%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: tensor.dvi %%Pages: 18 %%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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FJ(\016)1648 1262 y FI(V)1715 1250 y FQ(:)h FJ(V)1884 1203 y FE(\030)1856 1250 y FL(\000)-40 b(!)23 b FJ(V)2054 1215 y FE(\003\003)456 1250 y FQ(\(2.2.1\))456 1428 y(with)28 b(the)g(prop)r(erties:)1615 1605 y FJ(\016)1652 1617 y FI(V)14 b FE(\012)p FI(W)1856 1605 y FQ(=)22 b FJ(\016)1980 1617 y FI(V)2056 1605 y FL(\012)c FJ(\016)2176 1617 y FI(W)2252 1605 y FJ(;)-1819 b FQ(\(2.2.2\))1754 1735 y FJ(\016)1791 1747 y Fs(1)1856 1735 y FQ(=)22 b(id)q FJ(;)-1580 b FQ(\(2.2.3\))1703 1874 y FJ(\016)1740 1886 y FI(V)1794 1869 y Fx(\003)1856 1874 y FQ(=)22 b(\()p FJ(\016)2015 1839 y FE(\003)2012 1894 y FI(V)2070 1874 y FQ(\))2102 1839 y FE(\000)p FM(1)2192 1874 y FJ(;)-1759 b FQ(\(2.2.4\))456 2031 y(where)27 b(for)g FJ(f)k FL(2)24 b FQ(Hom\()p FJ(U;)14 b(V)k FQ(\),)28 b FJ(f)1472 2001 y FE(\003)1533 2031 y FL(2)c FQ(Hom\()p FJ(V)1884 2001 y FE(\003)1922 2031 y FJ(;)14 b(U)2025 2001 y FE(\003)2063 2031 y FQ(\))28 b(is)f(de\014ned)h(b)n(y)f(\(2.1.15\).)605 2131 y(The)e(existence)g(of)h(suc)n(h)f(isomorphisms)f FJ(\016)k FQ(do)r(es)d(not)g(follo)n(w)g(from)g(the)h(other)e(axioms)h (of)456 2230 y(a)i(rigid)g(BTC.)g(Let)h(us)f(in)n(tro)r(duce)h(the)g (last)f(formal)g(de\014nition)h(of)f(this)h(c)n(hapter.)605 2390 y FP(Definition)k FQ(2.2.1)p FP(.)40 b FQ(A)35 b FO(ribb)l(on)j(c)l(ate)l(gory)43 b FQ(is)35 b(a)g(rigid)g(braided)f (tensor)h(category)e(with)456 2490 y(functorial)27 b(isomorphisms)f (\(2.2.1\))h(satisfying)g(\(2.2.2|2.2.4\))e(for)i(all)h(ob)5 b(jects)27 b FJ(V)46 b FQ(and)28 b FJ(W)12 b FQ(.)605 2650 y(As)25 b(b)r(efore,)g(unless)f(otherwise)g(sp)r(eci\014ed)h(w)n (e)f(assume)g(that)h FL(C)30 b FQ(is)24 b(ab)r(elian,)h FL(\012)f FQ(is)h(bilinear,)456 2749 y FK(1)i FQ(is)g(simple)h(and)g (End)o(\()p FK(1)p FQ(\))c(=)e FJ(k)s FQ(.)605 2849 y(The)29 b(term)g(\\ribb)r(on",)f(in)n(tro)r(duced)h(b)n(y)g(Reshetikhin)g(and)g (T)-7 b(uraev)28 b([)p FK(R)-8 b(T1)p FQ(],)29 b(will)h(b)r(e)f(ex-)456 2948 y(plained)i(later.)46 b(Ribb)r(on)32 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FE(\003)2824 3594 y FM(id)11 b FE(\012)p FI(e)2796 3641 y FL(\000)-26 b(\000)-19 b(\000)-26 b(!)23 b FJ(V)5 b(:)-2646 b FQ(\(2.2.6\))456 3799 y(Ho)n(w)n(ev)n(er,)25 b FJ( )31 b FQ(do)r(es)c(not)h(satisfy)f(the)h(prop)r(ert)n(y)g (\(2.2.2\))f(unless)g FL(C)32 b FQ(is)c(symmetric.)605 3959 y FP(Lemma)j FQ(2.2.2)p FP(.)40 b FO(In)29 b(any)i(rigid)g(br)l (aide)l(d)g(tensor)f(c)l(ate)l(gory)g FL(C)5 b FO(,)30 b(we)g(have)6 b FQ(:)1379 4111 y FJ( )1433 4123 y FI(V)14 b FE(\012)p FI(W)1637 4111 y FQ(=)22 b FJ(\033)1771 4123 y FI(W)9 b(V)1901 4111 y FJ(\033)1948 4123 y FI(V)15 b(W)2077 4111 y FQ(\()p FJ( )2163 4123 y FI(V)2240 4111 y FL(\012)j FJ( )2377 4123 y FI(W)2452 4111 y FQ(\))p FJ(;)1518 4236 y( )1572 4248 y Fs(1)1637 4236 y FQ(=)k(id)14 b FJ(:)456 4414 y FO(If,)30 b(in)g(addition,)i FL(C)i FO(is)d(ribb)l(on,)f(then)1467 4591 y FJ( )1521 4603 y FI(V)1575 4587 y Fx(\003)1637 4591 y FQ(=)22 b FJ(\016)1764 4557 y FE(\003)1761 4612 y FI(V)1819 4591 y FJ( )1876 4557 y FE(\003)1873 4612 y FI(V)1931 4591 y FJ(\016)1971 4557 y FE(\003)1968 4612 y FI(V)2025 4591 y FJ(:)605 4752 y FP(Pr)n(oof.)41 b FQ(Left)25 b(as)f(an)g(exercise,)g(whic)n(h)h (is)f(b)r(est)h(done)f(using)g(the)h(pictorial)f(tec)n(hnique)h(of)456 4851 y(Section)i(2.3.)p 3384 4851 V 3388 4798 50 4 v 3388 4851 V 3437 4851 4 57 v 605 5026 a(Let)h FL(C)k FQ(b)r(e)c(a)f(ribb)r(on)h(category)-7 b(.)35 b(W)-7 b(e)28 b(de\014ne)f(functorial)h(isomorphisms)1292 5182 y FJ(\022)1331 5194 y FI(V)1412 5182 y FQ(=)23 b FJ( )1554 5194 y FI(V)1611 5182 y FJ(\016)1648 5194 y FI(V)1715 5182 y FQ(:)28 b FJ(V)1884 5134 y FE(\030)1856 5182 y FL(\000)-40 b(!)23 b FJ(V)5 b(;)180 b(V)42 b FL(2)24 b FQ(Ob)13 b FL(C)5 b FJ(:)-2152 b FQ(\(2.2.7\))p eop %%Page: 34 6 34 37 bop 456 226 a FM(34)1032 b(2.)29 b(RIBBON)h(CA)-5 b(TEGORIES)456 425 y FQ(The)24 b(isomorphisms)e FJ(\022)27 b FQ(are)c(usually)g(called)h FO(b)l(alancing)j(isomorphisms)p FQ(,)h(or)23 b FO(twists)p FQ(.)35 b(It)24 b(follo)n(ws)456 525 y(from)j(Lemma)g(2.2.2)g(that)g(they)h(satisfy)f(the)h FO(b)l(alancing)j(axiom)p FQ(:)1401 679 y FJ(\022)1440 691 y FI(V)15 b FE(\012)p FI(W)1644 679 y FQ(=)23 b FJ(\033)1779 691 y FI(W)9 b(V)1908 679 y FJ(\033)1955 691 y FI(V)15 b(W)2085 679 y FQ(\()p FJ(\022)2156 691 y FI(V)2232 679 y FL(\012)j FJ(\022)2354 691 y FI(W)2430 679 y FQ(\))p FJ(;)-2029 b FQ(\(2.2.8\))1540 809 y FJ(\022)1579 821 y Fs(1)1644 809 y FQ(=)23 b(id)p FJ(;)-1368 b FQ(\(2.2.9\))1490 938 y FJ(\022)1529 950 y FI(V)1582 934 y Fx(\003)1644 938 y FQ(=)23 b(\()p FJ(\022)1803 950 y FI(V)1861 938 y FQ(\))1893 904 y FE(\003)1931 938 y FJ(:)-1498 b FQ(\(2.2.10\))456 1098 y(Con)n(v)n(ersely)-7 b(,)18 b(a)g(system)g(of)g(t)n(wists)g (satisfying)g(the)h(balancing)e(axioms)h(\(2.2.8{2.2.10\))d(uniquely) 456 1198 y(de\014nes)27 b FJ(\016)765 1210 y FI(V)850 1198 y FQ(satisfying)g(\(2.2.2{2.2.4\).)605 1361 y FP(Cor)n(ollar)-6 b(y)33 b FQ(2.2.3)p FP(.)39 b FO(If)27 b FL(C)k FO(is)c(a)g(symmetric)g (tensor)f(c)l(ate)l(gory,)j(then)d FJ(\016)2857 1373 y FI(V)2938 1361 y FQ(=)d FJ( )3083 1325 y FE(\000)p FM(1)3080 1385 y FI(V)3198 1361 y FO(de\014nes)456 1460 y(a)30 b(structur)l(e)e(of)i(a)h(ribb)l(on)f(c)l(ate)l(gory)h(on)e FL(C)5 b FO(.)605 1621 y FQ(A)19 b(non-trivial)e(example)h(of)g(a)g (ribb)r(on)g(category)f(is)h(pro)n(vided)f(b)n(y)h(the)h(category)e FL(R)p FJ(ep)3251 1633 y FI(f)3293 1621 y FJ(U)3350 1633 y FI(q)3387 1621 y FQ(\()p FA(g)p FQ(\))456 1721 y(of)30 b(\014nite)i(dimensional)e(represen)n(tations)f(of)i(the)g(quan)n(tum)g (group)e FJ(U)2701 1733 y FI(q)2738 1721 y FQ(\()p FA(g)p FQ(\))i(\(see)g(Section)f(1.3)456 1821 y(and)i(Example)h(2.1.4\).)52 b(Here)33 b FJ(q)j FQ(will)d(b)r(e)g(a)g(formal)f(v)-5 b(ariable.)52 b(Recalling)34 b(\(2.1.11\))o(,)g(w)n(e)f(see)456 1920 y(that)27 b FJ(V)702 1890 y FE(\003\003)798 1920 y FL(')22 b FJ(V)47 b FQ(as)27 b(a)g(v)n(ector)f(space,)h(but)h(has)f (a)h(di\013eren)n(t)f(action)g(of)h FJ(U)2743 1932 y FI(q)2779 1920 y FQ(\()p FA(g)p FQ(\),)g(namely)1288 2079 y FJ(\031)1335 2091 y FI(V)1389 2075 y Fx(\003\003)1459 2079 y FQ(\()p FJ(a)p FQ(\))23 b(=)g FJ(\031)1725 2091 y FI(V)1783 2079 y FQ(\()p FJ(\015)1863 2045 y FM(2)1900 2079 y FQ(\()p FJ(a)p FQ(\)\))p FJ(;)181 b(a)22 b FL(2)i FJ(U)2446 2091 y FI(q)2482 2079 y FQ(\()p FA(g)p FQ(\))p FJ(:)-2155 b FQ(\(2.2.11\))456 2234 y(These)27 b(t)n(w)n(o)g(represen)n (tations)e(are)i(isomorphic:)605 2395 y FP(Pr)n(oposition)k FQ(2.2.4)p FP(.)40 b FO(The)28 b(c)l(ate)l(gory)g FL(R)p FJ(ep)2017 2407 y FI(f)2059 2395 y FJ(U)2116 2407 y FI(q)2153 2395 y FQ(\()p FA(g)p FQ(\))f FO(is)h(a)f(ribb)l(on)h(c)l(ate)l(gory)g (with)g(b)l(alanc-)456 2505 y(ing)35 b FJ(\016)638 2517 y FI(V)729 2505 y FQ(=)f FJ(q)868 2474 y FM(2)p FI(\032)949 2505 y FQ(:)29 b FJ(V)1130 2457 y FE(\030)1102 2505 y FL(\000)-40 b(!)34 b FJ(V)1311 2474 y FE(\003\003)1383 2505 y FO(,)j(wher)l(e)f FJ(\032)g FO(is)g(half)h(of)f(the)g(sum)f(of)h (p)l(ositive)h(r)l(o)l(ots)e(in)h FA(g)p FO(.)56 b FQ(\()p FO(On)456 2607 y(the)34 b(weight)h(subsp)l(ac)l(e)g FJ(V)1264 2577 y FI(\025)1339 2607 y FL(\032)c FJ(V)19 b FO(,)36 b FJ(\025)31 b FL(2)h FA(h)1772 2577 y FE(\003)1811 2607 y FO(,)k FJ(q)1912 2577 y FM(2)p FI(\032)2018 2607 y FO(acts)e(as)h(a)f(multiplic)l(ation)i(by)f FJ(q)3063 2577 y FE(h)-11 b(h)p FM(2)p FI(\032;\025)p FE(i)g(i)3313 2607 y FO(and)456 2707 y FL(h)d(h)p FJ(\032;)14 b(\013)639 2719 y FI(i)667 2707 y FL(i)-14 b(i)24 b FQ(:=)e FL(h)-14 b(h)p FJ(\013)954 2719 y FI(i)983 2707 y FJ(;)14 b(\013)1073 2719 y FI(i)1101 2707 y FL(i)-14 b(i)p FJ(=)p FQ(2)22 b(=)h FJ(d)1388 2719 y FI(i)1446 2707 y FO(.)p FQ(\))605 2868 y FP(Pr)n(oof.)41 b FQ(Quite)27 b(trivial)g(and)h(based)f(on)g (the)h(iden)n(tit)n(y)1346 3022 y FJ(\015)1394 2988 y FM(2)1431 3022 y FQ(\()p FJ(a)p FQ(\))23 b(=)g FJ(q)1690 2988 y FM(2)p FI(\032)1762 3022 y FJ(aq)1846 2988 y FE(\000)p FM(2)p FI(\032)1969 3022 y FJ(;)180 b(a)23 b FL(2)g FJ(U)2374 3034 y FI(q)2411 3022 y FQ(\()p FA(g)p FQ(\))p FJ(;)456 3182 y FQ(whic)n(h)k(can)g(b)r(e)h(easily)f(c)n(hec)n(k)n(ed)g(on)g (the)h(generators)d(of)j FJ(U)2304 3194 y FI(q)2340 3182 y FQ(\()p FA(g)p FQ(\))g(\(see)f(\(1.3.11\))o(\).)p 3384 3182 4 57 v 3388 3129 50 4 v 3388 3182 V 3437 3182 4 57 v 605 3360 a(Let)22 b(us)f(calculate)f FJ( )25 b FQ(and)c FJ(\022)j FQ(in)d(the)h(category)d FL(C)28 b FQ(=)22 b FL(R)p FJ(ep)2349 3372 y FI(f)2392 3360 y FJ(U)2449 3372 y FI(q)2486 3360 y FQ(\()p FA(g)p FQ(\).)35 b(Let)21 b FJ(R)h FQ(b)r(e)g(the)g(univ)n(ersal)456 3460 y(R-matrix)k(of)i FJ(U)966 3472 y FI(q)1002 3460 y FQ(\()p FA(g)p FQ(\))g(\(see)g (Example)f(1.2.8\(iii\))g(and)g(\(1.3.12\))o(\).)37 b(W)-7 b(rite)1336 3634 y FJ(R)1400 3598 y FE(\000)p FM(1)1399 3656 y(21)1512 3634 y FQ(=)1600 3555 y Fy(X)1734 3634 y FJ(a)1778 3646 y FI(i)1823 3634 y FL(\012)19 b FJ(b)1943 3646 y FI(i)1970 3634 y FJ(;)97 b(R)2153 3646 y FM(21)2246 3634 y FQ(=)23 b FJ(P)12 b FQ(\()p FJ(R)q FQ(\))p FJ(;)-2094 b FQ(\(2.2.12\))456 3816 y(where)28 b FJ(P)12 b FQ(\()p FJ(a)19 b FL(\012)g FJ(b)p FQ(\))25 b(:=)f FJ(b)19 b FL(\012)g FJ(a)p FQ(.)40 b(Recall)28 b(that)h FJ(\033)g FQ(=)24 b FJ(P)12 b(R)26 b FQ(=)e FJ(R)2295 3828 y FM(21)2366 3816 y FJ(P)12 b FQ(,)29 b(so)f FJ(\033)2636 3786 y FE(\000)p FM(1)2750 3816 y FQ(=)d FJ(P)12 b(R)2969 3780 y FE(\000)p FM(1)2968 3838 y(21)3057 3816 y FQ(.)41 b(Let)28 b FL(f)p FJ(v)3352 3828 y FI(i)3380 3816 y FL(g)p FQ(,)456 3915 y FL(f)p FJ(v)541 3885 y FI(i)568 3915 y FL(g)23 b FQ(b)r(e)h(dual)f (bases)g(in)h FJ(V)19 b FQ(,)24 b FJ(V)1410 3885 y FE(\003)1448 3915 y FQ(,)h(and)e(let)h FJ(x)g FL(2)f FJ(V)1985 3885 y FE(\003\003)2057 3915 y FQ(.)36 b(Then)23 b(w)n(e)g(compute)h FJ( )s FQ(\()p FJ(x)p FQ(\))h(using)f(\(2.1.9\))o(,)456 4015 y(\(2.2.6\))o(:)539 4211 y FJ(x)f FL(7!)715 4132 y Fy(X)763 4309 y FI(i)849 4211 y FJ(v)889 4223 y FI(i)935 4211 y FL(\012)18 b FJ(v)1061 4177 y FI(i)1108 4211 y FL(\012)g FJ(x)23 b FL(7!)1367 4132 y Fy(X)1390 4309 y FI(i;j)1501 4211 y FJ(v)1541 4223 y FI(i)1587 4211 y FL(\012)18 b FJ(b)1706 4223 y FI(j)1741 4211 y FQ(\()p FJ(x)p FQ(\))h FL(\012)f FJ(a)1998 4223 y FI(j)2033 4211 y FQ(\()p FJ(v)2108 4177 y FI(i)2136 4211 y FQ(\))24 b FL(7!)850 4449 y(7!)956 4370 y Fy(X)979 4547 y FI(i;j)1090 4449 y FJ(v)1130 4461 y FI(i)1171 4449 y FQ(\()p FJ(b)1239 4461 y FI(j)1274 4449 y FQ(\()p FJ(x)p FQ(\))q FJ(;)14 b(a)1467 4461 y FI(j)1502 4449 y FQ(\()p FJ(v)1577 4415 y FI(i)1605 4449 y FQ(\)\))24 b(=)1780 4370 y Fy(X)1803 4547 y FI(i;j)1914 4449 y FJ(v)1954 4461 y FI(i)1996 4449 y FQ(\()p FJ(\015)2076 4415 y FE(\000)p FM(1)2165 4449 y FQ(\()p FJ(a)2241 4461 y FI(j)2276 4449 y FQ(\))p FJ(b)2344 4461 y FI(j)2379 4449 y FQ(\()p FJ(x)p FQ(\))q FJ(;)14 b(v)2571 4415 y FI(i)2599 4449 y FQ(\))23 b(=)2742 4370 y Fy(X)2786 4547 y FI(j)2876 4449 y FJ(\015)2924 4415 y FE(\000)p FM(1)3012 4449 y FQ(\()p FJ(a)3088 4461 y FI(j)3123 4449 y FQ(\))p FJ(b)3191 4461 y FI(j)3226 4449 y FQ(\()p FJ(x)p FQ(\))p FJ(:)456 4687 y FQ(Th)n(us)k(w)n(e)g (obtained:)605 4848 y FP(Lemma)k FQ(2.2.5)p FP(.)40 b FJ(\022)30 b FQ(=)e FJ(u)1363 4818 y FE(\000)p FM(1)1451 4848 y FJ(q)1491 4818 y FM(2)p FI(\032)1563 4848 y FO(,)34 b(wher)l(e)f FJ(u)1907 4818 y FE(\000)p FM(1)2023 4848 y FQ(:=)2139 4786 y Fy(P)2226 4873 y FI(j)2275 4848 y FJ(\015)2323 4818 y FE(\000)p FM(1)2412 4848 y FQ(\()p FJ(a)2488 4860 y FI(j)2523 4848 y FQ(\))p FJ(b)2591 4860 y FI(j)2658 4848 y FO(lies)g(in)g(a)g(c)l(ertain)f(c)l(om-)456 4955 y(pletion)e(of)h FJ(U)884 4967 y FI(q)920 4955 y FQ(\()p FA(g)p FQ(\))p FO(.)605 5116 y FP(Exer)n(cise)h FQ(2.2.6)p FP(.)40 b FQ(Deduce)27 b(from)f(the)h(previous)f(lemma)h (that)g FJ(\022)i FQ(acts)d(as)g(a)h(m)n(ultiplica-)456 5216 y(tion)g(b)n(y)h FJ(q)782 5185 y FE(h)-11 b(h)p FI(\025;\025)p FM(+2)p FI(\032)p FE(i)g(i)1115 5216 y FQ(on)28 b(the)f(irreducible)g FJ(U)1841 5228 y FI(q)1878 5216 y FQ(\()p FA(g)p FQ(\)-mo)r(dule)h FJ(V)2353 5228 y FI(\025)2424 5216 y FQ(with)g(highest)g(w)n(eigh)n(t)f FJ(\025)p FQ(.)p eop %%Page: 35 7 35 38 bop 1194 226 a FM(2.3.)29 b(GRAPHICAL)h(CALCULUS)g(F)n(OR)f (MORPHISMS)648 b(35)605 425 y FQ(The)35 b(cen)n(tral)g(elemen)n(t)g FJ(\022)k FQ(=)c FJ(u)1606 395 y FE(\000)p FM(1)1695 425 y FJ(q)1735 395 y FM(2)p FI(\032)1842 425 y FQ(is)h(called)f(a)g (\(univ)n(ersal\))f FO(Casimir)39 b(op)l(er)l(ator)45 b FQ(for)456 525 y FJ(U)513 537 y FI(q)549 525 y FQ(\()p FA(g)p FQ(\).)37 b(The)28 b(elemen)n(t)g FJ(u)f FQ(w)n(as)f(in)n(tro)r (duced)i(and)f(studied)h(b)n(y)g(Drinfeld)g([)p FK(Dr2)p FQ(].)605 624 y(Our)j(next)i(example)e(of)h(a)g(ribb)r(on)f(category)f (will)j(b)r(e)f(the)g(Drinfeld)h(category)d FL(D)r FQ(\()p FA(g)p FJ(;)14 b FH({)s FQ(\))456 724 y(\(see)27 b(Section)h(1.4\).)605 897 y FP(Theorem)k FQ(2.2.7)e(\(Drinfeld)f([)p FK(Dr2)p FQ(]\))p FP(.)41 b FQ(\(i\))g FO(The)h(c)l(ate)l(gory)f FL(D)r FQ(\()p FA(g)p FJ(;)14 b FH({)s FQ(\))42 b FO(is)f(a)g(ribb)l (on)h(c)l(ate-)456 997 y(gory,)31 b(with)f FJ(V)914 967 y FE(\003)952 997 y FJ(;)14 b(e;)g(i;)g(\016)32 b FO(the)e(same)g(as)g (in)g(the)g(c)l(ate)l(gory)g(of)h(ve)l(ctor)f(sp)l(ac)l(es.)605 1097 y FQ(\(ii\))i FO(In)e(this)i(c)l(ate)l(gory,)g(the)f(universal)h (twist)e(is)i(given)f(by)h FJ(\022)27 b FQ(=)e FJ(e)2676 1067 y FI(\031)r FM(i)p FI(D)r(=)p Fz({)2908 1097 y FO(wher)l(e)32 b FJ(D)h FO(is)e(the)456 1196 y(Casimir)g(op)l(er)l(ator)g(of)f FA(g)g FO(de\014ne)l(d)g(by)37 b FQ(\(1.4.4\))p FO(.)605 1370 y FQ(Note)30 b(that)f(this)h(theorem)f(is)h(not)f(immediately)h (ob)n(vious)e(since)i(b)r(oth)g(the)g(rigidit)n(y)e(ax-)456 1469 y(ioms)41 b(and)g(the)h(de\014nition)f(of)h FJ(\022)h FQ(in)n(v)n(olv)n(e)d(the)i(Drinfeld)g(asso)r(ciator)d FJ(\013)j FQ(\(1.4.8\))o(,)j(i.e.,)g(the)456 1569 y(asymptotics)26 b(of)i(solutions)f(of)g(KZ)g(equations.)36 b(Ho)n(w)n(ev)n(er,)26 b(it)i(is)g(not)f(to)r(o)h(di\016cult.)605 1669 y(The)22 b(form)n(ula)g(for)f(the)i(univ)n(ersal)e(t)n(wist)h(giv)n(en)f(b)n(y)h (this)h(theorem)e(agrees)g(with)h FJ(q)3138 1639 y FE(h)-11 b(h)p FI(\025;\025)p FM(+2)p FI(\032)q FE(i)g(i)456 1768 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FP(.)40 b FQ(A)h FO(ribb)l(on)j (tangle)k FQ(\(or)40 b(a)h FO(fr)l(ame)l(d)j(tangle)6 b FQ(\))42 b(of)g FJ(n)f FQ(strands)f(is)i(an)456 3187 y(isotop)n(y)e(class)g(of)h(a)g(union)g(of)g FJ(n)g FQ(non-in)n (tersecting)f(ribb)r(ons)g(in)i FH(R)2695 3157 y FM(2)2766 3187 y FL(\002)27 b FQ([0)p FJ(;)14 b FQ(1],)43 b(suc)n(h)e(that)456 3286 y(the)30 b(bases)e(of)i(all)f(ribb)r(ons)g(lie)h(on)f(the)h(lines) g FH(R)25 b FL(\002)20 b(f)p FQ(0)p FL(g)e(\002)h(f)p FQ(0)p FJ(;)14 b FQ(1)p FL(g)p FQ(,)28 b(and)i(near)e(these)i(lines,)g (the)456 3386 y(ribb)r(ons)d(are)f(turned)i(with)g(their)g(face)f(side) g(up)n(w)n(ard.)36 b(A)28 b(ribb)r(on)f(tangle)g(comp)r(osed)g(only)h (of)456 3486 y(ann)n(uli)i(is)h(called)f(a)g FO(ribb)l(on)38 b FQ(\(or)30 b FO(fr)l(ame)l(d)9 b FQ(\))31 b FO(link)9 b FQ(,)32 b(and)f(a)f(ribb)r(on)h(link)f(consisting)g(of)h(a)f(single) 456 3585 y(ann)n(ulus)d(is)g(called)g(a)h FO(fr)l(ame)l(d)i(knot)p FQ(.)605 3749 y(Examples)19 b(of)h(ribb)r(on)g(tangles)f(are)g(sho)n 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y(thing)j(for)h(the)g(top)f(edge)h(of)f (the)h(coup)r(on,)g(where)f(no)n(w)g(w)n(e)h(put)g FJ(")c FQ(=)f(+)k(for)h(the)g(\\outgoing")456 2196 y(ribb)r(ons,)36 b(and)e(get)g(an)h(ob)5 b(ject)34 b FJ(Y)19 b FQ(.)58 b(Then)35 b(w)n(e)f(require)f(that)i(the)g(coup)r(on)f(is)h(lab)r(eled) g(b)n(y)f(a)456 2295 y(morphism)27 b FJ(f)18 b FQ(:)27 b FJ(X)j FL(!)23 b FJ(Y)18 b FQ(.)605 2434 y(Note)26 b(that)g(an)n(y)f(plane)h(pro)5 b(jection)25 b(of)g(a)h(usual)f(tangle) h(can)f(b)r(e)h(considered)f(as)g(a)h(ribb)r(on)456 2533 y(tangle)f(whic)n(h)g(is)h(\\lying)f(\015at",)g(i.e.,)h(alw)n(a)n(ys)e (with)i(the)g(face)g(side)f(up,)i(as)e(in)g(Example)g(2.3.10)456 2633 y(\(this)g(is)g(called)g(the)h(\\blac)n(kb)r(oard)d(framing"\).)35 b(F)-7 b(or)24 b(tec)n(hnical)h(reasons,)f(w)n(e)h(will)g(often)h(dra)n (w)456 2732 y(lines)35 b(instead)f(of)h(ribb)r(ons,)i(alw)n(a)n(ys)c (assuming)h(the)h(blac)n(kb)r(oard)f(framing.)58 b(Also,)37 b(w)n(e)d(will)456 2832 y(often)27 b(omit)h(the)g(arro)n(ws)d(p)r(oin)n (ting)j(up)g(when)f(there)h(is)f(no)h(am)n(biguit)n(y)-7 b(.)605 2932 y(Note,)27 b(ho)n(w)n(ev)n(er,)e(that)j(di\013eren)n(t)f (pro)5 b(jections)25 b(of)i(the)h(same)e(\(non-framed\))h(tangle)f(no)n (w)456 3031 y(giv)n(e)e(rise)i(to)f(di\013eren)n(t)h(framed)g(tangles.) 35 b(F)-7 b(or)25 b(example,)h(the)g(ribb)r(on)g(tangle)f(corresp)r (onding)456 3131 y(to)33 b(the)g(t)n(wist)g FJ(\022)961 3143 y FI(V)1052 3131 y FQ(is)g(no)g(longer)f(isotopic)g(to)h(the)g (trivial)g(tangle.)53 b(W)-7 b(e)33 b(will)h(insert)e(a)h(circle)456 3231 y(con)n(taining)26 b(the)i(letter)g FJ(\022)i FQ(\(or)d FJ(\022)1468 3200 y FE(\000)p FM(1)1557 3231 y FQ(\))h(to)g(represen)n (t)e(t)n(wists:)1313 4048 y @beginspecial 0 @llx 0 @lly 153 @urx 81 @ury 1530 @rwi @setspecial %%BeginDocument: figures/mod1a.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: mod1a.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Tue Oct 13 13:58:42 1998 %%For: shurik@localhost.localdomain (Alexander Kirillov,,,) %%Orientation: Portrait %%BoundingBox: 0 0 153 81 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind 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FQ(,)g(Theorem)e(2.5]\).)605 4548 y(Let)c FL(C)27 b FQ(b)r(e)22 b(a)g(ribb)r(on)f(category)-7 b(.)33 b(Fix)23 b(ob)5 b(jects)21 b FJ(V)2077 4560 y FM(1)2115 4548 y FJ(;)14 b(:)g(:)g(:)f(;)h(V)2347 4560 y FI(n)2415 4548 y FQ(in)22 b FL(C)27 b FQ(and)21 b(consider)g(all)h(p)r(ossible)456 4647 y(expressions)j(of)j(the)g(form)1067 4782 y FJ(X)i FQ(=)22 b(\(\()p FJ(V)1365 4794 y FM(1)1422 4782 y FL(\012)c FJ(V)1553 4794 y FM(2)1591 4782 y FQ(\))g FL(\012)g FJ(V)1772 4794 y FM(4)1810 4782 y FQ(\))h FL(\012)f FQ(\(\()p FJ(V)2075 4748 y FE(\003\003\003)2056 4802 y FM(1)2200 4782 y FL(\012)g FK(1)p FQ(\))g FL(\012)2464 4748 y FE(\003)2503 4782 y FJ(V)2551 4794 y FM(2)2588 4782 y FQ(\))h FL(\012)f(\001)c(\001)g (\001)-2363 b FQ(\(2.3.16\))456 4917 y(where)29 b(w)n(e)h(tak)n(e)f (the)h(tensor)f(pro)r(duct)h(of)g FJ(V)1861 4929 y FM(1)1899 4917 y FJ(;)14 b(:)g(:)g(:)f(;)h(V)2131 4929 y FI(n)2207 4917 y FQ(in)30 b(arbitrary)e(order)g(and)i(allo)n(w)f(rep)r(e-)456 5016 y(titions,)e(arbitrary)f(n)n(um)n(b)r(er)h(of)h(left)g(and)g(righ) n(t)e(stars)h(and)g FK(1)p FQ('s.)605 5116 y(T)-7 b(o)24 b(eac)n(h)f(expression)f FJ(X)31 b FQ(as)23 b(ab)r(o)n(v)n(e)f(w)n(e)i (assign)f(a)g(sequence)h FJ(F)12 b FQ(\()p FJ(X)7 b FQ(\))24 b(of)f(arro)n(ws)f(and)i(lab)r(els)456 5216 y(b)n(y)32 b(the)g(follo)n(wing)f(rule:)46 b(to)33 b(an)f(ob)5 b(ject)1769 5185 y FE(\003\001\001\001)o(\003)1900 5216 y FJ(V)1967 5185 y FE(\003\001\001\001)o(\003)2131 5216 y FQ(w)n(e)31 b(assign)h FL(#)2550 5228 y FI(V)2639 5216 y FQ(if)h(the)f(total)g(n)n (um)n(b)r(er)h(of)p eop %%Page: 42 14 42 45 bop 456 226 a FM(42)1032 b(2.)29 b(RIBBON)h(CA)-5 b(TEGORIES)456 425 y FQ(stars)27 b(is)i(o)r(dd)g(and)g FL(")1111 437 y FI(V)1197 425 y FQ(if)g(it)g(is)g(ev)n(en.)40 b(All)29 b FK(1)p FQ('s)g(are)f(skipp)r(ed.)40 b(F)-7 b(or)29 b(example,)f(to)h(the)g(elemen)n(t)456 525 y(\(2.3.16\))d(w)n (e)h(assign)g(the)h(sequence)1550 728 y @beginspecial 0 @llx 0 @lly 96 @urx 16 @ury 960 @rwi @setspecial 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FL(i)k FQ(with)g(the)h(m)n(ultiplication)f(rule)f FJ(x)2844 4163 y FI(i)2872 4151 y FJ(x)2919 4163 y FI(j)2978 4151 y FQ(=)3066 4089 y Fy(P)3153 4176 y FI(k)3208 4151 y FJ(N)3284 4121 y FI(k)3275 4172 y(ij)3333 4151 y FJ(x)3380 4163 y FI(k)3421 4151 y FQ(.)605 4250 y(3.)37 b(Since)27 b(End)14 b FJ(V)1134 4262 y FI(i)1185 4250 y FQ(=)23 b FJ(k)s FQ(,)k(w)n(e)g(ha)n(v)n(e)1424 4400 y FJ(\022)1463 4412 y FI(V)1502 4420 y FG(i)1556 4400 y FQ(=)c FJ(\022)1683 4412 y FI(i)1724 4400 y FQ(id)1793 4412 y FI(V)1832 4420 y FG(i)1863 4400 y FJ(;)180 b FQ(dim)14 b FJ(V)2266 4412 y FI(i)2317 4400 y FQ(=)23 b FJ(d)2448 4412 y FI(i)456 4400 y FQ(\(2.4.4\))456 4550 y(for)37 b(some)g FJ(\022)850 4562 y FI(i)878 4550 y FJ(;)14 b(d)958 4562 y FI(i)1025 4550 y FL(2)41 b FJ(k)s FQ(.)67 b(The)37 b(n)n(um)n(b)r(er)h FJ(d)1793 4562 y FI(i)1859 4550 y FQ(is)f(called)h(the)g FO(quantum)g(dimension)45 b FQ(of)38 b FJ(V)3268 4562 y FI(i)3296 4550 y FQ(,)i(cf.)456 4650 y(Exercise)26 b(2.3.4.)35 b(W)-7 b(e)28 b(ha)n(v)n(e)1297 4799 y FJ(\022)1336 4811 y FM(0)1396 4799 y FQ(=)23 b(1)p FJ(;)59 b(\022)1647 4811 y FI(i)1670 4795 y Fx(\003)1733 4799 y FQ(=)22 b FJ(\022)1859 4811 y FI(i)1887 4799 y FJ(;)-1454 b FQ(\(2.4.5\))1297 4941 y FJ(d)1340 4953 y FM(0)1400 4941 y FQ(=)23 b(1)p FJ(;)60 b(d)1656 4953 y FI(i)1679 4937 y Fx(\003)1741 4941 y FQ(=)23 b FJ(d)1872 4953 y FI(i)1900 4941 y FJ(;)60 b(d)2026 4953 y FI(i)2053 4941 y FJ(d)2096 4953 y FI(j)2155 4941 y FQ(=)2242 4879 y Fy(P)2330 4966 y FI(k)2371 4941 y FJ(N)2447 4907 y FI(k)2438 4962 y(ij)2496 4941 y FJ(d)2539 4953 y FI(k)2580 4941 y FJ(:)-2147 b FQ(\(2.4.6\))605 5100 y FP(Lemma)31 b FQ(2.4.1)p FP(.)40 b FO(In)32 b(a)g(semisimple)i (ribb)l(on)e(c)l(ate)l(gory,)i(al)t(l)f(dimensions)g FJ(d)2969 5112 y FI(i)3024 5100 y FQ(=)27 b(dim)14 b FJ(V)3316 5112 y FI(i)3377 5100 y FO(of)456 5199 y(simple)30 b(obje)l(cts)h(ar)l(e)f(non-zer)l(o.)p eop %%Page: 45 17 45 48 bop 1453 226 a FM(2.4.)29 b(SEMISIMPLE)e(CA)-5 b(TEGORIES)908 b(45)605 425 y FP(Pr)n(oof.)41 b FQ(Since)28 b FJ(N)1226 395 y FM(0)1217 447 y FI(ii)1263 430 y Fx(\003)1328 425 y FQ(=)c FJ(N)1493 395 y FI(i)1484 447 y(i)p FM(0)1569 425 y FQ(=)g(1,)k(w)n(e)g(can)g(write)h FJ(V)2289 437 y FI(i)2336 425 y FL(\012)19 b FJ(V)2486 395 y FE(\003)2468 447 y FI(i)2549 425 y FQ(=)24 b FJ(X)i FL(\010)18 b FJ(X)2892 395 y FE(0)2915 425 y FQ(,)29 b(where)f FJ(X)j FL(')24 b FK(1)456 525 y FQ(\(not)39 b(canonically\),)j(and)d(Hom\()p FK(1)p FJ(;)14 b(X)1688 495 y FE(0)1711 525 y FQ(\))43 b(=)g(0.)72 b(Th)n(us,)42 b(the)e(maps)f FJ(i)9 b FQ(:)32 b FK(1)43 b FL(!)g FJ(V)3021 537 y FI(i)3075 525 y FL(\012)26 b FJ(V)3233 495 y FE(\003)3214 546 y FI(i)3311 525 y FQ(and)456 624 y FJ(e)9 b FQ(:)27 b FJ(V)602 636 y FI(i)634 624 y FL(\012)t FJ(V)769 594 y FE(\003)751 646 y FI(i)830 624 y FL(!)c FK(1)d FQ(can)g(b)r(e)h(considered)e(as)g(maps)h FK(1)j FL(!)g FJ(X)7 b FQ(,)21 b FJ(X)29 b FL(!)23 b FK(1)p FQ(.)35 b(Since)20 b(b)r(oth)h(of)f(these)g(maps)456 724 y(are)26 b(non-zero)g(\(this)i(follo)n(ws)f(from)g(rigidit)n(y\),)g (their)h(comp)r(osition)f(is)g(also)g(non-zero.)p 3384 724 4 57 v 3388 671 50 4 v 3388 724 V 3437 724 4 57 v 605 899 a FP(Remark)32 b FQ(2.4.2)p FP(.)39 b FQ(It)28 b(is)e(tempting)h(\(and)g(man)n(y)f(authors)g(do)g(so\))g(to)h(c)n(ho)r (ose)e(some)i(iden-)456 1005 y(ti\014cation)35 b FJ(V)881 975 y FE(\003)862 1027 y FI(i)983 958 y FE(\030)955 1005 y FL(\000)-40 b(!)36 b FJ(V)1147 1017 y FI(i)1170 1001 y Fx(\003)1245 1005 y FQ(to)f(simplify)h(the)f(form)n(ulas.)59 b(Ho)n(w)n(ev)n(er,)35 b(this)g(should)g(b)r(e)h(a)n(v)n(oided.)456 1105 y(Not)29 b(only)g(this)h(cannot)e(b)r(e)i(done)f(canonically)-7 b(,)29 b(but)h(in)f(man)n(y)g(cases)f(it)i(cannot)f(b)r(e)h(done)f(at) 456 1205 y(all!)56 b(Indeed,)37 b(in)d(order)f(for)h(this)g(to)h(b)r(e) f(useful)h(w)n(e)f(need)g(to)g(c)n(ho)r(ose)g(these)g(isomorphisms)456 1304 y FJ(\036)505 1316 y FI(i)542 1304 y FQ(:)28 b FJ(V)660 1274 y FE(\003)641 1326 y FI(i)721 1304 y FL(!)23 b FJ(V)875 1316 y FI(i)898 1300 y Fx(\003)965 1304 y FQ(in)28 b(suc)n(h)f(a)h(w)n (a)n(y)e(that)i(the)g(comp)r(osition)1588 1512 y FJ(V)1636 1524 y FI(i)1716 1450 y(\036)1756 1423 y Fx(\000)p FF(1)1756 1472 y FG(i)1778 1460 y Fx(\003)1687 1512 y FL(\000)-19 b(\000)f(!)23 b FJ(V)1951 1478 y FE(\003)1932 1533 y FI(i)1955 1516 y Fx(\003)2047 1456 y FI(\036)2087 1431 y Fx(\003)2087 1472 y FG(i)2018 1512 y FL(\000)-16 b(!)23 b FJ(V)2240 1478 y FE(\003\003)2221 1533 y FI(i)456 1664 y FQ(coincides)31 b(with)h FJ(\016)1041 1676 y FI(V)1080 1684 y FG(i)1111 1664 y FQ(.)49 b(W)-7 b(e)32 b(lea)n(v)n(e)f(it)h(as)f (an)g(exercise)g(to)h(the)g(reader)e(that)i(for)f(the)h(category)456 1764 y FL(R)p FJ(ep)607 1776 y FI(f)649 1764 y FQ(\()p FA(sl)742 1776 y FM(2)779 1764 y FQ(\))f(\(with)g(the)f(standard)g (balancing\),)g(considered)f(in)i(Example)e(2.2.8,)h(this)g(is)g(im-) 456 1873 y(p)r(ossible:)49 b(regardless)32 b(of)i(the)h(c)n(hoice)e(of) h FJ(\036)1858 1885 y FI(i)1887 1873 y FQ(,)i(the)e(comp)r(osition)g (ab)r(o)n(v)n(e)f(will)h(giv)n(e)3152 1852 y(~)3148 1873 y FJ(\016)k FQ(rather)456 1973 y(than)27 b FJ(\016)s FQ(.)605 2133 y(When)i FL(C)h FQ(=)24 b FL(R)p FJ(ep)1162 2145 y FI(f)1205 2133 y FJ(G)p FQ(,)30 b FJ(G)p FQ(|a)e(\014nite)i (group,)d(an)i(imp)r(ortan)n(t)f(role)g(in)h(the)g(represen)n(tation) 456 2232 y(theory)35 b(of)h FJ(G)g FQ(is)g(pla)n(y)n(ed)e(b)n(y)i(the)g (regular)e(represen)n(tation)g FJ(R)k FQ(=)e FJ(F)12 b(un)p FQ(\()p FJ(G)p FQ(\).)62 b(This)36 b(is)g(a)f FJ(G)p FQ(-)456 2335 y(bimo)r(dule)c(\(i.e.,)g(an)f(elemen)n(t)h(of)f FL(C)1587 2304 y Fv(\002)p FM(2)1676 2335 y FQ(\);)i(as)e(a)g(bimo)r (dule,)i(it)f(is)f(isomorphic)g(to)3029 2272 y Fy(L)3135 2335 y FJ(V)3183 2347 y FI(i)3231 2335 y Fw(\002)20 b FJ(V)3383 2304 y FE(\003)3364 2356 y FI(i)3421 2335 y FQ(.)456 2434 y(More)32 b(generally)-7 b(,)33 b(an)n(y)f(rigid)h (semisimple)g(ab)r(elian)g(category)e(automatically)h(giv)n(es)g(rise)h (to)456 2534 y(an)27 b(ob)5 b(ject)27 b FJ(R)d FL(2)f FQ(ind)p FL(\000C)1216 2504 y Fv(\002)p FM(2)1333 2534 y FQ(de\014ned)28 b(b)n(y)1633 2698 y FJ(R)c FQ(=)1808 2619 y Fy(M)1819 2797 y FI(i)p FE(2)p FI(I)1947 2698 y FJ(V)1995 2710 y FI(i)2042 2698 y Fw(\002)18 b FJ(V)2192 2664 y FE(\003)2173 2718 y FI(i)2230 2698 y FJ(;)-1797 b FQ(\(2.4.7\))456 2944 y(where)31 b(w)n(e)g(denote)h(b)n(y)f(ind)p FL(\000C)1446 2914 y Fv(\002)p FM(2)1567 2944 y FQ(the)h(category)d (whose)i(ob)5 b(jects)32 b(are)e(in\014nite)j(sums)e(of)h(the)456 3044 y(form)655 2981 y Fy(P)756 3044 y FJ(A)818 3056 y FI(i)866 3044 y Fw(\002)20 b FJ(B)1014 3056 y FI(i)1073 3044 y FQ(with)31 b FJ(A)1327 3056 y FI(i)1354 3044 y FJ(;)14 b(B)1454 3056 y FI(i)1510 3044 y FL(2)28 b(C)5 b FQ(.)45 b(The)31 b(ob)5 b(ject)30 b FJ(R)h FQ(do)r(es)f(not)h(dep)r (end)g(on)f(the)h(c)n(hoice)f(of)456 3143 y(represen)n(tativ)n(es)d FJ(V)1072 3155 y FI(i)1131 3143 y FQ(of)j(the)g(isomorphism)f(classes:) 40 b(if)31 b(w)n(e)f(c)n(ho)r(ose)e(another)h(represen)n(tativ)n(e)468 3224 y(~)456 3245 y FJ(V)504 3257 y FI(i)532 3245 y FQ(,)g(then)g(one)g (has)f(a)h(canonical)e(isomorphism)h FJ(R)2117 3198 y FE(\030)2089 3245 y FL(\000)-39 b(!)2241 3224 y FQ(~)2223 3245 y FJ(R)q FQ(.)40 b(In)29 b(particular,)f(if)i FL(C)j FQ(is)c(balanced,)456 3345 y(then)f FJ(R)h FQ(is)f(symmetric)f(in)h (the)h(follo)n(wing)e(sense:)37 b(w)n(e)27 b(sa)n(y)g(that)h(an)g(ob)5 b(ject)28 b FJ(R)c FL(2)g FQ(ind)p FL(\000C)3272 3315 y Fv(\002)p FM(2)3389 3345 y FQ(is)456 3447 y FO(symmetric)29 b FQ(if)g(w)n(e)f(are)g(giv)n(en)g(an)g(isomorphism)g FJ(s)9 b FQ(:)28 b FJ(R)2183 3417 y FM(op)2310 3400 y FE(\030)2282 3447 y FL(\000)-40 b(!)25 b FJ(R)30 b FQ(suc)n(h)e(that)h FJ(ss)2955 3417 y FM(op)3054 3447 y FQ(=)24 b(id)q(.)40 b(Here)456 3547 y(the)28 b(functor)f(op)g(is)h(de\014ned)g(b)n(y)f(\()p FJ(A)19 b Fw(\002)f FJ(B)t FQ(\))1782 3517 y FM(op)1880 3547 y FQ(=)k FJ(B)h Fw(\002)18 b FJ(A)p FQ(.)605 3646 y(F)-7 b(or)29 b(the)h(ob)5 b(ject)29 b FJ(R)h FQ(de\014ned)g(b)n(y)g (\(2.4.7\))o(,)g FJ(s)g FQ(can)f(b)r(e)h(written)f(explicitly)h(as)e (follo)n(ws:)40 b(for)456 3753 y(ev)n(ery)25 b FJ(i)p FQ(,)i(c)n(ho)r(ose)e(an)h(isomorphism)g FJ(')1668 3765 y FI(i)1705 3753 y FQ(:)i FJ(V)1823 3723 y FE(\003)1804 3775 y FI(i)1912 3706 y FE(\030)1884 3753 y FL(\000)-40 b(!)23 b FJ(V)2063 3765 y FI(i)2086 3749 y Fx(\003)2126 3753 y FQ(,)k(and)g(de\014ne)g FJ( )2630 3765 y FI(i)2667 3753 y FQ(:)g FJ(V)2784 3723 y FE(\003)2765 3775 y FI(i)2839 3753 y Fw(\002)16 b FJ(V)2968 3765 y FI(i)3019 3753 y FL(!)23 b FJ(V)3173 3765 y FI(i)3196 3749 y Fx(\003)3253 3753 y Fw(\002)16 b FJ(V)3400 3723 y FE(\003)3382 3775 y FI(i)3405 3758 y Fx(\003)456 3853 y FQ(b)n(y)26 b FJ( )624 3865 y FI(i)675 3853 y FQ(=)d FJ(')817 3865 y FI(i)861 3853 y Fw(\002)17 b FJ(\016)980 3865 y FI(V)1019 3873 y FG(i)1049 3853 y FQ(\()p FJ(')1135 3823 y FE(\003)1135 3874 y FI(i)1174 3853 y FQ(\))1206 3823 y FE(\000)p FM(1)1296 3853 y FQ(.)36 b(This)27 b(ob)n(viously)e(do)r(es)i(not)g(dep)r(end)g (on)g(the)g(c)n(hoice)f(of)h FJ(')3182 3865 y FI(i)3236 3853 y FQ(\(since)456 3952 y(End)13 b FJ(V)666 3964 y FI(i)717 3952 y FQ(=)23 b FJ(k)s FQ(\).)37 b(No)n(w,)27 b(de\014ne)1545 4121 y FJ(s)c FQ(=)1695 4042 y Fy(M)1834 4121 y FJ( )1888 4133 y FI(i)1925 4121 y FQ(:)28 b FJ(R)2091 4074 y FE(\030)2063 4121 y FL(\000)-40 b(!)23 b FJ(R)2258 4087 y FM(op)2332 4121 y FJ(:)-1899 b FQ(\(2.4.8\))605 4285 y(W)-7 b(e)28 b(will)g(also)e(frequen)n(tly)i(use)f(the)h(follo)n (wing)f(ob)5 b(ject)27 b(of)h(ind)p FL(\000)o(C)33 b FQ(asso)r(ciated)26 b(with)i FJ(R)g FQ(:)1483 4449 y FJ(H)h FQ(=)23 b FL(\012)p FQ(\()p FJ(R)q FQ(\))g(=)1973 4370 y Fy(M)2112 4449 y FJ(V)2160 4461 y FI(i)2206 4449 y FL(\012)18 b FJ(V)2356 4415 y FE(\003)2337 4470 y FI(i)2394 4449 y FJ(:)-1961 b FQ(\(2.4.9\))456 4618 y(The)26 b(previous)f (argumen)n(ts)g(sho)n(w)g(that)i FJ(H)33 b FQ(is)26 b(also)f (canonically)g(isomorphic)g(to)3038 4556 y Fy(L)3144 4618 y FJ(V)3211 4588 y FE(\003)3192 4639 y FI(i)3265 4618 y FL(\012)15 b FJ(V)3393 4630 y FI(i)3421 4618 y FQ(.)456 4717 y(Also,)25 b FJ(H)31 b FQ(is)25 b(canonically)e (isomorphic)h(to)h FJ(H)1866 4687 y FE(\003)1927 4717 y FQ(=)2014 4655 y Fy(L)2120 4717 y FJ(V)2187 4687 y FE(\003\003)2168 4739 y FI(i)2272 4717 y FL(\012)13 b FJ(V)2417 4687 y FE(\003)2398 4739 y FI(i)2455 4717 y FQ(.)36 b(Both)25 b(these)f(prop)r(erties)g(will)456 4817 y(b)r(e)k(used)f(man)n(y)g(times)h(in)g(the)g(sequel.)605 4917 y(One)c(of)h(the)g(main)f(goals)f(of)i(the)g(next)f(lectures)g (will)h(b)r(e)g(to)g(answ)n(er)e(the)i(follo)n(wing)e(ques-)456 5016 y(tion:)36 b(giv)n(en)26 b(a)g(semisimple)g(ab)r(elian)g(category) f FL(C)31 b FQ(and)26 b(a)g(symmetric)g(ob)5 b(ject)27 b FJ(R)c FL(2)h FQ(ind)p FL(\000C)3333 4986 y Fv(\002)p FM(2)3421 5016 y FQ(,)456 5116 y(what)g(extra)g(data)g(are)g(needed)h (to)g(reconstruct)e(the)j(structure)e(of)g(a)h(ribb)r(on)f(category)f (on)i FL(C)5 b FQ(?)456 5216 y(W)-7 b(e)28 b(will)f(giv)n(e)g(an)g (answ)n(er)f(to)i(this)g(question)f(in)h(Chapter)f(5.)p eop %%Page: 46 18 46 49 bop 456 226 a FM(46)1032 b(2.)29 b(RIBBON)h(CA)-5 b(TEGORIES)605 425 y FQ(Finally)e(,)34 b(when)e(w)n(orking)f(with)i(an) f(ob)5 b(ject)32 b FJ(R)g FL(2)g FQ(ind)p FL(\000)p FJ(C)2447 395 y Fv(\002)p FM(2)2569 425 y FQ(\(not)g(necessarily)f(the)i(one)456 525 y(giv)n(en)26 b(b)n(y)i(\(2.4.7\)\),)f(it)h(is)f(con)n(v)n(enien)n (t)g(to)g(use)g(the)h(follo)n(wing)e(notation:)36 b(if)28 b FJ(R)c FQ(=)3034 463 y Fy(L)3140 525 y FJ(A)3202 537 y FI(i)3248 525 y Fw(\002)18 b FJ(B)3394 537 y FI(i)3421 525 y FQ(,)456 628 y(then)30 b(w)n(e)f(write)h(Hom\()p FJ(X)r(;)14 b(R)1363 597 y FM(\(1\))1471 628 y FL(\012)20 b FJ(Y)38 b FL(\012)20 b FJ(R)1791 597 y FM(\(2\))1880 628 y FQ(\))30 b(instead)f(of)2327 565 y Fy(L)2419 653 y FI(i)2461 628 y FQ(Hom\()p FJ(X)r(;)14 b(A)2836 640 y FI(i)2883 628 y FL(\012)20 b FJ(Y)38 b FL(\012)20 b FJ(B)3202 640 y FI(i)3230 628 y FQ(\),)30 b(etc.)456 727 y(If)h FJ(R)g FQ(is)f(symmetric,)h(then)h(the)f(sup)r(erscripts)e (\\\(1\)",)i(\\\(2\)")f(can)g(b)r(e)h(omitted,)h(and)e(w)n(e)g(will)456 827 y(just)e(write)f(Hom\()p FJ(X)r(;)14 b(R)19 b FL(\012)f FJ(Y)37 b FL(\012)18 b FJ(R)q FQ(\).)37 b(Similarly)-7 b(,)27 b(w)n(e)h(can)f(use)g(sev)n(eral)f(copies)h FJ(R)3002 839 y FM(1)3039 827 y FJ(;)14 b(:)g(:)g(:)g(;)g(R)3287 839 y FI(m)3378 827 y FQ(of)456 926 y FJ(R)28 b FQ(and)f(write,)h(for)f (example,)907 1073 y FJ(R)970 1085 y FM(1)1026 1073 y FL(\012)18 b FJ(X)25 b FL(\012)18 b FJ(R)1349 1085 y FM(2)1405 1073 y FL(\012)g FJ(R)1551 1085 y FM(2)1606 1073 y FL(\012)g FJ(R)1752 1085 y FM(1)1813 1073 y FQ(=)k FJ(R)1964 1029 y FM(\(1\))1963 1095 y(1)2072 1073 y FL(\012)c FJ(X)24 b FL(\012)18 b FJ(R)2395 1029 y FM(\(1\))2394 1095 y(2)2503 1073 y FL(\012)g FJ(R)2650 1029 y FM(\(2\))2649 1095 y(2)2757 1073 y FL(\012)g FJ(R)2904 1029 y FM(\(2\))2903 1095 y(1)1813 1227 y FQ(=)1900 1148 y Fy(M)1926 1325 y FI(i;j)2040 1227 y FJ(A)2102 1239 y FI(i)2148 1227 y FL(\012)g FJ(X)25 b FL(\012)18 b FJ(A)2470 1239 y FI(j)2523 1227 y FL(\012)g FJ(B)2669 1239 y FI(j)2723 1227 y FL(\012)g FJ(B)2869 1239 y FI(i)2896 1227 y FJ(:)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF