%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: KaplConjPrep4Vers.dvi %%CreationDate: Wed Sep 30 15:57:39 1998 %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips KaplConjPrep4Vers.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.07.08:2349 %%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/KaplConj/KaplConjPrep4Vers.dvi) @start /Fa 1 83 df<1FFC000603000601800601800C03000C06000FF8000C0C00180C 00180C00180C00180C00300C80FC0700110E7E8D15>82 D E /Fb 2 89 df80 D88 D E /Fc 2 4 df0 D<040004000400C460E4E03F800E003F80E4E0C4 600400040004000B0D7E8D11>3 D E /Fd 11 62 df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e 10 121 df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f 41 121 df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g 10 121 df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h 3 91 df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i 24 120 df<78FCFCFCFC7806067D850D>46 D<00600001E0000FE000FFE000F3E0 0003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0 0003E00003E00003E00003E00003E00003E00003E00003E0007FFF807FFF80111B7D9A18 >49 D<07F8001FFE00383F80780FC0FC07C0FC07E0FC03E0FC03E07803E00007E00007C0 0007C0000F80001F00001E0000380000700000E0000180600300600600600800E01FFFC0 3FFFC07FFFC0FFFFC0FFFFC0131B7E9A18>I<03F8001FFE003C1F003C0F807C07C07E07 C07C07C03807C0000F80000F80001E00003C0003F800001E00000F800007C00007C00007 E03007E07807E0FC07E0FC07E0FC07C0780F80781F001FFE0007F800131B7E9A18>I<00 0180000380000780000F80001F80003F80006F8000CF80008F80018F80030F80060F800C 0F80180F80300F80600F80C00F80FFFFF8FFFFF8000F80000F80000F80000F80000F8000 0F8001FFF801FFF8151B7F9A18>I<1801801FFF001FFE001FFC001FF8001FC000180000 18000018000018000019F8001E0E00180F801007800007C00007E00007E00007E07807E0 F807E0F807E0F807C0F007C0600F80381F001FFE0007F000131B7E9A18>I<007E0003FF 000781800F03C01E07C03C07C03C0380780000780000F80000F8F800FB0E00FA0780FC03 80FC03C0F803E0F803E0F803E0F803E07803E07803E07803C03C03C03C07801E0F0007FE 0003F800131B7E9A18>I<6000007FFFE07FFFE07FFFC07FFF807FFF80E00300C00600C0 0C00C0180000300000300000600000E00000E00001E00001C00003C00003C00003C00003 C00007C00007C00007C00007C00007C00007C000038000131C7D9B18>I<03F8000FFE00 1E0F803807803803C07803C07803C07E03C07F83807FC7003FFE001FFC000FFE0007FF80 1DFF80387FC0781FE0F007E0F003E0F001E0F001E0F001E07801C07803803E07801FFE00 03F800131B7E9A18>I<00038000000380000007C0000007C0000007C000000FE000000F E000001FF000001BF000001BF0000031F8000031F8000061FC000060FC0000E0FE0000C0 7E0000C07E0001803F0001FFFF0003FFFF8003001F8003001F8006000FC006000FC00E00 0FE00C0007E0FFC07FFEFFC07FFE1F1C7E9B24>65 D<7FFFFFE07FFFFFE0781F81E0701F 80E0601F8060E01F8070C01F8030C01F8030C01F8030C01F8030001F8000001F8000001F 8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F 8000001F8000001F8000001F8000001F800007FFFE0007FFFE001C1C7E9B21>84 D<03FC000E0E001C1F003C1F00781F00780E00F80000F80000F80000F80000F80000F800 007800007801803C01801C03000E0E0003F80011127E9115>99 D<000FF0000FF00001F0 0001F00001F00001F00001F00001F00001F00001F00001F001F9F00F07F01C03F03C01F0 7801F07801F0F801F0F801F0F801F0F801F0F801F0F801F07801F07801F03C01F01C03F0 0F0FFE03F9FE171D7E9C1B>I<01FC000F07001C03803C01C07801C07801E0F801E0F801 E0FFFFE0F80000F80000F800007800007C00603C00601E00C00F038001FC0013127F9116 >I<03F8F00E0F381E0F381C07303C07803C07803C07803C07801C07001E0F000E0E001B F8001000001800001800001FFF001FFFC00FFFE01FFFF07801F8F00078F00078F0007870 00707800F01E03C007FF00151B7F9118>103 DI107 DIII<01FC000F07801C01C03C01E07800F07800F0F800F8F800F8F800F8F8 00F8F800F8F800F87800F07800F03C01E01E03C00F078001FC0015127F9118>I114 D<030003000300030007000700 0F000F003FFCFFFC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F08 079803F00E1A7F9913>116 D119 D E /Fj 32 121 df<0001FF0000001FFFC000007F81E00000FC01E00001F807F00003F807F00007F007F0 0007F007F00007F007F00007F007F00007F001C00007F000000007F000000007F0000000 07F03FF800FFFFFFF800FFFFFFF800FFFFFFF80007F003F80007F003F80007F003F80007 F003F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003F80007F0 03F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003 F80007F003F80007F003F80007F003F8007FFF3FFF807FFF3FFF807FFF3FFF80212A7FA9 25>12 D<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE007FFFFE7FFFFE7FFFFE17277BA622>49 D<00FF800007FFF0000FFFFC001E03FE 003800FF807C003F80FE003FC0FF001FC0FF001FE0FF000FE0FF000FE07E000FE03C001F E000001FE000001FC000001FC000003F8000003F0000007E000000FC000000F8000001F0 000003E00000078000000F0000001E0000003C00E0007000E000E000E001C001C0038001 C0060001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF801B277D A622>I<007F800003FFF00007FFFC000F80FE001F007F003F807F003F803F803F803F80 3F803F801F803F801F003F8000007F0000007F0000007E000000FC000001F8000007F000 00FFC00000FFC0000001F80000007E0000003F0000003F8000001FC000001FC000001FE0 00001FE03C001FE07E001FE0FF001FE0FF001FE0FF001FC0FF003FC0FE003F807C007F00 3F00FE001FFFFC0007FFF00000FF80001B277DA622>I<00000E0000001E0000003E0000 007E000000FE000000FE000001FE000003FE0000077E00000E7E00000E7E00001C7E0000 387E0000707E0000E07E0000E07E0001C07E0003807E0007007E000E007E000E007E001C 007E0038007E0070007E00E0007E00FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE0000 00FE000000FE000000FE000000FE000000FE000000FE00007FFFF8007FFFF8007FFFF81D 277EA622>I<180003001F801F001FFFFE001FFFFC001FFFF8001FFFF0001FFFC0001FFF 00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C7FC0001DFF F8001F80FC001E003F0008003F0000001F8000001FC000001FC000001FE000001FE01800 1FE07C001FE0FE001FE0FE001FE0FE001FE0FE001FC0FC001FC078003F8078003F803C00 7F001F01FE000FFFFC0003FFF00000FF80001B277DA622>I<0007F800003FFE0000FFFF 0001FC078003F00FC007C01FC00F801FC01F801FC01F001FC03F000F803F0000007E0000 007E0000007E000000FE020000FE1FF000FE3FFC00FE603E00FE801F00FF801F80FF000F C0FF000FC0FE000FE0FE000FE0FE000FE0FE000FE07E000FE07E000FE07E000FE07E000F E03E000FE03F000FC01F000FC01F001F800F801F0007E07E0003FFFC0001FFF800003FC0 001B277DA622>I<380000003E0000003FFFFFF03FFFFFF03FFFFFF07FFFFFE07FFFFFC0 7FFFFF807FFFFF0070000E0070000E0070001C00E0003800E0007000E000E0000001E000 0001C000000380000007800000070000000F0000001F0000001E0000003E0000003E0000 007E0000007C0000007C000000FC000000FC000000FC000000FC000001FC000001FC0000 01FC000001FC000001FC000001FC000001FC000000F80000007000001C297CA822>I<00 3FC00001FFF00003FFFC0007C07E000F003F001E001F001E000F803E000F803E000F803F 000F803F800F803FC00F803FF01F001FFC1E001FFE3C000FFFF8000FFFE00007FFF80001 FFFC0001FFFE0007FFFF000F0FFF801E03FFC03E01FFC07C007FE07C001FE0F8000FE0F8 0007E0F80003E0F80003E0F80003E0F80003C07C0003C07E0007803F000F001FC03F000F FFFC0003FFF800007FC0001B277DA622>I<007F800001FFF00007FFF8000FC0FC001F80 3E003F001F007E001F807E001F807E000F80FE000FC0FE000FC0FE000FC0FE000FE0FE00 0FE0FE000FE0FE000FE0FE000FE07E001FE07E001FE03F003FE01F002FE00F80CFE007FF 8FE001FF0FE000080FE000000FC000000FC000000FC000001F803E001F807F001F807F00 3F007F003E007F007E007E00FC003E03F8001FFFE0000FFF800001FE00001B277DA622> I<00000780000000000780000000000FC0000000000FC0000000000FC0000000001FE000 0000001FE0000000003FF0000000003FF0000000003FF00000000077F80000000077F800 000000F7FC00000000E3FC00000000E3FC00000001C1FE00000001C1FE00000003C1FF00 00000380FF0000000380FF00000007007F80000007007F8000000F007FC000000E003FC0 00000E003FC000001C001FE000001C001FE000003FFFFFF000003FFFFFF000003FFFFFF0 0000700007F80000700007F80000F00007FC0000E00003FC0000E00003FC0001C00001FE 0001C00001FE0003C00001FF00FFFE003FFFFCFFFE003FFFFCFFFE003FFFFC2E297EA833 >65 D73 D82 D<7FFFFFFFFF807FFFFFFFFF807FFFFFFFFF807F807F807F807C007F800F8078007F8007 8078007F80078070007F800380F0007F8003C0F0007F8003C0E0007F8001C0E0007F8001 C0E0007F8001C0E0007F8001C0E0007F8001C000007F80000000007F80000000007F8000 0000007F80000000007F80000000007F80000000007F80000000007F80000000007F8000 0000007F80000000007F80000000007F80000000007F80000000007F80000000007F8000 0000007F80000000007F80000000007F80000000007F80000000007F80000000007F8000 0000007F80000000FFFFFFC00000FFFFFFC00000FFFFFFC0002A287EA72F>84 D<03FF80000FFFF0001F01FC003F80FE003F807F003F803F003F803F801F003F8000003F 8000003F8000003F8000003F80003FFF8001FC3F800FE03F801F803F803F003F807E003F 80FC003F80FC003F80FC003F80FC003F80FC005F807E00DF803F839FFC1FFE0FFC03F803 FC1E1B7E9A21>97 D<003FF00001FFFC0003F03E000FC07F001F807F003F007F003F007F 007F003E007E0000007E000000FE000000FE000000FE000000FE000000FE000000FE0000 00FE0000007E0000007E0000007F0000003F0003803F8003801F8007000FE00E0003F83C 0001FFF800003FC000191B7E9A1E>99 D<00007FF000007FF000007FF0000007F0000007 F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007 F0000007F0003F87F001FFF7F007F03FF00FC00FF01F8007F03F0007F03F0007F07E0007 F07E0007F07E0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007 F0FE0007F07E0007F07E0007F03F0007F03F0007F01F800FF00FC01FF007E07FFF01FFE7 FF007F87FF202A7EA925>I<003FC00001FFF00003E07C000F803E001F801F001F001F00 3F000F807E000F807E000FC07E000FC0FE0007C0FE0007C0FFFFFFC0FFFFFFC0FE000000 FE000000FE0000007E0000007E0000007F0000003F0001C01F0001C00F80038007C00700 03F01E0000FFFC00003FE0001A1B7E9A1F>I<0007F8003FFC007E3E01FC7F03F87F03F0 7F07F07F07F03E07F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FFFF C007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F0 0007F00007F00007F00007F00007F00007F00007F00007F00007F0007FFF807FFF807FFF 80182A7EA915>I<007F80F001FFE3F807C0FE1C0F807C7C1F003E7C1F003E103F003F00 3F003F003F003F003F003F003F003F003F003F001F003E001F003E000F807C0007C0F800 05FFE0000C7F8000180000001C0000001C0000001E0000001FFFF8001FFFFF000FFFFFC0 07FFFFE003FFFFF00FFFFFF03E0007F07C0001F8F80000F8F80000F8F80000F8F80000F8 7C0001F07C0001F03F0007E00FC01F8007FFFF00007FF0001E287E9A22>II<07000F801FC03FE03FE0 3FE01FC00F8007000000000000000000000000000000FFE0FFE0FFE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFE FFFEFFFE0F2B7EAA12>I<000700000F80001FC0003FE0003FE0003FE0001FC0000F8000 070000000000000000000000000000000000000000000001FFE001FFE001FFE0000FE000 0FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE000 0FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE000 0FE0000FE0000FE07C0FE0FE0FE0FE0FC0FE1F80FE1F007C3E003FFC000FF000133784AA 15>I110 D<003FE00001FFFC0003F07E000FC01F801F800FC03F0007E0 3F0007E07E0003F07E0003F07E0003F0FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8 FE0003F8FE0003F8FE0003F87E0003F07E0003F03F0007E03F0007E01F800FC00FC01F80 07F07F0001FFFC00003FE0001D1B7E9A22>II114 D<03FE300FFFF03E03F07800F07000F0F0 0070F00070F80070FE0000FFE0007FFF007FFFC03FFFE01FFFF007FFF800FFF80007FC00 00FCE0007CE0003CF0003CF00038F80038FC0070FF01E0E7FFC0C1FF00161B7E9A1B>I< 00700000700000700000700000F00000F00000F00001F00003F00003F00007F0001FFFE0 FFFFE0FFFFE007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F07007F07007F07007F07007F07007F07007F07003F0E001F8C0 00FFC0003F0014267FA51A>III120 D E /Fk 25 122 df<60F0F868080808101020C0050B7D990B>39 D<60F0F06004047D830B>46 D72 D75 D87 D<3F8070C070E020700070007007F01C7030707070E070E071E071E0F171FB1E3C10107E 8F13>97 DI<07F80C1C381C30087000E000E000E000E000 E000E0007000300438080C1807E00E107F8F11>I<07C01C3030187018600CE00CFFFCE0 00E000E000E0006000300438080C1807E00E107F8F11>101 D<01F0031807380E100E00 0E000E000E000E000E00FFC00E000E000E000E000E000E000E000E000E000E000E000E00 0E000E007FE00D1A80990C>I<0FCE187330307038703870387038303018602FC0200060 0070003FF03FFC1FFE600FC003C003C003C0036006381C07E010187F8F13>II<018003C003C001800000000000000000000000000FC001C001C001 C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C041C0E180E3 007E000A2182990C>106 DII110 D<07E01C38300C700E6006E007E007E007E007E007E0076006700E381C1C3807E010107F 8F13>II114 D<1F2060E04020C020C020F0007F003FC01FE000F0807080 30C030C020F0408F800C107F8F0F>I<0400040004000C000C001C003C00FFC01C001C00 1C001C001C001C001C001C001C201C201C201C201C200E4003800B177F960F>IIII121 D E /Fl 7 117 df<00030000000780000007800000078000000FC000000FC00000 1BE000001BE000001BE0000031F0000031F0000060F8000060F80000E0FC0000C07C0000 C07C0001803E0001FFFE0003FFFF0003001F0003001F0006000F8006000F800E000FC0FF C07FFCFFC07FFC1E1A7F9921>65 D<0FF0001C3C003E1E003E0E003E0F001C0F00000F00 00FF000FCF003E0F007C0F00F80F00F80F00F80F00F817007C27E01FC3E013117F9015> 97 DI<03FC000F0E001C1F003C1F00781F00780E00F800 00F80000F80000F80000F800007800007800003C01801C03000F060003FC0011117F9014 >I114 D<1FB020704030C030C030F000FF807FE03FF807F8003CC00CC00CE0 0CE008F830CFE00E117F9011>I<06000600060006000E000E001E003FF0FFF01E001E00 1E001E001E001E001E001E001E181E181E181E181E180F3003E00D187F9711>I E /Fm 91 128 df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n 13 128 df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o 17 122 df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end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 510 369 a Fo(On)21 b(Kaplansky's)e(conjectures)684 522 y Fn(Y)l(orc)o(k)d(Sommerh\177)-24 b(auser)453 680 y Fm(Extended)16 b(v)o(ersion)e(of)f(a)h(talk)f(giv)o(en)g(at)h(the)g (conference)281 730 y(`In)o(teractions)h(b)q(et)o(w)o(een)g(Ring)e (Theory)h(and)g(Represen)o(tations)h(of)e(Algebras',)587 780 y(Murcia,)g(Spain,)g(Jan)o(uary)h(12-17,)e(1998)825 892 y Fl(Abstract)327 968 y Fk(W)m(e)h(surv)o(ey)g(the)g(kno)o(wn)g (results)h(on)f(Kaplansky's)i(ten)e(conjectures)h(on)f(Hopf)328 1014 y(algebras.)224 1182 y Fj(1)67 b(In)n(tro)r(duction)224 1306 y Fm(In)12 b(the)h(autumn)e(of)g(1973,)g(I.)g(Kaplansky)h(ga)o(v)o (e)g(a)f(course)j(on)e(bialgebras)f(in)h(Chicago.)224 1356 y(F)m(or)j(this)g(course,)h(he)g(prepared)h(some)d(lecture)i (notes)g(that)f(he)h(originally)d(in)o(tended)224 1405 y(to)k(turn)g(in)o(to)e(a)h(comprehensiv)o(e)h(accoun)o(t)g(on)f(the)h (sub)r(ject.)h(In)e(1975,)f(he)i(c)o(hanged)224 1455 y(his)e(mind)e(and)i(published)g(these)h(lecture)h(notes)e(without)g (larger)g(additions.)e(These)224 1505 y(lecture)i(notes)f(con)o(tain,)e (b)q(esides)j(a)e(fairly)e(comprehensiv)o(e)j(bibliograph)o(y)d(of)i (the)g(lit-)224 1555 y(erature)h(a)o(v)n(ailable)d(at)h(that)h(time,)e (t)o(w)o(o)h(app)q(endices.)h(The)h(\014rst)f(of)f(these)i(app)q (endices)224 1605 y(is)g(concerned)i(with)d(bialgebras)g(of)h(lo)o(w)e (dimension,)g(whereas)j(the)g(second)f(one)g(con-)224 1654 y(tains)j(a)f(list)g(of)g(ten)i(conjectures)g(on)f(Hopf)f (algebras)g(whic)o(h)h(are)g(kno)o(wn)f(to)q(da)o(y)g(as)224 1704 y(Kaplansky's)e(conjectures.)224 1787 y(Kaplansky's)c(conjectures) h(did)f(not)f(arise)h(as)g(the)h(pro)q(duct)f(of)f(a)h(long)f(in)o(v)o (estigation)f(in)224 1836 y(the)13 b(\014eld)f(of)g(Hopf)g(algebras;)f (also,)g(Kaplansky)h(did)g(not)g(mak)o(e)e(man)o(y)g(con)o(tributions) 224 1886 y(to)21 b(the)g(solution)f(of)g(his)h(conjectures.)h(He)f (only)f(in)o(tended)h(to)g(list)f(a)g(n)o(um)o(b)q(er)g(of)224 1936 y(in)o(teresting)d(problems)e(at)h(the)g(end)h(of)e(his)h(lecture) h(notes)f(-)g(lecture)h(notes)g(that)f(he)224 1986 y(himself)d(called)h (informal.)d(Because)16 b(of)e(this,)g(it)g(happ)q(ened)i(that)e(one)h (conjecture)h(in)224 2036 y(the)i(list)e(w)o(as)h(already)f(solv)o(ed)h (at)g(the)g(time)f(of)g(publication,)f(another)i(one)g(is)g(v)o(ery)224 2085 y(simple.)9 b(That)h(the)h(conjectures)i(nev)o(ertheless)f(gained) e(considerable)h(imp)q(ortance)f(for)224 2135 y(the)j(\014eld)g(is)f (due)h(to)g(the)g(fact)f(that)h(Kaplansky)f(ac)o(hiev)o(ed)g(to)h(touc) o(h)f(up)q(on)h(a)f(n)o(um)o(b)q(er)224 2185 y(of)i(questions)g(of)f (fundamen)o(tal)f(c)o(haracter.)899 2310 y(1)p eop %%Page: 2 2 2 1 bop 224 195 a Fm(This)19 b(article)g(tries)g(to)f(summarize)f(the)i (presen)o(t)h(kno)o(wledge)e(ab)q(out)h(Kaplansky's)224 245 y(conjectures.)14 b(Brief)d(surv)o(eys)i(can)f(b)q(e)g(found)f(in)g ([53)o(])g(and)g([65)o(],)g(some)f(conjectures)k(are)224 295 y(also)h(discussed)i(in)e([38)o(])g(and)g([43)o(].)f(Here,)i(the)g (exp)q(osition)f(shall)g(b)q(e)h(more)e(detailed,)224 344 y(but)f(nev)o(ertheless)h(not)e(comprehensiv)o(e.)g(Since)h (Kaplansky's)f(lecture)h(notes)g(are)g(not)224 394 y(alw)o(a)o(ys)e (easily)h(accessible,)g(w)o(e)h(ha)o(v)o(e)e(repro)q(duced)j(the)f (conjectures)h(in)d(their)h(original)224 444 y(form)o(ulation)17 b(in)h(an)i(app)q(endix.)e(The)i(reader)h(should)e(note)g(that,)g (except)i(for)e(the)224 494 y(app)q(endix,)i(the)g(form)o(ulation)d(of) i(results)i(do)q(es)f(not)g(follo)o(w)d(literally)i(the)h(quoted)224 544 y(sources.)14 b(In)f(addition,)e(usually)h(not)g(all)f(imp)q(ortan) o(t)g(results)j(of)e(a)g(quoted)h(article)g(are)224 594 y(men)o(tioned.)18 b(W)m(e)g(also)g(note)i(that)f(Kaplansky)f(p)q(osed) i(v)n(arious)e(other)h(conjectures)224 643 y(concerning)c(di\013eren)o (t)g(\014elds)f(of)g(mathematics;)d(these)k(are)f(not)g(discussed)i (here.)224 812 y Fj(2)67 b(The)23 b(\014rst)f(conjecture)224 936 y Fi(2.1)48 b Fm(Kaplansky's)10 b(\014rst)i(conjecture)g(states)g (that)f(a)f(Hopf)g(algebra)h(is)f(a)g(free)i(mo)q(dule)224 986 y(o)o(v)o(er)f(an)o(y)g(Hopf)f(subalgebra.)g(This)h(assertion)h (should)e(b)q(e)i(compared)e(with)g(the)i(group)224 1036 y(ring)h(situation,)e(where)j(a)e(system)g(of)g(represen)o(tativ)o(es)j (for)e(the)g(cosets)h(forms)d(a)h(basis)224 1086 y(of)g(the)h(group)f (ring)g(o)o(v)o(er)g(the)h(group)f(ring)g(of)f(a)h(subgroup.)g(A)o(t)h (the)g(time)e(of)g(its)h(publi-)224 1136 y(cation,)h(it)g(w)o(as)g (already)g(kno)o(wn)g(that)g(the)h(conjecture)h(is)e(false,)g(b)q (ecause)i(U.)e(Ob)q(erst)224 1185 y(and)j(H.-J.)f(Sc)o(hneider)i(had,)e (in)g(an)g(article)h(quoted)g(b)o(y)g(Kaplansky)m(,)e(constructed)k(a) 224 1235 y(coun)o(terexample)12 b(\(cf.)g([51)o(],)e(Prop.)i(10,)f(p.)g (31,)g(see)i(also)e([38)o(],)g(Example)f(3.5.2,)g(p.)h(38\).)224 1285 y(In)i(this)f(coun)o(terexample,)g(Ob)q(erst)h(and)g(Sc)o(hneider) g(construct)h(an)e(extension)h(of)e(real)224 1335 y(Hopf)j(algebras)g (via)f(Galois)f(descen)o(t)k(from)c(the)i(extension)791 1420 y Fh(C)6 b Fm([2)p Fh(Z)-11 b Fm(])8 b Fg(\032)j Fh(C)6 b Fm([)p Fh(Z)-10 b Fm(])224 1505 y(of)17 b(the)h(complex)e (group)i(ring)f(of)g(the)h(in)o(tegers)g(o)o(v)o(er)f(the)i(group)e (ring)g(of)g(the)h(ev)o(en)224 1554 y(n)o(um)o(b)q(ers.)13 b(In)f(fact,)h(instead)g(of)g(the)g(extension)h Fh(R)5 b Fg(\032)12 b Fh(C)6 b Fm(,)15 b(they)f(consider)g(more)e(gener-)224 1604 y(ally)i(arbitrary)h(quadratic)g(Galois)e(extensions)j(of)e (\014elds.)h(Later,)g(Sc)o(hneider)h(ga)o(v)o(e)f(a)224 1654 y(coun)o(terexample)h(o)o(v)o(er)f(arbitrary)g(\014elds)h(\(cf.)f ([66)o(]\):)g(Supp)q(ose)h(that)f Ff(H)k Fm(is)c(the)h(Hopf)224 1704 y(algebra)g(represen)o(ting)i(the)f(a\016ne)g(group)f(sc)o(heme)g (of)g(the)h(sp)q(ecial)g(linear)f(group)g(of)224 1754 y(degree)g Ff(n)e Fm(and)g Ff(A)e Fm(=)h Ff(K)s Fm([)p Ff(Z)640 1760 y Fe(n)662 1754 y Fm(])h(the)h(group)f(ring)g(of)f(the)i (cyclic)g(group)f(of)f(order)i Ff(n)p Fm(,)f(rep-)224 1803 y(resen)o(ting)i(the)f(a\016ne)g(group)f(sc)o(heme)h(of)f Ff(n)p Fm(-th)h(ro)q(ots)g(of)f(unit)o(y)m(.)g(The)h(ro)q(ots)g(of)f (unit)o(y)224 1853 y(are)g(realized)f(via)f(diagonal)e(matrices)j(as)f (a)h(normal)d(subgroup)k(of)e(the)h(sp)q(ecial)g(linear)224 1903 y(group.)g(In)g(this)g(situation,)f(w)o(e)h(can)h(lo)q(ok)e(at)h (the)g(quotien)o(t)g(sc)o(heme,)g(whic)o(h)g(is)g(repre-)224 1953 y(sen)o(ted)k(b)o(y)e(a)g(Hopf)g(algebra)g Ff(B)r Fm(.)h(No)o(w)f(Sc)o(hneider)h(pro)o(v)o(es)g(that)g Ff(H)i Fm(is)d(not)g(free)h(o)o(v)o(er)224 2003 y Ff(B)h Fm(if)c Ff(n)h Fm(is)f(ev)o(en.)224 2135 y Fi(2.2)48 b Fm(Ho)o(w)o(ev)o(er,)17 b(sev)o(eral)h(p)q(ositiv)o(e)f(results)h(w)o (ere)h(pro)o(v)o(ed)e(in)g(the)h(follo)o(wing)c(y)o(ears,)224 2185 y(sho)o(wing)e(that)h(the)g(conjecture)h(holds)f(under)g (additional)e(assumptions.)g(Kaplansky)899 2310 y(2)p eop %%Page: 3 3 3 2 bop 224 195 a Fm(already)11 b(men)o(tioned)e(that)i(W.)f(D.)f(Nic)o (hols)i(pro)o(v)o(ed)f(that)h(the)g(conjecture)i(holds)d(if)g(the)224 245 y(coradical)j(of)f(the)i(large)f(Hopf)f(algebra)h(is)g(con)o (tained)g(in)g(the)g(Hopf)g(subalgebra,)f(this)224 295 y(is)j(also)f(a)g(direct)i(consequence)h(of)d(a)h(result)g(b)o(y)f(D.)g (E.)h(Radford)f(\(cf.)g([58)o(],)g(Cor.)g(2.3,)224 344 y(p.)j(146\).)f(In)h(the)g(same)f(y)o(ear,)h(Radford)f(pro)o(v)o(ed)h (that)g(the)g(conjecture)i(holds)d(for)h(a)224 394 y(large)d(class)g (of)g(Hopf)f(algebras,)g(namely)g(the)h(p)q(oin)o(ted)g(ones)g(\(cf.)g ([57)o(],)f(p.)g(271\):)224 494 y Fi(Theorem)36 b Fm(Supp)q(ose)11 b(that)f Ff(A)f Fm(is)h(a)g(Hopf)f(subalgebra)h(of)f(the)i(Hopf)e (algebra)g Ff(H)s Fm(.)g(Sup-)224 544 y(p)q(ose)16 b(that)e Ff(H)k Fm(is)c(p)q(oin)o(ted,)h(i.)e(e.,)h(ev)o(ery)i(simple)d(sub)q (coalgebra)i(is)f(one-dimensional.)224 594 y(Then)h Ff(H)i Fm(is)c(a)h(free)h(left)e(and)h(righ)o(t)f Ff(A)p Fm(-mo)q(dule.)224 712 y(F)m(urthermore,)18 b(Radford)f(pro)o(v)o(ed)h(in)g([59)o(])g (that)g(comm)o(utativ)o(e)d(Hopf)j(algebras)g(are)224 762 y(free)e(o)o(v)o(er)f(\014nite-dimensional)e(Hopf)h(subalgebras.)h (He)g(also)g(pro)o(v)o(ed)g(that)g(here)h(the)224 812 y(assumption)9 b(that)i(the)g(Hopf)f(subalgebra)g(b)q(e)h (\014nite-dimensional)d(could)i(b)q(e)h(w)o(eak)o(end)224 862 y(to)k(the)g(assumption)f(that)g(only)g(the)i(coradical)e(of)g(the) h(Hopf)f(subalgebra)h(b)q(e)g(\014nite-)224 912 y(dimensional)i(b)o(y)j (imp)q(osing)d(the)j(additional)e(assumption)g(that)h(the)h(coradical)f (of)224 961 y(the)c(large)f(Hopf)g(algebra)g(is)g(a)g(Hopf)f (subalgebra,)h(whic)o(h)g(is)g(the)h(case)g(o)o(v)o(er)f(\014elds)h(of) 224 1011 y(c)o(haracteristic)f(zero)f(b)o(y)g(a)f(result)h(of)f(M.)f(T) m(ak)o(euc)o(hi)h(\(cf.)g([80)o(]\).)g(He)h(also)f(pro)o(v)o(ed)g (there)224 1061 y(that)17 b(Hopf)e(algebras)h(are)h(free)g(o)o(v)o(er)f (\014nite-dimensional)e(Hopf)i(subalgebras)g(if)g(the)224 1111 y(coradical)e(of)f(the)i(large)e(Hopf)h(algebra)f(is)h(co)q(comm)o (utativ)o(e.)224 1246 y Fi(2.3)48 b Fm(W.)12 b(D.)g(Nic)o(hols)g(had)h (b)q(een)h(concerned)h(with)d(Kaplansky's)g(conjectures)j(from)224 1296 y(the)g(v)o(ery)f(b)q(eginning.)f(In)h(the)h(eigh)o(ties,)e(he)i (and)e(his)h(former)f(Ph.)h(D.)f(studen)o(t)i(M.)f(B.)224 1346 y(Zo)q(eller)i(w)o(ork)o(ed)f(in)g(particular)g(on)g(Kaplansky's)g (\014rst)h(conjecture.)h(They)e(summa-)224 1396 y(rized)20 b(their)f(e\013orts)h(in)f(a)f(series)j(of)d(articles)h(\(cf.)g([89)o (],)f([44)o(],)g([46)o(]\))g(of)h(subsequen)o(t)224 1445 y(generalizations,)g(culminating)d(in)i(what)h(is)g(no)o(w)g(kno)o(wn)f (as)h(the)g(Nic)o(hols-Zo)q(eller)224 1495 y(theorem:)224 1595 y Fi(Theorem)36 b Fm(A)13 b(\014nite-dimensional)d(Hopf)j(algebra) f(is)h(free)g(o)o(v)o(er)g(ev)o(ery)g(Hopf)g(subal-)224 1645 y(gebra.)224 1763 y(In)18 b(fact,)g(Nic)o(hols)g(and)f(Zo)q(eller) i(pro)o(v)o(e)f(the)h(more)e(general)h(theorem)g(that)g(relativ)o(e)224 1813 y(Hopf)c(mo)q(dules)f(are)i(free.)f(Their)g(theorem)g(implies)e (that)i(the)h(dimension)e(of)g(a)h(Hopf)224 1863 y(subalgebra)20 b(divides)f(the)g(dimension)f(of)g(the)i(large)f(Hopf)g(algebra,)f (whic)o(h)h(is)g(the)224 1913 y(Hopf)13 b(algebra)g(v)o(ersion)g(of)f (Lagrange's)h(theorem)g(on)f(the)i(order)g(of)e(subgroups.)i(This)224 1963 y(theorem)h(has)g(man)o(y)f(consequences)k(and)d(is)g(commonl)o(y) d(considered)17 b(as)e(one)g(of)g(the)224 2013 y(most)c(imp)q(ortan)o (t)e(theorems)j(of)f(the)h(theory)g(of)f(Hopf)g(algebras.)g(Nic)o(hols) g(and)g(Zo)q(eller)224 2062 y(also)j(exhibited)h(an)f(example)f(that)i (in\014nite)f(dimensional)f(Hopf)h(algebras)g(need)i(not)224 2112 y(b)q(e)h(free)g(o)o(v)o(er)e(\014nite)h(dimensional)e(grouplik)o (e)h(Hopf)g(subalgebras)i(\(cf.)e([45)o(]\).)g(These)224 2162 y(authors)g(ha)o(v)o(e)e(also)g(sho)o(wn)h(in)f(a)h(more)f(recen)o (t)i(article)f(that)g(a)f(Hopf)h(algebra)f(is)h(free)899 2310 y(3)p eop %%Page: 4 4 4 3 bop 224 195 a Fm(o)o(v)o(er)18 b(a)f(semisimple)e(Hopf)j (subalgebra)f(\(cf.)h([47)o(]\).)f(\(Due)g(to)h(marriage,)d(the)k(name) 224 245 y(`Zo)q(eller')14 b(has)g(c)o(hanged)g(to)g(`Ric)o(hmond'.)o (\))224 328 y(The)d(w)o(ork)f(of)f(Nic)o(hols)h(and)f(Zo)q(eller)i(has) f(b)q(een)h(ampli\014ed)d(b)o(y)i(H.-J.)f(Sc)o(hneider.)i(Giv)o(en)224 378 y(a)h(Hopf)g(subalgebra)g(of)f(a)h(\014nite-dimensional)d(Hopf)j (algebra,)f(it)g(is)h(p)q(ossible)g(to)g(form)224 428 y(a)17 b(quotien)o(t)h(whic)o(h)f(is)g(not)g(alw)o(a)o(ys)f(a)h(Hopf)g (algebra,)f(but)i(alw)o(a)o(ys)e(is)h(a)g(coalgebra.)224 477 y(The)c(original)d(Hopf)i(algebra)g(then)h(b)q(ecomes)g(a)f(como)q (dule)f(o)o(v)o(er)h(this)g(quotien)o(t.)g(No)o(w)224 527 y(Sc)o(hneider)h(sho)o(ws)f(\(cf.)g([67)o(]\))f(that)g(the)i (original)c(Hopf)j(algebra)f(can)h(b)q(e)g(decomp)q(osed,)224 577 y(as)j(a)g(mo)q(dule)e(and)i(a)f(como)q(dule,)f(in)o(to)h(a)h (tensor)h(pro)q(duct)f(of)g(the)g(Hopf)f(subalgebra)224 627 y(and)g(the)g(quotien)o(t.)f(In)h(a)f(di\013eren)o(t)i(article,)e (Sc)o(hneider)i(pro)o(v)o(es)f(that)g(Hopf)g(algebras)224 677 y(are)h(free)f(o)o(v)o(er)g(\014nite)g(dimensional)e(normal)g(Hopf) h(subalgebras)h(\(cf.)g([68)o(]\).)224 760 y(The)h(Nic)o(hols-Zo)q (eller)f(theorem)g(has)g(b)q(een)i(extended)f(to)f(coideal)g (subalgebras)g(that)224 810 y(are)h(quasi-F)m(rob)q(enius)e(algebras)h (b)o(y)f(A.)g(Masuok)n(a)g(\(cf.)h([29)o(]\).)f(There,)h(he)g(also)f (deter-)224 859 y(mines)h(su\016cien)o(t)h(conditions)g(for)f(a)g (coideal)h(subalgebra)f(to)h(b)q(e)g(a)g(quasi-F)m(rob)q(enius)224 909 y(algebra)f(\(cf.)f(also)h([30)o(]\).)224 1042 y Fi(2.4)48 b Fm(Although)14 b(no)o(w)o(ada)o(ys)f(quite)h(a)g(lot)f(is)h (kno)o(wn)g(ab)q(out)g(Kaplansky's)g(\014rst)g(con-)224 1092 y(jecture,)k(the)g(fact)f(that)g(it)f(do)q(es)i(not)f(hold)f(in)h (the)g(in\014nite)g(dimensional)d(case)k(has)224 1142 y(giv)o(en)e(rise)h(to)g(the)g(question)f(whether)i(w)o(eak)o(er)f (prop)q(erties)h(than)f(freeness)h(hold)e(in)224 1192 y(all)d(cases.)j(One)f(p)q(ossible)f(v)n(arian)o(t)f(of)h(this)g (question)h(has)f(b)q(een)i(stated)f(b)o(y)f(S.)g(Mon)o(t-)224 1242 y(gomery)f(in)g([38)o(],)g(Question)i(3.5.4,)d(p.)h(39:)224 1325 y(Is)i Ff(H)h Fm(alw)o(a)o(ys)d(left)h(and)g(righ)o(t)f (faithfully)f(\015at)h(o)o(v)o(er)h(an)o(y)g(subHopfalgebra)f Ff(K)s Fm(?)224 1408 y(This)19 b(question)g(migh)o(t)d(b)q(e)j (considered)h(as)f(a)f(p)q(erp)q(etuation)i(of)e(Kaplansky's)g(\014rst) 224 1458 y(conjecture.)e(It)f(pla)o(ys)f(an)h(imp)q(ortan)o(t)e(role)h (for)h(the)g(quotien)o(t)g(theory)g(of)f(Hopf)g(alge-)224 1507 y(bras.)224 1591 y(Sev)o(eral)i(things)f(are)g(kno)o(wn)g(ab)q (out)g(this)g(question)g(to)q(da)o(y)m(.)f(It)h(has)g(b)q(een)h(kno)o (wn)f(for)224 1640 y(quite)g(a)f(time)g(from)f(the)i(theory)g(of)f (algebraic)g(groups)h(that)g(this)g(is)f(the)h(case)h(if)e(the)224 1690 y(Hopf)19 b(algebra)f(is)h(comm)o(utati)o(v)o(e)d(\(cf.)j([7)o(],) f(Chap.)g(I)q(I)q(I,)h Fg(x)g Fm(3,)f(no.)g(7,)g(Theorem)g(7.2,)224 1740 y([84)o(],)d(sec.)i(14.1\).)d(This)i(result)h(can)f(also)f(b)q(e)i (found)e(in)g(an)h(article)g(b)o(y)g(M.)f(T)m(ak)o(euc)o(hi)224 1790 y(\(cf.)20 b([79)o(],)e(Theorem)h(3.1\),)f(where)j(it)e(is)h(also) e(sho)o(wn)i(that)f(a)h(Hopf)f(algebra)g(with)224 1840 y(co)q(comm)o(utativ)o(e)10 b(coradical)h(is)h(faithfully)e(\015at)i(o) o(v)o(er)g(ev)o(ery)h(Hopf)f(subalgebra.)f(Later,)224 1890 y(T)m(ak)o(euc)o(hi)16 b(pro)o(v)o(ed)h(that)f(comm)o(utativ)o(e)d (Hopf)j(algebras)h(are)f(pro)r(jectiv)o(e)i(o)o(v)o(er)e(their)224 1939 y(Hopf)f(subalgebras)g(\(cf.)g([82)o(],)f(Cor.)h(1,)f(p.)h(460\).) f(Ho)o(w)o(ev)o(er,)g(as)i(w)o(as)f(sho)o(wn)g(later)g(b)o(y)224 1989 y(A.)h(Masuok)n(a)f(and)h(D.)g(Wigner)f(\(cf.)h([31)o(]\),)f (faithful)g(\015atness)i(and)f(pro)r(jectivit)o(y)g(are)224 2039 y(essen)o(tially)e(equiv)n(alen)o(t)f(conditions)h(in)f(this)h (situation:)224 2135 y Fi(Theorem)36 b Fm(Supp)q(ose)16 b(that)e Ff(H)k Fm(is)d(a)f(Hopf)h(algebra)f(with)h(bijectiv)o(e)f(an)o (tip)q(o)q(de,)h(and)224 2185 y(that)f Ff(A)g Fm(is)g(a)g(righ)o(t)g (coideal)f(subalgebra)h(of)f Ff(H)s Fm(.)h(Then)g(the)h(follo)o(wing)c (assertions)k(are)899 2310 y(4)p eop %%Page: 5 5 5 4 bop 224 195 a Fm(equiv)n(alen)o(t:)275 314 y(1.)20 b Ff(H)d Fm(is)d(faithfully)d(\015at)j(as)g(a)g(left)f Ff(A)p Fm(-mo)q(dule.)275 397 y(2.)20 b Ff(H)d Fm(is)d(a)f(pro)r (jectiv)o(e)i(generator)g(as)f(a)f(left)h Ff(A)p Fm(-mo)q(dule.)224 516 y(This)e(theorem)g(applies)f(in)h(particular)f(if)h Ff(A)f Fm(is)h(a)g(Hopf)f(subalgebra)h(of)g Ff(H)s Fm(.)f(Of)h(course,) 224 565 y(the)k(\014rst)g(statemen)o(t)f(can)g(easily)f(b)q(e)i (deduced)g(from)e(the)h(second.)h(In)f(addition,)e(H.-)224 615 y(J.)20 b(Sc)o(hneider)i(has)e(pro)o(v)o(ed)g(that)h(a)e(left)h(no) q(etherian)h(Hopf)f(algebra)g(is)g(faithfully)224 665 y(\015at)c(o)o(v)o(er)f(cen)o(tral)g(Hopf)g(subalgebras)h(\(cf.)f([68)o (]\).)f(S.)h(H.)g(Ng)g(has)g(sho)o(wn)h(that)f(Hopf)224 715 y(algebras)f(are)h(faithfully)d(\015at)h(o)o(v)o(er)h(grouplik)o(e) g(Hopf)f(subalgebras)h(if)g(the)g(c)o(haracter-)224 765 y(istic)h(of)f(the)h(base)g(\014eld)g(is)f(zero)h(\(cf.)g([41)o(]\);)e (he)i(also)f(considers)i(the)f(case)h(of)e(p)q(ositiv)o(e)224 814 y(c)o(haracteristic.)224 900 y(These)g(results)g(ha)o(v)o(e)f(b)q (een)h(partially)d(generalized)i(to)g(coideal)f(subalgebras:)h(A.)f (Ma-)224 950 y(suok)n(a)17 b(and)h(D.)f(Wigner)g(pro)o(v)o(e)g(that)h (comm)o(utativ)o(e)c(Hopf)k(algebras)f(are)h(\015at)f(o)o(v)o(er)224 1000 y(righ)o(t)h(coideal)g(subalgebras)h(\(cf.)f([31)o(]\),)g(they)h (are)g(not)f(faithfully)e(\015at)j(in)f(general.)224 1049 y(Moreo)o(v)o(er,)e(Masuok)n(a)f(has)g(sho)o(wn)h(that)f(Hopf)g (algebras)h(with)f(co)q(comm)o(utativ)o(e)d(co-)224 1099 y(radical)i(are)g(faithfully)e(\015at)h(o)o(v)o(er)h(certain)h(coideal) e(subalgebras)h(\(cf)h([28)o(]\).)224 1185 y(Ho)o(w)o(ev)o(er,)c(the)f (answ)o(er)h(to)f(the)h(ab)q(o)o(v)o(e)f(question)h(is)f(negativ)o(e)g (in)f(general:)h(P)m(.)g(Sc)o(hauen-)224 1235 y(burg)i(has)h (constructed)h(a)d(coun)o(terexample)h(\(cf.)g([63)o(]\))f(b)o(y)h(mo)q (difying)d(constructions)224 1284 y(of)14 b(W.)g(D.)f(Nic)o(hols)h(and) g(M.)g(T)m(ak)o(euc)o(hi)g(\(cf.)g([42)o(],)g([81)o(]\).)f(His)i (example)e(relies)i(on)f(the)224 1334 y(fact)d(that)f(the)h(Hopf)e (subalgebra)i(can)f(b)q(e)h(c)o(hosen)g(to)f(ha)o(v)o(e)g(a)g (non-bijectiv)o(e)g(an)o(tip)q(o)q(de,)224 1384 y(although)h(the)i(an)o (tip)q(o)q(de)f(of)f(the)h(large)g(Hopf)f(algebra)h(ma)o(y)e(b)q(e)i (bijectiv)o(e.)g(Therefore,)224 1434 y(as)h(Sc)o(hauen)o(burg)g(p)q (oin)o(ts)f(out,)f(the)i(remaining)e(question)h(is)g(whether)i(Hopf)e (algebras)224 1484 y(with)18 b(bijectiv)o(e)g(an)o(tip)q(o)q(des)h(are) g(faithfully)d(\015at)i(o)o(v)o(er)g(Hopf)g(subalgebras)g(with)g(bi-) 224 1534 y(jectiv)o(e)g(an)o(tip)q(o)q(des.)e(This)h(question)g(ma)o(y) e(b)q(e)i(considered)h(as)f(the)h(curren)o(t)g(form)d(of)224 1583 y(Kaplansky's)f(\014rst)g(conjecture.)224 1756 y Fj(3)67 b(The)23 b(second)e(and)i(the)g(fourth)g(conjecture)224 1883 y Fi(3.1)48 b Fm(Kaplansky's)14 b(second)h(conjecture)g(states)g (that)f(a)g(coalgebra)f(is)h(admissible)e(if)224 1933 y(and)19 b(only)e(if)h(ev)o(ery)h(\014nite)g(subset)g(is)g(con)o (tained)f(in)g(a)g(\014nite-dimensional)e(admis-)224 1983 y(sible)k(sub)q(coalgebra.)g(Here,)h(a)e(coalgebra)h(is)g(called)g (admissible)e(if)h(it)g(admits)g(an)224 2032 y(algebra)13 b(structure)j(making)11 b(it)i(a)g(Hopf)g(algebra.)g(This)g(question)h (should)f(b)q(e)h(view)o(ed)224 2082 y(as)k(a)f(v)n(arian)o(t)f(of)g (the)i(theorem)f(that)g(ev)o(ery)h(\014nite)f(subset)i(of)e(a)f (coalgebra)h(is)g(con-)224 2132 y(tained)f(in)g(a)f (\014nite-dimensional)f(sub)q(coalgebra)i(\(cf.)g([17)o(],)f(Theorem)h (2,)f(p.)g(7,)g([38)o(],)224 2182 y(Theorem)i(5.1.1,)e(p.)i(56\).)g(R.) f(G.)h(Larson)g(has)h(p)q(oin)o(ted)f(out)g(the)h(follo)o(wing)d (simple)899 2310 y(5)p eop %%Page: 6 6 6 5 bop 224 195 a Fm(coun)o(terexample)17 b(to)g(this)g(conjecture:)h (The)g(p)q(olynomial)13 b(ring)k Ff(K)s Fm([)p Ff(x)p Fm(])f(in)g(one)i(inde-)224 245 y(terminate)e(o)o(v)o(er)g(a)f(\014eld) h(of)g(c)o(haracteristic)h(zero)g(b)q(ecomes)f(a)g(Hopf)f(algebra)h(if) f Ff(x)h Fm(is)224 295 y(required)e(to)f(b)q(e)g(primitiv)o(e.)d(Ho)o (w)o(ev)o(er,)j Ff(x)f Fm(is)h(not)f(con)o(tained)h(in)f(a)h (\014nite-dimensional)224 344 y(admissible)k(sub)q(coalgebra.)i(This)f (is)h(b)q(ecause)h(a)e(Hopf)g(algebra)g Ff(H)k Fm(that)d(con)o(tains) 224 394 y(a)h(primitiv)o(e)e(elemen)o(t)h(admits)f(a)i(Hopf)f(algebra)h (homom)o(orphism)c(from)i Ff(K)s Fm([)p Ff(x)p Fm(])h(to)224 444 y Ff(H)g Fm(that)c(maps)f Ff(x)h Fm(to)g(this)h(primitiv)o(e)d (elemen)o(t.)h(Because)j(the)f(c)o(haracteristic)h(of)e(the)224 494 y(base)h(\014eld)f(is)g(zero,)g(the)h(\014rst)g(step)g(of)e(the)i (coradical)f(\014ltration)f(of)g(the)i(p)q(olynomial)224 544 y(ring)g(consists)g(only)f(of)g(1)g(and)g Ff(x)h Fm(\(cf.)f([38)o(],)f(Prop.)i(5.5.3\),)d(and)j(therefore)h(the)f(Hopf) 224 594 y(algebra)g(homomo)o(rphism)c(is)k(injectiv)o(e)g(on)f(this)h (subspace.)h(Therefore,)g(it)f(is)f(injec-)224 643 y(tiv)o(e)f(on)g (the)g(whole)g(p)q(olynomial)c(ring)k(\(cf.)f([38)o(],)g(Theorem)h (5.3.1\),)e(and)h Ff(H)k Fm(m)o(ust)c(b)q(e)224 693 y (in\014nite-dimensional.)224 829 y Fi(3.2)48 b Fm(Kaplansky's)17 b(fourth)g(conjecture)i(states)g(that)e(if)f(an)h(elemen)o(t)g Ff(x)g Fm(in)g(a)g(Hopf)224 878 y(algebra)d Ff(H)j Fm(satis\014es)726 928 y Ff(a)748 935 y Fd(\(1\))792 928 y Ff(xS)r Fm(\()p Ff(a)881 935 y Fd(\(2\))927 928 y Fm(\))11 b(=)h Ff(\017)p Fm(\()p Ff(a)p Fm(\))p Ff(x)224 1003 y Fm(for)19 b(ev)o(ery)h Ff(a)h Fg(2)f Ff(H)s Fm(,)e(then)i Ff(x)f Fm(is)g(con)o(tained)h(in)e (the)i(cen)o(ter)h(of)e Ff(H)s Fm(.)f(Here,)i(w)o(e)g(ha)o(v)o(e)224 1053 y(used)14 b(the)f(follo)o(wing)d(v)n(arian)o(t)h(of)h(the)i (Heyneman-Sw)o(eedler)e(sigma)f(notation)h(for)g(the)224 1103 y(copro)q(duct:)745 1152 y(\001\()p Ff(a)p Fm(\))g(=)g Ff(a)912 1159 y Fd(\(1\))965 1152 y Fg(\012)e Ff(a)1029 1159 y Fd(\(2\))224 1263 y Fm(This)k(is)g(ob)o(vious,)f(b)q(ecause)i (then)g(w)o(e)f(ha)o(v)o(e:)536 1354 y Ff(ax)d Fm(=)h Ff(a)659 1361 y Fd(\(1\))704 1354 y Ff(xS)r Fm(\()p Ff(a)793 1361 y Fd(\(2\))838 1354 y Fm(\))p Ff(a)876 1361 y Fd(\(3\))932 1354 y Fm(=)g Ff(\017)p Fm(\()p Ff(a)1031 1361 y Fd(\(1\))1075 1354 y Fm(\))p Ff(xa)1137 1361 y Fd(\(2\))1193 1354 y Fm(=)g Ff(xa)224 1445 y Fm(This)17 b(calculation)f(can)h(certainly)g(b) q(e)h(found)e(in)h(man)o(y)e(places,)i(among)d(them)i([38)o(],)224 1495 y(Lemma)c(5.7.2,)f(p.)j(83.)224 1668 y Fj(4)67 b(The)23 b(third)h(and)f(the)f(nin)n(th)j(conjecture)224 1795 y Fi(4.1)48 b Fm(Kaplansky's)17 b(third)g(conjecture)j(states)e(that)g (a)f(Hopf)g(algebra)g(o)o(v)o(er)g(a)g(\014eld)224 1845 y(of)g(c)o(haracteristic)h(zero)g(do)q(es)g(not)f(con)o(tain)g(nonzero) h(cen)o(tral)f(nilp)q(oten)o(t)g(elemen)o(ts.)224 1895 y(Although)f(is)h(seems)g(that)f(no)h(literature)g(on)f(this)h (conjecture)h(exists,)f(it)g(has)f(b)q(een)224 1944 y(ob)o(vious)e(to)f (most)g(researc)o(hers)k(for)c(some)g(time)g(that)h(this)g(is)g(false.) 224 2030 y(Consider)22 b(a)f(left)h(in)o(tegral)e(\003)715 2036 y Fe(H)768 2030 y Fm(in)h(a)g(\014nite-dimensional)e (non-semisimple)g(Hopf)224 2080 y(algebra.)i(By)h(the)g(generalized)g (Masc)o(hk)o(e)h(theorem)e(due)h(to)g(R.)e(G.)h(Larson)h(and)224 2130 y(M.)e(E.)f(Sw)o(eedler)i(\(cf.)e([26]\),)f(w)o(e)i(then)h(ha)o(v) o(e)e Ff(\017)1007 2136 y Fe(H)1039 2130 y Fm(\(\003)1084 2136 y Fe(H)1115 2130 y Fm(\))j(=)f(0,)e(and)h(therefore)h(also)224 2179 y(\003)253 2164 y Fd(2)253 2191 y Fe(H)304 2179 y Fm(=)f(0.)e(\003)436 2185 y Fe(H)485 2179 y Fm(is)h(therefore)h(a)e (nilp)q(oten)o(t)g(elemen)o(t.)g(No)o(w,)g(if)f(\003)1284 2185 y Fe(H)1334 2179 y Fm(is)i(also)f(a)g(righ)o(t)899 2310 y(6)p eop %%Page: 7 7 7 6 bop 224 195 a Fm(in)o(tegral,)16 b(i.)f(e.,)h(if)f Ff(H)20 b Fm(is)c(unimo)q(dular,)e(then)j(\003)989 201 y Fe(H)1036 195 y Fm(is)f(ob)o(viously)g(a)g(cen)o(tral)g(elemen)o(t.) 224 245 y(Kaplansky's)c(third)h(conjecture)h(w)o(ould)e(therefore)i (imply)c(that,)i(o)o(v)o(er)g(\014elds)h(of)f(c)o(har-)224 295 y(acteristic)i(zero,)f(all)e(\014nite-dimensional)f(unimo)q(dular)g (Hopf)i(algebras)g(are)h(semisim-)224 344 y(ple.)224 430 y(This)i(is,)e(ho)o(w)o(ev)o(er,)i(not)f(the)h(case.)f(There)i (exist)f(\014nite-dimensional)d(Hopf)i(algebras)224 480 y(that)k(are)g(not)g(semisimple,)c(for)k(example)e(the)i(T)m(aft)f (algebra)g(\(cf.)h([77)o(]\),)f(whic)o(h)g(is,)224 530 y(ho)o(w)o(ev)o(er,)i(not)g(unimo)q(dular.)e(But)i(ev)o(ery)h (\014nite-dimensional)d(Hopf)i(algebra)f(ma)o(y)224 579 y(b)q(e)e(em)o(b)q(edded)f(in)f(the)h(Drinfel'd)f(double)h Ff(D)q Fm(\()p Ff(H)s Fm(\),)g(and)f(this)h(is)g(a)f(unimo)q(dular)f (Hopf)224 629 y(algebra)f(b)o(y)f(a)g(result)h(of)f(D.)g(E.)h(Radford)e (\(cf.)i([61)o(],)e(Theorem)h(4,)g([71)o(],)g(Theorem)g(5.4\).)224 679 y(By)16 b(another)f(result)h(of)e(Radford)g(\(cf.)h([61)o(],)f (Prop.)g(7\),)h Ff(D)q Fm(\()p Ff(H)s Fm(\))g(is)g(semisimple)e(if)h (and)224 729 y(only)j(if)f Ff(H)j Fm(is)e(semisimple)e(and)h (cosemisimple.)f(Therefore,)i(the)h(Drinfel'd)e(double)224 779 y(of)d(the)g(T)m(aft)f(algebra)g(is)h(a)g(\014nite-dimensional)d (unimo)q(dular)h(Hopf)h(algebra)h(whic)o(h)g(is)224 829 y(not)h(semisimple.)224 964 y Fi(4.2)48 b Fm(W)m(e)17 b(no)o(w)g(consider)h(another)g(coun)o(terexample)f(to)g(Kaplansky's)g (third)g(con-)224 1014 y(jecture,)g(namely)d(the)i(F)m(rob)q (enius-Lusztig)g(k)o(ernel)h(of)e Ff(U)1135 1020 y Fe(q)1153 1014 y Fm(\()p Ff(sl)q Fm(\(2\)\),)i(whic)o(h)e(is)h(sligh)o(tly)224 1064 y(more)c(explicit)g(than)h(the)g(example)e(considered)j(ab)q(o)o (v)o(e)e(and,)g(as)h(w)o(e)g(shall)e(see)j(b)q(elo)o(w,)224 1113 y(also)f(pro)o(vides)g(a)g(coun)o(terexample)f(to)h(Kaplanksky's)f (nin)o(th)h(conjecture.)h(In)f(fact,)g(it)224 1163 y(can)f(b)q(e)g(sho) o(wn)f(that)h(it)f(is)g(a)g(quotien)o(t)g(of)g(the)h(Drinfel'd)e (double)h(of)g(the)h(T)m(aft)e(algebra.)224 1249 y(W)m(e)19 b(w)o(ork)f(o)o(v)o(er)g(an)g(algebraically)f(closed)i(\014eld)g(of)f (c)o(haracteristic)i(zero.)f(Supp)q(ose)224 1299 y(that)13 b Ff(q)h Fm(is)f(a)g(primitiv)o(e)d Ff(k)q Fm(-th)j(ro)q(ot)g(of)g (unit)o(y)m(,)e(where)j Ff(k)f(>)e Fm(2)i(is)g(o)q(dd)g(-)g(the)g(ev)o (en)h(case)g(is)224 1348 y(similar.)f(The)i(F)m(rob)q(enius-Lusztig)h (k)o(ernel)f(of)g Ff(U)999 1354 y Fe(q)1018 1348 y Fm(\()p Ff(sl)q Fm(\(2\)\))h(is)f(de\014ned)h(as)f(the)h(algebra)224 1398 y Ff(U)j Fm(with)14 b(generators)h Ff(K)q(;)7 b(K)660 1383 y Fc(\000)p Fd(1)704 1398 y Ff(;)g(E)15 b Fm(and)f Ff(F)19 b Fm(and)14 b(relations)760 1490 y Ff(K)s(K)836 1472 y Fc(\000)p Fd(1)893 1490 y Fm(=)24 b(1)11 b(=)h Ff(K)1063 1472 y Fc(\000)p Fd(1)1108 1490 y Ff(K)606 1557 y(K)s(E)h Fm(=)f Ff(q)752 1540 y Fd(2)771 1557 y Ff(E)r(K)86 b(K)s(F)17 b Fm(=)12 b Ff(q)1071 1540 y Fc(\000)p Fd(2)1116 1557 y Ff(F)6 b(K)743 1650 y(E)r(F)15 b Fg(\000)10 b Ff(F)c(E)13 b Fm(=)985 1622 y Ff(K)g Fg(\000)c Ff(K)1112 1607 y Fc(\000)p Fd(1)p 985 1641 172 2 v 1004 1679 a Ff(q)h Fg(\000)f Ff(q)1094 1667 y Fc(\000)p Fd(1)654 1741 y Ff(K)692 1724 y Fe(k)724 1741 y Fm(=)j(1)83 b Ff(E)905 1724 y Fe(k)937 1741 y Fm(=)12 b(0)82 b Ff(F)1117 1724 y Fe(k)1149 1741 y Fm(=)12 b(0)224 1832 y(The)j(copro)q(duct)g(is) e(determined)h(on)g(the)h(generators)g(b)o(y)e(the)i(form)o(ulas:)550 1923 y(\001)o(\()p Ff(K)s Fm(\))d(=)g Ff(K)h Fg(\012)c Ff(K)45 b Fm(\001\()p Ff(K)968 1906 y Fc(\000)p Fd(1)1012 1923 y Fm(\))12 b(=)g Ff(K)1122 1906 y Fc(\000)p Fd(1)1176 1923 y Fg(\012)d Ff(K)1255 1906 y Fc(\000)p Fd(1)441 1991 y Fm(\001\()p Ff(E)r Fm(\))i(=)p Ff(E)h Fg(\012)d Fm(1)g(+)h Ff(K)i Fg(\012)d Ff(E)44 b Fm(\001\()p Ff(F)6 b Fm(\))11 b(=)h Ff(F)i Fg(\012)c Ff(K)1179 1974 y Fc(\000)p Fd(1)1233 1991 y Fm(+)f(1)g Fg(\012)h Ff(F)224 2082 y Fm(This)g(Hopf)e(algebra)h(has)h(a)f(basis)g(consisting)g(of)g(the)h (elemen)o(ts)f Ff(E)1244 2067 y Fe(i)1258 2082 y Ff(K)1296 2067 y Fe(j)1314 2082 y Ff(F)1347 2067 y Fe(m)1387 2082 y Fm(for)g Ff(i;)e(j;)g(m)k Fm(=)224 2132 y(0)p Ff(;)c(:)g(:)g(:)12 b(;)7 b(k)k Fg(\000)f Fm(1.)15 b(If)g Ff(T)21 b Fm(:=)614 2101 y Fb(P)658 2111 y Fe(k)658 2144 y(i)p Fd(=1)721 2132 y Ff(q)741 2117 y Fd(2)p Fe(i)p Fd(\()p Fe(k)q Fc(\000)p Fd(1\))858 2132 y Ff(K)896 2117 y Fe(i)910 2132 y Fm(,)15 b(it)g(is)h(easy)g(to)f(see)i(that)f Ff(E)1363 2117 y Fe(k)q Fc(\000)p Fd(1)1426 2132 y Ff(T)6 b(F)1489 2117 y Fe(k)q Fc(\000)p Fd(1)1567 2132 y Fm(is)224 2182 y(a)18 b(left)g(and)g(righ)o(t)f(in)o(tegral,)g(and)h(therefore)i(a)d(cen)o (tral)i(elemen)o(t)e(\(cf.)h([74)o(],)f(p.)h(368,)899 2310 y(7)p eop %%Page: 8 8 8 7 bop 224 195 a Fm([71)o(],)15 b(Prop.)h(6.3,)e(note)i(a)f(sligh)o(t) g(di\013erence)j(in)d(the)i(de\014nitions\).)e(Since)h(the)g(counit)224 245 y(v)n(anishes)e(on)g(the)h(in)o(tegral,)d(its)i(square)h(is)f (zero.)224 330 y(The)f(in)o(tegral)f(is)g(not)g(the)h(only)f(nilp)q (oten)o(t)g(cen)o(tral)g(elemen)o(t)g(of)g Ff(U)5 b Fm(.)12 b(Its)g(cen)o(ter)i(can)f(b)q(e)224 380 y(describ)q(ed)j(completely)d (\(cf.)g([64)o(]\).)224 516 y Fi(4.3)48 b Fm(As)19 b(noted)g(b)o(y)f (H.-J.)f(Sc)o(hneider)j(\(cf.)e([65)o(]\),)f(the)i(F)m(rob)q (enius-Lusztig)g(k)o(ernel)224 565 y(considered)d(ab)q(o)o(v)o(e)d (also)h(pro)o(vides)g(a)f(coun)o(terexample)h(to)f(Kaplansky's)h(nin)o (th)f(con-)224 615 y(jecture.)21 b(This)f(conjecture)h(states)g(that)f (for)g(a)f(\014nite-dimensional)f(Hopf)h(algebra)224 665 y(o)o(v)o(er)d(an)g(algebraically)f(closed)h(\014eld,)g(the)h (dimension)d(of)h(the)i(Jacobson)g(radical)e(of)224 715 y(the)f(Hopf)f(algebra)f(equals)h(the)h(dimension)e(of)g(the)i (Jacobson)f(radical)g(of)f(the)i(dual)e(if)224 765 y(the)j(c)o (haracteristic)g(of)e(the)i(base)f(\014eld)g(do)q(es)h(not)f(divide)f (the)i(dimension.)d(In)h(partic-)224 814 y(ular,)g(semisimple)f(Hopf)h (algebras)h(should)f(b)q(e)i(cosemisimple,)c(a)i(problem)g(that)h(will) 224 864 y(b)q(e)h(considered)g(b)q(elo)o(w)f(in)f(conjunction)h(with)g (the)g(\014fth)g(conjecture.)224 950 y(W)m(e)h(no)o(w)f(explain)g (brie\015y)i(wh)o(y)e(Kaplansky's)g(nin)o(th)h(conjecture)i(is)d (refuted)i(b)o(y)f(the)224 1000 y(F)m(rob)q(enius-Lusztig)j(k)o(ernel)f Ff(U)5 b Fm(.)16 b(The)h(simple)e(mo)q(dules)h(of)g Ff(U)21 b Fm(are)d(kno)o(wn)e(\(cf.)g([18)o(],)224 1049 y(Theorem)h(VI.5.7,)f (p.)g(137,)h([15)o(],)f(sec.)i(2.13,)e(p.)g(24\);)h(in)g(the)g(o)q(dd)h (case)g(considered)224 1099 y(ab)q(o)o(v)o(e,)i(there)h(is)f(one)h (isomorphism)c(class)j(of)g(simple)f(mo)q(dules)g(for)h(eac)o(h)g(of)g (the)224 1149 y(dimensions)9 b(1)p Ff(;)e(:)g(:)g(:)k(;)c(k)q Fm(.)i(Since)i(the)f(Jacobson)g(radical)g Ff(J)j Fm(of)d Ff(U)k Fm(consists,)d(b)o(y)e(de\014nition,)224 1199 y(precisely)16 b(of)e(the)h(elemen)o(ts)f(that)h(annihilate)e(ev)o(ery) j(simple)d(mo)q(dule,)f Ff(U)r(=J)19 b Fm(has)c(the)224 1249 y(same)d(simple)e(mo)q(dules)i(as)g Ff(U)5 b Fm(.)12 b(Since)g Ff(U)r(=J)17 b Fm(is)12 b(a)g(semisimple)e(algebra,)h(its)h (dimension)224 1299 y(is)j(the)g(sum)e(of)h(the)i(squares)f(of)f(the)h (dimensions)f(of)f(the)j(simple)d(mo)q(dules,)g(i.)g(e.,)h(w)o(e)224 1348 y(ha)o(v)o(e:)546 1429 y(dim)5 b Ff(U)r(=J)16 b Fm(=)783 1377 y Fe(k)762 1389 y Fb(X)756 1477 y Fe(m)p Fd(=1)834 1429 y Ff(m)870 1412 y Fd(2)901 1429 y Fm(=)950 1401 y(1)p 950 1419 21 2 v 950 1457 a(6)976 1429 y Ff(k)q Fm(\()p Ff(k)10 b Fm(+)f(1\)\(2)p Ff(k)h Fm(+)g(1\))224 1575 y(On)19 b(the)f(other)h(hand,)f(the)g(Jacobson)h(radical)e(of)h Ff(U)1089 1560 y Fc(\003)1126 1575 y Fm(is)g(precisely)h(orthogonal)e (to)224 1625 y(the)e(coradical)f Ff(C)i Fm(of)e Ff(U)19 b Fm(\(cf.)14 b([38)o(],)f(Remark)f(5.1.7,)g(p.)i(58\).)f(Kaplansky's)h (conjecture)224 1675 y(therefore)19 b(w)o(ould)d(imply)f(that)i(the)h (coradical)e(has)h(the)h(same)e(dimension)g(as)h Ff(U)r(=J)t Fm(.)224 1724 y(But,)g(as)f(can)g(b)q(e)h(sho)o(wn)f(as)g(in)f([38)o (],)g(Lemma)f(5.5.5,)g(p.)h(76,)g(the)i(coradical)e(of)h Ff(U)21 b Fm(is)224 1774 y(spanned)15 b(b)o(y)f(the)g(p)q(o)o(w)o(ers)h (of)e Ff(K)s Fm(,)h(and)f(therefore)j(w)o(e)e(ha)o(v)o(e:)816 1866 y(dim)5 b Ff(C)14 b Fm(=)e Ff(k)224 1957 y Fm(Note)18 b(that)f(the)g(pro)q(of)g(in)f([38)o(])h(needs)h(a)f(sligh)o(t)f(mo)q (di\014cation.)e(F)m(urther)k(details)f(on)224 2007 y(the)g(Jacobson)f (radical)f(of)g Ff(U)21 b Fm(can)16 b(b)q(e)g(found)f(in)g([74)o(],)g (Cor.)g(3.8,)g(p.)g(367;)f(note)i(that)224 2057 y(the)f(setup)g(there)g (is)f(sligh)o(tly)e(di\013eren)o(t.)899 2310 y(8)p eop %%Page: 9 9 9 8 bop 224 195 a Fj(5)67 b(The)23 b(\014fth)f(and)h(the)g(sev)n(en)n (th)g(conjecture)224 322 y Fi(5.1)48 b Fm(It)20 b(is)f(easy)h(to)g(see) h(that)f(the)g(an)o(tip)q(o)q(de)f(of)g(a)h(Hopf)f(algebra)g(is)h(an)f (in)o(v)o(olu-)224 372 y(tion)13 b(if)f(the)i(Hopf)f(algebra)g(is)g (comm)o(utativ)o(e)d(or)j(co)q(comm)o(utativ)o(e.)e(Since)i(the)h (notion)224 421 y(of)j(an)g(an)o(tip)q(o)q(de)f(in)h(a)g(Hopf)f (algebra)h(generalizes)h(the)f(notion)f(of)h(an)g(in)o(v)o(erse)g(in)g (a)224 471 y(group,)c(it)h(is)f(reasonable)h(to)f(exp)q(ect)i(that)f (this)g(is)f(a)g(more)g(general)h(feature.)f(In)h(fact,)224 521 y(it)k(w)o(as)f(not)h(kno)o(wn)f(for)h(some)e(time)h(whether)i (there)g(exist)f(Hopf)f(algebras)h(whose)224 571 y(an)o(tip)q(o)q(des) 13 b(w)o(ere)h(not)e(in)o(v)o(olutions)f(-)h(at)h(least)f(in)h(the)g (\014nite-dimensional)d(case.)j(Suc)o(h)224 621 y(a)k(Hopf)e(algebra)h (w)o(as)h(then)g(constructed)h(b)o(y)e(E.)h(J.)f(T)m(aft)f(\(cf.)h([77) o(],)g([78)o(]\))g(and)g(also)224 670 y(b)o(y)f(M.)f(E.)g(Sw)o(eedler)i (\(unpublished\))f(and)g(D.)f(E.)g(Radford)g(\(cf.)g([54)o(]\).)g (Before)i(that,)224 720 y(R.)c(G.)f(Larson)i(as)f(w)o(ell)g(as)g(M.)g (E.)g(Sw)o(eedler)i(already)e(had)g(constructed)j(examples)c(of)224 770 y(in\014nite)k(dimensional)e(Hopf)h(algebras)h(with)f(an)o(tip)q(o) q(des)h(of)g(in\014nite)f(order)i(\(cf.)e([19)o(],)224 820 y([76)o(],)h(Chap.)g(IV,)g(Exercises\),)i(whereas)g(Radford)e(pro)o (v)o(ed)h(later)f(that)h(an)o(tip)q(o)q(des)g(of)224 870 y(\014nite-dimensional)c(Hopf)i(algebras)f(ha)o(v)o(e)h(\014nite)g (order)h(\(cf.)e([55)o(],)g([56)o(]\).)224 955 y(When)k(Kaplansky)f (wrote)g(do)o(wn)g(his)g(conjectures,)i(it)e(w)o(as)g(therefore)i(kno)o (wn)d(that)224 1005 y(the)i(an)o(tip)q(o)q(de)g(of)f(a)g(Hopf)g (algebra)g(w)o(as)g(not)g(alw)o(a)o(ys)g(an)g(in)o(v)o(olution.)e (Kaplansky's)224 1055 y(\014fth)k(conjecture)i(no)o(w)d(states)i(that)f (the)h(an)o(tip)q(o)q(de)e(of)h(a)f(\014nite-dimensional)f(Hopf)224 1105 y(algebra)d(o)o(v)o(er)g(an)f(algebraically)f(closed)j(\014eld)f (is)f(an)h(in)o(v)o(olution)e(if)h(the)h(Hopf)g(algebra)224 1154 y(is)j(semisimple)d(or)j(cosemisimple.)d(Here,)k(a)f(Hopf)f (algebra)g(is)h(called)g(cosemisimple)224 1204 y(if)d(the)i(dual)e (Hopf)h(algebra)f(is)h(semisimple.)224 1290 y(It)i(should)f(\014rst)h (b)q(e)f(observ)o(ed)i(that)e(the)h(question)f(of)g(algebraic)f (closure)i(is)f(not)h(im-)224 1340 y(p)q(ortan)o(t)h(here,)h(b)q (ecause)g(the)g(order)f(of)f(the)i(an)o(tip)q(o)q(de)f(remains)e(unc)o (hanged)j(under)224 1389 y(base)h(\014eld)g(extension.)g(As)f(will)f(b) q(e)i(explained)g(b)q(elo)o(w,)e(the)i(c)o(haracteristic)h(of)e(the)224 1439 y(base)d(\014eld)f(is)g(of)f(greater)i(imp)q(ortance.)224 1525 y(One)d(of)f(the)h(\014rst)g(results)g(on)g(this)f(conjecture)i(w) o(as)e(obtained)g(b)o(y)g(R.)f(G.)h(Larson,)g(who)224 1575 y(pro)o(v)o(ed)18 b(it)f(for)f(Hopf)h(algebras)g(o)o(v)o(er)g (\014elds)h(of)f(c)o(haracteristic)h(zero)g(or)g(su\016cien)o(tly)224 1624 y(large)f(c)o(haracteristic,)h(pro)o(vided)f(that)g(the)h(Hopf)f (algebra)f(is)h(semisimple)e(and)i(the)224 1674 y(irreducible)h(mo)q (dules)f(ha)o(v)o(e)g(dimension)f(one)h(or)h(t)o(w)o(o)f(\(cf.)g([22)o (]\).)g(His)g(results)i(w)o(ere)224 1724 y(generalized)13 b(b)o(y)f(D.)g(E.)f(Radford,)g(who)h(pro)o(v)o(ed)g(that)h(it)e (su\016ces)j(to)e(assume)g(that)g(the)224 1774 y(t)o(w)o(o-sided)17 b(ideals)g(corresp)q(onding)h(to)f(the)h(simple)d(mo)q(dules)h(of)h (dimension)e(one)i(or)224 1824 y(t)o(w)o(o)d(generate)h(the)f(Hopf)g (algebra)f Ff(H)k Fm(as)d(a)g(coalgebra)f(\(cf.)h([60)o(]\).)224 1909 y(Kaplansky's)d(\014fth)h(conjecture)h(is)e(b)q(etter)i(understo)q (o)q(d)f(if)f(it)g(is)g(split)g(in)o(to)g(t)o(w)o(o)f(partial)899 2310 y(9)p eop %%Page: 10 10 10 9 bop 224 195 a Fm(problems:)275 322 y(1.)20 b(Is)14 b(ev)o(ery)h(semisimple)c(Hopf)j(algebra)f(cosemisimple?)275 405 y(2.)20 b(Is)14 b(the)g(an)o(tip)q(o)q(de)g(of)f(a)g(semisimple,)d (cosemisimple)i(Hopf)h(algebra)g(an)g(in)o(v)o(olu-)328 455 y(tion?)224 582 y(In)h(what)g(follo)o(ws,)e(w)o(e)i(shall)f(treat)i (these)g(problems)e(separately)m(.)224 717 y Fi(5.2)48 b Fm(The)21 b(most)f(substan)o(tial)g(progress)i(on)f(Kaplansky's)f (\014fth)h(conjecture)i(w)o(as)224 767 y(made)11 b(b)o(y)h(R.)f(G.)g (Larson)h(and)g(D.)f(E.)h(Radford)f(in)h(t)o(w)o(o)f(closely)h(related) h(articles)f(pub-)224 817 y(lished)i(in)g(1988.)e(In)i(the)g(\014rst)h (one)f(of)f(these)j(they)e(answ)o(ered)h(a\016rmativ)o(ely)c(the)k (\014rst)224 867 y(question)h(ab)q(o)o(v)o(e)g(under)h(the)f (assumption)f(that)g(the)i(base)f(\014eld)g(has)g(c)o(haracteristic)224 917 y(zero)f(\(cf.)e([24)o(]\).)g(Their)h(pro)q(of)f(relies)h(on)f(a)h (trace)g(form)o(ula)d(that)j(expresses)i(the)e(trace)224 966 y(of)g(the)g(squared)h(an)o(tip)q(o)q(de)f(in)f(terms)h(of)f(the)i (in)o(tegrals)e(of)g(the)i(Hopf)e(algebra:)690 1102 y Ff(T)6 b(r)q Fm(\()p Ff(S)783 1085 y Fd(2)781 1112 y Fe(H)813 1102 y Fm(\))12 b(=)g Ff(\017)902 1108 y Fe(H)933 1102 y Fm(\(\000)975 1108 y Fe(H)1007 1102 y Fm(\))p Ff(\032)1044 1108 y Fe(H)1076 1102 y Fm(\(1\))224 1212 y(Here,)h Ff(\017)349 1218 y Fe(H)393 1212 y Fm(and)f Ff(S)497 1218 y Fe(H)541 1212 y Fm(denote)h(the)g(counit)f(and)g(the)g (an)o(tip)q(o)q(de)g(of)g(the)h(Hopf)e(algebra)h Ff(H)s Fm(,)224 1262 y(whereas)17 b(\000)409 1268 y Fe(H)455 1262 y Fg(2)c Ff(H)19 b Fm(and)c Ff(\032)653 1268 y Fe(H)699 1262 y Fg(2)e Ff(H)778 1247 y Fc(\003)813 1262 y Fm(denote)j(righ)o(t)f (in)o(tegrals)g(satisfying)f Ff(\032)1426 1268 y Fe(H)1458 1262 y Fm(\(\000)1500 1268 y Fe(H)1532 1262 y Fm(\))g(=)224 1312 y(1.)20 b(As)h(H.-J.)f(Sc)o(hneider)i(p)q(oin)o(ts)f(out,)f(the)h (immedia)o(te)e(consequence)k(dim)5 b Ff(H)48 b Fm(=)224 1362 y Ff(\017)241 1368 y Fe(H)273 1362 y Fm(\(\000)315 1368 y Fe(H)346 1362 y Fm(\))p Ff(\032)383 1368 y Fe(H)415 1362 y Fm(\(1\))16 b(if)f Ff(S)551 1347 y Fd(2)549 1373 y Fe(H)596 1362 y Fm(=)h Ff(id)680 1368 y Fe(H)727 1362 y Fm(w)o(as)g(already)f(observ)o(ed)i(in)f([50)o(].)f(He)h(has)h(later) f(used)224 1412 y(the)d(metho)q(ds)f(of)g(this)h(article)f(to)h(giv)o (e)e(simpli\014ed)g(pro)q(ofs)h(of)g(the)h(\014rst)h(trace)f(form)o (ula)224 1461 y(stated)k(ab)q(o)o(v)o(e)e(as)h(w)o(ell)f(as)g(the)h (second)h(trace)g(form)o(ula)c(discussed)k(b)q(elo)o(w,)e(and)g(also) 224 1511 y(for)f(Radford's)f(form)o(ula)e(for)j(the)g(fourth)g(p)q(o)o (w)o(er)g(of)f(the)i(an)o(tip)q(o)q(de)f(\(cf.)f([65)o(]\).)224 1597 y(Recen)o(tly)m(,)k(the)h(\014rst)g(problem)e(men)o(tioned)g(ab)q (o)o(v)o(e)h(has)g(also)g(b)q(een)h(solv)o(ed)f(in)g(large)224 1647 y(p)q(ositiv)o(e)e(c)o(haracteristic)h(\(cf.)f([70)o(]\).)f(The)h (tec)o(hniques)h(there)g(rely)f(on)g(the)g(one)h(hand)224 1696 y(on)e(the)g(adjunction)e(of)h(a)g(grouplik)o(e)g(elemen)o(t)g (making)e(the)j(square)g(of)f(the)h(an)o(tip)q(o)q(de)224 1746 y(an)e(inner)f(automorphism,)d(on)j(the)h(other)g(hand)f(on)g(the) h(P)o(erron-F)m(rob)q(enius)h(theorem)224 1796 y(to)h(pro)o(v)o(e)g (the)h(in)o(v)o(ertibilit)o(y)d(of)h(the)h(c)o(haracter)i(of)d(the)h (adjoin)o(t)f(represen)o(tation.)224 1931 y Fi(5.3)48 b Fm(W)m(e)18 b(no)o(w)h(discuss)g(the)g(second)h(problem)d(men)o (tioned)h(ab)q(o)o(v)o(e.)g(It)g(should)h(b)q(e)224 1981 y(observ)o(ed)c(that)f(for)g(this)g(problem,)e(w)o(e)i(already)g(kno)o (w)f(b)o(y)h(Radford's)f(form)o(ula)e(that)224 2031 y(the)18 b(fourth)g(p)q(o)o(w)o(er)f(of)g(the)h(an)o(tip)q(o)q(de)f(is)g(the)h (iden)o(tit)o(y)f(\(cf.)g([56)o(]\).)f(In)i(their)f(second)224 2081 y(article,)h(whic)o(h)g(w)o(as)g(also)g(published)g(in)g(1988,)f (Larson)h(and)g(Radford)f(also)h(made)224 2131 y(imp)q(ortan)o(t)10 b(progress)j(on)e(the)h(second)g(problem)e(\(cf.)h([23)o(]\).)g(There,) h(they)g(pro)o(v)o(ed)f(that)224 2180 y(the)16 b(an)o(tip)q(o)q(de)e (of)g(a)h(semisimple,)c(cosemisimple)h(Hopf)j(algebra)f(is)g(an)h(in)o (v)o(olution)d(if)889 2310 y(10)p eop %%Page: 11 11 11 10 bop 224 195 a Fm(the)18 b(c)o(haracteristic)g(of)e(the)h(base)h (\014eld)f(is)f(zero)i(or)e(larger)h(than)g(the)g(square)h(of)e(the)224 245 y(dimension)j(of)h(the)i(Hopf)e(algebra.)g(Their)g(tec)o(hnique)i (relies)f(on)f(a)h(second)g(trace)224 295 y(form)o(ula:)621 429 y Ff(T)6 b(r)q Fm(\()p Ff(S)714 412 y Fd(2)712 439 y Fe(H)744 429 y Fm(\))12 b(=)g(\(dim)5 b Ff(H)s Fm(\))p Ff(T)h(r)q Fm(\()p Ff(S)1055 412 y Fd(2)1053 439 y Fe(H)1096 429 y Fg(j)1108 435 y Fe(x)1127 439 y Fa(R)1150 435 y Fe(H)1182 429 y Fm(\))224 537 y(Here,)12 b Ff(x)355 543 y Fe(R)394 537 y Fg(2)f Ff(H)j Fm(denotes)e(the)f(c)o(haracter)i(of)d (the)i(regular)f(represen)o(tation)h(of)f Ff(H)1478 522 y Fc(\003)1497 537 y Fm(,)f(i.)g(e.,)224 587 y(the)17 b(elemen)o(t)e(satisfying)g Ff(\036)p Fm(\()p Ff(x)705 593 y Fe(R)732 587 y Fm(\))g(=)g Ff(T)6 b(r)q Fm(\()p Ff(L)904 593 y Fe(\036)926 587 y Fm(\),)16 b(where)h Ff(L)1120 593 y Fe(\036)1157 587 y Fm(:)d Ff(H)1221 572 y Fc(\003)1255 587 y Fg(!)g Ff(H)1349 572 y Fc(\003)1384 587 y Fm(denotes)j(left)224 636 y(m)o(ultiplication)12 b(b)o(y)i Ff(\036)p Fm(.)g(It)h(should)g(b)q(e)h(noted)f(that)g(it)f (is)h(reasonable)g(to)g(exp)q(ect)h(that)224 686 y(this)d(c)o(haracter) h(is)e(an)g(in)o(tegral)g(of)g Ff(H)s Fm(.)g(If)g(it)g(w)o(ere)i(p)q (ossible)e(to)h(pro)o(v)o(e)f(this)h(in)f(general,)224 736 y(the)j(ab)q(o)o(v)o(e)f(trace)g(form)o(ula)e(w)o(ould)h(read)h Ff(T)6 b(r)q Fm(\()p Ff(S)976 721 y Fd(2)974 748 y Fe(H)1006 736 y Fm(\))12 b(=)g(dim)5 b Ff(H)s Fm(,)13 b(and)h(it)f(w)o(ould)g(b)q (e)i(easy)224 786 y(to)f(deriv)o(e)g(strong)h(conclusions)f(on)g (Kaplansky's)f(\014fth)h(conjecture.)224 870 y(V)m(ery)e(recen)o(tly)m (,)f(P)m(.)f(Etingof)g(and)h(S.)g(Gelaki)e(ha)o(v)o(e)i(settled)h(the)g (second)g(problem)e(com-)224 920 y(pletely)h(\(cf.)g([11)o(]\).)f(In)h (this)g(v)o(ery)g(in)o(teresting)h(article,)e(they)i(pro)o(v)o(e)f (that)g(the)h(an)o(tip)q(o)q(de)224 970 y(of)k(a)f(semisimple,)e (cosemisimple)g(Hopf)j(algebra)f(o)o(v)o(er)h(an)o(y)f(\014eld)h(is)f (an)h(in)o(v)o(olution.)224 1020 y(Their)i(metho)q(d,)e(whic)o(h)h(has) h(sev)o(eral)f(other)h(applications)f(that)g(will)f(b)q(e)i(discussed) 224 1070 y(b)q(elo)o(w,)13 b(is)g(based)g(on)g(a)g(cohomology)d(v)n (anishing)i(result)i(of)e(D.)h(S)-21 b(\030)s(tefan)13 b(\(cf.)g([73)o(]\))f(that)224 1119 y(w)o(as)j(originally)e(in)o(v)o (en)o(ted)i(to)g(deal)f(with)h(Kaplansky's)g(ten)o(th)g(conjecture)i (that)e(will)224 1169 y(also)i(b)q(e)h(discussed)i(b)q(elo)o(w.)c (Etingof)h(and)g(Gelaki)g(use)h(this)g(result)g(to)f(pro)o(v)o(e)h (that,)224 1219 y(for)12 b(a)f(semisimple)f(and)h(cosemisimple)f(Hopf)h (algebra,)g(there)i(exists)g(a)e(complete)h(dis-)224 1269 y(crete)i(v)n(aluation)c(ring)h(of)g(c)o(haracteristic)i(zero,)f (the)g(ring)f(of)h(Witt)f(v)o(ectors,)h(suc)o(h)g(that)224 1319 y(the)k(Hopf)e(algebra)h(under)g(consideration)h(can)f(b)q(e)g (obtained)g(from)e(a)h(Hopf)h(algebra)224 1369 y(de\014ned)j(o)o(v)o (er)f(the)h(discrete)g(v)n(aluation)d(ring)i(b)o(y)f(reduction)i(mo)q (dulo)c(the)k(maxim)o(al)224 1418 y(ideal.)13 b(By)g(instead)h(passing) f(to)g(the)h(quotien)o(t)g(\014eld)f(of)g(the)h(discrete)h(v)n (aluation)c(ring,)224 1468 y(they)18 b(can)e(apply)g(the)h(c)o (haracteristic)h(zero)g(result)f(to)g(pro)o(v)o(e)f(that)h(the)g(an)o (tip)q(o)q(de)g(is)224 1518 y(an)d(in)o(v)o(olution.)224 1652 y Fi(5.4)48 b Fm(The)21 b(metho)q(ds)f(of)g(Larson)h(and)f (Radford)g(describ)q(ed)i(ab)q(o)o(v)o(e)e(also)g(enabled)224 1702 y(them)d(to)g(solv)o(e)g(Kaplansky's)f(sev)o(en)o(th)j (conjecture.)f(This)f(conjecture)i(states)f(that)224 1752 y(for)i(a)g(semisimple,)e(cosemisimple)f(Hopf)j(algebra)g(o)o(v)o (er)g(an)g(algebraically)f(closed)224 1802 y(\014eld,)h(the)g(c)o (haracteristic)h(of)e(the)h(\014eld)g(do)q(es)h(not)e(divide)g(the)i (dimension)d(of)h(the)224 1851 y(Hopf)11 b(algebra.)e(Once)j(again,)d (the)j(assumption)d(that)i(the)g(base)g(\014eld)g(b)q(e)g (algebraically)224 1901 y(closed)j(is)f(not)g(necessary)m(.)h(Using)f (the)g(ab)q(o)o(v)o(e)g(trace)h(form)o(ulas,)c(it)j(is)g(p)q(ossible)g (to)g(giv)o(e)224 1951 y(their)i(complete)e(pro)q(of)g(of)h(this)g (conjecture)h(in)f(a)f(few)h(lines:)224 2036 y(By)j(Larson)f(and)g(Sw)o (eedler's)h(generalization)f(of)g(Masc)o(hk)o(e's)g(theorem,)g(a)g (Hopf)f(al-)224 2085 y(gebra)i(is)f(semisimple)d(if)i(and)h(only)f(if)g (the)i(counit)f(do)q(es)h(not)f(v)n(anish)f(on)h(a)g(nonzero)224 2135 y(in)o(tegral.)f(Therefore)h(the)g(righ)o(t)f(hand)g(side)h(of)e (the)i(\014rst)g(trace)h(form)o(ula)c(is)i(nonzero)224 2185 y(if)f(the)h(Hopf)f(algebra)g(is)h(semisimple)c(and)k (cosemisimple.)c(Therefore,)16 b(also)e(the)h(left)889 2310 y(11)p eop %%Page: 12 12 12 11 bop 224 195 a Fm(hand)15 b(side)f(is)g(nonzero,)h(and)f(b)o(y)g (the)h(second)h(trace)f(form)o(ula)d(this)i(implies)e(that)j(the)224 245 y(dimension)i(of)g(the)h(Hopf)g(algebra)f(is)h(nonzero)g(as)g(an)g (elemen)o(t)f(of)g(the)h(base)h(\014eld,)224 295 y(i.)13 b(e.,)h(the)g(c)o(haracteristic)h(of)f(the)g(base)h(\014eld)f(do)q(es)g (not)g(divide)f(the)i(dimension.)224 429 y Fi(5.5)48 b Fm(The)17 b(results)h(on)f(Kaplansky's)f(\014fth)h(conjecture)i(men)o (tioned)c(so)i(far)g(can)g(b)q(e)224 479 y(summarized)12 b(in)i(the)g(follo)o(wing)e(theorem:)224 577 y Fi(Theorem)36 b Fm(Supp)q(ose)15 b(that)f Ff(H)i Fm(is)e(a)g(semisimple)d(Hopf)i (algebra.)275 693 y(1.)20 b(Supp)q(ose)e(that)e(the)h(c)o (haracteristic)h Ff(p)f Fm(of)e(the)j(base)f(\014eld)f(is)h(zero)g(or)f (satis\014es)328 743 y Ff(p)c(>)f(m)440 728 y Fe(m)p Fc(\000)p Fd(4)529 743 y Fm(where)k Ff(m)d Fm(=)g(2)d(\(dim)c Ff(H)s Fm(\))917 728 y Fd(2)935 743 y Fm(.)14 b(Then)g Ff(H)j Fm(is)d(cosemisimple.)275 825 y(2.)20 b(If)14 b Ff(H)i Fm(is)e(also)f(cosemisimple,)e(then)k(the)g(an)o(tip)q(o)q(de) e(of)h Ff(H)i Fm(is)e(an)g(in)o(v)o(olution.)224 941 y(There)e(are,)e(ho)o(w)o(ev)o(er,)h(additional)d(results)k(on)e(this)g (conjecture.)i(According)f(to)f(W.)f(D.)224 991 y(Nic)o(hols,)18 b(M.)f(Eb)q(erw)o(ein)i(has)f(pro)o(v)o(ed)g(that)g(the)h(an)o(tip)q(o) q(de)e(of)h(a)f(semisimple)f(Hopf)224 1041 y(algebra)e(is)g(an)g(in)o (v)o(olution,)d(pro)o(vided)j(that)g(the)h(c)o(haracteristic)h(of)d (the)i(base)f(\014eld)h(is)224 1091 y(larger)h(than)h(the)f(squared)h (dimension)e(of)g(the)i(Hopf)f(algebra)f(and)h(the)h(irreducible)224 1140 y(mo)q(dules)c(are)i(of)e(dimension)g(1,)g(2)h(or)g(3)g(\(cf.)f ([8],)g([43)o(]\).)g(This)h(generalizes)h(the)g(ab)q(o)o(v)o(e)224 1190 y(men)o(tioned)f(result)i(of)e(Larson)h(\(cf.)g([22)o(]\).)f (Larson)h(and)f(Radford)g(ha)o(v)o(e)h(also)f(pro)o(v)o(ed)224 1240 y(a)h(n)o(um)o(b)q(er)f(of)g(additional)f(results;)i(they)h(ha)o (v)o(e)e(studied)h(the)h(question)f(whether)h(the)224 1290 y(prop)q(ert)o(y)j(that)e(the)h(an)o(tip)q(o)q(de)g(is)f(an)h(in)o (v)o(olution)d(can)j(b)q(e)g(extended)h(from)d(a)h(Hopf)224 1340 y(subalgebra)g(to)g(a)g(larger)g(Hopf)f(algebra,)g(and)h(they)h (ha)o(v)o(e)e(carried)i(out)f(computer-)224 1390 y(based)h(analysis)e (of)h(the)h(algebra)e(structures)k(for)d(a)f(p)q(oten)o(tial)h(coun)o (terexample)g(to)224 1439 y(Kaplansky's)d(\014fth)g(conjecture)h(\(cf.) f([25)o(]\).)224 1611 y Fj(6)67 b(The)23 b(sixth)g(conjecture)224 1737 y Fi(6.1)48 b Fm(Kaplansky's)11 b(sixth)f(conjecture)j(ma)o(y)c(b) q(e)i(in)o(terpreted)i(as)e(follo)o(ws:)e(F)m(or)h(a)h(Hopf)224 1787 y(algebra)g(o)o(v)o(er)f(an)h(algebraically)e(closed)i(\014eld,)g (the)g(dimension)e(of)h(ev)o(ery)i(simple)d(mo)q(d-)224 1836 y(ule)h(divides)g(the)g(dimension)f(of)g(the)h(Hopf)g(algebra.)e (In)i(this)g(generalit)o(y)m(,)f(it)g(w)o(as)h(kno)o(wn)224 1886 y(at)k(that)g(time)e(that)i(this)g(is)f(false)h(ev)o(en)g(for)g (group)f(rings.)h(F)m(or)f(example,)f(the)i(sp)q(ecial)224 1936 y(linear)h(groups)f Ff(S)r(L)p Fm(\(2)p Ff(;)7 b(p)p Fm(\))15 b(of)f(2)c Fg(\002)g Fm(2-matrices)j(o)o(v)o(er)i(a)f(\014eld) h(with)f Ff(p)g Fm(elemen)o(ts,)g(where)224 1986 y Ff(p)i Fm(is)f(an)h(o)q(dd)f(prime,)f(admit)g(simple)g(mo)q(dules)g(o)o(v)o (er)i(algebraically)e(closed)i(\014elds)g(of)224 2036 y(c)o(haracteristic)d Ff(p)f Fm(with)f(a)h(dimension)e(that)h(do)q(es)i (not)e(divide)h(the)g(order)g(of)f(the)i(group)224 2085 y(\(cf.)h([6)o(],)f(Example)g(\(17.17\),)f(p.)i(426\).)f(Ho)o(w)o(ev)o (er,)h(it)f(w)o(as)h(an)g(op)q(en)g(question)h(at)e(that)224 2135 y(time)i(whether,)j(for)e(an)g(absolutely)f(irreducible)i (represen)o(tation)h(in)e(c)o(haracteristic)224 2185 y Ff(p)e(>)g Fm(0,)h(the)g Ff(p)p Fm(-comp)q(onen)o(t)g(of)f(the)i (dimension)e(of)g(the)i(irreducible)g(mo)q(dule)e(divides)889 2310 y(12)p eop %%Page: 13 13 13 12 bop 224 195 a Fm(the)14 b(order)h(of)e(the)h(group;)e(this)i(w)o (as)f(Problem)g(17)f(in)h(a)h(list)f(of)f(op)q(en)i(problems)f(com-)224 245 y(piled)k(b)o(y)f(R.)f(Brauer)j(in)e(1963)g(\(cf.)g([3)o(],)g([12)o (],)f(p.)h(166\).)g(Ev)o(en)h(this)f(conjecture)j(w)o(as)224 295 y(refuted)12 b(when)f(J.)f(G.)f(Thac)o(kra)o(y)h(sho)o(w)o(ed)h(in) f(1981)f(that)h(McLaughlin's)g(simple)f(group)224 344 y(of)j(order)g(2)397 329 y Fd(7)421 344 y Fg(\001)5 b Fm(3)459 329 y Fd(6)482 344 y Fg(\001)g Fm(5)520 329 y Fd(3)544 344 y Fg(\001)g Fm(7)g Fg(\001)g Fm(11)11 b(has)h(a)f(simple)g(mo)q(dule)f(of)h(dimension)g(2)1294 329 y Fd(9)1317 344 y Fg(\001)5 b Fm(7)12 b(in)f(c)o(haracter-)224 394 y(istic)h(2)g(\(cf.)g([83)o(]\).)f(Of)h(course,)h(the)g(result)g (holds)f(for)f(group)h(rings)g(o)o(v)o(er)g(algebraically)224 444 y(closed)g(\014elds)f(of)g(c)o(haracteristic)h(zero,)f(or)g(if)f (the)i(c)o(haracteristic)g(of)e(the)i(\014eld)f(do)q(es)h(not)224 494 y(divide)i(the)h(order)f(of)g(the)h(group.)e(More)h(generally)m(,)f (it)h(holds)g(for)f Ff(p)p Fm(-solv)n(able)g(groups)224 544 y(\(cf.)h([6)o(],)f(Cor.)h(\(22.5\),)e(p.)h(518\).)224 628 y(The)23 b(\014rst)f(result)h(on)e(the)h(conjecture)i(for)d (arbitrary)h(Hopf)f(algebras)h(is)f(due)i(to)224 678 y(R.)15 b(G.)g(Larson.)h(This)g(result)g(also)g(sho)o(ws)g(wh)o(y)g (the)g(problem)f(is)h(more)f(di\016cult)g(for)224 727 y(Hopf)j(algebras)f(than)h(for)f(groups.)g(As)h(noted)g(ab)q(o)o(v)o (e,)f(the)i(conjecture)g(is)e(true)i(for)224 777 y(group)d(rings)f(of)g (\014nite)h(groups)f(o)o(v)o(er)h(algebraically)d(closed)j(\014elds)g (of)f(c)o(haracteristic)224 827 y(zero)e(\(cf.)e([6],)f(Prop.)i (\(9.32\),)e(p.)h(216\).)f(The)i(pro)q(of)g(of)f(this)g(result)i (exploits)e(prop)q(erties)224 877 y(of)j(algebraic)g(in)o(tegers,)h (and,)f(in)g(doing)g(so,)g(exploits)g(the)h(fact)g(that)f(the)i(group)e (ring)224 927 y Ff(K)s Fm([)p Ff(G)p Fm(])h(has)h(an)g(ob)o(vious)f(in) o(tegral)g(form,)f(namely)g Fh(Z)-13 b Fm([)o Ff(G)p Fm(])o(.)13 b(Larson's)j(result)g(sa)o(ys)g(that)224 976 y(the)j(same)d(holds)i(if)f(the)h(Hopf)f(algebra)g(under)i (consideration)e(is)h(de\014ned)h(o)o(v)o(er)e Fh(Z)-13 b Fm(.)224 1026 y(More)15 b(precisely)m(,)f(Larson)g(pro)o(v)o(es)g (\(cf.)g([21)o(],)f(Prop.)g(4.2,)g(p.)g(208\):)224 1124 y Fi(Theorem)36 b Fm(Supp)q(ose)13 b(that)f Ff(R)g Fm(is)h(a)f (Dedekind)g(domain)e(with)i(fraction)g(\014eld)g Ff(K)k Fm(and)224 1174 y(that)d Ff(H)j Fm(is)d(a)f(Hopf)h(algebra)f(o)o(v)o (er)h Ff(R)f Fm(that)h(is)g(\014nitely)f(generated)j(pro)r(jectiv)o(e)e (o)o(v)o(er)g Ff(R)p Fm(.)224 1223 y(Assume)i(that)h Ff(H)c Fg(\012)550 1229 y Fe(R)588 1223 y Ff(K)18 b Fm(is)d(split)g (semisimple)d(and)j(in)o(v)o(olutory)m(.)e(Denote)i(the)h(set)g(of)224 1273 y(left)11 b(ideals)f(in)g Ff(H)k Fm(b)o(y)c Ff(L)p Fm(.)g(Then,)h(for)f(ev)o(ery)i(simple)d Ff(H)d Fg(\012)1087 1279 y Fe(R)1116 1273 y Ff(K)s Fm(-mo)q(dule)k Ff(V)f Fm(,)h(the)h(principal)224 1323 y(ideal)j(\(dim)5 b Ff(V)k Fm(\))14 b(of)g Ff(R)f Fm(divides)h(the)g(ideal)g Ff(\017)900 1329 y Fe(H)931 1323 y Fm(\()p Ff(L)p Fm(\),)g(i.)f(e.,)g(w)o(e)h(ha)o (v)o(e)g(\(dim)5 b Ff(V)10 b Fm(\))h Fg(\023)h Ff(\017)1480 1329 y Fe(H)1511 1323 y Fm(\()p Ff(L)p Fm(\).)224 1439 y(V)m(arian)o(ts)i(of)f(this)h(theorem)f(can)i(also)e(b)q(e)h(found)g (in)f([62)o(])h(and)f([11)o(].)224 1573 y Fi(6.2)48 b Fm(A)14 b(v)o(ery)f(imp)q(ortan)o(t)f(result)i(on)g(Kaplansky's)f (sixth)g(conjecture)i(is)f(a)f(theorem)224 1623 y(of)e(W.)f(D.)g(Nic)o (hols)h(and)g(M.)f(B.)h(Ric)o(hmond)e(that)i(an)o(w)o(ers)g(the)h (question)f(a\016rmativ)o(ely)224 1673 y(if)i(the)i(simple)d(mo)q(dule) h(is)h(t)o(w)o(o-dimensional)c(\(cf.)k([48)o(],)f(Cor.)g(12\):)224 1770 y Fi(Theorem)36 b Fm(The)10 b(dimension)d(of)i(a)g(semisimple)e (Hopf)h(algebra)h(o)o(v)o(er)h(an)f(algebraically)224 1820 y(closed)15 b(\014eld)f(is)g(ev)o(en)g(if)f(the)i(Hopf)e(algebra)h (has)g(a)f(simple)g(mo)q(dule)f(of)h(dimension)f(2.)224 1936 y(The)18 b(tec)o(hnique)h(of)d(the)i(authors)g(is)f(to)h(analyse)f (the)h(p)q(ossible)g(decomp)q(ositions)e(of)224 1986 y(the)d(tensor)g(pro)q(duct)g(of)f(the)g(t)o(w)o(o-dimensional)e (simple)g(mo)q(dule)h(and)h(its)g(dual.)f(More)224 2036 y(precisely)m(,)16 b(the)f(argumen)o(t)f(pro)q(ceeds)k(in)c(the)i(w)o (a)o(y)e(that)i(either)g(this)f(decomp)q(osition)224 2085 y(b)q(eha)o(v)o(es)g(similar)d(to)i(the)h(decomp)q(osition)e(of)g (the)i(regular)f(mo)q(dule)f(of)g(the)i(Lie)f(alge-)224 2135 y(bra)g Ff(sl)q Fm(\(2\),)g(in)g(whic)o(h)f(case)i(the)g(Hopf)e (algebra)h(considered)h(is)f(in\014nite-dimensional,)224 2185 y(or)g(the)h(corresp)q(onding)f(Hopf)g(algebra)f(admits)f(certain) i(Hopf)g(algebra)f(quotien)o(ts)h(of)889 2310 y(13)p eop %%Page: 14 14 14 13 bop 224 195 a Fm(dimension)14 b(2,)h(12,)f(24)h(or)g(60)g(whose)h (represen)o(tations)h(are)f(similar)d(to)i(those)h(of)f(the)224 245 y(dihedral)j(groups)h Ff(D)563 251 y Fe(n)586 245 y Fm(,)f(the)h(tetrahedral)g(group)f Ff(A)1064 251 y Fd(4)1083 245 y Fm(,)g(the)g(o)q(ctahedral)h(\(resp.)g(hex-)224 295 y(ahedral\))f(group)f Ff(S)538 301 y Fd(4)557 295 y Fm(,)f(or)i(the)f(icosahedral)g(\(resp.)h(do)q(decahedral\))h(group)e Ff(A)1476 301 y Fd(5)1494 295 y Fm(.)g(The)224 344 y(authors)e(ha)o(v)o (e)e(con)o(tin)o(ued)h(their)h(in)o(v)o(estigation)d(in)i([49)o(].)224 480 y Fi(6.3)48 b Fm(Kaplansky's)19 b(sixth)h(conjecture)h(has)e(also)g (b)q(een)i(considered)g(b)o(y)e(S.)g(Mon)o(t-)224 530 y(gomery)f(and)g(S.)g(J.)g(Withersp)q(o)q(on)h(\(cf.)f([40)o(]\).)g (They)h(pro)o(v)o(e)f(that,)g(in)h(certain)g(sit-)224 579 y(uations,)f(Kaplansky's)g(conjecture)i(holds)e(for)g(a)g(Hopf)g (algebra)g(if)g(it)g(holds)g(for)g(a)224 629 y(subalgebra.)c(More)g (precisely)m(,)g(they)h(pro)o(v)o(e:)224 729 y Fi(Theorem)36 b Fm(Supp)q(ose)21 b(that)g Ff(A)g Fm(is)f(a)h(\014nite-dimensional)d (algebra)i(o)o(v)o(er)h(an)f(alge-)224 779 y(braically)c(closed)h (\014eld)f(suc)o(h)h(that)g(the)g(dimensions)e(of)h(the)h(simple)e(mo)q (dules)g(of)h Ff(A)224 829 y Fm(divide)g(the)h(dimension)e(of)h Ff(A)p Fm(.)g(Supp)q(ose)h(that)f Ff(G)g Fm(is)h(a)f(\014nite)g(group)g (suc)o(h)h(that)g(the)224 878 y(c)o(haracteristic)e(of)d(the)i(base)g (\014eld)f(do)q(es)g(not)g(divide)g(the)h(order)f(of)g Ff(G)p Fm(.)f(If)h Ff(B)i Fm(is)e(a)g(cros-)224 928 y(sed)f(pro)q(duct) h(of)d Ff(A)h Fm(and)h(the)f(group)g(ring)g Ff(K)s Fm([)p Ff(G)p Fm(],)f(then)i(the)g(dimensions)e(of)g(the)i(simple)224 978 y(mo)q(dules)k(of)f Ff(B)k Fm(divide)d(the)h(dimension)e(of)h Ff(B)r Fm(.)g(This)g(also)g(holds)g(if)g Ff(B)i Fm(is)f(a)f(crossed)224 1028 y(pro)q(duct)f(of)e Ff(A)h Fm(and)g(the)h(dual)e(group)h(ring)f Ff(K)s Fm([)p Ff(G)p Fm(])1015 1013 y Fc(\003)1033 1028 y Fm(.)224 1147 y(Using)h(results)h(of)f(G.)f(I.)g(Kac)i(and)e(A.)h (Masuok)n(a)f(\(cf.)h([16)o(],)f([34)o(]\),)g(Mon)o(tgomery)g(and)224 1196 y(Withersp)q(o)q(on)j(then)g(sho)o(w)f(that)g(Hopf)g(algebras)g (of)g(prime)f(p)q(o)o(w)o(er)i(dimension)d(o)o(v)o(er)224 1246 y(algebraically)e(closed)h(\014elds)h(of)f(c)o(haracteristic)h (zero)g(can)f(b)q(e)h(constructed)h(b)o(y)e(iterat-)224 1296 y(ing)j(the)h(crossed)h(pro)q(duct)f(constructions)h(describ)q(ed) g(in)e(the)h(theorem.)f(Therefore,)224 1346 y(Kaplansky's)f(sixth)g (conjecture)h(holds)f(for)f(these)j(Hopf)d(algebras.)224 1481 y Fi(6.4)48 b Fm(Kaplansky's)17 b(sixth)h(conjecture)h(has)f(also) f(b)q(een)i(v)o(eri\014ed)f(for)g(other)g(classes)224 1531 y(of)c(Hopf)g(algebras.)f(Y.)h(Zh)o(u)h(has)f(pro)o(v)o(ed)g (that,)g(for)g(a)g(semisimple)d(Hopf)j(algebra)g Ff(H)224 1581 y Fm(o)o(v)o(er)e(an)g(algebraically)e(closed)j(\014eld)f(of)f(c)o (haracteristic)j(zero,)e(the)h(dimensions)d(of)i(the)224 1631 y(simple)j Ff(D)q Fm(\()p Ff(H)s Fm(\)-submo)q(dules)h(of)f Ff(H)k Fm(divide)c(the)h(dimension)f(of)g Ff(H)k Fm(\(cf.)c([88)o(]\),) g(where)224 1680 y Ff(D)q Fm(\()p Ff(H)s Fm(\))j(denotes)g(the)g (Drinfel'd)e(double)h(of)f Ff(H)s Fm(.)g(His)h(argumen)o(t)f(relies,)h (b)q(esides)i(the)224 1730 y(so-called)e(class)h(equation)f(that)h (will)d(b)q(e)j(discussed)h(b)q(elo)o(w,)e(on)g(the)h(fact)f(that)h (the)224 1780 y(action)f(of)f(the)i(Drinfel'd)e(double)h(on)f Ff(H)k Fm(precisely)e(cen)o(tralizes)g(the)g(action)e(of)h(the)224 1830 y(c)o(haracter)j(algebra)e(on)f Ff(H)s Fm(.)h(Similar)d(results)k (and)f(additional)f(consequences)k(w)o(ere)224 1880 y(obtained)15 b(indep)q(enden)o(tly)m(,)f(but)h(later)g(in)f(the)h(author's)f (dissertation,)h(among)d(them)224 1930 y(the)f(semisimpli)o(cit)o(y)c (of)i(the)i(c)o(haracter)g(ring)f(in)f(p)q(ositiv)o(e)h(c)o (haracteristic.)h(Zh)o(u's)f(result)224 1979 y(has)h(b)q(een)h (generalized)g(b)o(y)e(P)m(.)g(Etingof)g(and)h(S.)f(Gelaki)g(to)h (arbitrary)f(simple)g(mo)q(dules)224 2029 y(of)16 b(the)h(Drinfel'd)e (double)h Ff(D)q Fm(\()p Ff(H)s Fm(\).)g(As)h(a)f(consequence,)i(they)f (w)o(ere)g(able)f(to)g(obtain)889 2310 y(14)p eop %%Page: 15 15 15 14 bop 224 195 a Fm(the)15 b(follo)o(wing)c(theorem)j(\(cf.)f([9],)g (Theorem)g(1.5,)f([11)o(],)h(Cor.)h(3.6\):)224 295 y Fi(Theorem)36 b Fm(Supp)q(ose)16 b(that)f Ff(H)j Fm(is)d(a)g (semisimple,)d(cosemisimple,)h(quasitriangular)224 344 y(Hopf)g(algebra)g(o)o(v)o(er)g(an)g(algebraically)e(closed)j(\014eld.) f(Then)h(the)g(dimension)d(of)i(ev)o(ery)224 394 y(simple)g Ff(H)s Fm(-mo)q(dule)f(divides)i(the)g(dimension)e(of)i Ff(H)s Fm(.)224 567 y Fj(7)67 b(The)23 b(eigh)n(th)g(conjecture)224 694 y Fi(7.1)48 b Fm(Kaplansky's)20 b(eigh)o(th)h(conjecture)h(states)g (that)f(a)g(Hopf)f(algebra)g(of)h(prime)224 744 y(dimension)15 b(o)o(v)o(er)h(an)g(algebraically)e(closed)i(\014eld)g(is)g(comm)o (utativ)o(e)d(and)j(co)q(comm)o(u-)224 794 y(tativ)o(e.)f(Since)h(comm) o(utativ)o(it)o(y)c(prop)q(erties)17 b(remain)d(unc)o(hanged)h(under)h (extensions)224 843 y(of)h(the)g(base)h(\014eld,)e(the)h(assumption)f (of)g(algebraic)h(closure)g(is)g(again)f(not)g(relev)n(an)o(t)224 893 y(here.)e(Ho)o(w)o(ev)o(er,)f(in)f(the)i(algebraically)d(closed)i (case,)g(something)f(more)g(can)h(b)q(e)g(said.)224 943 y(As)j(observ)o(ed)h(b)o(y)e(P)m(.)g(Cartier)h(and)f(D.)g(Harrison)h (\(cf.)f([38)o(],)f(Theorem)h(2.3.1\))f(\014nite-)224 993 y(dimensional)i(co)q(comm)o(utativ)o(e)f(cosemisimple)h(Hopf)h (algebras)h(o)o(v)o(er)g(algebraically)224 1043 y(closed)e(\014elds)g (are)g(group)f(rings.)g(As)h(a)f(consequence)j(of)d(the)h(Nic)o (hols-Zo)q(eller)f(theo-)224 1092 y(rem,)c(it)g(w)o(as)g(sho)o(wn)h(b)o (y)f(R.)g(G.)f(Larson)i(and)f(D.)g(E.)g(Radford)g(that)h(a)f(Hopf)g (algebra)g(of)224 1142 y(prime)j(dimension)e(is)i(semisimple)e(and)i (cosemisimple)e(if)i(the)g(c)o(haracteristic)i(of)e(the)224 1192 y(base)g(\014eld)f(is)g(zero)h(or)f(greater)h(than)f(the)g (dimension)f(of)g(the)i(Hopf)e(algebra)h(\(cf.)f([25)o(],)224 1242 y(Theorem)g(2.3\).)g(Therefore,)h(the)g(conjecture)h(w)o(ould)e (imply)e(in)i(these)i(cases)g(that)f(the)224 1292 y(only)h(example)f (of)g(a)h(Hopf)g(algebra)g(of)g(prime)f(dimension)f(is)i(the)h(group)f (ring)g(of)g(the)224 1342 y(cyclic)e(group)f(of)f(prime)g(order.)i(Of)f (course,)h(this)f(do)q(es)h(not)f(hold)f(if)g(the)i(c)o(haracteristic) 224 1391 y(of)j(the)h(base)f(\014eld)g(coincides)h(with)f(the)g (dimension)f(of)g(the)i(Hopf)f(algebra,)f(b)q(ecause)224 1441 y(then)e(the)f(restricted)i(en)o(v)o(eloping)d(algebra)g(of)g(the) i(base)f(\014eld,)f(considered)j(an)d(ab)q(elian)224 1491 y(restricted)15 b(Lie)e(algebra)g(via)f(the)h(F)m(rob)q(enius)h (homom)o(orphism)o(,)c(is)i(another)i(example)224 1541 y(of)g(a)f(comm)o(utativ)o(e)e(and)j(co)q(comm)o(utativ)o(e)d(Hopf)j (algebra)f(of)g(dimension)g Ff(p)p Fm(.)224 1626 y(The)i(history)f(of)g (Kaplansky's)g(eigh)o(th)g(conjecture)h(is)g(sligh)o(tly)d(strange.)j (Kaplansky)224 1676 y(himself)c(had)h(considered)i(the)f(dimensions)f (2,)f(3)h(and)h(5)f(in)g(the)h(\014rst)g(app)q(endix)g(of)f(his)224 1726 y(notes.)h(He)g(added)g(as)f(a)g(remark)g(to)g(the)h(conjecture)i (that)d(G.)g(I.)f(Kac)i(had)g(partial)e(re-)224 1776 y(sults)g(on)g(this)f(conjecture,)i(and)f(his)f(bibliograph)o(y)f(also) h(con)o(tains)g(the)i(corresp)q(onding)224 1826 y(reference)k([16)o(].) c(The)h(conjecture)h(w)o(as)f(then)h(considered)g(as)f(op)q(en)g(for)g (ab)q(out)f(t)o(w)o(en)o(t)o(y)224 1875 y(y)o(ears;)17 b(the)f(only)g(con)o(tribution)g(w)o(as)g(giv)o(en)g(b)o(y)f(C.)h(R.)f (Cai)h(and)g(H.)g(X.)f(Chen,)i(who)224 1925 y(pro)o(v)o(ed)11 b(the)h(conjecture)g(for)e(dimension)f(7)i(and)f(13)h(o)o(v)o(er)f (\014elds)h(of)f(c)o(haracteristic)j(zero)224 1975 y(\(cf.)j([4]\).)f (A)i(y)o(ear)f(later,)g(Y.)g(Zh)o(u)h(w)o(as)f(able)g(to)g(pro)o(v)o(e) h(the)g(conjecture)h(completely)224 2025 y(o)o(v)o(er)i(\014elds)f(of)g (c)o(haracteristic)i(zero,)e(using)g(a)g(rather)h(di\013eren)o(t)h (argumen)o(t)d(based)224 2075 y(on)d(the)h(Drinfel'd)e(double)h (construction)h(\(cf.)f([86)o(]\).)f(According)i(to)f(Zh)o(u,)g(E.)g (E\013ros)224 2125 y(then)f(p)q(oin)o(ted)f(out)f(Kac's)h(w)o(ork)g(to) g(him,)d(and)j(up)q(on)f(reading)h(it)f(he)i(disco)o(v)o(ered)f(that) 224 2174 y(Kac')i(argumen)o(t)f(could)h(b)q(e)g(extended)i(comparativ)o (ely)12 b(easily)j(to)f(the)i(general)f(case.)889 2310 y(15)p eop %%Page: 16 16 16 15 bop 224 195 a Fm(Zh)o(u)18 b(then)g(decided)h(to)e(follo)o(w)e (Kac')j(argumen)o(t)e(in)h(the)h(published)f(v)o(ersion)h(of)f(his)224 245 y(man)o(uscript)c(\(cf.)h([87)o(]\).)224 330 y(The)k(argumen)o(t)d (of)h(Kac)h(and)g(Zh)o(u)g(is)g(based)g(on)f(a)h(theorem)f(whic)o(h)h (is)f(no)o(w)h(called)224 380 y(the)d(class)g(equation,)e(and)h(ma)o(y) e(b)q(e)i(considered)i(as)e(ev)o(en)h(more)e(imp)q(ortan)o(t)f(than)i (the)224 430 y(solution)g(of)h(the)g(conjecture)i(itself:)224 530 y Fi(Theorem)36 b Fm(Supp)q(ose)22 b(that)g Ff(H)i Fm(is)e(a)f(semisimple)e(Hopf)i(algebra)g(o)o(v)o(er)h(an)f(alge-)224 579 y(braically)10 b(closed)i(\014eld)f(of)f(c)o(haracteristic)i(zero.) g(Supp)q(ose)g(that)f Ff(e)1257 585 y Fd(1)1275 579 y Ff(;)c(:)g(:)g(:)12 b(;)7 b(e)1394 585 y Fe(m)1436 579 y Fm(is)k(a)g(com-)224 629 y(plete)16 b(set)g(of)f(primitiv)o(e)d (orthogonal)i(idemp)q(oten)o(ts)h(of)f(the)i(c)o(haracter)h(ring)d Ff(C)s(h)p Fm(\()p Ff(H)s Fm(\),)224 679 y(where)h Ff(e)363 685 y Fd(1)396 679 y Fm(is)f(an)g(in)o(tegral)f(of)g Ff(H)732 664 y Fc(\003)751 679 y Fm(.)g(Then)i(w)o(e)f(ha)o(v)o(e)708 801 y(dim)5 b Ff(H)15 b Fm(=)893 750 y Fe(m)878 762 y Fb(X)881 850 y Fe(i)p Fd(=1)944 801 y Fm(dim)6 b Ff(e)1040 807 y Fe(i)1054 801 y Ff(H)1092 784 y Fc(\003)224 929 y Fm(dim)g Ff(e)320 935 y Fd(1)338 929 y Ff(H)376 914 y Fc(\003)407 929 y Fm(=)12 b(1)h(and)h(dim)5 b Ff(e)661 935 y Fe(i)675 929 y Ff(H)713 914 y Fc(\003)746 929 y Fm(divides)14 b(dim)5 b Ff(H)s Fm(.)224 1048 y(Note)16 b(that)f(the)g(non)o(trivial)f(part)h(is)g(the)g(divisibilit)o(y)e (statemen)o(t.)h(The)h(theorem)g(re-)224 1098 y(duces)i(to)e(the)h (ordinary)e(class)i(equation)f(if)f(applied)g(to)h(the)h(group)f(ring)g (of)f(a)h(\014nite)224 1147 y(group,)f(b)q(ecause)i(the)e(idemp)q(oten) o(ts)g(ab)q(o)o(v)o(e)g(then)g(b)q(ecome)g(the)h(c)o(haracteristic)g (func-)224 1197 y(tions)d(of)g(the)h(conjugacy)f(classes.)g(If)g (applied)g(to)g(the)g(dual)g(group)g(ring,)f(the)i(theorem)224 1247 y(implies)i(that)i(the)h(dimensions)d(of)i(the)g(simple)e(mo)q (dules)h(divide)h(the)g(order)h(of)e(the)224 1297 y(group.)d(A)g (simpli\014ed)e(pro)q(of)i(of)g(the)g(class)h(equation)f(has)g(b)q(een) h(giv)o(en)f(b)o(y)g(M.)g(Lorenz)224 1347 y(\(cf.)j([27)o(]\).)f(S.)g (J.)h(Withersp)q(o)q(on)g(has)g(related)h(the)f(class)h(equation)e(to)h (analogues)f(of)224 1396 y(conjugacy)f(class)g(sums)g(\(cf.)f([85)o (]\).)224 1482 y(As)k(a)f(consequence)j(of)d(their)g(lifting)f(theorem) h(already)g(discussed)i(in)d(conjunction)224 1532 y(with)e(the)h (\014fth)e(conjecture,)j(P)m(.)d(Etingof)g(and)h(S.)f(Gelaki)g(v)o(ery) h(recen)o(tly)h(pro)o(v)o(ed)f(o)o(v)o(er)224 1582 y(an)o(y)h(\014eld)g (that)g(a)f(semisimple)f(and)h(cosemisimple)f(Hopf)h(algebra)h(of)f (prime)g(dimen-)224 1631 y(sion)d(is)f(comm)o(utativ)o(e)e(and)i(co)q (comm)o(utativ)o(e)e(\(cf.)j([11)o(],)e(Theorem)h(3.3\).)g(By)h(the)g (ab)q(o)o(v)o(e)224 1681 y(men)o(tioned)g(result)i(of)e(Larson)h(and)g (Radford,)e(the)j(semisimpli)o(cit)o(y)c(assumptions)i(will)224 1731 y(b)q(e)15 b(often)f(satis\014ed.)224 1866 y Fi(7.2)48 b Fm(No)o(w)o(ada)o(ys,)16 b(Kaplansky's)g(eigh)o(th)h(conjecture)h(ma) o(y)d(b)q(e)j(understo)q(o)q(d)g(as)f(the)224 1916 y(\014rst)i(step)h (in)d(the)i(program)e(to)h(understand)h(the)g(structure)i(of)c (semisimple)f(Hopf)224 1966 y(algebras)k(in)g(terms)g(of)f(the)i(prime) d(factors)j(of)e(their)i(dimension,)c(similar)h(to)i(the)224 2016 y(situation)15 b(in)g(\014nite)g(group)g(theory)m(.)g(This)g (program,)e(also)h(called)h(the)h(classi\014cation)224 2066 y(program)10 b(for)g(semisimple)f(Hopf)h(algebras,)h(has)g(recen)o (tly)h(exp)q(erienced)h(considerable)224 2116 y(progress,)k(the)g (class)f(equation)g(b)q(eing)g(an)g(imp)q(ortan)o(t)e(instrumen)o(t)h (in)h(this)g(in)o(v)o(esti-)224 2165 y(gation.)11 b(One)j(of)e(the)h (\014rst)g(consequences)i(of)d(the)h(class)g(equation)f(is)h(the)g (observ)n(ation)889 2310 y(16)p eop %%Page: 17 17 17 16 bop 224 195 a Fm(that)15 b(a)f(Hopf)g(algebra)g(of)g(prime)f(p)q (o)o(w)o(er)i(dimension)e(con)o(tains)h(a)g(non)o(trivial)f(cen)o(tral) 224 245 y(grouplik)o(e)j(elemen)o(t)f(\(cf.)h([16)o(],)g([34)o(]\).)f (Here,)i(the)g(pro)q(of)f(is)g(analogous)f(to)h(the)h(pro)q(of)224 295 y(that)h(a)g Ff(p)p Fm(-group)f(has)h(non)o(trivial)e(cen)o(ter.)j (It)f(w)o(as)f(recen)o(tly)i(extended)g(to)f(\014elds)g(of)224 344 y(p)q(ositiv)o(e)f(c)o(haracteristic)g(b)o(y)g(P)m(.)e(Etingof)h (and)g(S.)g(Gelaki)f(\(cf.)h([11)o(],)g(Theorem)g(3.7\).)224 394 y(As)f(a)e(consequence,)j(A.)d(Masuok)n(a)g(has)h(pro)o(v)o(ed)f (that)h(o)o(v)o(er)g(a)f(\014eld)h(of)f(c)o(haracteristic)224 444 y(zero,)18 b(Hopf)e(algebras)h(of)f(dimension)f Ff(p)866 429 y Fd(2)902 444 y Fm(are)i(comm)o(utativ)o(e)d(and)j(co)q(comm)o (utativ)o(e.)224 494 y(A.)11 b(Masuok)n(a)f(has)h(also)g(con)o (tributed)g(sev)o(eral)h(other)f(results)i(\(cf.)d([32)o(],)g([33)o(],) g([35)o(],)g([36)o(],)224 544 y([37)o(]\).)j(F)m(urther)i(results)g (can)e(b)q(e)i(found)e(in)g([14)o(],)g([10)o(],)g([72)o(].)f(Since)j (this)e(dev)o(elopmen)o(t)224 594 y(is)h(review)o(ed)h(in)e([39)o(],)g (w)o(e)h(shall)f(not)h(giv)o(e)g(further)g(details)g(here.)224 767 y Fj(8)67 b(The)23 b(ten)n(th)g(conjecture)224 893 y Fi(8.1)48 b Fm(Kaplansky's)14 b(ten)o(th)h(conjecture)h(states)f (that,)f(o)o(v)o(er)g(an)g(algebraically)f(closed)224 943 y(\014eld,)20 b(the)h(n)o(um)o(b)q(er)e(of)h(isomorphism)d(classes) k(of)e(Hopf)h(algebras)g(of)g(a)f(giv)o(en)h(di-)224 993 y(mension)g(is)h(\014nite,)g(pro)o(vided)h(that)f(the)h(c)o (haracteristic)g(of)f(the)h(base)f(\014eld)h(do)q(es)224 1043 y(not)d(divide)e(the)i(dimension.)e(This)h(w)o(as)g(pro)o(v)o(ed)h (b)o(y)f(D.)f(S)-21 b(\030)s(tefan)19 b(for)f(isomorphism)224 1092 y(classes)g(of)d(semisimple)e(and)j(cosemisimple)d(Hopf)j (algebras)g(\(cf.)f([73)o(]\).)h(Note)g(that,)224 1142 y(due)f(to)f(the)h(p)q(ositiv)o(e)f(solution)g(of)f(the)i(sev)o(en)o (th)g(conjecture,)h(for)e(a)g(semisimple)e(and)224 1192 y(cosemisimple)f(Hopf)h(algebra)h(the)g(c)o(haracteristic)h(do)q(es)g (not)f(divide)f(the)i(dimension.)224 1242 y(H.-J.)g(Sc)o(hneider)h(has) f(giv)o(en)f(a)h(simpli\014ed)e(pro)q(of)h(of)h(this)f(theorem)h(\(cf.) g([69)o(]\).)224 1377 y Fi(8.2)48 b Fm(Ho)o(w)o(ev)o(er,)14 b(the)h(conjecture)h(is)f(false)f(in)g(general.)g(This)g(w)o(as)g (realized)h(more)e(or)224 1427 y(less)18 b(sim)o(ultaneously)c(b)o(y)i (three)i(groups)f(of)f(researc)o(hers)j(in)d(1997,)g(namely)e(N.)i(An-) 224 1477 y(druskiewitsc)o(h)11 b(and)f(H.-J.)f(Sc)o(hneider)i(on)e(the) i(one)f(hand,)f(M.)g(Beattie,)h(S.)g(D\025)-21 b(asc\025)g(alescu)224 1527 y(and)17 b(L.)f(Gr)q(\177)-22 b(unenfelder)17 b(on)f(the)h(other)g (hand,)f(and)h(also)f(b)o(y)g(S.)g(Gelaki)f(\(cf.)h([1],)f([2)o(],)224 1577 y([13)o(]\).)h(In)h(the)g(approac)o(h)f(of)g(the)i(\014rst)f(t)o (w)o(o)f(groups,)g(the)h(conjecture)i(is)d(refuted)i(b)o(y)224 1626 y(constructing)f(p)q(oin)o(ted)f(Hopf)g(algebras)g(of)f(dimension) f Ff(p)1152 1611 y Fd(4)1187 1626 y Fm(for)h(a)h(prime)f(n)o(um)o(b)q (er)g Ff(p)p Fm(,)224 1676 y(whereas)21 b(in)e(Gelaki's)e(approac)o(h)i (in\014nite)g(families)e(of)i(Hopf)f(algebras)h(of)g(dimen-)224 1726 y(sion)f Ff(mn)375 1711 y Fd(2)413 1726 y Fm(are)h(constructed,)h (where)f Ff(m)h(>)f Fm(2)f(and)h Ff(n)g(>)g Fm(1)f(are)h(natural)f(n)o (um)o(b)q(ers)224 1776 y(suc)o(h)c(that)e Ff(n)g Fm(divides)g Ff(m)p Fm(.)h(The)g(approac)o(h)f(of)g(Andruskiewitsc)o(h)h(and)g(Sc)o (hneider)g(also)224 1826 y(leads)i(to)g(the)g(classi\014cation)g(of)f (p)q(oin)o(ted)h(Hopf)f(algebras)h(of)f(dimension)g Ff(p)1428 1811 y Fd(3)1461 1826 y Fm(o)o(v)o(er)h(an)224 1875 y(algebraically)d (closed)j(\014eld)f(of)f(c)o(haracteristic)i(zero,)f(where)h Ff(p)f Fm(is)g(a)f(prime.)g(This)g(has)224 1925 y(also)h(b)q(e)g (obtained)g(b)o(y)f(S.)h(Caenep)q(eel)h(and)f(S.)f(D\025)-21 b(asc\025)g(alescu)15 b(\(cf.)f([5)o(]\).)889 2310 y(17)p eop %%Page: 18 18 18 17 bop 224 195 a Fj(9)67 b(App)r(endix)224 322 y Fm(In)11 b(this)g(app)q(endix,)g(w)o(e)g(repro)q(duce)i(literally)d(app)q(endix) h(2)f(from)g([17)o(],)g(whic)o(h)h(con)o(tains)224 372 y(Kaplansky's)21 b(conjectures.)h(It)f(ma)o(y)d(b)q(e)k(helpful)e(for)g (the)h(understanding)h(of)e(the)224 421 y(conjectures)f(to)c(kno)o(w)h (that)g(Kaplansky's)g(lecture)h(notes)g(con)o(tain)f(a)f(bibliograph)o (y)224 471 y(whic)o(h,)i(among)e(man)o(y)h(other)h(references,)j(lists) d(the)h(items)e(lab)q(eled)i([16)o(],)e([20)o(])h(and)224 521 y([51)o(])d(in)f(the)i(presen)o(t)g(article.)f(Kaplansky's)f (conjectures)j(are:)275 648 y(1.)k(If)11 b Ff(C)k Fm(is)c(a)h(Hopf)f (subalgebra)h(of)f(the)h(Hopf)f(algebra)g Ff(B)k Fm(then)d Ff(B)i Fm(is)e(a)f(free)i(left)e Ff(C)s Fm(-)328 698 y(mo)q(dule.)i(\(Remark.)g(Nic)o(hols)h(has)h(pro)o(v)o(ed)g(this)f(if) g Ff(B)j Fm(con)o(tains)e(the)g(coradical)328 748 y(of)e Ff(C)s Fm(.\))275 831 y(2.)20 b(Call)11 b(a)h(coalgebra)g Ff(C)j Fm(admissible)p 675 838 188 2 v 11 w(if)c(it)h(admits)f(an)h (algebra)f(structure)k(making)328 881 y(it)i(a)h(Hopf)f(algebra.)g(The) h(conjecture)h(states)g(that)e Ff(C)k Fm(is)c(admissible)f(if)h(and)328 930 y(only)e(if)h(ev)o(ery)g(\014nite)h(subset)g(of)f Ff(C)i Fm(lies)e(in)g(a)f(\014nite-dimensional)f(admissible)328 980 y(sub)q(coalgebra.)i(\(Remarks.)f(1.)g(Both)h(implications)e(seem)h (hard.)h(2.)f(There)j(is)328 1030 y(a)13 b(corresp)q(onding)h (conjecture)h(where)f(\\Hopf)e(algebra")g(is)h(replaced)h(b)o(y)f (\\bial-)328 1080 y(gebra".)h(3.)f(There)i(is)e(a)h(dual)f(conjecture)j (for)d(lo)q(cally)g(\014nite)h(algebras.\))275 1163 y(3.)20 b(A)e(Hopf)f(algebra)g(of)g(c)o(haracteristic)i(0)e(has)g(no)h (non-zero)g(cen)o(tral)g(nilp)q(oten)o(t)328 1213 y(elemen)o(ts.)275 1296 y(4.)i(\(Nic)o(hols\).)14 b(Let)h Ff(x)g Fm(b)q(e)g(an)g(elemen)o (t)f(in)g(a)g(Hopf)h(algebra)f Ff(H)j Fm(with)e(an)o(tip)q(o)q(de)f Ff(S)r Fm(.)328 1345 y(Assume)g(that)g(for)f(an)o(y)h Ff(a)g Fm(in)f Ff(H)k Fm(w)o(e)d(ha)o(v)o(e)779 1403 y Fb(X)846 1443 y Ff(b)864 1449 y Fe(i)878 1443 y Ff(xS)r Fm(\()p Ff(c)963 1449 y Fe(i)977 1443 y Fm(\))e(=)g Ff(\017)p Fm(\()p Ff(a)p Fm(\))p Ff(x)328 1542 y Fm(where)j(\001)p Ff(a)c Fm(=)560 1511 y Fb(P)611 1542 y Ff(b)629 1548 y Fe(i)652 1542 y Fg(\012)e Ff(c)711 1548 y Fe(i)725 1542 y Fm(.)k(Conjecture:)i Ff(x)f Fm(is)f(in)h(the)g(cen)o(ter)i(of)d Ff(H)s Fm(.)224 1669 y(In)g(the)h(remaining)d(six)h(conjectures)j Ff(H)h Fm(is)c(a)h(\014nite-dimensional)e(Hopf)h(algebra)g(o)o(v)o(er) 224 1719 y(an)i(algebraically)e(closed)j(\014eld.)275 1846 y(5.)20 b(If)h Ff(H)i Fm(is)e(semisimple)d(on)j(either)g(side)g (\(i.e.)f(either)i Ff(H)i Fm(or)c(the)i(dual)e Ff(H)1527 1831 y Fc(\003)1567 1846 y Fm(is)328 1896 y(semisimple)12 b(as)h(an)h(algebra\))g(the)g(square)h(of)e(the)i(an)o(tip)q(o)q(de)e (is)h(the)h(iden)o(tit)o(y)m(.)275 1979 y(6.)20 b(The)c(size)g(of)e (the)i(matrices)e(o)q(ccuring)i(in)e(an)o(y)h(full)f(matrix)f (constituen)o(t)j(of)f Ff(H)328 2029 y Fm(divides)f(the)g(dimension)f (of)g Ff(H)s Fm(.)275 2112 y(7.)20 b(If)13 b Ff(H)j Fm(is)d(semisimple) d(on)j(b)q(oth)g(sides)h(the)g(c)o(haracteristic)g(do)q(es)g(not)f (divide)g(the)328 2162 y(dimension.)889 2310 y(18)p eop %%Page: 19 19 19 18 bop 275 195 a Fm(8.)20 b(If)13 b(the)i(dimension)d(of)h Ff(H)j Fm(is)e(prime)e(then)j Ff(H)h Fm(is)e(comm)o(utativ)n(e)d(and)j (co)q(comm)o(u-)328 245 y(tativ)o(e.)224 366 y(Remark.)e(Kac,)i (Larson,)g(and)f(Sw)o(eedler)i(ha)o(v)o(e)f(partial)f(results)i(on)f(5) f(-)h(8.)224 449 y(In)i(the)h(t)o(w)o(o)e(\014nal)h(conjectures)i (assume)d(that)h(the)h(c)o(haracteristic)g(do)q(es)g(not)f(divide)224 499 y(the)f(dimension)d(of)h Ff(H)s Fm(.)275 620 y(9.)20 b(The)15 b(dimension)d(of)h(the)i(radical)e(is)g(the)i(same)e(on)h(b)q (oth)g(sides.)254 701 y(10.)20 b(There)c(are)g(only)e(a)g(\014nite)i(n) o(um)o(b)q(er)e(\(up)h(to)g(isomorphism\))d(of)i(Hopf)h(algebras)328 751 y(of)e(a)h(giv)o(en)f(dimension.)224 872 y Fi(Ac)o(kno)o(wledgemen) o(t)6 b Fm(The)k(author)f(thanks)h(R.)e(G.)h(Larson)g(for)g(pro)o (viding)f(the)i(coun-)224 922 y(terexample)g(to)g(Kaplansky's)g(second) h(conjecture)h(used)f(in)e(the)i(text.)f(He)h(also)e(thanks)224 972 y(S.)14 b(Sc)o(hmidt-Samoa)c(for)k(discussions)h(that)e(led)h(to)g (impro)o(v)o(emen)o(ts)e(in)h(Section)h(4.)224 1142 y Fj(References)245 1249 y Fm([1])20 b(N.)c(Andruskiewitsc)o(h/H.-J.)h (Sc)o(hneider:)g(Lifting)e(of)h(quan)o(tum)f(linear)h(spaces)310 1299 y(and)d(p)q(oin)o(ted)h(Hopf)g(algebras)g(of)f(order)i Ff(p)981 1284 y Fd(3)999 1299 y Fm(,)e(Preprin)o(t,)i(1997)245 1380 y([2])20 b(M.)14 b(Beattie/S.)h(D\025)-21 b(asc\025)g(alescu/L.)15 b(Gr)q(\177)-22 b(unenfelder:)15 b(On)g(the)h(n)o(um)o(b)q(er)e(of)g(t) o(yp)q(es)i(of)310 1430 y(\014nite)e(dimensional)d(Hopf)j(algebras,)f (Preprin)o(t,)h(1997)245 1511 y([3])20 b(R.)f(Brauer:)h(Represen)o (tations)h(of)e(\014nite)h(groups.)g(In:)f(Lectures)j(on)d(mo)q(dern) 310 1561 y(mathematics,)11 b(V)m(ol.)h(I,)i(133-175,)d(Wiley)m(,)h(New) j(Y)m(ork,)e(1963)245 1642 y([4])20 b(C.)15 b(R.)g(Cai/H.)g(X.)h(Chen:) g(Prime)f(dimensional)f(Hopf)h(algebras.)h(In:)g(Pro)q(c.)g(of)310 1692 y(the)10 b(\014rst)h(China-Japan)f(in)o(ternational)e(symp)q (osium)g(on)i(ring)f(theory)m(,)h(Ok)n(a)o(y)o(ama)310 1742 y(Univ.,)i(Ok)n(a)o(y)o(ama,)f(1992)245 1823 y([5])20 b(S.)g(Caenep)q(eel/S.)h(D\025)-21 b(asc\025)g(alescu:)20 b(P)o(oin)o(ted)h(Hopf)f(algebras)g(of)g(dimension)e Ff(p)1564 1808 y Fd(3)1583 1823 y Fm(,)310 1873 y(Preprin)o(t,)c(1997) 245 1954 y([6])20 b(C.)11 b(W.)h(Curtis/I.)g(Reiner:)g(Metho)q(ds)h(of) f(represen)o(tation)i(theory)f(with)f(applica-)310 2004 y(tions)h(to)h(\014nite)g(groups)g(and)g(orders,)h(V)m(ol.)d(I,)h (Wiley)g(In)o(terscience,)j(New)e(Y)m(ork,)310 2054 y(1981)245 2135 y([7])20 b(M.)14 b(Demazure/P)m(.)f(Gabriel:)g(Group)q(es)i(alg)o (\023)-20 b(ebriques)14 b(I,)g(North-Holland,)f(Ams-)310 2185 y(terdam,)f(1970)889 2310 y(19)p eop %%Page: 20 20 20 19 bop 245 195 a Fm([8])20 b(M.)11 b(Eb)q(erw)o(ein:)h(Cosemisimple) d(Hopf)i(algebras,)g(Ph.)g(D.)g(Dissertation,)g(Florida)310 245 y(State)j(Univ)o(ersit)o(y)m(,)f(T)m(allahassee,)g(1992)245 328 y([9])20 b(P)m(.)14 b(Etingof/S.)f(Gelaki:)g(Some)g(prop)q(erties)j (of)f(\014nite-dimensional)d(semisimple)310 378 y(Hopf)h(algebras,)h (Preprin)o(t,)g(1997,)e(to)i(app)q(ear)g(in:)f(Math.)h(Res.)f(Lett.)224 461 y([10])20 b(P)m(.)f(Etingof/S.)g(Gelaki:)f(Semisimple)f(Hopf)j (algebras)g(of)g(dimension)e Ff(pq)j Fm(are)310 511 y(trivial,)12 b(Preprin)o(t,)i(1998,)e(to)i(app)q(ear)g(in:)f(J.)h(Algebra)224 594 y([11])20 b(P)m(.)13 b(Etingof/S.)f(Gelaki:)g(On)i (\014nite-dimensional)d(semisimple)g(and)j(cosemisim-)310 643 y(ple)g(Hopf)f(algebras)h(in)f(p)q(ositiv)o(e)h(c)o(haracteristic,) h(Preprin)o(t,)f(1998)224 726 y([12])20 b(W.)9 b(F)m(eit:)h(The)g (represen)o(tation)i(theory)f(of)f(\014nite)g(groups,)g(North-Holland,) f(Am-)310 776 y(sterdam,)k(1982)224 859 y([13])20 b(S.)14 b(Gelaki:)f(On)h(p)q(oin)o(ted)h(Hopf)f(algebras)h(and)f(Kaplansky's)g (10th)g(conjecture,)310 909 y(Preprin)o(t,)g(1998)224 992 y([14])20 b(S.)14 b(Gelaki/S.)f(W)m(estreic)o(h:)i(On)h(semisimple) c(Hopf)j(algebras)f(of)h(dimension)e Ff(pq)q Fm(,)310 1042 y(Preprin)o(t,)h(1998)224 1125 y([15])20 b(J.)9 b(C.)h(Jan)o(tzen:)g(Lectures)i(of)e(quan)o(tum)e(groups,)i(Grad.)f (Stud.)g(in)h(Math.,)f(V)m(ol.)f(6,)310 1175 y(Amer.)13 b(Math.)g(So)q(c.,)g(Pro)o(vidence,)i(1996)224 1258 y([16])20 b(G.)d(I.)h(Kac:)h(Certain)f(arithmetic)g(prop)q(erties)i(of)e(ring)g (groups,)g(F)m(unktsional.)310 1308 y(Anal.)h(i)g(Prilozhen.)h(6)g (\(1972\),)f(88-90.)f(English)i(translation:)f(F)m(unct.)g(Anal.)310 1357 y(Appl.)13 b(6)g(\(1972\),)g(158-160)224 1440 y([17])20 b(I.)f(Kaplansky:)g(Bialgebras,)g(Lecture)i(Notes,)f(Univ.)f(of)g (Chicago,)g(Chicago,)310 1490 y(1975)224 1573 y([18])h(C.)14 b(Kassel:)g(Quan)o(tum)g(groups,)g(Grad.)f(T)m(exts)i(in)f(Math.,)f(V)m (ol.)g(155,)h(Springer,)310 1623 y(Berlin,)f(1995)224 1706 y([19])20 b(R.)h(G.)g(Larson:)h(The)h(order)g(of)f(the)h(an)o(tip) q(o)q(de)f(of)g(a)g(Hopf)g(algebra,)f(Pro)q(c.)310 1756 y(Amer.)13 b(Math.)g(So)q(c.)h(21)f(\(1969\),)g(167-170)224 1839 y([20])20 b(R.)14 b(G.)g(Larson:)h(Characters)i(of)e(Hopf)g (algebras,)f(J.)h(Algebra)h(17)e(\(1971\),)g(352-)310 1889 y(368)224 1972 y([21])20 b(R.)13 b(G.)g(Larson:)g(Orders)j(in)d (Hopf)h(algebras,)f(J.)h(Algebra)f(22)h(\(1972\),)e(201-210)224 2055 y([22])20 b(R.)14 b(G.)h(Larson:)h(Cosemisimpl)o(e)e(Hopf)h (algebras)g(with)h(small)d(simple)i(sub)q(coal-)310 2105 y(gebras)f(are)h(in)o(v)o(olutory)m(,)c(Comm.)f(Algebra)k(11)f (\(1983\),)g(1175-1186)889 2310 y(20)p eop %%Page: 21 21 21 20 bop 224 195 a Fm([23])20 b(R.)15 b(G.)g(Larson/D.)g(E.)g (Radford:)g(Semisimple)e(cosemisimple)h(Hopf)h(algebras,)310 245 y(Amer.)e(J.)g(Math.)h(110)f(\(1988\),)f(187-195)224 328 y([24])20 b(R.)12 b(G.)h(Larson/D.)f(E.)i(Radford:)e(Finite)h (dimensional)e(cosemisimple)g(Hopf)i(al-)310 378 y(gebras)e(in)f(c)o (haracteristic)i(0)f(are)g(semisimple,)d(J.)i(Algebra)h(117)f (\(1988\),)f(267-289)224 461 y([25])20 b(R.)11 b(G.)h(Larson/D.)g(E.)g (Radford:)f(Semisimple)f(Hopf)i(algebras,)g(J.)h(Algebra)f(171)310 511 y(\(1995\),)g(5-35)224 594 y([26])20 b(R.)14 b(G.)g(Larson/M.)g(E.) h(Sw)o(eedler:)g(An)g(asso)q(ciativ)o(e)g(orthogonal)f(bilinear)g(form) 310 643 y(for)f(Hopf)h(algebras,)f(Amer.)g(J.)h(Math.)f(91)g(\(1969\),) g(75-93)224 726 y([27])20 b(M.)14 b(Lorenz:)g(On)h(the)g(class)g (equation)f(for)f(Hopf)h(algebras,)g(Preprin)o(t,)h(1996,)e(to)310 776 y(app)q(ear)h(in:)f(Pro)q(c.)h(Amer.)f(Math.)g(So)q(c.)224 859 y([28])20 b(A.)11 b(Masuok)n(a:)f(On)h(Hopf)g(algebras)g(with)g (comm)o(utativ)o(e)e(coradicals,)h(J.)i(Algebra)310 909 y(144)h(\(1991\),)g(415-466)224 992 y([29])20 b(A.)28 b(Masuok)n(a:)g(F)m(reeness)i(of)f(Hopf)f(algebras)h(o)o(v)o(er)f (coideal)h(subalgebras,)310 1042 y(Comm.)10 b(Algebra)k(20)f(\(1992\),) g(1353-1373)224 1125 y([30])20 b(A.)11 b(Masuok)n(a:)f(Coideal)g (subalgebras)i(in)f(\014nite)h(Hopf)f(algebras,)g(J.)g(Algebra)g(163) 310 1175 y(\(1994\),)h(819-831)224 1258 y([31])20 b(A.)15 b(Masuok)n(a/D.)f(Wigner:)i(F)m(aithful)e(\015atness)j(of)e(Hopf)h (algebras,)f(J.)h(Algebra)310 1308 y(170)d(\(1994\),)g(156-164)224 1391 y([32])20 b(A.)28 b(Masuok)n(a:)f(Semisimpl)o(e)f(Hopf)i(algebras) g(of)g(dimension)e(6,)i(8,)f(Israel)310 1440 y(J.)13 b(Math.)h(92)f(\(1995\),)g(361-373)224 1523 y([33])20 b(A.)9 b(Masuok)n(a:)g(Semisimple)e(Hopf)j(algebras)g(of)f(dimension)g (2)p Ff(p)p Fm(,)g(Comm.)e(Algebra)310 1573 y(23)13 b(\(1995\),)g (1931-1940)224 1656 y([34])20 b(A.)10 b(Masuok)n(a:)f(The)i Ff(p)647 1641 y Fe(n)669 1656 y Fm(-theorem)f(for)g(semisimple)e(Hopf)i (algebras,)g(Pro)q(c.)h(Amer.)310 1706 y(Math.)i(So)q(c.)h(124)f (\(1996\),)g(735-737)224 1789 y([35])20 b(A.)13 b(Masuok)n(a:)f (Semisimple)e(Hopf)j(algebras)g(of)g(dimension)f Ff(p)1292 1774 y Fd(3)1324 1789 y Fm(obtained)h(b)o(y)g(an)310 1839 y(extension,)h(J.)g(Algebra)f(178)g(\(1995\),)g(791-806)224 1922 y([36])20 b(A.)14 b(Masuok)n(a:)f(Some)g(further)i (classi\014cation)f(results)h(on)f(semisimple)e(Hopf)h(al-)310 1972 y(gebras,)h(Comm.)c(Algebra)k(24)f(\(1996\),)g(307-329)224 2055 y([37])20 b(A.)e(Masuok)n(a:)g(Calculation)f(of)i(some)f(groups)h (of)f(Hopf)g(algebra)h(extensions,)310 2105 y(J.)13 b(Algebra)h(191)f (\(1997\),)g(568-588)889 2310 y(21)p eop %%Page: 22 22 22 21 bop 224 195 a Fm([38])20 b(S.)13 b(Mon)o(tgomery:)e(Hopf)i (Algebras)g(and)g(their)h(Actions)g(on)f(Rings,)f(CBMS)i(Re-)310 245 y(gional)c(Conf.)g(Ser.)i(in)f(Math.,)g(V)m(ol.)f(82,)h(Amer.)f (Math.)h(So)q(c.,)g(Pro)o(vidence,)h(1993)224 327 y([39])20 b(S.)g(Mon)o(tgomery:)e(Classifying)h(\014nite-dimensional)f (semisimple)g(Hopf)i(alge-)310 376 y(bras,)13 b(Preprin)o(t,)g(1997,)e (to)i(app)q(ear)g(in:)f(Con)o(temp.)f(Math.,)h(Pro)q(c.)i(of)e(the)h (AMS-)310 426 y(IMS-SIAM)21 b(Summer)e(Researc)o(h)k(Conference)f(on)f (\014nite-dimensional)e(alge-)310 476 y(bras,)14 b(Seattle)224 558 y([40])20 b(S.)13 b(Mon)o(tgomery/S.)f(J.)i(Withersp)q(o)q(on:)g (Irreducible)h(represen)o(tations)h(of)e(cros-)310 607 y(sed)g(pro)q(ducts,)h(J.)f(Pure)h(Appl.)e(Algebra,)g(in)h(press)224 689 y([41])20 b(S.)i(H.)g(Ng:)h(On)g(the)g(pro)r(jectivit)o(y)g(of)f (mo)q(dule)f(coalgebras,)i(to)f(app)q(ear)h(in:)310 739 y(Pro)q(c.)14 b(Amer.)f(Math.)g(So)q(c.)224 821 y([42])20 b(W.)c(D.)f(Nic)o(hols:)h(Quotien)o(ts)h(of)f(Hopf)h(algebras,)f(Comm.) d(Algebra)j(6)h(\(1978\),)310 870 y(1789-1800)224 952 y([43])j(W.)15 b(D.)g(Nic)o(hols:)h(Cosemisimpl)o(e)e(Hopf)i(algebras.) f(In:)h(Adv)n(ances)h(in)f(Hopf)f(al-)310 1002 y(gebras,)f(Dekk)o(er,)g (New)g(Y)m(ork,)f(1994)224 1083 y([44])20 b(W.)12 b(D.)g(Nic)o(hols/M.) g(B.)h(Zo)q(eller:)g(Finite)g(dimensional)e(Hopf)h(algebras)h(are)h (free)310 1133 y(o)o(v)o(er)g(grouplik)o(e)f(subalgebras,)g(J.)h(Pure)h (Appl.)e(Algebra)h(56)f(\(1989\),)g(51-57)224 1215 y([45])20 b(W.)14 b(D.)g(Nic)o(hols/M.)g(B.)h(Zo)q(eller:)g(F)m(reeness)i(of)d (in\014nite)h(dimensional)e(Hopf)h(al-)310 1265 y(gebras)g(o)o(v)o(er)g (grouplik)o(e)f(subalgebras,)h(Comm.)c(Algebra)k(17)f(\(1989\),)g (413-424)224 1346 y([46])20 b(W.)14 b(D.)g(Nic)o(hols/M.)g(B.)g(Zo)q (eller:)h(A)g(Hopf)f(algebra)h(freeness)i(theorem,)d(Amer.)310 1396 y(J.)f(Math.)h(111)f(\(1989\),)g(381-385)224 1478 y([47])20 b(W.)14 b(D.)g(Nic)o(hols/M.)g(B.)h(Ric)o(hmond:)e(F)m (reeness)k(of)d(in\014nite)h(dimensional)e(Hopf)310 1528 y(algebras,)g(Comm.)d(Algebra)k(20)f(\(1992\),)g(1489-1492)224 1609 y([48])20 b(W.)e(D.)g(Nic)o(hols/M.)g(B.)h(Ric)o(hmond:)d(The)k (Grothendiec)o(k)f(group)g(of)f(a)h(Hopf)310 1659 y(algebra,)13 b(J.)g(Pure)i(Appl.)e(Algebra)h(106)f(\(1996\),)g(297-306)224 1741 y([49])20 b(W.)15 b(D.)g(Nic)o(hols/M.)h(B.)g(Ric)o(hmond:)d(The)k (Grothendiec)o(k)g(algebra)e(of)h(a)g(Hopf)310 1791 y(algebra)d(I,)g (Comm.)e(Algebra)j(26)f(\(1998\),)g(1081-1095)224 1872 y([50])20 b(U.)13 b(Ob)q(erst/H.-J.)i(Sc)o(hneider:)818 1862 y(\177)813 1872 y(Ub)q(er)g(Un)o(tergrupp)q(en)h(endlic)o(her)e (algebraisc)o(her)310 1922 y(Grupp)q(en,)g(Man)o(uscripta)g(Math.)f(8)h (\(1973\),)f(217-241)224 2004 y([51])20 b(U.)14 b(Ob)q(erst/H.-J.)i(Sc) o(hneider:)g(Un)o(tergrupp)q(en)h(formeller)d(Grupp)q(en)h(v)o(on)g (end-)310 2054 y(lic)o(hem)d(Index,)i(J.)g(Algebra)f(31)h(\(1974\),)e (10-44)224 2135 y([52])20 b(B.)11 b(P)o(areigis:)f(Endlic)o(he)h (Hopf-Algebren,)g(Algebra-Ber.,)g(Uni-Druc)o(k,)f(Munic)o(h,)310 2185 y(1973)889 2310 y(22)p eop %%Page: 23 23 23 22 bop 224 195 a Fm([53])20 b(B.)12 b(P)o(areigis:)h(Lectures)h(on)f (quan)o(tum)e(groups)i(and)g(non-comm)o(utativ)n(e)e(geome-)310 245 y(try)m(,)i(Lecture)j(notes,)e(Univ.)f(M)q(\177)-22 b(unc)o(hen,)14 b(Munic)o(h,)g(1998)224 327 y([54])20 b(D.)c(E.)h(Radford:)f(A)i(free)g(rank)f(4)g(Hopf)g(algebra)g(with)g (an)o(tip)q(o)q(de)g(of)g(order)h(4,)310 376 y(Pro)q(c.)c(Nat.)f(Acad.) h(Sci.)g(USA)g(30)f(\(1971\),)g(55-58)224 458 y([55])20 b(D.)13 b(E.)i(Radford:)e(The)i(an)o(tip)q(o)q(de)f(of)g(a)g (\014nite-dimensional)e(Hopf)i(algebra)g(o)o(v)o(er)310 508 y(a)f(\014eld)h(has)g(\014nite)g(order,)h(Bull.)d(Amer.)h(Math.)h (So)q(c.)f(81)h(\(1975\),)e(1103-1105)224 589 y([56])20 b(D.)14 b(E.)h(Radford:)e(The)j(order)g(of)e(the)i(an)o(tip)q(o)q(de)f (of)f(a)h(\014nite-dimensional)e(Hopf)310 639 y(algebra)g(is)h (\014nite,)f(Amer.)g(J.)h(Math.)f(98)h(\(1976\),)e(333-355)224 721 y([57])20 b(D.)15 b(E.)h(Radford:)f(P)o(oin)o(ted)h(Hopf)g (algebras)g(are)h(free)g(o)o(v)o(er)g(Hopf)f(subalgebras,)310 771 y(J.)d(Algebra)h(45)g(\(1977\),)e(266-273)224 852 y([58])20 b(D.)12 b(E.)i(Radford:)e(Op)q(erators)j(on)e(Hopf)g (algebras,)g(Amer.)f(J.)i(Math.)f(99)g(\(1977\),)310 902 y(139-158)224 984 y([59])20 b(D.)c(E.)h(Radford:)f(F)m(reeness)j (\(Pro)r(jectivit)o(y\))f(criteria)g(for)e(Hopf)h(algebras)g(o)o(v)o (er)310 1034 y(Hopf)c(subalgebras,)h(J.)g(Pure)h(Appl.)e(Algebra)h(11)f (\(1977\),)g(15-28)224 1115 y([60])20 b(D.)f(E.)h(Radford:)e(On)i(the)h (an)o(tip)q(o)q(de)f(of)f(a)h(cosemisimple)d(Hopf)j(algebra,)f(J.)310 1165 y(Algebra)14 b(88)f(\(1984\),)g(68-88)224 1247 y([61])20 b(D.)14 b(E.)h(Radford:)e(Minimal)f(quasitriangular)i(Hopf)g(algebras,) h(J.)f(Algebra)h(157)310 1297 y(\(1993\),)d(285-315)224 1378 y([62])20 b(D.)13 b(Rum)o(ynin:)d(W)m(eak)j(in)o(tegral)g(forms)g (and)g(the)h(sixth)g(Kaplansky)f(conjecture,)310 1428 y(Preprin)o(t,)h(1998)224 1510 y([63])20 b(P)m(.)11 b(Sc)o(hauen)o (burg:)i(F)m(aithful)d(\015atness)k(o)o(v)o(er)e(Hopf)g(subalgebras)h (-)f(coun)o(terexam-)310 1560 y(ples,)h(submitted)f(to:)h(Pro)q (ceedings)h(of)f(the)h(conference)h(`In)o(teractions)e(b)q(et)o(w)o (een)310 1609 y(Ring)f(theory)j(and)f(Represen)o(tations)h(of)e (algebras',)g(Murcia,)g(Spain,)g(1998)224 1691 y([64])20 b(S.)d(Sc)o(hmidt-Samoa:)d(Ein)j(Quotien)o(t)h(der)h(Quan)o(tengrupp)q (e)g Ff(U)1335 1697 y Fe(q)1354 1691 y Fm(\()p Ff(sl)1401 1697 y Fd(2)1420 1691 y Fm(\))f(im)e(Ein-)310 1741 y(heitswurzelfall,)d (Diplomarb)q(eit,)e(Univ.)i(G\177)-21 b(ottingen,)13 b(G\177)-21 b(ottingen,)12 b(1995)224 1822 y([65])20 b(H.-J.)e(Sc)o(hneider:)i(Lectures)i(on)d(Hopf)f(algebras,)h(Univ)o (ersidad)g(de)h(Cordoba)310 1872 y(T)m(raba)r(jos)e(de)h(Matematica,)e (Serie)j(\\B",)e(No.)g(31/95,)f(Cordoba,)h(Argen)o(tina,)310 1922 y(1995)224 2004 y([66])i(H.-J.)28 b(Sc)o(hneider:)h(Zerlegbare)i (Un)o(tergrupp)q(en)f(a\016ner)f(Grupp)q(en,)g(Math.)310 2054 y(Ann.)13 b(255)g(\(1981\),)g(139-158)224 2135 y([67])20 b(H.-J.)15 b(Sc)o(hneider:)i(Normal)e(basis)h(and)g(transitivit)o(y)g (of)f(crossed)j(pro)q(ducts)g(for)310 2185 y(Hopf)13 b(algebras,)h(J.)f(Algebra)h(152)f(\(1992\),)g(196-231)889 2310 y(23)p eop %%Page: 24 24 24 23 bop 224 195 a Fm([68])20 b(H.-J.)13 b(Sc)o(hneider:)i(Some)e (remarks)h(on)g(exact)h(sequences)i(of)c(quan)o(tum)g(groups,)310 245 y(Comm.)d(Algebra)k(21)f(\(1993\),)g(3337-3357)224 326 y([69])20 b(H.-J.)13 b(Sc)o(hneider:)i(In)e(preparation)224 406 y([70])20 b(Y.)i(Sommerh\177)-21 b(auser:)21 b(On)i(Kaplansky's)f (\014fth)g(conjecture,)i(J.)f(Algebra)f(204)310 456 y(\(1998\),)12 b(202-224)224 537 y([71])20 b(Y.)d(Sommerh\177)-21 b(auser:)17 b(Ribb)q(on)g(transformations,)f(In)o(tegrals,)i(and)g(T)m(riangular) 310 587 y(Decomp)q(ositions,)7 b(Preprin)o(t)k(gk-mp-9707/5)o(2,)6 b(1997,)i(to)i(app)q(ear)g(in:)e(J.)i(Algebra)224 668 y([72])20 b(Y.)13 b(Sommerh\177)-21 b(auser:)12 b(Y)m(etter-Drinfel'd)i (Hopf)f(algebras)h(o)o(v)o(er)f(groups)h(of)g(prime)310 717 y(order,)g(in)f(preparation)224 798 y([73])20 b(D.)11 b(S)-21 b(\030)s(tefan:)11 b(The)i(set)g(of)e(t)o(yp)q(es)i(of)e (n-dimensional)f(semisimple)f(and)j(cosemisim-)310 848 y(ple)i(Hopf)f(algebras)h(is)g(\014nite,)f(J.)h(Algebra)g(193)f (\(1997\),)f(571-580)224 929 y([74])20 b(R.)9 b(Suter:)h(Mo)q(dules)g (o)o(v)o(er)g Ff(U)755 935 y Fe(q)773 929 y Fm(\()p Ff(sl)820 935 y Fd(2)840 929 y Fm(\),)f(Comm.)e(Math.)i(Ph)o(ys.)h(163)f (\(1994\),)f(359-393)224 1010 y([75])20 b(M.)9 b(E.)h(Sw)o(eedler:)g (In)o(tegrals)g(for)g(Hopf)g(algebras,)f(Ann.)h(of)f(Math.)g(\(2\))h (89)g(\(1969\),)310 1059 y(323-335)224 1140 y([76])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)224 1221 y([77])20 b(E.)11 b(J.)g(T)m(aft:)f (The)i(order)g(of)e(the)i(an)o(tip)q(o)q(de)g(of)e (\014nite-dimensional)f(Hopf)i(algebra,)310 1271 y(Pro)q(c.)j(Nat.)f (Acad.)h(Sci.)g(USA)g(68)f(\(1971\),)g(2631-2633)224 1352 y([78])20 b(E.)c(J.)h(T)m(aft/R.)e(L.)h(Wilson:)f(There)j(exist)g (\014nite-dimensional)c(Hopf)j(algebras)310 1401 y(with)c(an)o(tip)q(o) q(des)h(of)g(arbitrary)g(ev)o(en)g(order,)g(J.)g(Algebra)g(62)f (\(1980\),)g(283-291)224 1482 y([79])20 b(M.)15 b(T)m(ak)o(euc)o(hi:)g (A)h(corresp)q(ondence)j(b)q(et)o(w)o(een)e(Hopf)e(ideals)h(and)g (sub-Hopf)f(al-)310 1532 y(gebras,)f(Man)o(uscripta)g(Math.)f(7)h (\(1972\),)f(251-270)224 1613 y([80])20 b(M.)c(T)m(ak)o(euc)o(hi:)f(On) i(a)g(semidirect)f(pro)q(duct)i(decomp)q(osition)d(of)h(a\016ne)g (groups)310 1663 y(o)o(v)o(er)e(a)f(\014eld)h(of)f(c)o(haracteristic)j (0,)d(T^)-21 b(ohoku)13 b(Math.)h(J.)f(24)h(\(1972\),)e(453-456)224 1743 y([81])20 b(M.)g(T)m(ak)o(euc)o(hi:)g(F)m(ree)i(Hopf)f(algebras)g (generated)h(b)o(y)f(coalgebras,)g(J.)g(Math.)310 1793 y(So)q(c.)14 b(Japan)f(23)h(\(1971\),)e(561-582)224 1874 y([82])20 b(M.)10 b(T)m(ak)o(euc)o(hi:)f(Relativ)o(e)h(Hopf)g(mo)q (dules)f(-)i(equiv)n(alences)g(and)f(freeness)j(criteria,)310 1924 y(J.)g(Algebra)h(60)g(\(1979\),)e(452-471)224 2005 y([83])20 b(J.)13 b(G.)g(Thac)o(kra)o(y:)h(Mo)q(dular)f(represen)o (tations)j(of)d(some)g(\014nite)h(groups,)g(Disser-)310 2054 y(tation,)e(Univ.)h(Cam)o(bridge,)f(Cam)o(bridge,)g(1981)224 2135 y([84])20 b(W.)15 b(W)m(aterhouse:)i(In)o(tro)q(duction)f(to)h (a\016ne)f(group)g(sc)o(hemes,)h(Grad.)e(T)m(exts)i(in)310 2185 y(Math.,)c(V)m(ol.)f(66,)h(Springer,)h(Berlin,)f(1979)889 2310 y(24)p eop %%Page: 25 25 25 24 bop 224 195 a Fm([85])20 b(S.)d(J.)h(Withersp)q(o)q(on:)f(The)h (represen)o(tation)i(ring)d(and)h(the)g(cen)o(tre)h(of)e(a)h(Hopf)310 245 y(algebra,)13 b(Preprin)o(t,)h(1998)224 328 y([86])20 b(Y.)12 b(Zh)o(u:)h(Quan)o(tum)e(double)i(construction)h(of)e (quasitriangular)f(Hopf)i(algebras)310 378 y(and)g(Kaplansky's)h (conjecture,)h(Preprin)o(t,)f(1993)224 461 y([87])20 b(Y.)e(Zh)o(u:)g(Hopf)f(algebras)i(of)e(prime)g(dimension,)f(In)o (ternat.)j(Math.)f(Res.)g(No-)310 511 y(tices)d(1)e(\(1994\),)g(53-59) 224 594 y([88])20 b(Y.)9 b(Zh)o(u:)h(A)g(comm)o(uting)d(pair)i(in)h (Hopf)f(algebras,)h(Pro)q(c.)g(Amer.)f(Math.)g(So)q(c.)h(125)310 643 y(\(1997\),)i(2847-2851)224 726 y([89])20 b(M.)12 b(B.)g(Zo)q(eller:)h(F)m(reeness)h(of)e(Hopf)g(algebras)h(o)o(v)o(er)f (semisimple)e(grouplik)o(e)i(sub-)310 776 y(algebras,)h(J.)h(Algebra)g (118)f(\(1988\),)f(102-108)889 2310 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF