%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: ln2.dvi %%Pages: 38 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips ln2 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.11.03:0755 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 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(tered)h(quan)m(tum)e(monoids)e(and)i(studied)g(their)0 1662 y(role)33 b(as)g(monoids)f(op)s(erating)h(on)g(quan)m(tum)h (spaces.)48 b(The)34 b(\\elemen)m(ts")f(of)g(quan)m(tum)h(monoids)0 1778 y(op)s(erating)21 b(on)i(quan)m(tum)g(spaces)i(should)d(b)s(e)h (understo)s(o)s(d)g(as)g(endomorphisms)f(of)g(the)i(quan)m(tum)0 1895 y(spaces.)62 b(In)38 b(the)h(construction)f(of)g(the)g(m)m (ultiplication)c(for)j(univ)m(ersal)h(quan)m(tum)h(monoids)d(of)0 2011 y(quan)m(tum)27 b(spaces)i(w)m(e)f(ha)m(v)m(e)g(seen)g(that)f (this)f(m)m(ultiplication)d(is)j(essen)m(tially)g(the)i(\\comp)s (osition")0 2127 y(of)k(endomorphisms.)149 2243 y(W)-8 b(e)33 b(are,)g(ho)m(w)m(ev)m(er,)i(primarily)29 b(in)m(terested)34 b(in)e(automorphisms)f(and)h(w)m(e)i(kno)m(w)f(that)g(auto-)0 2360 y(morphisms)23 b(should)h(form)e(a)i(group)g(under)h(comp)s (osition.)38 b(This)24 b(c)m(hapter)h(is)f(dev)m(oted)i(to)d(\014nding) 0 2476 y(group)32 b(structures)j(on)d(quan)m(tum)h(monoids,)e(i.e.)43 b(to)33 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b(w)m(e)j(will)e(b)s(e)i(able)e(to)i(p)s(erform)e(one)i(of)f(the)h (most)f(imp)s(ortan)m(t)f(and)i(most)0 3522 y(basic)f(constructions)i (in)d(group)i(theory)-8 b(,)30 b(the)g(formation)d(of)i(the)h(group)g (of)f(in)m(v)m(ertible)g(elemen)m(ts)0 3638 y(of)44 b(a)h(monoid.)78 b(In)45 b(the)h(case)f(of)g(a)f(quan)m(tum)h(monoid)e(acting)h(univ)m (ersally)g(on)h(a)g(quan)m(tum)0 3754 y(space)36 b(this)e(will)e(lead)i (to)g(the)h(go)s(o)s(d)e(de\014nition)h(of)g(a)g(quan)m(tum)h (automorphism)d(group)i(of)h(the)0 3871 y(quan)m(tum)e(space.)149 3987 y(In)41 b(order)f(to)g(ha)m(v)m(e)i(the)e(appropriate)g(to)s(ols)f (for)g(in)m(tro)s(ducing)g(quan)m(tum)i(groups)f(w)m(e)h(\014rst)0 4103 y(in)m(tro)s(duce)34 b(Hopf)h(algebras)f(whic)m(h)h(will)d(b)s(e)i (the)h(represen)m(ting)h(algebras)e(of)g(quan)m(tum)h(groups.)0 4219 y(F)-8 b(urthermore)24 b(w)m(e)i(need)g(the)f(notion)f(of)g(a)g (monoid)f(and)i(of)g(a)f(group)h(in)f(a)g(category)-8 b(.)41 b(W)-8 b(e)25 b(will)e(see,)0 4336 y(ho)m(w)m(ev)m(er,)35 b(that)d(quan)m(tum)g(groups)h(are)f(in)f(general)h(not)g(groups)g(in)g (the)g(category)h(of)f(quan)m(tum)0 4452 y(spaces.)149 4568 y(W)-8 b(e)40 b(\014rst)f(study)h(the)f(simple)f(cases)i(of)e (a\016ne)i(algebraic)d(groups)i(and)g(of)f(formal)f(groups.)0 4684 y(They)42 b(will)c(ha)m(v)m(e)k(Hopf)f(algebras)f(as)g(represen)m (ting)i(ob)5 b(jects)42 b(and)f(will)d(indeed)j(b)s(e)g(groups)g(in)0 4800 y(reasonable)h(categories.)72 b(Then)43 b(w)m(e)g(come)f(to)f (quan)m(tum)i(groups,)h(and)f(construct)g(quan)m(tum)0 4917 y(automorphism)30 b(groups)j(of)f(quan)m(tum)h(spaces.)149 5033 y(A)m(t)g(the)g(end)h(of)e(the)h(c)m(hapter)g(y)m(ou)g(should)175 5216 y Fr(\017)42 b Fu(kno)m(w)34 b(what)e(a)h(Hopf)f(algebra)g(is,) 1762 5365 y Fq(31)p eop %%Page: 32 2 32 1 bop 0 236 a Fq(32)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)175 425 y Fr(\017)42 b Fu(kno)m(w)34 b(what)e(a)h(group)f(in)g(a)g(category)h(is,)175 541 y Fr(\017)42 b Fu(kno)m(w)30 b(the)g(de\014nition)e(and)h(examples) h(of)e(a\016ne)i(algebraic)e(groups)h(and)h(formal)d(groups,)175 658 y Fr(\017)42 b Fu(kno)m(w)32 b(the)g(de\014nition)e(and)i(examples) f(of)g(quan)m(tum)h(groups)f(and)h(b)s(e)f(able)g(to)g(construct)267 774 y(quan)m(tum)i(automorphism)d(groups)j(for)f(small)e(quan)m(tum)j (spaces,)175 890 y Fr(\017)42 b Fu(understand)28 b(wh)m(y)h(a)e(Hopf)g (algebra)f(is)h(a)g(reasonable)g(represen)m(ting)h(algebra)e(for)h(a)g (quan-)267 1006 y(tum)32 b(group.)p eop %%Page: 33 3 33 2 bop 1425 236 a Fq(1.)35 b(HOPF)e(ALGEBRAS)1321 b(33)1375 425 y Fs(1.)56 b(Hopf)38 b(Algebras)149 600 y Fu(The)28 b(di\013erence)g(b)s(et)m(w)m(een)h(a)e(monoid)e(and)i(a)g(group)f (lies)g(in)h(the)g(existence)i(of)d(an)h(additional)0 716 y(map)38 b Fp(S)45 b Fu(:)38 b Fp(G)h Fr(3)g Fp(g)j Fr(7!)c Fp(g)892 680 y Fo(\000)p Fn(1)1024 716 y Fr(2)h Fp(G)g Fu(for)f(a)h(group)g Fp(G)g Fu(that)f(allo)m(ws)g(forming)f(in)m (v)m(erses.)65 b(This)39 b(map)0 832 y(satis\014es)33 b(the)g(equation)g Fp(S)6 b Fu(\()p Fp(g)t Fu(\))p Fp(g)30 b Fu(=)d(1)32 b(or)h(in)e(a)i(diagrammatic)c(form)1252 991 y Fp(G)375 b Fr(f)p Fu(1)p Fr(g)p 1358 968 317 4 v 1592 966 a Fm(-)1500 947 y Fl(")2227 991 y Fp(G)p 1881 968 V 2115 966 a Fm(-)2022 948 y Fn(1)p 1289 1288 4 254 v 1291 1288 a Fm(?)1193 1184 y Fn(\001)1153 1386 y Fp(G)22 b Fr(\002)g Fp(G)700 b(G)22 b Fr(\002)g Fp(G)p 1457 1358 642 4 v 2016 1356 a Fm(-)1698 1330 y Fl(S)t Fo(\002)p Fn(id)p 2264 1288 4 254 v 2266 1118 a Fm(6)2305 1184 y Fn(m)n(ult)0 1513 y Fu(W)-8 b(e)41 b(w)m(an)m(t)g(to)f(carry)h(this)f (prop)s(ert)m(y)h(o)m(v)m(er)h(to)e(a)g(de\014nition)f(of)h(quan)m(tum) h(groups.)67 b(W)-8 b(e)41 b(kno)m(w)0 1629 y(already)e(that)g(quan)m (tum)g(monoids)f Fp(G)h Fu(are)g(represen)m(ted)i(b)m(y)f(bialgebras)e Fp(H)8 b Fu(.)63 b(So)39 b(an)g(\\in)m(v)m(erse)0 1746 y(map")c(should)h(b)s(e)g(a)f(morphism)f Fp(S)39 b Fu(:)34 b Fp(G)f Fr(\000)-60 b(!)33 b Fp(G)j Fu(with)f(a)h(certain)f(prop)s (ert)m(y)-8 b(,)38 b(if)c Fp(G)i Fu(is)f(to)h(b)s(ecome)0 1862 y(a)41 b(quan)m(tum)h(group,)i(or)d(an)g(algebra)g(homomorphism)e Fp(S)48 b Fu(:)43 b Fp(H)51 b Fr(\000)-60 b(!)43 b Fp(H)49 b Fu(for)41 b(the)h(represen)m(ting)0 1978 y(bialgebra)29 b Fp(H)38 b Fu(of)31 b Fp(G)p Fu(.)43 b(W)-8 b(e)31 b(need)h(a)e(sligh) m(tly)g(more)g(general)g(de\014nition)g(of)h(Hopf)f(algebras.)43 b(They)0 2094 y(will)30 b(then)j(b)s(e)g(the)g(represen)m(ting)h (algebras)e(for)g(quan)m(tum)h(groups.)149 2257 y Fs(De\014nition)k (2.1.1.)49 b Fu(A)42 b Fk(left)i(Hopf)g(algebr)-5 b(a)41 b Fp(H)50 b Fu(is)42 b(a)g(bialgebra)f Fp(H)49 b Fu(together)43 b(with)f(a)g Fk(left)0 2373 y(antip)-5 b(o)g(de)30 b Fp(S)j Fu(:)28 b Fp(H)35 b Fr(\000)-60 b(!)28 b Fp(H)8 b Fu(,)31 b(i.e.)42 b(a)31 b(linear)e(map)h Fp(S)36 b Fu(suc)m(h)d(that)d(the)h(follo)m(wing)d(diagram)h(comm)m(utes:)1268 2536 y Fp(H)412 b Fj(K)p 1386 2503 348 4 v 1651 2501 a Fm(-)1543 2483 y Fl(")2243 2536 y Fp(H)p 1829 2503 386 4 v 2132 2501 a Fm(-)2002 2470 y Fl(\021)p 1311 2824 4 254 v 1313 2824 a Fm(?)1215 2719 y Fn(\001)1163 2922 y Fp(H)30 b Fr(\012)22 b Fp(H)684 b(H)30 b Fr(\012)22 b Fp(H)p 1491 2894 619 4 v 2027 2892 a Fm(-)1720 2866 y Fl(S)t Fo(\012)p Fn(id)p 2286 2824 4 254 v 2287 2653 a Fm(6)2326 2719 y Fo(r)0 3049 y Fu(Symmetrically)35 b(w)m(e)j(de\014ne)g(a)f Fk(right)j(Hopf)f(algebr)-5 b(a)36 b Fp(H)8 b Fu(.)57 b(A)38 b Fk(Hopf)h(algebr)-5 b(a)36 b Fu(is)h(a)g(left)f(and)i(righ)m(t)0 3165 y(Hopf)32 b(algebra.)43 b(The)33 b(map)f Fp(S)38 b Fu(is)32 b(called)g(a)g (\(left,)g(righ)m(t,)g(t)m(w)m(o-sided\))h Fk(antip)-5 b(o)g(de)p Fu(.)149 3327 y(Using)33 b(the)g(Sw)m(eedler)g(notation)f (\(A.6.3\))g(the)h(comm)m(utativ)m(e)e(diagram)g(ab)s(o)m(v)m(e)i(can)g (also)f(b)s(e)0 3444 y(expressed)j(b)m(y)f(the)f(equation)1326 3481 y Fi(X)1486 3576 y Fp(S)6 b Fu(\()p Fp(a)1641 3591 y Fn(\(1\))1735 3576 y Fu(\))p Fp(a)1824 3591 y Fn(\(2\))1946 3576 y Fu(=)28 b Fp(\021)t(")p Fu(\()p Fp(a)p Fu(\))0 3728 y(for)45 b(all)f Fp(a)50 b Fr(2)h Fp(H)8 b Fu(.)83 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(the)f(order)f(of)h(the)g(m)m(ultiplication)27 b(and)33 b(the)g(com)m(ultiplication".)149 4588 y(3.)61 b(Let)39 b Fp(H)46 b Fu(and)39 b Fp(K)45 b Fu(b)s(e)39 b(Hopf)f(algebras)g(and)h (let)f Fp(f)48 b Fu(:)38 b Fp(H)45 b Fr(\000)-60 b(!)38 b Fp(K)46 b Fu(b)s(e)38 b(a)h(homomorphism)c(of)0 4704 y(bialgebras.)42 b(Then)34 b Fp(f)11 b(S)873 4719 y Fl(H)967 4704 y Fu(=)28 b Fp(S)1131 4719 y Fl(K)1199 4704 y Fp(f)11 b Fu(,)33 b(i.e.)43 b Fp(f)g Fu(is)32 b(compatible)f(with)h(the)h(an)m (tip)s(o)s(de.)149 4867 y Fs(De\014nition)k(2.1.2.)49 b Fu(Because)39 b(of)e(Problem)g(2.1.1)g(3.)58 b(ev)m(ery)39 b(homomorphism)c(of)i(bialge-)0 4983 y(bras)d(b)s(et)m(w)m(een)i(Hopf)e (algebras)f(is)h(compatible)e(with)h(the)h(an)m(tip)s(o)s(des.)47 b(So)34 b(w)m(e)h(de\014ne)g(a)f Fk(homo-)0 5099 y(morphism)42 b(of)g(Hopf)h(algebr)-5 b(as)41 b Fu(to)g(b)s(e)h(a)f(homomorphism)d (of)j(bialgebras.)69 b(The)43 b(category)e(of)0 5216 y(Hopf)32 b(algebras)g(will)f(b)s(e)h(denoted)i(b)m(y)f Fj(K)18 b Fu(-)5 b Fs(Hopf)q Fu(.)p eop %%Page: 34 4 34 3 bop 0 236 a Fq(34)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fs(Prop)s(osition)k (2.1.3.)49 b Fk(L)-5 b(et)38 b Fp(H)45 b Fk(b)-5 b(e)37 b(a)g(bialgebr)-5 b(a)36 b(with)h(an)g(algebr)-5 b(a)37 b(gener)-5 b(ating)36 b(set)i Fp(X)8 b Fk(.)52 b(L)-5 b(et)0 541 y Fp(S)41 b Fu(:)35 b Fp(H)42 b Fr(\000)-58 b(!)36 b Fp(H)530 505 y Fl(op)642 541 y Fk(b)-5 b(e)39 b(an)f(algebr)-5 b(a)38 b(homomorphism)f(such)h(that)2359 467 y Fi(P)2481 541 y Fp(S)6 b Fu(\()p Fp(x)2640 557 y Fn(\(1\))2735 541 y Fu(\))p Fp(x)2828 557 y Fn(\(2\))2957 541 y Fu(=)35 b Fp(\021)t(")p Fu(\()p Fp(x)p Fu(\))k Fk(for)f(al)5 b(l)0 658 y Fp(x)28 b Fr(2)g Fp(X)8 b Fk(.)45 b(Then)34 b Fp(S)40 b Fk(is)35 b(a)g(left)g(antip)-5 b(o)g(de)34 b(of)g Fp(H)8 b Fk(.)149 847 y Fh(Pr)n(oof.)42 b Fu(Assume)g Fp(a;)17 b(b)43 b Fr(2)f Fp(H)49 b Fu(suc)m(h)42 b(that)1769 772 y Fi(P)1891 847 y Fp(S)6 b Fu(\()p Fp(a)2046 862 y Fn(\(1\))2140 847 y Fu(\))p Fp(a)2229 862 y Fn(\(2\))2366 847 y Fu(=)42 b Fp(\021)t(")p Fu(\()p Fp(a)p Fu(\))e(and)2947 772 y Fi(P)3069 847 y Fp(S)6 b Fu(\()p Fp(b)3214 862 y Fn(\(1\))3308 847 y Fu(\))p Fp(b)3387 862 y Fn(\(2\))3524 847 y Fu(=)0 963 y Fp(\021)t(")p Fu(\()p Fp(b)p Fu(\).)43 b(Then)375 1069 y Fi(P)497 1144 y Fp(S)6 b Fu(\(\()p Fp(ab)p Fu(\))769 1160 y Fn(\(1\))863 1144 y Fu(\)\()p Fp(ab)p Fu(\))1069 1160 y Fn(\(2\))1197 1144 y Fu(=)1300 1069 y Fi(P)1422 1144 y Fp(S)g Fu(\()p Fp(a)1577 1160 y Fn(\(1\))1671 1144 y Fp(b)1712 1160 y Fn(\(1\))1807 1144 y Fu(\))p Fp(a)1896 1160 y Fn(\(2\))1990 1144 y Fp(b)2031 1160 y Fn(\(2\))2154 1144 y Fu(=)2257 1069 y Fi(P)2379 1144 y Fp(S)g Fu(\()p Fp(b)2524 1160 y Fn(\(1\))2619 1144 y Fu(\))p Fp(S)g Fu(\()p Fp(a)2812 1160 y Fn(\(1\))2906 1144 y Fu(\))p Fp(a)2995 1160 y Fn(\(2\))3089 1144 y Fp(b)3130 1160 y Fn(\(2\))1197 1260 y Fu(=)1300 1186 y Fi(P)1422 1260 y Fp(S)g Fu(\()p Fp(b)1567 1276 y Fn(\(1\))1661 1260 y Fu(\))p Fp(\021)t(")p Fu(\()p Fp(a)p Fu(\))p Fp(b)1965 1276 y Fn(\(2\))2087 1260 y Fu(=)28 b Fp(\021)t(")p Fu(\()p Fp(a)p Fu(\))p Fp(\021)t(")p Fu(\()p Fp(b)p Fu(\))f(=)g Fp(\021)t(")p Fu(\()p Fp(ab)p Fu(\))p Fp(:)0 1439 y Fu(Since)h(ev)m (ery)j(elemen)m(t)d(of)f Fp(H)36 b Fu(is)28 b(a)g(\014nite)g(sum)g(of)g (\014nite)g(pro)s(ducts)h(of)f(elemen)m(ts)g(in)g Fp(X)8 b Fu(,)29 b(for)e(whic)m(h)0 1555 y(the)33 b(equalit)m(y)f(holds,)h (this)f(equalit)m(y)g(extends)j(to)d(all)e(of)i Fp(H)40 b Fu(b)m(y)34 b(induction.)p 3529 1555 4 66 v 3533 1493 59 4 v 3533 1555 V 3591 1555 4 66 v 149 1744 a Fs(Example)j(2.1.4.)49 b Fu(1.)42 b(Let)28 b Fp(V)49 b Fu(b)s(e)28 b(a)g(v)m(ector)h(space)f (and)g Fp(T)14 b Fu(\()p Fp(V)22 b Fu(\))27 b(the)h(tensor)h(algebra)d (o)m(v)m(er)j Fp(V)22 b Fu(.)0 1861 y(W)-8 b(e)33 b(ha)m(v)m(e)g(seen)h (in)d(Problem)h(A.5.6)g(that)g Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))32 b(is)g(a)g(bialgebra)e(and)j(that)f Fp(V)53 b Fu(generates)34 b Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))0 1977 y(as)31 b(an)f(algebra.)42 b(De\014ne)31 b Fp(S)j Fu(:)27 b Fp(V)50 b Fr(\000)-60 b(!)27 b Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))1554 1941 y Fl(op)1659 1977 y Fu(b)m(y)31 b Fp(S)6 b Fu(\()p Fp(v)t Fu(\))27 b(:=)h Fr(\000)p Fp(v)35 b Fu(for)30 b(all)e Fp(v)k Fr(2)c Fp(V)21 b Fu(.)43 b(By)31 b(the)g(univ)m(ersal)0 2093 y(prop)s(ert)m(y)42 b(of)f(the)h(tensor)g (algebra)e(this)h(map)f(extends)k(to)d(an)g(algebra)f(homomorphism)f Fp(S)48 b Fu(:)0 2209 y Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))28 b Fr(\000)-60 b(!)28 b Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))623 2173 y Fl(op)696 2209 y Fu(.)42 b(Since)28 b(\001\()p Fp(v)t Fu(\))f(=)h Fp(v)15 b Fr(\012)c Fu(1)g(+)g(1)g Fr(\012)g Fp(v)33 b Fu(w)m(e)c(ha)m(v)m(e)2237 2135 y Fi(P)2359 2209 y Fp(S)6 b Fu(\()p Fp(v)2510 2225 y Fn(\(1\))2604 2209 y Fu(\))p Fp(v)2689 2225 y Fn(\(2\))2811 2209 y Fu(=)28 b Fr(r)p Fu(\()p Fp(S)17 b Fr(\012)11 b Fu(1\)\001\()p Fp(v)t Fu(\))28 b(=)0 2326 y Fr(\000)p Fp(v)i Fu(+)c Fp(v)42 b Fu(=)37 b(0)h(=)f Fp(\021)t(")p Fu(\()p Fp(v)t Fu(\))h(for)g(all)e Fp(v)41 b Fr(2)e Fp(V)21 b Fu(,)40 b(hence)g Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))39 b(is)e(a)i(Hopf)f (algebra)f(b)m(y)j(the)f(preceding)0 2442 y(prop)s(osition.)149 2558 y(2.)82 b(Let)45 b Fp(V)67 b Fu(b)s(e)46 b(a)f(v)m(ector)h(space)h (and)e Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))46 b(the)f(symmetric)g(algebra) f(o)m(v)m(er)j Fp(V)66 b Fu(\(that)45 b(is)0 2674 y(comm)m(utativ)m (e\).)j(W)-8 b(e)34 b(ha)m(v)m(e)i(seen)g(in)d(Problem)h(A.5.7)g(that)g Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))34 b(is)g(a)g(bialgebra)f(and)h(that)g Fp(V)0 2790 y Fu(generates)g Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))33 b(as)g(an)g(algebra.)44 b(De\014ne)33 b Fp(S)h Fu(:)29 b Fp(V)50 b Fr(\000)-60 b(!)28 b Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))33 b(b)m(y)h Fp(S)6 b Fu(\()p Fp(v)t Fu(\))28 b(:=)g Fr(\000)p Fp(v)37 b Fu(for)32 b(all)f Fp(v)h Fr(2)d Fp(V)22 b Fu(.)44 b Fp(S)0 2907 y Fu(extends)33 b(to)e(an)g(algebra)f(homomorphism)e Fp(S)33 b Fu(:)28 b Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))28 b Fr(\000)-60 b(!)28 b Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\).)43 b(Since)31 b(\001\()p Fp(v)t Fu(\))d(=)f Fp(v)c Fr(\012)c Fu(1)g(+)g(1)g Fr(\012)g Fp(v)0 3023 y Fu(w)m(e)38 b(ha)m(v)m(e)378 2948 y Fi(P)499 3023 y Fp(S)6 b Fu(\()p Fp(v)650 3038 y Fn(\(1\))745 3023 y Fu(\))p Fp(v)830 3038 y Fn(\(2\))960 3023 y Fu(=)35 b Fr(r)p Fu(\()p Fp(S)c Fr(\012)26 b Fu(1\)\001\()p Fp(v)t Fu(\))35 b(=)g Fr(\000)p Fp(v)30 b Fu(+)25 b Fp(v)39 b Fu(=)c(0)g(=)h Fp(\021)t(")p Fu(\()p Fp(v)t Fu(\))g(for)g(all)g Fp(v)j Fr(2)d Fp(V)21 b Fu(,)39 b(hence)0 3139 y Fp(S)6 b Fu(\()p Fp(V)21 b Fu(\))33 b(is)f(a)g(Hopf)h(algebra)e(b)m(y)i(the)g (preceding)g(prop)s(osition.)149 3328 y Fs(Example)k(2.1.5.)49 b Fu(\()p Fs(Group)37 b(Algebras\))30 b Fu(F)-8 b(or)31 b(eac)m(h)h(algebra)e Fp(A)h Fu(w)m(e)i(can)e(form)f(the)i Fk(gr)-5 b(oup)0 3444 y(of)38 b(units)f Fp(U)10 b Fu(\()p Fp(A)p Fu(\))35 b(:=)g Fr(f)p Fp(a)f Fr(2)h Fp(A)p Fr(j9)p Fp(a)1205 3408 y Fo(\000)p Fn(1)1335 3444 y Fr(2)g Fp(A)p Fr(g)h Fu(with)g(the)i(m)m(ultiplication)31 b(of)37 b Fp(A)f 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(diagram)i(comm)m(utes)1460 4321 y Fp(G)296 b(U)10 b Fu(\()p Fj(K)18 b Fp(G)p Fu(\))p 1566 4298 238 4 v 1721 4296 a Fm(-)1668 4278 y Fp(\023)1623 4565 y(f)1567 4448 y Fm(@)1650 4531 y(@)1733 4614 y(@)1816 4697 y(@)1835 4716 y(@)-83 b(R)1860 4809 y Fp(U)10 b Fu(\()p Fp(A)p Fu(\))p Fp(:)p 1985 4716 4 351 v 1986 4716 a Fm(?)2025 4552 y Fp(g)149 4983 y Fu(The)35 b(group)e(algebra)f Fj(K)18 b Fp(G)39 b Fu(is)33 b(\(if)f(it)h(exists\))h(unique)g(up)g(to) f(isomorphism.)44 b(It)33 b(is)g(generated)0 5099 y(as)k(an)g(algebra)e (b)m(y)j(the)f(image)e(of)i Fp(G)p Fu(.)56 b(The)37 b(map)f Fp(\023)g Fu(:)f Fp(G)g Fr(\000)-60 b(!)34 b Fp(U)10 b Fu(\()p Fj(K)19 b Fp(G)p Fu(\))40 b Fr(\022)c Fj(K)17 b Fp(G)43 b Fu(is)36 b(injectiv)m(e)h(and)0 5216 y(the)c(image)e(of)h Fp(G)g Fu(in)g Fj(K)18 b Fp(G)38 b Fu(is)32 b(a)h(basis.)p eop %%Page: 35 5 35 4 bop 1425 236 a Fq(1.)35 b(HOPF)e(ALGEBRAS)1321 b(35)149 425 y Fu(The)37 b(group)e(algebra)f(can)i(b)s(e)g(constructed)h(as)f (the)g(free)g(v)m(ector)g(space)h Fj(K)17 b Fp(G)42 b Fu(with)35 b(basis)g Fp(G)0 541 y Fu(and)f(the)f(algebra)g(structure)h (of)f Fj(K)18 b Fp(G)39 b Fu(is)33 b(giv)m(en)h(b)m(y)g Fj(K)18 b Fp(G)28 b Fr(\012)23 b Fj(K)18 b Fp(G)35 b Fr(3)30 b Fp(g)c Fr(\012)d Fp(h)29 b Fr(7!)g Fp(g)t(h)g Fr(2)g Fj(K)18 b Fp(G)39 b Fu(and)34 b(the)0 658 y(unit)e Fp(\021)f Fu(:)d Fj(K)51 b Fr(3)28 b Fp(\013)h Fr(7!)e Fp(\013)q(e)h Fr(2)g Fj(K)17 b Fp(G)p Fu(.)149 774 y(The)41 b(group)e(algebra)g Fj(K)17 b Fp(G)46 b Fu(is)39 b(a)g(Hopf)h(algebra.) 63 b(The)41 b(com)m(ultiplication)35 b(is)k(giv)m(en)h(b)m(y)h(the)0 890 y(diagram)1430 999 y Fp(G)371 b Fj(K)18 b Fp(G)p 1535 966 314 4 v 1766 964 a Fm(-)1680 946 y Fl(\023)1610 1225 y(f)1536 1116 y Fm(@)1619 1199 y(@)1702 1282 y(@)1785 1365 y(@)1804 1384 y(@)-83 b(R)1740 1482 y Fj(K)18 b Fp(G)28 b Fr(\012)22 b Fj(K)c Fp(G)p 1954 1384 4 351 v 1956 1384 a Fm(?)1995 1231 y Fn(\001)0 1593 y Fu(with)39 b Fp(f)11 b Fu(\()p Fp(g)t Fu(\))38 b(:=)h Fp(g)30 b Fr(\012)d Fp(g)42 b Fu(whic)m(h)e(de\014nes)h(a)e(group)g(homomorphism) e Fp(f)49 b Fu(:)40 b Fp(G)f Fr(\000)-60 b(!)39 b Fp(U)10 b Fu(\()p Fj(K)18 b Fp(G)32 b Fr(\012)c Fj(K)17 b Fp(G)p Fu(\).)0 1710 y(The)34 b(counit)e(is)g(giv)m(en)g(b)m(y)1498 1818 y Fp(G)372 b Fj(K)18 b Fp(G)p 1604 1786 314 4 v 1835 1784 a Fm(-)1749 1765 y Fl(\023)1679 2044 y(f)1605 1935 y Fm(@)1688 2018 y(@)1771 2101 y(@)1854 2184 y(@)1873 2203 y(@)-83 b(R)1985 2306 y Fj(K)p 2023 2203 4 351 v 2024 2203 a Fm(?)2063 2042 y Fl(")0 2409 y Fu(where)33 b Fp(f)11 b Fu(\()p Fp(g)t Fu(\))27 b(=)g(1)32 b(for)g(all)e Fp(g)h Fr(2)d Fp(G)p Fu(.)43 b(One)33 b(sho)m(ws)g(easily)f(b)m(y)h (using)f(the)g(univ)m(ersal)g(prop)s(ert)m(y)-8 b(,)33 b(that)0 2525 y(\001)39 b(is)g(coasso)s(ciativ)m(e)g(and)g(has)g (counit)g Fp(")p Fu(.)62 b(De\014ne)40 b(an)f(algebra)e(homomorphism)g Fp(S)44 b Fu(:)39 b Fj(K)17 b Fp(G)45 b Fr(\000)-60 b(!)0 2642 y Fu(\()p Fj(K)17 b Fp(G)p Fu(\))224 2605 y Fl(op)337 2642 y Fu(b)m(y)1461 2756 y Fp(G)372 b Fj(K)17 b Fp(G)p 1567 2723 314 4 v 1798 2721 a Fm(-)1711 2703 y Fl(\023)1642 2982 y(f)1568 2873 y Fm(@)1651 2956 y(@)1734 3039 y(@)1817 3122 y(@)1836 3141 y(@)-83 b(R)1835 3234 y Fu(\()p Fj(K)18 b Fp(G)p Fu(\))2060 3198 y Fl(op)p 1986 3141 4 351 v 1987 3141 a Fm(?)2026 2988 y Fl(S)0 3366 y Fu(with)47 b Fp(f)11 b Fu(\()p Fp(g)t Fu(\))51 b(:=)h Fp(g)680 3330 y Fo(\000)p Fn(1)821 3366 y Fu(whic)m(h)c(is)e(a)h(group)g (homomorphism)e Fp(f)63 b Fu(:)52 b Fp(G)h Fr(\000)-60 b(!)52 b Fp(U)10 b Fu(\(\()p Fj(K)18 b Fp(G)p Fu(\))3146 3330 y Fl(op)3226 3366 y Fu(\).)87 b(Then)0 3483 y(Prop)s(osition)31 b(1.3)h(sho)m(ws)i(that)f Fj(K)17 b Fp(G)39 b Fu(is)32 b(a)g(Hopf)g(algebra.)149 3654 y(The)h(preceding)f(example)g(of)f(a)g (Hopf)h(algebra)e(giv)m(es)j(rise)e(to)h(the)g(de\014nition)e(of)i (particular)0 3771 y(elemen)m(ts)h(in)e(arbitrary)g(Hopf)h(algebras,)g (that)g(share)h(certain)f(prop)s(erties)g(with)g(elemen)m(ts)g(of)g(a)0 3887 y(group.)43 b(W)-8 b(e)33 b(will)e(use)i(and)g(study)h(these)f 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b(Let)42 b Fj(K)65 b Fu(b)s(e)42 b(a)f(\014eld.)70 b(Sho)m(w)42 b(that)f(an)h(elemen)m(t)f Fp(x)i Fr(2)g Fj(K)18 b Fp(G)47 b Fu(satis\014es)0 4983 y(\001\()p Fp(x)p Fu(\))28 b(=)g Fp(x)22 b Fr(\012)h Fp(x)33 b Fu(and)g Fp(")p Fu(\()p Fp(x)p Fu(\))27 b(=)h(1)k(if)g(and)g(only)g(if)g Fp(x)c Fu(=)f Fp(g)k Fr(2)d Fp(G)p Fu(.)149 5099 y(2.)91 b(Sho)m(w)49 b(that)f(the)h(group-lik)m(e)e(elemen)m(ts)i(of)e(a)h(Hopf)h(algebra)e (form)g(a)h(group)g(under)0 5216 y(m)m(ultiplication)28 b(of)k(the)h(Hopf)f(algebra.)p eop %%Page: 36 6 36 5 bop 0 236 a Fq(36)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fs(Example)37 b(2.1.7.)49 b Fu(\()p Fs(Univ)m(ersal)39 b(En)m(v)m(eloping)g (Algebras\))34 b Fu(A)h Fk(Lie)i(algebr)-5 b(a)34 b Fu(consists)h(of)0 541 y(a)j(v)m(ector)h(space)g Fg(g)f Fu(together)g(with)g(a)f (\(linear\))f(m)m(ultiplication)e Fg(g)26 b Fr(\012)g Fg(g)37 b Fr(3)g Fp(x)27 b Fr(\012)f Fp(y)40 b Fr(7!)c Fu([)p Fp(x;)17 b(y)t Fu(])37 b 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Fg(g)p Fu(\))23 b Fr(\012)f Fp(U)10 b Fu(\()p Fg(g)p Fu(\))p 1923 5131 4 351 v 1924 5131 a Fm(?)1963 4978 y Fn(\001)p eop %%Page: 37 7 37 6 bop 1425 236 a Fq(1.)35 b(HOPF)e(ALGEBRAS)1321 b(37)0 425 y Fu(with)43 b Fp(f)11 b Fu(\()p Fp(x)p Fu(\))45 b(:=)g Fp(x)30 b Fr(\012)g Fu(1)f(+)g(1)g Fr(\012)g Fp(x)44 b Fu(whic)m(h)f(de\014nes)i(a)e(Lie)f(algebra)g(homomorphism)e Fp(f)56 b Fu(:)45 b Fg(g)h Fr(\000)-60 b(!)0 541 y Fu(\()p Fp(U)10 b Fu(\()p Fg(g)p Fu(\))23 b Fr(\012)f Fp(U)10 b Fu(\()p Fg(g)p Fu(\)\))602 505 y Fl(L)655 541 y Fu(.)44 b(The)33 b(counit)f(is)g(giv)m(en)h(b)m(y)1493 713 y Fg(g)361 b Fp(U)10 b Fu(\()p Fg(g)p Fu(\))p 1572 690 305 4 v 1794 688 a Fm(-)1711 670 y Fl(\023)1660 949 y(f)1586 840 y Fm(@)1670 923 y(@)1753 1006 y(@)1836 1089 y(@)1854 1108 y(@)-83 b(R)1967 1210 y Fj(K)p 2004 1108 4 351 v 2006 1108 a Fm(?)2045 946 y Fl(")0 1357 y Fu(with)35 b Fp(f)11 b Fu(\()p Fp(x)p Fu(\))34 b(=)f(0)i(for)g(all)f Fp(x)g Fr(2)f Fg(g)p Fu(.)53 b(One)36 b(sho)m(ws)i(easily)d(b)m(y)i (using)e(the)h(univ)m(ersal)g(prop)s(ert)m(y)-8 b(,)37 b(that)0 1473 y(\001)g(is)e(coasso)s(ciativ)m(e)i(and)f(has)h(counit)f Fp(")p Fu(.)54 b(De\014ne)37 b(an)f(algebra)f(homomorphism)f Fp(S)40 b Fu(:)34 b Fp(U)10 b Fu(\()p Fg(g)p Fu(\))35 b Fr(\000)-60 b(!)0 1589 y Fu(\()p Fp(U)10 b Fu(\()p Fg(g)p Fu(\)\))278 1553 y Fl(op)385 1589 y Fu(b)m(y)1456 1721 y Fg(g)361 b Fp(U)10 b Fu(\()p Fg(g)p Fu(\))p 1534 1697 305 4 v 1756 1695 a Fm(-)1674 1677 y Fl(\023)1623 1956 y(f)1549 1847 y Fm(@)1632 1930 y(@)1715 2013 y(@)1798 2096 y(@)1817 2115 y(@)-83 b(R)1792 2208 y Fu(\()p Fp(U)10 b Fu(\()p Fg(g)p Fu(\)\))2070 2172 y Fl(op)p 1967 2115 4 351 v 1968 2115 a Fm(?)2007 1962 y Fl(S)0 2366 y Fu(with)39 b Fp(f)11 b Fu(\()p Fp(x)p Fu(\))40 b(:=)g Fr(\000)p Fp(x)g Fu(whic)m(h)h(is)e(a)g(Lie)g(algebra)g(homomorphism)e Fp(f)50 b Fu(:)40 b Fg(g)g Fr(\000)-60 b(!)40 b Fu(\()p Fp(U)10 b Fu(\()p Fg(g)p Fu(\))3122 2330 y Fl(op)3196 2366 y Fu(\))3234 2330 y Fl(L)3286 2366 y Fu(.)65 b(Then)0 2483 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b(Hopf)h(algebra)e(of)h(c)m(haracteristic)h(0)f(has)h(no)g (non-zero)f(cen)m(tral)h(nilp)s(oten)m(t)e(elemen)m(ts.)416 4050 y(\(First)h(coun)m(ter)i(example)e(giv)m(en)h(b)m(y)g([Sc)m (hmidt-Samoa].)42 b(If)32 b Fp(H)41 b Fu(is)32 b(unimo)s(dular)e(and) 267 4167 y(not)35 b(semisimple,)e(e.g.)51 b(a)34 b(Drinfel'd)g(double)g (of)h(a)f(not)h(semisimple)e(\014nite)h(dimensional)267 4283 y(Hopf)f(algebra,)g(then)h(the)g(in)m(tegral)e(\003)h(satis\014es) h(\003)29 b Fr(6)p Fu(=)g(0,)34 b(\003)2465 4247 y Fn(2)2533 4283 y Fu(=)29 b Fp(")p Fu(\(\003\)\003)g(=)g(0)k(since)h Fp(D)s Fu(\()p Fp(H)8 b Fu(\))267 4399 y(is)37 b(not)h(semisimple,)f (and)h Fp(a)p Fu(\003)e(=)h Fp(")p Fu(\()p Fp(a)p Fu(\)\003)f(=)h(\003) p Fp(")p Fu(\()p Fp(a)p Fu(\))f(=)g(\003)p Fp(a)i Fu(since)g Fp(D)s Fu(\()p Fp(H)8 b Fu(\))37 b(is)h(unimo)s(dular)267 4515 y([Sommerh\177)-49 b(auser].\))149 4632 y(4.)42 b(\(Nic)m(hols\).)h(Let)32 b Fp(x)h Fu(b)s(e)g(an)f(elemen)m(t)g(in)g (a)g(Hopf)h(algebra)e Fp(H)40 b Fu(with)32 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5216 y Fu(=)h Fp(xa:)p Fu(\))p eop %%Page: 39 9 39 8 bop 1425 236 a Fq(1.)35 b(HOPF)e(ALGEBRAS)1321 b(39)416 425 y Fu(In)28 b(the)f(remaining)e(six)i(conjectures)i Fp(H)34 b Fu(is)27 b(a)f(\014nite-dimensional)e(Hopf)j(algebra)f(o)m(v) m(er)267 541 y(an)32 b(algebraically)e(closed)i(\014eld.)149 658 y(5.)42 b(If)30 b Fp(H)38 b Fu(is)30 b(semisimple)e(on)j(either)f (side)g(\(i.e.)43 b(either)30 b Fp(H)38 b Fu(or)30 b(the)h(dual)e Fp(H)2855 622 y Fo(\003)2925 658 y Fu(is)h(semisimple)e(as)267 774 y(an)k(algebra\))g(the)h(square)g(of)g(the)g(an)m(tip)s(o)s(de)e (is)h(the)h(iden)m(tit)m(y)-8 b(.)416 890 y(\(Y)g(es)35 b(if)e(c)m(har\()p Fj(K)18 b Fu(\))40 b(=)34 b(0)g([Larson-Radford],)g (y)m(es)i(if)d(c)m(har\()p Fj(K)18 b Fu(\))40 b(is)34 b(large)f([Sommerh\177)-49 b(au-)267 1006 y(ser]\))149 1123 y(6.)42 b(The)35 b(size)g(of)f(the)h(matrices)f(o)s(ccurring)g(in) g(an)m(y)h(full)e(matrix)g(constituen)m(t)i(of)f Fp(H)42 b Fu(divides)267 1239 y(the)33 b(dimension)e(of)h Fp(H)8 b Fu(.)416 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Fu(.)149 2634 y(9.)42 b(The)33 b(dimension)f(of)g(the)h(radical)d(is)i(the)h(same)g (on)f(b)s(oth)h(sides.)416 2750 y(\(Coun)m(terexample)c(b)m(y)f([Nic)m (hols];)h(coun)m(terexample)g(in)e(F)-8 b(rob)s(enius-)27 b(Lusztig)g(k)m(ernel)267 2866 y(of)32 b Fp(U)444 2881 y Fl(q)482 2866 y Fu(\()p Fp(sl)r Fu(\(2\)\))g([Sc)m(hneider]\))101 2982 y(10.)41 b(There)29 b(are)g(only)e(a)h(\014nite)g(n)m(um)m(b)s(er) h(\(up)f(to)g(isomorphism\))e(of)i(Hopf)g(algebras)f(of)h(a)g(giv)m(en) 267 3099 y(dimension.)416 3215 y(\(Y)-8 b(es)34 b(for)e(semisimple,)e (cosemisimple)g(Hopf)j(algebras:)43 b(Stefan)32 b(1997\))416 3331 y(\(Coun)m(terexamples:)45 b([Andruskiewitsc)m(h,)34 b(Sc)m(hneider],)g([Beattie,)f(others])g(1997\))p eop %%Page: 40 10 40 9 bop 0 236 a Fq(40)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)841 425 y Fs(2.)93 b(Monoids)38 b(and)g(Groups)g(in)f(a)h(Category)149 600 y Fu(Before)d(w)m(e)g(use)g(Hopf)f(algebras)g(to)g(describ)s(e)h(quan)m 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b(an)g(arbitrary)f(category)-8 b(.)51 b(Let)36 b Fp(G)31 b Fr(2)i(C)41 b Fu(b)s(e)35 b(an)g(ob)5 b(ject.)52 b(W)-8 b(e)0 1693 y(use)44 b(the)f(notation)f Fp(G)p Fu(\()p Fp(X)8 b Fu(\))45 b(:=)g(Mor)1369 1708 y Fo(C)1414 1693 y Fu(\()p Fp(X)r(;)17 b(G)p Fu(\))43 b(for)f(all)f Fp(X)53 b Fr(2)46 b(C)6 b Fu(,)46 b Fp(G)p Fu(\()p Fp(f)11 b Fu(\))45 b(:=)g(Mor)3000 1708 y Fo(C)3045 1693 y Fu(\()p Fp(f)5 b(;)17 b(G)p Fu(\))43 b(for)f(all)0 1810 y(morphisms)29 b Fp(f)38 b Fu(:)28 b Fp(X)35 b Fr(\000)-60 b(!)28 b Fp(Y)51 b Fu(in)30 b Fr(C)6 b Fu(,)30 b(and)h Fp(f)11 b Fu(\()p Fp(X)d Fu(\))27 b(:=)g(Mor)1974 1825 y Fo(C)2019 1810 y Fu(\()p Fp(X)r(;)17 b(f)11 b Fu(\))30 b(for)g(all)d(morphisms)i Fp(f)39 b Fu(:)27 b Fp(G)h Fr(\000)-60 b(!)28 b Fp(G)3550 1773 y Fo(0)3573 1810 y Fu(.)149 1926 y Fp(G)43 b Fu(together)g(with)f (a)h(natural)e(transformation)g Fp(m)k Fu(:)g Fp(G)p Fu(\(-)o(\))29 b Fr(\002)h Fp(G)p Fu(\(-)o(\))45 b Fr(\000)-60 b(!)45 b Fp(G)p Fu(\(-)o(\))e(is)f(called)f(a)0 2042 y Fk(gr)-5 b(oup)37 b Fu(\()p Fk(monoid)p Fu(\))g Fk(in)g(the)h(c)-5 b(ate)g(gory)37 b Fr(C)6 b Fu(,)37 b(if)d(the)i(sets)h Fp(G)p Fu(\()p Fp(X)8 b Fu(\))35 b(together)g(with)h(the)g(m)m (ultiplication)0 2158 y Fp(m)p Fu(\()p Fp(X)8 b Fu(\))28 b(:)f Fp(G)p Fu(\()p Fp(X)8 b Fu(\))22 b Fr(\002)h Fp(G)p Fu(\()p Fp(X)8 b Fu(\))27 b Fr(\000)-60 b(!)28 b Fp(G)p Fu(\()p Fp(X)8 b Fu(\))32 b(are)h(groups)f(\(monoids\))g(for)g(all)e Fp(X)36 b Fr(2)28 b(C)6 b Fu(.)149 2275 y(Let)34 b(\()p Fp(G;)17 b(m)p Fu(\))32 b(and)h(\()p Fp(G)944 2238 y Fo(0)967 2275 y Fp(;)17 b(m)1096 2238 y Fo(0)1120 2275 y Fu(\))33 b(b)s(e)g(groups)g(in)f Fr(C)6 b Fu(.)45 b(A)33 b(morphism)e Fp(f)39 b Fu(:)28 b Fp(G)g Fr(\000)-60 b(!)29 b Fp(G)2915 2238 y Fo(0)2971 2275 y Fu(in)j Fr(C)39 b Fu(is)32 b(called)g(a)0 2391 y Fk(homomorphism)h(of)h(gr)-5 b(oups)33 b Fu(in)e Fr(C)6 b Fu(,)33 b(if)f(the)h(diagrams)1215 2570 y Fp(G)p Fu(\()p Fp(X)8 b Fu(\))22 b Fr(\002)g Fp(G)p Fu(\()p Fp(X)8 b Fu(\))356 b Fp(G)p Fu(\()p Fp(X)8 b Fu(\))p 1849 2547 300 4 v 2066 2545 a Fm(-)1908 2509 y Fl(m)p Fn(\()p Fl(X)d Fn(\))1191 3057 y Fp(G)1268 3021 y Fo(0)1292 3057 y Fu(\()p Fp(X)j Fu(\))22 b Fr(\002)g Fp(G)1655 3021 y Fo(0)1678 3057 y Fu(\()p Fp(X)8 b Fu(\))322 b Fp(G)2242 3021 y Fo(0)2265 3057 y Fu(\()p Fp(X)8 b Fu(\))p 1872 3034 265 4 v 2054 3032 a Fm(-)1902 2997 y Fl(m)1964 2974 y Ff(0)1987 2997 y Fn(\()p Fl(X)d Fn(\))p 1516 2964 4 351 v 1517 2964 a Fm(?)1105 2805 y Fl(f)i Fn(\()p Fl(X)e Fn(\))p Fo(\002)p Fl(f)i Fn(\()p Fl(X)e Fn(\))p 2296 2964 V 2297 2964 a Fm(?)2336 2805 y Fl(f)i Fn(\()p Fl(X)e Fn(\))0 3200 y Fu(comm)m(ute)32 b(for)g(all)f Fp(X)k Fr(2)28 b(C)6 b Fu(.)149 3317 y(Let)34 b(\()p Fp(G;)17 b(m)p Fu(\))33 b(and)h(\()p Fp(G)946 3280 y Fo(0)969 3317 y Fp(;)17 b(m)1098 3280 y Fo(0)1121 3317 y Fu(\))34 b(b)s(e)f(monoids)f(in)h Fr(C)6 b Fu(.)46 b(A)34 b(morphism)d Fp(f)40 b Fu(:)29 b Fp(G)g Fr(\000)-60 b(!)29 b Fp(G)2994 3280 y Fo(0)3051 3317 y Fu(in)k Fr(C)39 b Fu(is)33 b(called)0 3433 y(a)f Fk(homomorphism)h(of)i(monoids)c Fu(in)h Fr(C)6 b Fu(,)32 b(if)g(the)h(diagrams)1215 3612 y Fp(G)p Fu(\()p Fp(X)8 b Fu(\))22 b Fr(\002)g Fp(G)p Fu(\()p Fp(X)8 b Fu(\))356 b Fp(G)p Fu(\()p Fp(X)8 b Fu(\))p 1849 3589 300 4 v 2066 3587 a Fm(-)1908 3551 y Fl(m)p Fn(\()p Fl(X)d Fn(\))1191 4099 y Fp(G)1268 4063 y Fo(0)1292 4099 y Fu(\()p Fp(X)j Fu(\))22 b Fr(\002)g Fp(G)1655 4063 y Fo(0)1678 4099 y Fu(\()p Fp(X)8 b Fu(\))322 b Fp(G)2242 4063 y Fo(0)2265 4099 y Fu(\()p Fp(X)8 b Fu(\))p 1872 4076 265 4 v 2054 4074 a Fm(-)1902 4039 y Fl(m)1964 4016 y Ff(0)1987 4039 y Fn(\()p Fl(X)d Fn(\))p 1516 4006 4 351 v 1517 4006 a Fm(?)1105 3847 y Fl(f)i Fn(\()p Fl(X)e Fn(\))p Fo(\002)p Fl(f)i Fn(\()p Fl(X)e Fn(\))p 2296 4006 V 2297 4006 a Fm(?)2336 3847 y Fl(f)i Fn(\()p Fl(X)e Fn(\))0 4242 y Fu(and)1719 4325 y Fr(f\003g)1572 4558 y Fl(u)1719 4452 y Fm(\001)1677 4535 y(\001)1635 4618 y(\001)1594 4701 y(\001)1585 4719 y(\001)-42 b(\013)1975 4570 y Fl(u)2016 4546 y Ff(0)1828 4452 y Fm(A)1870 4535 y(A)1911 4618 y(A)1953 4701 y(A)1962 4719 y(A)g(U)1430 4813 y Fp(G)p Fu(\()p Fp(X)8 b Fu(\))233 b Fp(G)1982 4777 y Fo(0)2006 4813 y Fu(\()p Fp(X)8 b Fu(\))p 1700 4790 178 4 v 1795 4788 a Fm(-)1709 4752 y Fl(f)f Fn(\()p Fl(X)e Fn(\))0 4935 y Fu(comm)m(ute)32 b(for)g(all)f Fp(X)k Fr(2)28 b(C)6 b Fu(.)149 5099 y Fs(Problem)36 b(2.2.4.)49 b Fu(1\))29 b(If)h(a)f(set)h Fp(Z)36 b Fu(together)30 b(with)f(a)g(m)m(ultiplication)c Fp(m)j Fu(:)g Fp(Z)22 b Fr(\002)16 b Fp(Z)35 b Fr(\000)-60 b(!)28 b Fp(Z)36 b Fu(is)29 b(a)0 5216 y(monoid,)d(then)i(the)f(unit)f(elemen)m(t)h Fp(e)h Fr(2)g Fp(Z)34 b Fu(is)26 b(uniquely)h(determined.)41 b(If)27 b(it)f(is)g(a)h(group)f(then)i(also)p eop %%Page: 41 11 41 10 bop 931 236 a Fq(2.)76 b(MONOIDS)32 b(AND)g(GR)n(OUPS)g(IN)h(A)g (CA)-6 b(TEGOR)g(Y)827 b(41)0 425 y Fu(the)37 b(in)m(v)m(erse)g Fp(i)d Fu(:)g Fp(Z)40 b Fr(\000)-60 b(!)34 b Fp(Z)43 b Fu(is)36 b(uniquely)g(determined.)54 b(Unit)35 b(elemen)m(t)h(and)g (in)m(v)m(erses)i(of)e(groups)0 541 y(are)d(preserv)m(ed)i(b)m(y)e (maps)f(that)h(are)f(compatible)f(with)h(the)h(m)m(ultiplication.)149 658 y(2\))i(Find)g(an)g(example)f(of)h(monoids)f Fp(Y)56 b Fu(and)36 b Fp(Z)42 b Fu(and)35 b(a)g(map)f Fp(f)43 b Fu(:)32 b Fp(Y)53 b Fr(\000)-60 b(!)33 b Fp(Z)42 b Fu(with)34 b Fp(f)11 b Fu(\()p Fp(y)3328 673 y Fn(1)3367 658 y Fp(y)3415 673 y Fn(2)3454 658 y Fu(\))32 b(=)0 774 y Fp(f)11 b Fu(\()p Fp(y)145 789 y Fn(1)184 774 y Fu(\))p Fp(f)g Fu(\()p Fp(y)367 789 y Fn(2)405 774 y Fu(\))33 b(for)f(all)e Fp(y)808 789 y Fn(1)847 774 y Fp(;)17 b(y)939 789 y Fn(2)1006 774 y Fr(2)28 b Fp(Y)21 b Fu(,)33 b(but)f Fp(f)11 b Fu(\()p Fp(e)1558 789 y Fl(Y)1619 774 y Fu(\))28 b Fr(6)p Fu(=)f Fp(e)1833 789 y Fl(Z)1890 774 y Fu(.)149 890 y(3\))39 b(If)g(\()p Fp(G;)17 b(m)p Fu(\))39 b(is)f(a)h(group)g(in)f Fr(C)45 b Fu(and)39 b Fp(i)1621 905 y Fl(X)1727 890 y Fu(:)g Fp(G)p Fu(\()p Fp(X)8 b Fu(\))38 b Fr(\000)-60 b(!)39 b Fp(G)p Fu(\()p Fp(X)8 b Fu(\))38 b(is)h(the)g(in)m(v)m(erse,)j(then)e Fp(i)f 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b Fr(\000)-58 b(!)29 b Fp(G)p Fk(,)35 b Fp(u)28 b Fu(:)g Fp(E)35 b Fr(\000)-58 b(!)28 b Fp(G)p Fk(,)35 b(and)g Fp(S)f Fu(:)28 b Fp(G)h Fr(\000)-58 b(!)28 b Fp(G)36 b Fk(such)f(that)0 1692 y(the)g(diagr)-5 b(ams)564 1941 y Fp(G)23 b Fr(\002)f Fp(G)g Fr(\002)h Fp(G)210 b(G)22 b Fr(\002)h Fp(G)p 1068 1912 153 4 v 1138 1910 a Fm(-)1068 1885 y Fl(m)p Fo(\002)p Fn(1)664 2428 y Fp(G)f Fr(\002)h Fp(G)408 b(G)p 968 2400 351 4 v 1236 2398 a Fm(-)1113 2379 y Fl(m)p 800 2330 4 351 v 802 2330 a Fm(?)610 2172 y Fn(1)p Fo(\002)p Fl(m)p 1385 2330 V 1387 2330 a Fm(?)1426 2169 y Fl(m)1719 1941 y Fp(E)28 b Fr(\002)22 b Fp(G)2023 1913 y Fr(\030)2024 1945 y Fu(=)2128 1941 y Fp(G)2233 1913 y Fr(\030)2234 1945 y Fu(=)2338 1941 y Fp(G)g Fr(\002)h Fp(E)151 b(G)22 b Fr(\002)h Fp(G)p 2643 1912 88 4 v 2648 1910 a Fm(-)2622 1885 y Fn(1)p Fo(\002)p Fl(u)1736 2428 y Fp(G)f Fr(\002)h Fp(G)847 b(G)p 2041 2400 790 4 v 2748 2398 a Fm(-)2404 2379 y Fl(m)p 1872 2330 4 351 v 1874 2330 a Fm(?)1704 2172 y Fl(u)p Fo(\002)p Fn(1)p 2896 2330 V 2898 2330 a Fm(?)2937 2169 y Fl(m)2594 2176 y Fn(1)2269 2034 y Fm(Q)2352 2090 y(Q)2435 2145 y(Q)2518 2200 y(Q)2601 2256 y(Q)2684 2311 y(Q)2712 2330 y(Q)-83 b(s)1372 2730 y Fp(G)312 b(E)p 1477 2697 255 4 v 1649 2695 a Fm(-)2152 2730 y Fp(G)p 1868 2697 V 2040 2695 a Fm(-)1975 2677 y Fl(u)1272 3213 y Fp(G)22 b Fr(\002)h Fp(G)504 b(G)22 b Fr(\002)h Fp(G)p 1577 3185 447 4 v 1941 3183 a Fm(-)1732 3158 y Fn(1)p Fo(\002)p Fl(S)1732 3247 y(S)t Fo(\002)p Fn(1)p 1408 3115 4 351 v 1410 3115 a Fm(?)1312 2962 y Fn(\001)p 2188 3115 V 2190 2847 a Fm(6)2229 2954 y Fl(m)0 3433 y Fk(c)-5 b(ommute)32 b(wher)-5 b(e)33 b Fu(\001)g Fk(is)f(the)h (morphism)e(de\014ne)-5 b(d)32 b(in)h(A.2.)44 b(The)32 b(multiplic)-5 b(ations)32 b(ar)-5 b(e)32 b(r)-5 b(elate)g(d)33 b(by)0 3549 y Fp(m)85 3564 y Fl(X)180 3549 y Fu(=)28 b(Mor)460 3564 y Fo(C)505 3549 y Fu(\()p Fp(X)r(;)17 b(m)p Fu(\))28 b(=)g Fp(m)p Fu(\()p Fp(X)8 b Fu(\))p Fk(.)149 3665 y(A)n(n)35 b(analo)-5 b(gous)34 b(statement)h(holds)f (for)g(monoids.)149 3887 y Fh(Pr)n(oof.)42 b Fu(The)i(Y)-8 b(oneda)43 b(Lemma)e(de\014nes)j(a)f(bijection)e(b)s(et)m(w)m(een)k (the)e(set)g(of)f(morphisms)0 4003 y Fp(f)48 b Fu(:)37 b Fp(X)45 b Fr(\000)-60 b(!)37 b Fp(Y)60 b Fu(and)38 b(the)h(set)f(of)g(natural)f(transformations)g Fp(f)11 b Fu(\(-)o(\))37 b(:)g Fp(X)8 b Fu(\(-)o(\))37 b Fr(\000)-60 b(!)38 b Fp(Y)21 b Fu(\(-)o(\))38 b(b)m(y)h Fp(f)11 b Fu(\()p Fp(Z)c Fu(\))37 b(=)0 4119 y(Mor)176 4134 y Fo(C)221 4119 y Fu(\()p Fp(Z)r(;)17 b(f)11 b Fu(\).)42 b(In)33 b(particular)e(w)m(e)j(ha)m(v)m(e)g Fp(m)1564 4134 y Fl(X)1659 4119 y Fu(=)28 b(Mor)1939 4134 y Fo(C)1984 4119 y Fu(\()p Fp(X)r(;)17 b(m)p Fu(\))28 b(=)f Fp(m)p Fu(\()p Fp(X)8 b Fu(\).)44 b(The)33 b(diagram)1039 4382 y Fp(G)p Fu(\(-)o(\))22 b Fr(\002)h Fp(G)p Fu(\(-)o(\))f Fr(\002)h Fp(G)p Fu(\(-)o(\))232 b Fp(G)p Fu(\(-)o(\))22 b Fr(\002)h Fp(G)p Fu(\(-)o(\))p 1867 4358 174 4 v 1958 4356 a Fm(-)1859 4317 y Fl(m)1921 4337 y Fu(-)1958 4317 y Fo(\002)p Fn(1)1192 4869 y Fp(G)p Fu(\(-)o(\))g Fr(\002)f Fp(G)p Fu(\(-\))538 b Fp(G)p Fu(\(-)o(\))p 1713 4846 481 4 v 2111 4844 a Fm(-)1904 4805 y Fl(m)1966 4826 y Fu(-)p 1437 4776 4 351 v 1438 4776 a Fm(?)1210 4612 y Fn(1)p Fo(\002)p Fl(m)1362 4632 y Fu(-)p 2314 4776 V 2316 4776 a Fm(?)2355 4605 y Fl(m)2417 4625 y Fu(-)0 5099 y(comm)m(utes)30 b(if)f(and)h(only)f(if)g(Mor)1204 5114 y Fo(C)1249 5099 y Fu(\(-)p Fp(;)17 b(m)p Fu(\()p Fp(m)g Fr(\002)g Fu(1\)\))28 b(=)f(Mor)2115 5114 y Fo(C)2161 5099 y Fu(\(-)o Fp(;)17 b(m)p Fu(\)\(Mor)2612 5114 y Fo(C)2657 5099 y Fu(\(-)o Fp(;)g(m)p Fu(\))g Fr(\002)h Fu(1\))27 b(=)h Fp(m)3309 5114 y Fu(-)3346 5099 y(\()p Fp(m)3469 5114 y Fu(-)3523 5099 y Fr(\002)0 5216 y Fu(1\))36 b(=)g Fp(m)320 5231 y Fu(-)357 5216 y(\(1)25 b Fr(\002)i Fp(m)658 5231 y Fu(-)695 5216 y(\))36 b(=)g(Mor)1057 5231 y Fo(C)1102 5216 y Fu(\(-)p Fp(;)17 b(m)p Fu(\)\(1)25 b Fr(\002)h Fu(Mor)1731 5231 y Fo(C)1776 5216 y Fu(\(-)p Fp(;)17 b(m)p Fu(\)\))36 b(=)g(Mor)2376 5231 y Fo(C)2421 5216 y Fu(\(-)p Fp(;)17 b(m)p Fu(\(1)25 b Fr(\002)h Fp(m)p Fu(\)\))38 b(if)e(and)i(only)f(if)p eop %%Page: 42 12 42 11 bop 0 236 a Fq(42)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fp(m)p Fu(\()p Fp(m)23 b Fr(\002)f Fu(1\))28 b(=)f Fp(m)p Fu(\(1)22 b Fr(\002)h Fp(m)p Fu(\))33 b(if)e(and)i(only)f(if)f(the)i(diagram)1320 613 y Fp(G)22 b Fr(\002)h Fp(G)f Fr(\002)g Fp(G)210 b(G)22 b Fr(\002)h Fp(G)p 1823 585 153 4 v 1893 583 a Fm(-)1823 557 y Fl(m)p Fo(\002)p Fn(1)1419 1100 y Fp(G)f Fr(\002)h Fp(G)409 b(G)p 1724 1073 351 4 v 1992 1071 a Fm(-)1868 1052 y Fl(m)p 1556 1002 4 351 v 1557 1002 a Fm(?)1366 845 y Fn(1)p Fo(\002)p Fl(m)p 2141 1002 V 2142 1002 a Fm(?)2181 841 y Fl(m)0 1248 y Fu(comm)m(utes.)43 b(In)33 b(a)g(similar)c(w)m(a)m(y)34 b(one)f(sho)m(ws)h(the)f(equiv)-5 b(alence)33 b(of)f(the)h(other)f(diagram\(s\).)p 3529 1248 4 66 v 3533 1185 59 4 v 3533 1248 V 3591 1248 4 66 v 149 1431 a Fs(Problem)k(2.2.5.)49 b Fu(Let)26 b Fr(C)32 b Fu(b)s(e)26 b(a)f(category)h(with)f(\014nite)g(pro)s(ducts.) 42 b(Sho)m(w)26 b(that)g(a)f(morphism)0 1547 y Fp(f)38 b Fu(:)28 b Fp(G)g Fr(\000)-60 b(!)28 b Fp(G)468 1511 y Fo(0)523 1547 y Fu(in)k Fr(C)39 b Fu(is)32 b(a)g(homomorphism)e(of)i (groups)h(if)e(and)i(only)f(if)1466 1708 y Fp(G)22 b Fr(\002)h Fp(G)311 b(G)p 1771 1680 254 4 v 1942 1678 a Fm(-)1866 1659 y Fl(m)1443 2198 y Fp(G)1520 2162 y Fo(0)1565 2198 y Fr(\002)23 b Fp(G)1742 2162 y Fo(0)2041 2198 y Fp(G)2118 2162 y Fo(0)p 1794 2168 219 4 v 1930 2166 a Fm(-)1861 2146 y Fl(m)1923 2123 y Ff(0)p 1602 2098 4 351 v 1604 2098 a Fm(?)1428 1939 y Fl(f)7 b Fo(\002)p Fl(f)p 2090 2098 V 2092 2098 a Fm(?)2131 1939 y Fl(f)0 2338 y Fu(comm)m(utes.)149 2522 y Fs(De\014nition)37 b(2.2.3.)49 b Fu(A)35 b Fk(c)-5 b(o)g(gr)g(oup)34 b Fu(\()p Fk(c)-5 b(omonoid)p Fu(\))33 b Fp(G)i Fu(in)f Fr(C)41 b Fu(is)34 b(a)g(group)h(\(monoid\))e(in)h Fr(C)3339 2486 y Fl(op)3413 2522 y Fu(,)h(i.e.)0 2638 y(an)28 b(ob)5 b(ject)29 b Fp(G)f Fr(2)g Fu(Ob)16 b Fr(C)34 b Fu(=)28 b(Ob)16 b Fr(C)1161 2602 y Fl(op)1264 2638 y Fu(together)28 b(with)g(a)f(natural)g(transformation)f Fp(m)p Fu(\()p Fp(X)8 b Fu(\))28 b(:)g Fp(G)p Fu(\()p Fp(X)8 b Fu(\))13 b Fr(\002)0 2754 y Fp(G)p Fu(\()p Fp(X)8 b Fu(\))27 b Fr(\000)-60 b(!)28 b Fp(G)p Fu(\()p Fp(X)8 b Fu(\))22 b(where)h Fp(G)p Fu(\()p Fp(X)8 b Fu(\))27 b(=)h(Mor)1498 2769 y Fo(C)1539 2750 y Fe(op)1610 2754 y Fu(\()p Fp(X)r(;)17 b(G)p Fu(\))27 b(=)h(Mor)2197 2769 y Fo(C)2242 2754 y Fu(\()p Fp(G;)17 b(X)8 b Fu(\),)23 b(suc)m(h)h(that)e(\()p Fp(G)p Fu(\()p Fp(X)8 b Fu(\))p Fp(;)17 b(m)p Fu(\()p Fp(X)8 b Fu(\)\))0 2870 y(is)32 b(a)g(group)h(\(monoid\))d(for)j(eac)m (h)g Fp(X)i Fr(2)29 b(C)6 b Fu(.)149 3054 y Fs(Remark)37 b(2.2.4.)50 b Fu(Let)c Fr(C)52 b Fu(b)s(e)46 b(a)f(category)i(with)e (\014nite)h(\(categorical\))d(copro)s(ducts.)85 b(An)0 3170 y(ob)5 b(ject)35 b Fp(G)f Fu(in)f Fr(C)40 b Fu(carries)34 b(the)h(structure)g Fp(m)c Fu(:)f Fp(G)p Fu(\(-)o(\))23 b Fr(\002)h Fp(G)p Fu(\(-)o(\))30 b Fr(\000)-60 b(!)31 b Fp(G)p Fu(\(-)o(\))j(of)g(a)f(cogroup)h(in)g Fr(C)40 b Fu(if)33 b(and)0 3286 y(only)h(if)f(there)i(are)g(morphisms)e(\001)e (:)g Fp(G)g Fr(\000)-60 b(!)31 b Fp(G)23 b Fr(q)h Fp(G)p Fu(,)35 b Fp(")30 b Fu(:)h Fp(G)g Fr(\000)-60 b(!)31 b Fp(I)8 b Fu(,)35 b(and)f Fp(S)j Fu(:)31 b Fp(G)g Fr(\000)-60 b(!)31 b Fp(G)j Fu(suc)m(h)i(that)0 3402 y(the)d(diagrams)458 4072 y Fp(G)22 b Fr(q)f Fp(G)215 b(G)22 b Fr(q)f Fp(G)i Fr(q)e Fp(G)p 759 4040 157 4 v 833 4038 a Fm(-)763 4019 y Fn(\001)p Fo(q)p Fn(1)556 3585 y Fp(G)410 b(G)22 b Fr(q)f Fp(G)p 661 3552 353 4 v 931 3550 a Fm(-)808 3532 y Fn(\001)p 1177 3970 4 351 v 1179 3970 a Fm(?)1218 3817 y Fn(1)p Fo(q)p Fn(\001)p 592 3970 V 594 3970 a Fm(?)496 3817 y Fn(\001)1822 4070 y Fp(G)i Fr(q)e Fp(G)177 b(I)30 b Fr(q)22 b Fp(G)2546 4043 y Fr(\030)2547 4074 y Fu(=)2652 4070 y Fp(G)2756 4043 y Fr(\030)2757 4074 y Fu(=)2861 4070 y Fp(G)h Fr(q)e Fp(I)p 2124 4040 120 4 v 2161 4038 a Fm(-)2123 4019 y Fl(")p Fo(q)p Fn(1)1920 3585 y Fp(G)849 b(G)22 b Fr(q)g Fp(G)p 2026 3552 792 4 v 2735 3550 a Fm(-)2393 3532 y Fn(\001)p 2981 3970 4 351 v 2983 3970 a Fm(?)3022 3817 y Fn(1)p Fo(q)p Fl(")p 1957 3970 V 1959 3970 a Fm(?)1861 3817 y Fn(\001)2386 3816 y(1)2061 3674 y Fm(Q)2144 3730 y(Q)2227 3785 y(Q)2310 3840 y(Q)2393 3896 y(Q)2476 3951 y(Q)2505 3970 y(Q)-83 b(s)1372 4236 y Fp(G)326 b(I)p 1477 4203 269 4 v 1663 4201 a Fm(-)2152 4236 y Fp(G)p 1855 4203 V 2041 4201 a Fm(-)1972 4183 y Fl(")1274 4723 y Fp(G)22 b Fr(q)g Fp(G)507 b(G)22 b Fr(q)f Fp(G)p 1575 4691 451 4 v 1943 4689 a Fm(-)1733 4670 y Fn(1)p Fo(q)p Fl(S)1733 4754 y(S)t Fo(q)p Fn(1)p 1408 4621 4 351 v 1410 4621 a Fm(?)1312 4468 y Fn(\001)p 2188 4621 V 2190 4353 a Fm(6)2229 4468 y Fo(r)0 4867 y Fu(comm)m(ute)28 b(where)i Fr(r)e Fu(is)g(dual)g(to)g(the)h(morphism) e(\001)i(de\014ned)h(in)e(A.2.)42 b(The)30 b(m)m(ultiplications)24 b(are)0 4983 y(related)32 b(b)m(y)i(\001)542 4998 y Fl(X)637 4983 y Fu(=)28 b(Mor)917 4998 y Fo(C)962 4983 y Fu(\(\001)p Fp(;)17 b(X)8 b Fu(\))27 b(=)h(\001\()p Fp(X)8 b Fu(\).)149 5099 y(Let)40 b Fr(C)46 b Fu(b)s(e)39 b(a)g(category)h(with)f(\014nite) g(copro)s(ducts)h(and)g(let)e Fp(G)i Fu(and)f Fp(G)2778 5063 y Fo(0)2841 5099 y Fu(b)s(e)g(cogroups)h(in)f Fr(C)6 b Fu(.)0 5216 y(Then)37 b(a)e(homomorphism)e(of)i(groups)h Fp(f)43 b Fu(:)33 b Fp(G)1694 5179 y Fo(0)1750 5216 y Fr(\000)-60 b(!)32 b Fp(G)k Fu(is)f(a)g(morphism)f Fp(f)43 b Fu(:)33 b Fp(G)g Fr(\000)-60 b(!)32 b Fp(G)3143 5179 y Fo(0)3202 5216 y Fu(in)j Fr(C)41 b Fu(suc)m(h)p eop %%Page: 43 13 43 12 bop 931 236 a Fq(2.)76 b(MONOIDS)32 b(AND)g(GR)n(OUPS)g(IN)h(A)g (CA)-6 b(TEGOR)g(Y)827 b(43)0 425 y Fu(that)32 b(the)h(diagram)1470 547 y Fp(G)311 b(G)22 b Fr(\002)h Fp(G)p 1576 519 254 4 v 1747 517 a Fm(-)1673 498 y Fn(\001)1458 1037 y Fp(G)1535 1001 y Fo(0)1835 1037 y Fp(G)1912 1001 y Fo(0)1957 1037 y Fr(\002)g Fp(G)2134 1001 y Fo(0)p 1587 1007 219 4 v 1723 1005 a Fm(-)1656 985 y Fn(\001)1715 962 y Ff(0)p 1994 936 4 351 v 1996 936 a Fm(?)2035 778 y Fl(f)7 b Fo(\002)p Fl(f)p 1507 936 V 1508 936 a Fm(?)1428 778 y Fl(f)0 1142 y Fu(comm)m(utes.)43 b(An)33 b(analogous)f(result)g (holds)g(for)g(comonoids.)149 1317 y Fs(Remark)37 b(2.2.5.)50 b Fu(Ob)m(viously)33 b(similar)e(observ)-5 b(ations)34 b(and)g(statemen)m(ts)h(can)f(b)s(e)g(made)f(for)0 1433 y(other)38 b(algebraic)f(structures)j(in)d(a)h(category)h Fr(C)6 b Fu(.)60 b(So)39 b(one)f(can)h(in)m(tro)s(duce)f(v)m(ector)h (spaces)h(and)0 1549 y(co)m(v)m(ector)46 b(spaces,)i(monoids)43 b(and)h(comonoids,)i(rings)e(and)g(corings)g(in)f(a)h(category)h Fr(C)6 b Fu(.)78 b(The)0 1666 y(structures)44 b(can)e(b)s(e)g(describ)s (ed)h(b)m(y)g(morphisms)e(in)g Fr(C)48 b Fu(if)41 b Fr(C)48 b Fu(is)42 b(a)g(category)g(with)g(\014nite)f(\(co-\))0 1782 y(pro)s(ducts.)149 1956 y Fs(Problem)36 b(2.2.6.)49 b Fu(Determine)37 b(the)h(structure)i(of)d(a)g(co)m(v)m(ector)j(space)f (on)e(a)h(v)m(ector)h(space)0 2072 y Fp(V)54 b Fu(from)31 b(the)i(fact)g(that)f(Hom)o(\()p Fp(V)5 b(;)17 b(W)d Fu(\))32 b(is)g(a)h(v)m(ector)g(space)h(for)e(all)f(v)m(ector)i(spaces) h Fp(W)14 b Fu(.)149 2247 y Fs(Prop)s(osition)36 b(2.2.6.)49 b Fk(L)-5 b(et)39 b Fp(G)34 b Fr(2)h(C)45 b Fk(b)-5 b(e)38 b(a)g(gr)-5 b(oup)38 b(with)g(multiplic)-5 b(ation)38 b Fp(a)25 b Fr(\003)g Fp(b)p Fk(,)39 b(unit)g Fp(e)p Fk(,)h(and)0 2363 y(inverse)k Fp(a)396 2327 y Fo(\000)p Fn(1)536 2363 y Fk(in)h Fp(G)p Fu(\()p Fp(X)8 b Fu(\))p Fk(.)76 b(Then)45 b(the)g(morphisms)f Fp(m)j Fu(:)h Fp(G)29 b Fr(\002)i Fp(G)47 b Fr(\000)-58 b(!)47 b Fp(G)p Fk(,)h Fp(u)f Fu(:)g Fp(E)53 b Fr(\000)-58 b(!)48 b Fp(G)p Fk(,)g(and)0 2479 y Fp(S)33 b Fu(:)28 b Fp(G)g Fr(\000)-58 b(!)28 b Fp(G)35 b Fk(ar)-5 b(e)35 b(given)f(by)968 2636 y Fp(m)28 b Fu(=)f Fp(p)1233 2651 y Fn(1)1295 2636 y Fr(\003)22 b Fp(p)1416 2651 y Fn(2)1455 2636 y Fp(;)216 b(u)27 b Fu(=)h Fp(e)1930 2651 y Fl(E)1990 2636 y Fp(;)216 b(S)33 b Fu(=)28 b(id)2511 2593 y Fo(\000)p Fn(1)2511 2661 y Fl(G)2605 2636 y Fp(:)149 2810 y Fh(Pr)n(oof.)42 b Fu(By)35 b(the)f(Y)-8 b(oneda)34 b(Lemma)e(A.9.1)i(these)h(morphisms)e(can)h(b)s (e)g(constructed)h(from)0 2927 y(the)28 b(natural)f(transformation)f (as)i(follo)m(ws.)40 b(Under)29 b(Mor)2098 2942 y Fo(C)2143 2927 y Fu(\()p Fp(G)12 b Fr(\002)g Fp(G;)17 b(G)12 b Fr(\002)g Fp(G)p Fu(\))30 b(=)d Fp(G)12 b Fr(\002)g Fp(G)p Fu(\()p Fp(G)g Fr(\002)g Fp(G)p Fu(\))3523 2899 y Fr(\030)3523 2931 y Fu(=)0 3057 y Fp(G)p Fu(\()p Fp(G)j Fr(\002)g Fp(G)p Fu(\))g Fr(\002)g Fp(G)p Fu(\()p Fp(G)g Fr(\002)g Fp(G)p Fu(\))1024 3001 y Fo(\003)961 3057 y Fr(\000)-16 b(!)27 b Fp(G)p Fu(\()p Fp(G)15 b Fr(\002)g Fp(G)p Fu(\))27 b(=)h(Mor)1870 3072 y Fo(C)1915 3057 y Fu(\()p Fp(G)15 b Fr(\002)g Fp(G;)i(G)p Fu(\))28 b(the)h(iden)m(tit)m(y)g(id)3003 3072 y Fl(G)p Fo(\002)p Fl(G)3200 3057 y Fu(=)f(\()p Fp(p)3391 3072 y Fn(1)3430 3057 y Fp(;)17 b(p)3523 3072 y Fn(2)3562 3057 y Fu(\))0 3173 y(is)44 b(mapp)s(ed)h(to)f Fp(m)49 b Fu(=)f Fp(p)932 3188 y Fn(1)1002 3173 y Fr(\003)30 b Fp(p)1131 3188 y Fn(2)1171 3173 y Fu(.)80 b(Under)45 b(Mor)1761 3188 y Fo(C)1806 3173 y Fu(\()p Fp(E)6 b(;)17 b(E)6 b Fu(\))48 b(=)h Fp(E)6 b Fu(\()p Fp(E)g Fu(\))48 b Fr(\000)-60 b(!)49 b Fp(G)p Fu(\()p Fp(E)6 b Fu(\))48 b(=)g(Mor)3280 3188 y Fo(C)3325 3173 y Fu(\()p Fp(E)6 b(;)17 b(G)p Fu(\))0 3290 y(the)40 b(iden)m(tit)m(y)g(of)g Fp(E)46 b Fu(is)39 b(mapp)s(ed)h(to)f(the)i(neutral)e(elemen)m(t)h Fp(u)g Fu(=)g Fp(e)2534 3305 y Fl(E)2594 3290 y Fu(.)65 b(Under)41 b(Mor)3165 3305 y Fo(C)3210 3290 y Fu(\()p Fp(G;)17 b(G)p Fu(\))40 b(=)0 3409 y Fp(G)p Fu(\()p Fp(G)p Fu(\))27 b Fr(\000)-60 b(!)28 b Fp(G)p Fu(\()p Fp(G)p Fu(\))g(=)f(Mor)939 3424 y Fo(C)984 3409 y Fu(\()p Fp(G;)17 b(G)p Fu(\))32 b(the)h(iden)m(tit)m(y)g(is)f(mapp)s(ed)g(to)g(its)g Fr(\003)p Fu(-in)m(v)m(erse)i Fp(S)f Fu(=)28 b(id)3227 3366 y Fo(\000)p Fn(1)3227 3434 y Fl(G)3321 3409 y Fu(.)p 3529 3409 4 66 v 3533 3346 59 4 v 3533 3409 V 3591 3409 4 66 v 149 3583 a Fs(Corollary)36 b(2.2.7.)50 b Fk(L)-5 b(et)40 b Fp(G)d Fr(2)g(C)46 b Fk(b)-5 b(e)40 b(a)f(c)-5 b(o)g(gr)g(oup)40 b(with)g(multiplic)-5 b(ation)39 b Fp(a)26 b Fr(\003)g Fp(b)p Fk(,)41 b(unit)f Fp(e)p Fk(,)i(and)0 3699 y(inverse)k Fp(a)398 3663 y Fo(\000)p Fn(1)540 3699 y Fk(in)g Fp(G)p Fu(\()p Fp(X)8 b Fu(\))p Fk(.)81 b(Then)46 b(the)h(morphisms)e Fu(\001)50 b(:)h Fp(G)e Fr(\000)-58 b(!)51 b Fp(G)31 b Fr(q)g Fp(G)p Fk(,)50 b Fp(")f Fu(:)h Fp(G)g Fr(\000)-58 b(!)51 b Fp(I)8 b Fk(,)50 b(and)0 3815 y Fp(S)33 b Fu(:)28 b Fp(G)g Fr(\000)-58 b(!)28 b Fp(G)35 b Fk(ar)-5 b(e)35 b(given)f(by)999 3972 y Fu(\001)28 b(=)f Fp(\023)1245 3987 y Fn(1)1308 3972 y Fr(\003)21 b Fp(\023)1413 3987 y Fn(2)1453 3972 y Fp(;)216 b(")28 b Fu(=)f Fp(e)1918 3987 y Fl(I)1958 3972 y Fp(;)216 b(S)34 b Fu(=)27 b(id)2480 3929 y Fo(\000)p Fn(1)2480 3998 y Fl(G)2574 3972 y Fp(:)p eop %%Page: 44 14 44 13 bop 0 236 a Fq(44)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)1114 425 y Fs(3.)94 b(A\016ne)37 b(Algebraic)f(Groups)149 600 y Fu(W)-8 b(e)33 b(apply)g(the)g(preceding)g(considerations)f(to)g(the)h(categories)f Fj(K)18 b Fu(-)5 b Fs(cAlg)32 b Fu(and)g Fj(K)18 b Fu(-)5 b Fs(cCoalg)p Fu(.)149 716 y(Consider)35 b Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(,)35 b(the)f(category)g(of)g(comm)m(utativ)m(e)f Fj(K)17 b Fu(-algebras.)53 b(Let)34 b Fp(A;)17 b(B)35 b Fr(2)30 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(.)0 832 y(Then)38 b Fp(A)26 b Fr(\012)f Fp(B)43 b Fu(is)36 b(again)g(a)h(comm)m(utativ)m (e)f Fj(K)17 b Fu(-algebra)42 b(with)37 b(comp)s(onen)m(t)m(wise)g(m)m (ultiplication.)0 948 y(In)k(fact)g(this)g(holds)g(also)f(for)h (non-comm)m(utativ)m(e)f Fj(K)17 b Fu(-algebras)46 b(\(A.5.3\),)d(but)f (in)e Fj(K)18 b Fu(-)5 b Fs(cAlg)40 b Fu(w)m(e)0 1064 y(ha)m(v)m(e)149 1239 y Fs(Prop)s(osition)c(2.3.1.)49 b Fk(The)28 b(tensor)h(pr)-5 b(o)g(duct)29 b(in)g Fj(K)17 b Fk(-)6 b Fs(cAlg)27 b Fk(is)i(the)g(\(c)-5 b(ate)g(goric)g(al\))27 b(c)-5 b(opr)g(o)g(duct.)149 1414 y Fh(Pr)n(oof.)42 b Fu(Let)33 b Fp(f)39 b Fr(2)28 b Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(A;)17 b(Z)7 b Fu(\))p Fp(;)17 b(g)31 b Fr(2)d Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(B)f(;)17 b(Z)7 b Fu(\).)43 b(The)34 b(map)e([)p Fp(f)5 b(;)17 b(g)t Fu(])27 b(:)h Fp(A)22 b Fr(\012)h Fp(B)32 b Fr(\000)-60 b(!)28 b Fp(Z)0 1531 y Fu(de\014ned)40 b(b)m(y)f([)p Fp(f)5 b(;)17 b(g)t Fu(]\()p Fp(a)26 b Fr(\012)g Fp(b)p Fu(\))38 b(:=)g Fp(f)11 b Fu(\()p Fp(a)p Fu(\))p Fp(g)t Fu(\()p Fp(b)p Fu(\))38 b(is)g(the)h(unique)f(algebra)g(homomorphism)d (suc)m(h)40 b(that)0 1647 y([)p Fp(f)5 b(;)17 b(g)t Fu(]\()p Fp(a)22 b Fr(\012)g Fu(1\))28 b(=)f Fp(f)11 b Fu(\()p Fp(a)p Fu(\))33 b(and)f([)p Fp(f)5 b(;)17 b(g)t Fu(]\(1)22 b Fr(\012)g Fp(b)p Fu(\))28 b(=)g Fp(g)t Fu(\()p Fp(b)p Fu(\))k(or)g(suc)m(h)i(that)f(the)g(diagram)1372 1810 y Fp(A)217 b(A)22 b Fr(\012)g Fp(B)p 1474 1782 159 4 v 1550 1780 a Fm(-)1515 1750 y Fl(\023)1540 1761 y Fe(A)2149 1810 y Fp(B)p 1964 1782 156 4 v 1964 1780 a Fm(\033)2003 1750 y Fl(\023)2028 1761 y Fe(B)1502 1992 y Fl(f)1477 1932 y Fm(@)1560 2015 y(@)1643 2098 y(@)1647 2102 y(@)-83 b(R)2086 1983 y Fl(g)2037 1932 y Fm(\000)1954 2015 y(\000)1871 2098 y(\000)1867 2102 y(\000)g(\011)1762 2204 y Fp(Z)p 1797 2102 4 254 v 1799 2102 a Fm(?)1838 1919 y Fn([)o Fl(f)s(;g)r Fn(])0 2328 y Fu(comm)m(utes,)33 b(where)g Fp(\023)801 2343 y Fl(A)859 2328 y Fu(\()p Fp(a)p Fu(\))28 b(=)f Fp(a)c Fr(\012)f Fu(1)32 b(and)h Fp(\023)1595 2343 y Fl(B)1656 2328 y Fu(\()p Fp(b)p Fu(\))28 b(=)g(1)22 b Fr(\012)g Fp(b)33 b Fu(are)g(algebra)e(homomorphisms.)p 3529 2328 4 66 v 3533 2266 59 4 v 3533 2328 V 3591 2328 4 66 v 149 2496 a(So)i(the)g(category)g Fj(K)17 b Fu(-)6 b Fs(cAlg)31 b Fu(has)i(\014nite)f(copro)s(ducts)i(and)e(also)g(an)g (initial)d(ob)5 b(ject)34 b Fj(K)17 b Fu(.)149 2612 y(A)35 b(more)f(general)g(prop)s(ert)m(y)h(of)f(the)h(tensor)h(pro)s(duct)e (of)g(arbitrary)g(algebras)g(w)m(as)h(already)0 2728 y(considered)e(in)f(1.2.13.)149 2844 y(Observ)m(e)j(that)d(the)h(follo) m(wing)d(diagram)h(comm)m(utes)1373 3000 y Fp(A)220 b(A)22 b Fr(\012)h Fp(A)p 1475 2972 162 4 v 1554 2970 a Fm(-)1526 2942 y Fl(\023)1551 2951 y Fd(1)2153 3000 y Fp(A)p 1963 2972 V 1963 2970 a Fm(\033)2014 2942 y Fl(\023)2039 2951 y Fd(2)1459 3181 y Fn(1)1494 3192 y Fe(A)1478 3122 y Fm(@)1561 3205 y(@)1644 3288 y(@)1649 3292 y(@)-83 b(R)2088 3181 y Fn(1)2123 3192 y Fe(A)2039 3122 y Fm(\000)1956 3205 y(\000)1873 3288 y(\000)1868 3292 y(\000)g(\011)1763 3394 y Fp(A)p 1798 3292 4 254 v 1800 3292 a Fm(?)1839 3115 y Fo(r)0 3518 y Fu(where)40 b Fr(r)f Fu(is)f(the)h(m)m (ultiplication)34 b(of)k(the)h(algebra)f(and)h(b)m(y)g(the)h(diagram)c (the)j(co)s(diagonal)d(of)0 3635 y(the)d(copro)s(duct.)149 3809 y Fs(De\014nition)k(2.3.2.)49 b Fu(An)39 b Fk(a\016ne)h(algebr)-5 b(aic)40 b(gr)-5 b(oup)39 b Fu(is)f(a)h(group)g(in)f(the)h(category)h (of)e(com-)0 3926 y(m)m(utativ)m(e)32 b(geometric)g(spaces.)149 4101 y(By)f(the)f(dualit)m(y)f(b)s(et)m(w)m(een)j(the)f(categories)e (of)h(comm)m(utativ)m(e)f(geometric)f(spaces)k(and)e(com-)0 4217 y(m)m(utativ)m(e)j(algebras,)g(an)g(a\016ne)h(algebraic)e(group)h (is)g(represen)m(ted)j(b)m(y)e(a)f(cogroup)h(in)e(the)i(cate-)0 4333 y(gory)e(of)h Fj(K)17 b Fu(-)6 b Fs(cAlg)31 b Fu(of)h(comm)m (utativ)m(e)f(algebras.)149 4449 y(F)-8 b(or)32 b(an)h(arbitrary)e (a\016ne)i(algebraic)e(group)i Fp(H)40 b Fu(w)m(e)33 b(get)g(b)m(y)g(Corollary)e(2.2.7)1082 4607 y(\001)d(=)f Fp(\023)1328 4622 y Fn(1)1390 4607 y Fr(\003)22 b Fp(\023)1496 4622 y Fn(2)1564 4607 y Fr(2)28 b Fj(K)17 b Fu(-)6 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b(H)29 b Fr(\012)23 b Fp(H)8 b Fu(\))p Fp(;)482 4767 y(")27 b Fu(=)g Fp(e)h Fr(2)g Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b Fj(K)g Fu(\))p Fp(;)251 b Fu(and)32 b Fp(S)i Fu(=)27 b(\(id)o(\))2252 4726 y Fo(\000)p Fn(1)2374 4767 y Fr(2)h Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b(H)8 b Fu(\))p Fp(:)0 4905 y Fu(These)34 b(maps)f(and)f(Corollary)f(2.2.7)h(lead)g(to)149 5080 y Fs(Prop)s(osition)k(2.3.3.)49 b Fk(L)-5 b(et)40 b Fp(H)j Fr(2)37 b Fj(K)17 b Fk(-)6 b Fs(cAlg)o Fk(.)58 b Fp(H)46 b Fk(is)39 b(a)g(r)-5 b(epr)g(esenting)39 b(obje)-5 b(ct)39 b(for)g(a)g(functor)0 5196 y Fj(K)17 b Fk(-)6 b Fs(cAlg)27 b Fr(\000)-58 b(!)28 b Fs(Gr)35 b Fk(if)f(and)h(only)f(if) h Fp(H)42 b Fk(is)35 b(a)g(Hopf)f(algebr)-5 b(a.)p eop %%Page: 45 15 45 14 bop 1166 236 a Fq(3.)76 b(AFFINE)33 b(ALGEBRAIC)g(GR)n(OUPS)1062 b(45)149 425 y Fh(Pr)n(oof.)42 b Fu(Both)28 b(statemen)m(ts)h(are)f (equiv)-5 b(alen)m(t)27 b(to)h(the)g(existence)h(of)f(morphisms)e(in)h Fj(K)18 b Fu(-)5 b Fs(cAlg)859 583 y Fu(\001)28 b(:)f Fp(H)36 b Fr(\000)-60 b(!)27 b Fp(H)j Fr(\012)23 b Fp(H)105 b(")27 b Fu(:)h Fp(H)35 b Fr(\000)-60 b(!)28 b Fj(K)121 b Fp(S)33 b Fu(:)28 b Fp(H)35 b Fr(\000)-60 b(!)28 b Fp(H)0 740 y Fu(suc)m(h)34 b(that)e(the)h(follo)m(wing)d(diagrams)h (comm)m(ute)1823 903 y Fp(H)594 b(H)30 b Fr(\012)23 b Fp(H)p 1941 875 529 4 v 2387 873 a Fm(-)2176 854 y Fn(\001)p 1866 1195 4 254 v 1868 1195 a Fm(?)1770 1091 y Fn(\001)698 1068 y Fu(\(coasso)s(ciativit)m(y\))p 2646 1195 V 2648 1195 a Fm(?)2687 1087 y Fn(\001)p Fo(\012)p Fn(1)1718 1293 y Fp(H)30 b Fr(\012)23 b Fp(H)383 b(H)30 b Fr(\012)22 b Fp(H)30 b Fr(\012)23 b Fp(H)p 2046 1265 318 4 v 2281 1263 a Fm(-)2131 1238 y Fn(1)p Fo(\012)p Fn(\001)1231 1445 y Fp(H)1276 b(H)30 b Fr(\012)23 b Fp(H)p 1349 1417 1211 4 v 2477 1415 a Fm(-)1925 1396 y Fn(\001)1126 1836 y Fp(H)30 b Fr(\012)22 b Fp(H)564 b Fj(K)45 b Fr(\012)23 b Fp(H)2296 1808 y Fr(\030)2297 1840 y Fu(=)2401 1836 y Fp(H)2517 1808 y Fr(\030)2518 1840 y Fu(=)2623 1836 y Fp(H)29 b Fr(\012)23 b Fj(K)p 1454 1808 499 4 v 1870 1806 a Fm(-)1641 1780 y Fl(")p Fo(\012)p Fn(1)691 1611 y Fu(\(counit\))p 1274 1737 4 254 v 1275 1737 a Fm(?)1178 1633 y Fn(\001)p 2736 1737 V 2738 1737 a Fm(?)2777 1628 y Fn(1)p Fo(\012)p Fl(")1957 1632 y Fn(1)1480 1511 y Fm(P)1563 1539 y(P)1646 1567 y(P)1729 1594 y(P)1812 1622 y(P)1895 1650 y(P)1978 1677 y(P)2061 1705 y(P)2144 1733 y(P)2158 1737 y(P)-83 b(q)897 2129 y Fu(\(coin)m(v)m(erse\))1535 1968 y Fp(H)412 b Fj(K)p 1652 1935 348 4 v 1917 1933 a Fm(-)1812 1915 y Fl(\017)2510 1968 y Fp(H)p 2096 1935 386 4 v 2399 1933 a Fm(-)2269 1901 y Fl(\021)p 1578 2255 4 254 v 1579 2255 a Fm(?)1481 2151 y Fn(\001)1430 2353 y Fp(H)29 b Fr(\012)23 b Fp(H)684 b(H)29 b Fr(\012)23 b Fp(H)p 1758 2326 619 4 v 2294 2324 a Fm(-)1986 2297 y Fl(S)t Fo(\012)p Fn(id)1981 2389 y(id)11 b Fo(\012)p Fl(S)p 2552 2255 4 254 v 2554 2085 a Fm(6)2593 2151 y Fo(r)p 3529 2491 4 66 v 3533 2428 59 4 v 3533 2491 V 3591 2491 4 66 v 149 2657 a Fu(This)35 b(Prop)s(osition)e(sa)m(ys)j(t)m (w)m(o)f(things.)48 b(First)34 b(of)g(all)e(eac)m(h)j(comm)m(utativ)m (e)f(Hopf)g(algebra)f Fp(H)0 2774 y Fu(de\014nes)40 b(a)f(functor)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(H)r(;)17 b Fu(-\))38 b(:)g Fj(K)17 b Fu(-)6 b Fs(cAlg)37 b Fr(\000)-60 b(!)38 b Fs(Set)h Fu(that)f(factors)h(through)f(the)i(category)0 2890 y(of)d(groups)h(or)f(simply)f(a)h(functor)h Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(H)r(;)17 b Fu(-\))36 b(:)g Fj(K)17 b Fu(-)6 b Fs(cAlg)34 b Fr(\000)-60 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b(ring)i(in)f(one)h(variable)f Fp(x)p Fk(.)149 4751 y Fh(Pr)n(oof.)42 b Fj(G)606 4766 y Fl(a)691 4751 y Fu(is)37 b(an)g(underlying)g(functor)g(that)h(forgets)f(the)h(m) m(ultiplicativ)m(e)c(structure)k(of)0 4867 y(the)22 b(algebra)f(and)h (only)f(preserv)m(es)k(the)d(additiv)m(e)f(group)h(of)f(the)i(algebra.) 38 b(W)-8 b(e)23 b(ha)m(v)m(e)g(to)e(determine)0 4983 y(natural)28 b(isomorphisms)e(\(natural)i(in)g Fp(A)p Fu(\))h Fj(G)1639 4998 y Fl(a)1687 4983 y Fu(\()p Fp(A)p Fu(\))1864 4955 y Fr(\030)1864 4987 y Fu(=)1969 4983 y Fj(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q Fu(])p Fp(;)g(A)p Fu(\).)48 b(Eac)m(h)29 b(elemen)m(t)g Fp(a)f Fr(2)g Fp(A)3541 4947 y Fn(+)0 5099 y Fu(is)33 b(mapp)s(ed)h(to)g(the)g(homomorphism)d(of)j(algebras)f Fp(a)1997 5114 y Fo(\003)2067 5099 y Fu(:)d Fj(K)17 b Fu([)p Fp(x)q Fu(])36 b Fr(3)30 b Fp(p)p Fu(\()p Fp(x)p Fu(\))h Fr(7!)e Fp(p)p Fu(\()p Fp(a)p Fu(\))h Fr(2)h Fp(A)p Fu(.)48 b(This)34 b(is)f(a)0 5216 y(homomorphism)c(of)j (algebras)f(since)i Fp(a)1468 5231 y Fo(\003)1507 5216 y Fu(\()p Fp(p)p Fu(\()p Fp(x)p Fu(\))22 b(+)f Fp(q)t Fu(\()p Fp(x)p Fu(\)\))28 b(=)f Fp(p)p Fu(\()p Fp(a)p Fu(\))21 b(+)g Fp(q)t Fu(\()p Fp(a)p Fu(\))28 b(=)f Fp(a)2841 5231 y Fo(\003)2881 5216 y Fu(\()p Fp(p)p Fu(\()p Fp(x)p Fu(\)\))21 b(+)g Fp(a)3306 5231 y Fo(\003)3346 5216 y Fu(\()p Fp(q)t Fu(\()p Fp(x)p Fu(\)\))p eop %%Page: 46 16 46 15 bop 0 236 a Fq(46)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fu(and)37 b Fp(a)245 440 y Fo(\003)285 425 y Fu(\()p Fp(p)p Fu(\()p Fp(x)p Fu(\))p Fp(q)t Fu(\()p Fp(x)p Fu(\)\))f(=)g Fp(p)p Fu(\()p Fp(a)p Fu(\))p Fp(q)t Fu(\()p Fp(a)p Fu(\))g(=)g Fp(a)1416 440 y Fo(\003)1455 425 y Fu(\()p Fp(p)p Fu(\()p Fp(x)p Fu(\)\))p Fp(a)1762 440 y Fo(\003)1802 425 y Fu(\()p Fp(q)t Fu(\()p Fp(x)p Fu(\)\).)58 b(Another)38 b(reason)g(to)f(see)h (this)f(is)g(that)0 541 y Fj(K)17 b Fu([)q Fp(x)p Fu(])49 b(is)42 b(the)h(free)g(\(comm)m(utativ)m(e\))f Fj(K)18 b 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Fu(-)5 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)q Fp(x)p Fu(])p Fp(;)g(A)p Fu(\))p 1220 1036 453 4 v 1590 1034 a Fm(-)1412 1007 y Fu(-)1444 1015 y Ff(\003)p 1153 1454 4 351 v 1155 1454 a Fm(?)1080 1286 y Fl(g)p 2030 1454 V 2032 1454 a Fm(?)2071 1295 y Fn(K)p Fu(-)o Fn(Alg\()p Fl(K)5 b Fn([)o Fl(x)p Fn(])p Fl(;g)r Fn(\))0 1716 y Fu(comm)m(utes)32 b(for)h(all)d Fp(g)h Fr(2)d Fj(K)18 b Fu(-)5 b Fs(Alg)o Fu(\()p Fp(A;)17 b(B)5 b Fu(\).)p 3529 1716 4 66 v 3533 1653 59 4 v 3533 1716 V 3591 1716 4 66 v 149 1890 a Fs(Remark)37 b(2.3.6.)50 b Fu(Since)35 b Fp(A)1215 1854 y Fn(+)1310 1890 y Fu(has)g(the)h(structure)h(of)e(an)g(additiv)m(e)g(group)g(the)h (sets)h(of)e(ho-)0 2007 y(momorphisms)30 b(of)i(algebras)g Fj(K)18 b Fu(-)5 b Fs(Alg)o Fu(\()p Fj(K)18 b Fu([)p Fp(x)p Fu(])p Fp(;)f(A)p Fu(\))38 b(are)33 b(also)f(additiv)m(e)g (groups.)149 2181 y Fs(Lemma)38 b(2.3.7.)49 b Fk(The)30 b(functor)h Fj(G)1459 2196 y Fl(m)1559 2181 y Fu(=)d Fp(U)38 b Fu(:)28 b Fj(K)18 b Fk(-)5 b Fs(Alg)27 b Fr(\000)-58 b(!)28 b Fs(Gr)j Fk(de\014ne)-5 b(d)29 b(by)i Fj(G)2965 2196 y Fl(m)3037 2181 y Fu(\()p Fp(A)p Fu(\))d(:=)f Fp(U)10 b Fu(\()p Fp(A)p Fu(\))p Fk(,)0 2297 y(the)28 b(underlying)f(multiplic) -5 b(ative)28 b(gr)-5 b(oup)27 b(of)h(units)g(of)g(the)f(algebr)-5 b(a)27 b Fp(A)p Fk(,)j(is)d(a)h(r)-5 b(epr)g(esentable)27 b(functor)0 2413 y(r)-5 b(epr)g(esente)g(d)27 b(by)h(the)g(algebr)-5 b(a)26 b Fj(K)18 b Fu([)p Fp(x;)g(x)1352 2377 y Fo(\000)p Fn(1)1452 2413 y Fu(])28 b(=)f Fj(K)18 b Fu([)p Fp(x;)f(y)t Fu(])p Fp(=)p Fu(\()p Fp(xy)10 b Fr(\000)c Fu(1\))33 b Fk(the)28 b(ring)f(of)g(L)-5 b(aur)g(ent)28 b(p)-5 b(olynomials)0 2530 y(in)35 b(one)f(variable)g Fp(x)p Fk(.)149 2704 y Fh(Pr)n(oof.)42 b Fu(W)-8 b(e)44 b(ha)m(v)m(e)g(to)f (determine)g(natural)f(isomorphisms)f(\(natural)h(in)h Fp(A)p Fu(\))g Fj(G)3255 2719 y Fl(m)3328 2704 y Fu(\()p Fp(A)p Fu(\))3523 2676 y Fr(\030)3523 2708 y Fu(=)0 2820 y Fj(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q(;)g(x)573 2784 y Fo(\000)p Fn(1)673 2820 y Fu(])p Fp(;)g(A)p Fu(\).)54 b(Eac)m(h)37 b(elemen)m(t)e Fp(a)f Fr(2)g Fj(G)1802 2835 y Fl(m)1874 2820 y Fu(\()p Fp(A)p Fu(\))i(is)g(mapp)s(ed)f(to)h(the)g(homomorphism)d(of)0 2937 y(algebras)39 b Fp(a)438 2952 y Fo(\003)517 2937 y Fu(:=)g(\()p Fj(K)18 b Fu([)p Fp(x;)g(x)951 2900 y Fo(\000)p Fn(1)1051 2937 y Fu(])39 b Fr(3)h Fp(x)g Fr(7!)f Fp(a)h Fr(2)g Fp(A)p Fu(\).)64 b(This)40 b(de\014nes)h(a)f(unique)g (homomorphism)c(of)0 3053 y(algebras)h(since)h(eac)m(h)h(homomorphism) 34 b(of)k(algebras)f Fp(f)48 b Fu(from)36 b Fj(K)18 b Fu([)p Fp(x;)f(x)2633 3017 y Fo(\000)p Fn(1)2734 3053 y Fu(])36 b(=)g Fj(K)18 b Fu([)p Fp(x;)f(y)t Fu(])p Fp(=)p Fu(\()p Fp(xy)34 b Fr(\000)26 b Fu(1\))0 3169 y(to)j Fp(A)f Fu(is)h(completely)f(determined)g(b)m(y)i(the)f(images)f(of)g Fp(x)h Fu(and)g(of)g Fp(y)j Fu(but)d(in)f(addition)f(the)i(images)0 3285 y(ha)m(v)m(e)d(to)f(satisfy)g Fp(f)11 b Fu(\()p Fp(x)p Fu(\))p Fp(f)g Fu(\()p Fp(y)t Fu(\))26 b(=)i(1,)e(i.e.)40 b Fp(f)11 b Fu(\()p Fp(x)p Fu(\))25 b(m)m(ust)g(b)s(e)g(in)m(v)m (ertible)f(and)h Fp(f)11 b Fu(\()p Fp(y)t Fu(\))24 b(m)m(ust)h(b)s(e)g (the)h(in)m(v)m(erse)0 3402 y(to)32 b Fp(f)11 b Fu(\()p Fp(x)p Fu(\).)44 b(This)32 b(mapping)f(is)i(bijectiv)m(e.)43 b(F)-8 b(urthermore)32 b(it)g(is)g(natural)f(in)h Fp(A)g Fu(since)1044 4060 y Fp(B)413 b Fj(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q(;)g(x)2104 4024 y Fo(\000)p Fn(1)2204 4060 y Fu(])p Fp(;)g(B)5 b Fu(\))p 1152 4035 351 4 v 1420 4033 a Fm(-)1292 4095 y Fu(-)1325 4103 y Ff(\003)1047 3573 y Fp(A)414 b Fj(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q(;)g(x)2107 3537 y Fo(\000)p Fn(1)2207 3573 y Fu(])p Fp(;)g(A)p Fu(\))p 1149 3547 356 4 v 1422 3545 a Fm(-)1292 3518 y Fu(-)1325 3526 y Ff(\003)p 1082 3965 4 351 v 1084 3965 a Fm(?)1009 3797 y Fl(g)p 1960 3965 V 1961 3965 a Fm(?)2000 3809 y Fn(K)p Fu(-)p Fn(Alg)o(\()p Fl(K)5 b Fn([)p Fl(x;x)2407 3786 y Ff(\000)p Fd(1)2488 3809 y Fn(])p Fl(;g)r Fn(\))0 4227 y Fu(for)32 b(all)f Fp(g)f Fr(2)f Fj(K)17 b Fu(-)6 b Fs(Alg)n Fu(\()p Fp(A;)17 b(B)5 b Fu(\))33 b(comm)m(ute.)p 3529 4227 4 66 v 3533 4165 59 4 v 3533 4227 V 3591 4227 4 66 v 149 4402 a Fs(Remark)k(2.3.8.)50 b Fu(Since)28 b Fp(U)10 b Fu(\()p Fp(A)p Fu(\))29 b(has)f(the)h(structure)h(of)d(a)h (\(m)m(ultiplicativ)m(e\))d(group)j(the)h(sets)0 4518 y Fj(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q(;)g(x)573 4482 y Fo(\000)p Fn(1)673 4518 y Fu(])p Fp(;)g(A)p Fu(\))33 b(are)f(also)g(groups.)149 4692 y Fs(Lemma)38 b(2.3.9.)49 b Fk(The)41 b(functor)g Fj(M)1497 4707 y Fl(n)1590 4692 y Fu(:)f Fj(K)17 b Fk(-)6 b Fs(Alg)39 b Fr(\000)-58 b(!)40 b Fj(K)18 b Fk(-)5 b Fs(Alg)41 b Fk(with)g Fj(M)2772 4707 y Fl(n)2825 4692 y Fu(\()p Fp(A)p Fu(\))g Fk(the)h(algebr)-5 b(a)40 b(of)0 4809 y Fp(n)22 b Fr(\002)g Fp(n)p Fk(-matric)-5 b(es)34 b(with)h(entries)f(in)h Fp(A)f Fk(is)h(r)-5 b(epr)g(esentable)33 b(by)i(the)g(algebr)-5 b(a)34 b Fj(K)17 b Fr(h)p Fp(x)2896 4824 y Fn(1)q(1)2977 4809 y Fp(;)g(x)3076 4824 y Fn(12)3151 4809 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(x)3441 4824 y Fl(nn)3531 4809 y Fr(i)p Fk(,)0 4925 y(the)35 b(non)f(c)-5 b(ommutative)35 b(p)-5 b(olynomialring)33 b(in)h(the)h(variables)f Fp(x)2329 4940 y Fl(ij)2390 4925 y Fk(.)149 5099 y Fh(Pr)n(oof.)42 b Fu(The)37 b(p)s(olynomial)32 b(ring)j Fj(K)18 b Fr(h)p Fp(x)1615 5114 y Fl(ij)1681 5099 y Fr(i)36 b Fu(is)f(free)h(o)m(v)m(er)h(the)f(set)h Fr(f)p Fp(x)2691 5114 y Fl(ij)2752 5099 y Fr(g)e Fu(in)g(the)h (category)g(of)0 5216 y(\(non)h(comm)m(utativ)m(e\))f(algebras,)i(i.e.) 56 b(for)37 b(eac)m(h)h(algebra)d(and)j(for)e(eac)m(h)i(map)e Fp(f)46 b Fu(:)36 b Fr(f)p Fp(x)3229 5231 y Fl(ij)3289 5216 y Fr(g)g(\000)-60 b(!)35 b Fp(A)p eop %%Page: 47 17 47 16 bop 1166 236 a Fq(3.)76 b(AFFINE)33 b(ALGEBRAIC)g(GR)n(OUPS)1062 b(47)0 425 y Fu(there)38 b(exists)g(a)e(unique)i(homomorphism)c(of)i (algebras)h Fp(g)h Fu(:)e Fj(K)17 b Fr(h)p Fp(x)2441 440 y Fn(1)q(1)2522 425 y Fp(;)g(x)2621 440 y Fn(12)2696 425 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(x)2986 440 y Fl(nn)3076 425 y Fr(i)35 b(\000)-60 b(!)35 b Fp(A)i Fu(suc)m(h)0 541 y(that)32 b(the)h(diagram)1435 648 y Fr(f)p Fp(x)1540 663 y Fl(ij)1601 648 y Fr(g)244 b Fj(K)17 b Fr(h)p Fp(x)2060 663 y Fl(ij)2127 648 y Fr(i)p 1679 626 188 4 v 1784 624 a Fm(-)1760 605 y Fl(\023)1685 885 y(f)1611 776 y Fm(@)1694 859 y(@)1777 942 y(@)1860 1025 y(@)1879 1043 y(@)-83 b(R)1994 1146 y Fp(A)p 2028 1044 4 351 v 2030 1044 a Fm(?)2069 876 y Fl(g)0 1253 y Fu(comm)m(utes.)54 b(So)36 b(eac)m(h)g(matrix)f(in)g Fp(M)1400 1268 y Fl(n)1447 1253 y Fu(\()p Fp(A)p Fu(\))h(de\014nes)i(a)e(unique)g(a)g (homomorphism)d(of)j(algebras)0 1369 y Fj(K)17 b Fr(h)p Fp(x)166 1384 y Fn(11)246 1369 y Fp(;)g(x)345 1384 y Fn(12)420 1369 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)711 1384 y Fl(nn)801 1369 y Fr(i)27 b(\000)-60 b(!)28 b Fp(A)k Fu(and)h(con)m(v)m(ersely)-8 b(.)p 3529 1369 4 66 v 3533 1307 59 4 v 3533 1369 V 3591 1369 4 66 v 149 1546 a Fs(Example)37 b(2.3.10.)49 b Fu(1.)44 b(The)33 b(a\016ne)g(algebraic)e(group)h (called)g Fk(additive)i(gr)-5 b(oup)1374 1707 y Fj(G)1445 1722 y Fl(a)1521 1707 y Fu(:)27 b Fj(K)18 b Fu(-)5 b Fs(cAlg)27 b Fr(\000)-60 b(!)27 b Fs(Ab)0 1869 y Fu(with)37 b Fj(G)298 1884 y Fl(a)346 1869 y Fu(\()p Fp(A)p Fu(\))f(:=)g Fp(A)743 1833 y Fn(+)839 1869 y Fu(from)g(Lemma)g(2.3.5)h(is)g (represen)m(ted)i(b)m(y)g(the)e(Hopf)h(algebra)e Fj(K)17 b Fu([)p Fp(x)q Fu(].)64 b(W)-8 b(e)0 1985 y(determine)32 b(com)m(ultiplication,)c(counit,)33 b(and)f(an)m(tip)s(o)s(de.)149 2101 y(By)46 b(Corollary)d(2.2.7)i(the)h(com)m(ultiplication)40 b(is)45 b(\001)k(=)g Fp(\023)2315 2116 y Fn(1)2386 2101 y Fr(\003)30 b Fp(\023)2500 2116 y Fn(2)2589 2101 y Fr(2)49 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)q Fu(])p Fp(;)g Fj(K)23 b Fu([)p Fp(x)q Fu(])37 b Fr(\012)0 2218 y Fj(K)17 b Fu([)q Fp(x)p Fu(]\))253 2190 y Fr(\030)254 2222 y Fu(=)358 2218 y Fj(G)429 2233 y Fl(a)477 2218 y Fu(\()p Fj(K)g Fu([)q Fp(x)p Fu(])28 b Fr(\012)23 b Fj(K)17 b Fu([)p Fp(x)q Fu(]\).)49 b(Hence)998 2379 y(\001\()p Fp(x)p Fu(\))28 b(=)f Fp(\023)1375 2394 y Fn(1)1415 2379 y Fu(\()p Fp(x)p Fu(\))c(+)f Fp(\023)1701 2394 y Fn(2)1741 2379 y Fu(\()p Fp(x)p Fu(\))28 b(=)f Fp(x)c Fr(\012)f Fu(1)g(+)g(1)g Fr(\012)h Fp(x:)149 2540 y Fu(The)34 b(counit)e(is)g Fp(")27 b Fu(=)h Fp(e)968 2555 y Fc(K)1052 2540 y Fu(=)f(0)h Fr(2)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fj(K)18 b Fu([)p Fp(x)p Fu(])p Fp(;)g Fj(K)23 b Fu(\))2069 2513 y Fr(\030)2070 2544 y Fu(=)2174 2540 y Fj(G)2246 2555 y Fl(a)2294 2540 y Fu(\()p Fj(K)17 b Fu(\))38 b(hence)1608 2702 y Fp(")p Fu(\()p Fp(x)p Fu(\))28 b(=)f(0)p Fp(:)149 2866 y Fu(The)34 b(an)m(tip)s(o)s(de)e(is)g Fp(S)h Fu(=)28 b(id)1130 2823 y Fo(\000)p Fn(1)1130 2896 y Fc(K)5 b Fn([)p Fl(x)p Fn(])1293 2866 y Fr(2)28 b Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fj(K)18 b Fu([)p Fp(x)p Fu(])p Fp(;)f Fj(K)23 b Fu([)q Fp(x)p Fu(]\))2240 2838 y Fr(\030)2241 2870 y Fu(=)2345 2866 y Fj(G)2416 2881 y Fl(a)2464 2866 y Fu(\()p Fj(K)17 b Fu([)q Fp(x)p Fu(]\))38 b(hence)1556 3040 y Fp(S)6 b Fu(\()p Fp(x)p Fu(\))28 b(=)f Fr(\000)p Fp(x:)149 3202 y Fu(2.)44 b(The)33 b(a\016ne)g (algebraic)e(group)i(called)e Fk(multiplic)-5 b(ative)34 b(gr)-5 b(oup)1361 3363 y Fj(G)1433 3378 y Fl(m)1533 3363 y Fu(:)28 b Fj(K)17 b Fu(-)6 b Fs(cAlg)26 b Fr(\000)-60 b(!)28 b Fs(Ab)0 3525 y Fu(with)41 b Fj(G)302 3540 y Fl(m)375 3525 y Fu(\()p Fp(A)p Fu(\))h(:=)h Fp(A)785 3489 y Fo(\003)867 3525 y Fu(=)f Fp(U)10 b Fu(\()p Fp(A)p Fu(\))42 b(from)e(Lemma)g(2.3.7)h(is)f(represen)m(ted)k(b)m(y)e(the)g (Hopf)f(algebra)0 3641 y Fj(K)17 b Fu([)q Fp(x;)g(x)253 3605 y Fo(\000)p Fn(1)353 3641 y Fu(])28 b(=)g Fj(K)17 b Fu([)p Fp(x)q(;)g(y)t Fu(])p Fp(=)p Fu(\()p Fp(xy)30 b Fr(\000)23 b Fu(1\).)43 b(W)-8 b(e)33 b(determine)f(com)m (ultiplication,)c(counit,)33 b(and)f(an)m(tip)s(o)s(de.)149 3757 y(By)i(Corollary)c(2.2.7)i(the)h(com)m(ultiplication)28 b(is)23 3919 y(\001)g(=)g Fp(\023)270 3934 y Fn(1)332 3919 y Fr(\003)22 b Fp(\023)438 3934 y Fn(2)505 3919 y Fr(2)28 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fj(K)17 b Fu([)q Fp(x;)g(x)1222 3877 y Fo(\000)p Fn(1)1322 3919 y Fu(])p Fp(;)g Fj(K)h Fu([)p Fp(x;)f(x)1646 3877 y Fo(\000)p Fn(1)1747 3919 y Fu(])22 b Fr(\012)h Fj(K)17 b Fu([)p Fp(x)q(;)g(x)2149 3877 y Fo(\000)p Fn(1)2249 3919 y Fu(]\))2342 3891 y Fr(\030)2343 3923 y Fu(=)2447 3919 y Fj(G)2518 3934 y Fl(m)2591 3919 y Fu(\()p Fj(K)g Fu([)p Fp(x)q(;)g(x)2882 3877 y Fo(\000)p Fn(1)2982 3919 y Fu(])23 b Fr(\012)f Fj(K)c Fu([)p Fp(x;)f(x)3384 3877 y Fo(\000)p Fn(1)3485 3919 y Fu(]\))p Fp(:)0 4080 y Fu(Hence)1191 4201 y(\001\()p Fp(x)p Fu(\))29 b(=)e Fp(\023)1569 4216 y Fn(1)1609 4201 y Fu(\()p Fp(x)p Fu(\))c Fr(\001)e Fp(\023)1846 4216 y Fn(2)1886 4201 y Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(x)22 b Fr(\012)h Fp(x:)149 4342 y Fu(The)34 b(counit)e(is)g Fp(")27 b Fu(=)h Fp(e)968 4357 y Fc(K)1052 4342 y Fu(=)f(1)h Fr(2)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fj(K)18 b Fu([)p Fp(x;)f(x)1948 4306 y Fo(\000)p Fn(1)2049 4342 y Fu(])p Fp(;)g Fj(K)g Fu(\))2263 4314 y Fr(\030)2264 4346 y Fu(=)2368 4342 y Fj(G)2439 4357 y Fl(m)2512 4342 y Fu(\()p Fj(K)g Fu(\))39 b(hence)1608 4503 y Fp(")p Fu(\()p Fp(x)p Fu(\))28 b(=)f(1)p Fp(:)149 4667 y Fu(The)45 b(an)m(tip)s(o)s(de)f(is)f Fp(S)53 b Fu(=)47 b(id)1203 4624 y Fo(\000)p Fn(1)1203 4698 y Fc(K)5 b Fn([)p Fl(x;x)1369 4679 y Ff(\000)p Fd(1)1456 4698 y Fn(])1527 4667 y Fr(2)47 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fj(K)17 b Fu([)q Fp(x;)g(x)2263 4631 y Fo(\000)p Fn(1)2363 4667 y Fu(])p Fp(;)g Fj(K)h Fu([)p Fp(x;)f(x)2687 4631 y Fo(\000)p Fn(1)2788 4667 y Fu(]\))2900 4640 y Fr(\030)2901 4671 y Fu(=)3024 4667 y Fj(G)3096 4682 y Fl(a)3144 4667 y Fu(\()p Fj(K)g Fu([)p Fp(x)q(;)g(x)3435 4631 y Fo(\000)p Fn(1)3535 4667 y Fu(]\))0 4792 y(hence)1547 4913 y Fp(S)6 b Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(x)1931 4872 y Fo(\000)p Fn(1)2025 4913 y Fp(:)149 5054 y Fu(3.)44 b(The)33 b(a\016ne)g (algebraic)e(group)i(called)e Fk(additive)j(matrix)h(gr)-5 b(oup)1343 5216 y Fj(M)1431 5174 y Fn(+)1431 5240 y Fl(n)1524 5216 y Fu(:)28 b Fj(K)17 b Fu(-)6 b Fs(cAlg)26 b Fr(\000)-60 b(!)28 b Fs(Ab)p Fp(;)p eop %%Page: 48 18 48 17 bop 0 236 a Fq(48)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fu(with)c Fj(M)306 389 y Fn(+)306 450 y Fl(n)371 425 y Fu(\()p Fp(A)p Fu(\))h(the)f(additiv)m(e)g(group)g(of)g Fp(n)14 b Fr(\002)g Fp(n)p Fu(-matrices)28 b(with)g(co)s(e\016cien)m(ts)h(in)f Fp(A)g Fu(is)g(represen)m(ted)0 541 y(b)m(y)36 b(the)f(comm)m(utativ)m (e)f(algebra)g Fp(M)1341 505 y Fn(+)1331 566 y Fl(n)1432 541 y Fu(=)e Fj(K)17 b Fu([)p Fp(x)1694 556 y Fl(ij)1760 541 y Fr(j)p Fu(1)32 b Fr(\024)g Fp(i;)17 b(j)37 b Fr(\024)32 b Fp(n)p Fu(])k(\(Lemma)d(2.3.9\).)50 b(This)35 b(algebra)0 658 y(m)m(ust)e(b)s(e)f(a)h(Hopf)f(algebra.)149 774 y(The)f(com)m (ultiplication)25 b(is)k(\001)f(=)g Fp(\023)1404 789 y Fn(1)1460 774 y Fr(\003)16 b Fp(\023)1560 789 y Fn(2)1627 774 y Fr(2)28 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(M)2195 738 y Fn(+)2185 799 y Fl(n)2255 774 y Fp(;)17 b(M)2403 738 y Fn(+)2393 799 y Fl(n)2478 774 y Fr(\012)f Fp(M)2675 738 y Fn(+)2665 799 y Fl(n)2736 774 y Fu(\))2801 746 y Fr(\030)2802 778 y Fu(=)2907 774 y Fj(M)2995 738 y Fn(+)2995 799 y Fl(n)3060 774 y Fu(\()p Fp(M)3202 738 y Fn(+)3192 799 y Fl(n)3278 774 y Fr(\012)g Fp(M)3475 738 y Fn(+)3465 799 y Fl(n)3535 774 y Fu(\).)0 890 y(Hence)846 1023 y(\001\()p Fp(x)1020 1038 y Fl(ij)1081 1023 y Fu(\))28 b(=)g Fp(\023)1285 1038 y Fn(1)1325 1023 y Fu(\()p Fp(x)1418 1038 y Fl(ij)1478 1023 y Fu(\))22 b(+)h Fp(\023)1671 1038 y Fn(2)1710 1023 y Fu(\()p Fp(x)1803 1038 y Fl(ij)1864 1023 y Fu(\))28 b(=)f Fp(x)2088 1038 y Fl(ij)2172 1023 y Fr(\012)22 b Fu(1)g(+)g(1)g Fr(\012)h Fp(x)2666 1038 y Fl(ij)2727 1023 y Fp(:)149 1176 y Fu(The)34 b(counit)e(is)g Fp(")27 b Fu(=)h Fp(e)968 1191 y Fc(K)1052 1176 y Fu(=)f(\(0\))h Fr(2)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(M)1875 1140 y Fn(+)1865 1201 y Fl(n)1935 1176 y Fp(;)17 b Fj(K)g Fu(\))2122 1148 y Fr(\030)2123 1180 y Fu(=)2227 1176 y Fj(M)2315 1140 y Fn(+)2315 1201 y Fl(n)2380 1176 y Fu(\()p Fj(K)h Fu(\))38 b(hence)1578 1349 y Fp(")p Fu(\()p Fp(x)1717 1364 y Fl(ij)1777 1349 y Fu(\))28 b(=)f(0)p Fp(:)149 1529 y Fu(The)34 b(an)m(tip)s(o)s(de)e(is)g Fp(S)h Fu(=)28 b(id)1130 1486 y Fo(\000)p Fn(1)1130 1570 y Fl(M)1205 1540 y Fd(+)1198 1581 y Fe(n)1288 1529 y Fr(2)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(M)1855 1492 y Fn(+)1845 1553 y Fl(n)1915 1529 y Fp(;)17 b(M)2063 1492 y Fn(+)2053 1553 y Fl(n)2123 1529 y Fu(\))2188 1501 y Fr(\030)2189 1533 y Fu(=)2293 1529 y Fj(M)2381 1492 y Fn(+)2381 1553 y Fl(n)2446 1529 y Fu(\()p Fp(M)2588 1492 y Fn(+)2578 1553 y Fl(n)2648 1529 y Fu(\))33 b(hence)1495 1721 y Fp(S)6 b Fu(\()p Fp(x)1654 1736 y Fl(ij)1715 1721 y Fu(\))28 b(=)f Fr(\000)p Fp(x)2016 1736 y Fl(ij)2078 1721 y Fp(:)149 1894 y Fu(4.)43 b(The)31 b(matrix)e(algebra)g Fj(M)1208 1909 y Fl(n)1261 1894 y Fu(\()p Fp(A)p Fu(\))h(also)f(has)i (a)e(noncomm)m(utativ)m(e)h(m)m(ultiplication,)c(the)k(ma-)0 2011 y(trix)j(m)m(ultiplication,)c(de\014ning)k(a)h(monoid)d(structure) k Fj(M)2148 1974 y Fo(\002)2148 2035 y Fl(n)2213 2011 y Fu(\()p Fp(A)p Fu(\).)46 b(Th)m(us)35 b Fj(K)18 b Fu([)p Fp(x)2837 2026 y Fl(ij)2904 2011 y Fu(])33 b(carries)h(another)0 2127 y(coalgebra)40 b(structure)h(whic)m(h)g(de\014nes)i(a)d(bialgebra) f Fp(M)2107 2091 y Fo(\002)2097 2151 y Fl(n)2208 2127 y Fu(=)i Fj(K)18 b Fu([)p Fp(x)2479 2142 y Fl(ij)2546 2127 y Fu(].)67 b(Ob)m(viously)41 b(there)g(is)f(no)0 2243 y(an)m(tip)s(o)s(de.)149 2359 y(The)31 b(com)m(ultiplication)25 b(is)k(\001)f(=)g Fp(\023)1404 2374 y Fn(1)1460 2359 y Fr(\003)16 b Fp(\023)1560 2374 y Fn(2)1627 2359 y Fr(2)28 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(M)2195 2323 y Fo(\002)2185 2384 y Fl(n)2255 2359 y Fp(;)17 b(M)2403 2323 y Fo(\002)2393 2384 y Fl(n)2478 2359 y Fr(\012)f Fp(M)2675 2323 y Fo(\002)2665 2384 y Fl(n)2736 2359 y Fu(\))2801 2332 y Fr(\030)2802 2363 y Fu(=)2907 2359 y Fj(M)2995 2323 y Fo(\002)2995 2384 y Fl(n)3060 2359 y Fu(\()p Fp(M)3202 2323 y Fo(\002)3192 2384 y Fl(n)3278 2359 y Fr(\012)g Fp(M)3475 2323 y Fo(\002)3465 2384 y Fl(n)3535 2359 y Fu(\).)0 2476 y(Hence)34 b(\001\(\()p Fp(x)502 2491 y Fl(ij)563 2476 y Fu(\)\))28 b(=)f Fp(\023)804 2491 y Fn(1)844 2476 y Fu(\(\()p Fp(x)975 2491 y Fl(ij)1036 2476 y Fu(\)\))22 b Fr(\001)g Fp(\023)1218 2491 y Fn(2)1258 2476 y Fu(\(\()p Fp(x)1389 2491 y Fl(ij)1449 2476 y Fu(\)\))28 b(=)g(\()p Fp(x)1750 2491 y Fl(ij)1810 2476 y Fu(\))23 b Fr(\012)f Fu(\()p Fp(x)2063 2491 y Fl(ij)2124 2476 y Fu(\))33 b(or)1317 2674 y(\001\()p Fp(x)1491 2689 y Fl(ik)1558 2674 y Fu(\))28 b(=)1727 2579 y Fi(X)1783 2789 y Fl(j)1888 2674 y Fp(x)1943 2689 y Fl(ij)2026 2674 y Fr(\012)23 b Fp(x)2181 2689 y Fl(j)t(k)2256 2674 y Fp(:)149 2954 y Fu(The)34 b(counit)e(is)g Fp(")27 b Fu(=)h Fp(e)968 2969 y Fc(K)1052 2954 y Fu(=)f Fp(E)34 b Fr(2)28 b Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(M)1829 2917 y Fo(\002)1819 2978 y Fl(n)1888 2954 y Fp(;)17 b Fj(K)h Fu(\))2075 2926 y Fr(\030)2076 2958 y Fu(=)2180 2954 y Fj(M)2268 2917 y Fo(\002)2268 2978 y Fl(n)2334 2954 y Fu(\()p Fj(K)f Fu(\))38 b(hence)1550 3127 y Fp(")p Fu(\()p Fp(x)1689 3142 y Fl(ij)1750 3127 y Fu(\))27 b(=)h Fp(\016)1962 3142 y Fl(ij)2023 3127 y Fp(:)149 3300 y Fu(5.)65 b(Let)40 b Fj(K)63 b Fu(b)s(e)40 b(a)f(\014eld)h(of)f(c)m (haracteristic)h Fp(p)p Fu(.)64 b(The)41 b(algebra)d Fp(H)48 b Fu(=)39 b Fj(K)18 b Fu([)p Fp(x)p Fu(])q Fp(=)p Fu(\()p Fp(x)3025 3264 y Fl(p)3070 3300 y Fu(\))40 b(carries)g(the)0 3417 y(structure)f(of)e(a)g(Hopf)g(algebra)g(with)g(\001\()p Fp(x)p Fu(\))g(=)f Fp(x)26 b Fr(\012)f Fu(1)h(+)f(1)g Fr(\012)h Fp(x)p Fu(,)40 b Fp(")p Fu(\()p Fp(x)p Fu(\))c(=)g(0,)i(and)g Fp(S)6 b Fu(\()p Fp(x)p Fu(\))36 b(=)g Fr(\000)p Fp(x)p Fu(.)0 3533 y(T)-8 b(o)37 b(sho)m(w)h(that)e(\001)h(is)f(w)m(ell)g (de\014ned)i(w)m(e)g(ha)m(v)m(e)g(to)e(sho)m(w)i(\001\()p Fp(x)p Fu(\))2329 3497 y Fl(p)2404 3533 y Fu(=)c(0.)56 b(But)37 b(this)f(is)g(ob)m(vious)h(b)m(y)0 3649 y(the)d(rules)g(for)f (computing)f Fp(p)p Fu(-th)i(p)s(o)m(w)m(ers)h(in)e(c)m(haracteristic)g Fp(p)p Fu(.)47 b(W)-8 b(e)34 b(ha)m(v)m(e)h(\()p Fp(x)24 b Fr(\012)f Fu(1)g(+)f(1)h Fr(\012)g Fp(x)p Fu(\))3454 3613 y Fl(p)3524 3649 y Fu(=)0 3765 y Fp(x)55 3729 y Fl(p)117 3765 y Fr(\012)g Fu(1)f(+)g(1)g Fr(\012)g Fp(x)611 3729 y Fl(p)679 3765 y Fu(=)28 b(0.)149 3882 y(Th)m(us)35 b(the)e(algebra)e Fp(H)40 b Fu(represen)m(ts)35 b(an)e(a\016ne)g (algebraic)e(group:)890 4055 y Fp(\013)952 4070 y Fl(p)991 4055 y Fu(\()p Fp(A)p Fu(\))d(:=)g Fj(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fp(H)r(;)17 b(A)p Fu(\))1934 4027 y Fr(\030)1935 4059 y Fu(=)2039 4055 y Fr(f)p Fp(a)28 b Fr(2)g Fp(A)p Fr(j)p Fp(a)2414 4014 y Fl(p)2481 4055 y Fu(=)g(0)p Fr(g)p Fp(:)0 4228 y Fu(The)45 b(group)f(m)m (ultiplication)39 b(is)44 b(the)g(addition)e(of)i Fp(p)p Fu(-nilp)s(oten)m(t)e(elemen)m(ts.)78 b(So)44 b(w)m(e)h(ha)m(v)m(e)h (the)0 4345 y Fk(gr)-5 b(oup)35 b(of)f Fp(p)p Fk(-nilp)-5 b(otent)35 b(elements)p Fu(.)149 4461 y(6.)65 b(The)40 b(algebra)f Fp(H)47 b Fu(=)39 b Fj(K)18 b Fu([)p Fp(x)p Fu(])p Fp(=)p Fu(\()p Fp(x)1412 4425 y Fl(n)1492 4461 y Fr(\000)28 b Fu(1\))39 b(is)g(a)g(Hopf)h(algebra)e(with)h(the)h(com)m (ultiplication)0 4577 y(\001\()p Fp(x)p Fu(\))33 b(=)e Fp(x)25 b Fr(\012)f Fp(x)p Fu(,)36 b(the)g(counit)e Fp(")p Fu(\()p Fp(x)p Fu(\))e(=)g(1,)k(and)f(the)g(an)m(tip)s(o)s(de)g Fp(S)6 b Fu(\()p Fp(x)p Fu(\))32 b(=)g Fp(x)2712 4541 y Fl(n)p Fo(\000)p Fn(1)2849 4577 y Fu(.)51 b(These)37 b(maps)e(are)0 4693 y(w)m(ell)30 b(de\014ned)i(since)f(w)m(e)g(ha)m(v)m (e)h(for)e(example)g(\001\()p Fp(x)p Fu(\))1870 4657 y Fl(n)1946 4693 y Fu(=)d(\()p Fp(x)19 b Fr(\012)f Fp(x)p Fu(\))2349 4657 y Fl(n)2424 4693 y Fu(=)28 b Fp(x)2583 4657 y Fl(n)2648 4693 y Fr(\012)19 b Fp(x)2799 4657 y Fl(n)2874 4693 y Fu(=)27 b(1)18 b Fr(\012)g Fu(1.)43 b(Observ)m(e)0 4810 y(that)29 b(this)h(Hopf)f(algebra)f(is)h (isomorphic)f(to)h(the)h(group)g(algebra)e Fj(K)17 b Fp(C)2589 4825 y Fl(n)2672 4810 y Fu(of)29 b(the)h(cyclic)f(group)g Fp(C)3553 4825 y Fl(n)0 4926 y Fu(of)j(order)h Fp(n)p Fu(.)149 5042 y(Th)m(us)i(the)e(algebra)e Fp(H)40 b Fu(represen)m(ts)35 b(an)e(a\016ne)g(algebraic)e(group:)884 5216 y Fp(\026)943 5231 y Fl(n)990 5216 y Fu(\()p Fp(A)p Fu(\))c(:=)h Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b(A)p Fu(\))1932 5188 y Fr(\030)1933 5220 y Fu(=)2037 5216 y Fr(f)p Fp(a)28 b Fr(2)g Fp(A)p Fr(j)p Fp(a)2412 5174 y Fl(n)2487 5216 y Fu(=)f(1)p Fr(g)p Fp(;)p eop %%Page: 49 19 49 18 bop 1166 236 a Fq(3.)76 b(AFFINE)33 b(ALGEBRAIC)g(GR)n(OUPS)1062 b(49)0 425 y Fu(that)43 b(is)f(the)h Fk(gr)-5 b(oup)45 b(of)f Fp(n)p Fk(-th)g(r)-5 b(o)g(ots)45 b(of)f(unity)p Fu(.)75 b(The)44 b(group)f(m)m(ultiplication)38 b(is)k(the)i(ordinary)0 541 y(m)m(ultiplication)28 b(of)k(ro)s(ots)g(of)g(unit)m(y)-8 b(.)149 658 y(7.)41 b(The)27 b(linear)d(groups)i(or)f(matrix)f(groups)i Fj(G)c(L)17 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\),)27 b Fj(SL)5 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\))26 b(and)g(other)g(suc)m(h)h(groups)0 774 y(are)36 b(further)g(examples)g (of)f(a\016ne)i(algebraic)d(groups.)54 b(W)-8 b(e)36 b(will)e(discuss)j(them)e(in)g(the)i(section)0 890 y(on)32 b(quan)m(tum)h(groups.)149 1064 y Fs(Problem)j(2.3.7.)49 b Fu(1.)44 b(The)33 b(construction)g(of)f(the)h(general)f(linear)f (group)848 1221 y Fj(G)22 b(L)17 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\))28 b(=)g Fr(f)p Fu(\()p Fp(a)1546 1236 y Fl(ij)1606 1221 y Fu(\))g Fr(2)g Fj(M)1854 1236 y Fl(n)1907 1221 y Fu(\()p Fp(A)p Fu(\))p Fr(j)p Fu(\()p Fp(a)2173 1236 y Fl(ij)2233 1221 y Fu(\))33 b(in)m(v)m(ertible)o Fr(g)0 1378 y Fu(de\014nes)h(an)f(a\016ne)g(algebraic)e(group.)43 b(Describ)s(e)33 b(the)g(represen)m(ting)g(Hopf)g(algebra.)149 1495 y(2.)62 b(The)39 b(sp)s(ecial)f(linear)f(group)h Fj(S)o(L)6 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\))38 b(is)g(an)h(a\016ne)g(algebraic)e(group.)61 b(What)38 b(is)h(the)0 1611 y(represen)m(ting)34 b(Hopf)e(algebra?)149 1727 y(3.)67 b(The)41 b(real)f(unit)f(circle)h Fj(S)1235 1691 y Fn(1)1269 1727 y Fu(\()p Fj(R)t Fu(\))47 b(carry)40 b(the)h(structure)h(of)e(a)g(group)g(b)m(y)h(the)g(addition)d(of)0 1843 y(angles.)48 b(Is)34 b(it)f(p)s(ossible)h(to)f(mak)m(e)i Fj(S)1349 1807 y Fn(1)1416 1843 y Fu(with)f(the)h(a\016ne)f(algebra)f Fj(K)18 b Fu([)p Fp(c;)f(s)p Fu(])p Fp(=)p Fu(\()p Fp(s)2805 1807 y Fn(2)2873 1843 y Fu(+)23 b Fp(c)3014 1807 y Fn(2)3076 1843 y Fr(\000)h Fu(1\))34 b(in)m(to)f(an)0 1959 y(a\016ne)38 b(algebraic)e(group?)59 b(\(Hin)m(t:)54 b(Ho)m(w)38 b(can)g(y)m(ou)h (add)f(t)m(w)m(o)g(p)s(oin)m(ts)g(\()p Fp(x)2717 1974 y Fn(1)2757 1959 y Fp(;)17 b(y)2849 1974 y Fn(1)2887 1959 y Fu(\))38 b(and)g(\()p Fp(x)3251 1974 y Fn(2)3291 1959 y Fp(;)17 b(y)3383 1974 y Fn(2)3421 1959 y Fu(\))38 b(on)0 2076 y(the)33 b(unit)f(circle,)g(suc)m(h)i(that)e(y)m(ou)h(get)g (the)g(addition)e(of)h(the)h(asso)s(ciated)f(angles?\))p eop %%Page: 50 20 50 19 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 Fs(4.)57 b(F)-9 b(ormal)36 b(Groups)149 600 y Fu(Consider)25 b(no)m(w)g Fj(K)18 b Fu(-)5 b Fs(cCoalg)24 b Fu(the)h(category)g(of)f(co)s(comm)m (utativ)m(e)f Fj(K)18 b Fu(-coalgebras.)45 b(Let)25 b Fp(C)r(;)17 b(D)30 b Fr(2)0 716 y Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(.)51 b(Then)37 b Fp(C)30 b Fr(\012)25 b Fp(D)37 b Fu(is)e(again)f(a)g(co)s(comm)m(utativ)m(e)g Fj(K)18 b Fu(-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 Fj(K)17 b Fu(-algebras,)38 b(but)33 b(in)e Fj(K)18 b Fu(-)5 b Fs(cCoalg)32 b Fu(w)m(e)i(ha)m(v)m(e)149 984 y Fs(Prop)s(osition)i(2.4.1.)49 b Fk(The)26 b(tensor)h(pr)-5 b(o)g(duct)27 b(in)f Fj(K)17 b Fk(-)6 b Fs(cCoalg)26 b Fk(is)h(the)g(\(c)-5 b(ate)g(goric)g(al\))25 b(pr)-5 b(o)g(duct.)149 1136 y Fh(Pr)n(oof.)42 b Fu(Let)37 b Fp(f)45 b Fr(2)35 b Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(Z)r(;)17 b(C)7 b Fu(\))p Fp(;)17 b(g)36 b Fr(2)f Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(Z)r(;)17 b(D)s Fu(\).)54 b(The)38 b(map)d(\()p Fp(f)5 b(;)17 b(g)t Fu(\))33 b(:)i Fp(Z)41 b Fr(\000)-60 b(!)0 1252 y Fp(C)36 b Fr(\012)29 b Fp(D)45 b Fu(de\014ned)f(b)m(y)g(\()p Fp(f)5 b(;)17 b(g)t Fu(\)\()p Fp(z)t Fu(\))44 b(:=)1370 1177 y Fi(P)1492 1252 y Fp(f)11 b Fu(\()p Fp(z)1634 1268 y Fn(\(1\))1728 1252 y Fu(\))29 b Fr(\012)h Fp(g)t Fu(\()p Fp(z)2036 1268 y Fn(\(2\))2130 1252 y Fu(\))42 b(is)g(the)h(unique)g (homomorphism)c(of)0 1368 y(coalgebras)g(suc)m(h)h(that)f(\(1)26 b Fr(\012)h Fp(")1186 1383 y Fl(D)1250 1368 y Fu(\)\()p Fp(f)5 b(;)17 b(g)t Fu(\)\()p Fp(z)t Fu(\))38 b(=)h Fp(f)11 b Fu(\()p Fp(z)t Fu(\))39 b(and)h(\()p Fp(")2294 1383 y Fl(C)2379 1368 y Fr(\012)27 b Fu(1\)\()p Fp(f)5 b(;)17 b(g)t Fu(\)\()p Fp(z)t Fu(\))38 b(=)h Fp(g)t Fu(\()p Fp(z)t Fu(\))g(or)f(suc)m(h)0 1485 y(that)32 b(the)h(diagram)1370 1965 y Fp(C)217 b(C)29 b Fr(\012)23 b Fp(D)p 1476 1937 154 4 v 1476 1935 a Fm(\033)1508 1903 y Fl(p)1544 1914 y Fe(C)2147 1965 y Fp(D)p 1968 1937 151 4 v 2036 1935 a Fm(-)1998 1903 y Fl(p)2034 1914 y Fe(D)1502 1757 y Fl(f)1647 1697 y Fm(\000)1564 1780 y(\000)1481 1863 y(\000)1477 1867 y(\000)-83 b(\011)2054 1748 y Fl(g)1867 1697 y Fm(@)1950 1780 y(@)2033 1863 y(@)2037 1867 y(@)g(R)1762 1580 y Fp(Z)p 1797 1867 4 254 v 1799 1867 a Fm(?)1838 1830 y Fn(\()p Fl(f)s(;g)r Fn(\))0 2185 y Fu(comm)m(utes,)31 b(where)g Fp(p)812 2200 y Fl(C)871 2185 y Fu(\()p Fp(c)17 b Fr(\012)h Fp(d)p Fu(\))27 b(=)h(\(1)17 b Fr(\012)h Fp(")p Fu(\)\()p Fp(c)f Fr(\012)g Fp(d)p Fu(\))28 b(=)f Fp(c")p Fu(\()p Fp(d)p Fu(\))i(and)i Fp(p)2458 2200 y Fl(D)2522 2185 y Fu(\()p Fp(c)17 b Fr(\012)h Fp(d)p Fu(\))27 b(=)h(\()p Fp(")16 b Fr(\012)i Fu(1\)\()p Fp(c)f Fr(\012)h Fp(d)p Fu(\))27 b(=)0 2301 y Fp(")p Fu(\()p Fp(c)p Fu(\))p Fp(d)32 b Fu(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 Fj(K)17 b Fu(-)6 b Fs(cCoalg)32 b Fu(has)h(\014nite)f(pro)s(ducts)h(and)g(also)f(a)g (\014nal)g(ob)5 b(ject)33 b Fj(K)18 b Fu(.)149 2619 y Fs(De\014nition)37 b(2.4.2.)49 b Fu(A)d Fk(formal)g(gr)-5 b(oup)46 b Fu(is)f(a)h(group)f(in)g(the)i(category)f(of)f Fj(K)18 b Fu(-)5 b Fs(cCoalg)45 b Fu(of)0 2736 y(co)s(comm)m(utativ)m (e)31 b(coalgebras.)149 2888 y(A)38 b(formal)e(group)i Fp(G)g Fu(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 Fj(K)17 b Fu(-)6 b Fs(cCoalg)0 3004 y Fu(to)32 b Fs(Gr)p Fu(,)h(the)g(category)g(of)f (groups.)149 3156 y Fs(Prop)s(osition)k(2.4.3.)49 b Fk(L)-5 b(et)40 b Fp(H)j Fr(2)36 b Fj(K)17 b Fk(-)6 b Fs(cCoalg)p Fk(.)57 b Fp(H)47 b Fk(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 Fj(K)17 b Fk(-)6 b Fs(cCoalg)809 3429 y Fr(r)27 b Fu(:)h Fp(H)i Fr(\012)22 b Fp(H)36 b Fr(\000)-58 b(!)28 b Fp(H)r(;)116 b(u)28 b Fu(:)f Fj(K)51 b Fr(\000)-58 b(!)29 b Fp(H)r(;)116 b(S)34 b Fu(:)27 b Fp(H)36 b Fr(\000)-58 b(!)28 b Fp(H)0 3586 y Fk(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 Fp(H)30 b Fr(\012)22 b Fp(H)594 b(H)p 2099 4116 529 4 v 2545 4114 a Fm(-)2334 4095 y Fo(r)p 2699 4046 4 254 v 2700 4046 a Fm(?)2739 3942 y Fo(r)750 3919 y Fk(\(asso)-5 b(ciativity\))p 1919 4046 V 1920 4046 a Fm(?)1732 3939 y Fo(r\012)p Fn(1)1666 3754 y Fp(H)30 b Fr(\012)22 b Fp(H)30 b Fr(\012)22 b Fp(H)384 b(H)30 b Fr(\012)22 b Fp(H)p 2204 3726 318 4 v 2439 3724 a Fm(-)2289 3699 y Fn(1)p Fo(\012r)1137 4689 y Fp(H)30 b Fr(\012)23 b Fp(H)1276 b(H)p 1465 4662 1211 4 v 2593 4660 a Fm(-)2041 4640 y Fo(r)1114 4300 y Fj(K)46 b Fr(\012)23 b Fp(H)1430 4272 y Fr(\030)1431 4304 y Fu(=)1535 4300 y Fp(H)1651 4272 y Fr(\030)1652 4304 y Fu(=)1756 4300 y Fp(H)30 b Fr(\012)23 b Fj(K)579 b Fp(H)29 b Fr(\012)23 b Fp(H)p 2072 4271 499 4 v 2488 4269 a Fm(-)2256 4244 y Fl(u)p Fo(\012)p Fn(1)702 4465 y Fk(\(unit\))p 2747 4591 4 254 v 2749 4591 a Fm(?)2788 4487 y Fo(r)p 1285 4591 V 1287 4591 a Fm(?)1117 4482 y Fn(1)p Fo(\012)p Fl(u)2263 4486 y Fn(1)1784 4365 y Fm(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 Fk(\(inverse\))1486 4821 y Fp(H)412 b Fj(K)p 1604 4788 348 4 v 1869 4786 a Fm(-)1761 4768 y Fl(")2461 4821 y Fp(H)p 2047 4788 386 4 v 2350 4786 a Fm(-)2220 4755 y Fl(\021)p 1529 5108 4 254 v 1530 5108 a Fm(?)1433 5004 y Fn(\001)1381 5207 y Fp(H)30 b Fr(\012)22 b Fp(H)684 b(H)30 b Fr(\012)22 b Fp(H)p 1709 5179 619 4 v 2245 5177 a Fm(-)1938 5150 y Fl(S)t Fo(\012)p Fn(id)1932 5242 y(id)11 b Fo(\012)p Fl(S)p 2504 5108 4 254 v 2505 4938 a Fm(6)2544 5004 y Fo(r)p eop %%Page: 51 21 51 20 bop 1416 236 a Fq(4.)35 b(F)n(ORMAL)d(GR)n(OUPS)1312 b(51)149 425 y Fh(Pr)n(oof.)42 b Fu(F)-8 b(or)36 b(an)g(arbitrary)f (formal)f(group)i Fp(H)43 b Fu(w)m(e)37 b(get)f Fr(r)e Fu(=)g Fp(p)2573 440 y Fn(1)2637 425 y Fr(\003)24 b Fp(p)2760 440 y Fn(2)2833 425 y Fr(2)34 b Fj(K)18 b Fu(-)5 b Fs(cCoalg)p Fu(\()p Fp(H)32 b Fr(\012)0 541 y Fp(H)r(;)17 b(H)8 b Fu(\),)35 b Fp(u)d Fu(=)g Fp(e)h Fr(2)g Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fj(K)17 b Fp(;)h(H)8 b Fu(\),)41 b(and)35 b Fp(S)j Fu(=)32 b(\(id\))2032 505 y Fo(\000)p Fn(1)2158 541 y Fr(2)h Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(H)r(;)17 b(H)8 b Fu(\).)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 Fs(Remark)k(2.4.4.)50 b Fu(In)23 b(particular)e(the)i (represen)m(ting)h(ob)5 b(ject)24 b(\()p Fp(H)r(;)17 b Fr(r)p Fp(;)g(u;)g Fu(\001)p Fp(;)g(";)g(S)6 b Fu(\))20 b(of)j(a)f(formal)0 937 y(group)34 b Fp(G)g Fu(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 Fs(Corollary)j(2.4.5.)50 b Fk(A)40 b(c)-5 b(o)g(algebr)g(a)40 b Fp(H)45 b Fr(2)38 b Fj(K)18 b Fk(-)6 b Fs(cCoalg)40 b Fk(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 Fp(H)43 b Fk(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 Fs(Corollary)36 b(2.4.6.)50 b Fk(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 Fu(1.)42 b Fk(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 Fu(2.)42 b Fk(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 Fu(3.)42 b Fk(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 Fs(Example)37 b(2.4.7.)199 b Fu(1.)41 b(Group)32 b(algebras)g Fj(K)18 b Fp(G)38 b Fu(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 Fp(U)10 b Fu(\()p Fg(g)p Fu(\))33 b(of)f(Lie)g(algebras)g Fg(g)h Fu(are)f(formal)e(groups.)149 2610 y(3.)42 b(The)33 b(tensor)h(algebra)d Fp(T)14 b Fu(\()p Fp(V)21 b Fu(\))33 b(and)f(the)h(symmetric)f(algebra)f Fp(S)6 b Fu(\()p Fp(V)22 b Fu(\))32 b(are)h(formal)d(groups.)149 2726 y(4.)42 b(Let)48 b Fp(C)55 b Fu(b)s(e)48 b(a)f(co)s(comm)m(utativ) m(e)g(coalgebra)g(and)h Fp(G)g Fu(b)s(e)g(a)f(group.)90 b(Then)49 b(the)f(group)267 2842 y Fj(K)17 b Fp(G)p Fu(\()p Fp(C)7 b Fu(\))36 b(=)30 b Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(C)r(;)17 b Fj(K)g Fp(G)p Fu(\))39 b(is)34 b(isomorphic)e(to)h(the)i(set)f(of)g(families)c(\()p Fp(h)3128 2806 y Fo(\003)3128 2867 y Fl(g)3169 2842 y Fr(j)p Fp(g)i Fr(2)f Fp(G)p Fu(\))i(of)267 2958 y(decomp)s(ositions)26 b(of)g(the)i(unit)e(of)h Fp(C)1594 2922 y Fo(\003)1660 2958 y Fu(in)m(to)g(a)g(sum)g(of)f(orthogonal)g(idemp)s(oten)m(ts)h Fp(h)3322 2922 y Fo(\003)3322 2983 y Fl(g)3390 2958 y Fr(2)h Fp(C)3561 2922 y Fo(\003)267 3075 y Fu(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 Fj(K)18 b Fu(-)5 b Fs(cCoalg)p Fu(\()p Fp(C)r(;)17 b Fj(K)f Fp(G)p Fu(\))55 b Fr(\022)50 b Fu(Hom)o(\()p Fp(C)r(;)17 b Fj(K)g Fp(G)p Fu(\))51 b(and)45 b(em)m(b)s(edthe)h(set)267 3307 y(Hom)o(\()p Fp(C)r(;)17 b Fj(K)g Fp(G)p Fu(\))37 b(in)m(to)31 b(the)h(set)h(\()p Fp(C)1478 3271 y Fo(\003)1517 3307 y Fu(\))1555 3271 y Fl(G)1646 3307 y Fu(of)e Fp(G)p Fu(-families)d(of)j(elemen)m(ts)h(in)f(the)h(algebra)f Fp(C)3426 3271 y Fo(\003)3497 3307 y Fu(b)m(y)267 3423 y Fp(h)j Fr(7!)f Fu(\()p Fp(h)584 3387 y Fo(\003)584 3448 y Fl(g)624 3423 y Fu(\))j(with)g Fp(h)p Fu(\()p Fp(c)p Fu(\))e(=)1242 3349 y Fi(P)1347 3452 y Fl(g)r Fo(2)p Fl(G)1506 3423 y Fp(h)1562 3387 y Fo(\003)1562 3448 y Fl(g)1602 3423 y Fu(\()p Fp(c)p Fu(\))p Fp(g)t Fu(.)53 b(The)37 b(linear)e(map)g Fp(h)h Fu(is)g(a)g(homomorphism)d(of) 267 3548 y(coalgebras)h(i\013)f(\()p Fp(h)23 b Fr(\012)h Fp(h)p Fu(\)\001)31 b(=)f(\001)p Fp(h)35 b Fu(and)g Fp("h)30 b Fu(=)g Fp(")k Fu(i\013)2190 3474 y Fi(P)2311 3548 y Fp(h)p Fu(\()p Fp(c)2447 3564 y Fn(\(1\))2542 3548 y Fu(\))23 b Fr(\012)h Fp(h)p Fu(\()p Fp(c)2840 3564 y Fn(\(2\))2934 3548 y Fu(\))31 b(=)3109 3474 y Fi(P)3231 3548 y Fp(h)p Fu(\()p Fp(c)p Fu(\))3405 3564 y Fn(\(1\))3523 3548 y Fr(\012)267 3665 y Fp(h)p Fu(\()p Fp(c)p Fu(\))441 3680 y Fn(\(2\))562 3665 y Fu(and)c Fp(")p Fu(\()p Fp(h)p Fu(\()p Fp(c)p Fu(\)\))g(=)h Fp(")p Fu(\()p Fp(c)p Fu(\))e(for)h(all)e Fp(c)i Fr(2)h Fp(C)34 b Fu(i\013)2015 3590 y Fi(P)2137 3665 y Fp(h)2193 3628 y Fo(\003)2193 3689 y Fl(g)2233 3665 y Fu(\()p Fp(c)2313 3680 y Fn(\(1\))2407 3665 y Fu(\))p Fp(g)14 b Fr(\012)d Fp(h)2650 3628 y Fo(\003)2650 3691 y Fl(l)2690 3665 y Fu(\()p Fp(c)2770 3680 y Fn(\(2\))2864 3665 y Fu(\))p Fp(l)30 b Fu(=)3065 3590 y Fi(P)3186 3665 y Fp(h)3242 3628 y Fo(\003)3242 3689 y Fl(g)3283 3665 y Fu(\()p Fp(c)p Fu(\))p Fp(g)14 b Fr(\012)d Fp(g)267 3785 y Fu(and)457 3710 y Fi(P)579 3785 y Fp(h)635 3749 y Fo(\003)635 3810 y Fl(g)676 3785 y Fu(\()p Fp(c)p Fu(\))29 b(=)g Fp(")p Fu(\()p Fp(c)p Fu(\))k(i\013)1242 3710 y Fi(P)1364 3785 y Fp(h)1420 3749 y Fo(\003)1420 3810 y Fl(g)1460 3785 y Fu(\()p Fp(c)1540 3801 y Fn(\(1\))1635 3785 y Fu(\))p Fp(h)1729 3749 y Fo(\003)1729 3811 y Fl(l)1768 3785 y Fu(\()p Fp(c)1848 3801 y Fn(\(2\))1942 3785 y Fu(\))d(=)f Fp(\016)2158 3800 y Fl(g)r(l)2220 3785 y Fp(h)2276 3749 y Fo(\003)2276 3810 y Fl(g)2316 3785 y Fu(\()p Fp(c)p Fu(\))34 b(and)2658 3710 y Fi(P)2780 3785 y Fp(h)2836 3749 y Fo(\003)2836 3810 y Fl(g)2906 3785 y Fu(=)29 b Fp(")k Fu(i\013)g Fp(h)3264 3749 y Fo(\003)3264 3810 y Fl(g)3327 3785 y Fr(\003)22 b Fp(h)3455 3749 y Fo(\003)3455 3811 y Fl(l)3524 3785 y Fu(=)267 3906 y Fp(\016)310 3921 y Fl(g)r(l)372 3906 y Fp(h)428 3870 y Fo(\003)428 3930 y Fl(g)501 3906 y Fu(and)692 3831 y Fi(P)814 3906 y Fp(h)870 3870 y Fo(\003)870 3930 y Fl(g)939 3906 y Fu(=)29 b(1)1093 3921 y Fl(C)1148 3902 y Ff(\003)1187 3906 y Fu(.)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 Fp(c)28 b Fr(2)g Fp(C)40 b Fu(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 Fp(C)31 b Fu(b)s(e)25 b(a)f(co)s(comm)m(utativ)m(e)f(coalgebra)h(and)h Fj(K)17 b Fu([)p Fp(x)q Fu(])30 b(b)s(e)25 b(the)g(Hopf)f(algebra)g (with)g(\001\()p Fp(x)p Fu(\))k(=)267 4254 y Fp(x)k Fr(\012)h Fu(1)f(+)g(1)f Fr(\012)i Fp(x)48 b Fu(\(the)f(symmetric)f(algebra)g(of) h(the)h(one)f(dimensional)e(v)m(ector)j(space)267 4374 y Fj(K)17 b Fp(x)q Fu(\).)81 b(W)-8 b(e)44 b(em)m(b)s(ed)g(as)f(b)s (efore)h Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(C)r(;)17 b Fj(K)g Fu([)p Fp(x)p Fu(]\))52 b Fr(\022)47 b Fu(Hom)o(\()p Fp(C)r(;)17 b Fj(K)g Fu([)p Fp(x)p Fu(]\))52 b(=)46 b(\()p Fp(C)3323 4337 y Fo(\003)3362 4374 y Fu(\))3400 4337 y Fo(f)p Fb(N)3498 4346 y Fd(0)3533 4337 y Fo(g)3573 4374 y Fu(,)267 4490 y(the)41 b(set)g(of)f(lo)s(cally)e(\014nite)j Fs(N)1385 4505 y Fn(0)1424 4490 y Fu(-families)c(in)j Fp(C)2019 4454 y Fo(\003)2099 4490 y Fu(b)m(y)h Fp(h)p Fu(\()p Fp(c)p Fu(\))h(=)2575 4415 y Fi(P)2680 4441 y Fo(1)2680 4519 y Fl(i)p Fn(=0)2815 4490 y Fp(h)2871 4454 y Fo(\003)2871 4514 y Fl(i)2911 4490 y Fu(\()p Fp(c)p Fu(\))p Fp(x)3084 4454 y Fl(i)3112 4490 y Fu(.)68 b(The)42 b(map)267 4607 y Fp(h)d Fu(is)f(a)g(homomorphism)e(of)i(coalgebras)h (i\013)e(\001\()p Fp(h)p Fu(\()p Fp(c)p Fu(\)\))i(=)2450 4533 y Fi(P)2571 4607 y Fp(h)2627 4571 y Fo(\003)2627 4632 y Fl(i)2667 4607 y Fu(\()p Fp(c)p Fu(\)\()p Fp(x)26 b Fr(\012)h Fu(1)f(+)h(1)f Fr(\012)g Fp(x)p Fu(\))3457 4571 y Fl(i)3524 4607 y Fu(=)267 4654 y Fi(P)389 4729 y Fp(h)445 4692 y Fo(\003)445 4753 y Fl(i)484 4729 y Fu(\()p Fp(c)p Fu(\))602 4648 y Fi(\000)648 4684 y Fl(i)649 4763 y(l)672 4648 y Fi(\001)717 4729 y Fp(x)772 4692 y Fl(l)799 4729 y Fr(\012)p Fp(x)931 4692 y Fl(i)p Fo(\000)p Fl(l)1065 4729 y Fu(=)h(\()p Fp(h)p Fr(\012)p Fp(h)p Fu(\)\001\()p Fp(c)p Fu(\))i(=)1765 4654 y Fi(P)1886 4729 y Fp(h)1942 4692 y Fo(\003)1942 4753 y Fl(i)1982 4729 y Fu(\()p Fp(c)2062 4744 y Fn(\(1\))2156 4729 y Fu(\))p Fp(h)2250 4692 y Fo(\003)2250 4753 y Fl(j)2290 4729 y Fu(\()p Fp(c)2370 4744 y Fn(\(2\))2464 4729 y Fu(\))p Fp(x)2557 4692 y Fl(i)2586 4729 y Fr(\012)p Fp(x)2718 4692 y Fl(j)2777 4729 y Fu(and)22 b Fp(")p Fu(\()3040 4654 y Fi(P)3161 4729 y Fp(h)3217 4692 y Fo(\003)3217 4753 y Fl(i)3257 4729 y Fu(\()p Fp(c)p Fu(\))p Fp(x)3430 4692 y Fl(i)3459 4729 y Fu(\))27 b(=)267 4863 y Fp(")p Fu(\()p Fp(c)p Fu(\))32 b(i\013)f Fp(h)635 4826 y Fo(\003)635 4887 y Fl(i)697 4863 y Fr(\003)22 b Fp(h)825 4826 y Fo(\003)825 4887 y Fl(j)892 4863 y Fu(=)996 4782 y Fi(\000)1042 4818 y Fl(i)p Fn(+)p Fl(j)1085 4897 y(i)1153 4782 y Fi(\001)1198 4863 y Fp(h)1254 4826 y Fo(\003)1254 4887 y Fl(i)p Fn(+)p Fl(j)1403 4863 y Fu(and)32 b Fp(h)1648 4826 y Fo(\003)1648 4887 y Fn(0)1716 4863 y Fu(=)27 b Fp(")g Fu(=)h(1)2045 4878 y Fl(C)2100 4859 y Ff(\003)2140 4863 y Fu(.)416 4983 y(No)m(w)36 b(let)e Fj(K)58 b Fu(b)s(e)35 b(a)f(\014eld)h(of)f(c)m (haracteristic)g(zero.)51 b(Let)35 b Fp(p)2519 4998 y Fl(i)2578 4983 y Fu(:=)c Fp(h)2768 4947 y Fo(\003)2768 5008 y Fl(i)2808 4983 y Fp(=i)p Fu(!.)50 b(Then)36 b(the)f(con-)267 5099 y(ditions)c(simplify)g(to)h Fp(p)1125 5114 y Fl(i)1153 5099 y Fp(p)1202 5114 y Fl(j)1267 5099 y Fu(=)27 b Fp(p)1419 5114 y Fl(i)p Fn(+)p Fl(j)1567 5099 y Fu(and)33 b Fp(p)1806 5114 y Fn(0)1874 5099 y Fu(=)28 b(1)p Fp(:)k Fu(Hence)i(the)f(series)h (for)e Fp(h)h Fu(is)f(completely)267 5216 y(determined)g(b)m(y)g(the)h (term)e Fp(p)c Fu(:=)h Fp(p)1566 5231 y Fn(1)1637 5216 y Fu(since)33 b Fp(p)1925 5231 y Fl(n)1999 5216 y Fu(=)28 b Fp(p)2152 5179 y Fl(n)2152 5240 y Fn(1)2199 5216 y Fu(.)43 b(Since)32 b(the)g(series)h(m)m(ust)f(b)s(e)g(lo)s(cally)p eop %%Page: 52 22 52 21 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 Fu(\014nite)f(w)m(e) h(get)f(that)g(for)g(eac)m(h)h Fp(c)c Fr(2)g Fp(C)38 b Fu(there)32 b(m)m(ust)f(b)s(e)h(an)f Fp(n)d Fr(2)g Fs(N)2688 440 y Fn(0)2758 425 y Fu(suc)m(h)33 b(that)e Fp(p)3236 389 y Fl(m)3302 425 y Fu(\()p Fp(c)p Fu(\))d(=)f(0)267 541 y(for)32 b(all)e Fp(m)e Fr(\025)h Fp(n)p Fu(.)43 b(Hence)34 b(the)f(elemen)m(t)f Fp(p)h Fu(is)f Fk(top)-5 b(olo)g(gic)g(al)5 b(ly)34 b(nilp)-5 b(otent)32 b Fu(and)1144 698 y Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(C)r(;)17 b Fj(K)g Fu([)p Fp(x)p Fu(])q(\))1988 671 y Fr(\030)1989 702 y Fu(=)2093 698 y(rad)2234 713 y Fl(t)2264 698 y Fu(\()p Fp(C)2379 657 y Fo(\003)2418 698 y Fu(\))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 Fp(C)2401 819 y Fo(\003)2440 855 y Fu(.)416 971 y(It)c(is)g(easy)h(to)e(see)i(that)f(rad)1436 986 y Fl(t)1465 971 y Fu(\()p Fp(C)1580 935 y Fo(\003)1620 971 y Fu(\))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 Fj(K)17 b Fu(-)6 b Fs(cCoalg)o Fu(\()p Fp(C)r(;)17 b Fj(K)g Fu([)p Fp(x)p Fu(]\).)149 1262 y Fs(Remark)37 b(2.4.8.)50 b Fu(Let)32 b Fp(H)40 b Fu(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 Fp(H)912 1342 y Fo(\003)995 1378 y Fu(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 Fp(H)3460 1458 y Fo(\003)3534 1495 y Fu(is)0 1611 y(again)41 b(a)h(Hopf)h(algebra.)72 b(Hence)44 b(the)f(functor)f(-)1878 1575 y Fo(\003)1962 1611 y Fu(:)i Fs(v)m(ec)h Fr(\000)-60 b(!)45 b Fs(v)m(ec)d Fu(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 Fo(\003)1936 1727 y Fu(:)53 b Fj(K)17 b Fu(-)6 b Fs(hopfalg)53 b Fr(\000)-60 b(!)53 b Fj(K)18 b Fu(-)5 b Fs(hopfalg)48 b Fu(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 Fk(\014nite)i Fu(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 Fk(\014nite)h Fu(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 %%Page: 53 23 53 22 bop 1378 236 a Fq(5.)35 b(QUANTUM)d(GR)n(OUPS)1274 b(53)1301 425 y Fs(5.)56 b(Quan)m(tum)38 b(Groups)149 600 y(De\014nition)f(2.5.1.)49 b Fu(\(Drinfel'd\))28 b(A)j Fk(quantum)i(gr)-5 b(oup)30 b Fu(is)g(a)g(noncomm)m(utativ)m(e)f (nonco)s(com-)0 716 y(m)m(utativ)m(e)j(Hopf)h(algebra.)149 887 y Fs(Remark)k(2.5.2.)50 b Fu(W)-8 b(e)45 b(shall)f(consider)h(all)f (Hopf)h(algebras)f(as)h(quan)m(tum)h(groups.)81 b(Ob-)0 1003 y(serv)m(e,)54 b(ho)m(w)m(ev)m(er,)g(that)48 b(the)h(comm)m (utativ)m(e)e(Hopf)h(algebras)f(ma)m(y)h(b)s(e)g(considered)h(as)f (a\016ne)0 1119 y(algebraic)40 b(groups)i(and)g(that)f(the)h(co)s(comm) m(utativ)m(e)f(Hopf)g(algebras)g(ma)m(y)g(b)s(e)h(considered)h(as)0 1236 y(formal)27 b(groups.)43 b(Their)30 b(prop)s(ert)m(y)g(as)g(a)g (quan)m(tum)f(space)i(or)f(as)f(a)h(quan)m(tum)g(monoid)e(will)f(pla)m (y)0 1352 y(some)33 b(role.)43 b(But)33 b(often)g(the)g(\(p)s(ossibly)f (nonexisting\))g(dual)g(Hopf)h(algebra)f(will)e(ha)m(v)m(e)k(the)g (geo-)0 1468 y(metrical)24 b(meaning.)40 b(The)27 b(follo)m(wing)d (examples)i Fj(SL)1935 1483 y Fl(q)1973 1468 y Fu(\(2\))g(and)h Fj(G)21 b(L)2452 1483 y Fl(q)2490 1468 y Fu(\(2\))k(will)g(ha)m(v)m(e)i (a)f(geometrical)0 1584 y(meaning.)149 1755 y Fs(Example)37 b(2.5.3.)49 b Fu(The)34 b(smallest)d(prop)s(er)i(quan)m(tum)g(group,)f (i.e.)44 b(the)33 b(smallest)e(noncom-)0 1872 y(m)m(utativ)m(e)h(nonco) s(comm)m(utativ)m(e)g(Hopf)g(algebra,)g(is)g(the)h(4-dimensional)c (algebra)1059 2029 y Fp(H)1140 2044 y Fn(4)1207 2029 y Fu(:=)f Fj(K)17 b Fr(h)p Fp(g)t(;)g(x)p Fr(i)p Fp(=)p Fu(\()p Fp(g)1775 1987 y Fn(2)1841 2029 y Fr(\000)22 b Fu(1)p Fp(;)17 b(x)2088 1987 y Fn(2)2128 2029 y Fp(;)g(xg)25 b Fu(+)d Fp(g)t(x)p Fu(\))0 2185 y(whic)m(h)33 b(w)m(as)h(\014rst)f (describ)s(ed)g(b)m(y)h(M.)f(Sw)m(eedler.)44 b(The)34 b(coalgebra)d(structure)j(is)e(giv)m(en)h(b)m(y)936 2335 y(\001\()p Fp(g)t Fu(\))27 b(=)h Fp(g)d Fr(\012)e Fp(g)t(;)263 b Fu(\001\()p Fp(x)p Fu(\))28 b(=)f Fp(x)c Fr(\012)g Fu(1)f(+)g Fp(g)j Fr(\012)e Fp(x;)1041 2451 y(")p Fu(\()p Fp(g)t Fu(\))j(=)i(1)p Fp(;)648 b(")p Fu(\()p Fp(x)p Fu(\))27 b(=)h(0)p Fp(;)868 2568 y(S)6 b Fu(\()p Fp(g)t Fu(\))27 b(=)g Fp(g)1242 2531 y Fo(\000)p Fn(1)1336 2568 y Fu(\(=)g Fp(g)t Fu(\))p Fp(;)398 b(S)6 b Fu(\()p Fp(x)p Fu(\))27 b(=)h Fr(\000)p Fp(g)t(x:)0 2721 y Fu(Since)33 b(it)f(is)h(\014nite)g(dimensional)d(its)j(linear)e(dual)h Fp(H)1953 2685 y Fo(\003)1945 2745 y Fn(4)2025 2721 y Fu(is)h(also)f(a)h(noncomm)m(utativ)m(e)f(nonco)s(com-)0 2837 y(m)m(utativ)m(e)g(Hopf)g(algebra.)42 b(It)32 b(is)f(isomorphic)g (as)h(a)g(Hopf)g(algebra)f(to)g Fp(H)2700 2852 y Fn(4)2739 2837 y Fu(.)44 b(In)32 b(fact)g Fp(H)3204 2852 y Fn(4)3275 2837 y Fu(is)g(up)g(to)0 2953 y(isomorphism)24 b(the)k(only)e(noncomm)m (utativ)m(e)g(nonco)s(comm)m(utativ)m(e)g(Hopf)g(algebra)g(of)g (dimension)0 3069 y(4.)149 3241 y Fs(Example)37 b(2.5.4.)49 b Fu(The)40 b(a\016ne)g(algebraic)d(group)h Fj(SL)5 b Fu(\()p Fp(n)p Fu(\))39 b(:)f Fj(K)18 b Fu(-)5 b Fs(cAlg)37 b Fr(\000)-60 b(!)39 b Fs(Gr)g Fu(de\014ned)h(b)m(y)0 3357 y Fj(S)o(L)6 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\),)36 b(the)f(group)g(of)g Fp(n)24 b Fr(\002)g Fp(n)p Fu(-matrices)34 b(with)h(co)s(e\016cien)m(ts)h(in)e(the)i(comm)m(utativ)m(e)e(algebra)0 3473 y Fp(A)49 b Fu(and)g(with)f(determinan)m(t)g(1,)53 b(is)48 b(represen)m(ted)j(b)m(y)f(the)f(algebra)f Fr(O)s Fu(\()p Fj(S)o(L)5 b Fu(\()p Fp(n)p Fu(\)\))56 b(=)f Fp(S)6 b(L)p Fu(\()p Fp(n)p Fu(\))55 b(=)0 3589 y Fj(K)17 b Fu([)q Fp(x)154 3604 y Fl(ij)221 3589 y Fu(])p Fp(=)p Fu(\(det\()p Fp(x)563 3604 y Fl(ij)624 3589 y Fu(\))22 b Fr(\000)g Fu(1\))33 b(i.e.)820 3746 y Fj(SL)5 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\))1253 3718 y Fr(\030)1254 3750 y Fu(=)1358 3746 y Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fj(K)17 b Fu([)p Fp(x)1882 3761 y Fl(ij)1948 3746 y Fu(])p Fp(=)p Fu(\(det\()p Fp(x)2290 3761 y Fl(ij)2351 3746 y Fu(\))22 b Fr(\000)h Fu(1\))p Fp(;)17 b(A)p Fu(\))p Fp(:)0 3903 y Fu(Since)25 b Fj(SL)5 b Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\))25 b(has)h(a)e(group)h(structure)i(b)m(y)f(the)f (m)m(ultiplication)c(of)j(matrices,)i(the)f(represen)m(t-)0 4019 y(ing)33 b(comm)m(utativ)m(e)g(algebra)g(has)i(a)f(Hopf)g(algebra) f(structure)j(with)d(the)i(diagonal)d(\001)f(=)f Fp(\023)3390 4034 y Fn(1)3453 4019 y Fr(\003)23 b Fp(\023)3560 4034 y Fn(2)0 4135 y Fu(hence)1317 4252 y(\001\()p Fp(x)1491 4267 y Fl(ik)1558 4252 y Fu(\))28 b(=)1727 4157 y Fi(X)1888 4252 y Fp(x)1943 4267 y Fl(ij)2026 4252 y Fr(\012)23 b Fp(x)2181 4267 y Fl(j)t(k)2256 4252 y Fp(;)0 4410 y Fu(the)h(counit)g Fp(")p Fu(\()p Fp(x)588 4425 y Fl(ij)648 4410 y Fu(\))k(=)f Fp(\016)860 4425 y Fl(ij)945 4410 y Fu(and)d(the)g(an)m(tip)s(o)s(de)f Fp(S)6 b Fu(\()p Fp(x)1839 4425 y Fl(ij)1900 4410 y Fu(\))27 b(=)h Fp(ad)-11 b(j)6 b Fu(\()p Fp(X)i Fu(\))2371 4425 y Fl(ij)2455 4410 y Fu(where)25 b Fp(ad)-11 b(j)6 b Fu(\()p Fp(X)i Fu(\))23 b(is)h(the)g(adjoin)m(t)0 4526 y(matrix)31 b(of)h Fp(X)k Fu(=)27 b(\()p Fp(x)741 4541 y Fl(ij)802 4526 y Fu(\).)43 b(W)-8 b(e)33 b(lea)m(v)m(e)h(the)f(v)m(eri\014cation)f(of)g(these)i (facts)e(to)h(the)g(reader.)149 4654 y(W)-8 b(e)29 b(consider)g Fj(SL)5 b Fu(\()p Fp(n)p Fu(\))28 b Fr(\022)g(M)1198 4669 y Fl(n)1272 4654 y Fu(=)g Fj(A)1442 4618 y Fl(n)1485 4594 y Fd(2)1558 4654 y Fu(as)h(a)f(subspace)i(of)e(the)h Fp(n)2483 4618 y Fn(2)2523 4654 y Fu(-dimensional)d(a\016ne)j(space.) 149 4887 y Fs(Example)37 b(2.5.5.)49 b Fu(Let)36 b Fp(M)1194 4902 y Fl(q)1233 4887 y Fu(\(2\))c(=)h Fj(K)1588 4746 y Fi(\034)5 b(\022)1741 4827 y Fp(a)88 b(b)1746 4943 y(c)f(d)1926 4746 y Fi(\023\035)2091 4887 y Fp(=I)43 b Fu(as)36 b(in)f(1.3.6)g(with)g Fp(I)44 b Fu(the)36 b(ideal)e(gen-)0 5059 y(erated)f(b)m(y)98 5216 y Fp(ab)23 b Fr(\000)g Fp(q)360 5174 y Fo(\000)p Fn(1)454 5216 y Fp(ba;)17 b(ac)23 b Fr(\000)f Fp(q)852 5174 y Fo(\000)p Fn(1)947 5216 y Fp(ca;)17 b(bd)22 b Fr(\000)g Fp(q)1344 5174 y Fo(\000)p Fn(1)1439 5216 y Fp(db;)17 b(cd)k Fr(\000)i Fp(q)1836 5174 y Fo(\000)p Fn(1)1930 5216 y Fp(dc;)17 b Fu(\()p Fp(ad)k Fr(\000)i Fp(q)2375 5174 y Fo(\000)p Fn(1)2469 5216 y Fp(bc)p Fu(\))g Fr(\000)f Fu(\()p Fp(da)g Fr(\000)h Fp(q)t(cb)p Fu(\))p Fp(;)17 b(bc)22 b Fr(\000)h Fp(cb:)p eop %%Page: 54 24 54 23 bop 0 236 a Fq(54)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fu(W)-8 b(e)34 b(\014rst)g(de\014ne)h(the)f Fk(quantum)i(determinant)c Fu(det)2057 440 y Fl(q)2124 425 y Fu(=)d Fp(ad)23 b Fr(\000)g Fp(q)2501 389 y Fo(\000)p Fn(1)2595 425 y Fp(bc)30 b Fu(=)f Fp(da)22 b Fr(\000)h Fp(q)t(cb)34 b Fu(in)f Fp(M)3410 440 y Fl(q)3448 425 y Fu(\(2\).)0 541 y(It)e(is)e(a)i(cen)m(tral)f (elemen)m(t.)43 b(T)-8 b(o)30 b(sho)m(w)i(this,)e(it)g(su\016ces)i(to)e (sho)m(w)i(that)e(det)2720 556 y Fl(q)2788 541 y Fu(comm)m(utes)g(with) h(the)0 658 y(generators)i Fp(a;)17 b(b;)g(c;)g(d)p Fu(:)506 825 y(\()p Fp(ad)22 b Fr(\000)h Fp(q)815 789 y Fo(\000)p Fn(1)909 825 y Fp(bc)p Fu(\))p Fp(a)28 b Fu(=)g Fp(a)p Fu(\()p Fp(da)22 b Fr(\000)h Fp(q)t(bc)p Fu(\))p Fp(;)121 b Fu(\()p Fp(ad)22 b Fr(\000)h Fp(q)2151 789 y Fo(\000)p Fn(1)2245 825 y Fp(bc)p Fu(\))p Fp(b)28 b Fu(=)g Fp(b)p Fu(\()p Fp(ad)22 b Fr(\000)h Fp(q)2889 789 y Fo(\000)p Fn(1)2983 825 y Fp(bc)p Fu(\))p Fp(;)468 941 y Fu(\()p Fp(ad)f Fr(\000)h Fp(q)777 905 y Fo(\000)p Fn(1)871 941 y Fp(bc)p Fu(\))p Fp(c)28 b Fu(=)g Fp(c)p Fu(\()p Fp(ad)22 b Fr(\000)g Fp(q)1516 905 y Fo(\000)p Fn(1)1611 941 y Fp(bc)p Fu(\))p Fp(;)121 b Fu(\()p Fp(da)22 b Fr(\000)h Fp(q)t(bc)p Fu(\))p Fp(d)k Fu(=)h Fp(d)p Fu(\()p Fp(ad)21 b Fr(\000)i Fp(q)2851 905 y Fo(\000)p Fn(1)2945 941 y Fp(bc)p Fu(\))p Fp(:)0 1111 y Fu(W)-8 b(e)33 b(can)g(form)e(the)i(quan) m(tum)g(determinan)m(t)f(of)g(an)h(arbitrary)e(quan)m(tum)i(matrix)e (in)h Fp(A)h Fu(b)m(y)655 1341 y(det)790 1356 y Fl(q)845 1201 y Fi(\022)918 1282 y Fp(a)969 1245 y Fo(0)1080 1282 y Fp(b)1121 1245 y Fo(0)923 1398 y Fp(c)965 1362 y Fo(0)1076 1398 y Fp(d)1127 1362 y Fo(0)1150 1201 y Fi(\023)1251 1341 y Fu(:=)27 b Fp(a)1432 1300 y Fo(0)1456 1341 y Fp(d)1507 1300 y Fo(0)1552 1341 y Fr(\000)22 b Fp(q)1698 1300 y Fo(\000)p Fn(1)1793 1341 y Fp(b)1834 1300 y Fo(0)1857 1341 y Fp(c)1899 1300 y Fo(0)1950 1341 y Fu(=)28 b Fp(d)2105 1300 y Fo(0)2128 1341 y Fp(a)2179 1300 y Fo(0)2224 1341 y Fr(\000)23 b Fp(q)t(c)2413 1300 y Fo(0)2436 1341 y Fp(b)2477 1300 y Fo(0)2529 1341 y Fu(=)k Fp(')p Fu(\(det)2869 1356 y Fl(q)2908 1341 y Fu(\))0 1577 y(if)j Fp(')d Fu(:)h Fp(M)328 1592 y Fl(q)366 1577 y Fu(\(2\))g Fr(\000)-60 b(!)27 b Fp(A)k Fu(is)f(the)i(algebra)d(homomorphism)f(asso)s(ciated)j (with)f(the)h(quan)m(tum)g(matrix)0 1617 y Fi(\022)73 1698 y Fp(a)124 1662 y Fo(0)235 1698 y Fp(b)276 1662 y Fo(0)78 1814 y Fp(c)120 1778 y Fo(0)231 1814 y Fp(d)282 1778 y Fo(0)305 1617 y Fi(\023)378 1758 y Fu(.)149 2005 y(Giv)m(en)25 b(t)m(w)m(o)g(comm)m(uting)e(quan)m(tum)i(2)6 b Fr(\002)g Fu(2-matrices)2105 1865 y Fi(\022)2179 1945 y Fp(a)2230 1909 y Fo(0)2341 1945 y Fp(b)2382 1909 y Fo(0)2183 2061 y Fp(c)2225 2025 y Fo(0)2336 2061 y Fp(d)2387 2025 y Fo(0)2410 1865 y Fi(\023)2500 2005 y Fp(;)2544 1865 y Fi(\022)2617 1945 y Fp(a)2668 1909 y Fo(00)2798 1945 y Fp(b)2839 1909 y Fo(0)q(0)2622 2061 y Fp(c)2664 2025 y Fo(00)2794 2061 y Fp(d)2845 2025 y Fo(00)2887 1865 y Fi(\023)2960 2005 y Fu(.)41 b(The)26 b(quan)m(tum)0 2177 y(determinan)m(t)32 b(preserv)m(es)j(the)e(pro)s(duct,)g(since)504 2407 y(det)640 2422 y Fl(q)678 2407 y Fu(\()716 2267 y Fi(\022)789 2347 y Fp(a)840 2311 y Fo(0)951 2347 y Fp(b)992 2311 y Fo(0)794 2464 y Fp(c)836 2428 y Fo(0)947 2464 y Fp(d)998 2428 y Fo(0)1021 2267 y Fi(\023)17 b(\022)1184 2347 y Fp(a)1235 2311 y Fo(00)1365 2347 y Fp(b)1406 2311 y Fo(0)q(0)1189 2464 y Fp(c)1231 2428 y Fo(00)1361 2464 y Fp(d)1412 2428 y Fo(0)o(0)1454 2267 y Fi(\023)1527 2407 y Fu(\))28 b(=)f(det)1832 2422 y Fl(q)1886 2267 y Fi(\022)1960 2347 y Fp(a)2011 2311 y Fo(0)2034 2347 y Fp(a)2085 2311 y Fo(0)q(0)2150 2347 y Fu(+)22 b Fp(b)2289 2311 y Fo(0)2313 2347 y Fp(c)2355 2311 y Fo(00)2480 2347 y Fp(a)2531 2311 y Fo(0)2555 2347 y Fp(b)2596 2311 y Fo(00)2661 2347 y Fu(+)g Fp(b)2800 2311 y Fo(0)2823 2347 y Fp(d)2874 2311 y Fo(00)1960 2464 y Fp(c)2002 2428 y Fo(0)2025 2464 y Fp(a)2076 2428 y Fo(00)2141 2464 y Fu(+)g Fp(d)2290 2428 y Fo(0)2313 2464 y Fp(c)2355 2428 y Fo(00)2480 2464 y Fp(c)2522 2428 y Fo(0)2545 2464 y Fp(b)2586 2428 y Fo(0)q(0)2651 2464 y Fu(+)h Fp(d)2801 2428 y Fo(0)2823 2464 y Fp(d)2874 2428 y Fo(00)2917 2267 y Fi(\023)678 2583 y Fu(=)k(\()p Fp(a)870 2547 y Fo(0)894 2583 y Fp(a)945 2547 y Fo(00)1010 2583 y Fu(+)22 b Fp(b)1149 2547 y Fo(0)1172 2583 y Fp(c)1214 2547 y Fo(0)q(0)1257 2583 y Fu(\)\()p Fp(c)1375 2547 y Fo(0)1398 2583 y Fp(b)1439 2547 y Fo(0)q(0)1504 2583 y Fu(+)g Fp(d)1653 2547 y Fo(0)1676 2583 y Fp(d)1727 2547 y Fo(00)1769 2583 y Fu(\))g Fr(\000)h Fp(q)1976 2547 y Fo(\000)p Fn(1)2070 2583 y Fu(\()p Fp(a)2159 2547 y Fo(0)2182 2583 y Fp(b)2223 2547 y Fo(0)q(0)2288 2583 y Fu(+)f Fp(b)2427 2547 y Fo(0)2451 2583 y Fp(d)2502 2547 y Fo(00)2544 2583 y Fu(\)\()p Fp(c)2662 2547 y Fo(0)2686 2583 y Fp(a)2737 2547 y Fo(00)2801 2583 y Fu(+)g Fp(d)2950 2547 y Fo(0)2973 2583 y Fp(c)3015 2547 y Fo(00)3058 2583 y Fu(\))678 2699 y(=)27 b Fp(a)832 2663 y Fo(0)856 2699 y Fp(c)898 2663 y Fo(0)921 2699 y Fp(a)972 2663 y Fo(0)q(0)1015 2699 y Fp(b)1056 2663 y Fo(00)1121 2699 y Fu(+)22 b Fp(b)1260 2663 y Fo(0)1284 2699 y Fp(c)1326 2663 y Fo(0)1349 2699 y Fp(c)1391 2663 y Fo(00)1433 2699 y Fp(b)1474 2663 y Fo(0)q(0)1539 2699 y Fu(+)g Fp(a)1688 2663 y Fo(0)1712 2699 y Fp(d)1763 2663 y Fo(0)1786 2699 y Fp(a)1837 2663 y Fo(00)1880 2699 y Fp(d)1931 2663 y Fo(0)o(0)1995 2699 y Fu(+)g Fp(b)2134 2663 y Fo(0)2158 2699 y Fp(d)2209 2663 y Fo(0)2232 2699 y Fp(c)2274 2663 y Fo(00)2316 2699 y Fp(d)2367 2663 y Fo(00)775 2816 y Fr(\000)p Fp(q)899 2780 y Fo(\000)p Fn(1)994 2816 y Fu(\()p Fp(a)1083 2780 y Fo(0)1106 2816 y Fp(c)1148 2780 y Fo(0)1172 2816 y Fp(b)1213 2780 y Fo(00)1256 2816 y Fp(a)1307 2780 y Fo(00)1371 2816 y Fu(+)g Fp(b)1510 2780 y Fo(0)1534 2816 y Fp(c)1576 2780 y Fo(0)1600 2816 y Fp(d)1651 2780 y Fo(0)o(0)1693 2816 y Fp(a)1744 2780 y Fo(00)1808 2816 y Fu(+)g Fp(a)1957 2780 y Fo(0)1981 2816 y Fp(d)2032 2780 y Fo(0)2055 2816 y Fp(b)2096 2780 y Fo(0)q(0)2139 2816 y Fp(c)2181 2780 y Fo(00)2245 2816 y Fu(+)g Fp(b)2384 2780 y Fo(0)2408 2816 y Fp(d)2459 2780 y Fo(0)2482 2816 y Fp(d)2533 2780 y Fo(00)2575 2816 y Fp(c)2617 2780 y Fo(00)2660 2816 y Fu(\))678 2932 y(=)27 b Fp(b)822 2896 y Fo(0)846 2932 y Fp(c)888 2896 y Fo(0)911 2932 y Fp(c)953 2896 y Fo(0)q(0)996 2932 y Fp(b)1037 2896 y Fo(00)1102 2932 y Fu(+)22 b Fp(a)1251 2896 y Fo(0)1274 2932 y Fp(d)1325 2896 y Fo(0)1348 2932 y Fp(a)1399 2896 y Fo(0)q(0)1442 2932 y Fp(d)1493 2896 y Fo(00)1557 2932 y Fr(\000)h Fp(q)1704 2896 y Fo(\000)p Fn(1)1798 2932 y Fp(b)1839 2896 y Fo(0)1863 2932 y Fp(c)1905 2896 y Fo(0)1928 2932 y Fp(d)1979 2896 y Fo(00)2021 2932 y Fp(a)2072 2896 y Fo(0)q(0)2137 2932 y Fr(\000)g Fp(q)2284 2896 y Fo(\000)p Fn(1)2378 2932 y Fp(a)2429 2896 y Fo(0)2452 2932 y Fp(d)2503 2896 y Fo(0)2526 2932 y Fp(b)2567 2896 y Fo(0)q(0)2610 2932 y Fp(c)2652 2896 y Fo(00)678 3048 y Fu(=)k Fp(b)822 3012 y Fo(0)846 3048 y Fp(c)888 3012 y Fo(0)911 3048 y Fp(c)953 3012 y Fo(0)q(0)996 3048 y Fp(b)1037 3012 y Fo(00)1102 3048 y Fu(+)22 b Fp(a)1251 3012 y Fo(0)1274 3048 y Fp(d)1325 3012 y Fo(0)1348 3048 y Fp(a)1399 3012 y Fo(0)q(0)1442 3048 y Fp(d)1493 3012 y Fo(00)1557 3048 y Fr(\000)h Fp(q)1704 3012 y Fo(\000)p Fn(1)1798 3048 y Fp(b)1839 3012 y Fo(0)1863 3048 y Fp(c)1905 3012 y Fo(0)1928 3048 y Fp(d)1979 3012 y Fo(00)2021 3048 y Fp(a)2072 3012 y Fo(0)q(0)2137 3048 y Fr(\000)g Fp(q)2284 3012 y Fo(\000)p Fn(1)2378 3048 y Fp(a)2429 3012 y Fo(0)2452 3048 y Fp(d)2503 3012 y Fo(0)2526 3048 y Fp(b)2567 3012 y Fo(0)q(0)2610 3048 y Fp(c)2652 3012 y Fo(00)775 3164 y Fr(\000)p Fp(q)899 3128 y Fo(\000)p Fn(1)994 3164 y Fp(b)1035 3128 y Fo(0)1059 3164 y Fp(c)1101 3128 y Fo(0)1124 3164 y Fu(\()p Fp(a)1213 3128 y Fo(00)1256 3164 y Fp(d)1307 3128 y Fo(00)1371 3164 y Fr(\000)g Fp(d)1522 3128 y Fo(0)o(0)1564 3164 y Fp(a)1615 3128 y Fo(00)1679 3164 y Fr(\000)g Fp(q)1826 3128 y Fo(\000)p Fn(1)1920 3164 y Fp(b)1961 3128 y Fo(0)q(0)2004 3164 y Fp(c)2046 3128 y Fo(00)2111 3164 y Fu(+)f Fp(q)t(c)2298 3128 y Fo(00)2340 3164 y Fp(b)2381 3128 y Fo(0)q(0)2424 3164 y Fu(\))678 3281 y(=)27 b Fp(a)832 3244 y Fo(0)856 3281 y Fp(d)907 3244 y Fo(0)930 3281 y Fp(a)981 3244 y Fo(00)1024 3281 y Fp(d)1075 3244 y Fo(0)o(0)1139 3281 y Fr(\000)22 b Fp(q)1285 3244 y Fo(\000)p Fn(1)1380 3281 y Fp(a)1431 3244 y Fo(0)1454 3281 y Fp(d)1505 3244 y Fo(0)1528 3281 y Fp(b)1569 3244 y Fo(0)q(0)1612 3281 y Fp(c)1654 3244 y Fo(00)1719 3281 y Fr(\000)g Fp(q)1865 3244 y Fo(\000)p Fn(1)1959 3281 y Fp(b)2000 3244 y Fo(0)2024 3281 y Fp(c)2066 3244 y Fo(0)2089 3281 y Fu(\()p Fp(a)2178 3244 y Fo(0)q(0)2221 3281 y Fp(d)2272 3244 y Fo(00)2336 3281 y Fr(\000)h Fp(q)2483 3244 y Fo(\000)p Fn(1)2577 3281 y Fp(b)2618 3244 y Fo(0)q(0)2661 3281 y Fp(c)2703 3244 y Fo(00)2745 3281 y Fu(\))678 3397 y(=)k(\()p Fp(a)870 3361 y Fo(0)894 3397 y Fp(d)945 3361 y Fo(0)990 3397 y Fr(\000)c Fp(q)1137 3361 y Fo(\000)p Fn(1)1231 3397 y Fp(b)1272 3361 y Fo(0)1296 3397 y Fp(c)1338 3361 y Fo(0)1361 3397 y Fu(\)\()p Fp(a)1488 3361 y Fo(00)1530 3397 y Fp(d)1581 3361 y Fo(00)1646 3397 y Fr(\000)f Fp(q)1792 3361 y Fo(\000)p Fn(1)1886 3397 y Fp(b)1927 3361 y Fo(0)q(0)1970 3397 y Fp(c)2012 3361 y Fo(00)2055 3397 y Fu(\))678 3576 y(=)27 b(det)917 3591 y Fl(q)972 3436 y Fi(\022)1045 3516 y Fp(a)1096 3480 y Fo(0)1207 3516 y Fp(b)1248 3480 y Fo(0)1049 3633 y Fp(c)1091 3596 y Fo(0)1202 3633 y Fp(d)1253 3596 y Fo(0)1276 3436 y Fi(\023)1366 3576 y Fu(det)1502 3591 y Fl(q)1556 3436 y Fi(\022)1630 3516 y Fp(a)1681 3480 y Fo(00)1811 3516 y Fp(b)1852 3480 y Fo(0)q(0)1634 3633 y Fp(c)1676 3596 y Fo(0)q(0)1806 3633 y Fp(d)1857 3596 y Fo(00)1900 3436 y Fi(\023)1990 3576 y Fp(:)0 2992 y Fu(\(2\))0 3812 y(In)33 b(particular)e(w)m(e)j(ha)m(v)m (e)g(\001\(det)1195 3827 y Fl(q)1234 3812 y Fu(\))27 b(=)h(det)1539 3827 y Fl(q)1593 3812 y Fr(\012)17 b Fu(det)1823 3827 y Fl(q)1894 3812 y Fu(and)32 b Fp(")p Fu(\(det)2302 3827 y Fl(q)2340 3812 y Fu(\))c(=)g(1.)44 b(The)33 b(quan)m(tum)g (determi-)0 3928 y(nan)m(t)g(is)f(a)g(group)h(lik)m(e)f(elemen)m(t)g (\(see)i(2.1.6\).)149 4044 y(No)m(w)g(w)m(e)f(de\014ne)h(an)f(algebra) 1197 4218 y Fp(S)6 b(L)1329 4233 y Fl(q)1367 4218 y Fu(\(2\))28 b(:=)f Fp(M)1744 4233 y Fl(q)1783 4218 y Fu(\(2\))p Fp(=)p Fu(\(det)2129 4233 y Fl(q)2190 4218 y Fr(\000)22 b Fu(1\))p Fp(:)0 4393 y Fu(The)34 b(algebra)d Fp(S)6 b(L)674 4408 y Fl(q)712 4393 y Fu(\(2\))32 b(represen)m(ts)j(the)e(functor)601 4631 y Fj(S)o(L)722 4646 y Fl(q)760 4631 y Fu(\(2\)\()p Fp(A)p Fu(\))28 b(=)f Fr(f)1215 4491 y Fi(\022)1289 4571 y Fp(a)1340 4535 y Fo(0)1451 4571 y Fp(b)1492 4535 y Fo(0)1293 4688 y Fp(c)1335 4652 y Fo(0)1446 4688 y Fp(d)1497 4652 y Fo(0)1520 4491 y Fi(\023)1621 4631 y Fr(2)h(M)1835 4646 y Fl(q)1873 4631 y Fu(\(2\)\()p Fp(A)p Fu(\))p Fr(j)p Fu(det)2310 4646 y Fl(q)2364 4491 y Fi(\022)2438 4571 y Fp(a)2489 4535 y Fo(0)2600 4571 y Fp(b)2641 4535 y Fo(0)2442 4688 y Fp(c)2484 4652 y Fo(0)2595 4688 y Fp(d)2646 4652 y Fo(0)2669 4491 y Fi(\023)2770 4631 y Fu(=)g(1)p Fr(g)p Fp(:)0 4867 y Fu(There)42 b(is)d(a)h(surjectiv)m(e)i (homomorphism)c(of)h(algebras)h Fp(M)2226 4882 y Fl(q)2265 4867 y Fu(\(2\))g Fr(\000)-60 b(!)41 b Fp(S)6 b(L)2720 4882 y Fl(q)2758 4867 y Fu(\(2\))40 b(and)h Fj(S)o(L)3242 4882 y Fl(q)3280 4867 y Fu(\(2\))f(is)g(a)0 4983 y(subfunctor)34 b(of)e Fr(M)717 4998 y Fl(q)754 4983 y Fu(\(2\).)149 5099 y(Let)45 b Fp(X)r(;)17 b(Y)65 b Fu(b)s(e)44 b(comm)m(uting)e(quan) m(tum)i(matrices)f(satisfying)g(det)2654 5114 y Fl(q)2692 5099 y Fu(\()p Fp(X)8 b Fu(\))47 b(=)g(1)f(=)h(det)3381 5114 y Fl(q)3419 5099 y Fu(\()p Fp(Y)21 b Fu(\).)0 5216 y(Since)66 b(det)424 5231 y Fl(q)462 5216 y Fu(\()p Fp(X)8 b Fu(\))17 b(det)779 5231 y Fl(q)817 5216 y Fu(\()p Fp(Y)k Fu(\))85 b(=)g(det)1352 5231 y Fl(q)1390 5216 y Fu(\()p Fp(X)8 b(Y)21 b Fu(\))66 b(for)g(comm)m(uting)f(quan)m(tum)h(matrices)g (w)m(e)h(get)p eop %%Page: 55 25 55 24 bop 1378 236 a Fq(5.)35 b(QUANTUM)d(GR)n(OUPS)1274 b(55)0 425 y Fu(det)135 440 y Fl(q)174 425 y Fu(\()p Fp(X)8 b(Y)21 b Fu(\))47 b(=)g(1,)g(hence)e Fj(SL)1114 440 y Fl(q)1152 425 y Fu(\(2\))f(is)g(a)g(quan)m(tum)g(submonoid)f(of)h Fr(M)2695 440 y Fl(q)2733 425 y Fu(\(2\))f(and)i Fp(S)6 b(L)3235 440 y Fl(q)3273 425 y Fu(\(2\))44 b(is)f(a)0 541 y(bialgebra)31 b(with)h(diagonal)1105 769 y(\001)1203 629 y Fi(\022)1277 710 y Fp(a)87 b(b)1281 826 y(c)h(d)1461 629 y Fi(\023)1562 769 y Fu(=)1666 629 y Fi(\022)1739 710 y Fp(a)g(b)1744 826 y(c)g(d)1924 629 y Fi(\023)2020 769 y Fr(\012)2119 629 y Fi(\022)2193 710 y Fp(a)g(b)2197 826 y(c)g(d)2378 629 y Fi(\023)2468 769 y Fp(;)0 997 y Fu(and)1352 1165 y Fp(")1415 1025 y Fi(\022)1488 1105 y Fp(a)f(b)1492 1222 y(c)h(d)1673 1025 y Fi(\023)1774 1165 y Fu(=)1877 1025 y Fi(\022)1950 1105 y Fu(1)83 b(0)1950 1222 y(0)g(1)2131 1025 y Fi(\023)2221 1165 y Fp(:)149 1378 y Fu(T)-8 b(o)30 b(sho)m(w)g(that)f Fp(S)6 b(L)868 1393 y Fl(q)907 1378 y Fu(\(2\))29 b(has)g(an)h(an)m(tip)s(o)s(de)e(w)m (e)j(\014rst)e(de\014ne)i(a)e(homomorphism)e(of)i(algebras)0 1494 y Fp(T)41 b Fu(:)28 b Fp(M)247 1509 y Fl(q)286 1494 y Fu(\(2\))f Fr(\000)-60 b(!)28 b Fp(M)677 1509 y Fl(q)715 1494 y Fu(\(2\))840 1458 y Fl(op)946 1494 y Fu(b)m(y)1161 1731 y Fp(T)1249 1590 y Fi(\022)1322 1671 y Fp(a)88 b(b)1327 1787 y(c)f(d)1507 1590 y Fi(\023)1608 1731 y Fu(:=)1739 1590 y Fi(\022)1917 1671 y Fp(d)188 b Fr(\000)p Fp(q)t(b)1812 1787 y Fr(\000)p Fp(q)1936 1751 y Fo(\000)p Fn(1)2031 1787 y Fp(c)140 b(a)2321 1590 y Fi(\023)2411 1731 y Fp(:)0 2034 y Fu(W)-8 b(e)33 b(c)m(hec)m(k)i(that)d Fp(T)42 b Fu(:)27 b Fj(K)884 1894 y Fi(\034)5 b(\022)1037 1974 y Fp(a)88 b(b)1042 2091 y(c)g(d)1222 1894 y Fi(\023)q(\035)1398 2034 y Fr(\000)-60 b(!)28 b Fp(M)1637 2049 y Fl(q)1675 2034 y Fu(\(2\))1800 1998 y Fl(op)1906 2034 y Fu(v)-5 b(anishes)33 b(on)g(the)g(ideal)e Fp(I)8 b Fu(.)442 2277 y Fp(T)14 b Fu(\()p Fp(ab)23 b Fr(\000)f Fp(q)812 2236 y Fo(\000)p Fn(1)906 2277 y Fp(ba)p Fu(\))29 b(=)e Fp(T)14 b Fu(\()p Fp(b)p Fu(\))p Fp(T)g Fu(\()p Fp(a)p Fu(\))22 b Fr(\000)h Fp(q)1723 2236 y Fo(\000)p Fn(1)1817 2277 y Fp(T)14 b Fu(\()p Fp(a)p Fu(\))p Fp(T)g Fu(\()p Fp(b)p Fu(\))28 b(=)f Fr(\000)p Fp(q)t(bd)c Fu(+)f Fp(q)2718 2236 y Fo(\000)p Fn(1)2812 2277 y Fp(q)t(db)28 b Fu(=)f(0)p Fp(:)0 2449 y Fu(W)-8 b(e)46 b(lea)m(v)m(e)h(the)g(c)m(hec)m(k)h(of)e (the)g(other)g(de\014ning)g(relations)f(to)h(the)g(reader.)85 b(F)-8 b(urthermore)45 b Fp(T)0 2565 y Fu(restricts)e(to)f(a)h (homomorphism)c(of)k(algebras)e Fp(S)51 b Fu(:)45 b Fp(S)6 b(L)2130 2580 y Fl(q)2168 2565 y Fu(\(2\))44 b Fr(\000)-60 b(!)45 b Fp(S)6 b(L)2631 2580 y Fl(q)2669 2565 y Fu(\(2\))2794 2529 y Fl(op)2910 2565 y Fu(since)43 b Fp(T)14 b Fu(\(det)3403 2580 y Fl(q)3441 2565 y Fu(\))45 b(=)0 2681 y Fp(T)14 b Fu(\()p Fp(ad)30 b Fr(\000)h Fp(q)396 2645 y Fo(\000)p Fn(1)490 2681 y Fp(bc)p Fu(\))49 b(=)g Fp(T)14 b Fu(\()p Fp(d)p 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(o)s(de)h(b)m(y)h(2.1.3.)51 b(Symmetrically)33 b Fp(S)41 b Fu(is)35 b(a)g(righ)m(t)g(an)m(tip)s(o)s(de.)0 3146 y(Th)m(us)f(the)f(bialgebra)e Fp(S)6 b(L)970 3161 y Fl(q)1008 3146 y Fu(\(2\))32 b(is)g(a)h(Hopf)f(algebra)f(or)i(a)f(quan)m(tum)h (group.)149 3329 y Fs(Example)k(2.5.6.)49 b Fu(The)38 b(a\016ne)g(algebraic)e(group)h Fj(G)21 b(L)c Fu(\()p Fp(n)p Fu(\))36 b(:)g Fj(K)17 b Fu(-)6 b Fs(cAlg)34 b Fr(\000)-60 b(!)35 b Fs(Gr)i Fu(de\014ned)i(b)m(y)0 3445 y Fj(G)21 b(L)d Fu(\()p Fp(n)p Fu(\)\()p Fp(A)p Fu(\),)28 b(the)f(group)f(of)h(in)m(v)m(ertible)f Fp(n)10 b Fr(\002)g Fp(n)p Fu(-matrices)26 b(with)g(co)s(e\016cien)m(ts)i(in)e(the)h(comm)m (utativ)m(e)0 3562 y(algebra)c Fp(A)p Fu(,)i(is)f(represen)m(ted)i(b)m (y)f(the)f(algebra)f Fr(O)s Fu(\()p Fj(G)e(L)d Fu(\()p Fp(n)p Fu(\)\))27 b(=)h Fp(GL)p Fu(\()p Fp(n)p Fu(\))g(=)f Fj(K)18 b Fu([)p Fp(x)2805 3577 y Fl(i)q(j)2872 3562 y Fp(;)f(t)p Fu(])p Fp(=)p Fu(\(det\()p Fp(x)3293 3577 y Fl(ij)3354 3562 y Fu(\))p Fp(t)t Fr(\000)t Fu(1\))0 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Fl(ij)680 4498 y Fu(\))g(=)h Fp(\016)916 4513 y Fl(ij)1016 4498 y Fu(and)g(the)g(an)m(tip)s(o)s(de)e Fp(S)6 b Fu(\()p Fp(x)1957 4513 y Fl(ij)2018 4498 y Fu(\))40 b(=)f Fp(t)27 b Fr(\001)g Fp(ad)-11 b(j)6 b Fu(\()p Fp(X)i Fu(\))2630 4513 y Fl(ij)2729 4498 y Fu(where)41 b Fp(ad)-11 b(j)6 b Fu(\()p Fp(X)i Fu(\))39 b(is)h(the)0 4614 y(adjoin)m(t)k(matrix)g(of) g Fp(X)57 b Fu(=)49 b(\()p Fp(x)1152 4629 y Fl(ij)1213 4614 y Fu(\).)80 b(W)-8 b(e)46 b(lea)m(v)m(e)g(the)f(v)m(eri\014cation) f(of)h(these)h(facts)g(from)e(linear)0 4730 y(algebra)31 b(to)i(the)g(reader.)44 b(The)33 b(diagonal)d(applied)i(to)g Fp(t)h Fu(giv)m(es)1529 4902 y(\001\()p Fp(t)p Fu(\))28 b(=)f Fp(t)c Fr(\012)f Fp(t:)0 5074 y Fu(Hence)h Fp(t)p Fu(\(=)28 b(det\()p Fp(X)8 b Fu(\))756 5038 y Fo(\000)p Fn(1)850 5074 y Fu(\))22 b(is)f(a)h(grouplik)m(e)e(elemen)m(t)i(in)f Fp(GL)p Fu(\()p Fp(n)p Fu(\).)40 b(This)22 b(re\015ects)h(the)g(rule)e (det\()p Fp(AB)5 b Fu(\))28 b(=)0 5191 y(det\()p Fp(A)p Fu(\))17 b(det)q(\()p Fp(B)5 b Fu(\))32 b(hence)i(det)q(\()p Fp(AB)5 b Fu(\))1259 5155 y Fo(\000)p Fn(1)1381 5191 y Fu(=)27 b(det)q(\()p Fp(A)p Fu(\))1769 5155 y Fo(\000)p Fn(1)1880 5191 y Fu(det\()p Fp(B)5 b Fu(\))2170 5155 y Fo(\000)p Fn(1)2264 5191 y Fu(.)p eop %%Page: 56 26 56 25 bop 0 236 a Fq(56)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fs(Example)37 b(2.5.7.)49 b Fu(Let)33 b Fp(M)1191 440 y Fl(q)1229 425 y Fu(\(2\))g(b)s(e)f(as)h(in)f(the)h(example)f(2.5.5.)43 b(W)-8 b(e)33 b(de\014ne)1339 589 y Fp(GL)1482 604 y Fl(q)1520 589 y Fu(\(2\))27 b(:=)h Fp(M)1897 604 y Fl(q)1935 589 y Fu(\(2\)[)p Fp(t)p Fu(])p Fp(=J)0 752 y Fu(with)j Fp(J)40 b Fu(generated)32 b(b)m(y)g(the)f(elemen)m(ts)h Fp(t)19 b Fr(\001)g Fu(\()p Fp(ad)g Fr(\000)g Fp(q)1859 716 y Fo(\000)p Fn(1)1953 752 y Fp(bc)p Fu(\))h Fr(\000)g Fu(1)p Fp(:)30 b Fu(The)j(algebra)c Fp(GL)2979 767 y Fl(q)3018 752 y Fu(\(2\))h(represen)m(ts)0 868 y(the)j(functor)346 1068 y Fj(G)22 b(L)490 1083 y Fl(q)528 1068 y Fu(\(2\)\()p Fp(A)p Fu(\))27 b(=)h Fr(f)983 928 y Fi(\022)1056 1009 y Fp(a)1107 972 y Fo(0)1218 1009 y Fp(b)1259 972 y Fo(0)1061 1125 y Fp(c)1103 1089 y Fo(0)1214 1125 y Fp(d)1265 1089 y Fo(0)1288 928 y Fi(\023)1389 1068 y Fr(2)g(M)1603 1083 y Fl(q)1640 1068 y Fu(\(2\)\()p Fp(A)p Fu(\))p Fr(j)p Fu(det)2077 1083 y Fl(q)2132 928 y Fi(\022)2205 1009 y Fp(a)2256 972 y Fo(0)2367 1009 y Fp(b)2408 972 y Fo(0)2210 1125 y Fp(c)2252 1089 y Fo(0)2363 1125 y Fp(d)2414 1089 y Fo(0)2437 928 y Fi(\023)2559 1068 y Fu(in)m(v)m(ertible)k(in)g Fp(A)p Fr(g)p Fp(:)0 1293 y Fu(In)45 b(fact)g(there)h(is)e(a)h (canonical)e(homomorphism)f(of)j(algebras)f Fp(M)2556 1308 y Fl(q)2594 1293 y Fu(\(2\))k Fr(\000)-60 b(!)49 b Fp(GL)3076 1308 y Fl(q)3115 1293 y Fu(\(2\).)80 b(A)45 b(ho-)0 1409 y(momorphism)40 b(of)h(algebras)h Fp(')i Fu(:)h Fp(M)1380 1424 y Fl(q)1418 1409 y Fu(\(2\))f Fr(\000)-60 b(!)44 b Fp(A)f Fu(has)g(a)f(unique)g(con)m(tin)m(uation)g(to)g Fp(GL)3311 1424 y Fl(q)3349 1409 y Fu(\(2\))g(i\013)0 1590 y(det)135 1605 y Fl(q)174 1590 y Fu(\()p Fp(')293 1450 y Fi(\022)365 1530 y Fp(a)88 b(b)370 1647 y(c)g(d)550 1450 y Fi(\023)624 1590 y Fu(\))32 b(is)g(in)m(v)m(ertible:)1188 1814 y Fp(M)1282 1829 y Fl(q)1321 1814 y Fu(\(2\))185 b Fp(M)1725 1829 y Fl(q)1764 1814 y Fu(\(2\)[)p Fp(t)p Fu(])p 1474 1793 129 4 v 1520 1791 a Fm(-)2172 1814 y Fp(G)2249 1829 y Fl(q)2287 1814 y Fu(\(2\))p 2006 1793 138 4 v 2061 1791 a Fm(-)1768 2312 y Fp(A)1385 1942 y Fm(@)1468 2025 y(@)1551 2108 y(@)1634 2191 y(@)1653 2210 y(@)-83 b(R)p 1803 2210 4 351 v 68 w(?)2141 1942 y(\000)2058 2025 y(\000)1975 2108 y(\000)1892 2191 y(\000)1873 2210 y(\000)g(\011)0 2528 y Fu(with)37 b Fp(t)e Fr(7!)g Fu(det)568 2543 y Fl(q)622 2388 y Fi(\022)696 2468 y Fp(a)747 2432 y Fo(0)858 2468 y Fp(b)899 2432 y Fo(0)700 2585 y Fp(c)742 2548 y Fo(0)853 2585 y Fp(d)904 2548 y Fo(0)927 2388 y Fi(\023)1000 2410 y Fo(\000)p Fn(1)1111 2528 y Fp(:)i Fu(Th)m(us)i Fj(G)21 b(L)1571 2543 y Fl(q)1609 2528 y Fu(\(2\)\()p 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Fp(ad)22 b Fr(\000)h Fp(q)3287 3483 y Fo(\000)p Fn(1)3381 3524 y Fp(bc:)0 3744 y Fu(As)43 b(in)f(2.5.5)g Fp(T)56 b Fu(de\014nes)45 b(a)d(homomorphism)e(of)i (algebras.)73 b(W)-8 b(e)43 b(obtain)f(an)g(induced)h(homo-)0 3860 y(morphism)36 b(of)i(algebras)f Fp(S)43 b Fu(:)38 b Fp(GL)1274 3875 y Fl(q)1312 3860 y Fu(\(2\))f Fr(\000)-60 b(!)37 b Fp(GL)1771 3875 y Fl(q)1809 3860 y Fu(\(2\))1934 3824 y Fl(op)2046 3860 y Fu(or)h(a)f Fp(GL)2400 3875 y Fl(q)2439 3860 y Fu(\(2\))2564 3824 y Fl(op)2637 3860 y Fu(-p)s(oin)m(t)g(in)g Fp(GL)3192 3875 y Fl(q)3231 3860 y Fu(\(2\))h(since)0 3976 y Fp(S)6 b Fu(\()p Fp(t)p Fu(\()p Fp(ad)19 b Fr(\000)i Fp(q)443 3940 y Fo(\000)p Fn(1)537 3976 y Fp(bc)p Fu(\))f Fr(\000)g Fu(1\))27 b(=)h(\()p Fp(S)6 b Fu(\()p Fp(d)p Fu(\))p Fp(S)g Fu(\()p Fp(a)p Fu(\))19 b Fr(\000)h Fp(q)1580 3940 y Fo(\000)p Fn(1)1674 3976 y Fp(S)6 b Fu(\()p Fp(c)p Fu(\))p Fp(S)g Fu(\()p Fp(b)p Fu(\)\))p Fp(S)g Fu(\()p Fp(t)p Fu(\))20 b Fr(\000)g Fp(S)6 b Fu(\(1\))27 b(=)g(\()p Fp(t)2767 3940 y Fn(2)2807 3976 y Fp(ad)20 b Fr(\000)g Fp(q)3073 3940 y Fo(\000)p Fn(1)3167 3976 y Fp(t)3202 3940 y Fn(2)3242 3976 y Fp(cb)p Fu(\)\()p Fp(ad)g Fr(\000)0 4092 y Fp(q)47 4056 y Fo(\000)p Fn(1)141 4092 y Fp(bc)p Fu(\))j Fr(\000)f Fu(1)28 b(=)f Fp(t)599 4056 y Fn(2)639 4092 y Fu(\()p Fp(ad)22 b Fr(\000)h Fp(q)948 4056 y Fo(\000)p Fn(1)1042 4092 y Fp(bc)p Fu(\))1163 4056 y Fn(2)1225 4092 y Fr(\000)f Fu(1)28 b(=)f(0.)149 4209 y(Since)k Fp(S)36 b Fu(satis\014es)856 4134 y Fi(P)978 4209 y Fp(S)6 b Fu(\()p Fp(x)1137 4224 y Fn(\(1\))1231 4209 y Fu(\))p Fp(x)1324 4224 y Fn(\(2\))1446 4209 y Fu(=)28 b Fp(")p Fu(\()p Fp(x)p Fu(\))i(for)h(all)d(giv)m(en)j (generators,)g Fp(S)37 b Fu(is)30 b(a)g(left)g(an)m(tip)s(o)s(de)0 4325 y(b)m(y)35 b(2.1.3.)49 b(Symmetrically)31 b Fp(S)40 b Fu(is)34 b(a)g(righ)m(t)g(an)m(tip)s(o)s(de.)48 b(Th)m(us)36 b(the)e(bialgebra)f Fp(GL)3014 4340 y Fl(q)3052 4325 y Fu(\(2\))h(is)g(a)g(Hopf)0 4441 y(algebra)d(or)i(a)f(quan)m(tum)h (group.)149 4619 y Fs(Example)k(2.5.8.)49 b Fu(Let)42 b Fp(sl)r Fu(\(2\))f(b)s(e)h(the)g(3-dimensional)c(v)m(ector)43 b(space)g(generated)f(b)m(y)h(the)0 4735 y(matrices)669 4895 y Fp(X)35 b Fu(=)889 4754 y Fi(\022)962 4835 y Fu(0)83 b(1)962 4951 y(0)g(0)1142 4754 y Fi(\023)1232 4895 y Fp(;)212 b(Y)49 b Fu(=)1681 4754 y Fi(\022)1754 4835 y Fu(0)83 b(0)1754 4951 y(1)g(0)1935 4754 y Fi(\023)2025 4895 y Fp(;)211 b(H)36 b Fu(=)2483 4754 y Fi(\022)2556 4835 y Fu(1)122 b(0)2556 4951 y(0)83 b Fr(\000)p Fu(1)2814 4754 y Fi(\023)2904 4895 y Fp(:)0 5099 y Fu(Then)26 b Fp(sl)r Fu(\(2\))e(is)h(a)f(subspace)j(of)d(the)h(algebra)f Fp(M)10 b Fu(\(2\))25 b(of)f(2)6 b Fr(\002)g Fu(2-matrices)24 b(o)m(v)m(er)i Fj(K)18 b Fu(.)46 b(W)-8 b(e)26 b(easily)e(v)m(erify)0 5216 y([)p Fp(X)r(;)17 b(Y)k Fu(])28 b(=)g Fp(X)8 b(Y)25 b Fr(\000)t Fp(Y)d(X)36 b Fu(=)27 b Fp(H)8 b Fu(,)25 b([)p Fp(H)r(;)17 b(X)8 b Fu(])28 b(=)f Fp(H)8 b(X)k Fr(\000)t Fp(X)c(H)35 b Fu(=)28 b(2)p Fp(X)8 b Fu(,)25 b(and)f([)p Fp(H)r(;)17 b(Y)k Fu(])28 b(=)g Fp(H)8 b(Y)25 b Fr(\000)t Fp(Y)d(H)35 b Fu(=)27 b Fr(\000)p Fu(2)p Fp(Y)22 b Fu(,)p eop %%Page: 57 27 57 26 bop 1378 236 a Fq(5.)35 b(QUANTUM)d(GR)n(OUPS)1274 b(57)0 425 y Fu(so)29 b(that)f Fp(sl)r Fu(\(2\))g(b)s(ecomes)i(a)e(Lie) g(subalgebra)g(of)g Fp(M)10 b Fu(\(2\))1995 389 y Fl(L)2048 425 y Fu(,)29 b(whic)m(h)g(is)f(the)h(Lie)f(algebra)g(of)g(matrices)0 541 y(of)33 b(trace)g(zero.)46 b(The)34 b(univ)m(ersal)f(en)m(v)m (eloping)h(algebra)e Fp(U)10 b Fu(\()p Fp(sl)r Fu(\(2\)\))33 b(is)g(a)g(Hopf)g(algebra)f(generated)0 658 y(as)h(an)f(algebra)g(b)m (y)h(the)g(elemen)m(ts)g Fp(X)r(;)17 b(Y)5 b(;)17 b(H)40 b Fu(with)32 b(the)h(relations)852 875 y([)p Fp(X)r(;)17 b(Y)22 b Fu(])28 b(=)f Fp(H)r(;)114 b Fu([)p Fp(H)r(;)17 b(X)8 b Fu(])28 b(=)f(2)p Fp(X)r(;)115 b Fu([)p Fp(H)r(;)17 b(Y)k Fu(])28 b(=)f Fr(\000)p Fu(2)p Fp(Y)5 b(:)149 1092 y Fu(As)39 b(a)f(consequence)j(of)c(the)i(P)m(oincar)m(\023)-46 b(e-Birkho\013-Witt)36 b(Theorem)i(\(that)g(w)m(e)h(don't)f(pro)m(v)m (e\))0 1208 y(the)d(Hopf)f(algebra)f Fp(U)10 b Fu(\()p Fp(sl)r Fu(\(2\)\))34 b(has)g(the)h(basis)f Fr(f)p Fp(X)1866 1172 y Fl(i)1894 1208 y Fp(Y)1972 1172 y Fl(j)2009 1208 y Fp(H)2098 1172 y Fl(k)2140 1208 y Fr(j)p Fp(i;)17 b(j;)g(k)33 b Fr(2)e Fj(N)9 b Fr(g)p Fu(.)54 b(F)-8 b(urthermore)33 b(one)i(can)0 1325 y(pro)m(v)m(e)f(that)e(all)f(\014nite)h(dimensional) e Fp(U)10 b Fu(\()p Fp(sl)r Fu(\(2\)\)-mo)s(dules)32 b(are)g(semisimple.)149 1533 y Fs(Example)37 b(2.5.9.)49 b Fu(W)-8 b(e)27 b(de\014ne)g(the)f(so-called)f Fp(q)t Fu(-deformed)g(v)m(ersion)i Fp(U)2799 1548 y Fl(q)2837 1533 y Fu(\()p Fp(sl)r Fu(\(2\)\))f(of)g Fp(U)10 b Fu(\()p Fp(sl)r Fu(\(2\)\))0 1650 y(for)33 b(an)m(y)h Fp(q)e Fr(2)e Fj(K)17 b Fu(,)40 b Fp(q)32 b Fr(6)p Fu(=)d(1)p Fp(;)17 b Fr(\000)p Fu(1)33 b(and)g Fp(q)k Fu(in)m(v)m(ertible.)45 b(Let)33 b Fp(U)2058 1665 y Fl(q)2097 1650 y Fu(\()p Fp(sl)r Fu(\(2\)\))g(b)s(e)g(the)h(algebra)e(generated)i(b)m(y)0 1766 y(the)f(elemen)m(ts)g Fp(E)6 b(;)17 b(F)s(;)g(K)r(;)g(K)1018 1730 y Fo(0)1073 1766 y Fu(with)32 b(the)h(relations)1427 1976 y Fp(K)7 b(K)1607 1940 y Fo(0)1659 1976 y Fu(=)27 b Fp(K)1852 1940 y Fo(0)1876 1976 y Fp(K)34 b Fu(=)28 b(1)p Fp(;)1069 2092 y(K)7 b(E)f(K)1327 2056 y Fo(0)1379 2092 y Fu(=)27 b Fp(q)1529 2056 y Fn(2)1568 2092 y Fp(E)6 b(;)212 b(K)7 b(F)14 b(K)2142 2056 y Fo(0)2193 2092 y Fu(=)28 b Fp(q)2344 2056 y Fo(\000)p Fn(2)2438 2092 y Fp(F)s(;)1333 2267 y(E)6 b(F)35 b Fr(\000)23 b Fp(F)14 b(E)34 b Fu(=)1905 2200 y Fp(K)29 b Fr(\000)23 b Fp(K)2207 2164 y Fo(0)p 1905 2244 326 4 v 1913 2336 a Fp(q)j Fr(\000)c Fp(q)2128 2307 y Fo(\000)p Fn(1)2240 2267 y Fp(:)0 2543 y Fu(Since)41 b Fp(K)353 2507 y Fo(0)417 2543 y Fu(is)f(the)h(in)m(v)m (erse)i(of)d Fp(K)48 b Fu(in)40 b Fp(U)1469 2558 y Fl(q)1507 2543 y Fu(\()p Fp(sl)r Fu(\(2\)\))g(w)m(e)i(write)f Fp(K)2325 2507 y Fo(\000)p Fn(1)2461 2543 y Fu(=)g Fp(K)2668 2507 y Fo(0)2692 2543 y Fu(.)68 b(The)41 b(represen)m(tation)0 2660 y(theory)33 b(of)g(this)f(algebra)f(is)i(fundamen)m(tally)e (di\013eren)m(t)i(dep)s(ending)g(on)f(whether)j Fp(q)h Fu(is)c(a)h(ro)s(ot)e(of)0 2776 y(unit)m(y)i(or)f(not.)149 2892 y(W)-8 b(e)33 b(sho)m(w)h(that)e Fp(U)836 2907 y Fl(q)875 2892 y Fu(\()p Fp(sl)r Fu(\(2\)\))g(is)g(a)g(Hopf)h(algebra)e (or)h(quan)m(tum)h(group.)44 b(W)-8 b(e)33 b(de\014ne)599 3110 y(\001\()p Fp(E)6 b Fu(\))28 b(=)g(1)22 b Fr(\012)g Fp(E)29 b Fu(+)22 b Fp(E)28 b Fr(\012)22 b Fp(K)r(;)212 b Fu(\001\()p Fp(F)14 b Fu(\))28 b(=)f Fp(K)2313 3074 y Fo(\000)p Fn(1)2430 3110 y Fr(\012)22 b Fp(F)36 b Fu(+)22 b Fp(F)36 b Fr(\012)23 b Fu(1)p Fp(;)1449 3227 y Fu(\001\()p Fp(K)7 b Fu(\))28 b(=)f Fp(K)j Fr(\012)22 b Fp(K)r(;)1117 3343 y(")p Fu(\()p Fp(E)6 b Fu(\))27 b(=)h Fp(")p Fu(\()p Fp(F)14 b Fu(\))26 b(=)i(0)p Fp(;)212 b(")p Fu(\()p Fp(K)7 b Fu(\))27 b(=)g(1)p Fp(;)734 3459 y(S)6 b Fu(\()p Fp(E)g Fu(\))28 b(=)f Fr(\000)p Fp(E)6 b(K)1330 3423 y Fo(\000)p Fn(1)1425 3459 y Fp(;)115 b(S)6 b Fu(\()p Fp(F)14 b Fu(\))27 b(=)g Fr(\000)p Fp(K)7 b(F)s(;)115 b(S)6 b Fu(\()p Fp(K)h Fu(\))27 b(=)h Fp(K)2744 3423 y Fo(\000)p Fn(1)2838 3459 y Fp(:)0 3613 y Fu(.)149 3729 y(First)33 b(w)m(e)i(sho)m(w)f(that)g (\001)g(can)f(b)s(e)h(expanded)h(in)e(a)g(unique)i(w)m(a)m(y)f(to)g(an) f(algebra)f(homomor-)0 3845 y(phism)40 b(\001)h(:)h Fp(U)553 3860 y Fl(q)591 3845 y Fu(\()p Fp(sl)r Fu(\(2\)\))f Fr(\000)-60 b(!)41 b Fp(U)1134 3860 y Fl(q)1173 3845 y Fu(\()p Fp(sl)r Fu(\(2\)\))27 b Fr(\012)h Fp(U)1649 3860 y Fl(q)1687 3845 y Fu(\()p Fp(sl)r Fu(\(2\)\).)67 b(W)-8 b(rite)40 b Fp(U)2404 3860 y Fl(q)2443 3845 y Fu(\()p Fp(sl)r Fu(\(2\)\))g(as)h (the)g(residue)g(class)0 3962 y(algebra)31 b Fj(K)18 b Fr(h)p Fp(E)6 b(;)17 b(F)s(;)g(K)r(;)g(K)903 3926 y Fo(\000)p Fn(1)1002 3962 y Fr(i)p Fp(=I)40 b Fu(where)34 b Fp(I)40 b Fu(is)32 b(generated)h(b)m(y)1222 4177 y Fp(K)7 b(K)1402 4141 y Fo(\000)p Fn(1)1519 4177 y Fr(\000)22 b Fu(1)p Fp(;)212 b(K)1996 4141 y Fo(\000)p Fn(1)2090 4177 y Fp(K)30 b Fr(\000)22 b Fu(1)p Fp(;)1008 4293 y(K)7 b(E)f(K)1266 4257 y Fo(\000)p Fn(1)1382 4293 y Fr(\000)23 b Fp(q)1529 4257 y Fn(2)1568 4293 y Fp(E)6 b(;)212 b(K)7 b(F)14 b(K)2142 4257 y Fo(\000)p Fn(1)2259 4293 y Fr(\000)22 b Fp(q)2405 4257 y Fo(\000)p Fn(2)2499 4293 y Fp(F)s(;)1302 4475 y(E)6 b(F)36 b Fr(\000)22 b Fp(F)14 b(E)28 b Fr(\000)1865 4407 y Fp(K)h Fr(\000)23 b Fp(K)2167 4371 y Fo(\000)p Fn(1)p 1865 4452 397 4 v 1908 4543 a Fp(q)j Fr(\000)d Fp(q)2124 4514 y Fo(\000)p Fn(1)2271 4475 y Fp(:)0 4751 y Fu(Since)34 b Fp(K)346 4714 y Fo(\000)p Fn(1)473 4751 y Fu(m)m(ust)g(b)s(e)g(mapp)s(ed)f(to)g(the)g(in)m(v)m(erse)i(of)e (\001\()p Fp(K)7 b Fu(\))34 b(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h(\001\()p Fp(K)3049 4714 y Fo(\000)p Fn(1)3143 4751 y Fu(\))29 b(=)g Fp(K)3405 4714 y Fo(\000)p Fn(1)3523 4751 y Fr(\012)0 4867 y Fp(K)90 4831 y Fo(\000)p Fn(1)184 4867 y Fu(.)40 b(No)m(w)23 b(\001)f(can)h(b)s(e)f(expanded)i(in)d(a)h(unique)g(w)m(a)m (y)i(to)e(the)g(free)g(algebra)f(\001)28 b(:)g Fp(K)7 b Fr(h)p Fp(E)f(;)17 b(F)s(;)g(K)r(;)g(K)3468 4831 y Fo(\000)p Fn(1)3561 4867 y Fr(i)0 4983 y(\000)-60 b(!)46 b Fp(U)229 4998 y Fl(q)268 4983 y Fu(\()p Fp(sl)r Fu(\(2\)\))29 b Fr(\012)h Fp(U)748 4998 y Fl(q)787 4983 y Fu(\()p Fp(sl)r Fu(\(2\)\).)76 b(W)-8 b(e)44 b(ha)m(v)m(e)h(\001\()p Fp(K)7 b(K)1882 4947 y Fo(\000)p Fn(1)1976 4983 y Fu(\))47 b(=)f(\001\()p Fp(K)7 b Fu(\)\001\()p Fp(K)2639 4947 y Fo(\000)p Fn(1)2734 4983 y Fu(\))46 b(=)h(1)c(and)h(similarly)0 5099 y(\001\()p Fp(K)209 5063 y Fo(\000)p Fn(1)304 5099 y Fp(K)7 b Fu(\))28 b(=)h(1.)45 b(F)-8 b(urthermore)32 b(w)m(e)j(ha)m(v)m(e)f(\001\()p Fp(K)7 b(E)f(K)1998 5063 y Fo(\000)p Fn(1)2093 5099 y Fu(\))28 b(=)h(\001\()p Fp(K)7 b Fu(\)\001\()p Fp(E)f Fu(\)\001\()p Fp(K)2955 5063 y Fo(\000)p Fn(1)3050 5099 y Fu(\))29 b(=)f(\()p Fp(K)i Fr(\012)23 b Fp(K)7 b Fu(\))0 5216 y(\(1)16 b Fr(\012)i Fp(E)k Fu(+)17 b Fp(E)23 b Fr(\012)17 b Fp(K)7 b Fu(\)\()p Fp(K)830 5179 y Fo(\000)p Fn(1)941 5216 y Fr(\012)17 b Fp(K)1125 5179 y Fo(\000)p Fn(1)1220 5216 y Fu(\))27 b(=)h Fp(K)7 b(K)1569 5179 y Fo(\000)p Fn(1)1680 5216 y Fr(\012)18 b Fp(K)7 b(E)f(K)2033 5179 y Fo(\000)p Fn(1)2144 5216 y Fu(+)17 b Fp(K)7 b(E)f(K)2495 5179 y Fo(\000)p Fn(1)2606 5216 y Fr(\012)17 b Fp(K)2790 5179 y Fn(2)2830 5216 y Fp(K)2920 5179 y Fo(\000)p Fn(1)3042 5216 y Fu(=)27 b Fp(q)3192 5179 y Fn(2)3232 5216 y Fu(\(1)16 b Fr(\012)h Fp(E)23 b Fu(+)p eop %%Page: 58 28 58 27 bop 0 236 a Fq(58)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fp(E)27 b Fr(\012)21 b Fp(K)7 b Fu(\))27 b(=)h Fp(q)503 389 y Fn(2)542 425 y Fu(\001\()p Fp(E)6 b Fu(\))28 b(=)g(\001\()p Fp(q)1075 389 y Fn(2)1114 425 y Fp(E)6 b Fu(\))32 b(and)g(similarly)c (\001\()p Fp(K)7 b(F)14 b(K)2225 389 y Fo(\000)p Fn(1)2319 425 y Fu(\))28 b(=)f(\001\()p Fp(q)2654 389 y Fo(\000)p Fn(2)2749 425 y Fp(F)14 b Fu(\).)43 b(Finally)29 b(w)m(e)k(ha)m(v)m(e) 414 580 y(\001\()p Fp(E)6 b(F)36 b Fr(\000)23 b Fp(F)14 b(E)6 b Fu(\))32 b(=)27 b(\(1)22 b Fr(\012)h Fp(E)28 b Fu(+)22 b Fp(E)28 b Fr(\012)23 b Fp(K)7 b Fu(\)\()p Fp(K)2001 544 y Fo(0)2047 580 y Fr(\012)22 b Fp(F)36 b Fu(+)22 b Fp(F)36 b Fr(\012)22 b Fu(1\))1230 697 y Fr(\000)p Fu(\()p Fp(K)1435 661 y Fo(0)1481 697 y Fr(\012)h Fp(F)35 b Fu(+)22 b Fp(F)36 b Fr(\012)23 b Fu(1\)\(1)f Fr(\012)g Fp(E)28 b Fu(+)22 b Fp(E)29 b Fr(\012)22 b Fp(K)7 b Fu(\))1035 813 y(=)27 b Fp(K)1228 777 y Fo(0)1274 813 y Fr(\012)c Fp(E)6 b(F)36 b Fu(+)22 b Fp(F)35 b Fr(\012)23 b Fp(E)28 b Fu(+)22 b Fp(E)6 b(K)2213 777 y Fo(0)2259 813 y Fr(\012)22 b Fp(K)7 b(F)36 b Fu(+)22 b Fp(E)6 b(F)36 b Fr(\012)23 b Fp(K)1230 929 y Fr(\000)p Fp(K)1397 893 y Fo(0)1443 929 y Fr(\012)g Fp(F)14 b(E)28 b Fr(\000)22 b Fp(K)1909 893 y Fo(0)1933 929 y Fp(E)28 b Fr(\012)22 b Fp(F)14 b(K)29 b Fr(\000)23 b Fp(F)36 b Fr(\012)23 b Fp(E)28 b Fr(\000)22 b Fp(F)14 b(E)28 b Fr(\012)23 b Fp(K)1035 1045 y Fu(=)k Fp(K)1228 1009 y Fo(0)1274 1045 y Fr(\012)c Fu(\()p Fp(E)6 b(F)35 b Fr(\000)23 b Fp(F)14 b(E)6 b Fu(\))22 b(+)g(\()p Fp(E)6 b(F)36 b Fr(\000)22 b Fp(F)14 b(E)6 b Fu(\))22 b Fr(\012)h Fp(K)1035 1222 y Fu(=)1148 1155 y Fp(K)1238 1119 y Fo(0)1284 1155 y Fr(\012)g Fu(\()p Fp(K)29 b Fr(\000)22 b Fp(K)1723 1119 y Fo(0)1747 1155 y Fu(\))g(+)g(\()p Fp(K)29 b Fr(\000)23 b Fp(K)2245 1119 y Fo(0)2268 1155 y Fu(\))f Fr(\012)h Fp(K)p 1148 1199 1370 4 v 1678 1291 a(q)j Fr(\000)d Fp(q)1894 1262 y Fo(\000)p Fn(1)1035 1455 y Fu(=)k(\001)1236 1314 y Fi(\022)1320 1387 y Fp(K)i Fr(\000)23 b Fp(K)1622 1351 y Fo(0)p 1320 1432 326 4 v 1328 1523 a Fp(q)i Fr(\000)e Fp(q)1543 1494 y Fo(\000)p Fn(1)1655 1314 y Fi(\023)0 1667 y Fu(hence)j(\001)f(v)-5 b(anishes)26 b(on)f Fp(I)32 b Fu(and)25 b(can)h(b)s(e)f(factorized)f(through)h(a)g(unique)g (algebra)f(homomorphism)984 1824 y(\001)k(:)g Fp(U)1214 1839 y Fl(q)1253 1824 y Fu(\()p Fp(sl)r Fu(\(2\)\))f Fr(\000)-60 b(!)28 b Fp(U)1769 1839 y Fl(q)1807 1824 y Fu(\()p Fp(sl)r Fu(\(2\)\))22 b Fr(\012)g Fp(U)2272 1839 y Fl(q)2311 1824 y Fu(\()p Fp(sl)r Fu(\(2\)\))p Fp(:)149 1981 y Fu(In)33 b(a)g(similar)c(w)m(a)m(y)-8 b(,)34 b(actually)d(m)m(uc)m(h)i(simpler,)e(one)i(gets)g(an)g(algebra)e (homomorphism)1406 2138 y Fp(")d Fu(:)g Fp(U)1601 2153 y Fl(q)1639 2138 y Fu(\()p Fp(sl)r Fu(\(2\)\))f Fr(\000)-60 b(!)28 b Fj(K)17 b Fp(:)149 2295 y Fu(T)-8 b(o)33 b(c)m(hec)m(k)j(that) c(\001)i(is)e(coasso)s(ciativ)m(e)h(it)e(su\016ces)36 b(to)c(c)m(hec)m(k)j(this)e(for)f(the)i(generators)f(of)g(the)0 2411 y(algebra.)49 b(W)-8 b(e)35 b(ha)m(v)m(e)h(\(\001)24 b Fr(\012)g Fu(1\)\001\()p Fp(E)6 b Fu(\))31 b(=)g(\(\001)24 b Fr(\012)g Fu(1\)\(1)f Fr(\012)h Fp(E)30 b Fu(+)23 b Fp(E)30 b Fr(\012)24 b Fp(K)7 b Fu(\))31 b(=)g(1)24 b Fr(\012)g Fu(1)f Fr(\012)h Fp(E)30 b Fu(+)23 b(1)h Fr(\012)g Fp(E)30 b Fr(\012)0 2527 y Fp(K)37 b Fu(+)29 b Fp(E)35 b Fr(\012)30 b Fp(K)37 b Fr(\012)29 b Fp(K)53 b Fu(=)46 b(\(1)29 b Fr(\012)h Fu(\001\)\(1)f Fr(\012)h Fp(E)36 b Fu(+)29 b Fp(E)35 b Fr(\012)30 b Fp(K)7 b Fu(\))46 b(=)g(\(1)29 b Fr(\012)h Fu(\001\)\001\()p Fp(E)6 b Fu(\).)75 b(Similarly)40 b(w)m(e)k(get)0 2644 y(\(\001)15 b Fr(\012)g 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y(rically)40 b Fp(S)49 b Fu(is)42 b(a)g(righ)m(t)g(an) m(tip)s(o)s(de.)72 b(Th)m(us)45 b(the)e(bialgebra)d Fp(U)2264 4146 y Fl(q)2303 4131 y Fu(\()p Fp(sl)r Fu(\(2\)\))i(is)g(a)g(Hopf)h (algebra)e(or)h(a)0 4248 y(quan)m(tum)33 b(group.)149 4364 y(This)27 b(quan)m(tum)g(group)f(is)g(of)h(cen)m(tral)f(in)m (terest)h(in)f(theoretical)f(ph)m(ysics.)43 b(Its)27 b(represen)m(tation)0 4480 y(theory)45 b(is)f(w)m(ell)g(understo)s(o)s (d.)79 b(If)45 b Fp(q)j Fu(is)c(not)g(a)g(ro)s(ot)g(of)g(unit)m(y)g (then)h(the)g(\014nite)f(dimensional)0 4596 y Fp(U)66 4611 y Fl(q)104 4596 y Fu(\()p Fp(sl)r Fu(\(2\)\)-mo)s(dules)c(are)i (semisimple.)67 b(Man)m(y)43 b(more)e(prop)s(erties)g(can)h(b)s(e)g (found)f(in)g([Kassel:)0 4712 y(Quan)m(tum)32 b(Groups].)p eop %%Page: 59 29 59 28 bop 1028 236 a Fq(6.)35 b(QUANTUM)e(A)n(UTOMORPHISM)e(GR)n(OUPS) 925 b(59)915 425 y Fs(6.)56 b(Quan)m(tum)37 b(Automorphism)f(Groups)149 600 y(Lemma)i(2.6.1.)49 b Fk(The)34 b(c)-5 b(ate)g(gory)35 b Fj(K)17 b Fk(-)6 b Fs(Alg)34 b 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Fl(k)2878 1565 y Fu(:)f Fp(A)3008 1580 y Fl(k)3079 1565 y Fr(\000)-60 b(!)30 b Fp(B)5 b Fr(j)p Fp(k)32 b Fr(2)d Fp(I)8 b Fu(\))0 1681 y(that)32 b(the)h(follo)m(wing)d(diagram)h(giv)m(es)i(the)g (required)g(univ)m(ersal)f(prop)s(ert)m(y)698 1855 y Fp(A)771 1870 y Fl(k)1284 1781 y Fi(L)1411 1855 y Fp(A)1484 1870 y Fl(i)p 843 1832 404 4 v 1163 1830 a Fm(-)1011 1798 y Fl(j)1040 1810 y Fe(k)1836 1855 y Fp(T)14 b Fu(\()1945 1781 y Fi(L)2072 1855 y Fp(A)2145 1870 y Fl(i)2173 1855 y Fu(\))p 1533 1832 274 4 v 1724 1830 a Fm(-)1657 1812 y Fl(\023)2412 1855 y Fp(T)g Fu(\()2521 1781 y Fi(L)2648 1855 y Fp(A)2721 1870 y Fl(i)2749 1855 y Fu(\))p Fp(=L)p 2240 1832 144 4 v 2301 1830 a Fm(-)2291 1812 y Fl(\027)2617 2498 y Fp(B)1570 2164 y Fl(f)1604 2176 y Fe(k)961 1926 y Fm(P)1044 1954 y(P)1127 1982 y(P)1210 2009 y(P)1293 2037 y(P)1376 2065 y(P)1459 2093 y(P)1542 2120 y(P)1625 2148 y(P)1708 2176 y(P)1791 2203 y(P)1874 2231 y(P)1957 2259 y(P)2040 2286 y(P)2123 2314 y(P)2206 2342 y(P)2289 2369 y(P)2369 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Fr(2)h Fj(N)9 b Fu(\))39 b(b)m(y)1349 1772 y Fp(B)1423 1787 y Fn(0)1490 1772 y Fu(:=)28 b Fp(B)5 b(;)1349 1888 y(B)1423 1903 y Fl(i)p Fn(+1)1569 1888 y Fu(:=)28 b Fp(B)1779 1841 y Fl(opcop)1774 1914 y(i)1953 1888 y Fp(;)17 b(i)28 b Fr(2)g Fj(N)9 b Fp(:)0 2113 y Fu(Let)46 b Fp(B)267 2077 y Fo(0)336 2113 y Fu(b)s(e)g(the)g(copro)s (duct)h(of)e(the)h(family)d(\()p Fp(B)1863 2128 y Fl(i)1892 2113 y Fr(j)p Fp(i)50 b Fr(2)g Fj(N)9 b Fu(\))52 b(with)46 b(injections)f Fp(\023)2995 2128 y Fl(i)3073 2113 y Fu(:)51 b Fp(B)3225 2128 y Fl(i)3303 2113 y Fr(\000)-60 b(!)50 b Fp(B)3549 2077 y Fo(0)3573 2113 y Fu(.)0 2230 y(Because)27 b Fp(B)445 2193 y Fo(0)494 2230 y Fu(is)e(a)g(copro)s(duct)h(of)f (bialgebras)f(there)i(is)f(a)h(unique)g(homomorphism)c(of)j(bialgebras) 0 2346 y Fp(S)66 2310 y Fo(0)117 2346 y Fu(:)j Fp(B)251 2310 y Fo(0)302 2346 y Fr(\000)-60 b(!)27 b Fp(B)525 2310 y Fo(0)549 2299 y Fl(opcop)755 2346 y Fu(suc)m(h)34 b(that)f(the)g(diagrams)1375 3047 y Fp(B)1454 2999 y Fl(opcop)1449 3072 y(i)p Fn(+1)1948 3047 y Fp(B)2027 3010 y Fo(0)2051 3000 y Fl(opcop)p 1657 3027 264 4 v 1837 3025 a Fm(-)1724 3073 y Fl(\023)1749 3083 y Fe(i)p Fd(+1)1451 2566 y Fp(B)1525 2581 y Fl(i)2035 2566 y Fp(B)2114 2530 y Fo(0)p 1582 2539 426 4 v 1924 2537 a Fm(-)1768 2509 y Fl(\023)1793 2519 y Fe(i)p 1500 2957 4 351 v 1502 2957 a Fm(?)1404 2804 y Fn(id)p 2085 2957 V 2087 2957 a Fm(?)2126 2807 y Fl(S)2173 2783 y Ff(0)0 3266 y Fu(comm)m(ute.)149 3382 y(No)m(w)h(let)e Fp(I)40 b Fu(b)s(e)33 b(the)g(t)m(w)m(o)g(sided)g (ideal)e(in)h Fp(B)1757 3346 y Fo(0)1813 3382 y Fu(generated)h(b)m(y) 652 3606 y Fr(f)p Fu(\()p Fp(S)806 3565 y Fo(0)851 3606 y Fr(\003)22 b Fu(1)g Fr(\000)h Fp(u")p Fu(\)\()p Fp(x)1327 3621 y Fl(i)1354 3606 y Fu(\))p Fp(;)17 b Fu(\(1)22 b Fr(\003)g Fp(S)1683 3565 y Fo(0)1728 3606 y Fr(\000)h Fp(u")p Fu(\)\()p Fp(x)2061 3621 y Fl(i)2088 3606 y Fu(\))p Fr(j)p Fp(x)2209 3621 y Fl(i)2265 3606 y Fr(2)28 b Fp(\023)2393 3621 y Fl(i)2422 3606 y Fu(\()p Fp(B)2534 3621 y Fl(i)2562 3606 y Fu(\))p Fp(;)17 b(i)28 b Fr(2)g Fj(N)9 b Fr(g)p Fp(:)0 3830 y(I)40 b Fu(is)32 b(a)h(coideal,)e(i.e.)43 b Fp(")829 3845 y Fl(B)885 3826 y Ff(0)912 3830 y Fu(\()p Fp(I)8 b Fu(\))27 b(=)h(0)k(and)h(\001)1522 3845 y Fl(B)1578 3826 y Ff(0)1605 3830 y Fu(\()p Fp(I)8 b Fu(\))27 b Fr(\022)h Fp(I)i Fr(\012)23 b Fp(B)2116 3793 y Fo(0)2162 3830 y Fu(+)f Fp(B)2339 3793 y Fo(0)2384 3830 y Fr(\012)h Fp(I)8 b Fu(.)149 3946 y(Since)45 b Fp(")462 3961 y Fl(B)518 3942 y Ff(0)588 3946 y Fu(and)f(\001)870 3961 y Fl(B)926 3942 y Ff(0)998 3946 y Fu(are)g(homomorphisms)e(of)h(algebras)h(it)f (su\016ces)j(to)d(c)m(hec)m(k)k(this)c(for)0 4062 y(the)36 b(generating)e(elemen)m(ts)i(of)f Fp(I)8 b Fu(.)51 b(Let)36 b Fp(x)d Fr(2)f Fp(B)1733 4077 y Fl(i)1797 4062 y Fu(b)s(e)j(giv)m(en.) 52 b(Then)37 b(w)m(e)f(ha)m(v)m(e)h Fp(")p Fu(\(\(1)23 b Fr(\003)h Fp(S)3199 4026 y Fo(0)3222 4062 y Fu(\))p Fp(\023)3294 4077 y Fl(i)3323 4062 y Fu(\()p Fp(x)p Fu(\)\))32 b(=)0 4178 y Fp(")p Fu(\()p Fr(r)p Fu(\(1)7 b Fr(\012)g 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y(\001\(\(1)22 b Fr(\003)g Fp(S)758 4480 y Fo(0)781 4517 y Fu(\))p Fp(\023)853 4532 y Fl(i)882 4517 y Fu(\()p Fp(x)p Fu(\)\))392 4633 y(=)28 b(\001)p Fr(r)p Fu(\(1)22 b Fr(\012)h Fp(S)935 4597 y Fo(0)958 4633 y Fu(\)\001)p Fp(\023)1111 4648 y Fl(i)1140 4633 y Fu(\()p Fp(x)p Fu(\))392 4749 y(=)28 b(\()p Fr(r)22 b(\012)h(r)p Fu(\)\(1)e Fr(\012)i Fp(\034)34 b Fr(\012)22 b Fu(1\)\(\001)g Fr(\012)h Fu(\001\)\(1)f Fr(\012)h Fp(S)1965 4713 y Fo(0)1988 4749 y Fu(\)\()p Fp(\023)2098 4764 y Fl(i)2148 4749 y Fr(\012)g Fp(\023)2282 4764 y Fl(i)2311 4749 y Fu(\)\001)2430 4764 y Fl(i)2458 4749 y Fu(\()p Fp(x)p Fu(\))392 4865 y(=)28 b(\()p Fr(r)22 b(\012)h(r)p Fu(\)\(1)e Fr(\012)i Fp(\034)34 b Fr(\012)22 b Fu(1\)\(\001)g Fr(\012)h Fp(\034)11 b Fu(\()p Fp(S)1728 4829 y Fo(0)1774 4865 y Fr(\012)22 b Fp(S)1939 4829 y Fo(0)1962 4865 y Fu(\)\001\)\()p Fp(\023)2191 4880 y Fl(i)2242 4865 y Fr(\012)h Fp(\023)2376 4880 y Fl(i)2405 4865 y Fu(\)\001)2524 4880 y Fl(i)2552 4865 y Fu(\()p Fp(x)p Fu(\))392 4981 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Fu(\()p Fp(x)1938 5113 y Fn(\(3\))2033 5098 y Fu(\))392 5214 y(=)496 5139 y Fi(P)618 5214 y Fp(\023)652 5229 y Fl(i)680 5214 y Fu(\()p Fp(x)773 5229 y Fn(\(1\))868 5214 y Fu(\))p Fp(S)972 5178 y Fo(0)995 5214 y Fp(\023)1029 5229 y Fl(i)1058 5214 y Fu(\()p Fp(x)1151 5229 y Fn(\(3\))1245 5214 y Fu(\))f Fr(\012)h Fu(\(1)f Fr(\003)g Fp(S)1652 5178 y Fo(0)1675 5214 y Fu(\))p Fp(\023)1747 5229 y Fl(i)1776 5214 y Fu(\()p Fp(x)1869 5229 y Fn(\(2\))1963 5214 y Fu(\))p Fp(:)p eop %%Page: 61 31 61 30 bop 1028 236 a Fq(6.)35 b(QUANTUM)e(A)n(UTOMORPHISM)e(GR)n(OUPS) 925 b(61)0 425 y Fu(Hence)34 b(w)m(e)g(ha)m(v)m(e)552 618 y(\001\(\(1)22 b Fr(\003)g Fp(S)918 582 y Fo(0)941 618 y Fr(\000)p Fp(u")p Fu(\))p Fp(\023)1192 633 y Fl(i)1220 618 y Fu(\()p Fp(x)p Fu(\)\))941 735 y(=)1045 660 y Fi(P)1166 735 y Fp(\023)1200 750 y Fl(i)1229 735 y Fu(\()p Fp(x)1322 750 y Fn(\(1\))1417 735 y Fu(\))p Fp(S)1521 699 y Fo(0)1544 735 y Fp(\023)1578 750 y Fl(i)1606 735 y Fu(\()p Fp(x)1699 750 y Fn(\(3\))1794 735 y 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967 y Fu(\))f Fr(\000)h Fu(\001)p Fp(u"\023)2866 982 y Fl(i)2894 967 y Fu(\()p Fp(x)p Fu(\))941 1083 y(=)1045 1009 y Fi(P)1166 1083 y Fp(\023)1200 1098 y Fl(i)1229 1083 y Fu(\()p Fp(x)1322 1099 y Fn(\(1\))1417 1083 y Fu(\))p Fp(S)1521 1047 y Fo(0)1544 1083 y Fp(\023)1578 1098 y Fl(i)1606 1083 y Fu(\()p Fp(x)1699 1099 y Fn(\(3\))1794 1083 y Fu(\))f Fr(\012)h Fu(\(1)f Fr(\003)g Fp(S)2201 1047 y Fo(0)2246 1083 y Fr(\000)g Fp(u")p Fu(\))p Fp(\023)2519 1098 y Fl(i)2547 1083 y Fu(\()p Fp(x)2640 1099 y Fn(\(2\))2735 1083 y Fu(\))1136 1200 y(+)1229 1125 y Fi(P)1350 1200 y Fp(\023)1384 1215 y Fl(i)1413 1200 y Fu(\()p Fp(x)1506 1215 y Fn(\(1\))1601 1200 y Fu(\))p Fp(S)1705 1163 y Fo(0)1728 1200 y Fp(\023)1762 1215 y Fl(i)1790 1200 y Fu(\()p Fp(x)1883 1215 y Fn(\(2\))1978 1200 y Fu(\))g Fr(\012)h Fu(1)2187 1215 y Fl(B)2243 1196 y Ff(0)2292 1200 y Fr(\000)f Fp(u"\023)2527 1215 y Fl(i)2555 1200 y Fu(\()p Fp(x)p Fu(\))h Fr(\012)f Fu(1)2857 1215 y Fl(B)2913 1196 y Ff(0)941 1316 y Fu(=)1045 1241 y Fi(P)1166 1316 y Fp(\023)1200 1331 y Fl(i)1229 1316 y Fu(\()p Fp(x)1322 1331 y Fn(\(1\))1417 1316 y Fu(\))p Fp(S)1521 1280 y Fo(0)1544 1316 y Fp(\023)1578 1331 y Fl(i)1606 1316 y Fu(\()p Fp(x)1699 1331 y Fn(\(3\))1794 1316 y Fu(\))g Fr(\012)h Fu(\(1)f Fr(\003)g Fp(S)2201 1280 y Fo(0)2246 1316 y Fr(\000)g Fp(u")p Fu(\))p Fp(\023)2519 1331 y Fl(i)2547 1316 y Fu(\()p Fp(x)2640 1331 y Fn(\(2\))2735 1316 y Fu(\))1136 1432 y(+\(1)g Fr(\003)g Fp(S)1459 1396 y Fo(0)1504 1432 y Fr(\000)h Fp(u")p Fu(\))p Fp(\023)1778 1447 y Fl(i)1805 1432 y Fu(\()p Fp(x)p Fu(\))g Fr(\012)g Fu(1)2108 1447 y Fl(B)2164 1428 y Ff(0)941 1548 y Fr(2)28 b Fp(B)1114 1512 y Fo(0)1160 1548 y Fr(\012)22 b Fp(I)30 b Fu(+)22 b Fp(I)30 b Fr(\012)23 b Fp(B)1682 1512 y Fo(0)1705 1548 y Fp(:)0 1768 y Fu(Th)m(us)34 b Fp(I)40 b Fu(is)33 b(a)f(coideal)f(and)i(a)f(biideal)e(of)i Fp(B)1610 1732 y Fo(0)1634 1768 y Fu(.)149 1884 y(No)m(w)40 b(let)f Fp(H)8 b Fu(\()p Fp(B)d Fu(\))39 b(:=)g Fp(B)1030 1848 y Fo(0)1054 1884 y Fp(=I)47 b Fu(and)39 b(let)g Fp(\027)46 b Fu(:)39 b Fp(B)1776 1848 y Fo(0)1839 1884 y Fr(\000)-60 b(!)40 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))38 b(b)s(e)i(the)g(residue)g (class)f(homomor-)0 2001 y(phism.)55 b(W)-8 b(e)37 b(sho)m(w)g(that)g Fp(H)8 b Fu(\()p Fp(B)d Fu(\))36 b(is)g(a)g(bialgebra)e(and)j Fp(\027)43 b Fu(is)36 b(a)g(homomorphism)e(of)i(bialgebras.)0 2117 y Fp(H)8 b Fu(\()p Fp(B)d Fu(\))29 b(is)g(an)h(algebra)e(and)i Fp(\027)36 b Fu(is)29 b(a)g(homomorphism)e(of)i(algebras)g(since)h Fp(I)38 b Fu(is)29 b(a)g(t)m(w)m(o)i(sided)e(ideal.)0 2233 y(Since)k Fp(I)i Fr(\022)28 b Fu(Ker)q(\()p Fp(")p Fu(\))k(there)h(is)f(a)g(unique)h(factorization)1480 2455 y Fp(B)1559 2419 y Fo(0)1918 2455 y Fp(B)1997 2419 y Fo(0)2020 2455 y Fp(=I)p 1611 2432 278 4 v 1806 2430 a Fm(-)1723 2412 y Fp(\027)1646 2711 y(")1692 2674 y Fo(0)1600 2582 y Fm(@)1683 2665 y(@)1766 2748 y(@)1849 2831 y(@)1868 2849 y(@)-83 b(R)1980 2952 y Fj(K)p 2017 2849 4 351 v 2019 2849 a Fm(?)2058 2695 y Fp(")0 3138 y Fu(where)31 b Fp(")c Fu(:)h Fp(B)486 3102 y Fo(0)509 3138 y Fp(=I)35 b Fr(\000)-60 b(!)28 b Fj(K)53 b Fu(is)29 b(a)g(homomorphism)e(of)i(algebras.)41 b(Since)30 b(\001\()p Fp(I)8 b Fu(\))28 b Fr(\022)g Fp(B)2943 3102 y Fo(0)2982 3138 y Fr(\012)16 b Fp(I)23 b Fu(+)16 b Fp(I)23 b Fr(\012)16 b Fp(B)3471 3102 y Fo(0)3523 3138 y Fr(\022)0 3255 y Fu(Ker\()p Fp(\027)29 b Fr(\012)24 b Fp(\027)35 b Fu(:)30 b Fp(B)592 3219 y Fo(0)638 3255 y Fr(\012)23 b Fp(B)817 3219 y Fo(0)870 3255 y Fr(\000)-60 b(!)29 b Fp(B)1095 3219 y Fo(0)1118 3255 y Fp(=I)i Fr(\012)23 b Fp(B)1420 3219 y Fo(0)1443 3255 y Fp(=I)8 b Fu(\))33 b(and)h(th)m(us)h Fp(I)i Fr(\022)29 b Fu(Ker\(\001\()p Fp(\027)h Fr(\012)23 b Fp(\027)6 b Fu(\)\))34 b(w)m(e)g(ha)m(v)m(e)h(a)e(unique)0 3371 y(factorization)1202 4004 y Fp(B)1281 3968 y Fo(0)1326 4004 y Fr(\012)23 b Fp(B)1505 3968 y Fo(0)1882 4004 y Fp(B)1961 3968 y Fo(0)1985 4004 y Fp(=I)29 b Fr(\012)23 b Fp(B)2285 3968 y Fo(0)2308 4004 y Fp(=I)p 1557 3981 297 4 v 1771 3979 a Fm(-)1639 4039 y Fl(\027)t Fo(\012)p Fl(\027)1314 3517 y Fp(B)1393 3481 y Fo(0)2044 3517 y Fp(B)2123 3481 y Fo(0)2147 3517 y Fp(=I)p 1445 3493 571 4 v 1933 3491 a Fm(-)1710 3473 y Fl(\027)p 1364 3911 4 351 v 1365 3911 a Fm(?)1192 3749 y Fn(\001)1251 3768 y Fe(B)1299 3754 y Ff(0)p 2143 3911 V 2145 3911 a Fm(?)2184 3758 y Fn(\001)0 4212 y Fu(b)m(y)31 b(an)f(algebra)f(homomorphism)e (\001)h(:)g Fp(B)1534 4176 y Fo(0)1557 4212 y Fp(=I)35 b Fr(\000)-60 b(!)28 b Fp(B)1908 4176 y Fo(0)1932 4212 y Fp(=I)c Fr(\012)17 b Fp(B)2221 4176 y Fo(0)2245 4212 y Fp(=I)8 b Fu(.)42 b(No)m(w)31 b(it)e(is)g(easy)i(to)f(v)m(erify)g (that)0 4328 y Fp(B)79 4292 y Fo(0)102 4328 y Fp(=I)41 b Fu(b)s(ecomes)33 b(a)f(bialgebra)f(and)h Fp(\027)39 b Fu(a)33 b(bialgebra)d(homomorphism.)149 4444 y(W)-8 b(e)39 b(sho)m(w)g(that)e(the)h(map)f Fp(\027)6 b(S)1301 4408 y Fo(0)1361 4444 y Fu(:)37 b Fp(B)1504 4408 y Fo(0)1564 4444 y Fr(\000)-60 b(!)37 b Fp(B)1797 4408 y Fo(0)1820 4444 y Fp(=I)46 b Fu(can)38 b(b)s(e)g(factorized)f(through)h Fp(B)3185 4408 y Fo(0)3208 4444 y Fp(=I)45 b Fu(in)38 b(the)0 4561 y(comm)m(utativ)m(e)32 b(diagram)1478 4736 y Fp(B)1557 4700 y Fo(0)1916 4736 y Fp(B)1995 4700 y Fo(0)2018 4736 y Fp(=I)p 1609 4713 278 4 v 1804 4711 a Fm(-)1721 4693 y Fp(\027)1569 4992 y(\027)6 b(S)1689 4956 y Fo(0)1598 4863 y Fm(@)1681 4946 y(@)1764 5029 y(@)1847 5112 y(@)1866 5131 y(@)-83 b(R)1916 5224 y Fp(B)1995 5188 y Fo(0)2018 5224 y Fp(=I)p 2015 5131 4 351 v 2017 5131 a Fm(?)2056 4989 y Fp(S)p eop %%Page: 62 32 62 31 bop 0 236 a Fq(62)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fu(This)h(holds)f(if)f Fp(I)36 b Fr(\022)28 b Fu(Ker\()p Fp(\027)6 b(S)1066 389 y Fo(0)1090 425 y Fu(\).)43 b(Since)33 b(Ker\()p Fp(\027)6 b Fu(\))28 b(=)g Fp(I)40 b Fu(it)31 b(su\016ces)k(to)d(sho)m (w)i Fp(S)2811 389 y Fo(0)2834 425 y Fu(\()p Fp(I)8 b Fu(\))28 b Fr(\022)g Fp(I)8 b Fu(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)1122 597 y Fp(S)1188 561 y Fo(0)1211 597 y Fu(\(\()p Fp(S)1353 561 y Fo(0)1398 597 y Fr(\003)22 b Fu(1\))p Fp(\023)1591 612 y Fl(i)1619 597 y Fu(\()p Fp(x)p Fu(\)\))28 b(=)1122 721 y(=)f Fr(r)p Fp(\034)11 b Fu(\()p Fp(S)1465 685 y Fo(0)1489 675 y Fn(2)1528 721 y Fp(\023)1562 736 y Fl(i)1613 721 y Fr(\012)23 b Fp(S)1779 685 y Fo(0)1802 721 y Fp(\023)1836 736 y Fl(i)1865 721 y Fu(\)\001)1984 736 y Fl(i)2012 721 y Fu(\()p Fp(x)p Fu(\))1122 838 y(=)k Fr(r)p Fp(\034)11 b Fu(\()p Fp(S)1465 802 y Fo(0)1511 838 y Fr(\012)23 b Fu(1\)\()p Fp(\023)1770 853 y Fl(i)p Fn(+1)1910 838 y Fr(\012)g Fp(\023)2044 853 y Fl(i)p Fn(+1)2163 838 y Fu(\)\001)2282 853 y Fl(i)2310 838 y Fu(\()p Fp(x)p Fu(\))1122 954 y(=)k Fr(r)p Fu(\(1)22 b Fr(\012)h Fp(S)1583 918 y Fo(0)1606 954 y Fu(\)\()p Fp(\023)1716 969 y Fl(i)p Fn(+1)1857 954 y Fr(\012)f Fp(\023)1990 969 y Fl(i)p Fn(+1)2109 954 y Fu(\))p Fp(\034)11 b Fu(\001)2281 969 y Fl(i)2310 954 y Fu(\()p Fp(x)p Fu(\))1122 1070 y(=)27 b Fr(r)p Fu(\(1)22 b Fr(\012)h Fp(S)1583 1034 y Fo(0)1606 1070 y Fu(\)\()p Fp(\023)1716 1085 y Fl(i)p Fn(+1)1857 1070 y Fr(\012)f Fp(\023)1990 1085 y Fl(i)p Fn(+1)2109 1070 y Fu(\)\001)2228 1085 y Fl(i)p Fn(+1)2347 1070 y Fu(\()p Fp(x)p Fu(\))1122 1186 y(=)27 b(\(1)22 b Fr(\003)g Fp(S)1472 1150 y Fo(0)1495 1186 y Fu(\))p Fp(\023)1567 1201 y Fl(i)p Fn(+1)1686 1186 y Fu(\()p Fp(x)p Fu(\))0 1356 y(and)634 1489 y Fp(S)700 1448 y Fo(0)723 1489 y Fu(\()p Fp(u"\023)897 1504 y Fl(i)925 1489 y Fu(\()p Fp(x)p Fu(\)\))28 b(=)f Fp(S)1291 1448 y Fo(0)1314 1489 y Fu(\(1\))p Fp(")1485 1504 y Fl(i)1513 1489 y Fu(\()p Fp(x)p Fu(\))h(=)f Fp(S)1841 1448 y Fo(0)1865 1489 y Fu(\(1\))p Fp(")2036 1504 y Fl(i)p Fn(+1)2153 1489 y Fu(\()p Fp(x)p Fu(\))h(=)g Fp(S)2482 1448 y Fo(0)2505 1489 y Fu(\()p Fp(u"\023)2679 1504 y Fl(i)p Fn(+1)2797 1489 y Fu(\()p Fp(x)p Fu(\)\))0 1642 y(hence)34 b(w)m(e)g(get)782 1775 y Fp(S)848 1733 y Fo(0)871 1775 y Fu(\(\()p Fp(S)1013 1733 y Fo(0)1058 1775 y Fr(\003)22 b Fu(1)g Fr(\000)h Fp(u")p Fu(\))p Fp(\023)1475 1790 y Fl(i)1503 1775 y Fu(\()p Fp(x)p Fu(\)\))28 b(=)f(\(1)22 b Fr(\003)g Fp(S)2050 1733 y Fo(0)2095 1775 y Fr(\000)h Fp(u")p Fu(\))p Fp(\023)2369 1790 y Fl(i)p Fn(+1)2487 1775 y Fu(\()p Fp(x)p Fu(\))28 b Fr(2)g Fp(I)8 b(:)0 1928 y Fu(This)28 b(sho)m(ws)h Fp(S)559 1892 y Fo(0)582 1928 y Fu(\()p Fp(I)8 b Fu(\))27 b Fr(\022)i Fp(I)8 b Fu(.)41 b(So)28 b(there)g(is)f(a)g(unique)h (homomorphism)d(of)i(bialgebras)f Fp(S)34 b Fu(:)28 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))27 b Fr(\000)-60 b(!)0 2044 y Fp(H)8 b Fu(\()p Fp(B)d Fu(\))244 2008 y Fl(opcop)450 2044 y Fu(suc)m(h)34 b(that)e(the)h(diagram)1334 2697 y Fp(B)1413 2661 y Fo(0)1436 2650 y Fl(opcop)1848 2697 y Fp(H)8 b Fu(\()p Fp(B)d Fu(\))2092 2661 y Fl(opcop)p 1639 2674 181 4 v 1736 2672 a Fm(-)1709 2719 y Fl(\027)1421 2209 y Fp(B)1500 2173 y Fo(0)1935 2209 y Fp(H)j Fu(\()p Fp(B)d Fu(\))p 1552 2186 355 4 v 1824 2184 a Fm(-)1710 2165 y Fl(\027)p 1470 2603 4 351 v 1472 2603 a Fm(?)1364 2453 y Fl(S)1411 2430 y Ff(0)p 2055 2603 V 2057 2603 a Fm(?)2096 2451 y Fl(S)0 2848 y Fu(comm)m(utes.)149 2964 y(No)m(w)30 b(w)m(e)h(sho)m(w)f(that)f Fp(H)8 b Fu(\()p Fp(B)d Fu(\))29 b(is)g(a)g(Hopf)g(algebra)f(with)h(an)m(tip)s (o)s(de)f Fp(S)6 b Fu(.)42 b(By)30 b(Prop)s(osition)d(2.1.3)0 3081 y(it)34 b(su\016ces)j(to)e(test)g(on)g(generators)h(of)e Fp(H)8 b Fu(\()p Fp(B)d Fu(\))35 b(hence)h(on)f(images)f Fp(\027)6 b(\023)2575 3096 y Fl(i)2604 3081 y Fu(\()p Fp(x)p Fu(\))35 b(of)g(elemen)m(ts)g Fp(x)d Fr(2)h Fp(B)3545 3096 y Fl(i)3573 3081 y Fu(.)0 3197 y(W)-8 b(e)33 b(ha)m(v)m(e)485 3344 y(\(1)22 b Fr(\003)g Fp(S)6 b Fu(\))p Fp(\027)g(\023)858 3359 y Fl(i)887 3344 y Fu(\()p Fp(x)p Fu(\))28 b(=)f Fr(r)p Fu(\()p Fp(\027)i Fr(\012)22 b Fp(S)6 b(\027)g Fu(\)\001)p Fp(\023)1719 3359 y Fl(i)1749 3344 y Fu(\()p Fp(x)p Fu(\))28 b(=)f Fr(r)p Fu(\()p Fp(\027)i Fr(\012)22 b Fp(\027)6 b Fu(\)\(1)23 b Fr(\012)f Fp(S)2675 3308 y Fo(0)2698 3344 y Fu(\)\001)p Fp(\023)2851 3359 y Fl(i)2880 3344 y Fu(\()p Fp(x)p Fu(\))28 b(=)976 3460 y(=)f Fp(\027)6 b Fu(\(1)23 b Fr(\003)f Fp(S)1381 3424 y Fo(0)1404 3460 y Fu(\))p Fp(\023)1476 3475 y Fl(i)1504 3460 y Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(\027)6 b(u"\023)1957 3475 y Fl(i)1985 3460 y Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(u"\027)6 b(\023)2438 3475 y Fl(i)2466 3460 y Fu(\()p Fp(x)p Fu(\))p Fp(:)0 3635 y Fu(By)33 b(Prop)s(osition)e(2.1.3)h Fp(S)38 b Fu(is)32 b(an)h(an)m(tip)s(o)s(de)f(for)g Fp(H)8 b Fu(\()p Fp(B)d Fu(\).)149 3751 y(W)-8 b(e)44 b(pro)m(v)m(e)g(no)m(w)f (that)g Fp(H)8 b Fu(\()p Fp(B)d Fu(\))42 b(together)i(with)e Fp(\023)k Fu(:=)f Fp(\027)6 b(\023)2266 3766 y Fn(0)2352 3751 y Fu(:)45 b Fp(B)50 b Fr(\000)-60 b(!)45 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))43 b(is)f(a)h(free)g(Hopf)0 3868 y(algebra)30 b(o)m(v)m(er)j Fp(B)5 b Fu(.)43 b(Let)32 b Fp(H)39 b Fu(b)s(e)32 b(a)f(Hopf)g(algebra)g(and)g(let)g Fp(f)38 b Fu(:)28 b Fp(B)33 b Fr(\000)-60 b(!)28 b Fp(H)39 b Fu(b)s(e)31 b(a)h(homomorphism)c(of)0 3984 y(bialgebras.)42 b(W)-8 b(e)33 b(will)c(sho)m(w)34 b(that)d(there)i(is)f(a)g(unique)g (homomorphism)2756 3957 y(\026)2735 3984 y Fp(f)38 b Fu(:)28 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))27 b Fr(\000)-60 b(!)28 b Fp(H)39 b Fu(suc)m(h)0 4100 y(that)1476 4199 y Fp(B)331 b(H)8 b Fu(\()p Fp(B)d Fu(\))p 1584 4176 270 4 v 1771 4174 a Fm(-)1706 4156 y Fl(\023)1657 4435 y(f)1584 4326 y Fm(@)1667 4409 y(@)1750 4492 y(@)1833 4575 y(@)1852 4594 y(@)-83 b(R)1959 4696 y Fp(H)p 2001 4594 4 351 v 2003 4594 a Fm(?)2057 4422 y Fn(\026)2042 4440 y Fl(f)0 4815 y Fu(comm)m(utes.)149 4931 y(W)-8 b(e)33 b(de\014ne)h(a)f(family)d (of)i(homomorphisms)e(of)i(bialgebras)f Fp(f)2437 4946 y Fl(i)2493 4931 y Fu(:)d Fp(B)2622 4946 y Fl(i)2678 4931 y Fr(\000)-60 b(!)28 b Fp(H)40 b Fu(b)m(y)1387 5098 y Fp(f)1435 5113 y Fn(0)1502 5098 y Fu(:=)28 b Fp(f)5 b(;)1387 5214 y(f)1435 5229 y Fl(i)p Fn(+1)1581 5214 y Fu(:=)28 b Fp(S)1772 5229 y Fl(H)1839 5214 y Fp(f)1887 5229 y Fl(i)1915 5214 y Fp(;)17 b(i)28 b Fr(2)g Fj(N)9 b Fp(:)p eop %%Page: 63 33 63 32 bop 1028 236 a Fq(6.)35 b(QUANTUM)e(A)n(UTOMORPHISM)e(GR)n(OUPS) 925 b(63)0 425 y Fu(W)-8 b(e)32 b(ha)m(v)m(e)h(in)d(particular)g Fp(f)1000 440 y Fl(i)1056 425 y Fu(=)e Fp(S)1226 389 y Fl(i)1220 450 y(H)1287 425 y Fp(f)42 b Fu(for)31 b(all)e Fp(i)f Fr(2)g Fj(N)9 b Fu(.)49 b(Th)m(us)33 b(there)g(is)e(a)g(unique)h (homomorphism)0 541 y(of)g(bialgebras)f Fp(f)631 505 y Fo(0)682 541 y Fu(:)d Fp(B)816 505 y Fo(0)867 541 y Fu(=)970 467 y Fi(`)1081 541 y Fp(B)1155 556 y Fl(i)1211 541 y Fr(\000)-60 b(!)28 b Fp(H)40 b Fu(suc)m(h)34 b(that)e Fp(f)1967 505 y Fo(0)1990 541 y Fp(\023)2024 556 y Fl(i)2081 541 y Fu(=)27 b Fp(f)2232 556 y Fl(i)2293 541 y Fu(for)32 b(all)e Fp(i)e Fr(2)g Fj(N)9 b Fu(.)149 658 y(W)-8 b(e)33 b(sho)m(w)h(that)e Fp(f)829 622 y Fo(0)852 658 y Fu(\()p Fp(I)8 b Fu(\))28 b(=)f(0.)44 b(Let)32 b Fp(x)d Fr(2)f Fp(B)1656 673 y Fl(i)1684 658 y Fu(.)43 b(Then)756 890 y Fp(f)815 854 y Fo(0)838 890 y Fu(\(\(1)22 b Fr(\003)g Fp(S)1123 854 y Fo(0)1146 890 y Fu(\))p Fp(\023)1218 905 y Fl(i)1247 890 y Fu(\()p Fp(x)p Fu(\)\)=)28 b Fp(f)1579 854 y Fo(0)1602 890 y Fu(\()p Fr(r)p Fu(\(1)21 b Fr(\012)i Fp(S)1997 854 y Fo(0)2020 890 y Fu(\)\()p Fp(\023)2130 905 y Fl(i)2181 890 y Fr(\012)f Fp(\023)2314 905 y Fl(i)2343 890 y Fu(\)\001)2462 905 y Fl(i)2491 890 y Fu(\()p Fp(x)p Fu(\)\))1416 1006 y(=)1520 932 y Fi(P)1641 1006 y Fp(f)1700 970 y Fo(0)1723 1006 y Fp(\023)1757 1021 y Fl(i)1786 1006 y Fu(\()p Fp(x)1879 1022 y Fn(\(1\))1974 1006 y Fu(\))p Fp(f)2071 970 y Fo(0)2094 1006 y Fp(S)2160 970 y Fo(0)2183 1006 y Fp(\023)2217 1021 y Fl(i)2245 1006 y Fu(\()p Fp(x)2338 1022 y Fn(\(2\))2433 1006 y Fu(\))1416 1123 y(=)1520 1048 y Fi(P)1641 1123 y Fp(f)1700 1086 y Fo(0)1723 1123 y Fp(\023)1757 1138 y Fl(i)1786 1123 y Fu(\()p Fp(x)1879 1138 y Fn(\(1\))1974 1123 y Fu(\))p Fp(f)2071 1086 y Fo(0)2094 1123 y Fp(\023)2128 1138 y Fl(i)p Fn(+1)2247 1123 y Fu(\()p Fp(x)2340 1138 y Fn(\(2\))2434 1123 y Fu(\))1416 1239 y(=)1520 1164 y Fi(P)1641 1239 y Fp(f)1689 1254 y Fl(i)1718 1239 y Fu(\()p Fp(x)1811 1254 y Fn(\(1\))1905 1239 y Fu(\))p Fp(f)1991 1254 y Fl(i)p Fn(+1)2110 1239 y Fu(\()p Fp(x)2203 1254 y Fn(\(2\))2297 1239 y Fu(\))1416 1355 y(=)1520 1280 y Fi(P)1641 1355 y Fp(f)1689 1370 y Fl(i)1718 1355 y Fu(\()p Fp(x)1811 1371 y Fn(\(1\))1905 1355 y Fu(\))p Fp(S)6 b(f)2057 1370 y Fl(i)2085 1355 y Fu(\()p Fp(x)2178 1371 y Fn(\(2\))2273 1355 y Fu(\))1416 1471 y(=)28 b(\(1)21 b Fr(\003)h Fp(S)6 b Fu(\))p Fp(f)1852 1486 y Fl(i)1880 1471 y Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(u"f)2293 1486 y Fl(i)2320 1471 y Fu(\()p Fp(x)p Fu(\))g(=)g Fp(u")2685 1486 y Fl(i)2712 1471 y Fu(\()p Fp(x)p Fu(\))1416 1588 y(=)g Fp(f)1579 1551 y Fo(0)1602 1588 y Fu(\()p Fp(u"\023)1776 1603 y Fl(i)1803 1588 y Fu(\()p Fp(x)p Fu(\)\))p Fp(:)0 1823 y Fu(This)i(together)h (with)f(the)g(symmetric)g(statemen)m(t)g(giv)m(es)h Fp(f)2207 1787 y Fo(0)2230 1823 y Fu(\()p Fp(I)8 b Fu(\))27 b(=)h(0.)42 b(Hence)32 b(there)f(is)f(a)f(unique)0 1940 y(factorization)24 b(through)i(a)f(homomorphism)f(of)h(algebras)2178 1913 y(\026)2157 1940 y Fp(f)38 b Fu(:)28 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))27 b Fr(\000)-60 b(!)28 b Fp(H)33 b Fu(suc)m(h)28 b(that)d Fp(f)3305 1903 y Fo(0)3356 1940 y Fu(=)3481 1913 y(\026)3460 1940 y Fp(f)10 b(\027)c Fu(.)149 2056 y(The)38 b(homomorphism)1069 2030 y(\026)1047 2056 y Fp(f)46 b Fu(:)35 b Fp(H)8 b Fu(\()p Fp(B)d Fu(\))35 b Fr(\000)-60 b(!)35 b Fp(H)44 b Fu(is)37 b(a)f(homomorphism)e(of)j (bialgebras)e(since)j(the)0 2172 y(diagram)1914 2481 y Fp(B)1993 2445 y Fo(0)2016 2481 y Fp(=I)p 1510 2458 376 4 v 1803 2456 a Fm(-)1678 2504 y Fl(\027)p 1510 2361 1017 4 v 2444 2359 a Fm(-)1986 2327 y Fl(f)2027 2304 y Ff(0)p 2145 2458 383 4 v 2444 2456 a Fm(-)2330 2514 y Fn(\026)2315 2531 y Fl(f)p 1622 2848 800 4 v 2339 2846 a Fm(-)1931 2916 y Fl(f)1972 2893 y Ff(0)1994 2916 y Fo(\012)p Fl(f)2090 2893 y Ff(0)1752 2774 y Fp(B)1831 2738 y Fo(0)1854 2774 y Fp(=I)30 b Fr(\012)23 b Fp(B)2155 2738 y Fo(0)2178 2774 y Fp(=I)p 1622 2751 102 4 v 1641 2749 a Fm(-)1606 2723 y Fl(\027)t Fo(\012)p Fl(\027)p 2306 2751 116 4 v 2339 2749 a Fm(-)2310 2699 y Fn(\026)2295 2717 y Fl(f)8 b Fo(\012)2406 2699 y Fn(\026)2392 2717 y Fl(f)2556 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b(a)h(homomorphism)e(of)h(algebr)-5 b(as.)149 4689 y Fu(2.)78 b Fk(The)45 b(biline)-5 b(ar)44 b(form)i Fu(ev)j(:)f Fp(L)31 b Fr(\012)f Fp(H)56 b Fr(\000)-58 b(!)48 b Fj(K)69 b Fk(satis\014es)45 b(\(4\))g(if)h(and)f(only)g(if)h Fp( )51 b Fu(:)d Fp(H)55 b Fr(\000)-58 b(!)0 4805 y Fu(Hom)o(\()p Fp(L;)17 b Fj(K)h Fu(\))40 b Fk(is)35 b(a)g(homomorphism)d(of)j(algebr) -5 b(as.)149 4983 y Fh(Pr)n(oof.)42 b Fu(ev)30 b(:)e Fp(L)7 b Fr(\012)g Fp(H)37 b Fr(\000)-60 b(!)28 b Fj(K)48 b Fu(satis\014es)26 b(the)g(righ)m(t)f(equation)g(of)g(\(3\))f(i\013)h Fp(')p Fu(\()p Fp(ab)p Fu(\)\()p Fp(h)p Fu(\))j(=)f Fr(h)p Fp(ab;)17 b(h)p Fr(i)28 b Fu(=)0 5099 y Fr(h)p Fp(a)16 b Fr(\012)g Fp(b;)284 5025 y Fi(P)407 5099 y Fp(h)463 5115 y Fn(\(1\))573 5099 y Fr(\012)g Fp(h)722 5115 y Fn(\(2\))817 5099 y Fr(i)28 b Fu(=)987 5025 y Fi(P)1093 5099 y Fr(h)p Fp(a;)17 b(h)1283 5115 y Fn(\(1\))1377 5099 y Fr(ih)p Fp(b;)g(h)1596 5115 y Fn(\(2\))1690 5099 y Fr(i)27 b Fu(=)1860 5025 y Fi(P)1982 5099 y Fp(')p Fu(\()p Fp(a)p Fu(\)\()p Fp(h)2267 5115 y Fn(\(1\))2361 5099 y Fu(\))p Fp(')p Fu(\()p Fp(b)p Fu(\)\()p Fp(h)2674 5115 y Fn(\(2\))2769 5099 y Fu(\))g(=)h(\()p Fp(')p Fu(\()p Fp(a)p Fu(\))16 b Fr(\003)g Fp(')p Fu(\()p Fp(b)p Fu(\)\)\()p Fp(h)p Fu(\))0 5216 y(b)m(y)33 b(the)g(de\014nition)f(of)g(the)h (algebra)f(structure)h(on)g(Hom)o(\()p Fp(H)r(;)17 b Fj(K)g Fu(\).)p eop %%Page: 66 36 66 35 bop 0 236 a Fq(66)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fu(ev)e(:)e Fp(L)23 b Fr(\012)f Fp(H)35 b Fr(\000)-60 b(!)28 b Fj(K)56 b Fu(satis\014es)33 b(the)g(left)f(equation)g(of)h(\(3\))f(i\013)f Fp(')p Fu(\(1\)\()p Fp(h)p Fu(\))d(=)f Fr(h)p Fu(1)p Fp(;)17 b(h)p Fr(i)27 b Fu(=)g Fp(")p Fu(\()p Fp(h)p Fu(\).)149 541 y(The)34 b(second)g(part)e(of)g(the)h(Lemma)e(follo)m (ws)h(b)m(y)h(symmetry)-8 b(.)p 3529 541 4 66 v 3533 479 59 4 v 3533 541 V 3591 541 4 66 v 149 717 a Fs(Example)37 b(2.7.5.)49 b Fu(There)44 b(is)d(a)h(w)m(eak)i(dualit)m(y)d(b)s(et)m(w) m(een)j(the)f(quan)m(tum)f(groups)h Fj(S)o(L)3437 732 y Fl(q)3475 717 y Fu(\(2\))0 833 y(and)33 b Fp(U)256 848 y Fl(q)294 833 y Fu(\()p Fp(sl)r Fu(\(2\)\).)43 b(\(Kassel:)h (Chapter)33 b(VI)s(I.4\).)149 1009 y Fs(Prop)s(osition)j(2.7.6.)49 b Fk(L)-5 b(et)33 b Fu(ev)d(:)e Fp(L)17 b Fr(\012)g Fp(H)36 b Fr(\000)-58 b(!)28 b Fj(K)56 b Fk(b)-5 b(e)32 b(a)g(we)-5 b(ak)32 b(duality)h(of)f(Hopf)h(algebr)-5 b(as.)43 b(L)-5 b(et)0 1125 y Fp(I)41 b Fu(:=)33 b(Ker\()p Fp(')g Fu(:)h Fp(L)f Fr(\000)-58 b(!)34 b Fu(Hom)o(\()p Fp(H)r(;)17 b Fj(K)g Fu(\)\))44 b Fk(and)37 b Fp(J)43 b Fu(:=)33 b(Ker\()p Fp( )k Fu(:)c Fp(H)41 b Fr(\000)-58 b(!)34 b Fu(Hom)o(\()p Fp(L;)17 b Fj(K)h Fu(\)\))p Fk(.)59 b(L)-5 b(et)p 3198 1045 67 4 v 38 w Fp(L)34 b Fu(:=)f Fp(L=I)0 1247 y Fk(and)p 200 1167 89 4 v 45 w Fp(H)55 b Fu(:=)48 b Fp(H)r(=J)9 b Fk(.)77 b(Then)p 1053 1167 67 4 v 44 w Fp(L)46 b Fk(and)p 1366 1167 89 4 v 46 w Fp(H)53 b Fk(ar)-5 b(e)45 b(Hopf)h(algebr)-5 b(as)44 b(and)h(the)h(induc)-5 b(e)g(d)45 b(biline)-5 b(ar)44 b(form)p 0 1313 97 4 v 0 1368 a Fu(ev)30 b(:)p 179 1288 67 4 v 28 w Fp(L)22 b Fr(\012)p 367 1288 89 4 v 23 w Fp(H)35 b Fr(\000)-58 b(!)28 b Fj(K)59 b Fk(is)34 b(a)h(duality.)149 1544 y Fh(Pr)n(oof.)42 b Fu(First)48 b(observ)m(e)i(that)e Fp(I)57 b Fu(and)48 b Fp(J)58 b Fu(are)48 b(t)m(w)m(o)h(sided)g(ideals)e(hence) p 3013 1464 67 4 v 50 w Fp(L)i Fu(and)p 3333 1464 89 4 v 48 w Fp(H)56 b Fu(are)0 1660 y(algebras.)51 b(Then)36 b(ev)e(:)e Fp(L)24 b Fr(\012)h Fp(H)39 b Fr(\000)-60 b(!)32 b Fj(K)59 b Fu(can)35 b(b)s(e)h(factored)f(through)p 2511 1605 97 4 v 35 w(ev)f(:)p 2698 1580 67 4 v 32 w Fp(L)25 b Fr(\012)p 2890 1580 89 4 v 24 w Fp(H)40 b Fr(\000)-60 b(!)32 b Fj(K)58 b Fu(and)36 b(the)0 1776 y(equations)d(\(3\))f(and)h (\(4\))f(are)g(still)f(satis\014ed)i(for)f(the)h(residue)g(classes.)149 1892 y(The)f(ideals)d Fp(I)37 b Fu(and)31 b Fp(J)39 b Fu(are)30 b(biideals.)41 b(In)30 b(fact,)h(let)e Fp(x)g Fr(2)f Fp(I)38 b Fu(then)30 b Fr(h)p Fu(\001\()p Fp(x)p Fu(\))p Fp(;)17 b(a)h Fr(\012)f Fp(b)p Fr(i)28 b Fu(=)g Fr(h)p Fp(x;)17 b(ab)p Fr(i)28 b Fu(=)f(0)0 2009 y(hence)43 b(\001\()p Fp(x)p Fu(\))h Fr(2)g Fu(Ker\()p Fp(')28 b Fr(\012)h Fp(')43 b Fu(:)g Fp(L)29 b Fr(\012)g Fp(L)44 b Fr(\000)-60 b(!)43 b Fu(Hom)o(\()p Fp(H)36 b Fr(\012)29 b Fp(H)r(;)17 b Fj(K)g Fu(\))49 b(=)43 b Fp(I)37 b Fr(\012)29 b Fp(L)f Fu(+)g Fp(L)h Fr(\012)g Fp(I)50 b Fu(\(the)42 b(last)0 2125 y(equalit)m(y)33 b(is)g(an)h(easy)h(exercise)g(in)d (linear)g(algebra\))h(and)h Fp(")p Fu(\()p Fp(x)p Fu(\))29 b(=)g Fr(h)p Fp(x;)17 b Fu(1)p Fr(i)29 b Fu(=)h(0.)46 b(Hence)35 b(as)f(in)f(the)0 2246 y(pro)s(of)39 b(of)h(Theorem)g(2.6.3) f(w)m(e)j(get)e(that)p 1580 2166 67 4 v 39 w Fp(L)h Fu(=)f Fp(L=I)48 b Fu(and)p 2206 2166 89 4 v 40 w Fp(H)g Fu(=)40 b Fp(H)r(=J)49 b Fu(are)40 b(bialgebras.)65 b(Since)0 2367 y Fr(h)p Fp(S)6 b Fu(\()p Fp(x)p Fu(\))p Fp(;)17 b(a)p Fr(i)47 b Fu(=)h Fr(h)p Fp(x;)17 b(S)6 b Fu(\()p Fp(a)p Fu(\))p Fr(i)48 b Fu(=)g(0)c(w)m(e)h(ha)m(v)m(e)h(an)f(induced)g (homomorphism)p 2791 2287 66 4 v 41 w Fp(S)54 b Fu(:)p 2980 2287 67 4 v 48 w Fp(L)48 b Fr(\000)-60 b(!)p 3259 2287 V 48 w Fp(L)q Fu(.)79 b(The)0 2488 y(iden)m(tities)33 b(satis\014ed)h(in)f Fp(L)i Fu(hold)e(also)g(for)g(the)i(residue)f (classes)h(in)p 2508 2408 V 33 w Fp(L)f Fu(so)h(that)e Fp(L)i Fu(and)f(similarly)p 0 2525 89 4 v 0 2605 a Fp(H)42 b Fu(b)s(ecome)34 b(Hopf)f(algebras.)48 b(Finally)31 b(w)m(e)k(ha)m(v)m(e)h(b)m(y)f(de\014nition)e(of)g Fp(I)42 b Fu(that)34 b Fr(h)p 2862 2550 56 4 v Fp(x;)p 2961 2550 52 4 v 17 w(a)p Fr(i)c Fu(=)g Fr(h)p Fp(x;)17 b(a)p Fr(i)30 b Fu(=)g(0)0 2726 y(for)35 b(all)f Fp(a)g Fr(2)f Fp(H)44 b Fu(i\013)35 b Fp(a)e Fr(2)h Fp(I)44 b Fu(or)p 1113 2671 V 35 w Fp(a)34 b Fu(=)f(0.)53 b(Th)m(us)37 b(the)g(bilinear)c (form)p 2447 2671 97 4 v 35 w(ev)i(:)p 2637 2646 67 4 v 34 w Fp(L)25 b Fr(\012)p 2829 2646 89 4 v 24 w Fp(H)41 b Fr(\000)-60 b(!)34 b Fp(K)43 b Fu(de\014nes)37 b(a)0 2842 y(dualit)m(y)-8 b(.)p 3529 2842 4 66 v 3533 2780 59 4 v 3533 2842 V 3591 2842 4 66 v 149 3018 a Fs(Problem)36 b(2.7.8.)49 b Fu(\(in)32 b(Linear)g(Algebra\))149 3156 y(1.)42 b(F)-8 b(or)34 b Fp(U)42 b Fr(\022)33 b Fp(V)56 b Fu(de\014ne)36 b Fp(U)1135 3120 y Fo(?)1227 3156 y Fu(:=)c Fr(f)p Fp(f)42 b Fr(2)32 b Fp(V)1679 3120 y Fo(\003)1718 3156 y Fr(j)p Fp(f)11 b Fu(\()p Fp(U)f Fu(\))32 b(=)f(0)p Fr(g)p Fu(.)51 b(F)-8 b(or)34 b Fp(Z)k Fr(\022)33 b Fp(V)2743 3120 y Fo(\003)2818 3156 y Fu(de\014ne)j Fp(Z)3176 3120 y Fo(?)3267 3156 y Fu(:=)31 b Fr(f)p Fp(v)36 b Fr(2)267 3272 y Fp(V)21 b Fr(j)p Fp(Z)7 b Fu(\()p Fp(v)t Fu(\))27 b(=)h(0)p Fr(g)p Fu(.)43 b(Sho)m(w)33 b(that)g(the)g(follo)m(wing)c (hold:)343 3388 y(\(a\))41 b Fp(U)d Fr(\022)28 b Fp(V)54 b Fu(=)-17 b Fr(\))33 b Fp(U)38 b Fu(=)28 b Fp(U)1305 3352 y Fo(??)1419 3388 y Fu(;)337 3504 y(\(b\))42 b Fp(Z)34 b Fr(\022)28 b Fp(V)794 3468 y Fo(\003)866 3504 y Fu(and)33 b(dim)15 b Fp(Z)34 b(<)28 b Fr(1)k Fu(=)-17 b Fr(\))33 b Fp(Z)h Fu(=)28 b Fp(Z)2043 3468 y Fo(??)2157 3504 y Fu(;)348 3621 y(\(c\))42 b Fr(f)p Fp(U)61 b Fr(\022)51 b Fp(V)22 b Fr(j)17 b Fu(dim)d Fp(V)d(=U)61 b(<)51 b Fr(1g)1687 3593 y(\030)1688 3625 y Fu(=)1815 3621 y Fr(f)p Fp(Z)57 b Fr(\022)52 b Fp(V)2196 3584 y Fo(\003)2236 3621 y Fr(j)17 b Fu(dim)d Fp(Z)58 b(<)50 b Fr(1g)c Fu(under)h(the)f (maps)509 3737 y Fp(U)38 b Fr(7!)27 b Fp(U)816 3701 y Fo(?)909 3737 y Fu(and)32 b Fp(Z)j Fr(7!)27 b Fp(Z)1401 3701 y Fo(?)1460 3737 y Fu(.)149 3853 y(2.)42 b(Let)d Fp(V)59 b Fu(=)678 3778 y Fi(L)789 3805 y Fo(1)789 3882 y Fl(i)p Fn(=1)924 3853 y Fj(K)17 b Fp(x)1051 3868 y Fl(i)1123 3853 y Fu(b)s(e)39 b(an)g(in\014nite-dimensional)34 b(v)m(ector)40 b(space.)63 b(Find)38 b(an)g(elemen)m(t)267 3969 y Fp(g)31 b Fr(2)d Fu(\()p Fp(V)43 b Fr(\012)23 b Fp(V)f Fu(\))794 3933 y Fo(\003)866 3969 y Fu(that)32 b(is)g(not)g(in)g Fp(V)1541 3933 y Fo(\003)1602 3969 y Fr(\012)23 b Fp(V)1781 3933 y Fo(\003)1918 3969 y Fu(\()p Fr(\022)28 b Fu(\()p Fp(V)43 b Fr(\012)23 b Fp(V)e Fu(\))2415 3933 y Fo(\003)2455 3969 y Fu(\).)149 4145 y Fs(De\014nition)37 b(2.7.7.)49 b Fu(Let)34 b Fp(A)f Fu(b)s(e)h(an)f(algebra.)45 b(W)-8 b(e)34 b(de\014ne)h Fp(A)2442 4109 y Fl(o)2509 4145 y Fu(:=)30 b Fr(f)p Fp(f)39 b Fr(2)30 b Fp(A)2948 4109 y Fo(\003)2988 4145 y Fr(j9)i Fu(ideal)3336 4160 y Fl(A)3393 4145 y Fp(I)3436 4160 y Fl(A)3523 4145 y Fr(\022)0 4261 y Fp(A)c Fu(:)g(dim)n(\()p Fp(A=I)8 b Fu(\))27 b Fp(<)h Fr(1)k Fu(and)h Fp(f)11 b Fu(\()p Fp(I)d Fu(\))27 b(=)g(0)p Fr(g)p Fu(.)149 4437 y Fs(Lemma)38 b(2.7.8.)49 b Fk(L)-5 b(et)35 b Fp(A)g Fk(b)-5 b(e)35 b(an)f(algebr)-5 b(a)34 b(and)g Fp(f)39 b Fr(2)28 b Fp(A)2165 4401 y Fo(\003)2204 4437 y Fk(.)45 b(The)34 b(fol)5 b(lowing)34 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)149 4575 y Fu(1.)42 b Fp(f)c Fr(2)28 b Fp(A)520 4539 y Fl(o)559 4575 y Fk(;)149 4691 y Fu(2.)42 b Fk(ther)-5 b(e)35 b(exists)f Fp(I)819 4706 y Fl(A)904 4691 y Fr(\022)28 b Fp(A)35 b Fk(such)g(that)g Fu(dim)15 b Fp(A=I)36 b(<)27 b Fr(1)35 b Fk(and)f Fp(f)11 b Fu(\()p Fp(I)d Fu(\))27 b(=)h(0)p Fk(;)149 4807 y Fu(3.)42 b Fp(A)22 b Fr(\001)g Fp(f)38 b Fr(\022)603 4822 y Fl(A)677 4807 y Fu(Hom)880 4822 y Fc(K)936 4807 y Fu(\()p Fp(:A)1074 4822 y Fl(A)1131 4807 y Fp(;)17 b(:)p Fj(K)h Fu(\))40 b Fk(is)35 b(\014nite)g(dimensional;)149 4924 y Fu(4.)42 b Fp(A)22 b Fr(\001)g Fp(f)33 b Fr(\001)22 b Fp(A)35 b Fk(is)f(\014nite)h(dimensional;)149 5040 y Fu(5.)42 b Fr(r)350 5004 y Fo(\003)389 5040 y Fu(\()p Fp(f)11 b Fu(\))28 b Fr(2)g Fp(A)719 5004 y Fo(\003)780 5040 y Fr(\012)23 b Fp(A)953 5004 y Fo(\003)993 5040 y Fk(.)149 5216 y Fh(Pr)n(oof.)42 b Fu(1.)76 b(=)-17 b Fr(\))65 b Fu(2.)43 b(and)33 b(4.)76 b(=)-17 b Fr(\))65 b Fu(3.)43 b(are)33 b(trivial.)p eop %%Page: 67 37 67 36 bop 1155 236 a Fq(7.)35 b(DUALITY)d(OF)h(HOPF)g(ALGEBRAS)1052 b(67)149 425 y Fu(2.)79 b(=)-17 b Fr(\))66 b Fu(3.)46 b(Let)34 b Fp(I)870 440 y Fl(A)956 425 y Fr(\022)c Fp(A)k Fu(with)f Fp(f)11 b Fu(\()p Fp(I)d Fu(\))29 b(=)g(0)k(and)h(dim)15 b Fp(A=I)37 b(<)29 b Fr(1)p Fu(.)46 b(W)-8 b(rite)33 b Fp(A)2990 389 y Fo(\003)3053 425 y Fr(\012)23 b Fp(A)29 b Fr(\000)-60 b(!)30 b Fj(K)57 b Fu(as)0 541 y Fr(h)p Fp(g)t(;)17 b(a)p Fr(i)p Fu(.)42 b(Then)34 b Fr(h)p Fp(af)5 b(;)17 b(i)p Fr(i)27 b Fu(=)h Fr(h)p Fp(f)5 b(;)17 b(ia)p Fr(i)28 b Fu(=)f(0)33 b(hence)h Fp(Af)k Fr(\032)28 b Fp(I)1996 505 y Fo(?)2088 541 y Fu(and)k(dim)15 b Fp(Af)39 b(<)28 b Fr(1)p Fu(.)149 658 y(3.)77 b(=)-17 b Fr(\))65 b Fu(2.)44 b(Let)33 b(dim)15 b Fp(Af)39 b(<)28 b Fr(1)p Fu(.)44 b(Then)34 b Fp(I)1733 673 y Fl(A)1818 658 y Fu(:=)28 b(\()p Fp(Af)11 b Fu(\))2157 622 y Fo(?)2248 658 y Fu(is)33 b(an)f(ideal)g(of)g(\014nite)g(co)s(dimension)0 774 y(in)g Fp(A)g Fu(and)h Fp(f)11 b Fu(\()p Fp(I)d Fu(\))27 b(=)h(0)k(holds.)149 890 y(2.)112 b(=)-17 b Fr(\))76 b Fu(1.)j(Let)44 b Fp(I)956 905 y Fl(A)1061 890 y Fr(\032)k Fp(A)d Fu(with)f(dim)15 b Fp(A=I)1882 905 y Fl(A)1986 890 y Fp(<)48 b Fr(1)c Fu(and)g Fp(f)11 b Fu(\()p Fp(I)d Fu(\))48 b(=)f(0)d(b)s(e)h(giv)m(en.) 79 b(Then)0 1006 y(righ)m(t)37 b(m)m(ultiplication)d(induces)k Fp(')f Fu(:)g Fp(A)g Fr(\000)-60 b(!)37 b Fu(Hom)1857 1021 y Fc(K)1913 1006 y Fu(\()p Fp(A=I)8 b(:;)17 b(A=I)8 b(:)p Fu(\))37 b(and)h(dim)15 b(End)3019 1021 y Fc(K)3075 1006 y Fu(\()p Fp(A=I)8 b Fu(\))37 b Fp(<)f Fr(1)p Fu(.)0 1123 y(Th)m(us)41 b Fp(J)48 b Fu(=)39 b(Ker\()p Fp(')p Fu(\))f Fr(\022)i Fp(A)f Fu(is)g(a)g(t)m(w)m(o)h(sided)f(ideal)f(of)g (\014nite)h(co)s(dimension)f(and)h Fp(J)48 b Fr(\032)40 b Fp(I)47 b Fu(\(since)0 1239 y Fp(')p Fu(\()p Fp(j)6 b Fu(\)\()224 1218 y(\026)224 1239 y(1)o(\))28 b(=)g(0)f(=)622 1218 y(\026)622 1239 y(1)22 b Fr(\001)f Fp(j)34 b Fu(=)918 1216 y(\026)920 1239 y Fp(j)k Fu(implies)30 b Fp(j)k Fr(2)28 b Fp(I)8 b Fu(\).)43 b(F)-8 b(urthermore)32 b(w)m(e)i(ha)m(v)m (e)g Fp(f)11 b Fu(\()p Fp(J)e Fu(\))27 b Fr(\022)h Fp(f)11 b Fu(\()p Fp(I)d Fu(\))27 b(=)h(0.)149 1355 y(1.)76 b(=)-17 b Fr(\))65 b Fu(4.)43 b Fr(h)p Fp(af)11 b(b;)17 b(i)p Fr(i)28 b Fu(=)f Fr(h)p Fp(f)5 b(;)17 b(bia)p Fr(i)29 b Fu(=)e(0)32 b(implies)f Fp(A)22 b Fr(\001)g Fp(f)32 b Fr(\001)22 b Fp(A)28 b Fr(\022)2407 1380 y Fl(A)2466 1355 y Fp(I)2517 1319 y Fo(?)2509 1380 y Fl(A)2609 1355 y Fu(hence)34 b(dim)15 b Fp(Af)c(A)27 b(<)h Fr(1)p Fu(.)149 1471 y(3.)97 b(=)-17 b Fr(\))72 b Fu(5.)65 b(W)-8 b(e)40 b(observ)m(e)h(that)e Fr(r)1524 1435 y Fo(\003)1564 1471 y Fu(\()p Fp(f)11 b Fu(\))39 b(=)h Fp(f)11 b Fr(r)39 b(2)h Fu(\()p Fp(A)27 b Fr(\012)h Fp(A)p Fu(\))2495 1435 y Fo(\003)2534 1471 y Fu(.)65 b(W)-8 b(e)40 b(w)m(an)m(t)g(to)f(sho)m (w)i(that)0 1588 y Fr(r)83 1551 y Fo(\003)122 1588 y Fu(\()p Fp(f)11 b Fu(\))29 b Fr(2)g Fp(A)454 1551 y Fo(\003)516 1588 y Fr(\012)23 b Fp(A)689 1551 y Fo(\003)728 1588 y Fu(.)45 b(Let)33 b Fp(g)1022 1603 y Fn(1)1061 1588 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(g)1344 1603 y Fl(n)1423 1588 y Fu(b)s(e)34 b(a)e(basis)h(of)g Fp(Af)11 b Fu(.)45 b(Then)34 b(there)g(exist)f Fp(h)2986 1603 y Fn(1)3026 1588 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(h)3317 1603 y Fl(n)3392 1588 y Fr(2)29 b Fp(A)3560 1551 y Fo(\003)0 1704 y Fu(suc)m(h)39 b(that)f Fp(bf)48 b Fu(=)692 1629 y Fi(P)814 1704 y Fp(h)870 1719 y Fl(i)898 1704 y Fu(\()p Fp(b)p Fu(\))p Fp(g)1062 1719 y Fl(i)1090 1704 y Fu(.)60 b(Let)38 b Fp(a;)17 b(b)37 b Fr(2)h Fp(A)p Fu(.)60 b(Then)39 b Fr(hr)2176 1668 y Fo(\003)2215 1704 y Fu(\()p Fp(f)11 b Fu(\))p Fp(;)17 b(a)25 b Fr(\012)h Fp(b)p Fr(i)38 b Fu(=)e Fr(h)p Fp(f)5 b(;)17 b(ab)p Fr(i)37 b Fu(=)g Fr(h)p Fp(bf)5 b(;)17 b(a)p Fr(i)37 b Fu(=)0 1745 y Fi(P)122 1820 y Fp(h)178 1835 y Fl(i)206 1820 y Fu(\()p Fp(b)p Fu(\))p Fp(g)370 1835 y Fl(i)398 1820 y Fu(\()p Fp(a)p Fu(\))28 b(=)g Fr(h)696 1745 y Fi(P)817 1820 y Fp(g)864 1835 y Fl(i)914 1820 y Fr(\012)23 b Fp(h)1070 1835 y Fl(i)1098 1820 y Fp(;)17 b(a)22 b Fr(\012)h Fp(b)p Fr(i)33 b Fu(so)f(that)h Fr(r)1842 1784 y Fo(\003)1881 1820 y Fu(\()p Fp(f)11 b Fu(\))27 b(=)2147 1745 y Fi(P)2269 1820 y Fp(g)2316 1835 y Fl(i)2366 1820 y Fr(\012)22 b Fp(h)2521 1835 y Fl(i)2577 1820 y Fr(2)28 b Fp(A)2744 1784 y Fo(\003)2806 1820 y Fr(\012)23 b Fp(A)2979 1784 y Fo(\003)3018 1820 y Fu(.)149 1936 y(5.)77 b(=)-17 b Fr(\))66 b Fu(3.)44 b(Let)33 b Fr(r)905 1900 y Fo(\003)945 1936 y Fu(\()p Fp(f)11 b Fu(\))28 b(=)1212 1861 y Fi(P)1334 1936 y Fp(g)1381 1951 y Fl(i)1431 1936 y Fr(\012)23 b Fp(h)1587 1951 y Fl(i)1644 1936 y Fr(2)29 b Fp(A)1812 1900 y Fo(\003)1874 1936 y Fr(\012)22 b Fp(A)2046 1900 y Fo(\003)2086 1936 y Fu(.)45 b(Then)34 b Fp(bf)39 b Fu(=)2646 1861 y Fi(P)2767 1936 y Fp(h)2823 1951 y Fl(i)2852 1936 y Fu(\()p Fp(b)p Fu(\))p Fp(g)3016 1951 y Fl(i)3077 1936 y Fu(for)32 b(all)f Fp(b)e Fr(2)g Fp(A)0 2052 y Fu(as)k(b)s(efore.)43 b(Th)m(us)34 b Fp(Af)44 b Fu(is)32 b(generated)h(b)m(y)h(the)f Fp(g)1755 2067 y Fn(1)1794 2052 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(g)2076 2067 y Fl(n)2123 2052 y Fu(.)p 3529 2052 4 66 v 3533 1990 59 4 v 3533 2052 V 3591 2052 4 66 v 149 2227 a Fs(Prop)s(osition)36 b(2.7.9.)49 b Fk(L)-5 b(et)26 b Fu(\()p Fp(A;)17 b(m;)g(u)p Fu(\))24 b Fk(b)-5 b(e)24 b(an)h(algebr)-5 b(a.)41 b(Then)24 b(we)g(have)h Fp(m)2910 2191 y Fo(\003)2949 2227 y Fu(\()p Fp(A)3060 2191 y Fl(o)3099 2227 y Fu(\))i Fr(\022)i Fp(A)3343 2191 y Fl(o)3381 2227 y Fr(\012)p Fp(A)3531 2191 y Fl(o)3570 2227 y Fk(.)0 2343 y(F)-7 b(urthermor)i(e)34 b Fu(\()p Fp(A)674 2307 y Fl(o)712 2343 y Fp(;)17 b Fu(\001)p Fp(;)g(")p Fu(\))34 b Fk(is)h(a)g(c)-5 b(o)g(algebr)g(a)33 b(with)i Fu(\001)28 b(=)g Fp(m)2113 2307 y Fo(\003)2187 2343 y Fk(and)35 b Fp(")27 b Fu(=)g Fp(u)2609 2307 y Fo(\003)2648 2343 y Fk(.)149 2517 y Fh(Pr)n(oof.)42 b Fu(Let)33 b Fp(f)38 b Fr(2)28 b Fp(A)962 2481 y Fl(o)1033 2517 y Fu(and)33 b(let)e Fp(g)1410 2532 y Fn(1)1449 2517 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(g)1732 2532 y Fl(n)1811 2517 y Fu(b)s(e)32 b(a)g(basis)g(for)g Fp(Af)11 b Fu(.)44 b(Then)33 b(w)m(e)g(ha)m(v)m(e)h Fp(m)3322 2481 y Fo(\003)3362 2517 y Fu(\()p Fp(f)11 b Fu(\))27 b(=)0 2559 y Fi(P)122 2634 y Fp(g)169 2649 y Fl(i)199 2634 y Fr(\012)s Fp(h)335 2649 y Fl(i)387 2634 y Fu(for)22 b(suitable)g Fp(h)936 2649 y Fl(i)992 2634 y Fr(2)29 b Fp(A)1160 2597 y Fo(\003)1222 2634 y Fu(as)23 b(in)g(the)g(pro)s(of)f(of)h(the)g(previous)h(prop)s (osition.)38 b(Since)23 b Fp(g)3318 2649 y Fl(i)3374 2634 y Fr(2)28 b Fp(Af)0 2750 y Fu(w)m(e)33 b(get)e Fp(Ag)424 2765 y Fl(i)480 2750 y Fr(\022)d Fp(Af)43 b Fu(and)32 b(dim)n(\()p Fp(Ag)1258 2765 y Fl(i)1286 2750 y Fu(\))c Fp(<)f Fr(1)32 b Fu(and)f(hence)i Fp(g)2092 2765 y Fl(i)2148 2750 y Fr(2)28 b Fp(A)2315 2714 y Fl(o)2353 2750 y Fu(.)44 b(Cho)s(ose)32 b Fp(a)2813 2765 y Fn(1)2853 2750 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(a)3139 2765 y Fl(n)3214 2750 y Fr(2)28 b Fp(A)j Fu(suc)m(h)0 2866 y(that)i Fp(g)259 2881 y Fl(i)287 2866 y Fu(\()p Fp(a)376 2881 y Fl(j)412 2866 y Fu(\))c(=)f Fp(\016)626 2881 y Fl(ij)687 2866 y Fu(.)44 b(Then)34 b(\()p Fp(f)11 b(a)1161 2881 y Fl(j)1198 2866 y Fu(\)\()p Fp(a)p Fu(\))28 b(=)g Fp(f)11 b Fu(\()p Fp(a)1643 2881 y Fl(j)1680 2866 y Fp(a)p Fu(\))28 b(=)h Fr(h)p Fp(m)2026 2830 y Fo(\003)2065 2866 y Fu(\()p Fp(f)11 b Fu(\))p Fp(;)17 b(a)2295 2881 y Fl(j)2354 2866 y Fr(\012)22 b Fp(a)p Fr(i)29 b Fu(=)2676 2791 y Fi(P)2798 2866 y Fp(g)2845 2881 y Fl(i)2873 2866 y Fu(\()p Fp(a)2962 2881 y Fl(j)2998 2866 y Fu(\))p Fp(h)3092 2881 y Fl(i)3121 2866 y Fu(\()p Fp(a)p Fu(\))f(=)g Fp(h)3436 2881 y Fl(j)3473 2866 y Fu(\()p Fp(a)p Fu(\))0 2982 y(implies)34 b Fp(f)11 b(a)445 2997 y Fl(j)516 2982 y Fu(=)34 b Fp(h)682 2997 y Fl(j)752 2982 y Fr(2)h Fp(f)11 b(A)p Fu(.)55 b(Observ)m(e)38 b(that)e(dim)o(\()p Fp(f)11 b(A)p Fu(\))33 b Fp(<)h Fr(1)i Fu(hence)i(dim)o(\()p Fp(h)2842 2997 y Fl(j)2878 2982 y Fp(A)p Fu(\))d Fp(<)f Fr(1)p Fu(,)j(so)f(that)0 3099 y Fp(h)56 3114 y Fl(j)120 3099 y Fr(2)28 b Fp(A)287 3062 y Fl(o)326 3099 y Fu(.)43 b(This)33 b(pro)m(v)m(es)h Fp(m)1005 3062 y Fo(\003)1045 3099 y Fu(\()p Fp(f)11 b Fu(\))27 b Fr(2)h Fp(A)1374 3062 y Fl(o)1435 3099 y Fr(\012)23 b Fp(A)1608 3062 y Fl(o)1646 3099 y Fu(.)149 3215 y(One)33 b(c)m(hec)m(ks)j(easily)c(that)g(counit)g(la)m(w)g(and)h (coasso)s(ciativit)m(y)e(hold.)p 3529 3215 V 3533 3152 59 4 v 3533 3215 V 3591 3215 4 66 v 149 3389 a Fs(Theorem)38 b(2.7.10.)49 b(\(The)57 b(Sw)m(eedler)f(dual:\))76 b Fk(L)-5 b(et)51 b Fu(\()p Fp(B)5 b(;)17 b(m;)g(u;)g Fu(\001)p Fp(;)g(")p Fu(\))49 b Fk(b)-5 b(e)50 b(a)g(bialgebr)-5 b(a.)0 3505 y(Then)48 b Fu(\()p Fp(B)385 3469 y Fl(o)424 3505 y Fp(;)17 b Fu(\001)549 3469 y Fo(\003)588 3505 y Fp(;)g(")678 3469 y Fo(\003)717 3505 y Fp(;)g(m)846 3469 y Fo(\003)885 3505 y Fp(;)g(u)985 3469 y Fo(\003)1024 3505 y Fu(\))49 b Fk(again)f(is)h(a)g(bialgebr)-5 b(a.)86 b(If)48 b Fp(B)59 b Fu(=)54 b Fp(H)i Fk(is)49 b(a)g(Hopf)g(algebr)-5 b(a)48 b(with)0 3622 y(antip)-5 b(o)g(de)34 b Fp(S)6 b Fk(,)35 b(then)f Fp(S)799 3585 y Fo(\003)873 3622 y Fk(is)h(an)f(antip)-5 b(o)g(de)35 b(for)f Fp(B)1738 3585 y Fl(o)1804 3622 y Fu(=)28 b Fp(H)1997 3585 y Fl(o)2035 3622 y Fk(.)149 3796 y Fh(Pr)n(oof.)42 b Fu(W)-8 b(e)35 b(kno)m(w)h(that)e(\()p Fp(B)1291 3760 y Fo(\003)1331 3796 y Fp(;)17 b Fu(\001)1456 3760 y Fo(\003)1495 3796 y Fp(;)g(")1585 3760 y Fo(\003)1624 3796 y Fu(\))34 b(is)g(an)g (algebra)g(and)g(that)h(\()p Fp(B)2799 3760 y Fl(o)2837 3796 y Fp(;)17 b(m)2966 3760 y Fo(\003)3005 3796 y Fp(;)g(u)3105 3760 y Fo(\003)3144 3796 y Fu(\))34 b(is)g(a)g(coal-)0 3912 y(gebra.)52 b(W)-8 b(e)36 b(sho)m(w)g(no)m(w)g(that)g Fp(B)1227 3876 y Fl(o)1298 3912 y Fr(\022)c Fp(B)1486 3876 y Fo(\003)1561 3912 y Fu(is)j(a)g(subalgebra.)52 b(Let)36 b Fp(f)5 b(;)17 b(g)36 b Fr(2)d Fp(B)2817 3876 y Fl(o)2890 3912 y Fu(with)i(dim)o(\()p Fp(B)5 b(f)11 b Fu(\))32 b Fp(<)0 4028 y Fr(1)52 b Fu(and)g(dim)o(\()p Fp(B)5 b(g)t Fu(\))60 b Fp(<)h Fr(1)p Fu(.)102 b(Let)53 b Fp(a)61 b Fr(2)h Fp(B)5 b Fu(.)103 b(Then)53 b(w)m(e)g(ha)m(v)m(e)h (\()p Fp(a)p Fu(\()p Fp(f)11 b(g)t Fu(\)\)\()p Fp(b)p Fu(\))60 b(=)i(\()p Fp(f)11 b(g)t Fu(\)\()p Fp(ba)p Fu(\))60 b(=)0 4070 y Fi(P)122 4145 y Fp(f)11 b Fu(\()p Fp(b)260 4160 y Fn(\(1\))354 4145 y Fp(a)405 4160 y Fn(\(1\))500 4145 y Fu(\))p Fp(g)t Fu(\()p Fp(b)668 4160 y Fn(\(2\))762 4145 y Fp(a)813 4160 y Fn(\(2\))907 4145 y Fu(\))69 b(=)1160 4070 y Fi(P)1265 4145 y Fu(\()p Fp(a)1354 4160 y Fn(\(1\))1448 4145 y Fp(f)11 b Fu(\)\()p Fp(b)1624 4160 y Fn(\(1\))1719 4145 y Fu(\)\()p Fp(a)1846 4160 y Fn(\(2\))1940 4145 y Fp(g)t Fu(\)\()p Fp(b)2108 4160 y Fn(\(2\))2202 4145 y Fu(\))69 b(=)2454 4070 y Fi(P)2560 4145 y Fu(\(\()p Fp(a)2687 4160 y Fn(\(1\))2781 4145 y Fp(f)11 b Fu(\)\()p Fp(a)2967 4160 y Fn(\(2\))3061 4145 y Fp(g)t Fu(\)\)\()p Fp(b)p Fu(\))57 b(hence)0 4261 y Fp(a)p Fu(\()p Fp(f)11 b(g)t Fu(\))82 b(=)478 4186 y Fi(P)583 4261 y Fu(\()p Fp(a)672 4276 y Fn(\(1\))767 4261 y Fp(f)11 b Fu(\)\()p Fp(a)953 4276 y Fn(\(2\))1047 4261 y Fp(g)t Fu(\))83 b Fr(2)g Fu(\()p Fp(B)5 b(f)11 b Fu(\)\()p Fp(B)5 b(g)t Fu(\).)140 b(Since)66 b(dim)n(\()p Fp(B)5 b(f)11 b Fu(\)\()p Fp(B)5 b(g)t Fu(\))83 b Fp(<)g Fr(1)64 b Fu(w)m(e)j(ha)m(v)m(e)0 4377 y(dim)o(\()p Fp(B)5 b Fu(\()p Fp(f)11 b(g)t Fu(\)\))49 b Fp(<)i Fr(1)45 b Fu(so)h(that)g Fp(f)11 b(g)54 b Fr(2)d Fp(B)1539 4341 y Fl(o)1577 4377 y Fu(.)84 b(F)-8 b(urthermore)45 b(w)m(e)i(ha)m(v)m(e)h Fp(")i Fr(2)h Fp(B)2954 4341 y Fl(o)2992 4377 y Fu(,)f(since)c(Ker\()p Fp(")p Fu(\))0 4493 y(has)32 b(co)s(dimension)f(1.)43 b(Th)m(us)33 b Fp(B)1173 4457 y Fl(o)1239 4493 y Fr(\022)28 b Fp(B)1423 4457 y Fo(\003)1495 4493 y Fu(is)j(a)h(subalgebra.)43 b(It)32 b(is)g(no)m(w)g(easy)h(to)f(see)h(that)e Fp(B)3383 4457 y Fl(o)3454 4493 y Fu(is)g(a)0 4610 y(bialgebra.)149 4726 y(No)m(w)39 b(let)d Fp(S)44 b Fu(b)s(e)38 b(the)f(an)m(tip)s(o)s (de)g(of)g Fp(H)8 b Fu(.)58 b(W)-8 b(e)38 b(sho)m(w)h Fp(S)2121 4690 y Fo(\003)2160 4726 y Fu(\()p Fp(H)2287 4690 y Fl(o)2325 4726 y Fu(\))d Fr(\022)h Fp(H)2602 4690 y Fl(o)2639 4726 y Fu(.)59 b(Let)37 b Fp(a)g Fr(2)f Fp(H)8 b Fu(,)39 b Fp(f)47 b Fr(2)36 b Fp(H)3535 4690 y Fl(o)3573 4726 y Fu(.)0 4842 y(Then)52 b Fr(h)p Fp(aS)429 4806 y Fo(\003)468 4842 y Fu(\()p Fp(f)11 b Fu(\))p Fp(;)17 b(b)p Fr(i)58 b Fu(=)g Fr(h)p Fp(S)1024 4806 y Fo(\003)1063 4842 y Fu(\()p Fp(f)11 b Fu(\))p Fp(;)17 b(ba)p Fr(i)58 b Fu(=)g Fr(h)p Fp(f)5 b(;)17 b(S)6 b Fu(\()p Fp(ba)p Fu(\))p Fr(i)59 b Fu(=)f Fr(h)p Fp(f)5 b(;)17 b(S)6 b Fu(\()p Fp(a)p Fu(\))p Fp(S)g Fu(\()p Fp(b)p Fu(\))p Fr(i)58 b Fu(=)g Fr(h)p Fp(f)11 b(S)6 b Fu(\()p Fp(a)p Fu(\))p Fp(;)17 b(S)6 b Fu(\()p Fp(b)p Fu(\))p Fr(i)57 b Fu(=)0 4958 y Fr(h)p Fp(S)105 4922 y Fo(\003)144 4958 y Fu(\()p Fp(f)11 b(S)6 b Fu(\()p Fp(a)p Fu(\)\))p Fp(;)17 b(b)p Fr(i)p Fu(.)95 b(This)50 b(implies)d Fp(aS)1423 4922 y Fo(\003)1463 4958 y Fu(\()p Fp(f)11 b Fu(\))56 b(=)h Fp(S)1853 4922 y Fo(\003)1893 4958 y Fu(\()p Fp(f)11 b(S)6 b Fu(\()p Fp(a)p Fu(\)\))49 b(and)h Fp(H)8 b(S)2632 4922 y Fo(\003)2671 4958 y Fu(\()p Fp(f)j Fu(\))56 b(=)h Fp(S)3061 4922 y Fo(\003)3101 4958 y Fu(\()p Fp(f)11 b(S)6 b Fu(\()p Fp(H)i Fu(\)\))56 b Fr(\022)0 5074 y Fp(S)66 5038 y Fo(\003)105 5074 y Fu(\()p Fp(f)11 b(H)d Fu(\).)90 b(Since)48 b Fp(f)65 b Fr(2)55 b Fp(H)1039 5038 y Fl(o)1125 5074 y Fu(w)m(e)50 b(get)e(dim)n(\()p Fp(f)11 b(H)d Fu(\))54 b Fp(<)g Fr(1)48 b Fu(so)g(that)h(dim)n(\()p Fp(S)2810 5038 y Fo(\003)2849 5074 y Fu(\()p Fp(f)11 b(H)d Fu(\)\))54 b Fp(<)g Fr(1)48 b Fu(and)0 5191 y(dim)o(\()p Fp(H)8 b(S)356 5155 y Fo(\003)394 5191 y Fu(\()p Fp(f)j Fu(\)\))27 b Fp(<)h Fr(1)p Fu(.)43 b(This)31 b(sho)m(ws)i Fp(S)1434 5155 y Fo(\003)1473 5191 y Fu(\()p Fp(f)11 b Fu(\))27 b Fr(2)h Fp(H)1818 5155 y Fl(o)1856 5191 y Fu(.)43 b(The)32 b(rest)g(of)f(the)g(pro)s(of)g(is)f(no)m(w)i(trivial.) p 3529 5191 V 3533 5128 59 4 v 3533 5191 V 3591 5191 4 66 v eop %%Page: 68 38 68 37 bop 0 236 a Fq(68)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,) h(F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fs(De\014nition)37 b(2.7.11.)49 b Fu(Let)35 b Fp(G)c Fu(=)g Fj(K)17 b Fu(-)6 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b Fu(-)o(\))34 b(b)s(e)h(an)g(a\016ne)g (group)g(and)f Fp(R)f Fr(2)e Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(.)0 541 y(W)-8 b(e)28 b(de\014ne)h Fp(G)12 b Fr(\012)606 556 y Fc(K)673 541 y Fp(R)29 b Fu(:=)f Fp(G)p Fr(j)1012 576 y Fl(R)p Fu(-)o Fs(cAlg)1350 541 y Fu(to)g(b)s(e)f(the)h (restriction)f(to)g(comm)m(utativ)m(e)f Fp(R)q Fu(-algebras.)41 b(The)0 673 y(functor)33 b Fp(G)22 b Fr(\012)515 688 y Fc(K)593 673 y Fp(R)34 b Fu(is)e(represen)m(ted)j(b)m(y)e Fp(H)d Fr(\012)23 b Fp(R)29 b Fr(2)f Fp(R)q Fu(-)o Fs(cAlg)o Fu(:)664 830 y Fp(G)p Fr(j)769 865 y Fl(R)p Fu(-)p Fs(cAlg)1081 830 y Fu(\()p Fp(A)p Fu(\))f(=)h Fj(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b(A)p Fu(\))1996 802 y Fr(\030)1997 834 y Fu(=)2101 830 y Fp(R)q Fu(-)p Fs(cAlg)o Fu(\()p Fp(H)29 b Fr(\012)23 b Fp(R)q(;)17 b(A)p Fu(\))p Fp(:)149 1025 y Fs(Theorem)38 b(2.7.12.)49 b(\(The)36 b(Cartier)f(dual:\))44 b Fk(L)-5 b(et)34 b Fp(H)41 b Fk(b)-5 b(e)34 b(a)f(\014nite)h (dimensional)e(c)-5 b(ommu-)0 1142 y(tative)43 b(c)-5 b(o)g(c)g(ommutative)42 b(Hopf)g(algebr)-5 b(a.)67 b(L)-5 b(et)43 b Fp(G)f Fu(=)g Fj(K)17 b Fk(-)6 b Fs(cAlg)o Fu(\()p Fp(H)r(;)17 b Fk(-)p Fu(\))42 b Fk(b)-5 b(e)42 b(the)h(asso)-5 b(ciate)g(d)42 b(a\016ne)0 1258 y(gr)-5 b(oup)35 b(and)f(let)h Fp(D)s Fu(\()p Fp(G)p Fu(\))27 b(:=)h Fj(K)17 b Fk(-)6 b Fs(cAlg)o Fu(\()p Fp(H)1450 1222 y Fo(\003)1489 1258 y Fp(;)17 b Fk(-)o Fu(\))35 b Fk(b)-5 b(e)35 b(the)g(dual)f(gr)-5 b(oup.)45 b(Then)34 b(we)g(have)1390 1415 y Fp(D)s Fu(\()p Fp(G)p Fu(\))27 b(=)h Fr(G)6 b Fp(r)s Fu(\()p Fp(G;)17 b(G)2106 1430 y Fl(m)2172 1415 y Fu(\))0 1572 y Fk(wher)-5 b(e)39 b Fr(G)6 b Fp(r)s Fu(\()p Fp(G;)17 b(G)628 1587 y Fl(m)694 1572 y Fu(\)\()p Fp(R)q Fu(\))38 b(=)f(Gr)o(\()p Fp(G)26 b Fr(\012)1366 1587 y Fc(K)1449 1572 y Fp(R)q(;)17 b(G)1645 1587 y Fl(m)1737 1572 y Fr(\012)1814 1587 y Fc(K)1896 1572 y Fp(R)q Fu(\))40 b Fk(is)g(the)g(set)g(of)g(gr)-5 b(oup)40 b(\(-functor\))g(homo-)0 1688 y(morphisms)33 b(and)i Fp(G)761 1703 y Fl(m)862 1688 y Fk(is)g(the)g(multiplic)-5 b(ative)34 b(gr)-5 b(oup.)149 1862 y Fh(Pr)n(oof.)42 b Fu(W)-8 b(e)23 b(ha)m(v)m(e)g Fr(G)6 b Fp(r)r Fu(\()p Fp(G;)17 b(G)1253 1877 y Fl(m)1320 1862 y Fu(\)\()p Fp(R)q Fu(\))27 b(=)h(Gr)o(\()p Fp(G)p Fr(\012)1946 1877 y Fc(K)2002 1862 y Fp(R)q(;)17 b(G)2198 1877 y Fl(M)2277 1862 y Fr(\012)2354 1877 y Fc(K)2410 1862 y Fp(R)q Fu(\))2551 1835 y Fr(\030)2552 1866 y Fu(=)2656 1862 y Fp(R)q Fu(-Hopf-Alg)n(\()p Fj(K)h Fu([)p Fp(t;)f(t)3402 1826 y Fo(\000)p Fn(1)3502 1862 y Fu(])p Fr(\012)0 1979 y Fp(R)q(;)g(H)h Fr(\012)11 b Fp(R)q Fu(\))447 1951 y Fr(\030)448 1983 y Fu(=)552 1979 y Fp(R)q Fu(-)o(Hopf-Alg)o(\()p Fp(R)q Fu([)p Fp(t;)17 b(t)1301 1942 y Fo(\000)p Fn(1)1395 1979 y Fu(])p Fp(;)g(H)h Fr(\012)11 b Fp(R)q Fu(\))1794 1951 y Fr(\030)1795 1983 y Fu(=)1899 1979 y Fr(f)p Fp(x)28 b Fr(2)g Fp(U)10 b Fu(\()p Fp(H)19 b Fr(\012)11 b Fp(R)q Fu(\))p Fr(j)p Fu(\001\()p Fp(x)p Fu(\))28 b(=)g Fp(x)11 b Fr(\012)g Fp(x;)17 b(")p Fu(\()p Fp(x)p Fu(\))28 b(=)f(1)p Fr(g)p Fu(,)0 2095 y(since)33 b(\001\()p Fp(x)p Fu(\))28 b(=)g Fp(x)22 b Fr(\012)h Fp(x)33 b Fu(and)g Fp(")p Fu(\()p Fp(x)p Fu(\))27 b(=)h(1)k(imply)f Fp(xS)6 b Fu(\()p Fp(x)p Fu(\))28 b(=)g Fp(")p Fu(\()p Fp(x)p Fu(\))f(=)h(1.)149 2211 y(Consider)37 b Fp(x)e Fr(2)g Fu(Hom)954 2226 y Fl(R)1012 2211 y Fu(\(\()p Fp(H)d Fr(\012)26 b Fp(R)q Fu(\))1417 2175 y Fo(\003)1456 2211 y Fp(;)17 b(R)q Fu(\))34 b(=)g(Hom)1961 2226 y Fl(R)2018 2211 y Fu(\()p Fp(H)2145 2175 y Fo(\003)2209 2211 y Fr(\012)25 b Fp(R)q(;)17 b(R)q Fu(\).)56 b(Then)37 b(\001\()p Fp(x)p Fu(\))e(=)f Fp(x)26 b Fr(\012)f Fp(x)37 b Fu(i\013)0 2327 y Fp(x)p Fu(\()p Fp(v)144 2291 y Fo(\003)184 2327 y Fp(w)257 2291 y Fo(\003)295 2327 y Fu(\))44 b(=)g Fr(h)p Fp(x;)17 b(v)686 2291 y Fo(\003)725 2327 y Fp(w)798 2291 y Fo(\003)837 2327 y Fr(i)44 b Fu(=)g Fr(h)p Fu(\001\()p Fp(x)p Fu(\))p Fp(;)17 b(v)1386 2291 y Fo(\003)1454 2327 y Fr(\012)29 b Fp(w)1633 2291 y Fo(\003)1672 2327 y Fr(i)43 b Fu(=)h Fp(x)p Fu(\()p Fp(v)2018 2291 y Fo(\003)2058 2327 y Fu(\))p Fp(x)p Fu(\()p Fp(w)2262 2291 y Fo(\003)2301 2327 y Fu(\))e(and)g Fp(")p Fu(\()p Fp(x)p Fu(\))i(=)g(1)e(i\013)f Fr(h)p Fp(x;)17 b(")p Fr(i)43 b Fu(=)h(1.)0 2443 y(Hence)34 b Fp(x)28 b Fr(2)g Fp(R)q Fu(-)p Fs(cAlg)n Fu(\(\()p Fp(H)i Fr(\012)22 b Fp(R)q Fu(\))1195 2407 y Fo(\003)1235 2443 y Fp(;)17 b(R)q Fu(\))1419 2416 y Fr(\030)1420 2448 y Fu(=)1524 2443 y Fj(K)h Fu(-)5 b Fs(cAlg)o Fu(\()p Fp(H)1983 2407 y Fo(\003)2022 2443 y Fp(;)17 b(R)q Fu(\))27 b(=)h Fp(D)s Fu(\()p Fp(G)p Fu(\)\()p Fp(R)q Fu(\).)p 3529 2443 4 66 v 3533 2381 59 4 v 3533 2443 V 3591 2443 4 66 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF