%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Endomorp.dvi %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -O 0cm,2cm Endomorp %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1994.03.09:1637 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 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put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform 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w(?)674 742 y Fp(!)r Fr(\()p Fp(I)t Fr(\))212 b(cohom)o(\()p Fp(!)1171 724 y Fh(0)1183 742 y Fp(;)8 b(!)r Fr(\))j Fn(\012)g Fp(!)1349 724 y Fh(0)1361 742 y Fr(\()p Fp(I)t Fr(\))p Fp(:)p 784 730 184 2 v 926 729 a Ff(-)0 843 y Fr(It)16 b(is)g(easy)g(to)h(v)o(erify)d(the)i(algebra)h(la)o(ws)f(and)h (the)f(additional)g(claims)f(of)h(the)g(theorem.)p 1926 816 24 2 v 1926 841 2 25 v 1948 841 V 1926 843 24 2 v 0 1003 a(PR)o(OPOSITION)f(2.14)25 b(\(the)16 b(pro)q(duct)h(in)f(the)g (algebra)h(cohom\))0 1063 y Fo(The)h(pr)n(o)n(duct)e(in)i Fr(cohom)o(\()p Fp(!)521 1045 y Fh(0)533 1063 y Fp(;)8 b(!)r Fr(\))17 b Fo(is)h(given)h(by)624 1173 y Fr(\()p 643 1133 113 2 v Fp(x)11 b Fn(\012)g Fp(\030)s Fr(\))g Fn(\001)g Fr(\()p 830 1139 V Fp(y)h Fn(\012)f Fp(\021)r Fr(\))j(=)p 1027 1133 286 2 v 14 w Fp(x)c Fn(\012)h Fp(y)i Fn(\012)e Fp(\030)i Fn(\012)e Fp(\021)r(:)0 1287 y Fr(Pro)q(of:)21 b(W)l(e)12 b(apply)h(the)g(diagram)f(de\014ning)h(the)f(m)o(ultiplic)o (ation)f(of)i(cohom)o(\()p Fp(!)1441 1269 y 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(assumptions)f(of)g(Pr)n(op)n(osition2.4)f(b)n(e)h(satis\014e)n(d)h (and)f(let)i(al)r(l)f(diagr)n(ams)e Fr(\()p Fn(D)q Fp(;)8 b(!)r Fr(\))p Fo(,)21 b Fr(\()p Fn(D)q Fp(;)8 b(!)1807 1844 y Fh(0)1819 1862 y Fr(\))p Fo(,)20 b(and)0 1922 y Fr(\()p Fn(D)q Fp(;)8 b(!)112 1904 y Fh(00)134 1922 y Fr(\))17 b Fo(b)n(e)h(monoidal.)23 b(Then)18 b(the)g(morphism)423 2032 y Fr(\001)c(:)f(cohom)o(\()p Fp(!)694 2012 y Fh(00)716 2032 y Fp(;)8 b(!)r Fr(\))14 b Fn(\000)-9 b(!)14 b Fr(cohom)o(\()p Fp(!)1086 2012 y Fh(0)1098 2032 y Fp(;)8 b(!)r Fr(\))j Fn(\012)f Fr(cohom\()p Fp(!)1421 2012 y Fh(00)1442 2032 y Fp(;)e(!)1496 2012 y Fh(0)1508 2032 y Fr(\))0 2142 y Fo(in)18 b(2.4)f(is)g(an)h(algebr)n(a)g(homomorphism.)p 1926 2116 V 1926 2140 2 25 v 1948 2140 V 1926 2142 24 2 v 0 2356 a Fr(COR)o(OLLAR)l(Y)e(2.16)25 b(\(co)q(end\()p Fp(!)r Fr(\))17 b(is)f(a)g(bialgebra\))0 2416 y Fo(L)n(et)25 b Fr(\()p Fn(D)q Fp(;)8 b(!)r Fr(\))26 b Fo(and)g Fr(\()p Fn(D)q Fp(;)8 b(!)464 2398 y Fh(0)476 2416 y Fr(\))26 b 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b(:)g(:)g(:)f(;)h(')322 2622 y Fm(m)372 2615 y Fr(are)16 b(morphisms.)100 2713 y(If)h(there)h(are)h(additional)f(comm)o(utativi)o(t)o(y)d(relations)j Fp(r)1145 2720 y Fi(1)1165 2713 y Fp(;)8 b(:)g(:)g(:)f(;)h(r)1296 2720 y Fm(k)1336 2713 y Fr(for)18 b(the)g(morphisms)f(expressed)0 2773 y(b)o(y)c(the)h Fp(')179 2780 y Fm(i)193 2773 y Fr(,)g(they)f(can)h(b)q(e)g(added)g(to)h(the)e(de\014ning)h(congruence) g(relations)f(to)i(de\014ne)e(the)h(free)f(monoidal)0 2834 y(category)j Fn(C)s Fr([)p Fp(X)279 2841 y Fi(1)299 2834 y Fp(;)8 b(:)g(:)g(:)g(;)g(X)449 2841 y Fm(n)473 2834 y Fr(;)g Fp(')527 2841 y Fi(1)546 2834 y Fp(;)g(:)g(:)g(:)f(;)h(') 687 2841 y Fm(m)721 2834 y Fr(;)g Fp(r)765 2841 y Fi(1)784 2834 y Fp(;)g(:)g(:)g(:)g(;)g(r)916 2841 y Fm(k)937 2834 y Fr(])15 b(b)o(y)0 2932 y(\()p Fp(R)56 2939 y Fi(4)76 2932 y Fr(\))24 b Fp(r)141 2939 y Fi(1)161 2932 y Fp(;)8 b(:)g(:)g(:)f(;)h(r)292 2939 y Fm(k)330 2932 y Fr(are)16 b(in)g(the)g(congruence)g(relation.)p eop %%Page: 22 22 22 21 bop 1901 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Fp(B)s Fr(\)\()p Fp(f)16 b Fn(\012)10 b Fp(f)5 b Fr(\)\()p Fp(R)p Fr(\))15 b Fn(\022)f Fp(C)g Fn(\012)d Fp(S)19 b Fr(that)e(is)f(if)466 1641 y Fp(R)891 b(C)14 b Fn(\012)d Fp(S)p 518 1627 862 2 v 1338 1626 a Ff(-)934 1607 y Fp(f)p 484 1835 2 176 v 485 1835 a Ff(?)p 1459 1835 V 933 w(?)418 1885 y Fp(A)f Fn(\012)h Fp(A)251 b(C)14 b Fn(\012)d Fp(C)k Fn(\012)c Fp(B)i Fn(\012)e Fp(B)p 566 1871 223 2 v 747 1870 a Ff(-)618 1850 y Fp(f)16 b Fn(\012)11 b Fp(f)1340 1885 y(C)k Fn(\012)10 b Fp(B)k Fn(\012)d Fp(B)p 1156 1871 170 2 v 1284 1870 a Ff(-)1177 1856 y Fp(m)g Fn(\012)f Fr(1)0 1950 y(comm)o(utes.)24 b(The)18 b(set)h(of)f(all)g(quadratic)g(homomorphisms)e(from)h(\()p Fp(A;)8 b(R)p Fr(\))18 b(to)g Fp(C)e Fn(\012)c Fr(\()p Fp(B)s(;)c(S)s Fr(\))18 b(is)g(denoted)0 2011 y(b)o(y)c Fp(K)t Fr(-Alg)202 1989 y Fm(q)221 2011 y Fr(\(\()p Fp(A;)8 b(R)p Fr(\))p Fp(;)g(C)j Fn(\012)d Fr(\()p Fp(B)s(;)g(S)s Fr(\)\).)19 b(Then)c(one)g(pro)o(v)o(es)f(as)h(in)f(Theorem)g(3.4)g (that)i Fp(K)t Fr(-Alg)1671 1989 y 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y Fp(\016)848 2648 y(')699 2568 y Ff(P)741 2582 y(P)782 2596 y(P)824 2610 y(P)865 2624 y(P)907 2637 y(P)948 2651 y(P)990 2665 y(P)1031 2679 y(P)1073 2693 y(P)1114 2707 y(P)1156 2720 y(P)1184 2730 y(P)-42 b(q)1212 2776 y Fp(C)15 b Fn(\012)10 b Fr(\()p Fp(B)s(;)e(S)s Fr(\))p 1327 2730 2 176 v 1328 2730 a Ff(?)1355 2654 y Fr(~)-32 b Fp(')11 b Fn(\012)g Fr(1)0 2845 y Fo(c)n(ommutes.)p 1926 2818 24 2 v 1926 2843 2 25 v 1948 2843 V 1926 2845 24 2 v eop %%Page: 27 27 27 26 bop 1901 -9 a Fr(27)0 136 y Fl(3.4)70 b(Complete)20 b(quadratic)j(quan)n(tum)g(spaces)0 228 y Fr(The)17 b(most)e(in)o (teresting)h(diagram)g(for)g(constructing)h(como)q(dule)e(algebras)j(o) o(v)o(er)d(bialgebras)i(is)f(de\014ned)0 289 y(o)o(v)o(er)e(the)h(free) f(monoidal)g(category)h Fn(D)g Fr(=)f Fn(C)s Fr([)p Fp(X)q(;)8 b(\032)p Fr(])15 b(with)g Fp(\032)f Fr(:)f Fp(X)g Fn(\012)8 b Fp(X)18 b Fn(\000)-9 b(!)14 b Fp(X)f Fn(\012)8 b Fp(X)t Fr(.)21 b(If)14 b Fp(!)i Fr(:)e Fn(D)h(\000)-8 b(!)13 b(A)i Fr(is)g(a)0 349 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b(Certainly)21 b(all)g(v)o(ector)g(spaces)h Fp(V)804 740 y Fh(\012)p Fm(n)876 758 y Fr(are)g Fp(B)1000 765 y Fi(\()p Fm(V)r(;f)t Fi(\))1082 758 y Fr(-como)q(dules)g(and)g Fp(f)27 b Fr(is)22 b(a)g(como)q(dule)e(homo-)0 818 y(morphism.)43 b(Th)o(us)24 b(Im)o(\()p Fp(f)5 b Fr(\))28 b Fn(\022)f Fp(V)h Fn(\012)16 b Fp(V)35 b Fr(is)24 b(a)h(sub)q(como)q(dule.)45 b(But)24 b(then)g(one)g(c)o(hec)o(ks)f(easily)h(that)0 878 y Fp(A)37 886 y Fi(\()p Fm(V)r(;f)t Fi(\))133 878 y Fr(:=)13 b Fp(T)7 b Fr(\()p Fp(V)k Fr(\))p Fp(=)p Fr(\(Im)n(\()p Fp(f)5 b Fr(\)\))17 b(is)f(a)g(como)q(dule)g(as)g(w)o(ell)f(and)i(that) g(it)e(is)h(in)g(fact)h(a)f(como)q(dule)f(algebra)i(o)o(v)o(er)0 938 y Fp(B)37 946 y Fi(\()p Fm(V)r(;f)t Fi(\))119 938 y Fr(.)p 1926 912 24 2 v 1926 937 2 25 v 1948 937 V 1926 939 24 2 v 100 1098 a(No)o(w)i(the)g(algebra)g Fp(A)511 1106 y Fi(\()p Fm(V)r(;f)t Fi(\))613 1098 y Fr(can)g(b)q(ecome)f(v)o (ery)g(small,)g(in)h(fact)g(degenerate,)g(namely)e(if)i Fp(f)24 b Fr(:)19 b Fp(V)24 b Fn(\012)0 1158 y Fp(V)h Fn(\000)-8 b(!)13 b Fp(V)22 b Fn(\012)10 b Fp(V)27 b Fr(is)15 b(bijectiv)o(e.)k(Then)d Fp(A)d Fr(=)h Fp(K)g Fn(\010)c Fp(V)27 b Fr(where)16 b(the)g(m)o(ultipli)o(cation)d(on)j Fp(V)28 b Fr(is)15 b(the)h(zero)f(map.)0 1218 y(This)g(happ)q(ens)h(in) f("most")g(cases,)g(since)f(for)i Fp(f)k Fr(to)15 b(b)q(e)h(bijectiv)o (e)c(it)j(su\016ces)f(that)i(det)o(\()p Fp(f)5 b Fr(\))15 b Fn(6)p Fr(=)e(0.)22 b(But)14 b(the)0 1279 y(next)i(prop)q(osition)h (sho)o(ws)g(that)g(ev)o(en)e(in)h(the)g(degenerate)g(case)g(w)o(e)g (still)f(ha)o(v)o(e)g(ro)q(om)h(to)h(mo)o(v)o(e.)0 1393 y(PR)o(OPOSITION)e(3.10)25 b(\(c)o(hange)16 b(of)h(the)f Fp(R)p Fr(-matrix\))0 1453 y Fo(L)n(et)h Fr(\()p Fp(V)s(;)8 b(f)19 b Fr(:)13 b Fp(V)22 b Fn(\012)10 b Fp(V)25 b Fn(\000)-8 b(!)13 b Fp(V)22 b Fn(\012)10 b Fp(V)h Fr(\))17 b Fo(with)h Fp(V)28 b Fo(\014nite)19 b(dimensional)f(b)n(e)f(given.)24 b(Then)18 b(for)e(every)i Fp(\025)c Fn(2)g Fp(K)21 b Fo(we)0 1513 y(have)d Fp(B)149 1525 y Fi(\()p Fm(V)r(;f)t Fh(\000)p Fm(\025)p Fh(\001)p Fe(id)n Fi(\))338 1513 y Fr(=)c Fp(B)427 1521 y Fi(\()p Fm(V)r(;f)t Fi(\))509 1513 y Fo(.)0 1627 y Fr(Pro)q(of:)22 b(By)15 b(Theorem)g(3.3)h(w)o(e)g (kno)o(w)g(that)g Fp(B)853 1635 y Fi(\()p Fm(V)r(;f)t Fi(\))949 1627 y Fr(=)e Fp(T)7 b Fr(\()p Fp(V)21 b Fn(\012)10 b Fp(V)1194 1609 y Fh(\003)1213 1627 y Fr(\))p Fp(=I)t Fr(.)21 b(W)l(e)16 b(lo)q(ok)g(at)g(the)g(relations.)21 b(The)0 1688 y(ideal)16 b Fp(I)j Fr(is)d(generated)g(b)o(y)g(elemen)o (ts)e(of)i(the)g(form)582 1798 y Fp(x)11 b Fn(\012)g Fp(y)h Fn(\012)f Fp(f)786 1777 y Fh(\003)806 1798 y Fr(\()p Fp(\030)j Fn(\012)d Fp(\021)r Fr(\))f Fn(\000)h Fp(f)5 b Fr(\()p Fp(x)11 b Fn(\012)g Fp(y)r Fr(\))g Fn(\012)g Fp(\030)i Fn(\012)e Fp(\021)0 1908 y Fr(with)16 b Fp(x;)8 b(y)r(;)g Fn(2)13 b Fp(V)s(;)8 b(\030)r(;)g(\021)16 b Fn(2)e Fp(V)480 1889 y Fh(\003)499 1908 y Fr(.)22 b(But)16 b(then)g(the)g(same)f(ideal)g(is)h(also)h(generated)f(b)o(y)g(elemen)o (ts)e(of)i(the)g(form)361 2018 y Fp(x)11 b Fn(\012)g Fp(y)i Fn(\012)d Fr(\()p Fp(f)584 1997 y Fh(\003)616 2018 y Fn(\000)g Fp(\025)i Fn(\001)f Fr(id)770 1996 y Fh(\003)790 2018 y Fr(\)\()p Fp(\030)j Fn(\012)d Fp(\021)r Fr(\))g Fn(\000)f Fr(\()p Fp(f)17 b Fn(\000)11 b Fp(\025)g Fn(\001)g Fr(id\)\()p Fp(x)f Fn(\012)h Fp(y)r Fr(\))g Fn(\012)g Fp(\030)i Fn(\012)e Fp(\021)r(;)0 2128 y Fr(since)k(the)h Fp(\025)p Fr(-terms)g(simply)e(cancel.)20 b(Th)o(us)d Fp(B)866 2139 y Fi(\()p Fm(V)r(;f)t Fh(\000)p Fm(\025)p Fh(\001)p Fe(id)n Fi(\))1055 2128 y Fr(=)c Fp(B)1143 2135 y Fi(\()p Fm(V)r(;f)t Fi(\))1226 2128 y Fr(.)p 1926 2101 V 1926 2126 2 25 v 1948 2126 V 1926 2128 24 2 v 0 2287 a(COR)o(OLLAR)l(Y)j(3.11)25 b(\(the)16 b(sp)q(ectrum)f(of)h (quan)o(tum)f(spaces)i(for)f(an)h Fp(R)p Fr(-matrix\))0 2348 y Fo(L)n(et)f Fr(\()p Fp(V)s(;)8 b(f)20 b Fr(:)13 b Fp(V)21 b Fn(\012)9 b Fp(V)25 b Fn(\000)-9 b(!)14 b Fp(V)20 b Fn(\012)10 b Fp(V)h Fr(\))16 b Fo(with)i Fp(V)28 b Fo(\014nite)18 b(dimensional)f(b)n(e)g(given.)24 b(Then)17 b(for)f(every)h Fp(\025)e Fn(2)f Fp(K)20 b Fo(the)0 2408 y(algebr)n(a)g Fp(A)204 2419 y Fi(\()p Fm(V)r(;f)t Fh(\000)p Fm(\025)p Fh(\001)p Fe(id)n Fi(\))398 2408 y Fo(is)f(a)g Fp(B)533 2415 y Fi(\()p Fm(V)r(;f)t Fi(\))616 2408 y Fo(-c)n(omo)n(dule)h(algebr)n(a.)28 b(It)20 b(is)f(non-de)n(gener)n (ate)j(if)d(and)h(only)f(of)h Fp(\025)f Fo(is)h(an)0 2468 y(eigenvalue)g(of)e Fp(f)5 b Fo(.)0 2582 y Fr(Pro)q(of:)22 b(Instead)16 b(of)h(c)o(hanging)f Fp(f)22 b Fr(in)15 b(the)h(de\014nition)g(of)g Fp(B)1079 2590 y Fi(\()p Fm(V)r(;f)t Fi(\))1178 2582 y Fr(b)o(y)f(a)i(m)o(ultiple)c(of)j(the)g (iden)o(tit)o(y)l(,)d(w)o(e)j(can)0 2642 y(as)h(w)o(ell)e(c)o(hange)h (it)g(in)g(the)g(de\014nition)f(of)i Fp(A)819 2650 y Fi(\()p Fm(V)r(;f)t Fi(\))901 2642 y Fr(.)100 2702 y(So)e(for)h Fp(B)277 2710 y Fi(\()p Fm(V)r(;f)t Fi(\))374 2702 y Fr(w)o(e)f(ha)o(v)o(e)f(obtained)i(a)f(one-parameter)g(family)e Fp(A)1302 2714 y Fi(\()p Fm(V)r(;f)t Fh(\000)p Fm(\025)p Fh(\001)p Fe(id)m Fi(\))1491 2702 y Fr(of)j(como)q(dule)e(algebras.)0 2763 y(Since)h Fp(V)27 b Fr(is)15 b(\014nite)g(dimensional)f(and)i Fp(f)21 b Fr(has)16 b(only)g(\014nitely)e(man)o(y)g(eigen)o(v)m(alues,) g(all)h(but)h(\014nitely)e(man)o(y)0 2823 y(of)j(these)e(como)q(dule)h (algebras)h(are)f(degenerate)g(b)o(y)f(Lemma)g(3.9.)p 1926 2796 V 1926 2821 2 25 v 1948 2821 V 1926 2823 24 2 v eop %%Page: 28 28 28 27 bop 1901 -9 a Fr(28)0 136 y(EXAMPLE)16 b(3.12)25 b(\(t)o(w)o(o)16 b(parameter)f(quan)o(tum)g(matrices\))0 196 y(Let)i(us)g(tak)o(e)g Fp(V)26 b Fr(=)15 b Fp(K)t(x)d Fn(\010)f Fp(K)t(y)18 b Fr(t)o(w)o(o)f(dimensional)f(and)h Fp(f)k Fr(:)14 b Fp(V)23 b Fn(\012)11 b Fp(V)27 b Fn(\000)-9 b(!)15 b Fp(V)23 b Fn(\012)11 b Fp(V)28 b Fr(giv)o(en)17 b(b)o(y)f(the)h(matrix)0 256 y(\(with)f Fp(q)r(;)8 b(p)14 b Fn(6)p Fr(=)f(0\))619 406 y Fp(R)i Fr(=)722 283 y Fc(0)722 356 y(B)722 381 y(B)722 406 y(B)722 432 y(@)779 315 y Fr(1)66 b(0)167 b(0)143 b(0)779 375 y(0)66 b(0)144 b Fp(q)1061 357 y Fh(\000)p Fi(1)1227 375 y Fr(0)779 436 y(0)42 b Fp(p)869 417 y Fh(\000)p Fi(1)958 436 y Fr(1)12 b Fn(\000)f Fp(q)1068 417 y Fh(\000)p Fi(1)1114 436 y Fp(p)1138 417 y Fh(\000)p Fi(1)1227 436 y Fr(0)779 496 y(0)66 b(0)167 b(0)143 b(1)1273 283 y Fc(1)1273 356 y(C)1273 381 y(C)1273 406 y(C)1273 432 y(A)1317 406 y Fp(:)0 580 y Fr(The)12 b(bialgebra)h(generated)f(b)o(y)g(this)h(matrix)d(is)i (generated)h(b)o(y)f(the)g(elemen)o(ts)e Fp(a)j Fr(=)h Fp(x)s Fn(\012)s Fp(\030)r(;)21 b(b)13 b Fr(=)h Fp(x)s Fn(\012)s Fp(\021)r(;)20 b(c)14 b Fr(=)0 640 y Fp(y)f Fn(\012)d Fp(\030)r(;)25 b(d)14 b Fr(=)g Fp(y)f Fn(\012)e Fp(\021)r Fr(,)32 b(where)15 b Fp(\030)r(;)25 b(\021)18 b Fr(is)e(the)g(dual)h(basis)g(to)f Fp(x;)24 b(y)r Fr(.)d(The)16 b(relations)g(are)0 750 y Fp(ac)d Fr(=)h Fp(q)136 729 y Fh(\000)p Fi(1)183 750 y Fp(ca;)k(bd)c Fr(=)g Fp(q)398 729 y Fh(\000)p Fi(1)445 750 y Fp(db;)19 b(ad)q Fn(\000)q Fp(q)640 729 y Fh(\000)p Fi(1)686 750 y Fp(cb)13 b Fr(=)h Fp(da)q Fn(\000)q Fp(q)r(bc;)k(ab)13 b Fr(=)h Fp(p)1119 729 y Fh(\000)p Fi(1)1166 750 y Fp(ba;)19 b(cd)14 b Fr(=)g Fp(p)1382 729 y Fh(\000)p Fi(1)1429 750 y Fp(dc;)20 b(ad)q Fn(\000)q Fp(p)1625 729 y Fh(\000)p Fi(1)1672 750 y Fp(bc)13 b Fr(=)h Fp(da)q Fn(\000)q Fp(pcb:)0 860 y Fr(F)l(rom)d(this)h(follo)o (ws)g Fp(q)r(bc)h Fr(=)g Fp(pcb)p Fr(.)20 b(This)13 b(is)e(the)h(t)o(w) o(o)g(parameter)f(v)o(ersion)g(of)h(a)h(quan)o(tum)e(matrix)f (bialgebra)0 920 y(constructed)17 b(in)g([6])f(Chap.)25 b(4,)17 b(4.10.)25 b(The)17 b(matrix)e Fp(R)j Fr(has)g(t)o(w)o(o)f (eigen)o(v)m(alues)f Fp(\025)1499 927 y Fi(1)1535 920 y Fr(=)f(1)i(\(of)h(m)o(ultipli)o(cit)n(y)0 980 y(three\))e(and)g Fp(\025)266 987 y Fi(2)300 980 y Fr(=)e Fn(\000)p Fp(q)415 962 y Fh(\000)p Fi(1)462 980 y Fp(p)486 962 y Fh(\000)p Fi(1)549 980 y Fr(\(of)j(m)o(ultiplic)o(it)n(y)c(one\))k(whic)o(h)e (lead)h(to)h(algebras)684 1090 y Fp(A)721 1097 y Fi(1)755 1090 y Fr(=)c Fp(K)t Fn(h)p Fp(x;)8 b(y)r Fn(i)p Fp(=)p Fr(\()p Fp(xy)13 b Fn(\000)e Fp(q)1147 1070 y Fh(\000)p Fi(1)1193 1090 y Fp(y)r(x)p Fr(\))0 1200 y(and)617 1260 y Fp(A)654 1267 y Fi(2)687 1260 y Fr(=)i Fp(K)t Fn(h)p Fp(x;)8 b(y)r Fn(i)p Fp(=)p Fr(\()p Fp(x)968 1240 y Fi(2)988 1260 y Fp(;)24 b(y)1052 1240 y Fi(2)1072 1260 y Fp(;)g(xy)12 b Fr(+)f Fp(py)r(x)p Fr(\))p Fp(:)0 1347 y Fr(These)j(are)h(the)f(quan) o(tum)f(plane)h(with)g(parameter)f Fp(q)j Fr(and)f(the)f(dual)g(quan)o (tum)f(plane)h(with)g(parameter)0 1408 y Fp(p)p Fr(.)0 1522 y(EXAMPLE)i(3.13)25 b(\(t)o(w)o(o)16 b(further)g(quan)o(tum)f(2)d Fn(\002)e Fr(2-matrices\))0 1582 y(In)16 b(w)o(e)g(replace)f(the)h (generating)h(matrix)d(in)i(previous)g(example)e(b)o(y)659 1778 y Fp(R)h Fr(=)762 1656 y Fc(0)762 1729 y(B)762 1754 y(B)762 1778 y(B)762 1805 y(@)820 1688 y Fr(1)61 b(0)127 b(0)107 b(0)820 1748 y(0)61 b(0)104 b Fp(q)1057 1730 y Fh(\000)p Fi(1)1187 1748 y Fr(0)820 1808 y(0)61 b(1)g(1)12 b Fn(\000)e Fp(q)1099 1790 y Fh(\000)p Fi(1)1187 1808 y Fr(0)820 1868 y(0)41 b Fn(\000)p Fr(1)85 b Fp(q)1057 1850 y Fh(\000)p Fi(1)1187 1868 y Fr(1)1233 1656 y Fc(1)1233 1729 y(C)1233 1754 y(C)1233 1778 y(C)1233 1805 y(A)1277 1778 y Fp(;)0 1975 y Fr(then)16 b(the)g(relations)g(for)h(the)f (corresp)q(onding)h(bialgebra)f(are)h(giv)o(en)e(b)o(y)385 2077 y Fp(ac)e Fr(=)h Fp(q)521 2059 y Fh(\000)p Fi(1)568 2077 y Fp(ca;)23 b(bd)14 b Fr(=)g Fp(q)788 2059 y Fh(\000)p Fi(1)835 2077 y Fp(db;)24 b(ad)11 b Fn(\000)g Fp(q)1055 2059 y Fh(\000)p Fi(1)1101 2077 y Fp(cb)j Fr(=)g Fp(da)d Fn(\000)f Fp(q)r(bc;)124 2149 y(ba)j Fr(=)h Fp(ab)d Fr(+)g Fp(b)364 2131 y Fi(2)383 2149 y Fp(;)24 b(cb)11 b Fr(+)g Fp(cd)g Fr(+)g Fp(d)654 2131 y Fi(2)688 2149 y Fr(=)j Fp(da)d Fn(\000)g Fp(db)g Fr(+)g Fp(dc;)24 b(ad)11 b Fn(\000)g Fp(b)p Fr(\()p Fp(c)g Fn(\000)g Fp(d)p Fr(\))j(=)g Fp(da)d Fn(\000)g Fr(\()p Fp(c)g Fr(+)g Fp(d)p Fr(\))p Fp(b:)0 2251 y Fr(The)16 b(bialgebra)h(coacts)f(on)h(the)f(algebras)684 2361 y Fp(A)721 2368 y Fi(1)755 2361 y Fr(=)d Fp(K)t Fn(h)p Fp(x;)8 b(y)r Fn(i)p Fp(=)p Fr(\()p Fp(xy)13 b Fn(\000)e Fp(q)1147 2340 y Fh(\000)p Fi(1)1193 2361 y Fp(y)r(x)p Fr(\))0 2471 y(and)572 2531 y Fp(A)609 2538 y Fi(2)642 2531 y Fr(=)j Fp(K)t Fn(h)p Fp(x;)8 b(y)r Fn(i)p Fp(=)p Fr(\()p Fp(x)924 2511 y Fi(2)943 2531 y Fp(;)25 b(xy)12 b Fr(+)f Fp(y)r(x;)23 b(xy)13 b Fn(\000)d Fp(y)1326 2511 y Fi(2)1346 2531 y Fr(\))p Fp(:)0 2618 y Fr(Finally)15 b(if)h(w)o(e)f(tak)o(e)706 2765 y Fp(R)f Fr(=)809 2642 y Fc(0)809 2715 y(B)809 2740 y(B)809 2765 y(B)809 2792 y(@)866 2674 y Fr(1)61 b(0)g(0)h(0)866 2735 y(0)f(0)g(1)42 b Fn(\000)p Fr(1)866 2795 y(0)61 b(1)g(0)h(1)866 2855 y(0)42 b Fn(\000)p Fr(1)f(1)62 b(0)1186 2642 y Fc(1)1186 2715 y(C)1186 2740 y(C)1186 2765 y(C)1186 2792 y(A)1231 2765 y Fp(;)p eop %%Page: 29 29 29 28 bop 1901 -9 a Fr(29)0 136 y(then)16 b(the)g(bialgebra)g (satis\014es)h(the)f(relations)124 229 y Fp(ca)e Fr(=)f Fp(ac)e Fr(+)g Fp(c)364 211 y Fi(2)384 229 y Fp(;)24 b(bc)11 b Fr(+)g Fp(bd)g Fr(+)g Fp(d)655 211 y Fi(2)689 229 y Fr(=)j Fp(da)d Fr(+)g Fp(db)g Fn(\000)g Fp(dc;)24 b(ad)11 b Fn(\000)g Fp(c)p Fr(\()p Fp(b)g Fn(\000)f Fp(d)p Fr(\))15 b(=)e Fp(da)e 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Fp(eg)k Fr(=)f Fp(hd)d Fn(\000)g Fp(g)r(e;)259 2935 y(g)16 b Fr(=)d Fp(cd)f Fn(\000)f Fp(af)18 b Fr(=)c Fp(dc)d Fn(\000)g Fp(f)5 b(a;)26 b(h)13 b Fr(=)h Fp(bf)i Fn(\000)11 b Fp(ce)j Fr(=)f Fp(f)5 b(b)12 b Fn(\000)e Fp(ec;)25 b(i)14 b Fr(=)g Fp(ae)c Fn(\000)h Fp(db)j Fr(=)f Fp(ea)e Fn(\000)f Fp(bd:)p eop %%Page: 31 31 31 30 bop 1901 -9 a Fr(31)0 144 y Fq(4)83 b(Automorphisms)26 b(and)h(Hopf)h(Algebras)0 254 y Fr(In)13 b(this)g(last)h(short)g (section)f(w)o(e)g(w)o(an)o(t)h(to)f(extend)g(our)h(tec)o(hniques)e(of) i(using)f(diagrams)h(for)f(determining)0 314 y(bialgebras)k(to)f(Hopf)g (algebras.)100 374 y(Hopf)j(algebras)i(arise)e(as)i(function)e(rings)i (of)f(a\016ne)f(algebraic)h(groups)h(whic)o(h)e(act)h(as)g(group)h(of)0 434 y(automorphisms)11 b(on)h(algebraic)g(v)m(arieties.)19 b(The)12 b(problem)e(in)i(non-comm)o(utativ)o(e)d(geometry)h(is)i(that) g(the)0 495 y(de\014nition)18 b(of)g(an)g(automorphism)f(group)i(is)f (not)g(that)h(clear.)25 b(So)19 b(one)f(de\014nes)g(the)g 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Fr(is)d(the)g(co)q(endomorphism)e (bialgebra)j(of)f(the)g(diagram)g(of)g(all)g(\014nite-dimen-)0 2884 y(sional)i Fp(B)s Fr(-como)q(dules)e(b)o(y)h([10].)24 b(So)18 b(the)f(construction)g(with)h(diagrams)f(giv)o(en)f(ab)q(o)o(v) o(e)h(giv)o(es)g(the)g(Hopf)0 2945 y(en)o(v)o(elop)q(e)e(of)h(an)h (arbitrary)f(bialgebra.)p eop %%Page: 34 34 34 33 bop 1901 -9 a Fr(34)100 136 y(If)16 b(a)h(\014nite-dimensional)e (non-comm)o(utativ)o(e)e(algebra)18 b Fp(A)e Fr(is)g(giv)o(en)g(then)h (the)f(rigidization)g(of)h(the)0 196 y(\014nite)f(monoidal)g(diagram)g Fp(!)557 203 y Fm(A)600 196 y Fr(:)f Fn(D)h(\000)-9 b(!)15 b(A)h Fr(with)h Fn(D)f Fr(:=)e Fn(C)s Fr([)p Fp(X)t Fr(;)8 b Fp(m;)g(u)p Fr(])16 b(generated)g(b)o(y)g Fp(A)h Fr(as)g(in)f(3.2)h (has)h(a)0 256 y(co)q(endomorphism)d(bialgebra)h Fp(H)21 b Fr(whic)o(h)16 b(is)g(the)h(Hopf)f(en)o(v)o(elop)q(e)f(of)i(the)f(co) q(endomorphism)f(bialgebra)0 317 y(of)i Fp(A)p Fr(.)j(A)c(similar)e (remark)h(holds)i(for)f(quadratic)g(algebras.)100 377 y(In)h(the)g(category)h(of)g(quan)o(tum)e(spaces)i(the)f(endomorphism)e (quan)o(tum)i(space)g Fn(E)23 b Fr(of)17 b(a)h(quan)o(tum)0 437 y(space)d Fn(X)22 b Fr(can)15 b(b)q(e)h(restricted)e(to)h(the)g ("automorphism")f(quan)o(tum)g(space)h Fn(A)g Fr(b)o(y)g(using)g(the)g (homomor-)0 497 y(phism)j(from)g(the)h(represen)o(ting)f(bialgebra)h (of)h Fn(E)k Fr(to)19 b(its)g(Hopf)g(en)o(v)o(elop)q(e.)28 b(Th)o(us)20 b Fn(A)f Fr(acts)g(also)h(on)g Fn(X)7 b Fr(.)0 557 y(The)20 b(existence)e(and)j(the)f(construction)g(in)f(the)h (relev)m(an)o(t)f(cases)i(can)f(b)q(e)g(obtained)g(from)f(asso)q (ciated)0 618 y(rigid)d(monoidal)f(diagrams)h(as)h(in)f(the)g(ab)q(o)o (v)o(e)g(theorem.)0 784 y Fq(References)24 893 y Fr([1])24 b(Miriam)e(Cohen,)j(Sara)f(W)l(estreic)o(h:)34 b Fg(F)l(rom)23 b(Sup)q(ersymmetry)d(to)k(Quan)o(tum)e(Comm)o(utativit)o(y)-5 b(.)100 954 y Fr(Preprin)o(t)16 b(1991.)24 1055 y([2])24 b(P)l(.)e(Deligne,)f(J.)g(S.)h(Milne:)30 b Fg(T)l(annakian)23 b(Categories.)e(Ho)q(dge)h(Cycles,)g(Motiv)o(es)e(and)j(Shim)o(ura)100 1116 y(V)l(arieties.)15 b Fr(Springer)h(LN)g(Math)g(900,)h(1982,)h (101-228.)24 1217 y([3])24 b(Saunders)18 b(Mac)f(Lane:)24 b Fg(Categories)18 b(for)f(the)g(W)l(orking)h(Mathematician.)d Fr(Springer-V)l(erlag)i(New)100 1277 y(Y)l(ork,)f(Heidelb)q(erg,)e (Berlin,)g(1971.)24 1379 y([4])24 b(Shahn)35 b(Ma)s(jid:)56 b Fg(Rank)34 b(of)h(Quan)o(tum)e(Groups)i(and)f(Braided)g(Groups)h(in)e (Dual)i(F)l(orm.)100 1439 y Fr(D)o(AMTP/90-44,)18 b(Euler)e(Inst.)f (Programme)g(on)i(Quan)o(tum)e(Groups,)i(Leningrad,)f(1990.)24 1541 y([5])24 b(Y)l(uri)c(I.)g(Manin:)31 b Fg(Quan)o(tum)20 b(Groups)i(and)f(Non-Comm)o(utativ)o(e)d(Geometry)l(.)h Fr(Les)i(publications)100 1601 y(CRM,)16 b(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(Mon)o(tr)o(\023)-23 b(eal,)15 b(1988.)24 1703 y([6])24 b(Y)l(uri)j(I.)f(Manin:)43 b Fg(T)l(opics)27 b(in)g(Noncomm)o(utativ)n(e)d(Geometry)p Fr(.)h(Princeton)i(Univ)o (ersit)o(y)d(Press,)100 1763 y(Princeton,)16 b(N.)f(J.)h(,)g(1991.)24 1865 y([7])24 b(Bo)q(do)17 b(P)o(areigis:)k Fg(Categories)c(and)g(F)l (unctors.)f Fr(Academic)d(Press)k(1970.)24 1967 y([8])24 b(Bo)q(do)19 b(P)o(areigis:)25 b Fg(Non-additiv)o(e)17 b(Ring)h(and)g(Mo)q(dule)g(Theory)g(I.)g(General)f(Theory)i(of)f (Monoids.)100 2027 y Fr(Publ.)e(Math.)g(\(Debrecen\))f(24,)i(1977,)g (189-204.)24 2128 y([9])24 b(Bo)q(do)32 b(P)o(areigis:)51 b Fg(Non-additiv)o(e)30 b(Ring)h(and)h(Mo)q(dule)f(Theory)h(I)q(I)q(I.) e(Morita)h(equiv)m(alences.)100 2189 y Fr(Publ.)16 b(Math.)g (\(Debrecen\))f(25,)i(1978,)g(177-186.)0 2290 y([10])24 b(Bo)q(do)d(P)o(areigis:)28 b Fg(A)19 b(Non-Comm)o(utativ)o(e)d(Non-Co) q(comm)o(utativ)o(e)i(Hopf)h(Algebra)h(in)f("Nature".)100 2351 y Fr(J.)d(Alg.)f(70,)i(1981,)g(356-374.)0 2452 y([11])24 b(Bo)q(do)14 b(P)o(areigis:)20 b Fg(F)l(our)13 b(Lectures)g(on)h(Hopf)f (Algebras)p Fr(.)g(Cen)o(tre)g(de)g(Recerca)f(Matem\022)-24 b(atica)12 b(Institut)100 2512 y(d'Estudis)17 b(Catalan,)f(No)h(6,)f (Octubre)f(1984,)j(47)f(pp.)0 2614 y([12])24 b(P)o(eter)13 b(Sc)o(hauen)o(burg:)20 b Fg(T)l(annak)m(a)15 b(Dualit)o(y)e(for)h (Arbitrary)f(Hopf)h(Algebras.)f Fr(Algebra)h(Beric)o(h)o(te)d(No)100 2674 y(66,)17 b(V)l(erlag)f(Reinhard)g(Fisc)o(her,)e(M)q(\177)-26 b(unc)o(hen,)16 b(1992,)h(57)g(pp.)0 2776 y([13])24 b(P)o(eter)h(Sc)o (hauen)o(burg:)39 b Fg(On)26 b(Co)q(quasitriangular)g(Hopf)g(Algebras)f (and)h(the)f(Quan)o(tum)f(Y)l(ang-)100 2836 y(Baxter)d(Equation.)g Fr(Algebra)g(Beric)o(h)o(te)d(No)j(67,)i(V)l(erlag)d(Reinhard)h(Fisc)o (her,)g(M)q(\177)-26 b(unc)o(hen,)22 b(1992,)100 2896 y(78)17 b(pp.)p eop %%Page: 35 35 35 34 bop 1901 -9 a Fr(35)0 136 y([14])24 b(M.)c(T)l(ak)o(euc)o(hi:)29 b(F)l(ree)20 b(Hopf)h(Algebras)g(generated)f(b)o(y)h(Coalgebras.)g(J.)g (Math.)f(So)q(c.)h(Japan)h(23,)100 196 y(No.4,)16 b(1971.)0 298 y([15])24 b(D.)14 b(T)l(am)o(bara:)20 b Fg(The)14 b(co)q(endomorphism)e(bialgebra)i(of)g(an)h(algebra.)f Fr(J.)f(F)l(ac.)g(Sci.)g(Univ.)g(T)l(oky)o(o)h(37,)100 358 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