%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: KaplConjRev.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips KaplConjRev.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.06.21:1731 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 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b(\014elds)g(of)f(c)o (haracteristic)h(zero,)g(a)f(semisimple)e(Hopf)i(algebra)f(of)h(prime)f (p)q(o)o(w)o(er)257 2354 y(dimension)e(con)o(tains)h(a)g(non)o(trivial) f(cen)o(tral)i(grouplik)o(e)e(elemen)o(t)h(\(cf.)g([27)o(],)f([46)o (]\).)h(Here,)257 2403 y(the)10 b(pro)q(of)f(is)g(analogous)f(to)i(the) g(pro)q(of)f(that)g(a)g Ff(p)p Fj(-group)g(has)g(non)o(trivial)f(cen)o (ter.)i(As)g(ab)q(o)o(v)o(e,)257 2453 y(the)16 b(lifting)e(theorem)h (of)g(Etingof)f(and)i(Gelaki)e(can)h(b)q(e)i(used)f(to)f(generalize)h (this)g(result)257 2503 y(to)f(semisimple,)d(cosemisimple)h(Hopf)h (algebras)h(o)o(v)o(er)g(\014elds)g(of)f(p)q(ositiv)o(e)h(c)o (haracteristic)953 2628 y(14)p eop %%Page: 15 15 15 14 bop 257 262 a Fj(\(cf.)17 b([20)o(],)f(Thm.)e(3.10,)i(p.)g (861\).)f(As)j(a)e(consequence,)j(A.)d(Masuok)n(a)g(has)h(pro)o(v)o(ed) g(that,)257 311 y(o)o(v)o(er)j(a)f(\014eld)g(of)g(c)o(haracteristic)i (zero,)f(semisimple)c(Hopf)j(algebras)h(of)e(dimension)g Ff(p)1671 296 y Fd(2)257 361 y Fj(are)d(comm)o(utativ)o(e)c(and)j(co)q (comm)o(utativ)o(e.)d(A.)j(Masuok)n(a)f(has)i(also)e(con)o(tributed)i (sev)o(eral)257 411 y(other)g(results)f(\(cf.)g([44)o(],)f([45)o(],)f ([47)o(],)h([48)o(],)g([49)o(]\).)g(F)m(urther)h(results)h(can)f(b)q(e) g(found)f(in)g([23)o(],)257 461 y([19)o(],)g([86)o(],)g(and)g([53)o(].) g(Since)h(this)g(dev)o(elopmen)o(t)f(is)g(review)o(ed)i(in)e([51)o(],)g (w)o(e)h(shall)e(not)i(giv)o(e)257 511 y(further)h(details)f(here.)257 676 y Fk(8)67 b(The)22 b(ten)n(th)i(conjecture)257 798 y Fi(8.1)48 b Fj(Kaplansky's)9 b(ten)o(th)i(conjecture)g(states)h (that,)d(o)o(v)o(er)h(an)f(algebraically)f(closed)j(\014eld,)257 848 y(the)20 b(n)o(um)o(b)q(er)e(of)h(isomorphism)d(classes)k(of)f (Hopf)f(algebras)h(of)g(a)g(giv)o(en)f(dimension)g(is)257 898 y(\014nite,)i(pro)o(vided)f(that)h(the)g(c)o(haracteristic)h(of)e (the)h(base)g(\014eld)g(do)q(es)g(not)f(divide)g(this)257 947 y(dimension.)c(This)h(w)o(as)h(pro)o(v)o(ed)f(b)o(y)h(D.)e(S)-20 b(\030)r(tefan)17 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Ff(m)g(>)g(n)f Fj(are)i(natural)e(n)o(um)o(b)q(ers)h(suc)o(h)g(that)g Ff(n)g Fj(divides)g Ff(m)p Fj(.)f(Suc)o(h)i(a)257 1825 y(construction,)f(from)d(whic)o(h)h(the)i(Hopf)e(algebras)h(of)f (dimension)f Ff(p)1338 1810 y Fd(4)1372 1825 y Fj(are)i(obtained)g(as)g (a)257 1875 y(sp)q(ecial)i(case,)f(can)g(also)f(b)q(e)i(found)f(in)f (the)i(article)f(b)o(y)f(Andruskiewitsc)o(h)i(and)f(Sc)o(hnei-)257 1925 y(der;)i(their)g(metho)q(d)e(also)h(leads)g(to)g(the)h (classi\014cation)f(of)g(p)q(oin)o(ted)g(Hopf)g(algebras)g(of)257 1974 y(dimension)e Ff(p)478 1959 y Fd(3)514 1974 y Fj(o)o(v)o(er)i(an)f (algebraically)f(closed)i(\014eld)f(of)g(c)o(haracteristic)i(zero,)f (where)h Ff(p)257 2024 y Fj(is)d(an)g(o)q(dd)g(prime.)f(This)h (classi\014cation)g(has)g(also)g(b)q(e)h(obtained)f(b)o(y)f(S.)h (Caenep)q(eel)i(and)257 2074 y(S.)c(D\025)-21 b(asc\025)g(alescu)15 b(as)f(w)o(ell)f(as)h(b)o(y)f(D.)g(S)-20 b(\030)r(tefan)14 b(and)g(F.)g(v)n(an)f(Oystaey)o(en)i(\(cf.)e([11],)f([88)o(]\).)257 2154 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y(program)f(ev)o(en)i(of)f(p)q(oin)o (ted)h(Hopf)f(algebras)g(of)g(dimension)e Ff(p)1239 595 y Fd(5)1271 610 y Fj(\(cf.)j([24)o(],)e([25)o(]\).)h(Also)g(the)257 660 y(other)18 b(groups)f(men)o(tioned)e(ab)q(o)o(v)o(e)i(ha)o(v)o(e)g (con)o(tin)o(ued)g(their)g(in)o(v)o(estigations)f(\(cf.)g([7)o(],)g([8) o(],)257 710 y([12)o(]\).)257 877 y Fk(9)67 b(App)r(endix)257 1000 y Fj(In)17 b(this)g(app)q(endix,)f(w)o(e)h(repro)q(duce)h (literally)d(app)q(endix)i(2)f(from)f([28)o(],)h(whic)o(h)h(con)o (tains)257 1050 y(Kaplansky's)g(conjectures.)j(It)d(ma)o(y)f(b)q(e)i (helpful)f(for)g(the)h(understanding)g(of)f(the)h(con-)257 1099 y(jectures)f(to)f(kno)o(w)e(that)i(Kaplansky's)e(lecture)j(notes)f (con)o(tain)f(a)g(bibliograph)o(y)e(whic)o(h,)257 1149 y(among)g(man)o(y)f(other)j(references,)i(lists)e(the)g(items)e(lab)q (elled)h([27)o(],)g([32)o(],)f(and)i([64)o(])f(in)g(the)257 1199 y(presen)o(t)i(article.)d(Kaplansky's)h(conjectures)i(are:)308 1312 y(1.)k(If)f Ff(C)j Fj(is)d(a)g(Hopf)f(subalgebra)i(of)e(the)i (Hopf)f(algebra)g Ff(B)i Fj(then)f Ff(B)i Fj(is)d(a)g(free)h(left)361 1362 y Ff(C)s Fj(-mo)q(dule.)14 b(\(Remark.)g(Nic)o(hols)i(has)g(pro)o (v)o(ed)h(this)f(if)f Ff(B)k Fj(con)o(tains)d(the)h(coradical)361 1412 y(of)c Ff(C)s Fj(.\))308 1491 y(2.)20 b(Call)13 b(a)h(coalgebra)g Ff(C)i Fj(admissible)p 715 1498 188 2 v 13 w(if)d(it)h(admits)e(an)i(algebra)g(structure)i(making)c(it)361 1541 y(a)j(Hopf)g(algebra.)g(The)g(conjecture)i(states)g(that)e Ff(C)j Fj(is)d(admissible)f(if)h(and)g(only)f(if)361 1591 y(ev)o(ery)c(\014nite)g(subset)h(of)e Ff(C)j Fj(lies)d(in)g(a)g (\014nite-dimensional)e(admissible)h(sub)q(coalgebra.)361 1641 y(\(Remarks.)17 b(1.)g(Both)h(implications)d(seem)j(hard.)f(2.)g (There)i(is)f(a)g(corresp)q(onding)361 1691 y(conjecture)d(where)g (\\Hopf)e(algebra")f(is)i(replaced)g(b)o(y)f(\\bialgebra".)f(3.)h (There)h(is)g(a)361 1740 y(dual)f(conjecture)j(for)e(lo)q(cally)e (\014nite)i(algebras.\))308 1820 y(3.)20 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Fj(is)e(semisimple)d(on)i(either)h(side)g(\(i.e.)f (either)h Ff(H)i Fj(or)e(the)g(dual)e Ff(H)1425 2438 y Fc(\003)1455 2453 y Fj(is)h(semisimple)361 2503 y(as)k(an)g (algebra\))f(the)i(square)f(of)g(the)g(an)o(tip)q(o)q(de)g(is)g(the)g (iden)o(tit)o(y)m(.)953 2628 y(16)p eop %%Page: 17 17 17 16 bop 308 262 a Fj(6.)20 b(The)f(size)h(of)e(the)i(matrices)e(o)q (ccurring)i(in)e(an)o(y)h(full)e(matrix)g(constituen)o(t)j(of)f Ff(H)361 311 y Fj(divides)14 b(the)g(dimension)f(of)g Ff(H)s Fj(.)308 394 y(7.)20 b(If)e Ff(H)j Fj(is)d(semisimple)e(on)i(b)q (oth)g(sides)h(the)g(c)o(haracteristic)h(do)q(es)f(not)f(divide)g(the) 361 444 y(dimension.)308 527 y(8.)i(If)10 b(the)h(dimension)e(of)h Ff(H)j Fj(is)d(prime)f(then)i Ff(H)j Fj(is)c(comm)o(utativ)o(e)d(and)k (co)q(comm)o(utativ)o(e.)257 654 y(Remark.)h(Kac,)i(Larson,)g(and)f(Sw) o(eedler)i(ha)o(v)o(e)f(partial)f(results)i(on)f(5)f(-)h(8.)257 740 y(In)h(the)h(t)o(w)o(o)f(\014nal)f(conjectures)j(assume)e(that)g 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y([2])20 b(N.)11 b(Andruskiewitsc)o(h/H.-J.)h(Sc)o(hneider:)g (Hopf)f(algebras)g(of)g(order)h Ff(p)1445 1724 y Fd(2)1475 1739 y Fj(and)f(braided)343 1789 y(Hopf)i(algebras)h(of)f(order)i Ff(p)p Fj(,)e(J.)h(Algebra)g(199)f(\(1998\),)f(430-454)278 1872 y([3])20 b(N.)13 b(Andruskiewitsc)o(h/H.-J.)h(Sc)o(hneider:)h (Lifting)d(of)i(quan)o(tum)e(linear)h(spaces)j(and)343 1922 y(p)q(oin)o(ted)e(Hopf)f(algebras)h(of)f(order)i Ff(p)933 1907 y Fd(3)951 1922 y Fj(,)f(J.)f(Algebra)h(209)f(\(1998\),)g (658-691)278 2005 y([4])20 b(N.)d(Andruskiewitsc)o(h/H.-J.)i(Sc)o (hneider:)f(Finite)g(quan)o(tum)e(groups)j(and)f(Cartan)343 2055 y(matrices,)13 b(Preprin)o(t,)h(1998,)e(to)i(app)q(ear)g(in:)f (Adv.)h(Math.)278 2138 y([5])20 b(N.)12 b(Andruskiewitsc)o(h/H.-J.)g (Sc)o(hneider:)i(Lifting)d(of)h(Nic)o(hols)g(algebras)g(of)g(t)o(yp)q (e)h Ff(A)1671 2144 y Fd(2)343 2188 y Fj(and)g(p)q(oin)o(ted)h(Hopf)g (algebras)g(of)f(order)h Ff(p)1013 2173 y Fd(4)1032 2188 y Fj(,)f(Preprin)o(t,)i(1999)278 2271 y([6])20 b(M.)10 b(Beattie/S.)g(D\025)-21 b(asc\025)g(alescu/L.)11 b(Gr)q(\177)-22 b(unenfelder:)10 b(On)h(the)g(n)o(um)o(b)q(er)f(of)f(t)o(yp)q(es)j(of)d (\014nite)343 2321 y(dimensional)i(Hopf)j(algebras,)f(In)o(v)o(en)o(t.) h(Math.)f(136)g(\(1999\),)g(1-7)278 2404 y([7])20 b(M.)14 b(Beattie/S.)h(D\025)-21 b(asc\025)g(alescu/L.)15 b(Gr)q(\177)-22 b(unenfelder:)15 b(Constructing)g(p)q(oin)o(ted)g(Hopf)f(al-)343 2453 y(gebras)g(b)o(y)g(Ore)h(extensions,)f(Preprin)o(t,)g(1997)953 2628 y(17)p eop %%Page: 18 18 18 17 bop 278 262 a Fj([8])20 b(M.)e(Beattie/S.)g(D\025)-21 b(asc\025)g(alescu/L.)19 b(Gr)q(\177)-22 b(unenfelder:)19 b(On)g(p)q(oin)o(ted)f(Hopf)h(algebras)f(of)343 311 y(dimension)12 b Ff(p)560 296 y Fe(n)582 311 y Fj(,)i(Preprin)o(t,)g(1997,)e(to)i(app) q(ear)g(in:)f(Pro)q(c.)i(Am.)d(Math.)h(So)q(c.)278 394 y([9])20 b(R.)12 b(Brauer:)i(Represen)o(tations)g(of)f(\014nite)g (groups.)g(In:)g(Lectures)i(on)e(mo)q(dern)f(math-)343 444 y(ematics,)g(V)m(ol.)h(I,)g(133-175,)f(Wiley)m(,)g(New)i(Y)m(ork,)f (1963)257 527 y([10])20 b(C.)14 b(R.)h(Cai/H.)f(X.)g(Chen:)i(Prime)e (dimensional)f(Hopf)i(algebras.)f(In:)h(Pro)q(ceedings)343 577 y(of)c(the)i(\014rst)g(China-Japan)e(in)o(ternational)g(symp)q (osium)f(on)i(ring)f(theory)m(,)h(Ok)n(a)o(y)o(ama)343 627 y(Univ.,)g(Ok)n(a)o(y)o(ama,)f(1992)257 710 y([11])20 b(S.)13 b(Caenep)q(eel/S.)h(D\025)-21 b(asc\025)g(alescu:)14 b(P)o(oin)o(ted)f(Hopf)h(algebras)f(of)g(dimension)f Ff(p)1543 695 y Fd(3)1562 710 y Fj(,)h(J.)g(Al-)343 760 y(gebra)h(209)f(\(1998\),)g(622-634)257 843 y([12])20 b(S.)e(Caenep)q(eel/S.)h(D\025)-21 b(asc\025)g(alescu:)20 b(On)f(p)q(oin)o(ted)f(Hopf)h(algebras)f(of)g(dimension)g(2)1656 828 y Fe(n)1678 843 y Fj(,)343 892 y(Bull.)13 b(Lond.)g(Math.)g(So)q (c.)h(31)f(\(1999\),)g(17-24)257 976 y([13])20 b(S.)c(Caenep)q(eel/S.)h (D\025)-21 b(asc\025)g(alescu/S.)16 b(Raian)o(u:)f(Classifying)f(p)q (oin)o(ted)j(Hopf)f(algebras)343 1025 y(of)d(dimension)f(16,)h(Preprin) o(t,)h(1998)257 1108 y([14])20 b(P)m(.)f(Cartier:)h(Group)q(es)h(alg)o (\023)-20 b(ebriques)21 b(et)g(group)q(es)g(formels.)d(In:)i(Collo)q (que)f(sur)i(la)343 1158 y(th)o(\023)-20 b(eorie)14 b(des)h(group)q(es) g(alg)o(\023)-20 b(ebriques,)13 b(Bruxelles,)h(CBRM,)g(1962)257 1241 y([15])20 b(C.)10 b(W.)g(Curtis/I.)h(Reiner:)g(Metho)q(ds)g(of)g (represen)o(tation)h(theory)g(with)e(applications)343 1291 y(to)j(\014nite)i(groups)f(and)f(orders,)i(V)m(ol.)d(I,)i(Wiley)m (,)e(New)i(Y)m(ork,)f(1981)257 1374 y([16])20 b(M.)11 b(Demazure/P)m(.)g(Gabriel:)f(Group)q(es)j(alg)o(\023)-20 b(ebriques,)12 b(V)m(ol.)e(I,)h(North-Holland,)g(Am-)343 1424 y(sterdam,)i(1970)257 1507 y([17])20 b(M.)f(Eb)q(erw)o(ein:)h (Cosemisimpl)o(e)d(Hopf)i(algebras,)g(Ph.)g(D.)g(Dissertation,)g (Florida)343 1557 y(State)14 b(Univ.,)f(T)m(allahassee,)g(1992)257 1640 y([18])20 b(P)m(.)9 b(Etingof/S.)f(Gelaki:)g(Some)g(prop)q(erties) j(of)e(\014nite-dimensional)f(semisimple)f(Hopf)343 1689 y(algebras,)13 b(Math.)g(Res.)h(Lett.)g(5)g(\(1998\),)e(191-197)257 1773 y([19])20 b(P)m(.)11 b(Etingof/S.)g(Gelaki:)f(Semisimple)g(Hopf)h (algebras)h(of)g(dimension)f Ff(pq)i Fj(are)f(trivial,)343 1822 y(J.)h(Algebra)h(210)f(\(1998\),)g(664-669)257 1905 y([20])20 b(P)m(.)c(Etingof/S.)f(Gelaki:)h(On)h(\014nite-dimensional)e (semisimple)f(and)j(cosemisimple)343 1955 y(Hopf)h(algebras)g(in)g(p)q (ositiv)o(e)h(c)o(haracteristic,)g(In)o(t.)f(Math.)g(Res.)h(Not.)f(16)g (\(1998\),)343 2005 y(851-864)257 2088 y([21])i(W.)14 b(F)m(eit:)h(The)h(represen)o(tation)g(theory)g(of)f(\014nite)h (groups,)f(North-Holland,)f(Ams-)343 2138 y(terdam,)e(1982)257 2221 y([22])20 b(S.)15 b(Gelaki:)f(P)o(oin)o(ted)h(Hopf)g(algebras)h (and)f(Kaplansky's)g(10th)h(conjecture,)h(J.)e(Al-)343 2271 y(gebra)f(209)f(\(1998\),)g(635-657)257 2354 y([23])20 b(S.)h(Gelaki/S.)f(W)m(estreic)o(h:)i(On)g(semisimple)d(Hopf)j (algebras)g(of)f(dimension)f Ff(pq)q Fj(,)343 2403 y(Preprin)o(t,)14 b(1998,)e(to)i(app)q(ear)g(in:)f(Pro)q(c.)i(Am.)d(Math.)h(So)q(c.)257 2487 y([24])20 b(M.)13 b(Gra)q(~)-22 b(na:)13 b(P)o(oin)o(ted)h(Hopf)f (algebras)h(of)f(dimension)f(32,)h(Preprin)o(t)953 2628 y(18)p eop %%Page: 19 19 19 18 bop 257 262 a Fj([25])20 b(M.)13 b(Gra)q(~)-22 b(na:)13 b(On)h(p)q(oin)o(ted)g(Hopf)f(algebras)h(of)g(dimension)e Ff(p)1292 246 y Fd(5)1310 262 y Fj(,)i(Preprin)o(t)257 345 y([26])20 b(J.)f(C.)g(Jan)o(tzen:)h(Lectures)i(of)d(quan)o(tum)f (groups,)h(Grad.)g(Stud.)h(Math.,)e(V)m(ol.)g(6,)343 394 y(Am.)12 b(Math.)h(So)q(c.,)g(Pro)o(vidence,)i(1996)257 477 y([27])20 b(G.)10 b(I.)g(Kac:)h(Certain)g(arithmetic)e(prop)q (erties)k(of)d(ring)g(groups,)h(F)m(unktsional.)e(Anal.)h(i)343 527 y(Prilozhen.)g(6)g(\(1972\),)f(88-90.)g(English)h(translation:)f(F) m(unct.)h(Anal.)g(Appl.)g(6)g(\(1972\),)343 577 y(158-160)257 660 y([28])20 b(I.)13 b(Kaplansky:)g(Bialgebras,)g(Lecture)j(Notes,)e (Univ.)f(of)g(Chicago,)g(Chicago,)g(1975)257 743 y([29])20 b(C.)11 b(Kassel:)h(Quan)o(tum)f(groups,)h(Grad.)f(T)m(exts)h(Math.,)f (V)m(ol.)f(155,)h(Springer,)h(Berlin,)343 793 y(1995)257 876 y([30])20 b(R.)e(G.)h(Larson:)h(Co)q(comm)o(utati)o(v)o(e)d(Hopf)j (algebras,)f(Can.)g(J.)g(Math.)g(19)h(\(1967\),)343 926 y(350-360)257 1009 y([31])g(R.)9 b(G.)g(Larson:)h(The)h(order)g(of)e (the)i(an)o(tip)q(o)q(de)f(of)g(a)g(Hopf)g(algebra,)f(Pro)q(c.)i(Am.)d (Math.)343 1059 y(So)q(c.)13 b(21)h(\(1969\),)f(167-170)257 1142 y([32])20 b(R.)13 b(G.)g(Larson:)g(Characters)i(of)f(Hopf)f (algebras,)g(J.)h(Algebra)g(17)f(\(1971\),)g(352-368)257 1225 y([33])20 b(R.)13 b(G.)g(Larson:)g(Orders)j(in)d(Hopf)h(algebras,) f(J.)h(Algebra)f(22)h(\(1972\),)e(201-210)257 1308 y([34])20 b(R.)10 b(G.)g(Larson:)h(Cosemisimple)e(Hopf)i(algebras)g(with)g(small) e(simple)h(sub)q(coalgebras)343 1357 y(are)k(in)o(v)o(olutory)m(,)e (Comm)n(un.)f(Algebra)j(11)f(\(1983\),)g(1175-1186)257 1440 y([35])20 b(R.)11 b(G.)h(Larson/D.)f(E.)h(Radford:)g(Semisimpl)o (e)e(cosemisimple)g(Hopf)i(algebras,)g(Am.)343 1490 y(J.)h(Math.)h(110) f(\(1988\),)g(187-195)257 1573 y([36])20 b(R.)15 b(G.)g(Larson/D.)g(E.) h(Radford:)e(Finite)i(dimensional)e(cosemisimple)g(Hopf)h(alge-)343 1623 y(bras)f(in)f(c)o(haracteristic)j(0)d(are)i(semisimple,)10 b(J.)k(Algebra)g(117)f(\(1988\),)g(267-289)257 1706 y([37])20 b(R.)e(G.)f(Larson/D.)h(E.)h(Radford:)e(Semisimple)f(Hopf)j(algebras,)f (J.)h(Algebra)f(171)343 1756 y(\(1995\),)12 b(5-35)257 1839 y([38])20 b(R.)13 b(G.)h(Larson/M.)g(E.)h(Sw)o(eedler:)g(An)g (asso)q(ciativ)o(e)g(orthogonal)e(bilinear)h(form)f(for)343 1889 y(Hopf)g(algebras,)g(Am.)g(J.)g(Math.)h(91)f(\(1969\),)g(75-93)257 1972 y([39])20 b(M.)g(Lorenz:)i(On)f(the)h(class)f(equation)g(for)f (Hopf)h(algebras,)f(Pro)q(c.)i(Am.)d(Math.)343 2022 y(So)q(c.)13 b(126)h(\(1998\),)e(2841-2844)257 2105 y([40])20 b(A.)e(Masuok)n(a:)f (On)h(Hopf)g(algebras)g(with)g(comm)o(utativ)n(e)e(coradicals,)i(J.)g (Algebra)343 2154 y(144)13 b(\(1991\),)f(415-466)257 2237 y([41])20 b(A.)h(Masuok)n(a:)f(F)m(reeness)j(of)e(Hopf)g(algebras) h(o)o(v)o(er)f(coideal)g(subalgebras,)g(Com-)343 2287 y(m)o(un.)12 b(Algebra)h(20)h(\(1992\),)e(1353-1373)257 2370 y([42])20 b(A.)d(Masuok)n(a:)f(Coideal)h(subalgebras)h(in)f (\014nite)h(Hopf)f(algebras,)g(J.)h(Algebra)f(163)343 2420 y(\(1994\),)12 b(819-831)953 2628 y(19)p eop %%Page: 20 20 20 19 bop 257 262 a Fj([43])20 b(A.)14 b(Masuok)n(a/D.)e(Wigner:)i(F)m (aithful)e(\015atness)k(of)d(Hopf)h(algebras,)g(J.)g(Algebra)g(170)343 311 y(\(1994\),)e(156-164)257 394 y([44])20 b(A.)14 b(Masuok)n(a:)f (Semisimpl)o(e)f(Hopf)i(algebras)g(of)g(dimension)f(6,)g(8,)h(Isr.)g (J.)g(Math.)g(92)343 444 y(\(1995\),)e(361-373)257 527 y([45])20 b(A.)11 b(Masuok)n(a:)g(Semisimple)e(Hopf)j(algebras)g(of)f (dimension)f(2)p Ff(p)p Fj(,)h(Comm)o(un.)e(Algebra)343 577 y(23)k(\(1995\),)g(1931-1940)257 660 y([46])20 b(A.)g(Masuok)n(a:)g (The)i Ff(p)712 645 y Fe(n)734 660 y Fj(-theorem)f(for)g(semisimple)d (Hopf)j(algebras,)g(Pro)q(c.)g(Am.)343 710 y(Math.)13 b(So)q(c.)h(124)f(\(1996\),)g(735-737)257 793 y([47])20 b(A.)f(Masuok)n(a:)f(Semisimple)f(Hopf)i(algebras)g(of)g(dimension)f Ff(p)1368 778 y Fd(3)1406 793 y Fj(obtained)h(b)o(y)h(an)343 843 y(extension,)14 b(J.)f(Algebra)h(178)f(\(1995\),)g(791-806)257 926 y([48])20 b(A.)c(Masuok)n(a:)g(Some)f(further)j(classi\014cation)e (results)i(on)f(semisimple)d(Hopf)i(alge-)343 976 y(bras,)e(Comm)n(un.) d(Algebra)j(24)f(\(1996\),)g(307-329)257 1059 y([49])20 b(A.)12 b(Masuok)n(a:)g(Calculation)g(of)g(some)g(groups)h(of)g(Hopf)f (algebra)h(extensions,)g(J.)g(Al-)343 1108 y(gebra)h(191)f(\(1997\),)g (568-588)257 1191 y([50])20 b(S.)d(Mon)o(tgomery:)f(Hopf)h(Algebras)h (and)f(their)h(Actions)g(on)g(Rings,)e(CBMS)i(Reg.)343 1241 y(Conf.)13 b(Ser.)h(Math.,)f(V)m(ol.)f(82,)h(Am.)f(Math.)h(So)q (c.,)h(Pro)o(vidence,)g(1993)257 1324 y([51])20 b(S.)e(Mon)o(tgomery:)e (Classifying)h(\014nite-dimensional)f(semisimple)g(Hopf)i(algebras.)343 1374 y(In:)11 b(T)m(rends)i(in)e(the)i(represen)o(tation)h(theory)e(of) g(\014nite)g(dimensional)d(algebras,)j(Con-)343 1424 y(temp.)g(Math.,)h(V)m(ol.)g(229,)f(Am.)g(Math.)i(So)q(c.,)f(Pro)o (vidence,)h(1998)257 1507 y([52])20 b(S.)c(Mon)o(tgomery/S.)e(J.)j (Withersp)q(o)q(on:)f(Irreducible)i(represen)o(tations)h(of)d(crossed) 343 1557 y(pro)q(ducts,)e(J.)g(Pure)h(Appl.)e(Algebra)h(129)f (\(1998\),)g(315-326)257 1640 y([53])20 b(S.)13 b(Natale:)g(On)h (semisimple)e(Hopf)h(algebras)h(of)f(dimension)g Ff(pq)1363 1625 y Fd(2)1381 1640 y Fj(,)g(Preprin)o(t,)i(1998)257 1723 y([54])20 b(S.)h(H.)g(Ng:)g(On)g(the)h(pro)r(jectivit)o(y)g(of)e (mo)q(dule)g(coalgebras,)i(Pro)q(c.)f(Am.)f(Math.)343 1773 y(So)q(c.)13 b(126)h(\(1998\),)e(3191-3198)257 1856 y([55])20 b(W.)d(D.)h(Nic)o(hols:)f(Quotien)o(ts)i(of)f(Hopf)g (algebras,)g(Comm)n(un.)d(Algebra)k(6)f(\(1978\),)343 1905 y(1789-1800)257 1988 y([56])i(W.)10 b(D.)g(Nic)o(hols:)g (Cosemisimple)e(Hopf)j(algebras.)f(In:)h(Adv)n(ances)h(in)e(Hopf)h (algebras,)343 2038 y(Lect.)j(Notes)h(Pure)g(Appl.)e(Math.,)g(V)m(ol.)f (158,)h(Dekk)o(er,)h(New)g(Y)m(ork,)f(1994)257 2121 y([57])20 b(W.)10 b(D.)f(Nic)o(hols/M.)h(B.)h(Zo)q(eller:)f(Finite)h(dimensional) d(Hopf)i(algebras)h(are)g(free)g(o)o(v)o(er)343 2171 y(grouplik)o(e)i(subalgebras,)g(J.)h(Pure)h(Appl.)e(Algebra)h(56)f (\(1989\),)g(51-57)257 2254 y([58])20 b(W.)10 b(D.)g(Nic)o(hols/M.)g (B.)h(Zo)q(eller:)g(F)m(reeness)i(of)d(in\014nite)h(dimensional)e(Hopf) i(algebras)343 2304 y(o)o(v)o(er)j(grouplik)o(e)f(subalgebras,)g(Comm)o (un.)e(Algebra)i(17)h(\(1989\),)e(413-424)257 2387 y([59])20 b(W.)e(D.)h(Nic)o(hols/M.)f(B.)h(Zo)q(eller:)g(A)h(Hopf)f(algebra)f (freeness)k(theorem,)c(Am.)g(J.)343 2437 y(Math.)13 b(111)g(\(1989\),)g (381-385)953 2628 y(20)p eop %%Page: 21 21 21 20 bop 257 262 a Fj([60])20 b(W.)14 b(D.)g(Nic)o(hols/M.)h(B.)g(Ric) o(hmond:)d(F)m(reeness)17 b(of)e(in\014nite)g(dimensional)e(Hopf)h(al-) 343 311 y(gebras,)g(Comm)n(un.)d(Algebra)j(20)f(\(1992\),)g(1489-1492) 257 394 y([61])20 b(W.)9 b(D.)g(Nic)o(hols/M.)g(B.)g(Ric)o(hmond:)f (The)i(Grothendiec)o(k)g(group)g(of)f(a)h(Hopf)f(algebra,)343 444 y(J.)k(Pure)i(Appl.)e(Algebra)h(106)f(\(1996\),)g(297-306)257 527 y([62])20 b(W.)11 b(D.)h(Nic)o(hols/M.)g(B.)g(Ric)o(hmond:)e(The)j (Grothendiec)o(k)h(algebra)e(of)g(a)g(Hopf)g(alge-)343 577 y(bra)i(I,)f(Comm)o(un.)d(Algebra)k(26)f(\(1998\),)g(1081-1095)257 660 y([63])20 b(U.)j(Ob)q(erst/H.-J.)i(Sc)o(hneider:)882 650 y(\177)877 660 y(Ub)q(er)g(Un)o(tergrupp)q(en)h(endlic)o(her)f (algebraisc)o(her)343 710 y(Grupp)q(en,)14 b(Man)o(uscr.)g(Math.)f(8)h (\(1973\),)f(217-241)257 793 y([64])20 b(U.)9 b(Ob)q(erst/H.-J.)h(Sc)o (hneider:)g(Un)o(tergrupp)q(en)h(formeller)d(Grupp)q(en)i(v)o(on)e (endlic)o(hem)343 843 y(Index,)14 b(J.)f(Algebra)h(31)f(\(1974\),)g (10-44)257 926 y([65])20 b(B.)c(P)o(areigis:)g(Endlic)o(he)g (Hopf-Algebren,)g(Algebra-Ber.,)g(Uni-Druc)o(k,)g(M)q(\177)-22 b(unc)o(hen,)343 976 y(1973)257 1059 y([66])20 b(B.)14 b(P)o(areigis:)g(Lectures)j(on)d(quan)o(tum)f(groups)h(and)h(non-comm)o (utati)o(v)o(e)d(geometry)m(,)343 1108 y(Lecture)j(notes,)f(Univ.)f(M)q (\177)-22 b(unc)o(hen,)15 b(M)q(\177)-22 b(unc)o(hen,)14 b(1998)257 1191 y([67])20 b(D.)h(E.)h(Radford:)f(A)h(free)h(rank)f(4)g (Hopf)f(algebra)h(with)g(an)o(tip)q(o)q(de)g(of)f(order)i(4,)343 1241 y(Pro)q(c.)14 b(Natl.)f(Acad.)h(Sci.)f(USA)h(30)g(\(1971\),)e (55-58)257 1324 y([68])20 b(D.)c(E.)g(Radford:)g(The)h(an)o(tip)q(o)q (de)g(of)g(a)f(\014nite-dimensional)f(Hopf)h(algebra)h(o)o(v)o(er)g(a) 343 1374 y(\014eld)d(has)g(\014nite)g(order,)g(Bull.)f(Am.)f(Math.)h (So)q(c.)h(81)f(\(1975\),)g(1103-1105)257 1457 y([69])20 b(D.)g(E.)g(Radford:)f(The)i(order)h(of)e(the)h(an)o(tip)q(o)q(de)g(of) f(a)g(\014nite-dimensional)f(Hopf)343 1507 y(algebra)13 b(is)h(\014nite,)f(Am.)g(J.)g(Math.)h(98)f(\(1976\),)g(333-355)257 1590 y([70])20 b(D.)10 b(E.)h(Radford:)f(P)o(oin)o(ted)h(Hopf)g (algebras)h(are)f(free)h(o)o(v)o(er)g(Hopf)f(subalgebras,)g(J.)g(Al-) 343 1640 y(gebra)j(45)f(\(1977\),)g(266-273)257 1723 y([71])20 b(D.)13 b(E.)h(Radford:)f(Op)q(erators)i(on)f(Hopf)g (algebras,)f(Am.)f(J.)i(Math.)g(99)g(\(1977\),)e(139-)343 1773 y(158)257 1856 y([72])20 b(D.)12 b(E.)h(Radford:)e(F)m(reeness)k (\(Pro)r(jectivit)o(y\))f(criteria)f(for)f(Hopf)h(algebras)g(o)o(v)o (er)g(Hopf)343 1905 y(subalgebras,)h(J.)f(Pure)i(Appl.)e(Algebra)h(11)f (\(1977\),)g(15-28)257 1988 y([73])20 b(D.)14 b(E.)i(Radford:)e(On)h (the)h(an)o(tip)q(o)q(de)g(of)f(a)g(cosemisimple)e(Hopf)i(algebra,)g (J.)g(Alge-)343 2038 y(bra)f(88)f(\(1984\),)g(68-88)257 2121 y([74])20 b(D.)h(E.)g(Radford:)g(Minimal)e(quasitriangular)h(Hopf) i(algebras,)f(J.)g(Algebra)h(157)343 2171 y(\(1993\),)12 b(285-315)257 2254 y([75])20 b(D.)f(Rum)o(ynin:)f(W)m(eak)i(in)o (tegral)g(forms)f(and)h(the)i(sixth)e(Kaplansky)g(conjecture,)343 2304 y(Preprin)o(t,)14 b(1998)257 2387 y([76])20 b(P)m(.)11 b(Sc)o(hauen)o(burg:)h(F)m(aithful)e(\015atness)i(o)o(v)o(er)g(Hopf)f (subalgebras)h(-)g(coun)o(terexamples.)343 2437 y(In:)17 b(Pro)q(ceedings)i(of)e(the)i(conference)h(`In)o(teractions)e(b)q(et)o (w)o(een)h(Ring)d(theory)j(and)343 2487 y(Represen)o(tations)c(of)e (algebras',)g(Murcia,)g(Spain,)g(1998)g(\(this)h(v)o(olume\))953 2628 y(21)p eop %%Page: 22 22 22 21 bop 257 262 a Fj([77])20 b(S.)25 b(Sc)o(hmidt-Samo)o(a:)d(Ein)k (Quotien)o(t)f(der)h(Quan)o(tengrupp)q(e)h Ff(U)1414 268 y Fe(q)1433 262 y Fj(\()p Ff(sl)1480 268 y Fd(2)1500 262 y Fj(\))e(im)f(Ein-)343 311 y(heitswurzelfall,)13 b(Diplomarb)q(eit,)e(Univ.)i(G\177)-21 b(ottingen,)13 b(G\177)-21 b(ottingen,)12 b(1995)257 394 y([78])20 b(H.-J.)14 b(Sc)o(hneider:)j(Lectures)g(on)e(Hopf)g(algebras,)g(Univ)o(ersidad)g (de)h(C\023)-21 b(ordoba)15 b(T)m(ra-)343 444 y(ba)r(jos)e(de)i (Matematica,)d(Serie)i(\\B",)g(No.)f(31/95,)f(C\023)-21 b(ordoba,)13 b(Argen)o(tina,)g(1995)257 527 y([79])20 b(H.-J.)10 b(Sc)o(hneider:)h(Zerlegbare)h(Un)o(tergrupp)q(en)g (a\016ner)e(Grupp)q(en,)h(Math.)f(Ann.)g(255)343 577 y(\(1981\),)i(139-158)257 660 y([80])20 b(H.-J.)11 b(Sc)o(hneider:)i (Normal)d(basis)i(and)g(transitivit)o(y)f(of)h(crossed)i(pro)q(ducts)f (for)f(Hopf)343 710 y(algebras,)h(J.)h(Algebra)f(152)h(\(1992\),)e (196-231)257 793 y([81])20 b(H.-J.)g(Sc)o(hneider:)i(Some)e(remarks)g (on)h(exact)h(sequences)i(of)c(quan)o(tum)g(groups,)343 843 y(Comm)n(un.)11 b(Algebra)j(21)f(\(1993\),)g(3337-3357)257 926 y([82])20 b(H.-J.)13 b(Sc)o(hneider:)i(Finiteness)h(results)f(for)f (semisimple)e(Hopf)i(algebras,)f(in)h(prepa-)343 976 y(ration)257 1059 y([83])20 b(H.-J.)9 b(Sc)o(hneider:)h(Some)e(remarks) h(on)h(factorizable)f(Hopf)g(algebras,)g(in)g(preparation)257 1142 y([84])20 b(Y.)12 b(Sommerh\177)-21 b(auser:)11 b(On)j(Kaplansky's)e(\014fth)h(conjecture,)h(J.)f(Algebra)g(204)f (\(1998\),)343 1191 y(202-224)257 1274 y([85])20 b(Y.)e(Sommerh\177)-21 b(auser:)16 b(Ribb)q(on)i(transformations,)f(in)o(tegrals,)g(and)i (triangular)e(de-)343 1324 y(comp)q(ositions,)12 b(Preprin)o(t)i (gk-mp-9707/52,)c(1997,)i(to)i(app)q(ear)g(in:)f(J.)h(Algebra)257 1407 y([86])20 b(Y.)13 b(Sommerh\177)-21 b(auser:)11 b(Y)m(etter-Drinfel'd)j(Hopf)f(algebras)g(o)o(v)o(er)h(groups)f(of)g (prime)g(or-)343 1457 y(der,)h(Preprin)o(t)h(gk-mp-9905/)o(59,)10 b(1999)257 1540 y([87])20 b(D.)13 b(S)-21 b(\030)s(tefan:)14 b(The)h(set)g(of)e(t)o(yp)q(es)j(of)d(n-dimensional)f(semisimple)g(and) i(cosemisimple)343 1590 y(Hopf)f(algebras)h(is)g(\014nite,)f(J.)h (Algebra)g(193)f(\(1997\),)g(571-580)257 1673 y([88])20 b(D.)13 b(S)-21 b(\030)s(tefan/F.)13 b(v)n(an)g(Oystaey)o(en:)h(Ho)q(c) o(hsc)o(hild)g(cohomology)d(and)j(the)g(coradical)f(\014l-)343 1723 y(tration)f(of)g(p)q(oin)o(ted)g(coalgebras:)h(Applications,)e(J.) h(Algebra)h(210)f(\(1998\),)f(535-556)257 1806 y([89])20 b(R.)10 b(Suter:)i(Mo)q(dules)g(o)o(v)o(er)g Ff(U)795 1812 y Fe(q)813 1806 y Fj(\()p Ff(sl)860 1812 y Fd(2)880 1806 y Fj(\),)f(Comm)o(un.)d(Math.)j(Ph)o(ys.)g(163)g(\(1994\),)f (359-393)257 1889 y([90])20 b(M.)11 b(E.)h(Sw)o(eedler:)h(In)o(tegrals) f(for)f(Hopf)h(algebras,)f(Ann.)h(Math.,)f(I)q(I.)h(Ser.)g(89)g (\(1969\),)343 1939 y(323-335)257 2022 y([91])20 b(M.)13 b(E.)h(Sw)o(eedler:)g(Hopf)g(algebras,)f(W.)g(A.)h(Benjamin,)d(New)k(Y) m(ork,)e(1969)257 2105 y([92])20 b(E.)c(J.)h(T)m(aft:)f(The)h(order)h (of)e(the)i(an)o(tip)q(o)q(de)f(of)f(\014nite-dimensional)f(Hopf)h (algebra,)343 2154 y(Pro)q(c.)e(Natl.)f(Acad.)h(Sci.)f(USA)h(68)g (\(1971\),)e(2631-2633)257 2237 y([93])20 b(E.)13 b(J.)g(T)m(aft/R.)e (L.)i(Wilson:)f(There)i(exist)g(\014nite-dimensional)d(Hopf)i(algebras) g(with)343 2287 y(an)o(tip)q(o)q(des)h(of)f(arbitrary)h(ev)o(en)g (order,)h(J.)e(Algebra)h(62)f(\(1980\),)g(283-291)257 2370 y([94])20 b(M.)9 b(T)m(ak)o(euc)o(hi:)g(A)i(corresp)q(ondence)i(b) q(et)o(w)o(een)e(Hopf)f(ideals)g(and)g(sub-Hopf)g(algebras,)343 2420 y(Man)o(uscr.)k(Math.)f(7)h(\(1972\),)f(251-270)953 2628 y(22)p eop %%Page: 23 23 23 22 bop 257 262 a Fj([95])20 b(M.)12 b(T)m(ak)o(euc)o(hi:)f(On)h(a)g (semi-direct)g(pro)q(duct)h(decomp)q(osition)e(of)h(a\016ne)g(groups)g (o)o(v)o(er)343 311 y(a)h(\014eld)h(of)g(c)o(haracteristic)h(0,)e(T^) -21 b(ohoku)13 b(Math.)h(J.)f(24)h(\(1972\),)e(453-456)257 394 y([96])20 b(M.)27 b(T)m(ak)o(euc)o(hi:)g(F)m(ree)i(Hopf)f(algebras) f(generated)j(b)o(y)e(coalgebras,)f(J.)h(Math.)343 444 y(So)q(c.)13 b(Japan)h(23)g(\(1971\),)e(561-582)257 527 y([97])20 b(M.)d(T)m(ak)o(euc)o(hi:)f(Relativ)o(e)g(Hopf)h(mo)q(dules)g (-)g(equiv)n(alences)h(and)f(freeness)j(criteria,)343 577 y(J.)13 b(Algebra)h(60)g(\(1979\),)e(452-471)257 660 y([98])20 b(J.)c(G.)g(Thac)o(kra)o(y:)g(Mo)q(dular)g(represen)o (tations)j(of)d(some)g(\014nite)h(groups,)f(Disserta-)343 710 y(tion,)d(Univ.)g(Cam)o(bridge,)e(Cam)o(bridge,)h(1981)257 793 y([99])20 b(Y.)11 b(Tsang/Y.)f(Zh)o(u:)h(On)h(the)g(Drinfeld)f (double)g(of)f(a)i(Hopf)e(algebra,)h(Preprin)o(t,)g(1998)257 876 y([100])19 b(W.)11 b(W)m(aterhouse:)g(In)o(tro)q(duction)h(to)f (a\016ne)g(group)h(sc)o(hemes,)f(Grad.)g(T)m(exts)g(Math.,)343 926 y(V)m(ol.)h(66,)h(Springer,)h(Berlin,)f(1979)257 1009 y([101])19 b(S.)13 b(J.)f(Withersp)q(o)q(on:)h(The)g(represen)o (tation)h(ring)e(and)g(the)h(cen)o(tre)i(of)d(a)g(Hopf)g(alge-)343 1059 y(bra,)h(Preprin)o(t,)h(1998)257 1142 y([102])19 b(E.)27 b(Witt:)g(Zyklisc)o(he)g(K\177)-21 b(orp)q(er)29 b(und)e(Algebren)h(der)g(Charakteristik)f Ff(p)g Fj(v)o(om)343 1191 y(Grad)13 b Ff(p)470 1176 y Fe(n)493 1191 y Fj(,)g(J.)h(Reine)g (Angew.)f(Math.)h(176)f(\(1937\),)g(126-140)257 1274 y([103])19 b(Y.)f(Zh)o(u:)g(Quan)o(tum)e(double)i(construction)h(of)e (quasitriangular)g(Hopf)h(algebras)343 1324 y(and)13 b(Kaplansky's)h(conjecture,)h(Preprin)o(t,)f(1993)257 1407 y([104])19 b(Y.)13 b(Zh)o(u:)g(Hopf)g(algebras)g(of)f(prime)g (dimension,)f(In)o(t.)i(Math.)f(Res.)h(Not.)g(1)g(\(1994\),)343 1457 y(53-59)257 1540 y([105])19 b(Y.)d(Zh)o(u:)g(A)h(comm)o(uting)c (pair)j(in)g(Hopf)f(algebras,)h(Pro)q(c.)h(Am.)d(Math.)i(So)q(c.)g(125) 343 1590 y(\(1997\),)c(2847-2851)257 1673 y([106])19 b(M.)13 b(B.)h(Zo)q(eller:)f(F)m(reeness)j(of)d(Hopf)g(algebras)g(o)o (v)o(er)g(semisimple)e(grouplik)o(e)i(subal-)343 1723 y(gebras,)h(J.)f(Algebra)h(118)f(\(1988\),)g(102-108)257 1850 y(T)o(yp)q(eset)j(using)d Fg(A)550 1859 y(M)595 1850 y(S)h Fj(-)g(L)673 1841 y Fd(A)691 1850 y Fj(T)714 1859 y(E)737 1850 y(X)953 2628 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF