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b(2)147 1089 y Fr(Hopf)45 b(Algebras,)h(Algebraic,)g(F)-11 b(ormal,)45 b(and)g(Quan)l(tum)g (Groups)1762 5365 y Fq(49)p eop %%Page: 50 2 50 1 bop 0 236 a Fq(50)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)1359 425 y Fp(4.)57 b(F)-9 b(ormal)36 b(Groups)149 600 y Fs(Consider)25 b(no)m(w)g Fo(K)18 b Fs(-)5 b Fp(cCoalg)24 b Fs(the)h(category)g(of)f(co)s(comm)m (utativ)m(e)f Fo(K)18 b Fs(-coalgebras.)45 b(Let)25 b Fn(C)r(;)17 b(D)30 b Fm(2)0 716 y Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(.)51 b(Then)37 b Fn(C)30 b Fm(\012)25 b Fn(D)37 b Fs(is)e(again)f(a)g(co)s(comm)m(utativ)m(e)g Fo(K)18 b Fs(-coalgebra)39 b(b)m(y)d(Problem)e(A.11.4.)0 832 y(In)f(fact)f(this)h(holds)f(also)f(for)h(non-comm)m(utativ)m(e)g Fo(K)17 b Fs(-algebras,)38 b(but)33 b(in)e Fo(K)18 b Fs(-)5 b Fp(cCoalg)32 b Fs(w)m(e)i(ha)m(v)m(e)149 984 y Fp(Prop)s(osition)i(2.4.1.)49 b Fl(The)26 b(tensor)h(pr)-5 b(o)g(duct)27 b(in)f Fo(K)17 b Fl(-)6 b Fp(cCoalg)26 b Fl(is)h(the)g(\(c)-5 b(ate)g(goric)g(al\))25 b(pr)-5 b(o)g(duct.)149 1136 y Fk(Pr)n(oof.)42 b Fs(Let)37 b Fn(f)45 b Fm(2)35 b Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(Z)r(;)17 b(C)7 b Fs(\))p Fn(;)17 b(g)36 b Fm(2)f Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(Z)r(;)17 b(D)s Fs(\).)54 b(The)38 b(map)d(\()p Fn(f)5 b(;)17 b(g)t Fs(\))33 b(:)i Fn(Z)41 b Fm(\000)-60 b(!)0 1252 y Fn(C)36 b Fm(\012)29 b Fn(D)45 b Fs(de\014ned)f(b)m(y)g(\()p Fn(f)5 b(;)17 b(g)t Fs(\)\()p Fn(z)t Fs(\))44 b(:=)1370 1177 y Fj(P)1492 1252 y Fn(f)11 b Fs(\()p Fn(z)1634 1268 y Fi(\(1\))1728 1252 y Fs(\))29 b Fm(\012)h Fn(g)t Fs(\()p Fn(z)2036 1268 y Fi(\(2\))2130 1252 y Fs(\))42 b(is)g(the)h(unique)g (homomorphism)c(of)0 1368 y(coalgebras)g(suc)m(h)h(that)f(\(1)26 b Fm(\012)h Fn(")1186 1383 y Fh(D)1250 1368 y Fs(\)\()p Fn(f)5 b(;)17 b(g)t Fs(\)\()p Fn(z)t Fs(\))38 b(=)h Fn(f)11 b Fs(\()p Fn(z)t Fs(\))39 b(and)h(\()p Fn(")2294 1383 y Fh(C)2379 1368 y Fm(\012)27 b Fs(1\)\()p Fn(f)5 b(;)17 b(g)t Fs(\)\()p Fn(z)t Fs(\))38 b(=)h Fn(g)t Fs(\()p Fn(z)t Fs(\))g(or)f(suc)m(h)0 1485 y(that)32 b(the)h(diagram)1370 1965 y Fn(C)217 b(C)29 b Fm(\012)23 b Fn(D)p 1476 1937 154 4 v 1476 1935 a Fg(\033)1508 1903 y Fh(p)1544 1914 y Ff(C)2147 1965 y Fn(D)p 1968 1937 151 4 v 2036 1935 a Fg(-)1998 1903 y Fh(p)2034 1914 y Ff(D)1502 1757 y Fh(f)1647 1697 y Fg(\000)1564 1780 y(\000)1481 1863 y(\000)1477 1867 y(\000)-83 b(\011)2054 1748 y Fh(g)1867 1697 y Fg(@)1950 1780 y(@)2033 1863 y(@)2037 1867 y(@)g(R)1762 1580 y Fn(Z)p 1797 1867 4 254 v 1799 1867 a Fg(?)1838 1830 y Fi(\()p Fh(f)s(;g)r Fi(\))0 2185 y Fs(comm)m(utes,)31 b(where)g Fn(p)812 2200 y Fh(C)871 2185 y Fs(\()p Fn(c)17 b Fm(\012)h Fn(d)p Fs(\))27 b(=)h(\(1)17 b Fm(\012)h Fn(")p Fs(\)\()p Fn(c)f Fm(\012)g Fn(d)p Fs(\))28 b(=)f Fn(c")p Fs(\()p Fn(d)p Fs(\))i(and)i Fn(p)2458 2200 y Fh(D)2522 2185 y Fs(\()p Fn(c)17 b Fm(\012)h Fn(d)p Fs(\))27 b(=)h(\()p Fn(")16 b Fm(\012)i Fs(1\)\()p Fn(c)f Fm(\012)h Fn(d)p Fs(\))27 b(=)0 2301 y Fn(")p Fs(\()p Fn(c)p Fs(\))p Fn(d)32 b Fs(are)g(homomorphisms)f(of)h(coalgebras.)p 3529 2301 4 66 v 3533 2239 59 4 v 3533 2301 V 3591 2301 4 66 v 149 2467 a(So)h(the)g(category)g Fo(K)17 b Fs(-)6 b Fp(cCoalg)32 b Fs(has)h(\014nite)f(pro)s(ducts)h(and)g(also)f(a)g (\014nal)g(ob)5 b(ject)33 b Fo(K)18 b Fs(.)149 2619 y Fp(De\014nition)37 b(2.4.2.)49 b Fs(A)d Fl(formal)g(gr)-5 b(oup)46 b Fs(is)f(a)h(group)f(in)g(the)i(category)f(of)f Fo(K)18 b Fs(-)5 b Fp(cCoalg)45 b Fs(of)0 2736 y(co)s(comm)m(utativ)m (e)31 b(coalgebras.)149 2888 y(A)38 b(formal)e(group)i Fn(G)g Fs(de\014nes)h(a)f(con)m(tra)m(v)-5 b(arian)m(t)37 b(represen)m(table)j(functor)e(from)f Fo(K)17 b Fs(-)6 b Fp(cCoalg)0 3004 y Fs(to)32 b Fp(Gr)p Fs(,)h(the)g(category)g(of)f (groups.)149 3156 y Fp(Prop)s(osition)k(2.4.3.)49 b Fl(L)-5 b(et)40 b Fn(H)j Fm(2)36 b Fo(K)17 b Fl(-)6 b Fp(cCoalg)p Fl(.)57 b Fn(H)47 b Fl(a)39 b(r)-5 b(epr)g(esents)38 b(a)h(formal)g(gr)-5 b(oup)39 b(if)g(and)0 3272 y(only)c(if)f(ther)-5 b(e)35 b(ar)-5 b(e)35 b(given)f(morphisms)f(in)i Fo(K)17 b Fl(-)6 b Fp(cCoalg)809 3429 y Fm(r)27 b Fs(:)h Fn(H)i Fm(\012)22 b Fn(H)36 b Fm(\000)-58 b(!)28 b Fn(H)r(;)116 b(u)28 b Fs(:)f Fo(K)51 b Fm(\000)-58 b(!)29 b Fn(H)r(;)116 b(S)34 b Fs(:)27 b Fn(H)36 b Fm(\000)-58 b(!)28 b Fn(H)0 3586 y Fl(such)35 b(that)g(the)g(fol)5 b(lowing)33 b(diagr)-5 b(ams)34 b(c)-5 b(ommute)1771 4144 y Fn(H)30 b Fm(\012)22 b Fn(H)594 b(H)p 2099 4116 529 4 v 2545 4114 a Fg(-)2334 4095 y Fe(r)p 2699 4046 4 254 v 2700 4046 a Fg(?)2739 3942 y Fe(r)750 3919 y Fl(\(asso)-5 b(ciativity\))p 1919 4046 V 1920 4046 a Fg(?)1732 3939 y Fe(r\012)p Fi(1)1666 3754 y Fn(H)30 b Fm(\012)22 b Fn(H)30 b Fm(\012)22 b Fn(H)384 b(H)30 b Fm(\012)22 b Fn(H)p 2204 3726 318 4 v 2439 3724 a Fg(-)2289 3699 y Fi(1)p Fe(\012r)1137 4689 y Fn(H)30 b Fm(\012)23 b Fn(H)1276 b(H)p 1465 4662 1211 4 v 2593 4660 a Fg(-)2041 4640 y Fe(r)1114 4300 y Fo(K)46 b Fm(\012)23 b Fn(H)1430 4272 y Fm(\030)1431 4304 y Fs(=)1535 4300 y Fn(H)1651 4272 y Fm(\030)1652 4304 y Fs(=)1756 4300 y Fn(H)30 b Fm(\012)23 b Fo(K)579 b Fn(H)29 b Fm(\012)23 b Fn(H)p 2072 4271 499 4 v 2488 4269 a Fg(-)2256 4244 y Fh(u)p Fe(\012)p Fi(1)702 4465 y Fl(\(unit\))p 2747 4591 4 254 v 2749 4591 a Fg(?)2788 4487 y Fe(r)p 1285 4591 V 1287 4591 a Fg(?)1117 4482 y Fi(1)p Fe(\012)p Fh(u)2263 4486 y Fi(1)1784 4365 y Fg(P)1867 4393 y(P)1950 4421 y(P)2033 4449 y(P)2116 4476 y(P)2199 4504 y(P)2282 4532 y(P)2365 4559 y(P)2448 4587 y(P)2461 4591 y(P)-83 b(q)945 4982 y Fl(\(inverse\))1486 4821 y Fn(H)412 b Fo(K)p 1604 4788 348 4 v 1869 4786 a Fg(-)1761 4768 y Fh(")2461 4821 y Fn(H)p 2047 4788 386 4 v 2350 4786 a Fg(-)2220 4755 y Fh(\021)p 1529 5108 4 254 v 1530 5108 a Fg(?)1433 5004 y Fi(\001)1381 5207 y Fn(H)30 b Fm(\012)22 b Fn(H)684 b(H)30 b Fm(\012)22 b Fn(H)p 1709 5179 619 4 v 2245 5177 a Fg(-)1938 5150 y Fh(S)t Fe(\012)p Fi(id)1932 5242 y(id)11 b Fe(\012)p Fh(S)p 2504 5108 4 254 v 2505 4938 a Fg(6)2544 5004 y Fe(r)p eop %%Page: 51 3 51 2 bop 1416 236 a Fq(4.)35 b(F)n(ORMAL)d(GR)n(OUPS)1312 b(51)149 425 y Fk(Pr)n(oof.)42 b Fs(F)-8 b(or)36 b(an)g(arbitrary)f (formal)f(group)i Fn(H)43 b Fs(w)m(e)37 b(get)f Fm(r)e Fs(=)g Fn(p)2573 440 y Fi(1)2637 425 y Fm(\003)24 b Fn(p)2760 440 y Fi(2)2833 425 y Fm(2)34 b Fo(K)18 b Fs(-)5 b Fp(cCoalg)p Fs(\()p Fn(H)32 b Fm(\012)0 541 y Fn(H)r(;)17 b(H)8 b Fs(\),)35 b Fn(u)d Fs(=)g Fn(e)h Fm(2)g Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fo(K)17 b Fn(;)h(H)8 b Fs(\),)41 b(and)35 b Fn(S)j Fs(=)32 b(\(id\))2032 505 y Fe(\000)p Fi(1)2158 541 y Fm(2)h Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(H)r(;)17 b(H)8 b Fs(\).)51 b(These)37 b(maps,)0 658 y(the)c(Y)-8 b(oneda)33 b(Lemma)e(and)i(Remark)f(2.2.6)g(lead)g(to) g(the)h(pro)s(of)f(of)g(the)h(prop)s(osition.)p 3529 658 4 66 v 3533 595 59 4 v 3533 658 V 3591 658 4 66 v 149 821 a Fp(Remark)k(2.4.4.)50 b Fs(In)23 b(particular)e(the)i (represen)m(ting)h(ob)5 b(ject)24 b(\()p Fn(H)r(;)17 b Fm(r)p Fn(;)g(u;)g Fs(\001)p Fn(;)g(";)g(S)6 b Fs(\))20 b(of)j(a)f(formal)0 937 y(group)34 b Fn(G)g Fs(is)g(a)g(co)s(comm)m (utativ)m(e)f(Hopf)i(algebra)e(and)h(ev)m(ery)i(suc)m(h)g(Hopf)e (algebra)f(represen)m(ts)k(a)0 1053 y(formal)e(group.)59 b(Hence)39 b(the)f(category)g(of)f(formal)e(groups)j(is)g(equiv)-5 b(alen)m(t)37 b(to)g(the)h(category)g(of)0 1170 y(co)s(comm)m(utativ)m (e)31 b(Hopf)i(algebras.)149 1333 y Fp(Corollary)j(2.4.5.)50 b Fl(A)40 b(c)-5 b(o)g(algebr)g(a)40 b Fn(H)45 b Fm(2)38 b Fo(K)18 b Fl(-)6 b Fp(cCoalg)40 b Fl(r)-5 b(epr)g(esents)39 b(a)i(formal)e(gr)-5 b(oup)40 b(if)h(and)0 1449 y(only)35 b(if)f Fn(H)43 b Fl(is)34 b(a)h(c)-5 b(o)g(c)g(ommutative)34 b(Hopf)h(algebr)-5 b(a.)149 1565 y(The)25 b(c)-5 b(ate)g(gory)26 b(of)f(c)-5 b(o)g(c)g(ommutative)25 b(Hopf)h(algebr)-5 b(as)24 b(is)h(e)-5 b(quivalent)26 b(to)f(the)h(c)-5 b(ate)g(gory)25 b(of)h(formal)0 1682 y(gr)-5 b(oups.)149 1845 y Fp(Corollary)36 b(2.4.6.)50 b Fl(The)34 b(fol)5 b(lowing)33 b(c)-5 b(ate)g(gories)34 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)149 1982 y Fs(1.)42 b Fl(The)34 b(c)-5 b(ate)g(gory)35 b(of)f(c)-5 b(ommutative,)35 b(c)-5 b(o)g(c)g(ommutative)34 b(Hopf)h(algebr)-5 b(as.)149 2098 y Fs(2.)42 b Fl(The)34 b(c)-5 b(ate)g(gory)35 b(of)f(c)-5 b(ommutative)35 b(formal)f(gr)-5 b(oups.)149 2214 y Fs(3.)42 b Fl(The)34 b(dual)h(of)f(the)h(c)-5 b(ate)g(gory)35 b(of)g(c)-5 b(ommutative)34 b(a\016ne)g(algebr)-5 b(aic)34 b(gr)-5 b(oups.)149 2377 y Fp(Example)37 b(2.4.7.)199 b Fs(1.)41 b(Group)32 b(algebras)g Fo(K)18 b Fn(G)38 b Fs(are)33 b(formal)d(groups.)149 2494 y(2.)42 b(Univ)m(ersal)32 b(en)m(v)m(eloping)h(algebras)f Fn(U)10 b Fs(\()p Fd(g)p Fs(\))33 b(of)f(Lie)g(algebras)g Fd(g)h Fs(are)f(formal)e(groups.)149 2610 y(3.)42 b(The)33 b(tensor)h(algebra)d Fn(T)14 b Fs(\()p Fn(V)21 b Fs(\))33 b(and)f(the)h(symmetric)f(algebra)f Fn(S)6 b Fs(\()p Fn(V)22 b Fs(\))32 b(are)h(formal)d(groups.)149 2726 y(4.)42 b(Let)48 b Fn(C)55 b Fs(b)s(e)48 b(a)f(co)s(comm)m(utativ) m(e)g(coalgebra)g(and)h Fn(G)g Fs(b)s(e)g(a)f(group.)90 b(Then)49 b(the)f(group)267 2842 y Fo(K)17 b Fn(G)p Fs(\()p Fn(C)7 b Fs(\))36 b(=)30 b Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(C)r(;)17 b Fo(K)g Fn(G)p Fs(\))39 b(is)34 b(isomorphic)e(to)h(the)i(set)f(of)g(families)c(\()p Fn(h)3128 2806 y Fe(\003)3128 2867 y Fh(g)3169 2842 y Fm(j)p Fn(g)i Fm(2)f Fn(G)p Fs(\))i(of)267 2958 y(decomp)s(ositions)26 b(of)g(the)i(unit)e(of)h Fn(C)1594 2922 y Fe(\003)1660 2958 y Fs(in)m(to)g(a)g(sum)g(of)f(orthogonal)g(idemp)s(oten)m(ts)h Fn(h)3322 2922 y Fe(\003)3322 2983 y Fh(g)3390 2958 y Fm(2)h Fn(C)3561 2922 y Fe(\003)267 3075 y Fs(that)k(are)h(lo)s(cally)d (\014nite.)416 3191 y(T)-8 b(o)46 b(see)g(this)f(em)m(b)s(ed)g Fo(K)18 b Fs(-)5 b Fp(cCoalg)p Fs(\()p Fn(C)r(;)17 b Fo(K)f Fn(G)p Fs(\))55 b Fm(\022)50 b Fs(Hom)o(\()p Fn(C)r(;)17 b Fo(K)g Fn(G)p Fs(\))51 b(and)45 b(em)m(b)s(edthe)h(set)267 3307 y(Hom)o(\()p Fn(C)r(;)17 b Fo(K)g Fn(G)p Fs(\))37 b(in)m(to)31 b(the)h(set)h(\()p Fn(C)1478 3271 y Fe(\003)1517 3307 y Fs(\))1555 3271 y Fh(G)1646 3307 y Fs(of)e Fn(G)p Fs(-families)d(of)j(elemen)m(ts)h(in)f(the)h(algebra)f Fn(C)3426 3271 y Fe(\003)3497 3307 y Fs(b)m(y)267 3423 y Fn(h)j Fm(7!)f Fs(\()p Fn(h)584 3387 y Fe(\003)584 3448 y Fh(g)624 3423 y Fs(\))j(with)g Fn(h)p Fs(\()p Fn(c)p Fs(\))e(=)1242 3349 y Fj(P)1347 3452 y Fh(g)r Fe(2)p Fh(G)1506 3423 y Fn(h)1562 3387 y Fe(\003)1562 3448 y Fh(g)1602 3423 y Fs(\()p Fn(c)p Fs(\))p Fn(g)t Fs(.)53 b(The)37 b(linear)e(map)g Fn(h)h Fs(is)g(a)g(homomorphism)d(of) 267 3548 y(coalgebras)h(i\013)f(\()p Fn(h)23 b Fm(\012)h Fn(h)p Fs(\)\001)31 b(=)f(\001)p Fn(h)35 b Fs(and)g Fn("h)30 b Fs(=)g Fn(")k Fs(i\013)2190 3474 y Fj(P)2311 3548 y Fn(h)p Fs(\()p Fn(c)2447 3564 y Fi(\(1\))2542 3548 y Fs(\))23 b Fm(\012)h Fn(h)p Fs(\()p Fn(c)2840 3564 y Fi(\(2\))2934 3548 y Fs(\))31 b(=)3109 3474 y Fj(P)3231 3548 y Fn(h)p Fs(\()p Fn(c)p Fs(\))3405 3564 y Fi(\(1\))3523 3548 y Fm(\012)267 3665 y Fn(h)p Fs(\()p Fn(c)p Fs(\))441 3680 y Fi(\(2\))562 3665 y Fs(and)c Fn(")p Fs(\()p Fn(h)p Fs(\()p Fn(c)p Fs(\)\))g(=)h Fn(")p Fs(\()p Fn(c)p Fs(\))e(for)h(all)e Fn(c)i Fm(2)h Fn(C)34 b Fs(i\013)2015 3590 y Fj(P)2137 3665 y Fn(h)2193 3628 y Fe(\003)2193 3689 y Fh(g)2233 3665 y Fs(\()p Fn(c)2313 3680 y Fi(\(1\))2407 3665 y Fs(\))p Fn(g)14 b Fm(\012)d Fn(h)2650 3628 y Fe(\003)2650 3691 y Fh(l)2690 3665 y Fs(\()p Fn(c)2770 3680 y Fi(\(2\))2864 3665 y Fs(\))p Fn(l)30 b Fs(=)3065 3590 y Fj(P)3186 3665 y Fn(h)3242 3628 y Fe(\003)3242 3689 y Fh(g)3283 3665 y Fs(\()p Fn(c)p Fs(\))p Fn(g)14 b Fm(\012)d Fn(g)267 3785 y Fs(and)457 3710 y Fj(P)579 3785 y Fn(h)635 3749 y Fe(\003)635 3810 y Fh(g)676 3785 y Fs(\()p Fn(c)p Fs(\))29 b(=)g Fn(")p Fs(\()p Fn(c)p Fs(\))k(i\013)1242 3710 y Fj(P)1364 3785 y Fn(h)1420 3749 y Fe(\003)1420 3810 y Fh(g)1460 3785 y Fs(\()p Fn(c)1540 3801 y Fi(\(1\))1635 3785 y Fs(\))p Fn(h)1729 3749 y Fe(\003)1729 3811 y Fh(l)1768 3785 y Fs(\()p Fn(c)1848 3801 y Fi(\(2\))1942 3785 y Fs(\))d(=)f Fn(\016)2158 3800 y Fh(g)r(l)2220 3785 y Fn(h)2276 3749 y Fe(\003)2276 3810 y Fh(g)2316 3785 y Fs(\()p Fn(c)p Fs(\))34 b(and)2658 3710 y Fj(P)2780 3785 y Fn(h)2836 3749 y Fe(\003)2836 3810 y Fh(g)2906 3785 y Fs(=)29 b Fn(")k Fs(i\013)g Fn(h)3264 3749 y Fe(\003)3264 3810 y Fh(g)3327 3785 y Fm(\003)22 b Fn(h)3455 3749 y Fe(\003)3455 3811 y Fh(l)3524 3785 y Fs(=)267 3906 y Fn(\016)310 3921 y Fh(g)r(l)372 3906 y Fn(h)428 3870 y Fe(\003)428 3930 y Fh(g)501 3906 y Fs(and)692 3831 y Fj(P)814 3906 y Fn(h)870 3870 y Fe(\003)870 3930 y Fh(g)939 3906 y Fs(=)29 b(1)1093 3921 y Fh(C)1148 3902 y Fc(\003)1187 3906 y Fs(.)46 b(F)-8 b(urthermore)33 b(the)g(families)e(m)m(ust)i(b)s(e)g(lo)s(cally)e(\014nite,)i(i.e.)45 b(for)267 4022 y(eac)m(h)33 b Fn(c)28 b Fm(2)g Fn(C)40 b Fs(only)32 b(\014nitely)f(man)m(y)i(of)f(them)g(giv)m(e)h(non-zero)g (v)-5 b(alues.)149 4138 y(5.)42 b(Let)25 b Fn(C)31 b Fs(b)s(e)25 b(a)f(co)s(comm)m(utativ)m(e)f(coalgebra)h(and)h Fo(K)17 b Fs([)p Fn(x)q Fs(])30 b(b)s(e)25 b(the)g(Hopf)f(algebra)g (with)g(\001\()p Fn(x)p Fs(\))k(=)267 4254 y Fn(x)k Fm(\012)h Fs(1)f(+)g(1)f Fm(\012)i Fn(x)48 b Fs(\(the)f(symmetric)f(algebra)g(of) h(the)h(one)f(dimensional)e(v)m(ector)j(space)267 4374 y Fo(K)17 b Fn(x)q Fs(\).)81 b(W)-8 b(e)44 b(em)m(b)s(ed)g(as)f(b)s (efore)h Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(C)r(;)17 b Fo(K)g Fs([)p Fn(x)p Fs(]\))52 b Fm(\022)47 b Fs(Hom)o(\()p Fn(C)r(;)17 b Fo(K)g Fs([)p Fn(x)p Fs(]\))52 b(=)46 b(\()p Fn(C)3323 4337 y Fe(\003)3362 4374 y Fs(\))3400 4337 y Fe(f)p Fb(N)3498 4346 y Fa(0)3533 4337 y Fe(g)3573 4374 y Fs(,)267 4490 y(the)41 b(set)g(of)f(lo)s(cally)e(\014nite)j Fp(N)1385 4505 y Fi(0)1424 4490 y Fs(-families)c(in)j Fn(C)2019 4454 y Fe(\003)2099 4490 y Fs(b)m(y)h Fn(h)p Fs(\()p Fn(c)p Fs(\))h(=)2575 4415 y Fj(P)2680 4441 y Fe(1)2680 4519 y Fh(i)p Fi(=0)2815 4490 y Fn(h)2871 4454 y Fe(\003)2871 4514 y Fh(i)2911 4490 y Fs(\()p Fn(c)p Fs(\))p Fn(x)3084 4454 y Fh(i)3112 4490 y Fs(.)68 b(The)42 b(map)267 4607 y Fn(h)d Fs(is)f(a)g(homomorphism)e(of)i(coalgebras)h (i\013)e(\001\()p Fn(h)p Fs(\()p Fn(c)p Fs(\)\))i(=)2450 4533 y Fj(P)2571 4607 y Fn(h)2627 4571 y Fe(\003)2627 4632 y Fh(i)2667 4607 y Fs(\()p Fn(c)p Fs(\)\()p Fn(x)26 b Fm(\012)h Fs(1)f(+)h(1)f Fm(\012)g Fn(x)p Fs(\))3457 4571 y Fh(i)3524 4607 y Fs(=)267 4654 y Fj(P)389 4729 y Fn(h)445 4692 y Fe(\003)445 4753 y Fh(i)484 4729 y Fs(\()p Fn(c)p Fs(\))602 4648 y Fj(\000)648 4684 y Fh(i)649 4763 y(l)672 4648 y Fj(\001)717 4729 y Fn(x)772 4692 y Fh(l)799 4729 y Fm(\012)p Fn(x)931 4692 y Fh(i)p Fe(\000)p Fh(l)1065 4729 y Fs(=)h(\()p Fn(h)p Fm(\012)p Fn(h)p Fs(\)\001\()p Fn(c)p Fs(\))i(=)1765 4654 y Fj(P)1886 4729 y Fn(h)1942 4692 y Fe(\003)1942 4753 y Fh(i)1982 4729 y Fs(\()p Fn(c)2062 4744 y Fi(\(1\))2156 4729 y Fs(\))p Fn(h)2250 4692 y Fe(\003)2250 4753 y Fh(j)2290 4729 y Fs(\()p Fn(c)2370 4744 y Fi(\(2\))2464 4729 y Fs(\))p Fn(x)2557 4692 y Fh(i)2586 4729 y Fm(\012)p Fn(x)2718 4692 y Fh(j)2777 4729 y Fs(and)22 b Fn(")p Fs(\()3040 4654 y Fj(P)3161 4729 y Fn(h)3217 4692 y Fe(\003)3217 4753 y Fh(i)3257 4729 y Fs(\()p Fn(c)p Fs(\))p Fn(x)3430 4692 y Fh(i)3459 4729 y Fs(\))27 b(=)267 4863 y Fn(")p Fs(\()p Fn(c)p Fs(\))32 b(i\013)f Fn(h)635 4826 y Fe(\003)635 4887 y Fh(i)697 4863 y Fm(\003)22 b Fn(h)825 4826 y Fe(\003)825 4887 y Fh(j)892 4863 y Fs(=)996 4782 y Fj(\000)1042 4818 y Fh(i)p Fi(+)p Fh(j)1085 4897 y(i)1153 4782 y Fj(\001)1198 4863 y Fn(h)1254 4826 y Fe(\003)1254 4887 y Fh(i)p Fi(+)p Fh(j)1403 4863 y Fs(and)32 b Fn(h)1648 4826 y Fe(\003)1648 4887 y Fi(0)1716 4863 y Fs(=)27 b Fn(")g Fs(=)h(1)2045 4878 y Fh(C)2100 4859 y Fc(\003)2140 4863 y Fs(.)416 4983 y(No)m(w)36 b(let)e Fo(K)58 b Fs(b)s(e)35 b(a)f(\014eld)h(of)f(c)m (haracteristic)g(zero.)51 b(Let)35 b Fn(p)2519 4998 y Fh(i)2578 4983 y Fs(:=)c Fn(h)2768 4947 y Fe(\003)2768 5008 y Fh(i)2808 4983 y Fn(=i)p Fs(!.)50 b(Then)36 b(the)f(con-)267 5099 y(ditions)c(simplify)g(to)h Fn(p)1125 5114 y Fh(i)1153 5099 y Fn(p)1202 5114 y Fh(j)1267 5099 y Fs(=)27 b Fn(p)1419 5114 y Fh(i)p Fi(+)p Fh(j)1567 5099 y Fs(and)33 b Fn(p)1806 5114 y Fi(0)1874 5099 y Fs(=)28 b(1)p Fn(:)k Fs(Hence)i(the)f(series)h (for)e Fn(h)h Fs(is)f(completely)267 5216 y(determined)g(b)m(y)g(the)h (term)e Fn(p)c Fs(:=)h Fn(p)1566 5231 y Fi(1)1637 5216 y Fs(since)33 b Fn(p)1925 5231 y Fh(n)1999 5216 y Fs(=)28 b Fn(p)2152 5179 y Fh(n)2152 5240 y Fi(1)2199 5216 y Fs(.)43 b(Since)32 b(the)g(series)h(m)m(ust)f(b)s(e)g(lo)s(cally)p eop %%Page: 52 4 52 3 bop 0 236 a Fq(52)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)267 425 y Fs(\014nite)f(w)m(e)h (get)f(that)g(for)g(eac)m(h)h Fn(c)c Fm(2)g Fn(C)38 b Fs(there)32 b(m)m(ust)f(b)s(e)h(an)f Fn(n)d Fm(2)g Fp(N)2688 440 y Fi(0)2758 425 y Fs(suc)m(h)33 b(that)e Fn(p)3236 389 y Fh(m)3302 425 y Fs(\()p Fn(c)p Fs(\))d(=)f(0)267 541 y(for)32 b(all)e Fn(m)e Fm(\025)h Fn(n)p Fs(.)43 b(Hence)34 b(the)f(elemen)m(t)f Fn(p)h Fs(is)f Fl(top)-5 b(olo)g(gic)g(al)5 b(ly)34 b(nilp)-5 b(otent)32 b Fs(and)1144 698 y Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(C)r(;)17 b Fo(K)g Fs([)p Fn(x)p Fs(])q(\))1988 671 y Fm(\030)1989 702 y Fs(=)2093 698 y(rad)2234 713 y Fh(t)2264 698 y Fs(\()p Fn(C)2379 657 y Fe(\003)2418 698 y Fs(\))267 855 y(the)33 b(radial)d(of)i(top)s(ologically)d(nilp)s(oten)m(t)i (elemen)m(ts)i(of)f Fn(C)2401 819 y Fe(\003)2440 855 y Fs(.)416 971 y(It)c(is)g(easy)h(to)e(see)i(that)f(rad)1436 986 y Fh(t)1465 971 y Fs(\()p Fn(C)1580 935 y Fe(\003)1620 971 y Fs(\))f(is)h(a)f(group)h(under)h(addition)d(and)i(that)g(this)f (group)267 1088 y(structure)34 b(coincides)e(with)g(the)h(one)g(on)g Fo(K)17 b Fs(-)6 b Fp(cCoalg)o Fs(\()p Fn(C)r(;)17 b Fo(K)g Fs([)p Fn(x)p Fs(]\).)149 1262 y Fp(Remark)37 b(2.4.8.)50 b Fs(Let)32 b Fn(H)40 b Fs(b)s(e)32 b(a)g(\014nite)g (dimensional)e(Hopf)i(algebra.)42 b(Then)34 b(b)m(y)f(A.6.6)f(and)0 1378 y(A.6.8)44 b(w)m(e)i(get)e(that)h Fn(H)912 1342 y Fe(\003)995 1378 y Fs(is)f(an)g(algebra)g(and)g(a)h(coalgebra.)78 b(The)45 b(comm)m(utativ)m(e)f(diagrams)0 1495 y(de\014ning)35 b(the)h(bialgebra)e(prop)s(ert)m(y)i(and)g(the)g(an)m(tip)s(o)s(de)f (can)h(b)s(e)f(transferred)i(easily)-8 b(,)35 b(so)h Fn(H)3460 1458 y Fe(\003)3534 1495 y Fs(is)0 1611 y(again)41 b(a)h(Hopf)h(algebra.)72 b(Hence)44 b(the)f(functor)f(-)1878 1575 y Fe(\003)1962 1611 y Fs(:)i Fp(v)m(ec)h Fm(\000)-60 b(!)45 b Fp(v)m(ec)d Fs(from)f(\014nite)h(dimensional)0 1727 y(v)m(ector)49 b(spaces)g(to)e(itself)f(induces)i(a)f(dualit)m(y)g (-)1843 1691 y Fe(\003)1936 1727 y Fs(:)53 b Fo(K)17 b Fs(-)6 b Fp(hopfalg)53 b Fm(\000)-60 b(!)53 b Fo(K)18 b Fs(-)5 b Fp(hopfalg)48 b Fs(from)f(the)0 1843 y(category)33 b(of)f(\014nite)g(dimensional)e(Hopf)j(algebras)f(to)g(itself.)149 1959 y(An)i(a\016ne)h(algebraic)d(group)i(is)f(called)f Fl(\014nite)i Fs(if)f(the)h(represen)m(ting)h(Hopf)e(algebra)g(is)g (\014nite)0 2076 y(dimensional.)43 b(A)33 b(formal)e(group)i(is)g (called)f Fl(\014nite)h Fs(if)f(the)i(represen)m(ting)g(Hopf)f(algebra) f(is)h(\014nite)0 2192 y(dimensional.)149 2308 y(Th)m(us)g(the)e (category)g(of)f(\014nite)g(a\016ne)h(algebraic)e(groups)i(is)g(equiv) -5 b(alen)m(t)30 b(to)g(the)h(category)g(of)0 2424 y(\014nite)h(formal) e(groups.)149 2541 y(The)37 b(category)f(of)f(\014nite)g(comm)m(utativ) m(e)f(a\016ne)i(algebraic)e(groups)i(is)f(self)g(dual.)51 b(The)36 b(cat-)0 2657 y(egory)42 b(of)f(\014nite)g(comm)m(utativ)m(e)g (a\016ne)h(algebraic)e(groups)i(is)f(equiv)-5 b(alen)m(t)42 b(\(and)f(dual\))g(to)h(the)0 2773 y(category)33 b(of)f(\014nite)g (comm)m(utativ)m(e)g(formal)e(groups.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF