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(follo)o(wing)g(diagram)f(comm)o(utes)664 1984 y Fm(A)10 b Fo(\012)h Fm(A)198 b(A)10 b Fo(\012)h Fm(A)p 810 1970 174 2 v 942 1969 a Fe(-)886 1961 y Fl(\033)878 2152 y Fm(A)759 2064 y Fj(r)760 2040 y Fe(@)801 2081 y(@)826 2106 y(@)-42 b(R)1005 2064 y Fj(r)992 2040 y Fe(\000)951 2081 y(\000)926 2106 y(\000)g(\011)0 2276 y Ff(R)n(emark)22 b Fp(1.2)p Ff(.)f Fp(Let)16 b Fm(M)k Fo(2)14 b(C)511 2283 y Fl(A)539 2276 y Fp(.)22 b(Then)16 b(it)g(is)g(easy)g(to)h(see)f (that)g(the)g(morphism)558 2409 y Fm(\025)586 2416 y Fl(M)640 2409 y Fp(:)d Fm(A)e Fo(\012)g Fm(M)860 2382 y Fl(\033)831 2409 y Fo(\000)-9 b(!)14 b Fm(M)i Fo(\012)11 b Fm(A)1103 2379 y Fl(\032)1121 2385 y Fd(M)1089 2409 y Fo(\000)-9 b(!)14 b Fm(M)0 2541 y Fp(de\014nes)k(an)g(\()p Fm(A;)8 b(A)p Fp(\)-bimo)q(dule)15 b(structure)i(on)h Fm(M)5 b Fp(.)26 b(The)18 b(compatibilit)o(y)c(of)k(the)g(left)e(and)j (righ)o(t)p eop %%Page: 4 4 4 3 bop 0 92 a Ft(4)750 b(Bo)q(do)14 b(P)o(areigis)0 200 y Fm(A)p 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y(@)1033 448 y(@)1058 473 y(@)g(R)843 613 y Fl(\032)861 619 y Fd(M)895 613 y Fj(\012)p Fg(1)743 573 y Fe(@)784 614 y(@)826 656 y(@)851 681 y(@)g(R)1258 610 y Fl(\032)1276 616 y Fd(M)1158 573 y Fe(@)1199 614 y(@)1241 656 y(@)1265 681 y(@)g(R)p 505 681 2 357 v -801 w(?)393 510 y Fg(1)p Fj(\012)p Fl(\032)456 516 y Fd(M)p 1335 681 V 1336 681 a Fe(?)1353 506 y Fl(\032)1371 512 y Fd(M)50 826 y Fp(The)14 b(category)g(of)h(righ)o(t)f Fm(A)p Fp(-mo)q(dules)f Fo(C)780 833 y Fl(A)823 826 y Fp(th)o(us)h(can)h(b)q(e)f(view)o(ed)f (as)i(a)f(full)g(sub)q(category)h(of)f Fm(A)p Fp(-)0 884 y Fm(A)p Fp(-bimo)q(dules)285 891 y Fl(A)313 884 y Fo(C)339 891 y Fl(A)368 884 y Fp(.)22 b(Actually)15 b(this)h(is)g(p)q(ossible)h(in)f(t)o(w)o(o)g(distinct)g(w)o(a)o(ys,)g (namely)e(b)o(y)i(de\014ning)0 956 y(the)i(left)g(structure)g(b)o(y)f Fm(A)12 b Fo(\012)h Fm(M)654 930 y Fl(\033)624 956 y Fo(\000)-8 b(!)17 b Fm(M)h Fo(\012)12 b Fm(A)923 927 y Fl(\032)892 956 y Fo(\000)-8 b(!)17 b Fm(M)23 b Fp(or)c(b)o(y)f Fm(A)12 b Fo(\012)g Fm(M)1371 930 y Fl(\033)1392 918 y Fc(\000)p Fb(1)1362 956 y Fo(\000)-9 b(!)18 b Fm(M)g Fo(\012)12 b Fm(A)1661 927 y Fl(\032)1630 956 y Fo(\000)-9 b(!)17 b Fm(M)5 b Fp(.)0 1014 y(So)18 b(w)o(e)g(get)f(the)h(t)o(w)o(o)g (full)e(em)o(b)q(eddings)h(\006)794 1021 y Fl(l)823 1014 y Fp(:)f Fo(C)879 1021 y Fl(A)925 1014 y Fo(\000)-9 b(!)1021 1021 y Fl(A)1050 1014 y Fo(C)1076 1021 y Fl(A)1122 1014 y Fp(and)19 b(\006)1254 1021 y Fl(r)1289 1014 y Fp(:)d Fo(C)1345 1021 y Fl(A)1390 1014 y Fo(\000)-8 b(!)1487 1021 y Fl(A)1515 1014 y Fo(C)1541 1021 y Fl(A)1570 1014 y Fp(.)26 b(W)l(e)17 b(shall)0 1073 y(restrict)e(our)g(considerations)h (to)g(\006)664 1080 y Fl(l)691 1073 y Fp(:)e Fo(C)745 1080 y Fl(A)787 1073 y Fo(\000)-8 b(!)881 1080 y Fl(A)910 1073 y Fo(C)936 1080 y Fl(A)980 1073 y Fp(de\014ned)15 b(b)o(y)g(the)g(left)g(action)h(as)g(in)f(remark)0 1131 y(1.2.)50 1192 y(W)l(e)c(no)o(w)g(in)o(v)o(estigate)f(the)h(tensor)h (pro)q(duct)f(o)o(v)o(er)g Fm(A)g Fp(in)1060 1199 y Fl(A)1089 1192 y Fo(C)1115 1199 y Fl(A)1144 1192 y Fp(.)19 b(A)11 b(straigh)o(tforw)o(ard)h(calculation)0 1250 y(giv)o(es)0 1367 y Fu(Lemma)k(1.3.)j Ff(L)n(et)e Fm(M)r(;)8 b(N)19 b Fo(2)14 b(C)592 1374 y Fl(A)621 1367 y Ff(.)22 b(Then)c(the)f(left)g (mo)n(dule)g(structur)n(e)g(on)g Fm(M)d Fo(\012)1486 1374 y Fl(A)1524 1367 y Fm(N)22 b Ff(as)16 b(de\014ne)n(d)0 1425 y(in)21 b Fp(1.2)d Ff(c)n(oincides)g(with)g(the)g(induc)n(e)n(d)f (left)i(mo)n(dule)e(structur)n(e)h(of)f Fm(M)f Fo(\012)1328 1432 y Fl(A)1367 1425 y Fm(N)23 b Ff(in)1489 1432 y Fl(A)1517 1425 y Fo(C)1543 1432 y Fl(A)1572 1425 y Ff(,)17 b(henc)n(e)i Fo(C)1765 1432 y Fl(A)0 1483 y Ff(is)d(a)h(monoidal)g(c)n(ate)n(gory)f (with)g(tensor)h(pr)n(o)n(duct)f(over)h Fm(A)f Ff(and)h Fp(\006)1206 1490 y Fl(l)1233 1483 y Fp(:)c Fo(C)1286 1490 y Fl(A)1329 1483 y Fo(\000)-9 b(!)1423 1490 y Fl(A)1451 1483 y Fo(C)1477 1490 y Fl(A)1522 1483 y Ff(de\014nes)18 b(a)f(ful)r(l)0 1541 y(monoidal)h(emb)n(e)n(dding.)50 1658 y 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a Fe(-)877 2264 y Fl(\033)898 2252 y Fb(2)878 2455 y Fm(A)759 2367 y Fj(r)760 2343 y Fe(@)801 2384 y(@)826 2409 y(@)-42 b(R)1005 2367 y Fj(r)992 2343 y Fe(\000)951 2384 y(\000)926 2409 y(\000)g(\011)0 2541 y Fp(comm)o(utes)13 b(or)k(equiv)m(alen)o (tly)d(if)i(\()p Fo(r)p Fm(\033)e Fp(:)g Fm(A)c Fo(\012)h Fm(A)j Fo(\000)-9 b(!)14 b Fm(A)p Fp(\))f(=)h(\()p Fo(r)p Fm(\033)1190 2523 y Fj(\000)p Fg(1)1250 2541 y Fp(:)f Fm(A)e Fo(\012)g Fm(A)i Fo(\000)-9 b(!)14 b Fm(A)p Fp(\).)p eop %%Page: 5 5 5 4 bop 660 92 a Ft(On)14 b(Braiding)f(and)h(Dyslexia)638 b(5)50 200 y Fp(A)16 b(mo)q(dule)f Fm(M)21 b Fp(in)16 b Fo(C)426 207 y Fl(A)471 200 y Fp(is)g(called)f Fi(dyslectic)g Fp(if)h(the)g(follo)o(wing)g(diagram)f(comm)o(utes)649 307 y Fm(M)h Fo(\012)11 b Fm(A)182 b(M)16 b Fo(\012)11 b Fm(A)p 811 293 158 2 v 927 292 a Fe(-)871 284 y Fl(\033)892 272 y Fb(2)864 475 y Fm(M)763 379 y Fl(\032)753 363 y Fe(@)794 404 y(@)819 429 y(@)-42 b(R)999 379 y Fl(\032)985 363 y Fe(\000)944 404 y(\000)919 429 y(\000)g(\011)1130 458 y Fm(:)50 562 y Fp(A)20 b(comm)o(utativ)n(e)d(algebra)k Fm(A)f Fp(is)h(clearly)e(dyslectic.)32 b(Ho)o(w)o(ev)o(er,)19 b(not)i(all)f Fm(A)p Fp(-mo)q(dules)f(o)o(v)o(er)0 620 y(a)f(comm)o(utativ)n(e)c(algebra)k Fm(A)g Fp(are)f(dyslectic.)23 b(In)17 b(fact)h(the)f(category)h(of)g(dyslectic)d Fm(A)p Fp(-mo)q(dules)0 678 y(dys)8 b Fo(C)106 685 y Fl(A)151 678 y Fp(is)16 b(the)g(equalizer)f(of)i(the)f(t)o(w)o(o)g(em)o(b)q (edding)e(functors)j(\006)1191 685 y Fl(l)1204 678 y Fm(;)8 b Fp(\006)1261 685 y Fl(r)1294 678 y Fp(:)13 b Fo(C)1347 685 y Fl(A)1390 678 y Fo(\000)-9 b(!)1484 685 y Fl(A)1513 678 y Fo(C)1539 685 y Fl(A)1567 678 y Fp(.)50 739 y(F)l(or)15 b(a)g(comm)o(utativ)n(e)d(algebra)j Fm(A)f Fp(there)g(is)h(a)g(braiding)g(morphism)d(for)j(the)g(tensor)g(pro)q (duct.)0 851 y Fu(Prop)r(osition)j(2.2.)i Ff(L)n(et)c Fm(A)h Ff(b)n(e)g(c)n(ommutative)h(and)f Fm(M)i Fo(2)14 b(C)1125 858 y Fl(A)1171 851 y Ff(b)n(e)j(dysle)n(ctic.)23 b(Then)18 b(the)f(fol)r(low-)0 909 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1304 y Fd(M)1407 1347 y Fm(N)16 b Fo(\012)1501 1354 y Fl(A)1541 1347 y Fm(M)p 1251 1334 144 2 v 1353 1333 a Fe(-)1313 1325 y Fl(\027)211 1193 y Fm(M)g Fo(\012)11 b Fm(N)17 b Fo(\012)10 b Fm(A)p 338 1149 2 98 v 338 1149 a Fe(?)255 1109 y Fg(1)p Fj(\012)p Fl(\033)p 338 1304 V 338 1304 a Fe(?)186 1257 y Fl(\033)206 1263 y Fd(M)q(;N)s Fc(\012)p Fd(A)p 1167 1304 2 254 v 1168 1304 a Fe(?)1185 1185 y Fl(\033)p 1499 1304 V 1500 1304 a Fe(?)1516 1193 y Fh(e)-22 b Fl(\033)1593 1354 y Fm(:)0 1477 y Ff(Pr)n(o)n(of.)19 b Fp(The)j(lo)o(w)o(er)g(left)f(hand)i("square")g(\()p Fm(\032)860 1484 y Fl(N)909 1477 y Fo(\012)14 b Fp(1\))p Fm(\033)1033 1484 y Fl(M)r(;N)t Fj(\012)p Fl(A)1166 1477 y Fp(\(1)i Fo(\012)f Fm(\033)r Fp(\))23 b(=)h Fm(\033)r Fp(\(1)15 b Fo(\012)g Fm(\032)1580 1484 y Fl(N)1614 1477 y Fp(\)\(1)g Fo(\012)g Fm(\033)r Fp(\))0 1535 y(comm)o(utes)j(b)o(y)j (functorialit)o(y)f(of)h Fm(\033)r Fp(.)36 b(F)l(urthermore)19 b(if)i Fm(M)26 b Fp(is)21 b(dyslectic)f(then)h(the)g(follo)o(wing)0 1593 y(diagram)16 b(comm)o(ute)o(s)271 1697 y Fm(M)h Fo(\012)11 b Fm(A)f Fo(\012)h Fm(N)82 b(M)17 b Fo(\012)10 b Fm(N)17 b Fo(\012)11 b Fm(A)p 539 1683 53 2 v 550 1682 a Fe(-)532 1670 y Fg(1)p Fj(\012)p Fl(\033)401 1767 y(\033)421 1773 y Fd(M)s Fc(\012)p Fd(A;N)457 1731 y Fe(H)498 1752 y(H)540 1773 y(H)581 1794 y(H)623 1814 y(H)631 1819 y(H)-42 b(j)923 1767 y Fl(\033)943 1773 y Fd(M)q(;N)s Fc(\012)p Fd(A)789 1731 y Fe(H)830 1752 y(H)872 1773 y(H)913 1794 y(H)955 1814 y(H)963 1819 y(H)g(j)p 398 1985 2 274 v 399 1985 a(?)285 1855 y Fl(\032)303 1861 y Fd(M)337 1855 y Fj(\012)p Fg(1)p 730 1819 2 108 v 731 1819 a Fe(?)747 1774 y Fl(\033)q Fj(\012)p Fg(1)603 1863 y Fm(N)17 b Fo(\012)10 b Fm(M)17 b Fo(\012)11 b Fm(A)76 b(N)16 b Fo(\012)11 b Fm(A)g Fo(\012)g Fm(M)p 870 1848 53 2 v 881 1847 a Fe(-)863 1836 y Fg(1)p Fj(\012)p Fl(\033)1267 1863 y Fm(N)16 b Fo(\012)11 b Fm(M)17 b Fo(\012)10 b Fm(A)p 1202 1848 V 1213 1847 a Fe(-)1195 1836 y Fg(1)p Fj(\012)p Fl(\033)p 730 1985 2 108 v 731 1985 a Fe(?)747 1938 y Fg(1)p Fj(\012)p Fl(\032)810 1944 y Fd(M)p 1394 1985 V 1394 1985 a Fe(?)1411 1938 y Fg(1)p Fj(\012)p Fl(\032)1474 1944 y Fd(M)320 2029 y Fm(M)16 b Fo(\012)11 b Fm(N)180 b(N)16 b Fo(\012)11 b Fm(M)p 490 2015 151 2 v 599 2014 a Fe(-)554 2005 y Fl(\033)1316 2029 y Fm(N)16 b Fo(\012)11 b Fm(M)p 822 2015 483 2 v 1263 2014 a Fe(-)1054 2006 y Fg(1)0 2114 y Fp(whic)o(h)16 b(is)g(the)g(upp)q(er)g(left)g (hand)g("square".)p 868 2087 33 2 v 868 2116 2 30 v 899 2116 V 868 2118 33 2 v 0 2218 a Ff(R)n(emark)22 b Fp(2.3)p Ff(.)f Fp(There)d(is)g(a)g(second)h(w)o(a)o(y)f(to)g(de\014ne)g(a)h (quasi-symmetry)c(map)i(in)h Fo(C)1579 2225 y Fl(A)1608 2218 y Fp(,)g(namely)0 2276 y(with)e Fm(\033)141 2255 y Fj(\000)p Fg(1)139 2288 y Fl(M)r(;N)234 2276 y Fp(instead)g(of)h Fm(\033)486 2283 y Fl(M)r(;N)564 2276 y Fp(.)22 b(A)15 b(similar)g(pro)q(of)i(as)g(for)f(Prop)q(osition)i(2.2)e(giv)o(es:)50 2344 y(Let)h Fm(A)g Fp(b)q(e)g(comm)o(utativ)n(e)d(and)j Fm(N)k Fo(2)15 b(C)780 2351 y Fl(A)826 2344 y Fp(b)q(e)i(dyslectic.)22 b(Then)17 b Fm(\033)1272 2323 y Fj(\000)p Fg(1)1270 2356 y Fl(M)r(;N)1366 2344 y Fp(induces)f(a)i(morphism)0 2402 y(on)f(the)f(di\013erence)f(cok)o(ernels:)601 2398 y Fh(e)598 2402 y Fm(\033)g Fp(:)e Fm(M)k Fo(\012)771 2409 y Fl(A)810 2402 y Fm(N)i Fo(\000)-8 b(!)13 b Fm(N)k Fo(\012)1057 2409 y Fl(A)1096 2402 y Fm(M)5 b Fp(.)0 2532 y Fu(Prop)r(osition)18 b(2.4.)i Ff(If)d Fm(M)23 b Ff(and)17 b Fm(N)23 b Ff(in)18 b Fo(C)774 2539 y Fl(A)820 2532 y Ff(ar)n(e)f(dysle)n(ctic)h(then)g(so) g(is)f Fm(M)f Fo(\012)1422 2539 y Fl(A)1462 2532 y Fm(N)5 b Ff(.)p eop %%Page: 6 6 6 5 bop 0 92 a Ft(6)750 b(Bo)q(do)14 b(P)o(areigis)0 200 y Ff(Pr)n(o)n(of.)19 b Fp(W)l(e)d(ha)o(v)o(e)g(to)g(sho)o(w)h(that) 528 319 y(\()p Fm(M)g Fo(\012)650 326 y Fl(A)689 319 y Fm(N)5 b Fp(\))12 b Fo(\012)e Fm(A)93 b Fp(\()p Fm(M)17 b Fo(\012)1065 326 y Fl(A)1104 319 y Fm(N)5 b Fp(\))11 b Fo(\012)g Fm(A)p 862 307 69 2 v 889 306 a Fe(-)877 298 y Fl(\033)898 286 y Fb(2)870 531 y Fm(M)749 414 y 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2545 y Fg(\(0\))1168 2537 y Fp(\))1187 2545 y Fg(\(0\))1236 2537 y Fo(\001)r Fm(h)1280 2517 y Fj(\003)1280 2550 y Fl(i)1300 2537 y Fp(\()p Fm(f)g Fp(\()p Fo(\000)1406 2545 y Fg(\(0\))1453 2537 y Fp(\))1472 2545 y Fg(\(1\))1519 2537 y Fm(S)s Fp(\()p Fo(\000)1610 2545 y Fg(\(1\))1657 2537 y Fp(\)\))r Fo(\012)r Fm(h)1766 2544 y Fl(i)1780 2537 y Fm(:)p eop %%Page: 8 8 8 7 bop 0 92 a Ft(8)750 b(Bo)q(do)14 b(P)o(areigis)50 200 y Fp(Then)26 b(the)g(canonical)g(morphism)d Fo(M)p Fp(\()p Fm(M)g Fo(\012)18 b Fm(N)r(;)8 b(P)f Fp(\))1099 186 y Fo(\030)1099 202 y Fp(=)1168 200 y Fo(M)p Fp(\()p Fm(M)r(;)h Fp(Hom)o(\()p Fm(N)r(;)g(P)f Fp(\)\))27 b(giv)o(en)e(b)o(y)0 258 y Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)p Fp(\))j(=)g Fm(g)r Fp(\()p Fm(m)p Fp(\)\()p Fm(n)p Fp(\))i(restricts)f(to)468 364 y Fo(M)528 343 y Fl(H)562 364 y Fp(\()p Fm(M)h Fo(\012)11 b Fm(N)r(;)d(P)f Fp(\))828 350 y Fo(\030)829 366 y Fp(=)881 364 y Fo(M)941 343 y Fl(H)974 364 y Fp(\()p Fm(M)r(;)h Fp(Hom)o(\()p Fm(N)r(;)g(P)f 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Fg(\(1\))1277 635 y Fp(\))583 694 y(=)662 660 y Fh(P)714 694 y Fm(f)5 b Fp(\()p Fm(m)805 701 y Fg(\(0\))864 694 y Fo(\012)10 b Fm(n)942 701 y Fg(\(0\))990 694 y Fp(\))h Fo(\012)g Fm(m)1113 701 y Fg(\(1\))1159 694 y Fm(n)1188 701 y Fg(\(1\))1236 694 y Fm(S)s Fp(\()p Fm(n)1317 701 y Fg(\(2\))1364 694 y Fp(\))583 752 y(=)662 718 y Fh(P)714 752 y Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)895 759 y Fg(\(0\))943 752 y Fp(\))962 759 y Fg(\(0\))1020 752 y Fo(\012)g Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)f Fm(n)1250 759 y Fg(\(0\))1298 752 y Fp(\))1317 759 y Fg(\(1\))1364 752 y Fm(S)s Fp(\()p Fm(n)1445 759 y Fg(\(1\))1492 752 y Fp(\))583 810 y(=)662 777 y Fh(P)714 810 y Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)895 817 y Fg(\(0\))943 810 y Fp(\))962 817 y Fg(\(0\))1020 810 y Fo(\012)g Fm(h)1098 792 y Fj(\003)1098 822 y Fl(i)1117 810 y Fp(\()p Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)1317 817 y Fg(\(0\))1364 810 y Fp(\))1383 817 y Fg(\(1\))1431 810 y Fm(S)s Fp(\()p Fm(n)1512 817 y Fg(\(1\))1559 810 y Fp(\)\))p Fm(h)1625 817 y Fl(i)583 868 y Fp(=)662 835 y Fh(P)714 868 y Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)895 876 y Fg(\(0\))943 868 y Fp(\))962 876 y Fg(\(0\))1020 868 y Fo(\001)g Fm(h)1073 850 y Fj(\003)1073 880 y Fl(i)1092 868 y Fp(\()p Fm(f)5 b Fp(\()p Fm(m)11 b Fo(\012)g Fm(n)1292 876 y Fg(\(0\))1340 868 y Fp(\))1359 876 y Fg(\(1\))1406 868 y Fm(S)s Fp(\()p Fm(n)1487 876 y Fg(\(1\))1534 868 y Fp(\)\))g Fo(\012)g Fm(h)1661 875 y Fl(i)583 926 y Fp(=)662 893 y Fh(P)714 926 y Fm(g)r Fp(\()p Fm(m)p Fp(\)\()p Fm(n)868 934 y Fg(\(0\))915 926 y Fp(\))934 934 y Fg(\(0\))993 926 y Fo(\001)f Fm(h)1045 908 y Fj(\003)1045 938 y Fl(i)1065 926 y Fp(\()p Fm(g)r Fp(\()p Fm(m)p Fp(\)\()p Fm(n)1238 934 y Fg(\(0\))1285 926 y Fp(\))1304 934 y Fg(\(1\))1351 926 y Fm(S)s Fp(\()p Fm(n)1432 934 y Fg(\(1\))1479 926 y Fp(\)\))h Fo(\012)g Fm(h)1606 933 y Fl(i)583 984 y Fp(=)662 951 y Fh(P)714 984 y Fm(g)r Fp(\()p Fm(m)p Fp(\))820 992 y Fg(\(0\))867 984 y Fp(\()p Fm(n)p Fp(\))g Fo(\012)g Fm(g)r Fp(\()p Fm(m)p Fp(\))1101 992 y Fg(\(1\))1148 984 y Fm(:)0 1088 y Fp(In)24 b(a)g(similar)e(w)o(a)o(y)i(one)h(sho)o (ws)g(that)f(the)g(in)o(v)o(erse)f(map)g(also)i(restricts)e(to)i (morphisms)d(in)0 1146 y Fo(M)60 1128 y Fl(H)93 1146 y Fp(.)p 156 1119 33 2 v 156 1148 2 30 v 187 1148 V 156 1150 33 2 v 0 1255 a Ff(R)n(emark)g Fp(3.2)p Ff(.)f Fp(In)16 b(general)g Fo(M)573 1237 y Fl(H)623 1255 y Fp(will)f(not)h(b)q(e)h (left)e(closed.)50 1339 y(If)d Fo(M)i Fp(=)g Fn(K)s Fp(-)q(V)l(ek)h (then)d Fo(M)536 1321 y Fl(H)583 1339 y Fp(has)h(k)o(ernels.)19 b(So)13 b(for)h(a)f(\014nite)f(co)q(quasitriangular)i(Hopf)f(algebra)0 1397 y Fm(H)22 b Fp(o)o(v)o(er)17 b(a)i(\014eld)e Fn(K)25 b Fp(the)18 b(category)g(of)h Fm(H)t Fp(-como)q(dules)e Fo(M)1076 1379 y Fl(H)1128 1397 y Fp(satis\014es)h(the)g(prop)q(erties) g(for)g Fo(C)k Fp(as)0 1455 y(required)15 b(at)i(the)f(b)q(eginning)g (of)h(this)f(section.)50 1582 y(W)l(e)h(return)h(no)o(w)g(to)g(the)f (general)g(case.)26 b(If)17 b Fo(C)k Fp(is)c(righ)o(t)g(closed)h(and)g (has)g(di\013erence)f(k)o(ernels)0 1640 y(then)111 1647 y Fl(A)140 1640 y Fo(C)166 1647 y Fl(A)211 1640 y Fp(is)f(also)g(righ)o (t)g(closed)g(with)g([)p Fm(M)r(;)8 b(N)d Fp(])873 1647 y Fl(A)918 1640 y Fp(the)16 b(di\013erence)f(k)o(ernel)g(in)455 1745 y([)p Fm(M)r(;)8 b(N)d Fp(])598 1752 y Fl(A)640 1745 y Fo(\000)-9 b(!)14 b Fp([)p Fm(M)r(;)8 b(N)d Fp(])p 889 1721 150 2 v 997 1720 a Fe(-)p 889 1738 V 997 1737 a(-)1082 1745 y Fp([)p Fm(M)16 b Fo(\012)11 b Fm(A;)d(N)d Fp(])p Fm(:)0 1851 y Fp(T)l(ec)o(hniques)15 b(as)i(in)f([9])f(can)i(b)q (e)f(used)h(to)f(pro)o(v)o(e)g(this.)50 1913 y(W)l(e)e(consider)g(the)g (pair)g(of)h(adjoin)o(t)f(functors)h(-)7 b Fo(\012)g Fm(A)14 b Fp(and)h([)p Fm(A;)8 b Fp(-)o(].)20 b(Let)14 b Fm(\021)i Fp(:)e Fm(M)19 b Fo(\000)-9 b(!)14 b Fp([)p Fm(A;)8 b(M)j Fo(\012)c Fm(A)p Fp(])0 1971 y(b)q(e)22 b(the)h(unit)f(and)g Fm(")i Fp(:)g([)p Fm(A;)8 b(M)d Fp(])14 b Fo(\012)h Fm(A)24 b Fo(\000)-9 b(!)24 b Fm(M)k Fp(b)q(e)23 b(the)f(counit)g(of)g(the)g(adjoin)o(t)h(pair.)39 b(Then)0 2029 y(the)18 b(isomorphism)e(Mor)461 2036 y Fj(C)484 2029 y Fp(\()p Fm(M)i Fo(\012)12 b Fm(A;)c(N)d Fp(\))758 2015 y Fo(\030)759 2031 y Fp(=)814 2029 y(Mor)903 2036 y Fj(C)925 2029 y Fp(\()p Fm(M)r(;)j Fp([)p Fm(A;)g(M)d Fp(]\))18 b(is)g(giv)o(en)g(b)o(y)f Fm(f)23 b Fo(7!)18 b Fp([)p Fm(A;)8 b(f)d Fp(])p Fm(\021)19 b Fp(and)0 2087 y Fm(g)d Fo(7!)e Fm(")p Fp([)p Fm(g)e Fo(\012)f Fp(1)249 2094 y Fl(A)278 2087 y Fp(],)k(in)h(particular)g(w)o(e)g(get)g Fm(")p Fp(\([)p Fm(A;)8 b(f)d Fp(])p Fm(\021)12 b Fo(\012)f Fp(1)1024 2094 y Fl(A)1052 2087 y Fp(\))j(=)g Fm(f)5 b Fp(.)50 2149 y(Let)16 b Fm(M)22 b Fp(b)q(e)16 b(a)h(righ)o(t)f Fm(A)p Fp(-mo)q(dule)f(with)h(structure)g(morphism)e Fm(\032)1229 2156 y Fl(M)1282 2149 y Fp(:)f Fm(M)k Fo(\012)11 b Fm(A)i Fo(\000)-8 b(!)13 b Fm(M)5 b Fp(.)22 b(Let)516 2277 y Fm(K)p 574 2261 176 2 v 708 2260 a Fe(-)635 2244 y Fm(\023)652 2251 y Fl(K)p 826 2252 290 2 v 392 w Fe(-)908 2235 y Fg([)p Fl(A;\032)972 2241 y Fd(M)1006 2235 y Fg(])o Fl(\021)762 2272 y Fm(M)325 b Fp([)p Fm(A;)8 b(M)d Fp(])p 826 2269 V 1074 2268 a Fe(-)866 2304 y Fg([)p Fl(A;\032)930 2310 y Fd(M)963 2304 y Fl(\033)984 2292 y Fb(2)983 2316 y Fd(M)q(;A)1048 2304 y Fg(])p Fl(\021)0 2411 y Fp(b)q(e)16 b(a)h(di\013erence)e(k)o(ernel)g(\(equalizer\).)k(\(W)l(e)d(abbreviate) g Fm(\033)1103 2393 y Fg(2)1101 2424 y Fl(M)r(;A)1188 2411 y Fp(:=)e Fm(\033)1282 2418 y Fl(A;M)1357 2411 y Fm(\033)1385 2418 y Fl(M)r(;A)1459 2411 y Fp(.\))0 2532 y Fu(Lemma)i(3.3.)j Fm(K)j Ff(is)17 b(an)h Fm(A)p Ff(-submo)n(dule)g (of)f Fm(M)5 b Ff(.)p eop %%Page: 9 9 9 8 bop 660 92 a Ft(On)14 b(Braiding)f(and)h(Dyslexia)638 b(9)0 200 y Ff(Pr)n(o)n(of.)19 b Fp(W)l(e)d(ha)o(v)o(e)70 296 y Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)207 303 y Fl(M)246 296 y Fp(])p Fm(\021)r(\032)311 303 y Fl(M)350 296 y Fp(\()p Fm(\023)386 303 y Fl(K)431 296 y Fo(\012)j Fp(1)505 303 y Fl(A)534 296 y Fp(\))g Fo(\012)g Fp(1)638 303 y Fl(A)667 296 y Fp(\))70 355 y(=)j Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)259 362 y Fl(M)297 355 y Fp(])p Fm(\021)13 b Fo(\012)e Fp(1)422 362 y Fl(A)451 355 y Fp(\)\()p Fm(\032)514 362 y Fl(M)564 355 y Fo(\012)g Fp(1)638 362 y Fl(A)667 355 y Fp(\)\()p Fm(\023)722 362 y Fl(K)767 355 y Fo(\012)g Fp(1)841 362 y Fl(A)881 355 y Fo(\012)g Fp(1)955 362 y Fl(A)984 355 y Fp(\))70 413 y(=)j Fm(\032)147 420 y Fl(M)187 413 y Fp(\()p Fm(\032)231 420 y Fl(M)281 413 y Fo(\012)d Fp(1)355 420 y Fl(A)384 413 y Fp(\)\()p Fm(\023)439 420 y Fl(K)485 413 y Fo(\012)f Fp(1)558 420 y Fl(A)598 413 y Fo(\012)h Fp(1)672 420 y Fl(A)701 413 y Fp(\))70 471 y(=)j Fm(\032)147 478 y Fl(M)187 471 y Fp(\(1)230 478 y Fl(A)270 471 y Fo(\012)d(r)p Fp(\)\()p Fm(\023)417 478 y Fl(K)461 471 y Fo(\012)g Fp(1)535 478 y Fl(A)575 471 y Fo(\012)g Fp(1)649 478 y Fl(A)678 471 y Fp(\))70 529 y(=)j Fm(\032)147 536 y Fl(M)187 529 y Fp(\(1)230 536 y Fl(A)270 529 y Fo(\012)d(r)p Fp(\)\(1)424 536 y Fl(M)474 529 y Fo(\012)g Fm(\033)552 536 y Fl(A;A)616 529 y Fp(\)\()p Fm(\023)671 536 y Fl(K)717 529 y Fo(\012)f Fp(1)790 536 y Fl(A)830 529 y Fo(\012)h Fp(1)904 536 y Fl(A)933 529 y Fp(\))65 b(since)16 b Fm(A)f Fp(is)h(comm)o(utativ)o (e)70 587 y(=)e Fm(\032)147 594 y Fl(M)187 587 y Fp(\()p Fm(\032)231 594 y Fl(M)281 587 y Fo(\012)d Fp(1)355 594 y Fl(A)384 587 y Fp(\)\()p Fm(\023)439 594 y Fl(K)485 587 y Fo(\012)f Fp(1)558 594 y Fl(A)598 587 y Fo(\012)h Fp(1)672 594 y Fl(A)701 587 y Fp(\)\(1)763 594 y Fl(M)814 587 y Fo(\012)g Fm(\033)892 594 y Fl(A;A)956 587 y Fp(\))70 645 y(=)j Fm(\032)147 652 y Fl(M)187 645 y Fp(\()p Fm(\032)231 652 y Fl(M)281 645 y Fo(\012)d Fp(1)355 652 y Fl(A)384 645 y Fp(\)\()p Fm(\033)452 627 y Fg(2)450 657 y Fl(M)r(;A)535 645 y Fo(\012)g Fp(1)609 652 y Fl(A)637 645 y Fp(\)\()p Fm(\023)692 652 y Fl(K)738 645 y Fo(\012)g Fp(1)812 652 y Fl(A)852 645 y Fo(\012)f Fp(1)925 652 y Fl(A)954 645 y Fp(\)\(1)1016 652 y Fl(M)1067 645 y Fo(\012)h Fm(\033)1145 652 y Fl(A;A)1209 645 y Fp(\))65 b(\(b)o(y)16 b(de\014nition)g(of)h Fm(\023)1670 652 y Fl(K)1704 645 y Fp(\))70 705 y(=)d Fm(\032)147 712 y Fl(M)187 705 y Fp(\(1)230 712 y Fl(A)270 705 y Fo(\012)d(r)p Fp(\)\(1)424 712 y Fl(M)474 705 y Fo(\012)g Fm(\033)552 712 y Fl(A;A)616 705 y Fp(\)\()p Fm(\033)684 686 y Fg(2)682 717 y Fl(M)r(;A)767 705 y Fo(\012)g Fp(1)841 712 y Fl(A)869 705 y Fp(\)\(1)931 712 y Fl(M)982 705 y Fo(\012)g Fm(\033)1060 712 y Fl(A;A)1125 705 y Fp(\)\()p Fm(\023)1180 712 y Fl(K)1225 705 y Fo(\012)g Fp(1)1299 712 y Fl(A)1339 705 y Fo(\012)g Fp(1)1413 712 y Fl(A)1441 705 y Fp(\))70 764 y(=)j Fm(\032)147 771 y Fl(M)187 764 y Fp(\()p Fm(\032)231 771 y Fl(M)281 764 y Fo(\012)d Fp(1)355 771 y Fl(A)384 764 y Fp(\))p Fm(\033)433 746 y Fg(2)431 776 y Fl(M)t Fj(\012)p Fl(A;A)560 764 y Fp(\()p Fm(\023)596 771 y Fl(K)642 764 y Fo(\012)g Fp(1)716 771 y Fl(A)756 764 y Fo(\012)g Fp(1)830 771 y Fl(A)858 764 y Fp(\))65 b(\(prop)q(ert)o(y)17 b(of)f Fm(\033)r Fp(\))70 823 y(=)e Fm(\032)147 830 y Fl(M)187 823 y Fm(\033)217 805 y Fg(2)215 836 y Fl(M)r(;A)288 823 y Fp(\()p Fm(\032)332 830 y Fl(M)383 823 y Fo(\012)d Fp(1)457 830 y Fl(A)486 823 y Fp(\)\()p Fm(\023)541 830 y Fl(K)586 823 y Fo(\012)g Fp(1)660 830 y Fl(A)700 823 y Fo(\012)g Fp(1)774 830 y Fl(A)803 823 y Fp(\))70 883 y(=)j Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)259 890 y Fl(M)297 883 y Fm(\033)327 865 y Fg(2)325 895 y Fl(M)r(;A)399 883 y Fp(])p Fm(\021)k Fo(\012)f Fp(1)523 890 y Fl(A)552 883 y Fp(\)\()p Fm(\032)615 890 y Fl(M)666 883 y Fo(\012)g Fp(1)740 890 y Fl(A)768 883 y Fp(\)\()p Fm(\023)823 890 y Fl(K)869 883 y Fo(\012)g Fp(1)943 890 y Fl(A)983 883 y Fo(\012)f Fp(1)1056 890 y Fl(A)1085 883 y Fp(\))70 942 y(=)k Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)259 949 y Fl(M)297 942 y Fm(\033)327 924 y Fg(2)325 955 y Fl(M)r(;A)399 942 y Fp(])p Fm(\021)r(\032)464 949 y Fl(M)503 942 y Fp(\()p Fm(\023)539 949 y Fl(K)584 942 y Fo(\012)j Fp(1)658 949 y Fl(A)687 942 y Fp(\))g Fo(\012)g Fp(1)791 949 y Fl(A)820 942 y Fp(\))0 1047 y(whic)o(h)j(implies)e([)p Fm(A;)c(\032)400 1054 y Fl(M)439 1047 y Fp(])p Fm(\021)r(\032)504 1054 y Fl(M)543 1047 y Fp(\()p Fm(\023)579 1054 y Fl(K)622 1047 y Fo(\012)g Fp(1)693 1054 y Fl(A)722 1047 y Fp(\))13 b(=)h([)p Fm(A;)8 b(\032)904 1054 y Fl(M)943 1047 y Fm(\033)973 1028 y Fg(2)971 1059 y Fl(M)r(;A)1044 1047 y Fp(])p Fm(\021)r(\032)1109 1054 y Fl(M)1148 1047 y Fp(\()p Fm(\023)1184 1054 y Fl(K)1227 1047 y Fo(\012)g Fp(1)1298 1054 y Fl(A)1327 1047 y Fp(\))p Fm(:)14 b Fp(So)i(there)e(is)h(a)g(unique)0 1105 y(morphism)10 b Fm(\032)249 1112 y Fl(K)298 1105 y Fp(:)j Fm(K)8 b Fo(\012)t Fm(A)13 b Fo(\000)-8 b(!)13 b Fm(K)k Fp(suc)o(h)c(that)g Fm(\023)845 1112 y Fl(K)879 1105 y Fm(\032)904 1112 y Fl(K)953 1105 y Fp(=)g Fm(\032)1029 1112 y Fl(M)1069 1105 y Fp(\()p Fm(\023)1105 1112 y Fl(K)1143 1105 y Fo(\012)t Fp(1)1210 1112 y Fl(A)1239 1105 y Fp(\),)g(since)f Fm(\023)1418 1112 y Fl(K)1465 1105 y Fp(is)h(the)f(di\013erence)0 1163 y(k)o(ernel)j(of)h([)p Fm(A;)8 b(\032)297 1170 y Fl(M)336 1163 y Fp(])p Fm(\021)17 b Fp(and)g([)p Fm(A;)8 b(\032)584 1170 y Fl(M)623 1163 y Fm(\033)653 1145 y Fg(2)651 1175 y Fl(M)r(;A)724 1163 y Fp(])p Fm(\021)r Fp(.)p 826 1136 33 2 v 826 1165 2 30 v 857 1165 V 826 1167 33 2 v 0 1265 a Fu(Lemma)16 b(3.4.)j Fm(K)j Ff(is)17 b(dysle)n(ctic.)0 1376 y(Pr)n(o)n(of.)i Fp(W)l(e)12 b(ha)o(v)o(e)e Fm(\023)353 1383 y Fl(K)388 1376 y Fm(\032)413 1383 y Fl(K)447 1376 y Fm(\033)477 1358 y Fg(2)475 1389 y Fl(K;A)557 1376 y Fp(=)k Fm(\032)634 1383 y Fl(M)674 1376 y Fp(\()p Fm(\023)710 1383 y Fl(K)746 1376 y Fo(\012)r Fp(1)811 1383 y Fl(A)839 1376 y Fp(\))p Fm(\033)888 1358 y Fg(2)886 1389 y Fl(K;A)968 1376 y Fp(=)g Fm(\032)1045 1383 y Fl(M)1085 1376 y Fm(\033)1115 1358 y Fg(2)1113 1389 y Fl(M)r(;A)1186 1376 y Fp(\()p Fm(\023)1222 1383 y Fl(K)1258 1376 y Fo(\012)r Fp(1)1323 1383 y Fl(A)1352 1376 y Fp(\))g(=)g Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)1574 1383 y Fl(M)1612 1376 y Fm(\033)1642 1358 y Fg(2)1640 1389 y Fl(M)r(;A)1713 1376 y Fp(])p Fm(\021)t Fo(\012)0 1439 y Fp(1)24 1446 y Fl(A)53 1439 y Fp(\)\()p Fm(\023)108 1446 y Fl(K)157 1439 y Fo(\012)15 b Fp(1)235 1446 y Fl(A)264 1439 y Fp(\))24 b(=)g Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)506 1446 y Fl(M)544 1439 y Fm(\033)574 1421 y Fg(2)572 1451 y Fl(M)r(;A)646 1439 y Fp(])p Fm(\021)r(\023)703 1446 y Fl(K)751 1439 y Fo(\012)15 b Fp(1)829 1446 y Fl(A)858 1439 y Fp(\))24 b(=)g Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)1100 1446 y Fl(M)1138 1439 y Fp(])p Fm(\021)r(\023)1195 1446 y Fl(K)1244 1439 y Fo(\012)15 b Fp(1)1322 1446 y Fl(A)1351 1439 y Fp(\))24 b(=)g Fm(\032)1481 1446 y Fl(M)1520 1439 y Fp(\()p Fm(\023)1556 1446 y Fl(K)1606 1439 y Fo(\012)15 b Fp(1)1684 1446 y Fl(A)1712 1439 y Fp(\))24 b(=)0 1497 y Fm(\023)17 1504 y Fl(K)51 1497 y Fm(\032)76 1504 y Fl(K)111 1497 y Fp(.)p 173 1470 V 173 1499 2 30 v 204 1499 V 173 1501 33 2 v 0 1600 a Fu(Lemma)16 b(3.5.)j Ff(If)f Fm(P)23 b Fo(2)15 b Fp(dys)8 b Fo(C)566 1607 y Fl(A)613 1600 y Ff(is)18 b(dysle)n(ctic,)h Fm(M)h Fo(2)c(C)1018 1607 y Fl(A)1064 1600 y Ff(and)j Fm(f)h Fp(:)15 b Fm(P)22 b Fo(\000)-9 b(!)15 b Fm(M)24 b Ff(an)18 b Fm(A)p Ff(-homomor-)0 1658 y(phism,)f(then)h Fm(f)23 b Ff(factors)17 b(uniquely)i(thr)n(ough) e Fm(\023)858 1665 y Fl(K)907 1658 y Fp(:)c Fm(K)18 b Fo(\000)-9 b(!)14 b Fm(M)5 b Ff(.)0 1769 y(Pr)n(o)n(of.)19 b Fp(W)l(e)d(ha)o(v)o(e)g(to)g(sho)o(w)h(that)g([)p Fm(A;)8 b(\032)730 1776 y Fl(M)768 1769 y Fp(])p Fm(\021)r(f)19 b Fp(=)14 b([)p Fm(A;)8 b(\032)1001 1776 y Fl(M)1039 1769 y Fm(\033)1069 1751 y Fg(2)1067 1781 y Fl(M)r(;A)1141 1769 y Fp(])p Fm(\021)r(f)d Fp(.)21 b(This)16 b(follo)o(ws)g(from)61 1874 y Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)198 1881 y Fl(M)237 1874 y Fp(])p Fm(\021)r(f)15 b Fo(\012)c Fp(1)390 1881 y Fl(A)419 1874 y Fp(\))j(=)g Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)641 1881 y Fl(M)679 1874 y Fp(])p Fm(\021)13 b Fo(\012)d Fp(1)803 1881 y Fl(A)832 1874 y Fp(\)\()p Fm(f)17 b Fo(\012)10 b Fp(1)984 1881 y Fl(A)1013 1874 y Fp(\))k(=)g Fm(\032)1123 1881 y Fl(M)1163 1874 y Fp(\()p Fm(f)i Fo(\012)11 b Fp(1)1296 1881 y Fl(A)1325 1874 y Fp(\))i(=)h Fm(f)5 b(\032)1463 1881 y Fl(P)1507 1874 y Fp(=)14 b Fm(f)5 b(\032)1613 1881 y Fl(P)1643 1874 y Fm(\033)1673 1856 y Fg(2)1671 1887 y Fl(P)q(;A)61 1934 y Fp(=)14 b Fm(\032)138 1941 y Fl(M)178 1934 y Fp(\()p Fm(f)i Fo(\012)11 b Fp(1)311 1941 y Fl(A)340 1934 y Fp(\))p Fm(\033)389 1916 y Fg(2)387 1946 y Fl(P)q(;A)462 1934 y Fp(=)j Fm(\032)539 1941 y Fl(M)579 1934 y Fm(\033)609 1916 y Fg(2)607 1946 y Fl(M)r(;A)680 1934 y Fp(\()p Fm(f)i Fo(\012)11 b Fp(1)813 1941 y Fl(A)842 1934 y Fp(\))j(=)g Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)1064 1941 y Fl(M)1102 1934 y Fm(\033)1132 1916 y Fg(2)1130 1946 y Fl(M)r(;A)1203 1934 y Fp(])p Fm(\021)13 b Fo(\012)e Fp(1)1328 1941 y Fl(A)1357 1934 y Fp(\)\()p Fm(f)16 b Fo(\012)11 b Fp(1)1509 1941 y Fl(A)1538 1934 y Fp(\))61 1993 y(=)j Fm(\017)p Fp(\([)p Fm(A;)8 b(\032)250 2000 y Fl(M)288 1993 y Fm(\033)318 1975 y Fg(2)316 2005 y Fl(M)r(;A)390 1993 y Fp(])p Fm(\021)r(f)16 b Fo(\012)11 b Fp(1)544 2000 y Fl(A)572 1993 y Fp(\))p Fm(:)p 654 1966 V 654 1995 2 30 v 685 1995 V 654 1997 33 2 v 0 2095 a Fu(Theorem)17 b(3.6.)i Ff(L)n(et)f Fo(C)j Ff(b)n(e)d(as)g(in)h(The)n (or)n(em)h Fp(2.5)p Ff(,)f(b)n(e)f(right)g(close)n(d)g(and)h(have)f (di\013er)n(enc)n(e)g(ker-)0 2153 y(nels.)23 b(L)n(et)16 b Fm(A)g Ff(b)n(e)h(a)f(c)n(ommutative)h(algebr)n(a)g(in)f Fo(C)s Ff(.)22 b(Then)17 b(the)g(c)n(ate)n(gory)f(of)g(dysle)n(ctic)h Fm(A)p Ff(-mo)n(dules)0 2211 y Fp(dys)8 b Fo(C)106 2218 y Fl(A)153 2211 y Ff(is)17 b(a)g(c)n(or)n(e\015exive)h(sub)n(c)n(ate)n (gory)f(of)g Fo(C)822 2218 y Fl(A)851 2211 y Ff(.)0 2322 y(Pr)n(o)n(of.)i Fp(W)l(e)12 b(ha)o(v)o(e)g(to)h(sho)o(w)h(that)f(the)f (construction)h(of)g Fm(K)j Fp(as)e(in)e(the)g(previous)h(Lemmas)d (de\014nes)0 2381 y(a)15 b(righ)o(t)g(adjoin)o(t)g(functor)g(to)g(the)g (em)o(b)q(edding)e(of)j(dys)8 b Fo(C)1035 2388 y Fl(A)1079 2381 y Fp(in)o(to)14 b Fo(C)1202 2388 y Fl(A)1231 2381 y Fp(.)21 b(But)14 b(this)h(is)g(demonstrated)0 2439 y(b)o(y)h(the)g(univ)o(ersal)f(prop)q(ert)o(y)h(giv)o(en)f(in)h(3.5.)p 866 2412 V 866 2441 2 30 v 897 2441 V 866 2443 33 2 v 50 2541 a(W)l(e)d(remark,)f(that)i(w)o(e)g(only)f(needed)g(a)h(righ)o (t)f(adjoin)o(t)h(functor)g([)p Fm(A;)8 b Fp(-)o(])13 b(for)h(-)6 b Fo(\012)g Fm(A)13 b Fp(in)g(the)h(pro)q(of.)p eop %%Page: 10 10 10 9 bop 0 92 a Ft(10)729 b(Bo)q(do)14 b(P)o(areigis)276 200 y Fp(4.)27 b Fk(Examples)17 b(of)h(suit)m(able)f(braided)i(base)f (ca)m(tegories)50 284 y Fp(W)l(e)13 b(close)g(with)h(some)e(sp)q(ecial) h(examples)f(of)i(braided)f(monoidal)g(categories)g Fo(C)k Fp(\(co)q(complete)0 342 y(suc)o(h)e(that)h(the)g(tensor)g(pro)q(duct)g (preserv)o(es)f(arbitrary)g(colimits\),)e(that)j(ma)o(y)e(b)q(e)i(used) g(as)g(base)0 401 y(categories.)35 b(One)20 b(sp)q(ecial)g(example)f (is)i(the)f(category)h(of)g(Y)l(etter-Drinfel'd)e(mo)q(dules)g Fo(Y)t Fm(D)1760 380 y Fl(H)1760 413 y(H)0 459 y Fp(o)o(v)o(er)e(a)h (Hopf)f(algebra)h Fm(H)t Fp(.)26 b(By)18 b([14])f(it)g(can)h(b)q(e)g (view)o(ed)e(as)j(a)f(category)f(of)h(como)q(dules)f(o)o(v)o(er)g(a)0 517 y(co)q(quasitriangular)g(Hopf)f(algebra.)50 576 y(F)l(or)i(an)h (arbitrary)f(co)q(comm)o(utativ)n(e)d(Hopf)j(algebra)h Fm(H)j Fp(suc)o(h)c(a)g(category)g(can)h(also)f(b)q(e)h(ob-)0 634 y(tained)k(in)h(the)f(follo)o(wing)g(w)o(a)o(y)l(.)43 b(Consider)23 b(the)h(category)g(of)f(righ)o(t)h Fm(H)t Fp(-mo)q(dules)f(whic)o(h)f(is)0 692 y(a)e(co)q(complete)d(symmetric)f (monoidal)j(category)g(suc)o(h)g(that)h(the)f(tensor)h(pro)q(duct)g (preserv)o(es)0 750 y(arbitrary)c(colimits.)50 810 y(Observ)o(e)f(that) i Fm(H)j Fp(acts)d(on)g(itself)e(b)o(y)g(the)h(adjoin)o(t)h(action)444 899 y Fm(\013)e Fp(:)e Fm(H)i Fo(\012)c Fm(H)18 b Fo(3)c Fm(h)d Fo(\012)g Fm(k)16 b Fo(7!)921 858 y Fh(X)989 899 y Fm(S)s Fp(\()p Fm(k)1066 907 y Fg(\(1\))1113 899 y Fp(\))p Fm(hk)1185 907 y Fg(\(2\))1247 899 y Fo(2)e Fm(H)q(:)0 989 y(H)21 b Fp(is)c(a)h(righ)o(t)f Fm(H)t Fp(-mo)q(dule)f(Hopf)h (algebra)h(b)o(y)e(the)h(adjoin)o(t)g(action)h(as)f(can)h(b)q(e)f (easily)f(c)o(hec)o(k)o(ed.)0 1047 y(Th)o(us)d Fm(H)k Fp(is)12 b(a)h(Hopf)g(algebra)g(in)f(the)g(category)h Fo(M)928 1054 y Fl(H)974 1047 y Fp(and)g(the)g(category)f Fo(M)1398 1026 y Fl(H)1427 1030 y Fc(\017)1398 1059 y Fl(H)1460 1047 y Fp(of)h Fm(H)t Fp(-como)q(dules)0 1105 y(in)j Fo(M)117 1112 y Fl(H)167 1105 y Fp(is)g(a)g(monoidal)g(category) l(.)21 b(A)16 b Fn(K)t Fp(-mo)q(dule)i Fm(M)k Fp(is)16 b(in)f Fo(M)1193 1084 y Fl(H)1222 1088 y Fc(\017)1193 1117 y Fl(H)1259 1105 y Fp(i\013)50 1164 y(a\))h(it)g(is)g(a)h(righ)o (t)f Fm(H)t Fp(-mo)q(dule)f Fm(\032)f Fp(:)g Fm(M)i Fo(\012)11 b Fm(H)18 b Fo(\000)-8 b(!)13 b Fm(M)22 b Fp(,)50 1224 y(b\))16 b(it)g(is)g(a)h(righ)o(t)e Fm(H)t Fp(-como)q(dule)h Fm(\016)f Fp(:)f Fm(M)19 b Fo(\000)-9 b(!)14 b Fm(M)j Fo(\012)10 b Fm(H)t Fp(,)16 b(and)50 1283 y(c\))11 b Fm(\016)r Fp(\()p Fm(mh)p Fp(\))i(=)h Fm(\016)r Fp(\()p Fm(m)p Fp(\))p Fm(h)d Fp(or)i Fm(\016)r Fp(\()p Fm(mh)p Fp(\))g(=)698 1250 y Fh(P)750 1283 y Fm(m)793 1291 y Fg(\(0\))840 1283 y Fm(h)868 1291 y Fg(\(1\))917 1283 y Fo(\012)r Fm(m)1001 1291 y Fg(\(1\))1051 1283 y Fo(\001)r Fm(h)1095 1291 y Fg(\(2\))1156 1283 y Fp(=)1208 1250 y Fh(P)1260 1283 y Fm(m)1303 1291 y Fg(\(0\))1350 1283 y Fm(h)1378 1291 y Fg(\(1\))1427 1283 y Fo(\012)r Fm(S)s Fp(\()p Fm(h)1548 1291 y Fg(\(2\))1596 1283 y Fp(\))p Fm(m)1658 1291 y Fg(\(1\))1704 1283 y Fm(h)1732 1291 y Fg(\(3\))1780 1283 y Fm(:)0 1341 y Fp(In)j(view)f(of)i(the)f(co)q (comm)o(utativit)n(y)d(of)k Fm(H)j Fp(the)c(last)h(condition)f(is)g (equiv)m(alen)o(t)f(to)457 1389 y Fh(X)517 1431 y Fp(\()p Fm(mh)607 1438 y Fg(1)626 1431 y Fp(\))645 1438 y Fg(0)676 1431 y Fo(\012)c Fm(h)754 1438 y Fg(2)773 1431 y Fp(\()p Fm(mh)863 1438 y Fg(1)883 1431 y Fp(\))902 1438 y Fg(1)935 1431 y Fp(=)987 1389 y Fh(X)1055 1431 y Fm(m)1098 1438 y Fg(0)1118 1431 y Fm(h)1146 1438 y Fg(1)1177 1431 y Fo(\012)f Fm(m)1269 1438 y Fg(1)1289 1431 y Fm(h)1317 1438 y Fg(2)0 1531 y Fp(whic)o(h)h(is)h(the)f(Y)l(etter-Drinfel'd)f (condition)i([13].)19 b(Th)o(us)12 b Fo(M)1106 1510 y Fl(H)1135 1514 y Fc(\017)1106 1543 y Fl(H)1169 1531 y Fp(=)i Fo(Y)t Fm(D)1300 1510 y Fl(H)1300 1543 y(H)1346 1531 y Fp(is)d(a)h(braided)g(monoidal)0 1589 y(category)17 b([2,)f(15)q(],)f(the)h(braiding)g(morphism)e(b)q(eing)j(de\014ned)f(b) o(y)370 1683 y Fm(\033)g Fp(:)d Fm(M)j Fo(\012)11 b Fm(N)19 b Fo(3)14 b Fm(m)d Fo(\012)g Fm(n)j Fo(7!)869 1642 y Fh(X)938 1683 y Fm(n)967 1691 y Fg(\(0\))1025 1683 y Fo(\012)d Fm(mn)1147 1691 y Fg(\(1\))1208 1683 y Fo(2)j Fm(N)i Fo(\012)11 b Fm(M)r(:)0 1784 y Fp(Ob)o(viously)17 b Fo(M)291 1762 y Fl(H)320 1766 y Fc(\017)291 1796 y Fl(H)359 1784 y Fp(is)h(not)h(symmet)o(ric)c(since)860 1750 y Fh(P)912 1784 y Fm(m)955 1791 y Fg(\(0\))1002 1784 y Fm(n)1031 1791 y Fg(\(1\))1090 1784 y Fo(\012)e Fm(n)1171 1791 y Fg(\(0\))1218 1784 y Fm(m)1261 1791 y Fg(\(1\))1308 1784 y Fm(n)1337 1791 y Fg(\(2\))1401 1784 y Fp(=)18 b Fm(m)12 b Fo(\012)g Fm(n)19 b Fp(do)q(es)g(not)0 1842 y(hold)d(in)g(general.)50 1901 y(A)e(sp)q(ecial)h(case)g(for)g (this)g(structure)g(is)g(obtained)g(b)o(y)g(c)o(ho)q(osing)g Fm(H)k Fp(=)13 b Fn(K)t Fm(G)q Fp(,)k(a)f(group)g(algebra)0 1959 y(with)24 b(a)h(\014nite)e(group)i Fm(G)p Fp(.)46 b(In)24 b(this)g(situation)h(the)f(name)f(of)h(dyslectic)f(algebra)h(w) o(as)h(\014rst)0 2017 y(in)o(tro)q(duced)16 b(b)o(y)g(Haran)g([4].)50 2076 y(No)o(w)f(w)o(e)g(can)h(giv)o(e)f(an)h(example)d(of)j(a)g(mo)q (dule)f(whic)o(h)f(is)i(not)g(dyslectic.)j(Let)d Fm(G)e Fp(=)g Fm(C)1661 2083 y Fg(2)1690 2076 y Fo(\002)c Fm(C)1774 2083 y Fg(2)0 2135 y Fp(b)q(e)22 b(the)f(Klein)f(four)i(group.)37 b(Let)22 b(c)o(har\()p Fn(K)s Fp(\))k Fo(6)p Fp(=)c(2.)37 b(Then)21 b(the)h(Hopf)f(algebra)h Fm(H)27 b Fp(:=)22 b Fn(K)s Fm(G)k Fp(=)0 2193 y Fn(K)t Fp([)p Fm(s;)8 b(t)p Fp(])p Fm(=)p Fp(\()p Fm(s)193 2175 y Fg(2)220 2193 y Fo(\000)d Fp(1)p Fm(;)j(t)328 2175 y Fg(2)353 2193 y Fo(\000)d Fp(1\))14 b(is)g(co)q(quasitriangular)g(with)f Fm(r)i Fp(:)f Fm(H)c Fo(\012)5 b Fm(H)18 b Fo(\000)-9 b(!)14 b Fn(K)20 b Fp(de\014ned)14 b(b)o(y)f Fm(r)q Fp(\()p Fm(s)5 b Fo(\012)g Fm(s)p Fp(\))14 b(=)0 2251 y Fm(r)q Fp(\()p Fm(t)8 b Fo(\012)g Fm(s)p Fp(\))14 b(=)f Fm(r)q Fp(\()p Fm(t)8 b Fo(\012)g Fm(t)p Fp(\))14 b(=)f(1,)i Fm(r)q Fp(\()p Fm(s)8 b Fo(\012)g Fm(t)p Fp(\))14 b(=)g Fo(\000)p Fp(1.)21 b(Observ)o(e)14 b(that)h Fm(r)q Fp(\()p Fm(st)8 b Fo(\012)g Fm(t)p Fp(\))13 b(=)h Fm(r)q Fp(\()p Fm(s)8 b Fo(\012)g Fm(t)p Fp(\))p Fm(r)q Fp(\()p Fm(t)g Fo(\012)g Fm(t)p Fp(\))13 b(=)h Fo(\000)p Fp(1.)0 2309 y(Let)j Fm(A)d Fp(=)g Fn(K)t Fp(1)h Fo(\010)c Fn(K)s Fm(x)20 b Fp(with)c Fm(\016)r Fp(\(1\))e(=)h(1)c Fo(\012)g Fp(1,)17 b Fm(\016)r Fp(\()p Fm(x)p Fp(\))d(=)g Fm(x)d Fo(\012)g Fm(t)p Fp(.)22 b(Then)17 b Fm(A)f Fp(b)q(ecomes)f(a)i(comm)o (utativ)o(e)0 2367 y(algebra)h(in)f Fo(M)290 2349 y Fl(H)340 2367 y Fp(b)o(y)g Fm(x)437 2349 y Fg(2)472 2367 y Fp(=)e(1.)25 b(Let)17 b Fm(M)k Fp(=)16 b Fn(K)s Fm(y)h Fo(\010)11 b Fn(K)t Fm(z)22 b Fp(with)17 b Fm(\016)r Fp(\()p Fm(y)r Fp(\))e(=)g Fm(y)e Fo(\012)f Fm(s)17 b Fp(and)h Fm(\016)r Fp(\()p Fm(z)r Fp(\))d(=)g Fm(z)f Fo(\012)d Fm(st)p Fp(.)0 2425 y(Then)17 b Fm(M)23 b 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