(onen)o(ts)g(\014rst.)257 2315 y Fn(Lemma)308 2365 y Fp(1.)20 b Fo(\022)381 2350 y Fm(k)q Fj(\000)p Fl(1)458 2365 y Fp(is)14 b(a)f(nonzero)i(left)e(and)h(righ)o (t)f(in)o(tegral)g(of)g Fo(A)p Fp(.)g(The)h(in)o(tegral)f(group)h (elemen)o(t)361 2419 y(of)f Fo(A)h Fp(is)g Fo(g)516 2404 y Fm(k)q Fj(\000)p Fl(1)579 2419 y Fp(.)g(The)g(in)o(tegral)f(c)o (haracter)i(of)f Fo(A)g Fp(is)f(giv)o(en)h(b)o(y)866 2493 y Fo(\023)881 2499 y Fm(A)907 2493 y Fp(\()p Fo(g)944 2476 y Fm(n)967 2493 y Fp(\))e(=)g Fo(q)1059 2476 y Fl(2)p Fm(n)p Fl(\()p Fm(k)q Fj(\000)p Fl(1\))953 2628 y Fp(57)p eop %%Page: 58 58 58 57 bop 308 262 a Fp(2.)20 b Fo(p)382 268 y Fm(k)q Fj(\000)p Fl(1)458 262 y Fp(is)14 b(a)f(nonzero)h(left)f(and)h(righ)o (t)f(in)o(tegral)g(of)f Fo(B)r Fp(.)i(The)g(in)o(tegral)f(group)g (elemen)o(t)361 311 y(of)g Fo(B)k Fp(is)d Fo(g)519 296 y Fj(\000)p Fl(\()p Fm(k)q Fj(\000)p Fl(1\))634 311 y Fp(.)f(The)h(in)o(tegral)g(c)o(haracter)h Fo(\023)1092 317 y Fm(B)1134 311 y Fp(of)e Fo(B)k Fp(is)c(equal)h(to)g Fo(\023)1446 317 y Fm(A)1472 311 y Fp(.)308 418 y(3.)20 b(The)14 b(sum)552 378 y Fm(k)539 386 y Fh(P)534 454 y Fm(i)p Fl(=1)595 418 y Fo(g)616 403 y Fm(i)644 418 y Fp(is)g(a)f(left)h(and)g(righ)o(t)f(in)o(tegral)g(of)h Fo(H)s Fp(.)257 539 y Fn(Pro)q(of.)36 b Fp(The)14 b(third)g(assertion)g (is)g(ob)o(vious)f(and)g(extremely)g(w)o(ell)g(kno)o(wn)g(\(cf.)h([19)o (],)f(Ex-)257 589 y(ample)h(2.1.2,)g(p.)h(17\).)g(The)h(\014rst)h (assertion)f(is)g(also)f(ob)o(vious.)f(It)i(remains)f(to)g(sho)o(w)h (the)257 639 y(second)f(assertion.)f(As)h(w)o(e)f(ha)o(v)o(e)g(already) f(p)q(oin)o(ted)h(out)g(ab)q(o)o(v)o(e,)f(w)o(e)h(ha)o(v)o(e:)662 753 y(\001)697 759 y Fm(A)724 753 y Fp(\()p Fo(\022)760 736 y Fm(n)783 753 y Fp(\))e(=)875 702 y Fm(n)855 714 y Fh(X)858 802 y Fm(i)p Fl(=0)922 753 y Fo(q)942 736 y Fm(i)p Fl(\()p Fm(n)p Fj(\000)p Fm(i)p Fl(\))1040 695 y Fh(\024)1067 725 y Fo(n)1072 782 y(i)1096 695 y Fh(\025)1118 753 y Fo(\022)1138 736 y Fm(i)1162 753 y Fk(\012)e Fo(\022)1224 736 y Fm(n)p Fj(\000)p Fm(i)257 874 y Fp(The)15 b(pro)q(duct)f(of)g(t)o (w)o(o)f(elemen)o(ts)h(of)f(the)i(dual)e(basis)h(is)g(therefore)h (equal)e(to:)778 980 y Fo(p)799 986 y Fm(i)812 980 y Fo(p)833 986 y Fm(j)862 980 y Fp(=)f Fo(q)926 963 y Fm(ij)955 922 y Fh(\024)982 952 y Fo(i)e Fp(+)f Fo(j)1017 1009 y(i)1072 922 y Fh(\025)1094 980 y Fo(p)1115 986 y Fm(i)p Fl(+)p Fm(j)257 1087 y Fp(In)14 b(addition,)f(w)o(e)h(ha)o(v)o(e)f(b)o (y)h(de\014nition:)617 1171 y Fo(\016)635 1177 y Fm(B)664 1171 y Fp(\()p Fo(p)701 1177 y Fm(n)724 1171 y Fp(\))d(=)h Fo(p)816 1177 y Fm(n)848 1171 y Fk(\012)e Fo(g)911 1154 y Fj(\000)p Fm(n)1042 1171 y Fo(p)1063 1177 y Fm(n)1097 1171 y Fk( )h Fo(g)i Fp(=)f Fo(q)1247 1154 y Fl(2)p Fm(n)1286 1171 y Fo(p)1307 1177 y Fm(n)257 1255 y Fp(The)j(second)g(assertion)f (is)g(no)o(w)f(clear.)h Fg(\003)257 1368 y Fp(By)h(Theorem)e(5.4,)f(w)o (e)i(no)o(w)g(can)g(obtain)f(immediately)e(the)j(in)o(tegral)f(of)g (our)h(quotien)o(t:)257 1463 y Fn(Prop)q(osition)33 b Fp(The)15 b(elemen)o(t)771 1534 y Fm(k)750 1546 y Fh(X)753 1635 y Fm(i)p Fl(=1)817 1586 y Fo(q)837 1569 y Fl(2)p Fm(i)p Fl(\()p Fm(k)q Fj(\000)p Fl(1\))954 1586 y Fo(E)987 1569 y Fm(k)q Fj(\000)p Fl(1)1050 1586 y Fo(K)1088 1569 y Fm(i)1102 1586 y Fo(F)1135 1569 y Fm(k)q Fj(\000)p Fl(1)257 1706 y Fp(is)f(a)g(nonzero)g(left)g(and)g(righ)o(t)f(in)o (tegral)g(of)h Fo(A)9 b Fk(\012)g Fo(H)k Fk(\012)c Fo(B)r Fp(.)257 1819 y(This)14 b(can)g(of)g(course)h(also)e(b)q(e)h(v)o (eri\014ed)h(directly)m(.)257 1988 y Fq(References)278 2094 y Fp([1])20 b(N.)k(Andruskiewitsc)o(h:)h(Notes)g(on)f(extensions)h (of)f(Hopf)g(algebras,)f(Canad.)h(J.)343 2144 y(Math.)15 b(48)h(\(1996\),)f(3-42)g(\(with)h(an)g(app)q(endix)g(b)o(y)g(N.)g (Andruskiewitsc)o(h)h(and)f(H.-)343 2194 y(J.)d(Sc)o(hneider\))278 2274 y([2])20 b(N.)11 b(Andruskiewitsc)o(h,)g(H.-J.)g(Sc)o(hneider:)h (Hopf)f(algebras)g(of)g(order)h Fo(p)1446 2259 y Fl(2)1475 2274 y Fp(and)f(braided)343 2323 y(Hopf)i(algebras)h(of)f(order)i Fo(p)p Fp(,)e(preprin)o(t,)h(1996)278 2403 y([3])20 b(V.)h(G.)g (Drinfel'd:)f(Quan)o(tum)h(groups,)g(in:)g(Pro)q(ceedings)j(of)d(the)h (In)o(ternational)343 2453 y(Congress)13 b(of)f(Mathematicians,)f(V)m (olume)f(I,)i(798-820,)f(Amer.)g(Math.)h(So)q(c.,)g(Berk)o(e-)343 2503 y(ley)m(,)g(1987)953 2628 y(58)p eop %%Page: 59 59 59 58 bop 278 262 a Fp([4])20 b(Y.)13 b(Doi:)g(Hopf)g(mo)q(dules)g(in)g (Y)m(etter-Drinfel'd)h(categories,)g(preprin)o(t,)g(1997)278 344 y([5])20 b(D.)39 b(Fisc)o(hman,)g(S.)h(Mon)o(tgomery)m(,)d(H.-J.)j (Sc)o(hneider:)h(F)m(rob)q(enius)g(exten-)343 393 y(sions)27 b(of)g(subalgebras)h(of)f(Hopf)g(algebras,)g(preprin)o(t,)h(1995,)e(to) h(app)q(ear)h(in:)343 443 y(T)m(rans.)13 b(Amer.)g(Math.)g(So)q(c.)278 525 y([6])20 b(P)m(.)12 b(J.)i(F)m(reyd,)f(D.)g(N.)g(Y)m(etter:)h (Braided)g(compact)e(closed)i(categories)g(with)g(applica-)343 575 y(tions)f(to)h(lo)o(w)f(dimensional)f(top)q(ology)m(,)f(Adv.)j (Math.)f(77)h(\(1989\),)e(156-182)278 657 y([7])20 b(M.)d(A.)g (Hennings:)g(In)o(v)n(arian)o(ts)f(of)h(links)g(and)g(3-manifol)o(ds)e (obtained)i(from)f(Hopf)343 707 y(algebras,)d(preprin)o(t,)h(1990)278 789 y([8])20 b(J.)c(C.)f(Jan)o(tzen:)i(Lectures)h(on)e(quan)o(tum)e (groups,)i(Grad.)f(Stud.)h(Math.)f(6,)h(Amer.)343 839 y(Math.)d(So)q(c.,)g(Pro)o(vidence,)i(R.)e(I.,)g(USA,)g(1996)278 921 y([9])20 b(A.)9 b(Jo)o(y)o(al,)g(R.)g(Street:)i(Braided)g(tensor)g (categories,)f(Adv.)g(Math.)f(102)h(\(1993\),)f(20-78)257 1003 y([10])20 b(F.)f(Kasc)o(h:)h(Grundlagen)g(einer)g(Theorie)h(der)f (F)m(rob)q(eniuserw)o(eiterungen,)i(Math.)343 1053 y(Ann.)13 b(127)g(\(1954\),)g(453-474)257 1135 y([11])20 b(M.)f(Koppinen:)g (Coideal)f(subalgebras)i(in)f(Hopf)g(algebras:)f(F)m(reeness,)j(in)o (tegrals,)343 1185 y(smash)13 b(pro)q(ducts,)h(Comm.)d(Algebra)i(21)h (\(1993\),)e(427-444)257 1267 y([12])20 b(R.)13 b(G.)g(Larson,)h(M.)g (E.)g(Sw)o(eedler:)g(An)h(asso)q(ciativ)o(e)f(orthogonal)f(bilinear)g (form)g(for)343 1316 y(Hopf)g(algebras,)g(Amer.)g(J.)h(Math.)f(91)h (\(1969\),)e(75-93)257 1398 y([13])20 b(G.)g(Lusztig:)h(In)o(tro)q (duction)g(to)g(Quan)o(tum)f(Groups,)g(Progr.)h(Math.)g(110,)f(Birk-) 343 1448 y(h\177)-21 b(auser,)14 b(Basel,)g(1993)257 1530 y([14])20 b(G.)15 b(Lusztig:)i(Finite)f(dimensional)f(Hopf)h (algebras)g(arising)g(from)f(quan)o(tized)i(uni-)343 1580 y(v)o(ersal)d(en)o(v)o(eloping)f(algebras,)g(J.)h(Amer.)f(Math.)g (So)q(c.)h(3)f(\(1990\),)g(257-296)257 1662 y([15])20 b(G.)13 b(Lusztig:)h(Quan)o(tum)f(groups)h(at)h(ro)q(ots)f(of)g(1,)f (Geom.)f(Dedicata)i(35)g(\(1990\),)f(89-)343 1712 y(114)257 1794 y([16])20 b(V.)13 b(Lyubashenk)o(o:)g(T)m(angles)g(and)h(Hopf)f (algebras)h(in)f(braided)h(categories,)g(J.)f(Pure)343 1844 y(Appl.)g(Algebra)h(98)f(\(1995\),)g(245-278)257 1926 y([17])20 b(V.)e(Lyubashenk)o(o:)g(Mo)q(dular)g(transformations)e (for)i(tensor)i(categories,)e(J.)g(Pure)343 1976 y(Appl.)13 b(Algebra)h(98)f(\(1995\),)g(279-327)257 2058 y([18])20 b(S.)c(Ma)r(jid:)f(Double)h(b)q(osonization)g(of)g(braided)g(groups)h (and)f(the)i(construction)f(of)343 2108 y Fa(U)376 2114 y Fm(q)391 2108 y Fp(\()p Fo(g)q Fp(\),)d(preprin)o(t)g(D)o (AMTP/95-57,)e(1995)257 2190 y([19])20 b(S.)9 b(Mon)o(tgomery:)f(Hopf)i (Algebras)g(and)g(their)h(Actions)f(on)g(Rings,)f(CBMS)h(Regional)343 2239 y(Conf.)j(Ser.)h(in)f(Math.)g(82,)g(Amer.)g(Math.)h(So)q(c.,)f (Pro)o(vidence,)h(R.)f(I.,)g(USA,)h(1993)257 2321 y([20])20 b(B.)i(P)o(areigis:)g(Einige)f(Bemerkungen)j(\177)-22 b(ub)q(er)23 b(F)m(rob)q(enius-Erw)o(eiterungen,)h(Math.)343 2371 y(Ann.)13 b(153)g(\(1964\),)g(1-13)257 2453 y([21])20 b(B.)h(P)o(areigis:)g(Endlic)o(he)g(Hopf-Algebren,)g(Algebra-Ber.,)g (Uni-Druc)o(k,)g(Munic)o(h,)343 2503 y(1973)953 2628 y(59)p eop %%Page: 60 60 60 59 bop 257 262 a Fp([22])20 b(D.)11 b(Radford:)g(The)i(order)g(of)f (the)h(an)o(tip)q(o)q(de)f(of)g(a)g(\014nite-dimensional)e(Hopf)i (algebra)343 311 y(is)h(\014nite,)h(Amer.)f(J.)h(Math)f(98)h(\(1976\),) e(333-355)257 391 y([23])20 b(D.)10 b(Radford:)f(The)i(structure)i(of)d (Hopf)g(algebras)h(with)f(a)g(pro)r(jection,)h(J.)f(Algebra)h(92)343 441 y(\(1985\),)h(322-347)257 520 y([24])20 b(D.)11 b(Radford:)g (Minimal)e(quasitriangular)i(Hopf)h(algebras,)f(J.)h(Algebra)g(157)g (\(1993\),)343 570 y(285-315)257 649 y([25])20 b(D.)d(Radford:)f (Generalized)i(double)g(crosspro)q(ducts)i(asso)q(ciated)e(with)g(the)g (quan-)343 699 y(tized)c(en)o(v)o(eloping)f(algebras,)h(preprin)o(t,)g (Chicago,)e(1991)257 779 y([26])20 b(D.)13 b(Radford:)g(The)i(trace)g (function)f(and)g(Hopf)g(algebras,)g(J.)g(Algebra)g(163)g(\(1994\),)343 828 y(583-622)257 908 y([27])20 b(D.)12 b(Radford,)g(L.)g(Kau\013man:)g (A)h(necessary)i(and)e(su\016cien)o(t)g(condition)g(for)g(a)f (\014nite-)343 958 y(dimensional)g(Drinfel'd)i(double)h(to)g(b)q(e)h(a) f(ribb)q(on)g(Hopf)f(algebra,)g(J.)h(Algebra)g(159)343 1008 y(\(1993\),)d(98-114)257 1087 y([28])20 b(A.)10 b(Rosen)o(b)q(erg:)g(Hopf)g(algebras)g(and)h(Lie)f(algebras)g(in)g (quasisymmetric)e(categories,)343 1137 y(preprin)o(t,)14 b(Mosco)o(w,)f(1978)257 1216 y([29])20 b(P)m(.)j(Sc)o(hauen)o(burg:)h (Hopf)f(mo)q(dules)f(and)i(Y)m(etter-Drinfel'd)f(mo)q(dules,)f(J.)i (Alge-)343 1266 y(bra)14 b(169)f(\(1994\),)f(874-890)257 1346 y([30])20 b(H.-J.)14 b(Sc)o(hneider:)j(Lectures)g(on)e(Hopf)g (algebras,)g(Univ)o(ersidad)g(de)h(Cordoba)f(T)m(ra-)343 1395 y(ba)r(jos)e(de)i(Matematica,)d(Ser.)i(B,)g(No.)f(31/95,)f (Cordoba,)h(Argen)o(tina,)g(1995)257 1475 y([31])20 b(Y.)11 b(Sommerh\177)-21 b(auser:)9 b(Deformierte)i(univ)o(erselle)h(Einh)q (\177)-22 b(ullende,)11 b(Diplomarb)q(eit,)e(Mu-)343 1525 y(nic)o(h,)k(1994)257 1604 y([32])20 b(Y.)e(Sommerh\177)-21 b(auser:)18 b(Deformed)g(en)o(v)o(eloping)g(algebras,)h(New)g(Y)m(ork)g (J.)g(Math.)g(2)343 1654 y(\(1996\),)12 b(35-58)257 1733 y([33])20 b(M.)13 b(E.)h(Sw)o(eedler:)g(Hopf)g(algebras,)f(W.)g(A.)h (Benjamin,)d(New)k(Y)m(ork,)e(1969)257 1813 y([34])20 b(M.)14 b(E.)h(Sw)o(eedler:)h(In)o(tegrals)f(for)g(Hopf)g(algebras,)f (Ann.)h(of)f(Math.)h(\(2\))g(89)g(\(1969\),)343 1863 y(323-335)257 1942 y([35])20 b(M.)13 b(T)m(ak)o(euc)o(hi:)f(Matc)o(hed) j(pairs)e(of)g(groups)h(and)f(bismash)f(pro)q(ducts)j(of)e(Hopf)g (alge-)343 1992 y(bras,)h(Comm)o(.)d(Algebra)j(9)f(\(1981\),)g(841-882) 257 2071 y([36])20 b(V.)c(G.)g(T)m(uraev:)g(Mo)q(dular)g(categories)h (and)g(3-manifol)o(d)d(in)o(v)n(arian)o(ts,)h(In)o(ternat.)i(J.)343 2121 y(Mo)q(dern)d(Ph)o(ys.)g(B)g(6)g(\(1992\),)f(1807-1824)257 2201 y([37])20 b(V.)c(G.)g(T)m(uraev:)g(Quan)o(tum)g(in)o(v)n(arian)o (ts)f(of)i(knots)g(and)f(3-manifolds,)e(de)j(Gruyter)343 2251 y(Stud.)c(Math.)h(18,)f(de)h(Gruyter,)g(Berlin,)g(1994)257 2330 y([38])20 b(D.)12 b(N.)g(Y)m(etter:)h(Quan)o(tum)f(groups)h(and)f (represen)o(tations)j(of)d(monoidal)e(categories,)343 2380 y(Math.)j(Pro)q(c.)h(Cam)o(bridge)e(Philos.)h(So)q(c.)h(108)f (\(1990\),)g(261-290)1239 2494 y(T)o(yp)q(eset)i(in)e Fk(A)1471 2503 y(M)1516 2494 y(S)h Fp(-)g(L)1594 2486 y Fl(A)1612 2494 y Fp(T)1635 2503 y(E)1658 2494 y(X)953 2628 y(60)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF