%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Alles.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Alles %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.06.12:1321 %%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 0 0]N /nn 0 N 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b(and)h(doubles)g(of)f(quan)n(tum)g(group)r(oids)g (\()p Fw(\002)1315 577 y Fv(R)1343 570 y Fx(-Hopf)h(algebras\))788 678 y Fu(P)o(eter)c(Sc)o(hauen)o(burg)377 791 y Ft(Abstra)o(ct.)21 b Fs(W)m(e)14 b(generalize)e(the)h(construction)e(of)j(Drinfeld's)e (quan)o(tum)g(double)g(to)377 832 y(the)i(theory)f(of)i(quan)o(tum)d (group)q(oids,)h(or,)i(more)f(precisely)m(,)f Fr(\002)1234 858 y Fq(R)1259 832 y Fs(-bialgebras)f(as)i(in)o(tro-)377 874 y(duced)c(b)o(y)h(T)m(ak)o(euc)o(hi.)813 1068 y Fp(1.)24 b(In)o(tro)q(duction)303 1142 y Fo(\002)335 1148 y Fn(R)362 1142 y Fm(-bialgebras)14 b(w)o(ere)i(in)o(tro)q(duced)f(b)o(y)g(M.)f(T) m(ak)o(euc)o(hi)g([)p Fp(19)p Fm(])g(as)h(a)f(generalization)g(of)h (the)228 1192 y(notion)e(of)h(bialgebra.)19 b(The)c(de\014nition)f(of)g (a)g Fo(\002)989 1198 y Fn(R)1016 1192 y Fm(-bialgebra)g(in)o(v)o(olv)o (es)f(a)h Fl(k)q Fm(-algebra)g Fl(R)g Fm(\(o)o(v)o(er)228 1242 y(some)i(comm)o(utativ)o(e)e(base)k(ring)f Fl(k)q Fm(\))g(and)g(usual)g Fl(k)q 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(seems)g(to)f(kno)o(w)h(T)m(ak)o(euc)o(hi's)f(previous)h(w)o(ork.\))k (A)c(predecessor)228 1640 y(in)f(this)h(line)g(is)f(Maltsiniotis)g([)p Fp(7)o Fm(])h(where)h(the)f(algebra)g Fl(R)f Fm(is)h(comm)o(utativ)o (e.)303 1690 y(The)i(motiv)n(atio)o(n)d(in)i(Lu's)h(and)f(Xu's)h(pap)q (ers)h(is)e(that)h(the)g(notion)f(of)g(a)h Fo(\002)1515 1696 y Fn(R)1542 1690 y Fm(-bialgebra)228 1740 y(\(p)q(erhaps)c(equipp) q(ed)f(with)f(some)g(sort)h(of)f(an)o(tip)q(o)q(de\))h(is)f(related)h (to)g(the)g(notion)f(of)g(a)g(group)q(oid)228 1790 y(in)k(m)o(uc)o(h)f (the)i(same)f(manner)g(as)h(Hopf)f(algebras)g(relate)h(to)g(groups.)20 b(F)m(or)14 b(this)h(reason)g(\(cer-)228 1840 y(tain\))e Fo(\002)362 1846 y Fn(R)389 1840 y Fm(-bialgebras)g(could)h(p)q(erhaps) h(righ)o(tly)e(b)q(e)h(called)g(quan)o(tum)e(group)q(oids.)303 1890 y(Recall)d(that)g(a)h(group)q(oid)f(consists)i(of)e(a)h(set)g Fl(X)k Fm(of)9 b(ob)r(jects,)i(a)f(set)g Fl(Y)20 b Fm(of)9 b(arro)o(ws,)h(t)o(w)o(o)f(maps)228 1939 y(\(source)18 b(and)e(target\))i Fl(s;)7 b(t)e Fm(:)13 b Fl(Y)26 b Fo(!)15 b Fl(X)s Fm(,)j(and)e(a)g(partially)g(de\014ned)h(pro)q(duct)h (from)d Fo(f)p Fm(\()p Fl(y)q(;)7 b(y)1650 1924 y Fk(0)1662 1939 y Fm(\))17 b Fo(2)228 1989 y Fl(Y)22 b Fo(\002)13 b Fl(Y)d Fo(j)p Fl(s)p Fm(\()p Fl(y)q Fm(\))22 b(=)f Fl(t)p Fm(\()p Fl(y)564 1974 y Fk(0)576 1989 y Fm(\))p Fo(g)f Fm(to)f Fl(Y)29 b Fm(\(satisfying)18 b(axioms)g(that)h(w)o(e'll) f(skip)i(altogether\).)35 b(If)19 b(w)o(e)228 2039 y(assume)11 b(that)g(w)o(e)h(are)g(talking)e(ab)q(out)h(an)h(a\016ne)f(group)q(oid) g(sc)o(heme)g(\(i.)g(e.)g(a)h(situation)e(as)i(just)228 2089 y(describ)q(ed,)i(but)f(with)f(all)g(sets)i(and)e(maps)g(replaced) i(b)o(y)e(a\016ne)h(sc)o(hemes)g(and)f(morphisms)f(of)228 2139 y(a\016ne)f(sc)o(hemes\),)h(this)f(corresp)q(onds)i(to)e(the)h (follo)o(wing)d(setup:)17 b(One)11 b(is)f(giv)o(en)f(a)h(comm)o(utativ) o(e)228 2188 y(algebra)17 b Fl(R)g Fm(\(the)i(algebra)e(of)g(functions) h(on)f(the)i(space)f(of)f(ob)r(jects\))i(and)f(a)f(comm)o(utativ)o(e) 228 2238 y(algebra)c Fl(L)h Fm(\(the)h(algebra)e(of)h(functions)g(on)f (the)i(space)g(of)e(arro)o(ws\),)h(and)g(t)o(w)o(o)f(algebra)g(maps)228 2288 y Fl(\013;)7 b(!)f Fm(:)13 b Fl(R)18 b Fo(!)e Fl(L)p Fm(;)j(w)o(e)f(write)f Fl(\013)p Fm(\()p Fl(r)q Fm(\))g(=)h Fl(r)g Fm(and)f Fl(!)q Fm(\()p Fl(r)q Fm(\))h(=)p 1086 2265 20 2 v 18 w Fl(r)q Fm(.)28 b(Comp)q(osition)15 b(corresp)q(onds)k (to)e(an)228 2338 y(algebra)c(map)309 2407 y(\001)5 b(:)13 b Fl(L)e Fo(!)g Fl(L)f Fo(\005)f Fl(L)j Fm(:=)f(\()645 2413 y Fn(!)669 2407 y Fl(L)p Fm(\))h Fo(\012)729 2437 y Fn(R)769 2407 y Fm(\()785 2413 y Fn(\013)809 2407 y Fl(L)p Fm(\))g(=)g Fl(L)d Fo(\012)h Fl(L=)p Fo(h)p 1053 2384 V Fl(r)q(`)f Fo(\012)h Fl(`)1158 2389 y Fk(0)1179 2407 y Fo(\000)f Fl(`)h Fo(\012)f Fl(r)q(`)1325 2389 y Fk(0)1337 2407 y Fo(j)p Fl(r)j Fo(2)f Fl(R;)c(`;)g(`)1523 2389 y Fk(0)1546 2407 y Fo(2)k Fl(L)p Fo(i)p Fl(:)p 228 2474 250 2 v 303 2521 a Fs(1991)g Fj(Mathematics)i(Subje)n(ct)h (Classi\014c)n(ation.)k Fs(16W30,18D10,2)o(0L05)o(,22)o(A22.)1378 2601 y Fy(c)1370 2602 y Fi(\015)p Fy(0000)e(\(cop)o(yrigh)o(t)f (holder\))967 2658 y Fh(1)p eop %%Page: 2 2 2 1 bop 228 113 a Fh(2)536 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(When)d(w)o(e)f(try)h(to)f(pass)h(from)e(this)i(setup)g(to)g (noncomm)o(utativ)n(e)d(algebraic)i(geometry)g(or)h(the)228 262 y(notion)g(of)g(a)g(quan)o(tum)g(group)q(oid,)f(w)o(e)i(w)o(ould)f (formally)e(lik)o(e)i(to)h(replace)g(b)q(oth)g Fl(R)g Fm(and)f Fl(L)h Fm(b)o(y)228 312 y(noncomm)o(utativ)n(e)e(algebras.)19 b(It)14 b(turns)h(out)f(that)g(it)g(is)g(reasonable)g(to)g(assume)g (that)g(one)h(of)228 362 y(the)h(t)o(w)o(o)f(maps)f(from)g Fl(R)h Fm(to)g Fl(L)p Fm(,)h(sa)o(y)f Fl(!)q Fm(,)h(is)f(an)g Fg(anti)t Fm(algebra)g(map,)f(and)h(that)h(the)g(images)e(of)228 412 y Fl(\013)i Fm(and)g Fl(!)h Fm(comm)o(ute.)23 b(Ho)o(w)o(ev)o(er,) 17 b(one)f(cannot)h(exp)q(ect)h(that)e Fl(L)11 b Fo(\005)f Fl(L)17 b Fm(de\014ned)g(as)f(ab)q(o)o(v)o(e)g(is)g(a)228 462 y(quotien)o(t)f(algebra)g(of)g Fl(L)c Fo(\012)f Fl(L)p Fm(,)16 b(and)g(th)o(us)g(one)f(cannot)h(de\014ne)h(a)e(com)o (ultiplication)d(on)k Fl(L)f Fm(to)228 511 y(b)q(e)g(an)g(algebra)f (map)f Fl(L)h Fo(!)e Fl(L)e Fo(\005)g Fl(L)p Fm(.)21 b(No)o(w)14 b(on)h(the)g(subspace)i Fl(L)p Fo(\002)1270 517 y Fn(R)1298 511 y Fl(L)c Fo(\032)g Fl(L)d Fo(\005)g Fl(L)15 b Fm(consisting)g(of)228 561 y(those)334 530 y Ff(P)385 561 y Fl(`)402 567 y Fn(i)422 561 y Fo(\012)5 b Fl(`)476 546 y Fk(0)476 572 y Fn(i)502 561 y Fo(2)11 b Fl(L)5 b Fo(\005)g Fl(L)13 b Fm(satisfying)824 530 y Ff(P)875 561 y Fl(`)892 567 y Fn(i)p 906 538 20 2 v 906 561 a Fl(r)6 b Fo(\012)f Fl(`)985 546 y Fk(0)985 572 y Fn(i)1011 561 y Fm(=)1055 530 y Ff(P)1106 561 y Fl(`)1123 567 y Fn(i)1142 561 y Fo(\012)g Fl(`)1196 546 y Fk(0)1196 572 y Fn(i)1211 561 y Fl(r)13 b Fm(for)e(all)g Fl(r)i Fo(2)e Fl(R)g Fm(one)i(do)q(es)f(ha)o(v)o(e)228 611 y(an)g(algebra)g(structure)i(induced)f(b)o(y)f(that)g(of)g Fl(L)6 b Fo(\012)g Fl(L)p Fm(,)13 b(and)f(the)h(de\014nition)f(of)g(a)g Fo(\002)1515 617 y Fn(R)1542 611 y Fm(-bialgebra)228 661 y(sa)o(ys)i(that)g(com)o(ultiplication)c(is)k(an)g(algebra)f(map)f Fl(L)g Fo(!)f Fl(L)p Fo(\002)1202 667 y Fn(R)1229 661 y Fl(L)p Fm(.)303 711 y(Indep)q(enden)o(t)18 b(evidence)h(that)e(the)h (\(quite)f(tec)o(hnical\))g(de\014nition)g(of)g(a)f(bialgebroid)g(is) 228 761 y(the)c(\\righ)o(t")g(one)g(is)g(giv)o(en)f(in)h([)p Fp(14)o Fm(]:)17 b(Recall)11 b(that)h(a)g Fl(k)q Fm(-algebra)f Fl(H)k Fm(is)d(a)g(bialgebra)f(if)g(and)h(only)228 810 y(if)e(the)i(category)496 816 y Fn(H)528 810 y Fo(M)f Fm(of)f(left)h Fl(H)s Fm(-mo)q(dules)f(is)h(a)g(monoidal)d(category)k (in)e(suc)o(h)i(a)f(w)o(a)o(y)f(that)i(the)228 860 y(underlying)j (functor)h(to)g(the)g(category)g(of)f Fl(k)q Fm(-mo)q(dules)g(is)g (monoidal)e(\(i.)i(e.)h(tensor)g(pro)q(duct)228 910 y(preserving\).)21 b(Stated)15 b(more)f(simply)m(,)d(the)16 b(k)o(ey)e(fact)h(ab)q(out)f (a)g(bialgebra)g(is)g(that)h(the)g(tensor)228 960 y(pro)q(duct)d(o)o(v) o(er)g Fl(k)g Fm(of)f(t)o(w)o(o)g(of)g(its)h(mo)q(dules)e(is)i(again)e (a)i(mo)q(dule.)j(This)d(is)f(p)q(erhaps)i(the)f(starting)228 1010 y(p)q(oin)o(t)i(of)g(T)m(annak)n(a)f(dualit)o(y)h(or)g (reconstruction)j(theory)e(for)f(quan)o(tum)f(groups,)i(where)h(one)228 1059 y(reconstructs)j(bialgebras)e(and)f(additional)f(structures)k (\(lik)o(e)d(an)o(tip)q(o)q(des)h(or)g(quasitriangu-)228 1109 y(lar)c(structures\))k(from)12 b(their)j(monoidal)c(mo)q(dule)h (categories)j(and)f(additional)e(structures)17 b(of)228 1159 y(those.)j(In)14 b(m)o(uc)o(h)f(the)i(same)e(w)o(a)o(y)h Fo(\002)810 1165 y Fn(R)837 1159 y Fm(-bialgebras)g(can)g(b)q(e)h(c)o (haracterized)h(b)o(y)e(the)h(fact)f(that)228 1209 y(the)i(category)f (of)g(mo)q(dules)f(o)o(v)o(er)h(a)g Fo(\002)841 1215 y Fn(R)868 1209 y Fm(-bialgebra)f Fl(L)i Fm(is)f(a)g(monoidal)d (category)m(.)22 b(Only)15 b(this)228 1259 y(time)d(the)i(underlying)f (functor)h(to)g(the)g(category)g(of)f Fl(k)q Fm(-mo)q(dules)f(is)i(not) f(monoidal,)d(but)k(one)228 1308 y(has)j(a)f(monoidal)e(underlying)i (functor)i(to)e(the)i(category)f(of)f Fl(R)p Fm(-bimo)q(dules.)25 b(Stated)17 b(more)228 1358 y(simply)m(,)11 b(the)j(tensor)h(pro)q (duct)g(o)o(v)o(er)f Fl(R)f Fm(of)h(t)o(w)o(o)f Fl(L)p Fm(-mo)q(dules)g(is)h(again)e(an)i Fl(L)p Fm(-mo)q(dule.)303 1408 y(F)m(urthermore,)i(the)h(notion)e(of)h(a)g Fo(\002)883 1414 y Fn(R)910 1408 y Fm(-bialgebra)f(con)o(tains)h(that)h(of)f(a)g (face)g(algebra)g(as)228 1458 y(de\014ned)e(b)o(y)e(Ha)o(y)o(ashi)g([)p Fp(3)o Fm(])g(as)h(a)f(sp)q(ecial)h(case)h([)p Fp(15)o Fm(].)j(Close)c(relations)f(to)h(the)g(notions)f(of)g(w)o(eak)228 1508 y(Hopf)h(algebra)g(or)h(quan)o(tum)e(group)q(oid)h(considered)i(b) o(y)f(sev)o(eral)g(authors)g([)p Fp(1)o(,)i(9,)g(8)p Fm(],)d(mainly)228 1558 y(in)g(connection)g(to)g(inclusions)g(of)g(v)o (on)g(Neumann)f(algebras,)h(should)g(b)q(e)h(exp)q(ected)h(but)e(ha)o (v)o(e)228 1607 y(not)h(b)q(een)h(w)o(ork)o(ed)f(out)f(in)h(detail.)303 1657 y(The)f(topic)g(of)f(the)h(presen)o(t)i(pap)q(er)e(are)g Fo(\002)966 1663 y Fn(R)993 1657 y Fm(-bialgebra)f(analogs)g(of)g(some) g(constructions)228 1707 y(and)i(facts)g(w)o(ell-kno)o(wn)e(in)i(the)g (theory)h(of)e(usual)h Fl(k)q Fm(-bialgebras.)303 1757 y(In)e(section)h(3)f(w)o(e)h(will)e(discuss)i(a)f(suitable)g(notion)g (of)g Fo(\002)1189 1763 y Fn(R)1216 1757 y Fm(-Hopf)g(algebra.)17 b(F)m(ormally)10 b(the)228 1807 y(de\014nition)h(will)f(sa)o(y)i(that)f (a)g Fo(\002)708 1813 y Fn(R)735 1807 y Fm(-bialgebra)f(is)i(a)f Fo(\002)1030 1813 y Fn(R)1058 1807 y Fm(-Hopf)g(algebra)g(if)g(and)g (only)g(if)f(it)i(is)f(a)g(left)228 1856 y(Hopf-Galois)f(ob)r(ject)i(o) o(v)o(er)g(itself.)17 b(Although)11 b(this)h(is)g(w)o(ell)f(kno)o(wn)g (to)h(c)o(haracterize)i(ordinary)228 1906 y Fl(k)q Fm(-Hopf)j(algebras) g(among)e(bialgebras,)j(w)o(e)f(will)f(pro)o(vide)i(b)q(etter)h (evidence)g(that)e(it)g(is)h(the)228 1956 y(\\righ)o(t")12 b(de\014nition)h(for)g(the)g Fo(\002)719 1962 y Fn(R)746 1956 y Fm(-bialgebra)f(case:)19 b(W)m(e)13 b(will)e(sho)o(w)i(that)h(a) f Fo(\002)1449 1962 y Fn(R)1476 1956 y Fm(-Hopf)g(algebra)228 2006 y(structure)j(on)e(a)g Fo(\002)531 2012 y Fn(R)558 2006 y Fm(-bialgebra)f Fl(L)h Fm(can)h(b)q(e)f(reconstructed)j(from)12 b(an)i(appropriate)g(prop)q(ert)o(y)228 2056 y(of)f(the)i(monoidal)c (category)j(of)g Fl(L)p Fm(-mo)q(dules.)j(One)e(v)o(ersion)f(of)f(this) h(approac)o(h)g(is)g(kno)o(wn)g(for)228 2106 y(ordinary)j Fl(k)q Fm(-Hopf)f(algebras)i(if)e Fl(k)i Fm(is)g(a)f(\014eld:)25 b(The)18 b(category)f(of)g(\014nite)h(dimensional)d(righ)o(t)228 2155 y Fl(H)s Fm(-como)q(dules)e(o)o(v)o(er)h(a)g Fl(k)q Fm(-bialgebra)f Fl(H)k Fm(is)d(rigid)f(if)h(and)g(only)f(if)g Fl(H)k Fm(is)d(a)g(Hopf)g(algebra)g([)p Fp(21)o Fm(].)228 2205 y(Here)j(a)g(monoidal)c(category)k(is)f(called)g(rigid)g(if)f(one) i(has)f(a)g(w)o(ell-b)q(eha)o(v)o(ed)g(notion)g(of)g(dual)228 2255 y(ob)r(jects)f(within)f(the)g(category)m(.)19 b(No)o(w)14 b(this)g(c)o(haracterization)h(of)e(Hopf)h(algebras)g(cannot)h(b)q(e) 228 2305 y(generalized)e(to)g(the)g Fo(\002)594 2311 y Fn(R)621 2305 y Fm(-bialgebra)e(situation:)17 b(Ev)o(en)c(for)g (ordinary)f(bialgebras)g(o)o(v)o(er)h(a)g(ring)228 2355 y Fl(k)i Fm(rather)f(than)g(a)g(\014eld,)f(one)i(cannot)f(extract)h (enough)f(information)d(from)h(just)i(the)h(\014nitely)228 2404 y(generated)f(pro)r(jectiv)o(e)g(como)q(dules)e(\(or)i(the)f (\014nitely)g(generated)i(pro)r(jectiv)o(e)e(mo)q(dules\),)f(and)228 2454 y(a)h(como)q(dule)f(\(or)h(mo)q(dule\))g(that)g(is)g(not)g (\014nitely)g(generated)i(pro)r(jectiv)o(e)f(o)o(v)o(er)g(the)g(base)g (ring)228 2504 y Fl(k)g Fm(can)g(nev)o(er)h(ha)o(v)o(e)f(a)f(dual)g(in) h(the)g(sense)h(of)f(the)g(de\014nition)f(of)h(a)f(rigid)g(monoidal)e (category)m(.)228 2554 y(Ho)o(w)o(ev)o(er,)16 b(if)f Fl(H)j Fm(is)e(a)g(Hopf)f(algebra)g(o)o(v)o(er)h(a)f(comm)o(utativ)o(e) e(ring)i Fl(k)q Fm(,)h(then)h(for)e(an)o(y)g(t)o(w)o(o)g Fl(H)s Fm(-)228 2604 y(mo)q(dules)10 b Fl(V)r(;)d(W)17 b Fm(one)11 b(can)g(endo)o(w)g(the)g(mo)q(dule)f(Hom)1058 2610 y Fn(k)1079 2604 y Fm(\()p Fl(V)r(;)d(W)f Fm(\))11 b(of)f Fl(k)q Fm(-linear)h(maps)e(with)i(a)g(go)q(o)q(d)p eop %%Page: 3 3 3 2 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)310 b(3)228 213 y Fl(H)s Fm(-mo)q(dule)13 b(structure.)25 b(In)15 b(particular,)g(the)h(category)g(of)f Fl(H)s Fm(-mo)q(dules)f(is)h(\(lik)o(e)g(ev)o(ery)h(rigid)228 262 y(category)c(as)g(w)o(ell\))f(closed)h(in)g(the)g(sense)h(of)f(Mac) g(Lane:)17 b(One)12 b(has,)g(within)f(the)i(category)m(,)e(so-)228 312 y(called)i(inner)h(hom-ob)r(jects)g(hom)n(\()p Fl(V)r(;)7 b(W)f Fm(\))14 b(for)f(all)g Fl(V)r(;)7 b(W)f Fm(,)13 b(satisfying)g(a)h(formal)d(analog)h(of)i(the)228 362 y(w)o(ell-kno)o(wn)i(adjoin)o(tness)i(b)q(et)o(w)o(een)h(tensor)g(pro)q (duct)f(and)g(hom-functors)f(for)g(bimo)q(dules.)228 412 y(No)o(w)e(while)h(rigidit)o(y)e(w)o(as)i(to)q(o)g(strong)g(for)g (our)g(purp)q(oses,)h(requiring)f(inner)g(hom-functors)228 462 y(is)i(to)q(o)f(w)o(eak:)26 b(It)19 b(turns)f(out)g(that)g(the)h (category)f(of)g Fl(L)p Fm(-mo)q(dules)f(is)g(alw)o(a)o(ys)g(closed,)i (ev)o(en)228 511 y(for)c(an)o(y)h Fo(\002)406 517 y Fn(R)433 511 y Fm(-bialgebra)f Fl(L)p Fm(.)24 b(But)16 b(our)g (\\reconstruction")h(theorem)e(for)h Fo(\002)1430 517 y Fn(R)1457 511 y Fm(-Hopf)f(algebras)228 561 y(will)e(sa)o(y)i(that)g (a)g Fo(\002)539 567 y Fn(R)566 561 y Fm(-bialgebra)f Fl(L)h Fm(is)g(a)g Fo(\002)915 567 y Fn(R)942 561 y Fm(-Hopf)f(algebra) h(if)f(and)h(only)f(if)g(the)i(underlying)228 611 y(functor)k(from)482 617 y Fn(L)507 611 y Fo(M)g Fm(to)f(the)i(category)f(of)f Fl(R)p Fm(-bimo)q(dules)f(is)i(compatible)e(with)h(the)i(inner)228 661 y(hom-functors.)j(In)16 b(the)h Fl(k)q Fm(-bialgebra)e(case)i(this) g(is)f(precisely)h(the)g(observ)n(ation)f(that)g(inner)228 711 y(hom-functors)e(in)532 717 y Fn(H)563 711 y Fo(M)i Fm(are)f(mo)q(delled)f(on)h(the)h(mo)q(dules)e(of)h Fl(k)q Fm(-linear)g(maps,)f(not)h(on)g(some)228 761 y(other)f(ob)r(ject.)303 810 y(In)h(the)g(rest)h(of)f(the)g(pap)q(er)h(w)o(e)f(will)f(construct) j(a)d Fo(\002)1143 816 y Fn(R)1170 810 y Fm(-bialgebra)g(analog)g(of)g (Drinfeld's)228 860 y(quan)o(tum)k(double.)37 b(Let)21 b(us)g(recall)f(\014rst)h(the)g(construction)g(of)f(the)h(double)f Fl(D)q Fm(\()p Fl(H)s Fm(\))h(of)e(a)228 910 y(\014nite)12 b(dimensional)e Fl(k)q Fm(-Hopf)i(algebra)f Fl(H)16 b Fm(\(a)c(reference)i(is)e(Kassel's)h(b)q(o)q(ok)f([)p Fp(6)p Fm(],)f(with)h(di\013eren)o(t)228 960 y(con)o(v)o(en)o(tions)20 b(from)f(ours\):)32 b(One)21 b(has)g Fl(D)q Fm(\()p Fl(H)s Fm(\))i(=)g Fl(H)17 b Fo(\012)d Fl(H)1193 945 y Fk(\003)1232 960 y Fm(as)21 b(coalgebras;)i(the)f(algebra)228 1010 y(structure)17 b(of)d Fl(D)q Fm(\()p Fl(H)s Fm(\))i(is)f(more)f (complicated)f(and)i(p)q(erhaps)h(explained)f(most)f(easily)g(b)o(y)h (Doi)228 1059 y(and)20 b(T)m(ak)o(euc)o(hi)g([)p Fp(2)o Fm(]:)31 b(In)21 b(an)o(y)f(bialgebra)g Fl(k)q Fm(,)h(m)o (ultiplication)c(can)k(b)q(e)g(altered)g(b)o(y)g(a)f(t)o(w)o(o-)228 1109 y(co)q(cycle)15 b Fl(\034)10 b Fm(:)j Fl(B)f Fo(\012)e Fl(B)15 b Fo(!)c Fl(k)q Fm(.)19 b(Suc)o(h)c(a)f(t)o(w)o(o-co)q(cycle)g (can)h(b)q(e)g(found)f(v)o(ery)g(easily)g(if)f Fl(B)i Fm(=)d Fl(H)h Fo(\012)c Fm(\003)228 1159 y(is)16 b(the)g(tensor)h(pro)q (duct)g(of)e(t)o(w)o(o)h(Hopf)f(algebras)h Fl(H)j Fm(and)c(\003,)h(and) g(one)g(has)g(a)g(sk)o(ew)g(pairing)228 1209 y Fl(\033)6 b Fm(:)13 b(\003)d Fo(\012)f Fl(H)16 b Fo(!)c Fl(k)q Fm(,)i(that)g(is,)g(a)g(bilinear)g(form)e(with)i(resp)q(ect)j(to)d (whic)o(h)h(m)o(ultipli)o(cation)c(\(resp.)228 1259 y(com)o (ultiplicatio)o(n\))h(of)h Fl(H)k Fm(and)d(com)o(ultiplication)d (\(resp.)k(m)o(ultiplication\))c(of)j(\003)g(are)g(adjoin)o(t)228 1308 y(to)h(eac)o(h)h(other)g(\(and)f(called)h(a)f Fg(skew)g Fm(pairing)f(b)q(ecause)j(in)e(one)h(of)f(the)h(equations)f(opp)q (osite)228 1358 y(com)o(ultiplicatio)o(n)c(is)i(used\).)18 b(The)c(equations)f(for)g(a)f(sk)o(ew)i(pairing)e(are)i(easier)f(to)g (treat)h(than)228 1408 y(those)i(of)f(a)h(t)o(w)o(o-co)q(cycle;)g(the)h (most)d(ob)o(vious)h(example)f(of)h(a)h(sk)o(ew)g(pairing)e(is)i(ev)n (aluation,)228 1458 y(as)h(a)g(sk)o(ew)h(pairing)f(b)q(et)o(w)o(een)i Fl(H)770 1443 y Fk(\003)p Fh(op)841 1458 y Fm(and)e Fl(H)s Fm(.)28 b(In)18 b(this)f(situation)g(the)h(t)o(wist)g(of)e(the)i(usual) 228 1508 y(tensor)c(pro)q(duct)g(Hopf)f(algebra)g Fl(H)e Fo(\012)e Fl(H)878 1493 y Fk(\003)p Fh(op)945 1508 y Fm(b)o(y)k(the)h(co)q(cycle)g(obtained)f(from)f(the)i(canonical)228 1558 y(sk)o(ew)f(pairing)f(is)g(the)i(Drinfeld)e(double)g Fl(D)q Fm(\()p Fl(H)s Fm(\).)19 b(A)13 b(categorical)f(explanation)g (of)g Fl(D)q Fm(\()p Fl(H)s Fm(\))h(is)g(as)228 1607 y(follo)o(ws:)18 b(T)m(o)c(an)o(y)g(monoidal)e(category)j(one)g(can)g (asso)q(ciate)h(a)e(braided)h(monoidal)d(category)228 1657 y(called)h(its)f(cen)o(ter.)20 b(The)13 b(cen)o(ter)i(of)d(the)i (category)1031 1663 y Fn(H)1063 1657 y Fo(M)e Fm(is)h(the)h(category) 1403 1642 y Fn(H)1403 1669 y(H)1435 1657 y Fo(Y)s(D)g Fm(of)e(\(left-left\))228 1707 y(Y)m(etter-Drinfeld)k(mo)q(dules;)g (these)i(are)e(b)o(y)g(de\014nition)g Fl(H)s Fm(-mo)q(dules)f(as)h(w)o (ell)g(as)g(como)q(dules)228 1757 y(satisfying)f(a)g(compatibilit)o(y)d (condition.)23 b(No)o(w)15 b(b)o(y)h(dualizing)e(the)i(como)q(dule)f (structure)i(to)228 1807 y(giv)o(e)c(an)h Fl(H)409 1792 y Fk(\003)p Fh(op)463 1807 y Fm(-mo)q(dule)f(structure,)j(a)e(Y)m (etter-Drinfeld)g(mo)q(dule)f(is)h(the)h(same)e(as)i(a)f(mo)q(dule)228 1856 y(o)o(v)o(er)g(the)g(Drinfeld)f(double.)303 1906 y(If)i(w)o(e)i(w)o(an)o(t)f(to)g(generalize)g(all)f(of)h(this)g(to)g (the)h Fo(\002)1112 1912 y Fn(R)1140 1906 y Fm(-bialgebra)e(setting,)h (the)h(\014rst)g(step)228 1956 y(is)e(to)g(de\014ne)i(dual)e Fo(\002)571 1962 y Fn(R)598 1956 y Fm(-bialgebras.)22 b(W)m(e)15 b(will)f(giv)o(e)h(a)g(de\014nition)g(of)g(sk)o(ew)h (pairings)f(of)g Fo(\002)1681 1962 y Fn(R)1708 1956 y Fm(-)228 2006 y(bialgebras)e(in)g(Section)h(5,)f(as)g(w)o(ell)g(as)h (the)g(de\014nition)f(of)g(a)g(\\sk)o(ew")h(dual)f Fo(\002)1449 2012 y Fn(R)1476 2006 y Fm(-bialgebra)g Fl(L)1698 1991 y Fk(_)228 2056 y Fm(of)g(a)h Fo(\002)342 2062 y Fn(R)369 2056 y Fm(-bialgebra)f Fl(L)h Fm(that)g(is)g(\014nite)g(pro)r(jectiv)o (e)h(as)f Fl(R)p Fm(-mo)q(dule.)i(The)f(sk)o(ew)f(dual)f(is)h(a)g Fo(\002)1681 2062 y Fn(R)1708 2056 y Fm(-)228 2106 y(bialgebra)h (analog)f(of)h(the)h(opp)q(osite)g(bialgebra)f(of)g(the)h(dual)f(of)g (an)h(ordinary)f Fl(k)q Fm(-bialgebra.)228 2155 y(Oddly)k(enough,)h (neither)g(Hopf)f(pairings)f(\(rather)i(than)f(sk)o(ew)h(ones\))g(nor)f (duals)g(\(rather)228 2205 y(than)c(opp)q(osite)h(duals\))f(seem)h(to)f (exist)h(for)f Fo(\002)972 2211 y Fn(R)999 2205 y Fm(-bialgebras)g (\(note)h(that)g(the)g(opp)q(osite)g(of)e(a)228 2255 y Fo(\002)260 2261 y Fn(R)287 2255 y Fm(-bialgebra)k(is)g(not)g(a)g Fo(\002)682 2261 y Fn(R)709 2255 y Fm(-bialgebra\).)31 b(A)18 b(predecessor)k(of)17 b(our)i(dualit)o(y)e(construction)228 2305 y(can)h(b)q(e)g(found)f(in)g([)p Fp(12)p Fm(],)g(where)i(t)o(w)o (o)e(sub)q(classes)j(of)d Fo(\002)1130 2311 y Fn(R)1157 2305 y Fm(-bialgebras)g(for)h(comm)o(utativ)n(e)d Fl(R)228 2355 y Fm(are)f(sho)o(wn)h(to)f(b)q(e)g(in)g(dualit)o(y)f(with)h(eac)o (h)h(other.)k(As)c(the)f(referee)j(kindly)c(p)q(oin)o(ted)h(out,)g(the) 228 2404 y(dualit)o(y)e(b)q(et)o(w)o(een)j Fo(\002)561 2410 y Fn(R)588 2404 y Fm(-coalgebras)e(and)h Fl(R)915 2389 y Fn(e)933 2404 y Fm(-rings)f(w)o(as)h(already)f(treated)i (completely)d(\(with)228 2454 y(a)h(trivial)f(v)n(ariation)g(in)h (notations)g(that)h(is)g(not)f(suited)h(w)o(ell)f(to)g(the)i Fo(\002)1354 2460 y Fn(R)1381 2454 y Fm(-bialgebra)e(case\))h(b)o(y)228 2504 y(T)m(ak)o(euc)o(hi)f(in)h([)p Fp(20)o Fm(].)303 2554 y(Luc)o(kily)c(a)h Fg(skew)g Fm(pairing)g(is)g(what)g(w)o(e)h (need)g(to)g(generalize)g(Doi-T)m(ak)o(euc)o(hi's)d(generalized)228 2604 y(Drinfeld)j(double)i(in)f(Section)h(6.)j(Here)e(another)f (di\016cult)o(y)e(arises.)19 b(Doi)12 b(and)i(T)m(ak)o(euc)o(hi)e (\014rst)p eop %%Page: 4 4 4 3 bop 228 113 a Fh(4)536 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(tak)o(e)i(the)h(ordinary)e(tensor)i(pro)q(duct)g(bialgebra,)f (and)g(then)g(t)o(wist)g(it)g(b)o(y)g(a)g(sk)o(ew)g(pairing)228 262 y(in)g(a)h(second)h(step.)34 b(Ho)o(w)o(ev)o(er,)20 b(the)g(\014rst)f(step)h(is)f(imp)q(ossible)e(in)i(our)g(setting.)33 b(F)m(or)19 b(t)o(w)o(o)228 312 y Fo(\002)260 318 y Fn(R)287 312 y Fm(-bialgebras)f Fl(L)h Fm(and)f(\003,)h(there)h(do)q(es)f(not)f (seem)g(to)h(b)q(e)g(a)f(reasonable)h(tensor)g(pro)q(duct)228 362 y Fo(\002)260 368 y Fn(R)287 362 y Fm(-bialgebra:)j(While)15 b Fl(L)h Fo(\012)701 375 y Fn(R)p Fk(\012)p 752 351 26 2 v Fn(R)795 362 y Fm(\003)g(do)q(es)h(not)f(ha)o(v)o(e)g(an)g(algebra) f(structure,)k Fl(L)p Fo(\002)1570 368 y Fn(R)1597 362 y Fm(\003)d(do)q(es)228 412 y(not)i(ha)o(v)o(e)g(a)f(com)o (ultiplication.)27 b(Luc)o(kily)m(,)18 b(once)g(w)o(e)h(are)f(giv)o(en) g(a)f(sk)o(ew)i(pairing)e(b)q(et)o(w)o(een)228 462 y(\003)i(and)h Fl(L)p Fm(,)h(with)f Fl(L)g Fm(a)f Fo(\002)645 468 y Fn(R)673 462 y Fm(-Hopf)g(algebra,)h(w)o(e)g(can)g(do)g(b)q(oth)g (steps)h(in)e(one)h(and)g(endo)o(w)228 511 y Fl(L)12 b(.)-7 b(/)11 b Fm(\003)g(:=)g Fl(L)h Fo(\012)481 524 y Fn(R)p Fk(\012)p 532 500 V Fn(R)571 511 y Fm(\003)g(with)f(a)g Fo(\002)768 517 y Fn(R)795 511 y Fm(-bialgebra)g(structure)i(that)f (generalizes)g(Doi-T)m(ak)o(euc)o(hi's)228 561 y(t)o(wist)i(of)f(the)h (tensor)h(pro)q(duct)g(bialgebra.)303 611 y(T)m(o)e(complete)h(the)h (picture,)h(w)o(e)e(sho)o(w)h(that)g(mo)q(dules)e(o)o(v)o(er)i(the)g (double)f Fl(D)q Fm(\()p Fl(L)p Fm(\))g(:=)f Fl(L)g(.)-7 b(/)228 661 y(L)256 646 y Fk(_)294 661 y Fm(of)12 b(a)h(\014nite)h(pro) r(jectiv)o(e)g Fo(\002)704 667 y Fn(R)731 661 y Fm(-Hopf)f(algebra)f Fl(L)i Fm(are)f(precisely)i(the)e(Y)m(etter-Drinfeld)h(mo)q(d-)228 711 y(ules)g(o)o(v)o(er)g Fl(L)p Fm(,)g(a)f(notion)g(w)o(e)h(ha)o(v)o (e)g(de\014ned)h(in)f(Section)g(4.)618 831 y Fp(2.)24 b(Bialgebroi)o(ds)13 b(and)i(their)e(mo)q(dules)303 906 y Fm(In)d(this)g(section)i(w)o(e)e(will)f(v)o(ery)i(brie\015y)f(recall) h(some)e(de\014nitions)i(and)f(notations)g(concern-)228 956 y(ing)16 b Fo(\002)332 962 y Fn(R)359 956 y Fm(-bialgebras,)h(and)g (the)g(monoidal)d(categories)k(of)e(their)h(mo)q(dules)f(and)g(como)q (dules.)228 1006 y(F)m(or)f(more)g(details)g(w)o(e)h(refer)h(to)f([)p Fp(18)o(,)i(19,)g(14)p Fm(].)23 b(W)m(e)16 b(will)e(not)i(mak)o(e)e (explicit)h(reference)j(to)228 1055 y(the)c(closely)g(related)h (notions)e(in)h([)p Fp(4)o(,)i(11,)g(10)p Fm(].)303 1105 y(F)m(or)d(the)i(rest)h(of)d(the)i(pap)q(er)g Fl(k)g Fm(denotes)g(a)f(comm)o(utativ)o(e)e(ring)h(and)h Fl(R)g Fm(denotes)i(a)e(\014xed)228 1155 y Fl(k)q Fm(-algebra.)j(W)m(e)c (denote)i(the)f(opp)q(osite)g(algebra)f(b)o(y)p 1070 1122 32 2 v 14 w Fl(R)p Fm(,)g(w)o(e)h(let)g Fl(R)d Fo(3)g Fl(r)i Fo(7!)p 1415 1132 20 2 v 11 w Fl(r)f Fo(2)p 1485 1122 32 2 v 11 w Fl(R)i Fm(denote)h(the)228 1205 y(ob)o(vious)h Fl(k)q Fm(-algebra)f(an)o(tiisomorphism,)e(and)k(abbreviate)g(the)g(en) o(v)o(eloping)f(algebra)g Fl(R)1645 1190 y Fn(e)1678 1205 y Fm(:=)228 1255 y Fl(R)9 b Fo(\012)p 310 1221 V 9 w Fl(R)p Fm(.)18 b(W)m(e)c(consider)g Fl(R)g Fm(and)p 732 1221 V 14 w Fl(R)g Fm(as)g(subalgebras)g(of)f Fl(R)1132 1240 y Fn(e)1164 1255 y Fm(in)g(the)h(ob)o(vious)g(w)o(a)o(y)m(.)303 1304 y(Let)j Fl(M)r(;)7 b(N)22 b Fo(2)541 1310 y Fn(R)566 1302 y Fe(e)585 1304 y Fo(M)635 1310 y Fn(R)660 1302 y Fe(e)678 1304 y Fm(.)28 b(W)m(e)17 b(use)h(the)g(follo)o(wing)d (notations,)i(due)h(to)f(Mac)h(Lane,)g(see)228 1354 y([)p Fp(18)o(,)e(19)p Fm(]:)h(W)m(e)d(let)383 1399 y Ff(Z)406 1493 y Fn(r)p 431 1445 19 2 v 431 1462 a(r)450 1455 y Fl(M)g Fo(\012)545 1461 y Fn(r)564 1455 y Fl(N)i Fm(:=)11 b Fl(M)j Fo(\012)c Fl(N)802 1422 y Ff(\016)826 1455 y Fo(h)p 842 1432 20 2 v Fl(r)q(m)g Fo(\012)f Fl(n)g Fo(\000)h Fl(m)f Fo(\012)h Fl(r)q(n)p Fo(j)p Fl(r)i Fo(2)f Fl(R;)c(m)k Fo(2)g Fl(M)r(;)c(n)12 b Fo(2)f Fl(N)5 b Fo(i)303 1586 y Fm(W)m(e)10 b(let)429 1530 y Ff(Z)470 1540 y Fn(r)495 1586 y Fl(M)p 535 1576 19 2 v 6 x Fn(r)558 1586 y Fo(\012)t Fl(N)627 1592 y Fn(r)657 1586 y Fo(\032)i Fl(M)d Fo(\012)t Fl(N)16 b Fm(denote)c(the)g Fl(k)q Fm(-submo)q(dule)e(consisting)h(of)f (all)g(elemen)o(ts)228 1626 y Ff(P)272 1657 y Fl(m)308 1663 y Fn(i)323 1657 y Fo(\012)q Fl(n)381 1663 y Fn(i)406 1657 y Fo(2)h Fl(M)5 b Fo(\012)q Fl(N)15 b Fm(satisfying)752 1626 y Ff(P)795 1657 y Fl(m)831 1663 y Fn(i)p 846 1634 20 2 v 846 1657 a Fl(r)q Fo(\012)q Fl(n)924 1663 y Fn(i)950 1657 y Fm(=)993 1626 y Ff(P)1037 1657 y Fl(m)1073 1663 y Fn(i)1088 1657 y Fo(\012)q Fl(n)1146 1663 y Fn(i)1160 1657 y Fl(r)10 b Fm(for)g(all)e Fl(r)k Fo(2)g Fl(R)p Fm(.)k(In)10 b(blatan)o(t)e(abuse)228 1707 y(of)h(notation,)h(w)o(e)g (will)f(most)g(of)h(the)g(time)f(write)i Fl(m)r Fo(\012)r Fl(n)f Fm(for)g(a)g(general)g(elemen)o(t)g(of)1511 1674 y Ff(R)1539 1684 y Fn(r)1564 1707 y Fl(M)p 1604 1697 19 2 v 7 x Fn(r)1625 1707 y Fo(\012)r Fl(N)1692 1713 y Fn(r)1711 1707 y Fm(,)228 1757 y(although)j(of)g(course)i(this)f Fl(k)q Fm(-mo)q(dule)f(needs)i(not)f(b)q(e)g(generated)i(b)o(y)d (decomp)q(osable)h(tensors)228 1807 y(in)f Fl(M)h Fo(\012)c Fl(N)5 b Fm(.)303 1857 y(In)14 b(the)i(instance)f(ab)q(o)o(v)o(e)g(the) 781 1823 y Ff(R)801 1871 y Fn(r)834 1857 y Fm(sym)o(b)q(ol)e(is)i(just) g(a)f(tensor)i(pro)q(duct)g(o)o(v)o(er)f Fl(R)p Fm(,)f(with)h(the)228 1907 y(adv)n(an)o(tage)k(of)g(indicating)f(precisely)j(the)f(mo)q(dule) f(structures)j(w)o(e)e(ha)o(v)o(e)g(in)f(mind.)34 b(The)228 1956 y(sym)o(b)q(ols)387 1923 y Ff(R)407 1971 y Fn(r)440 1956 y Fm(and)522 1923 y Ff(R)550 1933 y Fn(r)583 1956 y Fm(can)15 b(b)q(e)g(mixed)f(to)h(indicate)f(iterated)i(constructions) g(of)f(submo)q(dules)228 2006 y(and)h(quotien)o(t)g(mo)q(dules;)g(it)g (should)g(b)q(e)h(noted)g(that)f(the)h(t)o(w)o(o)f(op)q(erations)g (will)f(in)h(general)228 2056 y(not)e(comm)o(ute.)303 2106 y(F)m(or)g(t)o(w)o(o)g(left)h Fl(R)563 2091 y Fn(e)580 2106 y Fm(-mo)q(dules)f Fl(M)r(;)7 b(N)20 b Fm(w)o(e)15 b(abbreviate)g Fl(M)f Fo(\005)c Fl(N)18 b Fm(:=)1332 2072 y Ff(R)1351 2121 y Fn(r)p 1377 2095 V 1377 2112 a(r)1395 2106 y Fl(M)d Fo(\012)1492 2112 y Fn(r)1510 2106 y Fl(N)5 b Fm(,)15 b(whic)o(h)f(is)228 2161 y(again)d(in)384 2167 y Fn(R)409 2159 y Fe(e)427 2161 y Fo(M)i Fm(with)f(the)h(left)p 724 2128 32 2 v 12 w Fl(R)p Fm(-mo)q(dule)e(structure)j(induced)f(b)o (y)g(the)g(structure)h(of)e Fl(N)5 b Fm(,)12 b(and)228 2211 y(the)j(left)f Fl(R)p Fm(-mo)q(dule)e(structure)17 b(induced)e(b)o(y)f(the)h(structure)h(of)e Fl(M)5 b Fm(.)19 b(In)14 b(this)h(w)o(a)o(y)m(,)d(\()1573 2217 y Fn(R)1598 2209 y Fe(e)1617 2211 y Fo(M)p Fl(;)7 b Fo(\005)p Fm(\))228 2261 y(is)14 b(a)f(monoidal)e(category)m(,)i(naturally)g(isomorphic)f (to)i(\()1115 2267 y Fn(R)1143 2261 y Fo(M)1193 2267 y Fn(R)1220 2261 y Fl(;)7 b Fo(\012)1270 2267 y Fn(R)1298 2261 y Fm(\).)303 2311 y(F)m(or)13 b(t)o(w)o(o)g Fl(R)487 2296 y Fn(e)505 2311 y Fm(-bimo)q(dules)f Fl(M)19 b Fm(and)14 b Fl(N)k Fm(w)o(e)d(abbreviate)723 2404 y Fl(M)5 b Fo(\002)771 2434 y Fn(R)800 2404 y Fl(N)16 b Fm(:=)905 2347 y Ff(Z)946 2357 y Fn(s)971 2347 y Ff(Z)994 2442 y Fn(r)p 1019 2393 19 2 v 1019 2410 a(r)1037 2404 y Fl(M)p 1077 2393 18 2 v 6 x Fn(s)1105 2404 y Fo(\012)1146 2410 y Fn(r)1165 2404 y Fl(N)1198 2410 y Fn(s)1216 2404 y Fl(:)228 2507 y Fm(This)e(is)g(again)g(an)g Fl(R)567 2492 y Fn(e)585 2507 y Fm(-bimo)q(dule)e(with)i Fl(r)p 895 2484 20 2 v 1 w(s)q Fm(\()p Fl(m)c Fo(\012)g Fl(n)p Fm(\))p Fl(t)p 1075 2484 24 2 v(u)i Fm(=)h Fl(r)q(mt)d Fo(\012)p 1279 2484 20 2 v 10 w Fl(sn)p 1323 2484 24 2 v(u)k Fm(for)g Fl(r)o(;)7 b(s;)g(t;)g(u)k Fo(2)h Fl(R)i Fm(and)228 2558 y Fl(m)t Fo(\012)t Fl(n)f Fo(2)e Fl(M)5 b Fo(\002)458 2564 y Fn(R)485 2558 y Fl(N)g Fm(,)12 b(i.)e(e.)i Fl(R)f Fm(acts)h(on)f(the)h(left,)p 956 2525 32 2 v 12 w Fl(R)f Fm(on)g(the)h(righ)o(t)f(factor.)17 b(The)12 b(pro)q(duct)h(functor)228 2608 y Fo(\002)260 2614 y Fn(R)301 2608 y Fm(th)o(us)h(de\014ned)h(on)f (the)h(category)832 2614 y Fn(R)857 2606 y Fe(e)875 2608 y Fo(M)925 2614 y Fn(R)950 2606 y Fe(e)982 2608 y Fm(needs)h(neither)e (b)q(e)h(asso)q(ciativ)o(e)f(nor)g(unitary)m(.)p eop %%Page: 5 5 5 4 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)310 b(5)303 213 y Fm(F)m(or)13 b Fl(M)r(;)7 b(N)r(;)g(P)17 b Fo(2)575 219 y Fn(R)600 211 y Fe(e)618 213 y Fo(M)668 219 y Fn(R)693 211 y Fe(e)726 213 y Fm(one)d(de\014nes) 589 301 y Fl(M)5 b Fo(\002)638 331 y Fn(R)666 301 y Fl(P)h Fo(\002)702 331 y Fn(R)731 301 y Fl(N)16 b Fm(:=)836 245 y Ff(Z)877 255 y Fn(s;u)931 245 y Ff(Z)954 339 y Fn(r)o(;t)p 1000 291 19 2 v 1000 307 a(r)1019 301 y Fl(M)p 1059 291 18 2 v 6 x Fn(s)1086 301 y Fo(\012)1128 312 y Fn(r)o(;)p 1152 290 13 2 v(t)1167 301 y Fl(P)1194 307 y Fn(s;)p 1220 291 20 2 v(u)1250 301 y Fo(\012)1291 307 y Fn(t)1306 301 y Fl(N)1339 307 y Fn(u)228 402 y Fm(\(where)364 369 y Ff(R)392 379 y Fn(s;u)450 402 y Fm(:=)506 369 y Ff(R)533 379 y Fn(s)558 369 y Ff(R)586 379 y Fn(u)619 402 y Fm(=)663 369 y Ff(R)690 379 y Fn(u)719 369 y Ff(R)747 379 y Fn(s)765 402 y Fm(\).)i(There)d(are)f(asso)q(ciativit)o(y)f(maps) 747 477 y(\()p Fl(M)5 b Fo(\002)811 507 y Fn(R)840 477 y Fl(P)h Fm(\))p Fo(\002)893 507 y Fn(R)921 477 y Fl(N)980 453 y Fn(\013)970 477 y Fo(!)11 b Fl(M)5 b Fo(\002)1072 507 y Fn(R)1101 477 y Fl(P)h Fo(\002)1136 507 y Fn(R)1165 477 y Fl(N)747 579 y(M)f Fo(\002)795 610 y Fn(R)824 579 y Fm(\()p Fl(P)h Fo(\002)876 610 y Fn(R)905 579 y Fl(N)f Fm(\))975 556 y Fn(\013)997 543 y Fd(0)970 579 y Fo(!)11 b Fl(M)5 b Fo(\002)1072 610 y Fn(R)1101 579 y Fl(P)h Fo(\002)1136 610 y Fn(R)1165 579 y Fl(N)228 667 y Fm(whic)o(h)12 b(are)h(\(with)g(our)g(abuse)g(of)f(notations!\))17 b(giv)o(en)12 b(on)h(elemen)o(ts)f(b)o(y)h(the)g(form)o(ula)d Fl(m)d 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Fl(m:)228 1057 y Fm(W)m(e)i(will)g(write)h(E)e(:=)f(End\()p Fl(R)p Fm(\))j(b)q(elo)o(w.)303 1107 y(A)f Fo(\002)379 1113 y Fn(R)406 1107 y Fm(-coalgebra)g Fl(L)h Fm(is)f(an)g Fl(R)776 1092 y Fn(e)794 1107 y Fm(-bimo)q(dule)e(equipp)q(ed)j(with)f (a)g(com)o(ultiplication,)d(a)j(map)228 1157 y(\001)5 b(:)14 b Fl(L)22 b Fo(!)f Fl(L)p Fo(\002)468 1163 y Fn(R)495 1157 y Fl(L)f Fm(of)g Fl(R)629 1142 y Fn(e)646 1157 y Fm(-bimo)q(dules)f(o)o(v)o(er)h Fl(R)991 1142 y Fn(e)1008 1157 y Fm(,)h(and)f(a)g(counit,)h(a)e(map)f Fl(")k Fm(:)g Fl(L)f Fo(!)h Fm(E)e(of)228 1206 y Fl(R)260 1191 y Fn(e)277 1206 y Fm(-bimo)q(dules,)12 b(suc)o(h)j(that)617 1274 y Fl(\013)p Fm(\(\001)p Fo(\002)699 1304 y Fn(R)726 1274 y Fl(L)p Fm(\)\001)d(=)g Fl(\013)888 1257 y Fk(0)899 1274 y Fm(\()p Fl(L)p Fo(\002)947 1304 y Fn(R)976 1274 y Fm(\001\)\001)5 b(:)12 b Fl(L)g Fo(!)f Fl(L)p Fo(\002)1216 1304 y Fn(R)1244 1274 y Fl(L)p Fo(\002)1276 1304 y Fn(R)1305 1274 y Fl(L)-1105 b Fm(\(2.1\))717 1356 y Fl(\022)q Fm(\()p Fl(L)p Fo(\002)785 1386 y Fn(R)814 1356 y Fl(")p Fm(\)\001)12 b(=)g(id)974 1362 y Fn(L)1010 1356 y Fm(=)g Fl(\022)q Fm(\()p Fl(")p Fo(\002)1113 1386 y Fn(R)1143 1356 y Fl(L)p Fm(\)\001)p Fl(:)-1006 b Fm(\(2.2\))228 1441 y(Note)13 b(that)f(a)g(map)f(\001)5 b(:)13 b Fl(L)f Fo(!)f Fl(L)p Fo(\002)757 1447 y Fn(R)785 1441 y Fl(L)h Fm(in)872 1447 y Fn(R)897 1439 y Fe(e)916 1441 y Fo(M)966 1447 y Fn(R)991 1439 y Fe(e)1021 1441 y Fm(induces)i(a)e(map)f(\001)1327 1447 y Fh(0)1350 1441 y Fm(:)i Fl(L)f Fo(!)f Fl(L)6 b Fo(\005)g Fl(L)13 b Fm(in)1617 1447 y Fn(R)1642 1439 y Fe(e)1661 1441 y Fo(M)p Fm(.)228 1491 y(A)h(map)e Fl(")f Fm(:)h Fl(L)f Fo(!)g Fm(E)j(in)601 1497 y Fn(R)626 1489 y Fe(e)645 1491 y Fo(M)695 1497 y Fn(R)720 1489 y Fe(e)752 1491 y Fm(induces)g(a)f(map)g Fl(")1045 1497 y Fh(0)1075 1491 y Fm(:)e Fl(L)h Fo(3)f Fl(`)h Fo(7!)f Fl(")p Fm(\()p Fl(`)p Fm(\)\(1\))h Fo(2)g Fl(R)h Fm(in)1525 1497 y Fn(R)1550 1489 y Fe(e)1569 1491 y Fo(M)o Fm(.)18 b(One)228 1541 y(c)o(hec)o(ks)d(that)f(\001)f(and)h Fl(")g Fm(ful\014ll)f(the)h (equations)g(\(2.1\))f(and)h(\(2.2\))f(if)g(and)h(only)f(if)598 1608 y(\(\001)649 1614 y Fh(0)677 1608 y Fo(\005)c Fl(L)p Fm(\)\001)786 1614 y Fh(0)816 1608 y Fm(=)j(\()p Fl(L)d Fo(\005)g Fm(\001)978 1614 y Fh(0)997 1608 y Fm(\)\001)1048 1614 y Fh(0)1070 1608 y Fm(:)14 b Fl(L)e Fo(!)f Fl(L)e Fo(\005)g Fl(L)h Fo(\005)f Fl(L)228 1676 y Fm(and)19 b(\()p Fl(")349 1682 y Fh(0)382 1676 y Fo(\005)12 b Fl(L)p Fm(\)\001)494 1682 y Fh(0)534 1676 y Fm(=)22 b(id)622 1682 y Fn(L)669 1676 y Fm(=)f(\()p Fl(L)14 b Fo(\005)f Fl(")833 1682 y Fh(0)852 1676 y Fm(\))19 b(hold.)35 b(These)21 b(mean)e(that)h Fl(L)p Fm(,)g(considered)i(as)d(an)228 1725 y Fl(R)p Fm(-)p Fl(R)p Fm(-bimo)q(dule)11 b(via)i(the)i(left)e Fl(R)744 1710 y Fn(e)762 1725 y Fm(-mo)q(dule)f(structure,)k(is)e(an)f Fl(R)p Fm(-coring.)303 1775 y(By)18 b(an)g Fl(R)466 1760 y Fn(e)484 1775 y Fm(-ring)g(w)o(e)g(will)f(simply)g(mean)g(an)h (algebra)g Fl(L)g Fm(equipp)q(ed)h(with)f(an)h(algebra)228 1825 y(map)c Fl(i)5 b Fm(:)14 b Fl(R)400 1810 y Fn(e)435 1825 y Fo(!)i Fl(L)p Fm(;)i(a)f(map)e(of)h Fl(R)766 1810 y Fn(e)784 1825 y Fm(-rings)h(will)e(b)q(e)j(an)e(algebra)h(map)e(comm) o(uting)f(with)j(the)228 1875 y(resp)q(ectiv)o(e)h(maps)c(from)h Fl(R)666 1860 y Fn(e)683 1875 y Fm(.)25 b(Of)15 b(course,)j(an)o(y)d Fl(R)1034 1860 y Fn(e)1052 1875 y Fm(-ring)g(is)h(an)g Fl(R)1289 1860 y Fn(e)1307 1875 y Fm(-bimo)q(dule)d(in)j(a)g(natural) 228 1925 y(w)o(a)o(y)m(,)g(and)h(it)f(turns)i(out)f(that)g(if)f Fl(L;)7 b(M)22 b Fm(are)17 b Fl(R)988 1910 y Fn(e)1006 1925 y Fm(-rings,)g(then)g Fl(L)p Fo(\002)1295 1931 y Fn(R)1322 1925 y Fl(M)22 b Fm(is)17 b(an)f Fl(R)1521 1910 y Fn(e)1539 1925 y Fm(-ring)g(with)228 1974 y(\()p Fl(`)11 b Fo(\012)g Fl(`)332 1959 y Fk(0)344 1974 y Fm(\)\()p Fl(m)g Fo(\012)g Fl(m)502 1959 y Fk(0)514 1974 y Fm(\))k(=)g Fl(`m)d Fo(\012)f Fl(`)717 1959 y Fk(0)729 1974 y Fl(m)765 1959 y Fk(0)777 1974 y Fm(.)24 b(Similarly)l(,)13 b(the)k(triple)f Fo(\002)1219 1980 y Fn(R)1246 1974 y Fm(-pro)q(duct)h(of)e Fl(R)1499 1959 y Fn(e)1517 1974 y Fm(-rings)h(is)f(an)228 2024 y Fl(R)260 2009 y Fn(e)277 2024 y Fm(-ring.)303 2074 y(By)d(de\014nition)f(a)g Fo(\002)613 2080 y Fn(R)640 2074 y Fm(-bialgebra)g Fl(L)h Fm(is)f(an)h Fl(R)999 2059 y Fn(e)1017 2074 y Fm(-ring)f(equipp)q(ed)h(whic)o(h)g(is)g(a)f Fo(\002)1510 2080 y Fn(R)1538 2074 y Fm(-coalgebra)228 2124 y(suc)o(h)16 b(that)g(com)o(ultiplication)d(\001)5 b(:)13 b Fl(L)i Fo(!)f Fl(L)p Fo(\002)947 2130 y Fn(R)975 2124 y Fl(L)i Fm(and)f(counit)h Fl(")5 b Fm(:)14 b Fl(L)h Fo(!)f Fl(E)k Fm(are)e(maps)f(of)g Fl(R)1691 2109 y Fn(e)1708 2124 y Fm(-)228 2174 y(rings.)35 b(The)20 b(follo)o(wing)d(equiv)n (alen)o(t)i(de\014nition)h(w)o(as)f(p)q(oin)o(ted)h(out)f(b)o(y)h(the)g (referee:)32 b(The)228 2223 y(category)20 b(of)e Fo(\002)486 2229 y Fn(R)513 2223 y Fm(-coalgebras)h(is)g(a)h(monoidal)c(category)k (with)f(tensor)h(pro)q(duct)g Fo(\012)1603 2229 y Fn(R)1628 2221 y Fe(e)1646 2223 y Fm(,)g(b)o(y)228 2273 y([)p Fp(19)o Fm(,)15 b(Prop.)23 b(4.7].)e(The)15 b(neutral)h(ob)r(ject)g(is)f(the)h Fo(\002)1042 2279 y Fn(R)1070 2273 y Fm(-coalgebra)f Fl(R)1302 2258 y Fn(e)1334 2273 y Fm(with)g(com)o(ultiplication)228 2323 y(\001\()p Fl(r)7 b Fo(\012)p 343 2300 20 2 v 6 w Fl(s)p Fm(\))k(=)h(\()p Fl(r)7 b Fo(\012)p 513 2291 21 2 v 6 w Fm(1\))f Fo(\012)g Fm(\(1)g Fo(\012)p 675 2300 20 2 v 6 w Fl(s)p Fm(\))12 b(and)g(counit)g Fl(")p Fm(\()p Fl(r)7 b Fo(\012)p 1025 2300 V 6 w Fl(s)q Fm(\)\()p Fl(t)p Fm(\))12 b(=)g Fl(r)q(ts)p Fm(.)17 b(No)o(w)12 b(a)g Fo(\002)1405 2329 y Fn(R)1432 2323 y Fm(-bialgebra)g(is)g(the)228 2373 y(same)i(as)g(an)h(algebra)f(in)g(the)h(monoidal)d(category)j(of)f Fo(\002)1139 2379 y Fn(R)1166 2373 y Fm(-coalgebras)h(b)o(y)f([)p Fp(19)o Fm(,)h(Prop.)20 b(4.8].)228 2423 y(In)11 b(particular)g Fl(R)498 2408 y Fn(e)527 2423 y Fm(is)g(a)g Fo(\002)630 2429 y Fn(R)657 2423 y Fm(-bialgebra)g(with)f(com)o(ultiplication)e (\001\()p Fl(r)d Fo(\012)p 1354 2400 V 4 w Fl(s)p Fm(\))12 b(=)g(\()p Fl(r)5 b Fo(\012)p 1521 2391 21 2 v 4 w Fm(1\))t Fo(\012)t Fm(\(1)t Fo(\012)p 1675 2400 20 2 v 4 w Fl(s)q Fm(\).)303 2473 y(F)m(or)17 b Fo(\002)414 2479 y Fn(R)441 2473 y Fm(-bialgebras)h(w)o(e)g(will)f(mak)o(e)g(use)i(of)f(the)g (usual)h(Sw)o(eedler)g(notation,)f(writing)228 2522 y(\001\()p Fl(`)p Fm(\))12 b(=:)f Fl(`)396 2529 y Fh(\(1\))450 2522 y Fo(\012)e Fl(`)508 2529 y Fh(\(2\))565 2522 y Fo(2)i Fl(L)p Fo(\002)665 2528 y Fn(R)692 2522 y Fl(L)p Fm(,)i(and)573 2590 y Fl(\013)p Fm(\(\001)p Fo(\002)655 2620 y Fn(R)683 2590 y Fl(L)p Fm(\)\001\()p Fl(`)p Fm(\))f(=:)f Fl(`)895 2597 y Fh(\(1\))949 2590 y Fo(\012)f Fl(`)1008 2597 y Fh(\(2\))1062 2590 y Fo(\012)f Fl(`)1120 2597 y Fh(\(3\))1177 2590 y Fo(2)i Fl(L)p Fo(\002)1248 2620 y Fn(R)1277 2590 y Fl(L)p Fo(\002)1309 2620 y Fn(R)1337 2590 y Fl(L:)p eop %%Page: 6 6 6 5 bop 228 113 a Fh(6)536 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(Ho)o(w)o(ev)o(er,)10 b(a)f(w)o(ord)g(of)g(w)o(arning)g(is)g (necessary)m(.)18 b(When)10 b(using)f(Sw)o(eedler)h(notation)f(for)g (ordinary)228 262 y Fl(k)q Fm(-coalgebras,)14 b(one)h(can)g(substitute) h(expressions)g(and)e(equalities)g(lik)o(e)g Fl(c)1404 269 y Fh(\(1\))1458 262 y Fo(\012)c Fl(c)1518 269 y Fh(\(2\))1573 262 y Fo(\012)g Fl(c)1633 269 y Fh(\(3\))1690 262 y Fm(=)228 312 y Fl(c)246 319 y Fh(\(1\))q(\(1\))345 312 y Fo(\012)g Fl(c)405 319 y Fh(\(1\))r(\(2\))505 312 y Fo(\012)g Fl(c)565 319 y Fh(\(2\))625 312 y Fm(freely)15 b(in)o(to)f(m)o(ultilinear)f (expressions.)23 b(When)16 b(doing)e(calculations)228 362 y(with)f Fo(\002)355 368 y Fn(R)382 362 y Fm(-bialgebras,)g (complications)e(arise:)19 b(Although)13 b(coasso)q(ciativit)o(y)g (reads)400 439 y Fl(`)417 446 y Fh(\(1\))471 439 y Fo(\012)d Fl(`)530 446 y Fh(\(2\))q(\(1\))628 439 y Fo(\012)g Fl(`)687 446 y Fh(\(2\))r(\(2\))788 439 y Fm(=)h Fl(`)848 446 y Fh(\(1\))903 439 y Fo(\012)e Fl(`)961 446 y Fh(\(2\))1015 439 y Fo(\012)h Fl(`)1074 446 y Fh(\(3\))1130 439 y Fm(=)i Fl(`)1191 446 y Fh(\(1\))r(\(1\))1290 439 y Fo(\012)d Fl(`)1348 446 y Fh(\(1\))r(\(2\))1447 439 y Fo(\012)g Fl(`)1505 446 y Fh(\(2\))228 517 y Fm(\(as)20 b(elemen)o(ts)g(of)f Fl(L)13 b Fo(\005)g Fl(L)h Fo(\005)f Fl(L)p Fm(\),)21 b(the)g(elemen)o(ts)f Fl(`)1029 524 y Fh(\(1\))1087 517 y Fo(\012)13 b Fl(`)1149 524 y Fh(\(2\))r(\(1\))1252 517 y Fo(\012)g Fl(`)1314 524 y Fh(\(2\))r(\(2\))1425 517 y Fo(2)22 b Fl(L)p Fo(\002)1535 523 y Fn(R)1562 517 y Fm(\()p Fl(L)p Fo(\002)1639 523 y Fn(R)1666 517 y Fl(L)p Fm(\))p Fl(;)228 567 y(`)245 574 y Fh(\(1\))302 567 y Fo(\012)14 b Fl(`)365 574 y Fh(\(2\))422 567 y Fo(\012)f Fl(`)484 574 y Fh(\(3\))550 567 y Fo(2)20 b Fl(L)p Fo(\002)658 573 y Fn(R)686 567 y Fl(L)p Fo(\002)746 573 y Fn(R)773 567 y Fl(L;)f Fm(and)h Fl(`)936 574 y Fh(\(1\))q(\(1\))1038 567 y Fo(\012)13 b Fl(`)1100 574 y Fh(\(1\))r(\(2\))1202 567 y Fo(\012)g Fl(`)1264 574 y Fh(\(2\))1330 567 y Fo(2)20 b Fm(\()p Fl(L)p Fo(\002)1455 573 y Fn(R)1482 567 y Fl(L)p Fm(\))p Fo(\002)1558 573 y Fn(R)1586 567 y Fl(L)f Fm(m)o(ust)228 617 y(not)d(b)q(e)h(confused,)g(ev)o(en)g(if)f(\(as)h(is)f(true)h(in)f (man)o(y)f(in)o(teresting)h(cases\))i(b)q(oth)f Fl(\013)f Fm(and)g Fl(\013)1639 602 y Fk(0)1667 617 y Fm(are)228 667 y(isomorphisms.)303 747 y Fc(Example)h Fm(2.1)p Fc(.)j Fm(Let)c Fl(R)e Fm(=)h Fl(k)c Fo(\002)g Fl(k)17 b Fm(with)e(the)h (canonical)f(idemp)q(oten)o(ts)h Fl(e)1480 753 y Fh(1)1499 747 y Fl(;)7 b(e)1537 753 y Fh(2)1569 747 y Fo(2)15 b Fl(k)c Fo(\002)g Fl(k)q Fm(,)228 797 y(and)k(consider)i Fl(L)d Fm(=)h Fl(R)10 b Fo(\012)h Fl(R)k Fm(with)h(basis)f Fl(e)914 803 y Fn(ij)958 797 y Fm(:=)f Fl(e)1035 803 y Fn(i)1060 797 y Fo(\012)d Fl(e)1122 803 y Fn(j)1139 797 y Fm(.)23 b(W)m(e)16 b(let)f(the)i(action)e(of)g Fl(R)g Fm(on)h(the)228 850 y(left)g(tensor)i(factor)f(of)f Fl(L)h Fm(giv)o(e)g(rise)g(to)g(b)q(oth)g(the)g(left)g Fl(R)p Fm(-)f(and)p 1267 816 32 2 v 17 w Fl(R)p Fm(-mo)q(dule)f (structure,)k(and)228 899 y(similarly)11 b(on)i(the)i(righ)o(t.)i(In)d (this)g(w)o(a)o(y)m(,)e Fl(L)j Fm(is)e(an)h Fl(R)1042 884 y Fn(e)1060 899 y Fm(-bimo)q(dule.)303 949 y(Consider)g(the)g(map) 551 1027 y Fl(')578 1034 y Fh(\(2.3\))654 1027 y Fm(:)f Fl(L)c Fo(\012)h Fl(L)f Fo(\012)h Fl(L)i Fo(!)f Fl(k)-724 b Fm(\(2.3\))600 1089 y Fl(e)619 1095 y Fh(11)664 1089 y Fo(\012)9 b Fl(e)724 1095 y Fh(12)769 1089 y Fo(\012)h Fl(e)830 1095 y Fh(21)877 1089 y Fo(7!)h Fm(1)590 1151 y Fl(e)609 1157 y Fn(ij)648 1151 y Fo(\012)e Fl(e)708 1157 y Fn(k)q(`)752 1151 y Fo(\012)h Fl(e)813 1157 y Fn(mn)877 1151 y Fo(7!)h Fm(0)i(for)h(other)g(sets)h(of)f(indices)p Fl(:)228 1229 y Fm(W)m(e)f(claim)f(that)478 1306 y Fl(')505 1313 y Fh(\(2.4\))580 1306 y Fm(:)i(\()p Fl(L)p Fo(\002)654 1336 y Fn(R)682 1306 y Fl(L)p Fm(\))p Fo(\002)730 1336 y Fn(R)759 1306 y Fl(L)e Fo(!)f Fl(k)q Fm(;)18 b Fl(`)9 b Fo(\012)h Fl(`)990 1289 y Fk(0)1011 1306 y Fo(\012)g Fl(`)1070 1289 y Fk(0)o(0)1103 1306 y Fo(7!)h Fl(')1183 1313 y Fh(\(2.3\))1253 1306 y Fm(\()p Fl(`)f Fo(\012)g Fl(`)1355 1289 y Fk(0)1376 1306 y Fo(\012)f Fl(`)1434 1289 y Fk(00)1456 1306 y Fm(\))-1244 b(\(2.4\))228 1401 y(is)14 b(w)o(ell)f(de\014ned.)19 b(In)14 b(fact)g(it)f(is)h(immedia)o (te)e(to)i(c)o(hec)o(k)g(that)g Fl(')1212 1408 y Fh(\(2.3\))1297 1401 y Fm(factors)g(o)o(v)o(er)771 1478 y Fl(')798 1485 y Fh(\(2.5\))873 1478 y Fm(:)g Fl(L)9 b Fo(\005)g Fl(L)g Fo(\012)h Fl(L)i Fo(!)f Fl(k)q(;)-945 b Fm(\(2.5\))228 1556 y(and)14 b(the)g(restriction)h(of)e Fl(')653 1563 y Fh(\(2.5\))738 1556 y Fm(to)g Fl(L)p Fo(\002)849 1562 y Fn(R)876 1556 y Fl(L)d Fo(\012)f Fl(L)14 b Fm(v)n(anishes,)g(since)g Fl(e)1292 1562 y Fh(11)1337 1556 y Fo(\012)c Fl(e)1398 1562 y Fh(12)1445 1556 y Fo(62)h Fl(L)p Fo(\002)1544 1562 y Fn(R)1572 1556 y Fl(L)p Fm(.)303 1606 y(Clearly)18 b(the)h(canonical)f(morphisms)f Fl(\013)958 1591 y Fk(0)974 1606 y Fm(:)e Fl(L)p Fo(\002)1061 1612 y Fn(R)1088 1606 y Fm(\()p Fl(L)p Fo(\002)1165 1612 y Fn(R)1192 1606 y Fl(L)p Fm(\))20 b Fo(!)g Fl(L)p Fo(\002)1378 1612 y Fn(R)1405 1606 y Fl(L)p Fo(\002)1466 1612 y Fn(R)1493 1606 y Fl(L)f Fm(as)g(w)o(ell)f(as)228 1656 y Fl(\013)5 b Fm(:)14 b(\()p Fl(L)p Fo(\002)363 1662 y Fn(R)390 1656 y Fl(L)p Fm(\))p Fo(\002)466 1662 y Fn(R)494 1656 y Fl(L)k Fo(!)g Fl(L)p Fo(\002)660 1662 y Fn(R)688 1656 y Fl(L)p Fo(\002)748 1662 y Fn(R)775 1656 y Fl(L)g Fm(are)h(isomorphisms,)c(so)j(one)g(w)o (ould)g(exp)q(ect)h(that)f(one)228 1705 y(also)13 b(has)h(a)g(w)o(ell)f (de\014ned)i(map)478 1783 y Fl(')505 1790 y Fh(\(2.6\))580 1783 y Fm(:)f Fl(L)p Fo(\002)638 1813 y Fn(R)666 1783 y Fm(\()p Fl(L)p Fo(\002)714 1813 y Fn(R)743 1783 y Fl(L)p Fm(\))e Fo(!)f Fl(k)q Fm(;)18 b Fl(`)9 b Fo(\012)h Fl(`)990 1766 y Fk(0)1011 1783 y Fo(\012)g Fl(`)1070 1766 y Fk(0)o(0)1103 1783 y Fo(7!)h Fl(')1183 1790 y Fh(\(2.3\))1253 1783 y Fm(\()p Fl(`)f Fo(\012)g Fl(`)1355 1766 y Fk(0)1376 1783 y Fo(\012)f Fl(`)1434 1766 y Fk(00)1456 1783 y Fm(\))-1244 b(\(2.6\))228 1880 y(Ho)o(w)o(ev)o(er,)14 b(w)o(e)f(ha)o(v)o(e)h(0)d(=) h Fl(e)658 1886 y Fh(12)702 1880 y Fo(\012)e Fl(e)763 1886 y Fh(21)810 1880 y Fo(2)h Fl(L)e Fo(\005)g Fl(L)p Fm(,)k(hence)i(0)c(=)h Fl(e)1179 1886 y Fh(11)1224 1880 y Fo(\012)d Fl(e)1284 1886 y Fh(12)1328 1880 y Fo(\012)g Fl(e)1388 1886 y Fh(21)1435 1880 y Fo(2)j Fl(L)p Fo(\002)1535 1886 y Fn(R)1562 1880 y Fm(\()p Fl(L)p Fo(\002)1639 1886 y Fn(R)1666 1880 y Fl(L)p Fm(\),)228 1930 y(but)i Fl(')331 1937 y Fh(\(2.6\))415 1930 y Fm(w)o(ould)f(ha)o(v)o(e)h(to)g(map)e (this)i(to)g(1)d Fo(2)g Fl(k)q Fm(.)303 2010 y(The)i(example)f(sho)o (ws)i(that)g(whenev)o(er)h(w)o(e)e(use)h(equalities)g(lik)o(e)e(coasso) q(ciativit)o(y)h(within)228 2060 y(a)e(computation,)f(where)i(they)g(o) q(ccur)h(substituted)f(in)o(to)f(some)g(kind)g(of)g(m)o(ultili)o(near)e (form)o(ula,)228 2110 y(w)o(e)14 b(ha)o(v)o(e)f(to)h(mak)o(e)e(sure)i (that)g(all)e(the)j(expressions)g(w)o(e)f(consider)g(are)g(w)o(ell)f (de\014ned,)h(and)g(w)o(e)228 2159 y(ev)o(en)e(ha)o(v)o(e)f(to)g(do)g (this)h(when,)g(inside)f(a)g(w)o(ell-de\014ned)h(expression,)h(w)o(e)e (wish)g(to)h(replace)g(some)228 2209 y(tensor)17 b(b)o(y)f(another)h (one,)g(whic)o(h)f(is)g(\\equal".)25 b(Due)17 b(to)f(this)g(necessit)o (y)m(,)i(w)o(e)f(ha)o(v)o(e)f(to)g(use)h(a)228 2259 y(large)d(n)o(um)o (b)q(er)h(of)f(auxiliary)f(maps,)h(and)h(since)g(it)g(seems)g(p)q(oin)o (tless)g(to)g(in)o(v)o(en)o(t)g(meaningful)228 2309 y(names)e(for)h (all)f(of)g(them,)g(w)o(e)h(shall)g(often)g(tak)o(e)g(refuge)g(in)g (the)h(notation)e(already)h(emplo)o(y)o(ed)228 2359 y(ab)q(o)o(v)o(e:)k (The)c(auxiliary)e(map)g(de\014ned)j(in)f(equation)f(\(2.3\))g(is)h (called)g(simply)d Fl(')1491 2366 y Fh(\(2.3\))1562 2359 y Fm(.)303 2409 y(The)g(category)h(of)f(mo)q(dules)f(o)o(v)o(er)h(a)g Fo(\002)906 2415 y Fn(R)933 2409 y Fm(-bialgebra)f Fl(L)i Fm(has)f(the)h(structure)h(of)e(a)g(monoidal)228 2458 y(category)m(.)17 b(The)11 b(tensor)h(pro)q(duct)g(in)809 2464 y Fn(L)834 2458 y Fo(M)f Fm(of)f(t)o(w)o(o)h Fl(L)p Fm(-mo)q(dules)f Fl(M)15 b Fm(and)c Fl(N)16 b Fm(is)11 b Fl(M)d Fo(\005)s Fl(N)15 b Fm(equipp)q(ed)228 2508 y(with)e(the)h(left)f Fl(L)p Fm(-mo)q(dule)f(structure)k Fl(`)8 b(.)g Fm(\()p Fl(m)h Fo(\012)g Fl(n)p Fm(\))j(=)f Fl(`)1100 2515 y Fh(\(1\))1154 2508 y Fl(.)d(m)g Fo(\012)h Fl(`)1285 2515 y Fh(\(2\))1338 2508 y Fl(.)f(n)p Fm(.)18 b(In)13 b(particular)h(the)228 2558 y(underlying)i(functor)584 2564 y Fn(L)609 2558 y Fo(M)f(!)732 2564 y Fn(R)757 2556 y Fe(e)775 2558 y Fo(M)h Fm(is)h(a)f(monoidal)d(functor;)18 b(essen)o(tially)e Fo(\002)1499 2564 y Fn(R)1526 2558 y Fm(-bialgebras)228 2608 y(can)e(b)q(e)g(c)o(haracterized)i(b)o(y)e (this)g(b)q(eha)o(vior.)p eop %%Page: 7 7 7 6 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)310 b(7)303 213 y Fm(The)18 b(suitable)f(de\014nition)h(of)f (como)q(dules)g(o)o(v)o(er)h(a)f Fo(\002)1159 219 y Fn(R)1187 213 y Fm(-bialgebra)f Fl(L)i Fm(is)g(as)g(follo)o(ws:)24 b(A)228 262 y(left)19 b Fl(L)p Fm(-como)q(dule)f(is)h(an)h Fl(R)p Fm(-bimo)q(dule)d Fl(M)24 b Fm(together)c(with)f(a)g(map)f Fl(\025)5 b Fm(:)15 b Fl(M)26 b Fo(!)20 b Fl(L)p Fo(\002)1597 268 y Fn(R)1625 262 y Fl(M)k Fm(of)228 312 y Fl(R)p Fm(-bimo)q(dules)12 b(suc)o(h)j(that)607 385 y Fl(\013)634 368 y Fk(0)646 385 y Fm(\()p Fl(L)p Fo(\002)694 415 y Fn(R)723 385 y Fl(\025)p Fm(\))p Fl(\025)d Fm(=)f Fl(\013)p Fm(\(\001)p Fo(\002)924 415 y Fn(R)952 385 y Fl(M)5 b Fm(\))p Fl(\025)g Fm(:)13 b Fl(M)k Fo(!)11 b Fl(L)p Fo(\002)1209 415 y Fn(R)1237 385 y Fl(L)p Fo(\002)1269 415 y Fn(R)1298 385 y Fl(M)228 477 y Fm(and)17 b Fl(\022)332 462 y Fk(0)344 477 y Fm(\()p Fl(")p Fo(\002)412 483 y Fn(R)439 477 y Fl(M)5 b Fm(\))p Fl(\025)17 b Fm(=)g(id)624 483 y Fn(M)678 477 y Fm(hold.)27 b(If)17 b(w)o(e)g(denote)h(b)o(y)e Fl(\025)1126 483 y Fh(0)1150 477 y Fm(:)e Fl(M)22 b Fo(!)16 b Fl(L)c Fo(\005)f Fl(M)21 b Fm(the)d(comp)q(osition)228 527 y(of)c Fl(\025)g Fm(with)g(the)h(inclusion)f(of)f Fl(L)d Fo(\005)g Fl(M)19 b Fm(in)o(to)13 b Fl(L)p Fo(\002)975 533 y Fn(R)1002 527 y Fl(M)5 b Fm(,)14 b(then)h(coasso)q(ciativit)o(y)e (is)i(equiv)n(alen)o(t)e(to)228 577 y(\()p Fl(L)5 b Fo(\005)g Fl(\025)327 583 y Fh(0)346 577 y Fm(\))p Fl(\025)386 583 y Fh(0)416 577 y Fm(=)12 b(\(\001)511 583 y Fh(0)534 577 y Fo(\005)5 b Fl(M)g Fm(\))p Fl(\025)645 583 y Fh(0)669 577 y Fm(:)13 b Fl(M)j Fo(!)11 b Fl(L)5 b Fo(\005)g Fl(L)g Fo(\005)g Fl(M)17 b Fm(and)11 b(\()p Fl(")1091 583 y Fh(0)1115 577 y Fo(\005)5 b Fl(M)g Fm(\))p Fl(\025)1226 583 y Fh(0)1257 577 y Fm(=)11 b(id)1335 583 y Fn(M)1372 577 y Fm(.)17 b(W)m(e)11 b(will)g(denote)h(b)o(y)228 612 y Fn(L)253 627 y Fo(M)j Fm(the)h(category)g(of)f(left)g Fl(L)p Fm(-como)q(dules.)22 b(W)m(e)15 b(will)f(use)i(Sw)o(eedler)g (notation)f(in)g(the)h(form)228 677 y Fl(\025)p Fm(\()p Fl(m)p Fm(\))c(=)g Fl(m)412 684 y Fh(\()p Fk(\000)p Fh(1\))489 677 y Fo(\012)6 b Fl(m)563 684 y Fh(\(0\))621 677 y Fm(and)12 b Fl(\013)p Fm(\(\001)p Fo(\002)810 683 y Fn(R)837 677 y Fl(M)5 b Fm(\)\()p Fl(m)p Fm(\))12 b(=)g Fl(m)1058 684 y Fh(\()p Fk(\000)p Fh(2\))1135 677 y Fo(\012)6 b Fl(m)1209 684 y Fh(\()p Fk(\000)p Fh(1\))1287 677 y Fo(\012)g Fl(m)1361 684 y Fh(\(0\))1418 677 y Fm(for)12 b Fl(L)p Fm(-como)q(dules.)303 730 y(The)k(category)560 715 y Fn(L)584 730 y Fo(M)g Fm(of)g(left)g Fl(L)p Fm(-como)q(dules)f(o)o(v)o (er)h(a)g Fo(\002)1181 736 y Fn(R)1208 730 y Fm(-bialgebra)f(is)h (monoidal.)22 b(The)228 780 y(tensor)16 b(pro)q(duct)f(of)g Fl(M)r(;)7 b(N)18 b Fo(2)712 765 y Fn(L)736 780 y Fo(M)d Fm(is)g(their)g(tensor)h(pro)q(duct)g Fl(M)i Fo(\012)1317 786 y Fn(R)1357 780 y Fl(N)i Fm(o)o(v)o(er)15 b Fl(R)p Fm(,)f(equipp)q(ed)228 830 y(with)f(the)i(como)q(dule)d(structure)647 903 y Fl(M)k Fo(\012)706 933 y Fn(R)747 903 y Fl(N)g Fo(!)11 b Fl(L)p Fo(\002)881 933 y Fn(R)910 903 y Fm(\()p Fl(M)16 b Fo(\012)986 933 y Fn(R)1026 903 y Fl(N)5 b Fm(\))672 982 y Fl(m)10 b Fo(\012)g Fl(n)h Fo(7!)g Fl(m)885 989 y Fh(\()p Fk(\000)p Fh(1\))956 982 y Fl(n)981 989 y Fh(\()p Fk(\000)p Fh(1\))1061 982 y Fo(\012)e Fl(m)1138 989 y Fh(\(0\))1192 982 y Fo(\012)h Fl(n)1259 989 y Fh(\(0\))505 1082 y Fp(3.)24 b Fo(\002)598 1088 y Fn(R)625 1082 y Fp(-Hopf)16 b(algebras)e(and)h(dualit)o(y)e(for)i(mo)q(dules)303 1157 y Fm(W)m(e)e(will)g(giv)o(e)g(a)h(de\014nition)g(of)f Fo(\002)837 1163 y Fn(R)865 1157 y Fm(-Hopf)g(algebras)h(in)g(the)g (spirit)g(of)g(T)m(annak)n(a)e(dualit)o(y)m(,)228 1207 y(that)19 b(is,)h(w)o(e)f(will)f(try)h(to)g(\014nd)h(a)f(de\014nition)f (for)h(whic)o(h)g(the)h(prop)q(ert)o(y)g(of)f(b)q(eing)g(a)g(Hopf)228 1257 y(algebra)13 b(rather)h(than)g(just)f(a)g(bialgebra)g(re\015ects)j (prop)q(erties)e(of)f(the)h(monoidal)d(category)j(of)228 1307 y(represen)o(tations.)303 1356 y(It)g(is)h(w)o(ell)e(kno)o(wn)i (that)f(an)h(ordinary)f(bialgebra)f Fl(H)18 b Fm(o)o(v)o(er)c(a)h (\014eld)f Fl(k)i Fm(is)e(a)g(Hopf)h(algebra)228 1406 y(if)e(and)h(only)g(if)f(the)i(category)g(of)f(its)g(\014nite)h (dimensional)d(como)q(dules)h(is)h(rigid)g([)p Fp(21)o Fm(].)19 b(Here)c(a)228 1456 y(monoidal)9 b(category)14 b Fo(C)h Fm(is)d(said)g(to)h(b)q(e)g(rigid)f(if)g(ev)o(ery)h(ob)r(ject) h Fl(X)h Fo(2)c(C)k Fm(has)e(a)f(left)h(dual)f(ob)r(ject,)228 1506 y(whic)o(h)k(in)f(turn)i(is)f(a)g(triple)g(\()p Fl(X)741 1491 y Fk(\003)761 1506 y Fl(;)7 b Fm(ev)p Fl(;)g Fm(db\))16 b(where)h Fl(X)1076 1491 y Fk(\003)1112 1506 y Fm(is)f(an)g(ob)r(ject)h(of)e Fo(C)r Fm(,)i(ev)5 b(:)14 b Fl(X)1554 1491 y Fk(\003)1585 1506 y Fo(\012)d Fl(X)19 b Fo(!)228 1556 y Fl(I)j Fm(and)c(db)t(:)d Fl(I)23 b Fo(!)18 b Fl(X)e Fo(\012)c Fl(X)662 1541 y Fk(\003)700 1556 y Fm(are)19 b(morphisms)d(satisfying)h(\()p Fl(X)f Fo(\012)d Fm(ev\)\(db)8 b Fo(\012)p Fl(X)s Fm(\))20 b(=)f(id)1606 1562 y Fn(X)1655 1556 y Fm(and)228 1605 y(\(ev)8 b Fo(\012)p Fl(X)361 1590 y Fk(\003)381 1605 y Fm(\)\()p Fl(X)450 1590 y Fk(\003)479 1605 y Fo(\012)i Fm(db)q(\))i(=)h(id)676 1611 y Fn(X)705 1603 y Fd(\003)725 1605 y Fm(.)20 b(In)14 b(the)h(category)g(of)f(\014nite)h(dimensional)d(righ)o(t)i(como)q (dules)228 1655 y(o)o(v)o(er)e(a)f(Hopf)h(algebra)f Fl(H)k Fm(o)o(v)o(er)d(a)g(\014eld,)g(the)g(dual)g(v)o(ector)g(space,)h (equipp)q(ed)g(with)f(the)g(unique)228 1705 y(como)q(dule)k(structure)k (satisfying)d(ev\()p Fl(\036)869 1712 y Fh(\(0\))926 1705 y Fo(\012)12 b Fl(v)q Fm(\))p Fl(\036)1032 1712 y Fh(\(1\))1094 1705 y Fm(=)18 b(ev)q(\()p Fl(\036)12 b Fo(\012)g Fl(v)1303 1712 y Fh(\(0\))1348 1705 y Fm(\))p Fl(S)r Fm(\()p Fl(v)1428 1712 y Fh(\(1\))1474 1705 y Fm(\))18 b(is)f(a)h(dual)f(in)228 1755 y(the)k(ab)q(o)o(v)o(e)f(sense.) 38 b(Ho)o(w)o(ev)o(er,)22 b(in\014nite)e(dimensional)e Fl(H)s Fm(-como)q(dules)h(can)h(nev)o(er)h(ha)o(v)o(e)f(a)228 1805 y(dual)14 b(ob)r(ject.)23 b(In)16 b(particular,)e(since)j (\014nite)e(mo)q(dules)f(alone)h(are)h(not)f(enough)g(to)g(sa)o(y)g(m)o (uc)o(h)228 1854 y(ab)q(out)f(an)g(algebra)f(\(the)i(situation)f(b)q (eing)g(di\013eren)o(t)h(for)f(como)q(dules)f(due)i(to)f(the)h (\014niteness)228 1904 y(theorems\),)e(w)o(e)g(cannot)h(hop)q(e)g(to)f (cop)o(y)g(this)g(result)h(for)f(mo)q(dules)g(o)o(v)o(er)g(bialgebras,) f(or)h(ev)o(en)228 1954 y(less)h(for)g(mo)q(dules)f(o)o(v)o(er)h Fo(\002)652 1960 y Fn(R)679 1954 y Fm(-bialgebras.)303 2004 y(A)i(suitable)g(w)o(eak)o(er)h(concept)h(of)e(dualit)o(y)f(for)i (monoidal)c(categories)18 b(is)e(the)h(notion)f(of)228 2054 y(an)d(inner)i(hom-functor.)303 2132 y Fc(Definition)g Fm(3.1)p Fc(.)21 b Fm(Let)14 b Fo(C)i Fm(b)q(e)f(a)e(monoidal)e (category)m(.)303 2181 y(Let)j Fl(Y)23 b Fm(b)q(e)14 b(an)g(ob)r(ject)g(of)f Fo(C)r Fm(.)19 b(If)13 b(the)h(functor)h Fo(C)e(3)f Fl(X)j Fo(7!)c Fl(X)h Fo(\012)e Fl(Y)20 b Fo(2)12 b(C)k Fm(is)d(left)h(adjoin)o(t,)e(w)o(e)228 2231 y(denote)k(its)f(righ)o(t)g(adjoin)o(t)f(b)o(y)h(b)o(y)g Fo(C)h(3)d Fl(X)18 b Fo(7!)13 b Fm(hom)n(\()p Fl(Y)r(;)7 b(X)s Fm(\))15 b Fo(2)e(C)18 b Fm(and)d(call)f(it)h(a)g(\(righ)o(t\))g (inner)228 2281 y(hom-functor.)303 2331 y(If)d(the)h(functor)f({)c Fo(\012)p Fl(Y)22 b Fm(is)12 b(left)g(adjoin)o(t)g(for)g(ev)o(ery)h Fl(Y)21 b Fo(2)11 b(C)r Fm(,)h(then)h(w)o(e)g(sa)o(y)f(that)g Fo(C)j Fm(is)d(\(righ)o(t\))228 2381 y(closed.)303 2458 y(By)e(the)g(standard)h(mac)o(hinery)e(of)g(adjoin)o(t)g(functors,)i (an)f(inner)g(hom-functor)f(hom)n(\()p Fl(X)q(;)e Fm({)q(\))228 2508 y(comes)14 b(with)h(an)g(adjunction)g(morphism)d(ev)6 b(:)20 b(hom)o(\()p Fl(X)q(;)7 b(Y)i Fm(\))h Fo(\012)h Fl(X)17 b Fo(!)c Fl(Y)24 b Fm(with)15 b(the)h(follo)o(wing)228 2558 y(univ)o(ersal)f(prop)q(ert)o(y:)21 b(F)m(or)15 b(eac)o(h)h Fl(T)k Fo(2)13 b(C)18 b Fm(and)d Fl(e)5 b Fm(:)14 b Fl(T)i Fo(\012)10 b Fl(X)18 b Fo(!)13 b Fl(Y)25 b Fm(there)16 b(is)f(a)g(unique)h Fl(f)9 b Fm(:)14 b Fl(T)20 b Fo(!)228 2608 y Fm(hom)n(\()p Fl(X)q(;)7 b(Y)i Fm(\))14 b(suc)o(h)h(that)f Fl(e)e Fm(=)g(ev)q(\()p Fl(f)i Fo(\012)9 b Fl(X)s Fm(\).)p eop %%Page: 8 8 8 7 bop 228 113 a Fh(8)536 b(PETER)16 b(SCHA)o(UENBUR)o(G)303 213 y Fm(If)d Fl(X)18 b Fm(has)c(a)f(left)h(dual,)f(then)h({)8 b Fo(\012)p Fl(X)18 b Fm(is)13 b(left)h(adjoin)o(t)f(with)g(hom)o(\()p Fl(X)q(;)7 b(Y)i Fm(\))j(=)f Fl(Y)19 b Fo(\012)10 b Fl(X)1612 198 y Fk(\003)303 262 y Fm(Of)15 b(course,)h(the)g(notion)f(of)g(inner) h(hom-functor)e(is)h(mo)q(delled)f(on)h(the)h(standard)g(hom-)228 312 y(tensor)h(adjunction)e(for)g(\(bi\)mo)q(dules.)23 b(In)15 b(particular,)h(the)g(category)g(\()1397 318 y Fn(R)1422 310 y Fe(e)1441 312 y Fo(M)p Fl(;)7 b Fo(\005)p Fm(\),)15 b(whic)o(h)g(is)228 362 y(isomorphic)d(to)i(\()505 368 y Fn(R)532 362 y Fo(M)582 368 y Fn(R)609 362 y Fl(;)7 b Fo(\012)660 368 y Fn(R)687 362 y Fm(\),)14 b(is)f(closed.)19 b(W)m(e)14 b(note)g(the)g(details)g(for)f(later)h(use:)303 439 y Fc(Remark)i Fm(3.2)p Fc(.)k Fm(The)13 b(monoidal)c(category)k(\() 1013 445 y Fn(R)1038 437 y Fe(e)1056 439 y Fo(M)p Fl(;)7 b Fo(\005)p Fm(\))k(is)h(righ)o(t)g(closed)h(with)f(inner)g(hom-)228 489 y(functors)f(hom)463 500 y Fe(R)484 494 y(e)503 495 y Fk(M)544 489 y Fm(\()p Fl(N)r(;)c(P)f Fm(\))11 b(=)h(Hom)p 804 477 28 2 v 804 502 a Fn(R)832 489 y Fm(\()p Fl(N)r(;)7 b(P)f Fm(\))j(for)h Fl(N)r(;)d(P)17 b Fo(2)1157 495 y Fn(R)1182 487 y Fe(e)1201 489 y Fo(M)p Fm(,)10 b(where)h(the)g(left)f Fl(R)1558 474 y Fn(e)1576 489 y Fm(-mo)q(dule)228 539 y(structure)16 b(is)e(giv)o(en)f(b)o(y)h(\(\()p Fl(r)c Fo(\012)p 717 516 20 2 v 10 w Fl(s)p Fm(\))p Fl(f)t Fm(\)\()p Fl(n)p Fm(\))j(=)f Fl(r)q(f)t Fm(\()p Fl(sn)p Fm(\).)303 616 y(If)j Fl(H)k Fm(is)e(a)f(Hopf)f(algebra,)h(then)h(the)g Fl(k)q Fm(-mo)q(dule)e(Hom)1188 622 y Fn(k)1208 616 y Fm(\()p Fl(V)r(;)7 b(W)f Fm(\))17 b(of)e Fl(k)q Fm(-linear)h(maps)f (has)228 666 y(a)g(canonical)g Fl(H)s Fm(-mo)q(dule)e(structure,)18 b(de\014ned)e(b)o(y)f(\()p Fl(hf)t Fm(\)\()p Fl(v)q Fm(\))h(=)e Fl(h)1261 673 y Fh(\(1\))1306 666 y Fl(f)t Fm(\()p Fl(S)r Fm(\()p Fl(h)1413 673 y Fh(\(2\))1459 666 y Fm(\))p Fl(v)q Fm(\))i(for)f Fl(h)f Fo(2)g Fl(H)s Fm(,)228 716 y Fl(f)i Fo(2)11 b Fm(Hom)390 722 y Fn(k)410 716 y Fm(\()p Fl(V)r(;)c(W)f Fm(\))12 b(and)f Fl(v)i Fo(2)e Fl(V)e Fm(.)17 b(With)11 b(this)g(structure,)i(Hom)1213 722 y Fn(k)1233 716 y Fm(\()p Fl(V)r(;)7 b(W)f Fm(\))12 b(de\014nes)g(a)f(righ)o(t)g(inner) 228 765 y(Hom-functor)h(in)i(the)g(category)h(of)e(left)h Fl(H)s Fm(-mo)q(dules.)303 815 y(No)o(w)19 b(while)h(a)f(Hopf)h (algebra)g(allo)o(ws)e(us)j(to)f(construct)h(inner)g(hom-functors)e(in) g(its)228 865 y(mo)q(dule)12 b(category)m(,)g(the)i(con)o(v)o(erse)h (is)e(not)g(true:)18 b(a)13 b(bialgebra)g(needs)h(not)f(b)q(e)h(a)f (Hopf)g(algebra)228 915 y(for)j(its)h(mo)q(dule)f(category)h(to)g(b)q (e)h(closed.)27 b(On)18 b(the)f(con)o(trary)h(the)f(mo)q(dule)f (category)h(o)o(v)o(er)228 965 y(an)o(y)c(bialgebra)g(and)h(ev)o(en)g Fo(\002)694 971 y Fn(R)721 965 y Fm(-bialgebra)f(is)h(closed.)303 1042 y Fc(Pr)o(oposition)h Fm(3.3)p Fc(.)21 b Fg(L)n(et)d Fl(L)g Fg(b)n(e)h(a)g Fo(\002)905 1048 y Fn(R)932 1042 y Fg(-bialgebr)n(a.)29 b(Then)19 b(the)f(c)n(ate)n(gory)1497 1048 y Fn(L)1521 1042 y Fo(M)h Fg(is)f(right)228 1092 y(close)n(d)d(with)f(right)g(inner)h(hom-functors)669 1163 y Fm(hom)747 1173 y Fe(L)769 1169 y Fk(M)810 1163 y Fm(\()p Fl(N)r(;)7 b(P)f Fm(\))11 b(=)h(Hom)1070 1169 y Fn(L)1095 1163 y Fm(\()p Fl(L)e Fo(\005)f Fl(N)r(;)e(P)f Fm(\))228 1234 y Fg(wher)n(e)14 b Fl(L)8 b Fo(\005)g Fl(N)19 b Fg(is)14 b(an)h Fl(L)p Fg(-)p Fl(L)p Fg(-bimo)n(dule)g(with)e (the)i(left)e Fl(L)p Fg(-mo)n(dule)i(structur)n(e)f(as)g(ab)n(ove,)h (and)g(the)228 1284 y(right)f Fl(L)p Fg(-mo)n(dule)h(structur)n(e)f (induc)n(e)n(d)i(by)f(that)g(of)g(the)g(factor)f Fl(L)p Fg(.)303 1361 y Fc(Pr)o(oof.)21 b Fm(Using)14 b(that)564 1433 y Fl(M)g Fo(\005)9 b Fl(N)17 b Fo(3)11 b Fl(m)f Fo(\012)f Fl(n)j Fo(7!)f Fm(1)e Fo(\012)g Fl(n)g Fo(\012)h Fl(m)i Fo(2)f Fm(\()p Fl(L)f Fo(\005)f Fl(N)c Fm(\))11 b Fo(\012)1302 1463 y Fn(L)1341 1433 y Fl(M)228 1524 y Fm(is)j(an)f(isomorphism)e(with)i(in)o(v)o(erse)i(\()p Fl(`)10 b Fo(\012)f Fl(n)p Fm(\))g Fo(\012)h Fl(m)i Fo(7!)f Fl(`m)f Fo(\012)g Fl(n)p Fm(,)j(w)o(e)h(obtain)297 1595 y(Hom)383 1601 y Fn(L)408 1595 y Fm(\()p Fl(M)g Fo(\005)9 b Fl(N)r(;)e(P)f Fm(\))622 1584 y Fo(\030)622 1597 y Fm(=)666 1595 y(Hom)753 1601 y Fn(L)777 1595 y Fm(\(\()p Fl(L)k Fo(\005)f Fl(N)c Fm(\))12 b Fo(\012)947 1626 y Fn(L)986 1595 y Fl(M)r(;)7 b(P)f Fm(\))1107 1584 y Fo(\030)1107 1597 y Fm(=)1151 1595 y(Hom)1238 1601 y Fn(L)1263 1595 y Fm(\()p Fl(M)r(;)h Fm(Hom)1426 1601 y Fn(L)1451 1595 y Fm(\()p Fl(L)j Fo(\005)f Fl(N)r(;)e(P)f Fm(\)\))228 1684 y(from)12 b(the)i(standard)h(hom-tensor)e(adjunction.)p 1692 1684 2 29 v 1694 1658 25 2 v 1694 1684 V 1719 1684 2 29 v 303 1765 a(No)o(w)j(to)g(sa)o(v)o(e)h(the)g(idea)f(that)h(b)q (eing)f(a)h(Hopf)f(algebra)g(is)g(related)h(to)g(the)g(category)g(of) 228 1815 y(mo)q(dules)g(b)q(eing)h(closed,)h(w)o(e)f(need)i(to)e (observ)o(e)h(that)f(the)h(inner)f(hom-functors)f(for)h(mo)q(d-)228 1864 y(ules)c(o)o(v)o(er)f(a)g(Hopf)g(algebra)g(are)h(mo)q(delled)e(on) h(the)h(inner)g(hom-functors)e(in)h(the)i(underlying)228 1914 y(category)d(of)g Fl(k)q Fm(-mo)q(dules,)e(while)i(this)g(is)g (not)f(so)i(in)e(the)i(bialgebra)e(case.)18 b(In)12 b(a)g(more)f (categor-)228 1964 y(ical)j(language,)h(the)g(underlying)g(functor)918 1970 y Fn(H)949 1964 y Fo(M)f(!)1069 1970 y Fn(k)1089 1964 y Fo(M)h Fm(for)g(a)g(Hopf)g(algebra)g Fl(H)j Fm(preserv)o(es)228 2014 y(hom-functors,)12 b(in)i(the)g(follo)o(wing)d(sense)16 b(\(cf.)e([)p Fp(13)o Fm(,)f(I.4.3]\):)303 2091 y Fc(Definition)i Fm(3.4)p Fc(.)21 b Fm(Let)c Fo(F)9 b Fm(:)14 b Fo(C)20 b(!)c(D)j Fm(b)q(e)e(a)g(monoidal)e(functor)i(b)q(et)o(w)o(een)i(righ)o (t)e(closed)228 2141 y(monoidal)11 b(categories,)j(and)g(let)619 2212 y Fl(\020)8 b Fm(:)13 b Fo(F)t Fm(\(hom)n(\()p Fl(X)q(;)7 b(Y)j Fm(\)\))i Fo(!)f Fm(hom)n(\()p Fo(F)t Fm(\()p Fl(X)s Fm(\))p Fl(;)c Fo(F)t Fm(\()p Fl(Y)j Fm(\)\))228 2284 y(for)j Fl(X)q(;)7 b(Y)21 b Fo(2)11 b(C)16 b Fm(b)q(e)f(the)f(unique)g (morphism)d(for)j(whic)o(h)469 2585 y Fo(F)t Fm(\(hom)n(\()p Fl(X)q(;)7 b(Y)j Fm(\))f Fo(\012)g Fl(X)s Fm(\))397 b Fo(F)t Fm(\()p Fl(Y)10 b Fm(\))p 833 2576 373 2 v 1164 2575 a Fb(-)966 2614 y Fo(F)t Fm(\(ev)q(\))436 2378 y Fo(F)t Fm(\(hom)n(\()p Fl(X)q(;)d(Y)i Fm(\)\))h Fo(\012)f(F)t Fm(\()p Fl(X)s Fm(\))174 b(hom)o(\()p Fo(F)t Fm(\()p Fl(X)s Fm(\))p Fl(;)7 b Fo(F)t Fm(\()p Fl(Y)i Fm(\)\))h Fo(\012)f(F)t Fm(\()p Fl(X)s Fm(\))p 867 2368 145 2 v 970 2367 a Fb(-)851 2348 y Fl(\020)j Fo(\012)e(F)t Fm(\()p Fl(X)s Fm(\))p 644 2546 2 150 v 645 2546 a Fb(?)596 2472 y Fo(\030)596 2485 y Fm(=)p 1266 2546 V 1267 2546 a Fb(?)1284 2480 y Fm(ev)p eop %%Page: 9 9 9 8 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)310 b(9)228 213 y Fm(comm)o(utes.)17 b(W)m(e)d(sa)o(y)g(that)g Fo(F)k Fm(preserv)o(es)f(righ)o(t)d(inner)g(hom-functors)g(if)f Fl(\020)k Fm(an)d(isomorphism)228 262 y(for)f(all)g Fl(X)q(;)7 b(Y)21 b Fo(2)11 b(C)r Fm(.)303 338 y Fc(Theorem)16 b(and)h(Definition) e Fm(3.5)p Fc(.)21 b Fg(L)n(et)14 b Fl(L)g Fg(b)n(e)h(a)f Fo(\002)1151 344 y Fn(R)1179 338 y Fg(-bialgebr)n(a)f Fl(L)p Fg(.)19 b(Then)c(the)f(fol)r(low-)228 387 y(ing)h(ar)n(e)f(e)n (quivalent:)303 446 y Fm(\(1\))20 b Fg(The)15 b(underlying)g(functor) 810 452 y Fn(L)835 446 y Fo(M)c(!)949 452 y Fn(R)974 444 y Fe(e)993 446 y Fo(M)k Fg(pr)n(eserves)f(right)g(inner)h (hom-functors.)303 499 y Fm(\(2\))20 b Fg(The)15 b(map)558 488 y Fm(~)552 499 y Fl(\014)8 b Fm(:)13 b Fl(L)f Fo(\012)p 680 488 28 2 v 13 x Fn(R)719 499 y Fl(L)g Fo(3)f Fl(`)e Fo(\012)h Fl(m)i Fo(7!)f Fl(`)984 506 y Fh(\(1\))1038 499 y Fo(\012)f Fl(`)1097 506 y Fh(\(2\))1141 499 y Fl(m)i Fo(2)g Fl(L)d Fo(\005)g Fl(L)15 b Fg(is)g(a)g(bije)n(ction.)228 558 y(If)f(these)h(e)n(quivalent)g(c)n(onditions)h(hold,)f(we)f(say)h (that)g Fl(L)g Fg(is)g(a)g Fo(\002)1245 564 y Fn(R)1273 558 y Fg(-Hopf)f(algebr)n(a,)h(and)g(write)688 627 y Fl(`)705 633 y Fh(+)742 627 y Fo(\012)10 b Fl(`)801 633 y Fk(\000)841 627 y Fm(:=)902 616 y(~)896 627 y Fl(\014)921 610 y Fk(\000)p Fh(1)966 627 y Fm(\()p Fl(`)g Fo(\012)g Fm(1\))h Fo(2)g Fl(L)h Fo(\012)p 1182 641 26 2 v 1182 665 a Fn(R)1222 627 y Fl(L:)303 737 y Fc(Pr)o(oof.)21 b Fm(Note)15 b(\014rst)f(that)g(if)786 726 y(~)780 737 y Fl(\014)j Fm(is)d(a)f(bijection,)g(then)i(so)f(is)576 795 y(~)570 806 y Fl(\014)593 812 y Fn(N)630 806 y Fm(:)f Fl(L)f Fo(\012)p 699 819 V 699 843 a Fn(R)739 806 y Fl(N)k Fo(3)11 b Fl(`)f Fo(\012)g Fl(n)h Fo(7!)g Fl(`)1002 813 y Fh(\(1\))1056 806 y Fo(\012)f Fl(`)1115 813 y Fh(\(2\))1169 806 y Fl(.)f(n)i Fo(2)g Fl(L)f Fo(\005)f Fl(N)228 908 y Fm(for)k(ev)o(ery)i Fl(N)h Fo(2)490 914 y Fn(L)514 908 y Fo(M)p Fm(,)e(since)g(w)o(e)g(can)h(iden)o(tify)986 898 y(~)980 908 y Fl(\014)1003 914 y Fn(N)1048 908 y Fm(with)1149 898 y(~)1143 908 y Fl(\014)f Fo(\012)1212 914 y Fn(L)1249 908 y Fl(N)5 b Fm(.)303 958 y(Let)14 b Fl(N)r(;)7 b(P)17 b Fo(2)514 964 y Fn(L)539 958 y Fo(M)p Fm(.)h(One)c(c)o(hec)o(ks)i(that)e(the)g(ev)n(aluation)700 1026 y(ev)6 b(:)20 b(Hom)865 1032 y Fn(L)889 1026 y Fm(\()p Fl(L)10 b Fo(\005)f Fl(N)r(;)e(P)f Fm(\))j Fo(\005)g Fl(N)16 b Fo(!)11 b Fl(P)228 1094 y Fm(for)k(the)g(inner)h(hom-functor) d(in)760 1100 y Fn(L)785 1094 y Fo(M)i Fm(is)g(giv)o(en)f(b)o(y)h(ev)q (\()p Fl(f)g Fo(\012)10 b Fl(n)p Fm(\))k(=)g Fl(f)t Fm(\(1)c Fo(\012)h Fl(n)p Fm(\).)21 b(On)16 b(the)g(other)228 1144 y(hand,)c(w)o(e)g(can)g(iden)o(tify)f(the)h(inner)h(hom-functor)d (Hom)p 1119 1133 28 2 v 1119 1157 a Fn(R)1146 1144 y Fm(\()p Fl(N)r(;)d(P)f Fm(\))11 b(in)1323 1150 y Fn(R)1348 1142 y Fe(e)1366 1144 y Fo(M)h Fm(canonically)e(with)228 1194 y(Hom)314 1200 y Fn(L)339 1194 y Fm(\()p Fl(L)i Fo(\012)p 427 1182 V 12 x Fn(R)466 1194 y Fl(N)r(;)7 b(P)f Fm(\);)13 b(under)i(this)f(iden)o(ti\014cation,)e(the)j(ev)n (aluation)d(map)686 1262 y(ev)727 1245 y Fk(0)744 1262 y Fm(:)20 b(Hom)862 1268 y Fn(L)887 1262 y Fm(\()p Fl(L)12 b Fo(\012)p 947 1275 26 2 v 947 1299 a Fn(R)987 1262 y Fl(N)r(;)7 b(P)f Fm(\))j Fo(\005)g Fl(N)16 b Fo(!)11 b Fl(P)228 1357 y Fm(is)j(giv)o(en)f(b)o(y)h(ev)477 1342 y Fk(0)488 1357 y Fm(\()p Fl(f)g Fo(\012)c Fl(n)p Fm(\))h(=)h Fl(f)t Fm(\(1)e Fo(\012)g Fl(n)p Fm(\).)18 b(Then)c(the)h(unique)f(map) 604 1425 y Fl(T)c Fm(:)21 b(Hom)757 1431 y Fn(L)782 1425 y Fm(\()p Fl(L)10 b Fo(\005)f Fl(N)r(;)e(P)f Fm(\))11 b Fo(!)g Fm(Hom)1119 1431 y Fn(L)1144 1425 y Fm(\()p Fl(L)h Fo(\012)p 1203 1438 V 1203 1462 a Fn(R)1244 1425 y Fl(N)r(;)7 b(P)f Fm(\))228 1518 y(compatible)12 b(with)i(the)g(resp)q (ectiv)o(e)i(ev)n(aluation)c(maps)h(is)h(giv)o(en)f(b)o(y)269 1598 y Fl(T)6 b Fm(\()p Fl(f)t Fm(\)\()p Fl(`)11 b Fo(\012)f Fl(n)p Fm(\))h(=)h Fl(`)e(.)e(T)e Fm(\()p Fl(f)t Fm(\)\(1)11 b Fo(\012)e Fl(n)p Fm(\))j(=)g Fl(`)d(.)g(f)t Fm(\(1)h Fo(\012)g Fl(n)p Fm(\))h(=)h Fl(f)t Fm(\()p Fl(`)e(.)f Fm(\(1)h Fo(\012)f Fl(n)p Fm(\)\))1069 1670 y(=)j Fl(f)t Fm(\()p Fl(`)1170 1677 y Fh(\(1\))1225 1670 y Fo(\012)e Fl(`)1284 1677 y Fh(\(2\))1338 1670 y Fl(.)f(n)p Fm(\))i(=)h Fl(f)1495 1659 y Fm(~)1488 1670 y Fl(\014)1511 1676 y Fn(N)1544 1670 y Fm(\()p Fl(`)e Fo(\012)f Fl(n)p Fm(\))p Fl(:)228 1748 y Fm(Y)m(oneda's)17 b(Lemma)e(no)o(w)j(sa)o(ys)g(that)f Fl(T)24 b Fm(is)18 b(a)f(bijection)h(for)f(all)g Fl(P)23 b Fm(if)17 b(and)g(only)g(if)1589 1737 y(~)1583 1748 y Fl(\014)1606 1754 y Fn(N)1656 1748 y Fm(is)g(a)228 1798 y(bijection.)p 1692 1798 2 29 v 1694 1772 25 2 v 1694 1798 V 1719 1798 2 29 v 303 1874 a Fc(Example)g Fm(3.6)p Fc(.)j Fm(Surely)f(the)h(\(iden)o(tical\))f(underlying)g (functor)1362 1880 y Fn(R)1387 1872 y Fe(e)1406 1874 y Fo(M)h(!)1538 1880 y Fn(R)1563 1872 y Fe(e)1582 1874 y Fo(M)f Fm(pre-)228 1924 y(serv)o(es)d(inner)f(hom-functors.)j(Hence)e Fl(R)882 1909 y Fn(e)914 1924 y Fm(is)e(a)g Fo(\002)1024 1930 y Fn(R)1051 1924 y Fm(-Hopf)g(algebra;)f(one)i(c)o(hec)o(ks)h(\()p Fl(r)10 b Fo(\012)p 1617 1901 20 2 v 10 w Fl(s)q Fm(\))1653 1930 y Fh(+)1690 1924 y Fo(\012)228 1973 y Fm(\()p Fl(r)g Fo(\012)p 315 1951 V 10 w Fl(s)p Fm(\))350 1979 y Fk(\000)390 1973 y Fm(=)i(\()p Fl(r)e Fo(\012)p 520 1942 21 2 v 9 w Fm(1\))f Fo(\012)h Fm(\()p Fl(s)g Fo(\012)p 694 1942 V 9 w Fm(1\))k(for)g Fl(r)o(;)7 b(s)k Fo(2)g Fl(R)p Fm(.)303 2049 y(In)18 b(the)g(case)h Fl(R)g Fm(=)f Fl(k)h Fm(of)f(an)g(ordinary) f(bialgebra)g Fl(L)p Fm(,)i(it)f(is)g(in)f(fact)h(w)o(ell-kno)o(wn)f (that)228 2099 y Fl(L)g Fm(is)g(a)g(Hopf)f(algebra)h(if)f(and)h(only)f (if)875 2089 y(~)870 2099 y Fl(\014)j Fm(is)e(a)g(bijection.)27 b(The)18 b(in)o(v)o(erse)f(of)1476 2089 y(~)1470 2099 y Fl(\014)i Fm(in)e(this)g(case)228 2150 y(is)f(giv)o(en)g(b)o(y)448 2139 y(~)442 2150 y Fl(\014)467 2135 y Fk(\000)p Fh(1)513 2150 y Fm(\()p Fl(`)11 b Fo(\012)g Fl(`)617 2135 y Fk(0)629 2150 y Fm(\))k(=)h Fl(`)725 2157 y Fh(\(1\))781 2150 y Fo(\012)11 b Fl(S)r Fm(\()p Fl(`)884 2157 y Fh(\(2\))930 2150 y Fm(\))p Fl(`)963 2135 y Fk(0)975 2150 y Fm(,)16 b(where)h Fl(S)i Fm(denotes)f(the)e(an)o(tip)q(o)q(de)g(of)g Fl(L)p Fm(.)25 b(In)228 2200 y(particular,)15 b(w)o(e)h(ha)o(v)o(e)f Fl(`)610 2206 y Fh(+)648 2200 y Fo(\012)c Fl(`)708 2206 y Fk(\000)751 2200 y Fm(=)j Fl(`)814 2207 y Fh(\(1\))870 2200 y Fo(\012)c Fl(S)r Fm(\()p Fl(`)972 2207 y Fh(\(2\))1018 2200 y Fm(\).)23 b(The)16 b(an)o(tip)q(o)q(de)g(can)f(b)q(e)i(obtained) e(from)234 2247 y(~)228 2258 y Fl(\014)253 2243 y Fk(\000)p Fh(1)314 2258 y Fm(b)o(y)h(the)h(form)o(ula)c Fl(S)r Fm(\()p Fl(`)p Fm(\))k(=)f(\()p Fl(")11 b Fo(\012)g Fl(L)p Fm(\))882 2247 y(~)876 2258 y Fl(\014)901 2243 y Fk(\000)p Fh(1)947 2258 y Fm(\()p Fl(`)g Fo(\012)g Fm(1\).)25 b(In)16 b(the)h(general)f Fo(\002)1413 2264 y Fn(R)1441 2258 y Fm(-bialgebra)f(case,)228 2308 y(there)k(do)q(es)f(not)f(app)q(ear)h (to)f(b)q(e)i(an)e(analogous)f(w)o(a)o(y)h(to)g(obtain)g(an)g (endomorphism)e(of)i Fl(L)228 2359 y Fm(\(a)e(more)g(direct)h(analog)e (of)h(an)g(an)o(tip)q(o)q(de\))h(from)1042 2348 y(~)1036 2359 y Fl(\014)1061 2344 y Fk(\000)p Fh(1)1106 2359 y Fm(:)21 b(Applying)15 b Fl(")g Fm(to)h(the)g(\014rst)g(factor)g(of)228 2409 y Fl(`)245 2415 y Fh(+)282 2409 y Fo(\012)10 b Fl(`)341 2415 y Fk(\000)384 2409 y Fm(yields)15 b(an)f(elemen)o(t)g(of)g Fl(E)h Fo(\012)p 840 2397 28 2 v 12 x Fn(R)880 2409 y Fl(L)p Fm(,)g(whic)o(h)f(do)q(es)i(not)e(come)g(with)h(a)f(canonical)g (map)228 2458 y(to)j Fl(L)p Fm(.)29 b(Nev)o(ertheless)20 b(w)o(e)e(can)f(k)o(eep)i(in)e(mind)e(that)j Fl(`)1109 2464 y Fh(+)1148 2458 y Fo(\012)12 b Fl(`)1209 2464 y Fk(\000)1255 2458 y Fm(as)17 b(a)h Fo(\002)1380 2464 y Fn(R)1407 2458 y Fm(-bialgebra)f(analog)228 2508 y(of)c(the)i(Hopf)f (algebraic)g(expression)h Fl(`)842 2515 y Fh(\(1\))897 2508 y Fo(\012)9 b Fl(S)r Fm(\()p Fl(`)998 2515 y Fh(\(2\))1044 2508 y Fm(\);)14 b(at)h(an)o(y)e(rate,)i(this)f(analogy)f(will)g(mak)o (e)228 2558 y(the)k(form)o(ulas)d(for)i(\\an)o(ticom)o(ultipli)o (cation")d Fl(`)j Fo(7!)f Fl(`)1066 2564 y Fh(+)1105 2558 y Fo(\012)c Fl(`)1165 2564 y Fk(\000)1210 2558 y Fm(in)16 b(the)h(follo)o(wing)d(Prop)q(osition)228 2608 y(ob)o(viously)e(desirable.)18 b(More)c(generally)m(,)e(for)h(a)g(left) g Fl(L)p Fm(-como)q(dule)f(algebra)g Fl(A)i Fm(o)o(v)o(er)f(a)g Fl(k)q Fm(-Hopf)p eop %%Page: 10 10 10 9 bop 228 113 a Fh(10)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(algebra)c Fl(L)h Fm(bijectivit)o(y)e(of)h(a)h(map)e Fl(A)c Fo(\012)g Fl(A)k Fo(!)g Fl(L)c Fo(\012)g Fl(A)13 b Fm(analogous)e(to)1313 202 y(~)1307 213 y Fl(\014)16 b Fm(is)c(the)h(k)o(ey)g(part)g(of)f(the)228 262 y(de\014nition)g(of)f (Hopf-Galois)f(extensions;)k(one)e(can)h(compare)e(the)i(form)o(ulas)d (in)i(the)h(follo)o(wing)228 312 y(Prop)q(osition)g(to)h(the)g(ones)h (obtained)f(for)f(the)i(in)o(v)o(erse)f(of)f(the)i(Galois)d(map)h(in)g ([)p Fp(17)o Fm(].)303 396 y Fc(Pr)o(oposition)i Fm(3.7)p Fc(.)21 b Fg(L)n(et)14 b Fl(L)h Fg(b)n(e)g(a)g Fo(\002)891 402 y Fn(R)918 396 y Fg(-Hopf)f(algebr)n(a.)19 b(Then)c(we)g(have)692 479 y Fl(`)709 485 y Fh(+)737 486 y(\(1\))791 479 y Fo(\012)9 b Fl(`)849 485 y Fh(+)877 486 y(\(2\))922 479 y Fl(`)939 485 y Fk(\000)979 479 y Fm(=)j Fl(`)d Fo(\012)h Fm(1)h Fo(2)g Fl(L)f Fo(\005)e Fl(L)-1029 b Fm(\(3.1\))676 544 y Fl(`)693 551 y Fh(\(1\))738 550 y(+)774 544 y Fo(\012)10 b Fl(`)833 551 y Fh(\(1\))878 550 y Fk(\000)906 544 y Fl(`)923 551 y Fh(\(2\))979 544 y Fm(=)i Fl(`)d Fo(\012)h Fm(1)h Fo(2)g Fl(L)h Fo(\012)p 1206 557 26 2 v 1206 581 a Fn(R)1246 544 y Fl(L)-1046 b Fm(\(3.2\))732 661 y Fl(`)749 667 y Fh(+)786 661 y Fo(\012)10 b Fl(`)845 667 y Fk(\000)885 661 y Fo(2)924 604 y Ff(Z)965 614 y Fn(s)990 604 y Ff(Z)1013 698 y Fn(r)p 1038 650 18 2 v 1038 667 a(s)1056 661 y Fl(L)p 1084 650 19 2 v 6 x Fn(r)1112 661 y Fo(\012)p 1153 650 V 1153 667 a Fn(r)1172 661 y Fl(L)p 1200 650 18 2 v 6 x Fn(s)228 661 y Fm(\(3.3\))592 746 y(\()p Fl(`m)p Fm(\))677 752 y Fh(+)715 746 y Fo(\012)f Fm(\()p Fl(`m)p Fm(\))841 752 y Fk(\000)882 746 y Fm(=)j Fl(`)943 752 y Fh(+)971 746 y Fl(m)1007 752 y Fh(+)1044 746 y Fo(\012)d Fl(m)1121 752 y Fk(\000)1150 746 y Fl(`)1167 752 y Fk(\000)1207 746 y Fo(2)i Fl(L)h Fo(\012)p 1289 759 26 2 v 1289 783 a Fn(R)1330 746 y Fl(L)-1130 b Fm(\(3.4\))827 835 y(1)848 841 y Fh(+)885 835 y Fo(\012)9 b Fm(1)947 841 y Fk(\000)987 835 y Fm(=)i(1)e Fo(\012)h Fm(1)-895 b(\(3.5\))394 933 y Fl(`)411 939 y Fh(+)439 940 y(\(1\))492 933 y Fo(\012)10 b Fl(`)551 939 y Fh(+)579 940 y(\(2\))632 933 y Fo(\012)g Fl(`)691 939 y Fk(\000)731 933 y Fm(=)i Fl(`)792 940 y Fh(\(1\))846 933 y Fo(\012)d Fl(`)904 940 y Fh(\(2\))949 939 y(+)986 933 y Fo(\012)g Fl(`)1044 940 y Fh(\(2\))1089 939 y Fk(\000)1129 933 y Fo(2)1168 877 y Ff(Z)1210 887 y Fn(tu)1251 877 y Ff(Z)1274 971 y Fn(r)q(s)p 1315 923 19 2 v 1315 939 a(r)1333 933 y Fl(L)p 1361 922 15 2 v 11 x Fn(t)1385 933 y Fo(\012)1427 939 y Fn(r)p 1443 923 20 2 v 1 w(u)1465 933 y Fl(L)1493 939 y Fn(t)p 1506 923 16 2 v(s)1532 933 y Fo(\012)p 1574 923 18 2 v 1574 939 a Fn(s)1592 933 y Fl(L)p 1620 923 22 2 v 6 x Fn(u)228 933 y Fm(\(3.6\))419 1052 y Fl(`)436 1058 y Fh(+)473 1052 y Fo(\012)g Fl(`)531 1058 y Fk(\000)560 1059 y Fh(\(1\))613 1052 y Fo(\012)h Fl(`)672 1058 y Fk(\000)700 1059 y Fh(\(2\))756 1052 y Fm(=)i Fl(`)817 1058 y Fh(+)s(+)882 1052 y Fo(\012)d Fl(`)940 1058 y Fk(\000)978 1052 y Fo(\012)g Fl(`)1036 1058 y Fh(+)s Fk(\000)1104 1052 y Fo(2)1143 995 y Ff(Z)1184 1005 y Fn(tu)1226 995 y Ff(Z)1249 1089 y Fn(r)q(s)p 1290 1040 15 2 v 1290 1062 a(t)1304 1052 y Fl(L)p 1332 1041 19 2 v 6 x Fn(r)1360 1052 y Fo(\012)p 1402 1041 18 2 v 1402 1058 a Fn(s)1419 1052 y Fl(L)p 1447 1041 22 2 v 6 x Fn(u)1479 1052 y Fo(\012)1520 1058 y Fn(s)p 1536 1041 17 2 v(r)1554 1052 y Fl(L)1582 1062 y Fn(u)p 1602 1040 13 2 v(t)228 1052 y Fm(\(3.7\))841 1145 y Fl(`)858 1151 y Fh(+)p 886 1109 151 2 v 886 1145 a Fl(")p Fm(\()p Fl(`)938 1151 y Fk(\000)967 1145 y Fm(\)\(1\))j(=)f Fl(`)-880 b Fm(\(3.8\))841 1210 y Fl(`)858 1216 y Fh(+)886 1210 y Fl(`)903 1216 y Fk(\000)943 1210 y Fm(=)12 b Fl(")p Fm(\()p Fl(`)p Fm(\)\(1\))-880 b(\(3.9\))303 1304 y Fc(Pr)o(oof.)21 b Fm(\(3.1\))14 b(and)g(\(3.2\))f(are)i(just)f(the)h(de\014nition)e(of) h Fl(`)1219 1310 y Fh(+)1256 1304 y Fo(\012)c Fl(`)1315 1310 y Fk(\000)1343 1304 y Fm(.)19 b(F)m(or)14 b(\(3.3\))o(,)f(apply) 1680 1293 y(~)1674 1304 y Fl(\014)1697 1310 y Fn(L)228 1354 y Fm(to)p 281 1331 20 2 v 16 w Fl(s)q(`)318 1360 y Fh(+)356 1354 y Fo(\012)f Fl(`)417 1360 y Fk(\000)462 1354 y Fm(and)k Fl(`)562 1360 y Fh(+)601 1354 y Fo(\012)11 b Fl(`)661 1360 y Fk(\000)p 689 1331 V 689 1354 a Fl(s)17 b Fm(for)f Fl(s)h Fo(2)e Fl(R)i Fm(to)f(get)h Fl(`)11 b Fo(\012)p 1115 1331 V 11 w Fl(s)17 b Fm(eac)o(h)g(time.)25 b(Note)17 b(that)f(the)i(righ)o(t)228 1406 y(hand)c(side)h(of)i (\(3.4\))d(is)h(w)o(ell-de\014ned)g(b)o(y)g(\(3.3\))o(.)20 b(Apply)1145 1395 y(~)1139 1406 y Fl(\014)1162 1412 y Fn(L)1202 1406 y Fm(to)15 b(the)g(righ)o(t)f(hand)g(side)h(to)g(get)228 1456 y Fl(`m)c Fo(\012)f Fm(1.)21 b(\(3.5\))15 b(is)g(trivial.)20 b(F)m(or)15 b(\(3.6\))o(,)g(consider)h(b)q(oth)g(sides)g(as)f(elemen)o (ts)g(of)f Fl(L)d Fo(\005)e Fl(L)14 b Fo(\012)p 1642 1445 28 2 v 1642 1469 a Fn(R)1682 1456 y Fl(L)p Fm(,)228 1512 y(and)i(apply)f Fl(L)c Fo(\005)503 1501 y Fm(~)497 1512 y Fl(\014)520 1518 y Fn(L)562 1512 y Fm(\(note)16 b(that)g(this)g(is)g(w)o(ell-de\014ned!\))25 b(to)16 b(get,)g(for)g(the)h(righ)o(t)e(hand)h(side,)228 1561 y Fl(`)245 1568 y Fh(\(1\))299 1561 y Fo(\012)9 b Fl(`)357 1568 y Fh(\(2\))411 1561 y Fo(\012)h Fm(1)k(b)o(y)f(de\014nition,)g (and)h(for)g(the)g(left)g(hand)f(side)269 1657 y Fl(`)286 1663 y Fh(+)314 1664 y(\(1\))368 1657 y Fo(\012)c Fl(`)426 1663 y Fh(+)454 1664 y(\(2\))r(\(1\))553 1657 y Fo(\012)g Fl(`)611 1663 y Fh(+)639 1664 y(\(2\))r(\(2\))728 1657 y Fl(`)745 1663 y Fk(\000)785 1657 y Fm(=)j Fl(`)846 1663 y Fh(+)874 1664 y(\(1\))927 1657 y Fo(\012)e Fl(`)986 1663 y Fh(+)1014 1664 y(\(2\))1068 1657 y Fo(\012)f Fl(`)1126 1663 y Fh(+)1154 1664 y(\(3\))1199 1657 y Fl(`)1216 1663 y Fk(\000)831 1719 y Fm(=)j Fl(`)892 1725 y Fh(+)920 1726 y(\(1\))r(\(1\))1018 1719 y Fo(\012)e Fl(`)1077 1725 y Fh(+)1105 1726 y(\(1\))q(\(2\))1203 1719 y Fo(\012)g Fl(`)1262 1725 y Fh(+)1290 1726 y(\(2\))1334 1719 y Fl(`)1351 1725 y Fk(\000)1391 1719 y Fm(=)i Fl(`)1452 1726 y Fh(\(1\))1506 1719 y Fo(\012)d Fl(`)1564 1726 y Fh(\(2\))1618 1719 y Fo(\012)h Fm(1)228 1803 y(noting)j(that)h(the)g(second)h(expression)g (is)f(w)o(ell-de\014ned.)303 1852 y(F)m(or)h(\(3.7\))h(w)o(e)h (consider)g(the)g(left)f Fl(L)p Fm(-mo)q(dule)f Fl(L)c Fo(\005)f Fl(L)p Fm(.)26 b(The)17 b(left)f(hand)g(side)h(of)i(\(3.7\))c (is)234 1895 y(~)228 1906 y Fl(\014)253 1888 y Fk(\000)p Fh(1)251 1918 y Fn(L)p Fk(\005)p Fn(L)316 1906 y Fm(\()p Fl(`)10 b Fo(\012)f Fm(1)g Fo(\012)h Fm(1\),)j(and)275 1995 y(~)269 2006 y Fl(\014)292 2012 y Fn(L)p Fk(\005)p Fn(L)357 2006 y Fm(\()p Fl(`)390 2012 y Fh(+)s(+)455 2006 y Fo(\012)d Fl(`)514 2012 y Fk(\000)551 2006 y Fo(\012)g Fl(`)610 2012 y Fh(+)s Fk(\000)666 2006 y Fm(\))h(=)h Fl(`)754 2012 y Fh(+)s(+)810 2013 y(\(1\))863 2006 y Fo(\012)e Fl(`)922 2012 y Fh(+)s(+)977 2013 y(\(2\))r(\(1\))1066 2006 y Fl(`)1083 2012 y Fk(\000)1121 2006 y Fo(\012)f Fl(`)1179 2012 y Fh(+)s(+)1235 2013 y(\(2\))q(\(2\))1324 2006 y Fl(`)1341 2012 y Fh(+)s Fk(\000)316 2069 y Fm(=)j Fl(`)377 2075 y Fh(+)s(+)432 2076 y(\(1\))486 2069 y Fo(\012)e Fl(`)545 2075 y Fh(+)r(+)600 2076 y(\(2\))645 2069 y Fl(`)662 2075 y Fk(\000)699 2069 y Fo(\012)g Fl(`)758 2075 y Fh(+)r(+)813 2076 y(\(3\))858 2069 y Fl(`)875 2075 y Fh(+)r Fk(\000)942 2069 y Fm(=)i Fl(`)1003 2075 y Fh(+)s(+)1058 2076 y(\(1\))r(\(1\))1157 2069 y Fo(\012)d Fl(`)1215 2075 y Fh(+)s(+)1270 2076 y(\(1\))r(\(2\))1360 2069 y Fl(`)1377 2075 y Fk(\000)1414 2069 y Fo(\012)h Fl(`)1473 2075 y Fh(+)s(+)1528 2076 y(\(2\))1573 2069 y Fl(`)1590 2075 y Fh(+)r Fk(\000)1075 2131 y Fm(=)i Fl(`)1136 2137 y Fh(+)1164 2138 y(\(1\))1217 2131 y Fo(\012)e Fl(`)1276 2137 y Fh(+)1304 2138 y(\(2\))1348 2131 y Fl(`)1365 2137 y Fk(\000)1403 2131 y Fo(\012)f Fm(1)j(=)f Fl(`)f Fo(\012)f Fm(1)g Fo(\012)h Fm(1)228 2214 y(since)15 b(it)e(is)h(easy)g (to)g(see)h(that)f(the)g(third)g(expression)h(is)f(w)o(ell-de\014ned.) 303 2266 y(F)m(or)f(\(3.8\))o(,)h(apply)f(the)h(map)f Fl(`)c Fo(\012)h Fl(`)851 2251 y Fk(0)874 2266 y Fo(7!)p 927 2230 134 2 v 11 w Fl(")p Fm(\()p Fl(`)979 2254 y Fk(0)992 2266 y Fm(\)\(1\))p Fl(`)k Fm(to)g(\(3.1\))f(and)h(\014nd)269 2366 y Fl(`)e Fm(=)p 342 2330 126 2 v 12 w Fl(")p Fm(\(1\)\(1\))p Fl(`)g Fm(=)p 540 2330 240 2 v 12 w Fl(")p Fm(\()p Fl(`)592 2372 y Fh(+)620 2373 y(\(2\))665 2366 y Fl(`)682 2372 y Fk(\000)710 2366 y Fm(\)\(1\))p Fl(`)796 2372 y Fh(+)824 2373 y(\(1\))880 2366 y Fm(=)p 924 2330 324 2 v 12 w Fl(")p Fm(\()p Fl(`)976 2372 y Fh(+)1005 2373 y(\(2\))1049 2366 y Fm(\)\()p Fl(")p Fm(\()p Fl(`)1133 2372 y Fk(\000)1162 2366 y Fm(\)\(1\)\))q Fl(`)1265 2372 y Fh(+)1292 2373 y(\(1\))467 2439 y Fm(=)p 510 2403 375 2 v 11 w Fl(")p Fm(\()p Fl(`)562 2445 y Fh(+)591 2446 y(\(2\))635 2439 y Fm(\)\()p Fl(")p Fm(\()p Fl(`)719 2445 y Fk(\000)748 2439 y Fm(\)\(1\))e Fo(\001)f Fm(1\))o Fl(`)901 2445 y Fh(+)929 2446 y(\(1\))985 2439 y Fm(=)p 1029 2403 377 2 v 12 w(\()p Fl(")p Fm(\()p Fl(`)1097 2445 y Fh(+)1126 2446 y(\(2\))1170 2439 y Fm(\))p Fl(")p Fm(\()p Fl(`)1238 2445 y Fk(\000)1267 2439 y Fm(\)\(1\)\)\(1\))q Fl(`)1423 2445 y Fh(+)1450 2446 y(\(1\))453 2512 y Fm(=)p 497 2476 345 2 v 12 w Fl(")p Fm(\()p Fl(`)549 2518 y Fh(+)577 2519 y(\(2\))621 2512 y Fl(")p Fm(\()p Fl(`)673 2518 y Fk(\000)702 2512 y Fm(\)\(1\)\)\(1\))q Fl(`)858 2518 y Fh(+)886 2519 y(\(1\))942 2512 y Fm(=)p 985 2476 195 2 v 11 w Fl(")p Fm(\()p Fl(`)1037 2518 y Fh(+)1066 2519 y(\(2\))1110 2512 y Fm(\)\(1\))q Fl(`)1197 2518 y Fh(+)1224 2519 y(\(1\))p 1269 2476 151 2 v 1269 2512 a Fl(")p Fm(\()p Fl(`)1321 2518 y Fk(\000)1350 2512 y Fm(\)\(1\))i(=)h Fl(`)1491 2518 y Fh(+)p 1519 2476 V 1519 2512 a Fl(")p Fm(\()p Fl(`)1571 2518 y Fk(\000)1600 2512 y Fm(\)\(1\))p Fl(:)228 2597 y Fm(Finally)m(,)f(\(3.9\))i(follo)o(ws)g(from)f(\(3.1\)) h(b)o(y)h(applying)e Fl(`)e Fo(\012)f Fl(`)1122 2582 y Fk(0)1146 2597 y Fo(7!)i Fl(")p Fm(\()p Fl(`)p Fm(\)\(1\))p Fl(`)1337 2582 y Fk(0)1350 2597 y Fm(.)p 1692 2597 2 29 v 1694 2571 25 2 v 1694 2597 V 1719 2597 2 29 v eop %%Page: 11 11 11 10 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(11)303 213 y Fc(Remark)16 b Fm(3.8)p Fc(.)k Fm(The)12 b(iden)o(tities)f(\(3.3\))o(,)g(\(3.4\))f(and)h(\(3.5\))f (can)h(b)q(e)h(summarized)d(as)i(sa)o(ying)228 262 y(that)725 315 y Fo(S)d Fm(:)13 b Fl(L)f Fo(3)f Fl(`)h Fo(7!)p 944 273 141 2 v 11 w Fl(`)961 321 y Fk(\000)999 315 y Fo(\012)p 1040 282 45 2 v 9 w Fl(`)1057 321 y Fh(+)1097 315 y Fo(2)p 1136 274 89 2 v 11 w Fl(L)p Fo(\002)1168 346 y Fn(R)p 1197 282 29 2 v 1197 315 a Fl(L)228 395 y Fm(is)i(a)f(map)f(of)i Fl(R)476 380 y Fn(e)493 395 y Fm(-rings,)g(or,)f(equiv)n(alen)o(tly)m (,)710 473 y Fo(S)738 456 y Fk(0)755 473 y Fm(:)p 780 440 V 13 w Fl(L)f Fo(3)p 859 439 18 2 v 11 w Fl(`)g Fo(7!)p 941 431 141 2 v 941 439 46 2 v 11 w Fl(`)958 479 y Fk(\000)995 473 y Fo(\012)e Fl(`)1054 479 y Fh(+)1093 473 y Fo(2)p 1133 431 108 2 v 1133 440 29 2 v 12 w Fl(L)f Fo(\002)p 1174 486 26 2 v 1174 510 a Fn(R)1212 473 y Fl(L)228 578 y Fm(is)14 b(a)f(map)f(of)p 444 545 32 2 v 14 w Fl(R)475 552 y Fn(e)505 567 y Fo(\030)505 580 y Fm(=)549 578 y Fl(R)581 563 y Fn(e)598 578 y Fm(-rings.)303 628 y(The)18 b(equations)g(\(3.6\))o(,)g(\(3.7\))g(and)f(\(3.8\))h(then)g(sa)o(y)g (that)g Fo(S)1285 613 y Fk(0)1314 628 y Fm(is)g(an)g(Ess)h(map)d(for)i (the)228 679 y Fo(\002)p 231 692 26 2 v 231 716 a Fn(R)260 679 y Fm(-bialgebra)p 454 646 29 2 v 13 w Fl(L)c Fm(in)f(the)i(sense)g (of)f([)p Fp(19)o Fm(,)f(Def.)h(6.8])690 797 y Fp(4.)24 b(Y)l(etter-Drinf)o(el)o(d)13 b(mo)q(dules)303 872 y Fm(In)19 b(this)g(section)g(w)o(e)h(will)d(generalize)j(the)g(notion)e (of)h(a)f(Y)m(etter-Drinfeld)i(mo)q(dule)d(or)228 922 y(crossed)j(bimo)q(dule)c([)p Fp(22)o Fm(])i(and)g(its)g(close)h (connection)f(to)g(the)h(construction)g(of)f(the)h(cen)o(ter)228 972 y(of)c(a)g(monoidal)e(category)j([)p Fp(5)o Fm(],)f(see)i(also)e([) p Fp(6)p Fm(],)g(to)g(the)h(theory)h(of)e Fo(\002)1306 978 y Fn(R)1333 972 y Fm(-bialgebras.)23 b(After)16 b(w)o(e)228 1022 y(manage)h(to)h(sho)o(w)h(that)f(the)h(condition)f(for)g(a)h(mo)q (dule)e(and)h(como)q(dule)f(to)i(b)q(e)g(a)f(Y)m(etter-)228 1071 y(Drinfeld)11 b(mo)q(dule)f(is)i(w)o(ell-de\014ned)g(in)f(our)h (setting,)g(almost)e(ev)o(erything)i(will)e(b)q(e)j(\\as)e(usual",)228 1121 y(except)20 b(that)e(w)o(e)h(will)e(ha)o(v)o(e)i(to)f(replace)h (the)g(cen)o(ter)i(construction)e(b)o(y)f(the)i(w)o(eak)e(cen)o(ter,) 228 1171 y(where)d(the)f(braiding)f(maps)g(are)h(not)g(necessarily)h (isomorphisms.)303 1248 y Fc(Lemma)h Fm(4.1)p Fc(.)k Fg(L)n(et)f Fl(L)h Fg(b)n(e)f(a)h Fo(\002)796 1254 y Fn(R)824 1248 y Fg(-bialgebr)n(a)e(and)j Fl(V)28 b Fg(a)20 b(left)f Fl(L)p Fg(-c)n(omo)n(dule)g(as)h(wel)r(l)e(as)i(a)228 1297 y(left)15 b Fl(L)p Fg(-mo)n(dule)h(such)g(that)g(the)g(underlying) g(left)f Fl(R)p Fg(-mo)n(dule)h(structur)n(es)f(ar)n(e)h(the)f(same,)h (and)p 228 1324 20 2 v 228 1347 a Fl(r)q(v)d Fm(=)f Fl(v)q(r)k Fg(holds)f(for)f(al)r(l)h Fl(r)d Fo(2)f Fl(R)k Fg(and)h Fl(v)d Fo(2)e Fl(V)e Fg(.)19 b(Then)d(the)e(maps)494 1418 y Fl(')521 1425 y Fh(\(4.1\))596 1418 y Fm(:)g Fl(L)9 b Fo(\012)h Fl(V)21 b Fo(3)11 b Fl(`)e Fo(\012)h Fl(v)j Fo(7!)e Fl(`)956 1425 y Fh(\(1\))1001 1418 y Fl(v)1022 1425 y Fh(\()p Fk(\000)p Fh(1\))1102 1418 y Fo(\012)f Fl(`)1161 1425 y Fh(\(2\))1215 1418 y Fl(.)f(v)1266 1425 y Fh(\(0\))1323 1418 y Fo(2)i Fl(L)p Fo(\002)1394 1448 y Fn(R)1422 1418 y Fl(V)-1218 b Fm(\(4.1\))408 1500 y Fl(')435 1507 y Fh(\(4.2\))510 1500 y Fm(:)14 b Fl(L)9 b Fo(\012)h Fl(V)20 b Fo(3)12 b Fl(`)d Fo(\012)h Fl(v)j Fo(7!)e Fm(\()p Fl(`)886 1507 y Fh(\(1\))940 1500 y Fl(.)e(v)q Fm(\))1007 1507 y Fh(\()p Fk(\000)p Fh(1\))1078 1500 y Fl(`)1095 1507 y Fh(\(2\))1149 1500 y Fo(\012)h Fm(\()p Fl(`)1224 1507 y Fh(\(1\))1278 1500 y Fl(.)f(v)q Fm(\))1345 1507 y Fh(\(0\))1402 1500 y Fo(2)i Fl(L)f Fo(\005)f Fl(V)-1305 b Fm(\(4.2\))228 1570 y Fg(ar)n(e)14 b(wel)r(l-de\014ne)n(d.)303 1647 y Fc(Pr)o(oof.)21 b Fm(Clearly)14 b(the)g(map)458 1717 y Fl(')485 1724 y Fh(\(4.3\))560 1717 y Fm(:)f Fl(L)d Fo(\012)f Fl(L)h Fo(\012)f Fl(V)21 b Fo(3)12 b Fl(`)d Fo(\012)h Fl(m)f Fo(\012)h Fl(v)j Fo(7!)e Fl(`)1086 1724 y Fh(\(1\))1131 1717 y Fl(m)e Fo(\012)h Fl(`)1235 1724 y Fh(\(2\))1289 1717 y Fl(.)f(v)k Fo(2)e Fl(L)f Fo(\005)f Fl(V)-1255 b Fm(\(4.3\))228 1788 y(is)14 b(w)o(ell-de\014ned.)k(No)o(w) 269 1871 y Fl(')296 1878 y Fh(\(4.3\))367 1871 y Fm(\()p Fl(`)10 b Fo(\012)p 452 1848 V 10 w Fl(r)q(m)g Fo(\012)f Fl(v)q Fm(\))k(=)g Fl(`)671 1878 y Fh(\(1\))p 716 1848 V 716 1871 a Fl(r)q(m)d Fo(\012)f Fl(`)840 1878 y Fh(\(2\))895 1871 y Fl(.)g(v)14 b Fm(=)e Fl(`)1020 1878 y Fh(\(1\))1065 1871 y Fl(m)e Fo(\012)g Fl(`)1170 1878 y Fh(\(2\))1215 1871 y Fl(r)g(.)g(v)j Fm(=)g Fl(')1380 1878 y Fh(\(4.3\))1451 1871 y Fm(\()p Fl(`)d Fo(\012)f Fl(m)h Fo(\012)g Fl(r)q(v)q Fm(\))228 1944 y(holds)15 b(for)h Fl(`;)7 b(m)15 b Fo(2)f Fl(L)p Fm(,)i Fl(v)h Fo(2)d Fl(V)25 b Fm(and)16 b Fl(r)g Fo(2)e Fl(R)p Fm(,)i(whence)h Fl(')1111 1951 y Fh(\(4.3\))1197 1944 y Fm(factors)f(through)g Fl(L)11 b Fo(\012)g Fm(\()p Fl(L)g Fo(\005)f Fl(V)g Fm(\),)228 1994 y(and)15 b Fl(')337 2001 y Fh(\(4.3\))408 1994 y Fm(\()p Fl(`)10 b Fo(\012)h Fl(v)515 2001 y Fh(\()p Fk(\000)p Fh(1\))596 1994 y Fo(\012)g Fl(v)660 2001 y Fh(\(0\))705 1994 y Fm(\))j(=)g Fl(`)798 2001 y Fh(\(1\))843 1994 y Fl(v)864 2001 y Fh(\()p Fk(\000)p Fh(1\))945 1994 y Fo(\012)d Fl(`)1005 2001 y Fh(\(2\))1060 1994 y Fl(.)f(v)1112 2001 y Fh(\(0\))1171 1994 y Fm(=)k Fl(')1244 2001 y Fh(\(4.1\))1315 1994 y Fm(\()p Fl(`)d Fo(\012)f Fl(v)q Fm(\))16 b(is)f(w)o(ell-de\014ned.)228 2044 y(F)m(urther,)f Fl(v)411 2051 y Fh(\()p Fk(\000)p Fh(1\))491 2044 y Fo(\012)c Fl(v)554 2051 y Fh(\(0\))611 2044 y Fo(2)h Fl(L)p Fo(\002)710 2050 y Fn(R)738 2044 y Fl(V)23 b Fm(implies)12 b Fl(`)943 2051 y Fh(\(1\))988 2044 y Fl(v)1009 2051 y Fh(\()p Fk(\000)p Fh(1\))1089 2044 y Fo(\012)d Fl(`)1147 2051 y Fh(\(2\))1202 2044 y Fl(.)g(v)1253 2051 y Fh(\(0\))1309 2044 y Fo(2)i Fl(L)p Fo(\002)1409 2050 y Fn(R)1436 2044 y Fl(V)f Fm(.)303 2093 y(No)o(w)j(de\014ne)733 2164 y Fl(')760 2171 y Fh(\(4.4\))835 2164 y Fm(:)g Fl(L)d Fo(\012)f Fl(L)h Fo(\012)f Fl(V)21 b Fo(!)11 b Fl(L)f Fo(\005)f Fl(V)-980 b Fm(\(4.4\))228 2237 y(b)o(y)13 b Fl(')312 2244 y Fh(\(4.4\))383 2237 y Fm(\()p Fl(`)d Fo(\012)f Fl(m)h Fo(\012)f Fl(v)q Fm(\))k(=)f(\()p Fl(`)d(.)g(v)q Fm(\))757 2244 y Fh(\()p Fk(\000)p Fh(1\))828 2237 y Fl(m)h Fo(\012)g Fm(\()p Fl(`)f(.)g(v)q Fm(\))1025 2244 y Fh(\(0\))1071 2237 y Fl(:)k Fm(Then)269 2320 y Fl(')296 2327 y Fh(\(4.4\))367 2320 y Fm(\()p 383 2297 V Fl(r)q(`)d Fo(\012)f Fl(m)h Fo(\012)f Fl(v)q Fm(\))k(=)e(\()p 667 2297 V Fl(r)q(`)f(.)f(v)q Fm(\))781 2327 y Fh(\()p Fk(\000)p Fh(1\))852 2320 y Fl(m)h Fo(\012)f Fm(\()p 955 2297 V Fl(r)q(`)h(.)f(v)q Fm(\))1069 2327 y Fh(\(0\))446 2385 y Fm(=)j(\(\()p Fl(`)e(.)f(v)q Fm(\))p Fl(r)q Fm(\))652 2392 y Fh(\()p Fk(\000)p Fh(1\))723 2385 y Fl(m)h Fo(\012)f Fm(\(\()p Fl(`)h(.)f(v)q Fm(\))p Fl(r)q Fm(\))972 2392 y Fh(\(0\))1029 2385 y Fm(=)j(\()p Fl(`)d(.)g(v)q Fm(\))1182 2392 y Fh(\()p Fk(\000)p Fh(1\))1253 2385 y Fl(r)q(m)h Fo(\012)g Fm(\()p Fl(`)f(.)g(v)q Fm(\))1470 2392 y Fh(\(0\))1310 2449 y Fm(=)j Fl(')1381 2456 y Fh(\(4.4\))1452 2449 y Fm(\()p Fl(`)d Fo(\012)h Fl(r)q(m)f Fo(\012)h Fl(v)q Fm(\))228 2520 y(for)j Fl(`;)7 b(m)12 b Fo(2)f Fl(L)j Fm(and)g Fl(r)e Fo(2)g Fl(R)p Fm(,)h(so)h(that)g Fl(')833 2527 y Fh(\(4.4\))917 2520 y Fm(factors)h(through)e(a)h(map)733 2591 y Fl(')760 2598 y Fh(\(4.5\))835 2591 y Fm(:)f Fl(L)d Fo(\005)f Fl(L)g Fo(\012)h Fl(V)21 b Fo(!)11 b Fl(L)e Fo(\005)g Fl(V)r(;)-982 b Fm(\(4.5\))p eop %%Page: 12 12 12 11 bop 228 113 a Fh(12)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(and)f Fl(')337 220 y Fh(\(4.5\))408 213 y Fm(\()p Fl(`)441 220 y Fh(\(1\))496 213 y Fo(\012)c Fl(`)556 220 y Fh(\(2\))611 213 y Fo(\012)g Fl(v)q Fm(\))k(=)f(\()p Fl(`)785 220 y Fh(\(1\))841 213 y Fl(.)c(v)q Fm(\))909 220 y Fh(\()p Fk(\000)p Fh(1\))980 213 y Fl(`)997 220 y Fh(\(2\))1052 213 y Fo(\012)h Fm(\()p Fl(`)1128 220 y Fh(\(1\))1183 213 y Fl(.)f(v)q Fm(\))1251 220 y Fh(\(0\))1311 213 y Fm(=)k Fl(')1384 220 y Fh(\(4.2\))1455 213 y Fm(\()p Fl(`)d Fo(\012)f Fl(v)q Fm(\))17 b(is)e(w)o(ell-)228 262 y(de\014ned.)p 1692 262 2 29 v 1694 236 25 2 v 1694 262 V 1719 262 2 29 v 303 347 a Fc(Definition)g Fm(4.2)p Fc(.)21 b Fm(Let)g Fl(L)f Fm(b)q(e)i(a)e Fo(\002)891 353 y Fn(R)918 347 y Fm(-bialgebra.)37 b(A)21 b(\(left-left\))f(Y)m (etter-Drinfeld)h Fl(L)p Fm(-)228 397 y(mo)q(dule)8 b(is)h(a)g(left)g Fl(L)p Fm(-mo)q(dule)f(and)h(left)h Fl(L)p Fm(-como)q(dule)e Fl(V)19 b Fm(suc)o(h)10 b(that)f(the)h(resp)q(ectiv)o(e)i(underlying) 228 446 y(left)i Fl(R)p Fm(-mo)q(dule)e(structures)k(are)f(the)f(same,) p 944 424 20 2 v 13 w Fl(r)q(v)f Fm(=)f Fl(v)q(r)j Fm(for)f(all)f Fl(r)f Fo(2)g Fl(R)h Fm(and)h Fl(v)f Fo(2)f Fl(V)23 b Fm(holds,)13 b(and)228 496 y(w)o(e)h(ha)o(v)o(e)493 553 y Fl(`)510 560 y Fh(\(1\))554 553 y Fl(v)575 560 y Fh(\()p Fk(\000)p Fh(1\))656 553 y Fo(\012)9 b Fl(`)714 560 y Fh(\(2\))768 553 y Fl(.)g(v)819 560 y Fh(\(0\))876 553 y Fm(=)j(\()p Fl(`)953 560 y Fh(\(1\))1007 553 y Fl(.)d(v)q Fm(\))1074 560 y Fh(\()p Fk(\000)p Fh(1\))1145 553 y Fl(`)1162 560 y Fh(\(2\))1216 553 y Fo(\012)h Fm(\()p Fl(`)1291 560 y Fh(\(1\))1345 553 y Fl(.)f(v)q Fm(\))1412 560 y Fh(\(0\))228 618 y Fm(in)k Fl(L)d Fo(\005)f Fl(V)23 b Fm(for)13 b(all)g Fl(`)f Fo(2)f Fl(L)j Fm(and)g Fl(v)f Fo(2)e Fl(V)303 668 y Fm(W)m(e)i(denote)i(the)f(category)h(of)e(Y)m (etter-Drinfeld)h Fl(L)p Fm(-mo)q(dules)f(b)o(y)1345 653 y Fn(L)1345 679 y(L)1370 668 y Fo(Y)s(D)r Fm(.)303 747 y(One)20 b(can)f(c)o(hec)o(k)h(directly)g(that)f(the)h(category) 1096 732 y Fn(L)1096 759 y(L)1120 747 y Fo(Y)s(D)h Fm(is)e(monoidal:)26 b(When)19 b Fl(V)r(;)7 b(W)27 b Fo(2)228 783 y Fn(L)228 809 y(L)253 798 y Fo(Y)s(D)q Fm(,)15 b(then)h Fl(V)k Fo(\005)9 b Fl(W)20 b Fm(=)14 b Fl(V)23 b Fo(\012)699 804 y Fn(B)741 798 y Fl(W)6 b Fm(,)15 b(equipp)q(ed)h(with)f(the)h (diagonal)d(mo)q(dule)h(structure)j(and)228 847 y(co)q(diagonal)d(como) q(dule)h(structure,)k(is)d(also)f(a)h(Y)m(etter-Drinfeld)g Fl(L)p Fm(-mo)q(dule.)24 b(Ho)o(w)o(ev)o(er)16 b(this)228 897 y(will)c(also)h(follo)o(w)f(from)h(Prop)q(osition)g(4.4)g(b)q(elo)o (w.)303 976 y Fc(Definition)i Fm(4.3)p Fc(.)21 b Fm(Let)16 b Fo(C)j Fm(b)q(e)e(a)f(monoidal)e(category)m(.)26 b(The)17 b(\(left\))f(w)o(eak)g(cen)o(ter)j(of)c Fo(C)r Fm(,)228 1026 y(denoted)k Fo(Z)419 1032 y Fh(0)438 1026 y Fm(\()p Fo(C)r Fm(\))g(is)f(a)g(category)h(whose)g(ob)r(jects)h(are)e(pairs)h (\()p Fl(X)q(;)7 b(\033)1318 1032 y Fn(X)h Fh({)1372 1026 y Fm(\))19 b(in)f(whic)o(h)g Fl(X)23 b Fo(2)18 b(C)r Fm(,)228 1076 y(and)g Fl(\033)337 1082 y Fn(X)r(Y)400 1076 y Fm(:)d Fl(X)h Fo(\012)c Fl(Y)29 b Fo(!)19 b Fl(Y)j Fo(\012)12 b Fl(X)22 b Fm(is)d(natural)f(in)g Fl(Y)29 b Fo(2)19 b(C)i Fm(and)d(satis\014es)i Fl(\033)1441 1082 y Fn(X;Y)6 b Fk(\012)p Fn(Z)1576 1076 y Fm(=)20 b(\()p Fl(Y)i Fo(\012)228 1125 y Fl(\033)252 1131 y Fn(X)r(Z)308 1125 y Fm(\)\()p Fl(\033)364 1131 y Fn(X)r(Y)432 1125 y Fo(\012)10 b Fl(Z)s Fm(\))15 b(for)g(all)e Fl(Y)r(;)7 b(Z)16 b Fo(2)d(C)k Fm(and)e Fl(\033)934 1131 y Fn(X)r(I)995 1125 y Fm(=)e(id)1075 1131 y Fn(X)1106 1125 y Fm(.)21 b(The)15 b(category)g Fo(Z)1423 1131 y Fh(0)1442 1125 y Fm(\()p Fo(C)r Fm(\))h(is)e(monoidal)228 1175 y(with)c(\()p Fl(X)q(;)d(\033)413 1181 y Fn(X)h Fh({)467 1175 y Fm(\))s Fo(\012)s Fm(\()p Fl(Y)r(;)f(\033)606 1181 y Fn(Y)12 b Fh({)658 1175 y Fm(\))f(=)h(\()p Fl(X)6 b Fo(\012)s Fl(Y)r(;)h(\033)889 1181 y Fn(X)r Fk(\012)p Fn(Y)r(;)p Fh({)995 1175 y Fm(\),)k(where)h Fl(\033)1175 1181 y Fn(X)r Fk(\012)p Fn(Y)r(;Z)1299 1175 y Fm(=)g(\()p Fl(\033)1383 1181 y Fn(X)r(Z)1442 1175 y Fo(\012)s Fl(Y)d Fm(\)\()p Fl(X)d Fo(\012)s Fl(\033)1641 1181 y Fn(Y)h(Z)1695 1175 y Fm(\).)303 1255 y Fc(Pr)o(oposition)15 b Fm(4.4)p Fc(.)21 b Fg(A)15 b(c)n(ate)n(gory)g(e)n(quivalenc)n(e)g(of)1130 1240 y Fn(L)1130 1266 y(L)1154 1255 y Fo(Y)s(D)i Fg(with)e(the)g(we)n (ak)g(c)n(enter)g(of)g(the)228 1305 y(monoidal)k(c)n(ate)n(gory)579 1311 y Fn(L)603 1305 y Fo(M)g Fg(is)g(given)h(as)f(fol)r(lows:)27 b(T)m(o)19 b Fl(V)29 b Fo(2)1222 1290 y Fn(L)1222 1317 y(L)1247 1305 y Fo(Y)s(D)20 b Fg(assign)g(the)f(underlying)228 1355 y Fl(L)p Fg(-mo)n(dule)c(of)g Fl(V)24 b Fg(to)n(gether)15 b(with)f(the)h(natur)n(al)f(tr)n(ansformation)524 1429 y Fl(\033)548 1435 y Fn(V)7 b(W)617 1429 y Fm(:)13 b Fl(V)19 b Fo(\005)9 b Fl(W)17 b Fo(3)12 b Fl(v)f Fo(\012)e Fl(w)j Fo(7!)f Fl(v)999 1436 y Fh(\()p Fk(\000)p Fh(1\))1080 1429 y Fl(.)e(w)h Fo(\012)f Fl(v)1212 1436 y Fh(\(0\))1269 1429 y Fo(2)i Fl(W)k Fo(\005)9 b Fl(V)228 1507 y Fg(for)20 b Fl(W)28 b Fo(2)417 1513 y Fn(L)441 1507 y Fo(M)p Fg(.)36 b(In)21 b(p)n(articular,)g(the)g(c)n(ate)n(gory)1053 1492 y Fn(L)1053 1519 y(L)1077 1507 y Fo(Y)s(D)h Fg(is)f(a)g(monoidal)g (c)n(ate)n(gory)f(by)h(the)228 1558 y(usual)15 b(tensor)f(pr)n(o)n (ducts)h(for)f Fl(L)p Fg(-mo)n(dules)h(and)g Fl(L)p Fg(-c)n(omo)n (dules,)g(and)g(the)g(e)n(quivalenc)n(e)1589 1543 y Fn(L)1589 1569 y(L)1614 1558 y Fo(Y)s(D)1690 1547 y(\030)1690 1560 y Fm(=)228 1608 y Fo(Z)258 1614 y Fh(0)277 1608 y Fm(\()293 1614 y Fn(L)318 1608 y Fo(M)o Fm(\))g Fg(is)g(a)g(monoidal)g(e)n (quivalenc)n(e.)303 1686 y Fc(Pr)o(oof.)21 b Fm(W)m(e)12 b(omit)d(c)o(hec)o(king)j(that)g(one)g(has)f(a)h(functor)1203 1671 y Fn(L)1203 1698 y(L)1227 1686 y Fo(Y)s(D)h(!)e(Z)1386 1692 y Fh(0)1405 1686 y Fm(\()1421 1692 y Fn(L)1446 1686 y Fo(M)p Fm(\))h(as)f(claimed.)228 1736 y(Assume)17 b(\()p Fl(V)r(;)7 b(\033)470 1742 y Fn(V)r(;)p Fh({)521 1736 y Fm(\))18 b Fo(2)g(Z)631 1742 y Fh(0)649 1736 y Fm(\()665 1742 y Fn(L)690 1736 y Fo(M)p Fm(\).)30 b(Endo)o(w)17 b Fl(V)27 b Fm(with)18 b(the)g(righ)o(t)f Fl(R)p Fm(-mo)q(dule)f (structure)k(that)228 1789 y(iden)o(ti\014es)14 b(with)g(its)g (underlying)f(left)p 834 1755 32 2 v 14 w Fl(R)p Fm(-mo)q(dule)f (structure,)j(and)f(put)577 1863 y Fl(\025)601 1869 y Fh(0)620 1863 y Fm(\()p Fl(v)q Fm(\))e(:=)f Fl(v)761 1870 y Fh(\()p Fk(\000)p Fh(1\))842 1863 y Fo(\012)e Fl(v)904 1870 y Fh(\(0\))961 1863 y Fm(:=)i Fl(\033)1040 1869 y Fn(V)c(L)1092 1863 y Fm(\()p Fl(v)k Fo(\012)e Fm(1\))j Fo(2)f Fl(L)e Fo(\005)g Fl(V)r(:)228 1937 y Fm(Let)14 b Fl(\032)p 323 1926 19 2 v 6 x Fn(r)347 1937 y Fm(:)f Fl(L)f Fo(!)f Fl(L)j Fm(denote)h(righ)o(t)e(m)o(ultiplication) d(b)o(y)p 1065 1914 20 2 v 14 w Fl(r)15 b Fm(for)e Fl(r)g Fo(2)e Fl(R)p Fm(.)18 b(Then)269 2023 y Fl(v)290 2030 y Fh(\()p Fk(\000)p Fh(1\))p 361 2000 V 361 2023 a Fl(r)11 b Fo(\012)e Fl(v)453 2030 y Fh(\(0\))510 2023 y Fm(=)j(\()p Fl(\032)p 591 2013 19 2 v 6 x Fn(r)619 2023 y Fo(\005)d Fl(V)g Fm(\))p Fl(\033)722 2029 y Fn(V)e(L)774 2023 y Fm(\()p Fl(v)k Fo(\012)e Fm(1\))j(=)f Fl(\033)978 2029 y Fn(V)c(L)1030 2023 y Fm(\()p Fl(V)19 b Fo(\005)9 b Fl(\032)p 1140 2013 V 6 x Fn(r)1158 2023 y Fm(\)\()p Fl(v)i Fo(\012)f Fm(1\))h(=)h Fl(\033)1379 2029 y Fn(V)7 b(L)1431 2023 y Fm(\()p Fl(v)k Fo(\012)p 1519 2000 20 2 v 9 w Fl(r)q Fm(\))540 2088 y(=)g Fl(\033)607 2094 y Fn(V)c(L)659 2088 y Fm(\()p 675 2065 V Fl(r)q Fm(\()p Fl(v)k Fo(\012)e Fm(1\)\))j(=)p 892 2065 V 12 w Fl(r\033)935 2094 y Fn(V)7 b(L)987 2088 y Fm(\()p Fl(v)k Fo(\012)e Fm(1\))j(=)g Fl(v)1189 2095 y Fh(\()p Fk(\000)p Fh(1\))1269 2088 y Fo(\012)p 1310 2065 V 9 w Fl(r)q(v)1351 2095 y Fh(\(0\))1408 2088 y Fm(=)g Fl(v)1473 2095 y Fh(\()p Fk(\000)p Fh(1\))1553 2088 y Fo(\012)e Fl(v)1616 2095 y Fh(\(0\))1661 2088 y Fl(r)228 2162 y Fm(sho)o(ws)j(that)g Fl(\025)459 2168 y Fh(0)490 2162 y Fm(tak)o(es)g(v)n(alues)g(in)f Fl(L)p Fo(\002)826 2168 y Fn(R)853 2162 y Fl(V)d Fm(.)18 b(T)m(o)12 b(sho)o(w)h(that)f Fl(\025)1190 2168 y Fh(0)1222 2162 y Fm(mak)o(es)g Fl(V)22 b Fm(a)12 b(left)h Fl(L)p Fm(-como)q(dule,)228 2212 y(consider)i(the)f(left)g Fl(L)p Fm(-mo)q(dule)e(map)g(\001)849 2218 y Fh(0)872 2212 y Fm(:)i Fl(L)d Fo(!)h Fl(L)d Fo(\005)g Fl(L)p Fm(.)18 b(W)m(e)c(ha)o(v)o(e)269 2298 y Fl(\033)293 2304 y Fn(V)r(;L)p Fk(\005)p Fn(L)389 2298 y Fm(\()p Fl(v)d Fo(\012)e Fm(1)g Fo(\012)h Fm(1\))h(=)h(\()p Fl(L)e Fo(\005)f Fl(\033)749 2304 y Fn(V)d(L)800 2298 y Fm(\)\()p Fl(\033)856 2304 y Fn(V)h(L)917 2298 y Fo(\005)i Fl(L)p Fm(\)\()p Fl(v)i Fo(\012)f Fm(1)f Fo(\012)g Fm(1\))641 2363 y(=)j(\()p Fl(L)e Fo(\005)f Fl(\033)793 2369 y Fn(V)d(L)844 2363 y Fm(\)\()p Fl(v)897 2370 y Fh(\()p Fk(\000)p Fh(1\))977 2363 y Fo(\012)k Fl(v)1040 2370 y Fh(\(0\))1094 2363 y Fo(\012)g Fm(1\))h(=)h Fl(v)1249 2370 y Fh(\()p Fk(\000)p Fh(1\))1329 2363 y Fo(\012)e Fl(v)1392 2370 y Fh(\(0\))r(\()p Fk(\000)p Fh(1\))1517 2363 y Fo(\012)f Fl(v)1579 2370 y Fh(\(0\))s(\(0\))1669 2363 y Fl(:)228 2440 y Fm(Moreo)o(v)o(er,)14 b(w)o(e)g(ha)o(v)o(e)g(\(\001)630 2446 y Fh(0)657 2440 y Fo(\005)9 b Fl(V)h Fm(\))p Fl(\033)761 2446 y Fn(V)c(L)824 2440 y Fm(=)11 b Fl(\033)891 2446 y Fn(V)r(;L)p Fk(\005)p Fn(L)987 2440 y Fm(\()p Fl(V)19 b Fo(\005)9 b Fm(\001)1111 2446 y Fh(0)1129 2440 y Fm(\))14 b(b)o(y)g(naturalit)o(y)e(of)i Fl(\033)q Fm(,)f(so)h(that)269 2526 y Fl(v)290 2533 y Fh(\()p Fk(\000)p Fh(1\))371 2526 y Fo(\012)9 b Fl(v)433 2533 y Fh(\(0\))r(\()p Fk(\000)p Fh(1\))558 2526 y Fo(\012)h Fl(v)621 2533 y Fh(\(0\))r(\(0\))722 2526 y Fm(=)i Fl(\033)790 2532 y Fn(V)r(;L)p Fk(\005)p Fn(L)885 2526 y Fm(\()p Fl(V)19 b Fo(\005)9 b Fm(\001)1009 2532 y Fh(0)1027 2526 y Fm(\)\()p Fl(v)i Fo(\012)f Fm(1\))917 2591 y(=)h(\(\001)1011 2597 y Fh(0)1039 2591 y Fo(\005)e Fl(V)g Fm(\))p Fl(\033)1142 2597 y Fn(V)e(L)1194 2591 y Fm(\()p Fl(v)k Fo(\012)e Fm(1\))j(=)g(\001)1410 2597 y Fh(0)1428 2591 y Fm(\()p Fl(v)1465 2598 y Fh(\()p Fk(\000)p Fh(1\))1536 2591 y Fm(\))d Fo(\012)h Fl(v)1624 2598 y Fh(\(0\))1669 2591 y Fl(:)p eop %%Page: 13 13 13 12 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(13)303 213 y Fm(Next,)13 b(for)g(an)o(y)g Fl(W)k Fo(2)651 219 y Fn(L)676 213 y Fo(M)c Fm(and)g Fl(w)g Fo(2)e Fl(W)6 b Fm(,)13 b(consider)h Fl(\032)1153 219 y Fn(w)1185 213 y Fm(:)f Fl(L)f Fo(3)f Fl(`)h Fo(7!)f Fl(`)d(.)g(w)k Fo(2)f Fl(W)6 b Fm(,)13 b(whic)o(h)g(is)228 262 y(a)g(left)h Fl(L)p Fm(-mo)q(dule)f(map.)j(By)e(naturalit)o(y)f(of) g Fl(\033)q Fm(,)h(w)o(e)g(ha)o(v)o(e)268 330 y Fl(\033)292 336 y Fn(V)6 b(W)356 330 y Fm(\()p Fl(v)11 b Fo(\012)f Fl(w)q Fm(\))h(=)h Fl(\033)571 336 y Fn(V)6 b(W)635 330 y Fm(\()p Fl(V)19 b Fo(\005)9 b Fl(\032)745 336 y Fn(w)772 330 y Fm(\)\()p Fl(v)i Fo(\012)f Fm(1\))h(=)h(\()p Fl(\032)1006 336 y Fn(w)1043 330 y Fo(\005)d Fl(V)g Fm(\))p Fl(\033)1146 336 y Fn(V)e(W)1211 330 y Fm(\()p Fl(v)k Fo(\012)e Fm(1\))j(=)f Fl(v)1412 337 y Fh(\()p Fk(\000)p Fh(1\))1493 330 y Fl(.)e(w)h Fo(\012)f Fl(v)1625 337 y Fh(\(0\))1671 330 y Fl(:)228 398 y Fm(Finally)m(,)i Fl(\033)401 404 y Fn(V)c(L)466 398 y Fm(is)14 b(an)g Fl(L)p Fm(-linear)f(map,)f(so)i(that)269 479 y Fl(`)286 486 y Fh(\(1\))331 479 y Fl(v)352 486 y Fh(\()p Fk(\000)p Fh(1\))433 479 y Fo(\012)9 b Fl(`)491 486 y Fh(\(2\))545 479 y Fl(.)g(v)596 486 y Fh(\(0\))653 479 y Fm(=)j Fl(`)d(.)g(\033)777 485 y Fn(V)e(L)828 479 y Fm(\()p Fl(v)k Fo(\012)f Fm(1\))h(=)h Fl(\033)1033 485 y Fn(V)7 b(L)1085 479 y Fm(\()p Fl(`)i(.)g Fm(\()p Fl(v)i Fo(\012)f Fm(1\)\))564 543 y(=)i Fl(\033)632 549 y Fn(V)7 b(L)683 543 y Fm(\()p Fl(`)716 550 y Fh(\(1\))761 543 y Fl(v)782 550 y Fh(\()p Fk(\000)p Fh(1\))863 543 y Fo(\012)i Fl(`)921 550 y Fh(\(2\))975 543 y Fl(.)g(v)1026 550 y Fh(\(0\))1072 543 y Fm(\))i(=)h(\()p Fl(`)1176 550 y Fh(\(1\))1230 543 y Fl(.)d(v)q Fm(\))1297 550 y Fh(\()p Fk(\000)p Fh(1\))1369 543 y Fl(`)1386 550 y Fh(\(2\))1440 543 y Fo(\012)g Fm(\()p Fl(`)1514 550 y Fh(\(1\))1568 543 y Fl(.)g(v)q Fm(\))1635 550 y Fh(\(0\))228 612 y Fm(and)14 b Fl(V)23 b Fm(is)14 b(a)f(Y)m(etter-Drinfeld)h(mo)q(dule.) 303 661 y(It)j(is)g(clear)h(that)g(b)o(y)f(the)h(equiv)n(alence)g(with) f(the)h(monoidal)c(category)k Fo(Z)1509 667 y Fh(0)1528 661 y Fm(\()1544 667 y Fn(L)1569 661 y Fo(M)p Fm(\),)g(the)228 711 y(category)394 696 y Fn(L)394 723 y(L)419 711 y Fo(Y)s(D)c Fm(is)e(also)g(monoidal,)e(with)i(the)h(mo)q(dule)e(structure)k(of)d (the)i(tensor)f(pro)q(duct)h(of)228 762 y Fl(V)r(;)7 b(W)17 b Fo(2)368 747 y Fn(L)368 773 y(L)393 762 y Fo(Y)s(D)e Fm(as)f(in)571 768 y Fn(L)596 762 y Fo(M)p Fm(.)k(F)m(or)13 b(the)i(como)q(dule)d(structure)k(w)o(e)e(compute)269 842 y Fl(\025)293 848 y Fh(0)312 842 y Fm(\()p Fl(v)d Fo(\012)f Fl(w)q Fm(\))h(=)h Fl(\033)527 848 y Fn(V)6 b Fk(\005)p Fn(W)o(;L)635 842 y Fm(\()p Fl(v)11 b Fo(\012)f Fl(w)g Fo(\012)g Fm(1\))h(=)h(\()p Fl(\033)938 848 y Fn(V)6 b(L)998 842 y Fo(\005)j Fl(W)d Fm(\)\()p Fl(V)19 b Fo(\005)9 b Fl(\033)1202 848 y Fn(W)t(L)1263 842 y Fm(\)\()p Fl(v)i Fo(\012)e Fl(w)h Fo(\012)g Fm(1\))652 907 y(=)i Fl(\033)720 913 y Fn(V)7 b(L)771 907 y Fm(\()p Fl(v)k Fo(\012)f Fl(w)891 914 y Fh(\()p Fk(\000)p Fh(1\))970 907 y Fo(\012)g Fl(w)1043 914 y Fh(\(0\))1087 907 y Fm(\))i(=)g Fl(v)1180 914 y Fh(\()p Fk(\000)p Fh(1\))1251 907 y Fl(w)1282 914 y Fh(\()p Fk(\000)p Fh(1\))1362 907 y Fo(\012)d Fl(v)1424 914 y Fh(\(0\))1478 907 y Fo(\012)h Fl(w)1551 914 y Fh(\(0\))1595 907 y Fl(:)p 1650 907 2 29 v 1652 880 25 2 v 1652 907 V 1677 907 2 29 v 669 994 a Fp(5.)24 b(Sk)o(ew)16 b(pairings)d(and)i (Dualit)o(y)303 1068 y Fm(In)20 b(this)g(section)g(w)o(e)h(will)d (de\014ne)j(sk)o(ew)g(pairings)e(b)q(et)o(w)o(een)j Fo(\002)1338 1074 y Fn(R)1365 1068 y Fm(-bialgebras,)f(as)f(w)o(ell)228 1118 y(as)f(duals)f(of)g(a)h Fo(\002)521 1124 y Fn(R)548 1118 y Fm(-bialgebra)f Fl(L)p Fm(,)i(whic)o(h)e(will)f(ha)o(v)o(e)i(sk) o(ew)g(pairings)f(with)h Fl(L)p Fm(.)32 b(In)19 b(the)g Fl(k)q Fm(-)228 1168 y(bialgebra)d(case,)j(our)e(de\014nitions)h(sp)q (ecialize)g(to)f(\(trivial)f(v)n(ariations)g(of)s(\))h(the)h(w)o (ell-kno)o(wn)228 1218 y(de\014nition,)12 b(see)h([)p Fp(2)p Fm(].)k(It)12 b(is)h(not)f(surprising)h(that)f(the)h(duals)g(of) e Fo(\002)1256 1224 y Fn(R)1283 1218 y Fm(-bialgebras)h(are)h(mo)q (delled)228 1268 y(on)g(the)h(dual)f Fl(R)p Fm(-mo)q(dules)f(of)h(the)h (algebra)e(in)h(question,)h(but)f(it)g(is)h(a)f(curious)h(p)q(oin)o(t)e (that)i(w)o(e)228 1317 y(cannot)i(de\014ne)g(Hopf)g(algebra)f(pairings) g(\(not)h(sk)o(ew\))g(b)q(et)o(w)o(een)h Fo(\002)1300 1323 y Fn(R)1328 1317 y Fm(-bialgebras,)e(nor)g(duals)228 1367 y(that)e(w)o(ould)f(directly)h(generalize)h(the)f(duals)g(of)g Fl(k)q Fm(-bialgebras.)k(It)c(app)q(ears)g(that)g(taking)g(the)228 1417 y(opp)q(osite)h(algebra)f(or)h(coalgebra)g(structure)i(has)e(to)g (b)q(e)g(built)f(in)h(from)e(the)i(start.)303 1492 y Fc(Definition)h Fm(5.1)p Fc(.)21 b Fm(Let)16 b Fl(R)h Fm(b)q(e)f(a)h Fl(k)q Fm(-algebra,)e(\003)h(and)h Fl(L)f Fm(t)o(w)o(o)g Fo(\002)1328 1498 y Fn(R)1355 1492 y Fm(-bialgebras.)25 b(A)17 b(sk)o(ew)228 1542 y(pairing)c(b)q(et)o(w)o(een)i(\003)f(and)f Fl(L)h Fm(is)g(a)g(map)e Fl(\034)d Fm(:)14 b(\003)9 b Fo(\012)g Fl(L)j Fo(!)f Fl(R)j Fm(satisfying)578 1610 y Fl(\034)5 b Fm(\(\()p Fl(r)10 b Fo(\012)p 703 1587 20 2 v 9 w Fl(s)q Fm(\))p Fl(\030)r Fm(\()p Fl(t)f Fo(\012)p 841 1587 24 2 v 10 w Fl(u)p Fm(\))p Fo(j)p Fl(`)p Fm(\))p Fl(v)k Fm(=)f Fl(r)q(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fm(\()p Fl(t)k Fo(\012)p 1175 1587 22 2 v 9 w Fl(v)r Fm(\))p Fl(`)p Fm(\()p Fl(u)g Fo(\012)p 1321 1587 20 2 v 10 w Fl(s)p Fm(\)\))-1144 b(\(5.1\))533 1683 y Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(`m)p Fm(\))12 b(=)g Fl(\034)5 b Fm(\()p Fl(\030)788 1690 y Fh(\(1\))833 1683 y Fo(j)p Fl(`)p 862 1647 168 2 v(\034)g Fm(\()p Fl(\030)921 1690 y Fh(\(2\))965 1683 y Fo(j)p Fl(m)p Fm(\)\))12 b(=)g Fl(\034)5 b Fm(\()p 1140 1647 V Fl(\034)g Fm(\()p Fl(\030)1199 1690 y Fh(\(2\))1243 1683 y Fo(j)p Fl(m)p Fm(\))p Fl(\030)1327 1690 y Fh(\(1\))1372 1683 y Fo(j)p Fl(`)p Fm(\))-1189 b(\(5.2\))559 1748 y Fl(\034)5 b Fm(\()p Fl(\030)r(\020)s Fo(j)p Fl(`)p Fm(\))11 b(=)h Fl(\034)5 b Fm(\()p Fl(\030)r(\034)g Fm(\()p Fl(\020)s Fo(j)p Fl(`)887 1755 y Fh(\(1\))932 1748 y Fm(\))p Fo(j)p Fl(`)977 1755 y Fh(\(2\))1021 1748 y Fm(\))12 b(=)g Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(\034)g Fm(\()p Fl(\020)s Fo(j)p Fl(`)1253 1755 y Fh(\(1\))1297 1748 y Fm(\))p Fl(`)1330 1755 y Fh(\(2\))1375 1748 y Fm(\))-1163 b(\(5.3\))831 1812 y Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fm(1\))11 b(=)h Fl(")p Fm(\()p Fl(\030)r Fm(\)\(1\))-890 b(\(5.4\))834 1877 y Fl(\034)5 b Fm(\(1)p Fo(j)p Fl(`)p Fm(\))11 b(=)h Fl(")p Fm(\()p Fl(`)p Fm(\)\(1\))-887 b(\(5.5\))228 1945 y(for)13 b(all)g Fl(r)o(;)7 b(s;)g(t;)g(u;)g(v)k Fo(2)g Fl(R)p Fm(,)i Fl(\030)r(;)7 b(\020)14 b Fo(2)e Fm(\003)h(and)h Fl(`;)7 b(m)12 b Fo(2)f Fl(L)p Fm(.)303 2020 y Fc(Remark)16 b Fm(5.2)p Fc(.)k Fm(Assume)e Fl(\034)k Fm(satis\014es)c(\(5.1\))e(in)h(the)h(preceding)g(de\014nition.)28 b(Then)18 b(the)228 2070 y(righ)o(t)f(hand)g(sides)h(of)i(\(5.2\))c (and)h(\(5.3\))g(are)h(w)o(ell-de\014ned)f(\(and)g(the)h(second)h (equalities)e(in)228 2120 y(these)e(equations)f(are)g(satis\014ed.\)) 303 2195 y Fc(Pr)o(oof.)21 b Fm(F)m(or)14 b Fl(\030)r(;)7 b(\030)600 2180 y Fk(0)623 2195 y Fo(2)k Fm(\003,)i Fl(`;)7 b(m)12 b Fo(2)f Fl(L)j Fm(and)g Fl(r)e Fo(2)f Fl(R)j Fm(w)o(e)g(ha)o(v)o(e)371 2265 y Fl(\034)5 b Fm(\()p 410 2242 20 2 v Fl(r)q(\030)r Fo(j)p Fl(`)p 479 2229 135 2 v(\034)g Fm(\()p Fl(\030)538 2253 y Fk(0)550 2265 y Fo(j)p Fl(m)p Fm(\)\))11 b(=)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(`)p 773 2229 V(\034)g Fm(\()p Fl(\030)832 2253 y Fk(0)844 2265 y Fo(j)p Fl(m)p Fm(\))p 915 2242 20 2 v 7 w Fl(r)q Fm(\))11 b(=)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(`)p 1094 2229 155 2 v(r)q(\034)g Fm(\()p Fl(\030)1173 2253 y Fk(0)1185 2265 y Fo(j)p Fl(m)p Fm(\)\))11 b(=)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(`)p 1408 2229 V(\034)g Fm(\()p Fl(r)q(\030)1487 2253 y Fk(0)1499 2265 y Fo(j)p Fl(m)p Fm(\)\))228 2333 y(whic)o(h)13 b(sho)o(ws)g(the)h(middle)d(term)i(of)i(\(5.2\))e(to)g(b)q(e)g(w)o(ell) g(de\014ned.)19 b(The)13 b(reasoning)g(for)g(\(5.3\))f(is)228 2383 y(similar.)p 1692 2383 2 29 v 1694 2356 25 2 v 1694 2383 V 1719 2383 2 29 v 303 2458 a Fc(Lemma)k Fm(5.3)p Fc(.)k Fg(L)n(et)f Fl(L)g Fg(and)h Fm(\003)e Fg(b)n(e)h(two)g Fo(\002)968 2464 y Fn(R)995 2458 y Fg(-bialgebr)n(as,)g(and)h Fl(\034)9 b Fm(:)15 b(\003)d Fo(\012)g Fl(L)19 b Fo(!)g Fl(R)f Fg(a)h(skew)228 2508 y(p)n(airing.)30 b(Then)19 b(a)g(monoidal)g(functor)881 2493 y Fn(L)906 2508 y Fo(M)g(!)1035 2514 y Fh(\003)1059 2508 y Fo(M)g Fg(is)f(de\014ne)n(d)i(by)f(endowing) g(a)g(left)f Fl(L)p Fg(-)228 2558 y(c)n(omo)n(dule)f Fl(V)27 b Fg(with)17 b(the)g(left)f Fm(\003)p Fg(-mo)n(dule)h(structur) n(e)g(de\014ne)n(d)h(by)f Fl(\030)c Fo(\017)e Fl(v)17 b Fm(=)g Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(v)1501 2565 y Fh(\()p Fk(\000)p Fh(1\))1571 2558 y Fm(\))p Fl(v)1608 2565 y Fh(\(0\))1671 2558 y Fg(for)228 2608 y Fl(\030)17 b Fo(2)d Fm(\003)i Fg(and)i Fl(v)e Fo(2)e Fl(V)c Fg(.)23 b(The)17 b(underlying)g(left)f Fl(R)p Fg(-mo)n(dule)g(structur)n(e)g (of)g Fl(V)26 b Fg(as)17 b(a)g Fm(\003)p Fg(-mo)n(dule)f(is)p eop %%Page: 14 14 14 13 bop 228 113 a Fh(14)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fg(the)h(same)g(as)g(the)g(original)f(one,)i(and)g(the)f (underlying)g(left)p 1215 179 32 2 v 16 w Fl(R)p Fg(-mo)n(dule)g (structur)n(e)f(is)h(given)228 262 y(by)p 282 240 20 2 v 15 w Fl(r)q(v)c Fm(=)f Fl(v)q(r)q Fg(.)303 339 y Fc(Pr)o(oof.)21 b Fl(\030)12 b Fo(\017)d Fl(v)15 b Fm(is)f(w)o (ell-de\014ned)g(b)q(ecause)i(of)g(\(5.1\))o(.)i(W)m(e)c(ha)o(v)o(e)478 409 y Fl(\030)r(\020)f Fo(\017)c Fl(v)k Fm(=)f Fl(\034)5 b Fm(\()p Fl(\030)r(\020)s Fo(j)p Fl(v)749 416 y Fh(\()p Fk(\000)p Fh(1\))820 409 y Fm(\))p Fl(v)857 416 y Fh(\(0\))914 409 y Fm(=)11 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(\034)g Fm(\()p Fl(\020)s Fo(j)p Fl(v)1121 416 y Fh(\()p Fk(\000)p Fh(1\))r(\(1\))1236 409 y Fm(\))p Fl(v)1273 416 y Fh(\()p Fk(\000)p Fh(1\))s(\(2\))1389 409 y Fm(\))p Fl(v)1426 416 y Fh(\(0\))228 409 y Fm(\(5.6\))228 480 y(and)433 550 y Fl(\030)12 b Fo(\017)d Fl(\020)j Fo(\017)d Fl(v)k Fm(=)f Fl(\030)f Fo(\017)e Fl(\034)c Fm(\()p Fl(\020)s Fo(j)p Fl(v)782 557 y Fh(\()p Fk(\000)p Fh(1\))853 550 y Fm(\))p Fl(v)890 557 y Fh(\(0\))947 550 y Fm(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(\034)g Fm(\()p Fl(\020)s Fo(j)p Fl(v)1155 557 y Fh(\()p Fk(\000)p Fh(1\))1225 550 y Fm(\))p Fl(v)1262 557 y Fh(\(0\))s(\()p Fk(\000)p Fh(1\))1378 550 y Fm(\))p Fl(v)1415 557 y Fh(\(0\))r(\(0\))1505 550 y Fl(:)-1289 b Fm(\(5.7\))228 623 y(The)14 b(t)o(w)o(o)f (expressions)j(coincide)e(b)q(ecause)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(\034)g Fm(\()p Fl(\020)s Fo(j)p Fl(v)1084 630 y Fh(\()p Fk(\000)p Fh(2\))1155 623 y Fm(\))p Fl(v)1192 630 y Fh(\()p Fk(\000)p Fh(1\))1263 623 y Fm(\))p Fl(v)1300 630 y Fh(\(0\))1359 623 y Fm(is)13 b(w)o(ell-de\014ned)i(as)e(an)228 673 y(easy)h(consequence)i(of)h(\(5.1\))o(.)h(W)m(e)c(ha)o(v)o(e)269 756 y(\()p Fl(r)p 305 733 V 1 w(s)q Fm(\))9 b Fo(\017)g Fl(v)k Fm(=)f Fl(\034)5 b Fm(\()p Fl(r)p 516 733 V 1 w(s)p Fo(j)p Fl(v)568 763 y Fh(\()p Fk(\000)p Fh(1\))639 756 y Fm(\))p Fl(v)676 763 y Fh(\(0\))733 756 y Fm(=)12 b Fl(r)q(\034)5 b Fm(\(1)p Fo(j)p Fl(v)890 763 y Fh(\()p Fk(\000)p Fh(1\))p 960 733 V 960 756 a Fl(s)p Fm(\))p Fl(v)1016 763 y Fh(\(0\))1073 756 y Fm(=)12 b Fl(r)q(\034)5 b Fm(\(1)p Fo(j)p Fl(v)1230 763 y Fh(\()p Fk(\000)p Fh(1\))1300 756 y Fm(\))p Fl(v)1337 763 y Fh(\(0\))1383 756 y Fl(s)1218 821 y Fm(=)12 b Fl(r)q(")p Fm(\()p Fl(v)1338 828 y Fh(\()p Fk(\000)p Fh(1\))1410 821 y Fm(\)\(1\))p Fl(v)1500 828 y Fh(\(0\))1545 821 y Fl(s)g Fm(=)g Fl(r)q(v)q(s)228 891 y Fm(whic)o(h)e(sho)o(ws)h(unitalit)o(y)e(of)h(the)i(mo)q(dule)d (structure)k(as)d(w)o(ell)g(as)h(the)g(claim)e(on)h(the)i(underlying) 228 941 y Fl(R)260 926 y Fn(e)277 941 y Fm(-mo)q(dule)h(structure)j(of) d(the)i(left)e(\003-mo)q(dule)f Fl(V)e Fm(.)303 991 y(No)o(w)15 b(consider)i(t)o(w)o(o)f(left)g Fl(L)p Fm(-como)q(dules)f Fl(V)26 b Fm(and)16 b Fl(W)6 b Fm(.)25 b(Of)16 b(course)h Fl(V)25 b Fo(\012)1451 997 y Fn(R)1494 991 y Fl(W)c Fm(=)16 b Fl(V)k Fo(\005)10 b Fl(W)228 1041 y Fm(as)16 b(left)g Fl(R)388 1026 y Fn(e)406 1041 y Fm(-mo)q(dules.)25 b(The)16 b(left)h(\003-mo)q(dule)d(structure)19 b(induced)e(b)o(y)f(the)h(left)f Fl(L)p Fm(-como)q(dule)228 1090 y(structure)g(of)d Fl(V)21 b Fo(\012)530 1096 y Fn(R)569 1090 y Fl(W)f Fm(satis\014es)269 1173 y Fl(\030)12 b Fo(\017)d Fm(\()p Fl(v)i Fo(\012)e Fl(w)q Fm(\))j(=)f Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fm(\()p Fl(v)11 b Fo(\012)e Fl(w)q Fm(\))725 1180 y Fh(\()p Fk(\000)p Fh(1\))796 1173 y Fm(\)\()p Fl(v)i Fo(\012)f Fl(w)q Fm(\))948 1180 y Fh(\(0\))1004 1173 y Fm(=)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(v)1139 1180 y Fh(\()p Fk(\000)p Fh(1\))1210 1173 y Fl(w)1241 1180 y Fh(\()p Fk(\000)p Fh(1\))1312 1173 y Fm(\))p Fl(v)1349 1180 y Fh(\(0\))1403 1173 y Fo(\012)10 b Fl(w)1476 1180 y Fh(\(0\))635 1246 y Fm(=)h Fl(\034)5 b Fm(\()p Fl(\030)737 1253 y Fh(\(1\))782 1246 y Fo(j)p Fl(v)815 1253 y Fh(\()p Fk(\000)p Fh(1\))p 886 1210 233 2 v 886 1246 a Fl(\034)g Fm(\()p Fl(\030)945 1253 y Fh(\(2\))989 1246 y Fo(j)p Fl(w)1032 1253 y Fh(\()p Fk(\000)p Fh(1\))1102 1246 y Fm(\)\))p Fl(v)1155 1253 y Fh(\(0\))1210 1246 y Fo(\012)k Fl(w)1282 1253 y Fh(\(0\))622 1311 y Fm(=)j Fl(\034)5 b Fm(\()p Fl(\030)725 1318 y Fh(\(1\))770 1311 y Fo(j)p Fl(v)803 1318 y Fh(\()p Fk(\000)p Fh(1\))873 1311 y Fm(\))p Fl(v)910 1318 y Fh(\(0\))956 1311 y Fl(\034)g Fm(\()p Fl(\030)1015 1318 y Fh(\(2\))1059 1311 y Fo(j)p Fl(w)1102 1318 y Fh(\()p Fk(\000)p Fh(1\))1172 1311 y Fm(\))10 b Fo(\012)f Fl(w)1270 1318 y Fh(\(0\))1326 1311 y Fm(=)j Fl(\030)1390 1318 y Fh(\(1\))1444 1311 y Fo(\017)d Fl(v)h Fo(\012)g Fl(\030)1566 1318 y Fh(\(2\))1620 1311 y Fo(\017)f Fl(w)228 1382 y Fm(whic)o(h)j(sho)o(ws)g(that)g(the)g (functor)h(from)d(left)h Fl(L)p Fm(-como)q(dules)g(to)h(left)g(\003-mo) q(dules)e(that)i(w)o(e)g(ha)o(v)o(e)228 1431 y(constructed)k(is)e (indeed)g(monoidal.)p 1692 1431 2 29 v 1694 1405 25 2 v 1694 1431 V 1719 1431 2 29 v 303 1511 a(It)e(will)f(no)o(w)h(b)q(e)h (our)f(aim)e(to)i(construct)i(dual)e Fo(\002)1067 1517 y Fn(R)1094 1511 y Fm(-bialgebras)g(from)f(\(suitably)g(\\\014nite"\)) 228 1560 y Fo(\002)260 1566 y Fn(R)287 1560 y Fm(-bialgebras.)31 b(In)18 b(more)f(detail,)h(w)o(e)g(will)f(construct)j(a)e(\\left)g (dual")f Fo(\002)1444 1566 y Fn(R)1471 1560 y Fm(-bialgebra)h Fl(L)1698 1545 y Fk(_)228 1610 y Fm(from)10 b(a)i(\\left)f(\014nite")h Fo(\002)605 1616 y Fn(R)633 1610 y Fm(-bialgebra)e Fl(L)j Fm(whic)o(h)e(will)g(act)h(b)o(y)g(ev)n(aluation)e(on)i(the)h(left)e (of)h Fl(L)g Fm(lik)o(e)228 1660 y(\003)i(in)g(the)h(de\014nition)f(of) g(a)h(sk)o(ew)g(pairing;)e(from)g(a)h(\\righ)o(t)g(\014nite")g Fo(\002)1330 1666 y Fn(R)1357 1660 y Fm(-bialgebra)f(\003)i(in)f(turn) 228 1710 y(w)o(e)g(will)f(de\014ne)j(a)e(\\righ)o(t)f(dual")h Fo(\002)791 1716 y Fn(R)819 1710 y Fm(-bialgebra)1013 1695 y Fk(_)1037 1710 y Fm(\003)g(whic)o(h)h(will)e(act)h(b)o(y)g(ev)n (aluation)f(on)i(the)228 1760 y(righ)o(t)e(of)h(\003)f(lik)o(e)g Fl(L)h Fm(in)g(the)g(de\014nition)g(of)f(a)h(sk)o(ew)g(pairing.)303 1809 y(The)19 b(dualit)o(y)f(b)q(et)o(w)o(een)i Fl(R)736 1794 y Fn(e)754 1809 y Fm(-rings)f(and)g Fo(\002)993 1815 y Fn(R)1020 1809 y Fm(-coalgebras)g(that)g(underlies)h(our)f (dualit)o(y)228 1859 y(b)q(et)o(w)o(een)e Fo(\002)422 1865 y Fn(R)449 1859 y Fm(-bialgebras)e(w)o(as)h(w)o(ork)o(ed)f(out)h (already)f(b)o(y)g(T)m(ak)o(euc)o(hi)g(in)g([)p Fp(20)p Fm(].)22 b(In)16 b(fact,)f([)p Fp(20)p Fm(,)228 1909 y(Prop.)28 b(5.3])16 b(giv)o(es)h(a)g(dualit)o(y)f(b)q(et)o(w)o(een)j (the)f(categories)g(of)f(left)g Fl(R)p Fm(-\014nite)g(pro)r(jectiv)o(e) h Fl(S)r Fo(j)p Fl(R)p Fm(-)228 1960 y(corings)d(and)f(left)p 524 1927 32 2 v 14 w Fl(R)p Fm(-\014nite)h(pro)r(jectiv)o(e)g Fl(S)r Fo(j)p Fl(R)p Fm(-rings.)21 b(But)15 b(an)f Fl(R)1249 1945 y Fn(e)1281 1960 y Fm(ring)h(is)f(an)g Fl(R)p Fo(j)p Fl(R)p Fm(-ring,)f(and)228 2010 y(a)g Fo(\002)295 2016 y Fn(R)322 2010 y Fm(-coalgebra)g Fl(L)h Fm(is)g(the)h(same)e(as)h(an)f Fl(R)p Fo(j)p Fl(R)p Fm(-coring,)f(pro)o(vided)i(that)g(w)o(e)g(iden)o (tify)f(the)i(t)o(w)o(o)228 2060 y Fl(R)p Fm(-bimo)q(dule)i(structures) 22 b(needed)f(for)e(an)g Fl(R)p Fo(j)p Fl(R)p Fm(-coring)f(in)h(the)h (righ)o(t)f(w)o(a)o(y)g(with)g(the)h Fl(R)1691 2045 y Fn(e)1708 2060 y Fm(-)228 2110 y(bimo)q(dule)15 b(structure)k(of)d(the) h Fo(\002)750 2116 y Fn(R)777 2110 y Fm(-coalgebra:)23 b(The)18 b Fl(R)p Fm(-bimo)q(dule)c(structure)19 b(with)e(resp)q(ect) 228 2159 y(to)c(whic)o(h)g(an)h Fl(R)p Fo(j)p Fl(R)p Fm(-coring)e(is)h(an)g Fl(R)p Fm(-coring)g(has)h(to)f(b)q(e)h(obtained) f(from)f(the)i(left)f Fl(R)1558 2144 y Fn(e)1576 2159 y Fm(-mo)q(dule)228 2209 y(structure,)18 b(and)e(the)h(second)g Fl(R)p Fm(-bimo)q(dule)d(structure)k(of)e(the)h Fl(R)p Fo(j)p Fl(R)p Fm(-coring)e(from)f(the)j(righ)o(t)228 2259 y Fl(R)260 2244 y Fn(e)277 2259 y Fm(-mo)q(dule)11 b(structure)j(of)e(the)g Fo(\002)760 2265 y Fn(R)787 2259 y Fm(-coalgebra.)17 b(The)12 b(switc)o(hing)g(of)g(mo)q(dule)e (structures)15 b(is)d(of)228 2309 y(course)j(en)o(tirely)e(trivial,)f (but)h(essen)o(tial,)h(since)g(for)f Fo(\002)1088 2315 y Fn(R)1115 2309 y Fm(-bialgebras)g(the)h(mo)q(dule)e(structures)228 2359 y(to)h(b)q(e)i(used)f(in)g(the)g(de\014nition)f(of)g(the)i (coalgebra)e(structure)j(are)e(already)g(\014xed)g(b)o(y)f(the)i Fl(R)1691 2344 y Fn(e)1708 2359 y Fm(-)228 2409 y(ring)d(structure.)20 b(Instead)14 b(of)e(p)q(erforming)f(the)i(\(trivial)f(but)h (confusing\))g(task)g(of)f(translating)228 2458 y(the)i(dualit)o(y)f (from)f([)p Fp(20)o Fm(])i(to)f(our)h(situation,)f(w)o(e)h(will)f(dev)o (elop)h(it)f(again)g(from)f(scratc)o(h.)303 2508 y(W)m(e)g(start)h(b)o (y)g(de\014ning)g(dualit)o(y)e(on)i(the)g(lev)o(el)g(of)f(mere)h Fl(R)1223 2493 y Fn(e)1240 2508 y Fm(-bimo)q(dules.)j(The)e (de\014nition)228 2558 y(of)9 b(left)g(and)h(righ)o(t)f(dual)g Fl(R)632 2543 y Fn(e)650 2558 y Fm(-bimo)q(dules)f(will)h(b)q(e)h (guided)g(b)o(y)f(the)h(de\014nition)g(of)f(a)g(sk)o(ew)i(pairing:)228 2608 y(The)j(ev)n(aluations)f Fl(L)557 2593 y Fk(_)591 2608 y Fo(\012)c Fl(L)j Fo(!)f Fl(R)j Fm(as)g(w)o(ell)f(as)h(\003)9 b Fo(\012)1037 2593 y Fk(_)1061 2608 y Fm(\003)i Fo(!)g Fl(R)j Fm(should)g(satisfy)g(\(5.1\))o(.)p eop %%Page: 15 15 15 14 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(15)303 213 y Fc(Definition)15 b Fm(5.4)p Fc(.)21 b Fm(Let)16 b Fl(L)h Fm(and)f(\003)g(b)q(e)h(t)o(w)o(o)f Fl(R)1046 198 y Fn(e)1064 213 y Fm(-bimo)q(dules.)23 b(W)m(e)16 b(de\014ne)i(the)f(left)f(dual)228 262 y Fl(R)260 247 y Fn(e)277 262 y Fm(-bimo)q(dule)c(of)i Fl(L)g Fm(to)f(b)q(e)790 316 y Fl(L)818 298 y Fk(_)854 316 y Fm(:=)f(Hom)p 996 304 26 2 v 996 328 a Fn(R)p Fk(\000)1049 316 y Fm(\()p Fl(L;)7 b(R)p Fm(\))228 377 y(with)13 b(the)i Fl(R)426 362 y Fn(e)444 377 y Fm(-bimo)q(dule)c(structure)16 b(de\014ned)f(b)o (y)700 448 y(\(\()p Fl(r)c Fo(\012)p 803 425 20 2 v 9 w Fl(s)q Fm(\))p Fl(\030)r Fm(\()p Fl(t)e Fo(\012)p 941 425 24 2 v 10 w Fl(u)p Fm(\)\)\()p Fl(`)p Fm(\))j(=)g Fl(r)q(\030)r Fm(\()p Fl(t`u)p 1214 425 20 2 v(s)p Fm(\))228 519 y(for)h Fl(r)o(;)7 b(s;)g(t;)g(u)j Fo(2)h Fl(R)p Fm(,)i Fl(\030)h Fo(2)d Fl(L)629 504 y Fk(_)668 519 y Fm(and)i Fl(`)f Fo(2)g Fl(L)p Fm(.)303 569 y(F)m(urthermore,)h(w)o(e)h (de\014ne)h(the)f(righ)o(t)g(dual)f Fl(R)1032 553 y Fn(e)1050 569 y Fm(-bimo)q(dule)f(of)h(\003)g(to)h(b)q(e)789 622 y Fk(_)814 639 y Fm(\003)d(:=)h(Hom)996 645 y Fn(R)p Fk(\000)1049 639 y Fm(\(\003)p Fl(;)7 b(R)p Fm(\))228 710 y(with)13 b(the)i Fl(R)426 695 y Fn(e)444 710 y Fm(-bimo)q(dule)c (structure)16 b(de\014ned)f(b)o(y)700 780 y(\(\()p Fl(r)c Fo(\012)p 803 758 V 9 w Fl(s)q Fm(\))p Fl(`)p Fm(\()p Fl(t)f Fo(\012)p 938 758 24 2 v 9 w Fl(u)p Fm(\)\)\()p Fl(\030)r Fm(\))i(=)g Fl(`)p Fm(\()p 1135 758 V Fl(u\030)r(r)p 1199 750 15 2 v 1 w(t)p Fm(\))p Fl(s)228 851 y Fm(for)h Fl(r)o(;)7 b(s;)g(t;)g(u)j Fo(2)h Fl(R)p Fm(,)i Fl(\030)h Fo(2)d Fm(\003)j(and)g Fl(`)d Fo(2)h Fl(L)p Fm(.)303 901 y(If)h(no)g(confusion)h(is)f(lik)o(ely)m(,)f(w)o(e)i(use)g(the)h (notation)d Fo(h)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)g Fm(:=)g Fl(\030)r Fm(\()p Fl(`)p Fm(\))i(for)f Fl(`)f Fo(2)f Fl(L)j Fm(and)g Fl(\030)f Fo(2)f Fl(L)1698 886 y Fk(_)228 951 y Fm(as)i(w)o(ell)f(as)h Fo(h)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)e Fm(:=)f Fl(`)p Fm(\()p Fl(\030)r Fm(\))j(for)g Fl(`)e Fo(2)777 936 y Fk(_)801 951 y Fm(\003)i(and)g Fl(\030)f Fo(2)f Fm(\003.)303 1027 y(Next,)j(w)o(e)h(endo)o(w)g(the)g (left)f(and)g(righ)o(t)g(dual)g Fl(R)1067 1012 y Fn(e)1084 1027 y Fm(-bimo)q(dules)f(of)h(a)g Fo(\002)1414 1033 y Fn(R)1442 1027 y Fm(-coalgebra)f(with)228 1077 y Fl(R)260 1062 y Fn(e)277 1077 y Fm(-ring)g(structures.)20 b(Again,)13 b(w)o(e)h(are)g(guided)g(b)o(y)g(the)g(de\014nition)g(of)f(sk)o(ew)h (pairings.)303 1154 y Fc(Lemma)i Fm(5.5)p Fc(.)k Fg(L)n(et)f Fl(L)h Fg(b)n(e)f(a)g Fo(\002)796 1160 y Fn(R)823 1154 y Fg(-c)n(o)n(algebr)n(a.)32 b(Then)19 b(an)h Fl(R)1253 1139 y Fn(e)1271 1154 y Fg(-ring)e(structur)n(e)h(on)h Fl(L)1649 1139 y Fk(_)1692 1154 y Fg(is)228 1204 y(de\014ne)n(d)c(by) 610 1257 y Fo(h)p Fl(\030)r(\020)s Fo(j)p Fl(`)p Fo(i)c Fm(:=)f Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(j)p Fl(`)893 1264 y Fh(\(1\))938 1257 y Fo(i)p Fl(`)971 1264 y Fh(\(2\))1016 1257 y Fo(i)h Fm(=)f Fo(h)p Fl(\030)r Fo(h)p Fl(\020)s Fo(j)p Fl(`)1189 1264 y Fh(\(1\))1234 1257 y Fo(ij)p Fl(`)1279 1264 y Fh(\(2\))1324 1257 y Fo(i)228 1319 y Fg(for)j Fl(\030)r(;)7 b(\020)14 b Fo(2)e Fl(L)433 1304 y Fk(_)457 1319 y Fg(,)j(and)638 1372 y Fl(i)d Fm(:)f Fl(R)719 1355 y Fn(e)748 1372 y Fo(3)g Fl(r)g Fo(\012)p 858 1349 20 2 v 9 w Fl(s)h Fo(7!)f Fm(\()p Fl(`)h Fo(7!)f Fl(r)q(")p Fm(\()p Fl(`)p Fm(\)\()p Fl(s)p Fm(\)\))i Fo(2)f Fl(L)1276 1355 y Fk(_)1300 1372 y Fl(:)303 1449 y Fc(Pr)o(oof.)21 b Fm(W)m(e)13 b(ha)o(v)o(e)g(to)g(sho)o(w)g(that)g Fl(L)901 1434 y Fk(_)939 1449 y Fm(is)g(really)f(an)h(asso)q(ciativ)o (e)g Fl(R)1388 1434 y Fn(e)1406 1449 y Fm(-ring.)k(One)d(c)o(hec)o(ks) 228 1499 y(immedia)o(tely)9 b(that)j(m)o(ultlipli)o(cation)d(is)j(w)o (ell-de\014ned.)17 b(F)m(or)12 b Fl(\030)r(;)7 b(\020)s(;)g(\022)12 b Fo(2)f Fl(L)1358 1484 y Fk(_)1395 1499 y Fm(and)g Fl(`)h Fo(2)f Fl(L)h Fm(w)o(e)g(ha)o(v)o(e)269 1582 y Fo(h)p Fm(\()p Fl(\030)r(\020)s Fm(\))p Fl(\022)q Fo(j)p Fl(`)p Fo(i)h Fm(=)f Fo(h)p Fl(\030)r(\020)s Fo(jh)p Fl(\022)q Fo(j)p Fl(`)614 1589 y Fh(\(1\))659 1582 y Fo(i)p Fl(`)692 1589 y Fh(\(2\))737 1582 y Fo(i)g Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(jh)p Fl(\022)q Fo(j)p Fl(`)971 1589 y Fh(\(1\))1015 1582 y Fo(i)p Fl(`)1048 1589 y Fh(\(2\))r(\(1\))1138 1582 y Fo(i)p Fl(`)1171 1589 y Fh(\(2\))r(\(2\))1261 1582 y Fo(i)515 1633 y Fk(\003)508 1656 y Fm(=)f Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(jh)p Fl(\022)q Fo(j)p Fl(`)713 1663 y Fh(\(1\))758 1656 y Fo(i)p Fl(`)791 1663 y Fh(\(2\))836 1656 y Fo(i)p Fl(`)869 1663 y Fh(\(3\))914 1656 y Fo(i)h Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(jh)p Fl(\022)q Fo(j)p Fl(`)1148 1663 y Fh(\(1\))r(\(1\))1237 1656 y Fo(i)p Fl(`)1270 1663 y Fh(\(1\))r(\(2\))1360 1656 y Fo(i)p Fl(`)1393 1663 y Fh(\(2\))1438 1656 y Fo(i)1153 1721 y Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s(\022)q Fo(j)p Fl(`)1331 1728 y Fh(\(1\))1376 1721 y Fo(i)p Fl(`)1409 1728 y Fh(\(2\))1454 1721 y Fo(i)g Fm(=)f Fo(h)p Fl(\030)r Fm(\()p Fl(\020)s(\022)q Fm(\))p Fo(j)p Fl(`)p Fo(i)228 1792 y Fm(using)16 b(crucially)h(that)g(the)g(expression)h(on)f(the)h (righ)o(t)e(of)g(the)i(equalit)o(y)e(mark)o(ed)g Fo(\003)g Fm(is)h(w)o(ell-)228 1842 y(de\014ned,)j(whic)o(h)e(is)h(straigh)o (tforw)o(ard)f(to)g(c)o(hec)o(k.)33 b(F)m(or)18 b Fl(\030)j Fo(2)e Fl(L)1245 1827 y Fk(_)1269 1842 y Fl(;)7 b(`)19 b Fo(2)g Fl(L)g Fm(and)f Fl(r)o(;)7 b(s)19 b Fo(2)g Fl(R)f Fm(w)o(e)228 1892 y(ha)o(v)o(e)269 1975 y Fo(h)p Fl(i)p Fm(\()p Fl(r)11 b Fo(\012)p 387 1952 V 10 w Fl(s)p Fm(\))p Fl(\030)r Fo(j)p Fl(`)p Fo(i)h Fm(=)g Fo(h)p Fl(i)p Fm(\()p Fl(r)e Fo(\012)p 660 1952 V 10 w Fl(s)p Fm(\))p Fo(jh)p Fl(\030)r Fo(j)p Fl(`)772 1982 y Fh(\(1\))816 1975 y Fo(i)p Fl(`)849 1982 y Fh(\(2\))894 1975 y Fo(i)i Fm(=)g Fl(r)q(")p Fm(\()p Fo(h)p Fl(\030)r Fo(j)p Fl(`)1086 1982 y Fh(\(1\))1131 1975 y Fo(i)p Fl(`)1164 1982 y Fh(\(2\))1209 1975 y Fm(\)\()p Fl(s)p Fm(\))742 2048 y(=)g Fl(r)q Fo(h)p Fl(\030)r Fo(j)p Fl(`)871 2055 y Fh(\(1\))916 2048 y Fo(i)p Fl(")p Fm(\()p Fl(`)984 2055 y Fh(\(2\))p 1029 2025 V 1029 2048 a Fl(s)q Fm(\)\(1\))f(=)h Fl(r)q Fo(h)p Fl(\030)r Fo(j)p 1241 2012 187 2 v Fl(")p Fm(\()p Fl(`)1293 2055 y Fh(\(2\))p 1338 2025 20 2 v 1338 2048 a Fl(s)q Fm(\)\(1\))p Fl(`)1444 2055 y Fh(\(1\))1489 2048 y Fo(i)f Fm(=)h Fl(r)q Fo(h)p Fl(\030)r Fo(j)p Fl(`)p 1645 2025 V(s)q Fo(i)228 2119 y Fm(and)269 2202 y Fo(h)p Fl(\030)r(i)p Fm(\()p Fl(r)f Fo(\012)p 407 2179 V 10 w Fl(s)p Fm(\))p Fo(j)p Fl(`)p Fo(i)h Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(i)p Fm(\()p Fl(r)e Fo(\012)p 707 2179 V 9 w Fl(s)q Fm(\))p Fo(j)p Fl(`)772 2209 y Fh(\(1\))816 2202 y Fo(i)p Fl(`)849 2209 y Fh(\(2\))894 2202 y Fo(i)i Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(r)q(")p Fm(\()p Fl(`)1086 2209 y Fh(\(1\))1131 2202 y Fm(\)\()p Fl(s)p Fm(\))p Fl(`)1215 2209 y Fh(\(2\))1261 2202 y Fo(i)1118 2266 y Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(r)q(")p Fm(\()p Fl(`)1282 2273 y Fh(\(1\))1327 2266 y Fl(s)p Fm(\)\(1\))p Fl(`)1432 2273 y Fh(\(2\))1477 2266 y Fo(i)g Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(r)q(`s)p Fo(i)p Fl(:)228 2337 y Fm(This)i(enables)g(us)g(to)g (sho)o(w)g(that)g Fl(i)g Fm(is)g(an)f(algebra)h(map:)282 2408 y Fo(h)p Fl(i)p Fm(\()p Fl(r)c Fo(\012)p 399 2385 V 10 w Fl(s)p Fm(\))p Fl(i)p Fm(\()p Fl(t)g Fo(\012)p 531 2385 24 2 v 10 w Fl(u)o Fm(\))p Fo(j)p Fl(`)p Fo(i)i Fm(=)g Fl(r)q Fo(h)p Fl(i)p Fm(\()p Fl(t)d Fo(\012)p 803 2385 V 10 w Fl(u)p Fm(\))p Fo(j)p Fl(`)p 872 2385 20 2 v(s)p Fo(i)j Fm(=)g Fl(r)q(t")p Fm(\()p Fl(`)p 1050 2385 V(s)q Fm(\)\()p Fl(u)p Fm(\))f(=)h Fl(r)q(t")p Fm(\()p Fl(`)p Fm(\)\()p Fl(us)p Fm(\))h(=)f Fo(h)p Fl(i)p Fm(\()p Fl(r)q(t)e Fo(\012)p 1564 2385 44 2 v 9 w Fl(us)p Fm(\))p Fo(j)p Fl(`)p Fo(i)228 2478 y Fm(and)18 b(v)o(eri\014es)h(that)f(the)h (underlying)f Fl(R)869 2463 y Fn(e)887 2478 y Fm(-bimo)q(dule)e (structure)21 b(on)d Fl(L)1359 2463 y Fk(_)1402 2478 y Fm(is)g(the)h(same)e(as)i(in)228 2528 y(De\014nition)13 b(5.4.)p 1692 2528 2 29 v 1694 2502 25 2 v 1694 2528 V 1719 2528 2 29 v 303 2608 a(Similarl)o(y)e(one)j(pro)o(v)o(es)p eop %%Page: 16 16 16 15 bop 228 113 a Fh(16)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)303 213 y Fc(Lemma)g Fm(5.6)p Fc(.)k Fg(L)n(et)f Fm(\003)g Fg(b)n(e)g(a)h Fo(\002)796 219 y Fn(R)823 213 y Fg(-c)n(o)n(algebr)n (a.)32 b(Then)19 b(an)h Fl(R)1253 198 y Fn(e)1270 213 y Fg(-ring)f(structur)n(e)f(on)1620 198 y Fk(_)1645 213 y Fm(\003)g Fg(is)228 262 y(de\014ne)n(d)e(by)590 318 y Fo(h)p Fl(\030)r Fo(j)p Fl(`m)p Fo(i)c Fm(=)g Fo(h)p Fl(\030)799 325 y Fh(\(1\))844 318 y Fo(j)p Fl(`)p 873 282 145 2 v Fo(h)p Fl(\030)909 325 y Fh(\(2\))954 318 y Fo(j)p Fl(m)p Fo(ii)f Fm(=)h Fo(h)p 1105 282 V(h)p Fl(\030)1141 325 y Fh(\(2\))1186 318 y Fo(j)p Fl(m)p Fo(i)p Fl(\030)1270 325 y Fh(\(1\))1315 318 y Fo(j)p Fl(`)p Fo(i)228 381 y Fg(and)643 436 y Fl(i)5 b Fm(:)13 b Fl(R)719 419 y Fn(e)749 436 y Fo(3)e Fl(r)f Fo(\012)p 858 413 20 2 v 9 w Fl(s)i Fo(7!)g Fm(\()p Fl(\030)h Fo(7!)e Fl(")p Fm(\()p Fl(\030)r Fm(\)\()p Fl(r)q Fm(\))p Fl(s)p Fm(\))i Fo(2)1254 419 y Fk(_)1278 436 y Fm(\003)303 513 y Fc(Remark)j Fm(5.7)p Fc(.)k Fm(If)f Fl(R)h Fm(=)g Fl(k)q Fm(,)g(so)f(that)g(a)g Fo(\002)998 519 y Fn(R)1025 513 y Fm(-coalgebra)g(is)f(just)i(a)e Fl(k)q Fm(-coalgebra,)h(then)228 563 y Fl(L)256 548 y Fk(_)292 563 y Fm(=)12 b(\()p Fl(L)380 548 y Fk(\003)399 563 y Fm(\))415 548 y Fh(op)466 563 y Fm(and)547 548 y Fk(_)571 563 y Fm(\003)g(=)g(\003)685 548 y Fk(\003)717 563 y Fm(as)i(algebras.)303 614 y(F)m(or)h(an)o(y)h Fl(R)p Fm(,)f(w)o(e)i(ha)o(v)o(e)f(\()p Fl(R)729 599 y Fn(e)746 614 y Fm(\))762 599 y Fk(_)802 614 y Fm(=)f(Hom)p 936 602 26 2 v 936 627 a Fn(R)p Fk(\000)989 614 y Fm(\()p Fl(R)c Fo(\012)p 1091 581 32 2 v 11 w Fl(R)o(;)c(R)p Fm(\))1204 603 y Fo(\030)1204 616 y Fm(=)1251 614 y(End)q(\()p Fl(R)p Fm(\).)24 b(One)17 b(c)o(hec)o(ks)g(that)228 664 y(the)d(isomorphism)552 735 y Fl(')579 742 y Fh(\(5.8\))654 735 y Fm(:)f(End)q(\()p Fl(R)p Fm(\))e Fo(3)h Fl(f)k Fo(7!)11 b Fm(\()p Fl(r)f Fo(\012)p 1045 713 20 2 v 10 w Fl(s)i Fo(7!)f Fl(f)t Fm(\()p Fl(r)q Fm(\))p Fl(s)p Fm(\))i Fo(2)e Fm(\()p Fl(R)1340 718 y Fn(e)1358 735 y Fm(\))1374 718 y Fk(_)228 735 y Fm(\(5.8\))228 815 y(is)j(in)f(fact)h(an)f(isomorphism)e(of)i Fl(R)779 800 y Fn(e)797 815 y Fm(-rings.)18 b(Similarly)1105 800 y Fk(_)1130 815 y Fm(\()p Fl(R)1178 800 y Fn(e)1196 815 y Fm(\))1223 804 y Fo(\030)1223 817 y Fm(=)1267 815 y(End)q(\()p 1358 782 32 2 v Fl(R)p Fm(\))c(as)g(algebras.)303 892 y(It)d(will)g(require)i(suitable)e(\014niteness)i(conditions)f(to)g (also)f(dualize)h(an)f Fl(R)1443 877 y Fn(e)1461 892 y Fm(-ring)g(structure)228 942 y(to)j(giv)o(e)f(a)g Fo(\002)431 948 y Fn(R)458 942 y Fm(-coalgebra)g(structure,)j(and)e(to)f(dualize)h Fo(\002)1150 948 y Fn(R)1178 942 y Fm(-bialgebra)e(structures.)303 1019 y Fc(Definition)j Fm(5.8)p Fc(.)21 b Fm(Let)10 b Fl(M)16 b Fm(b)q(e)11 b(an)f Fl(R)890 1004 y Fn(e)908 1019 y Fm(-bimo)q(dule.)15 b(W)m(e)10 b(call)f Fl(M)16 b Fm(left)10 b(\014nite)g(if)g(it)g(is)h(\014nitely)228 1069 y(generated)20 b(pro)r(jectiv)o(e)g(as)g(a)f(left)p 795 1036 V 19 w Fl(R)p Fm(-mo)q(dule.)32 b(W)m(e)19 b(call)g Fl(M)24 b Fm(righ)o(t)19 b(\014nite)g(if)f(it)h(is)h(\014nitely)228 1119 y(generated)15 b(pro)r(jectiv)o(e)g(as)f(a)f(left)h Fl(R)p Fm(-mo)q(dule.)303 1196 y Fc(Remark)i Fm(5.9)p Fc(.)k Fm(As)12 b(a)e(\014rst)i(application)d(of)h(the)i(de\014nition,) e(the)i(categories)f(of)f(left)h(\014nite)228 1246 y(and)j(of)f(righ)o (t)g(\014nite)h Fl(R)594 1231 y Fn(e)612 1246 y Fm(-bimo)q(dules)e(are) j(dual)e(to)h(eac)o(h)g(other.)19 b(In)13 b(more)g(detail:)303 1296 y(If)g Fl(L)h Fm(and)g(\003)g(are)g Fl(R)611 1281 y Fn(e)628 1296 y Fm(-bimo)q(dules,)e(then)j Fl(R)978 1281 y Fn(e)995 1296 y Fm(-bimo)q(dule)d(maps)810 1368 y Fl(')837 1375 y Fh(\(5.9\))912 1368 y Fm(:)i Fl(L)d Fo(!)1031 1350 y Fk(_)1055 1368 y Fm(\()p Fl(L)1099 1350 y Fk(_)1124 1368 y Fm(\))-912 b(\(5.9\))801 1435 y Fl(')828 1442 y Fh(\(5.10\))920 1435 y Fm(:)13 b(\003)f Fo(!)f Fm(\()1055 1417 y Fk(_)1079 1435 y Fm(\003\))1124 1417 y Fk(_)228 1435 y Fm(\(5.10\))228 1509 y(are)k(de\014ned)g(b)o(y)g Fo(h)p Fl(\030)r Fo(j)p Fl(')575 1516 y Fh(\(5.9\))645 1509 y Fm(\()p Fl(`)p Fm(\))p Fo(i)e Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)i Fm(for)f Fl(\030)h Fo(2)d Fl(L)1029 1494 y Fk(_)1068 1509 y Fm(and)j Fl(`)e Fo(2)f Fl(L)p Fm(,)i(and)h Fo(h)p Fl(')1399 1516 y Fh(\(5.10\))1486 1509 y Fm(\()p Fl(\030)r Fm(\))p Fo(j)p Fl(`)p Fo(i)e Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)228 1559 y Fm(for)g Fl(\030)h Fo(2)d Fm(\003)j(and)g Fl(`)d Fo(2)554 1544 y Fk(_)578 1559 y Fm(\003.)303 1609 y(If)j Fl(L)h Fm(is)g(left)g(\014nite,)g(then)g Fl(L)747 1594 y Fk(_)787 1609 y Fm(is)g(righ)o(t)f(\014nite,)h(and)g Fl(')1159 1616 y Fh(\(5.9\))1244 1609 y Fm(is)g(an)g(isomorphism.)j(If)d(\003)f (is)228 1659 y(righ)o(t)f(\014nite,)h(then)540 1644 y Fk(_)565 1659 y Fm(\003)g(is)f(left)h(\014nite,)g(and)f Fl(')947 1666 y Fh(\(5.10\))1048 1659 y Fm(is)h(an)g(isomorphism.)303 1740 y Fc(Lemma)i Fm(5.10)p Fc(.)k Fg(L)n(et)15 b Fl(L)h Fg(b)n(e)g(a)f Fo(\002)802 1746 y Fn(R)829 1740 y Fg(-c)n(o)n(algebr)n (a.)20 b(Then)c(a)g(functor)1330 1725 y Fn(L)1355 1740 y Fo(M)d(!)1472 1746 y Fn(L)1495 1738 y Fd(_)1518 1740 y Fo(M)j Fg(is)f(given)228 1790 y(by)g(endowing)g(a)g(left)g Fl(L)p Fg(-c)n(omo)n(dule)g Fl(V)24 b Fg(with)14 b(the)h(mo)n(dule)g (structur)n(e)f(de\014ne)n(d)i(by)796 1862 y Fl(\030)11 b Fo(\017)e Fl(v)k Fm(=)f Fo(h)p Fl(\030)r Fo(j)p Fl(v)1001 1869 y Fh(\()p Fk(\000)p Fh(1\))1072 1862 y Fo(i)p Fl(v)1109 1869 y Fh(\(0\))228 1862 y Fm(\(5.11\))228 1936 y Fg(for)i Fl(v)f Fo(2)e Fl(V)25 b Fg(and)15 b Fl(\030)f Fo(2)d Fl(L)594 1921 y Fk(_)619 1936 y Fg(.)303 1986 y(If)j Fl(L)h Fg(is)g(left)f(\014nite,)h(the)g(functor)f(is)h(an)g(e)n (quivalenc)n(e.)303 2063 y Fc(Pr)o(oof.)21 b Fm(The)c(functor)f(is)g(w) o(ell-de\014ned)h(b)o(y)f(the)g(same)f(reasoning)i(as)f(in)f(the)i(pro) q(of)f(of)228 2113 y(Lemma)h(5.3.)34 b(No)o(w)19 b(assume)g(that)h Fl(L)g Fm(is)f(left)h(\014nite.)35 b(W)m(e)19 b(ha)o(v)o(e)h(to)f(sho)o (w)h(that)f(an)o(y)g(left)228 2162 y Fl(L)256 2147 y Fk(_)280 2162 y Fm(-mo)q(dule)12 b Fl(V)21 b Fm(arises)14 b(from)d(a)h(left)g Fl(L)p Fm(-como)q(dule)g(structure)j(on)d Fl(V)22 b Fm(in)12 b(the)h(w)o(a)o(y)f(describ)q(ed)j(in)228 2212 y(\(5.11\))o(.)i(W)m(e)11 b(start)h(b)o(y)e(endo)o(wing)h Fl(V)21 b Fm(with)11 b(the)g Fl(R)p Fm(-bimo)q(dule)e(structure)14 b(induced)d(canonically)228 2262 y(b)o(y)17 b(the)h(underlying)g Fl(R)607 2247 y Fn(e)624 2262 y Fm(-mo)q(dule)e(structure)k(of)d(the)i Fl(L)1125 2247 y Fk(_)1149 2262 y Fm(-mo)q(dule)d Fl(V)10 b Fm(.)29 b(Since)18 b Fl(L)g Fm(is)g(a)f(\014nite)228 2312 y(pro)r(jectiv)o(e)d(left)p 493 2279 V 14 w Fl(R)p Fm(-mo)q(dule,)e(w)o(e)i(ha)o(v)o(e,)f(for)h(an)o(y)f(left)h Fl(R)p Fm(-mo)q(dule)e Fl(W)6 b Fm(,)13 b(an)h(isomorphism)683 2384 y Fl(')710 2391 y Fh(\(5.12\))802 2384 y Fm(:)g Fl(L)9 b Fo(\005)g Fl(W)17 b Fo(!)12 b Fm(Hom)1091 2390 y Fn(R)1118 2384 y Fm(\()p Fl(L)1162 2366 y Fk(_)1187 2384 y Fl(;)7 b(W)f Fm(\))-1039 b(\(5.12\))228 2458 y(giv)o(en)16 b(b)o(y)g Fl(')426 2465 y Fh(\(5.12\))514 2458 y Fm(\()p Fl(`)11 b Fo(\012)g Fl(w)q Fm(\)\()p Fl(\030)r Fm(\))17 b(=)f Fo(h)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)p Fl(w)q Fm(.)26 b(Note)18 b Fl(')1046 2465 y Fh(\(5.12\))1149 2458 y Fm(is)f(left)f Fl(R)p Fm(-linear)g(when)h(w)o(e)g(endo)o(w)228 2508 y(Hom)314 2514 y Fn(R)342 2508 y Fm(\()p Fl(L)386 2493 y Fk(_)410 2508 y Fl(;)7 b(W)f Fm(\))18 b(with)h(the)g(left)f Fl(R)p Fm(-mo)q(dule)f(structure)k(induced)e(b)o(y)g(the)g(righ)o(t)f Fl(R)p Fm(-mo)q(dule)228 2558 y(structure)j(of)d Fl(L)491 2543 y Fk(_)516 2558 y Fm(.)33 b(No)o(w,)20 b(using)e Fl(')814 2565 y Fh(\(5.12\))920 2558 y Fm(for)h Fl(W)26 b Fm(=)20 b Fl(V)10 b Fm(,)19 b(w)o(e)g(de\014ne)h Fl(\025)1386 2564 y Fh(0)1410 2558 y Fm(:)15 b Fl(V)29 b Fo(!)20 b Fl(L)13 b Fo(\005)f Fl(V)28 b Fm(b)o(y)228 2608 y Fl(\025)252 2614 y Fh(0)271 2608 y Fm(\()p Fl(v)q Fm(\))12 b(=)g Fl(v)401 2615 y Fh(\()p Fk(\000)p Fh(1\))480 2608 y Fo(\012)d Fl(v)542 2615 y Fh(\(0\))587 2608 y Fm(,)k(where)i Fo(h)p Fl(\030)r Fo(j)p Fl(v)801 2615 y Fh(\()p Fk(\000)p Fh(1\))871 2608 y Fo(i)p Fl(v)908 2615 y Fh(\(0\))965 2608 y Fm(=)d Fl(\030)e(.)e(v)15 b Fm(for)e(all)g Fl(\030)g Fo(2)e Fl(L)1320 2593 y Fk(_)1345 2608 y Fm(.)18 b(One)c(c)o(hec)o(ks)h(that)e Fl(\025)1703 2614 y Fh(0)p eop %%Page: 17 17 17 16 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(17)228 213 y Fm(tak)o(es)14 b(v)n(alues)f(in)h Fl(L)p Fo(\002)566 219 y Fn(R)593 213 y Fl(V)c Fm(,)j(and)g(the)i (induced)f(map)e Fl(\025)5 b Fm(:)13 b Fl(V)21 b Fo(!)11 b Fl(L)p Fo(\002)1263 219 y Fn(R)1290 213 y Fl(V)f Fm(is)j(an)h Fl(R)1455 198 y Fn(e)1472 213 y Fm(-mo)q(dule)e(map.)228 262 y(T)m(o)h(see)i(coasso)q(ciativit)o(y)e(of)h Fl(\025)705 268 y Fh(0)723 262 y Fm(,)g(w)o(e)g(apply)f Fl(')951 269 y Fh(\(5.12\))1052 262 y Fm(t)o(wice)h(to)g(obtain)f(an)h (isomorphism)553 338 y Fl(')580 345 y Fh(\(5.13\))671 338 y Fm(:)g Fl(L)9 b Fo(\005)g Fl(L)h Fo(\005)f Fl(V)21 b Fo(!)11 b Fm(Hom)1016 344 y Fn(R)1043 338 y Fm(\()p Fl(L)1087 321 y Fk(_)1112 338 y Fl(;)c Fm(Hom)1217 344 y Fn(R)1244 338 y Fm(\()p Fl(L)1288 321 y Fk(_)1313 338 y Fl(;)g(V)i Fm(\)\))-1169 b(\(5.13\))228 414 y(with)16 b Fl(')352 421 y Fh(\(5.13\))439 414 y Fm(\()p Fl(`)c Fo(\012)f Fl(m)h Fo(\012)f Fl(v)q Fm(\)\()p Fl(\030)r Fm(\)\()p Fl(\020)s Fm(\))18 b(=)e Fo(h)p Fl(\020)s Fo(jh)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)p Fl(m)p Fo(i)p Fl(v)q Fm(.)27 b(The)17 b(calculations)f(\(5.6\))g(and)h(\(5.7\))e(for)228 464 y Fl(\034)h Fm(=)c Fo(h)p Fm({)c Fo(j)f Fm({)p Fo(i)14 b Fm(sho)o(w)533 537 y Fl(')560 544 y Fh(\(5.13\))648 537 y Fm(\()p Fl(v)685 544 y Fh(\()p Fk(\000)p Fh(1\))r(\(1\))810 537 y Fo(\012)9 b Fl(v)872 544 y Fh(\()p Fk(\000)p Fh(1\))r(\(2\))997 537 y Fo(\012)h Fl(v)1060 544 y Fh(\(0\))1105 537 y Fm(\)\()p Fl(\030)r Fm(\)\()p Fl(\020)s Fm(\))i(=)g(\()p Fl(\030)r(\020)s Fm(\))e Fl(.)f(v)228 610 y Fm(and)514 665 y Fl(')541 672 y Fh(\(5.13\))628 665 y Fm(\()p Fl(v)665 672 y Fh(\()p Fk(\000)p Fh(1\))746 665 y Fo(\012)g Fl(v)808 672 y Fh(\(0\))r(\()p Fk(\000)p Fh(1\))933 665 y Fo(\012)g Fl(v)995 672 y Fh(\(0\))s(\(0\)) 1085 665 y Fm(\)\()p Fl(\030)r Fm(\)\()p Fl(\020)s Fm(\))k(=)f Fl(\030)f(.)e Fm(\()p Fl(\020)j(.)d(v)q Fm(\))228 730 y(whic)o(h)14 b(sho)o(ws)g(coasso)q(ciativit)o(y)m(.)p 1692 730 2 29 v 1694 703 25 2 v 1694 730 V 1719 730 2 29 v 303 813 a(Before)i(dualizing)d Fl(R)644 798 y Fn(e)662 813 y Fm(-ring)h(structures)k(to)d(giv)o(e)f Fo(\002)1129 819 y Fn(R)1156 813 y Fm(-coalgebra)h(structures,)i(w)o(e)e(recall)228 863 y(a)f(standard)g(fact)h(from)d(ring)i(and)g(mo)q(dule)f(theory:)19 b(Let)c Fl(A)f Fm(and)g Fl(B)j Fm(b)q(e)e(t)o(w)o(o)f(rings,)g Fl(X)j Fm(a)d(left)228 913 y Fl(A)p Fm(-mo)q(dule,)e(and)h Fl(Y)24 b Fm(a)13 b Fl(B)r Fm(-)p Fl(A)p Fm(-bimo)q(dule.)k(Then)d(w)o (e)g(ha)o(v)o(e)g(a)g(map)440 986 y Fl(')467 993 y Fh(\(5.14\))559 986 y Fm(:)20 b(Hom)678 992 y Fn(A)705 986 y Fm(\()p Fl(X)q(;)7 b(A)p Fm(\))12 b Fo(\012)837 1016 y Fn(A)878 986 y Fm(Hom)964 992 y Fn(B)993 986 y Fm(\()p Fl(Y)r(;)7 b(B)r Fm(\))12 b Fo(!)f Fm(Hom)1254 992 y Fn(B)1283 986 y Fm(\()p Fl(Y)21 b Fo(\012)1348 1016 y Fn(A)1388 986 y Fl(X)q(;)7 b(B)r Fm(\))p Fl(;)-1275 b Fm(\(5.14\))228 1079 y(with)16 b Fl(')352 1086 y Fh(\(5.14\))439 1079 y Fm(\()p Fl(\030)e Fo(\012)d Fl(\020)s Fm(\)\()p Fl(y)j Fo(\012)d Fl(x)p Fm(\))17 b(=)f Fl(\020)s Fm(\()p Fl(y)q(\030)r Fm(\()p Fl(x)p Fm(\)\).)28 b(If)17 b Fl(X)j Fm(is)d(\014nitely)f (generated)j(pro)r(jectiv)o(e,)e(then)228 1132 y Fl(')255 1139 y Fh(\(5.14\))358 1132 y Fm(is)f(an)g(isomorphism;)e(its)i(in)o(v) o(erse)h(is)f(giv)o(en)g(b)o(y)g Fl(')1162 1139 y Fh(\(5.14\))1249 1117 y Fk(\000)p Fh(1)1294 1132 y Fm(\()p Fl(T)6 b Fm(\))15 b(=)1419 1101 y Ff(P)1470 1132 y Fl(x)1494 1117 y Fn(i)1518 1132 y Fo(\012)c Fl(T)6 b Fm(\({)i Fo(\012)p Fl(x)1692 1138 y Fn(i)1706 1132 y Fm(\))228 1185 y(when)15 b Fl(x)361 1191 y Fh(1)380 1185 y Fl(;)7 b(:)g(:)g(:)k(;)c(x)503 1191 y Fn(n)539 1185 y Fo(2)13 b Fl(X)19 b Fm(and)c Fl(x)739 1170 y Fh(1)757 1185 y Fl(;)7 b(:)g(:)g(:)12 b(;)7 b(x)881 1170 y Fn(n)916 1185 y Fo(2)13 b Fm(Hom)1044 1191 y Fn(A)1071 1185 y Fm(\()p Fl(X)q(;)7 b(A)p Fm(\))15 b(satisfy)1334 1154 y Ff(P)1384 1185 y Fl(x)1408 1170 y Fn(i)1422 1185 y Fm(\()p Fl(x)p Fm(\))p Fl(x)1502 1191 y Fn(i)1529 1185 y Fm(=)f Fl(x)h Fm(for)f(all)228 1235 y Fl(x)d Fo(2)g Fl(X)s Fm(.)303 1285 y(No)o(w)i(let)h Fl(L)g Fm(and)f Fl(M)19 b Fm(b)q(e)14 b(t)o(w)o(o)f Fl(R)804 1270 y Fn(e)822 1285 y Fm(-bimo)q(dules.)j(Then)f(a)e(v)o(ersion)h(of)f Fl(')1407 1292 y Fh(\(5.14\))1508 1285 y Fm(giv)o(es)g(a)h(map)629 1360 y Fl(')656 1367 y Fh(\(5.15\))748 1360 y Fm(:)f Fl(L)801 1343 y Fk(_)835 1360 y Fo(\005)c Fl(M)910 1343 y Fk(_)946 1360 y Fo(!)i Fm(Hom)p 1085 1349 28 2 v 1085 1373 a Fn(R)1112 1360 y Fm(\()p Fl(L)h Fo(\012)p 1172 1373 26 2 v 1172 1397 a Fn(R)1212 1360 y Fl(M)r(;)7 b(R)p Fm(\))-1093 b(\(5.15\))228 1458 y(whic)o(h)14 b(is)f(an)h(isomorphism)d (if)i Fl(M)18 b Fm(is)c(left)g(\014nite.)k(The)c(calculation)228 1545 y(\(5.16\))40 b Fl(')402 1552 y Fh(\(5.15\))490 1545 y Fm(\()p Fl(\030)p 526 1522 20 2 v 2 w(r)10 b Fo(\012)f Fl(\020)s(s)p Fm(\)\()p Fl(`)i Fo(\012)e Fl(m)p Fm(\))k(=)e Fo(h)p Fl(\030)p 881 1522 V 2 w(r)q Fo(j)p Fl(`)p 930 1509 121 2 v Fo(h)p Fl(\020)s(s)p Fo(j)p Fl(m)p Fo(i)q(i)h Fm(=)g Fo(h)p Fl(\030)r Fo(j)p Fl(`r)p 1208 1509 V 1 w Fo(h)p Fl(\020)s Fo(j)p Fl(sm)p Fo(i)q(i)1222 1610 y Fm(=)g Fl(')1293 1617 y Fh(\(5.15\))1380 1610 y Fm(\()p Fl(\030)g Fo(\012)d Fl(\020)s Fm(\)\()p Fl(`r)i Fo(\012)f Fl(sm)p Fm(\))228 1686 y(for)k Fl(\030)h Fo(2)d Fl(L)393 1671 y Fk(_)418 1686 y Fm(,)i Fl(\020)i Fo(2)c Fl(M)563 1671 y Fk(_)587 1686 y Fm(,)i Fl(`)f Fo(2)g Fl(L)p Fm(,)h Fl(m)f Fo(2)f Fl(M)5 b Fm(,)14 b(and)h Fl(r)o(;)7 b(s)12 b Fo(2)g Fl(R)i Fm(sho)o(ws)h(that)g(if)e Fl(\030)f Fo(\012)e Fl(\020)16 b Fo(2)c Fl(L)1557 1671 y Fk(_)1582 1686 y Fo(\002)1614 1692 y Fn(R)1641 1686 y Fl(M)1686 1671 y Fk(_)1711 1686 y Fm(,)228 1736 y(then)i Fl(')349 1743 y Fh(\(5.15\))437 1736 y Fm(\()p Fl(\030)d Fo(\012)f Fl(\020)s Fm(\))k(factors)g(o)o(v)o(er)g Fl(L)e Fo(\012)871 1742 y Fn(R)896 1733 y Fe(e)926 1736 y Fl(M)5 b Fm(,)13 b(and)h(th)o(us)696 1811 y Fl(')723 1818 y Fh(\(5.17\))815 1811 y Fm(:)g Fl(L)869 1794 y Fk(_)893 1811 y Fo(\002)897 1841 y Fn(R)926 1811 y Fl(M)971 1794 y Fk(_)1006 1811 y Fo(!)d Fm(\()p Fl(L)17 b Fo(\012)1115 1841 y Fn(R)1140 1833 y Fe(e)1168 1811 y Fl(M)5 b Fm(\))1229 1794 y Fk(_)228 1811 y Fm(\(5.17\))228 1912 y(with)15 b Fo(h)p Fl(')367 1919 y Fh(\(5.17\))455 1912 y Fm(\()p Fl(\030)e Fo(\012)d Fl(\020)s Fm(\))p Fo(j)p Fl(`)h Fo(\012)g Fl(m)p Fo(i)16 b Fm(=)f Fo(h)p Fl(\030)r Fo(j)p Fl(`)p 844 1876 102 2 v Fo(h)p Fl(\020)s Fo(j)p Fl(m)p Fo(ii)g Fm(=)h Fo(h)p 1040 1876 V(h)p Fl(\020)s Fo(j)p Fl(m)p Fo(i)p Fl(\030)r Fo(j)p Fl(`)p Fo(i)g Fm(is)g(w)o(ell-de\014ned.)24 b(If)16 b Fl(M)21 b Fm(is)15 b(left)228 1962 y(\014nite,)20 b(so)g(that)g Fl(')532 1969 y Fh(\(5.17\))638 1962 y Fm(is)g(an)f(isomorphism,)e (then)k(\(5.16\))d(sho)o(ws)i(more)f(precisely)h(that)228 2012 y Fl(')255 2019 y Fh(\(5.15\))342 2012 y Fm(\()p Fl(\030)9 b Fo(\012)f Fl(\020)s Fm(\))13 b(factors)g(o)o(v)o(er)g Fl(L)f Fo(\012)769 2018 y Fn(R)794 2010 y Fe(e)824 2012 y Fl(M)17 b Fm(if)12 b(and)h(only)f(if)g Fl(\030)d Fo(\012)e Fl(\020)15 b Fo(2)c Fl(L)1291 1997 y Fk(_)1316 2012 y Fo(\002)1348 2018 y Fn(R)1375 2012 y Fl(M)1420 1997 y Fk(_)1445 2012 y Fm(,)h(so)h(that)g Fl(')1635 2019 y Fh(\(5.17\))228 2062 y Fm(is)h(an)f(isomorphism.)i Fl(')613 2069 y Fh(\(5.17\))714 2062 y Fm(is)f(a)f(morphism)f(of)h Fl(R)1064 2047 y Fn(e)1081 2062 y Fm(-bimo)q(dules,)f(since)j(for)e Fl(r)o(;)7 b(s;)g(t;)g(u)j Fo(2)h Fl(R)269 2156 y Fo(h)p Fl(')312 2163 y Fh(\(5.17\))400 2156 y Fm(\()p Fl(r)p 436 2134 20 2 v 1 w(s)p Fm(\()p Fl(\030)h Fo(\012)d Fl(\020)s Fm(\))p Fl(t)p 594 2134 24 2 v(u)p Fm(\))p Fo(j)p Fl(`)g Fo(\012)h Fl(m)p Fo(i)i Fm(=)g Fo(h)p Fl(')865 2163 y Fh(\(5.17\))952 2156 y Fm(\()p Fl(r)q(\030)r(t)e Fo(\012)p 1074 2134 20 2 v 9 w Fl(s)q(\020)p 1115 2134 24 2 v 3 w(u)o Fm(\))p Fo(j)p Fl(`)g Fo(\012)f Fl(m)p Fo(i)j Fm(=)g Fo(h)p Fl(r)q(\030)r(t)p Fo(j)p Fl(`)p 1442 2120 145 2 v Fo(h)p 1458 2134 20 2 v Fl(s)q(\020)p 1499 2134 24 2 v 3 w(u)p Fo(j)p Fl(m)p Fo(ii)324 2229 y Fm(=)g Fl(r)q Fo(h)p Fl(\030)r Fo(j)p Fl(t`)p 468 2193 145 2 v Fo(h)p Fl(\020)s Fo(j)p Fl(mu)p 577 2207 20 2 v(s)p Fo(ii)g Fm(=)g Fl(r)q Fo(h)p Fl(')747 2236 y Fh(\(5.17\))834 2229 y Fm(\()p Fl(\030)g Fo(\012)d Fl(\020)s Fm(\))p Fo(j)p Fl(t`)h Fo(\012)f Fl(mu)p 1113 2207 V(s)q Fo(i)i Fm(=)h Fo(h)p Fl(r)p 1240 2207 V 1 w(s)q(')1287 2236 y Fh(\(5.17\))1374 2229 y Fm(\()p Fl(\030)f Fo(\012)f Fl(\020)s Fm(\))p Fl(t)p 1513 2207 24 2 v(u)p Fo(j)p Fl(`)f Fo(\012)h Fl(m)p Fo(i)p Fl(:)303 2308 y Fc(Theorem)16 b Fm(5.11)p Fc(.)k Fg(L)n(et)f Fl(L)g Fg(b)n(e)g(a)h(left)e(\014nite)i Fl(R)1051 2293 y Fn(e)1068 2308 y Fg(-ring.)31 b(Then)20 b(ther)n(e)f(is)g(a)g(unique)h Fo(\002)1680 2314 y Fn(R)1707 2308 y Fg(-)228 2358 y(c)n(o)n(algebr)n(a)14 b(structur)n(e)g(on)i Fl(L)667 2342 y Fk(_)706 2358 y Fg(satisfying)590 2432 y Fo(h)p Fl(\030)r Fo(j)p Fl(`m)p Fo(i)c Fm(=)g Fo(h)p Fl(\030)799 2439 y Fh(\(1\))844 2432 y Fo(j)p Fl(`)p 873 2396 145 2 v Fo(h)p Fl(\030)909 2439 y Fh(\(2\))954 2432 y Fo(j)p Fl(m)p Fo(ii)f Fm(=)h Fo(h)p 1105 2396 V(h)p Fl(\030)1141 2439 y Fh(\(2\))1186 2432 y Fo(j)p Fl(m)p Fo(i)p Fl(\030)1270 2439 y Fh(\(1\))1315 2432 y Fo(j)p Fl(`)p Fo(i)-1132 b Fm(\(5.18\))228 2508 y Fg(for)17 b(al)r(l)f Fl(\030)j Fo(2)c Fl(L)466 2493 y Fk(_)508 2508 y Fg(and)j Fl(`;)7 b(m)17 b Fo(2)e Fl(L)p Fg(.)27 b Fl(')818 2515 y Fh(\(5.9\))893 2508 y Fm(:)14 b Fl(L)j Fo(!)1021 2493 y Fk(_)1046 2508 y Fm(\()p Fl(L)1090 2493 y Fk(_)1115 2508 y Fm(\))g Fg(is)g(an)h(isomorphism)f(of)g Fl(R)1587 2493 y Fn(e)1605 2508 y Fg(-rings.)228 2558 y(The)c(c)n(ate)n(gories)g(of)g(left)g(\014nite)g Fl(R)754 2543 y Fn(e)772 2558 y Fg(-rings)f(and)i(right)f(\014nite)h Fo(\002)1204 2564 y Fn(R)1231 2558 y Fg(-c)n(o)n(algebr)n(as)f(ar)n(e)g (dual)g(to)h(e)n(ach)228 2608 y(other.)p eop %%Page: 18 18 18 17 bop 228 113 a Fh(18)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)303 213 y Fc(Pr)o(oof.)21 b Fm(Clearly)g(\(5.18\))f(is)h(equiv)n(alen)o(t)f (to)h Fl(')1083 220 y Fh(\(5.17\))1170 213 y Fm(\()p Fl(\030)1206 220 y Fh(\(1\))1265 213 y Fo(\012)14 b Fl(\030)1331 220 y Fh(\(2\))1376 213 y Fm(\))23 b(=)g Fo(r)1505 198 y Fk(_)1530 213 y Fm(\()p Fl(\030)r Fm(\),)f(where)228 262 y Fo(r)5 b Fm(:)14 b Fl(L)22 b Fo(\012)376 268 y Fn(R)401 260 y Fe(e)441 262 y Fl(L)g Fo(!)f Fl(L)f Fm(denotes)h(m)o (ultiplication.)33 b(This)20 b(lea)o(v)o(es)g(us)g(with)g(the)g(only)f (p)q(ossi-)228 312 y(bilit)o(y)e(to)i(de\014ne)i(\001)5 b(:)14 b Fl(L)617 297 y Fk(_)662 312 y Fo(!)20 b Fl(L)752 297 y Fk(_)776 312 y Fo(\002)809 318 y Fn(R)836 312 y Fl(L)864 297 y Fk(_)908 312 y Fm(b)o(y)e(\001)i(=)h Fl(')1105 319 y Fh(\(5.17\))1192 297 y Fk(\000)p Fh(1)1237 312 y Fo(r)1272 297 y Fk(_)1296 312 y Fm(.)33 b(With)19 b(this)g (de\014nition,)228 362 y(\(5.18\))g(is)i(satis\014ed.)38 b(T)m(o)20 b(c)o(hec)o(k)h(that)f(com)o(ultiplication)e(is)i(coasso)q (ciativ)o(e,)i(w)o(e)e(consider)228 412 y(\001)263 418 y Fh(0)286 412 y Fm(:)13 b Fl(L)339 397 y Fk(_)377 412 y Fo(3)g Fl(\030)j Fo(7!)c Fm(\001\()p Fl(\030)r Fm(\))i Fo(2)e Fl(L)675 397 y Fk(_)710 412 y Fo(\005)e Fl(L)769 397 y Fk(_)808 412 y Fm(and)15 b(v)o(erify)g(\(\001)1058 418 y Fh(0)1086 412 y Fo(\005)9 b Fl(L)p Fm(\)\001)1195 418 y Fh(0)1227 412 y Fm(=)14 b(\()p Fl(L)c Fo(\005)g Fm(\001)1393 418 y Fh(0)1411 412 y Fm(\)\001)1462 418 y Fh(0)1480 412 y Fm(.)21 b(T)m(o)15 b(this)g(end)228 462 y(w)o(e)f(use)h(\(5.14\))e(t)o(wice)h(to)g(obtain)f(the)h (isomorphism)557 530 y Fl(')584 537 y Fh(\(5.19\))675 530 y Fm(:)g Fl(L)729 513 y Fk(_)763 530 y Fo(\005)9 b Fl(L)821 513 y Fk(_)855 530 y Fo(\005)f Fl(L)912 513 y Fk(_)949 530 y Fo(!)j Fm(Hom)p 1088 519 28 2 v 1088 543 a Fn(R)1115 530 y Fm(\()p Fl(L)h Fo(\012)p 1175 544 26 2 v 1175 568 a Fn(R)1215 530 y Fl(L)g Fo(\012)p 1259 544 V 1259 568 a Fn(R)1299 530 y Fl(L;)7 b(R)p Fm(\))-1166 b(\(5.19\))228 643 y(with)13 b Fl(')349 650 y Fh(\(5.19\))437 643 y Fm(\()p Fl(\030)e Fo(\012)f Fl(\020)i Fo(\012)d Fl(\022)q Fm(\)\()p Fl(`)i Fo(\012)e Fl(m)h Fo(\012)g Fl(n)p Fm(\))h(=)h Fo(h)p Fl(\030)r Fo(j)p Fl(`)p 965 598 191 2 v Fo(h)p Fl(\020)s Fo(j)p Fl(m)p 1050 606 90 2 v Fo(h)p Fl(\022)q Fo(j)p Fl(n)p Fo(i)q(ii)p Fl(:)h Fm(No)o(w)269 724 y Fl(')296 731 y Fh(\(5.19\))384 724 y Fm(\(\001)435 730 y Fh(0)462 724 y Fo(\005)c Fl(L)p Fm(\)\001)571 730 y Fh(0)590 724 y Fm(\()p Fl(\030)r Fm(\)\()p Fl(`)h Fo(\012)f Fl(m)h Fo(\012)f Fl(n)p Fm(\))j(=)g Fl(')937 731 y Fh(\(5.19\))1024 724 y Fm(\()p Fl(\030)1060 731 y Fh(\(1\))r(\(1\))1159 724 y Fo(\012)d Fl(\030)1220 731 y Fh(\(1\))r(\(2\))1319 724 y Fo(\012)g Fl(\030)1380 731 y Fh(\(2\))1425 724 y Fm(\)\()p Fl(`)h Fo(\012)f Fl(m)h Fo(\012)f Fl(n)p Fm(\))438 805 y(=)j Fo(h)p Fl(\030)518 812 y Fh(\(1\))q(\(1\))607 805 y Fo(j)p Fl(`)p 636 761 323 2 v Fo(h)p Fl(\030)672 812 y Fh(\(1\))r(\(2\))761 805 y Fo(j)p Fl(m)p 809 769 134 2 v Fo(h)p Fl(\030)845 812 y Fh(\(2\))890 805 y Fo(j)p Fl(n)p Fo(iii)f Fm(=)h Fo(h)p Fl(\030)1066 812 y Fh(\(1\))1111 805 y Fo(j)p Fl(`m)p 1176 769 V Fo(h)p Fl(\030)1212 812 y Fh(\(2\))1257 805 y Fo(j)p Fl(n)p Fo(ii)f Fm(=)h Fo(h)p Fl(\030)r Fo(j)p Fl(`mn)p Fo(i)537 887 y Fm(=)g Fo(h)p Fl(\030)617 894 y Fh(\(1\))661 887 y Fo(j)p Fl(`)p 690 850 170 2 v Fo(h)p Fl(\030)726 894 y Fh(\(2\))771 887 y Fo(j)p Fl(mn)p Fo(ii)g Fm(=)g Fo(h)p Fl(\030)968 894 y Fh(\(1\))1012 887 y Fo(j)p Fl(`)p 1041 842 368 2 v Fo(h)p Fl(\030)1077 894 y Fh(\(2\))r(\(1\))1167 887 y Fo(j)p Fl(m)p 1215 850 178 2 v Fo(h)p Fl(\030)1251 894 y Fh(\(2\))q(\(2\))1340 887 y Fo(j)p Fl(n)p Fo(i)o(i)q(i)253 951 y Fm(=)g Fl(')324 958 y Fh(\(5.19\))411 951 y Fm(\()p Fl(\030)447 958 y Fh(\(1\))501 951 y Fo(\012)d Fl(\030)562 958 y Fh(\(2\))r(\(1\))661 951 y Fo(\012)g Fl(\030)722 958 y Fh(\(2\))r(\(2\))812 951 y Fm(\)\()p Fl(`)g Fo(\012)h Fl(m)g Fo(\012)f Fl(n)p Fm(\))j(=)g Fl(')1123 958 y Fh(\(5.19\))1210 951 y Fm(\()p Fl(L)d Fo(\005)g Fm(\001)1328 957 y Fh(0)1347 951 y Fm(\)\001)1398 957 y Fh(0)1416 951 y Fm(\()p Fl(\030)r Fm(\)\()p Fl(`)h Fo(\012)f Fl(m)h Fo(\012)g Fl(n)p Fm(\))228 1022 y(pro)o(v)o(es)16 b(coasso)q(ciativit)o(y)m(.)22 b(It)15 b(is)h(easy)g(to)f(c)o(hec)o(k)i(that)e Fl(")p Fm(\()p Fl(\030)r Fm(\)\()p Fl(r)q Fm(\))g(=)g Fo(h)p Fl(\030)p 1314 1000 20 2 v 2 w(r)q Fo(j)p Fm(1)p Fo(i)f Fm(=)g Fo(h)p Fl(\030)r(r)q Fo(j)p Fm(1)p Fo(i)h Fm(de\014nes)i(a)228 1072 y(counit)d Fl(")5 b Fm(:)13 b Fl(L)432 1057 y Fk(_)468 1072 y Fo(!)e Fm(E.)18 b(The)d(remaining)d(assertions)j(follo)o(w.)p 1692 1072 2 29 v 1694 1046 25 2 v 1694 1072 V 1719 1072 2 29 v 303 1149 a(Similarl)o(y)m(,)10 b(one)k(pro)o(v)o(es:)303 1224 y Fc(Theorem)i Fm(5.12)p Fc(.)k Fg(L)n(et)d Fm(\003)g Fg(b)n(e)g(a)g(right)g(\014nite)g Fl(R)1067 1209 y Fn(e)1085 1224 y Fg(-ring.)25 b(Then)17 b(ther)n(e)g(is)g(a)g(unique)h Fo(\002)1680 1230 y Fn(R)1707 1224 y Fg(-)228 1274 y(c)n(o)n(algebr)n (a)c(structur)n(e)g(on)639 1259 y Fk(_)663 1274 y Fm(\003)h Fg(satisfying)616 1343 y Fo(h)p Fl(\030)r(\020)s Fo(j)p Fl(`)p Fo(i)d Fm(=)g Fo(h)p Fl(\030)r Fo(h)p Fl(\020)s Fo(j)p Fl(`)876 1350 y Fh(\(1\))921 1343 y Fo(ij)p Fl(`)966 1350 y Fh(\(2\))1010 1343 y Fo(i)g Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(j)p Fl(`)1196 1350 y Fh(\(1\))1240 1343 y Fo(i)p Fl(`)1273 1350 y Fh(\(2\))1318 1343 y Fo(i)-1106 b Fm(\(5.20\))228 1414 y Fg(for)18 b(al)r(l)f Fl(\030)r(;)7 b(\020)21 b Fo(2)c Fm(\003)h Fg(and)h Fl(`)f Fo(2)g Fl(L)p Fg(.)29 b Fl(')792 1421 y Fh(\(5.10\))884 1414 y Fm(:)14 b(\003)k Fo(!)f Fm(\()1032 1399 y Fk(_)1057 1414 y Fm(\003\))1102 1399 y Fk(_)1144 1414 y Fg(is)h(an)h(isomorphism)f(of)g Fl(R)1587 1399 y Fn(e)1605 1414 y Fg(-rings.)228 1464 y(The)13 b(c)n(ate)n(gories)g(of)g(right)g(\014nite)g Fl(R)782 1449 y Fn(e)800 1464 y Fg(-rings)g(and)h(left)e(\014nite)i Fo(\002)1204 1470 y Fn(R)1231 1464 y Fg(-c)n(o)n(algebr)n(as)f(ar)n(e)g (dual)g(to)h(e)n(ach)228 1514 y(other.)303 1589 y Fc(Theorem)i Fm(5.13)p Fc(.)k Fg(L)n(et)11 b Fl(L)g Fg(\(r)n(esp.)f Fm(\003)p Fg(\))h(b)n(e)h(a)f(left)f(\(r)n(esp.)h(right\))f(\014nite)h Fo(\002)1399 1595 y Fn(R)1426 1589 y Fg(-bialgebr)n(a.)17 b(Then)228 1639 y Fl(L)256 1624 y Fk(_)296 1639 y Fg(is)d(a)i(right)e (\014nite)i Fo(\002)616 1645 y Fn(R)643 1639 y Fg(-bialgebr)n(a,)844 1624 y Fk(_)868 1639 y Fm(\003)f Fg(is)g(a)g(left)f(\014nite)i Fo(\002)1204 1645 y Fn(R)1231 1639 y Fg(-bialgebr)n(a,)f(and)h Fl(')1540 1646 y Fh(\(5.9\))1615 1639 y Fm(:)d Fl(L)g Fo(!)228 1674 y Fk(_)252 1689 y Fm(\()p Fl(L)296 1674 y Fk(_)321 1689 y Fm(\))i Fg(and)h Fl(')460 1696 y Fh(\(5.10\))552 1689 y Fm(:)d(\003)e Fo(!)g Fm(\()686 1674 y Fk(_)711 1689 y Fm(\003\))756 1674 y Fk(_)795 1689 y Fg(ar)n(e)k(isomorphisms)f (of)h Fo(\002)1208 1695 y Fn(R)1236 1689 y Fg(-bialgebr)n(as.)303 1739 y(The)c(c)n(ate)n(gories)g(of)h(left)f(\014nite)g(and)i(right)e (\014nite)g Fo(\002)1098 1745 y Fn(R)1125 1739 y Fg(-bialgebr)n(as)g (ar)n(e)g(dual)h(to)g(e)n(ach)g(other.)303 1814 y Fc(Pr)o(oof.)21 b Fm(W)m(e)15 b(only)f(sho)o(w)h(that)g Fl(L)854 1799 y Fk(_)894 1814 y Fm(is)g(a)g Fo(\002)1005 1820 y Fn(R)1032 1814 y Fm(-bialgebra.)20 b(T)m(o)15 b(do)g(so,)f(w)o(e)i(apply)e Fl(')1635 1821 y Fh(\(5.17\))228 1864 y Fm(to)f(\()p Fl(\030)r(\020)s Fm(\))351 1871 y Fh(\(1\))405 1864 y Fo(\012)d Fm(\()p Fl(\030)r(\020)s Fm(\))520 1871 y Fh(\(2\))576 1864 y Fo(2)i Fl(L)644 1849 y Fk(_)668 1864 y Fo(\002)701 1870 y Fn(R)728 1864 y Fl(L)756 1849 y Fk(_)794 1864 y Fm(and)i Fl(\030)895 1871 y Fh(\(1\))939 1864 y Fl(\020)960 1871 y Fh(\(1\))1014 1864 y Fo(\012)9 b Fl(\030)1075 1871 y Fh(\(2\))1120 1864 y Fl(\020)1141 1871 y Fh(\(2\))1197 1864 y Fo(2)i Fl(L)1264 1849 y Fk(_)1289 1864 y Fo(\002)1321 1870 y Fn(R)1349 1864 y Fl(L)1377 1849 y Fk(_)1415 1864 y Fm(for)i Fl(\030)r(;)7 b(\020)15 b Fo(2)c Fl(L)1617 1849 y Fk(_)1655 1864 y Fm(and)228 1914 y(get)269 1995 y Fo(h)p Fl(')312 2002 y Fh(\(5.17\))400 1995 y Fm(\(\()p Fl(\030)r(\020)s Fm(\))489 2002 y Fh(\(1\))543 1995 y Fo(\012)f Fm(\()p Fl(\030)r(\020)s Fm(\))658 2002 y Fh(\(2\))703 1995 y Fm(\))p Fo(j)p Fl(`)f Fo(\012)h Fl(m)p Fo(i)i Fm(=)f Fo(h)p Fl(\030)r(\020)s Fo(j)p Fl(`m)p Fo(i)i Fm(=)f Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(j)p Fm(\()p Fl(`m)p Fm(\))1283 2002 y Fh(\(1\))1328 1995 y Fo(i)p Fm(\()p Fl(`m)p Fm(\))1429 2002 y Fh(\(2\))1475 1995 y Fo(i)363 2068 y Fm(=)g Fo(h)p Fl(\030)r Fo(jh)p Fl(\020)s Fo(j)p Fl(`)521 2075 y Fh(\(1\))565 2068 y Fl(m)601 2075 y Fh(\(1\))646 2068 y Fo(i)p Fl(`)679 2075 y Fh(\(2\))724 2068 y Fl(m)760 2075 y Fh(\(2\))805 2068 y Fo(i)g Fm(=)g Fo(h)p Fl(\030)913 2075 y Fh(\(1\))958 2068 y Fo(jh)p Fl(\020)1007 2075 y Fh(\(1\))1051 2068 y Fo(j)p Fl(`)1080 2075 y Fh(\(1\))p 1124 2032 191 2 v 1124 2068 a Fo(h)p Fl(\020)1161 2075 y Fh(\(2\))1206 2068 y Fo(j)p Fl(m)1254 2075 y Fh(\(1\))1299 2068 y Fo(ii)p Fl(`)1348 2075 y Fh(\(2\))p 1393 2032 190 2 v 1393 2068 a Fo(h)p Fl(\030)1429 2075 y Fh(\(2\))1474 2068 y Fo(j)p Fl(m)1522 2075 y Fh(\(2\))1566 2068 y Fo(ii)598 2141 y Fm(=)g Fo(h)p Fl(\030)678 2148 y Fh(\(1\))723 2141 y Fo(jh)p Fl(\020)772 2148 y Fh(\(1\))816 2141 y Fo(j)p Fl(`)845 2148 y Fh(\(1\))889 2141 y Fo(i)p Fl(`)922 2148 y Fh(\(2\))967 2141 y Fo(h)p Fl(\020)1004 2148 y Fh(\(2\))1049 2141 y Fo(j)p Fl(m)1097 2148 y Fh(\(1\))1142 2141 y Fo(i)p 1158 2105 V(h)p Fl(\030)1194 2148 y Fh(\(2\))1239 2141 y Fo(j)p Fl(m)1287 2148 y Fh(\(2\))1331 2141 y Fo(ii)598 2214 y Fm(=)g Fo(h)p Fl(\030)678 2221 y Fh(\(1\))p 723 2178 191 2 v 723 2214 a Fo(h)p Fl(\020)760 2221 y Fh(\(2\))804 2214 y Fo(j)p Fl(m)852 2221 y Fh(\(1\))897 2214 y Fo(ijh)p Fl(\020)962 2221 y Fh(\(1\))1007 2214 y Fo(j)p Fl(`)1036 2221 y Fh(\(1\))1080 2214 y Fo(i)p Fl(`)1113 2221 y Fh(\(2\))p 1158 2178 190 2 v 1158 2214 a Fo(h)p Fl(\030)1194 2221 y Fh(\(2\))1239 2214 y Fo(j)p Fl(m)1287 2221 y Fh(\(2\))1331 2214 y Fo(ii)598 2287 y Fm(=)g Fo(h)p Fl(\030)678 2294 y Fh(\(1\))723 2287 y Fo(jh)p Fl(\020)772 2294 y Fh(\(1\))816 2287 y Fo(j)p Fl(`)845 2294 y Fh(\(1\))889 2287 y Fo(i)p Fl(`)922 2294 y Fh(\(2\))p 967 2251 381 2 v 967 2287 a Fo(h)p Fl(\030)1003 2294 y Fh(\(2\))1048 2287 y Fo(h)p Fl(\020)1085 2294 y Fh(\(2\))1130 2287 y Fo(j)p Fl(m)1178 2294 y Fh(\(1\))1223 2287 y Fo(ij)p Fl(m)1287 2294 y Fh(\(2\))1331 2287 y Fo(ii)561 2361 y Fm(=)g Fo(h)p Fl(\030)641 2368 y Fh(\(1\))686 2361 y Fl(\020)707 2368 y Fh(\(1\))752 2361 y Fo(j)p Fl(`)p 792 2324 211 2 v 11 w Fo(h)p Fl(\030)828 2368 y Fh(\(2\))873 2361 y Fl(\020)894 2368 y Fh(\(2\))939 2361 y Fo(j)p Fl(m)p Fo(ii)f Fm(=)h Fo(h)p Fl(')1117 2368 y Fh(\(5.17\))1205 2361 y Fm(\()p Fl(\030)1241 2368 y Fh(\(1\))1285 2361 y Fl(\020)1306 2368 y Fh(\(1\))1360 2361 y Fo(\012)e Fl(\030)1422 2368 y Fh(\(2\))1467 2361 y Fl(\020)1488 2368 y Fh(\(2\))1532 2361 y Fm(\))p Fo(j)p Fl(`)g Fo(\012)f Fl(m)p Fo(i)228 2429 y Fm(for)k Fl(`;)7 b(m)12 b Fo(2)f Fl(L)p Fm(.)p 1692 2429 2 29 v 1694 2403 25 2 v 1694 2429 V 1719 2429 2 29 v 303 2506 a Fc(Example)17 b Fm(5.14)p Fc(.)i Fm(If)f Fl(R)g Fm(is)g(a)f(\014nitely)h(generated)h (pro)r(jectiv)o(e)g Fl(k)q Fm(-mo)q(dule,)e(the)h(dualit)o(y)228 2556 y(theory)e(ab)q(o)o(v)o(e)g(applies)f(to)h(the)g Fo(\002)778 2562 y Fn(R)806 2556 y Fm(-bialgebra)e Fl(R)1033 2541 y Fn(e)1067 2556 y Fm(to)h(giv)o(e)h(a)f Fo(\002)1275 2562 y Fn(R)1303 2556 y Fm(-bialgebra)f(structure)k(on)228 2608 y Fl(E)g Fm(=)d(End)q(\()p Fl(R)p Fm(\).)25 b(F)m(or)16 b Fl(f)21 b Fo(2)15 b Fl(E)r Fm(,)i(write)840 2597 y(^)831 2608 y Fl(f)k Fm(:=)15 b Fl(')958 2615 y Fh(\(5.8\))1029 2608 y Fm(\()p Fl(f)t Fm(\))i Fo(2)e Fm(\()p Fl(R)1193 2593 y Fn(e)1211 2608 y Fm(\))1227 2593 y Fk(_)1252 2608 y Fm(.)25 b(Then)17 b(\001)1435 2614 y Fh(0)1453 2608 y Fm(\()p Fl(f)t Fm(\))g Fo(2)e Fl(E)e Fo(\005)e Fl(E)18 b Fm(is)p eop %%Page: 19 19 19 18 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(19)228 213 y Fm(determined)13 b(b)o(y)h Fl(')529 220 y Fh(\(5.15\))616 213 y Fm(\()641 202 y(^)632 213 y Fl(f)657 220 y Fh(\(1\))710 213 y Fo(\012)759 202 y Fm(^)750 213 y Fl(f)775 220 y Fh(\(2\))820 213 y Fm(\)\()p Fl(r)9 b Fo(\012)p 921 190 20 2 v 9 w Fl(s)g Fo(\012)f Fl(u)h Fo(\012)p 1062 190 22 2 v 8 w Fl(v)r Fm(\))i(=)1164 202 y(^)1155 213 y Fl(f)5 b Fm(\()p Fl(r)q(u)j Fo(\012)p 1289 190 42 2 v 9 w Fl(v)q(s)q Fm(\).)18 b(With)13 b(the)h(canonical) 228 262 y(isomorphism)537 346 y Fl(')564 353 y Fh(\(5.21\))656 346 y Fm(:)20 b(Hom)p 775 335 26 2 v 775 359 a Fn(R)p Fk(\000)828 346 y Fm(\()p Fl(R)876 329 y Fn(e)905 346 y Fo(\012)p 909 360 V 909 384 a Fn(R)949 346 y Fl(R)981 329 y Fn(e)999 346 y Fl(;)7 b(R)p Fm(\))k Fo(!)g Fm(Hom)n(\()p Fl(R)e Fo(\012)h Fl(R;)d(R)p Fm(\))-1186 b(\(5.21\))228 466 y(de\014ned)12 b(b)o(y)f Fl(')450 473 y Fh(\(5.21\))537 466 y Fm(\()p Fl(F)6 b Fm(\)\()p Fl(r)f Fo(\012)p 678 443 20 2 v 4 w Fl(s)g Fo(\012)t Fl(u)t Fo(\012)p 802 443 22 2 v 4 w Fl(v)r Fm(\))11 b(=)h Fl(F)6 b Fm(\()p Fl(r)q(u)p Fm(\))p Fl(v)q(s)p Fm(,)12 b(w)o(e)f(ha)o(v)o(e)g Fl(')1246 473 y Fh(\(5.21\))1334 466 y Fl(')1361 473 y Fh(\(5.15\))1448 466 y Fm(\()1473 455 y(^)1464 466 y Fl(f)e Fo(\012)c Fm(^)-22 b Fl(g)q Fm(\)\()p Fl(r)5 b Fo(\012)t Fl(s)p Fm(\))13 b(=)228 516 y Fl(f)t Fm(\()p Fl(r)q Fm(\))p Fl(g)q Fm(\()p Fl(s)p Fm(\),)i(so)g(that)f(\001)580 522 y Fh(0)598 516 y Fm(\()p Fl(f)t Fm(\))h(is)f(c)o(haracterized)i(b)o (y)e(the)h(equation)e Fl(f)1287 523 y Fh(\(1\))1333 516 y Fm(\()p Fl(r)q Fm(\))p Fl(f)1409 523 y Fh(\(2\))1454 516 y Fm(\()p Fl(s)p Fm(\))g(=)f Fl(f)t Fm(\()p Fl(r)q(s)p Fm(\))j(for)228 566 y(all)d Fl(r)o(;)7 b(s)k Fo(2)h Fl(R)p Fm(.)17 b(This)d Fo(\002)581 572 y Fn(R)608 566 y Fm(-bialgebra)f (structure)j(on)d(End)q(\()p Fl(R)p Fm(\))h(w)o(as)f(giv)o(en)h(b)o(y)f (Lu)h(in)g([)p Fp(4)o Fm(].)303 650 y Fc(Cor)o(ollar)m(y)k Fm(5.15)p Fc(.)i Fg(L)n(et)c Fl(L)f Fg(b)n(e)h(a)g(left)f(\014nite)h Fo(\002)1069 656 y Fn(R)1096 650 y Fg(-bialgebr)n(a.)21 b(Then)16 b(an)g(e)n(quivalenc)n(e)g(of)228 700 y(monoidal)e(c)n(ate)n (gories)f(b)n(etwe)n(en)747 685 y Fn(L)772 700 y Fo(M)g Fg(and)915 706 y Fn(L)938 698 y Fd(_)961 700 y Fo(M)h Fg(is)f(given)h(by)g(endowing)g(an)h Fl(L)p Fg(-c)n(omo)n(dule)f Fl(V)228 750 y Fg(with)f(the)g Fl(L)412 735 y Fk(_)437 750 y Fg(-mo)n(dule)g(structur)n(e)g(de\014ne)n(d)i(by)f Fl(\030)8 b Fo(\017)e Fl(v)13 b Fm(=)f Fo(h)p Fl(\030)r Fo(j)p Fl(v)1156 757 y Fh(\()p Fk(\000)p Fh(1\))1227 750 y Fo(i)p Fl(v)1264 757 y Fh(\(0\))1323 750 y Fg(for)g Fl(\030)i Fo(2)d Fl(L)1486 735 y Fk(_)1524 750 y Fg(and)k Fl(v)e Fo(2)e Fl(V)e Fg(.)303 834 y Fc(Remark)16 b Fm(5.16)p Fc(.)k Fm(If)h Fl(R)i Fm(=)g Fl(k)q Fm(,)f(so)f(that)f(a)h Fo(\002)1034 840 y Fn(R)1061 834 y Fm(-bialgebra)f(is)h(just)g(a)f (bialgebra,)h(then)228 884 y Fl(L)256 869 y Fk(_)294 884 y Fm(=)15 b(\()p Fl(L)385 869 y Fk(\003)404 884 y Fm(\))420 869 y Fh(op)473 884 y Fm(and)555 869 y Fk(_)579 884 y Fm(\003)f(=)g(\(\003)713 869 y Fh(op)750 884 y Fm(\))766 869 y Fk(\003)801 884 y Fm(as)h(bialgebras.)22 b(Note)16 b(that)f(a)g(direct)i(generalization)d(to)228 934 y Fo(\002)260 940 y Fn(R)287 934 y Fm(-bialgebras)i(of)g Fl(L)578 919 y Fk(\003)614 934 y Fm(is)g(not)g(de\014ned,)i(since)f (the)g(opp)q(osite)g(of)f(a)g Fo(\002)1357 940 y Fn(R)1384 934 y Fm(-bialgebra)g(is)g(not)g(a)228 984 y Fo(\002)260 990 y Fn(R)287 984 y Fm(-bialgebra.)631 1116 y Fp(6.)24 b(A)16 b(generalized)d(Drinfel)o(d)g(double)303 1191 y Fm(In)e(this)h(\014nal)f(section)i(w)o(e)f(will)e(construct)j(a)f Fo(\002)1038 1197 y Fn(R)1065 1191 y Fm(-bialgebra)f(analog)f(of)i (Drinfeld's)f(quan-)228 1241 y(tum)e(double,)i(and,)f(more)g(generally) m(,)g(a)g(v)o(ersion)g(of)g(the)i(t)o(wist)e(of)g(a)g(tensor)i(pro)q (duct)f(bialgebra)228 1290 y(b)o(y)j(a)g(sk)o(ew)g(pairing,)f(due)i(to) f(Doi)f(and)h(T)m(ak)o(euc)o(hi)f([)p Fp(2)p Fm(].)18 b(Up)c(to)g(trivial)f(mo)q(di\014cations)f(in)i(our)228 1340 y(con)o(v)o(en)o(tions,)f(the)h(outcome)f(will)f(b)q(e)j(the)f (same)f(as)g(in)g([)p Fp(2)p Fm(].)k(There,)e(giv)o(en)e(t)o(w)o(o)g Fl(k)q Fm(-bialgebras)228 1390 y Fl(L)j Fm(and)g(\003)f(and)h(a)g(con)o (v)o(olution)f(in)o(v)o(ertible)g(sk)o(ew)i(pairing)d Fl(\034)c Fm(:)k(\003)c Fo(\012)h Fl(L)k Fo(!)g Fl(k)q Fm(,)h(one)g(can)g(de\014ne)228 1440 y(a)f(t)o(wisted)h(tensor)h(pro)q (duct)f Fl(L)f(.)-7 b(/)772 1446 y Fn(\034)806 1440 y Fm(\003)15 b(to)h(b)q(e)g Fl(L)11 b Fo(\012)f Fm(\003)15 b(as)h(a)f(coalgebra,)g(with)h(m)o(ultiplicati)o(on)228 1490 y(go)o(v)o(erned)e(b)o(y)g(the)g(form)o(ula)505 1574 y(\(1)d Fl(.)-7 b(/)11 b(\030)r Fm(\)\()p Fl(`)h(.)-7 b(/)12 b Fm(1\))f(=)h Fl(\034)5 b Fm(\()p Fl(\030)878 1581 y Fh(\(1\))922 1574 y Fo(j)p Fl(`)951 1581 y Fh(\(1\))996 1574 y Fm(\))p Fl(`)1029 1581 y Fh(\(2\))1085 1574 y Fl(.)-7 b(/)11 b(\030)1151 1581 y Fh(\(2\))1196 1574 y Fl(\034)1219 1556 y Fk(\000)p Fh(1)1263 1574 y Fm(\()p Fl(\030)1299 1581 y Fh(\(3\))1344 1574 y Fo(j)p Fl(`)1373 1581 y Fh(\(3\))1418 1574 y Fm(\))p Fl(:)228 1660 y Fm(When)g Fl(L)g Fm(has)h(an)f(an)o(tip)q(o)q(de,)g(a)g(sk)o(ew)g(pairing)f(is)h (automatically)d(in)o(v)o(ertible)j(with)g Fl(\034)1553 1645 y Fk(\000)p Fh(1)1597 1660 y Fm(\()p Fl(\030)r Fo(j)p Fl(`)p Fm(\))h(=)228 1710 y Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(S)r Fm(\()p Fl(`)p Fm(\)\);)15 b(if)e(w)o(e)h(substitute)i (this)e(ab)q(o)o(v)o(e)g(and)g(further)h(write)f Fl(`)1255 1717 y Fh(\(1\))1310 1710 y Fo(\012)9 b Fl(S)r Fm(\()p Fl(`)1411 1717 y Fh(\(2\))1457 1710 y Fm(\))j(=)h Fl(`)1547 1716 y Fh(+)1584 1710 y Fo(\012)d Fl(`)1643 1716 y Fk(\000)1685 1710 y Fm(as)228 1760 y(suggested)i(b)o(y)e(our)h(discussion)g(of)f (the)i(notion)e(of)g Fo(\002)1043 1766 y Fn(R)1070 1760 y Fm(-Hopf)g(algebra)h(in)f(section)h(3,)g(w)o(e)g(obtain)499 1844 y(\(1)h Fl(.)-7 b(/)11 b(\030)r Fm(\)\()p Fl(`)h(.)-7 b(/)11 b Fm(1\))g(=)h Fl(\034)5 b Fm(\()p Fl(\030)872 1851 y Fh(\(1\))917 1844 y Fo(j)p Fl(`)946 1851 y Fh(\(1\))990 1844 y Fm(\))p Fl(`)1023 1851 y Fh(\(2\))1068 1850 y(+)1107 1844 y Fl(.)-7 b(/)11 b(\030)1173 1851 y Fh(\(2\))1218 1844 y Fl(\034)5 b Fm(\()p Fl(\030)1277 1851 y Fh(\(3\))1322 1844 y Fo(j)p Fl(`)1351 1851 y Fh(\(2\))1395 1850 y Fk(\000)1423 1844 y Fm(\))p Fl(:)228 1928 y Fm(It)17 b(will)e(b)q(e)j(p)q(ossible)f (to)g(generalize)g(this)g(form)o(ula)d(almost)h(directly)i(to)g(the)h Fo(\002)1515 1934 y Fn(R)1542 1928 y Fm(-bialgebra)228 1978 y(case.)26 b(Ho)o(w)o(ev)o(er,)17 b(it)f(will)f(tak)o(e)i(a)f(ma)r (jor)f(amoun)o(t)f(of)i(w)o(ork)g(to)g(sho)o(w)h(that)f(m)o (ultiplication,)228 2028 y(as)i(w)o(e)h(shall)e(de\014ne)i(it,)g(is)f (a)g(w)o(ell-de\014ned)g(map.)30 b(Also,)18 b(one)h(should)f(note)g (that)h(a)f(naiv)o(e)228 2077 y(generalization)c(of)f([)p Fp(2)p Fm(])g(meets)i(a)f(ma)r(jor)e(obstacle:)20 b(There)15 b(is)f(no)h(tensor)g(pro)q(duct)g(bialgebra)228 2127 y(of)k(t)o(w)o(o)g Fo(\002)397 2133 y Fn(R)424 2127 y Fm(-bialgebras)g(that)g(one)h(could)f(t)o(wist)h(to)f(obtain)g Fl(L)i(.)-7 b(/)20 b Fm(\003:)29 b(\\comp)q(onen)o(t)o(wise")228 2177 y(m)o(ultiplicatio)o(n)11 b(on)j(the)g(tensor)h(pro)q(duct)g Fl(L)d Fo(\012)976 2183 y Fn(R)1001 2175 y Fe(e)1031 2177 y Fm(\003)i(is)f(not)h(w)o(ell-de\014ned.)303 2261 y Fc(Lemma)i Fm(6.1)p Fc(.)k Fg(L)n(et)13 b Fl(L)g Fg(b)n(e)g(a)g Fo(\002)770 2267 y Fn(R)797 2261 y Fg(-Hopf)g(algebr)n(a,)f Fm(\003)h Fg(a)g Fo(\002)1170 2267 y Fn(R)1197 2261 y Fg(-bialgebr)n(a,)g(and)g Fl(\034)d Fm(:)j(\003)5 b Fo(\012)g Fl(L)11 b Fo(!)g Fl(R)228 2311 y Fg(a)k(skew)f(p)n(airing.)19 b(Then)c(the)g(map)414 2422 y Fl(')441 2429 y Fh(\(6.1\))516 2422 y Fm(:)549 2365 y Ff(Z)572 2460 y Fn(uv)p 618 2411 V 618 2428 a(u)640 2422 y Fm(\003)9 b Fo(\012)719 2428 y Fn(u)p 739 2411 18 2 v(v)758 2422 y Fm(\003)h Fo(\012)838 2428 y Fn(v)858 2422 y Fm(\003)f Fo(\012)937 2365 y Ff(Z)979 2375 y Fn(t)1000 2365 y Ff(Z)1023 2460 y Fn(r)q(s)p 1064 2411 V 1064 2428 a(s)1082 2422 y Fl(L)p 1110 2410 15 2 v 10 x Fn(t)1134 2422 y Fo(\012)1176 2428 y Fn(s)1193 2422 y Fl(L)1221 2428 y Fn(t)p 1234 2411 17 2 v(r)1262 2422 y Fo(\012)p 1303 2411 19 2 v 1303 2428 a Fn(r)1322 2422 y Fl(L)j Fo(!)f Fl(L)16 b Fo(\012)1454 2452 y Fn(R)1479 2444 y Fe(e)1507 2422 y Fm(\003)-1308 b(\(6.1\))489 2516 y Fl(\030)11 b Fo(\012)e Fl(\030)579 2498 y Fk(0)600 2516 y Fo(\012)h Fl(\030)662 2498 y Fk(0)o(0)692 2516 y Fo(\012)g Fl(m)g Fo(\012)f Fl(m)857 2498 y Fk(0)878 2516 y Fo(\012)h Fl(m)956 2498 y Fk(00)989 2516 y Fo(7!)h Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p Fm(\))p Fl(m)1201 2498 y Fk(0)1223 2516 y Fo(\012)p 1264 2479 166 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1323 2504 y Fk(00)1344 2516 y Fo(j)p Fl(m)1392 2504 y Fk(00)1414 2516 y Fm(\))p Fl(\030)1450 2498 y Fk(0)228 2600 y Fg(is)14 b(wel)r(l-de\014ne)n(d.)p eop %%Page: 20 20 20 19 bop 228 113 a Fh(20)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)303 213 y Fc(Pr)o(oof.)21 b Fm(First)15 b(de\014ne)720 296 y Fl(')747 303 y Fh(\(6.2\))822 296 y Fm(:)e(\003)876 279 y Fk(\012)p Fh(3)930 296 y Fo(\012)c Fl(L)999 279 y Fk(\012)p Fh(3)1056 296 y Fo(!)i Fl(L)16 b Fo(\012)1149 326 y Fn(R)1174 318 y Fe(e)1201 296 y Fm(\003)-1002 b(\(6.2\))483 387 y Fl(\030)11 b Fo(\012)f Fl(\030)574 369 y Fk(0)595 387 y Fo(\012)f Fl(\030)656 369 y Fk(00)687 387 y Fo(\012)g Fl(m)h Fo(\012)f Fl(m)851 369 y Fk(0)873 387 y Fo(\012)g Fl(m)950 369 y Fk(00)983 387 y Fo(7!)i Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p Fm(\))p Fl(m)1195 369 y Fk(0)1217 387 y Fo(\012)p 1258 350 166 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1317 375 y Fk(00)1339 387 y Fo(j)p Fl(m)1387 375 y Fk(0)o(0)1408 387 y Fm(\))p Fl(\030)1444 369 y Fk(0)1456 387 y Fl(:)228 470 y Fm(Then)14 b Fl(')363 477 y Fh(\(6.2\))448 470 y Fm(factors)g(through)595 579 y Fl(')622 586 y Fh(\(6.3\))697 579 y Fm(:)g(\003)752 562 y Fk(\012)p Fh(3)805 579 y Fo(\012)847 523 y Ff(Z)870 617 y Fn(r)q(s)p 911 569 18 2 v 911 586 a(s)928 579 y Fl(L)c Fo(\012)1007 585 y Fn(s)1025 579 y Fl(L)p 1053 569 19 2 v 7 x Fn(r)1081 579 y Fo(\012)p 1123 569 V 1123 586 a Fn(r)1141 579 y Fl(L)i Fo(!)f Fl(L)16 b Fo(\012)1273 610 y Fn(R)1298 601 y Fe(e)1326 579 y Fm(\003)-1127 b(\(6.3\))228 683 y(since)269 779 y Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p 340 756 20 2 v Fl(sm)p Fm(\))p Fl(m)447 762 y Fk(0)p 460 756 V 460 779 a Fl(r)10 b Fo(\012)p 531 743 166 2 v 10 w Fl(\034)5 b Fm(\()p Fl(\030)590 767 y Fk(0)o(0)611 779 y Fo(j)p Fl(m)659 767 y Fk(0)o(0)680 779 y Fm(\))p Fl(\030)716 762 y Fk(0)740 779 y Fm(=)11 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p Fm(\))p Fl(sm)961 762 y Fk(0)983 779 y Fo(\012)p 1025 743 186 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)1084 767 y Fk(0)o(0)1105 779 y Fo(j)p Fl(m)1153 767 y Fk(00)1174 779 y Fm(\))p Fl(r)q(\030)1230 762 y Fk(0)1167 852 y Fm(=)11 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p Fm(\))p Fl(sm)1388 835 y Fk(0)1410 852 y Fo(\012)p 1452 816 V 10 w Fl(\034)g Fm(\()p Fl(\030)1511 840 y Fk(00)1532 852 y Fo(j)p 1544 829 20 2 v Fl(r)q(m)1600 840 y Fk(00)1621 852 y Fm(\))p Fl(\030)1657 835 y Fk(0)1669 852 y Fl(:)228 935 y Fm(W)m(e)13 b(restrict)j Fl(')470 942 y Fh(\(6.3\))554 935 y Fm(to)e(giv)o(e)550 1052 y Fl(')577 1059 y Fh(\(6.4\))652 1052 y Fm(:)f(\003)706 1035 y Fk(\012)p Fh(3)760 1052 y Fo(\012)802 996 y Ff(Z)843 1006 y Fn(t)865 996 y Ff(Z)888 1090 y Fn(r)q(s)p 929 1042 18 2 v 929 1058 a(s)946 1052 y Fl(L)p 974 1041 15 2 v 11 x Fn(t)998 1052 y Fo(\012)1040 1058 y Fn(s)1058 1052 y Fl(L)1086 1058 y Fn(t)p 1099 1042 17 2 v(r)1126 1052 y Fo(\012)p 1168 1042 19 2 v 1168 1058 a Fn(r)1186 1052 y Fl(L)f Fo(!)f Fl(L)16 b Fo(\012)1319 1082 y Fn(R)1344 1074 y Fe(e)1371 1052 y Fm(\003)-1172 b(\(6.4\))228 1157 y(and)14 b(claim)d(that)j Fl(')536 1164 y Fh(\(6.4\))621 1157 y Fm(factors)g(to)g(giv)o(e)f(the)h(desired)h(map)e Fl(')1224 1164 y Fh(\(6.1\))1294 1157 y Fm(.)18 b(In)c(fact)269 1255 y Fl(')296 1262 y Fh(\(6.4\))367 1255 y Fm(\()p 383 1232 24 2 v Fl(u\030)d Fo(\012)p 478 1232 22 2 v 10 w Fl(v)q(\030)519 1238 y Fk(0)540 1255 y Fo(\012)f Fl(\030)602 1238 y Fk(0)o(0)632 1255 y Fo(\012)g Fl(m)f Fo(\012)h Fl(m)797 1238 y Fk(0)818 1255 y Fo(\012)g Fl(m)896 1238 y Fk(00)917 1255 y Fm(\))466 1328 y(=)i Fl(\034)5 b Fm(\()p 549 1305 24 2 v Fl(u\030)r Fo(j)p Fl(m)p Fm(\))p Fl(m)693 1311 y Fk(0)715 1328 y Fo(\012)p 756 1292 188 2 v 9 w Fl(v)q(\034)g Fm(\()p Fl(\030)836 1316 y Fk(00)858 1328 y Fo(j)p Fl(m)906 1316 y Fk(0)o(0)927 1328 y Fm(\))p Fl(\030)963 1311 y Fk(0)987 1328 y Fm(=)11 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p 1137 1305 24 2 v(u)p Fm(\))p Fl(m)1213 1311 y Fk(0)1235 1328 y Fo(\012)p 1276 1292 188 2 v 9 w Fl(\034)g Fm(\()p Fl(v)q(\030)1356 1316 y Fk(00)1378 1328 y Fo(j)p Fl(m)1426 1316 y Fk(00)1447 1328 y Fm(\))p Fl(\030)1483 1311 y Fk(0)379 1401 y Fm(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(m)p Fm(\))p Fl(m)582 1384 y Fk(0)594 1401 y Fl(u)k Fo(\012)p 668 1365 V 9 w Fl(\034)c Fm(\()p Fl(v)q(\030)748 1389 y Fk(0)q(0)770 1401 y Fo(j)p Fl(m)818 1389 y Fk(00)839 1401 y Fm(\))q Fl(\030)876 1384 y Fk(0)899 1401 y Fm(=)12 b Fl(')970 1408 y Fh(\(6.4\))1040 1401 y Fm(\()p Fl(\030)g Fo(\012)d Fl(u\030)1171 1384 y Fk(0)1192 1401 y Fo(\012)g Fl(v)q(\030)1274 1384 y Fk(0)q(0)1306 1401 y Fo(\012)g Fl(m)h Fo(\012)f Fl(m)1470 1384 y Fk(0)1492 1401 y Fo(\012)g Fl(m)1569 1384 y Fk(00)1591 1401 y Fm(\))p 1650 1401 2 29 v 1652 1375 25 2 v 1652 1401 V 1677 1401 2 29 v 303 1499 a Fc(Definition)15 b Fm(6.2)p Fc(.)21 b Fm(Let)16 b Fl(L)g Fm(b)q(e)h(a)f Fo(\002)873 1505 y Fn(R)900 1499 y Fm(-Hopf)g(algebra,)f(\003)h(a)g Fo(\002)1291 1505 y Fn(R)1318 1499 y Fm(-bialgebra)f(and)h Fl(\034)9 b Fm(:)14 b(\003)d Fo(\012)228 1549 y Fl(L)h Fo(!)f Fl(R)i Fm(a)f(sk)o(ew)i(pairing.)j(Then)c Fl(L)f(.)-7 b(/)837 1555 y Fn(\034)869 1549 y Fm(\003)13 b(is)g Fl(L)f Fo(\012)1024 1555 y Fn(R)1049 1547 y Fe(e)1079 1549 y Fm(\003)h(as)g(a)g Fo(\002)1237 1555 y Fn(R)1264 1549 y Fm(-coalgebra,)g(endo)o(w)o(ed)g(with)228 1599 y(the)h(m)o (ultiplication)d(de\014ned)k(b)o(y)459 1684 y(\()p Fl(`)10 b Fo(\012)f Fl(\030)r Fm(\)\()p Fl(m)h Fo(\012)g Fl(\020)s Fm(\))i(=)g Fl(`\034)5 b Fm(\()p Fl(\030)852 1691 y Fh(\(1\))897 1684 y Fo(j)p Fl(m)945 1691 y Fh(\(1\))989 1684 y Fm(\))p Fl(m)1041 1691 y Fh(\(2\))1086 1690 y(+)1123 1684 y Fo(\012)p 1165 1648 241 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)1224 1691 y Fh(\(3\))1268 1684 y Fo(j)p Fl(m)1316 1691 y Fh(\(2\))1361 1690 y Fk(\000)1389 1684 y Fm(\))p Fl(\030)1425 1691 y Fh(\(2\))1470 1684 y Fl(\020)732 1757 y Fm(=)12 b Fl(`\034)5 b Fm(\()p Fl(\030)852 1764 y Fh(\(1\))897 1757 y Fo(j)p Fl(m)945 1763 y Fh(+)972 1764 y(\(1\))1017 1757 y Fm(\))p Fl(m)1069 1763 y Fh(+)1097 1764 y(\(2\))1151 1757 y Fo(\012)p 1192 1721 196 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1251 1764 y Fh(\(3\))1296 1757 y Fo(j)p Fl(m)1344 1763 y Fk(\000)1372 1757 y Fm(\))p Fl(\030)1408 1764 y Fh(\(2\))1453 1757 y Fl(\020)228 1841 y Fm(for)13 b Fl(`;)7 b(m)12 b Fo(2)f Fl(L)j Fm(and)g Fl(\030)r(;)7 b(\020)14 b Fo(2)e Fm(\003.)303 1925 y Fc(Lemma)k Fm(6.3)p Fc(.)k Fg(Multiplic)n(ation)15 b(in)f Fl(L)e(.)-7 b(/)935 1931 y Fn(\034)967 1925 y Fm(\003)15 b Fg(is)f(a)h(wel)r(l-de\014ne)n(d)g Fl(R)1348 1910 y Fn(e)1366 1925 y Fg(-bimo)n(dule)g(map)669 2009 y Fm(\()p Fl(L)d(.)-7 b(/)760 2015 y Fn(\034)792 2009 y Fm(\003\))16 b Fo(\012)849 2039 y Fn(R)874 2030 y Fe(e)901 2009 y Fm(\()p Fl(L)c(.)-7 b(/)992 2015 y Fn(\034)1024 2009 y Fm(\003\))12 b Fo(!)f Fl(L)g(.)-7 b(/)1208 2015 y Fn(\034)1240 2009 y Fm(\003)p Fl(:)228 2110 y Fg(In)15 b(p)n(articular,)f Fl(i)5 b Fm(:)13 b Fl(R)561 2095 y Fn(e)590 2110 y Fo(!)e Fl(L)h(.)-7 b(/)718 2116 y Fn(\034)750 2110 y Fm(\003)14 b Fg(is)h(multiplic)n(ative)f(and)h Fm(1)9 b Fo(\012)h Fm(1)h Fo(2)g Fl(L)h(.)-7 b(/)1392 2116 y Fn(\034)1424 2110 y Fm(\003)15 b Fg(is)f(a)h(unit.)303 2194 y Fc(Pr)o(oof.)21 b Fm(Using)14 b Fl(')610 2201 y Fh(\(6.1\))695 2194 y Fm(from)e(Lemma)f(6.1)i(and)g(\(3.6\),)g(w)o(e) h(can)g(de\014ne)h(a)f(map)678 2277 y Fl(')705 2284 y Fh(\(6.5\))780 2277 y Fm(:)g Fl(L)9 b Fo(\012)h Fm(\003)f Fo(\012)g Fl(L)h Fo(\012)f Fm(\003)j Fo(!)f Fl(L)g(.)-7 b(/)1211 2283 y Fn(\034)1243 2277 y Fm(\003)-1044 b(\(6.5\))465 2350 y Fl(`)10 b Fo(\012)f Fl(\030)i Fo(\012)f Fl(m)g Fo(\012)f Fl(\020)15 b Fo(7!)c Fl(`\034)5 b Fm(\()p Fl(\030)853 2357 y Fh(\(1\))898 2350 y Fo(j)p Fl(m)946 2357 y Fh(\(1\))990 2350 y Fm(\))p Fl(m)1042 2357 y Fh(\(2\))1087 2356 y(+)1124 2350 y Fo(\012)p 1166 2314 241 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)1225 2357 y Fh(\(3\))1269 2350 y Fo(j)p Fl(m)1317 2357 y Fh(\(2\))1362 2356 y Fk(\000)1390 2350 y Fm(\))p Fl(\030)1426 2357 y Fh(\(2\))1471 2350 y Fl(\020)724 2424 y Fm(=)12 b Fl(`\034)5 b Fm(\()p Fl(\030)844 2431 y Fh(\(1\))889 2424 y Fo(j)p Fl(m)937 2430 y Fh(+)964 2431 y(\(1\))1009 2424 y Fm(\))p Fl(m)1061 2430 y Fh(+)1089 2431 y(\(2\))1143 2424 y Fo(\012)p 1184 2387 196 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1243 2431 y Fh(\(3\))1288 2424 y Fo(j)p Fl(m)1336 2430 y Fk(\000)1364 2424 y Fm(\))p Fl(\030)1400 2431 y Fh(\(2\))1444 2424 y Fl(\020)228 2507 y Fm(whic)o(h)14 b(factors)g(through)650 2591 y Fl(')677 2598 y Fh(\(6.6\))753 2591 y Fm(:)f Fl(L)f(.)-7 b(/)853 2597 y Fn(\034)885 2591 y Fm(\003)9 b Fo(\012)g Fl(L)j(.)-7 b(/)1039 2597 y Fn(\034)1071 2591 y Fm(\003)11 b Fo(!)g Fl(L)h(.)-7 b(/)1239 2597 y Fn(\034)1271 2591 y Fm(\003)-1072 b(\(6.6\))p eop %%Page: 21 21 21 20 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(21)228 213 y Fm(b)q(ecause)269 307 y Fl(')296 314 y Fh(\(6.5\))367 307 y Fm(\()p Fl(`)p Fm(\()p Fl(r)11 b Fo(\012)p 487 284 20 2 v 9 w Fl(s)q Fm(\))e Fo(\012)h Fl(\030)h Fo(\012)e Fl(m)h Fo(\012)g Fl(\020)s Fm(\))582 380 y(=)h Fl(`r)q(\034)5 b Fm(\()p Fl(\030)721 387 y Fh(\(1\))766 380 y Fo(j)p Fl(m)814 387 y Fh(\(1\))859 380 y Fm(\))p 875 357 V Fl(sm)930 387 y Fh(\(2\))975 386 y(+)1012 380 y Fo(\012)p 1054 344 241 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)1113 387 y Fh(\(3\))1157 380 y Fo(j)p Fl(m)1205 387 y Fh(\(2\))1250 386 y Fk(\000)1278 380 y Fm(\))p Fl(\030)1314 387 y Fh(\(2\))1359 380 y Fl(\020)582 453 y Fm(=)11 b Fl(`\034)5 b Fm(\()p Fl(r)q(\030)721 460 y Fh(\(1\))766 453 y Fo(j)p Fl(m)814 460 y Fh(\(1\))859 453 y Fm(\))p Fl(m)911 460 y Fh(\(2\))956 459 y(+)993 453 y Fo(\012)p 1034 417 260 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1093 460 y Fh(\(3\))1138 453 y Fo(j)p Fl(m)1186 460 y Fh(\(2\))1230 459 y Fk(\000)p 1258 430 20 2 v 1258 453 a Fl(s)q Fm(\))p Fl(\030)1314 460 y Fh(\(2\))1359 453 y Fl(\020)582 526 y Fm(=)11 b Fl(`\034)5 b Fm(\()p Fl(r)q(\030)721 533 y Fh(\(1\))766 526 y Fo(j)p Fl(m)814 533 y Fh(\(1\))859 526 y Fm(\))p Fl(m)911 533 y Fh(\(2\))956 532 y(+)993 526 y Fo(\012)p 1034 490 260 2 v 9 w Fl(\034)g Fm(\()p 1073 503 20 2 v Fl(s)q(\030)1113 533 y Fh(\(3\))1157 526 y Fo(j)p Fl(m)1205 533 y Fh(\(2\))1250 532 y Fk(\000)1278 526 y Fm(\))p Fl(\030)1314 533 y Fh(\(2\))1359 526 y Fl(\020)1137 591 y Fm(=)12 b Fl(')1208 598 y Fh(\(6.5\))1279 591 y Fm(\()p Fl(`)d Fo(\012)h Fm(\()p Fl(r)g Fo(\012)p 1450 568 V 10 w Fl(s)p Fm(\))p Fl(\030)i Fo(\012)d Fl(m)h Fo(\012)f Fl(\020)s Fm(\))228 673 y(and)269 768 y Fl(')296 775 y Fh(\(6.5\))367 768 y Fm(\()p Fl(`)h Fo(\012)f Fl(\030)i Fo(\012)f Fl(m)p Fm(\()p Fl(r)h Fo(\012)p 645 745 V 9 w Fl(s)q Fm(\))e Fo(\012)h Fl(\020)s Fm(\))582 841 y(=)h Fl(`\034)5 b Fm(\()p Fl(\030)701 848 y Fh(\(1\))746 841 y Fo(j)p Fl(m)794 848 y Fh(\(1\))839 841 y Fl(r)q Fm(\))p Fl(m)911 848 y Fh(\(2\))956 847 y(+)993 841 y Fo(\012)p 1034 805 260 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1093 848 y Fh(\(3\))1138 841 y Fo(j)p Fl(sm)1205 848 y Fh(\(2\))1250 847 y Fk(\000)1278 841 y Fm(\))p Fl(\030)1314 848 y Fh(\(2\))1359 841 y Fl(\020)582 914 y Fm(=)11 b Fl(`\034)5 b Fm(\()p Fl(\030)701 921 y Fh(\(1\))p 746 891 20 2 v 746 914 a Fl(r)q Fo(j)p Fl(m)814 921 y Fh(\(1\))859 914 y Fm(\))p Fl(m)911 921 y Fh(\(2\))956 920 y(+)993 914 y Fo(\012)p 1034 878 260 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1093 921 y Fh(\(3\))1138 914 y Fl(s)p Fo(j)p Fl(m)1205 921 y Fh(\(2\))1250 920 y Fk(\000)1278 914 y Fm(\))p Fl(\030)1314 921 y Fh(\(2\))1359 914 y Fl(\020)540 987 y Fm(=)12 b Fl(`\034)5 b Fm(\()p Fl(\030)660 994 y Fh(\(1\))705 987 y Fo(j)p Fl(m)753 994 y Fh(\(1\))797 987 y Fm(\))p Fl(m)849 994 y Fh(\(2\))895 993 y(+)931 987 y Fo(\012)p 973 951 241 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)1032 994 y Fh(\(3\))1076 987 y Fo(j)p Fl(m)1124 994 y Fh(\(2\))1169 993 y Fk(\000)1197 987 y Fm(\))p Fl(\030)1233 994 y Fh(\(2\))1278 987 y Fm(\()p Fl(r)10 b Fo(\012)p 1365 964 20 2 v 10 w Fl(s)p Fm(\))p Fl(\020)1137 1052 y Fm(=)i Fl(')1208 1059 y Fh(\(6.5\))1279 1052 y Fm(\()p Fl(`)d Fo(\012)h Fl(\030)h Fo(\012)f Fl(m)f Fo(\012)h Fm(\()p Fl(r)g Fo(\012)p 1608 1029 V 10 w Fl(s)p Fm(\))p Fl(\020)s Fm(\))228 1136 y(No)o(w)j Fl(')349 1143 y Fh(\(6.6\))434 1136 y Fm(in)g(turn)i(factors)f(through)g(\()p Fl(L)e(.)-7 b(/)958 1142 y Fn(\034)990 1136 y Fm(\003\))11 b Fo(\012)1078 1142 y Fn(R)1103 1134 y Fe(e)1133 1136 y Fm(\()p Fl(L)h(.)-7 b(/)1224 1142 y Fn(\034)1256 1136 y Fm(\003\),)13 b(b)q(ecause)269 1239 y Fl(')296 1246 y Fh(\(6.6\))367 1239 y Fm(\()p Fl(`)d Fo(\012)f Fl(\030)r Fm(\()p Fl(r)h Fo(\012)p 558 1216 V 10 w Fl(s)p Fm(\))g Fo(\012)f Fl(m)h Fo(\012)g Fl(\020)s Fm(\))h(=)h Fl(`\034)5 b Fm(\()p Fl(\030)900 1246 y Fh(\(1\))945 1239 y Fl(r)q Fo(j)p Fl(m)1013 1246 y Fh(\(1\))1058 1239 y Fm(\))p Fl(m)1110 1246 y Fh(\(2\))1155 1245 y(+)1192 1239 y Fo(\012)p 1233 1203 260 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1292 1246 y Fh(\(3\))p 1337 1216 20 2 v 1337 1239 a Fl(s)p Fo(j)p Fl(m)1404 1246 y Fh(\(2\))1449 1245 y Fk(\000)1477 1239 y Fm(\))p Fl(\030)1513 1246 y Fh(\(2\))1557 1239 y Fl(\020)316 1312 y Fm(=)12 b Fl(`\034)5 b Fm(\()p Fl(\030)436 1319 y Fh(\(1\))481 1312 y Fo(j)p Fl(r)q(m)549 1319 y Fh(\(1\))593 1312 y Fm(\))p Fl(m)645 1319 y Fh(\(2\))690 1318 y(+)727 1312 y Fo(\012)p 769 1276 260 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)828 1319 y Fh(\(3\))872 1312 y Fo(j)p Fl(m)920 1319 y Fh(\(2\))965 1318 y Fk(\000)993 1312 y Fl(s)p Fm(\))p Fl(\030)1048 1319 y Fh(\(2\))1093 1312 y Fl(\020)15 b Fm(=)d Fl(')1197 1319 y Fh(\(6.6\))1267 1312 y Fm(\()p Fl(`)e Fo(\012)f Fl(\030)j Fo(\012)d Fm(\()p Fl(r)h Fo(\012)p 1509 1289 20 2 v 10 w Fl(s)p Fm(\))p Fl(m)g Fo(\012)g Fl(\020)s Fm(\))p Fl(:)228 1394 y Fm(It)k(is)g(clear)g (that)g(m)o(ultiplicati)o(on)d(in)j Fl(L)d(.)-7 b(/)893 1400 y Fn(\034)925 1394 y Fm(\003)14 b(is)g(left)f(and)h(righ)o(t)f Fl(R)1295 1379 y Fn(e)1313 1394 y Fm(-linear.)p 1692 1394 2 29 v 1694 1368 25 2 v 1694 1394 V 1719 1394 2 29 v 303 1490 a Fc(Pr)o(oposition)i Fm(6.4)p Fc(.)21 b Fg(L)n(et)14 b Fl(L)g Fg(b)n(e)g(a)h Fo(\002)888 1496 y Fn(R)915 1490 y Fg(-Hopf)f(algebr)n(a,)g Fm(\003)g Fg(a)g Fo(\002)1294 1496 y Fn(R)1321 1490 y Fg(-bialgebr)n(a,)f(and)i Fl(\034)10 b Fm(:)j(\003)8 b Fo(\012)228 1540 y Fl(L)k Fo(!)f Fl(R)j Fg(a)h(skew)g(p)n(airing.)k(Then)c Fl(L)c(.)-7 b(/)841 1546 y Fn(\034)873 1540 y Fm(\003)15 b Fg(is)g(an)g Fl(R)1053 1525 y Fn(e)1071 1540 y Fg(-ring.)303 1623 y Fc(Pr)o(oof.)21 b Fm(W)m(e)13 b(ha)o(v)o(e)g(to)g(sho)o(w)g(that)g(m) o(ultipli)o(cation)d(is)j(asso)q(ciativ)o(e.)18 b(Note)13 b(\014rst)h(that)f(w)o(e)228 1673 y(ha)o(v)o(e)g(\()p Fl(`)8 b Fo(\012)g Fm(1\)\()p Fl(m)g Fo(\012)g Fl(\020)s Fm(\))k(=)g Fl(`m)d Fo(\012)f Fl(\020)16 b Fm(and)d(\()p Fl(`)8 b Fo(\012)g Fl(\030)r Fm(\)\(1)g Fo(\012)g Fl(\020)s Fm(\))k(=)g Fl(`)c Fo(\012)g Fl(\030)r(\020)16 b Fm(for)d Fl(`;)7 b(m)12 b Fo(2)f Fl(L)i Fm(and)g Fl(\030)r(;)7 b(\020)15 b Fo(2)c Fm(\003.)228 1723 y(F)m(urther)k(\(1)9 b Fo(\012)g Fl(\030)r Fm(\)\(\(1)h Fo(\012)f Fl(\020)s Fm(\)\()p Fl(`)h Fo(\012)g Fm(1\)\))h(=)h(\(1)d Fo(\012)h Fl(\030)r(\020)s Fm(\)\()p Fl(`)g Fo(\012)f Fm(1\))14 b(b)o(y)g(the)g(follo)o(wing)d(calculation:)270 1805 y(\(1)e Fo(\012)h Fl(\030)r Fm(\)\(\(1)f Fo(\012)h Fl(\020)s Fm(\)\()p Fl(`)g Fo(\012)f Fm(1\)\))318 1878 y(=)j(\(1)d Fo(\012)h Fl(\030)r Fm(\)\()p Fl(\034)5 b Fm(\()p Fl(\020)562 1885 y Fh(\(1\))607 1878 y Fo(j)p Fl(`)636 1885 y Fh(\(1\))680 1878 y Fm(\))p Fl(`)713 1885 y Fh(\(2\))758 1884 y(+)795 1878 y Fo(\012)p 837 1842 223 2 v 10 w Fl(\034)g Fm(\()p Fl(\020)897 1885 y Fh(\(3\))941 1878 y Fo(j)p Fl(`)970 1885 y Fh(\(2\))1015 1884 y Fk(\000)1043 1878 y Fm(\))p Fl(\020)1080 1885 y Fh(\(2\))1125 1878 y Fm(\))318 1951 y(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)421 1958 y Fh(\(1\))466 1951 y Fo(j)p Fl(\034)g Fm(\()p Fl(\020)538 1958 y Fh(\(1\))582 1951 y Fo(j)p Fl(`)611 1958 y Fh(\(1\))655 1951 y Fm(\))p Fl(`)688 1958 y Fh(\(2\))733 1957 y(+)761 1958 y(\(1\))806 1951 y Fm(\))p Fl(`)839 1958 y Fh(\(2\))884 1957 y(+)911 1958 y(\(2\))956 1957 y(+)993 1951 y Fo(\012)p 1034 1915 294 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1093 1958 y Fh(\(3\))1138 1951 y Fo(j)p Fl(`)1167 1958 y Fh(\(2\))1211 1957 y(+)1239 1958 y(\(2\))1283 1957 y Fk(\000)1311 1951 y Fm(\))p Fl(\030)1347 1958 y Fh(\(2\))p 1392 1915 223 2 v 1392 1951 a Fl(\034)g Fm(\()p Fl(\020)1452 1958 y Fh(\(3\))1497 1951 y Fo(j)p Fl(`)1526 1958 y Fh(\(2\))1570 1957 y Fk(\000)1598 1951 y Fm(\))p Fl(\020)1635 1958 y Fh(\(2\))318 2024 y Fm(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)421 2031 y Fh(\(1\))466 2024 y Fo(j)p Fl(\034)g Fm(\()p Fl(\020)538 2031 y Fh(\(1\))582 2024 y Fo(j)p Fl(`)611 2031 y Fh(\(1\))655 2024 y Fm(\))p Fl(`)688 2031 y Fh(\(2\))733 2030 y(+)761 2031 y(\(1\))806 2024 y Fm(\))p Fl(`)839 2031 y Fh(\(2\))884 2030 y(+)911 2031 y(\(2\))956 2030 y(+)993 2024 y Fo(\012)p 1034 1988 516 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)1093 2031 y Fh(\(3\))1138 2024 y Fl(\034)g Fm(\()p Fl(\020)1198 2031 y Fh(\(3\))1242 2024 y Fo(j)p Fl(`)1271 2031 y Fh(\(2\))1316 2030 y Fk(\000)1344 2024 y Fm(\))p Fo(j)p Fl(`)1389 2031 y Fh(\(2\))1433 2030 y(+)1461 2031 y(\(2\))1506 2030 y Fk(\000)1534 2024 y Fm(\))p Fl(\030)1570 2031 y Fh(\(2\))1614 2024 y Fl(\020)1635 2031 y Fh(\(2\))318 2097 y Fm(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)421 2104 y Fh(\(1\))466 2097 y Fo(j)p Fl(\034)g Fm(\()p Fl(\020)538 2104 y Fh(\(1\))582 2097 y Fo(j)p Fl(`)611 2104 y Fh(\(1\))655 2097 y Fm(\))p Fl(`)688 2104 y Fh(\(2\))733 2097 y Fm(\))p Fl(`)766 2104 y Fh(\(3\))811 2103 y(+)848 2097 y Fo(\012)p 890 2061 533 2 v 10 w Fl(\034)g Fm(\()p Fl(\030)949 2104 y Fh(\(3\))993 2097 y Fl(\034)g Fm(\()p Fl(\020)1053 2104 y Fh(\(3\))1098 2097 y Fo(j)p Fl(`)1127 2104 y Fh(\(3\))1172 2103 y Fk(\000)1200 2104 y Fh(\(1\))1244 2097 y Fm(\))p Fo(j)p Fl(`)1289 2104 y Fh(\(3\))1334 2103 y Fk(\000)1362 2104 y Fh(\(2\))1406 2097 y Fm(\))p Fl(\030)1442 2104 y Fh(\(2\))1487 2097 y Fl(\020)1508 2104 y Fh(\(2\))318 2170 y Fm(=)12 b Fl(\034)5 b Fm(\()p Fl(\030)421 2177 y Fh(\(1\))466 2170 y Fl(\020)487 2177 y Fh(\(1\))532 2170 y Fo(j)p Fl(`)561 2177 y Fh(\(1\))605 2170 y Fm(\))p Fl(`)638 2177 y Fh(\(2\))683 2176 y(+)720 2170 y Fo(\012)p 761 2134 287 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)820 2177 y Fh(\(3\))865 2170 y Fl(\020)886 2177 y Fh(\(3\))931 2170 y Fo(j)p Fl(`)960 2177 y Fh(\(2\))1004 2176 y Fk(\000)1032 2170 y Fm(\))p Fl(\030)1068 2177 y Fh(\(2\))1113 2170 y Fl(\020)1134 2177 y Fh(\(2\))1190 2170 y Fm(=)12 b(\(1)d Fo(\012)h Fl(\030)r(\020)s Fm(\)\()p Fl(`)g Fo(\012)f Fm(1\))228 2252 y(in)k(whic)o(h)h(the)g(fourth)g (equalit)o(y)f(needs)i(extra)g(justi\014cation.)i(W)m(e)d(ha)o(v)o(e) 269 2347 y Fl(`)286 2354 y Fh(\(1\))340 2347 y Fo(\012)c Fl(`)399 2354 y Fh(\(2\))444 2353 y(+)471 2354 y(\(1\))525 2347 y Fo(\012)g Fl(`)584 2354 y Fh(\(2\))628 2353 y(+)656 2354 y(\(2\))701 2353 y(+)737 2347 y Fo(\012)g Fl(`)796 2354 y Fh(\(2\))841 2353 y Fk(\000)878 2347 y Fo(\012)g Fl(`)937 2354 y Fh(\(2\))981 2353 y(+)1009 2354 y(\(2\))1054 2353 y Fk(\000)952 2409 y Fm(=)i Fl(`)1013 2416 y Fh(\(1\))1067 2409 y Fo(\012)d Fl(`)1125 2416 y Fh(\(2\))1179 2409 y Fo(\012)h Fl(`)1238 2416 y Fh(\(3\))1283 2415 y(+)1320 2409 y Fo(\012)f Fl(`)1378 2416 y Fh(\(3\))1423 2415 y Fk(\000)1451 2416 y Fh(\(1\))1505 2409 y Fo(\012)g Fl(`)1563 2416 y Fh(\(3\))1608 2415 y Fk(\000)1636 2416 y Fh(\(2\))228 2491 y Fm(as)14 b(elemen)o(ts)g(of)542 2587 y Fo(X)j Fm(:=)645 2530 y Ff(Z)686 2540 y Fn(v)q(w)738 2530 y Ff(Z)761 2624 y Fn(r)q(stu)p 834 2576 19 2 v 834 2593 a(r)852 2587 y Fl(L)p 880 2576 20 2 v 6 x Fn(v)909 2587 y Fo(\012)951 2593 y Fn(r)p 967 2576 16 2 v 1 w(s)985 2587 y Fl(L)1013 2593 y Fn(v)p 1031 2576 25 2 v 1 w(w)1067 2587 y Fo(\012)1109 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1255 1193 V 10 w Fl(\034)g Fm(\()p Fl(\030)1314 1220 y Fk(0)o(0)1335 1232 y Fl(\034)g Fm(\()p Fl(\020)1395 1220 y Fk(00)1416 1232 y Fo(j)p Fl(`)1445 1220 y Fk(00)o(0)1476 1232 y Fm(\))p Fo(j)p Fl(u)p 1528 1201 15 2 v(t)o(`)1559 1220 y Fk(000)o(0)1600 1232 y Fm(\))p Fl(\030)1636 1214 y Fk(0)1648 1232 y Fl(\020)1669 1214 y Fk(0)228 1307 y Fm(b)o(y)13 b(de\014nition)h(of)f(a)h(sk)o(ew)g(pairing.)j(One)e(c)o (hec)o(ks)g(that)269 1395 y Fl(')296 1402 y Fh(\(6.8\))367 1395 y Fm(\()p 383 1372 24 2 v Fl(x\030)c Fo(\012)p 478 1372 22 2 v 10 w Fl(y)q(\030)519 1378 y Fk(0)540 1395 y Fo(\012)f Fl(\030)602 1378 y Fk(00)632 1395 y Fo(\012)g Fl(\020)i Fo(\012)e Fl(\020)767 1378 y Fk(0)788 1395 y Fo(\012)g Fl(\020)851 1378 y Fk(0)o(0)881 1395 y Fo(\012)g Fl(`)f Fo(\012)h Fl(:)d(:)g(:)g Fo(\012)j Fl(`)1107 1378 y Fk(00)o(00)1148 1395 y Fm(\))742 1462 y(=)i Fl(')813 1469 y Fh(\(6.8\))884 1462 y Fm(\()p Fl(\030)f Fo(\012)f Fl(x\030)1015 1445 y Fk(0)1036 1462 y Fo(\012)f Fl(y)q(\030)1118 1445 y 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Fh(\(1\))492 2591 y Fo(j)p Fl(`)521 2598 y Fh(\(1\))565 2591 y Fl(m)601 2598 y Fh(\(1\))646 2591 y Fm(\))p Fl(`)679 2598 y Fh(\(2\))724 2597 y(+)752 2591 y Fl(m)788 2598 y Fh(\(2\))833 2597 y(+)870 2591 y Fo(\012)p 911 2555 331 2 v 9 w Fl(\034)g Fm(\()p Fl(\030)970 2598 y Fh(\(3\))1015 2591 y Fo(j)p Fl(m)1063 2598 y Fh(\(2\))1107 2597 y Fk(\000)1135 2591 y Fl(`)1152 2598 y Fh(\(2\))1197 2597 y Fk(\000)1225 2591 y Fm(\))p Fl(\030)1261 2598 y Fh(\(2\))p eop %%Page: 23 23 23 22 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(23)228 213 y Fm(where,)14 b(to)g(b)q(e)h(able)f(to)f (substitute)j(coasso)q(ciativit)o(y)d(of)g(\003)h(in)g(the)g(fourth)g (equalit)o(y)m(,)e(w)o(e)i(ha)o(v)o(e)228 262 y(to)g(sho)o(w)f(that)h (the)h(map)709 330 y Fl(')736 337 y Fh(\(6.10\))828 330 y Fm(:)f(\003)9 b Fo(\012)g Fl(L)h Fo(\012)f Fl(L)j Fo(!)f Fl(L)h(.)-7 b(/)1180 336 y Fn(\034)1212 330 y Fm(\003)-1013 b(\(6.10\))269 432 y Fl(')296 439 y Fh(\(6.10\))384 432 y Fm(\()p 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Fk(0)1049 1504 y Fm(\))823 1571 y(=)j Fl(')894 1578 y Fh(\(6.12\))981 1571 y Fm(\()p Fl(\030)f Fo(\012)f Fl(:)d(:)g(:)e(\030)1143 1554 y Fk(000)o(0)1193 1571 y Fo(\012)10 b Fl(`)f Fo(\012)h Fl(r)q(`)1340 1554 y Fk(0)1361 1571 y Fo(\012)p 1402 1548 V 9 w Fl(s)q(`)1439 1554 y Fk(0)o(0)1469 1571 y Fo(\012)g Fl(m)f Fo(\012)h Fl(m)1634 1554 y Fk(0)1646 1571 y Fm(\))p Fl(;)228 1639 y Fm(and)k(w)o(e)g(can)g(pro)q(ceed)h(to)f (de\014ne)704 1707 y Fl(')731 1714 y Fh(\(6.13\))823 1707 y Fm(:)g(\003)878 1690 y Fk(\012)p Fh(5)931 1707 y Fo(\012)c Fl(L)1001 1690 y Fk(\012)p Fh(3)1057 1707 y Fo(!)h Fl(L)h(.)-7 b(/)1185 1713 y Fn(\034)1217 1707 y Fm(\003)-1018 b(\(6.13\))269 1811 y Fl(\030)12 b Fo(\012)d Fl(:)e(:)g(:)h Fo(\012)h Fl(\030)459 1793 y Fk(00)o(00)509 1811 y Fo(\012)g Fl(`)h Fo(\012)f Fl(m)h Fo(\012)g Fl(m)742 1793 y Fk(0)660 1877 y Fo(7!)h Fl(')740 1884 y Fh(\(6.12\))827 1877 y Fm(\()p Fl(\030)h Fo(\012)d Fl(:)e(:)g(:)h Fo(\012)h Fl(\030)1033 1860 y Fk(00)o(00)1083 1877 y Fo(\012)g 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b(Then)d(w)o(e)f(ha)o(v)o(e)p 269 1242 V 269 1278 a Fl(\034)5 b Fm(\()p Fl(\030)328 1285 y Fh(\(2\))373 1278 y Fo(j)p Fl(`)402 1285 y Fh(\(2\))446 1278 y Fm(\))p Fl(g)q Fm(\()p Fl(\030)519 1285 y Fh(\(1\))565 1278 y Fm(\))p Fl(f)t Fm(\()p Fl(`)638 1285 y Fh(\(1\))683 1278 y Fm(\))398 1351 y(=)p 441 1315 V 11 w Fl(\034)g Fm(\()p Fl(\030)500 1358 y Fh(\(2\))545 1351 y Fo(j)p Fl(`)574 1358 y Fh(\(2\))618 1351 y Fm(\))q Fl(\034)g Fm(\()p Fl(\030)694 1358 y Fh(\(1\))q(\(1\))783 1351 y Fo(j)p Fl(`)812 1358 y Fh(\(1\))856 1357 y(+)884 1358 y(\(1\))928 1351 y Fm(\))p Fl(f)t Fm(\()p Fl(`)1001 1358 y Fh(\(1\))1047 1357 y(+)1075 1358 y(\(2\))1119 1351 y Fm(\))p 1135 1315 266 2 v Fl(\034)g Fm(\()p Fl(\030)1194 1358 y Fh(\(1\))r(\(3\))1284 1351 y Fo(j)p Fl(`)1313 1358 y Fh(\(1\))1357 1357 y Fk(\000)1385 1351 y Fm(\))p Fl(g)q Fm(\()p Fl(\030)1458 1358 y Fh(\(1\))r(\(2\))1548 1351 y Fm(\))398 1425 y(=)11 b Fl(\034)5 b Fm(\()p Fl(\030)500 1432 y Fh(\(1\))r(\(1\))590 1425 y Fo(j)p Fl(`)619 1432 y 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Fm(\()p Fl(\030)d Fo(\012)f Fl(`)p 717 2585 20 2 v(r)g Fo(\012)g Fl(`)805 2591 y Fk(0)817 2608 y Fl(s)f Fo(\012)h Fl(`)904 2591 y Fk(00)925 2608 y Fm(\))i(=)g Fl(')1024 2615 y Fh(\(6.18\))1111 2608 y Fm(\()p Fl(\030)g Fo(\012)d Fl(`)h Fo(\012)p 1266 2585 V 9 w Fl(r)q(`)1303 2591 y Fk(0)1324 2608 y Fo(\012)g Fl(s`)1402 2591 y Fk(00)1424 2608 y Fm(\))p Fl(;)p eop %%Page: 25 25 25 24 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(25)228 213 y Fm(the)14 b(expression)269 296 y Fl(')296 303 y Fh(\(6.18\))384 296 y Fm(\()p Fl(\030)d Fo(\012)e Fl(`)487 303 y Fh(\(1\))532 302 y(+)569 296 y Fo(\012)h Fl(`)628 303 y Fh(\(1\))672 302 y Fk(\000)710 296 y Fo(\012)f Fl(`)768 303 y Fh(\(2\))813 296 y Fm(\))648 378 y(=)j Fl(\034)5 b Fm(\()p Fl(\030)751 385 y Fh(\(1\))795 378 y Fo(j)p Fl(`)824 385 y Fh(\(1\))869 384 y(+)896 385 y(\(1\))941 378 y Fm(\))p Fl(f)t Fm(\()p Fl(`)1014 385 y Fh(\(1\))1060 384 y(+)1087 385 y(\(2\))1132 378 y Fm(\))p 1148 333 415 2 v Fl(\034)g 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1921 y Fm(\))p Fl(g)q Fm(\()p Fl(\030)1479 1928 y Fh(\(1\))1524 1921 y Fm(\))p Fl(f)t Fm(\()p Fl(\030)1600 1928 y Fh(\(1\))1646 1921 y Fm(\))p Fl(;)228 1992 y Fm(using)11 b(that)g(the)h(expression)g (on)f(the)h(righ)o(t)f(of)g(the)g(second)i(equalit)o(y)d(is)h(indeed)h (a)f(w)o(ell-de\014ned)228 2042 y(elemen)o(t)g(of)h(\()p Fl(L)g(.)-7 b(/)516 2048 y Fn(\034)547 2042 y Fm(\003\))5 b Fo(\005)g Fm(\()p Fl(L)13 b(.)-7 b(/)715 2048 y Fn(\034)747 2042 y Fm(\003\),)11 b(whic)o(h)h(is)g(easy)h(to)e(c)o(hec)o(k.)19 b(No)o(w)11 b(Prop)q(osition)h(6.5)f(sho)o(ws)228 2092 y(that)j(the)g(standard)h(com)o(ultiplicati)o(on)c(of)i Fl(L)f(.)-7 b(/)990 2098 y Fn(\034)1022 2092 y Fm(\003)14 b(is)f(indeed)i(an)f(algebra)f(map.)p 1692 2092 2 29 v 1694 2066 25 2 v 1694 2092 V 1719 2092 2 29 v 303 2172 a Fc(Cor)o(ollar)m(y)18 b Fm(6.8)p Fc(.)i Fg(L)n(et)d Fl(L)f Fg(b)n(e)h(a)f Fo(\002)871 2178 y Fn(R)899 2172 y Fg(-Hopf)g(algebr)n(a,)g Fm(\003)g Fg(a)h Fo(\002)1286 2178 y Fn(R)1313 2172 y 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b(left)g(\003-mo)q(dule)e(b)o(y)773 2591 y Fl(\030)f Fo(\017)e Fl(v)k Fm(:=)f Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(v)1013 2598 y Fh(\()p Fk(\000)p Fh(1\))1083 2591 y Fm(\))p Fl(v)1120 2598 y Fh(\(0\))1166 2591 y Fl(:)-950 b Fm(\(6.19\))p eop %%Page: 26 26 26 25 bop 228 113 a Fh(26)519 b(PETER)16 b(SCHA)o(UENBUR)o(G)228 213 y Fm(F)m(or)d Fl(\030)h Fo(2)d Fm(\003,)i Fl(`)f Fo(2)f Fl(L)j Fm(and)g Fl(v)f Fo(2)e Fl(V)24 b Fm(w)o(e)14 b(calculate)228 310 y(\(6.20\))40 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fl(`)463 317 y Fh(\(1\))508 310 y Fl(v)529 317 y Fh(\()p Fk(\000)p Fh(1\))600 310 y Fm(\)\()p Fl(`)649 317 y Fh(\(2\))703 310 y Fl(.)k(v)754 317 y Fh(\(0\))800 310 y Fm(\))i(=)h Fl(\034)5 b Fm(\()p Fl(\030)930 317 y Fh(\(1\))975 310 y Fo(j)p Fl(`)1004 317 y Fh(\(1\))p 1048 274 239 2 v 1048 310 a Fl(\034)g Fm(\()p Fl(\030)1107 317 y Fh(\(2\))1152 310 y Fo(j)p Fl(m)1200 317 y Fh(\()p Fk(\000)p Fh(1\))1270 310 y Fm(\)\)\()p Fl(`)1335 317 y Fh(\(2\))1390 310 y Fl(.)k(v)1441 317 y Fh(\(0\))1486 310 y Fm(\))590 375 y(=)j Fl(\034)5 b Fm(\()p Fl(\030)693 382 y Fh(\(1\))738 375 y Fo(j)p Fl(`)767 382 y Fh(\(1\))811 375 y Fm(\))p Fl(`)844 382 y Fh(\(2\))889 375 y Fl(\034)g Fm(\()p Fl(\030)948 382 y Fh(\(2\))993 375 y Fo(j)p Fl(v)1026 382 y Fh(\()p Fk(\000)p Fh(1\))1096 375 y Fm(\))10 b Fl(.)f(v)1173 382 y Fh(\(0\))1229 375 y Fm(=)j Fl(\034)5 b Fm(\()p Fl(\030)1332 382 y Fh(\(1\))1377 375 y Fo(j)p Fl(`)1406 382 y Fh(\(1\))1450 375 y Fm(\))p Fl(`)1483 382 y Fh(\(2\))1538 375 y Fl(.)k Fm(\()p Fl(\030)i Fo(\017)e Fl(v)q Fm(\))228 458 y(and)228 554 y(\(6.21\))40 b Fl(\034)5 b Fm(\()p Fl(\030)r Fo(j)p Fm(\()p Fl(`)479 561 y Fh(\(1\))533 554 y Fl(.)k(v)q Fm(\))600 561 y Fh(\()p Fk(\000)p Fh(1\))671 554 y Fl(`)688 561 y Fh(\(2\))733 554 y Fm(\)\()p Fl(`)782 561 y Fh(\(1\))837 554 y Fl(.)g(v)q Fm(\))904 561 y Fh(\(0\))584 627 y Fm(=)j Fl(\034)5 b Fm(\()p Fl(\030)687 634 y Fh(\(1\))732 627 y Fo(j)p Fm(\()p Fl(`)777 634 y Fh(\(1\))830 627 y Fl(.)k(v)q Fm(\))897 634 y Fh(\()p 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772 y Fh(\(1\))1496 765 y Fo(\017)k Fm(\()p Fl(`)1559 772 y Fh(\(1\))1613 765 y Fl(.)g(v)q Fm(\))228 848 y(Hence,)17 b(if)e Fl(V)25 b Fm(is)16 b(a)g(Y)m(etter-Drinfeld)g(mo)q(dule,)e (then,)i(b)o(y)g(Prop)q(osition)f(6.5,)g Fl(V)25 b Fm(is)16 b(a)g(mo)q(dule)228 898 y(o)o(v)o(er)e Fl(L)d(.)-7 b(/)391 904 y Fn(\034)423 898 y Fm(\003)14 b(in)f(the)i(claimed)d(w)o(a)o(y)m (.)p 1692 898 2 29 v 1694 872 25 2 v 1694 898 V 1719 898 2 29 v 303 995 a Fc(Definition)j Fm(6.9)p Fc(.)21 b Fm(Let)c Fl(L)h Fm(b)q(e)g(a)f(left)g(\014nite)g Fo(\002)1063 1001 y Fn(R)1091 995 y Fm(-Hopf)f(algebra.)28 b(Let)18 b Fl(L)1487 980 y Fk(_)1529 995 y Fm(b)q(e)g(the)f(left)228 1045 y(dual)c(of)g Fl(L)p Fm(,)h(and)g Fl(\034)9 b Fm(:)k Fl(L)582 1030 y Fk(_)616 1045 y Fo(\012)d Fl(L)i Fo(!)f Fl(R)i Fm(the)i(canonical)e(sk)o(ew)h(pairing.)j(Then)814 1128 y Fl(D)q Fm(\()p Fl(L)p Fm(\))12 b(:=)g Fl(L)g(.)-7 b(/)1052 1134 y Fn(\034)1083 1128 y Fl(L)1111 1111 y Fk(_)228 1212 y Fm(is)14 b(called)f(the)i(Drinfeld)e(double)h(of)f Fl(L)p Fm(.)303 1295 y Fc(Theorem)j Fm(6.10)p Fc(.)k Fg(L)n(et)14 b Fl(L)i Fg(b)n(e)e(a)i(left)e(\014nite)h Fo(\002)1025 1301 y Fn(R)1052 1295 y Fg(-Hopf)g(algebr)n(a.)k(Then)c (the)g(functor)834 1361 y Fn(L)834 1389 y(L)859 1379 y Fo(Y)s(D)d(!)988 1386 y Fn(D)q Fh(\()p Fn(L)p Fh(\))1066 1379 y Fo(M)228 1462 y Fg(de\014ne)n(d)19 b(by)f(endowing)h(a)f(Y)m (etter-Drinfeld)f(mo)n(dule)h Fl(V)27 b Fg(with)18 b(the)g Fl(D)q Fm(\()p Fl(L)p Fm(\))p Fg(-mo)n(dule)h(structur)n(e)228 1512 y Fm(\()p Fl(`)9 b Fo(\012)h Fl(\030)r Fm(\))p Fl(v)j Fm(=)f Fl(`)e(.)f Fo(h)p Fl(\030)r Fo(j)p Fl(v)551 1519 y Fh(\()p Fk(\000)p Fh(1\))622 1512 y Fo(i)p Fl(v)659 1519 y Fh(\(0\))719 1512 y Fg(is)14 b(a)h(monoidal)h(c)n(ate)n(gory)e (e)n(quivalenc)n(e.)303 1596 y Fc(Pr)o(oof.)21 b Fm(After)c(Lemma)d (5.10)h(and)h(Prop)q(osition)g(6.5,)f(it)h(remains)g(to)g(sho)o(w)g (that,)g(for)228 1645 y(a)g(left)g Fl(L)p Fm(-como)q(dule)f(and)g(left) h Fl(L)p Fm(-mo)q(dule)f Fl(V)26 b Fm(whose)16 b(underlying)g Fl(R)p Fm(-)p Fl(R)p Fm(-bimo)q(dule)e(and)i Fl(R)1691 1630 y Fn(e)1708 1645 y Fm(-)228 1695 y(mo)q(dule)c(structures)17 b(agree)d(up)g(to)g(natural)f(iden)o(ti\014cation,)g(the)i(follo)o (wing)c(are)j(equiv)n(alen)o(t:)303 1762 y(\(1\))20 b Fl(V)k Fm(is)13 b(a)h(Y)m(etter-Drinfeld)g(mo)q(dule.)303 1812 y(\(2\))20 b(The)15 b Fl(L)p Fm(-mo)q(dule)d(structure)k(and)e Fl(L)937 1796 y Fk(_)962 1812 y Fm(-mo)q(dule)e(structure)k(of)d Fl(V)23 b Fm(satisfy)518 1899 y Fo(h)p Fl(\030)554 1906 y Fh(\(1\))599 1899 y Fo(j)p Fl(`)628 1906 y Fh(\(1\))672 1899 y Fo(i)p Fl(`)705 1906 y Fh(\(2\))760 1899 y Fl(.)9 b Fm(\()p Fl(\030)826 1906 y Fh(\(2\))880 1899 y Fo(\017)g Fl(v)q Fm(\))j(=)p 1003 1863 171 2 v 12 w Fo(h)p Fl(\030)1039 1906 y Fh(\(2\))1083 1899 y Fo(j)p Fl(`)1112 1906 y Fh(\(2\))1157 1899 y Fo(i)p Fl(\030)1193 1906 y Fh(\(1\))1247 1899 y Fo(\017)d Fm(\()p Fl(`)1310 1906 y Fh(\(1\))1364 1899 y Fl(.)g(v)q Fm(\))-1203 b(\(6.22\))376 1985 y(for)14 b Fl(\030)f Fo(2)f Fl(L)539 1970 y Fk(_)563 1985 y Fm(,)i Fl(`)e Fo(2)f Fl(L)j Fm(and)g Fl(v)f Fo(2)e Fl(V)e Fm(.)228 2051 y(Note)20 b(that)h(the)g(de\014ning)f(equation)g(of)f(a)h(Y)m (etter-Drinfeld)h(mo)q(dule)d(is)j(an)f(equation)f(on)228 2101 y(elemen)o(ts)12 b(of)f Fl(L)6 b Fo(\005)g Fl(V)j Fm(.)17 b(When)c(w)o(e)f(apply)g Fl(')883 2108 y Fh(\(5.12\))982 2101 y Fm(to)g(it,)f(then,)i(b)o(y)f(the)h(calculations)e(\(6.20\))g (and)228 2151 y(\(6.21\))i(\(with,)g(of)g(course,)i Fl(\034)h Fm(=)c Fo(h)p Fm({)c Fo(j)f Fm({)p Fo(i)p Fm(\),)14 b(w)o(e)g(get)g (precisely)i(\(6.22\))o(.)p 1692 2151 2 29 v 1694 2125 25 2 v 1694 2151 V 1719 2151 2 29 v 303 2248 a Fc(Example)h Fm(6.11)p Fc(.)i Fm(If)12 b Fl(R)f Fm(is)h(a)f(\014nitely)g(generated)i (pro)r(jectiv)o(e)g Fl(k)q Fm(-mo)q(dule,)d(then)i Fl(D)q Fm(\()p Fl(R)1644 2233 y Fn(e)1662 2248 y Fm(\))g(=)228 2298 y Fl(R)260 2283 y Fn(e)289 2298 y Fl(.)-7 b(/)12 b Fm(End\()p Fl(R)p Fm(\))486 2287 y Fo(\030)486 2300 y Fm(=)531 2298 y(End\()p Fl(R)p Fm(\))i(as)g Fo(\002)767 2304 y Fn(R)794 2298 y Fm(-bialgebras,)f(so)h(that)g(w)o(e)h(ha)o(v)o (e)f(a)f(category)i(equiv)n(alence)228 2348 y Fo(Z)258 2354 y Fh(0)277 2348 y Fm(\()293 2354 y Fn(R)320 2348 y Fo(M)370 2354 y Fn(R)397 2348 y Fm(\))433 2337 y Fo(\030)433 2350 y Fm(=)486 2348 y Fo(Z)516 2354 y Fh(0)535 2348 y Fm(\()551 2354 y Fn(R)576 2346 y Fe(e)594 2348 y Fo(M)p Fm(\))681 2337 y Fo(\030)681 2350 y Fm(=)733 2355 y Fh(End)q(\()p Fn(R)p Fh(\))845 2348 y Fo(M)p Fm(.)34 b(Assume)19 b(further)h(that)f Fl(R)g Fm(is)g(a)g(faithfully)e(\015at)228 2398 y Fl(k)q Fm(-mo)q(dule.)30 b(Then)18 b(it)g(is)h(easy)f(to)h(see)g(that)g Fl(k)g Fm(is)f(a)g(direct)h(summand)d(of)i Fl(R)p Fm(,)h(and)f Fl(R)g Fm(is)g(a)228 2448 y(progenerator.)g(Hence)609 2455 y Fh(End)q(\()p Fn(R)p Fh(\))721 2448 y Fo(M)782 2437 y(\030)782 2450 y Fm(=)826 2454 y Fn(k)846 2448 y Fo(M)12 b Fm(b)o(y)f(Morita)f(theory)m(.)18 b(It)11 b(follo)o(ws)f(that)h(ev)o(ery)h(ob)r(ject)228 2498 y Fl(M)17 b Fo(2)12 b(Z)355 2504 y Fh(0)373 2498 y Fm(\()389 2504 y Fn(R)417 2498 y Fo(M)467 2504 y Fn(R)494 2498 y Fm(\))i(has)g(the)h(form)e Fl(M)826 2487 y Fo(\030)826 2500 y Fm(=)870 2498 y Fl(R)c Fo(\012)h Fl(V)24 b Fm(for)14 b(a)g Fl(k)q Fm(-mo)q(dule)e Fl(V)e Fm(,)k(with)f(the)i Fl(R)p Fm(-bimo)q(dule)228 2548 y(structure)g(induced)e(b)o(y)g(that)g (of)f Fl(R)p Fm(.)18 b(This)12 b(can)i(b)q(e)f(pro)o(v)o(ed)g(directly) g(b)o(y)g(using)g(faithfully)d(\015at)228 2597 y(descen)o(t,)15 b(cf.)f([)p Fp(16)o Fm(].)p eop %%Page: 27 27 27 26 bop 553 113 a Fh(DUALS)13 b(AND)h(DOUBLES)g(OF)i(QUANTUM)c(GR)o (OUPOIDS)293 b(27)863 213 y Fp(References)245 279 y Fs([1])17 b Ft(B)327 276 y(\177)326 279 y(ohm,)e(G.,)h(Nill,)f(F.,)f(and)i(Szla)o (ch)832 276 y(\023)831 279 y(anyi,)h(K.)d Fs(W)m(eak)f(Hopf)h(algebras) e(I:)i(In)o(tegral)e(theory)g(and)h Fv(C)1692 267 y Fi(\003)1710 279 y Fs(-)300 321 y(structure.)c Fj(J.)k(A)o(lgebr)n(a)i(221)i Fs(\(1999\),)9 b(385{438.)245 362 y([2])17 b Ft(Doi,)9 b(Y.,)g(and)g(T)m(akeuchi,)h(M.)d Fs(Multiplication)e(alteration)g(b)o (y)i(t)o(w)o(o-co)q(cycles)e(|)j(the)f(quan)o(tum)e(v)o(ersion.)300 404 y Fj(Comm.)13 b(in)g(A)o(lg.)g(22)k Fs(\(1994\),)10 b(5715{5732)o(.)245 445 y([3])17 b Ft(Ha)n(y)n(ashi,)c(T.)d Fs(F)m(ace)g(algebras)f(I{A)h(generalizatio)o(n)e(of)i(quan)o(tum)e (group)h(theory)m(.)g Fj(J.)i(Math.)h(So)n(c.)h(Jap)n(an)300 487 y(50)k Fs(\(1998\),)10 b(293{315.)245 528 y([4])17 b Ft(Jiang-Hua)d(Lu)p Fs(.)e(Hopf)f(algebroids)e(and)h(quan)o(tum)f (group)q(oids.)g Fj(Int.)j(J.)h(Math.)g(7)j Fs(\(1996\),)10 b(47{70.)245 570 y([5])17 b Ft(Jo)o(y)n(al,)11 b(A.,)f(and)h(Street,)e (R.)g Fs(T)m(ortile)f(Y)m(ang-Baxter)f(op)q(erators)g(in)i(tensor)f (categories.)e Fj(J.)11 b(Pur)n(e)g(Appl.)300 611 y(A)o(lgebr)n(a)k(71) i Fs(\(1991\),)10 b(43{51.)245 653 y([6])17 b Ft(Kassel,)c(C.)e Fj(Quantum)j(Gr)n(oups)p Fs(,)d(v)o(ol.)g(155)g(of)g Fj(GTM)p Fs(.)f(Springer,)f(1995.)245 694 y([7])17 b Ft(Mal)n(tsiniotis,)12 b(G.)f Fs(Group)q(o)l(\177)-14 b(\020des)8 b(quan)o(tiques.)g Fj(C.)k(R.)g(A)n(c)n(ad.)i(Sci.,)e (Paris,)g(Ser)h(I,)f(Math.)g(314)k Fs(\(1992\),)300 736 y(249{252.)245 777 y([8])h Ft(Nikshych,)23 b(D.,)d(and)h(V)l(ainerman,) h(L.)d Fs(Algebraic)e(v)o(ersions)g(of)i(a)g(\014nite-dimen)o(sion)o (al)d(quan)o(tum)300 819 y(group)q(oid.)9 b Fj(pr)n(eprint)16 b Fs(\(math.QA/980805)o(4\).)245 860 y([9])h Ft(Nill,)c(F.)e Fs(Axioms)f(for)h(w)o(eak)g(bialgebras.)e Fj(pr)n(eprint)16 b Fs(\(q-alg/9708)o(023)o(\).)228 902 y([10])g Ft(Ping)d(Xu)p Fs(.)f(Quan)o(tum)e(group)q(oids.)f Fj(pr)n(eprint)15 b Fs(\(math.QA/9905192)o(\).)228 943 y([11])h Ft(Ping)e(Xu)p Fs(.)g(Quan)o(tum)d(group)q(oids)f(and)i(deformation)d(quan)o(tization) o(.)h Fj(C.)k(R.)g(A)n(c)n(ad.)h(Sci.,)g(Paris,)f(Ser.)300 985 y(I,)f(Math.)g(326)j Fs(\(1998\),)10 b(289{294.)228 1026 y([12])16 b Ft(R)o(umynin,)d(D.)e Fs(Dualit)o(y)f(for)h(Hopf)g (algebroids.)e Fj(J.)j(A)o(lgebr)n(a)j(223)i Fs(\(2000\),)10 b(237{255.)228 1068 y([13])16 b Ft(Saa)m(vedra)f(Riv)m(ano,)g(N.)d Fj(Cat)o(\023)-17 b(egories)15 b(tannakiennes)p Fs(,)f(v)o(ol.)e(265)g (of)g Fj(L)n(e)n(ctur)n(e)k(Notes)f(in)f(Mathematics)p Fs(.)300 1109 y(Springer,)9 b(1972.)228 1151 y([14])16 b Ft(Scha)o(uenbur)o(g,)h(P.)12 b Fs(Bialgebras)f(o)o(v)o(er)g(noncomm) o(uta)o(tiv)o(e)e(rings)j(and)g(a)h(structure)e(theorem)f(for)j(Hopf) 300 1192 y(bimo)q(dules.)c Fj(Appl.)14 b(Cate)n(goric)n(al)g(Structur)n (es)f(6)k Fs(\(1998\),)9 b(193{222.)228 1234 y([15])16 b Ft(Scha)o(uenbur)o(g,)k(P.)c Fs(F)m(ace)f(algebras)f(are)h Fr(\002)917 1240 y Fq(R)943 1234 y Fs(-bialgebras.)d(In)k Fj(R)o(ings,)i(Hopf)f(algebr)n(as,)j(and)d(Br)n(auer)300 1275 y(gr)n(oups;)12 b(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)d(the)h(fourth)g (we)n(ek)g(on)g(algebr)n(a)h(and)f(algebr)n(aic)i(ge)n(ometry)h Fs(\(1998\),)7 b(S.)g(Caenep)q(eel)300 1317 y(and)i(A.)i(V)m(ersc)o (horen,)d(Eds.,)h(v)o(ol.)g(197)h(of)f Fj(L)n(e)n(ctur)n(e)k(Notes)f (in)f(Pur)n(e)i(and)e(Applie)n(d)i(Mathematics)p Fs(,)e(Marcel)300 1358 y(Dekk)o(er,)f(Inc.,)h(pp.)f(275{285.)228 1400 y([16])16 b Ft(Scha)o(uenbur)o(g,)g(P.)d Fs(The)g(monoidal)d(cen)o(ter)h (construction)f(and)i(bimo)q(dules.)e Fj(J.)j(Pur)n(e)i(Appl.)g(A)o (lgebr)n(a)300 1441 y Fs(\(to)c(app)q(ear\))228 1483 y([17])16 b Ft(Schneider,)h(H.-J.)e Fs(Represen)o(tation)d(theory)j(of) g(Hopf-Galois)f(extensions.)f Fj(Isr)n(ael)k(J.)g(of)g(Math.)g(72)300 1524 y Fs(\(1990\),)10 b(196{231.)228 1566 y([18])16 b Ft(Sweedler,)11 b(M.)h(E.)f Fs(Groups)f(of)h(simple)f(algebras.)f Fj(Publ.)14 b(Math.)f(I.H.E.S.)f(44)17 b Fs(\(1974\),)10 b(79{189.)228 1608 y([19])16 b Ft(T)m(akeuchi,)d(M.)e Fs(Groups)f(of)h(algebras)f(o)o(v)o(er)g Fv(A)f Fr(\012)p 979 1581 27 2 v 8 w Fv(A)p Fs(.)j Fj(J.)g(Math.)h(So)n(c.)h(Jap)n(an)f (29)k Fs(\(1977\),)9 b(459{492.)228 1652 y([20])16 b Ft(T)m(akeuchi,)d(M.)532 1624 y Fr(p)p 562 1624 105 2 v 562 1652 a Fs(Morita)d(theory)m(.)g Fj(J.)i(Math.)h(So)n(c.)h(Jap)n (an)f(39)k Fs(\(1987\),)9 b(301{336.)228 1694 y([21])16 b Ft(Ulbrich,)d(K.-H.)e Fs(On)g(Hopf)h(algebras)d(and)i(rigid)f (monoidal)f(categories.)g Fj(Isr)n(ael)k(J.)g(Math.)g(72)j Fs(\(1990\),)300 1735 y(252{256.)228 1777 y([22])g Ft(Yetter,)f(D.)g (N.)g Fs(Quan)o(tum)f(groups)f(and)h(represen)o(tation)o(s)e(of)j (monoidal)e(categories.)f Fj(Math.)k(Pr)n(o)n(c.)300 1818 y(Camb.)d(Phil.)g(So)n(c.)h(108)j Fs(\(1990\),)9 b(261{290.)303 1896 y Ft(Ma)n(thema)n(tisches)14 b(Institut)j(der)e (Universit)981 1893 y(\177)980 1896 y(at)h(M)1070 1893 y(\177)1069 1896 y(unchen,)h(Theresienstr.)d(39,)j(80333)h(M)1670 1893 y(\177)1669 1896 y(un-)228 1937 y(chen,)12 b(Germany)303 1979 y Fj(E-mail)h(addr)n(ess)s Fs(:)i Fa(schauen@rz.)o(mat)o(he)o(mat) o(ik.)o(uni)o(-m)o(uen)o(che)o(n.d)o(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF