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y(Consider)h(the)g(next)h(diagram.)149 4060 y(Square)g(\(1\))e (comm)m(utes)h(since)g Fq(M)43 b Fv(and)33 b Fq(N)43 b Fv(are)32 b(como)s(dules.)149 4176 y(Squares)j(\(2\))f(and)f(\(3\))g (comm)m(ute)g(since)h Fq(\034)41 b Fv(:)30 b Fq(M)j Fs(\012)24 b Fq(N)40 b Fs(\000)-60 b(!)29 b Fq(N)k Fs(\012)24 b Fq(M)44 b Fv(for)33 b Fp(K)17 b Fv(-mo)s(dules)39 b Fq(M)44 b Fv(and)0 4293 y Fq(N)f Fv(is)32 b(a)g(natural)g(transformation.)149 4409 y(Square)i(\(4\))e(represen)m(ts)j(an)e(in)m(teresting)f(prop)s (ert)m(y)h(of)f Fq(\034)44 b Fv(namely)466 4636 y(\(1)22 b Fs(\012)h Fv(1)e Fs(\012)i Fq(\034)11 b Fv(\)\()p Fq(\034)1016 4651 y Fn(B)s Fk(\012)p Fn(M)s(;B)1302 4636 y Fs(\012)22 b Fv(1\))28 b(=)f(\(1)22 b Fs(\012)h Fv(1)e Fs(\012)i Fq(\034)11 b Fv(\)\()p Fq(\034)34 b Fs(\012)23 b Fv(1)f Fs(\012)g Fv(1\)\(1)g Fs(\012)g Fq(\034)34 b Fs(\012)23 b Fv(1\))k(=)518 4752 y(\()p Fq(\034)33 b Fs(\012)23 b Fv(1)f Fs(\012)h Fv(1\)\(1)e Fs(\012)i Fv(1)f Fs(\012)g Fq(\034)11 b Fv(\)\(1)23 b Fs(\012)f 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4225 y Fq(B)k Fs(\012)c Fq(N)1395 4189 y Fk(0)107 4176 y Fq(M)211 4140 y Fk(0)257 4176 y Fs(\012)f Fq(N)444 4140 y Fk(0)p 497 4146 534 4 v 947 4144 a Fi(-)702 4117 y Fn(\016)r Fk(\012)p Fn(\016)2058 4125 y Fq(B)28 b Fs(\012)22 b Fq(B)5 b Fs(\012)2056 4225 y Fq(M)2160 4189 y Fk(0)2207 4225 y Fs(\012)22 b Fq(N)2394 4189 y Fk(0)p 1494 4146 536 4 v 1947 4144 a Fi(-)1652 4118 y Fl(1)p Fk(\012)p Fn(\034)8 b Fk(\012)p Fl(1)2931 4176 y Fq(B)27 b Fs(\012)c Fq(M)3236 4140 y Fk(0)3282 4176 y Fs(\012)g Fq(N)3470 4140 y Fk(0)p 2445 4146 458 4 v 2819 4144 a Fi(-)2553 4118 y Fk(r\012)p Fl(1)p Fk(\012)p Fl(1)0 4362 y Fv(comm)m(utes.)43 b(Th)m(us)35 b Fq(f)d Fs(\012)23 b Fq(g)36 b Fv(is)c(a)g(homomorphism)e(of)i Fq(B)5 b Fv(-como)s(dules.)p 3529 4362 4 66 v 3533 4300 59 4 v 3533 4362 V 3591 4362 4 66 v 149 4543 a Ft(Corollary)36 b(3.1.5.)50 b Fo(L)-5 b(et)41 b Fq(B)47 b Fo(b)-5 b(e)41 b(a)g(bialgebr)-5 b(a.)63 b(Then)40 b Fs(\012)g Fv(:)g Fq(B)5 b Fo(-)p Ft(Como)s(d)27 b Fs(\002)g Fq(B)5 b Fo(-)p 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Fs(\012)g Fq(M)3308 5114 y Fl(3)3348 5099 y Fv(\))37 b Fo(with)267 5216 y Fq(\013)q Fv(\(\()p Fq(m)27 b Fs(\012)h Fq(n)p Fv(\))g Fs(\012)f Fq(p)p Fv(\))41 b(=)g Fq(m)28 b Fs(\012)f Fv(\()p Fq(n)h Fs(\012)g Fq(p)p Fv(\))42 b Fo(is)f(a)h(natur)-5 b(al)42 b(tr)-5 b(ansformation)41 b(fr)-5 b(om)42 b(the)g(functor)p eop %%Page: 73 5 73 4 bop 916 236 a Fr(1.)77 b(REPRESENT)-6 b(A)g(TIONS)31 b(OF)i(HOPF)g(ALGEBRAS)846 b(73)267 425 y Fs(\012)26 b(\016)f Fv(\()p Fs(\012)h(\002)g Fv(Id\))39 b Fo(to)h(the)f(functor)g Fs(\012)26 b(\016)f Fv(\(Id)17 b Fs(\002\012)p Fv(\))40 b Fo(in)f(the)g(variables)g Fq(M)2835 440 y Fl(1)2874 425 y Fo(,)i Fq(M)3039 440 y Fl(2)3078 425 y Fo(,)g(and)d Fq(M)3436 440 y Fl(3)3515 425 y Fo(in)267 541 y Fq(B)5 b Fo(-)p Ft(Mo)s(d)q Fo(.)149 658 y Fv(2.)42 b Fo(The)30 b(c)-5 b(ounit)32 b(isomorphisms)d Fq(\025)f Fv(:)f Fp(K)32 b Fs(\012)15 b Fq(M)44 b Fs(\000)-58 b(!)28 b Fq(M)42 b Fo(with)32 b Fq(\025)p Fv(\()p Fq(\024)14 b Fs(\012)g Fq(m)p Fv(\))28 b(=)g Fq(\024m)k 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Fq(M)2286 1270 y Fl(4)p 1087 1232 158 4 v 1162 1230 a Fi(-)898 1146 y Fn(\013)p Fl(\()p Fn(M)1038 1155 y Fh(1)1073 1146 y Fn(;M)1161 1155 y Fh(2)1195 1146 y Fn(;M)1283 1155 y Fh(3)1317 1146 y Fl(\))p Fk(\012)p Fl(1)2541 1255 y Fq(M)2635 1270 y Fl(1)2697 1255 y Fs(\012)h Fv(\(\()p Fq(M)2967 1270 y Fl(2)3029 1255 y Fs(\012)f Fq(M)3222 1270 y Fl(3)3262 1255 y Fv(\))g Fs(\012)h Fq(M)3516 1270 y Fl(4)3556 1255 y Fv(\))p 2355 1232 V 2430 1230 a Fi(-)2132 1146 y Fn(\013)p Fl(\()p Fn(M)2272 1155 y Fh(1)2307 1146 y Fn(;M)2395 1155 y Fh(2)2429 1146 y Fk(\012)p Fn(M)2552 1155 y Fh(3)2586 1146 y Fn(;M)2674 1155 y Fh(4)2708 1146 y Fl(\))p 531 1552 4 254 v 533 1552 a Fi(?)572 1442 y Fn(\013)p Fl(\()p Fn(M)712 1451 y Fh(1)747 1442 y Fk(\012)p Fn(M)870 1451 y Fh(2)904 1442 y Fn(;M)992 1451 y Fh(3)1026 1442 y Fn(;M)1114 1451 y Fh(4)1148 1442 y Fl(\))p 3066 1552 V 3067 1552 a Fi(?)2492 1442 y Fl(1)p Fk(\012)p Fn(\013)p Fl(\()p Fn(M)2722 1451 y Fh(2)2757 1442 y Fn(;M)2845 1451 y Fh(3)2879 1442 y Fn(;M)2967 1451 y Fh(4)3001 1442 y Fl(\))6 1645 y Fv(\()p Fq(M)138 1660 y Fl(1)200 1645 y Fs(\012)g Fq(M)394 1660 y Fl(2)434 1645 y Fv(\))f Fs(\012)g Fv(\()p Fq(M)725 1660 y Fl(3)787 1645 y Fs(\012)h Fq(M)981 1660 y Fl(4)1021 1645 y Fv(\))1482 b Fq(M)2635 1660 y Fl(1)2697 1645 y Fs(\012)23 b Fv(\()p Fq(M)2929 1660 y Fl(2)2991 1645 y Fs(\012)f Fv(\()p Fq(M)3222 1660 y Fl(3)3284 1645 y Fs(\012)h Fq(M)3478 1660 y Fl(4)3518 1645 y Fv(\)\))p 1087 1622 1426 4 v 2430 1620 a Fi(-)1498 1584 y Fn(\013)p Fl(\()p Fn(M)1638 1593 y Fh(1)1673 1584 y Fn(;M)1761 1593 y Fh(2)1795 1584 y Fn(;M)1883 1593 y Fh(3)1917 1584 y Fk(\012)p Fn(M)2040 1593 y Fh(4)2074 1584 y Fl(\))883 1872 y Fv(\()p Fq(M)1015 1887 y Fl(1)1077 1872 y Fs(\012)f Fp(K)c Fv(\))28 b Fs(\012)22 b Fq(M)1507 1887 y Fl(2)2053 1872 y Fq(M)2147 1887 y Fl(1)2209 1872 y Fs(\012)g Fv(\()p Fp(K)46 b Fs(\012)22 b Fq(M)2639 1887 y Fl(2)2679 1872 y Fv(\))p 1576 1849 449 4 v 1942 1847 a Fi(-)1602 1811 y Fn(\013)p Fl(\()p Fn(M)1742 1820 y Fh(1)1777 1811 y Fn(;)p Fe(K)t Fn(;M)1930 1820 y Fh(2)1970 1811 y Fl(\))1605 2263 y Fq(M)1699 2278 y Fl(1)1761 2263 y Fs(\012)h Fq(M)1955 2278 y Fl(2)1163 2058 y Fn(\032)p Fl(\()p Fn(M)1294 2067 y Fh(1)1328 2058 y Fl(\))p Fk(\012)p Fl(1)1317 1970 y Fi(Q)1400 2026 y(Q)1483 2081 y(Q)1566 2136 y(Q)1615 2169 y(Q)-83 b(s)2154 2058 y Fl(1)p Fk(\012)p Fn(\025)p Fl(\()p Fn(M)2380 2067 y Fh(2)2415 2058 y Fl(\))2200 1970 y Fi(\021)2117 2026 y(\021)2034 2081 y(\021)1951 2136 y(\021)1902 2169 y(\021)g(+)149 2425 y Fj(Pr)n(oof.)42 b Fv(The)k(homomorphisms)d Fq(\013)q Fv(,)48 b Fq(\025)p Fv(,)f(and)e Fq(\032)h Fv(are)e(already)h(de\014ned)h(in)e(the)i (category)0 2542 y Fp(K)17 b Fv(-)6 b Ft(Mo)s(d)48 b Fv(and)g(satisfy)g(the)g(claimed)e(prop)s(erties.)88 b(So)48 b(w)m(e)h(ha)m(v)m(e)g(to)e(sho)m(w,)53 b(that)47 b(these)i(are)0 2658 y(homomorphisms)30 b(in)i Fq(B)5 b Fv(-)o Ft(Mo)s(d)33 b Fv(and)g(that)f Fp(K)55 b Fv(is)32 b(a)g Fq(B)5 b Fv(-mo)s(dule.)42 b Fp(K)56 b Fv(is)32 b(a)g Fq(B)5 b Fv(-mo)s(dule)30 b(b)m(y)j Fq(")22 b Fs(\012)g Fv(1)3490 2673 y Fe(K)3573 2658 y Fv(:)0 2774 y Fq(B)f Fs(\012)16 b Fp(K)51 b Fs(\000)-60 b(!)28 b Fp(K)18 b Fv(.)48 b(The)30 b(easy)h(v)m(eri\014cation)e(uses)i(the)f(coasso)s (ciativit)m(y)e(and)h(the)h(counital)e(prop)s(ert)m(y)0 2890 y(of)k Fq(B)5 b Fv(.)p 3529 2890 4 66 v 3533 2828 59 4 v 3533 2890 V 3591 2890 4 66 v 149 3082 a(Similarly)29 b(w)m(e)34 b(get)149 3269 y Ft(Prop)s(osition)i(3.1.7.)49 b Fo(L)-5 b(et)35 b Fq(B)40 b Fo(b)-5 b(e)35 b(a)g(bialgebr)-5 b(a.)43 b(Then)34 b(the)h(tensor)g(pr)-5 b(o)g(duct)873 3446 y Fs(\012)28 b Fv(:)g Fq(B)5 b Fo(-)p Ft(Como)s(d)22 b Fs(\002)g Fq(B)5 b Fo(-)p Ft(Como)s(d)28 b Fs(\000)-58 b(!)28 b Fq(B)5 b Fo(-)p Ft(Como)s(d)0 3624 y Fo(satis\014es)34 b(the)h(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)149 3772 y Fv(1.)42 b Fo(The)36 b(asso)-5 b(ciativity)37 b(isomorphism)e Fq(\013)d Fv(:)g(\()p Fq(M)1882 3787 y Fl(1)1945 3772 y Fs(\012)24 b Fq(M)2140 3787 y Fl(2)2180 3772 y Fv(\))g Fs(\012)g Fq(M)2437 3787 y Fl(3)2508 3772 y Fs(\000)-58 b(!)32 b Fq(M)2753 3787 y Fl(1)2817 3772 y Fs(\012)24 b Fv(\()p Fq(M)3050 3787 y Fl(2)3113 3772 y Fs(\012)g Fq(M)3308 3787 y Fl(3)3348 3772 y Fv(\))37 b Fo(with)267 3888 y Fq(\013)q Fv(\(\()p Fq(m)27 b Fs(\012)h Fq(n)p Fv(\))g Fs(\012)f Fq(p)p Fv(\))41 b(=)g Fq(m)28 b Fs(\012)f Fv(\()p Fq(n)h Fs(\012)g Fq(p)p Fv(\))42 b Fo(is)f(a)h(natur)-5 b(al)42 b(tr)-5 b(ansformation)41 b(fr)-5 b(om)42 b(the)g(functor)267 4004 y Fs(\012)26 b(\016)f Fv(\()p Fs(\012)h(\002)g Fv(Id\))39 b Fo(to)h(the)f(functor)g Fs(\012)26 b(\016)f Fv(\(Id)17 b Fs(\002\012)p Fv(\))40 b Fo(in)f(the)g(variables)g Fq(M)2835 4019 y Fl(1)2874 4004 y Fo(,)i Fq(M)3039 4019 y Fl(2)3078 4004 y Fo(,)g(and)d Fq(M)3436 4019 y Fl(3)3515 4004 y Fo(in)267 4120 y Fq(B)5 b Fo(-)p Ft(Como)s(d)p Fo(.)149 4236 y Fv(2.)42 b Fo(The)30 b(c)-5 b(ounit)32 b(isomorphisms)d Fq(\025)f Fv(:)f Fp(K)32 b Fs(\012)15 b Fq(M)44 b Fs(\000)-58 b(!)28 b Fq(M)42 b Fo(with)32 b Fq(\025)p Fv(\()p Fq(\024)14 b Fs(\012)g Fq(m)p Fv(\))28 b(=)g Fq(\024m)k Fo(and)e Fq(\032)e Fv(:)g Fq(M)d Fs(\012)14 b Fp(K)267 4353 y Fs(\000)-58 b(!)35 b Fq(M)49 b Fo(with)38 b Fq(\032)p Fv(\()p Fq(m)26 b Fs(\012)f Fq(\024)p Fv(\))34 b(=)h Fq(\024m)k Fo(ar)-5 b(e)38 b(natur)-5 b(al)38 b(tr)-5 b(ansformations)38 b(in)g(the)h(variable)e Fq(M)49 b Fo(in)267 4469 y Fq(B)5 b Fo(-)p Ft(Como)s(d)35 b Fo(fr)-5 b(om)34 b(the)h(functor)g Fp(K)45 b Fs(\012)23 b Fo(-)35 b(r)-5 b(esp.)44 b(-)22 b Fs(\012)h Fp(K)58 b Fo(to)35 b(the)g(identity)g(functor)g Fv(Id)q Fo(.)149 4585 y Fv(3.)42 b Fo(The)34 b(fol)5 b(lowing)34 b(diagr)-5 b(ams)33 b(of)i(natur)-5 b(al)35 b(tr)-5 b(ansformations)34 b(ar)-5 b(e)34 b(c)-5 b(ommutative)6 4834 y Fv(\(\()p Fq(M)176 4849 y Fl(1)238 4834 y Fs(\012)23 b Fq(M)432 4849 y Fl(2)472 4834 y Fv(\))f Fs(\012)g Fq(M)725 4849 y Fl(3)765 4834 y Fv(\))g Fs(\012)h Fq(M)1019 4849 y Fl(4)1274 4834 y Fv(\()p Fq(M)1406 4849 y Fl(1)1468 4834 y Fs(\012)f Fv(\()p Fq(M)1699 4849 y Fl(2)1761 4834 y Fs(\012)h Fq(M)1955 4849 y Fl(3)1995 4834 y Fv(\)\))f Fs(\012)g Fq(M)2286 4849 y Fl(4)p 1087 4811 158 4 v 1162 4809 a Fi(-)898 4724 y Fn(\013)p Fl(\()p Fn(M)1038 4733 y Fh(1)1073 4724 y Fn(;M)1161 4733 y Fh(2)1195 4724 y Fn(;M)1283 4733 y Fh(3)1317 4724 y Fl(\))p Fk(\012)p Fl(1)2541 4834 y Fq(M)2635 4849 y Fl(1)2697 4834 y Fs(\012)h Fv(\(\()p Fq(M)2967 4849 y Fl(2)3029 4834 y Fs(\012)f Fq(M)3222 4849 y Fl(3)3262 4834 y Fv(\))g Fs(\012)h Fq(M)3516 4849 y Fl(4)3556 4834 y Fv(\))p 2355 4811 V 2430 4809 a Fi(-)2132 4724 y Fn(\013)p Fl(\()p Fn(M)2272 4733 y Fh(1)2307 4724 y Fn(;M)2395 4733 y Fh(2)2429 4724 y Fk(\012)p Fn(M)2552 4733 y Fh(3)2586 4724 y Fn(;M)2674 4733 y Fh(4)2708 4724 y Fl(\))p 531 5131 4 254 v 533 5131 a Fi(?)572 5021 y Fn(\013)p Fl(\()p Fn(M)712 5030 y Fh(1)747 5021 y Fk(\012)p Fn(M)870 5030 y Fh(2)904 5021 y Fn(;M)992 5030 y Fh(3)1026 5021 y Fn(;M)1114 5030 y Fh(4)1148 5021 y Fl(\))p 3066 5131 V 3067 5131 a Fi(?)2492 5021 y Fl(1)p Fk(\012)p Fn(\013)p Fl(\()p Fn(M)2722 5030 y Fh(2)2757 5021 y Fn(;M)2845 5030 y Fh(3)2879 5021 y Fn(;M)2967 5030 y Fh(4)3001 5021 y Fl(\))6 5224 y Fv(\()p Fq(M)138 5239 y Fl(1)200 5224 y Fs(\012)g Fq(M)394 5239 y Fl(2)434 5224 y Fv(\))f Fs(\012)g Fv(\()p Fq(M)725 5239 y Fl(3)787 5224 y Fs(\012)h Fq(M)981 5239 y Fl(4)1021 5224 y Fv(\))1482 b Fq(M)2635 5239 y Fl(1)2697 5224 y Fs(\012)23 b Fv(\()p Fq(M)2929 5239 y Fl(2)2991 5224 y Fs(\012)f Fv(\()p Fq(M)3222 5239 y Fl(3)3284 5224 y Fs(\012)h Fq(M)3478 5239 y Fl(4)3518 5224 y Fv(\)\))p 1087 5201 1426 4 v 2430 5199 a Fi(-)1498 5163 y Fn(\013)p Fl(\()p Fn(M)1638 5172 y Fh(1)1673 5163 y Fn(;M)1761 5172 y Fh(2)1795 5163 y Fn(;M)1883 5172 y Fh(3)1917 5163 y Fk(\012)p Fn(M)2040 5172 y Fh(4)2074 5163 y Fl(\))p eop %%Page: 74 6 74 5 bop 0 236 a Fr(74)228 b(3.)35 b(REPRESENT)-6 b(A)g(TION)31 b(THEOR)-6 b(Y,)32 b(RECONSTR)n(UCTION)g(AND)g(T)-6 b(ANNAKA)31 b(DUALITY)883 453 y Fv(\()p Fq(M)1015 468 y Fl(1)1077 453 y Fs(\012)22 b Fp(K)c Fv(\))28 b Fs(\012)22 b Fq(M)1507 468 y Fl(2)2053 453 y Fq(M)2147 468 y Fl(1)2209 453 y Fs(\012)g Fv(\()p Fp(K)46 b Fs(\012)22 b Fq(M)2639 468 y Fl(2)2679 453 y Fv(\))p 1576 430 449 4 v 1942 428 a Fi(-)1602 392 y Fn(\013)p Fl(\()p Fn(M)1742 401 y Fh(1)1777 392 y Fn(;)p Fe(K)t Fn(;M)1930 401 y Fh(2)1970 392 y Fl(\))1605 845 y Fq(M)1699 860 y Fl(1)1761 845 y Fs(\012)h Fq(M)1955 860 y Fl(2)1163 640 y Fn(\032)p Fl(\()p Fn(M)1294 649 y Fh(1)1328 640 y Fl(\))p Fk(\012)p Fl(1)1317 552 y Fi(Q)1400 607 y(Q)1483 662 y(Q)1566 718 y(Q)1615 750 y(Q)-83 b(s)2154 640 y Fl(1)p Fk(\012)p Fn(\025)p Fl(\()p Fn(M)2380 649 y Fh(2)2415 640 y Fl(\))2200 552 y Fi(\021)2117 607 y(\021)2034 662 y(\021)1951 718 y(\021)1902 750 y(\021)g(+)149 1007 y Ft(Remark)37 b(3.1.8.)50 b Fv(W)-8 b(e)40 b(no)m(w)g(get)g(some) g(simple)e(prop)s(erties)i(of)g(the)g(underlying)f(functors)0 1123 y Fs(U)f Fv(:)28 b Fq(B)5 b Fv(-)o Ft(Mo)s(d)29 b Fs(\000)-60 b(!)27 b Fp(K)18 b Fv(-)5 b Ft(Mo)s(d)34 b Fv(resp.)44 b Fs(U)38 b Fv(:)28 b Fq(B)5 b Fv(-)o Ft(Como)s(d)28 b Fs(\000)-60 b(!)28 b Fp(K)17 b Fv(-)6 b Ft(Mo)s(d)33 b Fv(that)f(are)h(easily)f(v)m(eri\014ed.)1121 1295 y Fs(U)10 b Fv(\()p Fq(M)33 b Fs(\012)23 b Fq(N)10 b Fv(\))28 b(=)g Fs(U)10 b Fv(\()p Fq(M)g Fv(\))23 b Fs(\012)f(U)10 b Fv(\()p Fq(N)g Fv(\))p Fq(;)1121 1411 y Fs(U)g Fv(\()p Fq(f)33 b Fs(\012)23 b Fq(g)t Fv(\))k(=)g Fq(f)33 b Fs(\012)23 b Fq(g)t(;)1121 1527 y Fs(U)10 b Fv(\()p Fp(K)18 b Fv(\))34 b(=)27 b Fp(K)18 b Fq(;)1121 1644 y Fs(U)10 b Fv(\()p Fq(\013)q Fv(\))28 b(=)f Fq(\013)q(;)49 b Fs(U)10 b Fv(\()p Fq(\025)p Fv(\))28 b(=)f Fq(\025;)50 b Fs(U)10 b Fv(\()p Fq(\032)p Fv(\))28 b(=)f Fq(\032:)149 1827 y Ft(Problem)36 b(3.1.1.)49 b Fv(W)-8 b(e)28 b(ha)m(v)m(e)g(seen)g(that)f(in)f (represen)m(tation)h(theory)h(and)e(in)h(corepresen)m(ta-)0 1943 y(tion)g(theory)h(of)g(quan)m(tum)g(groups)g(suc)m(h)h(as)f Fp(K)18 b Fq(G)p Fv(,)35 b Fq(U)10 b Fv(\()p Fd(g)p Fv(\),)29 b Fq(S)6 b(L)2252 1958 y Fn(q)2290 1943 y Fv(\(2\),)29 b Fq(U)2537 1958 y Fn(q)2575 1943 y Fv(\()p Fq(sl)r Fv(\(2\)\))f(the)g (ordinary)f(ten-)0 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y Fl(\(1\))1034 5183 y Fq(m)19 b Fs(\012)h Fq(f)11 b Fv(\()p Fq(b)p Fv(\))1411 5199 y Fl(\(2\))1505 5183 y Fq(n)28 b Fv(=)1695 5109 y Fm(P)1816 5183 y Fq(f)11 b Fv(\()p Fq(b)1954 5199 y Fl(\(1\))2049 5183 y Fv(\))p Fq(m)19 b Fs(\012)g Fq(f)11 b Fv(\()p Fq(b)2425 5199 y Fl(\(2\))2520 5183 y Fv(\))p Fq(n)28 b Fv(=)2747 5109 y Fm(P)2869 5183 y Fq(b)2910 5199 y Fl(\(1\))3005 5183 y Fq(m)19 b Fs(\012)g Fq(b)3246 5199 y Fl(\(2\))3341 5183 y Fq(n)p Fv(.)p 3529 5183 4 66 v 3533 5121 59 4 v 3533 5183 V 3591 5183 4 66 v eop %%Page: 75 7 75 6 bop 916 236 a Fr(1.)77 b(REPRESENT)-6 b(A)g(TIONS)31 b(OF)i(HOPF)g(ALGEBRAS)846 b(75)149 425 y Ft(Remark)37 b(3.1.11.)50 b Fv(Let)28 b Fq(C)36 b Fv(and)29 b Fq(D)i Fv(b)s(e)e(coalgebras)f(o)m(v)m(er)i(a)e(comm)m(utativ)m(e)g(ring)f Fp(K)18 b Fv(.)48 b(Let)29 b Fq(f)38 b Fv(:)0 541 y Fq(C)e Fs(\000)-60 b(!)29 b Fq(D)36 b Fv(b)s(e)e(a)f(homomorphism)d(of)j (coalgebras.)46 b(Then)34 b(w)m(e)h(ha)m(v)m(e)f(a)f(functor)h Fs(U)3008 556 y Fn(f)3083 541 y Fv(:)29 b 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Fk(\003)1334 3850 y Fq(;)17 b Fv(ev)1474 3865 y Fn(M)1553 3850 y Fv(\))27 b(and)g(\()p Fq(N)1928 3813 y Fk(\003)1968 3850 y Fq(;)17 b Fv(ev)2108 3865 y Fn(N)2175 3850 y Fv(\))27 b(b)s(e)h(left)e(duals)h (for)f Fq(M)38 b Fv(resp.)43 b Fq(N)10 b Fv(.)42 b(F)-8 b(or)0 3966 y(eac)m(h)33 b(morphism)e Fq(f)39 b Fv(:)27 b Fq(M)39 b Fs(\000)-60 b(!)28 b Fq(N)43 b Fv(w)m(e)33 b(de\014ne)h(the)f Fo(tr)-5 b(ansp)g(ose)g(d)34 b(morphism)144 4179 y Fv(\()p Fq(f)241 4138 y Fk(\003)308 4179 y Fv(:)28 b Fq(N)451 4138 y Fk(\003)518 4179 y Fs(\000)-60 b(!)28 b Fq(M)767 4138 y Fk(\003)807 4179 y Fv(\))g(:=)f(\()p Fq(N)1129 4138 y Fk(\003)1197 4120 y Fl(1)p Fk(\012)p Fl(db)1366 4131 y Fg(M)1235 4179 y Fs(\000)-16 b(!)65 b Fq(N)1549 4138 y Fk(\003)1611 4179 y Fs(\012)22 b Fq(M)33 b Fs(\012)23 b Fq(M)2041 4138 y Fk(\003)2109 4119 y Fl(1)p Fk(\012)p Fn(f)7 b Fk(\012)p Fl(1)2139 4179 y Fs(\000)-16 b(!)58 b Fq(N)2446 4138 y Fk(\003)2508 4179 y Fs(\012)23 b Fq(N)32 b Fs(\012)23 b Fq(M)2922 4138 y Fk(\003)2990 4120 y Fl(ev)3059 4131 y Fg(N)3129 4120 y Fk(\012)p Fl(1)3024 4179 y Fs(\000)-16 b(!)62 b Fq(M)3351 4138 y Fk(\003)3391 4179 y Fv(\))p Fq(:)149 4370 y Fv(With)33 b(this)g(de\014nition)f(w)m (e)j(get)e(that)g(the)h(left)e(dual)h(is)g(a)g(con)m(tra)m(v)-5 b(arian)m(t)32 b(functor,)i(since)g(w)m(e)0 4486 y(ha)m(v)m(e)149 4676 y Ft(Lemma)k(3.3.8.)49 b Fo(L)-5 b(et)38 b Fv(\()p Fq(M)1170 4640 y Fk(\003)1210 4676 y Fq(;)17 b Fv(ev)1350 4691 y Fn(M)1429 4676 y Fv(\))p Fo(,)38 b Fv(\()p Fq(N)1661 4640 y Fk(\003)1701 4676 y Fq(;)17 b Fv(ev)1841 4691 y Fn(N)1908 4676 y Fv(\))p Fo(,)38 b(and)f Fv(\()p Fq(P)2321 4640 y Fk(\003)2360 4676 y Fq(;)17 b Fv(ev)2500 4691 y Fn(P)2558 4676 y Fv(\))38 b Fo(b)-5 b(e)37 b(left)h(duals)f(for)h Fq(M)10 b Fo(,)38 b Fq(N)0 4793 y Fo(and)c Fq(P)49 b Fo(r)-5 b(esp)g(e)g(ctively.)149 4909 y(1.)45 b(We)35 b(have)f Fv(\(1)762 4924 y Fn(M)841 4909 y Fv(\))879 4873 y Fk(\003)946 4909 y Fv(=)27 b(1)1098 4924 y Fn(M)1173 4905 y Ff(\003)1213 4909 y Fo(.)149 5025 y(2.)45 b(If)34 b Fq(f)39 b Fv(:)27 b Fq(M)39 b Fs(\000)-58 b(!)28 b Fq(N)46 b Fo(and)34 b Fq(g)d Fv(:)d Fq(N)38 b Fs(\000)-58 b(!)28 b Fq(P)48 b Fo(ar)-5 b(e)35 b(given)f(then)h Fv(\()p Fq(g)t(f)11 b Fv(\))2440 4989 y Fk(\003)2506 5025 y Fv(=)27 b Fq(f)2668 4989 y Fk(\003)2707 5025 y Fq(g)2758 4989 y Fk(\003)2832 5025 y Fo(holds.)149 5216 y Fj(Pr)n(oof.)42 b Fv(1.)i(\(1)741 5231 y Fn(M)819 5216 y Fv(\))857 5179 y Fk(\003)924 5216 y Fv(=)28 b(\(ev)19 b Fs(\012)p Fv(1\)\(1)j Fs(\012)g Fv(1)g Fs(\012)h Fv(1\)\(1)e Fs(\012)i Fv(db\))28 b(=)g(1)2344 5231 y Fn(M)2419 5212 y Ff(\003)2458 5216 y Fv(.)p eop %%Page: 86 18 86 17 bop 0 236 a Fr(86)228 b(3.)35 b(REPRESENT)-6 b(A)g(TION)31 b(THEOR)-6 b(Y,)32 b(RECONSTR)n(UCTION)g(AND)g(T)-6 b(ANNAKA)31 b(DUALITY)149 425 y Fv(2.)44 b(The)33 b(follo)m(wing)d(diagram)g(comm)m (utes)956 706 y Fq(M)456 b(N)33 b Fs(\012)23 b Fq(N)1805 670 y Fk(\003)1867 706 y Fs(\012)f Fq(M)p 1090 678 389 4 v 1396 676 a Fi(-)1165 646 y Fl(db)1243 657 y Fg(N)1313 646 y Fk(\012)p 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595 v 2554 3464 a(?)2593 3185 y Fl(ev)13 b Fk(\012)p Fl(1)p 310 3464 4 351 v 312 3464 a Fi(?)351 3307 y Fl(ev)g Fk(\012)p Fl(1)p 1578 3464 V 1579 3464 a Fi(?)1618 3307 y Fl(ev)h Fk(\012)p Fl(1)240 3562 y Fq(M)344 3526 y Fk(\003)1277 3562 y Fq(N)1365 3526 y Fk(\003)1427 3562 y Fs(\012)23 b Fq(N)32 b Fs(\012)23 b Fq(M)1841 3526 y Fk(\003)p 412 3535 837 4 v 412 3533 a Fi(\033)751 3507 y Fl(1)p Fk(\012)p Fl(ev)2244 3562 y Fq(N)2332 3526 y Fk(\003)2394 3562 y Fs(\012)g Fq(M)33 b Fs(\012)22 b Fq(M)2824 3526 y Fk(\003)p 1910 3535 306 4 v 1910 3533 a Fi(\033)1951 3500 y Fl(1)p Fk(\012)p Fn(f)7 b Fk(\012)p Fl(1)p 1578 2977 4 351 v 1579 2977 a Fi(?)1138 2818 y Fl(1)p Fk(\012)p Fn(g)r Fk(\012)p Fl(1)p Fk(\012)p Fn(f)g Fk(\012)p Fl(1)3465 2835 y Fq(N)3553 2799 y Fk(\003)3102 2454 y Fl(ev)13 b Fk(\012)p Fl(1)2645 2201 y Fi(Z)2728 2263 y(Z)2811 2325 y(Z)2894 2387 y(Z)2977 2450 y(Z)3060 2512 y(Z)3143 2574 y(Z)3226 2636 y(Z)3309 2699 y(Z)3355 2733 y(Z)-83 b(~)3102 3187 y Fl(1)p Fk(\012)p Fl(db)3355 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Fq(M)10 b Fv(.)55 b(Sho)m(w)37 b(that)f(db)e(:)g Fq(I)42 b Fs(\000)-60 b(!)33 b Fq(M)j Fs(\012)25 b Fq(M)3059 4692 y Fk(\003)3135 4729 y Fv(is)36 b(uniquely)0 4845 y(determined)c(b)m(y)i Fq(M)10 b Fv(,)33 b Fq(M)913 4809 y Fk(\003)954 4845 y Fv(,)f(and)h(ev)r(.)43 b(\(Uniqueness)35 b(of)d(the)h(dual)f (basis.\))149 4961 y(5.)56 b(Let)37 b(\()p Fq(M)602 4925 y Fk(\003)642 4961 y Fq(;)17 b Fv(ev)r(\))36 b(b)s(e)h(a)g(left)e(dual) h(for)g Fq(M)10 b Fv(.)57 b(Sho)m(w)38 b(that)e(ev)h(:)e Fq(M)2589 4925 y Fk(\003)2654 4961 y Fs(\012)25 b Fq(M)46 b Fs(\000)-60 b(!)34 b Fq(I)45 b Fv(is)36 b(uniquely)0 5077 y(determined)c(b)m(y)i Fq(M)10 b Fv(,)33 b Fq(M)913 5041 y Fk(\003)954 5077 y Fv(,)f(and)h(db.)p eop %%Page: 87 19 87 18 bop 1421 236 a Fr(3.)77 b(DUAL)32 b(OBJECTS)1353 b(87)149 425 y Ft(Corollary)36 b(3.3.9.)50 b Fo(L)-5 b(et)35 b Fq(M)5 b(;)17 b(N)46 b Fo(have)34 b(the)h(left)h(duals)e Fv(\()p Fq(M)2340 389 y Fk(\003)2381 425 y Fq(;)17 b Fv(ev)2521 440 y Fn(M)2600 425 y Fv(\))35 b 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y Fk(\003)526 5041 y Fo(is)35 b(an)f(algebr)-5 b(a)34 b(with)h(multiplic)-5 b(ation)632 5216 y Fs(r)28 b Fv(:=)f(1)922 5231 y Fn(M)1023 5216 y Fs(\012)c Fv(ev)c Fs(\012)p Fv(1)1362 5231 y Fn(M)1437 5212 y Ff(\003)1504 5216 y Fv(:)28 b Fq(M)33 b Fs(\012)23 b Fq(M)1890 5174 y Fk(\003)1952 5216 y Fs(\012)g Fq(M)33 b Fs(\012)22 b Fq(M)2382 5174 y Fk(\003)2450 5216 y Fs(\000)-58 b(!)28 b Fq(M)33 b Fs(\012)23 b Fq(M)2928 5174 y Fk(\003)p eop %%Page: 90 22 90 21 bop 0 236 a Fr(90)228 b(3.)35 b(REPRESENT)-6 b(A)g(TION)31 b(THEOR)-6 b(Y,)32 b(RECONSTR)n(UCTION)g(AND)g(T)-6 b(ANNAKA)31 b(DUALITY)0 425 y Fo(and)j(unit)1286 560 y Fq(u)27 b Fv(:=)h(db)g(:)g Fq(I)35 b Fs(\000)-58 b(!)28 b Fq(M)33 b Fs(\012)23 b Fq(M)2247 519 y Fk(\003)2287 560 y Fv(;)0 715 y(2.)45 b Fq(M)225 678 y Fk(\003)287 715 y Fs(\012)23 b Fq(M)45 b Fo(is)35 b(a)g(c)-5 b(o)g(algebr)g(a)33 b(with)i(c)-5 b(omultiplic)g(ation)627 890 y Fv(\001)28 b(:=)f(1)915 905 y Fn(M)990 886 y Ff(\003)1052 890 y Fs(\012)c Fv(db)17 b Fs(\012)p 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b(OBJECTS)1353 b(91)0 425 y Fv(Con)m(v)m(ersely)42 b(let)d Fq(g)k Fv(:)d Fq(A)g Fs(\000)-60 b(!)40 b Fq(M)e Fs(\012)27 b Fq(M)1416 389 y Fk(\003)1496 425 y Fv(b)s(e)40 b(an)g(algebra)e(homomorphism)f (and)j(consider)g Fm(e)-55 b Fq(g)43 b Fv(:=)0 541 y(\(1)22 b Fs(\012)h Fv(ev)r(\)\()p Fq(g)j Fs(\012)d Fv(1\))28 b(:)h Fq(A)22 b Fs(\012)h Fq(M)39 b Fs(\000)-60 b(!)29 b Fq(M)k Fs(\012)23 b Fq(M)1530 505 y Fk(\003)1593 541 y Fs(\012)g Fq(M)39 b Fs(\000)-60 b(!)29 b Fq(M)10 b Fv(.)45 b(Then)35 b Fq(M)43 b Fv(b)s(ecomes)34 b(a)f(left)f Fq(A)p Fv(-mo)s(dule)0 658 y(since)42 807 y Fq(A)22 b Fs(\012)g Fq(A)h Fs(\012)f Fq(M)2734 b(A)22 b Fs(\012)h Fq(M)p 565 778 2666 4 v 3148 776 a Fi(-)1823 751 y Fk(r\012)p Fl(1)283 1197 y Fq(A)f Fs(\012)h Fq(M)33 b Fs(\012)22 b Fq(M)808 1161 y Fk(\003)871 1197 y Fs(\012)g Fq(M)240 b(M)33 b Fs(\012)22 b Fq(M)1634 1161 y Fk(\003)1697 1197 y Fs(\012)g Fq(M)33 b Fs(\012)23 b Fq(M)2127 1161 y Fk(\003)2189 1197 y Fs(\012)g Fq(M)p 1104 1168 172 4 v 1193 1166 a 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4347 a Fi(?)1278 4233 y Fl(=)p 2150 4347 V 2152 4347 a Fi(?)1333 4445 y Fp(K)626 b(K)45 b Fs(\012)23 b Fq(B)p 1439 4417 545 4 v 1901 4415 a Fi(-)p 1370 4737 4 254 v 1372 4737 a(?)1292 4624 y Fn(u)p 2150 4737 V 2152 4737 a Fi(?)2191 4628 y Fn(u)p Fk(\012)p Fl(1)1335 4835 y Fq(A)593 b(A)23 b Fs(\012)f Fq(B)5 b(:)p 1437 4807 536 4 v 1890 4805 a Fi(-)1684 4773 y Fn(f)0 4942 y Fv(Hence)34 b Fq(f)k Fv(:)28 b Fq(A)g Fs(\000)-60 b(!)28 b Fq(A)22 b Fs(\012)g Fq(B)38 b Fv(is)32 b(a)h(homomorphism)c(of)j(algebras.)149 5058 y(Th)m(us)j(w)m(e)e(ha)m (v)m(e)h(de\014ned)g(an)f(isomorphism)988 5216 y Fp(K)18 b Ft(Alg)t Fv(\()p Fq(A;)f(A)22 b Fs(\012)h Fq(B)5 b Fv(\))1732 5188 y Fs(\030)1733 5220 y Fv(=)1837 5216 y(Nat)1997 5174 y Fk(\012)2056 5216 y Fv(\()p Fq(!)2155 5231 y Fn(A)2212 5216 y Fq(;)17 b(!)2317 5231 y Fn(A)2395 5216 y Fs(\012)23 b Fq(B)5 b Fv(\))p eop %%Page: 105 37 105 36 bop 791 236 a Fr(7.)35 b(THE)e(QUANTUM)g(MONOID)f(OF)h(A)g (QUANTUM)f(SP)-6 b(A)n(CE)650 b(105)0 425 y Fv(that)33 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Fk(\012)p Fn(\017)p Fk(\012)p Fl(1)p Fk(\012)p Fl(1)2289 1484 y Fi(Q)2372 1539 y(Q)2455 1594 y(Q)2538 1650 y(Q)2587 1682 y(Q)g(s)930 1964 y Fq(B)28 b Fs(\012)22 b Fq(B)1386 b(B)p 1239 1936 1323 4 v 2479 1934 a Fi(-)1871 1915 y Fk(r)930 2745 y Fq(B)28 b Fs(\012)22 b Fq(B)606 b Fp(K)p 1239 2716 545 4 v 1701 2714 a Fi(-)1455 2688 y Fn(\017)p Fk(\012)p Fn(\017)1610 2354 y Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fq(B)1523 2146 y Fn(\016)r Fl(\()p Fn(B)s Fk(\012)p Fn(B)s Fl(\))1207 2044 y Fi(H)1290 2086 y(H)1373 2127 y(H)1456 2169 y(H)1539 2210 y(H)1622 2252 y(H)1631 2256 y(H)-83 b(j)1976 2147 y Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fl(1)1987 2256 y Fi(\010)2070 2215 y(\010)2153 2173 y(\010)2236 2132 y(\010)2319 2090 y(\010)2402 2049 y(\010)2411 2044 y(\010)g(*)p 1069 2646 4 644 v 1070 2646 a(?)996 2346 y Fl(1)1523 2537 y(1)p Fk(\012)p Fl(1)p Fk(\012)p Fn(\017)1631 2434 y Fi(\010)1548 2476 y(\010)1465 2517 y(\010)1382 2559 y(\010)1299 2600 y(\010)1216 2642 y(\010)1207 2646 y(\010)g(\031)2301 2339 y Fn(\017)2479 2086 y Fi(\000)2396 2169 y(\000)2313 2252 y(\000)2230 2335 y(\000)2147 2418 y(\000)2064 2501 y(\000)1981 2584 y(\000)1918 2646 y(\000)g(\011)560 2909 y Fp(K)2285 b Fq(B)p 666 2876 2205 4 v 2788 2874 a Fi(-)1748 2842 y Fn(\021)1239 3685 y Fp(K)46 b Fs(\012)22 b Fq(B)407 b Fp(K)45 b Fs(\012)23 b Fq(B)k Fs(\012)c Fq(B)p 1546 3656 345 4 v 1808 3654 a Fi(-)1602 3618 y Fn(\016)r Fl(\()p Fe(K)6 b Fl(\))q Fk(\012)p Fl(1)2798 3685 y Fq(B)27 b Fs(\012)c Fq(B)p 2426 3656 344 4 v 2687 3654 a Fi(-)2571 3615 y Fk(\030)2571 3635 y Fl(=)559 4074 y Fq(B)1385 b(B)28 b Fs(\012)22 b Fq(B)p 667 4046 1323 4 v 1907 4044 a Fi(-)1264 4012 y Fn(\021)r Fk(\012)p Fl(1)2019 3295 y Fp(K)46 b Fs(\012)22 b Fq(B)1445 3086 y Fn(\016)r Fl(\()p Fe(K)5 b Fl(\))871 2963 y Fi(X)954 2984 y(X)1037 3005 y(X)1120 3026 y(X)1203 3046 y(X)1286 3067 y(X)1370 3088 y(X)1453 3109 y(X)1536 3129 y(X)1619 3150 y(X)1702 3171 y(X)1785 3192 y(X)1802 3196 y(X)-83 b(z)1049 3281 y Fn(\016)r Fl(\()p Fe(K)6 b Fl(\))667 3026 y Fi(@)750 3109 y(@)833 3192 y(@)916 3275 y(@)999 3358 y(@)1082 3441 y(@)1165 3524 y(@)1227 3586 y(@)-83 b(R)2611 3071 y Fk(\030)2611 3092 y Fl(=)2295 3196 y Fi(\010)2378 3155 y(\010)2461 3113 y(\010)2544 3072 y(\010)2627 3030 y(\010)2710 2989 y(\010)2719 2984 y(\010)g(*)1051 3851 y Fk(\030)1051 3872 y Fl(=)1159 3764 y Fi(\010)1076 3806 y(\010)993 3847 y(\010)910 3889 y(\010)827 3930 y(\010)744 3972 y(\010)735 3976 y(\010)g(\031)1831 3864 y Fn(\021)r Fk(\012)p Fl(1)1515 3764 y Fi(H)1598 3806 y(H)1681 3847 y(H)1764 3889 y(H)1847 3930 y(H)1930 3972 y(H)1939 3976 y(H)g(j)2611 3865 y Fl(1)2646 3876 y Fg(B)2719 3764 y Fi(\010)2636 3806 y(\010)2553 3847 y(\010)2470 3889 y(\010)2387 3930 y(\010)2304 3972 y(\010)2295 3976 y(\010)g(\031)p 597 3976 4 1034 v -1780 w(?)522 3467 y Fn(\021)p 2157 3586 4 254 v 2158 3586 a Fi(?)2197 3479 y Fl(1)p Fk(\012)p Fl(\001)p 2157 3976 V 2158 3976 a Fi(?)2197 3851 y Fk(\030)2197 3872 y Fl(=)p 2937 3586 4 644 v 2938 3586 a Fi(?)2977 3287 y Fl(\001)0 4197 y Fv(and)981 4317 y Fp(K)1506 b(K)p 1087 4284 1426 4 v 2430 4282 a Fi(-)1782 4264 y Fl(1)1661 4703 y Fp(K)45 b Fs(\012)23 b Fq(B)1472 4494 y Fn(\016)r Fl(\()p Fe(K)6 b Fl(\))1157 4392 y Fi(H)1240 4434 y(H)1323 4475 y(H)1406 4517 y(H)1489 4558 y(H)1572 4600 y(H)1580 4604 y(H)-83 b(j)2009 4496 y Fl(1)p Fk(\012)p Fn(\017)1936 4604 y Fi(\010)2020 4563 y(\010)2103 4521 y(\010)2186 4480 y(\010)2269 4438 y(\010)2352 4397 y(\010)2360 4392 y(\010)g(*)1312 4680 y Fn(\021)1088 4434 y Fi(@)1171 4517 y(@)1254 4600 y(@)1337 4683 y(@)1420 4766 y(@)1503 4849 y(@)1586 4932 y(@)1649 4994 y(@)g(R)2250 4687 y Fn(\017)1868 4994 y Fi(\000)1951 4911 y(\000)2034 4828 y(\000)2117 4745 y(\000)2200 4662 y(\000)2283 4579 y(\000)2366 4496 y(\000)2429 4434 y(\000)g(\022)1760 5097 y Fq(B)p 1798 4994 4 254 v 1800 4994 a Fi(?)1839 4870 y Fk(\030)1839 4891 y Fl(=)0 5216 y Fv(Hence)34 b Fq(\021)i Fv(and)d Fs(r)f Fv(are)h(coalgebra)e(homomorphisms.)p eop %%Page: 115 47 115 46 bop 980 236 a Fr(8.)35 b(RECONSTR)n(UCTION)d(AND)g Fa(C)t Fr(-CA)-6 b(TEGORIES)838 b(115)149 425 y Fv(T)-8 b(o)38 b(sho)m(w)g(the)g(asso)s(ciativit)m(y)e(of)h Fs(r)g Fv(w)m(e)i(iden)m(tify)d(along)g(the)i(maps)f Fq(\013)g Fv(:)f(\()p Fq(M)g Fs(\012)26 b Fq(N)10 b Fv(\))25 b Fs(\012)h Fq(P)3523 398 y Fs(\030)3523 429 y Fv(=)0 541 y Fq(M)15 b Fs(\012)t Fv(\()p Fq(N)g Fs(\012)t Fq(P)f Fv(\))24 b(and)g(furthermore)f(simplify)f(the)i(relev)-5 b(an)m(t)23 b(diagram)f(b)m(y)j(\014xing)e(that)h Fq(\033)j Fv(represen)m(ts)0 658 y(a)32 b(suitable)g(p)s(erm)m(utation)f(of)h (the)h(tensor)g(factors.)44 b(Then)34 b(the)f(follo)m(wing)d(comm)m (ute)146 842 y Fq(B)d Fs(\012)c Fq(B)k Fs(\012)c Fq(B)782 b(B)28 b Fs(\012)22 b Fq(B)28 b Fs(\012)22 b Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fq(B)k Fs(\012)c Fq(B)p 655 813 721 4 v 1293 811 a Fi(-)708 776 y Fn(\033)r Fl(\()p Fn(\016)r Fl(\()p Fn(B)s Fl(\))p Fk(\012)p Fn(\016)r Fl(\()p Fn(B)s Fl(\))p Fk(\012)p Fn(\016)r Fl(\()p Fn(B)s Fl(\))2973 842 y Fq(B)k Fs(\012)c Fq(B)k Fs(\012)c Fq(B)p 2517 813 428 4 v 2862 811 a Fi(-)2587 786 y Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fl(1)146 1329 y Fq(B)k Fs(\012)c Fq(B)k Fs(\012)c Fq(B)782 b(B)28 b Fs(\012)22 b Fq(B)28 b Fs(\012)22 b Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fq(B)k Fs(\012)c Fq(B)p 655 1272 721 4 v 1293 1270 a Fi(-)738 1234 y Fn(\033)r Fl(\()p Fn(\016)r Fl(\()p Fn(B)s Fk(\012)p Fn(B)s Fl(\))p Fk(\012)p Fn(\016)r Fl(\()p Fn(B)s Fl(\)\))p 655 1330 V 1293 1328 a Fi(-)738 1398 y Fn(\033)r Fl(\()p Fn(\016)r Fl(\()p Fn(B)s Fl(\))p Fk(\012)p Fn(\016)r Fl(\()p Fn(B)s Fk(\012)p Fn(B)s Fl(\)\))3074 1329 y Fq(B)k Fs(\012)c Fq(B)p 2517 1301 529 4 v 2963 1299 a Fi(-)2638 1273 y Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fl(1)146 1817 y Fq(B)k Fs(\012)c Fq(B)k Fs(\012)c Fq(B)983 b(B)28 b Fs(\012)22 b Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fq(B)p 655 1789 922 4 v 1494 1787 a Fi(-)932 1751 y Fn(\016)r Fl(\()p Fn(B)s Fk(\012)p Fn(B)s Fk(\012)p Fn(B)s Fl(\))3174 1817 y Fq(B)p 2316 1789 830 4 v 3063 1787 a Fi(-)2587 1761 y Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fn(\017)p Fk(\012)p Fl(1)p 385 1231 4 351 v 386 1231 a Fi(?)425 1077 y Fl(1)p 385 1719 V 386 1719 a Fi(?)425 1564 y Fl(1)p 1915 1231 V 1917 1231 a Fi(?)1584 1072 y Fl(1)p Fk(\012)p Fl(\()p Fk(r\012)p Fl(1\))p 1974 1231 V 1975 1231 a Fi(?)2014 1072 y Fl(1)p Fk(\012)p Fl(\(1)p Fk(\012r)p Fl(\))p 3183 1231 V 3184 1231 a Fi(?)2996 1075 y Fk(r\012)p Fl(1)p 3241 1231 V 3243 1231 a Fi(?)3282 1075 y Fl(1)p Fk(\012r)p 1945 1719 V 1946 1719 a Fi(?)1985 1562 y Fl(1)p Fk(\012r)p 3212 1719 V 3214 1719 a Fi(?)3253 1566 y Fk(r)0 1944 y Fv(The)34 b(upp)s(er)f(ro)m(w)g(is)f(the)h(iden)m(tit)m(y)f(hence)i (w)m(e)g(get)e(the)h(asso)s(ciativ)m(e)f(la)m(w.)149 2060 y(F)-8 b(or)35 b(the)g(pro)s(of)f(that)h Fq(\021)j Fv(has)e(the)f(prop)s(erties)g(of)f(a)h(unit)f(w)m(e)i(m)m(ust)f (explicitly)e(consider)j(the)0 2176 y(coherence)43 b(morphisms)c Fq(\025)i Fv(and)f Fq(\032)i Fv(By)f(reasons)g(of)g(symmetry)g(w)m(e)h (will)c(only)i(sho)m(w)i(one)f(half)0 2292 y(of)g(of)g(the)g(unit)g (axiom.)68 b(This)41 b(axiom)f(follo)m(ws)g(from)g(the)i(comm)m (utativit)m(y)d(of)i(the)h(follo)m(wing)0 2409 y(diagram)117 2595 y Fq(B)216 b(B)27 b Fs(\012)c Fq(B)p 225 2566 154 4 v 296 2564 a Fi(-)229 2529 y Fn(\016)r Fl(\()p Fn(B)s Fl(\))892 2595 y Fq(B)28 b Fs(\012)22 b Fq(B)27 b Fs(\012)c Fp(K)p 716 2566 149 4 v 782 2564 a Fi(-)730 2532 y Fn(\032)766 2509 y Ff(\000)p Fh(1)2844 2595 y Fq(B)28 b Fs(\012)22 b Fq(B)p 1400 2566 1416 4 v 2733 2564 a Fi(-)2090 2533 y Fn(\032)3578 2595 y Fq(B)p 3153 2566 397 4 v 3467 2564 a Fi(-)3292 2539 y Fn(\017)p Fk(\012)p Fl(1)18 2985 y Fq(B)27 b Fs(\012)c Fp(K)519 b Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fp(K)45 b Fs(\012)23 b Fq(B)p 325 2956 439 4 v 681 2954 a Fi(-)374 2919 y Fn(\016)r Fl(\()p Fn(B)s Fl(\))p Fk(\012)p Fn(\016)r Fl(\()p Fe(K)8 b Fl(\))1767 2985 y Fq(B)27 b Fs(\012)c Fp(K)45 b Fs(\012)23 b Fq(B)k Fs(\012)c Fq(B)p 1500 2956 238 4 v 1655 2954 a Fi(-)1510 2929 y Fl(1)p Fk(\012)p Fn(\034)8 b Fk(\012)p Fl(1)2744 2985 y Fq(B)27 b Fs(\012)c Fq(B)k Fs(\012)c Fq(B)p 2475 2956 240 4 v 2632 2954 a Fi(-)2487 2923 y Fn(\032)p Fk(\012)p Fl(1)p Fk(\012)p Fl(1)3478 2985 y Fq(B)k Fs(\012)c Fq(B)p 3253 2956 196 4 v 3366 2954 a Fi(-)3246 2929 y Fn(\017)p Fk(\012)p Fl(1)p Fk(\012)p Fl(1)18 3375 y Fq(B)k Fs(\012)c Fp(K)1594 b Fq(B)27 b Fs(\012)c Fp(K)46 b Fs(\012)22 b Fq(B)p 325 3346 1515 4 v 1757 3344 a Fi(-)955 3309 y Fn(\016)r Fl(\()p Fn(B)s Fk(\012)p Fe(K)7 b Fl(\))117 3764 y Fq(B)2653 b(B)28 b Fs(\012)22 b Fq(B)p 225 3737 2591 4 v 2733 3735 a Fi(-)1447 3699 y Fn(\016)r Fl(\()p Fn(B)s Fl(\))3578 3764 y Fq(B)p 3153 3737 397 4 v 3467 3735 a Fi(-)3292 3709 y Fn(\017)p Fk(\012)p Fl(1)p 155 2886 4 254 v 157 2886 a Fi(?)-1 2781 y Fn(\032)35 2758 y Ff(\000)p Fh(1)p 1130 2886 V 1132 2886 a Fi(?)1171 2776 y Fl(1)p Fk(\012)p Fl(1)p Fk(\012)p Fn(\016)r Fl(\()p Fe(K)6 b Fl(\))p 2983 2886 V 2984 2886 a Fi(?)2727 2775 y Fl(1)p Fk(\012)p Fl(1)p Fk(\012)p Fn(\021)p 3616 2886 V 3618 2886 a Fi(?)3451 2775 y Fl(1)p Fk(\012)p Fn(\021)p 155 3276 V 157 3276 a Fi(?)63 3162 y Fl(=)p 2105 3276 V 2107 3276 a Fi(?)2146 3169 y Fl(1)p Fk(\012)p Fl(1)p Fk(\012r)p 2983 3666 4 644 v 2984 3666 a Fi(?)2796 3364 y Fl(1)p Fk(\012r)p 3616 3666 V 3618 3666 a Fi(?)3520 3367 y Fk(r)p 155 3666 4 254 v 157 3666 a Fi(?)82 3547 y Fn(\032)347 2776 y(\016)r Fl(\()p Fn(B)s Fl(\))p Fk(\012)p Fl(1)391 2886 y Fi(\010)474 2845 y(\010)557 2803 y(\010)640 2762 y(\010)723 2720 y(\010)806 2679 y(\010)815 2674 y(\010)-83 b(*)2608 3555 y Fn(\032)p Fk(\012)p Fl(1)2292 3454 y Fi(H)2375 3496 y(H)2458 3537 y(H)2541 3579 y(H)2624 3620 y(H)2707 3662 y(H)2716 3666 y(H)g(j)p 3529 3892 4 66 v 3533 3829 59 4 v 3533 3892 V 3591 3892 4 66 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF