%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: arXiv:math.RT/0502055 v1 2 Feb 2005 %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -z -R -K1 certain_examples.dvi -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2005.02.02:1938 %%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 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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 /setpagedevice{pop}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 %%BeginProcSet: hps.pro /HPSdict 20 dict dup begin/braindeaddistill 50 def/rfch{dup length 1 sub 1 exch getinterval}bind def/splituri{dup(#)search{exch pop}{()exch} ifelse dup(file:)anchorsearch{pop exch pop 3 -1 roll pop false}{pop 3 -1 roll exch pop true}ifelse}bind def/lookuptarget{exch rfch dup /TargetAnchors where{pop TargetAnchors dup 3 -1 roll known{exch get true }{pop(target unknown:)print == false}ifelse}{pop pop (target dictionary unknown\012)print false}ifelse}bind def/savecount 0 def/stackstopped{count counttomark sub/savecount exch store stopped count savecount sub 1 sub dup 0 gt{{exch pop}repeat}{pop}ifelse}bind def /tempstring 256 string def/targetvalidate{1 index dup length 255 gt exch dup(/)search{pop pop pop exch pop true exch}{pop}ifelse cvn tempstring cvs token pop pop length 0 ne or not}bind def/targetdump-hook where{pop} {/targetdump-hook{dup mark exch gsave initmat setmatrix{{mark/Dest 4 2 roll targetvalidate{aload pop exch pop/Page 3 1 roll/View exch[exch /FitH exch]/DEST pdfmark}{cleartomark}ifelse}forall}stackstopped pop grestore}bind def}ifelse/baseurl{mark exch 1 dict dup 3 -1 roll/Base exch put/URI exch/DOCVIEW{pdfmark}stackstopped pop}bind def /externalhack systemdict/PDF known def/oldstyle true def/initmat matrix currentmatrix def/actiondict 2 dict dup/Subtype/URI put def /weblinkhandler{dup 3 1 roll mark 4 1 roll/Title 4 1 roll splituri 3 -1 roll dup length 0 gt{cvn/Dest exch 4 2 roll}{pop}ifelse{externalhack{ /HTTPFile exch}{actiondict dup 3 -1 roll/URI exch put/Action exch} ifelse}{externalhack{/HTTPFile exch}{/File exch/Action/GoToR}ifelse} ifelse counttomark 2 sub -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll /Color exch oldstyle{/LNK}{/Subtype/Link/ANN}ifelse gsave initmat setmatrix{pdfmark}stackstopped grestore}bind def/externalhandler where{ pop}{/externalhandler{2 copy{weblinkhandler}exec{/externalhack externalhack not store 2 copy{weblinkhandler}exec{/externalhack externalhack not store/oldstyle false store 2 copy{weblinkhandler}exec{ (WARNING: external refs disabled\012)print/externalhandler{pop pop}bind store externalhandler}{pop pop}ifelse}{pop pop/externalhack externalhack not store}ifelse}{pop pop/externalhandler{weblinkhandler pop}bind store} ifelse}bind def}ifelse/pdfmnew{dup type/stringtype eq{externalhandler}{ exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll/Color exch/LNK gsave initmat setmatrix pdfmark grestore}{pop pop}ifelse}ifelse}bind def/pdfmold{dup type/stringtype eq{externalhandler}{exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop pop 0 3 getinterval /Rect 3 1 roll/Border exch/LNK gsave initmat setmatrix pdfmark grestore} {pop pop}ifelse}ifelse}bind def/pdfm where{pop}{/pdfm /currentdistillerparams where{pop currentdistillerparams dup /CoreDistVersion known{/CoreDistVersion get}{0}ifelse dup braindeaddistill le{(WARNING: switching to old pdfm because version =) print ==/pdfmold}{pop/pdfmnew}ifelse load}{/pdfmark where{pop{dup type /stringtype eq{externalhandler}{2 copy mark 3 1 roll{pdfmnew} stackstopped{2 copy mark 3 1 roll{pdfmold}stackstopped{ (WARNING: pdfm disabled\012)print/pdfm{pop pop}store}{ (WARNING: new pdfm failed, switching to old pdfm\012)print/pdfm/pdfmold load store}ifelse}{/pdfm/pdfmnew load store}ifelse pop pop}ifelse}}{{ pop pop}}ifelse}ifelse bind def}ifelse end def %%EndProcSet TeXDict begin @defspecial /DvipsToPDF { 72.27 mul Resolution div } def /PDFToDvips { 72.27 div Resolution mul } def /HyperBorder { 1 PDFToDvips } def /H.V {pdf@hoff pdf@voff null} def /H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def /H.S { currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF /pdf@hoff exch def HyperBorder sub /pdf@llx exch def } def /H.L { 2 sub dup /HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def /pdf@urx exch def } def /H.A { H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def } def /H.R { currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def } def systemdict /pdfmark known not {userdict /pdfmark systemdict /cleartomark get put} if @fedspecial end %%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 <krisrose@brics.dk> % % 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 false def /UnderlinePosition -300 def /UnderlineThickness 150 def end readonly def /FontName /XYDASH10 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 50 /d50 put dup 64 /d64 put dup 124 /d124 put dup 126 /d126 put readonly def /FontBBox{-40 -520 503 520}readonly def /UniqueXX 5092844 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d743b8793c40476b99911a1be6c93ca a7ffc9533764a6a2a3ebcf0bebc6668e399d80ad8b0e5e21d556d8fa71b95a1e 01e6689c74f977a4bbec6795aec114d8507f237839f414ee4fbf8162c865260f 923a63721852c7bff69703f7e0ab99c3b85e83c62c13ea99442890e370376cce 7133ce8f3de2f4c1dc78fb55dff4eb737c195d266281adef5d56fbbc3b785b1b 59d6efeab3b93e713f4b9105cf1594c83472177c0f2b04c840760c92c094a0b9 2a720e4c7b03708d225531ac69324547d65009965f1c52d2be3112c67b6002b1 3d5f2c82505b7f0136cc926ff2bda0b53691b13e816817e913048ad033e0ff31 9d18776c4be80936c7449f316ff7f9026e5eeb9984867fc558bb18773e9a5390 d4490fb8e63a0ce175f52732043cba9d379d01ef25fc4be056d3206186b53195 63ee3d03fa580efa0ad7d3162f77878d348a841432fabedfebc8559530f6cbc1 59df0a77aacfa9f0974542a736680e064ac101c646442b0ca133c4701c206de9 6b70d341f9558a800520c2d32be3628b6df05a19538ec2596d2334f05d54e742 a1a18ebbc12f04c45b899f667d9e6f3a4eaa1854562506d0da4057c4bbfbbacc c1c208cc47b76226ef6d4d3da7d976b7a21a2cc7aa7cf0602fbd2a46022f7894 c0667e19a31cc10ca33811f882ca5cc140bd49eb62545ffe3f418e8cb9b223e3 b2630b486a3b948c74751c414e84334424a1eee8f20b1bd4eab9a0e0545c9bf2 f8cda548feb88b89e369f29f5318ee43b25672b275b05016b635dc656bca5b14 a28e91c516e3f5e99609f5a37a696fbb39379b8374a044e2fe6d4a193d5360d4 31229d74455ff8645ba7462da11460be68629c6a2b1b4b4f409c806cdaec4d3f 941ec5e5a1a6aaaf2c72de027d73b6d446b29f4a0504dfa9e100f273e0b8f54f 707a5a7e1e5f5f3734783960d641ff957f220cdff18bb2d536a406abc54e557f a1e9728df44ca1a17c233e052e050fcd4d771fc5fa346a7324c5f767ba6adfe7 33ef1f05382c604434b60e544b0e71cf354d228e2bae79b3607a479ed9620bec 642d5135e9d402e33b4634ae556651cc989092f47c679da8df2174dcf89b89fa a19c210f9b75904741e5649462ef944a340d82555e8f22027e4df6069eb22648 e7706c287417c8c31bdf6d95c5f8646e0c20d31ce4f7e888632e571297a2e080 2fa675970b5660e4fca8acb48ba7dae24b19eff6f6d84fead24933d8c3b2c0b4 6bb025dc155400217db0205eed8b610f38291e686d3c5e8223fd90515c06cd00 e268c953cdedde12cb372d3735bcd8fbbe97affb215da4baf94fd44de6348100 ea7664590e69f16628d05ab36288fd9548c1c44df03449fedaeaf120c5b3f8f5 2712ea06f1a845d564b378e976b1a69768f7c98ba6e904ee52b452871a73e7c8 4fb42ebec098c79fd15b059fb45efded174ea490584f6638dea53190b23f0f81 5b4c40d3ca9e8ade3ce23efe94687b0256522c0769e4ea76c2f5c0f626b57802 87124f61067d008900585b9d0e04fbb048f1e5fd2401d523523c87369f061279 d0cbf6f3e2064ab97210421163adbe6f56591c6fc5c41aa8a6104c4d8c9a196b c3c09a73efe043da8521312f421b4f43541cf5c74d90b606b32cc16a0e7abc07 96e4006e66d745adaad4efb2f8184397aec2b0d780770faea39178673d510a2c 10cce11ffc9f2f05e65324ae41eb8e60f0e49d7c3005225909055f603dc84bbe a8a0b23137cd21e88cf9f9ee4f2ae17b7cceca9db94e7d2abf862ad1aa56d5ea 8e077ed2901702 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 <krisrose@brics.dk> % % 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 15 /d15 put dup 47 /d47 put dup 79 /d79 put dup 89 /d89 put dup 111 /d111 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 0c37542986ec1e4613cc8a16521e5c2720d88fa18111a1083dcee82e855361ac 2b2c6f5fc5d91dda0ccd1b729396fe411b2e41ad22ec1e2acd7c415a5b7e6c30 a7f4468501f7903953663148e12722f13621b988d914097cc757bb53512e9b27 a349f19a44f9c7d41da06b4db6686de36640f242597232d46a1ecfb7022943ed ab468c20923ef52ccc721480c33b5be25a68c572a8c997ecd17ac53dc0c492b3 f20155db43c4c1b71b5878fa0c7fba169cd0b5baee935e67c871001de7f08314 40ab0934d7b9e7db714bf7e186024daae257bf1c3bdcf7367f5e40d4869fe167 f91fc243b2461f72357694a64a9587efe688621754620432f90ed56627f865fb e6453abf508a628194af8506f53eaadf54b0c941978bac31c6b5fdc9c91427db 60a91b8eb3931bf79c27d24696d63e0cd931f347d7278011efd85773edbd9bb9 ec5e58559fd90fa1f9f9933e12b5b7eec33b2ca0c6a05011bd6020761985ac50 ac2afe7a73fb21d4336a028d259a1f69246a21c2327624a33a0fa896bc743044 76346afb96df2e20134369d4dad8d3930b8f0e092e6d03f944a06c04e65276cf 068030fcda4b356886121d0011e9e4101e170df27b29e969824204a604005f82 dc727dcf855938cb0a28c76e721ad92cf460233d448b892a8f4e46438ae46984 357b9397ea321629b6aece454b0dd2ab6d1fc6382b0101567c682725024677ba 03478d4bb8d430f1f91e2305e5b4df84df340d572f9a6fcda1a34b6b270617cc d256d42daf40536d4e8ffe6b86f06c4033e36f7df8e44849b048708b4f8149fe 1b7703ed0882deb1a379a8d2daa6a10041baf59ff88c0a0f7650820893ac32c1 aae28e7e9fe595db3179ece0be83109504befe5e5cec22b5a1dc8ab8acb7eafb a20bde55dadab62cb1050a1e66a570ac9625af539e35a9fc6c8a0f5508962ae7 5e077ca8888388a4cd2db76a464525ae0b590c5ddeb15b35051624ceeb1c96ae 5773e679796c401ea1d8e9dd45b21cf7e6c64d6ffbaf03fc242cf0fd086508dc f0ac2b843db5b01bfd5f7d1ffc3a67259a21e9c71a49b43ebed89bac01269bd1 cb 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 <krisrose@brics.dk> % % 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 15 /d15 put dup 47 /d47 put dup 79 /d79 put dup 89 /d89 put dup 111 /d111 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 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2 1 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.2) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)711 b(2)0 133 y Fv(function)20 b(from)e Ft(I)23 b Fv(to)18 b(complex)i(n)o(um)o(b)q(ers.)32 b(Giv)o(en)20 b(w)o(eigh)o(t)f Ft(\025)f Fv(=)i(\()p Ft(\025)1219 140 y Fq(i)1232 133 y Fv(\))1250 140 y Fq(i)p Fm(2)p Fq(I)1324 133 y Fv(the)g(deformed)f (prepro)s(jectiv)o(e)0 187 y(algebra)c(of)g(w)o(eigh)o(t)g Ft(\025)g Fv(is)442 133 y SDict begin H.S end 442 133 a 442 133 a SDict begin 13 H.A end 442 133 a 442 133 a SDict begin [ /View [/XYZ H.V] /Dest (equation.1.1) cvn H.B /DEST pdfmark end 442 133 a 0 285 a Fv(\(1.1\))424 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/I /Border [0 0 12] /Subtype /Link /Dest (cite.CrB2) cvn H.B /ANN pdfmark end 718 1747 a Fv(],)f([)771 1747 y SDict begin H.S end 771 1747 a Fv(M2)835 1716 y SDict begin H.R end 835 1716 a 835 1747 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.MyWorkshop) cvn H.B /ANN pdfmark end 835 1747 a Fv(]\):)898 1693 y SDict begin H.S end 898 1693 a 898 1693 a SDict begin 13 H.A end 898 1693 a 898 1693 a SDict begin [ /View [/XYZ H.V] /Dest (equation.1.2) cvn H.B /DEST pdfmark end 898 1693 a 0 1876 a Fv(\(1.2\))204 b Ft(e)320 1883 y Fq(c)337 1876 y Fv(\005)371 1857 y Fq(\025)394 1876 y Fv(\()p Ft(Q)p Fv(\))p Ft(e)487 1883 y Fq(c)517 1876 y Fv(=)13 b Fo(C)6 b Fn(h)p Ft(x)638 1883 y Fs(1)661 1876 y Ft(;)i(:)g(:)g(:)d(;) j(x)789 1883 y Fq(n)811 1876 y Fn(j)p Ft(P)853 1883 y Fq(i)867 1876 y Fv(\()p Ft(x)911 1883 y Fq(i)925 1876 y Fv(\))k(=)h(0)8 b(\()p Ft(i)j Fv(=)i(1)p Ft(;)8 b(:)g(:)g(:)d(;)j(n)p Fv(\))p Ft(;)1338 1819 y Fq(n)1316 1833 y Fl(X)1321 1930 y Fq(i)p 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b(comm)o(utativ)o(e)g(ring)g Ft(e)1679 362 y Fs(0)1699 355 y Fv(\005)1733 338 y Fq(\025)1756 355 y Fv(\()p Ft(Q)p Fv(\))p Ft(e)1849 362 y Fs(0)0 409 y Fv(is)e(itself)h(isomorphic)g(to)e (the)h(co)q(ordinate)h(ring)f(of)f(some)h(\014b)q(er)h(of)e(the)h (semiuniv)o(ersal)i(deformation)d(of)h(the)0 463 y(quotien)o(t)j (singularit)o(y)g Fo(C)437 446 y Fs(2)460 463 y Ft(==)p Fv(\000)f(where)h(\000)f(is)h(the)g(\014nite)h(subgroup)e(of)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))19 b(corresp)q(onding)f(to)e Ft(Q)g Fv(b)o(y)0 517 y(the)f(McKa)o(y)g(corresp)q(ondence.)50 571 y(Recall)d(that)f(McKa)o(y)h(corresp)q(ondence)h(assigns)f(to)f (eac)o(h)h(v)o(ertex)f Ft(i)h Fv(of)f Ft(Q)h Fv(an)f(irreducible)k (represen)o(tation)0 625 y Ft(V)27 632 y Fq(i)53 625 y Fv(of)d(\000.)19 b(An)13 b(iden)o(tit)o(y)g(represen)o(tation)g(is)g (assigned)g(to)f(the)g(extending)i(v)o(ertex)e(0.)18 b Ft(\016)1457 632 y Fq(i)1484 625 y Fv(giv)o(es)13 b(the)f(dimension)0 679 y(of)h Ft(V)77 686 y Fq(i)91 679 y Fv(.)19 b(The)14 b(n)o(um)o(b)q(er)g(of)f(edges)h(b)q(et)o(w)o(een)g(an)o(y)f Ft(i)h Fv(and)g Ft(j)i Fv(|)d(v)o(ertices)h(of)f Ft(Q)g Fv(equals)i(to)e(the)g(n)o(um)o(b)q(er)h(of)g(times)0 733 y Ft(V)27 740 y Fq(i)52 733 y Fv(o)q(ccurs)e(in)g(the)g(decomp)q (osition)h(of)e Ft(V)h Fn(\012)s Ft(V)761 740 y Fq(j)791 733 y Fv(in)o(to)f(irreducibles.)22 b(W)l(e)11 b(denote)h(b)o(y)g Ft(V)21 b Fv(the)11 b(tautological)h(t)o(w)o(o-)0 787 y(dimensional)19 b(represen)o(tation)e(of)f(\000)h(as)g(a)f(subgroup)h (of)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\).)27 b(W)l(e)17 b(c)o(ho)q(ose)g(a)f(hermitian)i(structure)0 841 y(on)d(eac)o(h)g Ft(V)192 848 y Fq(i)221 841 y Fv(whic)o(h)h(mak)o(es)f(it)g(an)g (unitary)h(represen)o(tation.)50 899 y(Supp)q(ose)23 b Ft(Q)e Fv(is)i(bipartite)f(\(this)g(includes)i(all)f(cases)f(except) 1178 888 y Fl(f)1172 899 y Ft(A)1206 906 y Fq(n)1251 899 y Fv(with)h(o)q(dd)f(n)o(um)o(b)q(er)g(of)f(v)o(ertices\),)0 953 y(so)f(that)f(some)h(v)o(ertices)g(are)g(called)i(o)q(dd)e(and)h (some)e(are)h(ev)o(en.)35 b(F)l(or)19 b(the)i(group)e(\000)i(it)f (means)g(that)g(\000)0 1007 y(con)o(tains)e(negativ)o(e)g(iden)o(tit)o (y)h(of)e Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\).)30 b(Let)19 b(0)e(b)q(e)i(ev)o(en.)28 b(Supp)q(ose)19 b(all)g(arro)o(ws)d (are)i(directed)h(from)0 1061 y(o)q(dd)d(v)o(ertices)g(to)f(ev)o(en)h (ones,)g(and)g Ft(\025)646 1068 y Fq(i)673 1061 y Fv(=)e Fn(\000)p Ft(\016)777 1068 y Fq(i)807 1061 y Fv(for)h(o)q(dd)h(v)o (ertices)g(and)g Ft(\025)1248 1068 y Fq(i)1275 1061 y Fv(=)e Ft(\016)1344 1068 y Fq(i)1374 1061 y Fv(for)h(ev)o(en)h(ones.)22 b(Clearly)16 b Ft(\025)1854 1068 y Fq(i)0 1115 y Fv(equals)h(to)e(the)h (trace)g(of)g Fn(\000)p Ft(I)i Fn(2)c Fv(\000)j(in)g Ft(V)685 1122 y Fq(i)698 1115 y Fv(.)23 b(Let)16 b Ft(i)e Fn(2)g Ft(I)t Fv(.)22 b(Consider)17 b(End)1221 1122 y Fg(C)1247 1115 y Fv(\()p Ft(V)1292 1122 y Fq(i)1306 1115 y Fv(\))e(equipp)q(ed)k(with)d(\000-action)h(b)o(y)0 1169 y(conjugation.)28 b(Let)18 b Ft(S)384 1176 y Fg(R)428 1169 y Fv(b)q(e)g(the)g(real)g(unit)h(sphere)f(in)h Fo(R)999 1153 y Fs(3)1033 1169 y Fv(and)f Ft(S)1152 1176 y Fg(C)1196 1169 y Fv(b)q(e)h(the)f(a\016ne)g(v)m(ariet)o(y)g(consisting)g(of)0 1223 y(p)q(oin)o(ts)e(\()p Ft(x;)8 b(y)r(;)g(z)r Fv(\))k Fn(2)i Fo(C)374 1207 y Fs(3)397 1223 y Fv(,)h(whic)o(h)i(satisfy)f Ft(x)725 1207 y Fs(2)755 1223 y Fv(+)11 b Ft(y)825 1207 y Fs(2)855 1223 y Fv(+)g Ft(z)924 1207 y Fs(2)957 1223 y Fv(=)k(1.)21 b(T)l(o)16 b(de\014ne)h(\000-action)g(on)e Ft(S)1536 1230 y Fg(R)1578 1223 y Fv(and)h Ft(S)1695 1230 y Fg(C)1737 1223 y Fv(w)o(e)g(use)0 1277 y(the)h(w)o(ell-kno)o(wn) h(homomorphism)f(from)f(\000)g Fn(\032)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))20 b(to)c Ft(S)s(O)q Fv(\(3)p Ft(;)8 b Fo(R)l Fv(\))13 b Fn(\032)j Ft(S)s(O)q Fv(\(3)p Ft(;)8 b Fo(C)d Fv(\).)28 b(The)17 b(image)g(of)g(\000)0 1331 y(in)e Ft(S)s(O)q Fv(\(3)p Ft(;)8 b Fo(R)l Fv(\))i(is)15 b(isomorphic)g(to)e(\000)p Ft(=)p Fn(f)p Ft(I)t(;)8 b Fn(\000)p Ft(I)t Fn(g)k Fv(whic)o(h)j(w)o(e)e(denote)i(b)o(y)e(\000) 1213 1314 y Fm(0)1225 1331 y Fv(.)20 b(Note)13 b(that)h(\000)1492 1314 y Fm(0)1518 1331 y Fv(acts)f(on)h(End)1753 1338 y Fg(C)1779 1331 y Fv(\()p Ft(V)1824 1338 y Fq(i)1838 1331 y Fv(\).)0 1385 y(W)l(e)h(pro)o(v)o(e)g(the)g(follo)o(wing:)0 1394 y SDict begin H.S end 0 1394 a 0 1394 a SDict begin 13 H.A end 0 1394 a 0 1394 a SDict begin [ /View [/XYZ H.V] /Dest (thm.1) cvn H.B /DEST pdfmark end 0 1394 a 88 x FD(Theorem)i(1.)j Fk(The)c(algebr)n(a)g Ft(e)551 1489 y Fq(i)565 1482 y Fv(\005)599 1465 y Fq(\025)622 1482 y Fv(\()p Ft(Q)p Fv(\))p Ft(e)715 1489 y Fq(i)745 1482 y Fk(is)g(isomorphic)h(to)f(the)h(algebr)n(a)e(of)i(p)n(olynomial)f Fv(\000)1621 1465 y Fm(0)1633 1482 y Fk(-e)n(quivariant)0 1536 y(maps)i(for)h Ft(S)223 1543 y Fg(C)267 1536 y Fk(to)f Fv(End)404 1543 y Fg(C)430 1536 y Fv(\()p Ft(V)475 1543 y Fq(i)489 1536 y Fv(\))p Fk(.)25 b(The)18 b(involution)g(on)f(the)i (latter)f(algebr)n(a)f(given)h(by)g(the)g(formula)h Ft(f)1736 1519 y Fm(\003)1756 1536 y Fv(\()p Ft(x)p Fv(\))c(=)0 1590 y Ft(f)5 b Fv(\()p 45 1565 26 2 v Ft(x)p Fv(\))89 1573 y Fm(\003)108 1590 y Fk(,)16 b Ft(x)d Fn(2)g Ft(S)248 1597 y Fg(C)274 1590 y Fk(,)j(c)n(oincides)f(with)i(the)f(one)g(induc)n (e)n(d)g(fr)n(om)h(the)f(former)h(algebr)n(a.)50 1687 y Fv(If,)f(furthermore,)f Ft(V)401 1694 y Fq(i)431 1687 y Fv(is)i(not)e(exceptional)j(in)f(the)f(sense)h(of)e(the)h (de\014nition)1384 1687 y SDict begin H.S end 1384 1687 a Fv(5.1)1442 1658 y SDict begin H.R end 1442 1658 a 1442 1687 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.5.1) cvn H.B /ANN pdfmark end 1442 1687 a 16 w Fv(then)h(it)f(is)h(p)q (ossible)h(to)0 1741 y(construct)d(a)g Ft(C)274 1724 y Fm(\003)293 1741 y Fv(-en)o(v)o(eloping)i(algebra)0 1750 y SDict begin H.S end 0 1750 a 0 1750 a SDict begin 13 H.A end 0 1750 a 0 1750 a SDict begin [ /View [/XYZ H.V] /Dest (thm.2) cvn H.B /DEST pdfmark end 0 1750 a 88 x FD(Theorem)g(2.)k Fk(If)15 b Ft(V)358 1845 y Fq(i)388 1838 y Fk(is)h(not)g(exc)n(eptional)g(in)g(the)h(sense)e(of)h(the)h (de\014nition)1336 1838 y SDict begin H.S end 1336 1838 a Fk(5.1)1396 1809 y SDict begin H.R end 1396 1809 a 1396 1838 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.5.1) cvn H.B /ANN pdfmark end 1396 1838 a 17 w Fk(then)f(the)h Ft(C)1626 1822 y Fm(\003)1645 1838 y Fk(-enveloping)0 1892 y(algebr)n(a)i(of)h Ft(e)235 1899 y Fq(i)249 1892 y Fv(\005)283 1876 y Fq(\025)306 1892 y Fv(\()p Ft(Q)p Fv(\))p Ft(e)399 1899 y Fq(i)432 1892 y Fk(exists)e(and)i(is)f (isomorphic)h(to)g(the)f Ft(C)1111 1876 y Fm(\003)1131 1892 y Fk(-algebr)n(a)g(of)h(c)n(ontinuous)f Fv(\000)1621 1876 y Fm(0)1633 1892 y Fk(-e)n(quivariant)0 1946 y(maps)d(for)h Ft(S)219 1953 y Fg(R)261 1946 y Fk(to)g Fv(End)397 1953 y Fg(C)423 1946 y Fv(\()p Ft(V)468 1953 y Fq(i)482 1946 y Fv(\))p Fk(.)50 2043 y Fv(Exceptional)f(cases)f(are)g(listed)i(in)f (the)f(section)894 2043 y SDict begin H.S end 894 2043 a Fv(6)916 2014 y SDict begin H.R end 916 2014 a 916 2043 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (section.6) cvn H.B /ANN pdfmark end 916 2043 a Fv(.)50 2097 y(Supp)q(ose)i Ft(V)256 2104 y Fq(i)286 2097 y Fv(is)g(not)f(exceptional.)25 b(W)l(e)17 b(can)g(see)f(at)g(this)h(p)q(oin)o(t)g(that)f(the)g Ft(C)1396 2081 y Fm(\003)1416 2097 y Fv(-en)o(v)o(elop)q(e)h(of)f Ft(e)1690 2104 y Fq(i)1705 2097 y Fv(\005)1739 2081 y Fq(\025)1761 2097 y Fv(\()p Ft(Q)p Fv(\))p Ft(e)1854 2104 y Fq(i)0 2151 y Fv(de\014nes)23 b(a)e(bundle)i(of)e(algebras)h(on) f Ft(S)694 2158 y Fg(R)720 2151 y Ft(=)p Fv(\000)771 2135 y Fm(0)783 2151 y Fv(,)i(whic)o(h)f(is)g(homeomorphic)g(to)f Ft(S)1409 2158 y Fg(R)1435 2151 y Fv(.)39 b(It)22 b(o)q(ccurs)g(that)f (this)0 2205 y(bundle)f(can)e(b)q(e)h(trivialized)h(in)f(the)f(follo)o (wing)h(sence.)29 b(Let)18 b Ft(x)g Fn(2)g Ft(S)1200 2212 y Fg(R)1225 2205 y Ft(=)p Fv(\000)1276 2188 y Fm(0)1306 2205 y Fv(b)q(e)h(an)o(y)f(orbit)g(in)h Ft(S)1658 2212 y Fg(R)1684 2205 y Fv(.)28 b(Cho)q(ose)0 2259 y(some)14 b(represen)o(tativ)o(e)g Ft(x)430 2266 y Fs(0)464 2259 y Fv(for)g Ft(x)g Fv(in)h(the)g(fundamen)o(tal)f(region)h(and)f (consider)i(the)e(stabilizer)i(subgroup)e(of)0 2313 y(\000)28 2296 y Fm(0)54 2313 y Fv(for)g Ft(x)149 2320 y Fs(0)169 2313 y Fv(.)20 b(W)l(e)14 b(denote)h(the)f(set)h(of)f(all)h(elemen)o (ts)g(in)g(End)1005 2320 y Fg(C)1031 2313 y Fv(\()p Ft(V)1076 2320 y Fq(i)1089 2313 y Fv(\))f(whic)o(h)i(comm)o(ute)e(with)g(the)h (stabilizer)h(b)o(y)0 2367 y Ft(M)44 2374 y Fq(x)64 2379 y Fp(0)83 2367 y Fv(.)k(Then,)15 b(the)h(*-algebra)f Ft(M)567 2374 y Fq(x)587 2379 y Fp(0)621 2367 y Fv(has)g(the)h(form)349 2471 y Ft(M)393 2478 y Fq(x)413 2483 y Fp(0)445 2459 y Fn(\030)445 2473 y Fv(=)493 2471 y(End\()p Fo(C)622 2453 y Fq(d)640 2458 y Fp(1)662 2471 y Fv(\))10 b Fn(\002)h Fv(End\()p Fo(C)865 2453 y Fq(d)883 2458 y Fp(2)905 2471 y Fv(\))f Fn(\002)g(\001)e(\001)g(\001)h(\002)h Fv(End\()p Fo(C)1216 2453 y Fq(d)1234 2459 y Fz(k)1258 2471 y Fv(\))i Fn(\032)h Fv(End)q(\()p Fo(C)1465 2453 y Fq(d)1489 2471 y Fv(\))p Ft(;)0 2574 y Fv(with)h Ft(d)126 2581 y Fs(1)158 2574 y Fn(\025)f Ft(d)230 2581 y Fs(2)262 2574 y Fn(\025)f(\001)c(\001) g(\001)j(\025)i Ft(d)447 2581 y Fq(k)481 2574 y Fv(and)h Ft(d)e Fv(is)i(the)f(dimension)i(of)e Ft(V)1014 2581 y Fq(i)1028 2574 y Fv(.)19 b(Denote)13 b(b)o(y)g Ft(N)1312 2581 y Fq(x)1346 2574 y Fv(the)h(subalgebra)f(of)g(End)q(\()p Fo(C)1827 2557 y Fq(d)1850 2574 y Fv(\))0 2628 y(giv)o(en)j(b)o(y)g (the)g(righ)o(thand)g(side)h(of)e(the)h(expression)g(ab)q(o)o(v)o(e.)21 b(Clearly)c(if)f(the)g(stabilizer)h(lies)g(in)g(the)f(cen)o(ter)0 2682 y(of)e(\000)79 2665 y Fm(0)104 2682 y Fv(then)h Ft(M)251 2689 y Fq(x)287 2682 y Fv(equals)f(to)g(the)g(whole)g(End)764 2689 y Fg(C)790 2682 y Fv(\()p Ft(V)835 2689 y Fq(i)848 2682 y Fv(\).)19 b(It)14 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end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)711 b(6)50 133 y Fv(Let)19 b Ft(Q)g Fv(b)q(e)h(a)e(quiv)o(er)i(obtained)g (from)e Ft(G)h Fv(b)o(y)g(directing)h(ev)o(ery)f(edge)g(to)o(w)o(ards)f (an)g(ev)o(en)i(v)o(ertex.)31 b(Let)0 187 y Ft(\025)12 b Fv(=)h(\()p Ft(\025)132 194 y Fq(i)145 187 y Fv(\))i(b)q(e)h(giv)o (en)g(b)o(y)f Ft(\025)449 194 y Fq(i)475 187 y Fv(=)e Ft(\033)549 194 y Fq(i)563 187 y Ft(\016)583 194 y Fq(i)597 187 y Fv(.)50 241 y(By)k(the)g(McKa)o(y)g(corresp)q(ondence)i(there)e (exists)h(a)e(\014nite)j(group)e(\000)f Fn(\032)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))20 b(acting)d(on)h Ft(V)25 b Fv(=)16 b Fo(C)1845 224 y Fs(2)0 295 y Fv(and)e(a)f(bijection)i Ft(i)e Fn( )-8 b(!)12 b Ft(V)459 302 y Fq(i)486 295 y Fv(b)q(et)o(w)o(een)i Ft(I)j Fv(and)d(the)g(set)g(of)f(all)i (nonisomorphic)f(irreducible)j(represen)o(tations)0 349 y(for)e(\000)g(suc)o(h)h(that)0 362 y SDict begin H.S end 0 362 a 0 362 a SDict begin 13 H.A end 0 362 a 0 362 a SDict begin [ /View [/XYZ H.V] /Dest (Item.6) cvn H.B /DEST pdfmark end 0 362 a 73 416 a Fv(\(1\))j Ft(V)178 423 y Fs(0)213 416 y Fv(is)c(the)h(trivial)g(represen)o(tation.)0 427 y SDict begin H.S end 0 427 a 0 427 a SDict begin 13 H.A end 0 427 a 0 427 a SDict begin [ /View [/XYZ H.V] /Dest (Item.7) cvn H.B /DEST pdfmark end 0 427 a 73 470 a Fv(\(2\))j Ft(V)178 477 y Fq(i)198 470 y Fn(\012)6 b Ft(V)24 b Fv(is)14 b(isomorphic)g(to)f(the)g(direct)i(sum)e(of)g Ft(V)991 477 y Fq(j)1022 470 y Fv(where)g Ft(j)j Fv(ranges)d(o)o(v)o (er)g(v)o(ertices)g(of)g Ft(I)k Fv(connected)151 524 y(with)f Ft(i)f Fv(b)o(y)g(an)g(edge.)0 533 y SDict begin H.S end 0 533 a 0 533 a SDict begin 13 H.A end 0 533 a 0 533 a SDict begin [ /View [/XYZ H.V] /Dest (Item.8) cvn H.B /DEST pdfmark end 0 533 a 73 578 a Fv(\(3\))k(dim)9 b Ft(V)262 585 y Fq(i)288 578 y Fv(=)k Ft(\016)356 585 y Fq(i)370 578 y Fv(.)0 645 y(Note)j(that)f(for)h(the)g(cases)g(under)h (consideration)g(\000)g(con)o(tains)f(an)g(elemen)o(t)h Ft(\034)j Fv(=)14 b Fn(\000)p Ft(I)1485 652 y Fq(V)1532 645 y Fv(whic)o(h)j(b)q(elongs)h(to)0 699 y(the)d(cen)o(ter)h(of)e (\000)i(and)f(tr)433 706 y Fq(V)454 711 y Fz(i)476 699 y Ft(\034)j Fv(=)13 b Ft(\025)589 706 y Fq(i)617 699 y Fv(for)i(all)h Ft(i)c Fn(2)h Ft(I)t Fv(.)50 753 y(Consider)j(a)f(sk)o (ew)f(group)h(algebra)h Ft(S)f Fv(=)e Fo(C)7 b Fn(h)p Ft(V)12 b Fn(i)e(\003)f Fv(\000)16 b(where)g Fo(C)6 b Fn(h)p Ft(V)13 b Fn(i)i Fv(denotes)g(the)g(tensor)g(algebra)g(of)g Ft(V)1836 737 y Fm(\003)1855 753 y Fv(.)0 807 y(Denote)h(b)o(y)g Ft(x)p Fv(,)g Ft(y)i Fv(the)f(elemen)o(ts)g(of)f(the)g(standard)g (basis)g(of)g Ft(V)1108 791 y Fm(\003)1142 807 y Fv(=)f Fo(C)1222 791 y Fs(2)p Fm(\003)1262 807 y Fv(,)h(b)o(y)g Ft(")1376 814 y Fq(x)1398 807 y Fv(,)h Ft(")1449 814 y Fq(y)1486 807 y Fv(the)f(elemen)o(ts)h(of)f(the)0 861 y(dual)h(basis)g(of)f Ft(V)26 b Fv(and)16 b(b)o(y)g Ft(w)h Fv(the)g(elemen)o(t)g(of)f(\()p Ft(V)k Fn(\012)11 b Ft(V)f Fv(\))992 845 y Fm(\003)1026 849 y Fn(\030)1026 863 y Fv(=)1075 861 y Ft(V)1112 845 y Fm(\003)1143 861 y Fn(\012)h Ft(V)1225 845 y Fm(\003)1260 861 y Fn(\032)j Ft(S)19 b Fv(giv)o(en)e(b)o(y)f Ft(x)11 b Fn(\012)g Ft(y)i Fn(\000)e Ft(y)i Fn(\012)e Ft(x)16 b Fv(so)0 915 y(that)g Ft(w)q Fv(\()p Ft(")173 922 y Fq(x)195 915 y Ft(;)8 b(")237 922 y Fq(y)257 915 y Fv(\))15 b(=)h(1,)g Ft(w)q Fv(\()p Ft(")466 922 y Fq(y)487 915 y Ft(;)8 b(")529 922 y Fq(x)550 915 y Fv(\))15 b(=)h Fn(\000)p Fv(1)h(and)g Ft(w)q Fv(\()p Ft(")872 922 y Fq(x)893 915 y Ft(;)8 b(")935 922 y Fq(x)957 915 y Fv(\))15 b(=)h Ft(w)q Fv(\()p Ft(")1114 922 y Fq(y)1134 915 y Ft(;)8 b(")1176 922 y Fq(y)1196 915 y Fv(\))15 b(=)h(0.)25 b(Then)17 b Ft(S)i Fv(is)f(generated)f(b)o(y)f Ft(x)p Fv(,)0 969 y Ft(y)h Fv(and)f(elemen)o(ts)h(of)e(\000.)21 b Fo(Z)-12 b Fv(-g)o(rading)13 b(of)j Fo(C)6 b Fn(h)p Ft(V)13 b Fn(i)i Fv(induces)i(a)f Fo(Z)-13 b Fv(-gra)o(ding)14 b(of)h Ft(S)s Fv(,)g(whic)o(h)h(in)h(its)e(turn)h(induces)h(a)0 1023 y Fo(Z)-13 b Ft(=)p Fv(2)p Fo(Z)d Fv(-gra)o(ding.)18 b(So)d Ft(S)j Fv(is)e(a)f(sup)q(eralgebra.)50 1077 y(There)g(exists)h (a)f(unique)h(in)o(v)o(olution)h Fn(\016)e Fv(on)g Ft(S)i Fv(suc)o(h)f(that)0 1099 y SDict begin H.S end 0 1099 a 0 1099 a SDict begin 13 H.A end 0 1099 a 0 1099 a SDict begin [ /View [/XYZ H.V] /Dest (Item.9) cvn H.B /DEST pdfmark end 0 1099 a 73 1144 a Fv(\(1\))j Ft(g)175 1128 y Fm(\016)207 1144 y Fv(=)13 b Ft(g)279 1128 y Fm(\000)p Fs(1)341 1144 y Fv(for)h Ft(g)g Fn(2)f Fv(\000,)0 1156 y SDict begin H.S end 0 1156 a 0 1156 a SDict begin 13 H.A end 0 1156 a 0 1156 a SDict begin [ /View [/XYZ H.V] /Dest (Item.10) cvn H.B /DEST pdfmark end 0 1156 a 73 1198 a Fv(\(2\))19 b(\()p Fn(\001)p Ft(;)8 b(v)r Fv(\))245 1182 y Fm(\016)276 1198 y Fv(=)13 b FD(i)p Ft(w)q Fv(\()p Fn(\001)p Ft(;)8 b(v)r Fv(\))13 b(for)h Ft(v)h Fn(2)e Ft(V)c Fv(,)0 1266 y(here)16 b FD(i)d Fv(=)174 1231 y Fn(p)p 212 1231 59 2 v 35 x(\000)p Fv(1.)20 b(One)15 b(can)h(calculate)632 1347 y Ft(x)658 1328 y Fm(\016)690 1347 y Fv(=)d(\()p Fn(\001)p Ft(;)8 b(")811 1354 y Fq(x)832 1347 y Fv(\))850 1328 y Fm(\016)882 1347 y Fv(=)13 b FD(i)p Ft(w)q Fv(\()p Fn(\001)p Ft(;)8 b(")1051 1354 y Fq(x)1072 1347 y Fv(\))k(=)h Fn(\000)p FD(i)p Ft(y)r(;)651 1436 y(y)675 1417 y Fm(\016)707 1436 y Fv(=)g(\()p Fn(\001)p Ft(;)8 b(")828 1443 y Fq(y)847 1436 y Fv(\))865 1417 y Fm(\016)897 1436 y Fv(=)13 b FD(i)p Ft(w)q Fv(\()p Fn(\001)p Ft(;)8 b(")1066 1443 y Fq(y)1086 1436 y Fv(\))k(=)h FD(i)p Ft(x:)0 1507 y Fv(It)i(follo)o(ws)g(that)662 1569 y Ft(w)e Fv(=)g FD(i)p Ft(xx)822 1550 y Fm(\016)852 1569 y Fv(+)e FD(i)p Ft(y)r(y)960 1550 y Fm(\016)993 1569 y Fv(so)h Ft(w)1080 1550 y Fm(\016)1112 1569 y Fv(=)h Ft(w)q(:)50 1640 y Fv(T)l(ak)o(e)18 b(a)h(factor)f(algebra)h Ft(S)533 1623 y Fq(\025)574 1640 y Fv(=)h Ft(S=)p Fv(\()p Ft(xy)13 b Fn(\000)g Ft(y)r(x)g Fn(\000)g Ft(\034)5 b Fv(\).)31 b(Clearly)l(,)20 b(it)g(is)f(again)g(a)g(sup)q(eralgebra)h (and)f(the)0 1695 y(in)o(v)o(olution)d Fn(\016)f Fv(of)g Ft(S)j Fv(induces)f(an)e(in)o(v)o(olution)h(on)f Ft(S)881 1679 y Fq(\025)918 1695 y Fv(whic)o(h)i(w)o(e)d(will)j(denote)f(again)f (b)o(y)g Fn(\016)p Fv(.)50 1749 y(Next,)h(consider)h(the)f(path)g (algebra)g(of)f(the)h(double)i(of)d Ft(Q)h Fv(denoted)h(b)o(y)f(\005.) 22 b(It)16 b(is)h(generated)f(b)o(y)g(idem-)0 1803 y(p)q(oten)o(ts)f Ft(e)183 1810 y Fq(i)213 1803 y Fv(for)g(eac)o(h)g(v)o(ertex)g Ft(i)e Fn(2)g Ft(I)t Fv(,)i(arro)o(ws)f Ft(a)f Fn(2)h Ft(Q)h Fv(and)h(opp)q(osite)g(arro)o(ws)e Ft(a)1365 1786 y Fm(\003)1400 1803 y Fv(for)h Ft(a)e Fn(2)g Ft(Q)p Fv(.)21 b(This)16 b(algebra)0 1857 y(is)g(a)f(sup)q(eralgebra)h(in)g(an)f(ob)o (vious)g(w)o(a)o(y)f(and)i(w)o(e)f(de\014ne)h(an)f(in)o(v)o(olution)h (b)o(y)0 1879 y SDict begin H.S end 0 1879 a 0 1879 a SDict begin 13 H.A end 0 1879 a 0 1879 a SDict begin [ /View [/XYZ H.V] /Dest (Item.11) cvn H.B /DEST pdfmark end 0 1879 a 73 1924 a Fv(\(1\))j Ft(e)172 1908 y Fm(\016)172 1937 y Fq(i)205 1924 y Fv(=)13 b Ft(e)274 1931 y Fq(i)303 1924 y Fv(for)i Ft(i)d Fn(2)h Ft(I)t Fv(,)0 1937 y SDict begin H.S end 0 1937 a 0 1937 a SDict begin 13 H.A end 0 1937 a 0 1937 a SDict begin [ /View [/XYZ H.V] /Dest (Item.12) cvn H.B /DEST pdfmark end 0 1937 a 73 1978 a Fv(\(2\))19 b Ft(a)175 1962 y Fm(\016)208 1978 y Fv(=)13 b Fn(\000)p FD(i)p Ft(a)329 1962 y Fm(\003)364 1978 y Fv(for)i Ft(a)e Fn(2)g Ft(Q)p Fv(,)0 1990 y SDict begin H.S end 0 1990 a 0 1990 a SDict begin 13 H.A end 0 1990 a 0 1990 a SDict begin [ /View [/XYZ H.V] /Dest (Item.13) cvn H.B /DEST pdfmark end 0 1990 a 73 2032 a Fv(\(3\))19 b Ft(a)175 2016 y Fm(\003\016)225 2032 y Fv(=)13 b FD(i)p Ft(a)p Fv(.)50 2101 y(The)i(factor)f(algebra)i(\005)467 2084 y Fq(\025)505 2101 y Fv(is)f(de\014ned)i(b)o(y)572 2188 y(\005)606 2169 y Fq(\025)642 2188 y Fv(=)c(\005)p Ft(=)p Fv(\()767 2145 y Fl(X)765 2243 y Fq(a)p Fm(2)p Fq(Q)834 2188 y Fv(\()p Ft(aa)900 2169 y Fm(\003)930 2188 y Fn(\000)d Ft(a)999 2169 y Fm(\003)1019 2188 y Ft(a)p Fv(\))g Fn(\000)1116 2145 y Fl(X)1122 2243 y Fq(i)p Fm(2)p Fq(I)1189 2188 y Ft(\025)1216 2195 y Fq(i)1230 2188 y Ft(e)1251 2195 y Fq(i)1265 2188 y Fv(\))p Ft(;)0 2317 y Fv(and)15 b(is)h(again)f(a)g(sup)q(eralgebra)h(with)g(an)f (induced)i(in)o(v)o(olution.)50 2371 y(There)g(is)g(also)f(a)g (classical)j(in)o(v)o(olution)e Fn(\003)g Fv(on)f(\005)h(whic)o(h)g (induces)h(a)f(classical)h(in)o(v)o(olution)g(on)e(\005)1730 2354 y Fq(\025)1753 2371 y Fv(.)24 b(The)0 2425 y(action)15 b(of)g Fn(\003)g Fv(is)h(giv)o(en)f(b)o(y)643 2487 y Ft(e)664 2468 y Fm(\003)664 2498 y Fq(i)697 2487 y Fv(=)e Ft(e)766 2494 y Fq(i)780 2487 y Ft(;)20 b Fv(\()p Ft(a)p Fv(\))873 2468 y Fm(\003)904 2487 y Fv(=)13 b Ft(a)976 2468 y Fm(\003)996 2487 y Ft(;)20 b Fv(\()p Ft(a)1071 2468 y Fm(\003)1090 2487 y Fv(\))1108 2468 y Fm(\003)1140 2487 y Fv(=)13 b Ft(a:)0 2558 y SDict begin H.S end 0 2558 a 0 2558 a SDict begin 13 H.A end 0 2558 a 0 2558 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 0 2558 a 564 2601 a Fv(3.)22 b Fu(Connection)16 b(between)i Fv(\005)g Fu(and)f Ft(S)50 2682 y Fv(F)l(ollo)o(wing)11 b([)262 2682 y SDict begin H.S end 262 2682 a Fv(CBH)361 2651 y SDict begin H.R end 361 2651 a 361 2682 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.CrB) cvn H.B /ANN pdfmark end 361 2682 a Fv(])g(c)o(ho)q(ose)g(an)f(idemp)q(oten)o(t)i Ft(f)841 2689 y Fq(i)866 2682 y Fv(for)f(eac)o(h)g Ft(i)h Fn(2)h Ft(I)h Fv(suc)o(h)d(that)g Ft(V)1355 2689 y Fq(i)1381 2669 y Fn(\030)1381 2684 y Fv(=)1429 2682 y Fo(C)c Fv(\000)p Ft(f)1509 2689 y Fq(i)1526 2682 y Fv(.)19 b(W)l(e)11 b(additionally)0 2736 y(require)16 b Ft(f)175 2743 y Fq(i)204 2736 y Fv(to)f(b)q(e)g(self-adjoin)o(t)h(and)f(the)g (Hermitian)h(structure)f(on)g Ft(V)1222 2743 y Fq(i)1251 2736 y Fv(to)f(b)q(e)i(induced)h(from)d(that)h(of)f Fo(C)7 b Fv(\000.)0 2790 y(Put)15 b Ft(f)j Fv(=)177 2755 y Fl(P)225 2803 y Fq(i)p Fm(2)p Fq(I)288 2790 y Ft(f)310 2797 y Fq(i)324 2790 y Fv(.)i(Then)0 2805 y SDict begin H.S end 0 2805 a 0 2805 a SDict begin 13 H.A end 0 2805 a 0 2805 a SDict begin [ /View [/XYZ H.V] /Dest (prop.3.1) cvn H.B /DEST pdfmark end 0 2805 a eop %%Page: 7 7 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)711 b(7)0 133 y FD(Prop)q(osition)22 b(3.1)f Fv(\(see)e([)482 133 y SDict begin H.S end 482 133 a Fv(CBH)580 102 y SDict begin H.R end 580 102 a 580 133 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.CrB) cvn H.B /ANN pdfmark end 580 133 a Fv(,)611 133 y SDict begin H.S end 611 133 a Fv(M2)676 102 y SDict begin H.R end 676 102 a 676 133 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.MyWorkshop) cvn H.B /ANN pdfmark end 676 133 a Fv(]\))p FD(.)i Fk(Ther)n(e)e(exists)f(a)h (gr)n(ade)n(d)h(isomorphism)f Ft(\036)1489 140 y Fs(1)1527 133 y Fv(:)f Ft(f)5 b(S)s(f)1660 120 y Fn(\030)1660 135 y Fv(=)1714 133 y(\005)19 b Fk(\(with)0 187 y(r)n(esp)n(e)n(ct)c(to)i Fo(Z)-13 b Fk(-gr)n(ading\))13 b(such)j(that)0 211 y SDict begin H.S end 0 211 a 0 211 a SDict begin 13 H.A end 0 211 a 0 211 a SDict begin [ /View [/XYZ H.V] /Dest (Item.14) cvn H.B /DEST pdfmark end 0 211 a 73 253 a Fv(\(1\))j Ft(\036)178 260 y Fs(1)198 253 y Fv(\()p Ft(f)238 260 y Fq(i)252 253 y Fv(\))13 b(=)f Ft(e)351 260 y Fq(i)382 253 y Fk(for)17 b Ft(i)12 b Fn(2)h Ft(I)t Fk(,)0 264 y SDict begin H.S end 0 264 a 0 264 a SDict begin 13 H.A end 0 264 a 0 264 a SDict begin [ /View [/XYZ H.V] /Dest (Item.15) cvn H.B /DEST pdfmark end 0 264 a 73 307 a Fv(\(2\))19 b Ft(\036)178 314 y Fs(1)198 307 y Fv(\()216 273 y Fl(P)264 321 y Fq(i)p Fm(2)p Fq(I)327 307 y Ft(\016)347 314 y Fq(i)361 307 y Ft(f)383 314 y Fq(i)398 307 y Fv(\()p Ft(xy)11 b Fn(\000)g Ft(y)r(x)p Fv(\))p Ft(f)611 314 y Fq(i)625 307 y Fv(\))h(=)703 273 y Fl(P)751 321 y Fq(a)p Fm(2)p Fq(Q)823 307 y Fv(\()p Ft(aa)889 291 y Fm(\003)918 307 y Fn(\000)f Ft(a)988 291 y Fm(\003)1008 307 y Ft(a)p Fv(\))p Fk(.)0 383 y(This)16 b(induc)n(es)f(a)i(gr)n(ade)n(d)f (isomorphism)h Ft(f)5 b(S)774 367 y Fq(\025)796 383 y Ft(f)836 371 y Fn(\030)836 385 y Fv(=)884 383 y(\005)918 367 y Fq(\025)957 383 y Fk(\(with)16 b(r)n(esp)n(e)n(ct)f(to)i Fo(Z)-13 b Ft(=)p Fv(2)o Fo(Z)e Fk(-gr)n(ading\))o(.)50 467 y Fv(W)l(e)15 b(pro)o(v)o(e)g(a)g(stronger)f(result:)0 476 y SDict begin H.S end 0 476 a 0 476 a SDict begin 13 H.A end 0 476 a 0 476 a SDict begin [ /View [/XYZ H.V] /Dest (prop.3.2) cvn H.B /DEST pdfmark end 0 476 a 76 x FD(Prop)q(osition)j (3.2.)i Fk(Ther)n(e)c(exists)f(a)i(gr)n(ade)n(d)f(isomorphism)h Ft(\036)c Fv(:)h Ft(f)5 b(S)s(f)1244 539 y Fn(\030)1244 554 y Fv(=)1292 552 y(\005)15 b Fk(\(with)h(r)n(esp)n(e)n(ct)e(to)h Fo(Z)-12 b Fk(-gr)n(ading)o(\))0 606 y(such)16 b(that)0 618 y SDict begin H.S end 0 618 a 0 618 a SDict begin 13 H.A end 0 618 a 0 618 a SDict begin [ /View [/XYZ H.V] /Dest (Item.16) cvn H.B /DEST pdfmark end 0 618 a 73 672 a Fv(\(1\))j Ft(\036)e Fk(satis\014es)d(c)n(onditions)i(of)g(the)h(pr)n (op)n(osition)946 672 y SDict begin H.S end 946 672 a Fk(3.1)1006 643 y SDict begin H.R end 1006 643 a 1006 672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.3.1) cvn H.B /ANN pdfmark end 1006 672 a 16 w Fk(and)0 683 y SDict begin H.S end 0 683 a 0 683 a SDict begin 13 H.A end 0 683 a 0 683 a SDict begin [ /View [/XYZ H.V] /Dest (Item.17) cvn H.B /DEST pdfmark end 0 683 a 73 726 a Fv(\(2\))i Ft(\036)p Fv(\()p Ft(a)220 709 y Fm(\016)240 726 y Fv(\))12 b(=)h Ft(\036)p Fv(\()p Ft(a)p Fv(\))405 709 y Fm(\016)441 726 y Fk(for)j(any)g Ft(a)d Fn(2)g Ft(f)5 b(S)s(f)g Fk(.)0 793 y(This)16 b(induc)n(es)h(a)g(gr)n(ade)n(d)g(isomorphism)g Ft(f)5 b(S)777 777 y Fq(\025)800 793 y Ft(f)841 781 y Fn(\030)841 795 y Fv(=)890 793 y(\005)924 777 y Fq(\025)964 793 y Fk(\(with)17 b(r)n(esp)n(e)n(ct)f(to)h Fo(Z)-12 b Ft(=)o Fv(2)p Fo(Z)d Fk(-gr)n(adin)o(g\))14 b(which)k(r)n(esp)n(e)n (cts)0 847 y(involutions.)0 932 y(Pr)n(o)n(of.)i Fv(Supp)q(ose)j(a)d (collection)j(of)d(p)q(ositiv)o(e)i(real)g(n)o(um)o(b)q(ers)f(\()p Ft(c)1129 939 y Fq(a)1149 932 y Fv(\))1167 939 y Fq(a)p Fm(2)p Fq(Q)1260 932 y Fv(is)g(giv)o(en.)38 b(Than)21 b(there)g(exists)g(a)0 986 y(unique)c(graded)e(automorphism)g Ft(\036)625 993 y Fs(2)657 986 y Fv(:)d(\005)h Fn(\000)-7 b(!)12 b Fv(\005)j(suc)o(h)h(that)0 1007 y SDict begin H.S end 0 1007 a 0 1007 a SDict begin 13 H.A end 0 1007 a 0 1007 a SDict begin [ /View [/XYZ H.V] /Dest (Item.18) cvn H.B /DEST pdfmark end 0 1007 a 73 1052 a Fv(\(1\))j Ft(\036)178 1059 y Fs(2)198 1052 y Fv(\()p Ft(e)237 1059 y Fq(i)251 1052 y Fv(\))12 b(=)h Ft(e)350 1059 y Fq(i)380 1052 y Fv(for)h Ft(i)f Fn(2)g Ft(I)t Fv(,)0 1063 y SDict begin H.S end 0 1063 a 0 1063 a SDict begin 13 H.A end 0 1063 a 0 1063 a SDict begin [ /View [/XYZ H.V] /Dest (Item.19) cvn H.B /DEST pdfmark end 0 1063 a 73 1106 a Fv(\(2\))19 b Ft(\036)178 1113 y Fs(2)198 1106 y Fv(\()p Ft(a)p Fv(\))12 b(=)h Ft(c)338 1113 y Fq(a)359 1106 y Ft(a)i Fv(for)f Ft(a)f Fn(2)g Ft(Q)p Fv(,)0 1117 y SDict begin H.S end 0 1117 a 0 1117 a SDict begin 13 H.A end 0 1117 a 0 1117 a SDict begin [ /View [/XYZ H.V] /Dest (Item.20) cvn H.B /DEST pdfmark end 0 1117 a 73 1160 a Fv(\(3\))19 b Ft(\036)178 1167 y Fs(2)198 1160 y Fv(\()p Ft(a)240 1143 y Fm(\003)260 1160 y Fv(\))12 b(=)h Ft(c)358 1143 y Fm(\000)p Fs(1)358 1171 y Fq(a)405 1160 y Ft(a)429 1143 y Fm(\003)464 1160 y Fv(for)h Ft(a)f Fn(2)g Ft(Q)p Fv(.)0 1226 y(W)l(e)i(put)g Ft(\036)d Fv(=)h Ft(\036)275 1233 y Fs(2)304 1226 y Fn(\016)c Ft(\036)363 1233 y Fs(1)383 1226 y Fv(.)20 b(Clearly)15 b(the)g(\014rst)g(condition)h(is)f (satis\014ed.)20 b(W)l(e)15 b(need)h(to)e(pro)o(v)o(e)g(that)g(it)h(is) g(p)q(osible)0 1280 y(to)h(c)o(ho)q(ose)g(n)o(um)o(b)q(ers)h Ft(c)407 1287 y Fq(a)444 1280 y Fv(in)h(suc)o(h)e(a)h(w)o(a)o(y)e(that) h(the)h(second)g(condition)g(w)o(ould)g(b)q(e)h(satis\014ed.)24 b(W)l(e)17 b(denote)0 1334 y(the)e(in)o(v)o(olution)i(on)e(\005)g (induced)i(from)e Ft(f)5 b(S)s(f)20 b Fv(b)o(y)15 b Ft(\036)871 1341 y Fs(2)906 1334 y Fv(as)g Ft(?)p Fv(.)50 1388 y(Let)k Ft(a)h Fn(2)f Ft(Q)p Fv(.)32 b(Then)20 b Ft(a)456 1371 y Fq(?)495 1388 y Fv(=)g Ft(t)566 1395 y Fq(a)587 1388 y Ft(a)611 1371 y Fm(\003)631 1388 y Fv(,)g Ft(t)680 1395 y Fq(a)721 1388 y Fn(2)f Fo(C)29 b Fv(b)q(ecause)21 b Ft(?)e Fv(preserv)o(es)g Fo(Z)-13 b Fv(-gra)o(ding)18 b(and)h(there)g(is)h(at)f(most)0 1448 y(one)h(arro)o(w)f(b)q(et)o(w)o (een)h(eac)o(h)g(t)o(w)o(o)e(v)o(ertices.)35 b(Then)20 b Ft(a)942 1431 y Fq(??)1000 1448 y Fv(=)h Fn(\000)p Ft(a)f Fv(implies)i Ft(a)1318 1431 y Fm(\003)p Fq(?)1376 1448 y Fv(=)f Fn(\000)1488 1427 y Fv(\026)1467 1448 y Ft(t)1483 1429 y Fm(\000)p Fs(1)1483 1454 y Fq(a)1531 1448 y Ft(a)f Fv(and)g(since)h Ft(w)f Fv(is)0 1502 y(self-adjoin)o(t) 282 1555 y Fl(X)280 1653 y Fq(a)p Fm(2)p Fq(Q)350 1598 y Fv(\()p Ft(aa)416 1579 y Fm(\003)446 1598 y Fn(\000)10 b Ft(a)515 1579 y Fm(\003)535 1598 y Ft(a)p Fv(\))i(=)h(\()657 1555 y Fl(X)655 1653 y Fq(a)p Fm(2)p Fq(Q)725 1598 y Fv(\()p Ft(aa)791 1579 y Fm(\003)821 1598 y Fn(\000)d Ft(a)890 1579 y Fm(\003)910 1598 y Ft(a)p Fv(\)\))970 1579 y Fq(?)1001 1598 y Fv(=)1052 1555 y Fl(X)1049 1653 y Fq(a)p Fm(2)p Fq(Q)1127 1598 y Fn(\000)1167 1567 y Ft(t)1183 1574 y Fq(a)p 1167 1588 38 2 v 1175 1621 a Fv(\026)1167 1629 y Ft(t)1183 1636 y Fq(a)1210 1598 y Fv(\()p Ft(aa)1276 1579 y Fm(\003)1305 1598 y Fn(\000)h Ft(a)1375 1579 y Fm(\003)1394 1598 y Ft(a)p Fv(\))i(implies)771 1758 y Fn(\000)811 1727 y Ft(t)827 1734 y Fq(a)p 811 1747 V 818 1781 a Fv(\026)811 1789 y Ft(t)827 1796 y Fq(a)866 1758 y Fv(=)g(1)p Ft(;)19 b(a)13 b Fn(2)g Ft(Q:)0 1847 y Fv(Therefore)i Ft(t)222 1854 y Fq(a)256 1847 y Fv(=)e Ft(r)325 1854 y Fq(a)345 1847 y FD(i)j Fv(for)e(some)h Ft(r)579 1854 y Fq(a)612 1847 y Fn(2)e Fo(R)n Fv(.)k(Then)f(w)o(e)e (can)i(express)1148 1793 y SDict begin H.S end 1148 1793 a 1148 1793 a SDict begin 13 H.A end 1148 1793 a 1148 1793 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.1) cvn H.B /DEST pdfmark end 1148 1793 a 0 1947 a Fv(\(3.1\))367 1904 y Fl(X)365 2002 y Fq(a)p Fm(2)p Fq(Q)435 1947 y Fv(\()p Ft(aa)501 1928 y Fm(\003)531 1947 y Fn(\000)10 b Ft(a)600 1928 y Fm(\003)620 1947 y Ft(a)p Fv(\))i(=)724 1904 y Fl(X)722 2002 y Fq(a)p Fm(2)p Fq(Q)792 1947 y Fv(\()p Fn(\000)864 1916 y FD(i)p 850 1936 42 2 v 850 1978 a Ft(r)871 1985 y Fq(a)896 1947 y Ft(aa)944 1928 y Fq(?)974 1947 y Fn(\000)f FD(i)p Ft(r)1055 1954 y Fq(a)1076 1947 y Ft(a)1100 1928 y Fm(\003)1119 1947 y Ft(a)1143 1928 y Fm(\003)p Fq(?)1181 1947 y Fv(\))h(=)h Fn(\000)p FD(i)1319 1904 y Fl(X)1316 2007 y Fq(a)p Fm(2)1367 1998 y Fs(\026)1359 2007 y Fq(Q)1394 1947 y Ft(q)1414 1954 y Fq(a)1435 1947 y Ft(aa)1483 1928 y Fq(?)0 2084 y Fv(for)i(some)f (real)i(n)o(um)o(b)q(ers)f Ft(q)475 2091 y Fq(a)497 2084 y Fv(,)f Ft(a)f Fn(2)614 2073 y Fv(\026)604 2084 y Ft(Q)p Fv(.)20 b(On)15 b(the)h(other)f(hand)496 2126 y Fl(X)502 2225 y Fq(i)p Fm(2)p Fq(I)569 2169 y Ft(\016)589 2176 y Fq(i)603 2169 y Ft(f)625 2176 y Fq(i)640 2169 y Fv(\()p Ft(xy)c Fn(\000)g Ft(y)r(x)p Fv(\))p Ft(f)853 2176 y Fq(i)879 2169 y Fv(=)i FD(i)949 2126 y Fl(X)955 2225 y Fq(i)p Fm(2)p Fq(I)1023 2169 y Ft(\016)1043 2176 y Fq(i)1057 2169 y Ft(f)1079 2176 y Fq(i)1093 2169 y Fv(\()p Ft(xx)1163 2151 y Fm(\016)1193 2169 y Fv(+)d Ft(y)r(y)1286 2151 y Fm(\016)1306 2169 y Fv(\))p Ft(f)1346 2176 y Fq(i)1360 2169 y Ft(:)0 2292 y Fv(Consider)16 b(the)f(follo)o(wing)h(elemen)o(t)g (of)f(the)g(group)g(algebra)750 2390 y Ft(J)i Fv(=)861 2359 y(1)p 845 2379 54 2 v 845 2421 a Fn(j)p Fv(\000)p Fn(j)911 2346 y Fl(X)912 2445 y Fq(g)q Fm(2)p Fs(\000)985 2390 y Ft(g)r(f)5 b(g)1060 2371 y Fm(\000)p Fs(1)1105 2390 y Ft(:)0 2516 y Fv(Clearly)19 b(it)f(b)q(elongs)g(to)g(the)f(cen)o (ter)h(of)g(the)g(group)f(algebra)h(and)g(its)g(trace)f(on)h(eac)o(h)g (irreducible)j(repre-)0 2570 y(sen)o(tation)15 b(equals)i(to)e(1.)21 b(So)16 b Ft(J)541 2554 y Fm(\000)p Fs(1)604 2570 y Fv(is)h(cen)o(tral) e(and)h(p)q(ositiv)o(e,)h(hence)g(there)e(exists)h(cen)o(tral)g (self-adjoin)o(t)h Ft(J)1856 2554 y Fm(0)0 2624 y Fv(suc)o(h)f(that)e Ft(J)230 2608 y Fm(0)242 2624 y Ft(J)t(J)300 2608 y Fm(0)326 2624 y Fv(=)f(1)397 2631 y Fg(C)r Fs(\000)444 2624 y Fv(.)20 b(It)c(follo)o(ws)f(that)f(1)797 2631 y Fg(C)s Fs(\000)860 2624 y Fv(can)i(b)q(e)f(represen)o(ted)h(as)661 2746 y(1)684 2753 y Fg(C)s Fs(\000)745 2746 y Fv(=)810 2689 y Fq(K)793 2703 y Fl(X)793 2802 y Fq(k)q Fs(=1)866 2746 y Ft(\013)895 2753 y Fq(k)916 2746 y Ft(f)5 b(\013)972 2727 y Fm(\016)972 2758 y Fq(k)994 2746 y Ft(;)20 b(\013)1056 2753 y Fq(k)1090 2746 y Fn(2)13 b Fo(C)7 b Fv(\000)p Ft(:)p eop %%Page: 8 8 8 7 bop 0 0 a SDict begin /product where{pop 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Fv(+)c Ft(y)r(y)1066 495 y Fm(\016)1085 513 y Fv(\))p Ft(f)1125 520 y Fq(i)0 634 y Fv(is)16 b(mapp)q(ed)g(b)o(y)f Ft(\036)309 641 y Fs(1)344 634 y Fv(to)g(a)g(linear)h(com)o(bination)g (with)f(p)q(ositiv)o(e)i(co)q(e\016cien)o(ts)f(of)f(elemen)o(ts)h(of)e (the)i(form)e Ft(aa)1848 617 y Fq(?)0 688 y Fv(for)j Ft(a)g Fn(2)170 676 y Fv(\026)160 688 y Ft(Q)p Fv(.)27 b(Hence)19 b(the)e(n)o(um)o(b)q(ers)h Ft(q)660 695 y Fq(a)699 688 y Fv(in)754 688 y SDict begin H.S end 754 688 a Fv(3.1)813 658 y SDict begin H.R end 813 658 a 813 688 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 813 688 a 17 w Fv(are)f(negativ)o(e)h(and)g(it)g (follo)o(ws)g(that)f(the)g(n)o(um)o(b)q(ers)h Ft(r)1769 695 y Fq(a)1807 688 y Fv(are)0 741 y(negativ)o(e)d(to)q(o.)20 b(W)l(e)15 b(put)g Ft(c)458 748 y Fq(a)491 741 y Fv(=)539 709 y Fn(p)p 577 709 77 2 v 32 x(\000)p Ft(r)633 748 y Fq(a)669 741 y Fv(for)g Ft(a)d Fn(2)h Ft(Q)i Fv(and)h(obtain)524 819 y Ft(\036)551 826 y Fs(2)571 819 y Fv(\()p Ft(a)613 800 y Fq(?)632 819 y Fv(\))d(=)g FD(i)p Ft(r)746 826 y Fq(a)766 819 y Ft(c)786 800 y Fm(\000)p Fs(1)786 830 y Fq(a)833 819 y Ft(a)857 800 y Fm(\003)890 819 y Fv(=)g Fn(\000)p FD(i)p Ft(c)1007 826 y Fq(a)1028 819 y Ft(a)1052 800 y Fm(\003)1084 819 y Fv(=)g Ft(\036)1159 826 y Fs(2)1179 819 y Fv(\()p Ft(a)p Fv(\))1239 800 y Fm(\016)1271 819 y Fv(and)548 900 y Ft(\036)575 907 y Fs(2)595 900 y Fv(\()p Ft(a)637 881 y Fm(\003)p Fq(?)674 900 y Fv(\))f(=)h Fn(\000)p FD(i)p Ft(r)823 881 y Fm(\000)p Fs(1)822 911 y Fq(a)871 900 y Ft(c)891 907 y Fq(a)911 900 y Ft(a)g Fv(=)g FD(i)p Ft(c)1030 881 y Fm(\000)p Fs(1)1030 911 y Fq(a)1077 900 y Ft(a)g Fv(=)g Ft(\036)1189 907 y Fs(2)1209 900 y Fv(\()p Ft(a)1251 881 y Fm(\003)1270 900 y Fv(\))1288 881 y Fm(\016)1308 900 y Ft(;)0 967 y Fv(so)i(the)g(comp)q(osition)h Ft(\036)415 974 y Fs(2)445 967 y Fn(\016)10 b Ft(\036)505 974 y Fs(1)540 967 y Fv(satis\014es)15 b(the)h(second)f(condition.)705 b Ff(\003)0 1023 y SDict begin H.S end 0 1023 a 0 1023 a SDict begin 13 H.A end 0 1023 a 0 1023 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end 0 1023 a 739 1068 a Fv(4.)22 b Fu(The)c(even)f(p)m(ar)m(t)50 1149 y Fv(W)l(e)d(are)h(going)f(to)g(consider)i(algebras)e Ft(e)752 1156 y Fq(i)766 1149 y Fv(\005)800 1133 y Fq(\025)823 1149 y Ft(e)844 1156 y Fq(i)858 1149 y Fv(.)20 b(Since)c(the)f(graph)f (is)h(bipartite)g(w)o(e)g(can)f(replace)i(\005)1783 1133 y Fq(\025)1820 1149 y Fv(b)o(y)0 1203 y(its)f(ev)o(en)h(part,)e(i.e.)20 b(the)c(follo)o(wing)g(is)f(clear)0 1212 y SDict begin H.S end 0 1212 a 0 1212 a SDict begin 13 H.A end 0 1212 a 0 1212 a SDict begin [ /View [/XYZ H.V] /Dest (prop.4.1) cvn H.B /DEST pdfmark end 0 1212 a 74 x FD(Prop)q(osition)h(4.1.)j Fk(The)14 b(sup)n(er)n(algebr)n(as)g Ft(e)766 1293 y Fq(i)780 1286 y Fv(\005)p Ft(e)835 1293 y Fq(i)849 1286 y Fk(,)h Ft(e)899 1293 y Fq(i)913 1286 y Fv(\005)947 1270 y Fq(\025)970 1286 y Ft(e)991 1293 y Fq(i)1020 1286 y Fk(have)g(zer)n(o)f(o)n(dd)h(p)n(art.)20 b(So)15 b Ft(e)1496 1293 y Fq(i)1510 1286 y Fv(\005)p Ft(e)1565 1293 y Fq(i)1592 1286 y Fv(=)e Ft(e)1661 1293 y Fq(i)1675 1286 y Fv(\005)1709 1293 y Fq(ev)1746 1286 y Ft(e)1767 1293 y Fq(i)1796 1286 y Fk(and)0 1340 y Ft(e)21 1347 y Fq(i)35 1340 y Fv(\005)69 1324 y Fq(\025)92 1340 y Ft(e)113 1347 y Fq(i)140 1340 y Fv(=)g Ft(e)209 1347 y Fq(i)223 1340 y Fv(\005)257 1324 y Fq(\025)257 1351 y(ev)294 1340 y Ft(e)315 1347 y Fq(i)329 1340 y Fk(.)50 1423 y Fv(Recall)j(that)e(the)i(algebra)f(\005)g(has)g(a)g(classical)i (in)o(v)o(olution)f Fn(\003)p Fv(.)0 1432 y SDict begin H.S end 0 1432 a 0 1432 a SDict begin 13 H.A end 0 1432 a 0 1432 a SDict begin [ /View [/XYZ H.V] /Dest (prop.4.2) cvn H.B /DEST pdfmark end 0 1432 a 74 x FD(Prop)q(osition)j(4.2.)h Fk(The)c(r)n(estrictions)f(of)h Fn(\003)g Fk(and)g Fn(\016)g Fk(to)h Fv(\005)1024 1513 y Fq(ev)1077 1506 y Fk(c)n(oincide.)0 1589 y(Pr)n(o)n(of.)j Fv(It)e(is)g(enough)g(to)f(c)o(hec)o(k)g(the)h (statemen)o(t)e(for)h(elemen)o(ts)h(of)f(degree)h(2.)26 b(Since)19 b(eac)o(h)f(arro)o(w)e(of)h(the)0 1643 y(quiv)o(er)f(go)q (es)e(from)h(an)g(o)q(dd)g(v)o(ertex)g(to)f(an)h(ev)o(en)g(one)g(w)o(e) g(ha)o(v)o(e)g(that)f(pro)q(ducts)h Ft(ab)g Fv(and)g Ft(a)1568 1626 y Fm(\003)1588 1643 y Ft(b)1608 1626 y Fm(\003)1642 1643 y Fv(are)g(zero)g(for)0 1697 y Ft(a;)8 b(b)j Fn(2)i Ft(Q)p Fv(.)20 b(F)l(or)15 b Ft(a)294 1680 y Fm(\003)313 1697 y Ft(b)g Fv(and)g Ft(ab)480 1680 y Fm(\003)457 1773 y Fv(\()p Ft(a)499 1754 y Fm(\003)519 1773 y Ft(b)p Fv(\))557 1754 y Fm(\016)588 1773 y Fv(=)e Ft(b)656 1754 y Fm(\003)675 1773 y Ft(a)g Fv(=)g(\()p Ft(a)802 1754 y Fm(\003)822 1773 y Ft(b)p Fv(\))860 1754 y Fm(\003)891 1773 y Fv(and)g(\()p Ft(ab)1039 1754 y Fm(\003)1058 1773 y Fv(\))1076 1754 y Fm(\016)1108 1773 y Fv(=)g Ft(ba)1200 1754 y Fm(\003)1232 1773 y Fv(=)g(\()p Ft(ab)1342 1754 y Fm(\003)1361 1773 y Fv(\))1379 1754 y Fm(\003)1398 1773 y Ft(:)1833 1849 y Ff(\003)50 1932 y Fv(Consider)20 b(the)g(space)h Ft(W)26 b Fv(of)19 b(traceless)h(op)q (erators)f(on)h Ft(V)10 b Fv(.)34 b(It)20 b(is)g(a)g(complex)h (3-dimensional)h(v)o(ector)0 1986 y(space)15 b(with)h(symmetric)f (bilinear)j(form)c Ft(g)j Fv(giv)o(en)e(b)o(y)764 2062 y Ft(g)r Fv(\()p Ft(a;)8 b(b)p Fv(\))i(=)j(2)8 b(tr)o(\()p Ft(ab)p Fv(\))p Ft(:)0 2139 y Fv(Clearly)17 b(the)g(subspace)g Ft(W)473 2146 y Fg(R)516 2139 y Fv(of)f(traceless)g(hermitian)i(op)q (erators)e(on)g Ft(V)26 b Fv(is)17 b(a)f(real)h(3-dimensional)i(v)o (ector)0 2193 y(space)c(with)g(scalar)f(pro)q(duct)h(induced)i(b)o(y)d Ft(g)r Fv(.)19 b(The)c(group)f Ft(S)s(U)5 b Fv(\()p Ft(V)k Fv(\))14 b(acts)g(on)h Ft(W)20 b Fv(and)15 b Ft(W)1533 2200 y Fg(R)1574 2193 y Fv(\014xing)g(the)g(form)0 2247 y Ft(g)r Fv(.)k(This)d(pro)q(duces)g(a)f(w)o(ell-kno)o(wn)h (homomorphism)492 2323 y Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))682 2310 y Fn(\030)682 2325 y Fv(=)730 2323 y Ft(S)s(U)g Fv(\()p Ft(V)k Fv(\))j Fn(\000)-7 b(!)12 b Ft(S)s(O)q Fv(\()p Ft(W)1095 2330 y Fg(R)1120 2323 y Fv(\))1151 2310 y Fn(\030)1151 2325 y Fv(=)1198 2323 y Ft(S)s(O)q Fv(\(3)p Ft(;)c Fo(R)l Fv(\))0 2399 y(whose)15 b(k)o(ernel)i(is)f Fn(f\000)p Fv(1)p Ft(;)8 b Fv(1)p Fn(g)p Fv(.)19 b(The)c(sub)o(v)m(ariet)o(y)h Ft(W)849 2406 y Fs(1)882 2399 y Fn(\032)d Ft(W)22 b Fv(of)15 b(op)q(erators)f Ft(a)i Fv(suc)o(h)g(that)e Ft(g)r Fv(\()p Ft(a;)8 b(a)p Fv(\))j(=)j(1)h(is)h(again)0 2453 y(equipp)q(ed)h(with)f(\000-action.) 50 2507 y(Next,)f(w)o(e)f(iden)o(tify)j Ft(V)j Fn(\012)10 b Ft(V)25 b Fv(with)15 b Ft(V)20 b Fn(\012)11 b Ft(V)782 2490 y Fm(\003)814 2507 y Fv(=)i(End)944 2514 y Fg(C)970 2507 y Fv(\()p Ft(V)c Fv(\))15 b(b)o(y)g(the)g(map)h Ft( )g Fv(de\014ned)g(as)630 2583 y Ft( )r Fv(\()p Ft(\013)10 b Fn(\012)g Ft(\014)r Fv(\))j(=)g Ft(\013w)q Fv(\()p Ft(\014)r(;)8 b Fn(\001)p Fv(\))p Ft(;)18 b(\013;)8 b(\014)14 b Fn(2)f Ft(V)s(:)0 2659 y Fv(Clearly)18 b Ft( )g Fv(is)f(equiv)m (arian)o(t)h(and)g Ft( )616 2643 y Fm(\000)p Fs(1)679 2659 y Fv(restricts)f(to)f(an)h(em)o(b)q(edding)h Ft(W)1252 2666 y Fs(1)1288 2659 y Fn(\000)-8 b(!)15 b Ft(V)21 b Fn(\012)12 b Ft(V)27 b Fv(whic)o(h)17 b(induces)i(an)0 2713 y(epimorphism)14 b(of)e(co)q(ordinate)h(rings)g Fo(C)7 b Fv([)p Ft(V)17 b Fn(\012)5 b Ft(V)10 b Fv(])i Fn(\000)-7 b(!)12 b Fo(C)7 b Fv([)p Ft(W)1008 2720 y Fs(1)1030 2713 y Fv(])12 b(whic)o(h)i(in)f(its)g(turn)g(induces)h(an)f (epimorphism)707 2790 y(\011)742 2797 y Fs(1)775 2790 y Fv(:)f Fo(C)7 b Fn(h)p Ft(V)12 b Fn(i)905 2797 y Fq(ev)954 2790 y Fn(\000)-7 b(!)12 b Fo(C)7 b Fv([)p Ft(W)1126 2797 y Fs(1)1148 2790 y Fv(])p eop %%Page: 9 9 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)711 b(9)0 133 y Fv(whic)o(h)15 b(is)f(equiv)m(arian)o(t.)21 b(Denote)14 b(its)g(k)o(ernel)h(b)o(y)f Ft(K)s Fv(.)19 b(Let)14 b Ft(a)f Fv(=)g Ft(x)1104 116 y Fs(2)1124 133 y Fv(,)g Ft(b)g Fv(=)f Ft(y)1254 116 y Fs(2)1274 133 y Fv(,)i Ft(c)e Fv(=)h Ft(xy)d Fv(+)e Ft(y)r(x)14 b Fv(and)g Ft(d)e Fv(=)h Ft(xy)d Fn(\000)e Ft(y)r(x)0 187 y Fv(b)q(e)16 b(generators)e(of)h Fo(C)7 b Fn(h)p Ft(V)12 b Fn(i)439 194 y Fq(ev)475 187 y Fv(.)0 198 y SDict begin H.S end 0 198 a 0 198 a SDict begin 13 H.A end 0 198 a 0 198 a SDict begin [ /View [/XYZ H.V] /Dest (prop.4.3) cvn H.B /DEST pdfmark end 0 198 a 72 x FD(Prop)q(osition)19 b(4.3.)h Fk(The)c(ide)n(al)f Ft(K)20 b Fk(has)c(the)g(fol)r(lowing)g(set)g(of)h (gener)n(ators:)1356 216 y SDict begin H.S end 1356 216 a 1356 216 a SDict begin 13 H.A end 1356 216 a 1356 216 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.1) cvn H.B /DEST pdfmark end 1356 216 a 0 347 a Fv(\(4.1\))435 b Fn(f)p Ft(ab)9 b Fn(\000)h Ft(ba;)e(ac)h Fn(\000)h Ft(ca;)e(bc)h Fn(\000)h Ft(cb;)e(d;)g Fv(4)p Ft(ab)f Fn(\000)j Ft(c)1218 328 y Fs(2)1248 347 y Fv(+)g(1)p Fn(g)0 430 y Fk(Pr)n(o)n(of.)20 b Fv(The)c(k)o(ernel)h(of)e(the)h(epimorphism)h Fo(C)6 b Fn(h)p Ft(V)13 b Fn(i)876 437 y Fq(ev)926 430 y Fn(\000)-8 b(!)13 b Fo(C)7 b Fv([)p Ft(V)23 b Fn(\012)10 b Ft(V)g Fv(])15 b(is)h(generated)g(b)o(y)g(pairwise)g(comm)o(u-)0 484 y(tators)h(of)i(generators)f Ft(a)p Fv(,)i Ft(b)p Fv(,)f Ft(c)p Fv(,)g Ft(d)p Fv(.)31 b(In)o(tro)q(duce)19 b(a)g(co)q(ordinates)g Ft(m)1180 491 y Fq(ij)1230 484 y Fv(\()p Ft(i;)8 b(j)19 b Fv(=)h(1)p Ft(;)8 b Fv(2\))17 b(in)j Ft(W)25 b Fv(suc)o(h)19 b(that)g(for)0 538 y Ft(m)13 b Fn(2)f Ft(W)667 614 y(m)g Fv(=)767 550 y Fl(\022)801 587 y Ft(m)841 594 y Fs(11)878 587 y Fv(\()p Ft(m)p Fv(\))41 b Ft(m)1035 594 y Fs(12)1072 587 y Fv(\()p Ft(m)p Fv(\))801 641 y Ft(m)841 648 y Fs(21)878 641 y Fv(\()p Ft(m)p Fv(\))g Ft(m)1035 648 y Fs(22)1072 641 y Fv(\()p Ft(m)p Fv(\))1147 550 y Fl(\023)1188 614 y Ft(:)0 707 y Fv(Then)16 b(for)e Ft(t)f Fv(=)g Ft(\013)e Fn(\012)f Ft(\014)15 b Fn(2)e Ft(V)19 b Fn(\012)11 b Ft(V)287 811 y( )r Fv(\()p Ft(t)p Fv(\))h(=)h Ft(\013w)q Fv(\()p Ft(\014)r(;)8 b Fn(\001)p Fv(\))j(=)651 747 y Fl(\022)684 783 y Ft(x)p Fv(\()p Ft(\013)p Fv(\))685 837 y Ft(y)r Fv(\()p Ft(\013)p Fv(\))775 747 y Fl(\023)816 774 y(\000)837 810 y Fn(\000)p Ft(y)r Fv(\()p Ft(\014)r Fv(\))41 b Ft(x)p Fv(\()p Ft(\014)r Fv(\))1090 774 y Fl(\001)1124 811 y Fv(=)1172 747 y Fl(\022)1205 783 y Fn(\000)1245 766 y Fq(c)p Fs(+)p Fq(d)p 1245 773 61 2 v 1267 799 a Fs(2)1311 783 y Fv(\()p Ft(t)p Fv(\))65 b Ft(a)p Fv(\()p Ft(t)p Fv(\))1231 841 y Fn(\000)p Ft(b)p Fv(\()p Ft(t)p Fv(\))1410 823 y Fq(c)p Fm(\000)p Fq(d)p 1410 830 V 1431 856 a Fs(2)1475 841 y Fv(\()p Ft(t)p Fv(\))1527 747 y Fl(\023)1568 811 y Ft(:)0 915 y Fv(Th)o(us,)15 b(taking)g(the)g(in)o(v)o(erse)h(map)f(w)o(e)g(obtain)210 992 y(\011)245 999 y Fs(1)265 992 y Fv(\()p Ft(a)p Fv(\))d(=)h Ft(m)425 999 y Fs(12)463 992 y Ft(;)19 b Fv(\011)530 999 y Fs(1)550 992 y Fv(\()p Ft(b)p Fv(\))12 b(=)h Fn(\000)p Ft(m)741 999 y Fs(21)779 992 y Ft(;)19 b Fv(\011)846 999 y Fs(1)866 992 y Fv(\()p Ft(c)p Fv(\))12 b(=)h Ft(m)1022 999 y Fs(22)1069 992 y Fn(\000)e Ft(m)1155 999 y Fs(11)1192 992 y Ft(;)20 b Fv(\011)1260 999 y Fs(1)1280 992 y Fv(\()p Ft(d)p Fv(\))12 b(=)h Fn(\000)p Ft(m)1475 999 y Fs(11)1522 992 y Fn(\000)e Ft(m)1608 999 y Fs(22)1645 992 y Ft(:)0 1069 y Fv(The)k(equations)h(of)f Ft(W)392 1076 y Fs(1)427 1069 y Fv(are)479 1146 y Ft(m)519 1153 y Fs(11)566 1146 y Fv(+)c Ft(m)652 1153 y Fs(22)701 1146 y Fv(=)i(0)p Ft(;)20 b Fv(2)p Ft(m)868 1128 y Fs(2)868 1158 y(11)915 1146 y Fv(+)11 b(4)p Ft(m)1024 1153 y Fs(12)1061 1146 y Ft(m)1101 1153 y Fs(21)1148 1146 y Fv(+)f(2)p Ft(m)1256 1128 y Fs(2)1256 1158 y(22)1306 1146 y Fv(=)j(1)p Ft(;)0 1223 y Fv(so)i Ft(K)j Fv(is)e(generated)f(b)o(y)g(pairwise)h(comm)o (utators)d(of)i Ft(a)p Fv(,)g Ft(b)p Fv(,)f Ft(c)p Fv(,)h Ft(d)f Fv(and)i(t)o(w)o(o)e(more)g(elemen)o(ts:)375 1328 y Ft(d)f Fv(and)502 1297 y(\()p Ft(c)d Fv(+)g Ft(d)p Fv(\))637 1281 y Fs(2)p 502 1317 154 2 v 568 1359 a Fv(2)672 1328 y Fn(\000)g Fv(4)p Ft(ab)f Fv(+)844 1297 y(\()p Ft(c)g Fn(\000)i Ft(d)p Fv(\))979 1281 y Fs(2)p 844 1317 V 909 1359 a Fv(2)1013 1328 y Fn(\000)f Fv(1)j(=)g Ft(c)1162 1309 y Fs(2)1191 1328 y Fv(+)e Ft(d)1261 1309 y Fs(2)1290 1328 y Fn(\000)f Fv(4)p Ft(ab)g Fn(\000)g Fv(1)p Ft(;)0 1418 y Fv(and)15 b(it)h(can)f(b)q(e)h(easily)g(seen)g(that)f(the)g(set) 751 1418 y SDict begin H.S end 751 1418 a Fv(4.1)809 1388 y SDict begin H.R end 809 1388 a 809 1418 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.4.1) cvn H.B /ANN pdfmark end 809 1418 a 15 w Fv(generates)g(the)g(same)g(ideal.)512 b Ff(\003)50 1502 y Fv(Denote)13 b(b)o(y)h Ft(I)j Fv(the)d(ideal)h(in)f Ft(S)i Fv(generated)e(b)o(y)g Ft(xy)8 b Fn(\000)f Ft(y)r(x)g Fn(\000)g Ft(\034)20 b Fv(and)14 b(b)o(y)f Ft(I)1255 1509 y Fq(ev)1306 1502 y Fv(the)g(in)o(tersection)i Ft(I)10 b Fn(\\)d Ft(S)1720 1509 y Fq(ev)1757 1502 y Fv(.)20 b(The)0 1556 y(follo)o(wing)c(is)g(true:)0 1565 y SDict begin H.S end 0 1565 a 0 1565 a SDict begin 13 H.A end 0 1565 a 0 1565 a SDict begin [ /View [/XYZ H.V] /Dest (prop.4.4) cvn H.B /DEST pdfmark end 0 1565 a 74 x FD(Prop)q(osition)j(4.4.)h Fk(The)c(ide)n(al)f Ft(I)604 1646 y Fq(ev)658 1639 y Fk(is)g(gener)n(ate)n(d)h(by)507 1716 y Ft(ab)10 b Fn(\000)g Ft(ba;)e(ac)h Fn(\000)h Ft(ca;)e(bc)g Fn(\000)j Ft(cb;)d(d)g Fn(\000)i Ft(\034)s(;)e Fv(4)p Ft(ab)g Fn(\000)j Ft(c)1251 1697 y Fs(2)1280 1716 y Fv(+)g(1)p Ft(:)0 1799 y Fk(Pr)n(o)n(of.)20 b Fv(The)13 b(ideal)g Ft(I)356 1806 y Fq(ev)405 1799 y Fv(can)g(b)q(e)g(generated)f(b)o(y)g Ft(xy)6 b Fn(\000)t Ft(y)r(x)t Fn(\000)t Ft(\034)18 b Fv(and)13 b Ft(V)1155 1783 y Fm(\003)1175 1799 y Fv(\()p Ft(xy)6 b Fn(\000)t Ft(y)r(x)t Fn(\000)t Ft(\034)f Fv(\))p Ft(V)1459 1783 y Fm(\003)1479 1799 y Fv(,)12 b(so)g(b)o(y)g(\014v)o(e)g(elemen)o(ts)0 1853 y Ft(g)22 1860 y Fs(1)41 1853 y Fv(,)j Ft(g)91 1860 y Fs(2)111 1853 y Fv(,)f Ft(g)160 1860 y Fs(3)180 1853 y Fv(,)h Ft(g)230 1860 y Fs(4)249 1853 y Fv(,)g Ft(g)299 1860 y Fs(5)318 1853 y Fv(,)g(where)477 1799 y SDict begin H.S end 477 1799 a 477 1799 a SDict begin 13 H.A end 477 1799 a 477 1799 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.2) cvn H.B /DEST pdfmark end 477 1799 a 1271 1921 a Ft(xy)d Fn(\000)e Ft(y)r(x)g Fn(\000)g Ft(\034)18 b Fv(=)13 b Ft(d)d Fn(\000)g Ft(\034)18 b Fv(=)13 b Ft(g)1754 1928 y Fs(1)411 1991 y Fv(2)p Ft(x)p Fv(\()p Ft(xy)e Fn(\000)g Ft(y)r(x)f Fn(\000)g Ft(\034)5 b Fv(\))p Ft(x)13 b Fv(=)g Ft(a)p Fv(\()p Ft(c)c Fn(\000)h Ft(d)p Fv(\))g Fn(\000)g Fv(\()p Ft(c)g Fv(+)g Ft(d)p Fv(\))p Ft(a)g Fv(+)g(2)p Ft(a\034)18 b Fn(\021)1400 1998 y Fs(mo)q(d)11 b Fq(g)1497 2003 y Fp(1)1528 1991 y Ft(ac)f Fn(\000)h Ft(ca)h Fv(=)h Ft(g)1754 1998 y Fs(2)438 2061 y Fv(2)p Ft(y)r Fv(\()p Ft(xy)e Fn(\000)f Ft(y)r(x)g Fn(\000)h Ft(\034)5 b Fv(\))p Ft(y)14 b Fv(=)f(\()p Ft(c)d Fn(\000)g Ft(d)p Fv(\))p Ft(b)f Fn(\000)h Ft(b)p Fv(\()p Ft(c)f Fv(+)i Ft(d)p Fv(\))e(+)i(2)p Ft(b\034)17 b Fn(\021)1409 2068 y Fs(mo)q(d)11 b Fq(g)1506 2073 y Fp(1)1537 2061 y Ft(cb)f Fn(\000)g Ft(bc)i Fv(=)h Ft(g)1754 2068 y Fs(3)188 2137 y Fv(4)p Ft(x)p Fv(\()p Ft(xy)e Fn(\000)g Ft(y)r(x)f Fn(\000)g Ft(\034)5 b Fv(\))p Ft(y)15 b Fv(=)e(4)p Ft(ab)c Fn(\000)h Fv(\()p Ft(c)g Fv(+)g Ft(d)p Fv(\)\()p Ft(c)f Fv(+)i Ft(d)p Fv(\))e(+)h(2\()p Ft(c)g Fv(+)g Ft(d)p Fv(\))p Ft(\034)17 b Fn(\021)1304 2144 y Fs(mo)q(d)10 b Fq(g)1400 2149 y Fp(1)1432 2137 y Fv(4)p Ft(ab)f Fn(\000)i Ft(c)1574 2118 y Fs(2)1603 2137 y Fv(+)g(1)h(=)h Ft(g)1754 2144 y Fs(4)242 2213 y Fv(4)p Ft(y)r Fv(\()p Ft(xy)e Fn(\000)g Ft(y)r(x)f Fn(\000)g Ft(\034)5 b Fv(\))p Ft(x)13 b Fv(=)g(\()p Ft(c)c Fn(\000)i Ft(d)p Fv(\)\()p Ft(c)d Fn(\000)j Ft(d)p Fv(\))e Fn(\000)i Fv(4)p Ft(ba)e Fv(+)h(2\()p Ft(c)g Fn(\000)g Ft(d)p Fv(\))p Ft(\034)17 b Fn(\021)1358 2220 y Fs(mo)q(d)10 b Fq(g)1454 2225 y Fp(1)1486 2213 y Ft(c)1506 2194 y Fs(2)1535 2213 y Fn(\000)h Fv(4)p Ft(ba)e Fn(\000)i Fv(1)h(=)1249 2283 y Fn(\021)1320 2290 y Fs(mo)q(d)e Fq(g)1416 2295 y Fp(4)1448 2283 y Fv(4\()p Ft(ab)f Fn(\000)h Ft(ba)p Fv(\))i(=)h(4)p Ft(g)1754 2290 y Fs(5)0 2106 y Fv(\(4.2\))1833 2359 y Ff(\003)50 2443 y Fv(T)l(ak)o(e)i(an)g(automorphism)g(\011)559 2450 y Fs(2)594 2443 y Fv(of)g Ft(S)674 2450 y Fq(ev)726 2443 y Fv(whic)o(h)h(satis\014es)258 2520 y(\011)293 2527 y Fs(2)313 2520 y Fv(\()p Ft(g)r Fv(\))11 b(=)i Ft(g)h Fv(\()p Ft(g)g Fn(2)e Fv(\000\))p Ft(;)20 b Fv(\011)678 2527 y Fs(2)698 2520 y Fv(\()p Ft(a)p Fv(\))12 b(=)h Ft(a;)20 b Fv(\011)910 2527 y Fs(2)930 2520 y Fv(\()p Ft(b)p Fv(\))12 b(=)h Ft(b;)19 b Fv(\011)1133 2527 y Fs(2)1153 2520 y Fv(\()p Ft(c)p Fv(\))12 b(=)h Ft(c;)19 b Fv(\011)1356 2527 y Fs(2)1376 2520 y Fv(\()p Ft(d)p Fv(\))12 b(=)h Ft(d)d Fv(+)g Ft(\034)s(:)0 2597 y Fv(Since)15 b(b)q(oth)e Ft(d)g Fv(and)h Ft(\034)k Fv(comm)o(ute)13 b(with)h(elemen)o(ts)g(of)e(\000)i(suc)o(h)g(an)f(automorphism)g (exists)h(and)f(prop)q(ositions)0 2651 y SDict begin H.S end 0 2651 a Fv(4.3)58 2622 y SDict begin H.R end 58 2622 a 58 2651 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.4.3) cvn H.B /ANN pdfmark end 58 2651 a 15 w Fv(and)162 2651 y SDict begin H.S end 162 2651 a Fv(4.4)220 2622 y SDict begin H.R end 220 2622 a 220 2651 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.4.4) cvn H.B /ANN pdfmark end 220 2651 a 15 w Fv(imply)0 2660 y SDict begin H.S end 0 2660 a 0 2660 a SDict begin 13 H.A end 0 2660 a 0 2660 a SDict begin [ /View [/XYZ H.V] /Dest (cor.4.1) cvn H.B /DEST pdfmark end 0 2660 a 74 x FD(Corollary)20 b(4.1.)i Fk(The)17 b(epimorphism)i Fv(\011)740 2741 y Fs(0)776 2734 y Fv(:)c Ft(S)832 2741 y Fq(ev)885 2734 y Fn(\000)-7 b(!)15 b Fo(C)7 b Fv([)p Ft(W)1060 2741 y Fs(1)1082 2734 y Fv(])k Fn(\003)g Fv(\000)19 b Fk(de\014ne)n(d)e(as)h Fv(\011)1437 2741 y Fs(0)1472 2734 y Fv(=)f(\(\011)1577 2741 y Fs(1)1608 2734 y Fn(\012)11 b Ft(I)t(d)1702 2741 y Fg(C)s Fs(\000)1750 2734 y Fv(\))g Fn(\016)g Fv(\011)1848 2741 y Fs(2)0 2790 y Fk(has)16 b(kernel)g Ft(I)237 2797 y Fq(ev)273 2790 y Fk(.)21 b(Thus)16 b Fv(\011)458 2797 y Fs(0)495 2790 y Fk(induc)n(es)f(an)h(isomorphism)h Fv(\011)c(:)f Ft(S)1093 2773 y Fq(\025)1090 2801 y(ev)1139 2777 y Fn(\030)1139 2792 y Fv(=)1187 2790 y Fo(C)7 b Fv([)p Ft(W)1273 2797 y Fs(1)1296 2790 y Fv(])i Fn(\003)h Fv(\000)p Fk(.)p eop %%Page: 10 10 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(10)50 133 y Fv(De\014ne)16 b(a)e(classical)j(in)o(v)o(olution)g Fn(\003)d Fv(on)i Fo(C)6 b Fv([)p Ft(W)803 140 y Fs(1)826 133 y Fv(])14 b(b)o(y)i(form)o(ula)758 215 y(\()p Ft(f)803 196 y Fm(\003)822 215 y Fv(\)\()p Ft(m)p Fv(\))c(=)p 976 175 123 2 v 13 w Ft(f)5 b Fv(\()p Ft(m)1061 201 y Fm(\003)1080 215 y Fv(\))p Ft(:)0 292 y Fv(It)20 b(induces)h(a)e (classical)i(in)o(v)o(olution)g Fn(\003)e Fv(on)h Fo(C)7 b Fv([)p Ft(W)855 299 y Fs(1)877 292 y Fv(])13 b Fn(\003)f Fv(\000)20 b(since)h(the)f(op)q(eration)g(of)f(taking)g(the)h (hermitian)0 346 y(adjoin)o(t)15 b(comm)o(utes)g(with)g(the)g(group)g (action.)20 b(It)c(o)q(ccurs)f(that)0 354 y SDict begin H.S end 0 354 a 0 354 a SDict begin 13 H.A end 0 354 a 0 354 a SDict begin [ /View [/XYZ H.V] /Dest (cor.4.2) cvn H.B /DEST pdfmark end 0 354 a 75 x FD(Corollary)j(4.2.)i Fk(The)c(homomorphism)i Fv(\011)f Fk(r)n(esp)n(e)n(cts)d(involutions.)0 512 y(Pr)n(o)n(of.)20 b Fv(It)c(is)f(enough)h(to)f(pro)o(v)o(e)f(the)h (statemen)o(t)g(for)f(homomorphisms)h(\011)1303 519 y Fs(1)1338 512 y Fv(and)h(\011)1462 519 y Fs(2)1482 512 y Fv(.)k(Compute)515 589 y Ft(a)539 570 y Fm(\016)572 589 y Fv(=)13 b Fn(\000)p Ft(b;)19 b(b)727 570 y Fm(\016)759 589 y Fv(=)13 b Fn(\000)p Ft(a;)20 b(c)919 570 y Fm(\016)951 589 y Fv(=)13 b Ft(c;)20 b(d)1076 570 y Fm(\016)1108 589 y Fv(=)13 b Ft(d;)19 b(\034)1237 570 y Fm(\016)1270 589 y Fv(=)13 b Ft(\034)s(;)0 666 y Fv(and)711 724 y Ft(m)751 705 y Fm(\003)751 735 y Fq(ij)794 724 y Fv(=)g Ft(m)882 731 y Fq(j)r(i)925 724 y Fv(\()p Ft(i;)8 b(j)14 b Fv(=)f(1)p Ft(;)8 b Fv(2\))p Ft(:)0 792 y Fv(Using)16 b(de\014ning)g(form)o(uli)g(for)f(\011)561 799 y Fs(1)596 792 y Fv(and)g(\011)719 799 y Fs(2)755 792 y Fv(w)o(e)g(pro)o(v)o(e)f (the)h(statemen)o(t.)601 b Ff(\003)0 831 y SDict begin H.S end 0 831 a 0 831 a SDict begin 13 H.A end 0 831 a 0 831 a SDict begin [ /View [/XYZ H.V] /Dest (cor.4.3) cvn H.B /DEST pdfmark end 0 831 a 44 x FD(Corollary)18 b(4.3.)i Fk(Ther)n(e)c(is)g(an)g(isomorphism)h Ft(e)869 882 y Fq(i)883 875 y Fv(\005)917 859 y Fq(\025)940 875 y Ft(e)961 882 y Fq(i)987 863 y Fn(\030)987 877 y Fv(=)1035 875 y Ft(f)1057 882 y Fq(i)1072 875 y Fo(C)7 b Fv([)p Ft(W)1158 882 y Fs(1)1180 875 y Fv(])j Fn(\003)f Fv(\000)p Ft(f)1285 882 y Fq(i)1316 875 y Fk(which)17 b(r)n(esp)n(e)n(cts)e(involutions.)0 958 y(Pr)n(o)n(of.)20 b Fv(An)c(isomorphism)f(can)h(b)q(e)f (constructed)h(using)f(prop)q(osition)1228 958 y SDict begin H.S end 1228 958 a Fv(3.2)1286 929 y SDict begin H.R end 1286 929 a 1286 958 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.3.2) cvn H.B /ANN pdfmark end 1286 958 a 15 w Fv(and)h(corollary)1578 958 y SDict begin H.S end 1578 958 a Fv(4.1)1636 929 y SDict begin H.R end 1636 929 a 1636 958 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cor.4.1) cvn H.B /ANN pdfmark end 1636 958 a 15 w Fv(taking)f(in)o(to)0 1012 y(accoun)o(t)c(prop)q(osition)400 1012 y SDict begin H.S end 400 1012 a Fv(4.1)458 983 y SDict begin H.R end 458 983 a 458 1012 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.4.1) cvn H.B /ANN pdfmark end 458 1012 a Fv(.)19 b(The)12 b(fact)f(that)g(this)h(isomorphism)g(resp)q(ects)g(in)o(v)o (olution)h(follo)o(ws)f(from)f(prop)q(o-)0 1066 y(sitions)149 1066 y SDict begin H.S end 149 1066 a Fv(4.2)207 1037 y SDict begin H.R end 207 1037 a 207 1066 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.4.2) cvn H.B /ANN pdfmark end 207 1066 a Fv(,)243 1066 y SDict begin H.S end 243 1066 a Fv(3.2)301 1037 y SDict begin H.R end 301 1037 a 301 1066 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.3.2) cvn H.B /ANN pdfmark end 301 1066 a 22 w Fv(and)22 b(corollary)613 1066 y SDict begin H.S end 613 1066 a Fv(4.2)671 1037 y SDict begin H.R end 671 1037 a 671 1066 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cor.4.2) cvn H.B /ANN pdfmark end 671 1066 a Fv(.)39 b(All)24 b(in)o(v)o(olv)o(ed)e(algebras)g(can)g(b) q(e)g(displa)o(y)o(ed)i(on)d(the)h(follo)o(wing)0 1120 y(diagram:)362 1177 y Ft(e)383 1184 y Fq(i)397 1177 y Fv(\005)431 1160 y Fq(\025)454 1177 y Ft(e)475 1184 y Fq(i)599 1177 y Fn(\032)103 b Fv(\005)771 1160 y Fq(\025)771 1188 y(ev)910 1177 y Ff(\021)140 b Fv(\005)1130 1184 y Fq(ev)1307 1177 y Ft(,)-8 b Fn(!)112 b Fv(\005)408 1231 y Fn(ko)311 b(ko)322 b(ko)e(ko)363 1287 y Ft(f)385 1294 y Fq(i)399 1287 y Ft(S)430 1270 y Fq(\025)452 1287 y Ft(f)474 1294 y Fq(i)599 1287 y Fn(\032)79 b Ft(f)5 b(S)771 1270 y Fq(\025)768 1298 y(ev)805 1287 y Ft(f)83 b Ff(\021)116 b Ft(f)5 b(S)1127 1294 y Fq(ev)1163 1287 y Ft(f)122 b(,)-8 b Fn(!)86 b Ft(f)5 b(S)s(f)408 1341 y Fn(ko)313 b(\\)329 b(\\)d(\\)357 1396 y Ft(f)379 1403 y Fq(i)393 1396 y Ft(S)424 1380 y Fq(\025)421 1408 y(ev)458 1396 y Ft(f)480 1403 y Fq(i)599 1396 y Fn(\032)106 b Ft(S)771 1380 y Fq(\025)768 1408 y(ev)910 1396 y Ff(\021)143 b Ft(S)1127 1403 y Fq(ev)1307 1396 y Ft(,)-8 b Fn(!)114 b Ft(S)408 1450 y Fn(ko)311 b(ko)322 b(ko)e(ko)293 1504 y Ft(f)315 1511 y Fq(i)330 1504 y Fo(C)6 b Fv([)p Ft(W)415 1511 y Fs(1)438 1504 y Fv(])j Fn(\003)h Fv(\000)p Ft(f)543 1511 y Fq(i)599 1504 y Fn(\032)42 b Fo(C)7 b Fv([)p Ft(W)762 1511 y Fs(1)784 1504 y Fv(])j Fn(\003)g Fv(\000)42 b Ff(\021)f Fo(C)7 b Fn(h)p Ft(V)22 b Fn(\012)11 b Ft(V)e Fn(i)h(\003)g Fv(\000)42 b Ft(,)-8 b Fn(!)41 b Fo(C)7 b Fn(h)p Ft(V)12 b Fn(i)e(\003)f Fv(\000)0 1570 y(On)20 b(the)f(diagram)f(relation)i Ft(A)f Fn(\032)g Ft(B)i Fv(means)e(that)f(the)h(algebra)g Ft(A)g Fv(is)g(con)o(tained)h(in)g Ft(B)r Fv(,)f(but)g(can)g(ha)o(v)o(e)0 1624 y(di\013eren)o(t)g(unit)g (form)f(the)g(unit)i(of)e Ft(B)r Fv(.)29 b(Sym)o(b)q(ols)20 b(")p Ff(\020)p Fv(")d(and)i(")p Ft(,)-8 b Fn(!)p Fv(")17 b(denote)i(unit)g(preserving)g(surjectiv)o(e)0 1678 y(and)c(injectiv)o (e)i(homomorphisms)e(of)g(algebras.)998 b Ff(\003)50 1762 y Fv(Cho)q(ose)15 b(an)h(orthonormal)f(basis)h(with)g(resp)q(ect)g (to)f(the)h(form)f Ft(g)h Fv(in)h Ft(W)1283 1769 y Fg(R)1309 1762 y Fv(.)k(This)16 b(giv)o(es)g(an)g(orthonormal)0 1816 y(basis)e(in)g Ft(W)19 b Fv(and,)14 b(in)g(this)f(basis,)h(the)f (equation)h(of)f Ft(W)936 1823 y Fs(1)968 1816 y Fv(coincides)j(with)d (the)h(equation)f(of)g Ft(S)1598 1823 y Fg(C)1637 1816 y Fv(giv)o(en)h(in)g(the)0 1870 y(in)o(tro)q(duction,)i(so)f Ft(W)372 1877 y Fs(1)404 1857 y Fn(\030)404 1872 y Fv(=)452 1870 y Ft(S)480 1877 y Fg(C)521 1870 y Fv(and)g(the)h(theorem)f(1)f (follo)o(ws)i(from)e(the)h(follo)o(wing)h(general)g(observ)m(ation:)0 1879 y SDict begin H.S end 0 1879 a 0 1879 a SDict begin 13 H.A end 0 1879 a 0 1879 a SDict begin [ /View [/XYZ H.V] /Dest (lem.4.1) cvn H.B /DEST pdfmark end 0 1879 a 74 x FD(Lemma)c(4.1.)k Fk(If)c(a)g(\014nite)f(gr)n(oup)h Fv(\000)h Fk(acts)f(on)f(an)h(a\016ne)g(variety)g Ft(X)j Fk(over)d Fo(C)22 b Fk(and)12 b Ft(p)g Fk(is)g(an)f(idemp)n(otent)h(in) g(the)0 2007 y(gr)n(oup)h(algebr)n(a)g(of)f Fv(\000)i Fk(then)e(the)h(algebr)n(a)f Ft(p)p Fo(C)7 b Fv([)p Ft(X)t Fv(])r Fn(\003)r Fv(\000)p Ft(p)15 b Fk(is)d(isomorphic)h(to)g(the)g (algebr)n(a)g Ft(F)1473 2014 y Fs(\000)1497 2007 y Fv(\()p Ft(X)q(;)8 b Fv(End)1655 2014 y Fg(C)1681 2007 y Fv(\()p Fo(C)f Fv(\000)p Ft(p)p Fv(\)\))15 b Fk(of)0 2061 y(r)n(e)n(gular)e Fv(\000)p Fk(-e)n(quivariant)i(maps)e(fr)n(om)h Ft(X)j Fk(to)d Fv(End)834 2068 y Fg(C)860 2061 y Fv(\()p Fo(C)7 b Fv(\000)p Ft(p)p Fv(\))p Fk(,)17 b(wher)n(e)d Fv(\000)g Fk(acts)f(on)g Fv(End)1411 2068 y Fg(C)1437 2061 y Fv(\()p Fo(C)6 b Fv(\000)q Ft(p)p Fv(\))16 b Fk(by)d(c)n(onjugation.)0 2115 y(If,)22 b(mor)n(e)n(over,)h Ft(p)e Fk(is)g(self-adjoint)g(and)h Ft(X)j Fk(is)c(de\014ne)n(d)f(over)i Fo(R)n Fk(,)e(then)h(the)h (involution)f(on)g Fo(C)7 b Fv([)p Ft(X)t Fv(])23 b Fk(given)0 2174 y(by)e Ft(f)91 2157 y Fm(\003)110 2174 y Fv(\()p Ft(x)p Fv(\))f(=)p 248 2134 89 2 v 21 w Ft(f)5 b Fv(\()p 293 2149 26 2 v Ft(x)p Fv(\))20 b Fk(induc)n(es)g(an)g(involution)g(on) g Ft(p)p Fo(C)7 b Fv([)p Ft(X)t Fv(])15 b Fn(\003)e Fv(\000)p Ft(p)p Fk(,)22 b(and)e(the)h(c)n(orr)n(esp)n(onding)e(involution)h(on)0 2228 y Ft(F)29 2235 y Fs(\000)53 2228 y Fv(\()p Ft(X)q(;)8 b Fv(End)211 2235 y Fg(C)237 2228 y Fv(\()p Fo(C)f Fv(\000)p Ft(p)p Fv(\)\))19 b Fk(is)d(given)f(by)413 2305 y Ft(f)440 2286 y Fm(\003)460 2305 y Fv(\()p Ft(x)p Fv(\))d(=)h(\()p Ft(f)5 b Fv(\()p 645 2280 V Ft(x)o Fv(\)\))706 2286 y Fm(\003)725 2305 y Ft(;)20 b Fk(for)d Ft(x)12 b Fn(2)h Ft(X)t Fk(,)j Ft(f)i Fn(2)12 b Ft(F)1095 2312 y Fs(\000)1120 2305 y Fv(\()p Ft(X)q(;)c Fv(End)1278 2312 y Fg(C)1304 2305 y Fv(\()p Fo(C)e Fv(\000)q Ft(p)p Fv(\)\))p Fk(.)50 2388 y Fv(F)l(or)14 b(our)h(case)h(one)f(should)h(set)f Ft(X)h Fv(=)d Ft(S)737 2395 y Fg(C)778 2388 y Fv(and)i Ft(p)e Fv(=)g Ft(f)972 2395 y Fq(i)1001 2388 y Fv(so)i(that)g Fo(C)6 b Fv(\000)q Ft(f)1236 2395 y Fq(i)1266 2376 y Fn(\030)1266 2390 y Fv(=)1314 2388 y Ft(V)1341 2395 y Fq(i)1354 2388 y Fv(.)0 2447 y SDict begin H.S end 0 2447 a 0 2447 a SDict begin 13 H.A end 0 2447 a 0 2447 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 0 2447 a 722 2492 a Fv(5.)22 b Fu(Real)c(str)o(ucture)50 2573 y Fv(W)l(e)g(denote)h(b)o(y)f Ft(A)g Fv(the)h(*-algebra)f Ft(F)710 2580 y Fs(\000)734 2573 y Fv(\()p Ft(S)780 2580 y Fg(C)806 2573 y Ft(;)8 b Fv(End)908 2580 y Fg(C)934 2573 y Fv(\()p Ft(V)979 2580 y Fq(i)992 2573 y Fv(\)\))18 b(|)f(the)i(*-algebra)f(of)g(regular)g(\000-equiv)m(arian)o(t)0 2627 y(maps)c(from)f Ft(S)252 2634 y Fg(C)292 2627 y Fv(to)g(End)428 2634 y Fg(C)454 2627 y Fv(\()p Ft(V)499 2634 y Fq(i)512 2627 y Fv(\))h(where)g Ft(S)702 2634 y Fg(C)742 2627 y Fv(is)h(an)e(a\016ne)i(v)m(ariet)o(y)f(in)h Fo(C)1199 2610 y Fs(3)1236 2627 y Fv(giv)o(en)g(b)o(y)f(the)g(equation) g Ft(\013)1706 2610 y Fs(2)1734 2627 y Fv(+)8 b Ft(\014)1805 2610 y Fs(2)1833 2627 y Fv(+)0 2681 y Ft(\015)27 2664 y Fs(2)65 2681 y Fv(=)19 b(1)f(and)h Ft(V)279 2688 y Fq(i)312 2681 y Fv(is)g(a)g(unitary)g(irreducible)j(represen)o(tation)d (of)f(\000.)31 b(The)19 b(in)o(v)o(olution)i(in)e Ft(A)g Fv(is)h(giv)o(en)f(b)o(y)0 2734 y Ft(f)27 2718 y Fm(\003)47 2734 y Fv(\()p Ft(x)p Fv(\))13 b(=)h Ft(f)5 b Fv(\()p 216 2709 26 2 v Ft(x)p Fv(\))260 2718 y Fm(\003)295 2734 y Fv(for)15 b Ft(x)e Fn(2)h Ft(S)476 2741 y Fg(C)502 2734 y Fv(,)i Ft(f)j Fn(2)13 b Ft(A)p Fv(.)22 b(As)16 b(it)g(w)o(as)f(pro)o(v)o(ed)g(in)i(the)f(previous)g(section)h Ft(A)f Fv(is)g(isomorphic)h(to)0 2790 y Ft(e)21 2797 y Fq(i)35 2790 y Fv(\005)69 2773 y Fq(\025)92 2790 y Ft(e)113 2797 y Fq(i)142 2790 y Fv(as)e(a)g(*-algebra.)p eop %%Page: 11 11 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(11)50 133 y Fv(F)l(or)14 b(an)o(y)h(p)q(oin)o(t)h Ft(x)d Fn(2)f Ft(S)445 140 y Fg(C)487 133 y Fv(w)o(e)i(denote)i(b)o(y)f Ft(S)s(tab)854 140 y Fq(x)888 133 y Fn(\032)e Fv(\000)j(the)f (stabilizer)i(of)d Ft(x)p Fv(.)20 b(Put)96 213 y Ft(M)140 220 y Fq(x)175 213 y Fv(=)13 b Fn(f)p Ft(m)f Fn(2)h Fv(End)422 220 y Fg(C)448 213 y Fv(\()p Ft(V)493 220 y Fq(i)507 213 y Fv(\))p Fn(j)p Ft(mg)g Fv(=)g Ft(g)r(m;)h Fv(for)g(an)o(y)8 b Ft(g)14 b Fn(2)f Ft(S)s(tab)1070 220 y Fq(x)1091 213 y Fn(g)f Fv(|)j(the)g(cen)o(tralizer)h(of)f Ft(S)s(tab)1629 220 y Fq(x)1666 213 y Fv(in)h Ft(V)1746 220 y Fq(i)1759 213 y Fv(.)0 293 y Ft(M)44 300 y Fq(x)81 293 y Fv(is)f(a)f (*-subalgebra)h(of)g(End)561 300 y Fg(C)587 293 y Fv(\()p Ft(V)632 300 y Fq(i)645 293 y Fv(\))g(and)g(it)g(is)g(clear)g(that)f (for)g(an)o(y)h Ft(x)d Fn(2)h Ft(S)1326 300 y Fg(C)1367 293 y Ft(f)5 b Fv(\()p Ft(x)p Fv(\))12 b Fn(2)h Ft(M)1555 300 y Fq(x)1577 293 y Fv(.)19 b(The)c(opp)q(osite)0 347 y(is)h(true:)0 347 y SDict begin H.S end 0 347 a 0 347 a SDict begin 13 H.A end 0 347 a 0 347 a SDict begin [ /View [/XYZ H.V] /Dest (lem.5.1) cvn H.B /DEST pdfmark end 0 347 a 85 x FD(Lemma)f(5.1.)j Fk(F)m(or)c(any)g(tuple)g(of)h(p)n (oints)e Ft(x)769 439 y Fs(1)789 432 y Fk(,)i Ft(x)844 439 y Fs(2)863 432 y Fk(,)g(.)7 b(.)g(.)g(,)14 b Ft(x)1009 439 y Fq(n)1047 432 y Fk(with)h(p)n(airwise)f(disjoint)f(orbits)h(the)h (map)g(fr)n(om)0 486 y Ft(A)h Fk(to)h Ft(M)149 493 y Fq(x)169 498 y Fp(1)198 486 y Fn(\002)11 b Ft(M)288 493 y Fq(x)308 498 y Fp(2)337 486 y Fn(\002)g(\001)d(\001)g(\001)g(\002)i Ft(M)535 493 y Fq(x)555 497 y Fz(n)595 486 y Fk(given)15 b(by)629 566 y Ft(f)i Fn(7\000)-7 b(!)12 b Fv(\()p Ft(f)5 b Fv(\()p Ft(x)843 573 y Fs(1)863 566 y Fv(\))p Ft(;)j(f)d Fv(\()p Ft(x)973 573 y Fs(2)991 566 y Fv(\))p Ft(;)j(:)g(:)g(:)t(;)g(f) d Fv(\()p Ft(x)1181 573 y Fq(n)1204 566 y Fv(\)\))0 647 y Fk(is)16 b(surje)n(ctive.)0 732 y(Pr)n(o)n(of.)k Fv(F)l(or)e(an)o(y)f Ft(y)335 739 y Fs(1)372 732 y Fn(2)h Ft(M)464 739 y Fq(x)484 744 y Fp(1)503 732 y Fv(,)g Ft(y)556 739 y Fs(2)593 732 y Fn(2)g Ft(M)685 739 y Fq(x)705 744 y Fp(2)724 732 y Fv(,)g(.)8 b(.)f(.)g(,)18 b Ft(y)869 739 y Fq(n)910 732 y Fn(2)f Ft(M)1001 739 y Fq(x)1021 743 y Fz(n)1062 732 y Fv(tak)o(e)h(an)o(y)f(regular)h(map)g(from)f Ft(S)1651 739 y Fg(C)1695 732 y Fv(whic)o(h)i(at)0 786 y(p)q(oin)o(ts)g(of)g(the) g(form)f Ft(g)r(x)439 793 y Fq(j)475 786 y Fv(accept)i(v)m(alue)g Ft(g)r(y)788 793 y Fq(j)806 786 y Ft(g)830 769 y Fm(\000)p Fs(1)876 786 y Fv(.)31 b(Clearly)20 b(suc)o(h)f(a)g(map)f(exists)i(b)q (ecause)g(the)f(n)o(um)o(b)q(er)0 844 y(of)h(p)q(oin)o(ts)h(of)f(the)g (form)g Ft(g)r(x)502 851 y Fq(j)540 844 y Fv(is)h(\014nite)g(and)g(if)g Ft(g)875 851 y Fs(1)894 844 y Ft(x)920 851 y Fq(j)960 844 y Fv(=)g Ft(g)1038 851 y Fs(2)1058 844 y Ft(x)1084 851 y Fq(k)1125 844 y Fv(then)g Ft(j)j Fv(=)d Ft(k)h Fv(and)f Ft(g)1495 851 y Fs(1)1514 844 y Ft(y)1536 851 y Fq(j)1554 844 y Ft(g)1578 825 y Fm(\000)p Fs(1)1576 857 y(1)1646 844 y Fv(=)h Ft(g)1725 851 y Fs(2)1744 844 y Ft(y)1766 851 y Fq(j)1785 844 y Ft(g)1809 825 y Fm(\000)p Fs(1)1807 857 y(2)1855 844 y Fv(.)0 898 y(Av)o(eraging)d(this)h(map)f (with)h(resp)q(ect)g(to)e(all)j(elemen)o(ts)f(of)e(\000)i(giv)o(es)g (an)f(elemen)o(t)h(of)f Ft(A)g Fv(whic)o(h)i(maps)e(to)0 952 y(\()p Ft(y)40 959 y Fs(1)60 952 y Ft(;)8 b(y)103 959 y Fs(2)122 952 y Ft(;)g(:)g(:)g(:)d(;)j(y)246 959 y Fq(n)269 952 y Fv(\).)1533 b Ff(\003)50 1040 y Fv(There)19 b(is)f(a)h(cen)o(tral)f(*-subalgebra)h Ft(A)731 1047 y Fq(Z)778 1040 y Fv(of)f Ft(A)g Fv(whic)o(h)i(consists)e(of)g(suc)o(h) h Ft(f)k Fn(2)c Ft(A)f Fv(that)g Ft(f)5 b Fv(\()p Ft(x)p Fv(\))18 b(is)h(scalar)0 1094 y(for)c(all)i Ft(x)c Fn(2)h Ft(S)245 1101 y Fg(C)271 1094 y Fv(.)21 b(The)16 b(algebra)f Ft(A)592 1101 y Fq(Z)636 1094 y Fv(is)h(the)g(algebra)g(of)f(regular)h (functions)g(on)g Ft(S)1416 1101 y Fg(C)1442 1094 y Ft(=)p Fv(\000.)21 b(Ev)o(ery)15 b(irreducible)0 1148 y(*-represen)o(tation)j Ft(\032)f Fv(of)h Ft(A)g Fv(giv)o(es)g(a)g(c)o(haracter)f(on)h Ft(A)942 1155 y Fq(Z)989 1148 y Fv(b)o(y)g(the)g(Sc)o(h)o(ur's)f (lemma,)i(th)o(us)f(a)g(p)q(oin)o(t)g Ft(x)f Fn(2)h Ft(S)1842 1155 y Fg(C)0 1202 y Fv(suc)o(h)e(that)e(for)h Ft(f)i Fn(2)c Ft(A)387 1209 y Fq(Z)431 1202 y Ft(\032)p Fv(\()p Ft(f)5 b Fv(\))11 b(=)i Ft(f)5 b Fv(\()p Ft(x)p Fv(\).)20 b(The)15 b(p)q(oin)o(t)h Ft(x)f Fv(is)h(suc)o(h)f(that)393 1287 y Ft(f)420 1268 y Fm(\003)440 1287 y Fv(\()p Ft(x)p Fv(\))d(=)h Ft(\032)p Fv(\()p Ft(f)631 1268 y Fm(\003)650 1287 y Fv(\))f(=)p 728 1247 87 2 v 13 w Ft(\032)p Fv(\()p Ft(f)5 b Fv(\))12 b(=)p 874 1247 89 2 v 12 w Ft(f)5 b Fv(\()p Ft(x)p Fv(\))13 b(=)g Ft(f)1051 1268 y Fm(\003)1070 1287 y Fv(\()p 1088 1262 26 2 v Ft(x)p Fv(\))p Ft(;)20 b Fv(for)14 b(an)o(y)e Ft(f)18 b Fn(2)13 b Ft(A)1434 1294 y Fq(Z)1463 1287 y Ft(;)0 1367 y Fv(so)p 56 1342 V 15 w Ft(x)f Fv(=)h Ft(g)r(x)p Fv(,)h(some)h Ft(g)f Fn(2)f Fv(\000.)0 1376 y SDict begin H.S end 0 1376 a 0 1376 a SDict begin 13 H.A end 0 1376 a 0 1376 a SDict begin [ /View [/XYZ H.V] /Dest (prop.5.1) cvn H.B /DEST pdfmark end 0 1376 a 76 x FD(Prop)q(osition)19 b(5.1.)h Fk(If)15 b Ft(x)e Fn(2)g Ft(S)542 1459 y Fg(C)584 1452 y Fk(and)p 672 1427 26 2 v 16 w Ft(x)g Fv(=)g Ft(g)r(x)j Fk(for)g Ft(g)e Fn(2)f Fv(\000)k Fk(then)f(we)g(ar)n(e)g(in)g(the)h(one)f(of)g (two)h(c)n(ases:)0 1474 y SDict begin H.S end 0 1474 a 0 1474 a SDict begin 13 H.A end 0 1474 a 0 1474 a SDict begin [ /View [/XYZ H.V] /Dest (Item.21) cvn H.B /DEST pdfmark end 0 1474 a 73 1519 a Fv(\(1\))i Ft(x)13 b Fv(=)p 238 1494 26 2 v 13 w Ft(x)p Fk(,)j Ft(x)d Fn(2)g Ft(S)404 1526 y Fg(R)430 1519 y Fk(.)0 1531 y SDict begin H.S end 0 1531 a 0 1531 a SDict begin 13 H.A end 0 1531 a 0 1531 a SDict begin [ /View [/XYZ H.V] /Dest (Item.22) cvn H.B /DEST pdfmark end 0 1531 a 73 1573 a Fv(\(2\))19 b Ft(x)d Fk(and)p 280 1548 26 2 v 15 w Ft(x)f Fk(ar)n(e)h(line)n(arly)e (indep)n(endent)g(over)i Fo(C)6 b Fk(,)19 b Ft(S)s(tab)1063 1580 y Fq(x)1097 1573 y Fv(=)13 b Fn(f\000)p Fv(1)p Ft(;)8 b Fv(1)p Fn(g)14 b Fk(and)h Ft(g)i Fk(is)d(an)i(element)e(of)i(or)n (der)151 1627 y Fv(4)g Fk(in)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))p Fk(.)0 1713 y(Pr)n(o)n(of.)20 b Fv(If)14 b(w)o(e)g(supp)q(ose)g(that)f Ft(x)h Fv(and)p 642 1688 V 13 w Ft(x)g Fv(are)f(linearly)j(dep)q(enden)o(t,)f(i.e.)20 b Ft(x)12 b Fv(=)h Ft(c)p 1333 1688 V(x)g Fv(for)g Ft(c)g Fn(2)g Fo(C)23 b Fv(then)14 b(expressing)0 1766 y Ft(x)h Fv(in)h(co)q(ordinates)g Ft(x)c Fv(=)h(\()p Ft(x)466 1773 y Fs(1)486 1766 y Ft(;)8 b(x)533 1773 y Fs(2)552 1766 y Ft(;)g(x)599 1773 y Fs(3)617 1766 y Fv(\))15 b(giv)o(es)520 1854 y(1)d(=)h Ft(x)629 1836 y Fs(2)629 1866 y(1)659 1854 y Fv(+)e Ft(x)731 1836 y Fs(2)731 1866 y(2)761 1854 y Fv(+)f Ft(x)832 1836 y Fs(2)832 1866 y(3)864 1854 y Fv(=)j Ft(c)932 1836 y Fs(2)952 1854 y Fv(\()p 970 1812 46 2 v Ft(x)996 1839 y Fs(2)996 1867 y(1)1025 1854 y Fv(+)p 1071 1812 V 11 w Ft(x)1097 1839 y Fs(2)1097 1867 y(2)1126 1854 y Fv(+)p 1172 1812 V 11 w Ft(x)1198 1839 y Fs(2)1198 1867 y(3)1218 1854 y Fv(\))f(=)h Ft(c)1316 1836 y Fs(2)1335 1854 y Ft(:)0 1936 y Fv(If)g Ft(c)g Fv(=)g Fn(\000)p Fv(1)g(then)g Ft(x)322 1919 y Fs(2)322 1948 y Fq(j)355 1936 y Fv(=)g Fn(\000)p Ft(x)464 1943 y Fq(j)p 482 1911 45 2 v 482 1936 a Ft(x)508 1943 y Fq(j)539 1936 y Fn(\024)g Fv(0)g(|)g(con)o(tradiction,)g(so)g Ft(c)f Fv(=)h(1)g(and)g Ft(x)g Fv(=)p 1312 1911 26 2 v 13 w Ft(x)p Fv(.)19 b(Supp)q(ose)14 b(that)f Ft(x)g Fv(and)p 1768 1911 V 13 w Ft(x)g Fv(are)0 1991 y(linearly)18 b(indep)q(enden)o(t)h(o)o(v)o(er)c Fo(C)7 b Fv(.)26 b(W)l(e)17 b(obtain,)f(since)i Ft(g)f Fv(is)g(de\014ned)h(o)o(v)o(er)d Fo(R)c Ft(x)k Fv(=)g Ft(g)p 1444 1966 V 2 w(x)o Fv(,)h(whic)o(h)h (implies)i(that)0 2045 y Ft(g)24 2028 y Fs(2)60 2045 y Fv(is)f(the)g(iden)o(tit)o(y)g(on)f Fo(C)454 2028 y Fs(3)494 2045 y Fv(since)i(it)e(has)g(at)g(least)h(t)o(w)o(o)e(eigen)o (v)m(alues)j(1.)26 b(If)18 b Ft(S)s(tab)1431 2052 y Fq(x)1469 2045 y Fv(con)o(tains)g(an)f(elemen)o(t)0 2099 y Ft(h)e Fn(2)g Fv(\000)i(whic)o(h)h(acts)e(non-trivially)j(on)d Ft(S)712 2106 y Fg(C)755 2099 y Fv(then)h(it's)f(eigenspace)i(for)e (eigen)o(v)m(alue)j(1)d(is)h(one-dimensional,)0 2152 y(so)e Ft(x)g Fv(and)p 185 2127 V 15 w Ft(x)g Fv(cannot)g(b)q(e)h (linearly)h(indep)q(enden)o(t.)980 b Ff(\003)50 2241 y Fv(Let)16 b Ft(m)172 2248 y Fq(x)209 2241 y Fv(b)q(e)h(the)f(t)o(w)o (o-sided)f(ideal)j(in)e Ft(A)g Fv(of)f(maps)h(v)m(anishing)h(at)e Ft(x)p Fv(,)h Ft(m)1290 2225 y Fm(\003)1290 2252 y Fq(x)1325 2241 y Fv(=)e Ft(m)1414 2248 y Fq(x)1436 2241 y Fv(.)21 b(Clearly)l(,)c Ft(\032)e Fv(is)h(zero)g(on)0 2295 y Ft(m)40 2302 y Fq(x)72 2295 y Fn(\\)10 b Ft(A)146 2302 y Fq(Z)175 2295 y Fv(.)20 b(In)15 b(general)h Ft(A)p Fv(\()p Ft(m)513 2302 y Fq(x)545 2295 y Fn(\\)10 b Ft(A)619 2302 y Fq(Z)648 2295 y Fv(\))i Fn(6)p Fv(=)h Ft(m)766 2302 y Fq(x)788 2295 y Fv(,)i(but)g(the)g(follo)o(wing)h(holds:)0 2306 y SDict begin H.S end 0 2306 a 0 2306 a SDict begin 13 H.A end 0 2306 a 0 2306 a SDict begin [ /View [/XYZ H.V] /Dest (prop.5.2) cvn H.B /DEST pdfmark end 0 2306 a 74 x FD(Prop)q(osition)f(5.2.)j Fk(L)n(et)12 b Ft(\032)h Fk(b)n(e)g(a)h(*-r)n(epr)n(esentation)f(of)h Ft(A)f Fk(and)h Ft(x)f Fn(2)g Ft(S)1208 2387 y Fg(C)1247 2380 y Fk(such)h(that)p 1438 2355 26 2 v 14 w Ft(x)e Fv(=)h Ft(g)r(x)g Fk(.)20 b(If)13 b Ft(\032)g Fk(vanishes)0 2434 y(on)j Ft(m)105 2441 y Fq(x)137 2434 y Fn(\\)10 b Ft(A)211 2441 y Fq(Z)256 2434 y Fk(than)16 b Ft(\032)g Fk(vanishes)f(on)h Ft(m)685 2441 y Fq(x)707 2434 y Fk(.)0 2519 y(Pr)n(o)n(of.)k Fv(Let)c Ft(a)d Fn(2)f Ft(m)341 2526 y Fq(x)363 2519 y Fv(,)j Ft(a)e Fv(=)g Ft(a)500 2503 y Fm(\003)519 2519 y Fv(.)20 b(Consider)c(the)f(c)o(haracteristic)h(p)q(olynomial)365 2600 y Ft(p)p Fv(\()p Ft(t)p Fv(;)8 b Ft(X)t Fv(\))i(=)j(det)8 b Ft(X)13 b Fn(\000)e Ft(a)p Fv(\()p Ft(t)p Fv(\))h(=)h Ft(X)907 2582 y Fq(k)938 2600 y Fv(+)d Ft(p)1006 2607 y Fq(k)q Fm(\000)p Fs(1)1073 2600 y Fv(\()p Ft(t)p Fv(\))p Ft(X)1167 2582 y Fq(k)q Fm(\000)p Fs(1)1242 2600 y Fv(+)h Fn(\001)d(\001)g(\001)g Fv(+)i Ft(p)1419 2607 y Fs(0)1439 2600 y Fv(\()p Ft(t)p Fv(\))p Ft(:)0 2682 y Fv(Its)j(co)q(e\016cien)o (ts)h Ft(p)316 2689 y Fq(j)334 2682 y Fv(\()p Ft(t)p Fv(\))f(b)q(elong)h(to)e Ft(A)629 2689 y Fq(Z)671 2682 y Fv(and)h Ft(a)p Fv(\()p Ft(x)p Fv(\))f(=)h(0,)g(so)g Ft(p)1029 2689 y Fq(j)1059 2682 y Fn(2)g Ft(m)1142 2689 y Fq(x)1170 2682 y Fn(\\)6 b Ft(A)1240 2689 y Fq(Z)1269 2682 y Fv(,)13 b(and)g(it)g(follo)o(ws)g(that)g Ft(\032)p Fv(\()p Ft(a)p Fv(\))1752 2665 y Fq(k)1785 2682 y Fv(=)g(0.)0 2736 y(Since)g Ft(\032)p Fv(\()p Ft(a)p Fv(\))d(is)h(self-adjoin)o(t)h Ft(\032)p Fv(\()p Ft(a)p Fv(\))g(=)g(0.)19 b(Since)12 b(ev)o(ery)f(elemen)o(t)h Ft(b)g Fn(2)h Ft(m)1187 2743 y Fq(x)1220 2736 y Fv(can)e(b)q(e)h(represen)o(ted)g(as)f Ft(b)h Fv(=)h Ft(a)1751 2743 y Fs(1)1772 2736 y Fv(+)r FD(i)p Ft(a)1847 2743 y Fs(2)0 2790 y Fv(whith)j Ft(a)153 2797 y Fs(1)173 2790 y Ft(;)8 b(a)218 2797 y Fs(2)252 2790 y Fv(|)14 b(self-adjoin)o(t)i(elemen)o(ts)g(of)f Ft(m)824 2797 y Fq(x)861 2790 y Fv(w)o(e)g(obtain)g Ft(\032)p Fv(\()p Ft(b)p Fv(\))c(=)i(0.)589 b Ff(\003)p eop %%Page: 12 12 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(12)50 133 y Fv(Th)o(us)15 b(ev)o(ery)g(irreducible)i(*-represen)o (tation)e(of)g Ft(A)g Fv(is)g(induced)i(from)e(a)f(represen)o(tation)h (of)g Ft(M)1697 140 y Fq(x)1719 133 y Fv(.)20 b(Let)15 b Ft(\032)1857 116 y Fm(0)0 187 y Fv(b)q(e)h(the)f(corresp)q(onding)h (represen)o(tation)g(of)e Ft(M)825 194 y Fq(x)847 187 y Fv(.)20 b(Then,)15 b(for)g(an)o(y)g Ft(a)e Fn(2)g Ft(A)146 267 y(\032)170 248 y Fm(0)181 267 y Fv(\()p Ft(a)p Fv(\()p Ft(x)p Fv(\)\))303 248 y Fm(\003)334 267 y Fv(=)g Ft(\032)p Fv(\()p Ft(a)p Fv(\))466 248 y Fm(\003)497 267 y Fv(=)g Ft(\032)p Fv(\()p Ft(a)611 248 y Fm(\003)630 267 y Fv(\))f(=)h Ft(\032)732 248 y Fm(0)744 267 y Fv(\()p Ft(a)786 248 y Fm(\003)805 267 y Fv(\()p Ft(x)p Fv(\)\))f(=)h Ft(\032)969 248 y Fm(0)980 267 y Fv(\()p Ft(a)p Fv(\()p 1040 242 26 2 v Ft(x)o Fv(\))1083 248 y Fm(\003)1102 267 y Fv(\))g(=)g Ft(\032)1205 248 y Fm(0)1216 267 y Fv(\()p Ft(a)p Fv(\()p Ft(g)r(x)p Fv(\))1344 248 y Fm(\003)1362 267 y Fv(\))f(=)h Ft(\032)1464 248 y Fm(0)1475 267 y Fv(\()p Ft(g)r(a)p Fv(\()p Ft(x)p Fv(\))1603 248 y Fm(\003)1621 267 y Ft(g)1645 248 y Fm(\000)p Fs(1)1692 267 y Fv(\))p Ft(:)0 348 y Fv(In)j(fact)e(w)o(e)h(ha)o(v)o(e)0 348 y SDict begin H.S end 0 348 a 0 348 a SDict begin 13 H.A end 0 348 a 0 348 a SDict begin [ /View [/XYZ H.V] /Dest (prop.5.3) cvn H.B /DEST pdfmark end 0 348 a 85 x FD(Prop)q(osition)22 b(5.3.)g Fk(F)m(or)c(al)r(l)h Ft(a)e Fn(2)h Ft(M)678 440 y Fq(x)719 433 y Ft(\032)743 416 y Fm(0)754 433 y Fv(\()p Ft(g)r(ag)844 416 y Fm(\000)p Fs(1)889 433 y Fv(\))g(=)f Ft(\032)1001 416 y Fm(0)1012 433 y Fv(\()p Ft(a)p Fv(\))p Fk(.)29 b(Thus)18 b Ft(\032)1256 416 y Fm(0)1267 433 y Fv(\()p Ft(a)p Fv(\()p Ft(x)p Fv(\)\))1389 416 y Fm(\003)1425 433 y Fv(=)g Ft(\032)1502 416 y Fm(0)1513 433 y Fv(\()p Ft(a)p Fv(\()p Ft(x)p Fv(\))1617 416 y Fm(\003)1636 433 y Fv(\))p Fk(.)28 b(So)19 b(every)0 487 y(irr)n(e)n(ducible)d(*-r)n (epr)n(esentation)f(of)i Ft(A)f Fk(is)g(induc)n(e)n(d)g(fr)n(om)g(a)h (*-r)n(epr)n(esentation)e(of)i Ft(M)1453 494 y Fq(x)1475 487 y Fk(,)f Ft(x)d Fn(2)g Ft(S)1615 494 y Fg(C)1641 487 y Fk(.)0 572 y(Pr)n(o)n(of.)20 b Fv(Consider)d(the)g(op)q(erator)f Ft(\036)g Fv(on)h Ft(M)748 579 y Fq(x)786 572 y Fv(sending)h Ft(a)e Fv(to)g Ft(g)r(ag)1121 556 y Fm(\000)p Fs(1)1167 572 y Fv(.)23 b(Since)18 b Ft(g)r(x)c Fv(=)p 1437 547 26 2 v 15 w Ft(x)j Fv(and)f(the)h(action)g(of)f(\000)0 626 y(on)i Ft(S)94 633 y Fg(C)139 626 y Fv(is)g(de\014ned)i(o)o(v)o(er) e Fo(R)13 b Ft(S)s(tab)590 633 y Fq(x)629 626 y Fv(=)18 b Ft(S)s(tab)p 773 614 22 2 v 7 x Fq(x)813 626 y Fv(and)h(it)f(follo)o (ws)h(that)e Ft(g)j Fv(comm)o(utes)e(with)g Ft(S)s(tab)1663 633 y Fq(x)1685 626 y Fv(,)h(th)o(us)f(its)0 680 y(image)f(in)h(End)270 687 y Fg(C)296 680 y Fv(\()p Ft(V)341 687 y Fq(i)354 680 y Fv(\))f(b)q(elongs)h(to)e Ft(M)655 687 y Fq(x)677 680 y Fv(.)26 b(Then)17 b Ft(\036)863 664 y Fs(2)900 680 y Fv(is)h(the)f(iden)o(tit)o(y)h(since)g Ft(g)1335 664 y Fs(2)1354 680 y Ft(x)e Fv(=)g Ft(x)h Fv(and)g(so)g Ft(g)1662 664 y Fs(2)1697 680 y Fn(2)f Ft(S)s(tab)1834 687 y Fq(x)1855 680 y Fv(.)0 734 y(It)i(follo)o(ws)h(that)e Ft(M)350 741 y Fq(x)390 734 y Fv(can)i(b)q(e)f(split)i(in)o(to)e(t)o(w) o(o)f(eigenspaces)i(corresp)q(onding)g(to)f(eigen)o(v)m(alues)i(1)e (and)g Fn(\000)p Fv(1.)0 788 y(Supp)q(ose)e(there)g(is)f Ft(a)e Fn(2)g Ft(M)464 795 y Fq(x)501 788 y Fv(suc)o(h)j(that)e Ft(\036)p Fv(\()p Ft(a)p Fv(\))e(=)h Fn(\000)p Ft(a)p Fv(.)20 b(So)560 870 y Ft(\032)584 851 y Fm(0)595 870 y Fv(\()p Ft(a)p Fv(\))p Ft(\032)679 851 y Fm(0)690 870 y Fv(\()p Ft(a)p Fv(\))750 851 y Fm(\003)781 870 y Fv(=)13 b Ft(\032)853 851 y Fm(0)864 870 y Fv(\()p Ft(ag)r(a)954 851 y Fm(\003)973 870 y Ft(g)997 851 y Fm(\000)p Fs(1)1044 870 y Fv(\))f(=)h Fn(\000)p Ft(\032)1181 851 y Fm(0)1192 870 y Fv(\()p Ft(aa)1258 851 y Fm(\003)1278 870 y Fv(\))p Ft(;)0 951 y Fv(but)19 b Ft(aa)135 934 y Fm(\003)173 951 y Fv(has)f(sp)q(ectrum)h(con)o(tained)g(in)g Fn(f)p Ft(r)g Fn(2)f Fo(R)n Fn(j)p Ft(r)e Fn(\025)i Fv(0)p Fn(g)p Fv(,)g(so)g(the)h(op)q(erator)e(on)i(the)f(lefthand)i(side)f(has)0 1005 y(sp)q(ectrum)g(con)o(tained)g(in)h Fn(f)p Ft(r)e Fn(2)h Fo(R)n Fn(j)p Ft(r)d Fn(\024)i Fv(0)p Fn(g)p Fv(.)30 b(On)19 b(the)g(other)f(hand)h Ft(\032)1231 988 y Fm(0)1242 1005 y Fv(\()p Ft(a)p Fv(\))p Ft(\032)1326 988 y Fm(0)1337 1005 y Fv(\()p Ft(a)p Fv(\))1397 988 y Fm(\003)1434 1005 y Fn(\025)g Fv(0,)f(so)h Ft(\032)1626 988 y Fm(0)1637 1005 y Fv(\()p Ft(a)p Fv(\))e(=)i(0.)30 b(It)0 1059 y(follo)o(ws)18 b(that)g(the)g(eigenspace)i(with)e(eigen)o(v)m(alue)i Fn(\000)p Fv(1)e(b)q(elongs)i(to)d(the)h(k)o(ernel)h(of)f Ft(\032)1485 1042 y Fm(0)1515 1059 y Fv(whic)o(h)h(implies)h(the)0 1113 y(statemen)o(t.)1623 b Ff(\003)0 1147 y SDict begin H.S end 0 1147 a 0 1147 a SDict begin 13 H.A end 0 1147 a 0 1147 a SDict begin [ /View [/XYZ H.V] /Dest (definition.5.1) cvn H.B /DEST pdfmark end 0 1147 a 51 x FD(De\014nition)17 b(5.1.)i Fk(We)c(say)g(that)g Ft(V)621 1205 y Fq(i)650 1198 y Fk(is)f(exc)n(eptional)g(if)h(ther)n(e)f(is)h(an)f(element)g(of) h Fv(\000)g Fk(of)g(or)n(der)g Fv(4)g Fk(which)g(acts)0 1252 y(as)h(a)h(sc)n(alar)e(in)h Ft(V)311 1259 y Fq(i)325 1252 y Fk(.)0 1290 y SDict begin H.S end 0 1290 a 0 1290 a SDict begin 13 H.A end 0 1290 a 0 1290 a SDict begin [ /View [/XYZ H.V] /Dest (prop.5.4) cvn H.B /DEST pdfmark end 0 1290 a 47 x FD(Prop)q(osition)d(5.4.)j Fk(If)c Ft(V)446 1344 y Fq(i)471 1337 y Fk(is)g(not)f(exc)n(eptional)h(then)f(every)h (irr)n(e)n(ducible)g(*-r)n(epr)n(esentation)f(of)h Ft(A)g Fk(is)f(induc)n(e)n(d)0 1391 y(fr)n(om)16 b(a)h(*-r)n(epr)n(esentation) f(of)g Ft(M)580 1398 y Fq(x)618 1391 y Fk(for)h Ft(x)c Fn(2)g Ft(S)801 1398 y Fg(R)826 1391 y Fk(.)0 1476 y(Pr)n(o)n(of.)20 b Fv(Supp)q(ose)25 b(w)o(e)f(are)g(giv)o(en)g(an)g(irreducible)j (*-represen)o(tation)d Ft(\032)1289 1460 y Fm(0)1324 1476 y Fv(of)f Ft(M)1428 1483 y Fq(x)1474 1476 y Fv(whic)o(h)i(induces) g(a)f(*-)0 1530 y(represen)o(tation)19 b(of)g Ft(A)g Fv(and)g Ft(x)g Fn(6)p Fv(=)p 601 1505 26 2 v 19 w Ft(x)p Fv(.)32 b(By)19 b(the)g(prop)q(osition)1072 1530 y SDict begin H.S end 1072 1530 a Fv(5.1)1130 1501 y SDict begin H.R end 1130 1501 a 1130 1530 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.5.1) cvn H.B /ANN pdfmark end 1130 1530 a 1149 1505 26 2 v 19 w Ft(x)g Fv(=)g Ft(g)r(x)f Fv(for)h Ft(g)h Fv(of)f(order)f(4.)31 b(Then,)21 b(for)0 1584 y(all)16 b Ft(a)d Fn(2)g Ft(A)167 1665 y(\032)191 1646 y Fm(0)202 1665 y Fv(\()p Ft(a)p Fv(\()p Ft(x)p Fv(\)\))f(=)h Ft(\032)p Fv(\()p Ft(a)p Fv(\))e(=)i Ft(\032)p Fv(\()p Ft(a)593 1646 y Fm(\003)612 1665 y Fv(\))630 1646 y Fm(\003)662 1665 y Fv(=)g Ft(\032)734 1646 y Fm(0)745 1665 y Fv(\()p Ft(a)787 1646 y Fm(\003)807 1665 y Fv(\()p Ft(x)p Fv(\)\))887 1646 y Fm(\003)918 1665 y Fv(=)g Ft(\032)990 1646 y Fm(0)1001 1665 y Fv(\()p Ft(a)p Fv(\()p 1061 1640 V Ft(x)p Fv(\))1105 1646 y Fm(\003)1124 1665 y Fv(\))1142 1646 y Fm(\003)1174 1665 y Fv(=)g Ft(\032)1246 1646 y Fm(0)1257 1665 y Fv(\()p Ft(a)p Fv(\()p 1317 1640 V Ft(x)o Fv(\)\))f(=)h Ft(\032)1462 1646 y Fm(0)1474 1665 y Fv(\()p Ft(g)r(a)p Fv(\()p Ft(x)p Fv(\))p Ft(g)1626 1646 y Fm(\000)p Fs(1)1670 1665 y Fv(\))p Ft(:)0 1745 y Fv(Since)18 b Ft(S)s(tab)211 1752 y Fq(x)247 1745 y Fv(=)d Fn(f\000)p Fv(1)p Ft(;)8 b Fv(1)p Fn(g)p Fv(,)15 b Ft(M)517 1752 y Fq(x)553 1745 y Fv(=)g(End)685 1752 y Fg(C)711 1745 y Fv(\()p Ft(V)756 1752 y Fq(i)769 1745 y Fv(\))h(and)g(is)h(simple.)25 b(Th)o(us)17 b Ft(\032)1243 1729 y Fm(0)1270 1745 y Fv(has)f(zero)h(k)o (ernel,)g(whic)o(h)g(implies)0 1799 y(that)d Ft(g)j Fv(comm)o(utes)e (with)g(elemen)o(ts)h(of)f(End)773 1806 y Fg(C)799 1799 y Fv(\()p Ft(V)844 1806 y Fq(i)858 1799 y Fv(\).)k(Hence)d Ft(g)h Fv(is)e(scalar)g(in)h Ft(V)1336 1806 y Fq(i)1350 1799 y Fv(,)f(so)g Ft(V)1461 1806 y Fq(i)1489 1799 y Fv(is)h(exceptional.)63 b Ff(\003)50 1894 y Fv(Let)15 b(us)f(in)o(tro)q(duce)i(a)e Ft(C)463 1877 y Fm(\003)482 1894 y Fv(-algebra)667 1882 y Fl(e)656 1894 y Ft(A)g Fv(of)g(all)i(con)o(tin)o(uous)f(\000-equiv)m(arian)o(t)h(maps)e(from)g Ft(S)1577 1901 y Fg(R)1617 1894 y Fv(to)g(End)1753 1901 y Fg(C)1779 1894 y Fv(\()p Ft(V)1824 1901 y Fq(i)1838 1894 y Fv(\).)0 1954 y(W)l(e)20 b(consider)i(a)e(natural)g(map)g Ft( )i Fv(:)f Ft(A)g Fn(\000)-7 b(!)825 1942 y Fl(e)814 1954 y Ft(A)21 b Fv(giv)o(en)g(b)o(y)f(restriction)h(from)e Ft(S)1424 1961 y Fg(C)1470 1954 y Fv(to)h Ft(S)1559 1961 y Fg(R)1585 1954 y Fv(.)35 b(The)20 b(map)g(is)0 2008 y(an)f(inclusion)j(since)e Ft(S)404 2015 y Fg(R)449 2008 y Fv(is)g(algebraically)h(dense)f(in)g Ft(S)980 2015 y Fg(C)1006 2008 y Fv(.)31 b(Then,)21 b(an)o(y)d(irreducible)k (*-represen)o(tation)d Ft(\032)0 2062 y Fv(of)g Ft(A)h Fv(b)o(y)f(prop)q(osition)420 2062 y SDict begin H.S end 420 2062 a Fv(5.4)479 2032 y SDict begin H.R end 479 2032 a 479 2062 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.5.4) cvn H.B /ANN pdfmark end 479 2062 a 19 w Fv(is)h(induced)i(from)c (a)i(*-represen)o(tation)f Ft(\032)1239 2045 y Fm(0)1270 2062 y Fv(of)g Ft(M)1370 2069 y Fq(x)1392 2062 y Fv(,)h Ft(x)g Fn(2)g Fo(R)n Fv(,)d(whic)o(h)j(induces)0 2119 y(a)e(*-represen)o(tation)h Ft(\032)404 2126 y Fs(0)442 2119 y Fv(of)508 2107 y Fl(e)497 2119 y Ft(A)g Fv(suc)o(h)g(that)f Ft(\032)g Fv(=)g Ft(\032)877 2126 y Fs(0)909 2119 y Fn(\016)12 b Ft( )r Fv(.)30 b(Because)19 b(of)f(the)h(lemma)1483 2119 y SDict begin H.S end 1483 2119 a Fv(5.1)1541 2090 y SDict begin H.R end 1541 2090 a 1541 2119 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (lem.5.1) cvn H.B /ANN pdfmark end 1541 2119 a 19 w Fv(w)o(e)f(can)h(apply)g(a)0 2173 y(Stone-W)l(eierstrass)c(theorem)g(to)g(the)g(image)g(of)g Ft( )i Fv(to)d(sho)o(w)h(that)g(it)g(is)h(dense.)21 b(Giv)o(en)15 b(this)h(the)f(theorem)0 2227 y(2)g(follo)o(ws.)0 2286 y SDict begin H.S end 0 2286 a 0 2286 a SDict begin 13 H.A end 0 2286 a 0 2286 a SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn H.B /DEST pdfmark end 0 2286 a 567 2340 a Fv(6.)22 b Fu(Ex)o(ceptional)17 b(represent)m(a)m(tions)50 2421 y Fv(Here)e(w)o(e)f(are)g(going)h(to)f(list)i(all)f(exceptional)h (represen)o(tations)f(of)f(\000)h(according)g(to)f(the)h(de\014nition) 1797 2421 y SDict begin H.S end 1797 2421 a Fv(5.1)1855 2392 y SDict begin H.R end 1855 2392 a 1855 2421 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.5.1) cvn H.B /ANN pdfmark end 1855 2421 a Fv(.)0 2430 y SDict begin H.S end 0 2430 a 0 2430 a SDict begin 13 H.A end 0 2430 a 0 2430 a SDict begin [ /View [/XYZ H.V] /Dest (prop.6.1) cvn H.B /DEST pdfmark end 0 2430 a 76 x FD(Prop)q(osition)25 b(6.1.)f Fk(Supp)n(ose)d Fv(\000)h Fk(is)g(a)g(\014nite)e(sub)n(gr)n (oup)i(of)g Ft(S)s(U)5 b Fv(\(2)p Ft(;)j Fo(C)d Fv(\))p Fk(,)26 b Ft(g)d Fk(b)n(e)e(an)g(element)g(of)h(or)n(der)g Fv(4)0 2560 y Fk(and)g Ft(V)31 b Fk(|)21 b(an)h(irr)n(e)n(ducible)f(r)n (epr)n(esentation)f(of)i Fv(\000)g Fk(such)g(that)g Ft(g)h Fk(is)e(sc)n(alar)h(in)f Ft(V)9 b Fk(.)37 b(Then)21 b(either)h Ft(V)31 b Fk(is)0 2614 y(one-dimensional,)15 b(or)i(one)f(of)g(the)h (fol)r(lowing)f(holds:)0 2636 y SDict begin H.S end 0 2636 a 0 2636 a SDict begin 13 H.A end 0 2636 a 0 2636 a SDict begin [ /View [/XYZ H.V] /Dest (Item.23) cvn H.B /DEST pdfmark end 0 2636 a 73 2682 a Fv(\(1\))j Fk(The)d Fv(\000)h Fk(is)e(a)h(binary)g(dihe)n(dr)n(al)g(gr)n(oup)h(whose)f(image)g(in)g Ft(S)s(O)q Fv(\(3)p Ft(;)8 b Fo(R)l Fv(\))k Fk(is)k(a)g(gr)n(oup)h(of)f (symmetries)g(of)151 2736 y(\015at)i(p)n(olygon)g(of)g Ft(n)g Fk(numb)n(er)g(of)g(vertic)n(es,)g Ft(n)g Fk(even,)g Ft(n)e Fn(\025)g Fv(4)p Fk(.)26 b(The)17 b(image)i(of)f Ft(g)h Fk(is)f(the)g(symmetry)151 2790 y(with)f(r)n(esp)n(e)n(ct)f(to)h (a)f(line)g(p)n(assing)f(thr)n(ough)j(the)f(c)n(enter)e(of)i(the)g(p)n (olygon)f(ortho)n(gonal)h(to)g(the)f(plane)p eop %%Page: 13 13 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(13)151 133 y Fk(of)19 b(the)g(p)n(olygon.)27 b Ft(V)h Fk(is)19 b(any)f(r)n(epr)n(esentation)f(c)n(orr)n(esp)n(onding)h(to)h (one)f(of)h(the)f(black)h(vertic)n(es)e(on)151 187 y(the)g(fol)r (lowing)f(pictur)n(e)g(by)h(the)f(McKay)g(c)n(orr)n(esp)n(ondenc)n(e:) 310 262 y Fn(\016)1202 b(\016)459 409 y(\016)p 496 400 100 2 v 345 276 a Fh(@)360 290 y(@)374 304 y(@)389 318 y(@)403 333 y(@)418 347 y(@)432 361 y(@)346 522 y(~)360 508 y(~)374 494 y(~)389 479 y(~)403 465 y(~)418 451 y(~)432 437 y(~)609 409 y Fn(\017)p 645 400 V 126 w(\016)p 794 400 V 126 w(\017)p 944 400 V 1057 401 a Ft(:)8 b(:)g(:)p 1123 400 V 1236 409 a Fn(\017)p 1273 400 V 127 w(\016)1508 290 y Fh(~)1493 305 y(~)1479 319 y(~)1464 333 y(~)1450 347 y(~)1436 361 y(~)1421 375 y(~)1508 507 y(@)1493 493 y(@)1479 479 y(@)1464 465 y(@)1450 451 y(@)1436 437 y(@)1421 423 y(@)310 556 y Fn(\016)1202 b(\016)0 592 y SDict begin H.S end 0 592 a 0 592 a SDict begin 13 H.A end 0 592 a 0 592 a SDict begin [ /View [/XYZ H.V] /Dest (Item.24) cvn H.B /DEST pdfmark end 0 592 a 73 630 a Fv(\(2\))19 b Fk(The)e Fv(\000)g Fk(is)f(a)h(binary)g(o)n(ctahe)n(dr)n(al)f(gr)n (oup)i(whose)f(image)f(in)h Ft(S)s(O)q Fv(\(3)p Ft(;)8 b Fo(R)l Fv(\))13 b Fk(is)j(a)h(gr)n(oup)g(of)g(symmetries)151 684 y(of)e(the)f(r)n(e)n(gular)g(o)n(ctahe)n(dr)n(on.)20 b(The)14 b(image)h(of)f Ft(g)h Fk(is)f(a)h(symmetry)f(with)h(r)n(esp)n (e)n(ct)e(to)i(the)f(line)f(p)n(assing)151 738 y(thr)n(ough)20 b(opp)n(osite)e(vertic)n(es)g(of)g(the)h(o)n(ctahe)n(dr)n(on.)27 b Ft(V)h Fk(is)18 b(the)h(r)n(epr)n(esentation)e(which)i(c)n(orr)n(esp) n(ond)151 792 y(to)e(the)f(black)g(vertex)h(on)f(the)g(pictur)n(e:)840 738 y SDict begin H.S end 840 738 a 840 738 a SDict begin 13 H.A end 840 738 a 840 738 a SDict begin [ /View [/XYZ H.V] /Dest (Item.25) cvn H.B /DEST pdfmark end 840 738 a 0 869 a Fv(\(6.1\))923 868 y Fn(\017)p 933 981 2 100 v 475 1014 a(\016)p 511 1005 100 2 v 126 w(\016)p 660 1005 V 126 w(\016)p 810 1005 V 127 w(\016)p 959 1005 V 126 w(\016)p 1108 1005 V 126 w(\016)p 1258 1005 V 127 w(\016)0 1093 y Fk(Pr)n(o)n(of.)k 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SDict begin 13 H.A end 0 1320 a 0 1320 a SDict begin [ /View [/XYZ H.V] /Dest (Item.26) cvn H.B /DEST pdfmark end 0 1320 a 73 1374 a Fv(\(1\))j(On)d(the)f(iden)o(tit)o(y)h(represen)o(tation)g(the)f (trace)g(is)g(1.)0 1385 y SDict begin H.S end 0 1385 a 0 1385 a SDict begin 13 H.A end 0 1385 a 0 1385 a SDict begin [ /View [/XYZ H.V] /Dest (Item.27) cvn H.B /DEST pdfmark end 0 1385 a 73 1428 a Fv(\(2\))k(On)f(the)f(tautological)h(t)o(w) o(o-dimensional)g(represen)o(tation)f Ft(V)27 b Fv(the)17 b(trace)g(is)h(0,)f(so)g(for)f(an)o(y)h(other)151 1482 y(represen)o(tation)f Ft(V)476 1489 y Fq(i)506 1482 y Fv(the)g(trace)f(on)h Ft(V)k Fn(\012)11 b Ft(V)883 1489 y Fq(i)913 1482 y Fv(is)16 b(zero.)22 b(Th)o(us)16 b(the)g(sum)g(of)f (lab)q(els)j(of)d(neigh)o(b)q(ours)i(of)151 1536 y Ft(i)e Fv(is)h(zero.)0 1536 y SDict begin H.S end 0 1536 a 0 1536 a SDict begin 13 H.A end 0 1536 a 0 1536 a SDict begin [ /View [/XYZ H.V] /Dest (Item.28) cvn H.B /DEST pdfmark end 0 1536 a 73 1590 a Fv(\(3\))j(F)l(or)14 b(an)o(y)g(ev)o(en)h(v)o (ertex)f(w)o(e)g(ha)o(v)o(e)g Ft(g)751 1573 y Fs(2)785 1590 y Fv(acts)g(iden)o(tically)j(in)e(the)g(corresp)q(onding)g (represen)o(tation,)f(so)151 1644 y(its)i(sp)q(ectrum)f(con)o(tains)h (only)f(1)g(and)h Fn(\000)p Fv(1.)0 1653 y SDict begin H.S end 0 1653 a 0 1653 a SDict begin 13 H.A end 0 1653 a 0 1653 a SDict begin [ /View [/XYZ H.V] /Dest (Item.29) cvn H.B /DEST pdfmark end 0 1653 a 73 1698 a Fv(\(4\))j(F)l(or)14 b(an)o(y)h(o)q(dd)g(v)o(ertex)f(w)o(e)g(ha)o(v)o(e)g Ft(g)738 1681 y Fs(2)772 1698 y Fv(acts)g(as)g(negativ)o(e)h(iden)o (tit)o(y)g(in)h(the)f(corresp)q(onding)g(represen-)151 1752 y(tation,)g(so)g(its)g(sp)q(ectrum)h(con)o(tains)f(only)h FD(i)f Fv(and)h Fn(\000)p FD(i)p Fv(.)0 1817 y(W)l(e)f(obtain)h(only)f (t)o(w)o(o)f(p)q(ossible)j(lab)q(elings)h(whic)o(h)e(satisfy)f(the)g (conditions)h(ab)q(o)o(v)o(e:)923 1897 y(0)p 933 2012 2 102 v 439 2055 a(1)p 475 2041 100 2 v 127 w(0)p 625 2041 V 738 2053 a Fn(\000)p Fv(1)p 810 2041 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1021 y(A)1071 1241 y(C)934 1490 y(A)968 1473 y Fm(0)0 1544 y Fv(where)22 b Ft(A)f Fv(and)g Ft(A)321 1527 y Fm(0)354 1544 y Fv(are)g(cen)o(ters)h (of)e(faces,)j Ft(B)g Fv(is)f(the)f(cen)o(ter)g(of)g(an)g(edge)h(and)f Ft(C)j Fv(is)e(a)f(v)o(ertex)g(of)g(the)0 1598 y(corresp)q(o)q(ding)e (dihedra,)f(tetrahedra,)f(o)q(ctahedra)h(or)f(icosahedra.)27 b(Denote)18 b(this)g(fundamen)o(tal)g(domain)0 1651 y(b)o(y)f Fn(F)5 b Fv(.)24 b(The)17 b(group)f(is)i(generated)f(b)o(y)f(three)h (elemen)o(ts)h Ft(a)p Fv(,)f Ft(b)p Fv(,)f Ft(c)p Fv(,)h(where)g Ft(a)f Fv(is)i(a)e(rotation)g(around)h Ft(A)p Fv(,)g Ft(b)f Fv(is)0 1705 y(a)i(rotation)f(around)h Ft(B)i Fv(and)e Ft(c)f Fv(is)i(a)f(rotation)f(around)h Ft(C)s Fv(.)28 b(Elemen)o(ts)18 b Ft(a)p Fv(,)g Ft(b)p Fv(,)g Ft(c)g Fv(can)g(b)q(e)h(c)o(ho)q(osen)f(in)h(suc)o(h)0 1759 y(a)f(w)o(a)o(y)f(that)g Ft(cA)g Fv(=)h Ft(A)395 1743 y Fm(0)406 1759 y Fv(,)h Ft(bA)492 1743 y Fm(0)520 1759 y Fv(=)f Ft(A)g Fv(and)g Ft(abc)e Fv(=)i Ft(e)p Fv(,)h Ft(e)f Fv(denotes)g(the)g(iden)o(tit)o(y)l(.)29 b(Then)19 b(the)f(stabilizer)h(of)f Ft(A)0 1813 y Fv(is)i(generated)f (b)o(y)g Ft(a)p Fv(,)g(the)h(stabilizer)g(of)f Ft(B)i Fv(b)o(y)e Ft(b)p Fv(,)g(the)g(stabilizer)i(of)e Ft(C)j Fv(b)o(y)d Ft(c)f Fv(and)i(the)f(stabilizer)i(of)d Ft(A)1856 1797 y Fm(0)0 1867 y Fv(b)o(y)e Ft(cac)128 1851 y Fm(\000)p Fs(1)174 1867 y Fv(.)22 b(Other)16 b(p)q(oin)o(ts)g(of)g(the)g (fundamen)o(tal)g(domain)g(ha)o(v)o(e)g(trivial)h(stabilizers.)23 b(Ev)o(ery)16 b(orbit)f(of)h(the)0 1921 y(group)i(in)o(tersects)g(the)g (fundamen)o(tal)h(domain)g(in)g(exactly)f(one)g(p)q(oin)o(t)h(except)g (orbits)f(passing)g(through)0 1975 y(the)g(b)q(oundary)l(,)h(i.e.)28 b(the)19 b(elemen)o(t)f Ft(b)g Fv(maps)f(p)q(oin)o(ts)i(of)e(the)h (segmen)o(t)g Ft(B)r(A)1320 1959 y Fm(0)1349 1975 y Fv(to)f Ft(B)r(A)h Fv(and)g Ft(c)g Fv(maps)g(p)q(oin)o(ts)0 2029 y(of)e Ft(C)s(A)g Fv(to)f Ft(C)s(A)265 2013 y Fm(0)277 2029 y Fv(.)22 b(It)17 b(follo)o(ws)f(that)f(the)i(quotien)o(t)f Ft(S)900 2036 y Fg(R)926 2029 y Ft(=)p Fv(\000)977 2013 y Fm(0)1005 2029 y Fv(can)g(b)q(e)h(obtained)g(b)o(y)f(gluing)h(the)g (fundamen)o(tal)0 2083 y(domain)f(along)f(actions)g(of)g Ft(b)g Fv(and)g Ft(c)p Fv(.)k(W)l(e)d(are)f(going)g(to)f(consider)j (the)e(follo)o(wing)h(class)f(of)g Ft(C)1637 2067 y Fm(\003)1657 2083 y Fv(-algebras.)0 2092 y SDict begin H.S end 0 2092 a 0 2092 a SDict begin 13 H.A end 0 2092 a 0 2092 a SDict begin [ /View [/XYZ H.V] /Dest (definition.7.1) cvn H.B /DEST pdfmark end 0 2092 a 73 x FD(De\014nition)k(7.1.)h Fk(Supp)n(ose)c(we)h (ar)n(e)f(given)f(the)i(fol)r(lowing)f(data:)0 2184 y SDict begin H.S end 0 2184 a 0 2184 a SDict begin 13 H.A end 0 2184 a 0 2184 a SDict begin [ /View [/XYZ H.V] /Dest (Item.30) cvn H.B /DEST pdfmark end 0 2184 a 73 2229 a Fv(\(1\))j Fk(a)e(\014nite)e(dimensional)g(hermitian)i(ve)n(ctor)f (sp)n(ac)n(e)g Ft(H)t Fk(,)0 2241 y SDict begin H.S end 0 2241 a 0 2241 a SDict begin 13 H.A end 0 2241 a 0 2241 a SDict begin [ /View [/XYZ H.V] /Dest (Item.31) cvn H.B /DEST pdfmark end 0 2241 a 73 2283 a Fv(\(2\))j Fk(a)e(c)n(ontinuous)e (map)i Ft(m)559 2290 y Fq(b)589 2283 y Fv(:)12 b Ft(B)r(A)685 2267 y Fm(0)709 2283 y Fn(\000)-7 b(!)12 b Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))14 b Fk(which)j(is)f(c)n(onstant)f(in)g(the)i (neighb)n(ourho)n(o)n(ds)e(of)i(the)151 2337 y(endp)n(oints)f(of)g Ft(B)r(A)479 2321 y Fm(0)491 2337 y Fk(,)0 2346 y SDict begin H.S end 0 2346 a 0 2346 a SDict begin 13 H.A end 0 2346 a 0 2346 a SDict begin [ /View [/XYZ H.V] /Dest (Item.32) cvn H.B /DEST pdfmark end 0 2346 a 73 2391 a Fv(\(3\))j Fk(a)e(c)n(ontinuous)g(map)g Ft(m)561 2398 y Fq(c)592 2391 y Fv(:)d Ft(C)s(A)f Fn(\000)-7 b(!)13 b Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))15 b Fk(which)j(is)e(c)n(onstant)g(in)g(the)h (neighb)n(ourho)n(o)n(ds)g(of)g(the)151 2445 y(endp)n(oints)f(of)g Ft(C)s(A)p Fk(,)0 2454 y SDict begin H.S end 0 2454 a 0 2454 a SDict begin 13 H.A end 0 2454 a 0 2454 a SDict begin [ /View [/XYZ H.V] /Dest (Item.33) cvn H.B /DEST pdfmark end 0 2454 a 73 2499 a Fv(\(4\))j Fk(a)e(*-sub)n(algebr)n(a)f Ft(M)490 2506 y Fq(P)532 2499 y Fn(\032)d Fv(End\()p Ft(H)t Fv(\))i Fk(for)i(every)f(p)n(oint)g Ft(P)23 b Fk(in)16 b(the)g(set)g Fn(f)p Ft(A;)8 b(B)r(;)g(C)q(;)g(A)1541 2483 y Fm(0)1549 2499 y Fn(g)p Fk(,)0 2563 y(which)17 b(satisfy)e(the)i(fol)r(lowing)f(c)n(onditions:)0 2583 y SDict begin H.S end 0 2583 a 0 2583 a SDict begin 13 H.A end 0 2583 a 0 2583 a SDict begin [ /View [/XYZ H.V] /Dest (Item.34) cvn H.B /DEST pdfmark end 0 2583 a 73 2628 a Fv(\(1\))j Ft(m)191 2635 y Fq(b)208 2628 y Fv(\()p Ft(B)r Fv(\))d Fk(c)n(ommutes)g(with)h Ft(M)651 2635 y Fq(B)681 2628 y Fk(,)0 2639 y SDict begin H.S end 0 2639 a 0 2639 a SDict begin 13 H.A end 0 2639 a 0 2639 a SDict begin [ /View [/XYZ H.V] /Dest (Item.35) cvn H.B /DEST pdfmark end 0 2639 a 73 2682 a Fv(\(2\))i Ft(m)191 2689 y Fq(c)209 2682 y Fv(\()p Ft(C)s Fv(\))c Fk(c)n(ommutes)h(with)h Ft(M)650 2689 y Fq(C)680 2682 y Fk(,)0 2693 y SDict begin H.S end 0 2693 a 0 2693 a SDict begin 13 H.A end 0 2693 a 0 2693 a SDict begin [ /View [/XYZ H.V] /Dest (Item.36) cvn H.B /DEST pdfmark end 0 2693 a 73 2736 a Fv(\(3\))i Ft(m)191 2743 y Fq(b)208 2736 y Fv(\()p Ft(A)260 2719 y Fm(0)272 2736 y Fv(\))p Ft(m)330 2743 y Fq(c)347 2736 y Fv(\()p Ft(A)p Fv(\))d Fk(c)n(ommutes)g(with)h Ft(M)787 2743 y Fq(A)815 2736 y Fk(,)0 2747 y SDict begin H.S end 0 2747 a 0 2747 a SDict begin 13 H.A end 0 2747 a 0 2747 a SDict begin [ /View [/XYZ H.V] /Dest (Item.37) cvn H.B /DEST pdfmark end 0 2747 a 73 2790 a Fv(\(4\))i Ft(m)191 2797 y Fq(c)209 2790 y Fv(\()p Ft(A)p Fv(\))p Ft(M)323 2797 y Fq(A)351 2790 y Ft(m)391 2797 y Fq(c)408 2790 y Fv(\()p Ft(A)p Fv(\))478 2773 y Fm(\000)p Fs(1)537 2790 y Fv(=)13 b Ft(M)629 2797 y Fq(A)655 2788 y Fw(0)669 2790 y Fk(.)p eop %%Page: 15 15 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(15)0 133 y Fk(Then)16 b(we)i(denote)f(by)g Fn(C)s Fv(\()p Ft(H)q(;)8 b(m)539 140 y Fq(b)554 133 y Ft(;)g(m)615 140 y Fq(c)632 133 y Ft(;)g Fv(\()p Ft(M)715 140 y Fq(P)743 133 y Fv(\)\))17 b Fk(the)g Ft(C)908 116 y Fm(\003)928 133 y Fk(-algebr)n(a)g(of)g(c)n(ontinuous)g(maps)g Ft(f)j Fv(:)14 b Fn(F)k(\000)-7 b(!)14 b Fv(End\()p Ft(H)t Fv(\))0 187 y Fk(such)i(that:)0 199 y SDict begin H.S end 0 199 a 0 199 a SDict begin 13 H.A end 0 199 a 0 199 a SDict begin [ /View [/XYZ H.V] /Dest (Item.38) cvn H.B /DEST pdfmark end 0 199 a 73 253 a Fv(\(1\))j Ft(f)5 b Fv(\()p Ft(P)h Fv(\))13 b Fn(2)g Ft(M)349 260 y Fq(P)395 253 y Fk(for)j Ft(P)j Fn(2)13 b(f)p Ft(A;)8 b(B)r(;)g(C)q(;)g(A)783 237 y Fm(0)791 253 y Fn(g)p Fk(,)0 265 y SDict begin H.S end 0 265 a 0 265 a SDict begin 13 H.A end 0 265 a 0 265 a SDict begin [ /View [/XYZ H.V] /Dest (Item.39) cvn H.B /DEST pdfmark end 0 265 a 73 307 a Fv(\(2\))19 b Ft(f)5 b Fv(\()p Ft(bx)p Fv(\))12 b(=)h Ft(m)360 314 y Fq(b)377 307 y Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(m)568 314 y Fq(b)584 307 y Fv(\()p Ft(x)p Fv(\))646 291 y Fm(\000)p Fs(1)709 307 y Fk(for)16 b Ft(x)d Fn(2)g Ft(B)r(A)934 291 y Fm(0)945 307 y Fk(,)0 319 y SDict begin H.S end 0 319 a 0 319 a SDict begin 13 H.A end 0 319 a 0 319 a SDict begin [ /View [/XYZ H.V] /Dest (Item.40) cvn H.B /DEST pdfmark end 0 319 a 73 361 a Fv(\(3\))19 b Ft(f)5 b Fv(\()p Ft(cx)p Fv(\))12 b(=)h Ft(m)360 368 y Fq(c)377 361 y Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(m)568 368 y Fq(c)584 361 y Fv(\()p Ft(x)p Fv(\))646 345 y Fm(\000)p Fs(1)709 361 y Fk(for)17 b Ft(x)12 b Fn(2)h Ft(C)s(A)p Fk(.)50 452 y Fv(It)g(is)h(clear)g(that)f(the)g(algebra)g Fn(A)h Fv(is)g(isomorphic)g(to)f(the)g(algebra)h Fn(C)s Fv(\()p Ft(V)1260 459 y Fq(i)1273 452 y Ft(;)1291 440 y Fl(e)1294 452 y Ft(b)n(;)7 b Fl(e)-24 b Ft(c)o(;)8 b Fv(\()p Ft(M)1435 459 y Fq(P)1464 452 y Fv(\)\),)k(where)1652 440 y Fl(e)1655 452 y Ft(b)h Fv(and)g Fl(e)-25 b Ft(c)13 b Fv(are)0 506 y(constan)o(t)h(maps)h(with)575 578 y Fl(e)578 590 y Ft(b)d Fv(=)737 559 y Ft(V)764 566 y Fq(i)778 559 y Fv(\()p Ft(b)p Fv(\))p 663 579 243 2 v 676 610 a Fz(d)666 589 y Fl(p)p 712 589 194 2 v 38 x Fv(det\()p Ft(V)820 634 y Fq(i)834 627 y Fv(\()p Ft(b)p Fv(\)\))911 590 y Ft(;)20 b Fl(e)-25 b Ft(c)12 b Fv(=)1103 559 y Ft(V)1130 566 y Fq(i)1143 559 y Fv(\()p Ft(c)p Fv(\))p 1029 579 243 2 v 1042 610 a Fz(d)1032 589 y Fl(p)p 1078 589 195 2 v 38 x Fv(det\()p Ft(V)1186 634 y Fq(i)1199 627 y Fv(\()p Ft(c)p Fv(\)\))1277 590 y Ft(;)0 695 y Fv(here)k Ft(V)126 702 y Fq(i)139 695 y Fv(\()p Ft(b)p Fv(\))e(denotes)i(the)f(action)g (of)g Ft(b)g Fv(on)g Ft(V)765 702 y Fq(i)794 695 y Fv(and)g(the)g(same) g(for)g Ft(V)1171 702 y Fq(i)1184 695 y Fv(\()p Ft(c)p Fv(\).)50 749 y(Supp)q(ose)i(w)o(e)g(are)f(giv)o(en)h(t)o(w)o(o)e (algebras)h Fn(C)787 733 y Fs(1)821 749 y Fv(=)g Fn(C)s Fv(\()p Ft(H)q(;)8 b(m)1017 733 y Fs(1)1017 763 y Fq(b)1035 749 y Ft(;)g(m)1096 733 y Fs(1)1096 760 y Fq(c)1114 749 y Ft(;)g Fv(\()p Ft(M)1202 733 y Fs(1)1197 762 y Fq(P)1226 749 y Fv(\)\))16 b(and)g Fn(C)1394 733 y Fs(2)1428 749 y Fv(=)f Fn(C)s Fv(\()p Ft(H)q(;)8 b(m)1623 733 y Fs(2)1623 763 y Fq(b)1641 749 y Ft(;)g(m)1702 733 y Fs(2)1702 760 y Fq(c)1721 749 y Ft(;)g Fv(\()p Ft(M)1809 733 y Fs(2)1804 762 y Fq(P)1833 749 y Fv(\)\))0 803 y(with)16 b(the)f(same)g(space)g Ft(H)t Fv(.)0 812 y SDict begin H.S end 0 812 a 0 812 a SDict begin 13 H.A end 0 812 a 0 812 a SDict begin [ /View [/XYZ H.V] /Dest (definition.7.2) cvn H.B /DEST pdfmark end 0 812 a 75 x FD(De\014nition)i(7.2.)h Fk(A)d(map)g Ft(t)e Fv(:)f Fn(F)17 b(\000)-7 b(!)12 b Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))13 b Fk(is)h(c)n(al)r(le)n(d)f(a)i(morphism)g(b)n(etwe)n (en)f Fn(C)1447 871 y Fs(1)1480 887 y Fk(and)h Fn(C)1594 871 y Fs(2)1628 887 y Fk(if)f(it)h(satis\014es)0 941 y(the)i(fol)r(lowing)e(pr)n(op)n(erties:)0 963 y SDict begin H.S end 0 963 a 0 963 a SDict begin 13 H.A end 0 963 a 0 963 a SDict begin [ /View [/XYZ H.V] /Dest (Item.41) cvn H.B /DEST pdfmark end 0 963 a 73 1008 a Fv(\(1\))k Ft(t)e Fk(is)f(c)n(ontinuous)g(in)f(al)r(l)h(p)n(oints)g(exc)n(ept,)g (p)n(ossibly,)f Ft(A)p Fk(,)h Ft(B)r Fk(,)g Ft(C)j Fk(and)e Ft(A)1352 991 y Fm(0)1363 1008 y Fk(.)0 1019 y SDict begin H.S end 0 1019 a 0 1019 a SDict begin 13 H.A end 0 1019 a 0 1019 a SDict begin [ /View [/XYZ H.V] /Dest (Item.42) cvn H.B /DEST pdfmark end 0 1019 a 73 1062 a Fv(\(2\))i Fk(If)d Ft(P)23 b Fk(is)16 b(any)h(p)n(oint)f(among)g Ft(A)p Fk(,)h Ft(B)r Fk(,)f Ft(C)j Fk(and)e Ft(A)956 1045 y Fm(0)984 1062 y Fk(then)f(ther)n(e)h(is)f(a)g(neighb)n(ourho)n (o)n(d)h(of)g Ft(P)23 b Fk(such)16 b(that)151 1116 y(for)h(any)f(p)n (oint)g Ft(x)g Fk(fr)n(om)h(the)f(neighb)n(ourho)n(o)n(d)h(and)f Ft(u)d Fn(2)g Ft(M)1165 1099 y Fs(1)1160 1129 y Fq(P)1205 1062 y SDict begin H.S end 1205 1062 a 1205 1062 a SDict begin 13 H.A end 1205 1062 a 1205 1062 a SDict begin [ /View [/XYZ H.V] /Dest (Item.43) cvn H.B /DEST pdfmark end 1205 1062 a 0 1199 a Fv(\(7.1\))500 b Ft(t)p Fv(\()p Ft(x)p Fv(\))p Ft(ut)p Fv(\()p Ft(x)p Fv(\))777 1180 y Fm(\000)p Fs(1)836 1199 y Fv(=)13 b Ft(t)p Fv(\()p Ft(P)6 b Fv(\))p Ft(ut)p Fv(\()p Ft(P)g Fv(\))1084 1180 y Fm(\000)p Fs(1)1144 1199 y Fn(2)13 b Ft(M)1236 1180 y Fs(2)1231 1210 y Fq(P)1261 1199 y Ft(:)0 1235 y SDict begin H.S end 0 1235 a 0 1235 a SDict begin 13 H.A end 0 1235 a 0 1235 a SDict begin [ /View [/XYZ H.V] /Dest (Item.44) cvn H.B /DEST pdfmark end 0 1235 a 73 1277 a Fv(\(3\))19 b Ft(m)191 1261 y Fs(2)191 1291 y Fq(b)211 1277 y Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))12 b(=)h Ft(t)p Fv(\()p Ft(bx)p Fv(\))p Ft(m)549 1261 y Fs(1)549 1291 y Fq(b)568 1277 y Fv(\()p Ft(x)p Fv(\))i Fk(for)i(al)r(l)f Ft(x)c Fn(2)h Ft(B)r(A)935 1261 y Fm(0)947 1277 y Fk(,)j(exc)n(ept,)g (p)n(ossibly,)f Ft(B)j Fk(and)e Ft(A)1482 1261 y Fm(0)1494 1277 y Fk(.)0 1291 y SDict begin H.S end 0 1291 a 0 1291 a SDict begin 13 H.A end 0 1291 a 0 1291 a SDict begin [ /View [/XYZ H.V] /Dest (Item.45) cvn H.B /DEST pdfmark end 0 1291 a 73 1333 a Fv(\(4\))j Ft(m)191 1317 y Fs(2)191 1345 y Fq(c)211 1333 y Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))12 b(=)h Ft(t)p Fv(\()p Ft(cx)p Fv(\))p Ft(m)549 1317 y Fs(1)549 1345 y Fq(c)568 1333 y Fv(\()p Ft(x)p Fv(\))i Fk(for)i(al)r(l)f Ft(x)d Fn(2)g Ft(C)s(A)p Fk(,)j(exc)n(ept,)g(p)n(ossibly,)f Ft(C)k Fk(and)d Ft(A)p Fk(.)0 1375 y SDict begin H.S end 0 1375 a 0 1375 a SDict begin 13 H.A end 0 1375 a 0 1375 a SDict begin [ /View [/XYZ H.V] /Dest (prop.7.2) cvn H.B /DEST pdfmark end 0 1375 a 46 x FD(Prop)q(osition)d(7.2.)j Fk(If)c Ft(t)g Fk(is)f(a)h(morphism)h(then)f(ther)n(e)g(is)f(a)h(homomorphism)i (of)e Ft(C)1419 1404 y Fm(\003)1438 1421 y Fk(-algebr)n(as)1622 1412 y Fl(e)1623 1421 y Ft(t)h Fv(:)f Fn(C)1704 1404 y Fs(1)1736 1421 y Fn(\000)-7 b(!)12 b(C)1849 1404 y Fs(2)0 1475 y Fk(de\014ne)n(d)j(by)457 1535 y Fv(\()474 1526 y Fl(e)475 1535 y Ft(tf)5 b Fv(\)\()p Ft(x)p Fv(\))12 b(=)h Ft(t)p Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))903 1516 y Fm(\000)p Fs(1)949 1535 y Ft(;)20 b Fk(for)c(al)r(l)g Ft(x)d Fn(2)g(F)5 b Fk(,)15 b Ft(f)j Fn(2)13 b(C)1378 1516 y Fs(1)1397 1535 y Fk(.)0 1619 y(Pr)n(o)n(of.)20 b Fv(First)15 b(w)o(e)f(pro)o(v)o(e)g(that)g(con)o(tin)o(uous)h(maps)g (are)f(mapp)q(ed)h(to)f(con)o(tin)o(uous)h(maps.)20 b(This)15 b(prop)q(ert)o(y)f(is)0 1673 y(ob)o(vious)j(in)h(all)f(p)q(oin)o(ts)h (except)f Ft(A;)8 b(B)r(;)g(C)q(;)g(A)770 1657 y Fm(0)778 1673 y Fv(.)25 b(If)17 b Ft(P)24 b Fv(is)17 b(one)g(of)f Ft(A)p Fv(,)h Ft(B)r Fv(,)g Ft(C)s Fv(,)f Ft(A)1331 1657 y Fm(0)1360 1673 y Fv(and)h Ft(f)22 b Fv(is)17 b(an)g(elemen)o(t)g(of)g Fn(C)s Fv(,)0 1727 y(then)191 1806 y(lim)171 1836 y Fq(x)p Fm(\000)-6 b(!)p Fq(P)283 1806 y Ft(t)p Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))528 1787 y Fm(\000)p Fs(1)584 1806 y Fn(\000)10 b Ft(t)p Fv(\()p Ft(P)c Fv(\))p Ft(f)f Fv(\()p Ft(P)h Fv(\))p Ft(t)p Fv(\()p Ft(P)g Fv(\))901 1787 y Fm(\000)p Fs(1)962 1806 y Fv(=)33 b(lim)1009 1836 y Fq(x)p Fm(\000)-5 b(!)p Fq(P)1121 1806 y Ft(t)p Fv(\()p Ft(x)p Fv(\)\()p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))k Fn(\000)i Ft(f)5 b Fv(\()p Ft(P)h Fv(\)\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))1555 1787 y Fm(\000)p Fs(1)1614 1806 y Fv(=)13 b(0)p Ft(;)0 1904 y Fv(the)19 b(\014rst)f(equalit)o(y)h(follo)o(ws)g(from)f(the)g (condition)i(\(2\))e(of)g(the)g(de\014nition)1322 1904 y SDict begin H.S end 1322 1904 a Fv(7.2)1380 1875 y SDict begin H.R end 1380 1875 a 1380 1904 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.2) cvn H.B /ANN pdfmark end 1380 1904 a 18 w Fv(and)h(the)g(last)f(equalit)o(y)h(is)0 1958 y(true)14 b(b)q(ecause)h Ft(t)p Fv(\()p Ft(x)p Fv(\))f(and)h Ft(t)p Fv(\()p Ft(x)p Fv(\))520 1942 y Fm(\000)p Fs(1)581 1958 y Fv(are)f(b)q(ounded.)20 b(T)l(o)14 b(pro)o(v)o(e)g(the)g (statemen)o(t)f(w)o(e)h(m)o(ust)g(c)o(ho)q(ose)g(an)o(y)g Ft(f)k Fn(2)13 b(C)1849 1942 y Fs(1)0 2015 y Fv(and)i(sho)o(w)g(that)f Ft(g)g Fv(=)383 2007 y Fl(e)383 2015 y Ft(t)q(f)k Fn(2)13 b(C)510 1998 y Fs(2)529 2015 y Fv(.)20 b(W)l(e)15 b(c)o(hec)o(k)g (conditions)i(of)e(the)g(de\014nition)1312 2015 y SDict begin H.S end 1312 2015 a Fv(7.1)1370 1986 y SDict begin H.R end 1370 1986 a 1370 2015 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.1) cvn H.B /ANN pdfmark end 1370 2015 a 15 w Fv(one)g(b)o(y)h(one.)0 2036 y SDict begin H.S end 0 2036 a 0 2036 a SDict begin 13 H.A end 0 2036 a 0 2036 a SDict begin [ /View [/XYZ H.V] /Dest (Item.46) cvn H.B /DEST pdfmark end 0 2036 a 73 2081 a Fv(\(1\))j(F)l(or)11 b Ft(P)20 b Fn(2)12 b(f)p Ft(A;)c(B)r(;)g(C)q(;)g(A)545 2065 y Fm(0)554 2081 y Fn(g)j Ft(g)r Fv(\()p Ft(P)6 b Fv(\))12 b(=)h Ft(t)p Fv(\()p Ft(P)6 b Fv(\))p Ft(f)f Fv(\()p Ft(P)h Fv(\))p Ft(t)p Fv(\()p Ft(P)g Fv(\))1015 2065 y Fm(\000)p Fs(1)1075 2081 y Fn(2)13 b Ft(M)1167 2065 y Fs(2)1162 2095 y Fq(P)1203 2081 y Fv(since)g Ft(f)5 b Fv(\()p Ft(P)h Fv(\))12 b Fn(2)h Ft(M)1513 2065 y Fs(1)1508 2095 y Fq(P)1549 2081 y Fv(b)o(y)f(the)g(condition)151 2135 y(\(2\))j(of)f(the)i(de\014nition)557 2135 y SDict begin H.S end 557 2135 a Fv(7.2)615 2106 y SDict begin H.R end 615 2106 a 615 2135 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.2) cvn H.B /ANN pdfmark end 615 2135 a Fv(.)0 2147 y SDict begin H.S end 0 2147 a 0 2147 a SDict begin 13 H.A end 0 2147 a 0 2147 a SDict begin [ /View [/XYZ H.V] /Dest (Item.47) cvn H.B /DEST pdfmark end 0 2147 a 73 2189 a Fv(\(2\))j(F)l(or)c Ft(x)e Fn(2)f Ft(B)r(A)385 2173 y Fm(0)397 2189 y Fv(,)j Ft(x)d Fn(6)p Fv(=)h Ft(B)r Fv(,)i Ft(x)d Fn(6)p Fv(=)h Ft(A)696 2173 y Fm(0)300 2270 y Ft(g)r Fv(\()p Ft(bx)p Fv(\))p Ft(m)446 2251 y Fs(2)446 2282 y Fq(b)464 2270 y Fv(\()p Ft(x)p Fv(\))f(=)h Ft(t)p Fv(\()p Ft(bx)p Fv(\))p Ft(f)5 b Fv(\()p Ft(bx)p Fv(\))p Ft(t)p Fv(\()p Ft(bx)p Fv(\))891 2251 y Fm(\000)p Fs(1)935 2270 y Ft(m)975 2251 y Fs(2)975 2282 y Fq(b)995 2270 y Fv(\()p Ft(x)p Fv(\))12 b(=)h Ft(t)p Fv(\()p Ft(bx)p Fv(\))p Ft(f)5 b Fv(\()p Ft(bx)p Fv(\))p Ft(m)1364 2251 y Fs(1)1364 2282 y Fq(b)1382 2270 y Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))1522 2251 y Fm(\000)p Fs(1)337 2346 y Fv(=)13 b Ft(t)p Fv(\()p Ft(bx)p Fv(\))p Ft(m)523 2327 y Fs(1)523 2358 y Fq(b)542 2346 y Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))771 2327 y Fm(\000)p Fs(1)829 2346 y Fv(=)13 b Ft(m)917 2327 y Fs(2)917 2358 y Fq(b)937 2346 y Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))p Ft(f)5 b Fv(\()p Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(x)p Fv(\))1244 2327 y Fm(\000)p Fs(1)1302 2346 y Fv(=)13 b Ft(m)1390 2327 y Fs(2)1390 2358 y Fq(b)1409 2346 y Fv(\()p Ft(x)p Fv(\))p Ft(g)r Fv(\()p Ft(x)p Fv(\))p Ft(:)151 2428 y Fv(Since)k Ft(g)g Fv(is)e(already)h(kno)o(wn)f(to)f(b)q(e)i(con)o(tin)o (uous)g(the)f(statemen)o(t)f(follo)o(ws)h(for)g Ft(B)i Fv(and)e Ft(A)1686 2412 y Fm(0)1698 2428 y Fv(.)0 2437 y SDict begin H.S end 0 2437 a 0 2437 a SDict begin 13 H.A end 0 2437 a 0 2437 a SDict begin [ /View [/XYZ H.V] /Dest (Item.48) cvn H.B /DEST pdfmark end 0 2437 a 73 2482 a Fv(\(3\))k(Can)c(b)q(e)h(pro)o(v)o(ed)f(analogously)h(to)e(\(2\).) 1833 2549 y Ff(\003)50 2636 y Fv(Using)i(morphisms)f(w)o(e)g(can)g(sho) o(w)g(that)0 2645 y SDict begin H.S end 0 2645 a 0 2645 a SDict begin 13 H.A end 0 2645 a 0 2645 a SDict begin [ /View [/XYZ H.V] /Dest (thm.4) cvn H.B /DEST pdfmark end 0 2645 a 75 x FD(Theorem)24 b(4.)g Fk(If)e Fn(C)375 2704 y Fs(1)417 2720 y Fv(=)i Fn(C)s Fv(\()p Ft(H)q(;)8 b(m)621 2704 y Fs(1)621 2734 y Fq(b)639 2720 y Ft(;)g(m)700 2704 y Fs(1)700 2731 y Fq(c)718 2720 y Ft(;)g Fv(\()p Ft(M)806 2704 y Fs(1)801 2734 y Fq(P)830 2720 y Fv(\)\))21 b Fk(and)h Fn(C)1008 2704 y Fs(2)1051 2720 y Fv(=)h Fn(C)s Fv(\()p Ft(H)q(;)8 b(m)1254 2704 y Fs(2)1254 2734 y Fq(b)1272 2720 y Ft(;)g(m)1333 2704 y Fs(2)1333 2731 y Fq(c)1352 2720 y Ft(;)g Fv(\()p Ft(M)1440 2704 y Fs(2)1435 2734 y Fq(P)1463 2720 y Fv(\)\))22 b Fk(ar)n(e)g(such)g(that)h(for)0 2776 y Ft(P)c Fv(=)13 b Ft(A;)8 b(B)r(;)g(C)14 b Fk(the)g(*-algebr)n(a) f Ft(M)568 2760 y Fs(1)563 2790 y Fq(P)606 2776 y Fk(is)g(isomorphic)g (to)h Ft(M)982 2760 y Fs(2)977 2790 y Fq(P)1020 2776 y Fk(then)f Fn(C)1145 2760 y Fs(1)1177 2776 y Fk(and)g Fn(C)1289 2760 y Fs(2)1322 2776 y Fk(ar)n(e)g(isomorphic)h Ft(C)1662 2760 y Fm(\003)1682 2776 y Fk(-algebr)n(as.)p eop %%Page: 16 16 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 33 a SDict begin H.S end 0 33 a 0 33 a SDict begin H.R end 0 33 a 0 33 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end 0 33 a 723 w Fs(CER)m(T)m(AIN)17 b(EXAMPLES.)7 b(.)e(.)694 b(16)0 133 y Fk(Pr)n(o)n(of.)20 b Fv(Cho)q(ose)f Ft(u)328 140 y Fq(P)376 133 y Fn(2)g Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))17 b(suc)o(h)i(that)f Ft(M)844 116 y Fs(2)839 146 y Fq(P)887 133 y Fv(=)g Ft(u)966 140 y Fq(P)996 133 y Ft(M)1045 116 y Fs(1)1040 146 y Fq(P)1069 133 y Ft(u)1095 114 y Fm(\000)p Fs(1)1095 147 y Fq(P)1161 133 y Fv(for)g Ft(P)25 b Fn(2)19 b(f)p Ft(A;)8 b(B)r(;)g(C)s Fn(g)p Fv(.)28 b(W)l(e)19 b(\014rst)f(de\014ne)0 189 y Ft(t)p Fv(\()p Ft(P)6 b Fv(\))13 b(=)g Ft(u)174 196 y Fq(P)204 189 y Fv(,)i Ft(t)p Fv(\()p Ft(A)300 173 y Fm(0)312 189 y Fv(\))d(=)i Ft(m)431 173 y Fs(2)431 200 y Fq(c)450 189 y Fv(\()p Ft(A)p Fv(\))p Ft(u)546 196 y Fq(A)574 189 y Ft(m)614 173 y Fs(1)614 200 y Fq(c)634 189 y Fv(\()p Ft(A)p Fv(\))704 173 y Fm(\000)p Fs(1)750 189 y Fv(.)21 b(Then)16 b(w)o(e)f(c)o(ho)q(ose)g Ft(t)h Fv(on)f Ft(AB)i Fv(suc)o(h)f(that)f Ft(t)g Fv(equals)h(to)f Ft(u)1748 196 y Fq(A)1792 189 y Fv(in)h(a)0 243 y(neigh)o(b)q(ourho)q(o)q(d)e(of) e Ft(A)p Fv(,)h(to)f Ft(u)496 250 y Fq(B)539 243 y Fv(in)i(a)e(neigh)o (b)q(ourho)q(o)q(d)i(of)e Ft(B)j Fv(and)d(is)i(con)o(tin)o(uous.)19 b(Since)14 b Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))11 b(is)i(linearly)0 297 y(connected)k(it)g(is)g(p)q(ossible)i(to)d(do)g(so.)24 b(Then)17 b(for)f Ft(AC)j Fv(w)o(e)d(do)h(analogously)l(.)24 b(F)l(or)16 b Ft(B)r(A)1507 281 y Fm(0)1535 297 y Fv(and)h Ft(C)s(A)1695 281 y Fm(0)1723 297 y Fv(\(except)0 351 y Ft(B)r Fv(,)e Ft(C)s Fv(,)f Ft(A)162 335 y Fm(0)174 351 y Fv(\))h(w)o(e)g(put)102 426 y Ft(t)p Fv(\()p Ft(x)p Fv(\))d(=)h Ft(m)280 407 y Fs(2)280 437 y Fq(b)300 426 y Fv(\()p Ft(x)p Fv(\))362 407 y Fm(\000)p Fs(1)408 426 y Ft(t)p Fv(\()p Ft(bx)p Fv(\))p Ft(m)546 407 y Fs(1)546 437 y Fq(b)565 426 y Fv(\()p Ft(x)p Fv(\))f(and)h Ft(t)p Fv(\()p Ft(x)p Fv(\))g(=)g Ft(m)904 407 y Fs(2)904 437 y Fq(c)923 426 y Fv(\()p Ft(c)961 407 y Fm(\000)p Fs(1)1008 426 y Ft(x)p Fv(\))p Ft(t)p Fv(\()p Ft(c)1106 407 y Fm(\000)p Fs(1)1152 426 y Ft(x)p Fv(\))p Ft(m)1236 407 y Fs(1)1236 437 y Fq(c)1256 426 y Fv(\()p Ft(c)1294 407 y Fm(\000)p Fs(1)1340 426 y Ft(x)p Fv(\))1384 407 y Fm(\000)p Fs(1)1443 426 y Fv(corresp)q(ondingly)l(.)0 499 y(In)j(suc)o(h)f(a)g(w)o(a)o(y)f (w)o(e)h(ac)o(hiev)o(e)h(prop)q(erties)g(\(3\))e(and)i(\(4\))e(of)h (the)g(de\014nition)1295 499 y SDict begin H.S end 1295 499 a Fv(7.2)1353 470 y SDict begin H.R end 1353 470 a 1353 499 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.2) cvn H.B /ANN pdfmark end 1353 499 a Fv(.)50 553 y(Next)i(w)o(e)g (consider)h(small)g(disks)f(around)h(p)q(oin)o(ts)f Ft(A)p Fv(,)g Ft(B)r Fv(,)g Ft(C)s Fv(,)g Ft(A)1172 536 y Fm(0)1184 553 y Fv(.)26 b(F)l(or)16 b(small)i(disk)g(around)f Ft(A)g Fv(w)o(e)g(put)0 607 y Ft(t)p Fv(\()p Ft(x)p Fv(\))c(=)h Ft(u)166 614 y Fq(A)210 607 y Fv(and)i(the)g(condition)h(\(2\))e(will)i (b)q(e)f(satis\014ed.)22 b(F)l(or)15 b(small)i(disk)f(around)g Ft(B)h Fv(w)o(e)f(already)g(de\014ned)0 661 y Ft(t)h Fv(on)g(p)q(oin)o(ts)g(of)f(segmen)o(t)g Ft(B)r(A)g Fv(to)g(b)q(e)i Ft(u)700 668 y Fq(B)746 661 y Fv(and)f(on)g(p)q(oin)o(ts)g(of)f(segmen) o(t)g Ft(B)r(A)1340 644 y Fm(0)1368 661 y Fv(to)g(b)q(e)h Ft(m)1528 644 y Fs(2)1528 675 y Fq(b)1548 661 y Fv(\()p Ft(B)r Fv(\))1621 644 y Fm(\000)p Fs(1)1667 661 y Ft(u)1693 668 y Fq(B)1724 661 y Ft(m)1764 644 y Fs(1)1764 675 y Fq(b)1783 661 y Fv(\()p Ft(B)r Fv(\).)0 717 y(Since)g Ft(m)159 700 y Fs(1)159 731 y Fq(b)178 717 y Fv(\()p Ft(B)r Fv(\))d(comm)o(utes)h(with)g(elemen)o(ts)h(of)f Ft(M)869 700 y Fs(1)864 730 y Fq(B)909 717 y Fv(and)g Ft(m)1037 700 y Fs(2)1037 731 y Fq(b)1057 717 y Fv(\()p Ft(B)r Fv(\))1130 700 y Fm(\000)p Fs(1)1191 717 y Fv(comm)o(utes)g (with)g(elemen)o(ts)h(of)f Ft(M)1795 700 y Fs(2)1790 730 y Fq(B)1833 717 y Fv(=)0 775 y Ft(u)26 782 y Fq(B)56 775 y Ft(M)105 759 y Fs(1)100 789 y Fq(B)131 775 y Ft(u)157 756 y Fm(\000)p Fs(1)157 790 y Fq(B)217 775 y Fv(w)o(e)e(get)f Ft(u)381 756 y Fm(\000)p Fs(1)381 790 y Fq(B)428 775 y Ft(m)468 759 y Fs(2)468 789 y Fq(b)488 775 y Fv(\()p Ft(B)r Fv(\))561 759 y Fm(\000)p Fs(1)607 775 y Ft(u)633 782 y Fq(B)677 775 y Fv(and,)h(hence)h Ft(u)926 756 y Fm(\000)p Fs(1)926 790 y Fq(B)973 775 y Ft(m)1013 759 y Fs(2)1013 789 y Fq(b)1033 775 y Fv(\()p Ft(B)r Fv(\))1106 759 y Fm(\000)p Fs(1)1152 775 y Ft(u)1178 782 y Fq(B)1208 775 y Ft(m)1248 759 y Fs(1)1248 789 y Fq(b)1268 775 y Fv(\()p Ft(B)r Fv(\))e(comm)o(ute)g(with)i(elemen)o(ts)f(of)0 834 y Ft(M)49 818 y Fs(1)44 848 y Fq(B)74 834 y Fv(.)24 b(Then)17 b(there)f(exists)h(a)f(path)h Ft(p)d Fv(:)g([0)p Ft(;)8 b Fv(1])13 b Fn(\000)-7 b(!)14 b Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))15 b(connecting)i Ft(I)1288 841 y Fq(H)1338 834 y Fv(and)g Ft(u)1454 815 y Fm(\000)p Fs(1)1454 849 y Fq(B)1501 834 y Ft(m)1541 818 y Fs(2)1541 848 y Fq(b)1561 834 y Fv(\()p Ft(B)r Fv(\))1634 818 y Fm(\000)p Fs(1)1680 834 y Ft(u)1706 841 y Fq(B)1736 834 y Ft(m)1776 818 y Fs(1)1776 848 y Fq(b)1796 834 y Fv(\()p Ft(B)r Fv(\))0 891 y(suc)o(h)j(that)f(ev)o(ery)h(p)q(oin)o(t)g(of)f (the)h(path)f(comm)o(utes)g(with)h(elemen)o(ts)h(of)e Ft(M)1329 874 y Fs(1)1324 904 y Fq(B)1354 891 y Fv(.)34 b(T)l(ak)o(e)19 b(the)h(path)f Ft(u)1734 898 y Fq(B)1765 891 y Ft(p)p Fv(.)33 b(It)0 946 y(connects)21 b Ft(u)216 953 y Fq(B)267 946 y Fv(with)f Ft(m)415 930 y Fs(2)415 960 y Fq(b)435 946 y Fv(\()p Ft(B)r Fv(\))508 930 y Fm(\000)p Fs(1)554 946 y Ft(u)580 953 y Fq(B)611 946 y Ft(m)651 930 y Fs(1)651 960 y Fq(b)670 946 y Fv(\()p Ft(B)r Fv(\),)h(so)f(w)o(e) g(can)h(use)f(this)h(path)g(to)e(de\014ne)j Ft(t)f Fv(on)f(arcs)g(of)g (circles)0 1000 y(cen)o(tered)i(at)g Ft(B)h Fv(with)f(endp)q(oin)o(ts)h (on)f Ft(B)r(A)g Fv(and)g Ft(B)r(A)961 984 y Fm(0)972 1000 y Fv(.)40 b(Th)o(us)22 b(w)o(e)f(obtain)i(a)e(map)h(de\014ned)h (in)g(a)e(disk)0 1054 y(cen)o(tered)16 b(at)e Ft(B)r Fv(,)h(whic)o(h)h(is)g(con)o(tin)o(uous)f(at)g(all)h(p)q(oin)o(ts)f (except)h Ft(B)h Fv(and)e(for)g Ft(u)e Fn(2)f Ft(M)1442 1038 y Fs(1)1437 1068 y Fq(B)1468 1054 y Fv(,)j Ft(x)g Fv(in)h(this)f(disk)165 1131 y Ft(t)p Fv(\()p Ft(x)p Fv(\))p Ft(ut)p Fv(\()p Ft(x)p Fv(\))347 1113 y Fm(\000)p Fs(1)406 1131 y Fv(=)e Ft(u)480 1138 y Fq(B)510 1131 y Ft(p)p Fv(\()p Ft(\034)5 b Fv(\))p Ft(up)p Fv(\()p Ft(\034)g Fv(\))p Ft(u)730 1112 y Fm(\000)p Fs(1)730 1146 y Fq(B)789 1131 y Fv(=)13 b Ft(u)863 1138 y Fq(B)893 1131 y Ft(uu)945 1112 y Fm(\000)p Fs(1)945 1146 y Fq(B)1005 1131 y Fv(=)g Ft(t)p Fv(\()p Ft(B)r Fv(\))p Ft(ut)p Fv(\()p Ft(B)r Fv(\))1257 1113 y Fm(\000)p Fs(1)1303 1131 y Ft(;)20 b Fv(for)14 b(some)h Ft(\034)j Fn(2)13 b Fv([0)p Ft(;)8 b Fv(1])n Ft(:)0 1204 y Fv(The)15 b(same)g(can)h(b)q(e)g(done)f(for)g Ft(C)j Fv(and)d Ft(A)704 1188 y Fm(0)731 1204 y Fv(since)h(for)f Ft(C)s Fv(:)450 1278 y Ft(u)476 1258 y Fm(\000)p Fs(1)476 1292 y Fq(C)524 1278 y Ft(m)564 1259 y Fs(2)564 1289 y Fq(c)583 1278 y Fv(\()p Ft(C)s Fv(\))p Ft(t)p Fv(\()p Ft(C)s Fv(\))p Ft(m)783 1259 y Fs(1)783 1289 y Fq(c)801 1278 y Fv(\()p Ft(C)s Fv(\))873 1259 y Fm(\000)p Fs(1)932 1278 y Fv(=)e Ft(u)1006 1258 y Fm(\000)p Fs(1)1006 1292 y Fq(C)1053 1278 y Ft(m)1093 1259 y Fs(2)1093 1289 y Fq(c)1113 1278 y Fv(\()p Ft(C)s Fv(\))p Ft(u)1211 1285 y Fq(C)1240 1278 y Ft(m)1280 1259 y Fs(1)1280 1289 y Fq(c)1299 1278 y Fv(\()p Ft(C)s Fv(\))1371 1259 y Fm(\000)p Fs(1)0 1353 y Fv(comm)o(utes)g(with)g(elemen)o(ts)h(of)f Ft(M)596 1337 y Fs(1)591 1367 y Fq(C)621 1353 y Fv(.)19 b(In)14 b(the)f(neigh)o(b)q(ourho)q(o)q(d)i(of)e Ft(A)1177 1337 y Fm(0)1202 1353 y Fv(w)o(e)g(ha)o(v)o(e)f(v)m(alue)j Ft(m)1525 1337 y Fs(2)1525 1367 y Fq(b)1544 1353 y Fv(\()p Ft(A)1596 1337 y Fm(0)1608 1353 y Fv(\))1626 1337 y Fm(\000)p Fs(1)1673 1353 y Ft(u)1699 1360 y Fq(A)1727 1353 y Ft(m)1767 1337 y Fs(1)1767 1367 y Fq(b)1787 1353 y Fv(\()p Ft(A)1839 1337 y Fm(0)1850 1353 y Fv(\))0 1409 y(for)g(p)q(oin)o(ts)g(of)g Ft(B)r(A)329 1392 y Fm(0)356 1409 y Fv(and)g Ft(m)484 1392 y Fs(2)484 1420 y Fq(c)504 1409 y Fv(\()p Ft(A)p Fv(\))p Ft(u)600 1416 y Fq(A)628 1409 y Ft(m)668 1392 y Fs(1)668 1420 y Fq(c)687 1409 y Fv(\()p Ft(A)p Fv(\))757 1392 y Fm(\000)p Fs(1)819 1409 y Fv(for)g(p)q(oin)o(ts)g(of)g Ft(C)s(A)1147 1392 y Fm(0)1174 1409 y Fv(and)g(for)g Ft(A)1366 1392 y Fm(0)1392 1409 y Fv(itself.)21 b(So)15 b(w)o(e)g(c)o(hec)o(k)h(that)110 1484 y(\()p Ft(m)168 1465 y Fs(2)168 1495 y Fq(c)187 1484 y Fv(\()p Ft(A)p Fv(\))p Ft(u)283 1491 y Fq(A)311 1484 y Ft(m)351 1465 y Fs(1)351 1495 y Fq(c)371 1484 y Fv(\()p Ft(A)p Fv(\))441 1465 y Fm(\000)p Fs(1)488 1484 y Fv(\))506 1465 y Fm(\000)p Fs(1)552 1484 y Ft(m)592 1465 y Fs(2)592 1496 y Fq(b)612 1484 y Fv(\()p Ft(A)664 1465 y Fm(0)676 1484 y Fv(\))694 1465 y Fm(\000)p Fs(1)740 1484 y Ft(u)766 1491 y Fq(A)795 1484 y Ft(m)835 1465 y Fs(1)835 1496 y Fq(b)855 1484 y Fv(\()p Ft(A)907 1465 y Fm(0)918 1484 y Fv(\))c(=)h Ft(m)1036 1465 y Fs(1)1036 1495 y Fq(c)1056 1484 y Fv(\()p Ft(A)p Fv(\))p Ft(u)1152 1465 y Fm(\000)p Fs(1)1152 1498 y Fq(A)1198 1484 y Ft(m)1238 1465 y Fs(2)1238 1495 y Fq(c)1258 1484 y Fv(\()p Ft(A)p Fv(\))1328 1465 y Fm(\000)p Fs(1)1375 1484 y Ft(m)1415 1465 y Fs(2)1415 1496 y Fq(b)1434 1484 y Fv(\()p Ft(A)1486 1465 y Fm(0)1498 1484 y Fv(\))1516 1465 y Fm(\000)p Fs(1)1563 1484 y Ft(u)1589 1491 y Fq(A)1617 1484 y Ft(m)1657 1465 y Fs(1)1657 1496 y Fq(b)1677 1484 y Fv(\()p Ft(A)1729 1465 y Fm(0)1740 1484 y Fv(\))590 1561 y(=)g Ft(m)678 1542 y Fs(1)678 1572 y Fq(c)698 1561 y Fv(\()p Ft(A)p Fv(\))776 1524 y Fl(\002)793 1561 y Ft(u)819 1541 y Fm(\000)p Fs(1)819 1575 y Fq(A)867 1561 y Ft(m)907 1542 y Fs(2)907 1572 y Fq(c)926 1561 y Fv(\()p Ft(A)p Fv(\))996 1542 y Fm(\000)p Fs(1)1043 1561 y Ft(m)1083 1542 y Fs(2)1083 1572 y Fq(b)1103 1561 y Fv(\()p Ft(A)1155 1542 y Fm(0)1166 1561 y Fv(\))1184 1542 y Fm(\000)p Fs(1)1231 1561 y Ft(u)1257 1568 y Fq(A)1285 1561 y Ft(m)1325 1542 y Fs(1)1325 1572 y Fq(b)1345 1561 y Fv(\()p Ft(A)1397 1542 y Fm(0)1408 1561 y Fv(\))p Ft(m)1466 1542 y Fs(1)1466 1572 y Fq(c)1486 1561 y Fv(\()p Ft(A)p Fv(\))1556 1524 y Fl(\003)1582 1561 y Ft(m)1622 1542 y Fs(1)1622 1572 y Fq(c)1641 1561 y Fv(\()p Ft(A)p Fv(\))1711 1542 y Fm(\000)p Fs(1)0 1640 y Fv(comm)o(utes)18 b(with)h(elemen)o(ts)g(of)g Ft(M)618 1623 y Fs(1)613 1654 y Fq(A)639 1645 y Fw(0)671 1640 y Fv(since)h Ft(M)835 1623 y Fs(1)830 1654 y Fq(A)856 1645 y Fw(0)888 1640 y Fv(=)f Ft(m)982 1623 y Fs(1)982 1651 y Fq(c)1002 1640 y Fv(\()p Ft(A)p Fv(\))p Ft(M)1121 1623 y Fs(1)1116 1653 y Fq(A)1144 1640 y Ft(m)1184 1623 y Fs(1)1184 1651 y Fq(c)1203 1640 y Fv(\()p Ft(A)p Fv(\))1273 1623 y Fm(\000)p Fs(1)1339 1640 y Fv(and)f(the)h(elemen)o(t)h(in)f (square)0 1696 y(brac)o(k)o(ets)f(comm)o(utes)g(with)h(elemen)o(ts)g (of)f Ft(M)799 1680 y Fs(1)794 1710 y Fq(A)823 1696 y Fv(.)29 b(Indeed,)21 b Ft(m)1070 1680 y Fs(1)1070 1710 y Fq(b)1090 1696 y Fv(\()p Ft(A)1142 1680 y Fm(0)1153 1696 y Fv(\))p Ft(m)1211 1680 y Fs(1)1211 1707 y Fq(c)1230 1696 y Fv(\()p Ft(A)p Fv(\))d(comm)o(utes)g(with)h(elemen)o(ts)g(of)0 1752 y Ft(M)49 1735 y Fs(1)44 1765 y Fq(A)90 1752 y Fv(b)q(ecause)g(of) f(the)g(prop)q(ert)o(y)f(\(3\))g(of)h(the)f(de\014nition)1001 1752 y SDict begin H.S end 1001 1752 a Fv(7.1)1059 1723 y SDict begin H.R end 1059 1723 a 1059 1752 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.1) cvn H.B /ANN pdfmark end 1059 1752 a Fv(,)h(and)g Ft(m)1221 1735 y Fs(2)1221 1763 y Fq(c)1241 1752 y Fv(\()p Ft(A)p Fv(\))1311 1735 y Fm(\000)p Fs(1)1358 1752 y Ft(m)1398 1735 y Fs(2)1398 1766 y Fq(b)1417 1752 y Fv(\()p Ft(A)1469 1735 y Fm(0)1481 1752 y Fv(\))1499 1735 y Fm(\000)p Fs(1)1563 1752 y Fv(comm)o(utes)g (with)0 1810 y(elemen)o(ts)e(of)f Ft(M)287 1794 y Fs(2)282 1824 y Fq(A)325 1810 y Fv(b)o(y)h(the)f(same)g(reason.)k(Using)d Ft(M)917 1794 y Fs(1)912 1824 y Fq(A)953 1810 y Fv(=)d Ft(u)1027 1791 y Fm(\000)p Fs(1)1027 1825 y Fq(A)1074 1810 y Ft(M)1123 1794 y Fs(2)1118 1824 y Fq(A)1147 1810 y Ft(u)1173 1817 y Fq(A)1217 1810 y Fv(w)o(e)i(obtain)g(the)g(required) h(prop)q(ert)o(y)l(.)50 1864 y(Then)k(using)g(the)f(fact)g(that)g Ft(S)s(U)5 b Fv(\()p Ft(H)t Fv(\))17 b(is)j(simply)h(connected)f(w)o(e) f(can)h(extend)g Ft(t)g Fv(to)e(the)i(whole)g(of)f Fn(F)0 1918 y Fv(whic)o(h)c(will)h(giv)o(e)e(a)g(morphism)g(in)h(the)f(sense)h (of)f(the)g(de\014nition)1128 1918 y SDict begin H.S end 1128 1918 a Fv(7.2)1186 1889 y SDict begin H.R end 1186 1889 a 1186 1918 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.2) cvn H.B /ANN pdfmark end 1186 1918 a Fv(.)19 b(By)c(the)f(prop)q (osition)1604 1918 y SDict begin H.S end 1604 1918 a Fv(7.2)1662 1889 y SDict begin H.R end 1662 1889 a 1662 1918 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (prop.7.2) cvn H.B /ANN pdfmark end 1662 1918 a 14 w Fv(w)o(e)g(obtain)0 1972 y(a)j(homomorphism)g(of)g Ft(C)453 1956 y Fm(\003)472 1972 y Fv(-algebras.)27 b(T)l(aking)17 b(the)h(in)o(v)o(erse)f(of)g Ft(t)h Fv(at)e(eac)o(h)i(p)q(oin)o(t)g(of)e Fn(F)22 b Fv(giv)o(es)17 b(an)h(in)o(v)o(erse)0 2026 y(homomorphism.)i(Th)o(us)15 b(the)g(theorem)g(is)h(pro)o(v)o(ed.)932 b Ff(\003)50 2106 y Fv(The)21 b(theorem)g(3)f(is)i(a)f(trivial)h(consequence)g(of)e (the)i(theorem)e(ab)q(o)o(v)o(e)h(since)h(b)q(oth)f(algebras)g(in)h (the)0 2160 y(statemen)o(t)16 b(of)h(the)g(theorem)g(3)f(can)i(b)q(e)f (represen)o(ted)h(according)f(to)g(the)g(de\014nition)1506 2160 y SDict begin H.S end 1506 2160 a Fv(7.1)1564 2130 y SDict begin H.R end 1564 2130 a 1564 2160 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (definition.7.1) cvn H.B /ANN pdfmark end 1564 2160 a Fv(,)g(they)g(ha)o(v)o(e)g(the)0 2213 y(same)e(space)g Ft(H)h Fv(=)d Ft(V)364 2220 y Fq(i)391 2213 y Fv(=)g Fo(C)468 2197 y Fq(d)507 2213 y Fv(and)i(isomorphic)h (subalgebras)g Ft(M)1115 2220 y Fq(P)1160 2213 y Fv(at)e(p)q(oin)o(ts)i Ft(A)p Fv(,)f Ft(B)r Fv(,)f Ft(C)s Fv(.)801 2306 y Fu(References)0 2333 y SDict begin H.S end 0 2333 a 0 2333 a SDict begin 13 H.A end 0 2333 a 0 2333 a SDict begin [ /View [/XYZ H.V] 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