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y Fl(\003)3391 4179 y Fr(\))p Fn(:)149 4370 y Fr(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 Fo(Lemma)k(3.3.8.)49 b Fk(L)-5 b(et)38 b Fr(\()p Fn(M)1170 4640 y Fl(\003)1210 4676 y Fn(;)17 b Fr(ev)1350 4691 y Fi(M)1429 4676 y Fr(\))p Fk(,)38 b Fr(\()p Fn(N)1661 4640 y Fl(\003)1701 4676 y Fn(;)17 b Fr(ev)1841 4691 y Fi(N)1908 4676 y Fr(\))p Fk(,)38 b(and)f Fr(\()p Fn(P)2321 4640 y Fl(\003)2360 4676 y Fn(;)17 b Fr(ev)2500 4691 y Fi(P)2558 4676 y Fr(\))38 b Fk(b)-5 b(e)37 b(left)h(duals)f(for)h Fn(M)10 b Fk(,)38 b Fn(N)0 4793 y Fk(and)c Fn(P)49 b Fk(r)-5 b(esp)g(e)g(ctively.)149 4909 y(1.)45 b(We)35 b(have)f Fr(\(1)762 4924 y Fi(M)841 4909 y Fr(\))879 4873 y Fl(\003)946 4909 y Fr(=)27 b(1)1098 4924 y Fi(M)1173 4905 y Fh(\003)1213 4909 y Fk(.)149 5025 y(2.)45 b(If)34 b Fn(f)39 b Fr(:)27 b Fn(M)39 b Fm(\000)-58 b(!)28 b Fn(N)46 b Fk(and)34 b Fn(g)d Fr(:)d Fn(N)38 b Fm(\000)-58 b(!)28 b Fn(P)48 b Fk(ar)-5 b(e)35 b(given)f(then)h Fr(\()p Fn(g)t(f)11 b Fr(\))2440 4989 y Fl(\003)2506 5025 y Fr(=)27 b Fn(f)2668 4989 y Fl(\003)2707 5025 y Fn(g)2758 4989 y Fl(\003)2832 5025 y Fk(holds.)149 5216 y Fg(Pr)n(oof.)42 b Fr(1.)i(\(1)741 5231 y Fi(M)819 5216 y Fr(\))857 5179 y Fl(\003)924 5216 y Fr(=)28 b(\(ev)19 b Fm(\012)p Fr(1\)\(1)j Fm(\012)g Fr(1)g Fm(\012)h Fr(1\)\(1)e Fm(\012)i Fr(db\))28 b(=)g(1)2344 5231 y Fi(M)2419 5212 y Fh(\003)2458 5216 y Fr(.)p eop %%Page: 86 4 86 3 bop 0 236 a Fp(86)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 Fr(2.)44 b(The)33 b(follo)m(wing)d(diagram)g(comm)m(utes)956 706 y Fn(M)456 b(N)33 b Fm(\012)23 b Fn(N)1805 670 y Fl(\003)1867 706 y Fm(\012)f Fn(M)p 1090 678 389 4 v 1396 676 a Ff(-)1165 646 y Fj(db)1243 657 y Fe(N)1313 646 y Fl(\012)p Fj(1)p 1007 998 4 254 v 1009 998 a Ff(?)1048 888 y Fi(f)p 1787 998 V 1789 998 a Ff(?)1828 888 y Fj(1)p Fl(\012)p Fj(1)p Fl(\012)p Fi(f)965 1096 y Fn(N)472 b(N)32 b Fm(\012)23 b Fn(N)1813 1060 y Fl(\003)1875 1096 y Fm(\012)g Fn(N)p 1082 1068 404 4 v 1403 1066 a Ff(-)1165 1036 y Fj(db)1243 1047 y Fe(N)1313 1036 y Fl(\012)p Fj(1)2524 1096 y Fn(N)p 2092 1068 V 2413 1066 a Ff(-)2185 1036 y Fj(1)p Fl(\012)p Fj(ev)2345 1047 y Fe(N)p 1787 1388 4 254 v 1789 1388 a Ff(?)1828 1276 y Fi(g)r Fl(\012)p Fj(1)p Fl(\012)p Fj(1)p 2567 1388 V 2569 1388 a Ff(?)2608 1269 y Fi(g)1521 1486 y Fn(P)35 b Fm(\012)23 b Fn(N)1807 1450 y Fl(\003)1869 1486 y Fm(\012)f Fn(N)485 b(P)p 2086 1458 416 4 v 2419 1456 a Ff(-)2185 1426 y Fj(1)p Fl(\012)p Fj(ev)2345 1437 y Fe(N)0 1724 y Fr(Hence)45 b(w)m(e)f(ha)m(v)m(e)g Fn(g)t(f)56 b Fr(=)46 b(\(1)29 b Fm(\012)h Fr(ev)1286 1739 y Fi(N)1353 1724 y Fr(\)\()p Fn(g)j Fm(\012)d Fr(1)f Fm(\012)h Fn(f)11 b Fr(\)\(db)2044 1739 y Fi(N)2128 1724 y Fm(\012)p Fr(1\).)75 b(Th)m(us)45 b(the)f(follo)m(wing)c(diagram)0 1841 y(comm)m(utes)254 2100 y Fn(P)331 2064 y Fl(\003)1291 2100 y Fn(P)1368 2064 y Fl(\003)1429 2100 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y Fl(\003)2612 2831 y Fm(\012)g Fn(M)32 b Fm(\012)23 b Fn(M)3042 2795 y Fl(\003)2133 2694 y Fj(1)p Fl(\012)p Fi(g)r Fl(\012)p Fj(1)1852 2647 y Ff(X)1935 2667 y(X)2018 2688 y(X)2101 2709 y(X)2184 2730 y(X)2198 2733 y(X)-83 b(z)p 310 2977 4 351 v 312 2977 a(?)351 2818 y Fj(1)p Fl(\012)p Fi(g)r(f)7 b Fl(\012)p Fj(1)22 3075 y Fn(P)99 3039 y Fl(\003)160 3075 y Fm(\012)22 b Fn(P)36 b Fm(\012)23 b Fn(M)562 3039 y Fl(\003)1059 3075 y Fn(P)1136 3039 y Fl(\003)1197 3075 y Fm(\012)g Fn(P)35 b Fm(\012)23 b Fn(N)1583 3039 y Fl(\003)1645 3075 y Fm(\012)g Fn(N)32 b Fm(\012)23 b Fn(M)2059 3039 y Fl(\003)p 631 3047 400 4 v 631 3045 a Ff(\033)700 3019 y Fj(1)p Fl(\012)p Fj(ev)13 b Fl(\012)p Fj(1)2133 2940 y(1)p Fl(\012)p Fi(f)7 b Fl(\012)p Fj(1)2198 2890 y Ff(\030)2115 2911 y(\030)2032 2932 y(\030)1949 2953 y(\030)1866 2973 y(\030)1852 2977 y(\030)-83 b(9)p 2552 3464 4 595 v 2554 3464 a(?)2593 3185 y Fj(ev)13 b Fl(\012)p Fj(1)p 310 3464 4 351 v 312 3464 a Ff(?)351 3307 y Fj(ev)g Fl(\012)p Fj(1)p 1578 3464 V 1579 3464 a Ff(?)1618 3307 y Fj(ev)h Fl(\012)p Fj(1)240 3562 y Fn(M)344 3526 y Fl(\003)1277 3562 y Fn(N)1365 3526 y Fl(\003)1427 3562 y Fm(\012)23 b Fn(N)32 b Fm(\012)23 b Fn(M)1841 3526 y Fl(\003)p 412 3535 837 4 v 412 3533 a Ff(\033)751 3507 y Fj(1)p Fl(\012)p Fj(ev)2244 3562 y Fn(N)2332 3526 y Fl(\003)2394 3562 y Fm(\012)g Fn(M)33 b Fm(\012)22 b Fn(M)2824 3526 y Fl(\003)p 1910 3535 306 4 v 1910 3533 a Ff(\033)1951 3500 y Fj(1)p Fl(\012)p Fi(f)7 b Fl(\012)p Fj(1)p 1578 2977 4 351 v 1579 2977 a Ff(?)1138 2818 y Fj(1)p Fl(\012)p Fi(g)r Fl(\012)p Fj(1)p Fl(\012)p Fi(f)g Fl(\012)p Fj(1)3465 2835 y Fn(N)3553 2799 y Fl(\003)3102 2454 y Fj(ev)13 b Fl(\012)p Fj(1)2645 2201 y Ff(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 Fj(1)p Fl(\012)p Fj(db)3355 2932 y Ff(\032)3272 2994 y(\032)3189 3056 y(\032)3106 3119 y(\032)3023 3181 y(\032)2940 3243 y(\032)2857 3305 y(\032)2774 3368 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4729 y Fr(is)36 b(uniquely)0 4845 y(determined)c(b)m(y)i Fn(M)10 b Fr(,)33 b Fn(M)913 4809 y Fl(\003)954 4845 y Fr(,)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(\()p Fn(M)602 4925 y Fl(\003)642 4961 y Fn(;)17 b Fr(ev)r(\))36 b(b)s(e)h(a)g(left)e(dual) h(for)g Fn(M)10 b Fr(.)57 b(Sho)m(w)38 b(that)e(ev)h(:)e Fn(M)2589 4925 y Fl(\003)2654 4961 y Fm(\012)25 b Fn(M)46 b Fm(\000)-60 b(!)34 b Fn(I)45 b Fr(is)36 b(uniquely)0 5077 y(determined)c(b)m(y)i Fn(M)10 b Fr(,)33 b Fn(M)913 5041 y Fl(\003)954 5077 y Fr(,)f(and)h(db.)p eop %%Page: 87 5 87 4 bop 1421 236 a Fp(3.)77 b(DUAL)32 b(OBJECTS)1353 b(87)149 425 y Fo(Corollary)36 b(3.3.9.)50 b Fk(L)-5 b(et)35 b Fn(M)5 b(;)17 b(N)46 b Fk(have)34 b(the)h(left)h(duals)e Fr(\()p Fn(M)2340 389 y Fl(\003)2381 425 y Fn(;)17 b Fr(ev)2521 440 y Fi(M)2600 425 y Fr(\))35 b Fk(and)f Fr(\()p Fn(N)2988 389 y Fl(\003)3028 425 y Fn(;)17 b Fr(ev)3168 440 y Fi(N)3235 425 y Fr(\))35 b Fk(and)g(let)0 541 y Fn(f)j 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Fr(\))30 b Fk(and)f(let)0 2972 y Fn(f)38 b Fr(:)28 b Fn(M)39 b Fm(\000)-58 b(!)28 b Fn(N)45 b Fk(b)-5 b(e)35 b(a)f(morphism)g(in)h Fm(C)6 b Fk(.)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(diagr)-5 b(am)34 b(c)-5 b(ommutes)1334 3651 y Fn(N)1422 3615 y Fl(\003)1484 3651 y Fm(\012)23 b Fn(N)401 b(I)8 b(:)p 1701 3623 307 4 v 1925 3621 a Ff(-)1790 3669 y Fj(ev)1860 3680 y Fe(N)1326 3164 y Fn(N)1414 3127 y Fl(\003)1476 3164 y Fm(\012)23 b Fn(M)233 b(M)2007 3127 y Fl(\003)2069 3164 y Fm(\012)23 b Fn(M)p 1709 3135 165 4 v 1791 3133 a Ff(-)1707 3102 y Fi(f)1748 3078 y Fh(\003)1784 3102 y Fl(\012)p Fj(1)p 1501 3553 4 351 v 1503 3553 a Ff(?)1333 3394 y Fj(1)p Fl(\012)p Fi(f)p 2086 3553 V 2088 3553 a Ff(?)2127 3386 y Fj(ev)2197 3397 y Fe(M)149 3824 y Fg(Pr)n(oof.)42 b Fr(This)31 b(statemen)m(t)g(follo)m(ws)e(immediately) e(from)i(the)i(symmetry)f(of)g(the)h(de\014nition)0 3940 y(of)h(a)g(left)g(dual.)p 3529 3940 4 66 v 3533 3878 59 4 v 3533 3940 V 3591 3940 4 66 v 149 4123 a Fo(Example)37 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b(there)j(are)e(elemen)m(ts)h Fn(m)985 4603 y Fj(1)1025 4588 y Fn(;)17 b(:)g(:)g(:)32 b(;)17 b(m)1345 4603 y Fi(n)1420 4588 y Fm(2)28 b Fn(M)43 b Fr(und)34 b Fn(m)1932 4552 y Fj(1)1971 4588 y Fn(;)17 b(:)g(:)g(:)33 b(;)17 b(m)2292 4552 y Fi(n)2366 4588 y Fm(2)28 b Fr(Hom)2664 4603 y Fi(R)2721 4588 y Fr(\()p Fn(M)5 b(:;)17 b(R)q(:)p Fr(\))33 b(suc)m(h)h(that)1179 4841 y Fm(8)p Fn(m)28 b Fm(2)g Fn(M)39 b Fr(:)1679 4716 y Fi(n)1629 4746 y Fc(X)1644 4956 y Fi(i)p Fj(=1)1789 4841 y Fn(m)1874 4856 y Fi(i)1903 4841 y Fn(m)1988 4799 y Fi(i)2016 4841 y Fr(\()p Fn(m)p Fr(\))28 b(=)g Fn(m:)149 5099 y Fr(Actually)43 b(this)g(is)g(a)g(consequence)k(of)c(the)h(dual)f (basis)g(lemma.)74 b(But)44 b(this)f(de\014nition)f(is)0 5216 y(equiv)-5 b(alen)m(t)32 b(to)h(the)g(usual)f(de\014nition.)p eop %%Page: 88 6 88 5 bop 0 236 a Fp(88)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 Fr(Let)g Fn(M)38 b Fm(2)549 440 y Fi(R)607 425 y Fm(M)727 440 y Fi(R)785 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Fn(m)3259 1023 y Fi(i)3287 1008 y Fn(m)3372 971 y Fi(i)3401 1008 y Fr(\()p Fn(m)p Fr(\)\))0 1124 y(=)28 b Fn(f)11 b Fr(\()p Fn(m)p Fr(\))32 b(for)g(all)f Fn(m)d Fm(2)g Fn(M)43 b Fr(hence)34 b(\(ev)19 b Fm(\012)1484 1139 y Fi(R)1542 1124 y Fr(1\)\(1)i Fm(\012)1814 1139 y Fi(R)1895 1124 y Fr(db\)\()p Fn(f)11 b Fr(\))27 b(=)h Fn(f)11 b Fr(.)149 1240 y(Con)m(v)m(ersely)42 b(if)c Fn(M)50 b Fr(has)39 b(a)g(left)g(dual)f Fn(M)1663 1204 y Fl(\003)1743 1240 y Fr(then)h(ev)j(:)d Fn(M)2277 1204 y Fl(\003)2344 1240 y Fm(\012)2421 1255 y Fi(R)2505 1240 y Fn(M)50 b Fm(\000)-60 b(!)39 b Fn(R)i Fr(de\014nes)g(a)d(homo-)0 1356 y(morphism)28 b Fn(\023)h Fr(:)e Fn(M)674 1320 y Fl(\003)742 1356 y Fm(\000)-60 b(!)28 b Fr(Hom)1090 1371 y Fi(R)1148 1356 y Fr(\()p Fn(M)5 b(:;)17 b(R)q(:)p Fr(\))30 b(in)1638 1371 y Fi(R)1695 1356 y Fm(M)1815 1371 y Fi(R)1903 1356 y Fr(b)m(y)h Fn(\023)p Fr(\()p Fn(m)2193 1320 y Fl(\003)2233 1356 y Fr(\)\()p Fn(m)p Fr(\))d(=)f(ev)s(\()p Fn(m)2783 1320 y Fl(\003)2840 1356 y Fm(\012)2917 1371 y Fi(R)2992 1356 y Fn(m)p Fr(\).)43 b(W)-8 b(e)31 b(de\014ne)0 1398 y Fc(P)105 1424 y Fi(n)105 1502 y(i)p Fj(=1)240 1472 y Fn(m)325 1487 y Fi(i)371 1472 y Fm(\012)18 b Fn(m)551 1436 y Fi(i)607 1472 y Fr(:=)27 b(db)q(\(1\))g Fm(2)h Fn(M)g Fm(\012)18 b Fn(M)1413 1436 y Fl(\003)1453 1472 y Fr(,)31 b(then)g Fn(m)d Fr(=)f(\(1)17 b Fm(\012)h Fr(ev)r(\)\(db)f Fm(\012)p Fr(1\)\()p Fn(m)p Fr(\))28 b(=)f(\(1)17 b Fm(\012)h Fr(ev)r(\)\()3270 1398 y Fc(P)3392 1472 y Fn(m)3477 1487 y Fi(i)3523 1472 y Fm(\012)0 1590 y Fn(m)85 1554 y Fi(i)153 1590 y Fm(\012)40 b Fn(m)p Fr(\))71 b(=)611 1515 y Fc(P)733 1590 y Fn(m)818 1605 y Fi(i)846 1590 y Fn(\023)p Fr(\()p Fn(m)1003 1554 y Fi(i)1032 1590 y Fr(\)\()p Fn(m)p Fr(\))58 b(so)h(that)e Fn(m)1756 1605 y Fj(1)1796 1590 y Fn(;)17 b(:)g(:)g(:)32 b(;)17 b(m)2116 1605 y Fi(n)2234 1590 y Fm(2)72 b Fn(M)d Fr(and)58 b Fn(\023)p Fr(\()p Fn(m)2907 1554 y Fj(1)2947 1590 y Fr(\))p Fn(;)17 b(:)g(:)g(:)32 b(;)17 b(\023)p Fr(\()p Fn(m)3377 1554 y 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g(dual.)149 2539 y Fo(De\014nition)37 b(3.3.12.)49 b Fr(Giv)m(en)j(ob)5 b(jects)54 b Fn(M)5 b(;)17 b(N)64 b Fr(in)52 b Fm(C)6 b Fr(.)104 b(An)53 b(ob)5 b(ject)53 b([)p Fn(M)5 b(;)17 b(N)10 b Fr(])54 b(is)e(called)f(a)0 2656 y Fk(left)44 b(inner)g(Hom)e Fr(of)h Fn(M)53 b Fr(and)43 b Fn(N)53 b Fr(if)42 b(there)h(is)f(a)h(natural)f(isomorphism)e(Mor) 2957 2671 y Fl(C)3002 2656 y Fr(\(-)28 b Fm(\012)i Fn(M)5 b(;)17 b(N)10 b Fr(\))3523 2628 y Fm(\030)3523 2660 y Fr(=)0 2772 y(Mor)176 2787 y Fl(C)221 2772 y Fr(\(-)p Fn(;)17 b Fr([)p Fn(M)5 b(;)17 b(N)10 b Fr(]\),)33 b(i.)e(e.)44 b(if)32 b(it)f(represen)m(ts)k(the)e(functor)g(Mor)2248 2787 y Fl(C)2293 2772 y Fr(\(-)22 b Fm(\012)g Fn(M)5 b(;)17 b(N)10 b Fr(\).)149 2888 y(If)43 b(there)g(is)f(an)g (isomorphism)e(Mor)1525 2903 y Fl(C)1570 2888 y Fr(\()p Fn(P)i Fm(\012)30 b Fn(M)5 b(;)17 b(N)10 b Fr(\))2133 2860 y Fm(\030)2134 2892 y Fr(=)2255 2888 y(Mor)2431 2903 y Fl(C)2476 2888 y Fr(\()p Fn(P)s(;)17 b Fr([)p Fn(M)5 b(;)17 b(N)10 b Fr(]\))43 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Fl(\003)799 3537 y Fr(.)43 b(In)33 b(particular)e(left)g(rigid)g(monoidal)f(categories)i(are)h (left)e(closed.)149 3721 y Fo(Example)37 b(3.3.13.)199 b Fr(1.)41 b(The)49 b(category)g(of)f(\014nite)g(dimensional)d(v)m (ector)50 b(spaces)f(is)f(a)267 3837 y(monoidal)35 b(category)k(where)g (eac)m(h)h(ob)5 b(ject)39 b(has)f(a)g(\(left)g(and)g(righ)m(t\))g (dual.)60 b(Hence)39 b(it)e(is)267 3954 y(\(left)31 b(and)i(righ)m(t\)) f(closed)h(and)f(rigid.)149 4070 y(2.)42 b(Let)g Fo(Ban)h Fr(b)s(e)g(the)g(category)f(of)g(\(complex\))g(Banac)m(h)h(spaces)h (where)f(the)g(morphisms)267 4186 y(satisfy)27 b Fm(k)h Fn(f)11 b Fr(\()p Fn(m)p Fr(\))27 b Fm(k\024k)h Fn(m)g Fm(k)f Fr(i.)f(e.)42 b(the)28 b(maps)f(are)g(b)s(ounded)h(b)m(y)h(1)e (or)f(con)m(tracting.)42 b Fo(Ban)27 b Fr(is)267 4307 y(a)32 b(monoidal)d(category)k(b)m(y)g(the)g(complete)f(tensor)h(pro)s (duct)f Fn(M)2652 4291 y Fc(b)2640 4307 y Fm(\012)r Fn(N)10 b Fr(.)44 b(In)32 b Fo(Ban)h Fr(exists)g(an)267 4423 y(inner)c(Hom)f(functor)h([)p Fn(M)5 b(;)17 b(N)10 b Fr(])30 b(that)f(consists)h(of)f(the)g(set)h(of)f(b)s(ounded)h(linear)d (maps)i(from)267 4539 y Fn(M)46 b Fr(to)34 b Fn(N)46 b Fr(made)34 b(in)m(to)g(a)h(Banac)m(h)h(space)g(b)m(y)g(an)e (appropriate)h(top)s(ology)-8 b(.)49 b(Th)m(us)36 b Fo(Ban)f Fr(is)267 4655 y(a)d(monoidal)e(closed)j(category)-8 b(.)149 4771 y(3.)42 b(Let)23 b Fn(H)31 b Fr(b)s(e)23 b(a)g(Hopf)h(algebra.)39 b(The)24 b(category)g Fn(H)8 b Fr(-)o Fo(Mo)s(d)24 b Fr(of)e(left)h Fn(H)8 b Fr(-mo)s(dules)21 b(is)i(a)g(monoidal)267 4888 y(category)39 b(\(see)g(Example)f(3.2.4)g (2.\).)61 b(Then)40 b(Hom)2183 4903 y Fb(K)2239 4888 y Fr(\()p Fn(M)5 b(;)17 b(N)10 b Fr(\))39 b(is)f(an)g(ob)5 b(ject)39 b(in)f Fn(H)8 b Fr(-)o Fo(Mo)s(d)267 5004 y Fr(b)m(y)33 b(the)g(m)m(ultiplication)1152 5194 y(\()p Fn(hf)11 b Fr(\)\()p Fn(m)p Fr(\))28 b(:=)1663 5099 y Fc(X)1823 5194 y Fn(h)1879 5209 y Fj(\(1\))1974 5194 y Fn(f)11 b Fr(\()p Fn(mS)6 b Fr(\()p Fn(h)2316 5209 y Fj(\(2\))2410 5194 y Fr(\))p eop %%Page: 89 7 89 6 bop 1421 236 a Fp(3.)77 b(DUAL)32 b(OBJECTS)1353 b(89)267 425 y Fr(as)33 b(in)e(Prop)s(osition)g(3.1.14.)416 541 y(Hom)619 556 y Fb(K)675 541 y Fr(\()p Fn(M)5 b(;)17 b(N)10 b Fr(\))39 b(is)e(an)h(inner)g(Hom)f(functor)h(in)f(the)i (monoidal)c(category)j Fn(H)8 b Fr(-)o Fo(Mo)s(d)q Fr(.)267 658 y(The)38 b(isomorphism)c Fn(\036)h Fr(:)g(Hom)1404 673 y Fb(K)1460 658 y Fr(\()p Fn(P)s(;)17 b Fr(Hom)1810 673 y Fb(K)1866 658 y Fr(\()p Fn(M)5 b(;)17 b(N)10 b Fr(\)\))2246 630 y Fm(\030)2247 662 y Fr(=)2359 658 y(Hom)2562 673 y Fb(K)2618 658 y Fr(\()p Fn(P)38 b Fm(\012)26 b Fn(M)5 b(;)17 b(N)10 b Fr(\))37 b(can)g(b)s(e)g(re-)267 774 y(stricted)c(to)f(an)g(isomorphism)821 948 y(Hom)1024 963 y Fi(H)1091 948 y Fr(\()p Fn(P)s(;)17 b Fr(Hom)1442 963 y Fb(K)1498 948 y Fr(\()p Fn(M)5 b(;)17 b(N)10 b Fr(\)\))1871 920 y Fm(\030)1872 952 y Fr(=)1976 948 y(Hom)2179 963 y Fi(H)2246 948 y Fr(\()p Fn(P)36 b Fm(\012)22 b Fn(M)5 b(;)17 b(N)10 b Fr(\))p Fn(;)267 1122 y Fr(b)s(ecause)53 b Fn(\036)p Fr(\()p Fn(f)11 b Fr(\)\()p Fn(h)p Fr(\()p Fn(p)35 b Fm(\012)g Fn(m)p Fr(\)\))61 b(=)f Fn(\036)p Fr(\()p Fn(f)11 b Fr(\)\()1757 1048 y Fc(P)1878 1122 y Fn(h)1934 1138 y Fj(\(1\))2029 1122 y Fn(p)35 b Fm(\012)h Fn(h)2282 1138 y Fj(\(2\))2376 1122 y Fn(m)p Fr(\))61 b(=)2696 1048 y Fc(P)2818 1122 y Fn(f)11 b Fr(\()p Fn(h)2971 1138 y Fj(\(1\))3065 1122 y Fn(p)p Fr(\)\()p Fn(h)3246 1138 y Fj(\(2\))3340 1122 y Fn(m)p Fr(\))61 b(=)267 1164 y Fc(P)372 1239 y Fr(\()p Fn(h)466 1254 y Fj(\(1\))560 1239 y Fr(\()p Fn(f)11 b Fr(\()p Fn(p)p Fr(\)\)\)\()p Fn(h)952 1254 y Fj(\(2\))1046 1239 y Fn(m)p Fr(\))51 b(=)1347 1164 y Fc(P)1468 1239 y Fn(h)1524 1254 y Fj(\(1\))1619 1239 y Fr(\()p Fn(f)11 b Fr(\()p Fn(p)p Fr(\)\()p Fn(S)6 b Fr(\()p Fn(h)2039 1254 y Fj(\(2\))2133 1239 y Fr(\))p Fn(h)2227 1254 y Fj(\(3\))2321 1239 y Fn(m)p Fr(\)\))51 b(=)g Fn(h)p Fr(\()p Fn(f)11 b Fr(\()p Fn(p)p Fr(\)\()p Fn(m)p Fr(\)\))50 b(=)h Fn(h)p Fr(\()p Fn(\036)p Fr(\()p Fn(f)11 b Fr(\))267 1355 y(\()p Fn(p)j Fm(\012)g Fn(m)p Fr(\)\))29 b(and)g(con)m(v)m(ersely)h(\()p Fn(h)p Fr(\()p Fn(f)11 b Fr(\()p Fn(p)p Fr(\)\)\)\()p Fn(m)p Fr(\))28 b(=)1981 1280 y Fc(P)2103 1355 y Fn(h)2159 1370 y Fj(\(1\))2254 1355 y Fr(\()p Fn(f)11 b Fr(\()p Fn(p)p Fr(\)\()p Fn(S)6 b Fr(\()p Fn(h)2674 1370 y Fj(\(2\))2767 1355 y Fr(\))p Fn(m)p Fr(\)\))28 b(=)3097 1280 y Fc(P)3219 1355 y Fn(h)3275 1370 y Fj(\(1\))3370 1355 y Fr(\()p Fn(\036)p Fr(\()p Fn(f)11 b Fr(\))267 1471 y(\()p Fn(p)23 b Fm(\012)h Fn(S)6 b Fr(\()p Fn(h)638 1486 y Fj(\(2\))733 1471 y Fr(\))p Fn(m)p Fr(\)\))31 b(=)1070 1396 y Fc(P)1192 1471 y Fn(\036)p Fr(\()p Fn(f)11 b Fr(\)\()p Fn(h)1479 1486 y Fj(\(1\))1573 1471 y Fn(p)23 b Fm(\012)h Fn(h)1802 1486 y Fj(\(2\))1897 1471 y Fn(S)6 b Fr(\()p Fn(h)2057 1486 y Fj(\(3\))2151 1471 y Fr(\))p Fn(m)p Fr(\))31 b(=)g Fn(\036)p Fr(\()p Fn(f)11 b Fr(\)\()p Fn(hp)23 b Fm(\012)h Fn(m)p Fr(\))32 b(=)f Fn(f)11 b Fr(\()p Fn(hp)p Fr(\)\()p Fn(m)p Fr(\).)267 1587 y(Th)m(us)34 b Fn(H)8 b Fr(-)o Fo(Mo)s(d)33 b Fr(is)f(left)g (closed.)416 1703 y(If)39 b Fn(M)50 b Fm(2)39 b Fn(H)8 b Fr(-)o Fo(Mo)s(d)39 b Fr(is)g(a)f(\014nite)h(dimensional)e(v)m(ector) j(space)g(then)f(the)h(dual)e(v)m(ector)267 1820 y(space)43 b Fn(M)641 1784 y Fl(\003)725 1820 y Fr(:=)h(Hom)1076 1835 y Fb(K)1131 1820 y Fr(\()p Fn(M)5 b(;)17 b Fd(K)h Fr(\))48 b(again)41 b(is)g(an)h Fn(H)8 b Fr(-mo)s(dule)40 b(b)m(y)j(\()p Fn(hf)11 b Fr(\)\()p Fn(m)p Fr(\))44 b(:=)g Fn(f)11 b Fr(\()p Fn(S)6 b Fr(\()p Fn(h)p Fr(\))p Fn(m)p Fr(\))p Fn(:)267 1936 y Fr(F)-8 b(urthermore)32 b Fn(M)936 1900 y Fl(\003)1009 1936 y Fr(is)g(a)g(left)g(dual)f(for)h Fn(M)44 b Fr(with)32 b(the)h(morphisms)1002 2126 y(db)28 b(:)g Fd(K)51 b Fm(3)28 b Fr(1)g Fm(7!)1596 2032 y Fc(X)1656 2241 y Fi(i)1757 2126 y Fn(m)1842 2141 y Fi(i)1892 2126 y Fm(\012)23 b Fn(m)2077 2085 y Fi(i)2133 2126 y Fm(2)28 b Fn(M)33 b Fm(\012)23 b Fn(M)2558 2085 y Fl(\003)267 2387 y Fr(and)1044 2520 y(ev)30 b(:)e Fn(M)1327 2479 y Fl(\003)1389 2520 y Fm(\012)23 b Fn(M)39 b Fm(3)28 b Fn(f)k Fm(\012)23 b Fn(m)28 b Fm(7!)f Fn(f)11 b Fr(\()p Fn(m)p Fr(\))28 b Fm(2)g Fd(K)267 2674 y Fr(where)45 b Fn(m)645 2689 y Fi(i)717 2674 y Fr(and)e Fn(m)1002 2638 y Fi(i)1074 2674 y Fr(are)h(a)f(dual)g(basis)h(of)f(the)h(v)m (ector)g(space)h Fn(M)10 b Fr(.)77 b(Clearly)43 b(w)m(e)i(ha)m(v)m(e) 267 2790 y(\(1)6 b Fm(\012)g Fr(ev)s(\)\(db)17 b Fm(\012)p Fr(1\))28 b(=)f(1)1085 2805 y Fi(M)1189 2790 y Fr(and)e(\(ev)19 b Fm(\012)p Fr(1\)\(1)6 b Fm(\012)g Fr(db)q(\))28 b(=)f(1)2189 2805 y Fi(M)2264 2787 y Fh(\003)2329 2790 y Fr(since)e Fn(M)36 b Fr(is)24 b(a)h(\014nite)f(dimensional)267 2907 y(v)m(ector)f(space.)41 b(W)-8 b(e)22 b(ha)m(v)m(e)i(to)d(sho)m(w)i (that)44 b(db)g(and)g(ev)j(are)22 b Fn(H)8 b Fr(-mo)s(dule)19 b(homomorphisms.)267 3023 y(W)-8 b(e)33 b(ha)m(v)m(e)358 3171 y(\()p Fn(h)17 b Fr(db\(1\)\)\()p Fn(m)p Fr(\))28 b(=)f(\()p Fn(h)p Fr(\()1164 3096 y Fc(P)1286 3171 y Fn(m)1371 3186 y Fi(i)1422 3171 y Fm(\012)22 b Fn(m)1606 3134 y Fi(i)1635 3171 y Fr(\)\)\()p Fn(m)p Fr(\))28 b(=)f(\()2041 3096 y Fc(P)2163 3171 y Fn(h)2219 3186 y Fj(\(1\))2313 3171 y Fn(m)2398 3186 y Fi(i)2449 3171 y Fm(\012)c Fn(h)2605 3186 y Fj(\(2\))2699 3171 y Fn(m)2784 3134 y Fi(i)2813 3171 y Fr(\)\()p Fn(m)p Fr(\))k(=)358 3212 y Fc(P)463 3287 y Fr(\()p Fn(h)557 3302 y Fj(\(1\))652 3287 y Fn(m)737 3302 y Fi(i)765 3287 y Fr(\)\(\()p Fn(h)935 3302 y Fj(\(2\))1030 3287 y Fn(m)1115 3251 y Fi(i)1143 3287 y Fr(\)\()p Fn(m)p Fr(\)\))h(=)1511 3212 y Fc(P)1616 3287 y Fr(\()p Fn(h)1710 3302 y Fj(\(1\))1805 3287 y Fn(m)1890 3302 y Fi(i)1918 3287 y Fr(\)\()p Fn(m)2079 3251 y Fi(i)2108 3287 y Fr(\()p Fn(S)6 b Fr(\()p Fn(h)2306 3302 y Fj(\(2\))2400 3287 y Fr(\))p Fn(m)p Fr(\)\))28 b(=)358 3328 y Fc(P)480 3403 y Fn(h)536 3419 y Fj(\(1\))630 3403 y Fn(S)6 b Fr(\()p Fn(h)790 3419 y Fj(\(2\))885 3403 y Fr(\))p Fn(m)28 b Fr(=)f Fn(")p Fr(\()p Fn(h)p Fr(\))p Fn(m)h Fr(=)f Fn(")p Fr(\()p Fn(h)p Fr(\)\()1749 3328 y Fc(P)1871 3403 y Fn(m)1956 3418 y Fi(i)2006 3403 y Fm(\012)c Fn(m)2191 3367 y Fi(i)2219 3403 y Fr(\)\()p Fn(m)p Fr(\))28 b(=)g Fn(")p Fr(\()p Fn(h)p Fr(\))17 b(db)o(\(1\)\()p Fn(m)p Fr(\))28 b(=)358 3519 y(db)q(\()p Fn(")p Fr(\()p Fn(h)p Fr(\)1\)\()p Fn(m)p Fr(\))f(=)g(db)q(\()p Fn(h)p Fr(1\)\()p Fn(m)p Fr(\))p Fn(;)267 3695 y Fr(hence)34 b Fn(h)17 b Fr(db\(1\))27 b(=)h(db\()p Fn(h)p Fr(1\).)43 b(F)-8 b(urthermore)32 b(w)m(e)i(ha)m(v)m(e)340 3868 y Fn(h)17 b Fr(ev)r(\()p Fn(f)32 b Fm(\012)23 b Fn(m)p Fr(\))28 b(=)g Fn(hf)11 b Fr(\()p Fn(m)p Fr(\))27 b(=)1389 3793 y Fc(P)1511 3868 y Fn(h)1567 3883 y Fj(\(1\))1661 3868 y Fn(f)11 b Fr(\()p Fn(S)6 b Fr(\()p Fn(h)1918 3883 y Fj(\(2\))2012 3868 y Fr(\))p Fn(h)2106 3883 y Fj(\(3\))2200 3868 y Fn(m)p Fr(\))28 b(=)2455 3793 y Fc(P)2560 3868 y Fr(\()p Fn(h)2654 3883 y Fj(\(1\))2748 3868 y Fn(f)11 b Fr(\)\()p Fn(h)2939 3883 y Fj(\(2\))3033 3868 y Fn(m)p Fr(\))28 b(=)340 3909 y Fc(P)461 3984 y Fr(ev)s(\()p Fn(h)652 3999 y Fj(\(1\))746 3984 y Fn(f)33 b Fm(\012)23 b Fn(h)983 3999 y Fj(\(2\))1077 3984 y Fn(m)p Fr(\))28 b(=)g(ev)r(\()p Fn(h)p Fr(\()p Fn(f)33 b Fm(\012)22 b Fn(m)p Fr(\)\))p Fn(:)149 4160 y Fr(4.)42 b(Let)28 b Fn(H)36 b Fr(b)s(e)28 b(a)g(Hopf)f(algebra.)41 b(Then)29 b(the)g(category)f(of)g(left)f Fn(H)8 b Fr(-como)s(dules)26 b(\(see)j(Example)267 4276 y(3.2.4)34 b(3.\))50 b(is)35 b(a)g(monoidal)d(category)-8 b(.)51 b(Let)35 b Fn(M)42 b Fm(2)32 b Fn(H)8 b Fr(-)o Fo(Como)s(d)35 b Fr(b)s(e)g(a)g(\014nite)f (dimensional)267 4392 y(v)m(ector)47 b(space.)86 b(Let)46 b Fn(m)1186 4407 y Fi(i)1261 4392 y Fr(b)s(e)g(a)g(basis)g(for)g Fn(M)57 b Fr(and)46 b(let)f(the)i(com)m(ultiplication)42 b(of)k(the)267 4508 y(como)s(dule)29 b(b)s(e)i Fn(\016)t Fr(\()p Fn(m)1002 4523 y Fi(i)1030 4508 y Fr(\))d(=)1199 4434 y Fc(P)1321 4508 y Fn(h)1377 4523 y Fi(ij)1456 4508 y Fm(\012)19 b Fn(m)1637 4523 y Fi(j)1674 4508 y Fr(.)42 b(Then)32 b(w)m(e)g(ha)m(v)m(e)g(\001\()p Fn(h)2536 4523 y Fi(ik)2603 4508 y Fr(\))27 b(=)2772 4434 y Fc(P)2894 4508 y Fn(h)2950 4523 y Fi(ij)3029 4508 y Fm(\012)18 b Fn(h)3180 4523 y Fi(j)t(k)3255 4508 y Fr(.)43 b Fn(M)3429 4472 y Fl(\003)3497 4508 y Fr(:=)267 4625 y(Hom)470 4640 y Fb(K)526 4625 y Fr(\()p Fn(M)5 b(;)17 b Fd(K)g Fr(\))35 b(b)s(ecomes)29 b(a)g(left)e Fn(H)8 b Fr(-como)s(dule)27 b Fn(\016)t Fr(\()p Fn(m)2203 4589 y Fi(j)2240 4625 y Fr(\))g(:=)2436 4550 y Fc(P)2558 4625 y Fn(S)6 b Fr(\()p Fn(h)2718 4640 y Fi(ij)2778 4625 y Fr(\))14 b Fm(\012)g Fn(m)3006 4589 y Fi(i)3036 4625 y Fr(.)42 b(One)29 b(v)m(eri\014es)267 4741 y(that)j Fn(M)582 4705 y Fl(\003)655 4741 y Fr(is)g(a)g(left)g (dual)g(for)g Fn(M)10 b Fr(.)149 4925 y Fo(Lemma)38 b(3.3.14.)49 b Fk(L)-5 b(et)35 b Fn(M)k Fm(2)28 b(C)41 b Fk(b)-5 b(e)34 b(an)h(obje)-5 b(ct)34 b(with)h(left)g(dual)g Fr(\()p Fn(M)2669 4889 y Fl(\003)2709 4925 y Fn(;)17 b Fr(ev)r(\))p Fk(.)44 b(Then)0 5041 y Fr(1.)h Fn(M)33 b Fm(\012)22 b Fn(M)451 5005 y Fl(\003)526 5041 y Fk(is)35 b(an)f(algebr)-5 b(a)34 b(with)h(multiplic)-5 b(ation)632 5216 y Fm(r)28 b Fr(:=)f(1)922 5231 y Fi(M)1023 5216 y Fm(\012)c Fr(ev)c Fm(\012)p Fr(1)1362 5231 y Fi(M)1437 5212 y Fh(\003)1504 5216 y Fr(:)28 b Fn(M)33 b Fm(\012)23 b Fn(M)1890 5174 y Fl(\003)1952 5216 y Fm(\012)g Fn(M)33 b Fm(\012)22 b Fn(M)2382 5174 y Fl(\003)2450 5216 y Fm(\000)-58 b(!)28 b Fn(M)33 b Fm(\012)23 b Fn(M)2928 5174 y Fl(\003)p eop %%Page: 90 8 90 7 bop 0 236 a Fp(90)333 b(3.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fk(and)i(unit)1286 560 y Fn(u)27 b Fr(:=)h(db)g(:)g Fn(I)35 b Fm(\000)-58 b(!)28 b Fn(M)33 b Fm(\012)23 b Fn(M)2247 519 y Fl(\003)2287 560 y Fr(;)0 715 y(2.)45 b Fn(M)225 678 y Fl(\003)287 715 y Fm(\012)23 b Fn(M)45 b Fk(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 Fr(\001)28 b(:=)f(1)915 905 y Fi(M)990 886 y Fh(\003)1052 890 y Fm(\012)c Fr(db)17 b Fm(\012)p Fr(1)1403 905 y Fi(M)1510 890 y Fr(:)27 b Fn(M)1668 849 y Fl(\003)1731 890 y Fm(\012)22 b Fn(M)39 b Fm(\000)-58 b(!)28 b Fn(M)2214 849 y Fl(\003)2276 890 y Fm(\012)23 b Fn(M)33 b Fm(\012)23 b Fn(M)2707 849 y Fl(\003)2769 890 y Fm(\012)g Fn(M)0 1065 y Fk(and)34 b(c)-5 b(ounit)1297 1199 y Fn(")27 b Fr(:=)h(ev)i(:)d Fn(M)1783 1158 y Fl(\003)1846 1199 y Fm(\012)22 b Fn(M)39 b Fm(\000)-58 b(!)28 b Fn(I)8 b(:)149 1384 y Fg(Pr)n(oof.)42 b Fr(1.)74 b(The)44 b(asso)s(ciativit)m(y)d(is)h(giv)m(en)h(b)m(y)h(\() p Fm(r)29 b(\012)g Fr(1\))p Fm(r)45 b Fr(=)g(\(1)2662 1399 y Fi(M)2770 1384 y Fm(\012)29 b Fr(ev)19 b Fm(\012)p Fr(1)3115 1399 y Fi(M)3190 1380 y Fh(\003)3259 1384 y Fm(\012)30 b Fr(1)3415 1399 y Fi(M)3523 1384 y Fm(\012)0 1501 y Fr(1)49 1516 y Fi(M)124 1497 y Fh(\003)164 1501 y Fr(\)\(1)289 1516 y Fi(M)397 1501 y Fm(\012)h Fr(ev)19 b Fm(\012)p Fr(1)744 1516 y Fi(M)819 1497 y Fh(\003)859 1501 y Fr(\))47 b(=)h(1)1117 1516 y Fi(M)1226 1501 y Fm(\012)30 b Fr(ev)19 b Fm(\012)e Fr(ev)i Fm(\012)p Fr(1)1779 1516 y Fi(M)1854 1497 y Fh(\003)1941 1501 y Fr(=)48 b(\(1)2152 1516 y Fi(M)2261 1501 y Fm(\012)30 b Fr(1)2417 1516 y Fi(M)2492 1497 y Fh(\003)2562 1501 y Fm(\012)g Fr(1)2718 1516 y Fi(M)2827 1501 y Fm(\012)h Fr(ev)19 b Fm(\012)p Fr(1)3174 1516 y Fi(M)3249 1497 y Fh(\003)3289 1501 y Fr(\)\(1)3414 1516 y Fi(M)3523 1501 y Fm(\012)0 1617 y Fr(ev)g Fm(\012)p Fr(1)239 1632 y Fi(M)314 1613 y Fh(\003)354 1617 y Fr(\))28 b(=)f(\(1)9 b Fm(\012)g(r)p Fr(\))p Fm(r)p Fr(.)42 b(The)27 b(axiom)e(for)g(the)i(left)f(unit)f(is)h Fm(r)p Fr(\()p Fn(u)9 b Fm(\012)g Fr(1\))28 b(=)f(\(1)2793 1632 y Fi(M)2881 1617 y Fm(\012)9 b Fr(ev)19 b Fm(\012)p Fr(1)3206 1632 y Fi(M)3281 1613 y Fh(\003)3322 1617 y Fr(\)\(db)e Fm(\012)0 1733 y Fr(1)49 1748 y Fi(M)150 1733 y Fm(\012)23 b Fr(1)299 1748 y Fi(M)374 1729 y Fh(\003)413 1733 y Fr(\))28 b(=)f(1)631 1748 y Fi(M)732 1733 y Fm(\012)c Fr(1)881 1748 y Fi(M)956 1729 y Fh(\003)996 1733 y Fr(.)149 1849 y(2.)44 b(is)32 b(dual)g(to)g(the)h(statemen)m(t)g(for)f (algebras.)p 3529 1849 4 66 v 3533 1787 59 4 v 3533 1849 V 3591 1849 4 66 v 149 2034 a Fo(Lemma)38 b(3.3.15.)49 b Fr(1.)44 b Fk(L)-5 b(et)33 b Fn(A)g Fk(b)-5 b(e)32 b(an)h(algebr)-5 b(a)32 b(in)g Fm(C)39 b Fk(and)32 b(left)h Fn(M)38 b Fm(2)28 b(C)39 b Fk(b)-5 b(e)33 b(a)f(left)h(rigid)f(obje)-5 b(ct)0 2150 y(with)34 b(left)h(dual)f Fr(\()p Fn(M)731 2114 y Fl(\003)771 2150 y Fn(;)17 b Fr(ev)r(\))p Fk(.)45 b(Ther)-5 b(e)34 b(is)g(a)g(bije)-5 b(ction)34 b(b)-5 b(etwe)g(en)34 b(the)h(set)f(of)g(morphisms)g Fn(f)k Fr(:)28 b Fn(A)21 b Fm(\012)h Fn(M)0 2282 y Fm(\000)-58 b(!)28 b Fn(M)41 b Fk(making)28 b Fn(M)40 b Fk(a)30 b(left)f Fn(A)p Fk(-mo)-5 b(dule)29 b(and)g(the)g(set)h(of)f(algebr)-5 b(a)29 b(morphisms)2852 2255 y Fc(e)2834 2282 y Fn(f)38 b Fr(:)28 b Fn(A)g Fm(\000)-58 b(!)28 b Fn(M)21 b Fm(\012)10 b Fn(M)3529 2246 y Fl(\003)3570 2282 y Fk(.)0 2398 y Fr(2.)42 b Fk(L)-5 b(et)26 b Fn(C)33 b Fk(b)-5 b(e)25 b(a)h(c)-5 b(o)g(algebr)g(a)24 b(in)i Fm(C)32 b Fk(and)25 b(left)h Fn(M)38 b Fm(2)28 b(C)k Fk(b)-5 b(e)26 b(a)g(left)f(rigid)h (obje)-5 b(ct)25 b(with)h(left)f(dual)h Fr(\()p Fn(M)3352 2362 y Fl(\003)3392 2398 y Fn(;)17 b Fr(ev)r(\))p Fk(.)0 2514 y(Ther)-5 b(e)31 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y(H)-83 b(j)0 4719 y Fr(The)34 b(unit)d(axiom)g(is)h(giv)m(en)h(b) m(y)1039 4899 y Fn(I)479 b(M)33 b Fm(\012)23 b Fn(M)1892 4863 y Fl(\003)p 1119 4871 414 4 v 1449 4869 a Ff(-)1286 4851 y Fj(db)1028 5241 y Fn(A)363 b(A)22 b Fm(\012)h Fn(M)33 b Fm(\012)22 b Fn(M)1989 5204 y Fl(\003)p 1130 5213 306 4 v 1352 5211 a Ff(-)1198 5185 y Fj(1)p Fl(\012)p Fj(db)2244 5241 y Fn(M)33 b Fm(\012)22 b Fn(M)2574 5204 y Fl(\003)p 2059 5213 156 4 v 2132 5211 a Ff(-)2071 5179 y Fi(f)7 b Fl(\012)p Fj(1)p 1063 5143 4 205 v 1064 5143 a Ff(?)985 5054 y Fi(u)p 1745 5143 V 1747 5143 a Ff(?)1577 5058 y Fi(u)p Fl(\012)p Fj(1)442 b(1)p Fl(\012)p Fj(1)1883 4979 y Ff(H)1966 5021 y(H)2049 5062 y(H)2132 5104 y(H)2210 5143 y(H)-83 b(j)p eop %%Page: 91 9 91 8 bop 1421 236 a Fp(3.)77 b(DUAL)32 b(OBJECTS)1353 b(91)0 425 y Fr(Con)m(v)m(ersely)42 b(let)d Fn(g)k Fr(:)d Fn(A)g Fm(\000)-60 b(!)40 b Fn(M)e Fm(\012)27 b Fn(M)1416 389 y Fl(\003)1496 425 y Fr(b)s(e)40 b(an)g(algebra)e(homomorphism)f (and)j(consider)g Fc(e)-55 b Fn(g)43 b Fr(:=)0 541 y(\(1)22 b Fm(\012)h Fr(ev)r(\)\()p Fn(g)j Fm(\012)d Fr(1\))28 b(:)h Fn(A)22 b Fm(\012)h Fn(M)39 b Fm(\000)-60 b(!)29 b Fn(M)k Fm(\012)23 b Fn(M)1530 505 y Fl(\003)1593 541 y Fm(\012)g Fn(M)39 b Fm(\000)-60 b(!)29 b Fn(M)10 b Fr(.)45 b(Then)35 b Fn(M)43 b Fr(b)s(ecomes)34 b(a)f(left)f Fn(A)p Fr(-mo)s(dule)0 658 y(since)42 807 y Fn(A)22 b Fm(\012)g Fn(A)h Fm(\012)f Fn(M)2734 b(A)22 b Fm(\012)h Fn(M)p 565 778 2666 4 v 3148 776 a Ff(-)1823 751 y Fl(r\012)p Fj(1)283 1197 y Fn(A)f Fm(\012)h Fn(M)33 b Fm(\012)22 b Fn(M)808 1161 y Fl(\003)871 1197 y Fm(\012)g Fn(M)240 b(M)33 b Fm(\012)22 b Fn(M)1634 1161 y Fl(\003)1697 1197 y Fm(\012)g Fn(M)33 b Fm(\012)23 b Fn(M)2127 1161 y Fl(\003)2189 1197 y Fm(\012)g Fn(M)p 1104 1168 172 4 v 1193 1166 a Ff(-)1036 1277 y Fi(g)r Fl(\012)p Fj(1)p Fl(\012)p Fj(1)p Fl(\012)p Fj(1)2720 1197 y Fn(M)33 b Fm(\012)23 b Fn(M)3051 1161 y Fl(\003)3113 1197 y Fm(\012)g Fn(M)p 2422 1168 270 4 v 2609 1166 a Ff(-)2381 1092 y Fj(1)p Fl(\012)p Fj(ev)13 b Fl(\012)p Fj(1)p Fl(\012)p Fj(1)139 1977 y Fn(A)22 b Fm(\012)h Fn(M)2928 b(M)p 467 1949 2861 4 v 3245 1947 a Ff(-)1880 1915 y Fa(e)-39 b Fi(g)p 287 1879 4 1034 v 289 1879 a Ff(?)124 1379 y Fj(1)p Fl(\012)p Fa(e)g Fi(g)1550 1587 y Fn(M)33 b Fm(\012)23 b Fn(M)1881 1551 y Fl(\003)1943 1587 y Fm(\012)g Fn(M)p 1847 1489 4 254 v 1849 1489 a Ff(?)1888 1380 y Fj(1)p Fl(\012)p Fj(1)p Fl(\012)p Fj(1)p Fl(\012)p Fj(ev)p 3407 1879 4 1034 v 3409 1879 a Ff(?)3448 1379 y Fa(e)-39 b Fi(g)544 987 y Fj(1)p Fl(\012)p Fi(g)r Fl(\012)p Fj(1)357 928 y Ff(@)440 1011 y(@)523 1094 y(@)528 1099 y(@)-83 b(R)543 1575 y Fj(1)p Fl(\012)p Fj(1)p Fl(\012)p Fj(ev)603 1318 y Ff(\001)562 1401 y(\001)520 1484 y(\001)479 1567 y(\001)437 1650 y(\001)396 1733 y(\001)354 1816 y(\001)323 1878 y(\001)-42 b(\013)1135 987 y Fi(g)r Fl(\012)p Fi(g)r Fl(\012)p Fj(1)562 866 y Ff(X)645 887 y(X)728 907 y(X)811 928 y(X)894 949 y(X)977 970 y(X)1060 990 y(X)1143 1011 y(X)1226 1032 y(X)1309 1053 y(X)1392 1073 y(X)1475 1094 y(X)1493 1099 y(X)-83 b(z)876 1767 y Fi(g)r Fl(\012)p Fj(1)562 1879 y Ff(\030)645 1858 y(\030)728 1837 y(\030)811 1816 y(\030)894 1795 y(\030)977 1775 y(\030)1060 1754 y(\030)1143 1733 y(\030)1226 1712 y(\030)1309 1692 y(\030)1392 1671 y(\030)1475 1650 y(\030)1493 1646 y(\030)g(:)2695 1770 y Fj(1)p Fl(\012)p Fj(ev)2122 1646 y Ff(X)2205 1667 y(X)2288 1687 y(X)2371 1708 y(X)2454 1729 y(X)2537 1750 y(X)2620 1770 y(X)2703 1791 y(X)2786 1812 y(X)2869 1833 y(X)2952 1853 y(X)3035 1874 y(X)3053 1879 y(X)g(z)2994 1575 y Fj(1)p Fl(\012)p Fj(ev)3053 1318 y Ff(A)3094 1401 y(A)3136 1484 y(A)3177 1567 y(A)3219 1650 y(A)3260 1733 y(A)3302 1816 y(A)3333 1878 y(A)-42 b(U)3027 987 y Fi(g)r Fl(\012)p Fj(1)3257 928 y Ff(\000)3174 1011 y(\000)3091 1094 y(\000)3087 1099 y(\000)-83 b(\011)0 2104 y Fr(and)929 2565 y Fn(A)22 b Fm(\012)h Fn(M)342 b(M)33 b Fm(\012)23 b Fn(M)1891 2529 y Fl(\003)1953 2565 y Fm(\012)g Fn(M)p 1257 2537 274 4 v 1448 2535 a Ff(-)1331 2597 y Fi(g)r Fl(\012)p Fj(1)2587 2565 y Fn(M)p 2187 2537 372 4 v 2476 2535 a Ff(-)2292 2597 y Fj(1)p Fl(\012)p Fj(ev)1027 2179 y Fn(M)p 1077 2467 4 254 v 1079 2467 a Ff(?)909 2358 y Fi(u)p Fl(\012)p Fj(1)1226 2360 y(db)12 b Fl(\012)p Fj(1)1216 2255 y Ff(H)1299 2296 y(H)1382 2338 y(H)1465 2379 y(H)1548 2421 y(H)1631 2462 y(H)1639 2467 y(H)-83 b(j)1925 2362 y Fj(1)1352 2234 y Ff(X)1435 2255 y(X)1518 2276 y(X)1601 2296 y(X)1684 2317 y(X)1767 2338 y(X)1850 2359 y(X)1933 2379 y(X)2016 2400 y(X)2099 2421 y(X)2182 2442 y(X)2265 2462 y(X)2283 2467 y(X)g(z)0 2699 y Fr(comm)m(ute.)149 2815 y(2.)59 b(\()p Fn(M)426 2779 y Fl(\003)466 2815 y Fn(;)17 b Fr(ev)r(\))37 b(is)g(a)g(left)g(dual)g(for)g Fn(M)48 b Fr(in)37 b(the)h(category)g Fm(C)43 b Fr(if)37 b(and)g(only)g(if)g(\()p Fn(M)3094 2779 y Fl(\003)3134 2815 y Fn(;)17 b Fr(db\))37 b(is)h(the)0 2932 y(righ)m(t)d(dual)g(for)h Fn(M)47 b Fr(in)35 b(the)h(dual)g(category)g Fm(C)1716 2895 y Fi(op)1790 2932 y Fr(.)54 b(So)36 b(if)f(w)m(e)i(dualize)e(the)h (result)g(of)g(part)f(1.)54 b(w)m(e)0 3048 y(ha)m(v)m(e)34 b(to)e(c)m(hange)i(sides,)f(hence)h(2.)p 3529 3048 4 66 v 3533 2985 59 4 v 3533 3048 V 3591 3048 4 66 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF