%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: MilSchFalda.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips MilSchFalda.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.05.10:2002 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 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1906 y(for)e(stim)o(ulating)f(discussions)i(on)g(the)g(sub)r(ject)i(of) d(this)h(pap)q(er.)303 1956 y(Both)g(authors)g(w)o(ould)f(lik)o(e)g(to) h(thank)g(D.)f(Fisc)o(hman)g(and)g(S.)h(W)m(estreic)o(h)h(whose)f(sk)o (etc)o(h)228 2006 y(of)i(a)g(\014nite-dimensional,)f(link-indecomp)q (osable,)g(p)q(oin)o(ted)i(Hopf)f(algebra)g Fw(H)k Fx(with)c(group)228 2056 y Fw(S)253 2062 y Fu(3)289 2056 y Fx(of)g(group-lik)o(e)g(elemen)o (ts)h(sev)o(eral)g(y)o(ears)h(ago)e(motiv)n(ated)f(the)j(w)o(ork)e (presen)o(ted)k(in)c(this)228 2106 y(pap)q(er.)775 2229 y Fy(2.)24 b(Nic)o(hols)14 b(algebras)303 2303 y Fx(The)g (ground-\014eld)g(will)e(b)q(e)i(denoted)h(b)o(y)f Fv(|)-19 b Fx(,)10 b(and)k Ft(\012)e Fx(=)g Ft(\012)1217 2309 y Fm(|)1235 2303 y Fx(.)18 b(Let)d Fw(G)e Fx(b)q(e)h(a)g(group.)k (Recall)228 2353 y(that)10 b(a)h(Y)m(etter-Drinfeld)f(mo)q(dule)f Fw(V)20 b Fx(o)o(v)o(er)11 b Fv(|)-19 b Fw(G)7 b Fx(is)j(a)g Fw(G)p Fx(-graded)h(v)o(ector)g(space)g Fw(V)21 b Fx(=)1536 2322 y Fl(L)1582 2366 y Fs(g)q Fn(2)p Fs(G)1667 2353 y Fw(V)1691 2359 y Fs(g)1711 2353 y Fx(,)228 2404 y(whic)o(h)d(is)h(a)f Fw(G)p Fx(-mo)q(dule)f(suc)o(h)i(that)g Fw(g)14 b Ft(\001)e Fw(V)910 2410 y Fs(h)951 2404 y Ft(\032)20 b Fw(V)1027 2412 y Fs(g)q(hg)1080 2404 y Fk(\000)p Fj(1)k Fx(for)19 b(all)e Fw(g)q(;)7 b(h)19 b Ft(2)h Fw(G)p Fx(.)32 b(The)19 b(category)228 2442 y Fs(G)228 2468 y(G)256 2457 y Ft(Y)s(D)h Fx(of)f Fv(|)-19 b Fw(G)o Fx(-Y)m(etter-Drinfeld)16 b(mo)q(dules)i(is)h (braided:)29 b(F)m(or)18 b Fw(V)r(;)7 b(W)26 b Ft(2)1384 2442 y Fs(G)1384 2468 y(G)1412 2457 y Ft(Y)s(D)21 b Fx(the)e(braiding) 228 2507 y Fw(c)c Fx(:)h Fw(V)k Ft(\012)11 b Fw(W)22 b Ft(!)15 b Fw(W)i Ft(\012)11 b Fw(V)26 b Fx(is)16 b(de\014ned)i(b)o(y) e Fw(c)p Fx(\()p Fw(v)d Ft(\012)e Fw(w)q Fx(\))16 b(=)g(\()p Fw(g)c Ft(\001)e Fw(w)q Fx(\))h Ft(\012)g Fw(v)q Fx(,)18 b Fw(v)f Ft(2)e Fw(V)1440 2513 y Fs(g)1460 2507 y Fw(;)7 b(w)16 b Ft(2)f Fw(W)6 b Fx(.)26 b(The)228 2556 y Fw(G)p Fx(-gradation)12 b(and)i Fw(G)p Fx(-action)f(on)h Fw(V)19 b Ft(\012)9 b Fw(W)20 b Fx(are)14 b(de\014ned)i(in)d(the)i(usual)e(w)o (a)o(y)h(b)o(y)f(\()p Fw(V)19 b Ft(\012)10 b Fw(W)c Fx(\))1659 2562 y Fs(g)1690 2556 y Fx(=)228 2575 y Fl(L)274 2619 y Fs(ab)p Fu(=)p Fs(g)358 2606 y Fw(V)382 2612 y Fs(a)412 2606 y Ft(\012)j Fw(W)492 2612 y Fs(b)509 2606 y Fw(;)20 b Fx(and)14 b Fw(g)d Ft(\001)d Fx(\()p Fw(v)j Ft(\012)f Fw(w)q Fx(\))h(=)h Fw(g)f Ft(\001)e Fw(v)h Ft(\012)g Fw(g)g Ft(\001)f Fw(w)q(;)e(g)12 b Ft(2)f Fw(G;)c(v)13 b Ft(2)e Fw(V)r(;)c(w)12 b Ft(2)f Fw(W)6 b Fx(.)p eop %%Page: 3 3 3 2 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d(O) o(VER)i(CO)o(XETER)g(GR)o(OUPS)112 b(3)303 213 y Fx(Let)18 b Fw(V)28 b Fx(b)q(e)19 b(a)f(Y)m(etter-Drinfeld)h(mo)q(dule)e(o)o(v)o (er)h Fw(G)p Fx(,)h(and)f(let)h Fw(T)6 b Fx(\()p Fw(V)j Fx(\))20 b(=)1451 181 y Fl(L)1497 225 y Fs(n)p Fn(\025)p Fu(0)1569 213 y Fw(T)6 b Fx(\()p Fw(V)k Fx(\)\()p Fw(n)p Fx(\))228 268 y(denote)16 b(the)h(tensor)g(algebra)e(of)g(the)i(v)o (ector)f(space)h Fw(V)9 b Fx(,)16 b(that)g(is)g Fw(T)6 b Fx(\()p Fw(V)j Fx(\)\()p Fw(n)p Fx(\))15 b(=)g Fw(V)1518 253 y Fn(\012)p Fs(n)1582 268 y Fx(and)h(the)228 317 y(m)o(ultiplicatio)o(n)d Fw(\026)i Fx(is)h(de\014ned)g(as)g(tensor)h (pro)q(duct.)24 b(W)m(e)15 b(consider)h Fw(T)6 b Fx(\()p Fw(V)k Fx(\))16 b(as)f(an)h(algebra)f(in)228 352 y Fs(G)228 379 y(G)256 367 y Ft(Y)s(D)k Fx(in)f(the)g(ob)o(vious)f(w)o(a)o(y:)26 b Fw(g)13 b Ft(\001)f Fx(\()p Fw(v)822 373 y Fu(1)853 367 y Ft(\012)g(\001)7 b(\001)g(\001)j(\012)j Fw(v)1022 373 y Fs(n)1044 367 y Fx(\))19 b(:=)f(\()p Fw(g)c Ft(\001)d Fw(v)1234 373 y Fu(1)1253 367 y Fx(\))h Ft(\012)g(\001)7 b(\001)g(\001)k(\012)h Fx(\()p Fw(g)h Ft(\001)f Fw(v)1523 373 y Fs(n)1546 367 y Fx(\),)18 b Fw(g)i Ft(2)e Fw(G)p Fx(,)228 417 y Fw(v)248 423 y Fu(1)267 417 y Fw(;)7 b(:)g(:)g(:)t(;)g (v)379 423 y Fs(n)414 417 y Ft(2)13 b Fw(V)c Fx(;)15 b(if)e Fw(v)573 423 y Fs(i)600 417 y Ft(2)f Fw(V)664 423 y Fs(g)680 427 y Fi(i)696 417 y Fw(;)7 b Fx(1)12 b Ft(\024)h Fw(i)g 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y(ha)o(v)o(e)h(\001\()p Fw(v)q(w)q Fx(\))g(=)g(1)6 b Ft(\012)g Fw(v)q(w)h Fx(+)f Fw(v)h Ft(\012)f Fw(w)h Fx(+)f(\()p Fw(g)i Ft(\001)e Fw(w)q Fx(\))g Ft(\012)g Fw(v)h Fx(+)f Fw(v)q(w)h Ft(\012)f Fx(1)12 b(for)g Fw(v)h Ft(2)e Fw(V)1303 622 y Fs(g)1323 616 y Fw(;)c(w)12 b Ft(2)f Fw(V)e Fx(.)18 b(W)m(e)12 b(note)g(that)228 666 y(the)i(canonical)e(\014ltration)g(of)h Fw(T)6 b Fx(\()p Fw(V)k Fx(\))j(is)g(a)g(coalgebra)g(\014ltration.)k (Therefore)d(the)g(coradical)f(of)228 716 y(the)h(coalgebra)g Fw(T)6 b Fx(\()p Fw(V)j Fx(\))14 b(equals)g Fv(|)-19 b Fx(,)10 b(and)k Fw(T)6 b Fx(\()p Fw(V)j Fx(\))14 b(and)f(all)g(its)h (coalgebra)f(quotien)o(ts)h(are)g(p)q(oin)o(ted)228 766 y(irreducible)e(coalgebras)g([)p Fy(M1)p Fx(,)f(5.3.4,)f(5.3.5].)15 b(In)d(particular)f(this)h(implies)e(that)h(the)i(iden)o(tit)o(y)228 816 y(of)f Fw(T)6 b Fx(\()p Fw(V)j Fx(\))k(is)f(con)o(v)o(olution)f(in) o(v)o(ertible)h(b)o(y)h([)p Fy(M1)o Fx(,)g(5.2.10].)i(Hence)f Fw(T)6 b Fx(\()p Fw(V)j Fx(\))k(and)f(all)g(its)g(bialgebra)228 865 y(quotien)o(ts)k(in)460 850 y Fs(G)460 877 y(G)488 865 y Ft(Y)s(D)h Fx(are)f(Hopf)g(algebras)f(in)957 850 y Fs(G)957 877 y(G)985 865 y Ft(Y)s(D)q Fx(.)24 b(Note)16 b(that)g Fw(T)6 b Fx(\()p Fw(V)j Fx(\))16 b(is)g(a)f Fw(G)10 b Ft(\002)h Fv(N)p Fx(-graded)228 915 y(v)o(ector)j(space)h(as) f(eac)o(h)h Fw(T)6 b Fx(\()p Fw(V)j Fx(\)\()p Fw(n)p Fx(\))15 b(has)f(a)f(basis)h(consisting)g(of)f Fw(G)p Fx(-homogenous)f(elemen)o(ts.)303 970 y(W)m(e)h(consider)h(the)h(set) 679 960 y Fl(e)671 970 y Ft(S)i Fx(of)c(all)f(ideals)i(and)f(coideals)h Fw(I)j Fx(of)c Fw(T)6 b Fx(\()p Fw(V)j Fx(\))14 b(whic)o(h)g(are)g (generated)228 1025 y(as)d(ideals)f(b)o(y)g Fv(N)p Fx(-homo)o(geneous)f (elemen)o(ts)i(of)f(degree)i Ft(\025)g Fx(2.)k(Let)11 b Ft(S)j Fx(b)q(e)e(the)f(subset)h(of)e(all)g Fw(I)15 b Ft(2)1702 1015 y Fl(e)1694 1025 y Ft(S)228 1075 y Fx(whic)o(h)d(are)g (Y)m(etter-Drinfeld)h(submo)q(dules)e(of)h Fw(T)6 b Fx(\()p Fw(V)k 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Ft(\032)11 b Fw(P)6 b Fx(\()p Fw(T)g Fx(\()p Fw(V)k Fx(\))p Fw(=I)s Fx(\).)303 1436 y Fg(Lemma)16 b Fx(2.1)p Fg(.)k Ff(\(1\))15 b(L)n(et)g Fw(I)g Ft(2)c(S)s Ff(.)19 b(Then)c Fw(I)g Fx(=)d Fw(I)s Fx(\()p Fw(V)e Fx(\))15 b Ff(if)g(and)g(only)g(if)g Fw(V)21 b Fx(=)12 b Fw(P)6 b Fx(\()p Fw(T)g Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(\))p Ff(.)303 1496 y(\(2\))15 b Fw(I)s Fx(\()p Fw(V)10 b Fx(\))h(=)519 1485 y Fl(e)515 1496 y Fw(I)t Fx(\()p Fw(V)e Fx(\))p Ff(.)303 1594 y Fg(Pr)o(oof.)21 b Fx(\(1\))c(W)m(e)g(\014rst)h(sho)o(w)f(the)h(equalit)o(y)e Fw(V)26 b Fx(=)17 b Fw(P)6 b Fx(\()p Fw(T)g Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(\()p Fw(V)h Fx(\)\).)28 b(Let)18 b Fw(n)e Ft(\025)i Fx(2)e(and)228 1654 y(consider)d(the)h(in)o(v)o(erse)f (image)e Fw(X)16 b Fx(in)d Fw(T)6 b Fx(\()p Fw(V)j Fx(\)\()p Fw(n)p Fx(\))k(of)f(all)g(primitiv)o(e)e(elemen)o(ts)j(of)f Fw(T)6 b Fx(\()p Fw(V)k Fx(\))p Fw(=I)s Fx(\()p Fw(V)g Fx(\))j(in)228 1713 y(degree)j Fw(n)p Fx(.)22 b(Then)16 b Fw(X)j Fx(is)c(a)g(Y)m(etter-Drinfeld)g(submo)q(dule)g(of)f Fw(T)6 b Fx(\()p Fw(V)k Fx(\)\()p Fw(n)p Fx(\),)15 b(and)g(for)g(all)f Fw(x)g Ft(2)f Fw(X)s Fx(,)228 1773 y(\001\()p Fw(x)p Fx(\))e Ft(2)g Fw(x)e Ft(\012)f Fx(1)h(+)g(1)f Ft(\012)h Fw(x)g Fx(+)f Fw(T)e Fx(\()p Fw(V)k Fx(\))f Ft(\012)g Fw(I)s Fx(\()p Fw(V)h Fx(\))f(+)g Fw(I)s Fx(\()p Fw(V)h Fx(\))e Ft(\012)h Fw(T)d Fx(\()p Fw(V)k Fx(\))p Fw(:)j Fx(Hence)i(the)g(ideal)d(generated)j(b)o(y)228 1833 y Fw(I)s Fx(\()p Fw(V)10 b Fx(\))j(and)f Fw(X)17 b Fx(is)12 b(in)h Ft(S)s Fx(,)f(and)h Fw(X)i Ft(\032)d Fw(I)s Fx(\()p Fw(V)e Fx(\))j(b)o(y)f(the)i(maxim)o(al)o(it)o(y)9 b(of)k Fw(I)s Fx(\()p Fw(V)d Fx(\).)17 b(Hence)e(the)e(image)e(of)228 1893 y Fw(X)19 b Fx(in)14 b Fw(T)6 b Fx(\()p Fw(V)k Fx(\))p Fw(=I)s Fx(\()p Fw(V)g Fx(\))15 b(is)g(zero.)23 b(This)15 b(pro)o(v)o(es)g(our)g(claim)e(since)j(the)g(primitiv)o(e)d(elemen)o (ts)i(form)228 1952 y(a)e(graded)i(submo)q(dule.)303 2012 y(No)o(w)21 b(assume)h(that)g Fw(V)34 b Fx(=)25 b Fw(P)6 b Fx(\()p Fw(T)g Fx(\()p Fw(V)k Fx(\))p Fw(=I)s Fx(\).)42 b(Then)23 b(the)f(surjectiv)o(e)h(map)e Fw(T)6 b Fx(\()p Fw(V)j Fx(\))p Fw(=I)29 b Ft(!)228 2072 y Fw(T)6 b Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(\()p Fw(V)h Fx(\))i(of)f (irreducible)i(coalgebras)f(is)f(injectiv)o(e)h(b)o(y)f([)p Fy(M1)p Fx(,)h(5.3.3])d(since)k(b)o(y)e(assumption)228 2132 y(it)i(is)h(injectiv)o(e)g(on)g(the)g(primitiv)o(e)e(elemen)o(ts.) 18 b(Hence)d Fw(I)g Fx(=)d Fw(I)s Fx(\()p Fw(V)e Fx(\).)303 2192 y(\(2\))19 b(As)h(in)e(\(1\))i(w)o(e)f(see)i(that)e Fw(T)6 b Fx(\()p Fw(V)k Fx(\))p Fw(=I)s Fx(\()p Fw(V)g Fx(\))20 b Ft(!)g Fw(T)6 b Fx(\()p Fw(V)k Fx(\))p Fw(=)1209 2181 y Fl(e)1205 2192 y Fw(I)s Fx(\()p Fw(V)g Fx(\))19 b(is)h(bijectiv)o(e)f(since)h Fw(V)30 b Fx(=)228 2251 y Fw(P)6 b Fx(\()p Fw(T)g Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(\()p Fw(V)h Fx(\)\))k(b)o(y)g(\(1\).)p 1692 2251 2 29 v 1694 2225 25 2 v 1694 2251 V 1719 2251 2 29 v 303 2343 a(As)j(in)g([)p Fy(AS1)o Fx(],)g(w)o(e)h(call)e Fo(B)p Fx(\()p Fw(V)10 b Fx(\))17 b(:=)g Fw(T)6 b Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(\()p Fw(V)h Fx(\))18 b(the)g Ff(Nichols)f(algebr)n(a)g Fx(of)g Fw(V)27 b Ft(2)1618 2328 y Fs(G)1618 2355 y(G)1646 2343 y Ft(Y)s(D)r Fx(.)228 2394 y(More)11 b(generally)m(,)g(if)f Fw(U)16 b Fx(is)11 b(a)g(braided)g(subspace)i(of)d Fw(V)21 b Ft(2)1111 2379 y Fs(G)1111 2405 y(G)1139 2394 y Ft(Y)s(D)r Fx(,)11 b(that)g(is)g(a)g (subspace)h(suc)o(h)g(that)228 2450 y Fw(c)p Fx(\()p Fw(U)i Ft(\012)c Fw(U)5 b Fx(\))11 b Ft(\032)h Fw(U)i Ft(\012)9 b Fw(U)c Fx(,)14 b(where)h Fw(c)f Fx(is)f(the)i(braiding)e (of)g Fw(V)c Fx(,)14 b(w)o(e)g(de\014ne)h Fo(B)p Fx(\()p Fw(U)5 b Fx(\))11 b(:=)g Fw(T)6 b Fx(\()p Fw(U)f Fx(\))p Fw(=)1596 2440 y Fl(e)1592 2450 y Fw(I)t Fx(\()p Fw(U)g Fx(\).)303 2548 y Fg(Remark)16 b Fx(2.2)p Fg(.)k Fx(1\))d(By)f(Lemma)e (2.1,)i Fo(B)p Fx(\()p Fw(V)9 b Fx(\))16 b(as)h(an)f(algebra)g(and)g (coalgebra)g(only)g(de-)228 2608 y(p)q(ends)f(on)e(the)h(braiding)f(of) g Fw(V)c Fx(.)18 b(The)c(relations)f(in)h(eac)o(h)g(degree)h(of)e Fw(I)s Fx(\()p Fw(V)d Fx(\))j(can)h(b)q(e)g(describ)q(ed)p eop %%Page: 4 4 4 3 bop 228 119 a Fu(4)262 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)228 213 y Fx(b)o(y)17 b(the)g(action)g(of)f(the)i(braid)f(group)g(\(see)h(for)f (example)e([)p Fy(Sb)o Fx(,)i(2.9,)g(2.12]\).)26 b(Ho)o(w)o(ev)o(er)17 b(this)228 272 y(do)q(es)d(not)g(mean)f(that)h(the)g(relations)g(are)g (kno)o(wn)g(explicitly)m(.)303 332 y(2\))k(Let)h Fw(R)f Fx(=)539 301 y Fl(L)585 345 y Fs(n)p Fn(\025)p Fu(0)657 332 y Fw(R)p Fx(\()p Fw(n)p Fx(\))g(b)q(e)h(a)f(graded)h(Hopf)f (algebra)g(in)1314 317 y Fs(G)1314 344 y(G)1342 332 y Ft(Y)s(D)h Fx(with)f Fw(R)p Fx(\(0\))h(=)g Fv(|)-19 b Fx(1)228 392 y(and)16 b Fw(R)p Fx(\(1\))g(=)g Fw(P)6 b Fx(\()p Fw(R)p Fx(\))16 b(=)g Fw(V)9 b Fx(.)26 b(Then)17 b(b)o(y)g(Lemma)d(2.1,)h Fo(B)p Fx(\()p Fw(V)10 b Fx(\))16 b(is)h(the)g(subalgebra)f Fv(|)-19 b Fx([)p Fw(V)6 b Fx(])16 b(of)g Fw(R)228 452 y Fx(generated)f(b)o(y)f Fw(V)9 b Fx(,)k(since)i Fv(|)-19 b Fx([)p Fw(V)6 b Fx(])14 b(is)f(of)h(the)g(form)e Fw(T)6 b Fx(\()p Fw(V)k Fx(\))p Fw(=I)17 b Fx(with)d(primitiv)o(e)d(elemen)o(ts)j Fw(V)9 b Fx(.)303 511 y(3\))i(Let)h Fw(U)k Fx(b)q(e)c(a)f(braided)h(subspace)h (of)e Fw(V)21 b Ft(2)1001 496 y Fs(G)1001 523 y(G)1029 511 y Ft(Y)s(D)q Fx(.)d(Then)12 b(the)g(canonical)e(map)g Fo(B)p Fx(\()p Fw(U)5 b Fx(\))12 b Ft(!)228 571 y Fo(B)p Fx(\()p Fw(V)d Fx(\))14 b(is)g(injectiv)o(e)g(since)g(it)g(is)g (injectiv)o(e)g(on)f(the)i(primitiv)o(e)c(elemen)o(ts)j(b)o(y)g(Lemma)d (2.1.)303 631 y(4\))16 b(Let)h Fw(R)f Fx(=)530 600 y Fl(L)576 643 y Fs(n)p Fn(\025)p Fu(0)648 631 y Fw(R)p Fx(\()p Fw(n)p Fx(\))g(b)q(e)i(a)e(\014nite-dimensional)e(graded)j (Hopf)g(algebra)f(in)1630 616 y Fs(G)1630 642 y(G)1658 631 y Ft(Y)s(D)228 691 y Fx(with)f Fw(R)p Fx(\(0\))e(=)h Fv(|)-19 b Fx(1)12 b(whic)o(h)j(is)g(generated)h(as)f(an)g(algebra)g(b) o(y)g Fw(R)p Fx(\(1\),)f(and)h(let)h Fw(x)1476 697 y Fu(1)1494 691 y Fw(;)7 b(:)g(:)g(:)e(;)i(x)1611 697 y Fs(\022)1644 691 y Fx(b)q(e)15 b(a)228 751 y(basis)f(of)f Fw(R)p Fx(\(1\).)18 b(Then)c(there)h(is)f(a)f(non-zero)h(monomia)o(l)d (\003)g(=)h Fw(x)1250 757 y Fs(i)1262 761 y Fj(1)1286 751 y Ft(\001)7 b(\001)g(\001)f Fw(x)1366 757 y Fs(i)1378 761 y Fi(N)1406 751 y Fx(,)13 b(1)f Ft(\024)f Fw(i)1521 757 y Fu(1)1540 751 y Fw(;)c(:)g(:)g(:)e(;)i(i)1647 757 y Fs(N)1690 751 y Ft(\024)228 810 y Fw(\022)q Fx(,)16 b(of)g(maxim)o(al)d(length)j Fw(N)5 b Fx(.)24 b(Hence)17 b Fw(x)850 816 y Fs(i)864 810 y Fx(\003)e(=)g(0)g(=)h(\003)p Fw(x)1092 816 y Fs(i)1121 810 y Fx(for)g(all)e(1)h Ft(\024)h Fw(i)f Ft(\024)g Fw(\022)q Fx(,)i(and)f(\003)g(is)g(a)f(left)228 870 y(and)h(righ)o(t)g(in)o(tegral)g(of)g Fw(R)p Fx(.)26 b(Since)17 b Fw(R)g Fx(is)f(a)h(F)m(rob)q(enius)g(algebra)f([)p Fy(FMS)o Fx(],)g(the)i(space)f(of)f(left)228 930 y(\(or)g(righ)o(t\))g (in)o(tegrals)f(is)h(one-dimensional.)23 b(Hence)17 b Fw(R)e Fx(=)1177 899 y Fl(L)1223 909 y Fs(N)1223 942 y(n)p Fu(=0)1294 930 y Fw(R)p Fx(\()p Fw(n)p Fx(\),)h(and)g Fw(R)p Fx(\()p Fw(N)5 b Fx(\))15 b(=)h Fv(|)-19 b Fx(\003.)228 990 y(Since)17 b Fw(R)f Fx(is)g(a)g(graded)g(coalgebra,)h(the)f(linear) g(function)g Fw(\025)g Fx(:)f Fw(R)g Ft(!)g Fv(|)-5 b Fx(with)16 b Fw(\025)p Fx(\(\003\))g(=)f(1)h(and)228 1049 y Fw(\025)p Fx(\()p Fw(R)p Fx(\()p Fw(n)p Fx(\)\))d(=)h(0)g(for)g (all)g(0)f Ft(\024)g Fw(n)g Ft(\024)g Fw(N)i Ft(\000)10 b Fx(1)k(is)h(a)g(t)o(w)o(o-sided)f(in)o(tegral)g(of)g Fw(R)1373 1034 y Fn(\003)1392 1049 y Fx(.)21 b(Therefore)16 b(for)e(all)228 1109 y(0)d Ft(\024)h Fw(i)g Ft(\024)g Fw(N)5 b Fx(,)616 1174 y Fw(R)p Fx(\()p Fw(i)p Fx(\))10 b Ft(\012)f Fw(R)p Fx(\()p Fw(N)14 b Ft(\000)c Fw(i)p Fx(\))i Ft(!)f Fv(|)-19 b Fw(;)45 b(x)9 b Ft(\012)g Fw(y)k Ft(7!)f Fw(\025)p Fx(\()p Fw(xy)q Fx(\))p Fw(;)228 1247 y Fx(is)i(a)f(non-degenerate)j(pairing)c(\(cf.)19 b([)p Fy(N)o Fx(,)14 b(1.5]\).)j(In)c(particular,)h(dim)n Fw(R)p Fx(\()p Fw(i)p Fx(\))e(=)g(dim)n Fw(R)p Fx(\()p Fw(N)i Ft(\000)9 b Fw(i)p Fx(\).)303 1330 y(The)19 b(crucial)g(prop)q(ert)o(y) h Fw(V)30 b Fx(=)20 b Fw(P)6 b Fx(\()p Fw(B)r Fx(\()p Fw(V)k Fx(\)\))19 b(can)h(b)q(e)f(c)o(haracterized)i(as)f(follo)o(ws.) 32 b(F)m(or)18 b(a)228 1380 y(graded)h(coalgebra)g Fw(R)h Fx(=)663 1349 y Fl(L)709 1392 y Fs(n)p Fn(\025)p Fu(0)781 1380 y Fw(R)p Fx(\()p Fw(n)p Fx(\))f(with)g(pro)r(jections)h Fw(\031)1232 1386 y Fs(n)1274 1380 y Fx(:)f Fw(R)h Ft(!)g Fw(R)p Fx(\()p Fw(n)p Fx(\))p Fw(;)7 b(n)19 b Ft(\025)i Fx(0)p Fw(;)d Fx(w)o(e)228 1431 y(denote)d(b)o(y)492 1498 y(\001)527 1504 y Fs(i;j)577 1498 y Fx(:)c Fw(R)p Fx(\()p Fw(i)f Fx(+)f Fw(j)r Fx(\))775 1474 y Fu(\001)760 1498 y Ft(\000)-17 b(!)11 b Fw(R)e Ft(\012)g Fw(R)968 1470 y Fs(\031)987 1474 y Fi(i)1000 1470 y Fn(\012)p Fs(\031)1045 1474 y Fi(j)954 1498 y Ft(\000)-16 b(\000)-9 b(\000)g(\000)-14 b(!)11 b Fw(R)p Fx(\()p Fw(i)p Fx(\))f Ft(\012)f Fw(R)p Fx(\()p Fw(j)r Fx(\))p Fw(;)e(i;)g(j)14 b Ft(\025)e Fx(0)p Fw(;)228 1561 y Fx(the)i(\()p Fw(i;)7 b(j)r Fx(\)-th)15 b(comp)q(onen)o(t)e(of)g(the)i(com)o(ultiplicatio)o (n)c(\001.)18 b(More)c(generally)m(,)407 1633 y(\001)442 1639 y Fs(i)454 1643 y Fj(1)469 1639 y Fs(;:::;i)531 1643 y Fi(k)562 1633 y Fx(:)d Fw(R)p Fx(\()p Fw(i)647 1639 y Fu(1)675 1633 y Fx(+)f Ft(\001)d(\001)g(\001)g Fx(+)j Fw(i)830 1639 y Fs(k)851 1633 y Fx(\))h Ft(!)g Fw(R)p Fx(\()p Fw(i)993 1639 y Fu(1)1012 1633 y Fx(\))f Ft(\012)f(\001)e(\001)g(\001)h(\012)h Fw(R)p Fx(\()p Fw(i)1240 1639 y Fs(k)1261 1633 y Fx(\))p Fw(;)e(i)1310 1639 y Fu(1)1329 1633 y Fw(;)g(:)g(:)g(:)t(;)g(i)1435 1639 y Fs(k)1467 1633 y Ft(\025)12 b Fx(0)p Fw(;)228 1705 y Fx(denotes)j(the)f(\()p Fw(i)479 1711 y Fu(1)498 1705 y Fw(;)7 b(:)g(:)g(:)e(;)i(i)605 1711 y Fs(k)625 1705 y Fx(\)-th)14 b(comp)q(onen)o(t)g(of)f(the)h(\()p Fw(k)d Ft(\000)e Fx(1\)-fold)k(iteration)g(of)h(\001.)303 1798 y Fg(Lemma)i Fx(2.3)p Fg(.)k Ff(L)n(et)13 b Fw(R)e Fx(=)701 1767 y Fl(L)747 1810 y Fs(n)p Fn(\025)p Fu(0)819 1798 y Fw(R)p Fx(\()p Fw(n)p Fx(\))i Ff(b)n(e)g(a)g(gr)n(ade)n(d)g(c)n (o)n(algebr)n(a)g(with)f(gr)n(oup-like)g(element)228 1858 y(1)j(and)g Fw(R)p Fx(\(0\))d(=)g Fv(|)-19 b Fx(1)p Ff(.)15 b(Then)h(the)f(fol)r(lowing)f(ar)n(e)g(e)n(quivalent:)303 1917 y(\(1\))h(P\(R\))g(=)f(R\(1\).)303 1977 y(\(2\))h(F)m(or)f(al)r(l) g Fw(n)e Ft(\025)g Fx(2)p Fw(;)i Fx(\001)672 1983 y Fu(1)p Fs(;:::)n(;)p Fu(1)767 1977 y Fx(:)d Fw(R)p Fx(\()p Fw(n)p Fx(\))h Ft(!)f Fw(R)p Fx(\(1\))1029 1962 y Fn(\012)p Fs(n)1092 1977 y Ff(is)k(inje)n(ctive.)303 2037 y(\(3\))g(F)m(or)f(al)r (l)g Fw(i;)7 b(j)14 b Ft(\025)e Fx(0)p Ff(,)i Fx(\001)700 2043 y Fs(i;j)751 2037 y Fx(:)d Fw(R)p Fx(\()p Fw(i)e Fx(+)h Fw(j)r Fx(\))i Ft(!)f Fw(R)p Fx(\()p Fw(i)p Fx(\))e Ft(\012)h Fw(R)p Fx(\()p Fw(j)r Fx(\))15 b Ff(is)g(inje)n(ctive.)303 2097 y(\(4\))g(F)m(or)f(al)r(l)g Fw(n)e Ft(\025)g Fx(2)p Ff(,)i Fx(\001)673 2103 y Fs(n)p Fn(\000)p Fu(1)p Fs(;)p Fu(1)776 2097 y Fx(:)d Fw(R)p Fx(\()p Fw(n)p Fx(\))g Ft(!)g Fw(R)p Fx(\()p Fw(n)e Ft(\000)h Fx(1\))f Ft(\012)h Fw(R)p Fx(\(1\))k Ff(is)h(inje)n(ctive.)303 2190 y Fg(Pr)o(oof.)21 b Fx(Let)d Fw(x)f Ft(2)h Fw(R)p Fx(\()p Fw(n)p Fx(\))p Fw(;)7 b(n)16 b Ft(\025)i Fx(2)p Fw(;)f Fx(and)g(\001\()p Fw(x)p Fx(\))g(=)1122 2159 y Fl(P)1165 2169 y Fs(n)1165 2202 y(i)p Fu(=0)1228 2190 y Fw(y)1248 2196 y Fs(i)1280 2190 y Fx(where)h Fw(y)1423 2196 y Fs(i)1455 2190 y Fx(=)g(\001)1540 2196 y Fs(i;n)p Fn(\000)p Fs(i)1621 2190 y Fx(\()p Fw(x)p Fx(\))g Ft(2)228 2249 y Fw(R)p Fx(\()p Fw(i)p Fx(\))13 b Ft(\012)f Fw(R)p Fx(\()p Fw(n)h Ft(\000)f Fw(i)p Fx(\))19 b(for)g(all)e(0)i Ft(\024)h Fw(i)g Ft(\024)g Fw(n)p Fx(.)32 b(F)m(or)18 b(all)g Fw(m)i Ft(\025)f Fx(1,)h(let)e(\010)1316 2255 y Fs(m)1366 2249 y Fx(b)q(e)i(the)f(comp)q(osition)228 2324 y(\010)258 2330 y Fs(m)301 2324 y Fx(:)11 b Fw(R)381 2301 y Fu(\001)408 2288 y Fi(m)p Fk(\000)p Fj(1)367 2324 y Ft(\000)-16 b(\000)-9 b(\000)g(\000)-15 b(!)11 b Fw(R)531 2309 y Fn(\012)p Fs(m)614 2295 y(\031)634 2282 y Fk(\012)p Fi(m)633 2304 y Fj(1)599 2324 y Ft(\000)-15 b(\000)-9 b(\000)-14 b(!)11 b Fw(R)p Fx(\(1\))795 2309 y Fn(\012)p Fs(m)852 2324 y Fw(:)i Fx(Hence)h(\001)1034 2330 y Fu(1)p Fs(;:::)n(;)p Fu(1)1131 2324 y Fx(is)f(the)g(restriction)h(of)e(\010) 1516 2330 y Fs(m)1560 2324 y Fx(to)h Fw(R)p Fx(\()p Fw(m)p Fx(\).)228 2384 y(Then)h(for)g(all)e(1)g Ft(\024)f Fw(i)h Ft(\024)g Fw(n)d Ft(\000)h Fx(1,)548 2466 y(\010)578 2472 y Fs(n)600 2466 y Fx(\()p Fw(x)p Fx(\))i(=)g(\(\010)758 2472 y Fs(i)781 2466 y Ft(\012)d Fx(\010)852 2472 y Fs(n)p Fn(\000)p Fs(i)913 2466 y Fx(\)\(\001\()p Fw(x)p Fx(\)\))i(=)h(\(\010) 1153 2472 y Fs(i)1176 2466 y Ft(\012)e Fx(\010)1248 2472 y Fs(n)p Fn(\000)p Fs(i)1308 2466 y Fx(\)\()p Fw(y)1360 2472 y Fs(i)1374 2466 y Fx(\))p Fw(:)228 2548 y Fx(This)16 b(form)o(ula)d(sho)o(ws)j(that)g(\(1\))g Ft(\))g Fx(\(2\))f(b)o(y)h (induction)g(on)f Fw(n)h Fx(\(as)g(in)g([)p Fy(AS2)o Fx(,)g(Lemma)d(5.5]\),)228 2608 y(and)h(also)f(\(2\))h Ft(\))f Fx(\(3\).)p eop %%Page: 5 5 5 4 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d(O) o(VER)i(CO)o(XETER)g(GR)o(OUPS)112 b(5)303 213 y Fx(\(3\))15 b Ft(\))g Fx(\(4\))h(is)f(trivial,)f(and)i(\(4\))f Ft(\))g Fx(\(1\))h(is)f(clear)h(since)h(b)o(y)e(\(4\),)g Fw(R)p Fx(\()p Fw(n)p Fx(\))c Ft(\\)f Fw(P)c Fx(\()p Fw(R)p Fx(\))14 b(=)g(0)h(for)228 272 y(all)d Fw(n)g Ft(\025)g Fx(2)p Fw(:)p 1692 272 2 29 v 1694 246 25 2 v 1694 272 V 1719 272 2 29 v 303 366 a Fx(All)j(the)h(previous)g(results)h(hold)e (more)g(generally)h(for)f(Y)m(etter-Drinfeld)h(mo)q(dules)f(o)o(v)o(er) 228 416 y(Hopf)e(algebras)h(with)g(bijectiv)o(e)g(an)o(tip)q(o)q(de)g (instead)g(of)f(group)h(algebras.)k(W)m(e)c(no)o(w)f(recall)h(an)228 466 y(imp)q(ortan)o(t)8 b(to)q(ol)h(in)o(tro)q(duced)h(b)o(y)g(Nic)o (hols)f([)p Fy(N)p Fx(,)h(3.3])e(to)i(deal)f(with)h Fo(B)p Fx(\()p Fw(V)f Fx(\))h(o)o(v)o(er)g(group)f(algebras)228 516 y(without)16 b(kno)o(wing)f(the)i(relations)f(explicitly)m(.)24 b(Let)17 b Fw(V)25 b Ft(2)1165 501 y Fs(G)1165 527 y(G)1193 516 y Ft(Y)s(D)18 b Fx(b)q(e)f(of)f(\014nite)g(dimension)f Fw(\022)q Fx(.)228 566 y(W)m(e)c(c)o(ho)q(ose)i(a)e(basis)h Fw(x)583 572 y Fs(i)608 566 y Ft(2)g Fw(V)672 572 y Fs(g)688 576 y Fi(i)715 566 y Fx(with)g Fw(g)828 572 y Fs(i)853 566 y Ft(2)f Fw(G;)c Fx(1)j Ft(\024)i Fw(i)g Ft(\024)g Fw(\022)q(;)g Fx(of)f Fw(G)p Fx(-homogeneous)f(elemen)o(ts.)18 b(Let)228 617 y Fw(I)d Ft(2)c(S)17 b Fx(and)c Fw(R)f Fx(=)g Fw(T)6 b Fx(\()p Fw(V)j Fx(\))p Fw(=I)s Fx(.)19 b(Then)14 b Fw(R)f Fx(is)h(a)f(graded)h(Hopf)f(algebra)h(in)1338 602 y Fs(G)1338 628 y(G)1366 617 y Ft(Y)s(D)h Fx(with)e Fw(R)p Fx(\(0\))f(=)f Fv(|)-19 b Fx(1)228 666 y(and)12 b Fw(R)p Fx(\(1\))f(=)h Fw(V)d Fx(.)18 b(F)m(or)11 b(all)g(1)g Ft(\024)h Fw(i)g Ft(\024)g Fw(\022)h Fx(let)g Fw(\033)899 672 y Fs(i)924 666 y Fx(:)e Fw(R)g Ft(!)g Fw(R)h Fx(b)q(e)h(the)f (algebra)g(automorphism)d(giv)o(en)228 716 y(b)o(y)k(the)i(action)e(of) h Fw(g)549 722 y Fs(i)562 716 y Fx(.)303 815 y Fg(Pr)o(oposition)h Fx(2.4)p Fg(.)21 b Ff(L)n(et)14 b Fw(R;)7 b(x)803 821 y Fs(i)816 815 y Fw(;)g(\033)859 821 y Fs(i)872 815 y Fw(;)g Fx(1)k Ft(\024)h Fw(i)f Ft(\024)h Fw(\022)17 b Ff(as)e(ab)n(ove.)303 875 y(1\))20 b(F)m(or)f(al)r(l)h Fx(1)g Ft(\024)h Fw(i)g Ft(\024)g Fw(\022)q Ff(,)h(ther)n(e)d(exists)h (a)g(uniquely)g(determine)n(d)g Fx(\()p Fw(id;)7 b(\033)1496 881 y Fs(i)1509 875 y Fx(\))p Ff(-derivation)228 935 y Fw(D)262 941 y Fs(i)288 935 y Fx(:)k Fw(R)g Ft(!)g Fw(R)k Ff(with)f Fw(D)578 941 y Fs(i)592 935 y Fx(\()p Fw(x)632 941 y Fs(j)649 935 y Fx(\))e(=)g Fw(\016)739 941 y Fs(i;j)793 935 y Ff(\(Kr)n(one)n(cker)j Fw(\016)r Ff(\))g(for)f(al)r(l)g Fw(j)r Ff(.)303 994 y(2\))h Fw(I)g Fx(=)c Fw(I)s Fx(\()p Fw(V)g Fx(\))p Ff(,)j(that)h(is)g Fw(R)c Fx(=)h Fo(B)p Fx(\()p Fw(V)d Fx(\))p Ff(,)15 b(if)f(and)i(only)f (if)1145 963 y Fl(T)1180 974 y Fs(\022)1180 1007 y(i)p Fu(=1)1243 994 y Ff(ker)o Fx(\()p Fw(D)1348 1000 y Fs(i)1363 994 y Fx(\))c(=)h Fv(|)-19 b Fx(1)p Fw(:)303 1093 y Fg(Pr)o(oof.)21 b Fx(1\))14 b(F)m(or)g(all)f(1)e Ft(\024)h Fw(i)g Ft(\024)g Fw(\022)q Fx(,)i(let)g Fw(D)936 1099 y Fs(i)950 1093 y Fx(\(1\))e(=)g(0,)h(and)h(de\014ne)h(for)e(all)g Fw(x)f Ft(2)f Fw(R)p Fx(\()p Fw(n)p Fx(\))p Fw(;)c(n)k Ft(\025)h Fx(1)p Fw(;)228 1153 y Fx(elemen)o(ts)i Fw(D)432 1159 y Fs(i)446 1153 y Fx(\()p Fw(x)p Fx(\))d Ft(2)g Fw(R)p Fx(\()p Fw(n)f Ft(\000)f Fx(1\))14 b(b)o(y)721 1275 y(\001)756 1281 y Fs(n)p Fn(\000)p Fu(1)p Fs(;)p Fu(1)847 1275 y Fx(\()p Fw(x)p Fx(\))d(=)980 1223 y Fs(\022)958 1235 y Fl(X)961 1324 y Fs(i)p Fu(=1)1025 1275 y Fw(D)1059 1281 y Fs(i)1073 1275 y Fx(\()p Fw(x)p Fx(\))f Ft(\012)f Fw(x)1204 1281 y Fs(i)1218 1275 y Fw(:)228 1394 y Fx(Then)14 b(for)g(all)e Fw(n;)7 b(m)12 b Ft(\025)g Fx(1)p Fw(;)7 b(x)j Ft(2)h Fw(R)p Fx(\()p Fw(n)p Fx(\))p Fw(;)c(y)13 b Ft(2)e Fw(R)p Fx(\()p Fw(m)p Fx(\))p Fw(;)299 1485 y Fx(\001\()p Fw(xy)q Fx(\))h(=)g(\001\()p Fw(x)p Fx(\)\001\()p Fw(y)q Fx(\))f(=)h(\()p Fw(x)d Ft(\012)h Fx(1)f(+)g Fw(D)897 1491 y Fs(i)912 1485 y Fx(\()p Fw(x)p Fx(\))g Ft(\012)g Fw(x)1042 1491 y Fs(i)1065 1485 y Fx(+)h Fw(:)d(:)g(:)e Fx(\)\()p Fw(y)11 b Ft(\012)f Fx(1)f(+)g Fw(D)1372 1491 y Fs(i)1387 1485 y Fx(\()p Fw(y)q Fx(\))h Ft(\012)f Fw(x)1515 1491 y Fs(i)1538 1485 y Fx(+)h Fw(:)d(:)g(:)e Fx(\))657 1557 y(=)12 b Fw(xy)f Ft(\012)f Fx(1)f(+)g Fw(xD)927 1563 y Fs(i)941 1557 y Fx(\()p Fw(y)q Fx(\))h Ft(\012)g Fw(x)1070 1563 y Fs(i)1093 1557 y Fx(+)f Fw(D)1168 1563 y Fs(i)1182 1557 y Fx(\()p Fw(x)p Fx(\)\()p Fw(g)1274 1563 y Fs(i)1297 1557 y Ft(\001)g Fw(y)q Fx(\))h Ft(\012)g Fw(x)1431 1563 y Fs(i)1454 1557 y Fx(+)f Fw(:)e(:)g(:)228 1648 y Fx(b)o(y)12 b(the)h(de\014nition)g(of)f(the)h(braided)g(m)o (ultipli)o(cation)d(in)i Fw(R)6 b Ft(\012)h Fw(R)p Fx(.)18 b(Hence)c Fw(D)1410 1654 y Fs(i)1424 1648 y Fx(\()p Fw(xy)q Fx(\))e(=)g Fw(xD)1615 1654 y Fs(i)1629 1648 y Fx(\()p Fw(y)q Fx(\))7 b(+)228 1708 y Fw(D)262 1714 y Fs(i)276 1708 y Fx(\()p Fw(x)p Fx(\))p Fw(\033)356 1714 y Fs(i)370 1708 y Fx(\()p Fw(y)q Fx(\))p Fw(:)14 b Fx(Since)h(the)f(elemen)o(ts)g Fw(x)823 1714 y Fs(j)854 1708 y Fx(are)h(primitiv)o(e,)c Fw(D)1147 1714 y Fs(i)1161 1708 y Fx(\()p Fw(x)1201 1714 y Fs(j)1219 1708 y Fx(\))h(=)g Fw(\016)1309 1714 y Fs(ij)1352 1708 y Fx(for)i(all)f Fw(i;)7 b(j)r Fx(,)13 b(and)h Fw(D)1666 1714 y Fs(i)1694 1708 y Fx(is)228 1768 y(uniquely)f(determined)h(b)o(y) g(this)g(equation)f(since)i Fw(R)f Fx(is)f(generated)i(b)o(y)f Fw(x)1381 1774 y Fu(1)1399 1768 y Fw(;)7 b(:)g(:)g(:)e(;)i(x)1516 1774 y Fs(\022)1534 1768 y Fx(.)303 1827 y(2\))18 b(By)g(Lemma)e(2.1,)i Fw(I)k Fx(=)e Fw(I)s Fx(\()p Fw(V)10 b Fx(\))18 b(if)g(and)g(only)f(if) h Fw(P)6 b Fx(\()p Fw(R)p Fx(\))18 b(=)h Fw(R)p Fx(\(1\).)32 b(Hence)19 b(the)g(claim)228 1887 y(follo)o(ws)12 b(from)h(Lemma)e (2.3,)i(since)792 1856 y Fl(T)827 1866 y Fs(\022)827 1900 y(i)p Fu(=1)889 1887 y Fx(k)o(er)q(\()p Fw(D)995 1893 y Fs(i)1009 1887 y Fx(\))f(=)h Fv(|)-19 b Fx(1)11 b(if)i(and)h(only)g(if)f(\001)1424 1893 y Fs(n)p Fn(\000)p Fu(1)p Fs(;)p Fu(1)1529 1887 y Fx(is)h(injectiv)o(e)228 1947 y(for)f(all)g Fw(n)e Ft(\025)h Fx(2.)p 1692 1947 V 1694 1920 25 2 v 1694 1947 V 1719 1947 2 29 v 303 2041 a(F)m(or)h(later)h(use)h(w)o(e)f(note)303 2140 y Fg(Lemma)i Fx(2.5)p Fg(.)k Ff(L)n(et)15 b Fw(V)r(;)7 b(x)685 2146 y Fs(i)698 2140 y Fw(;)g(D)751 2146 y Fs(i)765 2140 y Fw(;)g Fx(1)k Ft(\024)h Fw(i)f Ft(\024)h Fw(\022)q(;)j Ff(b)n(e)g(as)g(in)g(Pr)n(op)n(osition)g(2.4.)303 2200 y(1\))i(F)m(or)h(al)r(l)f Fx(1)f Ft(\024)h Fw(i)g Ft(\024)g Fw(\022)q(;)g Ff(let)h Fo(B)p Fx(\()p Fw(V)9 b Fx(\))880 2185 y Fu(\()p Fs(i)p Fu(\))937 2200 y Ff(b)n(e)18 b(the)g(sub)n (algebr)n(a)f(of)h Fo(B)p Fx(\()p Fw(V)9 b Fx(\))18 b Ff(gener)n(ate)n(d)g(by)g(al)r(l)228 2259 y Fw(x)252 2265 y Fs(j)269 2259 y Fw(;)7 b Fx(1)16 b Ft(\024)h Fw(i)g Ft(\024)f Fw(\022)q(;)7 b(j)19 b Ft(6)p Fx(=)e Fw(i)p Ff(.)28 b(Assume)18 b Fw(a;)7 b(b)15 b Ft(2)i Fo(B)p Fx(\()p Fw(V)9 b Fx(\))1011 2244 y Fu(\()p Fs(i)p Fu(\))1068 2259 y Ff(with)17 b Fw(a)11 b Fx(+)h Fw(bx)1280 2265 y Fs(i)1310 2259 y Fx(=)17 b(0)p Ff(.)27 b(Then)18 b Fw(a)e Fx(=)h(0)g Ff(and)228 2319 y Fw(b)11 b Fx(=)h(0)p Ff(.)303 2379 y(2\))h(L)n(et)f Fx(1)g Ft(\024)g Fw(i)514 2385 y Fu(1)533 2379 y Fw(;)7 b(:)g(:)g(:)t(;)g(i)639 2385 y Fs(n)673 2379 y Ft(\024)12 b Fw(\022)q Ff(,)i Fw(n)d Ft(\025)h Fx(1)p Ff(,)h(and)h Fw(i)984 2385 y Fs(k)1016 2379 y Ft(6)p Fx(=)e Fw(i)1074 2385 y Fs(l)1100 2379 y Ff(for)g(al)r(l)h Fw(k)f Ft(6)p Fx(=)g Fw(l)q Ff(.)18 b(Then)c Fw(x)1475 2385 y Fs(i)1487 2389 y Fj(1)1504 2379 y Fw(x)1528 2385 y Fs(i)1540 2389 y Fj(2)1565 2379 y Fw(:)7 b(:)g(:)e(x)1644 2385 y Fs(i)1656 2389 y Fi(n)1690 2379 y Ft(6)p Fx(=)228 2439 y(0)14 b Ff(in)h Fo(B)p Fx(\()p Fw(V)10 b Fx(\))p Ff(.)303 2538 y Fg(Pr)o(oof.)21 b Fx(T)m(o)15 b(pro)o(v)o(e)g(1\))g(note)h(that)f Fw(D)913 2544 y Fs(i)928 2538 y Fx(\()p Fo(B)p Fx(\()p Fw(V)9 b Fx(\))1046 2523 y Fu(\()p Fs(i)p Fu(\))1086 2538 y Fx(\))14 b(=)g(0.)22 b(Hence)17 b Fw(b)c Fx(=)i Fw(D)1454 2544 y Fs(i)1468 2538 y Fx(\()p Fw(a)10 b Fx(+)g Fw(bx)1600 2544 y Fs(i)1614 2538 y Fx(\))k(=)g(0.)228 2597 y(2\))g(follo)o(ws)e(from)g(1\))i(b)o(y) f(induction)h(on)f Fw(n)p Fx(.)p 1692 2597 V 1694 2571 25 2 v 1694 2597 V 1719 2597 2 29 v eop %%Page: 6 6 6 5 bop 228 119 a Fu(6)262 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)397 213 y Fy(3.)24 b(Non-categorical)14 b(splitti)o(ng)f(of)i(Y)l(etter-Drinf)o (el)o(d)e(mo)q(dules)303 287 y Fx(In)i(this)h(section)h(w)o(e)f(use)g (a)g(braided)g(v)o(ersion)f(of)h(the)g(fundamen)o(tal)e(theorem)h(on)h (Hopf)228 337 y(mo)q(dules)d(to)g(pro)o(v)o(e)h(freeness)i(of)e (certain)g(extensions)h(of)e(Y)m(etter-Drinfeld)h(Hopf)g(algebras.)303 387 y(Let)k Fw(L)f Fx(b)q(e)h(a)g(Hopf)f(algebra)g(with)g(bijectiv)o(e) g(an)o(tip)q(o)q(de,)h Fw(H)j Fx(a)c(Hopf)g(algebra)g(in)1633 372 y Fs(L)1633 398 y(L)1658 387 y Ft(Y)s(D)228 437 y Fx(\(see)f([)p Fy(M1)p Fx(,)e(10.6.10])e(for)i(the)h(general)g (de\014nition\))g(and)f Fw(V)24 b Fx(a)15 b(righ)o(t)f Fw(H)s Fx(-mo)q(dule)f(and)h(a)h(righ)o(t)228 487 y Fw(H)s Fx(-como)q(dule)d(with)h(structure)j(maps)c Fw(\026)870 493 y Fs(V)911 487 y Fx(:)f Fw(V)18 b Ft(\012)9 b Fw(H)14 b Ft(!)d Fw(V)23 b Fx(and)14 b Fw(\032)1268 493 y Fs(V)1309 487 y Fx(:)d Fw(V)21 b Ft(!)11 b Fw(V)18 b Ft(\012)9 b Fw(H)s Fx(.)18 b(Then)c Fw(V)228 536 y Fx(is)g(a)f(righ)o(t)h Fw(H)s Ff(-Hopf)g(mo)n(dule)p Fx(,)g(if)f(the)h(braided)g(Hopf)g(mo)q (dule)e(axiom)f(holds,)j(that)f(is)h(if)f Fw(\032)1651 542 y Fs(V)1694 536 y Fx(is)228 586 y(a)g(righ)o(t)f Fw(H)s Fx(-linear)h(map)f(where)i(the)g Fw(H)s Fx(-mo)q(dule)d (structure)16 b(of)c Fw(V)17 b Ft(\012)9 b Fw(H)16 b Fx(is)d(\()p Fw(\026)1446 592 y Fs(V)1483 586 y Ft(\012)8 b Fw(\026)p Fx(\)\()p Fw(id)g Ft(\012)g Fw(c)g Ft(\012)228 636 y Fw(id)p Fx(\)\()p Fw(id)h Ft(\012)h Fw(id)f Ft(\012)h Fx(\001\).)18 b(Here,)d Fw(\026)f Fx(and)g(\001)g(denote)h(the)f(m)o (ultiplication)d(and)j(com)o(ultiplication)d(of)228 686 y Fw(H)s Fx(.)18 b(Note)c(that)g(w)o(e)g(do)g(not)f(assume)h(that)g Fw(V)23 b Fx(is)14 b(a)f(Y)m(etter-Drinfeld)i(mo)q(dule.)303 776 y Fg(Lemma)h Fx(3.1)p Fg(.)k Ff(L)n(et)14 b Fw(L)h Ff(b)n(e)f(a)g(Hopf)g(algebr)n(a)g(with)g(bije)n(ctive)f(antip)n(o)n (de,)i Fw(H)i Ff(a)d(Hopf)g(algebr)n(a)228 836 y(in)279 821 y Fs(L)279 847 y(L)304 836 y Ft(Y)s(D)r Ff(,)h(and)i Fw(V)25 b Ff(a)16 b(right)e Fw(H)s Ff(-Hopf)i(mo)n(dule.)21 b(De\014ne)c Fw(V)1145 821 y Fs(coH)1220 836 y Fx(=)c Ft(f)p Fw(v)i Ft(2)d Fw(V)23 b Ft(j)12 b Fw(\032)1453 842 y Fs(V)1482 836 y Fx(\()p Fw(v)q Fx(\))i(=)f Fw(v)f Ft(\012)e Fx(1)p Ft(g)p Ff(.)228 896 y(Then)698 956 y Fw(V)731 939 y Fs(coH)803 956 y Ft(\012)g Fw(H)k Ft(!)d Fw(V)r(;)49 b(v)11 b Ft(\012)f Fw(h)h Ft(7!)g Fw(v)q(h;)228 1026 y Ff(is)j(bije)n(ctive.)303 1116 y Fg(Pr)o(oof.)21 b Fx(This)14 b(is)g(sho)o(wn)g(as)g(for)f(usual)h(Hopf)f(algebras)h (\(see)h([)p Fy(M1)p Fx(,)f(1.9.4]\).)p 1692 1116 2 29 v 1694 1090 25 2 v 1694 1116 V 1719 1116 2 29 v 303 1207 a Fg(Theorem)i Fx(3.2)p Fg(.)k Ff(L)n(et)c Fw(L)693 1192 y Fn(0)720 1207 y Ff(and)h Fw(L)f Ff(b)n(e)g(Hopf)g(algebr)n(as)f(with) g(bije)n(ctive)h(antip)n(o)n(de,)g(and)h Fw(\013)c Fx(:)228 1266 y Fw(L)256 1251 y Fn(0)283 1266 y Ft(!)i Fw(L)i Ff(a)g(Hopf)g(algebr)n(a)f(map.)25 b(L)n(et)17 b Fw(R)g Ff(b)n(e)g(a)g(bialgebr)n(a)f(in)1223 1251 y Fs(L)1223 1278 y(L)1248 1266 y Ft(Y)s(D)q Ff(,)h Fw(R)1374 1251 y Fn(0)1403 1266 y Ff(a)g(Hopf)g(algebr)n(a)f(in)228 1311 y Fs(L)251 1299 y Fk(0)228 1338 y Fs(L)251 1329 y Fk(0)264 1326 y Ft(Y)s(D)q Ff(,)g(and)h Fw(i)c Fx(:)g Fw(R)523 1311 y Fn(0)548 1326 y Ft(!)f Fw(R)p Ff(,)k Fw(\036)d Fx(:)g Fw(R)g Ft(!)g Fw(R)858 1311 y Fn(0)885 1326 y Ff(algebr)n(a)i(and)i(c)n(o)n(algebr)n(a)e(maps)i(such)f(that)g Fw(\036i)d Fx(=)h Fw(id)p Ff(.)228 1386 y(Assume)i(that)h Fw(i)f Ff(is)g(an)h Fw(L)636 1371 y Fn(0)648 1386 y Ff(-line)n(ar)f (map,)h(and)g(that)f Fw(i;)7 b(\036)16 b Ff(ar)n(e)g Fw(L)p Ff(-c)n(oline)n(ar,)g(wher)n(e)g Fw(R)g Ff(is)g(a)h(left)228 1446 y Fw(L)256 1431 y Fn(0)268 1446 y Ff(-mo)n(dule)e(and)h Fw(R)537 1431 y Fn(0)563 1446 y Ff(a)f(left)g Fw(L)p Ff(-c)n(omo)n(dule)g(by)h(r)n(estriction)d(via)i Fw(\013)p Ff(.)k(De\014ne)e Fw(K)e Fx(=)d Fw(R)1533 1431 y Fs(co\036)1598 1446 y Fx(=)g Ft(f)p Fw(r)h Ft(2)228 1506 y Fw(R)e Ft(j)g Fx(\()p Fw(id)f Ft(\012)f Fw(\036)p Fx(\)\(\001\()p Fw(r)q Fx(\)\))i(=)h Fw(r)e Ft(\012)g Fx(1)p Ft(g)p Ff(.)18 b(Then)d(the)g(multiplic)n(ation)f(map)719 1584 y Fw(K)f Ft(\012)c Fw(R)840 1567 y Fn(0)863 1584 y Ft(!)i Fw(R;)c(x)h Ft(\012)i Fw(y)j Ft(7!)e Fw(xi)p Fx(\()p Fw(y)q Fx(\))p Fw(;)228 1662 y Ff(is)j(bije)n(ctive,)g(and)i Fw(K)i Ff(is)d(a)g(sub)n(algebr)n(a)f(of)h Fw(R)g Ff(with)f Fx(\001\()p Fw(K)s Fx(\))e Ft(\032)f Fw(R)f Ft(\012)f Fw(K)s Ff(.)303 1752 y Fg(Pr)o(oof.)21 b Fx(W)m(e)12 b(consider)g Fw(R)g Fx(as)g(a)f(righ)o(t)g Fw(R)950 1737 y Fn(0)962 1752 y Fx(-mo)q(dule)f(b)o(y)h(restriction)i(via)e Fw(i)p Fx(,)h(and)f(as)h(a)g(righ)o(t)228 1814 y Fw(R)260 1799 y Fn(0)271 1814 y Fx(-como)q(dule)k(with)g(structure)j(map)854 1804 y(\026)847 1814 y(\001)c(:)h Fw(R)987 1791 y Fu(\001)973 1814 y Ft(\000)-18 b(!)16 b Fw(R)11 b Ft(\012)g Fw(R)1193 1789 y Fu(id)h Fn(\012)p Fs(\036)1179 1814 y Ft(\000)-8 b(\000)f(\000)i(!)15 b Fw(R)c Ft(\012)h Fw(R)1427 1799 y Fn(0)1438 1814 y Fx(.)26 b(Here)18 b(w)o(e)f(used)228 1874 y(that)f Fw(i)g Fx(is)g(an)f(algebra)h(map)e(and)i Fw(\036)f Fx(is)h(a)f(coalgebra)h(map.)22 b(Then)17 b(one)f(c)o(hec)o (ks)h(that)f Fw(R)g Fx(is)f(a)228 1934 y(righ)o(t)d Fw(R)359 1919 y Fn(0)370 1934 y Fx(-Hopf)g(mo)q(dule)f(using)h(that)h Fw(i)f Fx(is)g(a)h(coalgebra)f(map)e(and)j Fw(L)1306 1919 y Fn(0)1318 1934 y Fx(-linear,)e Fw(\036)h Fx(is)g(an)h(algebra) 228 1994 y(map)h(and)j Fw(L)p Fx(-colinear,)f(and)g Fw(\036i)f Fx(=)h Fw(id)p Fx(.)25 b(Hence)18 b(the)f(m)o(ultiplicati)o(on)d(map)g Fw(K)g Ft(\012)d Fw(R)1562 1979 y Fn(0)1589 1994 y Ft(!)k Fw(R)h Fx(is)228 2053 y(bijectiv)o(e)d(b)o(y)g(3.1.)k(Moreo)o(v)o(er)d (it)f(follo)o(ws)f(from)g(the)i(de\014nition)f(of)f Fw(K)17 b Fx(that)c(\001\()p Fw(K)s Fx(\))f Ft(\032)g Fw(R)c Ft(\012)g Fw(K)s Fx(.)228 2113 y(T)m(o)14 b(see)i(that)g Fw(K)i Fx(is)d(a)g(subalgebra)g(of)f Fw(R)p Fx(,)h(one)g(c)o(hec)o(ks)i (that)e(\()p Fw(id)10 b Ft(\012)g Fw(i)p Fx(\))1338 2103 y(\026)1330 2113 y(\001)k(:)f Fw(R)g Ft(!)g Fw(R)d Ft(\012)h Fw(R)j Fx(is)h(an)228 2173 y(algebra)f(map,)f(where)i(the)g(algebra)f (structure)j(on)e Fw(R)9 b Ft(\012)h Fw(R)k Fx(is)h(giv)o(en)f(b)o(y)g (the)h(braiding)f(of)g Fw(R)p Fx(,)228 2233 y(using)f(that)h Fw(i;)7 b(\036)14 b Fx(are)g Fw(L)p Fx(-colinear)g(algebra)f(maps.)p 1692 2233 V 1694 2206 25 2 v 1694 2233 V 1719 2233 2 29 v 303 2309 a(Theorem)j(3.2)g(is)h(true)h(in)e(a)h(more)f(general)i (v)o(ersion)f(whic)o(h)g(only)f(uses)i(the)g(braidings)228 2359 y Fw(c)13 b Fx(:)f Fw(R)d Ft(\012)h Fw(R)j Ft(!)f Fw(R)e Ft(\012)g Fw(R)k Fx(and)h Fw(c)695 2344 y Fn(0)719 2359 y Fx(:)d Fw(R)775 2344 y Fn(0)796 2359 y Ft(\012)e Fw(R)870 2344 y Fn(0)895 2359 y Ft(!)i Fw(R)981 2344 y Fn(0)1002 2359 y Ft(\012)e Fw(R)1076 2344 y Fn(0)1102 2359 y Fx(of)k Fw(R)g Fx(and)h Fw(R)1310 2344 y Fn(0)1321 2359 y Fx(.)21 b(It)14 b(su\016ces)i(to)f(assume)228 2409 y(that)g Fw(i;)7 b(\036)14 b Fx(are)h(algebra)f(and)h(coalgebra)f (maps)g(with)g Fw(\036i)f Fx(=)g Fw(id)p Fx(,)h(whic)o(h)h(satisfy)f (the)i(follo)o(wing)228 2458 y(iden)o(tities:)i(\()p Fw(i)10 b Ft(\012)f Fw(id)p Fx(\))p Fw(c)574 2443 y Fn(0)586 2458 y Fx(\()p Fw(\036)g Ft(\012)g Fw(id)p Fx(\))j(=)g(\()p Fw(id)d Ft(\012)h Fw(\036)p Fx(\))p Fw(c)p Fx(\()p Fw(id)f Ft(\012)g Fw(i)p Fx(\),)14 b(and)g Fw(c)p Fx(\()p Fw(i\036)9 b Ft(\012)h Fw(id)p Fx(\))h(=)h(\()p Fw(id)d Ft(\012)h Fw(i\036)p Fx(\))p Fw(c:)303 2508 y Fx(If)20 b Fw(X)27 b Fx(=)468 2477 y Fl(L)514 2487 y Fn(1)514 2521 y Fs(n)p Fu(=0)585 2508 y Fw(X)619 2514 y Fs(n)663 2508 y Fx(is)21 b(a)f(graded)i(v)o(ector)f(space)h(with)f(\014nite-dimensional)e(comp)q (o-)228 2558 y(nen)o(ts)j Fw(X)379 2564 y Fs(n)401 2558 y Fx(,)h Fw(n)g Ft(\025)g Fx(0,)f(the)f Ff(Hilb)n(ert)f(series)g Fx(of)g Fw(X)25 b Fx(is)20 b(the)i(formal)c(p)q(o)o(w)o(er)j(series)h Fw(P)1588 2564 y Fs(X)1620 2558 y Fx(\()p Fw(t)p Fx(\))h(=)228 2577 y Fl(P)272 2587 y Fn(1)272 2620 y Fs(i)p Fu(=0)334 2608 y Fx(dim)o(\()p Fw(X)454 2614 y Fs(n)477 2608 y Fx(\))p Fw(t)508 2593 y Fs(n)530 2608 y Fx(.)p eop %%Page: 7 7 7 6 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d(O) o(VER)i(CO)o(XETER)g(GR)o(OUPS)112 b(7)303 213 y Fg(Cor)o(ollar)m(y)18 b Fx(3.3)p Fg(.)i Ff(L)n(et)e Fw(L)f Ff(b)n(e)h(a)g(Hopf)f(algebr)n(a)g (with)g(bije)n(ctive)g(antip)n(o)n(de,)i Fw(V)25 b Ft(2)1620 198 y Fs(L)1620 224 y(L)1645 213 y Ft(Y)s(D)q Ff(,)228 272 y(and)16 b Fw(L)337 257 y Fn(0)360 272 y Ft(\032)c Fw(L)k Ff(a)f(Hopf)g(sub)n(algebr)n(a)g(with)f(bije)n(ctive)g(antip)n (o)n(de,)i(and)f Fw(V)1324 257 y Fn(0)1347 272 y Ft(\032)d Fw(V)25 b Ff(a)15 b(subsp)n(ac)n(e)h(such)228 332 y(that)c Fw(L)338 317 y Fn(0)350 332 y Fw(V)383 317 y Fn(0)407 332 y Ft(\032)f Fw(V)484 317 y Fn(0)495 332 y Ff(,)i Fw(\016)r Fx(\()p Fw(V)591 317 y Fn(0)602 332 y Fx(\))f Ft(\032)g Fw(L)702 317 y Fn(0)717 332 y Ft(\012)t Fw(V)787 317 y Fn(0)798 332 y Ff(,)h(wher)n(e)e Fw(\016)j Fx(:)d Fw(V)21 b Ft(!)11 b Fw(L)t Ft(\012)t Fw(V)21 b Ff(is)12 b(the)h(c)n(o)n(action)g(of)f Fw(V)d Ff(.)18 b(Then)13 b Fw(V)228 392 y Ff(is)g(a)h(left)f Fw(L)404 377 y Fn(0)416 392 y Ff(-mo)n(dule)g(by)h(r)n(estriction)e(and)i Fw(V)935 377 y Fn(0)960 392 y Ff(is)f(a)h(left)f Fw(L)p Ff(-c)n(omo)n(dule)h (via)g Fw(\016)r Ff(.)k(L)n(et)13 b Fw(')f Fx(:)f Fw(V)21 b Ft(!)11 b Fw(V)1711 377 y Fn(0)228 452 y Ff(b)n(e)k(an)g Fw(L)366 437 y Fn(0)378 452 y Ff(-line)n(ar)f(and)i Fw(L)p Ff(-c)n(oline)n(ar)e(map)h(with)g Fw(')c Ft(j)g Fw(V)1066 437 y Fn(0)1089 452 y Fx(=)h Fw(id)p Ff(.)303 511 y(Then)k Fw(V)445 496 y Fn(0)472 511 y Ff(is)g(a)g(Y)m(etter-Drinfeld)e(mo)n (dule)i(over)f Fw(L)1107 496 y Fn(0)1119 511 y Ff(,)h Fo(B)p Fx(\()p Fw(V)1234 496 y Fn(0)1246 511 y Fx(\))g Ff(is)f(a)h(gr)n(ade)n(d)g(sub)n(algebr)n(a)f(of)228 571 y Fo(B)p Fx(\()p Fw(V)9 b Fx(\))p Ff(,)20 b(and)f Fo(B)p Fx(\()p Fw(V)9 b Fx(\))19 b Ff(is)g(a)f(fr)n(e)n(e)g(right)g Fo(B)p Fx(\()p Fw(V)929 556 y Fn(0)940 571 y Fx(\))p Ff(-mo)n(dule.)31 b(If)18 b Fw(V)28 b Ff(is)19 b(\014nite-dimensional,) h(then)228 631 y Fw(P)255 638 y Fe(B)p Fu(\()p Fs(V)324 630 y Fk(0)335 638 y Fu(\))365 631 y Ff(divides)15 b Fw(P)530 638 y Fe(B)p Fu(\()p Fs(V)7 b Fu(\))614 631 y Ff(.)303 714 y Fg(Pr)o(oof.)21 b Fx(The)c(tensor)f(algebras)g Fw(T)6 b Fx(\()p Fw(V)k Fx(\))16 b(and)f Fw(T)6 b Fx(\()p Fw(V)1118 699 y Fn(0)1130 714 y Fx(\))16 b(are)g(Hopf)f(algebras)h(in) 1550 699 y Fs(L)1550 725 y(L)1575 714 y Ft(Y)s(D)h Fx(and)228 758 y Fs(L)251 746 y Fk(0)228 785 y Fs(L)251 777 y Fk(0)264 773 y Ft(Y)s(D)h Fx(as)f(in)g(the)h(\014rst)g(section)g(where)g(the)g (elemen)o(ts)f(of)f Fw(V)27 b Fx(and)16 b Fw(V)1346 758 y Fn(0)1375 773 y Fx(are)h(primitiv)o(e.)26 b(The)228 833 y(natural)13 b(map)f Fw(\036)g Fx(:)f Fw(T)6 b Fx(\()p Fw(V)j Fx(\))j Ft(!)f Fw(T)6 b Fx(\()p Fw(V)765 818 y Fn(0)776 833 y Fx(\))14 b(giv)o(en)g(b)o(y)g Fw(')d Fx(:)g Fw(V)21 b Ft(!)11 b Fw(V)1166 818 y Fn(0)1191 833 y Fx(is)j(an)g (algebra)f(and)h(a)f(coalgebra)228 893 y(map)d(with)h Fw(\036i)h Fx(=)f Fw(id)p Fx(,)h(where)h Fw(i)e Fx(:)g Fw(T)6 b Fx(\()p Fw(V)810 878 y Fn(0)822 893 y Fx(\))11 b Ft(!)g Fw(T)6 b Fx(\()p Fw(V)k Fx(\))i(denotes)h(the)f(inclusion)f (map.)k(Moreo)o(v)o(er,)d Fw(\036)228 953 y Fx(is)e Fw(L)p Fx(-colinear)f(and)h Fw(L)563 938 y Fn(0)575 953 y Fx(-linear,)g(where) h Fw(T)6 b Fx(\()p Fw(V)909 938 y Fn(0)920 953 y Fx(\))k(is)g(an)g Fw(L)p Fx(-como)q(dule)f(and)h Fw(T)c Fx(\()p Fw(V)j Fx(\))h(is)g(an)g Fw(L)1564 938 y Fn(0)1576 953 y Fx(-mo)q(dule)228 1013 y(via)k(the)i(inclusion)e(map)g Fw(\013)f Fx(:)g Fw(L)731 997 y Fn(0)757 1013 y Ft(!)g Fw(L)p Fx(.)22 b(Therefore)16 b Fw(\036)f Fx(induces)h(an)f Fw(L)p Fx(-colinear)g (algebra)f(and)228 1072 y(coalgebra)f(map)f Fo(B)p Fx(\()p Fw(V)e Fx(\))h Ft(!)g Fo(B)p Fx(\()p Fw(V)757 1057 y Fn(0)769 1072 y Fx(\))j(whic)o(h)f(will)g(again)f(b)q(e)i(denoted)h(b)o (y)e Fw(\036)p Fx(.)18 b(Since)c Fo(B)p Fx(\()p Fw(V)c Fx(\))j(and)228 1132 y Fo(B)p Fx(\()p Fw(V)314 1117 y Fn(0)326 1132 y Fx(\))f(are)g(graded)g(algebras)f(and)h(coalgebras)f (and)h Fw(\036)f Fx(is)h(a)f(graded)h(algebra)g(map)e(\(of)h(degree)228 1192 y(0\),)f Fw(K)15 b Fx(=)c Fo(B)p Fx(\()p Fw(V)f Fx(\))483 1177 y Fs(co\036)546 1192 y Fx(is)f(a)h(graded)g(subalgebra)g (of)f Fo(B)p Fx(\()p Fw(V)g Fx(\).)17 b(Hence)11 b(w)o(e)f(conclude)h (from)d(Theorem)228 1252 y(3.2)j(with)i Fw(R)e Fx(=)h Fo(B)p Fx(\()p Fw(V)d Fx(\))k(and)f Fw(R)700 1237 y Fn(0)723 1252 y Fx(=)g Fo(B)p Fx(\()p Fw(V)853 1237 y Fn(0)865 1252 y Fx(\))g(that)h Fw(K)d Ft(\012)d Fo(B)p Fx(\()p Fw(V)1152 1237 y Fn(0)1164 1252 y Fx(\))1191 1241 y Ft(\030)1191 1254 y Fx(=)1235 1252 y Fo(B)p Fx(\()p Fw(V)i Fx(\).)18 b(In)13 b(particular,)f Fo(B)p Fx(\()p Fw(V)d Fx(\))228 1311 y(is)14 b(free)g(o)o(v)o(er)g Fo(B)p Fx(\()p Fw(V)524 1296 y Fn(0)536 1311 y Fx(\),)g(and)f(if)g Fw(V)24 b Fx(is)13 b(\014nite-dimensional,)f Fw(P)1159 1318 y Fe(B)p Fu(\()p Fs(V)1227 1311 y Fk(0)1239 1318 y Fu(\))1254 1311 y Fw(P)1281 1317 y Fs(K)1324 1311 y Fx(=)g Fw(P)1395 1318 y Fe(B)p Fu(\()p Fs(V)6 b Fu(\))1479 1311 y Fx(.)p 1692 1311 2 29 v 1694 1285 25 2 v 1694 1311 V 1719 1311 2 29 v 303 1386 a(When)15 b Fo(B)p Fx(\()p Fw(V)511 1371 y Fn(0)522 1386 y Fx(\))g(is)g(\014nite-dimensional,)e(Corollary)g(3.3) h(also)g(follo)o(ws)f(from)h([)p Fy(G1)o Fx(,)g(Theo-)228 1436 y(rem)h(3.11])f(with)h(a)h(completely)e(di\013eren)o(t)j(pro)q (of.)23 b(But)16 b(the)h(argumen)o(t)d(in)h(Theorem)h(3.2)e(is)228 1486 y(quite)g(general)g(and)g(do)q(es)g(not)g(dep)q(end)h(on)f(Nic)o (hols)f(algebras.)303 1536 y(An)j(example)f(of)h(the)h(situation)f(of)f (Corollary)h(3.3)f(will)g(b)q(e)i(giv)o(en)f(in)g(Section)g(5.)26 b(The)228 1586 y(next)16 b(remark)e(sho)o(ws)i(that)g(the)g(existence)h (of)e(the)h(splitting)f(map)f Fw(')h Fx(in)g(Corollary)g(3.3)f(is)h(a) 228 1636 y(rather)g(mild)c(assumption.)303 1718 y Fg(Remark)16 b Fx(3.4)p Fg(.)k Fx(Let)15 b Fw(G)e Fx(b)q(e)h(a)g(group,)f Fw(G)943 1703 y Fn(0)966 1718 y Ft(\032)f Fw(G)h Fx(a)g(\014nite)h (subgroup)g(suc)o(h)h(that)f(the)g(c)o(har-)228 1778 y(acteristic)d(of)f Fv(|)-11 b Fx(do)q(es)11 b(not)g(divide)f(the)h (order)g(of)f Fw(G)1011 1763 y Fn(0)1023 1778 y Fx(.)16 b(Let)c Fw(V)21 b Ft(2)1207 1763 y Fs(G)1207 1789 y(G)1235 1778 y Ft(Y)s(D)12 b Fx(and)e Fw(V)1420 1763 y Fn(0)1444 1778 y Ft(\032)h Fw(V)20 b Fx(a)11 b(subspace)228 1838 y(suc)o(h)i(that)g Fw(G)442 1823 y Fn(0)454 1838 y Fw(V)487 1823 y Fn(0)510 1838 y Ft(\032)f Fw(V)588 1823 y Fn(0)599 1838 y Fx(,)h(and)g Fw(\016)r Fx(\()p Fw(V)773 1823 y Fn(0)785 1838 y Fx(\))f Ft(\032)f Fv(|)-19 b Fw(G)915 1823 y Fn(0)931 1838 y Ft(\012)8 b Fw(V)1004 1823 y Fn(0)1016 1838 y Fx(,)k(where)i Fw(\016)g Fx(:)d Fw(V)21 b Ft(!)11 b Fv(|)-19 b Fw(G)t Ft(\012)8 b Fw(V)22 b Fx(is)13 b(the)g(coaction)228 1897 y(of)j Fw(V)10 b Fx(.)27 b(Then)18 b(there)g(is)f(alw)o(a)o(ys)g (a)f Fv(|)-19 b Fw(G)850 1882 y Fn(0)859 1897 y Fx(-linear)16 b(and)h Fv(|)-19 b Fw(G)p Fx(-col)o(inear)15 b(map)g Fw(')i Fx(:)g Fw(V)26 b Ft(!)16 b Fw(V)1613 1882 y Fn(0)1641 1897 y Fx(with)228 1957 y Fw(')h Ft(j)g Fw(V)335 1942 y Fn(0)364 1957 y Fx(=)g Fw(id)p Fx(.)29 b(Hence)19 b(w)o(e)e(are)h(in) f(the)h(situation)f(of)f(Corollary)g(3.3)h(with)g Fw(L)1492 1942 y Fn(0)1521 1957 y Fx(=)h Fv(|)-19 b Fw(G)1629 1942 y Fn(0)1655 1957 y Fx(and)228 2017 y Fw(L)12 b Fx(=)f Fv(|)-19 b Fw(G)p Fx(.)303 2100 y Fg(Pr)o(oof.)21 b Fx(Since)13 b Fv(|)-19 b Fw(G)8 b Fx(is)k(cosemisimple,)d(there)k(is)f(a)f Fv(|)-19 b Fw(G)p Fx(-col)o(inear)9 b(map)h Fw(f)17 b Fx(:)11 b Fw(V)21 b Ft(!)11 b Fw(V)1618 2085 y Fn(0)1641 2100 y Fx(with)228 2159 y Fw(f)19 b Ft(j)14 b Fw(V)327 2144 y Fn(0)353 2159 y Fx(=)h Fw(id)p Fx(.)23 b(De\014ne)16 b Fw(')f Fx(:)f Fw(V)24 b Ft(!)14 b Fw(V)806 2144 y Fn(0)834 2159 y Fx(b)o(y)h Fw(')p Fx(\()p Fw(v)q Fx(\))h(=)1088 2143 y Fu(1)p 1040 2150 111 2 v 1040 2174 a(ord\()p Fs(G)1127 2165 y Fk(0)1138 2174 y Fu(\))1163 2128 y Fl(P)1207 2172 y Fs(g)q Fn(2)p Fs(G)1272 2164 y Fk(0)1293 2159 y Fw(g)q(f)t Fx(\()p Fw(g)1375 2144 y Fn(\000)p Fu(1)1421 2159 y Fw(v)q Fx(\))g(for)f(all)g Fw(v)h Ft(2)e Fw(V)c Fx(.)228 2219 y(Then)h Fw(')g Fx(is)g(a)f Fw(G)474 2204 y Fn(0)486 2219 y Fx(-linear)g(map)f(with)i Fw(')g Ft(j)g Fw(V)888 2204 y Fn(0)911 2219 y Fx(=)h Fw(id)p Fx(,)f(and)f Fw(')h Fx(is)g Fv(|)-19 b Fw(G)p Fx(-)o(colinear,)8 b(since)k(for)e(all)g Fw(h)h Ft(2)h Fw(G)228 2279 y Fx(and)h Fw(v)g Ft(2)e Fw(V)404 2285 y Fs(h)426 2279 y Fx(,)h(b)o(y)h(the)h(Y)m (etter-Drinfeld)f(condition)f Fw(g)1068 2264 y Fn(\000)p Fu(1)1113 2279 y Fw(v)h Ft(2)f Fw(V)1210 2287 y Fs(g)1227 2279 y Fk(\000)p Fj(1)1266 2287 y Fs(hg)1317 2279 y Fx(and)h Fw(g)q(f)t Fx(\()p Fw(g)1479 2264 y Fn(\000)p Fu(1)1525 2279 y Fw(v)q Fx(\))g Ft(2)e Fw(V)1647 2264 y Fn(0)1638 2291 y Fs(h)1672 2279 y Fx(for)228 2339 y(all)h Fw(g)h Ft(2)e Fw(G)390 2324 y Fn(0)402 2339 y Fx(.)p 1692 2339 2 29 v 1694 2312 25 2 v 1694 2339 V 1719 2339 2 29 v 499 2433 a Fy(4.)24 b(Link-indecomp)o(sabl)o(e)13 b(p)q(oin)o(ted)g (Hopf)i(algebras)303 2508 y Fx(Let)k Fw(A)g Fx(b)q(e)g(a)f(p)q(oin)o (ted)h(Hopf)g(algebra)f(with)g(group-lik)o(e)g(elemen)o(ts)h Fw(G)g Fx(=)h Fw(G)p Fx(\()p Fw(A)p Fx(\).)33 b(F)m(or)228 2558 y Fw(g)q(;)7 b(h)k Ft(2)g Fw(G)p Fx(,)i Fw(P)427 2564 y Fs(g)q(;h)475 2558 y Fx(\()p Fw(A)p Fx(\))f(=)g Ft(f)p Fw(x)f Ft(2)g Fw(A)h Ft(j)f Fx(\001\()p Fw(x)p Fx(\))g(=)h Fw(g)d Ft(\012)f Fw(x)f Fx(+)h Fw(x)g Ft(\012)g Fx(1)p Ft(g)k Fx(will)g(denote)i(the)g(\()p Fw(g)q(;)7 b(h)p Fx(\))p Ff(-primitive)228 2608 y(elements)14 b Fx(of)f Fw(A)p Fx(.)18 b Fw(P)533 2614 y Fs(g)q(;h)581 2608 y Fx(\()p Fw(A)p Fx(\))c(is)g(called)g Ff(non-trivial)f Fx(if)g Fv(|)-19 b Fx(\()p Fw(g)8 b Ft(\000)h Fw(h)p Fx(\))j Fv($)g Fw(P)1294 2614 y Fs(g)q(;h)1342 2608 y Fx(\()p Fw(A)p Fx(\).)p eop %%Page: 8 8 8 7 bop 228 119 a Fu(8)262 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)303 213 y Fx(W)m(e)h(recall)h(the)g(de\014nition)g(of)f(the)i Ff(quiver)f Fx(\000)1010 219 y Fs(A)1052 213 y Ff(of)h Fw(A)f Fx(in)g([)p Fy(M2)o Fx(].)k(The)c(v)o(ertices)i(of)d(\000)1626 219 y Fs(A)1667 213 y Fx(are)228 262 y(the)18 b(elemen)o(ts)g(of)f Fw(G)p Fx(;)h(for)g Fw(g)q(;)7 b(h)17 b Ft(2)h Fw(G)p Fx(,)g(there)h(exists)f(an)g(arro)o(w)f(from)f Fw(h)i Fx(to)f Fw(g)i Fx(if)e Fw(P)1565 268 y Fs(g)q(;h)1613 262 y Fx(\()p Fw(A)p Fx(\))h(is)228 312 y(non-trivial.)303 362 y(Note)13 b(that)h Fw(P)519 368 y Fs(g)q(;h)567 362 y Fx(\()p Fw(A)p Fx(\))g(is)f(non-trivial)f(if)h(and)g(only)g(if)f Fw(P)1162 370 y Fs(g)q(h)1198 362 y Fk(\000)p Fj(1)1237 370 y Fs(;)p Fu(1)1266 362 y Fx(\()p Fw(A)p Fx(\))h(or)h Fw(P)1420 370 y Fs(h)1439 362 y Fk(\000)p Fj(1)1478 370 y Fs(g)q(;)p Fu(1)1523 362 y Fx(\()p Fw(A)p Fx(\))g(is)f(non-)228 412 y(trivial.)20 b(Then)c Fw(g)q(;)7 b(h)13 b Ft(2)g Fw(G)i Fx(are)h Ff(c)n(onne)n(cte)n(d)g Fx(\(or)f Fw(g)g Fv(s)f Fw(h)p Fx(\))h(if)f(they)i(are)g(in)e(the)i(same)e(connected)228 462 y(comp)q(onen)o(t)20 b(of)g(\000)527 468 y Fs(A)575 462 y Fx(as)h(an)f(undirected)i(graph.)39 b(The)21 b(Hopf)f(algebra)h Fw(A)f Fx(is)h(called)g Ff(link-)228 511 y(inde)n(c)n(omp)n(osable)14 b Fx(if)f(an)o(y)h(t)o(w)o(o)f(elemen)o(ts)h(in)f Fw(G)h Fx(are)g(connected.)303 561 y(W)m(e)20 b(call)g(Supp\()p Fw(A)p Fx(\))k(=)f Ft(f)p Fw(g)h Ft(2)f Fw(G)f Ft(j)h Fw(P)931 567 y Fs(g)q(;)p Fu(1)976 561 y Fx(\()p Fw(A)p Fx(\))15 b(is)e(non-trivial)o Ft(g)20 b Fx(the)i Ff(supp)n(ort)f Fx(of)f Fw(A)p Fx(.)39 b(If)228 611 y Fw(V)25 b Fx(=)325 580 y Fl(L)371 624 y Fs(g)q Fn(2)p Fs(G)446 611 y Fw(V)470 617 y Fs(g)506 611 y Fx(is)16 b(a)h(left)f Fv(|)-19 b Fw(G)p Fx(-)o(com)o(o)q(dule)14 b(with)i(homogeneous)f(comp)q(onen)o (ts)i Fw(V)1530 617 y Fs(g)1549 611 y Fx(,)g(then)g(w)o(e)228 664 y(de\014ne)e(Supp\()p Fw(V)10 b Fx(\))h(=)h Ft(f)p Fw(g)h Ft(2)e Fw(G)g Ft(j)g Fw(V)745 670 y Fs(g)776 664 y Ft(6)p Fx(=)h(0)p Ft(g)p Fx(.)303 714 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Fy(AS1)o Fx(]\).)303 1056 y Fg(Lemma)h Fx(4.1)p Fg(.)k Ff(L)n(et)c Fw(A)g Ff(b)n(e)g(a)g(p)n(ointe)n(d)g(Hopf)g(algebr)n(a)g (with)f Fw(G)e Fx(=)g Fw(G)p Fx(\()p Fw(A)p Fx(\))j Ff(and)h(diagr)n (am)f Fw(R)p Ff(.)228 1116 y(De\014ne)g Fw(N)g Fx(=)c Ft(f)p Fw(g)h Ft(2)e Fw(G)g Ft(j)g Fw(g)i Fv(s)f Fx(1)p Ft(g)p Ff(,)i(and)h Fw(V)21 b Fx(=)12 b Fw(P)6 b Fx(\()p Fw(R)p Fx(\))p Ff(.)19 b(Then)303 1176 y(1\))d Fw(N)21 b Ff(is)16 b(a)h(normal)f(sub)n(gr)n(oup)g(of)h Fw(G)f Ff(and)h(the)f(c)n(onne)n(cte)n(d)h(c)n(omp)n(onents)h(of)e Fx(\000)1553 1182 y Fs(A)1596 1176 y Ff(ar)n(e)g(the)228 1235 y(c)n(osets)f(of)f Fw(N)5 b Ff(.)303 1295 y(2\))15 b Fw(N)k Ff(is)c(gener)n(ate)n(d)g(by)g(Supp)q Fx(\()p Fw(A)p Fx(\))p Ff(.)303 1355 y(3\))g(Supp\(A\))g(=)g(Supp\(V\).)303 1449 y Fg(Pr)o(oof.)21 b Fx(1\))15 b(is)f(noted)h(in)f([)p Fy(M2)o Fx(,)h(3.2].)j(By)c(de\014nition)g(there)i(exists)f(an)g(arro)o (w)f(b)q(et)o(w)o(een)228 1509 y Fw(g)q(;)7 b(h)19 b Ft(2)f Fw(G)g Fx(if)g(and)g(only)g(if)g 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Fx(is)16 b(generated)g(b)o(y)g(Supp\()p Fw(A)p Fx(\).)23 b(Con)o(v)o(ersely)m(,)15 b(if)228 1688 y Fw(u)252 1694 y Fu(1)270 1688 y Fw(;)7 b(:)g(:)g(:)e(;)i(u)387 1694 y Fs(n)p Fn(\000)p Fu(1)463 1688 y Ft(2)k Fx(Supp\()p Fw(A)p Fx(\))i(or)g(\(Supp)q(\()p Fw(A)p Fx(\)\))908 1673 y Fn(\000)p Fu(1)953 1688 y Fx(,)f Fw(n)g Ft(\025)f Fx(1,)i(and)f Fw(g)h Fx(=)f Fw(u)1283 1694 y Fu(1)1308 1688 y Fw(:)7 b(:)g(:)e(u)1387 1694 y Fs(n)p Fn(\000)p Fu(1)1452 1688 y Fx(,)13 b(de\014ne)h Fw(g)1616 1694 y Fu(1)1646 1688 y Fx(=)d Fw(g)q Fx(,)228 1748 y(and)i Fw(g)328 1754 y Fs(i)p Fu(+1)397 1748 y Fx(b)o(y)h Fw(u)479 1754 y Fs(i)504 1748 y Fx(=)e Fw(g)568 1754 y Fs(i)581 1748 y Fw(g)602 1730 y Fn(\000)p Fu(1)601 1759 y Fs(i)p Fu(+1)657 1748 y Fw(;)7 b Fx(1)k Ft(\024)g Fw(i)h Ft(\024)g Fw(n)d Ft(\000)f Fx(1.)18 b(Then)c Fw(g)h Fx(is)e(connected)j(to)d Fw(g)1394 1754 y Fs(n)1428 1748 y Fx(=)f(1.)17 b(This)d(sho)o(ws)228 1808 y(2\).)22 b(F)m(or)15 b(3\),)g(note)g(that)h Fw(P)6 b Fx(\()p Fw(R)p Fx(\)#)p Fv(|)-20 b Fw(G)824 1797 y Ft(\030)824 1810 y Fx(=)870 1808 y Fw(A)901 1814 y Fu(1)920 1808 y Fw(=)n(A)970 1814 y Fu(0)1003 1808 y Fx(b)o(y)15 b([)p Fy(AS1)o Fx(,)h(Lemma)c(2.4].)21 b(F)m(or)15 b(all)f Fw(g)q(;)7 b(h)13 b Ft(2)h Fw(G)228 1867 y Fx(c)o(ho)q(ose)i(a)f (decomp)q(osition)g(of)g(v)o(ector)h(spaces)h Fw(P)1004 1873 y Fs(g)q(;h)1052 1867 y Fx(\()p Fw(A)p Fx(\))e(=)f Fv(|)-18 b Fx(\()p Fw(g)8 b Ft(\000)j Fw(h)p Fx(\))f Ft(\010)h Fw(P)1410 1873 y Fs(g)q(;h)1458 1867 y Fx(\()p Fw(A)p Fx(\))1521 1852 y Fn(0)1533 1867 y Fx(.)23 b(Then)16 b(b)o(y)228 1927 y(the)e(Theorem)g(of)g(T)m(aft)f(and)h(Wilson)f([)p Fy(M1)o Fx(,)h(5.4.1],)1062 1896 y Fl(L)1109 1940 y Fs(g)q(;h)p Fn(2)p Fs(G)1212 1927 y Fw(P)1239 1933 y Fs(g)q(;h)1288 1927 y Fx(\()p Fw(A)p Fx(\))1351 1912 y Fn(0)1374 1916 y Ft(\030)1374 1929 y Fx(=)1419 1927 y Fw(A)1450 1933 y Fu(1)1468 1927 y Fw(=)n(A)1518 1933 y Fu(0)1537 1927 y Fx(.)k(Th)o(us)d(for)228 1987 y(all)d Fw(g)h Ft(2)e Fw(G)j Fx(w)o(e)g(obtain)f(an)h(isomorphism)d Fw(P)6 b Fx(\()p Fw(R)p Fx(\))992 1993 y Fs(g)1010 1987 y Fx(#1)1077 1976 y Ft(\030)1077 1989 y Fx(=)1121 1987 y Fw(P)1148 1993 y Fs(g)q(;)p Fu(1)1194 1987 y Fx(\()p Fw(A)p Fx(\))p Fw(=)p Fv(|)-19 b Fx(\(1)6 b Ft(\000)j Fw(g)q Fx(\).)p 1692 1987 2 29 v 1694 1961 25 2 v 1694 1987 V 1719 1987 2 29 v 303 2071 a(As)k(suggested)i(b)o(y)e(the)g(previous)h(Lemma,)c(w) o(e)k(sa)o(y)f(a)f(Y)m(etter-Drinfeld)i(mo)q(dule)e Fw(V)22 b Fx(o)o(v)o(er)228 2121 y Fw(G)13 b Fx(is)h Ff(link-inde)n(c)n(omp)n (osable)g Fx(if)f Fw(G)h Fx(is)f(generated)j(b)o(y)d(Supp\()p Fw(V)d Fx(\).)303 2215 y Fg(Pr)o(oposition)15 b Fx(4.2)p Fg(.)21 b Ff(L)n(et)14 b Fw(G)h Ff(b)n(e)g(a)g(\014nite)g(gr)n(oup.)k (Then)c(the)g(fol)r(lowing)f(ar)n(e)h(e)n(quivalent:)303 2275 y(\(1\))g(Ther)n(e)f(exists)h(a)g(\014nite-dimensional)h(p)n (ointe)n(d)f(and)h(link-inde)n(c)n(omp)n(osable)f(Hopf)h(al-)228 2334 y(gebr)n(a)f Fw(A)g Ff(such)g(that)g Fw(G)606 2323 y Ft(\030)606 2336 y Fx(=)650 2334 y Fw(G)p Fx(\()p Fw(A)p Fx(\))p Ff(.)303 2394 y(\(2\))h(Ther)n(e)g(exists)g(a)g(link-inde)n(c)n (omp)n(osable)h(Y)m(etter-Drinfeld)d(mo)n(dule)j Fw(V)26 b Ff(over)16 b Fw(G)g Ff(such)228 2454 y(that)f Fo(B)p Fx(\()p Fw(V)9 b Fx(\))15 b Ff(is)f(\014nite-dimensional.)303 2548 y Fg(Pr)o(oof.)21 b Fx(Assume)15 b(\(1\))g(and)g(let)g Fw(R)g Fx(b)q(e)g(the)h(diagram)d(of)h Fw(A)p Fx(.)21 b(Then)16 b Fw(V)22 b Fx(=)14 b Fw(P)6 b Fx(\()p Fw(R)p Fx(\))15 b(is)f(link-)228 2608 y(indecomp)q(osable)f(b)o(y)g(Lemma)f (4.1.)17 b(By)d(2.2)f(2\),)g Fv(|)-19 b Fx([)o Fw(V)7 b Fx(])1101 2597 y Ft(\030)1101 2610 y Fx(=)1145 2608 y Fo(B)p Fx(\()p Fw(V)i Fx(\))14 b(is)g(\014nite-dimensional.)p eop %%Page: 9 9 9 8 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d(O) o(VER)i(CO)o(XETER)g(GR)o(OUPS)112 b(9)303 213 y Fx(Assume)13 b(\(2\).)18 b(De\014ne)d Fw(R)c Fx(=)h Fo(B)p Fx(\()p Fw(V)d Fx(\),)14 b(and)f(let)h Fw(A)e Fx(=)g Fw(R)p Fx(#)p Fv(|)-20 b Fw(G)10 b Fx(b)q(e)15 b(the)f(Radford)f(bipro)q(duct.)228 272 y(It)18 b(is)f(w)o(ell-kno)o(wn)g(and)g(not)h(di\016cult)f(to)h (see)h(that)f Fw(P)1107 278 y Fs(g)q(;)p Fu(1)1152 272 y Fx(\()p Fw(A)p Fx(\))g(=)h Fw(P)6 b Fx(\()p Fw(R)p Fx(\))1381 278 y Fs(g)1400 272 y Fx(#1)11 b(+)h Fv(|)-19 b Fx(\()p Fw(g)10 b Ft(\000)i Fx(1\))18 b(\(if)228 332 y Fw(x)13 b Ft(2)h Fw(P)334 338 y Fs(g)q(;)p Fu(1)379 332 y Fx(\()p Fw(A)p Fx(\),)i(write)f(\()p Fw(")612 338 y Fs(R)650 332 y Ft(\012)c Fw(id)p Fx(\)\()p Fw(x)p Fx(\))i(=)h Fw(\013)p Fx(\()p Fw(g)e Ft(\000)e Fx(1\))p Fw(;)d(\013)13 b Ft(2)g Fv(|)-18 b Fw(;)11 b Fx(and)k(c)o(hec)o(k)i(that)e Fw(x)10 b Ft(\000)g Fw(\013)p Fx(\()p Fw(g)h Ft(\000)g Fx(1\))j Ft(2)228 392 y Fw(P)6 b Fx(\()p Fw(A)p Fx(\))324 377 y Fu(co)o Fm(|)-50 b Fs(G)324 402 y(g)399 392 y Fx(\).)18 b(Hence)d(Supp\()p Fw(A)p Fx(\))f(=)g(Supp\()p Fw(P)6 b Fx(\()p Fw(R)p Fx(\)\))13 b(=)h(Supp\()p Fw(V)c Fx(\))j(generates)i Fw(G)p Fx(,)e(and)g Fw(A)h Fx(is)f(link-)228 452 y(indecomp)q(osable)g (b)o(y)g(4.1.)p 1692 452 2 29 v 1694 425 25 2 v 1694 452 V 1719 452 2 29 v 267 569 a Fy(5.)24 b(Link-indecomposabl)o(e)13 b(p)q(oin)o(ted)g(Hopf)i(algebras)g(o)o(v)o(er)g(Co)o(xeter)g(groups) 303 644 y Fx(Let)10 b Fw(W)16 b Fx(b)q(e)11 b(a)f(group)g(and)g Fw(T)17 b Ft(\032)12 b Fw(W)k Fx(a)10 b(subset)i(suc)o(h)e(that)h(for)e (all)g Fw(g)k Ft(2)e Fw(W)o(;)c(t)k Ft(2)h Fw(T)s(;)7 b(g)q(tg)1585 629 y Fn(\000)p Fu(1)1641 644 y Ft(2)12 b Fw(T)6 b Fx(.)228 694 y(Let)14 b Fw(\037)e Fx(:)f Fw(W)k Ft(\002)9 b Fw(T)18 b Ft(!)11 b Fv(|)-13 b Ft(n)9 b(f)p Fx(0)p Ft(g)k Fx(b)q(e)i(a)f(function)f(suc)o(h)i(that)f(for)f(all)g Fw(g)q(;)7 b(h)k Ft(2)g Fw(W)20 b Fx(and)14 b Fw(t)d Ft(2)g Fw(T)6 b Fx(,)737 769 y Fw(\037)p Fx(\(1)p Fw(;)h(t)p Fx(\))k(=)h(1)p Fw(;)-710 b Fx(\(5.1\))713 838 y Fw(\037)p Fx(\()p Fw(g)q(h;)7 b(t)p Fx(\))k(=)h Fw(\037)p Fx(\()p Fw(g)q(;)7 b(hth)1050 821 y Fn(\000)p Fu(1)1094 838 y Fx(\))p Fw(\037)p Fx(\()p Fw(h;)g(t)p Fx(\))p Fw(:)-1010 b Fx(\(5.2\))228 916 y(W)m(e)13 b(can)h(then)g(de\014ne)h(a)f(Y)m (etter-Drinfeld)f(mo)q(dule)g Fw(V)21 b Fx(=)11 b Fw(V)f Fx(\()p Fw(W)o(;)d(T)s(;)g(\037)p Fx(\))13 b(o)o(v)o(er)h Fw(W)20 b Fx(with)13 b Fv(|)-19 b Fx(-basis)228 966 y Fw(x)252 972 y Fs(t)266 966 y Fw(;)7 b(t)k Ft(2)g Fw(T)s(;)j Fx(and)g(action)g(and)f(coaction)h(of)f Fw(W)20 b Fx(giv)o(en)14 b(b)o(y)812 1041 y Fw(g)q(x)857 1047 y Fs(t)883 1041 y Fx(=)e Fw(\037)p Fx(\()p Fw(g)q(;)7 b(t)p Fx(\))p Fw(x)1064 1049 y Fs(g)q(tg)1111 1040 y Fk(\000)p Fj(1)1151 1041 y Fw(;)-935 b Fx(\(5.3\))781 1106 y Fw(\016)r Fx(\()p Fw(x)841 1112 y Fs(t)855 1106 y Fx(\))12 b(=)g Fw(t)d Ft(\012)g Fw(x)1016 1112 y Fs(t)228 1106 y Fx(\(5.4\))228 1181 y(for)k(all)g Fw(g)g Ft(2)e Fw(W)o(;)c(t)k Ft(2)g Fw(T)6 b Fx(.)303 1231 y(Con)o(v)o(ersely)m(,)18 b(if)e(the)j(function) e Fw(\037)h Fx(de\014nes)h(a)e(Y)m(etter-Drinfeld)h(mo)q(dule)f(on)g (the)h(v)o(ector)228 1281 y(space)d Fw(V)23 b Fx(b)o(y)14 b(\(5.3\),)f(\(5.4\))o(,)h(then)g Fw(\037)g Fx(satis\014es)h(\(5.1\))o (,)f(\(5.2\))o(.)303 1330 y(Note)g(that)g(the)g(braiding)f Fw(c)h Fx(of)f Fw(V)d Fx(\()p Fw(W)o(;)d(T)s(;)g(\037)p Fx(\))13 b(is)h(determined)g(b)o(y)564 1406 y Fw(c)p Fx(\()p Fw(x)622 1412 y Fs(s)648 1406 y Ft(\012)c Fw(x)714 1412 y Fs(t)728 1406 y Fx(\))i(=)g Fw(\037)p Fx(\()p Fw(s;)7 b(t)p Fx(\))p Fw(x)935 1412 y Fs(sts)980 1416 y Fk(\000)p Fj(1)1029 1406 y Ft(\012)i Fw(x)1094 1412 y Fs(t)1122 1406 y Fx(for)14 b(all)e Fw(s;)7 b(t)12 b Ft(2)f Fw(T)s(;)228 1481 y Fx(hence)k(b)o(y)f(the)g(v)n(alues)g(of)f Fw(\037)h Fx(on)g Fw(T)h Ft(\002)9 b Fw(T)d Fx(.)303 1531 y(Our)11 b(main)e(example)g(comes)i(from)e(the)i(theory)g(of)g(Co) o(xeter)g(groups)g(\([)p Fy(B)p Fx(,)g(Chapitre)g(IV]\).)228 1580 y(Let)g Fw(S)j Fx(b)q(e)e(a)e(subset)j(of)d(the)i(group)f Fw(W)16 b Fx(of)11 b(elemen)o(ts)g(of)f(order)i(2.)17 b(F)m(or)10 b(all)g Fw(s;)d(s)1426 1565 y Fn(0)1449 1580 y Ft(2)12 b Fw(S)h Fx(let)e Fw(m)p Fx(\()p Fw(s;)c(s)1693 1565 y Fn(0)1706 1580 y Fx(\))228 1630 y(b)q(e)14 b(the)g(order)g(of)f Fw(ss)548 1615 y Fn(0)560 1630 y Fx(.)18 b(\()p Fw(W)o(;)7 b(S)r Fx(\))14 b(is)f(called)g(a)g Ff(Coxeter)h(system)g Fx(and)f Fw(W)19 b Fx(a)13 b Ff(Coxeter)h(gr)n(oup)g Fx(if)e Fw(W)228 1684 y Fx(is)i(generated)i(b)o(y)e Fw(S)j Fx(with)d(de\014ning)g(relations)h(\()p Fw(ss)1036 1669 y Fn(0)1048 1684 y Fx(\))1064 1669 y Fs(m)p Fu(\()p Fs(s;s)1148 1656 y Fk(0)1159 1669 y Fu(\))1187 1684 y Fx(=)d(1)i(for)g(all)f Fw(s;)7 b(s)1445 1669 y Fn(0)1470 1684 y Ft(2)12 b Fw(S)17 b Fx(suc)o(h)e(that)228 1734 y Fw(m)p Fx(\()p Fw(s;)7 b(s)337 1719 y Fn(0)349 1734 y Fx(\))14 b(is)g(\014nite.)303 1784 y(Let)e(\()p Fw(W)o(;)7 b(S)r Fx(\))13 b(b)q(e)f(a)g(Co)o(xeter)h (system.)k(F)m(or)11 b(an)o(y)h Fw(g)h Ft(2)e Fw(W)18 b Fx(there)13 b(is)f(a)g(sequence)i(\()p Fw(s)1557 1790 y Fu(1)1576 1784 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)1688 1790 y Fs(q)1706 1784 y Fx(\))228 1833 y(of)k(elemen)o(ts)g(in)g Fw(S)k Fx(with)c Fw(g)i Fx(=)e Fw(s)713 1839 y Fu(1)737 1833 y Ft(\001)5 b(\001)i(\001)g(\001)r(\001)e Fw(s)842 1839 y Fs(q)860 1833 y Fx(.)17 b(If)12 b Fw(q)g Fx(is)g(minim)o(al)c (among)h(all)h(suc)o(h)j(represen)o(tations,)228 1883 y(then)j Fw(q)f Fx(=)f Fw(l)q Fx(\()p Fw(g)q Fx(\))j(is)e(called)g(the) h Ff(length)f Fx(of)g Fw(g)q Fx(,)h(and)f(\()p Fw(s)1059 1889 y Fu(1)1078 1883 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)1190 1889 y Fs(q)1208 1883 y Fx(\))16 b(is)f(a)g Ff(r)n(e)n(duc)n(e)n(d)i(r) n(epr)n(esentation)228 1933 y Fx(of)c Fw(g)q Fx(.)303 2028 y Fg(Definition)i Fx(5.1)p Fg(.)21 b Fx(Let)11 b(\()p Fw(W)o(;)c(S)r Fx(\))12 b(b)q(e)g(a)f(Co)o(xeter)h(system,)g(and)f Fw(T)17 b Fx(=)12 b Ft(f)p Fw(g)q(sg)1455 2013 y Fn(\000)p Fu(1)1512 2028 y Ft(j)f Fw(g)i Ft(2)e Fw(W)o(;)c(s)12 b Ft(2)228 2088 y Fw(S)r Ft(g)p Fx(.)18 b(De\014ne)d Fw(\037)c Fx(:)h Fw(W)j Ft(\002)9 b Fw(T)18 b Ft(!)11 b Fv(|)-13 b Ft(n)9 b(f)p Fx(0)p Ft(g)k Fx(b)o(y)636 2173 y Fw(\037)p Fx(\()p Fw(g)q(;)7 b(t)p Fx(\))k(=)h(\()p Ft(\000)p Fx(1\))889 2156 y Fs(l)p Fu(\()p Fs(g)q Fu(\))959 2173 y Fx(for)i(all)f Fw(g)f Ft(2)g Fw(W)o(;)7 b(t)k Ft(2)g Fw(T)s(:)-1086 b Fx(\(5.5\))228 2258 y(Let)18 b Fw(V)28 b Fx(=)19 b Fw(V)9 b Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\))18 b Ft(2)667 2243 y Fs(W)667 2270 y(W)705 2258 y Ft(Y)s(D)r Fx(.)30 b(The)18 b(asso)q(ciated)h(Hopf)f(algebras)f(will) g(b)q(e)h(denoted)h(b)o(y)228 2318 y Fw(R)p Fx(\()p Fw(W)o(;)7 b(S)r Fx(\))12 b(=)g Fo(B)p Fx(\()p Fw(V)d Fx(\))14 b(and)g Fw(A)p Fx(\()p Fw(W)o(;)7 b(S)r Fx(\))12 b(:=)f Fo(B)p Fx(\()p Fw(V)f Fx(\)#)p Fv(|)-20 b Fw(W)6 b Fx(.)303 2403 y(By)19 b(Prop)q(osition)g(4.2,)g Fw(A)p Fx(\()p Fw(W)o(;)7 b(S)r Fx(\))21 b(is)e(a)g(p)q(oin)o(ted)g(link-indecomp)q (osable)f(Hopf)h(algebra)228 2453 y(with)13 b(group-lik)o(e)g(elemen)o (ts)h Fw(W)6 b Fx(.)303 2548 y Fg(Remark)16 b Fx(5.2)p Fg(.)k Fx(W)m(e)11 b(assume)g Fw(W)o(;)c(T)s(;)g(\037)j Fx(as)h(in)g(the)h(b)q(eginning)e(of)g(this)h(section,)h(and)f (de\014ne)228 2608 y Fw(V)21 b Fx(=)12 b Fw(V)d Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\).)p eop %%Page: 10 10 10 9 bop 228 119 a Fu(10)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)303 213 y Fx(1\))19 b(By)g(de\014nition,)h Fw(T)26 b Fx(is)19 b(a)g(union)g(of)f(conjugacy)i(classes)g(of)f Fw(W)6 b Fx(.)34 b(Let)20 b Fw(t)1518 219 y Fu(1)1558 213 y Ft(2)g Fw(T)25 b Fx(and)228 272 y Fw(T)252 278 y Fu(1)287 272 y Fx(the)17 b(conjugacy)f(class)h(con)o(taining)e Fw(t)876 278 y Fu(1)894 272 y Fx(.)26 b(Let)16 b Fw(s)1027 278 y Fs(i)1042 272 y Fw(;)7 b Fx(1)14 b Ft(\024)i Fw(i)g Ft(\024)g Fw(\022)q(;)g Fx(b)q(e)h(represen)o(tativ)o(es)i(of)d(the)228 332 y(righ)o(t)i(residue)j(classes)f(of)e(the)i(cen)o(tralizer)g Fw(C)983 338 y Fs(W)1021 332 y Fx(\()p Fw(t)1052 338 y Fu(1)1070 332 y Fx(\))p Fw(;)f(W)26 b Fx(=)1234 301 y Fl(S)1269 311 y Fs(\022)1269 345 y(i)p Fu(=1)1332 332 y Fw(s)1351 338 y Fs(i)1365 332 y Fw(C)1395 338 y Fs(W)1433 332 y Fx(\()p Fw(t)1464 338 y Fu(1)1482 332 y Fx(\),)20 b(and)f(de\014ne)228 392 y Fw(t)243 398 y Fs(i)269 392 y Fx(:=)11 b Fw(s)343 398 y Fs(i)358 392 y Fw(t)373 398 y Fu(1)391 392 y Fw(s)410 374 y Fn(\000)p Fu(1)410 403 y Fs(i)455 392 y Fw(;)c Fx(1)12 b Ft(\024)g Fw(i)g Ft(\024)g Fw(\022)q(:)j Fx(Then)f Fw(W)q(=C)867 398 y Fs(W)905 392 y Fx(\()p Fw(t)936 398 y Fu(1)955 392 y Fx(\))e Ft(!)f Fw(T)1060 398 y Fu(1)1079 392 y Fw(;)i Fx(\026)-27 b Fw(s)1117 398 y Fs(i)1143 392 y Ft(7!)12 b Fw(t)1212 398 y Fs(i)1225 392 y Fw(;)7 b Fx(1)12 b Ft(\024)g Fw(i)g Ft(\024)g Fw(\022)q(;)j Fx(is)f(bijectiv)o(e.)19 b(Let)232 452 y Fl(b)-27 b Fw(\037)14 b Fx(:)f Fw(C)323 458 y Fs(W)361 452 y Fx(\()p Fw(t)392 458 y Fu(1)410 452 y Fx(\))h Ft(!)g Fv(|)-12 b Ft(n)10 b(f)p Fx(0)p Ft(g)15 b Fx(b)q(e)h(the)g(group)f (homom)o(orphism)c(de\014ned)17 b(b)o(y)i Fl(b)-27 b Fw(\037)p Fx(\()p Fw(g)q Fx(\))14 b(=)h Fw(\037)p Fx(\()p Fw(g)q(;)7 b(t)1623 458 y Fu(1)1641 452 y Fx(\))15 b(for)228 511 y(all)d Fw(g)h Ft(2)e Fw(C)387 517 y Fs(W)425 511 y Fx(\()p Fw(t)456 517 y Fu(1)475 511 y Fx(\).)18 b(W)m(e)13 b(no)o(w)h(c)o(hange)g(the)g(basis)g(to)727 595 y Fw(x)751 601 y Fs(i)776 595 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b(resp)q(ect)k(to)c(the)i(new)f(basis)g Fw(y)794 1041 y Fs(t)807 1045 y Fi(i)834 1035 y Fx(=)d Fw(x)901 1041 y Fs(i)915 1035 y Fw(;)c Fx(1)k Ft(\024)h Fw(i)f Ft(\024)h Fw(\022)q Fx(.)303 1129 y Fg(Pr)o(oof.)21 b Fx(F)m(rom)14 b Fw(t)592 1135 y Fs(j)624 1129 y Fx(=)h Fw(g)q(t)707 1135 y Fs(i)721 1129 y Fw(g)742 1113 y Fn(\000)p Fu(1)802 1129 y Fx(and)h(the)g(de\014nition)f(of)g Fw(t)1208 1135 y Fs(i)1222 1129 y Fw(;)7 b(t)1256 1135 y Fs(j)1289 1129 y Fx(it)15 b(follo)o(ws)f(that)i Fw(s)1581 1111 y Fn(\000)p Fu(1)1581 1140 y Fs(j)1626 1129 y Fw(g)q(s)1666 1135 y Fs(i)1695 1129 y Ft(2)228 1188 y Fw(C)258 1194 y Fs(W)295 1188 y Fx(\()p Fw(t)326 1194 y Fu(1)345 1188 y Fx(\))p Fw(:)h Fx(Hence)j(b)o(y)e(\(5.2\))o(,)g Fw(\037)p Fx(\()p Fw(s)756 1194 y Fs(j)774 1188 y Fx(\()p Fw(s)809 1171 y Fn(\000)p Fu(1)809 1200 y Fs(j)854 1188 y Fw(g)q(s)894 1194 y Fs(i)909 1188 y Fx(\))p Fw(;)7 b(t)959 1194 y Fu(1)977 1188 y Fx(\))18 b(=)g Fw(\037)p Fx(\()p Fw(s)1122 1194 y Fs(j)1141 1188 y Fw(;)7 b(t)1175 1194 y Fu(1)1193 1188 y Fx(\))p Fw(\037)p Fx(\()p Fw(s)1270 1171 y Fn(\000)p Fu(1)1270 1200 y Fs(j)1315 1188 y Fw(g)q(s)1355 1194 y Fs(i)1369 1188 y Fw(;)g(t)1403 1194 y Fu(1)1422 1188 y Fx(\).)29 b(On)18 b(the)h(other)228 1248 y(hand)14 b(b)o(y)g(\(5.2\))o(,)g Fw(\037)p Fx(\()p Fw(g)q(s)583 1254 y Fs(i)597 1248 y Fw(;)7 b(t)631 1254 y Fu(1)649 1248 y Fx(\))12 b(=)g Fw(\037)p Fx(\()p Fw(g)q(;)7 b(t)818 1254 y Fs(i)831 1248 y Fx(\))p Fw(\037)p Fx(\()p Fw(s)908 1254 y Fs(i)923 1248 y Fw(;)g(t)957 1254 y Fu(1)975 1248 y Fx(\),)14 b(and)f(\(5.7\))g(and)h(\(5.8\))f(follo)o(w.)p 1692 1248 2 29 v 1694 1222 25 2 v 1694 1248 V 1719 1248 2 29 v 228 1341 a(2\))f(Con)o(v)o(ersely)m(,)f(an)o(y)h(c)o(haracter)17 b Fl(b)-27 b Fw(\037)11 b Fx(:)g Fw(C)839 1347 y Fs(W)877 1341 y Fx(\()p Fw(t)908 1347 y Fu(1)927 1341 y Fx(\))g Ft(!)g Fv(|)-14 b Ft(n)5 b(f)p Fx(0)p Ft(g)k Fx(de\014nes)k(a)f(Y)m (etter-Drinfeld)g(mo)q(dule)228 1370 y Fl(L)274 1413 y Fs(t)p Fn(2)p Fs(T)329 1417 y Fj(1)354 1401 y Fv(|)-19 b Fw(x)403 1407 y Fs(t)428 1401 y Fx(o)o(v)o(er)12 b Fw(W)19 b Fx(b)o(y)12 b(\(5.8\))g(and)h(\(5.6\))o(,)f(and)h(b)o(y)f (\(5.7\))g(a)h(function)f Fw(\037)f Fx(:)g Fw(W)i Ft(\002)7 b Fw(T)1520 1407 y Fu(1)1550 1401 y Ft(!)k Fv(|)-12 b Ft(n)6 b(f)p Fx(0)o Ft(g)228 1461 y Fx(satisfying)19 b(\(5.1\))g(and)h(\(5.2\))f(where)i Fw(\037)p Fx(\()p Fw(s)903 1467 y Fs(i)917 1461 y Fw(;)7 b(t)951 1467 y Fu(1)969 1461 y Fx(\))p Fw(;)g Fx(1)21 b Ft(\024)g Fw(i)h Ft(\024)g Fw(\022)q(;)e Fx(can)g(b)q(e)g(arbitrary)g(non-zero)228 1520 y(scalars.)31 b(This)18 b(construction)h(is)f(a)g(sp)q(ecial)h (case)g(of)e(the)i(w)o(ell-kno)o(wn)e(description)i(of)f(the)228 1580 y(simple)12 b(Y)m(etter-Drinfeld)i(mo)q(dules)f(o)o(v)o(er)h(a)g (\014nite)g(group)f(\(cf.)19 b([)p Fy(CR)o Fx(],)13 b([)p Fy(A)o(G)o Fx(]\).)303 1640 y(3\))20 b(Assume)g(that)g Fo(B)p Fx(\()p Fw(V)9 b Fx(\))20 b(is)g(\014nite-dimensional.)35 b(Then)21 b(for)e(all)g Fw(t)j Ft(2)g Fw(T)k Fx(there)21 b(is)f(a)228 1700 y(natural)15 b(n)o(um)o(b)q(er)g Fw(n)f Ft(\025)h Fx(2)g(suc)o(h)h(that)g(1)10 b(+)g Fw(\037)p Fx(\()p Fw(t;)d(t)p Fx(\))k(+)f Ft(\001)d(\001)g(\001)i Fx(+)i(\()p Fw(\037)p Fx(\()p Fw(t;)c(t)p Fx(\)\))1311 1685 y Fs(n)p Fn(\000)p Fu(1)1390 1700 y Fx(=)15 b(0,)g(and)g(if)g Fw(t)1622 1685 y Fu(2)1655 1700 y Fx(=)f(1)228 1760 y(and)g(c)o(har\()p Fv(|)-18 b Fx(\))9 b(=)k(0,)h(then)h Fw(\037)p Fx(\()p Fw(t;)7 b(t)p Fx(\))12 b(=)g Ft(\000)p Fx(1.)20 b(Note)15 b(that)f(in)g(1\),)g Fw(\037)p Fx(\()p Fw(t)1249 1766 y Fs(i)1263 1760 y Fw(;)7 b(t)1297 1766 y Fs(i)1310 1760 y Fx(\))13 b(=)j Fl(b)-26 b Fw(\037)p Fx(\()p Fw(t)1441 1766 y Fs(i)1454 1760 y Fx(\))13 b(=)g Fw(\021)q Fx(\()p Fw(t)1581 1766 y Fs(i)1595 1760 y Fw(;)7 b(t)1629 1766 y Fs(i)1642 1760 y Fx(\))14 b(for)228 1819 y(all)e(1)g Ft(\024)g Fw(i)f Ft(\024)h Fw(\022)q(:)303 1912 y Fg(Pr)o(oof.)21 b Fx(Let)11 b Fw(D)571 1918 y Fs(t)586 1912 y Fw(;)c(t)k Ft(2)g Fw(T)16 b Fx(b)q(e)10 b(the)h(sk)o(ew-deriv)n(ations)e(corresp)q (onding)i(to)f(the)g(basis)g Fw(x)1635 1918 y Fs(t)1650 1912 y Fw(;)d(t)k Ft(2)228 1972 y Fw(T)6 b Fx(.)25 b(Let)17 b Fw(t)f Ft(2)f Fw(T)22 b Fx(and)16 b Fw(q)h Fx(=)f Fw(\037)p Fx(\()p Fw(t;)7 b(t)p Fx(\).)25 b(It)16 b(follo)o(ws)f(easily)h(b)o(y)g (induction)g(on)g Fw(n)g Fx(that)h Fw(D)1581 1978 y Fs(t)1596 1972 y Fx(\()p Fw(x)1636 1957 y Fs(n)1636 1982 y(t)1658 1972 y Fx(\))f(=)228 2032 y(\(1)10 b(+)g Fw(q)i Fx(+)e Ft(\001)d(\001)g(\001)i Fx(+)h Fw(q)511 2017 y Fs(n)p Fn(\000)p Fu(1)576 2032 y Fx(\))p Fw(x)616 2014 y Fs(n)p Fn(\000)p Fu(1)616 2042 y Fs(t)697 2032 y Fx(for)15 b(all)f Fw(n)g Ft(\025)g Fx(1.)22 b(Since)16 b Fw(x)1095 2017 y Fs(n)1095 2042 y(t)1131 2032 y Ft(2)e Fo(B)p Fx(\()p Fw(V)9 b Fx(\)\()p Fw(n)p Fx(\),)15 b Fw(x)1383 2017 y Fs(n)1383 2042 y(t)1421 2032 y Fx(m)o(ust)f(b)q(e)i(zero)h(for)228 2092 y(large)f Fw(n)p Fx(,)h(hence)g(1)11 b(+)g Fw(q)h Fx(+)g Ft(\001)7 b(\001)g(\001)i Fx(+)i Fw(q)775 2077 y Fs(n)p Fn(\000)p Fu(1)856 2092 y Fx(=)17 b(0,)f(if)g Fw(x)1019 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Fw(\037)e Fx(is)h(de\014ned)h(as)e(in)h(\(5.5\))e (or)i(for)f(all)228 272 y Fw(g)f Ft(2)e Fw(S)325 278 y Fs(n)348 272 y Fw(;)c Fx(1)k Ft(\024)g Fw(i)h(<)g(j)i Ft(\024)e Fw(n;)655 395 y(\037)p Fx(\()p Fw(g)q(;)7 b Fx(\()p Fw(ij)r Fx(\)\))12 b(=)874 310 y Fl(8)874 347 y(<)874 422 y(:)911 363 y Fx(1)73 b Fw(;)21 b Fx(if)13 b Fw(g)q Fx(\()p Fw(i)p Fx(\))f Fw(<)g(g)q Fx(\()p Fw(j)r Fx(\))p Fw(;)911 434 y Ft(\000)p Fx(1)41 b Fw(;)21 b Fx(if)13 b Fw(g)q Fx(\()p Fw(i)p Fx(\))f Fw(>)g(g)q Fx(\()p Fw(j)r Fx(\))p Fw(:)228 395 y Fx(\(5.9\))303 530 y Fg(Pr)o(oof.)21 b Fx(Let)g Fw(t)562 536 y Fu(1)604 530 y Fx(=)h(\(12\).)38 b(The)21 b(cen)o(tralizer)h(of)e Fw(t)1152 536 y Fu(1)1191 530 y Fx(is)g Ft(f)p Fw(g)k Ft(2)e Fw(S)1379 536 y Fs(n)1425 530 y Ft(j)g(f)p Fw(g)q Fx(\(1\))p Fw(;)7 b(g)q Fx(\(2\))p Ft(g)22 b Fx(=)228 590 y Ft(f)p Fx(1)p Fw(;)7 b Fx(2)p Ft(gg)18 b Fx(=)j Ft(h)p Fx(\(34\))p Fw(;)7 b Fx(\(45\))p Fw(;)g(:)g(:)g(:)t(;)g Fx(\()p Fw(n)12 b Ft(\000)i Fx(1)p Fw(;)7 b(n)p Fx(\))p Ft(i)12 b([)g(h)p Fx(\(34\))p Fw(;)7 b Fx(\(45\))p Fw(;)g(:)g(:)g(:)t(;)g Fx(\()p Fw(n)13 b Ft(\000)g Fx(1)p Fw(;)7 b(n)p Fx(\))p Ft(i)p Fx(\(12\).)33 b(Hence)21 b(the)228 649 y(c)o(haracters)c Fl(b)-27 b Fw(\037)12 b Fx(:)f Fw(C)514 655 y Fs(S)534 659 y Fi(n)556 649 y Fx(\()p Fw(t)587 655 y Fu(1)606 649 y Fx(\))h Ft(!)f Fv(|)-15 b Ft(n)5 b(f)o Fx(0)p Ft(g)k Fx(are)j Fw(\037)907 655 y Fs(")923 659 y Fj(1)938 655 y Fs(;")964 659 y Fj(2)982 649 y Fw(;)f Fx(with)h Fw(")1117 655 y Fu(1)1135 649 y Fw(;)7 b(")1173 655 y Fu(2)1203 649 y Ft(2)12 b(f)p Fx(1)p Fw(;)7 b Ft(\000)p Fx(1)p Ft(g)p Fx(,)j(and)h Fw(\037)1504 655 y Fs(")1520 659 y Fj(1)1536 655 y Fs(;")1562 659 y Fj(2)1580 649 y Fx(\(\()p Fw(ij)r Fx(\)\))i(=)228 709 y Fw(")247 715 y Fu(1)282 709 y Fx(if)i(3)g Ft(\024)h Fw(i)g(<)f(j)j Ft(\024)e Fw(n)p Fx(,)g(and)g Fw(\037)728 715 y Fs(")744 719 y Fj(1)760 715 y Fs(;")786 719 y Fj(2)803 709 y Fx(\(\(12\)\))g(=)g Fw(")992 715 y Fu(2)1011 709 y Fx(.)24 b(Moreo)o(v)o(er)17 b Fw(")1251 715 y Fu(2)1285 709 y Fx(=)f Ft(\000)p Fx(1)g(since)h(w)o(e)g(assumed)228 769 y Fw(\021)q Fx(\()p Fw(t)281 775 y Fu(1)300 769 y Fw(;)7 b(t)334 775 y Fu(1)352 769 y Fx(\))k(=)h Ft(\000)p Fx(1.)18 b(The)12 b(c)o(haracters)i Fw(\037)811 775 y Fn(\000)p Fu(1)p Fs(;)p Fn(\000)p Fu(1)920 769 y Fx(and)e Fw(\037)1025 775 y Fu(1)p Fs(;)p Fn(\000)p Fu(1)1108 769 y Fx(de\014ne)h(the)g(functions)f Fw(\037)g Fx(in)g(\(5.5\))f(and) 228 829 y(\(5.9\))o(.)p 1692 829 2 29 v 1694 802 25 2 v 1694 829 V 1719 829 2 29 v 303 923 a Fg(Example)17 b Fx(5.4)p Fg(.)j Fx(Let)c Fw(W)22 b Fx(=)16 b Fw(D)805 929 y Fs(m)853 923 y Fx(=)f Ft(h)p Fw(t;)7 b(t)965 908 y Fn(0)977 923 y Ft(i)p Fw(;)g(m)15 b Ft(\025)h Fx(1,)h(b)q(e)f(the)h (dihedral)f(group)h(of)e(order)228 983 y(2)p Fw(m)p Fx(,)f(where)h Fw(t;)7 b(t)480 968 y Fn(0)503 983 y Ft(2)k Fw(T)21 b Fx(are)14 b(generators)h(of)f(order)h(2)f(and)f Fw(tt)1159 968 y Fn(0)1185 983 y Fx(is)h(of)g(order)g Fw(m)p Fx(.)20 b(Let)14 b Fw(S)h Fx(=)d Ft(f)p Fw(t;)7 b(t)1679 968 y Fn(0)1690 983 y Ft(g)p Fx(.)228 1043 y(Then)14 b(\()p Fw(W)o(;)7 b(S)r Fx(\))15 b(is)f(a)g(Co)o(xeter)h(sytem.)j(De\014ne)c Fw(t)980 1049 y Fs(i)1006 1043 y Fx(=)e(\()p Fw(tt)1096 1028 y Fn(0)1108 1043 y Fx(\))1124 1028 y Fs(i)p Fn(\000)p Fu(1)1180 1043 y Fw(t)i Fx(for)g(all)f Fw(i)f Ft(2)g Fv(Z)-14 b Fx(.)16 b(Then)e Fw(T)k Fx(=)12 b Ft(f)p Fw(t)1685 1049 y Fs(i)1711 1043 y Ft(j)228 1103 y Fx(1)h Ft(\024)g Fw(i)h Ft(\024)f Fw(m)p Ft(g)p Fx(,)i(and)g(for)g(all)e Fw(i;)7 b(j)r Fx(,)15 b Fw(t)763 1109 y Fs(i)777 1103 y Fw(t)792 1109 y Fs(j)809 1103 y Fw(t)824 1109 y Fs(i)851 1103 y Fx(=)f Fw(t)912 1109 y Fu(2)p Fs(i)p Fn(\000)p Fs(j)984 1103 y Fx(,)g(and)h Fw(t)1107 1109 y Fs(i)1134 1103 y Fx(=)f Fw(t)1195 1109 y Fs(j)1227 1103 y Fx(if)g(and)h(only)f (if)g Fw(i)g Ft(\021)f Fw(j)h Fx(mo)q(d)c Fw(m)p Fx(.)228 1163 y(If)k Fw(m)g Fx(is)g(o)q(dd,)g(then)h Fw(T)20 b Fx(is)14 b(one)g(conjugacy)g(class.)20 b(F)m(or)13 b Fw(m)g Fx(=)f(2)p Fw(l)j Fx(ev)o(en,)f Fw(T)21 b Fx(is)14 b(the)g(union)g(of)g(the)228 1222 y(t)o(w)o(o)h(conjugacy)h(classes)i Fw(T)663 1228 y Fu(0)697 1222 y Fx(=)d Ft(f)p Fw(t)780 1228 y Fu(2)p Fs(k)832 1222 y Ft(j)g Fx(1)g Ft(\024)h Fw(k)g Ft(\024)f Fw(l)q Ft(g)h Fx(and)g Fw(T)1185 1228 y Fu(1)1219 1222 y Fx(=)g Ft(f)p Fw(t)1303 1228 y Fu(2)p Fs(k)q Fn(\000)p Fu(1)1397 1222 y Ft(j)f Fx(1)g Ft(\024)h Fw(k)g Ft(\024)f Fw(l)q Ft(g)p Fx(,)h(and)228 1282 y Fw(z)d Fx(=)f(\()p Fw(tt)350 1267 y Fn(0)362 1282 y Fx(\))378 1267 y Fs(l)405 1282 y Fx(is)i(a)f(cen)o(tral)h(elemen)o(t)g(of)f (order)i(2.)303 1342 y(Let)f Fw(\037)g Fx(b)q(e)g(a)g(function)f (satisfying)h(\(5.1\))o(,)g(\(5.2\))f(and)g Fw(\037)p Fx(\()p Fw(t;)7 b(t)p Fx(\))12 b(=)g Ft(\000)p Fx(1)h(for)h(all)f Fw(t)e Ft(2)g Fw(T)6 b Fx(.)303 1402 y(a\))19 b(If)h Fw(m)h Fx(is)f(o)q(dd,)h(then)f(the)h(cen)o(tralizer)g(of)f Fw(t)1068 1408 y Fu(1)1106 1402 y Fx(is)g Fw(C)1184 1408 y Fs(W)1222 1402 y Fx(\()p Fw(t)1253 1408 y Fu(1)1272 1402 y Fx(\))i(=)g Ft(f)p Fx(1)p Fw(;)7 b(t)1440 1408 y Fu(1)1457 1402 y Ft(g)p Fx(.)37 b(W)m(e)19 b(c)o(ho)q(ose)228 1461 y(represen)o(tativ)o(es)h Fw(s)534 1467 y Fs(i)548 1461 y Fx(,)e(1)g Ft(\024)g Fw(i)g Ft(\024)h Fw(m)p Fx(,)f(of)f(the)i (residue)f(classes)h(of)f Fw(C)1305 1467 y Fs(W)1342 1461 y Fx(\()p Fw(t)1373 1467 y Fu(1)1392 1461 y Fx(\))g(b)o(y)f Fw(s)1506 1467 y Fs(i)1538 1461 y Fx(=)h Fw(t)1603 1467 y Fs(k)1641 1461 y Fx(with)228 1521 y(2)p Fw(k)10 b Ft(\000)g Fx(1)j Ft(\021)h Fw(i)e Fx(mo)q(d)e Fw(m)p Fx(,)15 b(hence)h Fw(s)718 1527 y Fs(i)732 1521 y Fw(t)747 1527 y Fu(1)766 1521 y Fw(s)785 1503 y Fn(\000)p Fu(1)785 1533 y Fs(i)843 1521 y Fx(=)d Fw(t)903 1527 y Fs(i)917 1521 y Fw(:)h Fx(In)h(\(5.8\))f(w)o(e)h(then)h(ha)o(v)o(e)f Fw(\021)q Fx(\()p Fw(t)1404 1527 y Fs(k)1424 1521 y Fw(;)7 b(t)1458 1527 y Fs(i)1471 1521 y Fx(\))14 b(=)f Ft(\000)p Fx(1)i(for)f(all)228 1581 y Fw(k)q(;)7 b(i)p Fx(,)13 b(that)h(is)g(w)o(e)g(are)g(in)f(case)i (\(5.5\).)303 1676 y Fg(Pr)o(oof.)21 b Fx(By)15 b(\(5.8\))e(with)h Fw(g)f Fx(=)f Fw(t)818 1682 y Fs(k)838 1676 y Fx(,)h Fw(\021)q Fx(\()p Fw(t)916 1682 y Fs(k)937 1676 y Fw(;)7 b(t)971 1682 y Fs(i)984 1676 y Fx(\))12 b(=)g Fw(\037)p Fx(\()p Fw(s)1117 1659 y Fn(\000)p Fu(1)1117 1688 y Fs(j)1162 1676 y Fw(t)1177 1682 y Fs(k)1197 1676 y Fw(s)1216 1682 y Fs(i)1231 1676 y Fw(;)7 b(t)1265 1682 y Fu(1)1283 1676 y Fx(\))k(=)h Ft(\000)p Fx(1,)i(since)g Fw(\037)p Fx(\()p Fw(t)1591 1682 y Fu(1)1610 1676 y Fw(;)7 b(t)1644 1682 y Fu(1)1662 1676 y Fx(\))12 b(=)228 1736 y Ft(\000)p Fx(1)19 b(b)o(y)h(assumption,)f(and)g Fw(s)707 1718 y Fn(\000)p Fu(1)707 1748 y Fs(j)752 1736 y Fw(t)767 1742 y Fs(k)788 1736 y Fw(s)807 1742 y Fs(i)842 1736 y Fx(=)i Fw(t)910 1742 y Fu(1)948 1736 y Fx(\(since)g Fw(s)1091 1718 y Fn(\000)p Fu(1)1091 1748 y Fs(j)1136 1736 y Fw(t)1151 1742 y Fs(k)1171 1736 y Fw(s)1190 1742 y Fs(i)1225 1736 y Ft(6)p Fx(=)g(1)f(as)f(a)g(pro)q(duct)i(of)e(three)228 1796 y(re\015ections\).)p 1692 1796 V 1694 1770 25 2 v 1694 1796 V 1719 1796 2 29 v 228 1891 a(b\))i(If)f Fw(m)k Fx(=)f(2)p Fw(l)f Fx(is)e(ev)o(en,)j(w)o(e)e(pic)o(k)g Fw(t)847 1897 y Fu(0)886 1891 y Fx(in)f(the)i(conjugacy)e(class)i Fw(T)1349 1897 y Fu(0)1368 1891 y Fx(.)38 b(Then)22 b Fw(C)1564 1897 y Fs(W)1601 1891 y Fx(\()p Fw(t)1632 1897 y Fu(0)1651 1891 y Fx(\))h(=)228 1951 y Ft(f)p Fx(1)p Fw(;)7 b(t)304 1957 y Fu(0)321 1951 y Fw(;)g(z)r(;)g(t)395 1957 y Fu(0)413 1951 y Fw(z)r Ft(g)p Fx(.)31 b(W)m(e)18 b(c)o(ho)q(ose)h(represen)o(tativ)o(es)h Fw(s)1016 1957 y Fs(i)1031 1951 y Fx(,)f(1)f Ft(\024)h Fw(i)g Ft(\024)g Fw(l)q(;)f Fx(of)g(the)h(residue)g(classes)h(of)228 2011 y Fw(C)258 2017 y Fs(W)295 2011 y Fx(\()p Fw(t)326 2017 y Fu(0)345 2011 y Fx(\))f(b)o(y)g Fw(s)462 2017 y Fs(i)497 2011 y Fx(=)i Fw(t)565 2017 y Fs(i)579 2011 y Fx(,)f(1)g Ft(\024)h Fw(i)g Ft(\024)f Fw(l)q Fx(.)35 b(Then)19 b Fw(s)984 2017 y Fs(i)999 2011 y Fw(t)1014 2017 y Fu(0)1032 2011 y Fw(s)1051 1993 y Fn(\000)p Fu(1)1051 2022 y Fs(i)1117 2011 y Fx(=)h Fw(t)1184 2017 y Fu(2)p Fs(i)1215 2011 y Fx(,1)g Ft(\024)g Fw(i)h Ft(\024)g Fw(l)q(;)e Fx(and)g(for)g(clarit)o (y)228 2071 y(w)o(e)f(denote)i(the)f(elemen)o(ts)f(of)f Fw(T)757 2077 y Fu(0)795 2071 y Fx(b)o(y)h Fw(\034)875 2077 y Fs(i)908 2071 y Fx(=)h Fw(t)974 2077 y Fu(2)p Fs(i)1004 2071 y Fx(,)g(1)g Ft(\024)g Fw(i)g Ft(\024)h Fw(l)q(:)e Fx(Let)g Fw(g)j Fx(=)e Fw(t)1439 2077 y Fs(k)1459 2071 y Fx(,)g(1)g Ft(\024)g Fw(k)h Ft(\024)f Fw(m)p Fx(,)228 2130 y(and)g(1)i Ft(\024)h Fw(i)g Ft(\024)f Fw(l)q Fx(.)36 b(De\014ne)20 b(1)h Ft(\024)h Fw(j)i Ft(\024)d Fw(l)g Fx(b)o(y)e Fw(\034)998 2136 y Fs(j)1037 2130 y Fx(=)j Fw(g)q(\034)1130 2136 y Fs(i)1144 2130 y Fw(g)1165 2115 y Fn(\000)p Fu(1)1210 2130 y Fw(;)d Fx(that)h(is)g Fw(t)1400 2136 y Fu(2)p Fs(j)1455 2130 y Fx(=)i Fw(t)1524 2137 y Fu(2\()p Fs(k)q Fn(\000)p Fs(i)p Fu(\))1624 2130 y Fw(;)d Fx(and)228 2190 y Fw(j)e Ft(\021)f Fw(k)11 b Ft(\000)g Fw(i)h Fx(mo)q(d)e Fw(l)q(:)16 b Fx(In)g(\(5.6\))f(w)o(e)h(write)h Fw(x)895 2196 y Fu(2)p Fs(i)941 2190 y Fx(instead)f(of)f Fw(x)1159 2196 y Fs(i)1189 2190 y Fx(and)h(extend)h(this)f (de\014nition)f(b)o(y)228 2250 y Fw(x)252 2256 y Fu(2)p Fs(i)293 2250 y Fx(=)d Fw(\037)p Fx(\()p Fw(t)394 2257 y Fs(\036)p Fu(\()p Fs(i)p Fu(\))454 2250 y Fw(;)7 b(t)488 2256 y Fu(0)506 2250 y Fx(\))p Fw(x)546 2256 y Fu(2)p Fs(i)576 2250 y Fw(;)13 b Fx(where)h Fw(\036)p Fx(\()p Fw(i)p Fx(\))f(is)g(the)g(in)o(teger)h(with)e(1)g Ft(\024)f Fw(\036)p Fx(\()p Fw(i)p Fx(\))h Ft(\024)g Fw(l)i Fx(and)f Fw(\036)p Fx(\()p Fw(i)p Fx(\))f Ft(\021)g Fw(i)f Fx(mo)q(d)f Fw(l)q Fx(.)228 2310 y(W)m(e)18 b(then)h(ha)o(v)o(e)g Fw(t)519 2316 y Fs(k)539 2310 y Fw(x)563 2316 y Fu(2)p Fs(i)612 2310 y Fx(=)h Fw(\021)q Fx(\()p Fw(t)717 2316 y Fs(k)738 2310 y Fw(;)7 b(t)772 2316 y Fu(2)p Fs(i)801 2310 y Fx(\))p Fw(x)841 2317 y Fu(2\()p Fs(k)q Fn(\000)p Fs(i)p Fu(\))942 2310 y Fx(.)32 b(De\014ne)19 b Fw(")1138 2316 y Fu(0)1176 2310 y Ft(2)g(f)p Fx(1)p Fw(;)7 b Ft(\000)p Fx(1)p Ft(g)18 b Fx(b)o(y)g Fw(\037)p Fx(\()p Fw(z)r(;)7 b(t)1535 2316 y Fu(0)1553 2310 y Fx(\))20 b(=)f Ft(\000)p Fw(")1691 2316 y Fu(0)1711 2310 y Fx(.)228 2370 y(Then)14 b(for)g(all)e(1)g Ft(\024)f Fw(k)i Ft(\024)f Fw(m;)7 b Fx(1)k Ft(\024)h Fw(i)g Ft(\024)f Fw(l)q Fx(,)642 2543 y Fw(\021)q Fx(\()p Fw(t)695 2549 y Fs(k)715 2543 y Fw(;)c(t)749 2549 y Fu(2)p Fs(i)779 2543 y Fx(\))12 b(=)851 2458 y Fl(8)851 2496 y(<)851 2570 y(:)887 2512 y Ft(\000)p Fx(1)42 b Fw(;)20 b Fx(if)13 b(1)f Ft(\024)f Fw(k)g Ft(\000)e Fw(i)j Ft(\024)g Fw(l)q(;)887 2583 y(")906 2589 y Fu(0)996 2583 y Fx(otherwise)q Fw(:)228 2543 y Fx(\(5.10\))p eop %%Page: 12 12 12 11 bop 228 119 a Fu(12)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)303 213 y Fg(Pr)o(oof.)21 b Fx(By)15 b(\(5.8\))o(,)396 290 y Fw(\021)q Fx(\()p Fw(t)449 296 y Fs(k)470 290 y Fw(;)7 b(\034)507 296 y Fs(i)520 290 y Fx(\))12 b(=)i Fl(c)-44 b Fw(\037)618 296 y Fu(0)637 290 y Fx(\(\()p Fw(tt)699 273 y Fn(0)710 290 y Fx(\))726 273 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)826 290 y Fw(t)841 273 y Fn(0)852 290 y Fx(\))p Fw(;)21 b Fx(since)14 b Fw(s)1021 272 y Fn(\000)p Fu(1)1021 302 y Fs(j)1066 290 y Fw(t)1081 296 y Fs(k)1102 290 y Fw(s)1121 296 y Fs(i)1147 290 y Fx(=)d Fw(t)1205 296 y Fs(j)1223 290 y Fw(t)1238 296 y Fs(k)1258 290 y Fw(t)1273 296 y Fs(i)1299 290 y Fx(=)h(\()p Fw(tt)1389 273 y Fn(0)1400 290 y Fx(\))1416 273 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)1515 290 y Fw(t)1530 273 y Fn(0)1542 290 y Fw(:)228 367 y Fx(The)i(homomorphism)729 427 y Fl(c)-44 b Fw(\037)753 433 y Fu(0)783 427 y Fx(:)11 b Ft(f)p Fx(1)p Fw(;)c(t)882 433 y Fu(0)900 427 y Fw(;)g(z)r(;)g(t)974 433 y Fu(0)992 427 y Fw(z)r Ft(g)k(!)g Fv(|)-13 b Ft(n)9 b(f)p Fx(0)p Ft(g)228 496 y Fx(is)15 b(giv)o(en)h(b)o(y)h Fl(c)-44 b Fw(\037)467 502 y Fu(0)486 496 y Fx(\()p Fw(t)517 502 y Fu(0)535 496 y Fx(\))15 b(=)g Fw(\037)p Fx(\()p Fw(t)670 502 y Fu(0)689 496 y Fw(;)7 b(t)723 502 y Fu(0)741 496 y Fx(\))15 b(=)g Ft(\000)p Fx(1)g(and)i Fl(c)-43 b Fw(\037)996 502 y Fu(0)1014 496 y Fx(\()p Fw(z)r Fx(\))16 b(=)e Fw(\037)p Fx(\()p Fw(z)r(;)7 b(t)1226 502 y Fu(0)1245 496 y Fx(\))15 b(=)f Ft(\000)p Fw(")1373 502 y Fu(0)1393 496 y Fw(:)h Fx(If)g(1)g Ft(\024)g Fw(k)c Ft(\000)g Fw(i)k Ft(\024)f Fw(l)q Fx(,)228 556 y(then)d Fw(j)j Fx(=)e Fw(k)t Ft(\000)r Fw(i)p Fx(,)e(and)g(\()p Fw(tt)613 541 y Fn(0)625 556 y Fx(\))641 541 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)740 556 y Fw(t)755 541 y Fn(0)778 556 y Fx(=)i Fw(t)837 541 y Fn(0)860 556 y Fx(=)g Fw(t)919 562 y Fu(0)938 556 y Fx(,)e(and)h Fw(\021)q Fx(\()p Fw(t)1091 562 y Fs(k)1111 556 y Fw(;)c(\034)1148 562 y Fs(i)1162 556 y Fx(\))k(=)j Fl(c)-44 b Fw(\037)1259 562 y Fu(0)1278 556 y Fx(\()p Fw(t)1309 562 y Fu(0)1328 556 y Fx(\))11 b(=)h Ft(\000)p Fx(1)p Fw(:)e Fx(If)g Fw(k)t Ft(\000)r Fw(i)h Fx(=)h Fw(j)5 b Ft(\006)r Fw(l)q(;)228 616 y Fx(then)14 b(\()p Fw(tt)368 601 y Fn(0)380 616 y Fx(\))396 601 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)495 616 y Fw(t)510 601 y Fn(0)533 616 y Fx(=)e Fw(z)r(t)613 622 y Fu(0)632 616 y Fx(,)h(and)h Fw(\021)q Fx(\()p Fw(t)791 622 y Fs(k)812 616 y Fw(;)7 b(\034)849 622 y Fs(i)862 616 y Fx(\))12 b(=)h Fl(c)-43 b Fw(\037)960 622 y Fu(0)978 616 y Fx(\()p Fw(z)r(t)1030 622 y Fu(0)1049 616 y Fx(\))12 b(=)g Fw(")1140 622 y Fu(0)1159 616 y Fx(.)p 1692 616 2 29 v 1694 589 25 2 v 1694 616 V 1719 616 2 29 v 303 701 a(c\))f(If)f Fw(m)i Fx(=)g(2)p Fw(l)q Fx(,)e(w)o(e)h(pic)o(k)f Fw(t)691 707 y Fu(1)720 701 y Fx(in)h(the)g(conjugacy)f(class)h Fw(T)1142 707 y Fu(1)1161 701 y Fx(.)17 b(Then)11 b Fw(C)1325 707 y Fs(W)1362 701 y Fx(\()p Fw(t)1393 707 y Fu(1)1412 701 y Fx(\))h(=)g Ft(f)p Fx(1)p Fw(;)7 b(t)1560 707 y Fu(1)1577 701 y Fw(;)g(z)r(;)g(t)1651 707 y Fu(1)1669 701 y Fw(z)r Ft(g)p Fw(:)228 760 y Fx(W)m(e)14 b(c)o(ho)q(ose)i (represen)o(tativ)o(es)i Fw(s)737 766 y Fs(i)751 760 y Fw(;)7 b Fx(1)13 b Ft(\024)g Fw(i)h Ft(\024)g Fw(l)q Fx(,)h(b)o(y)g Fw(s)1041 766 y Fs(i)1068 760 y Fx(=)f Fw(t)1129 766 y Fs(i)1143 760 y Fx(.)21 b(Then)16 b Fw(s)1305 766 y Fs(i)1319 760 y Fw(t)1334 766 y Fu(1)1353 760 y Fw(s)1372 743 y Fn(\000)p Fu(1)1372 772 y Fs(i)1430 760 y Fx(=)e Fw(t)1491 766 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1564 760 y Fx(,)h(and)f(for)228 820 y(clarit)o(y)g(w)o(e)i(set)g Fw(\034)505 826 y Fs(i)533 820 y Fx(=)e Fw(t)594 826 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)666 820 y Fw(;)7 b Fx(1)13 b Ft(\024)h Fw(i)g Ft(\024)g Fw(l)q Fx(.)23 b(Let)15 b Fw(g)g Fx(=)f Fw(t)1058 826 y Fs(k)1079 820 y Fw(;)7 b Fx(1)13 b Ft(\024)h Fw(k)g Ft(\024)g Fw(m)p Fx(,)i(and)f(1)e Ft(\024)h Fw(i)g Ft(\024)g Fw(l)q Fx(.)22 b(De\014ne)228 880 y(1)14 b Ft(\024)h Fw(j)h Ft(\024)f Fw(l)i Fx(b)o(y)e Fw(\034)496 886 y Fs(j)528 880 y Fx(=)g Fw(g)q(\034)614 886 y Fs(i)628 880 y Fw(g)649 865 y Fn(\000)p Fu(1)694 880 y Fw(;)g Fx(that)h(is)f Fw(t)871 886 y Fu(2)p Fs(j)r Fn(\000)p Fu(1)962 880 y Fx(=)g Fw(t)1024 887 y Fu(2\()p Fs(k)q Fn(\000)p Fs(i)p Fu(+1\))p Fn(\000)p Fu(1)1209 880 y Fx(,)g(or)h Fw(j)h Ft(\021)d Fw(k)e Ft(\000)e Fw(i)h Fx(+)f(1)i(mo)q(d)e Fw(l)q Fx(.)23 b(In)228 940 y(\(5.6\))14 b(w)o(e)g(write)h Fw(x)520 946 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)608 940 y Fx(instead)f(of)g Fw(x)823 946 y Fs(i)837 940 y Fx(.)20 b(As)15 b(in)f(b\))h(w)o(e)g(de\014ne)g Fw(x)1240 946 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1326 940 y Fx(=)e Fw(\037)p Fx(\()p Fw(t)1428 947 y Fs(\036)p Fu(\()p Fs(i)p Fu(\))1488 940 y Fw(;)7 b(t)1522 946 y Fu(1)1540 940 y Fx(\))p Fw(x)1580 946 y Fs(t)1593 950 y Fj(2)p Fi(i)p Fk(\000)p Fj(1)1673 940 y Fx(for)228 1000 y(all)13 b(in)o(tegers)i Fw(i)p Fx(.)j(W)m(e)c(then)h(ha)o(v)o(e)f Fw(t)762 1006 y Fs(k)782 1000 y Fw(x)806 1006 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)891 1000 y Fx(=)e Fw(\021)q Fx(\()p Fw(t)988 1006 y Fs(k)1008 1000 y Fw(;)7 b(t)1042 1006 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1115 1000 y Fx(\))p Fw(x)1155 1007 y Fu(2\()p Fs(k)q Fn(\000)p Fs(i)p Fu(+1\))p Fn(\000)p Fu(1)1340 1000 y Fx(.)18 b(De\014ne)d Fw(")1518 1006 y Fu(1)1549 1000 y Ft(2)c(f)p Fx(1)p Fw(;)c Ft(\000)p Fx(1)p Ft(g)228 1059 y Fx(b)o(y)13 b Fw(\037)p Fx(\()p Fw(z)r(;)7 b(t)382 1065 y Fu(1)401 1059 y Fx(\))k(=)h Ft(\000)p Fw(")523 1065 y Fu(1)543 1059 y Fx(.)18 b(Then)c(for)f(all)g (1)e Ft(\024)h Fw(k)h Ft(\024)e Fw(m;)c Fx(1)12 b Ft(\024)f Fw(i)h Ft(\024)g Fw(l)q(;)585 1226 y(\021)q Fx(\()p Fw(t)638 1232 y Fs(k)658 1226 y Fw(;)7 b(t)692 1232 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)765 1226 y Fx(\))k(=)836 1141 y Fl(8)836 1178 y(<)836 1253 y(:)873 1194 y Ft(\000)p Fx(1)42 b Fw(;)20 b Fx(if)13 b(1)e Ft(\024)h Fx(1)d(+)g Fw(k)i Ft(\000)e Fw(i)j Ft(\024)g Fw(l)q(;)873 1266 y(")892 1272 y Fu(1)981 1266 y Fx(otherwise)q Fw(:)228 1226 y Fx(\(5.11\))303 1355 y Fg(Pr)o(oof.)21 b Fx(By)h(\(5.8\))e(with)h Fw(g)j Fx(=)f Fw(t)861 1361 y Fs(k)882 1355 y Fx(,)f Fw(\021)q Fx(\()p Fw(t)969 1361 y Fs(k)989 1355 y Fw(;)7 b(\034)1026 1361 y Fs(i)1040 1355 y Fx(\))23 b(=)i Fl(c)-44 b Fw(\037)1160 1361 y Fu(1)1179 1355 y Fx(\(\()p Fw(tt)1241 1340 y Fn(0)1253 1355 y Fx(\))1269 1340 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)1368 1355 y Fw(t)1383 1340 y Fn(0)1394 1355 y Fx(\))21 b(as)g(b)q(efore.)40 b(The)228 1415 y(homomo)o(rphism) 729 1475 y Fl(c)-44 b Fw(\037)753 1481 y Fu(1)783 1475 y Fx(:)11 b Ft(f)p Fx(1)p Fw(;)c(t)882 1481 y Fu(1)900 1475 y Fw(;)g(z)r(;)g(t)974 1481 y Fu(1)992 1475 y Fw(z)r Ft(g)k(!)g Fv(|)-13 b Ft(n)9 b(f)p Fx(0)p Ft(g)228 1543 y Fx(is)17 b(giv)o(en)f(b)o(y)i Fl(c)-43 b Fw(\037)471 1549 y Fu(1)489 1543 y Fx(\()p Fw(t)520 1549 y Fu(1)539 1543 y Fx(\))17 b(=)g Ft(\000)p Fx(1)p Fw(;)8 b Fl(c)-43 b Fw(\037)719 1549 y Fu(1)737 1543 y Fx(\()p Fw(z)r Fx(\))17 b(=)g Ft(\000)p Fw(")907 1549 y Fu(1)926 1543 y Fw(:)f Fx(If)h(1)f Ft(\024)h Fw(k)12 b Ft(\000)g Fw(i)f Fx(+)h(1)k Ft(\024)h Fw(l)q Fx(,)h(then)f Fw(j)i Fx(=)e Fw(k)12 b Ft(\000)g Fw(i)f Fx(+)h(1)p Fw(;)228 1603 y Fx(and)j(\()p Fw(tt)356 1588 y Fn(0)367 1603 y Fx(\))383 1588 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)482 1603 y Fw(t)497 1588 y Fn(0)522 1603 y Fx(=)f Fw(t)f Fx(=)h Fw(t)657 1609 y Fu(1)675 1603 y Fx(,)h(hence)h Fw(\021)q Fx(\()p Fw(t)871 1609 y Fs(k)892 1603 y Fw(;)7 b(\034)929 1609 y Fs(i)943 1603 y Fx(\))13 b(=)i Fl(c)-43 b Fw(\037)1044 1609 y Fu(1)1062 1603 y Fx(\()p Fw(t)1093 1609 y Fu(1)1112 1603 y Fx(\))13 b(=)h Ft(\000)p Fx(1)p Fw(:)g Fx(If)h Fw(k)c Ft(\000)f Fw(i)g Fx(+)g(1)j(=)h Fw(j)e Ft(\006)f Fw(l)q Fx(,)j(then)228 1663 y(\()p Fw(tt)274 1648 y Fn(0)286 1663 y Fx(\))302 1648 y Fs(j)r Fn(\000)p Fs(k)q Fu(+)p Fs(i)401 1663 y Fw(t)416 1648 y Fn(0)439 1663 y Fx(=)e Fw(z)r Fx(\()p Fw(tt)550 1648 y Fn(0)562 1663 y Fx(\))p Fw(t)593 1648 y Fn(0)616 1663 y Fx(=)g Fw(z)r(t)696 1669 y Fu(1)714 1663 y Fx(,)i(hence)h Fw(\021)q Fx(\()p Fw(t)908 1669 y Fs(k)929 1663 y Fw(;)7 b(\034)966 1669 y Fs(i)979 1663 y Fx(\))12 b(=)g Fw(")1070 1669 y Fu(1)1089 1663 y Fw(:)p 1692 1663 V 1694 1637 25 2 v 1694 1663 V 1719 1663 2 29 v 303 1748 a Fx(d\))k(Con)o(v)o(ersely)m(,)g(if)f Fw(m)h Fx(=)g(2)p Fw(l)h Fx(is)f(ev)o(en)h(and)f Fw(")1012 1754 y Fu(0)1031 1748 y Fw(;)7 b(")1069 1754 y Fu(1)1103 1748 y Fx(are)17 b(arbitrary)f(elemen)o(ts)g(in)g Ft(f)p Fx(1)p Fw(;)7 b Ft(\000)p Fx(1)p Ft(g)p Fx(,)228 1808 y(then)13 b(\(5.10\))o(,)g(\(5.11\))f(de\014ne)i(a)e(function)h Fw(\021)h Fx(satisfying)e(\(5.1\),)g(\(5.2\))g(and)h Fw(\021)q Fx(\()p Fw(t;)7 b(t)p Fx(\))k(=)h Ft(\000)p Fx(1)h(for)f(all)228 1868 y Fw(t)f Ft(2)h Fw(T)6 b Fx(.)303 1927 y(Note)14 b(that)g(the)g(case)h Fw(")671 1933 y Fu(0)702 1927 y Fx(=)c Ft(\000)p Fx(1)h(=)g Fw(")873 1933 y Fu(1)906 1927 y Fx(is)h(the)i(example)d(\(5.5\))i(for)f Fw(W)18 b Fx(=)12 b Fw(D)1479 1933 y Fs(m)1510 1927 y Fx(.)303 2017 y Fg(Lemma)k Fx(5.5)p Fg(.)k Ff(Assume)12 b Fw(W)o(;)7 b(T)18 b Ff(and)12 b Fw(\037)g Ff(as)g(in)g(the)g(b)n(e)n (ginning)h(of)e(this)h(se)n(ction)g(and)h(let)e Fw(V)21 b Fx(=)228 2077 y Fw(V)9 b Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\))p Ff(.)18 b(Then)c(the)f(antip)n(o)n(de)h Fw(S)g Fx(:)d Fo(B)p Fx(\()p Fw(V)e Fx(\))j Ft(!)f Fo(B)p Fx(\()p Fw(V)e Fx(\))14 b Ff(c)n(an)f(b)n(e)g(explicitly)g(c)n(ompute)n(d)h(as) f(fol-)228 2137 y(lows.)k(L)n(et)10 b Fw(s)421 2143 y Fu(1)440 2137 y Fw(;)d(:)g(:)g(:)e(;)i(s)552 2143 y Fs(q)582 2137 y Ft(2)k Fw(T)16 b Ff(and)c(de\014ne)g Fw(t)870 2143 y Fu(1)888 2137 y Fw(;)7 b(:)g(:)g(:)e(;)i(t)996 2143 y Fs(q)1025 2137 y Ft(2)k Fw(T)17 b Ff(by)11 b Fw(t)1170 2143 y Fs(i)1195 2137 y Fx(=)h(\()p Fw(s)1274 2143 y Fu(1)1300 2137 y Ft(\001)7 b(\001)g(\001)f Fw(s)1375 2143 y Fs(i)p Fn(\000)p Fu(1)1431 2137 y Fx(\))p Fw(s)1466 2143 y Fs(i)1481 2137 y Fx(\()p Fw(s)1516 2143 y Fu(1)1542 2137 y Ft(\001)h(\001)g(\001)e Fw(s)1616 2143 y Fs(i)p Fn(\000)p Fu(1)1673 2137 y Fx(\))1689 2122 y Fn(\000)p Fu(1)1734 2137 y Fw(;)228 2196 y Fx(1)11 b Ft(\024)h Fw(i)g Ft(\024)g Fw(q)q(:)i Ff(Then)542 2274 y Fw(S)r Fx(\()p Fw(x)609 2280 y Fs(s)625 2284 y Fj(1)644 2274 y Fw(x)668 2280 y Fs(s)684 2284 y Fj(2)709 2274 y Ft(\001)7 b(\001)g(\001)e Fw(x)788 2280 y Fs(s)804 2284 y Fi(q)822 2274 y Fx(\))12 b(=)g Fw(")p Fx(\()p Fw(s)948 2280 y Fu(1)967 2274 y Fw(;)7 b(s)1005 2280 y Fu(2)1024 2274 y Fw(;)g(:)g(:)g(:)t(;)g(s)1135 2280 y Fs(q)1154 2274 y Fx(\))p Fw(x)1194 2280 y Fs(t)1207 2284 y Fi(q)1232 2274 y Ft(\001)g(\001)g(\001)e Fw(x)1311 2280 y Fs(t)1324 2284 y Fj(2)1341 2274 y Fw(x)1365 2280 y Fs(t)1378 2284 y Fj(1)1396 2274 y Fw(;)228 2351 y Ff(wher)n(e)14 b Fw(")p Fx(\()p Fw(s)399 2357 y Fu(1)419 2351 y Fw(;)7 b(:)g(:)g(:)t(;)g(s)530 2357 y Fs(q)549 2351 y Fx(\))k(=)h(\()p Ft(\000)p Fx(1\))705 2336 y Fs(q)731 2320 y Fl(Q)770 2330 y Fs(q)770 2364 y(i)p Fu(=2)833 2351 y Fw(\037)p Fx(\()p Fw(s)894 2357 y Fu(1)913 2351 y Fw(s)932 2357 y Fu(2)958 2351 y Ft(\001)7 b(\001)g(\001)e Fw(s)1032 2357 y Fs(i)p Fn(\000)p Fu(1)1089 2351 y Fw(;)i(s)1127 2357 y Fs(i)1141 2351 y Fx(\))p Fw(:)303 2441 y Fg(Pr)o(oof.)21 b Fx(F)m(or)14 b(all)f Fw(x;)7 b(y)12 b Ft(2)f Fo(B)p Fx(\()p Fw(V)f Fx(\),)j Fw(S)r Fx(\()p Fw(xy)q Fx(\))g(=)f Fw(S)r Fx(\()p Fw(x)1069 2447 y Fn(\000)p Fu(1)1124 2441 y Ft(\001)c Fw(y)q Fx(\))p Fw(S)r Fx(\()p Fw(x)1248 2447 y Fu(0)1269 2441 y Fx(\))p Fw(:)13 b Fx(Hence)447 2519 y Fw(S)r Fx(\()p Fw(x)514 2525 y Fs(s)530 2529 y Fj(1)555 2519 y Ft(\001)7 b(\001)g(\001)f Fw(x)635 2525 y Fs(s)651 2529 y Fi(q)q Fk(\000)p Fj(1)705 2519 y Fw(x)729 2525 y Fs(s)745 2529 y Fi(q)763 2519 y Fx(\))12 b(=)g Fw(S)r Fx(\(\()p Fw(s)913 2525 y Fu(1)940 2519 y Ft(\001)7 b(\001)g(\001)e Fw(s)1014 2525 y Fs(q)q Fn(\000)p Fu(1)1076 2519 y Fx(\))k Ft(\001)g Fw(x)1146 2525 y Fs(s)1162 2529 y Fi(q)1180 2519 y Fx(\))p Fw(S)r Fx(\()p Fw(x)1263 2525 y Fs(s)1279 2529 y Fj(1)1304 2519 y Ft(\001)e(\001)g(\001)f Fw(x)1384 2525 y Fs(s)1400 2529 y Fi(q)q Fk(\000)p Fj(1)1454 2519 y Fx(\))791 2591 y(=)12 b Ft(\000)p Fw(\037)p Fx(\()p Fw(s)928 2597 y Fu(1)954 2591 y Ft(\001)7 b(\001)g(\001)f Fw(s)1029 2597 y Fs(q)q Fn(\000)p Fu(1)1090 2591 y Fw(;)h(s)1128 2597 y Fs(q)1146 2591 y Fx(\))p Fw(x)1186 2597 y Fs(t)1199 2601 y Fi(q)1217 2591 y Fw(S)r Fx(\()p Fw(x)1284 2597 y Fs(s)1300 2601 y Fj(1)1326 2591 y Ft(\001)g(\001)g(\001)e Fw(x)1405 2597 y Fs(s)1421 2601 y Fi(q)q Fk(\000)p Fj(1)1476 2591 y Fx(\))p Fw(;)p eop %%Page: 13 13 13 12 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d (O)o(VER)i(CO)o(XETER)g(GR)o(OUPS)95 b(13)228 213 y Fx(and)14 b(the)g(claim)e(follo)o(ws)g(b)o(y)i(induction)f(on)h Fw(q)q Fx(.)p 1692 213 2 29 v 1694 186 25 2 v 1694 213 V 1719 213 2 29 v 303 301 a(W)m(e)i(no)o(w)g(describ)q(e)j(some)d(of)g (the)i(relations)f(of)f Fo(B)p Fx(\()p Fw(V)9 b Fx(\).)27 b(In)17 b(general)g(w)o(e)g(do)g(not)g(kno)o(w)228 351 y(whether)e(these)g(are)g(de\014ning)f(relations.)k(W)m(e)13 b(b)q(egin)h(with)g(the)g(dihedral)g(group)g(of)f(order)i(4.)228 401 y(In)f(the)h(terminology)d(of)i([)p Fy(AS2)o Fx(],)f(the)i(Hopf)f (algebras)g(are)h(then)g(quan)o(tum)e(linear)g(spaces)j(or)228 451 y(of)d(t)o(yp)q(e)h Fw(A)399 457 y Fu(2)418 451 y Fx(.)k(F)m(or)c(completeness)g(w)o(e)g(giv)o(e)g(a)f(pro)q(of)h(in)f (the)i(con)o(text)f(of)g(this)g(pap)q(er.)303 547 y Fg(Lemma)i Fx(5.6)p Fg(.)k Ff(L)n(et)e Fw(W)24 b Fx(=)18 b Fw(D)766 553 y Fu(2)804 547 y Ff(b)n(e)g(gener)n(ate)n(d)g(by)h Fw(t)f Fx(=)g Fw(t)1201 553 y Fu(1)1219 547 y Fw(;)7 b(t)1253 532 y Fn(0)1282 547 y Fx(=)18 b Fw(t)1347 553 y Fu(2)1384 547 y Ff(as)h(in)f(Example)h(5.4)228 607 y(with)c Fw(T)21 b Fx(=)14 b Ft(f)p Fw(t)446 613 y Fu(1)464 607 y Fw(;)7 b(t)498 613 y Fu(2)516 607 y Ft(g)p Ff(.)23 b(L)n(et)16 b Fw(")665 613 y Fu(1)684 607 y Fw(;)7 b(")722 613 y Fu(0)755 607 y Ft(2)13 b(f)p Fx(1)p Fw(;)7 b Ft(\000)p Fx(1)p Ft(g)16 b Ff(and)h(de\014ne)g Fw(\037)d Fx(:)g Fw(W)i Ft(\002)10 b Fw(T)20 b Ft(!)14 b Fv(|)-12 b Ft(n)10 b(f)p Fx(0)p Ft(g)16 b Ff(by)k Fx(\(5.10\))228 667 y Ff(and)g Fx(\(5.11\))o Ff(.)g(L)n(et)15 b Fw(V)22 b Fx(=)13 b Fw(V)c Fx(\()p Fw(D)698 673 y Fu(2)717 667 y Fw(;)e(T)s(;)g(\021)q Fx(\))p Ff(,)15 b(and)h Fw(x)c Fx(=)h Fw(x)1034 673 y Fs(t)1048 667 y Fw(;)7 b(y)14 b Fx(=)e Fw(x)1169 673 y Fs(t)1182 665 y Fk(0)1195 667 y Ff(.)20 b(Thus)15 b(the)h(action)f (is)g(given)h(by)228 727 y Fw(t)9 b Ft(\001)g Fw(x)i Fx(=)h Ft(\000)p Fw(x;)7 b(t)442 712 y Fn(0)462 727 y Ft(\001)i Fw(y)k Fx(=)f Ft(\000)p Fw(y)17 b Ff(and)f Fw(t)725 712 y Fn(0)746 727 y Ft(\001)8 b Fw(x)k Fx(=)f Fw(")864 733 y Fu(1)883 727 y Fw(x;)c(t)i Ft(\001)g Fw(y)k Fx(=)f Fw(")1067 733 y Fu(0)1086 727 y Fw(y)q(:)303 786 y Ff(1\))j(Assume)g Fw(")530 792 y Fu(1)560 786 y Fx(=)d Fw(")623 792 y Fu(0)657 786 y Ff(and)j(let)g Fw(")d Fx(=)f Fw(")889 792 y Fu(1)920 786 y Fx(=)h Fw(")983 792 y Fu(0)1002 786 y Ff(.)19 b(Then)841 873 y Fw(x)865 856 y Fu(2)895 873 y Fx(=)11 b(0)p Fw(;)c(y)999 856 y Fu(2)1030 873 y Fx(=)k(0)p Fw(;)-878 b Fx(\(5.12\))837 946 y Fw(xy)14 b Fx(=)d Fw("y)q(x;)-786 b Fx(\(5.13\))228 1033 y Ff(ar)n(e)14 b(de\014ning)i(r)n(elations)f(of)f Fo(B)p Fx(\()p Fw(V)c Fx(\))p Ff(,)k(and)i Fx(1)p Fw(;)7 b(x;)g(y)q(;)g(xy)15 b Ff(is)g(a)g(b)n(asis)f(of)h Fo(B)p Fx(\()p Fw(V)9 b Fx(\))p Ff(.)303 1093 y(2\))15 b(Assume)g Fw(")530 1099 y Fu(1)560 1093 y Fx(=)d Ft(\000)p Fw(")655 1099 y Fu(0)674 1093 y Ff(,)j(and)g(char)p Fx(\()p Fv(|)-18 b Fx(\))8 b Ft(6)p Fx(=)k(2)p Fw(:)i Ff(Then)867 1180 y Fw(x)891 1163 y Fu(2)921 1180 y Fx(=)e(0)p Fw(;)7 b(y)1026 1163 y Fu(2)1056 1180 y Fx(=)12 b(0)p Fw(;)-905 b Fx(\(5.14\))811 1252 y Fw(xy)q(xy)12 b Fx(+)d Fw(y)q(xy)q(x)k Fx(=)f(0)p Fw(;)-905 b Fx(\(5.15\))228 1339 y Ff(ar)n(e)11 b(de\014ning)i(r)n (elations)e(of)g Fo(B)p Fx(\()p Fw(V)f Fx(\))p Ff(,)i(and)g Fx(1)p Fw(;)7 b(x;)g(y)q(;)g(xy)q(;)g(y)q(x;)g(xy)q(x;)g(y)q(xy)q(;)g (xy)q(xy)13 b Ff(is)e(a)h(b)n(asis)g(of)f Fo(B)p Fx(\()p Fw(V)e Fx(\))p Ff(.)303 1435 y Fg(Pr)o(oof.)21 b Fx(Let)14 b Fw(D)574 1441 y Fu(1)593 1435 y Fw(;)7 b(D)646 1441 y Fu(2)677 1435 y Fx(b)q(e)14 b(the)g(sk)o(ew-deriv)n(ations)f(in)f (Prop)q(osition)h(2.4)f(with)h(resp)q(ect)i(to)228 1495 y(the)h(basis)f Fw(x;)7 b(y)16 b Fx(of)f Fw(V)9 b Fx(.)22 b(By)15 b(Prop)q(osition)g(2.4)f(it)h(su\016ces)h(to)f(sho)o(w)h(that)f (the)h(iden)o(tities)f(hold)228 1555 y(after)f(applying)e Fw(D)529 1561 y Fs(i)557 1555 y Fx(for)i(all)f Fw(i)p Fx(.)303 1615 y(The)22 b(relations)g(\(5.12\))f(and)h(\(5.14\))f(follo) o(w)f(from)g Fw(\021)q Fx(\()p Fw(t;)7 b(t)p Fx(\))25 b(=)g Fw(\021)q Fx(\()p Fw(t)1390 1600 y Fn(0)1402 1615 y Fw(;)7 b(t)1436 1600 y Fn(0)1447 1615 y Fx(\))25 b(=)h Ft(\000)p Fx(1,)d(since)228 1674 y Fw(D)262 1680 y Fu(1)281 1674 y Fx(\()p Fw(x)321 1659 y Fu(2)339 1674 y Fx(\))g(=)g Fw(x)13 b Fx(+)h Fw(\021)q Fx(\()p Fw(t;)7 b(t)p Fx(\))p Fw(x)22 b Fx(=)h(0,)e Fw(D)808 1680 y Fu(2)827 1674 y Fx(\()p Fw(x)867 1659 y Fu(2)886 1674 y Fx(\))i(=)g(0)p Fw(;)c Fx(and)i(b)o(y)f(the)h(same)f(argumen)o(t)f Fw(D)1599 1680 y Fs(i)1613 1674 y Fx(\()p Fw(y)q Fx(\))24 b(=)228 1734 y(0)p Fw(;)7 b(i)k Fx(=)h(1)p Fw(;)7 b Fx(2.)303 1794 y(W)m(e)i(compute)h Fw(D)569 1800 y Fu(1)588 1794 y Fx(\()p Fw(xy)q Fx(\))i(=)g Fw(\021)q Fx(\()p Fw(t)774 1800 y Fu(1)793 1794 y Fw(;)7 b(t)827 1800 y Fu(2)845 1794 y Fx(\))p Fw(y)14 b Fx(=)d Fw(")957 1800 y Fu(0)976 1794 y Fw(y)q(;)c(D)1050 1800 y Fu(2)1070 1794 y Fx(\()p Fw(xy)q Fx(\))12 b(=)g Fw(x;)7 b(D)1280 1800 y Fu(1)1298 1794 y Fx(\()p Fw(y)q(x)p Fx(\))12 b(=)g Fw(y)k Fx(and)e Fw(D)1582 1800 y Fu(2)1601 1794 y Fx(\()p Fw(y)q(x)p Fx(\))e(=)228 1854 y Fw(\021)q Fx(\()p Fw(t)281 1860 y Fu(2)300 1854 y Fw(;)7 b(t)334 1860 y Fu(1)352 1854 y Fx(\))p Fw(x)19 b Fx(=)h Fw(")482 1860 y Fu(1)501 1854 y Fw(x)p Fx(.)32 b(Then)19 b Fw(D)716 1860 y Fu(1)735 1854 y Fx(\(\()p Fw(xy)q Fx(\))828 1839 y Fu(2)848 1854 y Fx(\))g(=)h Ft(\000)p Fw(y)q(xy)q(;)7 b(D)1086 1860 y Fu(2)1106 1854 y Fx(\(\()p Fw(xy)q Fx(\))1199 1839 y Fu(2)1219 1854 y Fx(\))20 b(=)f Fw(xy)q(x;)i Fx(and)14 b Fw(D)1523 1860 y Fu(1)1542 1854 y Fx(\(\()p Fw(y)q(x)p Fx(\))1635 1839 y Fu(2)1654 1854 y Fx(\))20 b(=)228 1914 y Fw(y)q(xy)q(;)7 b(D)347 1920 y Fu(2)367 1914 y Fx(\(\()p Fw(y)q(x)p Fx(\))460 1899 y Fu(2)479 1914 y Fx(\))12 b(=)g Ft(\000)p Fw(xy)q(x;)i Fx(since)g Fw(x)803 1899 y Fu(2)833 1914 y Fx(=)e(0)p Fw(;)7 b(y)938 1899 y Fu(2)968 1914 y Fx(=)12 b(0.)18 b(This)c(implies)e(\(5.13\))g(and)i(\(5.15\))o (.)303 1973 y(It)f(is)f(clear)i(from)d(the)i(relations)g(that)g(the)g (ab)q(o)o(v)o(e)g(elemen)o(ts)g(generate)h(the)f(v)o(ector)h(space)228 2033 y Fo(B)p Fx(\()p Fw(V)9 b Fx(\).)32 b(Hence)20 b(linear)e(indep)q (endency)i(of)e(these)i(elemen)o(ts)f(\(of)f(the)h(same)f(degree)h(in)g (the)228 2093 y(v)n(ariables)12 b Fw(x;)7 b(y)q Fx(\))13 b(implies)e(that)i(the)g(relations)g(are)g(de\014ning)g(relations.)18 b(This)13 b(is)f(clear)i(in)e(case)228 2153 y(1\))i(since)h Fw(D)415 2159 y Fu(1)434 2153 y Fx(\()p Fw(xy)q Fx(\))f Ft(6)p Fx(=)e(0,)i(hence)i Fw(xy)e Ft(6)p Fx(=)f(0)h(in)g Fo(B)p Fx(\()p Fw(V)9 b Fx(\),)14 b(and)h(follo)o(ws)e(easily)h(in)f (case)j(2\))e(b)o(y)g(using)228 2212 y(the)c(v)n(alues)g(of)f(the)i (deriv)n(ations)e Fw(D)767 2218 y Fs(i)791 2212 y Fx(whic)o(h)h(w)o(e)g (ha)o(v)o(e)f(just)h(computed.)17 b(F)m(or)9 b(example,)g(supp)q(ose) 228 2272 y Fw(\013xy)g Fx(+)g Fw(\014)r(y)q(x)k Fx(=)e(0)i(in)g(case)i (2\),)e(where)h Fw(\013;)7 b(\014)14 b Ft(2)d Fv(|)-19 b Fx(.)15 b(Then)e(0)f(=)g Fw(D)1219 2278 y Fu(1)1238 2272 y Fx(\()p Fw(\013xy)d Fx(+)g Fw(\014)r(y)q(x)p Fx(\))k(=)e(\()p Fw(\013")1579 2278 y Fu(0)1607 2272 y Fx(+)d Fw(\014)r Fx(\))p Fw(y)q Fx(,)228 2332 y(and)14 b(0)d(=)h Fw(D)419 2338 y Fu(2)438 2332 y Fx(\()p Fw(\013xy)e Fx(+)g Fw(\014)r(y)q(x)p Fx(\))j(=)e(\()p Fw(\013)f Fx(+)f Fw(\014)r(")857 2338 y Fu(1)877 2332 y Fx(\))p Fw(x)p Fx(,)k(hence)i Fw(\013)d Fx(=)f Fw(\014)k Fx(=)c(0)j(since)h(c)o(har\()p Fv(|)-10 b Ft(6)p Fx(=)12 b(2)p Fw(:)p 1692 2332 V 1694 2306 25 2 v 1694 2332 V 1719 2332 2 29 v 303 2428 a Fg(Theorem)k Fx(5.7)p Fg(.)k Ff(L)n(et)15 b Fw(W)22 b Ff(b)n(e)15 b(a)h(gr)n(oup,)f Fw(T)k Ft(\032)13 b Fw(W)21 b Ff(a)16 b(subset)f(which)h(is)f(stable)g(under)h(c)n(on-)228 2488 y(jugation)e(with)e(elements)i(in)f Fw(W)6 b Ff(,)13 b Fw(\037)f Fx(:)f Fw(W)h Ft(\002)6 b Fw(T)18 b Ft(!)11 b Fv(|)-13 b Ft(n)5 b(f)p Fx(0)p Ft(g)10 b Ff(a)k(function)f (satisfying)k Fx(\(5.1\))o Ff(,)d Fx(\(5.2\))228 2548 y Ff(and)i Fw(V)21 b Fx(=)13 b Fw(V)c Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\))12 b Ft(2)645 2533 y Fs(W)645 2559 y(W)682 2548 y Ft(Y)s(D)r Ff(.)19 b(L)n(et)c Fw(t;)7 b(t)900 2533 y Fn(0)924 2548 y Ft(2)k Fw(T)21 b Ff(b)n(e)16 b(elements)f(of)g(or)n (der)f(2)i(and)g(assume)f(that)228 2608 y Fw(tt)258 2593 y Fn(0)284 2608 y Ff(is)g(of)g(or)n(der)f Fw(m)e Ft(\025)g Fx(1)p Ff(.)p eop %%Page: 14 14 14 13 bop 228 119 a Fu(14)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)303 213 y Ff(Then)j(the)f(sub)n(gr)n(oup)h Fw(D)e Ft(\032)f Fw(W)21 b Ff(gener)n(ate)n(d)15 b(by)g Fw(t;)7 b(t)1087 198 y Fn(0)1112 213 y Ff(is)15 b(the)g(dihe)n(dr)n(al)f(gr)n(oup)h(of)f(or)n (der)g Fx(2)p Fw(m)p Ff(.)228 272 y(F)m(or)i(al)r(l)f Fw(i)g Ft(2)e Fv(Z)-13 b Ff(,)13 b(de\014ne)18 b Fw(t)632 278 y Fs(i)659 272 y Fx(=)d(\()p Fw(tt)752 257 y Fn(0)763 272 y Fx(\))779 257 y Fs(i)p Fn(\000)p Fu(1)836 272 y Fw(t)h Ff(as)g(in)h(Example)f(5.4.)23 b(Then)17 b Fw(t)1359 278 y Fs(i)1387 272 y Fx(=)d Fw(t)1448 278 y Fs(j)1482 272 y Ff(if)h(and)i(only)g(if)228 332 y Fw(i)12 b Ft(\021)f Fw(j)j Fx(mo)q(d)d Fw(m)p Ff(,)k(and)g(the)g(se)n(quenc)n(e)h Fw(t)817 338 y Fu(1)835 332 y Fw(;)7 b(:)g(:)g(:)e(;)i(t)943 338 y Fs(m)989 332 y Ff(is)15 b(de\014ne)n(d)h(inductively)f(by)539 409 y Fw(t)554 415 y Fu(0)584 409 y Fx(=)d Fw(t)643 392 y Fn(0)654 409 y Fw(;)7 b(t)688 415 y Fu(1)718 409 y Fx(=)12 b Fw(t)j Ff(and)g Fw(t)887 415 y Fs(i)901 409 y Fw(t)916 415 y Fs(i)p Fn(\000)p Fu(1)972 409 y Fw(t)987 415 y Fs(i)1013 409 y Fx(=)d Fw(t)1072 415 y Fs(i)p Fu(+1)1127 409 y Fw(;)7 b Fx(1)k Ft(\024)h Fw(i)g Ft(\024)g Fw(m)d Ft(\000)h Fx(1)p Fw(:)228 487 y Ff(Assume)15 b Fw(\037)p Fx(\()p Fw(t)440 493 y Fs(i)454 487 y Fw(;)7 b(t)488 493 y Fs(i)501 487 y Fx(\))12 b(=)f Ft(\000)p Fx(1)k Ff(for)g(al)r(l)f Fw(i)p Ff(.)303 547 y(If)i Fw(m)h Fx(=)f(2)p Fw(l)11 b Fx(+)h(1)k Ff(is)h(o)n(dd,)h(de\014ne)h Fw(x)861 553 y Fs(i)890 547 y Fx(=)d Fw(\037)p Fx(\()p Fw(t)995 553 y Fs(k)1015 547 y Fw(;)7 b(t)1049 553 y Fu(1)1067 547 y Fx(\))p Fw(x)1107 553 y Fs(t)1120 557 y Fi(i)1152 547 y Ff(for)17 b(al)r(l)f Fw(i)g Ft(2)g Fv(Z)q Ff(wher)n(e)g Fw(k)h Ft(2)e Fv(Z)q Ff(with)228 606 y Fw(i)h Ft(\021)h Fx(2)p Fw(k)12 b Ft(\000)f Fx(1)h(mo)q(d)e Fw(m:)17 b Ff(If)g Fw(m)g Fx(=)g(2)p Fw(l)h Ff(is)f(even,)i(de\014ne)f Fw(x)1099 612 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1188 606 y Fx(=)f Fw(\037)p Fx(\()p Fw(t)1294 613 y Fs(\036)p Fu(\()p Fs(i)p Fu(\))1354 606 y Fw(;)7 b(t)1388 612 y Fu(1)1406 606 y Fx(\))p Fw(x)1446 612 y Fs(t)1459 616 y Fj(2)p Fi(i)p Fk(\000)p Fj(1)1539 606 y Ff(and)16 b Fw(x)1644 612 y Fu(2)p Fs(i)1690 606 y Fx(=)228 666 y Fw(\037)p Fx(\()p Fw(t)285 673 y Fs(\036)p Fu(\()p Fs(i)p Fu(\))345 666 y Fw(;)7 b(t)379 672 y Fu(1)397 666 y Fx(\))p Fw(x)437 672 y Fs(t)450 676 y Fj(2)p Fi(i)492 666 y Ff(for)14 b(al)r(l)f Fw(i)f Ft(2)f Fv(Z)-13 b Ff(,)11 b(wher)n(e)i Fw(\036)p Fx(\()p Fw(i)p Fx(\))h Ff(is)g(the)g(natur)n(al)f(numb)n(er)h (with)g Fx(1)d Ft(\024)h Fw(\036)p Fx(\()p Fw(i)p Fx(\))g Ft(\024)f Fw(l)k Ff(and)228 726 y Fw(\036)p Fx(\()p Fw(i)p Fx(\))d Ft(\021)f Fw(i)h Fx(mo)q(d)e Fw(l)q(:)303 786 y Ff(Then)15 b Fw(x)435 792 y Fs(a)466 786 y Fx(=)d Fw(x)534 792 y Fs(b)565 786 y Ff(if)i Fw(a;)7 b(b)k Ft(2)g Fv(Z)-13 b Fw(;)7 b(a)h Ft(\021)k Fw(b)f Fx(mo)q(d)f Fw(m;)15 b Ff(and)h(for)e(al)r(l)g Fw(i;)7 b(j)14 b Ft(2)d Fv(Z)-13 b Ff(,)798 863 y Fw(t)813 869 y Fs(i)826 863 y Fw(x)850 869 y Fs(j)879 863 y Fx(=)12 b Fw(\021)q Fx(\()p Fw(t)976 869 y Fs(i)990 863 y Fw(;)7 b(t)1024 869 y Fs(j)1041 863 y Fx(\))p Fw(x)1081 869 y Fu(2)p Fs(i)p Fn(\000)p Fs(j)228 940 y Ff(wher)n(e)13 b(the)h(function)g Fw(\021)h Ff(is)e(de\014ne)n(d)i(by)j Fx(\(5.5\))13 b Ff(if)g Fw(m)h Ff(is)f(o)n(dd,)i(and)f(by)k Fx(\(5.10\))o Ff(,)13 b Fx(\(5.11\))g Ff(if)g Fw(m)f Fx(=)g(2)p Fw(l)228 1000 y Ff(is)i(even,)i(and)f(wher)n(e)f Fw(")597 1006 y Fu(0)616 1000 y Fw(;)7 b(")654 1006 y Fu(1)684 1000 y Ft(2)12 b(f)p Fx(1)p Fw(;)7 b Ft(\000)p Fx(1)p Ft(g)13 b Ff(ar)n(e)i(given)g (by)g Fw(\037)p Fx(\(\()p Fw(tt)1193 985 y Fn(0)1205 1000 y Fx(\))1221 985 y Fs(l)1234 1000 y Fw(;)7 b(t)1268 1006 y Fs(i)1281 1000 y Fx(\))12 b(=)g Ft(\000)p Fw(")1404 1006 y Fs(i)1418 1000 y Fw(;)7 b Fx(0)k Ft(\024)h Fw(i)f Ft(\024)h Fx(1)p Ff(.)303 1060 y(The)i(fol)r(lowing)g(r)n(elations)h (ar)n(e)f(satis\014e)n(d)h(in)g Fo(B)p Fx(\()p Fw(V)9 b Fx(\))p Ff(:)833 1137 y Fw(x)857 1120 y Fu(2)857 1148 y Fs(i)886 1137 y Fx(=)j(0)j Ff(for)f(al)r(l)h Fw(i:)-890 b Fx(\(5.16\))303 1215 y Ff(If)14 b Fw(m)h Ff(is)g(even,)g(assume)g Fw(")714 1221 y Fu(0)745 1215 y Fx(=)d Fw(")808 1221 y Fu(1)827 1215 y Fw(;)i Ff(and)i(de\014ne)g Fw(")c Fx(=)f Fw(")1148 1221 y Fu(0)1179 1215 y Fx(=)h Fw(")1242 1221 y Fu(1)1261 1215 y Fw(:)i Ff(Then)587 1292 y Fw(x)611 1298 y Fu(1)629 1292 y Fw(x)653 1298 y Fu(2)681 1292 y Fx(+)9 b Fw(x)746 1298 y Fu(2)765 1292 y Fw(x)789 1298 y Fu(3)816 1292 y Fx(+)h Ft(\001)d(\001)g(\001)g Fx(+)j Fw(x)981 1298 y Fs(m)p Fn(\000)p Fu(1)1055 1292 y Fw(x)1079 1298 y Fs(m)1122 1292 y Fx(=)h Fw("x)1208 1298 y Fs(m)1240 1292 y Fw(x)1264 1298 y Fu(1)1282 1292 y Fw(;)-1066 b Fx(\(5.17\))950 1364 y Fw(x)974 1370 y Fu(1)999 1364 y Ft(\001)7 b(\001)g(\001)f Fw(x)1079 1370 y Fs(m)1122 1364 y Fx(=)11 b Fw("x)1208 1370 y Fs(m)1247 1364 y Ft(\001)c(\001)g (\001)e Fw(x)1326 1370 y Fu(1)1345 1364 y Fw(;)-1129 b Fx(\(5.18\))967 1436 y(\()p Fw(x)1007 1442 y Fs(m)1039 1436 y Fw(x)1063 1442 y Fu(1)1081 1436 y Fx(\))1097 1419 y Fs(l)1122 1436 y Fx(=)11 b Fw(")p Fx(\()p Fw(x)1224 1442 y Fu(1)1243 1436 y Fw(x)1267 1442 y Fs(m)1298 1436 y Fx(\))1314 1419 y Fs(l)1327 1436 y Fw(:)-1111 b Fx(\(5.19\))303 1514 y Ff(If)14 b Fw(m)e Fx(=)g(2)p Fw(l)e Fx(+)g(1)k Ff(is)h(o)n(dd,)g(then)474 1591 y Fw(x)498 1597 y Fu(1)516 1591 y Fw(x)540 1597 y Fu(2)568 1591 y Fx(+)9 b Fw(x)633 1597 y Fu(2)651 1591 y Fw(x)675 1597 y Fu(3)703 1591 y Fx(+)g Ft(\001)e(\001)g(\001)h Fx(+)i Fw(x)868 1597 y Fs(m)p Fn(\000)p Fu(1)941 1591 y Fw(x)965 1597 y Fs(m)1006 1591 y Fx(+)f Fw(x)1071 1597 y Fs(m)1103 1591 y Fw(x)1127 1597 y Fu(1)1157 1591 y Fx(=)i(0)p Fw(;)-1005 b Fx(\(5.20\))943 1663 y Fw(x)967 1669 y Fu(1)985 1663 y Fw(x)1009 1669 y Fu(2)1034 1663 y Ft(\001)7 b(\001)g(\001)f Fw(x)1114 1669 y Fs(m)1157 1663 y Fx(=)11 b Fw(x)1224 1669 y Fs(m)1256 1663 y Fw(x)1280 1669 y Fs(m)p Fn(\000)p Fu(1)1360 1663 y Ft(\001)c(\001)g(\001)f Fw(x)1440 1669 y Fu(1)1458 1663 y Fw(;)-1242 b Fx(\(5.21\))947 1736 y(\()p Fw(x)987 1742 y Fs(m)1018 1736 y Fw(x)1042 1742 y Fu(1)1061 1736 y Fx(\))1077 1718 y Fs(l)1090 1736 y Fw(x)1114 1742 y Fs(m)1157 1736 y Fx(=)11 b(\()p Fw(x)1240 1742 y Fu(1)1259 1736 y Fw(x)1283 1742 y Fs(m)1314 1736 y Fx(\))1330 1718 y Fs(l)1343 1736 y Fw(x)1367 1742 y Fu(1)1385 1736 y Fw(:)-1169 b Fx(\(5.22\))303 1813 y Ff(Assume)14 b Fw(m)e Fx(=)g(2)p Fw(l)k Ff(and)g Fw(l)g Ff(ar)n(e)e(even)i(and)f Fw(")972 1819 y Fu(0)1003 1813 y Fx(=)d Fw(")1066 1819 y Fu(1)1096 1813 y Fx(=)g Ft(\000)p Fx(1)p Ff(.)303 1873 y(L)n(et)i Fw(x)398 1879 y Fn(\006)437 1873 y Fx(=)481 1842 y Fl(P)525 1852 y Fs(l)525 1885 y(i)p Fu(=1)581 1873 y Fx(\()p Ft(\006)p Fx(1\))666 1858 y Fs(i)p Fn(\000)p Fu(1)723 1873 y Fw(x)747 1879 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)819 1873 y Fw(;)h Ff(and)g Fw(y)946 1879 y Fn(\006)986 1873 y Fx(=)1030 1842 y Fl(P)1074 1852 y Fs(l)1074 1885 y(i)p Fu(=1)1130 1873 y Fx(\()p Ft(\006)p Fx(1\))1215 1858 y Fs(i)p Fn(\000)p Fu(1)1271 1873 y Fw(x)1295 1879 y Fu(2)p Fs(i)1325 1873 y Fw(:)303 1932 y Ff(If)f Fw(x)d Fx(=)h Fw(x)449 1938 y Fu(+)476 1932 y Fw(;)7 b(y)13 b Fx(=)f Fw(y)592 1938 y Fn(\000)635 1932 y Ff(or)j Fw(x)c Fx(=)h Fw(x)792 1938 y Fn(\000)820 1932 y Fw(;)7 b(y)13 b Fx(=)e Fw(y)935 1938 y Fu(+)963 1932 y Ff(,)k(then)867 2010 y Fw(x)891 1993 y Fu(2)921 2010 y Fx(=)d(0)p Fw(;)7 b(y)1026 1993 y Fu(2)1056 2010 y Fx(=)12 b(0)p Fw(;)-905 b Fx(\(5.23\))811 2082 y Fw(xy)q(xy)12 b Fx(+)d Fw(y)q(xy)q(x)k Fx(=)f(0)p Fw(;)-905 b Fx(\(5.24\))228 2159 y Ff(and)15 b(if)g(char)p Fx(\()p Fv(|)-19 b Fx(\))9 b Ft(6)p Fx(=)i(2)p Ff(,)k(then)g(the)g(sub)n(algebr)n(a)g(gener)n(ate)n(d)g(by)g Fw(x;)7 b(y)16 b Ff(has)f(dimension)h(8.)303 2249 y Fg(Pr)o(oof.)21 b Fx(Let)c Fw(U)k Ft(\032)15 b Fw(V)26 b Fx(b)q(e)17 b(the)g(braided)f(subspace)i(with)e(basis)g Fw(x)1375 2255 y Fu(1)1394 2249 y Fw(;)7 b(:)g(:)g(:)e(;)i(x)1511 2255 y Fs(m)1541 2249 y Fx(.)26 b(Then)16 b Fw(U)228 2309 y Fx(is)f(a)f(Y)m(etter-Drinfeld)h(mo)q(dule)f(o)o(v)o(er)h Fw(D)h Fx(and)f Fw(U)j Fx(=)c Fw(V)9 b Fx(\()p Fw(D)q(;)e Ft(f)p Fw(t)1197 2315 y Fu(1)1215 2309 y Fw(;)g(:)g(:)g(:)e(;)i(t)1323 2315 y Fs(m)1354 2309 y Ft(g)p Fw(;)g(\021)q Fx(\))15 b(with)f(resp)q(ect)j(to)228 2369 y(the)e(basis)f Fw(y)422 2375 y Fs(t)435 2379 y Fi(i)463 2369 y Fx(=)e Fw(x)531 2375 y Fs(i)545 2369 y Fw(;)7 b Fx(1)12 b Ft(\024)g Fw(i)h Ft(\024)g Fw(m;)h Fx(as)g(sho)o(wn)h(in)f(Example)f(5.4.)18 b(Since)d(the)g(ob)o(vious)f(natural)228 2428 y(map)e Fo(B)p Fx(\()p Fw(U)5 b Fx(\))11 b Ft(!)g Fo(B)p Fx(\()p Fw(V)f Fx(\))k(is)f(an)h(algebra)g(map,)d(w)o(e)j(can)h(assume)e(that)h Fw(U)i Fx(=)c Fw(V)e Fx(.)303 2488 y(F)m(or)h(all)f(1)h Ft(\024)h Fw(i)g Ft(\024)g Fw(m)p Fx(,)g(let)g Fw(D)728 2494 y Fs(i)753 2488 y Fx(b)q(e)h(the)f(sk)o(ew-deriv)n(ation)f (de\014ned)i(in)e(Prop)q(osition)g(2.4)f(with)228 2548 y(resp)q(ect)k(to)e Fw(x)441 2554 y Fs(i)466 2548 y Ft(2)f Fw(V)529 2554 y Fs(t)542 2558 y Fi(i)557 2548 y Fx(.)17 b(As)c(in)e(the)i(previous)f(Lemma)d(it)j(su\016ces)h(to)f(sho)o(w)g (that)g(the)g(iden)o(tities)228 2608 y(hold)h(after)h(applying)f Fw(D)622 2614 y Fs(i)650 2608 y Fx(for)g(all)g Fw(i)p Fx(.)p eop %%Page: 15 15 15 14 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d (O)o(VER)i(CO)o(XETER)g(GR)o(OUPS)95 b(15)303 213 y Fx(The)18 b(relations)g(\(5.16\))g(follo)o(w)e(from)g Fw(\021)q Fx(\()p Fw(t)969 219 y Fs(i)983 213 y Fw(;)7 b(t)1017 219 y Fs(i)1031 213 y Fx(\))18 b(=)i Fw(\037)p Fx(\()p Fw(t)1174 219 y Fs(i)1187 213 y Fw(;)7 b(t)1221 219 y Fs(i)1235 213 y Fx(\))19 b(=)g Ft(\000)p Fx(1)f(as)g(in)g(the)h(pro)q (of)f(of)228 272 y(\(5.12\))o(.)303 355 y(T)m(o)11 b(sho)o(w)i (\(5.17\))e(and)h(\(5.20\))o(,)g(let)h Fw(a)e Fx(=)h Fw(x)951 361 y Fu(1)969 355 y Fw(x)993 361 y Fu(2)1018 355 y Fx(+)6 b Fw(x)1080 361 y Fu(2)1098 355 y Fw(x)1122 361 y Fu(3)1147 355 y Fx(+)g Ft(\001)h(\001)g(\001)e Fx(+)h Fw(x)1302 361 y Fs(m)p Fn(\000)p Fu(1)1376 355 y Fw(x)1400 361 y Fs(m)1437 355 y Ft(\000)g Fw("x)1518 361 y Fs(m)1550 355 y Fw(x)1574 361 y Fu(1)1592 355 y Fw(;)12 b Fx(where)228 415 y Fw(")g Fx(=)g Ft(\000)p Fx(1)h(when)i Fw(m)f Fx(is)g(o)q(dd.)k(Then)518 497 y Fw(D)552 503 y Fu(1)571 497 y Fx(\()p Fw(a)p Fx(\))12 b(=)g Fw(t)696 503 y Fu(1)724 497 y Ft(\001)c Fw(x)768 503 y Fu(2)796 497 y Ft(\000)h Fw("x)880 503 y Fs(m)924 497 y Fx(=)i(\()p Fw(\021)q Fx(\()p Fw(t)1036 503 y Fu(1)1055 497 y Fw(;)c(t)1089 503 y Fu(2)1107 497 y Fx(\))j Ft(\000)f Fw(")p Fx(\))p Fw(x)1233 503 y Fu(0)1252 497 y Fw(;)518 569 y(D)552 575 y Fu(2)571 569 y Fx(\()p Fw(a)p Fx(\))j(=)g Fw(x)705 575 y Fu(1)732 569 y Fx(+)e Fw(t)789 575 y Fu(2)817 569 y Ft(\001)e Fw(x)861 575 y Fu(3)891 569 y Fx(=)k(\()p Fw(\021)q Fx(\()p Fw(t)1004 575 y Fu(2)1023 569 y Fw(;)7 b(t)1057 575 y Fu(3)1075 569 y Fx(\))j(+)f(1\))p Fw(x)1203 575 y Fu(1)1221 569 y Fw(;)632 641 y(:)e(:)g(:)463 714 y(D)497 720 y Fs(m)p Fn(\000)p Fu(1)571 714 y Fx(\()p Fw(a)p Fx(\))12 b(=)g Fw(x)705 720 y Fs(m)p Fn(\000)p Fu(2)788 714 y Fx(+)d Fw(t)844 720 y Fs(m)p Fn(\000)p Fu(1)927 714 y Ft(\001)g Fw(x)972 720 y Fs(m)1015 714 y Fx(=)j(\()p Fw(\021)q Fx(\()p Fw(t)1128 720 y Fs(m)p Fn(\000)p Fu(1)1202 714 y Fw(;)7 b(t)1236 720 y Fs(m)1267 714 y Fx(\))j(+)f(1\))p Fw(x)1395 720 y Fs(m)p Fn(\000)p Fu(2)1469 714 y Fw(;)505 786 y(D)539 792 y Fs(m)571 786 y Fx(\()p Fw(a)p Fx(\))j(=)g Fw(x)705 792 y Fs(m)p Fn(\000)p Fu(1)788 786 y Ft(\000)d Fw("t)863 792 y Fs(m)904 786 y Ft(\001)g Fw(x)949 792 y Fu(1)979 786 y Fx(=)j(\()p Ft(\000)p Fw("\021)q Fx(\()p Fw(t)1143 792 y Fs(m)1175 786 y Fw(;)7 b(t)1209 792 y Fu(1)1237 786 y Fx(+)i(1\))p Fw(x)1339 792 y Fs(m)p Fn(\000)p Fu(1)1413 786 y Fw(:)228 868 y Fx(Assume)18 b Fw(m)i Fx(is)e(ev)o(en.)33 b(F)m(rom)17 b(\(5.10\))h(and)g(\(5.11\))g(w)o(e)h(see)h(that)f Fw(\021)q Fx(\()p Fw(t)1348 874 y Fs(i)1362 868 y Fw(;)7 b(t)1396 874 y Fs(i)p Fu(+1)1451 868 y Fx(\))20 b(=)f Ft(\000)p Fx(1)g(for)f(all)228 928 y(2)d Ft(\024)h Fw(i)g Ft(\024)g Fw(m)c Ft(\000)f Fx(1,)16 b(and)h Fw(\021)q Fx(\()p Fw(t)667 934 y Fu(1)685 928 y Fw(;)7 b(t)719 934 y Fu(2)738 928 y Fx(\))15 b(=)i Fw(")837 934 y Fu(0)855 928 y Fw(;)7 b(\021)q Fx(\()p Fw(t)927 934 y Fs(m)959 928 y Fw(;)g(t)993 934 y Fu(1)1011 928 y Fx(\))16 b(=)g Fw(")1110 934 y Fu(1)1129 928 y Fx(.)25 b(Th)o(us)17 b Fw(D)1308 934 y Fs(i)1322 928 y Fx(\()p Fw(a)p Fx(\))f(=)g(0)g(for)g(all)g Fw(i)g Fx(since)228 988 y Fw(")247 994 y Fu(0)277 988 y Fx(=)c Fw(")340 994 y Fu(1)359 988 y Fx(.)303 1047 y(If)h Fw(m)h Fx(is)g(o)q(dd,)f(then)i Fw(D)658 1053 y Fs(i)672 1047 y Fx(\()p Fw(a)p Fx(\))d(=)g(0)h(for)h(all)f Fw(i)p Fx(,)g(since)i Ft(\000)p Fx(1)d(=)f Fw(")h Fx(=)g Fw(\021)q Fx(\()p Fw(t)1315 1053 y Fs(k)1336 1047 y Fw(;)7 b(t)1370 1053 y Fs(i)1383 1047 y Fx(\))14 b(for)f(all)g Fw(k)q(;)7 b(i)p Fx(.)303 1130 y(T)m(o)13 b(pro)o(v)o(e)h(\(5.18\))f (and)g(\(5.21\))o(,)h(w)o(e)g(\014rst)g(note)h(that)320 1212 y Fw(D)354 1218 y Fu(1)373 1212 y Fx(\()p Fw(x)413 1218 y Fu(1)431 1212 y Fx(\()p Fw(x)471 1218 y Fu(2)497 1212 y Ft(\001)7 b(\001)g(\001)e Fw(x)576 1218 y Fs(m)607 1212 y Fx(\)\))12 b(=)g Fw(t)710 1218 y Fu(1)738 1212 y Ft(\001)c Fx(\()p Fw(x)798 1218 y Fu(2)824 1212 y Ft(\001)f(\001)g (\001)e Fw(x)903 1218 y Fs(m)934 1212 y Fx(\))12 b(=)g Fw(\021)q Fx(\()p Fw(t)1059 1218 y Fu(1)1078 1212 y Fw(;)7 b(t)1112 1218 y Fu(2)1130 1212 y Fx(\))g Fw(:)g(:)g(:)e(\021)q Fx(\()p Fw(t)1261 1218 y Fu(1)1280 1212 y Fw(;)i(t)1314 1218 y Fs(m)1345 1212 y Fx(\))p Fw(x)1385 1218 y Fs(m)1416 1212 y Fw(x)1440 1218 y Fs(m)p Fn(\000)p Fu(1)1521 1212 y Ft(\001)g(\001)g(\001)e Fw(x)1600 1218 y Fu(2)1619 1212 y Fw(:)228 1295 y Fx(On)14 b(the)g(other)h(hand,)e Fw(D)626 1301 y Fu(1)645 1295 y Fx(\(\()p Fw(x)701 1301 y Fs(m)740 1295 y Ft(\001)7 b(\001)g(\001)e Fw(x)819 1301 y Fu(2)837 1295 y Fx(\))p Fw(x)877 1301 y Fu(1)896 1295 y Fx(\))12 b(=)f Fw(x)991 1301 y Fs(m)1030 1295 y Ft(\001)c(\001)g(\001)e Fw(x)1109 1301 y Fu(2)1127 1295 y Fw(:)303 1354 y Fx(Assume)20 b Fw(m)i Fx(=)h(2)p Fw(l)q Fx(.)37 b(By)21 b(\(5.10\))e(and)h(\(5.11\))f(w)o(e)i(compute)e Fw(\021)q Fx(\()p Fw(t)1365 1360 y Fu(1)1384 1354 y Fw(;)7 b(t)1418 1360 y Fu(2)1436 1354 y Fx(\))g Fw(:)g(:)g(:)f(\021)q Fx(\()p Fw(t)1568 1360 y Fu(1)1587 1354 y Fw(;)h(t)1621 1360 y Fs(m)1652 1354 y Fx(\))22 b(=)228 1414 y Fw(")247 1399 y Fs(l)247 1424 y Fu(0)266 1414 y Fw(")285 1396 y Fs(l)p Fn(\000)p Fu(1)285 1425 y(1)357 1414 y Fx(=)16 b Fw(":)h Fx(Hence)h Fw(D)614 1420 y Fu(1)633 1414 y Fx(\()p Fw(x)673 1420 y Fu(1)698 1414 y Ft(\001)7 b(\001)g(\001)f Fw(x)778 1420 y Fs(m)809 1414 y Fx(\))16 b(=)h Fw(D)924 1420 y Fu(1)943 1414 y Fx(\()p Fw("x)1002 1420 y Fs(m)1040 1414 y Ft(\001)7 b(\001)g(\001)f Fw(x)1120 1420 y Fu(1)1138 1414 y Fx(\))p Fw(:)17 b Fx(Similarl)o(y)d(w)o(e)j(obtain)f(the)h(same) 228 1474 y(equalit)o(y)c(for)g Fw(D)483 1480 y Fs(m)529 1474 y Fx(since)i Fw(\021)q Fx(\()p Fw(t)684 1480 y Fs(m)716 1474 y Fw(;)7 b(t)750 1480 y Fs(m)p Fn(\000)p Fu(1)823 1474 y Fx(\))g Fw(:)g(:)g(:)e(\021)q Fx(\()p Fw(t)954 1480 y Fs(m)986 1474 y Fw(;)i(t)1020 1480 y Fu(1)1038 1474 y Fx(\))12 b(=)g Fw(")1129 1456 y Fs(l)p Fn(\000)p Fu(1)1129 1485 y(0)1184 1474 y Fw(")1203 1459 y Fs(l)1203 1484 y Fu(1)1234 1474 y Fx(=)g Fw(":)303 1534 y Fx(If)i Fw(m)i Fx(is)f(o)q(dd,)f(then)i Fw(\021)q Fx(\()p Fw(t)683 1540 y Fs(k)704 1534 y Fw(;)7 b(t)738 1540 y Fs(i)751 1534 y Fx(\))14 b(=)f Ft(\000)p Fx(1)i(for)g(all)f Fw(k)q(;)7 b(i)p Fx(,)14 b(and)h Fw(D)1216 1540 y Fs(i)1230 1534 y Fx(\()p Fw(x)1270 1540 y Fu(1)1296 1534 y Ft(\001)7 b(\001)g(\001)e Fw(x)1375 1540 y Fs(m)1406 1534 y Fx(\))14 b(=)g Fw(D)1516 1540 y Fs(i)1530 1534 y Fx(\()p Fw(x)1570 1540 y Fs(m)1608 1534 y Ft(\001)7 b(\001)g(\001)f Fw(x)1688 1540 y Fu(1)1706 1534 y Fx(\))228 1593 y(for)13 b Fw(i)f Fx(=)g(1)i(and)f Fw(i)f Fx(=)g Fw(m)p Fx(.)303 1653 y(Finally)m(,)f(if) i(2)e Ft(\024)h Fw(i)g Ft(\024)g Fw(m)d Ft(\000)h Fx(1,)j(then)685 1736 y Fw(D)719 1742 y Fs(i)734 1736 y Fx(\(\()p Fw(x)790 1742 y Fu(1)815 1736 y Ft(\001)7 b(\001)g(\001)e Fw(x)894 1742 y Fs(i)p Fn(\000)p Fu(1)950 1736 y Fw(x)974 1742 y Fs(i)988 1736 y Fx(\)\()p Fw(x)1044 1742 y Fs(i)p Fu(+1)1107 1736 y Ft(\001)i(\001)g(\001)e Fw(x)1186 1742 y Fs(m)1217 1736 y Fx(\)\))12 b(=)613 1808 y Fw(D)647 1814 y Fs(i)661 1808 y Fx(\(\()p Fw(x)717 1814 y Fu(1)742 1808 y Ft(\001)7 b(\001)g(\001)f Fw(x)822 1814 y Fs(i)p Fn(\000)p Fu(1)878 1808 y Fx(\))p Fw(x)918 1814 y Fs(i)932 1808 y Fx(\))13 b Fw(t)976 1814 y Fs(i)1000 1808 y Ft(\001)8 b Fx(\()p Fw(x)1060 1814 y Fs(i)p Fu(+1)1123 1808 y Ft(\001)f(\001)g(\001)e Fw(x)1202 1814 y Fs(m)1234 1808 y Fx(\))11 b(=)731 1880 y(\()p Fw(x)771 1886 y Fu(1)796 1880 y Ft(\001)c(\001)g(\001)e Fw(x)875 1886 y Fs(i)p Fn(\000)p Fu(1)932 1880 y Fx(\))13 b Fw(t)976 1886 y Fs(i)1000 1880 y Ft(\001)8 b Fx(\()p Fw(x)1060 1886 y Fs(i)p Fu(+1)1123 1880 y Ft(\001)f(\001)g(\001)e Fw(x)1202 1886 y Fs(m)1234 1880 y Fx(\))11 b(=)h(0)p Fw(;)228 1962 y Fx(since)j Fw(t)345 1968 y Fs(i)368 1962 y Ft(\001)8 b Fx(\()p Fw(x)428 1968 y Fs(i)p Fu(+1)491 1962 y Ft(\001)f(\001)g(\001)e Fw(x)570 1968 y Fs(m)602 1962 y Fx(\))14 b(b)q(egins)g(with)f(the)i(elemen)o(t)e Fw(x)1103 1968 y Fs(i)p Fn(\000)p Fu(1)1159 1962 y Fx(,)h(and)f Fw(x)1289 1947 y Fu(2)1289 1973 y Fs(i)p Fn(\000)p Fu(1)1357 1962 y Fx(=)f(0)h(b)o(y)h(\(5.16\).)303 2022 y(Similarl)o(y)j(w)o(e)j (ha)o(v)o(e)f Fw(D)687 2028 y Fs(i)701 2022 y Fx(\()p Fw(x)741 2028 y Fs(m)779 2022 y Ft(\001)7 b(\001)g(\001)f Fw(x)859 2028 y Fu(1)877 2022 y Fx(\))21 b(=)g(\()p Fw(x)1007 2028 y Fs(m)1046 2022 y Ft(\001)7 b(\001)g(\001)e Fw(x)1125 2028 y Fs(i)p Fu(+1)1181 2022 y Fx(\))14 b Fw(t)1226 2028 y Fs(i)1253 2022 y Ft(\001)e Fx(\()p Fw(x)1317 2028 y Fs(i)p Fn(\000)p Fu(1)1380 2022 y Ft(\001)7 b(\001)g(\001)e Fw(x)1459 2028 y Fs(m)1491 2022 y Fx(\))21 b(=)g(0,)f(since)228 2082 y Fw(x)252 2067 y Fu(2)252 2093 y Fs(i)p Fu(+1)319 2082 y Fx(=)12 b(0.)303 2164 y(T)m(o)h(sho)o(w)i(\(5.19\))o(,)f (consider)h(the)f(sequence)j(\()p Fw(s)1041 2170 y Fu(1)1060 2164 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)1172 2170 y Fu(2)p Fs(l)1201 2164 y Fx(\))12 b(=)h(\()p Fw(t;)7 b(t)1339 2149 y Fn(0)1350 2164 y Fw(;)g(t;)g(t)1418 2149 y Fn(0)1428 2164 y Fw(;)g(:)g(:)g(:)e(;)i(t;)g(t)1570 2149 y Fn(0)1581 2164 y Fx(\).)19 b(Then)228 2224 y Fw(s)247 2230 y Fu(1)280 2224 y Fx(=)14 b Fw(t)h Fx(=)f Fw(t)417 2230 y Fu(1)436 2224 y Fw(;)7 b(s)474 2230 y Fu(1)492 2224 y Fw(s)511 2230 y Fu(2)530 2224 y Fw(s)549 2206 y Fn(\000)p Fu(1)549 2235 y(1)609 2224 y Fx(=)14 b Fw(tt)685 2209 y Fn(0)697 2224 y Fw(t)g Fx(=)g Fw(t)787 2230 y Fu(2)806 2224 y Fw(;)7 b(s)844 2230 y Fu(1)862 2224 y Fw(s)881 2230 y Fu(2)900 2224 y Fw(s)919 2230 y Fu(3)938 2224 y Fx(\()p Fw(s)973 2230 y Fu(1)993 2224 y Fw(s)1012 2230 y Fu(2)1031 2224 y Fx(\))1047 2209 y Fn(\000)p Fu(1)1106 2224 y Fx(=)14 b Fw(tt)1182 2209 y Fn(0)1194 2224 y Fw(tt)1224 2209 y Fn(0)1235 2224 y Fw(t)h Fx(=)f Fw(t)1326 2230 y Fu(3)1345 2224 y Fw(;)7 b(:)g(:)g(:)t(:)15 b Fx(Th)o(us)h Fw(t)1567 2230 y Fu(1)1586 2224 y Fw(;)7 b(:)g(:)g(:)t(;)g(t)1693 2230 y Fu(2)p Fs(l)228 2284 y Fx(coincide)13 b(with)g(the)h Fw(t)566 2290 y Fs(i)580 2284 y Fx('s)f(corresp)q(onding)h(to)f Fw(s)956 2290 y Fu(1)976 2284 y Fw(;)7 b(:)g(:)g(:)t(;)g(s)1087 2290 y Fu(2)p Fs(l)1130 2284 y Fx(in)13 b(Lemma)d(5.5.)17 b(Hence)e(b)o(y)e(Lemma)228 2344 y(5.5,)579 2426 y Fw(S)r Fx(\(\()p Fw(x)662 2432 y Fu(1)682 2426 y Fw(x)706 2432 y Fs(m)737 2426 y Fx(\))753 2409 y Fs(l)766 2426 y Fx(\))f(=)f Fw(")p Fx(\()p Fw(t;)c(t)921 2409 y Fn(0)933 2426 y Fw(;)g(:)g(:)g(:)e (;)i(t;)g(t)1075 2409 y Fn(0)1085 2426 y Fx(\))p Fw(x)1125 2432 y Fs(m)1157 2426 y Fw(x)1181 2432 y Fs(m)p Fn(\000)p Fu(1)1261 2426 y Ft(\001)g(\001)g(\001)f Fw(x)1341 2432 y Fu(1)1359 2426 y Fw(:)-1143 b Fx(\(5.25\))242 2508 y(Changing)12 b(the)j(roles)f(of)f Fw(t;)7 b(t)693 2493 y Fn(0)718 2508 y Fx(w)o(e)14 b(obtain)g(in)f(the)i(same)e(w)o(a)o(y) 604 2591 y Fw(S)r Fx(\(\()p Fw(x)687 2597 y Fs(m)719 2591 y Fw(x)743 2597 y Fu(1)761 2591 y Fx(\))777 2574 y Fs(l)790 2591 y Fx(\))f(=)g Fw(")p Fx(\()p Fw(t)912 2574 y Fn(0)924 2591 y Fw(;)7 b(t;)g(:)g(:)g(:)t(;)g(t)1065 2574 y Fn(0)1076 2591 y Fw(;)g(t)p Fx(\))p Fw(x)1150 2597 y Fu(1)1168 2591 y Fw(x)1192 2597 y Fu(2)1217 2591 y Ft(\001)g(\001)g(\001)f Fw(x)1297 2597 y Fs(m)1328 2591 y Fw(;)-1112 b Fx(\(5.26\))p eop %%Page: 16 16 16 15 bop 228 119 a Fu(16)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)228 213 y Fx(since)j(\()p Fw(t)361 198 y Fn(0)372 213 y Fw(t)p Fx(\))403 198 y Fs(m)p Fn(\000)p Fs(i)473 213 y Fw(t)488 198 y Fn(0)511 213 y Fx(=)d(\()p Fw(tt)601 198 y Fn(0)612 213 y Fx(\))628 198 y Fs(i)642 213 y Fw(t)657 198 y Fn(0)681 213 y Fx(=)f(\()p Fw(tt)770 198 y Fn(0)782 213 y Fx(\))798 198 y Fs(i)p Fn(\000)p Fu(1)855 213 y Fw(t)g Fx(=)h Fw(t)940 219 y Fs(i)968 213 y Fx(for)h(all)g Fw(i)p Fx(.)18 b(By)c(the)h (de\014nition)e(in)h(Lemma)d(5.5)288 326 y Fw(")p Fx(\()p Fw(t;)c(t)372 309 y Fn(0)384 326 y Fw(;)g(t;)g(t)452 309 y Fn(0)462 326 y Fw(;)g(:)g(:)g(:)e(;)i(t;)g(t)604 309 y Fn(0)614 326 y Fx(\))12 b(=)699 274 y Fu(2)p Fs(l)686 286 y Fl(Y)686 375 y Fs(i)p Fu(=2)747 326 y Fw(\021)q Fx(\()p Fw(s)804 332 y Fu(1)830 326 y Ft(\001)7 b(\001)g(\001)e Fw(s)904 332 y Fs(i)p Fn(\000)p Fu(1)961 326 y Fw(;)i(s)999 332 y Fs(i)1013 326 y Fx(\))k(=)h Fw(\021)q Fx(\()p Fw(t;)7 b(t)1171 309 y Fn(0)1183 326 y Fx(\))p Fw(\021)q Fx(\()p Fw(tt)1267 309 y Fn(0)1279 326 y Fw(;)g(t)p Fx(\))g Ft(\001)g(\001)g (\001)t Fw(\021)q Fx(\()p Fw(tt)1458 309 y Fn(0)1477 326 y Ft(\001)g(\001)g(\001)f Fw(tt)1563 309 y Fn(0)1574 326 y Fw(t;)h(t)1623 309 y Fn(0)1634 326 y Fx(\))p Fw(:)228 436 y Fx(F)m(rom)12 b(\(5.10\))o(,)h(\(5.11\))o(,)553 519 y Fw(\021)q Fx(\(\()p Fw(tt)637 501 y Fn(0)649 519 y Fx(\))665 501 y Fs(i)p Fn(\000)p Fu(1)721 519 y Fw(t;)7 b(t)770 501 y Fn(0)781 519 y Fx(\))12 b(=)g Fw(\021)q Fx(\()p Fw(t)906 525 y Fs(i)920 519 y Fw(;)7 b(t)954 525 y Fu(2)p Fs(l)982 519 y Fx(\))12 b(=)g Fw(")1073 525 y Fu(0)1106 519 y Fx(for)h(all)g(1)e Ft(\024)h Fw(i)g Ft(\024)g Fw(l)q(;)228 601 y Fx(and)388 665 y Fw(\021)q Fx(\(\()p Fw(t)457 648 y Fn(0)469 665 y Fw(t)p Fx(\))500 648 y Fs(i)514 665 y Fw(;)7 b(t)p Fx(\))k(=)h Fw(\021)q Fx(\()p Fw(t)672 671 y Fs(i)p Fu(+1)728 665 y Fw(t;)7 b(t)p Fx(\))k(=)h Fw(\021)q Fx(\()p Fw(t)901 671 y Fs(i)p Fu(+1)957 665 y Fw(;)7 b(t)p Fx(\))p Fw(\021)q Fx(\()p Fw(t;)g(t)p Fx(\))k(=)g(1)j(for)g(all)e(1)g Ft(\024)f Fw(i)h Ft(\024)g Fw(l)e Ft(\000)g Fx(1)p Fw(:)228 739 y Fx(Hence)15 b(w)o(e)f(ha)o(v)o(e)392 821 y Fw(")p Fx(\()p Fw(t;)7 b(t)476 804 y Fn(0)487 821 y Fw(;)g(t;)g(t)555 804 y Fn(0)566 821 y Fw(;)g(:)g(:)g(:)e(;)i(t;)g(t)708 804 y Fn(0)718 821 y Fx(\))12 b(=)g Fw(")809 804 y Fs(l)809 831 y Fu(0)828 821 y Fw(;)20 b Fx(and)14 b(similarly)d Fw(")p Fx(\()p Fw(t)1161 804 y Fn(0)1173 821 y Fw(;)c(t;)g(t)1241 804 y Fn(0)1251 821 y Fw(;)g(t;)g(:)g(:)g(:)e(;)i(t)1393 804 y Fn(0)1404 821 y Fw(;)g(t)p Fx(\))k(=)h Fw(")1528 804 y Fs(l)1528 831 y Fu(1)1547 821 y Fw(:)228 903 y Fx(Therefore)j(w)o(e)f(obtain)f(from)f(\(5.25\))o(,)i(\(5.26\))f(and)g (\(2\))h(the)h(equalit)o(y)729 985 y Fw(S)r Fx(\(\()p Fw(x)812 991 y Fu(1)832 985 y Fw(x)856 991 y Fs(m)887 985 y Fx(\))903 968 y Fs(l)916 985 y Fx(\))c(=)h Fw(S)r Fx(\()p Fw(")p Fx(\()p Fw(x)1089 991 y Fs(m)1122 985 y Fw(x)1146 991 y Fu(1)1164 985 y Fx(\))1180 968 y Fs(l)1193 985 y Fx(\))p Fw(;)228 1067 y Fx(and)i(the)g(claim)e(follo)o(ws)g(from) g(the)j(bijectivit)o(y)e(of)g(the)h(an)o(tip)q(o)q(de)g Fw(S)r Fx(.)303 1149 y(T)m(o)f(pro)o(v)o(e)h(\(5.22\))o(,)g(w)o(e)g (pro)q(ceed)i(as)f(in)e(the)i(previous)g(case.)20 b(Let)14 b(\()p Fw(s)1375 1155 y Fu(1)1394 1149 y Fw(;)7 b(:)g(:)g(:)e(;)i(s) 1506 1155 y Fu(2)p Fs(l)p Fu(+1)1577 1149 y Fx(\))15 b(b)q(e)g(the)228 1209 y(sequence)g(\()p Fw(t;)7 b(t)464 1194 y Fn(0)475 1209 y Fw(;)g(t;)g(t)543 1194 y Fn(0)554 1209 y Fw(;)g(:)g(:)g(:)e(;)i(t;)g(t)696 1194 y Fn(0)706 1209 y Fw(;)g(t)p Fx(\).)17 b(Then)d Fw(t)908 1215 y Fu(1)927 1209 y Fw(;)7 b(:)g(:)g(:)t(;)g(t)1034 1215 y Fu(2)p Fs(l)p Fu(+1)1118 1209 y Fx(coincide)14 b(with)e(the)i Fw(t)1456 1215 y Fs(i)1470 1209 y Fx('s)f(corresp)q(ond-)228 1269 y(ing)g(to)h Fw(s)367 1275 y Fu(1)386 1269 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)498 1275 y Fu(2)p Fs(l)p Fu(+1)583 1269 y Fx(in)13 b(Lemma)e(5.5.)17 b(By)d(Lemma)e(5.5,)581 1351 y Fw(S)r Fx(\(\()p Fw(x)664 1357 y Fu(1)683 1351 y Fw(x)707 1357 y Fs(m)738 1351 y Fx(\))754 1334 y Fs(l)767 1351 y Fw(x)791 1357 y Fu(1)810 1351 y Fx(\))f(=)h Fw(")p Fx(\()p Fw(t;)7 b(t)965 1334 y Fn(0)977 1351 y Fw(;)g(:)g(:)g(:)e(;)i (t)p Fx(\))p Fw(x)1125 1357 y Fs(m)1155 1351 y Fw(x)1179 1357 y Fs(m)p Fn(\000)p Fu(1)1260 1351 y Ft(\001)g(\001)g(\001)e Fw(x)1339 1357 y Fu(1)1358 1351 y Fw(;)228 1433 y Fx(where)15 b Fw(")p Fx(\()p Fw(t;)7 b(t)432 1418 y Fn(0)444 1433 y Fw(;)g(:)g(:)g(:)t(;)g(t)p Fx(\))12 b(=)g Ft(\000)p Fw(\021)q Fx(\()p Fw(t;)7 b(t)742 1418 y Fn(0)754 1433 y Fx(\))p Fw(\021)q Fx(\()p Fw(tt)838 1418 y Fn(0)850 1433 y Fw(;)g(t)p Fx(\))g Ft(\001)g(\001)g(\001)e Fw(\021)q Fx(\()p Fw(tt)1030 1418 y Fn(0)1049 1433 y Ft(\001)i(\001)g(\001)e Fw(tt)1134 1418 y Fn(0)1146 1433 y Fw(;)i(t)p Fx(\))k(=)h(\()p Ft(\000)p Fx(1\))1336 1418 y Fu(1+)p Fs(l)1392 1433 y Fx(.)18 b(By)d(c)o(hanging)e Fw(t;)7 b(t)1711 1418 y Fn(0)228 1493 y Fx(w)o(e)14 b(obtain)641 1558 y Fw(S)r Fx(\(\()p Fw(x)724 1564 y Fs(m)757 1558 y Fw(x)781 1564 y Fu(1)799 1558 y Fx(\))815 1540 y Fs(l)828 1558 y Fw(x)852 1564 y Fs(m)883 1558 y Fx(\))e(=)g(\()p Ft(\000)p Fx(1\))1040 1540 y Fu(1+)p Fs(l)1095 1558 y Fw(x)1119 1564 y Fu(1)1137 1558 y Fw(x)1161 1564 y Fu(2)1186 1558 y Ft(\001)7 b(\001)g(\001)f Fw(x)1266 1564 y Fs(m)1297 1558 y Fw(;)228 1631 y Fx(since)15 b(\()p Fw(t)361 1616 y Fn(0)372 1631 y Fw(t)p Fx(\))403 1616 y Fs(i)p Fn(\000)p Fu(1)460 1631 y Fw(t)475 1616 y Fn(0)498 1631 y Fx(=)d Fw(t)557 1637 y Fs(m)p Fn(\000)p Fs(i)p Fu(+1)668 1631 y Fx(,)h(1)f Ft(\024)f Fw(i)h Ft(\024)g Fx(2)p Fw(l)e Fx(+)g(1.)18 b(Therefore)d(the)f(claim)e(follo)o(ws)g (from)g(\(5.21\))o(.)303 1713 y(The)h(last)g(part)h(of)e(the)i(Theorem) f(follo)o(ws)f(from)f(the)j(fact)f(that)h(the)g(elemen)o(ts)f Fw(x;)7 b(y)14 b Fx(span)228 1773 y(a)g(braided)g(subspace)i(of)e(t)o (yp)q(e)h Fw(A)760 1779 y Fu(2)779 1773 y Fx(.)k(W)m(rite)14 b Fw(x)f Fx(=)1007 1742 y Fl(P)1051 1752 y Fs(l)1051 1785 y(i)p Fu(=1)1114 1773 y Fw(\013)1141 1758 y Fs(i)p Fn(\000)p Fu(1)1197 1773 y Fw(x)1221 1779 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1308 1773 y Fx(and)h Fw(y)g Fx(=)1468 1742 y Fl(P)1512 1752 y Fs(l)1512 1785 y(i)p Fu(=1)1575 1773 y Fw(\014)1600 1758 y Fs(i)p Fn(\000)p Fu(1)1657 1773 y Fw(x)1681 1779 y Fu(2)p Fs(i)1711 1773 y Fx(,)228 1833 y(where)h Fw(\013;)7 b(\014)13 b Ft(2)e(f\006)p Fx(1)p Ft(g)j Fx(and)g Fw(\013\014)f Fx(=)f Ft(\000)p Fx(1.)18 b(W)m(e)c(compute)f(the)i(braiding)d Fw(c)i Fx(with)g(resp)q(ect)i(to)e Fw(x;)7 b(y)q Fx(:)623 1957 y Fw(c)p Fx(\()p Fw(x)i Ft(\012)g Fw(y)q Fx(\))k(=)825 1918 y Fl(X)836 2006 y Fs(i;j)892 1957 y Fw(\013)919 1940 y Fs(i)p Fn(\000)p Fu(1)975 1957 y Fw(\014)1000 1940 y Fs(j)r Fn(\000)p Fu(1)1060 1957 y Fw(c)p Fx(\()p Fw(x)1118 1963 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1200 1957 y Ft(\012)d Fw(x)1266 1963 y Fu(2)p Fs(j)1299 1957 y Fx(\))p Fw(;)228 2065 y Fx(where)15 b(the)f(indices)h Fw(i;)7 b(j)16 b Fx(are)e(in)f Fv(Z)-13 b Fw(=)p Fx(\()p Fw(l)q Fx(\))11 b(\(since)k Fw(l)g Fx(is)f(ev)o(en\).)19 b(Since)640 2147 y Fw(c)p Fx(\()p Fw(x)698 2153 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)780 2147 y Ft(\012)9 b Fw(x)845 2153 y Fu(2)p Fs(j)879 2147 y Fx(\))j(=)f Ft(\000)p Fw(x)1006 2154 y Fu(2\(2)p Fs(i)p Fn(\000)p Fs(j)r Fn(\000)p Fu(1\))1172 2147 y Ft(\012)f Fw(x)1238 2153 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)228 2229 y Fx(,)j(w)o(e)h(substitute)h Fw(k)e Fx(=)f(2)p Fw(i)d Ft(\000)h Fw(j)h Ft(\000)f Fx(1)j(and)h(obtain)372 2311 y Fw(c)p Fx(\()p Fw(x)9 b Ft(\012)h Fw(y)q Fx(\))i(=)g Ft(\000)613 2272 y Fl(X)623 2361 y Fs(i;k)680 2311 y Fw(\013)707 2294 y Fs(i)p Fn(\000)p Fu(1)763 2311 y Fw(\014)788 2294 y Fs(k)809 2311 y Fw(x)833 2317 y Fu(2)p Fs(k)879 2311 y Ft(\012)e Fw(x)945 2317 y Fu(2)p Fs(i)p Fn(\000)p Fu(1)1029 2311 y Fx(=)i Fw(b)1091 2317 y Fu(12)1126 2311 y Fx(\()p Fw(y)f Ft(\012)f Fw(x)p Fx(\))j(,where)i Fw(b)1418 2317 y Fu(12)1464 2311 y Fx(=)d Ft(\000)p Fw(\014)r(:)228 2428 y Fx(In)h(the)i(same)d(w)o(a)o(y)h(one)h(sho)o(ws)g Fw(c)p Fx(\()p Fw(y)c Ft(\012)f Fw(x)p Fx(\))i(=)h Fw(b)951 2434 y Fu(21)986 2428 y Fx(\()p Fw(x)d Ft(\012)g Fw(y)q Fx(\))p Fw(;)e(c)p Fx(\()p Fw(x)h Ft(\012)h Fw(x)p Fx(\))j(=)f Fw(b)1352 2434 y Fu(11)1387 2428 y Fx(\()p Fw(x)e Ft(\012)g Fw(x)p Fx(\))p Fw(;)e(c)p Fx(\()p Fw(y)j Ft(\012)e Fw(y)q Fx(\))13 b(=)228 2488 y Fw(b)246 2494 y Fu(22)281 2488 y Fx(\()p Fw(y)h Ft(\012)f Fw(y)q Fx(\))p Fw(;)21 b Fx(where)15 b Fw(b)584 2494 y Fu(21)639 2488 y Fx(=)20 b Ft(\000)p Fw(\013;)7 b(b)787 2494 y Fu(11)840 2488 y Fx(=)20 b Fw(b)910 2494 y Fu(22)965 2488 y Fx(=)g Ft(\000)p Fx(1)p Fw(:)e Fx(Hence)i(the)f(braiding)f(on)g Fw(x;)7 b(y)20 b Fx(is)f(the)228 2548 y(braiding)13 b(of)g(case)i(2\))f(in)g(Lemma)d (5.6.)17 b(Since)e(the)f(relations)g(only)g(dep)q(end)h(on)f(the)g (braiding)228 2608 y(\(see)h(Remark)d(2.2,)h(1\)\),)g(the)i(claim)c (follo)o(ws)i(from)f(Lemma)f(5.6.)p 1692 2608 2 29 v 1694 2581 25 2 v 1694 2608 V 1719 2608 2 29 v eop %%Page: 17 17 17 16 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d (O)o(VER)i(CO)o(XETER)g(GR)o(OUPS)95 b(17)303 213 y Fx(In)12 b(the)i(next)f(theorem)g(w)o(e)g(assume)f(that)h(for)f(all)g(elemen)o (ts)h Fw(t;)7 b(t)1305 198 y Fn(0)1328 213 y Fx(suc)o(h)14 b(that)f(the)g(pro)q(duct)228 262 y Fw(tt)258 247 y Fn(0)284 262 y Fx(has)h(ev)o(en)h(order)g(2)p Fw(l)q Fx(,)f(w)o(e)h(ha)o(v)o(e)f Fw(")799 268 y Fu(0)831 262 y Fx(=)e Fw(")894 268 y Fu(1)928 262 y Fx(in)i(the)h(sense)h(of)d(Theorem)h(5.7.)19 b(Note)c(that)f (this)228 312 y(condition)i(is)h(satis\014ed)h(in)e(the)i(example)e (\(5.5\))g(where)i Fw(")1157 318 y Fu(0)1193 312 y Fx(=)f Fw(")1261 318 y Fu(1)1296 312 y Fx(=)g Ft(\000)p Fx(1,)h(and)f(also)f (for)h(the)228 362 y(symmetric)12 b(group)i(in)f(\(5.9\))g(where)i Fw(")834 368 y Fu(0)865 362 y Fx(=)d Fw(")928 368 y Fu(1)958 362 y Fx(=)g(1)i(if)f Fw(l)f Fx(=)g(1.)303 453 y Fg(Theorem)k Fx(5.8)p Fg(.)k Ff(L)n(et)15 b Fx(\()p Fw(W)o(;)7 b(S)r Fx(\))16 b Ff(b)n(e)f(a)g(Coxeter)g(system,)g Fw(T)21 b Ff(the)16 b(set)f(of)g(al)r(l)g Fw(W)6 b Ff(-c)n(onjugates)228 513 y(of)18 b(elements)h(in)g Fw(S)r Ff(,)h Fw(\037)f Fx(:)f Fw(W)g Ft(\002)12 b Fw(T)24 b Ft(!)18 b Fv(|)-9 b Ft(n)11 b(f)p Fx(0)p Ft(g)18 b Ff(a)h(function)g(satisfying)k Fx(\(5.1\))o Ff(,)c Fx(\(5.2\))p Ff(,)g Fw(V)28 b Fx(=)228 572 y Fw(V)9 b Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\))p Ff(,)16 b(and)h Fw(R)c Fx(=)h Fo(B)p Fx(\()p Fw(V)9 b Fx(\))p Ff(.)22 b(Assume)16 b(for)g(al)r(l)f Fw(t;)7 b(t)1094 557 y Fn(0)1119 572 y Ft(2)13 b Fw(T)22 b Ff(such)16 b(that)g(the)g(pr)n(o)n(duct)g Fw(tt)1635 557 y Fn(0)1663 572 y Ff(has)228 632 y(even)f(or)n(der)g Fx(2)p Fw(l)g Ff(that)g Fw(\037)p Fx(\(\()p Fw(tt)654 617 y Fn(0)666 632 y Fx(\))682 617 y Fs(l)695 632 y Fw(;)7 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871 y Ff(.)18 b(If)13 b Fw(x)f Ff(is)g(a)g Fw(W)6 b Ff(-homo)n(gene)n(ous)14 b(element)e(in)h Fw(R)1322 877 y Fs(g)1341 871 y Ff(,)f Fw(g)h Ft(2)e Fw(W)6 b Ff(,)13 b(then)g Fw(a)e Ft(2)g Fw(R)1703 877 y Fs(g)228 931 y Ff(and)k Fw(b)d Ft(2)f Fw(R)409 937 y Fs(g)q(t)440 931 y Ff(.)303 991 y(2\))k(F)m(or)g(al)r(l) f Fw(g)f Ft(2)f Fw(W)6 b Ff(,)15 b(the)g Fw(W)6 b Ff(-homo)n(gene)n (ous)16 b(c)n(omp)n(onents)h Fw(R)1274 997 y Fu(1)1307 991 y Ff(and)f Fw(R)1420 997 y Fs(g)1454 991 y Ff(ar)n(e)e(isomorphic) 228 1051 y(as)h(ve)n(ctor)f(sp)n(ac)n(es.)303 1110 y(3\))g(F)m(or)f(al) r(l)h Fw(g)f Ft(2)e Fw(W)6 b Ff(,)14 b(cho)n(ose)g(a)h(r)n(e)n(duc)n(e) n(d)f(r)n(epr)n(esentation)f Fw(g)g Fx(=)f Fw(s)1310 1116 y Fu(1)1336 1110 y Ft(\001)7 b(\001)g(\001)e Fw(s)1410 1116 y Fs(q)1429 1110 y Ff(,)14 b Fw(s)1475 1116 y Fu(1)1494 1110 y Fw(;)7 b Ft(\001)g(\001)g(\001)12 b Fw(;)7 b(s)1613 1116 y Fs(q)1642 1110 y Ft(2)12 b Fw(S)r Ff(,)228 1170 y(of)j Fw(g)q Ff(,)f(and)i(de\014ne)831 1232 y Fw(x)855 1238 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Ff(is)f(a)h(r)n(e)n(duc)n(e)n(d)g(r)n(epr)n(esentation)g(of)g Fw(g)679 1522 y Fx(0)p Fw(;)96 b Ff(otherwise)p Fw(:)228 1482 y Fx(\(5.27\))303 1613 y Fg(Pr)o(oof.)21 b Fx(1\))c(T)m(o)f(pro)o (v)o(e)h(the)g(existence)h(of)e(the)i(represen)o(tation)g(w)o(e)f(ha)o (v)o(e)f(to)h(sho)o(w)g(for)228 1673 y(an)o(y)e Fw(s)327 1679 y Fu(1)346 1673 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)458 1679 y Fs(q)491 1673 y Ft(2)14 b Fw(T)21 b Fx(and)16 b Fw(x)e Fx(=)h Fw(x)770 1679 y Fs(s)786 1683 y Fj(1)804 1673 y Fw(;)7 b Ft(\001)g(\001)g(\001)t Fw(x)901 1679 y Fs(s)917 1683 y Fi(q)951 1673 y Fx(the)16 b(existence)i(of)d Fw(W)6 b Fx(-homogeneous)14 b(elemen)o(ts)228 1732 y Fw(a;)7 b(b)k Ft(2)h Fw(R)370 1717 y Fu(\()p Fs(t)p Fu(\))424 1732 y Fx(with)i Fw(x)d Fx(=)i Fw(a)c Fx(+)h Fw(bx)714 1738 y Fs(t)728 1732 y Fx(.)18 b(This)c(follo)o(ws)f(easily)h(b)o(y)f (induction)h(on)g(the)h(length)f Fw(q)h Fx(from)228 1792 y(\(5.17\))o(,)h(\(5.20\))o(,)g(since)g(these)h(relations)f(are)g Fw(W)6 b Fx(-homogeneous)15 b(since)h(all)f(their)h(monomia)o(ls)228 1852 y(ha)o(v)o(e)d(the)h(same)f(degree)i Fw(t)642 1858 y Fs(i)p Fn(\000)p Fu(1)698 1852 y Fw(t)713 1858 y Fs(i)739 1852 y Fx(=)d Fw(t)798 1858 y Fs(i)811 1852 y Fw(t)826 1858 y Fs(i)p Fu(+1)882 1852 y Fx(.)18 b(The)c(uniqueness)h(of)e(the)h (represen)o(tation)h(is)f(clear)228 1912 y(from)e(Lemma)f(2.5)i(1\).) 303 1972 y(2\))g(Fix)h(an)f(elemen)o(t)h Fw(t)d Ft(2)g Fw(T)6 b Fx(.)18 b(By)d(1\))e(there)i(is)f(w)o(ell-de\014ned)g(linear)g (isomorphism)454 2051 y Fw(\036)479 2057 y Fs(t)505 2051 y Fx(:)d Fw(R)h Ft(!)f Fw(R;)20 b Fx(with)13 b Fw(\036)808 2057 y Fs(t)823 2051 y Fx(\()p Fw(a)c Fx(+)g Fw(bx)953 2057 y Fs(t)968 2051 y Fx(\))i(=)h Fw(b)d Fx(+)h Fw(ax)1154 2057 y Fs(t)1182 2051 y Fx(for)j(all)g Fw(a;)7 b(b)k Ft(2)g Fw(R)1444 2033 y Fu(\()p Fs(t)p Fu(\))1484 2051 y Fw(:)228 2130 y Fx(F)m(or)17 b(an)o(y)h Fw(g)i Ft(2)e Fw(W)6 b Fx(,)18 b(the)h(restriction)g(of)e Fw(\036)905 2136 y Fs(t)938 2130 y Fx(de\014nes)i(an)f(isomorphism)d Fw(R)1419 2136 y Fs(g)1456 2118 y Ft(\030)1456 2132 y Fx(=)1507 2130 y Fw(R)1539 2136 y Fs(g)q(t)1570 2130 y Fx(,)j(and)g(2\))228 2189 y(follo)o(ws,)12 b(since)i Fw(T)20 b Fx(generates)c(the)e(group)g Fw(W)6 b Fx(.)303 2249 y(3\))12 b(De\014ne)i(the)f(sequence)i(\()p Fw(t)752 2255 y Fu(1)771 2249 y Fw(;)7 b(:)g(:)g(:)e(;)i(t)879 2255 y Fs(q)896 2249 y Fx(\))13 b(of)g(elemen)o(ts)f(in)h Fw(T)19 b Fx(with)12 b(resp)q(ect)j(to)e(\()p Fw(s)1550 2255 y Fu(1)1569 2249 y Fw(;)7 b Ft(\001)g(\001)g(\001)12 b Fw(;)7 b(s)1688 2255 y Fs(q)1706 2249 y Fx(\))228 2309 y(as)15 b(in)f(Lemma)f(5.5.)20 b(Since)c(\()p Fw(s)710 2315 y Fu(1)729 2309 y Fw(;)7 b(:)g(:)g(:)e(;)i(s)841 2315 y Fs(q)859 2309 y Fx(\))15 b(is)g(a)g(reduced)i(represen)o(tation) f(of)f Fw(g)q Fx(,)g(the)g(elemen)o(ts)228 2369 y Fw(t)243 2375 y Fs(i)270 2369 y Fx(are)f(pairwise)g(distinct)g(b)o(y)f([)p Fy(B)p Fx(,)g(1.4,)f(Lemma)f(2].)17 b(Hence)e Fw(S)r Fx(\()p Fw(x)1257 2375 y Fs(g)1277 2369 y Fx(\))d Ft(6)p 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Fx(+)g(1)p Fw(:)228 387 y Fx(\(5.28\))228 504 y(This)k(follo)o(ws)e(directly)i(from)e(\(5.19\))o(,)i(\(5.22\))o(.)303 564 y(Finally)j(w)o(e)i(ha)o(v)o(e)f(to)h(sho)o(w)g(that)f Fw(x)897 570 y Fs(t)910 574 y Fj(1)935 564 y Ft(\001)7 b(\001)g(\001)e Fw(x)1014 570 y Fs(t)1027 574 y Fi(p)1066 564 y Fx(=)20 b(0)e(if)g(the)h(length)g(of)f Fw(g)i Fx(is)f Fw(<)h(p)p Fx(.)33 b(W)m(e)228 624 y(ma)o(y)16 b(assume)h(that)h(\()p Fw(t)594 630 y Fu(2)613 624 y Fw(;)7 b Ft(\001)g(\001)g(\001)k Fw(;)c(t)727 630 y Fs(p)746 624 y Fx(\))18 b(is)g(reduced.)31 b(Then)19 b(b)o(y)e(the)i(exc)o(hange)f(prop)q(ert)o(y)h(in)e([)p Fy(B)p Fx(,)228 684 y(1.4,)d(Lemma)e(3])j(there)i(exists)e(2)f Ft(\024)g Fw(j)i Ft(\024)f Fw(p)g Fx(suc)o(h)h(that)f Fw(t)1126 690 y Fu(1)1145 684 y Fw(t)1160 690 y Fu(2)1185 684 y Ft(\001)7 b(\001)g(\001)e Fw(t)1255 690 y Fs(j)r Fn(\000)p Fu(1)1329 684 y Fx(=)14 b Fw(t)1390 690 y Fu(2)1416 684 y Ft(\001)7 b(\001)g(\001)e Fw(t)1486 690 y Fs(j)r Fn(\000)p Fu(1)1546 684 y Fw(t)1561 690 y Fs(j)1579 684 y Fx(.)22 b(Hence)228 743 y(\()p Fw(x)268 749 y Fs(t)281 753 y Fj(1)305 743 y Ft(\001)7 b(\001)g(\001)f Fw(x)385 749 y Fs(t)398 753 y Fi(j)q Fk(\000)p Fj(1)451 743 y Fx(\)\()p Fw(x)507 749 y Fs(t)520 753 y Fi(j)544 743 y Ft(\001)h(\001)g(\001)f Fw(x)624 749 y Fs(t)637 753 y Fi(p)655 743 y Fx(\))14 b(=)g Ft(\006)p Fx(\()p Fw(x)803 749 y Fs(t)816 753 y Fj(2)841 743 y Ft(\001)7 b(\001)g(\001)e Fw(x)920 749 y Fs(t)933 753 y Fi(j)950 743 y Fx(\)\()p Fw(x)1006 749 y Fs(t)1019 753 y Fi(j)1043 743 y Ft(\001)i(\001)g(\001)f Fw(x)1123 749 y Fs(t)1136 753 y Fi(p)1154 743 y Fx(\))14 b(=)g(0)h(b)o(y)g(what)g(w)o(e)g(ha)o(v)o(e)g(already)228 803 y(sho)o(wn,)e(since)i(\()p Fw(x)507 809 y Fs(t)520 813 y Fi(j)537 803 y Fx(\))553 788 y Fu(2)583 803 y Fx(=)d(0)i(b)o(y)g (\(5.16\))o(.)p 1692 803 2 29 v 1694 777 25 2 v 1694 803 V 1719 803 2 29 v 303 878 a(The)g(follo)o(wing)d(Corollary)i(is)g (an)h(immediate)d(consequence)16 b(of)e(Theorem)f(5.8.)303 959 y Fg(Cor)o(ollar)m(y)18 b Fx(5.9)p Fg(.)i Ff(Assume)e(the)g (situation)f(of)h(The)n(or)n(em)f(5.8.)27 b(Then)18 b(the)f(sub)n (algebr)n(a)228 1019 y(of)e Fw(R)f Ff(gener)n(ate)n(d)h(by)g(al)r(l)g Fw(x)645 1025 y Fs(s)662 1019 y Fw(;)7 b(s)12 b Ft(2)f Fw(S)18 b Ff(has)d(the)g Fv(|)-19 b Ff(-b)n(asis)12 b Fw(x)1099 1025 y Fs(g)1118 1019 y Fw(;)7 b(g)12 b Ft(2)f Fw(W)6 b Ff(.)303 1079 y(If)18 b Fw(R)h Ff(is)f(\014nite-dimensional,)i (then)g Fw(W)k Ff(is)19 b(\014nite)g(and)h Fw(dim)p Fx(\()p Fw(R)p Fx(\))f(=)g Fw(or)q(d)p Fx(\()p Fw(W)6 b Fx(\))p Fw(dim)p Fx(\()p Fw(R)1675 1085 y Fu(1)1693 1079 y Fx(\))p Ff(.)228 1138 y(Her)n(e,)14 b Fw(R)372 1144 y Fu(1)405 1138 y Ff(is)g(the)h Fw(W)6 b Ff(-homo)n(gene)n(ous)16 b(c)n(omp)n(onent)g(of)f Fw(R)g Ff(of)g(the)g(unit)g(element)f(in)h Fw(W)6 b Ff(.)303 1210 y Fx(Finally)13 b(w)o(e)i(note)g(that)g(the)h (results)g(of)e(Section)i(3)e(giv)o(e)h(some)f(information)e(ab)q(out)j (the)228 1260 y(Hopf)g(algebras)h(considered)h(in)e(this)h(Section,)h 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Fx(\))18 b Ff(and)228 1630 y Fw(V)261 1615 y Fn(0)284 1630 y Fx(=)12 b Fw(V)e Fx(\()p Fw(W)423 1615 y Fn(0)434 1630 y Fw(;)d(T)483 1615 y Fn(0)495 1630 y Fw(;)g(\037)540 1615 y Fn(0)551 1630 y Fx(\))p Ff(.)19 b(Then)303 1690 y(\(1\))e Fo(B)p Fx(\()p Fw(V)9 b Fx(\))17 b Ff(is)g(a)g(fr)n(e)n(e)g(right)f Fo(B)p Fx(\()p Fw(V)848 1675 y Fn(0)860 1690 y Fx(\))p Ff(-mo)n(dule,)i(and)g(the)f(Hilb)n(ert)e(series)i Fw(P)1487 1697 y Fe(B)p Fu(\()p Fs(V)1556 1689 y Fk(0)1567 1697 y Fu(\))1599 1690 y Ff(divides)228 1749 y Fw(P)255 1756 y Fe(B)p Fu(\()p Fs(V)7 b Fu(\))339 1749 y Ff(.)303 1809 y(\(2\))16 b(As)g(in)g(The)n(or)n(em)j Fx(\(5.8\))c Ff(assume)i(for)e (al)r(l)h Fw(t;)7 b(t)1092 1794 y Fn(0)1117 1809 y Ft(2)14 b Fw(T)22 b Ff(such)16 b(that)h(the)f(pr)n(o)n(duct)g Fw(tt)1635 1794 y Fn(0)1663 1809 y Ff(has)228 1869 y(even)i(or)n(der)f Fx(2)p Fw(l)h Ff(that)f Fw(\037)p Fx(\(\()p Fw(tt)664 1854 y Fn(0)676 1869 y Fx(\))692 1854 y Fs(l)705 1869 y Fw(;)7 b(t)p Fx(\))16 b(=)g Fw(\037)p Fx(\(\()p Fw(tt)907 1854 y Fn(0)919 1869 y Fx(\))935 1854 y Fs(l)948 1869 y Fw(;)7 b(t)982 1854 y Fn(0)993 1869 y Fx(\))p Ff(.)26 b(L)n(et)1129 1858 y Fl(e)1122 1869 y Fo(B)p Fx(\()p Fw(V)10 b Fx(\))17 b Ff(r)n(esp.)1361 1858 y Fl(e)1354 1869 y Fo(B)p Fx(\()p Fw(V)1440 1854 y Fn(0)1452 1869 y Fx(\))g Ff(b)n(e)h(the)f(tensor)228 1929 y(algebr)n(a)i Fw(T)6 b Fx(\()p Fw(V)k Fx(\))20 b Ff(r)n(esp.)34 b Fw(T)6 b Fx(\()p Fw(V)689 1914 y Fn(0)700 1929 y Fx(\))21 b Ff(mo)n(dulo)f(al)r(l)f(the)i(quadr)n(atic)f(r)n(elations)i Fx(\(5.16\))d Ff(and)25 b Fx(\(5.17\))o Ff(,)228 1988 y Fx(\(5.20\))o Ff(.)19 b(Then)481 1978 y Fl(e)474 1988 y Fo(B)p Fx(\()p Fw(V)9 b Fx(\))15 b Ff(is)g(a)g(fr)n(e)n(e)f(right)857 1978 y Fl(e)850 1988 y Fo(B)p Fx(\()p Fw(V)936 1973 y Fn(0)948 1988 y Fx(\))p Ff(-mo)n(dule,)h(and)g Fw(P)1245 1995 y Fh(e)1240 2003 y Fe(B)q Fu(\()p Fs(V)1309 1994 y Fk(0)1320 2003 y Fu(\))1350 1988 y Ff(divides)g Fw(P)1520 1995 y Fh(e)1515 2003 y Fe(B)p Fu(\()p Fs(V)7 b Fu(\))1599 1988 y Ff(.)303 2070 y 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Fn(0)977 2249 y Fx(\))20 b(b)q(e)g(the)h(map) d(induced)j(b)o(y)e Fw(')p Fx(.)36 b(Since)21 b(the)228 2309 y(quadratic)15 b(relations)h(de\014ne)h(in)e(fact)h(primitiv)o(e)e (elemen)o(ts)h(in)h(the)g(tensor)h(algebras,)1627 2298 y Fl(e)1620 2309 y Fo(B)o Fx(\()p Fw(V)10 b Fx(\))228 2369 y(and)315 2358 y Fl(e)309 2369 y Fo(B)o Fx(\()p Fw(V)395 2354 y Fn(0)406 2369 y Fx(\))15 b(are)f(braided)g(Hopf)f (algebras.)19 b(The)14 b(map)e Fw(\036)i Fx(induces)h(a)e(map)g(on)g (the)i(quotien)o(ts)228 2428 y(since)e(the)h(quadratic)f(relations)f (are)h(preserv)o(ed.)20 b(This)13 b(is)g(clear)g(for)f(\(5.16\))o(,)h (and)f(is)h(also)f(true)228 2488 y(for)h(\(5.17\))o(,)h(\(5.20\))o(,)f (since)i(all)d(the)j Fw(t)800 2494 y Fs(i)814 2488 y Fw(;)7 b Fx(1)j Ft(\024)i Fw(i)g Ft(\024)g Fw(m;)i Fx(are)g(in)f Fw(T)1187 2473 y Fn(0)1199 2488 y Fx(,)g(if)g(for)h(one)g Fw(i)g Fx(b)q(oth)g Fw(t)1543 2494 y Fs(i)1571 2488 y Fx(and)f Fw(t)1666 2494 y Fs(i)p Fu(+1)228 2548 y Fx(are)i(in)g Fw(T)378 2533 y Fn(0)389 2548 y Fx(.)21 b(The)16 b(claim)c(no)o(w)j (follo)o(ws)e(from)g(Theorem)i(3.2)e(in)i(the)g(same)f(w)o(a)o(y)g(as)h (Corollary)228 2608 y(3.3.)p 1692 2608 V 1694 2581 25 2 v 1694 2608 V 1719 2608 2 29 v eop %%Page: 19 19 19 18 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d (O)o(VER)i(CO)o(XETER)g(GR)o(OUPS)95 b(19)844 213 y Fy(6.)24 b(Examples)303 287 y Fx(W)m(e)13 b(consider)i(some)e(sp)q(ecial)h (cases)h(of)e(Theorem)h(5.8.)303 385 y Fg(Example)j Fx(6.1)p Fg(.)j Fx(Let)15 b Fw(W)k Fx(=)14 b Fw(S)790 391 y Fs(n)813 385 y Fw(;)7 b(n)13 b Ft(\025)g Fx(2,)i(and)g Fw(T)k Fx(=)13 b Ft(f)p Fx(\()p Fw(ij)r Fx(\))i Ft(j)d Fx(1)i Ft(\024)f Fw(i)h(<)f(j)j Ft(\024)e Fw(n)p Ft(g)g Fx(the)i(set)g(of)228 445 y(all)e(transp)q(ositions.)24 b(De\014ne)16 b Fw(\037)g Fx(b)o(y)g(\(5.5\))f(and)g(let)h Fw(V)24 b Fx(=)15 b Fw(V)9 b Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\).)23 b(Then)17 b(the)f(follo)o(wing)228 505 y(relations)d(hold)h(in)f Fo(B)p Fx(\()p Fw(V)c Fx(\))14 b(for)g(all)f(1)e Ft(\024)h Fw(i)g(<)f(j)j Ft(\024)e Fw(n;)7 b Fx(1)k Ft(\024)h Fw(k)g(<)g(l)h Ft(\024)e Fw(n)p Fx(:)1355 594 y Fw(x)1379 577 y Fu(2)1379 605 y(\()p Fs(ij)r Fu(\))1446 594 y Fx(=)g(0)p Fw(:)-1294 b Fx(\(6.1\))501 666 y(If)13 b Ft(f)p Fw(i;)7 b(j)r Ft(g)i(\\)g(f)p Fw(k)q(;)e(l)q Ft(g)k Fx(=)g Ft(;)p Fw(;)20 b Fx(then)98 b Fw(x)1088 673 y Fu(\()p Fs(ij)r Fu(\))1143 666 y Fw(x)1167 673 y Fu(\()p Fs(k)q(l)p Fu(\))1233 666 y Fx(+)9 b Fw(x)1298 673 y Fu(\()p Fs(k)q(l)p Fu(\))1355 666 y Fw(x)1379 673 y Fu(\()p Fs(ij)r Fu(\))1446 666 y Fx(=)i(0)p Fw(:)-1294 b Fx(\(6.2\))421 739 y(If)14 b Fw(i)e(<)g(j)i(<)d(k)q(;)21 b Fx(then)97 b Fw(x)864 746 y Fu(\()p Fs(ij)r Fu(\))919 739 y Fw(x)943 746 y Fu(\()p Fs(j)r(k)q Fu(\))1014 739 y Fx(+)9 b Fw(x)1079 746 y Fu(\()p Fs(j)r(k)q Fu(\))1141 739 y Fw(x)1165 746 y Fu(\()p Fs(ik)q Fu(\))1232 739 y Fx(+)g Fw(x)1297 746 y Fu(\()p Fs(ik)q Fu(\))1355 739 y Fw(x)1379 746 y Fu(\()p Fs(ij)r Fu(\))1446 739 y Fx(=)i(0)p Fw(;)-1294 b Fx(\(6.3\))840 811 y Fw(x)864 818 y Fu(\()p Fs(j)r(k)q Fu(\))926 811 y Fw(x)950 818 y Fu(\()p Fs(ij)r Fu(\))1014 811 y Fx(+)9 b Fw(x)1079 818 y Fu(\()p Fs(ik)q Fu(\))1137 811 y Fw(x)1161 818 y Fu(\()p Fs(j)r(k)q Fu(\))1232 811 y Fx(+)g Fw(x)1297 818 y Fu(\()p Fs(ij)r Fu(\))1352 811 y Fw(x)1376 818 y Fu(\()p Fs(ik)q Fu(\))1446 811 y Fx(=)i(0)p Fw(:)303 909 y Fg(Example)17 b Fx(6.2)p Fg(.)j Fx(Let)15 b Fw(W)k Fx(=)14 b Fw(S)790 915 y Fs(n)813 909 y Fw(;)7 b(n)13 b Ft(\025)g Fx(2,)i(and)g Fw(T)k Fx(=)13 b Ft(f)p Fx(\()p Fw(ij)r Fx(\))i Ft(j)d Fx(1)i Ft(\024)f Fw(i)h(<)f(j)j Ft(\024)e Fw(n)p Ft(g)g Fx(the)i(set)g(of)228 969 y(all)e(transp)q(ositions.)24 b(De\014ne)16 b Fw(\037)g Fx(b)o(y)g(\(5.9\))f(and)g(let)h Fw(V)24 b Fx(=)15 b Fw(V)9 b Fx(\()p Fw(W)o(;)e(T)s(;)g(\037)p Fx(\).)23 b(Then)17 b(the)f(follo)o(wing)228 1028 y(relations)d(hold)h(in)f Fo(B)p Fx(\()p Fw(V)c Fx(\))14 b(for)g(all)f(1)e Ft(\024)h Fw(i)g(<)f(j)j Ft(\024)e Fw(n;)7 b Fx(1)k Ft(\024)h Fw(k)g(<)g(l)h Ft(\024)e Fw(n)p Fx(:)1343 1118 y Fw(x)1367 1101 y Fu(2)1367 1129 y(\()p Fs(ij)r Fu(\))1433 1118 y Fx(=)h(0)p Fw(:)-1282 b Fx(\(6.4\))513 1190 y(If)14 b Ft(f)p Fw(i;)7 b(j)r Ft(g)i(\\)g(f)p Fw(k)q(;)e(l)q Ft(g)j Fx(=)i Ft(;)p Fw(;)7 b(then)82 b(x)1075 1197 y Fu(\()p Fs(ij)r Fu(\))1130 1190 y Fw(x)1154 1197 y Fu(\()p Fs(k)q(l)p Fu(\))1220 1190 y Ft(\000)10 b Fw(x)1286 1197 y Fu(\()p Fs(k)q(l)p Fu(\))1343 1190 y Fw(x)1367 1197 y Fu(\()p Fs(ij)r Fu(\))1433 1190 y Fx(=)i(0)p Fw(:)-1282 b Fx(\(6.5\))434 1262 y(If)13 b Fw(i)f(<)g(j)i(<)e(k)q(;)7 b(then)82 b(x)851 1269 y Fu(\()p Fs(ij)r Fu(\))906 1262 y Fw(x)930 1269 y Fu(\()p Fs(j)r(k)q Fu(\))1001 1262 y Ft(\000)10 b Fw(x)1067 1269 y Fu(\()p Fs(j)r(k)q Fu(\))1128 1262 y Fw(x)1152 1269 y Fu(\()p Fs(ik)q Fu(\))1219 1262 y Ft(\000)g Fw(x)1285 1269 y Fu(\()p Fs(ik)q Fu(\))1343 1262 y Fw(x)1367 1269 y Fu(\()p Fs(ij)r Fu(\))1433 1262 y Fx(=)i(0)p Fw(;)-1282 b Fx(\(6.6\))827 1334 y Fw(x)851 1341 y Fu(\()p Fs(j)r(k)q Fu(\))913 1334 y Fw(x)937 1341 y Fu(\()p Fs(ij)r Fu(\))1001 1334 y Ft(\000)10 b Fw(x)1067 1341 y Fu(\()p Fs(ik)q Fu(\))1124 1334 y Fw(x)1148 1341 y Fu(\()p Fs(j)r(k)q Fu(\))1219 1334 y Ft(\000)g Fw(x)1285 1341 y Fu(\()p Fs(ij)r Fu(\))1340 1334 y Fw(x)1364 1341 y Fu(\()p Fs(ik)q Fu(\))1433 1334 y Fx(=)i(0)p Fw(:)303 1426 y Fx(The)h(algebras)g (generated)i(b)o(y)d(all)g Fw(x)874 1433 y Fu(\()p Fs(ij)r Fu(\))929 1426 y Fw(;)7 b Fx(1)k Ft(\024)h Fw(i)g(<)f(j)j Ft(\024)e Fw(n;)h Fx(with)g(the)g(quadratic)g(relations)228 1485 y(in)k(the)h(examples)e(6.1)h(and)g(6.2)g(are)g(examples)g(of)g (the)h(braided)f(Hopf)g(algebras)1575 1475 y Fl(e)1568 1485 y Fo(B)o Fx(\()p Fw(V)10 b Fx(\))18 b(in)228 1541 y(Corollary)f(\(5.10\))o(.)572 1530 y Fl(e)566 1541 y Fo(B)o Fx(\()p Fw(V)10 b Fx(\))18 b(in)g(example)f(6.2)g(is)h(the)h (algebra)f Ft(E)1271 1547 y Fs(n)1312 1541 y Fx(in)o(tro)q(duced)h(in)f ([)p Fy(FK)p Fx(])f(to)228 1591 y(describ)q(e)g(the)f(cohomology)c (ring)j(of)g(the)h(\015ag)f(v)n(ariet)o(y)m(.)21 b(W)m(e)15 b(b)q(eliev)o(e)g(that)h(indeed)g(the)g(qua-)228 1640 y(dratic)d(relations)g(in)f(the)i(examples)e(6.1)g(and)h(6.2)f(are)i (de\014ning)f(relations)f(for)h Fo(B)p Fx(\()p Fw(V)c Fx(\),)k(that)g(is)235 1685 y Fl(e)228 1696 y Fo(B)o Fx(\()p Fw(V)d Fx(\))j(=)h Fo(B)p Fx(\()p Fw(V)9 b Fx(\))15 b(in)f(these)i(cases.)22 b(Since)16 b(the)f(algebras)g Ft(E)1152 1702 y Fs(n)1189 1696 y Fx(are)g(braided)g(Hopf)f(algebras)h (w)o(e)228 1745 y(immedia)o(tely)c(obtain)i(the)303 1843 y Fg(Cor)o(ollar)m(y)18 b Fx(6.3)p Fg(.)i Ff(L)n(et)15 b Fw(n)c Ft(\025)h Fx(1)p Fw(:)303 1903 y Ff(\(1\))j Ft(E)395 1909 y Fs(n)p Fu(+1)474 1903 y Ff(is)f(a)i(fr)n(e)n(e)e(mo)n (dule)h(over)f Ft(E)888 1909 y Fs(n)911 1903 y Ff(,)h(and)g Fw(P)1046 1909 y Fn(E)1064 1913 y Fi(n)1101 1903 y Ff(divides)g Fw(P)1266 1909 y Fn(E)1284 1913 y Fi(n)p Fj(+1)1341 1903 y Ff(.)303 1963 y(\(2\))i(If)g Ft(E)443 1969 y Fs(n)483 1963 y Ff(is)g(\014nite-dimensional)h(with)f(highest)h(non-zer)n(o)f (homo)n(gene)n(ous)i(c)n(omp)n(onent)228 2023 y Ft(E)250 2029 y Fs(n)272 2023 y Fx(\()p Fw(N)5 b Fx(\))15 b Ff(,)g(then)g(for)f (al)r(l)h Fx(0)c Ft(\024)h Fw(i)g Ft(\024)g Fw(N)5 b Ff(,)14 b(dim)p Ft(E)904 2029 y Fs(n)927 2023 y Fx(\()p Fw(i)p Fx(\))e(=)f Ff(dim)p Ft(E)1118 2029 y Fs(n)1141 2023 y Fx(\()p Fw(N)j Ft(\000)c Fw(i)p Fx(\))p Ff(.)303 2121 y Fg(Pr)o(oof.)21 b Fx(\(1\))c(follo)o(ws)e(from)f(Corollary)h (5.10)h(\(2\))g(with)g Fw(W)22 b Fx(=)16 b Fw(S)1352 2127 y Fs(n)p Fu(+1)1433 2121 y Fx(and)g Fw(W)1561 2106 y Fn(0)1589 2121 y Fx(=)g Ft(f)p Fw(g)h Ft(2)228 2181 y Fw(W)g Ft(j)11 b Fw(g)q Fx(\()p Fw(n)f Fx(+)f(1\))j(=)g Fw(n)d Fx(+)g(1)p Ft(g)i Fx(=)h Fw(S)710 2187 y Fs(n)733 2181 y Fx(,)i(and)f(\(2\))h(is)g(a)g(sp)q(ecial)g(case)h(of)e(Remark)f (2.2)h(4\).)p 1692 2181 2 29 v 1694 2154 25 2 v 1694 2181 V 1719 2181 2 29 v 303 2273 a(The)19 b(`P)o(oincar)o(\023)-20 b(e-dualit)o(y")18 b(in)h(6.3)f(w)o(as)h(a)g(conjecture)i(in)e([)p Fy(FK)p Fx(],)h(and)f(6.3)f(\(1\))i(is)f(also)228 2322 y(sho)o(wn)14 b(in)f([)p Fy(FP)o Fx(])h(b)o(y)f(di\013eren)o(t)i(metho) q(ds.)303 2420 y Fg(Example)i Fx(6.4)p Fg(.)j Fx(Let)e Fo(B)p Fx(\()p Fw(V)741 2426 y Fs(n)763 2420 y Fx(\))g(b)q(e)g(the)h (Hopf)e(algebra)g(of)g(Example)f(6.1.)29 b(If)17 b Fw(n)h Fx(=)g(3)f(or)228 2480 y(4,)j(then)g(\(6.1\))o(,)g(\(6.2\))f(and)g (\(6.3\))g(are)g(de\014ning)h(relations)f(of)f Fw(B)r Fx(\()p Fw(V)1330 2486 y Fs(n)1354 2480 y Fx(\),)i(and)f(dim)o Fo(B)p Fx(\()p Fw(V)1635 2486 y Fu(3)1653 2480 y Fx(\))i(=)228 2540 y(12)p Fw(;)7 b Fx(dim)m Fo(B)p Fx(\()p Fw(V)434 2546 y Fu(4)453 2540 y Fx(\))16 b(=)g(24)575 2525 y Fu(2)593 2540 y Fx(.)26 b(In)17 b(b)q(oth)f(cases)i(the)f(in)o(tegral)f(can)h(b) q(e)g(describ)q(ed)h(in)e(terms)h(of)f(the)228 2600 y(longest)e(elemen) o(t)f(in)g(the)i(Co)o(xeter)g(groups)f Fw(S)956 2606 y Fu(3)975 2600 y Fw(;)7 b(S)1019 2606 y Fu(4)1037 2600 y Fx(.)p eop %%Page: 20 20 20 19 bop 228 119 a Fu(20)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)303 213 y Fg(Pr)o(oof.)21 b Fx(1\))c(W)m(e)f(\014rst)h(consider)h Fw(R)e 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321 y Fj(3)1471 317 y Fu(\))1502 332 y Fx(b)o(y)g(Theorem)228 392 y(5.8)g(1\).)23 b(On)16 b(the)h(other)f(hand)g(it)f(is)h(not)g(di\016cult)f(to)h(c)o (hec)o(k)g(\(using)g(the)h(diamond)c(lemma\))228 452 y(that)18 b(the)h(algebra)f(with)h(relations)f(\(6.1\))o(,)h(\(6.2\))f (and)g(\(6.3\))g(in)g(this)h(case)g(has)g(dimension)228 511 y(12.)h(The)c(longest)f(elemen)o(t)f(is)h Fw(s)747 517 y Fu(1)766 511 y Fw(s)785 517 y Fu(2)804 511 y Fw(s)823 517 y Fu(1)842 511 y Fx(.)21 b(Therefore)16 b Fw(x)1088 517 y Fu(1)1107 511 y Fw(x)1131 517 y Fu(2)1149 511 y Fw(x)1173 517 y Fu(1)1191 511 y Fw(x)1215 517 y Fu(3)1248 511 y Fx(is)f(a)g(non-zero)g(monomia)o(l)d(of)228 571 y(maxim)o(al)e(length,)k(hence)h(an)f(in)o(tegral.)303 631 y(2\))19 b(No)o(w)h(let)g Fw(R)i Fx(=)g Fo(B)p Fx(\()p Fw(V)711 637 y Fu(4)730 631 y Fx(\).)36 b(W)m(e)20 b(denote)h Fw(s)1031 637 y Fu(1)1071 631 y Fx(=)i(\(12\))p Fw(;)7 b(s)1238 637 y Fu(2)1278 631 y Fx(=)22 b(\(23\))p Fw(;)7 b(s)1444 637 y Fu(3)1484 631 y Fx(=)22 b(\(34\))p Fw(;)7 b(s)1650 637 y Fu(4)1690 631 y Fx(=)228 691 y(\(24\))p Fw(;)g(s)340 697 y Fu(5)375 691 y Fx(=)17 b(\(14\))p Fw(;)7 b(s)536 697 y Fu(6)571 691 y Fx(=)17 b(\(13\),)h(and)f Fw(x)832 697 y Fs(i)862 691 y Fx(=)g Fw(x)935 697 y Fs(s)951 701 y Fi(i)966 691 y Fw(;)7 b Fx(1)16 b Ft(\024)h Fw(i)g Ft(\024)g Fx(6.)28 b(Hence)18 b Fw(S)i Fx(=)d Ft(f)p Fw(s)1472 697 y Fu(1)1491 691 y Fw(;)7 b(s)1529 697 y Fu(2)1547 691 y Fw(;)g(s)1585 697 y Fu(3)1604 691 y Ft(g)p Fw(;)g(T)22 b Fx(=)228 751 y Ft(f)p Fw(s)268 757 y Fu(1)287 751 y Fw(;)7 b(:)g(:)g(:)t(;)g(s)398 757 y Fu(6)417 751 y Ft(g)p Fx(.)18 b(The)c(relations)g(are)g Fw(x)815 736 y Fu(2)815 761 y Fs(i)845 751 y Fx(=)e(0,)h(for)g(all)g(1)e Ft(\024)h Fw(i)g Ft(\024)g Fx(6,)h(and)640 832 y Fw(x)664 838 y Fu(1)682 832 y Fw(x)706 838 y Fu(3)733 832 y Fx(+)d Fw(x)799 838 y Fu(3)817 832 y Fw(x)841 838 y Fu(1)871 832 y Fx(=)i(0)p Fw(;)640 905 y(x)664 911 y Fu(2)682 905 y Fw(x)706 911 y Fu(5)733 905 y Fx(+)e Fw(x)799 911 y Fu(5)817 905 y Fw(x)841 911 y Fu(2)871 905 y Fx(=)i(0)p Fw(;)640 977 y(x)664 983 y Fu(4)682 977 y Fw(x)706 983 y Fu(6)733 977 y Fx(+)e Fw(x)799 983 y Fu(6)817 977 y Fw(x)841 983 y Fu(4)871 977 y Fx(=)i(0)p Fw(;)504 1049 y(x)528 1055 y Fu(1)546 1049 y Fw(x)570 1055 y Fu(2)598 1049 y Fx(+)e Fw(x)664 1055 y Fu(2)682 1049 y Fw(x)706 1055 y Fu(6)733 1049 y Fx(+)g Fw(x)799 1055 y Fu(6)817 1049 y Fw(x)841 1055 y Fu(1)871 1049 y Fx(=)i(0)p Fw(;)48 b(x)1020 1055 y Fu(2)1038 1049 y Fw(x)1062 1055 y Fu(1)1090 1049 y Fx(+)9 b Fw(x)1155 1055 y Fu(1)1174 1049 y Fw(x)1198 1055 y Fu(6)1225 1049 y Fx(+)h Fw(x)1291 1055 y Fu(6)1309 1049 y Fw(x)1333 1055 y Fu(2)1363 1049 y Fx(=)i(0)p Fw(;)504 1121 y(x)528 1127 y Fu(1)546 1121 y Fw(x)570 1127 y Fu(4)598 1121 y Fx(+)e Fw(x)664 1127 y Fu(4)682 1121 y Fw(x)706 1127 y Fu(5)733 1121 y Fx(+)g Fw(x)799 1127 y Fu(5)817 1121 y Fw(x)841 1127 y Fu(1)871 1121 y Fx(=)i(0)p Fw(;)48 b(x)1020 1127 y Fu(4)1038 1121 y Fw(x)1062 1127 y Fu(1)1090 1121 y Fx(+)9 b Fw(x)1155 1127 y Fu(1)1174 1121 y Fw(x)1198 1127 y Fu(5)1225 1121 y Fx(+)h Fw(x)1291 1127 y Fu(5)1309 1121 y Fw(x)1333 1127 y Fu(4)1363 1121 y Fx(=)i(0)p Fw(;)504 1194 y(x)528 1200 y Fu(2)546 1194 y Fw(x)570 1200 y Fu(3)598 1194 y Fx(+)e Fw(x)664 1200 y Fu(3)682 1194 y Fw(x)706 1200 y Fu(4)733 1194 y Fx(+)g Fw(x)799 1200 y Fu(4)817 1194 y Fw(x)841 1200 y Fu(2)871 1194 y Fx(=)i(0)p Fw(;)48 b(x)1020 1200 y Fu(3)1038 1194 y Fw(x)1062 1200 y Fu(2)1090 1194 y Fx(+)9 b Fw(x)1155 1200 y Fu(2)1174 1194 y Fw(x)1198 1200 y Fu(4)1225 1194 y Fx(+)h Fw(x)1291 1200 y Fu(4)1309 1194 y Fw(x)1333 1200 y Fu(3)1363 1194 y Fx(=)i(0)p Fw(;)504 1266 y(x)528 1272 y Fu(3)546 1266 y Fw(x)570 1272 y Fu(5)598 1266 y Fx(+)e Fw(x)664 1272 y Fu(5)682 1266 y Fw(x)706 1272 y Fu(6)733 1266 y Fx(+)g Fw(x)799 1272 y Fu(6)817 1266 y Fw(x)841 1272 y Fu(3)871 1266 y Fx(=)i(0)p Fw(;)48 b(x)1020 1272 y Fu(5)1038 1266 y Fw(x)1062 1272 y Fu(3)1090 1266 y Fx(+)9 b Fw(x)1155 1272 y Fu(3)1174 1266 y Fw(x)1198 1272 y Fu(6)1225 1266 y Fx(+)h Fw(x)1291 1272 y Fu(6)1309 1266 y Fw(x)1333 1272 y Fu(5)1363 1266 y Fx(=)i(0)p Fw(:)228 1348 y Fx(F)m(rom)e(these)j(relations)f(one)g(can)g(see)h(that)f Fo(B)p Fx(\()p Fw(V)984 1354 y Fu(4)1003 1348 y Fx(\))g(=)f Fv(|)-19 b Fx([)p Fw(x)1136 1354 y Fu(1)1151 1348 y Fw(;)7 b(x)1194 1354 y Fu(2)1212 1348 y Fw(;)g(x)1255 1354 y Fu(3)1273 1348 y Fx(])e Ft(\001)g Fv(|)-19 b Fx([)o Fw(x)1368 1354 y Fu(4)1383 1348 y Fw(;)7 b(x)1426 1354 y Fu(5)1444 1348 y Fw(;)g(x)1487 1354 y Fu(6)1505 1348 y Fx(].)17 b(By)12 b(Corol-)228 1408 y(lary)h(5.9,)g Fv(|)-19 b Fx([)o Fw(x)452 1414 y Fu(1)467 1408 y Fw(;)7 b(x)510 1414 y Fu(2)528 1408 y Fw(;)g(x)571 1414 y Fu(3)589 1408 y Fx(])13 b(has)h(dimension)f(ord)p Fw(S)970 1414 y Fu(4)1000 1408 y Fx(=)f(24.)303 1467 y(Let)k Fw(\033)g Fx(=)g(\(1243\))f(in)g Fw(S)673 1473 y Fu(4)692 1467 y Fx(.)25 b(Then)16 b Fw(\033)q(s)883 1473 y Fu(1)902 1467 y Fw(\033)927 1452 y Fn(\000)p Fu(1)987 1467 y Fx(=)f Fw(s)1053 1473 y Fu(4)1073 1467 y Fw(;)7 b(\033)q(s)1136 1473 y Fu(2)1154 1467 y Fw(\033)1179 1452 y Fn(\000)p Fu(1)1239 1467 y Fx(=)15 b Fw(s)1305 1473 y Fu(5)1324 1467 y Fw(;)7 b(\033)q(s)1387 1473 y Fu(3)1406 1467 y Fw(\033)1431 1452 y Fn(\000)p Fu(1)1491 1467 y Fx(=)15 b Fw(s)1557 1473 y Fu(6)1576 1467 y Fx(.)25 b(Hence)228 1527 y(the)19 b(action)e(of)h Fw(\033)h Fx(de\014nes)h(an)e (algebra)f(isomorphism)e Fw(\033)q Ft(\001)k Fx(:)f Fv(|)-19 b Fx([)o Fw(x)1273 1533 y Fu(1)1289 1527 y Fw(;)7 b(x)1332 1533 y Fu(2)1350 1527 y Fw(;)g(x)1393 1533 y Fu(3)1410 1527 y Fx(])18 b Ft(!)h Fv(|)-19 b Fx([)o Fw(x)1562 1533 y Fu(4)1578 1527 y Fw(;)7 b(x)1621 1533 y Fu(5)1638 1527 y Fw(;)g(x)1681 1533 y Fu(6)1699 1527 y Fx(].)228 1587 y(Hence)13 b(dim)o Fo(B)p Fx(\()p Fw(V)496 1593 y Fu(4)514 1587 y Fx(\))f Ft(\024)g Fx(24)628 1572 y Fu(2)646 1587 y Fx(.)17 b(Using)12 b(the)h(metho)q(d)e(of)h(the)h(pro)q(of)e(of)h (Lemma)d(5.6,)i(it)h(is)g(p)q(ossible)228 1647 y(to)17 b(sho)o(w)h(that)g(dim)n Fo(B)p Fx(\()p Fw(V)629 1653 y Fu(4)648 1647 y Fx(\))g(=)g(24)774 1632 y Fu(2)792 1647 y Fx(.)30 b(On)18 b(the)g(other)h(hand)e(it)h(follo)o(ws)e(from)g (the)i(diamond)228 1706 y(lemma)11 b(that)j(the)g(algebra)f(with)h(the) g(ab)q(o)o(v)o(e)g(realations)g(also)f(has)h(dimension)e(24)1537 1691 y Fu(2)1555 1706 y Fx(.)303 1766 y(The)20 b(longest)g(elemen)o(t)f (is)h Fw(w)778 1772 y Fu(0)818 1766 y Fx(=)i Fw(s)891 1772 y Fu(1)910 1766 y Fw(s)929 1772 y Fu(2)948 1766 y Fw(s)967 1772 y Fu(1)986 1766 y Fw(s)1005 1772 y Fu(3)1024 1766 y Fw(s)1043 1772 y Fu(2)1063 1766 y Fw(s)1082 1772 y Fu(1)1101 1766 y Fx(.)36 b(Since)20 b Fw(\033)q(w)1318 1772 y Fu(0)1337 1766 y Fw(\033)1362 1751 y Fn(\000)p Fu(1)1428 1766 y Fx(=)i Fw(s)1501 1772 y Fu(4)1520 1766 y Fw(s)1539 1772 y Fu(5)1558 1766 y Fw(s)1577 1772 y Fu(4)1596 1766 y Fw(s)1615 1772 y Fu(6)1635 1766 y Fw(s)1654 1772 y Fu(5)1673 1766 y Fw(s)1692 1772 y Fu(4)1711 1766 y Fx(,)228 1826 y Fw(x)252 1832 y Fu(1)270 1826 y Fw(x)294 1832 y Fu(2)312 1826 y Fw(x)336 1832 y Fu(1)355 1826 y Fw(x)379 1832 y Fu(3)397 1826 y Fw(x)421 1832 y Fu(2)440 1826 y Fw(x)464 1832 y Fu(1)482 1826 y Fw(x)506 1832 y Fu(4)524 1826 y Fw(x)548 1832 y Fu(5)567 1826 y Fw(x)591 1832 y Fu(4)609 1826 y Fw(x)633 1832 y Fu(6)651 1826 y Fw(x)675 1832 y Fu(5)694 1826 y Fw(x)718 1832 y Fu(4)752 1826 y Fx(is)16 b(a)g(non-zero)h(monomi)o(al)c(of)j(maxim)o(al)c (length,)17 b(hence)g(an)228 1886 y(in)o(tegral.)p 1692 1886 2 29 v 1694 1859 25 2 v 1694 1886 V 1719 1886 2 29 v 303 1967 a(According)11 b(to)h([)p Fy(FK)p Fx(,)f(\(2.8\)])f(also) h Ft(E)859 1973 y Fu(5)889 1967 y Fx(is)g(\014nite-dimensional.)k(Ho)o (w)o(ev)o(er)d(w)o(e)f(do)g(not)h(kno)o(w)228 2017 y(whether)j Ft(E)409 2023 y Fu(5)439 2017 y Fx(=)d Fo(B)p Fx(\()p Fw(V)560 2023 y Fu(5)579 2017 y Fx(\).)303 2110 y Fg(Example)17 b Fx(6.5)p Fg(.)j Fx(Assume)11 b(c)o(har\()p Fv(|)-18 b Fx(\))8 b Ft(6)p Fx(=)k(2.)17 b(W)m(e)11 b(consider)i(the)f(dihedral) f(group)h Fw(D)1582 2116 y Fu(4)1612 2110 y Fx(in)f(Ex-)228 2170 y(ample)g(5.4.)17 b(Then)c Fw(T)18 b Fx(=)12 b Ft(f)p Fw(t)661 2176 y Fu(1)679 2170 y Fw(;)7 b(:)g(:)g(:)e(;)i(t)787 2176 y Fu(4)805 2170 y Ft(g)p Fx(.)17 b(Let)d Fw(\037)f Fx(b)q(e)g(de\014ned)h(b)o(y)g(\(5.5\))o(,)f(and)g Fw(V)21 b Fx(=)11 b Fw(V)f Fx(\()p Fw(D)1585 2176 y Fu(4)1604 2170 y Fw(;)d(T)s(;)g(\037)p Fx(\).)228 2229 y(Then)18 b Fw(R)g Fx(=)g Fo(B)p Fx(\()p Fw(V)9 b Fx(\))18 b(is)g(generated)h(b)o (y)e(the)i(elemen)o(ts)e Fw(x)1133 2235 y Fs(i)1165 2229 y Fx(=)h Fw(x)1239 2235 y Fs(t)1252 2239 y Fi(i)1266 2229 y Fw(;)7 b Fx(1)18 b Ft(\024)g Fw(i)g Ft(\024)g Fx(4)g(and)f(de\014ning)228 2289 y(relations)c(are)i(the)f(quadratic)g (relations)g Fw(x)916 2274 y Fu(2)916 2300 y Fs(i)946 2289 y Fx(=)d(0)p Fw(;)c Fx(1)k Ft(\024)h Fw(i)g Ft(\024)f Fx(4)j(and)953 2371 y Fw(x)977 2377 y Fu(1)995 2371 y Fw(x)1019 2377 y Fu(3)1047 2371 y Fx(+)c Fw(x)1113 2377 y Fu(3)1131 2371 y Fw(x)1155 2377 y Fu(1)1185 2371 y Fx(=)i(0)p Fw(;)953 2443 y(x)977 2449 y Fu(2)995 2443 y Fw(x)1019 2449 y Fu(4)1047 2443 y Fx(+)e Fw(x)1113 2449 y Fu(4)1131 2443 y Fw(x)1155 2449 y Fu(2)1185 2443 y Fx(=)i(0)p Fw(;)682 2516 y(x)706 2522 y Fu(1)725 2516 y Fw(x)749 2522 y Fu(2)776 2516 y Fx(+)e Fw(x)842 2522 y Fu(2)860 2516 y Fw(x)884 2522 y Fu(3)912 2516 y Fx(+)f Fw(x)977 2522 y Fu(3)995 2516 y Fw(x)1019 2522 y Fu(4)1047 2516 y Fx(+)h Fw(x)1113 2522 y Fu(4)1131 2516 y Fw(x)1155 2522 y Fu(1)1185 2516 y Fx(=)i(0)p Fw(;)682 2588 y(x)706 2594 y Fu(1)725 2588 y Fw(x)749 2594 y Fu(4)776 2588 y Fx(+)e Fw(x)842 2594 y Fu(4)860 2588 y Fw(x)884 2594 y Fu(3)912 2588 y Fx(+)f Fw(x)977 2594 y Fu(3)995 2588 y Fw(x)1019 2594 y Fu(2)1047 2588 y Fx(+)h Fw(x)1113 2594 y Fu(2)1131 2588 y Fw(x)1155 2594 y Fu(1)1185 2588 y Fx(=)i(0)p Fw(;)p eop %%Page: 21 21 21 20 bop 355 113 a Fu(POINTED)15 b(INDECOMPOSABLE)g(HOPF)h(ALGEBRAS)d (O)o(VER)i(CO)o(XETER)g(GR)o(OUPS)95 b(21)228 213 y Fx(and)14 b(the)g(relations)g(of)f(degree)i(4)631 296 y Fw(xy)q(xy)c Fx(+)e Fw(y)q(xy)q(x)k Fx(=)f(0)p Fw(;)48 b(uv)q(uv)11 b Fx(+)e Fw(v)q(uv)q(u)j Fx(=)g(0)p Fw(;)228 380 y Fx(where)j Fw(x)c Fx(=)h Fw(x)451 386 y Fu(1)478 380 y Fx(+)e Fw(x)544 386 y Fu(3)562 380 y Fw(;)d(y)13 b Fx(=)f Fw(x)682 386 y Fu(2)710 380 y Ft(\000)d Fw(x)775 386 y Fu(4)793 380 y Fw(;)e(u)k Fx(=)h Fw(x)915 386 y Fu(1)943 380 y Ft(\000)d Fw(x)1008 386 y Fu(3)1026 380 y Fw(;)e(v)13 b Fx(=)f Fw(x)1146 386 y Fu(2)1173 380 y Fx(+)e Fw(x)1239 386 y Fu(4)1257 380 y Fx(.)303 439 y(The)17 b(dimension)e(of)i Fw(R)g Fx(is)f(64,)h(and)g(the)h(in)o(tegral)e(can)h(b)q(e)h(describ)q (ed)h(in)d(terms)h(of)g(the)228 499 y(longest)d(elemen)o(t)f(of)g(the)i (Co)o(xeter)f(group)g Fw(D)947 505 y Fu(4)966 499 y Fx(.)303 593 y Fg(Pr)o(oof.)21 b Fx(These)c(relations)e(are)g(part)h(of)e(the)i (relations)f(of)f(Theorem)h(5.7.)21 b(Hence)16 b Fw(R)f Fx(is)228 653 y(an)e(epimorphic)g(image)f(of)h(the)i(algebra)892 643 y Fl(e)884 653 y Fw(R)f Fx(generated)h(b)o(y)f Fw(x;)7 b(y)q(;)g(u;)g(v)14 b Fx(with)f(the)i(relations)631 737 y Fw(xy)q(xy)c Fx(+)e Fw(y)q(xy)q(x)k Fx(=)f(0)p Fw(;)48 b(uv)q(uv)11 b Fx(+)e Fw(v)q(uv)q(u)j Fx(=)g(0)p Fw(;)228 820 y Fx(and)e Fw(x)329 805 y Fu(2)358 820 y Fx(=)i(0)p Fw(;)7 b(y)463 805 y Fu(2)493 820 y Fx(=)12 b(0)p Fw(;)7 b(u)601 805 y Fu(2)630 820 y Fx(=)12 b(0)p Fw(;)7 b(v)735 805 y Fu(2)765 820 y Fx(=)12 b(0)p Fw(;)7 b(xu)q Fx(+)q Fw(ux)j Fx(=)i(0)p Fw(;)7 b(xv)s Fx(+)q Fw(v)q(x)k Fx(=)h(0)p Fw(;)7 b(y)q(u)q Ft(\000)q Fw(uy)14 b Fx(=)e(0)p Fw(;)7 b(y)q(v)s Fx(+)q Fw(v)q(y)14 b Fx(=)e(0,)228 880 y(whic)o(h)h(follo)o (w)e(from)g(the)i(quadratic)g(relations)g(ab)q(o)o(v)o(e.)k(Using)c (Lemma)d(5.6)i(it)h(is)g(easy)g(to)g(see)228 940 y(that)h(dim)395 929 y Fl(e)387 940 y Fw(R)d Fx(=)h(64.)303 999 y(On)22 b(the)h(other)h(hand,)g(dim)n Fv(|)-19 b Fx([)o Fw(x)834 1005 y Fu(1)850 999 y Fw(;)7 b(x)893 1005 y Fu(2)911 999 y Fx(])25 b(=)h(ord)p Fw(D)1100 1005 y Fu(4)1145 999 y Fx(=)g(8,)e(b)o(y)f(Corollary)e(5.9.)43 b(Since)228 1059 y Fw(t)243 1065 y Fu(1)261 1059 y Fw(t)276 1065 y Fu(2)295 1059 y Fw(t)310 1065 y Fu(1)329 1059 y Fw(t)344 1065 y Fu(2)362 1059 y Fw(t)377 1065 y Fu(1)418 1059 y Fx(=)22 b Fw(t)487 1065 y Fu(3)506 1059 y Fx(,)f(and)f Fw(t)641 1065 y Fu(1)660 1059 y Fw(t)675 1065 y Fu(2)693 1059 y Fw(t)708 1065 y Fu(2)727 1059 y Fw(t)742 1065 y Fu(2)760 1059 y Fw(t)775 1065 y Fu(1)816 1059 y Fx(=)i Fw(t)885 1065 y Fu(4)904 1059 y Fx(,)f(the)g(action)f(of)f Fw(t)1214 1065 y Fu(1)1233 1059 y Fw(t)1248 1065 y Fu(2)1287 1059 y Fx(de\014nes)i(an)f(isomorphism)228 1119 y Fv(|)-19 b Fx([)o Fw(x)289 1125 y Fu(1)305 1119 y Fw(;)7 b(x)348 1125 y Fu(2)366 1119 y Fx(])k Ft(!)g Fv(|)-19 b Fx([)o Fw(x)503 1125 y Fu(3)519 1119 y Fw(;)7 b(x)562 1125 y Fu(4)579 1119 y Fx(].)17 b(Again)12 b(one)h(can)g(sho)o(w)g(similarly)c (to)k(the)g(pro)q(of)f(of)g(Lemma)e(5.6)i(that)228 1179 y Fv(|)-19 b Fx([)o Fw(x)289 1185 y Fu(1)305 1179 y Fw(;)7 b(x)348 1185 y Fu(2)366 1179 y Fx(])g Ft(\001)g Fv(|)-18 b Fx([)o Fw(x)466 1185 y Fu(3)481 1179 y Fw(;)7 b(x)524 1185 y Fu(4)542 1179 y Fx(])13 b(has)g(dimension)f(8)857 1164 y Fu(2)875 1179 y Fx(.)18 b(Hence)d Fw(R)c Fx(=)h Fv(|)-19 b Fx([)p Fw(x)1176 1185 y Fu(1)1192 1179 y Fw(;)7 b(x)1235 1185 y Fu(2)1253 1179 y Fx(])g Ft(\001)h Fv(|)-19 b Fx([)o Fw(x)1353 1185 y Fu(3)1369 1179 y Fw(;)7 b(x)1412 1185 y Fu(4)1429 1179 y Fx(],)13 b(and)g(w)o(e)g(ha)o(v)o(e)g(a)228 1239 y(monomi)o(al)d(basis)k(in)g(the)g(elemen)o(ts)g Fw(x)835 1245 y Fs(i)848 1239 y Fx(.)303 1298 y(The)h(longest)h(elemen) o(t)f(in)g Fw(D)771 1304 y Fu(4)805 1298 y Fx(is)h Fw(t)864 1304 y Fu(1)882 1298 y Fw(t)897 1304 y Fu(2)916 1298 y Fw(t)931 1304 y Fu(1)950 1298 y Fw(t)965 1304 y Fu(2)983 1298 y Fx(.)23 b(Therefore)17 b Fw(x)1232 1304 y Fu(1)1250 1298 y Fw(x)1274 1304 y Fu(2)1292 1298 y Fw(x)1316 1304 y Fu(1)1335 1298 y Fw(x)1359 1304 y Fu(2)1377 1298 y Fw(x)1401 1304 y Fu(3)1419 1298 y Fw(x)1443 1304 y Fu(4)1462 1298 y Fw(x)1486 1304 y Fu(3)1504 1298 y Fw(x)1528 1304 y Fu(4)1562 1298 y Fx(is)e(a)g(non-)228 1358 y(zero)g(in)o(tegral.)p 1692 1358 2 29 v 1694 1332 25 2 v 1694 1358 V 1719 1358 2 29 v 303 1442 a(In)g(an)h(earlier)g(v)o(ersion)g(of)g(this)g(pap)q (er)g(w)o(e)h(sho)o(w)o(ed)f(dim)n Fw(R)f Fx(=)g(64)h(in)f(the)i(last)e (example)228 1492 y(in)f(a)h(di\013eren)o(t)h(w)o(a)o(y)m(.)21 b(In)15 b(addition)e(to)i(the)h(quadratic)f(relations,)g(w)o(e)g(pro)o (v)o(ed)g(the)h(follo)o(wing)228 1542 y(relations:)i(If)13 b Fw(y)474 1548 y Fu(1)505 1542 y Fx(=)e Fw(x)572 1548 y Fu(1)591 1542 y Fw(x)615 1548 y Fu(2)642 1542 y Fx(+)f Fw(x)708 1548 y Fu(4)726 1542 y Fw(x)750 1548 y Fu(1)769 1542 y Fw(;)d(y)808 1548 y Fu(2)838 1542 y Fx(=)k Fw(x)905 1548 y Fu(1)924 1542 y Fw(x)948 1548 y Fu(4)975 1542 y Fx(+)f Fw(x)1041 1548 y Fu(2)1059 1542 y Fw(x)1083 1548 y Fu(1)1101 1542 y Fx(,)k(then)693 1615 y Fw(y)714 1598 y Fu(2)713 1625 y(1)745 1615 y Fx(=)e(0)p Fw(;)7 b(y)850 1598 y Fu(2)849 1625 y(2)880 1615 y Fx(=)12 b(0)p Fw(;)7 b(y)984 1621 y Fu(1)1002 1615 y Fw(y)1022 1621 y Fu(2)1050 1615 y Fx(+)j Fw(y)1112 1621 y Fu(2)1131 1615 y Fw(y)1151 1621 y Fu(1)1181 1615 y Fx(=)i(0)p Fw(:)228 1689 y Fx(It)f(is)h(not)f(di\016cult)g(to)h(see)g(that)g(these)h (relations)e(de\014ne)i(an)e(algebra)g(of)g(dimension)f(64.)17 b(Then)228 1739 y(Gra)q(~)-22 b(na)16 b([)p Fy(G2)n Fx(,)h(5.2.1])d (noted)j(that)f(the)h(braiding)f(in)g(the)h(basis)f Fw(x;)7 b(y)q(;)g(u;)g(v)17 b Fx(of)e Fw(V)26 b Fx(reduces)19 b(the)228 1788 y(computation)12 b(of)h Fo(B)p Fx(\()p Fw(V)d Fx(\))j(to)h(the)h(case)g(of)e Fw(A)922 1794 y Fu(2)954 1788 y Fx(in)h(Lemma)d(5.6.)303 1882 y Fg(Remark)16 b Fx(6.6)p Fg(.)k Fx(The)i(examples)d(w)o(e)i(ha)o(v)o(e)g(constructed) i(sho)o(w)e(that)g(the)g(symmetric)228 1942 y(groups)14 b Fw(S)387 1948 y Fu(3)406 1942 y Fw(;)7 b(S)450 1948 y Fu(4)483 1942 y Fx(and)15 b Fw(S)590 1948 y Fu(5)623 1942 y Fx(and)f(the)h(dihedral)f(group)h Fw(D)1091 1948 y Fu(4)1124 1942 y Fx(o)q(ccur)g(as)g(the)g(groups)f(of)g(group-lik)o (e)228 2002 y(elemen)o(ts)h(of)g(\014nite-dimensional)e(p)q(oin)o(ted)i (and)g(link-indecomp)q(osable)e(Hopf)i(algebras.)22 b(In)228 2062 y(the)12 b(other)g(examples)e(of)h(link-indecomp)q(osable)e(Hopf)i (algebras)g(o)o(v)o(er)g(\014nite)h(Co)o(xeter)g(groups)228 2121 y(w)o(e)i(do)g(not)f(kno)o(w)h(whether)h(the)f(algebras)g Fo(B)p Fx(\()p Fw(V)c Fx(\))k(are)g(\014nite-dimensional.)863 2234 y Fy(References)228 2309 y Fr([A)o(G])j Fd(N.)h(Andr)o (uskiewitsch)h(and)g(M.)f(Gra)889 2306 y(~)888 2309 y(na)p Fr(,)g Fq(Br)n(aide)n(d)i(Hopf)f(algebr)n(as)h(over)f(non)g(ab)n(elian) h(gr)n(oups)p Fr(,)300 2359 y Fq(Bol.A)n(c)n(ad.Ciencias)15 b(\(C\023)-18 b(or)n(dob)n(a\))12 b Fc(63)h Fr(\(1999\),)c(45-78.)228 2409 y([AS1])17 b Fd(N.)12 b(Andr)o(uskiewitsch)g(and)h(H.-J.)e (Schneider)p Fr(,)f Fq(Lifting)j(of)g(quantum)g(line)n(ar)h(sp)n(ac)n (es)f(and)g(p)n(ointe)n(d)300 2458 y(Hopf)h(algebr)n(as)g(of)f(or)n (der)h Fb(p)676 2447 y FB(3)693 2458 y Fr(,)e Fq(J.)g(A)o(lgebr)n(a)i Fc(209)f Fr(\(1998\),)c(658-691.)228 2508 y([AS2])17 b Fd(N.)11 b(Andr)o(uskiewitsch)h(and)g(H.-J.)f(Schneider)p Fr(,)f Fq(Finite)j(quantum)f(gr)n(oups)h(and)g(Cartan)f(matric)n(es)p Fr(,)300 2558 y Fq(A)n(dv.)i(in)f(Math.)p Fr(,)f(to)f(app)q(ear.)228 2608 y([B])27 b Fd(N.)13 b(Bourbaki)p Fr(,)f Fq(Gr)n(oup)n(es)h(et)g (alg)q Fr(\022)-19 b Fb(e)q Fq(br)n(es)14 b(de)f(Lie)g(Chap.)g(IV,V,VI) p Fr(,)f(Hermann,)e(P)o(aris,)h(1968.)p eop %%Page: 22 22 22 21 bop 228 119 a Fu(22)245 b(ALEXANDER)13 b(MILINSKI)h(AND)g(HANS-J) 1117 112 y(\177)1113 119 y(UR)o(GEN)e(SCHNEIDER)228 213 y Fr([CR])17 b Fd(C.)g(Cibils)h(and)g(M.)f(R)o(osso)p Fr(,)g Fq(A)o(lg)q Fr(\022)-19 b Fb(e)q Fq(br)n(es)18 b(des)f(chemins)h(quantiques)p Fr(,)f Fq(A)n(dv.)i(in)e(Math.)g Fc(125)g Fr(\(1997\),)300 262 y(171-199.)228 312 y([FMS])g Fd(D.)12 b(Fischman,)g(S.)h(Montgomer)n(y,)f(H.-J.)f(Schneider)p Fr(,)g Fq(F)m(r)n(ob)n(enius)j(extensions)g(of)e(sub)n(algebr)n(as)j (of)300 362 y(Hopf)f(algebr)n(as)p Fr(,)f Fq(T)m(r)n(ans.AMS)f Fc(349)h Fr(\(1997\),)c(4857-4895.)228 412 y([FK])17 b Fd(S.)12 b(F)o(omin)h(and)f(A.)g(N.)g(Kirillo)o(v)p Fr(,)g Fq(Quadr)n(atic)h(algebr)n(as,)h(Dunkl)f(elements,)g(and)g (Schub)n(ert)h(c)n(alculus)p Fr(,)300 462 y Fq(Pr)n(o)n(gr)n(ess)g(in)f (Mathematics)g(172)p Fr(,)f(Birkh\177)-18 b(auser,)9 b(Basel,)i(1999,)f(146-182.)228 511 y([FP])17 b Fd(S.)g(F)o(omin)g(and) g(C.)f(Pr)o(ocesi)p Fr(,)g Fq(Fib)n(er)n(e)n(d)i(quadr)n(atic)h(Hopf)e (algebr)n(as)i(r)n(elate)n(d)f(to)f(Schub)n(ert)h(c)n(alculus)p Fr(,)300 561 y(preprin)o(t,)9 b(1998.)228 611 y([G1])16 b Fd(M.)c(Gra)435 608 y(~)434 611 y(na)p Fr(,)h Fq(A)h(fr)n(e)n(eness)f (the)n(or)n(em)h(for)f(Nichols)h(algebr)n(as)p Fr(,)f Fq(J.)f(A)o(lgebr)n(a)p Fr(,)i(to)d(app)q(ear.)228 661 y([G2])16 b Fd(M.)c(Gra)435 658 y(~)434 661 y(na)p Fr(,)h Fq(On)g(Nichols)h(algebr)n(as)g(of)f(low)g(dimension)p Fr(,)f Fq(Contemp.)h(Math.)p Fr(,)f(this)f(v)o(olume.)228 711 y([M1])17 b Fd(S.)f(Montgomer)n(y)p Fr(,)g Fq(Hopf)h(A)o(lgebr)n (as)h(and)f(Their)g(A)n(ctions)g(on)g(R)o(ings)p Fr(,)e(CBMS)i(Lecture) d(Notes)h Fc(82)p Fr(,)300 761 y(AMS,)d(1993.)228 810 y([M2])17 b Fd(S.)12 b(Montgomer)n(y)p Fr(,)f Fq(Inde)n(c)n(omp)n (osable)k(c)n(o)n(algebr)n(as,)f(simple)f(c)n(omo)n(dules,)h(and)e(p)n (ointe)n(d)i(Hopf)f(algebr)n(as)p Fr(,)300 860 y Fq(Pr)n(o)n(c.)h(AMS)e Fc(123)h Fr(\(1995\),)d(2343-2351.)228 910 y([N])26 b Fd(W.)12 b(Nichols)p Fr(,)g Fq(Bialgebr)n(as)i(of)f(typ)n(e)h(one)p Fr(,)e Fq(Comm.)h(A)o(lgebr)n(a)h Fc(6)e Fr(\(1978\),)e(1521-1552.)228 960 y([R])26 b Fd(D.)9 b(Radf)o(ord)p Fr(,)g Fq(The)h(structur)n(e)h (of)g(Hopf)g(algebr)n(as)h(with)e(a)g(pr)n(oje)n(ction)p Fr(,)h Fq(J.)f(A)o(lgebr)n(a)h Fc(92)f Fr(\(1985\),)d(322-347.)228 1010 y([Ro])17 b Fd(M.)12 b(R)o(osso)p Fr(,)f Fq(Quantum)j(algebr)n(as) g(and)g(quantum)f(shu\017es)p Fr(,)f Fq(Invent.math.)g Fc(133)h Fr(\(1998\),)c(399-416.)228 1059 y([Sb])16 b Fd(P.)c(Scha)o(uenbur)o(g)p Fr(,)i Fq(A)f(Char)n(acterization)h(of)e (the)i(Bor)n(el-like)g(Sub)n(algebr)n(as)h(of)e(Quantum)g(Enveloping) 300 1109 y(A)o(lgebr)n(as)p Fr(,)h Fq(Comm.)e(A)o(lgebr)n(a)i Fc(24)f Fr(\(1996\),)c(2811-2823.)228 1159 y([W])17 b Fd(S.)g(Westreich)p Fr(,)d Fq(Quasitriangular)k(Hopf)g(algebr)n(as)g (whose)f(gr)n(oup-like)h(elements)g(form)f(an)g(ab)n(elian)300 1209 y(gr)n(oup)p Fr(,)c Fq(Pr)n(o)n(c.)h(AMS)e Fc(124)h Fr(\(1996\),)c(1023-1026.)228 1259 y([W)m(o])17 b Fd(S.)c(W)o(or)o(ono) o(wicz)p Fr(,)f Fq(Di\013er)n(ential)i(c)n(alculus)g(on)g(c)n(omp)n (act)h(matrix)f(pseudo)n(gr)n(oups)g(\(quantum)g(gr)n(oups\))p Fr(,)300 1308 y Fq(Comm.)f(Math.)g(Phys.)f Fc(122)h Fr(\(1989\),)c (125-170.)303 1402 y Fd(Ma)n(thema)n(tisches)e(Institut,)j(Universit) 903 1399 y(\177)902 1402 y(at)17 b(M)993 1399 y(\177)992 1402 y(unchen,)10 b(Theresienstra\031e)e(39,)i(D-80333)f(M)1663 1399 y(\177)1662 1402 y(unchen,)228 1452 y(Germany)303 1502 y Fq(E-mail)g(addr)n(ess)s Fr(:)14 b Fa(milinski@)o(rz.)o(mat)o (he)o(mat)o(ik.)o(uni)o(-mu)o(en)o(che)o(n.d)o(e,)g(hanssch@rz.)o(mat)o (hem)o(at)o(ik.)o(uni)o(-mu)o(enc)o(he)o(n.d)o(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF