%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: tensor.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips tensor.dvi -o tensor.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.05.02:1439 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginFont: XYDASH10 %!PS-AdobeFont-1.1: XYDASH10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Aug 28 05:34:12 %%RevisionDate: 1997 Sep 18 10:23:31 % % XYDASH10: line segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYDASH10) readonly def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch 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1ac6cf81a8eb952a043b1340a7fc883b28638897d15bb4c394d70df7df3d1312 2629b5a8c236d9e91fe466b264d6e5018581bb79e23dd527875cb97ef7962d44 2fb47682afcf9a3869ef323af9fec2d18e8a3613c10d546970216927e6740be8 0e272580db4cdc1b8fece17f94fc78640294a0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYBTIP10 %!PS-AdobeFont-1.1: XYBTIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 46 /d46 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 110 /d110 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092839 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139c8eef1237e57525cbc853d7cbe3c9a8b54c378e8af02257a8daa736c3d9ae fb18fd198a33681c334984d81e2d783d32adf54549f5bea0bf351b1016032908 81685bde8d44703654d97063c8ebfb896e029b2383f5754d467163ec07f3398a d88196c720fd98b9a2260de8d7d3aa6453f831ce18233cfbf6cb098bc3ca2cd1 495386a279ced386537228ec08f3b3e400cc040ab2e763b0cd93c9a2c5ee0436 f0a2f033ba5d3e4231aacc9b0aa820f7ad72a3cec593a1153ee5527693ad3bc8 eaef55ac2f52fdf27146c04dcc825181a275e632e75a94cb9b3d3f7d17c1c08b 83bbf5c681f864e234d10b0f7c64839aa1671931f39a001e4134030b91d9a473 6c7d5e101e04feb20a04907ab46ab24902c1844b018beefd9014c8b629674e57 f1f0d63ad79dfa8ce4d1fffabeb4315386d494a3ab66cc9f291a714ef0ee4f9f 1687f0ecbcd2acea0e98dd5f94dfd700e546599e58d1f25bc54ef6ec0f12b91e 6690287b7c527a51724cea71da655f2b2974633ba5484cc6c2300ba28dff89e2 0c37542986ec1e4613cc8a16521e5c2720d88fa18111a1083dcee82b011400a2 8b4124ab1a5bdde460e2589f2872b0436396df646b0eb176e75de9af54d7c4ab f628a596d7e1ead5815ec6bb58786913f0125dbe4a6328ea358185ce03fdf5b2 8d3e5cee90066a67f548590d69d197b1503ae8f993a1a7aa4248f0be3be623c0 fe29e1772ade5b00f22b228152d197d6b3e1ddf4dece5c7cbadfec3e7bb38696 91becf5079caf4c910b4a25a1b19e3c64eeb79e5223d56f7aae7411259fda2b2 2e2a4652323e63b97e76d22ca5fa800398fb0b4636201037f11794493ce10d4a 80fc85a0a26686c096d560a3a77cb2cf4fdcd105e98e7e88454412c6a107c5dc 7605eef7932c093879753c20b5397cdeba78d0a798f789304d63956f0f471a10 ad93e8ade240e9185f8028420ecc26e436fcff5421bcd39aeb8ac91800241d34 133cc54e557b68947337eb889fb494948d932519e0e19f3dd891220c04935f13 96f8196a907180e760be47484d144c1039ee7934d0f08397e9b640062ffd670c 1c7fc029acbe5f5c7982e68820e9140313eb390f785901879c614f5461f3e0e6 908f053b64f7f0ec48d567b4125934d21158a2ba50e361b6966e857922618b72 06d412a748706d7eac5fff41ee5c52f4142fce1ce9ba67a3c97ec186b187b80c 41b8ee73ba38e3bf2bc6741de19ef652832faba07c55e7bd7ae5fb49d86b1808 8bb5e80ed536552a211ce36d2961583aff648edd54f1de84ae741b98bbd57e37 577e55ca0f97c4ac1fed15732a0a072a176b0c5f14dbcee98f6d60fe3b3df180 2d93213c7eb7a37ff92ff4f97619596aa0771d0cbc82c5cf55b5be4f44fba75e 631ae71cbc9afbc42312cecdd86b7aec3152ddcf7fa1c6f5e051581597864746 2e975089ad928df03a4073bf06bfe1e262a551f24a12bca29408aacf7ed2e43a 3b7f7ff8e40af2ad2b131622058405cdf249568c2653b0be457fed5352e113fc 1107d6ed67a6ffeef5608109c13e5d5bc2432e65b82d3c3fe6d012e7a84e7730 13ae1201e1ad5ac9c2c11163be1d85f342956954f92651fcde7efbd58975fd01 37afbc60f68065b6b4af9856ef3a425a57a25b3ce8e5375aeccfc1146b78fb7e 6aa315ea438e2826d160cb12ae97fbd9c4704114f4269ae8ed6a114882c3fe34 9aa90531189c9e3b9fd1bb8f1772fdd681245ccfcb8b42165e018ad8af8f2440 954f0390129495317eab98a4d43da3e8b81c4269b19e4069c61c0ecb28a57873 610c578517a8e08f9762dce236f12bdb99718602cba86a56166ddcd42497ebf9 5b52ebedd9f09a6109ae30158fbe7d998aff28df96cb38f706e1b05b685c7df4 8410ed3d5558666c7c85dbae18e6e50330f7fd850d55e2c7a2277fa9fc2372b5 5e8102cd124121243810279c8d7773e43cc5233bdab4d5379ae5bf2f2ef82d78 9621ea5c345dba373f989f82a3f1d6edeb69dc7c9abb0de83996d732589a642c b63d80a8dd4420f3593380d56dabe56ffbf30f6d1bc7b8e3d8da03de5eb33aca 4b1dd4b2ae4b385c05b16d65d336149a68d790bfe240b23f6d38c19945037652 4b54fc9e0a60df4b23b9054d0c05b7081ed97195ef3fa26beecf5ea3fe6cae8b ebea88ed339f7e59494275fa923c8e6d1f6d34d82dd65764941e4578de48351c b2bdd01bcb8f95fbd8adc59d14b5212aa0f4243d1284c6c9325eb8331f11a031 c3c72668b913f9dc2b818ffa189229145614d11cc645beb56e006f2eddb395be f40407977ed4c2018231bdb1b3975e3471c917838f156ce50b6b7881cd5a924e f908a4246961874932ef7694a89836e8b2bdccf9d5e37465d72a051b28eacaa9 0c0499a2caf6f048e16a2d538f93db84d5aa1f5ca2bf5f59e688483eb6f7b76b 86fa39f5bb91ac2cb449490d02cacfae6eeda6cc9f81dd1f23f75dfaa147ed8a 8409b1df8fc6ab93b5e0faa71e6ccd142f1d922463b33e72adb48edb72841e71 6e736726f7502bb9acc792946494e722e21bf0fd127e18a5a841c4eb8e5da923 4b33a1ca992a972d9e9be9ab9ac03490a1ac894a14d0d968505d2e85e031c7ba 214a73d474109de5438f3d6c6fcf7d7c3328c23bb891d9a032b826499fef9f6a 6d2f4b931dac8a9e52f0cdc71237f90c0cc8d3ea2b77300e8eaf805a1c96a531 29116f4d980e6b5d2255be1b414827ec5f1c704957686a2a15eb97163c52aa2e 7ff983316875f3254b75360a5eb964fbbe2512639537c9d747ec797438cc6a1f c29c4ece2df509aca9f4432f2802404525ec3ccdd00c16d5e4b1d988e05a32d0 b3109c438bb4d96461ec7a01f31397d05613041ce4e7a058c76db266c85186ea c120fc8ded67831b7b6eaedc2697dbe21aa0be8c23376566c5c2368571454153 c6376da88f348eba5f988877fe62b7173a34031cf1a438f1d0681e252ba36789 ff08228ba054542b40c4d51023bbcc575acd9b5e48212cbdd008e48b5b455ae4 5e8f7abaac5079262c2f68dab05f1a2bfaf0c701cd9f6212786d56080b547abe de0a0c45b69b6735611e712287a2a2dd7c48bd41b61d5a521465d0056ad19950 f3b69224cceb5e44f4434014634678fad7c159c53fe7f1641358a043c397a25d dfcaaa727be1061288fd97407d8ca2e249e551e25cd28a4013136d80e56a9f03 7b1cf2a97c9d3e555c87c7442f038cda997d27c059816266af66e5af51445134 21ba39ffb81e96b689e4b1e8e6a31a64146f292793e8eef0390cb8cb8ef6ee21 95ffc8dc3707c1b5edd3e55958bafd54976526eaa15c355b4ff695b5dea821e4 78100344d5aa80c8968e88f8084a6c3d4f8498773f922ca08b95e293f39a66c0 7cf7a6471e41f1d213e63b2fc4ed95aaaff8523974de592f8781313988ee1fa1 da857e896e84708b791ce58ef30be9ab9d02ced5d8d6a72833dffeb44db429e3 a9c41e2c05da263b966233f2f52870aff1f5814a7460361ebc8146d9addcde2a 44347bd9eaeffcc9761fdc17a89ea5e8f445f95006e017e9276e079f74c8df0e c40a1c57cd36402eca43f744fa2586a1159bebcb5cdfb5cc7d85a50dcfa3a62d 3ee6c5bfb1867e002e8aa07a2acf3e66b535885f1e226169f48a0128d65481cf dcb8c0c147570b40a573595e69a63d1b7ea70ea2512bf2a3ec433d10ccbd63fb 692e642cb7397d926de8e29fa5320505e0023dc37be3ed06cc5d08b0efd62382 92c8a6fafb49304e2bb94b5a9603c34c7f30fe91bc74b9ba972013ac477ccfac 93541acea4d89fd121931f93bedede6bca4bb76959c2bae0655474de2364b0f2 3f0837b0dfb1ffd40fef2cad7cbd9f4e483227ec15a96a23badbf56d9c46a0b6 71771274545d4e81f830000c86c88934f4eda292bd7f13d93f158f446364831a 887ec603d4bb1308fed4f6effe728ed2460280640addf145df15c70d87970054 f86abf37ce50926a7d7f3a8dafb63ed484128a581d72a90c9ead1890f6551444 8fb113f29934bd2da3fc2a33a83bbcb5ea207c71cad63b05f82d1a55fc81280d 548451d254f5df3c97cd3bf42181bb306f365f419cf90ce7778ff5ffea51ea09 4fab9198f70afaac4dd6a703065f723b7d6e53db961c9699bfe7d79280e2f62f 4a05d7a1c0bbe55ae3be7e851f8bdc5206de3f6f1ea55f26248ca6fbdb061417 eb7689dd9d59d4a6d2e42d8a40b606ff7d4ff5fe0b6f6bf498528e0693b398f6 c74708bcec71b15d2cea5140321727a091677ac067f787001ea089276558c727 51a1e8644e499199d2f896154a8275d8b9b01aa32dec10a0f0196e4474e4b188 ab9a89e29a8edf5c449d7c25ea1091646d2c246599e5dcc4f0167f7ee1081ad8 c267205b3b256b96a3b54281236eb39a700fa43acf8517abc72a845bc110e1ec 153fc374710975248e3401bc8d1e66458fa2198d8885374f1feb57e7f77bf4ab 5edec4d089a223380cdd01d9ce39fdaf30ea1e3512db78a713a40159e3f7f578 fc119c61b32b73c6de5dbdf2899d6e9ccfd5c563887bc57122bc0dbac2091844 6b1ab98b5786c8f18477cb06b0df72dddc8c343deea9161c526b7210c5397fe9 faf8d92b0f886b1cda38491d477c1c8d082a8b542e505c6a8bc5bed72b14f1c3 9f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYATIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 102 /d102 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092838 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139d5cb0db6f50b26bd8ab173859c9c94db82970311d7eb0a02bee1be5f0d126 f9079e67107eda14b460e46b03b0422eb45a4f4afa382841c35b0bc0c8639b73 43c819afc69838ce781c2de7d22bf503ef6eec27c83cfd52a77bbc754b4f2050 55341700991f3ef4b5b5a54c21283034b38c8b6a6e65abccc94c0c26836806ec 4df8d8c64f595841395072f8a2d289b7fe5497bc0e061810b16a1e48653bb092 42ffa3ef0bba4e37a5aa4ddc31f3138aaf10998fd66f3817b77012eac677ed2d 7447908c771fbaba4dfcfeca5374a6b87e5657809a82f0ae068c5384c12f4653 2a82645a512212140d815e80ec76b3370c382e9f6d29ec6afc178a622249fda7 775d4f6a554f04748ed4210257fa6e376188a175db3c00f88421820b063a985c 07ad665eff7e2d32a27015c528227c2805aa8df134f4abad9958b841aa4263ec 9ec6d907308b0a5a51049de002cfe60ef35bf33fc863ba14ff361749554abd65 47426fbf3958ddb506ea3e303c932edec2896d3017a57913cde60cbdafeddbd3 cd5ef6dd49299783a92fe9deafb24e7f74a6ca6c0198b2fcda46b51445a2800f d1dc6a092b3192cfb314892e52753b5c7b94edd34c8213b032f8aab5d08753cb 65bd1a1225ba43194efa625ee5dc6eec6d0353d06ef3bef9c0d7df78fe189482 779c9276e83ccd71b50e87ba92cf092d65498e5b43cdc89436019e306c0d628e c7d1470ab322ca2d6adc3e9ff4196c0f7792ca0b20f741ad7bbd4391b0218511 14f0e97a44fd7d03e7003d1fe2c70d6266740dad7b07b3794dc53871887c8eba 2910d6fa654346318753cc752a7d25c1ef970e7dda8d8bbe249e9edf2c8c2ae6 abb93bab2dd8560466a7d08ce3e200d7cc488c6045871a8033ec1dc3a53e7056 3fe265f58c6ac98754d399b47b817b7aed7ebbbb503268e7324d6c0a0931978b 51a7d37187a99f234ae4c46b585383938e7cf59d2dfd20717031966eaa4317e7 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8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /Times-Italic /Times-Italic-iso isovec ReEncode /Helvetica /Helvetica-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2512 m -1000 -1000 l 3808 -1000 l 3808 2512 l cp clip 0.06000 0.06000 sc /Helvetica-iso ff 120.00 scf sf 1920 705 m gs 1 -1 sc (-1) col-1 sh gr 7.500 slw % Ellipse n 1935 720 150 150 0 360 DrawEllipse gs col-1 s gr % Ellipse n 375 735 120 120 0 360 DrawEllipse gs col-1 s gr /Symbol ff 180.00 scf sf 330 795 m gs 1 -1 sc (q) col-1 sh gr /Symbol ff 180.00 scf sf 2560 780 m gs 1 -1 sc (q) col-1 sh gr /Helvetica-iso ff 120.00 scf sf 2635 705 m gs 1 -1 sc (-1) col-1 sh gr % Ellipse n 2650 720 150 150 0 360 DrawEllipse gs col-1 s gr % Arc gs n 809.0 730.3 461.2 -68.9 163.4 arc gs col-1 s gr gr % Arc gs n 789.0 715.5 430.9 -166.0 -105.3 arc gs col-1 s gr gr % Polyline n 750 1125 m 750 75 l gs col-1 s gr % Polyline n 900 1125 m 900 75 l gs col-1 s gr % Polyline n 900 1260 m 900 1500 l gs col-1 s gr /Symbol ff 180.00 scf sf 1845 780 m gs 1 -1 sc (q) col-1 sh gr % Polyline n 750 1260 m 750 1500 l gs col-1 s gr /Times-Italic-iso ff 180.00 scf sf 2700 1500 m gs 1 -1 sc (k) col-1 sh gr % Polyline n 790 285 m 855 285 l gs col0 s gr % Polyline n 2365 165 m 2625 75 l gs col-1 s gr % Polyline n 2350 1270 m 2351 1270 l 2354 1268 l 2362 1263 l 2375 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FQ(\(3.1.1\))29 b FO(and)1751 1923 y FJ(t)1781 1935 y FI(ij)1862 1923 y FQ(=)23 b FJ(\016)1987 1935 y FI(ij)2046 1923 y FJ(\022)2085 1935 y FI(i)2112 1923 y FJ(;)-1679 b FQ(\(3.1.8\))1751 2052 y FJ(c)1787 2064 y FI(ij)1868 2052 y FQ(=)23 b FJ(\016)1993 2064 y FI(ij)2046 2048 y Fx(\003)2086 2052 y FJ(:)-1653 b FQ(\(3.1.9\))456 2217 y FO(Then)30 b(we)g(have)6 b FQ(:)1547 2380 y(\()s(~)-45 b FJ(s)q(t)p FQ(\))1681 2346 y FM(3)1741 2380 y FQ(=)23 b FJ(p)1871 2346 y FM(+)1929 2380 y FQ(~)-45 b FJ(s)1965 2346 y FM(2)2002 2380 y FJ(;)-1569 b FQ(\(3.1.10\))1458 2519 y(\()s(~)-45 b FJ(st)1559 2485 y FE(\000)p FM(1)1649 2519 y FQ(\))1681 2485 y FM(3)1741 2519 y FQ(=)23 b FJ(p)1871 2485 y FE(\000)1930 2519 y FQ(~)-45 b FJ(s)1966 2485 y FM(2)2003 2519 y FJ(c;)-1606 b FQ(\(3.1.11\))1453 2658 y FJ(ct)23 b FQ(=)f FJ(tc;)85 b(c)s FQ(~)-45 b FJ(s)23 b FQ(=)j(~)-45 b FJ(s)o(c;)99 b(c)2221 2624 y FM(2)2281 2658 y FQ(=)23 b(1)p FJ(;)-1978 b FQ(\(3.1.12\))456 2856 y FO(wher)l(e)30 b FJ(p)732 2826 y FE(\006)818 2856 y FO(ar)l(e)g(de\014ne)l(d)g(by)37 b FQ(\(3.1.7\))o FO(.)i(Mor)l(e)l (over,)32 b(when)h FQ(~)-45 b FJ(s)29 b FO(is)i(invertible,)g(we)f (have)1645 3046 y FQ(~)-45 b FJ(s)1681 3012 y FM(2)1741 3046 y FQ(=)23 b FJ(p)1871 3012 y FM(+)1926 3046 y FJ(p)1968 3012 y FE(\000)2024 3046 y FJ(c:)-1627 b FQ(\(3.1.13\))605 3217 y FP(Pr)n(oof.)41 b FQ(The)21 b(fact)g(that)h FJ(c)f FQ(comm)n(utes)g(with)k(~)-45 b FJ(s)21 b FQ(and)g FJ(t)g FQ(follo)n(ws)g(from)f(\(3.1.3\))h(and)g(\(2.4.5\))o(;)456 3317 y(and)36 b FJ(c)662 3287 y FM(2)737 3317 y FQ(=)i(1)e(b)r(ecause)g FJ(i)1263 3287 y FE(\003\003)1373 3317 y FQ(=)h FJ(i)p FQ(.)64 b(T)-7 b(o)36 b(pro)n(v)n(e)f(the)i(non-trivial)e(relations)g (\(3.1.10,)j(3.1.11\),)456 3416 y(consider)26 b(\014rst)i(the)f(iden)n (tit)n(y)1242 4263 y @beginspecial 0 @llx 0 @lly 170 @urx 91 @ury 1700 @rwi @setspecial %%BeginDocument: figures/pftmtc1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: pftmtc1.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 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b(is)h(one)f(of)h(the)g(reasons)d(wh)n(y)j(suc)n(h)f (CFTs)h(are)e(called)h FO(r)l(ational)p FQ(.)605 885 y(One)g(can)h(also)e(easily)h(pro)n(v)n(e)f(the)i(follo)n(wing)e (result.)605 1046 y FP(Theorem)32 b FQ(3.1.21)p FP(.)39 b FO(A)n(l)t(l)30 b(the)g(numb)l(ers)f FJ(s)1933 1058 y FI(ij)1991 1046 y FJ(=s)2072 1058 y FM(0)p FI(j)2163 1046 y FQ(=)d(~)-45 b FJ(s)2290 1058 y FI(ij)2348 1046 y FJ(=d)2433 1058 y FI(j)2497 1046 y FO(ar)l(e)30 b(algebr)l(aic)i (inte)l(gers.)605 1207 y FP(Pr)n(oof.)41 b FQ(By)22 b(V)-7 b(erlinde)23 b(form)n(ula)f(\(3.1.26\))o(,)i(these)e(n)n(um)n(b)r(ers)h (are)f(the)h(eigen)n(v)-5 b(alues)21 b(of)i(the)456 1307 y(matrix)k FJ(N)793 1319 y FI(i)848 1307 y FQ(with)h(in)n(teger)e(en)n (tries.)p 3384 1307 4 57 v 3388 1254 50 4 v 3388 1307 V 3437 1307 4 57 v 934 1534 a FK(3.2.)46 b(Example:)40 b(Quan)m(tum)31 b(double)g(of)h(a)g(\014nite)g(group)605 1684 y FQ(W)-7 b(e)27 b(will)h(giv)n(e)e(the)h(simplest)g(example)g(of) 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FQ(is)e(a)f(Hopf)h(algebra)e(with)i(a)f FJ(k)s FQ(-basis)f FL(f)p FJ(x)p FL(g)2739 2393 y FI(x)p FE(2)p FI(G)2905 2381 y FQ(and)894 2536 y(m)n(ultiplication)516 b FJ(x)19 b FL(\012)f FJ(y)26 b FL(7!)d FJ(xy)s(;)180 b(x;)14 b(y)26 b FL(2)e FJ(G;)894 2666 y FQ(unit)874 b FJ(e)166 b FQ(\(the)28 b(unit)h(elemen)n(t)e(of)h FJ(G)p FQ(\))p FJ(;)894 2795 y FQ(com)n(ultiplication)437 b(\001\()p FJ(x)p FQ(\))25 b(=)d FJ(x)d FL(\012)f FJ(x;)181 b(x)23 b FL(2)g FJ(G;)894 2925 y FQ(counit)795 b FJ(")p FQ(\()p FJ(x)p FQ(\))24 b(=)f(1)p FJ(;)894 3064 y FQ(an)n(tip)r(o)r(de)705 b FJ(\015)5 b FQ(\()p FJ(x)p FQ(\))24 b(=)f FJ(x)2233 3029 y FE(\000)p FM(1)2322 3064 y FJ(:)456 3224 y FQ(This)41 b(Hopf)h(algebra)d(is)i(co)r(comm)n(utativ)n(e.)77 b(A)42 b(represen)n(tation)e(of)h FJ(k)s FQ([)p FJ(G)p FQ(])g(is)h(the)f(same) g(as)456 3323 y(a)c(represen)n(tation)g(of)h FJ(G)p FQ(.)68 b(By)38 b(Masc)n(hk)n(e's)e(theorem,)k(the)f(category)d FL(R)p FJ(ep)2903 3335 y FI(f)2960 3323 y FJ(k)s FQ([)p FJ(G)p FQ(])i(of)g(\014nite)456 3423 y(dimensional)27 b(represen)n(tations)e(is)j(semisimple.)605 3522 y(The)k(Hopf)f (algebra)f(dual)h(to)h FJ(k)s FQ([)p FJ(G)p FQ(])g(is)f(isomorphic)f (to)i(the)f(function)i(algebra)c FJ(F)12 b FQ(\()p FJ(G)p FQ(\))33 b(of)456 3622 y(the)28 b(group)e FJ(G)p FQ(.)37 b(It)28 b(has)f(a)g FJ(k)s FQ(-basis)g FL(f)p FJ(\016)1623 3634 y FI(g)1661 3622 y FL(g)1703 3634 y FI(g)r FE(2)p FI(G)1865 3622 y FQ(consisting)g(of)g(delta)h(functions:)1387 3855 y FJ(\016)1424 3867 y FI(g)1462 3855 y FQ(\()p FJ(x)p FQ(\))c(=)f FJ(\016)1722 3867 y FI(g)r(;x)1841 3855 y FQ(=)1929 3713 y Fy(\()1996 3798 y FQ(1)82 b(for)46 b FJ(g)25 b FQ(=)e FJ(x;)1996 3918 y FQ(0)82 b(for)46 b FJ(g)25 b FL(6)p FQ(=)e FJ(x:)456 4080 y FQ(It)28 b(has)790 4234 y(m)n(ultiplication)412 b FJ(\016)1744 4246 y FI(g)1783 4234 y FJ(\016)1820 4246 y FI(h)1886 4234 y FQ(=)22 b FJ(\016)2010 4246 y FI(g)r(;h)2108 4234 y FJ(\016)2145 4246 y FI(g)2183 4234 y FJ(;)180 b(g)s(;)14 b(h)22 b FL(2)i FJ(G;)790 4364 y FQ(unit)770 b(1)23 b(=)1860 4302 y Fy(P)1947 4389 y FI(g)r FE(2)p FI(G)2096 4364 y FJ(\016)2133 4376 y FI(g)2172 4364 y FJ(;)790 4496 y FQ(com)n(ultiplication)333 b(\001\()p FJ(\016)1845 4508 y FI(g)1884 4496 y FQ(\))24 b(=)2027 4433 y Fy(P)2115 4521 y FI(g)2147 4529 y FF(1)2180 4521 y FI(g)2212 4529 y FF(2)2245 4521 y FM(=)p FI(g)2348 4496 y FJ(\016)2385 4508 y FI(g)2417 4516 y FF(1)2472 4496 y FL(\012)18 b FJ(\016)2592 4508 y FI(g)2624 4516 y FF(2)2661 4496 y FJ(;)180 b(g)26 b FL(2)d FJ(G;)790 4627 y FQ(counit)691 b FJ(")p FQ(\()p FJ(\016)1815 4639 y FI(g)1854 4627 y FQ(\))23 b(=)g FJ(\016)2034 4639 y FI(g)r(;e)2124 4627 y FJ(;)790 4757 y FQ(an)n(tip)r(o)r(de)601 b FJ(\015)5 b FQ(\()p FJ(\016)1824 4769 y FI(g)1863 4757 y FQ(\))23 b(=)g FJ(\016)2043 4773 y FI(g)2077 4756 y Fx(\000)p FF(1)2159 4757 y FJ(:)456 4917 y FQ(A)33 b(represen)n(tation) e(of)i FJ(F)12 b FQ(\()p FJ(G)p FQ(\))33 b(is)g(the)g(same)f(as)g(a)h FJ(G)p FQ(-graded)e(v)n(ector)h(space)g(\(since)h FL(f)p FJ(\016)3230 4929 y FI(g)3268 4917 y FL(g)3310 4929 y FI(g)r FE(2)p FI(G)456 5016 y FQ(are)26 b(pro)5 b(jectors\).)605 5116 y(Applying)31 b(Drinfeld's)f(quan)n(tum)h(double)f(construction)f ([)p FK(Dr3)q FQ(])h(it)h(is)f(easy)g(to)g(describ)r(e)456 5216 y(explicitly)e(the)h(quan)n(tum)f(double)g FJ(D)r FQ(\()p FJ(G)p FQ(\))h(of)f FJ(k)s FQ([)p FJ(G)p FQ(].)40 b(As)28 b(a)g(v)n(ector)f(space,)h FJ(D)r FQ(\()p FJ(G)p FQ(\))d(=)e FJ(F)12 b FQ(\()p FJ(G)p FQ(\))20 b FL(\012)3404 5228 y FI(k)p eop %%Page: 60 14 60 63 bop 456 226 a FM(60)838 b(3.)29 b(MODULAR)g(TENSOR)g(CA)-5 b(TEGORIES)456 425 y FJ(k)s FQ([)p FJ(G)p FQ(].)37 b(It)28 b(is)f(a)g(Hopf)h(algebra)e(with)574 606 y(m)n(ultiplication)197 b(\()p FJ(\016)1345 618 y FI(g)1403 606 y FL(\012)18 b FJ(x)p FQ(\)\()p FJ(\016)1634 618 y FI(h)1696 606 y FL(\012)g FJ(y)s FQ(\))23 b(=)g FJ(\016)2003 618 y FI(g)r(x;xh)2175 606 y FQ(\()p FJ(\016)2244 618 y FI(g)2301 606 y FL(\012)18 b FJ(xy)s FQ(\))p FJ(;)181 b(x;)14 b(y)s(;)g(g)s(;)g(h)22 b FL(2)h FJ(G;)574 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1138 y FL(\012)18 b FJ(x)2178 1104 y FE(\000)p FM(1)2267 1138 y FJ(:)456 1349 y FQ(The)27 b(Hopf)h(algebra)e FJ(D)r FQ(\()p FJ(G)p FQ(\))j(is)e(quasitriangular)e(with)574 1554 y(R-matrix)370 b FJ(R)24 b FQ(=)1451 1492 y Fy(P)1538 1579 y FI(g)r FE(2)p FI(G)1673 1554 y FQ(\()p FJ(\016)1742 1566 y FI(g)1800 1554 y FL(\012)18 b FJ(e)p FQ(\))g FL(\012)g FQ(\(1)g FL(\012)g FJ(g)s FQ(\))p FJ(:)456 1742 y FQ(\(Of)26 b(course,)g(once)g (w)n(e)g(kno)n(w)g(the)h(ab)r(o)n(v)n(e)e(form)n(ulas,)h(they)g(can)g (b)r(e)h(easily)f(c)n(hec)n(k)n(ed)f(directly)-7 b(.\))605 1842 y(Note)40 b(that)g FJ(F)12 b FQ(\()p FJ(G)p FQ(\))41 b(and)e FJ(k)s FQ([)p FJ(G)p FQ(])h(em)n(b)r(ed)g(in)g FJ(D)r FQ(\()p FJ(G)p FQ(\))h(as)e FJ(k)s FQ(-algebras)e(and)j FJ(D)r FQ(\()p FJ(G)p FQ(\))h(is)f(their)456 1942 y(semidirect)27 b(pro)r(duct:)1549 2122 y FJ(D)r FQ(\()p FJ(G)p FQ(\))d(=)f FJ(F)12 b FQ(\()p FJ(G)p FQ(\))19 b FH(o)f FJ(k)s FQ([)p FJ(G)p FQ(])p FJ(;)-1881 b FQ(\(3.2.1\))456 2304 y(with)1378 2484 y FJ(x\016)1462 2496 y FI(g)1501 2484 y FJ(x)1548 2450 y FE(\000)p FM(1)1661 2484 y FQ(=)22 b FJ(\016)1785 2500 y FI(xg)r(x)1895 2484 y Fx(\000)p FF(1)92 b FQ(for)45 b FJ(g)s(;)14 b(x)23 b FL(2)h FJ(G:)-2066 b FQ(\(3.2.2\))605 2670 y(Let)32 b FL(R)p FJ(ep)909 2682 y FI(f)952 2670 y FJ(D)r FQ(\()p FJ(G)p FQ(\))h(b)r(e)f(the)g(category)e(of)i(\014nite) g(dimensional)g(represen)n(tations)e(of)i FJ(D)r FQ(\()p FJ(G)p FQ(\))456 2770 y(as)d(a)g FJ(k)s FQ(-algebra.)40 b(By)30 b(the)f(ab)r(o)n(v)n(e)g(remarks,)f(a)h(represen)n(tation)f FJ(V)49 b FQ(of)29 b FJ(D)r FQ(\()p FJ(G)p FQ(\))i(is)e(the)h(same)f (as)456 2870 y(a)34 b FJ(G)p FQ(-mo)r(dule)h(with)g(a)f FJ(G)p FQ(-grading)f FJ(V)53 b FQ(=)1797 2807 y Fy(L)1890 2894 y FI(g)r FE(2)p FI(G)2038 2870 y FJ(V)2086 2882 y FI(g)2160 2870 y FQ(satisfying)34 b FJ(xV)2632 2882 y FI(g)2706 2870 y FL(\032)g FJ(V)2853 2885 y FI(xg)r(x)2963 2869 y Fx(\000)p FF(1)3045 2870 y FQ(,)j FJ(x;)14 b(g)37 b FL(2)e FJ(G)p FQ(.)456 2976 y(In)d(other)g(w)n(ords,)h(ob)5 b(jects)32 b(of)h FL(R)p FJ(ep)1594 2988 y FI(f)1637 2976 y FJ(D)r FQ(\()p FJ(G)p FQ(\))g(are)f(\014nite)h(dimensional)f FJ(G)p FQ(-equiv)-5 b(arian)n(t)32 b(v)n(ector)456 3076 y(bundles)c(o)n(v)n(er)d FJ(G)p FQ(.)38 b(W)-7 b(e)28 b(will)g(sho)n(w)e(that)i(the)g(category)e FL(R)p FJ(ep)2378 3088 y FI(f)2421 3076 y FJ(D)r FQ(\()p FJ(G)p FQ(\))j(is)e(semisimple)h (and)f(will)456 3176 y(describ)r(e)g(its)h(simple)f(ob)5 b(jects.)605 3275 y(F)-7 b(or)27 b FJ(V)42 b FL(2)23 b FQ(Ob)14 b FL(R)p FJ(ep)1198 3287 y FI(f)1241 3275 y FJ(D)r FQ(\()p FJ(G)p FQ(\))28 b(and)g FJ(v)e FL(2)d FJ(V)47 b FQ(the)28 b(submo)r(dule)g(generated)e(b)n(y)i FJ(v)i FQ(is)1108 3476 y FJ(D)r FQ(\()p FJ(G)p FQ(\))p FJ(v)d FQ(=)1468 3397 y Fy(X)1463 3575 y FI(g)r FE(2)p FI(G)1608 3476 y FJ(k)s FQ([)p FJ(G)p FQ(])p FJ(\016)1802 3488 y FI(g)1840 3476 y FJ(v)g FQ(=)2000 3397 y Fy(X)1994 3575 y FI(g)r FE(2)p FI(G)2209 3397 y Fy(M)2139 3579 y FI(xg)r(x)2249 3563 y Fx(\000)p FF(1)2326 3579 y FE(2)p 2371 3546 35 3 v FI(g)2419 3476 y FJ(xZ)6 b FQ(\()p FJ(g)s FQ(\))p FJ(\016)2673 3488 y FI(g)2712 3476 y FJ(v)s(;)456 3751 y FQ(where)p 696 3705 43 4 v 27 w FJ(g)31 b FQ(denotes)d(the)h (conjugasy)e(class)g(and)h FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))28 b(the)h(cen)n(tralizer)e(of)h FJ(g)j FQ(in)e FJ(G)p FQ(.)39 b(Note)28 b(that)456 3850 y FJ(k)s FQ([)p FJ(Z)6 b FQ(\()p FJ(g)s FQ(\)])p FJ(\016)755 3862 y FI(g)793 3850 y FJ(v)31 b FQ(is)c(an)h(irreducible)e(represen)n(tation)g FJ(\031)31 b FQ(of)d FJ(Z)6 b FQ(\()p FJ(g)s FQ(\).)37 b(Hence)1409 4051 y FJ(V)p 1457 4030 35 3 v 12 x FI(g)r(;\031)1579 4051 y FQ(:=)23 b FJ(k)s FQ([)p FJ(G)p FQ(])p FJ(\016)1884 4063 y FI(g)1923 4051 y FJ(v)j FQ(=)2147 3972 y Fy(M)2077 4155 y FI(xg)r(x)2187 4138 y Fx(\000)p FF(1)2264 4155 y FE(2)p 2309 4121 V FI(g)2357 4051 y FJ(x\031)s(;)-2021 b FQ(\(3.2.3\))456 4326 y(is)33 b(an)g(irreducible)g FJ(D)r FQ(\()p FJ(G)p FQ(\)-mo)r(dule)i(whic)n(h)f(dep)r(ends)g(only)f (on)g(the)h(conjugacy)f(class)p 3234 4280 43 4 v 32 w FJ(g)k FQ(and)456 4426 y(the)d(isomorphism)e(class)h(of)h(the)g (irreducible)f(represen)n(tation)f FJ(\031)37 b FQ(of)d FJ(Z)6 b FQ(\()p FJ(g)s FQ(\).)55 b(The)33 b(action)h(of)456 4525 y FJ(D)r FQ(\()p FJ(G)p FQ(\))28 b(on)f FJ(V)p 847 4504 35 3 v 13 x FI(g)882 4537 y(;\031)975 4525 y FQ(is)g(giv)n(en)g (explicitly)h(b)n(y:)931 4706 y(\()p FJ(\016)1000 4718 y FI(f)1062 4706 y FL(\012)18 b FJ(h)p FQ(\)\()p FJ(xv)s FQ(\))24 b(=)f FJ(\016)1528 4722 y FI(f)s(;hxg)r(h)1733 4705 y Fx(\000)p FF(1)1810 4722 y FI(x)1848 4705 y Fx(\000)p FF(1)1943 4706 y FJ(hxv)87 b FQ(for)45 b FJ(f)t(;)14 b(h;)g(x)23 b FL(2)h FJ(G;)37 b(v)26 b FL(2)e FJ(\031)s(:)-2513 b FQ(\(3.2.4\))605 4892 y(This)40 b(sho)n(ws)f(that)h(the)g(category)e FL(R)p FJ(ep)1903 4904 y FI(f)1946 4892 y FJ(D)r FQ(\()p FJ(G)p FQ(\))j(is)f(semisimple)f(with)i(simple)f(ob)5 b(jects)456 5016 y FJ(V)p 504 4995 V 13 x FI(g)539 5028 y(;\031)631 5016 y FQ(lab)r(eled)27 b(b)n(y)g(pairs)g(\()p 1265 4971 43 4 v FJ(g)s(;)14 b(\031)s FQ(\),)28 b(where)p 1718 4971 V 27 w FJ(g)d FL(2)p 1862 4950 66 4 v 24 w FJ(G)j FQ(is)f(a)g(conjugacy)f(class)h(in)g FJ(G)h FQ(and)g FJ(\031)e FL(2)3198 4992 y FH([)3191 5016 y FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))28 b(is)456 5116 y(an)i(isomorphism)f(class)g(of)i (irreducible)f(represen)n(tation)e(of)j(the)g(cen)n(tralizer)d FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))31 b(of)f(some)456 5216 y(elemen)n(t)d FJ(g)f FL(2)p 906 5170 43 4 v 23 w FJ(g)31 b FQ(\()p FJ(\031)g FQ(is)c(indep)r(enden)n(t)i(of)e(the)h(c)n(hoice)f (of)h FJ(g)s FQ(\).)p eop %%Page: 61 15 61 64 bop 1005 226 a FM(3.2.)29 b(EXAMPLE:)f(QUANTUM)h(DOUBLE)g(OF)g(A) g(FINITE)g(GR)n(OUP)461 b(61)605 425 y FQ(In)25 b(what)f(follo)n(ws)g (w)n(e)g(will)h(use)g(the)g(orthogonalit)n(y)d(relations)h(of)i (irreducible)f(c)n(haracters)456 525 y(of)j(a)g(\014nite)i(group)d FJ(G)p FQ(:)1064 670 y(1)p 1029 707 112 4 v 1029 783 a FL(j)p FJ(G)p FL(j)1172 647 y Fy(X)1164 826 y FI(h)p FE(2)p FI(G)1313 726 y FQ(tr)1378 738 y FI(\031)1419 722 y Fx(\003)1458 726 y FQ(\()p FJ(h)p FQ(\))14 b(tr)1649 738 y FI(\031)1690 722 y Fx(0)1716 726 y FQ(\()p FJ(hg)s FQ(\))23 b(=)1992 670 y(tr)2057 682 y FI(\031)2102 670 y FQ(\()p FJ(g)s FQ(\))p 1992 707 218 4 v 1994 783 a(tr)2059 795 y FI(\031)2104 783 y FQ(\()p FJ(e)p FQ(\))2219 726 y FJ(\016)2256 738 y FI(\031)r(;\031)2358 722 y Fx(0)2384 726 y FJ(;)180 b(\031)s(;)14 b(\031)2724 692 y FE(0)2771 726 y FL(2)2866 705 y Fy(b)2849 726 y FJ(G)q(;)37 b(g)25 b FL(2)f FJ(G;)-2751 b FQ(\(3.2.5\))1012 926 y(1)p 924 963 216 4 v 924 1039 a FL(j)p FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))p FL(j)1173 903 y Fy(X)1164 1098 y FI(\031)r FE(2)1262 1083 y Fu(b)1250 1098 y FI(G)1315 982 y FQ(tr)1380 994 y FI(\031)1421 978 y Fx(\003)1460 982 y FQ(\()p FJ(g)s FQ(\))14 b(tr)1646 994 y FI(\031)1691 982 y FQ(\()p FJ(h)p FQ(\))23 b(=)g FJ(\016)p 1951 974 35 3 v 26 x FI(g)r(;)p 2005 959 39 3 v(h)2048 982 y FJ(;)180 b(h;)14 b(g)25 b FL(2)f FJ(G:)-2112 b FQ(\(3.2.6\))456 1240 y(Also)27 b(recall)f(that)i FL(j)p 1067 1194 43 4 v FJ(g)s FL(jj)p FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))p FL(j)23 b FQ(=)g FL(j)p FJ(G)p FL(j)p FQ(.)605 1403 y FP(Theorem)32 b FQ(3.2.1)p FP(.)40 b FL(R)p FJ(ep)1400 1415 y FI(f)1442 1403 y FJ(D)r FQ(\()p FJ(G)p FQ(\))35 b FO(is)f(a)h(mo)l(dular)f(tensor)g(c)l(ate)l (gory)g(with)h(simple)g(obje)l(cts)456 1527 y FJ(V)p 504 1506 35 3 v 13 x FI(g)539 1539 y(;\031)633 1527 y FO(lab)l(ele)l(d)c(by)f FQ(\()p 1036 1482 43 4 v FJ(g)s(;)14 b(\031)s FQ(\))p FO(,)p 1254 1482 V 31 w FJ(g)25 b FL(2)p 1398 1461 66 4 v 24 w FJ(G)p FO(,)30 b FJ(\031)d FL(2)1677 1503 y FH([)1670 1527 y FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))30 b(\()p FJ(g)25 b FL(2)p 2046 1482 43 4 v 24 w FJ(g)r FQ(\))p FO(.)39 b(We)30 b(have)6 b FQ(:)832 1686 y FJ(V)898 1651 y FE(\003)p 880 1673 35 3 v 880 1706 a FI(g)s(;\031)1002 1686 y FL(')23 b FJ(V)p 1138 1663 112 3 v 29 x FI(g)1172 1699 y Fx(\000)p FF(1)1250 1715 y FI(;\031)1311 1699 y Fx(\003)1350 1686 y FJ(;)-917 b FQ(\(3.2.7\))832 1878 y FJ(t)862 1904 y FM(\()p 888 1870 35 3 v FI(g)r(;\031)r FM(\))p FI(;)p FM(\()p 1055 1855 57 3 v FI(g)1089 1887 y Fx(0)1111 1904 y FI(;\031)1172 1887 y Fx(0)1194 1904 y FM(\))1247 1878 y FQ(=)23 b FJ(\016)1372 1904 y FM(\()p 1398 1870 35 3 v FI(g)r(;\031)r FM(\))p FI(;)p FM(\()p 1565 1855 57 3 v FI(g)1599 1887 y Fx(0)1622 1904 y FI(;\031)1683 1887 y Fx(0)1705 1904 y FM(\))1758 1822 y FQ(tr)1823 1834 y FI(\031)1868 1822 y FQ(\()p FJ(g)s FQ(\))p 1758 1859 218 4 v 1760 1935 a(tr)1825 1947 y FI(\031)1870 1935 y FQ(\()p FJ(e)p FQ(\))1985 1878 y FJ(;)-1552 b FQ(\(3.2.8\))832 2099 y FJ(s)871 2124 y FM(\()p 897 2091 35 3 v FI(g)r(;\031)r FM(\))p FI(;)p FM(\()p 1064 2076 57 3 v FI(g)1098 2108 y Fx(0)1120 2124 y FI(;\031)1181 2108 y Fx(0)1203 2124 y FM(\))1256 2099 y FQ(=)1561 2043 y(1)p 1354 2080 456 4 v 1354 2156 a FL(j)p FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))p FL(jj)p FJ(Z)g FQ(\()p FJ(g)1731 2132 y FE(0)1754 2156 y FQ(\))p FL(j)1969 2020 y Fy(X)1961 2199 y FI(h)p FE(2)p FI(G)1833 2264 y(hg)1906 2239 y Fx(0)1929 2264 y FI(h)1968 2239 y Fx(\000)p FF(1)2045 2264 y FE(2)p FI(Z)t FM(\()p FI(g)r FM(\))2239 2099 y FQ(tr)2304 2111 y FI(\031)2349 2099 y FQ(\()p FJ(hg)2472 2065 y FE(0)2495 2053 y(\000)p FM(1)2584 2099 y FJ(h)2632 2065 y FE(\000)p FM(1)2721 2099 y FQ(\))14 b(tr)2832 2111 y FI(\031)2873 2095 y Fx(0)2899 2099 y FQ(\()p FJ(h)2979 2065 y FE(\000)p FM(1)3069 2099 y FJ(g)3112 2065 y FE(\000)p FM(1)3200 2099 y FJ(h)p FQ(\))p FJ(:)-2847 b FQ(\(3.2.9\))456 2420 y FO(The)30 b(numb)l(ers)f FJ(p)997 2390 y FE(\006)1083 2420 y FO(fr)l(om)36 b FQ(\(3.1.7\))29 b FO(ar)l(e)h(e)l(qual)g(to)g (the)g(or)l(der)h(of)f FJ(G)p FO(.)605 2583 y FQ(The)i FJ(s)p FQ(-matrix)g(\(3.2.9\))f(w)n(as)g(\014rst)h(in)n(tro)r(duced)g (b)n(y)g(Lusztig)g([)p FK(L5)p FQ(])h(\(see)f(also)f([)p FK(L6,)37 b(L7)p FQ(]\))456 2683 y(under)27 b(the)i(names)e(\\non-ab)r (elian)g(F)-7 b(ourier)27 b(transform")f(and)i(\\exotic)f(F)-7 b(ourier)27 b(transform".)456 2783 y(Then)e(it)h(app)r(eared)f(in)h([)p FK(D)m(VVV)q FQ(])g(and)g([)p FK(KT)p FQ(])g(in)g(connection)f(with)h (\\orbifolds".)34 b(Dijkgraaf,)456 2882 y(P)n(asquier)28 b(and)i(Ro)r(c)n(he)g([)p FK(DPR)p FQ(])h(considered)e(a)h (generalization)e(of)j(the)f(ab)r(o)n(v)n(e)f(construction)456 2982 y(whic)n(h)34 b(is)g(also)f(related)g(to)h(orbifolds.)56 b(They)34 b(in)n(tro)r(duced)g(a)g(quasi-Hopf)f(algebra)g FJ(D)3258 2952 y FI(c)3291 2982 y FQ(\()p FJ(G)p FQ(\),)456 3081 y(dep)r(ending)h(on)f(a)g(cohomology)e(class)i FJ(c)g FL(2)g FQ(H)1937 3051 y FM(3)1975 3081 y FQ(\()p FJ(G;)14 b FQ(U\(1\)\),)36 b(whic)n(h)d(reduces)g(to)h FJ(D)r FQ(\()p FJ(G)p FQ(\))g(when)456 3181 y FJ(c)23 b FQ(=)f(1.)605 3365 y FP(Pr)n(oof)31 b(of)h(Theorem)g FQ(3.2.1)p FP(.)39 b FQ(Eq.)18 b(\(3.2.7\))g(follo)n(ws)f(easily)h(from)g(the)h (de\014nitions)f(\(note)456 3464 y(that)27 b FJ(Z)6 b FQ(\()p FJ(g)773 3434 y FE(\000)p FM(1)862 3464 y FQ(\))23 b(=)g FJ(Z)6 b FQ(\()p FJ(g)s FQ(\))28 b(and)f(tr)1429 3476 y FI(\031)1470 3460 y Fx(\003)1509 3464 y FQ(\()p FJ(h)p FQ(\))c(=)g(tr)1796 3476 y FI(\031)1841 3464 y FQ(\()p FJ(h)1921 3434 y FE(\000)p FM(1)2011 3464 y FQ(\)\).)605 3564 y(T)-7 b(o)24 b(pro)n(v)n(e)f(\(3.2.8\))o(,)j(w)n(e)e(compute)h (the)g(t)n(wists)f FJ(\022)j FQ(using)d(the)h(results)f(of)h(Prop)r (osition)e(2.2.4)456 3663 y(and)k(Lemma)g(2.2.5.)36 b(Since)28 b FJ(\015)1409 3633 y FM(2)1469 3663 y FQ(=)22 b(id,)28 b(it)g(follo)n(ws)f(that)h FJ(\016)2249 3675 y FI(V)2329 3663 y FQ(=)23 b(id,)28 b(cf.)g(\(2.2.11\))o(.)37 b(Hence,)1543 3838 y FJ(\022)25 b FQ(=)e FJ(u)1743 3804 y FE(\000)p FM(1)1854 3838 y FQ(=)1950 3759 y Fy(X)1942 3938 y FI(h)p FE(2)p FI(G)2105 3838 y FJ(\016)2142 3850 y FI(h)2203 3838 y FL(\012)18 b FJ(h:)-1901 b FQ(\(3.2.10\))456 4080 y(As)26 b FJ(g)k FQ(is)c(cen)n(tral)g(in)g FJ(Z)6 b FQ(\()p FJ(g)s FQ(\),)27 b(it)g(acts)f(as)g(a)g(constan)n(t)g(=)d(tr)2226 4092 y FI(\031)2271 4080 y FQ(\()p FJ(g)s FQ(\))p FJ(=)14 b FQ(tr)2498 4092 y FI(\031)2543 4080 y FQ(\()p FJ(e)p FQ(\))27 b(on)f(the)h(represen)n(tation)456 4180 y FJ(\031)s FQ(;)h(hence)f(b)n(y)i(\(3.2.4\))o(,)f FJ(\022)p 1229 4159 35 3 v 12 x FI(g)r(;\031)1351 4180 y FQ(=)23 b(tr)1503 4192 y FI(\031)1548 4180 y FQ(\()p FJ(g)s FQ(\))p FJ(=)14 b FQ(tr)1776 4192 y FI(\031)1821 4180 y FQ(\()p FJ(e)p FQ(\).)605 4279 y(T)-7 b(o)24 b(pro)n(v)n(e)e(\(3.2.9\))o(,)j(w)n(e)f (will)g(use)g(\(3.1.2\))o(.)36 b(W)-7 b(e)24 b(compute)g(for)g FJ(x;)14 b(x)2660 4249 y FE(0)2707 4279 y FL(2)24 b FJ(G)p FQ(,)h FJ(v)h FL(2)d FJ(\031)3093 4249 y FE(\003)3132 4279 y FQ(,)i FJ(v)3223 4249 y FE(0)3269 4279 y FL(2)f FJ(\031)3398 4249 y FE(0)3421 4279 y FQ(:)813 4439 y FJ(\022)852 4451 y FI(V)906 4431 y Fx(\003)p 891 4447 31 3 v 891 4470 a FG(g)r(;\031)981 4451 y FE(\012)p FI(V)p 1072 4435 54 3 v 26 x FG(g)1102 4465 y Fx(0)1125 4477 y FG(;\031)1180 4465 y Fx(0)1211 4439 y FQ(\()p FJ(xv)f FL(\012)18 b FJ(x)1483 4405 y FE(0)1506 4439 y FJ(v)1549 4405 y FE(0)1573 4439 y FQ(\))23 b(=)g(\001\()p FJ(u)1865 4405 y FE(\000)p FM(1)1954 4439 y FQ(\)\()p FJ(xv)g FL(\012)18 b FJ(x)2258 4405 y FE(0)2282 4439 y FJ(v)2325 4405 y FE(0)2348 4439 y FQ(\))1234 4605 y(=)1378 4526 y Fy(X)1370 4705 y FI(h)p FE(2)p FI(G)1322 4758 y(h)1361 4766 y FF(1)1393 4758 y FI(h)1432 4766 y FF(2)1464 4758 y FM(=)p FI(h)1554 4605 y FQ(\()p FJ(\016)1623 4617 y FI(h)1662 4625 y FF(1)1717 4605 y FL(\012)g FJ(h)p FQ(\)\()p FJ(xv)s FQ(\))i FL(\012)e FQ(\()p FJ(\016)2206 4617 y FI(h)2245 4625 y FF(2)2300 4605 y FL(\012)g FJ(h)p FQ(\)\()p FJ(x)2542 4571 y FE(0)2566 4605 y FJ(v)2609 4571 y FE(0)2633 4605 y FQ(\))1234 4895 y(=)1378 4816 y Fy(X)1370 4995 y FI(h)p FE(2)p FI(G)1322 5048 y(h)1361 5056 y FF(1)1393 5048 y FI(h)1432 5056 y FF(2)1464 5048 y FM(=)p FI(h)1568 4895 y FJ(\016)1605 4911 y FI(h)1644 4919 y FF(1)1676 4911 y FI(;hxg)1807 4894 y Fx(\000)p FF(1)1884 4911 y FI(x)1922 4894 y Fx(\000)p FF(1)1999 4911 y FI(h)2038 4894 y Fx(\000)p FF(1)2120 4895 y FJ(hxv)k FL(\012)c FJ(\016)2397 4913 y FI(h)2436 4921 y FF(2)2468 4913 y FI(;hx)2565 4897 y Fx(0)2587 4913 y FI(g)2621 4897 y Fx(0)2644 4913 y FI(x)2682 4897 y Fx(0)2704 4894 y(\000)p FF(1)2781 4913 y FI(h)2820 4897 y Fx(\000)p FF(1)2901 4895 y 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FQ(The)i(pro)r(of)f(of)h(this)h(theorem)e(is)h(straigh)n(tforw)n (ard.)31 b(In)20 b(particular,)h(this)f(theorem)f(implies)456 3414 y(that)975 3552 y(F)-7 b(or)27 b(0)c FL(\024)g FJ(n)g FL(\024)f FH({)g FL(\000)c FQ(2,)28 b FJ(V)1738 3564 y FI(n)1811 3552 y FQ(is)f(irreducible)g(and)h(dim)2605 3564 y FI(q)2656 3552 y FJ(V)2704 3564 y FI(n)2772 3552 y FL(6)p FQ(=)23 b(0,)-2469 b(\(3.3.1\))456 3694 y(whic)n(h)26 b(is)g(ob)n(vious)f(b)r(ecause)g(in)i(this)f(case)g(all)g FJ(q)s FQ(-factorials)e(are)h(non-zero.)35 b(\(In)27 b(fact,)f(one)g(has)456 3794 y(a)h(stronger)f(statemen)n(t:)36 b FJ(V)1314 3806 y FI(n)1388 3794 y FQ(is)27 b(irreducible)g(i\013)h FJ(n)23 b(<)g FH({)31 b FQ(or)c FJ(n)c FQ(=)f FJ(l)r FH({)g FL(\000)c FQ(1,)27 b FJ(l)e FL(2)e FH(Z)2959 3806 y FM(+)3008 3794 y FQ(,)k(see)h([)p FK(AP)p FQ(].\))605 3893 y(W)-7 b(e)26 b(will)g(need)f(a)g(similar)g(result)g(for)g(an)g (arbitrary)f(semisimple)h(\014nite)i(dimensional)e(Lie)456 3993 y(algebra)32 b FA(g)p FQ(.)58 b(Recall)34 b(the)h(n)n(um)n(b)r(er) f FJ(m)g FQ(from)h(\(1.3.17\))n(.)58 b(W)-7 b(e)35 b(let)g FJ(q)i FQ(=)d FJ(e)2749 3963 y FI(\031)r FM(i)p FI(=m)p Fz({)2953 3993 y FQ(,)j FH({)h FL(2)d FH(Z)o FQ(,)c(and)456 4093 y(assume)g(that)h FH({)i FL(\025)29 b FJ(h)1162 4063 y FE(_)1211 4093 y FQ(,)k(where)e FJ(h)1559 4063 y FE(_)1638 4093 y FQ(=)f FL(h)p FJ(\032;)14 b(\022)r FL(i)22 b FQ(+)e(1)32 b(is)g(the)g(dual)f(Co)n(xeter)g(n)n(um)n(b)r (er,)i FJ(\032)e FQ(is)h(the)456 4192 y(half)27 b(sum)h(of)g(p)r (ositiv)n(e)f(ro)r(ots,)f(and)i FJ(\022)i FQ(is)d(the)h(highest)g(ro)r (ot)f(of)g FA(g)p FQ(.)605 4343 y FP(Theorem)32 b FQ(3.3.5)p FP(.)40 b FQ(dim)1387 4355 y FI(q)1438 4343 y FJ(V)1486 4355 y FI(\025)1561 4343 y FQ(=)32 b(0)i FO(if)i(and)f(only)g(if)g FJ(\025)23 b FQ(+)e FJ(\032)32 b FL(2)h FJ(H)2645 4355 y FI(\013;l)2767 4343 y FO(for)j(some)f FJ(\013)d FL(2)h FQ(\001)3364 4355 y FM(+)3419 4343 y FO(,)456 4443 y FJ(l)24 b FL(2)f FH(Z)p FO(,)h(wher)l(e)30 b FJ(H)997 4455 y FI(\013;l)1116 4443 y 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FO(ther)l(e)j(exists)f(a)h(unique)f(inde)l(c)l(omp)l(osable)j (tilting)d(mo)l(dule)h FJ(T)3401 5128 y FI(\025)456 5216 y FO(such)29 b(that)h(its)g(weight)h(subsp)l(ac)l(e)f FQ(\()p FJ(T)1602 5228 y FI(\025)1645 5216 y FQ(\))1677 5185 y FI(\026)1752 5216 y FO(is)g FQ(0)f FO(unless)g FJ(\026)23 b FL(\024)g FJ(\025)30 b FO(and)g FQ(\()p FJ(T)2641 5228 y FI(\025)2685 5216 y FQ(\))2717 5185 y FI(\025)2783 5216 y FQ(=)23 b FH(C)15 b FO(.)p eop %%Page: 67 21 67 70 bop 1187 226 a FM(3.3.)30 b(QUANTUM)f(GR)n(OUPS)g(A)-5 b(T)29 b(R)n(OOTS)h(OF)f(UNITY)643 b(67)605 425 y FQ(\(iii\))30 b FO(F)-6 b(or)29 b FJ(\025)24 b FL(2)f FJ(C)35 b FO(we)30 b(have)g FJ(T)1528 437 y FI(\025)1594 425 y FQ(=)23 b FJ(V)1730 437 y FI(\025)1774 425 y FO(,)30 b(while)g(for)g FJ(\025)24 b FL(62)f FJ(C)35 b FO(we)30 b(have)g FQ(dim)2867 437 y FI(q)2918 425 y FJ(T)2967 437 y FI(\025)3033 425 y FQ(=)23 b(0)p FO(.)38 b(Henc)l(e)456 525 y FQ(dim)594 537 y FI(q)644 525 y FJ(T)d FL(\025)22 b FQ(0)29 b FO(for)i(al)t(l)g FJ(T)j 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b FQ(0)k(if)h FJ(V)42 b FL(6')23 b FQ(0.)456 3860 y(These)d(prop)r(erties)g(sho)n(w)g(that)h FL(C)1494 3830 y FM(in)n(t)1598 3860 y FQ(is)g(the)g(category)e(w)n(e)h(w)n(an)n (ted.)34 b(It)22 b(is)e(a)h(semisimple)g(ribb)r(on)456 3960 y(category)h(with)k(a)e(\014nite)h(n)n(um)n(b)r(er)g(of)f(simple)h (ob)5 b(jects.)36 b(A)25 b(natural)f(question)g(is)h(whether)f(this)456 4059 y(category)h(is)j(mo)r(dular.)36 b(W)-7 b(e)28 b(will)g(sho)n(w)f (that)g(the)h(answ)n(er)f(is)g(p)r(ositiv)n(e.)605 4205 y FP(Theorem)32 b FQ(3.3.20)p FP(.)39 b FL(C)1339 4175 y FM(in)n(t)1446 4205 y FO(is)25 b(a)f(mo)l(dular)h(tensor)f(c)l(ate)l (gory)h(with)f(simple)i(obje)l(cts)e FJ(V)3217 4217 y FI(\025)3285 4205 y FQ(\()p FJ(\025)g FL(2)456 4304 y FJ(C)6 b FQ(\))p FO(,)980 4455 y FJ(s)1019 4467 y FI(\025\026)1126 4455 y FQ(=)22 b FL(j)p FJ(P)7 b(=)p FH({)s FJ(Q)1462 4421 y FE(_)1511 4455 y FL(j)1534 4421 y FE(\000)p FM(1)p FI(=)p FM(2)1704 4455 y FQ(i)1727 4421 y FE(j)p FM(\001)1802 4429 y FF(+)1849 4421 y 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FM(2)p FI(\031)r FM(i)p FI(c=)p FM(24)1617 5182 y FJ(;)98 b(c)23 b FQ(=)g(\()p FH({)f FL(\000)c FJ(h)2125 5147 y FE(_)2174 5182 y FQ(\))c(dim)g FA(g)p FJ(=)p FH({)s FJ(:)-2081 b FQ(\(3.3.10\))p eop %%Page: 68 22 68 71 bop 456 226 a FM(68)838 b(3.)29 b(MODULAR)g(TENSOR)g(CA)-5 b(TEGORIES)605 425 y FP(Pr)n(oof.)41 b FQ(The)27 b(calculations)g(in)h (the)g(pro)r(of)f(of)g(Theorem)g(3.3.9)f(and)i(Eq.)f(\(3.1.5\))g(giv)n (e)930 609 y(~)-46 b FJ(s)965 621 y FI(\025\026)1072 609 y FQ(=)23 b FJ(\037)1212 621 y FI(\026)1256 609 y FQ(\()p FJ(q)1328 575 y FM(2\()p FI(\025)p FM(+)p FI(\032)p FM(\))1542 609 y FQ(\))14 b(dim)1727 621 y FI(q)1777 609 y FJ(V)1825 621 y FI(\025)1893 609 y FQ(=)2078 553 y(1)p 1990 590 217 4 v 1990 666 a FJ(\016)s FQ(\()p FJ(q)2102 642 y FM(2)p FI(\032)2174 666 y FQ(\))2253 530 y Fy(X)2230 708 y FI(w)r FE(2)p FI(W)2396 609 y FQ(\()p FL(\000)p FQ(1\))2567 575 y FI(l)p FM(\()p FI(w)r FM(\))2694 609 y FJ(q)2734 575 y FM(2)p FE(h)-11 b(h)p FI(w)r FM(\()p FI(\025)p FM(+)p FI(\032)p FM(\))o FI(;\026)p FM(+)p FI(\032)q FE(i)g(i)3227 609 y FJ(:)456 842 y FQ(T)k(o)37 b(sho)n(w)g(that)h(det)17 b(~)-45 b FJ(s)40 b FL(6)p FQ(=)f(0,)h(w)n(e)e(will)g(calculate)f(the)h(matrix)i(~)-45 b FJ(s)2539 812 y FM(2)2576 842 y FQ(.)67 b(First)38 b(note)g(that)g(if)g(w)n(e)456 941 y(use)d(the)h(form)n(ula)f(ab)r(o)n (v)n(e)f(to)i(extend)j(~)-45 b FJ(s)1741 953 y FI(\025\026)1860 941 y FQ(for)36 b FJ(\025;)14 b(\026)36 b FL(2)h FJ(P)12 b FQ(,)38 b(this)e(extended)g(matrix)f(will)h(b)r(e)456 1041 y(an)n(tisymmetric)27 b(with)h(resp)r(ect)f(to)g(the)h(shifted)h (action)e(of)g(the)h(a\016ne)g(W)-7 b(eyl)28 b(group)e FJ(W)3232 1011 y FI(a)3272 1041 y FQ(:)1312 1188 y(~)-45 b FJ(s)1348 1200 y FI(w)r(:\025;\026)1544 1188 y FQ(=)23 b(\()p FL(\000)p FQ(1\))1803 1154 y FI(l)p FM(\()p FI(w)r FM(\))1933 1188 y FQ(~)-46 b FJ(s)1968 1200 y FI(\025;\026)2072 1188 y FJ(;)180 b(w)25 b FL(2)f FJ(W)2528 1154 y FI(a)2568 1188 y FJ(:)-2135 b FQ(\(3.3.11\))456 1334 y(In)27 b(particular,)j(~) -45 b FJ(s)1005 1346 y FI(\025\026)1112 1334 y FQ(=)22 b(0)27 b(when)h FJ(\025)g FQ(or)f FJ(\026)h FQ(are)e(on)i(the)g(w)n (alls)e(of)i FJ(C)6 b FQ(.)605 1433 y(Since)825 1371 y Fy(P)913 1458 y FI(\026)p FE(2)p FI(C)1071 1433 y FQ(~)-45 b FJ(s)1107 1445 y FI(\025\026)1194 1433 y FQ(~)g FJ(s)1230 1445 y FI(\026\027)1343 1433 y FQ(is)31 b(symmetric)g(with)h(resp)r (ect)f(to)g(the)g(shifted)h(action)f(of)g FJ(W)3285 1403 y FI(a)3357 1433 y FQ(on)456 1535 y FJ(\026)26 b FQ(and)h FJ(C)32 b FQ(is)27 b(the)g(fundamen)n(tal)f(domain)g(for)g(the)h (action)f(of)h FJ(W)2481 1505 y FI(a)2547 1535 y FQ(on)g FJ(P)12 b FQ(,)26 b(w)n(e)g(can)h(replace)e(the)456 1635 y(range)h(of)h(summation)h(with)g FJ(P)7 b(=W)1587 1605 y FI(a)1627 1635 y FQ(.)37 b(Since)27 b FJ(W)1993 1605 y FI(a)2057 1635 y FL(')22 b FJ(W)31 b FH(n)18 b({)s FJ(Q)2460 1605 y FE(_)2509 1635 y FQ(,)28 b(this)g(sum)f(equals)596 1788 y(1)p 549 1825 137 4 v 549 1901 a FL(j)p FJ(W)12 b FL(j)803 1765 y Fy(X)708 1947 y FI(\026)p FE(2)p FI(P)5 b(=)p Fz({)s FI(Q)974 1930 y Fx(_)1036 1844 y FQ(~)-46 b FJ(s)1071 1856 y FI(\025\026)1158 1844 y FQ(~)h FJ(s)1194 1856 y FI(\026\027)855 2112 y FQ(=)1000 2056 y(1)p 953 2093 V 953 2169 a FL(j)p FJ(W)12 b FL(j)1181 2034 y Fy(X)1113 2212 y FI(w)r(;w)1233 2195 y Fx(0)1254 2212 y FE(2)p FI(W)1479 2034 y Fy(X)1384 2216 y FI(\026)p FE(2)p FI(P)5 b(=)p Fz({)s FI(Q)1650 2199 y Fx(_)1708 2112 y FJ(\016)s FQ(\()p FJ(q)1820 2078 y FM(2)p FI(\032)1892 2112 y FQ(\))1924 2078 y FE(\000)p FM(2)2013 2112 y FQ(\()p FL(\000)p FQ(1\))2184 2078 y FI(l)p FM(\()p FI(w)r FM(\)+)p FI(l)p FM(\()p FI(w)2455 2053 y Fx(0)2477 2078 y FM(\))2507 2112 y FJ(q)2547 2078 y FM(2)p FE(h)-11 b(h)p FI(\026)p FM(+)p FI(\032;w)r FM(\()p FI(\025)p FM(+)p FI(\032)p FM(\)+)p FI(w)3095 2053 y Fx(0)3117 2078 y FM(\()p FI(\027)t FM(+)p FI(\032)p FM(\))q FE(i)g(i)3338 2112 y FJ(:)456 2349 y FQ(No)n(w)27 b(w)n(e)g(need)h(an)f(ob)n(vious)f(lemma.)605 2559 y FP(Lemma)31 b FQ(3.3.21)p FP(.)1290 2480 y Fy(X)1195 2662 y FI(\026)p FE(2)p FI(P)5 b(=)p Fz({)s FI(Q)1461 2645 y Fx(_)1518 2559 y FJ(q)1558 2524 y FM(2)p FE(h)-11 b(h)p FI(\026;a)p FE(i)g(i)1800 2559 y FQ(=)1887 2417 y Fy(\()1954 2502 y FQ(0)364 b FO(for)47 b FJ(a)23 b FL(62)g FH({)s FJ(Q)2778 2472 y FE(_)2828 2502 y FJ(;)1954 2622 y FL(j)p FJ(P)7 b(=)p FH({)s FJ(Q)2203 2591 y FE(_)2252 2622 y FL(j)85 b FO(for)47 b FJ(a)23 b FL(2)g FH({)s FJ(Q)2778 2591 y FE(_)2828 2622 y FJ(:)605 2811 y FQ(Note)29 b(that)f FJ(w)r FQ(\()p FJ(\025)21 b FQ(+)d FJ(\032)p FQ(\))i(+)e FJ(w)1471 2781 y FE(0)1495 2811 y FQ(\()p FJ(\027)25 b FQ(+)18 b FJ(\032)p FQ(\))25 b(=)f FJ(w)r FQ(\()p FJ(\025)d FQ(+)d FJ(\032)p FQ(\))h FL(\000)g FJ(w)2349 2781 y FE(0)2373 2811 y FJ(w)2432 2823 y FM(0)2470 2811 y FQ(\()p FJ(\027)2548 2781 y FE(\003)2606 2811 y FQ(+)f FJ(\032)p FQ(\))25 b FL(2)g FH({)s FJ(Q)2993 2781 y FE(_)3070 2811 y FQ(i\013)30 b FJ(\025)19 b FQ(+)g FJ(\032)24 b FL(2)456 2911 y FJ(w)517 2880 y FE(\000)p FM(1)606 2911 y FJ(w)667 2880 y FE(0)691 2911 y FJ(w)750 2923 y FM(0)788 2911 y FQ(\()p FJ(\027)866 2880 y FE(\003)926 2911 y FQ(+)c FJ(\032)p FQ(\))h(+)f FH({)s FJ(Q)1316 2880 y FE(_)1397 2911 y FQ(where)30 b FJ(w)1699 2923 y FM(0)1768 2911 y FQ(is)h(the)h(longest)e(elmen)n(t)h(in)h FJ(W)12 b FQ(.)47 b(But)32 b(since)e(b)r(oth)i FJ(\025)456 3010 y FQ(and)j FJ(\027)671 2980 y FE(\003)746 3010 y FQ(are)f(in)i FJ(C)6 b FQ(,)39 b(whic)n(h)c(is)h(a)f(fundamen)n(tal)h (domain)g(of)f FJ(W)2522 2980 y FI(a)2562 3010 y FQ(,)j(this)e(is)g (only)g(p)r(ossible)f(if)456 3110 y FJ(\025)18 b FQ(+)h FJ(\032)j FQ(=)h FJ(\027)805 3080 y FE(\003)862 3110 y FQ(+)18 b FJ(\032)p FQ(,)28 b FJ(w)1100 3080 y FE(\000)p FM(1)1190 3110 y FJ(w)1251 3080 y FE(0)1298 3110 y FQ(=)22 b FJ(w)1444 3122 y FM(0)1482 3110 y FQ(.)37 b(Therefore)1274 3219 y Fy(X)1265 3397 y FI(\026)p FE(2)p FI(C)1419 3298 y FQ(~)-45 b FJ(s)1455 3310 y FI(\025\026)1542 3298 y FQ(~)g FJ(s)1578 3310 y FI(\026\027)1682 3298 y FQ(=)1780 3242 y FL(j)p FJ(P)7 b(=)p FH({)s FJ(Q)2029 3212 y FE(_)2078 3242 y FL(j)p 1780 3279 322 4 v 1814 3355 a FJ(\016)s FQ(\()p FJ(q)1926 3331 y FM(2)p FI(\032)1998 3355 y FQ(\))2030 3331 y FM(2)2111 3298 y FQ(\()p FL(\000)p FQ(1\))2282 3264 y FI(l)p FM(\()p FI(w)2377 3272 y FF(0)2409 3264 y FM(\))2440 3298 y FJ(\016)2477 3310 y FI(\025;\027)2573 3294 y Fx(\003)2612 3298 y FJ(:)456 3528 y FQ(This)27 b(n)n(um)n(b)r(er)h(is)f(non-zero,)f(hence)i(det)17 b(~)-45 b FJ(s)23 b FL(6)p FQ(=)f(0.)605 3627 y(This)27 b(also)f(giv)n(es)g FJ(D)j FQ(since)e(\()s(~)-45 b FJ(s)1535 3597 y FM(2)1573 3627 y FQ(\))1605 3639 y FI(\025\027)1709 3627 y FQ(=)22 b FJ(D)1867 3597 y FM(2)1905 3627 y FJ(\016)1942 3639 y FI(\025;\027)2038 3623 y Fx(\003)2077 3627 y FQ(.)37 b(F)-7 b(orm)n(ula)26 b(\(3.3.8\))g(for)h(the)g(t)n(wist)g(follo)n(ws) 456 3727 y(directly)g(from)g(Example)f(2.2.6.)35 b(The)28 b(rest)f(of)g(the)g(pro)r(of)g(is)g(straigh)n(tforw)n(ard)d(and)k(is)f (left)h(to)456 3827 y(the)g(reader.)p 3384 3827 4 57 v 3388 3774 50 4 v 3388 3827 V 3437 3827 4 57 v 605 3991 a FP(Example)j FQ(3.3.22)p FP(.)39 b FQ(When)28 b FA(g)23 b FQ(=)g FA(sl)1732 4003 y FM(2)1769 3991 y FQ(,)28 b(w)n(e)f(ha)n(v)n (e:)1053 4197 y FJ(s)1092 4209 y FI(\025\026)1199 4197 y FQ(=)1287 4067 y Fy(r)p 1370 4067 79 4 v 1388 4141 a FQ(2)p 1380 4178 59 4 v 1380 4254 a FH({)1462 4197 y FQ(sin)1578 4080 y Fy(\022)1639 4197 y FJ(\031)1699 4141 y FQ(\()p FJ(\025)19 b FQ(+)f(1\)\()p FJ(\026)h FQ(+)f(1\))p 1699 4178 514 4 v 1927 4254 a FH({)2223 4080 y Fy(\023)2298 4197 y FJ(;)180 b FQ(0)22 b FL(\024)h FJ(\025;)14 b(\026)23 b FL(\024)g FH({)f FL(\000)c FQ(2)p FJ(:)605 4414 y FQ(The)28 b(argumen)n(ts)e(of)i(Theorem)f(3.3.20)f(can) h(b)r(e)i(rep)r(eated)e(for)g FJ(q)g FQ(=)c FJ(e)2789 4384 y FI(\031)r FM(i)p FI(=m)p Fz({)2993 4414 y FQ(,)28 b FH({)f FL(2)c FH(Q)6 b FQ(,)34 b(but)456 4513 y(in)27 b(this)h(case)f(the)h(matrix)i(~)-45 b FJ(s)28 b FQ(ma)n(y)f(b)r(e)h (degenerate.)605 4613 y(Note)36 b(that)h(the)f(form)n(ulas)f(for)h(the) g(matrices)g FJ(s;)14 b(t)36 b FQ(coincide)g(with)g(the)h(Kac{P)n (eterson)456 4713 y(form)n(ula)20 b([)p FK(KP)p FQ(])h(for)f(the)i(mo)r (dular)e(transformations)f(of)i(c)n(haracters)d(of)j(the)h(a\016ne)e (Lie)h(algebra)456 4811 y(^)456 4815 y FA(g)i FQ(when)g FJ(q)j FQ(=)d FJ(e)923 4785 y FI(\031)r FM(i)p FI(=m)p Fz({)1150 4815 y FQ(\(their)h(matrix)e FJ(T)35 b FQ(corresp)r(onds)21 b(to)i(the)h(matrix)f FJ(t=\020)29 b FQ(in)23 b(our)g(notations\).)456 4915 y(This)k(fact)h(will)g(b)r(e)g(explained)f(later.)605 5015 y(Finally)-7 b(,)23 b(let)g(us)f(discuss)f(the)i(V)-7 b(erlinde)22 b(algebra)e(for)i FL(C)2303 4985 y FM(in)n(t)2386 5015 y FQ(.)36 b(Let)22 b FL(V)30 b FQ(=)22 b FJ(K)6 b FQ(\()p FL(R)p FJ(ep)3016 5027 y FI(f)3059 5015 y FQ(\()p FA(g)p FQ(\)\))h FL(\012)g FH(C)44 b FQ(b)r(e)456 5116 y(the)26 b(complexi\014ed)f(Grothendiec)n(k)g(ring)g(of)h FL(R)p FJ(ep)1996 5128 y FI(f)2039 5116 y FQ(\()p FA(g)p FQ(\);)h(similarly)-7 b(,)25 b(denote)h FL(V)2866 5128 y FI(k)2929 5116 y FQ(=)d FJ(K)6 b FQ(\()p FL(C)3175 5086 y FM(in)n(t)3258 5116 y FQ(\))15 b FL(\012)f FH(C)456 5216 y FQ(\(where,)27 b(as)g(b)r(efore,)g FH({)g FQ(=)c FJ(k)e FQ(+)d FJ(h)1490 5185 y FE(_)1539 5216 y FQ(\).)p eop %%Page: 69 23 69 72 bop 1187 226 a FM(3.3.)30 b(QUANTUM)f(GR)n(OUPS)g(A)-5 b(T)29 b(R)n(OOTS)h(OF)f(UNITY)643 b(69)605 425 y FP(Pr)n(oposition)31 b FQ(3.3.23)p FP(.)39 b FO(The)h(V)-6 b(erlinde)39 b(algebr)l(a)g FL(V)2279 437 y FI(k)2359 425 y FO(is)f(the)h(quotient)f(of)h FL(V)7 b FO(,)41 b(namely,)456 525 y FL(V)507 537 y FI(k)588 525 y FQ(=)g FL(V)7 b FJ(=)p FL(I)839 537 y FI(k)879 525 y FO(,)43 b(wher)l(e)d FL(I)1236 537 y FI(k)1319 525 y FL(\032)g(V)47 b FO(is)40 b(the)g(line)l(ar)g(sp)l(an)g(of)g FL(h)p FJ(V)2401 537 y FI(\025)2446 525 y FL(i)26 b(\000)f FQ(\()p FL(\000)p FQ(1\))2765 495 y FI(l)p FM(\()p FI(w)r FM(\))2891 525 y FL(h)p FJ(V)2971 537 y FI(w)r(:\025)3085 525 y FL(i)40 b FO(for)h FJ(\025)g FL(2)456 624 y FJ(P)509 636 y FM(+)564 624 y FJ(;)14 b(w)26 b FL(2)d FJ(W)854 594 y FI(a)894 624 y FJ(;)14 b(w)r(:\025)24 b FL(2)g FJ(P)1219 636 y FM(+)1274 624 y FO(.)605 774 y FP(Pr)n(oof.)41 b FQ(The)c(construction)g(giv)n(en)g(in)h(Theorem)f(3.1.11)f(de\014nes) h(a)h(surjectiv)n(e)f(map)456 874 y FJ(\026)9 b FQ(:)28 b FL(V)j(!)25 b(V)807 886 y FI(k)848 874 y FQ(.)40 b(It)29 b(follo)n(ws)e(from)i(W)-7 b(eyl)29 b(c)n(haracter)d(form)n(ula)i(that) h FL(I)2579 886 y FI(k)2645 874 y FL(\032)24 b FQ(k)n(er)13 b FJ(\026)p FQ(.)40 b(On)28 b(the)h(other)456 973 y(hand,)e(it)h(follo) n(ws)f(from)g(Theorem)g(3.3.6\(iii\))g(that)h(dim)14 b FL(V)7 b FJ(=)p FL(I)2398 985 y FI(k)2461 973 y FQ(=)23 b FL(j)p FJ(C)6 b FL(j)23 b FQ(=)g(dim)14 b FL(V)2974 985 y FI(k)3015 973 y FQ(.)p 3384 973 4 57 v 3388 920 50 4 v 3388 973 V 3437 973 4 57 v 605 1123 a FP(Exer)n(cise)32 b FQ(3.3.24)p FP(.)39 b FQ(\(i\))31 b(Sho)n(w)e(that)h(for)g FA(g)d FQ(=)f FJ(A)2155 1135 y FI(n)2201 1123 y FQ(,)k(the)h(ideal)e FL(I)2645 1135 y FI(k)2717 1123 y FQ(is)g(the)i(linear)e(span)h(of)456 1222 y FL(h)p FJ(V)536 1234 y FI(\025)580 1222 y FL(i)e FQ(for)f FJ(\025)c FL(2)h FJ(P)970 1234 y FM(+)1025 1222 y FJ(;)14 b FQ(\()p FJ(\025)19 b FQ(+)f FJ(\032;)c(\022)1365 1192 y FE(_)1414 1222 y FQ(\))24 b(=)e FH({)s FQ(.)605 1322 y(\(ii\))28 b(Sho)n(w)g(that)f(for)g FA(g)c FQ(=)g FJ(E)1483 1334 y FM(8)1548 1322 y FQ(this)28 b(is)g(not)f(so.)605 1425 y(\(iii\))h(Sho)n(w)g(that)f(the)h(fusion)g(rules)f(for)g FJ(U)1934 1437 y FI(q)1970 1425 y FQ(\()p FA(sl)2063 1437 y FM(2)2100 1425 y FQ(\))h(for)f FJ(q)f FQ(=)d FJ(e)2477 1394 y FI(\031)r FM(i)p FI(=)p FM(\()p FI(k)q FM(+2\))2775 1425 y FQ(are)j(giv)n(en)h(b)n(y)1488 1659 y FL(h)p FJ(V)1568 1671 y FI(m)1631 1659 y FL(ih)p FJ(V)1743 1671 y FI(n)1790 1659 y FL(i)c FQ(=)1933 1580 y Fy(X)1982 1759 y FI(l)2066 1659 y FJ(N)2142 1624 y FI(l)2133 1679 y(mn)2237 1659 y FL(h)p FJ(V)2317 1671 y FI(l)2343 1659 y FL(i)p FJ(;)456 1868 y FQ(where)604 2156 y FJ(N)680 2122 y FI(l)671 2176 y(mn)798 2156 y FQ(=)885 2014 y Fy(\()952 2099 y FQ(1)83 b(for)45 b FL(j)p FJ(m)19 b FL(\000)f FJ(n)p FL(j)23 b(\024)f FJ(l)j FL(\024)d FJ(m)d FQ(+)f FJ(n;)36 b(l)25 b FL(\024)e FQ(2)p FJ(k)d FL(\000)e FQ(\()p FJ(m)h FQ(+)f FJ(n)p FQ(\))p FJ(;)37 b(l)20 b FQ(+)e FJ(m)g FQ(+)g FJ(n)23 b FL(2)h FQ(2)p FH(Z)n FJ(;)952 2219 y FQ(0)83 b(otherwise)456 2365 y(\(cf.)28 b(Example)f(2.1.10\).)p eop %%Page: 70 24 70 73 bop 456 226 a FM(70)838 b(3.)29 b(MODULAR)g(TENSOR)g(CA)-5 b(TEGORIES)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF