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%%Title: main.dvi
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%%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
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/FullName (XYDASH10) readonly def
/FamilyName (XYDASH) readonly def
/Weight (Medium) readonly def
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/UnderlineThickness 150 def
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/PaintType 0 def
/FontType 1 def
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%!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
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dup 47 /d47 put
dup 53 /d53 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueXX 5092839 def
currentdict end
currentfile eexec
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%%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 53 /d53 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueXX 5092838 def
currentdict end
currentfile eexec
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 @endspecial 387 2277 a FL(Figure)31 b(2.1:)44 b(Dynkin)32
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2562 y(mid)j(1980s)h(with)g(the)h(in)m(v)m(en)m(tion)g(of)f(Quan)m(tum)
g(groups.)56 b(In)36 b(con)m(trast)h(to)g(the)f(usual)257
2682 y(en)m(v)m(eloping)i(algebras)g(of)f(Lie)g(algebras,)i(Quan)m(tum)
f(groups,)h(b)s(eing)f(generally)f(non-)257 2802 y(comm)m(utativ)m(e)k
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3164 y(w)m(e)34 b(refer)f(to)f([Kac].)257 3392 y FJ(Def)10
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b(,)34 b(i.e.)257 3298 y(with)39 b(the)g(same)g(generators)h(and)f
(similar)d(relations.)61 b(The)40 b(only)e(c)m(hanges)j(p)s(ossible)257
3419 y(are)35 b(in)e(the)i(\\quan)m(tum)f(Serre")h(relations)e
(\(3.14\))g(and)i(in)e(the)i(ro)s(ot)e(v)m(ector)j(relations)257
3539 y(\(3.15\).)65 b(Before)40 b(w)m(e)h(giv)m(e)f(an)g(explicit)e
(form)m(ulation,)h(w)m(e)i(need)g(some)f(fundamen)m(tal)257
3659 y(terminology)-8 b(.)257 3888 y FJ(Def)10 b(inition)36
b(3.3)98 b FE(L)-5 b(et)27 b FL(\()p FI(a)1284 3903 y
FG(ij)1345 3888 y FL(\))g FE(b)-5 b(e)27 b(a)g(gener)-5
b(alize)g(d)25 b FL(\()p FI(\022)8 b FC(\002)d FI(\022)s
FL(\))p FE(-Cartan)28 b(matrix)e(\(cf.)42 b([Kac]\).)501
4008 y(The)35 b(c)-5 b(orr)g(esp)g(onding)33 b(Dynkin)h(diagr)-5
b(am)34 b(with)h(a)g(numb)-5 b(er)34 b(of)h(additional)f(e)-5
b(dges,)501 4128 y(dr)g(awn)31 b(as)h(dotte)-5 b(d)32
b(e)-5 b(dges)30 b(that)j(do)e(not)h(shar)-5 b(e)31 b(vertic)-5
b(es,)32 b(wil)5 b(l)31 b(b)-5 b(e)31 b(denote)-5 b(d)31
b FI(D)k FE(and)501 4249 y(c)-5 b(al)5 b(le)-5 b(d)36
b(a)44 b FL(link)-5 b(able)32 b(Dynkin)i(diagram)p FE(.)47
b(Two)36 b(vertic)-5 b(es)36 b FI(i)h FE(and)e FI(j)i
FC(6)p FL(=)31 b FI(i)36 b FE(c)-5 b(onne)g(cte)g(d)501
4369 y(by)35 b(such)g(dotte)-5 b(d)35 b(e)-5 b(dges)34
b(ar)-5 b(e)35 b(c)-5 b(al)5 b(le)-5 b(d)44 b FL(link)-5
b(able)p FE(.)42 b(This)35 b(is)f(written)h FI(i)17 b
FC(\001)g(\001)g(\001)e FI(j:)257 4598 y FJ(Def)10 b(inition)36
b(3.4)98 b FE(A)30 b FL(link)-5 b(able)25 b(braiding)g(matrix)g(of)i
FI(D)s FL(-Cartan)f(t)m(yp)s(e)31 b FE(for)f(a)f(linkable)501
4718 y(Dynkin)g(diagr)-5 b(am)28 b FI(D)33 b FE(is)c(a)g
FL(\()p FI(\022)13 b FC(\002)d FI(\022)s FL(\))30 b FE(matrix)g
FL(\()p FI(b)2183 4733 y FG(ij)2244 4718 y FL(\))f FE(with)g(the)h(fol)
5 b(lowing)28 b(pr)-5 b(op)g(erties)1090 4938 y FI(b)1131
4953 y FG(ii)1211 4938 y FC(6)p FL(=)28 b(1)p FI(;)1855
b FL(\(3.18\))979 5083 y FI(b)1020 5098 y FG(ij)1081
5083 y FI(b)1122 5098 y FG(j)t(i)1211 5083 y FL(=)28
b FI(b)1356 5029 y FG(a)1393 5039 y Fo(ij)1356 5109 y
FG(ii)1452 5083 y FI(;)1767 b FL(\(3.19\))839 5249 y
FI(b)880 5195 y FH(1)p Fx(\000)p FG(a)1007 5205 y Fo(ij)880
5277 y FG(k)r(i)1067 5249 y FI(b)1108 5264 y FG(k)r(j)1211
5249 y FL(=)28 b(1)p FI(;)116 b(k)30 b FL(=)e(1)p FI(;)17
b(:)g(:)g(:)f(;)h(\022)s(;)68 b FE(if)34 b FI(i)h FE(is)g(linkable)f
(to)h FI(j:)338 b FL(\(3.20\))1828 5637 y(20)p eop
%%Page: 21 23
21 22 bop 257 573 a FJ(Def)10 b(inition)36 b(3.5)98 b
FE(A)48 b(linkable)f(br)-5 b(aiding)47 b(matrix)g FL(\()p
FI(b)2314 588 y FG(ij)2375 573 y FL(\))h FE(is)g(c)-5
b(al)5 b(le)-5 b(d)57 b FL(realizable)45 b(o)m(v)m(er)501
693 y(the)28 b(ab)s(elian)e(group)i FI(\000)44 b FE(if)30
b(ther)-5 b(e)31 b(ar)-5 b(e)30 b(elements)f FI(g)2307
708 y FH(1)2347 693 y FI(;)17 b(:)g(:)g(:)e(;)i(g)2612
708 y FG(\022)2679 693 y FC(2)28 b FI(\000)44 b FE(and)30
b(char)-5 b(acters)501 814 y FI(\037)562 829 y FH(1)602
814 y FI(;)17 b(:)g(:)g(:)f(\037)838 829 y FG(\022)904
814 y FC(2)1019 788 y FL(^)999 814 y FI(\000)48 b FE(such)35
b(that)1271 994 y FI(b)1312 1009 y FG(ij)1401 994 y FL(=)27
b FI(\037)1565 1009 y FG(j)1602 994 y FL(\()p FI(g)1687
1009 y FG(i)1715 994 y FL(\))p FI(;)250 b FE(for)35 b(al)5
b(l)34 b FI(i;)17 b(j)141 b FE(and)508 b FL(\(3.21\))1259
1160 y FI(\037)1320 1106 y FH(1)p Fx(\000)p FG(a)1447
1116 y Fo(ij)1320 1185 y FG(i)1507 1160 y FI(\037)1568
1175 y FG(j)1632 1160 y FL(=)28 b(1)p FI(;)250 b FE(whenever)34
b FI(i)17 b FC(\001)g(\001)g(\001)e FI(j:)509 b FL(\(3.22\))257
1343 y FJ(Def)10 b(inition)36 b(3.6)98 b FE(A)37 b FL(linking)32
b(datum)i(of)h(Cartan)f(t)m(yp)s(e)k FE(is)f(a)f(c)-5
b(ol)5 b(le)-5 b(ction)36 b(of)h(the)g(fol-)501 1463
y(lowing)d(ingr)-5 b(e)g(dients:)617 1655 y FC(\017)49
b FE(an)34 b(ab)-5 b(elian)34 b(gr)-5 b(oup)35 b FI(\000)14
b FE(,)617 1805 y FC(\017)49 b FE(elements)34 b FI(h)1173
1820 y FH(1)1213 1805 y FI(;)17 b(:)g(:)g(:)e(h)1443
1820 y FG(t)1501 1805 y FC(2)29 b FI(\000)s(;)35 b FE(such)g(that)g
FI(\000)42 b FL(=)p FI(<)28 b(h)2480 1820 y FH(1)2548
1805 y FI(>)g FC(\010)g FI(<)g(h)2917 1820 y FH(2)2985
1805 y FI(>)g FC(\010)17 b FI(:)g(:)g(:)f FC(\010)28
b FI(<)716 1926 y(h)772 1941 y FG(t)829 1926 y FI(>)p
FE(,)35 b FI(M)1064 1941 y FG(k)1135 1926 y FL(:=)27
b(ord\()p FI(h)1500 1941 y FG(k)1543 1926 y FL(\))p FE(,)35
b FI(k)30 b FL(=)e(1)p FI(;)17 b(:)g(:)g(:)f(;)h(t)p
FE(,)617 2076 y FC(\017)49 b FE(a)41 b(linkable)e(Dynkin)i(diagr)-5
b(am)39 b FI(D)44 b FE(with)d FI(\022)j FE(vertic)-5
b(es)41 b(and)f(Cartan)h(matrix)716 2197 y FL(\()p FI(a)805
2212 y FG(ij)866 2197 y FL(\))p FE(,)617 2347 y FC(\017)49
b FE(a)44 b(linkable)g(br)-5 b(aiding)43 b(matrix)i(of)f
FI(D)s FE(-Cartan)g(typ)-5 b(e,)47 b(which)d(is)g(r)-5
b(e)g(alizable)716 2467 y(over)34 b FI(\000)49 b FE(with)35
b(elements)f(and)g(char)-5 b(acters)34 b FI(g)2357 2482
y FG(i)2413 2467 y FC(2)28 b FI(\000)s(;)35 b(\037)2693
2482 y FG(i)2749 2467 y FC(2)2864 2442 y FL(^)2843 2467
y FI(\000)14 b(;)35 b(i)28 b FL(=)g(1)p FI(;)17 b(:)g(:)g(:)e(;)i(\022)
s(;)617 2618 y FC(\017)49 b FE(p)-5 b(ar)g(ameters)39
b FI(\025)1276 2633 y FG(ij)1373 2618 y FC(2)e FF(|)-9
b FI(;)34 b FL(1)i FC(\024)h FI(i)g FC(6)p FL(=)g FI(j)42
b FC(\024)37 b FI(\022)s(;)j FE(such)g(that)g FI(\025)2779
2633 y FG(ij)2876 2618 y FL(=)c(0)j FE(if)h FI(i)g FE(is)f(not)716
2738 y(linkable)34 b(to)h FI(j)40 b FE(and)35 b FI(\025)1514
2753 y FG(ij)1602 2738 y FL(=)27 b FC(\000)p FI(\037)1843
2753 y FG(j)1881 2738 y FL(\()p FI(g)1966 2753 y FG(i)1993
2738 y FL(\))p FI(\025)2088 2753 y FG(j)t(i)2184 2738
y FE(when)34 b FI(a)2484 2753 y FG(ij)2572 2738 y FL(=)28
b(0)p FI(:)501 2930 y FE(V)-7 b(ertic)i(es)41 b FI(i)g
FE(and)f FI(j)46 b FE(with)41 b FI(\025)1510 2945 y FG(ij)1609
2930 y FC(6)p FL(=)d(0)i FE(ar)-5 b(e)41 b(c)-5 b(al)5
b(le)-5 b(d)50 b FL(link)m(ed)p FE(.)62 b(If)40 b FI(i)h
FE(and)f FI(j)47 b FE(ar)-5 b(e)40 b(linke)-5 b(d)501
3051 y(and)42 b(lie)g(in)g(the)g(same)f(c)-5 b(onne)g(cte)g(d)41
b(c)-5 b(omp)g(onent)41 b(of)h(the)g(Dynkin)g(diagr)-5
b(am,)43 b(we)501 3171 y(talk)30 b(of)g(a)38 b FL(self-linking)p
FE(.)i(A)31 b FL(linking)25 b(datum)i(of)g(\014nite)g(Cartan)g(t)m(yp)s
(e)k FE(is)f(a)g(linking)501 3291 y(datum)41 b(wher)-5
b(e)41 b(the)g(diagonal)f(elements)h(of)g(the)g(br)-5
b(aiding)40 b(matrix)h(have)g(\014nite)501 3412 y(or)-5
b(der)39 b(and)g(the)g(Cartan)g(matrix)g(is)g(of)g(\014nite)g(typ)-5
b(e,)41 b(i.e.)57 b(it)40 b(c)-5 b(orr)g(esp)g(onds)38
b(to)h(a)501 3532 y(\014nite)c(dimensional)e(semisimple)g(Lie)i(algebr)
-5 b(a.)257 3715 y FJ(Remark:)49 b FL(The)33 b(notion)e(of)g
FE(linkable)j(vertic)-5 b(es)40 b FL(is)31 b(simpler)g(and)h(hence)h
(more)f(general)501 3835 y(than)37 b(the)g(one)g(giv)m(en)g(in)f([AS3,)
i(De\014nition)d(5.1.].)55 b(Ho)m(w)m(ev)m(er,)40 b(the)e(notions)e
FE(do)501 3956 y FL(mainly)31 b(coincide)i(when)h(w)m(e)h(require)e
(the)h(existence)h(of)d(a)h(linking)e(datum)i(and)501
4076 y(demand)g(that)f(link)-5 b(able)31 b(v)m(ertices)j(lie)d(in)h
(di\013eren)m(t)h(connected)h(comp)s(onen)m(ts)g(of)501
4196 y(the)46 b(Dynkin)f(diagram.)79 b(W)-8 b(e)46 b(note)f(that)g(a)g
(link)-5 b(able)43 b(braiding)g(matrix)h(of)g(a)501 4317
y(linking)26 b(datum)h(of)g(\014nite)h(Cartan)g(t)m(yp)s(e)g(is)g(a)f
(braiding)f(matrix)g(of)h(Cartan)h(t)m(yp)s(e)501 4437
y(as)38 b(de\014ned)h(in)e(De\014nition)f(3.2,)j(and)e(the)h(t)m(w)m(o)
h(notions)e(coincide)g(when)h(there)501 4557 y(are)33
b(no)f(link)-5 b(able)31 b(v)m(ertices.)257 4740 y FJ(Def)10
b(inition)36 b(3.7)98 b FE(The)32 b(algebr)-5 b(a)33
b Fs(U)p FL(\()p FC(D)s FL(\))g FE(for)g(a)g(linking)f(datum)i
FC(D)i FE(of)d(Cartan)g(typ)-5 b(e)33 b(is)501 4860 y(given)h(by)h
(gener)-5 b(ators)35 b FI(y)1404 4875 y FG(k)1446 4860
y FI(;)g FL(1)27 b FC(\024)h FI(k)j FC(\024)d FI(t;)35
b(x)2028 4875 y FG(i)2057 4860 y FI(;)g FL(1)27 b FC(\024)i
FI(i)f FC(\024)g FI(\022)s(;)35 b FE(and)f(r)-5 b(elations)1095
5040 y FI(y)1147 4994 y FG(M)1215 5006 y Fo(k)1143 5068
y FG(k)1283 5040 y FL(=)28 b(1)p FI(;)116 b(y)1627 5055
y FG(k)1669 5040 y FI(y)1717 5055 y FG(l)1771 5040 y
FL(=)27 b FI(y)1922 5055 y FG(l)1948 5040 y FI(y)1996
5055 y FG(k)2038 5040 y FI(;)172 b FL(1)27 b FC(\024)h
FI(k)s(;)17 b(l)30 b FC(\024)e FI(t)p FL(;)504 b(\(3.23\))1082
5186 y FI(y)1130 5201 y FG(k)1172 5186 y FI(x)1227 5201
y FG(i)1283 5186 y FL(=)28 b FI(\037)1448 5201 y FG(i)1476
5186 y FL(\()p FI(h)1570 5201 y FG(k)1613 5186 y FL(\))p
FI(x)1706 5201 y FG(i)1735 5186 y FI(y)1783 5201 y FG(k)1825
5186 y FI(;)385 b FL(1)27 b FC(\024)h FI(k)j FC(\024)d
FI(t;)17 b FL(1)28 b FC(\024)g FI(i)g FC(\024)g FI(\022)s
FL(;)139 b(\(3.24\))640 5352 y(\()p FI(adx)835 5367 y
FG(i)863 5352 y FL(\))901 5311 y FH(1)p Fx(\000)p FG(a)1028
5321 y Fo(ij)1088 5352 y FL(\()p FI(x)1181 5367 y FG(j)1218
5352 y FL(\))27 b(=)h FI(\025)1444 5367 y FG(ij)1504
5352 y FL(\(1)22 b FC(\000)h FI(g)1764 5298 y FH(1)p
Fx(\000)p FG(a)1891 5308 y Fo(ij)1760 5378 y FG(i)1950
5352 y FI(g)1997 5367 y FG(j)2033 5352 y FL(\))p FI(;)139
b FL(1)27 b FC(\024)h FI(i)g FC(6)p FL(=)g FI(j)33 b
FC(\024)c FI(\022)s(:)409 b FL(\(3.25\))1828 5637 y(21)p
eop
%%Page: 22 24
22 23 bop 501 573 a FE(So)35 b(far,)f FL(ad)h FE(is)g(only)f(a)h(symb)
-5 b(ol)34 b(and)h(has)f(the)h(fol)5 b(lowing)33 b(explicit)i(form:)632
888 y FL(\(ad)17 b FI(x)845 903 y FG(i)874 888 y FL(\))912
847 y FH(1)p Fx(\000)p FG(a)1039 857 y Fo(ij)1098 888
y FL(\()p FI(x)1191 903 y FG(j)1228 888 y FL(\))28 b(:=)1424
753 y FH(1)p Fx(\000)p FG(a)1551 763 y Fo(ij)1443 794
y Fw(X)1451 1006 y FG(k)r FH(=0)1607 888 y FL(\()p FC(\000)p
FL(1\))1809 847 y FG(k)1851 748 y Fw(\022)1925 821 y
FL(1)22 b FC(\000)g FI(a)2146 836 y FG(ij)2039 957 y
FI(k)2207 748 y Fw(\023)2237 987 y FG(q)2269 997 y Fo(i)2299
888 y FI(q)2346 829 y FL(\()2384 790 y Fo(k)2386 847
y Fy(2)2418 829 y FL(\))2342 914 y FG(i)2460 888 y FI(b)2501
847 y FG(k)2501 913 y(ij)2562 888 y FI(x)2617 834 y FH(1)p
Fx(\000)p FG(a)2744 844 y Fo(ij)2800 834 y Fx(\000)p
FG(k)2617 914 y(i)2898 888 y FI(x)2953 903 y FG(j)2990
888 y FI(x)3045 847 y FG(k)3045 913 y(i)3088 888 y FI(;)131
b FL(\(3.26\))501 1199 y FE(wher)-5 b(e)30 b FI(q)815
1214 y FG(i)871 1199 y FL(:=)e FI(b)1043 1214 y FG(ii)1123
1199 y FL(=)g FI(\037)1288 1214 y FG(i)1316 1199 y FL(\()p
FI(g)1401 1214 y FG(i)1429 1199 y FL(\))p FI(:)j FE(The)f(gr)-5
b(oup)31 b(elements)f FI(g)2428 1214 y FG(i)2486 1199
y FE(ar)-5 b(e)31 b(interpr)-5 b(ete)g(d)30 b(as)h(wor)-5
b(ds)501 1319 y(in)35 b(the)g(gener)-5 b(ators)34 b FI(y)1304
1334 y FG(k)1346 1319 y FI(:)501 1481 y FE(Now)f(let)g
FC(D)i FE(b)-5 b(e)32 b(a)h(linking)e(datum)i(of)f(\014nite)g(Cartan)h
(typ)-5 b(e.)44 b(We)33 b(de\014ne)e(the)i(r)-5 b(o)g(ot)501
1601 y(ve)g(ctors)41 b FI(x)890 1616 y FG(\013)981 1601
y FE(for)f(al)5 b(l)41 b(p)-5 b(ositive)40 b(r)-5 b(o)g(ots)41
b FI(\013)f FC(2)f FL(\010)2166 1565 y FH(+)2267 1601
y FE(in)h(the)h(same)g(way)f(as)h(sketche)-5 b(d)501
1722 y(on)33 b(p)-5 b(age)32 b(18.)44 b(Given)33 b(a)g(family)f
Fq(u)c FL(=)g(\()p FI(u)1984 1737 y FG(\013)2032 1722
y FL(\))p FI(;)34 b FE(wher)-5 b(e)32 b FI(u)2460 1737
y FG(\013)2542 1722 y FE(is)h(an)g(expr)-5 b(ession)31
b(in)i(the)501 1842 y(gener)-5 b(ators)29 b FI(x)1024
1857 y FG(i)1053 1842 y FI(;)17 b(y)1145 1857 y FG(k)1216
1842 y FE(for)30 b(every)f(p)-5 b(ositive)29 b(r)-5 b(o)g(ot)30
b FI(\013)q(;)f FE(we)h(de\014ne)e(the)i(algebr)-5 b(a)29
b Fs(u)p FL(\()p FC(D)s FI(;)17 b Fq(u)p FL(\))501 1962
y FE(in)35 b(the)g(same)f(way)h(as)f Fs(U)p FL(\()p FC(D)s
FL(\))p FE(,)h(but)g(with)g(the)g(extr)-5 b(a)34 b(r)-5
b(elations)1362 2182 y FI(x)1417 2141 y FG(N)1473 2152
y Fo(I)1417 2207 y FG(\013)1540 2182 y FL(=)28 b FI(u)1700
2197 y FG(\013)1749 2182 y FI(;)216 b(\013)28 b FC(2)g
FL(\010)2246 2141 y FH(+)2246 2209 y FG(I)2306 2182 y
FI(;)17 b(I)35 b FC(2)28 b(X)15 b FI(:)611 b FL(\(3.27\))501
2402 y FC(X)61 b FE(is)46 b(again)f(the)h(set)g(of)g(c)-5
b(onne)g(cte)g(d)45 b(c)-5 b(omp)g(onents)45 b(of)h(the)g(Dynkin)f
(diagr)-5 b(am,)501 2523 y(and)43 b FI(N)777 2538 y FG(I)861
2523 y FE(denotes)g(the)h(c)-5 b(ommon)42 b(or)-5 b(der)43
b(of)g(those)h(diagonal)e(elements)h(of)g(the)501 2643
y(br)-5 b(aiding)41 b(matrix)h(that)h(c)-5 b(orr)g(esp)g(ond)41
b(to)i(the)f(c)-5 b(omp)g(onent)41 b FI(I)8 b FE(.)67
b(We)42 b(c)-5 b(al)5 b(l)42 b Fq(u)g FL(ro)s(ot)501
2764 y(v)m(ector)34 b(parameters)p FE(.)257 2992 y FJ(Remark:)49
b FL(When)33 b FI(a)1060 3007 y FG(ij)1149 2992 y FL(=)27
b(0,)33 b(\(3.25\))e(simpli\014es)g(to)1297 3212 y FI(x)1352
3227 y FG(i)1381 3212 y FI(x)1436 3227 y FG(j)1495 3212
y FC(\000)23 b FI(\037)1656 3227 y FG(j)1692 3212 y FL(\()p
FI(g)1777 3227 y FG(i)1805 3212 y FL(\))p FI(x)1898 3227
y FG(j)1935 3212 y FI(x)1990 3227 y FG(i)2046 3212 y
FL(=)28 b FI(\025)2207 3227 y FG(ij)2267 3212 y FL(\(1)22
b FC(\000)h FI(g)2523 3227 y FG(i)2551 3212 y FI(g)2598
3227 y FG(j)2634 3212 y FL(\))p FI(:)547 b FL(\(3.28\))501
3432 y(When)48 b(there)f(are)g(no)f(self-linkings,)i(then)f
FE(al)5 b(l)56 b FL(relations)46 b(\(3.25\))f(with)i(non-)501
3552 y(v)-5 b(anishing)32 b(righ)m(t)f(hand)i(side)g(are)f(of)h(the)g
(form)e(\(3.28\).)257 3781 y FJ(Prop)s(osition)36 b(3.3)98
b FE(Ther)-5 b(e)48 b(exists)g(a)g(unique)h(Hopf)f(algebr)-5
b(a)48 b(structur)-5 b(e)49 b(on)f Fs(U)p FL(\()p FC(D)s
FL(\))501 3901 y FE(determine)-5 b(d)34 b(by)1002 4121
y FL(\001\()p FI(y)1169 4136 y FG(k)1212 4121 y FL(\))27
b(=)h FI(y)1429 4136 y FG(k)1493 4121 y FC(\012)23 b
FI(y)1641 4136 y FG(k)1683 4121 y FI(;)853 b FL(1)28
b FC(\024)g FI(k)j FC(\024)d FI(t)p FL(;)252 b(\(3.29\))1009
4266 y(\001\()p FI(x)1183 4281 y FG(i)1212 4266 y FL(\))27
b(=)h FI(g)1428 4281 y FG(i)1478 4266 y FC(\012)22 b
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1629 y(cases)h(for)e(the)g(lifting)d(pro)s(cedure)29
b(and)e(the)h(ones)g(men)m(tioned)e(in)h(Theorem)g(3.1)g(do)g(not)257
1750 y(in)m(terfere.)257 1978 y FJ(Theorem)37 b(3.4)98
b FE([AS3,)35 b(The)-5 b(or)g(em)34 b(1.1.])501 2098
y(\(a\).)60 b(L)-5 b(et)41 b FI(p)c(>)h FL(17)h FE(b)-5
b(e)40 b(a)g(prime)g(and)f Fv(H)i FE(a)f(p)-5 b(ointe)g(d)39
b(\014nite)h(dimensional)e(Hopf)501 2219 y(algebr)-5
b(a)39 b(such)g(that)g FI(G)p FL(\()p Fv(H)q FL(\))c
FC(')i FI(\000)49 b FL(:=)36 b(\()p FF(Z)p FI(=)p FL(\()p
FI(p)p FL(\)\))2224 2183 y FG(s)2258 2219 y FE(.)58 b(Then)38
b(ther)-5 b(e)39 b(exists)g(a)g(linking)501 2339 y(datum)g
FC(D)h FE(of)e(\014nite)g(Cartan)g(typ)-5 b(e)39 b(with)f(gr)-5
b(oup)38 b FI(\000)52 b FE(and)38 b(no)g(self-linkings)e(such)501
2460 y(that)f Fv(H)29 b FC(')f Fs(u)p FL(\()p FC(D)s
FI(;)17 b FJ(0)p FL(\))p FE(.)501 2621 y(\(b\).)71 b(Conversely,)45
b(given)e(a)g(linking)g(datum)h FC(D)i FE(of)e(\014nite)f(Cartan)g(typ)
-5 b(e)44 b(with)501 2742 y(gr)-5 b(oup)35 b FI(\000)s(;)g
FE(the)h(algebr)-5 b(a)34 b Fv(H)28 b FL(:=)g Fs(u)p
FL(\()p FC(D)s FI(;)17 b FJ(0)p FL(\))34 b FE(is)h(p)-5
b(ointe)g(d,)34 b FI(G)p FL(\()p Fv(H)q FL(\))28 b FC(')g
FI(\000)49 b FE(and)35 b FL(dim)15 b Fv(H)28 b FL(=)501
2862 y FI(p)550 2826 y FG(s)p FH(+)p Fx(j)p FH(\010)709
2803 y Fy(+)759 2826 y Fx(j)783 2862 y FI(:)404 3090
y FL(Although)d(this)i(result)f(pro)m(vides)h(a)g(go)s(o)s(d)e(answ)m
(er)j(to)e(the)h(classi\014cation)e(problem,)257 3211
y(there)39 b(are)e(still)e(a)i(few)h(di\016culties)f(when)i(w)m(e)f(w)m
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3331 y(this)c(kind)f(explicitly)-8 b(.)42 b(This)32 b(is)g(the)h
(starting)f(p)s(oin)m(t)g(of)g(this)g(Ph.D.)h(thesis.)404
3452 y(One)i(asp)s(ect)h(needing)f(clari\014cation)e(is)i(the)g
(linking)e(parameters.)51 b(Ha)m(ving)35 b(\014xed)257
3572 y(the)d(Dynkin)f(diagram,)f(the)h(group)g(and)h(c)m(haracter)g
(elemen)m(ts,)g(what)f(p)s(ossible)g FI(\025)g FL(can)257
3692 y(app)s(ear?)87 b(It)48 b(is)e(not)h(at)g(all)e(ob)m(vious)i(whic)
m(h)h(v)m(ertices)g(can)g(b)s(e)f(link)m(ed.)87 b(\\Exotic")257
3813 y(linkings)42 b(lik)m(e)g([AS3,)k(Example)c(5.13.],)j(where)f(4)f
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(in)m(to)g(a)257 3933 y(circle)36 b(are)h(p)s(ossible.)54
b(The)38 b(general)e(picture)h(w)m(as)g(presen)m(ted)i(in)d([D1)o(])h
(and)f(the)h(next)257 4054 y(c)m(hapter)d(is)e(dev)m(oted)i(to)e(this)h
(problem.)404 4174 y(Giv)m(en)28 b(a)h(\014xed)h(prime)e
FI(p)g FL(and)h(an)g FI(s;)g FL(what)g(Dynkin)g(diagrams)e(are)i
(realizable?)40 b(In)257 4294 y(other)35 b(w)m(ords,)g(for)f(whic)m(h)g
(diagrams)f(can)h(w)m(e)h(\014nd)f(group)g(elemen)m(ts)h(and)f(c)m
(haracters)257 4415 y(suc)m(h)42 b(that)e(\(3.19\))g(can)h(b)s(e)f
(ful\014lled?)65 b(This)41 b(question)g(has)g(b)s(een)g(addressed)h(so)
f(far)257 4535 y(only)32 b(for)g FI(s)c FL(=)g(1)k(in)g([AS2].)43
b(W)-8 b(e)33 b(will)e(presen)m(t)j(the)f(answ)m(ers)h(for)e
FI(s)c FL(=)g(2)k(in)g(Chapter)h(5.)404 4655 y(In)g(the)g(last)f(c)m
(hapter)i(w)m(e)f(will)e(b)s(e)i(concerned)h(with)e(analyzing)g(ho)m(w)
h(di\013eren)m(t)g(all)257 4776 y(these)h(new)g(Hopf)e(algebras)g
(actually)f(are.)1828 5637 y(25)p eop
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26 27 bop 257 1237 a FK(Chapter)78 b(4)257 1652 y(The)g(structure)g(of)
f(link)-13 b(able)257 1901 y(Dynkin)77 b(diagrams)257
2354 y FL(In)38 b(this)f(c)m(hapter)h(w)m(e)h(w)m(an)m(t)f(to)f
(address)h(the)g(problem)e(of)h(determining)f(all)f(p)s(ossible)257
2474 y(linkings.)42 b(W)-8 b(e)33 b(will)d(b)s(e)j(mainly)d(concerned)k
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2594 y(a)j(link)-5 b(able)32 b(braiding)g(matrix)h(of)h(a)g(giv)m(en)h
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b(will)d(lead)257 2715 y(to)44 b(a)f(c)m(haracterization)g(of)g(the)h
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b(W)-8 b(e)257 2835 y(sho)m(w)47 b(ho)m(w)f(these)h(ideas)f(are)g
(related)f(to)g(the)h(usual)g(quan)m(tized)g(en)m(v)m(eloping)g(alge-)
257 2956 y(bras)30 b(and)g(to)f(the)h(\014nite)g(dimensional)d(Hopf)i
(algebras)g(constructed)i(in)e([AS3],)i(whic)m(h)257
3076 y(are)42 b(themselv)m(es)g(v)-5 b(ariations)40 b(of)g(the)i
(\014nite)f(dimensional)e(Quan)m(tum)i(groups)g(called)257
3196 y(F)-8 b(rob)s(enius-Lusztig)32 b(k)m(ernels)h([Lus3)q(].)404
3317 y(T)-8 b(o)32 b(get)g(a)g(nice)g(result,)g(w)m(e)h(sligh)m(tly)e
(sp)s(ecialize)g(some)g(of)h(our)g(earlier)f(de\014nitions.)257
3437 y(W)-8 b(e)33 b(will)e(discuss)i(generalisations)e(later.)501
3615 y(F)-8 b(rom)22 b(no)m(w)i(on)f(all)e(link)-5 b(able)21
b(Dynkin)j(diagrams)d(are)j(assumed)g(to)e(b)s(e)i(link-)501
3736 y(connected,)34 b(i.e.)43 b(when)32 b(view)m(ed)h(as)f(a)g(graph)f
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b(w)m(e)f(will)d(restrict)i(our)g(considerations)g(to)g(diagrams)e
(where)501 3976 y(t)m(w)m(o)45 b(v)m(ertices)g(are)g(link)-5
b(able)42 b(only)h(if)g(they)i(lie)e(in)g(di\013eren)m(t)i(connected)
501 4097 y(comp)s(onen)m(ts)25 b(of)f(the)h(original)c(diagram,)j(i.e.)
40 b(there)25 b(are)f(no)h(self-linkings.)501 4217 y(Finally)-8
b(,)37 b(all)f(diagonal)g(elemen)m(ts)j(of)e(the)i(braiding)d(matrix)h
(ha)m(v)m(e)j(\014nite)501 4338 y(order)26 b(and)g(the)g(base)g
(\014eld)f FF(|)11 b FL(is)25 b(required)h(to)g(con)m(tain)f(a)g
FI(p)2627 4301 y FH(th)2723 4338 y FL(ro)s(ot)g(of)g(unit)m(y)501
4458 y(for)32 b(a)h(prime)e FI(p)d(>)f FL(3)p FI(:)257
4636 y FL(F)-8 b(or)38 b(t)m(w)m(o)h(v)m(ertices)h FI(i;)17
b(j)44 b FL(of)38 b(the)h(Dynkin)f(diagram)f(with)h FI(a)2455
4651 y FG(ij)2553 4636 y FC(6)p FL(=)g(0)p FI(;)g FL(the)h(symmetry)g
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4823 y FG(a)1737 4833 y Fo(ij)1700 4902 y FG(ii)1825
4877 y FL(=)27 b FI(b)1969 4823 y FG(a)2006 4833 y Fo(j)s(i)1969
4902 y FG(j)t(j)2066 4877 y FI(:)1202 b FL(\(4.1\))257
5039 y(F)-8 b(or)44 b FI(i)17 b FC(\001)g(\001)g(\001)e
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5039 y FL(=)i(0,)f(as)e(w)m(e)h(required)f(the)g(v)m(ertices)h(to)e
(lie)f(in)h(di\013eren)m(t)257 5159 y(connection)33 b(comp)s(onen)m
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5365 y FG(ii)1554 5350 y FL(=)27 b FI(b)1698 5308 y Fx(\000)p
FH(1)1698 5375 y FG(ij)1821 5350 y FL(=)g FI(b)1965 5365
y FG(j)t(i)2054 5350 y FL(=)g FI(b)2198 5308 y Fx(\000)p
FH(1)2198 5375 y FG(j)t(j)2293 5350 y FI(:)975 b FL(\(4.2\))1828
5637 y(26)p eop
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27 28 bop 257 573 a FD(4.1)161 b(The)53 b(\014nite)h(case)257
792 y FL(First)41 b(w)m(e)i(will)c(only)i(consider)h(Dynkin)f(diagrams)
f(of)h(\014nite)h(t)m(yp)s(e,)i(i.e.)70 b(the)42 b(corre-)257
912 y(sp)s(onding)48 b(Lie)f(algebras)h(are)g(\014nite)g(dimensional.)
87 b(In)49 b(order)f(to)g(get)g(in)m(teresting)257 1033
y(applications)40 b(in)h(regard)g(of)g([AS3)q(])g(w)m(e)i(further)f
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1153 y(matrix)31 b(has)i(the)g(follo)m(wing)d(prop)s(ert)m(y:)538
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b(than)g(2)h(and)538 1471 y(not)i(divisible)f(by)h(3)g(if)g(the)g
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b(a)538 1592 y(c)-5 b(omp)g(onent)33 b(of)i(typ)-5 b(e)35
b FI(G)1419 1607 y FH(2)1459 1592 y FE(.)3295 1471 y
FL(\(4.3\))404 1740 y(The)27 b(\014rst)g(prop)s(erties)g(are)f(presen)m
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1860 y(5.6.)43 b(in)32 b([AS3].)44 b(Ho)m(w)m(ev)m(er,)35
b(w)m(e)f(form)m(ulate)d(it)g(on)i(the)g(lev)m(el)f(of)g(the)h
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b(am)31 b FI(D)k FE(and)d(a)g(c)-5 b(orr)g(esp)g(ond-)501
2189 y(ing)45 b(linkable)e(br)-5 b(aiding)44 b(matrix)h
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2309 y(linkable)34 b(to)h FI(k)j FE(and)c FI(l)r FE(,)h(r)-5
b(esp)g(e)g(ctively.)44 b(Then)34 b FI(a)2200 2324 y
FG(ij)2289 2309 y FL(=)27 b FI(a)2443 2324 y FG(k)r(l)2508
2309 y FI(:)257 2518 y FJ(Pro)s(of:)49 b FL(If)33 b FI(a)758
2533 y FG(il)836 2518 y FC(6)p FL(=)28 b(0)33 b(or)f
FI(a)1192 2533 y FG(j)t(k)1296 2518 y FC(6)p FL(=)c(0)k(then)i(w)m(e)g
(immediately)29 b(get)k FI(a)2617 2533 y FG(ij)2706 2518
y FL(=)28 b FI(a)2861 2533 y FG(k)r(l)2954 2518 y FL(=)g(0,)33
b(b)s(ecause)501 2638 y(link)-5 b(able)23 b(v)m(ertices)j(m)m(ust)g
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FI(D)s(:)h FL(So)g(w)m(e)501 2759 y(no)m(w)31 b(tak)m(e)f
FI(a)961 2774 y FG(il)1039 2759 y FL(=)d FI(a)1193 2774
y FG(j)t(k)1296 2759 y FL(=)h(0)p FI(:)h FL(Without)g(loss)h(of)f
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2759 y FC(\024)e FI(a)3403 2774 y FG(k)r(l)3468 2759
y FI(:)501 2879 y FL(Using)k(\(3.19\))g(and)h(\(3.20\))f(alternately)-8
b(,)31 b(w)m(e)j(get)876 3082 y FI(b)917 3028 y FG(a)954
3038 y Fo(ij)917 3107 y FG(ii)1041 3082 y FL(=)28 b FI(b)1186
3097 y FG(ij)1247 3082 y FI(b)1288 3097 y FG(j)t(i)1377
3082 y FL(=)f FI(b)1521 3041 y Fx(\000)p FH(1)1521 3110
y FG(il)1616 3082 y FI(b)1657 3041 y Fx(\000)p FH(1)1657
3110 y FG(j)t(k)1779 3082 y FL(=)h FI(b)1924 3097 y FG(l)q(i)1974
3082 y FI(b)2015 3097 y FG(k)r(j)2119 3082 y FL(=)f FI(b)2263
3041 y Fx(\000)p FH(1)2263 3110 y FG(l)q(k)2358 3082
y FI(b)2399 3041 y Fx(\000)p FH(1)2399 3110 y FG(k)r(l)2521
3082 y FL(=)h FI(b)2666 3036 y Fx(\000)p FG(a)2758 3048
y Fo(k)q(l)2666 3110 y FG(k)r(k)2849 3082 y FL(=)f FI(b)2993
3036 y FG(a)3030 3048 y Fo(k)q(l)2993 3107 y FG(ii)3094
3082 y FI(:)501 3285 y FL(In)32 b(the)h(last)e(step)h(w)m(e)h(used)g
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3525 y FI(a)552 3540 y FG(ij)647 3525 y FL(=)34 b FC(\000)p
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b FI(b)689 3661 y FG(ii)788 3646 y FL(=)j(3)c(is)g(a)h(con)m
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3766 y(diagonal)30 b(elemen)m(ts.)2008 b FJ(qed.)404
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4095 y(grams)25 b(that)g(admit)f(a)h(corresp)s(onding)g(braiding)e
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4216 y(w)m(e)34 b(ha)m(v)m(e)g(to)e(in)m(tro)s(duce)h(some)f
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b(4.1)98 b FE(F)-7 b(or)34 b(every)i(cycle)1683 4368
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b(by)g(the)35 b FL(w)m(eigh)m(t)27 b FI(w)1364 4539 y
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b(double)g(e)-5 b(dges)25 b(in)h(that)g(cycle)g(with)g(the)g(arr)-5
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4765 y(and)38 b(against)h(it.)57 b(The)45 b FL(length)39
b FI(l)1728 4780 y FG(c)1801 4765 y FE(of)g(the)f(cycle)h(is)g
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5637 y FL(27)p eop
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28 29 bop 257 573 a FL(In)35 b(preparation)e(for)h(some)g(tec)m
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b(er)39 b FI(h)1844 978 y FG(i)1844 1039 y(j)1881 1014
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(all)g(the)g(corre-)257 5337 y(sp)r(onding)28 b(edges.)1828
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2616 y FH(th)2076 2653 y FL(ro)s(ot)h(of)g(unit)m(y)-8
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y(No)m(w)26 b(w)m(e)g(c)m(ho)s(ose)g(a)e(v)m(ertex)j
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3296 y(for)32 b(ev)m(ery)h(v)m(ertex)h FI(j)g FC(6)p
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3416 y(b)m(y)38 b(the)g(sequence)i(of)c(its)h(v)m(ertices)h(\()p
FI(i)e FL(=)f FI(p)2103 3431 y FH(0)2143 3416 y FI(;)17
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FI(b)1722 3552 y FG(p)1758 3564 y Fo(k)1797 3552 y FG(p)1833
3564 y Fo(k)1874 3537 y FI(;)45 b(k)30 b FL(=)e(1)p FI(;)17
b(:)g(:)g(:)f(;)h(t;)32 b FL(recursiv)m(ely:)1064 3850
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3865 y FG(p)1292 3877 y Fo(k)q Fy(+1)1439 3850 y FL(=)1542
3679 y Fw(\()1623 3774 y FI(b)1664 3737 y Fx(\000)p FH(1)1664
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y FG(p)2522 3801 y Fo(k)2560 3789 y FG(p)2596 3801 y
Fo(k)q Fy(+1)2743 3774 y FL(=)c(0)p FI(;)1623 3934 y(b)1664
3867 y FG(a)1701 3875 y Fo(p)1733 3893 y(k)1772 3875
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3956 y Fo(k)2221 3934 y FI(;)98 b FL(otherwise.)3295
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(v)m(e)h(to)e(tak)m(e)501 4649 y(third)40 b(ro)s(ots)g(when)h(w)m(e)g
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(yp)s(e)501 4769 y FI(G)578 4784 y FH(2)618 4769 y FI(:)32
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k(3.)501 4931 y(W)-8 b(e)33 b(no)m(w)g(sho)m(w)h(that)f(this)f(pro)s
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693 y FI(;)17 b(p)683 708 y FH(1)722 693 y FI(;)g(:)g(:)g(:)f(;)h(p)990
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b(q)1899 708 y FH(1)1938 693 y FI(;)g(:)g(:)g(:)f(;)h(q)2200
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1175 y FL(:=)c FI(P)3292 1190 y FG(q)3324 1198 y Fo(n)3366
1190 y FG(q)3398 1198 y Fo(m)3452 1213 y Fy(2)501 1295
y FL(leads)i(to)f(the)h(same)f(v)-5 b(alue)1620 1509
y FI(b)1661 1524 y FG(p)1697 1532 y Fo(m)1751 1547 y
Fy(1)1790 1524 y FG(p)1826 1532 y Fo(m)1880 1547 y Fy(1)1951
1509 y FL(=)27 b FI(b)2095 1524 y FG(q)2127 1532 y Fo(m)2181
1547 y Fy(2)2221 1524 y FG(q)2253 1532 y Fo(m)2307 1547
y Fy(2)2349 1509 y FI(:)919 b FL(\(4.6\))501 1722 y(As)30
b(triple)d(edges)j(are)f(not)g(part)g(of)f(cycles,)j(w)m(e)f(easily)e
(obtain)g(a)h(closed)g(form)m(ula)501 1842 y(for)j(the)h(desired)g(v)-5
b(alues:)777 2056 y FI(b)818 2071 y FG(p)854 2079 y Fo(m)908
2094 y Fy(1)947 2071 y FG(p)983 2079 y Fo(m)1037 2094
y Fy(1)1107 2056 y FL(=)28 b FI(Q)1288 2014 y FH(\()p
Fx(\000)p FH(1\))1432 1991 y Fo(l)1452 2006 y Fy(1)1492
2014 y FH(2)1527 1991 y Fo(w)1571 2006 y Fy(1)1810 2056
y FL(and)195 b FI(b)2203 2071 y FG(q)2235 2079 y Fo(m)2289
2094 y Fy(2)2328 2071 y FG(q)2360 2079 y Fo(m)2414 2094
y Fy(2)2485 2056 y FL(=)27 b FI(Q)2665 2014 y FH(\()p
Fx(\000)p FH(1\))2809 1991 y Fo(l)2829 2006 y Fy(2)2870
2014 y FH(2)2905 1991 y Fo(w)2949 2006 y Fy(2)2992 2056
y FI(:)276 b FL(\(4.7\))501 2269 y(Here)30 b FI(Q)e FL(:=)g
FI(b)1005 2284 y FG(p)1041 2292 y Fo(n)1083 2284 y FG(p)1119
2292 y Fo(n)1165 2269 y FL(,)i FI(l)1251 2284 y FG(i)1307
2269 y FC(\025)e FL(0)h(denotes)h(the)g(n)m(um)m(b)s(er)f(of)g(dotted)g
(edges)h(in)f(the)g(path)501 2389 y FI(P)564 2404 y FG(i)631
2389 y FL(and)38 b FI(w)896 2404 y FG(i)962 2389 y FC(2)g
FF(Z)e FL(is)i(the)h(di\013erence)g(of)f(the)g(n)m(um)m(b)s(ers)i(of)e
(double)g(edges)h(in)f FI(P)3467 2404 y FG(i)501 2510
y FL(that)27 b(ha)m(v)m(e)g(the)g(arro)m(w)g(p)s(oin)m(ting)e(against)g
(the)i(path's)g(orien)m(tation)e(and)i(with)f(it.)501
2630 y(Without)32 b(loss)g(of)h(generalit)m(y)e(w)m(e)j(assume)f
FI(w)2191 2645 y FH(1)2258 2630 y FC(\025)28 b FI(w)2433
2645 y FH(2)2505 2630 y FL(and)33 b(ha)m(v)m(e)g FI(w)2989
2645 y FG(c)3052 2630 y FL(=)27 b FI(w)3225 2645 y FH(1)3286
2630 y FC(\000)c FI(w)3456 2645 y FH(2)501 2751 y FL(and)33
b FI(l)720 2766 y FG(c)782 2751 y FL(=)28 b FI(l)915
2766 y FH(1)977 2751 y FL(+)22 b FI(l)1104 2766 y FH(2)1143
2751 y FL(.)501 2911 y(As)28 b FI(Q)f FL(is)g(a)g FI(d)964
2875 y FH(th)1061 2911 y FL(ro)s(ot)f(of)h(unit)m(y)g(and)g
FI(d)g FL(divides)g(all)e(cycle)j(genera)f(w)m(e)h(get)f
FI(Q)3245 2875 y FG(g)3279 2883 y Fo(c)3343 2911 y FL(=)g(1)501
3032 y(or)1514 3152 y FI(Q)1591 3111 y FH(2)1626 3088
y Fy(\()p Fo(w)1694 3103 y Fy(1)1730 3088 y Fu(\000)p
Fo(w)1822 3103 y Fy(2)1857 3088 y(\))1917 3152 y FL(=)h
FI(Q)2098 3111 y FH(\()p Fx(\000)p FH(1\))2242 3088 y
Fy(\()p Fo(l)2286 3103 y Fy(1)2321 3088 y(+)p Fo(l)2388
3103 y Fy(2)2423 3088 y(\))2455 3152 y FI(:)501 3324
y FL(T)-8 b(aking)36 b(b)s(oth)g(sides)h(to)f(the)h(p)s(o)m(w)m(er)h
(of)e(\()p FC(\000)p FL(1\))2203 3288 y FG(l)2224 3297
y Fy(1)2263 3324 y FL(2)2312 3288 y FG(w)2363 3297 y
Fy(2)2401 3324 y FI(;)g FL(w)m(e)i(arriv)m(e)e(at)g(\(4.6\).)54
b(Here)501 3444 y(w)m(e)29 b(w)m(ould)f(lik)m(e)g(to)f(remind)g(the)i
(reader)f(that)g(all)e(v)-5 b(alues)28 b(are)g FI(d)2825
3408 y FH(th)2924 3444 y FL(ro)s(ots)f(of)h(unit)m(y)501
3564 y(and)33 b(hence)h(there)f(is)f(no)h(am)m(biguit)m(y)e(regarding)g
(signs.)501 3725 y(The)55 b(so)e(sp)s(eci\014ed)i(diagonal)c(en)m
(tries)j(of)f(the)h(braiding)e(matrix)g(ful\014ll)f(the)501
3846 y(requiremen)m(ts)40 b(for)e(their)g(orders)h(b)s(ecause)i
FI(d)d FL(is)g(o)s(dd)h(and)g(not)f(divisible)f(b)m(y)i(3)501
3966 y(when)34 b(there)f(are)g(comp)s(onen)m(ts)g(of)f(t)m(yp)s(e)h
FI(G)2112 3981 y FH(2)2184 3966 y FL(in)f FI(D)s(:)501
4248 y FL(W)-8 b(e)29 b(no)m(w)g(giv)m(e)g(the)g(remaining)d(en)m
(tries)j(for)f(the)h(link)-5 b(able)26 b(braiding)h(matrix,)h(i.e.)501
4368 y(w)m(e)41 b(sp)s(ecify)f FI(b)1023 4383 y FG(ij)1124
4368 y FL(for)f FI(i)h FC(6)p FL(=)g FI(j:)g FL(F)-8
b(or)39 b(this)g(w)m(e)i(divide)e(the)h(set)g FC(f)p
FL(\()p FI(i;)17 b(j)6 b FL(\))40 b(:)g FI(i)g FC(6)p
FL(=)f FI(j)6 b FC(g)40 b FL(of)501 4488 y(pairs)32 b(of)g(v)m(ertices)
i(in)m(to)e(4)g(classes:)501 4690 y FJ(None)38 b(of)f(the)h(t)m(w)m(o)e
(v)m(ertices)h(is)f(link)-6 b(able)36 b(to)h(an)m(y)h(other)f(one.)50
b FL(W)-8 b(e)33 b(set)1546 4903 y FI(b)1587 4918 y FG(j)t(i)1676
4903 y FL(:=)28 b FI(z)t(;)212 b(b)2136 4918 y FG(ij)2225
4903 y FL(:=)28 b FI(b)2397 4849 y FG(a)2434 4859 y Fo(ij)2397
4929 y FG(ii)2493 4903 y FI(z)2542 4862 y Fx(\000)p FH(1)2638
4903 y FI(:)501 5136 y FJ(The)38 b(t)m(w)m(o)f(v)m(ertices)f(are)i
(link)-6 b(able)35 b(to)i(eac)m(h)h(other.)49 b FL(W)-8
b(e)33 b(set)1576 5350 y FI(b)1617 5365 y FG(ij)1706
5350 y FL(:=)28 b FI(b)1878 5308 y Fx(\000)p FH(1)1878
5375 y FG(ii)1973 5350 y FI(;)211 b(b)2252 5365 y FG(j)t(i)2341
5350 y FL(:=)28 b FI(b)2513 5308 y Fx(\000)p FH(1)2513
5375 y FG(j)t(j)2608 5350 y FI(:)1828 5637 y FL(31)p
eop
%%Page: 32 34
32 33 bop 501 573 a FJ(Only)37 b(one)h(of)g(the)f(t)m(w)m(o)g(v)m
(ertices)f(is)h(link)-6 b(able)36 b(to)h(another)g(v)m(ertex.)716
693 y FL(W)-8 b(e)33 b(assume)g FI(i)g FL(is)f(link)-5
b(able)30 b(to)i FI(k)s FL(.)44 b(W)-8 b(e)33 b(set)1248
908 y FI(b)1289 923 y FG(j)t(i)1378 908 y FL(:=)28 b
FI(z)t(;)823 b(b)2449 923 y FG(ij)2537 908 y FL(:=)28
b FI(b)2709 854 y FG(a)2746 864 y Fo(ij)2709 934 y FG(ii)2806
908 y FI(z)2855 867 y Fx(\000)p FH(1)2950 908 y FI(;)1234
1062 y(b)1275 1077 y FG(j)t(k)1378 1062 y FL(:=)g FI(z)1558
1021 y Fx(\000)p FH(1)1653 1062 y FI(;)713 b(b)2434 1077
y FG(k)r(j)2537 1062 y FL(:=)28 b FI(b)2709 1006 y FG(a)2746
1018 y Fo(k)q(j)2709 1089 y FG(k)r(k)2818 1062 y FI(z)t(:)501
1297 y FJ(Both)37 b(v)m(ertices)f(are)i(link)-6 b(able)36
b(to)h(other)g(v)m(ertices.)48 b FL(W)-8 b(e)93 b(assume)g
FI(i)g FL(is)716 1417 y(link)-5 b(able)42 b(to)i FI(k)k
FL(and)d FI(j)50 b FL(is)44 b(link)-5 b(able)43 b(to)h
FI(l)r(:)h FL(F)-8 b(or)43 b FI(i)i FL(and)g FI(k)i FL(to)e(b)s(e)f
(link)-5 b(able)716 1537 y(w)m(e)31 b(can)f(not)f(ha)m(v)m(e)i
FI(a)1476 1552 y FG(ij)1565 1537 y FC(6)p FL(=)c(0)j(and)g
FI(a)1985 1552 y FG(j)t(k)2088 1537 y FC(6)p FL(=)d(0)p
FI(:)j FL(So)f(after)h(a)g(p)s(ossible)f(renaming)716
1658 y(of)34 b(the)i(indices)f FI(i)g FL(and)g FI(k)s(;)g
FL(w)m(e)i(can)e(assume)g(that)g FI(a)2631 1673 y FG(j)t(k)2739
1658 y FL(=)c(0)p FI(:)k FL(By)h(the)f(same)716 1778
y(reasoning)d(w)m(e)i(tak)m(e)f FI(a)1556 1793 y FG(il)1634
1778 y FL(=)27 b(0)p FI(:)33 b FL(No)m(w)g(w)m(e)h(set)1082
1993 y FI(b)1123 2008 y FG(j)t(i)1212 1993 y FL(:=)27
b FI(b)1383 2008 y FG(k)r(j)1486 1993 y FL(:=)h FI(z)t(;)631
b(b)2365 2008 y FG(ij)2454 1993 y FL(:=)28 b FI(b)2626
2008 y FG(l)q(i)2704 1993 y FL(:=)g FI(b)2876 1939 y
FG(a)2913 1949 y Fo(ij)2876 2019 y FG(ii)2972 1993 y
FI(z)3021 1952 y Fx(\000)p FH(1)3117 1993 y FI(;)1067
2157 y(b)1108 2172 y FG(j)t(k)1212 2157 y FL(:=)f FI(b)1383
2172 y FG(k)r(l)1476 2157 y FL(:=)h FI(z)1656 2116 y
Fx(\000)p FH(1)1751 2157 y FI(;)557 b(b)2376 2172 y FG(il)2454
2157 y FL(:=)28 b FI(b)2626 2172 y FG(l)q(k)2718 2157
y FL(:=)g FI(b)2890 2103 y Fx(\000)p FG(a)2982 2113 y
Fo(ij)2890 2182 y FG(ii)3042 2157 y FI(z)t(:)501 2392
y FL(In)k(all)d(the)i(cases,)i FI(z)g FC(6)p FL(=)27
b(0)k(can)g(b)s(e)h(c)m(hosen)g(freely)g(from)d(the)j(\014eld)f
FF(|)16 b FL(and)32 b(can)f(b)s(e)501 2512 y(di\013eren)m(t)i(for)f(ev)
m(ery)i(class)f(and)g(pair)e(of)h(v)m(ertices.)501 2673
y(W)-8 b(e)28 b(w)m(ould)f(lik)m(e)g(to)g(p)s(oin)m(t)g(out)g(that)g
(all)e(pairs)i(of)g(indices)g(fall)e(in)m(to)i(one)h(of)e(those)501
2793 y(classes)38 b(and)g(that)f(there)h(are)f(no)g(o)m(v)m(erlapping)g
(cases,)i(i.e.)57 b(eac)m(h)38 b(o\013-diagonal)501 2914
y(elemen)m(t)33 b(is)f(only)g(set)h(in)f(one)h(of)f(these.)501
3075 y(In)41 b(this)e(w)m(a)m(y)j(w)m(e)f(ha)m(v)m(e)g(explicitly)d
(constructed)k(the)e(matrix)f Fq(b)h FL(=)g(\()p FI(b)3168
3090 y FG(ij)3229 3075 y FL(\).)66 b(W)-8 b(e)501 3195
y(are)39 b(left)f(to)g(sho)m(w)i(that)e(\(3.19\))g(and)h(\(3.20\))e
(are)i(ful\014lled.)60 b(F)-8 b(or)37 b(the)j(diagonal)501
3316 y(en)m(tries)29 b(this)g(has)g(already)f(b)s(een)i(done.)42
b(F)-8 b(or)28 b(the)h(en)m(tries)g(b)s(eing)f(set)i(in)e(the)h
(\014rst)501 3436 y(three)34 b(classes)f(it)f(is)g(immediately)d(clear)
j(from)g(the)h(de\014nition.)501 3597 y(In)g(the)g(last)f(class)h(only)
f(the)h(relation)d FI(b)1985 3612 y FG(k)r(l)2050 3597
y FI(b)2091 3612 y FG(l)q(k)2184 3597 y FL(=)e FI(b)2329
3551 y FG(a)2366 3563 y Fo(k)q(l)2329 3625 y FG(k)r(k)2461
3597 y FL(m)m(ust)33 b(still)d(b)s(e)j(c)m(hec)m(k)m(ed.)501
3758 y(W)-8 b(e)43 b(note)g(that)g(neither)g FI(i)f FL(and)h
FI(j)49 b FL(nor)43 b FI(k)i FL(and)e FI(l)i FL(can)e(form)e(a)i(comp)s
(onen)m(t)g(of)501 3879 y(t)m(yp)s(e)35 b FI(G)799 3894
y FH(2)839 3879 y FL(,)f(as)h(this)f(w)m(ould)g(con)m(tradict)g(part)g
(1)g(of)g(the)g(assumption.)48 b(F)-8 b(rom)33 b(the)501
3999 y(construction)g(w)m(e)g(get)g FI(b)1407 4014 y
FG(k)r(l)1472 3999 y FI(b)1513 4014 y FG(l)q(k)1605 3999
y FL(=)28 b FI(b)1750 3945 y Fx(\000)p FG(a)1842 3955
y Fo(ij)1750 4025 y FG(ii)1902 3999 y FI(:)k FL(As)h(v)m(ertex)h
FI(i)f FL(is)f(link)-5 b(able)30 b(to)i FI(k)k FL(w)m(e)d(kno)m(w)501
4120 y FI(b)542 4135 y FG(k)r(k)652 4120 y FL(=)27 b
FI(b)796 4078 y Fx(\000)p FH(1)796 4145 y FG(ii)891 4120
y FI(:)33 b FL(W)-8 b(e)32 b(will)f(sho)m(w)i(that)g
FI(a)1807 4135 y FG(k)r(l)1899 4120 y FL(=)28 b FI(a)2054
4135 y FG(ij)2115 4120 y FI(:)501 4281 y FL(If)41 b FI(a)658
4296 y FG(ij)759 4281 y FL(=)g(0)f(then)h(w)m(e)h(immediately)37
b(get)j FI(a)2131 4296 y FG(k)r(l)2237 4281 y FL(=)h(0,)h(b)s(ecause)g
FI(a)2892 4296 y FG(k)r(l)2957 4281 y FI(a)3008 4296
y FG(l)q(k)3113 4281 y FL(=)f(1)f(or)g(2)501 4401 y(is)33
b(not)h(p)s(ermitted)e(b)m(y)j(part)e(2)g(of)g(the)h(assumption.)46
b(Analogously)33 b(w)m(e)h(get)g(the)501 4522 y(result)f(if)e(w)m(e)j
(assume)f FI(a)1394 4537 y FG(k)r(l)1486 4522 y FL(=)28
b(0)p FI(:)501 4683 y FL(The)34 b(case)f(where)h(the)f(4)f(indices)h
(form)e(a)h(sub-diagram)f(of)h(the)h(kind)1744 5388 y
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 @endspecial 1828 5637 a(32)p eop
%%Page: 33 35
33 34 bop 501 573 a FL(is)36 b(excluded,)i(as)e(the)h(cycle)g(gen)m(us)
g(for)f(this)f(diagram)f(is)i(1)d(=)h(2)2922 537 y FH(1)2986
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2508 y FG(c)3129 2493 y FL(and)i(part)898 2613 y(2)j(of)f(this)g(Lemma)
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FI(:)39 b FL(W)-8 b(e)39 b(pic)m(k)h(in)e FI(c)2242 2869
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3060 y FH(2)3408 3037 y Fo(w)3452 3052 y Fy(2)3203 3137
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b(the)h(argumen)m(t)g(for)f(all)f(the)i(other)g(cycles)h(with)f(cycle)
898 3954 y(gen)m(us)28 b(bigger)e(than)h(2,)h(w)m(e)f(conclude)g(that)g
(for)f(ev)m(ery)j(Lev)m(el)e(0)f(v)m(ertex)898 4074 y
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 @endspecial 257 5069 a(then)41 b(Lemma)e(4.1)g(already)h(giv)m(es)g
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b(to)i FE(at)h(most)257 5189 y FL(one)c(other)f(v)m(ertex.)55
b(One)36 b(just)g(tak)m(es)i FI(i)33 b FL(=)g FI(j)42
b FL(and)36 b(\014nds)h FI(a)2438 5204 y FG(k)r(l)2536
5189 y FL(=)c(2)j(or)f FI(a)2903 5204 y FG(k)r(l)3002
5189 y FL(=)e FI(a)3162 5204 y FG(l)q(k)3260 5189 y FL(=)g
FC(\000)p FL(1)257 5310 y(and)38 b(the)g(order)g(of)f
FI(b)1042 5325 y FG(ii)1132 5310 y FL(is)g(3.)58 b(If)37
b(the)h(order)g(of)f FI(b)2061 5325 y FG(ii)2151 5310
y FL(is)g(t)m(w)m(o,)j(then)e(one)g(could)f(consider)1828
5637 y(39)p eop
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%%Page: 50 52
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%%Page: 51 53
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3028 y FH(2)1767 3013 y FI(;)17 b(A)1884 3028 y FH(2)1946
3013 y FC([)22 b FI(B)2108 3028 y FH(2)2148 3013 y FI(;)17
b(A)2265 3028 y FH(2)2326 3013 y FC([)23 b FI(G)2492
3028 y FH(2)2531 3013 y FI(;)17 b(A)2648 3028 y FH(2)2710
3013 y FC([)22 b FI(A)2871 3028 y FH(1)2933 3013 y FC([)g
FI(A)3094 3028 y FH(1)3134 3013 y FI(:)501 3218 y FL(W)-8
b(e)41 b(\014x)g(the)g(en)m(umeration)e(of)h(the)h(v)m(ertices)h(so)e
(that)g(it)g(follo)m(ws)f(the)i(w)m(a)m(y)g(w)m(e)501
3338 y(wrote)31 b(the)f(diagrams)e(in)m(tuitiv)m(ely)-8
b(.)41 b(With)30 b(this)g(w)m(e)h(mean)e(that)h(the)h(arro)m(ws)f(are)
501 3459 y(dra)m(wn)35 b(in)e(the)i(directions)e(giv)m(en)h(b)m(y)h
(Figure)e(2.1,)i(and)f(the)g(order)g(of)g(the)g(con-)501
3579 y(nected)f(comp)s(onen)m(ts)e(is)g(the)g(same)g(as)g(in)f(the)i
(written)f(notation.)41 b(The)32 b(double)501 3700 y(and)f(triple)f
(edges)i(are)f(alw)m(a)m(ys)g(last)f(in)h(eac)m(h)g(comp)s(onen)m(t.)43
b(As)32 b(an)f(example)f(w)m(e)501 3820 y(mak)m(e)i(this)g(precise)h
(for)f(the)g(diagram)e FI(C)2030 3835 y FH(3)2090 3820
y FC([)22 b FI(A)2251 3835 y FH(1)2290 3820 y FL(.)44
b(The)33 b(\014rst)f(t)m(w)m(o)h(v)m(ertices)g(form)501
3940 y(an)41 b FI(A)718 3955 y FH(2)799 3940 y FL(sub)s(diagram.)67
b(The)42 b(arro)m(w)f(p)s(oin)m(ts)g(from)f(v)m(ertex)j(3)e(to)f(v)m
(ertex)j(2)e(and)501 4061 y(v)m(ertex)36 b(4)e(corresp)s(onds)i(to)e
(the)g FI(A)1781 4076 y FH(1)1855 4061 y FL(comp)s(onen)m(t.)49
b(The)35 b(c)m(haracteristic)f(feature)501 4181 y(of)k(this)g(class)g
(of)g(diagrams)f(is)g(that)h(the)h(\014rst)g(t)m(w)m(o)g(v)m(ertices)g
(alw)m(a)m(ys)g(form)e(an)501 4301 y FI(A)574 4316 y
FH(2)646 4301 y FL(sub)s(diagram)31 b(and)i FI(a)1416
4316 y FH(24)1519 4301 y FL(=)27 b(0.)501 4461 y(No)m(w,)46
b(assume)e(that)e(the)h(diagrams)e(are)i(realizable)e(o)m(v)m(er)j
FI(\000)s(;)f FL(i.e.)74 b(there)43 b(are)501 4582 y
FI(g)548 4597 y FH(1)587 4582 y FI(;)17 b(:)g(:)g(:)f(g)809
4597 y FH(4)876 4582 y FC(2)28 b FI(\000)46 b FL(and)31
b(c)m(haracters)i FI(\037)1788 4597 y FH(1)1828 4582
y FI(;)17 b(:)g(:)g(:)f(\037)2064 4597 y FH(4)2131 4582
y FC(2)2246 4556 y FL(^)2225 4582 y FI(\000)46 b FL(so)31
b(that)h(the)g(braiding)e(matrix)501 4702 y(\()p FI(\037)600
4717 y FG(j)637 4702 y FL(\()p FI(g)722 4717 y FG(i)750
4702 y FL(\)\))826 4717 y FG(ij)923 4702 y FL(is)36 b(of)h(the)g
(corresp)s(onding)g(\014nite)f(Cartan)h(t)m(yp)s(e.)58
b(W)-8 b(e)37 b(assume)g(that)501 4822 y FI(g)548 4837
y FH(1)627 4822 y FL(and)i FI(g)870 4837 y FH(2)949 4822
y FL(generate)h FI(\000)14 b FL(.)64 b(Then)40 b FI(g)1819
4837 y FH(3)1898 4822 y FL(=)f FI(g)2064 4786 y FG(n)2060
4847 y FH(1)2110 4822 y FI(g)2161 4786 y FG(m)2157 4847
y FH(2)2267 4822 y FL(and)g FI(g)2510 4837 y FH(4)2589
4822 y FL(=)g FI(g)2755 4786 y FG(k)2751 4847 y FH(1)2797
4822 y FI(g)2848 4786 y FG(l)2844 4847 y FH(2)2922 4822
y FL(for)g(some)h(0)f FC(\024)501 4943 y FI(n;)17 b(m;)g(k)s(;)g(l)30
b(<)d(p:)c FL(W)-8 b(e)22 b(set)h FI(q)32 b FL(:=)27
b FI(\037)1656 4958 y FH(1)1696 4943 y FL(\()p FI(g)1781
4958 y FH(1)1820 4943 y FL(\))22 b(and)g(kno)m(w)i(that)e
FI(q)2552 4907 y FG(p)2619 4943 y FL(=)27 b(1)p FI(:)22
b FL(Hence,)k FI(\037)3190 4958 y FH(1)3230 4943 y FL(\()p
FI(g)3315 4958 y FH(2)3354 4943 y FL(\))h(=)501 5063
y FI(q)548 5027 y FG(t)611 5063 y FL(for)33 b(some)g(0)28
b FC(\024)i FI(t)f(<)f(p:)33 b FL(W)-8 b(e)34 b(can)g(no)m(w)f(express)
j FI(\037)2421 5078 y FH(1)2460 5063 y FL(\()p FI(g)2545
5078 y FH(3)2585 5063 y FL(\))d(and)g FI(\037)2907 5078
y FH(1)2947 5063 y FL(\()p FI(g)3032 5078 y FH(4)3071
5063 y FL(\))g(in)f(terms)501 5183 y(of)g FI(q)t(;)17
b(t;)g(m;)g(n;)g(k)35 b FL(and)e FI(l)r FL(.)44 b(Using)32
b(the)h(Cartan)g(condition)e(\(3.3\))1274 5388 y FI(\037)1335
5403 y FG(i)1363 5388 y FL(\()p FI(g)1448 5403 y FG(j)1484
5388 y FL(\))p FI(\037)1583 5403 y FG(j)1620 5388 y FL(\()p
FI(g)1705 5403 y FG(i)1733 5388 y FL(\))c(=)h FI(\037)1963
5403 y FG(i)1991 5388 y FL(\()p FI(g)2076 5403 y FG(i)2104
5388 y FL(\))2142 5347 y FG(a)2179 5357 y Fo(ij)2266
5388 y FL(=)g FI(\037)2431 5403 y FG(j)2467 5388 y FL(\()p
FI(g)2552 5403 y FG(j)2588 5388 y FL(\))2626 5347 y FG(a)2663
5357 y Fo(j)s(i)3295 5388 y FL(\(5.1\))1828 5637 y(51)p
eop
%%Page: 52 54
52 53 bop 501 573 a FL(w)m(e)44 b(can)f(also)f(write)h(do)m(wn)h
FI(\037)1639 588 y FH(2)1679 573 y FL(\()p FI(g)1764
588 y FH(1)1803 573 y FL(\))p FI(;)f(\037)1972 588 y
FH(3)2011 573 y FL(\()p FI(g)2096 588 y FH(1)2135 573
y FL(\))g(and)g FI(\037)2477 588 y FH(4)2517 573 y FL(\()p
FI(g)2602 588 y FH(1)2641 573 y FL(\))g(explicitly)-8
b(.)72 b(As)44 b(the)501 693 y(\014rst)35 b(t)m(w)m(o)f(v)m(ertices)h
(form)e(an)h FI(A)1685 708 y FH(2)1758 693 y FL(diagram)e(and)i(the)g
(third)g(and)g(fourth)f(v)m(ertex)501 814 y(are)e(not)g(connected)h(to)
f(the)g(\014rst)g(v)m(ertex)i(directly)d(in)g(all)f(the)i(diagrams)e
(of)h(our)501 934 y(class,)44 b(all)c(the)h(braiding)f(matrix)g(en)m
(tries)i(determined)f(so)h(far)f(are)g(the)h(same)501
1054 y(for)36 b(all)e(diagrams.)52 b FI(\037)1320 1069
y FH(2)1359 1054 y FL(\()p FI(g)1444 1069 y FH(2)1483
1054 y FL(\))34 b(=)f FI(q)40 b FL(b)s(ecause)d(of)e(\(5.1\),)i(and)f
(this)g(lets)f(us)i(express)501 1175 y FI(\037)562 1190
y FH(2)602 1175 y FL(\()p FI(g)687 1190 y FH(3)726 1175
y FL(\))g(and)h FI(\037)1057 1190 y FH(2)1096 1175 y
FL(\()p FI(g)1181 1190 y FH(4)1221 1175 y FL(\).)58 b(Using)37
b(\(5.1\))g(again,)g(w)m(e)i(no)m(w)f(\014nd)g FI(\037)2772
1190 y FH(3)2811 1175 y FL(\()p FI(g)2896 1190 y FH(2)2936
1175 y FL(\))f(and)h FI(\037)3267 1190 y FH(4)3306 1175
y FL(\()p FI(g)3391 1190 y FH(2)3430 1175 y FL(\).)501
1295 y(With)33 b(those)h(v)-5 b(alues,)33 b(all)e(remaining)g(en)m
(tries)i(of)g(the)h(braiding)d(matrix)h(can)h(b)s(e)501
1416 y(calculated.)43 b(As)33 b(a)f(\014nal)g(result)h(w)m(e)g(get)g
(\()p FI(b)2099 1431 y FG(ij)2160 1416 y FL(\))27 b(=)h(\()p
FI(\037)2428 1431 y FG(j)2464 1416 y FL(\()p FI(g)2549
1431 y FG(i)2577 1416 y FL(\)\))g(=)572 1543 y Fw(0)572
1718 y(B)572 1778 y(B)572 1842 y(@)754 1614 y FI(q)265
b(q)1109 1578 y Fx(\000)p FG(t)p Fx(\000)p FH(1)1781
1614 y FI(q)1828 1578 y Fx(\000)p FG(n)p Fx(\000)p FG(tm)2766
1614 y FI(q)2813 1578 y Fx(\000)p FG(k)r Fx(\000)p FG(tl)739
1734 y FI(q)786 1698 y FG(t)1150 1734 y FI(q)488 b(q)1728
1698 y FG(tn)p FH(+)p FG(n)p Fx(\000)p FG(m)p FH(+)p
FG(a)2103 1707 y Fy(23)2747 1734 y FI(q)2794 1698 y FG(tk)r
FH(+)p FG(k)r Fx(\000)p FG(l)659 1862 y FI(q)706 1826
y FG(n)p FH(+)p FG(tm)979 1862 y FI(q)1026 1826 y Fx(\000)p
FG(tn)p Fx(\000)p FG(n)p FH(+)p FG(m)1569 1862 y FI(q)1616
1826 y Fx(\000)p FG(n)1714 1803 y Fy(2)1748 1826 y FH(+)p
FG(nm)p Fx(\000)p FG(m)2025 1803 y Fy(2)2060 1826 y FH(+)p
FG(a)2152 1835 y Fy(23)2218 1826 y FG(m)2485 1862 y FI(q)2532
1826 y Fx(\000)p FG(k)r(n)p Fx(\000)p FG(tl)q(n)p FH(+)p
FG(tk)r(m)p FH(+)p FG(k)r(m)p Fx(\000)p FG(l)q(m)681
1990 y FI(q)728 1954 y FG(k)r FH(+)p FG(tl)1003 1990
y FI(q)1050 1954 y Fx(\000)p FG(tk)r Fx(\000)p FG(k)r
Fx(\000)p FG(l)1451 1990 y FI(q)1498 1954 y Fx(\000)p
FG(nk)r Fx(\000)p FG(tmk)r FH(+)p FG(tnl)q FH(+)p FG(nl)q
Fx(\000)p FG(ml)q FH(+)p FG(a)2312 1963 y Fy(23)2376
1954 y FG(l)2687 1990 y FI(q)2734 1954 y Fx(\000)p FG(k)2828
1930 y Fy(2)2862 1954 y FH(+)p FG(k)r(l)q Fx(\000)p FG(l)3055
1930 y Fy(2)3294 1543 y Fw(1)3294 1718 y(C)3294 1778
y(C)3294 1842 y(A)3398 1803 y FI(:)501 2182 y FL(W)-8
b(e)42 b(note)g(that)f(the)h(parameter)f FI(t)g FL(is)g(cancelled)g(in)
g(the)h(expressions)h(for)e(the)501 2302 y(diagonal)28
b(elemen)m(ts.)43 b(Some)29 b(of)h(the)g(en)m(tries)h(can)f(b)s(e)h
(calculated)e(using)g(\(5.1\))h(in)501 2423 y(a)24 b(di\013eren)m(t)g
(w)m(a)m(y)g(whic)m(h)h(leads)e(to)h(three)g(imp)s(ortan)m(t)e
(compatibilit)m(y)e(conditions.)501 2581 y(First,)39
b FI(\037)834 2596 y FH(3)874 2581 y FL(\()p FI(g)959
2596 y FH(3)998 2581 y FL(\))f(can)h(b)s(e)f(determined)h(from)e
FI(\037)2210 2596 y FH(2)2249 2581 y FL(\()p FI(g)2334
2596 y FH(2)2374 2581 y FL(\))h(directly)g(b)m(y)h(the)g(last)e(part)
501 2702 y(of)46 b(\(5.1\).)85 b(W)-8 b(e)47 b(\014nd)g
FI(\037)1391 2717 y FH(3)1431 2702 y FL(\()p FI(g)1516
2717 y FH(3)1555 2702 y FL(\))k(=)g FI(q)1818 2666 y
FG(x)1908 2702 y FL(where)d FI(x)k FL(:=)f FI(a)2516
2717 y FH(23)2591 2702 y FI(=a)2691 2717 y FH(32)2812
2702 y FL(unless)d FI(a)3166 2717 y FH(32)3292 2702 y
FL(=)j(0)p FI(;)501 2822 y FL(as)42 b(is)e(the)i(case)g(for)f(the)g
(last)g(four)f(diagrams)g(of)g(our)h(class.)70 b(Then)42
b FI(x)g FL(is)e(not)501 2943 y(further)c(determined.)50
b(Comparing)34 b(this)h(with)f(the)i(expression)g(found)f(so)h(far,)501
3063 y(w)m(e)27 b(get)f(the)g(condition)e(\(5.2\).)40
b(Next,)28 b(w)m(e)f(can)f(determine)f FI(\037)2722 3078
y FH(4)2762 3063 y FL(\()p FI(g)2847 3078 y FH(4)2886
3063 y FL(\))g(directly)g(from)501 3183 y FI(\037)562
3198 y FH(3)602 3183 y FL(\()p FI(g)687 3198 y FH(3)726
3183 y FL(\),)37 b(when)g(v)m(ertices)g(3)f(and)g(4)g(are)g(connected.)
55 b(In)37 b(this)e(case)i FI(\037)3064 3198 y FH(4)3104
3183 y FL(\()p FI(g)3189 3198 y FH(4)3228 3183 y FL(\))c(=)h
FI(q)3456 3147 y FG(z)501 3304 y FL(with)23 b FI(z)33
b FL(:=)27 b FI(xa)1028 3319 y FH(34)1104 3304 y FI(=a)1204
3319 y FH(43)1278 3304 y FI(:)d FL(When)g FI(a)1655 3319
y FH(43)1758 3304 y FL(=)j(0)d(w)m(e)g(again)e(lea)m(v)m(e)i
FI(z)k FL(not)23 b(further)h(sp)s(eci\014ed.)501 3424
y(This)38 b(time)f(the)h(compatibilit)m(y)c(b)s(et)m(w)m(een)40
b(the)f(di\013eren)m(t)f(w)m(a)m(ys)h(of)f(calculating)501
3544 y(the)24 b(matrix)f(elemen)m(ts)h(leads)g(us)g(to)f(\(5.3\).)41
b(Finally)-8 b(,)22 b(w)m(e)j(can)f(apply)g(\(5.1\))f(to)h(\014nd)501
3665 y(that)33 b FI(\037)774 3680 y FH(3)814 3665 y FL(\()p
FI(g)899 3680 y FH(3)938 3665 y FL(\))p FI(\037)1037
3680 y FH(4)1077 3665 y FL(\()p FI(g)1162 3680 y FH(4)1201
3665 y FL(\))c(=)g FI(q)1420 3629 y FG(xa)1497 3638 y
Fy(34)1566 3665 y FL(.)46 b(Comparing)32 b(this)h(to)g(the)h
(expressions)h(found)f(so)501 3785 y(far)e(giv)m(es)h(us)g(\(5.4\).)43
b(These)35 b(are)d FE(al)5 b(l)43 b FL(relations.)501
3944 y(W)-8 b(e)35 b(list)f(the)h(p)s(ossible)f(v)-5
b(alues)35 b(for)f(the)h(ab)s(o)m(v)m(e)h(parameters)e(for)h(eac)m(h)g
(diagram)501 4064 y(explicitly)d(in)g(T)-8 b(able)33
b(5.1.)45 b(Let)33 b(us)h(no)m(w)g(lo)s(ok)e(at)h(the)h(system)g(of)f
(equations)g(w)m(e)501 4185 y(ha)m(v)m(e)h(found.)1208
4383 y FC(\000)p FI(n)1343 4342 y FH(2)1405 4383 y FL(+)22
b FI(nm)h FC(\000)f FI(m)1853 4342 y FH(2)1915 4383 y
FL(+)g FI(a)2064 4398 y FH(23)2139 4383 y FI(m)28 b FL(=)g
FI(x)197 b FL(mo)s(d)33 b FI(p)417 b FL(\(5.2\))1656
4529 y FC(\000)p FI(k)1787 4487 y FH(2)1849 4529 y FL(+)22
b FI(k)s(l)i FC(\000)f FI(l)2185 4487 y FH(2)2252 4529
y FL(=)28 b FI(z)202 b FL(mo)s(d)32 b FI(p)423 b FL(\(5.3\))993
4674 y FI(k)s FL(\()p FI(m)23 b FC(\000)f FL(2)p FI(n)p
FL(\))g(+)g FI(l)r FL(\()p FI(n)h FC(\000)f FL(2)p FI(m)h
FL(+)f FI(a)2112 4689 y FH(23)2187 4674 y FL(\))27 b(=)h
FI(xa)2462 4689 y FH(34)2734 4674 y FL(mo)s(d)k FI(p)292
b FL(\(5.4\))501 4872 y(F)-8 b(rom)31 b(no)m(w)j(on)e(all)e
(calculations)h(will)f(b)s(e)j(made)f(mo)s(dulo)f FI(p)p
FL(.)501 5031 y(A)f(linear)d(transformation)g(helps)j(to)e(bring)h(the)
g(ab)s(o)m(v)m(e)h(equations)g(in)m(to)e(a)h(nicer)501
5152 y(form.)43 b(W)-8 b(e)32 b(set)623 5370 y FI(a)c
FL(:=)842 5302 y FI(m)23 b FL(+)f FI(n)g FC(\000)h FI(a)1279
5317 y FH(23)p 842 5347 512 4 v 1074 5438 a FL(2)1364
5370 y FI(;)212 b(b)28 b FL(:=)1812 5302 y(3)p FI(m)23
b FC(\000)f FL(3)p FI(n)g FC(\000)h FI(a)2348 5317 y
FH(23)p 1812 5347 611 4 v 2093 5438 a FL(2)2530 5370
y(and)98 b FI(y)31 b FL(:=)d(3)p FI(x)22 b FC(\000)h
FI(a)3272 5328 y FH(2)3272 5394 y(23)3346 5370 y FI(:)1828
5637 y FL(52)p eop
%%Page: 53 55
53 54 bop 501 573 a FL(Then)34 b(\(5.2\)-\(5.4\))d(can)i(b)s(e)g
(rewritten)f(as)1926 793 y(3)p FI(a)2026 752 y FH(2)2087
793 y FL(+)22 b FI(b)2226 752 y FH(2)2289 793 y FL(+)g
FI(y)30 b FL(=)e(0)p FI(;)650 b FL(\(5.5\))1776 938 y
FI(k)1830 897 y FH(2)1892 938 y FC(\000)23 b FI(k)s(l)h
FL(+)e FI(l)2228 897 y FH(2)2290 938 y FL(+)g FI(z)32
b FL(=)c(0)p FI(;)650 b FL(\(5.6\))1352 1083 y FI(k)s
FL(\()p FI(a)22 b FC(\000)h FI(b)p FL(\))f(+)g FI(l)r
FL(\()p FI(a)h FL(+)f FI(b)p FL(\))g(+)g FI(xa)2362 1098
y FH(34)2465 1083 y FL(=)28 b(0)p FI(:)650 b FL(\(5.7\))501
1303 y(The)44 b(\014rst)f(thing)f(to)g(notice)h(is)f(that)g(\(5.5\))h
(alw)m(a)m(ys)g(has)g(a)f(solution)3123 1267 y FH(1)3204
1303 y FL(.)74 b(This)501 1424 y(can)38 b(easily)e(b)s(e)h(seen)i(b)m
(y)f(the)f(follo)m(wing)d(argumen)m(t.)57 b(As)38 b FI(a)f
FL(runs)h(through)f(the)501 1544 y(n)m(um)m(b)s(ers)32
b(0)e(till)d FI(p)18 b FC(\000)g FL(1)p FI(;)31 b FL(the)g(expression)g
(3)p FI(a)2123 1508 y FH(2)2193 1544 y FL(tak)m(es)2451
1500 y FG(p)p FH(+1)p 2451 1521 126 4 v 2496 1579 a(2)2617
1544 y FL(di\013eren)m(t)g(v)-5 b(alues.)42 b(The)501
1673 y(expression)47 b FC(\000)p FI(b)1102 1637 y FH(2)1174
1673 y FC(\000)32 b FI(y)48 b FL(has)f(also)1785 1630
y FG(p)p FH(+1)p 1785 1651 V 1831 1708 a(2)1967 1673
y FL(di\013eren)m(t)f(v)-5 b(alues)46 b(for)f(all)f(the)i(p)s(ossible)
501 1794 y(v)-5 b(alues)30 b(for)g FI(b)p FL(.)43 b(Assume)32
b(that)e(all)e(these)k(v)-5 b(alues)30 b(are)g(di\013eren)m(t.)43
b(This)31 b(means)f(w)m(e)501 1914 y(ha)m(v)m(e)40 b(altogether)e(2)
1258 1870 y FG(p)p FH(+1)p 1258 1891 V 1303 1949 a(2)1432
1914 y FL(=)g FI(p)26 b FL(+)h(1)38 b(v)-5 b(alues.)62
b(But)39 b(mo)s(dulo)e FI(p)i FL(there)g(are)g(only)f
FI(p)501 2035 y FL(n)m(um)m(b)s(ers.)44 b(Hence)31 b(there)g(is)e(at)h
(least)g(one)g(solution.)41 b(Actually)-8 b(,)30 b(there)h(are)f(ev)m
(en)501 2155 y(6)p FI(d)f FL(solutions)f(for)h(ev)m(ery)i(prime)d
FI(p)h FL(of)g(the)h(form)e FI(p)g FL(=)f(6)p FI(d)15
b FC(\006)g FL(1.)43 b(This)30 b(can)f(b)s(e)h(seen)501
2275 y(b)m(y)k(considering)e(the)h(homogeneous)f(equation)1598
2495 y(3)q(~)-50 b FI(a)1698 2454 y FH(2)1760 2495 y
FL(+)1854 2469 y(~)1858 2495 y FI(b)1899 2454 y FH(2)1961
2495 y FL(+)22 b FI(y)t(c)2153 2454 y FH(2)2219 2495
y FL(=)28 b(0)p FI(;)896 b FL(\(5.8\))501 2715 y(whic)m(h)32
b(w)m(e)g(get)e(b)m(y)i(m)m(ultiplying)c(\(5.5\))i(b)m(y)i
FI(c)2141 2679 y FH(2)2211 2715 y FL(and)f(setting)h(~)-50
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b(suc)m(h)i(equations)f(there)g(is)f(a)h(nice)f(theory)-8
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2956 y(solutions)37 b(from)g([BS)q(,)i(Theorem)g(2,)g(p.25].)61
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(trivial)d(one\))j(with)f FI(c)e FL(=)f(0.)50 b(This)35
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(the)h(original)c(problem)i(\(5.5\),)h(and)h(all)d(other)i(solutions)g
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FI(p)g FL(of)h(the)g(form)e FI(p)c FL(=)h(6)p FI(d)22
b FL(+)g(1)p FI(:)501 4585 y FL(As)28 b FI(p)g FC(\025)g
FL(5)f(w)m(e)i(kno)m(w)f(that)f(\(5.5\))g(has)h(at)f(least)g(6)g
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y(alw)m(a)m(ys)j(pic)m(k)f(a)g(solution)e(with)h FI(a)e
FC(6)p FL(=)g FI(b)p FL(,)k(as)f(this)f(corresp)s(onds)i(to)f(2)g
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h(\(5.7\))e(for)h FI(k)j FL(and)d(substitute)g(the)h(result)e(in)h
(\(5.6\).)p 257 4894 1296 4 v 370 4955 a FB(1)407 4985
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h(tips)257 5085 y(concerning)e(this)h(equation.)1828
5637 y FL(53)p eop
%%Page: 54 56
54 55 bop 640 557 a FL(Diagram)p 1152 594 4 121 v 193
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557 y FI(x)194 b(y)202 b(z)p 2348 594 V 182 w(D)30 b
FL(=)e(4)p FI(y)t(z)e FC(\000)c FL(3)p FI(x)3016 521
y FH(2)3056 557 y FI(a)3107 521 y FH(2)3107 582 y(34)p
495 597 2764 4 v 768 681 a FI(A)841 696 y FH(4)p 1152
717 4 121 v 1203 681 a FC(\000)p FL(1)100 b FC(\000)p
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FL(3)k(=)h(5)767 802 y FI(B)841 817 y FH(4)p 1152 838
V 1203 802 a FC(\000)p FL(1)100 b FC(\000)p FL(1)149
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2348 1078 V 288 w(16)21 b FC(\000)i FL(12)k(=)h(2)3033
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b FL(3)p FI(x)2831 1608 y FH(2)2899 1644 y FL(=)27 b(3)p
FI(x)3106 1608 y FH(2)654 1765 y FI(A)727 1780 y FH(2)789
1765 y FC([)22 b FI(G)954 1780 y FH(2)p 1152 1801 V 1242
1765 a FL(0)138 b FC(\000)p FL(1)145 b FI(x)167 b FL(3)p
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1728 y FH(2)2923 1765 y FL(=)k FI(x)3081 1728 y FH(2)545
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1152 1921 V 1242 1885 a FL(0)177 b(0)183 b FI(x)167 b
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495 1924 2764 4 v 948 2193 a FL(T)-8 b(able)32 b(5.1:)43
b(P)m(arameters)33 b(for)f(Dynkin)h(diagrams)501 2478
y(After)g(some)f(reordering)g(w)m(e)i(get)803 2609 y
Fw(")861 2639 y(\022)946 2712 y FI(a)22 b FL(+)g FI(b)p
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y Fw(\023)1243 2661 y FH(2)1304 2780 y FL(+)1413 2712
y FI(a)f FL(+)g FI(b)p 1412 2757 V 1412 2848 a(a)g FC(\000)h
FI(b)1659 2780 y FL(+)f(1)1806 2609 y Fw(#)1880 2780
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FL(Multiplying)34 b(b)m(y)j(the)f(common)f(denominator,)g(simplifying)e
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3759 y FI(y)t(l)1122 3718 y FH(2)1183 3759 y FC(\000)22
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FL(2)p FI(y)1200 4127 y Fw(\023)1273 4149 y FH(2)1335
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b(our)d(system)i(of)e(equations)h(has)g(a)f(solution)f(mo)s(dulo)g
FI(p)h FL(if)g(and)g(only)g(if)501 4660 y FI(D)f FL(:=)d(4)p
FI(y)t(z)c FC(\000)e FL(3)p FI(x)1125 4624 y FH(2)1164
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4780 y(eters)k(for)f(our)g(class)g(of)g(Dynkin)g(diagrams)e(are)i(giv)m
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4900 y(cases)j(where)g FI(a)1095 4915 y FH(23)1211 4900
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FL(=)h(0)f(w)m(e)i(can)f(freely)f(c)m(ho)s(ose)i(the)e(v)-5
b(alues)41 b(of)f FI(x)501 5021 y FL(or)35 b FI(z)k FL(resp)s(ectiv)m
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5141 y(diagrams)d(except)k(for)d FI(A)1448 5156 y FH(4)1521
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b(these)j(w)m(e)f(need)g(5)e(or)h(3)f(resp)s(ec-)501
5262 y(tiv)m(ely)e(to)g(b)s(e)h(a)f(square)i(mo)s(dulo)c
FI(p)p FL(.)43 b(According)30 b(to)g(the)h(quadratic)f(recipro)s(cit)m
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FL(=)h(12)p FI(d)16 b FC(\006)i FL(1)p FI(;)29 b FL(resp)s(ectiv)m(ely)
-8 b(.)44 b(After)1828 5637 y(54)p eop
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55 56 bop 501 573 a FL(c)m(ho)s(osing)35 b(a)h(solution)e(of)h(\(5.5\))
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976 y(calculation)f(are)j(to)s(o)e(sp)s(ecial)h(and)h(other)f(c)m
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1096 y(realize)32 b FI(A)878 1111 y FH(4)950 1096 y FL(and)g
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b(let)f(us)h(consider)g(this)f(more)g(closely)-8 b(.)501
1258 y(First)30 b(w)m(e)h(see)h(that)e(w)m(e)h(can)g(not)f(ha)m(v)m(e)i
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2623 y(Then)25 b FI(g)794 2638 y FH(2)861 2623 y FL(=)i
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FI(g)1539 3000 y FH(4)1602 2985 y FL(generate)g FI(\000)14
b FL(.)40 b(In)24 b(this)f(case)i(w)m(e)f(simply)f(en)m(umerate)501
3105 y(the)39 b(v)m(ertices)h(of)e FI(A)1224 3120 y FH(4)1302
3105 y FL(from)g(the)h(other)g(end)g(and)g(get)f(bac)m(k)i(the)f
(original)c(case,)501 3225 y(b)s(ecause)49 b FI(A)950
3240 y FH(4)1038 3225 y FL(is)e(symmetric.)89 b(But)47
b(if)g FI(g)2070 3240 y FH(4)2163 3225 y FL(=)53 b FI(g)2343
3189 y FG(b)2339 3250 y FH(3)2426 3225 y FL(for)48 b(some)f
FI(b)54 b FC(2)g FF(Z)p FI(=)p FL(\()p FI(p)p FL(\),)49
b(w)m(e)501 3346 y(argue)44 b(in)f(the)h(follo)m(wing)d(w)m(a)m(y)-8
b(.)78 b(Set)44 b FI(q)50 b FL(:=)d FI(\037)2249 3361
y FH(1)2288 3346 y FL(\()p FI(g)2373 3361 y FH(1)2412
3346 y FL(\))p FI(:)d FL(Then)h FI(\037)2848 3361 y FH(1)2887
3346 y FL(\()p FI(g)2972 3361 y FH(3)3011 3346 y FL(\))i(=)f
FI(q)3265 3310 y FG(t)3338 3346 y FL(and)501 3466 y FI(\037)562
3481 y FH(1)602 3466 y FL(\()p FI(g)687 3481 y FH(4)726
3466 y FL(\))27 b(=)h FI(q)942 3430 y FG(tb)1031 3466
y FL(for)g(some)h FI(t)f FC(2)g FF(Z)p FI(=)p FL(\()p
FI(p)p FL(\))p FI(:)e FL(Using)j(\(5.1\))g(w)m(e)h(kno)m(w)g(that)f
FI(\037)3031 3481 y FH(3)3071 3466 y FL(\()p FI(g)3156
3481 y FH(1)3195 3466 y FL(\))e(=)h FI(q)3411 3430 y
Fx(\000)p FG(t)501 3587 y FL(and)37 b FI(\037)756 3602
y FH(4)795 3587 y FL(\()p FI(g)880 3602 y FH(1)919 3587
y FL(\))e(=)f FI(q)1149 3550 y Fx(\000)p FG(tb)1300 3587
y FL(and)i(hence)i FI(\037)1829 3602 y FH(3)1869 3587
y FL(\()p FI(g)1954 3602 y FH(2)1993 3587 y FL(\))c(=)g
FI(q)2222 3550 y Fx(\000)p FG(at)2380 3587 y FL(and)j
FI(\037)2635 3602 y FH(4)2674 3587 y FL(\()p FI(g)2759
3602 y FH(2)2798 3587 y FL(\))e(=)f FI(q)3028 3550 y
Fx(\000)p FG(atb)3180 3587 y FI(:)i FL(Again)501 3707
y(w)m(e)46 b(use)f(\(5.1\))f(to)h(calculate)e FI(\037)1693
3722 y FH(2)1733 3707 y FL(\()p FI(g)1818 3722 y FH(3)1857
3707 y FL(\))48 b(=)g FI(q)2114 3671 y FG(at)p Fx(\000)p
FH(1)2315 3707 y FL(and)d FI(\037)2578 3722 y FH(2)2618
3707 y FL(\()p FI(g)2703 3722 y FH(4)2742 3707 y FL(\))j(=)g
FI(q)2999 3671 y FG(atb)3096 3707 y FL(.)79 b(But)45
b(as)501 3827 y FI(g)548 3842 y FH(4)623 3827 y FL(=)36
b FI(g)786 3791 y FG(b)782 3852 y FH(3)858 3827 y FL(w)m(e)i(see)h
(that)e FI(b)h FL(m)m(ust)f(b)s(e)g(zero.)58 b(On)38
b(the)f(other)h(hand,)h(w)m(e)f(ha)m(v)m(e)h(the)501
3948 y(condition)31 b FI(b)970 3912 y FH(2)1031 3948
y FL(+)21 b FI(b)h FL(+)f(1)27 b(=)h(0)j(b)s(ecause)j(v)m(ertices)f(3)f
(and)g(4)g(form)f(an)h FI(A)3050 3963 y FH(2)3121 3948
y FL(diagram,)501 4068 y(cf.)64 b([AS2,)41 b(page)e(25].)63
b(This)39 b(is)g(a)g(con)m(tradiction.)62 b(So)39 b(w)m(e)h(ha)m(v)m(e)
h(fully)d(pro)m(v)m(ed)501 4188 y(that)33 b FI(A)786
4203 y FH(4)858 4188 y FL(can)g(b)s(e)f(realized)g(o)m(v)m(er)i
FI(\000)46 b FL(only)32 b(if)g(5)g(is)g(a)g(square)i(mo)s(dulo)d
FI(p)p FL(.)501 4350 y FJ(P)m(art)37 b(\(b\))256 b FL(W)-8
b(e)33 b(consider)g(most)f(of)g(the)h(remaining)d(4-v)m(ertex)k
(diagrams:)501 4570 y FI(B)575 4585 y FH(2)628 4570 y
FC([)13 b FI(B)781 4585 y FH(2)821 4570 y FI(;)k(B)939
4585 y FH(2)992 4570 y FC([)c FI(G)1148 4585 y FH(2)1188
4570 y FI(;)k(G)1309 4585 y FH(2)1361 4570 y FC([)c FI(G)1517
4585 y FH(2)1557 4570 y FI(;)k(B)1675 4585 y FH(2)1727
4570 y FC([)c FI(A)1879 4585 y FH(1)1933 4570 y FC([)g
FI(A)2085 4585 y FH(1)2125 4570 y FI(;)k(G)2246 4585
y FH(2)2298 4570 y FC([)c FI(A)2450 4585 y FH(1)2504
4570 y FC([)g FI(A)2656 4585 y FH(1)2696 4570 y FI(;)k(A)2813
4585 y FH(1)2865 4570 y FC([)c FI(A)3017 4585 y FH(1)3071
4570 y FC([)g FI(A)3223 4585 y FH(1)3276 4570 y FC([)g
FI(A)3428 4585 y FH(1)3468 4570 y FI(:)501 4790 y FL(These)40
b(diagrams)35 b(are)j(c)m(haracterized)g(b)m(y)h FI(a)2169
4805 y FH(23)2280 4790 y FL(=)d(0)p FI(;)h FL(the)h(en)m(umeration)f
(of)g(the)501 4911 y(v)m(ertices)32 b(is)e(lik)m(e)g(b)s(efore,)h(but)f
(all)f(the)i(arro)m(ws)g(are)f(assumed)h(to)f(b)s(e)h(p)s(oin)m(ting)e
(at)501 5031 y(v)m(ertices)34 b(1)e(or)h(3.)501 5193
y(Again)28 b(w)m(e)h(assume)g(that)g(these)g(diagrams)e(are)i
(realizable)d(o)m(v)m(er)k FI(\000)42 b FL(and)29 b(that)f
FI(g)3456 5208 y FH(1)501 5313 y FL(and)34 b FI(g)739
5328 y FH(2)812 5313 y FL(generate)g FI(\000)14 b FL(.)47
b(W)-8 b(e)34 b(then)g(carry)g(out)g(the)g(completely)e(analogous)h
(steps)1828 5637 y(55)p eop
%%Page: 56 58
56 57 bop 501 573 a FL(as)37 b(in)e(part)h(\(a\).)55
b(W)-8 b(e)36 b(in)m(tro)s(duce)h(the)f(parameters)h
FI(x;)17 b(y)39 b FL(and)e FI(z)k FL(to)36 b(denote)h(the)501
693 y(exp)s(onen)m(ts)e(of)d(the)h(diagonal)d(braiding)h(matrix)g
(elemen)m(ts,)i(so)g(that)829 912 y FI(\037)890 927 y
FH(1)930 912 y FL(\()p FI(g)1015 927 y FH(1)1054 912
y FL(\))27 b(=)h FI(q)t(;)114 b(\037)1472 927 y FH(2)1511
912 y FL(\()p FI(g)1596 927 y FH(2)1636 912 y FL(\))27
b(=)h FI(q)1852 871 y FG(x)1895 912 y FI(;)115 b(\037)2098
927 y FH(3)2137 912 y FL(\()p FI(g)2222 927 y FH(3)2261
912 y FL(\))28 b(=)f FI(q)2477 871 y FG(y)2519 912 y
FI(;)114 b(\037)2721 927 y FH(4)2760 912 y FL(\()p FI(g)2845
927 y FH(4)2885 912 y FL(\))27 b(=)h FI(q)3101 871 y
FG(z)3140 912 y FI(:)501 1132 y FL(With)g FI(g)796 1147
y FH(3)862 1132 y FL(=)g FI(g)1017 1096 y FG(n)1013 1156
y FH(1)1063 1132 y FI(g)1114 1096 y FG(m)1110 1156 y
FH(2)1208 1132 y FL(and)g FI(g)1440 1147 y FH(4)1507
1132 y FL(=)f FI(g)1661 1096 y FG(k)1657 1156 y FH(1)1703
1132 y FI(g)1754 1096 y FG(l)1750 1156 y FH(2)1817 1132
y FL(the)h(three)h(compatibilit)m(y)c(conditions)h(come)501
1252 y(out)33 b(as)1729 1471 y FI(n)1787 1430 y FH(2)1849
1471 y FL(+)22 b FI(a)1998 1486 y FH(12)2073 1471 y FI(nm)h
FL(+)f FI(xm)2477 1430 y FH(2)2539 1471 y FL(+)g FI(y)31
b FL(=)c(0)p FI(;)400 b FL(\(5.9\))1847 1617 y FI(k)1901
1576 y FH(2)1963 1617 y FL(+)22 b FI(a)2112 1632 y FH(12)2187
1617 y FI(k)s(l)i FL(+)e FI(xl)2478 1576 y FH(2)2540
1617 y FL(+)g FI(z)33 b FL(=)27 b(0)p FI(;)351 b FL(\(5.10\))1101
1762 y FI(k)s FL(\(2)p FI(n)23 b FL(+)f FI(a)1472 1777
y FH(12)1546 1762 y FI(m)p FL(\))h(+)f FI(l)r FL(\(2)p
FI(xm)h FL(+)f FI(a)2220 1777 y FH(12)2295 1762 y FI(n)p
FL(\))g(+)g FI(a)2562 1777 y FH(34)2637 1762 y FI(y)31
b FL(=)c(0)p FI(:)351 b FL(\(5.11\))501 1981 y(As)31
b(v)m(ertex)h(2)e(and)h(3)f(are)g(not)g(directly)g(connected)i(in)e
(all)e(the)j(diagrams)d(of)i(our)501 2102 y(class,)38
b(w)m(e)f(can)g(c)m(ho)s(ose)g FI(y)g FC(6)p FL(=)e(0)h(freely)-8
b(.)54 b(Hence,)39 b(w)m(e)f(can)e(alw)m(a)m(ys)h(arrange)f(that)501
2222 y(\(5.9\))f(has)h(a)f(solution.)51 b(Assume)36 b(that)f(2)p
FI(n)24 b FL(+)g FI(a)2274 2237 y FH(12)2349 2222 y FI(m)33
b FC(6)p FL(=)f(0.)52 b(Then)37 b(w)m(e)f(can)g(solv)m(e)501
2342 y(\(5.11\))31 b(for)g FI(k)k FL(and)d(plug)f(the)h(result)g(in)m
(to)f(\(5.10\).)42 b(After)32 b(some)g(reordering)f(and)501
2463 y(simplifying)e(w)m(e)34 b(see)g(that)e(w)m(e)i(can)f(apply)f
(\(5.9\))g(to)g(get)629 2682 y(\(4)p FI(x)23 b FC(\000)f
FI(a)944 2641 y FH(2)944 2707 y(12)1019 2682 y FL(\))p
FI(y)t(l)1140 2641 y FH(2)1201 2682 y FC(\000)h FI(ma)1437
2697 y FH(34)1512 2682 y FL(\(4)p FI(x)f FC(\000)h FI(a)1827
2641 y FH(2)1827 2707 y(12)1902 2682 y FL(\))p FI(y)t(l)g
FC(\000)g FL(\()p FI(a)2233 2641 y FH(2)2233 2707 y(34)2308
2682 y FI(y)2360 2641 y FH(2)2421 2682 y FL(+)f FI(z)t
FL(\(2)p FI(n)g FL(+)g FI(a)2884 2697 y FH(12)2959 2682
y FI(m)p FL(\))3082 2641 y FH(2)3122 2682 y FL(\))28
b(=)f(0)p FI(:)501 2901 y FL(F)-8 b(or)32 b(solving)f(this)i(quadratic)
f(equation)g(w)m(e)i(consider)f(the)g(discriminan)m(t)565
3078 y Fw(\020)635 3121 y FI(ma)771 3136 y FH(34)p 635
3166 212 4 v 716 3257 a FL(2)856 3078 y Fw(\021)915 3101
y FH(2)977 3189 y FL(+)1085 3121 y FI(a)1136 3085 y FH(2)1136
3146 y(34)1211 3121 y FI(y)1263 3085 y FH(2)1324 3121
y FL(+)22 b FI(z)t FL(\(2)p FI(n)g FL(+)g FI(a)1787 3136
y FH(12)1862 3121 y FI(m)p FL(\))1985 3085 y FH(2)p 1085
3166 941 4 v 1316 3257 a FL(\(4)p FI(x)g FC(\000)g FI(a)1630
3223 y FH(2)1630 3282 y(12)1705 3257 y FL(\))p FI(y)2063
3189 y FL(=)2166 3048 y Fw(\022)2250 3121 y FL(2)p FI(n)g
FL(+)g FI(a)2528 3136 y FH(12)2603 3121 y FI(m)p 2250
3166 439 4 v 2444 3257 a FL(2)2698 3048 y Fw(\023)2771
3071 y FH(2)2827 3048 y Fw(\024)2931 3121 y FL(4)p FI(z)k
FC(\000)d FI(a)3202 3085 y FH(2)3202 3146 y(34)3277 3121
y FI(y)p 2890 3166 479 4 v 2890 3257 a FL(\(4)p FI(x)f
FC(\000)h FI(a)3205 3223 y FH(2)3205 3282 y(12)3280 3257
y FL(\))p FI(y)3379 3048 y Fw(\025)501 3465 y FL(and)31
b(see)h(that)e(w)m(e)h(ha)m(v)m(e)h(a)e(solution)f(to)i(the)g(system)g
(of)f(equations)h(\(5.9\)-\(5.11\))501 3606 y(if)g FI(D)f
FL(:=)872 3554 y FH(4)p FG(z)s Fx(\000)p FG(a)1035 3531
y Fy(2)1035 3575 y(34)1100 3554 y FG(y)p 842 3583 325
4 v 842 3642 a FH(\(4)p FG(x)p Fx(\000)p FG(a)1036 3619
y Fy(2)1036 3663 y(12)1102 3642 y FH(\))p FG(y)1209 3606
y FL(is)h(a)h(square)h(mo)s(dulo)d FI(p)p FL(.)43 b(All)30
b(the)j(parameters)f(are)g(listed)f(in)501 3740 y(T)-8
b(able)34 b(5.2.)48 b(When)35 b FI(x)f FL(and)h FI(z)j
FL(can)d(b)s(e)f(c)m(hosen)i(freely)-8 b(,)34 b(w)m(e)h(can)g(alw)m(a)m
(ys)f(arrange)501 3861 y(for)39 b FI(D)k FL(to)c(b)s(e)h(a)g(square.)66
b(And)40 b(so)g(w)m(e)h(see)g(that)e(all)f(these)j(diagrams)d(can)i(b)s
(e)501 3981 y(realized)27 b(o)m(v)m(er)i FI(\000)42 b
FL(except)30 b FI(B)1536 3996 y FH(2)1589 3981 y FC([)13
b FI(G)1745 3996 y FH(2)1784 3981 y FL(,)29 b(where)g(w)m(e)g(need)h(3)
d(to)h(b)s(e)g(a)g(square)h(mo)s(dulo)501 4101 y FI(p)p
FL(.)501 4263 y(Again)45 b(w)m(e)i(sho)m(w)g(that)f(our)f(assumptions)h
(are)g(not)g(to)s(o)f(sp)s(ecial.)82 b(W)-8 b(e)47 b(only)501
4384 y(consider)28 b(the)g(problematic)d(diagram)g FI(B)2018
4399 y FH(2)2069 4384 y FC([)11 b FI(G)2223 4399 y FH(2)2263
4384 y FI(:)28 b FL(If)f(2)p FI(n)11 b FL(+)g FI(a)2666
4399 y FH(12)2741 4384 y FI(m)28 b FL(=)g(0)f(and)g(2)p
FI(xm)11 b FL(+)501 4504 y FI(a)552 4519 y FH(12)627
4504 y FI(n)28 b FL(=)g(0,)j(w)m(e)h(\014nd)g(that)f(4)p
FI(x)20 b FC(\000)f FI(a)1741 4468 y FH(2)1741 4529 y(12)1844
4504 y FL(=)28 b(0.)42 b(But)32 b(that)f(is)f(not)i(the)f(case)h(as)g
(w)m(e)g(see)501 4624 y(from)38 b(T)-8 b(able)39 b(5.2.)62
b(So)39 b(w)m(e)h(assume)g(2)p FI(n)27 b FL(+)f FI(a)2150
4639 y FH(12)2225 4624 y FI(m)39 b FL(=)g(0)f(and)i(solv)m(e)f
(\(5.11\))f(for)h FI(l)r FL(.)501 4745 y(F)-8 b(rom)32
b(\(5.9\))g(w)m(e)i(\014nd)g FI(y)e FL(=)c FC(\000)p
FL(\(4)p FI(x)23 b FC(\000)g FI(a)1908 4709 y FH(2)1908
4769 y(12)1983 4745 y FL(\))p FI(m)2106 4709 y FH(2)2146
4745 y FI(=)p FL(4)32 b(and)i(so)f FI(l)e FL(=)d FI(a)2802
4760 y FH(34)2877 4745 y FI(m=)p FL(2)p FI(:)33 b FL(Plugging)501
4865 y(this)h(in)m(to)g(\(5.10\))f(w)m(e)j(solv)m(e)e(the)h(quadratic)f
(equation)g(and)h(\014nd)f(that)g(it)g(has)g(a)501 4986
y(solution)j(if)h(3)g(is)g(a)h(square)h(mo)s(dulo)c FI(p)p
FL(.)62 b(Note)39 b(that)f FI(z)43 b FL(=)38 b(3)p FI(y)t(;)g(x)g
FL(=)g(2)g(=)g FC(\000)p FI(a)3420 5001 y FH(12)501 5106
y FL(and)33 b FI(a)742 5121 y FH(34)845 5106 y FL(=)27
b FC(\000)p FL(3)p FI(:)501 5268 y FL(No)m(w)44 b(w)m(e)h(will)c(sho)m
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5268 y FL(is)e(a)g(p)s(o)m(w)m(er)h(of)f FI(g)3041 5283
y FH(1)3124 5268 y FL(do)s(es)g(not)501 5388 y(lead)i(to)g(an)m(y)h
(simpli\014cation)c(of)i(the)i(condition)e(that)h(3)g(m)m(ust)h(b)s(e)f
(a)g(square)1828 5637 y(56)p eop
%%Page: 57 59
57 58 bop 960 580 a FL(Diagram)p 1583 658 4 186 v 305
w FI(a)1686 595 y FH(12)1860 580 y FI(a)1911 595 y FH(34)2086
580 y FI(x)125 b(z)p 2389 658 V 130 w(D)30 b FL(=)2695
529 y FH(4)p FG(z)s Fx(\000)p FG(a)2858 505 y Fy(2)2858
549 y(34)2923 529 y FG(y)p 2666 557 325 4 v 2666 616
a FH(\(4)p FG(x)p Fx(\000)p FG(a)2860 593 y Fy(2)2860
637 y(12)2926 616 y FH(\))p FG(y)p 703 662 2348 4 v 975
748 a FI(B)1049 763 y FH(2)1110 748 y FC([)23 b FI(B)1273
763 y FH(2)p 1583 826 4 165 v 1634 748 a FC(\000)p FL(2)100
b FC(\000)p FL(2)j(2)g(2)p FI(y)p 2389 826 V 2594 704
a FH(4)p FG(y)p 2594 725 73 4 v 2594 783 a FH(4)p FG(y)2704
748 y FL(=)28 b(1)973 913 y FI(B)1047 928 y FH(2)1109
913 y FC([)23 b FI(G)1275 928 y FH(2)p 1583 991 4 165
v 1634 913 a FC(\000)p FL(2)100 b FC(\000)p FL(3)j(2)g(3)p
FI(y)p 2389 991 V 2574 869 a FH(3)p FG(y)p 2574 890 73
4 v 2574 947 a FH(4)p FG(y)2684 913 y FL(=)2815 874 y
FH(3)p 2797 890 70 4 v 2797 947 a(2)2832 928 y Fy(2)972
1078 y FI(G)1049 1093 y FH(2)1110 1078 y FC([)23 b FI(G)1276
1093 y FH(2)p 1583 1156 4 165 v 1634 1078 a FC(\000)p
FL(3)100 b FC(\000)p FL(3)j(3)g(3)p FI(y)p 2389 1156
V 2594 1034 a FH(3)p FG(y)p 2594 1055 73 4 v 2594 1112
a FH(3)p FG(y)2704 1078 y FL(=)28 b(1)864 1240 y FI(B)938
1255 y FH(2)999 1240 y FC([)23 b FI(A)1161 1255 y FH(1)1222
1240 y FC([)g FI(A)1384 1255 y FH(1)p 1583 1318 4 163
v 1634 1240 a FC(\000)p FL(2)139 b(0)i(2)128 b FI(z)p
2389 1318 V 2591 1201 a FH(4)p FG(z)p 2590 1217 73 4
v 2590 1274 a FH(4)p FG(y)2700 1240 y FL(=)2815 1201
y FG(z)p 2814 1217 38 4 v 2814 1274 a(y)862 1402 y FI(G)939
1417 y FH(2)1001 1402 y FC([)22 b FI(A)1162 1417 y FH(1)1224
1402 y FC([)g FI(A)1385 1417 y FH(1)p 1583 1480 4 163
v 1634 1402 a FC(\000)p FL(3)139 b(0)i(3)128 b FI(z)p
2389 1480 V 2529 1363 a FH(4)p FG(z)p 2528 1379 73 4
v 2528 1436 a FH(3)p FG(y)2638 1402 y FL(=)28 b(2)2791
1366 y FH(2)2859 1363 y FG(z)p 2840 1379 V 2840 1436
a FH(3)p FG(y)752 1564 y FI(A)825 1579 y FH(1)887 1564
y FC([)23 b FI(A)1049 1579 y FH(1)1110 1564 y FC([)g
FI(A)1272 1579 y FH(1)1334 1564 y FC([)f FI(A)1495 1579
y FH(1)p 1583 1642 4 163 v 1673 1564 a FL(0)177 b(0)138
b FI(x)125 b(z)p 2389 1642 V 2571 1525 a FH(4)p FG(z)p
2550 1542 113 4 v 2550 1599 a FH(4)p FG(xy)2700 1564
y FL(=)2834 1525 y FG(z)p 2814 1542 77 4 v 2814 1599
a(xy)p 703 1646 2348 4 v 826 1914 a FL(T)-8 b(able)33
b(5.2:)43 b(More)33 b(parameters)f(for)g(Dynkin)h(diagrams)501
2199 y(for)j FI(B)728 2214 y FH(2)793 2199 y FC([)25
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1302 y(C)2074 1365 y(A)2178 1267 y FI(:)2417 947 y Fw(2)2417
1122 y(6)2417 1182 y(6)2417 1242 y(6)2417 1302 y(6)2417
1365 y(4)2639 1025 y FI(g)2686 1040 y FH(1)2639 1146
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FH(4)2530 1291 y(1)2573 1266 y FI(g)2624 1230 y FH(2)2620
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1386 y FI(g)2534 1350 y FH(3)2530 1411 y(1)2573 1386
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1507 y FI(g)2622 1471 y FH(4)2618 1531 y(2)2689 1507
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y(7)2880 1182 y(7)2880 1242 y(7)2880 1302 y(7)2880 1365
y(5)1828 5637 y FL(59)p eop
%%Page: 60 62
60 61 bop 257 1237 a FK(Chapter)78 b(6)257 1652 y(Quasi-isomorphisms)
257 2105 y FL(W)-8 b(e)30 b(will)c(sho)m(w)k(that)e(a)h(large)f(class)g
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2225 y(quasi-isomorphic)21 b(to)h(their)g(asso)s(ciated)g(graded)h(v)m
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3308 y(sho)m(w)m(ed)49 b(in)e([Mas1])g(and)h(in)e(a)h(priv)-5
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b(This)31 b(w)m(as)h(dispro)m(v)m(ed)g(in)e([EG])h(for)f(families)e(of)
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b(Ho)m(w)m(ev)m(er,)32 b(our)d(results)h(supp)s(ort)g(Masuok)-5
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b FL(if)46 b(their)h(categories)g(of)g(como)s(dules)g(are)h(monoidally)
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b(alen)m(t)35 b(to)g(the)h(Hopf)f(algebras)f(b)s(eing)h(2-co)s(cycle)f
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b(Section)32 b(2.5.)1828 5637 y(60)p eop
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61 62 bop 404 573 a FL(W)-8 b(e)31 b(\014x)h(a)f(linking)e(datum)h
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FL(\()p FC(D)s FI(;)17 b Fq(u)p FL(\))29 b(is)g(a)h(p)s(oin)m(ted)f
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b(ro)s(ot)f(v)m(ector)i(parameters)e(are)h(kno)m(wn)h(explicitly)d(and)
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2379 y(t)m(yp)s(e)28 b(ha)m(v)m(e)g(b)s(een)g(determined)f(explicitly)
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2499 y(the)33 b(same)g(result)f(as)h(for)f(the)h FI(A)1451
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b(of)257 3292 y Fs(u)p FL(\()p FC(D)s FI(;)17 b FJ(0)p
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b(e)38 b(of)g(the)h FI(\025)1979 3655 y FG(ij)2078 3640
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4124 y FG(ij)3365 4109 y FL(are)501 4230 y(0.)62 b(This)39
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4726 y FG(ij)2564 4711 y FC(6)p FL(=)e(0)h(for)f(some)h
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4952 y(are)39 b(in)g(the)g(same)g(connected)h(comp)s(onen)m(t)f
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y FG(j)2154 5234 y FL(\))2192 5249 y FH(1)p Fx(\024)p
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5249 y FG(ij)2607 5234 y FL(\))2645 5249 y FH(1)p Fx(\024)p
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eop
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62 63 bop 501 573 a FL(there)42 b(is)f(a)g FI(j)47 b
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693 y(that)35 b FI(\025)772 708 y FG(k)r(l)869 693 y
FL(=)d(0)j(for)g(all)e FI(k)i FC(2)e FI(I)8 b FL(.)51
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971 y(No)m(w,)47 b(from)42 b(the)h(pro)s(of)g(of)g(Theorem)g(5.17.)g
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FC(U)d(\012)16 b(B)s FL(\))1433 1107 y FG(\033)1480 1092
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1731 y(\001)17 b(\001)g(\001)28 b(\010)h FI(<)11 b(Z)913
1741 y FH(~)907 1759 y FG(\022)957 1731 y FI(>;)43 b
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1851 y(ord)17 b FI(g)706 1866 y FG(i)758 1851 y FL(and)24
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b FI(g)2844 1987 y FG(i)2909 1972 y FL(divides)38 b(ord)16
b FI(Z)3467 1987 y FG(i)501 2092 y FL(for)32 b(all)f
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FC(D)1763 2265 y FH(1)1835 2250 y FL(with)g(the)h(original)c(group)k
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Fs(u)p FL(\()p FC(D)1169 3025 y FH(1)1208 3010 y FL(\))i(and)g
FC(U)46 b FL(:=)35 b Fs(u)p FL(\()p FC(D)1890 3025 y
FH(2)1929 3010 y FL(\))p FI(:)i FL(W)-8 b(e)37 b(denote)h(the)g
(generators)f(of)g FC(B)j FL(b)m(y)501 3130 y FI(b)548
3140 y FH(~)542 3158 y FG(\022)s FH(+1)672 3130 y FI(;)17
b(:)g(:)g(:)f(;)h(b)932 3145 y FG(\022)1011 3130 y FL(and)40
b FI(y)1256 3145 y FH(1)1295 3130 y FI(;)17 b(:)g(:)g(:)f(;)h(y)1562
3145 y FG(s)1598 3130 y FI(;)40 b FL(and)g(the)g(generators)h(of)e
FC(U)50 b FL(b)m(y)41 b FI(u)2946 3145 y FH(1)2985 3130
y FI(;)17 b(:)g(:)g(:)f(;)h(u)3266 3140 y FH(~)3260 3158
y FG(\022)3338 3130 y FL(and)501 3251 y FI(z)546 3266
y FH(1)586 3251 y FI(;)g(:)g(:)g(:)f(;)h(z)856 3261 y
FH(~)850 3278 y FG(\022)889 3251 y FI(:)501 3408 y FL(The)34
b(cen)m(tral)e(Hopf)h(subalgebra)f FC(K)i FL(of)e FC(U)g(\012)23
b(B)36 b FL(is)c FF(|)-9 b FL([)p FI(z)2498 3423 y FG(i)2543
3408 y FC(\012)23 b FI(g)2694 3367 y Fx(\000)p FH(1)2690
3434 y FG(i)2815 3408 y FL(:)28 b(1)f FC(\024)h FI(i)g
FC(\024)3225 3382 y FL(~)3217 3408 y FI(\022)t FL(])p
FI(:)501 3566 y FL(The)34 b(co)s(cycle)f FI(\033)j FL(for)c
FC(U)h(\012)22 b(B)36 b FL(is)c(de\014ned)i(b)m(y)1226
3756 y FI(\033)t FL(\()p FI(u)21 b FC(\012)i FI(a;)17
b(v)26 b FC(\012)c FI(b)p FL(\))28 b(:=)g FI(")2051 3771
y Fx(U)2106 3756 y FL(\()p FI(u)p FL(\))p FI(\034)11
b FL(\()p FI(v)t(;)17 b(a)p FL(\))g FI(")2574 3771 y
Fx(B)2626 3756 y FL(\()p FI(b)p FL(\))p FI(:)501 3946
y(\034)39 b FL(:)28 b FC(U)33 b(\012)22 b(B)31 b(!)d
FF(|)17 b FL(is)33 b(a)f(linear)f(map)h(with)g(the)h(follo)m(wing)d
(list)h(of)h(prop)s(erties;)591 4141 y(1.)49 b FI(\034)11
b FL(\()p FI(uv)t(;)17 b(a)p FL(\))27 b(=)h FI(\034)11
b FL(\()p FI(u;)17 b(a)1420 4156 y Fy(\(1\))1502 4141
y FL(\))p FI(\034)11 b FL(\()p FI(v)t(;)17 b(a)1777 4156
y Fy(\(2\))1860 4141 y FL(\))p FI(;)591 4294 y FL(2.)49
b FI(\034)11 b FL(\()p FI(u;)17 b(ab)p FL(\))28 b(=)f
FI(\034)11 b FL(\()p FI(u)1315 4309 y Fy(\(1\))1398 4294
y FI(;)17 b(b)p FL(\))p FI(\034)11 b FL(\()p FI(u)1668
4309 y Fy(\(2\))1750 4294 y FI(;)17 b(a)p FL(\))p FI(;)591
4448 y FL(3.)49 b FI(\034)11 b FL(\(1)p FI(;)17 b(a)p
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FI(a)p FL(\))p FI(;)591 4601 y FL(4.)49 b FI(\034)11
b FL(\()p FI(u;)17 b FL(1\))27 b(=)g FI(")1170 4616 y
Fx(U)1225 4601 y FL(\()p FI(u)p FL(\))p FI(:)501 4796
y FL(It)33 b(is)f(giv)m(en)h(b)m(y)1575 4916 y FI(\034)11
b FL(\()p FI(u)22 b FC(\012)h FI(b)p FL(\))28 b(:=)g
FI(')p FL(\()p FI(u)p FL(\)\()p FI(b)p FL(\))p FI(;)501
5078 y FL(where)i FI(')e FL(:)f FC(U)38 b(!)28 b FL(\()p
FC(B)1259 5042 y Fx(\003)1298 5078 y FL(\))1336 5042
y FG(cop)1469 5078 y FL(is)g(a)g(Hopf)g(algebra)g(homomorphism)d
(de\014ned)30 b(on)f(the)501 5198 y(generators)k(of)f
FC(U)43 b FL(b)m(y)1251 5388 y FI(')p FL(\()p FI(z)1398
5403 y FG(i)1426 5388 y FL(\))28 b(:=)f FI(\015)1673
5403 y FG(i)1896 5388 y FL(and)196 b FI(')p FL(\()p FI(u)2407
5403 y FG(j)2442 5388 y FL(\))28 b(:=)g FI(\016)2682
5403 y FG(j)2718 5388 y FI(:)1828 5637 y FL(62)p eop
%%Page: 63 65
63 64 bop 501 573 a FL(Here)34 b FI(\015)783 588 y FG(i)838
573 y FL(:)28 b FC(B)j(!)c FF(|)18 b FL(is)32 b(a)g(c)m(haracter)i
(de\014ned)g(on)e(the)h(generators)g(of)f FC(B)k FL(b)m(y)1167
793 y FI(\015)1218 808 y FG(i)1246 793 y FL(\()p FI(y)1332
808 y FG(k)1374 793 y FL(\))28 b(:=)f FI(\037)1631 808
y FG(i)1660 793 y FL(\()p FI(y)1746 808 y FG(k)1788 793
y FL(\))195 b(and)g FI(\015)2424 808 y FG(i)2452 793
y FL(\()p FI(b)2531 808 y FG(l)2557 793 y FL(\))28 b(:=)f(0)p
FI(;)501 1013 y FL(and)33 b FI(\016)734 1028 y FG(j)798
1013 y FL(:)28 b FC(B)j(!)d FF(|)17 b FL(is)32 b(a)h(\()p
FI(")1427 1028 y Fx(B)1479 1013 y FI(;)17 b(\015)1574
1028 y FG(j)1610 1013 y FL(\)-deriv)-5 b(ation)31 b(de\014ned)j(b)m(y)
1074 1233 y FI(\016)1117 1248 y FG(j)1154 1233 y FL(\()p
FI(y)1240 1248 y FG(k)1282 1233 y FL(\))28 b(:=)g(0)194
b(and)i FI(\016)2118 1248 y FG(j)2154 1233 y FL(\()p
FI(b)2233 1248 y FG(l)2260 1233 y FL(\))27 b(:=)h FC(\000)p
FI(\037)2594 1248 y FG(j)2631 1233 y FL(\()p FI(g)2716
1248 y FG(l)2742 1233 y FL(\))p FI(\025)2837 1248 y FG(j)t(l)2895
1233 y FI(:)501 1453 y FL(In)39 b(all)e(the)i(ab)s(o)m(v)m(e)g(form)m
(ulas)e(1)h FC(\024)g FI(i;)17 b(j)44 b FC(\024)2115
1427 y FL(~)2107 1453 y FI(\022)e FL(and)2397 1427 y(~)2390
1453 y FI(\022)f(<)c(k)s(;)17 b(l)40 b FC(\024)e FI(\022)s(:)h
FL(The)h(co)s(cycle)506 1573 y(~)-54 b FI(\033)44 b FL(is)c(no)m(w)h
(de\014ned)g(in)e(exactly)i(the)f(same)g(w)m(a)m(y)i(as)e
FI(\033)n(;)h FL(only)e(in)h(the)g(last)f(part)506 1667
y(~)501 1694 y FI(\016)544 1709 y FG(j)581 1694 y FL(\()p
FI(b)660 1709 y FG(l)686 1694 y FL(\))k(:=)g(0)e(as)h(all)d(the)j
FI(\025)1510 1709 y FG(j)t(l)1611 1694 y FL(=)h(0)p FI(:)f
FL(This)f(is)g(b)s(ecause)i(for)2738 1668 y(~)2714 1694
y FC(D)s FI(;)e FL(w)m(e)i(w)m(an)m(ted)g(the)501 1814
y(comp)s(onen)m(t)33 b FI(I)i FL(=)28 b FC(f)p FL(1)p
FI(;)17 b(:)g(:)g(:)e(;)1507 1788 y FL(~)1499 1814 y
FI(\022)s FC(g)33 b FL(not)f(to)g(b)s(e)h(link)m(ed)f(to)g(an)m(y)i
(other)e(comp)s(onen)m(t.)501 1976 y(The)i(in)m(v)m(erse)g
FI(\033)1084 1940 y Fx(\000)p FH(1)1211 1976 y FL(is)e(giv)m(en)g(in)g
(the)h(same)g(w)m(a)m(y)g(b)m(y)h FI(\034)2476 1940 y
Fx(\000)p FH(1)2571 1976 y FI(;)e FL(where)1227 2196
y FI(\034)1280 2155 y Fx(\000)p FH(1)1375 2196 y FL(\()p
FI(u;)17 b(b)p FL(\))27 b(:=)h FI(')p FL(\()p Fv(S)16
b FI(u)p FL(\)\()p FI(b)p FL(\))28 b(=)f FI(')p FL(\()p
FI(u)p FL(\)\()p Fv(S)2552 2152 y Fx(\000)p FH(1)2663
2196 y FI(b)p FL(\))p FI(:)501 2416 y FL(If)i FI(\033)n(;)23
b FL(~)-55 b FI(\033)34 b FL(are)29 b(t)m(w)m(o)g(2-co)s(cycles)h(for)e
(a)h(Hopf)g(algebra)f FC(A)p FI(;)g FL(then)i FI(\032)e
FL(:=)33 b(~)-54 b FI(\033)t(\033)3050 2380 y Fx(\000)p
FH(1)3173 2416 y FL(is)29 b(again)501 2536 y(a)24 b(2-co)s(cycle,)i
(but)f(for)f(the)h(Hopf)f(algebra)g FC(A)2128 2551 y
FG(\033)2174 2536 y FI(:)h FL(In)g(our)f(case)h(this)f(means)h
FI(\032)g FL(is)f(a)g(2-)501 2657 y(co)s(cycle)k(for)e(the)i(Hopf)e
(algebra)g FC(A)i FL(:=)f(\()p FC(U)21 b(\012)11 b(B)s
FL(\))2259 2672 y FG(\033)2307 2657 y FI(:)27 b FL(Then)h(\()p
FC(U)21 b(\012)11 b(B)s FL(\))2929 2672 y FH(~)-39 b
FG(\033)3000 2657 y FL(=)28 b FC(A)3184 2672 y FG(\032)3224
2657 y FL(.)41 b(If)27 b(w)m(e)501 2777 y(can)j(sho)m(w)h(that)f
FI(\032)g FL(passes)h(do)m(wn)g(naturally)d(to)h(a)h(2-co)s(cycle)f
FI(\032)2827 2741 y Fx(0)2881 2777 y FL(on)g FC(A)p FI(=)p
FC(K)3219 2741 y FH(+)3308 2777 y FL(suc)m(h)501 2897
y(that)k(\()p FC(A)p FI(=)p FC(K)957 2861 y FH(+)1015
2897 y FL(\))1053 2912 y FG(\032)1089 2894 y Fu(0)1148
2897 y FL(is)f(isomorphic)f(to)h FC(A)1939 2912 y FG(\032)1979
2897 y FI(=)p FC(K)2105 2861 y FH(+)2164 2897 y FI(;)h
FL(then)g(the)g(statemen)m(t)g(is)f(clear:)1193 3117
y Fs(u)p FL(\()1306 3092 y(~)1283 3117 y FC(D)r FL(\))c
FC(')g(A)1613 3132 y FG(\032)1653 3117 y FI(=)p FC(K)1779
3076 y FH(+)1865 3117 y FC(')h FL(\()p FC(A)p FI(=)p
FC(K)2215 3076 y FH(+)2273 3117 y FL(\))2311 3132 y FG(\032)2347
3114 y Fu(0)2402 3117 y FC(')f Fs(u)p FL(\()p FC(D)s
FL(\))2715 3132 y FG(\032)2751 3114 y Fu(0)2776 3117
y FI(:)492 b FL(\(6.1\))501 3337 y(Therefore,)34 b(w)m(e)g(w)m(an)m(t)f
(the)g(follo)m(wing)d(situation.)1587 3586 y FC(A)22
b(\012)h(A)2482 3518 y FG(\032)3219 3556 y Fd(/)p Fc(/)p
1893 3558 1326 4 v 1753 3857 a FG(\031)1728 4055 y Fd(\017)p
Fc(\017)p 1726 4055 4 436 v 3244 3590 a FF(|)697 4156
y FL(\()p FC(A)f(\012)h(A)p FL(\))p FI(=)p FL(\()p FC(A)e(\012)h(K)1419
4120 y FH(+)1501 4156 y FL(+)g FC(K)1676 4120 y FH(+)1757
4156 y FC(\012)h(A)p FL(\))k FC(')h(A)p FI(=)p FC(K)2313
4120 y FH(+)2394 4156 y FC(\012)23 b(A)p FI(=)p FC(K)2700
4120 y FH(+)2450 3805 y FG(\032)2486 3782 y Fu(0)3219
3576 y Fd(5)p Fc(5)3180 3590 y Fb(k)3102 3619 y(k)3024
3648 y(k)2946 3677 y(k)2868 3706 y(k)2790 3735 y(k)2712
3764 y(k)2633 3793 y(k)2555 3822 y(k)2477 3851 y(k)2399
3880 y(k)2321 3909 y(k)2243 3939 y(k)2165 3968 y(k)2087
3997 y(k)2009 4026 y(k)1931 4055 y(k)501 4383 y FL(F)-8
b(or)44 b(this)h(it)f(su\016ces)j(to)e(sho)m(w)h(that)f
FI(\032)g FL(is)g(0)f(on)h(the)h(k)m(ernel)f(of)g(the)g(natural)501
4503 y(pro)5 b(jection)36 b FI(\031)k FL(and)c(w)m(e)h(get)f(the)h
(factorization)c(and)j(\()p FC(A)p FI(=)p FC(K)2747 4467
y FH(+)2806 4503 y FL(\))2844 4518 y FG(\032)2880 4499
y Fu(0)2940 4503 y FC(')e(A)3131 4518 y FG(\032)3171
4503 y FI(=)p FC(K)3297 4467 y FH(+)3392 4503 y FL(b)m(y)501
4624 y(de\014nition)k(of)g FI(\032)1108 4587 y Fx(0)1132
4624 y FI(:)h FL(So)f(w)m(e)i(see)g(that)f(it)e(is)i(enough)g(to)f(sho)
m(w)i FI(\032)p FL(\()p FC(A)p FI(;)17 b FC(K)3083 4587
y FH(+)3142 4624 y FL(\))38 b(=)g(0)g(=)501 4744 y FI(\032)p
FL(\()p FC(K)666 4708 y FH(+)726 4744 y FI(;)17 b FC(A)p
FL(\))p FI(:)32 b FL(This)g(means)h(that)f(for)g(all)f
FI(u)c FC(2)h(U)10 b FI(;)33 b(b)28 b FC(2)g(B)36 b FL(and)d(1)27
b FC(\024)i FI(i)f FC(\024)3083 4718 y FL(~)3075 4744
y FI(\022)35 b FL(w)m(e)f(need)1251 4964 y FI(\032)p
FL(\()p FI(u)22 b FC(\012)g FI(b;)17 b(z)1646 4979 y
FG(i)1697 4964 y FC(\012)23 b FI(g)1848 4923 y Fx(\000)p
FH(1)1844 4989 y FG(i)1941 4964 y FL(\))28 b(=)g FI(\032)p
FL(\()p FI(u)21 b FC(\012)i FI(b;)17 b FL(1)22 b FC(\012)h
FL(1\))p FI(;)1251 5109 y(\032)p FL(\()p FI(z)1384 5124
y FG(i)1435 5109 y FC(\012)f FI(g)1585 5068 y Fx(\000)p
FH(1)1581 5135 y FG(i)1679 5109 y FI(;)17 b(u)k FC(\012)i
FI(b)p FL(\))28 b(=)g FI(\032)p FL(\(1)22 b FC(\012)g
FL(1)p FI(;)17 b(u)k FC(\012)i FI(b)p FL(\))p FI(:)1828
5637 y FL(63)p eop
%%Page: 64 66
64 65 bop 501 573 a FL(W)-8 b(e)46 b(calculate)e(using)g(the)i
(de\014nition)e(of)g FI(\032)p FL(,)49 b(the)c(con)m(v)m(olution)g(pro)
s(duct)g(and)501 693 y(prop)s(ert)m(y)34 b(3)e(of)g FI(\034)44
b FL(and)38 b(~)-54 b FI(\034)39 b FL(:)505 878 y FI(\032)p
FL(\()p FI(u)22 b FC(\012)g FI(b;)17 b(z)900 893 y FG(i)951
878 y FC(\012)23 b FI(g)1102 837 y Fx(\000)p FH(1)1098
903 y FG(i)1196 878 y FL(\))k(=)33 b(~)-54 b FI(\033)t
FL(\()p FI(u)1518 893 y Fy(\(1\))1622 878 y FC(\012)22
b FI(b)1762 893 y Fy(\(1\))1845 878 y FI(;)17 b(z)1934
893 y FG(i)1985 878 y FC(\012)22 b FI(g)2135 837 y Fx(\000)p
FH(1)2131 903 y FG(i)2229 878 y FL(\))p FI(\033)2326
837 y Fx(\000)p FH(1)2420 878 y FL(\()p FI(u)2514 893
y Fy(\(2\))2618 878 y FC(\012)h FI(b)2759 893 y Fy(\(2\))2842
878 y FI(;)17 b(z)2931 893 y FG(i)2981 878 y FC(\012)23
b FI(g)3132 837 y Fx(\000)p FH(1)3128 903 y FG(i)3226
878 y FL(\))1261 1023 y(=)28 b FI(")1410 1038 y Fx(U)1465
1023 y FL(\()p FI(u)1559 1038 y Fy(\(1\))1641 1023 y
FL(\))5 b(~)-54 b FI(\034)12 b FL(\()p FI(z)1816 1038
y FG(i)1844 1023 y FI(;)17 b(b)1929 1038 y Fy(\(1\))2012
1023 y FL(\))g FI(")2112 1038 y Fx(B)2164 1023 y FL(\()p
FI(g)2253 982 y Fx(\000)p FH(1)2249 1049 y FG(i)2347
1023 y FL(\))g FI(")2447 1038 y Fx(U)2502 1023 y FL(\()p
FI(u)2596 1038 y Fy(\(2\))2678 1023 y FL(\))p FI(\034)11
b FL(\()p Fv(S)17 b FI(z)2923 1038 y FG(i)2951 1023 y
FI(;)g(b)3036 1038 y Fy(\(2\))3119 1023 y FL(\))g FI(")3219
1038 y Fx(B)3271 1023 y FL(\()p FI(g)3360 982 y Fx(\000)p
FH(1)3356 1049 y FG(i)3454 1023 y FL(\))1261 1168 y(=)28
b FI(")1410 1183 y Fx(U)1465 1168 y FL(\()p FI(u)p FL(\))5
b(~)-54 b FI(\034)11 b FL(\()p FI(z)1733 1183 y FG(i)1761
1168 y FI(;)17 b(b)1846 1183 y Fy(\(1\))1929 1168 y FL(\))p
FI(\034)11 b FL(\()p FI(z)2107 1127 y Fx(\000)p FH(1)2103
1194 y FG(i)2203 1168 y FI(;)17 b(b)2288 1183 y Fy(\(2\))2370
1168 y FL(\))p FI(;)625 1314 y(\032)p FL(\()p FI(u)22
b FC(\012)h FI(b;)17 b FL(1)22 b FC(\012)h FL(1\))k(=)33
b(~)-54 b FI(\033)t FL(\()p FI(u)1518 1329 y Fy(\(1\))1622
1314 y FC(\012)22 b FI(b)1762 1329 y Fy(\(1\))1845 1314
y FI(;)17 b FL(1)22 b FC(\012)h FL(1\))p FI(\033)2206
1273 y Fx(\000)p FH(1)2300 1314 y FL(\()p FI(u)2394 1329
y Fy(\(2\))2498 1314 y FC(\012)g FI(b)2639 1329 y Fy(\(2\))2722
1314 y FI(;)17 b FL(1)k FC(\012)i FL(1\))1261 1459 y(=)28
b FI(")1410 1474 y Fx(U)1465 1459 y FL(\()p FI(u)1559
1474 y Fy(\(1\))1641 1459 y FL(\))5 b(~)-54 b FI(\034)12
b FL(\(1)p FI(;)17 b(b)1905 1474 y Fy(\(1\))1987 1459
y FL(\))g FI(")2087 1474 y Fx(B)2139 1459 y FL(\(1\))g
FI(")2326 1474 y Fx(U)2381 1459 y FL(\()p FI(u)2475 1474
y Fy(\(2\))2557 1459 y FL(\))p FI(\034)11 b FL(\()p Fv(S)17
b FL(1)p FI(;)g(b)2891 1474 y Fy(\(2\))2974 1459 y FL(\))g
FI(")3074 1474 y Fx(B)3126 1459 y FL(\(1\))1261 1604
y(=)28 b FI(")1410 1619 y Fx(U)1465 1604 y FL(\()p FI(u)p
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FI(b)p FL(\))p FI(:)501 1789 y FL(As)42 b FI(z)699 1804
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b FI(\034)53 b FL(that)41 b(it)g(is)501 1909 y(enough)33
b(to)f(v)m(erify)1424 2094 y(~)-54 b FI(\034)12 b FL(\()p
FI(z)1556 2109 y FG(i)1584 2094 y FI(;)17 b(b)1669 2109
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2052 y Fx(\000)p FH(1)1926 2119 y FG(i)2025 2094 y FI(;)17
b(b)2110 2109 y Fy(\(2\))2193 2094 y FL(\))28 b(=)f FI(")2408
2109 y Fx(B)2460 2094 y FL(\()p FI(b)p FL(\))718 b(\(6.2\))501
2278 y(on)33 b(the)g(generators)g(of)f FC(B)s FI(:)h
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2399 y(4)k(this)g(time,)f(sho)m(ws)j(that)e(for)f(the)i(second)g
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2519 y(the)c(generators)g(of)f FC(U)1387 2704 y FL(~)-54
b FI(\034)12 b FL(\()p FI(u)1530 2719 y Fy(\(1\))1612
2704 y FI(;)17 b(g)1707 2662 y Fx(\000)p FH(1)1703 2729
y FG(i)1800 2704 y FL(\))p FI(\034)11 b FL(\()p FI(u)1985
2719 y Fy(\(2\))2068 2704 y FI(;)17 b(g)2159 2719 y FG(i)2186
2704 y FL(\))28 b(=)f FI(")2401 2719 y Fx(U)2456 2704
y FL(\()p FI(u)p FL(\))p FI(:)680 b FL(\(6.3\))501 2888
y(The)34 b(v)m(eri\014cation)e(go)s(es)g(as)h(follo)m(ws)e(\(1)d
FC(\024)g FI(j)34 b FC(\024)2267 2862 y FL(~)2259 2888
y FI(\022)s(;)2358 2862 y FL(~)2351 2888 y FI(\022)c(<)e(k)j
FC(\024)d FI(\022)s FL(\):)501 3137 y FI(b)g FL(=)g FI(y)722
3152 y FG(k)797 3137 y FL(in)j(\(6.2\))1365 3346 y(~)-54
b FI(\034)11 b FL(\()p FI(z)1496 3361 y FG(i)1525 3346
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b FL(\()p FI(z)1837 3305 y Fx(\000)p FH(1)1833 3372 y
FG(i)1932 3346 y FI(;)17 b(y)2024 3361 y FG(k)2066 3346
y FL(\))28 b(=)j(~)-53 b FI(\015)2286 3361 y FG(i)2314
3346 y FL(\()p FI(y)2400 3361 y FG(k)2442 3346 y FL(\))p
FI(\015)2536 3305 y Fx(\000)p FH(1)2531 3372 y FG(i)2630
3346 y FL(\()p FI(y)2716 3361 y FG(k)2759 3346 y FL(\))2132
3492 y(=)27 b FI(\037)2296 3507 y FG(i)2324 3492 y FL(\()p
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3450 y Fx(\000)p FH(1)2552 3517 y FG(i)2646 3492 y FL(\()p
FI(y)2732 3507 y FG(k)2774 3492 y FL(\))h(=)f(1)h(=)f
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4500 y FG(i)1736 4475 y FL(\))p FI(\034)11 b FL(\()p
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Fx(U)2685 5184 y FL(\()p FI(u)2779 5199 y FG(j)2814 5184
y FL(\))p FI(:)3292 5369 y FJ(qed.)1828 5637 y FL(64)p
eop
%%Page: 65 67
65 66 bop 257 573 a FD(6.2)161 b(A)54 b(sp)t(ecial)h(ro)t(ot)e(v)l
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(algebras)g(where)i(the)f(Dynkin)f(diagram)e(is)j(just)f(one)h(cop)m(y)
257 912 y(of)41 b FI(A)450 927 y FG(n)497 912 y FI(;)h(n)h
FC(\025)g FL(1)p FI(;)e FL(and)g(hence)i(there)f(can)g(b)s(e)f(no)h
(linking.)67 b(T)-8 b(o)42 b(\014x)g(the)f(order)h(of)f(the)257
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FC(D)j FL(of)d FI(A)1555 1168 y FG(n)1636 1153 y FL(t)m(yp)s(e)h(with)f
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b(with)i(certain)f(cen)m(tral)g(elemen)m(ts)h FI(u)1933
1288 y FG(\013)2015 1273 y FL(from)e(the)i(group)f(algebra.)404
1394 y(In)44 b([AS5],)k(the)c(explicit)f(form)g(of)g(the)i(ro)s(ot)e(v)
m(ectors)j FI(x)2539 1409 y FG(\013)2633 1394 y FL(and)e(all)e(the)j(p)
s(ossible)257 1514 y(families)35 b FI(u)674 1529 y FG(\013)760
1514 y FL(suc)m(h)j(that)f Fs(u)p FL(\()p FC(D)s FI(;)17
b Fq(u)p FL(\))36 b(is)h(a)g(Hopf)g(algebra)f(ha)m(v)m(e)i(b)s(een)g
(determined.)58 b(W)-8 b(e)257 1635 y(rep)s(eat)33 b(these)h(here)f
(for)f(later)g(calculations.)404 1755 y(Set)h FI(q)615
1770 y FG(i;j)723 1755 y FL(:=)27 b FI(\037)914 1770
y FG(j)951 1755 y FL(\()p FI(g)1036 1770 y FG(i)1064
1755 y FL(\))p FI(;)44 b FL(1)28 b FC(\024)g FI(i;)17
b(j)33 b FC(\024)c FI(n;)j(N)39 b FL(:=)27 b(ord)17 b
FI(\037)2194 1770 y FG(i)2222 1755 y FL(\()p FI(g)2307
1770 y FG(i)2335 1755 y FL(\))32 b(and)1415 1962 y FI(B)1494
1920 y FG(i;j)1489 1986 y(p;r)1610 1962 y FL(:=)1889
1867 y Fw(Y)1741 2079 y FG(i)p Fx(\024)p FG(l)q(<j;p)p
Fx(\024)p FG(h<r)2181 1962 y FI(q)2224 1977 y FG(l)q(;h)2310
1962 y FI(:)257 2251 y(N)39 b FL(is)28 b(de\014ned)i(indep)s(enden)m
(tly)f(of)f(the)h(c)m(hoice)f(of)g FI(i)h FL(as)f FI(\037)2328
2266 y FG(i)2357 2251 y FL(\()p FI(g)2442 2266 y FG(i)2470
2251 y FL(\))f(=)h FI(\037)2700 2266 y FG(j)2736 2251
y FL(\()p FI(g)2821 2266 y FG(j)2858 2251 y FL(\))g(due)h(to)f
(\(3.19\).)257 2371 y(Then)704 2492 y FI(C)781 2444 y
FG(j)774 2517 y(i;p)885 2492 y FL(:=)f(\(1)22 b FC(\000)h
FI(q)1271 2450 y Fx(\000)p FH(1)1365 2492 y FL(\))1403
2450 y FG(N)1470 2492 y FL(\()p FI(B)1587 2444 y FG(p;j)1582
2517 y(i;p)1679 2492 y FL(\))1717 2456 y(\()1755 2417
y Fo(N)1766 2474 y Fy(2)1808 2456 y FL(\))1850 2492 y
FI(;)212 b FL(1)28 b FC(\024)g FI(i)g(<)f(p)h(<)g(j)33
b FC(\024)28 b FI(n)23 b FL(+)f(1)p FI(:)257 2651 y FL(The)36
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2958 y FI(e)631 2973 y FG(i;i)p FH(+1)821 2958 y FL(:=)27
b FI(x)1006 2973 y FG(i)1035 2958 y FI(;)1471 b FL(1)28
b FC(\024)g FI(i)g FC(\024)g FI(n)p FL(;)329 b(\(6.4\))668
3119 y FI(e)713 3134 y FG(i;j)821 3119 y FL(:=)27 b FI(e)996
3134 y FG(i;j)t Fx(\000)p FH(1)1167 3119 y FI(e)1212
3134 y FG(j)t Fx(\000)p FH(1)p FG(;j)1413 3119 y FC(\000)c
FI(B)1592 3071 y FG(i;j)t Fx(\000)p FH(1)1587 3144 y
FG(j)t Fx(\000)p FH(1)p FG(;j)1765 3119 y FI(e)1810 3134
y FG(j)t Fx(\000)p FH(1)p FG(;j)1989 3119 y FI(e)2034
3134 y FG(i;j)t Fx(\000)p FH(1)2533 3119 y FI(j)28 b
FC(\000)23 b FI(i)28 b FC(\025)g FL(2)p FI(:)352 b FL(\(6.5\))257
3305 y(W)-8 b(e)33 b(set)828 3425 y FI(\037)889 3440
y FG(i;j)997 3425 y FL(:=)1158 3330 y Fw(Y)1128 3543
y FG(i)p Fx(\024)p FG(l)q(<j)1333 3425 y FI(\037)1394
3440 y FG(l)1420 3425 y FI(;)114 b(g)1608 3440 y FG(i;j)1715
3425 y FL(:=)1876 3330 y Fw(Y)1846 3543 y FG(i)p Fx(\024)p
FG(l)q(<j)2051 3425 y FI(g)2098 3440 y FG(l)2221 3425
y FL(and)98 b FI(h)2532 3440 y FG(i;j)2640 3425 y FL(:=)27
b FI(g)2821 3384 y FG(N)2817 3450 y(i;j)2897 3425 y FI(:)257
3693 y FL(Then)40 b FI(B)597 3657 y FG(i;j)592 3718 y(p;r)724
3693 y FL(=)e FI(\037)899 3708 y FG(p;r)992 3693 y FL(\()p
FI(g)1077 3708 y FG(i;j)1157 3693 y FL(\))p FI(:)h FL(Next)g(w)m(e)h
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FL(The)39 b(elemen)m(ts)f FI(u)2777 3829 y FG(i;j)2893
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3934 y(de\014ned)c(inductiv)m(ely:)394 4132 y FI(u)450
4147 y FG(i;j)530 4132 y FL(\()p FI(\015)5 b FL(\))27
b(:=)h FI(\015)871 4147 y FG(i;j)951 4132 y FL(\(1)21
b FC(\000)i FI(h)1215 4147 y FG(i;j)1295 4132 y FL(\))f(+)1482
4038 y Fw(X)1453 4248 y FG(i<p<j)1672 4132 y FI(C)1749
4085 y FG(j)1742 4158 y(i;p)1825 4132 y FI(\015)1876
4147 y FG(i;p)1959 4132 y FI(u)2015 4147 y FG(p;j)2106
4132 y FL(\()p FI(\015)5 b FL(\))p FI(;)114 b FL(1)27
b FC(\024)i FI(i)f(<)f(j)34 b FC(\024)28 b FI(n)22 b
FL(+)g(1)p FI(:)137 b FL(\(6.6\))257 4414 y(W)-8 b(e)33
b(require)g(that)g(for)f(all)e FI(i)e(<)g(j;)1185 4600
y(\015)1236 4615 y FG(i;j)1343 4600 y FL(=)g(0)97 b(if)g
FI(\037)1809 4559 y FG(N)1809 4625 y(i;j)1917 4600 y
FC(6)p FL(=)27 b FI(")44 b FL(or)27 b FI(h)2280 4615
y FG(i;j)2388 4600 y FL(=)h(1)p FI(:)727 b FL(\(6.7\))257
4786 y(W)-8 b(e)38 b(will)d(call)g(suc)m(h)k(families)34
b FE(admissible)p FL(.)56 b(In)37 b([AS5)q(,)h(Lemma)e(7.20.])57
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4922 y FG(i;j)2826 4907 y FI(;)g FL(and)g(pro)m(v)m(es)h(all)257
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5147 y FL(F)-8 b(or)43 b(an)g(admissible)f(family)f FI(\015)49
b FL(w)m(e)44 b(simply)f(write)g FC(A)p FL(\()p FI(\015)5
b FL(\))43 b(instead)h(of)f Fs(u)p FL(\()p FC(D)s FI(;)17
b Fq(u)45 b FL(=)257 5268 y(\()p FI(u)351 5283 y FG(i;j)431
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5388 y(sional)d(p)s(oin)m(ted)i(Hopf)f(algebra.)42 b(Our)33
b(result)f(is:)1828 5637 y(65)p eop
%%Page: 66 68
66 67 bop 257 573 a FJ(Theorem)37 b(6.2)98 b FE(F)-7
b(or)44 b(a)h(linking)f(datum)h(of)f FI(A)2095 588 y
FG(n)2187 573 y FE(typ)-5 b(e)46 b(and)e(two)h(admissible)e(fami-)501
693 y(lies)f FI(\015)48 b FE(and)42 b FI(\015)1035 657
y Fx(0)1101 693 y FE(of)g(sc)-5 b(alars)42 b(the)g(Hopf)h(algebr)-5
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FC(A)p FL(\()p FI(\015)2967 657 y Fx(0)2990 693 y FL(\))g
FE(ar)-5 b(e)43 b(quasi-)501 814 y(isomorphic.)257 1013
y FL(F)-8 b(or)25 b(the)i(pro)s(of)e(of)g(this)h(theorem)f(w)m(e)i
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1133 y([AS5)q(].)404 1253 y(Let)g FI(\000)38 b FL(b)s(e)25
b(an)g(ab)s(elian)d(group)j(and)g FI(H)1784 1268 y FH(0)1848
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FC(f)p FL(1)p FI(;)17 b FL(2)p FI(;)g(:)g(:)g(:)e(;)i(P)d
FC(g)p FI(;)31 b FL(in)m(tegers)257 1494 y FI(N)335 1509
y FG(j)403 1494 y FI(>)f FL(1)35 b(where)g FI(j)i FC(2)31
b FI(Z)41 b FL(and)35 b(elemen)m(ts)g FI(x)1806 1509
y FG(i)1865 1494 y FC(2)c FI(H)2043 1509 y FH(0)2082
1494 y FI(;)k(h)2200 1509 y FG(i)2259 1494 y FC(2)c FI(\000)s(;)k(\021)
2529 1509 y FG(i)2588 1494 y FC(2)2706 1469 y FL(^)2685
1494 y FI(\000)14 b(;)35 b FL(1)30 b FC(\024)h FI(i)g
FC(\024)h FI(P)s(;)i FL(suc)m(h)257 1615 y(that)594 1830
y FI(g)t(x)700 1845 y FG(i)756 1830 y FL(=)27 b FI(\021)907
1845 y FG(i)936 1830 y FL(\()p FI(g)t FL(\))p FI(x)1118
1845 y FG(i)1145 1830 y FI(g)101 b FL(for)32 b(all)f
FI(g)g FC(2)d FI(\000)s(;)44 b FL(1)28 b FC(\024)g FI(i)g
FC(\024)g FI(P)s(:)496 1998 y(x)551 2013 y FG(i)580 1998
y FI(x)635 1944 y FG(N)691 1954 y Fo(j)635 2024 y FG(j)756
1998 y FL(=)f FI(\021)911 1944 y FG(N)967 1954 y Fo(j)907
2024 y FG(j)1004 1998 y FL(\()p FI(h)1098 2013 y FG(i)1126
1998 y FL(\))p FI(x)1219 1944 y FG(N)1275 1954 y Fo(j)1219
2024 y FG(j)1312 1998 y FI(x)1367 2013 y FG(i)1493 1998
y FL(for)32 b(all)f(1)c FC(\024)h FI(i)g FC(\024)g FI(P)s(;)44
b(j)34 b FC(2)28 b FI(Z)r(:)414 2154 y FL(The)33 b(elemen)m(ts)g
FI(x)1068 2111 y FG(a)1105 2120 y Fy(1)1068 2179 y FH(1)1161
2154 y FI(:)17 b(:)g(:)f(x)1347 2109 y FG(a)1384 2120
y Fo(P)1347 2181 y FG(P)1441 2154 y FI(g)t(;)43 b(a)1613
2169 y FH(1)1653 2154 y FI(;)17 b(:)g(:)g(:)f(a)1879
2169 y FG(P)1965 2154 y FC(\025)28 b FL(0)p FI(;)17 b(g)31
b FC(2)d FI(\000)46 b FL(form)32 b(a)g FF(|)-9 b FL(-basi)o(s)27
b(of)33 b FI(H)3273 2169 y FH(0)3312 2154 y FI(:)257
2568 y FJ(Theorem)k(6.3)98 b FE(L)-5 b(et)31 b FI(u)1193
2583 y FG(j)1229 2568 y FI(;)44 b(j)34 b FC(2)28 b FI(Z)r(;)i
FE(b)-5 b(e)30 b(a)g(family)g(of)g(elements)f(in)h FF(|)-9
b FI(\000)s(;)25 b FE(and)30 b FI(I)38 b FE(the)30 b(ide)-5
b(al)501 2704 y(in)37 b FI(H)704 2719 y FH(0)781 2704
y FE(gener)-5 b(ate)g(d)36 b(by)i(al)5 b(l)37 b FI(x)1542
2650 y FG(N)1598 2660 y Fo(j)1542 2730 y FG(j)1659 2704
y FC(\000)24 b FI(u)1816 2719 y FG(j)1852 2704 y FI(;)38
b(j)g FC(2)33 b FI(Z)r(:)j FE(L)-5 b(et)38 b FI(A)32
b FL(=)g FI(H)2691 2719 y FH(0)2731 2704 y FI(=I)45 b
FE(b)-5 b(e)37 b(the)g(quotient)501 2824 y(algebr)-5
b(a.)501 2986 y(If)35 b FI(u)660 3001 y FG(j)730 2986
y FE(is)g(c)-5 b(entr)g(al)35 b(in)f FI(H)1353 3001 y
FH(0)1427 2986 y FE(for)h(al)5 b(l)34 b FI(j)g FC(2)28
b FI(Z)42 b FE(and)34 b FI(u)2244 3001 y FG(j)2308 2986
y FL(=)28 b(0)34 b FE(if)h FI(\021)2642 2932 y FG(N)2698
2942 y Fo(j)2638 3011 y FG(j)2762 2986 y FC(6)p FL(=)27
b FI(")p FE(,)501 3106 y(then)k(the)g(r)-5 b(esidue)31
b(classes)f(of)h FI(x)1682 3062 y FG(a)1719 3071 y Fy(1)1682
3130 y FH(1)1776 3106 y FI(:)17 b(:)g(:)f(x)1962 3061
y FG(a)1999 3072 y Fo(P)1962 3133 y FG(P)2055 3106 y
FI(g)t(;)30 b(a)2214 3121 y FG(i)2270 3106 y FC(\025)f
FL(0)p FI(;)i(a)2534 3121 y FG(j)2598 3106 y FI(<)d(N)2780
3121 y FG(j)2847 3106 y FE(if)j FI(j)j FC(2)28 b FI(Z)r(;)j(g)g
FC(2)d FI(\000)s(;)501 3226 y FE(form)35 b(a)f FF(|)-8
b FE(-b)i(asis)29 b(of)34 b FI(A:)257 3425 y FL(The)c(pro)s(of)f(of)f
(this)h(theorem)g(is)f(exactly)i(the)f(same)g(as)g(in)g(the)g(original)
d(pap)s(er,)k(where)257 3546 y FI(Z)40 b FL(included)32
b(all)e(indices)j(from)e(1)h(to)h FI(P)s(:)404 3787 y
FL(T)-8 b(o)38 b(see)h(ho)m(w)f(this)g(can)g(b)s(e)g(applied)f(in)g
(our)h(situation,)g(w)m(e)h(recall)e(t)m(w)m(o)h(more)f(re-)257
3907 y(sults)45 b(\(Theorem)h(7.21.)e(and)h(Lemma)f(7.22.\))80
b(from)43 b([AS5)q(].)80 b(First,)47 b(the)f(elemen)m(ts)257
4037 y FI(e)302 3993 y FG(b)332 4002 y Fy(12)302 4061
y FH(1)p FG(;)p FH(2)402 4037 y FI(e)447 3993 y FG(b)477
4002 y Fy(13)447 4061 y FH(1)p FG(;)p FH(3)563 4037 y
FI(:)17 b(:)g(:)f(e)739 3986 y FG(b)769 3995 y Fo(nn)p
Fy(+1)739 4061 y FG(n;n)p FH(+1)939 4037 y FI(g)t(;)32
b(g)f FC(2)d FI(\000)s(;)33 b(b)1385 4052 y FG(ij)1474
4037 y FC(\025)28 b FL(0)p FI(;)k FL(where)i(the)f(ro)s(ot)f(v)m
(ectors)i(are)e(arranged)h(in)f(the)257 4157 y(lexicographic)25
b(order,)i(form)e(a)h FF(|)-9 b FL(-basi)o(s)21 b(of)k
FI(H)1920 4172 y FH(0)1959 4157 y FI(:)h FL(F)-8 b(urthermore,)27
b(w)m(e)g(ha)m(v)m(e)g(the)g(follo)m(wing)257 4278 y(crucial)32
b(comm)m(utation)e(rule)i(for)g(all)f(1)c FC(\024)h FI(i)g(<)g(j)33
b FC(\024)c FI(n)22 b FL(+)g(1)p FI(;)32 b FL(1)c FC(\024)g
FI(s)g(<)f(t)h FC(\024)g FI(N)33 b FL(+)22 b(1)p FI(;)1372
4493 y(e)1417 4508 y FG(i;j)1497 4493 y FI(e)1542 4451
y FG(N)1542 4517 y(s;t)1652 4493 y FL(=)28 b FI(\037)1817
4451 y FG(N)1817 4517 y(s;t)1898 4493 y FL(\()p FI(g)1983
4508 y FG(i;j)2063 4493 y FL(\))p FI(e)2146 4451 y FG(N)2146
4517 y(s;t)2228 4493 y FI(e)2273 4508 y FG(i;j)2354 4493
y FI(:)914 b FL(\(6.8\))257 4708 y(W)-8 b(e)41 b(set)h(no)m(w)f
FI(H)886 4723 y FH(0)966 4708 y FL(:=)h Fs(U)p FL(\()p
FC(D)s FL(\))e(and)h(kno)m(w)g(that)g(the)g(coradical)e(is)h(a)g
(subalgebra.)67 b(So)257 4828 y(the)37 b(last)e(theorem)h(giv)m(es)g
(us,)i(for)d(instance,)j(a)e(basis)g(of)f FI(A)p FL(\()p
FI(\015)5 b FL(\))p FI(;)36 b FL(as)g(w)m(e)i(get)e Fs(u)f
FL(from)g Fs(U)257 4948 y FL(exactly)e(b)m(y)h(dividing)d(out)h(the)h
(ro)s(ot)f(v)m(ector)h(relations.)404 5069 y(W)-8 b(e)33
b(are)f(no)m(w)h(ready)h(to)e(giv)m(e)g(the)h(pro)s(of)f(of)g(our)h
(Theorem.)257 5268 y FJ(Pro)s(of)38 b(of)f(Theorem)g(6.2:)49
b FL(As)36 b(in)f(the)h(linking)d(case,)k(it)d(is)h(enough)h(to)f(pro)m
(v)m(e)h(that)501 5388 y(for)h(an)m(y)i(admissible)c(family)h
FI(\015)5 b FL(,)39 b FI(A)p FL(\()p FI(\015)5 b FL(\))37
b(is)h(quasi-isomorphic)d(to)i FI(A)p FL(\()p FI(\015)3131
5403 y FH(0)3170 5388 y FL(\))h(where)1828 5637 y(66)p
eop
%%Page: 67 69
67 68 bop 501 573 a FI(\015)552 588 y FH(0)624 573 y
FL(denotes)34 b(the)f(family)d(in)i(whic)m(h)h(all)d(the)j
FI(\015)2190 588 y FG(i;j)2302 573 y FL(are)g(zero.)44
b(Again,)31 b(this)i(will)d(b)s(e)501 693 y(ac)m(hiev)m(ed)k(b)m(y)g
(follo)m(wing)c(a)i(step)m(wise)i(pro)s(cedure.)501 852
y FC(\017)29 b FL(Fix)g FI(i)785 867 y FH(0)854 852 y
FL(with)g(1)e FC(\024)h FI(i)1287 867 y FH(0)1355 852
y FC(\024)g FI(n)i FL(suc)m(h)g(that)g(for)e(the)i(giv)m(en)g
(admissible)d(family)g FI(\015)34 b FL(w)m(e)501 973
y(ha)m(v)m(e)1065 1093 y FI(\015)1116 1108 y FG(i;j)1223
1093 y FL(=)28 b(0)k(for)g(all)f(1)c FC(\024)h FI(i)g(<)g(i)2072
1108 y FH(0)2144 1093 y FL(and)33 b FI(i)28 b(<)f(j)34
b FC(\024)28 b FI(n)23 b FL(+)f(1)p FI(:)501 1260 y FL(Set)42
b(~)-52 b FI(\015)726 1275 y FG(i;j)842 1260 y FL(:=)37
b FI(\015)1033 1275 y FG(i;j)1151 1260 y FL(if)g FI(i)g
FC(6)p FL(=)g FI(i)1462 1275 y FH(0)1540 1260 y FL(and)42
b(~)-53 b FI(\015)1786 1275 y FG(i)1810 1284 y Fy(0)1844
1275 y FG(;j)1937 1260 y FL(:=)37 b(0)h(for)f(all)f FI(i)2492
1275 y FH(0)2569 1260 y FI(<)h(j)43 b FC(\024)37 b FI(n)26
b FL(+)g(1)p FI(:)38 b FL(Then)43 b(~)-53 b FI(\015)501
1380 y FL(is)37 b(again)f(admissible.)57 b(It)38 b(is)f(no)m(w)h
(su\016cien)m(t)h(to)e(pro)m(v)m(e)i(that)e FC(A)p FL(\()p
FI(\015)5 b FL(\))37 b(and)h FC(A)p FL(\()t(~)-53 b FI(\015)t
FL(\))501 1500 y(are)41 b(quasi-isomorphic)d(and)i(then)h(to)g(rep)s
(eat)f(this)g(step)i(with)e(increasing)f FI(i)3428 1515
y FH(0)3468 1500 y FI(;)501 1621 y FL(replacing)32 b
FI(\015)37 b FL(b)m(y)g(~)-52 b FI(\015)5 b(:)32 b FL(Set)572
1822 y FI(H)j FL(=)28 b FI(H)873 1837 y FH(0)912 1822
y FI(=I)8 b(;)44 b(I)36 b FL(:=)27 b(\()p FI(e)1375 1781
y FG(N)1375 1847 y(i;j)1478 1822 y FC(\000)22 b FI(u)1633
1837 y FG(i;j)1757 1822 y FL(:)45 b(1)27 b FC(\024)h
FI(i)g(<)g(j)33 b FC(\024)c FI(n)22 b FL(+)g(1)p FI(;)44
b(i)28 b FC(6)p FL(=)g FI(i)2850 1837 y FH(0)2922 1822
y FL(or)k FI(\037)3102 1781 y FG(N)3102 1847 y(i;j)3210
1822 y FC(6)p FL(=)c FI(")o FL(\))p FI(:)501 2024 y FL(Note)33
b(that)f FI(u)1004 2039 y FG(i;j)1084 2024 y FL(\()p
FI(\015)5 b FL(\))28 b(=)f FI(u)1403 2039 y FG(i;j)1483
2024 y FL(\()t(~)-53 b FI(\015)5 b FL(\))32 b(for)g(all)f(the)i
FI(u)2156 2039 y FG(i;j)2268 2024 y FL(app)s(earing)e(in)h(the)h(ideal)
e FI(I)8 b(:)501 2183 y FC(\017)33 b FI(H)40 b FL(is)32
b(a)g(Hopf)g(algebra.)501 2304 y(T)-8 b(o)33 b(see)h(this,)e(w)m(e)h
(recall)f(the)h(com)m(ultiplication)28 b(on)k(the)h(ro)s(ot)f(v)m
(ectors)629 2499 y(\001\()p FI(e)793 2458 y FG(N)793
2523 y(i;j)896 2499 y FC(\000)22 b FI(u)1051 2514 y FG(i;j)1131
2499 y FL(\))28 b(=)f(\()p FI(e)1383 2458 y FG(N)1383
2523 y(i;j)1486 2499 y FC(\000)22 b FI(u)1641 2514 y
FG(i;j)1721 2499 y FL(\))g FC(\012)h FL(1)f(+)g FI(h)2106
2514 y FG(i;j)2208 2499 y FC(\012)h FL(\()p FI(e)2391
2458 y FG(N)2391 2523 y(i;j)2493 2499 y FC(\000)g FI(u)2649
2514 y FG(i;j)2728 2499 y FL(\)+)635 2674 y(+)761 2580
y Fw(X)733 2790 y FG(i<p<j)951 2674 y FI(C)1028 2627
y FG(j)1021 2700 y(i;p)1104 2674 y FI(e)1149 2633 y FG(N)1149
2699 y(i;p)1232 2674 y FI(h)1288 2689 y FG(p;j)1402 2674
y FC(\012)g FL(\()p FI(e)1585 2633 y FG(N)1585 2699 y(p;j)1699
2674 y FC(\000)g FI(u)1855 2689 y FG(p;j)1946 2674 y
FL(\))f(+)2133 2580 y Fw(X)2104 2790 y FG(i<p<j)2322
2674 y FI(C)2399 2627 y FG(j)2392 2700 y(i;p)2475 2674
y FL(\()p FI(e)2558 2633 y FG(N)2558 2699 y(i;p)2664
2674 y FC(\000)g FI(u)2819 2689 y FG(i;p)2902 2674 y
FL(\))p FI(h)2996 2689 y FG(p;j)3110 2674 y FC(\012)h
FI(u)3266 2689 y FG(p;j)3357 2674 y FI(:)3295 2515 y
FL(\(6.9\))501 2971 y(If)36 b FI(i)d(<)g(i)810 2986 y
FH(0)885 2971 y FL(w)m(e)j(ha)m(v)m(e)h FI(\015)1310
2986 y FG(i;j)1423 2971 y FL(=)c(0)i(for)g(all)e FI(i)g(<)g(j)39
b FC(\024)33 b FI(n)25 b FL(+)f(1)35 b(and)h(hence)h
FI(u)3060 2986 y FG(i;j)3172 2971 y FL(=)c(0)p FI(:)i
FL(So)501 3091 y(all)30 b(the)h(summands)h(in)e(\(6.9\))h(are)g(in)g
FI(H)1984 3106 y FH(0)2043 3091 y FC(\012)20 b FI(I)39
b FL(or)31 b FI(I)d FC(\012)20 b FI(H)2589 3106 y FH(0)2628
3091 y FI(;)32 b FL(b)s(ecause)g FI(e)3091 3055 y FG(N)3091
3116 y(i;p)3202 3091 y FC(2)d FI(I)39 b FL(for)501 3212
y(all)31 b FI(i)d(<)f(p)h(<)f(j:)501 3332 y FL(The)d(case)f
FI(i)28 b(>)g(i)1086 3347 y FH(0)1148 3332 y FL(is)22
b(ob)m(vious.)40 b(When)23 b FI(i)28 b FL(=)g FI(i)2095
3347 y FH(0)2157 3332 y FL(then,)d(according)d(to)g(the)h(de\014nition)
501 3453 y(of)i FI(I)8 b FL(,)27 b FI(e)755 3416 y FG(N)755
3477 y(i)779 3486 y Fy(0)814 3477 y FG(;j)878 3453 y
FC(\000)8 b FI(u)1019 3468 y FG(i)1043 3477 y Fy(0)1078
3468 y FG(;j)1160 3453 y FL(is)25 b(in)g FI(I)33 b FL(only)25
b(if)g FI(\037)1785 3416 y FG(N)1785 3477 y(i)1809 3486
y Fy(0)1844 3477 y FG(;j)1927 3453 y FC(6)p FL(=)j FI(")16
b(:)26 b FL(F)-8 b(rom)24 b(\(6.7\))h(follo)m(ws)g(then)h
FI(\015)3201 3468 y FG(i)3225 3477 y Fy(0)3259 3468 y
FG(;j)3343 3453 y FL(=)h(0)501 3573 y(and)33 b FI(u)747
3588 y FG(i)771 3597 y Fy(0)805 3588 y FG(;j)889 3573
y FL(=)27 b(0)p FI(:)33 b FL(W)-8 b(e)33 b(ha)m(v)m(e)1213
3774 y FI(\037)1274 3733 y FG(N)1274 3799 y(i)1298 3808
y Fy(0)1333 3799 y FG(;j)1417 3774 y FL(=)27 b FI(\037)1581
3733 y FG(N)1581 3799 y(i)1605 3808 y Fy(0)1640 3799
y FG(;p)1699 3774 y FI(\037)1760 3733 y FG(N)1760 3799
y(p;j)2047 3774 y FL(for)32 b(all)f FI(i)2365 3789 y
FH(0)2432 3774 y FI(<)d(p)f(<)h(j:)501 3976 y FL(And)33
b(hence)g FI(\037)1046 3940 y FG(N)1046 4001 y(i)1070
4010 y Fy(0)1105 4001 y FG(;p)1192 3976 y FC(6)p FL(=)28
b FI(")j FL(and)i FI(u)1619 3991 y FG(i)1643 4000 y Fy(0)1677
3991 y FG(;p)1764 3976 y FL(=)27 b(0)32 b(or)g FI(\037)2128
3940 y FG(N)2128 4001 y(p;j)2247 3976 y FC(6)p FL(=)c
FI(")k FL(and)g FI(u)2674 3991 y FG(p;j)2793 3976 y FL(=)27
b(0)p FI(:)32 b FL(This)g(pro)m(v)m(es)501 4096 y(that)1461
4217 y(\001)q(\()p FI(I)8 b FL(\))27 b FC(\032)h FI(H)1883
4232 y FH(0)1945 4217 y FC(\012)22 b FI(I)30 b FL(+)22
b FI(I)30 b FC(\012)23 b FI(H)2469 4232 y FH(0)2508 4217
y FI(:)501 4383 y FL(A)34 b(simple)e(calculation)f(sho)m(ws)k
FI(")p FL(\()p FI(e)1817 4398 y FG(i;j)1897 4383 y FL(\))30
b(=)f(0)g(=)h FI(")o FL(\()p FI(u)2393 4398 y FG(i;j)2473
4383 y FL(\))p FI(:)j FL(So)h(w)m(e)h(can)e(get)h(t)m(w)m(o)g(re-)501
4504 y(cursion)27 b(form)m(ulas)f(for)h(the)h(an)m(tip)s(o)s(de)e(of)h
FI(e)2079 4467 y FG(N)2079 4528 y(i;j)2170 4504 y FC(\000)11
b FI(u)2314 4519 y FG(i;j)2422 4504 y FL(from)26 b(the)i(com)m
(ultiplication)501 4624 y(form)m(ula,)40 b(as)f Fv(S)p
FL(\()p FI(a)1164 4639 y Fy(\(1\))1247 4624 y FL(\))p
FI(a)1336 4639 y Fy(\(2\))1458 4624 y FL(=)h FI(")o FL(\()p
FI(a)p FL(\))g(=)f FI(a)1952 4639 y Fy(\(1\))2052 4624
y Fv(S)p FL(\()p FI(a)2195 4639 y Fy(\(2\))2277 4624
y FL(\))p FI(:)h FL(Using)f(the)h(second)h(of)e(these)501
4744 y(form)m(ulas)h(for)h(the)h(case)h FI(i)g(<)g(i)1685
4759 y FH(0)1766 4744 y FL(w)m(e)f(immediately)d(ha)m(v)m(e)j
Fv(S)p FL(\()p FI(e)2853 4708 y FG(N)2853 4769 y(i;j)2934
4744 y FL(\))h FC(2)g FI(I)8 b(:)41 b FL(When)501 4865
y FI(i)28 b(>)g(i)699 4880 y FH(0)738 4865 y FI(;)33
b FL(an)g(inductiv)m(e)f(argumen)m(t)h(using)f(the)h(\014rst)g(form)m
(ula)e(and)1158 5066 y Fv(S)p FL(\()p FI(e)1295 5025
y FG(N)1295 5091 y(i;i)p FH(+1)1480 5066 y FC(\000)22
b FI(u)1635 5081 y FG(i;i)p FH(+1)1797 5066 y FL(\))27
b(=)h FC(\000)p FI(g)2094 5025 y Fx(\000)p FG(N)2090
5092 y(i)2216 5066 y FL(\()p FI(e)2299 5025 y FG(N)2299
5091 y(i;i)p FH(+1)2483 5066 y FC(\000)23 b FI(u)2639
5081 y FG(i;i)p FH(+1)2800 5066 y FL(\))501 5268 y(giv)m(es)32
b(again)e Fv(S)p FL(\()p FI(e)1135 5232 y FG(N)1135 5292
y(i;j)1215 5268 y FL(\))e FC(2)g FI(I)8 b(:)31 b FL(F)-8
b(or)31 b FI(i)d FL(=)f FI(i)1855 5283 y FH(0)1895 5268
y FI(;)k FL(a)h(com)m(bination)d(of)i(the)h(reasoning)e(from)501
5388 y(the)f(discussion)f(of)f(the)h(com)m(ultiplication)c(and)k(the)g
(inductiv)m(e)g(argumen)m(t)f(from)1828 5637 y(67)p eop
%%Page: 68 70
68 69 bop 501 573 a FL(the)36 b(last)f(case)h(giv)m(e)f(the)h(desired)g
(result,)g(establishing)e Fv(S)p FL(\()p FI(I)8 b FL(\))32
b FC(\032)h FI(I)8 b(:)36 b FL(Hence)g FI(I)44 b FL(is)501
693 y(a)33 b(Hopf)f(ideal.)501 855 y FC(\017)48 b FL(Next)g(w)m(e)h
(tak)m(e)f(as)g FI(K)55 b FL(the)48 b(subalgebra)f(of)g
FI(H)55 b FL(generated)49 b(b)m(y)f(the)g(group)501 976
y FI(\000)56 b FL(and)43 b(the)g(remaining)d FI(e)1505
939 y FG(N)1505 1000 y(i)1529 1009 y Fy(0)1563 1000 y
FG(;j)1620 976 y FI(;)i FL(i.e.)g FI(i)1889 991 y FH(0)1973
976 y FI(<)i(j)50 b FC(\024)45 b FI(n)29 b FL(+)g(1)p
FI(;)42 b FL(suc)m(h)i(that)e FI(\037)3127 939 y FG(N)3127
1000 y(i)3151 1009 y Fy(0)3185 1000 y FG(;j)3286 976
y FL(=)i FI(")16 b(:)501 1096 y FL(A)46 b(similar)d(calculation)g(to)j
(the)g(one)g(ab)s(o)m(v)m(e)h(rev)m(eals)g(that)e FI(K)53
b FL(is)46 b(actually)e(a)501 1216 y(Hopf)49 b(subalgebra)f(of)g
FI(H)r(:)h FL(This)g(time,)i(one)e(has)g(to)f(use)i(the)f(fact)f(that)h
(for)501 1337 y(an)m(y)43 b FI(p)e FL(b)s(et)m(w)m(een)j
FI(i)1204 1352 y FH(0)1285 1337 y FL(and)e FI(j;)g FL(either)g
FI(\037)1940 1301 y FG(N)1940 1361 y(i)1964 1370 y Fy(0)1998
1361 y FG(;p)2101 1337 y FL(=)h FI(")e FL(or)g(that)h
FI(\037)2717 1301 y FG(N)2717 1361 y(p;j)2852 1337 y
FC(6)p FL(=)h FI(")e FL(and)h(hence)501 1457 y FI(u)557
1472 y FG(p;j)676 1457 y FL(=)28 b(0)f(=)h FI(e)1005
1421 y FG(N)1005 1482 y(p;j)1129 1457 y FL(in)k FI(H)r(:)501
1619 y FL(As)g(all)d(the)i FI(u)1000 1634 y FG(i;j)1111
1619 y FL(ful\014ll)e(the)i(conditions)f(of)h(Theorem)g(6.3)g(w)m(e)h
(immediately)c(get)501 1739 y(a)40 b(basis)f(of)h FI(H)r(:)f
FL(The)i(comm)m(utation)d(relations)g(\(6.8\))h(for)g(the)i
FI(N)2976 1703 y FH(th)3087 1739 y FL(p)s(o)m(w)m(ers)g(of)501
1860 y(the)33 b(ro)s(ot)e(v)m(ectors)j(sho)m(w)f(that)f(the)g
(generators)h(of)f FI(K)39 b FL(all)30 b(comm)m(ute)i(with)f(eac)m(h)
501 1980 y(other,)42 b(b)s(ecause)e(the)g(factor)f(is)g(1)p
FI(;)g FL(as)h FI(\037)2028 1944 y FG(N)2028 2005 y(i)2052
2014 y Fy(0)2087 2005 y FG(;j)2182 1980 y FL(=)f FI(")g
FL(for)g(all)e(generators.)65 b(Hence,)501 2100 y(an)m(y)31
b(monomial)26 b(in)k FI(K)37 b FL(can)30 b(b)s(e)h(reordered)g(and)f
(is)g(then)g(a)g(basis)g(elemen)m(t)g(of)g FI(H)r(:)501
2221 y FL(So)j FI(K)39 b FL(is)32 b(just)h(the)g(p)s(olynomial)c
(algebra)j(on)g(its)g(generators.)501 2383 y FC(\017)h
FL(W)-8 b(e)33 b(de\014ne)g(an)g(algebra)e(map)h FI(f)38
b FL(:)28 b FI(K)35 b FC(!)27 b FF(|)18 b FL(b)m(y)34
b(setting)561 2603 y FI(f)11 b FL(\()p FI(e)703 2562
y FG(N)703 2627 y(i)727 2636 y Fy(0)762 2627 y FG(;j)818
2603 y FL(\))28 b(:=)f FI(\015)1065 2618 y FG(i)1089
2627 y Fy(0)1124 2618 y FG(;j)1180 2603 y FI(;)211 b(f)11
b FL(\()p FI(g)t FL(\))27 b(:=)h(1)195 b(on)32 b(all)f(the)i
(generators)g(of)f FI(K)r(;)h(g)e FC(2)d FI(\000)s(:)501
2823 y FL(Algebra)34 b(maps,)i(from)e(a)g(Hopf)h(algebra)f
FI(K)42 b FL(to)35 b(the)g(base)h(\014eld,)g(form)e(a)g(group)501
2943 y(under)i(the)f(con)m(v)m(olution)g(pro)s(duct)g(where)h(the)f(in)
m(v)m(erse)i(is)d(giv)m(en)h(b)m(y)h(the)f(com-)501 3064
y(p)s(osition)40 b(with)h(the)h(an)m(tip)s(o)s(de.)70
b(This)41 b(group)h(acts)g(on)f(the)h(Hopf)g(algebra)e
FI(K)501 3184 y FL(from)32 b(the)h(left)e(and)i(the)g(righ)m(t)f(b)m(y)
1185 3404 y FI(f)5 b(:x)28 b FL(=)g FI(x)1507 3419 y
Fy(\(1\))1590 3404 y FI(f)11 b FL(\()p FI(x)1742 3419
y Fy(\(2\))1825 3404 y FL(\))p FI(;)211 b(x:f)39 b FL(=)28
b FI(f)11 b FL(\()p FI(x)2526 3419 y Fy(\(1\))2608 3404
y FL(\))p FI(x)2701 3419 y Fy(\(2\))2784 3404 y FI(:)501
3624 y FC(\017)43 b FL(T)-8 b(o)43 b(b)s(e)g(able)f(to)h(apply)f
(Theorem)h(2.)g(of)f([Mas1)q(],)j(whic)m(h)f(will)c(giv)m(e)j(us)h(the)
501 3744 y(desired)27 b(quasi-isomorphism,)d(w)m(e)j(ha)m(v)m(e)h(to)d
(calculate)g FI(f)5 b(:e)2663 3708 y FG(N)2663 3769 y(i)2687
3778 y Fy(0)2723 3769 y FG(;j)2779 3744 y FI(:f)2865
3708 y Fx(\000)p FH(1)2959 3744 y FI(:)26 b FL(As)h(a)e(prepa-)501
3865 y(ration)32 b(for)h(this)g(w)m(e)h(\014rst)g(calculate)e
FI(f)11 b FL(\()p FI(u)2036 3880 y FG(i;i)p FH(+1)2197
3865 y FL(\))29 b(=)f(0)33 b(for)g(all)e(1)e FC(\024)g
FI(i)g FC(\024)g FI(n)34 b FL(and)f(see)501 3985 y(then)38
b(from)d(the)j(inductiv)m(e)f(de\014nition)f(\(6.6\))g(that)h
FI(f)11 b FL(\()p FI(u)2606 4000 y FG(i;j)2685 3985 y
FL(\))35 b(=)g(0)i(for)f(all)f FI(i)h(<)f(j:)501 4105
y FL(So)e(for)f(the)h(generators)g(of)f FI(K)40 b FL(w)m(e)33
b(ha)m(v)m(e)776 4337 y FI(f)5 b(:e)901 4296 y FG(N)901
4362 y(i)925 4371 y Fy(0)961 4362 y FG(;j)1044 4337 y
FL(=)28 b FI(e)1193 4296 y FG(N)1193 4362 y(i)1217 4371
y Fy(0)1252 4362 y FG(;j)1308 4337 y FI(f)11 b FL(\(1\))21
b(+)h FI(h)1667 4352 y FG(i)1691 4361 y Fy(0)1726 4352
y FG(;j)1782 4337 y FI(f)11 b FL(\()p FI(e)1924 4296
y FG(N)1924 4362 y(i)1948 4371 y Fy(0)1983 4362 y FG(;j)2039
4337 y FL(\))22 b(+)2243 4243 y Fw(X)2197 4453 y FG(i)2221
4462 y Fy(0)2256 4453 y FG(<p<j)2450 4337 y FI(C)2527
4290 y FG(j)2520 4363 y(i)2544 4372 y Fy(0)2578 4363
y FG(;p)2638 4337 y FI(e)2683 4296 y FG(N)2683 4362 y(i)2707
4371 y Fy(0)2741 4362 y FG(;p)2801 4337 y FI(h)2857 4352
y FG(p;j)2949 4337 y FI(f)11 b FL(\()p FI(e)3091 4296
y FG(N)3091 4362 y(p;j)3182 4337 y FL(\))1044 4595 y(=)28
b FI(e)1193 4554 y FG(N)1193 4620 y(i)1217 4629 y Fy(0)1252
4620 y FG(;j)1330 4595 y FL(+)22 b FI(\015)1479 4610
y FG(i)1503 4619 y Fy(0)1537 4610 y FG(;j)1593 4595 y
FI(h)1649 4610 y FG(i)1673 4619 y Fy(0)1708 4610 y FG(;j)1764
4595 y FI(;)501 4823 y FL(as)36 b FI(e)669 4787 y FG(N)669
4848 y(p;j)794 4823 y FL(=)c FI(u)958 4838 y FG(p;j)1085
4823 y FL(in)j FI(H)r(:)g FL(The)i(recursiv)m(e)g(form)m(ula)c(for)i
(the)h(an)m(tip)s(o)s(de)f(of)g(the)h FI(N)3424 4787
y FH(th)501 4943 y FL(p)s(o)m(w)m(ers)e(of)e(the)h(ro)s(ot)f(v)m
(ectors)i(is)990 5175 y Fv(S)p FL(\()p FI(e)1127 5134
y FG(N)1127 5200 y(i)1151 5209 y Fy(0)1186 5200 y FG(;j)1242
5175 y FL(\))28 b(=)f FC(\000)p FI(h)1544 5134 y Fx(\000)p
FH(1)1544 5201 y FG(i)1568 5210 y Fy(0)1604 5201 y FG(;j)1660
5175 y FI(e)1705 5134 y FG(N)1705 5200 y(i)1729 5209
y Fy(0)1764 5200 y FG(;j)1842 5175 y FC(\000)1988 5081
y Fw(X)1942 5291 y FG(i)1966 5300 y Fy(0)2000 5291 y
FG(<p<j)2194 5175 y FI(C)2271 5128 y FG(j)2264 5201 y(i)2288
5210 y Fy(0)2323 5201 y FG(;p)2382 5175 y FI(h)2438 5134
y Fx(\000)p FH(1)2438 5201 y FG(p;j)2549 5175 y Fv(S)p
FL(\()p FI(e)2686 5134 y FG(N)2686 5200 y(i)2710 5209
y Fy(0)2745 5200 y FG(;p)2804 5175 y FL(\))p FI(e)2887
5134 y FG(N)2887 5200 y(p;j)2979 5175 y FI(:)1828 5637
y FL(68)p eop
%%Page: 69 71
69 70 bop 501 573 a FL(So)33 b(w)m(e)g(get)668 787 y(\()p
FI(f)5 b(:e)831 746 y FG(N)831 811 y(i)855 820 y Fy(0)890
811 y FG(;j)946 787 y FL(\))p FI(:f)1070 746 y Fx(\000)p
FH(1)1192 787 y FL(=)27 b FI(\015)1346 802 y FG(i)1370
811 y Fy(0)1404 802 y FG(;j)1460 787 y FI(f)11 b FL(\()p
FI(h)1613 745 y Fx(\000)p FH(1)1613 812 y FG(i)1637 821
y Fy(0)1672 812 y FG(;j)1728 787 y FL(\))p FI(h)1822
802 y FG(i)1846 811 y Fy(0)1881 802 y FG(;j)1959 787
y FL(+)22 b FI(f)11 b FL(\()p Fv(S)p FL(\()p FI(e)2291
746 y FG(N)2291 811 y(i)2315 820 y Fy(0)2349 811 y FG(;j)2406
787 y FL(\)\))22 b(+)g FI(f)11 b FL(\()p FI(h)2755 745
y Fx(\000)p FH(1)2755 812 y FG(i)2779 821 y Fy(0)2813
812 y FG(;j)2869 787 y FL(\))p FI(e)2952 746 y FG(N)2952
811 y(i)2976 820 y Fy(0)3011 811 y FG(;j)3067 787 y FL(+)1393
963 y(+)1520 868 y Fw(X)1491 1078 y FG(i<p<j)1710 963
y FI(C)1787 916 y FG(j)1780 988 y(i;p)1863 963 y FI(f)g
FL(\()p FI(h)2016 922 y Fx(\000)p FH(1)2016 988 y FG(i)2040
997 y Fy(0)2074 988 y FG(;j)2130 963 y FL(\))p FI(f)g
FL(\()p Fv(S)p FL(\()p FI(e)2402 922 y FG(N)2402 988
y(i)2426 997 y Fy(0)2461 988 y FG(;p)2520 963 y FL(\)\))p
FI(e)2641 922 y FG(N)2641 988 y(p;j)814 1221 y FL(=)27
b FI(e)962 1180 y FG(N)962 1245 y(i)986 1254 y Fy(0)1021
1245 y FG(;j)1099 1221 y FL(+)22 b FI(\015)1248 1236
y FG(i)1272 1245 y Fy(0)1306 1236 y FG(;j)1363 1221 y
FI(h)1419 1236 y FG(i)1443 1245 y Fy(0)1477 1236 y FG(;j)1533
1221 y FL(+)906 1466 y(+)g FI(f)1079 1296 y Fw( )1158
1466 y FC(\000)p FI(h)1291 1425 y Fx(\000)p FH(1)1291
1491 y FG(i)1315 1500 y Fy(0)1350 1491 y FG(;j)1406 1466
y FI(e)1451 1425 y FG(N)1451 1491 y(i)1475 1500 y Fy(0)1510
1491 y FG(;j)1588 1466 y FC(\000)1717 1371 y Fw(X)1671
1581 y FG(i)1695 1590 y Fy(0)1730 1581 y FG(<p<j)1924
1466 y FI(C)2001 1419 y FG(j)1994 1491 y(i)2018 1500
y Fy(0)2053 1491 y FG(;p)2112 1466 y FI(h)2168 1425 y
Fx(\000)p FH(1)2168 1491 y FG(p;j)2279 1466 y Fv(S)p
FL(\()p FI(e)2416 1425 y FG(N)2416 1491 y(i)2440 1500
y Fy(0)2475 1491 y FG(;p)2534 1466 y FL(\))p FI(e)2617
1425 y FG(N)2617 1491 y(p;j)2709 1296 y Fw(!)2804 1466
y FL(+)906 1812 y(+)1032 1717 y Fw(X)1004 1927 y FG(i<p<j)1222
1812 y FI(C)1299 1764 y FG(j)1292 1837 y(i;p)1375 1812
y FI(f)1450 1641 y Fw( )1529 1812 y FC(\000)p FI(h)1662
1770 y Fx(\000)p FH(1)1662 1837 y FG(i)1686 1846 y Fy(0)1721
1837 y FG(;p)1781 1812 y FI(e)1826 1771 y FG(N)1826 1836
y(i)1850 1845 y Fy(0)1885 1836 y FG(;p)1966 1812 y FC(\000)2096
1717 y Fw(X)2049 1927 y FG(i)2073 1936 y Fy(0)2108 1927
y FG(<q)r(<p)2303 1812 y FI(C)2380 1764 y FG(p)2373 1837
y(i)2397 1846 y Fy(0)2432 1837 y FG(;q)2489 1812 y FI(h)2545
1771 y Fx(\000)p FH(1)2545 1836 y FG(q)r(;p)2657 1812
y Fv(S)p FL(\()p FI(e)2794 1771 y FG(N)2794 1836 y(i)2818
1845 y Fy(0)2852 1836 y FG(;q)2910 1812 y FL(\))p FI(e)2993
1771 y FG(N)2993 1836 y(q)r(;p)3086 1641 y Fw(!)3182
1812 y FI(u)3238 1827 y FG(p;j)814 2088 y FL(=)27 b FI(e)962
2047 y FG(N)962 2112 y(i)986 2121 y Fy(0)1021 2112 y
FG(;j)1099 2088 y FL(+)22 b FI(\015)1248 2103 y FG(i)1272
2112 y Fy(0)1306 2103 y FG(;j)1363 2088 y FI(h)1419 2103
y FG(i)1443 2112 y Fy(0)1477 2103 y FG(;j)1556 2088 y
FC(\000)g FI(f)11 b FL(\()p FI(e)1797 2047 y FG(N)1797
2112 y(i)1821 2121 y Fy(0)1856 2112 y FG(;j)1912 2088
y FL(\))22 b FC(\000)2084 1993 y Fw(X)2055 2203 y FG(i<p<j)2273
2088 y FI(C)2350 2040 y FG(j)2343 2113 y(i;p)2426 2088
y FI(f)11 b FL(\()p FI(e)2568 2047 y FG(N)2568 2112 y(i)2592
2121 y Fy(0)2627 2112 y FG(;p)2686 2088 y FL(\))p FI(u)2780
2103 y FG(p;j)814 2364 y FL(=)27 b FI(e)962 2323 y FG(N)962
2388 y(i)986 2397 y Fy(0)1021 2388 y FG(;j)1099 2364
y FC(\000)c FI(\015)1250 2379 y FG(i)1274 2388 y Fy(0)1308
2379 y FG(;j)1364 2364 y FL(\(1)f FC(\000)h FI(h)1629
2379 y FG(i)1653 2388 y Fy(0)1687 2379 y FG(;j)1744 2364
y FL(\))f FC(\000)1916 2269 y Fw(X)1887 2479 y FG(i<p<j)2105
2364 y FI(C)2182 2316 y FG(j)2175 2389 y(i;p)2258 2364
y FI(\015)2309 2379 y FG(i)2333 2388 y Fy(0)2367 2379
y FG(;p)2427 2364 y FI(u)2483 2379 y FG(p;j)814 2622
y FL(=)27 b FI(e)962 2580 y FG(N)962 2646 y(i)986 2655
y Fy(0)1021 2646 y FG(;j)1099 2622 y FC(\000)c FI(u)1255
2637 y FG(i)1279 2646 y Fy(0)1313 2637 y FG(;j)1564 2622
y FL(b)m(y)34 b(\(6.6\))o FI(:)3246 2752 y FL(\(6.10\))501
2977 y(Tw)m(o)j(parts)f(of)f(the)h(sum)g(v)-5 b(anish)35
b(as)h(they)h(are)f(m)m(ultiples)d(of)j FI(f)11 b FL(\()p
FI(e)2968 2941 y FG(N)2968 3002 y(p;j)3092 2977 y FL(=)33
b FI(u)3257 2992 y FG(p;j)3348 2977 y FL(\))g(=)501 3098
y(0)28 b(=)f FI(f)11 b FL(\()p FI(e)823 3062 y FG(N)823
3122 y(q)r(;p)944 3098 y FL(=)27 b FI(u)1103 3113 y FG(q)r(;p)1196
3098 y FL(\))p FI(:)j FL(Note)f(that)g(in)g(the)h(second)h(to)e(last)f
(step,)j(if)e FI(j)k(>)28 b(i)3144 3113 y FH(0)3213 3098
y FL(is)h(suc)m(h)501 3218 y(that)h FI(\037)771 3182
y FG(N)771 3243 y(i)795 3252 y Fy(0)830 3243 y FG(;j)914
3218 y FC(6)p FL(=)e FI(")o(;)j FL(then)g FI(e)1386 3182
y FG(N)1386 3243 y(i)1410 3252 y Fy(0)1445 3243 y FG(;j)1531
3218 y FL(is)f(not)g(a)h(generator)f(of)g FI(K)38 b FL(and)30
b(w)m(e)i(can)e(not)h(simply)501 3338 y(apply)38 b(the)g(de\014nition)f
(of)h FI(f)5 b(:)38 b FL(But)g(in)f(this)h(case,)i FI(u)2432
3353 y FG(i)2456 3362 y Fy(0)2490 3353 y FG(;j)2583 3338
y FL(=)d(0)g(=)g FI(e)2940 3302 y FG(N)2940 3363 y(i)2964
3372 y Fy(0)2998 3363 y FG(;j)3092 3338 y FL(in)h FI(H)45
b FL(and)501 3459 y FI(\015)552 3474 y FG(i)576 3483
y Fy(0)610 3474 y FG(;j)701 3459 y FL(is)33 b(zero)i(as)f(w)m(ell,)g
(as)g(this)g(is)g(required)h(for)e(an)h(admissible)e(family)-8
b(.)46 b(So)34 b(w)m(e)501 3579 y(still)c(ha)m(v)m(e)k
FI(f)11 b FL(\()p FI(e)1058 3543 y FG(N)1058 3604 y(i)1082
3613 y Fy(0)1117 3604 y FG(;j)1173 3579 y FL(\))28 b(=)f
FI(\015)1393 3594 y FG(i)1417 3603 y Fy(0)1451 3594 y
FG(;j)1507 3579 y FI(:)501 3741 y FC(\017)67 b FL(Let)f
FI(J)76 b FL(b)s(e)67 b(the)g(Hopf)g(ideal)e(of)h FI(K)74
b FL(generated)67 b(b)m(y)h(all)c(the)j(genera-)501 3861
y(tors)58 b FI(e)767 3825 y FG(N)767 3886 y(i)791 3895
y Fy(0)826 3886 y FG(;j)940 3861 y FL(of)g FI(K)r(:)f
FL(Then,)66 b(according)57 b(to)h([Mas1,)64 b(Theorem)59
b(2.],)64 b FI(H)r(=)p FL(\()p FI(f)5 b(:J)k FL(\))501
3982 y(is)59 b(an)g(\()p FI(H)r(=)p FL(\()p FI(J)9 b
FL(\))p FI(;)17 b(H)r(=)p FL(\()p FI(f)5 b(:J)n(:f)1529
3946 y Fx(\000)p FH(1)1623 3982 y FL(\)\)-bi-Galois)56
b(ob)5 b(ject)60 b(and)f(hence)i FI(H)r(=)p FL(\()p FI(J)9
b FL(\))59 b(and)501 4102 y FI(H)r(=)p FL(\()p FI(f)5
b(:J)n(:f)889 4066 y Fx(\000)p FH(1)984 4102 y FL(\))23
b(are)h(quasi-isomorphic)d(if)h(the)i(bi-Galois)c(ob)5
b(ject)25 b(is)e(not)g(zero.)41 b(W)-8 b(e)501 4223 y(see)36
b(that)e FI(u)930 4238 y FG(i)954 4247 y Fy(0)988 4238
y FG(;j)1044 4223 y FL(\()t(~)-53 b FI(\015)5 b FL(\))31
b(=)g(0)j(and)h(so)f FI(A)p FL(\()t(~)-53 b FI(\015)5
b FL(\))31 b(=)g FI(H)r(=)p FL(\()p FI(J)9 b FL(\))p
FI(:)34 b FL(Calculation)e(\(6.10\))i(sho)m(w)m(ed)501
4343 y(that)39 b FI(A)p FL(\()p FI(\015)5 b FL(\))39
b(=)g FI(H)r(=)p FL(\()p FI(f)5 b(:J)n(:f)1466 4307 y
Fx(\000)p FH(1)1561 4343 y FL(\))p FI(:)39 b FL(W)-8
b(e)40 b(are)f(left)g(to)f(sho)m(w)j(that)e FI(B)44 b
FL(:=)39 b FI(H)r(=)p FL(\()p FI(f)5 b(:J)k FL(\))40
b(is)501 4463 y(not)33 b(zero.)501 4625 y FC(\017)42
b FI(B)50 b FL(=)44 b FI(H)918 4640 y FH(0)957 4625 y
FI(=)p FL(\()p FI(I)8 b(;)17 b(f)5 b(:J)k FL(\))43 b(b)m(y)g
(construction.)73 b(W)-8 b(e)43 b(ha)m(v)m(e)h(a)e(basis)g(of)g
FI(H)3089 4640 y FH(0)3171 4625 y FL(and)g(see)501 4746
y(that)28 b(w)m(e)g(could)g(apply)f(Theorem)h(6.3)f(to)g(get)h(a)f
(basis)h(of)f FI(B)5 b FL(.)42 b(It)28 b(just)g(remains)f(to)501
4866 y(c)m(hec)m(k)i(that)c(the)i(elemen)m(ts)f FI(\015)1567
4881 y FG(i)1591 4890 y Fy(0)1626 4881 y FG(;j)1682 4866
y FI(h)1738 4881 y FG(i)1762 4890 y Fy(0)1796 4881 y
FG(;j)1879 4866 y FL(app)s(earing)f(in)g FI(f)5 b(:J)36
b FL(satisfy)26 b(the)g(conditions)501 4986 y(of)32 b(the)h(theorem.)
501 5148 y(If)40 b FI(\037)667 5112 y FG(N)667 5173 y(i)691
5182 y Fy(0)725 5173 y FG(;j)821 5148 y FC(6)p FL(=)g
FI(")f FL(then)h FI(\015)1302 5163 y FG(i)1326 5172 y
Fy(0)1360 5163 y FG(;j)1456 5148 y FL(=)f(0,)i(b)s(ecause)g
FI(\015)k FL(is)39 b(admissible.)62 b FI(h)2844 5163
y FG(i)2868 5172 y Fy(0)2903 5163 y FG(;j)2999 5148 y
FL(is)39 b(in)g FI(\000)53 b FL(and)501 5269 y(so)46
b(comm)m(utes)f(with)g(all)e(group)i(elemen)m(ts.)82
b(W)-8 b(e)46 b(will)d(sho)m(w)j(no)m(w)g(that)f FI(h)3380
5284 y FG(i)3404 5293 y Fy(0)3439 5284 y FG(;j)1828 5637
y FL(69)p eop
%%Page: 70 72
70 71 bop 501 573 a FL(comm)m(utes)33 b(also)e(with)i(all)d(the)j
(generators)g(of)f FI(H)2345 588 y FH(0)2384 573 y FI(:)h
FL(F)-8 b(or)32 b(this)g(w)m(e)i(calculate)1171 779 y
FI(h)1227 794 y FG(i)1251 803 y Fy(0)1285 794 y FG(;j)1342
779 y FI(x)1397 794 y FG(k)1467 779 y FL(=)28 b FI(\037)1632
794 y FG(k)1675 779 y FL(\()p FI(h)1769 794 y FG(i)1793
803 y Fy(0)1827 794 y FG(;j)1884 779 y FL(\))p FI(x)1977
794 y FG(k)2020 779 y FI(h)2076 794 y FG(i)2100 803 y
Fy(0)2134 794 y FG(;j)2190 779 y FI(;)1089 953 y(\037)1150
968 y FG(k)1193 953 y FL(\()p FI(h)1287 968 y FG(i)1311
977 y Fy(0)1346 968 y FG(;j)1402 953 y FL(\))f(=)h FI(\037)1632
968 y FG(k)1675 953 y FL(\()1767 858 y Fw(Y)1713 1069
y FG(i)1737 1078 y Fy(0)1771 1069 y Fx(\024)p FG(p<j)1966
953 y FI(g)2017 912 y FG(N)2013 977 y(p)2083 953 y FL(\))g(=)2307
858 y Fw(Y)2252 1069 y FG(i)2276 1078 y Fy(0)2311 1069
y Fx(\024)p FG(p<j)2505 953 y FI(\037)2566 912 y FG(N)2566
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3420 y(section.)44 b(Then)34 b(the)f(monomials)610 3635
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Fy(+1)1563 3662 y FG(S)1606 3670 y Fo(t)1634 3662 y FH(+)p
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257 5174 y(the)i(ro)s(ot)f(v)m(ector)i(relations)561
5388 y FI(e)606 5342 y FG(N)662 5354 y Fo(k)606 5415
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%%Page: 71 73
71 72 bop 257 573 a FL(Here)32 b FI(N)564 588 y FG(k)637
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FL(in)257 693 y(the)i FI(k)479 657 y FH(th)583 693 y
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FI(e)649 1231 y FG(b)679 1240 y Fy(1)p Fo(;)p Fy(2)649
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1225 y FG(b)1588 1236 y Fo(S)1626 1249 y(t)1654 1236
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y(t)1852 1236 y Fy(+)p Fo(n)1937 1249 y(t)1964 1236 y
Fy(+1)1558 1311 y FG(S)1601 1319 y Fo(t)1629 1311 y FH(+)p
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1284 y FI(g)t(;)116 b FL(0)27 b FC(\024)i FI(b)2482 1299
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1311 y Fo(k)2743 1299 y FH(+)p FG(j)2862 1284 y FI(<)e(N)3043
1299 y FG(k)3086 1284 y FI(;)17 b(g)31 b FC(2)d FI(\000)s(:)3246
1404 y FL(\(6.11\))257 1622 y FJ(Pro)s(of:)49 b FL(The)39
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(considerations)g(in)g(ev)m(ery)i(comp)s(onen)m(t,)f(that)f(the)i
(ideal)d(is)501 1863 y(a)40 b(Hopf)h(ideal.)65 b(F)-8
b(or)40 b(the)h(dimension)e(and)h(basis)h(w)m(e)g(will)d(use)k(Theorem)
e(6.3.)501 1983 y(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e(c)m(hec)m(k)j(all)c
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(v)m(ectors)i(of)501 2264 y(one)f(comp)s(onen)m(t)f(and)g
FI(N)1428 2228 y FH(th)1522 2264 y FL(p)s(o)m(w)m(ers)i(of)d(ro)s(ot)h
(v)m(ectors)h(of)f(the)g(other)h(comp)s(onen)m(ts.)501
2384 y(Secondly)-8 b(,)39 b(w)m(e)e(ha)m(v)m(e)i(to)d(sho)m(w)i(that)f
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2520 y FG(a)1103 2505 y FL(of)f Fs(U)p FL(\()p FC(D)s
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2754 y Fo(l)1270 2742 y FH(+)p FG(s)1389 2715 y FL(=)c
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2754 y Fo(l)1792 2742 y FH(+)p FG(s)1884 2715 y FL(\()p
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2754 y Fo(k)3040 2742 y FH(+)p FG(j)3131 2715 y FI(;)88
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2888 y FG(S)1689 2900 y Fo(k)1726 2888 y FH(+)p FG(i;S)1868
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b FL(\(6.13\))501 3084 y(where)38 b(1)33 b FC(\024)h
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FC(\024)g FI(S)3165 3379 y FG(k)r FH(+1)3298 3364 y FI(;)j
FL(the)501 3485 y(ab)s(o)m(v)m(e)f(equations)e(follo)m(w)f(already)h
(from)g(the)h(original)c(pap)s(er)k([AS5].)501 3605 y(The)28
b(ro)s(ot)e(v)m(ectors)i(are)f(linear)f(com)m(binations)f(of)h
(homogeneous)h(monomials)d(in)501 3726 y(the)30 b(generators)f
FI(x)1189 3741 y FG(a)1261 3726 y FL(of)f Fs(U)p FL(\()p
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FL(=)c FI(l)r FL(.)501 4007 y(\(6.13\))e(is)g(sho)m(wn)i(b)m(y)f
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b(of)g(the)h FI(u)1069 4142 y FG(i;j)1149 4127 y FL(,)g(the)g(crucial)e
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y Fo(i;j)864 4338 y FL(\(1)22 b FC(\000)h FI(h)1129 4353
y FG(S)1172 4365 y Fo(k)1210 4353 y FH(+)p FG(i;S)1352
4365 y Fo(k)1389 4353 y FH(+)p FG(j)1481 4338 y FL(\))p
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4353 y FG(k)1835 4363 y Fo(i;j)1912 4338 y FI(x)1967
4353 y FG(a)2009 4338 y FL(\(1)22 b FC(\000)h FI(\037)2279
4353 y FG(a)2320 4338 y FL(\()p FI(h)2414 4353 y FG(S)2457
4365 y Fo(k)2496 4353 y FH(+)p FG(i;S)2638 4365 y Fo(k)2675
4353 y FH(+)p FG(j)2767 4338 y FL(\))p FI(h)2861 4353
y FG(S)2904 4365 y Fo(k)2942 4353 y FH(+)p FG(i;S)3084
4365 y Fo(k)3121 4353 y FH(+)p FG(j)3213 4338 y FL(\))1644
4498 y(=)k FI(\015)1798 4513 y FG(k)1835 4523 y Fo(i;j)1912
4498 y FI(x)1967 4513 y FG(a)2009 4498 y FL(\(1)22 b
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4525 y FG(S)2322 4537 y Fo(k)2360 4525 y FH(+)p FG(i;S)2502
4537 y Fo(k)2539 4525 y FH(+)p FG(j)2631 4498 y FL(\()p
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4498 y FL(\))p FI(h)2907 4513 y FG(S)2950 4525 y Fo(k)2988
4513 y FH(+)p FG(i;S)3130 4525 y Fo(k)3168 4513 y FH(+)p
FG(j)3259 4498 y FL(\))1644 4656 y(=)k FI(x)1802 4671
y FG(a)1844 4656 y FI(\015)1895 4671 y FG(k)1932 4681
y Fo(i;j)2009 4656 y FL(\(1)22 b FC(\000)h FI(h)2274
4671 y FG(S)2317 4683 y Fo(k)2355 4671 y FH(+)p FG(i;S)2497
4683 y Fo(k)2534 4671 y FH(+)p FG(j)2626 4656 y FL(\))p
FI(:)501 4866 y FL(In)29 b(the)g(second)h(step)f(w)m(e)g(used)h
(\(3.19\))d(and)i(as)g FI(a)f FL(is)g(not)g(in)g(the)h
FI(k)2903 4830 y FH(th)3002 4866 y FL(comp)s(onen)m(t,)501
4987 y(all)i(the)j(corresp)s(onding)f(en)m(tries)g(of)g(the)h(Cartan)f
(matrix)e(are)i(zero.)46 b(The)34 b(third)501 5107 y(step)g(uses)g(the)
f(premise)f(that)g FI(\015)38 b FL(is)32 b(admissible.)501
5268 y(No)m(w)88 b(w)m(e)h(can)e(apply)h(Theorem)f(6.3)g(and)h(the)g
(pro)s(of)e(is)h(\014nished.)3292 5388 y FJ(qed.)1828
5637 y FL(71)p eop
%%Page: 72 74
72 73 bop 257 573 a FJ(Lemma)37 b(6.5)98 b FE(F)-7 b(or)33
b(al)5 b(l)34 b(indic)-5 b(es)33 b FI(i)h FE(not)g(in)f(the)h
FI(k)2109 537 y Ff(th)2208 573 y FE(c)-5 b(omp)g(onent)32
b(of)i(the)g(diagr)-5 b(am)33 b(and)501 693 y FI(S)561
708 y FG(k)632 693 y FI(<)27 b(j)34 b FC(\024)28 b FI(l)i
FC(\024)e FI(S)1138 708 y FG(k)1203 693 y FL(+)22 b FI(n)1359
708 y FG(k)1437 693 y FE(we)34 b(have)g(with)h FI(q)c
FL(=)d FI(\037)2256 708 y FG(j)2293 693 y FL(\()p FI(g)2378
708 y FG(j)2414 693 y FL(\))f(=)h FI(\037)2644 708 y
FG(l)2670 693 y FL(\()p FI(g)2755 708 y FG(l)2781 693
y FL(\))501 976 y FI(i)p FL(\))100 b FI(x)727 991 y FG(i)756
976 y FI(e)801 991 y FG(j;l)q FH(+1)993 976 y FL(=)528
1333 y(=)649 1129 y Fw(8)649 1218 y(<)649 1398 y(:)748
1186 y FI(\037)809 1201 y FG(j)846 1186 y FL(\()p FI(g)931
1201 y FG(i)959 1186 y FL(\))p FI(x)1052 1201 y FG(j)1089
1186 y FI(x)1144 1201 y FG(i)1194 1186 y FL(+)22 b FI(\025)1349
1201 y FG(ij)1410 1186 y FL(\(1)g FC(\000)g FI(g)1665
1201 y FG(i)1693 1186 y FI(g)1740 1201 y FG(j)1776 1186
y FL(\))p FI(;)440 b FE(if)35 b FI(j)f FL(=)27 b FI(l)r(;)586
b FL(\(6.14a\))748 1331 y FI(\037)809 1346 y FG(j;l)q
FH(+1)974 1331 y FL(\()p FI(g)1059 1346 y FG(i)1086 1331
y FL(\))p FI(e)1169 1346 y FG(j;l)q FH(+1)1334 1331 y
FI(x)1389 1346 y FG(i)1440 1331 y FL(+)22 b FI(\025)1595
1346 y FG(ij)1655 1331 y FL(\(1)g FC(\000)g FI(q)1910
1290 y Fx(\000)p FH(1)2005 1331 y FL(\))p FI(e)2088 1346
y FG(j)t FH(+1)p FG(;l)q FH(+1)2346 1331 y FI(;)108 b
FE(if)34 b FI(\025)2632 1346 y FG(il)2710 1331 y FL(=)27
b(0)p FI(;)45 b(j)33 b(<)28 b(l)r(;)23 b FL(\(6.14b\))748
1477 y FI(\037)809 1492 y FG(j;l)q FH(+1)974 1477 y FL(\()p
FI(g)1059 1492 y FG(i)1086 1477 y FL(\))p FI(e)1169 1492
y FG(j;l)q FH(+1)1334 1477 y FI(x)1389 1492 y FG(i)1440
1477 y FC(\000)f FI(\025)1596 1492 y FG(il)1646 1477
y FL(\(1)g FC(\000)h FI(q)1902 1435 y Fx(\000)p FH(1)1996
1477 y FL(\))p FI(\037)2095 1492 y FG(j;l)2169 1477 y
FL(\()p FI(g)2254 1492 y FG(i)2282 1477 y FL(\))p FI(e)2365
1492 y FG(j;l)2439 1477 y FI(g)2486 1492 y FG(i)2514
1477 y FI(g)2561 1492 y FG(l)2586 1477 y FI(;)67 b FE(otherwise.)95
b FL(\(6.14c\))501 1704 y FI(ii)p FL(\))100 b FI(x)760
1719 y FG(i)789 1704 y FI(e)834 1658 y FG(N)890 1670
y Fo(k)834 1732 y FG(j;l)q FH(+1)1026 1704 y FL(=)28
b FI(\037)1191 1658 y FG(N)1247 1670 y Fo(k)1191 1732
y FG(j;l)q FH(+1)1355 1704 y FL(\()p FI(g)1440 1719 y
FG(i)1468 1704 y FL(\))p FI(e)1551 1658 y FG(N)1607 1670
y Fo(k)1551 1732 y FG(j;l)q FH(+1)1715 1704 y FI(x)1770
1719 y FG(i)1799 1704 y FI(:)257 1932 y FJ(Pro)s(of:)49
b FI(i)p FL(\))33 b FC(\017)f FL(The)i(case)f FI(j)h
FL(=)27 b FI(l)35 b FL(is)d(simply)f(the)i(de\014ning)g(relation)e
(\(3.28\).)501 2094 y(F)-8 b(rom)31 b(no)m(w)j(on)e FI(j)i(<)27
b(l)r FL(.)44 b(W)-8 b(e)33 b(consider)g(all)d(p)s(ossible)i(linkings.)
501 2256 y FC(\017)j FL(If)h FI(i)g FL(is)f(not)g(link)m(ed)g(to)g(an)m
(y)h(v)m(ertex)i FI(p)d FL(with)g FI(j)k FC(\024)33 b
FI(p)f FC(\024)h FI(l)r FL(,)k(then)f FI(\025)2997 2271
y FG(ij)3090 2256 y FL(=)c FI(\025)3255 2271 y FG(il)3338
2256 y FL(=)g(0)501 2376 y(and)h(a)f(rep)s(eated)i(use)f(of)f(\(3.28\))
g(giv)m(es)h(\(6.14b\).)501 2538 y FC(\017)c FL(If)f
FI(i)h FL(is)f(link)m(ed)g(to)g FI(j)6 b FL(,)30 b(then)f(it)e(can)i
(not)f(b)s(e)h(link)m(ed)f(to)h FI(l)h FL(as)f(w)m(ell.)41
b(Hence)30 b FI(\025)3265 2553 y FG(il)3343 2538 y FL(=)d(0)501
2659 y(and)h(w)m(e)g(ha)m(v)m(e)h(to)e(sho)m(w)i(the)f(second)g(case.)
43 b(W)-8 b(e)28 b(pro)s(ceed)g(b)m(y)g(induction)f(on)g
FI(l)14 b FC(\000)e FI(j)501 2779 y FL(and)33 b(use)g(the)g(recursiv)m
(e)h(de\014nition)e(\(6.5\))g(of)g(the)h(ro)s(ot)f(v)m(ectors.)501
2941 y(F)-8 b(or)32 b FI(l)24 b FC(\000)f FI(j)34 b FL(=)27
b(1)32 b(w)m(e)i(ha)m(v)m(e)643 3161 y FI(x)698 3176
y FG(i)726 3161 y FI(e)771 3176 y FG(j;l)q FH(+1)963
3161 y FL(=)28 b FI(x)1122 3176 y FG(i)1151 3161 y FL([)p
FI(x)1233 3176 y FG(j)1270 3161 y FI(x)1325 3176 y FG(l)1373
3161 y FC(\000)23 b FI(\037)1534 3176 y FG(l)1560 3161
y FL(\()p FI(g)1645 3176 y FG(j)1681 3161 y FL(\))p FI(x)1774
3176 y FG(l)1800 3161 y FI(x)1855 3176 y FG(j)1892 3161
y FL(])963 3306 y(=)28 b FI(\037)1128 3321 y FG(j)1165
3306 y FL(\()p FI(g)1250 3321 y FG(i)1277 3306 y FL(\))p
FI(x)1370 3321 y FG(j)1407 3306 y FI(x)1462 3321 y FG(i)1491
3306 y FI(x)1546 3321 y FG(l)1594 3306 y FL(+)22 b FI(\025)1749
3321 y FG(ij)1810 3306 y FL(\(1)g FC(\000)g FI(g)2065
3321 y FG(i)2093 3306 y FI(g)2140 3321 y FG(j)2177 3306
y FL(\))p FI(x)2270 3321 y FG(l)2318 3306 y FC(\000)h
FI(\037)2479 3321 y FG(l)2505 3306 y FL(\()p FI(g)2590
3321 y FG(j)2626 3306 y FL(\))p FI(\037)2725 3321 y FG(l)2751
3306 y FL(\()p FI(g)2836 3321 y FG(i)2864 3306 y FL(\))p
FI(x)2957 3321 y FG(l)2983 3306 y FI(x)3038 3321 y FG(i)3067
3306 y FI(x)3122 3321 y FG(j)963 3451 y FL(=)28 b FI(\037)1128
3466 y FG(j)1165 3451 y FL(\()p FI(g)1250 3466 y FG(i)1277
3451 y FL(\))p FI(\037)1376 3466 y FG(l)1402 3451 y FL(\()p
FI(g)1487 3466 y FG(i)1515 3451 y FL(\))p FI(x)1608 3466
y FG(j)1645 3451 y FI(x)1700 3466 y FG(l)1727 3451 y
FI(x)1782 3466 y FG(i)1832 3451 y FL(+)22 b FI(\025)1987
3466 y FG(ij)2048 3451 y FI(x)2103 3466 y FG(l)2151 3451
y FC(\000)h FI(\025)2308 3466 y FG(ij)2368 3451 y FI(\037)2429
3466 y FG(l)2456 3451 y FL(\()p FI(g)2541 3466 y FG(i)2568
3451 y FI(g)2615 3466 y FG(j)2652 3451 y FL(\))p FI(x)2745
3466 y FG(l)2771 3451 y FI(g)2818 3466 y FG(i)2846 3451
y FI(g)2893 3466 y FG(j)1055 3597 y FC(\000)g FI(\037)1216
3612 y FG(l)1242 3597 y FL(\()p FI(g)1327 3612 y FG(j)1363
3597 y FL(\))p FI(\037)1462 3612 y FG(l)1488 3597 y FL(\()p
FI(g)1573 3612 y FG(i)1601 3597 y FL(\))p FI(\037)1700
3612 y FG(j)1737 3597 y FL(\()p FI(g)1822 3612 y FG(i)1850
3597 y FL(\))p FI(x)1943 3612 y FG(l)1969 3597 y FI(x)2024
3612 y FG(j)2061 3597 y FI(x)2116 3612 y FG(i)2167 3597
y FC(\000)f FI(\037)2327 3612 y FG(l)2353 3597 y FL(\()p
FI(g)2438 3612 y FG(j)2475 3597 y FL(\))p FI(\037)2574
3612 y FG(l)2600 3597 y FL(\()p FI(g)2685 3612 y FG(i)2712
3597 y FL(\))p FI(x)2805 3612 y FG(l)2832 3597 y FI(\025)2889
3612 y FG(ij)2949 3597 y FL(\(1)g FC(\000)h FI(g)3205
3612 y FG(i)3233 3597 y FI(g)3280 3612 y FG(j)3316 3597
y FL(\))963 3742 y(=)28 b FI(\037)1128 3757 y FG(j;l)q
FH(+1)1292 3742 y FL(\()p FI(g)1377 3757 y FG(i)1405
3742 y FL(\)[)p FI(x)1525 3757 y FG(j)1562 3742 y FI(x)1617
3757 y FG(l)1666 3742 y FC(\000)22 b FI(\037)1826 3757
y FG(l)1852 3742 y FL(\()p FI(g)1937 3757 y FG(j)1974
3742 y FL(\))p FI(x)2067 3757 y FG(l)2093 3742 y FI(x)2148
3757 y FG(j)2185 3742 y FL(])p FI(x)2267 3757 y FG(i)2318
3742 y FL(+)g FI(\025)2473 3757 y FG(ij)2533 3742 y FL(\(1)g
FC(\000)h FI(\037)2803 3757 y FG(l)2829 3742 y FL(\()p
FI(g)2914 3757 y FG(i)2942 3742 y FL(\))p FI(\037)3041
3757 y FG(l)3067 3742 y FL(\()p FI(g)3152 3757 y FG(j)3188
3742 y FL(\)\))p FI(x)3319 3757 y FG(l)963 3896 y FL(=)28
b FI(\037)1128 3911 y FG(j;l)q FH(+1)1292 3896 y FL(\()p
FI(g)1377 3911 y FG(i)1405 3896 y FL(\))p FI(e)1488 3911
y FG(j;l)q FH(+1)1652 3896 y FI(x)1707 3911 y FG(i)1758
3896 y FL(+)22 b FI(\025)1913 3911 y FG(ij)1974 3896
y FL(\(1)g FC(\000)g FI(\037)2243 3911 y FG(i)2271 3896
y FL(\()p FI(g)2360 3854 y Fx(\000)p FH(1)2356 3923 y
FG(l)2454 3896 y FL(\))p FI(\037)2553 3911 y FG(j)2590
3896 y FL(\()p FI(g)2679 3854 y Fx(\000)p FH(1)2675 3923
y FG(l)2772 3896 y FL(\))2182 3960 y Fw(|)p 2227 3960
225 12 v 225 w({z)p 2542 3960 V 225 w(})2451 4045 y FH(=1)2827
3896 y FI(\037)2888 3840 y FG(x)2928 3852 y Fo(j)s(l)2888
3921 y FG(j)2984 3896 y FL(\()p FI(g)3069 3911 y FG(j)3105
3896 y FL(\))2827 3971 y Fw(|)p 2872 3971 69 12 v 69
w({z)p 3031 3971 V 69 w(})2899 4063 y FH(=)p FG(q)2988
4045 y Fu(\000)p Fy(1)3143 3896 y FL(\))p FI(x)3236 3911
y FG(l)3263 3896 y FI(:)501 4269 y FL(W)-8 b(e)26 b(used)g(\(3.19\))e
(and)h(the)g(condition)f FI(\037)1975 4284 y FG(i)2003
4269 y FI(\037)2064 4284 y FG(j)2128 4269 y FL(=)k(1)c(as)h
FI(\025)2474 4284 y FG(ij)2563 4269 y FC(6)p FL(=)i(0)p
FI(:)e FL(F)-8 b(or)24 b(the)h(induction)501 4389 y(step)34
b(w)m(e)f(use)h(an)e(analogue)g(calculation.)41 b(The)33
b(last)f(steps)i(are)f(as)g(follo)m(ws)501 4609 y FI(x)556
4624 y FG(i)585 4609 y FI(e)630 4624 y FG(j;l)q FH(+1)822
4609 y FL(=)28 b FI(\037)987 4624 y FG(j;l)q FH(+1)1151
4609 y FL(\()p FI(g)1236 4624 y FG(i)1264 4609 y FL(\))p
FI(e)1347 4624 y FG(j;l)q FH(+1)1511 4609 y FI(x)1566
4624 y FG(i)1617 4609 y FL(+)22 b FI(\025)1772 4624 y
FG(ij)1832 4609 y FL(\(1)g FC(\000)h FI(q)2088 4568 y
Fx(\000)p FH(1)2182 4609 y FL(\)[)p FI(e)2292 4624 y
FG(j)t FH(+1)p FG(;l)2460 4609 y FI(x)2515 4624 y FG(l)2564
4609 y FC(\000)f FI(\037)2724 4624 y FG(l)2751 4609 y
FL(\()p FI(g)2836 4624 y FG(j;l)2909 4609 y FL(\))p FI(\037)3008
4624 y FG(l)3034 4609 y FL(\()p FI(g)3119 4624 y FG(i)3147
4609 y FL(\))p FI(x)3240 4624 y FG(l)3267 4609 y FI(e)3312
4624 y FG(j)t FH(+1)p FG(;l)3480 4609 y FL(])822 4755
y(=)28 b FI(\037)987 4770 y FG(j;l)q FH(+1)1151 4755
y FL(\()p FI(g)1236 4770 y FG(i)1264 4755 y FL(\))p FI(e)1347
4770 y FG(j;l)q FH(+1)1511 4755 y FI(x)1566 4770 y FG(i)1595
4755 y FL(+)1012 4900 y(+)22 b FI(\025)1167 4915 y FG(ij)1227
4900 y FL(\(1)g FC(\000)g FI(q)1482 4859 y Fx(\000)p
FH(1)1577 4900 y FL(\)[)p FI(e)1687 4915 y FG(j)t FH(+1)p
FG(;l)1855 4900 y FI(x)1910 4915 y FG(l)1959 4900 y FC(\000)g
FI(\037)2119 4915 y FG(l)2145 4900 y FL(\()p FI(g)2230
4915 y FG(j)t FH(+1)p FG(;l)2398 4900 y FL(\))f FI(\037)2518
4915 y FG(l)2544 4900 y FL(\()p FI(g)2629 4915 y FG(j)2666
4900 y FL(\))p FI(\037)2765 4915 y FG(l)2791 4900 y FL(\()p
FI(g)2876 4915 y FG(i)2903 4900 y FL(\))2457 4965 y Fw(|)p
2502 4965 153 12 v 153 w({z)p 2745 4965 V 153 w(})2453
5068 y FG(\037)2497 5078 y Fo(j)2529 5068 y FH(\()p FG(g)2592
5038 y Fu(\000)p Fy(1)2590 5092 y Fo(l)2675 5068 y FH(\))p
FG(\037)2746 5078 y Fo(i)2773 5068 y FH(\()p FG(g)2836
5038 y Fu(\000)p Fy(1)2834 5092 y Fo(l)2919 5068 y FH(\))2963
4900 y FI(x)3018 4915 y FG(l)3044 4900 y FI(e)3089 4915
y FG(j)t FH(+1)p FG(;l)3257 4900 y FL(])822 5219 y(=)28
b FI(\037)987 5234 y FG(j;l)q FH(+1)1151 5219 y FL(\()p
FI(g)1236 5234 y FG(i)1264 5219 y FL(\))p FI(e)1347 5234
y FG(j;l)q FH(+1)1511 5219 y FI(x)1566 5234 y FG(i)1617
5219 y FL(+)22 b FI(\025)1772 5234 y FG(ij)1832 5219
y FL(\(1)g FC(\000)h FI(q)2088 5178 y Fx(\000)p FH(1)2182
5219 y FL(\))p FI(e)2265 5234 y FG(j)t FH(+1)p FG(;l)q
FH(+1)2523 5219 y FI(:)1828 5637 y FL(72)p eop
%%Page: 73 75
73 74 bop 501 573 a FC(\017)30 b FL(If)g FI(i)g FL(is)g(link)m(ed)f(to)
h FI(l)i FL(w)m(e)f(ha)m(v)m(e)g FI(\025)1717 588 y FG(il)1795
573 y FC(6)p FL(=)c(0)j(and)g(hence)i(w)m(e)f(need)g(to)e(pro)m(v)m(e)i
(\(6.14c\).)501 693 y(A)d(direct)g(calculation)e(using)h(the)i
(de\014nition)e(of)g(the)i(ro)s(ot)e(v)m(ectors)j(and)e(\(6.14b\))501
814 y(giv)m(es)524 1003 y FI(x)579 1018 y FG(i)608 1003
y FI(e)653 1018 y FG(j;l)q FH(+1)845 1003 y FL(=)g FI(x)1004
1018 y FG(i)1032 1003 y FL([)p FI(e)1104 1018 y FG(j;l)1179
1003 y FI(x)1234 1018 y FG(l)1282 1003 y FC(\000)23 b
FI(\037)1443 1018 y FG(l)1469 1003 y FL(\()p FI(g)1554
1018 y FG(j;l)1628 1003 y FL(\))p FI(x)1721 1018 y FG(l)1747
1003 y FI(e)1792 1018 y FG(j;l)1866 1003 y FL(])845 1148
y(=)28 b FI(\037)1010 1163 y FG(j;l)1084 1148 y FL(\()p
FI(g)1169 1163 y FG(i)1197 1148 y FL(\))p FI(e)1280 1163
y FG(j;l)1354 1148 y FI(x)1409 1163 y FG(i)1437 1148
y FI(x)1492 1163 y FG(l)1541 1148 y FC(\000)23 b FI(\037)1702
1163 y FG(l)1728 1148 y FL(\()p FI(g)1813 1163 y FG(j;l)1886
1148 y FL(\))p FI(\037)1985 1163 y FG(l)2011 1148 y FL(\()p
FI(g)2096 1163 y FG(i)2124 1148 y FL(\))p FI(x)2217 1163
y FG(l)2244 1148 y FI(x)2299 1163 y FG(i)2327 1148 y
FI(e)2372 1163 y FG(j;l)2469 1148 y FC(\000)f FI(\037)2629
1163 y FG(l)2655 1148 y FL(\()p FI(g)2740 1163 y FG(j;l)2814
1148 y FL(\))p FI(\025)2909 1163 y FG(il)2959 1148 y
FL(\(1)g FC(\000)g FI(g)3214 1163 y FG(i)3242 1148 y
FI(g)3289 1163 y FG(l)3315 1148 y FL(\))p FI(e)3398 1163
y FG(j;l)845 1294 y FL(=)28 b FI(\037)1010 1309 y FG(j;l)1084
1294 y FL(\()p FI(g)1169 1309 y FG(i)1197 1294 y FL(\))p
FI(e)1280 1309 y FG(j;l)1354 1294 y FI(\037)1415 1309
y FG(l)1441 1294 y FL(\()p FI(g)1526 1309 y FG(i)1554
1294 y FL(\))p FI(x)1647 1309 y FG(l)1673 1294 y FI(x)1728
1309 y FG(i)1779 1294 y FL(+)22 b FI(\037)1938 1309 y
FG(j;l)2012 1294 y FL(\()p FI(g)2097 1309 y FG(i)2125
1294 y FL(\))p FI(e)2208 1309 y FG(j;l)2282 1294 y FI(\025)2339
1309 y FG(il)2389 1294 y FL(\(1)g FC(\000)g FI(g)2644
1309 y FG(i)2672 1294 y FI(g)2719 1309 y FG(l)2745 1294
y FL(\))937 1439 y FC(\000)h FI(\037)1098 1454 y FG(l)1124
1439 y FL(\()p FI(g)1209 1454 y FG(j;l)1282 1439 y FL(\))p
FI(\037)1381 1454 y FG(l)1407 1439 y FL(\()p FI(g)1492
1454 y FG(i)1520 1439 y FL(\))p FI(x)1613 1454 y FG(l)1640
1439 y FI(\037)1701 1454 y FG(j;l)1775 1439 y FL(\()p
FI(g)1860 1454 y FG(i)1888 1439 y FL(\))p FI(e)1971 1454
y FG(j;l)2045 1439 y FI(x)2100 1454 y FG(i)2151 1439
y FC(\000)f FI(\037)2311 1454 y FG(l)2337 1439 y FL(\()p
FI(g)2422 1454 y FG(j;l)2496 1439 y FL(\))p FI(\025)2591
1454 y FG(il)2641 1439 y FI(e)2686 1454 y FG(j;l)937
1584 y FL(+)g FI(\037)1096 1599 y FG(l)1122 1584 y FL(\()p
FI(g)1207 1599 y FG(j;l)1281 1584 y FL(\))p FI(\025)1376
1599 y FG(il)1426 1584 y FI(\037)1487 1599 y FG(j;l)1561
1584 y FL(\()p FI(g)1646 1599 y FG(i)1674 1584 y FI(g)1721
1599 y FG(l)1746 1584 y FL(\))p FI(e)1829 1599 y FG(j;l)1904
1584 y FI(g)1951 1599 y FG(i)1978 1584 y FI(g)2025 1599
y FG(l)845 1729 y FL(=)28 b FI(\037)1010 1744 y FG(j;l)q
FH(+1)1174 1729 y FL(\()p FI(g)1259 1744 y FG(i)1287
1729 y FL(\)[)p FI(e)1397 1744 y FG(j;l)1471 1729 y FI(x)1526
1744 y FG(l)1575 1729 y FC(\000)22 b FI(\037)1735 1744
y FG(l)1761 1729 y FL(\()p FI(g)1846 1744 y FG(j;l)1920
1729 y FL(\))p FI(x)2013 1744 y FG(l)2039 1729 y FI(e)2084
1744 y FG(j;l)2159 1729 y FL(])p FI(x)2241 1744 y FG(i)2291
1729 y FL(+)h([)p FI(\037)2478 1744 y FG(j;l)2552 1729
y FL(\()p FI(g)2637 1744 y FG(i)2665 1729 y FL(\))f FC(\000)g
FI(\037)2885 1744 y FG(l)2911 1729 y FL(\()p FI(g)2996
1744 y FG(j;l)3070 1729 y FL(\)])p FI(\025)3192 1744
y FG(il)3242 1729 y FI(e)3287 1744 y FG(j;l)937 1875
y FL(+)g([)p FI(\037)1123 1890 y FG(l)1149 1875 y FL(\()p
FI(g)1234 1890 y FG(j;l)1308 1875 y FL(\))p FI(\037)1407
1890 y FG(j;l)1481 1875 y FL(\()p FI(g)1566 1890 y FG(i)1594
1875 y FI(g)1641 1890 y FG(l)1667 1875 y FL(\))g FC(\000)g
FI(\037)1887 1890 y FG(j;l)1961 1875 y FL(\()p FI(g)2046
1890 y FG(i)2074 1875 y FL(\)])p FI(\025)2196 1890 y
FG(il)2246 1875 y FI(e)2291 1890 y FG(j;l)2365 1875 y
FI(g)2412 1890 y FG(i)2440 1875 y FI(g)2487 1890 y FG(l)2513
1875 y FI(:)501 2064 y FL(As)33 b FI(i)g FL(is)f(not)h(in)e(the)i
FI(k)1318 2028 y FH(th)1422 2064 y FL(comp)s(onen)m(t)f(w)m(e)i(ha)m(v)
m(e)g(b)m(y)f(\(3.19\))f(and)h FI(\037)2957 2079 y FG(i)2985
2064 y FI(\037)3046 2079 y FG(l)3100 2064 y FL(=)28 b(1)1393
2254 y FI(\037)1454 2269 y FG(j;l)1528 2254 y FL(\()p
FI(g)1613 2269 y FG(i)1641 2254 y FL(\))f(=)h FI(\037)1871
2212 y Fx(\000)p FH(1)1871 2279 y FG(i)1965 2254 y FL(\()p
FI(g)2050 2269 y FG(j;l)2124 2254 y FL(\))f(=)h FI(\037)2354
2269 y FG(l)2380 2254 y FL(\()p FI(g)2465 2269 y FG(j;l)2539
2254 y FL(\))p FI(:)501 2443 y FL(Hence)d(the)f(second)g(term)f(in)f
(the)i(last)f(step)h(of)f(the)g(ab)s(o)m(v)m(e)i(calculation)20
b(v)-5 b(anishes,)501 2563 y(and)33 b(for)f(the)h(brac)m(k)m(et)h(of)e
(the)h(third)f(term)g(w)m(e)i(calculate)858 2753 y FI(\037)919
2768 y FG(l)945 2753 y FL(\()p FI(g)1030 2768 y FG(j;l)1103
2753 y FL(\))p FI(\037)1202 2768 y FG(j;l)1277 2753 y
FL(\()p FI(g)1362 2768 y FG(i)1389 2753 y FI(g)1436 2768
y FG(l)1462 2753 y FL(\))22 b FC(\000)h FI(\037)1683
2768 y FG(j;l)1757 2753 y FL(\()p FI(g)1842 2768 y FG(i)1870
2753 y FL(\))k(=)h FI(\037)2100 2768 y FG(j;l)2174 2753
y FL(\()p FI(g)2259 2768 y FG(i)2287 2753 y FL(\)\()p
FI(\037)2424 2768 y FG(l)2450 2753 y FL(\()p FI(g)2535
2768 y FG(j;l)2609 2753 y FL(\))p FI(\037)2708 2768 y
FG(j;l)2782 2753 y FL(\()p FI(g)2867 2768 y FG(l)2892
2753 y FL(\))22 b FC(\000)h FL(1\))1340 2898 y FI(\037)1401
2913 y FG(l)1427 2898 y FL(\()p FI(g)1512 2913 y FG(j;l)1586
2898 y FL(\))p FI(\037)1685 2913 y FG(j;l)1759 2898 y
FL(\()p FI(g)1844 2913 y FG(l)1870 2898 y FL(\))k(=)h
FI(\037)2100 2913 y FG(l)2126 2898 y FL(\()p FI(g)2211
2913 y FG(j)2247 2898 y FL(\))p FI(\037)2346 2913 y FG(l)2372
2898 y FL(\()p FI(g)2457 2913 y FG(j)t FH(+1)2584 2898
y FL(\))17 b FC(\001)g(\001)g(\001)d FI(\037)2832 2913
y FG(l)2858 2898 y FL(\()p FI(g)2943 2913 y FG(l)q Fx(\000)p
FH(1)3059 2898 y FL(\))2027 3043 y FC(\001)22 b FI(\037)2138
3058 y FG(j)2175 3043 y FL(\()p FI(g)2260 3058 y FG(l)2286
3043 y FL(\))p FI(\037)2385 3058 y FG(j)t FH(+1)2511
3043 y FL(\()p FI(g)2596 3058 y FG(l)2622 3043 y FL(\))17
b FC(\001)g(\001)g(\001)d FI(\037)2870 3058 y FG(l)q
Fx(\000)p FH(1)2987 3043 y FL(\()p FI(g)3072 3058 y FG(l)3097
3043 y FL(\))1935 3189 y(=)28 b(1)22 b FC(\001)g FL(1)17
b FC(\001)g(\001)g(\001)d FL(1)22 b FC(\001)g FI(\037)2540
3204 y FG(l)2566 3189 y FL(\()p FI(g)2651 3204 y FG(l)2676
3189 y FL(\))2714 3148 y Fx(\000)p FH(1)2809 3189 y FI(:)501
3378 y FC(\017)37 b FL(F)-8 b(or)35 b(the)i(last)f(case)h(where)h
FI(i)f FL(is)f(link)m(ed)g(to)g(a)g(v)m(ertex)j FI(p)d
FL(with)g FI(j)41 b(<)34 b(p)g(<)g(l)r(;)j FL(w)m(e)501
3499 y(again)31 b(pro)s(ceed)h(b)m(y)h(induction)e(on)h
FI(l)23 b FC(\000)e FI(p:)32 b FL(As)h FI(\025)2282 3514
y FG(ij)2370 3499 y FL(=)27 b FI(\025)2530 3514 y FG(j)t(l)2616
3499 y FL(=)h(0)j(w)m(e)i(ha)m(v)m(e)g(to)f(sho)m(w)501
3619 y(\(6.14b\).)501 3776 y(If)e FI(l)20 b FC(\000)e
FI(p)28 b FL(=)f(1)j(w)m(e)i(use)f(the)g(recursiv)m(e)g(de\014nition)e
(of)h(the)h(ro)s(ot)e(v)m(ectors)j(and)f(then)501 3897
y(\(6.14c\).)43 b(W)-8 b(e)33 b(set)h FI(F)41 b FL(:=)28
b FI(\025)1477 3912 y FG(i)p FH(\()p FG(l)q Fx(\000)p
FH(1\))1672 3897 y FL(\(1)21 b FC(\000)i FI(q)1927 3861
y Fx(\000)p FH(1)2021 3897 y FL(\))p FI(\037)2120 3912
y FG(j;l)q Fx(\000)p FH(1)2285 3897 y FL(\()p FI(g)2370
3912 y FG(i)2397 3897 y FL(\))33 b(and)f(ha)m(v)m(e)546
4086 y FI(x)601 4101 y FG(i)629 4086 y FI(e)674 4101
y FG(j;l)q FH(+1)866 4086 y FL(=)c FI(x)1025 4101 y FG(i)1054
4086 y FL([)p FI(e)1126 4101 y FG(j;l)1200 4086 y FI(x)1255
4101 y FG(l)1304 4086 y FC(\000)22 b FI(\037)1464 4101
y FG(l)1490 4086 y FL(\()p FI(g)1575 4101 y FG(i;l)1645
4086 y FL(\))p FI(x)1738 4101 y FG(l)1764 4086 y FI(e)1809
4101 y FG(j;l)1883 4086 y FL(])866 4232 y(=)28 b([)p
FI(\037)1058 4247 y FG(j;l)1132 4232 y FL(\()p FI(g)1217
4247 y FG(i)1245 4232 y FL(\))p FI(e)1328 4247 y FG(j;l)1402
4232 y FI(x)1457 4247 y FG(i)1508 4232 y FL(+)22 b FI(F)36
b FC(\001)22 b FI(e)1800 4247 y FG(j;l)q Fx(\000)p FH(1)1964
4232 y FI(g)2011 4247 y FG(i)2039 4232 y FI(g)2086 4247
y FG(l)q Fx(\000)p FH(1)2202 4232 y FL(])p FI(x)2284
4247 y FG(l)958 4377 y FC(\000)h FI(\037)1119 4392 y
FG(l)1145 4377 y FL(\()p FI(g)1230 4392 y FG(i;l)1299
4377 y FL(\))p FI(\037)1398 4392 y FG(l)1425 4377 y FL(\()p
FI(g)1510 4392 y FG(i)1537 4377 y FL(\))p FI(x)1630 4392
y FG(l)1657 4377 y FL([)p FI(\037)1745 4392 y FG(j;l)1819
4377 y FL(\()p FI(g)1904 4392 y FG(i)1932 4377 y FL(\))p
FI(e)2015 4392 y FG(j;l)2089 4377 y FI(x)2144 4392 y
FG(i)2195 4377 y FL(+)f FI(F)36 b FC(\001)21 b FI(e)2486
4392 y FG(j;l)q Fx(\000)p FH(1)2651 4377 y FI(g)2698
4392 y FG(i)2726 4377 y FI(g)2773 4392 y FG(l)q Fx(\000)p
FH(1)2889 4377 y FL(])866 4522 y(=)28 b FI(\037)1031
4537 y FG(j;l)1105 4522 y FL(\()p FI(g)1190 4537 y FG(i)1218
4522 y FL(\))p FI(e)1301 4537 y FG(j;l)1375 4522 y FI(\037)1436
4537 y FG(l)1462 4522 y FL(\()p FI(g)1547 4537 y FG(i)1575
4522 y FL(\))p FI(x)1668 4537 y FG(l)1694 4522 y FI(x)1749
4537 y FG(i)1800 4522 y FC(\000)23 b FI(\037)1961 4537
y FG(l)1987 4522 y FL(\()p FI(g)2072 4537 y FG(i;l)2141
4522 y FL(\))p FI(\037)2240 4537 y FG(l)2266 4522 y FL(\()p
FI(g)2351 4537 y FG(i)2379 4522 y FL(\))p FI(\037)2478
4537 y FG(j;l)2552 4522 y FL(\()p FI(g)2637 4537 y FG(i)2665
4522 y FL(\))p FI(x)2758 4537 y FG(l)2784 4522 y FI(e)2829
4537 y FG(j;l)2904 4522 y FI(x)2959 4537 y FG(i)958 4667
y FL(+)f FI(F)36 b FC(\001)22 b FI(\037)1266 4682 y FG(l)1292
4667 y FL(\()p FI(g)1377 4682 y FG(i)1405 4667 y FI(g)1452
4682 y FG(l)q Fx(\000)p FH(1)1568 4667 y FL(\))p FI(e)1651
4682 y FG(j;l)q Fx(\000)p FH(1)1815 4667 y FI(x)1870
4682 y FG(l)1897 4667 y FI(g)1944 4682 y FG(i)1972 4667
y FI(g)2019 4682 y FG(l)q Fx(\000)p FH(1)2157 4667 y
FC(\000)g FI(F)36 b FC(\001)22 b FI(\037)2466 4682 y
FG(l)2492 4667 y FL(\()p FI(g)2577 4682 y FG(i;l)2646
4667 y FL(\))p FI(\037)2745 4682 y FG(l)2771 4667 y FL(\()p
FI(g)2856 4682 y FG(i)2884 4667 y FL(\))p FI(x)2977 4682
y FG(l)3004 4667 y FI(e)3049 4682 y FG(j;l)q Fx(\000)p
FH(1)3213 4667 y FI(g)3260 4682 y FG(i)3288 4667 y FI(g)3335
4682 y FG(l)q Fx(\000)p FH(1)866 4813 y FL(=)28 b FI(\037)1031
4828 y FG(j;l)q FH(+1)1195 4813 y FL(\()p FI(g)1280 4828
y FG(i)1308 4813 y FL(\))p FI(e)1391 4828 y FG(j;l)q
FH(+1)1556 4813 y FI(x)1611 4828 y FG(i)958 4958 y FL(+)22
b FI(F)36 b FC(\001)22 b FI(\037)1266 4973 y FG(l)1292
4958 y FL(\()p FI(g)1377 4973 y FG(i)1405 4958 y FI(g)1452
4973 y FG(l)q Fx(\000)p FH(1)1568 4958 y FL(\)[)p FI(e)1678
4973 y FG(j;l)q Fx(\000)p FH(1)1842 4958 y FI(x)1897
4973 y FG(l)1946 4958 y FC(\000)h FI(\037)2107 4973 y
FG(l)2133 4958 y FL(\()p FI(g)2218 4973 y FG(i;l)q Fx(\000)p
FH(1)2377 4958 y FL(\))p FI(x)2470 4973 y FG(l)2497 4958
y FI(e)2542 4973 y FG(j;l)q Fx(\000)p FH(1)2706 4958
y FL(])p FI(g)2780 4973 y FG(i)2808 4958 y FI(g)2855
4973 y FG(l)q Fx(\000)p FH(1)2971 4958 y FI(:)501 5147
y FL(Ho)m(w)m(ev)m(er,)52 b(the)47 b(last)e(square)i(brac)m(k)m(et)h
(is)d(zero)i(according)e(to)h([AS5,)k(\(7.17\)].)501
5268 y(The)34 b(induction)d(step)j(is)e(no)m(w)h(simple.)1828
5637 y(73)p eop
%%Page: 74 76
74 75 bop 501 573 a FI(ii)p FL(\))33 b FC(\017)g FL(The)g(case)h
FI(\025)1185 588 y FG(ij)1273 573 y FL(=)27 b FI(\025)1433
588 y FG(il)1511 573 y FL(=)g(0)33 b(is)f(trivial.)501
735 y FC(\017)37 b FL(So)g(let)f(no)m(w)i FI(j)j FL(=)35
b FI(l)k FL(and)e FI(\025)1592 750 y FG(ij)1688 735 y
FC(6)p FL(=)e(0.)56 b(Then)38 b FI(\037)2251 750 y FG(j)2288
735 y FL(\()p FI(g)2373 750 y FG(i)2401 735 y FL(\))d(=)g
FI(\037)2646 694 y Fx(\000)p FH(1)2646 760 y FG(i)2740
735 y FL(\()p FI(g)2825 750 y FG(j)2861 735 y FL(\))g(=)h
FI(\037)3107 750 y FG(j)3143 735 y FL(\()p FI(g)3228
750 y FG(j)3264 735 y FL(\))f(=)g FI(q)501 855 y FL(and)e(using)f
(\(6.14a\))g(w)m(e)h(ha)m(v)m(e)517 1075 y FI(x)572 1090
y FG(i)601 1075 y FI(x)656 1029 y FG(N)712 1041 y Fo(k)656
1101 y FG(j)782 1075 y FL(=)28 b FI(q)933 1034 y FG(N)989
1046 y Fo(k)1031 1075 y FI(x)1086 1029 y FG(N)1142 1041
y Fo(k)1086 1101 y FG(j)1185 1075 y FI(x)1240 1090 y
FG(i)1290 1075 y FL(+)22 b FI(\025)1445 1090 y FG(ij)1506
1075 y FL(\(1)g(+)g FI(q)k FL(+)c FI(q)1927 1034 y FH(2)1988
1075 y FL(+)g FC(\001)17 b(\001)g(\001)j FL(+)i FI(q)2369
1034 y FG(N)2425 1046 y Fo(k)2463 1034 y Fx(\000)p FH(1)2558
1075 y FL(\))p FI(x)2651 1029 y FG(N)2707 1041 y Fo(k)2745
1029 y Fx(\000)p FH(1)2651 1101 y FG(j)2839 1075 y FL(\(1)g
FC(\000)h FI(q)3095 1034 y FG(N)3151 1046 y Fo(k)3189
1034 y Fx(\000)p FH(1)3283 1075 y FI(g)3330 1090 y FG(i)3358
1075 y FI(g)3405 1090 y FG(j)3441 1075 y FL(\))782 1238
y(=)28 b FI(x)941 1192 y FG(N)997 1204 y Fo(k)941 1264
y FG(j)1039 1238 y FI(x)1094 1253 y FG(i)1123 1238 y
FI(:)501 1458 y FL(Here)34 b(w)m(e)f(used)h(the)f(fact)f(that)h
FI(N)1748 1473 y FG(k)1823 1458 y FL(is)f(the)h(order)g(of)f
FI(q)t FL(.)501 1579 y(F)-8 b(rom)31 b(no)m(w)j(on)e(again)f
FI(j)j(<)27 b(l)r(:)501 1740 y FC(\017)39 b FL(If)g FI(\025)751
1755 y FG(ij)851 1740 y FC(6)p FL(=)g(0)p FI(;)g FL(w)m(e)h(set)g
FI(x)g FL(=)e FI(e)1644 1755 y FG(j;l)q FH(+1)1809 1740
y FI(;)h(y)j FL(=)d FI(x)2135 1755 y FG(i)2164 1740 y
FI(;)g(z)44 b FL(=)39 b FI(\025)2491 1755 y FG(ij)2551
1740 y FL(\(1)27 b FC(\000)g FI(q)2816 1704 y Fx(\000)p
FH(1)2910 1740 y FL(\))p FI(e)2993 1755 y FG(j)t FH(+1)p
FG(;l)q FH(+1)3251 1740 y FI(;)40 b(\013)f FL(=)501 1861
y FI(\037)562 1876 y FG(j;l)q FH(+1)727 1861 y FL(\()p
FI(g)812 1876 y FG(i)839 1861 y FL(\))33 b(and)g FI(\014)h
FL(=)28 b FI(\037)1354 1820 y Fx(\000)p FH(1)1354 1889
y FG(j)t FH(+1)p FG(;l)q FH(+1)1612 1861 y FL(\()p FI(g)1697
1876 y FG(j;l)q FH(+1)1861 1861 y FL(\))p FI(:)33 b FL(Then,)h(b)s
(ecause)g(of)f([AS5,)g(\(7.24\)])f FI(z)t(x)d FL(=)501
1981 y FI(\014)6 b(xz)t(:)34 b FL(Hence,)g(cf.)43 b([AS4,)33
b(Lemma)e(3.4)h(\(1\)],)991 2297 y FI(y)t(x)1098 2256
y FG(N)1154 2268 y Fo(k)1224 2297 y FL(=)c FI(\013)1391
2256 y FG(N)1447 2268 y Fo(k)1488 2297 y FI(x)1543 2256
y FG(N)1599 2268 y Fo(k)1642 2297 y FI(y)d FL(+)1813
2126 y Fw( )1892 2169 y FG(N)1948 2181 y Fo(k)1986 2169
y Fx(\000)p FH(1)1913 2202 y Fw(X)1908 2411 y FG(m)p
FH(=0)2093 2297 y FI(\013)2156 2256 y FG(m)2222 2297
y FI(\014)2283 2256 y FG(N)2339 2268 y Fo(k)2377 2256
y Fx(\000)p FH(1)p Fx(\000)p FG(m)2589 2126 y Fw(!)2684
2297 y FI(x)2739 2256 y FG(N)2795 2268 y Fo(k)2834 2256
y Fx(\000)p FH(1)2928 2297 y FI(z)t(:)501 2613 y FL(Using)32
b FI(\037)836 2628 y FG(i)864 2613 y FI(\037)925 2628
y FG(j)989 2613 y FL(=)c(1)k(and)g(\(3.19\))f(w)m(e)i(see)g(that)e
FI(\013)e FL(=)e FI(\037)2409 2572 y Fx(\000)p FH(1)2409
2639 y FG(i)2504 2613 y FL(\()p FI(g)2589 2628 y FG(j;l)q
FH(+1)2753 2613 y FL(\))g(=)h FI(\037)2983 2628 y FG(j)3019
2613 y FL(\()p FI(g)3104 2628 y FG(j;l)q FH(+1)3268 2613
y FL(\))k(and)501 2734 y(so)661 2954 y FI(\013)724 2913
y FG(m)790 2954 y FI(\014)851 2913 y FG(N)907 2925 y
Fo(k)945 2913 y Fx(\000)p FH(1)p Fx(\000)p FG(m)1184
2954 y FL(=)c FI(\014)1349 2913 y FG(N)1405 2925 y Fo(k)1442
2913 y Fx(\000)p FH(1)1537 2954 y FI(\037)1598 2913 y
FG(m)1598 2978 y(j)1664 2954 y FL(\()p FI(g)1749 2969
y FG(j;l)q FH(+1)1913 2954 y FL(\))p FI(\037)2012 2913
y FG(m)2012 2978 y(j)t FH(+1)p FG(;l)q FH(+1)2271 2954
y FL(\()p FI(g)2356 2969 y FG(j;l)q FH(+1)2519 2954 y
FL(\))1184 3118 y(=)g FI(\014)1349 3077 y FG(N)1405 3089
y Fo(k)1442 3077 y Fx(\000)p FH(1)1537 3118 y FI(\037)1598
3077 y FG(m)1598 3142 y(j;l)q FH(+1)1762 3118 y FL(\()p
FI(g)1847 3133 y FG(j;l)q FH(+1)2011 3118 y FL(\))g(=)f
FI(\014)2241 3077 y FG(N)2297 3089 y Fo(k)2335 3077 y
Fx(\000)p FH(1)2429 3118 y FL(\()p FI(B)2546 3071 y FG(j;l)q
FH(+1)2541 3146 y FG(j;l)q FH(+1)2710 3118 y FL(\))2748
3077 y FG(m)2843 3118 y FL(=)g FI(\014)3007 3077 y FG(N)3063
3089 y Fo(k)3101 3077 y Fx(\000)p FH(1)3195 3118 y FI(q)3242
3077 y FG(m)3309 3118 y FI(:)501 3338 y FL(The)39 b(last)f(equalit)m(y)
f(follo)m(ws)g(from)g([AS5,)j(\(7.5\)].)60 b(The)39 b(geometric)e(sum)h
(giv)m(es)501 3458 y(zero)33 b(again.)501 3620 y FC(\017)j
FL(The)g(\014nal)f(case)i FI(\025)1277 3635 y FG(il)1359
3620 y FC(6)p FL(=)c(0)i(is)h(treated)g(similarly)c(to)j(the)h
(previous)g(one.)53 b(This)501 3740 y(time)45 b FI(z)56
b FL(=)51 b FC(\000)p FI(\025)1099 3755 y FG(il)1150
3740 y FL(\(1)31 b FC(\000)h FI(q)1424 3704 y Fx(\000)p
FH(1)1518 3740 y FL(\))p FI(\037)1617 3755 y FG(j;l)1691
3740 y FL(\()p FI(g)1776 3755 y FG(i)1804 3740 y FL(\))p
FI(e)1887 3755 y FG(j;l)1961 3740 y FI(g)2008 3755 y
FG(i)2036 3740 y FI(g)2083 3755 y FG(l)2155 3740 y FL(and)47
b FI(\014)56 b FL(=)51 b FI(\037)2658 3755 y FG(j;l)q
FH(+1)2823 3740 y FL(\()p FI(g)2908 3755 y FG(i)2936
3740 y FI(g)2983 3755 y FG(l)3008 3740 y FL(\))p FI(\037)3107
3755 y FG(j;l)q FH(+1)3271 3740 y FL(\()p FI(g)3356 3755
y FG(j;l)3430 3740 y FL(\))p FI(;)501 3861 y FL(b)s(ecause)34
b(of)e([AS5,)h(\(7.23\)].)43 b(So)32 b(w)m(e)i(ha)m(v)m(e)928
4081 y FI(\013)991 4040 y FG(m)1057 4081 y FI(\014)1118
4040 y FG(N)1174 4052 y Fo(k)1212 4040 y Fx(\000)p FH(1)p
Fx(\000)p FG(m)1451 4081 y FL(=)28 b FI(\014)1616 4040
y FG(N)1672 4052 y Fo(k)1710 4040 y Fx(\000)p FH(1)1804
4081 y FI(\037)1865 4040 y FG(m)1865 4106 y(j;l)q FH(+1)2029
4081 y FL(\()p FI(g)2114 4096 y FG(i)2142 4081 y FL(\))p
FI(\037)2241 4040 y Fx(\000)p FG(m)2241 4109 y(j;l)q
FH(+1)2405 4081 y FL(\()p FI(g)2490 4096 y FG(i)2518
4081 y FL(\))p FI(\037)2617 4040 y Fx(\000)p FG(m)2617
4109 y(j;l)q FH(+1)2782 4081 y FL(\()p FI(g)2867 4096
y FG(j;l)q FH(+1)3030 4081 y FL(\))1451 4248 y(=)g FI(\014)1616
4207 y FG(N)1672 4219 y Fo(k)1710 4207 y Fx(\000)p FH(1)1804
4248 y FL(\()p FI(B)1921 4201 y FG(j;l)q FH(+1)1916 4276
y FG(j;l)q FH(+1)2085 4248 y FL(\))2123 4207 y Fx(\000)p
FG(m)2272 4248 y FL(=)g FI(\014)2437 4207 y FG(N)2493
4219 y Fo(k)2531 4207 y Fx(\000)p FH(1)2625 4248 y FI(q)2672
4207 y Fx(\000)p FG(m)2793 4248 y FI(:)3292 4468 y FJ(qed.)257
4696 y FL(Here)34 b(is)e(the)h(\014nal)f(result.)257
4900 y FJ(Theorem)37 b(6.6)98 b FE(L)-5 b(et)46 b FC(D)i
FE(and)d FC(D)1557 4864 y Fx(0)1625 4900 y FE(b)-5 b(e)46
b(two)f(linking)f(data)i(as)f(ab)-5 b(ove)45 b(and)f
FI(\015)51 b FE(and)45 b FI(\015)3472 4864 y Fx(0)501
5020 y FE(two)i(admissible)f(p)-5 b(ar)g(ameter)46 b(families.)80
b(Then)47 b FC(A)p FL(\()p FC(D)s FI(;)17 b(\015)5 b
FL(\))46 b FE(and)g FC(A)p FL(\()p FC(D)3134 4984 y Fx(0)3156
5020 y FI(;)17 b(\015)3256 4984 y Fx(0)3279 5020 y FL(\))47
b FE(ar)-5 b(e)501 5141 y(quasi-isomorphic.)1828 5637
y FL(74)p eop
%%Page: 75 77
75 76 bop 257 573 a FJ(Pro)s(of:)49 b FL(The)23 b(pro)s(of)f(is)f(just)
i(a)f(com)m(bination)e(of)i(the)g(results)h(obtained)f(in)f(the)i
(previous)501 693 y(sections.)43 b(First)30 b(w)m(e)h(sho)m(w)g(that)e
(the)i(Hopf)f(algebras)f(are)h(quasi-isomorphic)e(to)501
814 y(ones)35 b(where)h(all)c(the)j(parameters)f FI(\015)39
b FL(are)c(zero.)49 b(Hence,)36 b FC(A)p FL(\()p FC(D)s
FI(;)17 b(\015)5 b FL(\))33 b(is)h(a)g(co)s(cycle)501
934 y(deformation)k(of)g Fs(u)p FL(\()p FC(D)s FL(\))h(and)g
FC(A)p FL(\()p FC(D)1811 898 y Fx(0)1833 934 y FI(;)17
b(\015)1933 898 y Fx(0)1956 934 y FL(\))39 b(is)g(a)g(co)s(cycle)h
(deformation)d(of)i Fs(u)p FL(\()p FC(D)3408 898 y Fx(0)3430
934 y FL(\).)501 1054 y(Then)d(w)m(e)f(can)f(use)i(Theorem)e(6.1)g(to)g
(see)h(that)g Fs(u)p FL(\()p FC(D)s FL(\))e(is)h(quasi-isomorphic)e(to)
501 1175 y Fs(u)p FL(\()p FC(D)671 1139 y Fx(0)694 1175
y FL(\).)67 b(Because)42 b(of)e(the)h(transitivit)m(y)e(of)h(the)h
(quasi-isomorphism)d(relation,)501 1295 y(this)32 b(is)h(the)g(desired)
g(result.)501 1450 y(T)-8 b(o)31 b(sho)m(w)i(that)e(the)g
FI(\015)36 b FL(can)c(all)d(b)s(e)i(set)h(zero,)g(w)m(e)g(pro)s(ceed)g
(again)d(step)m(wise.)45 b(Let)501 1570 y(1)35 b FC(\024)h
FI(i)731 1585 y FH(0)806 1570 y FC(\024)g FI(S)979 1585
y FG(t)1033 1570 y FL(+)25 b FI(n)1192 1585 y FG(t)1259
1570 y FL(b)s(e)37 b(suc)m(h)i(that)e FI(\015)1888 1585
y FG(i;j)2003 1570 y FL(=)e(0)i(for)f(all)f FI(i)g(<)g(i)2705
1585 y FH(0)2782 1570 y FL(and)i(all)e FI(j)41 b(>)35
b(i)j FL(for)501 1691 y(whic)m(h)i(there)g(is)e(a)h(ro)s(ot)f(v)m
(ector)i FI(e)1794 1706 y FG(i;j)1875 1691 y FL(.)62
b(W)-8 b(e)40 b(set)j(~)-52 b FI(\015)2349 1706 y FG(i;j)2467
1691 y FL(:=)39 b FI(\015)2660 1706 y FG(i;j)2779 1691
y FL(when)h FI(i)f FC(6)p FL(=)f FI(i)3259 1706 y FH(0)3338
1691 y FL(and)501 1811 y(zero)33 b(otherwise.)45 b(It)33
b(is)g(enough)g(to)g(pro)m(v)m(e)h(that)e FC(A)p FL(\()p
FC(D)s FI(;)17 b(\015)5 b FL(\))32 b(is)g(quasi-isomorphic)501
1932 y(to)d FC(A)p FL(\()p FC(D)s FI(;)20 b FL(~)-52
b FI(\015)t FL(\).)42 b(Rep)s(eating)29 b(the)h(last)e(step)j(with)e
(increased)h FI(i)2705 1947 y FH(0)2774 1932 y FL(and)j(~)-53
b FI(\015)35 b FL(as)29 b(the)h(new)501 2052 y FI(\015)5
b(;)36 b FL(w)m(e)i(\014nd)e(that)g FC(A)p FL(\()p FC(D)s
FI(;)17 b(\015)5 b FL(\))35 b(is)g(quasi-isomorphic)f(to)i
FC(A)p FL(\()p FC(D)s FI(;)17 b FL(0\))32 b(=)i Fs(u)p
FL(\()p FC(D)s FL(\))h(for)g(all)501 2172 y(admissible)c
FI(\015)5 b FL(.)501 2327 y(W)-8 b(e)33 b(set)g FI(H)902
2342 y FH(0)969 2327 y FL(:=)28 b Fs(U)p FL(\()p FC(D)s
FL(\))k(and)h(consider)g(for)f(1)27 b FC(\024)i FI(k)h
FC(\024)f FI(t)j FL(its)g(ideals)699 2498 y FI(I)742
2513 y FG(k)812 2498 y FL(:=)c(\()p FI(e)1026 2452 y
FG(N)1082 2464 y Fo(k)1026 2523 y FG(i;j)1146 2498 y
FC(\000)23 b FI(u)1302 2513 y FG(i;j)1426 2498 y FL(:)44
b FI(S)1557 2513 y FG(k)1628 2498 y FI(<)27 b(i)h(<)g(j)33
b FC(\024)28 b FI(S)2134 2513 y FG(k)r FH(+1)2289 2498
y FL(+)22 b(1)p FI(;)45 b(i)27 b FC(6)p FL(=)h FI(i)2705
2513 y FH(0)2777 2498 y FL(or)k FI(\037)2957 2452 y FG(N)3013
2464 y Fo(k)2957 2523 y FG(i;j)3084 2498 y FC(6)p FL(=)27
b FI(")p FL(\))p FI(:)501 2677 y FL(When)42 b FI(i)827
2692 y FH(0)907 2677 y FL(is)e(not)h(in)f(the)h FI(k)1547
2641 y FH(th)1658 2677 y FL(comp)s(onen)m(t,)i(i.e.)67
b FI(i)2419 2692 y FH(0)2501 2677 y FC(\024)41 b FI(S)2679
2692 y FG(k)2763 2677 y FL(or)f FI(i)2923 2692 y FH(0)3004
2677 y FI(>)i(S)3182 2692 y FG(k)r FH(+1)3314 2677 y
FL(,)h(w)m(e)501 2798 y(kno)m(w)k(that)e FI(I)1036 2813
y FG(k)1124 2798 y FL(is)g(a)g(Hopf)g(ideal)f(from)g(the)h(pro)s(of)g
(of)g(Theorem)g(7.25.\(i\))f(in)501 2918 y([AS5].)53
b(F)-8 b(or)35 b FI(S)1049 2933 y FG(k)1086 2942 y Fy(0)1158
2918 y FI(<)e(i)1300 2933 y FH(0)1372 2918 y FC(\024)h
FI(S)1543 2933 y FG(k)1580 2942 y Fy(0)1614 2933 y FH(+1)1744
2918 y FL(the)i(considerations)g(that)f(pro)m(v)m(e)i
FI(I)3077 2933 y FG(k)3114 2942 y Fy(0)3188 2918 y FL(to)f(b)s(e)f(a)
501 3038 y(Hopf)28 b(ideal)f(ha)m(v)m(e)i(b)s(een)g(carried)e(out)h
(explicitly)e(in)i(the)g(pro)s(of)f(of)h(our)g(Theorem)501
3159 y(6.2.)41 b(Hence)27 b FI(H)35 b FL(:=)28 b FI(H)1305
3174 y FH(0)1344 3159 y FI(=I)34 b FL(is)25 b(a)g(Hopf)h(algebra,)g
(where)h FI(I)33 b FL(denotes)27 b(the)f(sum)g(of)f(all)501
3279 y(the)30 b(ideals)d FI(I)976 3294 y FG(k)1019 3279
y FI(:)i FL(As)h(b)s(efore,)f(de\014ne)i FI(K)36 b FL(as)29
b(the)g(Hopf)g(subalgebra)f(of)h FI(H)36 b FL(whic)m(h)30
b(is)501 3413 y(generated)h(b)m(y)f FI(\000)44 b FL(and)30
b(the)g(remaining)d FI(e)2028 3352 y FG(N)2084 3364 y
Fo(k)2117 3379 y Fy(0)2028 3439 y FG(i)2052 3448 y Fy(0)2087
3439 y FG(;j)2160 3413 y FI(;)j(S)2277 3428 y FG(k)2314
3437 y Fy(0)2380 3413 y FI(<)d(i)2516 3428 y FH(0)2584
3413 y FI(<)g(j)34 b FC(\024)28 b FI(S)2926 3428 y FG(k)2963
3437 y Fy(0)2998 3428 y FH(+1)3108 3413 y FL(+)16 b(1)p
FI(;)30 b FL(with)501 3566 y FI(\037)562 3505 y FG(N)618
3517 y Fo(k)651 3532 y Fy(0)562 3592 y FG(i;j)722 3566
y FL(=)d FI(")17 b(:)32 b FL(Using)g(\(6.8\))g(w)m(e)h(see)h(that)e
FI(K)39 b FL(is)32 b(comm)m(utativ)m(e.)43 b(Because)34
b(Lemma)501 3687 y(6.5)i(establishes)h(\(6.12\),)f(w)m(e)i(can)e(apply)
g(Theorem)h(6.3)f(to)g(\014nd)h(a)f(basis)g(of)g FI(H)501
3807 y FL(and)28 b(see)g(that)g FI(K)34 b FL(is)27 b(just)h(the)g(p)s
(olynomial)c(algebra)i(on)i(its)e(generators.)43 b(Hence,)501
3941 y(the)37 b(algebra)e(map)g FI(f)45 b FL(:)34 b FI(K)41
b FC(!)34 b FF(|)21 b FL(is)36 b(w)m(ell)g(de\014ned)h(b)m(y)g(setting)
f FI(f)11 b FL(\()p FI(e)2989 3880 y FG(N)3045 3892 y
Fo(k)3078 3907 y Fy(0)2989 3967 y FG(i)3013 3976 y Fy(0)3048
3967 y FG(;j)3121 3941 y FL(\))34 b(:=)g FI(\015)3381
3956 y FG(i)3405 3965 y Fy(0)3439 3956 y FG(;j)501 4062
y FL(on)k(all)d(the)j(generators)h(of)e(K)g(and)h FI(f)11
b FL(\()p FI(g)t FL(\))35 b(:=)h(1)i(for)f(all)e FI(g)40
b FC(2)d FI(\000)s(:)h FL(The)g(analogue)501 4201 y(of)d(computation)g
(\(6.10\))f(establishes)j FI(f)5 b(:e)2085 4140 y FG(N)2141
4152 y Fo(k)2174 4167 y Fy(0)2085 4227 y FG(i)2109 4236
y Fy(0)2144 4227 y FG(;j)2217 4201 y FI(:f)2303 4165
y Fx(\000)p FH(1)2430 4201 y FL(=)33 b FI(e)2584 4140
y FG(N)2640 4152 y Fo(k)2673 4167 y Fy(0)2584 4227 y
FG(i)2608 4236 y Fy(0)2643 4227 y FG(;j)2740 4201 y FC(\000)25
b FI(u)2898 4216 y FG(i)2922 4225 y Fy(0)2956 4216 y
FG(;j)3012 4201 y FI(:)36 b FL(W)-8 b(e)36 b(de\014ne)501
4354 y FI(J)43 b FL(as)35 b(the)f(Hopf)g(ideal)e(of)i
FI(K)41 b FL(generated)35 b(b)m(y)g(all)d(the)i FI(e)2534
4293 y FG(N)2590 4305 y Fo(k)2623 4320 y Fy(0)2534 4380
y FG(i)2558 4389 y Fy(0)2593 4380 y FG(;j)2700 4354 y
FL(in)f FI(K)7 b FL(.)48 b(No)m(w)35 b(w)m(e)g(can)501
4475 y(apply)43 b([Mas1)q(,)j(Theorem)d(2])g(to)g(pro)m(v)m(e)h(that)f
FC(A)p FL(\()p FC(D)s FI(;)17 b(\015)5 b FL(\))45 b(=)g
FI(H)r(=)p FL(\()p FI(f)5 b(:J)n(:f)3162 4438 y Fx(\000)p
FH(1)3257 4475 y FL(\))43 b(and)501 4595 y FC(A)p FL(\()p
FC(D)s FI(;)20 b FL(~)-52 b FI(\015)t FL(\))35 b(=)f
FI(H)r(=)p FL(\()p FI(J)9 b FL(\))36 b(are)h(quasi-isomorphic)d(if)i
FI(B)k FL(:=)34 b FI(H)r(=)p FL(\()p FI(f)5 b(:J)k FL(\))35
b FC(6)p FL(=)f(0)p FI(:)j FL(F)-8 b(or)36 b(this)501
4735 y(last)e(step)i(w)m(e)f(calculate)f FI(f)5 b(:e)1576
4674 y FG(N)1632 4686 y Fo(k)1665 4701 y Fy(0)1576 4760
y FG(i)1600 4769 y Fy(0)1635 4760 y FG(;j)1739 4735 y
FL(=)31 b FI(e)1891 4674 y FG(N)1947 4686 y Fo(k)1980
4701 y Fy(0)1891 4760 y FG(i)1915 4769 y Fy(0)1950 4760
y FG(;j)2047 4735 y FL(+)23 b FI(\015)2197 4750 y FG(i)2221
4759 y Fy(0)2255 4750 y FG(;j)2311 4735 y FI(h)2367 4750
y FG(i)2391 4759 y Fy(0)2426 4750 y FG(;j)2482 4735 y
FI(:)35 b FL(As)g FI(\015)k FL(is)34 b(assumed)i(to)e(b)s(e)501
4855 y(admissible,)28 b(w)m(e)i(see)g(from)e(Lemma)g(6.5)g(that)h(w)m
(e)h(can)g(use)g(Theorem)f(6.3)g(again)501 4975 y(to)34
b(\014nd)i(a)e(basis)h(of)f FI(B)i FL(=)31 b FI(H)1556
4990 y FH(0)1595 4975 y FI(=)p FL(\()p FI(I)8 b(;)17
b(f)5 b(:J)k FL(\))34 b(consisting)g(of)g(the)i(monomials)31
b(\(6.11\).)501 5096 y(So)i FI(B)k FL(is)32 b(not)h(zero)g(and)g(ev)m
(erything)g(is)f(pro)m(v)m(ed.)989 b FJ(qed.)404 5268
y FL(The)33 b(theorem)f(extends)i(the)e(original)d(results)k(of)e
([Mas1)q(],)h(whic)m(h)h(dealt)e(with)h(the)257 5388
y(case)38 b(of)f(copies)h(of)e FI(A)1065 5403 y FH(1)1142
5388 y FL(only)-8 b(,)38 b(and)f([BDR])g(that)g(includes)g(a)g(pro)s
(of)f(for)h(the)g(diagram)1828 5637 y(75)p eop
%%Page: 76 78
76 77 bop 257 573 a FI(A)330 588 y FH(2)370 573 y FL(.)42
b(F)-8 b(or)29 b(this)g(pro)s(of)g(the)h(authors)g(need)h(to)e(express)
j(the)e(Hopf)f(algebra,)g(ho)m(w)m(ev)m(er,)k(as)257
693 y(an)g(Ore)g(extension.)404 814 y(It)k(should)f(b)s(e)i(p)s
(ossible)e(to)g(extend)j(our)e(pro)s(of)f(to)g(arbitrary)g(Dynkin)h
(diagrams)257 934 y(of)42 b(\014nite)g(Cartan)g(t)m(yp)s(e.)74
b(The)43 b(only)f(crucial)f(part)h(is)g(the)g(comm)m(utation)f
(relations)257 1054 y(b)s(et)m(w)m(een)c(the)f(ro)s(ot)e(v)m(ectors)i
(and)f(their)g(p)s(o)m(w)m(ers.)52 b(They)36 b(should)f(b)s(e)g(c)m
(hec)m(k)m(ed)j(for)d(the)257 1175 y(other)k(diagrams.)59
b(An)39 b(analogue)e(of)h(Lemma)f(6.5)h(for)g(the)h(comm)m(utation)e
(relations)257 1295 y(b)s(et)m(w)m(een)31 b(ro)s(ot)d(v)m(ectors)j(b)s
(elonging)c(to)h(di\013eren)m(t)h(Cartan)g(t)m(yp)s(es)h(w)m(ould)f(b)s
(e)g(esp)s(ecially)257 1416 y(useful.)257 1644 y FJ(Remark:)49
b FL(The)25 b(Hopf)f(algebras)g(in)g(Section)g(4.6)g(coming)f(from)g
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1885 y(of)h(this)g(c)m(hapter)h(are)f(not)g(applicable.)49
b(The)36 b(step)m(wise)h(approac)m(h)e(used)h(in)f(the)501
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(quasi-isomorphism)c(is)i(not)h(p)s(ossible)f(there,)i(b)s(e-)501
2125 y(cause)29 b(the)f(linking)e(parameters)i(app)s(ear)f(in)g(the)h
(ro)s(ot)f(v)m(ector)i(parameters.)42 b(W)-8 b(e)501
2246 y(can)32 b(not)f(ev)m(en)i(apply)f(Theorem)f(6.3,)h(simply)e(b)s
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y(tor)36 b(parameters)g(are)g(not)g(in)f(the)i(group)f(algebra)f(and)h
(are)g(not)g(cen)m(tral.)54 b(But)501 2487 y(ha)m(ving)37
b(got)g(all)e(relations)g(from)h(felix,)h(w)m(e)h(kno)m(w)g(that)f(the)
h(diamond)d(lemma)501 2607 y(giv)m(es)e(us)g(a)g(basis)f(already)-8
b(.)501 2769 y(In)37 b(a)f(priv)-5 b(ate)35 b(note)i([Mas2],)g(ho)m(w)m
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501 2889 y(algebras)47 b(arising)f(from)g(self-linking)f
FI(A)2053 2904 y FH(2)2139 2889 y FE(ar)-5 b(e)55 b FL
(quasi-isomorphic.)86 b(F)-8 b(or)46 b(ter-)501 3010
y(minology)39 b(and)i(con)m(v)m(en)m(tions)h(w)m(e)g(refer)g(to)e
(Section)h(4.6.1.)69 b(His)40 b(approac)m(h)i(is)501
3130 y(as)d(follo)m(ws.)62 b(In)39 b(a)g(\014rst)h(step)g(he)f(deals)g
(with)g(the)g(linking)e(relations)h(and)h(the)501 3250
y(ro)s(ot)f(v)m(ector)i(relations)d(for)h(the)h(simple)f(ro)s(ots)g
(sim)m(ultaneously)-8 b(.)60 b(F)-8 b(or)38 b(this)h(he)501
3371 y(sets)k FI(I)50 b FL(:=)43 b(\()p FI(x)1033 3335
y FH(3)1033 3395 y FG(i)1073 3371 y FI(;)17 b(z)t(x)1221
3386 y FG(i)1278 3371 y FC(\000)29 b FI(q)1431 3335 y
FG(i)1459 3371 y FI(x)1514 3386 y FG(i)1542 3371 y FI(z)t
FL(;)17 b FI(i)44 b FL(=)f(1)p FI(;)17 b FL(2\))40 b(and)i
FI(H)48 b FL(is)41 b Fs(U)p FL(\()p FI(A)2661 3386 y
FH(2)2701 3371 y FL(\),)j(but)e FE(without)51 b FL(the)501
3491 y(Serre)46 b(relations)e(\(3.25\).)81 b(Then)47
b(he)f(sho)m(ws)h(using)e(his)g([Mas1,)k(Theorem)c(2])501
3611 y(that)39 b FI(T)51 b FL(:=)38 b FI(H)r(=)p FL(\()p
FI(I)8 b(:f)1275 3575 y Fx(\000)p FH(1)1369 3611 y FL(\))38
b(is)g(a)h(bi-Galois)c(ob)5 b(ject)39 b(for)f FI(L)2578
3575 y Fx(0)2640 3611 y FL(:=)f FI(H)r(=)p FL(\()p FI(f)5
b(:I)j(:f)3167 3575 y Fx(\000)p FH(1)3262 3611 y FL(\))38
b(and)501 3732 y FI(L)30 b FL(:=)g FI(H)r(=)p FL(\()p
FI(I)8 b FL(\).)46 b(Here)35 b FI(f)44 b FL(is)34 b(the)g(algebra)e
(map)h(sending)h(the)g(generators)h(of)e FI(I)42 b FL(to)501
3852 y FC(\000)p FI(\026)637 3867 y FG(i)698 3852 y FL(and)33
b FC(\000)p FI(\015)1016 3867 y FG(i)1077 3852 y FL(resp)s(ectiv)m(ely)
-8 b(.)44 b(De\014ning)899 4072 y FI(u)27 b FL(:=)h FI(z)1162
4031 y FH(3)1224 4072 y FC(\000)23 b FL(\()p FI(q)i FC(\000)e
FL(1\))1617 4031 y FH(3)1656 4072 y FI(\026)1715 4087
y FH(1)1754 4072 y FI(\026)1813 4087 y FH(2)1875 4072
y FC(\000)f FL(\(1)g FC(\000)h FI(q)t FL(\))p FI(\015)2319
4087 y FH(1)2357 4072 y FI(\015)2408 4087 y FH(2)2470
4072 y FL(+)f FI(\025)195 b FL(in)31 b FI(T)8 b(;)904
4218 y(v)31 b FL(:=)d FI(z)1162 4176 y FH(3)1224 4218
y FL(+)22 b(\()p FI(q)k FC(\000)c FL(1\))1615 4176 y
FH(3)1655 4218 y FI(\026)1714 4233 y FH(1)1753 4218 y
FI(\026)1812 4233 y FH(2)1851 4218 y FL(\()p FI(g)1940
4176 y FH(3)1936 4242 y(2)2001 4218 y FC(\000)g FL(1\)+)1074
4363 y(+)g(\(1)g FC(\000)g FI(q)t FL(\))p FI(\015)1516
4378 y FH(1)1555 4363 y FI(\015)1606 4378 y FH(2)1645
4363 y FL(\()p FI(g)1730 4378 y FH(1)1769 4363 y FI(g)1820
4322 y FH(2)1816 4387 y(2)1881 4363 y FC(\000)h FL(1\))f
FC(\000)g FI(\025)p FL(\()p FI(g)2335 4322 y FH(3)2331
4387 y(1)2374 4363 y FI(g)2425 4322 y FH(3)2421 4387
y(2)2486 4363 y FC(\000)g FL(1\))195 b(in)32 b FI(L)3047
4322 y Fx(0)3071 4363 y FI(;)501 4583 y FL(giv)m(es)38
b(that)f Fs(u)p FL(\()p FI(A)1124 4598 y FH(2)1163 4583
y FL(\))e(=)h FI(L)1414 4547 y Fx(0)1438 4583 y FI(=)p
FL(\()p FI(v)t FL(\))g(and)h Fs(u)p FL(\()p FC(D)2011
4598 y FH(0)2050 4583 y FI(;)17 b FJ(0)p FL(\))35 b(=)h
FI(L=)p FL(\()p FI(z)2537 4547 y FH(3)2577 4583 y FL(\))h(with)g
FC(D)2956 4598 y FH(0)3033 4583 y FL(the)g(linking)501
4703 y(datum)24 b(with)h(no)g(linking.)39 b(With)24 b(ingenious)g
(insigh)m(t)g(and)h(explicit)e(calculations)501 4824
y(he)34 b(then)g(sho)m(ws)g(that)f FI(Q)c FL(:=)g FI(T)11
b(=)p FL(\()p FI(u)p FL(\))32 b(is)h(actually)f(a)h(bi-Galois)c(ob)5
b(ject)34 b(for)f(these)501 4944 y(t)m(w)m(o)k(Hopf)f(algebras)f(and)i
(hence)g(they)g(are)f(quasi-isomorphic.)52 b(The)37 b(calcula-)501
5064 y(tions)d(in)g(this)g(case)h(are)g(sligh)m(tly)e(tric)m(kier)h
(than)g(for)g([BDR,)h(Prop)s(osition)d(3.3],)501 5185
y(as)d(the)g(structure)h(maps)f(for)f FI(T)42 b FL(map)28
b FI(T)43 b FL(in)m(to)27 b FI(L)2263 5149 y Fx(0)2301
5185 y FC(\012)14 b FI(T)43 b FL(and)29 b FI(T)f FC(\012)14
b FI(L;)30 b FL(resp)s(ectiv)m(ely)-8 b(.)501 5305 y(When)36
b(calculating)d(the)j(image)d(of)i FI(u)f FL(under)i(these)h(maps,)f
(one)f(therefore)h(has)1828 5637 y(76)p eop
%%Page: 77 79
77 78 bop 501 573 a FL(to)36 b(apply)h(di\013eren)m(t)f(comm)m(utation)
f(rules,)i(dep)s(ending)g(on)g(whic)m(h)g(tensor)g(fac-)501
693 y(tor)42 b(one)g(is)f(in.)71 b(All)41 b(tec)m(hnical)g
(di\016culties)g(can)h(b)s(e)g(dealt)g(with)f(directly)-8
b(,)44 b(as)501 814 y(the)31 b(comm)m(utation)d(relations)h(are)h
(explicit)f(and)i(the)g(diamond)d(lemma)g(can)j(b)s(e)501
934 y(applied.)501 1096 y(The)46 b(pro)s(of)f(for)g FI(A)1217
1111 y FH(2)1301 1096 y FL(should)h(b)s(e)f(easily)g(transferable)g(to)
g(the)g(self-linking)e(of)501 1216 y FI(B)575 1231 y
FH(2)615 1216 y FL(,)c(as)g(giv)m(en)f(in)g(Figure)f(4.4.)60
b(Ho)m(w)m(ev)m(er,)42 b(the)d(calculations)d(are)i(m)m(uc)m(h)h(more)
501 1337 y(in)m(v)m(olv)m(ed.)62 b(Ev)m(en)41 b(if)c(w)m(e)j(could)e
(guess)i(the)f(righ)m(t)f(\\)p FI(u)p FL(",)h(giving)e(us)i(a)g
(bi-Galois)501 1457 y(ob)5 b(ject)47 b(for)d(the)i(algebras)f(that)h
(incorp)s(orate)e(the)i(ro)s(ot)f(v)m(ector)h(relation)e(for)501
1577 y FI(z)t FL(,)h(pro)m(ving)d(it)f(directly)g(seems)i(hop)s(eless.)
72 b(The)43 b(expressions)h(for)d(\001\()p FI(z)3245
1541 y FH(5)3285 1577 y FL(\))h(are)501 1698 y(just)34
b(to)s(o)f(messy)-8 b(.)46 b(And)34 b(a)f(computer)h(algebra)e(program)
g(lik)m(e)h(felix)f(can)i(not)f(b)s(e)501 1818 y(applied)k(directly)-8
b(,)38 b(as)g(the)g(separate)g(tensor)g(factors)g(ha)m(v)m(e)h
(di\013eren)m(t)f(comm)m(u-)501 1939 y(tation)33 b(rules.)47
b(Besides,)35 b(after)e(dealing)f(with)i(the)g(relations)e(for)h
FI(z)2997 1902 y FH(5)3037 1939 y FL(,)h(w)m(e)h(w)m(ould)501
2059 y(then)k(still)d(ha)m(v)m(e)j(to)f(incorp)s(orate)f(the)i
(relations)e(for)g FI(u)2589 2023 y FH(5)2628 2059 y
FL(.)61 b(So,)39 b(again)e(w)m(e)i(need)501 2179 y(a)d(deep)s(er)h
(insigh)m(t)e(in)m(to)g(the)h(general)f(structure)i(theory)g(for)e
(these)i(self-link)m(ed)501 2300 y(algebras)32 b(\014rst,)h(b)s(efore)g
(w)m(e)h(attempt)e(a)g(quasi-isomorphism)d(theorem.)1828
5637 y(77)p eop
%%Page: 78 80
78 79 bop 257 1237 a FK(App)6 b(endix)78 b(A)257 1652
y(F)-19 b(elix)76 b(programs)257 2105 y FL(Here)37 b(w)m(e)h(giv)m(e)e
(the)g(programs)g(used)h(for)f(the)h(calculations)d(in)h(Section)i
(4.6.)54 b(T)-8 b(o)36 b(run)257 2225 y(these)43 b(scripts)g(one)f(has)
g(to)g(ha)m(v)m(e)h(a)e(w)m(orking)h(cop)m(y)h(of)e FJ(felix)g
FL([AK],)j(whic)m(h)f(can)f(b)s(e)257 2345 y(do)m(wnloaded)28
b(at)f Fe(http://felix.hgb-leipzig.de)q(/)p FL(.)48 b(F)-8
b(or)26 b(the)h(results)h(concerning)257 2466 y(the)e(copro)s(duct)f
(of)g(higher)g(p)s(o)m(w)m(ers)h(of)f(the)g(ro)s(ot)f(v)m(ectors)j(one)
e(needs)i(a)e(tensor)h(mo)s(dule.)257 2586 y(Istv\023)-49
b(an)26 b(Hec)m(k)m(en)m(b)s(erger)h(\(hec)m(k)m(en)m(b)s
(e@mathematik.uni-leipzig.de\))c(w)m(as)i(kind)f(enough)257
2707 y(to)41 b(pro)m(vide)g(his)g(extension)g(mo)s(dule)f
Fe(tensor.cmp)p FL(,)46 b(of)40 b(whic)m(h)h(w)m(e)h(use)g(the)g
Fe(ttimes)257 2827 y FL(function.)75 b(Sa)m(ving)43 b(one)h(of)e(the)i
(scripts)f(b)s(elo)m(w)g(in)g(a)g FE(\014le)p FL(,)i(one)f(can)f
(execute)i(it)d(b)m(y)257 2947 y(calling)78 b Fe(felix)53
b(<)e FE(\014le)29 b FL(in)f(a)h(Unix-lik)m(e)e(system)j(en)m(vironmen)
m(t.)43 b(The)30 b(tensor)g(mo)s(dule)257 3068 y(is)i(compiled)f(in)h
(the)h(same)g(w)m(a)m(y)-8 b(.)404 3188 y(W)g(e)33 b(w)m(an)m(t)g(to)f
(giv)m(e)h(some)f(commen)m(ts)h(on)f(these)i(scripts.)403
3383 y FC(\017)48 b FL(If)e(the)g(tensor)h(mo)s(dule)d(is)h(not)h(a)m
(v)-5 b(ailable,)47 b(then)f(only)f(the)h(Gro)s(ebner)g(basis)501
3503 y(calculation)23 b(is)i(p)s(ossible.)40 b(In)26
b(this)f(case)h(remo)m(v)m(e)g(the)f(\014rst)h(line)e(from)g(the)i
(scripts)501 3623 y(and)33 b(the)g(\014rst)g(\045-sign)f(in)g(fron)m(t)
g(of)g(the)h(\014rst)g Fe(bye)p FL(.)403 3824 y FC(\017)48
b FL(Eac)m(h)42 b(script)e(starts)h(b)m(y)h(de\014ning)e(a)h(domain)d
(with)j(its)f(parameters)g(and)h(the)501 3944 y(v)-5
b(ariables.)50 b(The)36 b(sym)m(b)s(ol)e Fe(ixi)h FL(is)g(de\014ned)h
(as)g(a)e(v)-5 b(ariable)33 b(and)i(treated)h(b)m(y)g(the)501
4065 y(tensor)d(mo)s(dule)f(as)g(the)h(tensor)h(sign)e
FC(\012)p FL(.)403 4265 y FC(\017)48 b FL(Then)34 b(a)e(matrix)f(is)h
(giv)m(en)h(whic)m(h)g(giv)m(es)g(an)g(appropriate)e(term-ordering.)403
4465 y FC(\017)48 b FL(The)37 b(ideal)d(has)h(all)f(the)i(de\014ning)f
(relations)f(of)h(the)h(algebra.)51 b(Because)37 b(of)e(the)501
4586 y(sp)s(ecial)f(prop)s(ert)m(y)i FI(q)1269 4550 y
FG(p)1340 4586 y FL(=)31 b(1)p FI(;)k FL(w)m(e)h(need)g(to)e(treat)h
FI(q)j FL(formally)32 b(as)j(a)g(v)-5 b(ariable)33 b(and)501
4706 y(not)f(as)h(a)f(parameter.)43 b(By)32 b(assigning)f(a)h(zero)h
(to)f(its)f(p)s(osition)g(in)g(the)i(ordering)501 4827
y(matrix)e(and)i(giving)e(all)f(the)j(comm)m(utation)e(relations)g(for)
h FI(q)t FL(,)g(felix)g(treats)h FI(q)j FL(in)501 4947
y(e\013ect)e(lik)m(e)e(a)g(parameter.)403 5147 y FC(\017)48
b FL(The)e(function)f Fe(standard)j FL(computes)d(the)h(Gro)s(ebner)f
(basis.)82 b(As)46 b(the)f(algo-)501 5268 y(rithm)50
b(for)h(the)h(non-comm)m(utativ)m(e)e(case)i(is)f(non-deterministic,)j
(the)e(righ)m(t)501 5388 y(term-ordering)43 b(can)i(b)s(e)g(essen)m
(tial.)79 b(This)45 b(is)f(esp)s(ecially)g(imp)s(ortan)m(t)f(for)h(the)
1828 5637 y(78)p eop
%%Page: 79 81
79 80 bop 501 573 a FI(G)578 588 y FH(2)647 573 y FL(case.)43
b(W)-8 b(e)29 b(w)m(an)m(t)h(to)f(thank)g(Istv\023)-49
b(an)30 b(Hec)m(k)m(en)m(b)s(erger)i(for)d(sho)m(wing)g(us)h(ho)m(w)g
(to)501 693 y(determine)i(a)h(useful)f(ordering.)403
897 y FC(\017)48 b FL(The)34 b(copro)s(duct)f(for)f(the)h(v)-5
b(ariables)31 b(is)h(denoted)i(b)m(y)f Fe(del)p FL(.)403
1100 y FC(\017)48 b FL(After)24 b(calculating)e(the)j(higher)e(p)s(o)m
(w)m(ers)j(of)d(the)i(copro)s(duct)f(of)g(the)g(ro)s(ot)g(v)m(ectors,)
501 1220 y(the)k(coun)m(ter)f(terms)g(are)g(guessed)i(from)d(the)h
(output)g(and)g(the)g(result)g(is)g(already)501 1341
y(incorp)s(orated)c(in)g(the)i(scripts.)41 b(The)24 b(summands)g(are)g
(all)e(on)h(separate)i(lines)e(and)501 1461 y(the)33
b(co)s(e\016cien)m(ts)h(are)f(also)e(separately)i(de\014ned.)403
1665 y FC(\017)48 b FL(The)38 b(last)f(part)f(of)h(the)g(calculations)f
(is)g(alw)m(a)m(ys)i(a)f(test,)h(c)m(hec)m(king)h(if)d(the)h(new)501
1785 y(expressions)46 b(are)e(sk)m(ew-primitiv)m(e)g(and)g(if)f(the)i
(p)s(o)m(w)m(ers)g(of)f(the)g(ro)s(ot)g(v)m(ectors)501
1905 y(are)34 b(cen)m(tral.)46 b(T)-8 b(o)34 b(b)s(e)g(able)e(to)i(see)
g(these)h(tests)g(b)s(etter)f(in)f(the)h(output,)g(a)f(short)501
2026 y(message)46 b(is)e(prin)m(ted)i(b)s(efore)f(them.)81
b(This)45 b(causes)i(felix)d(to)g(ev)-5 b(aluate)45 b(these)501
2146 y Fe(print)h FL(commands)d(and)i(pro)s(duces)g Fe(FALSE)g
FL(in)f(the)h(output.)78 b(This)45 b(is)e(not)h(a)501
2267 y(problem.)f(Successful)34 b(tests)f(will)e(giv)m(e)h
Fe(@)52 b(:=)f(0)33 b FL(as)g(output)g(afterw)m(ards.)257
2599 y FD(A.1)161 b(Listing)54 b(for)g(self-linking)i(of)e
Fp(A)2581 2621 y FJ(2)257 2818 y Fe(link\("tensor.mdl"\)$)257
3059 y(select)f(int\(lam12,lam21,mu1,mu2\))q(<z,x)q(1,x2)q(,g1)q(,g2,)q
(ixi,)q(q;)257 3179 y({{0,0,0,0,0,1,0},)309 3300 y({1,1,1,0,0,0,0},)309
3420 y({1,0,0,0,0,0,0},)309 3541 y({0,1,0,0,0,0,0},)309
3661 y({0,0,0,1,1,0,0},)309 3781 y({0,0,0,1,0,0,0},})257
3902 y(>$)257 4143 y(si:=ideal\()257 4263 y(q*x1-x1*q,)h(q*x2-x2*q,)g
(q*z-z*q,)257 4383 y(q*g1-g1*q,)g(q*g2-g2*q,)g(q*ixi-ixi*q,)257
4504 y(q^2+q+1,)g(g1*g2-g2*g1,)257 4744 y(g1*x1-q*x1*g1,)h
(g2*x1-q*x1*g2,)257 4865 y(g1*x2-q*x2*g1,)g(g2*x2-q*x2*g2,)257
4985 y(g1*z-q^2*z*g1,)g(g2*z-q^2*z*g2,)257 5226 y(x1*x2-q*x2*x1-z,)257
5346 y(x1*z-q^2*z*x1-lam12*\(1-g)q(1^2)q(*g2\))q(,)1828
5637 y FL(79)p eop
%%Page: 80 82
80 81 bop 257 573 a Fe(x2*\(x2*x1-q*x1*x2\)-q^2*\()q(x2*)q(x1-q)q(*x1*)
q(x2\)*)q(x2-)q(lam2)q(1*\(1)q(-g2^)q(2*g)q(1\),)257
814 y(x1^3-mu1*\(1-g1^3\),)257 934 y(x2^3-mu2*\(1-g2^3\))257
1054 y(\)$)257 1295 y(si:=standard\(si\)$)257 1416 y(\045bye$\045)257
1656 y(delx1:=g1*ixi*x1+x1*ixi$)257 1777 y(delx2:=g2*ixi*x2+x2*ixi$)257
1897 y(delz:=remainder\(ttimes\(d)q(elx)q(1,de)q(lx2\))821
2017 y(-ttimes\(q*ixi,ttimes\(del)q(x2,d)q(elx)q(1\)\),)q(si\)$)257
2138 y(delz2:=remainder\(ttimes\()q(del)q(z,de)q(lz\),)q(si\)$)257
2258 y(delz3:=remainder\(ttimes\()q(del)q(z2,d)q(elz\))q(,si\))q($)257
2499 y(f0:=\(1-q\)^3*mu1*mu2$)257 2619 y(f1:=\(1-q\)*q*lam12*lam21$)257
2860 y(v:=z^3)257 2980 y(+f0*\(1-g2^3\))257 3101 y(+f1*\(1-g2^2*g1\)$)
257 3342 y(delv:=delz3)257 3462 y(+f0*\(ixi-g2^3*ixi*g2^3\))257
3582 y(+f1*\(ixi-g2^2*g1*ixi*g2^)q(2*g)q(1\)$)257 3823
y(print\("TEST)55 b(IF)c(v)h(SKEW-PRIMITIVE"\)$)257 3944
y(remainder\(delv-v*ixi-g1^)q(3*g)q(2^3*)q(ixi*)q(v,si)q(\)$)257
4064 y(print\("TEST)j(IF)c(z^3)h(CENTRAL"\)$)257 4184
y(remainder\(z^3*x1-x1*z^3,)q(si\))q($)257 4305 y
(remainder\(z^3*x2-x2*z^3,)q(si\))q($)257 4425 y(bye$)257
4758 y FD(A.2)161 b(Listing)54 b(for)g(self-linking)i(of)e
Fp(B)2580 4779 y FJ(2)257 4977 y Fe(link\("tensor.mdl"\)$)257
5218 y(select)f(int\(lam12,lam21,mu1,mu2\))q(<u,z)q(,x1,)q(x2,)q(g1,g)q
(2,ix)q(i,q;)257 5338 y({{0,0,0,0,0,0,1,0},)1828 5637
y FL(80)p eop
%%Page: 81 83
81 82 bop 309 573 a Fe({1,1,1,1,0,0,0,0},)309 693 y({1,0,0,0,0,0,0,0},)
309 814 y({0,1,0,0,0,0,0,0},)309 934 y({0,0,1,0,0,0,0,0},)309
1054 y({0,0,0,0,1,1,0,0},)309 1175 y({0,0,0,0,1,0,0,0}})257
1295 y(>$)257 1536 y(si:=ideal\()257 1777 y(q*u-u*q,)54
b(q*z-z*q,)f(q*x1-x1*q,)h(q*x2-x2*q,)257 1897 y(q*g1-g1*q,)g
(q*g2-g2*q,)g(q*ixi-ixi*q,)257 2017 y(q^4+q^3+q^2+q+1,)i(g1*g2-g2*g1,)
257 2258 y(g1*u-q^4*u*g1,)f(g2*u-q^4*u*g2,)257 2379 y(g1*z-q^3*z*g1,)g
(g2*z-q^3*z*g2,)257 2499 y(g1*x2-q*x2*g1,)g(g2*x2-q*x2*g2,)257
2619 y(g1*x1-q^2*x1*g1,)h(g2*x1-q^2*x1*g2,)257 2860 y
(x2*x1-q^2*x1*x2-z,)257 2980 y(x2*z-q^3*z*x2-u,)257 3221
y(x1*\(x1*x2-q*x2*x1\)-q^3*\()q(x1*)q(x2-q)q(*x2*)q(x1\)*)q(x1-)q(lam1)
q(2*\(1)q(-g1^)q(2*g)q(2\),)257 3342 y(x2*u-q^4*u*x2-lam21*\(1-g)q(2^3)
q(*g1\))q(,)257 3582 y(x1^5-mu1*\(1-g1^5\),)257 3703
y(x2^5-mu2*\(1-g2^5\))257 3823 y(\)$)257 4064 y(si:=standard\(si\)$)257
4184 y(\045bye$\045)257 4425 y(delx1:=g1*ixi*x1+x1*ixi*)q(1$)257
4545 y(delx2:=g2*ixi*x2+x2*ixi*)q(1$)257 4666 y
(delz:=remainder\(ttimes\(d)q(elx)q(2,de)q(lx1\))873
4786 y(-ttimes\(q^2*ixi,ttimes\(delx)q(1,d)q(elx2)q(\)\),s)q(i\)$)257
4907 y(delz2:=remainder\(ttimes\()q(del)q(z,de)q(lz\),)q(si\)$)257
5027 y(delz3:=remainder\(ttimes\()q(del)q(z2,d)q(elz\))q(,si\))q($)257
5147 y(delz4:=remainder\(ttimes\()q(del)q(z3,d)q(elz\))q(,si\))q($)257
5268 y(delz5:=remainder\(ttimes\()q(del)q(z4,d)q(elz\))q(,si\))q($)1828
5637 y FL(81)p eop
%%Page: 82 84
82 83 bop 257 573 a Fe(f0:=-mu1*mu2*\(q^2-1\)^5$)257
693 y(f1:=\(-q^3+q^2+q-1\)*lam12)q(^2*)q(lam2)q(1$)257
814 y(f2:=\(2*q^3+2*q^2+1\)*lam1)q(2^2)q(*lam)q(21$)257
934 y(f3:=\(q^3+2*q^2+3*q-1\)*la)q(m12)q(*lam)q(21$)257
1054 y(f4:=-q*\(q^2+3*q+1\)*lam12)q(^2*)q(lam2)q(1$)257
1295 y(v:=z^5)257 1416 y(+f0*\(1-g1^5\))257 1536 y(+f1*g1^5*g2^5)257
1656 y(+f2*g1^4*g2^2)257 1777 y(+f3*z*x1*g1^2*g2)257
1897 y(+f4*g1^2*g2$)257 2138 y(delv:=remainder\(delz5)257
2258 y(+f0*\(ixi-g1^5*ixi*g1^5\))257 2379 y(+f1*g1^5*g2^5*ixi*g1^5*g)q
(2^5)257 2499 y(+f2*g1^4*g2^2*ixi*g1^4*g)q(2^2)257 2619
y(+f3*ttimes\(ttimes\(delz,d)q(elx)q(1\),g)q(1^2*)q(g2*i)q(xi*)q(g1^2)q
(*g2\))257 2740 y(+f4*g1^2*g2*ixi*g1^2*g2)257 2860 y(,si\)$)257
3101 y(print\("TEST)55 b(IF)c(v)h(SKEWPRIMITIVE"\)$)257
3221 y(remainder\(delv-v*ixi-g1^)q(5*g)q(2^5*)q(ixi*)q(v,si)q(\)$)257
3342 y(print\("TEST)j(IF)c(z^5)h(CENTRAL"\)$)257 3462
y(remainder\(z^5*x1-x1*z^5,)q(si\))q($)257 3582 y
(remainder\(z^5*x2-x2*z^5,)q(si\))q($)257 3703 y(\045bye$\045)257
3944 y(delu:=remainder\(ttimes\(d)q(elx)q(2,de)q(lz\))873
4064 y(-ttimes\(q^3*ixi,ttimes\(delz)q(,de)q(lx2\))q(\),si)q(\)$)257
4184 y(delu2:=remainder\(ttimes\()q(del)q(u,de)q(lu\),)q(si\)$)257
4305 y(delu3:=remainder\(ttimes\()q(del)q(u2,d)q(elu\))q(,si\))q($)257
4425 y(delu4:=remainder\(ttimes\()q(del)q(u3,d)q(elu\))q(,si\))q($)257
4545 y(delu5:=remainder\(ttimes\()q(del)q(u4,d)q(elu\))q(,si\))q($)257
4786 y(k0:=-2*\(q-1\)^5*mu2$)257 4907 y(k1:=-\(q^2-1\)^5*\(q-1\)^5*m)q
(u2^)q(2$)257 5027 y(k2:=-5*\(2*q^3-q^2+q-2\)*l)q(am1)q(2^2*)q(lam2)q
(1$)257 5147 y(k3:=-\(3*q^3+q^2-q+2\)*lam)q(12*)q(lam2)q(1^3$)257
5268 y(k4:=\(3*q^3+q^2+4*q+2\)*la)q(m12)q(*lam)q(21^3)q($)257
5388 y(k5:=-5*\(q^3+1\)*lam12*lam)q(21^)q(2$)1828 5637
y FL(82)p eop
%%Page: 83 85
83 84 bop 257 573 a Fe(k6:=5*\(q^3-1\)*lam12*lam2)q(1$)257
693 y(k7:=-2*\(q^3+2*q^2+3*q-1\))q(*la)q(m12*)q(lam2)q(1^3$)257
814 y(k8:=-5*\(q^3+q^2-1\)*lam12)q(*la)q(m21^)q(2$)257
934 y(k9:=\(2*q^3+4*q^2+q-2\)*la)q(m12)q(*lam)q(21^3)q($)257
1175 y(w:=u^5)257 1295 y(+k0*v)257 1416 y(+k1*x1^5)257
1536 y(+k2*x2^5*g1^5*g2^5)257 1656 y(+k3*g1^5*g2^10)257
1777 y(+k4*g1^4*g2^7)257 1897 y(+k5*u*x2*g1^3*g2^4)257
2017 y(+k6*u^2*x2^2*g1^2*g2)257 2138 y(+k7*g1^3*g2^4)257
2258 y(+k8*u*x2*g1^2*g2)257 2379 y(+k9*g1^2*g2$)257 2619
y(delw:=remainder\(delu5)257 2740 y(+k0*delv)257 2860
y(+k1*\(x1^5*ixi+g1^5*ixi*x)q(1^5)q(\))257 2980 y
(+k2*ttimes\(x2^5*ixi+g2^5)q(*ix)q(i*x2)q(^5,g)q(1^5*)q(g2^)q(5*ix)q
(i*g1)q(^5*g)q(2^5)q(\))257 3101 y(+k3*g1^5*g2^10*ixi*g1^5*)q(g2^)q(10)
257 3221 y(+k4*g1^4*g2^7*ixi*g1^4*g)q(2^7)257 3342 y
(+k5*ttimes\(ttimes\(delu,d)q(elx)q(2\),g)q(1^3*)q(g2^4)q(*ix)q(i*g1)q
(^3*g)q(2^4\))257 3462 y(+k6*ttimes\(ttimes\(ttimes)q(\(tt)q(imes)q
(\(del)q(u,de)q(lu\))q(,del)q(x2\),)q(delx)q(2\))821
3582 y(,g1^2*g2*ixi*g1^2*g2\))257 3703 y(+k7*g1^3*g2^4*ixi*g1^3*g)q
(2^4)257 3823 y(+k8*ttimes\(ttimes\(delu,d)q(elx)q(2\),g)q(1^2*)q(g2*i)
q(xi*)q(g1^2)q(*g2\))257 3944 y(+k9*g1^2*g2*ixi*g1^2*g2)257
4064 y(,si\)$)257 4305 y(print\("TEST)55 b(IF)c(w)h(SKEW-PRIMITIVE"\)$)
257 4425 y(remainder\(delw-w*ixi-g1^)q(5*g)q(2^10)q(*ixi)q(*w,s)q(i\)$)
257 4545 y(print\("TEST)j(IF)c(u^5)h(CENTRAL"\)$)257
4666 y(remainder\(u^5*x1-x1*u^5,)q(si\))q($)257 4786
y(remainder\(u^5*x2-x2*u^5,)q(si\))q($)257 4907 y(bye$)1828
5637 y FL(83)p eop
%%Page: 84 86
84 85 bop 257 573 a FD(A.3)161 b(Listing)54 b(for)g(self-linking)i(of)e
Fp(G)2583 594 y FJ(2)257 792 y Fe(link\("tensor.mdl"\)$)257
1033 y(select)f(int\(lam12,lam21,mu1,mu2\))q(<w,v)q(,u,z)q(,x1)q(,x2,)q
(g1,g)q(2,ix)q(i,q)q(;)257 1153 y({{0,0,0,0,0,0,0,0,1,0},)309
1273 y({5,4,3,2,1,1,0,0,0,0},)309 1394 y({2,2,1,1,1,1,0,0,0,0},)309
1514 y({0,0,0,0,1,1,0,0,0,0},)309 1635 y({0,0,0,0,1,0,0,0,0,0},)309
1755 y({0,0,0,1,0,0,0,0,0,0},)309 1875 y({0,0,1,0,0,0,0,0,0,0},)309
1996 y({0,1,0,0,0,0,0,0,0,0},)309 2116 y({0,0,0,0,0,0,1,1,0,0},)309
2236 y({0,0,0,0,0,0,1,0,0,0}})257 2357 y(>$)257 2598
y(si:=ideal\()257 2718 y(q*w-w*q,)h(q*v-v*q,)f(q*u-u*q,)h(q*z-z*q,)257
2838 y(q*x2-x2*q,)g(q*x1-x1*q,)g(q*g2-g2*q,)g(q*g1-g1*q,)257
2959 y(q^6+q^5+q^4+q^3+q^2+q+1,)k(g1*g2-g2*g1,)c(q*ixi-ixi*q,)257
3200 y(g1*w-q^2*w*g1,)h(g2*w-q^2*w*g2,)257 3320 y(g1*v-q^6*v*g1,)g
(g2*v-q^6*v*g2,)257 3440 y(g1*u-q^5*u*g1,)g(g2*u-q^5*u*g2,)257
3561 y(g1*z-q^4*z*g1,)g(g2*z-q^4*z*g2,)257 3681 y(g1*x2-q^3*x2*g1,)h
(g2*x2-q^3*x2*g2,)257 3801 y(g1*x1-q*x1*g1,)f(g2*x1-q*x1*g2,)257
4042 y(z-x2*x1+q*x1*x2,)257 4163 y(u-z*x1+q^2*x1*z,)257
4283 y(v-u*x1+q^3*x1*u,)257 4403 y(w-z*u+q^3*u*z,)257
4644 y(x1*\(x1*\(x1*\(x1*x2-q^3*x2)q(*x1)q(\))262 b
(-q^4*\(x1*x2-q^3*x2*x1\)*x)q(1\))616 4764 y
(-q^5*\(x1*\(x1*x2-q^3*x2*x)q(1\)-q)q(^4*\()q(x1*)q(x2-q)q(^3*x)q(2*x1)
q(\)*x)q(1\)*x)q(1\))411 4885 y(-q^6*\(x1*\(x1*\(x1*x2-q^3*)q(x2*x)q
(1\)-q)q(^4*\()q(x1*)q(x2-q)q(^3*x)q(2*x1)q(\)*x)q(1\))616
5005 y(-q^5*\(x1*\(x1*x2-q^3*x2*x)q(1\)-q)q(^4*\()q(x1*)q(x2-q)q(^3*x)q
(2*x1)q(\)*x)q(1\)*x)q(1\)*x)q(1)309 5126 y(-lam12*\(1-g1^4*g2\),)257
5246 y(x2*z-q^4*z*x2-lam21*\(1-g)q(1*g)q(2^2\))q(,)1828
5637 y FL(84)p eop
%%Page: 85 87
85 86 bop 257 573 a Fe(x1^7-mu1*\(1-g1^7\),)257 693 y
(x2^7-mu2*\(1-g2^7\))257 814 y(\)$)257 1054 y(si:=standard\(si\)$)257
1175 y(\045bye$\045)257 1416 y(delx1:=g1*ixi*x1+x1*ixi*)q(1$)257
1536 y(delx2:=g2*ixi*x2+x2*ixi*)q(1$)257 1777 y
(delz:=remainder\(ttimes\(d)q(elx)q(2,de)q(lx1\))873
1897 y(-ttimes\(q*ixi,ttimes\(delx1,)q(del)q(x2\)\))q(,si\))q($)257
2017 y(delz2)53 b(:=remainder\(ttimes\(delz,d)q(elz\))q(,si\))q($)257
2138 y(delz3)g(:=remainder\(ttimes\(delz2,)q(delz)q(\),si)q(\)$)257
2258 y(delz4)g(:=remainder\(ttimes\(delz3,)q(delz)q(\),si)q(\)$)257
2379 y(delz5)g(:=remainder\(ttimes\(delz4,)q(delz)q(\),si)q(\)$)257
2499 y(delz6)g(:=remainder\(ttimes\(delz5,)q(delz)q(\),si)q(\)$)257
2619 y(delz7)g(:=remainder\(ttimes\(delz6,)q(delz)q(\),si)q(\)$)257
2860 y(f0:=\(1-q^3\)^7*mu1*mu2$)257 2980 y(f1:=\(-2*q^5-4*q^4+q^3-q^)q
(2+4)q(*q+2)q(\)*la)q(m12*)q(lam)q(21$)257 3101 y
(f2:=\(6*q^5+8*q^4+6*q^3-3)q(*q-)q(3\)*l)q(am12)q(*lam)q(21^)q(2$)257
3221 y(f3:=\(-q^4-3*q^3+q^2-3*q-)q(1\)*)q(lam1)q(2*la)q(m21^)q(2$)257
3342 y(f4:=\(2*q^5+2*q^4+4*q^3+5)q(*q^)q(2+2*)q(q-1\))q(*lam)q(12*)q
(lam2)q(1^3$)257 3462 y(f5:=\(q^5-2*q^4-4*q^3-7*q)q(^2-)q(6*q-)q(3\)*l)
q(am12)q(*la)q(m21^)q(3$)257 3582 y(f6:=\(-4*q^5-2*q^4-2*q^3+)q(2*q)q
(^2+2)q(*q+4)q(\)*la)q(m12)q(*lam)q(21^3)q($)257 3703
y(f7:=\(q^5+2*q^4+2*q^3+2*q)q(\)*l)q(am12)q(*lam)q(21^3)q($)257
3944 y(zz:=z^7)257 4064 y(+f0*\(1-g1^7\))257 4184 y
(+f1*z^2*x2^2*g1^4*g2)257 4305 y(+f2*z*x2*g1^5*g2^3)257
4425 y(+f3*z*x2*g1^4*g2)257 4545 y(+f4*g1^7*g2^7)257
4666 y(+f5*g1^6*g2^5)257 4786 y(+f6*g1^5*g2^3)257 4907
y(+f7*g1^4*g2$)257 5147 y(delzz:=delz7)257 5268 y
(+f0*\(ixi-g1^7*ixi*g1^7\))257 5388 y(+f1*ttimes\(ttimes\(delz2,)q(tti)
q(mes\()q(delx)q(2,de)q(lx2)q(\)\),g)q(1^4*)q(g2*i)q(xi*)q(g1^4)q
(*g2\))1828 5637 y FL(85)p eop
%%Page: 86 88
86 87 bop 257 573 a Fe(+f2*ttimes\(ttimes\(delz,d)q(elx)q(2\),g)q(1^5*)
q(g2^3)q(*ix)q(i*g1)q(^5*g)q(2^3\))257 693 y
(+f3*ttimes\(ttimes\(delz,d)q(elx)q(2\),g)q(1^4*)q(g2*i)q(xi*)q(g1^4)q
(*g2\))257 814 y(+f4*g1^7*g2^7*ixi*g1^7*g)q(2^7)257 934
y(+f5*g1^6*g2^5*ixi*g1^6*g)q(2^5)257 1054 y(+f6*g1^5*g2^3*ixi*g1^5*g)q
(2^3)257 1175 y(+f7*g1^4*g2*ixi*g1^4*g2$)257 1416 y(print\("TEST)55
b(IF)c(zz)h(SKEW_PRIMITIVE"\)$)257 1536 y(remainder\(delzz-zz*ixi-g)q
(1^7)q(*g2^)q(7*ix)q(i*zz)q(,si)q(\)$)257 1656 y(print\("TEST)j(IF)c
(z^7)h(CENTRAL"\)$)257 1777 y(remainder\(z^7*x1-x1*z^7,)q(si\))q($)257
1897 y(remainder\(z^7*x2-x2*z^7,)q(si\))q($)257 2017
y(bye$)1828 5637 y FL(86)p eop
%%Page: 87 89
87 88 bop 257 1237 a FK(Bibliograph)-6 b(y)257 1689 y
FL([A)m(CM])104 b(G.)35 b(Amelino-Camelia,)e(S.)j(Ma)5
b(jid,)37 b FE(Waves)h(on)g(Nonc)-5 b(ommutative)37 b(Sp)-5
b(ac)g(e-)644 1810 y(time)27 b(and)g(Gamma-R)-5 b(ay)26
b(Bursts)p FL(,)h(In)m(t.)e(J.)f(Mo)s(d.)h(Ph)m(ys.)h(A)e
FJ(15)h FL(\(2000\),)g(4301-)644 1930 y(4323.)257 2134
y([A)m(G])186 b(N.)45 b(Andruskiewitsc)m(h,)51 b(M.)46
b(Gra)s(~)-51 b(na,)47 b FE(Br)-5 b(aide)g(d)46 b(Hopf)h(algebr)-5
b(as)46 b(over)g(non-)644 2254 y(ab)-5 b(elian)33 b(gr)-5
b(oups)p FL(,)33 b(Bol.)e(Acad.)i(Ciencias)g(\(C\023)-49
b(ordoba\))32 b FJ(63)h FL(\(1999\),)e(45-78.)257 2457
y([AK])184 b(J.)46 b(Ap)s(el,)j(U.)e(Klaus,)i FE(F)-7
b(elix)p FL(,)48 b(A)e(sp)s(ecial)f(computer)h(algebra)g(system)h(for)
644 2578 y(the)41 b(computation)f(in)g(comm)m(utativ)m(e)g(and)i
(non-comm)m(utativ)m(e)d(rings)i(and)644 2698 y(mo)s(dules,)31
b(a)m(v)-5 b(ailable)30 b(at)j Fe(http://felix.hgb-leipzig.d)q(e/)p
FL(.)257 2902 y([And])152 b(N.)34 b(Andruskiewitsc)m(h,)i
FE(A)n(b)-5 b(out)36 b(\014nite)g(dimensional)e(Hopf)i(algebr)-5
b(as)p FL(,)34 b(Notes)644 3022 y(of)39 b(a)g(course)i(giv)m(en)f(at)f
(the)h(CIMP)-8 b(A)41 b(Sc)m(ho)s(ol)e("Quan)m(tum)h(symmetries)f(in)
644 3142 y(theoretical)45 b(ph)m(ysics)j(and)e(mathematics",)j(Barilo)s
(c)m(he)c(2000,)k(Con)m(temp.)644 3263 y(Math)32 b FJ(294)h
FL(\(2002\),)f(1-57.)257 3466 y([AS1])157 b(N.)47 b(Andruskiewitsc)m
(h,)53 b(H.-J.)47 b(Sc)m(hneider,)52 b FE(Lifting)c(of)f(Quantum)i
(Line)-5 b(ar)644 3587 y(Sp)g(ac)g(es)53 b(and)g(Pointe)-5
b(d)54 b(Hopf)g(algebr)-5 b(as)53 b(of)g(Or)-5 b(der)54
b FI(p)2685 3550 y FH(3)2724 3587 y FL(,)59 b(J.)54 b(Algebra)e
FJ(209)644 3707 y FL(\(1998\),)31 b(658-691.)257 3910
y([AS2])157 b(N.)37 b(Andruskiewitsc)m(h,)j(H.-J.)c(Sc)m(hneider,)k
FE(Finite)e(Quantum)h(Gr)-5 b(oups)39 b(and)644 4031
y(Cartan)34 b(Matric)-5 b(es)p FL(,)33 b(Adv.)g(Math.)g
FJ(154)g FL(\(2000\),)e(1-45.)257 4234 y([AS3])157 b(N.)41
b(Andruskiewitsc)m(h,)46 b(H.-J.)c(Sc)m(hneider,)j FE(Finite)d(quantum)
i(gr)-5 b(oups)43 b(over)644 4355 y(ab)-5 b(elian)40
b(gr)-5 b(oups)42 b(of)g(prime)f(exp)-5 b(onent)p FL(,)41
b(Ann.)g(Sci.)f(Ec.)h(Norm.)e(Sup)s(er.)i FJ(35)644 4475
y FL(\(2002\),)31 b(1-26.)257 4678 y([AS4])157 b(N.)32
b(Andruskiewitsc)m(h,)i(H.-J.)e(Sc)m(hneider,)i FE(Lifting)g(of)g
(Nichols)g(A)n(lgebr)-5 b(as)34 b(of)644 4799 y(T)-7
b(yp)i(e)37 b FI(A)954 4814 y FH(2)1031 4799 y FE(and)g(Pointe)-5
b(d)37 b(Hopf)h(A)n(lgebr)-5 b(as)36 b(of)i(or)-5 b(der)37
b FI(p)2641 4763 y FH(4)2680 4799 y FE(,)i(in)e(\\Hopf)g(algebr)-5
b(as)644 4919 y(and)34 b(quantum)g(gr)-5 b(oups",)34
b(Pr)-5 b(o)g(c)g(e)g(e)g(dings)33 b(of)i(the)f(Brussels)g(Confer)-5
b(enc)g(e,)33 b(e)-5 b(ds.)644 5039 y(S.)36 b(Caenep)-5
b(e)g(el,)35 b(F.)h(V)-7 b(an)37 b(Oystaeyen)p FL(,)d(Lecture)i(Notes)f
(in)f(Pure)h(and)g(Appl.)644 5160 y(Math.,)e(Marcel)f(Dekk)m(er,)i(New)
g(Y)-8 b(ork)32 b FJ(209)h FL(\(2000\),)f(1-14.)1828
5637 y(87)p eop
%%Page: 88 90
88 89 bop 257 573 a FL([AS5])157 b(N.)55 b(Andruskiewitsc)m(h,)64
b(H.-J.)55 b(Sc)m(hneider,)63 b FE(Pointe)-5 b(d)56 b(Hopf)g(algebr)-5
b(as)p FL(,)60 b(in)644 693 y("New)36 b(directions)g(in)f(Hopf)h
(algebras",)g(MSRI)h(series,)h(Cam)m(bridge)d(Univ.)644
814 y(Press,)f(2002.)257 1017 y([AS6])157 b(N.)42 b(Andruskiewitsc)m
(h,)47 b(H.-J.)c(Sc)m(hneider,)j FE(A)e(char)-5 b(acterization)43
b(of)h(quan-)644 1137 y(tum)35 b(gr)-5 b(oups)p FL(,)32
b(preprin)m(t)65 b(\(2002\),)32 b(a)m(v)-5 b(ailable)30
b(at)644 1258 y Fe(www.mathematik.uni-muenche)q(n.de)q(/~ha)q(nss)q
(ch/P)q(ubli)q(cati)q(ons)q(.htm)q(l)p FL(.)257 1461
y([BDR])117 b(M.)32 b(Beattie,)g(S.)h(D\025)-49 b(asc\025)g(alescu,)33
b(S.)f(Raian)m(u,)g FE(Lifting)i(of)h(Nichols)f(algebr)-5
b(as)34 b(of)644 1582 y(typ)-5 b(e)35 b FI(B)922 1597
y FH(2)961 1582 y FL(,)e(preprin)m(t)65 b(\(2001\),)32
b(a)m(v)-5 b(ailable)30 b(as)j Fe(math.QA/0204075)p FL(.)257
1785 y([BS])210 b(Z.)26 b(I.)g(Borevic,)i(I.)e(Shafarevic,)i
FE(Numb)-5 b(er)29 b(the)-5 b(ory)p FL(,)28 b(Russian)e(edition)f(of)g
(\\T)-8 b(eo-)644 1905 y(ria)31 b(Chisel")64 b(\(1964\),)32
b(Izdatelstv)m(o)h(Nauk)-5 b(a,)33 b(Mosco)m(w.)257 2109
y([CD])188 b(S.)27 b(Caenep)s(eel,)h(S.)f(D\025)-49 b(asc\025)g
(alescu,)28 b FE(Pointe)-5 b(d)29 b(Hopf)h(algebr)-5
b(as)28 b(of)h(dimension)f FI(p)3429 2073 y FH(3)3468
2109 y FL(,)644 2229 y(J.)k(Algebra)g FJ(209)h FL(\(1998\),)e(622-634.)
257 2433 y([CK])187 b(A.)43 b(Connes,)k(D.)c(Kreimer,)i
FE(R)-5 b(enormalization)43 b(in)h(quantum)h(\014eld)f(the)-5
b(ory)644 2553 y(and)42 b(the)i(R)n(iemann-Hilb)-5 b(ert)42
b(pr)-5 b(oblem)42 b(I:)h(the)g(Hopf)h(algebr)-5 b(a)42
b(structur)-5 b(e)45 b(of)644 2673 y(gr)-5 b(aphs)39
b(and)h(the)g(main)f(the)-5 b(or)g(em)p FL(,)39 b(Comm)m(un.)f(Math.)h
(Ph)m(ys.)h FJ(210)e FL(\(2000\),)644 2794 y(249-273.)257
2997 y([CP])197 b(V.)31 b(Chari,)g(A.)h(N.)f(Pressley)-8
b(,)33 b FE(A)h(Guide)g(to)g(Quantum)g(Gr)-5 b(oups)p
FL(,)32 b(Cam)m(bridge)644 3117 y(Univ.)g(Press,)i(1994.)257
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(arise)g(from)f(the)i(lifting)257 4666 y(metho)s(d)30
b(using)g(only)g(Dynkin)h(diagrams)d(of)i(t)m(yp)s(e)i
FI(A)2234 4681 y FG(n)2311 4666 y FL(displa)m(y)e(the)h(same)f(b)s(eha)
m(viour,)257 4786 y(Theorem)54 b(6.6.)106 b(This)53 b(means)h(that)f
(all)e(the)j(\014nite)f(dimensional)e(p)s(oin)m(ted)i(Hopf)257
4907 y(algebras)63 b(constructed)j(in)d(this)g(w)m(a)m(y)-8
b(,)73 b(whic)m(h)64 b(only)f(di\013er)g(in)g(their)h(c)m(hoice)g(of)
257 5027 y(parameters)52 b(are)h(2-co)s(cycle)e(deformations)g(of)h
(eac)m(h)h(other.)102 b(Our)52 b(pro)s(of)f(should)257
5147 y(b)s(e)46 b(easily)g(adaptable)f(to)g(the)h(Hopf)g(algebras)f
(asso)s(ciated)h(with)f(the)i(other)f(t)m(yp)s(es)257
5268 y(of)g(\014nite)h(Dynkin)f(diagrams,)j(once)e(all)d(parameters)j
(ha)m(v)m(e)h(b)s(een)f(determined)g(for)257 5388 y(these)37
b(algebras)d(explicitly)-8 b(.)49 b(This)35 b(raises)g(the)g(hop)s(e)g
(that)g(Masuok)-5 b(a's)36 b(conjecture)g(in)1828 5637
y(92)p eop
%%Page: 93 95
93 94 bop 257 573 a FL([Mas1)q(])35 b(can)g(b)s(e)g(sa)m(v)m(ed)i(in)d
(spite)h(of)f(the)i(coun)m(ter-example)f(in)f([EG])h(b)m(y)h(sp)s
(ecializing)257 693 y(it)c(sligh)m(tly)f(\(page)h(60\).)257
934 y FD(Zusammenfassung)404 1175 y FL(In)37 b(dieser)g(Dissertation)e
(w)m(ollen)h(wir)g(zu)h(Klassi\014zierungsresultaten)f(f)s(\177)-51
b(ur)36 b(punk-)257 1295 y(tierte)h(Hopfalgebren)f(mit)f(ab)s(elsc)m
(hem)h(Koradik)-5 b(al)34 b(b)s(eitragen,)j(die)f(v)m(or)h(kurzer)h
(Zeit)257 1416 y(v)m(on)j(Andruskiewitsc)m(h)h(und)f(Sc)m(hneider)h
([AS1,)e(AS3,)h(AS5,)f(AS6])g(gew)m(onnen)i(wur-)257
1536 y(den.)63 b(Deren)39 b(Liftingmetho)s(de)d(erzeugt)j(neue)h
(Klassen)f(v)m(on)g(Hopfalgebren.)61 b(Diese)257 1656
y(Algebren)f(w)m(erden)i(ausgehend)g(v)m(on)e(einem)g(V)-8
b(erbindungs-Datum)59 b(k)m(onstruiert,)257 1777 y(w)m(elc)m(hes)26
b(aus)f(einer)f(Grupp)s(e,)i(einem)e(Dynkin-Diagramm,)e(einigen)h(P)m
(arametern)i(und)257 1897 y(einer)35 b(Reihe)f(v)m(on)h(Grupp)s
(enelemen)m(ten)g(und)g(-c)m(harakteren,)h(die)e(gewisse)i(Kompati-)257
2017 y(bilit\177)-49 b(atsb)s(edingungen)35 b(erf)s(\177)-51
b(ullen,)37 b(b)s(esteh)m(t.)59 b(Diese)37 b(Bedingungen)h(sind)f
(ziemlic)m(h)e(im-)257 2138 y(plizit)21 b(gegeb)s(en)j(und)g(erlaub)s
(en)f(daher)h(meist)e(k)m(eine)i(einfac)m(he)g(explizite)e(Besc)m
(hreibung)257 2258 y(dieser)27 b(Hopfalgebren.)41 b(In)27
b(dieser)f(Arb)s(eit)g(b)s(ehandeln)h(wir)e(ausf)s(\177)-51
b(uhrlic)m(h)25 b(v)m(ersc)m(hiedene)257 2379 y(Asp)s(ekte)35
b(solc)m(h)d(einer)h(Besc)m(hreibung.)404 2499 y(Einer)46
b(unserer)j(w)m(esen)m(tlic)m(hen)f(Beitr\177)-49 b(age)46
b(ist)g(die)h(explizite)f(Ausarb)s(eitung)g(des)257 2619
y(V)-8 b(erbindungsk)m(onzepts.)80 b(Aufbauend)44 b(auf)f(der)h
(Originalarb)s(eit)c([AS3])k(f)s(\177)-51 b(uhren)43
b(wir)257 2740 y(zun\177)-49 b(ac)m(hst)28 b(eine)f(passende)h(T)-8
b(erminologie)23 b(ein,)k(De\014nitionen)e(3.3-3.7.)40
b(Danac)m(h)27 b(geb)s(en)257 2860 y(wir)46 b(ein)g(einfac)m(h)g(an)m
(w)m(endbares)j(Kriterium)44 b(an,)50 b(Theorem)d(4.2,)i(das)e(en)m
(tsc)m(heiden)257 2980 y(hilft,)24 b(w)m(elc)m(he)h(V)-8
b(erbindungen)24 b(zu)h(endlic)m(hdimensionalen)c(Hopfalgebren)i(f)s
(\177)-51 b(uhren)24 b(und)257 3101 y(w)m(as)40 b(f)s(\177)-51
b(ur)37 b(Bedingungen)i(an)f(das)h(Koradik)-5 b(al)36
b(gestellt)h(w)m(erden)j(m)s(\177)-51 b(ussen.)61 b(Dazu)39
b(m)m(u\031)257 3221 y(man)c(n)m(ur,)i(gewisse)f(Ob)5
b(jekte)37 b(in)e(Graphen)g(z\177)-49 b(ahlen)36 b(und)g(aus)f(diesen)h
(Daten)g(das)f(so-)257 3342 y(genann)m(te)41 b(Gesc)m(hlec)m(h)m(t)g(b)
s(erec)m(hnen.)66 b(Wir)39 b(erw)m(eitern)i(unser)f(Resultat)f(in)g
(Theorem)257 3462 y(4.5)h(auc)m(h)g(auf)g(a\016ne)g(Dynkin-Diagramme.)
62 b(Beispiele)39 b(exotisc)m(her)i(V)-8 b(erbindungen)257
3582 y(gibt)42 b(Abbildung)g(4.2.)73 b(Einige)41 b(Ausnahmef\177)-49
b(alle,)44 b(w)m(elc)m(he)g(normalerw)m(eise)e(v)m(on)h(den)257
3703 y(Klassi\014zierungsresultaten)38 b(ausgenommen)g(w)m(erden)i(m)s
(\177)-51 b(ussen,)40 b(en)m(tstehen)h(in)c(Situ-)257
3823 y(ationen,)48 b(die)c(wir)g(Selbstv)m(erbindungen)i(nennen.)82
b(Wir)44 b(pr\177)-49 b(asen)m(tieren)46 b(die)e(Proto-)257
3944 y(t)m(yp)s(en)32 b(v)m(on)g(Hopfalgebren)e(in)g(solc)m(h)h(einer)f
(Situation)f(in)h(Absc)m(hnitt)i(4.6.)42 b(Die)30 b(neuen)257
4064 y(Hopfalgebren,)i(die)g(v)m(om)g(Diagramm)d FI(B)1803
4079 y FH(2)1875 4064 y FL(stammen)i(und)i(w)m(elc)m(he)h(wir)d
(mittels)g(eines)257 4184 y(Computeralgebraprogramms)f(b)s(erec)m
(hnen,)35 b(sind)e(in)e(Abbildung)h(4.4)g(angegeb)s(en.)404
4305 y(Eine)25 b(w)m(eitere)h(o\013ene)f(F)-8 b(rage)25
b(b)s(etri\013t)f(die)h(Kompatibilit\177)-49 b(at)20
b(der)26 b(Grupp)s(en)f(mit)e(den)257 4425 y(Dynkin-Diagrammen)37
b(in)i(einem)h(V)-8 b(erbindungs-Datum.)64 b(Ob)m(w)m(ohl)41
b(eine)f(generelle)257 4545 y(An)m(t)m(w)m(ort)28 b(au\031er)f(Reic)m
(h)m(w)m(eite)g(sc)m(hein)m(t,)i(k\177)-49 b(onnen)28
b(wir)e(mit)e(Theorem)j(5.1)f(eine)h(An)m(t)m(w)m(ort)257
4666 y(f)s(\177)-51 b(ur)30 b(die)g(Grupp)s(en)g(\()p
FF(Z)p FI(=)p FL(\()p FI(p)p FL(\)\))1291 4630 y FH(2)1358
4666 y FL(geb)s(en.)43 b(Wir)30 b(b)s(ew)m(eisen,)i(da\031,)f(bis)f
(auf)g(w)m(enige)h(Ausnah-)257 4786 y(men,)26 b(alle)c(Diagramme)f(mit)
h(maximal)f(vier)i(Ec)m(k)m(en)k(f)s(\177)-51 b(ur)22
b(die)i(Konstruktion)g(endlic)m(hdi-)257 4907 y(mensionaler)29
b(punktierter)j(Hopfalgebren)e(mit)f(diesen)i(Grupp)s(en)g(als)e
(Koradik)-5 b(al)28 b(b)s(e-)257 5027 y(n)m(utzt)34 b(w)m(erden)g
(k\177)-49 b(onnen.)404 5147 y(Das)28 b(letzte)g(gro\031e)g(Thema)g
(dieser)h(Arb)s(eit)e(ist)h(die)g(Un)m(tersuc)m(h)m(ung)j(der)e(Bezieh)
m(un-)257 5268 y(gen)e(zwisc)m(hen)h(den,)h(mittels)24
b(der)j(Liftingmetho)s(de)d(k)m(onstruierten,)29 b(neuen)f(Hopfalge-)
257 5388 y(bren.)42 b(Es)26 b(stellt)e(sic)m(h)i(heraus,)i(da\031)c(v)m
(ersc)m(hiedene)29 b(V)-8 b(erbindungsparameter)25 b(zu)h(quasi-)1828
5637 y(93)p eop
%%Page: 94 96
94 95 bop 257 573 a FL(isomorphen)34 b(Hopfalgebren)g(f)s(\177)-51
b(uhren,)34 b(Theorem)h(6.1.)48 b(Alle)33 b(Hopfalgebren,)h(die)g(mit-)
257 693 y(tels)e(der)f(Liftingmetho)s(de)e(n)m(ur)j(aus)g(Diagrammen)c
(des)33 b(T)m(yps)g FI(A)2694 708 y FG(n)2772 693 y FL(en)m(tstehen,)h
(zeigen)257 814 y(das)44 b(gleic)m(he)e(V)-8 b(erhalten,)45
b(Theorem)f(6.6.)74 b(Das)42 b(hei\031t,)j(da\031)e(alle)e(so)i(k)m
(onstruierten,)257 934 y(endlic)m(hdimensionalen)27 b(punktierten)j
(Hopfalgebren,)f(die)g(sic)m(h)g(n)m(ur)g(durc)m(h)i(die)d(W)-8
b(ahl)257 1054 y(ihrer)27 b(P)m(arameter)g(un)m(tersc)m(heiden,)k
(2-Kozyklus-Deformationen)24 b(v)m(oneinander)k(sind.)257
1175 y(Unser)d(Bew)m(eis)g(sollte)e(sic)m(h)h(einfac)m(h)g(auf)g
(Hopfalgebren,)h(die)f(mit)e(den)j(anderen)g(T)m(yp)s(en)257
1295 y(endlic)m(her)38 b(Dynkin-Diagramme)c(assoziiert)j(sind,)42
b(\177)-51 b(ub)s(ertragen)38 b(lassen,)i(sobald)d(alle)257
1416 y(P)m(arameter)e(f)s(\177)-51 b(ur)34 b(diese)h(Algebren)g
(explizit)e(b)s(estimm)m(t)g(w)m(orden)j(sind.)50 b(Dies)34
b(b)s(est\177)-49 b(arkt)257 1536 y(die)37 b(Ho\013n)m(ung,)i(da\031)f
(Masuok)-5 b(as)38 b(V)-8 b(erm)m(utung)37 b(in)g([Mas1)q(])g(trotz)g
(des)i(Gegen)m(b)s(eispiels)257 1656 y(in)32 b([EG])h(gerettet)g(w)m
(erden)h(k)-5 b(ann,)33 b(indem)f(man)f(sie)i(et)m(w)m(as)h(sp)s
(ezialisiert)c(\(Seite)j(60\).)1828 5637 y(94)p eop
%%Page: 95 97
95 96 bop 1660 593 a Fa(Leb)5 b(enslauf)640 1016 y Fn(P)l(ers\177)-67
b(onlic)l(he)46 b(Daten)737 1220 y FL(Daniel)31 b(Didt)737
1502 y(gebren)j(am)d(3.)i(August)g(1974)f(in)f(Rosto)m(w)i(am)f(Don)g
(\(Ru\031land\))737 1622 y(v)m(erheiratet;)i(4)e(Kinder)494
1946 y Fn(Sc)l(h)l(ulausbildung)737 2149 y FJ(1981-1989)87
b FL(Besuc)m(h)30 b(einer)d(P)m(olytec)m(hnisc)m(hen)j(Ob)s(ersc)m(h)m
(ule)f(in)e(Leipzig)952 2270 y(und)e(ab)f(1983)f(einer)h(Klasse)g(mit)f
(erw)m(eitertem)h(Russisc)m(h)m(un)m(terric)m(h)m(t)737
2432 y FJ(1989-1993)87 b FL(Nac)m(h)59 b(erfolgreic)m(her)f(Aufnahmepr)
s(\177)-51 b(ufung)58 b(Besuc)m(h)i(der)952 2552 y(Sp)s(ezialsc)m(h)m
(ule)47 b(math.-naturwiss.-tec)m(hn.)85 b(Ric)m(h)m(tung)46
b(\\Wilhelm)952 2672 y(Ost)m(w)m(ald")33 b(in)f(Leipzig)737
2834 y FJ(Juli)k(1993)87 b FL(Abitur)494 3038 y Fn(Ho)t(c)l(hsc)l(h)l
(ulausbildung)737 3241 y FJ(Okt.)38 b(1993)86 b FL(Beginn)29
b(eines)g(Dopp)s(elstudiums)f(\(Diplom-Mathematik)952
3362 y(und)33 b(Diplom-Ph)m(ysik\))d(an)j(der)g(Univ)m(ersit\177)-49
b(at)32 b(Leipzig)737 3523 y FJ(Sept.)38 b(1995)87 b
FL(V)-8 b(ordiplom)30 b(in)j(Mathematik)f(\(Neb)s(enfac)m(h)j
(Informatik\))952 3644 y(und)e(Ph)m(ysik)h(\(Neb)s(enfac)m(h)g
(Chemie\))737 3806 y FJ(F)-9 b(eb.)38 b(1996)g(-)g(Jan.)g(1999)86
b FL(Mitglied)111 b(der)h(Studienstiftung)h(des)952 3926
y(Deutsc)m(hen)34 b(V)-8 b(olk)m(es)737 4088 y FJ(Sept.)51
b(1996)37 b(-)h(Juli)e(1998)86 b FL(Leiter)23 b(eines)h(Mathematikzirk)
m(els)f(f)s(\177)-51 b(ur)22 b(die)952 4208 y(Klassen)30
b(11/12)e(der)i(\\Leipziger)f(Sc)m(h)s(\177)-51 b(ulergesellsc)m(haft)
29 b(f)s(\177)-51 b(ur)28 b(Mathe-)952 4329 y(matik")737
4491 y FJ(Jan.)39 b(1999)86 b FL(Absc)m(hluss)34 b(des)g(Ph)m
(ysikstudiums)g(mit)c(Diplom)737 4653 y FJ(M)o(\177)-55
b(arz)38 b(2000)87 b FL(Absc)m(hluss)34 b(des)f(Mathematikstudiums)f
(mit)f(Diplom)737 4814 y FJ(Jan.)39 b(2000)e(-)h(Dez.)f(2002)49
b FL(Promotion)44 b(am)g(Mathematisc)m(hen)i(Insti-)952
4935 y(tut)37 b(der)g(LMU)g(M)s(\177)-51 b(unc)m(hen)38
b(im)d(Graduiertenk)m(olleg)g(\\Mathematik)952 5055 y(im)c(Bereic)m(h)i
(ihrer)f(W)-8 b(ec)m(hselwirkung)34 b(mit)d(der)i(Ph)m(ysik")p
eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF