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b(that)e(gr)16 b Ft(A)672 591 y Fm(\030)673 622 y Fu(=)777 618 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000.)-129 779 y(\(3\))41 b(Generation)e(in)g(degree)i(one:)58 b(Decide)40 b(whic)m(h)g(Hopf)g(algebras)f Ft(A)h Fu(are)g(generated)g(b)m(y)h (group-lik)m(e)e(and)37 895 y(sk)m(ew-primitiv)m(e)32 b(elemen)m(ts,)h(that)f(is)g(gr)16 b Ft(A)33 b Fu(is)f(generated)h(in)f (degree)i(one.)-180 1033 y(W)-8 b(e)34 b(conjecture)h(that)f(all)e (\014nite-dimensional)f(p)s(oin)m(ted)i(Hopf)h(algebras)f(o)m(v)m(er)i (an)f(algebraically)d(closed)j(\014eld)-180 1150 y(of)e(c)m (haracteristic)g(0)h(are)f(indeed)h(generated)h(b)m(y)f(group-lik)m(e)e (and)i(sk)m(ew-primitiv)m(e)f(elemen)m(ts.)-80 1310 y(In)38 b(the)g(sequel,)h(w)m(e)g(describ)s(e)f(the)g(steps)h(of)f(this)f (program)f(in)h(detail)f(and)i(explain)e(the)i(p)s(ositiv)m(e)f (results)-180 1426 y(obtained)e(so)g(far)g(in)g(this)g(direction.)50 b(It)36 b(is)f(not)g(our)g(in)m(ten)m(tion)g(to)g(giv)m(e)g(a)g 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y Ft(;)17 b(:)g(:)g(:)e(;)i(x)2378 2227 y Fs(\022)2450 2212 y Fu(is)32 b(a)h(basis)f(of)g Ft(V)5 b(:)-180 2390 y Fu(Moreo)m(v)m(er)36 b(there)f(are)f(elemen)m (ts)g Ft(g)1112 2405 y Fn(1)1151 2390 y Ft(;)17 b(:)g(:)g(:)f(;)h(g) 1417 2405 y Fs(\022)1486 2390 y Fm(2)31 b Fu(\000,)j(and)h(c)m (haracters)g Ft(\037)2423 2405 y Fn(1)2463 2390 y Ft(;)17 b(:)g(:)g(:)e(;)i(\037)2742 2405 y Fs(\022)2812 2390 y Fm(2)2911 2365 y Fl(b)2908 2390 y Fu(\000)34 b(suc)m(h)i(that)e Ft(q)3481 2405 y Fs(ij)3572 2390 y Fu(=)c Ft(\037)3739 2405 y Fs(j)3776 2390 y Fu(\()p Ft(g)3861 2405 y Fs(i)3889 2390 y Fu(\))p Ft(:)-180 2506 y Fu(The)g(group)f(acts)h(on)f Ft(x)675 2521 y Fs(i)733 2506 y Fu(via)g(the)h(c)m(haracter)g Ft(\037)1538 2521 y Fs(i)1566 2506 y Fu(,)g(and)f Ft(x)1864 2521 y Fs(i)1922 2506 y Fu(is)g(a)g Ft(g)2142 2521 y Fs(i)2170 2506 y Fu(-homogeneous)g(elemen)m(t)g(with)g(resp)s(ect)h(to) f(the)-180 2623 y(coaction)j(of)g(\000.)43 b(W)-8 b(e)33 b(in)m(tro)s(duced)g(braidings)e(of)h(Cartan)h(t)m(yp)s(e)g([AS2)q(])f (where)475 2783 y Ft(q)518 2798 y Fs(ij)579 2783 y Ft(q)622 2798 y Fs(j)t(i)711 2783 y Fu(=)27 b Ft(q)861 2729 y Fs(a)898 2739 y Fi(ij)857 2809 y Fs(ii)957 2783 y Ft(;)17 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)c Ft(\022)s(;)49 b Fu(and)33 b(\()p Ft(a)1842 2798 y Fs(ij)1902 2783 y Fu(\))g(is)f(a)g(generalized)g(Cartan)h(matrix.)-180 2944 y(If)j(\()p Ft(a)10 2959 y Fs(ij)70 2944 y Fu(\))g(is)g(a)f (Cartan)h(matrix)e(of)i(\014nite)f(t)m(yp)s(e,)j(then)e(the)h(algebras) e Fj(B)p Fu(\()p Ft(V)21 b Fu(\))36 b(can)g(b)s(e)g(understo)s(o)s(d)h (as)f(t)m(wisting)-180 3060 y(of)c(the)h(F)-8 b(rob)s(enius-Lusztig)31 b(k)m(ernels)j Fj(u)1269 3024 y Fk(\025)p Fn(0)1363 3060 y Fu(\()p Fj(g)p Fu(\),)f Fj(g)f Fu(a)h(semisimple)d(Lie)i(algebra.)-80 3177 y(By)f(deforming)f(the)i(quan)m(tum)g(Serre)g(relations)e(for)g (simple)g(ro)s(ots)h(whic)m(h)h(lie)e(in)h(t)m(w)m(o)h(di\013eren)m(t)f (connected)-180 3293 y(comp)s(onen)m(ts)38 b(of)f(the)h(Dynkin)g (diagram,)f(w)m(e)i(de\014ne)g(\014nite-dimensional)34 b(p)s(oin)m(ted)j(Hopf)h(algebras)f Fj(u)p Fu(\()p Fm(D)s Fu(\))g(in)-180 3409 y(terms)32 b(of)f(a)g("linking)f(datum)h Fm(D)j Fu(of)d(\014nite)g(Cartan)h(t)m(yp)s(e\\)g([AS4)q(].)43 b(They)33 b(generalize)e(the)h(F)-8 b(rob)s(enius-Lusztig)-180 3525 y(k)m(ernels)34 b Fj(u)p Fu(\()p Fj(g)p Fu(\))e(and)g(are)h (liftings)d(of)i Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000.)-80 3642 y(In)32 b(some)f(cases)j(linking)29 b(data)j(of)f(\014nite)h(Cartan)f(t)m(yp)s(e)i(are)f(general)f(enough)i (to)e(obtain)g(complete)g(classi\014-)-180 3758 y(cation)h(results.)-80 3874 y(F)-8 b(or)34 b(example,)i(if)e(\000)e(=)g(\()p Fp(Z)p Ft(=)p Fu(\()p Ft(p)p Fu(\)\))1122 3838 y Fs(s)1155 3874 y Fu(,)k Ft(p)g Fu(a)f(prime)f Ft(>)e Fu(17)i(and)i Ft(s)c Fm(\025)g Fu(1,)k(w)m(e)g(ha)m(v)m(e)h(determined)e(the)h (structure)g(of)-180 3990 y(all)30 b(\014nite-dimensional)g(Hopf)i (algebras)g Ft(A)h Fu(with)f Ft(G)p Fu(\()p Ft(A)p Fu(\))c Fm(')g Fu(\000.)43 b(They)34 b(are)f(all)d(of)i(the)h(form)f Fj(u)p Fu(\()p Fm(D)s Fu(\))f([AS4].)-80 4106 y(Similar)d(data)j(allo)m (w)f(a)i(classi\014cation)e(of)h(in\014nite-dimensional)d(p)s(oin)m (ted)j(Hopf)h(algebras)f Ft(A)g Fu(with)h(ab)s(elian)-180 4223 y(group)41 b Ft(G)p Fu(\()p Ft(A)p Fu(\),)j(without)e(zero)g (divisors)f(and)g(\014nite)h(Gelfand-Kirillo)m(v)36 b(dimension,)43 b(in)e(the)h(case)h(when)f(the)-180 4339 y(in\014nitesimal)29 b(braiding)i(is)h("p)s(ositiv)m(e\\)f([AS5].)43 b(But)33 b(the)g(general)f(case)h(is)f(more)f(in)m(v)m(olv)m(ed.)44 b(W)-8 b(e)33 b(also)e(ha)m(v)m(e)j(to)-180 4455 y(deform)e(the)h(ro)s (ot)f(v)m(ector)h(relations)e(of)h(the)h Fj(u)p Fu(\()p Fj(g)p Fu(\))1670 4419 y Fk(0)1693 4455 y Ft(s)p Fu(.)-80 4616 y(The)44 b(structure)g(of)e(p)s(oin)m(ted)h(Hopf)g(algebras)f Ft(A)h Fu(with)f Fo(non-ab)-5 b(elian)41 b Fu(group)i Ft(G)p Fu(\()p Ft(A)p Fu(\))g(is)f(widely)h(unkno)m(wn.)-180 4732 y(One)34 b(basic)g(op)s(en)g(problem)f(is)g(to)h(decide)g(whic)m (h)h(\014nite)e(groups)h(app)s(ear)g(as)g(groups)h(of)e(group-lik)m(e)g (elemen)m(ts)-180 4848 y(of)38 b(\014nite-dimensional)d(p)s(oin)m(ted)i (Hopf)h(algebras)g(whic)m(h)g(are)h(link-indecomp)s(osable)c(in)i(the)i (sense)h(of)d([M2)q(].)-180 4964 y(In)h(our)g(form)m(ulation,)f(this)h (problem)f(is)h(the)g(main)f(part)h(of)f(follo)m(wing)f(question:)55 b(giv)m(en)38 b(a)g(\014nite)g(group)g(\000,)-180 5081 y(determine)30 b(all)e(Y)-8 b(etter-Drinfeld)28 b(mo)s(dules)h Ft(V)51 b Fu(o)m(v)m(er)31 b(\000)f(suc)m(h)i(that)d Fj(B)p Fu(\()p Ft(V)22 b Fu(\))30 b(is)f(\014nite)h(dimensional.)40 b(On)30 b(the)h(one)p eop %%Page: 4 4 4 3 bop -180 0 a Fq(4)1285 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER) -180 203 y Fu(hand,)g(there)h(are)f(a)f(n)m(um)m(b)s(er)h(of)g(sev)m (ere)i(constrain)m(ts)e(on)f Ft(V)54 b Fu([G)s(~)-51 b(n3)n(].)44 b(See)32 b(also)f(the)h(exp)s(osition)f(in)g([A,)h (5.3.10].)-180 319 y(On)k(the)g(other)f(hand,)i(it)d(is)h(v)m(ery)j (hard)d(to)g(pro)m(v)m(e)i(the)f(\014niteness)h(of)e(the)h(dimension,)f (and)h(in)e(fact)i(this)f(w)m(as)-180 435 y(done)43 b(only)f(for)g(a)g (few)h(examples)f([MiS],)j([FK],)g([FP])d(whic)m(h)h(are)g(again)e (related)h(to)g(ro)s(ot)f(systems.)75 b(The)-180 552 y(examples)34 b(o)m(v)m(er)h(the)f(symmetric)f(groups)h(in)f([FK])g(w)m (ere)i(in)m(tro)s(duced)f(to)g(describ)s(e)g(the)h(cohomology)d(ring)g (of)-180 668 y(the)i(\015ag)e(v)-5 b(ariet)m(y)d(.)45 b(A)m(t)34 b(this)e(stage,)i(the)f(main)f(di\016cult)m(y)h(is)g(to)f (decide)i(when)g(certain)f(Nic)m(hols)f(algebras)h(o)m(v)m(er)-180 784 y(non-ab)s(elian)d(groups,)j(for)f(example)g(the)h(symmetric)f (groups)h Fp(S)2240 799 y Fs(n)2281 784 y Fu(,)g(are)g (\014nite-dimensional.)-80 951 y(The)44 b(last)f(Chapter)i(can)f(serv)m (e)h(as)f(a)g(concrete)h(illustration)40 b(of)j(the)h(theory)h (explained)e(in)g(this)h(pap)s(er.)-180 1068 y(W)-8 b(e)35 b(describ)s(e)g(explicitly)d(all)g(\014nite-dimensional)f(p)s(oin)m (ted)j(Hopf)g(algebras)g(with)g(ab)s(elian)e(group)j Ft(G)p Fu(\()p Ft(A)p Fu(\))f(and)-180 1184 y(in\014nitesimal)25 b(braiding)h(of)h(t)m(yp)s(e)i Ft(A)1143 1199 y Fs(n)1218 1184 y Fu(\(up)f(to)f(some)h(exceptional)f(cases\).)43 b(The)29 b(main)d(results)i(in)g(this)f(Chapter)-180 1300 y(are)39 b(new,)i(and)f(complete)e(pro)s(ofs)g(are)h(giv)m(en.)63 b(The)40 b(only)e(cases)j(whic)m(h)e(w)m(ere)h(kno)m(wn)g(b)s(efore)f (are)g(the)h(easy)-180 1416 y(case)33 b Ft(A)99 1431 y Fn(1)172 1416 y Fu([AS1],)f(and)h Ft(A)724 1431 y Fn(2)796 1416 y Fu([AS3].)-80 1533 y(The)39 b(new)g(relations)e(concern)j(the)f (ro)s(ot)e(v)m(ectors)j Ft(e)1868 1548 y Fs(i;j)1949 1533 y Ft(;)17 b Fu(1)37 b Fm(\024)h Ft(i)g(<)f(j)44 b Fm(\024)38 b Ft(n)27 b Fu(+)f(1.)60 b(The)40 b(relations)d Ft(e)3554 1496 y Fs(N)3554 1557 y(i;j)3672 1533 y Fu(=)g(0)i(in)-180 1659 y Fj(u)-128 1623 y Fk(\025)p Fn(0)-128 1684 y Fs(q)-34 1659 y Fu(\()p Ft(sl)79 1674 y Fs(n)126 1659 y Fu(\),)32 b Ft(q)37 b Fu(a)32 b(ro)s(ot)g(of)g(unit)m(y)g(of)h(order)f Ft(N)10 b Fu(,)33 b(are)g(replaced)g(b)m(y)812 1842 y Ft(e)857 1801 y Fs(N)857 1867 y(i;j)965 1842 y Fu(=)28 b Ft(u)1125 1857 y Fs(i;j)1237 1842 y Fu(for)k(a)g(family)e Ft(u)1821 1857 y Fs(i;j)1929 1842 y Fm(2)e Fp(|)-9 b Fu(\000)p Ft(;)17 b Fu(1)k Fm(\024)28 b Ft(i)g(<)g(j)33 b Fm(\024)c Ft(n)22 b Fu(+)g(1)p Ft(;)-180 2010 y Fu(dep)s(ending)33 b(on)f(a)h(family)d(of)i(free)h(parameters)f(in)g Fp(|)-9 b Fu(.)-80 2177 y(T)h(o)44 b(study)i(the)e(relations)f(b)s(et)m(w)m (een)k(a)d(\014ltered)g(ob)5 b(ject)46 b(and)e(its)g(asso)s(ciated)g (graded)h(ob)5 b(ject)45 b(is)f(a)g(basic)-180 2293 y(tec)m(hnique)26 b(in)e(mo)s(dern)g(algebra.)39 b(W)-8 b(e)25 b(w)m(ould)g(lik)m(e)f(to) g(stress)i(that)e(\014nite)h(dimensional)d(p)s(oin)m(ted)i(Hopf)g (algebras)-180 2409 y(enjo)m(y)h(a)e(remark)-5 b(able)23 b(rigidit)m(y;)i(it)d(is)i(seldom)f(the)h(case)h(that)e(one)i(is)e 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y(Braided)i(categories)g(w)m(ere)i(in)m(tro)s(duced)f(in)f([JS)q (].)53 b(W)-8 b(e)36 b(refer)g(to)f([Ka,)h(Ch.)54 b(XI,)36 b(Ch.)53 b(XI)s(I)s(I])36 b(for)f(a)h(detailed)-180 4604 y(exp)s(osition.)43 b(There)33 b(is)f(a)h(hierarc)m(h)m(y)g(of)f (categories)h(with)f(a)g(tensor)h(pro)s(duct)g(functor:)-80 4772 y(a\).)43 b(A)32 b Fo(monoidal)f Fu(or)i Fo(tensor)f Fu(category)h(is)f(a)g(collection)f(\()p Fm(C)6 b Ft(;)17 b Fm(\012)p Ft(;)g(a;)g Fp(I)-7 b Ft(;)16 b(`;)h(r)s Fu(\),)26 b(where)-54 4913 y Fm(\017)41 b(C)e Fu(is)32 b(a)g(category)h(and)g Fm(\012)28 b Fu(:)g Fm(C)g(\002)23 b(C)34 b(!)27 b(C)39 b Fu(is)32 b(a)g(functor,)-54 5081 y Fm(\017)41 b Fp(I)19 b Fu(is)32 b(an)h(ob)5 b(ject)33 b(of)f Fm(C)6 b Fu(,)33 b(and)p eop %%Page: 5 5 5 4 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1488 b(5)-54 203 y Fm(\017)41 b Ft(a)28 b Fu(:)g Ft(V)43 b Fm(\012)23 b Fu(\()p Ft(W)36 b Fm(\012)22 b Ft(U)10 b Fu(\))29 b Fm(!)e Fu(\()p Ft(V)44 b Fm(\012)22 b Ft(W)14 b Fu(\))22 b Fm(\012)g Ft(U)10 b 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Fk(\003)1537 3805 y Fu(giv)m(en)32 b(b)m(y)482 3962 y(\010\()p Ft(')23 b Fm(\012)f Ft( )t Fu(\)\()p Ft(v)k Fm(\012)d Ft(w)s Fu(\))k(=)g Ft( )t Fu(\()p Ft(v)t Fu(\))p Ft(')p Fu(\()p Ft(w)s Fu(\))p Ft(;)211 b(')27 b Fm(2)h Ft(W)2269 3921 y Fk(\003)2308 3962 y Ft(;)17 b( )32 b Fm(2)c Ft(V)2619 3921 y Fk(\003)2659 3962 y Ft(;)17 b(v)31 b Fm(2)d Ft(V)5 b(;)17 b(w)30 b Fm(2)e Ft(W)m(:)-3471 b Fu(\(1.6\))-180 4161 y FD(Remark)37 b(1.3.)49 b Fu(W)-8 b(e)44 b(see)h(that)f(a)f(Y)-8 b(etter-Drinfeld)42 b(mo)s(dule)h(is)g(a)h(braided)f(v)m(ector)i(space.)78 b(Con)m(v)m(ersely)-8 b(,)49 b(a)-180 4300 y(braided)29 b(v)m(ector)i(space)g(\()p Ft(V)5 b(;)17 b(c)p Fu(\))30 b(can)g(b)s(e)f(realized)g(as)h(a)g(Y)-8 b(etter-Drinfeld)28 b(mo)s(dule)g(o)m(v)m(er)j(some)e(Hopf)h(algebra)e Ft(H)-180 4440 y Fu(if)j(and)i(only)f(if)g Ft(c)g Fu(is)g Fo(rigid)g Fu([Tk1)q(].)44 b(If)32 b(this)g(is)g(the)h(case,)h(it)e(can)g(b)s(e)h (realized)f(in)g(man)m(y)g(di\013eren)m(t)h(w)m(a)m(ys.)-80 4616 y(W)-8 b(e)48 b(recall)f(that)h(a)g(Hopf)g(bimo)s(dule)e(o)m(v)m (er)j(a)f(Hopf)g(algebra)f Ft(H)56 b Fu(is)48 b(sim)m(ultaneously)f(a)h (bimo)s(dule)e(and)-180 4732 y(a)e(bicomo)s(dule)e(satisfying)i(all)e (p)s(ossible)i(compatibilit)m(y)d(conditions.)77 b(The)46 b(category)3179 4696 y Fs(H)3179 4757 y(H)3247 4732 y Fm(M)3367 4696 y Fs(H)3367 4757 y(H)3478 4732 y Fu(of)e(all)e(Hopf)-180 4848 y(bimo)s(dules)24 b(o)m(v)m(er)j Ft(H)33 b Fu(is)25 b(a)h(braided)f(category)-8 b(.)42 b(The)26 b(category)2104 4812 y Fs(H)2104 4873 y(H)2172 4848 y Fm(Y)8 b(D)28 b Fu(is)d(equiv)-5 b(alen)m(t,)27 b(as)f(a)f(braided)h(category)-8 b(,)27 b(to)-180 4964 y(the)e(category)f(of)g(Hopf)g(bimo)s(dules.)39 b(This)25 b(w)m(as)g(essen)m(tially)f(\014rst)h(observ)m(ed)h(in)d([W) -8 b(o])25 b(and)f(then)h(indep)s(enden)m(tly)-180 5081 y(in)32 b([AnDe,)h(App)s(endix],)g([Sbg],)g([Ro1)o(].)p eop %%Page: 7 7 7 6 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1488 b(7)-80 203 y Fu(If)32 b Ft(H)40 b Fu(is)32 b(a)g(\014nite)g (dimensional)e(Hopf)i(algebra,)f(then)i(the)g(category)2492 167 y Fs(H)2492 228 y(H)2559 203 y Fm(Y)8 b(D)35 b Fu(is)d(equiv)-5 b(alen)m(t)32 b(to)g(the)g(category)-180 319 y(of)i(mo)s(dules)g(o)m(v) m(er)i(the)g(double)e(of)h Ft(H)42 b Fu([Mj1].)51 b(The)36 b(braiding)d(in)2259 283 y Fs(H)2259 344 y(H)2326 319 y Fm(Y)8 b(D)37 b Fu(corresp)s(onds)g(to)d(the)i(braiding)d(giv)m(en) -180 435 y(b)m(y)g(the)g(\\canonical")d Ft(R)q Fu(-matrix)h(of)h(the)g (double.)44 b(In)32 b(particular,)f(if)g Ft(H)40 b Fu(is)32 b(a)g(semisimple)e(Hopf)i(algebra)f(then)-180 516 y Fs(H)-180 577 y(H)-113 552 y Fm(Y)8 b(D)36 b Fu(is)c(a)h(semisimple)d(category)-8 b(.)45 b(Indeed,)35 b(it)d(is)g(kno)m(wn)i(that)f(the)h(double)e(of)h (a)g(semisimple)d(Hopf)j(algebra)-180 668 y(is)f(again)f(semisimple.) -80 854 y(The)36 b(case)f(of)g(Y)-8 b(etter-Drinfeld)33 b(mo)s(dules)h(o)m(v)m(er)i(group)f(algebras)f(is)g(esp)s(ecially)g (imp)s(ortan)m(t)f(for)i(the)g(appli-)-180 970 y(cations)f(to)g(p)s (oin)m(ted)g(Hopf)g(algebras.)48 b(If)34 b Ft(H)k Fu(=)31 b Fp(|)-9 b Fu(\000,)29 b(where)35 b(\000)f(is)g(a)g(group,)h(then)g (an)f Ft(H)8 b Fu(-como)s(dule)32 b Ft(V)56 b Fu(is)33 b(just)-180 1086 y(a)j(\000-graded)h(v)m(ector)g(space:)53 b Ft(V)j Fu(=)34 b Fm(\010)1226 1101 y Fs(g)r Fk(2)p Fn(\000)1358 1086 y Ft(V)1415 1101 y Fs(g)1455 1086 y Fu(,)k(where)f Ft(V)1862 1101 y Fs(g)1937 1086 y Fu(=)d Fm(f)p Ft(v)k Fm(2)d Ft(V)56 b Fm(j)35 b Ft(\016)t Fu(\()p Ft(v)t Fu(\))f(=)g Ft(g)28 b Fm(\012)e Ft(v)t Fm(g)p Fu(.)55 b(W)-8 b(e)37 b(will)d(write)3751 1050 y Fn(\000)3751 1112 y(\000)3799 1086 y Fm(Y)8 b(D)-180 1204 y Fu(for)34 b(the)h(category)g(of)f(Y)-8 b(etter-Drinfeld)33 b(mo)s(dules)h(o)m(v)m (er)h Fp(|)-8 b Fu(\000,)29 b(and)35 b(sa)m(y)g(that)g Ft(V)52 b Fm(2)2887 1168 y Fn(\000)2887 1229 y(\000)2935 1204 y Fm(Y)8 b(D)37 b Fu(is)d(a)h(Y)-8 b(etter-Drinfeld)-180 1320 y(mo)s(dule)31 b(o)m(v)m(er)j(\000)e(\(when)i(the)f(\014eld)f(is)g (\014xed\).)-180 1549 y FD(Remark)37 b(1.4.)49 b Fu(Let)35 b(\000)f(b)s(e)h(a)f(group,)h Ft(V)56 b Fu(a)34 b(left)g Fp(|)-9 b Fu(\000-m)o(o)s(dul)o(e,)29 b(and)35 b(a)f(left)g Fp(|)-9 b Fu(\000-com)o(o)s(dul)o(e)29 b(with)34 b(grading)f Ft(V)53 b Fu(=)-180 1689 y Fm(\010)-103 1704 y Fs(g)r Fk(2)p Fn(\000)29 1689 y Ft(V)86 1704 y Fs(g)126 1689 y Fu(.)43 b(W)-8 b(e)33 b(de\014ne)h(a)e(linear)f(isomorphism)f Ft(c)e Fu(:)g Ft(V)43 b Fm(\012)23 b Ft(V)49 b Fm(!)28 b Ft(V)43 b Fm(\012)23 b Ft(V)54 b Fu(b)m(y)868 1898 y Ft(c)p Fu(\()p Ft(x)22 b Fm(\012)h Ft(y)t Fu(\))j(=)i Ft(g)t(y)d Fm(\012)d Ft(x;)50 b Fu(for)32 b(all)e Ft(x)e Fm(2)h Ft(V)2219 1913 y Fs(g)2259 1898 y Ft(;)49 b(g)31 b Fm(2)d Fu(\000)p Ft(;)49 b(y)31 b Fm(2)d Ft(V)5 b(:)-3086 b Fu(\(1.7\))-180 2107 y(Then)-91 2282 y(a\))41 b Ft(V)49 b Fm(2)237 2246 y Fn(\000)237 2307 y(\000)286 2282 y Fm(Y)8 b(D)34 b Fu(if)e(and)h(only)f(if)f Ft(g)t(V)1163 2297 y Fs(h)1235 2282 y Fm(\032)d Ft(V)1397 2298 y Fs(g)r(hg)1510 2280 y Fh(\000)p Fg(1)1629 2282 y Fu(for)k(all)e Ft(g)t(;)17 b(h)27 b Fm(2)h Fu(\000)p Ft(:)-97 2491 y Fu(b\))42 b(If)33 b Ft(V)49 b Fm(2)335 2455 y Fn(\000)335 2516 y(\000)383 2491 y Fm(Y)8 b(D)r Fu(,)33 b(then)g(\()p Ft(V)5 b(;)17 b(c)p Fu(\))33 b(is)f(a)g(braided)g(v)m(ector)i(space.)-86 2700 y(c\))42 b(Con)m(v)m(ersely)-8 b(,)35 b(if)d(V)i(is)e(a)h (faithful)e(\000-mo)s(dule)h(\(that)g(is,)h(if)f(for)h(all)e Ft(g)h Fm(2)d Fu(\000)p Ft(;)17 b(g)t(v)32 b Fu(=)c Ft(v)37 b Fu(for)c(all)e Ft(v)h Fm(2)d Ft(V)22 b Fu(,)33 b(implies)37 2840 y Ft(g)e Fu(=)d(1\),)k(and)h(if)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))32 b(is)g(a)h(braided)f(v)m(ector)i(space,)f(then)h Ft(V)49 b Fm(2)2435 2804 y Fn(\000)2435 2865 y(\000)2483 2840 y Fm(Y)8 b(D)r Fu(.)-180 3065 y Fo(Pr)-5 b(o)g(of.)41 b Fu(a\))32 b(is)g(clear)g(from)g(the)h(de\014nition.)-80 3204 y(By)40 b(applying)f(b)s(oth)g(sides)i(of)e(the)h(braid)f (equation)h(to)g(elemen)m(ts)g(of)f(the)h(form)f Ft(x)28 b Fm(\012)f Ft(y)j Fm(\012)e Ft(z)t(;)17 b(x)41 b Fm(2)g Ft(V)3713 3219 y Fs(g)3753 3204 y Ft(;)17 b(y)42 b Fm(2)-180 3344 y Ft(V)-123 3359 y Fs(h)-78 3344 y Ft(;)17 b(z)32 b Fm(2)c Ft(V)5 b(;)33 b Fu(it)e(is)h(easy)i(to)e(see)i(that)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))33 b(is)f(a)g(braided)g(v)m(ector)i(space)g (if)d(and)i(only)f(if)720 3553 y Ft(c)p Fu(\()p Ft(g)t(y)25 b Fm(\012)d Ft(g)t(z)t Fu(\))28 b(=)g Ft(g)t(hz)e Fm(\012)d Ft(g)t(y)t(;)47 b Fu(for)32 b(all)f Ft(g)t(;)17 b(h)27 b Fm(2)h Fu(\000)p Ft(;)49 b(y)31 b Fm(2)d Ft(V)2672 3568 y Fs(h)2716 3553 y Ft(;)50 b(z)32 b Fm(2)c Ft(V)5 b(:)-3233 b Fu(\(1.8\))-180 3762 y(Let)30 b(us)h(write)e Ft(g)t(y)h Fu(=)594 3687 y Fl(P)699 3791 y Fs(a)p Fk(2)p Fn(\000)848 3762 y Ft(x)903 3777 y Fs(a)946 3762 y Fu(,)g(where)h Ft(x)1337 3777 y Fs(a)1407 3762 y Fm(2)d Ft(V)1558 3777 y Fs(a)1629 3762 y Fu(for)i(all)e Ft(a)g Fm(2)g Fu(\000.)42 b(Then)31 b Ft(c)p Fu(\()p Ft(g)t(y)19 b Fm(\012)f Ft(g)t(z)t Fu(\))27 b(=)3026 3687 y Fl(P)3131 3791 y Fs(a)p Fk(2)p Fn(\000)3280 3762 y Ft(ag)t(z)21 b Fm(\012)c Ft(x)3597 3777 y Fs(a)3639 3762 y Ft(:)30 b Fu(Hence)-180 3902 y(\(1.8\))h(means)g(that)g Ft(ag)t(z)h Fu(=)27 b Ft(g)t(hz)t Fu(,)32 b(for)f(all)e Ft(z)j Fm(2)c Ft(V)53 b Fu(and)31 b Ft(a)d Fm(2)g Fu(\000)j(suc)m(h)i(that)e(the)h(homogeneous)f(comp)s (onen)m(t)g Ft(x)3815 3917 y Fs(a)3888 3902 y Fu(is)-180 4041 y(not)h(zero.)44 b(This)33 b(pro)m(v)m(es)h(b\))f(and)g(c\).)p 3883 4041 4 66 v 3887 3978 59 4 v 3887 4041 V 3945 4041 4 66 v -180 4270 a FD(Remark)k(1.5.)49 b Fu(If)36 b(\000)f(is)h(ab)s (elian,)e(a)i(Y)-8 b(etter-Drinfeld)34 b(mo)s(dule)g(o)m(v)m(er)j Ft(H)k Fu(=)33 b Fp(|)-9 b Fu(\000)30 b(is)35 b(nothing)g(but)h(a)f (\000-graded)-180 4410 y(\000-mo)s(dule.)-80 4549 y(Assume)d(that)g (\000)g(is)f(ab)s(elian)f(and)i(furthermore)f(that)h(the)g(action)f(of) g(\000)h(is)f(diagonalizable)e(\(this)i(is)g(alw)m(a)m(ys)-180 4689 y(the)38 b(case)g(if)e(\000)h(is)f(\014nite\).)57 b(That)38 b(is,)g Ft(V)57 b Fu(=)35 b Fm(\010)1519 4719 y Fs(\037)p Fk(2)1612 4702 y Ff(b)1610 4719 y Fn(\000)1659 4689 y Ft(V)1737 4652 y Fs(\037)1785 4689 y Fu(,)k(where)f Ft(V)2215 4652 y Fs(\037)2299 4689 y Fu(=)d Fm(f)p Ft(v)k Fm(2)d Ft(V)57 b Fm(j)35 b Ft(g)t(v)k Fu(=)c Ft(\037)p Fu(\()p Ft(g)t Fu(\))p Ft(v)h Fu(for)c(all)e Ft(g)39 b Fm(2)d Fu(\000)p Fm(g)p Fu(.)-180 4828 y(Then)1515 5037 y Ft(V)50 b Fu(=)27 b Fm(\010)1802 5068 y Fs(g)r Fk(2)p Fn(\000)p Fs(;\037)p Fk(2)2042 5051 y Ff(b)2040 5068 y Fn(\000)2088 5037 y Ft(V)2167 4996 y Fs(\037)2145 5062 y(g)2215 5037 y Ft(;)-2422 b Fu(\(1.9\))p eop %%Page: 8 8 8 7 bop -180 0 a Fq(8)1285 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER) -180 203 y Fu(where)e Ft(V)176 167 y Fs(\037)155 228 y(g)252 203 y Fu(=)d Ft(V)434 167 y Fs(\037)495 203 y Fm(\\)13 b Ft(V)631 218 y Fs(g)672 203 y Fu(.)42 b(Con)m(v)m(ersely)-8 b(,)31 b(an)m(y)e(v)m(ector)g(space)h(with)e(a)g(decomp)s(osition)e (\(1.9\))i(is)g(a)g(Y)-8 b(etter-Drinfeld)-180 342 y(mo)s(dule)31 b(o)m(v)m(er)j(\000.)43 b(The)34 b(braiding)c(is)j(giv)m(en)f(b)m(y)607 553 y Ft(c)p Fu(\()p Ft(x)22 b Fm(\012)h Ft(y)t Fu(\))k(=)g Ft(\037)p Fu(\()p Ft(g)t Fu(\))p Ft(y)e Fm(\012)e Ft(x;)49 b Fu(for)32 b(all)f Ft(x)d Fm(2)g Ft(V)2095 568 y Fs(g)2135 553 y Ft(;)49 b(g)31 b Fm(2)d Fu(\000)p Ft(;)49 b(y)31 b Fm(2)d Ft(V)2772 512 y Fs(\037)2820 553 y Ft(;)49 b(\037)28 b Fm(2)3082 528 y Fl(b)3079 553 y Fu(\000)p Ft(:)-80 760 y Fu(It)46 b(is)g(useful)g(to)g(c)m(haracterize)h(abstractly)f (those)h(braided)f(v)m(ector)h(spaces)h(whic)m(h)f(come)f(from)f(Y)-8 b(etter-)-180 876 y(Drinfeld)34 b(mo)s(dules)g(o)m(v)m(er)j(groups)e (or)g(ab)s(elian)f(groups.)53 b(The)36 b(\014rst)g(part)f(of)g(the)h (follo)m(wing)d(de\014nition)h(is)h(due)-180 992 y(to)d(M.)h(T)-8 b(ak)m(euc)m(hi.)-180 1222 y FD(De\014nition)36 b(1.6.)49 b Fu(Let)33 b(\()p Ft(V)5 b(;)17 b(c)p Fu(\))32 b(b)s(e)h(a)f(\014nite) h(dimensional)d(braided)i(v)m(ector)i(space.)-54 1397 y Fm(\017)41 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))36 b(is)g(of)g Fo(gr)-5 b(oup)38 b(typ)-5 b(e)36 b Fu(if)f(there)i(exists)g(a)f(basis) g Ft(x)1992 1412 y Fn(1)2032 1397 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2306 1412 y Fs(\022)2381 1397 y Fu(of)36 b Ft(V)58 b Fu(and)36 b(elemen)m(ts)h Ft(g)3254 1412 y Fs(i)3281 1397 y Fu(\()p Ft(x)3374 1412 y Fs(j)3411 1397 y Fu(\))d Fm(2)g Ft(V)58 b Fu(for)36 b(all)37 1536 y Ft(i;)17 b(j)38 b Fu(suc)m(h)c(that)1006 1747 y Ft(c)p Fu(\()p Ft(x)1141 1762 y Fs(i)1191 1747 y Fm(\012)23 b Ft(x)1346 1762 y Fs(j)1383 1747 y Fu(\))28 b(=)f Ft(g)1599 1762 y Fs(i)1627 1747 y Fu(\()p Ft(x)1720 1762 y Fs(j)1757 1747 y Fu(\))22 b Fm(\012)h Ft(x)1972 1762 y Fs(i)2000 1747 y Ft(;)212 b Fu(1)27 b Fm(\024)i Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(\022)s Fu(;)-2931 b(\(1.10\))37 1957 y(necessarily)33 b Ft(g)570 1972 y Fs(i)626 1957 y Fm(2)28 b Ft(GL)p Fu(\()p Ft(V)22 b Fu(\).)-54 2168 y Fm(\017)41 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))25 b(is)f(of)g Fo(\014nite)k(gr)-5 b(oup)27 b(typ)-5 b(e)25 b Fu(\(resp.)42 b(of)24 b Fo(ab)-5 b(elian)27 b(gr)-5 b(oup)28 b(typ)-5 b(e)p Fu(\))25 b(if)e(it)h(is)g(of)h(group)f(t)m(yp)s(e)i(and)f(the)g (subgroup)37 2307 y(of)32 b Ft(GL)p Fu(\()p Ft(V)22 b Fu(\))32 b(generated)i(b)m(y)f Ft(g)1105 2322 y Fn(1)1144 2307 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)1410 2322 y Fs(\022)1481 2307 y Fu(is)32 b(\014nite)h(\(resp.)44 b(ab)s(elian\).)-54 2518 y Fm(\017)d Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))32 b(is)g(of)h Fo(diagonal)g(typ)-5 b(e)33 b Fu(if)e Ft(V)54 b Fu(has)33 b(a)g(basis)f Ft(x)1843 2533 y Fn(1)1883 2518 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2157 2533 y Fs(\022)2229 2518 y Fu(suc)m(h)34 b(that)1029 2728 y Ft(c)p Fu(\()p Ft(x)1164 2743 y Fs(i)1215 2728 y Fm(\012)22 b Ft(x)1369 2743 y Fs(j)1406 2728 y Fu(\))28 b(=)g Ft(q)1619 2743 y Fs(ij)1679 2728 y Ft(x)1734 2743 y Fs(j)1793 2728 y Fm(\012)23 b Ft(x)1948 2743 y Fs(i)1977 2728 y Ft(;)211 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(;)-2908 b Fu(\(1.11\))37 2938 y(for)32 b(some)g Ft(q)473 2953 y Fs(ij)567 2938 y Fu(in)g Fp(|)-9 b Fu(.)37 b(The)d(matrix)d(\()p Ft(q)1405 2953 y Fs(ij)1466 2938 y Fu(\))h(is)g(called)g(the)h Fo(matrix)f Fu(of)g(the)h(braiding.)-54 3149 y Fm(\017)41 b Fu(If)f(\()p Ft(V)5 b(;)17 b(c)p Fu(\))40 b(is)g(of)g(diagonal)d(t)m(yp)s(e,)44 b(then)c(w)m(e)i(sa)m(y) f(that)f(it)f(is)h Fo(inde)-5 b(c)g(omp)g(osable)37 b Fu(if)i(for)h(all)e Ft(i)j Fu(and)f Ft(j)6 b Fu(,)42 b(there)37 3288 y(exists)35 b(a)e(sequence)k Ft(i)30 b Fu(=)f Ft(i)997 3303 y Fn(1)1037 3288 y Fu(,)34 b Ft(i)1131 3303 y Fn(2)1171 3288 y Fu(,)f Ft(:)17 b(:)g(:)33 b Fu(,)h Ft(i)1473 3303 y Fs(t)1533 3288 y Fu(=)c Ft(j)39 b Fu(of)33 b(elemen)m(ts)i(of)e Fm(f)p Fu(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(\022)s Fm(g)34 b Fu(suc)m(h)h(that)e Ft(q)3268 3303 y Fs(i)3292 3311 y Fi(s)3326 3303 y Fs(;i)3370 3312 y Fi(s)p Fg(+1)3484 3288 y Ft(q)3527 3303 y Fs(i)3551 3312 y Fi(s)p Fg(+1)3662 3303 y Fs(;i)3706 3311 y Fi(s)3772 3288 y Fm(6)p Fu(=)d(1,)37 3428 y(1)i Fm(\024)h Ft(s)f Fm(\024)h Ft(t)24 b Fm(\000)h Fu(1.)51 b(Otherwise,)37 b(w)m(e)f(sa)m(y)g(that)f(the)h(matrix)e(is)h (decomp)s(osable.)51 b(W)-8 b(e)36 b(can)f(also)g(refer)g(then)37 3567 y(to)d(the)h(comp)s(onen)m(ts)g(of)f(the)h(matrix.)-80 3774 y(If)38 b Ft(V)58 b Fm(2)241 3738 y Fn(\000)241 3799 y(\000)289 3774 y Fm(Y)8 b(D)41 b Fu(is)c(\014nite-dimensional)e (with)i(braiding)f Ft(c)p Fu(,)k(then)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))38 b(is)g(of)f(group)h(t)m(yp)s(e)h(b)m(y)h(\(1.4\))o(.)60 b(Con-)-180 3890 y(v)m(ersely)-8 b(,)30 b(assume)f(that)f(\()p Ft(V)5 b(;)17 b(c)p Fu(\))28 b(is)g(a)g(\014nite-dimensional)d(braided) j(v)m(ector)h(space)h(of)d(group)h(t)m(yp)s(e.)43 b(Let)29 b(\000)f(b)s(e)g(the)-180 4006 y(subgroup)j(of)e Ft(GL)p Fu(\()p Ft(V)22 b Fu(\))30 b(generated)g(b)m(y)h Ft(g)1298 4021 y Fn(1)1337 4006 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)1603 4021 y Fs(\022)1642 4006 y Fu(.)42 b(De\014ne)31 b(a)e(coaction)g(b)m (y)i Ft(\016)t Fu(\()p Ft(x)2744 4021 y Fs(i)2773 4006 y Fu(\))c(=)h Ft(g)2989 4021 y Fs(i)3034 4006 y Fm(\012)17 b Ft(x)3183 4021 y Fs(i)3241 4006 y Fu(for)30 b(all)d Ft(i)p Fu(.)43 b(Then)31 b Ft(V)-180 4123 y Fu(is)h(a)g(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)j(\000)e(with)h(braiding)d Ft(c)j Fu(b)m(y)g(Remark)f(1.4)h(,)f(c\).)-80 4310 y(A)39 b(braided)g(v)m(ector)i(space)f(of)f(diagonal)e(t)m(yp)s(e)k(is)d (clearly)h(of)g(ab)s(elian)e(group)j(t)m(yp)s(e;)k(it)38 b(is)h(of)g(\014nite)g(group)-180 4426 y(t)m(yp)s(e)33 b(if)f(the)h Ft(q)340 4441 y Fs(ij)401 4426 y Fu('s)g(are)f(ro)s(ots)h (of)f(one.)-180 4732 y(1.3.)56 b FD(Braided)37 b(Hopf)g(algebras.)-80 4848 y Fu(The)k(notion)f(of)g(\\braided)g(Hopf)g(algebra")f(is)h(one)h (of)f(the)h(basic)g(features)g(of)f(braided)g(categories.)67 b(W)-8 b(e)-180 4964 y(will)32 b(deal)i(in)g(this)g(pap)s(er)h(only)f (with)g(braided)g(Hopf)h(algebras)f(in)f(categories)i(of)f(Y)-8 b(etter-Drinfeld)33 b(mo)s(dules,)-180 5081 y(mainly)d(o)m(v)m(er)k(a)e (group)h(algebra.)p eop %%Page: 9 9 9 8 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1488 b(9)-80 203 y Fu(Let)44 b Ft(H)52 b Fu(b)s(e)45 b(a)f(Hopf)h(algebra.) 78 b(First,)47 b(the)e(tensor)g(pro)s(duct)g(in)2411 167 y Fs(H)2411 228 y(H)2479 203 y Fm(Y)8 b(D)46 b Fu(allo)m(ws)e(to)g (de\014ne)i(algebras)e(and)-180 319 y(coalgebras)39 b(in)419 283 y Fs(H)419 344 y(H)486 319 y Fm(Y)8 b(D)r Fu(.)63 b(Namely)-8 b(,)40 b(an)f(algebra)f(in)g(the)i(category)2299 283 y Fs(H)2299 344 y(H)2366 319 y Fm(Y)8 b(D)42 b Fu(is)c(an)h(asso)s (ciativ)m(e)g(algebra)f(\()p Ft(R)q(;)17 b(m)p Fu(\),)-180 435 y(where)41 b Ft(m)g Fu(:)g Ft(R)28 b Fm(\012)g Ft(R)42 b Fm(!)e Ft(R)h Fu(is)f(the)g(pro)s(duct,)i(with)e(unit)g Ft(u)g Fu(:)g Fp(|)26 b Fm(!)40 b Ft(R)q Fu(,)j(suc)m(h)e(that)f Ft(R)h Fu(is)f(a)g(Y)-8 b(etter-Drinfeld)-180 552 y(mo)s(dule)31 b(o)m(v)m(er)j Ft(H)40 b Fu(and)32 b(b)s(oth)h Ft(m)g Fu(and)f Ft(u)g Fu(are)h(morphisms)e(in)2080 516 y Fs(H)2080 577 y(H)2147 552 y Fm(Y)8 b(D)s Fu(.)-80 720 y(Similarly)-8 b(,)27 b(a)k(coalgebra)f(in)g(the)i(category)1535 684 y Fs(H)1535 745 y(H)1602 720 y Fm(Y)8 b(D)33 b Fu(is)e(a)g(coasso)s (ciativ)m(e)g(coalgebra)f(\()p Ft(R)q(;)17 b Fu(\001\),)31 b(where)i(\001)28 b(:)f Ft(R)i Fm(!)-180 836 y Ft(R)c Fm(\012)f Ft(R)35 b Fu(is)f(the)h(copro)s(duct,)h(with)e(counit)g Ft(")c Fu(:)i Ft(R)g Fm(!)e Fp(|)-8 b Fu(,)29 b(suc)m(h)36 b(that)e Ft(R)i Fu(is)e(a)g(Y)-8 b(etter-Drinfeld)33 b(mo)s(dule)g(o)m(v)m(er)i Ft(H)-180 952 y Fu(and)e(b)s(oth)f(\001)h (and)f Ft(")h Fu(are)f(morphisms)f(in)1392 916 y Fs(H)1392 978 y(H)1460 952 y Fm(Y)8 b(D)r Fu(.)-80 1121 y(Let)44 b(no)m(w)h Ft(R)q Fu(,)i Ft(S)j Fu(b)s(e)44 b(t)m(w)m(o)h(algebras)f (in)1437 1085 y Fs(H)1437 1146 y(H)1504 1121 y Fm(Y)8 b(D)s Fu(.)78 b(Then)45 b(the)g(braiding)d Ft(c)47 b Fu(:)h Ft(S)36 b Fm(\012)30 b Ft(R)49 b Fm(!)e Ft(R)31 b Fm(\012)f Ft(S)50 b Fu(allo)m(ws)43 b(to)-180 1237 y(pro)m(vide)34 b(the)h(Y)-8 b(etter-Drinfeld)33 b(mo)s(dule)f Ft(R)25 b Fm(\012)e Ft(S)40 b Fu(with)34 b(a)g("t)m(wisted")h(v)m (ersion)f(of)g(algebra)f(in)3313 1201 y Fs(H)3313 1262 y(H)3380 1237 y Fm(Y)8 b(D)r Fu(.)49 b(Namely)-8 b(,)-180 1353 y(the)33 b(pro)s(duct)g(in)f Ft(R)23 b Fm(\012)f Ft(S)39 b Fu(is)32 b Ft(m)946 1368 y Fs(R)p Fk(\012)p Fs(S)1133 1353 y Fu(:=)c(\()p Ft(m)1387 1368 y Fs(R)1467 1353 y Fm(\012)22 b Ft(m)1651 1368 y Fs(S)1703 1353 y Fu(\)\(id)32 b Fm(\012)p Ft(c)23 b Fm(\012)f Fu(id)16 b(\):)1206 1506 y Ft(R)23 b Fm(\012)g Ft(S)28 b Fm(\012)22 b Ft(R)h Fm(\012)g Ft(S)97 b Fm(\000)-30 b(\000)-22 b(\000)-30 b(!)102 b Ft(R)23 b Fm(\012)f Ft(S)1203 1687 y Fn(id)h Fk(\012)p Fs(c)p Fk(\012)p Fn(id)1496 1577 y Fl(?)1496 1637 y(?)1496 1697 y(y)2393 1577 y(?)2393 1637 y(?)2393 1697 y(y)2459 1680 y Fn(=)1206 1893 y Ft(R)g Fm(\012)g Ft(R)g Fm(\012)g Ft(S)k Fm(\012)c Ft(S)1931 1834 y Fs(m)1993 1845 y Fi(R)2044 1834 y Fk(\012)p Fs(m)2161 1845 y Fi(S)1902 1893 y Fm(\000)-43 b(\000)-22 b(\000)g(\000)h(\000)-44 b(!)49 b Ft(R)23 b Fm(\012)g Ft(S:)-80 2031 y Fu(W)-8 b(e)34 b(shall)e(denote)j(this)e(algebra)f(b)m(y)j Ft(R)q Fm(\012)p 1379 2055 78 4 v 1 w Ft(S)6 b Fu(.)46 b(The)35 b(di\013erence)f(with)g(the)g(usual)f(tensor)i(pro)s(duct)f(algebra)e (is)-180 2147 y(the)h(presence)i(of)d(the)h(braiding)e Ft(c)h Fu(instead)h(of)f(the)h(usual)f(transp)s(osition)f Ft(\034)11 b Fu(.)-180 2364 y FD(De\014nition)36 b(1.7.)49 b Fu(A)33 b Fo(br)-5 b(aide)g(d)34 b(bialgebr)-5 b(a)34 b(in)1531 2328 y Fs(H)1531 2389 y(H)1598 2364 y Fm(Y)8 b(D)35 b Fu(is)d(a)h(collection)d(\()p Ft(R)q(;)17 b(m;)g(u;)g Fu(\001)p Ft(;)g(")p Fu(\),)31 b(where)-54 2530 y Fm(\017)41 b Fu(\()p Ft(R)q(;)17 b(m;)g(u)p Fu(\))32 b(is)g(an)g(algebra)g(in)1138 2493 y Fs(H)1138 2555 y(H)1205 2530 y Fm(Y)8 b(D)r Fu(.)-54 2721 y Fm(\017)41 b Fu(\()p Ft(R)q(;)17 b Fu(\001)p Ft(;)g(")p Fu(\))32 b(is)g(a)g(coalgebra)g(in)1162 2685 y Fs(H)1162 2746 y(H)1229 2721 y Fm(Y)8 b(D)r Fu(.)-54 2913 y Fm(\017)41 b Fu(\001)28 b(:)g Ft(R)g Fm(!)g Ft(R)q Fm(\012)p 506 2937 V Ft(R)34 b Fu(is)e(a)g(morphism)f(of)h(algebras.)-54 3104 y Fm(\017)41 b Ft(u)27 b Fu(:)h Fp(|)13 b Fm(!)27 b Ft(R)34 b Fu(and)f Ft(")27 b Fu(:)h Ft(R)g Fm(!)g Fp(|)18 b Fu(are)32 b(morphisms)f(of)i(algebras.)-80 3270 y(W)-8 b(e)33 b(sa)m(y)g(that)f(it)g(is)g(a)g Fo(br)-5 b(aide)g(d)34 b(Hopf)h(algebr)-5 b(a)34 b(in)1766 3233 y Fs(H)1766 3295 y(H)1833 3270 y Fm(Y)8 b(D)35 b Fu(if)c(in)h(addition:)-54 3435 y Fm(\017)41 b Fu(The)34 b(iden)m(tit)m(y)e(is)g(con)m(v)m (olution)g(in)m(v)m(ertible)g(in)g(End)17 b(\()p Ft(R)q Fu(\);)32 b(its)g(in)m(v)m(erse)i(is)e(the)h(an)m(tip)s(o)s(de)f(of)g Ft(R)q Fu(.)-80 3601 y(A)k Fo(gr)-5 b(ade)g(d)36 b Fu(braided)g(Hopf)g (algebra)g(in)1397 3564 y Fs(H)1397 3626 y(H)1465 3601 y Fm(Y)8 b(D)39 b Fu(is)d(a)g(braided)g(Hopf)g(algebra)g Ft(R)h Fu(in)3016 3564 y Fs(H)3016 3626 y(H)3083 3601 y Fm(Y)8 b(D)39 b Fu(pro)m(vided)e(with)f(a)-180 3740 y(grading)30 b Ft(R)f Fu(=)e Fm(\010)454 3755 y Fs(n)p Fk(\025)p Fn(0)592 3740 y Ft(R)q Fu(\()p Ft(n)p Fu(\))k(of)g(Y)-8 b(etter-Drinfeld)29 b(mo)s(dules,)i(suc)m(h)i(that)e Ft(R)h Fu(is)f(a)g(graded)g(algebra)f(and)i(a)f(graded)-180 3880 y(coalgebra.)-180 4097 y FD(Remark)37 b(1.8.)49 b Fu(There)27 b(is)f(a)f(non-categorical)f(v)m(ersion)i(of)g(braided)g (Hopf)f(algebras,)i(see)g([Tk1)q(].)41 b(An)m(y)27 b(braided)-180 4236 y(Hopf)33 b(algebra)g(in)516 4200 y Fs(H)516 4261 y(H)584 4236 y Fm(Y)8 b(D)35 b Fu(giv)m(es)f(rise)g(to)f(a)g(braided)g (Hopf)g(algebra)g(in)f(the)i(sense)i(of)d([Tk1])h(b)m(y)g(forgetting)e (the)-180 4375 y(action)40 b(and)i(coaction,)h(and)e(preserving)h(the)g (m)m(ultiplication,)d(com)m(ultiplication)e(and)42 b(braiding.)68 b(F)-8 b(or)40 b(the)-180 4515 y(con)m(v)m(erse)k(see)e([Tk1)q(,)i(Th.) 70 b(5.7].)f(Analogously)-8 b(,)43 b(one)e(can)h(de\014ne)g(graded)g (braided)f(Hopf)g(algebras)g(in)f(the)-180 4654 y(spirit)31 b(of)h([Tk1)q(].)-80 4848 y(Let)45 b Ft(R)i Fu(b)s(e)f(a)f (\014nite-dimensional)e(Hopf)i(algebra)g(in)2001 4812 y Fs(H)2001 4873 y(H)2068 4848 y Fm(Y)8 b(D)r Fu(.)83 b(The)47 b(dual)d Ft(S)56 b Fu(=)50 b Ft(R)3092 4812 y Fk(\003)3177 4848 y Fu(is)45 b(a)h(braided)f(Hopf)-180 4964 y(algebra)35 b(in)283 4928 y Fs(H)283 4990 y(H)350 4964 y Fm(Y)8 b(D)38 b Fu(with)e(m)m(ultiplication)c(\001)1477 4928 y Fk(\003)1477 4990 y Fs(R)1535 4964 y Fu(\010)k(and)h(com)m (ultiplication)32 b(\010)2627 4928 y Fk(\000)p Fn(1)2721 4964 y Ft(\026)2780 4928 y Fk(\003)2780 4990 y Fs(R)2838 4964 y Fu(,)37 b Fo(cf.)54 b Fu(\(1.6\))o(;)38 b(this)e(is)g Ft(R)3697 4928 y Fk(\003)p Fs(bop)3873 4964 y Fu(in)-180 5081 y(the)d(notation)e(of)h([A)m(G,)h(Section)f(2].)p eop %%Page: 10 10 10 9 bop -180 0 a Fq(10)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(In)46 b(the)h(same)g(w)m(a)m(y)-8 b(,)51 b(if)45 b Ft(R)53 b Fu(=)e Fm(\010)1164 218 y Fs(n)p Fk(\025)p Fn(0)1302 203 y Ft(R)q Fu(\()p Ft(n)p Fu(\))c(is)f(a)g(graded)h(braided)f(Hopf)g(algebra)g(in)3200 167 y Fs(H)3200 228 y(H)3268 203 y Fm(Y)8 b(D)48 b Fu(with)f(\014nite-) -180 319 y(dimensional)36 b(homogeneous)j(comp)s(onen)m(ts,)i(then)e (the)g(graded)g(dual)f Ft(S)44 b Fu(=)38 b Ft(R)2782 283 y Fk(\003)2859 319 y Fu(=)g Fm(\010)3050 334 y Fs(n)p Fk(\025)p Fn(0)3188 319 y Ft(R)q Fu(\()p Ft(n)p Fu(\))3397 283 y Fk(\003)3475 319 y Fu(is)g(a)h(graded)-180 435 y(braided)32 b(Hopf)h(algebra)e(in)866 399 y Fs(H)866 461 y(H)933 435 y Fm(Y)8 b(D)s Fu(.)-180 699 y(1.4.)56 b FD(Examples.)49 b(The)38 b(quan)m(tum)f(binomial)f(form)m(ula.)-80 815 y Fu(W)-8 b(e)29 b(shall)f(pro)m(vide)h(man)m(y)g(examples)h(of)e (braided)h(Hopf)g(algebras)g(in)f(Chapter)i(2.)42 b(Here)30 b(w)m(e)g(discuss)g(a)f(v)m(ery)-180 931 y(simple)i(class)i(of)f (braided)g(Hopf)h(algebras.)-80 1047 y(W)-8 b(e)44 b(\014rst)g(recall)f (the)h(w)m(ell-kno)m(wn)g(quan)m(tum)g(binomial)c(form)m(ula.)76 b(Let)44 b Ft(U)54 b Fu(and)44 b Ft(V)65 b Fu(b)s(e)44 b(elemen)m(ts)h(of)e(an)-180 1164 y(asso)s(ciativ)m(e)32 b(algebra)g(o)m(v)m(er)h Fp(|)-8 b Fu([)p Ft(q)s Fu(],)27 b Ft(q)37 b Fu(an)32 b(indeterminate,)f(suc)m(h)j(that)f Ft(V)21 b(U)39 b Fu(=)27 b Ft(q)t(U)10 b(V)22 b Fu(.)44 b(Then)918 1398 y(\()p Ft(U)33 b Fu(+)22 b Ft(V)f Fu(\))1269 1357 y Fs(n)1344 1398 y Fu(=)1481 1303 y Fl(X)1447 1514 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(n)1676 1258 y Fl(\022)1749 1331 y Ft(n)1762 1466 y(i)1808 1258 y Fl(\023)1881 1497 y Fs(q)1919 1398 y Ft(U)1995 1357 y Fs(i)2024 1398 y Ft(V)2102 1357 y Fs(n)p Fk(\000)p Fs(i)2228 1398 y Ft(;)244 b Fu(if)32 b Ft(n)c Fm(\025)g Fu(1)p Ft(:)-3036 b Fu(\(1.12\))-180 1668 y(Here)323 1697 y Fl(\022)396 1770 y Ft(n)409 1905 y(i)454 1697 y Fl(\023)528 1936 y Fs(q)593 1837 y Fu(=)872 1770 y(\()p Ft(n)p Fu(\))1006 1785 y Fs(q)1044 1770 y Fu(!)p 707 1814 529 4 v 707 1905 a(\()p Ft(i)p Fu(\))816 1920 y Fs(q)854 1905 y Fu(!\()p Ft(n)22 b Fm(\000)h Ft(i)p Fu(\))1170 1920 y Fs(q)1208 1905 y Fu(!)1245 1837 y Ft(;)147 b Fu(where)34 b(\()p Ft(n)p Fu(\))1835 1852 y Fs(q)1873 1837 y Fu(!)28 b(=)2074 1742 y Fl(Y)2031 1953 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(n)2243 1837 y Fu(\()p Ft(i)p Fu(\))2352 1852 y Fs(q)2391 1837 y Ft(;)146 b Fu(and)33 b(\()p Ft(i)p Fu(\))2863 1852 y Fs(q)2929 1837 y Fu(=)3106 1742 y Fl(X)3032 1953 y Fn(0)p Fk(\024)p Fs(j)t Fk(\024)p Fs(i)p Fk(\000)p Fn(1)3341 1837 y Ft(q)3388 1796 y Fs(j)3424 1837 y Ft(:)-180 2095 y Fu(By)g(sp)s(ecialization,)e(\(1.12\))h(holds)h(for)f Ft(q)g Fm(2)d Fp(|)-9 b Fu(.)39 b(In)33 b(particular,)f(if)g Ft(U)43 b Fu(and)33 b Ft(V)55 b Fu(are)33 b(elemen)m(ts)g(of)f(an)h (asso)s(ciativ)m(e)-180 2211 y(algebra)e(o)m(v)m(er)j Fp(|)-9 b Fu(,)27 b(and)33 b Ft(q)j Fu(is)c(a)g(ro)s(ot)g(of)g(1)g(of)h (order)f(dividing)f Ft(n)p Fu(,)i(suc)m(h)h(that)e Ft(V)22 b(U)38 b Fu(=)28 b Ft(q)t(U)10 b(V)54 b Fu(then)1424 2384 y(\()p Ft(U)33 b Fu(+)22 b Ft(V)f Fu(\))1775 2343 y Fs(n)1850 2384 y Fu(=)27 b Ft(U)2029 2343 y Fs(n)2099 2384 y Fu(+)22 b Ft(V)2276 2343 y Fs(n)2323 2384 y Ft(:)-2530 b Fu(\(1.13\))-180 2605 y FD(Example)36 b(1.9.)50 b Fu(Let)32 b(\()p Ft(q)761 2620 y Fs(ij)822 2605 y Fu(\))860 2620 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)1153 2605 y Fu(b)s(e)g(a)h(matrix)e(suc)m(h)j(that)1226 2801 y Ft(q)1269 2816 y Fs(ij)1329 2801 y Ft(q)1372 2816 y Fs(j)t(i)1461 2801 y Fu(=)27 b(1)p Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(;)49 b(i)28 b Fm(6)p Fu(=)g Ft(j:)-2728 b Fu(\(1.14\))-180 2997 y(Let)33 b Ft(N)73 3012 y Fs(i)134 2997 y Fu(b)s(e)f(the)h(order)g(of)f Ft(q)843 3012 y Fs(ii)896 2997 y Fu(,)g(when)i(this)e(is)g(\014nite.) -80 3137 y(Let)g Ft(R)i Fu(b)s(e)f(the)g(algebra)e(presen)m(ted)k(b)m (y)e(generators)h Ft(x)1944 3152 y Fn(1)1983 3137 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2257 3152 y Fs(\022)2329 3137 y Fu(with)32 b(relations)119 3333 y Ft(x)174 3288 y Fs(N)230 3298 y Fi(i)174 3358 y Fs(i)289 3333 y Fu(=)27 b(0)p Ft(;)114 b Fu(if)48 b(ord)17 b Ft(q)889 3348 y Fs(ii)969 3333 y Ft(<)28 b Fm(1)p Ft(:)-1380 b Fu(\(1.15\))119 3501 y Ft(x)174 3516 y Fs(i)203 3501 y Ft(x)258 3516 y Fs(j)322 3501 y Fu(=)28 b Ft(q)469 3516 y Fs(ij)530 3501 y Ft(x)585 3516 y Fs(j)622 3501 y Ft(x)677 3516 y Fs(i)705 3501 y Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(i)g(<)g(j)33 b Fm(\024)28 b Ft(\022)s(:)-1626 b Fu(\(1.16\))-180 3698 y(Giv)m(en)34 b(a)g(group)h(\000)f(and)g(elemen)m(ts)h Ft(g)1200 3713 y Fn(1)1239 3698 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)1505 3713 y Fs(\022)1578 3698 y Fu(in)33 b(the)i(cen)m(ter)h(of)e(\000,)h(and)f (c)m(haracters)i Ft(\037)3109 3713 y Fn(1)3148 3698 y Ft(;)17 b(:)g(:)g(:)f(;)h(\037)3428 3713 y Fs(\022)3501 3698 y Fu(of)34 b(\000,)h(there)-180 3837 y(exists)e(a)g(unique)g (structure)h(of)e(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)h(o)m(v)m(er)j (\000)f(on)f Ft(R)q Fu(,)h(suc)m(h)h(that)1379 4033 y Ft(x)1434 4048 y Fs(i)1490 4033 y Fm(2)28 b Ft(R)1659 3992 y Fs(\037)1703 4002 y Fi(i)1658 4058 y Fs(g)1692 4068 y Fi(i)1733 4033 y Ft(;)212 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)s(:)-180 4229 y Fu(Note)33 b(that)f(the)h (braiding)e(is)h(determined)g(b)m(y)657 4426 y Ft(c)p Fu(\()p Ft(x)792 4441 y Fs(i)843 4426 y Fm(\012)22 b Ft(x)997 4441 y Fs(j)1034 4426 y Fu(\))28 b(=)g Ft(q)1247 4441 y Fs(ij)1324 4426 y Ft(x)1379 4441 y Fs(j)1438 4426 y Fm(\012)23 b Ft(x)1593 4441 y Fs(i)1621 4426 y Ft(;)50 b Fu(where)33 b Ft(q)2022 4441 y Fs(ij)2111 4426 y Fu(=)27 b Ft(\037)2275 4441 y Fs(j)2312 4426 y Fu(\()p Ft(g)2397 4441 y Fs(i)2425 4426 y Fu(\))p Ft(;)114 b Fu(1)27 b Fm(\024)i Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(\022)s(:)-180 4622 y Fu(F)-8 b(urthermore,)25 b Ft(R)g Fu(is)f(a)g(braided)g(Hopf)g (algebra)f(with)g(the)i(com)m(ultiplication)20 b(giv)m(en)k(b)m(y)h (\001\()p Ft(x)3204 4637 y Fs(i)3233 4622 y Fu(\))i(=)h Ft(x)3457 4637 y Fs(i)3490 4622 y Fm(\012)5 b Fu(1)g(+)g(1)g Fm(\012)g Ft(x)3898 4637 y Fs(i)3927 4622 y Fu(.)-180 4761 y(T)-8 b(o)37 b(c)m(hec)m(k)j(that)d(the)g(com)m(ultiplication)c (preserv)m(es)40 b(\(1.15\))c(one)i(uses)g(\(1.13\);)h(the)f(v)m (eri\014cation)e(for)h(\(1.16\))f(is)-180 4901 y(easy)-8 b(.)44 b(W)-8 b(e)30 b(kno)m(w)i([AS1])e(that)g(dim)15 b Ft(R)32 b Fu(is)e(in\014nite)f(unless)i(all)d(the)j(orders)g(of)e Ft(q)2687 4916 y Fs(ii)2740 4901 y Fu('s)i(are)f(\014nite;)h(in)e(this) h(last)g(case,)-180 5040 y(dim)15 b Ft(R)29 b Fu(=)205 4966 y Fl(Q)300 5069 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(\022)524 5040 y Ft(N)602 5055 y Fs(i)631 5040 y Fu(.)43 b(W)-8 b(e)33 b(also)f(ha)m(v)m(e)i Ft(P)14 b Fu(\()p Ft(R)q Fu(\))27 b(=)g Fm(\010)1725 5055 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(\022)1934 5040 y Fp(|)-9 b Ft(x)2050 5055 y Fs(i)2073 5040 y Fu(.)p eop %%Page: 11 11 11 10 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(11)-180 203 y Fu(1.5.)56 b FD(Bipro)s(ducts,)36 b(or)i(b)s (osonizations.)-80 319 y Fu(Let)j Ft(A)p Fu(,)i Ft(H)48 b Fu(b)s(e)42 b(Hopf)e(algebras)h(and)g Ft(\031)46 b Fu(:)c Ft(A)g Fm(!)f Ft(H)49 b Fu(and)41 b Ft(\023)h Fu(:)g Ft(H)50 b Fm(!)41 b Ft(A)g Fu(Hopf)g(algebra)f(homomorphisms.) -180 435 y(Assume)g(that)g Ft(\031)t(\023)g Fu(=)f(id)753 450 y Fs(H)820 435 y Fu(,)i(so)f(that)f Ft(\031)44 b Fu(is)39 b(surjectiv)m(e,)j(and)e Ft(\023)g Fu(is)f(injectiv)m(e.)64 b(By)40 b(analogy)f(with)g(elemen)m(tary)-180 552 y(group)28 b(theory)-8 b(,)30 b(one)f(w)m(ould)g(lik)m(e)e(to)i(reconstruct)h Ft(A)e Fu(from)g Ft(H)36 b Fu(and)28 b(the)h(k)m(ernel)g(of)f Ft(\031)33 b Fu(as)c(a)f(semidirect)f(pro)s(duct.)-180 668 y(Ho)m(w)m(ev)m(er,)35 b(the)e(natural)e(candidate)i(for)f(the)h(k) m(ernel)g(of)f Ft(\031)37 b Fu(is)32 b(the)h(algebra)e(of)h(coin)m(v)-5 b(arian)m(ts)941 836 y Ft(R)29 b Fu(:=)f Ft(A)1248 795 y Fn(co)c Fs(\031)1413 836 y Fu(=)j Fm(f)p Ft(a)h Fm(2)g Ft(A)g Fu(:)g(\(id)k Fm(\012)p Ft(\031)t Fu(\)\001\()p Ft(a)p Fu(\))c(=)g Ft(a)22 b Fm(\012)h Fu(1)p Fm(g)-180 1005 y Fu(whic)m(h)36 b(is)g Fo(not)p Fu(,)g(in)f(general,)i(a)e(Hopf)h (algebra.)52 b(Instead,)38 b Ft(R)f Fu(is)e(a)h(braided)f(Hopf)h (algebra)e(in)3336 969 y Fs(H)3336 1030 y(H)3403 1005 y Fm(Y)8 b(D)38 b Fu(with)d(the)-180 1121 y(follo)m(wing)30 b(structure:)-54 1264 y Fm(\017)41 b Fu(The)34 b(action)d Fm(\001)h Fu(of)g Ft(H)41 b Fu(on)32 b Ft(R)i Fu(is)e(the)h (restriction)e(of)h(the)h(adjoin)m(t)f(action)g(\(comp)s(osed)g(with)g Ft(\023)p Fu(\).)-54 1432 y Fm(\017)41 b Fu(The)34 b(coaction)d(is)h (\()p Ft(\031)26 b Fm(\012)d Fu(id)16 b(\)\001.)-54 1601 y Fm(\017)41 b Ft(R)34 b Fu(is)e(a)g(subalgebra)g(of)g Ft(A)p Fu(.)-54 1769 y Fm(\017)41 b Fu(The)34 b(com)m(ultiplication)28 b(is)k(\001)1135 1784 y Fs(R)1193 1769 y Fu(\()p Ft(r)s Fu(\))27 b(=)g Ft(r)1490 1785 y Fn(\(1\))1585 1769 y Ft(\023\031)t Fm(S)7 b Fu(\()p Ft(r)1827 1785 y Fn(\(2\))1922 1769 y Fu(\))22 b Fm(\012)h Ft(r)2126 1785 y Fn(\(3\))2220 1769 y Fu(,)33 b(for)f(all)e Ft(r)h Fm(2)d Ft(R)q Fu(.)-80 1917 y(Giv)m(en)35 b(a)h(braided)g(Hopf)f(algebra)g Ft(R)i Fu(in)1460 1881 y Fs(H)1460 1942 y(H)1528 1917 y Fm(Y)8 b(D)r Fu(,)37 b(one)f(can)g(consider)g(the)h Fo(b)-5 b(osonization)34 b Fu(or)i Fo(bipr)-5 b(o)g(duct)35 b Fu(of)h Ft(R)-180 2034 y Fu(b)m(y)31 b Ft(H)37 b Fu([Ra],)30 b([Mj2].)43 b(This)30 b(is)f(a)h(usual)f(Hopf)h(algebra)e Ft(R)q Fu(#)p Ft(H)8 b Fu(,)31 b(with)e(underlying)g(v)m(ector)i(space) g Ft(R)18 b Fm(\012)e Ft(H)8 b Fu(,)31 b(whose)-180 2150 y(m)m(ultiplication)d(and)33 b(com)m(ultiplication)28 b(are)k(giv)m(en)h(b)m(y)853 2401 y(\()p Ft(r)s Fu(#)p Ft(h)p Fu(\)\()p Ft(s)p Fu(#)p Ft(f)11 b Fu(\))28 b(=)f Ft(r)s Fu(\()p Ft(h)1647 2417 y Fn(\(1\))1764 2401 y Fm(\001)21 b Ft(s)p Fu(\)#)p Ft(h)2034 2417 y Fn(\(2\))2129 2401 y Ft(f)5 b(;)1034 2569 y Fu(\001\()p Ft(r)s Fu(#)p Ft(h)p Fu(\))28 b(=)f Ft(r)1553 2528 y Fn(\(1\))1647 2569 y Fu(#\()p Ft(r)1813 2528 y Fn(\(2\))1908 2569 y Fu(\))1946 2584 y Fn(\()p Fk(\000)p Fn(1\))2095 2569 y Ft(h)2151 2584 y Fn(\(1\))2267 2569 y Fm(\012)c Fu(\()p Ft(r)2452 2528 y Fn(\(2\))2546 2569 y Fu(\))2584 2584 y Fn(\(0\))2678 2569 y Fu(#)p Ft(h)2815 2584 y Fn(\(2\))2910 2569 y Ft(:)-180 2492 y Fu(\(1.17\))-80 2743 y(The)36 b(maps)f Ft(\031)h Fu(:)c Ft(R)q Fu(#)p Ft(H)40 b Fm(!)32 b Ft(H)43 b Fu(and)35 b Ft(\023)e Fu(:)f Ft(H)40 b Fm(!)32 b Ft(R)q Fu(#)p Ft(H)8 b Fu(,)35 b Ft(\031)t Fu(\()p Ft(r)s Fu(#)p Ft(h)p Fu(\))e(=)f Ft(\017)p Fu(\()p Ft(r)s Fu(\))p Ft(h)p Fu(,)k Ft(\023)p Fu(\()p Ft(h)p Fu(\))c(=)g(1#)p Ft(h)p Fu(,)37 b(are)e(Hopf)g(algebra)-180 2859 y(homomorphisms;)30 b(w)m(e)k(ha)m(v)m(e)g Ft(R)29 b Fu(=)e Fm(f)p Ft(a)h Fm(2)g Ft(R)q Fu(#)p Ft(H)36 b Fu(:)27 b(\(id)33 b Fm(\012)p Ft(\031)t Fu(\)\001\()p Ft(a)p Fu(\))28 b(=)g Ft(a)22 b Fm(\012)h Fu(1)p Fm(g)p Fu(.)-80 3028 y(Con)m(v)m(ersely)-8 b(,)34 b(if)e Ft(A)g Fu(and)h Ft(H)40 b Fu(are)33 b(Hopf)f(algebras)g (as)h(ab)s(o)m(v)m(e)g(and)g Ft(R)c Fu(=)e Ft(A)2584 2992 y Fn(co)d Fs(\031)2754 3028 y Fu(then)33 b Ft(A)28 b Fm(')g Ft(R)q Fu(#)p Ft(H)8 b Fu(.)-80 3196 y(Let)32 b Ft(#)d Fu(:)e Ft(A)h Fm(!)f Ft(R)34 b Fu(b)s(e)f(the)g(map)f(giv)m (en)g(b)m(y)i Ft(#)p Fu(\()p Ft(a)p Fu(\))28 b(=)g Ft(a)1845 3212 y Fn(\(1\))1939 3196 y Ft(\023\031)t Fm(S)7 b Fu(\()p Ft(a)2188 3212 y Fn(\(2\))2284 3196 y Fu(\).)43 b(Then)1335 3365 y Ft(#)p Fu(\()p Ft(ab)p Fu(\))29 b(=)e Ft(a)1743 3380 y Fn(\(1\))1838 3365 y Ft(#)p Fu(\()p Ft(b)p Fu(\))p Ft(\023\031)t Fm(S)7 b Fu(\()p Ft(a)2261 3380 y Fn(\(2\))2357 3365 y Fu(\))p Ft(;)-2602 b Fu(\(1.18\))-180 3539 y(for)42 b(all)f Ft(a;)17 b(b)45 b Fm(2)g Ft(A)p Fu(,)h(and)c Ft(#)p Fu(\()p Ft(\023)p Fu(\()p Ft(h)p Fu(\)\))k(=)f Ft(")p Fu(\()p Ft(h)p Fu(\))d(for)g(all)f Ft(h)k Fm(2)g Ft(H)8 b Fu(;)47 b(therefore,)f(for)c(all)f Ft(a)k Fm(2)g Ft(A)p Fu(,)h Ft(h)f Fm(2)g Ft(H)8 b Fu(,)45 b(w)m(e)e(ha)m(v)m(e)-180 3655 y Ft(#)p Fu(\()p Ft(a\023)p Fu(\()p Ft(h)p Fu(\)\))29 b(=)e Ft(#)p Fu(\()p Ft(a)p Fu(\))p Ft(")p Fu(\()p Ft(h)p Fu(\))33 b(and)1476 3824 y Ft(#)p Fu(\()p Ft(\023)p Fu(\()p Ft(h)p Fu(\))p Ft(a)p Fu(\))c(=)e Ft(h)22 b Fm(\001)g Ft(#)p Fu(\()p Ft(a)p Fu(\))p Ft(:)-2477 b Fu(\(1.19\))-180 4000 y(Notice)39 b(also)g(that)g Ft(#)i Fu(induces)f(a)f(coalgebra)g (isomorphism)e Ft(A=)-5 b(A\023)p Fu(\()p Ft(H)8 b Fu(\))2501 3964 y Fn(+)2600 4000 y Fm(')40 b Ft(R)q Fu(.)65 b(In)40 b(fact,)h(the)f(isomorphism)-180 4116 y Ft(A)28 b Fm(!)f Ft(R)q Fu(#)p Ft(H)41 b Fu(can)32 b(b)s(e)h(expressed)i(explicitly)c (as)1208 4285 y Ft(a)c Fm(7!)h Ft(#)p Fu(\()p Ft(a)1560 4300 y Fn(\(1\))1655 4285 y Fu(\)#)p Ft(\031)t Fu(\()p Ft(a)1922 4300 y Fn(\(2\))2016 4285 y Fu(\))p Ft(;)212 b(a)28 b Fm(2)g Ft(A:)-80 4453 y Fu(If)k Ft(A)h Fu(is)f(a)g(Hopf)h (algebra,)e(the)i(adjoin)m(t)f(represen)m(tation)h(ad)49 b(of)32 b Ft(A)h Fu(on)f(itself)f(is)h(giv)m(en)h(b)m(y)1410 4622 y(ad)16 b Ft(x)p Fu(\()p Ft(y)t Fu(\))27 b(=)h Ft(x)1898 4637 y Fn(\(1\))1992 4622 y Ft(y)t Fm(S)7 b Fu(\()p Ft(x)2204 4637 y Fn(\(2\))2299 4622 y Fu(\))p Ft(:)-180 4796 y Fu(If)32 b Ft(R)h Fu(is)f(a)f(braided)h(Hopf)g(algebra)f(in)1246 4760 y Fs(H)1246 4821 y(H)1313 4796 y Fm(Y)8 b(D)s Fu(,)32 b(then)h(there)f(is)g(also)f(a)h(braided)g(adjoin)m(t)f(represen)m (tation)h(ad)3809 4811 y Fs(c)3875 4796 y Fu(of)-180 4912 y Ft(R)i Fu(on)e(itself)f(de\014ned)j(b)m(y)899 5081 y(ad)1018 5096 y Fs(c)1053 5081 y Ft(x)p Fu(\()p Ft(y)t Fu(\))27 b(=)g Ft(\026)p Fu(\()p Ft(\026)22 b Fm(\012)g(S)1703 5096 y Fs(R)1761 5081 y Fu(\)\(id)33 b Fm(\012)p Ft(c)p Fu(\)\(\001)2228 5096 y Fs(R)2308 5081 y Fm(\012)23 b Fu(id)16 b(\)\()p Ft(x)22 b Fm(\012)h Ft(y)t Fu(\))p Ft(;)p eop %%Page: 12 12 12 11 bop -180 0 a Fq(12)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(where)38 b Ft(\026)e Fu(is)f(the)i(m)m(ultiplication)32 b(and)k Ft(c)e Fm(2)h Fu(End)17 b(\()p Ft(R)26 b Fm(\012)f Ft(R)q Fu(\))36 b(is)g(the)h(braiding.)53 b(Note)36 b(that)g(if)f Ft(x)g Fm(2)f(P)8 b Fu(\()p Ft(R)q Fu(\))37 b(then)-180 319 y(the)c(braided)f(adjoin)m(t)g(represen)m(tation)h(of)f Ft(x)h Fu(is)f(just)1104 478 y(ad)1224 493 y Fs(c)1258 478 y Ft(x)p Fu(\()p Ft(y)t Fu(\))c(=)f Ft(\026)p Fu(\(id)32 b Fm(\000)p Ft(c)p Fu(\)\()p Ft(x)23 b Fm(\012)g Ft(y)t Fu(\))j(=:)i([)p Ft(x;)17 b(y)t Fu(])2608 493 y Fs(c)2642 478 y Ft(:)-2849 b Fu(\(1.20\))-180 643 y(F)-8 b(or)32 b(an)m(y)h Ft(x;)17 b(y)31 b Fm(2)d Ft(R)q Fu(,)33 b(w)m(e)g(call)e([)p Ft(x;)17 b(y)t Fu(])1113 658 y Fs(c)1175 643 y Fu(:=)28 b Ft(\026)p Fu(\(id)k Fm(\000)p Ft(c)p Fu(\)\()p Ft(x)23 b Fm(\012)f Ft(y)t Fu(\))32 b(a)g Fo(br)-5 b(aide)g(d)34 b(c)-5 b(ommutator)p Fu(.)-80 801 y(When)33 b Ft(A)28 b Fu(=)f Ft(R)q Fu(#)p Ft(H)8 b Fu(,)33 b(then)g(for)f(all)f Ft(b;)17 b(d)27 b Fm(2)h Ft(R)q Fu(,)1264 960 y(ad)1383 976 y Fn(\()p Fs(b)p Fn(#1\))1567 960 y Fu(\()p Ft(d)p Fu(#1\))f(=)h(\(ad)2112 975 y Fs(c)2146 960 y Ft(b)p Fu(\()p Ft(d)p Fu(\)\)#1)p Ft(:)-2689 b Fu(\(1.21\))-180 1138 y(1.6.)56 b FD(Some)37 b(prop)s(erties)g(of)g(braided)h(Hopf)g (algebras.)-80 1255 y Fu(In)31 b(this)g(Section,)h(w)m(e)g(\014rst)g (collect)f(sev)m(eral)h(useful)f(facts)h(ab)s(out)f(braided)g(Hopf)g (algebras)g(in)g(the)g(category)-180 1371 y(of)38 b(Y)-8 b(etter-Drinfeld)37 b(mo)s(dules)h(o)m(v)m(er)h(an)g(ab)s(elian)e (group)h(\000.)62 b(W)-8 b(e)39 b(b)s(egin)f(with)g(some)h(iden)m (tities)f(on)g(braided)-180 1487 y(comm)m(utators.)-80 1646 y(In)e(the)h(follo)m(wing)c(t)m(w)m(o)k(Lemmata,)e Ft(R)i Fu(denotes)h(a)d(braided)h(Hopf)g(algebra)f(in)2898 1610 y Fn(\000)2898 1671 y(\000)2946 1646 y Fm(Y)8 b(D)s Fu(.)54 b(Let)36 b Ft(a)3411 1661 y Fn(1)3451 1646 y Ft(;)17 b(a)3546 1661 y Fn(2)3585 1646 y Ft(;)g Fm(\001)g(\001)g(\001) 32 b(2)i Ft(R)-180 1777 y Fu(b)s(e)f(elemen)m(ts)g(suc)m(h)h(that)e Ft(a)834 1792 y Fs(i)890 1777 y Fm(2)c Ft(R)1059 1740 y Fs(\037)1103 1750 y Fi(i)1058 1801 y Fs(g)1092 1811 y Fi(i)1134 1777 y Fu(,)k(for)g(some)h Ft(\037)1648 1792 y Fs(i)1704 1777 y Fm(2)1800 1751 y Fl(b)1798 1777 y Fu(\000,)f Ft(g)1965 1792 y Fs(i)2021 1777 y Fm(2)c Fu(\000.)-180 1987 y FD(Lemma)37 b(1.10.)49 b Fo(\(a\).)483 2169 y Fu([[)p Ft(a)588 2184 y Fn(1)628 2169 y Ft(;)17 b(a)723 2184 y Fn(2)763 2169 y Fu(])790 2184 y Fs(c)824 2169 y Ft(;)g(a)919 2184 y Fn(3)959 2169 y Fu(])986 2184 y Fs(c)1043 2169 y Fu(+)22 b Ft(\037)1202 2184 y Fn(2)1241 2169 y Fu(\()p Ft(g)1326 2184 y Fn(1)1365 2169 y Fu(\))p Ft(a)1454 2184 y Fn(2)1494 2169 y Fu([)p Ft(a)1572 2184 y Fn(1)1612 2169 y Ft(;)17 b(a)1707 2184 y Fn(3)1746 2169 y Fu(])1773 2184 y Fs(c)1835 2169 y Fu(=)28 b([)p Ft(a)2017 2184 y Fn(1)2057 2169 y Ft(;)17 b Fu([)p Ft(a)2179 2184 y Fn(2)2218 2169 y Ft(;)g(a)2313 2184 y Fn(3)2353 2169 y Fu(])2380 2184 y Fs(c)2414 2169 y Fu(])2441 2184 y Fs(c)2498 2169 y Fu(+)22 b Ft(\037)2657 2184 y Fn(3)2697 2169 y Fu(\()p Ft(g)2782 2184 y Fn(2)2821 2169 y Fu(\)[)p Ft(a)2937 2184 y Fn(1)2977 2169 y Ft(;)17 b(a)3072 2184 y Fn(3)3111 2169 y Fu(])3138 2184 y Fs(c)3173 2169 y Ft(a)3224 2184 y Fn(2)3263 2169 y Ft(:)-3470 b Fu(\(1.22\))-80 2351 y Fo(\(b\).)44 b(If)34 b Fu([)p Ft(a)299 2366 y Fn(1)339 2351 y Ft(;)17 b(a)434 2366 y Fn(2)473 2351 y Fu(])500 2366 y Fs(c)563 2351 y Fu(=)27 b(0)35 b Fo(and)f Fu([)p Ft(a)1017 2366 y Fn(1)1057 2351 y Ft(;)17 b(a)1152 2366 y Fn(3)1191 2351 y Fu(])1218 2366 y Fs(c)1281 2351 y Fu(=)27 b(0)35 b Fo(then)g Fu([)p Ft(a)1763 2366 y Fn(1)1802 2351 y Ft(;)17 b Fu([)p Ft(a)1924 2366 y Fn(2)1964 2351 y Ft(;)g(a)2059 2366 y Fn(3)2098 2351 y Fu(])2125 2366 y Fs(c)2160 2351 y Fu(])2187 2366 y Fs(c)2250 2351 y Fu(=)27 b(0)p Fo(.)-80 2533 y(\(c\).)44 b(If)34 b Fu([)p Ft(a)299 2548 y Fn(1)339 2533 y Ft(;)17 b(a)434 2548 y Fn(3)473 2533 y Fu(])500 2548 y Fs(c)563 2533 y Fu(=)27 b(0)35 b Fo(and)f Fu([)p Ft(a)1017 2548 y Fn(2)1057 2533 y Ft(;)17 b(a)1152 2548 y Fn(3)1191 2533 y Fu(])1218 2548 y Fs(c)1281 2533 y Fu(=)27 b(0)35 b Fo(then)g Fu([[)p Ft(a)1790 2548 y Fn(1)1830 2533 y Ft(;)17 b(a)1925 2548 y Fn(2)1964 2533 y Fu(])1991 2548 y Fs(c)2026 2533 y Ft(;)g(a)2121 2548 y Fn(3)2160 2533 y Fu(])2187 2548 y Fs(c)2250 2533 y Fu(=)27 b(0)p Fo(.)-80 2715 y(\(d\).)44 b(Assume)35 b(that)g Ft(\037)748 2730 y Fn(1)787 2715 y Fu(\()p Ft(g)872 2730 y Fn(2)912 2715 y Fu(\))p Ft(\037)1011 2730 y Fn(2)1050 2715 y Fu(\()p Ft(g)1135 2730 y Fn(1)1174 2715 y Fu(\))p Ft(\037)1273 2730 y Fn(2)1313 2715 y Fu(\()p Ft(g)1398 2730 y Fn(2)1437 2715 y Fu(\))27 b(=)h(1)p Fo(.)44 b(Then)974 3037 y Fu([[)p Ft(a)1079 3052 y Fn(1)1119 3037 y Ft(;)17 b(a)1214 3052 y Fn(2)1254 3037 y Fu(])1281 3052 y Fs(c)1315 3037 y Ft(;)g(a)1410 3052 y Fn(2)1450 3037 y Fu(])1477 3052 y Fs(c)1539 3037 y Fu(=)28 b Ft(\037)1704 3052 y Fn(2)1743 3037 y Fu(\()p Ft(g)1828 3052 y Fn(1)1867 3037 y Fu(\))1905 2995 y Fk(\000)p Fn(1)2000 3037 y Ft(\037)2061 3052 y Fn(1)2100 3037 y Fu(\()p Ft(g)2185 3052 y Fn(2)2224 3037 y Fu(\)[)p Ft(a)2340 3052 y Fn(2)2380 3037 y Ft(;)17 b Fu([)p Ft(a)2502 3052 y Fn(2)2541 3037 y Ft(;)g(a)2636 3052 y Fn(1)2676 3037 y Fu(])2703 3052 y Fs(c)2738 3037 y Fu(])2765 3052 y Fs(c)-180 3037 y Fu(\(1.23\))-180 3247 y Fo(Pr)-5 b(o)g(of.)41 b Fu(Left)33 b(to)f(the)h(reader.)p 3883 3247 4 66 v 3887 3185 59 4 v 3887 3247 V 3945 3247 4 66 v -80 3415 a(The)g(follo)m(wing)d(tec)m(hnical)i(Lemma)f(will)g(b) s(e)h(used)i(at)e(a)h(crucial)e(p)s(oin)m(t)h(in)g(Section)g(6.1.)-180 3626 y FD(Lemma)37 b(1.11.)49 b Fo(Assume)35 b(that)g Ft(\037)1120 3641 y Fn(2)1160 3626 y Fu(\()p Ft(g)1245 3641 y Fn(2)1284 3626 y Fu(\))28 b Fm(6)p Fu(=)f Fm(\000)p Fu(1)35 b Fo(and)1381 3808 y Ft(\037)1442 3823 y Fn(1)1482 3808 y Fu(\()p Ft(g)1567 3823 y Fn(2)1606 3808 y Fu(\))p Ft(\037)1705 3823 y Fn(2)1744 3808 y Fu(\()p Ft(g)1829 3823 y Fn(1)1868 3808 y Fu(\))p Ft(\037)1967 3823 y Fn(2)2007 3808 y Fu(\()p Ft(g)2092 3823 y Fn(2)2131 3808 y Fu(\))28 b(=)f(1)p Ft(;)-2556 b Fu(\(1.24\))1381 3976 y Ft(\037)1442 3991 y Fn(2)1482 3976 y Fu(\()p Ft(g)1567 3991 y Fn(3)1606 3976 y Fu(\))p Ft(\037)1705 3991 y Fn(3)1744 3976 y Fu(\()p Ft(g)1829 3991 y Fn(2)1868 3976 y Fu(\))p Ft(\037)1967 3991 y Fn(2)2007 3976 y Fu(\()p Ft(g)2092 3991 y Fn(2)2131 3976 y Fu(\))28 b(=)f(1)p Ft(:)-2556 b Fu(\(1.25\))-180 4158 y Fo(If)1534 4341 y Fu([)p Ft(a)1612 4356 y Fn(2)1651 4341 y Ft(;)17 b(a)1746 4356 y Fn(2)1786 4341 y Fu(])1813 4356 y Fs(c)1847 4341 y Ft(;)g(a)1942 4356 y Fn(1)1982 4341 y Fu(])2009 4356 y Fs(c)2071 4341 y Fu(=)28 b(0)p Ft(;)-2431 b Fu(\(1.26\))1507 4509 y([[)p Ft(a)1612 4524 y Fn(2)1651 4509 y Ft(;)17 b(a)1746 4524 y Fn(2)1786 4509 y Fu(])1813 4524 y Fs(c)1847 4509 y Ft(;)g(a)1942 4524 y Fn(3)1982 4509 y Fu(])2009 4524 y Fs(c)2071 4509 y Fu(=)28 b(0)p Ft(;)-2431 b Fu(\(1.27\))1730 4678 y([)p Ft(a)1808 4693 y Fn(1)1847 4678 y Ft(;)17 b(a)1942 4693 y Fn(3)1982 4678 y Fu(])2009 4693 y Fs(c)2071 4678 y Fu(=)28 b(0)p Ft(;)-2431 b Fu(\(1.28\))-180 4860 y Fo(then)1403 5042 y Fu([[[)p Ft(a)1535 5057 y Fn(1)1575 5042 y Ft(;)17 b(a)1670 5057 y Fn(2)1709 5042 y Fu(])1736 5057 y Fs(c)1771 5042 y Ft(;)g(a)1866 5057 y Fn(3)1906 5042 y Fu(])1933 5057 y Fs(c)1967 5042 y Ft(;)g(a)2062 5057 y Fn(2)2102 5042 y Fu(])2129 5057 y Fs(c)2191 5042 y Fu(=)28 b(0)p Ft(:)-2551 b Fu(\(1.29\))p eop %%Page: 13 13 13 12 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(13)-180 203 y Fo(Pr)-5 b(o)g(of.)41 b Fu(W)-8 b(e)33 b(compute:)-17 419 y([[[)p Ft(a)115 434 y Fn(1)155 419 y Ft(;)17 b(a)250 434 y Fn(2)289 419 y Fu(])316 434 y Fs(c)351 419 y Ft(;)g(a)446 434 y Fn(3)485 419 y Fu(])512 434 y Fs(c)547 419 y Ft(;)g(a)642 434 y Fn(2)681 419 y Fu(])708 434 y Fs(c)771 419 y Fu(=)27 b Ft(a)925 434 y Fn(1)965 419 y Ft(a)1016 434 y Fn(2)1056 419 y Ft(a)1107 434 y Fn(3)1146 419 y Ft(a)1197 434 y Fn(2)1259 419 y Fm(\000)c Ft(\037)1420 434 y Fn(2)1459 419 y Fu(\()p Ft(g)1544 434 y Fn(1)1583 419 y Fu(\))17 b Ft(a)1689 434 y Fn(2)1729 419 y Ft(a)1780 434 y Fn(1)1819 419 y Ft(a)1870 434 y Fn(3)1910 419 y Ft(a)1961 434 y Fn(2)2023 419 y Fm(\000)22 b Ft(\037)2183 434 y Fn(3)2223 419 y Fu(\()p Ft(g)2308 434 y Fn(1)2347 419 y Fu(\))p Ft(\037)2446 434 y Fn(3)2485 419 y Fu(\()p Ft(g)2570 434 y Fn(2)2609 419 y Fu(\))17 b Ft(a)2715 434 y Fn(3)2755 419 y Ft(a)2806 434 y Fn(1)2845 419 y Ft(a)2896 378 y Fn(2)2896 444 y(2)765 588 y Fu(+)22 b Ft(\037)924 603 y Fn(3)964 588 y Fu(\()p Ft(g)1049 603 y Fn(1)1088 588 y Fu(\))p Ft(\037)1187 603 y Fn(3)1226 588 y Fu(\()p Ft(g)1311 603 y Fn(2)1350 588 y Fu(\))p Ft(\037)1449 603 y Fn(2)1489 588 y Fu(\()p Ft(g)1574 603 y Fn(1)1613 588 y Fu(\))17 b Ft(a)1719 603 y Fn(3)1758 588 y Ft(a)1809 603 y Fn(2)1849 588 y Ft(a)1900 603 y Fn(1)1940 588 y Ft(a)1991 603 y Fn(2)2052 588 y Fm(\000)23 b Ft(\037)2213 603 y Fn(2)2252 588 y Fu(\()p Ft(g)2337 603 y Fn(1)2377 588 y Fu(\))p Ft(\037)2476 603 y Fn(2)2515 588 y Fu(\()p Ft(g)2600 603 y Fn(2)2639 588 y Fu(\))p Ft(\037)2738 603 y Fn(2)2778 588 y Fu(\()p Ft(g)2863 603 y Fn(3)2902 588 y Fu(\))17 b Ft(a)3008 603 y Fn(2)3047 588 y Ft(a)3098 603 y Fn(1)3138 588 y Ft(a)3189 603 y Fn(2)3228 588 y Ft(a)3279 603 y Fn(3)765 756 y Fu(+)22 b Ft(\037)924 771 y Fn(2)964 756 y Fu(\()p Ft(g)1049 771 y Fn(1)1088 756 y Fu(\))1126 715 y Fn(2)1165 756 y Ft(\037)1226 771 y Fn(2)1266 756 y Fu(\()p Ft(g)1351 771 y Fn(2)1390 756 y Fu(\))p Ft(\037)1489 771 y Fn(2)1528 756 y Fu(\()p Ft(g)1613 771 y Fn(3)1652 756 y Fu(\))17 b Ft(a)1758 715 y Fn(2)1758 781 y(2)1798 756 y Ft(a)1849 771 y Fn(1)1888 756 y Ft(a)1939 771 y Fn(3)2001 756 y Fu(+)22 b Ft(\037)2160 771 y Fn(2)2200 756 y Fu(\()p Ft(g)2285 771 y Fn(1)2324 756 y Fu(\))p Ft(\037)2423 771 y Fn(2)2462 756 y Fu(\()p Ft(g)2547 771 y Fn(2)2586 756 y Fu(\))p Ft(\037)2685 771 y Fn(2)2725 756 y Fu(\()p Ft(g)2810 771 y Fn(3)2849 756 y Fu(\))p Ft(\037)2948 771 y Fn(3)2987 756 y Fu(\()p Ft(g)3072 771 y Fn(1)3111 756 y Fu(\))p Ft(\037)3210 771 y Fn(3)3250 756 y Fu(\()p Ft(g)3335 771 y Fn(2)3374 756 y Fu(\))17 b Ft(a)3480 771 y Fn(2)3519 756 y Ft(a)3570 771 y Fn(3)3610 756 y Ft(a)3661 771 y Fn(1)3700 756 y Ft(a)3751 771 y Fn(2)765 925 y Fm(\000)23 b Ft(\037)926 940 y Fn(2)965 925 y Fu(\()p Ft(g)1050 940 y Fn(1)1090 925 y Fu(\))1128 884 y Fn(2)1167 925 y Ft(\037)1228 940 y Fn(2)1267 925 y Fu(\()p Ft(g)1352 940 y Fn(2)1392 925 y Fu(\))p Ft(\037)1491 940 y Fn(2)1530 925 y Fu(\()p Ft(g)1615 940 y Fn(3)1654 925 y Fu(\))p Ft(\037)1753 940 y Fn(3)1793 925 y Fu(\()p Ft(g)1878 940 y Fn(1)1917 925 y Fu(\))p Ft(\037)2016 940 y Fn(3)2055 925 y Fu(\()p Ft(g)2140 940 y Fn(2)2179 925 y Fu(\))17 b Ft(a)2285 940 y Fn(2)2324 925 y Ft(a)2375 940 y Fn(3)2415 925 y Ft(a)2466 940 y Fn(2)2506 925 y Ft(a)2557 940 y Fn(1)2596 925 y Ft(:)-180 1141 y Fu(W)-8 b(e)45 b(index)h(consecutiv)m (ely)g(the)g(terms)f(in)f(the)i(righ)m(t-hand)e(side)h(b)m(y)h(roman)e (n)m(um)m(b)s(ers:)70 b(\()p Ft(I)8 b Fu(\))p Ft(;)17 b(:)g(:)g(:)e(;)i Fu(\()p Ft(V)k(I)8 b(I)g(I)g Fu(\).)-180 1281 y(Then)34 b(\()p Ft(I)8 b(I)g Fu(\))21 b(+)h(\()p Ft(V)g(I)8 b(I)g Fu(\))27 b(=)h(0,)k(b)m(y)j(\(1.25\))d(and)h(\(1.28\)) o(.)43 b(No)m(w,)627 1533 y(\()p Ft(I)8 b Fu(\))28 b(=)1256 1466 y(1)p 895 1510 770 4 v 895 1601 a Ft(\037)956 1616 y Fn(3)996 1601 y Fu(\()p Ft(g)1081 1616 y Fn(2)1120 1601 y Fu(\)\(1)22 b(+)g Ft(\037)1426 1616 y Fn(2)1465 1601 y Fu(\()p Ft(g)1550 1616 y Fn(2)1589 1601 y Fu(\)\))1675 1533 y Ft(a)1726 1548 y Fn(1)1766 1533 y Ft(a)1817 1492 y Fn(2)1817 1558 y(2)1856 1533 y Ft(a)1907 1548 y Fn(3)1969 1533 y Fu(+)2077 1466 y Ft(\037)2138 1481 y Fn(2)2178 1466 y Fu(\()p Ft(g)2263 1481 y Fn(2)2302 1466 y Fu(\))p Ft(\037)2401 1481 y Fn(3)2440 1466 y Fu(\()p Ft(g)2525 1481 y Fn(2)2564 1466 y Fu(\))p 2077 1510 526 4 v 2124 1601 a(1)g(+)g Ft(\037)2354 1616 y Fn(2)2393 1601 y Fu(\()p Ft(g)2478 1616 y Fn(2)2517 1601 y Fu(\))2612 1533 y Ft(a)2663 1548 y Fn(1)2703 1533 y Ft(a)2754 1548 y Fn(3)2793 1533 y Ft(a)2844 1492 y Fn(2)2844 1558 y(2)782 1951 y Fu(=)1256 1884 y(1)p 895 1928 770 4 v 895 2019 a Ft(\037)956 2034 y Fn(3)996 2019 y Fu(\()p Ft(g)1081 2034 y Fn(2)1120 2019 y Fu(\)\(1)g(+)g Ft(\037)1426 2034 y Fn(2)1465 2019 y Fu(\()p Ft(g)1550 2034 y Fn(2)1589 2019 y Fu(\)\))1675 1951 y Ft(a)1726 1966 y Fn(1)1766 1951 y Ft(a)1817 1910 y Fn(2)1817 1976 y(2)1856 1951 y Ft(a)1907 1966 y Fn(3)1969 1951 y Fu(+)2077 1884 y Ft(\037)2138 1899 y Fn(2)2178 1884 y Fu(\()p Ft(g)2263 1899 y Fn(2)2302 1884 y Fu(\))p Ft(\037)2401 1899 y Fn(3)2440 1884 y Fu(\()p Ft(g)2525 1899 y Fn(2)2564 1884 y Fu(\))p Ft(\037)2663 1899 y Fn(3)2703 1884 y Fu(\()p Ft(g)2788 1899 y Fn(1)2827 1884 y Fu(\))p 2077 1928 788 4 v 2255 2019 a(1)g(+)g Ft(\037)2485 2034 y Fn(2)2525 2019 y Fu(\()p Ft(g)2610 2034 y Fn(2)2649 2019 y Fu(\))2875 1951 y Ft(a)2926 1966 y Fn(3)2965 1951 y Ft(a)3016 1966 y Fn(1)3056 1951 y Ft(a)3107 1910 y Fn(2)3107 1976 y(2)782 2319 y Fu(=)27 b(\()p Ft(I)e(a)p Fu(\))d(+)g(\()p Ft(I)i(b)p Fu(\))p Ft(;)-180 2535 y Fu(b)m(y)35 b(\(1.27\))d(and)g(\(1.28\).)43 b(By)33 b(the)g(same)g (equations)f(\(1.27\))g(and)h(\(1.28\))o(,)g(w)m(e)g(also)f(ha)m(v)m(e) 36 2792 y(\()p Ft(V)21 b(I)8 b(I)g(I)g Fu(\))27 b(=)h Fm(\000)561 2724 y Ft(\037)622 2739 y Fn(2)662 2724 y Fu(\()p Ft(g)747 2739 y Fn(1)786 2724 y Fu(\))824 2688 y Fn(2)863 2724 y Ft(\037)924 2739 y Fn(2)964 2724 y Fu(\()p Ft(g)1049 2739 y Fn(2)1088 2724 y Fu(\))p Ft(\037)1187 2739 y Fn(2)1226 2724 y Fu(\()p Ft(g)1311 2739 y Fn(3)1350 2724 y Fu(\))p Ft(\037)1449 2739 y Fn(3)1489 2724 y Fu(\()p Ft(g)1574 2739 y Fn(1)1613 2724 y Fu(\))p 561 2769 1090 4 v 890 2860 a(1)22 b(+)g Ft(\037)1120 2875 y Fn(2)1160 2860 y Fu(\()p Ft(g)1245 2875 y Fn(2)1284 2860 y Fu(\))1661 2792 y Ft(a)1712 2751 y Fn(2)1712 2816 y(2)1752 2792 y Ft(a)1803 2807 y Fn(3)1842 2792 y Ft(a)1893 2807 y Fn(1)1955 2792 y Fm(\000)2065 2724 y Ft(\037)2126 2739 y Fn(2)2165 2724 y Fu(\()p Ft(g)2250 2739 y Fn(1)2289 2724 y Fu(\))2327 2688 y Fn(2)2367 2724 y Ft(\037)2428 2739 y Fn(2)2467 2724 y Fu(\()p Ft(g)2552 2739 y Fn(2)2591 2724 y Fu(\))p Ft(\037)2690 2739 y Fn(2)2730 2724 y Fu(\()p Ft(g)2815 2739 y Fn(3)2854 2724 y Fu(\))p Ft(\037)2953 2739 y Fn(3)2992 2724 y Fu(\()p Ft(g)3077 2739 y Fn(1)3116 2724 y Fu(\))p Ft(\037)3215 2739 y Fn(3)3255 2724 y Fu(\()p Ft(g)3340 2739 y Fn(2)3379 2724 y Fu(\))3417 2688 y Fn(2)p 2065 2769 1392 4 v 2545 2860 a Fu(1)g(+)g Ft(\037)2775 2875 y Fn(2)2814 2860 y Fu(\()p Ft(g)2899 2875 y Fn(2)2938 2860 y Fu(\))3466 2792 y Ft(a)3517 2807 y Fn(3)3557 2792 y Ft(a)3608 2751 y Fn(2)3608 2816 y(2)3648 2792 y Ft(a)3699 2807 y Fn(1)370 3214 y Fu(=)28 b Fm(\000)561 3147 y Ft(\037)622 3162 y Fn(2)662 3147 y Fu(\()p Ft(g)747 3162 y Fn(1)786 3147 y Fu(\))824 3110 y Fn(2)863 3147 y Ft(\037)924 3162 y Fn(2)964 3147 y Fu(\()p Ft(g)1049 3162 y Fn(2)1088 3147 y Fu(\))p Ft(\037)1187 3162 y Fn(2)1226 3147 y Fu(\()p Ft(g)1311 3162 y Fn(3)1350 3147 y Fu(\))p 561 3191 828 4 v 759 3282 a(1)22 b(+)g Ft(\037)989 3297 y Fn(2)1029 3282 y Fu(\()p Ft(g)1114 3297 y Fn(2)1153 3282 y Fu(\))1398 3214 y Ft(a)1449 3173 y Fn(2)1449 3239 y(2)1489 3214 y Ft(a)1540 3229 y Fn(1)1580 3214 y Ft(a)1631 3229 y Fn(3)1692 3214 y Fm(\000)1802 3147 y Ft(\037)1863 3162 y Fn(2)1903 3147 y Fu(\()p Ft(g)1988 3162 y Fn(1)2027 3147 y Fu(\))2065 3110 y Fn(2)2104 3147 y Ft(\037)2165 3162 y Fn(2)2205 3147 y Fu(\()p Ft(g)2290 3162 y Fn(2)2329 3147 y Fu(\))p Ft(\037)2428 3162 y Fn(2)2467 3147 y Fu(\()p Ft(g)2552 3162 y Fn(3)2591 3147 y Fu(\))p Ft(\037)2690 3162 y Fn(3)2730 3147 y Fu(\()p Ft(g)2815 3162 y Fn(1)2854 3147 y Fu(\))p Ft(\037)2953 3162 y Fn(3)2992 3147 y Fu(\()p Ft(g)3077 3162 y Fn(2)3116 3147 y Fu(\))3154 3110 y Fn(2)p 1802 3191 1392 4 v 2282 3282 a Fu(1)g(+)g Ft(\037)2512 3297 y Fn(2)2552 3282 y Fu(\()p Ft(g)2637 3297 y Fn(2)2676 3282 y Fu(\))3204 3214 y Ft(a)3255 3229 y Fn(3)3294 3214 y Ft(a)3345 3173 y Fn(2)3345 3239 y(2)3385 3214 y Ft(a)3436 3229 y Fn(1)370 3582 y Fu(=)28 b(\()p Ft(V)21 b(I)8 b(I)g(I)24 b(a)p Fu(\))f(+)f(\()p Ft(V)f(I)8 b(I)g(I)24 b(b)p Fu(\))p Ft(:)-180 3798 y Fu(W)-8 b(e)33 b(next)g(use)h(\(1.26\))e(to)g(sho)m(w) i(that)1158 4015 y(\()p Ft(I)24 b(a)p Fu(\))e(+)g(\()p Ft(V)g Fu(\))g(+)g(\()p Ft(V)f(I)8 b Fu(\))22 b(+)g(\()p Ft(V)g(I)8 b(I)g(I)24 b(a)p Fu(\))k(=)f(0)p Ft(;)1103 4352 y Fu(\()p Ft(I)d(b)p Fu(\))f(+)f(\()p Ft(I)8 b(I)g(I)g Fu(\))22 b(+)g(\()p Ft(I)8 b(V)21 b Fu(\))h(+)g(\()p Ft(V)g(I)8 b(I)g(I)24 b(b)p Fu(\))k(=)f(0)p Ft(:)-180 4568 y Fu(In)k(the)g(course)g(of)f(the)h(pro)s(of)e(of)h(these)i (equalities,)e(w)m(e)h(need)h(\(1.24\))d(and)i(\(1.25\))o(.)43 b(This)31 b(\014nishes)g(the)g(pro)s(of)e(of)-180 4708 y(\(1.29\))o(.)p 3883 4708 4 66 v 3887 4645 59 4 v 3887 4708 V 3945 4708 4 66 v -80 4941 a(Let)i Ft(H)38 b Fu(b)s(e)31 b(a)f(Hopf)h(algebra.)41 b(Then)32 b(the)f(existence)i(of)d(an)h(in)m (tegral)e(for)h(\014nite-dimensional)e(braided)i(Hopf)-180 5081 y(algebras)i(implies)p eop %%Page: 14 14 14 13 bop -180 0 a Fq(14)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 214 y FD(Lemma)37 b(1.12.)49 b Fo(L)-5 b(et)38 b Ft(R)32 b Fu(=)880 139 y Fl(L)991 165 y Fs(N)991 243 y(n)p Fn(=0)1145 214 y Ft(R)q Fu(\()p Ft(n)p Fu(\))37 b Fo(b)-5 b(e)36 b(a)h(\014nite-dimensional)e(gr)-5 b(ade)g(d)36 b(br)-5 b(aide)g(d)36 b(Hopf)h(algebr)-5 b(a)36 b(in)3732 178 y Fs(H)3732 239 y(H)3799 214 y Fm(Y)8 b(D)-180 353 y Fo(with)34 b Ft(R)q Fu(\()p Ft(N)10 b Fu(\))28 b Fm(6)p Fu(=)g(0)p Fo(.)44 b(Ther)-5 b(e)33 b(exists)h Ft(\025)28 b Fm(2)g Ft(R)q Fu(\()p Ft(N)10 b Fu(\))34 b Fo(which)f(is)h(a)g(left)g(inte)-5 b(gr)g(al)34 b(on)g Ft(R)h Fo(and)e(such)h(that)h Ft(R)q Fu(\()p Ft(i)p Fu(\))20 b Fm(\012)h Ft(R)q Fu(\()p Ft(N)31 b Fm(\000)-180 493 y Ft(i)p Fu(\))d Fm(!)f Fp(|)-8 b Ft(;)110 b(x)22 b Fm(\012)h Ft(y)31 b Fm(7!)c Ft(\025)p Fu(\()p Ft(xy)t Fu(\))p Fo(,)34 b(is)h(a)g(non-de)-5 b(gener)g(ate)33 b(p)-5 b(airing,)34 b(for)g(al)5 b(l)35 b Fu(0)27 b Fm(\024)h Ft(i)g Fm(\024)h Ft(N)10 b Fo(.)45 b(In)34 b(p)-5 b(articular,)1359 680 y(dim)o Ft(R)q Fu(\()p Ft(i)p Fu(\))28 b(=)g Fo(dim)o Ft(R)q Fu(\()p Ft(N)33 b Fm(\000)23 b Ft(i)p Fu(\))p Ft(:)-180 894 y Fo(Pr)-5 b(o)g(of.)41 b Fu(This)36 b(is)f(essen)m (tially)f(due)j(to)e(Nic)m(hols)f([N,)j(1.5].)51 b(In)36 b(this)f(form)m(ulation,)e(one)j(needs)h(the)e(existence)i(of)-180 1033 y(non-zero)c(in)m(tegrals)e(on)i Ft(R)q Fu(;)f(this)g(follo)m(ws)g (from)f([FMS].)44 b(See)33 b([A)m(G,)g(Prop.)44 b(3.2.2])32 b(for)g(details.)p 3883 1033 4 66 v 3887 971 59 4 v 3887 1033 V 3945 1033 4 66 v -180 1222 a(1.7.)56 b FD(The)27 b(in\014nitesimal)d(braiding)j(of)g(Hopf)g(algebras)h(whose)g (coradical)e(is)g(a)i(Hopf)f(subalgebra.)-80 1338 y Fu(F)-8 b(or)41 b(the)h(con)m(v)m(enience)j(of)c(the)i(reader,)i(w)m(e)e (\014rst)f(recall)f(in)g(this)h(Section)g(some)g(basic)g(de\014nitions) f(from)-180 1454 y(coalgebra)32 b(theory)-8 b(.)-180 1668 y FD(De\014nition)36 b(1.13.)49 b Fu(Let)33 b Ft(C)39 b Fu(b)s(e)33 b(a)g(coalgebra.)-54 1832 y Fm(\017)41 b Ft(G)p Fu(\()p Ft(C)7 b Fu(\))27 b(:=)h Fm(f)p Ft(x)g Fm(2)g Ft(C)h Fm(n)22 b(f)p Fu(0)p Fm(g)27 b(j)h Fu(\001\()p Ft(x)p Fu(\))g(=)f Ft(x)c Fm(\012)g Ft(x)p Fm(g)32 b Fu(is)g(the)h(set)h(of)e(all)e(group-lik)m(e)i(elemen)m(ts)g(of)h Ft(C)7 b Fu(.)-54 2019 y Fm(\017)41 b Fu(If)32 b Ft(g)t(;)17 b(h)27 b Fm(2)h Ft(G)p Fu(\()p Ft(C)7 b Fu(\),)32 b(then)g Ft(x)c Fm(2)h Ft(C)38 b Fu(is)32 b(\()p Ft(g)t(;)17 b(h)p Fu(\))p Fo(-skew)32 b(primitive)g Fu(if)f(\001\()p Ft(x)p Fu(\))d(=)f Ft(x)22 b Fm(\012)f Ft(h)g Fu(+)g Ft(g)j Fm(\012)e Ft(x)p Fu(.)43 b(The)33 b(space)g(of)f(all)37 2158 y(\()p Ft(g)t(;)17 b(h)p Fu(\)-sk)m(ew)36 b(primitiv)m(e)e(elemen) m(ts)i(of)f Ft(C)42 b Fu(is)35 b(denoted)i(b)m(y)g Fm(P)8 b Fu(\()p Ft(C)f Fu(\))2423 2173 y Fs(g)r(;h)2523 2158 y Fu(.)53 b(If)36 b Ft(C)42 b Fu(is)35 b(a)h(bialgebra)d(or)j(a)f (braided)37 2298 y(bialgebra,)c(and)i Ft(g)d Fu(=)e Ft(h)g Fu(=)f(1,)33 b(then)g Ft(P)14 b Fu(\()p Ft(C)7 b Fu(\))27 b(=)g Fm(P)8 b Fu(\()p Ft(C)f Fu(\))1966 2313 y Fn(1)p Fs(;)p Fn(1)2094 2298 y Fu(is)32 b(the)h(space)g(of)f Fo(primitive)g Fu(elemen)m(ts.)-54 2485 y Fm(\017)41 b Fu(The)35 b Fo(c)-5 b(or)g(adic)g(al)34 b Fu(of)g Ft(C)41 b Fu(is)34 b Ft(C)1038 2500 y Fn(0)1108 2485 y Fu(:=)1242 2410 y Fl(P)1363 2485 y Ft(D)s Fu(,)h(where)h Ft(D)h Fu(runs)e(through)f(all)f(the)i(simple)d(sub)s(coalgebras)j(of)f Ft(C)7 b Fu(;)37 2624 y(it)32 b(is)g(the)h(largest)f(cosemisimple)e (sub)s(coalgebra)i(of)g Ft(C)7 b Fu(.)44 b(In)32 b(particular,)f Fp(|)-8 b Ft(G)p Fu(\()p Ft(C)6 b Fu(\))22 b Fm(\022)28 b Ft(C)3230 2639 y Fn(0)3270 2624 y Fu(.)-54 2811 y Fm(\017)41 b Ft(C)e Fu(is)33 b Fo(p)-5 b(ointe)g(d)31 b Fu(if)h Fp(|)-9 b Ft(G)p Fu(\()p Ft(C)7 b Fu(\))22 b(=)27 b Ft(C)1154 2826 y Fn(0)1193 2811 y Fu(.)-54 2999 y Fm(\017)41 b Fu(The)d Fo(c)-5 b(or)g(adic)g(al)38 b(\014ltr)-5 b(ation)37 b Fu(of)f Ft(C)44 b Fu(is)37 b(the)g(ascending)h(\014ltration)d Ft(C)2488 3014 y Fn(0)2562 2999 y Fm(\022)h Ft(C)2745 3014 y Fn(1)2820 2999 y Fm(\022)f(\001)17 b(\001)g(\001)34 b(\022)i Ft(C)3267 3014 y Fs(j)3338 2999 y Fm(\022)g Ft(C)3521 3014 y Fs(j)t Fn(+1)3683 2999 y Fm(\022)g Ft(:)17 b(:)g(:)f(;)37 3138 y Fu(de\014ned)44 b(b)m(y)g Ft(C)599 3153 y Fs(j)t Fn(+1)771 3138 y Fu(:=)i Fm(f)p Ft(x)g Fm(2)f Ft(C)53 b Fm(j)45 b Fu(\001\()p Ft(x)p Fu(\))h Fm(2)g Ft(C)1818 3153 y Fs(j)1883 3138 y Fm(\012)30 b Ft(C)36 b Fu(+)30 b Ft(C)36 b Fm(\012)29 b Ft(C)2484 3153 y Fn(0)2524 3138 y Fm(g)p Fu(.)74 b(This)44 b(is)e(a)h(coalgebra)f (\014ltration:)37 3278 y(\001)p Ft(C)188 3293 y Fs(j)252 3278 y Fm(\022)357 3203 y Fl(P)463 3307 y Fn(0)p Fk(\024)p Fs(i)p Fk(\024)p Fs(j)685 3278 y Ft(C)755 3293 y Fs(i)805 3278 y Fm(\012)23 b Ft(C)975 3293 y Fs(j)t Fk(\000)p Fs(i)1090 3278 y Fu(;)32 b(and)h(it)f(is)g(exhaustiv)m(e:)45 b Ft(C)34 b Fu(=)2261 3203 y Fl(S)2344 3307 y Fs(n)p Fk(\025)p Fn(0)2498 3278 y Ft(C)2568 3293 y Fs(n)2615 3278 y Fu(.)-54 3465 y Fm(\017)41 b Fu(A)48 b Fo(gr)-5 b(ade)g(d)48 b(c)-5 b(o)g(algebr)g(a)47 b Fu(is)g(a)h(coalgebra)e Ft(G)i Fu(pro)m(vided)g(with)g(a)g(grading)e Ft(G)54 b Fu(=)f Fm(\010)3143 3480 y Fs(n)p Fk(\025)p Fn(0)3281 3465 y Ft(G)p Fu(\()p Ft(n)p Fu(\))48 b(suc)m(h)h(that)37 3604 y(\001)p Ft(G)p Fu(\()p Ft(j)6 b Fu(\))28 b Fm(\022)450 3529 y Fl(P)555 3633 y Fn(0)p Fk(\024)p Fs(i)p Fk(\024)p Fs(j)777 3604 y Ft(G)p Fu(\()p Ft(i)p Fu(\))23 b Fm(\012)f Ft(G)p Fu(\()p Ft(j)28 b Fm(\000)23 b Ft(i)p Fu(\))32 b(for)h(all)d Ft(j)k Fm(\025)28 b Fu(0.)-54 3791 y Fm(\017)41 b Fu(A)47 b Fo(c)-5 b(or)g(adic)g(al)5 b(ly)46 b(gr)-5 b(ade)g(d)46 b Fu(coalgebra)f([CM)q(])i(is)f(a)g(graded)g(coalgebra)g Ft(G)51 b Fu(=)g Fm(\010)2996 3806 y Fs(n)p Fk(\025)p Fn(0)3134 3791 y Ft(G)p Fu(\()p Ft(n)p Fu(\))46 b(suc)m(h)i(that)e(its) 37 3931 y(coradical)29 b(\014ltration)f(coincides)j(with)f(the)h (standard)g(ascending)g(\014ltration)d(arising)h(from)g(the)i(grading:) 37 4070 y(\(gr)16 b Ft(C)7 b Fu(\))293 4085 y Fs(n)376 4070 y Fu(=)37 b Fm(\010)566 4085 y Fs(m)p Fk(\024)p Fs(n)730 4070 y Fu(gr)16 b Ft(C)7 b Fu(\()p Ft(m)p Fu(\))p Ft(:)38 b Fu(A)g Fo(strictly)i(gr)-5 b(ade)g(d)37 b Fu(coalgebra)g([Sw) q(])h(is)f(a)g(coradically)f(graded)i(coalgebra)37 4210 y Ft(G)32 b Fu(suc)m(h)j(that)d Ft(G)p Fu(\(0\))g(is)g (one-dimensional.)-54 4397 y Fm(\017)41 b Fu(The)25 b(graded)g (coalgebra)e(asso)s(ciated)i(to)f(the)h(coalgebra)e(\014ltration)f(of)i Ft(C)32 b Fu(is)24 b(gr)16 b Ft(C)34 b Fu(=)28 b Fm(\010)3200 4412 y Fs(n)p Fk(\025)p Fn(0)3337 4397 y Fu(gr)16 b Ft(C)7 b Fu(\()p Ft(n)p Fu(\),)27 b(where)37 4536 y(gr)16 b Ft(C)7 b Fu(\()p Ft(n)p Fu(\))28 b(:=)f Ft(C)579 4551 y Fs(n)626 4536 y Ft(=C)745 4551 y Fs(n)p Fk(\000)p Fn(1)882 4536 y Fu(,)32 b Ft(n)c(>)g Fu(0,)k(gr)16 b Ft(C)7 b Fu(\(0\))28 b(:=)f Ft(C)1772 4551 y Fn(0)1811 4536 y Fu(.)44 b(It)32 b(is)h(a)f(coradically)e(graded)j(coalgebra.)-80 4727 y(W)-8 b(e)33 b(shall)e(need)i(a)g(basic)f(tec)m(hnical)g(fact)h (on)f(p)s(oin)m(ted)g(coalgebras.)-180 4941 y FD(Lemma)37 b(1.14.)49 b Fu([M1)q(,)34 b(5.3.3])p Fo(.)47 b(A)36 b(morphism)f(of)h(p)-5 b(ointe)g(d)35 b(c)-5 b(o)g(algebr)g(as)34 b(which)i(is)f(inje)-5 b(ctive)35 b(in)h(the)g(\014rst)g(term)-180 5081 y(of)f(the)f(c)-5 b(o)g(algebr)g(a)34 b(\014ltr)-5 b(ation,)35 b(is)f(inje)-5 b(ctive.)p 3882 5081 4 68 v 3886 5017 60 4 v 3886 5081 V 3945 5081 4 68 v eop %%Page: 15 15 15 14 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(15)-80 203 y Fu(Let)28 b(no)m(w)h Ft(A)f Fu(b)s(e)g(a)g(Hopf)g (algebra.)41 b(W)-8 b(e)28 b(shall)f(assume)i(in)e(what)h(follo)m(ws)f (that)h(the)h(coradical)d Ft(A)3442 218 y Fn(0)3510 203 y Fu(is)h(not)h(only)-180 319 y(a)k(sub)s(coalgebra)g(but)h(a)g(Hopf)f (subalgebra)g(of)g Ft(A)p Fu(;)h(this)f(is)g(the)h(case)h(if)d Ft(A)i Fu(is)f(p)s(oin)m(ted.)-80 493 y(T)-8 b(o)43 b(study)i(the)e (structure)i(of)e Ft(A)p Fu(,)j(w)m(e)e(consider)g(its)f(coradical)e (\014ltration;)47 b(b)s(ecause)d(of)f(our)g(assumption)-180 609 y(on)c Ft(A)p Fu(,)i(it)d(is)g(also)g(an)h(algebra)f(\014ltration)f ([M1].)63 b(Therefore,)41 b(the)f(asso)s(ciated)f(graded)g(coalgebra)f (gr)16 b Ft(A)39 b Fu(is)f(a)-180 726 y(graded)k(Hopf)g(algebra.)70 b(F)-8 b(urthermore,)44 b Ft(H)51 b Fu(:=)43 b Ft(A)1759 741 y Fn(0)1842 726 y Fm(')h Fu(gr)16 b Ft(A)p Fu(\(0\))42 b(is)f(a)h(Hopf)g(subalgebra)f(of)h(gr)16 b Ft(A)p Fu(;)46 b(and)c(the)-180 842 y(pro)5 b(jection)38 b Ft(\031)j Fu(:)c(gr)16 b Ft(A)38 b Fm(!)e Fu(gr)16 b Ft(A)p Fu(\(0\))38 b(with)g(k)m(ernel)h Fm(\010)1736 857 y Fs(n>)p Fn(0)1874 842 y Fu(gr)16 b Ft(A)p Fu(\()p Ft(n)p Fu(\),)39 b(is)f(a)g(Hopf)g (algebra)f(map)g(and)i(a)f(retraction)-180 958 y(of)c(the)h(inclusion.) 48 b(W)-8 b(e)36 b(can)e(then)i(apply)e(the)h(general)f(remarks)h(of)f (Section)h(1.5.)49 b(Let)35 b Ft(R)h Fu(b)s(e)f(the)g(algebra)e(of)-180 1074 y(coin)m(v)-5 b(arian)m(ts)30 b(of)g Ft(\031)t Fu(;)h Ft(R)h Fu(is)e(a)g(braided)g(Hopf)h(algebra)e(in)1903 1038 y Fs(H)1903 1100 y(H)1970 1074 y Fm(Y)8 b(D)33 b Fu(and)e(gr)16 b Ft(A)31 b Fu(can)f(b)s(e)h(reconstructed)i(from)c Ft(R)j Fu(and)-180 1191 y Ft(H)40 b Fu(as)33 b(a)f(b)s(osonization)f (gr)16 b Ft(A)28 b Fm(')g Ft(R)q Fu(#)p Ft(H)8 b Fu(.)-80 1364 y(The)27 b(braided)f(Hopf)g(algebra)g Ft(R)h Fu(is)f(graded,)i (since)f(it)f(inherits)f(the)i(gradation)e(from)g(gr)16 b Ft(A)p Fu(:)41 b Ft(R)28 b Fu(=)g Fm(\010)3580 1379 y Fs(n)p Fk(\025)p Fn(0)3718 1364 y Ft(R)q Fu(\()p Ft(n)p Fu(\),)-180 1481 y(where)34 b Ft(R)q Fu(\()p Ft(n)p Fu(\))28 b(=)f(gr)16 b Ft(A)p Fu(\()p Ft(n)p Fu(\))23 b Fm(\\)f Ft(R)q Fu(.)44 b(F)-8 b(urthermore,)32 b Ft(R)h Fu(is)f(strictly)g (graded;)h(this)f(means,)-173 1626 y FD(\(a\))p Fu(.)41 b Ft(R)q Fu(\(0\))27 b(=)h Fp(|)-9 b Fu(1)27 b(\(hence)33 b(the)g(coradical)e(is)h(trivial,)f Fo(cf.)42 b Fu([Sw)q(,)33 b(Chapter)g(11]\).)-181 1800 y FD(\(b\))p Fu(.)41 b Ft(R)q Fu(\(1\))27 b(=)h Ft(P)14 b Fu(\()p Ft(R)q Fu(\))32 b(\(the)h(space)g (of)g(primitiv)m(e)d(elemen)m(ts)j(of)f Ft(R)q Fu(\).)-80 1945 y(It)g(is)g(in)g(general)g(not)h(true)g(that)f(a)g(braided)g(Hopf) h(algebra)e Ft(R)j Fu(satisfying)e FD(\(a\))g Fu(and)g FD(\(b\))p Fu(,)g(also)g(satis\014es)-169 2090 y FD(\(c\))p Fu(.)41 b Ft(R)34 b Fu(is)e(generated)h(as)g(an)f(algebra)g(o)m(v)m(er) h Fp(|)18 b Fu(b)m(y)34 b Ft(R)q Fu(\(1\).)-80 2235 y(A)i(braided)g (graded)g(Hopf)g(algebra)f(satisfying)g FD(\(a\))p Fu(,)i FD(\(b\))e Fu(and)i FD(\(c\))e Fu(is)g(called)g(a)h(Nic)m(hols)g (algebra.)53 b(In)36 b(the)-180 2351 y(next)h(c)m(hapter)g(w)m(e)f (will)e(discuss)j(this)e(notion)g(in)g(detail.)52 b(Notice)35 b(that)h(the)g(subalgebra)g Ft(R)3258 2315 y Fk(0)3317 2351 y Fu(of)f Ft(R)i Fu(generated)-180 2467 y(b)m(y)c Ft(R)q Fu(\(1\),)g(a)f(Hopf)h(subalgebra)f(of)g Ft(R)q Fu(,)h(is)f(indeed)h(a)f(Nic)m(hols)g(algebra.)-180 2688 y FD(De\014nition)k(1.15.)49 b Fu(The)34 b(braiding)1463 2845 y Ft(c)28 b Fu(:)f Ft(V)44 b Fm(\012)23 b Ft(V)49 b Fm(!)27 b Ft(V)44 b Fm(\012)23 b Ft(V)5 b(;)-180 3022 y Fu(of)32 b Ft(V)49 b Fu(:=)28 b Ft(R)q Fu(\(1\))f(=)h Ft(P)14 b Fu(\()p Ft(R)q Fu(\))32 b(is)g(called)f(the)i Fo(in\014nitesimal)h(br)-5 b(aiding)31 b Fu(of)h Ft(A)p Fu(.)-80 3161 y(The)h(graded)g(braided)f(Hopf)h(algebra)e Ft(R)j Fu(is)e(called)f(the)i Fo(diagr)-5 b(am)32 b Fu(of)g Ft(A)p Fu(.)-80 3301 y(The)h(dimension)e(of)h Ft(V)50 b Fu(=)27 b Ft(P)14 b Fu(\()p Ft(R)q Fu(\))32 b(is)g(called)g(the)h Fo(r)-5 b(ank)32 b Fu(of)g Ft(A)p Fu(.)1379 3570 y(2.)55 b Fv(Nichols)38 b(algebras)-80 3744 y Fu(Let)d Ft(H)42 b Fu(b)s(e)35 b(a)f(Hopf)h(algebra.)49 b(In)35 b(this)g(Chapter,)h(w)m (e)g(discuss)g(a)e(functor)h Fj(B)g Fu(from)e(the)j(category)3611 3708 y Fs(H)3611 3770 y(H)3678 3744 y Fm(Y)8 b(D)37 b Fu(to)-180 3861 y(the)k(category)f(of)g(braided)f(Hopf)h(algebras)g(in) 1629 3824 y Fs(H)1629 3886 y(H)1697 3861 y Fm(Y)8 b(D)r Fu(;)44 b(giv)m(en)c(a)g(Y)-8 b(etter-Drinfeld)38 b(mo)s(dule)h Ft(V)21 b Fu(,)43 b(the)d(braided)-180 3977 y(Hopf)32 b(algebra)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(is)f(called)f(the)i Fo(Nichols)i(algebr)-5 b(a)31 b Fu(of)h Ft(V)22 b Fu(.)-80 4151 y(The)36 b(structure)g(of)f(a)g(Nic)m(hols)f(algebra)h(app)s (eared)g(\014rst)h(in)e(the)i(pap)s(er)f("Bialgebras)f(of)g(t)m(yp)s(e) j(one\\)e([N])g(of)-180 4267 y(Nic)m(hols)c(and)g(w)m(as)i(redisco)m(v) m(ered)g(later)d(b)m(y)j(sev)m(eral)f(authors.)43 b(In)32 b(our)f(language,)g(a)g(bialgebra)e(of)i(t)m(yp)s(e)i(one)e(is)-180 4383 y(just)25 b(a)f(b)s(osonization)e Fj(B)p Fu(\()p Ft(V)g Fu(\)#)p Ft(H)8 b Fu(.)40 b(Hence)26 b(Nic)m(hols)e(algebras)f (are)i(the)g Ft(H)8 b Fu(-coin)m(v)-5 b(arian)m(t)22 b(elemen)m(ts)i(of)g(bialgebras)-180 4499 y(of)j(t)m(yp)s(e)i(one,)g (also)d(called)h(quan)m(tum)h(symmetric)e(algebras)h(in)g([Ro2].)42 b(Sev)m(eral)28 b(y)m(ears)g(after)g([N],)h(W)-8 b(orono)m(wicz)-180 4616 y(de\014ned)48 b(Nic)m(hols)d(algebras)h(in)g(his)g(approac)m(h)h (to)f("quan)m(tum)g(di\013eren)m(tial)e(calculus")i([W)-8 b(o];)53 b(again,)c(they)-180 4732 y(app)s(eared)30 b(as)g(the)h(in)m (v)-5 b(arian)m(t)28 b(part)i(of)f(his)h("algebra)f(of)g(quan)m(tum)h (di\013eren)m(tial)e(forms".)42 b(Lusztig's)30 b(algebras)f Fj(f)-180 4848 y Fu([L3],)35 b(de\014ned)h(b)m(y)f(the)g (non-degeneracy)h(of)e(a)g(certain)g(in)m(v)-5 b(arian)m(t)34 b(bilinear)e(form,)i(are)g(Nic)m(hols)g(algebras.)48 b(In)-180 4964 y(fact)37 b(Nic)m(hols)f(algebras)g(can)h(alw)m(a)m(ys)g (b)s(e)h(de\014ned)g(b)m(y)g(the)f(non-degeneracy)h(of)e(an)h(in)m(v)-5 b(arian)m(t)36 b(bilinear)e(form)-180 5081 y([A)m(G].)44 b(The)33 b(algebras)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))32 b(are)h(called)e(bitensor)h(algebras)g(in)g([Sbg].)p eop %%Page: 16 16 16 15 bop -180 0 a Fq(16)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(In)40 b(a)f(sense,)k(Nic)m(hols)c (algebras)g(are)h(similar)c(to)j(symmetric)g(algebras;)j(indeed,)g(b)s (oth)d(notions)g(coincide)-180 319 y(in)34 b(the)h(trivial)d(braided)i (category)h(of)f(v)m(ector)i(spaces,)h(or)d(more)g(generally)f(in)h(an) m(y)h(symmetric)f(category)h(\()p Fo(e.)-180 435 y(g.)54 b Fu(in)36 b(the)h(category)f(of)g(sup)s(er)h(v)m(ector)h(spaces\).)56 b(But)36 b(when)i(the)e(braiding)f(is)h(not)g(a)g(symmetry)-8 b(,)37 b(a)f(Nic)m(hols)-180 552 y(algebra)e(could)h(ha)m(v)m(e)i(a)e (m)m(uc)m(h)g(ric)m(her)h(structure.)53 b(W)-8 b(e)35 b(hop)s(e)h(that)f(this)g(will)e(b)s(e)i(clari\014ed)g(in)f(the)i (examples.)-180 668 y(On)h(the)g(other)f(hand,)i(Nic)m(hols)e(algebras) g(are)h(also)e(similar)e(to)k(univ)m(ersal)f(en)m(v)m(eloping)h (algebras.)54 b(Ho)m(w)m(ev)m(er,)-180 784 y(in)31 b(spite)h(of)f(the)h (e\013orts)g(of)g(sev)m(eral)g(authors,)g(it)f(is)g(not)h(clear)f(to)g (us)i(ho)m(w)f(to)g(ac)m(hiev)m(e)h(a)e(compact,)h(functorial)-180 900 y(de\014nition)g(of)g(a)g("braided)g(Lie)g(algebra")f(from)g(a)i (Nic)m(hols)f(algebra.)-80 1064 y(W)-8 b(e)33 b(b)s(eliev)m(e)f(that)g (Nic)m(hols)g(algebras)g(are)h(v)m(ery)h(in)m(teresting)e(ob)5 b(jects)34 b(of)e(an)g(essen)m(tially)h(new)g(nature.)-180 1297 y(2.1.)56 b FD(De\014nition)36 b(of)i(Nic)m(hols)d(algebras.)-80 1414 y Fu(W)-8 b(e)33 b(no)m(w)g(presen)m(t)h(one)f(of)f(the)h(main)e (notions)h(of)g(this)g(surv)m(ey)-8 b(.)-180 1627 y FD(De\014nition)36 b(2.1.)49 b Fu(Let)c Ft(V)67 b Fu(b)s(e)45 b(a)f(Y)-8 b(etter-Drinfeld)43 b(mo)s(dule)h(o)m(v)m(er)i Ft(H)8 b Fu(.)80 b(A)45 b(braided)f(graded)h(Hopf)g(algebra)-180 1767 y Ft(R)29 b Fu(=)e Fm(\010)103 1782 y Fs(n)p Fk(\025)p Fn(0)241 1767 y Ft(R)q Fu(\()p Ft(n)p Fu(\))33 b(in)596 1731 y Fs(H)596 1792 y(H)664 1767 y Fm(Y)8 b(D)35 b Fu(is)d(called)f(a) h Fo(Nichols)j(algebr)-5 b(a)31 b Fu(of)h Ft(V)55 b Fu(if)31 b Fp(|)13 b Fm(')28 b Ft(R)q Fu(\(0\))k(and)h Ft(V)49 b Fm(')28 b Ft(R)q Fu(\(1\))33 b(in)3459 1731 y Fs(H)3459 1792 y(H)3527 1767 y Fm(Y)8 b(D)r Fu(,)33 b(and)70 1953 y Ft(P)14 b Fu(\()p Ft(R)q Fu(\))27 b(=)h Ft(R)q Fu(\(1\))p Ft(;)-836 b Fu(\(2.1\))70 2122 y Ft(R)34 b Fu(is)e(generated)h(as)g(an) g(algebra)e(b)m(y)j Ft(R)q Fu(\(1\))p Ft(:)-1860 b Fu(\(2.2\))-180 2308 y(The)34 b(dimension)d(of)h Ft(V)54 b Fu(will)30 b(b)s(e)j(called)e(the)i Fo(r)-5 b(ank)32 b Fu(of)g Ft(R)q Fu(.)-80 2498 y(W)-8 b(e)35 b(need)h(some)f(preliminaries)d(to)i(sho)m (w)j(the)e(existence)i(and)e(uniqueness)i(of)d(the)i(Nic)m(hols)e (algebra)g(of)g Ft(V)-180 2615 y Fu(in)-66 2578 y Fs(H)-66 2640 y(H)1 2615 y Fm(Y)8 b(D)r Fu(.)-80 2778 y(Let)39 b Ft(V)60 b Fu(b)s(e)39 b(a)f(Y)-8 b(etter-Drinfeld)37 b(mo)s(dule)g(o)m(v)m(er)j Ft(H)8 b Fu(.)61 b(Then)40 b(the)f(tensor)g(algebra)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))38 b(=)3330 2703 y Fl(L)3441 2807 y Fs(n)p Fk(\025)p Fn(0)3595 2778 y Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\)\()p Ft(n)p Fu(\))-180 2908 y(of)41 b(the)h(v)m(ector)h(space)g Ft(V)63 b Fu(admits)40 b(a)h(natural)g(structure)i(of)e(a)g(Y)-8 b(etter-Drinfeld)40 b(mo)s(dule,)j(since)3535 2872 y Fs(H)3535 2933 y(H)3602 2908 y Fm(Y)8 b(D)44 b Fu(is)d(a)-180 3024 y(braided)33 b(category)-8 b(.)47 b(It)34 b(is)f(then)i(an)e (algebra)g(in)1623 2988 y Fs(H)1623 3049 y(H)1690 3024 y Fm(Y)8 b(D)r Fu(.)47 b(There)35 b(exists)f(a)g(unique)g(algebra)e (map)h(\001)d(:)g Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))29 b Fm(!)-180 3140 y Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))p Fm(\012)p 45 3165 78 4 v 1 w Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))32 b(suc)m(h)i(that)f(\001\()p Ft(v)t Fu(\))27 b(=)h Ft(v)e Fm(\012)c Fu(1)g(+)g(1)g Fm(\012)h Ft(v)t Fu(,)32 b(for)g(all)f Ft(v)g Fm(2)d Ft(V)22 b Fu(.)43 b(F)-8 b(or)32 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Fu(\()p Ft(V)21 b Fu(\).)42 b(Let)28 b(then)-166 4532 y Fl(e)-180 4558 y Fj(S)j Fu(b)s(e)h(the)g(subset)h(of)e Fj(S)f Fu(consisting)h(of)g(all)f Ft(I)35 b Fm(2)28 b Fj(S)j Fu(whic)m(h)h(are)g(Y)-8 b(etter-Drinfeld)29 b(submo)s(dules)j (of)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\).)43 b(The)-180 4674 y(ideals)1211 4797 y Ft(I)8 b Fu(\()p Ft(V)21 b Fu(\))28 b(=)1547 4702 y Fl(X)1549 4914 y Fs(I)5 b Fk(2)p Fe(S)1708 4797 y Ft(I)j(;)2006 4772 y Fl(e)1998 4797 y Ft(I)g Fu(\()p Ft(V)21 b Fu(\))28 b(=)2337 4702 y Fl(X)2334 4934 y Fs(J)6 b Fk(2)2435 4916 y Ff(e)2426 4934 y Fe(S)2500 4797 y Ft(J)-180 5081 y Fu(are)33 b(the)g(largest)e(elemen)m(ts)i(in)f Fj(S)p Fu(,)g(resp)s(ectiv)m(ely)1666 5055 y Fl(e)1652 5081 y Fj(S)o Fu(.)p eop %%Page: 17 17 17 16 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(17)-80 203 y Fu(If)32 b Ft(I)k Fm(2)28 b Fj(S)k Fu(then)h Ft(R)c Fu(:=)e Ft(T)14 b Fu(\()p Ft(V)22 b Fu(\))p Ft(=I)35 b Fu(=)27 b Fm(\010)1293 218 y Fs(n)p Fk(\025)p Fn(0)1431 203 y Ft(R)q Fu(\()p Ft(n)p Fu(\))33 b(is)f(a)g(graded)h(algebra)e(and) i(a)f(graded)h(coalgebra)f(with)1175 368 y Ft(R)q Fu(\(0\))c(=)f Fp(|)-9 b Ft(;)206 b(V)49 b Fm(')29 b Ft(R)q Fu(\(1\))e Fm(\032)h Ft(P)14 b Fu(\()p Ft(R)q Fu(\))p Ft(:)-180 548 y Fu(If)33 b(actually)e Ft(I)k Fm(2)475 522 y Fl(e)461 548 y Fj(S)p Fu(,)e(then)g Ft(R)g Fu(is)f(a)h(graded)f(braided)h(Hopf)f (algebra)g(in)2478 512 y Fs(H)2478 574 y(H)2546 548 y Fm(Y)8 b(D)r Fu(.)-80 714 y(W)-8 b(e)33 b(can)f(sho)m(w)i(no)m(w)f (existence)h(and)f(uniqueness)i(of)d(Nic)m(hols)g(algebras.)-180 928 y FD(Prop)s(osition)k(2.2.)49 b Fo(L)-5 b(et)35 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))28 b(:=)g Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))p Ft(=)1501 903 y Fl(e)1492 928 y Ft(I)7 b Fu(\()p Ft(V)22 b Fu(\))p Fo(.)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(hold:)-80 1092 y Fu(1.)41 b Ft(V)49 b Fu(=)28 b Ft(P)14 b Fu(\()p Fj(B)p Fu(\()p Ft(V)21 b Fu(\)\))p Fo(,)34 b(henc)-5 b(e)34 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))35 b Fo(is)f(a)h(Nichols)f(algebr)-5 b(a)34 b(of)h 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b Fu(1.)56 b(W)-8 b(e)37 b(ha)m(v)m(e)h(to)e(sho)m(w)i(the)f(equalit)m(y)g Ft(V)56 b Fu(=)35 b Ft(P)14 b Fu(\()p Fj(B)p Fu(\()p Ft(V)21 b Fu(\)\).)55 b(Let)37 b(us)h(consider)f(the)g(in)m(v)m(erse)h (image)d Ft(X)45 b Fu(in)-180 2758 y Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))28 b(of)g(all)e(homogeneous)i(primitiv)m(e)d(elemen)m (ts)k(of)e Fj(B)p Fu(\()p Ft(V)22 b Fu(\))28 b(in)f(degree)i Ft(n)f Fm(\025)g Fu(2.)42 b(Then)29 b Ft(X)35 b Fu(is)28 b(a)g(graded)g(Y)-8 b(etter-)-180 2898 y(Drinfeld)25 b(submo)s(dule)h(of)g Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\),)28 b(and)f(for)f(all)f Ft(x)j Fm(2)g Ft(X)8 b Fu(,)28 b(\001\()p Ft(x)p Fu(\))g Fm(2)g Ft(x)10 b Fm(\012)g Fu(1)g(+)g(1)g Fm(\012)g Ft(x)g Fu(+)g Ft(T)k Fu(\()p Ft(V)24 b Fu(\))10 b Fm(\012)3106 2873 y Fl(e)3096 2898 y Ft(I)f Fu(\()p Ft(V)21 b Fu(\))10 b(+)3407 2873 y Fl(e)3398 2898 y Ft(I)e Fu(\()p Ft(V)22 b Fu(\))10 b Fm(\012)g Ft(T)k Fu(\()p Ft(V)22 b Fu(\))p Ft(:)-180 3037 y Fu(Hence)41 b(the)g(ideal)d (generated)j(b)m(y)1136 3012 y Fl(e)1128 3037 y Ft(I)7 b Fu(\()p Ft(V)22 b Fu(\))40 b(and)g Ft(X)47 b Fu(is)40 b(in)1939 3011 y Fl(e)1925 3037 y Fj(S)p Fu(,)h(and)f Ft(X)48 b Fm(\032)2529 3012 y Fl(e)2520 3037 y Ft(I)8 b Fu(\()p Ft(V)22 b Fu(\))39 b(b)m(y)i(the)g(maximalit)m(y)36 b(of)3730 3012 y Fl(e)3721 3037 y Ft(I)8 b Fu(\()p Ft(V)22 b Fu(\).)-180 3177 y(Hence)32 b(the)g(image)d(of)h Ft(X)39 b Fu(in)30 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))31 b(is)g(zero.)43 b(This)31 b(pro)m(v)m(es)i(our)d(claim)f(since)i(the)h(primitiv)m(e)c (elemen)m(ts)j(form)f(a)-180 3316 y(graded)j(submo)s(dule.)-80 3456 y(2.)42 b(W)-8 b(e)29 b(ha)m(v)m(e)h(to)f(sho)m(w)h(that)f(the)g (surjectiv)m(e)i(map)d Fj(B)p Fu(\()p Ft(V)21 b Fu(\))28 b Fm(!)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))p Ft(=I)8 b Fu(\()p Ft(V)22 b Fu(\))28 b(is)h(bijectiv)m(e.)42 b(This)29 b(follo)m(ws)f(from)-180 3595 y(1.)43 b(and)33 b(Lemma)e(1.14.)-80 3735 y(3.)43 b(The)33 b(k)m(ernel)h Ft(I)40 b Fu(of)32 b(the)h(canonical)e(pro)5 b(jection)33 b Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))27 b Fm(!)h Ft(R)33 b Fu(b)s(elongs)f(to)2751 3709 y Fl(e)2738 3735 y Fj(S)o Fu(;)h(hence)h Ft(I)h Fm(\022)3343 3710 y Fl(e)3334 3735 y Ft(I)8 b Fu(\()p Ft(V)22 b Fu(\).)-80 3874 y(4.)43 b(follo)m(ws)31 b(again)g(from)h(Lemma)f(1.14,)h(as)h(in)e(2.)-80 4014 y(5.)43 b(follo)m(ws)31 b(from)g(4.)p 3883 4014 4 66 v 3887 3951 59 4 v 3887 4014 V 3945 4014 4 66 v -80 4190 a(If)c Ft(U)38 b Fu(is)27 b(a)h(braided)f(subspace)j(of)d Ft(V)49 b Fm(2)1342 4154 y Fs(H)1342 4215 y(H)1409 4190 y Fm(Y)8 b(D)s Fu(,)28 b(that)g(is)f(a)g(subspace)j(suc)m(h)f(that)e Ft(c)p Fu(\()p Ft(U)c Fm(\012)12 b Ft(U)e Fu(\))28 b Fm(\032)h Ft(U)22 b Fm(\012)12 b Ft(U)e Fu(,)30 b(where)f Ft(c)-180 4306 y Fu(is)i(the)h(braiding)d(of)i Ft(V)22 b Fu(,)31 b(w)m(e)h(can)g(de\014ne)h Fj(B)p Fu(\()p Ft(U)10 b Fu(\))28 b(:=)g Ft(T)14 b Fu(\()p Ft(U)c Fu(\))p Ft(=I)e Fu(\()p Ft(U)i Fu(\))31 b(with)g(the)h(ob)m(vious)f(meaning)f(of)h Ft(I)8 b Fu(\()p Ft(U)i Fu(\).)44 b(Then)-180 4422 y(the)33 b(description)f(stated)h(in)f(Prop)s(osition)f(2.8)h(also)g(applies)f (to)i Fj(B)p Fu(\()p Ft(U)10 b Fu(\).)-180 4637 y FD(Corollary)37 b(2.3.)50 b Fo(The)41 b(assignment)f Ft(V)62 b Fm(7!)39 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))41 b Fo(is)h(a)f(functor)h(fr)-5 b(om)2590 4601 y Fs(H)2590 4662 y(H)2657 4637 y Fm(Y)8 b(D)44 b Fo(to)e(the)g(c)-5 b(ate)g(gory)41 b(of)h(br)-5 b(aide)g(d)-180 4777 y(Hopf)35 b(algebr)-5 b(as)34 b(in)546 4741 y Fs(H)546 4802 y(H)613 4777 y Fm(Y)8 b(D)s Fo(.)-80 4916 y(If)39 b Ft(U)51 b Fo(is)40 b(a)g(Y)-7 b(etter-Drinfeld)40 b(submo)-5 b(dule)39 b(of)h Ft(V)22 b Fo(,)41 b(or)f(mor)-5 b(e)40 b(gener)-5 b(al)5 b(ly)40 b(if)g Ft(U)51 b Fo(is)40 b(a)g(br)-5 b(aide)g(d)39 b(subsp)-5 b(ac)g(e)39 b(of)h Ft(V)22 b Fo(,)-180 5056 y(then)35 b(the)g(c)-5 b(anonic)g(al)33 b(map)h Fj(B)p Fu(\()p Ft(U)10 b Fu(\))28 b Fm(!)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))35 b Fo(is)g(inje)-5 b(ctive.)p eop %%Page: 18 18 18 17 bop -180 0 a Fq(18)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fo(Pr)-5 b(o)g(of.)41 b Fu(If)33 b Ft(\036)28 b Fu(:)h Ft(U)39 b Fm(!)29 b Ft(V)54 b Fu(is)33 b(a)g(morphism)e(in)1454 167 y Fs(H)1454 228 y(H)1522 203 y Fm(Y)8 b(D)r Fu(,)33 b(then)h Ft(T)14 b Fu(\()p Ft(\036)p Fu(\))28 b(:)h Ft(T)14 b Fu(\()p Ft(U)c Fu(\))28 b Fm(!)h Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))33 b(is)f(a)h(morphism)e(of)i(braided)-180 342 y(Hopf)f(algebras.)43 b(Since)32 b Ft(T)14 b Fu(\()p Ft(\036)p Fu(\)\()p Ft(I)8 b Fu(\()p Ft(U)i Fu(\)\))32 b(is)g(a)g(coideal)f(and)i(a)f(Y)-8 b(etter-Drinfeld)30 b(submo)s(dule)i(of)g Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\),)33 b(the)f(ideal)-180 482 y(generated)h(b)m(y)h Ft(T)14 b Fu(\()p Ft(\036)p Fu(\)\()652 457 y Fl(e)643 482 y Ft(I)7 b Fu(\()p Ft(U)j Fu(\)\))33 b(is)f(con)m(tained)h(in)1578 457 y Fl(e)1570 482 y Ft(I)7 b Fu(\()p Ft(V)22 b Fu(\).)43 b(Hence)34 b(b)m(y)g(Prop)s(osition)d(2.2,)h Fj(B)g Fu(is)g(a)h (functor.)-80 621 y(The)g(second)h(part)e(of)g(the)h(claim)e(follo)m (ws)g(from)g(Prop)s(osition)g(2.2,)h(5.)p 3883 621 4 66 v 3887 559 59 4 v 3887 621 V 3945 621 4 66 v -80 787 a(The)f(dualit)m(y)f(b)s(et)m(w)m(een)j(conditions)d(\(2.1\))h(and)g (\(2.2\))f(in)g(the)i(de\014nition)e(of)g(Nic)m(hols)g(algebra,)h (emphasized)-180 904 y(b)m(y)i(P)m(arts)h(3)e(and)h(5)f(of)g(Prop)s (osition)f(2.2,)h(is)g(explicitly)f(stated)i(in)f(the)h(follo)m(wing) -180 1099 y FD(Lemma)k(2.4.)49 b Fo(L)-5 b(et)52 b Ft(R)60 b Fu(=)f Fm(\010)971 1114 y Fs(n)p Fk(\025)p Fn(0)1108 1099 y Ft(R)q Fu(\()p Ft(n)p Fu(\))52 b Fo(b)-5 b(e)52 b(a)f(gr)-5 b(ade)g(d)51 b(br)-5 b(aide)g(d)51 b(Hopf)h(algebr)-5 b(a)50 b(in)3018 1063 y Fs(H)3018 1125 y(H)3085 1099 y Fm(Y)8 b(D)r Fo(;)60 b(supp)-5 b(ose)51 b(\014nite-)-180 1239 y(dimensional)41 b(homo)-5 b(gene)g(ous)41 b(c)-5 b(omp)g(onents)42 b(and)g Ft(R)q Fu(\(0\))g(=)g Fp(|)-8 b Fu(1)o Fo(.)63 b(L)-5 b(et)43 b Ft(S)48 b Fu(=)43 b Fm(\010)2738 1254 y Fs(n)p Fk(\025)p Fn(0)2875 1239 y Ft(R)q Fu(\()p Ft(n)p Fu(\))3084 1203 y Fk(\003)3167 1239 y Fo(b)-5 b(e)42 b(the)h(gr)-5 b(ade)g(d)42 b(dual)-180 1378 y(of)35 b Ft(R)q Fo(.)44 b(Then)34 b Ft(R)q Fu(\(1\))28 b(=)f Ft(P)14 b Fu(\()p Ft(R)q Fu(\))35 b Fo(if)f(and)h(only)f(if)h Ft(S)41 b Fo(is)34 b(gener)-5 b(ate)g(d)34 b(as)h(an)f(algebr)-5 b(a)34 b(by)h Ft(S)6 b Fu(\(1\))p Ft(:)-180 1574 y Fo(Pr)-5 b(o)g(of.)41 b Fu(See)34 b(for)e(instance)h([AS2,)f(Lemma)f(5.5].)p 3883 1574 V 3887 1511 59 4 v 3887 1574 V 3945 1574 4 66 v -180 1770 a FD(Example)36 b(2.5.)50 b Fu(Let)38 b Ft(F)52 b Fu(b)s(e)38 b(a)g(\014eld)g(of)g(p)s(ositiv)m(e)f(c)m (haracteristic)h Ft(p)p Fu(.)61 b(Let)38 b Ft(S)44 b Fu(b)s(e)39 b(the)f(\(usual\))g(Hopf)g(algebra)-180 1909 y Ft(F)14 b Fu([)p Ft(x)p Fu(])p Ft(=)p Fm(h)p Ft(x)149 1873 y Fs(p)185 1850 y Fg(2)224 1909 y Fm(i)32 b Fu(with)g Ft(x)c Fm(2)g Ft(P)14 b Fu(\()p Ft(S)6 b Fu(\).)43 b(Then)33 b Ft(x)1292 1873 y Fs(p)1360 1909 y Fm(2)28 b Ft(P)14 b Fu(\()p Ft(S)6 b Fu(\).)43 b(Hence)34 b Ft(R)29 b Fu(=)e Ft(S)2305 1873 y Fk(\003)2377 1909 y Fu(satis\014es)33 b(\(2.2\))f(but)h(not)f(\(2.1\).)-180 2105 y FD(Example)k(2.6.)50 b Fu(Let)34 b Ft(S)i Fu(=)30 b Fp(|)-9 b Fu([)p Ft(X)8 b Fu(])25 b(=)30 b Fm(\010)1296 2120 y Fs(n)p Fk(\025)p Fn(0)1433 2105 y Ft(S)6 b Fu(\()p Ft(n)p Fu(\))34 b(b)s(e)h(a)f(p)s (olynomial)c(algebra)j(in)g(one)i(v)-5 b(ariable.)46 b(W)-8 b(e)34 b(consider)-180 2244 y Ft(S)k Fu(as)32 b(a)g(braided)g(Hopf)g(algebra)f(in)1162 2208 y Fs(H)1162 2269 y(H)1229 2244 y Fm(Y)8 b(D)s Fu(,)32 b(where)h Ft(H)j Fu(=)27 b Fp(|)-9 b Fu(\000,)27 b(\000)32 b(an)g(in\014nite)f(cyclic)h (group)g(with)f(generator)h Ft(g)t Fu(,)-180 2384 y(with)g(action,)g (coaction)g(and)g(com)m(ultiplication)c(giv)m(en)33 b(b)m(y)618 2564 y Ft(\016)t Fu(\()p Ft(X)792 2523 y Fs(n)839 2564 y Fu(\))28 b(=)f Ft(g)1059 2523 y Fs(n)1128 2564 y Fm(\012)22 b Ft(X)1316 2523 y Fs(n)1363 2564 y Ft(;)114 b(g)26 b Fm(\001)21 b Ft(X)36 b Fu(=)27 b Ft(q)t(X)r(;)115 b Fu(\001\()p Ft(X)8 b Fu(\))27 b(=)h Ft(X)i Fm(\012)23 b Fu(1)f(+)g(1)f Fm(\012)i Ft(X)r(:)-180 2744 y Fu(Here)31 b Ft(q)g Fm(2)d Fp(|)16 b Fu(is)29 b(a)h(ro)s(ot)f(of)h(1)g(of)g(order)g Ft(N)10 b Fu(.)43 b(That)30 b(is,)h Ft(S)k Fu(is)30 b(a)g(so-called)f (quan)m(tum)h(line.)41 b(Then)32 b Ft(S)j Fu(satis\014es)c(\(2.2\))-180 2884 y(but)e(not)g(\(2.1\))f(since)i Ft(X)719 2847 y Fs(N)815 2884 y Fu(is)e(also)g(primitiv)m(e.)40 b(Hence)31 b(the)e(graded)g(dual)f Ft(R)h Fu(=)f Ft(S)2808 2847 y Fs(d)2876 2884 y Fu(=)f Fm(\010)3056 2899 y Fs(n)p Fk(\025)p Fn(0)3194 2884 y Ft(S)6 b Fu(\()p Ft(n)p Fu(\))3394 2847 y Fk(\003)3462 2884 y Fu(is)28 b(a)h(braided)-180 3023 y(Hopf)j(algebra)g(satisfying)h(\(2.1\))f(but)h(not)f(\(2.2\).)-80 3195 y(Ho)m(w)m(ev)m(er,)j(in)e(c)m(haracteristic)g(0)g(w)m(e)h(do)f (not)g(kno)m(w)h(an)m(y)g(\014nite)f(dimensional)e(example)h(of)h(a)g (braided)g(Hopf)-180 3312 y(algebra)e(satisfying)i(\(2.1\))f(but)h(not) g(\(2.2\))o(.)-180 3507 y FD(Conjecture)38 b(2.7.)49 b Fu([AS2,)j(Conjecture)d(1.4])f Fo(A)n(ny)h(\014nite)g(dimensional)e (br)-5 b(aide)g(d)48 b(Hopf)h(algebr)-5 b(a)48 b(in)3732 3471 y Fs(H)3732 3533 y(H)3799 3507 y Fm(Y)8 b(D)-180 3647 y Fo(satisfying)43 b Fu(\(2.1\))35 b Fo(also)f(satis\014es)42 b Fu(\(2.2\))p Fo(.)i(\(R)-5 b(e)g(c)g(al)5 b(l)34 b(that)i(the)f(b)-5 b(ase)34 b(\014eld)g Fp(|)20 b Fo(has)35 b(char)-5 b(acteristic)34 b(zer)-5 b(o.\))-80 3819 y Fu(The)43 b(compact)f(description)g(of)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))42 b(in)g(Lemma)f(2.2)h(sho)m(ws)i(that) e(it)f(dep)s(ends)j(only)e(on)g(the)h(algebra)-180 3935 y(and)f(coalgebra)f(structure)i(of)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\).)71 b(Since)42 b(the)h(com)m(ultiplication)37 b(of)42 b(the)g(tensor)h(algebra)d(w)m(as)j(de\014ned)-180 4052 y(using)36 b(the)g("t)m(wisted")h(m)m(ultiplication)31 b(of)36 b Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))p Fm(\012)p 1660 4076 78 4 v 1 w Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\),)37 b(w)m(e)g(see)g(that)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))36 b Fo(dep)-5 b(ends)37 b(as)g(an)h(algebr)-5 b(a)37 b(and)-180 4168 y(c)-5 b(o)g(algebr)g(a)36 b(only)i(on)f(the)h(br)-5 b(aiding)36 b(of)i Ft(V)21 b Fu(.)53 b(The)36 b(explicit)e(form)m(ula)g (for)h(the)h(com)m(ultiplication)31 b(of)k Ft(T)14 b Fu(\()p Ft(V)22 b Fu(\))35 b(leads)-180 4284 y(to)d(the)h(follo)m(wing) d(alternativ)m(e)i(description)g(of)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\).)-180 4499 y(2.2.)56 b FD(The)37 b(braid)h(group.)-80 4615 y Fu(Let)32 b(us)i(recall)d(that)h(the)h(braid)f(group)g Fp(B)1451 4630 y Fs(n)1536 4615 y Fu(is)g(presen)m(ted)j(b)m(y)e (generators)g Ft(\033)2733 4630 y Fn(1)2773 4615 y Ft(;)17 b(:)g(:)g(:)f(;)h(\033)3047 4630 y Fs(n)p Fk(\000)p Fn(1)3217 4615 y Fu(with)32 b(relations)868 4772 y Ft(\033)923 4787 y Fs(i)951 4772 y Ft(\033)1006 4787 y Fs(i)p Fn(+1)1125 4772 y Ft(\033)1180 4787 y Fs(i)1236 4772 y Fu(=)c Ft(\033)1395 4787 y Fs(i)p Fn(+1)1514 4772 y Ft(\033)1569 4787 y Fs(i)1597 4772 y Ft(\033)1652 4787 y Fs(i)p Fn(+1)1771 4772 y Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(n)22 b Fm(\000)h Fu(2)p Ft(;)1033 4923 y(\033)1088 4938 y Fs(i)1117 4923 y Ft(\033)1172 4938 y Fs(j)1236 4923 y Fu(=)28 b Ft(\033)1395 4938 y Fs(j)1432 4923 y Ft(\033)1487 4938 y Fs(i)1515 4923 y Ft(;)17 b(;)114 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(n)22 b Fm(\000)h Fu(2)p Ft(;)49 b Fm(j)p Ft(i)22 b Fm(\000)h Ft(j)6 b Fm(j)27 b Ft(>)g Fu(1)p Ft(:)-180 5081 y Fu(Here)33 b(are)g(some)f(basic)h(w)m(ell-kno)m (wn)f(facts)h(ab)s(out)f(the)h(braid)f(group.)p eop %%Page: 19 19 19 18 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(19)-80 203 y Fu(There)34 b(is)e(a)h(natural)f(pro)5 b(jection)33 b Ft(\031)f Fu(:)c Fp(B)1391 218 y Fs(n)1472 203 y Fm(!)g Fp(S)1661 218 y Fs(n)1735 203 y Fu(sending)34 b Ft(\033)2144 218 y Fs(i)2205 203 y Fu(to)f(the)g(transp)s(osition)f Ft(\034)3125 218 y Fs(i)3182 203 y Fu(:=)c(\()p Ft(i;)17 b(i)23 b Fu(+)f(1\))32 b(for)h(all)-180 319 y Ft(i)p Fu(.)44 b(The)33 b(pro)5 b(jection)33 b Ft(\031)j Fu(admits)31 b(a)i(set-theoretical)e(section)i Ft(s)28 b Fu(:)f Fp(S)2231 334 y Fs(n)2299 319 y Fm(!)h Fp(B)2487 334 y Fs(n)2573 319 y Fu(determined)k(b)m(y)1020 512 y Ft(s)p Fu(\()p Ft(\034)1146 527 y Fs(i)1175 512 y Fu(\))27 b(=)h Ft(\033)1399 527 y Fs(i)1428 512 y Ft(;)114 b Fu(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(n)22 b Fm(\000)h Fu(1)p Ft(;)973 663 y(s)p Fu(\()p Ft(\034)11 b(!)t Fu(\))27 b(=)h Ft(s)p Fu(\()p Ft(\034)11 b Fu(\))p Ft(s)p Fu(\()p Ft(!)t Fu(\))p Ft(;)114 b Fu(if)31 b Ft(`)p Fu(\()p Ft(\034)11 b(!)t Fu(\))27 b(=)h Ft(`)p Fu(\()p Ft(\034)11 b Fu(\))22 b(+)g Ft(`)p Fu(\()p Ft(!)t Fu(\))p Ft(:)-180 856 y Fu(Here)j Ft(`)g Fu(denotes)h(the)f(lengh)m(t)g(of)f(an)h(elemen)m(t)f(of)g Fp(S)1649 871 y Fs(n)1714 856 y Fu(with)h(resp)s(ect)h(to)e(the)h(set)h (of)e(generators)h Ft(\034)3280 871 y Fn(1)3320 856 y Ft(;)17 b(:)g(:)g(:)f(;)h(\034)3581 871 y Fs(n)p Fk(\000)p Fn(1)3718 856 y Fu(.)41 b(The)-180 972 y(map)34 b Ft(s)g Fu(is)h(called)e(the)i(Matsumoto)f(section.)50 b(In)35 b(other)g(w)m(ords,)h(if)e Ft(!)g Fu(=)d Ft(\034)2584 987 y Fs(i)2608 996 y Fg(1)2664 972 y Ft(:)17 b(:)g(:)f(\034)2837 987 y Fs(i)2861 998 y Fi(M)2968 972 y Fu(is)34 b(a)g(reduced)j (expression)-180 1088 y(of)32 b Ft(!)f Fm(2)d Fp(S)179 1103 y Fs(n)219 1088 y Fu(,)33 b(then)g Ft(s)p Fu(\()p Ft(!)t Fu(\))27 b(=)h Ft(\033)874 1103 y Fs(i)898 1112 y Fg(1)953 1088 y Ft(:)17 b(:)g(:)g(\033)1140 1103 y Fs(i)1164 1114 y Fi(M)1236 1088 y Fu(.)-80 1281 y(Let)32 b Ft(q)g Fm(2)c Fp(|)-8 b Fu(,)26 b Ft(q)32 b Fm(6)p Fu(=)27 b(0.)44 b(The)33 b(quotien)m(t)g(of)f(the)h(group)g(algebra)e Fp(|)-8 b Fu(\()p Fp(B)2312 1296 y Fs(n)2365 1281 y Fu(\))33 b(b)m(y)g(the)g(t)m(w)m(o-sided)g(ideal)e(generated)j(b)m(y)-180 1397 y(the)f(relations)1186 1549 y(\()p Ft(\033)1279 1564 y Fs(i)1330 1549 y Fm(\000)23 b Ft(q)t Fu(\)\()p Ft(\033)1608 1564 y Fs(i)1658 1549 y Fu(+)f(1\))p Ft(;)114 b Fu(1)27 b Fm(\024)i Ft(i)f Fm(\024)g Ft(n)22 b Fm(\000)h Fu(1)p Ft(;)-180 1721 y Fu(is)32 b(the)h(so-called)e Fo(He)-5 b(cke)35 b(algebr)-5 b(a)32 b Fu(of)g(t)m(yp)s(e)h Ft(A)1495 1736 y Fs(n)1542 1721 y Fu(,)g(denoted)g(b)m(y)h Fm(H)2190 1736 y Fs(q)2228 1721 y Fu(\()p Ft(n)p Fu(\).)-80 1913 y(Using)e(the)h(section)f Ft(s)p Fu(,)h(the)g(follo)m(wing)c (distinguished)j(elemen)m(ts)h(of)f(the)h(group)f(algebra)g Fp(|)-9 b(B)3375 1928 y Fs(n)3461 1913 y Fu(are)32 b(de\014ned:)1057 2181 y Fj(S)1140 2196 y Fs(n)1214 2181 y Fu(:=)1357 2087 y Fl(X)1345 2299 y Fs(\033)r Fk(2)p Fd(S)1477 2307 y Fi(n)1530 2181 y Ft(s)p Fu(\()p Ft(\033)t Fu(\))p Ft(;)212 b Fj(S)2033 2196 y Fs(i;j)2140 2181 y Fu(:=)2310 2087 y Fl(X)2271 2298 y Fs(\033)r Fk(2)p Fs(X)2418 2308 y Fi(i;j)2509 2181 y Ft(s)p Fu(\()p Ft(\033)t Fu(\);)-180 2467 y(here)33 b Ft(X)112 2482 y Fs(i;j)220 2467 y Fm(\032)28 b Fp(S)386 2482 y Fs(n)459 2467 y Fu(is)k(the)g(set)h(of)e(all)f(\()p Ft(i;)17 b(j)6 b Fu(\)-sh)m(u\017es.)45 b(The)33 b(elemen)m(t)f Fj(S)2373 2482 y Fs(n)2451 2467 y Fu(is)g(called)f(the)h(quan)m(tum)g (symmetrizer.)-80 2660 y(Giv)m(en)39 b(a)g(braided)g(v)m(ector)h(space) h(\()p Ft(V)5 b(;)17 b(c)p Fu(\),)41 b(there)f(are)f(represen)m (tations)i(of)e(the)h(braid)e(groups)i Ft(\032)3549 2675 y Fs(n)3635 2660 y Fu(:)g Fp(B)3762 2675 y Fs(n)3854 2660 y Fm(!)-180 2776 y Fu(Aut)17 b(\()p Ft(V)118 2740 y Fk(\012)p Fs(n)220 2776 y Fu(\))32 b(for)g(an)m(y)i Ft(n)28 b Fm(\025)g Fu(0,)k(giv)m(en)h(b)m(y)1006 2968 y Ft(\032)1056 2983 y Fs(n)1103 2968 y Fu(\()p Ft(\033)1196 2983 y Fs(i)1225 2968 y Fu(\))27 b(=)h(id)k Fm(\012)17 b(\001)g(\001)g(\001)k(\012)i Fu(id)32 b Fm(\012)p Ft(c)23 b Fm(\012)f Fu(id)33 b Fm(\012)17 b(\001)g(\001)g(\001)j(\012)j Fu(id)16 b Ft(;)-180 3161 y Fu(where)35 b Ft(c)f Fu(acts)g(in)f(the)h (tensor)g(pro)s(duct)g(of)f(the)h Ft(i)g Fu(and)g Ft(i)23 b Fu(+)g(1)33 b(copies)h(of)f Ft(V)22 b Fu(.)47 b(By)34 b(abuse)h(of)e(notation,)f(w)m(e)j(shall)-180 3277 y(denote)e(b)m(y)h Fj(S)353 3292 y Fs(n)399 3277 y Fu(,)f Fj(S)542 3292 y Fs(i;j)654 3277 y Fu(also)f(the)h(corresp)s(onding)f(endomorphisms)g Ft(\032)p Fu(\()p Fj(S)2510 3292 y Fs(n)2557 3277 y Fu(\),)g Ft(\032)p Fu(\()p Fj(S)2825 3292 y Fs(i;j)2905 3277 y Fu(\))h(of)f Ft(V)3165 3241 y Fk(\012)p Fs(n)3295 3277 y Fu(=)c Ft(T)3470 3241 y Fs(n)3516 3277 y Fu(\()p Ft(V)22 b Fu(\).)-80 3470 y(If)j Ft(C)35 b Fu(=)218 3395 y Fl(L)329 3499 y Fs(n)p Fk(\025)p Fn(0)483 3470 y Ft(C)7 b Fu(\()p Ft(n)p Fu(\))25 b(is)h(a)f(graded)h(coalgebra)e(with)i(com)m (ultiplication)21 b(\001,)27 b(w)m(e)g(denote)f(b)m(y)h(\001)3340 3485 y Fs(i;j)3448 3470 y Fu(:)h Ft(C)7 b Fu(\()p Ft(i)h Fu(+)g Ft(j)e Fu(\))27 b Fm(!)-180 3593 y Ft(C)7 b Fu(\()p Ft(i)p Fu(\))22 b Fm(\012)h Ft(C)7 b Fu(\()p Ft(j)f Fu(\),)32 b Ft(i;)17 b(j)34 b Fm(\025)28 b Fu(0,)k(the)h(\()p Ft(i;)17 b(j)6 b Fu(\)-graded)33 b(comp)s(onen)m(t)f(of)g(the)h(map)f(\001.)-180 3826 y FD(Prop)s(osition)k(2.8.)49 b Fo(L)-5 b(et)35 b Ft(V)49 b Fm(2)1017 3790 y Fs(H)1017 3852 y(H)1084 3826 y Fm(Y)8 b(D)s Fo(.)45 b(Then)2008 4042 y Fu(\001)2089 4057 y Fs(i;j)2197 4042 y Fu(=)28 b Fj(S)2384 4057 y Fs(i;j)2463 4042 y Ft(;)-2670 b Fu(\(2.3\))1267 4223 y Fj(B)p Fu(\()p Ft(V)21 b Fu(\))28 b(=)1641 4129 y Fl(M)1649 4339 y Fs(n)p Fk(\025)p Fn(0)1808 4223 y Ft(T)1879 4182 y Fs(n)1925 4223 y Fu(\()p Ft(V)22 b Fu(\))p Ft(=k)s(er)s Fu(\()p Fj(S)2396 4238 y Fs(n)2442 4223 y Fu(\))p Ft(:)-2687 b Fu(\(2.4\))-180 4537 y Fo(Pr)-5 b(o)g(of.)41 b Fu(See)34 b(for)e(instance)h([Sbg].)p 3883 4537 4 66 v 3887 4475 59 4 v 3887 4537 V 3945 4537 4 66 v -80 4747 a(This)28 b(description)f(of)g(the)h(relation)e(of)h Fj(B)p Fu(\()p Ft(V)22 b Fu(\))27 b(do)s(es)i(not)e(mean)g(that)h(the)g(relations)e (are)i(kno)m(wn.)43 b(In)28 b(general)-180 4863 y(it)k(is)h(v)m(ery)i (hard)e(to)g(compute)g(the)g(k)m(ernels)i(of)d(the)i(maps)f Fj(S)2090 4878 y Fs(n)2170 4863 y Fu(in)f(concrete)i(terms.)46 b(F)-8 b(or)32 b(an)m(y)i(braided)f(v)m(ector)-180 4979 y(space)h(\()p Ft(V)5 b(;)17 b(c)p Fu(\),)32 b(w)m(e)i(ma)m(y)e (de\014ne)i Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(b)m(y)i(\(2.4\))o(.)p eop %%Page: 20 20 20 19 bop -180 0 a Fq(20)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(2.3.)56 b FD(In)m(v)-6 b(ariance)37 b(under)h(t)m(wisting.)-80 319 y Fu(Twisting)44 b(is)h(a)h(metho)s(d)e(to)i(construct)g(new)h(Hopf)e(algebras)g(b)m(y)h ("deforming")d(the)j(com)m(ultiplication;)-180 435 y(originally)29 b(due)k(to)f(Drinfeld)f([Dr2)o(],)i(it)f(w)m(as)h(adapted)g(to)f(Hopf)h (algebras)f(in)f([Re)q(].)-80 634 y(Let)36 b Ft(A)f Fu(b)s(e)h(a)g (Hopf)f(algebra)g(and)h Ft(F)46 b Fm(2)34 b Ft(A)24 b Fm(\012)h Ft(A)36 b Fu(b)s(e)g(an)f(in)m(v)m(ertible)g(elemen)m(t.)53 b(Let)36 b(\001)3098 649 y Fs(F)3190 634 y Fu(:=)d Ft(F)14 b Fu(\001)p Ft(F)3561 597 y Fk(\000)p Fn(1)3688 634 y Fu(:)33 b Ft(A)g Fm(!)-180 750 y Ft(A)22 b Fm(\012)h Ft(A)p Fu(;)33 b(it)e(is)h(again)f(an)i(algebra)e(map.)43 b(If)961 948 y(\(1)22 b Fm(\012)g Ft(F)14 b Fu(\)\(id)32 b Fm(\012)p Fu(\001\)\()p Ft(F)14 b Fu(\))28 b(=)g(\()p Ft(F)36 b Fm(\012)22 b Fu(1\)\(\001)g Fm(\012)h Fu(id)16 b(\)\()p Ft(F)e Fu(\))p Ft(;)-2977 b Fu(\(2.5\))1320 1099 y(\(id)32 b Fm(\012)p Ft(")p Fu(\)\()p Ft(F)14 b Fu(\))27 b(=)h(\()p Ft(")22 b Fm(\012)g Fu(id)16 b(\)\()p Ft(F)e Fu(\))27 b(=)h(1)p Ft(;)-2798 b Fu(\(2.6\))-180 1303 y(then)34 b Ft(A)116 1318 y Fs(F)208 1303 y Fu(\(the)f(same)g (algebra,)f(but)h(with)g(com)m(ultiplication)c(\001)2230 1318 y Fs(F)2289 1303 y Fu(\))k(is)f(again)g(a)h(Hopf)g(algebra.)44 b(W)-8 b(e)33 b(shall)f(sa)m(y)-180 1419 y(that)g Ft(A)104 1434 y Fs(F)196 1419 y Fu(is)g(obtained)g(from)f Ft(A)i Fu(via)f(t)m(wisting)f(b)m(y)j Ft(F)14 b Fu(;)32 b Ft(F)46 b Fu(is)32 b(a)h(co)s(cycle)g(in)f(a)g(suitable)g(sense.)-80 1617 y(There)d(is)e(a)g(dual)g(v)m(ersion)i(of)e(the)h(t)m(wisting)f (op)s(eration,)g(whic)m(h)i(amoun)m(ts)e(to)h(a)f(t)m(wist)h(of)f(the)h (m)m(ultiplication)-180 1733 y([DT].)44 b(Let)32 b Ft(A)h Fu(b)s(e)g(a)f(Hopf)g(algebra)g(and)h(let)f Ft(\033)f Fu(:)d Ft(A)22 b Fm(\002)h Ft(A)28 b Fm(!)f Fp(|)18 b Fu(b)s(e)33 b(an)f(in)m(v)m(ertible)g(2-co)s(cycle)3231 1697 y Fn(1)3274 1733 y Fu(,)h(that)f(is)817 1931 y Ft(\033)t Fu(\()p Ft(x)969 1947 y Fn(\(1\))1063 1931 y Ft(;)17 b(y)1155 1947 y Fn(\(1\))1249 1931 y Fu(\))p Ft(\033)t Fu(\()p Ft(x)1439 1947 y Fn(\(2\))1533 1931 y Ft(y)1581 1947 y Fn(\(2\))1675 1931 y Ft(;)g(z)t Fu(\))28 b(=)g Ft(\033)t Fu(\()p Ft(y)2083 1947 y Fn(\(1\))2177 1931 y Ft(;)17 b(z)2266 1947 y Fn(\(1\))2360 1931 y Fu(\))p Ft(\033)t Fu(\()p Ft(x;)g(y)2642 1947 y Fn(\(2\))2736 1931 y Ft(z)2781 1947 y Fn(\(2\))2876 1931 y Fu(\))p Ft(;)1531 2082 y(\033)t Fu(\(1)p Ft(;)g Fu(1\))26 b(=)i(1)p Ft(;)-180 2281 y Fu(for)33 b(all)f Ft(x;)17 b(y)t(;)g(z)33 b Fm(2)d Ft(A)p Fu(.)47 b(Then)35 b Ft(A)952 2296 y Fs(\033)1033 2281 y Fu({)e(the)i(same)e Ft(A)h Fu(but)g(with)f(the)i(m)m (ultiplication)29 b Fm(\001)2865 2296 y Fs(\033)2945 2281 y Fu(b)s(elo)m(w)k({)h(is)f(again)f(a)i(Hopf)-180 2397 y(algebra,)e(where)1032 2554 y Ft(x)22 b Fm(\001)1137 2569 y Fs(\033)1206 2554 y Ft(y)31 b Fu(=)c Ft(\033)t Fu(\()p Ft(x)1540 2570 y Fn(\(1\))1635 2554 y Ft(;)17 b(y)1727 2570 y Fn(\(1\))1821 2554 y Fu(\))p Ft(x)1914 2570 y Fn(\(2\))2008 2554 y Ft(y)2056 2570 y Fn(\(2\))2150 2554 y Ft(\033)2209 2513 y Fk(\000)p Fn(1)2304 2554 y Fu(\()p Ft(x)2397 2570 y Fn(\(3\))2491 2554 y Ft(;)g(y)2583 2570 y Fn(\(3\))2677 2554 y Fu(\))p Ft(:)-80 2758 y Fu(Assume)36 b(no)m(w)h(that)f Ft(H)43 b Fu(is)36 b(a)f(Hopf)h(algebra,)g Ft(R)h Fu(is)e(a)h(braided)g(Hopf)f(algebra)g(in)2987 2722 y Fs(H)2987 2783 y(H)3055 2758 y Fm(Y)8 b(D)r Fu(,)37 b(and)f Ft(A)e Fu(=)f Ft(R)q Fu(#)p Ft(H)8 b Fu(.)-180 2874 y(Let)34 b Ft(\031)f Fu(:)d Ft(A)f Fm(!)g Ft(H)41 b Fu(and)34 b Ft(\023)c Fu(:)f Ft(H)37 b Fm(!)29 b Ft(A)34 b Fu(b)s(e)g(the)g(canonical)e(pro)5 b(jection)33 b(and)h(injection.)45 b(Let)34 b Ft(\033)g Fu(:)29 b Ft(H)h Fm(\002)24 b Ft(H)37 b Fm(!)29 b Fp(|)19 b Fu(b)s(e)-180 2991 y(an)32 b(in)m(v)m(ertible)g (2-co)s(cycle,)h(and)f(de\014ne)i Ft(\033)1357 3006 y Fs(\031)1432 2991 y Fu(:)28 b Ft(A)22 b Fm(\002)h Ft(A)28 b Fm(!)f Fp(|)18 b Fu(b)m(y)1556 3189 y Ft(\033)1611 3204 y Fs(\031)1686 3189 y Fu(:=)28 b Ft(\033)t Fu(\()p Ft(\031)e Fm(\012)c Ft(\031)t Fu(\);)-180 3387 y Ft(\033)-125 3402 y Fs(\031)-46 3387 y Fu(is)31 b(an)g(in)m(v)m(ertible)g(2-co)s (cycle,)h(with)f(in)m(v)m(erse)i(\()p Ft(\033)1699 3350 y Fk(\000)p Fn(1)1793 3387 y Fu(\))1831 3402 y Fs(\031)1878 3387 y Fu(.)43 b(The)32 b(maps)f Ft(\031)h Fu(:)c Ft(A)2616 3402 y Fs(\033)2656 3410 y Fi(\031)2730 3387 y Fm(!)g Ft(H)2939 3402 y Fs(\033)2985 3387 y Fu(,)k Ft(\023)c Fu(:)g Ft(H)3242 3402 y Fs(\033)3316 3387 y Fm(!)g Ft(A)3517 3402 y Fs(\033)3557 3410 y Fi(\031)3635 3387 y Fu(are)j(still)-180 3503 y(Hopf)g(algebra)f(maps.)43 b(Because)32 b(the)g(com)m (ultiplication)26 b(is)31 b(not)g(c)m(hanged,)h(the)g(space)g(of)e (coin)m(v)-5 b(arian)m(ts)31 b(of)f Ft(\031)35 b Fu(is)-180 3619 y Ft(R)q Fu(;)e(this)f(is)g(a)g(subalgebra)h(of)f Ft(A)997 3634 y Fs(\033)1037 3642 y Fi(\031)1116 3619 y Fu(that)g(w)m(e)i(denote)f Ft(R)1859 3634 y Fs(\033)1906 3619 y Fu(;)g(the)g(m)m(ultiplication)28 b(in)k Ft(R)2948 3634 y Fs(\033)3027 3619 y Fu(is)g(giv)m(en)h(b)m(y)863 3817 y Ft(x:)945 3832 y Fs(\033)992 3817 y Ft(y)e Fu(=)c Ft(\033)t Fu(\()p Ft(x)1326 3833 y Fn(\()p Fk(\000)p Fn(1\))1476 3817 y Ft(;)17 b(y)1568 3833 y Fn(\()p Fk(\000)p Fn(1\))1716 3817 y Fu(\))p Ft(x)1809 3833 y Fn(\(0\))1904 3817 y Ft(y)1952 3833 y Fn(\(0\))2046 3817 y Ft(;)212 b(x;)17 b(y)31 b Fm(2)d Ft(R)h Fu(=)e Ft(R)2837 3832 y Fs(\033)2884 3817 y Ft(:)-3091 b Fu(\(2.7\))-180 4034 y(Equation)30 b(\(2.7\))f(follo)m(ws)g(easily)g(using)i(\(1.17\))o(.)43 b(Clearly)-8 b(,)29 b Ft(R)2055 4049 y Fs(\033)2133 4034 y Fu(is)g(a)h(Y)-8 b(etter-Drinfeld)28 b(Hopf)i(algebra)e(in)3669 3992 y Fs(H)3727 4000 y Fi(\033)3669 4061 y Fs(H)3727 4069 y Fi(\033)3772 4034 y Fm(Y)8 b(D)s Fu(.)-180 4151 y(The)35 b(coaction)e(of)g Ft(H)601 4166 y Fs(\033)681 4151 y Fu(on)h Ft(R)892 4166 y Fs(\033)973 4151 y Fu(is)f(the)h(same)g (as)g(the)g(coaction)f(of)g Ft(H)41 b Fu(on)34 b Ft(R)q Fu(,)g(since)g(the)h(com)m(ultiplication)29 b(w)m(as)-180 4267 y(not)38 b(altered.)59 b(The)40 b(explicit)c(form)m(ula)g(for)i (the)g(action)f(of)h Ft(H)2121 4282 y Fs(\033)2206 4267 y Fu(on)g Ft(R)2421 4282 y Fs(\033)2506 4267 y Fu(can)g(b)s(e)g (written)g(do)m(wn;)k(w)m(e)d(shall)e(do)-180 4383 y(this)32 b(only)g(in)g(the)h(setting)f(w)m(e)i(are)f(in)m(terested)g(in.)-80 4581 y(Let)j Ft(H)k Fu(=)33 b Fp(|)-9 b Fu(\000)30 b(b)s(e)36 b(a)f(group)h(algebra;)g(an)g(in)m(v)m(ertible)f(co)s(cycle)h Ft(\033)h Fu(:)c Ft(H)f Fm(\002)25 b Ft(H)40 b Fm(!)33 b Fp(|)21 b Fu(is)35 b(uniquely)h(determined)-180 4697 y(b)m(y)d(its)g(restriction)e Ft(\033)h Fu(:)c(\000)22 b Fm(\002)g Fu(\000)28 b Fm(!)f Fp(|)1160 4661 y Fk(\002)1213 4697 y Fu(,)32 b(a)h(group)f(co)s(cycle)h(with)f(resp)s(ect)i(to)e(the) h(trivial)d(action.)p -180 4867 499 4 v -76 4928 a Fz(1)-1 4958 y FC(Here)38 b("in)n(v)n(ertible")e(means)i(that)h(the)f(asso)r (ciated)f(linear)g(map)h FA(\033)44 b FC(:)d FA(A)26 b Fx(\012)f FA(A)41 b Fx(!)f Fy(|)22 b FC(is)38 b(in)n(v)n(ertible)f (with)i(resp)r(ect)f(to)g(the)-180 5078 y(con)n(v)n(olution)26 b(pro)r(duct.)p eop %%Page: 21 21 21 20 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(21)-180 203 y FD(Lemma)37 b(2.9.)49 b Fo(L)-5 b(et)44 b Fu(\000)f Fo(b)-5 b(e)43 b(an)f(ab)-5 b(elian)43 b(gr)-5 b(oup)43 b(and)f(let)h Ft(R)i Fo(b)-5 b(e)43 b(a)f(br)-5 b(aide)g(d)43 b(Hopf)g(algebr)-5 b(a)42 b(in)3346 167 y Fn(\000)3346 228 y(\000)3394 203 y Fm(Y)8 b(D)s Fo(.)69 b(L)-5 b(et)44 b Ft(\033)j Fu(:)-180 342 y(\000)17 b Fm(\002)h Fu(\000)27 b Fm(!)h Fp(|)270 306 y Fk(\002)356 342 y Fo(b)-5 b(e)32 b(a)g(c)-5 b(o)g(cycle.)44 b(L)-5 b(et)33 b Ft(S)38 b Fo(b)-5 b(e)32 b(the)h(sub)-5 b(algebr)g(a)32 b(of)g Ft(R)i Fo(gener)-5 b(ate)g(d)32 b(by)h Ft(P)14 b Fu(\()p Ft(R)q Fu(\))p Fo(.)43 b(In)32 b(the)g(c)-5 b(ase)32 b Ft(y)f Fm(2)d Ft(S)3728 295 y Fs(\021)3722 370 y(h)3770 342 y Fo(,)33 b(for)-180 482 y(some)h Ft(h)28 b Fm(2)g Fu(\000)35 b Fo(and)f Ft(\021)d Fm(2)708 457 y Fl(b)706 482 y Fu(\000)p Fo(,)j(the)h(action)g(of)f Ft(H)h Fu(=)28 b Ft(H)1705 497 y Fs(\033)1786 482 y Fo(on)35 b Ft(R)2000 497 y Fs(\033)2082 482 y Fo(is)973 681 y Ft(g)30 b(*)1148 696 y Fs(\033)1223 681 y Ft(y)g Fu(=)e Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(\033)1750 640 y Fk(\000)p Fn(1)1843 681 y Fu(\()p Ft(h;)g(g)t Fu(\))p Ft(\021)t Fu(\()p Ft(g)t Fu(\))p Ft(y)t(;)213 b(g)31 b Fm(2)d Fu(\000)p Ft(:)-2981 b Fu(\(2.8\))-180 880 y Fo(Henc)-5 b(e,)34 b(the)h(br)-5 b(aiding)34 b Ft(c)720 895 y Fs(\033)802 880 y Fo(in)g Ft(R)995 895 y Fs(\033)1077 880 y Fo(is)h(given)f(in)h(this)f(c)-5 b(ase)35 b(by)662 1079 y Ft(c)704 1094 y Fs(\033)751 1079 y Fu(\()p Ft(x)22 b Fm(\012)h Ft(y)t Fu(\))k(=)g Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(\033)1531 1038 y Fk(\000)p Fn(1)1625 1079 y Fu(\()p Ft(h;)g(g)t Fu(\))p Ft(\021)t Fu(\()p Ft(g)t Fu(\))g Ft(y)22 b Fm(\012)h Ft(x;)216 b(x)29 b Fm(2)f Ft(R)2768 1094 y Fs(g)2808 1079 y Ft(;)17 b(g)31 b Fm(2)d Fu(\000)p Ft(:)-3292 b Fu(\(2.9\))-180 1278 y Fo(Ther)-5 b(efor)g(e,)34 b(for)g(such)h Ft(x)g Fo(and)g Ft(y)t Fo(,)f(we)g(have)1405 1477 y Fu([)p Ft(x;)17 b(y)t Fu(])1610 1492 y Fs(c)1641 1500 y Fi(\033)1714 1477 y Fu(=)27 b Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\)[)p Ft(x;)g(y)t Fu(])2308 1492 y Fs(c)2342 1477 y Ft(:)-2549 b Fu(\(2.10\))-180 1699 y Fo(Pr)-5 b(o)g(of.)41 b Fu(T)-8 b(o)33 b(pro)m(v)m(e)h(\(2.8\))o(,)f(it)e(is)h(enough)h(to)g(assume)g Ft(y)e Fm(2)d Ft(P)14 b Fu(\()p Ft(R)q Fu(\))2174 1652 y Fs(\021)2174 1727 y(h)2218 1699 y Fu(.)-80 1838 y(Let)32 b Ft(A)c Fu(=)g Ft(R)q Fu(#)p Ft(H)8 b Fu(;)32 b(in)g Ft(A)790 1853 y Fs(\033)830 1861 y Fi(\031)877 1838 y Fu(,)g(w)m(e)i(ha)m(v)m(e)316 2037 y Ft(g)t(:)394 2052 y Fs(\033)440 2037 y Ft(y)d Fu(=)c Ft(\033)t Fu(\()p Ft(g)t(;)17 b(\031)t Fu(\()p Ft(y)t Fu(\)\))g Ft(g)h(\033)1180 1996 y Fk(\000)p Fn(1)1274 2037 y Fu(\()p Ft(g)t(;)f Fu(1\))k(+)h Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))g Ft(g)t(y)g(\033)2091 1996 y Fk(\000)p Fn(1)2186 2037 y Fu(\()p Ft(g)t(;)g Fu(1\))j(+)i Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))g Ft(g)t(h)g(\033)3010 1996 y Fk(\000)p Fn(1)3102 2037 y Fu(\()p Ft(g)t(;)g(\031)t Fu(\()p Ft(y)t Fu(\)\))519 2206 y(=)27 b Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(g)t(y)t Fu(;)316 2374 y Ft(y)t(:)395 2389 y Fs(\033)441 2374 y Ft(g)31 b Fu(=)c Ft(\033)t Fu(\()p Ft(\031)t Fu(\()p Ft(y)t Fu(\))p Ft(;)17 b(g)t Fu(\))g Ft(g)h(\033)1180 2333 y Fk(\000)p Fn(1)1274 2374 y Fu(\(1)p Ft(;)f(g)t Fu(\))k(+)h Ft(\033)t Fu(\()p Ft(h;)17 b(g)t Fu(\))g Ft(y)t(g)g(\033)2091 2333 y Fk(\000)p Fn(1)2186 2374 y Fu(\(1)p Ft(;)g(g)t Fu(\))j(+)i Ft(\033)t Fu(\()p Ft(h;)17 b(g)t Fu(\))g Ft(hg)i(\033)3008 2333 y Fk(\000)p Fn(1)3102 2374 y Fu(\()p Ft(\031)t Fu(\()p Ft(y)t Fu(\))p Ft(;)e(g)t Fu(\))519 2543 y(=)27 b Ft(\033)t Fu(\()p Ft(h;)17 b(g)t Fu(\))p Ft(y)t(g)t Fu(;)-180 2742 y(hence)605 2900 y Ft(g)t(:)683 2915 y Fs(\033)729 2900 y Ft(y)31 b Fu(=)c Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(g)t(y)29 b Fu(=)f Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(\021)t Fu(\()p Ft(g)t Fu(\))p Ft(y)t(g)28 b Fu(=)f Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(\033)2469 2859 y Fk(\000)p Fn(1)2563 2900 y Fu(\()p Ft(h;)g(g)t Fu(\))p Ft(\021)t Fu(\()p Ft(g)t Fu(\))p Ft(y)t(:)3048 2915 y Fs(\033)3092 2900 y Ft(g)t(;)-180 3079 y Fu(whic)m(h)33 b(is)f(equiv)-5 b(alen)m(t)33 b(to)f(\(2.8\))o(.)44 b(No)m(w)34 b(\(2.9\))e(follo)m(ws)g(at)g(once,)h(and)g(\(2.10\))f(follo)m(ws)f (from)g(\(2.7\))h(and)h(\(2.9\))o(:)132 3278 y([)p Ft(x;)17 b(y)t Fu(])337 3293 y Fs(c)368 3301 y Fi(\033)441 3278 y Fu(=)27 b Ft(x:)626 3293 y Fs(\033)674 3278 y Ft(y)e Fm(\000)d Ft(:)873 3293 y Fs(\033)920 3278 y Ft(c)962 3293 y Fs(\033)1009 3278 y Fu(\()p Ft(x)h Fm(\012)f Ft(y)t Fu(\))27 b(=)h Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(xy)24 b Fm(\000)f Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft(\033)2303 3237 y Fk(\000)p Fn(1)2396 3278 y Fu(\()p Ft(h;)g(g)t Fu(\))p Ft(\021)t Fu(\()p Ft(g)t Fu(\))g Ft(\033)t Fu(\()p Ft(h;)g(g)t Fu(\))p Ft(y)t(x)24 b Fu(=)k Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\)[)p Ft(x;)g(y)t Fu(])3831 3293 y Fs(c)3864 3278 y Ft(:)p 3883 3477 4 66 v 3887 3414 59 4 v 3887 3477 V 3945 3477 4 66 v -80 3666 a Fu(The)36 b(pro)s(of)e(of)h(the)h(follo)m(wing)d (Lemma)h(is)h(clear,)g(since)h(the)g(com)m(ultiplication)31 b(of)k(a)g(Hopf)g(algebra)f(is)h(not)-180 3782 y(c)m(hanged)f(b)m(y)f (t)m(wisting.)-180 4004 y FD(Lemma)k(2.10.)49 b Fo(L)-5 b(et)43 b Ft(H)50 b Fo(b)-5 b(e)42 b(a)g(Hopf)g(algebr)-5 b(a)41 b(and)h(let)h Ft(R)g Fo(b)-5 b(e)42 b(a)g(br)-5 b(aide)g(d)41 b(Hopf)i(algebr)-5 b(a)41 b(in)3332 3968 y Fs(H)3332 4029 y(H)3399 4004 y Fm(Y)8 b(D)s Fo(.)67 b(L)-5 b(et)42 b Ft(\033)k Fu(:)-180 4144 y Ft(H)27 b Fm(\002)20 b Ft(H)36 b Fm(!)27 b Fp(|)19 b Fo(b)-5 b(e)33 b(an)h(invertible)f(2-c)-5 b(o)g(cycle.)43 b(Then)33 b Ft(R)i Fo(is)f(a)f(Nichols)h(algebr)-5 b(a)32 b(if)i(and)f(only)h(if) f Ft(R)3374 4159 y Fs(\033)3455 4144 y Fo(is)h(a)f(Nichols)-180 4283 y(algebr)-5 b(a)34 b(in)270 4240 y Fs(H)328 4248 y Fi(\033)270 4310 y Fs(H)328 4318 y Fi(\033)373 4283 y Fm(Y)8 b(D)s Fo(.)p 3882 4283 4 68 v 3886 4219 60 4 v 3886 4283 V 3945 4283 4 68 v 1135 4558 a Fu(3.)55 b Fv(Types)37 b(of)h(Nichols)g(algebras)-80 4732 y Fu(W)-8 b(e)24 b(no)m(w)h(discuss)h(sev)m(eral)f(examples)g(of)f(Nic)m(hols)f (algebras.)40 b(W)-8 b(e)25 b(are)g(in)m(terested)g(in)f(explicit)f (presen)m(tations,)-180 4848 y Fo(e.)78 b(g.)i Fu(b)m(y)46 b(generators)f(and)g(relations,)h(of)e Fj(B)p Fu(\()p Ft(V)22 b Fu(\),)48 b(for)c(braided)g(v)m(ector)i(spaces)h(in)d (suitable)f(classes,)49 b(for)-180 4964 y(instance,)34 b(those)g(of)f(group)h(t)m(yp)s(e.)47 b(W)-8 b(e)34 b(w)m(ould)g(also)e (lik)m(e)h(to)g(determine)g(when)i Fj(B)p Fu(\()p Ft(V)21 b Fu(\))34 b(has)g(\014nite)f(dimension,)-180 5081 y(or)f(p)s (olynomial)d(gro)m(wth.)p eop %%Page: 22 22 22 21 bop -180 0 a Fq(22)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(3.1.)56 b FD(Symmetries)35 b(and)j(braidings)e(of)i(Hec)m(k)m(e)e(t)m(yp)s(e.)49 b Fu(W)-8 b(e)33 b(b)s(egin)e(with)h(the)h(simplest)e(class)h(of)g (braided)-180 319 y(v)m(ector)i(spaces.)-180 514 y FD(Example)i(3.1.)50 b Fu(Let)42 b Ft(\034)55 b Fu(:)44 b Ft(V)50 b Fm(\012)29 b Ft(V)65 b Fm(!)43 b Ft(V)51 b Fm(\012)29 b Ft(V)63 b Fu(b)s(e)42 b(the)h(usual)e(transp)s(osition;)46 b(the)c(braided)g(v) m(ector)h(space)-180 653 y(\()p Ft(V)5 b(;)17 b(\034)11 b Fu(\))38 b(can)g(b)s(e)g(realized)f(as)h(a)g(Y)-8 b(etter-Drinfeld)36 b(mo)s(dule)g(o)m(v)m(er)j(an)m(y)f(Hopf)g(algebra)e Ft(H)8 b Fu(,)39 b(with)f(trivial)d(action)-180 793 y(and)e(coaction.) 42 b(Then)34 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))27 b(=)h(Sym)16 b(\()p Ft(V)22 b Fu(\),)32 b(the)h(symmetric)f(algebra)f(of)h Ft(V)22 b Fu(.)-80 973 y(The)34 b(braided)g(v)m(ector)h(space)g(\()p Ft(V)5 b(;)17 b Fm(\000)p Ft(\034)11 b Fu(\),)35 b(whic)m(h)f(can)g(b)s (e)g(realized)f Fo(e.)48 b(g.)f Fu(in)2718 937 y Fd(Z)2718 998 y(Z)2766 973 y Fm(Y)8 b(D)s Fu(,)34 b(has)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))29 b(=)h(\003\()p Ft(V)21 b Fu(\),)34 b(the)-180 1112 y(exterior)e(algebra)g(of)g Ft(V)21 b Fu(.)-180 1307 y FD(Example)36 b(3.2.)50 b Fu(Let)36 b Ft(V)54 b Fu(=)33 b Fm(\010)981 1322 y Fs(i)p Fk(2)p Fd(Z)-5 b Fs(=)p Fn(2)1172 1307 y Ft(V)21 b Fu(\()p Ft(i)p Fu(\))36 b(b)s(e)g(a)g(sup)s(er)g(v)m(ector)h(space)g(and)f(let)f Ft(c)e Fu(:)g Ft(V)46 b Fm(\012)25 b Ft(V)55 b Fm(!)32 b Ft(V)46 b Fm(\012)25 b Ft(V)57 b Fu(b)s(e)36 b(the)-180 1446 y(sup)s(ersymmetry:)850 1586 y Ft(c)p Fu(\()p Ft(v)26 b Fm(\012)c Ft(w)s Fu(\))27 b(=)h(\()p Fm(\000)p Fu(1\))1546 1545 y Fs(i:j)1626 1586 y Ft(w)d Fm(\012)d Ft(v)199 b(v)31 b Fm(2)e Ft(V)21 b Fu(\()p Ft(i)p Fu(\))p Ft(;)49 b(w)31 b Fm(2)d Ft(V)21 b Fu(\()p Ft(j)6 b Fu(\))p Ft(:)-180 1745 y Fu(Clearly)-8 b(,)40 b Ft(V)60 b Fu(can)39 b(b)s(e)g(realized)g (as)g(a)f(Y)-8 b(etter-Drinfeld)37 b(mo)s(dule)h(o)m(v)m(er)i Fp(Z)p Ft(=)p Fu(2.)59 b(Then)41 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))39 b Fm(')g Fu(Sym)16 b(\()p Ft(V)21 b Fu(\(0\)\))26 b Fm(\012)-180 1885 y Fu(\003\()p Ft(V)21 b Fu(\(1\)\),)32 b(the)h(sup)s(er-symmetric)g(algebra)e(of)h Ft(V)22 b Fu(.)-80 2056 y(The)33 b(simple)e(form)h(of)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(in)e(these)j(examples)f(can)g(b)s(e)f (explained)h(in)e(the)i(follo)m(wing)d(con)m(text.)-180 2251 y FD(De\014nition)36 b(3.3.)49 b Fu(W)-8 b(e)27 b(sa)m(y)f(that)g(a)g(braided)f(v)m(ector)i(space)h(\()p Ft(V)5 b(;)17 b(c)p Fu(\))25 b(is)h(of)f Fo(He)-5 b(cke-typ)g(e)26 b Fu(with)g(lab)s(el)e Ft(q)31 b Fm(2)d Fp(|)-8 b Fu(,)21 b Ft(q)31 b Fm(6)p Fu(=)d(0,)-180 2390 y(if)1497 2529 y(\()p Ft(c)22 b Fm(\000)g Ft(q)t Fu(\)\()p Ft(c)g Fu(+)g(1\))28 b(=)f(0)p Ft(:)-180 2689 y Fu(In)39 b(this)g(case,)i(the)e(represen)m (tation)h(of)e(the)h(braid)f(group)h Ft(\032)2089 2704 y Fs(n)2174 2689 y Fu(:)g Fp(B)2300 2704 y Fs(n)2392 2689 y Fm(!)e Fu(Aut)17 b(\()p Ft(V)2827 2653 y Fk(\012)p Fs(n)2929 2689 y Fu(\))39 b(factorizes)g(through)f(the)-180 2829 y(Hec)m(k)m(e)d(algebra)c Fm(H)527 2844 y Fs(q)565 2829 y Fu(\()p Ft(n)p Fu(\),)i(for)f(all)f Ft(n)d Fm(\025)g Fu(0;)k Fo(cf.)43 b Fu(Section)32 b(2.2.)-80 3009 y(If)c Ft(q)k Fu(=)27 b(1,)i(one)g(sa)m(ys)h(that)e Ft(c)g Fu(is)g(a)h Fo(symmetry)p Fu(.)42 b(Then)30 b Ft(\032)1905 3024 y Fs(n)1980 3009 y Fu(factorizes)e(through)h(the)g(symmetric)e(group)i Fp(S)3740 3024 y Fs(n)3781 3009 y Fu(,)g(for)-180 3148 y(all)h Ft(n)e Fm(\025)h Fu(0.)43 b(The)33 b(categorical)e(v)m(ersion)i (of)f(symmetries)g(is)h(that)f(of)g(symmetric)g(categories,)g(see)i (Section)e(1.1.)-180 3343 y FD(Prop)s(osition)k(3.4.)49 b Fo(L)-5 b(et)41 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))40 b Fo(b)-5 b(e)41 b(a)f(br)-5 b(aide)g(d)40 b(ve)-5 b(ctor)40 b(sp)-5 b(ac)g(e)40 b(of)h(He)-5 b(cke-typ)g(e)40 b(with)h(lab)-5 b(el)40 b Ft(q)t Fo(,)i(wich)e(is)g(either)-180 3482 y(1)g(or)h(not)g(a)f(r)-5 b(o)g(ot)41 b(of)f(1.)62 b(Then)40 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))41 b Fo(is)f(a)g(quadr)-5 b(atic)41 b(algebr)-5 b(a;)42 b(that)f(is,)h(the)f(ide)-5 b(al)40 b Ft(I)8 b Fu(\()p Ft(V)21 b Fu(\))40 b Fo(is)h(gener)-5 b(ate)g(d)40 b(by)-180 3622 y Ft(I)8 b Fu(\()p Ft(V)21 b Fu(\)\(2\))28 b(=)f(Ker)17 b Fj(S)538 3637 y Fn(2)577 3622 y Fo(.)-80 3761 y(Mor)-5 b(e)g(over,)42 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))41 b Fo(is)h(a)f(Koszul)g(algebr)-5 b(a)41 b(and)g(its)h(Koszul)f(dual)g(is)g(the)h(Nichols)f(algebr)-5 b(a)41 b Fj(B)p Fu(\()p Ft(V)3588 3725 y Fk(\003)3628 3761 y Fu(\))g Fo(c)-5 b(orr)g(e-)-180 3901 y(sp)g(onding)33 b(to)j(the)e(br)-5 b(aide)g(d)34 b(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(e)34 b Fu(\()p Ft(V)1496 3865 y Fk(\003)1535 3901 y Ft(;)17 b(q)1626 3865 y Fk(\000)p Fn(1)1720 3901 y Ft(c)1762 3865 y Fs(t)1792 3901 y Fu(\))p Fo(.)-80 4072 y Fu(A)32 b(nice)h(exp)s(osition)e(on)i(Koszul)f(algebras)g(is)g([BGS,) h(Chapter)g(2].)-180 4261 y Fo(Pr)-5 b(o)g(of.)41 b Fu(The)25 b(argumen)m(t)f(for)f(the)i(\014rst)g(claim)c(is)j(tak)m(en)h(from)e ([AA,)j(Prop.)41 b(3.3.1].)f(The)25 b(image)d(of)i(the)g(quan)m(tum) -180 4401 y(symmetrizer)k Fj(S)455 4416 y Fs(n)530 4401 y Fu(in)g(the)h(Hec)m(k)m(e)i(algebra)c Fm(H)1503 4416 y Fs(q)1541 4401 y Fu(\()p Ft(n)p Fu(\))i(is)f([)p Ft(n)p Fu(])1910 4416 y Fs(q)1949 4401 y Fu(!)p Ft(M)2070 4416 y Fs(")2107 4401 y Fu(,)h(where)h Ft(M)2535 4416 y Fs(")2601 4401 y Fu(sats\014es)g(the)f(follo)m(wing)d(prop)s(erties:)770 4581 y Ft(M)874 4540 y Fn(2)864 4605 y Fs(")942 4581 y Fu(=)i Ft(M)1140 4596 y Fs(")1177 4581 y Ft(;)114 b(M)1412 4596 y Fs(")1449 4581 y Ft(c)1491 4596 y Fs(i)1547 4581 y Fu(=)28 b Ft(c)1693 4596 y Fs(i)1721 4581 y Ft(M)1815 4596 y Fs(")1880 4581 y Fu(=)f Ft(q)t(M)2124 4596 y Fs(")2161 4581 y Ft(;)212 b Fu(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(n)22 b Fm(\000)h Fu(1)p Ft(:)-180 4761 y Fu(See)38 b(for)f(instance)h([HKW].)58 b(No)m(w,)40 b(w)m(e)e(ha)m(v)m(e)h(to)e (sho)m(w)i(that)e(Ker)16 b Fj(S)2402 4776 y Fs(n)2485 4761 y Fu(=)36 b Ft(T)2668 4725 y Fs(n)2714 4761 y Fu(\()p Ft(V)22 b Fu(\))j Fm(\\)h Ft(I)8 b Fu(,)39 b(where)f Ft(I)46 b Fu(is)36 b(the)i(ideal)-180 4900 y(generated)33 b(b)m(y)h(Ker)16 b Fj(S)656 4915 y Fn(2)723 4900 y Fu(=)27 b(Ker)17 b(\()p Ft(c)22 b Fu(+)g(1\))27 b(=)h(Im)16 b(\()p Ft(c)22 b Fm(\000)h Ft(q)t Fu(\);)32 b(but)h(clearly)e Ft(T)2458 4864 y Fs(n)2505 4900 y Fu(\()p Ft(V)22 b Fu(\))g Fm(\\)g Ft(I)36 b Fu(=)2952 4826 y Fl(P)3058 4930 y Fs(i)3102 4900 y Ft(I)3153 4864 y Fs(n;i)3244 4900 y Fu(,)c(where)724 5081 y Ft(I)775 5039 y Fs(n;i)893 5081 y Fu(=)27 b Ft(T)1067 5039 y Fs(i)p Fk(\000)p Fn(1)1185 5081 y Fu(\()p Ft(V)22 b Fu(\))g Fm(\012)h Fu(Im)16 b(\()p Ft(c)22 b Fm(\000)g Ft(q)t Fu(\))g Fm(\012)h Ft(T)2074 5039 y Fs(n)p Fk(\000)p Fs(i)p Fk(\000)p Fn(1)2290 5081 y Fu(\()p Ft(V)e Fu(\))28 b(=)f(Im)16 b(\()p Ft(c)2788 5096 y Fs(i)2839 5081 y Fm(\000)22 b Ft(q)t Fu(\))p Ft(:)p eop %%Page: 23 23 23 22 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(23)-180 203 y Fu(It)38 b(follo)m(ws)e(that)i Ft(T)544 167 y Fs(n)590 203 y Fu(\()p Ft(V)22 b Fu(\))k Fm(\\)g Ft(I)44 b Fm(\022)37 b Fu(Ker)16 b Fj(S)1320 218 y Fs(n)1367 203 y Fu(,)39 b(a)e(fact)h(that)f(w)m(e)i(already)e(kno)m(w)i(from)e (the)h(general)f(theory)-8 b(.)60 b(But)-180 342 y(moreo)m(v)m(er,)33 b Ft(T)338 306 y Fs(n)385 342 y Fu(\()p Ft(V)21 b Fu(\))h Fm(\\)h Ft(I)40 b Fu(is)32 b(a)h Fm(H)997 357 y Fs(q)1035 342 y Fu(\()p Ft(n)p Fu(\)-submo)s(dule)e(of)i Ft(T)1875 306 y Fs(n)1921 342 y Fu(\()p Ft(V)22 b Fu(\))32 b(since)1052 523 y Ft(c)1094 538 y Fs(j)1130 523 y Fu(\()p Ft(c)1210 538 y Fs(i)1260 523 y Fm(\000)23 b Ft(q)t Fu(\))k(=)h(\()p Ft(c)1656 538 y Fs(j)1715 523 y Fm(\000)22 b Ft(q)t Fu(\)\()p Ft(c)1979 538 y Fs(i)2029 523 y Fm(\000)h Ft(q)t Fu(\))f(+)g Ft(q)t Fu(\()p Ft(c)2461 538 y Fs(i)2511 523 y Fm(\000)g Ft(q)t Fu(\))p Ft(:)-180 703 y Fu(This)33 b(computation)f(also)g(sho)m (ws)j(that)e(the)g(action)f(of)h Fm(H)1959 718 y Fs(q)1997 703 y Fu(\()p Ft(n)p Fu(\))g(on)g(the)h(quotien)m(t)f(mo)s(dule)f Ft(T)3270 667 y Fs(n)3316 703 y Fu(\()p Ft(V)22 b Fu(\))p Ft(=T)3591 667 y Fs(n)3637 703 y Fu(\()p Ft(V)g Fu(\))g Fm(\\)h Ft(I)-180 842 y Fu(is)32 b(via)g(the)h(c)m(haracter)g(that)f (sends)j Ft(\033)1199 857 y Fs(i)1260 842 y Fu(to)d Ft(q)t Fu(;)g(hence)i Ft(M)1850 857 y Fs(")1920 842 y Fu(acts)f(on)f Ft(T)2327 806 y Fs(n)2374 842 y Fu(\()p Ft(V)21 b Fu(\))p Ft(=T)2648 806 y Fs(n)2695 842 y Fu(\()p Ft(V)g Fu(\))h Fm(\\)h Ft(I)40 b Fu(b)m(y)33 b(an)g(automorphism,)-180 982 y(and)h Fo(a)h(fortiori)f Ft(T)505 946 y Fs(n)552 982 y Fu(\()p Ft(V)21 b Fu(\))i Fm(\\)g Ft(I)38 b Fm(\023)30 b Fu(Ker)16 b Fj(S)1262 997 y Fs(n)1309 982 y Fu(.)46 b(Ha)m(ving)34 b(sho)m(wn)h(the)f(\014rst)g(claim,)e(the)i(second)h (claim)c(is)i(essen)m(tially)-180 1121 y(a)f(result)h(from)e([Gu,)i(W) -8 b(a)o(];)33 b(see)h(also)e(the)h(exp)s(osition)e(in)h([AA,)h (Sections)g(3.3)f(and)h(3.4].)p 3883 1121 4 66 v 3887 1059 59 4 v 3887 1121 V 3945 1121 4 66 v -180 1328 a FD(Example)j(3.5.)50 b Fu(Let)32 b Ft(q)g Fm(2)c Fp(|)911 1291 y Fk(\002)964 1328 y Fu(,)33 b Ft(q)j Fu(is)c(not)h(a)f(ro)s(ot)g (of)g(1.)44 b(The)33 b(braided)g(v)m(ector)g(space)h(\()p Ft(V)5 b(;)17 b(q)t(\034)11 b Fu(\))33 b(can)g(b)s(e)g(realized)-180 1467 y(in)-62 1431 y Fd(Z)-62 1493 y(Z)-13 1467 y Fm(Y)8 b(D)r Fu(.)58 b(It)37 b(can)h(b)s(e)f(sho)m(wn)i(that)e Fj(B)p Fu(\()p Ft(V)21 b Fu(\))36 b(=)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\),)39 b(the)e(tensor)h(algebra)e(of)h Ft(V)22 b Fu(,)38 b(for)f(all)e Ft(q)41 b Fu(in)36 b(an)i(op)s(en)f(set.)-180 1607 y(Problem:)42 b(Determine)32 b(this)g(op)s(en)h(set.)-80 1790 y(It)48 b(w)m(ould)g(b)s(e)g(in)m(teresting)f(to)h(kno)m(w)h (whether)g(other)f(conditions)f(on)h(the)h(minimal)43 b(p)s(olynomial)i(of)i(a)-180 1906 y(braiding)f(ha)m(v)m(e)k (consequences)h(on)d(the)h(structure)g(of)f(the)h(corresp)s(onding)f (Nic)m(hols)f(algebra.)89 b(The)49 b(\014rst)-180 2022 y(candidate)32 b(should)h(b)s(e)g(a)f(braiding)f(of)h(BMW-t)m(yp)s(e.) -180 2237 y(3.2.)56 b FD(Braidings)36 b(of)i(diagonal)f(t)m(yp)s(e.)-80 2353 y Fu(In)f(this)f(Section,)h(\()p Ft(V)5 b(;)17 b(c)p Fu(\))35 b(denotes)i(a)f(\014nite)f(dimensional)e(braided)i(v)m(ector)i (space)g(of)e(diagonal)e(t)m(yp)s(e;)38 b(that)-180 2470 y(is,)c Ft(V)56 b Fu(has)35 b(a)f(basis)g Ft(x)615 2485 y Fn(1)655 2470 y Ft(;)17 b(:)g(:)g(:)e(;)i(x)928 2485 y Fs(\022)1002 2470 y Fu(suc)m(h)35 b(that)f(\(1.11\))g(holds)g(for)f (some)i(non-zero)f Ft(q)2814 2485 y Fs(ij)2909 2470 y Fu(in)f Fp(|)-8 b Fu(.)42 b(Our)34 b(\014rst)h(goal)d(is)i(to)-180 2586 y(determine)i(p)s(olynomial)c(relations)j(on)h(the)g(generators)g Ft(x)2026 2601 y Fn(1)2066 2586 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2340 2601 y Fs(\022)2415 2586 y Fu(that)36 b(should)g(hold)f(in)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\).)54 b(W)-8 b(e)36 b(lo)s(ok)-180 2702 y(at)30 b(p)s(olynomial)c(expressions)32 b(in)d(these)i (generators)f(whic)m(h)h(are)f(homogeneous)g(of)f(degree)i Fm(\025)d Fu(2,)i(and)g(giv)m(e)g(rise)-180 2818 y(to)g(primitiv)m(e)f (elemen)m(ts)h(in)g(an)m(y)i(braided)e(Hopf)g(algebra)g(con)m(taining)f Ft(V)52 b Fu(inside)30 b(its)g(primitiv)m(e)e(elemen)m(ts.)44 b(F)-8 b(or)-180 2934 y(related)32 b(material,)e(see)k([Kh].)-180 3141 y FD(Lemma)j(3.6.)49 b Fo(L)-5 b(et)49 b Ft(R)h Fo(b)-5 b(e)48 b(a)g(br)-5 b(aide)g(d)48 b(Hopf)h(algebr)-5 b(a)47 b(in)2054 3105 y Fs(H)2054 3166 y(H)2121 3141 y Fm(Y)8 b(D)r Fo(,)52 b(for)d(some)e(Hopf)i(algebr)-5 b(a)48 b Ft(H)8 b Fo(,)51 b(such)d(that)-180 3280 y Ft(V)h Fm(\022)28 b Ft(P)14 b Fu(\()p Ft(R)q Fu(\))35 b Fo(as)f(br)-5 b(aide)g(d)34 b(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(es.)-156 3440 y Fu(\(a\).)41 b Fo(If)34 b Ft(q)182 3455 y Fs(ii)270 3440 y Fo(is)g(a)h(r)-5 b(o)g(ot)35 b(of)g(1)f(of)h(or)-5 b(der)34 b Ft(N)k(>)28 b Fu(1)35 b Fo(for)f(some)g Ft(i)28 b Fm(2)g(f)p Fu(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(\022)s Fm(g)p Fo(,)34 b(then)h Ft(x)2842 3404 y Fs(N)2842 3465 y(i)2937 3440 y Fm(2)28 b Ft(P)14 b Fu(\()p Ft(R)q Fu(\))p Fo(.)-162 3620 y Fu(\(b\).)42 b Fo(L)-5 b(et)33 b Fu(1)27 b Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(;)17 b(i)28 b Fm(6)p Fu(=)g Ft(j;)33 b Fo(such)f(that)h Ft(q)1456 3635 y Fs(ij)1517 3620 y Ft(q)1560 3635 y Fs(j)t(i)1648 3620 y Fu(=)28 b Ft(q)1799 3584 y Fs(r)1795 3645 y(ii)1847 3620 y Fo(,)33 b(wher)-5 b(e)32 b Fu(0)27 b Fm(\024)h(\000)p Ft(r)j(<)d Fu(ord)16 b Ft(q)2820 3635 y Fs(ii)2905 3620 y Fo(\(which)31 b(c)-5 b(ould)33 b(b)-5 b(e)32 b(in\014nite\).)37 3760 y(Then)i Fu(\(ad)448 3775 y Fs(c)483 3760 y Ft(x)538 3775 y Fs(i)567 3760 y Fu(\))605 3724 y Fn(1)p Fk(\000)p Fs(r)733 3760 y Fu(\()p Ft(x)826 3775 y Fs(j)863 3760 y Fu(\))g Fo(is)h(primitive)f(in)h Ft(R)q Fo(.)-180 3966 y(Pr)-5 b(o)g(of.)41 b Fu(\(a\))32 b(and)h(\(b\))f(are)h(consequences)i (of)d(the)h(quan)m(tum)g(binomial)c(form)m(ula,)h(see)k Fo(e.)45 b(g.)e Fu([AS2,)32 b(App)s(endix])-180 4106 y(for)g(\(b\).)p 3883 4106 V 3887 4043 59 4 v 3887 4106 V 3945 4106 4 66 v -80 4272 a(W)-8 b(e)39 b(apply)g(these)i(\014rst)f (remarks)f(to)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))39 b(and)g(see)i(ho)m(w) e(conditions)g(on)g(the)h(Nic)m(hols)e(algebra)g(induce)-180 4388 y(conditions)32 b(on)g(the)h(braiding.)-180 4594 y FD(Lemma)k(3.7.)49 b Fo(L)-5 b(et)35 b Ft(R)29 b Fu(=)f Fj(B)p Fu(\()p Ft(V)21 b Fu(\))p Fo(.)-156 4754 y Fu(\(a\).)41 b Fo(If)30 b Ft(q)178 4769 y Fs(ii)261 4754 y Fo(is)g(a)g(r)-5 b(o)g(ot)31 b(of)f(1)g(of)g(or)-5 b(der)30 b Ft(N)38 b(>)28 b Fu(1)i Fo(then)g Ft(x)1753 4718 y Fs(N)1753 4779 y(i)1849 4754 y Fu(=)d(0)p Fo(.)43 b(In)30 b(p)-5 b(articular,)31 b(if)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))30 b Fo(is)g(an)g(inte)-5 b(gr)g(al)30 b(domain,)37 4893 y(then)35 b Ft(q)297 4908 y Fs(hh)410 4893 y Fu(=)28 b(1)34 b Fo(or)h(it)g(is)g(not)g(a)f(r)-5 b(o)g(ot)35 b(of)g(1,)g(for)f(al)5 b(l)35 b Ft(h)p Fo(.)-162 5074 y Fu(\(b\).)42 b Fo(If)34 b Ft(i)28 b Fm(6)p Fu(=)g Ft(j)6 b Fo(,)35 b(then)f Fu(\(ad)788 5089 y Fs(c)823 5074 y Ft(x)878 5089 y Fs(i)907 5074 y Fu(\))945 5037 y Fs(r)983 5074 y Fu(\()p Ft(x)1076 5089 y Fs(j)1112 5074 y Fu(\))28 b(=)g(\()p Ft(r)s Fu(\)!)1432 5089 y Fs(q)1464 5099 y Fi(ii)1532 4999 y Fl(Q)1626 5103 y Fn(0)p Fk(\024)p Fs(k)r Fk(\024)p Fs(r)1864 4993 y Fl(\000)1910 5074 y Fu(1)22 b Fm(\000)h Ft(q)2128 5037 y Fs(k)2124 5098 y(ii)2176 5074 y Ft(q)2219 5089 y Fs(ij)2280 5074 y Ft(q)2323 5089 y Fs(j)t(i)2384 4993 y Fl(\001)2446 5074 y Ft(x)2501 5037 y Fs(r)2501 5098 y(i)2539 5074 y Ft(x)2594 5089 y Fs(j)2631 5074 y Ft(:)p eop %%Page: 24 24 24 23 bop -180 0 a Fq(24)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-151 203 y Fu(\(c\).)42 b Fo(If)24 b Ft(i)k Fm(6)p Fu(=)g Ft(j)j Fo(and)24 b Ft(q)587 218 y Fs(ij)648 203 y Ft(q)691 218 y Fs(j)t(i)779 203 y Fu(=)k Ft(q)930 167 y Fs(r)926 228 y(ii)978 203 y Fo(,)f(wher)-5 b(e)24 b Fu(0)k Fm(\024)g(\000)p Ft(r)j(<)c Fu(ord)17 b Ft(q)1938 218 y Fs(ii)2015 203 y Fo(\(which)24 b(c)-5 b(ould)25 b(b)-5 b(e)24 b(in\014nite\),)i(then)f Fu(\(ad)3432 218 y Fs(c)3467 203 y Ft(x)3522 218 y Fs(i)3550 203 y Fu(\))3588 167 y Fn(1)p Fk(\000)p Fs(r)3717 203 y Fu(\()p Ft(x)3810 218 y Fs(j)3846 203 y Fu(\))j(=)37 342 y(0)p Fo(.)-162 524 y Fu(\(d\).)42 b Fo(If)31 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))31 b Fo(has)g(\014nite)g(gr)-5 b(owth,)31 b(then)h(for)f(al)5 b(l)30 b Ft(i)e Fm(6)p Fu(=)g Ft(j)6 b Fo(,)32 b(ther)-5 b(e)31 b(exists)g Ft(r)2487 539 y Fs(ij)2575 524 y Ft(>)c Fu(0)k Fo(such)g(that)h Fu(\(ad)3329 539 y Fs(c)3364 524 y Ft(x)3419 539 y Fs(i)3447 524 y Fu(\))3485 488 y Fs(r)3517 498 y Fi(ij)3576 524 y Fu(\()p Ft(x)3669 539 y Fs(j)3706 524 y Fu(\))c(=)f(0)p Fo(.)-151 706 y Fu(\(e\).)42 b Fo(If)37 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))38 b Fo(is)f(a)g(domain)g(of)g(\014nite)g(gr)-5 b(owth)37 b(and)g(ther)-5 b(e)38 b(exists)f Ft(k)j Fo(such)e(that)f Ft(q)2921 721 y Fs(k)r(k)3035 706 y Fu(=)c(1)p Fo(,)38 b(then)f Ft(q)3523 721 y Fs(ii)3608 706 y Fu(=)32 b(1)p Fo(,)38 b(for)37 846 y(al)5 b(l)34 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)s Fo(,)35 b(and)f Ft(q)869 861 y Fs(ij)930 846 y Ft(q)973 861 y Fs(j)t(i)1062 846 y Fu(=)27 b(1)35 b Fo(for)g(al)5 b(l)34 b Fu(1)27 b Fm(\024)i Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(\022)s(;)17 b(i)28 b Fm(6)p Fu(=)g Ft(j)6 b Fo(.)-144 1028 y Fu(\(f)h(\).)41 b Fo(If)34 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))35 b Fo(has)f(\014nite)h (gr)-5 b(owth)34 b(and)g Ft(i)28 b Fm(6)p Fu(=)g Ft(j)6 b Fo(,)35 b(then)37 1167 y Fu(min)o Fm(f)p Ft(r)30 b(>)d Fu(0)h Fm(j)f Fu(\(ad)716 1182 y Fs(c)751 1167 y Ft(x)806 1182 y Fs(i)835 1167 y Fu(\))873 1131 y Fs(r)910 1167 y Fu(\()p Ft(x)1003 1182 y Fs(j)1040 1167 y Fu(\))h(=)g(0)p Fm(g)f Fu(=)g(min)1619 1087 y Fl(\000)1664 1167 y Fm(f)p Fu(ord)16 b Ft(N)1949 1182 y Fs(ii)2002 1167 y Fm(g)2069 1093 y Fl(S)2151 1167 y Fm(f)p Ft(k)31 b(>)c Fu(0)h Fm(j)f Ft(q)2561 1182 y Fs(ij)2622 1167 y Ft(q)2665 1182 y Fs(j)t(i)2726 1167 y Ft(q)2773 1131 y Fs(k)2769 1192 y(ii)2849 1167 y Fu(=)g(1)p Fm(g)3051 1087 y Fl(\001)3113 1167 y Ft(:)-180 1378 y Fo(Pr)-5 b(o)g(of.)41 b Fu(P)m(arts)35 b(\(a\))f(and)h(\(c\))f (follo)m(w)f(from)g(Lemma)g(3.6.)48 b(P)m(art)35 b(\(b\))f(is)g (apparen)m(tly)g(w)m(ell-kno)m(wn)h(and)f(app)s(ears)-180 1517 y(in)e([Ro2)o(])h(without)f(pro)s(of.)43 b(P)m(art\(d\))33 b(is)f([Ro2)o(,)h(Lemma)e(20].)43 b(The)33 b(rest)h(follo)m(ws)d(from)g (these)j(parts.)p 3883 1517 4 66 v 3887 1455 59 4 v 3887 1517 V 3945 1517 4 66 v -80 1685 a(W)-8 b(e)30 b(no)m(w)g(discuss)h(ho) m(w)g(the)f(t)m(wisting)f(op)s(eration,)g Fo(cf.)42 b Fu(Section)30 b(2.3,)g(a\013ects)h(Nic)m(hols)e(algebras)g(of)g (diagonal)-180 1802 y(t)m(yp)s(e.)-180 2012 y FD(De\014nition)36 b(3.8.)49 b Fu(W)-8 b(e)35 b(shall)f(sa)m(y)i(that)e(t)m(w)m(o)i (braided)e(v)m(ector)i(spaces)h(\()p Ft(V)5 b(;)17 b(c)p Fu(\))35 b(and)f(\()p Ft(W)m(;)17 b(d)p Fu(\))35 b(of)f(diagonal)f(t)m (yp)s(e,)-180 2151 y(with)f(matrices)g(\()p Ft(q)514 2166 y Fs(ij)575 2151 y Fu(\))g(and)h(\()t Fl(b)-59 b Ft(q)916 2166 y Fs(ij)977 2151 y Fu(\),)32 b(are)h Fo(twist-e)-5 b(quivalent)32 b Fu(if)f(dim)15 b Ft(V)50 b Fu(=)27 b(dim)15 b Ft(W)46 b Fu(and,)33 b(for)f(all)f Ft(i;)17 b(j)6 b Fu(,)32 b Ft(q)3444 2166 y Fs(ii)3524 2151 y Fu(=)g Fl(b)-59 b Ft(q)3671 2166 y Fs(ii)3756 2151 y Fu(and)1600 2333 y Ft(q)1643 2348 y Fs(ij)1704 2333 y Ft(q)1747 2348 y Fs(j)t(i)1835 2333 y Fu(=)32 b Fl(b)-59 b Ft(q)1982 2348 y Fs(ij)2047 2333 y Fl(b)g Ft(q)2086 2348 y Fs(j)t(i)2147 2333 y Ft(:)-2354 b Fu(\(3.1\))-180 2544 y FD(Prop)s(osition)36 b(3.9.)49 b Fo(L)-5 b(et)37 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))36 b Fo(and)h Fu(\()p Ft(W)m(;)17 b(d)p Fu(\))36 b Fo(b)-5 b(e)36 b(two)h(twist-e)-5 b(quivalent)36 b(br)-5 b(aide)g(d)36 b(ve)-5 b(ctor)37 b(sp)-5 b(ac)g(es)36 b(of)g(diagonal)-180 2683 y(typ)-5 b(e,)36 b(with)g(matric)-5 b(es)35 b Fu(\()p Ft(q)742 2698 y Fs(ij)803 2683 y Fu(\))h Fo(and)f Fu(\()t Fl(b)-59 b Ft(q)1148 2698 y Fs(ij)1209 2683 y Fu(\))p Fo(;)36 b(say)g(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)h(b)-5 b(asis)35 b Ft(x)2425 2698 y Fn(1)2465 2683 y Ft(;)17 b(:)g(:)g(:)f(x)2695 2698 y Fs(\022)2735 2683 y Fo(,)36 b(r)-5 b(esp.)50 b Fl(b)-58 b Ft(x)3104 2698 y Fn(1)3144 2683 y Ft(;)17 b(:)g(:)g(:)h Fl(b)-57 b Ft(x)3374 2698 y Fs(\022)3413 2683 y Fo(.)48 b(Then)35 b(ther)-5 b(e)-180 2823 y(exists)35 b(a)f(line)-5 b(ar)34 b(isomorphism)g Ft( )d Fu(:)d Fj(B)p Fu(\()p Ft(V)21 b Fu(\))28 b Fm(!)f Fj(B)p Fu(\()p Ft(W)14 b Fu(\))35 b Fo(such)g(that)1333 3005 y Ft( )t Fu(\()p Ft(x)1493 3020 y Fs(i)1522 3005 y Fu(\))27 b(=)k Fl(b)-58 b Ft(x)1746 3020 y Fs(i)1775 3005 y Ft(;)216 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s(:)-2620 b Fu(\(3.2\))-180 3215 y Fo(Pr)-5 b(o)g(of.)41 b Fu(Let)k(\000)h(b)s(e)f(the)g(free)h(ab)s(elian)d(group)i(of)g(rank)g Ft(\022)s Fu(,)k(with)c(basis)g Ft(g)2596 3230 y Fn(1)2635 3215 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)2901 3230 y Fs(\022)2939 3215 y Fu(.)82 b(W)-8 b(e)45 b(de\014ne)h(c)m(haracters)-180 3355 y Ft(\037)-119 3370 y Fn(1)-79 3355 y Ft(;)17 b(:)g(:)g(:)e(;)i (\037)200 3370 y Fs(\022)239 3355 y Fu(,)41 b Fl(b)-63 b Ft(\037)360 3370 y Fn(1)399 3355 y Ft(;)17 b(:)g(:)g(:)f(;)25 b Fl(b)-63 b Ft(\037)679 3370 y Fs(\022)751 3355 y Fu(of)32 b(\000)g(b)m(y)908 3537 y Ft(\037)969 3552 y Fs(i)998 3537 y Fu(\()p Ft(g)1083 3552 y Fs(j)1119 3537 y Fu(\))27 b(=)h Ft(q)1331 3552 y Fs(j)t(i)1392 3537 y Ft(;)220 b Fl(b)-63 b Ft(\037)1692 3552 y Fs(i)1720 3537 y Fu(\()p Ft(g)1805 3552 y Fs(j)1841 3537 y Fu(\))28 b(=)j Fl(b)-59 b Ft(q)2053 3552 y Fs(j)t(i)2114 3537 y Ft(;)212 b Fu(1)27 b Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(:)-180 3718 y Fu(W)-8 b(e)39 b(consider)h Ft(V)21 b Fu(,)41 b Ft(W)52 b Fu(as)40 b(Y)-8 b(etter-Drinfeld)37 b(mo)s(dules)h(o)m(v)m (er)i(\000)f(b)m(y)g(declaring)f Ft(x)2807 3733 y Fs(i)2874 3718 y Fm(2)h Ft(V)3058 3682 y Fs(\037)3102 3692 y Fi(i)3036 3743 y Fs(g)3070 3753 y Fi(i)3132 3718 y Fu(,)k Fl(b)-57 b Ft(x)3255 3733 y Fs(i)3322 3718 y Fm(2)39 b Ft(V)3512 3682 y Ff(b)-46 b Fs(\037)3549 3692 y Fi(i)3484 3743 y Fs(g)3518 3753 y Fi(i)3580 3718 y Fu(.)62 b(Hence,)-180 3858 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))p Ft(;)17 b Fj(B)p Fu(\()p Ft(W)d Fu(\))31 b(are)i(braided)f(Hopf)h(algebras)e(in)1655 3822 y Fn(\000)1655 3883 y(\000)1703 3858 y Fm(Y)8 b(D)s Fu(.)-80 3997 y(Let)32 b Ft(\033)g Fu(:)c(\000)22 b Fm(\002)h Fu(\000)k Fm(!)g Fp(|)696 3961 y Fk(\002)781 3997 y Fu(b)s(e)33 b(the)g(unique)g(bilinear)d(form)i(suc)m(h)i(that)1187 4272 y Ft(\033)t Fu(\()p Ft(g)1331 4287 y Fs(i)1358 4272 y Ft(;)17 b(g)1449 4287 y Fs(j)1485 4272 y Fu(\))28 b(=)1654 4068 y Fl(8)1654 4157 y(<)1654 4337 y(:)1844 4197 y(b)-58 b Ft(q)1884 4212 y Fs(ij)1944 4197 y Ft(q)1991 4155 y Fk(\000)p Fn(1)1987 4222 y Fs(ij)2085 4197 y Ft(;)212 b(i)28 b Fm(\024)g Ft(j;)1841 4364 y Fu(1)p Ft(;)406 b(i)28 b(>)g(j)6 b Fu(;)-180 4272 y(\(3.3\))-180 4552 y(it)28 b(is)h(a)g(group)g(co)s(cycle.)43 b(W)-8 b(e)30 b(claim)d(that)i Ft(')f Fu(:)f Ft(W)42 b Fm(!)27 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))2013 4567 y Fs(\033)2059 4552 y Fu(\(1\),)30 b Ft(')p Fu(\()s Fl(b)-58 b Ft(x)2398 4567 y Fs(i)2426 4552 y Fu(\))28 b(=)g Ft(x)2651 4567 y Fs(i)2679 4552 y Fu(,)i(1)e Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)s Fu(,)i(is)f(an)g(isomorphism)-180 4691 y(in)-66 4655 y Fn(\000)-66 4716 y(\000)-18 4691 y Fm(Y)8 b(D)r Fu(.)44 b(It)33 b(clearly)e(preserv)m(es)36 b(the)d(coaction;)e(for)h (the)h(action,)f(w)m(e)i(assume)f Ft(i)28 b Fm(\024)g Ft(j)38 b Fu(and)33 b(compute)1129 4873 y Ft(g)1176 4888 y Fs(j)1234 4873 y Fm(\001)1262 4888 y Fs(\033)1331 4873 y Ft(x)1386 4888 y Fs(i)1442 4873 y Fu(=)27 b Ft(\033)t Fu(\()p Ft(g)1689 4888 y Fs(j)1726 4873 y Ft(;)17 b(g)1817 4888 y Fs(i)1844 4873 y Fu(\))p Ft(\033)1941 4832 y Fk(\000)p Fn(1)2036 4873 y Fu(\()p Ft(g)2121 4888 y Fs(i)2148 4873 y Ft(;)g(g)2239 4888 y Fs(j)2275 4873 y Fu(\))p Ft(\037)2374 4888 y Fs(i)2403 4873 y Fu(\()p Ft(g)2488 4888 y Fs(j)2524 4873 y Fu(\))p Ft(x)2617 4888 y Fs(i)1442 5042 y Fu(=)27 b(\()t Fl(b)-59 b Ft(q)1626 5057 y Fs(ij)1687 5042 y Fu(\))1725 5000 y Fk(\000)p Fn(1)1819 5042 y Ft(q)1862 5057 y Fs(ij)1923 5042 y Ft(q)1966 5057 y Fs(j)t(i)2027 5042 y Ft(x)2082 5057 y Fs(i)2138 5042 y Fu(=)32 b Fl(b)-59 b Ft(q)2285 5057 y Fs(j)t(i)2346 5042 y Ft(x)2401 5057 y Fs(i)2429 5042 y Ft(;)p eop %%Page: 25 25 25 24 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(25)-180 203 y Fu(and)33 b(also)1124 401 y Ft(g)1171 416 y Fs(i)1221 401 y Fm(\001)1249 416 y Fs(\033)1318 401 y Ft(x)1373 416 y Fs(j)1438 401 y Fu(=)27 b Ft(\033)t Fu(\()p Ft(g)1685 416 y Fs(i)1713 401 y Ft(;)17 b(g)1804 416 y Fs(j)1840 401 y Fu(\))p Ft(\033)1937 360 y Fk(\000)p Fn(1)2031 401 y Fu(\()p Ft(g)2116 416 y Fs(j)2153 401 y Ft(;)g(g)2244 416 y Fs(i)2271 401 y Fu(\))p Ft(\037)2370 416 y Fs(j)2407 401 y Fu(\()p Ft(g)2492 416 y Fs(i)2520 401 y Fu(\))p Ft(x)2613 416 y Fs(j)1438 569 y Fu(=)31 b Fl(b)-59 b Ft(q)1584 584 y Fs(ij)1645 569 y Ft(q)1692 528 y Fk(\000)p Fn(1)1688 595 y Fs(ij)1786 569 y Ft(q)1829 584 y Fs(ij)1890 569 y Ft(x)1945 584 y Fs(j)2010 569 y Fu(=)31 b Fl(b)-59 b Ft(q)2156 584 y Fs(ij)2217 569 y Ft(x)2272 584 y Fs(j)2309 569 y Ft(;)-180 767 y Fu(where)43 b(w)m(e)f(ha)m(v)m(e)h(used)f(\(2.8\))f(and)h(the)g(h)m(yp)s(othesis)g (\(3.1\).)70 b(This)41 b(pro)m(v)m(es)i(the)f(claim.)68 b(By)42 b(Lemma)e(3.6,)j Ft(')-180 906 y Fu(extends)35 b(to)d(an)g(isomorphism)e Ft(')e Fu(:)g Fj(B)p Fu(\()p Ft(W)14 b Fu(\))27 b Fm(!)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))1814 921 y Fs(\033)1861 906 y Fu(;)32 b Ft( )g Fu(=)27 b Ft(')2182 870 y Fk(\000)p Fn(1)2309 906 y Fu(is)32 b(the)h(map)f(w)m(e)i(are)e(lo)s(oking)f(for.)p 3883 906 4 66 v 3887 844 59 4 v 3887 906 V 3945 906 4 66 v -180 1127 a FD(Remarks)37 b(3.10.)49 b Fu(\(i\).)42 b(The)32 b(map)e Ft( )35 b Fu(de\014ned)d(in)e(the)i(pro)s(of)e(is)g(m)m(uc)m(h) i(more)e(than)h(just)g(linear;)f(b)m(y)j(\(2.7\))e(and)-180 1267 y(\(2.10\))o(,)i(w)m(e)h(ha)m(v)m(e)752 1604 y Ft( )t Fu(\()p Ft(xy)t Fu(\))27 b(=)h Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\))p Ft( )t Fu(\()p Ft(x)p Fu(\))p Ft( )t Fu(\()p Ft(y)t Fu(\))p Ft(;)210 b(x)28 b Fm(2)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))2469 1619 y Fs(g)2509 1604 y Ft(;)114 b(y)31 b Fm(2)d Fj(B)p Fu(\()p Ft(V)22 b Fu(\))3066 1619 y Fs(h)3110 1604 y Fu(;)-3317 b(\(3.4\))620 1773 y Ft( )t Fu(\([)p Ft(x;)17 b(y)t Fu(])930 1788 y Fs(c)964 1773 y Fu(\))27 b(=)h Ft(\033)t Fu(\()p Ft(g)t(;)17 b(h)p Fu(\)[)p Ft( )t Fu(\()p Ft(x)p Fu(\))p Ft(;)g( )t Fu(\()p Ft(y)t Fu(\)])1910 1788 y Fs(c)1943 1773 y Ft(;)212 b(x)28 b Fm(2)g Ft(V)2438 1731 y Fs(\037)2416 1797 y(g)2485 1773 y Ft(;)115 b(y)31 b Fm(2)d Fj(B)p Fu(\()p Ft(V)21 b Fu(\))3042 1725 y Fs(\021)3042 1800 y(h)3087 1773 y Ft(:)-3294 b Fu(\(3.5\))-80 1970 y(\(ii\).)68 b(A)42 b(braided)f(v)m(ector)i(space)g(\()p Ft(V)5 b(;)17 b(c)p Fu(\))41 b(of)h(diagonal)d(t)m(yp)s(e,)45 b(with)c(matrix)f(\()p Ft(q)2876 1985 y Fs(ij)2937 1970 y Fu(\),)k(is)d(t)m(wist-equiv)-5 b(alen)m(t)41 b(to)-180 2110 y(\()p Ft(W)m(;)17 b(d)p Fu(\),)32 b(with)g(a)h Fo(symmetric)f Fu(matrix)f(\()t Fl(b)-59 b Ft(q)1313 2125 y Fs(ij)1374 2110 y Fu(\).)-180 2308 y(3.3.)56 b FD(Braidings)36 b(of)i(diagonal)f(t)m(yp)s(e)g(but)h (not)f(Cartan.)-80 2424 y Fu(In)24 b(the)h(next)g(Chapter,)i(w)m(e)f (shall)d(concen)m(trate)i(on)g(braidings)e(of)h(Cartan)g(t)m(yp)s(e.)42 b(There)25 b(are)g(a)f(few)h(examples)-180 2540 y(of)i(Nic)m(hols)h (algebras)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))28 b(of)f(\014nite)h(group)g (t)m(yp)s(e)g(and)h(rank)f(2,)h(whic)m(h)f(are)g Fo(not)g Fu(of)f(Cartan)h(t)m(yp)s(e,)i(but)f(where)-180 2656 y(w)m(e)e(kno)m(w)h(that)e(the)g(dimension)f(is)h(\014nite.)41 b(W)-8 b(e)27 b(no)m(w)f(list)f(them)h(all,)g(follo)m(wing)d([N,)k(G)s (~)-51 b(n3)n(].)42 b(The)27 b(braided)f(v)m(ector)-180 2773 y(space)37 b(is)e(necessarily)h(of)f(diagonal)e(t)m(yp)s(e;)38 b(w)m(e)f(shall)d(giv)m(e)i(the)g(matrix)e Ft(Q)i Fu(of)f(the)h (braiding,)e(the)i(constrain)m(ts)-180 2889 y(on)f(their)f(en)m(tries)h (and)g(the)g(dimension)f Ft(d)g Fu(of)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\).)50 b(Belo)m(w,)35 b Ft(!)t Fu(,)g(resp.)51 b Ft(\020)8 b Fu(,)35 b(denotes)h(an)e(arbitrary)g(primitiv)m(e)-180 3005 y(third)e(ro)s(ot)g(of)g(1,)g(resp.)44 b(di\013eren)m(t)33 b(from)f(1.)136 3192 y Fl(\022)209 3272 y Ft(q)252 3287 y Fn(11)414 3272 y Ft(q)457 3287 y Fn(12)209 3389 y Ft(q)252 3404 y Fn(21)410 3389 y Fm(\000)p Fu(1)536 3192 y Fl(\023)626 3332 y Fu(;)114 b Ft(q)814 3291 y Fk(\000)p Fn(1)810 3357 y(11)936 3332 y Fu(=)28 b Ft(q)1083 3347 y Fn(12)1158 3332 y Ft(q)1201 3347 y Fn(21)1303 3332 y Fm(6)p Fu(=)g(1;)800 b Ft(d)27 b Fu(=)h(4)17 b(ord)o(\()p Ft(q)2752 3347 y Fn(12)2827 3332 y Ft(q)2870 3347 y Fn(21)2945 3332 y Fu(\))p Ft(:)-3190 b Fu(\(3.6\))136 3613 y Fl(\022)209 3694 y Ft(q)252 3709 y Fn(11)410 3694 y Ft(q)453 3709 y Fn(12)209 3810 y Ft(q)252 3825 y Fn(21)437 3810 y Ft(!)528 3613 y Fl(\023)618 3754 y Fu(;)114 b Ft(q)806 3713 y Fk(\000)p Fn(1)802 3779 y(11)928 3754 y Fu(=)28 b Ft(q)1075 3769 y Fn(12)1150 3754 y Ft(q)1193 3769 y Fn(21)1295 3754 y Fm(6)p Fu(=)g Fm(\006)p Fu(1)p Ft(;)17 b(!)1634 3713 y Fk(\000)p Fn(1)1727 3754 y Fu(;)529 b Ft(d)27 b Fu(=)h(9)17 b(ord)o(\()p Ft(q)2752 3769 y Fn(11)2827 3754 y Fu(\))g(ord)o(\()p Ft(q)3103 3769 y Fn(12)3178 3754 y Ft(q)3221 3769 y Fn(21)3296 3754 y Ft(!)t Fu(\))p Ft(:)-3606 b Fu(\(3.7\))136 4035 y Fl(\022)209 4116 y Fm(\000)p Fu(1)83 b Ft(q)461 4131 y Fn(12)213 4232 y Ft(q)256 4247 y Fn(21)445 4232 y Ft(!)536 4035 y Fl(\023)626 4176 y Fu(;)114 b Ft(q)810 4191 y Fn(12)885 4176 y Ft(q)928 4191 y Fn(21)1031 4176 y Fu(=)28 b Fm(\000)p Fu(1;)995 b Ft(d)27 b Fu(=)h(108)p Ft(:)-2819 b Fu(\(3.8\))136 4457 y Fl(\022)209 4538 y Fm(\000)p Fu(1)83 b Ft(q)461 4553 y Fn(12)213 4654 y Ft(q)256 4669 y Fn(21)445 4654 y Ft(!)536 4457 y Fl(\023)626 4597 y Fu(;)114 b Ft(q)810 4612 y Fn(12)885 4597 y Ft(q)928 4612 y Fn(21)1031 4597 y Fu(=)28 b Ft(!)t Fu(;)1056 b Ft(d)27 b Fu(=)h(72)p Ft(:)-2770 b Fu(\(3.9\))136 4879 y Fl(\022)209 4959 y Fm(\000)p Fu(1)83 b Ft(q)461 4974 y Fn(12)213 5076 y Ft(q)256 5091 y Fn(21)445 5076 y Ft(!)536 4879 y Fl(\023)626 5019 y Fu(;)114 b Ft(q)810 5034 y Fn(12)885 5019 y Ft(q)928 5034 y Fn(21)1031 5019 y Fu(=)28 b Fm(\000)p Ft(!)t Fu(;)979 b Ft(d)27 b Fu(=)h(36)p Ft(:)-2770 b Fu(\(3.10\))p eop %%Page: 26 26 26 25 bop -180 0 a Fq(26)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)136 283 y Fl(\022)209 364 y Fm(\000)p Fu(1)83 b Ft(q)461 379 y Fn(12)213 480 y Ft(q)256 495 y Fn(21)452 480 y Ft(\020)536 283 y Fl(\023)626 424 y Fu(;)114 b Ft(q)810 439 y Fn(12)885 424 y Ft(q)928 439 y Fn(21)1031 424 y Fu(=)28 b Ft(\020)1186 383 y Fk(\000)p Fn(2)1279 424 y Fu(;)977 b Ft(d)27 b Fu(=)h(4)17 b(ord)o(\()p Ft(\020)8 b Fu(\))17 b(ord)n(\()p Fm(\000)p Ft(\020)3120 383 y Fk(\000)p Fn(1)3214 424 y Fu(\))p Ft(:)-3459 b Fu(\(3.11\))-180 896 y(3.4.)56 b FD(Braidings)36 b(of)i(\014nite)f (non-ab)s(elian)h(group)g(t)m(yp)s(e.)-80 1012 y Fu(W)-8 b(e)33 b(b)s(egin)e(with)i(a)f(class)h(of)f(examples)g(studied)h(in)f ([MiS].)-80 1128 y(Let)37 b(\000)h(b)s(e)f(a)g(group)h(and)f Ft(T)50 b Fm(\032)37 b Fu(\000)g(a)g(subset)i(suc)m(h)g(that)e(for)g (all)f Ft(g)j Fm(2)d Fu(\000)p Ft(;)17 b(t)36 b Fm(2)h Ft(T)8 b(;)17 b(g)t(tg)3055 1092 y Fk(\000)p Fn(1)3184 1128 y Fm(2)36 b Ft(T)14 b Fu(.)58 b(Th)m(us)39 b Ft(T)51 b Fu(is)37 b(a)-180 1245 y(union)c(of)h(conjugacy)h(classes)g(of)e (\000.)48 b(Let)35 b Ft(\036)30 b Fu(:)g(\000)23 b Fm(\002)h Ft(T)44 b Fm(!)29 b Fp(|)9 b Fm(n)23 b(f)p Fu(0)p Fm(g)33 b Fu(b)s(e)i(a)e(function)h(suc)m(h)i(that)d(for)h(all)e Ft(g)t(;)17 b(h)29 b Fm(2)i Fu(\000)-180 1361 y(and)i Ft(t)28 b Fm(2)g Ft(T)14 b Fu(,)177 1543 y Ft(\036)p Fu(\(1)p Ft(;)j(t)p Fu(\))27 b(=)g(1)p Ft(;)-825 b Fu(\(3.12\))119 1703 y Ft(\036)p Fu(\()p Ft(g)t(h;)17 b(t)p Fu(\))27 b(=)g Ft(\036)p Fu(\()p Ft(g)t(;)17 b(hth)907 1662 y Fk(\000)p Fn(1)1001 1703 y Fu(\))p Ft(\036)p Fu(\()p Ft(h;)g(t)p Fu(\))p Ft(:)-1515 b Fu(\(3.13\))-180 1891 y(W)-8 b(e)37 b(can)f(then)h(de\014ne)g(a)f(Y)-8 b(etter-Drinfeld)34 b(mo)s(dule)h Ft(V)55 b Fu(=)34 b Ft(V)21 b Fu(\(\000)p Ft(;)c(T)8 b(;)17 b(\036)p Fu(\))36 b(o)m(v)m(er)h(\000)f(with)g Fp(|)-9 b Fu(-basi)o(s)31 b Ft(x)3405 1906 y Fs(t)3435 1891 y Fu(,)37 b Ft(t)d Fm(2)g Ft(T)8 b(;)37 b Fu(and)-180 2007 y(action)32 b(and)g(coaction)g(of)g Ft(W)46 b Fu(giv)m(en)33 b(b)m(y)192 2189 y Ft(g)t(x)298 2204 y Fs(t)355 2189 y Fu(=)27 b Ft(\036)p Fu(\()p Ft(g)t(;)17 b(t)p Fu(\))p Ft(x)777 2206 y Fs(g)r(tg)874 2187 y Fh(\000)p Fg(1)961 2189 y Ft(;)-1168 b Fu(\(3.14\))119 2340 y Ft(\016)t Fu(\()p Ft(x)259 2355 y Fs(t)289 2340 y Fu(\))28 b(=)f Ft(t)c Fm(\012)f Ft(x)670 2355 y Fs(t)-180 2340 y Fu(\(3.15\))-180 2522 y(for)32 b(all)f Ft(g)f Fm(2)f Fu(\000)p Ft(;)17 b(t)27 b Fm(2)h Ft(T)14 b Fu(.)-80 2639 y(Con)m(v)m(ersely)-8 b(,)34 b(if)c(the)i(function)f Ft(\036)h Fu(de\014nes)h(a)e(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)g(on)i(the)g(v)m(ector)h(space)f Ft(V)54 b Fu(b)m(y)33 b(\(3.14\),)-180 2755 y(\(3.15\))o(,)g(then)g Ft(\036)f Fu(satis\014es)i(\(3.12\))o(,)f(\(3.13\))o(.)-80 2871 y(Note)f(that)h(the)g(braiding)d Ft(c)j Fu(of)f Ft(V)22 b Fu(\(\000)p Ft(;)17 b(T)8 b(;)17 b(\036)p Fu(\))31 b(is)h(determined)h(b)m(y)936 3053 y Ft(c)p Fu(\()p Ft(x)1071 3068 y Fs(s)1130 3053 y Fm(\012)23 b Ft(x)1285 3068 y Fs(t)1315 3053 y Fu(\))k(=)h Ft(\036)p Fu(\()p Ft(s;)17 b(t)p Fu(\))p Ft(x)1798 3068 y Fs(sts)1889 3077 y Fh(\000)p Fg(1)1997 3053 y Fm(\012)23 b Ft(x)2152 3068 y Fs(t)2215 3053 y Fu(for)32 b(all)e Ft(s;)17 b(t)28 b Fm(2)g Ft(T)8 b(;)-180 3235 y Fu(hence)34 b(b)m(y)f(the)g(v)-5 b(alues)33 b(of)f Ft(\036)g Fu(on)h Ft(T)j Fm(\002)22 b Ft(T)14 b Fu(.)-80 3417 y(The)38 b(main)e(examples)h(come)g(from)f(the)i (theory)g(of)f(Co)m(xeter)i(groups)e(\([BL,)i(Chapitre)e(IV]\).)h(Let)f Ft(S)44 b Fu(b)s(e)37 b(a)-180 3533 y(subset)j(of)f(the)g(group)g Ft(W)52 b Fu(of)39 b(elemen)m(ts)g(of)f(order)i(2.)62 b(F)-8 b(or)38 b(all)f Ft(s;)17 b(s)2343 3497 y Fk(0)2404 3533 y Fm(2)39 b Ft(S)44 b Fu(let)39 b Ft(m)p Fu(\()p Ft(s;)17 b(s)3020 3497 y Fk(0)3043 3533 y Fu(\))39 b(b)s(e)g(the)g (order)g(of)g Ft(ss)3904 3497 y Fk(0)3927 3533 y Fu(.)-180 3649 y(\()p Ft(W)m(;)17 b(S)6 b Fu(\))37 b(is)h(called)e(a)i Fo(Coxeter)h(system)f Fu(and)g Ft(W)51 b Fu(a)38 b Fo(Coxeter)h(gr)-5 b(oup)37 b Fu(if)g Ft(W)51 b Fu(is)38 b(generated)g(b)m(y)h Ft(S)k Fu(with)38 b(de\014ning)-180 3770 y(relations)31 b(\()p Ft(ss)346 3734 y Fk(0)369 3770 y Fu(\))407 3734 y Fs(m)p Fn(\()p Fs(s;s)582 3710 y Fh(0)604 3734 y Fn(\))664 3770 y Fu(=)c(1)32 b(for)g(all)f Ft(s;)17 b(s)1269 3734 y Fk(0)1320 3770 y Fm(2)28 b Ft(S)38 b Fu(suc)m(h)c(that)e Ft(m)p Fu(\()p Ft(s;)17 b(s)2202 3734 y Fk(0)2225 3770 y Fu(\))33 b(is)f(\014nite.)-80 3952 y(Let)h(\()p Ft(W)m(;)17 b(S)6 b Fu(\))34 b(b)s(e)g(a)f(Co)m(xeter)i(system.)48 b(F)-8 b(or)33 b(an)m(y)h Ft(g)f Fm(2)d Ft(W)48 b Fu(there)34 b(is)f(a)h(sequence)i(\()p Ft(s)2947 3967 y Fn(1)2986 3952 y Ft(;)17 b(:)g(:)g(:)f(;)h(s)3251 3967 y Fs(q)3289 3952 y Fu(\))33 b(of)h(elemen)m(ts)g(in)-180 4068 y Ft(S)k Fu(with)33 b Ft(g)e Fu(=)d Ft(s)369 4083 y Fn(1)431 4068 y Fm(\001)22 b(\001)17 b(\001)g(\001)j(\001)i Ft(s)715 4083 y Fs(q)753 4068 y Fu(.)44 b(If)33 b Ft(q)j Fu(is)c(minimal)d (among)i(all)g(suc)m(h)j(represen)m(tations,)g(then)g Ft(q)e Fu(=)27 b Ft(l)r Fu(\()p Ft(g)t Fu(\))33 b(is)f(called)f(the) -180 4184 y Fo(length)h Fu(of)g Ft(g)t Fu(,)g(and)h(\()p Ft(s)599 4199 y Fn(1)638 4184 y Ft(;)17 b(:)g(:)g(:)f(;)h(s)903 4199 y Fs(q)941 4184 y Fu(\))32 b(is)g(a)g Fo(r)-5 b(e)g(duc)g(e)g(d)35 b(r)-5 b(epr)g(esentation)32 b Fu(of)g Ft(g)t Fu(.)-180 4411 y FD(De\014nition)k(3.11.)49 b Fu(Let)43 b(\()p Ft(W)m(;)17 b(S)6 b Fu(\))41 b(b)s(e)i(a)f(Co)m(xeter)h(system,)j(and)c Ft(T)58 b Fu(=)43 b Fm(f)p Ft(g)t(sg)2723 4375 y Fk(\000)p Fn(1)2860 4411 y Fm(j)h Ft(g)j Fm(2)d Ft(W)m(;)17 b(s)44 b Fm(2)h Ft(S)6 b Fm(g)p Fu(.)71 b(De\014ne)-180 4550 y Ft(\036)27 b Fu(:)h Ft(W)36 b Fm(\002)23 b Ft(T)41 b Fm(!)27 b Fp(|)8 b Fm(n)22 b(f)p Fu(0)p Fm(g)32 b Fu(b)m(y)1095 4755 y Ft(\036)p Fu(\()p Ft(g)t(;)17 b(t)p Fu(\))26 b(=)i(\()p Fm(\000)p Fu(1\))1691 4714 y Fs(l)q Fn(\()p Fs(g)r Fn(\))1840 4755 y Fu(for)k(all)f Ft(g)g Fm(2)d Ft(W)m(;)17 b(t)28 b Fm(2)g Ft(T)8 b(:)-2859 b Fu(\(3.16\))-80 4964 y(This)35 b Ft(\036)g Fu(satis\014es)h(\(3.12\))f(and)g(\(3.13\))o(.)52 b(Th)m(us)37 b(w)m(e)f(ha)m(v)m(e)h(asso)s(ciated)e(to)g(eac)m(h)h(Co)m (xeter)h(group)e(the)h(Y)-8 b(etter-)-180 5081 y(Drinfeld)31 b(mo)s(dule)g Ft(V)21 b Fu(\()p Ft(W)m(;)c(T)8 b(;)17 b(\036)p Fu(\))28 b Fm(2)1121 5044 y Fs(W)1121 5106 y(W)1201 5081 y Fm(Y)8 b(D)s Fu(.)p eop %%Page: 27 27 27 26 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(27)-80 203 y Fu(The)28 b(functions)f Ft(\036)g Fu(satisfying)h (\(3.12\),)g(\(3.13\))f(can)g(b)s(e)h(constructed)h(up)f(to)f(a)g (diagonal)e(c)m(hange)j(of)f(the)h(basis)-180 319 y(from)34 b(c)m(haracters)j(of)d(the)i(cen)m(tralizers)f(of)g(elemen)m(ts)g(in)g (the)h(conjugacy)f(classes.)53 b(This)35 b(is)g(a)g(sp)s(ecial)f(case)i (of)-180 435 y(the)30 b(description)f(of)g(the)h(simple)d(mo)s(dules)i (in)1545 399 y Fn(\000)1545 461 y(\000)1593 435 y Fm(Y)8 b(D)32 b Fu(\(see)e([W])g(and)f(also)g([L4]\);)h(the)g(equiv)-5 b(alen)m(t)29 b(classi\014cation)-180 552 y(of)36 b(the)g(simple)f (Hopf)h(bimo)s(dules)e(o)m(v)m(er)j(\000)f(w)m(as)h(obtained)f(in)f ([DPR])h(\(o)m(v)m(er)h Fp(|)-8 b Fu(\))30 b(and)36 b(then)h(in)e([Ci]) h(\(o)m(v)m(er)h(an)m(y)-180 668 y(\014eld\).)-80 842 y(Let)28 b Ft(t)f Fu(b)s(e)h(an)g(elemen)m(t)g(in)f(\000.)41 b(W)-8 b(e)29 b(denote)f(b)m(y)h Fm(O)1688 857 y Fs(t)1745 842 y Fu(and)f(\000)1991 806 y Fs(t)2049 842 y Fu(the)g(conjugacy)g (class)g(and)g(the)g(cen)m(tralizer)f(of)h Ft(t)g Fu(in)-180 958 y(\000.)43 b(Let)30 b Ft(U)41 b Fu(b)s(e)31 b(an)m(y)g(left)e Fp(|)-9 b Fu(\000)833 922 y Fs(t)857 958 y Fu(-mo)s(dule.)41 b(It)30 b(is)g(easy)h(to)f(see)i(that)e(the)h(induced)f(represen)m (tation)i Ft(V)49 b Fu(=)27 b Fp(|)-8 b Fu(\000)11 b Fm(\012)3746 973 y Fd(|)-13 b Fn(\000)3834 954 y Fi(t)3877 958 y Ft(U)-180 1074 y Fu(is)32 b(a)g(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)j(\000)e(with)h(the)g(induced)g(action)e(of)h (\000)h(and)g(the)g(coaction)522 1248 y Ft(\016)f Fu(:)c Ft(V)49 b Fm(!)27 b Fp(|)-9 b Fu(\000)17 b Fm(\012)22 b Ft(V)5 b(;)115 b(\016)t Fu(\()p Ft(g)25 b Fm(\012)e Ft(u)p Fu(\))k(=)g Ft(g)t(tg)1945 1207 y Fk(\000)p Fn(1)2061 1248 y Fm(\012)22 b Ft(g)k Fm(\012)c Ft(u)32 b Fu(for)g(all)f Ft(g)g Fm(2)d Fu(\000)p Ft(;)17 b(u)27 b Fm(2)h Ft(U:)-180 1422 y Fu(W)-8 b(e)33 b(will)d(denote)k(this)e(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)i(\000)g(b)m(y)g Ft(M)10 b Fu(\()p Ft(t;)17 b(U)10 b Fu(\).)-80 1538 y(Assume)33 b(that)f(\000)g(is)f (\014nite.)43 b(Then)34 b Ft(V)49 b Fu(=)27 b Ft(M)10 b Fu(\()p Ft(t;)17 b(U)10 b Fu(\))34 b(is)d(a)h(simple)f(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)h(if)g Ft(U)43 b Fu(is)31 b(a)h(simple)-180 1654 y(represen)m(tation)25 b(of)f(\000)613 1618 y Fs(t)643 1654 y Fu(,)i(and)f(eac)m(h)h(simple)d(mo)s(dule)g(in) 1828 1618 y Fn(\000)1828 1680 y(\000)1876 1654 y Fm(Y)8 b(D)27 b Fu(has)e(this)f(form.)40 b(If)24 b(w)m(e)i(tak)m(e)f(from)f (eac)m(h)h(conjugacy)-180 1771 y(class)h(one)f(elemen)m(t)h Ft(t)f Fu(and)h(non-isomorphic)d(simple)h(\000)1844 1734 y Fs(t)1873 1771 y Fu(-mo)s(dules,)i(an)m(y)g(t)m(w)m(o)g(of)f(these)i (simple)d(Y)-8 b(etter-Drinfeld)-180 1887 y(mo)s(dules)32 b(are)g(non-isomorphic.)-80 2003 y(Let)d Ft(s)137 2018 y Fs(i)165 2003 y Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s Fu(,)i(b)s(e)g(a)e(complete)h(system)h(of)e (represen)m(tativ)m(es)k(of)c(the)i(residue)g(classes)g(of)e(\000)3441 1967 y Fs(t)3471 2003 y Fu(.)42 b(W)-8 b(e)30 b(de\014ne)-180 2119 y Ft(t)-145 2134 y Fs(i)-89 2119 y Fu(=)e Ft(s)61 2134 y Fs(i)89 2119 y Ft(ts)170 2078 y Fk(\000)p Fn(1)170 2145 y Fs(i)297 2119 y Fu(for)k(all)e(1)e Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)s Fu(.)44 b(Th)m(us)1159 2293 y(\000)p Ft(=)p Fu(\000)1330 2252 y Fs(t)1387 2293 y Fm(!)28 b(O)1594 2308 y Fs(t)1624 2293 y Ft(;)114 b(s)1811 2308 y Fs(i)1839 2293 y Fu(\000)1900 2252 y Fs(t)1957 2293 y Fm(7!)27 b Ft(t)2119 2308 y Fs(i)2148 2293 y Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s(;)-180 2467 y Fu(is)37 b(bijectiv)m(e,)h(and)f(as)g(a)g(v)m(ector)h(space,)h Ft(V)57 b Fu(=)1562 2392 y Fl(L)1673 2496 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(\022)1897 2467 y Ft(s)1943 2482 y Fs(i)1997 2467 y Fm(\012)25 b Ft(U)10 b Fu(.)58 b(F)-8 b(or)36 b(all)f Ft(g)j Fm(2)e Fu(\000)h(and)g(1)e Fm(\024)g Ft(i)h Fm(\024)g Ft(\022)s Fu(,)i(there)g(is)e(a)-180 2600 y(uniquely)c(determined)f(1)d Fm(\024)g Ft(j)34 b Fm(\024)28 b Ft(\022)35 b Fu(with)c Ft(s)1431 2558 y Fk(\000)p Fn(1)1431 2625 y Fs(j)1525 2600 y Ft(g)t(s)1622 2615 y Fs(i)1677 2600 y Fm(2)d Fu(\000)1832 2563 y Fs(t)1862 2600 y Fu(,)k(and)g(the)g(action)f(of)g Ft(g)k Fu(on)c Ft(s)2941 2615 y Fs(i)2990 2600 y Fm(\012)21 b Ft(u;)c(u)26 b Fm(2)i Ft(U)10 b Fu(,)33 b(is)e(giv)m(en)h(b)m(y)1357 2788 y Ft(g)t(s)1454 2803 y Fs(i)1503 2788 y Fm(\012)23 b Ft(u)k Fu(=)h Ft(s)1836 2803 y Fs(j)1894 2788 y Fm(\012)23 b Fu(\()p Ft(s)2078 2747 y Fk(\000)p Fn(1)2078 2814 y Fs(j)2172 2788 y Ft(g)t(s)2269 2803 y Fs(i)2297 2788 y Fu(\))p Ft(u:)-180 2969 y Fu(In)42 b(particular,)h(if)d Ft(U)53 b Fu(is)41 b(a)g(one-dimensional)e(\000)1640 2933 y Fs(t)1670 2969 y Fu(-mo)s(dule)h(with)h(basis)h Ft(u)f Fu(and)h(action)f Ft(hu)i Fu(=)g Ft(\037)p Fu(\()p Ft(g)t Fu(\))p Ft(u)e Fu(for)g(all)-180 3086 y Ft(h)31 b Fm(2)h Fu(\000)66 3049 y Fs(t)130 3086 y Fu(de\014ned)k(b)m(y)f(the)g (c)m(haracter)h Ft(\037)31 b Fu(:)g(\000)1414 3049 y Fs(t)1475 3086 y Fm(!)f Fp(|)9 b Fm(n)23 b(f)p Fu(0)p Fm(g)p Fu(,)35 b(then)g Ft(V)56 b Fu(has)35 b(a)f(basis)h Ft(x)2861 3101 y Fs(i)2920 3086 y Fu(=)c Ft(s)3073 3101 y Fs(i)3125 3086 y Fm(\012)24 b Ft(u;)17 b Fu(1)30 b Fm(\024)i Ft(i)f Fm(\024)g Ft(\022)s Fu(,)36 b(and)-180 3202 y(the)d(action)f(and)g(coaction)g(of)g(\000)h(are)f(giv)m(en)h(b)m (y)1119 3376 y Ft(g)t(x)1225 3391 y Fs(i)1281 3376 y Fu(=)27 b Ft(\037)p Fu(\()p Ft(s)1529 3334 y Fk(\000)p Fn(1)1529 3401 y Fs(j)1624 3376 y Ft(g)t(s)1721 3391 y Fs(i)1748 3376 y Fu(\))p Ft(x)1841 3391 y Fs(j)2008 3376 y Fu(and)33 b Ft(\016)t Fu(\()p Ft(x)2338 3391 y Fs(i)2366 3376 y Fu(\))28 b(=)f Ft(x)2590 3391 y Fs(j)2627 3376 y Ft(;)-180 3564 y Fu(if)j Ft(s)-46 3523 y Fk(\000)p Fn(1)-46 3590 y Fs(j)49 3564 y Ft(g)t(s)146 3579 y Fs(i)201 3564 y Fm(2)e Fu(\000)356 3528 y Fs(t)385 3564 y Fu(.)44 b(Note)31 b(that)h Ft(g)t(t)987 3579 y Fs(i)1014 3564 y Ft(g)1065 3528 y Fk(\000)p Fn(1)1187 3564 y Fu(=)27 b Ft(t)1325 3579 y Fs(j)1362 3564 y Fu(.)43 b(Hence)33 b(the)f(mo)s(dule)e(w)m(e)j(ha)m(v)m(e)g(constructed)g(is)e Ft(V)21 b Fu(\(\000)p Ft(;)c(T)8 b(;)17 b(\036)p Fu(\),)32 b(where)-180 3695 y Ft(T)46 b Fu(is)32 b(the)h(conjugacy)h(class)e(of)g Ft(t)p Fu(,)h(and)g Ft(\036)f Fu(is)g(giv)m(en)h(b)m(y)g Ft(\036)p Fu(\()p Ft(g)t(;)17 b(t)2073 3710 y Fs(i)2100 3695 y Fu(\))28 b(=)g Ft(\037)p Fu(\()p Ft(s)2415 3654 y Fk(\000)p Fn(1)2415 3720 y Fs(j)2509 3695 y Ft(g)t(s)2606 3710 y Fs(i)2633 3695 y Fu(\).)-80 3874 y(W)-8 b(e)33 b(no)m(w)g(construct)g(another)g(example)f(of)g(a)h(function)f Ft(\036)g Fu(satisfying)h(\(3.12\))o(,)g(\(3.13\))o(.)-180 4095 y FD(De\014nition)j(3.12.)49 b Fu(Let)39 b Ft(T)52 b Fu(b)s(e)39 b(the)g(set)g(of)f(all)f(transp)s(ositions)g(in)h(the)h (symmetric)f(group)g Fp(S)3394 4110 y Fs(n)3435 4095 y Fu(.)61 b(De\014ne)39 b Ft(\036)f Fu(:)-180 4234 y Fp(S)-119 4249 y Fs(n)-56 4234 y Fm(\002)23 b Ft(T)41 b Fm(!)28 b Fp(|)7 b Fm(n)22 b(f)p Fu(0)p Fm(g)32 b Fu(for)g(all)f Ft(g)g Fm(2)d Fp(S)1118 4249 y Fs(n)1159 4234 y Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(j)33 b Fm(\024)c Ft(n;)j Fu(b)m(y)1131 4529 y Ft(\037)p Fu(\()p Ft(g)t(;)17 b Fu(\()p Ft(ij)6 b Fu(\)\))27 b(=)1648 4325 y Fl(8)1648 4415 y(<)1648 4594 y(:)1737 4454 y Fu(1)175 b Ft(;)49 b Fu(if)31 b Ft(g)t Fu(\()p Ft(i)p Fu(\))c Ft(<)h(g)t Fu(\()p Ft(j)6 b Fu(\))p Ft(;)1737 4622 y Fm(\000)p Fu(1)98 b Ft(;)49 b Fu(if)31 b Ft(g)t Fu(\()p Ft(i)p Fu(\))c Ft(>)h(g)t Fu(\()p Ft(j)6 b Fu(\))p Ft(:)-180 4529 y Fu(\(3.17\))-80 4848 y(Let)44 b Ft(t)j Fu(=)g(\(12\).)76 b(The)45 b(cen)m(tralizer)f(of)f Ft(t)h Fu(in)f Fp(S)1674 4863 y Fs(n)1759 4848 y Fu(is)g Fm(h)p Fu(\(34\))p Ft(;)17 b Fu(\(45\))p Ft(;)g(:)g(:)g(:)d(;)j Fu(\()p Ft(n)30 b Fm(\000)g Fu(1)p Ft(;)17 b(n)p Fu(\))p Fm(i)29 b([)i(h)p Fu(\(34\))p Ft(;)17 b Fu(\(45\))p Ft(;)g(:)g(:)g(:)d(;)j Fu(\()p Ft(n)29 b Fm(\000)-180 4964 y Fu(1)p Ft(;)17 b(n)p Fu(\))p Fm(i)p Fu(\(12\).)39 b(Let)24 b Ft(\037)f Fu(b)s(e)g(the)h(c)m(haracter)g(of)f(\()p Fp(S)1437 4979 y Fs(n)1478 4964 y Fu(\))1516 4928 y Fs(t)1569 4964 y Fu(with)g Ft(\037)p Fu(\(\()p Ft(ij)6 b Fu(\)\))28 b(=)f(1)c(for)g(all) e(3)28 b Fm(\024)g Ft(i)g(<)f(j)34 b Fm(\024)28 b Ft(n)p Fu(,)d(and)f Ft(\037)p Fu(\(\(12\)\))j(=)g Fm(\000)p Fu(1.)-180 5081 y(Then)34 b(the)f(function)f Ft(\036)g Fu(de\014nied)h(b)m(y)i(\(3.17\))d(is)g(giv)m(en)h(b)m(y)g(the)g(c)m (haracter)h Ft(\037)e Fu(as)h(describ)s(ed)g(ab)s(o)m(v)m(e.)p eop %%Page: 28 28 28 27 bop -180 0 a Fq(28)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(Up)j(to)f(base)i(c)m(hange)f(w)m(e)h(ha) m(v)m(e)g(found)f(all)e(functions)i Ft(\036)f Fu(satisfying)i(\(3.12\)) o(,)f(\(3.13\))f(for)h(\000)c(=)g Fp(S)3496 218 y Fs(n)3537 203 y Fu(,)k(where)h Ft(T)-180 319 y Fu(is)e(the)i(conjugacy)f(class)g (of)g(all)e(transp)s(ositions,)h(and)h Ft(\036)p Fu(\()p Ft(t;)17 b(t)p Fu(\))32 b(=)f Fm(\000)p Fu(1)k(for)g(all)d Ft(t)g Fu(=)p Fm(2)g Ft(T)14 b Fu(.)50 b(The)36 b(case)g Ft(\036)p Fu(\()p Ft(t;)17 b(t)p Fu(\))32 b(=)f(1)-180 435 y(for)h(some)g Ft(t)c Fm(2)g Ft(T)47 b Fu(w)m(ould)32 b(lead)g(to)g(a)h(Nic)m(hols)f(algebra)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))32 b(of)g(in\014nite)g(dimension.)-80 599 y(T)-8 b(o)30 b(determine)g(the)g(structure)i(of)e Fj(B)p Fu(\()p Ft(V)21 b Fu(\))30 b(for)g(the)h(Y)-8 b(etter-Drinfeld)28 b(mo)s(dules)h(de\014ned)j(b)m(y)f(the)g(functions) f Ft(\036)-180 715 y Fu(in)f(\(3.16\))h(and)g(\(3.17\))f(seems)i(to)f (b)s(e)g(a)g(fundamen)m(tal)f(and)i(v)m(ery)g(hard)g(com)m(binatorial) 26 b(problem.)42 b(Only)29 b(a)h(few)-180 831 y(partial)g(results)j (are)g(kno)m(wn)h([MiS],)e([FK],)h([FP].)-80 994 y(W)-8 b(e)47 b(w)m(an)m(t)i(to)d(describ)s(e)i(some)g(of)e(the)i(relations)e (of)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\).)88 b(Let)47 b(us)h(\014rst)g (recall)e(an)h(imp)s(ortan)m(t)f(to)s(ol)-180 1111 y(in)m(tro)s(duced) 40 b(b)m(y)g(Nic)m(hols)f([N,)i(3.3])e(to)h(deal)e(with)h Fj(B)p Fu(\()p Ft(V)22 b Fu(\))39 b(o)m(v)m(er)i(group)e(algebras)g Fp(|)-9 b Fu(\000)34 b(without)39 b(kno)m(wing)g(the)-180 1227 y(relations)f(explicitly)f(\(see)k([MiS,)g(2.4]\).)63 b(Let)39 b Ft(V)61 b Fm(2)1806 1191 y Fn(\000)1806 1252 y(\000)1854 1227 y Fm(Y)8 b(D)42 b Fu(b)s(e)d(of)g(\014nite)g (dimension)f Ft(\022)s Fu(.)64 b(W)-8 b(e)40 b(c)m(ho)s(ose)g(a)f (basis)-180 1343 y Ft(x)-125 1358 y Fs(i)-59 1343 y Fm(2)e Ft(V)101 1358 y Fs(g)135 1368 y Fi(i)203 1343 y Fu(with)h Ft(g)478 1358 y Fs(i)543 1343 y Fm(2)f Fu(\000)p Ft(;)17 b Fu(1)37 b Fm(\024)g Ft(i)h Fm(\024)f Ft(\022)s(;)i Fu(of)e(\000-homogeneous)h(elemen)m(ts.)60 b(Let)39 b Ft(I)45 b Fm(2)37 b Fj(S)h Fu(and)g Ft(R)g Fu(=)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))p Ft(=I)46 b Fu(\(see)-180 1461 y(Section)37 b(2.1\).)58 b(Then)39 b Ft(R)f Fu(is)f(a)h(graded)f (Hopf)h(algebra)e(in)2009 1425 y Fn(\000)2009 1486 y(\000)2057 1461 y Fm(Y)8 b(D)40 b Fu(with)d Ft(R)q Fu(\(0\))f(=)g Fp(|)-9 b Fu(1)31 b(and)38 b Ft(R)q Fu(\(1\))d(=)h Ft(V)22 b Fu(.)58 b(F)-8 b(or)37 b(all)-180 1577 y(1)27 b Fm(\024)i Ft(i)f Fm(\024)g Ft(\022)35 b Fu(let)d Ft(\033)444 1592 y Fs(i)501 1577 y Fu(:)27 b Ft(R)i Fm(!)e Ft(R)34 b Fu(b)s(e)f(the)g (algebra)e(automorphism)g(giv)m(en)h(b)m(y)i(the)f(action)f(of)g Ft(g)3190 1592 y Fs(i)3218 1577 y Fu(.)-180 1791 y FD(Prop)s(osition)k (3.13.)49 b Fo(1\))41 b(F)-7 b(or)41 b(al)5 b(l)41 b Fu(1)f Fm(\024)g Ft(i)g Fm(\024)h Ft(\022)s Fo(,)i(ther)-5 b(e)42 b(exists)f(a)g(uniquely)h(determine)-5 b(d)41 b Fu(\()p Ft(id;)17 b(\033)3426 1806 y Fs(i)3454 1791 y Fu(\))p Fo(-derivation)-180 1930 y Ft(D)-99 1945 y Fs(i)-43 1930 y Fu(:)28 b Ft(R)g Fm(!)g Ft(R)36 b Fo(with)e Ft(D)644 1945 y Fs(i)672 1930 y Fu(\()p Ft(x)765 1945 y Fs(j)802 1930 y Fu(\))28 b(=)g Ft(\016)1015 1945 y Fs(i;j)1130 1930 y Fo(\(Kr)-5 b(one)g(cker)34 b Ft(\016)t Fo(\))g(for)h(al)5 b(l)34 b Ft(j)6 b Fo(.)-80 2069 y(2\))34 b Ft(I)i Fu(=)27 b Ft(I)8 b Fu(\()p Ft(V)22 b Fu(\))p Fo(,)34 b(that)i(is)e Ft(R)29 b Fu(=)e Fj(B)p Fu(\()p Ft(V)22 b Fu(\))p Fo(,)35 b(if)f(and)h(only)f(if)1904 1995 y Fl(T)1987 2021 y Fs(\022)1987 2099 y(i)p Fn(=1)2122 2069 y Fo(ker)p Fu(\()p Ft(D)2372 2084 y Fs(i)2400 2069 y Fu(\))27 b(=)h Fp(|)-9 b Fu(1)p Ft(:)p 3882 2069 4 68 v 3886 2006 60 4 v 3886 2069 V 3945 2069 4 68 v -80 2260 a Fu(These)44 b(sk)m(ew)g(deriv)-5 b(ations)41 b(are)h(v)m(ery)i (useful)e(to)g(\014nd)g(relations)f(of)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\).)73 b(W)-8 b(e)42 b(consider)h(some)f(sp)s(ecial)-180 2376 y(cases.)-180 2590 y FD(Example)36 b(3.14.)50 b Fu(Let)32 b Ft(W)42 b Fu(=)27 b Fp(S)1035 2605 y Fs(n)1076 2590 y Ft(;)17 b(n)27 b Fm(\025)i Fu(2,)j(and)h Ft(T)41 b Fu(=)28 b Fm(f)p Fu(\()p Ft(ij)6 b Fu(\))27 b Fm(j)h Fu(1)f Fm(\024)i Ft(i)f(<)f(j)34 b Fm(\024)28 b Ft(n)p Fm(g)33 b Fu(the)g(set)g(of)f(all)f(transp)s(ositions.)-180 2729 y(De\014ne)41 b Ft(\037)g Fu(b)m(y)i(\(3.16\))d(and)h(let)f Ft(V)63 b Fu(=)41 b Ft(V)22 b Fu(\()p Ft(W)m(;)17 b(T)8 b(;)17 b(\036)p Fu(\).)67 b(Then)42 b(the)f(follo)m(wing)d(relations)i (hold)g(in)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))41 b(for)f(all)-180 2868 y(1)27 b Fm(\024)i Ft(i)f(<)f(j)34 b Fm(\024)28 b Ft(n;)17 b Fu(1)27 b Fm(\024)h Ft(k)j(<)d(l)i Fm(\024)e Ft(n)p Fu(:)2751 3055 y Ft(x)2806 3014 y Fn(2)2806 3080 y(\()p Fs(ij)t Fn(\))2950 3055 y Fu(=)f(0)p Ft(:)-3309 b Fu(\(3.18\))776 3223 y(If)33 b Fm(f)p Ft(i;)17 b(j)6 b Fm(g)22 b(\\)g(f)p Ft(k)s(;)17 b(l)r Fm(g)27 b Fu(=)h Fm(;)p Ft(;)49 b Fu(then)228 b Ft(x)2165 3239 y Fn(\()p Fs(ij)t Fn(\))2281 3223 y Ft(x)2336 3239 y Fn(\()p Fs(k)r(l)q Fn(\))2478 3223 y Fu(+)22 b Ft(x)2631 3239 y Fn(\()p Fs(k)r(l)q Fn(\))2751 3223 y Ft(x)2806 3239 y Fn(\()p Fs(ij)t Fn(\))2950 3223 y Fu(=)27 b(0)p Ft(:)-3309 b Fu(\(3.19\))628 3392 y(If)33 b Ft(i)28 b(<)f(j)34 b(<)27 b(k)s(;)50 b Fu(then)228 b Ft(x)1670 3407 y Fn(\()p Fs(ij)t Fn(\))1786 3392 y Ft(x)1841 3407 y Fn(\()p Fs(j)t(k)r Fn(\))1993 3392 y Fu(+)22 b Ft(x)2146 3407 y Fn(\()p Fs(j)t(k)r Fn(\))2277 3392 y Ft(x)2332 3407 y Fn(\()p Fs(ik)r Fn(\))2476 3392 y Fu(+)g Ft(x)2629 3407 y Fn(\()p Fs(ik)r Fn(\))2751 3392 y Ft(x)2806 3407 y Fn(\()p Fs(ij)t Fn(\))2950 3392 y Fu(=)27 b(0)p Ft(;)-3309 b Fu(\(3.20\))1615 3561 y Ft(x)1670 3576 y Fn(\()p Fs(j)t(k)r Fn(\))1800 3561 y Ft(x)1855 3576 y Fn(\()p Fs(ij)t Fn(\))1993 3561 y Fu(+)22 b Ft(x)2146 3576 y Fn(\()p Fs(ik)r Fn(\))2268 3561 y Ft(x)2323 3576 y Fn(\()p Fs(j)t(k)r Fn(\))2476 3561 y Fu(+)g Ft(x)2629 3576 y Fn(\()p Fs(ij)t Fn(\))2745 3561 y Ft(x)2800 3576 y Fn(\()p Fs(ik)r Fn(\))2950 3561 y Fu(=)27 b(0)p Ft(:)-180 3774 y FD(Example)36 b(3.15.)50 b Fu(Let)32 b Ft(W)42 b Fu(=)27 b Fp(S)1035 3789 y Fs(n)1076 3774 y Ft(;)17 b(n)27 b Fm(\025)i Fu(2,)j(and)h Ft(T)41 b Fu(=)28 b Fm(f)p Fu(\()p Ft(ij)6 b Fu(\))27 b Fm(j)h Fu(1)f Fm(\024)i Ft(i)f(<)f(j)34 b Fm(\024)28 b Ft(n)p Fm(g)33 b Fu(the)g(set)g(of)f(all)f(transp)s(ositions.)-180 3914 y(De\014ne)41 b Ft(\036)g Fu(b)m(y)i(\(3.17\))d(and)h(let)f Ft(V)64 b Fu(=)41 b Ft(V)22 b Fu(\()p Ft(W)m(;)17 b(T)8 b(;)17 b(\036)p Fu(\).)68 b(Then)42 b(the)f(follo)m(wing)e(relations)g (hold)h(in)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))41 b(for)f(all)-180 4053 y(1)27 b Fm(\024)i Ft(i)f(<)f(j)34 b Fm(\024)28 b Ft(n;)17 b Fu(1)27 b Fm(\024)h Ft(k)j(<)d(l)i Fm(\024)e Ft(n)p Fu(:)2723 4239 y Ft(x)2778 4198 y Fn(2)2778 4264 y(\()p Fs(ij)t Fn(\))2921 4239 y Fu(=)g(0)p Ft(:)-3281 b Fu(\(3.21\))806 4408 y(If)33 b Fm(f)p Ft(i;)17 b(j)6 b Fm(g)21 b(\\)i(f)p Ft(k)s(;)17 b(l)r Fm(g)27 b Fu(=)h Fm(;)p Ft(;)17 b(then)195 b(x)2135 4424 y Fn(\()p Fs(ij)t Fn(\))2251 4408 y Ft(x)2306 4424 y Fn(\()p Fs(k)r(l)q Fn(\))2448 4408 y Fm(\000)23 b Ft(x)2603 4424 y Fn(\()p Fs(k)r(l)q Fn(\))2723 4408 y Ft(x)2778 4424 y Fn(\()p Fs(ij)t Fn(\))2921 4408 y Fu(=)28 b(0)p Ft(:)-3281 b Fu(\(3.22\))657 4577 y(If)32 b Ft(i)c(<)g(j)33 b(<)28 b(k)s(;)17 b(then)196 b(x)1639 4592 y Fn(\()p Fs(ij)t Fn(\))1754 4577 y Ft(x)1809 4592 y Fn(\()p Fs(j)t(k)r Fn(\))1962 4577 y Fm(\000)23 b Ft(x)2117 4592 y Fn(\()p Fs(j)t(k)r Fn(\))2247 4577 y Ft(x)2302 4592 y Fn(\()p Fs(ik)r Fn(\))2446 4577 y Fm(\000)g Ft(x)2601 4592 y Fn(\()p Fs(ik)r Fn(\))2723 4577 y Ft(x)2778 4592 y Fn(\()p Fs(ij)t Fn(\))2921 4577 y Fu(=)28 b(0)p Ft(;)-3281 b Fu(\(3.23\))1584 4745 y Ft(x)1639 4761 y Fn(\()p Fs(j)t(k)r Fn(\))1769 4745 y Ft(x)1824 4761 y Fn(\()p Fs(ij)t Fn(\))1962 4745 y Fm(\000)23 b Ft(x)2117 4761 y Fn(\()p Fs(ik)r Fn(\))2239 4745 y Ft(x)2294 4761 y Fn(\()p Fs(j)t(k)r Fn(\))2446 4745 y Fm(\000)g Ft(x)2601 4761 y Fn(\()p Fs(ij)t Fn(\))2717 4745 y Ft(x)2772 4761 y Fn(\()p Fs(ik)r Fn(\))2921 4745 y Fu(=)28 b(0)p Ft(:)-80 4964 y Fu(The)39 b(algebras)528 4938 y Fl(e)512 4964 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))38 b(generated)i(b)m(y)f(all)e Ft(x)1582 4980 y Fn(\()p Fs(ij)t Fn(\))1698 4964 y Ft(;)17 b Fu(1)37 b Fm(\024)h Ft(i)h(<)e(j)44 b Fm(\024)39 b Ft(n;)g Fu(with)f(the)h (quadratic)f(relations)g(in)f(the)-180 5081 y(examples)j(3.14)f(resp.) 66 b(3.15)40 b(are)f(braided)h(Hopf)g(algebras)f(in)g(the)h(category)g (of)g(Y)-8 b(etter-Drinfeld)38 b(mo)s(dules)p eop %%Page: 29 29 29 28 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(29)-180 218 y Fu(o)m(v)m(er)41 b Fp(S)98 233 y Fs(n)139 218 y Fu(.)248 192 y Fl(e)232 218 y Fj(B)p Fu(\()p Ft(V)21 b Fu(\))41 b(in)e(example)h(3.15)f(is)h(the)h(algebra)e Fm(E)1923 233 y Fs(n)2009 218 y Fu(in)m(tro)s(duced)i(b)m(y)g(F)-8 b(omin)38 b(and)i(Kirillo)m(v)d(in)i([FK])h(to)-180 334 y(describ)s(e)29 b(the)h(cohomology)d(ring)h(of)g(the)h(\015ag)g(v)-5 b(ariet)m(y)d(.)42 b(W)-8 b(e)29 b(b)s(eliev)m(e)g(that)f(indeed)h(the) h(quadratic)e(relations)g(in)-180 460 y(the)33 b(examples)g(3.14)e(and) i(3.15)f(are)h(de\014ning)f(relations)f(for)h Fj(B)p Fu(\()p Ft(V)22 b Fu(\),)33 b(that)f(is)2714 434 y Fl(e)2698 460 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))27 b(=)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(in)e(these)j(cases.)-80 618 y(It)25 b(w)m(as)h(noted)g(in)e([MiS])h(that)g(the)h(conjecture)g (in)f([FK)o(])g(ab)s(out)g(the)h("P)m(oincar)m(\023)-46 b(e-dualit)m(y\\)23 b(of)i(the)h(dimensions)-180 734 y(of)38 b(the)h(homogeneous)g(comp)s(onen)m(ts)g(of)f(the)h(algebras)f Fm(E)1987 749 y Fs(n)2072 734 y Fu(\(in)f(case)j(they)f(are)g (\014nite-dimensional\))c(follo)m(ws)-180 850 y(from)c(the)i(braided)g (Hopf)f(algebra)f(structure)j(as)f(a)f(sp)s(ecial)g(case)h(of)f(Lemma)g (1.12.)-80 1008 y(Another)c(result)g(ab)s(out)f(the)i(algebras)e Fm(E)1430 1023 y Fs(n)1504 1008 y Fu(b)m(y)i(F)-8 b(omin)26 b(and)i(Pro)s(cesi)g([FP])g(sa)m(ys)h(that)f Fm(E)3125 1023 y Fs(n)p Fn(+1)3290 1008 y Fu(is)f(a)h(free)g(mo)s(dule)-180 1125 y(o)m(v)m(er)37 b Fm(E)85 1140 y Fs(n)132 1125 y Fu(,)g(and)f Ft(P)452 1140 y Fk(E)489 1148 y Fi(n)572 1125 y Fu(divides)f Ft(P)966 1140 y Fk(E)1003 1149 y Fi(n)p Fg(+1)1127 1125 y Fu(,)i(where)h Ft(P)1540 1140 y Fs(A)1632 1125 y Fu(denotes)g(the)e(Hilb)s(ert)f(series)h(of)g(a)g (graded)g(algebra)f Ft(A)p Fu(.)54 b(The)-180 1241 y(pro)s(of)32 b(in)g([FP])g(used)i(the)f(relations)e(in)h(Example)g(3.15.)-80 1399 y(This)d(result)f(is)h(in)f(fact)g(a)h(sp)s(ecial)f(case)h(of)g(a) f(v)m(ery)j(general)d(splitting)e(theorem)j(for)f(braided)h(Hopf)f (algebras)-180 1515 y(in)g([MiS,)h(Theorem3.2])g(whic)m(h)g(is)f(an)g (application)e(of)i(the)h(fundamen)m(tal)f(theorem)g(for)h(Hopf)f(mo)s (dules)g(in)f(the)-180 1631 y(braided)32 b(situation.)42 b(This)33 b(splitting)d(theorem)j(generalizes)f(the)h(main)e(result)h (of)g([G)s(~)-51 b(n2)o(].)-80 1790 y(In)27 b([MiS])g(some)f(partial)f (results)i(are)g(obtained)g(ab)s(out)f(the)h(structure)i(of)d(the)h (Nic)m(hols)g(algebras)f(of)g(Co)m(xeter)-180 1906 y(groups.)44 b(In)33 b(particular)-180 2116 y FD(Theorem)k(3.16.)49 b Fu([MiS])32 b Fo(L)-5 b(et)32 b Fu(\()p Ft(W)m(;)17 b(S)6 b Fu(\))32 b Fo(b)-5 b(e)31 b(a)h(Coxeter)g(system,)g Ft(T)45 b Fo(the)32 b(set)g(of)g(al)5 b(l)31 b Ft(W)14 b Fo(-c)-5 b(onjugates)31 b(of)h(elements)-180 2255 y(in)43 b Ft(S)6 b Fo(,)45 b Ft(\036)e Fo(de\014ne)-5 b(d)42 b(by)53 b Fu(\(3.16\))o Fo(,)45 b Ft(V)65 b Fu(=)43 b Ft(V)22 b Fu(\()p Ft(W)m(;)17 b(T)8 b(;)17 b(\036)p Fu(\))43 b Fo(and)g Ft(R)h Fu(=)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))p Fo(.)70 b(F)-7 b(or)42 b(al)5 b(l)43 b Ft(g)k Fm(2)d Ft(W)14 b Fo(,)45 b(cho)-5 b(ose)42 b(a)h(r)-5 b(e)g(duc)g(e)g(d)-180 2395 y(r)g(epr)g(esentation)34 b Ft(g)d Fu(=)d Ft(s)683 2410 y Fn(1)739 2395 y Fm(\001)17 b(\001)g(\001)d Ft(s)917 2410 y Fs(q)955 2395 y Fo(,)35 b Ft(s)1066 2410 y Fn(1)1106 2395 y Ft(;)17 b Fm(\001)g(\001)g(\001)31 b Ft(;)17 b(s)1389 2410 y Fs(q)1454 2395 y Fm(2)28 b Ft(S)6 b Fo(,)35 b(of)f Ft(g)t Fo(,)g(and)h(de\014ne)1559 2576 y Ft(x)1614 2591 y Fs(g)1682 2576 y Fu(=)27 b Ft(x)1840 2591 y Fs(s)1873 2600 y Fg(1)1929 2576 y Fm(\001)17 b(\001)g(\001)e Ft(x)2117 2591 y Fs(s)2150 2599 y Fi(q)2188 2576 y Ft(:)-80 2757 y Fo(Then)35 b(the)i(sub)-5 b(algebr)g(a)36 b(of)g Ft(R)i Fo(gener)-5 b(ate)g(d)35 b(by)i(al)5 b(l)36 b Ft(x)1789 2772 y Fs(s)1827 2757 y Ft(;)17 b(s)30 b Fm(2)h Ft(S)43 b Fo(has)36 b(the)g Fp(|)-8 b Fo(-)o(b)j(asis)30 b Ft(x)2867 2772 y Fs(g)2908 2757 y Ft(;)17 b(g)34 b Fm(2)d Ft(W)14 b Fo(.)49 b(F)-7 b(or)36 b(al)5 b(l)36 b Ft(g)e Fm(2)d Ft(W)14 b Fo(,)-180 2897 y(the)35 b Ft(g)t Fo(-homo)-5 b(gene)g(ous)32 b(c)-5 b(omp)g(onent)34 b Ft(R)1212 2912 y Fs(g)1287 2897 y Fo(of)h Ft(R)h Fo(is)e(isomorphic)g(to)h Ft(R)2307 2912 y Fn(1)2347 2897 y Fo(.)-80 3036 y(If)f Ft(R)i Fo(is)e(\014nite-dimensional,)f(then)i Ft(W)48 b Fo(is)35 b(\014nite)f(and)h Ft(dim)p Fu(\()p Ft(R)q Fu(\))27 b(=)h Ft(or)s(d)p Fu(\()p Ft(W)14 b Fu(\))p Ft(dim)p Fu(\()p Ft(R)3023 3051 y Fn(1)3062 3036 y Fu(\))p Fo(.)p 3882 3036 4 68 v 3886 2972 60 4 v 3886 3036 V 3945 3036 4 68 v -80 3223 a Fu(This)32 b(theorem)h(holds)f(for)g(more)g (general)g(functions)g Ft(\036)p Fu(,)h(in)f(particular)e(for)i Ft(S)2816 3238 y Fs(n)2896 3223 y Fu(and)h Ft(\036)f Fu(de\014ned)i(in)e(\(3.17\))o(.)-80 3381 y(Let)40 b(\()p Ft(W)m(;)17 b(S)6 b Fu(\))39 b(b)s(e)h(a)g(Co)m(xeter)h(system)g(and)e Ft(V)62 b Fu(=)40 b Ft(V)21 b Fu(\()p Ft(W)m(;)c(T)8 b(;)17 b(\036)p Fu(\))40 b(as)g(in)f(Theorem)h([MiS].)65 b(Then)41 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))40 b(w)m(as)-180 3497 y(computed)33 b(in)f([MiS)o(])h(in)f(the)h(follo)m(wing)d(cases:) -54 3635 y Fm(\017)41 b Ft(W)56 b Fu(=)42 b Fp(S)364 3650 y Fn(3)398 3635 y Ft(;)17 b(S)47 b Fu(=)42 b Fm(f)p Fu(\(12\))p Ft(;)17 b Fu(\(23\))p Fm(g)p Fu(:)59 b(The)42 b(relations)e(of)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\))41 b(are)g(the)g(quadratic)g(relations)f(in)g(Example)37 3751 y(3.14,)32 b(and)h(dim)n Fj(B)p Fu(\()p Ft(V)22 b Fu(\))27 b(=)h(12.)-54 3909 y Fm(\017)41 b Ft(W)g Fu(=)28 b Fp(S)335 3924 y Fn(4)369 3909 y Ft(;)17 b(S)33 b Fu(=)27 b Fm(f)p Fu(\(12\))p Ft(;)17 b Fu(\(23\))p Ft(;)g Fu(\(34\))p Fm(g)p Fu(:)38 b(The)26 b(relations)f(of)g Fj(B)p Fu(\()p Ft(V)c Fu(\))26 b(are)g(the)g(quadratic)f(relations)f(in)h(Example)37 4025 y(3.14,)32 b(and)h(dim)n Fj(B)p Fu(\()p Ft(V)22 b Fu(\))27 b(=)h(24)22 b Fm(\001)g Fu(24.)-54 4183 y Fm(\017)41 b Ft(W)i Fu(=)30 b Ft(D)359 4198 y Fn(4)398 4183 y Fu(,)k(the)g(dihedral)f(group)g(of)g(order)h(8,)g Ft(S)h Fu(=)29 b Fm(f)p Ft(t;)17 b(t)2128 4147 y Fk(0)2152 4183 y Fm(g)p Fu(,)34 b(where)g Ft(t;)17 b(t)2659 4147 y Fk(0)2716 4183 y Fu(are)34 b(generators)g(of)f Ft(D)3546 4198 y Fn(4)3619 4183 y Fu(of)h(order)37 4300 y(2)h(suc)m(h)h(that)f Ft(tt)627 4263 y Fk(0)686 4300 y Fu(is)g(of)g(order)g(4.)51 b(There)36 b(are)f(quadratic)g(relations)f(and)h(relations)f(of)h (order)g(4)g(de\014ning)37 4416 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\),)32 b(and)h(dim)n Fj(B)p Fu(\()p Ft(V)22 b Fu(\))28 b(=)f(64.)-80 4553 y(In)35 b(all)e(three)i(cases)i(the)e(in)m(tegral,)f (whic)m(h)i(is)e(the)i(longest)e(non-zero)h(w)m(ord)h(in)e(the)h (generators)h Ft(x)3580 4568 y Fs(t)3610 4553 y Fu(,)f(can)h(b)s(e)-180 4669 y(describ)s(ed)k(in)e(terms)h(of)f(the)h(longest)g(elemen)m(t)f (in)g(the)i(Co)m(xeter)g(group.)62 b(In)39 b(all)e(the)j(other)e(cases) j(it)d(is)g(not)-180 4785 y(kno)m(wn)c(whether)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(is)f(\014nite-dimensional.)-80 4943 y(In)h([FK)o(])g(it)e(is)h(sho)m(wn)i(that)-54 5081 y Fm(\017)41 b Fu(dim\()p Fm(E)292 5096 y Fn(3)330 5081 y Fu(\))27 b(=)h(12)p Ft(:)p eop %%Page: 30 30 30 29 bop -180 0 a Fq(30)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-54 203 y Fm(\017)41 b Fu(dim\()p Fm(E)292 218 y Fn(4)330 203 y Fu(\))27 b(=)h(24)21 b Fm(\001)h Fu(24)p Ft(:)-54 364 y Fm(\017)41 b Fu(dim\()p Fm(E)292 379 y Fn(5)330 364 y Fu(\))32 b(is)g(\014nite)g(b)m(y)i(using)e(a)g (computer)h(program.)-180 502 y(Again,)f(for)g(the)h(other)f(cases)i Ft(n)28 b(>)g Fu(5)k(it)g(is)g(not)g(kno)m(wn)i(whether)g Fm(E)2324 517 y Fs(n)2403 502 y Fu(is)e(\014nite-dimensional.)-80 663 y(In)27 b([G)s(~)-51 b(n3)n(,)28 b(5.3.2])e(another)h(example)f(of) h(a)f(\014nite-dimensional)d(Nic)m(hols)j(algebra)g(of)g(a)h(braided)f (v)m(ector)i(space)-180 780 y(\()p Ft(V)5 b(;)17 b(c)p Fu(\))34 b(of)f(\014nite)h(group)f(t)m(yp)s(e)i(is)e(giv)m(en)h(with)g (dim\()p Ft(V)20 b Fu(\))30 b(=)f(4)34 b(and)g(dim\()p Fj(B)p Fu(\()p Ft(V)20 b Fu(\)\))30 b(=)f(72)p Ft(:)34 b Fu(The)g(de\014ning)g(relations)-180 896 y(of)e Fj(B)p Fu(\()p Ft(V)22 b Fu(\))32 b(are)h(quadratic)f(and)h(of)f(order)g(6.) -80 1057 y(By)h(a)f(result)h(of)f(Mon)m(tgomery)h([M2)q(],)f(an)m(y)i (p)s(oin)m(ted)e(Hopf)h(algebra)f Ft(B)37 b Fu(can)c(b)s(e)g(decomp)s (osed)h(as)f(a)f(crossed)-180 1173 y(pro)s(duct)1265 1293 y Ft(B)h Fm(')28 b Ft(A)p Fu(#)1631 1308 y Fs(\033)1678 1293 y Fp(|)-9 b Ft(G;)109 b(\033)36 b Fu(a)c(2-co)s(cycle)-180 1434 y(of)37 b(A,)h(its)f(link-indecomp)s(osable)e(comp)s(onen)m(t)i (con)m(taining)f(1)i(\(a)f(Hopf)g(subalgebra\))h(and)f(a)h(group)f (algebra)-180 1550 y Fp(|)-9 b Ft(G)p Fu(.)36 b(Ho)m(w)m(ev)m(er,)31 b(the)c(structure)i(of)d(suc)m(h)j(link-indecomp)s(osable)24 b(Hopf)j(algebras)g(A,)g(in)f(particular)g(in)g(the)i(case)-180 1666 y(when)34 b Ft(A)f Fu(is)f(\014nite-dimensional)e(and)i(the)i (group)e(of)g(its)h(group-lik)m(e)e(elemen)m(ts)i Ft(G)p Fu(\()p Ft(A)p Fu(\))g(is)f(non-ab)s(elian,)f(is)h(not)-180 1782 y(kno)m(wn.)44 b(T)-8 b(o)32 b(de\014ne)g Fo(link-inde)-5 b(c)g(omp)g(osable)31 b(p)-5 b(ointe)g(d)33 b(Hopf)h(algebr)-5 b(as)p Fu(,)30 b(w)m(e)j(recall)c(the)j(de\014nition)e(of)h(the)h Fo(quiver)-180 1899 y Fu(of)d Ft(A)h Fu(in)f([M2)q(].)42 b(The)31 b(v)m(ertices)g(of)f(the)g(quiv)m(er)h(of)e Ft(A)h Fu(are)g(the)g(elemen)m(ts)g(of)g(the)g(group)g Ft(G)p Fu(\()p Ft(A)p Fu(\);)h(for)e Ft(g)t(;)17 b(h)27 b Fm(2)h Ft(G)p Fu(\()p Ft(A)p Fu(\),)-180 2015 y(there)k(exists)f(an)g (arro)m(w)g(from)f Ft(h)h Fu(to)f Ft(g)k Fu(if)c Ft(P)1406 2030 y Fs(g)r(;h)1506 2015 y Fu(\()p Ft(A)p Fu(\))h(is)f(non-trivial,)f (that)i(is)f(if)g Fp(|)-9 b Fu(\()p Ft(g)16 b Fm(\000)j Ft(h)p Fu(\))28 b Fp($)g Ft(P)3230 2030 y Fs(g)r(;h)3330 2015 y Fu(\()p Ft(A)p Fu(\).)43 b(The)32 b(Hopf)-180 2131 y(algebra)f Ft(A)i Fu(is)f(called)g(link-indecomp)s(osable,)d(if)j (its)g(quiv)m(er)h(is)f(connected)j(as)d(an)h(undirected)g(graph.)-180 2343 y FD(De\014nition)j(3.17.)49 b Fu(Let)38 b(\000)f(b)s(e)h(a)f (\014nite)h(group)f(and)h Ft(V)57 b Fm(2)2072 2307 y Fn(\000)2072 2368 y(\000)2120 2343 y Fm(Y)8 b(D)r Fu(.)59 b(V)37 b(is)g(called)g Fo(link-inde)-5 b(c)g(omp)g(osable)34 b Fu(if)i(the)-180 2482 y(group)c(\000)h(is)f(generated)h(b)m(y)h(the)f (elemen)m(ts)g Ft(g)j Fu(with)c Ft(V)1797 2497 y Fs(g)1864 2482 y Fm(6)p Fu(=)c(0.)-80 2673 y(By)39 b([MiS,)i(4.2],)f Ft(V)60 b Fm(2)786 2636 y Fn(\000)786 2698 y(\000)835 2673 y Fm(Y)8 b(D)41 b Fu(is)d(link-indecomp)s(osable)f(if)h(and)h (only)f(if)g(the)h(Hopf)g(algebra)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000)33 b(is)-180 2789 y(link-indecomp)s (osable.)-80 2905 y(Th)m(us)49 b(b)m(y)g(the)f(examples)g(constructed)i (ab)s(o)m(v)m(e,)i(there)d(are)f(link-indecomp)s(osable,)h (\014nite-dimensional)-180 3021 y(p)s(oin)m(ted)32 b(Hopf)h(algebras)f Ft(A)g Fu(with)g Ft(G)p Fu(\()p Ft(A)p Fu(\))h(isomorphic)e(to)h Ft(S)2050 3036 y Fs(n)2097 3021 y Ft(;)17 b Fu(3)27 b Fm(\024)h Ft(n)g Fm(\024)g Fu(5,)33 b(or)f(to)g Ft(D)2941 3036 y Fn(4)2980 3021 y Fu(.)-180 3233 y FD(Question)37 b(3.18.)49 b Fu(Whic)m(h)41 b(\014nite)e(groups)i(are)f(isomorphic)e (to)h Ft(G)p Fu(\()p Ft(A)p Fu(\))h(for)g(some)g(\014nite-dimensional,) e(link-)-180 3373 y(indecomp)s(osable)45 b(p)s(oin)m(ted)g(Hopf)h (algebra)f Ft(A)p Fu(?)84 b(Are)46 b(there)h(\014nite)e(groups)i(whic)m (h)f(do)g(not)g(o)s(ccur)g(in)f(this)-180 3512 y(form?)-80 3701 y(Finally)-8 b(,)37 b(let)h(us)h(come)f(bac)m(k)i(to)e(the)h (simple)e(Y)-8 b(etter-Drinfeld)36 b(mo)s(dules)i Ft(V)59 b Fu(=)38 b Ft(M)10 b Fu(\()p Ft(t;)17 b(U)10 b Fu(\))39 b Fm(2)3435 3665 y Fn(\000)3435 3726 y(\000)3483 3701 y Fm(Y)8 b(D)s Fu(,)40 b(where)-180 3817 y Ft(t)i Fm(2)f Fu(\000)g(and)g Ft(U)51 b Fu(is)40 b(a)h(simple)e(left)g(\000)1168 3781 y Fs(t)1198 3817 y Fu(-mo)s(dule)g(of)h(dimension)f Ft(>)j Fu(1.)67 b(In)41 b(this)f(case,)k(strong)d(restrictions)f(are) -180 3933 y(kno)m(wn)34 b(for)e Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(to)f(b)s(e)h(\014nite-dimensional.)40 b(By)33 b(Sc)m(h)m(ur's)i(lemma,)30 b Ft(t)j Fu(acts)g(as)g(a)f(scalar)g Ft(q)k Fu(on)d Ft(U)10 b Fu(.)-180 4145 y FD(Prop)s(osition)36 b(3.19.)49 b Fu([G)s(~)-51 b(n3)n(,)35 b(3.1])h Fo(Assume)h(that)g Fu(dim)15 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))36 b Fo(is)g(\014nite.)50 b(If)36 b Fu(dim)15 b Ft(U)41 b Fm(\025)32 b Fu(3)p Fo(,)k(then)h Ft(q)d Fu(=)d Fm(\000)p Fu(1)p Fo(;)37 b(and)-180 4285 y(if)e Fu(dim)15 b Ft(U)38 b Fu(=)28 b(2)p Fo(,)34 b(then)h Ft(q)c Fu(=)d Fm(\000)p Fu(1)35 b Fo(or)g Ft(q)k Fo(is)34 b(a)h(r)-5 b(o)g(ot)35 b(of)f(unity)i(of)f(or)-5 b(der)34 b(thr)-5 b(e)g(e.)p 3882 4285 4 68 v 3886 4221 60 4 v 3886 4285 V 3945 4285 4 68 v -80 4474 a Fu(In)43 b(the)g(pro)s(of)f(of) h(Prop)s(osition)e(3.19,)k(a)d(result)h(of)g(Lusztig)f(on)h(braidings)f (of)g(Cartan)h(t)m(yp)s(e)h(\(see)g([AS2,)-180 4590 y(Theorem)33 b(3.1]\))f(is)g(used.)45 b(In)33 b(a)f(similar)d(w)m(a)m(y)34 b(Gra)s(~)-51 b(na)31 b(sho)m(w)m(ed)-180 4802 y FD(Prop)s(osition)36 b(3.20.)49 b Fu([G)s(~)-51 b(n3)n(,)35 b(3.2])h Fo(L)-5 b(et)37 b Fu(\000)g Fo(b)-5 b(e)36 b(a)g(\014nite)g(gr)-5 b(oup)37 b(of)f(o)-5 b(dd)36 b(or)-5 b(der,)36 b(and)g Ft(V)53 b Fm(2)3142 4766 y Fn(\000)3142 4827 y(\000)3190 4802 y Fm(Y)8 b(D)r Fo(.)50 b(Assume)36 b(that)-180 4941 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))32 b Fo(is)g(\014nite-dimensional.)42 b(Then)32 b(the)g(multiplicity)h(of)g(any)f(simple)g(Y)-7 b(etter-Drinfeld)31 b(mo)-5 b(dule)32 b(over)h Fu(\000)f Fo(as)-180 5081 y(a)j(dir)-5 b(e)g(ct)34 b(summand)g(in)h Ft(V)56 b Fo(is)35 b(at)g(most)g(2.)p eop %%Page: 31 31 31 30 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(31)-80 203 y Fo(In)29 b(p)-5 b(articular,)30 b(up)g(to)g(isomorphism) e(ther)-5 b(e)30 b(ar)-5 b(e)29 b(only)h(\014nitely)g(many)f(Y)-7 b(etter-Drinfeld)29 b(mo)-5 b(dules)29 b Ft(V)49 b Fm(2)3751 167 y Fn(\000)3751 228 y(\000)3799 203 y Fm(Y)8 b(D)-180 342 y Fo(such)35 b(that)g Fj(B)p Fu(\()p Ft(V)21 b Fu(\))35 b Fo(is)g(\014nite-dimensional.)p 3882 342 4 68 v 3886 279 60 4 v 3886 342 V 3945 342 4 68 v -80 535 a Fu(The)e(second)h (statemen)m(t)f(in)f(Prop)s(osition)f(3.20)h(w)m(as)h(a)f(conjecture)i (in)e(a)g(preliminary)e(v)m(ersion)j(of)g([AS2].)-180 778 y(3.5.)56 b FD(Braidings)36 b(of)i(\(in\014nite\))d(group)j(t)m(yp) s(e.)-80 894 y Fu(W)-8 b(e)42 b(brie\015y)f(men)m(tion)g(Nic)m(hols)g (algebras)g(o)m(v)m(er)h(a)f(free)h(ab)s(elian)e(group)h(of)g(\014nite) g(rank)h(with)f(a)h(braiding)-180 1010 y(whic)m(h)33 b(is)f(not)h(diagonal.)-180 1226 y FD(Example)j(3.21.)50 b Fu(Let)32 b(\000)c(=)f Fm(h)p Ft(g)t Fm(i)k Fu(b)s(e)i(a)f(free)g (group)h(in)e(one)i(generator.)43 b(Let)32 b Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\))32 b(b)s(e)h(the)g(Y)-8 b(etter-Drinfeld)-180 1365 y(mo)s(dule)37 b(of)i(dimension)e(2)h(suc)m(h)j(that)d Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\))38 b(=)g Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\))1984 1380 y Fs(g)2063 1365 y Fu(and)39 b(the)g(action)f(of)g Ft(g)k Fu(on)d Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\))38 b(is)h(giv)m(en,)h(in)e(a)-180 1505 y(basis)33 b Ft(x)115 1520 y Fn(1)154 1505 y Ft(;)17 b(x)253 1520 y Fn(2)293 1505 y Fu(,)33 b(b)m(y)1168 1654 y Ft(g)25 b Fm(\001)d Ft(x)1345 1669 y Fn(1)1413 1654 y Fu(=)27 b Ft(tx)1606 1669 y Fn(1)1646 1654 y Ft(;)212 b(g)26 b Fm(\001)c Ft(x)2063 1669 y Fn(2)2130 1654 y Fu(=)28 b Ft(tx)2324 1669 y Fn(2)2386 1654 y Fu(+)22 b Ft(x)2539 1669 y Fn(1)2579 1654 y Ft(:)-180 1823 y Fu(Here)33 b Ft(t)28 b Fm(2)g Fp(|)269 1787 y Fk(\002)322 1823 y Fu(.)43 b(Then:)-80 1962 y(\(a\).)g(If)32 b Ft(t)h Fu(is)f(not)g(a)h(ro)s(ot)e(of)i(1,)f(then)h Fj(B)p Fu(\()p Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\)\))28 b(=)f Ft(T)14 b Fu(\()p Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\)\).)-80 2102 y(\(b\).)41 b(If)27 b Ft(t)h Fu(=)g(1,)g(then)f Fj(B)p Fu(\()p Fm(V)8 b Fu(\(1)p Ft(;)17 b Fu(2\)\))27 b(=)h Fp(|)13 b Ft(<)27 b(x)1520 2117 y Fn(1)1560 2102 y Ft(;)17 b(x)1659 2117 y Fn(2)1699 2102 y Fm(j)p Ft(x)1782 2117 y Fn(1)1821 2102 y Ft(x)1876 2117 y Fn(2)1944 2102 y Fu(=)27 b Ft(x)2102 2117 y Fn(2)2142 2102 y Ft(x)2197 2117 y Fn(1)2248 2102 y Fu(+)11 b Ft(x)2390 2066 y Fn(2)2390 2126 y(1)2457 2102 y Ft(>)p Fu(;)29 b(this)e(is)f(the)i(w)m(ell-kno)m (wn)f(Jordanian)-180 2241 y(quan)m(tum)33 b(plane.)-180 2457 y FD(Example)j(3.22.)50 b Fu(More)30 b(generally)-8 b(,)30 b(if)g Ft(t)d Fm(2)i Fp(|)1549 2421 y Fk(\002)1602 2457 y Fu(,)i(let)f Fm(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\))30 b(b)s(e)h(the)g(Y)-8 b(etter-Drinfeld)28 b(mo)s(dule)h(of)h (dimension)-180 2596 y Ft(\022)h Fm(\025)d Fu(2)k(suc)m(h)i(that)f Fm(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\))28 b(=)f Fm(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\))1189 2611 y Fs(g)1262 2596 y Fu(and)33 b(the)g(action)e(of)h Ft(g)k Fu(on)d Fm(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\))32 b(is)g(giv)m(en,)h(in)f(a)g (basis)h Ft(x)3416 2611 y Fn(1)3456 2596 y Ft(;)17 b(:)g(:)g(:)f(;)h(x) 3730 2611 y Fs(\022)3769 2596 y Fu(,)32 b(b)m(y)852 2786 y Ft(g)26 b Fm(\001)c Ft(x)1030 2801 y Fn(1)1097 2786 y Fu(=)28 b Ft(tx)1291 2801 y Fn(1)1331 2786 y Ft(;)212 b(g)25 b Fm(\001)d Ft(x)1747 2801 y Fs(j)1812 2786 y Fu(=)27 b Ft(tx)2005 2801 y Fs(j)2064 2786 y Fu(+)22 b Ft(x)2217 2801 y Fs(j)t Fk(\000)p Fn(1)2345 2786 y Ft(;)114 b Fu(2)27 b Fm(\024)h Ft(j)34 b Fm(\024)28 b Ft(\022)s(:)-180 2975 y Fu(Note)j(there)g(is)f(an)h(inclusion)e(of)h(Y) -8 b(etter-Drinfeld)29 b(mo)s(dules)h Fm(V)8 b Fu(\()p Ft(t;)17 b Fu(2\))27 b Ft(,)-17 b Fm(!)28 b(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\);)32 b(hence,)g(if)e Ft(t)g Fu(is)h(not)f(a)h (ro)s(ot)e(of)-180 3115 y(1,)j Fj(B)p Fu(\()p Fm(V)8 b Fu(\()p Ft(t;)17 b(\022)s Fu(\)\))33 b(has)g(exp)s(onen)m(tial)f(gro) m(wth.)-180 3330 y FD(Question)37 b(3.23.)49 b Fu(Compute)33 b Fj(B)p Fu(\()p Fm(V)8 b Fu(\(1)p Ft(;)17 b(\022)s Fu(\)\);)32 b(do)s(es)h(it)f(ha)m(v)m(e)i(\014nite)e(gro)m(wth?)965 3578 y(4.)55 b Fv(Nichols)38 b(algebras)f(of)h(Car)-7 b(t)g(an)39 b(type)-80 3752 y Fu(W)-8 b(e)31 b(no)m(w)g(discuss)h (fundamen)m(tal)d(examples)i(of)f(Nic)m(hols)g(algebras)g(of)g (diagonal)e(t)m(yp)s(e)j(that)g(come)f(from)f(the)-180 3868 y(theory)k(of)f(quan)m(tum)h(groups.)-80 4035 y(W)-8 b(e)31 b(\014rst)h(need)g(to)f(\014x)h(some)f(notation.)42 b(Let)31 b Ft(A)d Fu(=)g(\()p Ft(a)1905 4050 y Fs(ij)1966 4035 y Fu(\))2004 4050 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)2295 4035 y Fu(b)s(e)j(a)g(generalized)g(symmetrizable)f (Cartan)-180 4151 y(matrix)j([K];)j(let)f(\()p Ft(d)565 4166 y Fn(1)604 4151 y Ft(;)17 b(:)g(:)g(:)e(;)i(d)873 4166 y Fs(\022)912 4151 y Fu(\))35 b(b)s(e)g(p)s(ositiv)m(e)f(in)m (tegers)h(suc)m(h)h(that)f Ft(d)2333 4166 y Fs(i)2361 4151 y Ft(a)2412 4166 y Fs(ij)2504 4151 y Fu(=)d Ft(d)2663 4166 y Fs(j)2699 4151 y Ft(a)2750 4166 y Fs(j)t(i)2811 4151 y Fu(.)50 b(Let)35 b Ft(g)j Fu(b)s(e)d(the)g(Kac-Mo)s(o)s(dy)-180 4267 y(algebra)41 b(corresp)s(onding)h(to)f(the)i(Cartan)f(matrix)e Ft(A)p Fu(.)72 b(Let)42 b Fm(X)57 b Fu(b)s(e)42 b(the)g(set)h(of)e (connected)j(comp)s(onen)m(ts)e(of)-180 4383 y(the)k(Dynkin)g(diagram)e (corresp)s(onding)i(to)g(it.)82 b(F)-8 b(or)46 b(eac)m(h)h Ft(I)58 b Fm(2)51 b(X)15 b Fu(,)49 b(w)m(e)e(let)e Fj(g)2864 4398 y Fs(I)2950 4383 y Fu(b)s(e)i(the)f(Kac-Mo)s(o)s(dy)f(Lie)-180 4499 y(algebra)31 b(corresp)s(onding)i(to)f(the)h(generalized)e(Cartan) i(matrix)e(\()p Ft(a)2315 4514 y Fs(ij)2375 4499 y Fu(\))2413 4514 y Fs(i;j)t Fk(2)p Fs(I)2609 4499 y Fu(and)h Fj(n)2850 4514 y Fs(I)2923 4499 y Fu(b)s(e)h(the)f(Lie)g(subalgebra)g(of)-180 4616 y Fj(g)-130 4631 y Fs(I)-52 4616 y Fu(spanned)40 b(b)m(y)f(all)d(its)i(p)s(ositiv)m(e)f(ro)s(ots.)60 b(W)-8 b(e)39 b(omit)d(the)j(subindex)g Ft(I)46 b Fu(when)40 b Ft(I)45 b Fu(=)37 b Fm(f)p Fu(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(\022)s Fm(g)p Fu(.)60 b(W)-8 b(e)38 b(assume)-180 4732 y(that)29 b(for)f(eac)m(h)i Ft(I)36 b Fm(2)28 b(X)15 b Fu(,)29 b(there)h(exist)g Ft(c)1220 4747 y Fs(I)1259 4732 y Ft(;)17 b(d)1354 4747 y Fs(I)1423 4732 y Fu(suc)m(h)30 b(that)f Ft(I)36 b Fu(=)27 b Fm(f)p Ft(j)34 b Fu(:)27 b Ft(c)2249 4747 y Fs(I)2317 4732 y Fm(\024)h Ft(j)34 b Fm(\024)28 b Ft(d)2652 4747 y Fs(I)2692 4732 y Fm(g)p Fu(;)i(that)f(is,)g(after)g (reordering)f(the)-180 4848 y(Cartan)33 b(matrix)f(is)h(a)g(matrix)f (of)h(blo)s(c)m(ks)h(corresp)s(onding)f(to)g(the)h(connected)h(comp)s (onen)m(ts.)47 b(Let)33 b Ft(I)k Fm(2)30 b(X)48 b Fu(and)-180 4964 y Ft(i)32 b Fm(\030)f Ft(j)41 b Fu(in)34 b Ft(I)8 b Fu(;)36 b(then)f Ft(N)606 4979 y Fs(i)666 4964 y Fu(=)c Ft(N)851 4979 y Fs(j)887 4964 y Fu(,)36 b(hence)g Ft(N)1301 4979 y Fs(I)1372 4964 y Fu(:=)31 b Ft(N)1584 4979 y Fs(i)1647 4964 y Fu(is)j(w)m(ell)g(de\014ned.)52 b(Let)35 b(\010)2575 4979 y Fs(I)2615 4964 y Fu(,)g(resp.)51 b(\010)2998 4923 y Fn(+)2998 4991 y Fs(I)3058 4964 y Fu(,)35 b(b)s(e)g(the)g(ro)s(ot)f (system,)-180 5081 y(resp.)41 b(the)25 b(subset)g(of)e(p)s(ositiv)m(e)h (ro)s(ots,)h(corresp)s(onding)f(to)f(the)h(Cartan)g(matrix)e(\()p Ft(a)2839 5096 y Fs(ij)2900 5081 y Fu(\))2938 5096 y Fs(i;j)t Fk(2)p Fs(I)3101 5081 y Fu(;)27 b(then)d(\010)k(=)3570 5006 y Fl(S)3653 5110 y Fs(I)5 b Fk(2X)3816 5081 y Fu(\010)3886 5096 y Fs(I)3927 5081 y Fu(,)p eop %%Page: 32 32 32 31 bop -180 0 a Fq(32)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(resp.)52 b(\010)142 167 y Fn(+)234 203 y Fu(=)341 128 y Fl(S)424 232 y Fs(I)5 b Fk(2X)588 203 y Fu(\010)658 162 y Fn(+)658 230 y Fs(I)753 203 y Fu(is)34 b(the)i(ro)s(ot)e(system,)j(resp.)52 b(the)35 b(subset)i(of)e(p)s(ositiv)m(e)f(ro)s(ots,)i(corresp)s(onding)f(to)f (the)-180 319 y(Cartan)f(matrix)e(\()p Ft(a)557 334 y Fs(ij)617 319 y Fu(\))655 334 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)915 319 y Ft(:)i Fu(Let)g Ft(\013)1212 334 y Fn(1)1251 319 y Ft(;)17 b(:)g(:)g(:)f(;)h(\013)1532 334 y Fs(\022)1603 319 y Fu(b)s(e)33 b(the)g(set)g(of)f(simple)f(ro)s (ots.)-80 483 y(Let)39 b Fm(W)199 498 y Fs(I)279 483 y Fu(b)s(e)h(the)g(W)-8 b(eyl)39 b(group)h(corresp)s(onding)f(to)g(the) h(Cartan)g(matrix)e(\()p Ft(a)2816 498 y Fs(ij)2876 483 y Fu(\))2914 498 y Fs(i;j)t Fk(2)p Fs(I)3077 483 y Fu(;)43 b(w)m(e)e(iden)m(tify)e(it)f(with)-180 599 y(a)j(subgroup)g(of)g(the)g (W)-8 b(eyl)41 b(group)g Fm(W)49 b Fu(corresp)s(onding)41 b(to)g(the)g(Cartan)g(matrix)f(\()p Ft(a)3017 614 y Fs(ij)3077 599 y Fu(\).)69 b(W)-8 b(e)41 b(\014x)h(a)e(reduced)-180 715 y(decomp)s(osition)31 b(of)h(the)h(longest)f(elemen)m(t)g Ft(!)1494 730 y Fn(0)p Fs(;I)1622 715 y Fu(of)g Fm(W)1831 730 y Fs(I)1904 715 y Fu(in)f(terms)i(of)f(simple)f(re\015ections.)44 b(Then)34 b(w)m(e)g(obtain)d(a)-180 831 y(reduced)38 b(decomp)s(osition)c(of)i(the)g(longest)g(elemen)m(t)g Ft(!)1879 846 y Fn(0)1952 831 y Fu(=)e Ft(s)2108 846 y Fs(i)2132 855 y Fg(1)2187 831 y Ft(:)17 b(:)g(:)f(s)2364 846 y Fs(i)2388 857 y Fi(P)2479 831 y Fu(of)36 b Fm(W)45 b Fu(from)35 b(the)h(expression)i(of)d Ft(!)3791 846 y Fn(0)3867 831 y Fu(as)-180 948 y(pro)s(duct)f(of)f(the)h Ft(!)529 963 y Fn(0)p Fs(;I)623 948 y Fu('s)g(in)f(some)h(\014xed)g (order)g(of)f(the)h(comp)s(onen)m(ts,)g(sa)m(y)h(the)f(order)g(arising) d(from)i(the)h(order)-180 1064 y(of)e(the)h(v)m(ertices.)45 b(Therefore)34 b Ft(\014)987 1079 y Fs(j)1051 1064 y Fu(:=)27 b Ft(s)1227 1079 y Fs(i)1251 1088 y Fg(1)1307 1064 y Ft(:)17 b(:)g(:)f(s)1484 1079 y Fs(i)1508 1089 y Fi(j)s Fh(\000)p Fg(1)1623 1064 y Fu(\()p Ft(\013)1723 1079 y Fs(i)1747 1089 y Fi(j)1784 1064 y Fu(\))32 b(is)g(a)g(n)m (umeration)g(of)g(\010)2732 1028 y Fn(+)2792 1064 y Fu(.)-180 1277 y FD(Example)k(4.1.)50 b Fu(Let)28 b Ft(q)k Fm(2)c Fp(|)-9 b Fu(,)23 b Ft(q)32 b Fm(6)p Fu(=)27 b(0,)i(and)f(consider)h (the)f(braided)g(v)m(ector)i(space)f(\()p Fp(V)q Ft(;)17 b(c)p Fu(\),)35 b(where)29 b Fp(V)35 b Fu(is)28 b(a)g(v)m(ector)-180 1417 y(space)34 b(with)e(a)g(basis)h Ft(x)679 1432 y Fn(1)719 1417 y Ft(;)17 b(:)g(:)g(:)e(;)i(x)992 1432 y Fs(\022)1064 1417 y Fu(and)33 b(the)g(braiding)e Ft(c)h Fu(is)g(giv)m(en)h(b)m(y)1340 1604 y Ft(c)p Fu(\()p Ft(x)1475 1619 y Fs(i)1526 1604 y Fm(\012)23 b Ft(x)1681 1619 y Fs(j)1718 1604 y Fu(\))k(=)h Ft(q)1934 1562 y Fs(d)1970 1572 y Fi(i)1996 1562 y Fs(a)2033 1572 y Fi(ij)2093 1604 y Ft(x)2148 1619 y Fs(j)2207 1604 y Fm(\012)22 b Ft(x)2361 1619 y Fs(i)2390 1604 y Ft(;)-2597 b Fu(\(4.1\))-180 1817 y FD(Theorem)37 b(4.2.)49 b Fu([L3])36 b Fo(L)-5 b(et)37 b Fu(\()p Fp(V)q Ft(;)17 b(c)p Fu(\))41 b Fo(b)-5 b(e)36 b(a)g(br)-5 b(aide)g(d)35 b(ve)-5 b(ctor)36 b(sp)-5 b(ac)g(e)36 b(with)g(br)-5 b(aiding)35 b(matrix)48 b Fu(\(4.1\))o Fo(.)h(If)35 b Ft(q)40 b Fo(is)c(not)g(a)-180 1957 y(r)-5 b(o)g(ot)35 b(of)g(1,)f(then)843 2103 y Fj(B)p Fu(\()p Fp(V)q Fu(\))f(=)28 b Fp(|)-9 b Fm(h)p Ft(x)1365 2118 y Fn(1)1399 2103 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)1673 2118 y Fs(\022)1712 2103 y Fm(j)p Fu(ad)1859 2118 y Fs(c)1893 2103 y Fu(\()p Ft(x)1986 2118 y Fs(i)2015 2103 y Fu(\))2053 2061 y Fn(1)p Fk(\000)p Fs(a)2180 2071 y Fi(ij)2239 2103 y Ft(x)2294 2118 y Fs(j)2359 2103 y Fu(=)28 b(0)p Ft(;)116 b(i)28 b Fm(6)p Fu(=)f Ft(j)6 b Fm(i)p Ft(:)p 3882 2269 4 68 v 3886 2205 60 4 v 3886 2269 V 3945 2269 4 68 v -80 2459 a Fu(The)23 b(Theorem)g(sa)m(ys)h(that)e Fj(B)p Fu(\()p Fp(V)q Fu(\))29 b(is)22 b(the)h(w)m(ell-kno)m(wn)g("p)s(ositiv) m(e)e(part")h Ft(U)2637 2423 y Fn(+)2627 2484 y Fs(q)2697 2459 y Fu(\()p Ft(g)t Fu(\))g(of)g(the)h(quan)m(tum)g(en)m(v)m(eloping) -180 2575 y(algebra)31 b(of)i Ft(g)t Fu(.)-80 2739 y(T)-8 b(o)44 b(state)h(the)g(follo)m(wing)d(imp)s(ortan)m(t)h(Theorem,)48 b(w)m(e)e(recall)d(the)i(de\014nition)e(of)i(braided)f(comm)m(utators) -180 2855 y(\(1.20\))o(.)f(Lusztig)31 b(de\014ned)i(ro)s(ot)d(v)m (ectors)j Ft(X)1434 2870 y Fs(\013)1511 2855 y Fm(2)28 b Fj(B)p Fu(\()p Fp(V)q Fu(\),)37 b Ft(\013)29 b Fm(2)f Fu(\010)2154 2819 y Fn(+)2244 2855 y Fu([L2].)43 b(One)32 b(can)f(see)h(from)e([L1,)h(L2])g(that,)g(up)-180 2971 y(to)i(a)g(non-zero)h(scalar,)f(eac)m(h)h(ro)s(ot)e(v)m(ector)j(can)f (b)s(e)f(written)g(as)h(an)f(iterated)g(braided)g(comm)m(utator)f(in)h (some)-180 3088 y(sequence)i Ft(X)305 3103 y Fs(`)334 3112 y Fg(1)373 3088 y Ft(;)17 b(:)g(:)g(:)f(;)h(X)673 3103 y Fs(`)702 3111 y Fi(a)776 3088 y Fu(of)32 b(simple)f(ro)s(ot)g(v) m(ectors)j(suc)m(h)g(as)f([[)p Ft(X)2205 3103 y Fs(`)2234 3112 y Fg(1)2273 3088 y Ft(;)17 b Fu([)p Ft(X)2425 3103 y Fs(`)2454 3112 y Fg(2)2492 3088 y Ft(;)g(X)2617 3103 y Fs(`)2646 3112 y Fg(3)2685 3088 y Fu(])2712 3103 y Fe(c)2743 3088 y Fu(])2770 3103 y Fe(c)2801 3088 y Ft(;)g Fu([)p Ft(X)2953 3103 y Fs(`)2982 3112 y Fg(4)3020 3088 y Ft(;)g(X)3145 3103 y Fs(`)3174 3112 y Fg(5)3212 3088 y Fu(])3239 3103 y Fe(c)3270 3088 y Fu(])3297 3103 y Fe(c)3329 3088 y Fu(.)43 b(See)33 b(also)f([Ri)o(].)-180 3301 y FD(Theorem)37 b(4.3.)49 b Fu([L1,)36 b(L2)o(,)g(L3,)f(Ro1,)g(Mu) q(])p Fo(.)53 b(L)-5 b(et)38 b Fu(\()p Fp(V)p Ft(;)18 b(c)p Fu(\))43 b Fo(b)-5 b(e)37 b(a)g(br)-5 b(aide)g(d)37 b(ve)-5 b(ctor)37 b(sp)-5 b(ac)g(e)37 b(with)h(br)-5 b(aiding)36 b(matrix)-180 3441 y Fu(\(4.1\))p Fo(.)55 b(Assume)38 b(that)h Ft(q)k Fo(is)38 b(a)g(r)-5 b(o)g(ot)39 b(of)f(1)g(of)g(o)-5 b(dd)38 b(or)-5 b(der)38 b Ft(N)10 b Fo(;)41 b(and)d(that)h(3)f(do)-5 b(es)38 b(not)g(divide)g Ft(N)49 b Fo(if)38 b(ther)-5 b(e)38 b(exists)-180 3580 y Ft(I)e Fm(2)28 b(X)49 b Fo(of)35 b(typ)-5 b(e)35 b Ft(G)509 3595 y Fn(2)548 3580 y Fo(.)-80 3720 y(The)f(algebr)-5 b(a)34 b Fj(B)p Fu(\()p Fp(V)q Fu(\))41 b Fo(is)34 b(\014nite)h (dimensional)e(if)h(and)h(only)f(if)h Fu(\()p Ft(a)2298 3735 y Fs(ij)2359 3720 y Fu(\))g Fo(is)f(a)h(\014nite)f(Cartan)h (matrix.)-80 3859 y(If)f(this)h(happ)-5 b(ens,)33 b(then)i Fj(B)p Fu(\()p Fp(V)q Fu(\))41 b Fo(c)-5 b(an)34 b(b)-5 b(e)35 b(pr)-5 b(esente)g(d)34 b(by)h(gener)-5 b(ators)34 b Ft(X)2502 3874 y Fs(i)2530 3859 y Fo(,)h Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)h Ft(\022)s Fo(,)35 b(and)f(r)-5 b(elations)1146 4046 y Fu(ad)1265 4061 y Fs(c)1299 4046 y Fu(\()p Ft(X)1418 4061 y Fs(i)1447 4046 y Fu(\))1485 4005 y Fn(1)p Fk(\000)p Fs(a)1612 4015 y Fi(ij)1671 4046 y Fu(\()p Ft(X)1790 4061 y Fs(j)1826 4046 y Fu(\))28 b(=)g(0)p Ft(;)215 b(i)28 b Fm(6)p Fu(=)g Ft(j;)-2699 b Fu(\(4.2\))1708 4214 y Ft(X)1797 4173 y Fs(N)1789 4239 y(\013)1892 4214 y Fu(=)28 b(0)p Ft(;)215 b(\013)29 b Fm(2)f Fu(\010)2542 4173 y Fn(+)2601 4214 y Ft(:)-2808 b Fu(\(4.3\))-180 4401 y Fo(Mor)-5 b(e)g(over,)34 b(the)h(fol)5 b(lowing)34 b(elements)g(c)-5 b(onstitute)35 b(a)g(b)-5 b(asis)34 b(of)h Fj(B)p Fu(\()p Fp(V)p Fu(\))p Fo(:)800 4587 y Ft(X)889 4544 y Fs(h)930 4553 y Fg(1)881 4615 y Fs(\014)921 4624 y Fg(1)968 4587 y Ft(X)1057 4544 y Fs(h)1098 4553 y Fg(2)1049 4615 y Fs(\014)1089 4624 y Fg(2)1153 4587 y Ft(:)17 b(:)g(:)f(X)1373 4542 y Fs(h)1414 4553 y Fi(P)1365 4615 y Fs(\014)1405 4626 y Fi(P)1469 4587 y Ft(;)216 b Fu(0)27 b Fm(\024)h Ft(h)1949 4602 y Fs(j)2013 4587 y Fm(\024)h Ft(N)j Fm(\000)23 b Fu(1)p Ft(;)116 b Fu(1)27 b Fm(\024)i Ft(j)k Fm(\024)28 b Ft(P)s(:)p 3882 4774 V 3886 4710 60 4 v 3886 4774 V 3945 4774 4 68 v -80 4964 a Fu(The)e(Theorem)g(sa)m(ys)g(that)g Fj(B)p Fu(\()p Fp(V)p Fu(\))32 b(is)25 b(the)g(w)m(ell-kno)m(wn)h("p)s(ositiv) m(e)e(part")h Fj(u)2641 4928 y Fn(+)2641 4989 y Fs(q)2700 4964 y Fu(\()p Ft(g)t Fu(\))f(of)h(the)h(so-called)e(F)-8 b(rob)s(enius-)-180 5081 y(Lusztig)32 b(k)m(ernel)i(of)e Ft(g)t Fu(.)p eop %%Page: 33 33 33 32 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(33)-80 203 y Fu(Motiv)-5 b(ated)31 b(b)m(y)i(the)f(preceding)g (Theorems)g(and)g(results,)h(w)m(e)g(in)m(tro)s(duced)f(the)g(follo)m (wing)d(notion)i(in)g([AS2])-180 319 y(\(see)j(also)d([F)m(G]\).)-180 530 y FD(De\014nition)36 b(4.4.)49 b Fu(Let)d(\()p Ft(V)5 b(;)17 b(c)p Fu(\))46 b(a)f(braided)h(v)m(ector)h(space)g(of)e (diagonal)f(t)m(yp)s(e)i(with)g(basis)g Ft(x)3368 545 y Fn(1)3408 530 y Ft(;)17 b(:)g(:)g(:)e(;)i(x)3681 545 y Fs(\022)3720 530 y Fu(,)50 b(and)-180 669 y(matrix)31 b(\()p Ft(q)218 684 y Fs(ij)279 669 y Fu(\),)h(that)h(is)967 811 y Ft(c)p Fu(\()p Ft(x)1102 826 y Fs(i)1152 811 y Fm(\012)23 b Ft(x)1307 826 y Fs(j)1344 811 y Fu(\))28 b(=)f Ft(q)1556 826 y Fs(i;j)1636 811 y Ft(x)1691 826 y Fs(j)1751 811 y Fm(\012)22 b Ft(x)1905 826 y Fs(i)1934 811 y Ft(;)49 b Fu(for)32 b(all)f(1)c Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(:)-180 972 y Fu(W)-8 b(e)29 b(shall)d(sa)m(y)j(that)f(\()p Ft(V)5 b(;)17 b(c)p Fu(\))28 b(is)g Fo(of)i(Cartan)h(typ)-5 b(e)28 b Fu(if)f(for)h(all)e Ft(i;)17 b(j)6 b Fu(,)29 b Ft(q)2143 987 y Fs(ij)2232 972 y Fu(is)e(a)h(ro)s(ot)g(of)f(unit)m(y)-8 b(,)29 b Ft(q)3026 987 y Fs(ii)3107 972 y Fm(6)p Fu(=)e(1)h(and)g(there)h (exists)-180 1112 y Ft(a)-129 1127 y Fs(ij)-41 1112 y Fm(2)g Fp(Z)g Fu(suc)m(h)34 b(that)1209 1294 y Ft(q)1252 1309 y Fs(ij)1313 1294 y Ft(q)1356 1309 y Fs(j)t(i)1445 1294 y Fu(=)27 b Ft(q)1595 1240 y Fs(a)1632 1250 y Fi(ij)1591 1320 y Fs(ii)1691 1294 y Ft(;)49 b Fu(for)32 b(all)f(1)c Fm(\024)i Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(\022)s(:)-2744 b Fu(\(4.4\))-80 1476 y(The)33 b(in)m(tegers)g Ft(a)532 1491 y Fs(ij)625 1476 y Fu(are)g(uniquely)g(determined)f(when)i(c)m (hosen)g(in)e(the)h(follo)m(wing)d(w)m(a)m(y:)70 1658 y Ft(a)121 1673 y Fs(ii)201 1658 y Fu(=)e(2;)-561 b(\(4.5\))92 1827 y Fm(\000)23 b Fu(ord)17 b Ft(q)393 1842 y Fs(ii)473 1827 y Ft(<)27 b(a)627 1842 y Fs(ij)716 1827 y Fm(\024)h Fu(0)p Ft(;)114 b(i)28 b Fm(6)p Fu(=)f Ft(j:)-1422 b Fu(\(4.6\))-180 2009 y(It)32 b(follo)m(ws)e(that)i(\()p Ft(a)544 2024 y Fs(ij)604 2009 y Fu(\))g(is)f(a)g(generalized)g(Cartan) h(matrix)e(\(GCM\))i(in)f(the)h(sense)h(of)f(the)g(b)s(o)s(ok)f([K].)43 b(W)-8 b(e)32 b(shall)-180 2148 y(adapt)42 b(the)h(terminology)e(from)g (generalized)h(Cartan)h(matrices)f(and)g(Dynkin)h(diagrams)e(to)h (braidings)f(of)-180 2288 y(Cartan)35 b(t)m(yp)s(e.)51 b(F)-8 b(or)34 b(instance,)i(w)m(e)g(shall)d(sa)m(y)j(that)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))35 b(is)f(of)h Fo(\014nite)h(Cartan)h(typ)-5 b(e)35 b Fu(if)f(it)f(is)i(of)f(Cartan)h(t)m(yp)s(e)-180 2427 y(and)c(the)g(corresp)s(onding)g(GCM)g(is)f(actually)f(of)h (\014nite)h(t)m(yp)s(e,)h Fo(i.)44 b(e.)e Fu(a)31 b(Cartan)g(matrix)e (asso)s(ciated)h(to)h(a)f(\014nite)-180 2567 y(dimensional)k (semisimple)g(Lie)h(algebra.)53 b(W)-8 b(e)37 b(shall)e(sa)m(y)i(that)f (a)g(Y)-8 b(etter-Drinfeld)34 b(mo)s(dule)h Ft(V)58 b Fu(is)35 b Fo(of)j(Cartan)-180 2706 y(typ)-5 b(e)33 b Fu(if)e(the)i(matrix)e(\()p Ft(q)677 2721 y Fs(ij)738 2706 y Fu(\))i(as)f(ab)s(o)m(v)m(e)i(is)e(of)g(Cartan)g(t)m(yp)s(e.)-80 2917 y(T)-8 b(o)32 b(form)m(ulate)f(our)h(\014rst)g(main)f(result,)h(w) m(e)i(need)f(one)f(more)g(de\014nition)f(from)g([AS2])i(for)e (braidings)g(whic)m(h)-180 3056 y(are)i(close)f(to)g(the)h(braidings)f (of)g(the)h(F)-8 b(rob)s(enius-Lusztig)31 b(k)m(ernels.)-180 3267 y FD(De\014nition)36 b(4.5.)49 b Fu(Let)40 b(\()p Ft(V)5 b(;)17 b(c)p Fu(\))39 b(b)s(e)h(a)f(braided)h(v)m(ector)g(space) h(of)e(Cartan)h(t)m(yp)s(e)g(with)f(Cartan)h(matrix)e(\()p Ft(a)3828 3282 y Fs(ij)3889 3267 y Fu(\).)-180 3406 y(W)-8 b(e)33 b(sa)m(y)h(that)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))32 b(is)g(of)g Fo(FL-typ)-5 b(e)33 b Fu(if)e(there)j(exist)e(p)s(ositiv)m (e)g(in)m(tegers)h Ft(d)2537 3421 y Fn(1)2576 3406 y Ft(;)17 b(:)g(:)g(:)f(;)h(d)2846 3421 y Fs(\022)2917 3406 y Fu(suc)m(h)34 b(that)733 3588 y(F)-8 b(or)31 b(all)g Ft(i;)17 b(j;)50 b(d)1288 3603 y Fs(i)1315 3588 y Ft(a)1366 3603 y Fs(ij)1455 3588 y Fu(=)27 b Ft(d)1609 3603 y Fs(j)1646 3588 y Ft(a)1697 3603 y Fs(j)t(i)1790 3588 y Fu(\(th)m(us)33 b(\()p Ft(a)2131 3603 y Fs(ij)2192 3588 y Fu(\))g(is)f (symmetrizable\).)-3223 b(\(4.7\))733 3757 y(There)33 b(exists)h Ft(q)d Fm(2)d Fp(|)18 b Fu(suc)m(h)34 b(that)f Ft(q)2015 3772 y Fs(ij)2103 3757 y Fu(=)28 b Ft(q)2254 3716 y Fs(d)2290 3726 y Fi(i)2316 3716 y Fs(a)2353 3726 y Fi(ij)2445 3757 y Fu(for)k(all)e Ft(i;)17 b(j:)-3053 b Fu(\(4.8\))-180 3939 y(W)-8 b(e)43 b(call)e(\()p Ft(V)5 b(;)17 b(c)p Fu(\))43 b Fo(lo)-5 b(c)g(al)5 b(ly)43 b(of)h(FL-typ)-5 b(e)43 b Fu(if)e(an)m(y)j(principal)c(2)29 b Fm(\002)h Fu(2)42 b(submatrix)g(of)g(\()p Ft(q)2917 3954 y Fs(ij)2978 3939 y Fu(\))h(de\014nes)h(a)e(braiding)f(of)-180 4078 y(FL-t)m(yp)s(e.)-80 4266 y(W)-8 b(e)37 b(no)m(w)h(\014x)f(for)g(eac)m (h)h Ft(\013)e Fm(2)f Fu(\010)1089 4230 y Fn(+)1186 4266 y Fu(suc)m(h)j(a)f(represen)m(tation)h(of)e Ft(X)2334 4281 y Fs(\013)2421 4266 y Fu(as)h(an)g(iterated)f(braided)h(comm)m (utator.)-180 4382 y(F)-8 b(or)37 b(a)h(general)f(braided)g(v)m(ector)i (space)g(\()p Ft(V)5 b(;)17 b(c)p Fu(\))38 b(of)f(\014nite)h(Cartan)f (t)m(yp)s(e,)k(w)m(e)d(de\014ne)h(ro)s(ot)e(v)m(ectors)j Ft(x)3612 4397 y Fs(\013)3699 4382 y Fu(in)d(the)-180 4498 y(tensor)26 b(algebra)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\),)27 b Ft(\013)i Fm(2)f Fu(\010)975 4462 y Fn(+)1034 4498 y Fu(,)f(as)f(the)g(same)g(formal)d(iteration)h(of)h(braided)h (comm)m(utators)e(in)h(the)i(elemen)m(ts)-180 4614 y Ft(x)-125 4629 y Fn(1)-85 4614 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)189 4629 y Fs(\022)267 4614 y Fu(instead)38 b(of)h Ft(X)808 4629 y Fn(1)847 4614 y Ft(;)17 b(:)g(:)g(:)f(;)h(X)1147 4629 y Fs(\022)1225 4614 y Fu(but)39 b(with)f(resp)s(ect)i(to)f(the)g (braiding)e Ft(c)i Fu(giv)m(en)g(b)m(y)h(the)f(general)f(matrix)-180 4731 y(\()p Ft(q)-99 4746 y Fs(ij)-38 4731 y Fu(\).)-180 4941 y FD(Theorem)f(4.6.)49 b Fu([AS2,)34 b(AS4])p Fo(.)47 b(L)-5 b(et)36 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))35 b Fo(b)-5 b(e)36 b(a)f(br)-5 b(aide)g(d)35 b(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(e)35 b(of)g(Cartan)g(typ)-5 b(e.)48 b(We)36 b(also)f(assume)-180 5081 y(that)g Ft(q)62 5096 y Fs(ij)158 5081 y Fo(has)f(o)-5 b(dd)35 b(or)-5 b(der)34 b(for)h(al)5 b(l)34 b Ft(i;)17 b(j)6 b Fo(.)p eop %%Page: 34 34 34 33 bop -180 0 a Fq(34)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fo(\(i\).)42 b(Assume)30 b(that)g Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))30 b Fo(is)f(lo)-5 b(c)g(al)5 b(ly)30 b(of)f(FL-typ)-5 b(e)30 b(and)f(that,)i(for)f(al)5 b(l)29 b Ft(i)p Fo(,)i(the)f(or)-5 b(der)30 b(of)f Ft(q)3117 218 y Fs(ii)3199 203 y Fo(is)h(r)-5 b(elatively)30 b(prime)-180 342 y(to)40 b(3)f(whenever)g Ft(a)512 357 y Fs(ij)609 342 y Fu(=)d Fm(\000)p Fu(3)k Fo(for)f(some)g Ft(j)6 b Fo(,)41 b(and)e(is)g(di\013er)-5 b(ent)39 b(fr)-5 b(om)39 b(3,)i(5,)f(7,)h(11,)f(13,)g(17.)59 b(If)39 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))40 b Fo(is)f(\014nite)-180 482 y(dimensional,)33 b(then)i Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))34 b Fo(is)h(of)f(\014nite)h(Cartan)g(typ)-5 b(e.)-80 669 y(\(ii\).)48 b(If)36 b Fu(\()p Ft(V)5 b(;)17 b(c)p Fu(\))36 b Fo(is)h(of)f(\014nite)g(Cartan)g(typ)-5 b(e,)37 b(then)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\))36 b Fo(is)g(\014nite)g (dimensional,)f(and)h(if)g(mor)-5 b(e)g(over)35 b(3)i(do)-5 b(es)-180 808 y(not)38 b(divide)g(the)g(or)-5 b(der)38 b(of)g Ft(q)869 823 y Fs(ii)959 808 y Fo(for)g(al)5 b(l)38 b Ft(i)h Fo(in)f(a)g(c)-5 b(onne)g(cte)g(d)37 b(c)-5 b(omp)g(onent)37 b(of)h(the)g(Dynkin)g(diagr)-5 b(am)37 b(of)h(typ)-5 b(e)39 b Ft(G)3885 823 y Fn(2)3924 808 y Fo(,)-180 948 y(then)1370 1094 y Fu(dim)16 b Fj(B)p Fu(\()p Ft(V)21 b Fu(\))28 b(=)1931 999 y Fl(Y)1923 1211 y Fs(I)5 b Fk(2X)2083 1094 y Ft(N)2171 1048 y Fn(dim)11 b Fe(n)2337 1059 y Fi(I)2161 1121 y Fs(I)2376 1094 y Ft(;)-180 1335 y Fo(wher)-5 b(e)31 b Ft(N)170 1350 y Fs(I)237 1335 y Fu(=)d(ord\()p Ft(q)563 1350 y Fs(ii)615 1335 y Fu(\))k Fo(for)f(al)5 b(l)31 b Ft(i)d Fm(2)g Ft(I)39 b Fo(and)31 b Ft(I)36 b Fm(2)28 b(X)15 b Fo(.)43 b(The)31 b(Nichols)g(algebr)-5 b(a)30 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))31 b Fo(is)g(pr)-5 b(esente)g(d)31 b(by)g(gener)-5 b(ators)-180 1474 y Ft(x)-125 1489 y Fs(i)-96 1474 y Fo(,)34 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)s Fo(,)35 b(and)f(r)-5 b(elations)1003 1661 y Fu(ad)1122 1676 y Fs(c)1157 1661 y Fu(\()p Ft(x)1250 1676 y Fs(i)1278 1661 y Fu(\))1316 1620 y Fn(1)p Fk(\000)p Fs(a)1443 1630 y Fi(ij)1503 1661 y Fu(\()p Ft(x)1596 1676 y Fs(j)1632 1661 y Fu(\))28 b(=)g(0)p Ft(;)215 b(i)28 b Fm(6)p Fu(=)g Ft(j;)-2505 b Fu(\(4.9\))1520 1830 y Ft(x)1575 1788 y Fs(N)1631 1799 y Fi(I)1575 1854 y Fs(\013)1698 1830 y Fu(=)28 b(0)p Ft(;)215 b(\013)29 b Fm(2)f Fu(\010)2348 1788 y Fn(+)2348 1857 y Fs(I)2407 1830 y Ft(;)52 b(I)35 b Fm(2)28 b(X)15 b Ft(:)-2951 b Fu(\(4.10\))-180 2016 y Fo(Mor)-5 b(e)g(over,)34 b(the)h(fol)5 b(lowing)34 b(elements)g(c)-5 b(onstitute)35 b(a)g(b)-5 b(asis)34 b(of)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\))p Fo(:)608 2203 y Ft(x)663 2160 y Fs(h)704 2169 y Fg(1)663 2231 y Fs(\014)703 2240 y Fg(1)743 2203 y Ft(x)798 2160 y Fs(h)839 2169 y Fg(2)798 2231 y Fs(\014)838 2240 y Fg(2)894 2203 y Ft(:)c(:)g(:)f(x)1080 2158 y Fs(h)1121 2169 y Fi(P)1080 2231 y Fs(\014)1120 2242 y Fi(P)1177 2203 y Ft(;)216 b Fu(0)27 b Fm(\024)h Ft(h)1657 2218 y Fs(j)1722 2203 y Fm(\024)g Ft(N)1905 2218 y Fs(I)1967 2203 y Fm(\000)23 b Fu(1)p Ft(;)51 b Fo(if)g Ft(\014)2360 2218 y Fs(j)2424 2203 y Fm(2)29 b Ft(I)8 b(;)116 b Fu(1)27 b Fm(\024)h Ft(j)34 b Fm(\024)28 b Ft(P)s(:)p 3882 2390 4 68 v 3886 2326 60 4 v 3886 2390 V 3945 2390 4 68 v -80 2580 a Fu(Let)120 2554 y Fl(b)104 2580 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))42 b(b)s(e)g(the)h(braided)f(Hopf)g(algebra)f(in)1794 2544 y Fn(\000)1794 2606 y(\000)1842 2580 y Fm(Y)8 b(D)45 b Fu(generated)e(b)m(y)g Ft(x)2693 2595 y Fn(1)2733 2580 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)3007 2595 y Fs(\022)3088 2580 y Fu(with)42 b(relations)f(\(4.9\),)-180 2712 y(where)g(the)e Ft(x)338 2727 y Fs(i)367 2712 y Fu('s)h(are)f(primitiv)m(e.)61 b(Let)40 b Fm(K)q Fu(\()p Ft(V)22 b Fu(\))39 b(b)s(e)g(the)h (subalgebra)f(of)2512 2686 y Fl(b)2496 2712 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))39 b(generated)h(b)m(y)g Ft(x)3426 2676 y Fs(N)3482 2687 y Fi(I)3426 2736 y Fs(\013)3522 2712 y Fu(,)h Ft(\013)f Fm(2)f Fu(\010)3867 2671 y Fn(+)3867 2739 y Fs(I)3927 2712 y Fu(,)-180 2845 y Ft(I)d Fm(2)28 b(X)15 b Fu(;)32 b(it)f(is)i(a)f(Y)-8 b(etter-Drinfeld)31 b(submo)s(dule)h(of)1712 2819 y Fl(b)1696 2845 y Fj(B)p Fu(\()p Ft(V)21 b Fu(\).)-180 3059 y FD(Theorem)37 b(4.7.)49 b Fu([AS4])35 b Fm(K)q Fu(\()p Ft(V)22 b Fu(\))35 b Fo(is)g(a)f(br)-5 b(aide)g(d)34 b(Hopf)h(sub)-5 b(algebr)g(a)34 b(in)2393 3023 y Fn(\000)2393 3084 y(\000)2441 3059 y Fm(Y)8 b(D)37 b Fo(of)2761 3033 y Fl(b)2745 3059 y Fj(B)p Fu(\()p Ft(V)21 b Fu(\))p Fo(.)p 3882 3059 V 3886 2995 60 4 v 3886 3059 V 3945 3059 4 68 v 194 3299 a Fu(5.)55 b Fv(Classifica)-7 b(tion)39 b(of)f(pointed)g(Hopf)g(algebras)e(by)i(the)f(lifting)h (method)-180 3473 y Fu(5.1.)56 b FD(Lifting)36 b(of)i(Cartan)f(t)m(yp)s (e.)-80 3589 y Fu(W)-8 b(e)29 b(prop)s(ose)f(to)h(sub)s(divide)f(the)h (classi\014cation)e(problem)h(for)g(\014nite-dimensional)d(p)s(oin)m (ted)j(Hopf)g(algebras)-180 3705 y(in)m(to)k(the)h(follo)m(wing)d (problems:)-156 3845 y(\(a\).)41 b(Determine)32 b(all)e(braided)i(v)m (ector)i(spaces)g Ft(V)54 b Fu(of)32 b(group)h(t)m(yp)s(e)g(suc)m(h)h (that)f Fj(B)p Fu(\()p Ft(V)21 b Fu(\))33 b(is)f(\014nite)g (dimensional.)-162 4009 y(\(b\).)42 b(Giv)m(en)g(a)g(\014nite)g(group)g (\000,)j(determine)c(all)g(realizations)f(of)h(braided)h(v)m(ector)h (spaces)h Ft(V)64 b Fu(as)42 b(in)g(\(a\))g(as)37 4125 y(Y)-8 b(etter-Drinfeld)31 b(mo)s(dules)g(o)m(v)m(er)j(\000.)-151 4289 y(\(c\).)42 b(The)i(lifting)39 b(problem:)62 b(F)-8 b(or)42 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))42 b(as)h(in)f(\(a\),)i (compute)f(all)d(Hopf)j(algebras)f Ft(A)g Fu(suc)m(h)i(that)f(gr)16 b Ft(A)44 b Fm(')37 4405 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Ft(H)40 b Fu(.)-162 4568 y(\(d\).)i(In)m(v)m(estigate)37 b(whether)g(an)m(y)g(\014nite)e(dimensional)f(p)s(oin)m(ted)h(Hopf)h (algebra)f(is)g(generated)i(as)f(an)g(algebra)37 4685 y(b)m(y)d(its)g(group-lik)m(e)e(and)h(sk)m(ew-primitiv)m(e)g(elemen)m (ts.)-80 4848 y(Problem)42 b(\(a\))h(w)m(as)i(discussed)g(in)e (Chapters)i(3)e(and)g(4.)76 b(W)-8 b(e)44 b(ha)m(v)m(e)h(seen)g(the)f (v)m(ery)h(imp)s(ortan)m(t)d(class)h(of)-180 4964 y(braidings)c(of)h (\014nite)g(Cartan)g(t)m(yp)s(e)h(and)f(some)g(isolated)f(examples)h (where)i(the)e(Nic)m(hols)g(algebra)f(is)h(\014nite-)-180 5081 y(dimensional.)h(But)33 b(the)g(general)f(case)h(of)f(problem)g (\(a\))g(seems)h(to)g(require)f(completely)g(new)h(ideas.)p eop %%Page: 35 35 35 34 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(35)-80 203 y Fu(Problem)26 b(\(b\))i(is)f(of)g(computational)e (nature.)42 b(F)-8 b(or)27 b(braidings)f(of)h(\014nite)g(Cartan)h(t)m (yp)s(e)h(with)e(Cartan)g(matrix)-180 319 y(\()p Ft(a)-91 334 y Fs(ij)-30 319 y Fu(\))8 334 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)307 319 y Fu(and)39 b(an)h(ab)s(elian)d (group)i(\000)h(w)m(e)g(ha)m(v)m(e)h(to)e(compute)g(elemen)m(ts)h Ft(g)2738 334 y Fn(1)2777 319 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)3043 334 y Fs(\022)3120 319 y Fm(2)40 b Fu(\000)f(and)g(c)m(haracters)-180 453 y Ft(\037)-119 468 y Fn(1)-79 453 y Ft(;)17 b(:)g(:)g(:)e(;)i(\037) 200 468 y Fs(\022)267 453 y Fm(2)364 427 y Fl(b)361 453 y Fu(\000)32 b(suc)m(h)i(that)968 615 y Ft(\037)1029 630 y Fs(i)1057 615 y Fu(\()p Ft(g)1142 630 y Fs(j)1178 615 y Fu(\))p Ft(\037)1277 630 y Fs(j)1314 615 y Fu(\()p Ft(g)1399 630 y Fs(i)1427 615 y Fu(\))27 b(=)h Ft(\037)1657 630 y Fs(i)1685 615 y Fu(\()p Ft(g)1770 630 y Fs(i)1798 615 y Fu(\))1836 574 y Fs(a)1873 584 y Fi(ij)1932 615 y Ft(;)50 b Fu(for)32 b(all)e(1)e Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)c Ft(\022)s(:)-2986 b Fu(\(5.1\))-180 779 y(T)-8 b(o)28 b(\014nd)g(these)h(elemen)m(ts)e(one)h(has)g(to)g (solv)m(e)g(a)f(system)h(of)g(quadratic)f(congruences)i(in)e(sev)m (eral)h(unkno)m(wns.)44 b(In)-180 895 y(man)m(y)31 b(cases)i(they)f(do) f(not)g(exist.)43 b(In)32 b(particular,)d(if)h Ft(\022)h(>)d Fu(2\(ord)o(\000\))2310 859 y Fn(2)2350 895 y Fu(,)j(then)h(the)g (braiding)d(cannot)i(b)s(e)h(realized)-180 1011 y(o)m(v)m(er)i(the)f (group)f(\000.)43 b(W)-8 b(e)33 b(refer)g(to)g([AS2,)f(Section)h(8])f (for)g(details.)-80 1174 y(Problem)f(\(d\))i(is)f(the)h(sub)5 b(ject)34 b(of)e(Section)h(5.4.)-80 1337 y(W)-8 b(e)33 b(will)d(no)m(w)j(discuss)h(the)f(lifting)c(problem)j(\(c\).)-80 1500 y(The)45 b(coradical)d(\014ltration)g Fp(|)-8 b Fu(\000)41 b(=)48 b Ft(A)1334 1515 y Fn(0)1421 1500 y Fm(\032)g Ft(A)1619 1515 y Fn(1)1706 1500 y Fm(\032)g Ft(:)17 b(:)g(:)60 b Fu(of)44 b(a)g(p)s(oin)m(ted)g(Hopf)g(algebra)f Ft(A)h Fu(is)g(stable)g(under)-180 1616 y(the)37 b(adjoin)m(t)f(action) g(of)h(the)g(group.)56 b(F)-8 b(or)36 b(ab)s(elian)f(groups)j(\000)e (and)i(\014nite-dimensional)33 b(Hopf)k(algebras,)g(the)-180 1732 y(follo)m(wing)29 b(stronger)k(result)f(holds.)43 b(It)32 b(is)g(the)g(starting)g(p)s(oin)m(t)f(of)h(the)g(lifting)d(pro) s(cedure,)k(and)g(w)m(e)g(will)d(use)j(it)-180 1849 y(sev)m(eral)g (times.)-80 1965 y(If)k Ft(M)47 b Fu(is)37 b(a)g Fp(|)-9 b Fu(\000-m)o(o)r(dule,)32 b(w)m(e)39 b(denote)e(b)m(y)h Ft(M)1588 1929 y Fs(\037)1672 1965 y Fu(=)e Fm(f)p Ft(m)f Fm(2)h Ft(M)46 b Fm(j)35 b Ft(g)t(m)g Fu(=)g Ft(\037)p Fu(\()p Ft(g)t Fu(\))p Ft(m)d Fu(for)g(all)f Ft(g)38 b Fm(2)e Fu(\000)p Fm(g)p Fu(,)i Ft(\037)d Fm(2)3695 1940 y Fl(b)3692 1965 y Fu(\000,)j(the)-180 2081 y(isot)m(ypic)32 b(comp)s(onen)m(t)h(of)f(t)m(yp)s(e)h Ft(\037)p Fu(.)-180 2294 y FD(Lemma)k(5.1.)49 b Fo(L)-5 b(et)26 b Ft(A)f Fo(b)-5 b(e)24 b(a)h(\014nite-dimensional)e(Hopf)i(algebr)-5 b(a)24 b(with)h(ab)-5 b(elian)24 b(gr)-5 b(oup)24 b Ft(G)p Fu(\()p Ft(A)p Fu(\))k(=)g(\000)c Fo(and)h(diagr)-5 b(am)-180 2434 y Ft(R)q Fo(.)45 b(L)-5 b(et)35 b Ft(V)49 b Fu(=)28 b Ft(R)q Fu(\(1\))f Fm(2)669 2398 y Fn(\000)669 2459 y(\000)717 2434 y Fm(Y)8 b(D)38 b Fo(with)c(b)-5 b(asis)35 b Ft(x)1408 2449 y Fs(i)1464 2434 y Fm(2)28 b Ft(V)1636 2398 y Fs(\037)1680 2408 y Fi(i)1615 2459 y Fs(g)1649 2469 y Fi(i)1711 2434 y Ft(;)17 b(g)1802 2449 y Fs(i)1857 2434 y Fm(2)28 b Fu(\000)p Ft(;)17 b(\037)2117 2449 y Fs(i)2173 2434 y Fm(2)2270 2409 y Fl(b)2267 2434 y Fu(\000)p Ft(;)g Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s(:)-80 2573 y Fo(\(a\).)43 b(The)32 b(isotypic)g(c)-5 b(omp)g(onent)32 b(of)g(trivial)g(typ)-5 b(e)33 b(of)f Ft(A)1944 2588 y Fn(1)2016 2573 y Fo(is)h Ft(A)2192 2588 y Fn(0)2231 2573 y Fo(.)44 b(Ther)-5 b(efor)g(e,)32 b Ft(A)2847 2588 y Fn(1)2914 2573 y Fu(=)c Ft(A)3091 2588 y Fn(0)3147 2573 y Fm(\010)18 b Fu(\()p Fm(\010)3357 2588 y Fs(\037)p Fk(6)p Fn(=)p Fs(")3493 2573 y Fu(\()p Ft(A)3604 2588 y Fn(1)3643 2573 y Fu(\))3681 2537 y Fs(\037)3729 2573 y Fu(\))32 b Fo(and)1198 2760 y Fm(\010)1275 2775 y Fs(\037)p Fk(6)p Fn(=)p Fs(")1410 2760 y Fu(\()p Ft(A)1521 2775 y Fn(1)1561 2760 y Fu(\))1599 2718 y Fs(\037)1704 2703 y Fk(')1675 2760 y Fm(\000)-63 b(!)27 b Ft(A)1889 2775 y Fn(1)1928 2760 y Ft(=)-5 b(A)2045 2775 y Fn(0)2151 2703 y Fk(')2112 2760 y Fm( )-44 b(\000)28 b Ft(V)22 b Fu(#)p Fp(|)-9 b Fu(\000)p Ft(:)-2762 b Fu(\(5.2\))-80 2951 y Fo(\(b\).)44 b(F)-7 b(or)34 b(al)5 b(l)34 b Ft(g)d Fm(2)d Fu(\000)p Fo(,)35 b Ft(\037)28 b Fm(2)924 2926 y Fl(b)921 2951 y Fu(\000)35 b Fo(with)f Ft(\037)28 b Fm(6)p Fu(=)g Ft(")p Fo(,)593 3137 y Fm(P)662 3152 y Fs(g)r(;)p Fn(1)758 3137 y Fu(\()p Ft(A)p Fu(\))907 3096 y Fs(\037)982 3137 y Fm(6)p Fu(=)g(0)55 b Fm(\()-17 b(\))90 b Fo(ther)-5 b(e)35 b(is)f(some)g Fu(1)28 b Fm(\024)g Ft(`)g Fm(\024)g Ft(\022)j Fu(:)c Ft(g)k Fu(=)d Ft(g)2774 3152 y Fs(`)2807 3137 y Ft(;)17 b(\037)27 b Fu(=)h Ft(\037)3104 3152 y Fs(`)3137 3137 y Fu(;)-3344 b(\(5.3\))605 3305 y Fm(P)674 3320 y Fs(g)r(;)p Fn(1)769 3305 y Fu(\()p Ft(A)p Fu(\))918 3264 y Fs(")982 3305 y Fu(=)28 b Fp(|)-9 b Fu(\(1)16 b Fm(\000)23 b Ft(g)t Fu(\))p Ft(:)-1646 b Fu(\(5.4\))-180 3519 y Fo(Pr)-5 b(o)g(of.)41 b Fu(\(a\))32 b(follo)m(ws)g(from)f([AS1,)i(Lemma)e(3.1])h(and)h(implies)d(\(b\).)43 b(See)34 b([AS1,)f(Lemma)e(5.3].)p 3883 3519 4 66 v 3887 3456 59 4 v 3887 3519 V 3945 3519 4 66 v -80 3692 a(W)-8 b(e)35 b(assume)g(that)f Ft(A)g Fu(is)g(a)h(\014nite-dimensional)c(p)s (oin)m(ted)j(Hopf)h(algebra)e(with)h(ab)s(elian)f(group)h Ft(G)p Fu(\()p Ft(A)p Fu(\))d(=)g(\000,)-180 3808 y(and)i(that)1507 3931 y(gr)p Ft(A)28 b Fm(')g Fj(B)p Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000)p Ft(;)-180 4073 y Fu(where)33 b Ft(V)49 b Fm(2)301 4037 y Fn(\000)301 4098 y(\000)350 4073 y Fm(Y)8 b(D)34 b Fu(is)d(a)h(giv)m(en)f(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)h(with)g(basis)h Ft(x)2504 4088 y Fs(i)2560 4073 y Fm(2)c Ft(V)2733 4037 y Fs(\037)2777 4047 y Fi(i)2711 4098 y Fs(g)2745 4108 y Fi(i)2807 4073 y Ft(;)17 b(g)2898 4088 y Fn(1)2954 4073 y Ft(:)g(:)g(:)f(;)h(g)3176 4088 y Fs(\022)3242 4073 y Fm(2)28 b Fu(\000)p Ft(;)17 b(\037)3502 4088 y Fn(1)3541 4073 y Ft(;)g(:)g(:)g(:)f(;)h(\037)3821 4088 y Fs(\022)3887 4073 y Fm(2)-177 4191 y Fl(b)-180 4216 y Fu(\000)p Ft(;)g Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s(:)-80 4332 y Fu(W)-8 b(e)32 b(\014rst)g(lift)d(the)k(basis)e (elemen)m(ts)h Ft(x)1299 4347 y Fs(i)1328 4332 y Fu(.)43 b(Using)33 b(\(5.2\))o(,)f(w)m(e)h(c)m(ho)s(ose)f Ft(a)2434 4347 y Fs(i)2490 4332 y Fm(2)c(P)8 b Fu(\()p Ft(A)p Fu(\))2810 4285 y Fs(\037)2854 4295 y Fi(i)2810 4357 y Fs(g)2844 4367 y Fi(i)2871 4357 y Fs(;)p Fn(1)2962 4332 y Fu(suc)m(h)33 b(that)e(the)h(canonical)-180 4453 y(image)24 b(of)i Ft(a)251 4468 y Fs(i)305 4453 y Fu(in)f Ft(A)485 4468 y Fn(1)525 4453 y Ft(=)-5 b(A)642 4468 y Fn(0)707 4453 y Fu(is)26 b Ft(x)854 4468 y Fs(i)908 4453 y Fu(\(whic)m(h)h(w)m(e)g (iden)m(tify)e(with)h Ft(x)1975 4468 y Fs(i)2003 4453 y Fu(#1\),)i(1)f Fm(\024)h Ft(i)g Fm(\024)h Ft(\022)s Fu(.)41 b(Since)26 b(the)h(elemen)m(ts)f Ft(x)3547 4468 y Fs(i)3602 4453 y Fu(together)-180 4569 y(with)31 b(\000)h(generate)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000,)26 b(it)31 b(follo)m(ws)f(from)h(a)g(standard)i(argumen)m(t)e(that)g Ft(a)2851 4584 y Fn(1)2891 4569 y Ft(;)17 b(:)g(:)g(:)f(;)h(a)3161 4584 y Fs(\022)3232 4569 y Fu(and)31 b(the)i(elemen)m(ts)-180 4685 y(in)f(\000)g(generate)h Ft(A)g Fu(as)g(an)f(algebra.)-80 4848 y(Our)d(aim)e(is)h(to)h(\014nd)h(relations)d(b)s(et)m(w)m(een)k (the)f Ft(a)1686 4812 y Fk(0)1686 4873 y Fs(i)1714 4848 y Ft(s)f Fu(and)g(the)h(elemen)m(ts)f(in)f(\000)h(whic)m(h)h(de\014ne)g (a)f(quotien)m(t)g(Hopf)-180 4964 y(algebra)g(of)g(the)h(correct)g (dimension)e(dim)o Fj(B)p Fu(\()p Ft(V)22 b Fu(\))16 b Fm(\001)g Fu(ord\(\000\).)42 b(The)31 b(idea)e(is)g(to)g("lift\\)e (the)j(relations)f(b)s(et)m(w)m(een)i(the)-180 5081 y Ft(x)-125 5044 y Fk(0)-125 5105 y Fs(i)-96 5081 y Ft(s)h Fu(and)h(the)g(elemen)m(ts)g(in)e(\000)i(in)f Fj(B)p Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000.)p eop %%Page: 36 36 36 35 bop -180 0 a Fq(36)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(W)-8 b(e)30 b(no)m(w)h(assume)g(moreo)m (v)m(er)f(that)g Ft(V)52 b Fu(is)30 b(of)f(\014nite)h(Cartan)g(t)m(yp)s (e)h(with)f(Cartan)g(matrix)f(\()p Ft(a)3304 218 y Fs(ij)3365 203 y Fu(\))h(with)g(resp)s(ect)-180 319 y(to)i(the)h(basis)g Ft(x)402 334 y Fn(1)442 319 y Ft(;)17 b(:)g(:)g(:)e(;)i(x)715 334 y Fs(\022)755 319 y Fu(,)32 b(that)h(is)f(\(5.1\))g(holds.)43 b(W)-8 b(e)33 b(also)f(assume)70 499 y(ord\()p Ft(\037)310 514 y Fs(j)347 499 y Fu(\()p Ft(g)432 514 y Fs(i)460 499 y Fu(\)\))g(is)g(o)s(dd)g(for)h(all)d Ft(i;)17 b(j;)-1467 b Fu(\(5.5\))70 651 y Ft(N)148 666 y Fs(i)204 651 y Fu(=)28 b(ord\()p Ft(\037)548 666 y Fs(i)576 651 y Fu(\()p Ft(g)661 666 y Fs(i)689 651 y Fu(\)\))k(is)g(prime)g(to)g(3)g(for)g(all)f Ft(i)d Fm(2)g Ft(I)8 b(;)17 b(I)35 b Fm(2)28 b(X)47 b Fu(of)32 b(t)m(yp)s(e)i Ft(G)2605 666 y Fn(2)2644 651 y Ft(:)-2851 b Fu(\(5.6\))-80 837 y(W)-8 b(e)41 b(\014x)h(a)f(presen)m (tation)h(\000)g(=)g Fm(h)p Ft(y)1206 852 y Fn(1)1245 837 y Fm(i)28 b(\010)g(\001)17 b(\001)g(\001)26 b(\010)j(h)p Ft(y)1754 852 y Fs(\033)1800 837 y Fm(i)p Fu(,)43 b(and)e(denote)h(b)m (y)h Ft(M)2669 852 y Fs(`)2743 837 y Fu(the)f(order)f(of)g Ft(y)3351 852 y Fs(`)3383 837 y Fu(,)j(1)e Fm(\024)h Ft(`)f Fm(\024)h Ft(\033)t Fu(.)-180 953 y(Then)35 b(Theorem)e(4.6)g (and)h(form)m(ulas)e(\(1.17\))h(imply)e(that)j Fj(B)p Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000)27 b(can)34 b(b)s(e)g(presen)m(ted)h(b)m(y)g(generators)f Ft(h)3894 968 y Fs(`)3927 953 y Fu(,)-180 1069 y(1)27 b Fm(\024)i Ft(`)e Fm(\024)h Ft(\033)t Fu(,)33 b(and)g Ft(x)539 1084 y Fs(i)567 1069 y Fu(,)g(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(\022)35 b Fu(with)e(de\014ning)f(relations)119 1365 y Ft(h)175 1319 y Fs(M)243 1331 y Fi(`)175 1393 y Fs(`)305 1365 y Fu(=)27 b(1)p Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(`)g Fm(\024)g Ft(\033)t Fu(;)-1220 b(\(5.7\))119 1517 y Ft(h)175 1532 y Fs(`)208 1517 y Ft(h)264 1532 y Fs(t)322 1517 y Fu(=)27 b Ft(h)481 1532 y Fs(t)511 1517 y Ft(h)567 1532 y Fs(`)600 1517 y Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(t)g(<)f(`)h Fm(\024)g Ft(\033)t Fu(;)-1529 b(\(5.8\))119 1668 y Ft(h)175 1683 y Fs(`)208 1668 y Ft(x)263 1683 y Fs(i)320 1668 y Fu(=)27 b Ft(\037)484 1683 y Fs(i)512 1668 y Fu(\()p Ft(y)598 1683 y Fs(`)631 1668 y Fu(\))p Ft(x)724 1683 y Fs(i)752 1668 y Ft(h)808 1683 y Fs(`)842 1668 y Ft(;)114 b Fu(1)27 b Fm(\024)h Ft(`)g Fm(\024)g Ft(\033)n(;)115 b Fu(1)27 b Fm(\024)i Ft(i)f Fm(\024)g Ft(\022)s Fu(;)-2136 b(\(5.9\))119 1831 y Ft(x)174 1789 y Fs(N)230 1800 y Fi(I)174 1855 y Fs(\013)298 1831 y Fu(=)27 b(0)p Ft(;)212 b(\013)28 b Fm(2)g Fu(\010)943 1789 y Fn(+)943 1858 y Fs(I)1003 1831 y Ft(;)17 b(I)35 b Fm(2)28 b(X)15 b Fu(;)-1512 b(\(5.10\))119 1991 y(ad)16 b(\()p Ft(x)331 2006 y Fs(i)360 1991 y Fu(\))398 1950 y Fn(1)p Fk(\000)p Fs(a)525 1960 y Fi(ij)584 1991 y Fu(\()p Ft(x)677 2006 y Fs(j)714 1991 y Fu(\))28 b(=)f(0)p Ft(;)212 b(i)28 b Fm(6)p Fu(=)f Ft(j;)-1582 b Fu(\(5.11\))-80 2171 y(and)32 b(where)i(the)f(Hopf)g(algebra)e(structure)j(is)e (determined)g(b)m(y)119 2352 y(\001\()p Ft(h)294 2367 y Fs(`)327 2352 y Fu(\))c(=)g Ft(h)553 2367 y Fs(`)608 2352 y Fm(\012)23 b Ft(h)764 2367 y Fs(`)797 2352 y Ft(;)114 b Fu(1)27 b Fm(\024)i Ft(`)e Fm(\024)h Ft(\033)t Fu(;)-1559 b(\(5.12\))119 2503 y(\001\()p Ft(x)293 2518 y Fs(i)322 2503 y Fu(\))28 b(=)f Ft(x)546 2518 y Fs(i)597 2503 y Fm(\012)22 b Fu(1)g(+)g Ft(g)912 2518 y Fs(i)962 2503 y Fm(\012)h Ft(x)1117 2518 y Fs(i)1146 2503 y Ft(;)114 b Fu(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(\022)s(:)-1890 b Fu(\(5.13\))-80 2683 y(Th)m(us)36 b Ft(A)f Fu(is)g(generated)g(b)m(y) h(the)g(elemen)m(ts)f Ft(a)1585 2698 y Fs(i)1613 2683 y Ft(;)17 b Fu(1)31 b Fm(\024)i Ft(i)f Fm(\024)g Ft(\022)s Fu(,)j(and)g Ft(h)2379 2698 y Fs(l)2406 2683 y Ft(;)17 b Fu(1)31 b Fm(\024)h Ft(l)i Fm(\024)e Ft(\033)t Fu(.)51 b(By)35 b(our)g(previous)g(c)m(hoice,)-180 2799 y(relations)c(\(5.7\),) h(\(5.8\),)g(\(5.9\))g(and)h(\(5.12\))o(,)g(\(5.13\))f(all)e(hold)i(in) g Ft(A)h Fu(with)f(the)h Ft(x)2767 2763 y Fk(0)2767 2824 y Fs(i)2795 2799 y Ft(s)g Fu(replaced)g(b)m(y)g(the)g Ft(a)3613 2763 y Fk(0)3613 2824 y Fs(i)3642 2799 y Ft(s)p Fu(.)-80 2980 y(The)26 b(remaining)e(problem)g(is)i(to)f(lift)f(the)i (quan)m(tum)g(Serre)g(relations)f(\(5.11\))g(and)g(the)i(ro)s(ot)d(v)m (ector)j(relations)-180 3096 y(\(5.10\))o(.)44 b(W)-8 b(e)33 b(will)d(do)i(this)h(in)e(the)i(next)h(t)m(w)m(o)f(Sections.) -180 3381 y(5.2.)56 b FD(Lifting)36 b(the)h(quan)m(tum)h(Serre)f (relations.)-80 3497 y Fu(W)-8 b(e)33 b(divide)f(the)h(problem)e(in)m (to)h(t)m(w)m(o)h(cases.)-54 3645 y Fm(\017)41 b Fu(Lifting)34 b(of)i(the)h(\\quan)m(tum)g(Serre)g(relations")e Ft(x)1868 3660 y Fs(i)1897 3645 y Ft(x)1952 3660 y Fs(j)2014 3645 y Fm(\000)25 b Ft(\037)2177 3660 y Fs(j)2214 3645 y Fu(\()p Ft(g)2299 3660 y Fs(i)2326 3645 y Fu(\))p Ft(x)2419 3660 y Fs(j)2456 3645 y Ft(x)2511 3660 y Fs(i)2574 3645 y Fu(=)35 b(0,)i(when)h Ft(i)c Fm(6)p Fu(=)h Ft(j)42 b Fu(are)37 b(in)f(di\013eren)m(t)37 3761 y(comp)s(onen)m(ts)d(of)f(the)h (Dynkin)g(diagram.)-54 3942 y Fm(\017)41 b Fu(Lifting)36 b(of)h(the)i(\\quan)m(tum)f(Serre)g(relations")f(ad)1941 3957 y Fs(c)1976 3942 y Fu(\()p Ft(x)2069 3957 y Fs(i)2097 3942 y Fu(\))2135 3905 y Fn(1)p Fk(\000)p Fs(a)2262 3915 y Fi(ij)2322 3942 y Fu(\()p Ft(x)2415 3957 y Fs(j)2452 3942 y Fu(\))f(=)h(0,)i(when)h Ft(i)d Fm(6)p Fu(=)f Ft(j)44 b Fu(are)38 b(in)g(the)g(same)37 4058 y(comp)s(onen)m(t)33 b(of)f(the)h(Dynkin)f(diagram.)-180 4206 y(The)i(\014rst)f(case)g(is)f (settled)h(in)f(the)h(next)g(Lemma)e(from)h([AS4].)-180 4431 y FD(Lemma)37 b(5.2.)49 b Fo(Assume)35 b(that)g Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(\022)s(;)17 b(i)28 b(<)f(j)41 b Fo(and)34 b Ft(i)28 b Fm(6\030)g Ft(j)6 b Fo(.)45 b(Then)1186 4635 y Ft(a)1237 4650 y Fs(i)1266 4635 y Ft(a)1317 4650 y Fs(j)1376 4635 y Fm(\000)22 b Ft(\037)1536 4650 y Fs(j)1573 4635 y Fu(\()p Ft(g)1658 4650 y Fs(i)1686 4635 y Fu(\))p Ft(a)1775 4650 y Fs(j)1811 4635 y Ft(a)1862 4650 y Fs(i)1918 4635 y Fu(=)28 b Ft(\025)2079 4650 y Fs(ij)2139 4635 y Fu(\(1)22 b Fm(\000)h Ft(g)2395 4650 y Fs(i)2423 4635 y Ft(g)2470 4650 y Fs(j)2506 4635 y Fu(\))p Ft(;)-2751 b Fu(\(5.14\))-180 4838 y Fo(wher)-5 b(e)34 b Ft(\025)152 4853 y Fs(ij)248 4838 y Fo(is)g(a)h(sc)-5 b(alar)34 b(in)h Fp(|)20 b Fo(which)34 b(c)-5 b(an)34 b(b)-5 b(e)35 b(chosen)e(such)i(that)119 5042 y Ft(\025)176 5057 y Fs(ij)271 5042 y Fo(is)g(arbitr)-5 b(ary)35 b(if)g Ft(g)925 5057 y Fs(i)953 5042 y Ft(g)1000 5057 y Fs(j)1064 5042 y Fm(6)p Fu(=)27 b(1)35 b Fo(and)f Ft(\037)1501 5057 y Fs(i)1529 5042 y Ft(\037)1590 5057 y Fs(j)1655 5042 y Fu(=)27 b Ft(";)52 b Fo(but)35 b(0)g(otherwise)o Ft(:)-2736 b Fu(\(5.15\))p eop %%Page: 37 37 37 36 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(37)-180 203 y Fo(Pr)-5 b(o)g(of.)41 b Fu(It)36 b(is)f(easy)i(to)e(c)m (hec)m(k)j(that)e Ft(a)1200 218 y Fs(i)1228 203 y Ft(a)1279 218 y Fs(j)1340 203 y Fm(\000)25 b Ft(\037)1503 218 y Fs(j)1540 203 y Fu(\()p Ft(g)1625 218 y Fs(i)1653 203 y Fu(\))p Ft(a)1742 218 y Fs(j)1778 203 y Ft(a)1829 218 y Fs(i)1891 203 y Fm(2)33 b(P)8 b Fu(\()p Ft(A)p Fu(\))2216 149 y Fs(\037)2260 159 y Fi(i)2287 149 y Fs(\037)2331 159 y Fi(j)2216 227 y Fs(g)2250 237 y Fi(i)2277 227 y Fs(g)2311 237 y Fi(j)2343 227 y Fs(;)p Fn(1)2402 203 y Fu(.)53 b(Supp)s(ose)36 b(that)g Ft(\037)3143 218 y Fs(i)3171 203 y Ft(\037)3232 218 y Fs(j)3302 203 y Fm(6)p Fu(=)d Ft(")i Fu(and)h Ft(a)3736 218 y Fs(i)3764 203 y Ft(a)3815 218 y Fs(j)3876 203 y Fm(\000)-180 342 y Ft(\037)-119 357 y Fs(j)-82 342 y Fu(\()p Ft(g)3 357 y Fs(i)30 342 y Fu(\))p Ft(a)119 357 y Fs(j)156 342 y Ft(a)207 357 y Fs(i)263 342 y Fm(6)p Fu(=)28 b(0.)43 b(Then)34 b(b)m(y)h(\(5.3\))o(,)e Ft(\037)1199 357 y Fs(i)1227 342 y Ft(\037)1288 357 y Fs(j)1352 342 y Fu(=)28 b Ft(\037)1517 357 y Fs(l)1575 342 y Fu(and)33 b Ft(g)1812 357 y Fs(i)1840 342 y Ft(g)1887 357 y Fs(j)1951 342 y Fu(=)27 b Ft(g)2101 357 y Fs(l)2160 342 y Fu(for)32 b(some)g(1)c Fm(\024)g Ft(l)i Fm(\024)e Ft(\022)s Fu(.)-80 482 y(Substituting)i Ft(g)524 497 y Fs(l)582 482 y Fu(and)i Ft(\037)832 497 y Fs(l)889 482 y Fu(in)f Ft(\037)1063 497 y Fs(i)1092 482 y Fu(\()p Ft(g)1177 497 y Fs(l)1202 482 y Fu(\))p Ft(\037)1301 497 y Fs(l)1327 482 y Fu(\()p Ft(g)1412 497 y Fs(i)1440 482 y Fu(\))d(=)f Ft(\037)1670 497 y Fs(i)1699 482 y Fu(\()p Ft(g)1784 497 y Fs(i)1811 482 y Fu(\))1849 446 y Fs(a)1886 458 y Fi(il)1969 482 y Fu(and)32 b(using)f Ft(\037)2473 497 y Fs(i)2502 482 y Fu(\()p Ft(g)2587 497 y Fs(j)2623 482 y Fu(\))p Ft(\037)2722 497 y Fs(j)2758 482 y Fu(\()p Ft(g)2843 497 y Fs(i)2871 482 y Fu(\))d(=)f(1)32 b(\(since)g Ft(a)3448 497 y Fs(ij)3536 482 y Fu(=)c(0,)j(since)-180 621 y Ft(i)i Fu(and)f Ft(j)39 b Fu(lie)31 b(in)h(di\013eren)m(t)h(comp)s(onen)m(ts\),)g(w)m(e)g(get)g Ft(\037)1752 636 y Fs(i)1780 621 y Fu(\()p Ft(g)1865 636 y Fs(i)1893 621 y Fu(\))1931 585 y Fn(2)1998 621 y Fu(=)28 b Ft(\037)2163 636 y Fs(i)2191 621 y Fu(\()p Ft(g)2276 636 y Fs(i)2304 621 y Fu(\))2342 585 y Fs(a)2379 597 y Fi(il)2430 621 y Ft(:)-80 761 y Fu(Th)m(us)34 b(w)m(e)g(ha)m(v)m (e)g(sho)m(wn)g(that)f Ft(a)1095 776 y Fs(il)1173 761 y Fm(\021)28 b Fu(2)67 b(mo)s(d)32 b Ft(N)1692 776 y Fs(i)1720 761 y Fu(,)h(and)g(in)f(the)h(same)g(w)m(a)m(y)h Ft(a)2746 776 y Fs(j)t(l)2832 761 y Fm(\021)29 b Fu(2)66 b(mo)s(d)32 b Ft(N)3351 776 y Fs(j)3388 761 y Fu(.)44 b(Since)33 b Ft(i)28 b Fm(6\030)h Ft(j)6 b Fu(,)-180 900 y Ft(a)-129 915 y Fs(il)-46 900 y Fu(or)32 b Ft(a)124 915 y Fs(j)t(l)215 900 y Fu(m)m(ust)h(b)s(e)g(0,)f(and)h(w)m(e)g (obtain)f(the)h(con)m(tradiction)e Ft(N)2174 915 y Fs(i)2230 900 y Fu(=)d(2)k(or)g Ft(N)2612 915 y Fs(i)2668 900 y Fu(=)c(2.)-80 1040 y(Therefore)33 b Ft(\037)422 1055 y Fs(i)450 1040 y Ft(\037)511 1055 y Fs(j)576 1040 y Fu(=)27 b Ft(")p Fu(,)32 b(and)g(the)h(claim)d(follo)m(ws)h(from)g (\(5.4\))o(,)h(or)g Ft(a)2380 1055 y Fs(i)2409 1040 y Ft(a)2460 1055 y Fs(j)2518 1040 y Fm(\000)22 b Ft(\037)2678 1055 y Fs(j)2714 1040 y Fu(\()p Ft(g)2799 1055 y Fs(i)2827 1040 y Fu(\))p Ft(a)2916 1055 y Fs(j)2953 1040 y Ft(a)3004 1055 y Fs(i)3060 1040 y Fu(=)27 b(0,)33 b(and)f(the)h(claim)c(is)-180 1179 y(trivial.)p 3883 1179 4 66 v 3887 1117 59 4 v 3887 1179 V 3945 1179 4 66 v -80 1356 a(Lemma)i(5.2)h(motiv)-5 b(ates)31 b(the)i(follo)m(wing)d(notion.)-180 1571 y FD(De\014nition)36 b(5.3.)49 b Fu([AS4])33 b(W)-8 b(e)33 b(sa)m(y)h(that)e(t)m(w)m(o)h(v)m(ertices)h Ft(i)f Fu(and)f Ft(j)39 b Fo(ar)-5 b(e)35 b(linkable)c Fu(\(or)h(that)g Ft(i)h Fo(is)i(linkable)e(to)g Ft(j)6 b Fu(\))33 b(if)119 1760 y Ft(i)28 b Fm(6\030)g Ft(j;)-532 b Fu(\(5.16\))119 1929 y Ft(g)166 1944 y Fs(i)194 1929 y Ft(g)241 1944 y Fs(j)305 1929 y Fm(6)p Fu(=)27 b(1)33 b(and)-827 b(\(5.17\))119 2097 y Ft(\037)180 2112 y Fs(i)208 2097 y Ft(\037)269 2112 y Fs(j)334 2097 y Fu(=)27 b Ft(":)-690 b Fu(\(5.18\))-80 2290 y(The)33 b(follo)m(wing)d(elemen)m(tary)j(prop)s(erties)f(are)h (easily)e(v)m(eri\014ed:)119 2455 y(If)i Ft(i)f Fu(is)h(link)-5 b(able)30 b(to)i Ft(j)6 b Fu(,)33 b(then)g Ft(\037)1249 2470 y Fs(i)1277 2455 y Fu(\()p Ft(g)1362 2470 y Fs(j)1398 2455 y Fu(\))p Ft(\037)1497 2470 y Fs(j)1534 2455 y Fu(\()p Ft(g)1619 2470 y Fs(i)1647 2455 y Fu(\))27 b(=)h(1)p Ft(;)114 b(\037)2067 2470 y Fs(j)2104 2455 y Fu(\()p Ft(g)2189 2470 y Fs(j)2225 2455 y Fu(\))27 b(=)h Ft(\037)2455 2470 y Fs(i)2483 2455 y Fu(\()p Ft(g)2568 2470 y Fs(i)2596 2455 y Fu(\))2634 2414 y Fk(\000)p Fn(1)2728 2455 y Ft(:)-2935 b Fu(\(5.19\))119 2624 y(If)33 b Ft(i)f Fu(and)h Ft(k)s Fu(,)g(resp.)44 b Ft(j)39 b Fu(and)32 b Ft(`)p Fu(,)h(are)f(link)-5 b(able,)31 b(then)i Ft(a)2022 2639 y Fs(ij)2111 2624 y Fu(=)27 b Ft(a)2265 2639 y Fs(k)r(`)2337 2624 y Fu(,)32 b Ft(a)2447 2639 y Fs(j)t(i)2536 2624 y Fu(=)27 b Ft(a)2690 2639 y Fs(`k)2762 2624 y Fu(.)-2969 b(\(5.20\))119 2792 y(A)33 b(v)m(ertex)h Ft(i)f Fu(can)g(not)f(b)s(e)h(link)-5 b(able)30 b(to)j(t)m(w)m(o)g(di\013eren)m(t)g(v)m(ertices)g Ft(j)39 b Fu(and)33 b Ft(h)p Fu(.)-3004 b(\(5.21\))-180 2971 y(A)34 b Fo(linking)g(datum)g Fu(is)f(a)g(collection)f(\()p Ft(\025)1255 2986 y Fs(ij)1315 2971 y Fu(\))1353 2986 y Fn(1)p Fk(\024)p Fs(i)28 b Fu(3)k(is)g(o)s(dd)g([AS3)q(].)-124 540 y(\(c\))42 b(The)34 b(Dynkin)e(diagram)e(is)i(arbitrary)-8 b(,)32 b(but)h(w)m(e)g(assume)h Ft(g)2205 496 y Fs(N)2261 506 y Fi(i)2201 566 y Fs(i)2318 540 y Fu(=)28 b(1)k(for)g(all)e Ft(i)j Fu([AS4].)-135 706 y(\(d\))42 b(The)34 b(Dynkin)e(diagram)e(is)i (of)h(t)m(yp)s(e)g Ft(A)1464 721 y Fs(n)1511 706 y Fu(,)g(an)m(y)g Ft(n)28 b Fm(\025)g Fu(2,)k(and)h Ft(N)38 b(>)28 b Fu(3)k(,)h(see)g (Section)g(6)f(of)g(this)h(pap)s(er.)-180 847 y(The)h(cases)h Ft(A)340 862 y Fn(2)380 847 y Ft(;)17 b(N)40 b Fu(=)29 b(3)k(and)h Ft(B)994 862 y Fn(2)1033 847 y Ft(;)17 b(N)43 b Fu(o)s(dd)32 b(and)62 b Fm(6)p Fu(=)30 b(5,)j(w)m(ere)i(recen)m(tly)g (done)f(in)e([BDR].)46 b(Here)35 b Ft(N)44 b Fu(denotes)34 b(the)-180 963 y(common)d(order)i(of)f Ft(\037)637 978 y Fs(i)665 963 y Fu(\()p Ft(g)750 978 y Fs(i)778 963 y Fu(\))h(for)f(all)e Ft(i)j Fu(when)h(the)f(Dynkin)f(diagram)e(is)j (connected.)-180 1203 y(5.4.)56 b FD(Generation)37 b(in)g(degree)h (one.)-80 1319 y Fu(Let)32 b(us)i(no)m(w)f(discuss)h(step)f(\(d\))g(of) f(the)h(Lifting)d(metho)s(d.)-80 1435 y(It)46 b(is)f(not)h(di\016cult)f (to)h(sho)m(w)h(that)f(our)g(conjecture)h(2.7)f(ab)s(out)f(Nic)m(hols)h (algebras,)j(in)c(the)i(setting)e(of)-180 1552 y Ft(H)35 b Fu(=)28 b Fp(|)-9 b Fu(\000,)27 b(is)32 b(equiv)-5 b(alen)m(t)32 b(to)-180 1767 y FD(Conjecture)38 b(5.7.)49 b Fu([AS3])p Fo(.)89 b(A)n(ny)50 b(p)-5 b(ointe)g(d)49 b(\014nite)g(dimensional)f(Hopf)h(algebr)-5 b(a)49 b(over)g Fp(|)35 b Fo(is)50 b(gener)-5 b(ate)g(d)49 b(by)-180 1906 y(gr)-5 b(oup-like)34 b(and)g(skew-primitive)g(elements.)-80 2098 y Fu(W)-8 b(e)30 b(ha)m(v)m(e)h(seen)g(in)e(Section)h(2.1)f(that)h (the)g(corresp)s(onding)g(conjecture)h(is)e(false)h(when)h(the)f(Hopf)g (algebra)e(is)-180 2214 y(in\014nite-dimensional)33 b(or)k(when)i(the)e (Hopf)g(algebra)g(is)f(\014nite-dimensional)e(and)k(the)f(c)m (haracteristic)h(of)e(the)-180 2330 y(\014eld)c(is)g Ft(>)c Fu(0.)43 b(A)33 b(strong)f(indication)f(that)h(the)h(conjecture) h(is)e(true)h(is)f(giv)m(en)h(b)m(y:)-180 2545 y FD(Theorem)k(5.8.)49 b Fu([AS4])p Fo(.)j(L)-5 b(et)37 b Ft(A)g Fo(b)-5 b(e)37 b(a)g(\014nite-dimensional)e(p)-5 b(ointe)g(d)36 b(Hopf)h(algebr)-5 b(a)36 b(with)h(c)-5 b(or)g(adic)g(al)36 b Fp(|)-9 b Fu(\000)31 b Fo(and)-180 2685 y(diagr)-5 b(am)34 b Ft(R)q Fo(,)h(that)g(is)1579 2833 y Fu(gr)16 b Fm(A)28 b(')g Ft(R)q Fu(#)p Fp(|)-9 b Fu(\000)p Ft(:)-180 3001 y Fo(Assume)34 b(that)g Ft(R)q Fu(\(1\))f Fo(is)h(a)g(Y)-7 b(etter-Drinfeld)33 b(mo)-5 b(dule)33 b(of)g(\014nite)h(Cartan)f(typ)-5 b(e)34 b(with)g(br)-5 b(aiding)33 b Fu(\()p Ft(q)3374 3016 y Fs(ij)3434 3001 y Fu(\))3472 3016 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)3732 3001 y Fo(.)45 b(F)-7 b(or)-180 3140 y(al)5 b(l)38 b(i,)h(let)f Ft(q)245 3155 y Fs(i)307 3140 y Fu(=)c Ft(q)460 3155 y Fs(ii)512 3140 y Ft(;)17 b(N)634 3155 y Fs(i)696 3140 y Fu(=)34 b Fo(or)-5 b(d)p Fu(\()p Ft(q)1023 3155 y Fs(i)1051 3140 y Fu(\))p Fo(.)54 b(Assume)39 b(that)f(or)-5 b(d)p Fu(\()p Ft(q)1960 3155 y Fs(ij)2021 3140 y Fu(\))38 b Fo(is)g(o)-5 b(dd)37 b(and)h Ft(N)2658 3155 y Fs(i)2724 3140 y Fo(is)g(not)h(divisible)d(by) j(3)f(and)f Ft(>)d Fu(7)-180 3280 y Fo(for)h(al)5 b(l)34 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)c Ft(\022)s Fo(.)-80 3444 y Fu(1.)41 b Fo(F)-7 b(or)44 b(any)g Fu(1)i Fm(\024)g Ft(i)g Fm(\024)g Ft(\022)i Fo(c)-5 b(ontaine)g(d)43 b(in)h(a)h(c)-5 b(onne)g(cte)g(d)43 b(c)-5 b(omp)g(onent)44 b(of)g(typ)-5 b(e)45 b Ft(B)2965 3459 y Fs(n)3012 3444 y Fo(,)i Ft(C)3159 3459 y Fs(n)3250 3444 y Fo(or)e Ft(F)3449 3459 y Fn(4)3533 3444 y Fo(r)-5 b(esp.)74 b Ft(G)3885 3459 y Fn(2)3924 3444 y Fo(,)37 3583 y(assume)34 b(that)h Ft(N)655 3598 y Fs(i)719 3583 y Fo(is)f(not)h(divisible)f(by)h(5)f(r)-5 b(esp.)45 b(by)35 b(5)f(or)h(7.)-80 3772 y Fu(2.)41 b Fo(If)34 b Ft(i)h Fo(and)g Ft(j)41 b Fo(b)-5 b(elong)33 b(to)j(di\013er)-5 b(ent)34 b(c)-5 b(omp)g(onents,)33 b(assume)h Ft(q)2215 3787 y Fs(i)2244 3772 y Ft(q)2287 3787 y Fs(j)2351 3772 y Fu(=)28 b(1)35 b Fo(or)f Fu(ord\()p Ft(q)2886 3787 y Fs(i)2915 3772 y Ft(q)2958 3787 y Fs(j)2994 3772 y Fu(\))28 b(=)g Ft(N)3242 3787 y Fs(i)3270 3772 y Fo(.)-180 3936 y(Then)i Ft(R)h Fo(is)f(gener)-5 b(ate)g(d)30 b(as)g(an)g(algebr)-5 b(a)29 b(by)i Ft(R)q Fu(\(1\))p Fo(,)g(that)f(is)g Ft(A)h Fo(is)f(gener)-5 b(ate)g(d)30 b(by)g(skew-primitive)f(and)h(gr)-5 b(oup-like)-180 4075 y(elements.)p 3882 4075 4 68 v 3886 4011 60 4 v 3886 4075 V 3945 4075 4 68 v -80 4267 a Fu(Let)31 b(us)h(discuss)h(the)f (idea)f(of)g(the)h(pro)s(of)e(of)h(Theorem)h(5.8.)43 b(A)m(t)31 b(one)h(decisiv)m(e)g(p)s(oin)m(t,)f(w)m(e)i(use)f(our)f (previous)-180 4383 y(results)i(ab)s(out)f(braidings)f(of)h(Cartan)h(t) m(yp)s(e)h(of)e(rank)h(2.)-80 4499 y(Let)c Ft(S)35 b Fu(b)s(e)30 b(the)g(graded)f(dual)g(of)g Ft(R)q Fu(.)42 b(By)30 b(the)g(dualit)m(y)f(principle)f(in)g(Lemma)g(2.4,)i Ft(S)35 b Fu(is)29 b(generated)h(in)f(degree)-180 4616 y(one)k(since)h Ft(P)14 b Fu(\()p Ft(R)q Fu(\))27 b(=)h Ft(R)q Fu(\(1\).)45 b(Our)33 b(problem)e(is)i(to)f(sho)m(w)i(that)f Ft(R)h Fu(is)f(generated)g(in)g(degree)g(one,)h(that)e(is)h Ft(S)39 b Fu(is)32 b(a)-180 4732 y(Nic)m(hols)g(algebra.)-80 4848 y(Since)j Ft(S)41 b Fu(is)34 b(generated)i(in)e(degree)i(one,)g (there)g(is)f(a)g(surjection)g(of)f(graded)h(braided)g(Hopf)g(algebras) g Ft(S)i Fm(!)-180 4964 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\),)41 b(where)h Ft(V)61 b Fu(=)41 b Ft(S)6 b Fu(\(1\))39 b(has)h(the)h(same)e (braiding)g(as)h Ft(R)q Fu(\(1\).)65 b(But)40 b(w)m(e)h(kno)m(w)g(the)g (de\014ning)f(relations)e(of)-180 5081 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\),)32 b(since)h(it)f(is)g(of)g(\014nite)g(Cartan)h(t)m (yp)s(e.)44 b(So)33 b(ha)m(v)m(e)h(to)e(sho)m(w)h(that)g(these)h (relations)d(also)g(hold)h(in)g Ft(S)6 b Fu(.)p eop %%Page: 40 40 40 39 bop -180 0 a Fq(40)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(In)40 b(the)h(case)h(of)e(a)g(quan)m (tum)g(Serre)i(relation)c(ad)1809 218 y Fs(c)1844 203 y Fu(\()p Ft(x)1937 218 y Fs(i)1966 203 y Fu(\))2004 167 y Fn(1)p Fk(\000)p Fs(a)2131 177 y Fi(ij)2190 203 y Fu(\()p Ft(x)2283 218 y Fs(j)2320 203 y Fu(\))j(=)g(0,)h Ft(i)f Fm(6)p Fu(=)g Ft(j)6 b Fu(,)43 b(w)m(e)e(consider)g(the)g(Y)-8 b(etter-)-180 319 y(Drinfeld)25 b(submo)s(dule)h Ft(W)40 b Fu(of)26 b Ft(S)33 b Fu(generated)27 b(b)m(y)h Ft(x)1633 334 y Fs(i)1688 319 y Fu(and)e(ad)1991 334 y Fs(c)2025 319 y Fu(\()p Ft(x)2118 334 y Fs(i)2147 319 y Fu(\))2185 283 y Fn(1)p Fk(\000)p Fs(a)2312 293 y Fi(ij)2371 319 y Fu(\()p Ft(x)2464 334 y Fs(j)2501 319 y Fu(\))h(and)g(assume)g(that)f (ad)3405 334 y Fs(c)3440 319 y Fu(\()p Ft(x)3533 334 y Fs(i)3562 319 y Fu(\))3600 283 y Fn(1)p Fk(\000)p Fs(a)3727 293 y Fi(ij)3786 319 y Fu(\()p Ft(x)3879 334 y Fs(j)3916 319 y Fu(\))-180 435 y Fm(6)p Fu(=)k(0.)49 b(The)35 b(assumptions)f (\(1\))g(and)h(\(2\))e(of)h(the)h(Theorem)g(guaran)m(tee)f(that)h Ft(W)47 b Fu(also)34 b(is)g(of)f(Cartan)i(t)m(yp)s(e,)h(but)-180 552 y(not)c(of)h(\014nite)f(Cartan)g(t)m(yp)s(e.)45 b(Th)m(us)34 b(ad)1308 567 y Fs(c)1343 552 y Fu(\()p Ft(x)1436 567 y Fs(i)1464 552 y Fu(\))1502 516 y Fn(1)p Fk(\000)p Fs(a)1629 526 y Fi(ij)1689 552 y Fu(\()p Ft(x)1782 567 y Fs(j)1819 552 y Fu(\))27 b(=)h(0)k(in)g Ft(S)6 b Fu(.)-80 691 y(Since)28 b(the)g(quan)m(tum)h(Serre)g(relations)d(hold)h(in)h Ft(S)6 b Fu(,)29 b(the)f(ro)s(ot)f(v)m(ector)i(relations)e(follo)m(w)f (automatically)f(from)-180 831 y(the)33 b(next)g(Lemma)f(whic)m(h)h(is) f(a)g(consequence)k(of)c(Theorem)h(4.7.)-180 1050 y FD(Lemma)k(5.9.)49 b Fu([AS4,)39 b(Lemma)e(7.5])i Fo(L)-5 b(et)40 b Ft(S)i Fu(=)36 b Fm(\010)1718 1065 y Fs(n)p Fk(\025)p Fn(0)1855 1050 y Ft(S)6 b Fu(\()p Ft(n)p Fu(\))40 b Fo(b)-5 b(e)39 b(a)g(\014nite-dimensional)f(gr)-5 b(ade)g(d)38 b(Hopf)i(algebr)-5 b(a)-180 1190 y(in)-56 1154 y Fn(\000)-56 1215 y(\000)-8 1190 y Fm(Y)8 b(D)42 b Fo(such)d(that)h Ft(S)6 b Fu(\(0\))36 b(=)g Fp(|)-9 b Fu(1)p Fo(.)52 b(Assume)40 b(that)g Ft(V)57 b Fu(=)36 b Ft(S)6 b Fu(\(1\))39 b Fo(is)g(of)g(Cartan)g(typ)-5 b(e)40 b(with)g(b)-5 b(asis)38 b Fu(\()p Ft(x)3498 1205 y Fs(i)3527 1190 y Fu(\))3565 1205 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)3864 1190 y Fo(as)-180 1329 y(describ)-5 b(e)g(d)34 b(in)g(the)h(b)-5 b(e)g(ginning)34 b(of)g(this)h(Se)-5 b(ction.)44 b(Assume)35 b(the)g(Serr)-5 b(e)34 b(r)-5 b(elations)737 1525 y Fu(\(ad)894 1540 y Fs(c)929 1525 y Ft(x)984 1540 y Fs(i)1013 1525 y Fu(\))1051 1484 y Fn(1)p Fk(\000)p Fs(a)1178 1494 y Fi(ij)1237 1525 y Ft(x)1292 1540 y Fs(j)1357 1525 y Fu(=)27 b(0)35 b Fo(for)g(al)5 b(l)34 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(\022)s(;)17 b(i)28 b Fm(6)p Fu(=)g Ft(j)41 b Fo(and)34 b Ft(i)28 b Fm(\030)g Ft(j:)-180 1720 y Fo(Then)34 b(the)h(r)-5 b(o)g(ot)35 b(ve)-5 b(ctor)35 b(r)-5 b(elations)1279 1915 y Ft(x)1334 1874 y Fs(N)1390 1885 y Fi(I)1334 1940 y Fs(\013)1457 1915 y Fu(=)28 b(0)p Ft(;)116 b(\013)28 b Fm(2)g Fu(\010)2007 1874 y Fn(+)2007 1942 y Fs(I)2067 1915 y Ft(;)116 b(I)36 b Fm(2)28 b(X)15 b Ft(;)-180 2111 y Fo(hold)34 b(in)h Ft(S)6 b Fo(.)p 3882 2111 4 68 v 3886 2047 60 4 v 3886 2111 V 3945 2111 4 68 v -180 2307 a Fu(5.5.)56 b FD(Applications.)-80 2447 y Fu(As)26 b(a)g(sp)s(ecial)g(case)h(of)f(the)g(theory)h (explained)f(ab)s(o)m(v)m(e)h(w)m(e)g(obtain)f(a)f(complete)h(answ)m (er)i(to)e(the)g(classi\014cation)-180 2586 y(problem)31 b(in)h(a)g(signi\014can)m(t)g(case.)-180 2806 y FD(Theorem)37 b(5.10.)49 b Fu([AS4])32 b Fo(L)-5 b(et)31 b Ft(p)g Fo(b)-5 b(e)31 b(a)g(prime)f Ft(>)e Fu(17)p Fo(,)j Ft(s)d Fm(\025)g Fu(1)p Fo(,)j(and)g Fu(\000)c(=)h(\()p Fp(Z)p Ft(=)p Fu(\()p Ft(p)p Fu(\)\))2802 2770 y Fs(s)2836 2806 y Fo(.)43 b(Up)31 b(to)h(isomorphism)d(ther)-5 b(e)-180 2945 y(ar)g(e)37 b(only)g(\014nitely)f(many)h(\014nite-dimensional)e(p)-5 b(ointe)g(d)36 b(Hopf)h(algebr)-5 b(as)36 b Ft(A)h Fo(with)g Ft(G)p Fu(\()p Ft(A)p Fu(\))31 b Fm(')h Fu(\000)p Fo(.)51 b(They)37 b(al)5 b(l)37 b(have)-180 3085 y(the)e(form)470 3239 y Ft(A)27 b Fm(')h Fj(u)p Fu(\()p Fm(D)s Fu(\))p Ft(;)51 b Fo(wher)-5 b(e)34 b Fm(D)k Fo(is)c(a)h(linking)f(datum)h(of)f (\014nite)h(Cartan)f(typ)-5 b(e)36 b(for)e Fu(\000)p Ft(:)p 3883 3436 4 66 v 3887 3373 59 4 v 3887 3436 V 3945 3436 4 66 v -80 3552 a Fu(If)25 b(w)m(e)h(really)f(w)m(an)m(t)h (to)f(write)g(do)m(wn)h(all)e(these)j(Hopf)e(algebras)g(w)m(e)h(still)d (ha)m(v)m(e)k(to)e(solv)m(e)h(the)g(follo)m(wing)d(serious)-180 3668 y(problems:)-54 3813 y Fm(\017)41 b Fu(Determine)32 b(all)e(Y)-8 b(etter-Drinfeld)31 b(mo)s(dules)g Ft(V)54 b Fu(o)m(v)m(er)34 b(\000)28 b(=)f(\()p Fp(Z)p Ft(=)p Fu(\()p Ft(p)p Fu(\)\))2540 3776 y Fs(s)2606 3813 y Fu(of)33 b(\014nite)f(Cartan)g(t)m(yp)s(e.)-54 3985 y Fm(\017)41 b Fu(Determine)32 b(all)e(the)j(p)s(ossible)f(linkings)f(for)h(the)h (mo)s(dules)f Ft(V)54 b Fu(o)m(v)m(er)33 b(\()p Fp(Z)p Ft(=)p Fu(\()p Ft(p)p Fu(\)\))2882 3948 y Fs(s)2948 3985 y Fu(in)f(\(a\).)-180 4177 y(By)25 b([AS2,)i(Prop)s(osition)c(8.3],)j (dim)n Ft(V)49 b Fm(\024)29 b Fu(2)p Ft(s)1423 4110 y(p)21 b Fm(\000)i Fu(1)p 1423 4154 220 4 v 1423 4246 a Ft(p)e Fm(\000)i Fu(2)1652 4177 y(,)j(for)e(all)f(the)i(p)s(ossible)f Ft(V)46 b Fu(in)24 b(\(a\).)40 b(This)25 b(pro)m(v)m(es)h(the)g (\014niteness)-180 4343 y(statemen)m(t)33 b(in)f(Theorem)h(5.10.)-80 4515 y(Note)f(that)h(w)m(e)g(ha)m(v)m(e)h(a)f(precise)g(information)c (ab)s(out)j(the)h(dimension)e(of)i(the)g(Hopf)f(algebras)g(in)g(5.10:) 1506 4695 y(dim)15 b Fj(u)p Fu(\()p Fm(D)s Fu(\))27 b(=)g Ft(p)2072 4654 y Fs(s)p Fk(j)p Fs(\036)2167 4630 y Fg(+)2217 4654 y Fk(j)2241 4695 y Ft(;)-180 4915 y Fu(where)43 b Fm(j)p Ft(\036)197 4879 y Fn(+)256 4915 y Fm(j)f Fu(is)g(the)g(n)m (um)m(b)s(er)h(of)f(the)g(p)s(ositiv)m(e)g(ro)s(ots)g(of)g(the)g(ro)s (ot)g(system)h(of)f(rank)g Ft(\022)48 b Fm(\024)c Fu(2)p Ft(s)3426 4848 y(p)22 b Fm(\000)h Fu(1)p 3426 4893 V 3426 4984 a Ft(p)f Fm(\000)h Fu(2)3698 4915 y(of)41 b(the)-180 5081 y(Cartan)33 b(matrix)e(of)h Fm(D)s Fu(.)p eop %%Page: 41 41 41 40 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(41)-80 203 y Fu(F)-8 b(or)31 b(arbitrary)h(\014nite)g(ab)s(elian)e (groups)j(\000,)f(there)h(usually)f(are)h(in\014nitely)e(man)m(y)h (non-isomorphic)e(p)s(oin)m(ted)-180 319 y(Hopf)h(algebras)g(of)g(the)g (same)g(\014nite)g(dimension.)42 b(The)32 b(\014rst)g(examples)f(w)m (ere)i(found)e(in)g(1997)f(indep)s(enden)m(tly)-180 435 y(in)37 b([AS1)q(],)i([BDG],)g([G].)61 b(No)m(w)38 b(it)g(is)f(v)m(ery) j(easy)f(to)f(construct)i(lots)d(of)h(examples)g(b)m(y)h(lifting.)58 b(Using)38 b([AS3,)-180 552 y(Lemma)31 b(1.2])h(it)g(is)g(p)s(ossible)g (to)g(decide)h(when)h(t)m(w)m(o)f(liftings)d(are)j(non-isomorphic.)-80 756 y(But)f(w)m(e)i(ha)m(v)m(e)g(a)e(b)s(ound)h(on)f(the)h(dimension)f (of)g Ft(A)p Fu(:)-180 983 y FD(Theorem)37 b(5.11.)49 b Fu([AS4])44 b Fo(F)-7 b(or)42 b(any)h(\014nite)g(\(not)g(ne)-5 b(c)g(essarily)42 b(ab)-5 b(elian\))42 b(gr)-5 b(oup)43 b Fu(\000)g Fo(of)g(o)-5 b(dd)42 b(or)-5 b(der)43 b(ther)-5 b(e)43 b(is)g(a)-180 1122 y(natur)-5 b(al)35 b(numb)-5 b(er)34 b Ft(n)p Fu(\(\000\))h Fo(such)g(that)1607 1286 y(dim)o Ft(A)28 b Fm(\024)g Ft(n)p Fu(\(\000\))-180 1471 y Fo(for)35 b(any)f(\014nite-dimensional)f(p)-5 b(ointe)g(d)34 b(Hopf)h(algebr)-5 b(a)34 b Ft(A)h Fo(with)g Ft(G)p Fu(\()p Ft(A)p Fu(\))27 b(=)h Fp(|)-9 b Fu(\000)p Fo(.)p 3882 1471 4 68 v 3886 1407 60 4 v 3886 1471 V 3945 1471 4 68 v -180 1697 a FD(Remark)37 b(5.12.)49 b Fu(As)31 b(a)f(corollary)e (of)i(the)h(Theorem)f(and)g(its)g(pro)s(of,)g(w)m(e)h(get)f(the)h (complete)f(classi\014cation)e(of)-180 1836 y(all)33 b(\014nite)i(dimensional)d(p)s(oin)m(ted)j(Hopf)g(algebras)f(with)h (coradical)e(of)h(prime)g(dimension)g Ft(p)p Fu(,)h Ft(p)d Fm(6)p Fu(=)g(2)p Ft(;)17 b Fu(5)p Ft(;)g Fu(7.)49 b(By)-180 1976 y([AS2,)35 b(Theorem)g(1.3],)g(the)g(only)f(p)s(ossibilities)e (for)i(the)h(Cartan)g(matrix)e(of)h Fm(D)j Fu(with)d(\000)h(of)f(o)s (dd)g(prime)g(order)-180 2115 y Ft(p)e Fu(are)-129 2287 y(\(a\))41 b Ft(A)110 2302 y Fn(1)182 2287 y Fu(and)33 b Ft(A)445 2302 y Fn(1)506 2287 y Fm(\002)23 b Ft(A)679 2302 y Fn(1)719 2287 y Fu(,)-135 2492 y(\(b\))42 b Ft(A)110 2507 y Fn(2)150 2492 y Fu(,)32 b(if)g Ft(p)27 b Fu(=)h(3)k(or)g Ft(p)c Fm(\021)g Fu(1)66 b(mo)s(d)32 b(3,)-124 2697 y(\(c\))42 b Ft(B)111 2712 y Fn(2)151 2697 y Fu(,)32 b(if)g Ft(p)27 b Fm(\021)h Fu(1)67 b(mo)s(d)32 b(4,)-135 2901 y(\(d\))42 b Ft(G)114 2916 y Fn(2)153 2901 y Fu(,)33 b(if)e Ft(p)d Fm(\021)g Fu(1)66 b(mo)s(d)33 b(3,)-124 3106 y(\(e\))42 b Ft(A)110 3121 y Fn(2)172 3106 y Fm(\002)22 b Ft(A)344 3121 y Fn(1)416 3106 y Fu(and)33 b Ft(A)679 3121 y Fn(2)741 3106 y Fm(\002)22 b Ft(A)913 3121 y Fn(1)953 3106 y Fu(,)33 b(if)e Ft(p)d Fu(=)f(3.)-180 3278 y(The)h(Nic)m(hols)e(algebras)g(o)m (v)m(er)i Fp(Z)p Ft(=)p Fu(\()p Ft(p)p Fu(\))c(for)i(these)i(Cartan)f (matrices)f(are)h(listed)f(in)g([AS2,)i(Theorem)f(1.3].)41 b(Hence)-180 3418 y(w)m(e)32 b(obtain)f(from)f(Theorem)h(5.10)g(for)g Ft(p)d Fm(6)p Fu(=)f(2)p Ft(;)17 b Fu(5)p Ft(;)g Fu(7)30 b(the)i(b)s(osonizations)e(of)h(the)h(Nic)m(hols)f(algebras,)g(the)g (liftings)-180 3557 y(in)h(case)h(\(a\),)f(that)g(is)g(quan)m(tum)g (lines)g(and)g(quan)m(tum)h(planes)f([AS1],)h(and)f(the)h(liftings)c (of)j(t)m(yp)s(e)h Ft(A)3538 3572 y Fn(2)3610 3557 y Fu([AS3])g(in)-180 3697 y(case)g(\(b\).)-80 3836 y(This)f(result)h(w)m (as)g(also)f(obtained)g(b)m(y)i(Musson)g([Mus)q(],)e(using)h(the)g (lifting)c(metho)s(d)j(and)h([AS2].)-80 3976 y(The)c(case)f Ft(p)g Fu(=)g(2)f(w)m(as)i(already)f(done)g(in)f([N].)42 b(In)29 b(this)e(case)i(the)g(dimension)d(of)i(the)g(p)s(oin)m(ted)g (Hopf)f(algebras)-180 4115 y(with)32 b(2-dimensional)e(coradical)g(is)j (not)f(b)s(ounded.)-80 4318 y(Let)k(us)i(men)m(tion)d(brie\015y)i(some) f(classi\014cation)g(results)h(for)f(Hopf)g(algebras)g(of)g(sp)s(ecial) g(order)g(whic)m(h)i(can)-180 4434 y(b)s(e)h(obtained)g(b)m(y)h(the)f (metho)s(ds)g(w)m(e)h(ha)m(v)m(e)h(describ)s(ed.)64 b(Let)39 b Ft(p)g(>)f Fu(2)h(b)s(e)g(a)g(prime.)62 b(Then)40 b(all)d(p)s(oin)m (ted)i(Hopf)-180 4550 y(algebras)34 b Ft(A)g Fu(of)g(dimension)f Ft(p)934 4514 y Fs(n)981 4550 y Ft(;)17 b Fu(1)30 b Fm(\024)h Ft(n)g Fm(\024)g Fu(5)k(are)f(kno)m(wn.)50 b(If)34 b(the)h(dimension)e (is)h Ft(p)g Fu(or)g Ft(p)3094 4514 y Fn(2)3134 4550 y Fu(,)h(then)g Ft(A)f Fu(is)g(a)g(group)-180 4667 y(algebra)27 b(or)h(a)h(T)-8 b(aft)28 b(Hopf)g(algebra.)41 b(The)30 b(cases)f(of)f(dimension)g Ft(p)2220 4631 y Fn(3)2287 4667 y Fu(and)h Ft(p)2522 4631 y Fn(4)2590 4667 y Fu(w)m(ere)h(treated) e(in)g([AS1])h(and)f([AS3)q(],)-180 4783 y(and)34 b(the)g (classi\014cation)e(of)i(dimension)e Ft(p)1375 4747 y Fn(5)1448 4783 y Fu(follo)m(ws)h(from)f([AS4])i(and)g([G)s(~)-51 b(n1)n(].)48 b(Indep)s(enden)m(tly)35 b(and)f(b)m(y)h(other)-180 4899 y(metho)s(ds,)e(the)g(case)g Ft(p)664 4863 y Fn(3)736 4899 y Fu(w)m(as)g(also)f(solv)m(ed)h(in)f([CD])h(and)f([SvO)q(].)-80 5081 y(See)h([A])g(for)f(a)g(discussion)h(of)f(what)h(is)f(kno)m(wn)i (on)f(classi\014cation)e(of)h(\014nite)g(dimensional)e(Hopf)j (algebras.)p eop %%Page: 42 42 42 41 bop -180 0 a Fq(42)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 203 y Fu(T)-8 b(o)26 b(form)m(ulate)e(a)i (classi\014cation)e(result)i(for)g(in\014nite-dimensional)c(Hopf)k (algebras,)g(w)m(e)i(no)m(w)e(assume)h(that)f Fp(|)-180 319 y Fu(is)c(the)h(\014eld)g(of)f(complex)g(n)m(um)m(b)s(ers)i(and)f (w)m(e)g(in)m(tro)s(duce)g(a)f(notion)g(from)f([AS5)q(].)40 b(The)24 b(collection)c Fm(D)25 b Fu(formed)d(b)m(y)i(a)-180 450 y Fo(fr)-5 b(e)g(e)28 b(ab)-5 b(elian)27 b(gr)-5 b(oup)25 b Fu(\000)g(of)g(\014nite)g(rank,)i(a)e(\014nite)g(Cartan)h (matrix)e(\()p Ft(a)2296 465 y Fs(ij)2356 450 y Fu(\))2394 465 y Fn(1)p Fk(\024)p Fs(i;j)t Fk(\024)p Fs(\022)2654 450 y Fu(,)j Ft(g)2755 465 y Fn(1)2794 450 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)3060 465 y Fs(\022)3126 450 y Fm(2)28 b Fu(\000)p Ft(;)17 b(\037)3386 465 y Fn(1)3425 450 y Ft(;)g(:)g(:)g(:)f(;)h(\037)3705 465 y Fs(\022)3772 450 y Fm(2)3869 424 y Fl(b)3866 450 y Fu(\000,)-180 566 y(and)33 b(a)f(linking)e(datum)i(\()p Ft(\025)817 581 y Fs(ij)878 566 y Fu(\))916 581 y Fn(1)p Fk(\024)p Fs(i)h Fu(0)p Ft(;)49 b Fu(for)32 b(all)e(1)e Fm(\024)g Ft(i;)17 b(j;)g Fm(\024)28 b Ft(\022)s(;)-180 896 y Fu(where)34 b Ft(d)153 911 y Fn(1)192 896 y Ft(;)17 b(:)g(:)g(:)f(;)h(d)462 911 y Fs(\022)533 896 y Fu(are)32 b(in)m(tegers)h(b)s(et)m(w)m(een)i(1) d(and)h(3)f(suc)m(h)i(that)e Ft(d)2267 911 y Fs(i)2295 896 y Ft(a)2346 911 y Fs(ij)2435 896 y Fu(=)27 b Ft(d)2589 911 y Fs(j)2626 896 y Ft(a)2677 911 y Fs(j)t(i)2737 896 y Fu(.)-80 1012 y(If)41 b Fm(D)j Fu(is)d(a)g(p)s(ositiv)m(e)g(datum)f (w)m(e)j(de\014ne)f(the)g(Hopf)f(algebra)g Ft(U)10 b Fu(\()p Fm(D)s Fu(\))41 b(b)m(y)h(generators)g Ft(a)3198 1027 y Fs(i)3227 1012 y Ft(;)17 b Fu(1)42 b Fm(\024)h Ft(i)g Fm(\024)g Ft(\022)s Fu(,)h(and)-180 1129 y Ft(h)-124 1087 y Fk(\006)-124 1156 y Fs(l)-65 1129 y Ft(;)17 b Fu(1)45 b Fm(\024)g Ft(l)j Fm(\024)d Ft(\033)i Fu(and)c(the)g (relations)e Ft(h)1336 1092 y Fk(\006)1336 1153 y Fs(m)1403 1129 y Ft(h)1459 1087 y Fk(\006)1459 1156 y Fs(l)1563 1129 y Fu(=)k Ft(h)1740 1087 y Fk(\006)1740 1156 y Fs(l)1799 1129 y Ft(h)1855 1087 y Fk(\006)1855 1156 y Fs(l)1915 1129 y Ft(;)17 b(h)2015 1087 y Fk(\006)2015 1156 y Fs(l)2073 1129 y Ft(h)2129 1087 y Fk(\007)2129 1156 y Fs(l)2234 1129 y Fu(=)45 b(1)p Ft(;)k Fu(for)32 b(all)e(1)45 b Fm(\024)g Ft(l)r(;)17 b(m)46 b Fm(\024)f Ft(\033)t Fu(,)h(de\014ning)c (the)-180 1245 y(free)37 b(ab)s(elian)e(group)i(of)f(rank)h Ft(\033)t Fu(,)h(and)f(\(5.9\),)h(the)f(quan)m(tum)g(Serre)h(relations) d(\(5.11\))h(for)g Ft(i)g Fm(6)p Fu(=)e Ft(j)43 b Fu(and)37 b Ft(i)e Fm(\030)h Ft(j)6 b Fu(,)-180 1361 y(\(5.12\))o(,\(5.13\))32 b(\(with)g Ft(a)689 1376 y Fs(i)750 1361 y Fu(instead)h(of)f Ft(x)1253 1376 y Fs(i)1281 1361 y Fu(\),)h(and)g(the)g(lifted)e(quan)m (tum)i(Serre)g(relations)e(\(5.14\))o(.)-80 1477 y(If)i(\()p Ft(V)5 b(;)17 b(c)p Fu(\))34 b(is)g(a)f(\014nite-dimensional)e(braided) i(v)m(ector)i(space,)h(w)m(e)f(will)c(sa)m(y)k(that)f(the)g Fo(br)-5 b(aiding)35 b(is)h(p)-5 b(ositive)33 b Fu(if)-180 1594 y(it)i(is)h(diagonal,)f(and)h(all)e(the)j(en)m(tries)f(of)g(the)h (matrix)d(\()p Ft(q)1952 1609 y Fs(ij)2013 1594 y Fu(\))i(of)g(the)g (braiding)f(are)h(p)s(ositiv)m(e,)g(and)h Ft(q)3557 1609 y Fs(ii)3643 1594 y Fm(6)p Fu(=)c(1)j(for)-180 1710 y(all)30 b Ft(i)p Fu(.)-80 1849 y(The)37 b(next)g(theorem)f(follo)m(ws)e(from)h (a)h(result)g(of)g(Rosso)g([Ro2)o(,)h(Theorem)g(21])f(and)g(the)g (theory)h(describ)s(ed)-180 1989 y(in)32 b(the)h(previous)g(Sections.) -180 2201 y FD(Theorem)k(5.13.)49 b Fu([AS5])42 b Fo(L)-5 b(et)43 b Ft(A)e Fo(b)-5 b(e)42 b(a)g(p)-5 b(ointe)g(d)41 b(Hopf)h(algebr)-5 b(a)40 b(with)i(ab)-5 b(elian)41 b(gr)-5 b(oup)42 b Fu(\000)e(=)g Ft(G)p Fu(\()p Ft(A)p Fu(\))i Fo(and)f(dia-)-180 2341 y(gr)-5 b(am)37 b Ft(R)q Fo(.)52 b(Assume)38 b(that)g Ft(R)q Fu(\(1\))f Fo(has)g(\014nite)g(dimension)e Ft(\022)41 b Fo(and)c(p)-5 b(ositive)37 b(br)-5 b(aiding.)51 b(Then)37 b(the)g(fol)5 b(lowing)36 b(ar)-5 b(e)-180 2480 y(e)g(quivalent:)-80 2620 y(\(a\).)44 b Ft(A)35 b Fo(is)f(a)h(domain)f(of)g(\014nite)h(Gelfand-Kiril)5 b(lov)33 b(dimension.)-80 2759 y(\(b\).)44 b(The)34 b(gr)-5 b(oup)35 b Fu(\000)g Fo(is)f(fr)-5 b(e)g(e)35 b(ab)-5 b(elian)34 b(of)g(\014nite)h(r)-5 b(ank,)34 b(and)892 2944 y Ft(A)28 b Fm(')g Ft(U)10 b Fu(\()p Fm(D)s Fu(\))p Ft(;)52 b Fo(wher)-5 b(e)34 b Fm(D)k Fo(is)c(a)h(p)-5 b(ositive)34 b(datum)h(for)g Fu(\000)p Ft(:)p 3883 3133 4 66 v 3887 3070 59 4 v 3887 3133 V 3945 3133 4 66 v -80 3249 a Fu(It)j(is)g(lik)m(ely)g(that)g(the)h(p)s(ositivit)m(y)e (assumption)h(on)h(the)g(in\014nitesimal)c(braiding)i(in)h(the)h(last)f (theorem)g(is)-180 3365 y(related)32 b(to)g(the)h(existence)i(of)d(a)g (compact)g(in)m(v)m(olution.)936 3600 y(6.)55 b Fv(Pointed)38 b(Hopf)g(algebras)f(of)h(type)f Ft(A)2791 3615 y Fs(n)-80 3890 y Fu(In)27 b(this)g(Chapter,)i(w)m(e)g(dev)m(elop)f(from)e(scratc) m(h,)j Fo(i.)43 b(e.)e Fu(without)27 b(using)g(Lusztig's)h(results,)h (the)e(classi\014cation)-180 4006 y(of)39 b(all)f(\014nite)i (dimensional)d(p)s(oin)m(ted)j(Hopf)g(algebras)f(whose)i (in\014nitesimal)c(braiding)h(is)h(of)h(t)m(yp)s(e)g Ft(A)3646 4021 y Fs(n)3693 4006 y Fu(.)66 b(The)-180 4123 y(main)31 b(results)i(of)f(this)g(Chapter)i(are)e(new.)-180 4306 y(6.1.)56 b FD(Nic)m(hols)35 b(algebras)j(of)g(t)m(yp)s(e)f Ft(A)1315 4321 y Fs(n)1362 4306 y FD(.)-80 4422 y Fu(Let)j Ft(N)51 b Fu(b)s(e)41 b(an)f(in)m(teger,)i Ft(N)52 b(>)41 b Fu(2,)h(and)f(let)e Ft(q)45 b Fu(b)s(e)40 b(a)g(ro)s(ot)g(of)g(1)g (of)g(order)h Ft(N)10 b Fu(.)67 b(F)-8 b(or)40 b(the)h(case)g Ft(N)52 b Fu(=)40 b(2,)j(see)-180 4539 y([AnDa].)-80 4700 y(Let)32 b Ft(q)137 4715 y Fs(ij)198 4700 y Fu(,)h(1)27 b Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(n)p Fu(,)33 b(b)s(e)g(ro)s(ots)f(of)g(1)g(suc)m(h)i(that)998 4956 y Ft(q)1041 4971 y Fs(ii)1121 4956 y Fu(=)27 b Ft(q)t(;)115 b(q)1456 4971 y Fs(ij)1516 4956 y Ft(q)1559 4971 y Fs(j)t(i)1648 4956 y Fu(=)1751 4786 y Fl(\()1832 4889 y Ft(q)1879 4853 y Fk(\000)p Fn(1)1973 4889 y Ft(;)195 b Fu(if)31 b Fm(j)p Ft(i)22 b Fm(\000)h Ft(j)6 b Fm(j)27 b Fu(=)h(1)p Ft(;)1832 5029 y Fu(1)p Ft(;)287 b Fu(if)31 b Fm(j)p Ft(i)22 b Fm(\000)h Ft(j)6 b Fm(j)27 b(\025)i Fu(2)p Ft(:)-180 4956 y Fu(\(6.1\))p eop %%Page: 43 43 43 42 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(43)-180 203 y Fu(for)32 b(all)f(1)c Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(n)p Fu(.)44 b(F)-8 b(or)31 b(con)m(v)m (enience,)k(w)m(e)f(denote)1321 388 y Ft(B)1400 347 y Fs(i;j)1395 413 y(p;r)1516 388 y Fu(:=)1896 294 y Fl(Y)1646 506 y Fs(i)p Fk(\024)p Fs(`)p Fk(\024)p Fs(j)t Fk(\000)p Fn(1)p Fs(;)11 b(p)p Fk(\024)p Fs(h)p Fk(\024)p Fs(r)r Fk(\000)p Fn(1)2289 388 y Ft(q)2332 403 y Fs(`;h)2426 388 y Ft(;)-180 659 y Fu(for)35 b(an)m(y)h(1)d Fm(\024)g Ft(i)g(<)f(j)39 b Fm(\024)33 b Ft(n)24 b Fu(+)g(1,)36 b(1)c Fm(\024)i Ft(p)e(<)h(r)i Fm(\024)e Ft(n)24 b Fu(+)g(1.)52 b(Then)37 b(w)m(e)f(ha)m(v)m(e)h(the)f(follo)m(wing)d(iden)m(tities,)i (whenev)m(er)-180 775 y Ft(i)28 b(<)f(s)h(<)g(j)6 b Fu(,)32 b Ft(p)c(<)f(t)h(<)g(r)s Fu(:)702 961 y Ft(B)781 919 y Fs(i;s)776 985 y(p;r)869 961 y Ft(B)948 919 y Fs(s;j)943 985 y(p;r)1065 961 y Fu(=)1412 866 y Fl(Y)1168 1078 y Fs(i)p Fk(\024)p Fs(`)p Fk(\024)p Fs(s)p Fk(\000)p Fn(1)p Fs(;p)p Fk(\024)p Fs(h)p Fk(\024)p Fs(r)r Fk(\000)p Fn(1)1800 961 y Ft(q)1843 976 y Fs(`;h)2201 866 y Fl(Y)1953 1078 y Fs(s)p Fk(\024)p Fs(`)p Fk(\024)p Fs(j)t Fk(\000)p Fn(1)p Fs(;p)p Fk(\024)p Fs(h)p Fk(\024)p Fs(r)r Fk(\000)p Fn(1)2593 961 y Ft(q)2636 976 y Fs(`;h)2757 961 y Fu(=)g Ft(B)2940 919 y Fs(i;j)2935 985 y(p;r)3028 961 y Fu(;)-3235 b(\(6.2\))719 1229 y Ft(B)798 1182 y Fs(i;j)793 1252 y(p;t)878 1229 y Ft(B)957 1182 y Fs(i;j)952 1252 y(t;r)1065 1229 y Fu(=)27 b Ft(B)1247 1188 y Fs(i;j)1242 1254 y(p;r)1336 1229 y Fu(;)-1543 b(\(6.3\))-180 1406 y(also,)964 1591 y Ft(B)1043 1544 y Fs(i;j)1038 1617 y(j;j)t Fn(+1)1213 1591 y Ft(B)1292 1544 y Fs(j;j)t Fn(+1)1287 1617 y Fs(i;j)1494 1591 y Fu(=)1677 1497 y Fl(Y)1598 1709 y Fs(i)p Fk(\024)p Fs(`)p Fk(\024)p Fs(j)t Fk(\000)p Fn(1)1899 1591 y Ft(q)1942 1606 y Fs(`;j)2129 1497 y Fl(Y)2044 1709 y Fs(i)p Fk(\024)p Fs(h)p Fk(\024)p Fs(j)t Fk(\000)p Fn(1)2358 1591 y Ft(q)2401 1606 y Fs(j;h)2522 1591 y Fu(=)27 b Ft(q)2672 1550 y Fk(\000)p Fn(1)2766 1591 y Fu(;)-2973 b(\(6.4\))1307 1860 y Ft(B)1386 1813 y Fs(i;j)1381 1886 y(i;j)1494 1860 y Fu(=)28 b Ft(q)t(:)-1852 b Fu(\(6.5\))-80 2037 y(W)-8 b(e)35 b(consider)g(in)g(this)f(Section)h(a)g(v)m(ector)h(space)g Ft(V)54 b Fu(=)31 b Ft(V)2042 2052 y Fs(n)2124 2037 y Fu(with)k(a)f(basis)h Ft(x)2729 2052 y Fn(1)2769 2037 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)3043 2052 y Fs(n)3125 2037 y Fu(and)35 b(braiding)f(deter-)-180 2153 y(mined)e(b)m(y:)1073 2281 y Ft(c)p Fu(\()p Ft(x)1208 2296 y Fs(i)1259 2281 y Fm(\012)22 b Ft(x)1413 2296 y Fs(j)1450 2281 y Fu(\))28 b(=)f Ft(q)1662 2296 y Fs(ij)1740 2281 y Ft(x)1795 2296 y Fs(j)1854 2281 y Fm(\012)22 b Ft(x)2008 2296 y Fs(i)2037 2281 y Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)c Ft(n)p Fu(;)-180 2430 y(that)j(is,)h Ft(V)54 b Fu(is)32 b(of)g(t)m(yp)s(e)h Ft(A)769 2445 y Fs(n)816 2430 y Fu(.)-180 2648 y FD(Remark)k(6.1.)49 b Fu(Let)33 b(\000)f(b)s(e)h(a)f(group,)h Ft(g)1301 2663 y Fn(1)1340 2648 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)1606 2663 y Fs(n)1685 2648 y Fu(in)31 b(the)i(cen)m(ter)h(of)f(\000,)f(and)h Ft(\037)2739 2663 y Fn(1)2778 2648 y Ft(;)17 b(:)g(:)g(:)f(;)h(\037) 3058 2663 y Fs(n)3137 2648 y Fu(in)3254 2623 y Fl(b)3251 2648 y Fu(\000)33 b(suc)m(h)h(that)1291 2840 y Ft(q)1334 2855 y Fs(ij)1422 2840 y Fu(=)28 b Fm(h)p Ft(\037)1626 2855 y Fs(j)1662 2840 y Ft(;)17 b(g)1753 2855 y Fs(i)1781 2840 y Fm(i)p Ft(;)114 b Fu(1)27 b Fm(\024)h Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(n:)-180 3033 y Fu(Then)34 b Ft(V)54 b Fu(can)33 b(b)s(e)f(realized)g(as)h(a)f(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)j(\000)e(b)m(y)i (declaring)1372 3225 y Ft(x)1427 3240 y Fs(i)1483 3225 y Fm(2)28 b Ft(V)1656 3184 y Fs(\037)1700 3194 y Fi(i)1634 3250 y Fs(g)1668 3260 y Fi(i)1730 3225 y Ft(;)212 b Fu(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(n:)-2582 b Fu(\(6.6\))-180 3418 y(F)-8 b(or)38 b(example,)i(w)m(e)f(could)f(consider)h(\000)f(=)g (\()p Fp(Z)p Ft(=P)14 b Fu(\))1703 3381 y Fs(n)1747 3418 y Fu(,)40 b(where)g Ft(P)52 b Fu(is)38 b(divisible)f(b)m(y)i(the)g (orders)h(of)e(all)e(the)j Ft(q)3800 3433 y Fs(ij)3861 3418 y Fu('s;)-180 3557 y(and)33 b(tak)m(e)g Ft(g)268 3572 y Fn(1)307 3557 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)573 3572 y Fs(n)652 3557 y Fu(as)33 b(the)g(canonical)e(basis)h(of)g(\000.) -80 3751 y(W)-8 b(e)29 b(shall)f(consider)h(a)g(braided)g(Hopf)g (algebra)f Ft(R)i Fu(pro)m(vided)g(with)f(an)g(inclusion)e(of)i (braided)g(v)m(ector)h(spaces)-180 3868 y Ft(V)52 b Fm(!)30 b Ft(P)14 b Fu(\()p Ft(R)q Fu(\).)47 b(W)-8 b(e)35 b(iden)m(tify)e(the) i(elemen)m(ts)f Ft(x)1512 3883 y Fn(1)1552 3868 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)1826 3883 y Fs(n)1907 3868 y Fu(with)33 b(their)h(images)f(in)g Ft(R)q Fu(.)48 b(Distinguished)33 b(examples)-180 3984 y(of)h(suc)m(h)i Ft(R)g Fu(are)e(the)h(tensor)g (algebra)f Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\))34 b(and)h(the)g(Nic)m (hols)f(algebra)f Fj(B)p Fu(\()p Ft(V)22 b Fu(\).)49 b(Additional)32 b(h)m(yp)s(otheses)37 b(on)-180 4100 y Ft(R)d Fu(will)c(b)s(e)j(stated)g(when)h(needed.)-80 4269 y(W)-8 b(e)33 b(in)m(tro)s(duce)f(the)h(family)d(\()p Ft(e)1070 4284 y Fs(ij)1131 4269 y Fu(\))1169 4284 y Fn(1)p Fk(\024)p Fs(i)e Fu(3)k(w)m(e)g(argue)g(b)m (y)g(induction)f(on)h Ft(r)27 b Fm(\000)d Ft(p)p Fu(.)53 b(If)35 b Ft(r)27 b Fm(\000)e Ft(p)33 b Fu(=)g(1,)j(then)g(there)g (exists)h(an)-180 2368 y(index)30 b Ft(h)g Fu(suc)m(h)h(that)f(either)g Ft(i)e(<)f(h)h(<)g(p)f(<)h(r)i Fu(=)e Ft(p)16 b Fu(+)h(1)27 b Ft(<)h(j)36 b Fu(or)29 b Ft(i)f(<)g(p)f(<)h(r)i Fu(=)e Ft(p)16 b Fu(+)h(1)27 b Ft(<)h(h)g(<)f(j)6 b Fu(.)43 b(In)30 b(the)g(\014rst)h(case,)-180 2507 y(b)m(y)k(\(6.19\))o(,)e(w)m (e)g(ha)m(v)m(e)1134 2679 y([)p Ft(e)1206 2694 y Fs(i;j)1287 2679 y Ft(;)17 b(e)1376 2694 y Fs(p;r)1469 2679 y Fu(])1496 2694 y Fs(c)1558 2679 y Fu(=)28 b([[)p Ft(e)1761 2694 y Fs(i;h)1849 2679 y Ft(;)17 b(e)1938 2694 y Fs(h;j)2035 2679 y Fu(])2062 2694 y Fs(c)2097 2679 y Ft(;)g(e)2186 2694 y Fs(p;p)p Fn(+1)2371 2679 y Fu(])2398 2694 y Fs(c)2460 2679 y Fu(=)28 b(0;)-180 2871 y(the)j(last)e(equalit)m(y)h(follo)m(ws)f (from)g(Lemma)f(1.10)i(\(c\),)h(b)s(ecause)g(of)37 b(\(6.17\))30 b(and)g(the)g(induction)g(h)m(yp)s(othesis.)43 b(In)-180 3011 y(the)33 b(second)h(case,)f(w)m(e)h(ha)m(v)m(e)1134 3182 y([)p Ft(e)1206 3197 y Fs(i;j)1287 3182 y Ft(;)17 b(e)1376 3197 y Fs(p;r)1469 3182 y Fu(])1496 3197 y Fs(c)1558 3182 y Fu(=)28 b([[)p Ft(e)1761 3197 y Fs(i;h)1849 3182 y Ft(;)17 b(e)1938 3197 y Fs(h;j)2035 3182 y Fu(])2062 3197 y Fs(c)2097 3182 y Ft(;)g(e)2186 3197 y Fs(p;p)p Fn(+1)2371 3182 y Fu(])2398 3197 y Fs(c)2460 3182 y Fu(=)28 b(0;)-180 3375 y(the)40 b(last)g(equalit)m(y)f(follo)m(ws)g(from)g (Lemma)f(1.10)i(\(c\),)i(b)s(ecause)f(of)f(the)g(induction)f(h)m(yp)s (othesis)i(and)f(\(6.18\).)-180 3514 y(Finally)-8 b(,)30 b(if)h Ft(r)25 b Fm(\000)e Ft(p)k(>)h Fu(1)k(then)1110 3686 y([)p Ft(e)1182 3701 y Fs(i;j)1263 3686 y Ft(;)17 b(e)1352 3701 y Fs(p;r)1445 3686 y Fu(])1472 3701 y Fs(c)1534 3686 y Fu(=)28 b([)p Ft(e)1710 3701 y Fs(i;j)1790 3686 y Ft(;)17 b Fu([)p Ft(e)1906 3701 y Fs(p;r)r Fk(\000)p Fn(1)2089 3686 y Ft(;)g(e)2178 3701 y Fs(r)r Fk(\000)p Fn(1)p Fs(;r)2360 3686 y Fu(])2387 3701 y Fs(c)2422 3686 y Fu(])2449 3701 y Fs(c)2511 3686 y Fu(=)28 b(0)-180 3878 y(b)m(y)33 b(Lemma)f(1.10)g(\(b\))g(and)h(the)g(induction)e(h)m (yp)s(othesis.)-80 4091 y(\(b\).)42 b(W)-8 b(e)29 b(pro)m(v)m(e)i (\(6.23\))d(b)m(y)i(induction)e(on)h Ft(r)17 b Fm(\000)e Ft(i)p Fu(.)43 b(If)29 b Ft(r)17 b Fm(\000)e Ft(i)29 b Fu(=)e(2,)j(then)f(the)h(claimed)d(equalit)m(y)i(is)f(just)h (\(6.13\).)-180 4230 y(If)k Ft(r)24 b Fm(\000)f Ft(i)28 b(>)f Fu(2)33 b(w)m(e)g(argue)g(b)m(y)g(induction)f(on)g Ft(j)d Fm(\000)22 b Ft(i)p Fu(.)44 b(If)32 b Ft(j)d Fm(\000)22 b Ft(i)28 b Fu(=)g(1)k(w)m(e)i(ha)m(v)m(e)518 4443 y([)p Ft(e)590 4458 y Fs(i;i)p Fn(+1)753 4443 y Ft(;)17 b(e)842 4458 y Fs(i;r)923 4443 y Fu(])950 4458 y Fs(c)1013 4443 y Fu(=)27 b([)p Ft(e)1188 4458 y Fs(i;i)p Fn(+1)1350 4443 y Ft(;)17 b Fu([)p Ft(e)1466 4458 y Fs(i;r)r Fk(\000)p Fn(1)1638 4443 y Ft(;)g(e)1727 4458 y Fs(r)r Fk(\000)p Fn(1)p Fs(;r)1909 4443 y Fu(])1936 4458 y Fs(c)1970 4443 y Fu(])1997 4458 y Fs(c)2060 4443 y Fu(=)27 b([[)p Ft(e)2262 4458 y Fs(i;i)p Fn(+1)2425 4443 y Ft(;)17 b(e)2514 4458 y Fs(i;r)r Fk(\000)p Fn(1)2685 4443 y Fu(])2712 4458 y Fs(c)2747 4443 y Ft(;)g(e)2836 4458 y Fs(r)n(;r)r Fk(\000)p Fn(1)3014 4443 y Fu(])3041 4458 y Fs(c)3103 4443 y Fu(=)28 b(0)-180 4655 y(b)m(y)35 b(\(1.22\))o(,)e(since)g([)p Ft(e)577 4670 y Fs(i;i)p Fn(+1)739 4655 y Ft(;)17 b(e)828 4670 y Fs(r)r Fk(\000)p Fn(1)p Fs(;r)1009 4655 y Fu(])1036 4670 y Fs(c)1099 4655 y Fu(=)28 b(0)k(b)m(y)j(\(6.17\))o(.)43 b(If)33 b Ft(j)28 b Fm(\000)23 b Ft(i)28 b(>)f Fu(2,)33 b(w)m(e)g(ha)m(v)m(e)1123 4868 y([)p Ft(e)1195 4883 y Fs(i;j)1276 4868 y Ft(;)17 b(e)1365 4883 y Fs(i;r)1446 4868 y Fu(])1473 4883 y Fs(c)1536 4868 y Fu(=)27 b([[)p Ft(e)1738 4883 y Fs(i;j)t Fk(\000)p Fn(1)1909 4868 y Ft(;)17 b(e)1998 4883 y Fs(j)t Fk(\000)p Fn(1)p Fs(;j)2177 4868 y Fu(])2204 4883 y Fs(c)2238 4868 y Ft(;)g(e)2327 4883 y Fs(i;r)2409 4868 y Fu(])2436 4883 y Fs(c)2498 4868 y Fu(=)28 b(0)-180 5081 y(b)m(y)33 b(Lemma)f(1.10)g(\(c\),)g(b)s (ecause)i(of)e(the)h(induction)f(h)m(yp)s(othesis)i(and)e(\(6.22\).)p eop %%Page: 47 47 47 46 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(47)-80 203 y Fu(The)33 b(pro)s(of)f(of)39 b(\(6.24\))32 b(is)g(analogous)f(to)h(the)h(pro)s(of)f(of)39 b(\(6.23\))o(,)33 b(using)h(\(6.14\))d(instead)i(of)39 b(\(6.13\))o(.)p 3883 203 4 66 v 3887 140 59 4 v 3887 203 V 3945 203 4 66 v -180 414 a FD(Lemma)e(6.8.)49 b Fo(Assume)35 b(that)45 b Fu(\(6.12\))o Fo(,)35 b Fu(\(6.13\))o Fo(,)g Fu(\(6.14\))f Fo(hold)g(in)h Ft(R)q Fo(.)45 b(Then)484 596 y Fu([)p Ft(e)556 611 y Fs(i;j)637 596 y Ft(;)17 b(e)726 611 y Fs(p;r)819 596 y Fu(])846 611 y Fs(c)908 596 y Fu(=)28 b Ft(B)1091 549 y Fs(pj)1086 621 y(j)t(r)1163 596 y Fu(\()p Ft(q)e Fm(\000)c Fu(1\))p Ft(e)1501 611 y Fs(ir)1563 596 y Ft(e)1608 611 y Fs(pj)1680 596 y Ft(;)251 b Fo(if)135 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(p)f(<)h(j)33 b(<)28 b(r)i Fm(\024)f Ft(n)22 b Fu(+)g(1)p Ft(:)-3470 b Fu(\(6.25\))-180 806 y Fo(Pr)-5 b(o)g(of.)41 b Fu(W)-8 b(e)33 b(compute:)463 989 y([)p Ft(e)535 1004 y Fs(i;j)616 989 y Ft(;)17 b(e)705 1004 y Fs(p;r)798 989 y Fu(])825 1004 y Fs(c)887 989 y Fu(=)28 b([[)p Ft(e)1090 1004 y Fs(i;p)1173 989 y Ft(;)17 b(e)1262 1004 y Fs(p;j)1354 989 y Fu(])1381 1004 y Fs(c)1416 989 y Ft(;)g(e)1505 1004 y Fs(p;r)1598 989 y Fu(])1625 1004 y Fs(c)887 1163 y Fu(=)28 b([)p Ft(e)1063 1178 y Fs(i;p)1146 1163 y Ft(;)17 b Fu([)p Ft(e)1262 1178 y Fs(p;j)1354 1163 y Ft(;)g(e)1443 1178 y Fs(p;r)1536 1163 y Fu(])1563 1178 y Fs(c)1598 1163 y Fu(])1625 1178 y Fs(c)1682 1163 y Fu(+)22 b Ft(B)1859 1122 y Fs(p;j)1854 1188 y(p;r)1951 1163 y Fu([)p Ft(e)2023 1178 y Fs(i;p)2106 1163 y Ft(;)17 b(e)2195 1178 y Fs(p;r)2288 1163 y Fu(])2315 1178 y Fs(c)2350 1163 y Ft(e)2395 1178 y Fs(p;j)2509 1163 y Fm(\000)23 b Ft(B)2688 1116 y Fs(i;p)2683 1189 y(p;j)2774 1163 y Ft(e)2819 1178 y Fs(p;j)2911 1163 y Fu([)p Ft(e)2983 1178 y Fs(i;p)3067 1163 y Ft(;)17 b(e)3156 1178 y Fs(p;r)3249 1163 y Fu(])3276 1178 y Fs(c)887 1338 y Fu(=)28 b Ft(B)1070 1296 y Fs(p;j)1065 1362 y(p;r)1162 1338 y Ft(e)1207 1353 y Fs(i;r)1288 1338 y Ft(e)1333 1353 y Fs(p;j)1447 1338 y Fm(\000)23 b Ft(B)1626 1290 y Fs(i;p)1621 1363 y(p;j)1713 1338 y Ft(e)1758 1353 y Fs(p;j)1850 1338 y Ft(e)1895 1353 y Fs(i;r)887 1512 y Fu(=)991 1431 y Fl(\000)1036 1512 y Ft(B)1115 1471 y Fs(p;j)1110 1537 y(p;r)1229 1512 y Fm(\000)g Ft(B)1408 1465 y Fs(i;p)1403 1538 y(p;j)1495 1512 y Fu(\()p Ft(B)1612 1465 y Fs(i;r)1607 1538 y(p;j)1698 1512 y Fu(\))1736 1471 y Fk(\000)p Fn(1)1831 1431 y Fl(\001)1893 1512 y Ft(e)1938 1527 y Fs(i;r)2020 1512 y Ft(e)2065 1527 y Fs(p;j)887 1681 y Fu(=)991 1600 y Fl(\000)1036 1681 y Ft(B)1115 1639 y Fs(p;j)1110 1705 y(p;r)1229 1681 y Fm(\000)g Fu(\()p Ft(B)1446 1633 y Fs(p;r)1441 1706 y(p;j)1539 1681 y Fu(\))1577 1639 y Fk(\000)p Fn(1)1671 1600 y Fl(\001)1734 1681 y Ft(e)1779 1696 y Fs(i;r)1860 1681 y Ft(e)1905 1696 y Fs(p;j)887 1855 y Fu(=)28 b Ft(B)1070 1808 y Fs(pj)1065 1881 y(j)t(r)1142 1855 y Fu(\()p Ft(q)e Fm(\000)c Fu(1\))p Ft(e)1480 1870 y Fs(ir)1542 1855 y Ft(e)1587 1870 y Fs(pj)1660 1855 y Ft(:)-180 2037 y Fu(Here,)37 b(the)f(\014rst)g(equalit)m(y)f(is) g(b)m(y)j(\(6.19\))o(;)f(the)f(second,)i(b)m(y)e(Lemma)f(1.10)f(\(a\);) j(the)f(third,)g(b)m(y)h(\(6.23\))e(and)h(b)m(y)-180 2177 y(\(6.19\))o(;)d(the)g(fourth,)f(b)m(y)j(\(6.22\))o(.)p 3883 2177 V 3887 2114 59 4 v 3887 2177 V 3945 2177 4 66 v -180 2387 a FD(Lemma)i(6.9.)49 b Fo(Assume)39 b(that)49 b Fu(\(6.12\))p Fo(,)40 b Fu(\(6.13\))e Fo(and)49 b Fu(\(6.14\))39 b Fo(hold)f(in)h Ft(R)q Fo(.)58 b(F)-7 b(or)38 b(any)h Fu(1)c Fm(\024)h Ft(i)g(<)g(j)41 b Fm(\024)c Ft(n)25 b Fu(+)g(1)39 b Fo(we)-180 2527 y(have)400 2709 y Fu(\001\()p Ft(e)564 2668 y Fs(N)564 2734 y(i;j)644 2709 y Fu(\))28 b(=)f Ft(e)858 2668 y Fs(N)858 2734 y(i;j)961 2709 y Fm(\012)c Fu(1)e(+)i(1)e Fm(\012)i Ft(e)1445 2668 y Fs(N)1445 2734 y(i;j)1548 2709 y Fu(+)f(\(1)f Fm(\000)i Ft(q)1901 2668 y Fk(\000)p Fn(1)1995 2709 y Fu(\))2033 2668 y Fs(N)2146 2614 y Fl(X)2117 2824 y Fs(i)f(j)6 b Fu(.)-80 351 y(By)34 b(\(6.27\))o(,)f(the)g(coaction)f(of)g Ft(Z)1114 366 y Fs(n)p Fk(\000)p Fn(1)1283 351 y Fu(on)h Ft(R)1493 366 y Fs(n)1572 351 y Fu(satis\014es)632 541 y Ft(\016)t Fu(\()p Ft(h)773 556 y Fs(i)801 541 y Fu(\))27 b(=)h Ft(g)1017 556 y Fs(i;n)p Fn(+1)1220 541 y Fm(\012)22 b Ft(e)1364 556 y Fs(i;n)p Fn(+1)1567 541 y Fu(+)g(\(1)g Fm(\000)h Ft(q)1921 500 y Fk(\000)p Fn(1)2015 541 y Fu(\))2149 447 y Fl(X)2070 657 y Fs(iF)-8 b(rom)36 b(the)-180 3346 y(preceding)d(claim,)d(w)m(e)k(deduce)g(that)344 3537 y(\001\()p Ft(h)519 3495 y Fc(m)592 3537 y Fu(\))27 b(=)h Ft(h)817 3495 y Fc(m)911 3537 y Fm(\012)23 b Fu(1)f(+)g(1)g Fm(\012)g Ft(h)1406 3495 y Fc(m)1500 3537 y Fu(+)1821 3442 y Fl(X)1598 3654 y Fn(0)p Fk(\024)p Fc(i)p Fk(\024)p Fc(m)p Fs(;)96 b Fn(0)p Fk(6)p Fn(=)p Fc(i)p Fk(6)p Fn(=)p Fc(m)2203 3537 y Ft(c)2245 3552 y Fc(m)p Fs(;)p Fc(i)2359 3537 y Ft(h)2415 3495 y Fc(i)2464 3537 y Fm(\012)23 b Ft(h)2620 3495 y Fc(m)p Fk(\000)p Fc(i)2770 3537 y Ft(;)114 b FD(m)27 b Fm(\024)h FD(N)22 b Fm(\000)g FD(1)p Fu(;)-180 3811 y(where)28 b Ft(c)138 3826 y Fc(m)p Fs(;)p Fc(i)280 3811 y Fm(6)p Fu(=)g(0)e(for)h(all)e FD(i)p Fu(.)41 b(W)-8 b(e)27 b(then)h(argue)f(recursiv)m(ely)h(as)f(in)g(the)g(pro)s(of)f(of) h([AS1,)h(Lemma)e(3.3])h(to)f(conclude)-180 3951 y(that)g(the)g(elemen) m(ts)g Ft(h)634 3915 y Fc(m)706 3951 y Fu(,)h FD(m)g Fm(\024)i FD(N)8 b Fm(\000)g FD(1)p Fu(,)28 b(are)d(linearly)f(indep)s (enden)m(t;)30 b(hence)d(the)f(dimension)e(of)i(the)g(subalgebra)-180 4090 y(of)32 b Ft(R)5 4105 y Fs(n)85 4090 y Fu(spanned)i(b)m(y)f Ft(h)656 4105 y Fn(1)696 4090 y Ft(;)17 b(:)g(:)g(:)f(;)h(h)971 4105 y Fs(n)1050 4090 y Fu(is)32 b Fm(\025)c Ft(N)1341 4054 y Fs(n)1389 4090 y Fu(,)k(as)h(claimed.)-80 4281 y(W)-8 b(e)43 b(can)g(no)m(w)g(\014nish)g(the)g(pro)s(of)f(of)h(the)g (Theorem.)74 b(Since)43 b(dim)15 b Ft(Z)2505 4296 y Fs(n)2597 4281 y Fm(\024)46 b Ft(N)2818 4211 y Fi(n)p Fg(\()p Fi(n)p Fg(+1\))p 2818 4229 202 3 v 2904 4271 a(2)3076 4281 y Fu(b)m(y)e(Lemma)e(6.12)g(and)-180 4421 y(dim)15 b Ft(Z)66 4436 y Fs(n)p Fk(\000)p Fn(1)238 4421 y Fu(=)35 b Ft(N)448 4350 y Fi(n)p Fg(\()p Fi(n)p Fh(\000)p Fg(1\))p 448 4369 204 3 v 534 4410 a(2)702 4421 y Fu(b)m(y)j(the)f(induction)f(h)m(yp)s (othesis,)k(w)m(e)e(ha)m(v)m(e)g(dim)15 b Ft(R)2596 4436 y Fs(n)2678 4421 y Fm(\024)36 b Ft(N)2879 4385 y Fs(n)2927 4421 y Fu(.)56 b(By)38 b(what)f(w)m(e)h(ha)m(v)m(e)g(just)-180 4560 y(seen,)33 b(this)d(dimension)g(is)h(exactly)g Ft(N)1225 4524 y Fs(n)1272 4560 y Fu(.)43 b(Therefore,)33 b(dim)15 b Ft(Z)2056 4575 y Fs(n)2130 4560 y Fu(=)28 b Ft(N)2333 4490 y Fi(n)p Fg(\()p Fi(n)p Fg(+1\))p 2333 4508 202 3 v 2418 4549 a(2)2548 4560 y Fu(;)k(in)e(presence)j(of)e(Lemma)e (6.12,)i(this)-180 4700 y(implies)f(the)j(Theorem.)p 3883 4700 4 66 v 3887 4637 59 4 v 3887 4700 V 3945 4700 4 66 v -180 4916 a FD(Theorem)k(6.14.)49 b Fo(The)35 b(Nichols)f(algebr)-5 b(a)34 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\))35 b Fo(c)-5 b(an)34 b(b)-5 b(e)35 b(pr)-5 b(esente)g(d)34 b(by)i(gener)-5 b(ators)34 b Ft(e)3100 4931 y Fs(i;i)p Fn(+1)3262 4916 y Fo(,)h Fu(1)28 b Fm(\024)g Ft(i)h Fm(\024)f Ft(n)p Fo(,)35 b(and)-180 5056 y(r)-5 b(elations)42 b Fu(\(6.12\))p Fo(,)35 b Fu(\(6.13\))o Fo(,)g Fu(\(6.14\))f Fo(and)44 b Fu(\(6.28\))p Fo(.)p eop %%Page: 51 51 51 50 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(51)-180 203 y Fo(Pr)-5 b(o)g(of.)41 b Fu(Let)27 b Fj(B)375 167 y Fk(0)425 203 y Fu(b)s(e)g(the)g(algebra)f(presen)m(ted)i(b)m(y)g (generators)f Ft(e)2121 218 y Fs(i;i)p Fn(+1)2283 203 y Fu(,)h(1)g Fm(\024)g Ft(i)g Fm(\024)g Ft(n)p Fu(,)g(and)f(relations)e (\(6.12\),)i(\(6.13\),)-180 342 y(\(6.14\))e(and)g(\(6.28\))o(.)41 b(W)-8 b(e)26 b(claim)d(that)i Fj(B)1300 306 y Fk(0)1349 342 y Fu(is)g(is)g(a)g(braided)g(Hopf)g(algebra)g(with)g(the)g Ft(e)2936 357 y Fs(i;i)p Fn(+1)3099 342 y Fu('s)g(primitiv)m(e.)39 b(Indeed,)-180 482 y(the)33 b(claim)d(follo)m(ws)h(without)i (di\016cult)m(y;)f(use)i(Lemma)d(6.9)h(for)g(relations)f(\(6.28\))o(.) -80 621 y(By)23 b(Lemma)f(6.12,)j(w)m(e)f(see)g(that)f(the)h(monomials) c(\(6.29\))i(span)i Fj(B)2329 585 y Fk(0)2376 621 y Fu(as)f(v)m(ector)h (space,)j(and)c(in)g(particular)e(that)-180 762 y(dim)15 b Fj(B)87 726 y Fk(0)145 762 y Fm(\024)35 b Ft(N)355 692 y Fi(n)p Fg(\()p Fi(n)p Fg(+1\))p 355 710 202 3 v 441 752 a(2)571 762 y Fu(.)56 b(By)37 b(Lemmas)e(6.3)h(and)h(6.11,)g (there)g(is)f(a)h(surjectiv)m(e)g(algebra)f(map)f Ft( )k Fu(:)34 b Fj(B)3492 726 y Fk(0)3550 762 y Fm(!)g Fj(B)p Fu(\()p Ft(V)22 b Fu(\).)-180 902 y(By)33 b(Theorem)g(6.13,)f Ft( )37 b Fu(is)32 b(an)g(isomorphism.)p 3883 902 4 66 v 3887 839 59 4 v 3887 902 V 3945 902 4 66 v -180 1077 a(6.2.)56 b FD(Lifting)36 b(of)i(Nic)m(hols)d(algebras)j(of)g(t)m(yp)s (e)f Ft(A)1806 1092 y Fs(n)1853 1077 y FD(.)-80 1193 y Fu(W)-8 b(e)41 b(\014x)h(in)e(this)h(Section)g(a)g(\014nite)g Fo(ab)-5 b(elian)40 b Fu(group)h(\000)g(suc)m(h)i(that)e(our)g(braided) g(v)m(ector)h(space)g Ft(V)63 b Fu(can)42 b(b)s(e)-180 1318 y(realized)37 b(in)302 1282 y Fn(\000)302 1343 y(\000)350 1318 y Fm(Y)8 b(D)r Fu(,)40 b(as)d(in)g(Remark)h(6.1.)58 b(That)38 b(is,)h(w)m(e)g(ha)m(v)m(e)g Ft(g)2204 1333 y Fn(1)2243 1318 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)2509 1333 y Fs(n)2593 1318 y Fu(in)37 b(\000,)i Ft(\037)2900 1333 y Fn(1)2940 1318 y Ft(;)17 b(:)g(:)g(:)e(;)i(\037)3219 1333 y Fs(n)3304 1318 y Fu(in)3426 1293 y Fl(b)3423 1318 y Fu(\000,)39 b(suc)m(h)g(that)-180 1434 y Ft(q)-137 1449 y Fs(ij)-49 1434 y Fu(=)28 b Fm(h)p Ft(\037)155 1449 y Fs(j)191 1434 y Ft(;)17 b(g)282 1449 y Fs(i)310 1434 y Fm(i)32 b Fu(for)g(all)f(i,j,)g(and)i Ft(V)54 b Fu(can)33 b(b)s(e)g(realized)e(as)i(a)f(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)j(\000)e(b)m(y)j(\(6.6\).)-80 1591 y(W)-8 b(e)41 b(also)f(\014x)i(a)f(\014nite)g(dimensional)e(p)s (oin)m(ted)i(Hopf)g(algebra)f Ft(A)h Fu(suc)m(h)i(that)e Ft(G)p Fu(\()p Ft(A)p Fu(\))g(is)g(isomorphic)e(to)i(\000,)-180 1708 y(and)31 b(the)h(in\014nitesimal)c(braiding)i(of)g Ft(A)i Fu(is)e(isomorphic)g(to)h Ft(V)53 b Fu(as)31 b(Y)-8 b(etter-Drinfeld)29 b(mo)s(dule)h(o)m(v)m(er)j(\000.)43 b(That)31 b(is,)-180 1824 y(gr)16 b Ft(A)29 b Fm(')h Ft(R)q Fu(#)p Fp(|)-9 b Fu(\000,)28 b(and)33 b(the)h(subalgebra)f Ft(R)1388 1788 y Fk(0)1445 1824 y Fu(of)g Ft(R)h Fu(generated)g(b)m(y)h Ft(R)q Fu(\(1\))e(is)f(isomorphic)g(to)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\).)46 b(W)-8 b(e)34 b(c)m(ho)s(ose)-180 1940 y(elemen)m(ts)f Ft(a)270 1955 y Fs(i)326 1940 y Fm(2)28 b Fu(\()p Ft(A)531 1955 y Fn(1)571 1940 y Fu(\))609 1904 y Fs(\037)653 1914 y Fi(i)609 1965 y Fs(g)643 1975 y Fi(i)715 1940 y Fu(suc)m(h)34 b(that)f Ft(\031)t Fu(\()p Ft(a)1295 1955 y Fs(i)1323 1940 y Fu(\))28 b(=)f Ft(x)1547 1955 y Fs(i)1576 1940 y Fu(,)32 b(1)c Fm(\024)g Ft(i)g Fm(\024)g Ft(n)p Fu(.)-80 2097 y(W)-8 b(e)33 b(shall)e(consider,)i (more)f(generally)-8 b(,)31 b(Hopf)i(algebras)f Ft(H)40 b Fu(pro)m(vided)33 b(with)-54 2234 y Fm(\017)41 b Fu(a)32 b(group)h(isomorphism)d(\000)d Fm(!)h Ft(G)p Fu(\()p Ft(H)8 b Fu(\);)-54 2391 y Fm(\017)41 b Fu(elemen)m(ts)33 b Ft(a)487 2406 y Fn(1)527 2391 y Ft(;)17 b(:)g(:)g(:)e(;)i(a)796 2406 y Fs(n)876 2391 y Fu(in)32 b Fm(P)8 b Fu(\()p Ft(H)g Fu(\))1232 2344 y Fs(\037)1276 2354 y Fi(i)1232 2416 y Fs(g)1266 2426 y Fi(i)1291 2416 y Fs(;)p Fn(1)1350 2391 y Fu(.)-180 2528 y(F)-8 b(urther)36 b(h)m(yp)s(otheses)j(on)d Ft(H)43 b Fu(will)34 b(b)s(e)j(stated)f(when)i(needed.)55 b(The)38 b(examples)e(of)f(suc)m(h)j Ft(H)44 b Fu(w)m(e)37 b(are)f(thinking)-180 2644 y(of)d(are)h(the)g(Hopf)g(algebra)f Ft(A)p Fu(,)h(and)g(an)m(y)g(b)s(osonization)e Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000,)28 b(where)35 b Ft(R)g Fu(is)e(an)m(y)i (braided)e(Hopf)h(algebra)f(in)-180 2724 y Fn(\000)-180 2786 y(\000)-132 2761 y Fm(Y)8 b(D)35 b Fu(pro)m(vided)e(with)g(a)f (monomorphism)e(of)i(Y)-8 b(etter-Drinfeld)31 b(mo)s(dules)h Ft(V)50 b Fm(!)27 b Ft(P)14 b Fu(\()p Ft(R)q Fu(\);)32 b(so)h(that)g Ft(a)3526 2776 y Fs(i)3582 2761 y Fu(:=)28 b Ft(x)3768 2776 y Fs(i)3797 2761 y Fu(#1,)-180 2892 y(1)f Fm(\024)i Ft(i)f Fm(\024)g Ft(n)p Fu(.)43 b(This)33 b(includes)g(notably)f(the)h(Hopf)f(algebras)g Ft(T)14 b Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000,)2534 2866 y Fl(b)2517 2892 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000,)27 b Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000.)-80 3023 y(Here)169 2997 y Fl(b)152 3023 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))35 b(b)s(e)g(the)g(braided)f (Hopf)h(algebra)f(in)1790 2987 y Fn(\000)1790 3048 y(\000)1838 3023 y Fm(Y)8 b(D)37 b Fu(generated)f(b)m(y)g Ft(x)2667 3038 y Fn(1)2707 3023 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2981 3038 y Fs(\022)3055 3023 y Fu(with)34 b(relations)f(\(6.12\),)-180 3139 y(\(6.13\))f(and)g(\(6.14\).)-80 3297 y(W)-8 b(e)33 b(in)m(tro)s(duce)f(inductiv)m(ely)g(the)h(follo)m(wing)d(elemen)m(ts)j (of)f Ft(H)8 b Fu(:)119 3454 y Ft(E)191 3469 y Fs(i;i)p Fn(+1)381 3454 y Fu(:=)28 b Ft(a)563 3469 y Fs(i)591 3454 y Fu(;)-798 b(\(6.30\))119 3675 y Ft(E)191 3690 y Fs(i;j)299 3675 y Fu(:=)28 b(ad)16 b(\()p Ft(E)659 3690 y Fs(i;j)t Fk(\000)p Fn(1)829 3675 y Fu(\)\()p Ft(E)977 3690 y Fs(j)t Fk(\000)p Fn(1)p Fs(;j)1156 3675 y Fu(\))p Ft(;)114 b Fu(1)28 b Fm(\024)g Ft(i)g(<)g(j)33 b Fm(\024)28 b Ft(n)23 b Fu(+)f(1)p Ft(;)33 b(j)28 b Fm(\000)23 b Ft(i)k Fm(\025)i Fu(2)p Ft(:)-2738 b Fu(\(6.31\))-180 3840 y(Assume)32 b(that)e Ft(H)35 b Fu(=)28 b Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)24 b(as)31 b(ab)s(o)m(v)m(e.)44 b(Then,)32 b(b)m(y)g(the)f(relations)e(b)s(et)m(w)m(een)k(braided)e (comm)m(utators)f(and)g(the)-180 3956 y(adjoin)m(t)i(\(1.21\))o(,)g (the)h(relations)f(\(6.12\))o(,)h(\(6.13\))e(and)i(\(6.14\))f (translate)g(resp)s(ectiv)m(ely)i(to)798 4116 y(ad)17 b Ft(E)990 4131 y Fs(i;i)p Fn(+1)1152 4116 y Fu(\()p Ft(E)1262 4131 y Fs(p;p)p Fn(+1)1447 4116 y Fu(\))28 b(=)f(0;)212 b(1)27 b Fm(\024)h Ft(i)g(<)g(p)f Fm(\024)i Ft(n;)212 b(p)22 b Fm(\000)g Ft(i)28 b Fm(\025)g Fu(2;)-3321 b(\(6.32\))616 4284 y(\(ad)16 b Ft(E)845 4299 y Fs(i;i)p Fn(+1)1007 4284 y Fu(\))1045 4243 y Fn(2)1085 4284 y Fu(\()p Ft(E)1195 4299 y Fs(i)p Fn(+1)p Fs(;i)p Fn(+2)1447 4284 y Fu(\))28 b(=)f(0)p Ft(;)212 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(n)p Fu(;)-2515 b(\(6.33\))616 4453 y(\(ad)16 b Ft(E)845 4468 y Fs(i)p Fn(+1)p Fs(;i)p Fn(+2)1097 4453 y Fu(\))1135 4412 y Fn(2)1175 4453 y Fu(\()p Ft(E)1285 4468 y Fs(i;i)p Fn(+1)1447 4453 y Fu(\))28 b(=)f(0)p Ft(;)212 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(n:)-2515 b Fu(\(6.34\))-180 4662 y FD(Remark)37 b(6.15.)49 b Fu(Relations)26 b(\(6.32\))o(,)j(\(6.33\))e(and)g(\(6.34\))g(can)h(b)s(e)g(considered,) h(more)e(generally)-8 b(,)28 b(in)e(an)m(y)j Ft(H)35 b Fu(as)-180 4802 y(ab)s(o)m(v)m(e.)49 b(If)34 b(these)h(relations)e (hold)g(in)h Ft(H)8 b Fu(,)34 b(then)g(w)m(e)h(ha)m(v)m(e)h(a)e(Hopf)g (algebra)f(map)g Ft(\031)2905 4817 y Fs(H)3003 4802 y Fu(:)3076 4776 y Fl(b)3060 4802 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000)24 b Fm(!)30 b Ft(H)8 b Fu(.)48 b(On)-180 4941 y(the)37 b(other)f(hand,)i(w)m(e)f(kno)m(w)g(b)m (y)g(Remark)f(6.10)g(that)g(the)h(com)m(ultiplication)31 b(of)36 b(the)h(elemen)m(ts)f Ft(E)3526 4905 y Fs(N)3520 4966 y(ij)3630 4941 y Fu(is)g(giv)m(en)-180 5081 y(b)m(y)c(\(6.27\))o (.)43 b(Hence,)31 b(the)f(same)g(form)m(ula)d(is)j(v)-5 b(alid)27 b(in)i Ft(H)8 b Fu(,)30 b(pro)m(vided)g(that)f(relations)g (\(6.32\))o(,)h(\(6.33\))f(and)h(\(6.34\))p eop %%Page: 52 52 52 51 bop -180 0 a Fq(52)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(hold)25 b(in)g(it.)40 b(In)26 b(particular,)g(the)g(subalgebra)f(of)h(H)g(generated)g(b)m(y)h (the)f(elemen)m(ts)g Ft(E)2900 167 y Fs(N)2894 228 y(ij)2967 203 y Fu(,)h Ft(g)3072 167 y Fs(N)3068 228 y(i;j)3148 203 y Fu(,)h(1)f Fm(\024)h Ft(i)g(<)g(j)33 b Fm(\024)28 b Ft(n)8 b Fu(+)g(1,)-180 342 y(is)32 b(a)g(Hopf)h(subalgebra)f(of)g Ft(H)8 b Fu(.)-180 552 y FD(Lemma)37 b(6.16.)49 b Fo(R)-5 b(elations)43 b Fu(\(6.32\))o Fo(,)35 b Fu(\(6.33\))f Fo(and)45 b Fu(\(6.34\))34 b Fo(hold)g(in)h Ft(A)g Fo(if)f Ft(N)k(>)28 b Fu(3)p Fo(.)-180 762 y(Pr)-5 b(o)g(of.)41 b Fu(This)31 b(is)g(a)f(particular)g(case)h(of)g(Theorem)g(5.6;)g(w)m (e)h(include)f(the)g(pro)s(of)f(for)g(completeness.)44 b(W)-8 b(e)32 b(kno)m(w,)-180 902 y(b)m(y)h(Lemma)f(2.10,)g(that)513 1083 y(ad)17 b Ft(E)705 1098 y Fs(i;i)p Fn(+1)867 1083 y Fu(\()p Ft(E)977 1098 y Fs(p;p)p Fn(+1)1162 1083 y Fu(\))27 b Fm(2)i(P)1391 1098 y Fs(g)1425 1108 y Fi(i)1451 1098 y Fs(g)1485 1106 y Fi(p)1520 1098 y Fs(;)p Fn(1)1580 1083 y Fu(\()p Ft(A)p Fu(\))1729 1041 y Fs(\037)1773 1051 y Fi(i)1799 1041 y Fs(\037)1843 1049 y Fi(p)1883 1083 y Ft(;)212 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(p)f Fm(\024)i Ft(n;)212 b(p)22 b Fm(\000)g Ft(i)28 b Fm(\025)g Fu(2)p Ft(;)398 1263 y Fu(\(ad)16 b Ft(E)627 1278 y Fs(i;i)p Fn(+1)789 1263 y Fu(\))827 1221 y Fn(2)867 1263 y Fu(\()p Ft(E)977 1278 y Fs(p;p)p Fn(+1)1162 1263 y Fu(\))27 b Fm(2)i(P)1391 1283 y Fs(g)1427 1260 y Fg(2)1425 1305 y Fi(i)1461 1283 y Fs(g)1495 1291 y Fi(p)1531 1283 y Fs(;)p Fn(1)1590 1263 y Fu(\()p Ft(A)p Fu(\))1739 1221 y Fs(\037)1783 1198 y Fg(2)1783 1243 y Fi(i)1818 1221 y Fs(\037)1862 1229 y Fi(p)1902 1263 y Ft(;)211 b Fu(1)28 b Fm(\024)g Ft(i;)17 b(p)28 b Fm(\024)g Ft(n;)212 b Fm(j)p Ft(p)22 b Fm(\000)g Ft(i)p Fm(j)28 b Fu(=)f(1)p Ft(:)-80 1444 y Fu(Assume)35 b(that)f(ad)17 b Ft(E)688 1459 y Fs(i;i)p Fn(+1)850 1444 y Fu(\()p Ft(E)960 1459 y Fs(p;p)p Fn(+1)1145 1444 y Fu(\))31 b Fm(6)p Fu(=)g(0,)k(and)f Ft(\037)1684 1459 y Fs(i)1713 1444 y Ft(\037)1774 1459 y Fs(p)1845 1444 y Fm(6)p Fu(=)c Ft(")p Fu(,)35 b(where)h(1)31 b Fm(\024)g Ft(i)h(<)f(p)g Fm(\024)g Ft(n)p Fu(,)36 b Ft(p)23 b Fm(\000)h Ft(i)31 b Fm(\025)h Fu(2.)49 b(By)35 b(Lemma)-180 1583 y(5.1,)d(there)i(exists)f Ft(`)p Fu(,)f(1)c Fm(\024)g Ft(`)g Fm(\024)g Ft(n)p Fu(,)33 b(suc)m(h)h(that)e Ft(g)1575 1598 y Fs(i)1603 1583 y Ft(g)1650 1598 y Fs(p)1717 1583 y Fu(=)c Ft(g)1868 1598 y Fs(`)1900 1583 y Fu(,)33 b Ft(\037)2021 1598 y Fs(i)2049 1583 y Ft(\037)2110 1598 y Fs(p)2178 1583 y Fu(=)27 b Ft(\037)2342 1598 y Fs(`)2375 1583 y Fu(.)44 b(But)32 b(then)982 1764 y Ft(q)g Fu(=)27 b Ft(\037)1221 1779 y Fs(`)1254 1764 y Fu(\()p Ft(g)1339 1779 y Fs(`)1372 1764 y Fu(\))g(=)h Ft(\037)1602 1779 y Fs(i)1630 1764 y Fu(\()p Ft(g)1715 1779 y Fs(i)1743 1764 y Fu(\))p Ft(\037)1842 1779 y Fs(i)1870 1764 y Fu(\()p Ft(g)1955 1779 y Fs(p)1995 1764 y Fu(\))p Ft(\037)2094 1779 y Fs(p)2133 1764 y Fu(\()p Ft(g)2218 1779 y Fs(i)2246 1764 y Fu(\))p Ft(\037)2345 1779 y Fs(p)2385 1764 y Fu(\()p Ft(g)2470 1779 y Fs(p)2509 1764 y Fu(\))g(=)f Ft(q)2725 1723 y Fn(2)2765 1764 y Ft(:)-180 1945 y Fu(Hence)34 b Ft(q)d Fu(=)d(1,)k(a)h(con)m(tradiction.)-80 2085 y(Assume)c(next)h (that)f(ad)16 b Ft(E)898 2048 y Fn(2)892 2109 y Fs(i;i)p Fn(+1)1054 2085 y Fu(\()p Ft(E)1164 2100 y Fs(p;p)p Fn(+1)1349 2085 y Fu(\))28 b Fm(6)p Fu(=)g(0,)h Fm(j)p Ft(p)15 b Fm(\000)g Ft(i)p Fm(j)27 b Fu(=)h(1.)58 b(and)29 b Ft(\037)2381 2048 y Fn(2)2381 2109 y Fs(i)2420 2085 y Ft(\037)2481 2100 y Fs(p)2549 2085 y Fm(6)p Fu(=)e Ft(")p Fu(.)42 b(By)30 b(Lemma)d(5.1)i(,)g(there)h(exists)-180 2224 y Ft(`)p Fu(,)j(1)27 b Fm(\024)h Ft(`)g Fm(\024)g Ft(n)p Fu(,)33 b(suc)m(h)h(that)e Ft(g)876 2188 y Fn(2)872 2249 y Fs(i)915 2224 y Ft(g)962 2239 y Fs(p)1029 2224 y Fu(=)27 b Ft(g)1179 2239 y Fs(`)1212 2224 y Fu(,)33 b Ft(\037)1333 2188 y Fn(2)1333 2249 y Fs(i)1372 2224 y Ft(\037)1433 2239 y Fs(p)1501 2224 y Fu(=)27 b Ft(\037)1665 2239 y Fs(`)1698 2224 y Fu(.)44 b(But)33 b(then)923 2405 y Ft(q)e Fu(=)d Ft(\037)1162 2420 y Fs(`)1195 2405 y Fu(\()p Ft(g)1280 2420 y Fs(`)1313 2405 y Fu(\))f(=)h Ft(\037)1543 2420 y Fs(i)1571 2405 y Fu(\()p Ft(g)1656 2420 y Fs(i)1684 2405 y Fu(\))1722 2364 y Fn(4)1761 2405 y Ft(\037)1822 2420 y Fs(i)1851 2405 y Fu(\()p Ft(g)1936 2420 y Fs(p)1975 2405 y Fu(\))2013 2364 y Fn(2)2052 2405 y Ft(\037)2113 2420 y Fs(p)2153 2405 y Fu(\()p Ft(g)2238 2420 y Fs(i)2266 2405 y Fu(\))2304 2364 y Fn(2)2343 2405 y Ft(\037)2404 2420 y Fs(p)2444 2405 y Fu(\()p Ft(g)2529 2420 y Fs(p)2568 2405 y Fu(\))g(=)g Ft(q)2785 2364 y Fn(3)2824 2405 y Ft(:)-180 2586 y Fu(Hence)34 b Ft(q)d Fu(=)d Fm(\006)p Fu(1,)33 b(a)f(con)m(tradiction)g(\(w)m(e)h(assumed)g Ft(N)39 b(>)27 b Fu(2\).)-80 2725 y(It)35 b(remains)f(to)h(exclude)h (the)f(cases)i Ft(\037)1347 2740 y Fs(i)1375 2725 y Ft(\037)1436 2740 y Fs(p)1508 2725 y Fu(=)32 b Ft(";)17 b Fm(j)p Ft(p)23 b Fm(\000)h Ft(i)p Fm(j)32 b(\025)g Fu(2,)k(and)f Ft(\037)2474 2689 y Fn(2)2474 2750 y Fs(i)2514 2725 y Ft(\037)2575 2740 y Fs(p)2647 2725 y Fu(=)c Ft(";)17 b Fm(j)p Ft(p)23 b Fm(\000)i Ft(i)p Fm(j)32 b Fu(=)f(1.)51 b(The)36 b(\014rst)g(case) -180 2865 y(leads)f(to)f(the)i(con)m(tradiction)d Ft(N)43 b Fu(=)31 b(3.)50 b(In)35 b(the)h(second)g(case)g(it)e(follo)m(ws)f (from)h(the)h(connectivit)m(y)h(of)e Ft(A)3693 2880 y Fs(n)3775 2865 y Fu(that)-180 3004 y Ft(N)43 b Fu(w)m(ould)33 b(divide)e(2)i(whic)m(h)g(is)f(also)g(imp)s(ossible.)p 3883 3004 4 66 v 3887 2942 59 4 v 3887 3004 V 3945 3004 4 66 v -180 3214 a FD(Lemma)37 b(6.17.)49 b Fo(If)35 b Ft(H)g Fu(=)27 b Ft(A)p Fo(,)35 b(then)g Ft(E)1251 3178 y Fs(N)1245 3239 y(i;j)1353 3214 y Fm(2)28 b Fp(|)-9 b Fu(\000)1569 3178 y Fs(N)1631 3214 y Fo(,)34 b(for)h(any)g Fu(1)27 b Fm(\024)h Ft(i)g(<)g(j)34 b Fm(\024)28 b Ft(n)22 b Fu(+)g(1)p Fo(.)-180 3424 y(Pr)-5 b(o)g(of.)41 b Fu(W)-8 b(e)36 b(\014rst)h(sho)m(w)g(that)e Ft(E)1031 3388 y Fs(N)1025 3449 y(i;j)1139 3424 y Fm(2)e Fp(|)-8 b Fu(\000,)30 b(1)j Fm(\024)h Ft(i)f(<)g(j)40 b Fm(\024)33 b Ft(n)25 b Fu(+)f(1.)53 b(\(F)-8 b(or)35 b(our)h(further)g(purp)s(oses,)i(this)d (is)g(what)-180 3564 y(w)m(e)f(really)d(need\).)-80 3703 y(Let)36 b Ft(i)e(<)g(j)6 b Fu(.)55 b(W)-8 b(e)37 b(claim)d(that)i (there)h(exists)g(no)f Ft(`)p Fu(,)h(1)d Fm(\024)h Ft(`)f Fm(\024)g Ft(n)p Fu(,)k(suc)m(h)f(that)f Ft(g)2817 3667 y Fs(N)2813 3728 y(i;j)2927 3703 y Fu(=)e Ft(g)3084 3718 y Fs(`)3117 3703 y Fu(,)j Ft(\037)3242 3667 y Fs(N)3242 3728 y(i;j)3357 3703 y Fu(=)d Ft(\037)3528 3718 y Fs(`)3561 3703 y Fu(.)54 b(Indeed,)-180 3842 y(otherwise)33 b(w)m(e)h(w)m(ould)e (ha)m(v)m(e)1141 4023 y Ft(q)g Fu(=)27 b Ft(\037)1380 4038 y Fs(`)1413 4023 y Fu(\()p Ft(g)1498 4038 y Fs(`)1531 4023 y Fu(\))g(=)h Ft(\037)1761 4038 y Fs(i;j)1841 4023 y Fu(\()p Ft(g)1926 4038 y Fs(i;j)2006 4023 y Fu(\))2044 3982 y Fs(N)2107 3959 y Fg(2)2173 4023 y Fu(=)g Ft(q)2324 3982 y Fs(N)2387 3959 y Fg(2)2453 4023 y Fu(=)g(1)p Ft(:)-80 4204 y Fu(By)33 b(Lemma)e(6.16)h(and)h(Remark)f(6.15,)g(w)m(e)h(ha)m(v) m(e)349 4388 y(\001\()p Ft(E)546 4347 y Fs(N)540 4413 y(i;j)620 4388 y Fu(\))28 b(=)f Ft(E)867 4347 y Fs(N)861 4413 y(i;j)964 4388 y Fm(\012)c Fu(1)e(+)h Ft(g)1283 4347 y Fs(N)1279 4413 y(i;j)1381 4388 y Fm(\012)h Ft(E)1559 4347 y Fs(N)1553 4413 y(i;j)1656 4388 y Fu(+)f(\(1)f Fm(\000)i Ft(q)2009 4347 y Fk(\000)p Fn(1)2103 4388 y Fu(\))2141 4347 y Fs(N)2254 4294 y Fl(X)2225 4504 y Fs(i)g Fu(1.)42 b(By)32 b(the)g(recursion)g(h)m(yp)s(othesis,)h(\001\()p Ft(E)3738 4901 y Fs(N)3732 4961 y(i;j)3812 4937 y Fu(\))28 b(=)-180 5076 y Ft(E)-102 5040 y Fs(N)-108 5101 y(i;j)-4 5076 y Fm(\012)c Fu(1)f(+)h Ft(g)320 5040 y Fs(N)316 5101 y(i;j)419 5076 y Fm(\012)g Ft(E)598 5040 y Fs(N)592 5101 y(i;j)696 5076 y Fu(+)g Ft(u)p Fu(,)34 b(for)h(some)f Ft(u)d Fm(2)h Fp(|)-9 b Fu(\000)18 b Fm(\012)24 b Fp(|)-9 b Fu(\000.)44 b(In)35 b(particular,)e(w)m(e)j(see)g(that)e Ft(E)3130 5040 y Fs(N)3124 5101 y(i;j)3236 5076 y Fm(2)e Fu(\()p Ft(A)3445 5091 y Fn(1)3484 5076 y Fu(\))3522 5040 y Fs(\037)3566 5017 y Fi(N)3566 5062 y(i)3628 5076 y Fu(.)50 b(Then,)p eop %%Page: 53 53 53 52 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(53)-180 203 y Fu(b)m(y)39 b(Lemma)e(5.1,)i(either)f Ft(\037)849 167 y Fs(N)849 228 y(i)953 203 y Fu(=)f Ft(")g Fu(and)h(hence)i Ft(E)1699 167 y Fs(N)1693 228 y(i;i)p Fn(+1)1892 203 y Fm(2)d Fp(|)-8 b Fu(\000,)33 b(or)38 b(else)g Ft(\037)2554 167 y Fs(N)2554 228 y(i)2659 203 y Fm(6)p Fu(=)e Ft(")p Fu(,)j(whic)m(h)g(implies)d Ft(u)g Fu(=)h(0)h(and)-180 342 y Ft(E)-102 306 y Fs(N)-108 367 y(i;i)p Fn(+1)82 342 y Fm(2)28 b(P)245 365 y Fs(g)281 342 y Fi(N)279 387 y(i)339 365 y Fs(;)p Fn(1)398 342 y Fu(\()p Ft(A)p Fu(\))547 306 y Fs(\037)591 283 y Fi(N)591 328 y(i)653 342 y Fu(.)43 b(Again,)32 b(this)g(last)g(p)s(ossibilit)m (y)e(con)m(tradicts)j(the)g(claim)d(ab)s(o)m(v)m(e.)-80 482 y(Finally)-8 b(,)35 b(let)h Ft(C)43 b Fu(b)s(e)37 b(the)h(subalgebra)e(of)g Ft(A)h Fu(generated)h(b)m(y)f(the)h(elemen)m (ts)f Ft(E)2803 446 y Fs(N)2797 507 y(ij)2870 482 y Fu(,)h Ft(g)2986 446 y Fs(N)2982 507 y(i;j)3062 482 y Fu(,)f(1)e Fm(\024)g Ft(i)g(<)g(j)41 b Fm(\024)35 b Ft(n)25 b Fu(+)g(1,)-180 621 y(whic)m(h)33 b(is)f(a)g(Hopf)h(subalgebra)f(of)g Ft(H)8 b Fu(.)43 b(Since)33 b Ft(E)1608 585 y Fs(N)1602 646 y(i;j)1710 621 y Fm(2)28 b Fp(|)-9 b Fu(\000)17 b Fm(\\)22 b Ft(C)7 b Fu(,)33 b(w)m(e)g(conclude)g(that)g Ft(E)3002 585 y Fs(N)2996 646 y(i;j)3104 621 y Fm(2)28 b Ft(C)3268 636 y Fn(0)3335 621 y Fu(=)f Fp(|)-8 b Fu(\000)3561 585 y Fs(N)3622 621 y Fu(.)p 3883 621 4 66 v 3887 559 59 4 v 3887 621 V 3945 621 4 66 v -80 793 a(T)g(o)22 b(solv)m(e)g(the)g(lifting)d(problem,)k(w)m(e)g(see)g(from)d(Lemma)h (6.17)g(that)h(w)m(e)g(\014rst)h(ha)m(v)m(e)g(to)e(answ)m(er)j(a)d(com) m(binatorial)-180 909 y(question)33 b(in)f(the)h(group)f(algebra)g(of)g (an)g(ab)s(elian)f(group.)43 b(T)-8 b(o)33 b(simplify)d(the)j(notation) e(w)m(e)j(de\014ne)876 1153 y Ft(h)932 1168 y Fs(ij)1021 1153 y Fu(=)27 b Ft(g)1175 1112 y Fs(N)1171 1177 y(i;j)1251 1153 y Ft(;)212 b(C)1567 1105 y Fs(j)1560 1178 y(i;p)1671 1153 y Fu(=)27 b(\(1)22 b Fm(\000)h Ft(q)2030 1112 y Fk(\000)p Fn(1)2124 1153 y Fu(\))2162 1112 y Fs(N)2246 1072 y Fl(\000)2291 1153 y Ft(B)2370 1105 y Fs(p;j)2365 1178 y(i;p)2462 1072 y Fl(\001)2508 1088 y Fs(N)7 b Fn(\()p Fs(N)g Fk(\000)p Fn(1\))p Fs(=)p Fn(2)2870 1153 y Ft(:)-80 1299 y Fu(W)-8 b(e)33 b(are)f(lo)s(oking)f(for)h(families)d(\()p Ft(u)1190 1314 y Fs(ij)1250 1299 y Fu(\))1288 1314 y Fn(1)p Fk(\024)p Fs(i)c Fu(2.)42 b(W)-8 b(e)30 b(assume)g(that)g Ft(\015)2118 736 y Fs(ab)2217 721 y Fm(2)e Fp(|)-9 b Ft(;)17 b Fu(1)k Fm(\024)28 b Ft(a)g(<)g(b)g Fm(\024)g Ft(n)16 b Fu(+)g(1)p Ft(;)h(b)f Fm(\000)g Ft(a)28 b Fm(\024)g Ft(k)19 b Fm(\000)d Fu(1)p Ft(;)30 b Fu(is)f(a)-180 861 y(family)f(of)i(scalars)h(with)f Ft(\015)810 876 y Fs(ab)909 861 y Fu(=)e(0,)i(if)g Ft(h)1263 876 y Fs(ab)1362 861 y Fu(=)e(1,)j(and)g(that)f(the)h(family)d Ft(u)2488 876 y Fs(ab)2587 861 y Fm(2)g Fp(|)-9 b Fu(\000)p Ft(;)17 b Fu(1)22 b Fm(\024)28 b Ft(a)g(<)f(b)h Fm(\024)g Ft(n)19 b Fu(+)f(1)p Ft(;)f(b)h Fm(\000)g Ft(a)28 b Fm(\024)-180 1000 y Ft(k)d Fm(\000)e Fu(1)p Ft(;)33 b Fu(de\014ned)h(inductiv)m(ely) e(b)m(y)i(the)f Ft(\015)1297 1015 y Fs(ab)1401 1000 y Fu(b)m(y)i(\(6.40\))d(is)h(a)f(solution)f(of)40 b(\(6.36\))o(.)k(Let)33 b(1)28 b Fm(\024)g Ft(i)g(<)g(j)34 b Fm(\024)28 b Ft(n)23 b Fu(+)f(1)p Ft(;)33 b Fu(and)-180 1140 y Ft(j)28 b Fm(\000)23 b Ft(i)28 b Fu(=)f Ft(k)s(:)33 b Fu(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e (sho)m(w)i(that)820 1379 y(\001\()p Ft(u)995 1394 y Fs(ij)1055 1379 y Fu(\))28 b(=)f Ft(u)1280 1394 y Fs(ij)1362 1379 y Fm(\012)c Fu(1)f(+)g Ft(h)1687 1394 y Fs(ij)1770 1379 y Fm(\012)g Ft(u)1925 1394 y Fs(ij)2008 1379 y Fu(+)2134 1284 y Fl(X)2106 1494 y Fs(i)41 b Fu(2,)h(a)e(ro)s(ot)f(of)-180 4091 y(unit)m(y)e Ft(q)j Fu(of)d(order)g Ft(N)10 b Fu(,)38 b Ft(g)737 4106 y Fn(1)776 4091 y Ft(;)17 b(:)g(:)g(:)f(;)h(g)1042 4106 y Fs(n)1123 4091 y Fm(2)35 b Fu(\000)p Ft(;)17 b(\037)1390 4106 y Fn(1)1429 4091 y Ft(;)g(:)g(:)g(:)f(;)h(\037)1709 4106 y Fs(n)1791 4091 y Fm(2)1895 4066 y Fl(b)1892 4091 y Fu(\000)37 b(suc)m(h)h(that)e Ft(q)2472 4106 y Fs(ij)2568 4091 y Fu(=)f Ft(\037)2740 4106 y Fs(j)2776 4091 y Fu(\()p Ft(g)2861 4106 y Fs(i)2889 4091 y Fu(\))i(for)f(all)f Ft(i;)17 b(j)42 b Fu(satisfy)c(\(6.1\),)-180 4209 y(and)33 b Ft(V)49 b Fm(2)210 4173 y Fn(\000)210 4234 y(\000)258 4209 y Fm(Y)8 b(D)35 b Fu(with)d(basis)h Ft(x)962 4224 y Fs(i)1018 4209 y Fm(2)28 b Ft(V)1191 4173 y Fs(\037)1235 4183 y Fi(i)1169 4234 y Fs(g)1203 4244 y Fi(i)1265 4209 y Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)h Ft(n)p Fu(.)-80 4353 y(W)-8 b(e)45 b(recall)f(that)613 4327 y Fl(b)596 4353 y Fj(B)q Fu(\()p Ft(V)21 b Fu(\))45 b(is)f(the)i (braided)f(Hopf)f(algebra)g(in)2271 4317 y Fn(\000)2271 4378 y(\000)2319 4353 y Fm(Y)8 b(D)47 b Fu(generated)f(b)m(y)g Ft(x)3178 4368 y Fn(1)3218 4353 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)3492 4368 y Fs(n)3584 4353 y Fu(with)44 b(the)-180 4469 y(quan)m(tum)33 b(Serre)g(relations)e(\(6.12\),)h(\(6.13\))g(and)h(\(6.14\))o(.)-80 4600 y(In)50 4574 y Fl(b)33 4600 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\))i(w)m(e)h(consider)g(the)g(iterated)f(braided)g(comm)m(utators) f Ft(e)2291 4615 y Fs(i;j)2371 4600 y Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g(<)g(j)34 b Fm(\024)28 b Ft(n)5 b Fu(+)g(1)24 b(de\014ned)i(inductiv)m(ely)-180 4716 y(b)m(y)35 b(\(6.8\))d(b)s (eginning)f(with)h Ft(e)904 4731 y Fs(i;i)p Fn(+1)1094 4716 y Fu(=)c Ft(x)1253 4731 y Fs(i)1314 4716 y Fu(for)k(all)e Ft(i)p Fu(.)-80 4833 y(Let)i Fp(A)61 b Fu(b)s(e)33 b(the)g(set)g(of)f (all)e(families)g(\()p Ft(a)1343 4848 y Fs(i;j)1423 4833 y Fu(\))1461 4848 y Fn(1)p Fk(\024)p Fs(i)g Fu(\()p Ft(s;)17 b(t)p Fu(\).)50 b(If)35 b Ft(s)c(<)h(i)j Fu(and)f Ft(t)e Fu(=)f Ft(i)24 b Fu(+)g(1,)35 b(resp.)51 b Ft(s)31 b(<)h(i)j Fu(and)-180 3678 y Ft(i)s Fu(+)s(1)28 b Ft(<)f(t)p Fu(,)f(resp.)41 b Ft(s)28 b Fu(=)f Ft(i)d Fu(and)f Ft(t)28 b(<)f(i)s Fu(+)s(1,)e(then)f Ft(e)1498 3693 y Fs(s;t)1580 3678 y Ft(e)1625 3693 y Fs(i;i)p Fn(+1)1815 3678 y Fu(=)k Ft(\037)1980 3693 y Fs(i)2008 3678 y Fu(\()p Ft(g)2093 3693 y Fs(s;t)2175 3678 y Fu(\))p Ft(e)2258 3693 y Fs(i;i)p Fn(+1)2420 3678 y Ft(e)2465 3693 y Fs(s;t)2570 3678 y Fu(b)m(y)e(\(6.24\))o(,)f(resp.\(6.22\),resp.\(6.23\))q(.)-180 3817 y(Hence)44 b(in)e(all)f(cases,)47 b Ft(e)720 3832 y Fs(i;i)p Fn(+1)882 3817 y Ft(e)927 3781 y Fs(N)927 3842 y(s;t)1055 3817 y Fu(=)e Ft(\037)1237 3776 y Fk(\000)p Fs(N)1237 3843 y(i)1359 3817 y Fu(\()p Ft(g)1444 3832 y Fs(s;t)1526 3817 y Fu(\))p Ft(e)1609 3781 y Fs(N)1609 3842 y(s;t)1691 3817 y Ft(e)1736 3832 y Fs(i;i)p Fn(+1)1898 3817 y Fu(.)74 b(This)43 b(pro)m(v)m(es)i(the)e(claim)e([)p Ft(e)3065 3832 y Fs(i;i)p Fn(+1)3227 3817 y Ft(;)17 b(e)3316 3781 y Fs(N)3316 3842 y(s;t)3398 3817 y Fu(])3425 3832 y Fs(c)3505 3817 y Fu(=)45 b(0,)g(since)-180 3957 y Ft(\037)-119 3915 y Fk(\000)p Fs(N)-119 3982 y(i)3 3957 y Fu(\()p Ft(g)88 3972 y Fs(s;t)170 3957 y Fu(\))27 b(=)h Ft(\037)400 3920 y Fs(N)400 3981 y(s;t)482 3957 y Fu(\()p Ft(g)567 3972 y Fs(i)595 3957 y Fu(\))p Ft(:)p 3883 3957 V 3887 3894 59 4 v 3887 3957 V 3945 3957 4 66 v -80 4150 a Fu(W)-8 b(e)35 b(w)m(an)m(t)h(to)e(compute)h(the)h(dimension)d(of)i(certain)f (quotien)m(t)h(algebras)f(of)h Fj(B)p Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000.)44 b(Since)35 b(this)g(part)-180 4266 y(of)e(the)h(theory)g(w)m(orks)h(for)e(an)m(y)h(\014nite)f(Cartan) g(t)m(yp)s(e,)i(w)m(e)g(no)m(w)f(consider)f(more)g(generally)g(a)g (left)f Fp(|)-8 b Fu(\000-)o(mo)r(dule)-180 4383 y(algebra)35 b Ft(R)h Fu(o)m(v)m(er)h(an)m(y)f(ab)s(elian)e(group)i(\000)f(and)h (assume)g(that)g(there)g(are)g(in)m(tegers)g Ft(P)49 b Fu(and)36 b Ft(N)3305 4398 y Fs(i)3366 4383 y Ft(>)d Fu(1,)k(elemen)m(ts)-180 4507 y Ft(y)-132 4522 y Fs(i)-77 4507 y Fm(2)29 b Ft(R)q Fu(,)j Ft(h)208 4522 y Fs(i)264 4507 y Fm(2)c Fu(\000,)33 b Ft(\021)527 4522 y Fs(i)583 4507 y Fm(2)680 4482 y Fl(b)677 4507 y Fu(\000,)f(1)c Fm(\024)g Ft(i)g Fm(\024)g Ft(P)14 b Fu(,)32 b(suc)m(h)i(that)119 4683 y Ft(g)26 b Fm(\001)21 b Ft(y)289 4698 y Fs(i)345 4683 y Fu(=)27 b Ft(\021)496 4698 y Fs(i)525 4683 y Fu(\()p Ft(g)t Fu(\))p Ft(y)700 4698 y Fs(i)727 4683 y Ft(;)17 b Fu(for)31 b(all)g Ft(g)g Fm(2)d Fu(\000)p Ft(;)17 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(P)s(:)-1952 b Fu(\(6.45\))119 4866 y Ft(y)167 4881 y Fs(i)195 4866 y Ft(y)247 4812 y Fs(N)303 4822 y Fi(j)243 4892 y Fs(j)366 4866 y Fu(=)28 b Ft(\021)522 4812 y Fs(N)578 4822 y Fi(j)518 4892 y Fs(j)614 4866 y Fu(\()p Ft(h)708 4881 y Fs(i)737 4866 y Fu(\))p Ft(y)827 4812 y Fs(N)883 4822 y Fi(j)823 4892 y Fs(j)918 4866 y Ft(y)966 4881 y Fs(i)1027 4866 y Fu(for)k(all)e(1)e Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)28 b Ft(P)s(:)-2021 b Fu(\(6.46\))119 5035 y(The)34 b(elemen)m(ts)e Ft(y)770 4991 y Fs(a)807 5000 y Fg(1)766 5059 y Fn(1)862 5035 y Ft(:)17 b(:)g(:)g(y)1046 4989 y Fs(a)1083 5000 y Fi(P)1042 5062 y Fs(P)1137 5035 y Ft(;)g(a)1232 5050 y Fn(1)1272 5035 y Ft(;)g(:)g(:)g(:)f(;)h(a)1542 5050 y Fs(P)1628 5035 y Fm(\025)28 b Fu(0)p Ft(;)49 b Fu(form)31 b(a)i Fp(|)7 b Fm(\000)23 b Fu(basis)32 b(of)g Ft(R)q(:)-2979 b Fu(\(6.47\))p eop %%Page: 58 58 58 57 bop -180 0 a Fq(58)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-180 203 y Fu(Let)46 b Fp(L)58 b Fu(b)s(e)46 b(the)g(set)h(of)f(all)e Ft(l)52 b Fu(=)f(\()p Ft(l)1163 218 y Fs(i)1191 203 y Fu(\))1229 218 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)1507 203 y Fm(2)g Fp(N)1690 167 y Fs(P)1801 203 y Fu(suc)m(h)d(that)d(0)51 b Fm(\024)g Ft(l)2516 218 y Fs(i)2595 203 y Ft(<)f(N)2799 218 y Fs(i)2873 203 y Fu(for)c(all)e(1)50 b Fm(\024)h Ft(i)g Fm(\024)g Ft(P)14 b Fu(.)84 b(F)-8 b(or)-180 319 y Ft(a)28 b Fu(=)f(\()p Ft(a)91 334 y Fs(i)120 319 y Fu(\))158 334 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)413 319 y Fm(2)h Fp(N)573 283 y Fs(P)638 319 y Fu(,)k(w)m(e)i(de\014ne)1039 493 y Ft(y)1091 452 y Fs(a)1159 493 y Fu(=)28 b Ft(y)1315 450 y Fs(a)1352 459 y Fg(1)1311 518 y Fn(1)1407 493 y Ft(:)17 b(:)g(:)f(y)1590 448 y Fs(a)1627 459 y Fi(P)1586 520 y Fs(P)1682 493 y Ft(;)49 b Fu(and)33 b Ft(aN)38 b Fu(=)28 b(\()p Ft(a)2308 508 y Fs(i)2336 493 y Ft(N)2414 508 y Fs(i)2442 493 y Fu(\))2480 508 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)2708 493 y Ft(:)-180 668 y Fu(Then)34 b(b)m(y)h(\(6.46\))o(,)d(\(6.47\),)g (the)h(elemen)m(ts)1338 846 y Ft(y)1390 805 y Fs(l)1415 846 y Ft(y)1467 805 y Fs(aN)1571 846 y Ft(;)212 b(l)30 b Fm(2)e Fp(L)5 b Ft(;)55 b(a)28 b Fm(2)g Fp(N)2344 805 y Fs(P)2409 846 y Ft(;)-180 1020 y Fu(form)j(a)i Fp(|)-9 b Fu(-)o(basis)27 b(of)32 b Ft(R)q Fu(.)-80 1136 y(In)i(the)h (application)d(to)i Fj(B)p Fu(\()p Ft(V)21 b Fu(\)#)p Fp(|)-8 b Fu(\000,)28 b Ft(P)48 b Fu(is)34 b(the)h(n)m(um)m(b)s(er)f (of)g(p)s(ositiv)m(e)g(ro)s(ots,)g(and)g(the)h Ft(y)3239 1151 y Fs(i)3301 1136 y Fu(pla)m(y)f(the)h(role)e(of)-180 1253 y(the)g(ro)s(ot)f(v)m(ectors)i Ft(e)573 1268 y Fs(i;j)653 1253 y Fu(.)-80 1369 y(T)-8 b(o)31 b(simplify)e(the)j(notation)e(in)h (the)h(smash)g(pro)s(duct)g(algebra)e Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000,)26 b(w)m(e)32 b(iden)m(tify)f Ft(r)g Fm(2)d Ft(R)k Fu(with)f Ft(r)s Fu(#1)h(and)-180 1485 y Ft(v)i Fm(2)c Fp(|)-9 b Fu(\000)28 b(with)33 b(1#)p Ft(v)t Fu(.)47 b(F)-8 b(or)33 b(1)d Fm(\024)g Ft(i)g Fm(\024)g Ft(P)14 b Fu(,)34 b(let)j Fl(e)-59 b Ft(\021)1485 1500 y Fs(i)1543 1485 y Fu(:)30 b Fp(|)-9 b Fu(\000)24 b Fm(!)29 b Fp(|)-8 b Fu(\000)27 b(b)s(e)34 b(the)h(algebra)d(map)h(de\014ned)j(b)m(y)i Fl(e)-59 b Ft(\021)3411 1500 y Fs(i)3440 1485 y Fu(\()p Ft(g)t Fu(\))28 b(=)i Ft(\021)3749 1500 y Fs(i)3777 1485 y Fu(\()p Ft(g)t Fu(\))p Ft(g)-180 1601 y Fu(for)i(all)f Ft(g)f Fm(2)f Fu(\000.)43 b(Then)1302 1735 y Ft(v)t(y)1401 1750 y Fs(i)1456 1735 y Fu(=)28 b Ft(y)1608 1750 y Fs(i)1639 1735 y Fl(e)-58 b Ft(\021)1684 1750 y Fs(i)1712 1735 y Fu(\()p Ft(v)t Fu(\))32 b(for)g(all)f Ft(v)g Fm(2)d Fp(|)-8 b Fu(\000)o Ft(:)-180 1889 y Fu(W)g(e)33 b(\014x)g(a)f(family)f Ft(u)562 1904 y Fs(i)589 1889 y Ft(;)17 b Fu(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(P)14 b Fu(,)32 b(of)g(elemen)m(ts)h(in)e Fp(|)-8 b Fu(\000,)26 b(and)33 b(denote)1072 2079 y Ft(u)1128 2038 y Fs(a)1197 2079 y Fu(:=)1376 1984 y Fl(Y)1327 2196 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)1568 2079 y Ft(u)1624 2034 y Fs(a)1661 2044 y Fi(i)1624 2104 y Fs(i)1691 2079 y Ft(;)49 b Fu(if)32 b Ft(a)c Fu(=)f(\()p Ft(a)2128 2094 y Fs(i)2157 2079 y Fu(\))2195 2094 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)2450 2079 y Fm(2)h Fp(N)2610 2038 y Fs(P)2675 2079 y Ft(:)-180 2358 y Fu(Let)33 b Ft(M)43 b Fu(b)s(e)33 b(a)f(free)h(righ)m(t)f Fp(|)-9 b Fu(\000-m)o(o)r(dule)27 b(with)32 b(basis)h Ft(m)p Fu(\()p Ft(l)r Fu(\))p Ft(;)17 b(l)30 b Fm(2)e Fp(L)5 b Fu(.)50 b(W)-8 b(e)33 b(then)g(de\014ne)h(a)e (righ)m(t)g Fp(|)-9 b Fu(\000-)o(l)o(inear)26 b(map)598 2536 y Ft(')h Fu(:)h Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)21 b Fm(!)28 b Ft(M)43 b Fu(b)m(y)34 b Ft(')p Fu(\()p Ft(y)1599 2495 y Fs(l)1624 2536 y Ft(y)1676 2495 y Fs(aN)1779 2536 y Fu(\))28 b(:=)g Ft(m)p Fu(\()p Ft(l)r Fu(\))p Ft(u)2224 2495 y Fs(a)2298 2536 y Fu(for)k(all)e Ft(l)g Fm(2)e Fp(L)6 b Ft(;)17 b(a)33 b Fm(2)28 b Fp(N)3084 2495 y Fs(P)3149 2536 y Ft(:)-180 2757 y FD(Lemma)37 b(6.23.)49 b Fo(Assume)35 b(that)-54 2926 y Fm(\017)41 b Ft(u)93 2941 y Fs(i)156 2926 y Fo(is)34 b(c)-5 b(entr)g(al)35 b(in)f Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)p Fo(,)29 b(for)34 b(al)5 b(l)35 b Fu(1)27 b Fm(\024)h Ft(i)g Fm(\024)h Ft(P)14 b Fo(,)34 b(and)-54 3123 y Fm(\017)41 b Ft(u)93 3138 y Fs(i)148 3123 y Fu(=)28 b(0)35 b Fo(if)f Ft(\021)482 3078 y Fs(N)538 3088 y Fi(i)478 3148 y Fs(i)568 3123 y Fu(\()p Ft(h)662 3138 y Fs(j)699 3123 y Fu(\))28 b Fm(6)p Fu(=)f(1)35 b Fo(for)g(some)f Fu(1)27 b Fm(\024)h Ft(j)34 b Fm(\024)28 b Ft(P)14 b Fo(.)-80 3291 y(Then)34 b(the)h(kernel)f(of)g Ft(')h Fo(is)g(a)f(right)h(ide)-5 b(al)34 b(of)h Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)28 b Fo(c)-5 b(ontaining)34 b Ft(y)2440 3247 y Fs(N)2496 3257 y Fi(i)2436 3317 y Fs(i)2548 3291 y Fm(\000)23 b Ft(u)2704 3306 y Fs(i)2766 3291 y Fo(for)35 b(al)5 b(l)34 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(P)14 b Fo(.)-180 3512 y(Pr)-5 b(o)g(of.)41 b Fu(By)33 b(de\014nition,)f Ft(')p Fu(\()p Ft(y)886 3468 y Fs(N)942 3478 y Fi(i)882 3538 y Fs(i)971 3512 y Fu(\))c(=)g Ft(m)p Fu(\(0\))p Ft(u)1407 3527 y Fs(i)1462 3512 y Fu(=)f Ft(')p Fu(\()p Ft(u)1723 3527 y Fs(i)1751 3512 y Fu(\))p Ft(:)-80 3652 y Fu(T)-8 b(o)32 b(sho)m(w)i(that)e(the)h(k)m(ernel)h(of)e Ft(')g Fu(is)g(a)g(righ)m(t)g (ideal,)f(let)624 3849 y Ft(z)i Fu(=)891 3755 y Fl(X)805 3974 y Fs(l)q Fk(2)p Fd(L)p Fs(;a)p Fk(2)p Fd(N)1064 3955 y Fi(P)1138 3849 y Ft(y)1190 3808 y Fs(l)1215 3849 y Ft(y)1267 3808 y Fs(aN)1371 3849 y Ft(v)1418 3864 y Fs(l)q(;a)1501 3849 y Ft(;)50 b Fu(where)33 b Ft(v)1906 3864 y Fs(l)q(;a)2017 3849 y Fm(2)28 b Fp(|)-8 b Fu(\000)p Ft(;)16 b Fu(for)26 b(all)31 b Ft(l)f Fm(2)e Fp(L)5 b Ft(;)17 b(a)34 b Fm(2)28 b Fp(N)3058 3808 y Fs(P)3123 3849 y Ft(;)-180 4137 y Fu(b)s(e)33 b(an)f(elemen)m(t)h(with)f Ft(')p Fu(\()p Ft(z)t Fu(\))c(=)g(0.)43 b(Then)34 b Ft(')p Fu(\()p Ft(z)t Fu(\))28 b(=)1686 4063 y Fl(P)1791 4167 y Fs(l)q(;a)1891 4137 y Ft(m)p Fu(\()p Ft(l)r Fu(\))p Ft(u)2139 4101 y Fs(a)2181 4137 y Ft(v)2228 4152 y Fs(l)q(;a)2339 4137 y Fu(=)f(0,)33 b(hence)1316 4256 y Fl(X)1296 4475 y Fs(a)p Fk(2)p Fd(N)1424 4456 y Fi(P)1496 4350 y Ft(u)1552 4309 y Fs(a)1593 4350 y Ft(v)1640 4365 y Fs(l)q(;a)1751 4350 y Fu(=)28 b(0)p Ft(;)17 b Fu(for)31 b(all)g Ft(l)e Fm(2)g Fp(L)5 b Ft(:)-180 4630 y Fu(Fix)32 b(1)27 b Fm(\024)h Ft(i)g Fm(\024)g Ft(P)14 b Fu(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e (sho)m(w)i(that)f Ft(')p Fu(\()p Ft(z)t(y)1654 4645 y Fs(i)1682 4630 y Fu(\))27 b(=)h(0.)-80 4769 y(F)-8 b(or)31 b(an)m(y)j Ft(l)c Fm(2)e Fp(L)5 b Fu(,)39 b(w)m(e)33 b(ha)m(v)m(e)h(the)f(basis)g(represen)m(tation)604 4967 y Ft(y)656 4925 y Fs(l)681 4967 y Ft(y)729 4982 y Fs(i)785 4967 y Fu(=)972 4872 y Fl(X)888 5091 y Fs(t)p Fk(2)p Fd(L)p Fs(;b)p Fk(2)p Fd(N)1144 5072 y Fi(P)1217 4967 y Ft(\013)1280 4925 y Fs(l)1279 4991 y(t;b)1359 4967 y Ft(y)1411 4925 y Fs(t)1440 4967 y Ft(y)1492 4925 y Fs(bN)1588 4967 y Ft(;)49 b Fu(where)34 b Ft(\013)2009 4925 y Fs(l)2008 4991 y(t;b)2115 4967 y Fm(2)28 b Fp(|)18 b Fu(for)32 b(all)f Ft(t)d Fm(2)g Fp(L)5 b Ft(;)17 b(b)34 b Fm(2)28 b Fp(N)3078 4925 y Fs(P)3143 4967 y Ft(:)p eop %%Page: 59 59 59 58 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(59)-180 203 y Fu(Since)33 b Ft(u)131 167 y Fs(a)204 203 y Fu(is)f(cen)m(tral)h(in)f Ft(R)q Fu(#)p Fp(|)-9 b Fu(\000,)1324 388 y Ft(u)1380 347 y Fs(a)1449 388 y Fu(=)31 b Fl(e)-58 b Ft(\021)1601 403 y Fs(i)1629 388 y Fu(\()p Ft(u)1723 347 y Fs(a)1764 388 y Fu(\))32 b(for)g(all)f Ft(a)d Fm(2)g Fp(N)2358 347 y Fs(P)2423 388 y Ft(:)-2630 b Fu(\(6.48\))-180 574 y(F)-8 b(or)42 b(an)m(y)h Ft(a)h Fu(=)h(\()p Ft(a)504 589 y Fs(i)532 574 y Fu(\))570 589 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)842 574 y Fm(2)g Fp(N)1019 538 y Fs(P)1126 574 y Fu(and)e(an)m(y)g(family)d(\()p Ft(g)1913 589 y Fs(i)1941 574 y Fu(\))1979 589 y Fn(1)p Fk(\024)p Fs(i)p Fk(\024)p Fs(P)2249 574 y Fu(of)i(elemen)m(ts)g(in)g (\000)g(w)m(e)i(de\014ne)f Ft(\021)3502 538 y Fs(aN)3607 574 y Fu(\(\()p Ft(g)3730 589 y Fs(i)3758 574 y Fu(\)\))h(=)-180 638 y Fl(Q)-86 742 y Fs(i)-41 713 y Ft(\021)7 728 y Fs(i)35 677 y(a)72 687 y Fi(i)99 677 y Fs(N)155 687 y Fi(i)186 713 y Fu(\()p Ft(g)271 728 y Fs(i)298 713 y Fu(\).)g(Then)33 b(b)m(y)i(\(6.46\),)d(for)g(all)f Ft(a;)17 b(b)28 b Fm(2)g Fp(N)1716 677 y Fs(P)1781 713 y Fu(,)749 898 y Ft(y)801 857 y Fs(aN)904 898 y Ft(y)952 913 y Fs(i)1008 898 y Fu(=)f Ft(y)1159 913 y Fs(i)1187 898 y Ft(y)1239 857 y Fs(aN)1343 898 y Ft(\021)1395 857 y Fs(aN)1499 898 y Fu(\()p Ft(g)1588 857 y Fs(a)1629 898 y Fu(\))p Ft(;)49 b Fu(and)33 b Ft(y)1985 857 y Fs(bN)2082 898 y Ft(y)2134 857 y Fs(aN)2265 898 y Fu(=)28 b Ft(y)2421 857 y Fn(\()p Fs(a)p Fn(+)p Fs(b)p Fn(\))p Fs(N)2665 898 y Ft(\021)2717 857 y Fs(aN)2821 898 y Fu(\()p Ft(g)2910 857 y Fs(b)2944 898 y Fu(\))p Ft(;)-3189 b Fu(\(6.49\))-180 1084 y(for)32 b(some)g(families)e(of)i(elemen)m(ts)h Ft(g)1130 1048 y Fs(a)1171 1084 y Ft(;)17 b(g)1266 1048 y Fs(b)1332 1084 y Fu(in)32 b(\000.)-80 1223 y(By)h(a)f(reform)m(ulation)e(of)i (our)g(assumption,)761 1409 y Ft(u)817 1367 y Fs(a)859 1409 y Ft(\021)911 1367 y Fs(aN)1015 1409 y Fu(\(\()p Ft(g)1138 1424 y Fs(i)1166 1409 y Fu(\)\))27 b(=)h Ft(u)1429 1367 y Fs(a)1503 1409 y Fu(for)k(an)m(y)h Ft(a)28 b Fm(2)g Fp(N)2075 1367 y Fs(P)2172 1409 y Fu(and)33 b(family)m(\()p Ft(g)2712 1424 y Fs(i)2740 1409 y Fu(\))g(in)e(\000)p Ft(:)-3192 b Fu(\(6.50\))-180 1594 y(Using)34 b(\(6.49\))d(w)m(e)j(no)m (w)f(can)g(compute)864 1779 y Ft(z)t(y)961 1794 y Fs(i)1017 1779 y Fu(=)1120 1685 y Fl(X)1153 1897 y Fs(l)q(;a)1281 1779 y Ft(y)1333 1738 y Fs(l)1358 1779 y Ft(y)1410 1738 y Fs(aN)1514 1779 y Ft(v)1561 1794 y Fs(l)q(;a)1644 1779 y Ft(y)1692 1794 y Fs(i)1748 1779 y Fu(=)1851 1685 y Fl(X)1884 1897 y Fs(l)q(;a)2012 1779 y Ft(y)2064 1738 y Fs(l)2089 1779 y Ft(y)2141 1738 y Fs(aN)2245 1779 y Ft(y)2293 1794 y Fs(i)2324 1779 y Fl(e)-59 b Ft(\021)2368 1794 y Fs(i)2397 1779 y Fu(\()p Ft(v)2482 1794 y Fs(l)q(;a)2565 1779 y Fu(\))1017 2062 y(=)1120 1967 y Fl(X)1153 2179 y Fs(l)q(;a)1281 2062 y Ft(y)1333 2020 y Fs(l)1358 2062 y Ft(y)1406 2077 y Fs(i)1434 2062 y Ft(y)1486 2020 y Fs(aN)1590 2062 y Ft(\021)1642 2020 y Fs(aN)1746 2062 y Fu(\()p Ft(g)1835 2020 y Fs(a)1876 2062 y Fu(\))t Fl(e)g Ft(\021)1962 2077 y Fs(i)1990 2062 y Fu(\()p Ft(v)2075 2077 y Fs(l)q(;a)2158 2062 y Fu(\))1017 2344 y(=)1120 2249 y Fl(X)1153 2462 y Fs(l)q(;a)1281 2249 y Fl(X)1315 2462 y Fs(t;b)1441 2344 y Ft(\013)1504 2303 y Fs(l)1503 2369 y(t;b)1583 2344 y Ft(y)1635 2303 y Fs(t)1664 2344 y Ft(y)1716 2303 y Fs(bN)1812 2344 y Ft(e)1857 2303 y Fs(aN)1962 2344 y Ft(\021)2014 2303 y Fs(aN)2119 2344 y Fu(\()p Ft(g)2208 2303 y Fs(a)2248 2344 y Fu(\))t Fl(e)g Ft(\021)2334 2359 y Fs(i)2363 2344 y Fu(\()p Ft(v)2448 2359 y Fs(l)q(;a)2531 2344 y Fu(\))1017 2626 y(=)1120 2532 y Fl(X)1153 2744 y Fs(l)q(;a)1281 2532 y Fl(X)1315 2744 y Fs(t;b)1441 2626 y Ft(\013)1504 2585 y Fs(l)1503 2651 y(t;b)1583 2626 y Ft(y)1635 2585 y Fs(t)1664 2626 y Ft(y)1716 2585 y Fn(\()p Fs(a)p Fn(+)p Fs(b)p Fn(\))p Fs(N)1959 2626 y Ft(\021)2011 2585 y Fs(aN)2116 2626 y Fu(\()p Ft(g)2205 2585 y Fs(a)2246 2626 y Fu(\))p Ft(\021)2336 2585 y Fs(aN)2440 2626 y Fu(\()p Ft(g)2529 2585 y Fs(b)2563 2626 y Fu(\))t Fl(e)g Ft(\021)2649 2641 y Fs(i)2677 2626 y Fu(\()p Ft(v)2762 2641 y Fs(l)q(;a)2845 2626 y Fu(\))p Ft(:)-180 2890 y Fu(Therefore)784 3075 y Ft(')p Fu(\()p Ft(z)t(y)983 3090 y Fs(i)1011 3075 y Fu(\))28 b(=)1180 2981 y Fl(X)1239 3190 y Fs(t)1341 3075 y Ft(m)p Fu(\()p Ft(t)p Fu(\))1554 2981 y Fl(X)1561 3193 y Fs(l)q(;a;b)1714 3075 y Ft(\013)1777 3034 y Fs(l)1776 3100 y(t;b)1856 3075 y Ft(u)1912 3034 y Fs(a)p Fn(+)p Fs(b)2038 3075 y Ft(\021)2090 3034 y Fs(aN)2194 3075 y Fu(\()p Ft(g)2283 3034 y Fs(a)2324 3075 y Fu(\))p Ft(\021)2414 3034 y Fs(aN)2518 3075 y Fu(\()p Ft(g)2607 3034 y Fs(b)2641 3075 y Fu(\))t Fl(e)-59 b Ft(\021)2727 3090 y Fs(i)2755 3075 y Fu(\()p Ft(v)2840 3090 y Fs(l)q(;a)2923 3075 y Fu(\))1077 3358 y(=)1180 3263 y Fl(X)1239 3472 y Fs(t)1341 3358 y Ft(m)p Fu(\()p Ft(t)p Fu(\))1554 3263 y Fl(X)1561 3475 y Fs(l)q(;a;b)1714 3358 y Ft(\013)1777 3317 y Fs(l)1776 3382 y(t;b)1856 3358 y Ft(u)1912 3317 y Fs(a)p Fn(+)p Fs(b)2041 3358 y Fl(e)h Ft(\021)2086 3373 y Fs(i)2114 3358 y Fu(\()p Ft(v)2199 3373 y Fs(l)q(;a)2282 3358 y Fu(\))p Ft(;)49 b Fu(b)m(y)34 b(\(6.50\))o Ft(;)1077 3640 y Fu(=)1180 3546 y Fl(X)1239 3755 y Fs(t)1341 3640 y Ft(m)p Fu(\()p Ft(t)p Fu(\))1554 3546 y Fl(X)1590 3758 y Fs(b;l)1714 3640 y Ft(\013)1777 3599 y Fs(l)1776 3665 y(t;b)1856 3640 y Ft(u)1912 3599 y Fs(b)1962 3546 y Fl(X)2015 3755 y Fs(a)2123 3640 y Ft(u)2179 3599 y Fs(a)2223 3640 y Fl(e)-58 b Ft(\021)2268 3655 y Fs(i)2296 3640 y Fu(\()p Ft(v)2381 3655 y Fs(l)q(;a)2464 3640 y Fu(\))1077 3923 y(=)1180 3828 y Fl(X)1239 4037 y Fs(t)1341 3923 y Ft(m)p Fu(\()p Ft(t)p Fu(\))1554 3828 y Fl(X)1590 4040 y Fs(b;l)1714 3923 y Ft(\013)1777 3881 y Fs(l)1776 3947 y(t;b)1856 3923 y Ft(u)1912 3881 y Fs(b)1949 3923 y Fl(e)f Ft(\021)1993 3938 y Fs(i)2022 3923 y Fu(\()2060 3828 y Fl(X)2113 4037 y Fs(a)2220 3923 y Ft(u)2276 3881 y Fs(a)2317 3923 y Ft(v)2364 3938 y Fs(l)q(;a)2448 3923 y Fu(\))p Ft(;)49 b Fu(b)m(y)33 b(\(6.48\))p Ft(;)1077 4205 y Fu(=)27 b(0)p Ft(;)49 b Fu(since)1561 4110 y Fl(X)1614 4319 y Fs(a)1721 4205 y Ft(u)1777 4164 y Fs(a)1818 4205 y Ft(v)1865 4220 y Fs(l)q(;a)1976 4205 y Fu(=)28 b(0)p Ft(:)p 3883 4449 4 66 v 3887 4387 59 4 v 3887 4449 V 3945 4449 4 66 v -180 4662 a FD(Theorem)37 b(6.24.)49 b Fo(L)-5 b(et)46 b Ft(u)809 4677 y Fs(i)837 4662 y Ft(;)17 b Fu(1)47 b Fm(\024)h Ft(i)g Fm(\024)g Ft(P)s(;)e Fo(b)-5 b(e)45 b(a)g(family)h(of)f(elements)g(in)g Fp(|)-9 b Fu(\000)p Fo(,)43 b(and)i Ft(I)53 b Fo(the)46 b(ide)-5 b(al)45 b(in)g Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)-180 4802 y Fo(gener)j(ate)g(d) 46 b(by)h(al)5 b(l)47 b Ft(y)607 4757 y Fs(N)663 4767 y Fi(i)603 4827 y Fs(i)724 4802 y Fm(\000)31 b Ft(u)888 4817 y Fs(i)916 4802 y Ft(;)17 b Fu(1)49 b Fm(\024)i Ft(i)f Fm(\024)h Ft(P)s(:)46 b Fo(L)-5 b(et)48 b Ft(A)i Fu(=)g(\()p Ft(R)q Fu(#)p Fp(|)-9 b Fu(\000\))p Ft(=I)49 b Fo(b)-5 b(e)46 b(the)h(quotient)h(algebr)-5 b(a.)80 b(Then)46 b(the)-180 4941 y(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)-80 5081 y(1\))34 b(The)g(r)-5 b(esidue)35 b(classes)f(of)g Ft(y)1061 5044 y Fs(l)1086 5081 y Ft(g)t(;)17 b(l)29 b Fm(2)f Fp(L)6 b Ft(;)17 b(g)37 b Fm(2)28 b Fu(\000)p Ft(;)35 b Fo(form)f(a)h Fp(|)-9 b Fo(-b)k(a)o(sis)29 b(of)34 b Ft(A)p Fo(.)p eop %%Page: 60 60 60 59 bop -180 0 a Fq(60)1247 b(ANDR)n(USKIEWITSCH)31 b(AND)h(SCHNEIDER)-80 210 y Fo(2\))i Ft(u)100 225 y Fs(i)163 210 y Fo(is)g(c)-5 b(entr)g(al)35 b(in)f Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)29 b Fo(for)35 b(al)5 b(l)34 b Fu(1)28 b Fm(\024)g Ft(i)g Fm(\024)g Ft(P)14 b Fo(,)34 b(and)g Ft(u)2041 225 y Fs(i)2097 210 y Fu(=)27 b(0)35 b Fo(if)g Ft(\021)2431 165 y Fs(N)2487 175 y Fi(i)2427 235 y Fs(i)2544 210 y Fm(6)p Fu(=)28 b Ft(":)-180 428 y Fo(Pr)-5 b(o)g(of.)41 b Fu(1\))d Fm(\))f Fu(2\))h(:)h(F)-8 b(or)38 b(all)e Ft(i)j Fu(and)g Ft(g)i Fm(2)d Fu(\000,)i Ft(g)t(y)1584 384 y Fs(N)1640 394 y Fi(i)1580 454 y Fs(i)1707 428 y Fu(=)e Ft(\021)1873 384 y Fs(N)1929 394 y Fi(i)1869 454 y Fs(i)1959 428 y Fu(\()p Ft(g)t Fu(\))p Ft(y)2138 384 y Fs(N)2194 394 y Fi(i)2134 454 y Fs(i)2223 428 y Ft(g)t Fu(,)i(hence)g Ft(u)2674 443 y Fs(i)2701 428 y Ft(g)i Fu(=)c Ft(g)t(u)3011 443 y Fs(i)3075 428 y Fm(\021)h Ft(\021)3243 384 y Fs(N)3299 394 y Fi(i)3239 454 y Fs(i)3329 428 y Fu(\()p Ft(g)t Fu(\))p Ft(u)3512 443 y Fs(i)3539 428 y Ft(g)70 b Fu(mo)s(d)32 b Ft(I)8 b Fu(.)-180 568 y(Since)28 b(b)m(y)g(assumption,)h Fp(|)-9 b Fu(\000)22 b(is)27 b(a)g(subspace)j(of)d Ft(A)p Fu(,)i(w)m(e)g(conclude)f(that)f Ft(u)2488 583 y Fs(i)2544 568 y Fu(=)g Ft(\021)2699 523 y Fs(N)2755 533 y Fi(i)2695 593 y Fs(i)2785 568 y Fu(\()p Ft(g)t Fu(\))p Ft(u)2968 583 y Fs(i)2995 568 y Fu(,)i(and)f Ft(u)3292 583 y Fs(i)3347 568 y Fu(=)g(0)f(if)g Ft(\021)3664 523 y Fs(N)3720 533 y Fi(i)3660 593 y Fs(i)3778 568 y Fm(6)p Fu(=)g Ft(")p Fu(.)-80 707 y(Similarly)-8 b(,)28 b(for)k(all)e(1)e Fm(\024)g Ft(i;)17 b(j)34 b Fm(\024)28 b Ft(n)p Fu(,)33 b Ft(y)1241 722 y Fs(i)1268 707 y Ft(y)1320 653 y Fs(N)1376 663 y Fi(j)1316 733 y Fs(j)1440 707 y Fu(=)27 b Ft(\021)1595 653 y Fs(N)1651 663 y Fi(j)1591 733 y Fs(j)1688 707 y Fu(\()p Ft(h)1782 722 y Fs(i)1810 707 y Fu(\))p Ft(y)1900 653 y Fs(N)1956 663 y Fi(j)1896 733 y Fs(j)1992 707 y Ft(y)2040 722 y Fs(i)2100 707 y Fu(b)m(y)35 b(\(6.46\))o(,)d(hence)i Ft(y)2864 722 y Fs(i)2892 707 y Ft(u)2948 722 y Fs(j)3012 707 y Fm(\021)28 b Ft(\021)3169 653 y Fs(N)3225 663 y Fi(j)3165 733 y Fs(j)3261 707 y Fu(\()p Ft(h)3355 722 y Fs(i)3384 707 y Fu(\))p Ft(u)3478 722 y Fs(j)3514 707 y Ft(y)3562 722 y Fs(i)3656 707 y Fu(mo)s(d)k Ft(I)8 b Fu(.)-180 854 y(Since)31 b(w)m(e)h(already)e(kno)m(w)i(that)e Ft(u)1075 869 y Fs(i)1130 854 y Fu(=)e(0)i(if)g Ft(\021)1453 799 y Fs(N)1509 809 y Fi(j)1449 879 y Fs(j)1573 854 y Fm(6)p Fu(=)d Ft(")p Fu(,)k(w)m(e)h(see)g(that)e Ft(y)2335 869 y Fs(i)2363 854 y Ft(u)2419 869 y Fs(j)2483 854 y Fm(\021)e Ft(u)2644 869 y Fs(j)2680 854 y Ft(y)2728 869 y Fs(i)2822 854 y Fu(mo)s(d)k Ft(I)8 b Fu(.)43 b(On)31 b(the)g(other)f(hand)-180 993 y Ft(u)-124 1008 y Fs(j)-88 993 y Ft(y)-40 1008 y Fs(i)15 993 y Fu(=)e Ft(y)167 1008 y Fs(i)198 993 y Fl(e)-58 b Ft(\021)243 1008 y Fs(i)271 993 y Fu(\()p Ft(u)365 1008 y Fs(j)401 993 y Fu(\).)43 b(Then)33 b(our)f(assumption)f(in)g (1\))h(implies)d(that)35 b Fl(e)-58 b Ft(\021)2272 1008 y Fs(i)2300 993 y Fu(\()p Ft(u)2394 1008 y Fs(j)2430 993 y Fu(\))28 b(=)f Ft(u)2655 1008 y Fs(j)2691 993 y Fu(.)43 b(In)33 b(other)f(w)m(ords,)h Ft(u)3499 1008 y Fs(j)3567 993 y Fu(is)e(cen)m(tral)-180 1132 y(in)h Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)o(.)-80 1272 y(2\))45 b Fm(\))g Fu(1\):)64 b(Let)43 b Ft(J)52 b Fu(b)s(e)43 b(the)h(righ)m(t)e(ideal)g(of)g Ft(R)q Fu(#)p Fp(|)-8 b Fu(\000)37 b(generated)44 b(b)m(y)f(all)e Ft(y)2713 1228 y Fs(N)2769 1238 y Fi(i)2709 1297 y Fs(i)2828 1272 y Fm(\000)30 b Ft(u)2991 1287 y Fs(i)3019 1272 y Ft(;)17 b Fu(1)45 b Fm(\024)h Ft(i)f Fm(\024)h Ft(P)s(:)d Fu(F)-8 b(or)42 b(an)m(y)-180 1411 y(1)27 b Fm(\024)i Ft(i)f Fm(\024)g Ft(P)46 b Fu(and)32 b Ft(g)f Fm(2)d Fu(\000)p Ft(;)787 1605 y(g)t Fu(\()p Ft(y)928 1561 y Fs(N)984 1571 y Fi(i)924 1631 y Fs(i)1035 1605 y Fm(\000)22 b Ft(u)1190 1620 y Fs(i)1218 1605 y Fu(\))28 b(=)f Ft(y)1439 1561 y Fs(N)1495 1571 y Fi(i)1435 1631 y Fs(i)1525 1605 y Ft(g)t(\021)1628 1561 y Fs(N)1684 1571 y Fi(i)1624 1631 y Fs(i)1713 1605 y Fu(\()p Ft(g)t Fu(\))22 b Fm(\000)g Ft(g)t(u)2068 1620 y Fs(i)2123 1605 y Fu(=)27 b(\()p Ft(y)2316 1561 y Fs(N)2372 1571 y Fi(i)2312 1631 y Fs(i)2424 1605 y Fm(\000)c Ft(u)2580 1620 y Fs(i)2607 1605 y Fu(\))p Ft(\021)2697 1561 y Fs(N)2753 1571 y Fi(i)2693 1631 y Fs(i)2784 1605 y Fu(\()p Ft(g)t Fu(\))p Ft(g)t(;)-180 1799 y Fu(since)33 b Ft(g)t(u)166 1814 y Fs(i)221 1799 y Fu(=)27 b Ft(u)380 1814 y Fs(i)408 1799 y Ft(\021)460 1754 y Fs(N)516 1764 y Fi(i)456 1824 y Fs(i)546 1799 y Fu(\()p Ft(g)t Fu(\))p Ft(g)35 b Fu(b)m(y)f(2\).)-80 1938 y(And)f(for)f(all)e(1)e Fm(\024)g Ft(i;)17 b(j)33 b Fm(\024)c Ft(P)14 b Fu(,)674 2132 y Ft(y)722 2147 y Fs(i)750 2132 y Fu(\()p Ft(y)840 2078 y Fs(N)896 2088 y Fi(j)836 2158 y Fs(j)953 2132 y Fm(\000)23 b Ft(u)1109 2147 y Fs(j)1145 2132 y Fu(\))28 b(=)f Ft(\021)1366 2078 y Fs(N)1422 2088 y Fi(j)1362 2158 y Fs(j)1459 2132 y Fu(\()p Ft(h)1553 2147 y Fs(i)1581 2132 y Fu(\))p Ft(y)1671 2078 y Fs(N)1727 2088 y Fi(j)1667 2158 y Fs(j)1763 2132 y Ft(y)1811 2147 y Fs(i)1861 2132 y Fm(\000)22 b Ft(y)2008 2147 y Fs(i)2036 2132 y Ft(u)2092 2147 y Fs(j)2156 2132 y Fu(=)27 b(\()p Ft(y)2349 2078 y Fs(N)2405 2088 y Fi(j)2345 2158 y Fs(j)2463 2132 y Fm(\000)c Ft(u)2619 2147 y Fs(j)2655 2132 y Fu(\))p Ft(\021)2745 2078 y Fs(N)2801 2088 y Fi(j)2741 2158 y Fs(j)2837 2132 y Fu(\()p Ft(h)2931 2147 y Fs(i)2959 2132 y Fu(\))p Ft(y)3045 2147 y Fs(i)3073 2132 y Ft(;)-180 2332 y Fu(since)33 b(b)m(y)g(2\))g Ft(y)362 2347 y Fs(i)389 2332 y Ft(u)445 2347 y Fs(j)509 2332 y Fu(=)28 b Ft(u)669 2347 y Fs(j)705 2332 y Ft(y)753 2347 y Fs(i)808 2332 y Fu(=)g Ft(u)968 2347 y Fs(j)1004 2332 y Ft(\021)1056 2278 y Fs(N)1112 2288 y Fi(j)1052 2358 y Fs(j)1148 2332 y Fu(\()p Ft(h)1242 2347 y Fs(i)1270 2332 y Fu(\))p Ft(y)1356 2347 y Fs(i)1384 2332 y Ft(:)-80 2472 y Fu(This)k(pro)m(v)m(es)j Ft(J)i Fu(=)27 b Ft(I)8 b Fu(.)-80 2611 y(It)34 b(is)g(clear)g(that)g (the)h(images)e(of)h(all)e Ft(y)1369 2575 y Fs(l)1394 2611 y Ft(g)t(;)17 b(l)32 b Fm(2)g Fp(L)5 b Ft(;)17 b(g)40 b Fm(2)31 b Fu(\000)p Ft(;)j Fu(generate)i(the)e(v)m(ector)i(space)f Ft(A)p Fu(.)49 b(T)-8 b(o)35 b(sho)m(w)g(linear)-180 2751 y(indep)s(endency)-8 b(,)34 b(supp)s(ose)855 2850 y Fl(X)797 3062 y Fs(l)q Fk(2)p Fd(L)p Fs(;g)r Fk(2)p Fn(\000)1074 2945 y Ft(\013)1136 2960 y Fs(l)q(;g)1217 2945 y Ft(y)1269 2903 y Fs(l)1294 2945 y Ft(g)d Fm(2)d Ft(I)8 b(;)50 b Fu(with)32 b Ft(\013)1878 2960 y Fs(l)q(;g)1987 2945 y Fm(2)c Fp(|)18 b Fu(for)32 b(all)e Ft(l)g Fm(2)e Fp(L)6 b Ft(;)17 b(g)37 b Fm(2)28 b Fu(\000)p Ft(:)-180 3222 y Fu(Since)33 b Ft(I)i Fu(=)28 b Ft(J)9 b Fu(,)33 b(w)m(e)g(obtain)f(from)f(Lemma)g(6.23)h(that)g Ft(')p Fu(\()p Ft(I)8 b Fu(\))28 b(=)f(0.)44 b(Therefore,)1043 3416 y(0)27 b(=)h Ft(')p Fu(\()1383 3321 y Fl(X)1325 3533 y Fs(l)q Fk(2)p Fd(L)p Fs(;g)r Fk(2)p Fn(\000)1601 3416 y Ft(\013)1663 3431 y Fs(l)q(;g)1745 3416 y Ft(y)1797 3375 y Fs(l)1822 3416 y Ft(g)t Fu(\))f(=)2099 3321 y Fl(X)2041 3533 y Fs(l)q Fk(2)p Fd(L)p Fs(;g)r Fk(2)p Fn(\000)2318 3416 y Ft(\013)2380 3431 y Fs(l)q(;g)2461 3416 y Ft(m)p Fu(\()p Ft(l)r Fu(\))p Ft(g)t(;)-180 3688 y Fu(hence)34 b Ft(\013)153 3703 y Fs(l)q(;g)262 3688 y Fu(=)28 b(0)k(for)g(all)f Ft(l)r(;)17 b(g)t Fu(.)p 3883 3688 4 66 v 3887 3625 59 4 v 3887 3688 V 3945 3688 4 66 v -80 3871 a(W)-8 b(e)33 b(come)f(bac)m(k)i(to)e Ft(A)757 3886 y Fs(n)804 3871 y Fu(.)43 b(Our)33 b(main)e(result)h(in)g (this)g(Chapter)i(is)-180 4089 y FD(Theorem)j(6.25.)49 b Fo(\(i\).)43 b(L)-5 b(et)32 b Ft(\015)h Fu(=)27 b(\()p Ft(\015)1198 4104 y Fs(i;j)1278 4089 y Fu(\))1316 4104 y Fn(1)p Fk(\024)p Fs(i)28 b Fu(3)p Ft(;)36 b Fo(or)f(if)g(the)g(in\014nitesimal)f(br)-5 b(aiding)34 b(of)h Ft(A)h Fo(is)f(of)g(typ)-5 b(e)35 b Ft(A)2193 4928 y Fs(n)2276 4913 y Fo(with)g Ft(N)k(>)28 b Fu(7)35 b Fo(and)g(not)g(divisible)f(by)i(3,)f(then)-180 5053 y Ft(A)g Fo(is)g(isomorphic)e(to)i(a)g(Hopf)g(algebr)-5 b(a)34 b Ft(A)1374 5068 y Fs(\015)1453 5053 y Fo(in)h(\(i\).)p eop %%Page: 61 61 61 60 bop 1347 0 a Fq(POINTED)33 b(HOPF)g(ALGEBRAS)1450 b(61)-180 218 y Fo(Pr)-5 b(o)g(of.)41 b Fu(\(i\).)g(By)27 b(Lemma)e(6.20,)j(the)f(elemen)m(ts)g Ft(u)1619 233 y Fs(i;j)1725 218 y Fu(are)g(cen)m(tral)g(in)2320 193 y Fl(b)2307 218 y Ft(U)38 b Fu(:=)2558 192 y Fl(b)2542 218 y Fj(B)p Fu(\()p Ft(V)22 b Fu(\)#)p Fp(|)-9 b Fu(\000,)22 b(and)27 b Ft(u)3277 233 y Fs(i;j)3384 218 y Fu(=)h(0)e(if)g Ft(g)3698 182 y Fs(N)3694 243 y(i;j)3802 218 y Fu(=)h(1)-180 357 y(or)35 b Ft(\037)3 321 y Fs(N)3 382 y(i;j)117 357 y Fm(6)p Fu(=)e Ft(":)i Fu(Hence)j(the)e(residue)g(classes)h(of)e(the)i (elemen)m(ts)f Ft(e)2180 321 y Fs(l)2206 357 y Ft(g)t(;)17 b(l)34 b Fm(2)g Fp(A)22 b Ft(;)17 b Fu(0)38 b Fm(\024)c Ft(l)2801 372 y Fs(i;j)2914 357 y Ft(<)g(N)43 b Fu(for)32 b(all)e(1)j Fm(\024)h Ft(i)f(<)g(j)39 b Fm(\024)-180 497 y Ft(n)23 b Fu(+)g(1)p Ft(;)17 b(g)34 b Fm(2)c Fu(\000)p Ft(;)k Fu(form)f(a)h(basis)g(of)g Ft(A)1134 512 y Fs(\015)1213 497 y Fu(b)m(y)h(Theorem)f(6.24.)48 b(By)34 b(Theorem)h(6.18,)f(the)g Ft(u)3040 512 y Fs(i;j)3154 497 y Fu(satisfy)h(\(6.36\).)48 b(The)-180 636 y(ideal)31 b Ft(I)40 b Fu(of)261 611 y Fl(b)248 636 y Ft(U)j Fu(generated)33 b(b)m(y)h(all)c Ft(e)1117 600 y Fs(N)1117 661 y(i;j)1220 636 y Fm(\000)22 b Ft(u)1375 651 y Fs(i;j)1487 636 y Fu(is)32 b(a)h(biideal,)d(since)-34 836 y(\001\()p Ft(e)130 795 y Fs(N)130 860 y(i;j)233 836 y Fm(\000)23 b Ft(u)389 851 y Fs(i;j)468 836 y Fu(\))28 b(=\()p Ft(e)693 795 y Fs(N)693 860 y(i;j)795 836 y Fm(\000)23 b Ft(u)951 851 y Fs(i;j)1031 836 y Fu(\))f Fm(\012)g Fu(1)g(+)g Ft(g)1410 795 y Fs(N)1406 860 y(i;j)1508 836 y Fm(\012)h Fu(\()p Ft(e)1691 795 y Fs(N)1691 860 y(i;j)1793 836 y Fm(\000)g Ft(u)1949 851 y Fs(i;j)2029 836 y Fu(\))632 1024 y(+)759 929 y Fl(X)730 1139 y Fs(i)e Fu(2,)j Ft(q)j Fu(a)35 b(ro)s(ot)h(of)f(unit)m(y)i(of)e(order)i Ft(N)10 b Fu(,)37 b(and)g Ft(m)2781 218 y Fn(1)2820 203 y Ft(;)17 b(:)g(:)g(:)f(;)h(m)3124 218 y Fs(n)3207 203 y Fu(in)m(tegers)37 b Ft(>)c Fu(1)j(suc)m(h)-180 342 y(that)28 b Ft(m)112 357 y Fs(i)168 342 y Fm(6)p Fu(=)f Ft(m)356 357 y Fs(j)421 342 y Fu(for)h(all)d Ft(i)j Fm(6)p Fu(=)g Ft(j)6 b Fu(.)42 b(Let)28 b(\000)g(b)s(e)g(the)g(comm)m(utativ)m(e)f(group)h(generated)h (b)m(y)f Ft(g)2993 357 y Fn(1)3033 342 y Ft(;)17 b(:)g(:)g(:)e(;)i(g) 3298 357 y Fs(n)3373 342 y Fu(with)27 b(relations)-180 482 y Ft(g)-129 438 y Fs(N)7 b(m)-4 448 y Fi(i)-133 508 y Fs(i)54 482 y Fu(=)28 b(1,)k(1)27 b Fm(\024)i Ft(i)e Fm(\024)i Ft(n:)j Fu(De\014ne)i Ft(\037)1095 497 y Fn(1)1134 482 y Ft(;)17 b(:)g(:)g(:)f(;)h(\037)1414 497 y Fs(n)1488 482 y Fm(2)1585 457 y Fl(b)1582 482 y Fu(\000)33 b(b)m(y)183 686 y Ft(\037)244 701 y Fs(j)280 686 y Fu(\()p Ft(g)365 701 y Fs(i)393 686 y Fu(\))28 b(=)f Ft(q)609 645 y Fs(a)646 655 y Fi(ij)706 686 y Ft(;)49 b Fu(where)34 b Ft(a)1115 701 y Fs(ii)1195 686 y Fu(=)27 b(2)33 b(for)f(all)e Ft(i;)17 b(a)1792 701 y Fs(ij)1880 686 y Fu(=)28 b Fm(\000)p Fu(1)33 b(if)e Fm(j)p Ft(i)22 b Fm(\000)h Ft(j)6 b Fm(j)27 b Fu(=)h(1)p Ft(;)17 b(a)2764 701 y Fs(ij)2852 686 y Fu(=)27 b(0)33 b(if)e Fm(j)p Ft(i)22 b Fm(\000)h Ft(j)6 b Fm(j)27 b(\025)h Fu(2)p Ft(:)-180 889 y Fu(Then)k Ft(\037)134 853 y Fs(N)134 914 y(i;j)242 889 y Fu(=)c Ft(")j Fu(and)g Ft(g)662 853 y Fs(N)658 914 y(i;j)766 889 y Fm(6)p Fu(=)c(1)k(for)g (all)e Ft(i)f(<)f(j:)32 b Fu(Th)m(us)h(for)e(an)m(y)h(family)d Ft(\015)j Fu(=)c(\()p Ft(\015)2644 904 y Fs(i;j)2723 889 y Fu(\))2761 904 y Fn(1)p Fk(\024)p Fs(i