%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Fact.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Fact.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.02.01:1837 %%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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(a)o(y:)228 910 y(\()p Fo(A;)c(R)p Fq(\))17 b(is)g(factorizable)g(if)g (and)h(only)f(if)f(the)i(double)f Fo(D)q Fq(\()p Fo(A)p Fq(\))g(is)h(isomorphic)e(\(in)228 968 y(a)h(sp)q(eci\014c)g(w)o(a)o (y\))g(to)g(a)h(2-co)q(cycle)e(t)o(wist)h(of)h(the)f(tensor)g(pro)q (duct)h Fo(A)11 b Fk(\012)h Fo(A)p Fq(.)23 b(The)228 1026 y(pro)q(of)17 b(of)g(4.3)f(follo)o(ws)g(from)g(the)g(theorem)e(on) j(Hopf)f(mo)q(dules.)278 1084 y(This)i(note)g(originated)g(on)g(the)g (o)q(ccasion)h(of)f(some)f(lectures)g(I)h(ga)o(v)o(e)f(at)h(the)228 1142 y(Univ)o(ersit)o(y)8 b(of)k(C\023)-24 b(ordoba,)13 b(Argen)o(tina,)e(in)g(the)g(fall)g(of)g(98.)21 b(I)11 b(w)o(ould)g(lik)o(e)e(to)j(thank)228 1201 y(Nicol\023)-24 b(as)15 b(Andruskiewitsc)o(h,)g(the)h(F)o(OMEC)f(and)i(F)l(aMAF)f(for)g (the)g(in)o(vitation.)512 1303 y(2.)28 b Fp(Central)17 b(and)h(gr)o(oup-like)f(elements)278 1390 y Fq(In)g(this)g(pap)q(er,)i (algebras)f(and)g(coalgebras)h(are)e(de\014ned)h(o)o(v)o(er)e(the)i (ground-)228 1448 y(\014eld)d Fo(k)r Fq(;)h(com)o(ultipli)o(cation)d (and)k(counit)e(of)i(a)f(coalgebra)g(and)h(the)e(an)o(tip)q(o)q(de)i (of)228 1506 y(a)g(Hopf)g(algebra)h(will)d(b)q(e)j(denoted)f(b)o(y)f (\001,)g Fo(\017)h Fq(and)h Fo(S)s Fq(.)23 b(If)16 b Fo(A)h Fq(is)g(an)g(algebra,)g(the)228 1564 y(dual)h(v)o(ector)f(space) h Fo(A)655 1546 y Fn(\003)692 1564 y Fq(is)g(an)h(\()p Fo(A;)8 b(A)p Fq(\)-bimo)q(dule)15 b(with)j(\()p Fo(af)5 b Fq(\)\()p Fo(b)p Fq(\))16 b(=)h Fo(f)5 b Fq(\()p Fo(ba)p Fq(\))18 b(and)228 1622 y(\()p Fo(f)5 b(a)p Fq(\)\()p Fo(b)p Fq(\))13 b(=)h Fo(f)5 b Fq(\()p Fo(ab)p Fq(\))16 b(for)g(all)g Fo(a;)8 b(b)13 b Fk(2)h Fo(A)p Fq(.)278 1680 y(Let)23 b Fo(A)f Fq(b)q(e)i(a)f(Hopf)g(algebra)h(and)f Fo(R)h Fq(an)f(in)o(v)o(ertible)e(elemen)o(t)f(in)i Fo(A)16 b Fk(\012)f Fo(A)p Fq(.)228 1739 y(Elemen)o(ts)9 b Fo(t)i Fq(in)g Fo(A)q Fk(\012)q Fo(A)h Fq(will)e(b)q(e)i(written)f(sym)o(b)q (olically)d(as)13 b Fo(t)g Fq(=)h Fo(t)1388 1721 y Fj(1)1408 1739 y Fk(\012)q Fo(t)1466 1721 y Fj(2)1486 1739 y Fq(.)20 b(F)l(ollo)o(wing)228 1797 y(Drinfeld,)15 b(the)h(pair)g(\()p Fo(A;)8 b(R)p Fq(\))16 b(is)g(called)f Fi(quasitriangular)23 b Fq(if)253 1866 y(\(1\))e Fo(x)364 1873 y Fj(2)384 1866 y Fo(R)421 1848 y Fj(1)452 1866 y Fk(\012)11 b Fo(x)530 1873 y Fj(1)549 1866 y Fo(R)586 1848 y Fj(2)620 1866 y Fq(=)j Fo(R)709 1848 y Fj(1)729 1866 y Fo(x)757 1873 y Fj(1)788 1866 y Fk(\012)d Fo(R)875 1848 y Fj(2)895 1866 y Fo(x)923 1873 y Fj(2)942 1866 y Fq(,)16 b(for)h(all)f Fo(x)d Fk(2)h Fo(A)p Fq(.)253 1924 y(\(2\))21 b(\001\()p Fo(R)433 1906 y Fj(1)453 1924 y Fq(\))11 b Fk(\012)g Fo(R)570 1906 y Fj(2)604 1924 y Fq(=)j Fo(R)693 1906 y Fj(1)724 1924 y Fk(\012)d Fo(r)797 1906 y Fj(1)828 1924 y Fk(\012)g Fo(R)915 1906 y Fj(2)935 1924 y Fo(r)958 1906 y Fj(2)978 1924 y Fo(:)253 1982 y Fq(\(3\))21 b Fo(R)373 1964 y Fj(1)405 1982 y Fk(\012)10 b Fq(\001\()p Fo(R)551 1964 y Fj(2)571 1982 y Fq(\))k(=)g Fo(R)693 1964 y Fj(1)713 1982 y Fo(r)736 1964 y Fj(1)767 1982 y Fk(\012)d Fo(r)840 1964 y Fj(2)871 1982 y Fk(\012)g Fo(R)958 1964 y Fj(2)978 1982 y Fo(:)228 2051 y Fq(Here)19 b Fo(R)h Fq(=)g Fo(r)h Fq(and)f(the)g(sym)o(b)q(olic)e(notations)j Fo(R)f Fq(=)g Fo(R)1267 2033 y Fj(1)1301 2051 y Fk(\012)13 b Fo(R)1390 2033 y Fj(2)1430 2051 y Fq(=)20 b Fo(r)1511 2033 y Fj(1)1545 2051 y Fk(\012)13 b Fo(r)1620 2033 y Fj(2)1660 2051 y Fq(\(to)228 2109 y(indicate)f(t)o(w)o(o)h(di\013eren)o (t)g(summation)e(indices\))i(and)h(\001\()p Fo(x)p Fq(\))f(=)h Fo(x)1406 2116 y Fj(1)1430 2109 y Fk(\012)5 b Fo(x)1502 2116 y Fj(2)1535 2109 y Fq(are)14 b(used.)228 2167 y(De\014ne)j Fo(R)417 2174 y Fj(21)470 2167 y Fq(=)f Fo(R)561 2149 y Fj(2)593 2167 y Fk(\012)11 b Fo(R)680 2149 y Fj(1)716 2167 y Fq(=)16 b Fo(\034)6 b Fq(\()p Fo(R)p Fq(\))17 b(where)g Fo(\034)k Fq(:)16 b Fo(R)c Fk(\012)f Fo(R)17 b Fk(!)e Fo(R)d Fk(\012)g Fo(R)18 b Fq(is)f(the)g(usual)228 2225 y(t)o(wist)f(map,)f(and)699 2284 y Fo(b)e Fq(:=)h Fo(R)836 2291 y Fj(21)873 2284 y Fo(R)g Fq(=)g Fo(r)999 2264 y Fj(2)1019 2284 y Fo(R)1056 2264 y Fj(1)1088 2284 y Fk(\012)c Fo(r)1160 2264 y Fj(1)1180 2284 y Fo(R)1217 2264 y Fj(2)1238 2284 y Fo(:)228 2354 y Fq(Note)17 b(that)i Fo(b)p Fq(\001\()p Fo(x)p Fq(\))d(=)g(\001\()p Fo(x)p Fq(\))p Fo(b)h Fq(for)i(all)e Fo(x)g Fk(2)g Fo(A)g Fq(since)g Fo(R)i Fq(and)g Fo(R)1404 2333 y Fn(\000)p Fj(1)1404 2366 y(21)1469 2354 y Fq(b)q(oth)g(satisfy)228 2412 y(\(1\).)278 2470 y(The)14 b(quasitriangular)h(Hopf)f(algebra)h(\()p Fo(A;)8 b(R)p Fq(\))14 b(is)h(called)e Fi(factorizable)19 b Fq([RS])14 b(if)228 2528 y(the)i(map)403 2608 y(\010)438 2615 y Fh(R)481 2608 y Fq(=)e(\010)g(:)f Fo(A)646 2587 y Fn(\003)679 2608 y Fk(!)h Fo(A;)105 b Fq(\010\()p Fo(f)5 b Fq(\))14 b(:=)f Fo(b)1101 2587 y Fj(1)1121 2608 y Fo(f)5 b Fq(\()p Fo(b)1190 2587 y Fj(2)1210 2608 y Fq(\))16 b(for)g(all)g Fo(f)j Fk(2)14 b Fo(A)1514 2590 y Fn(\003)1534 2608 y Fo(;)p eop %%Page: 3 3 3 2 bop 410 118 a Fl(SOME)17 b(PR)o(OPER)m(TIES)g(OF)g(F)l(A)o (CTORIZABLE)f(HOPF)g(ALGEBRAS)164 b(3)228 213 y Fq(is)13 b(an)g(isomorphism)e(of)i(v)o(ector)f(spaces;)i Fo(A)1013 195 y Fn(\003)1046 213 y Fq(is)e(the)h Fo(k)r Fq(-linear)g(dual)g(of)g Fo(A)p Fq(.)20 b(Imp)q(or-)228 271 y(tan)o(t)14 b(examples)f(of)i (factorizable)e(Hopf)i(algebras)g(are)f(the)h(Drinfeld)e(doubles)i(of) 228 329 y(arbitrary)e(\014nite-dimensional)e(Hopf)i(algebras)g([RS,)g (2.10])g(\(see)g(2.4)g(1\))h(b)q(elo)o(w\).)278 387 y(In)j([D],)g (Drinfeld)g(in)o(tro)q(duced)g(a)h(basic)g(construction)f(of)h(cen)o (tral)f(elemen)o(ts)228 445 y(of)f(a)h(quasitriangular)g(Hopf)f (algebra.)22 b(As)16 b(in)f([D])h(let)353 526 y Fo(C)t Fq(\()p Fo(A)p Fq(\))d(:=)g Fk(f)p Fo(f)19 b Fk(2)14 b Fo(A)697 505 y Fn(\003)731 526 y Fk(j)45 b Fq(for)16 b(all)31 b Fo(x;)8 b(y)15 b Fk(2)f Fo(A)g Fq(:)f Fo(f)5 b Fq(\()p Fo(xy)r Fq(\))13 b(=)h Fo(f)5 b Fq(\()p Fo(y)r(S)1454 505 y Fj(2)1474 526 y Fq(\()p Fo(x)p Fq(\)\))p Fk(g)p Fo(:)228 607 y Fq(Let)21 b Fo(Z)t Fq(\()p Fo(A)p Fq(\))g(denote)h(the)f (cen)o(ter)f(of)i Fo(A)p Fq(.)36 b(The)22 b(next)f(crucial)f(result)h (is)g(due)h(to)228 665 y(Drinfeld)g([D)o(,)i(1.2)g(and)f(3.3].)41 b(Ho)o(w)o(ev)o(er)21 b(the)h(form)o(ulation)f(b)q(elo)o(w)i(in)g (\(b\))f(is)228 723 y(more)17 b(general;)h([D,)g(3.3])g(just)g(sa)o(ys) h(that)f(the)g(map)f(\010)i(de\014nes)f(b)o(y)f(restriction)228 781 y(an)f(algebra)g(homomorphism)d(on)j Fo(C)t Fq(\()p Fo(A)p Fq(\).)k(F)l(or)c(completeness)d(the)j(short)g(pro)q(of)228 839 y(will)f(b)q(e)h(rep)q(eated)g(\(in)g(a)h(di\013eren)o(t)e (notation\).)228 927 y Ft(Theorem)h(2.1)j Fq(\(Drinfeld)c([D]\))p Ft(.)23 b Fi(L)n(et)15 b Fq(\()p Fo(A;)8 b(R)p Fq(\))15 b Fi(b)n(e)h(a)f(quasitriangular)h(Hopf)f(al-)228 985 y(gebr)n(a)i(and)h Fq(\010)c(=)g(\010)586 992 y Fh(R)629 985 y Fq(:)f Fo(A)693 967 y Fn(\003)726 985 y Fk(!)h Fo(A)p Fi(.)22 b(Then)c(for)f(al)r(l)i Fo(g)d Fk(2)e Fo(C)t Fq(\()p Fo(A)p Fq(\))i Fi(and)i Fo(f)h Fk(2)14 b Fo(A)1577 967 y Fn(\003)1596 985 y Fi(,)278 1044 y(\(a\))j Fq(\010\()p Fo(g)r Fq(\))d Fk(2)g Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fi(,)j(and)278 1102 y(\(b\))g Fq(\010\()p Fo(f)5 b(g)r Fq(\))15 b(=)f(\010\()p Fo(f)5 b Fq(\)\010\()p Fo(g)r Fq(\))p Fi(.)228 1190 y(Pr)n(o)n(of.)19 b Fq(\(a\))d(F)l(or)h (all)f Fo(g)g Fk(2)e Fo(C)t Fq(\()p Fo(A)p Fq(\))h(and)i Fo(x)c Fk(2)h Fo(A)p Fq(,)270 1271 y Fo(x)p Fq(\010\()p Fo(g)r Fq(\))g(=)g Fo(x)490 1278 y Fj(1)509 1271 y Fo(b)530 1250 y Fj(1)550 1271 y Fo(g)r Fq(\()p Fo(S)627 1250 y Fn(\000)p Fj(1)674 1271 y Fq(\()p Fo(x)721 1278 y Fj(3)740 1271 y Fq(\))p Fo(x)787 1278 y Fj(2)807 1271 y Fo(b)828 1250 y Fj(2)847 1271 y Fq(\))410 1351 y(=)g Fo(b)483 1331 y Fj(1)502 1351 y Fo(x)530 1358 y Fj(1)550 1351 y Fo(g)r Fq(\()p Fo(b)615 1331 y Fj(2)634 1351 y Fo(x)662 1358 y Fj(2)682 1351 y Fo(S)s Fq(\()p Fo(x)762 1358 y Fj(3)781 1351 y Fq(\)\))47 b(since)16 b Fo(g)g Fk(2)e Fo(C)t Fq(\()p Fo(A)p Fq(\))o Fo(;)24 b Fq(and)17 b(\001\()p Fo(x)p Fq(\))p Fo(b)12 b Fq(=)i Fo(b)p Fq(\001\()p Fo(x)p Fq(\))410 1427 y(=)g(\010\()p Fo(g)r Fq(\))p Fo(x:)278 1510 y Fq(\(b\))i(F)l(or)g(all)g Fo(f)j Fk(2)14 b Fo(A)641 1492 y Fn(\003)677 1510 y Fq(and)j Fo(g)f Fk(2)e Fo(C)t Fq(\()p Fo(A)p Fq(\),)236 1591 y(\010\()p Fo(f)5 b(g)r Fq(\))15 b(=)f Fo(R)467 1571 y Fj(2)487 1591 y Fo(r)510 1571 y Fj(1)530 1591 y Fq(\()p Fo(f)5 b(g)r Fq(\)\()p Fo(R)678 1571 y Fj(1)699 1591 y Fo(r)722 1571 y Fj(2)742 1591 y Fq(\))378 1671 y(=)14 b Fo(R)467 1651 y Fj(2)487 1671 y Fo(r)510 1651 y Fj(1)530 1671 y Fo(f)5 b Fq(\()p Fo(R)615 1651 y Fj(1)615 1684 y(1)635 1671 y Fo(r)658 1651 y Fj(2)657 1684 y(1)679 1671 y Fq(\))p Fo(g)r Fq(\()p Fo(R)779 1651 y Fj(1)779 1684 y(2)799 1671 y Fo(r)822 1651 y Fj(2)821 1684 y(2)842 1671 y Fq(\))378 1751 y(=)14 b Fo(R)467 1731 y Fj(2)487 1751 y Fo(r)510 1731 y Fj(2)530 1751 y Fo(s)553 1731 y Fj(1)573 1751 y Fo(t)591 1731 y Fj(1)610 1751 y Fo(f)5 b Fq(\()p Fo(R)695 1731 y Fj(1)716 1751 y Fo(t)734 1731 y Fj(2)753 1751 y Fq(\))p Fo(g)r Fq(\()p Fo(r)839 1731 y Fj(1)859 1751 y Fo(s)882 1731 y Fj(2)902 1751 y Fq(\))57 b(b)o(y)16 b(\(2\))h(and)g(\(3\))f(with)g(R) g(=)h(r)f(=)g(s)h(=)f(t)378 1832 y(=)e Fo(R)467 1811 y Fj(2)487 1832 y Fo(r)510 1811 y Fj(2)530 1832 y Fo(s)553 1811 y Fj(1)573 1832 y Fo(g)r Fq(\()p Fo(r)640 1811 y Fj(1)660 1832 y Fo(s)683 1811 y Fj(2)702 1832 y Fq(\))p Fo(t)739 1811 y Fj(1)759 1832 y Fo(f)5 b Fq(\()p Fo(R)844 1811 y Fj(1)864 1832 y Fo(t)882 1811 y Fj(2)902 1832 y Fq(\))378 1912 y(=)14 b Fo(R)467 1891 y Fj(2)487 1912 y Fq(\010\()p Fo(g)r Fq(\))p Fo(t)603 1891 y Fj(1)622 1912 y Fo(f)5 b Fq(\()p Fo(R)707 1891 y Fj(1)728 1912 y Fo(t)746 1891 y Fj(2)765 1912 y Fq(\))378 1987 y(=)14 b(\010\()p Fo(f)5 b Fq(\)\010\()p Fo(g)r Fq(\))496 b(since)15 b(\010\()p Fo(g)r Fq(\))i(is)f(cen)o(tral)f(b)o(y)h(\(a\))p Fo(:)p 1687 2141 2 33 v 1689 2109 30 2 v 1689 2141 V 1718 2141 2 33 v 228 2229 a Ft(Lemm)o(a)g(2.2.)24 b Fi(L)n(et)18 b Fo(A)h Fi(b)n(e)g(a)f(\014nite-dimensional)k(unimo)n(dular)c(Hopf)h (algebr)n(a.)228 2287 y(Then)f Fo(dim)p Fq(\()p Fo(C)t Fq(\()p Fo(A)p Fq(\)\))12 b(=)i Fo(dim)p Fq(\()p Fo(Z)t Fq(\()p Fo(A)p Fq(\)\))p Fi(.)228 2375 y(Pr)n(o)n(of.)19 b Fq(Let)e Fo(\025)e Fk(2)g Fo(A)593 2357 y Fn(\003)629 2375 y Fq(b)q(e)i(a)g(non-zero)g(left)f(in)o(tegral.)22 b(Then)17 b Fo(A)d Fk(!)h Fo(A)1500 2357 y Fn(\003)1519 2375 y Fq(,)i Fo(a)d Fk(7!)h Fo(\025a;)228 2433 y Fq(is)h(bijectiv)o (e.)21 b(Since)15 b Fo(A)i Fq(is)f(unimo)q(dular,)g(for)h(all)f Fo(a;)8 b(b)14 b Fk(2)h Fo(A)p Fq(,)h Fo(\025)p Fq(\()p Fo(ab)p Fq(\))e(=)h Fo(\025)p Fq(\()p Fo(bS)1620 2415 y Fj(2)1640 2433 y Fq(\()p Fo(a)p Fq(\)\))228 2492 y([OS)o(,)g(3.2,)h (2\)a\)].)21 b(Hence)14 b(for)h(all)g Fo(a;)8 b(x;)g(y)15 b Fk(2)f Fo(A)p Fq(,)g(\()p Fo(\025a)p Fq(\)\()p Fo(y)r(S)1265 2473 y Fj(2)1284 2492 y Fq(\()p Fo(x)p Fq(\)\))g(=)g Fo(\025)p Fq(\()p Fo(ay)r(S)1567 2473 y Fj(2)1586 2492 y Fq(\()p Fo(x)p Fq(\)\))f(=)228 2550 y Fo(\025)p Fq(\()p Fo(xay)r Fq(\))p Fo(:)k Fq(Th)o(us)i Fo(\025a)e Fk(2)h Fo(C)t Fq(\()p Fo(A)p Fq(\))g(if)f(and)i(only)g(if)f(for)g(all)g Fo(x)f Fk(2)h Fo(A)p Fq(,)g Fo(\025)p Fq(\()p Fo(xa)p Fq(\))f(=)h Fo(\025)p Fq(\()p Fo(ax)p Fq(\),)228 2608 y(that)e(is)h Fo(a)c Fk(2)h Fo(Z)t Fq(\()p Fo(A)p Fq(\).)p 1687 2608 V 1689 2577 30 2 v 1689 2608 V 1718 2608 2 33 v eop %%Page: 4 4 4 3 bop 228 118 a Fl(4)563 b(H.-J.)15 b(SCHNEIDER)278 213 y Fq(If)j Fo(C)k Fq(is)c(a)h(coalgebra,)h(then)e Fo(G)p Fq(\()p Fo(C)t Fq(\))g(=)g Fk(f)p Fo(c)g Fk(2)g Fo(C)j Fk(j)d Fq(\001\()p Fo(c)p Fq(\))g(=)f Fo(c)c Fk(\012)g Fo(c;)8 b(\017)p Fq(\()p Fo(c)p Fq(\))17 b(=)h(1)p Fk(g)228 271 y Fq(will)d(denote)h(the)g(set)g(of)h(all)f(group-lik)o(e)f(elemen) o(ts)f(of)i Fo(C)t Fq(.)228 359 y Ft(Theorem)g(2.3.)24 b Fi(L)n(et)19 b Fq(\()p Fo(A;)8 b(R)p Fq(\))20 b Fi(b)n(e)g(a)g (factorizable)g(Hopf)g(algebr)n(a.)30 b(Then)20 b Fq(\010)1661 366 y Fh(R)1709 359 y Fq(:)228 417 y Fo(A)265 399 y Fn(\003)298 417 y Fk(!)14 b Fo(A)j Fi(induc)n(es)h(by)f(r)n(estriction)278 475 y(\(a\))g(an)h(algebr)n(a)g(isomorphism)e Fo(C)t Fq(\()p Fo(A)p Fq(\))c Fk(!)i Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fi(,)j(and)278 534 y(\(b\))g(a)h(gr)n(oup)f(isomorphism)f Fo(G)p Fq(\()p Fo(A)915 515 y Fn(\003)934 534 y Fq(\))e Fk(!)g Fo(G)p Fq(\()p Fo(A)p Fq(\))d Fk(\\)g Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fo(:)228 622 y Fi(Pr)n(o)n(of.)19 b Fq(\(a\))c(By)f ([R94,)h(Prop)q(osition)h(3],)e Fo(A)g Fq(is)h(unimo)q(dular)f(\(for)h (another)g(pro)q(of)228 680 y(see)i(4.4)h(b)q(elo)o(w\).)26 b(By)18 b(Theorem)e(2.1,)i(\010)984 687 y Fh(R)1031 680 y Fq(de\014nes)g(an)g(injectiv)o(e)e(algebra)i(map)228 738 y Fo(C)t Fq(\()p Fo(A)p Fq(\))13 b Fk(!)g Fo(Z)t Fq(\()p Fo(A)p Fq(\))j(whic)o(h)g(is)g(bijectiv)o(e)d(b)o(y)j(Lemma)e (2.2.)278 796 y(\(b\))j(Let)h Fo(f)k Fk(2)16 b Fo(G)p Fq(\()p Fo(A)638 778 y Fn(\003)658 796 y Fq(\))g(=)g(Alg\(A,k\))n(.)26 b(Then)17 b Fo(f)22 b Fk(2)16 b Fo(C)t Fq(\()p Fo(A)p Fq(\))h(since)g Fo(f)5 b(S)1512 778 y Fj(2)1548 796 y Fq(=)16 b Fo(f)s(;)h(f)5 b(S)228 854 y Fq(b)q(eing)15 b(the)g(in)o(v)o(erse)f(of)i Fo(f)k Fq(in)15 b(the)g(group)i Fo(G)p Fq(\()p Fo(A)1072 836 y Fn(\003)1091 854 y Fq(\).)k(Hence)14 b(\010\()p Fo(f)5 b Fq(\))15 b Fk(2)f Fo(Z)t Fq(\()p Fo(A)p Fq(\))h(b)o(y)f(\(a\),)228 913 y(and)490 973 y(\010\()p Fo(f)5 b Fq(\))15 b(=)f Fo(R)696 952 y Fj(2)716 973 y Fo(r)739 952 y Fj(1)759 973 y Fo(f)5 b Fq(\()p Fo(R)844 952 y Fj(1)864 973 y Fo(r)887 952 y Fj(2)908 973 y Fq(\))13 b(=)h Fo(R)1029 952 y Fj(2)1049 973 y Fo(f)5 b Fq(\()p Fo(R)1134 952 y Fj(1)1155 973 y Fq(\))p Fo(r)1197 952 y Fj(1)1217 973 y Fo(f)g Fq(\()p Fo(r)1288 952 y Fj(2)1308 973 y Fq(\))14 b(=)g Fo(uv)r(;)228 1043 y Fq(where)19 b Fo(u)g Fq(:=)f Fo(R)526 1025 y Fj(2)546 1043 y Fo(f)5 b Fq(\()p Fo(R)631 1025 y Fj(1)652 1043 y Fq(\))p Fo(;)j(v)20 b Fq(:=)f Fo(r)831 1025 y Fj(1)851 1043 y Fo(f)5 b Fq(\()p Fo(r)922 1025 y Fj(2)942 1043 y Fq(\).)31 b(It)18 b(follo)o(ws)i (easily)e(from)g(\(3\))i(and)g(\(2\))228 1102 y(that)c Fo(u)f Fq(and)h Fo(v)h Fq(are)f(group-lik)o(e)f(elemen)o(ts)e(of)j Fo(A)p Fq(.)k(Th)o(us)c(\010)g(de\014nes)g(an)g(injectiv)o(e)228 1160 y(group)h(homomorphism)c(from)j Fo(G)p Fq(\()p Fo(A)920 1142 y Fn(\003)939 1160 y Fq(\))h(to)f Fo(G)p Fq(\()p Fo(A)p Fq(\))11 b Fk(\\)g Fo(Z)t Fq(\()p Fo(A)p Fq(\).)278 1218 y(T)l(o)g(pro)o(v)o(e)f(surjectivit)o(y)f(of)i(this)f(map)g(tak)o (e)h Fo(g)16 b Fk(2)e Fo(G)p Fq(\()p Fo(A)p Fq(\))p Fk(\\)p Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fo(:)c Fq(By)g(Lemma)e(2.2)228 1276 y Fo(\025g)281 1258 y Fn(\000)p Fj(1)347 1276 y Fq(is)19 b(in)f Fo(C)t Fq(\()p Fo(A)p Fq(\))g(since)g Fo(g)737 1258 y Fn(\000)p Fj(1)803 1276 y Fq(is)h(cen)o(tral.)28 b(Then)19 b(for)g(all)f Fo(p)h Fk(2)f Fo(A)1451 1258 y Fn(\003)1471 1276 y Fo(;)g(p)p Fq(\()p Fo(\025g)1599 1258 y Fn(\000)p Fj(1)1647 1276 y Fq(\))g(=)228 1334 y Fo(p)p Fq(\()p Fo(g)r Fq(\))p Fo(\025g)368 1316 y Fn(\000)p Fj(1)432 1334 y Fq(in)e Fo(A)526 1316 y Fn(\003)546 1334 y Fo(;)f Fq(since)h(for)g(all)g Fo(x)e Fk(2)g Fo(A;)297 1415 y Fq(\()p Fo(p)p Fq(\()p Fo(\025g)412 1395 y Fn(\000)p Fj(1)460 1415 y Fq(\)\)\()p Fo(x)p Fq(\))f(=)h Fo(p)p Fq(\()p Fo(x)700 1422 y Fj(1)720 1415 y Fq(\))p Fo(\025)p Fq(\()p Fo(g)811 1395 y Fn(\000)p Fj(1)858 1415 y Fo(x)886 1422 y Fj(2)906 1415 y Fq(\))g(=)f(\(\()p Fo(pg)r Fq(\))p Fo(\025)p Fq(\)\()p Fo(g)1187 1395 y Fn(\000)p Fj(1)1236 1415 y Fo(x)p Fq(\))g(=)h Fo(p)p Fq(\()p Fo(g)r Fq(\)\()p Fo(\025g)1507 1395 y Fn(\000)p Fj(1)1555 1415 y Fq(\)\()p Fo(x)p Fq(\))p Fo(;)228 1496 y Fq(since)k Fo(\025)h Fq(is)f(a)h(left)f (in)o(tegral.)27 b(Hence)17 b(it)h(follo)o(ws)h(from)e(Theorem)g(2.1)i (\(b\))g(that)228 1554 y(for)d(all)g Fo(p)e Fk(2)g Fo(A)492 1536 y Fn(\003)512 1554 y Fo(;)518 1635 y Fq(\010\()p Fo(p)p Fq(\)\010\()p Fo(\025g)722 1614 y Fn(\000)p Fj(1)771 1635 y Fq(\))f(=)h(\010\()p Fo(p)p Fq(\()p Fo(\025g)1005 1614 y Fn(\000)p Fj(1)1053 1635 y Fq(\)\))g(=)g Fo(p)p Fq(\()p Fo(g)r Fq(\)\010\()p Fo(\025g)1351 1614 y Fn(\000)p Fj(1)1400 1635 y Fq(\))p Fo(:)228 1715 y Fq(Therefore)j(the)g(left)f Fo(A)p Fq(-mo)q(dule)g(generated)h(b)o(y)g(\010\()p Fo(\025g)1246 1697 y Fn(\000)p Fj(1)1294 1715 y Fq(\))g(is)h(one-dimensional)228 1774 y(\(since)f(\010)h(:)f Fo(A)489 1756 y Fn(\003)526 1774 y Fk(!)g Fo(A)h Fq(is)g(bijectiv)o(e\).)26 b(Hence)17 b(there)g(is)i(an)g(algebra)f(homomor-)228 1832 y(phism)d Fo(\037)e Fq(:)g Fo(A)h Fk(!)g Fo(k)k Fq(with)e Fo(p)p Fq(\()p Fo(g)r Fq(\))e(=)g Fo(\037)p Fq(\(\010\()p Fo(p)p Fq(\)\))i(for)h(all)f Fo(p)e Fk(2)g Fo(A)1312 1814 y Fn(\003)1331 1832 y Fq(.)278 1890 y(It)g(remains)e(to)j(sho)o(w)g(that) g(\010\()p Fo(\037)p Fq(\))e(=)h Fo(g)j Fq(or)d(equiv)m(alen)o(tly)e Fo(\037)p Fq(\(\010\()p Fo(p)p Fq(\)\))i(=)g Fo(p)p Fq(\(\010\()p Fo(\037)p Fq(\)\))228 1948 y(for)i(all)g Fo(p)e Fk(2)g Fo(A)492 1930 y Fn(\003)512 1948 y Fq(.)21 b(De\014ne)16 b Fo(u)e Fq(:=)f Fo(R)842 1930 y Fj(1)862 1948 y Fo(\037)p Fq(\()p Fo(R)949 1930 y Fj(2)969 1948 y Fq(\))p Fo(;)8 b(v)15 b Fq(:=)e Fo(r)1137 1930 y Fj(2)1157 1948 y Fo(\037)p Fq(\()p Fo(r)1230 1930 y Fj(1)1250 1948 y Fq(\))p Fo(:)j Fq(F)l(or)g(all)g Fo(p)e Fk(2)g Fo(A)1576 1930 y Fn(\003)1595 1948 y Fo(;)510 2029 y(\037)p Fq(\(\010\()p Fo(p)p Fq(\)\))g(=)g Fo(\037)p Fq(\()p Fo(R)829 2009 y Fj(2)849 2029 y Fq(\))p Fo(\037)p Fq(\()p Fo(r)941 2009 y Fj(1)960 2029 y Fq(\))p Fo(p)p Fq(\()p Fo(R)1059 2009 y Fj(1)1080 2029 y Fo(r)1103 2009 y Fj(2)1123 2029 y Fq(\))g(=)g Fo(p)p Fq(\()p Fo(uv)r Fq(\))p Fo(;)23 b Fq(and)562 2113 y Fo(p)p Fq(\(\010\()p Fo(\037)p Fq(\)\))14 b(=)g Fo(p)p Fq(\()p Fo(R)874 2092 y Fj(2)894 2113 y Fo(r)917 2092 y Fj(1)937 2113 y Fo(\037)p Fq(\()p Fo(R)1024 2092 y Fj(1)1044 2113 y Fq(\))p Fo(\037)p Fq(\()p Fo(r)1136 2092 y Fj(2)1156 2113 y Fq(\)\))f(=)h Fo(p)p Fq(\()p Fo(v)r(u)p Fq(\))p Fo(:)228 2183 y Fq(Th)o(us)k(the)f (claim)f(follo)o(ws)h(from)g(the)g(equalit)o(y)f Fo(uv)h Fq(=)f Fo(v)r(u:)h Fq(Since)f(\010)i(de\014nes)g(an)228 2241 y(algebra)f(homomorphism)c Fo(C)t Fq(\()p Fo(A)p Fq(\))g Fk(!)g Fo(Z)t Fq(\()p Fo(A)p Fq(\),)j(\010\()p Fo(\037)p Fq(\))d(=)h Fo(S)s Fq(\(\010\()p Fo(\037S)s Fq(\)\);)h(hence)454 2322 y Fo(uv)g Fq(=)f(\010\()p Fo(\037)p Fq(\))g(=)f Fo(S)s Fq(\()p Fo(r)817 2302 y Fj(1)837 2322 y Fq(\))p Fo(\037)p Fq(\()p Fo(S)s Fq(\()p Fo(r)981 2302 y Fj(2)1001 2322 y Fq(\)\))p Fo(S)s Fq(\()p Fo(R)1128 2302 y Fj(2)1148 2322 y Fq(\))p Fo(\037)p Fq(\()p Fo(S)s Fq(\()p Fo(R)1306 2302 y Fj(1)1325 2322 y Fq(\)\))h(=)g Fo(v)r(u;)228 2403 y Fq(since)h Fo(R)g Fq(=)e Fo(S)s Fq(\()p Fo(R)539 2385 y Fj(1)560 2403 y Fq(\))e Fk(\012)f Fo(S)s Fq(\()p Fo(R)728 2385 y Fj(2)749 2403 y Fq(\))16 b(b)o(y)g([K)o(,)g(VI)q(I)q(I.2.4].)p 1687 2403 2 33 v 1689 2372 30 2 v 1689 2403 V 1718 2403 2 33 v 228 2492 a Ft(Remark)g(2.4.)24 b Fq(1\))15 b(Let)g Fo(H)20 b Fq(b)q(e)15 b(a)h(\014nite-dimensional)d(Hopf)i(algebra)g(and)h Fo(A)d Fq(=)228 2550 y Fo(D)q Fq(\()p Fo(H)t Fq(\))22 b(=)f Fo(H)476 2532 y Fn(\003)p Fh(cop)560 2550 y Fk(\012)14 b Fo(H)25 b Fq(the)20 b(Drinfeld)g(double)g(of)h Fo(H)t Fq(.)35 b(Elemen)o(ts)18 b Fo(p)c Fk(\012)g Fo(h;)20 b(p)i Fk(2)228 2608 y Fo(H)272 2590 y Fn(\003)292 2608 y Fo(;)8 b(h)15 b Fk(2)g Fo(H)21 b Fq(of)c Fo(D)q Fq(\()p Fo(H)t Fq(\))h(will)e(b)q(e)h(simply)e(denoted)i(b)o(y)f Fo(ph)p Fq(.)24 b(Then)17 b Fo(H)1495 2590 y Fn(\003)p Fh(cop)1582 2608 y Fq(and)h Fo(H)p eop %%Page: 5 5 5 4 bop 410 118 a Fl(SOME)17 b(PR)o(OPER)m(TIES)g(OF)g(F)l(A)o (CTORIZABLE)f(HOPF)g(ALGEBRAS)164 b(5)228 213 y Fq(are)13 b(sub)h(Hopf)f(algebras)h(of)f Fo(D)q Fq(\()p Fo(H)t Fq(\))i(and)f(com)o(ultipli)o(cation)d(and)i(m)o(ultiplication)228 271 y(in)j Fo(D)q Fq(\()p Fo(H)t Fq(\))h(are)f(giv)o(en)g(b)o(y)430 350 y(\001\()p Fo(ph)p Fq(\))d(=)h Fo(p)650 357 y Fj(2)670 350 y Fo(h)698 357 y Fj(1)729 350 y Fk(\012)d Fo(p)803 357 y Fj(1)823 350 y Fo(h)851 357 y Fj(2)871 350 y Fo(;)24 b Fq(and)17 b Fo(hp)d Fq(=)g Fo(p)p Fq(\()p Fo(S)1198 329 y Fn(\000)p Fj(1)1246 350 y Fq(\()p Fo(h)1293 357 y Fj(3)1313 350 y Fq(\))d Fk(\000)f Fo(h)1420 357 y Fj(1)1440 350 y Fq(\))p Fo(h)1487 357 y Fj(2)1507 350 y Fo(;)228 428 y Fq(for)16 b(all)g Fo(h)e Fk(2)g Fo(H)q(;)8 b(p)15 b Fk(2)f Fo(H)652 410 y Fn(\003)688 428 y Fq([K)o(,)i(IX.4.1].)278 486 y(Let)j(\()p Fo(e)410 493 y Fh(i)424 486 y Fq(\))h(and)g(\()p Fo(f)604 493 y Fh(i)618 486 y Fq(\))g(b)q(e)g(dual)f(bases)i(in)e Fo(H)24 b Fq(and)c Fo(H)1236 468 y Fn(\003)1256 486 y Fq(.)32 b(Then)20 b Fo(D)q Fq(\()p Fo(H)t Fq(\))g(is)g(qua-)228 544 y(sitriangular)k(with)g Fo(R)p Fq(-matrix)f Fo(R)28 b Fq(=)965 507 y Fg(P)1018 559 y Fh(i)1040 544 y Fo(e)1063 551 y Fh(i)1094 544 y Fk(\012)16 b Fo(f)1173 551 y Fh(i)1211 544 y Fq([K,)25 b(IX.4.2].)44 b(Th)o(us)24 b Fo(b)j Fq(=)228 565 y Fg(P)280 617 y Fh(i;j)329 602 y Fo(f)353 609 y Fh(j)371 602 y Fo(e)394 609 y Fh(i)419 602 y Fk(\012)11 b Fo(e)492 609 y Fh(j)510 602 y Fo(f)534 609 y Fh(i)564 602 y Fq(and)17 b(\010\()p Fo(F)7 b Fq(\))14 b(=)837 565 y Fg(P)890 617 y Fh(i;j)938 602 y Fo(f)962 609 y Fh(j)980 602 y Fo(e)1003 609 y Fh(i)1017 602 y Fo(F)7 b Fq(\()p Fo(e)1098 609 y Fh(j)1115 602 y Fo(f)1139 609 y Fh(i)1153 602 y Fq(\))17 b(for)g(all)f Fo(F)k Fk(2)14 b Fo(D)q Fq(\()p Fo(H)t Fq(\))1554 584 y Fn(\003)1575 602 y Fq(.)22 b(Since)228 665 y(the)16 b(elemen)o(ts)e Fo(f)536 672 y Fh(j)554 665 y Fo(e)577 672 y Fh(i)591 665 y Fq(,)i(1)f Fk(\024)f Fo(i;)8 b(j)17 b Fk(\024)e Fo(n;)h Fq(and)h Fo(S)s Fq(\()p Fo(f)1073 672 y Fh(i)1087 665 y Fq(\))p Fo(S)s Fq(\()p Fo(e)1181 672 y Fh(j)1199 665 y Fq(\),)f(1)f Fk(\024)f Fo(i;)8 b(j)17 b Fk(\024)e Fo(n;)h Fq(are)h(bases)228 723 y(of)k Fo(D)q Fq(\()p Fo(H)t Fq(\),)h(\010)p Fo(S)i Fq(is)d(injectiv)o(e.)32 b(Hence)20 b(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\))p Fo(;)8 b(R)p Fq(\))22 b(is)f(factorizable)f(\(cf.)34 b([RS,)228 781 y(Theorem)15 b(2.10],)h([R94,)g(p.)21 b(226]\).)278 839 y(2\))12 b(The)f(group)i(isomorphism)c(in)i(Theorem)g(2.3)h(can)f (b)q(e)h(describ)q(ed)f(explicitly)228 897 y(in)21 b(case)h Fo(A)i Fq(=)f Fo(D)q Fq(\()p Fo(H)t Fq(\))g(with)f Fo(R)p Fq(-matrix)f(as)i(in)e(1\).)39 b(F)l(or)22 b(an)o(y)g Fo(\037)h Fk(2)h Fo(G)p Fq(\()p Fo(A)1622 879 y Fn(\003)1642 897 y Fq(\))f(=)228 956 y(Alg\(A,k\))n(,)d(\010\()p Fo(\037)p Fq(\))g(=)f Fo(uv)i Fq(with)f Fo(u)f Fq(=)924 918 y Fg(P)977 970 y Fh(i)999 956 y Fo(f)1023 963 y Fh(i)1037 956 y Fo(\037)p Fq(\()p Fo(e)1110 963 y Fh(i)1123 956 y Fq(\))h(and)g Fo(v)i Fq(=)1363 918 y Fg(P)1416 970 y Fh(i)1438 956 y Fo(e)1461 963 y Fh(i)1475 956 y Fo(\037)p Fq(\()p Fo(f)1549 963 y Fh(i)1563 956 y Fq(\))p Fo(:)d Fq(Th)o(us)228 1014 y Fo(u)h Fk(2)g Fo(G)p Fq(\()p Fo(A)423 996 y Fn(\003)443 1014 y Fq(\))g(is)g(the)g(restriction)f(of)i Fo(\037)e Fq(on)i Fo(H)1086 996 y Fn(\003)1106 1014 y Fq(,)f(and)h(the)f (group-lik)o(e)f(elemen)o(t)228 1072 y Fo(v)24 b Fk(2)e Fo(G)p Fq(\()p Fo(H)t Fq(\))g(is)g(de\014ned)f(b)o(y)g Fo(p)p Fq(\()p Fo(v)r Fq(\))h(=)g Fo(\037)p Fq(\()p Fo(p)p Fq(\))g(for)f(all)g Fo(p)i Fk(2)g Fo(H)1358 1054 y Fn(\003)1378 1072 y Fq(.)37 b(Therefore)20 b(the)228 1130 y(isomorphism)12 b Fo(G)p Fq(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\))691 1112 y Fn(\003)712 1130 y Fq(\))i Fk(!)g Fo(G)p Fq(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\)\))8 b Fk(\\)g Fo(Z)t Fq(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\)\))17 b(of)e(Theorem)e(2.3)i(is)f(the)228 1188 y(isomorphism)g(constructed)i(b)o(y)g(Radford)h([R93,)f(Prop)q (ositions)h(9)g(and)g(10].)323 1287 y(3.)28 b Fp(A)18 b(direct)i(pr)o(oof)d(of)i(a)f(theorem)g(of)g(Etingof-Gelaki)278 1374 y Fq(Throughout)f(this)f(section)g(the)g(\014eld)f Fo(k)j Fq(is)e(algebraically)f(closed)h(of)g(c)o(harac-)228 1432 y(teristic)i(0.)33 b(The)20 b(Theorem)f(of)h(Etingof-Gelaki)g ([EG,)g(1.4])g(will)f(follo)o(w)g(from)228 1490 y(Drinfeld's)e(cen)o (tral)h(elemen)o(t)e(construction)i(in)h(section)f(2)h(and)g(the)g (follo)o(wing)228 1548 y(result)g(b)o(y)g(Kac-Zh)o(u)h(\(the)f(class)h (equation,)g(see)f([L])h(for)g(a)g(more)e(elemen)o(tary)228 1606 y(pro)q(of)s(\).)228 1688 y Ft(Theorem)e(3.1)j Fq(\(Kac-Zh)o(u\))p Ft(.)k Fi(L)n(et)c Fo(A)g Fi(b)n(e)h(a)g(semisimple)g(Hopf)f(algebr)n (a)h(and)g Fo(e)228 1746 y Fi(a)d(primitive)g(idemp)n(otent)g(in)g(the) h(char)n(acter)f(algebr)n(a)g Fo(C)t Fq(\()p Fo(A)p Fq(\))p Fi(.)k(Then)d Fo(dim)p Fq(\()p Fo(A)1662 1728 y Fn(\003)1681 1746 y Fo(e)p Fq(\))228 1805 y Fi(divides)g Fo(dim)p Fq(\()p Fo(A)p Fq(\))p Fo(:)228 1887 y Ft(Theorem)e(3.2.)24 b Fi(L)n(et)e Fq(\()p Fo(A;)8 b(R)p Fq(\))23 b Fi(b)n(e)g(a)g (factorizable)g(Hopf)g(algebr)n(a.)39 b(If)22 b Fo(A)h Fi(is)f(a)228 1945 y(semisimple)14 b(algebr)n(a)h(and)f Fo(V)25 b Fi(a)14 b(simple)g(left)h Fo(A)p Fi(-mo)n(dule,)g(then)f Fo(dim)p Fq(\()p Fo(V)d Fq(\))1544 1927 y Fj(2)1578 1945 y Fi(divides)228 2003 y Fo(dim)p Fq(\()p Fo(A)p Fq(\))p Fi(.)228 2085 y(Pr)n(o)n(of.)19 b Fq(By)e(assumption,)h Fo(A)782 2071 y Fk(\030)783 2087 y Fq(=)838 2085 y Fo(M)885 2092 y Fh(d)903 2097 y Ff(1)923 2085 y Fq(\()p Fo(k)r Fq(\))12 b Fk(\002)g(\001)c(\001)g(\001)k(\002)h Fo(M)1220 2092 y Fh(d)1238 2096 y Fe(t)1254 2085 y Fq(\()p Fo(k)r Fq(\))18 b(is)g(isomorphic)e(to)j(a)228 2143 y(pro)q(duct)c(of)g(full)f (matrix)f(rings)h Fo(M)868 2150 y Fh(d)886 2155 y Fe(i)902 2143 y Fq(\()p Fo(k)r Fq(\),)g(and)i Fo(Z)t Fq(\()p Fo(A)p Fq(\))1214 2129 y Fk(\030)1214 2145 y Fq(=)1266 2143 y Fo(k)10 b Fk(\002)e(\001)g(\001)g(\001)f(\002)h Fo(k)r Fq(.)21 b(Let)14 b Fo(E)1643 2150 y Fh(i)1672 2143 y Fq(b)q(e)228 2201 y(the)h(primitiv)o(e)d(idemp)q(oten)o(t)i(in)i Fo(Z)t Fq(\()p Fo(A)p Fq(\))f(corresp)q(onding)h(to)g(the)g(simple)d (mo)q(dule)228 2259 y Fo(V)e Fq(.)43 b(Th)o(us)23 b Fo(AE)527 2266 y Fh(i)567 2245 y Fk(\030)567 2261 y Fq(=)631 2259 y Fo(M)678 2266 y Fh(d)696 2271 y Fe(i)712 2259 y Fq(\()p Fo(k)r Fq(\))g(has)h(dimension)e Fo(d)1157 2241 y Fj(2)1157 2271 y Fh(i)1177 2259 y Fq(,)j(and)f Fo(d)1343 2266 y Fh(i)1383 2259 y Fq(=)i(dim)n(\()p Fo(V)11 b Fq(\).)43 b(By)228 2317 y(Theorem)12 b(2.1,)i(\010)556 2324 y Fh(R)599 2317 y Fq(induces)f(an)h(algebra)h(isomorphism)c Fo(C)t Fq(\()p Fo(A)p Fq(\))i Fk(!)g Fo(Z)t Fq(\()p Fo(A)p Fq(\))h(\(Note)228 2375 y(that)g(here)f(unimo)q(dularit)o(y)g(of)h Fo(A)f Fq(is)h(v)o(ery)f(easy)h(to)g(see)f(since)h Fo(A)f Fq(is)h(semisim)o (ple\).)228 2433 y(Let)i Fo(e)338 2440 y Fh(i)368 2433 y Fq(b)q(e)h(the)f(primitiv)n(e)d(idemp)q(oten)o(t)i(in)h Fo(C)t Fq(\()p Fo(A)p Fq(\))f(with)h(\010\()p Fo(e)1359 2440 y Fh(i)1373 2433 y Fq(\))e(=)g Fo(E)1494 2440 y Fh(i)1508 2433 y Fq(.)21 b(Then)c(b)o(y)228 2492 y(Theorem)h(2.1)h (\(b\),)h(\010\()p Fo(A)708 2473 y Fn(\003)728 2492 y Fo(e)751 2499 y Fh(i)764 2492 y Fq(\))f(=)g(\010\()p Fo(A)950 2473 y Fn(\003)970 2492 y Fq(\)\010\()p Fo(e)1066 2499 y Fh(i)1080 2492 y Fq(\))g(=)g Fo(AE)1248 2499 y Fh(i)1261 2492 y Fo(:)g Fq(Since)g(\010)g(is)g(bijectiv)o(e,)228 2550 y(dim)n(\()p Fo(A)365 2532 y Fn(\003)384 2550 y Fo(e)407 2557 y Fh(i)421 2550 y Fq(\))27 b(=)g(dim)o(\()p Fo(AE)706 2557 y Fh(i)719 2550 y Fq(\))g(=)g(\(dim)o(\()p Fo(V)11 b Fq(\)\))1027 2532 y Fj(2)1047 2550 y Fq(.)44 b(Therefore)24 b(\(dim)n(\()p Fo(V)12 b Fq(\)\))1531 2532 y Fj(2)1574 2550 y Fq(divides)228 2608 y(dim)n(\()p Fo(A)p Fq(\))k(b)o(y)g(Theorem)f(3.1.)p 1687 2608 2 33 v 1689 2577 30 2 v 1689 2608 V 1718 2608 2 33 v eop %%Page: 6 6 6 5 bop 228 118 a Fl(6)563 b(H.-J.)15 b(SCHNEIDER)228 213 y Ft(Corollary)k(3.3)f Fq(\(Etingof-Gelaki)e([EG]\))p Ft(.)23 b Fi(\(1\))d(If)f Fo(H)k Fi(is)d(a)f(semisimple)h(Hopf)228 271 y(algebr)n(a)13 b(and)f Fo(V)24 b Fi(a)13 b(simple)g(left)g Fo(D)q Fq(\()p Fo(H)t Fq(\))p Fi(-mo)n(dule,)i(then)e Fo(dim)p Fq(\()p Fo(V)e Fq(\))i Fi(divides)g Fo(dim)p Fq(\()p Fo(H)t Fq(\))p Fi(.)278 329 y(\(2\))i(If)h Fq(\()p Fo(H)q(;)8 b(R)p Fq(\))17 b Fi(is)f(a)g(quasitriangular)h(Hopf)f (algebr)n(a)h(and)f Fo(H)21 b Fi(is)16 b(semisimple,)228 387 y(then)i(for)f(any)g(simple)h(left)h Fo(H)t Fi(-mo)n(dule,)f Fo(dim)p Fq(\()p Fo(V)11 b Fq(\))17 b Fi(divides)h Fo(dim)p Fq(\()p Fo(H)t Fq(\))p Fi(.)228 482 y(Pr)n(o)n(of.)h Fq(\(1\))j(Since)f(the)h(c)o(haracteristic)f(of)h Fo(k)i Fq(is)e(0,)i Fo(D)q Fq(\()p Fo(H)t Fq(\))f(is)f(semisim)o(ple)d(b)o(y) 228 541 y([LR].)25 b(Hence)17 b(b)o(y)g(Theorem)g(3.2)h(applied)f(to)h (the)g(factorizable)f(Hopf)h(algebra)228 599 y(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\))p Fo(;)8 b(R)p Fq(\),)18 b(\(dim)n(\()p Fo(V)11 b Fq(\)\))676 581 y Fj(2)713 599 y Fq(divides)k(dim)o(\()p Fo(D)q Fq(\()p Fo(H)t Fq(\)\))g(=)g(\(dim)n(\()p Fo(H)t Fq(\)\))1389 581 y Fj(2)1409 599 y Fo(:)i Fq(Th)o(us)g(dim)n(\()p Fo(V)11 b Fq(\))228 657 y(divides)k(dim)n(\()p Fo(H)t Fq(\).)278 715 y(\(2\))20 b(follo)o(ws)g(from)f(\(1\))i(as)g(in)f([EG]) f(since)h Fo(H)k Fq(is)c(an)h(epimorphic)d(image)h(of)228 773 y Fo(D)q Fq(\()p Fo(H)t Fq(\).)p 1687 773 2 33 v 1689 742 30 2 v 1689 773 V 1718 773 2 33 v 228 869 a Ft(Remark)d(3.4.)24 b Fq(The)12 b(original)g(pro)q(of)h(of)g(Corollary) f(3.3)g(\(1\))h(uses)f(the)g(theory)g(of)228 927 y(mo)q(dular)k (categories.)21 b(In)c(particular,)e(in)h(the)g(\014rst)h(line)e(of)i (the)f(pro)q(of)i(of)f([EG,)228 985 y(Lemma)f(1.2],)j(the)g(authors)h (need)e(a)h(trace)g(form)o(ula)e(in)h(mo)q(dular)h(categories.)228 1043 y(F)l(or)i(a)g(pro)q(of)h(they)e(refer)g(to)h([Ki)o(])g(who)g (refers)f(to)h(T)l(uraev's)g(b)q(o)q(ok)h([T].)34 b(The)228 1101 y(luc)o(ky)14 b(reader)h(ma)o(y)e(\014nd)j(a)g(pro)q(of)g(in)f ([T,)g(I)q(I,3.2.2.\(ii\)])e(making)h(use)h(of)h(v)m(arious)228 1159 y(t)o(wistings)g(in)g(the)g(graphical)g(calculus.)278 1217 y(Ho)o(w)o(ev)o(er,)10 b(in)i(the)g(case)g(of)h(the)f(mo)q(dular)g (category)g(of)h(all)e(\014nite-dimensional)228 1275 y(represen)o(tations)22 b(of)h(a)g(semisimple)c(factorizable)j(Hopf)h (algebra)g(\()p Fo(A;)8 b(R)p Fq(\))23 b(\(in)228 1334 y([EG])c Fo(A)g Fq(is)h(the)f(Drinfeld)g(double)g(of)h(a)g(semisimpl)o (e)c(Hopf)k(algebra)g Fo(H)t Fq(\),)g(this)228 1392 y(form)o(ula)14 b(is)h(in)g(fact)h(a)f(sp)q(ecial)g(case)h(of)g(the)f(class)g(equation) h(3.1)g(in)f(the)g(v)o(ersion)228 1450 y(of)h(Lorenz)h([L].)278 1508 y(Let)e Fo(V)392 1515 y Fh(i)406 1508 y Fo(;)8 b Fq(0)14 b Fk(\024)g Fo(i)g Fk(\024)f Fo(m;)i Fq(with)g Fo(V)812 1515 y Fj(0)845 1508 y Fq(:=)f Fo(k)r Fq(,)h(b)q(e)g(a)h (complete)c(set)k(of)f(represen)o(tativ)o(es)228 1566 y(of)j(the)f(isomorphism)e(classes)j(of)g(the)f(simple)e(left)i Fo(A)p Fq(-mo)q(dules.)24 b(F)l(or)18 b(all)f Fo(i)p Fq(,)g(let)228 1624 y Fo(\037)259 1631 y Fh(i)287 1624 y Fq(b)q(e)d(the)g(c)o(haracter)f(of)h Fo(V)724 1631 y Fh(i)739 1624 y Fq(,)g(and)h Fo(\037)891 1631 y Fh(i)903 1622 y Fd(\003)936 1624 y Fq(=)f Fo(S)s Fq(\()p Fo(\037)1071 1631 y Fh(i)1084 1624 y Fq(\))g(b)q(e)h(the)e(c)o(haracter)h(of)g(\()p Fo(V)1574 1631 y Fh(i)1589 1624 y Fq(\))1608 1606 y Fn(\003)1627 1624 y Fq(.)21 b(By)228 1682 y(de\014nition,)15 b(the)h(matrix)f(\()p Fo(s)743 1689 y Fh(ij)773 1682 y Fq(\))h(in)g([EG])g(is)g(giv)o(en)f(b) o(y)656 1775 y Fo(s)679 1782 y Fh(ij)723 1775 y Fq(=)f Fo(\037)806 1782 y Fh(i)819 1775 y Fq(\(\010\()p Fo(\037)923 1782 y Fh(j)939 1773 y Fd(\003)959 1775 y Fq(\)\))p Fo(;)8 b Fq(0)14 b Fk(\024)g Fo(i;)8 b(j)16 b Fk(\024)e Fo(m;)228 1868 y Fq(where)h(\010)f(=)g(\010)504 1875 y Fh(R)533 1868 y Fq(.)21 b(The)16 b(trace)f(form)o(ula)f(needed)h(in)g(the)h(pro) q(of)g(of)g([EG,)f(Lemma)228 1927 y(1.2])h(then)g(is:)651 2010 y(F)l(or)g(all)g(j,)859 1963 y Fg(X)889 2068 y Fh(i)939 2010 y Fo(s)962 2017 y Fh(j)r(i)993 2010 y Fo(s)1016 2017 y Fh(ij)1044 2008 y Fd(\003)1078 2010 y Fq(=)e(dim)n(\()p Fo(A)p Fq(\))p Fo(:)278 2139 y Fq(T)l(o)i(pro)o(v)o(e)f(this)g(form)o (ula,)f(let)h Fo(E)872 2146 y Fh(i)902 2139 y Fq(in)g Fo(Z)t Fq(\()p Fo(A)p Fq(\))g(b)q(e)h(the)f(cen)o(tral)g(primitiv)o(e)d (idem-)228 2197 y(p)q(oten)o(t)18 b(of)g Fo(V)469 2204 y Fh(i)483 2197 y Fo(;)8 b Fq(0)17 b Fk(\024)f Fo(i)g Fk(\024)g Fo(m)p Fq(.)25 b(F)l(or)18 b(all)f Fo(i)p Fq(,)g(de\014ne)h Fo(e)1143 2204 y Fh(i)1173 2197 y Fq(:=)e(\010)1276 2179 y Fn(\000)p Fj(1)1323 2197 y Fq(\()p Fo(E)1378 2204 y Fh(i)1392 2197 y Fq(\),)i(and)g(let)f Fo(\026)1640 2204 y Fh(i)1672 2197 y Fq(b)q(e)228 2255 y(the)h(c)o(haracter)g(of)h(the)g (simple)d Fo(C)t Fq(\()p Fo(A)p Fq(\)-mo)q(dule)h Fo(C)t Fq(\()p Fo(A)p Fq(\))p Fo(e)1269 2262 y Fh(i)1282 2255 y Fq(.)28 b(Since)18 b(\010)h(de\014nes)g(an)228 2313 y(algebra)14 b(isomorphism)e(b)q(et)o(w)o(een)i Fo(C)t Fq(\()p Fo(A)p Fq(\))f(and)i Fo(Z)t Fq(\()p Fo(A)p Fq(\),)e Fo(\031)1253 2320 y Fh(i)1281 2313 y Fq(:=)g Fo(\026)1375 2320 y Fh(i)1390 2313 y Fq(\010)1425 2295 y Fn(\000)p Fj(1)1486 2313 y Fq(is)h(the)g(c)o(har-)228 2372 y(acter)g(of)h(the)f (simple)e Fo(Z)t Fq(\()p Fo(A)p Fq(\)-mo)q(dule)h Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fo(E)1080 2379 y Fh(i)1093 2372 y Fq(.)21 b(Note)14 b(that)h(for)f(an)o(y)h Fo(a)e Fk(2)h Fo(Z)t Fq(\()p Fo(A)p Fq(\),)228 2430 y(left)24 b(m)o(ultiplic)o(ation) f(with)h Fo(a)h Fq(on)g Fo(V)918 2437 y Fh(i)958 2430 y Fq(is)g(m)o(ultipli)o(cation)d(with)j Fo(\031)1485 2437 y Fh(i)1499 2430 y Fq(\()p Fo(a)p Fq(\),)h(hence)228 2488 y Fo(\037)259 2495 y Fh(i)272 2488 y Fq(\()p Fo(a)p Fq(\))14 b(=)g Fo(\031)430 2495 y Fh(i)444 2488 y Fq(\()p Fo(a)p Fq(\)dim)n(\()p Fo(V)636 2495 y Fh(i)650 2488 y Fq(\))j(\(F)l(or,)f(since)g Fo(V)954 2495 y Fh(i)985 2488 y Fq(is)g(simple,)e(left)i(m)o(ultplic)o(ation)e(with)j Fo(a)228 2550 y Fq(on)11 b Fo(V)318 2557 y Fh(i)344 2550 y Fq(is)g(m)o(ultipli)o(cation)d(with)j(some)f(scalar;)i(since)f Fo(AE)1254 2557 y Fh(i)1281 2536 y Fk(\030)1282 2552 y Fq(=)1334 2550 y Fo(V)1373 2527 y Fh(d)1391 2532 y Fe(i)1362 2562 y Fh(i)1406 2550 y Fo(;)d(d)1453 2557 y Fh(i)1482 2550 y Fq(:=)13 b(dim)n(\()p Fo(V)1675 2557 y Fh(i)1690 2550 y Fq(\))p Fo(;)228 2608 y Fq(and)k Fo(Z)t Fq(\()p Fo(A)p Fq(\))p Fo(E)471 2615 y Fh(i)501 2608 y Fq(is)f(one-dimensional,)e(this)i(scalar)h(is)f Fo(\031)1232 2615 y Fh(i)1245 2608 y Fq(\()p Fo(a)p Fq(\)\).)p eop %%Page: 7 7 7 6 bop 410 118 a Fl(SOME)17 b(PR)o(OPER)m(TIES)g(OF)g(F)l(A)o (CTORIZABLE)f(HOPF)g(ALGEBRAS)164 b(7)278 213 y Fq(Hence)15 b(for)h(all)g Fo(i;)8 b(j)s Fq(,)386 295 y Fo(s)409 302 y Fh(ij)453 295 y Fq(=)14 b Fo(\037)536 302 y Fh(i)550 295 y Fq(\(\010\()p Fo(\037)654 302 y Fh(j)670 293 y Fd(\003)690 295 y Fq(\)\))f(=)h Fo(\031)821 302 y Fh(i)835 295 y Fq(\(\010\()p Fo(\037)939 302 y Fh(j)955 293 y Fd(\003)975 295 y Fq(\)\)dim)n(\()p Fo(V)1141 302 y Fh(i)1156 295 y Fq(\))g(=)f Fo(\026)1269 302 y Fh(i)1284 295 y Fq(\()p Fo(\037)1334 302 y Fh(j)1350 293 y Fd(\003)1369 295 y Fq(\)dim)o(\()p Fo(V)1517 302 y Fh(i)1531 295 y Fq(\))p Fo(;)228 377 y Fq(since)i(\010\()p Fo(\037)432 384 y Fh(j)448 375 y Fd(\003)468 377 y Fq(\))i(is)f(in)f(the)h(cen)o (ter)f(of)i Fo(A)f Fq(b)o(y)g(Theorem)e(2.1.)278 435 y(Moreo)o(v)o(er)22 b(it)g(is)h(not)h(di\016cult)d(to)j(see)e(that)i (for)f(all)g Fo(i;)8 b(j)s Fq(,)24 b Fo(s)1433 442 y Fh(ij)1489 435 y Fq(=)h Fo(s)1575 442 y Fh(j)r(i)1605 435 y Fq(,)g(and)228 493 y Fo(s)251 500 y Fh(ij)295 493 y Fq(=)14 b Fo(s)370 500 y Fh(i)382 491 y Fd(\003)400 500 y Fh(j)416 491 y Fd(\003)436 493 y Fq(,)i(hence)f Fo(s)624 500 y Fh(ij)652 491 y Fd(\003)686 493 y Fq(=)f Fo(s)761 500 y Fh(j)r(i)789 491 y Fd(\003)826 493 y Fq(\(cf.)20 b([Ki)o(]\).)278 552 y(Therefore)15 b(one)i(obtains)407 587 y Fg(X)437 692 y Fh(i)487 634 y Fo(s)510 641 y Fh(j)r(i)540 634 y Fo(s)563 641 y Fh(ij)591 632 y Fd(\003)625 634 y Fq(=)677 587 y Fg(X)707 692 y Fh(i)757 634 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b(is)f(a)i(\014nite-dimensional)d(quasitriangular)i(Hopf)g(algebra,)g (de\014ne)228 1537 y Fo(\031)256 1544 y Fh(R)304 1537 y Fq(:)j Fo(D)q Fq(\()p Fo(A)p Fq(\))h Fk(!)f Fo(A)g Fq(b)o(y)g Fo(\031)697 1544 y Fh(R)726 1537 y Fq(\()p Fo(f)5 b(a)p Fq(\))19 b(:=)g Fo(f)5 b Fq(\()p Fo(R)994 1519 y Fj(1)1015 1537 y Fq(\))p Fo(R)1071 1519 y Fj(2)1091 1537 y Fo(a)19 b Fq(for)h(all)f Fo(f)25 b Fk(2)20 b Fo(A)1424 1519 y Fn(\003)1443 1537 y Fo(;)8 b(a)19 b Fk(2)h Fo(A:)f Fq(\(See)228 1595 y(2.4)i(for)g(notations)h(concerning)f Fo(D)q Fq(\()p Fo(A)p Fq(\)\).)36 b(This)21 b(map)f(app)q(ears)i(in)f ([D)o(,)h(Pro)q(of)228 1653 y(of)f(Prop)q(osition)h(6.2])f(where)g(it)g (w)o(as)g(observ)o(ed)g(that)h Fo(\031)1295 1660 y Fh(R)1344 1653 y Fq(is)f(a)h(Hopf)f(algebra)228 1719 y(homomorphism.)e(Since)729 1706 y Fg(e)720 1719 y Fo(R)c Fq(:=)f(\()p Fo(R)894 1726 y Fj(21)932 1719 y Fq(\))951 1701 y Fn(\000)p Fj(1)1012 1719 y Fq(=)h Fo(S)1098 1701 y Fn(\000)p Fj(1)1145 1719 y Fq(\()p Fo(R)1201 1701 y Fj(2)1221 1719 y Fq(\))d Fk(\012)f Fo(R)1339 1701 y Fj(1)1373 1719 y Fq(=)k Fo(R)1463 1701 y Fj(2)1495 1719 y Fk(\012)c Fo(S)s Fq(\()p Fo(R)1634 1701 y Fj(1)1654 1719 y Fq(\))16 b(is)228 1784 y(another)h Fo(R)p Fq(-matrix)f(for)i Fo(A)e Fq([K,)g(VI)q(I)q(I.2],)f(that)j(is)f (\()p Fo(A;)1248 1771 y Fg(e)1240 1784 y Fo(R)p Fq(\))f(is)h(again)h (quasitrian-)228 1842 y(gular,)e(there)g(is)g(alw)o(a)o(ys)g(a)g (second)h(pro)s(jection)f Fo(\031)1161 1849 y Fc(e)1154 1857 y Fh(R)1196 1842 y Fq(:)d Fo(D)q Fq(\()p Fo(A)p Fq(\))h Fk(!)g Fo(A)p Fq(.)278 1900 y(Finally)l(,)f(if)h Fo(\045)g Fq(:)g Fo(A)f Fk(!)h Fo(B)j Fq(is)e(a)h(homomorphism)c(of)j (bialgebras,)g(then)g Fo(A)1605 1882 y Fh(co\045)1671 1900 y Fq(:=)228 1958 y Fk(f)p Fo(x)e Fk(2)h Fo(A)g Fk(j)f Fo(x)447 1965 y Fj(1)478 1958 y Fk(\012)e Fo(\045)p Fq(\()p Fo(x)600 1965 y Fj(2)619 1958 y Fq(\))j(=)g Fo(x)d Fk(\012)f Fq(1)p Fk(g)p Fq(.)228 2047 y Ft(Lemm)o(a)16 b(4.1.)24 b Fi(L)n(et)19 b Fq(\()p Fo(A;)8 b(R)p Fq(\))19 b Fi(b)n(e)h(a)g 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Fq(\))e(:=)f Fo(F)7 b Fq(\001\()p Fo(x)p Fq(\))p Fo(F)1001 1489 y Fn(\000)p Fj(1)1063 1510 y Fq(for)16 b(all)g Fo(x)d Fk(2)h Fo(H)q(;)228 1590 y Fq(and)22 b(the)g(old)g (augmen)o(tation)f Fo(")p Fq(.)38 b(If)22 b Fo(H)k Fq(is)21 b(a)i(Hopf)f(algebra,)h(then)f Fo(H)1592 1572 y Fh(F)1643 1590 y Fq(is)g(a)228 1648 y(Hopf)d(algebra)h(with)f(an)o(tip)q(o)q(de)h Fo(S)873 1655 y Fh(F)902 1648 y Fq(\()p Fo(x)p Fq(\))f(:=)f Fo(v)r(S)s Fq(\()p Fo(x)p Fq(\))p Fo(v)1208 1630 y Fn(\000)p Fj(1)1273 1648 y Fq(for)h(all)g Fo(x)g Fk(2)g Fo(H)t Fq(,)h(where)228 1706 y Fo(v)15 b Fq(:=)e Fo(F)371 1688 y Fj(1)390 1706 y Fo(S)s Fq(\()p Fo(F)481 1688 y Fj(2)500 1706 y Fq(\))p Fo(;)8 b(v)567 1688 y Fn(\000)p Fj(1)628 1706 y Fq(=)13 b Fo(S)s Fq(\()p Fo(G)769 1688 y Fj(1)789 1706 y Fq(\))p Fo(G)846 1688 y Fj(2)866 1706 y Fo(;)8 b(G)15 b Fq(:=)e Fo(F)1045 1688 y Fn(\000)p Fj(1)1108 1706 y Fq(\(see)j([K)o(,)g(XV.3)f(and)i(XV.6]\).)278 1764 y(Theorem)f(2.9)i(in)g([RS)o(])g(describ)q(es)f(the)h(Drinfeld)f (double)h(of)g(a)g(factorizable)228 1822 y(Hopf)11 b(algebra)h Fo(A)g Fq(as)g(a)g(2-co)q(cycle)f(t)o(wist)g(of)h(the)f(usual)h(\(comp) q(onen)o(t)o(wise\))e(tensor)228 1880 y(pro)q(duct)17 b(bialgebra)f Fo(A)11 b Fk(\012)g Fo(A)k Fq(without)i(giving)f(an)h (explicit)d(pro)q(of.)278 1938 y(In)i(the)h(case)g(when)g Fo(A)d Fq(=)h Fo(D)q Fq(\()p Fo(H)t Fq(\))j(is)f(itself)f(the)h (Drinfeld)f(double)g(of)i(another)228 1997 y(Hopf)c(algebra)g Fo(H)t Fq(,)g(Tsang)i(and)e(Zh)o(u)g([TZ,)g(Theorem)e(2])i(recen)o(tly) e(redisco)o(v)o(ered)228 2055 y(this)j(result.)20 b(Their)15 b(pro)q(of)h(dep)q(ends)g(on)f(the)g(structure)g(of)g Fo(A)g Fq(as)h(the)f(double)g(of)228 2113 y Fo(H)t Fq(.)278 2171 y(In)g(the)h(next)f(theorem)f(a)j(stronger)f(result)f(is)h(sho)o (wn)h(b)o(y)e(using)h(the)g(concept)228 2229 y(of)f(Hopf)g(mo)q(dules)f (in)h(the)g(next)g(lemm)o(a.)j(F)l(actorizable)c(Hopf)i(algebras)f(can) h(in)228 2287 y(fact)j(b)q(e)f(c)o(haracterized)g(b)o(y)g(the)g(prop)q (ert)o(y)h(that)g(their)f(double)h(is)g(isomorphic)228 2345 y(\(in)j(a)i(sp)q(eci\014c)e(w)o(a)o(y\))h(to)g(a)g(2-co)q(cycle)g (t)o(wist)f(of)h(the)g(usual)g(tensor)h(pro)q(duct)228 2403 y(bialgebra.)228 2492 y Ft(Lemm)o(a)16 b(4.2.)24 b Fi(L)n(et)13 b Fo(A;)8 b(B)15 b Fi(and)e Fo(D)i Fi(b)n(e)e(bialgebr)n (as)h(and)g Fo(')f Fq(:)h Fo(D)i Fk(!)d Fo(A;)8 b( )15 b Fq(:)e Fo(D)j Fk(!)e Fo(B)228 2550 y Fi(bialgebr)n(a)20 b(maps.)28 b(De\014ne)21 b Fo(\016)f Fq(:)d Fo(D)i Fk(!)f Fo(A)12 b Fk(\012)g Fo(B)23 b Fi(by)c Fo(\016)r Fq(\()p Fo(x)p Fq(\))e(:=)g Fo(')p Fq(\()p Fo(x)1421 2557 y Fj(1)1440 2550 y Fq(\))c Fk(\012)f Fo( )r Fq(\()p Fo(x)1604 2557 y Fj(2)1623 2550 y Fq(\))20 b Fi(for)228 2608 y(al)r(l)f Fo(x)13 b Fk(2)h Fo(D)q Fi(.)p eop %%Page: 9 9 9 8 bop 410 118 a Fl(SOME)17 b(PR)o(OPER)m(TIES)g(OF)g(F)l(A)o (CTORIZABLE)f(HOPF)g(ALGEBRAS)164 b(9)278 213 y Fi(\(1\))17 b(If)g Fq(\()p Fo(D)q(;)8 b(R)p Fq(\))19 b Fi(is)e(a)g(quasitriangular) h(Hopf)g(algebr)n(a,)g(then)637 293 y Fo(F)i Fq(:=)14 b(1)d Fk(\012)g Fo( )r Fq(\()p Fo(S)s Fq(\()p Fo(R)982 272 y Fj(1)1002 293 y Fq(\)\))g Fk(\012)f Fo(')p Fq(\()p Fo(R)1188 272 y Fj(2)1209 293 y Fq(\))h Fk(\012)f Fq(1)228 373 y Fi(is)19 b(a)h(2-c)n(o)n(cycle)h(for)e Fo(A)12 b Fk(\012)g Fo(B)22 b Fi(with)f(c)n(omp)n(onentwise)f(bialgebr)n(a)h (structur)n(e,)f(and)228 431 y Fo(\016)15 b Fq(:)e Fo(D)j Fk(!)e Fq(\()p Fo(A)c Fk(\012)h Fo(B)s Fq(\))587 413 y Fh(F)634 431 y Fi(is)17 b(a)g(bialgebr)n(a)i(map.)278 489 y(\(2\))i(Assume)g(that)h Fo(B)i Fi(is)d(a)g(Hopf)h(algebr)n(a)f (and)h(that)g(ther)n(e)f(is)g(a)g(bialgebr)n(a)228 547 y(homomorphism)g Fo(\015)26 b Fq(:)c Fo(B)k Fk(!)d Fo(D)h Fi(with)f Fo( )r(\015)j Fq(=)d Fi(id)1158 554 y Fh(B)1189 547 y Fo(:)f Fi(Then)h Fo(\016)h Fq(:)f Fo(D)i Fk(!)e Fo(A)14 b Fk(\012)h Fo(B)228 605 y Fi(is)20 b(bije)n(ctive)i(if)e(and)h (only)f(if)h(the)f(r)n(estriction)h(of)f Fo(')g Fi(de\014nes)i(an)f (isomorphism)228 663 y(b)n(etwe)n(en)e Fo(D)447 645 y Fh(co )524 663 y Fi(and)e Fo(A)p Fi(.)228 751 y(Pr)n(o)n(of.)i Fq(\(1\))d(can)h(b)q(e)f(c)o(hec)o(k)o(ed)e(directly)l(.)278 809 y(\(2\))f(It)g(is)g(easy)g(to)g(see)g(that)g Fo(\016)j Fq(:)d Fo(D)j Fk(!)d Fo(A)5 b Fk(\012)g Fo(B)14 b Fq(is)f(a)h(map)e(of) h(righ)o(t)g(\()p Fo(B)s(;)8 b(B)s Fq(\)-Hopf)228 868 y(mo)q(dules.)19 b(Here)11 b Fo(D)j Fq(is)e(a)g(righ)o(t)g Fo(B)s Fq(-mo)q(dule)e(b)o(y)i(restriction)f(via)h Fo(\015)17 b Fq(:)c Fo(B)k Fk(!)c Fo(D)q Fq(,)h(and)228 926 y(a)e(righ)o(t)f Fo(B)s Fq(-como)q(dule)f(b)o(y)h(\(id)q Fk(\012)q Fo( )r Fq(\)\001;)h Fo(A)q Fk(\012)q Fo(B)j Fq(is)c(a)h(righ)o(t)f Fo(B)s Fq(-como)q(dule)g(b)o(y)g(id)q Fk(\012)q Fq(\001,)228 984 y(and)h(the)g(righ)o(t)f Fo(B)s Fq(-mo)q(dule)g(structure)g(is)h (giv)o(en)f(b)o(y)g(\()p Fo(a)r Fk(\012)r Fo(b)p Fq(\))r Fk(\001)r Fo(c)i Fq(:=)h Fo(a')p Fq(\()p Fo(\015)s Fq(\()p Fo(c)1561 991 y Fj(1)1580 984 y Fq(\)\))r Fk(\012)r Fo(bc)1703 991 y Fj(2)228 1042 y Fq(for)h(all)f Fo(a)f Fk(2)h Fo(A)h Fq(and)g Fo(b;)8 b(c)13 b Fk(2)h Fo(B)s Fq(.)21 b(Hence)13 b(b)o(y)h(the)h(theorem)e(on)i(Hopf)g(mo)q(dules)e([M,)228 1100 y(1.9],)j Fo(\016)j Fq(is)d(bijectiv)o(e)f(if)h(and)h(only)g(if)f Fo(D)953 1082 y Fh(coB)1032 1100 y Fq(=)e Fo(D)1125 1082 y Fh(co )1200 1100 y Fk(!)g Fq(\()p Fo(A)e Fk(\012)f Fo(B)s Fq(\))1441 1082 y Fh(coB)1518 1086 y Fk(\030)1518 1102 y Fq(=)1571 1100 y Fo(A;)d(x)14 b Fk(7!)228 1158 y Fo(')p Fq(\()p Fo(x)p Fq(\))p Fo(;)h Fq(is)h(bijectiv)o(e.)p 1687 1158 2 33 v 1689 1127 30 2 v 1689 1158 V 1718 1158 2 33 v 228 1246 a Ft(Theorem)g(4.3.)24 b Fi(L)n(et)13 b Fq(\()p Fo(A;)8 b(R)p Fq(\))k Fi(b)n(e)i(a)f(\014nite-dimensional)i (quasitriangular)f(Hopf)228 1304 y(algebr)n(a.)26 b(De\014ne)20 b Fo(F)603 1311 y Fh(R)647 1304 y Fq(:=)c(1)c Fk(\012)g Fo(R)839 1286 y Fj(2)871 1304 y Fk(\012)f Fo(R)958 1286 y Fj(1)990 1304 y Fk(\012)h Fq(1)18 b Fi(and)g Fo(\016)1200 1311 y Fh(R)1244 1304 y Fq(:)d Fo(D)q Fq(\()p Fo(A)p Fq(\))i Fk(!)e Fo(A)d Fk(\012)f Fo(A;)d(x)15 b Fk(7!)228 1362 y Fo(\016)250 1369 y Fh(R)278 1362 y Fq(\()p Fo(x)p Fq(\))26 b(:=)h Fg(e)-29 b Fo(\031)q Fq(\()p Fo(x)524 1369 y Fj(1)544 1362 y Fq(\))15 b Fk(\012)h Fo(\031)r Fq(\()p Fo(x)710 1369 y Fj(2)729 1362 y Fq(\))h Fi(with)h Fo(\031)28 b Fq(:=)e Fo(\031)1033 1369 y Fh(R)1079 1362 y Fi(and)18 b Fg(e)-29 b Fo(\031)28 b Fq(:=)d Fo(\031)1341 1369 y Fc(e)1334 1378 y Fh(R)1363 1362 y Fi(.)42 b(Then)25 b Fo(F)1586 1369 y Fh(R)1638 1362 y Fi(is)f(a)228 1420 y(2-c)n(o)n(cycle)18 b(for)f Fo(A)11 b Fk(\012)g Fo(A)17 b Fi(with)g(c)n(omp)n(onentwise)i(bialgebr)n(a)f(structur)n(e,)721 1500 y Fo(\016)743 1507 y Fh(R)786 1500 y Fq(:)13 b Fo(D)q Fq(\()p Fo(A)p Fq(\))h Fk(!)g Fq(\()p Fo(A)d Fk(\012)f Fo(A)p Fq(\))1179 1480 y Fh(F)1201 1486 y Fe(R)228 1580 y Fi(is)17 b(a)g(Hopf)h(algebr)n(a)g(homomorphism,)e(and)h(the)h(fol)r (lowing)i(ar)n(e)d(e)n(quivalent:)278 1638 y(\(1\))g Fo(\016)382 1645 y Fh(R)428 1638 y Fi(is)g(bije)n(ctive.)278 1696 y(\(2\))g Fq(\()p Fo(A;)8 b(R)p Fq(\))17 b Fi(is)g(factorizable.) 228 1784 y(Pr)n(o)n(of.)i Fq(Let)d(\()p Fo(e)506 1791 y Fh(i)520 1784 y Fq(\))g(and)h(\()p Fo(f)693 1791 y Fh(i)707 1784 y Fq(\))f(b)q(e)h(dual)f(bases)h(of)f Fo(A)g Fq(and)h Fo(A)1285 1766 y Fn(\003)1304 1784 y Fq(.)22 b(Apply)15 b(Lemma)f(4.2)228 1842 y(with)g Fo(D)h Fq(:=)f Fo(D)q Fq(\()p Fo(A)p Fq(\))g(and)h Fo(R)p Fq(-matrix)891 1805 y Fg(P)944 1857 y Fh(i)966 1842 y Fo(e)989 1849 y Fh(i)1010 1842 y Fk(\012)7 b Fo(f)1080 1849 y Fh(i)1093 1842 y Fq(,)14 b Fo(')g Fq(:=)g Fg(e)-29 b Fo(\031)r Fq(,)14 b Fo( )h Fq(:=)e Fo(\031)r Fq(,)h(and)h Fo(\015)i Fq(:)c Fo(A)g Fk(!)228 1901 y Fo(D)q Fq(\()p Fo(A)p Fq(\))j(the)g (inclusion)g(map.)k(The)d(2-co)q(cycle)e(of)i(Lemma)d(4.2)i(\(1\))h (then)f(is)306 1944 y Fg(X)336 2049 y Fh(i)386 1991 y Fq(1)c Fk(\012)e Fo(\031)r Fq(\()p Fo(S)s Fq(\()p Fo(e)595 1998 y Fh(i)608 1991 y Fq(\)\))h Fk(\012)h Fg(e)-29 b Fo(\031)r Fq(\()p Fo(f)780 1998 y Fh(i)794 1991 y Fq(\))11 b Fk(\012)g Fq(1)j(=)964 1944 y Fg(X)994 2049 y Fh(i)1044 1991 y Fq(1)d Fk(\012)g Fo(S)s Fq(\()p Fo(e)1204 1998 y Fh(i)1218 1991 y Fq(\))g Fk(\012)f Fo(f)1321 1998 y Fh(i)1336 1991 y Fq(\()p Fo(R)1392 1971 y Fj(2)1412 1991 y Fq(\))p Fo(S)s Fq(\()p Fo(R)1520 1971 y Fj(1)1540 1991 y Fq(\))h Fk(\012)g Fq(1)912 2125 y(=)j(1)d Fk(\012)g Fo(S)s Fq(\()1101 2078 y Fg(X)1131 2183 y Fh(i)1181 2125 y Fo(e)1204 2132 y Fh(i)1218 2125 y Fo(f)1242 2132 y Fh(i)1256 2125 y Fq(\()p Fo(R)1312 2104 y Fj(2)1332 2125 y Fq(\)\))g Fk(\012)g Fo(S)s Fq(\()p Fo(R)1520 2104 y Fj(1)1540 2125 y Fq(\))g Fk(\012)g Fq(1)912 2248 y(=)j(1)d Fk(\012)g Fo(S)s Fq(\()p Fo(R)1138 2228 y Fj(2)1158 2248 y Fq(\))g Fk(\012)g Fo(S)s Fq(\()p Fo(R)1327 2228 y Fj(1)1347 2248 y Fq(\))g Fk(\012)g Fq(1)912 2321 y(=)j Fo(F)996 2328 y Fh(R)1024 2321 y Fo(:)278 2404 y Fq(By)i(Lemma)f(4.2)i(\(2\))h (and)f(Lemma)e(4.1,)j(conditions)f(\(1\))g(and)h(\(2\))f(are)g(b)q(oth) 228 2462 y(equiv)m(alen)o(t)d(to)j(the)f(bijectivit)o(y)d(of)k(the)f (map)f Fo(D)q Fq(\()p Fo(A)p Fq(\))1197 2444 y Fh(co\031)1267 2462 y Fk(!)f Fo(A;)8 b(x)13 b Fk(7!)i Fg(e)-29 b Fo(\031)q Fq(\()p Fo(x)p Fq(\))p Fo(:)p 1687 2462 V 1689 2430 30 2 v 1689 2462 V 1718 2462 2 33 v 228 2550 a Ft(Remark)16 b(4.4.)24 b Fq(Radford)c([R94)q(,)41 b(Prop)q(osition)21 b(3])f(sho)o(w)o(ed)g(that)g(a)g(factoriz-)228 2608 y(able)e(Hopf)g (algebra)h(is)f(unimo)q(dular.)26 b(Another)18 b(pro)q(of)h(of)g(this)f (result)f(follo)o(ws)p eop %%Page: 10 10 10 9 bop 228 118 a Fl(10)544 b(H.-J.)15 b(SCHNEIDER)228 213 y Fq(from)f(the)h(isomorphism)e Fo(D)q Fq(\()p Fo(A)p Fq(\))839 199 y Fk(\030)839 215 y Fq(=)891 213 y(\()p Fo(A)c Fk(\012)f Fo(A)p Fq(\))1059 195 y Fh(F)1081 201 y Fe(R)1124 213 y Fq(in)14 b(Theorem)g(4.3)i(and)f(the)g(uni-)228 271 y(mo)q(dularit)o(y)g(of)i(the)g(double)f([R93)q(,)g(Theorem4].)22 b(Let)17 b(\003)f(b)q(e)h(a)h(left)d(in)o(tegral)i(in)228 329 y Fo(A)p Fq(.)i(Then)12 b(\003)r Fk(\012)r Fq(\003)f(is)h(a)g(left) f(in)o(tegral)g(in)h(\()p Fo(A)r Fk(\012)r Fo(A)p Fq(\))1086 311 y Fh(F)1108 317 y Fe(R)1134 329 y Fq(.)20 b(Since)11 b Fo(D)q Fq(\()p Fo(A)p Fq(\))h(is)f(unimo)q(dular,)228 387 y(\003)g Fk(\012)g Fq(\003)16 b(is)g(a)g(righ)o(t)g(in)o(tegral.)21 b(Hence)15 b(\003)h(is)g(a)g(righ)o(t)g(in)o(tegral)g(in)g Fo(A)p Fq(.)830 486 y Fp(References)228 565 y Fs([D])29 b(V.)15 b(G.)g(Drinfeld,)g Fm(On)h(almost)h(c)n(o)n(c)n(ommutative)f (Hopf)h(algebr)n(as)p Fs(,)e(Leningrad)g(Math.)g(J.)313 614 y Fb(1)f Fs(\(1990\),)f(321{342.)228 664 y([EG])19 b(P)m(.)e(Etingof)g(and)h(S.)f(Gelaki,)g Fm(Some)h(pr)n(op)n(erties)g (of)g(\014nite-dimensional)h(semisimple)313 714 y(Hopf)c(algebr)n(as)p Fs(,)e(Mathematical)f(Researc)o(h)j(Letters)h Fb(5)d Fs(\(1998\),)g(191{197.)228 764 y([K])29 b(C.)13 b(Kassel,)i Fm(Quantum)g(Gr)n(oups)p Fs(,)f(Springer,)f(GTM)h(155,)f(New)h(Y)m(ork) f(1995.)228 814 y([Ki])19 b(A.)13 b(A.)f(Kirillo)o(v)f(Jr.,)i Fm(On)h(an)h(inner)e(pr)n(o)n(duct)h(in)g(mo)n(dular)g(c)n(ate)n (gories)p Fs(,)e(J.)h(Amer.)f(Math.)313 863 y(So)q(c.)i Fb(9)g Fs(\(1996\),)e(1135{1169.)228 913 y([LR])19 b(R.)12 b(G.)g(Larson)g(and)h(D.)f(E.)g(Radford,)g Fm(Finite-dimensional)h(c)n (osemisimple)h(Hopf)f(alge-)313 963 y(br)n(as)i(in)g(char)n(acteristic) e(0)i(ar)n(e)g(semisimple)p Fs(,)e(J.)g(Alg.)g Fb(117)h Fs(\(1988\),)e(267{289.)228 1013 y([L])35 b(M.)16 b(Lorenz,)g Fm(On)h(the)g(class)f(e)n(quation)i(for)e(Hopf)g(algebr)n(as)p Fs(,)g(Pro)q(c.)g(Amer.)f(Math.)g(So)q(c.)313 1063 y Fb(126)f Fs(\(1998\),)e(2841{2844.)228 1113 y([M])23 b(S.)10 b(Mon)o(tgomery)m(,)f Fm(Hopf)j(algebr)n(as)f(and)i(their)e (actions)h(on)g(rings)p Fs(,)e(CBMS)h(Lecture)h(Notes)313 1162 y(82,)h(Amer.)g(Math.)g(So)q(c.,)h(1993.)228 1213 y([OS])20 b(U.)12 b(Ob)q(erst,)j(H.-J.)d(Sc)o(hneider,)839 1202 y Fm(\177)832 1213 y(Ub)n(er)h(Unter)n(grupp)n(en)h(end)r(licher)g (algebr)n(aischer)f(Grup-)313 1263 y(p)n(en)p Fs(,)h(man)o(uscripta)f (math.)f Fb(8)h Fs(\(1973\),)g(217{241.)228 1312 y([R93])19 b(D.)8 b(E.)i(Radford,)e Fm(Minimal)j(quasitriangular)f(Hopf)h(algebr)n (as)p Fs(,)f(J.)f(Algebra)g Fb(157)g Fs(\(1993\),)313 1362 y(285{315.)228 1412 y([R94])19 b(D.)32 b(E.)i(Radford,)j Fm(On)c(Kau\013man)l('s)h(knot)f(invariants)g(arising)g(fr)n(om)f (\014nite-)313 1462 y(dimensional)18 b(Hopf)g(algebr)n(as)p Fs(,)e(in:)24 b Fm(A)n(dvanc)n(es)18 b(in)g(Hopf)f(A)o(lgebr)n(as)p Fs(,)f(Lecture)j(Notes)f(in)313 1512 y(Pure)d(and)f(Applied)f (mathematics,)f(Dekk)o(er,)h(New)i(Y)m(ork,)e(1994,)f(205{266.)228 1562 y([RS])19 b(N.)14 b(Y.)f(Reshetikhin,)g(M.)g(A.)h(Semeno)o (v{Tian{Shansky)m(,)d Fm(Quantum)k(R{matric)n(es)f(and)313 1611 y(factorization)h(pr)n(oblems)p Fs(,)d(Journal)i(Geometry)f(Ph)o (ysics)i Fb(5)e Fs(\(1988\),)g(533{550.)228 1661 y([TZ])20 b(Y.)13 b(Tsang,)h(Y.)f(Zh)o(u,)h Fm(On)h(the)g(Drinfeld)f(double)i(of) f(a)g(Hopf)f(algebr)n(a)p Fs(,)g(preprin)o(t,)g(1998.)228 1711 y([T])31 b(V.)14 b(T)m(uraev,)f Fm(Quantum)j(invariants)e(of)h (knots)h(and)f(3-manifolds)p Fs(,)f(de)g(Gruyter)h(Studies)313 1761 y(in)f(Mathematics)f(18,)g(W)m(alter)g(de)h(Gruyter)h(&)f(Co.,)e (Berlin,)i(1994.)278 1854 y Fr(Ma)m(thema)m(tisches)24 b(Institut,)h(Universit)1055 1851 y(\177)1054 1854 y(at)f(M)1170 1851 y(\177)1169 1854 y(unchen,)j(Theresienstra\031e)228 1904 y(39,)16 b(D-80333)f(M)525 1901 y(\177)524 1904 y(unchen,)j(Germany)278 1954 y Fm(E-mail)c(addr)n(ess)s Fs(:)k Fa(hanssch@rz.mathemati)o(k.un)o(i-mue)o(nchen)o(.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF