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Fk(B)1155 5099 y Fs(=)27 b(\(1)22 b Fp(\012)h Fn(\034)11 b Fs(\)\()p Fn(\034)34 b Fp(\012)22 b Fs(1\).)149 5216 y(Square)34 b(\(5\))e(and)h(\(6\))f(comm)m(ute)g(b)m (y)h(the)g(prop)s(erties)g(of)f(the)h(tensor)g(pro)s(duct.)p eop %%Page: 72 4 72 3 bop 0 236 a Fo(72)333 b(3.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fs(Square)i(\(7\))e (comm)m(utes)h(since)g Fn(B)k Fs(is)32 b(a)h(bialgebra.)1071 579 y Fn(B)27 b Fp(\012)c Fn(M)10 b Fp(\012)1118 676 y Fn(B)27 b Fp(\012)c Fn(N)130 628 y(M)33 b Fp(\012)22 b Fn(N)p 474 599 569 4 v 960 597 a Ff(-)696 571 y Fk(\016)r Fh(\012)p Fk(\016)2058 579 y Fn(B)28 b Fp(\012)22 b Fn(B)5 b Fp(\012)2080 676 y Fn(M)33 b Fp(\012)22 b Fn(N)p 1483 599 547 4 v 1947 597 a Ff(-)1646 572 y Fi(1)p Fh(\012)p Fk(\034)8 b Fh(\012)p Fi(1)2954 628 y Fn(B)27 b Fp(\012)c Fn(M)33 b Fp(\012)23 b Fn(N)p 2445 599 481 4 v 2843 597 a Ff(-)2565 572 y Fh(r\012)p Fi(1)p Fh(\012)p Fi(1)p 286 1056 4 351 v 287 1056 a Ff(?)326 900 y 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b Fh(\012)p Fi(1)p 1261 1641 V 1262 1641 a Ff(?)1301 1476 y Fi(1)p Fh(\012)p Fk(\034)1422 1487 y Fd(B)r Fc(\012)p Fd(M)r(;B)1651 1476 y Fh(\012)p Fi(1)p Fh(\012)p Fi(1)p 2284 1641 V 2286 1641 a Ff(?)2325 1483 y Fi(1)p Fh(\012)p Fk(\034)g Fh(\012)p Fi(1)p Fh(\012)p Fi(1)p Fh(\012)p Fi(1)86 1749 y Fn(B)28 b Fp(\012)22 b Fn(B)28 b Fp(\012)517 b Fn(B)27 b Fp(\012)c Fn(B)k Fp(\012)c Fn(B)5 b Fp(\012)130 1846 y Fn(M)33 b Fp(\012)22 b Fn(N)570 b(M)33 b Fp(\012)23 b Fn(B)k Fp(\012)c Fn(N)p 495 1769 460 4 v 872 1767 a Ff(-)573 1741 y Fi(1)p Fh(\012)p Fi(1)p Fh(\012)p Fk(\016)r Fh(\012)p Fk(\016)2056 1749 y Fn(B)k Fp(\012)22 b Fn(B)28 b Fp(\012)22 b Fn(B)5 b Fp(\012)2077 1846 y Fn(B)27 b Fp(\012)c Fn(M)33 b Fp(\012)22 b Fn(N)p 1570 1769 457 4 v 1944 1767 a Ff(-)1599 1742 y Fi(1)p Fh(\012)p Fi(1)p Fh(\012)p Fi(1)p Fh(\012)p Fk(\034)8 b Fh(\012)p Fi(1)p 286 2226 4 351 v 287 2226 a Ff(?)326 2070 y Fh(r\012)p Fi(1)p Fh(\012)p Fi(1)p 1261 2226 V 1262 2226 a Ff(?)1301 2070 y Fh(r\012)p Fi(1)p 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Fk(")p Fh(\012)p Fk(")p Fh(\012)p Fi(1)p Fh(\012)p Fi(1)p 3211 3136 4 351 v 3212 3136 a Ff(?)3251 2978 y Fk(")p Fh(\012)p Fi(1)p Fh(\012)p Fi(1)0 3373 y Fs(where)34 b(the)f(last)f(square)h(comm)m(utes)g(since)g Fn(")f Fs(is)g(a)g(homomorphism)e(of)i(algebras.)149 3489 y(No)m(w)i(let)e Fn(f)43 b Fs(and)32 b Fn(g)k Fs(b)s(e)d (homomorphisms)d(of)i Fn(B)5 b Fs(-como)s(dules.)43 b(Then)33 b(the)g(diagram)1071 3637 y Fn(B)27 b Fp(\012)c Fn(M)10 b Fp(\012)1118 3735 y Fn(B)27 b Fp(\012)c Fn(N)130 3686 y(M)33 b Fp(\012)22 b Fn(N)p 474 3658 569 4 v 960 3656 a Ff(-)696 3630 y Fk(\016)r Fh(\012)p Fk(\016)2058 3637 y Fn(B)28 b Fp(\012)22 b Fn(B)5 b Fp(\012)2080 3735 y Fn(M)33 b Fp(\012)22 b Fn(N)p 1483 3658 547 4 v 1947 3656 a Ff(-)1646 3630 y Fi(1)p Fh(\012)p Fk(\034)8 b Fh(\012)p Fi(1)2954 3686 y Fn(B)27 b Fp(\012)c Fn(M)33 b Fp(\012)23 b Fn(N)p 2445 3658 481 4 v 2843 3656 a Ff(-)2565 3630 y Fh(r\012)p Fi(1)p Fh(\012)p Fi(1)p 286 4027 4 254 v 287 4027 a Ff(?)326 3916 y Fk(f)7 b Fh(\012)p Fk(g)p 1261 4027 V 1262 4027 a Ff(?)1301 3916 y Fi(1)p Fh(\012)p Fk(f)g Fh(\012)p Fi(1)p Fh(\012)p Fk(g)p 2236 4027 V 2237 4027 a Ff(?)2276 3916 y Fi(1)p Fh(\012)p Fi(1)p Fh(\012)p Fk(f)g Fh(\012)p Fk(g)p 3211 4027 V 3212 4027 a Ff(?)3251 3916 y Fi(1)p Fh(\012)p Fk(f)g Fh(\012)p Fk(g)1059 4127 y Fn(B)27 b Fp(\012)c Fn(M)1364 4091 y Fh(0)1388 4127 y Fp(\012)1106 4225 y Fn(B)k Fp(\012)c Fn(N)1395 4189 y Fh(0)107 4176 y Fn(M)211 4140 y Fh(0)257 4176 y Fp(\012)f Fn(N)444 4140 y Fh(0)p 497 4146 534 4 v 947 4144 a Ff(-)702 4117 y Fk(\016)r Fh(\012)p Fk(\016)2058 4125 y Fn(B)28 b Fp(\012)22 b Fn(B)5 b Fp(\012)2056 4225 y Fn(M)2160 4189 y Fh(0)2207 4225 y Fp(\012)22 b Fn(N)2394 4189 y Fh(0)p 1494 4146 536 4 v 1947 4144 a Ff(-)1652 4118 y Fi(1)p Fh(\012)p Fk(\034)8 b Fh(\012)p Fi(1)2931 4176 y Fn(B)27 b Fp(\012)c Fn(M)3236 4140 y Fh(0)3282 4176 y Fp(\012)g Fn(N)3470 4140 y Fh(0)p 2445 4146 458 4 v 2819 4144 a Ff(-)2553 4118 y Fh(r\012)p Fi(1)p Fh(\012)p Fi(1)0 4362 y Fs(comm)m(utes.)43 b(Th)m(us)35 b Fn(f)d Fp(\012)23 b Fn(g)36 b Fs(is)c(a)g(homomorphism)e(of)i Fn(B)5 b Fs(-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 Fq(Corollary)36 b(3.1.5.)50 b Fl(L)-5 b(et)41 b Fn(B)47 b Fl(b)-5 b(e)41 b(a)g(bialgebr)-5 b(a.)63 b(Then)40 b Fp(\012)g Fs(:)g Fn(B)5 b Fl(-)p Fq(Como)s(d)27 b Fp(\002)g Fn(B)5 b Fl(-)p Fq(Como)s(d)39 b Fp(\000)-58 b(!)0 4660 y Fn(B)5 b Fl(-)p Fq(Como)s(d)35 b Fl(with)f Fp(\012)p Fs(\()p Fn(M)5 b(;)17 b(N)10 b Fs(\))29 b(=)f Fn(M)k Fp(\012)23 b Fn(N)45 b Fl(and)35 b Fp(\012)p Fs(\()p Fn(f)5 b(;)17 b(g)t Fs(\))27 b(=)h Fn(f)33 b Fp(\012)22 b Fn(g)38 b Fl(is)d(a)g(functor.)149 4841 y Fq(Prop)s(osition)h(3.1.6.)49 b Fl(L)-5 b(et)30 b Fn(B)35 b Fl(b)-5 b(e)29 b(a)g(bialgebr)-5 b(a.)42 b(Then)29 b(the)h(tensor)f(pr)-5 b(o)g(duct)30 b Fp(\012)e Fs(:)f Fn(B)5 b Fl(-)p Fq(Mo)s(d)12 b Fp(\002)0 4957 y Fn(B)5 b Fl(-)p Fq(Mo)s(d)28 b Fp(\000)-58 b(!)29 b Fn(B)5 b Fl(-)p Fq(Mo)s(d)35 b Fl(satis\014es)f(the)h(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)149 5099 y Fs(1.)42 b Fl(The)36 b(asso)-5 b(ciativity)37 b(isomorphism)e Fn(\013)d Fs(:)g(\()p Fn(M)1882 5114 y Fi(1)1945 5099 y Fp(\012)24 b Fn(M)2140 5114 y Fi(2)2180 5099 y Fs(\))g Fp(\012)g Fn(M)2437 5114 y Fi(3)2508 5099 y Fp(\000)-58 b(!)32 b Fn(M)2753 5114 y Fi(1)2817 5099 y Fp(\012)24 b Fs(\()p Fn(M)3050 5114 y Fi(2)3113 5099 y Fp(\012)g Fn(M)3308 5114 y Fi(3)3348 5099 y Fs(\))37 b Fl(with)267 5216 y Fn(\013)q Fs(\(\()p Fn(m)27 b Fp(\012)h Fn(n)p Fs(\))g Fp(\012)f Fn(p)p Fs(\))41 b(=)g Fn(m)28 b Fp(\012)f Fs(\()p Fn(n)h Fp(\012)g Fn(p)p Fs(\))42 b Fl(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 Fo(1.)77 b(REPRESENT)-6 b(A)g(TIONS)31 b(OF)i(HOPF)g(ALGEBRAS)846 b(73)267 425 y Fp(\012)26 b(\016)f Fs(\()p Fp(\012)h(\002)g Fs(Id\))39 b Fl(to)h(the)f(functor)g Fp(\012)26 b(\016)f Fs(\(Id)17 b Fp(\002\012)p Fs(\))40 b Fl(in)f(the)g(variables)g Fn(M)2835 440 y Fi(1)2874 425 y Fl(,)i Fn(M)3039 440 y Fi(2)3078 425 y Fl(,)g(and)d Fn(M)3436 440 y Fi(3)3515 425 y Fl(in)267 541 y Fn(B)5 b Fl(-)p Fq(Mo)s(d)q Fl(.)149 658 y Fs(2.)42 b Fl(The)30 b(c)-5 b(ounit)32 b(isomorphisms)d Fn(\025)f Fs(:)f Fm(K)32 b Fp(\012)15 b Fn(M)44 b Fp(\000)-58 b(!)28 b Fn(M)42 b Fl(with)32 b Fn(\025)p Fs(\()p Fn(\024)14 b Fp(\012)g Fn(m)p Fs(\))28 b(=)g Fn(\024m)k Fl(and)e Fn(\032)e Fs(:)g Fn(M)d Fp(\012)14 b Fm(K)267 774 y Fp(\000)-58 b(!)35 b Fn(M)49 b Fl(with)38 b Fn(\032)p Fs(\()p Fn(m)26 b Fp(\012)f Fn(\024)p Fs(\))34 b(=)h Fn(\024m)k Fl(ar)-5 b(e)38 b(natur)-5 b(al)38 b(tr)-5 b(ansformations)38 b(in)g(the)h(variable)e Fn(M)49 b Fl(in)267 890 y Fn(B)5 b Fl(-)p Fq(Mo)s(d)35 b Fl(fr)-5 b(om)35 b(the)g(functor)g Fm(K)45 b Fp(\012)23 b Fl(-)35 b(r)-5 b(esp.)44 b(-)22 b Fp(\012)h Fm(K)58 b Fl(to)35 b(the)g(identity)g(functor)g Fs(Id)p Fl(.)149 1006 y Fs(3.)42 b Fl(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 1255 y Fs(\(\()p Fn(M)176 1270 y Fi(1)238 1255 y Fp(\012)23 b Fn(M)432 1270 y Fi(2)472 1255 y Fs(\))f Fp(\012)g Fn(M)725 1270 y Fi(3)765 1255 y Fs(\))g Fp(\012)h Fn(M)1019 1270 y Fi(4)1274 1255 y Fs(\()p Fn(M)1406 1270 y Fi(1)1468 1255 y Fp(\012)f Fs(\()p Fn(M)1699 1270 y Fi(2)1761 1255 y Fp(\012)h Fn(M)1955 1270 y Fi(3)1995 1255 y Fs(\)\))f Fp(\012)g Fn(M)2286 1270 y Fi(4)p 1087 1232 158 4 v 1162 1230 a Ff(-)898 1146 y Fk(\013)p Fi(\()p Fk(M)1038 1155 y Fe(1)1073 1146 y Fk(;M)1161 1155 y Fe(2)1195 1146 y Fk(;M)1283 1155 y Fe(3)1317 1146 y Fi(\))p Fh(\012)p Fi(1)2541 1255 y Fn(M)2635 1270 y Fi(1)2697 1255 y Fp(\012)h Fs(\(\()p Fn(M)2967 1270 y Fi(2)3029 1255 y Fp(\012)f Fn(M)3222 1270 y Fi(3)3262 1255 y Fs(\))g Fp(\012)h Fn(M)3516 1270 y Fi(4)3556 1255 y Fs(\))p 2355 1232 V 2430 1230 a Ff(-)2132 1146 y Fk(\013)p Fi(\()p Fk(M)2272 1155 y Fe(1)2307 1146 y Fk(;M)2395 1155 y Fe(2)2429 1146 y Fh(\012)p Fk(M)2552 1155 y Fe(3)2586 1146 y Fk(;M)2674 1155 y Fe(4)2708 1146 y Fi(\))p 531 1552 4 254 v 533 1552 a Ff(?)572 1442 y Fk(\013)p Fi(\()p Fk(M)712 1451 y Fe(1)747 1442 y Fh(\012)p Fk(M)870 1451 y Fe(2)904 1442 y Fk(;M)992 1451 y Fe(3)1026 1442 y Fk(;M)1114 1451 y Fe(4)1148 1442 y Fi(\))p 3066 1552 V 3067 1552 a Ff(?)2492 1442 y Fi(1)p Fh(\012)p Fk(\013)p Fi(\()p Fk(M)2722 1451 y Fe(2)2757 1442 y Fk(;M)2845 1451 y Fe(3)2879 1442 y Fk(;M)2967 1451 y Fe(4)3001 1442 y Fi(\))6 1645 y Fs(\()p Fn(M)138 1660 y Fi(1)200 1645 y Fp(\012)g Fn(M)394 1660 y Fi(2)434 1645 y Fs(\))f Fp(\012)g Fs(\()p Fn(M)725 1660 y Fi(3)787 1645 y Fp(\012)h Fn(M)981 1660 y Fi(4)1021 1645 y Fs(\))1482 b Fn(M)2635 1660 y Fi(1)2697 1645 y Fp(\012)23 b Fs(\()p Fn(M)2929 1660 y Fi(2)2991 1645 y Fp(\012)f Fs(\()p Fn(M)3222 1660 y Fi(3)3284 1645 y Fp(\012)h Fn(M)3478 1660 y Fi(4)3518 1645 y Fs(\)\))p 1087 1622 1426 4 v 2430 1620 a Ff(-)1498 1584 y Fk(\013)p Fi(\()p Fk(M)1638 1593 y Fe(1)1673 1584 y Fk(;M)1761 1593 y Fe(2)1795 1584 y Fk(;M)1883 1593 y Fe(3)1917 1584 y Fh(\012)p Fk(M)2040 1593 y Fe(4)2074 1584 y Fi(\))883 1872 y Fs(\()p Fn(M)1015 1887 y Fi(1)1077 1872 y Fp(\012)f Fm(K)c Fs(\))28 b Fp(\012)22 b Fn(M)1507 1887 y Fi(2)2053 1872 y Fn(M)2147 1887 y Fi(1)2209 1872 y Fp(\012)g Fs(\()p Fm(K)46 b Fp(\012)22 b Fn(M)2639 1887 y Fi(2)2679 1872 y Fs(\))p 1576 1849 449 4 v 1942 1847 a Ff(-)1602 1811 y Fk(\013)p Fi(\()p Fk(M)1742 1820 y Fe(1)1777 1811 y Fk(;)p Fb(K)t Fk(;M)1930 1820 y Fe(2)1970 1811 y Fi(\))1605 2263 y Fn(M)1699 2278 y Fi(1)1761 2263 y Fp(\012)h Fn(M)1955 2278 y Fi(2)1163 2058 y Fk(\032)p Fi(\()p Fk(M)1294 2067 y Fe(1)1328 2058 y Fi(\))p Fh(\012)p Fi(1)1317 1970 y Ff(Q)1400 2026 y(Q)1483 2081 y(Q)1566 2136 y(Q)1615 2169 y(Q)-83 b(s)2154 2058 y Fi(1)p Fh(\012)p Fk(\025)p Fi(\()p Fk(M)2380 2067 y Fe(2)2415 2058 y Fi(\))2200 1970 y Ff(\021)2117 2026 y(\021)2034 2081 y(\021)1951 2136 y(\021)1902 2169 y(\021)g(+)149 2425 y Fg(Pr)n(oof.)42 b Fs(The)k(homomorphisms)d Fn(\013)q Fs(,)48 b Fn(\025)p Fs(,)f(and)e Fn(\032)h Fs(are)e(already)h(de\014ned)h(in)e(the)i (category)0 2542 y Fm(K)17 b Fs(-)6 b Fq(Mo)s(d)48 b Fs(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 Fn(B)5 b Fs(-)o Fq(Mo)s(d)33 b Fs(and)g(that)f Fm(K)55 b Fs(is)32 b(a)g Fn(B)5 b Fs(-mo)s(dule.)42 b Fm(K)56 b Fs(is)32 b(a)g Fn(B)5 b Fs(-mo)s(dule)30 b(b)m(y)j Fn(")22 b Fp(\012)g Fs(1)3490 2673 y Fb(K)3573 2658 y Fs(:)0 2774 y Fn(B)f Fp(\012)16 b Fm(K)51 b Fp(\000)-60 b(!)28 b Fm(K)18 b Fs(.)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 Fn(B)5 b Fs(.)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 Fq(Prop)s(osition)i(3.1.7.)49 b Fl(L)-5 b(et)35 b Fn(B)40 b Fl(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 Fp(\012)28 b Fs(:)g Fn(B)5 b Fl(-)p Fq(Como)s(d)22 b Fp(\002)g Fn(B)5 b Fl(-)p Fq(Como)s(d)28 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y Fi(\))p 531 5131 4 254 v 533 5131 a Ff(?)572 5021 y Fk(\013)p Fi(\()p Fk(M)712 5030 y Fe(1)747 5021 y Fh(\012)p Fk(M)870 5030 y Fe(2)904 5021 y Fk(;M)992 5030 y Fe(3)1026 5021 y Fk(;M)1114 5030 y Fe(4)1148 5021 y Fi(\))p 3066 5131 V 3067 5131 a Ff(?)2492 5021 y Fi(1)p Fh(\012)p Fk(\013)p Fi(\()p Fk(M)2722 5030 y Fe(2)2757 5021 y Fk(;M)2845 5030 y Fe(3)2879 5021 y Fk(;M)2967 5030 y Fe(4)3001 5021 y Fi(\))6 5224 y Fs(\()p Fn(M)138 5239 y Fi(1)200 5224 y Fp(\012)g Fn(M)394 5239 y Fi(2)434 5224 y Fs(\))f Fp(\012)g Fs(\()p Fn(M)725 5239 y Fi(3)787 5224 y Fp(\012)h Fn(M)981 5239 y Fi(4)1021 5224 y Fs(\))1482 b Fn(M)2635 5239 y Fi(1)2697 5224 y Fp(\012)23 b Fs(\()p Fn(M)2929 5239 y Fi(2)2991 5224 y Fp(\012)f Fs(\()p Fn(M)3222 5239 y Fi(3)3284 5224 y Fp(\012)h Fn(M)3478 5239 y Fi(4)3518 5224 y Fs(\)\))p 1087 5201 1426 4 v 2430 5199 a Ff(-)1498 5163 y Fk(\013)p Fi(\()p Fk(M)1638 5172 y Fe(1)1673 5163 y Fk(;M)1761 5172 y Fe(2)1795 5163 y Fk(;M)1883 5172 y Fe(3)1917 5163 y Fh(\012)p Fk(M)2040 5172 y Fe(4)2074 5163 y Fi(\))p eop %%Page: 74 6 74 5 bop 0 236 a Fo(74)333 b(3.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)883 453 y Fs(\()p Fn(M)1015 468 y Fi(1)1077 453 y Fp(\012)22 b Fm(K)c Fs(\))28 b Fp(\012)22 b Fn(M)1507 468 y Fi(2)2053 453 y Fn(M)2147 468 y Fi(1)2209 453 y Fp(\012)g Fs(\()p Fm(K)46 b Fp(\012)22 b Fn(M)2639 468 y Fi(2)2679 453 y Fs(\))p 1576 430 449 4 v 1942 428 a Ff(-)1602 392 y Fk(\013)p Fi(\()p Fk(M)1742 401 y Fe(1)1777 392 y Fk(;)p Fb(K)t Fk(;M)1930 401 y Fe(2)1970 392 y Fi(\))1605 845 y Fn(M)1699 860 y Fi(1)1761 845 y Fp(\012)h Fn(M)1955 860 y Fi(2)1163 640 y Fk(\032)p Fi(\()p Fk(M)1294 649 y Fe(1)1328 640 y Fi(\))p Fh(\012)p Fi(1)1317 552 y Ff(Q)1400 607 y(Q)1483 662 y(Q)1566 718 y(Q)1615 750 y(Q)-83 b(s)2154 640 y Fi(1)p Fh(\012)p Fk(\025)p Fi(\()p Fk(M)2380 649 y Fe(2)2415 640 y Fi(\))2200 552 y Ff(\021)2117 607 y(\021)2034 662 y(\021)1951 718 y(\021)1902 750 y(\021)g(+)149 1007 y Fq(Remark)37 b(3.1.8.)50 b Fs(W)-8 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5082 y Fk(f)1772 5067 y Fs(\()p Fn(M)f Fp(\012)28 b Fn(N)10 b Fs(\))43 b(=)f Fn(M)d Fp(\012)28 b Fn(N)51 b Fs(w)m(e)42 b(ha)m(v)m(e)h Fn(b)p Fs(\()p Fn(m)28 b Fp(\012)h Fn(n)p Fs(\))42 b(=)0 5183 y Fn(f)11 b Fs(\()p Fn(b)p Fs(\)\()p Fn(m)19 b Fp(\012)h Fn(n)p Fs(\))27 b(=)642 5109 y Fj(P)764 5183 y Fn(f)11 b Fs(\()p Fn(b)p Fs(\))940 5199 y Fi(\(1\))1034 5183 y Fn(m)19 b Fp(\012)h Fn(f)11 b Fs(\()p Fn(b)p Fs(\))1411 5199 y Fi(\(2\))1505 5183 y Fn(n)28 b Fs(=)1695 5109 y Fj(P)1816 5183 y Fn(f)11 b Fs(\()p Fn(b)1954 5199 y Fi(\(1\))2049 5183 y Fs(\))p Fn(m)19 b Fp(\012)g Fn(f)11 b Fs(\()p Fn(b)2425 5199 y Fi(\(2\))2520 5183 y Fs(\))p Fn(n)28 b Fs(=)2747 5109 y Fj(P)2869 5183 y Fn(b)2910 5199 y Fi(\(1\))3005 5183 y Fn(m)19 b Fp(\012)g Fn(b)3246 5199 y Fi(\(2\))3341 5183 y Fn(n)p Fs(.)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 Fo(1.)77 b(REPRESENT)-6 b(A)g(TIONS)31 b(OF)i(HOPF)g(ALGEBRAS)846 b(75)149 425 y Fq(Remark)37 b(3.1.11.)50 b Fs(Let)28 b Fn(C)36 b Fs(and)29 b Fn(D)i Fs(b)s(e)e(coalgebras)f(o)m(v)m(er)i(a)e(comm)m(utativ)m(e)g(ring)f Fm(K)18 b Fs(.)48 b(Let)29 b Fn(f)38 b Fs(:)0 541 y Fn(C)e Fp(\000)-60 b(!)29 b Fn(D)36 b Fs(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 Fp(U)3008 556 y Fk(f)3083 541 y Fs(:)29 b Fn(C)7 b Fs(-)o Fq(Como)s(d)0 659 y Fp(\000)-60 b(!)50 b Fn(D)s Fs(-)o Fq(Como)s(d)c Fs(with)f Fp(U)978 674 y Fk(f)1024 659 y Fs(\()1062 623 y Fk(C)1121 659 y Fn(M)10 b Fs(\))51 b(=)1440 623 y Fk(D)1504 659 y Fn(M)57 b Fs(and)46 b Fp(U)1920 674 y Fk(f)1965 659 y Fs(\()p Fn(g)t Fs(\))k(=)g Fn(g)f Fs(where)e Fn(\016)2702 674 y Fk(D)2816 659 y Fs(=)j(\()p Fn(f)42 b Fp(\012)32 b Fs(1\))p Fn(\016)3309 674 y Fk(C)3418 659 y Fs(:)50 b Fn(M)0 775 y Fp(\000)-60 b(!)28 b Fn(C)f Fp(\012)21 b Fn(M)38 b Fp(\000)-60 b(!)28 b Fn(D)23 b Fp(\012)e Fn(M)10 b Fs(.)44 b(Again)30 b(the)i(functor)g Fp(U)1844 790 y Fk(f)1922 775 y Fs(is)f(called)f Fl(for)-5 b(getful)32 b Fs(or)f Fl(underlying)j(functor)p Fs(.)149 892 y(W)-8 b(e)33 b(denote)h(the)f(underlying)f(functors)h(previously)g (discussed)h(b)m(y)467 1049 y Fp(U)529 1064 y Fk(C)616 1049 y Fs(:)28 b Fn(C)7 b Fs(-)o Fq(Como)s(d)28 b Fp(\000)-60 b(!)27 b Fm(K)18 b Fs(-)5 b Fq(Mo)s(d)34 b Fs(resp.)44 b Fp(U)1981 1064 y Fk(D)2073 1049 y Fs(:)28 b Fn(D)s Fs(-)o Fq(Como)s(d)g Fp(\000)-60 b(!)27 b Fm(K)18 b Fs(-)5 b Fq(Mo)s(d)q Fn(:)149 1223 y Fq(Prop)s(osition)36 b(3.1.12.)49 b Fl(L)-5 b(et)41 b Fn(f)50 b Fs(:)39 b Fn(B)k Fp(\000)-58 b(!)40 b Fn(C)47 b Fl(b)-5 b(e)41 b(a)f(homomorphism)f(of)i(bialgebr)-5 b(as.)61 b(Then)0 1339 y Fp(U)62 1354 y Fk(f)135 1339 y Fs(:)28 b Fn(C)7 b Fl(-)p Fq(Como)s(d)27 b Fp(\000)-58 b(!)29 b Fn(D)s Fl(-)o Fq(Como)s(d)35 b Fl(satis\014es)f(the)h(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)1066 1497 y Fp(U)1128 1512 y Fk(f)1173 1497 y Fs(\()p Fn(M)33 b Fp(\012)23 b Fn(N)10 b Fs(\))28 b(=)g Fp(U)1758 1512 y Fk(f)1803 1497 y Fs(\()p Fn(M)10 b Fs(\))23 b Fp(\012)g(U)2168 1512 y Fk(f)2214 1497 y Fs(\()p Fn(N)10 b Fs(\))p Fn(;)1066 1614 y Fp(U)1128 1629 y Fk(f)1173 1614 y Fs(\()p Fn(g)26 b Fp(\012)c Fn(h)p Fs(\))28 b(=)g Fn(g)d Fp(\012)e Fn(h;)1066 1730 y Fp(U)1128 1745 y Fk(f)1173 1730 y Fs(\()p Fm(K)18 b Fs(\))34 b(=)27 b Fm(K)18 b Fn(;)1066 1846 y Fp(U)1128 1861 y Fk(f)1173 1846 y Fs(\()p Fn(\013)q Fs(\))28 b(=)f Fn(\013)q(;)52 b Fp(U)1647 1861 y Fk(f)1692 1846 y Fs(\()p Fn(\025)p Fs(\))28 b(=)f Fn(\025;)52 b Fp(U)2154 1861 y Fk(f)2199 1846 y Fs(\()p Fn(\032)p Fs(\))28 b(=)g Fn(\032;)1066 1962 y Fp(U)1128 1977 y Fk(C)1187 1962 y Fp(U)1249 1977 y Fk(f)1295 1962 y Fs(\()p Fn(M)10 b Fs(\))28 b(=)g Fp(U)1669 1977 y Fk(B)1730 1962 y Fs(\()p Fn(M)10 b Fs(\))p Fn(;)1066 2078 y Fp(U)1128 2093 y Fk(C)1187 2078 y Fp(U)1249 2093 y Fk(f)1295 2078 y Fs(\()p Fn(g)t Fs(\))27 b(=)g Fp(U)1614 2093 y Fk(B)1675 2078 y Fs(\()p Fn(g)t Fs(\))p Fn(:)149 2249 y Fg(Pr)n(oof.)42 b Fs(W)-8 b(e)33 b(lea)m(v)m(e)h(the)f(pro)s(of) e(to)i(the)g(reader.)p 3529 2249 4 66 v 3533 2186 59 4 v 3533 2249 V 3591 2249 4 66 v 149 2423 a Fq(Prop)s(osition)j (3.1.13.)49 b Fl(L)-5 b(et)37 b Fn(H)45 b Fl(b)-5 b(e)36 b(a)h(Hopf)g(algebr)-5 b(a.)49 b(L)-5 b(et)37 b Fn(M)48 b Fl(and)36 b Fn(N)48 b Fl(b)-5 b(e)36 b(b)-5 b(e)36 b Fn(H)8 b Fl(-mo)-5 b(dules.)0 2539 y(Then)43 b Fs(Hom)o(\()p Fn(M)5 b(;)17 b(N)10 b Fs(\))p Fl(,)46 b(the)e(set)g Fm(K)17 b Fl(-line)-5 b(ar)49 b(maps)43 b(fr)-5 b(om)44 b Fn(M)54 b Fl(to)44 b Fn(N)10 b Fl(,)46 b(b)-5 b(e)g(c)g(omes)43 b(an)g Fn(H)8 b Fl(-mo)-5 b(dule)43 b(by)0 2656 y Fs(\()p Fn(hf)11 b Fs(\)\()p Fn(m)p Fs(\))28 b(=)483 2581 y Fj(P)605 2656 y Fn(h)661 2671 y Fi(\(1\))755 2656 y Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(h)1012 2671 y Fi(\(2\))1106 2656 y Fn(m)p Fs(\))p Fl(.)45 b(This)34 b(structur)-5 b(e)36 b(makes)982 2813 y Fs(Hom)27 b(:)h Fn(H)8 b Fl(-)o Fq(Mo)s(d)23 b Fp(\002)g Fn(H)8 b Fl(-)o Fq(Mo)s(d)28 b Fp(\000)-58 b(!)29 b Fn(H)8 b Fl(-)o Fq(Mo)s(d)0 2970 y Fl(a)35 b(functor)g(c)-5 b(ontr)g(avariant)34 b(in)g(the)h(\014rst)g (variable)f(and)h(c)-5 b(ovariant)34 b(in)g(the)h(se)-5 b(c)g(ond)34 b(variable.)149 3144 y Fg(Pr)n(oof.)42 b Fs(The)50 b(main)c(part)i(to)g(b)s(e)h(pro)m(v)m(ed)g(is)f(that)g(the)h (action)e Fn(H)41 b Fp(\012)33 b Fs(Hom)o(\()p Fn(M)5 b(;)17 b(N)10 b Fs(\))55 b Fp(\000)-60 b(!)0 3260 y Fs(Hom)o(\()p Fn(M)5 b(;)17 b(N)10 b Fs(\))48 b(satis\014es)f(the)h(asso)s(ciativit)m (y)d(la)m(w.)87 b(Let)47 b Fn(f)63 b Fp(2)53 b Fs(Hom)o(\()p Fn(M)5 b(;)17 b(N)10 b Fs(\),)51 b Fn(h;)17 b(k)55 b Fp(2)e Fn(H)8 b Fs(,)50 b(and)0 3376 y Fn(m)28 b Fp(2)g Fn(M)10 b Fs(.)43 b(Then)30 b(\(\()p Fn(hk)s Fs(\))p Fn(f)11 b Fs(\)\()p Fn(m)p Fs(\))27 b(=)1245 3302 y Fj(P)1350 3376 y Fs(\()p Fn(hk)s Fs(\))1536 3392 y Fi(\(1\))1630 3376 y Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\(\()p Fn(hk)s Fs(\))2017 3392 y Fi(\(2\))2111 3376 y Fs(\))28 b(=)2280 3302 y Fj(P)2402 3376 y Fn(h)2458 3392 y Fi(\(1\))2553 3376 y Fn(k)2604 3392 y Fi(\(1\))2698 3376 y Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(k)2950 3392 y Fi(\(2\))3043 3376 y Fs(\))p Fn(S)g Fs(\()p Fn(h)3241 3392 y Fi(\(2\))3335 3376 y Fs(\))p Fn(m)p Fs(\))28 b(=)0 3418 y Fj(P)122 3493 y Fn(h)178 3508 y Fi(\(1\))272 3493 y Fs(\()p Fn(k)s(f)11 b Fs(\)\()p Fn(S)6 b Fs(\()p Fn(h)659 3508 y Fi(\(2\))753 3493 y Fs(\))p Fn(m)p Fs(\))28 b(=)f(\()p Fn(h)p Fs(\()p Fn(k)s(f)11 b Fs(\)\)\()p Fn(m)p Fs(\).)149 3609 y(W)-8 b(e)31 b(lea)m(v)m(e)g(the)g(pro)s(of)e(of)h(the)h(other)f(prop)s (erties,)h(in)e(particular)g(the)i(functorial)d(prop)s(erties,)0 3725 y(to)k(the)h(reader.)p 3529 3725 V 3533 3663 59 4 v 3533 3725 V 3591 3725 4 66 v 149 3899 a Fq(Corollary)j(3.1.14.)50 b Fl(L)-5 b(et)51 b Fn(M)62 b Fl(b)-5 b(e)50 b(an)h Fn(H)8 b Fl(-mo)-5 b(dule.)91 b(Then)50 b(the)h(dual)g Fm(K)17 b Fl(-mo)-5 b(dule)57 b Fn(M)3427 3863 y Fh(\003)3524 3899 y Fs(=)0 4016 y(Hom)o(\()p Fn(M)5 b(;)17 b Fm(K)h Fs(\))40 b Fl(b)-5 b(e)g(c)g(omes)34 b(an)h Fn(H)8 b Fl(-mo)-5 b(dule)33 b(by)i Fs(\()p Fn(hf)11 b Fs(\)\()p Fn(m)p Fs(\))28 b(=)f Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(h)p Fs(\))p Fn(m)p Fs(\))p Fl(.)149 4190 y Fg(Pr)n(oof.)42 b Fs(The)32 b(space)h Fm(K)54 b Fs(is)31 b(an)g Fn(H)8 b Fs(-mo)s(dule)29 b(via)h Fn(")d Fs(:)h Fn(H)35 b Fp(\000)-60 b(!)28 b Fm(K)18 b Fs(.)48 b(Hence)33 b(w)m(e)f(ha)m(v)m(e)h(\()p Fn(hf)11 b Fs(\)\()p Fn(m)p Fs(\))27 b(=)0 4231 y Fj(P)122 4306 y Fn(h)178 4322 y Fi(\(1\))272 4306 y Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(h)529 4322 y Fi(\(2\))623 4306 y Fn(m)p Fs(\))28 b(=)878 4231 y Fj(P)999 4306 y Fn(")p Fs(\()p Fn(h)1139 4322 y Fi(\(1\))1233 4306 y Fs(\))p Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(h)1528 4322 y Fi(\(2\))1622 4306 y Fn(m)p Fs(\))28 b(=)g Fn(f)11 b Fs(\()p Fn(S)6 b Fs(\()p Fn(h)p Fs(\))p Fn(m)p Fs(\).)p 3529 4306 V 3533 4244 59 4 v 3533 4306 V 3591 4306 4 66 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF