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Fi(V)1828 2572 y Fg(\003)1965 2608 y Fj(\()p Fh(\022)28 b Fj(\()p Fi(V)44 b Fh(\012)23 b Fi(V)e Fj(\))2463 2572 y Fg(\003)2503 2608 y Fj(\).)100 2806 y(\(35\))41 b(F)-8 b(or)33 b(morphisms)g Fi(f)42 b Fj(:)31 b Fi(I)39 b Fh(\000)-60 b(!)31 b Fi(M)45 b Fj(and)34 b Fi(g)g Fj(:)d Fi(I)39 b Fh(\000)-59 b(!)30 b Fi(N)45 b Fj(in)34 b(a)g(monoidal)d(category)k(w)m(e)h(de\014ne)315 2922 y(\()p Fi(f)i Fh(\012)28 b Fj(1)41 b(:)h Fi(N)52 b Fh(\000)-60 b(!)41 b Fi(M)e Fh(\012)28 b Fi(N)10 b Fj(\))42 b(:=)f(\()p Fi(f)d Fh(\012)29 b Fj(1)1820 2937 y Fe(I)1859 2922 y Fj(\))p Fi(\032)p Fj(\()p Fi(I)8 b Fj(\))2074 2886 y Fg(\000)p Fd(1)2209 2922 y Fj(and)41 b(\(1)27 b Fh(\012)i Fi(g)44 b Fj(:)e Fi(M)52 b Fh(\000)-60 b(!)42 b Fi(M)c Fh(\012)28 b Fi(N)10 b Fj(\))42 b(:=)315 3038 y(\(1)21 b Fh(\012)i Fi(g)t Fj(\))p Fi(\025)p Fj(\()p Fi(I)8 b Fj(\))796 3002 y Fg(\000)p Fd(1)889 3038 y Fj(.)44 b(Sho)m(w)33 b(that)f(the)h(diagram)1418 3707 y Fi(N)394 b(M)33 b Fh(\012)22 b Fi(N)p 1535 3679 326 4 v 1778 3677 a Fb(-)1632 3743 y Fe(f)7 b Fg(\012)p Fd(1)1437 3224 y Fi(I)515 b(M)p 1517 3192 450 4 v 1884 3190 a Fb(-)1720 3158 y Fe(f)p 1461 3609 4 351 v 1462 3609 a Fb(?)1387 3442 y Fe(g)p 2045 3609 V 2047 3609 a Fb(?)2086 3449 y Fd(1)p Fg(\012)p Fe(g)315 3866 y Fj(comm)m(utes.)100 4063 y(\(36\))41 b(Let)29 b Fi(G)g Fj(b)s(e)g(a)f(\014nite)h(group)g (and)g Fc(K)1574 4027 y Fe(G)1667 4063 y Fj(:=)f Fc(K)17 b Fj([)p Fi(G)p Fj(])2000 4027 y Fg(\003)2075 4063 y Fj(the)29 b(dual)f(of)h(the)g(group)g(algebra.)41 b(Sho)m(w)315 4179 y(that)35 b Fc(K)601 4143 y Fe(G)702 4179 y Fj(is)g(a)g(Hopf)h (algebra)f(and)h(that)f(eac)m(h)i(mo)s(dule)d(structure)j Fc(K)18 b Fj([)p Fi(G)p Fj(])30 b Fh(\012)25 b Fi(M)44 b Fh(\000)-60 b(!)33 b Fi(M)315 4296 y Fj(translates)26 b(to)h(the)g(structure)h(of)e(a)h(como)s(dule)e Fi(M)39 b Fh(\000)-60 b(!)27 b Fc(K)2402 4260 y Fe(G)2477 4296 y Fh(\012)10 b Fi(M)39 b Fj(and)27 b(con)m(v)m(ersely)-8 b(.)43 b(Sho)m(w)315 4412 y(that)32 b(this)g(de\014nes)i(a)f(monoidal)c (equiv)-5 b(alence)33 b(of)f(categories.)315 4528 y(Describ)s(e)j(the)g (group)g(v)-5 b(alued)35 b(functor)g Fc(K)17 b Fj(-)6 b Fa(c)g Fh(A)p Fa(lg)16 b Fj(\()p Fc(K)2251 4492 y Fe(G)2316 4528 y Fi(;)h Fh(\000)p Fj(\))35 b(in)f(terms)h(of)g(sets)h(and)f (their)315 4644 y(group)d(structure.)0 5159 y(Due)h(date:)43 b(T)-8 b(uesda)m(y)g(,)35 b(25.06.2002,)c(16:15)h(in)f(Lecture)j(Hall)d (E41)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF