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(primitiv)o(e)228 1271 y Fs(n)p Fu(-th)e(ro)q(ot)h(of)f(unit)o(y)l(.)20 b(They)14 b(satisfy)h(generalizations)f(of)h(the)g(\(an)o(ti-\)symmet)o (ry)228 1329 y(and)i(Jacabi)f(iden)o(tities.)278 1387 y(Our)h(main)f(aim)g(is)h(to)g(sho)o(w)h(that)g(these)f(Lie)g(algebras) h(ha)o(v)o(e)e(univ)o(ersal)g(en-)228 1445 y(v)o(eloping)i(algebras)i (whic)o(h)e(turn)h(out)h(to)f(b)q(e)g(Hopf)g(algebras)h(in)e Fr(Y)t(D)1540 1427 y Fq(K)1539 1458 y(K)1576 1445 y Fu(.)29 b(Con-)228 1504 y(v)o(ersely)14 b(the)i(set)f(of)i(primitiv)n(e)c (elemen)o(ts)g(of)j(a)h(Hopf)e(algebra)i(in)e Fr(Y)t(D)1528 1486 y Fq(K)1527 1516 y(K)1580 1504 y Fu(is)g(suc)o(h)228 1562 y(a)f(generalized)e(Lie)h(algebra.)21 b(W)l(e)13 b(also)h(giv)o(e)f(an)h(example)d(that)j(generalizes)e(the)228 1620 y(concept)k(of)g(orthogonal)i(or)f(symplectic)c(Lie)j(algebras.) 664 1740 y(2.)24 b Ft(Braid)19 b(Symmetriza)m(tion)278 1827 y Fu(W)l(e)g(b)q(egin)g(with)g(t)o(w)o(o)g(simple)e(mo)q(dule)h (theoretic)g(observ)m(ations.)31 b(The)19 b(fol-)228 1885 y(lo)o(wing)c(is)g(w)o(ell)f(kno)o(wn:)21 b(if)15 b Fs(A;)8 b(B)18 b Fu(are)d(algebras)h(and)h Fs(M)j Fu(is)c(an)g Fs(A)p Fu(-)p Fs(B)s Fu(-bimo)q(dule,)228 1943 y(then)21 b(Hom)446 1950 y Fq(A)474 1943 y Fu(\()p Fs(:P)q(;)8 b(:M)d Fu(\))21 b(is)g(a)h(righ)o(t)f Fs(B)s Fu(-mo)q(dule)f(for)h(ev)o (ery)f Fs(A)p Fu(-mo)q(dule)g Fs(P)7 b Fu(.)36 b(W)l(e)228 2001 y(need)16 b(a)g(como)q(dule)f(analogue)j(of)e(this.)278 2059 y(Let)26 b Fs(A)f Fu(b)q(e)h(an)h(algebra,)h Fs(C)h Fu(b)q(e)d(a)h(coalgebra,)h(and)1325 2066 y Fq(A)1353 2059 y Fs(M)1405 2041 y Fq(C)1461 2059 y Fu(b)q(e)e(an)h Fs(A)p Fu(-)p Fs(C)t Fu(-)228 2117 y(dimo)q(dule,)e(i.e.)46 b(a)26 b(left)e Fs(A)p Fu(-mo)q(dule)g(and)h(a)h(righ)o(t)e Fs(C)t Fu(-como)q(dule)g(suc)o(h)h(that)228 2175 y Fs(\016)r Fu(\()p Fs(am)p Fu(\))12 b(=)i(\()p Fs(a)d Fr(\012)g Fu(1\))p Fs(\016)r Fu(\()p Fs(m)p Fu(\).)228 2247 y FB(Prop)r(osition) 21 b(2.1.)g Fo(L)n(et)f Fs(P)27 b Fo(b)n(e)21 b(a)f(\014nitely)h(gener) n(ate)n(d)g(left)g Fs(A)p Fo(-mo)n(dule.)31 b(Then)228 2305 y Fu(Hom)329 2312 y Fq(A)358 2305 y Fu(\()p Fs(:P)q(;)8 b(:M)d Fu(\))13 b Fo(is)h(a)f(right)g Fs(C)t Fo(-c)n(omo)n(dule)g(with) h(the)g(c)n(anonic)n(al)g(c)n(omo)n(dule)g(struc-)228 2364 y(tur)n(e)j(such)h(that)228 2401 y Fm(\020)253 2449 y Fu(Hom)354 2456 y Fq(A)383 2449 y Fu(\()p Fs(P)q(;)8 b(M)d Fu(\))573 2423 y Fq(\016)542 2449 y Fr(\000)-9 b(!)14 b Fu(Hom)737 2456 y Fq(A)766 2449 y Fu(\()p Fs(P)q(;)8 b(M)d Fu(\))p Fr(\012)p Fs(C)18 b Fr(\000)-29 b(!)13 b Fu(Hom)1177 2456 y Fq(A)1205 2449 y Fu(\()p Fs(P)q(;)8 b(M)d Fr(\012)p Fs(C)t Fu(\))1427 2401 y Fm(\021)1466 2449 y Fu(=)14 b(Hom)1620 2456 y Fq(A)1648 2449 y Fu(\()p Fs(P)q(;)8 b(\016)r Fu(\))p Fs(:)228 2550 y Fo(Pr)n(o)n(of.)19 b Fu(Let)11 b Fs(p)483 2557 y Fl(1)503 2550 y Fs(;)d(:)g(:)g(:)16 b(;)8 b(p)645 2557 y Fq(n)679 2550 y Fu(b)q(e)j(a)h(generating)f(set)f (of)h Fs(P)19 b Fu(and)11 b(let)f Fs(f)19 b Fr(2)14 b Fu(Hom)1525 2557 y Fq(A)1553 2550 y Fu(\()p Fs(:P)q(;)8 b(:M)d Fu(\).)228 2608 y(Let)17 b Fs(m)359 2615 y Fq(i)387 2608 y Fu(:=)d Fs(f)5 b Fu(\()p Fs(p)525 2615 y Fq(i)540 2608 y Fu(\).)22 b(Then)17 b(b)o(y)f(the)h(structure)f(theorem)f(on)i (como)q(dules)f(the)g Fs(m)1708 2615 y Fq(i)p eop %%Page: 3 3 3 2 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)r(3)228 225 y Fu(are)i(con)o(tained)f(in)h(a)h(\014nite)e(dimensional)f(sub)q(como)q (dule)i Fs(M)1380 232 y Fl(0)1418 225 y Fr(\022)g Fs(M)24 b Fu(whic)o(h)18 b(is)228 283 y(ev)o(en)g(a)i(como)q(dule)f(o)o(v)o(er) g(a)h(\014nite)f(dimensional)f(sub)q(coalgebra)j Fs(C)1490 290 y Fl(0)1529 283 y Fr(\022)e Fs(C)t Fu(,)h(i.e.)228 341 y(the)c(diagram)791 419 y Fs(M)838 426 y Fl(0)977 419 y Fs(M)1024 426 y Fl(0)1055 419 y Fr(\012)10 b Fs(C)1139 426 y Fl(0)p 872 406 91 2 v 921 405 a Fk(-)905 396 y Fs(\016)798 615 y(M)147 b(M)16 b Fr(\012)11 b Fs(C)p 865 601 114 2 v 937 600 a Fk(-)909 591 y Fs(\016)p 823 566 2 127 v 824 566 a Fk(?)p 1067 566 V 202 w(?)228 675 y Fu(comm)o(utes.)32 b(F)l(urthermore)19 b Fs(M)824 682 y Fl(1)866 675 y Fu(:=)i Fs(AM)1023 682 y Fl(0)1063 675 y Fu(is)g(a)g Fs(C)1197 682 y Fl(0)1217 675 y Fu(-como)q(dule)f(con)o (tained)h(in)228 733 y Fs(M)5 b Fu(,)17 b(since)e Fs(M)22 b Fu(is)17 b(a)g(dimo)q(dule,)e(and)i Fs(f)j Fu(:)14 b Fs(P)22 b Fr(\000)-31 b(!)15 b Fs(M)22 b Fu(ob)o(viously)16 b(factors)h(through)228 791 y Fs(M)275 798 y Fl(1)295 791 y Fu(.)k(Since)15 b Fs(M)22 b Fu(and)17 b Fs(M)668 798 y Fl(1)704 791 y Fu(are)f(dimo)q(dules)f(the)h(diagram)251 896 y(Hom)353 903 y Fq(A)382 896 y Fu(\()p Fs(:P)q(;)8 b(:M)530 903 y Fl(1)549 896 y Fu(\))162 b(Hom)832 903 y Fq(A)860 896 y Fu(\()p Fs(:P)q(;)8 b(:M)1008 903 y Fl(1)1039 896 y Fr(\012)j Fs(C)1124 903 y Fl(0)1143 896 y Fu(\))p 582 884 134 2 v 674 883 a Fk(-)628 866 y Fs(\016)650 873 y Fj(\003)259 1091 y Fu(Hom)360 1098 y Fq(A)389 1091 y Fu(\()p Fs(:P)q(;)d(:M)d Fu(\))184 b(Hom)847 1098 y Fq(A)875 1091 y Fu(\()p Fs(:P)q(;)8 b(:M)17 b Fr(\012)10 b Fs(C)t Fu(\))p 575 1079 156 2 v 689 1078 a Fk(-)632 1061 y Fs(\016)654 1068 y Fj(\003)1266 896 y Fu(Hom)1368 903 y Fq(A)1396 896 y Fu(\()p Fs(:P)q(;)e(:M)1544 903 y Fl(1)1564 896 y Fu(\))j Fr(\012)g Fs(C)1679 903 y Fl(0)p 1177 884 76 2 v 1177 883 a Fk(\033)1195 858 y Fr(\030)1195 873 y Fu(=)1281 1091 y(Hom)1383 1098 y Fq(A)1412 1091 y Fu(\()p Fs(:P)q(;)d(:M)d Fu(\))11 b Fr(\012)g Fs(C)p 1161 1079 107 2 v 1161 1078 a Fk(\033)p 409 1044 2 127 v 410 1044 a(?)p 945 1044 V 494 w(?)p 1481 1044 V 494 w(?)228 1171 y Fu(comm)o(utes,)d(so)k(eac)o(h)e Fs(f)16 b Fu(has)c(a)f(uniquely)e(de\014ned)i(image)e Fs(\016)1294 1178 y Fj(\003)1314 1171 y Fu(\()p Fs(f)c Fu(\))14 b Fr(2)g Fu(Hom)1544 1178 y Fq(A)1572 1171 y Fu(\()p Fs(:P)q(;)8 b(:M)d Fu(\))p Fr(\012)228 1229 y Fs(C)t Fu(.)20 b(No)o(w)14 b(it)g(is)h(easy)f(to)h(c)o(hec)o(k)e(that)i(this)f(map)g(induces)g(a)h (como)q(dule)f(structure)228 1287 y(on)j(Hom)397 1294 y Fq(A)426 1287 y Fu(\()p Fs(P)q(;)8 b(M)d Fu(\).)1099 b Fi(\003)278 1378 y Fu(The)24 b(second)g(observ)m(ation)h(is)f(the)g (follo)o(wing.)44 b(W)l(e)24 b(consider)g Fs(k)r Fu(-algebras)228 1436 y Fs(A)f Fu(and)i Fs(B)s Fu(.)44 b(Let)24 b Fs(\013)j Fu(:)g Fs(B)i Fr(\000)-30 b(!)27 b Fs(A)c Fu(b)q(e)i(an)f(algebra)g (homomorphism.)42 b Fs(\013)24 b Fu(in-)228 1494 y(duces)16 b(an)h(underlying)f(functor)g Fs(V)869 1501 y Fq(\013)909 1494 y Fu(:)e Fs(A)p Fu(-)o(Mo)q(d)h Fr(\000)-30 b(!)14 b Fs(B)s Fu(-Mo)q(d)j(with)f(righ)o(t)g(adjoin)o(t)228 1552 y(Hom)329 1559 y Fq(B)360 1552 y Fu(\()p Fs(A;)8 b Fu(-)o(\))24 b(:)g Fs(B)s Fu(-)o(Mo)q(d)h Fr(\000)-30 b(!)23 b Fs(A)p Fu(-Mo)q(d)q(.)38 b(If)22 b Fs(\013)i Fu(:)g Fs(B)i Fr(\000)-30 b(!)23 b Fs(A)f Fu(is)g(surjectiv)o(e)e(then) 228 1611 y(Hom)329 1618 y Fq(B)360 1611 y Fu(\()p Fs(A;)8 b(M)d Fu(\))16 b Fr(\000)-31 b(!)17 b Fu(Hom)701 1618 y Fq(B)731 1611 y Fu(\()p Fs(B)s(;)8 b(M)d Fu(\))899 1597 y Fr(\030)900 1613 y Fu(=)954 1611 y Fs(M)24 b Fu(is)17 b(injectiv)o(e,)e(so)j(that)g(w)o(e)g(can)g(iden-)228 1669 y(tify)d(Hom)419 1676 y Fq(B)449 1669 y Fu(\()p Fs(A;)8 b(M)d Fu(\))14 b(=)f Fr(f)p Fs(m)h Fr(2)g Fs(M)5 b Fr(j)p Fu(Ker\()p Fs(\013)p Fu(\))p Fs(m)14 b Fu(=)f(0)p Fr(g)p Fu(.)278 1727 y(Let)i Fs(B)401 1734 y Fq(n)439 1727 y Fu(b)q(e)g(the)g(Artin)f(braid)h(group)h(with)f(generators)h Fs(\034)1346 1734 y Fq(i)1360 1727 y Fu(,)f Fs(i)e Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)h Fr(\000)f Fu(1)228 1785 y(and)17 b(relations)662 1856 y Fs(\034)683 1863 y Fq(i)697 1856 y Fs(\034)718 1863 y Fq(j)768 1856 y Fu(=)d Fs(\034)841 1863 y Fq(j)860 1856 y Fs(\034)881 1863 y Fq(i)1087 1856 y Fu(if)i Fr(j)p Fs(i)11 b Fr(\000)f Fs(j)s Fr(j)k(\025)g Fu(2;)585 1914 y Fs(\034)606 1921 y Fq(i)621 1914 y Fs(\034)642 1921 y Fq(i)p Fl(+1)701 1914 y Fs(\034)722 1921 y Fq(i)768 1914 y Fu(=)g Fs(\034)841 1921 y Fq(i)p Fl(+1)900 1914 y Fs(\034)921 1921 y Fq(i)936 1914 y Fs(\034)957 1921 y Fq(i)p Fl(+1)1016 1914 y Fs(:)1660 1886 y Fu(\(1\))228 2000 y(Let)e Fs(\020)17 b Fr(2)e Fs(k)e Fu(b)q(e)f(in)o(v)o(ertible.)17 b(Then)12 b Fs(k)r(B)917 2007 y Fq(n)954 2000 y Fr(3)i Fs(\034)1022 2007 y Fq(i)1050 2000 y Fr(7!)g Fs(\020)t(\034)1160 2007 y Fq(i)1188 2000 y Fr(2)g Fs(k)r(B)1299 2007 y Fq(n)1334 2000 y Fu(\(for)e(the)f (generators)228 2058 y Fs(\034)249 2065 y Fq(i)275 2058 y Fu(of)h Fs(B)363 2065 y Fq(n)387 2058 y Fu(\))g(is)g(an)g(algebra)h (automorphism)e(denoted)h(again)g(b)o(y)g Fs(\020)18 b Fu(:)13 b Fs(k)r(B)1511 2065 y Fq(n)1549 2058 y Fr(\000)-31 b(!)14 b Fs(k)r(B)1685 2065 y Fq(n)1709 2058 y Fu(.)228 2116 y(This)i(holds)h(true)f(since)f(the)h(relations)g(for)h Fs(B)1083 2123 y Fq(n)1123 2116 y Fu(are)f(homogeneous.)278 2174 y(\(Observ)o(e)k(that)h(this)g(construction)h(can)f(b)q(e)g(p)q (erformed)f(for)i(ev)o(ery)d(group)228 2232 y(algebra)c(if)g(the)g (group)h(is)e(giv)o(en)h(b)o(y)f(generators)i(and)g(homogeneous)f (relations.)228 2291 y(The)g(giv)o(en)f(construction)h(of)g(an)h (automorphism)d(for)j(ev)o(ery)d Fs(\020)18 b Fr(2)c Fs(U)5 b Fu(\()p Fs(k)r Fu(\))15 b(de\014nes)228 2349 y(a)h(group)i(homomorphism)13 b Fs(U)5 b Fu(\()p Fs(k)r Fu(\))14 b Fr(\000)-30 b(!)14 b Fu(Aut)o(\()p Fs(k)r(B)1106 2356 y Fq(n)1130 2349 y Fu(\))g Fr(\000)-31 b(!)14 b Fu(Aut\()p Fs(k)r(B)1401 2356 y Fq(n)1424 2349 y Fu(-Mo)q(d)q(\).\))278 2407 y(No)o(w)h(consider)g(the)g(canonical)g(quotien)o(t)f (homomorphism)f Fs(B)1445 2414 y Fq(n)1482 2407 y Fr(\000)-30 b(!)14 b Fs(S)1585 2414 y Fq(n)1623 2407 y Fu(from)228 2465 y(the)23 b(braid)h(group)g(on)o(to)g(the)f(symmetric)d(group.)44 b(It)23 b(induces)g(a)h(surjectiv)o(e)228 2523 y(homomorphism)13 b Fs(\015)k Fu(:)c Fs(k)r(B)705 2530 y Fq(n)743 2523 y Fr(\000)-30 b(!)13 b Fs(k)r(S)872 2530 y Fq(n)912 2523 y Fu(with)j(k)o(ernel)435 2608 y(Ker)o(\()p Fs(\015)s Fu(\))e(=)g Fr(h)p Fs( )r Fu(\()p Fs(\034)744 2587 y Fl(2)738 2620 y Fq(i)774 2608 y Fr(\000)d Fu(1\))p Fs(')p Fr(j)p Fs(';)d( )15 b Fr(2)f Fs(B)1098 2615 y Fq(n)1122 2608 y Fs(;)8 b(i)13 b Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)g Fu(1)p Fr(i)p Fs(:)p eop %%Page: 4 4 4 3 bop 228 125 a Fp(4)571 b(BODO)13 b(P)m(AREIGIS)278 240 y Fu(The)d(comp)q(osition)h Fs(\013)j Fu(:)f Fs(k)r(B)776 247 y Fq(n)845 211 y(\020)814 240 y Fr(\000)-9 b(!)14 b Fs(k)r(B)972 247 y Fq(n)1039 211 y(\015)1009 240 y Fr(\000)-8 b(!)13 b Fs(k)r(S)1160 247 y Fq(n)1195 240 y Fu(de\014nes)d(a)i(functor)e Fs(k)r(B)1612 247 y Fq(n)1636 240 y Fu(-Mo)q(d)228 298 y Fr(\000)-30 b(!)13 b Fs(k)r(S)357 305 y Fq(n)381 298 y Fu(-Mo)q(d)k(b)o(y)249 426 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b(:)41 b(=)14 b(Hom)586 433 y Fq(k)q(B)632 437 y Fh(n)655 426 y Fu(\()674 433 y Fq(\020)694 426 y Fs(k)r(S)751 433 y Fq(n)775 426 y Fs(;)8 b(M)d Fu(\))433 484 y(=)14 b Fr(f)p Fs(m)f Fr(2)h Fs(M)5 b Fr(j)p Fs(')711 466 y Fj(\000)p Fl(1)758 484 y Fs(\034)785 466 y Fl(2)779 497 y Fq(i)805 484 y Fs(')p Fu(\()p Fs(m)p Fu(\))13 b(=)h Fs(\020)1008 466 y Fl(2)1028 484 y Fs(m)48 b Fr(8)p Fs(')13 b Fr(2)h Fs(B)1277 491 y Fq(n)1300 484 y Fs(;)24 b(i)14 b Fu(=)g(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)g Fu(1)p Fr(g)p Fs(:)433 543 y Fu(=)j Fr(f)p Fs(m)f Fr(2)h Fs(M)5 b Fr(j)p Fs(\034)706 525 y Fl(2)700 555 y Fq(i)726 543 y Fs(')p Fu(\()p Fs(m)p Fu(\))13 b(=)h Fs(\020)929 525 y Fl(2)949 543 y Fs(')p Fu(\()p Fs(m)p Fu(\))48 b Fr(8)p Fs(')12 b Fr(2)j Fs(B)1268 550 y Fq(n)1291 543 y Fs(;)24 b(i)14 b Fu(=)f(1)p Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(n)j Fr(\000)f Fu(1)p Fr(g)p Fs(:)-32 b Fu(\(2\))228 627 y(This)24 b(holds)h(since)f(the)g(map)f Fs(\015)31 b Fu(:)c Fs(k)r(B)980 634 y Fq(n)1032 627 y Fr(\000)-31 b(!)28 b Fs(k)r(S)1175 634 y Fq(n)1223 627 y Fu(has)d(as)g(k)o(ernel)e(the)h(t)o(w)o(o-)228 685 y(sided)18 b(ideal)f(generated)h(as)h(a)f Fs(k)r Fu(-subspace)h(b)o(y)f Fr(f)p Fs( )r Fu(\()p Fs(\034)1225 667 y Fl(2)1219 697 y Fq(i)1256 685 y Fr(\000)12 b Fu(1\))p Fs(')18 b Fr(j)p Fs( )r(;)8 b(')16 b Fr(2)h Fs(B)1605 692 y Fq(n)1629 685 y Fs(;)8 b(i)16 b Fu(=)228 743 y(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)15 b Fr(\000)h Fu(1)p Fr(g)p Fu(.)41 b(So)23 b Fs(f)31 b Fr(2)25 b Fu(Hom)862 750 y Fq(k)q(B)908 754 y Fh(n)931 743 y Fu(\()950 750 y Fq(\020)970 743 y Fs(k)r(S)1027 750 y Fq(n)1050 743 y Fs(;)8 b(M)d Fu(\))23 b(with)g Fs(f)5 b Fu(\(1\))26 b(=)f Fs(m)f Fr(2)i Fs(M)5 b Fu(,)24 b(i\013)228 801 y Fs(\020)253 783 y Fj(\000)p Fl(1)300 801 y Fu(\()p Fs( )r Fu(\()p Fs(\034)399 783 y Fl(2)393 814 y Fq(i)428 801 y Fr(\000)11 b Fu(1\))p Fs(')p Fu(\))p Fs(m)i Fu(=)h(0)i(for)h(all) e Fs( )r(;)8 b(';)g(i)p Fu(,)14 b(i\013)i Fs(\020)1100 783 y Fj(\000)p Fl(1)1148 801 y Fu(\()p Fs(\034)1194 783 y Fl(2)1188 814 y Fq(i)1223 801 y Fr(\000)11 b Fu(1\))p Fs('m)i Fu(=)h(0)j(for)f(all)f Fs(';)8 b(i)p Fu(,)228 859 y(i\013)16 b Fs(\034)313 841 y Fl(2)307 872 y Fq(i)333 859 y Fs('m)d Fu(=)h Fs(\020)498 841 y Fl(2)518 859 y Fs('m)h Fu(for)i(all)f Fs(';)8 b(i)p Fu(,)15 b(i\013)h Fs(')941 841 y Fj(\000)p Fl(1)988 859 y Fs(\034)1015 841 y Fl(2)1009 872 y Fq(i)1035 859 y Fs(')p Fu(\()p Fs(m)p Fu(\))d(=)h Fs(\020)1238 841 y Fl(2)1258 859 y Fs(m)i Fu(for)g(all)g Fs(';)8 b(i)p Fu(.)278 917 y(If)j(the)h(action)h (of)f Fs(B)633 924 y Fq(n)669 917 y Fu(on)h Fs(M)18 b Fu(is)12 b(giv)o(en)f(b)o(y)h(an)h(action)f(of)h Fs(S)1318 924 y Fq(n)1353 917 y Fu(and)g(the)f(canonical)228 976 y(epimorphism)17 b Fs(B)559 983 y Fq(n)604 976 y Fr(\000)-30 b(!)21 b Fs(S)714 983 y Fq(n)737 976 y Fu(,)h(then)e(the)g (construction)h(of)g(the)f Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))21 b(b)q(ecomes)228 1034 y(trivial,)h(since)f Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))25 b(=)f Fr(f)p Fs(m)f Fr(2)h Fs(M)5 b Fr(j)p Fs(\034)962 1016 y Fl(2)956 1046 y Fq(i)982 1034 y Fs(')p Fu(\()p Fs(m)p Fu(\))24 b(=)f Fs(\020)1205 1016 y Fl(2)1225 1034 y Fs(')p Fu(\()p Fs(m)p Fu(\))h(=)g Fs(')p Fu(\()p Fs(m)p Fu(\))p Fr(g)f Fu(=)h(0)f(if)228 1092 y Fs(\020)253 1074 y Fl(2)295 1092 y Fr(6)p Fu(=)f(1)g(and)g Fs(M)5 b Fu(\()p Fr(\000)p Fu(1\))23 b(=)f Fs(M)5 b Fu(\(1\))23 b(=)g Fs(M)5 b Fu(.)37 b(Observ)o(e)20 b(that)i(the)f(mo)q(dule)f Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))228 1150 y(dep)q(ends)15 b(only)g(on)h Fs(\020)613 1132 y Fl(2)633 1150 y Fu(,)e(but)i(that)f (the)g(action)g(of)h Fs(k)r(S)1194 1157 y Fq(n)1232 1150 y Fu(on)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))16 b(dep)q(ends)f(on)h Fs(\020)t Fu(.)278 1208 y Fs(M)5 b Fu(\(1\))17 b(giv)o(es)e(a)i (solution)f(of)h(the)f(follo)o(wing)g(univ)o(ersal)f(problem.)228 1279 y FB(Prop)r(osition)j(2.2.)h Fo(F)l(or)e(every)h Fs(k)r(B)922 1286 y Fq(n)946 1279 y Fo(-mo)n(dule)g Fs(M)23 b Fo(the)18 b(subsp)n(ac)n(e)291 1362 y Fs(M)5 b Fu(\(1\))14 b(:=)g Fr(f)p Fs(m)f Fr(2)h Fs(M)5 b Fr(j)p Fs(')711 1342 y Fj(\000)p Fl(1)758 1362 y Fs(\034)785 1342 y Fl(2)779 1374 y Fq(i)805 1362 y Fs(')p Fu(\()p Fs(m)p Fu(\))13 b(=)h Fs(m)49 b Fr(8)p Fs(')13 b Fr(2)h Fs(B)1233 1369 y Fq(n)1257 1362 y Fs(;)25 b(i)13 b Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)g Fu(1)p Fr(g)228 1445 y Fo(is)21 b(a)f Fs(k)r(S)386 1452 y Fq(n)410 1445 y Fo(-mo)n(dule)i(and)f(the)g(inclusion)i Fs(M)5 b Fu(\(1\))21 b Fr(\000)-29 b(!)20 b Fs(M)26 b Fo(is)21 b(a)g Fs(k)r(B)1445 1452 y Fq(n)1468 1445 y Fo(-mo)n(dule)h(ho-)228 1504 y(momorphism,)14 b(such)h(that)g(for)f(every)h Fs(k)r(S)996 1511 y Fq(n)1020 1504 y Fo(-mo)n(dule)g Fs(T)22 b Fo(and)15 b(every)g Fs(k)r(B)1533 1511 y Fq(n)1557 1504 y Fo(-mo)n(dule)228 1562 y(homomorphism)j Fs(f)24 b Fu(:)18 b Fs(T)25 b Fr(\000)-29 b(!)18 b Fs(M)25 b Fo(ther)n(e)20 b(is)g(a)f(unique)j Fs(k)r(S)1296 1569 y Fq(n)1319 1562 y Fo(-mo)n(dule)f(homomor-)228 1620 y(phism)c Fs(g)f Fu(:)d Fs(T)20 b Fr(\000)-29 b(!)14 b Fs(M)5 b Fu(\(1\))18 b Fo(such)g(that)g(the)g(diagr)n(am)738 1886 y Fs(M)5 b Fu(\(1\))307 b Fs(M)p 867 1874 278 2 v 1103 1873 a Fk(-)998 1864 y Fs(\023)778 1695 y(T)p 795 1839 2 127 v 796 1839 a Fk(?)751 1782 y Fs(g)1022 1788 y(f)864 1733 y Fk(H)905 1754 y(H)947 1775 y(H)988 1795 y(H)1030 1816 y(H)1071 1837 y(H)1076 1839 y(H)-42 b(j)228 1958 y Fo(c)n(ommutes.)228 2029 y FB(De\014nition)18 b(2.3.)j Fu(W)l(e)c(call)f(the)h(functor)g(-\()p Fs(\020)t Fu(\))f(:)e Fs(k)r(B)1216 2036 y Fq(n)1240 2029 y Fu(-Mo)q(d)i Fr(\000)-30 b(!)15 b Fs(k)r(S)1500 2036 y Fq(n)1524 2029 y Fu(-Mo)q(d)i(the)228 2087 y Fs(\020)t Fu(-)p Fo(symmetrization)f Fu(of)h Fs(k)r(B)731 2094 y Fq(n)754 2087 y Fu(-mo)q(dules.)278 2158 y(The)f(de\014nition)g(giv)o(es)269 2241 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b(=)g Fr(f)p Fs(m)f Fr(2)i Fs(M)5 b Fr(j)p Fs(')677 2220 y Fj(\000)p Fl(1)724 2241 y Fs(\034)751 2220 y Fl(2)745 2253 y Fq(i)770 2241 y Fs(')p Fu(\()p Fs(m)p Fu(\))14 b(=)g Fs(\020)974 2220 y Fl(2)993 2241 y Fs(m)49 b Fr(8)p Fs(')12 b Fr(2)i Fs(B)1242 2248 y Fq(n)1266 2241 y Fs(;)24 b(i)13 b Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)g Fu(1)p Fr(g)p Fs(:)228 2324 y Fu(The)16 b(action)g(of)h Fs(S)560 2331 y Fq(n)600 2324 y Fu(on)f Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))17 b(is)f(giv)o(en)f(b) o(y)776 2408 y Fs(\033)804 2415 y Fq(i)818 2408 y Fu(\()p Fs(m)p Fu(\))e(=)h Fs(\020)989 2388 y Fj(\000)p Fl(1)1036 2408 y Fs(\034)1057 2415 y Fq(i)1072 2408 y Fu(\()p Fs(m)p Fu(\))p Fs(;)493 b Fu(\(3\))228 2492 y(where)19 b Fs(\033)400 2499 y Fq(i)433 2492 y Fu(resp.)30 b Fs(\034)585 2499 y Fq(i)619 2492 y Fu(are)19 b(the)h(canonical)f(generators)h(of)g Fs(S)1336 2499 y Fq(n)1378 2492 y Fu(resp.)31 b Fs(B)1547 2499 y Fq(n)1570 2492 y Fu(.)g(Th)o(us)228 2550 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))20 b(is)g(also)g(a)h Fs(k)r(B)626 2557 y Fq(n)649 2550 y Fu(-submo)q(dule)f(of)g Fs(M)5 b Fu(.)33 b(Since)19 b(the)g(functor)h Fs(M)26 b Fr(7!)20 b Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))228 2608 y(is)20 b(a)g(righ)o (tadjoin)o(t)g(functor,)h(it)f(preserv)o(es)f(limits.)30 b(Lik)o(e)19 b(for)i(eigenspaces)f(w)o(e)p eop %%Page: 5 5 5 4 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)r(5)228 225 y Fu(ha)o(v)o(e)f(that)i(the)f(sum)f(of)i(the)f(subspaces)h Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))17 b(for)h(all)f Fs(\020)k Fu(with)c(di\013eren)o(t)f Fs(\020)1652 207 y Fl(2)1689 225 y Fu(is)228 283 y(a)j(direct)e(sum.)26 b(On)19 b Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))18 b(w)o(e)g(ha)o(v)o(e)g(t)o(w)o(o)g (distinct)f Fs(k)r(S)1274 290 y Fq(n)1298 283 y Fu(-structures)h Fs(\033)1572 290 y Fq(i)1586 283 y Fu(\()p Fs(m)p Fu(\))f(=)228 341 y Fr(\000)p Fs(\020)292 323 y Fj(\000)p Fl(1)339 341 y Fs(\034)360 348 y Fq(i)374 341 y Fu(\()p Fs(m)p Fu(\))12 b(and)h Fs(\033)586 348 y Fq(i)613 341 y Fu(=)h Fs(\020)690 323 y Fj(\000)p Fl(1)737 341 y Fs(\034)758 348 y Fq(i)773 341 y Fu(\()p Fs(m)p Fu(\),)e(since)f Fs(\020)17 b Fu(and)c Fr(\000)p Fs(\020)j Fu(de\014ne)11 b(the)h(same)g(subspace)228 399 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b(=)g Fs(M)5 b Fu(\()p Fr(\000)p Fs(\020)t Fu(\))14 b Fr(\022)f Fs(M)5 b Fu(.)278 458 y(The)16 b Fs(\020)t Fu(-symmetriz)o(ation)e Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))17 b(of)f Fs(M)22 b Fu(can)16 b(also)h(b)q(e)f(calculated)g(b)o(y)228 531 y FB(Lemm)o(a)g(2.4.)228 619 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b(=)g Fr(f)p Fs(m)f Fr(2)h Fs(M)5 b Fr(j)p Fs(\034)630 598 y Fj(\000)p Fl(1)624 631 y Fq(i)677 619 y Fs(\034)704 598 y Fj(\000)p Fl(1)698 631 y Fq(i)p Fl(+1)766 619 y Fs(:)j(:)g(:)g(\034)859 598 y Fj(\000)p Fl(1)853 631 y Fq(j)r Fj(\000)p Fl(1)916 619 y Fs(\034)943 598 y Fl(2)937 631 y Fq(j)963 619 y Fs(\034)984 626 y Fq(j)r Fj(\000)p Fl(1)1055 619 y Fs(:)g(:)g(:)g(\034)1142 626 y Fq(i)p Fl(+1)1201 619 y Fs(\034)1222 626 y Fq(i)1237 619 y Fu(\()p Fs(m)p Fu(\))13 b(=)h Fs(\020)1408 598 y Fl(2)1428 619 y Fs(m)49 b Fr(8)p Fu(1)13 b Fr(\024)h Fs(i)f Fr(\024)h Fs(j)i Fr(\024)e Fs(n)p Fr(\000)p Fu(1)p Fr(g)228 707 y Fo(which)20 b(r)n(e)n(duc)n(es)f(the)i(numb)n(er)f(of)f(c)n (onditions)h(to)g(b)n(e)g(imp)n(ose)n(d)f(on)h(the)h Fs(m)c Fr(2)i Fs(M)228 765 y Fo(in)f(or)n(der)e(to)h(b)n(e)h(in)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))p Fo(.)228 860 y(Pr)n(o)n(of.)19 b Fu(Giv)o(en)c(in)h(App)q(endix.)886 b Fi(\003)278 955 y Fu(One)19 b(of)h(the)g(in)o(teresting)e Fs(k)r(B)838 962 y Fq(n)862 955 y Fu(-structures,)i(for)h(whic)o(h)e(w)o(e)g(will)g (apply)g(the)228 1013 y(previous)g(construction,)i(o)q(ccurs)f(on)h Fs(n)p Fu(-fold)f(tensor)g(pro)q(ducts)h Fs(M)1502 995 y Fq(n)1546 1013 y Fu(:=)f Fs(M)e Fr(\012)228 1071 y Fs(:)8 b(:)g(:)r Fr(\012)s Fs(M)18 b Fu(of)13 b(an)f(ob)s(ject)g Fs(M)18 b Fu(in)12 b(a)h(braided)f(monoidal)g(category)g(of)h(v)o (ector)e(spaces.)278 1129 y(Let)18 b Fs(K)23 b Fu(b)q(e)c(a)g(Hopf)g (algebra.)29 b(Let)19 b Fs(M)24 b Fu(b)q(e)19 b(an)g Fs(K)t Fu(-mo)q(dule)f(suc)o(h)g(that)h Fs(M)24 b Fu(is)228 1187 y(a)18 b Fs(k)r(B)334 1194 y Fq(n)358 1187 y Fu(-)p Fs(K)t Fu(-bimo)q(dule.)25 b(The)17 b(functorialit)o(y)g(of)h(our)g (construction)g(then)g(mak)o(es)228 1246 y Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))16 b(again)i(an)e Fs(K)t Fu(-mo)q(dule)g(and)g(in)g (fact)h(a)f Fs(k)r(S)1136 1253 y Fq(n)1160 1246 y Fu(-)p Fs(K)t Fu(-bimo)q(dule.)278 1304 y(Let)f Fs(M)21 b Fu(b)q(e)16 b(an)g Fs(K)t Fu(-como)q(dule)f(suc)o(h)g(that)h Fs(M)21 b Fu(is)16 b(a)f Fs(k)r(B)1278 1311 y Fq(n)1302 1304 y Fu(-)p Fs(K)t Fu(-dimo)q(dule.)20 b(Then)228 1362 y(b)o(y)26 b(Prop)q(osition)h(2.1)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))27 b(is)f(an)h Fs(K)t Fu(-como)q(dule)e(and)i(in)f(fact)g(a)h Fs(k)r(S)1621 1369 y Fq(n)1645 1362 y Fu(-)p Fs(K)t Fu(-)228 1420 y(dimo)q(dule.)278 1478 y(Let)17 b Fs(K)22 b Fu(b)q(e)c(a)g(Hopf)f (algebra)h(with)g(bijectiv)o(e)d(an)o(tip)q(o)q(de.)26 b(Let)17 b Fr(Y)t(D)1528 1460 y Fq(K)1527 1490 y(K)1581 1478 y Fu(denote)228 1536 y(the)11 b(category)g(of)h(Y)l (etter-Drinfeld)d(mo)q(dules)i(o)o(v)o(er)f Fs(K)t Fu(,)i(i.e.)18 b(of)12 b(righ)o(t)f Fs(K)t Fu(-mo)q(dules)228 1594 y(and)23 b(righ)o(t)g Fs(K)t Fu(-como)q(dules)f Fs(M)29 b Fu(suc)o(h)22 b(that)1064 1561 y Fm(P)1107 1594 y Fu(\()p Fs(x)16 b Fr(\001)f Fs(h)p Fu(\))1246 1601 y Fl(0)1281 1594 y Fr(\012)h Fu(\()p Fs(x)f Fr(\001)g Fs(h)p Fu(\))1474 1601 y Fl(1)1519 1594 y Fu(=)1583 1561 y Fm(P)1626 1594 y Fu(\()p Fs(x)1673 1601 y Fl(0)1708 1594 y Fr(\001)228 1652 y Fs(h)256 1659 y Fl(2)276 1652 y Fu(\))c Fr(\012)g Fs(S)s Fu(\()p Fs(h)436 1659 y Fl(1)455 1652 y Fu(\))p Fs(x)502 1659 y Fl(1)522 1652 y Fs(h)550 1659 y Fl(3)586 1652 y Fu(for)17 b(all)f Fs(x)d Fr(2)i Fs(M)5 b Fu(.)22 b(The)16 b(usual)h(tensor)g(pro)q(duct)g (mak)o(es)e Fr(Y)t(D)1687 1634 y Fq(K)1686 1665 y(K)228 1711 y Fu(a)24 b(monoidal)e(category)l(.)43 b Fr(Y)t(D)803 1692 y Fq(K)802 1723 y(K)861 1711 y Fu(has)24 b(a)g(braiding)g(giv)o (en)e(b)o(y)h Fs(\034)1433 1718 y Fq(X)q(;Y)1529 1711 y Fu(:)i Fs(X)20 b Fr(\012)c Fs(Y)228 1769 y Fr(\000)-30 b(!)24 b Fs(Y)i Fr(\012)16 b Fs(X)t Fu(,)24 b Fs(\034)6 b Fu(\()p Fs(x)14 b Fr(\012)i Fs(y)r Fu(\))24 b(=)776 1735 y Fm(P)828 1769 y Fs(y)852 1776 y Fl(0)887 1769 y Fr(\012)15 b Fs(xy)993 1776 y Fl(1)1012 1769 y Fu(.)41 b(W)l(e)22 b(assume)g(that)h(the)f(reader)g(is)228 1827 y(familiar)d(with)i(the)g(prop)q(erties)g(of)h(the)f Fs(B)1039 1834 y Fq(n)1063 1827 y Fu(-action)g(that)h(is)f(induced)g(b) o(y)f(the)228 1885 y(braiding)c Fs(\034)22 b Fu(on)17 b Fs(n)p Fu(-fold)g(tensor)f(pro)q(ducts)h(\([4])f(10.6\).)228 1958 y FB(Theorem)g(2.5.)k Fo(L)n(et)d Fs(K)22 b Fo(b)n(e)c(a)f(Hopf)h (algebr)n(a)g(with)f(bije)n(ctive)j(antip)n(o)n(de.)i(Then)228 2016 y(for)17 b(e)n(ach)h Fs(\020)h Fr(2)c Fs(k)531 1998 y Fj(\003)569 2016 y Fo(and)j(e)n(ach)g Fs(n)d Fr(\025)g Fu(2)j Fo(the)h(c)n(onstruction)g(given)g(ab)n(ove)g(de\014nes)g(a)228 2074 y(\(non-additive\))g(functor)539 2164 y Fr(Y)t(D)615 2144 y Fq(K)614 2176 y(K)664 2164 y Fr(3)14 b Fs(M)19 b Fr(7!)14 b Fu(\()p Fs(M)i Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(M)5 b Fu(\)\()p Fs(\020)t Fu(\))14 b Fr(2)g(Y)t(D)1362 2144 y Fq(K)1361 2176 y(K)1398 2164 y Fs(:)228 2259 y Fo(Pr)n(o)n(of.)19 b Fu(If)h Fs(M)26 b Fr(2)21 b(Y)t(D)633 2241 y Fq(K)632 2271 y(K)688 2259 y Fu(then)f(the)g Fs(n)p Fu(-fold)h(tensor)g(pro)q(duct)f Fs(M)g Fr(\012)13 b Fs(:)8 b(:)g(:)13 b Fr(\012)h Fs(M)25 b Fu(is)228 2317 y(a)c(Y)l(etter-Drinfeld)f(mo)q(dule)g(on)i(whic)o(h)e Fs(B)1049 2324 y Fq(n)1094 2317 y Fu(and)i(th)o(us)f Fs(k)r(B)1370 2324 y Fq(n)1415 2317 y Fu(acts)h(in)f(suc)o(h)g(a)228 2375 y(w)o(a)o(y)l(,)14 b(that)h Fs(M)k Fu(is)c(a)f(\()p Fs(k)r(B)674 2382 y Fq(n)698 2375 y Fs(;)8 b(K)t Fu(\)-bimo)q(dule)13 b(and)i(a)g(\()p Fs(k)r(B)1226 2382 y Fq(n)1249 2375 y Fs(;)8 b(K)t Fu(\)-dimo)q(dule.)19 b(The)c Fs(\020)t Fu(-)228 2433 y(symmetri)o(zation)c(functor)k(-\()p Fs(\020)t Fu(\))f(preserv)o(es)f(the)h(mo)q(dule)f(and)i(como)q(dule)e(struc-)228 2492 y(tures)j(hence)f(the)h(Y)l(etter-Drinfeld)f(structure.)278 2550 y(The)c(functor)h(is)g(not)g(additiv)o(e)e(since)h(the)g (\\diagonal")j(functor)d Fs(M)20 b Fr(7!)13 b Fs(M)7 b Fr(\012)r Fs(M)228 2608 y Fu(is)16 b(not)h(additiv)o(e.)1133 b Fi(\003)p eop %%Page: 6 6 6 5 bop 228 125 a Fp(6)571 b(BODO)13 b(P)m(AREIGIS)278 225 y Fu(W)l(e)k(abbreviate)g Fs(M)656 207 y Fq(n)680 225 y Fu(\()p Fs(\020)t Fu(\))g(:=)e Fs(M)879 207 y Fj(\012)p Fq(n)931 225 y Fu(\()p Fs(\020)t Fu(\))h(=)g Fs(M)i Fr(\012)11 b Fs(:)d(:)g(:)k Fr(\012)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\).)26 b(Then)17 b Fs(M)1635 207 y Fq(n)1659 225 y Fu(\()p Fs(\020)t Fu(\))228 283 y(is)g(a)g(submo)q(dule)g(of)g Fs(M)674 265 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e(additional)h(op)q(erations)h(ma)o(y)e(b)q(e)h(constructed)g(as)228 1198 y(follo)o(ws.)228 1269 y FB(Prop)r(osition)f(3.1.)h Fo(L)n(et)e Fs(\020)24 b Fo(b)n(e)c(a)f(primitive)h Fs(n)p Fo(-th)g(r)n(o)n(ots)e(of)i(unity.)29 b(Then)20 b(the)228 1327 y(op)n(er)n(ations)306 1410 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])434 1417 y Fq(n)455 1410 y Fu(])469 1417 y Fl(2)502 1410 y Fu(:)13 b Fs(P)567 1389 y Fq(n)p Fl(+1)636 1410 y Fu(\()p Fs(\020)t Fu(\))h Fr(3)g Fs(x)788 1417 y Fl(1)818 1410 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1014 1417 y Fq(n)p Fl(+1)1096 1410 y Fr(7!)j Fu([)p Fs(x)1202 1417 y Fl(1)1221 1410 y Fs(;)8 b Fu([)p Fs(x)1285 1417 y Fl(2)1304 1410 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(x)1450 1417 y Fq(n)p Fl(+1)1518 1410 y Fu(]])13 b Fr(2)h Fs(P)228 1493 y Fo(and)304 1575 y Fu([[)p Fs(:;)8 b(:)p Fu(])396 1582 y Fq(n)417 1575 y Fs(;)g(:)p Fu(])467 1582 y Fl(2)500 1575 y Fu(:)13 b Fs(P)565 1555 y Fq(n)p Fl(+1)634 1575 y Fu(\()p Fs(\020)t Fu(\))h Fr(3)g Fs(x)786 1582 y Fl(1)816 1575 y Fr(\012)d Fs(:)d(:)g(:)j Fr(\012)f Fs(x)1012 1582 y Fq(n)p Fl(+1)1094 1575 y Fr(7!)k Fu([[)p Fs(x)1214 1582 y Fl(1)1233 1575 y Fs(;)8 b(:)g(:)g(:)15 b(;)8 b(x)1378 1582 y Fq(n)1401 1575 y Fu(])p Fs(;)g(x)1465 1582 y Fq(n)p Fl(+1)1533 1575 y Fu(])13 b Fr(2)i Fs(P)228 1658 y Fo(ar)n(e)i(wel)r(l) i(de\014ne)n(d.)228 1747 y(Pr)n(o)n(of.)g Fu(Giv)o(en)c(in)h(App)q (endix.)886 b Fi(\003)278 1835 y Fu(W)l(e)16 b(will)f(ha)o(v)o(e)g(to)i (consider)f(ob)s(jects)228 1918 y Fs(P)266 1898 y Fq(n)p Fl(+1)335 1918 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))13 b(:=)h Fs(P)7 b Fr(\012)p Fs(P)677 1898 y Fq(n)700 1918 y Fu(\()p Fs(\020)t Fu(\))p Fr(\\f)p Fs(z)16 b Fr(2)e Fs(P)945 1898 y Fq(n)p Fl(+1)1014 1918 y Fr(j8)p Fs(')e Fr(2)i Fs(S)1178 1925 y Fq(n)1216 1918 y Fu(:)f(\(1)p Fr(\012)p Fs(')p Fu(\))1376 1898 y Fj(\000)p Fl(1)1423 1918 y Fs(\034)1450 1898 y Fl(2)1444 1930 y(1)1470 1918 y Fu(\(1)p Fr(\012)p Fs(')p Fu(\)\()p Fs(z)r Fu(\))g(=)h Fs(z)r Fr(g)p Fs(:)228 2001 y Fu(Since)g(this)h(is)g(a)h(k)o(ernel)d (\(limit\))g(construction)i(in)g Fr(C)s Fu(,)g Fs(P)1262 1983 y Fq(n)p Fl(+1)1331 2001 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))15 b(is)g(again)h(an)228 2059 y(ob)s(ject)f(in)h Fr(C)s Fu(.)228 2130 y FB(Prop)r(osition)k(3.2.)h Fo(L)n(et)e Fs(\020)24 b Fo(b)n(e)c(a)f(primitive)h Fs(n)p Fo(-th)g(r)n(o)n(ots)e (of)i(unity.)29 b(Then)20 b(the)228 2188 y(op)n(er)n(ations)282 2271 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])410 2278 y Fq(n)431 2271 y Fu(])445 2278 y Fl(2)478 2271 y Fu(:)13 b Fs(P)543 2250 y Fq(n)p Fl(+1)612 2271 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))14 b Fr(3)g Fs(x)d Fr(\012)f Fs(y)933 2278 y Fl(1)964 2271 y Fr(\012)h Fs(:)d(:)g(:)i Fr(\012)h Fs(y)1156 2278 y Fq(n)1193 2271 y Fr(7!)j Fu([)p Fs(x;)8 b Fu([)p Fs(y)1359 2278 y Fl(1)1377 2271 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(y)1519 2278 y Fq(n)1542 2271 y Fu(]])13 b Fr(2)h Fs(P)228 2354 y Fo(and)228 2436 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])356 2443 y Fl(2)373 2436 y Fs(;)g(:)p Fu(])423 2443 y Fq(n)445 2436 y Fs(\034)466 2443 y Fq(i)p Fj(\000)p Fl(1)534 2436 y Fs(:)g(:)g(:)f(\034)620 2443 y Fl(1)654 2436 y Fu(:)13 b Fs(P)719 2416 y Fq(n)p Fl(+1)788 2436 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))14 b Fr(3)g Fs(x)p Fr(\012)p Fs(y)1088 2443 y Fl(1)1107 2436 y Fr(\012)p Fs(:)8 b(:)g(:)o Fr(\012)p Fs(y)1266 2443 y Fq(n)1303 2436 y Fr(7!)13 b Fu([)p Fs(y)1404 2443 y Fl(1)1423 2436 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b Fu([)p Fs(x;)g(y)1629 2443 y Fq(i)1642 2436 y Fu(])p Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)1797 2443 y Fq(n)1820 2436 y Fu(])14 b Fr(2)g Fs(P)228 2519 y Fo(ar)n(e)j(wel)r(l)i(de\014ne)n(d.)228 2608 y(Pr)n(o)n(of.)g Fu(Giv)o(en)c(in)h(App)q(endix.)886 b Fi(\003)p eop %%Page: 7 7 7 6 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)r(7)278 225 y Fu(W)l(e)f(in)o(tro)q(duce)f(sp)q(ecial)h(brac)o(k)o(et)f(m)o(ultiplic) o(ations)f(whic)o(h)i(then)g(lead)g(to)g(the)228 283 y(de\014nition)g(of)g(a)h(Lie)f(algebra)g(on)h(a)g(Y)l(etter-Drinfeld)d (mo)q(dule.)228 357 y FB(De\014nition)20 b(3.3.)h Fu(Let)e Fs(A)f Fu(b)q(e)h(an)g(algebra)g(in)f Fr(C)j Fu(=)d Fr(Y)t(D)1296 339 y Fq(K)1295 369 y(K)1350 357 y Fu(and)h(let)f Fr(r)1562 339 y Fq(n)1603 357 y Fu(:)f Fs(A)12 b Fr(\012)228 415 y Fs(:)c(:)g(:)j Fr(\012)g Fs(A)k Fr(\000)-31 b(!)15 b Fs(A)i Fu(denote)g(the)g Fs(n)p Fu(-fold)g(m)o(ultiplic)o(ation.)k(W) l(e)c(de\014ne)g(a)g Fo(br)n(acket)h Fu(or)228 473 y Fo(symmetric)f(multiplic)n(ation)462 564 y Fu([)p Fs(:;)8 b(:)p Fu(])k(:)h Fs(A)616 543 y Fq(n)639 564 y Fu(\()p Fs(\020)t Fu(\))h Fr(\000)-30 b(!)13 b Fs(A)114 b Fu(b)o(y)f([)p Fs(z)r Fu(])13 b(:=)1249 522 y Fm(X)1235 614 y Fq(\033)q Fj(2)p Fq(S)1301 618 y Fh(n)1331 564 y Fr(r)1373 543 y Fq(n)1396 564 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))228 695 y(where)j(the)g(action)g(of)h Fs(S)685 702 y Fq(n)724 695 y Fu(on)g Fs(A)829 677 y Fq(n)852 695 y Fu(\()p Fs(\020)t Fu(\))f(is)g(giv)o(en)g(as)h(in)f(\(3\).)278 769 y(W)l(e)h(will)f(only) i(use)g(those)g(brac)o(k)o(et)e(op)q(erations)j(whic)o(h)e(are)g (de\014ned)h(with)f Fs(\020)228 827 y Fu(a)f(primitiv)o(e)d Fs(n)p Fu(-th)k(ro)q(ot)g(of)g(unit)o(y)e(\(for)i(all)e Fs(n)f Fr(2)g Fn(N)j Fu(and)g(all)f Fs(\020)t Fu(\).)278 885 y(W)l(e)g(consider)g(these)g(brac)o(k)o(et)g(op)q(erations)h(as)g (a)g(generalization)f(of)h(the)f(Lie-)228 943 y(brac)o(k)o(et)d([-)o Fs(;)8 b Fu(-)q(])13 b(:)g Fs(L)6 b Fr(\002)g Fs(L)15 b Fr(\000)-30 b(!)13 b Fs(L)i Fu(or)f([{])f(:)h Fs(L)6 b Fr(\012)g Fs(L)15 b Fr(\000)-31 b(!)14 b Fs(L)p Fu(.)21 b(Observ)o(e)13 b(that)h(our)g(brac)o(k)o(et)228 1001 y(op)q(eration)20 b(is)f(only)g(partially)f(de\014ned)h(and)h(should)f Fo(not)h Fu(b)q(e)f(considered)g(as)h(a)228 1059 y(m)o(ultili)o(near)15 b(op)q(eration,)j(since)f Fs(A)868 1041 y Fq(n)891 1059 y Fu(\()p Fs(\020)t Fu(\))f Fr(\022)g Fs(A)1062 1041 y Fq(n)1102 1059 y Fu(is)h(just)h(a)g(submo)q(dule)f(in)g Fr(C)k Fu(and)228 1117 y(do)q(es)i(not)g(necessarily)e(decomp)q(ose)h (in)o(to)g(an)h Fs(n)p Fu(-fold)g(tensor)g(pro)q(duct.)40 b(The)228 1175 y(elemen)o(ts)13 b(in)j Fs(A)521 1157 y Fq(n)544 1175 y Fu(\()p Fs(\020)t Fu(\))g(are,)g(ho)o(w)o(ev)o(er,)f (of)h(the)g(form)g Fs(z)f Fu(=)1264 1142 y Fm(P)1308 1186 y Fq(k)1337 1175 y Fs(x)1365 1182 y Fq(k)q(;)p Fl(1)1425 1175 y Fr(\012)c Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1621 1182 y Fq(k)q(;n)1673 1175 y Fu(.)278 1234 y(If)19 b(w)o(e)g(suppress)h(the) g(summation)d(index)i(and)h(the)g(summation)d(sign)j(then)228 1292 y(w)o(e)c(ma)o(y)f(write)h(the)g(brac)o(k)o(et)f(op)q(eration)j (on)f Fs(z)f Fu(=)e Fs(x)1193 1299 y Fl(1)1224 1292 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1420 1299 y Fq(n)1460 1292 y Fu(also)17 b(as)g([)p Fs(z)r Fu(])c(=)228 1350 y([)p Fs(x)270 1357 y Fl(1)289 1350 y Fs(;)8 b(:)g(:)g(:)15 b(;)8 b(x)434 1357 y Fq(n)458 1350 y Fu(].)20 b(If)c(w)o(e)g(de\014ne) 660 1439 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))e(=:)f Fs(x)860 1447 y Fq(\033)881 1438 y Fg(\000)p Ff(1)922 1447 y Fl(\(1\))980 1439 y Fr(\012)e Fs(:)d(:)g(:)j Fr(\012)f Fs(x)1176 1447 y Fq(\033)1197 1438 y Fg(\000)p Ff(1)1239 1447 y Fl(\()p Fq(n)p Fl(\))1660 1439 y Fu(\(4\))228 1528 y(then)16 b(w)o(e)g(get)603 1599 y([)p Fs(x)645 1606 y Fl(1)664 1599 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(x)810 1606 y Fq(n)833 1599 y Fu(])13 b(=)926 1557 y Fm(X)912 1649 y Fq(\033)q Fj(2)p Fq(S)978 1653 y Fh(n)1008 1599 y Fs(x)1036 1606 y Fq(\033)q Fl(\(1\))1115 1599 y Fr(\001)e Fs(:)d(:)g(:)i Fr(\001)h Fs(x)1261 1606 y Fq(\033)q Fl(\()p Fq(n)p Fl(\))1333 1599 y Fs(:)228 1717 y Fu(Observ)o(e)19 b(that)h(the)g(comp)q(onen)o (ts)f Fs(x)916 1724 y Fl(1)935 1717 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(x)1081 1724 y Fq(n)1124 1717 y Fu(in)20 b(this)f(expression)h (are)g(in)o(ter-)228 1775 y(c)o(hanged)13 b(according)g(to)g(the)f (action)h(of)g(the)g(braid)g(group)g(resp.)21 b(the)12 b(symmetric)228 1833 y(group)17 b(on)g Fs(A)471 1815 y Fq(n)494 1833 y Fu(\()p Fs(\020)t Fu(\),)f(so)g Fs(x)674 1841 y Fq(\033)q Fl(\(1\))754 1833 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(x)950 1841 y Fq(\033)q Fl(\()p Fq(n)p Fl(\))1038 1833 y Fu(is)16 b(only)g(a)h(sym)o(b)q(olic)d(expression.)278 1891 y(The)h(brac)o(k)o(et)f(op)q(eration)i(ob)o(viously)f(satis\014es) h(the)f(\\an)o(ti"-symmetry)e(iden-)228 1949 y(tit)o(y)666 2039 y([)p Fs(\033)r Fu(\()p Fs(z)r Fu(\)])g(=)g([)p Fs(z)r Fu(])195 b Fr(8)p Fs(\033)14 b Fr(2)g Fs(S)1247 2046 y Fq(n)1270 2039 y Fs(:)376 b Fu(\(5\))278 2128 y(W)l(e)17 b(apply)g(Prop)q(osition)i(3.1)f(to)g(an)g(algebra)g Fs(A)f Fu(in)h Fr(C)i Fu(with)e(the)f(op)q(erations)228 2186 y(giv)o(en)e(in)h(De\014nition)g(3.3)h(and)f(get)228 2260 y FB(Theorem)22 b(3.4.)h Fu(\(1.)e(Jacobi)g(iden)o(tit)o(y\))g Fo(F)l(or)g(al)r(l)j Fs(n)e Fr(2)h Fn(N)p Fo(,)h(for)e(al)r(l)h (primitive)228 2318 y Fs(n)p Fo(-th)18 b(r)n(o)n(ots)e(of)h(unity)h Fs(\020)t Fo(,)f(and)h(for)f(al)r(l)i Fs(z)c Fr(2)f Fs(A)1057 2300 y Fq(n)p Fl(+1)1125 2318 y Fu(\()p Fs(\020)t Fu(\))k Fo(we)g(have)356 2391 y Fq(n)p Fl(+1)359 2404 y Fm(X)361 2495 y Fq(i)p Fl(=1)422 2445 y Fu([)p Fs(x)464 2452 y Fq(i)478 2445 y Fs(;)8 b Fu([)p Fs(x)542 2452 y Fl(1)561 2445 y Fs(;)g(:)g(:)g(:)15 b(;)c Fu(^)-27 b Fs(x)706 2452 y Fq(i)720 2445 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(x)866 2452 y Fq(n)p Fl(+1)934 2445 y Fu(]])13 b(=)1027 2391 y Fq(n)p Fl(+1)1030 2404 y Fm(X)1031 2495 y Fq(i)p Fl(=1)1093 2445 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])1221 2452 y Fq(n)1242 2445 y Fu(])1256 2452 y Fl(2)1275 2445 y Fu(\(1)g Fs(:)g(:)g(:)g(i)p Fu(\)\()p Fs(z)r Fu(\))13 b(=)h(0)p Fs(;)1660 2503 y Fu(\(6\))228 2595 y Fo(wher)n(e)j(we)i(use)f (the)g(notation)g Fu(\(4\))p Fo(.)p eop %%Page: 8 8 8 7 bop 228 125 a Fp(8)571 b(BODO)13 b(P)m(AREIGIS)228 225 y Fo(Pr)n(o)n(of.)19 b Fu(W)l(e)k(de\014ne)g(\()p Fs(S)665 232 y Fq(n)p Fl(+1)733 225 y Fu(\))752 233 y Fl(\()p Fq(i)p Fl(\))819 225 y Fu(:=)i Fr(f)p Fs(\033)i Fr(2)f Fs(S)1065 232 y Fq(n)p Fl(+1)1134 225 y Fr(j)p Fs(\033)r Fu(\()p Fs(i)p Fu(\))e(=)i(1)p Fr(g)p Fu(.)42 b(Then)23 b Fs(S)1590 232 y Fq(n)p Fl(+1)1684 225 y Fu(=)228 250 y Fm(S)262 294 y Fq(i)276 283 y Fu(\()p Fs(S)325 290 y Fq(n)p Fl(+1)394 283 y Fu(\))413 291 y Fl(\()p Fq(i)p Fl(\))455 283 y Fu(.)d(F)l(or)15 b Fs(\033)g Fr(2)f Fu(\()p Fs(S)714 290 y Fq(n)p Fl(+1)783 283 y Fu(\))802 291 y Fl(\()p Fq(i)p Fl(\))858 283 y Fu(let)k(\026)-28 b Fs(\032)14 b Fu(:=)f Fs(\033)r Fu(\()p Fs(i)8 b(:)g(:)g(:)f Fu(1\).)21 b(Since)e(\026)-29 b Fs(\032)p Fu(\(1\))14 b(=)g(1)h(there)f(is)228 341 y(a)j(unique)f Fs(\032)e Fr(2)h Fs(S)545 348 y Fq(n)585 341 y Fu(with)21 b(\026)-28 b Fs(\032)14 b Fu(=)h(1)d Fr(\012)f Fs(\032)17 b Fu(and)g Fs(\033)f Fu(=)e(\(1)e Fr(\012)f Fs(\032)p Fu(\)\(1)d Fs(:)g(:)g(:)g(i)p Fu(\).)22 b(So)c(w)o(e)e(obtain)228 399 y(a)g(bijection)590 477 y Fs(S)620 484 y Fq(n)657 477 y Fr(3)e Fs(\032)g Fr(7!)g Fu(\(1)d Fr(\012)g Fs(\032)p Fu(\)\(1)d Fs(:)g(:)g(:)g(i)p Fu(\))14 b Fr(2)g Fu(\()p Fs(S)1218 484 y Fq(n)p Fl(+1)1286 477 y Fu(\))1305 485 y Fl(\()p Fq(i)p Fl(\))1347 477 y Fs(:)228 570 y Fu(Analogously)j(w)o (e)g(de\014ne)g(\()p Fs(S)769 577 y Fq(n)p Fl(+1)837 570 y Fu(\))856 552 y Fl(\()p Fq(i)p Fl(\))913 570 y Fu(:=)e Fr(f)p Fs(\033)i Fr(2)e Fs(S)1128 577 y Fq(n)p Fl(+1)1197 570 y Fr(j)p Fs(\033)r Fu(\()p Fs(i)p Fu(\))f(=)h Fs(n)d Fu(+)g(1)p Fr(g)17 b Fu(and)h(get)f(a)228 628 y(bijection)545 707 y Fs(S)575 714 y Fq(n)612 707 y Fr(3)d Fs(\032)g Fr(7!)g Fu(\()p Fs(\032)d Fr(\012)g Fu(1\)\()p Fs(n)h Fu(+)f(1)d Fs(:)g(:)g(:)g(i)p Fu(\))13 b Fr(2)h Fu(\()p Fs(S)1262 714 y Fq(n)p Fl(+1)1331 707 y Fu(\))1350 686 y Fl(\()p Fq(i)p Fl(\))1391 707 y Fs(:)228 795 y Fu(No)o(w)21 b(observ)o(e)h(that)g Fs(\034)656 802 y Fq(n)688 795 y Fs(:)8 b(:)g(:)f(\034)774 802 y Fl(1)794 795 y Fu(\()p Fs(z)r Fu(\))23 b(=)h Fs(\020)967 777 y Fq(n)990 795 y Fs(\033)1018 802 y Fq(n)1050 795 y Fs(:)8 b(:)g(:)f(\033)1143 802 y Fl(1)1163 795 y Fu(\()p Fs(z)r Fu(\))23 b(=)g(\()p Fs(n)15 b Fu(+)g(1)8 b Fs(:)g(:)g(:)g Fu(1\)\()p Fs(z)r Fu(\))22 b(\(b)o(y)228 853 y Fs(\020)253 835 y Fq(n)290 853 y Fu(=)14 b(1\))j(for)f Fs(z)g Fr(2)e Fs(P)600 835 y Fq(n)p Fl(+1)669 853 y Fu(\()p Fs(\020)t Fu(\))i(to)h(get)249 917 y Fm(P)292 930 y Fq(n)292 962 y(i)p Fl(=1)368 950 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(]])o(\(1)g Fs(:)g(:)g(:)e(i)p Fu(\)\()p Fs(z)r Fu(\))13 b(=)368 1008 y(=)420 975 y Fm(P)463 988 y Fq(n)463 1020 y(i)p Fl(=1)531 1008 y Fr(r)p Fu(\(1)e Fr(\012)g Fu([)p Fs(:;)d(:)p Fu(]\)\(1)g Fs(:)g(:)g(:)e(i)p Fu(\)\()p Fs(z)r Fu(\))k Fr(\000)h(r)p Fu(\([)p Fs(:;)d(:)p Fu(])h Fr(\012)i Fu(1\))p Fs(\034)1310 1015 y Fq(n)1342 1008 y Fs(:)d(:)g(:)f(\034)1428 1015 y Fl(1)1448 1008 y Fu(\(1)h Fs(:)g(:)g(:)h(i)p Fu(\)\()p Fs(z)r Fu(\))368 1066 y(=)420 1033 y Fm(P)463 1046 y Fq(n)463 1078 y(i)p Fl(=1)531 1066 y Fr(r)p Fu(\(1)i Fr(\012)g Fu([)p Fs(:;)d(:)p Fu(]\)\(1)g Fs(:)g(:)g(:)e(i)p Fu(\)\()p Fs(z)r Fu(\))k Fr(\000)h(r)p Fu(\([)p Fs(:;)d(:)p Fu(])h Fr(\012)i Fu(1\)\()p Fs(n)g Fu(+)g(1)d Fs(:)g(:)g(:)g(i)p Fu(\)\()p Fs(z)r Fu(\))368 1124 y(=)420 1091 y Fm(P)463 1104 y Fq(n)463 1136 y(i)p Fl(=1)531 1091 y Fm(P)575 1135 y Fq(\032)p Fj(2)p Fq(S)638 1139 y Fh(n)669 1124 y Fr(r)711 1106 y Fq(n)p Fl(+1)779 1124 y Fu(\(1)k Fr(\012)f Fs(\032)p Fu(\)\(1)d Fs(:)g(:)g(:)g(i)p Fu(\)\()p Fs(z)r Fu(\))j Fr(\000)f(r)1246 1106 y Fq(n)p Fl(+1)1314 1124 y Fu(\()p Fs(\032)i Fr(\012)e Fu(1\)\()p Fs(n)i Fu(+)f(1)d Fs(:)g(:)g(:)g(i)p Fu(\)\()p Fs(z)r Fu(\))368 1182 y(=)420 1149 y Fm(P)463 1193 y Fq(\033)q Fj(2)p Fq(S)529 1198 y Fh(n)p Ff(+1)600 1182 y Fr(r)642 1164 y Fq(n)p Fl(+1)710 1182 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))i Fr(\000)h(r)905 1164 y Fq(n)p Fl(+1)973 1182 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))i(=)h(0)p Fs(:)1683 1278 y Fi(\003)228 1385 y FB(Theorem)22 b(3.5.)h Fu(\(2.)e(Jacobi)g (iden)o(tit)o(y\))g Fo(F)l(or)g(al)r(l)j Fs(n)e Fr(2)h Fn(N)p Fo(,)h(for)e(al)r(l)h(primitive)228 1443 y Fs(n)p Fo(-th)c(r)n(o)n(ots)e(of)i(unity)g Fs(\020)t Fo(,)g(and)g(for)f(al)r (l)i Fs(z)e Fu(=)e Fs(x)11 b Fr(\012)h Fs(y)1154 1450 y Fl(1)1186 1443 y Fr(\012)f Fs(:)d(:)g(:)k Fr(\012)f Fs(y)1380 1450 y Fq(n)1420 1443 y Fr(2)16 b Fs(A)1506 1425 y Fq(n)p Fl(+1)1574 1443 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))228 1501 y Fo(we)18 b(have)522 1618 y Fu([)p Fs(x;)8 b Fu([)p Fs(y)624 1625 y Fl(1)643 1618 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)784 1625 y Fq(n)807 1618 y Fu(]])13 b(=)919 1564 y Fq(n)900 1576 y Fm(X)902 1668 y Fq(i)p Fl(=1)960 1618 y Fu([)p Fs(y)998 1625 y Fl(1)1017 1618 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b Fu([)p Fs(x;)g(y)1223 1625 y Fq(i)1236 1618 y Fu(])p Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)1391 1625 y Fq(n)1414 1618 y Fu(])232 b(\(7\))228 1753 y Fo(wher)n(e)17 b Fs(y)389 1760 y Fl(1)420 1753 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(y)612 1760 y Fq(i)p Fj(\000)p Fl(1)682 1753 y Fr(\012)g Fs(x)g Fr(\012)f Fs(y)844 1760 y Fq(i)869 1753 y Fr(\012)h Fs(:)d(:)g(:)i Fr(\012)h Fs(y)1061 1760 y Fq(n)1098 1753 y Fu(:=)j Fs(\034)1185 1760 y Fq(i)p Fj(\000)p Fl(1)1252 1753 y Fs(:)8 b(:)g(:)g(\034)1339 1760 y Fl(1)1359 1753 y Fu(\()p Fs(z)r Fu(\))17 b Fo(and)412 1851 y Fu([)p Fs(y)450 1858 y Fl(1)469 1851 y Fs(;)8 b(:)g(:)g(:)15 b(;)8 b(y)610 1858 y Fq(i)p Fj(\000)p Fl(1)669 1851 y Fs(;)g Fu([)p Fs(x;)g(y)779 1858 y Fq(i)792 1851 y Fu(])p Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)947 1858 y Fq(n)970 1851 y Fu(])14 b(=)g([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])1178 1858 y Fl(2)1194 1851 y Fs(;)g(:)p Fu(])1244 1858 y Fq(n)1267 1851 y Fs(\034)1288 1858 y Fq(i)p Fj(\000)p Fl(1)1355 1851 y Fs(:)g(:)g(:)g(\034)1442 1858 y Fl(1)1462 1851 y Fu(\()p Fs(z)r Fu(\))p Fs(:)121 b Fu(\(8\))228 1958 y Fo(Pr)n(o)n(of.)19 b Fu(The)d(equation)g(in)g (the)g(Theorem)f(can)i(also)f(b)q(e)h(written)e(as)546 2084 y([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])674 2091 y Fq(n)694 2084 y Fu(])708 2091 y Fl(2)727 2084 y Fu(\()p Fs(z)r Fu(\))14 b(=)875 2030 y Fq(n)856 2043 y Fm(X)857 2134 y Fq(i)p Fl(=1)916 2084 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])1044 2091 y Fl(2)1060 2084 y Fs(;)g(:)p Fu(])1110 2091 y Fq(n)1133 2084 y Fs(\034)1154 2091 y Fq(i)p Fj(\000)p Fl(1)1222 2084 y Fs(:)g(:)g(:)f(\034)1308 2091 y Fl(1)1328 2084 y Fu(\()p Fs(z)r Fu(\))p Fs(:)278 2219 y Fu(Lemma)14 b(8.1)i(together)h(with)24 b(~)-32 b Fs(')p Fu(\()p Fs(i)p Fu(\))13 b(=)h Fs(j)19 b Fu(sho)o(ws)319 2316 y Fr(r)361 2298 y Fq(n)384 2316 y Fs(')p Fu(\(1)12 b Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)g(r)k(\012)8 b Fs(:)g(:)g(:)j Fr(\012)g Fu(1\))p Fs(\034)926 2323 y Fq(i)p Fj(\000)p Fl(1)994 2316 y Fs(:)d(:)g(:)f(\034)1080 2323 y Fl(1)1100 2316 y Fu(\()p Fs(z)r Fu(\))696 2374 y(=)14 b Fr(r)790 2356 y Fq(n)813 2374 y Fu(\(1)e Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)f(r)h(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fu(1\))p Fs(')1331 2382 y Fl(\()p Fq(i)p Fl(\))1373 2374 y Fs(\034)1394 2381 y Fq(i)p Fj(\000)p Fl(1)1461 2374 y Fs(:)d(:)g(:)g(\034)1548 2381 y Fl(1)1568 2374 y Fu(\()p Fs(z)r Fu(\))696 2433 y(=)14 b Fr(r)790 2415 y Fq(n)p Fl(+1)858 2433 y Fs(\034)879 2440 y Fq(j)r Fj(\000)p Fl(1)951 2433 y Fs(:)8 b(:)g(:)g(\034)1038 2440 y Fl(1)1057 2433 y Fu(\(1)k Fr(\012)f Fs(')p Fu(\)\()p Fs(z)r Fu(\);)319 2491 y Fr(r)361 2473 y Fq(n)384 2491 y Fs(')p Fu(\(1)h Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)g(r)k(\012)8 b Fs(:)g(:)g(:)j Fr(\012)g Fu(1\))p Fs(\034)926 2498 y Fq(i)948 2491 y Fs(:)d(:)g(:)g(\034)1035 2498 y Fl(1)1055 2491 y Fu(\()p Fs(z)r Fu(\))696 2549 y(=)14 b Fr(r)790 2531 y Fq(n)813 2549 y Fu(\(1)e Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)f(r)h(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fu(1\))p Fs(')1331 2557 y Fl(\()p Fq(i)p Fl(\))1373 2549 y Fs(\034)1394 2556 y Fq(i)1416 2549 y Fs(:)d(:)g(:)g(\034)1503 2556 y Fl(1)1523 2549 y Fu(\()p Fs(z)r Fu(\))696 2607 y(=)14 b Fr(r)790 2589 y Fq(n)p Fl(+1)858 2607 y Fs(\034)879 2614 y Fq(j)906 2607 y Fs(:)8 b(:)g(:)f(\034)992 2614 y Fl(1)1012 2607 y Fu(\(1)12 b Fr(\012)f Fs(')p Fu(\)\()p Fs(z)r Fu(\);)p eop %%Page: 9 9 9 8 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)r(9)228 225 y Fu(hence)354 292 y Fr(r)396 273 y Fq(n)419 292 y Fs(')p Fu(\(1)11 b Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h(r)16 b(\012)8 b Fs(:)g(:)g(:)i Fr(\012)h Fu(1\))p Fs(\034)960 299 y Fq(i)p Fj(\000)p Fl(1)1028 292 y Fs(:)d(:)g(:)g(\034)1115 299 y Fl(1)1134 292 y Fu(\()p Fs(z)r Fu(\))731 350 y(=)14 b Fr(r)825 332 y Fq(n)p Fl(+1)893 350 y Fs(\034)914 357 y Fq(k)q Fj(\000)p Fl(1)989 350 y Fs(:)8 b(:)g(:)f(\034)1075 357 y Fl(1)1095 350 y Fu(\(1)k Fr(\012)g Fs(')p Fu(\)\()p Fs(z)r Fu(\))731 408 y(=)j Fr(r)825 390 y Fq(n)p Fl(+1)893 408 y Fs(\034)914 415 y Fq(l)935 408 y Fs(:)8 b(:)g(:)g(\034)1022 415 y Fl(1)1042 408 y Fu(\(1)j Fr(\012)g Fs(')p Fu(\)\()p Fs(z)r Fu(\))731 466 y(=)j Fr(r)825 448 y Fq(n)848 466 y Fs(')p Fu(\(1)d Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h(r)f(\012)h Fs(:)d(:)g(:)j Fr(\012)f Fu(1\))p Fs(\034)1386 473 y Fq(j)1413 466 y Fs(:)e(:)g(:)g(\034)1500 473 y Fl(1)1520 466 y Fu(\()p Fs(z)r Fu(\))p Fs(:)1660 438 y Fu(\(9\))228 549 y(for)17 b(all)f Fs(i;)8 b(j)18 b Fu(=)d(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)17 b Fu(with)24 b(~)-31 b Fs(')o Fu(\()p Fs(i)p Fu(\))15 b(=)23 b(~)-32 b Fs(')p Fu(\()p Fs(j)s Fu(\))11 b(+)h(1,)17 b(i.e.)22 b(for)17 b(all)f Fs(i)h Fu(except)24 b(~)-32 b Fs(')1613 531 y Fj(\000)p Fl(1)1660 549 y Fu(\(1\))228 607 y(and)12 b(all)f Fs(j)k Fu(except)k(~)-32 b Fs(')597 589 y Fj(\000)p Fl(1)644 607 y Fu(\()p Fs(n)p Fu(\).)20 b(The)11 b(other)h Fs(i)p Fu('s)f(and)i Fs(j)s Fu('s)e(used)h(in)f(\(9\))i(are)e(in)h(bijectiv)o (e)228 665 y(corresp)q(ondence.)278 723 y(T)l(o)18 b(pro)o(v)o(e)f(the) h(equation)g(of)g(the)g(theorem)e(w)o(e)i(write)f(eac)o(h)h Fs(\033)g Fr(2)f Fs(S)1543 730 y Fq(n)1585 723 y Fu(as)h Fs(\020)1671 705 y Fq(r)1690 723 y Fs(')228 781 y Fu(with)e(a)g (represen)o(tativ)o(e)e Fs(')g Fr(2)g Fs(B)822 788 y Fq(n)862 781 y Fu(and)j(a)f(suitable)g(p)q(o)o(w)o(er)g Fs(\020)1346 763 y Fq(r)1381 781 y Fu(according)g(to)h(\(3\))228 839 y(and)g(use)f(\(9\).)21 b(Then)c(w)o(e)f(get)274 885 y Fm(P)317 898 y Fq(n)317 930 y(i)p Fl(=1)377 918 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])505 925 y Fl(2)521 918 y Fs(;)g(:)p Fu(])571 925 y Fq(n)594 918 y Fs(\034)615 925 y Fq(i)p Fj(\000)p Fl(1)682 918 y Fs(:)g(:)g(:)g (\034)769 925 y Fl(1)789 918 y Fu(\()p 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Fq(n)778 1093 y Fs(\020)803 1074 y Fq(r)822 1093 y Fs(')p Fu(\(1)k Fr(\012)f Fs(:)d(:)g(:)i Fr(\012)h(r)f(\012)h Fs(:)d(:)g(:)i Fr(\012)h Fu(1\))p Fs(\034)1361 1100 y Fq(j)1380 1093 y Fs(\034)1401 1100 y Fq(j)r Fj(\000)p Fl(1)1473 1093 y Fs(:)d(:)g(:)f(\034)1559 1100 y Fl(1)1579 1093 y Fu(\()p Fs(z)r Fu(\))377 1152 y(=)428 1119 y Fm(P)472 1162 y Fq(\033)q Fj(2)p Fq(S)538 1166 y Fh(n)570 1152 y Fr(r)612 1134 y Fq(n)p Fl(+1)680 1152 y Fu(\(1)12 b Fr(\012)e Fs(\020)809 1134 y Fq(r)828 1152 y Fs(')p Fu(\)\()p Fs(z)r Fu(\))h Fr(\000)1003 1119 y Fm(P)1047 1162 y Fq(\033)q Fj(2)p Fq(S)1113 1166 y Fh(n)1145 1152 y Fr(r)1187 1134 y Fq(n)p Fl(+1)1255 1152 y Fs(\034)1276 1159 y Fq(n)1308 1152 y Fs(:)d(:)g(:)f(\034)1394 1159 y Fl(1)1414 1152 y Fu(\(1)12 b Fr(\012)f Fs(\020)1544 1134 y Fq(r)1563 1152 y Fs(')p Fu(\)\()p Fs(z)r Fu(\))377 1210 y(=)428 1177 y Fm(P)472 1220 y Fq(\033)q Fj(2)p Fq(S)538 1224 y Fh(n)570 1210 y Fr(r)612 1192 y Fq(n)p Fl(+1)680 1210 y Fu(\(1)h Fr(\012)e Fs(\020)809 1192 y Fq(r)828 1210 y Fs(')p Fu(\)\()p Fs(z)r Fu(\))h Fr(\000)1003 1177 y Fm(P)1047 1220 y Fq(\033)q Fj(2)p Fq(S)1113 1224 y Fh(n)1145 1210 y Fr(r)1187 1192 y Fq(n)p Fl(+1)1255 1210 y Fu(\()p Fs(\020)1299 1192 y Fq(r)1318 1210 y Fs(')g Fr(\012)g Fu(1\))p Fs(\034)1475 1217 y Fq(n)1507 1210 y Fs(:)d(:)g(:)g(\034)1594 1217 y Fl(1)1614 1210 y Fu(\()p Fs(z)r Fu(\))377 1268 y(=)13 b Fr(r)p Fu(\(1)e Fr(\012)574 1235 y Fm(P)618 1279 y Fq(\033)q Fj(2)p Fq(S)684 1283 y Fh(n)716 1268 y Fr(r)758 1250 y Fq(n)781 1268 y Fs(\033)r Fu(\)\()p Fs(z)r Fu(\))f Fr(\000)h(r)p Fu(\()1014 1235 y Fm(P)1057 1279 y Fq(\033)q Fj(2)p Fq(S)1123 1283 y Fh(n)1155 1268 y Fr(r)1197 1250 y Fq(n)1220 1268 y Fs(\033)i Fr(\012)e Fu(1\))p Fs(\034)1375 1275 y Fq(P)q(;P)1435 1266 y Fh(n)1459 1268 y Fu(\()p Fs(z)r Fu(\))377 1326 y(=)i(\()p Fr(r)e(\000)g(r)p Fs(\034)6 b Fu(\)\(1)k Fr(\012)h Fu([)p Fs(:;)d(:)p Fu(])819 1333 y Fq(n)841 1326 y Fu(\)\()p Fs(z)r Fu(\))377 1384 y(=)13 b([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])556 1391 y Fq(n)577 1384 y Fu(])591 1391 y Fl(2)610 1384 y Fu(\()p Fs(z)r Fu(\))p Fs(:)1683 1461 y Fi(\003)278 1546 y Fu(Clearly)15 b(there)h(are)g(symmetri)o(c)d(righ)o(t)j(sided)g (iden)o(tities.)392 1649 y(4.)25 b Ft(Lie)18 b(Algebras)f(on)h (Yetter-Drinfeld)g(Modules)278 1736 y Fu(No)o(w)j(w)o(e)g(can)h (de\014ne)f(the)h(notion)g(of)g(a)g(Lie)f(algebra)h(in)g(the)f (category)h(of)228 1794 y(Y)l(etter-Drinfeld)14 b(mo)q(dules.)228 1863 y FB(De\014nition)h(4.1.)j Fu(A)13 b(Y)l(etter-Drinfeld)f(mo)q (dule)h Fs(P)21 b Fu(together)14 b(with)g(op)q(erations)228 1921 y(in)i Fr(Y)t(D)361 1903 y Fq(K)360 1933 y(K)600 1980 y Fu([)p Fs(:;)8 b(:)p Fu(])k(:)h Fs(P)18 b Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(P)c Fu(\()p Fs(\020)t Fu(\))14 b(=)g Fs(P)1139 1960 y Fq(n)1163 1980 y Fu(\()p Fs(\020)t Fu(\))g Fr(\000)-31 b(!)14 b Fs(P)228 2050 y Fu(for)21 b(all)g Fs(n)i Fr(2)f Fn(N)g Fu(and)g(all)f(primitiv)o(e)d Fs(n)p Fu(-th)j(ro)q(ots)i(of)f(unit)o(y)e Fs(\020)25 b Fu(is)c(called)g(a)g Fo(Lie)228 2108 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Fq(n)p Fl(+1)955 2571 y Fu(]])13 b(=)1048 2517 y Fq(n)p Fl(+1)1051 2529 y Fm(X)1053 2620 y Fq(i)p Fl(=1)1115 2571 y Fu([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(]])o(\(1)g Fs(:)g(:)g(:)d(i)p Fu(\)\()p Fs(z)r Fu(\))14 b(=)f(0)p Fs(;)p eop %%Page: 10 10 10 9 bop 228 125 a Fp(10)552 b(BODO)13 b(P)m(AREIGIS)385 225 y Fu(where)j(w)o(e)g(use)g(the)g(notation)h(\(4\),)302 283 y(\(3\))k(for)16 b(all)g Fs(n)e Fr(2)g Fn(N)p Fu(,)j(for)f(all)g (primitiv)n(e)d Fs(n)p Fu(-th)k(ro)q(ots)g(of)g(unit)o(y)e Fs(\020)t Fu(,)h(and)g(for)h(all)385 341 y Fs(z)f Fu(=)e Fs(x)c Fr(\012)h Fs(y)588 348 y Fl(1)619 341 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(y)811 348 y Fq(n)848 341 y Fr(2)j Fs(P)933 323 y Fq(n)p Fl(+1)1002 341 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))16 b(w)o(e)g(ha)o(v)o(e)522 455 y([)p Fs(x;)8 b Fu([)p Fs(y)624 462 y Fl(1)643 455 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)784 462 y Fq(n)807 455 y Fu(]])13 b(=)919 401 y Fq(n)900 413 y Fm(X)902 504 y Fq(i)p Fl(=1)960 455 y Fu([)p Fs(y)998 462 y Fl(1)1017 455 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b Fu([)p Fs(x;)g(y)1223 462 y Fq(i)1236 455 y Fu(])p Fs(;)g(:)g(:)g(:)15 b(;)8 b(y)1391 462 y Fq(n)1414 455 y Fu(])385 575 y(where)16 b(w)o(e)g(use)g(the)g(notation)h(\(8\).)228 645 y FB(Corollary)27 b(4.2.)c Fo(L)n(et)h Fs(A)f Fo(b)n(e)i(an)f(algebr)n(a)h(in)f Fr(Y)t(D)1205 627 y Fq(K)1204 657 y(K)1240 645 y Fo(.)42 b(Then)25 b Fs(A)e Fo(c)n(arries)h(the)228 703 y(structur)n(e)17 b(of)h(a)f(Lie)g(algebr)n(a)h Fs(A)820 685 y Fq(L)863 703 y Fo(with)g(the)g(symmetric)f(multiplic)n(ations)466 787 y Fu([)p Fo({)o Fu(])d(:)f Fs(A)596 766 y Fq(n)619 787 y Fu(\()p Fs(\020)t Fu(\))h Fr(\000)-29 b(!)14 b Fs(A)116 b Fo(by)h Fu([)p Fs(z)r Fu(])13 b(:=)1231 745 y Fm(X)1217 837 y Fq(\033)q Fj(2)p Fq(S)1283 841 y Fh(n)1313 787 y Fr(r)1355 766 y Fq(n)1378 787 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))p Fs(:)228 909 y Fo(for)k(al)r(l)h Fs(n)c Fr(2)g Fn(N)19 b Fo(and)e(al)r(l)i(r)n(o)n(ots)d(of)h(unity)h Fs(\020)g Fr(2)c Fs(k)1098 891 y Fj(\003)1118 909 y Fo(.)228 997 y(Pr)n(o)n(of.)19 b Fu(This)13 b(is)g(a)h(rephrasing)f(of)h(the)f (\\an)o(ti"-symmetry)d(iden)o(tit)o(y)h(\(5\))j(and)f(the)228 1055 y(Jacobi)j(iden)o(tities)e(\(6\))j(and)g(\(7\))f(in)g(Theorems)g (3.4)g(and)h(3.5.)309 b Fi(\003)435 1166 y Fu(5.)25 b Ft(The)18 b(Lie)h(Algebra)e(of)h(Primitive)h(Elements)278 1253 y Fu(Let)13 b Fs(A)h Fu(b)q(e)f(an)i(algebra)f(in)f Fr(C)k Fu(=)d Fr(Y)t(D)935 1235 y Fq(K)934 1265 y(K)970 1253 y Fu(.)20 b(Then)14 b Fs(A)6 b Fr(\012)g Fs(A)12 b Fu(is)i(an)g(algebra)g(with)f(the)228 1323 y(m)o(ultipli)o(cation)e Fs(A)5 b Fr(\012)g Fs(A)g Fr(\012)g Fs(A)g Fr(\012)g Fs(A)846 1296 y Fl(1)p Fj(\012)p Fq(\034)t Fj(\012)p Fl(1)861 1323 y Fr(\000)-8 b(!)28 b Fs(A)5 b Fr(\012)g Fs(A)g Fr(\012)g Fs(A)g Fr(\012)g Fs(A)1278 1296 y Fj(r\012r)1281 1323 y Fr(\000)-8 b(!)16 b Fs(A)5 b Fr(\012)g Fs(A)p Fu(.)20 b(Let)14 b Fs(p)g Fu(:)g Fs(A)228 1381 y Fr(\000)-30 b(!)13 b Fs(A)e Fr(\012)f Fs(A)16 b Fu(b)q(e)h(the)f(map)f Fs(p)p Fu(\()p Fs(x)p Fu(\))f(:=)f Fs(x)e Fr(\012)g Fu(1)g(+)g(1)g Fr(\012)g Fs(x)p Fu(.)21 b(Then)16 b Fs(p)p Fu(\(=)e(1)e Fr(\012)f Fs(\021)h Fu(+)f Fs(\021)i Fr(\012)e Fu(1\))228 1439 y(is)16 b(in)g Fr(C)k Fu(but)d Fs(p)g Fu(is)f(not)h(an)g(algebra)g (morphism.)j(Let)d Fs(p)1246 1421 y Fq(n)1284 1439 y Fu(:)d Fs(A)1349 1421 y Fq(n)1387 1439 y Fr(\000)-31 b(!)15 b Fu(\()p Fs(A)10 b Fr(\012)h Fs(A)p Fu(\))1632 1421 y Fq(n)1672 1439 y Fu(b)q(e)228 1497 y(the)16 b Fs(n)p Fu(-fold)g(tensor)h(pro)q(duct)g(of)f Fs(p)h Fu(with)f(itself.) 228 1567 y FB(Lemm)o(a)23 b(5.1.)h Fo(L)n(et)e Fs(H)28 b Fo(b)n(e)23 b(a)g(Hopf)g(algebr)n(a)h(in)f Fr(C)s Fo(.)40 b(Then)23 b Fs(P)7 b Fu(\()p Fs(H)t Fu(\))25 b(:=)f Fr(f)p Fs(x)g Fr(2)228 1626 y Fs(H)t Fr(j)p Fu(\001\()p Fs(x)p Fu(\))13 b(=)h Fs(x)d Fr(\012)f Fu(1)i(+)f(1)g Fr(\012)g Fs(x)p Fr(g)17 b Fo(is)h(a)f(Y)l(etter-Drinfeld)i(submo)n(dule)f(of)g Fs(H)j Fo(in)d Fr(C)s Fo(.)228 1714 y(Pr)n(o)n(of.)h Fs(P)7 b Fu(\()p Fs(H)t Fu(\))14 b(=)g(Ker)o(\(\001)d Fr(\000)g Fs(p)p Fu(\).)864 b Fi(\003)278 1802 y Fu(In)22 b(particular)h(w)o(e)f(ha)o(v)o(e)g Fs(\016)r Fu(\()p Fs(x)p Fu(\))j Fr(2)g Fs(P)7 b Fu(\()p Fs(H)t Fu(\))16 b Fr(\012)f Fs(K)27 b Fu(and)d Fs(x\025)h Fr(2)g Fs(P)7 b Fu(\()p Fs(H)t Fu(\))24 b(for)f(all)228 1860 y Fs(x)13 b Fr(2)h Fs(P)7 b Fu(\()p Fs(H)t Fu(\))17 b(and)g(all)f Fs(\025)e Fr(2)g Fs(K)t Fu(.)228 1930 y FB(Lemm)o(a)i(5.2.)k Fs(p)558 1912 y Fq(n)582 1930 y Fu(\()p Fs(A)638 1912 y Fq(n)661 1930 y Fu(\()p Fs(\020)t Fu(\)\))14 b Fr(\022)g Fu(\()p Fs(A)c Fr(\012)h Fs(A)p Fu(\))982 1912 y Fq(n)1005 1930 y Fu(\()p Fs(\020)t Fu(\))p Fo(.)228 2018 y(Pr)n(o)n(of.)19 b Fu(By)g(Theorem)f(2.5)i Fs(p)g Fu(:)f Fs(A)g Fr(\000)-31 b(!)20 b Fs(A)13 b Fr(\012)g Fs(A)19 b Fu(induces)g Fs(p)1318 2000 y Fq(n)1361 2018 y Fu(:)g Fs(A)1431 2000 y Fq(n)1454 2018 y Fu(\()p Fs(\020)t Fu(\))h Fr(\000)-31 b(!)20 b Fu(\()p Fs(A)12 b Fr(\012)228 2076 y Fs(A)p Fu(\))284 2058 y Fq(n)307 2076 y Fu(\()p Fs(\020)t Fu(\).)1299 b Fi(\003)228 2164 y FB(Theorem)10 b(5.3.)16 b Fo(L)n(et)d Fs(\020)j Fo(b)n(e)d(a)g(primitive)g Fs(n)p Fo(-th)g(r)n(o)n(ot)e(of)i (unity)g(and)g(let)g Fs(z)j Fr(2)e Fs(A)1621 2146 y Fq(n)1644 2164 y Fu(\()p Fs(\020)t Fu(\))p Fo(.)228 2223 y(Then)809 2285 y Fu([)p Fs(p)847 2264 y Fq(n)871 2285 y Fu(\()p Fs(z)r Fu(\)])f(=)h Fs(p)p Fu(\([)p Fs(z)r Fu(]\))p Fs(:)228 2373 y Fo(Pr)n(o)n(of.)19 b Fu(If)i Fs(z)j Fu(=)539 2340 y Fm(P)582 2383 y Fq(k)612 2373 y Fs(x)640 2380 y Fq(k)q(;)p Fl(1)703 2373 y Fr(\012)14 b Fs(:)8 b(:)g(:)14 b Fr(\012)g Fs(x)909 2380 y Fq(k)q(;n)984 2373 y Fr(2)23 b Fs(A)1077 2355 y Fq(n)1100 2373 y Fu(\()p Fs(\020)t Fu(\))e(then)g(the)h (equation)f(of)g(the)228 2431 y(theorem)14 b(reads)j(as)388 2500 y([)402 2467 y Fm(P)446 2511 y Fq(k)467 2500 y Fu(\()p Fs(x)514 2507 y Fq(k)q(;)p Fl(1)573 2500 y Fr(\012)11 b Fu(1)h(+)f(1)g Fr(\012)g Fs(x)821 2507 y Fq(k)q(;)p Fl(1)869 2500 y Fu(\))g Fr(\012)g Fs(:)d(:)g(:)j Fr(\012)f Fu(\()p Fs(x)1114 2507 y Fq(k)q(;n)1178 2500 y Fr(\012)h Fu(1)g(+)g(1)g Fr(\012)g Fs(x)1425 2507 y Fq(k)q(;n)1478 2500 y Fu(\)])i(=)412 2558 y([)426 2525 y Fm(P)470 2569 y Fq(k)499 2558 y Fs(x)527 2565 y Fq(k)q(;)p Fl(1)587 2558 y Fr(\012)e Fs(:)d(:)g(:)i Fr(\012)h Fs(x)783 2565 y Fq(k)q(;n)835 2558 y Fu(])g Fr(\012)g Fu(1)g(+)g(1)h Fr(\012)e Fu([)1093 2525 y Fm(P)1137 2569 y Fq(k)1166 2558 y Fs(x)1194 2565 y Fq(k)q(;)p Fl(1)1254 2558 y Fr(\012)h Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1450 2565 y Fq(k)q(;n)1502 2558 y Fu(])d Fs(:)1635 2588 y Fu(\(10\))p eop %%Page: 11 11 11 10 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(11)228 225 y Fu(W)l(e)16 b(w)o(an)o(t)g(to)h(ev)m(aluate)249 304 y([)262 270 y Fm(P)306 314 y Fq(k)327 304 y Fu(\()p Fs(x)374 311 y Fq(k)q(;)p Fl(1)434 304 y Fr(\012)11 b Fu(1)g(+)g(1)h Fr(\012)e Fs(x)681 311 y Fq(k)q(;)p Fl(1)730 304 y Fu(\))h Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fu(\()p Fs(x)975 311 y Fq(k)q(;n)1038 304 y Fr(\012)g Fu(1)g(+)g(1)h Fr(\012)f Fs(x)1286 311 y Fq(k)q(;n)1338 304 y Fu(\)])346 362 y(=)398 329 y Fm(P)442 372 y Fq(\033)q Fj(2)p Fq(S)508 376 y Fh(n)540 362 y Fr(r)582 344 y Fq(n)605 362 y Fs(\033)r Fu(\()654 329 y Fm(P)697 372 y Fq(i)711 362 y Fu(\()p Fs(x)758 369 y Fq(k)q(;)p Fl(1)818 362 y Fr(\012)f Fu(1)i(+)f(1)g Fr(\012)g Fs(x)1065 369 y Fq(k)q(;)p Fl(1)1114 362 y Fu(\))g Fr(\012)f Fs(:)e(:)g(:)j Fr(\012)g Fu(\()p Fs(x)1359 369 y Fq(k)q(;n)1422 362 y Fr(\012)g Fu(1)g(+)g(1)h Fr(\012)e Fs(x)1669 369 y Fq(k)q(;n)1722 362 y Fu(\)\))228 454 y(where)16 b Fs(\033)g Fr(2)f Fs(S)491 461 y Fq(n)531 454 y Fu(op)q(erates)i(on)g Fs(p)817 436 y Fq(n)841 454 y Fu(\()p Fs(z)r Fu(\))e Fr(2)f Fu(\()p Fs(A)d Fr(\012)g Fs(A)p Fu(\))1139 436 y Fq(n)1162 454 y Fu(\()p Fs(\020)t Fu(\))17 b(as)g(describ)q(ed)f(in)h(section)228 513 y(2.)278 571 y(Let)i Fs(x;)8 b(y)21 b Fr(2)e Fs(A)g Fu(and)h Fs(\034)6 b Fu(\()p Fs(x)13 b Fr(\012)g Fs(y)r Fu(\))19 b(=)929 537 y Fm(P)973 581 y Fq(i)996 571 y Fs(u)1024 578 y Fq(i)1051 571 y Fr(\012)13 b Fs(v)1127 578 y Fq(i)1140 571 y Fu(.)31 b(Then)20 b(\(1)14 b Fr(\012)f Fs(x)p Fu(\))g Fr(\001)g Fu(\()p Fs(y)i Fr(\012)e Fu(1\))19 b(=)228 629 y(\()p Fr(r)5 b(\012)g(r)p Fu(\)\()418 596 y Fm(P)459 639 y Fq(i)482 629 y Fu(1)g Fr(\012)g Fs(u)583 636 y Fq(i)600 629 y Fr(\012)g Fs(v)668 636 y Fq(i)686 629 y Fr(\012)g Fu(1\))13 b(=)838 596 y Fm(P)882 639 y Fq(i)904 629 y Fs(u)932 636 y Fq(i)950 629 y Fr(\012)5 b Fs(v)1018 636 y Fq(i)1045 629 y Fu(=)14 b Fs(\034)6 b Fu(\()p Fs(x)f Fr(\012)g Fs(y)r Fu(\))11 b(=)1328 596 y Fm(P)1372 639 y Fq(i)1386 629 y Fu(\()p Fs(u)1433 636 y Fq(i)1451 629 y Fr(\012)5 b Fu(1\))g Fr(\001)g Fu(\(1)g Fr(\012)g Fs(v)1678 636 y Fq(i)1690 629 y Fu(\).)228 687 y(So)17 b(w)o(e)e(ha)o(v)o(e)696 765 y(\()p Fs(x)c Fr(\012)g Fu(1\)\()p Fs(y)i Fr(\012)d Fu(1\))15 b(=)e(\()p Fs(xy)g Fr(\012)e Fu(1\))p Fs(;)696 824 y Fu(\()p Fs(x)g Fr(\012)g Fu(1\)\(1)g Fr(\012)g Fs(y)r Fu(\))j(=)f(\()p Fs(x)e Fr(\012)g Fs(y)r Fu(\))p Fs(;)696 882 y Fu(\(1)h Fr(\012)e Fs(x)p Fu(\)\(1)h Fr(\012)g Fs(y)r Fu(\))j(=)f(\(1)f Fr(\012)f Fs(xy)r Fu(\))p Fs(;)696 940 y Fu(\(1)h Fr(\012)e Fs(x)p Fu(\)\()p Fs(y)j Fr(\012)d Fu(1\))15 b(=)e Fs(\034)6 b Fu(\()p Fs(x)11 b Fr(\012)g Fs(y)r Fu(\))p Fs(:)1635 853 y Fu(\(11\))278 1032 y(W)l(e)17 b(expand)i(a)f(pro)q(duct)h(\()p Fs(x)809 1039 y Fl(1)840 1032 y Fr(\012)12 b Fu(1)h(+)f(1)h Fr(\012)f Fs(x)1094 1039 y Fl(1)1114 1032 y Fu(\))g Fr(\001)g Fs(:)c(:)g(:)k Fr(\001)g Fu(\()p Fs(x)1314 1039 y Fq(n)1349 1032 y Fr(\012)g Fu(1)h(+)f(1)h Fr(\012)f Fs(x)1603 1039 y Fq(n)1626 1032 y Fu(\).)27 b(It)228 1091 y(pro)q(duces)14 b(after)g(m)o(ultipli)o (cation)d(2)877 1073 y Fq(n)915 1091 y Fu(summands,)h(eac)o(h)h(a)h (pro)q(duct)g(of)g Fs(n)g Fu(terms.)228 1149 y(A)k(t)o(ypical)f(pro)q (duct)j(is)e(\()p Fs(x)729 1156 y Fl(1)761 1149 y Fr(\012)13 b Fu(1\)\(1)g Fr(\012)g Fs(x)992 1156 y Fl(2)1011 1149 y Fu(\)\()p Fs(x)1077 1156 y Fl(3)1109 1149 y Fr(\012)g Fu(1\))8 b Fs(:)g(:)g(:)g Fu(,)19 b(some)e(of)i(the)g(factors)228 1207 y(b)q(eing)g(of)h(the)e(form)g Fs(x)653 1214 y Fq(j)684 1207 y Fr(\012)13 b Fu(1,)20 b(the)f(others)g(of)h(the)f(form)f(1)13 b Fr(\012)g Fs(x)1411 1214 y Fq(j)1429 1207 y Fu(.)30 b(T)l(o)19 b(ev)m(aluate)228 1265 y(suc)o(h)i(a)h(pro)q(duct)g(w)o(e)f (use)g(the)g(rule)g(of)h(m)o(ultipli)o(cation)d(in)i Fs(A)14 b Fr(\012)g Fs(A)21 b Fu(giv)o(en)g(b)o(y)228 1323 y Fr(r)270 1330 y Fq(A)p Fj(\012)p Fq(A)366 1323 y Fu(=)13 b(\()p Fr(r)e(\012)g(r)p Fu(\)\(1)g Fr(\012)f Fs(\034)17 b Fr(\012)11 b Fu(1\).)278 1381 y(T)l(o)18 b(explain)g(the)f(follo)o(wing)h(calculation)g(w)o(e)f(consider)h(as)h (an)g(example)c(the)228 1439 y(pro)q(duct)i(\()p Fs(x)458 1446 y Fl(1)488 1439 y Fr(\012)10 b Fu(1\)\(1)i Fr(\012)e Fs(x)712 1446 y Fl(2)732 1439 y Fu(\)\()p Fs(x)798 1446 y Fl(3)828 1439 y Fr(\012)g Fu(1\)\(1)i Fr(\012)e Fs(x)1052 1446 y Fl(4)1072 1439 y Fu(\)\()p Fs(x)1138 1446 y Fl(5)1168 1439 y Fr(\012)g Fu(1\).)22 b(It)15 b(is)h(calculated)f(with)228 1497 y(the)h(follo)o(wing)g(braid)g(diagram)p 975 1588 2 2 v 975 1589 V 975 1589 V 975 1590 V 975 1591 V 975 1591 V 975 1592 V 974 1593 V 974 1593 V 974 1594 V 974 1594 V 974 1595 V 973 1596 V 973 1596 V 973 1597 V 972 1597 V 972 1598 V 972 1598 V 971 1599 V 971 1599 V 970 1600 V 970 1600 V 969 1601 V 969 1601 V 968 1602 V 968 1602 V 967 1603 V 967 1603 V 966 1604 V 965 1604 V 965 1605 V 964 1605 V 963 1606 V 962 1606 V 962 1606 V 961 1607 V 928 1635 V 928 1634 V 928 1634 V 928 1633 V 928 1633 V 928 1632 V 928 1631 V 928 1631 V 929 1630 V 929 1629 V 929 1629 V 929 1628 V 929 1628 V 930 1627 V 930 1626 V 930 1626 V 931 1625 V 931 1625 V 932 1624 V 932 1624 V 932 1623 V 933 1623 V 933 1622 V 934 1622 V 934 1621 V 935 1621 V 936 1620 V 936 1620 V 937 1619 V 938 1619 V 938 1618 V 939 1618 V 940 1618 V 940 1617 V 941 1617 V 942 1616 V 928 1588 V 928 1589 V 928 1589 V 928 1590 V 928 1591 V 928 1591 V 929 1592 V 929 1593 V 929 1593 V 929 1594 V 930 1595 V 930 1595 V 931 1596 V 931 1597 V 932 1597 V 932 1598 V 933 1599 V 933 1599 V 934 1600 V 935 1601 V 936 1601 V 936 1602 V 937 1603 V 938 1603 V 939 1604 V 940 1605 V 941 1606 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1617 V 1036 1616 V 1022 1588 V 1022 1589 V 1022 1589 V 1022 1590 V 1022 1591 V 1023 1591 V 1023 1592 V 1023 1593 V 1023 1593 V 1024 1594 V 1024 1595 V 1025 1595 V 1025 1596 V 1025 1597 V 1026 1597 V 1027 1598 V 1027 1599 V 1028 1599 V 1028 1600 V 1029 1601 V 1030 1601 V 1031 1602 V 1032 1603 V 1032 1603 V 1033 1604 V 1034 1605 V 1035 1606 V 1036 1606 V 1037 1607 V 1038 1608 V 1040 1608 V 1041 1609 V 1042 1610 V 1043 1610 V 1044 1611 V 1046 1612 V 1069 1635 V 1069 1634 V 1069 1634 V 1069 1633 V 1069 1632 V 1069 1632 V 1069 1631 V 1068 1630 V 1068 1630 V 1068 1629 V 1067 1628 V 1067 1628 V 1067 1627 V 1066 1626 V 1066 1626 V 1065 1625 V 1064 1624 V 1064 1624 V 1063 1623 V 1062 1622 V 1062 1622 V 1061 1621 V 1060 1620 V 1059 1620 V 1058 1619 V 1057 1618 V 1056 1618 V 1055 1617 V 1054 1616 V 1053 1616 V 1052 1615 V 1051 1614 V 1050 1614 V 1048 1613 V 1047 1612 V 1046 1612 V 880 1682 2 95 v 1069 1682 2 48 v 927 1729 2 95 v 1045 1729 2 24 v 879 1689 2 7 v 879 1707 a Fe(\006)p 973 1689 V 61 w(\005)p 897 1707 62 2 v 1021 1689 2 7 v 15 w(\006)p 1068 1689 V 14 w(\005)p 1039 1707 14 2 v 228 1840 a Fu(The)f(second)g(and)h(fourth)f(factors)h(are)f(pulled)f (o)o(v)o(er)g(to)i(the)e(righ)o(t)h(and)h(then)f(all)228 1899 y(factors)20 b(are)h(m)o(ultipli)o(ed)c(according)j(to)h(\(11\).) 33 b(Th)o(us)21 b(w)o(e)e(ha)o(v)o(e)h(\()p Fs(x)1498 1906 y Fl(1)1531 1899 y Fr(\012)13 b Fu(1\)\(1)h Fr(\012)228 1957 y Fs(x)256 1964 y Fl(2)275 1957 y Fu(\)\()p Fs(x)341 1964 y Fl(3)367 1957 y Fr(\012)6 b Fu(1\)\(1)g Fr(\012)g Fs(x)577 1964 y Fl(4)598 1957 y Fu(\)\()p Fs(x)664 1964 y Fl(5)690 1957 y Fr(\012)g Fu(1\))14 b(=)g(\()p Fr(r)905 1939 y Fl(3)931 1957 y Fr(\012)6 b(r)1018 1939 y Fl(2)1037 1957 y Fu(\))p Fs(')p Fu(\()p Fs(x)1135 1964 y Fl(1)1161 1957 y Fr(\012)g Fs(x)1234 1964 y Fl(2)1260 1957 y Fr(\012)g Fs(x)1333 1964 y Fl(3)1359 1957 y Fr(\012)g Fs(x)1432 1964 y Fl(4)1458 1957 y Fr(\012)g Fs(x)1531 1964 y Fl(5)1551 1957 y Fu(\),)14 b(where)228 2015 y Fs(')g Fu(=)f Fs(\034)346 2022 y Fl(3)366 2015 y Fs(\034)387 2022 y Fl(4)407 2015 y Fs(\034)428 2022 y Fl(2)464 2015 y Fu(as)k(de\014ned)f(b)o(y)g(the)g (giv)o(en)f(braid)i(diagram.)278 2073 y(W)l(e)d(pro)o(v)o(e)g(no)o(w)h (b)o(y)f(induction)h(on)g Fs(n)g Fu(that)g(for)g(ev)o(ery)e(pro)q(duct) j(\()p Fs(x)1515 2080 y Fl(1)1542 2073 y Fr(\012)8 b Fu(1\)\(1)g Fr(\012)228 2131 y Fs(x)256 2138 y Fl(2)275 2131 y Fu(\)\()p Fs(x)341 2138 y Fl(3)369 2131 y Fr(\012)g Fu(1\))g Fs(:)g(:)g(:)22 b Fu(with)15 b Fs(i)f Fu(factors)h(of)g(the)g (form)e Fs(x)1123 2138 y Fq(j)1149 2131 y Fr(\012)8 b Fu(1)15 b(and)g Fs(n)8 b Fr(\000)g Fs(i)14 b Fu(factors)h(of)g(the)228 2189 y(form)g(1)c Fr(\012)g Fs(x)456 2196 y Fq(j)490 2189 y Fu(there)16 b(is)g(an)h(elemen)o(t)c Fs(')h Fr(2)g Fs(B)1042 2196 y Fq(n)1082 2189 y Fu(suc)o(h)i(that)343 2283 y(\()p Fs(x)390 2290 y Fl(1)421 2283 y Fr(\012)11 b Fu(1\)\(1)h Fr(\012)e Fs(x)646 2290 y Fl(2)666 2283 y Fu(\)\()p Fs(x)732 2290 y Fl(3)762 2283 y Fr(\012)h Fu(1\))d Fs(:)g(:)g(:)14 b Fu(=)g(\()p Fr(r)1048 2263 y Fq(i)1072 2283 y Fr(\012)d(r)1164 2263 y Fq(n)p Fj(\000)p Fq(i)1226 2283 y Fu(\))p Fs(')p Fu(\()p Fs(x)1324 2290 y Fl(1)1355 2283 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1551 2290 y Fq(n)1574 2283 y Fu(\))p Fs(:)228 2375 y Fu(F)l(urthermore)19 b(if)h Fs(t)h Fu(denotes)g(the)g(n)o(um)o(b)q (er)e(of)i(pairs)g(of)g(factors)h Fs(f)1487 2382 y Fl(1)1506 2375 y Fs(;)8 b(f)1552 2382 y Fl(2)1593 2375 y Fu(in)20 b(the)228 2433 y(pro)q(duct)15 b(\()p Fs(x)456 2440 y Fl(1)482 2433 y Fr(\012)7 b Fu(1\)\(1)g Fr(\012)g Fs(x)695 2440 y Fl(2)715 2433 y Fu(\)\()p Fs(x)781 2440 y Fl(3)808 2433 y Fr(\012)g Fu(1\))h Fs(:)g(:)g(:)22 b Fu(where)14 b Fs(f)1148 2440 y Fl(1)1182 2433 y Fu(is)g(to)h(the)f(left)f(of)i Fs(f)1530 2440 y Fl(2)1550 2433 y Fu(,)f Fs(f)1602 2440 y Fl(1)1636 2433 y Fu(is)g(of)228 2492 y(the)i(form)f(\(1)c Fr(\012)f Fs(x)558 2499 y Fq(j)576 2492 y Fu(\))16 b(and)h Fs(f)730 2499 y Fl(2)766 2492 y Fu(is)f(of)g(the)g(form)f(\()p Fs(x)1116 2499 y Fq(j)1145 2492 y Fr(\012)10 b Fu(1\),)16 b(or)h(brie\015y)e(the)h(n)o(um)o(b)q(er)228 2550 y(of)h(factors)g(in)g (rev)o(erse)e(p)q(osition,)j(then)e Fs(')h Fu(is)g(comp)q(osed)f(of)h Fs(t)g Fu(generators)h Fs(\034)1648 2557 y Fq(j)1683 2550 y Fu(of)228 2608 y Fs(B)265 2615 y Fq(n)288 2608 y Fu(.)k(Observ)o(e)15 b(that)h Fs(')g Fu(and)h(the)f(n)o(um)o(b)q(er)e Fs(t)i Fu(are)g(uniquely)e(determined)g(b)o(y)h(the)p eop %%Page: 12 12 12 11 bop 228 125 a Fp(12)552 b(BODO)13 b(P)m(AREIGIS)228 225 y Fu(prop)q(erties)g(of)g(the)f(m)o(ultiplication)e(of)j Fs(A)t Fr(\012)t Fs(A)f Fu(and)h(the)g(braid)g(group)g Fs(B)1548 232 y Fq(n)1572 225 y Fu(,)g(whic)o(h)228 283 y(has)k(homogeneous)f(relations.)278 341 y(F)l(or)23 b Fs(n)j Fu(=)g(1)e(w)o(e)f(ha)o(v)o(e)g(the)g(trivial)f(cases)i Fs(x)15 b Fr(\012)h Fu(1)27 b(=)e(\()p Fr(r)1384 323 y Fl(1)1420 341 y Fr(\012)15 b(r)1516 323 y Fl(0)1536 341 y Fu(\)\()p Fs(x)p Fu(\))23 b(and)228 399 y(1)12 b Fr(\012)f Fs(x)16 b Fu(=)f(\()p Fr(r)472 381 y Fl(0)503 399 y Fr(\012)c(r)595 381 y Fl(1)615 399 y Fu(\)\()p Fs(x)p Fu(\),)16 b(where)h Fr(r)914 381 y Fl(1)949 399 y Fu(=)e(id)i(and)h Fr(r)1198 381 y Fl(0)1233 399 y Fu(=)d(1.)24 b(F)l(or)18 b(the)f(induction)228 458 y(nothing)f(is)g(to)g(b)q(e)g (pro)o(v)o(ed)f(if)g Fs(i)f Fu(=)g Fs(n)i Fu(or)g Fs(i)d Fu(=)h(0.)22 b(In)15 b(these)h(cases)g(w)o(e)f(ha)o(v)o(e)g Fs(t)e Fu(=)h(0.)278 516 y(W)l(e)i(assume)f(no)o(w)i(that)g(the)f (claim)e(is)i(true)g(for)h Fs(n)p Fu(.)k(The)c(induction)f(step)g(for) 228 574 y Fs(i)d Fr(6)p Fu(=)h(0)p Fs(;)8 b(n)j Fu(+)g(1)17 b(is)f(giv)o(en)g(b)o(y)249 660 y(\()p Fs(x)296 667 y Fl(1)326 660 y Fr(\012)11 b Fu(1\))16 b Fr(\001)p Fu(\(1)c Fr(\012)f Fs(x)582 667 y Fl(2)601 660 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)703 667 y Fl(3)733 660 y Fr(\012)g Fu(1\))h Fr(\001)f Fs(:)d(:)g(:)i Fr(\001)h Fu(\(1)g Fr(\012)g Fs(x)1088 667 y Fq(n)p Fl(+1)1156 660 y Fu(\))j(=)435 718 y(=)g Fr(f)p Fu(\()p Fr(r)573 700 y Fq(i)598 718 y Fr(\012)d(r)690 700 y Fq(n)p Fj(\000)p Fq(i)752 718 y Fu(\))p Fs(')p Fu(\()p Fs(x)850 725 y Fl(1)880 718 y Fr(\012)g Fs(:)d(:)g(:)j Fr(\012)f Fs(x)1076 725 y Fq(n)1100 718 y Fu(\))p Fr(g)h(\001)f Fu(\(1)i Fr(\012)f Fs(x)1312 725 y Fq(n)p Fl(+1)1380 718 y Fu(\))435 776 y(=)j Fr(f)p Fu(\()p Fr(r)573 758 y Fq(i)598 776 y Fr(\012)d(r)690 758 y Fq(n)p Fj(\000)p Fq(i)752 776 y Fu(\))779 743 y Fm(P)823 786 y Fq(k)844 776 y Fu(\()p Fs(u)891 783 y Fq(k)q(;)p Fl(1)951 776 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(u)1147 783 y Fq(k)q(;n)1199 776 y Fu(\))p Fr(g)g(\001)g Fu(\(1)h Fr(\012)e Fs(x)1411 783 y Fq(n)p Fl(+1)1480 776 y Fu(\))435 834 y(=)k(\()506 801 y Fm(P)550 845 y Fq(k)580 834 y Fs(u)608 841 y Fq(k)q(;)p Fl(1)667 834 y Fr(\001)d Fs(:)d(:)g(:)i Fr(\001)h Fs(u)813 841 y Fq(k)q(;i)867 834 y Fr(\012)g Fs(u)945 841 y Fq(k)q(;i)p Fl(+1)1044 834 y Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fs(u)1190 841 y Fq(k)q(;n)1243 834 y Fu(\))g Fr(\001)g Fu(\(1)g Fr(\012)g Fs(x)1430 841 y Fq(n)p Fl(+1)1498 834 y Fu(\))435 892 y(=)487 859 y Fm(P)531 903 y Fq(k)552 892 y Fu(\()p Fs(u)599 899 y Fq(k)q(;)p Fl(1)659 892 y Fr(\012)g Fu(1\))g Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fu(\()p Fs(u)928 899 y Fq(k)q(;i)982 892 y Fr(\012)g Fu(1\))h Fr(\001)e Fu(\(1)i Fr(\012)f Fs(u)1244 899 y Fq(k)q(;i)p Fl(+1)1332 892 y Fu(\))g Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fu(\(1)g Fr(\012)g Fs(u)1612 899 y Fq(k)q(;n)1664 892 y Fu(\))g Fr(\001)g Fu(\(1)h Fr(\012)f Fs(x)1852 899 y Fq(n)p Fl(+1)1920 892 y Fu(\))435 950 y(=)j(\()p Fr(r)548 932 y Fq(i)573 950 y Fr(\012)d(r)665 932 y Fq(n)p Fj(\000)p Fq(i)p Fl(+1)772 950 y Fu(\))799 917 y Fm(P)843 961 y Fq(k)865 950 y Fu(\()p Fs(u)912 957 y Fq(k)q(;)p Fl(1)971 950 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(u)1167 957 y Fq(k)q(;n)1231 950 y Fr(\012)f Fs(x)1308 957 y Fq(n)p Fl(+1)1377 950 y Fu(\))435 1008 y(=)k(\()p Fr(r)548 990 y Fq(i)573 1008 y Fr(\012)d(r)665 990 y Fq(n)p Fj(\000)p Fq(i)p Fl(+1)772 1008 y Fu(\)\()p Fs(')g Fr(\012)g Fu(1\)\()p Fs(x)993 1015 y Fl(1)1024 1008 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1220 1015 y Fq(n)p Fl(+1)1288 1008 y Fu(\))228 1096 y(where)21 b Fs(t)p Fu(,)g(the)g(n)o(um)o(b)q(er)e(of)j(factors)f(in)g(rev)o(erse) f(p)q(osition,)i(do)q(es)g(not)g(c)o(hange,)228 1154 y(neither)15 b(do)q(es)j(the)e(n)o(um)o(b)q(er)f(of)i(generators)g Fs(\034)1079 1161 y Fq(i)1110 1154 y Fu(used)g(in)f(the)h(represen)o (tation)f(of)228 1212 y Fs(')11 b Fr(\012)g Fu(1.)21 b(The)c(second)f(p)q(ossibilit)o(y)f(is)249 1298 y(\()p Fs(x)296 1305 y Fl(1)326 1298 y Fr(\012)c Fu(1\))16 b Fr(\001)p Fu(\(1)c Fr(\012)f Fs(x)582 1305 y Fl(2)601 1298 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)703 1305 y Fl(3)733 1298 y Fr(\012)g Fu(1\))h Fr(\001)f Fs(:)d(:)g(:)i Fr(\001)h Fu(\()p Fs(x)1003 1305 y Fq(n)p Fl(+1)1082 1298 y Fr(\012)g Fu(1\))435 1357 y(=)j Fr(f)p Fu(\()p Fr(r)573 1339 y Fq(i)598 1357 y Fr(\012)d(r)690 1339 y Fq(n)p Fj(\000)p Fq(i)752 1357 y Fu(\))p Fs(')p Fu(\()p Fs(x)850 1364 y Fl(1)880 1357 y Fr(\012)g Fs(:)d(:)g(:)j Fr(\012)f Fs(x)1076 1364 y Fq(n)1100 1357 y Fu(\))p Fr(g)h(\001)f Fu(\()p Fs(x)1226 1364 y Fq(n)p Fl(+1)1306 1357 y Fr(\012)h Fu(1\))435 1415 y(=)j(\()506 1382 y Fm(P)550 1425 y Fq(k)580 1415 y Fs(u)608 1422 y Fq(k)q(;)p Fl(1)667 1415 y Fr(\001)d Fs(:)d(:)g(:)i Fr(\001)h Fs(u)813 1422 y Fq(k)q(;i)867 1415 y Fr(\012)g Fs(u)945 1422 y Fq(k)q(;i)p Fl(+1)1044 1415 y Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fs(u)1190 1422 y Fq(k)q(;n)1243 1415 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)1345 1422 y Fq(n)p Fl(+1)1424 1415 y Fr(\012)g Fu(1\))435 1473 y(=)j(\()506 1440 y Fm(P)550 1483 y Fq(k)580 1473 y Fs(u)608 1480 y Fq(k)q(;)p Fl(1)667 1473 y Fr(\001)d Fs(:)d(:)g(:)i Fr(\001)h Fs(u)813 1480 y Fq(k)q(;i)867 1473 y Fr(\012)g Fs(u)945 1480 y Fq(k)q(;i)p Fl(+1)1044 1473 y Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fs(u)1190 1480 y Fq(k)q(;n)p Fj(\000)p Fl(1)1288 1473 y Fu(\))g Fr(\001)g Fu(\(1)g Fr(\012)g Fs(u)1475 1480 y Fq(k)q(;n)1527 1473 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)1629 1480 y Fq(n)p Fl(+1)1709 1473 y Fr(\012)f Fu(1\))290 1602 y(=)j(\()360 1569 y Fm(P)404 1613 y Fq(k)434 1602 y Fs(u)462 1609 y Fq(k)q(;)p Fl(1)521 1602 y Fr(\001)e Fs(:)d(:)g(:)j Fr(\001)g Fs(u)668 1609 y Fq(k)q(;i)722 1602 y Fr(\012)f Fs(u)799 1609 y Fq(k)q(;i)p Fl(+1)898 1602 y Fr(\001)h Fs(:)d(:)g(:)j Fr(\001)g Fs(u)1045 1609 y Fq(k)q(;n)p Fj(\000)p Fl(1)1142 1602 y Fu(\))g Fr(\001)g Fu(\()p Fs(v)1240 1609 y Fq(k)q(;n)1303 1602 y Fr(\012)g Fu(1\))g Fr(\001)g Fu(\(1)h Fr(\012)e Fs(v)1560 1609 y Fq(k)q(;n)p Fl(+1)1658 1602 y Fu(\))290 1660 y(=)j(\()p Fr(r)402 1642 y Fq(i)p Fl(+1)472 1660 y Fr(\012)e(r)564 1642 y Fq(n)p Fj(\000)p Fq(i)p Fj(\000)p Fl(1)671 1660 y Fu(\))p Fs(\032)p Fu(\()734 1627 y Fm(P)778 1671 y Fq(k)808 1660 y Fs(u)836 1667 y Fq(k)q(;)p Fl(1)896 1660 y Fr(\012)f Fs(:)e(:)g(:)j Fr(\012)g Fs(u)1092 1667 y Fq(k)q(;n)p Fj(\000)p Fl(1)1200 1660 y Fr(\012)g Fs(v)1274 1667 y Fq(k)q(;n)1326 1660 y Fu(\))g Fr(\001)g Fu(\(1)g Fr(\012)g Fs(v)1509 1667 y Fq(k)q(;n)p Fl(+1)1606 1660 y Fu(\))290 1719 y(=)i(\()p Fr(r)402 1700 y Fq(i)p Fl(+1)472 1719 y Fr(\012)e(r)564 1700 y Fq(n)p Fj(\000)p Fq(i)626 1719 y Fu(\)\()p Fs(\032)h Fr(\012)e Fu(1\)\()812 1685 y Fm(P)857 1729 y Fq(k)886 1719 y Fs(u)914 1726 y Fq(k)q(;)p Fl(1)974 1719 y Fr(\012)h Fs(:)d(:)g(:)i Fr(\012)h Fs(u)1170 1726 y Fq(k)q(;n)p Fj(\000)p Fl(1)1278 1719 y Fr(\012)g Fs(v)1352 1726 y Fq(k)q(;n)1415 1719 y Fr(\012)g Fs(v)1489 1726 y Fq(k)q(;n)p Fl(+1)1586 1719 y Fu(\))290 1777 y(=)i(\()p Fr(r)402 1759 y Fq(i)p Fl(+1)472 1777 y Fr(\012)e(r)564 1759 y Fq(n)p Fj(\000)p Fq(i)626 1777 y Fu(\)\()p Fs(\032)h Fr(\012)e Fu(1\)\(1)836 1759 y Fq(n)p Fj(\000)p Fl(1)917 1777 y Fr(\012)h Fs(\034)6 b Fu(\)\()p Fs(')k Fr(\012)h Fu(1\)\()p Fs(x)1214 1784 y Fl(1)1245 1777 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1441 1784 y Fq(n)p Fl(+1)1509 1777 y Fu(\))p Fs(:)228 1867 y Fu(where)j Fs(')p Fu(\()p Fs(x)446 1874 y Fl(1)473 1867 y Fr(\012)7 b Fs(:)h(:)g(:)e Fr(\012)h Fs(x)657 1874 y Fq(n)680 1867 y Fu(\))14 b(=)765 1834 y Fm(P)809 1878 y Fq(k)838 1867 y Fs(u)866 1874 y Fq(k)q(;)p Fl(1)922 1867 y Fr(\012)7 b Fs(:)h(:)g(:)f Fr(\012)g Fs(u)1107 1874 y Fq(k)q(;n)1159 1867 y Fu(,)15 b Fs(\034)6 b Fu(\()p Fs(u)1262 1874 y Fq(k)q(;n)1321 1867 y Fr(\012)h Fs(x)1395 1874 y Fq(n)p Fl(+1)1463 1867 y Fu(\))14 b(=)1548 1834 y Fm(P)1600 1867 y Fs(v)1624 1874 y Fq(k)q(;n)1683 1867 y Fr(\012)228 1925 y Fs(v)252 1932 y Fq(k)q(;n)p Fl(+1)349 1925 y Fu(,)j(and)i(\(1)520 1907 y Fq(n)p Fj(\000)p Fl(1)600 1925 y Fr(\012)12 b Fs(\034)6 b Fu(\)\()p Fs(')12 b Fr(\012)f Fu(1\)\()p Fs(x)900 1932 y Fl(1)932 1925 y Fr(\012)h Fs(:)c(:)g(:)j Fr(\012)h Fs(x)1131 1932 y Fq(n)1166 1925 y Fr(\012)f Fs(x)1244 1932 y Fq(n)p Fl(+1)1313 1925 y Fu(\))16 b(=)1402 1892 y Fm(P)1445 1936 y Fq(k)1475 1925 y Fs(u)1503 1932 y Fq(k)q(;)p Fl(1)1564 1925 y Fr(\012)11 b Fs(:)d(:)g(:)j Fr(\012)228 1983 y Fs(u)256 1990 y Fq(k)q(;n)p Fj(\000)p Fl(1)368 1983 y Fr(\012)j Fs(v)445 1990 y Fq(k)q(;n)512 1983 y Fr(\012)g Fs(v)589 1990 y Fq(k)q(;n)p Fl(+1)686 1983 y Fu(.)37 b(W)l(e)21 b(determine)e(the)i(n)o(um)o(b)q(er)f Fs(t)p Fu(\()p Fs( )r Fu(\))h(of)g(generators)228 2042 y Fs(\034)249 2049 y Fq(i)282 2042 y Fu(o)q(ccurring)f(in)f Fs( )h Fu(=)f(\()p Fs(\032)13 b Fr(\012)g Fu(1\)\(1)865 2023 y Fq(n)p Fj(\000)p Fl(1)947 2042 y Fr(\012)g Fs(\034)6 b Fu(\)\()p Fs(')13 b Fr(\012)g Fu(1\).)30 b(W)l(e)19 b(ha)o(v)o(e)g(b)o(y)g(induction)228 2100 y Fs(t)p Fu(\()p Fs(')p Fu(\))14 b(=)h Fs(t)401 2107 y Fq(n)440 2100 y Fu(the)i(n)o(um)o(b)q(er)e(of)i(factors)g(in)g(\()p Fs(x)1023 2107 y Fl(1)1053 2100 y Fr(\012)11 b Fu(1\))h Fr(\001)f Fu(\(1)h Fr(\012)f Fs(x)1316 2107 y Fl(2)1336 2100 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)1438 2107 y Fl(3)1469 2100 y Fr(\012)g Fu(1\))h Fr(\001)f Fs(:)d(:)g(:)25 b Fu(in)228 2158 y(rev)o(erse)12 b(p)q(osition.)21 b(Also)14 b(w)o(e)f(ha)o(v)o(e)g Fs(t)897 2165 y Fq(n)p Fl(+1)979 2158 y Fu(=)h Fs(t)1049 2165 y Fq(n)1078 2158 y Fu(+)6 b(\()p Fs(n)g Fr(\000)g Fs(i)p Fu(\))13 b(the)h(n)o(um)o(b)q(er)e(of)i (factors)228 2216 y(in)g(\()p Fs(x)330 2223 y Fl(1)358 2216 y Fr(\012)9 b Fu(1\))f Fr(\001)h Fu(\(1)f Fr(\012)h Fs(x)607 2223 y Fl(2)626 2216 y Fu(\))g Fr(\001)f Fu(\()p Fs(x)723 2223 y Fl(3)751 2216 y Fr(\012)g Fu(1\))h Fr(\001)f Fs(:)g(:)g(:)g Fr(\001)g Fu(\()p Fs(x)1007 2223 y Fq(n)p Fl(+1)1084 2216 y Fr(\012)g Fu(1\))15 b(in)g(rev)o(erse)f(p)q(osition.) 21 b(Then)228 2274 y Fs(t)p Fu(\()p Fs( )r Fu(\))13 b(=)g Fs(t)p Fu(\(\()p Fs(\032)8 b Fr(\012)g Fu(1\)\(1)604 2256 y Fq(n)p Fj(\000)p Fl(1)681 2274 y Fr(\012)g Fs(\034)e Fu(\)\()p Fs(')i Fr(\012)g Fu(1\)\))13 b(=)h Fs(t)p Fu(\()p Fs(\032)8 b Fr(\012)g Fu(1\))g(+)g Fs(t)p Fu(\(1)1282 2256 y Fq(n)p Fj(\000)p Fl(1)1358 2274 y Fr(\012)g Fs(\034)e Fu(\))i(+)g Fs(t)p Fu(\()p Fs(')g Fr(\012)g Fu(1\))k(=)228 2332 y(\()p Fs(n)f Fr(\000)g Fs(i)f Fr(\000)h Fu(1\))h(+)f(1)g(+)g Fs(t)620 2339 y Fq(n)657 2332 y Fu(=)j Fs(t)727 2339 y Fq(n)p Fl(+1)795 2332 y Fu(.)278 2390 y(If)h(w)o(e)h(sum)f(up)i(w)o (e)f(obtain)228 2481 y(\()p Fs(x)275 2488 y Fl(1)296 2481 y Fr(\012)r Fu(1)r(+)r(1)r Fr(\012)r Fs(x)498 2488 y Fl(1)519 2481 y Fu(\))r Fr(\001)r Fs(:)8 b(:)g(:)q Fr(\001)r Fu(\()p Fs(x)678 2488 y Fq(n)703 2481 y Fr(\012)r Fu(1)r(+)r(1)r Fr(\012)r Fs(x)905 2488 y Fq(n)929 2481 y Fu(\))14 b(=)1014 2440 y Fm(X)1038 2531 y Fq(i)1082 2440 y Fm(X)1094 2527 y Fq(')1117 2532 y Fh(i)1142 2481 y Fu(\()p Fr(r)1203 2461 y Fq(i)1219 2481 y Fr(\012)r(r)1302 2461 y Fq(n)p Fj(\000)p Fq(i)1364 2481 y Fu(\))p Fs(')1415 2488 y Fq(i)1429 2481 y Fu(\()p Fs(x)1476 2488 y Fl(1)1497 2481 y Fr(\012)r Fs(:)8 b(:)g(:)r Fr(\012)r Fs(x)1667 2488 y Fq(n)1690 2481 y Fu(\))p Fs(;)228 2608 y Fu(for)16 b(certain)g Fs(')497 2615 y Fq(i)525 2608 y Fr(2)e Fs(B)609 2615 y Fq(n)649 2608 y Fu(whic)o(h)h(arise)h(in)g(the)g(ev)m(aluation)h (giv)o(en)e(ab)q(o)o(v)o(e.)p eop %%Page: 13 13 13 12 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(13)278 225 y Fu(No)o(w)16 b(let)g Fs(z)h Fr(2)e Fs(A)585 207 y Fq(n)608 225 y Fu(\()p Fs(\020)t Fu(\).)22 b(W)l(e)17 b(expand)g(the)f(pro)q (ducts)i(in)e Fr(r)1348 207 y Fq(n)1371 225 y Fs(p)1395 207 y Fq(n)1419 225 y Fs(z)h Fu(=)1511 192 y Fm(P)1555 235 y Fq(k)1577 225 y Fu(\()p Fs(x)1624 232 y Fq(k)q(;)p Fl(1)1683 225 y Fr(\012)228 283 y Fu(1)12 b(+)f(1)h Fr(\012)f Fs(x)427 290 y Fq(k)q(;)p Fl(1)475 283 y Fu(\))g Fr(\001)h Fs(:)c(:)g(:)i Fr(\001)h Fu(\()p Fs(x)671 290 y Fq(k)q(;n)735 283 y Fr(\012)g Fu(1)h(+)f(1)h Fr(\012)f Fs(x)984 290 y Fq(k)q(;n)1036 283 y Fu(\).)23 b(Eac)o(h)16 b(of)h(these)g(pro)q (ducts)g(in)f(the)228 341 y(sum)f(is)h(treated)g(in)g(the)g(same)f(w)o (a)o(y)h(as)h(describ)q(ed)f(ab)q(o)o(v)o(e.)21 b(Using)16 b(\(3\))h(w)o(e)e(get)318 430 y Fr(r)360 412 y Fq(n)383 430 y Fs(p)407 412 y Fq(n)431 430 y Fu(\()p Fs(z)r Fu(\))41 b(=)587 397 y Fm(P)631 440 y Fq(k)652 430 y Fu(\()p Fs(x)699 437 y Fq(k)q(;)p Fl(1)759 430 y Fr(\012)11 b Fu(1)g(+)g(1)g Fr(\012)g Fs(x)1006 437 y Fq(k)q(;)p Fl(1)1055 430 y Fu(\))g Fr(\001)g Fs(:)d(:)g(:)i Fr(\001)h Fu(\()p Fs(x)1250 437 y Fq(k)q(;n)1313 430 y Fr(\012)g Fu(1)h(+)f(1)g Fr(\012)g Fs(x)1561 437 y Fq(k)q(;n)1613 430 y Fu(\))535 488 y(=)587 455 y Fm(P)631 499 y Fq(k)661 455 y Fm(P)704 499 y Fq(i)727 455 y Fm(P)771 499 y Fq(')794 504 y Fh(i)809 488 y Fu(\()p Fr(r)870 470 y Fq(i)894 488 y Fr(\012)g(r)986 470 y Fq(n)p Fj(\000)p Fq(i)1049 488 y Fu(\))p Fs(')1100 495 y Fq(i)1114 488 y Fu(\()p Fs(x)1161 495 y Fq(k)q(;)p Fl(1)1220 488 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1416 495 y Fq(k)q(;n)1468 488 y Fu(\))535 546 y(=)587 513 y Fm(P)631 557 y Fq(i)653 513 y Fm(P)697 557 y Fq(')720 562 y Fh(i)735 546 y Fu(\()p Fr(r)796 528 y Fq(i)821 546 y Fr(\012)g(r)913 528 y Fq(n)p Fj(\000)p Fq(i)975 546 y Fu(\))p Fs(')1026 553 y Fq(i)1040 546 y Fu(\()p Fs(z)r Fu(\))535 607 y(=)587 574 y Fm(P)631 617 y Fq(i)653 574 y Fm(P)697 617 y Fq(')720 622 y Fh(i)735 607 y Fu(\()p Fr(r)796 589 y Fq(i)821 607 y Fr(\012)g(r)913 589 y Fq(n)p Fj(\000)p Fq(i)975 607 y Fu(\))p Fs(\020)1019 589 y Fj(\000)p Fq(t)p Fl(\()p Fq(')1096 594 y Fh(i)1109 589 y Fl(\))1125 607 y Fs(\032)1150 614 y Fq(i)1164 607 y Fu(\()p Fs(z)r Fu(\))228 697 y(where)k Fs(\032)393 704 y Fq(i)421 697 y Fr(2)f Fs(S)498 704 y Fq(n)537 697 y Fu(are)i(the)f(canonical)g(images)g(of)h(the)f Fs(')1244 704 y Fq(i)1272 697 y Fr(2)f Fs(B)1356 704 y Fq(n)1395 697 y Fu(and)i Fs(t)p Fu(\()p Fs(')1558 704 y Fq(i)1572 697 y Fu(\))f(is)g(the)228 755 y(n)o(um)o(b)q(er)f(of)j (factors)g Fs(\034)641 762 y Fq(j)675 755 y Fu(in)f(the)g(represen)o (tation)g(of)g Fs(')1222 762 y Fq(i)1236 755 y Fu(.)278 813 y(This)g(giv)o(es)f(us)473 899 y([)p Fs(p)511 881 y Fq(n)535 899 y Fu(\()p Fs(z)r Fu(\)])41 b(=)705 866 y Fm(P)748 910 y Fq(\033)q Fj(2)p Fq(S)814 914 y Fh(n)846 899 y Fr(r)888 881 y Fq(n)911 899 y Fs(p)935 881 y Fq(n)959 899 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))653 960 y(=)705 927 y Fm(P)748 970 y Fq(\033)780 927 y Fm(P)824 970 y Fq(i)846 927 y Fm(P)890 970 y Fq(')913 975 y Fh(i)937 960 y Fs(\020)962 942 y Fj(\000)p Fq(t)p Fl(\()p Fq(')1039 947 y Fh(i)1052 942 y Fl(\))1068 960 y Fu(\()p Fr(r)1129 942 y Fq(i)1153 960 y Fr(\012)11 b(r)1245 942 y Fq(n)p Fj(\000)p Fq(i)1307 960 y Fu(\))p Fs(\032)1351 967 y Fq(i)1366 960 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))653 1027 y(=)705 994 y Fm(P)748 1038 y Fq(\033)780 994 y Fm(P)824 1038 y Fq(i)846 979 y Fm(\020)879 994 y(P)923 1038 y Fq(')946 1043 y Fh(i)970 1027 y Fs(\020)995 1009 y Fj(\000)p Fq(t)p Fl(\()p Fq(')1072 1014 y Fh(i)1085 1009 y Fl(\))1101 979 y Fm(\021)1125 1027 y Fu(\()p Fr(r)1186 1009 y Fq(i)1211 1027 y Fr(\012)g(r)1303 1009 y Fq(n)p Fj(\000)p Fq(i)1365 1027 y Fu(\))p Fs(\033)r Fu(\()p Fs(z)r Fu(\))653 1093 y(=)705 1060 y Fm(P)748 1103 y Fq(i)771 1093 y Fs(c)792 1100 y Fq(i)806 1093 y Fu(\()p Fr(r)867 1075 y Fq(i)891 1093 y Fr(\012)g(r)983 1075 y Fq(n)p Fj(\000)p Fq(i)1046 1093 y Fu(\))1073 1060 y Fm(P)1117 1103 y Fq(\033)1149 1093 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))228 1189 y(where)19 b(the)g(factors)h Fs(c)642 1196 y Fq(i)675 1189 y Fu(=)732 1156 y Fm(P)775 1199 y Fq(')798 1204 y Fh(i)822 1189 y Fs(\020)847 1171 y Fj(\000)p Fq(t)p Fl(\()p Fq(')924 1176 y Fh(i)937 1171 y Fl(\))972 1189 y Fr(2)f Fs(k)r Fu(.)31 b(W)l(e)19 b(w)o(an)o(t)g(to)g(sho)o(w)h(that)g(the)f Fs(c)1708 1196 y Fq(i)228 1247 y Fu(are)d(zero)g(for)h(all)e(0)g Fs(<)e(i)h(<)g(n)p Fu(.)278 1305 y(So)f(\014x)g Fs(n)g Fu(and)h Fs(i)p Fu(.)20 b(Consider)13 b(one)h(pro)q(duct)f(\()p Fs(x)1106 1312 y Fl(1)1131 1305 y Fr(\012)5 b Fu(1\))g Fr(\001)g Fu(\(1)g Fr(\012)g Fs(x)1362 1312 y Fl(2)1380 1305 y Fu(\))g Fr(\001)g Fu(\()p Fs(x)1470 1312 y Fl(3)1493 1305 y Fr(\012)g Fu(1\))g Fr(\001)g Fs(:)j(:)g(:)20 b Fu(in)228 1363 y(the)12 b(dev)o(elopmen)o(t)e(of)j(\()p Fs(x)687 1370 y Fl(1)711 1363 y Fr(\012)t Fu(1)t(+)t(1)t Fr(\012)t Fs(x)923 1370 y Fl(1)942 1363 y Fu(\))t Fr(\001)t Fs(:)8 b(:)g(:)s Fr(\001)t Fu(\()p Fs(x)1109 1370 y Fq(n)1137 1363 y Fr(\012)t Fu(1)t(+)t(1)t Fr(\012)t Fs(x)1349 1370 y Fq(n)1372 1363 y Fu(\))14 b(=)1457 1330 y Fm(P)1500 1374 y Fq(i)1523 1330 y Fm(P)1567 1374 y Fq(')1590 1379 y Fh(i)1605 1363 y Fu(\()p Fr(r)1666 1345 y Fq(i)1683 1363 y Fr(\012)228 1424 y(r)270 1406 y Fq(n)p Fj(\000)p Fq(i)332 1424 y Fu(\))p Fs(')383 1431 y Fq(i)397 1424 y Fu(\()p Fs(x)444 1431 y Fl(1)465 1424 y Fr(\012)q Fs(:)8 b(:)g(:)p Fr(\012)q Fs(x)631 1431 y Fq(n)654 1424 y Fu(\))j(and)h(its)f (corresp)q(onding)i Fs(')1178 1431 y Fq(i)1192 1424 y Fu(.)19 b(The)12 b(c)o(hosen)f(summand)e(is)228 1482 y(completely)h(determined)g(b)o(y)i(giving)h(the)g(p)q(ositions)g(in)g Fr(f)p Fu(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)p Fr(g)13 b Fu(of)g(the)f Fs(n)t Fr(\000)t Fs(i)228 1540 y Fu(factors)19 b(of)h(the)e(form)g(\(1)c Fr(\012)e Fs(x)788 1547 y Fq(j)806 1540 y Fu(\).)30 b(The)19 b(\014rst)g(of)g(these)g(factors)h(has)f Fs(\025)1540 1547 y Fl(1)1580 1540 y Fu(factors)228 1599 y(of)h(the)h(form)e(\()p Fs(x)542 1606 y Fq(j)574 1599 y Fr(\012)13 b Fu(1\))21 b(to)g(its)f(righ)o(t)g(with)g(0)h Fr(\024)g Fs(\025)1196 1606 y Fl(1)1236 1599 y Fr(\024)g Fs(i)p Fu(.)33 b(So)21 b(it)f(con)o(tributes)228 1657 y Fs(\025)256 1664 y Fl(1)295 1657 y Fu(pairs)f(of)g(factors)g(in)g (rev)o(erse)e(p)q(osition.)30 b(The)19 b(second)g(factor)g(of)g(the)f (form)228 1715 y(\(1)t Fr(\012)t Fs(x)346 1722 y Fq(j)363 1715 y Fu(\))12 b(con)o(tributes)g Fs(\025)672 1722 y Fl(2)705 1715 y Fu(\(with)g(0)j Fr(\024)e Fs(\025)950 1722 y Fl(2)984 1715 y Fr(\024)h Fs(\025)1065 1722 y Fl(1)1099 1715 y Fr(\024)f Fs(i)p Fu(\))g(pairs)f(of)h(factors)g(in)g (rev)o(erse)228 1773 y(p)q(osition,)i(and)g(so)h(on.)21 b(W)l(e)15 b(obtain)g Fs(t)e Fu(=)h Fs(\025)1008 1780 y Fl(1)1036 1773 y Fu(+)8 b Fs(\025)1110 1780 y Fl(2)1139 1773 y Fu(+)g Fs(:)g(:)g(:)f Fu(+)h Fs(\025)1324 1780 y Fq(n)p Fj(\000)p Fq(i)1402 1773 y Fu(pairs)15 b(in)g(rev)o(erse)228 1831 y(p)q(osition.)28 b(If)18 b(w)o(e)g(kno)o(w)h(the)f Fs(\025)808 1838 y Fq(i)841 1831 y Fu(with)g(0)g Fr(\024)f Fs(\025)1080 1838 y Fq(n)p Fj(\000)p Fq(i)1161 1831 y Fr(\024)g Fs(:)8 b(:)g(:)17 b Fr(\024)g Fs(\025)1376 1838 y Fl(2)1414 1831 y Fr(\024)g Fs(\025)1498 1838 y Fl(1)1536 1831 y Fr(\024)g Fs(i)h Fu(then)228 1889 y(they)j(also)g (determine)e(uniquely)h(the)h(p)q(osition)h(of)f(the)g(factors)h(of)f (the)g(form)228 1947 y(\(1)14 b Fr(\012)h Fs(x)367 1954 y Fq(j)385 1947 y Fu(\).)35 b(Eac)o(h)21 b(partition)g(of)h Fs(t)f Fu(=)h Fs(\025)975 1954 y Fl(1)1010 1947 y Fu(+)14 b Fs(\025)1090 1954 y Fl(2)1124 1947 y Fu(+)h Fs(:)8 b(:)g(:)13 b Fu(+)h Fs(\025)1328 1954 y Fq(n)p Fj(\000)p Fq(i)1413 1947 y Fu(in)o(to)21 b(\(at)g(most\))228 2008 y Fs(n)11 b Fr(\000)g Fs(i)k Fu(parts)i(eac)o(h)f Fr(\024)e Fs(i)h Fu(giv)o(es)h(one)g(term)f Fs(\020)1021 1990 y Fj(\000)p Fq(t)1079 2008 y Fu(in)h Fs(c)1157 2015 y Fq(i)1185 2008 y Fu(=)1237 1975 y Fm(P)1281 2019 y Fq(')1304 2024 y Fh(i)1327 2008 y Fs(\020)1352 1990 y Fj(\000)p Fq(t)p Fl(\()p Fq(')1429 1995 y Fh(i)1442 1990 y Fl(\))1474 2008 y Fu(and)h(w)o(e)f(\014nd)228 2067 y Fs(p)p Fu(\()p Fs(i;)8 b(n)g Fr(\000)h Fs(i;)f(t)p Fu(\))14 b(partitions)h(of)g Fs(t)g Fu(in)o(to)g(at)g(most)f Fs(n)9 b Fr(\000)f Fs(i)15 b Fu(parts)h(eac)o(h)e Fr(\024)g Fs(i)p Fu(.)20 b(So)c(w)o(e)e(get)727 2160 y Fs(c)748 2167 y Fq(i)776 2160 y Fu(=)827 2119 y Fm(X)828 2210 y Fq(t)p Fj(\025)p Fl(0)896 2160 y Fs(p)p Fu(\()p Fs(i;)8 b(n)j Fr(\000)f Fs(i;)e(t)p Fu(\))p Fs(\020)1168 2140 y Fj(\000)p Fq(t)1210 2160 y Fs(:)228 2293 y Fu(By)15 b(a)i(theorem)e(of)h(Sylv)o(ester)f(\([1])g(Theorem)g(3.1\))i(w)o(e)f (ha)o(v)o(e)404 2373 y Fm(X)405 2465 y Fq(t)p Fj(\025)p Fl(0)472 2415 y Fs(p)p Fu(\()p Fs(i;)8 b(n)j Fr(\000)g Fs(i;)d(t)p Fu(\))p Fs(q)744 2394 y Fq(t)771 2415 y Fu(=)828 2381 y(\(1)j Fr(\000)g Fs(q)956 2363 y Fq(n)979 2381 y Fu(\)\(1)g Fr(\000)g Fs(q)1126 2363 y Fq(n)p Fj(\000)p Fl(1)1194 2381 y Fu(\))d Fs(:)g(:)g(:)g Fu(\(1)j Fr(\000)g Fs(q)1415 2363 y Fq(n)p Fj(\000)p Fq(i)p Fl(+1)1522 2381 y Fu(\))p 828 2403 714 2 v 891 2449 a(\(1)g Fr(\000)g Fs(q)1019 2435 y Fq(i)1033 2449 y Fu(\)\(1)g Fr(\000)g Fs(q)1180 2435 y Fq(i)p Fj(\000)p Fl(1)1239 2449 y Fu(\))d Fs(:)g(:)g(:)f Fu(\(1)12 b Fr(\000)f Fs(q)r Fu(\))228 2550 y(hence)j Fs(c)383 2557 y Fq(i)411 2550 y Fu(=)f(0)i(for)g(0)f Fs(<)g(i)g(<)f(n)i Fu(since)f Fs(\020)k Fu(and)e(also)f Fs(\020)1162 2532 y Fj(\000)p Fl(1)1224 2550 y Fu(are)f Fo(primitive)h Fs(n)p Fu(-th)g(ro)q(ots)228 2608 y(of)h(unit)o(y)l(.)p eop %%Page: 14 14 14 13 bop 228 125 a Fp(14)552 b(BODO)13 b(P)m(AREIGIS)278 225 y Fu(So)j(w)o(e)g(ha)o(v)o(e)g(sho)o(wn)511 295 y([)p Fs(p)549 277 y Fq(n)573 295 y Fu(\()p Fs(z)r Fu(\)])41 b(=)742 261 y Fm(P)786 305 y Fq(\033)q Fj(2)p Fq(S)852 309 y Fh(n)884 295 y Fr(r)926 277 y Fq(n)949 295 y Fs(p)973 277 y Fq(n)997 295 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))691 353 y(=)742 320 y Fm(P)786 363 y Fq(i)809 353 y Fs(c)830 360 y Fq(i)844 353 y Fu(\()p Fr(r)905 335 y Fq(i)929 353 y Fr(\012)11 b(r)1021 335 y Fq(n)p Fj(\000)p Fq(i)1084 353 y Fu(\))1111 320 y Fm(P)1155 363 y Fq(\033)1186 353 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))691 411 y(=)742 378 y Fm(P)786 421 y Fq(\033)818 411 y Fr(r)860 393 y Fq(n)883 411 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))f Fr(\012)h Fu(1)h(+)f(1)g Fr(\012)1206 378 y Fm(P)1250 421 y Fq(\033)1281 411 y Fr(r)1323 393 y Fq(n)1346 411 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))691 469 y(=)i Fs(p)p Fu([)p Fs(z)r Fu(])p Fs(:)1683 549 y Fi(\003)228 637 y FB(Corollary)h(5.4.)i Fo(L)n(et)d Fs(H)18 b Fo(b)n(e)c(a)f(Hopf)h(algebr)n(a)g(in)g Fr(C)s Fo(.)21 b(Then)14 b(the)g(set)g(of)f(primitive)228 695 y(elements)19 b Fs(P)7 b Fu(\()p Fs(H)t Fu(\))18 b Fo(forms)f(a)g(Lie)g (algebr)n(a)h(in)g Fr(C)s Fo(.)228 784 y(Pr)n(o)n(of.)h Fu(By)c(Lemma)f(5.1)j Fs(P)7 b Fu(\()p Fs(H)t Fu(\))16 b(is)g(a)h(Y)l(etter-Drinfeld)d(submo)q(dule)h(of)i Fs(H)t Fu(.)k(Let)228 842 y Fs(z)27 b Fr(2)e Fs(P)7 b Fu(\()p Fs(H)t Fu(\))456 824 y Fq(n)480 842 y Fu(\()p Fs(\020)t Fu(\).)41 b(Then)23 b Fs(p)p Fu(\([)p Fs(z)r Fu(]\))h(=)h([)p Fs(p)972 824 y Fq(n)996 842 y Fu(\()p Fs(z)r Fu(\)])f(=)h([\001)1215 824 y Fq(n)1238 842 y Fu(\()p Fs(z)r Fu(\)])f(=)h(\001\([)p Fs(z)r Fu(]\))c(since)i(\001)228 900 y(is)d(an)h(algebra)f (homomorphism.)31 b(Hence)19 b([)p Fs(z)r Fu(])h Fr(2)g Fs(P)7 b Fu(\()p Fs(H)t Fu(\).)34 b(So)21 b Fs(P)7 b Fu(\()p Fs(H)t Fu(\))21 b(is)f(a)g(Lie)228 958 y(subalgebra)d(of)f Fs(H)571 940 y Fq(L)598 958 y Fu(.)1071 b Fi(\003)228 1047 y FB(De\014nition)16 b(5.5.)j Fu(Let)d Fs(A)f Fu(b)q(e)g(an)h (algebra)g(in)f Fr(C)j Fu(and)e(let)f(end)o(\()p Fs(A)p Fu(\))g(b)q(e)h(the)f(inner)228 1105 y(endomorphism)9 b(ob)s(ject)h(of)i Fs(A)f Fu(in)f Fr(C)s Fu(,)i(i.e.)19 b(the)10 b(Y)l(etter-)h(Drinfeld)f(mo)q(dule)g(end\()p Fs(A)p Fu(\))228 1163 y(satisfying)16 b Fr(C)s Fu(\()p Fs(X)e Fr(\012)c Fs(A;)e(A)p Fu(\))724 1149 y Fr(\030)724 1165 y Fu(=)776 1163 y Fr(C)s Fu(\()p Fs(X)q(;)g Fu(end)q(\()p Fs(A)p Fu(\)\))15 b(for)h(all)f Fs(X)j Fr(2)c(C)s Fu(.)21 b(It)16 b(can)g(b)q(e)g(sho)o(wn)228 1221 y(that)249 1291 y(end)o(\()p Fs(A)p Fu(\))e(:=)f Fr(f)p Fs(f)19 b Fr(2)14 b Fu(Hom)o(\()p Fs(A;)8 b(A)p Fu(\))p Fr(j)o(9)878 1257 y Fm(P)930 1291 y Fs(f)954 1298 y Fl(\(0\))1012 1291 y Fr(\012)j Fs(f)1086 1298 y Fl(\(1\))1147 1291 y Fr(2)j Fu(Hom)n(\()p Fs(A;)8 b(A)p Fu(\))i Fr(\012)h Fs(K)t Fr(8)p Fs(a)h Fr(2)i Fs(A)g Fu(:)842 1315 y Fm(P)894 1349 y Fs(f)918 1356 y Fl(\(0\))965 1349 y Fu(\()p Fs(a)p Fu(\))d Fr(\012)f Fs(f)1113 1356 y Fl(\(1\))1174 1349 y Fu(=)1226 1315 y Fm(P)1278 1349 y Fs(f)5 b Fu(\()p Fs(a)1352 1356 y Fl(\(0\))1399 1349 y Fu(\))1418 1356 y Fl(\(0\))1477 1349 y Fr(\012)10 b Fs(f)5 b Fu(\()p Fs(a)1600 1356 y Fl(\(0\))1647 1349 y Fu(\))1666 1356 y Fl(\(1\))1714 1349 y Fs(S)s Fu(\()p Fs(a)1792 1356 y Fl(\(1\))1838 1349 y Fu(\))p Fr(g)228 1430 y Fu(is)24 b(the)g(Y)l(etter-Drinfeld)f(mo)q(dule)g(with)h(the)g(required)f(univ)o (ersal)h(prop)q(ert)o(y)l(.)228 1488 y(end\()p Fs(A)p Fu(\))f(op)q(erates)i(on)f Fs(A)f Fu(b)o(y)h(a)g(canonical)f(map)g(ev)k (:)g(end)o(\()p Fs(A)p Fu(\))16 b Fr(\012)g Fs(A)26 b Fr(\000)-30 b(!)27 b Fs(A)228 1546 y Fu(with)16 b(ev\()p Fs(f)g Fr(\012)11 b Fs(a)p Fu(\))j(=)f Fs(f)5 b Fu(\()p Fs(a)p Fu(\).)278 1604 y(A)19 b Fo(derivation)h Fu(from)f Fs(A)h Fu(to)g Fs(A)f Fu(is)h(a)g(linear)f(map)g(\()p Fs(d)i Fu(:)e Fs(A)h Fr(\000)-31 b(!)20 b Fs(A)p Fu(\))g Fr(2)g Fu(end\()p Fs(A)p Fu(\))228 1662 y(suc)o(h)c(that)538 1725 y Fs(d)p Fu(\()p Fs(ab)p Fu(\))d(=)h Fs(d)p Fu(\()p Fs(a)p Fu(\))p Fs(b)d Fu(+)g(\(1)g Fr(\012)g Fs(d)p Fu(\)\()p Fs(\034)17 b Fr(\012)11 b Fu(1\)\()p Fs(d)g Fr(\012)g Fs(a)g Fr(\012)g Fs(b)p Fu(\))228 1797 y(for)19 b(all)g Fs(a;)8 b(b)19 b Fr(2)g Fs(A)p Fu(.)30 b(Observ)o(e)19 b(that)h(in)f(the)g(symmetri)o(c)d(situation)k(this)f(means)228 1855 y Fs(d)p Fu(\()p Fs(ab)p Fu(\))13 b(=)h Fs(d)p Fu(\()p Fs(a)p Fu(\))p Fs(b)d Fu(+)g Fs(ad)p Fu(\()p Fs(b)p Fu(\).)278 1926 y(It)j(is)h(clear)g(that)g(all)g(deriv)m(ations)g(from)f Fs(A)h Fu(to)g Fs(A)g Fu(form)f(an)h(ob)s(ject)g(Der\()p Fs(A)p Fu(\))g(in)228 1984 y Fr(C)k Fu(and)e(that)g(there)e(is)h(an)h (op)q(eration)g(Der\()p Fs(A)p Fu(\))11 b Fr(\012)g Fs(A)i Fr(\000)-9 b(!)14 b Fs(A)p Fu(.)228 2054 y FB(Corollary)19 b(5.6.)g Fu(Der\()p Fs(A)p Fu(\))e Fo(is)h(a)f(Lie)g(algebr)n(a.)228 2143 y(Pr)n(o)n(of.)i Fu(Let)i Fs(m)f Fu(denote)g(the)g(m)o 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Fm(P)724 484 y Fs(m)p Fu(\()p Fr(r)828 466 y Fq(n)850 484 y Fs(p)874 466 y Fq(n)898 484 y Fs(\033)r Fu(\()p Fs(z)r Fu(\)\))g(=)1075 451 y Fm(P)1127 484 y Fr(r)1169 466 y Fq(n)1192 484 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))p Fs(m)g Fu(=)h([)p Fs(z)r Fu(])p Fs(m)228 570 y Fu(hence)h([)p Fs(z)r Fu(])e Fr(2)h Fu(Der\()p Fs(A)p Fu(\).)1040 b Fi(\003)260 689 y Fu(6.)24 b Ft(The)19 b(Universal)d(Enveloping)g (Algebra)h(of)h(a)h(Lie)g(Algebra)278 776 y Fu(As)c(in)g([5])g(w)o(e)g (can)g(no)o(w)h(construct)g(the)f(univ)o(ersal)f(en)o(v)o(eloping)g (algebra)i(of)g(a)228 834 y(Lie)c(algebra)g Fs(P)20 b Fu(in)12 b Fr(C)j Fu(as)e Fs(U)5 b Fu(\()p Fs(P)i Fu(\))14 b(:=)f Fs(T)7 b Fu(\()p Fs(P)g Fu(\))p Fs(=I)16 b Fu(where)c Fs(T)7 b Fu(\()p Fs(P)g Fu(\))k(is)h(the)g(tensor)h(algebra)228 893 y(o)o(v)o(er)e Fs(P)c Fu(,)13 b(whic)o(h)f(liv)o(es)f(again)i(in)f Fr(C)s Fu(,)h(and)g(where)f Fs(I)j Fu(is)e(the)f(ideal)f(generated)i(b) o(y)e(the)228 951 y(relations)i([)p Fs(z)r Fu(])5 b Fr(\000)525 917 y Fm(P)577 951 y Fr(r)619 933 y Fq(n)642 951 y Fs(\033)r Fu(\()p Fs(z)r Fu(\))12 b(for)i(all)f Fs(z)j Fr(2)e Fs(P)1008 933 y Fq(n)1031 951 y Fu(\()p Fs(\020)t Fu(\),)g(for)g(all)f Fs(n)g Fu(and)h(for)g(all)f(primitiv)o(e)228 1009 y Fs(n)p Fu(-th)i(ro)q(ots)h(of)g(unit)o(y)e Fs(\020)t Fu(.)20 b(Then)15 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))16 b(clearly)d(is)i(a)g(univ) o(ersal)f(solution)h(for)g(the)228 1067 y(follo)o(wing)h(univ)o(ersal)f (problem)742 1151 y Fs(P)321 b(U)5 b Fu(\()p Fs(P)i Fu(\))p 795 1139 285 2 v 1038 1138 a Fk(-)928 1129 y Fs(\023)895 1248 y(f)829 1193 y Fk(H)871 1214 y(H)912 1235 y(H)954 1255 y(H)995 1276 y(H)1037 1297 y(H)1041 1299 y(H)-42 b(j)1133 1350 y Fs(A)p 1150 1299 2 127 v 1151 1299 a Fk(?)1170 1242 y Fs(g)228 1419 y Fu(where)14 b(for)h(eac)o(h)g (morphism)d(of)j(Lie-algebras)g Fs(f)21 b Fu(there)14 b(is)g(a)i(unique)e(morphism)228 1477 y(of)i(algebras)h Fs(g)i Fu(suc)o(h)d(that)g(the)g(diagram)g(comm)o(utes.)228 1549 y FB(Theorem)k(6.1.)i Fo(L)n(et)e Fs(P)28 b Fo(b)n(e)21 b(a)f(Lie)h(algebr)n(a)g(in)g Fr(C)s Fo(.)32 b(Then)21 b(the)g(universal)h(en-)228 1607 y(veloping)d(algebr)n(a)f Fs(U)5 b Fu(\()p Fs(P)i Fu(\))18 b Fo(is)f(a)h(Hopf)f(algebr)n(a)h(in)g Fr(C)s Fo(.)228 1698 y(Pr)n(o)n(of.)h Fu(It)14 b(is)g(easily)g(seen)g (that)h Fs(\016)h Fu(:)d Fs(P)21 b Fr(\000)-30 b(!)13 b Fu(\()p Fs(U)5 b Fu(\()p Fs(P)i Fu(\))h Fr(\012)g Fs(U)d Fu(\()p Fs(P)i Fu(\)\))1326 1680 y Fq(L)1367 1698 y Fu(in)14 b Fr(M)1482 1680 y Fq(k)q(G)1545 1698 y Fu(giv)o(en)g(b)o(y)228 1757 y Fs(\016)r Fu(\()p Fs(x)p Fu(\))f(:=)p 396 1730 28 2 v 13 w Fs(x)c Fr(\012)g Fu(1)h(+)f(1)h Fr(\012)p 644 1730 V 9 w Fs(x)15 b Fu(where)p 827 1730 V 15 w Fs(x)g Fu(is)g(the)h(canonical)f(image)f(of)h Fs(x)f Fr(2)g Fs(P)23 b Fu(in)15 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))228 1815 y(and)18 b(the)f(counit)h Fs(")e Fu(:)g Fs(U)5 b Fu(\()p Fs(P)i Fu(\))16 b Fr(\000)-30 b(!)16 b Fs(k)k Fu(giv)o(en)c(b)o(y)i(the)f(zero)g(morphism)f(0)g(:)g Fs(P)23 b Fr(\000)-30 b(!)16 b Fs(k)228 1873 y Fu(de\014ne)g(the)g (structure)g(of)g(a)h(bialgebra)f(on)h Fs(U)5 b Fu(\()p Fs(P)i Fu(\))17 b(in)f Fr(C)s Fu(.)278 1931 y(No)o(w)h(w)o(e)g(w)o(an)o (t)h(to)g(de\014ne)f Fs(S)i Fu(:)d Fs(U)5 b Fu(\()p Fs(P)i Fu(\))16 b Fr(\000)-30 b(!)16 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))1185 1913 y Fq(op)p Fl(+)1268 1931 y Fu(b)o(y)17 b(the)g(Lie)g(homomor-)228 1989 y(phism)h Fs(S)23 b Fu(:)d Fs(P)27 b Fr(\000)-30 b(!)19 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))712 1971 y Fq(op)p Fl(+)777 1989 y Fu(,)20 b Fs(S)s Fu(\()p Fs(x)p Fu(\))g(=)g Fr(\000)s Fu(\026)-27 b Fs(x)o Fu(.)32 b(Here)19 b Fs(A)1256 1971 y Fq(op)p Fl(+)1340 1989 y Fu(is)h(the)f(algebra)h(ob-)228 2059 y(tained)11 b(from)g(the)g(algebra)h Fs(A)f Fu(b)o(y)g(the)g(m)o(ultiplication)e Fs(A)q Fr(\012)q Fs(A)1386 2032 y Fq(\034)1355 2059 y Fr(\000)-8 b(!)13 b Fs(A)q Fr(\012)q Fs(A)1603 2032 y Fj(r)1578 2059 y Fr(\000)-9 b(!)14 b Fs(A)p Fu(.)228 2117 y(Then)i(for)h Fs(z)e Fr(2)f Fs(P)553 2099 y Fq(n)577 2117 y Fu(\()p Fs(\020)t Fu(\))i(w)o(e)g(ha)o(v)o(e)249 2207 y Fs(S)s Fu(\([)p Fs(z)r Fu(]\))40 b(=)14 b Fr(\000)p 504 2164 52 2 v Fu([)p Fs(z)r Fu(])e(=)i Fr(\000)668 2174 y Fm(P)712 2217 y Fq(\033)744 2207 y Fr(r)786 2189 y Fq(n)809 2207 y Fs(\033)r Fu(\()p 858 2180 25 2 v Fs(z)q Fu(\))g(=)g Fr(\000)1014 2174 y Fm(P)1057 2217 y Fq(\033)1089 2207 y Fr(r)1131 2189 y Fq(n)1154 2207 y Fs(\031)r(\031)1214 2189 y Fj(\000)p Fl(1)1260 2207 y Fs(\033)r Fu(\()p 1309 2180 V Fs(z)r Fu(\))413 2278 y(=)g Fr(\000)512 2245 y Fm(P)556 2289 y Fq(\033)579 2278 y Fu(\()p Fr(r)640 2260 y Fq(op)676 2278 y Fu(\))695 2260 y Fq(n)719 2278 y Fs(\031)749 2260 y Fj(\000)p Fl(1)795 2278 y Fs(\033)r Fu(\()p 844 2252 V Fs(z)r Fu(\))i(\(b)o(y)g(\(3\)\))30 b(=)14 b Fr(\000)1201 2245 y Fm(P)1245 2289 y Fq(\033)1268 2278 y Fu(\()p Fr(r)1329 2260 y Fq(op)1366 2278 y Fu(\))1385 2260 y Fq(n)1408 2278 y Fs(\020)1438 2243 y Fh(n)p Ff(\()p Fh(n)p Fg(\000)p Ff(1\))p 1438 2252 102 2 v 1481 2273 a(2)1547 2278 y Fs(\032)1572 2260 y Fj(\000)p Fl(1)1619 2278 y Fs(\033)r Fu(\()p 1668 2252 25 2 v Fs(z)q Fu(\))413 2350 y(=)g Fr(\000)p Fs(\020)533 2314 y Fh(n)p Ff(\()p Fh(n)p Fg(\000)p Ff(1\))p 533 2324 102 2 v 577 2344 a(2)642 2350 y Fu([)p 656 2323 25 2 v Fs(z)q Fu(])g(=)g(\()p Fr(\000)p Fu(1\))861 2332 y Fq(n)884 2350 y Fu([)p 898 2323 V Fs(z)r Fu(])f(=)h([)p 1016 2307 120 2 v Fs(S)1049 2335 y Fq(n)1072 2350 y Fu(\()p Fs(z)r Fu(\))o(])228 2433 y(where)g Fs(\031)i Fr(2)e Fs(B)495 2440 y Fq(n)533 2433 y Fu(is)h(the)g(braid)g(map)f(giv)o(en)g (b)o(y)h(the)g(t)o(wist)f(of)i(all)e Fs(n)h Fu(strands)h(with)228 2491 y(source)11 b Fr(f)p Fu(1)p Fs(;)d(:)g(:)g(:)16 b(;)8 b(n)p Fr(g)i Fu(and)i(domain)e Fr(f)p Fs(n;)e(:)g(:)g(:)16 b(;)8 b Fu(1)p Fr(g)p Fu(,)j Fs(\031)k Fu(=)f(\()p Fs(\034)1242 2498 y Fl(1)1262 2491 y Fu(\))8 b Fs(:)g(:)g(:)g Fu(\()p Fs(\034)1395 2498 y Fq(n)p Fj(\000)p Fl(2)1472 2491 y Fs(:)g(:)g(:)f(\034)1558 2498 y Fl(2)1578 2491 y Fs(\034)1599 2498 y Fl(1)1619 2491 y Fu(\)\()p Fs(\034)1678 2498 y Fq(n)p Fj(\000)p Fl(1)1755 2491 y Fs(:)h(:)g(:)g(\034)1842 2498 y Fl(2)1862 2491 y Fs(\034)1883 2498 y Fl(1)1903 2491 y Fu(\))228 2550 y(and)228 2608 y Fs(\020)253 2589 y Fj(\000)285 2572 y Fh(n)p Ff(\()p Fh(n)p Fg(\000)p Ff(1\))p 285 2582 102 2 v 328 2602 a(2)394 2608 y Fs(\031)15 b Fu(=)f Fs(\032)g Fr(2)g Fs(S)605 2615 y Fq(n)628 2608 y Fu(.)p eop %%Page: 16 16 16 15 bop 228 125 a Fp(16)552 b(BODO)13 b(P)m(AREIGIS)278 225 y Fu(Hence)f Fs(S)17 b Fu(is)d(a)g(Lie)g(homomorphism)d(and)k (factorizes)e(through)i Fs(U)5 b Fu(\()p Fs(P)i Fu(\).)21 b(Since)228 283 y Fs(U)5 b Fu(\()p Fs(P)i Fu(\))16 b(is)g(generated)g (as)g(an)h(algebra)f(b)o(y)f Fs(P)24 b Fu(w)o(e)15 b(pro)o(v)o(e)g (that)h Fs(S)j Fu(is)d(the)g(an)o(tip)q(o)q(de)228 341 y(b)o(y)g(complete)d(induction:)605 427 y Fr(r)p Fu(\(1)e Fr(\012)g Fs(S)s Fu(\)\001\(1\))i(=)h(1)p Fs(S)s Fu(\(1\))h(=)e(1)i(=)e Fs(")p Fu(\(1\))p Fs(;)568 520 y Fr(r)p Fu(\(1)e Fr(\012)g Fs(S)s Fu(\)\001\()p 826 494 28 2 v Fs(x)o Fu(\))j(=)p 938 494 V 14 w Fs(x)d Fu(+)g Fs(S)s Fu(\()p 1078 494 V Fs(x)o Fu(\))j(=)g(0)g(=)g Fs(")p Fu(\()p 1322 494 V Fs(x)o Fu(\))p Fs(:)278 596 y Fu(Before)e(w)o(e)h(pro)o(v)o(e)f(the)h (general)g(induction)g(step)g(w)o(e)g(observ)o(e)f(that)i(\001)f(:)h Fs(U)5 b Fu(\()p Fs(P)i Fu(\))228 654 y Fr(\000)-30 b(!)26 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))17 b Fr(\012)f Fs(U)5 b Fu(\()p Fs(P)i Fu(\))25 b(is)f(a)g(morphism)e(in)i Fr(C)30 b Fu(=)d Fr(Y)t(D)1241 636 y Fq(K)1240 666 y(K)1300 654 y Fu(so)e(that)g(w)o(e)e(ha)o(v)o(e)h(in)228 712 y(particular)228 758 y Fm(X)288 799 y Fu(\()p Fs(a)333 779 y Fl(0)352 799 y Fu(\))371 806 y Fl(1)397 799 y Fr(\012)7 b Fu(\()p Fs(a)488 779 y Fl(0)507 799 y Fu(\))526 806 y Fl(2)552 799 y Fr(\012)g Fs(a)624 779 y Fl(1)656 799 y Fu(=)708 758 y Fm(X)768 799 y Fu(\()p Fs(a)813 806 y Fl(1)832 799 y Fu(\))851 779 y Fl(0)878 799 y Fr(\012)g Fu(\()p Fs(a)969 806 y Fl(2)987 799 y Fu(\))1006 779 y Fl(0)1032 799 y Fr(\012)g Fu(\()p Fs(a)1123 806 y Fl(1)1142 799 y Fu(\))1161 779 y Fl(1)1181 799 y Fu(\()p Fs(a)1226 806 y Fl(2)1245 799 y Fu(\))1264 779 y Fl(1)1298 799 y Fr(2)14 b Fs(U)5 b Fu(\()p Fs(P)i Fu(\))g Fr(\012)g Fs(U)e Fu(\()p Fs(P)i Fu(\))g Fr(\012)g Fs(K)228 892 y Fu(for)17 b Fs(a)e Fr(2)h Fs(U)5 b Fu(\()p Fs(P)i Fu(\).)25 b(\(Here)16 b(w)o(e)g(use)i Fs(\016)r Fu(\()p Fs(a)p Fu(\))c(=)995 858 y Fm(P)1047 892 y Fs(a)1073 873 y Fl(0)1104 892 y Fr(\012)d Fs(a)1180 873 y Fl(1)1217 892 y Fu(to)17 b(denote)g(the)g(como)q(dule)228 950 y(structure)g(in)g Fr(Y)t(D)572 932 y Fq(K)571 962 y(K)607 950 y Fu(.\))25 b(Assume)16 b(no)o(w)i(that)g Fs(a)f Fu(is)g(writte)g(as)h(a)g(pro)q (duct)g(of)f Fs(n)f Fr(\025)228 1008 y Fu(1)23 b(elemen)o(ts)e(in)h Fs(P)31 b Fu(and)23 b(that)821 975 y Fm(P)873 1008 y Fs(a)899 1015 y Fl(1)918 1008 y Fs(S)s Fu(\()p Fs(a)996 1015 y Fl(2)1015 1008 y Fu(\))j(=)f(0.)42 b(Then)24 b(for)f(all)f Fs(x)k Fr(2)f Fs(P)31 b Fu(w)o(e)228 1066 y(ha)o(v)o(e)342 1033 y Fm(P)386 1066 y Fu(\()p Fs(a)431 1073 y Fl(1)450 1066 y Fu(\))469 1048 y Fl(0)489 1066 y Fs(S)s Fu(\(\()p Fs(a)586 1073 y Fl(2)605 1066 y Fu(\))624 1048 y Fl(0)644 1066 y Fu(\))p Fs(S)s Fu(\()p Fs(x)p Fu(\()p Fs(a)788 1073 y Fl(1)807 1066 y Fu(\))826 1048 y Fl(1)845 1066 y Fu(\()p Fs(a)890 1073 y Fl(2)910 1066 y Fu(\))929 1048 y Fl(1)948 1066 y Fu(\))17 b(=)1040 1033 y Fm(P)1083 1066 y Fu(\()p Fs(a)1128 1048 y Fl(0)1148 1066 y Fu(\))1167 1073 y Fl(1)1186 1066 y Fs(S)s Fu(\(\()p Fs(a)1283 1048 y Fl(0)1303 1066 y Fu(\))1322 1073 y Fl(2)1341 1066 y Fu(\))p Fs(S)s Fu(\()p Fs(xa)1466 1048 y Fl(1)1485 1066 y Fu(\))g(=)g(0)i(since)228 1125 y Fs(\016)r Fu(\()p Fs(a)p Fu(\))13 b(=)380 1092 y Fm(P)433 1125 y Fs(a)459 1107 y Fl(0)489 1125 y Fr(\012)e Fs(a)565 1107 y Fl(1)598 1125 y Fr(2)p 645 1086 256 2 v 14 w Fs(P)18 b Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(P)18 b Fr(\012)11 b Fs(K)18 b Fr(\022)c Fs(U)5 b Fu(\()p Fs(P)i Fu(\))k Fr(\012)g Fs(K)t Fu(.)21 b(So)c(w)o(e)f(ha)o(v)o(e)297 1210 y Fr(r)p Fu(\(1)10 b Fr(\012)h Fs(S)s Fu(\)\001\()p Fs(xa)p Fu(\))41 b(=)13 b Fr(r)p Fu(\(1)e Fr(\012)g Fs(S)s Fu(\))925 1177 y Fm(P)969 1210 y Fu(\()p Fs(xa)1042 1217 y Fl(1)1072 1210 y Fr(\012)g Fs(a)1148 1217 y Fl(2)1178 1210 y Fu(+)g(\()p Fs(a)1272 1217 y Fl(1)1292 1210 y Fu(\))1311 1192 y Fl(0)1341 1210 y Fr(\012)g Fu(\()p Fs(x)p Fu(\()p Fs(a)1483 1217 y Fl(1)1502 1210 y Fu(\))1521 1192 y Fl(1)1541 1210 y Fu(\))p Fs(a)1586 1217 y Fl(2)1605 1210 y Fu(\))668 1268 y(=)719 1235 y Fm(P)771 1268 y Fs(xa)825 1275 y Fl(1)844 1268 y Fs(S)s Fu(\()p Fs(a)922 1275 y Fl(2)942 1268 y Fu(\))g(+)1021 1235 y Fm(P)1065 1268 y Fu(\()p Fs(a)1110 1275 y Fl(1)1129 1268 y Fu(\))1148 1250 y Fl(0)1167 1268 y Fs(S)s Fu(\(\()p Fs(x)p Fu(\()p Fs(a)1311 1275 y Fl(1)1330 1268 y Fu(\))1349 1250 y Fl(1)1369 1268 y Fu(\))p Fs(a)1414 1275 y Fl(2)1433 1268 y Fu(\))668 1326 y(=)719 1293 y Fm(P)763 1326 y Fu(\()p Fs(a)808 1333 y Fl(1)827 1326 y Fu(\))846 1308 y Fl(0)866 1326 y Fs(S)s Fu(\(\()p Fs(a)963 1333 y Fl(2)982 1326 y Fu(\))1001 1308 y Fl(0)1021 1326 y Fu(\))p Fs(S)s Fu(\()p Fs(x)p Fu(\()p Fs(a)1165 1333 y Fl(1)1184 1326 y Fu(\))1203 1308 y Fl(1)1222 1326 y Fu(\()p Fs(a)1267 1333 y Fl(2)1287 1326 y Fu(\))1306 1308 y Fl(1)1325 1326 y Fu(\))j(=)g(0)g(=)g Fs(\021)r(")p Fu(\()p Fs(xa)p Fu(\))p Fs(:)228 1413 y Fu(The)j(second)h(condition)f Fr(r)p Fu(\()p Fs(S)e Fr(\012)c Fu(1\)\001)17 b(=)f Fs(\021)r(")h Fu(is)g(pro)o(v)o(ed)g(in)g(a)h(similar)e(w)o(a)o(y)h(\(b)o(y)228 1471 y(using)i(elemen)o(ts)d(of)j(the)f(form)f Fs(ax)h Fu(and)i(the)e(equation)1279 1438 y Fm(P)1331 1471 y Fs(S)s Fu(\(\()p Fs(a\024)p Fu(\))1475 1478 y Fl(1)1494 1471 y Fu(\)\()p Fs(a\024)p Fu(\))1605 1478 y Fl(2)1642 1471 y Fu(=)g(0)228 1529 y(for)f Fs(a)f Fu(written)h(as)g(a)g(pro)q (duct)h(of)f Fs(n)g Fu(elemen)o(ts)d(in)i Fs(P)24 b Fu(and)17 b Fs(\024)e Fr(2)g Fs(K)t Fu(\).)23 b(So)17 b Fs(S)j Fu(is)d(an)228 1587 y(an)o(tip)q(o)q(de)g(and)g Fs(U)5 b Fu(\()p Fs(P)i Fu(\))16 b(is)g(a)h(Hopf)f(algebra)h(in)f Fr(C)s Fu(.)548 b Fi(\003)708 1707 y Fu(7.)24 b(\()p Fs(G;)8 b(\037)p Fu(\))19 b Ft(Lie)g(algebras)278 1794 y Fu(In)e([5])h(w)o(e)f(in)o(tro)q(duced)h(and)h(studied)e(the)h (concept)g(of)g Fs(G)p Fu(-graded)h(Lie)f(alge-)228 1852 y(bras)g(or)h(\()p Fs(G;)8 b(\037)p Fu(\)-Lie)17 b(algebras)i(for)f(an) h(ab)q(elian)f(group)h Fs(G)f Fu(with)g(a)g(bic)o(haracter)228 1910 y Fs(\037)j Fu(generalizing)g(the)h(concepts)g(of)g(Lie)g (algebras,)i(Lie)d(sup)q(er)i(algebras,)h(and)228 1969 y(Lie)c(color)g(algebras.)33 b(The)20 b(reader)g(ma)o(y)e(\014nd)i (examples)e(of)j(suc)o(h)e(\()p Fs(G;)8 b(\037)p Fu(\)-Lie)228 2027 y(algebras)24 b(in)f([5].)43 b(A)24 b(generalization)f(of)g(this)h (concept)f(of)h(Lie)g(algebras)g(to)228 2085 y(the)15 b(group)i(graded)g(case)e(for)h(a)h(noncomm)o(utativ)o(e)12 b(group)17 b(requires)e(the)g(use)h(of)228 2143 y(Y)l(etter-Drinfeld)j (mo)q(dules)h(o)o(v)o(er)g Fs(k)r(G)p Fu(.)36 b(W)l(e)21 b(sho)o(w)g(that)h(\()p Fs(G;)8 b(\037)p Fu(\)-Lie)21 b(algebras)228 2201 y(are)15 b(Lie)h(algebras)g(on)g(Y)l (etter-Drinfeld)d(mo)q(dules)i(in)g(the)g(sense)h(of)g(this)f(pap)q (er.)228 2259 y(W)l(e)h(use)g(the)g(notation)h(of)g([5].)278 2317 y(Let)e Fs(G)g Fu(b)q(e)g(an)h(ab)q(elian)f(group)h(with)e(a)i (bic)o(haracter)e Fs(\037)f Fu(:)h Fs(G)8 b Fr(\012)1412 2324 y Fd(Z)1445 2317 y Fs(G)14 b Fr(\000)-30 b(!)14 b Fs(k)1597 2299 y Fj(\003)1616 2317 y Fu(.)21 b(Let)228 2375 y Fs(P)i Fu(b)q(e)16 b(a)g Fs(k)r(G)p Fu(-como)q(dule.)k(Then)c Fs(P)23 b Fu(is)15 b(a)h(Y)l(etter-Drinfeld)e(mo)q(dule)h(o)o(v)o(er)g Fs(k)r(G)h Fu([4])228 2433 y(with)i(the)f(mo)q(dule)g(structure)h Fs(x)12 b Fr(\001)g Fs(g)19 b Fu(=)d Fs(\037)p Fu(\()p Fs(h;)8 b(g)r Fu(\))p Fs(x)18 b Fu(for)g(homogeneous)g(elemen)o(ts)228 2492 y Fs(x)24 b Fu(=)g Fs(x)370 2499 y Fq(h)417 2492 y Fr(2)h Fs(M)i Fu(with)c Fs(\016)r Fu(\()p Fs(x)p Fu(\))g(=)i Fs(x)15 b Fr(\012)g Fs(h)p Fu(.)40 b(The)22 b(braid)h(map)f(is)g Fs(\034)6 b Fu(\()p Fs(x)1506 2499 y Fq(h)1543 2492 y Fr(\012)15 b Fs(y)1621 2499 y Fq(g)1641 2492 y Fu(\))24 b(=)228 2550 y Fs(y)252 2557 y Fq(g)274 2550 y Fr(\012)s Fs(x)344 2557 y Fq(h)369 2550 y Fr(\001)s Fs(g)16 b Fu(=)d Fs(\037)p Fu(\()p Fs(h;)8 b(g)r Fu(\))p Fs(y)644 2557 y Fq(g)667 2550 y Fr(\012)s Fs(x)737 2557 y Fq(h)758 2550 y Fu(,)13 b(hence)f(the)f(braiding)i(giv)o(en)e(in)h([5])g(after)g (Example)228 2608 y(2.3.)p eop %%Page: 17 17 17 16 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(17)278 225 y Fu(Let)11 b Fs(\020)17 b Fr(2)e Fs(k)473 207 y Fj(\003)503 225 y Fu(b)q(e)c(giv)o(en.)19 b(Let)11 b(\()p Fs(g)833 232 y Fl(1)853 225 y Fs(;)d(:)g(:)g(:)15 b(;)8 b(g)993 232 y Fq(n)1017 225 y Fu(\))j(b)q(e)g(a)g Fs(\020)t Fu(-family)l(,)f (i.e.)18 b Fs(\037)p Fu(\()p Fs(g)1494 232 y Fq(i)1508 225 y Fs(;)8 b(g)1553 232 y Fq(j)1571 225 y Fu(\))p Fs(\037)p Fu(\()p Fs(g)1663 232 y Fq(j)1681 225 y Fs(;)g(g)1726 232 y Fq(i)1741 225 y Fu(\))13 b(=)228 283 y Fs(\020)253 265 y Fl(2)273 283 y Fu(.)21 b(Let)16 b Fs(Q)e Fu(:=)513 250 y Fm(P)557 294 y Fq(\033)q Fj(2)p Fq(S)623 298 y Fh(n)655 283 y Fs(P)686 290 y Fq(g)703 297 y Fh(\033)q Ff(\(1\))776 283 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(P)975 290 y Fq(g)992 297 y Fh(\033)q Ff(\()p Fh(n)p Ff(\))1058 283 y Fu(.)228 354 y FB(Lemm)o(a)16 b(7.1.)k Fs(Q)d Fo(is)h(a)f(right)g Fs(S)831 361 y Fq(n)855 354 y Fo(-mo)n(dule)h(by)403 436 y Fu(\()p Fs(x)450 443 y Fl(1)481 436 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(x)677 443 y Fq(n)700 436 y Fu(\))p Fs(\033)k Fu(=)f Fs(\032)p Fu(\()p Fs(\033)o(;)8 b Fu(\()p Fs(g)949 443 y Fl(1)969 436 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(g)1110 443 y Fq(n)1133 436 y Fu(\)\))p Fs(x)1199 444 y Fq(\033)q Fl(\(1\))1279 436 y Fr(\012)i Fs(:)e(:)g(:)j Fr(\012)g Fs(x)1475 444 y Fq(\033)q Fl(\()p Fq(n)p Fl(\))228 517 y Fo(for)17 b Fs(\033)e Fr(2)f Fs(S)426 524 y Fq(n)467 517 y Fo(and)k Fs(x)590 524 y Fl(1)620 517 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(x)816 524 y Fq(n)853 517 y Fr(2)j Fs(P)931 524 y Fq(g)948 529 y Ff(1)979 517 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(P)1178 524 y Fq(g)1195 528 y Fh(n)1218 517 y Fo(.)228 604 y(Pr)n(o)n(of.)19 b Fu(W)l(e)h(ha)o(v)o(e)f(to)i(sho)o(w)f(the)g(compatibilit)o(y)d(of)j (this)g(op)q(eration)h(with)f(the)228 662 y(comp)q(osition)15 b(of)h(p)q(erm)o(utations.)k(Let)c Fs(\033)o(;)8 b(\034)19 b Fr(2)14 b Fs(S)1125 669 y Fq(n)1148 662 y Fu(.)22 b(W)l(e)15 b(use)h(Lemma)d(2.2)j(of)g([5].)228 720 y(Then)285 788 y(\()p Fs(x)332 795 y Fl(1)351 788 y Fr(\012)p Fs(:)8 b(:)g(:)i Fr(\012)h Fs(x)536 795 y Fq(n)559 788 y Fu(\)\()p Fs(\033)r(\034)6 b Fu(\))13 b(=)390 846 y(=)h Fs(\032)p Fu(\()p Fs(\033)r(\034)s(;)8 b Fu(\()p Fs(g)604 853 y Fl(1)623 846 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(g)764 853 y Fq(n)788 846 y Fu(\)\))p Fs(x)854 854 y Fq(\033)q(\034)t Fl(\(1\))952 846 y Fr(\012)j Fs(:)d(:)g(:)j Fr(\012)f Fs(x)1148 854 y Fq(\033)q(\034)t Fl(\()p Fq(n)p Fl(\))390 905 y Fu(=)k Fs(\032)p Fu(\()p Fs(\033)o(;)8 b Fu(\()p Fs(g)577 912 y Fl(1)597 905 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(g)737 912 y Fq(n)761 905 y Fu(\)\))p Fs(\032)p Fu(\()p Fs(\034)s(;)g Fu(\()p Fs(g)931 912 y Fq(\033)q Fl(\(1\))1000 905 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(g)1140 912 y Fq(\033)q Fl(\()p Fq(n)p Fl(\))1213 905 y Fu(\)\))p Fs(x)1279 912 y Fq(\033)q(\034)t Fl(\(1\))1378 905 y Fr(\012)j Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1574 912 y Fq(\033)q(\034)t Fl(\()p Fq(n)p Fl(\))390 963 y Fu(=)j(\()p Fs(\032)p Fu(\()p Fs(\033)o(;)8 b Fu(\()p Fs(g)596 970 y Fl(1)616 963 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(g)756 970 y Fq(n)780 963 y Fu(\)\))p Fs(x)846 970 y Fq(\033)q Fl(\(1\))925 963 y Fr(\012)j Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1121 970 y Fq(\033)q Fl(\()p Fq(n)p Fl(\))1193 963 y Fu(\))p Fs(\034)390 1021 y Fu(=)j(\(\()p Fs(x)508 1028 y Fl(1)538 1021 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(x)734 1028 y Fq(n)757 1021 y Fu(\))p Fs(\033)r Fu(\))p Fs(\034)s(:)1683 1099 y Fi(\003)278 1186 y Fs(Q)f Fu(b)q(ecomes)g(a)h (left)f Fs(S)661 1193 y Fq(n)684 1186 y Fu(-mo)q(dule)g(b)o(y)h Fs(\033)r Fu(\()p Fs(x)1007 1193 y Fl(1)1025 1186 y Fr(\012)p Fs(:)d(:)g(:)o Fr(\012)p Fs(x)1188 1193 y Fq(n)1211 1186 y Fu(\))14 b(=)g Fs(\032)p Fu(\()p Fs(\033)1370 1168 y Fj(\000)p Fl(1)1417 1186 y Fs(;)8 b Fu(\()p Fs(g)1481 1193 y Fl(1)1501 1186 y Fs(;)g(:)g(:)g(:)15 b(;)8 b(g)1641 1193 y Fq(n)1665 1186 y Fu(\)\))p Fs(x)1731 1194 y Fq(\033)1752 1185 y Fg(\000)p Ff(1)1793 1194 y Fl(\(1\))1841 1186 y Fr(\012)228 1244 y Fs(:)g(:)g(:)g Fr(\012)h Fs(x)370 1252 y Fq(\033)391 1243 y Fg(\000)p Ff(1)432 1252 y Fl(\()p Fq(n)p Fl(\))483 1244 y Fu(.)21 b(Th)o(us)640 1211 y Fm(L)686 1258 y Fj(f)p Fl(\()p Fq(g)735 1263 y Ff(1)752 1258 y Fq(;:::)5 b(;g)824 1262 y Fh(n)845 1258 y Fl(\))16 b Fq(\020)r Fc(-family)p Fj(g)1059 1244 y Fs(P)1090 1251 y Fq(g)1107 1256 y Ff(1)1136 1244 y Fr(\012)8 b Fs(:)g(:)g(:)h Fr(\012)f Fs(P)1328 1251 y Fq(g)1345 1255 y Fh(n)1384 1244 y Fu(is)15 b(also)h(a)f(left)f Fs(S)1682 1251 y Fq(n)1706 1244 y Fu(-)228 1306 y(mo)q(dule.)278 1364 y(This)d(action)f(is)h(connected)f(with)h(the)g(action)f(of)i Fs(B)1203 1371 y Fq(n)1237 1364 y Fu(on)1299 1331 y Fm(L)1345 1379 y Fj(f)p Fl(\()p Fq(g)1394 1384 y Ff(1)1411 1379 y Fq(;:::)5 b(;g)1483 1383 y Fh(n)1504 1379 y Fl(\))16 b Fq(\020)r Fc(-)f(family)q Fj(g)1734 1364 y Fs(P)1765 1371 y Fq(g)1782 1376 y Ff(1)1801 1364 y Fr(\012)228 1426 y Fs(:)8 b(:)g(:)i Fr(\012)h Fs(P)377 1433 y Fq(g)394 1437 y Fh(n)434 1426 y Fu(b)o(y)552 1509 y Fs(\020)577 1489 y Fj(\000)p Fl(1)624 1509 y Fs(\034)645 1516 y Fq(i)659 1509 y Fu(\()p Fs(x)706 1516 y Fl(1)737 1509 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)933 1516 y Fq(n)956 1509 y Fu(\))j(=)g Fs(\033)1069 1516 y Fq(i)1082 1509 y Fu(\()p Fs(x)1129 1516 y Fl(1)1160 1509 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1356 1516 y Fq(n)1379 1509 y Fu(\))237 b(\(12\))228 1591 y(for)16 b(the)g(canonical)g(generators)h Fs(\034)857 1598 y Fq(i)888 1591 y Fu(of)f Fs(B)980 1598 y Fq(n)1020 1591 y Fu(resp.)21 b Fs(\033)1170 1598 y Fq(i)1201 1591 y Fu(of)16 b Fs(S)1286 1598 y Fq(n)1310 1591 y Fu(,)f(since)431 1669 y Fs(\020)456 1651 y Fj(\000)p Fl(1)503 1669 y Fs(\034)524 1676 y Fq(i)538 1669 y Fu(\()p Fs(x)585 1676 y Fl(1)616 1669 y Fr(\012)c Fs(:)d(:)g(:)i Fr(\012)h Fs(x)812 1676 y Fq(n)835 1669 y Fu(\))j(=)538 1727 y(=)g Fs(\020)615 1709 y Fj(\000)p Fl(1)662 1727 y Fs(\037)p Fu(\()p Fs(g)735 1734 y Fq(i)749 1727 y Fs(;)8 b(g)794 1734 y Fq(i)p Fl(+1)854 1727 y Fu(\))p Fs(x)901 1734 y Fl(1)931 1727 y Fr(\012)j Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1127 1734 y Fq(i)p Fl(+1)1197 1727 y Fr(\012)g Fs(x)1275 1734 y Fq(i)1300 1727 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1496 1734 y Fq(n)538 1786 y Fu(=)j Fs(\032)p Fu(\()p Fs(\033)664 1766 y Fj(\000)p Fl(1)662 1798 y Fq(i)711 1786 y Fs(;)8 b Fu(\()p Fs(g)775 1793 y Fl(1)795 1786 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(g)936 1793 y Fq(n)959 1786 y Fu(\)\))p Fs(x)1025 1800 y Fq(\033)1046 1785 y Fg(\000)p Ff(1)1045 1812 y Fh(i)1088 1800 y Fl(\(1\))1146 1786 y Fr(\012)j Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1342 1800 y Fq(\033)1363 1785 y Fg(\000)p Ff(1)1362 1812 y Fh(i)1404 1800 y Fl(\()p Fq(n)p Fl(\))538 1853 y Fu(=)j Fs(\033)618 1860 y Fq(i)632 1853 y Fu(\()p Fs(x)679 1860 y Fl(1)710 1853 y Fr(\012)c Fs(:)e(:)g(:)j Fr(\012)f Fs(x)905 1860 y Fq(n)929 1853 y Fu(\))p Fs(:)228 1932 y Fu(In)16 b(particular)g(w)o (e)f(ha)o(v)o(e)396 2011 y Fs(\034)423 1991 y Fj(\000)p Fl(1)417 2023 y Fq(i)470 2011 y Fs(\034)497 1991 y Fj(\000)p Fl(1)491 2023 y Fq(i)p Fl(+1)559 2011 y Fs(:)8 b(:)g(:)f(\034)651 1991 y Fj(\000)p Fl(1)645 2023 y Fq(j)r Fj(\000)p Fl(1)709 2011 y Fs(\034)736 1993 y Fl(2)730 2023 y Fq(j)755 2011 y Fs(\034)776 2018 y Fq(j)r Fj(\000)p Fl(1)848 2011 y Fs(:)h(:)g(:)g(\034)935 2018 y Fq(i)p Fl(+1)994 2011 y Fs(\034)1015 2018 y Fq(i)1029 2011 y Fu(\()p Fs(x)1076 2018 y Fl(1)1107 2011 y Fr(\012)j Fs(:)d(:)g(:)i Fr(\012)h Fs(x)1303 2018 y Fq(n)1326 2011 y Fu(\))j(=)470 2072 y(=)g Fs(\020)547 2054 y Fl(2)567 2072 y Fs(\033)597 2051 y Fj(\000)p Fl(1)595 2083 y Fq(i)643 2072 y Fs(\033)673 2051 y Fj(\000)p Fl(1)671 2083 y Fq(i)p Fl(+1)739 2072 y Fs(:)8 b(:)g(:)f(\033)834 2051 y Fj(\000)p Fl(1)832 2083 y Fq(j)r Fj(\000)p Fl(1)895 2072 y Fs(\033)925 2054 y Fl(2)923 2084 y Fq(j)944 2072 y Fs(\033)972 2079 y Fq(j)r Fj(\000)p Fl(1)1044 2072 y Fs(:)h(:)g(:)f(\033)1137 2079 y Fq(i)p Fl(+1)1196 2072 y Fs(\033)1224 2079 y Fq(i)1238 2072 y Fu(\()p Fs(x)1285 2079 y Fl(1)1316 2072 y Fr(\012)j Fs(:)e(:)g(:)j Fr(\012)g Fs(x)1512 2079 y Fq(n)1535 2072 y Fu(\))470 2131 y(=)j Fs(\020)547 2113 y Fl(2)567 2131 y Fu(\()p Fs(x)614 2138 y Fl(1)644 2131 y Fr(\012)d Fs(:)d(:)g(:)i Fr(\012)h Fs(x)840 2138 y Fq(n)863 2131 y Fu(\))p Fs(;)228 2214 y Fu(so)17 b(that)f Fs(x)421 2221 y Fl(1)452 2214 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(x)648 2221 y Fq(n)685 2214 y Fr(2)j Fs(P)770 2196 y Fq(n)794 2214 y Fu(\()p Fs(\020)t Fu(\))i(b)o(y)g(Lemma)e(2.4.)21 b(Th)o(us)c(w)o(e)f (ha)o(v)o(e)678 2258 y Fm(M)528 2358 y Fj(f)p Fl(\()p Fq(g)577 2363 y Ff(1)594 2358 y Fq(;:::)5 b(;g)666 2362 y Fh(n)686 2358 y Fl(\))16 b Fq(\020)r Fc(-family)q Fj(g)899 2299 y Fs(P)930 2306 y Fq(g)947 2311 y Ff(1)978 2299 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(P)1177 2306 y Fq(g)1194 2310 y Fh(n)1231 2299 y Fr(\022)j Fs(P)1322 2279 y Fq(n)1345 2299 y Fu(\()p Fs(\020)t Fu(\))p Fs(:)228 2433 y Fu(Con)o(v)o(ersely)g(let)543 2400 y Fm(P)596 2433 y Fs(x)624 2440 y Fl(1)652 2433 y Fr(\012)9 b Fs(:)f(:)g(:)h Fr(\012)g Fs(x)843 2440 y Fq(n)880 2433 y Fr(2)14 b Fs(P)965 2415 y Fq(n)1003 2433 y Fu(=)1054 2400 y Fm(L)1101 2444 y Fj(f)p Fl(\()p Fq(g)1150 2449 y Ff(1)1166 2444 y Fq(;:::)5 b(;g)1238 2448 y Fh(n)1259 2444 y Fl(\))p Fj(g)1301 2433 y Fs(P)1332 2440 y Fq(g)1349 2445 y Ff(1)1378 2433 y Fr(\012)k Fs(:)f(:)g(:)g Fr(\012)h Fs(P)1571 2440 y Fq(g)1588 2444 y Fh(n)1627 2433 y Fu(with)228 2492 y(homogeneous)k(summands)f (and)i(assume)f(that)g(one)h(of)f(the)g(summands)f(is)h(non-)228 2550 y(zero)i(in)g Fs(P)417 2557 y Fq(g)434 2562 y Ff(1)462 2550 y Fr(\012)9 b Fs(:)f(:)g(:)g Fr(\012)h Fs(P)655 2557 y Fq(g)672 2561 y Fh(n)711 2550 y Fu(where)15 b(\()p Fs(g)893 2557 y Fl(1)913 2550 y Fs(;)8 b(:)g(:)g(:)15 b(;)8 b(g)1053 2557 y Fq(n)1077 2550 y Fu(\))15 b(is)g(not)h(a)g Fs(\020)t Fu(-family)d(for)i(example)228 2608 y(b)o(y)k Fs(\037)p Fu(\()p Fs(g)372 2615 y Fq(i)386 2608 y Fs(;)8 b(g)431 2615 y Fq(i)p Fl(+1)490 2608 y Fu(\))p Fs(\037)p Fu(\()p Fs(g)582 2615 y Fq(i)p Fl(+1)641 2608 y Fs(;)g(g)686 2615 y Fq(i)701 2608 y Fu(\))19 b Fr(6)p Fu(=)g Fs(\020)821 2590 y Fl(2)841 2608 y Fu(.)31 b(Then)20 b(\()p Fs(\034)1063 2590 y Fl(2)1057 2620 y Fq(i)1096 2608 y Fr(\000)13 b Fs(\020)1173 2590 y Fl(2)1192 2608 y Fu(\)\()1230 2575 y Fm(P)1282 2608 y Fs(x)1310 2615 y Fl(1)1343 2608 y Fr(\012)g Fs(:)8 b(:)g(:)13 b Fr(\012)g Fs(x)1546 2615 y Fq(n)1569 2608 y Fu(\))20 b(has)g(a)p eop %%Page: 18 18 18 17 bop 228 125 a Fp(18)552 b(BODO)13 b(P)m(AREIGIS)228 225 y Fu(non-zero)18 b(comp)q(onen)o(t)f(in)g Fs(P)767 232 y Fq(g)784 237 y Ff(1)816 225 y Fr(\012)12 b Fs(:)c(:)g(:)j Fr(\012)h Fs(P)1018 232 y Fq(g)1035 236 y Fh(n)1058 225 y Fu(,)18 b(hence)1227 192 y Fm(P)1279 225 y Fs(x)1307 232 y Fl(1)1339 225 y Fr(\012)12 b Fs(:)c(:)g(:)j Fr(\012)h Fs(x)1538 232 y Fq(n)1579 225 y Fu(cannot)228 283 y(b)q(e)k(in)g Fs(P)389 265 y Fq(n)413 283 y Fu(\()p Fs(\020)t Fu(\).)21 b(This)16 b(pro)o(v)o(es)228 358 y FB(Prop)r(osition)i(7.2.)h Fo(L)n(et)f Fs(\020)g Fr(2)c Fs(k)836 340 y Fj(\003)873 358 y Fo(b)n(e)k(given.)23 b(Then)528 452 y Fs(P)566 431 y Fq(n)590 452 y Fu(\()p Fs(\020)t Fu(\))13 b(=)869 410 y Fm(M)718 511 y Fj(f)p Fl(\()p Fq(g)767 516 y Ff(1)784 511 y Fq(;:::)5 b(;g)856 515 y Fh(n)877 511 y Fl(\))17 b Fq(\020)r Fc(-family)p Fj(g)1090 452 y Fs(P)1121 459 y Fq(g)1138 464 y Ff(1)1169 452 y Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fs(P)1368 459 y Fq(g)1385 463 y Fh(n)1409 452 y Fs(:)278 600 y Fu(By)k(Lemma)e(7.1)j(and)g(\(12\))h(the)e(brac)o(k)o (et)g(m)o(ultipli)o(cation)e(of)j([5])f(is)h(a)g(sp)q(ecial)228 658 y(case)e(of)h(the)g(brac)o(k)o(et)e(m)o(ultiplic)o(ation)f(of)j (this)g(pap)q(er)g(and)g(\()p Fs(G;)8 b(\037)p Fu(\)-Lie)15 b(algebras)228 716 y(are)h(Lie)g(algebras)h(o)o(v)o(er)e(Y)l (etter-Drinfeld)g(mo)q(dules.)228 791 y FB(Example)f(7.3.)k Fu(As)c(a)h(new)f(example)e(of)j(Lie)f(algebras)g(w)o(e)g(giv)o(e)f (one)i(family)d(of)228 849 y(examples)k(of)i(\()p Fs(G;)8 b(\037)p Fu(\)-Lie)18 b(algebras.)27 b(Let)18 b Fs(G)g Fu(=)e Fs(C)1174 856 y Fl(3)1211 849 y Fu(=)h Fr(f)p Fu(0)p Fs(;)8 b Fu(1)p Fs(;)g Fu(2)p Fr(g)18 b Fu(b)q(e)h(the)e(cyclic) 228 907 y(group)i(with)f(3)g(elemen)o(ts.)24 b(De\014ne)17 b(the)h(structure)g(of)g(a)h(righ)o(t)e Fs(k)r(G)p Fu(-mo)q(dule)h(on) 228 966 y(a)e Fs(k)r(G)p Fu(-como)q(dule)f Fs(V)27 b Fu(\(i.e.)20 b(on)c(a)g Fs(C)867 973 y Fl(3)886 966 y Fu(-graded)h(v)o(ector)e(space)g Fs(V)25 b Fu(=)14 b Fs(V)1470 973 y Fl(0)1500 966 y Fr(\010)c Fs(V)1577 973 y Fl(1)1607 966 y Fr(\010)f Fs(V)1683 973 y Fl(2)1703 966 y Fu(\))228 1024 y(using)14 b(the)f(bic)o(haracter)g Fs(\037)g Fu(:)h Fs(C)792 1031 y Fl(3)817 1024 y Fr(\012)856 1031 y Fd(Z)885 1024 y Fs(C)920 1031 y Fl(3)954 1010 y Fr(\030)954 1026 y Fu(=)1006 1024 y Fs(C)1041 1031 y Fl(3)1075 1024 y Fr(\000)-30 b(!)13 b Fs(k)1174 1006 y Fj(\003)1194 1024 y Fu(,)h Fs(\037)p Fu(\(1)6 b Fr(\012)g Fu(1\))13 b(=)h Fs(\030)i Fu(a)e(primitiv)o(e)228 1082 y(3-rd)i(ro)q(ot)g(of)g(unit)o(y)l(,)e(b)o(y)h Fs(v)10 b Fr(\001)f Fs(g)16 b Fu(:=)d Fs(\037)p Fu(\(deg)q(\()p Fs(v)r Fu(\))8 b Fr(\012)h Fs(g)r Fu(\))p Fs(v)15 b Fu(=)f Fs(\037)p Fu(\(deg\()p Fs(v)r Fu(\))p Fs(;)8 b(g)r Fu(\))p Fs(v)16 b Fu(for)g Fs(g)g Fr(2)e Fs(G)228 1140 y Fu(and)e(homogeneous)f (elemen)o(ts)e Fs(v)15 b Fr(2)f Fs(V)d Fu(.)20 b(Then)12 b Fs(V)22 b Fu(is)11 b(a)h(Y)l(etter-Drinfeld)e(mo)q(dule.)278 1198 y(Let)h Fs(A)i Fu(:=)g(end\()p Fs(V)e Fu(\))g(b)q(e)g(the)g(inner) f(endomorphism)f(ob)s(ject)h(of)h Fs(V)22 b Fu(in)11 b Fs(k)r(G)p Fu(-como)q(d.)228 1256 y(By)18 b(Corollary)h(4.2)f Fs(A)h Fu(is)f(a)h(Lie)f(algebra.)29 b(One)19 b(v)o(eri\014es)e(easily) h(\(see)g([5]\))g(that)228 1314 y(the)e(only)h(non-zero)g(comp)q(onen)o (ts)f Fs(A)924 1296 y Fq(n)947 1314 y Fu(\()p Fs(\020)t Fu(\))g(for)h(the)g(partial)f(Lie)g(m)o(ultiplication)228 1372 y(are)446 1444 y Fs(A)483 1424 y Fl(2)502 1444 y Fu(\()p Fr(\000)p Fu(1\))e(=)g Fs(A)706 1451 y Fl(0)736 1444 y Fr(\012)d Fu(\()p Fs(A)842 1451 y Fl(1)872 1444 y Fr(\010)g Fs(A)959 1451 y Fl(2)978 1444 y Fu(\))h Fr(\010)e Fu(\()p Fs(A)1114 1451 y Fl(0)1145 1444 y Fr(\010)g Fs(A)1231 1451 y Fl(1)1262 1444 y Fr(\010)h Fs(A)1349 1451 y Fl(2)1368 1444 y Fu(\))g Fr(\012)g Fs(A)1485 1451 y Fl(0)228 1526 y Fu(and)555 1598 y Fs(A)592 1578 y Fl(3)612 1598 y Fu(\()p Fs(\030)r Fu(\))j(=)g Fs(A)776 1605 y Fl(1)806 1598 y Fr(\012)d Fs(A)893 1605 y Fl(1)923 1598 y Fr(\012)g Fs(A)1010 1605 y Fl(1)1041 1598 y Fr(\010)f Fs(A)1127 1605 y Fl(2)1158 1598 y Fr(\012)h Fs(A)1245 1605 y Fl(2)1275 1598 y Fr(\012)g Fs(A)1362 1605 y Fl(2)1381 1598 y Fs(:)278 1680 y Fu(No)o(w)16 b(let)f Fr(h)p Fs(:;)8 b(:)p Fr(i)14 b Fu(:)f Fs(V)22 b Fr(\012)11 b Fs(V)25 b Fr(\000)-30 b(!)14 b Fs(k)k Fu(b)q(e)e(a)h(bilinear)e(form)g(on)i Fs(V)28 b Fu(in)16 b Fr(C)s Fu(.)21 b(W)l(e)16 b(de\014ne)228 1767 y Fb(g)253 1772 y Fu(\()p Fs(V)11 b Fu(\))330 1779 y Fq(i)358 1772 y Fu(:=)i Fr(f)p Fs(f)19 b Fr(2)14 b Fs(A)575 1779 y Fq(i)589 1772 y Fr(j8)p Fs(v)r(;)8 b(w)13 b Fr(2)h Fs(V)s(;)8 b Fu(deg)q(\()p Fs(v)r Fu(\))13 b(=)h Fs(j)j Fu(:)24 b Fr(h)p Fs(f)5 b Fu(\()p Fs(v)r Fu(\))p Fs(;)j(w)q Fr(i)14 b Fu(=)g Fr(\000)p Fs(\037)p Fu(\()p Fs(i;)8 b(j)s Fu(\))p Fr(h)p Fs(v)r(;)g(f)d Fu(\()p Fs(w)q Fu(\))p Fr(ig)p Fs(:)228 1865 y Fu(This)17 b(space)g(is)g(the)g(homogeneous)h(comp)q (onen)o(t)e(of)1212 1860 y Fb(g)1237 1865 y Fu(\()p Fs(V)c Fu(\))j Fr(\022)g Fs(A)i Fu(that)g(b)q(ecomes)228 1923 y(a)f(Y)l(etter-Drinfeld)f(mo)q(dule.)278 1981 y(F)l(or)h Fs(f)j Fr(2)455 1976 y Fb(g)480 1981 y Fu(\()p Fs(V)12 b Fu(\))558 1988 y Fl(0)594 1981 y Fu(and)k Fs(g)g Fr(2)774 1976 y Fb(g)799 1981 y Fu(\()p Fs(V)c Fu(\))877 1988 y Fq(i)891 1981 y Fu(,)k Fs(i)d Fr(2)h Fs(C)1033 1988 y Fl(3)1053 1981 y Fu(,)i Fs(v)f Fr(2)f Fs(V)1197 1988 y Fq(j)1215 1981 y Fu(,)i Fs(w)f Fr(2)f Fs(V)28 b Fu(w)o(e)16 b(ha)o(v)o(e)444 2074 y Fr(h)p Fu([)p Fs(f)s(;)8 b(g)r Fu(]\()p Fs(v)r Fu(\))p Fs(;)g(w)q Fr(i)o Fu(=)13 b Fr(h)p Fu(\()p Fs(f)5 b(g)14 b Fr(\000)d Fs(g)r(f)5 b Fu(\)\()p Fs(v)r Fu(\))p Fs(;)j(w)q Fr(i)705 2132 y Fu(=)13 b Fr(h)p Fs(f)5 b(g)r Fu(\()p Fs(v)r Fu(\))p Fs(;)j(w)q Fr(i)k(\000)f(h)p Fs(g)r(f)5 b Fu(\()p Fs(v)r Fu(\))p Fs(;)j(w)q Fr(i)705 2190 y Fu(=)13 b Fs(\037)p Fu(\()p Fs(i;)8 b(j)s Fu(\))p Fr(h)p Fs(v)r(;)g(g)r(f)d Fu(\()p Fs(w)q Fu(\))p Fr(i)11 b(\000)g Fs(\037)p Fu(\()p Fs(i;)d(j)s Fu(\))p Fr(h)p Fs(v)r(;)g(f)d(g)r Fu(\()p Fs(w)q Fu(\))p Fr(i)705 2248 y Fu(=)13 b Fr(\000)p Fs(\037)p Fu(\()p Fs(i;)8 b(j)s Fu(\))p Fr(h)p Fs(v)r(;)g Fu([)p Fs(f)s(;)g(g)r Fu(]\()p Fs(w)q Fu(\))p Fr(i)p Fs(;)228 2341 y Fu(hence)15 b([)p Fs(f)s(;)8 b(g)r Fu(])13 b Fr(2)525 2336 y Fb(g)550 2341 y Fu(\()p Fs(V)e Fu(\))627 2348 y Fq(i)641 2341 y Fu(.)22 b(Analogously)16 b(one)h(sho)o(ws)g([)p Fs(g)r(;)8 b(f)d Fu(])13 b Fr(2)1347 2336 y Fb(g)1372 2341 y Fu(\()p Fs(V)e Fu(\))1449 2348 y Fq(i)1463 2341 y Fu(.)278 2399 y(F)l(or)16 b Fs(k)g Fu(=)e(1)p Fs(;)8 b Fu(2)p Fs(;)g Fu(3)17 b(let)e Fs(f)685 2406 y Fq(k)720 2399 y Fr(2)767 2394 y Fb(g)793 2399 y Fu(\()p Fs(V)c Fu(\))870 2406 y Fq(i)900 2399 y Fu(\()p Fs(i)j Fu(=)f(1)k(or)g Fs(i)c Fu(=)h(2\).)21 b(Then)249 2491 y Fr(h)p Fu([)p Fs(f)306 2498 y Fl(1)325 2491 y Fs(;)8 b(f)371 2498 y Fl(2)391 2491 y Fs(;)g(f)437 2498 y Fl(3)456 2491 y Fu(]\()p Fs(v)r Fu(\))p Fs(;)o(w)q Fr(i)14 b Fu(=)668 2458 y Fm(P)712 2501 y Fq(\033)q Fj(2)p Fq(S)778 2506 y Ff(3)798 2491 y Fr(h)p Fs(f)841 2498 y Fq(\033)q Fl(\(1\))910 2491 y Fs(f)934 2498 y Fq(\033)q Fl(\(2\))1002 2491 y Fs(f)1026 2498 y Fq(\033)q Fl(\(3\))1095 2491 y Fu(\()p Fs(v)r Fu(\))p Fs(;)8 b(w)q Fr(i)547 2549 y Fu(=)14 b(\()p Fr(\000)p Fu(1\))708 2516 y Fm(P)752 2559 y Fq(\033)q Fj(2)p Fq(S)818 2564 y Ff(3)846 2549 y Fs(\037)p Fu(\()p Fs(i;)8 b(i)i Fu(+)h Fs(i)f Fu(+)h Fs(j)s Fu(\))p Fs(\037)p Fu(\()p Fs(i;)d(i)i Fu(+)h Fs(j)s Fu(\))p Fs(\037)p Fu(\()p Fs(i;)d(j)s Fu(\))p Fr(h)p Fs(v)r(;)g(f)1558 2557 y Fq(\033)q Fl(\(3\))1625 2549 y Fs(f)1649 2557 y Fq(\033)q Fl(\(2\))1718 2549 y Fs(f)1742 2557 y Fq(\033)q Fl(\(1\))1810 2549 y Fu(\()p Fs(w)q Fu(\))p Fr(i)547 2607 y Fu(=)14 b Fr(\000h)p Fs(v)r(;)8 b Fu([)p Fs(f)743 2614 y Fl(1)761 2607 y Fs(;)g(f)807 2614 y Fl(2)827 2607 y Fs(;)g(f)873 2614 y Fl(3)892 2607 y Fu(]\()p Fs(w)q Fu(\))p Fr(i)p Fs(;)p eop %%Page: 19 19 19 18 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(19)228 225 y Fu(hence)17 b([)p Fs(f)403 232 y Fl(1)422 225 y Fs(;)8 b(f)468 232 y Fl(2)487 225 y Fs(;)g(f)533 232 y Fl(3)553 225 y Fu(])16 b Fr(2)632 220 y Fb(g)657 225 y Fu(\()p Fs(V)11 b Fu(\))734 232 y Fl(0)754 225 y Fu(.)25 b(Th)o(us)18 b(w)o(e)f(ha)o(v)o(e)f(a)i(Lie)f(algebra)1401 220 y Fb(g)1426 225 y Fu(\()p Fs(V)12 b Fu(\).)25 b(Dep)q(end-)228 283 y(ing)c(on)h(the)f(c)o(hoice)f(of)i(the)f(bilinear)f(form)g(this)h(is)g (a)h(generalization)f(of)g(the)228 341 y(orthogonal)d(or)e(the)g (symplectic)d(Lie)j(algebra.)828 446 y(8.)25 b Ft(Appendix)228 533 y Fo(Pr)n(o)n(of.)19 b Fu(of)d(Lemma)f(2.4:)278 591 y(De\014ne)h(actions)g Fs(\031)622 598 y Fq(i;j)678 591 y Fu(for)h(1)d Fr(\024)g Fs(i)f(<)h(j)j Fr(\024)c Fs(n)k Fu(on)f Fs(M)22 b Fu(b)o(y)566 673 y Fs(\031)594 680 y Fq(i;j)648 673 y Fu(:=)13 b Fs(\034)740 653 y Fj(\000)p Fl(1)734 686 y Fq(i)787 673 y Fs(\034)814 653 y Fj(\000)p Fl(1)808 686 y Fq(i)p Fl(+1)875 673 y Fs(:)8 b(:)g(:)g(\034)968 653 y Fj(\000)p Fl(1)962 686 y Fq(j)r Fj(\000)p Fl(2)1026 673 y Fs(\034)1047 680 y Fq(j)r Fj(\000)p Fl(1)1110 673 y Fs(\034)1131 680 y Fq(j)r Fj(\000)p Fl(2)1203 673 y Fs(:)g(:)g(:)f(\034)1289 680 y Fq(i)p Fl(+1)1349 673 y Fs(\034)1370 680 y Fq(i)1635 673 y Fu(\(13\))228 757 y(Observ)o(e)22 b(that)i Fs(\031)563 764 y Fq(i;i)p Fl(+1)670 757 y Fu(=)i Fs(\034)755 764 y Fq(i)769 757 y Fu(.)43 b(Since)23 b Fs(\034)982 764 y Fq(i)996 757 y Fs(\034)1017 764 y Fq(j)1062 757 y Fu(=)j Fs(\034)1147 764 y Fq(j)1165 757 y Fs(\034)1186 764 y Fq(i)1217 757 y Fu(if)16 b Fr(j)p Fs(i)f Fr(\000)h Fs(j)s Fr(j)26 b(\025)g Fu(2)e(a)f(simple)228 815 y(calculation)15 b(giv)o(es)503 892 y Fs(\031)531 899 y Fq(i;j)570 892 y Fs(\034)591 899 y Fq(k)627 892 y Fu(=)f Fs(\034)700 899 y Fq(k)721 892 y Fs(\031)749 899 y Fq(i;j)805 892 y Fu(for)j(all)e Fs(k)h(<)e(i)d Fr(\000)g Fu(1)16 b(and)h(all)f Fs(k)g(>)d(j;)765 950 y(\031)793 957 y Fq(i;j)833 950 y Fs(\034)854 957 y Fq(i)p Fj(\000)p Fl(1)927 950 y Fu(=)g Fs(\034)999 957 y Fq(i)p Fj(\000)p Fl(1)1059 950 y Fs(\031)1087 957 y Fq(i)p Fj(\000)p Fl(1)p Fq(;j)1172 950 y Fs(;)604 1008 y(\031)632 1015 y Fq(i;j)672 1008 y Fs(\034)693 1015 y Fq(k)728 1008 y Fu(=)h Fs(\034)801 1015 y Fq(k)823 1008 y Fs(\031)851 1015 y Fq(i;j)907 1008 y Fu(for)i(all)g Fs(i)d(<)h(k)i(<)e(j)g Fr(\000)d Fu(1)p Fs(;)586 1066 y(\031)614 1073 y Fq(i;j)654 1066 y Fs(\034)675 1073 y Fq(j)r Fj(\000)p Fl(1)752 1066 y Fu(=)j Fs(\034)825 1073 y Fq(j)r Fj(\000)p Fl(1)889 1066 y Fs(\031)917 1073 y Fq(i;j)r Fj(\000)p Fl(1)1018 1066 y Fu(if)h Fs(i)f(<)g(j)g Fr(\000)c Fu(1)17 b(and)589 1124 y Fs(\031)617 1131 y Fq(i;j)657 1124 y Fs(\034)678 1131 y Fq(j)r Fj(\000)p Fl(1)755 1124 y Fu(=)d Fs(\034)828 1131 y Fq(i)842 1124 y Fs(\034)863 1131 y Fq(i)892 1124 y Fu(=)f Fs(\034)964 1131 y Fq(j)r Fj(\000)p Fl(1)1028 1124 y Fs(\031)1056 1131 y Fq(i;j)1112 1124 y Fu(if)j Fs(i)d Fu(=)h Fs(j)g Fr(\000)d Fu(1)p Fs(:)1635 1009 y Fu(\(14\))278 1202 y(Let)18 b Fs(N)24 b Fr(\022)18 b Fs(M)24 b Fu(b)q(e)19 b(a)g Fs(k)r(B)734 1209 y Fq(n)776 1202 y Fu(submo)q(dule)f(of)h Fs(M)5 b Fu(.)29 b(Assume)18 b(furthermore)f(that)228 1261 y Fs(\034)255 1243 y Fl(2)249 1273 y Fq(i)274 1261 y Fs(\034)295 1268 y Fq(i)p Fl(+1)368 1261 y Fu(=)d Fs(\034)441 1268 y Fq(i)p Fl(+1)501 1261 y Fs(\034)528 1243 y Fl(2)522 1273 y Fq(i)560 1261 y Fu(on)f Fs(N)18 b Fu(for)c(all)e Fs(i)h Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)t Fr(\000)t Fu(2.)21 b(Then)13 b Fs(\034)1327 1243 y Fl(2)1321 1273 y Fq(i)p Fl(+1)1380 1261 y Fs(\034)1401 1268 y Fq(i)1429 1261 y Fu(=)h Fs(\034)1508 1243 y Fl(3)1502 1273 y Fq(i)1541 1261 y Fu(=)g Fs(\034)1614 1268 y Fq(i)1628 1261 y Fs(\034)1655 1243 y Fl(2)1649 1273 y Fq(i)p Fl(+1)1709 1261 y Fs(:)228 1319 y Fu(Consequen)o(tly)h(w) o(e)h(ha)o(v)o(e)860 1399 y Fs(\034)887 1379 y Fl(2)881 1411 y Fq(j)907 1399 y Fs(\034)928 1406 y Fq(i)956 1399 y Fu(=)e Fs(\034)1029 1406 y Fq(i)1043 1399 y Fs(\034)1070 1379 y Fl(2)1064 1411 y Fq(j)1635 1399 y Fu(\(15\))228 1485 y(on)i Fs(N)22 b Fu(for)16 b(all)g Fs(i;)8 b(j)16 b Fu(=)e(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)f Fu(1)p Fs(:)16 b Fu(Th)o(us)h(the)f Fs(\034)1145 1466 y Fl(2)1139 1497 y Fq(j)1180 1485 y Fu(comm)o(ute)d(with)j(all)g Fs(')e Fr(2)g Fs(B)1699 1492 y Fq(n)228 1543 y Fu(if)i(they)f(act)i(on) f Fs(N)5 b Fu(.)278 1602 y(W)l(e)16 b(in)o(tro)q(duce)f(the)h(v)o (ector)g(subspace)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))15 b Fr(\022)p 1194 1560 116 2 v 13 w Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))15 b Fr(\022)e Fs(M)22 b Fu(b)o(y)p 464 1644 V 464 1687 a Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b(:=)f Fr(f)p Fs(z)j Fr(2)e Fs(M)5 b Fr(j8)p Fu(1)13 b Fr(\024)h Fs(i)f(<)h(j)j Fr(\024)c Fs(n)h Fu(:)g Fs(\031)1226 1666 y Fl(2)1224 1699 y Fq(i;j)1263 1687 y Fu(\()p Fs(z)r Fu(\))g(=)g Fs(\020)1417 1666 y Fl(2)1437 1687 y Fs(z)r Fr(g)228 1778 y Fu(and)f(sho)o(w)h(that)p 539 1735 V 13 w Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))13 b(is)f(in)o(v)m(arian)o(t)h (under)f(the)h(action)g(of)g(the)f Fs(\034)1425 1785 y Fq(i)1452 1778 y Fu(and)i Fs(\034)1571 1760 y Fl(2)1565 1790 y Fq(i)1590 1778 y Fs(\034)1611 1785 y Fq(i)p Fl(+1)1684 1778 y Fu(=)228 1836 y Fs(\034)249 1843 y Fq(i)p Fl(+1)308 1836 y Fs(\034)335 1818 y Fl(2)329 1848 y Fq(i)371 1836 y Fu(on)j Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))16 b(for)h(all)f Fs(i)d Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)j Fr(\000)g Fu(2.)278 1894 y(F)l(or)18 b Fs(z)h Fr(2)e Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))18 b(and)h Fs(i)d(<)h(j)k Fu(w)o(e)d(ha)o(v)o(e)f Fs(\031)1036 1876 y Fl(2)1034 1906 y Fq(i;j)1074 1894 y Fs(\034)1095 1901 y Fq(k)1116 1894 y Fu(\()p Fs(z)r Fu(\))g(=)g Fs(\034)1272 1901 y Fq(k)1293 1894 y Fs(\031)1323 1876 y Fl(2)1321 1906 y Fq(i;j)1361 1894 y Fu(\()p Fs(z)r Fu(\))g(=)g Fs(\020)1521 1876 y Fl(2)1541 1894 y Fs(\034)1562 1901 y Fq(k)1583 1894 y Fu(\()p Fs(z)r Fu(\))h(for)228 1954 y(all)h Fs(k)j Fu(with)e(1)h Fr(\024)f Fs(k)i(<)f(i)13 b Fr(\000)g Fu(1)21 b(and)f Fs(j)k(<)c(k)i Fr(\024)e Fs(n)g Fu(b)o(y)g(\(14\))h(and)f(for)h(all)e Fs(k)j Fu(with)228 2012 y Fs(i)13 b(<)h(k)i(<)e(j)t Fr(\000)q Fu(1)e(b)o(y)e(\(14\).)21 b(F)l(urthermore)9 b(w)o(e)i(ha)o(v)o(e)g Fs(\031)1168 1994 y Fl(2)1166 2025 y Fq(i;j)1206 2012 y Fs(\034)1227 2019 y Fq(i)p Fj(\000)p Fl(1)1286 2012 y Fu(\()p Fs(z)r Fu(\))j(=)f Fs(\034)1435 2019 y Fq(i)p Fj(\000)p Fl(1)1495 2012 y Fs(\031)1525 1994 y Fl(2)1523 2025 y Fq(i)p Fj(\000)p Fl(1)p Fq(;j)1608 2012 y Fu(\()p Fs(z)r Fu(\))g(=)228 2075 y Fs(\020)253 2057 y Fl(2)273 2075 y Fs(\034)294 2082 y Fq(i)p Fj(\000)p Fl(1)353 2075 y Fu(\()p Fs(z)r Fu(\))i(b)o(y)f(\(14\),)i Fs(\031)643 2057 y Fl(2)641 2087 y Fq(i;j)680 2075 y Fs(\034)701 2082 y Fq(j)r Fj(\000)p Fl(1)765 2075 y Fu(\()p Fs(z)r Fu(\))d(=)h Fs(\034)914 2082 y Fq(j)r Fj(\000)p Fl(1)978 2075 y Fs(\031)1008 2057 y Fl(2)1006 2087 y Fq(i;j)r Fj(\000)p Fl(1)1091 2075 y Fu(\()p Fs(z)r Fu(\))f(=)h Fs(\020)1244 2057 y Fl(2)1264 2075 y Fs(\034)1285 2082 y Fq(j)r Fj(\000)p Fl(1)1348 2075 y Fu(\()p Fs(z)r Fu(\))h(\(for)g Fs(i)f(<)f(j)f Fr(\000)c Fu(1\))228 2137 y(b)o(y)k(\(14\),)h(and)g Fs(\031)526 2119 y Fl(2)524 2149 y Fq(i;j)564 2137 y Fs(\034)585 2144 y Fq(j)r Fj(\000)p Fl(1)648 2137 y Fu(\()p Fs(z)r Fu(\))h(=)g Fs(\034)798 2144 y Fq(j)r Fj(\000)p Fl(1)861 2137 y Fs(\031)891 2119 y Fl(2)889 2149 y Fq(i;j)929 2137 y Fu(\()p Fs(z)r Fu(\))g(=)g Fs(\020)1083 2119 y Fl(2)1102 2137 y Fs(\034)1123 2144 y Fq(j)r Fj(\000)p Fl(1)1187 2137 y Fu(\()p Fs(z)r Fu(\))e(\(for)h Fs(i)g Fu(=)h Fs(j)6 b Fr(\000)s Fu(1\))13 b(b)o(y)f(\(14\).)228 2199 y(So)j(there)g(remain)e(t)o(w)o(o)i(cases)g(to)h(in)o(v)o (estigate)d(for)i(whic)o(h)g(w)o(e)f(use)h Fs(\031)1486 2181 y Fl(2)1484 2212 y Fq(i;j)1524 2199 y Fu(\()p Fs(z)r Fu(\))f(=)g Fs(\020)1678 2181 y Fl(2)1697 2199 y Fs(z)228 2264 y Fu(and)j(symmetri)o(call)o(y)c Fs(\031)667 2244 y Fj(\000)p Fl(2)665 2276 y Fq(i;j)714 2264 y Fu(\()p Fs(z)r Fu(\))h(=)f Fs(\020)867 2246 y Fj(\000)p Fl(2)915 2264 y Fs(z)18 b Fu(for)e(all)g Fs(z)g Fr(2)e Fs(M)5 b Fu(\()p Fs(\020)t Fu(\).)278 2323 y(In)16 b(the)g(\014rst)g(case)g(w) o(e)g(get)346 2401 y Fs(\031)376 2383 y Fl(2)374 2413 y Fq(i;j)414 2401 y Fs(\034)435 2408 y Fq(i)449 2401 y Fu(\()p Fs(z)r Fu(\))41 b(=)14 b Fs(\034)632 2380 y Fj(\000)p Fl(1)626 2412 y Fq(i)687 2401 y Fs(:)8 b(:)g(:)g(\034)780 2383 y Fl(2)774 2413 y Fq(j)r Fj(\000)p Fl(1)846 2401 y Fs(:)g(:)g(:)f(\034)932 2408 y Fq(i)946 2401 y Fs(\034)967 2408 y Fq(i)982 2401 y Fu(\()p Fs(z)r Fu(\))13 b(=)h Fs(\034)1137 2380 y Fj(\000)p Fl(1)1131 2412 y Fq(i)1192 2401 y Fs(:)8 b(:)g(:)g(\034)1285 2383 y Fl(2)1279 2413 y Fq(j)r Fj(\000)p Fl(1)1351 2401 y Fs(:)g(:)g(:)f(\034)1437 2408 y Fq(i)p Fl(+1)1497 2401 y Fu(\()p Fs(\020)1541 2383 y Fl(2)1560 2401 y Fs(z)r Fu(\))553 2461 y(=)14 b Fs(\020)630 2443 y Fl(2)650 2461 y Fs(\034)677 2441 y Fj(\000)p Fl(1)671 2473 y Fq(i)732 2461 y Fs(:)8 b(:)g(:)f(\034)824 2443 y Fl(2)818 2474 y Fq(j)r Fj(\000)p Fl(1)890 2461 y Fs(:)h(:)g(:)g(\034)977 2468 y Fq(i)p Fl(+1)1036 2461 y Fu(\()p Fs(z)r Fu(\))14 b(=)g Fs(\020)1190 2443 y Fl(2)1209 2461 y Fs(\034)1236 2441 y Fj(\000)p Fl(1)1230 2473 y Fq(i)1283 2461 y Fs(\031)1313 2443 y Fl(2)1311 2474 y Fq(i)p Fl(+1)p Fq(;j)1396 2461 y Fu(\()p Fs(z)r Fu(\))553 2522 y(=)g Fs(\020)630 2504 y Fl(2)650 2522 y Fs(\034)677 2502 y Fj(\000)p Fl(1)671 2534 y Fq(i)724 2522 y Fu(\()p Fs(\020)768 2504 y Fl(2)787 2522 y Fs(z)r Fu(\))g(=)g Fs(\020)922 2504 y Fl(2)942 2522 y Fs(\034)969 2502 y Fj(\000)p Fl(1)963 2534 y Fq(i)1016 2522 y Fs(\034)1043 2504 y Fl(2)1037 2535 y Fq(i)1062 2522 y Fu(\()p Fs(z)r Fu(\))g(=)f Fs(\020)1215 2504 y Fl(2)1235 2522 y Fs(\034)1256 2529 y Fq(i)1270 2522 y Fu(\()p Fs(z)r Fu(\))228 2606 y(for)j Fs(i)11 b Fu(+)g(1)j Fs(<)g(j)19 b Fu(and)e Fs(\031)633 2588 y Fl(2)631 2618 y Fq(i;i)p Fl(+1)712 2606 y Fs(\034)733 2613 y Fq(i)747 2606 y Fu(\()p Fs(z)r Fu(\))d(=)f Fs(\034)902 2588 y Fl(3)896 2618 y Fq(i)922 2606 y Fu(\()p Fs(z)r Fu(\))h(=)f Fs(\020)1075 2588 y Fl(2)1095 2606 y Fs(\034)1116 2613 y Fq(i)1130 2606 y Fu(\()p Fs(z)r Fu(\).)p eop %%Page: 20 20 20 19 bop 228 125 a Fp(20)552 b(BODO)13 b(P)m(AREIGIS)278 225 y Fu(In)j(the)g(second)g(case)g(w)o(e)g(get)249 302 y Fs(\031)279 284 y Fl(2)277 315 y Fq(i;j)316 302 y Fs(\034)337 309 y Fq(j)356 302 y Fu(\()p Fs(z)r Fu(\))41 b(=)14 b Fs(\034)539 282 y Fj(\000)p Fl(1)533 314 y Fq(i)594 302 y Fs(:)8 b(:)g(:)g(\034)687 284 y Fl(2)681 315 y Fq(j)r Fj(\000)p Fl(1)752 302 y Fs(:)g(:)g(:)g(\034)839 309 y Fq(i)853 302 y Fs(\034)874 309 y Fq(j)893 302 y Fu(\()p Fs(z)r Fu(\))13 b(=)h Fs(\020)1046 284 y Fl(2)1066 302 y Fs(\034)1093 282 y Fj(\000)p Fl(1)1087 314 y Fq(i)1148 302 y Fs(:)8 b(:)g(:)g(\034)1241 284 y Fl(2)1235 315 y Fq(j)r Fj(\000)p Fl(1)1306 302 y Fs(:)g(:)g(:)g(\034)1393 309 y Fq(i)1407 302 y Fs(\034)1428 309 y Fq(j)1447 302 y Fs(\034)1474 282 y Fj(\000)p Fl(2)1468 314 y Fq(j)1521 302 y Fu(\()p Fs(z)r Fu(\))460 363 y(=)14 b Fs(\020)537 345 y Fl(2)557 363 y Fs(\034)584 343 y Fj(\000)p Fl(1)578 375 y Fq(j)631 363 y Fs(\034)652 370 y Fq(j)670 363 y Fs(\034)697 343 y Fj(\000)p Fl(1)691 375 y Fq(i)752 363 y Fs(:)8 b(:)g(:)g(\034)845 345 y Fl(2)839 376 y Fq(j)r Fj(\000)p Fl(1)911 363 y Fs(:)g(:)g(:)f(\034)997 370 y Fq(i)1011 363 y Fs(\034)1038 343 y Fj(\000)p Fl(1)1032 375 y Fq(j)1085 363 y Fu(\()p Fs(z)r Fu(\))14 b(=)g Fs(\020)1239 345 y Fl(2)1258 363 y Fs(\034)1285 343 y Fj(\000)p Fl(1)1279 375 y Fq(j)1332 363 y Fs(\034)1359 343 y Fj(\000)p Fl(1)1353 375 y Fq(i)1415 363 y Fs(:)8 b(:)g(:)f(\034)1501 370 y Fq(j)1520 363 y Fs(\034)1547 345 y Fl(2)1541 376 y Fq(j)r Fj(\000)p Fl(1)1604 363 y Fs(\034)1631 343 y Fj(\000)p Fl(1)1625 375 y Fq(j)1686 363 y Fs(:)h(:)g(:)g(\034)1773 370 y Fq(i)1787 363 y Fu(\()p Fs(z)r Fu(\))460 424 y(=)14 b Fs(\020)537 406 y Fl(2)557 424 y Fs(\034)584 403 y Fj(\000)p Fl(1)578 436 y Fq(j)631 424 y Fs(\034)658 403 y Fj(\000)p Fl(1)652 436 y Fq(i)713 424 y Fs(:)8 b(:)g(:)f(\034)805 403 y Fj(\000)p Fl(1)799 436 y Fq(j)r Fj(\000)p Fl(1)863 424 y Fs(\034)890 406 y Fl(2)884 436 y Fq(j)909 424 y Fs(\034)930 431 y Fq(j)r Fj(\000)p Fl(1)1002 424 y Fs(:)h(:)g(:)g(\034) 1089 431 y Fq(i)1103 424 y Fu(\()p Fs(z)r Fu(\))14 b(=)f Fs(\020)1256 406 y Fl(2)1276 424 y Fs(\034)1303 403 y Fj(\000)p Fl(1)1297 436 y Fq(j)1350 424 y Fs(\031)r(i;)8 b(j)13 b Fu(+)e(1)1525 405 y Fl(2)1545 424 y Fu(\()p Fs(z)r Fu(\))460 485 y(=)j Fs(\020)537 467 y Fl(2)557 485 y Fs(\034)584 464 y Fj(\000)p Fl(1)578 496 y Fq(j)631 485 y Fu(\()p Fs(\020)675 467 y Fl(2)694 485 y Fs(z)r Fu(\))g(=)g Fs(\020)829 467 y Fl(2)849 485 y Fs(\034)870 492 y Fq(j)888 485 y Fu(\()p Fs(z)r Fu(\))p Fs(:)278 566 y Fu(Hence)d(w)o(e)h(ha)o(v)o(e)f Fs(\034)616 573 y Fq(i)631 566 y Fu(\()p Fs(z)r Fu(\))i Fr(2)h Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))13 b(for)g(all)f Fs(z)k Fr(2)e Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))13 b(and)g(all)f Fs(i)h Fu(=)h(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)s Fr(\000)s Fu(1.)278 624 y(The)17 b(claim)e Fs(\034)537 606 y Fl(2)531 636 y Fq(i)556 624 y Fs(\034)577 631 y Fq(i)p Fl(+1)652 624 y Fu(=)g Fs(\034)726 631 y Fq(i)p Fl(+1)785 624 y Fs(\034)812 606 y Fl(2)806 636 y Fq(i)849 624 y Fu(is)i(clear)f(from)g (the)h(in)o(v)m(ariance)f(and)i(the)f(fact,)228 682 y(that)f Fs(\034)360 664 y Fl(2)354 694 y Fq(i)396 682 y Fu(on)h Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))17 b(is)f(m)o(ultipli)o(cation)e(b)o (y)h Fs(\020)1050 664 y Fl(2)1070 682 y Fu(.)278 744 y(Since)h(the)g Fs(\034)517 726 y Fl(2)511 757 y Fq(i)554 744 y Fu(comm)o(ute)d(in)k(their)f(action)h(on)p 1155 702 116 2 v 18 w Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))17 b(with)g(all)f Fs(')f Fr(2)g Fs(B)1599 751 y Fq(n)1640 744 y Fu(it)h(is)228 807 y(clear)f(that)p 450 764 V 17 w Fs(M)5 b Fu(\()p Fs(\020)t Fu(\))14 b Fr(\022)g Fs(M)5 b Fu(\()p Fs(\020)t Fu(\).)922 b Fi(\003)278 891 y Fu(W)l(e)16 b(no)o(w)g(study)h(sp)q(eci\014c)e(braids.)22 b(The)16 b(follo)o(wing)g(iden)o(tit)o(y)p 763 982 2 2 v 763 982 V 763 983 V 763 984 V 762 984 V 762 985 V 762 986 V 762 986 V 762 987 V 762 988 V 762 988 V 761 989 V 761 989 V 761 990 V 760 990 V 760 991 V 760 992 V 759 992 V 759 993 V 758 993 V 758 994 V 758 994 V 757 995 V 757 995 V 756 996 V 755 996 V 755 997 V 754 997 V 754 998 V 753 998 V 752 999 V 752 999 V 751 999 V 750 1000 V 749 1000 V 749 1001 V 715 1029 V 715 1028 V 716 1028 V 716 1027 V 716 1026 V 716 1026 V 716 1025 V 716 1024 V 716 1024 V 716 1023 V 717 1023 V 717 1022 V 717 1021 V 717 1021 V 718 1020 V 718 1020 V 718 1019 V 719 1019 V 719 1018 V 720 1018 V 720 1017 V 721 1016 V 721 1016 V 722 1015 V 722 1015 V 723 1015 V 723 1014 V 724 1014 V 725 1013 V 725 1013 V 726 1012 V 727 1012 V 727 1011 V 728 1011 V 729 1010 V 730 1010 V 715 982 V 716 982 V 716 983 V 716 984 V 716 984 V 716 985 V 716 986 V 716 986 V 717 987 V 717 988 V 717 989 V 718 989 V 718 990 V 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844 1094 V 844 1095 V 843 1095 V 810 1123 V 810 1123 V 810 1122 V 810 1121 V 810 1121 V 810 1120 V 810 1119 V 810 1119 V 811 1118 V 811 1118 V 811 1117 V 811 1116 V 812 1116 V 812 1115 V 812 1115 V 812 1114 V 813 1114 V 813 1113 V 814 1112 V 814 1112 V 814 1111 V 815 1111 V 815 1110 V 816 1110 V 817 1109 V 817 1109 V 818 1108 V 818 1108 V 819 1107 V 820 1107 V 820 1107 V 821 1106 V 822 1106 V 822 1105 V 823 1105 V 824 1104 V 810 1076 V 810 1077 V 810 1077 V 810 1078 V 810 1079 V 810 1080 V 811 1080 V 811 1081 V 811 1082 V 811 1082 V 812 1083 V 812 1084 V 813 1084 V 813 1085 V 814 1086 V 814 1086 V 815 1087 V 815 1088 V 816 1088 V 817 1089 V 818 1090 V 818 1090 V 819 1091 V 820 1092 V 821 1092 V 822 1093 V 823 1094 V 824 1094 V 825 1095 V 826 1096 V 827 1096 V 828 1097 V 830 1098 V 831 1098 V 832 1099 V 833 1100 V 857 1123 V 857 1123 V 857 1122 V 857 1121 V 857 1121 V 857 1120 V 856 1119 V 856 1119 V 856 1118 V 855 1117 V 855 1117 V 855 1116 V 854 1115 V 854 1115 V 853 1114 V 853 1113 V 852 1113 V 851 1112 V 851 1111 V 850 1111 V 849 1110 V 849 1109 V 848 1108 V 847 1108 V 846 1107 V 845 1106 V 844 1106 V 843 1105 V 842 1104 V 841 1104 V 840 1103 V 839 1102 V 837 1102 V 836 1101 V 835 1100 V 833 1100 V 904 1123 V 904 1124 V 904 1125 V 904 1125 V 904 1126 V 904 1127 V 904 1127 V 904 1128 V 903 1128 V 903 1129 V 903 1130 V 903 1130 V 903 1131 V 902 1131 V 902 1132 V 902 1133 V 901 1133 V 901 1134 V 900 1134 V 900 1135 V 900 1135 V 899 1136 V 899 1136 V 898 1137 V 898 1137 V 897 1138 V 896 1138 V 896 1139 V 895 1139 V 895 1140 V 894 1140 V 893 1141 V 892 1141 V 892 1141 V 891 1142 V 890 1142 V 857 1171 V 857 1170 V 857 1169 V 857 1169 V 857 1168 V 857 1167 V 857 1167 V 858 1166 V 858 1165 V 858 1165 V 858 1164 V 858 1164 V 859 1163 V 859 1162 V 859 1162 V 860 1161 V 860 1161 V 860 1160 V 861 1160 V 861 1159 V 862 1159 V 862 1158 V 863 1158 V 863 1157 V 864 1157 V 864 1156 V 865 1156 V 865 1155 V 866 1155 V 867 1154 V 867 1154 V 868 1153 V 869 1153 V 870 1152 V 870 1152 V 871 1152 V 857 1123 V 857 1124 V 857 1125 V 857 1125 V 857 1126 V 858 1127 V 858 1127 V 858 1128 V 858 1129 V 859 1129 V 859 1130 V 859 1131 V 860 1131 V 860 1132 V 861 1133 V 861 1133 V 862 1134 V 863 1135 V 863 1135 V 864 1136 V 865 1137 V 866 1137 V 866 1138 V 867 1139 V 868 1139 V 869 1140 V 870 1141 V 871 1142 V 872 1142 V 873 1143 V 874 1144 V 876 1144 V 877 1145 V 878 1146 V 879 1146 V 881 1147 V 904 1171 V 904 1170 V 904 1169 V 904 1168 V 904 1168 V 904 1167 V 904 1166 V 903 1166 V 903 1165 V 903 1164 V 902 1164 V 902 1163 V 901 1162 V 901 1162 V 900 1161 V 900 1160 V 899 1160 V 899 1159 V 898 1158 V 897 1158 V 897 1157 V 896 1156 V 895 1156 V 894 1155 V 893 1154 V 892 1154 V 891 1153 V 890 1152 V 889 1152 V 888 1151 V 887 1150 V 886 1150 V 885 1149 V 883 1148 V 882 1148 V 881 1147 V 904 1171 V 904 1171 V 904 1172 V 904 1172 V 904 1173 V 904 1174 V 904 1174 V 904 1175 V 903 1176 V 903 1176 V 903 1177 V 903 1177 V 903 1178 V 902 1179 V 902 1179 V 902 1180 V 901 1180 V 901 1181 V 900 1181 V 900 1182 V 900 1182 V 899 1183 V 899 1183 V 898 1184 V 898 1184 V 897 1185 V 896 1185 V 896 1186 V 895 1186 V 895 1187 V 894 1187 V 893 1188 V 892 1188 V 892 1189 V 891 1189 V 890 1189 V 857 1218 V 857 1217 V 857 1216 V 857 1216 V 857 1215 V 857 1214 V 857 1214 V 858 1213 V 858 1213 V 858 1212 V 858 1211 V 858 1211 V 859 1210 V 859 1210 V 859 1209 V 860 1208 V 860 1208 V 860 1207 V 861 1207 V 861 1206 V 862 1206 V 862 1205 V 863 1205 V 863 1204 V 864 1204 V 864 1203 V 865 1203 V 865 1202 V 866 1202 V 867 1201 V 867 1201 V 868 1200 V 869 1200 V 870 1200 V 870 1199 V 871 1199 V 857 1171 V 857 1171 V 857 1172 V 857 1173 V 857 1173 V 858 1174 V 858 1175 V 858 1175 V 858 1176 V 859 1177 V 859 1177 V 859 1178 V 860 1179 V 860 1179 V 861 1180 V 861 1181 V 862 1181 V 863 1182 V 863 1183 V 864 1183 V 865 1184 V 866 1185 V 866 1185 V 867 1186 V 868 1187 V 869 1187 V 870 1188 V 871 1189 V 872 1189 V 873 1190 V 874 1191 V 876 1191 V 877 1192 V 878 1193 V 879 1193 V 881 1194 V 904 1218 V 904 1217 V 904 1216 V 904 1216 V 904 1215 V 904 1214 V 904 1214 V 903 1213 V 903 1212 V 903 1212 V 902 1211 V 902 1210 V 901 1210 V 901 1209 V 900 1208 V 900 1208 V 899 1207 V 899 1206 V 898 1206 V 897 1205 V 897 1204 V 896 1204 V 895 1203 V 894 1202 V 893 1202 V 892 1201 V 891 1200 V 890 1199 V 889 1199 V 888 1198 V 887 1197 V 886 1197 V 885 1196 V 883 1195 V 882 1195 V 881 1194 V 1093 1123 V 1093 1124 V 1093 1125 V 1093 1125 V 1093 1126 V 1093 1127 V 1093 1127 V 1092 1128 V 1092 1128 V 1092 1129 V 1092 1130 V 1092 1130 V 1091 1131 V 1091 1131 V 1091 1132 V 1090 1133 V 1090 1133 V 1090 1134 V 1089 1134 V 1089 1135 V 1088 1135 V 1088 1136 V 1087 1136 V 1087 1137 V 1086 1137 V 1086 1138 V 1085 1138 V 1085 1139 V 1084 1139 V 1083 1140 V 1083 1140 V 1082 1141 V 1081 1141 V 1080 1141 V 1080 1142 V 1079 1142 V 1046 1171 V 1046 1170 V 1046 1169 V 1046 1169 V 1046 1168 V 1046 1167 V 1046 1167 V 1046 1166 V 1047 1165 V 1047 1165 V 1047 1164 V 1047 1164 V 1047 1163 V 1048 1162 V 1048 1162 V 1048 1161 V 1049 1161 V 1049 1160 V 1050 1160 V 1050 1159 V 1050 1159 V 1051 1158 V 1051 1158 V 1052 1157 V 1052 1157 V 1053 1156 V 1054 1156 V 1054 1155 V 1055 1155 V 1055 1154 V 1056 1154 V 1057 1153 V 1058 1153 V 1058 1152 V 1059 1152 V 1060 1152 V 1046 1123 V 1046 1124 V 1046 1125 V 1046 1125 V 1046 1126 V 1046 1127 V 1046 1127 V 1047 1128 V 1047 1129 V 1047 1129 V 1048 1130 V 1048 1131 V 1049 1131 V 1049 1132 V 1050 1133 V 1050 1133 V 1051 1134 V 1051 1135 V 1052 1135 V 1053 1136 V 1053 1137 V 1054 1137 V 1055 1138 V 1056 1139 V 1057 1139 V 1058 1140 V 1059 1141 V 1060 1142 V 1061 1142 V 1062 1143 V 1063 1144 V 1064 1144 V 1065 1145 V 1067 1146 V 1068 1146 V 1069 1147 V 1093 1171 V 1093 1170 V 1093 1169 V 1093 1168 V 1093 1168 V 1092 1167 V 1092 1166 V 1092 1166 V 1092 1165 V 1091 1164 V 1091 1164 V 1091 1163 V 1090 1162 V 1090 1162 V 1089 1161 V 1089 1160 V 1088 1160 V 1087 1159 V 1087 1158 V 1086 1158 V 1085 1157 V 1084 1156 V 1084 1156 V 1083 1155 V 1082 1154 V 1081 1154 V 1080 1153 V 1079 1152 V 1078 1152 V 1077 1151 V 1076 1150 V 1074 1150 V 1073 1149 V 1072 1148 V 1071 1148 V 1069 1147 V 1093 1171 V 1093 1171 V 1093 1172 V 1093 1172 V 1093 1173 V 1093 1174 V 1093 1174 V 1092 1175 V 1092 1176 V 1092 1176 V 1092 1177 V 1092 1177 V 1091 1178 V 1091 1179 V 1091 1179 V 1090 1180 V 1090 1180 V 1090 1181 V 1089 1181 V 1089 1182 V 1088 1182 V 1088 1183 V 1087 1183 V 1087 1184 V 1086 1184 V 1086 1185 V 1085 1185 V 1085 1186 V 1084 1186 V 1083 1187 V 1083 1187 V 1082 1188 V 1081 1188 V 1080 1189 V 1080 1189 V 1079 1189 V 1046 1218 V 1046 1217 V 1046 1216 V 1046 1216 V 1046 1215 V 1046 1214 V 1046 1214 V 1046 1213 V 1047 1213 V 1047 1212 V 1047 1211 V 1047 1211 V 1047 1210 V 1048 1210 V 1048 1209 V 1048 1208 V 1049 1208 V 1049 1207 V 1050 1207 V 1050 1206 V 1050 1206 V 1051 1205 V 1051 1205 V 1052 1204 V 1052 1204 V 1053 1203 V 1054 1203 V 1054 1202 V 1055 1202 V 1055 1201 V 1056 1201 V 1057 1200 V 1058 1200 V 1058 1200 V 1059 1199 V 1060 1199 V 1046 1171 V 1046 1171 V 1046 1172 V 1046 1173 V 1046 1173 V 1046 1174 V 1046 1175 V 1047 1175 V 1047 1176 V 1047 1177 V 1048 1177 V 1048 1178 V 1049 1179 V 1049 1179 V 1050 1180 V 1050 1181 V 1051 1181 V 1051 1182 V 1052 1183 V 1053 1183 V 1053 1184 V 1054 1185 V 1055 1185 V 1056 1186 V 1057 1187 V 1058 1187 V 1059 1188 V 1060 1189 V 1061 1189 V 1062 1190 V 1063 1191 V 1064 1191 V 1065 1192 V 1067 1193 V 1068 1193 V 1069 1194 V 1093 1218 V 1093 1217 V 1093 1216 V 1093 1216 V 1093 1215 V 1092 1214 V 1092 1214 V 1092 1213 V 1092 1212 V 1091 1212 V 1091 1211 V 1091 1210 V 1090 1210 V 1090 1209 V 1089 1208 V 1089 1208 V 1088 1207 V 1087 1206 V 1087 1206 V 1086 1205 V 1085 1204 V 1084 1204 V 1084 1203 V 1083 1202 V 1082 1202 V 1081 1201 V 1080 1200 V 1079 1199 V 1078 1199 V 1077 1198 V 1076 1197 V 1074 1197 V 1073 1196 V 1072 1195 V 1071 1195 V 1069 1194 V 1140 1218 V 1140 1218 V 1140 1219 V 1140 1220 V 1140 1220 V 1140 1221 V 1140 1222 V 1140 1222 V 1139 1223 V 1139 1223 V 1139 1224 V 1139 1225 V 1138 1225 V 1138 1226 V 1138 1226 V 1138 1227 V 1137 1227 V 1137 1228 V 1136 1229 V 1136 1229 V 1136 1230 V 1135 1230 V 1135 1231 V 1134 1231 V 1133 1232 V 1133 1232 V 1132 1233 V 1132 1233 V 1131 1234 V 1130 1234 V 1130 1234 V 1129 1235 V 1128 1235 V 1128 1236 V 1127 1236 V 1126 1237 V 1093 1265 V 1093 1264 V 1093 1264 V 1093 1263 V 1093 1262 V 1093 1262 V 1093 1261 V 1094 1260 V 1094 1260 V 1094 1259 V 1094 1259 V 1094 1258 V 1095 1257 V 1095 1257 V 1095 1256 V 1096 1256 V 1096 1255 V 1096 1255 V 1097 1254 V 1097 1253 V 1098 1253 V 1098 1252 V 1099 1252 V 1099 1251 V 1100 1251 V 1100 1250 V 1101 1250 V 1101 1249 V 1102 1249 V 1103 1249 V 1103 1248 V 1104 1248 V 1105 1247 V 1106 1247 V 1106 1246 V 1107 1246 V 1093 1218 V 1093 1218 V 1093 1219 V 1093 1220 V 1093 1220 V 1093 1221 V 1094 1222 V 1094 1222 V 1094 1223 V 1095 1224 V 1095 1224 V 1095 1225 V 1096 1226 V 1096 1226 V 1097 1227 V 1097 1228 V 1098 1228 V 1099 1229 V 1099 1230 V 1100 1230 V 1101 1231 V 1101 1232 V 1102 1233 V 1103 1233 V 1104 1234 V 1105 1235 V 1106 1235 V 1107 1236 V 1108 1237 V 1109 1237 V 1110 1238 V 1111 1239 V 1113 1239 V 1114 1240 V 1115 1241 V 1117 1241 V 1140 1265 V 1140 1264 V 1140 1264 V 1140 1263 V 1140 1262 V 1140 1262 V 1139 1261 V 1139 1260 V 1139 1259 V 1139 1259 V 1138 1258 V 1138 1257 V 1137 1257 V 1137 1256 V 1136 1255 V 1136 1255 V 1135 1254 V 1135 1253 V 1134 1253 V 1133 1252 V 1132 1251 V 1132 1251 V 1131 1250 V 1130 1249 V 1129 1249 V 1128 1248 V 1127 1247 V 1126 1247 V 1125 1246 V 1124 1245 V 1123 1245 V 1122 1244 V 1120 1243 V 1119 1243 V 1118 1242 V 1117 1241 V 1235 1312 V 1235 1313 V 1234 1313 V 1234 1314 V 1234 1315 V 1234 1315 V 1234 1316 V 1234 1317 V 1234 1317 V 1234 1318 V 1233 1318 V 1233 1319 V 1233 1320 V 1233 1320 V 1232 1321 V 1232 1321 V 1232 1322 V 1231 1322 V 1231 1323 V 1230 1323 V 1230 1324 V 1229 1325 V 1229 1325 V 1228 1326 V 1228 1326 V 1227 1327 V 1227 1327 V 1226 1327 V 1225 1328 V 1225 1328 V 1224 1329 V 1223 1329 V 1223 1330 V 1222 1330 V 1221 1331 V 1220 1331 V 1187 1359 V 1187 1358 V 1187 1357 V 1187 1357 V 1188 1356 V 1188 1356 V 1188 1355 V 1188 1354 V 1188 1354 V 1188 1353 V 1188 1352 V 1189 1352 V 1189 1351 V 1189 1351 V 1190 1351 V 1190 1350 V 1190 1349 V 1191 1349 V 1191 1348 V 1192 1348 V 1192 1347 V 1192 1347 V 1193 1346 V 1193 1346 V 1194 1345 V 1195 1345 V 1195 1344 V 1196 1344 V 1196 1343 V 1197 1343 V 1198 1342 V 1198 1342 V 1199 1342 V 1200 1341 V 1201 1341 V 1201 1340 V 1187 1312 V 1187 1313 V 1187 1313 V 1188 1314 V 1188 1315 V 1188 1315 V 1188 1316 V 1188 1317 V 1189 1317 V 1189 1318 V 1189 1319 V 1190 1319 V 1190 1320 V 1191 1321 V 1191 1321 V 1192 1322 V 1192 1323 V 1193 1324 V 1194 1324 V 1194 1325 V 1195 1326 V 1196 1326 V 1197 1327 V 1198 1328 V 1198 1328 V 1199 1329 V 1200 1330 V 1201 1330 V 1202 1331 V 1204 1332 V 1205 1332 V 1206 1333 V 1207 1334 V 1208 1334 V 1210 1335 V 1211 1336 V 1235 1359 V 1234 1358 V 1234 1357 V 1234 1357 V 1234 1356 V 1234 1355 V 1234 1355 V 1234 1354 V 1233 1353 V 1233 1353 V 1233 1352 V 1232 1351 V 1232 1351 V 1231 1350 V 1231 1350 V 1230 1349 V 1230 1348 V 1229 1348 V 1228 1347 V 1228 1346 V 1227 1346 V 1226 1345 V 1225 1344 V 1224 1344 V 1223 1343 V 1222 1342 V 1221 1342 V 1220 1341 V 1219 1340 V 1218 1340 V 1217 1339 V 1216 1338 V 1215 1338 V 1214 1337 V 1212 1336 V 1211 1336 V 763 1312 V 763 1313 V 763 1313 V 762 1314 V 762 1315 V 762 1315 V 762 1316 V 762 1317 V 761 1317 V 761 1318 V 761 1319 V 760 1319 V 760 1320 V 759 1321 V 759 1321 V 758 1322 V 758 1323 V 757 1324 V 756 1324 V 756 1325 V 755 1326 V 754 1326 V 753 1327 V 752 1328 V 752 1328 V 751 1329 V 750 1330 V 749 1330 V 748 1331 V 746 1332 V 745 1332 V 744 1333 V 743 1334 V 742 1334 V 740 1335 V 739 1336 V 715 1359 V 716 1358 V 716 1357 V 716 1357 V 716 1356 V 716 1355 V 716 1355 V 716 1354 V 717 1353 V 717 1353 V 717 1352 V 718 1351 V 718 1351 V 719 1350 V 719 1350 V 720 1349 V 720 1348 V 721 1348 V 722 1347 V 722 1346 V 723 1346 V 724 1345 V 725 1344 V 726 1344 V 727 1343 V 728 1342 V 729 1342 V 730 1341 V 731 1340 V 732 1340 V 733 1339 V 734 1338 V 735 1338 V 736 1337 V 738 1336 V 739 1336 V 715 1312 V 715 1313 V 716 1313 V 716 1314 V 716 1315 V 716 1315 V 716 1316 V 716 1317 V 716 1317 V 716 1318 V 717 1318 V 717 1319 V 717 1320 V 717 1320 V 718 1321 V 718 1321 V 718 1322 V 719 1322 V 719 1323 V 720 1323 V 720 1324 V 721 1325 V 721 1325 V 722 1326 V 722 1326 V 723 1327 V 723 1327 V 724 1327 V 725 1328 V 725 1328 V 726 1329 V 727 1329 V 727 1330 V 728 1330 V 729 1331 V 730 1331 V 763 1359 V 763 1358 V 763 1357 V 763 1357 V 762 1356 V 762 1356 V 762 1355 V 762 1354 V 762 1354 V 762 1353 V 762 1352 V 761 1352 V 761 1351 V 761 1351 V 760 1351 V 760 1350 V 760 1349 V 759 1349 V 759 1348 V 758 1348 V 758 1347 V 758 1347 V 757 1346 V 757 1346 V 756 1345 V 755 1345 V 755 1344 V 754 1344 V 754 1343 V 753 1343 V 752 1342 V 752 1342 V 751 1342 V 750 1341 V 749 1341 V 749 1340 V 857 1218 V 857 1218 V 857 1219 V 857 1220 V 857 1220 V 857 1221 V 856 1222 V 856 1222 V 856 1223 V 855 1224 V 855 1224 V 855 1225 V 854 1226 V 854 1226 V 853 1227 V 853 1228 V 852 1228 V 851 1229 V 851 1230 V 850 1230 V 849 1231 V 849 1232 V 848 1233 V 847 1233 V 846 1234 V 845 1235 V 844 1235 V 843 1236 V 842 1237 V 841 1237 V 840 1238 V 839 1239 V 837 1239 V 836 1240 V 835 1241 V 833 1241 V 810 1265 V 810 1264 V 810 1264 V 810 1263 V 810 1262 V 810 1262 V 811 1261 V 811 1260 V 811 1259 V 811 1259 V 812 1258 V 812 1257 V 813 1257 V 813 1256 V 814 1255 V 814 1255 V 815 1254 V 815 1253 V 816 1253 V 817 1252 V 818 1251 V 818 1251 V 819 1250 V 820 1249 V 821 1249 V 822 1248 V 823 1247 V 824 1247 V 825 1246 V 826 1245 V 827 1245 V 828 1244 V 830 1243 V 831 1243 V 832 1242 V 833 1241 V 810 1218 V 810 1218 V 810 1219 V 810 1220 V 810 1220 V 810 1221 V 810 1222 V 810 1222 V 811 1223 V 811 1223 V 811 1224 V 811 1225 V 812 1225 V 812 1226 V 812 1226 V 812 1227 V 813 1227 V 813 1228 V 814 1229 V 814 1229 V 814 1230 V 815 1230 V 815 1231 V 816 1231 V 817 1232 V 817 1232 V 818 1233 V 818 1233 V 819 1234 V 820 1234 V 820 1234 V 821 1235 V 822 1235 V 822 1236 V 823 1236 V 824 1237 V 857 1265 V 857 1264 V 857 1264 V 857 1263 V 857 1262 V 857 1262 V 857 1261 V 856 1260 V 856 1260 V 856 1259 V 856 1259 V 856 1258 V 855 1257 V 855 1257 V 855 1256 V 854 1256 V 854 1255 V 854 1255 V 853 1254 V 853 1253 V 852 1253 V 852 1252 V 851 1252 V 851 1251 V 850 1251 V 850 1250 V 849 1250 V 849 1249 V 848 1249 V 847 1249 V 847 1248 V 846 1248 V 845 1247 V 844 1247 V 844 1246 V 843 1246 V 1140 1076 V 1140 1077 V 1140 1077 V 1140 1078 V 1140 1079 V 1140 1080 V 1139 1080 V 1139 1081 V 1139 1082 V 1139 1082 V 1138 1083 V 1138 1084 V 1137 1084 V 1137 1085 V 1136 1086 V 1136 1086 V 1135 1087 V 1135 1088 V 1134 1088 V 1133 1089 V 1132 1090 V 1132 1090 V 1131 1091 V 1130 1092 V 1129 1092 V 1128 1093 V 1127 1094 V 1126 1094 V 1125 1095 V 1124 1096 V 1123 1096 V 1122 1097 V 1120 1098 V 1119 1098 V 1118 1099 V 1117 1100 V 1093 1123 V 1093 1123 V 1093 1122 V 1093 1121 V 1093 1121 V 1093 1120 V 1094 1119 V 1094 1119 V 1094 1118 V 1095 1117 V 1095 1117 V 1095 1116 V 1096 1115 V 1096 1115 V 1097 1114 V 1097 1113 V 1098 1113 V 1099 1112 V 1099 1111 V 1100 1111 V 1101 1110 V 1101 1109 V 1102 1108 V 1103 1108 V 1104 1107 V 1105 1106 V 1106 1106 V 1107 1105 V 1108 1104 V 1109 1104 V 1110 1103 V 1111 1102 V 1113 1102 V 1114 1101 V 1115 1100 V 1117 1100 V 1093 1076 V 1093 1077 V 1093 1077 V 1093 1078 V 1093 1079 V 1093 1079 V 1093 1080 V 1094 1081 V 1094 1081 V 1094 1082 V 1094 1082 V 1094 1083 V 1095 1084 V 1095 1084 V 1095 1085 V 1096 1085 V 1096 1086 V 1096 1086 V 1097 1087 V 1097 1088 V 1098 1088 V 1098 1089 V 1099 1089 V 1099 1090 V 1100 1090 V 1100 1091 V 1101 1091 V 1101 1092 V 1102 1092 V 1103 1092 V 1103 1093 V 1104 1093 V 1105 1094 V 1106 1094 V 1106 1095 V 1107 1095 V 1140 1123 V 1140 1123 V 1140 1122 V 1140 1121 V 1140 1121 V 1140 1120 V 1140 1119 V 1140 1119 V 1139 1118 V 1139 1118 V 1139 1117 V 1139 1116 V 1138 1116 V 1138 1115 V 1138 1115 V 1138 1114 V 1137 1114 V 1137 1113 V 1136 1112 V 1136 1112 V 1136 1111 V 1135 1111 V 1135 1110 V 1134 1110 V 1133 1109 V 1133 1109 V 1132 1108 V 1132 1108 V 1131 1107 V 1130 1107 V 1130 1107 V 1129 1106 V 1128 1106 V 1128 1105 V 1127 1105 V 1126 1104 V 1235 982 V 1234 982 V 1234 983 V 1234 984 V 1234 984 V 1234 985 V 1234 986 V 1234 986 V 1233 987 V 1233 988 V 1233 989 V 1232 989 V 1232 990 V 1231 991 V 1231 991 V 1230 992 V 1230 993 V 1229 993 V 1228 994 V 1228 995 V 1227 995 V 1226 996 V 1225 997 V 1224 997 V 1223 998 V 1222 999 V 1221 999 V 1220 1000 V 1219 1001 V 1218 1001 V 1217 1002 V 1216 1003 V 1215 1003 V 1214 1004 V 1212 1005 V 1211 1005 V 1187 1029 V 1187 1028 V 1187 1028 V 1188 1027 V 1188 1026 V 1188 1026 V 1188 1025 V 1188 1024 V 1189 1024 V 1189 1023 V 1189 1022 V 1190 1022 V 1190 1021 V 1191 1020 V 1191 1020 V 1192 1019 V 1192 1018 V 1193 1017 V 1194 1017 V 1194 1016 V 1195 1015 V 1196 1015 V 1197 1014 V 1198 1013 V 1198 1013 V 1199 1012 V 1200 1011 V 1201 1011 V 1202 1010 V 1204 1009 V 1205 1009 V 1206 1008 V 1207 1007 V 1208 1007 V 1210 1006 V 1211 1005 V 1187 982 V 1187 982 V 1187 983 V 1187 984 V 1188 984 V 1188 985 V 1188 986 V 1188 986 V 1188 987 V 1188 988 V 1188 988 V 1189 989 V 1189 989 V 1189 990 V 1190 990 V 1190 991 V 1190 992 V 1191 992 V 1191 993 V 1192 993 V 1192 994 V 1192 994 V 1193 995 V 1193 995 V 1194 996 V 1195 996 V 1195 997 V 1196 997 V 1196 998 V 1197 998 V 1198 999 V 1198 999 V 1199 999 V 1200 1000 V 1201 1000 V 1201 1001 V 1235 1029 V 1235 1028 V 1234 1028 V 1234 1027 V 1234 1026 V 1234 1026 V 1234 1025 V 1234 1024 V 1234 1024 V 1234 1023 V 1233 1023 V 1233 1022 V 1233 1021 V 1233 1021 V 1232 1020 V 1232 1020 V 1232 1019 V 1231 1019 V 1231 1018 V 1230 1018 V 1230 1017 V 1229 1016 V 1229 1016 V 1228 1015 V 1228 1015 V 1227 1015 V 1227 1014 V 1226 1014 V 1225 1013 V 1225 1013 V 1224 1012 V 1223 1012 V 1223 1011 V 1222 1011 V 1221 1010 V 1220 1010 V 715 1311 2 284 v 809 1217 2 95 v 856 1358 V 856 1075 V 903 1358 2 142 v 903 1122 V 1045 1358 V 1045 1122 V 1092 1358 2 95 v 1092 1075 V 1139 1217 V 1234 1311 2 284 v 794 1353 a Fs(:::)289 b(:::)794 995 y(:::)g(:::)956 1179 y Fu(=)p 810 1076 2 2 v 810 1075 V 810 1075 V 810 1074 V 810 1073 V 809 1072 V 809 1071 V 809 1071 V 809 1070 V 808 1069 V 808 1068 V 808 1068 V 807 1067 V 807 1066 V 806 1066 V 806 1065 V 805 1064 V 804 1063 V 804 1063 V 803 1062 V 802 1061 V 801 1061 V 801 1060 V 800 1060 V 799 1059 V 798 1058 V 797 1058 V 796 1057 V 795 1057 V 794 1056 V 793 1055 V 791 1055 V 790 1054 V 789 1054 V 788 1053 V 786 1053 V 786 1053 V 785 1052 V 784 1051 V 782 1051 V 781 1050 V 780 1050 V 779 1049 V 778 1049 V 777 1048 V 776 1047 V 775 1047 V 774 1046 V 773 1046 V 772 1045 V 771 1044 V 770 1044 V 770 1043 V 769 1042 V 768 1042 V 768 1041 V 767 1040 V 766 1040 V 766 1039 V 765 1038 V 765 1037 V 765 1037 V 764 1036 V 764 1035 V 764 1034 V 763 1034 V 763 1033 V 763 1032 V 763 1031 V 763 1031 V 763 1030 V 763 1029 V 763 1312 V 763 1311 V 763 1310 V 763 1310 V 763 1309 V 763 1308 V 763 1307 V 764 1307 V 764 1306 V 764 1305 V 765 1304 V 765 1304 V 765 1303 V 766 1302 V 766 1301 V 767 1301 V 768 1300 V 768 1299 V 769 1299 V 770 1298 V 770 1297 V 771 1297 V 772 1296 V 773 1295 V 774 1295 V 775 1294 V 776 1294 V 777 1293 V 778 1292 V 779 1292 V 780 1291 V 781 1291 V 782 1290 V 784 1290 V 785 1289 V 786 1288 V 786 1288 V 788 1288 V 789 1287 V 790 1287 V 791 1286 V 793 1286 V 794 1285 V 795 1285 V 796 1284 V 797 1283 V 798 1283 V 799 1282 V 800 1281 V 801 1281 V 801 1280 V 802 1280 V 803 1279 V 804 1278 V 804 1278 V 805 1277 V 806 1276 V 806 1275 V 807 1275 V 807 1274 V 808 1273 V 808 1273 V 808 1272 V 809 1271 V 809 1270 V 809 1270 V 809 1269 V 810 1268 V 810 1267 V 810 1266 V 810 1266 V 810 1265 V 1187 1312 V 1187 1311 V 1187 1310 V 1187 1310 V 1187 1309 V 1187 1308 V 1187 1307 V 1186 1307 V 1186 1306 V 1186 1305 V 1185 1304 V 1185 1304 V 1185 1303 V 1184 1302 V 1184 1301 V 1183 1301 V 1182 1300 V 1182 1299 V 1181 1299 V 1180 1298 V 1180 1297 V 1179 1297 V 1178 1296 V 1177 1295 V 1176 1295 V 1175 1294 V 1174 1294 V 1173 1293 V 1172 1292 V 1171 1292 V 1170 1291 V 1169 1291 V 1168 1290 V 1166 1290 V 1165 1289 V 1164 1288 V 1164 1288 V 1162 1288 V 1161 1287 V 1160 1287 V 1159 1286 V 1157 1286 V 1156 1285 V 1155 1285 V 1154 1284 V 1153 1283 V 1152 1283 V 1151 1282 V 1150 1281 V 1149 1281 V 1149 1280 V 1148 1280 V 1147 1279 V 1146 1278 V 1146 1278 V 1145 1277 V 1144 1276 V 1144 1275 V 1143 1275 V 1143 1274 V 1142 1273 V 1142 1273 V 1142 1272 V 1141 1271 V 1141 1270 V 1141 1270 V 1141 1269 V 1140 1268 V 1140 1267 V 1140 1266 V 1140 1266 V 1140 1265 V 1140 1076 V 1140 1075 V 1140 1075 V 1140 1074 V 1140 1073 V 1141 1072 V 1141 1071 V 1141 1071 V 1141 1070 V 1142 1069 V 1142 1068 V 1142 1068 V 1143 1067 V 1143 1066 V 1144 1066 V 1144 1065 V 1145 1064 V 1146 1063 V 1146 1063 V 1147 1062 V 1148 1061 V 1149 1061 V 1149 1060 V 1150 1060 V 1151 1059 V 1152 1058 V 1153 1058 V 1154 1057 V 1155 1057 V 1156 1056 V 1157 1055 V 1159 1055 V 1160 1054 V 1161 1054 V 1162 1053 V 1164 1053 V 1164 1053 V 1165 1052 V 1166 1051 V 1168 1051 V 1169 1050 V 1170 1050 V 1171 1049 V 1172 1049 V 1173 1048 V 1174 1047 V 1175 1047 V 1176 1046 V 1177 1046 V 1178 1045 V 1179 1044 V 1180 1044 V 1180 1043 V 1181 1042 V 1182 1042 V 1182 1041 V 1183 1040 V 1184 1040 V 1184 1039 V 1185 1038 V 1185 1037 V 1185 1037 V 1186 1036 V 1186 1035 V 1186 1034 V 1187 1034 V 1187 1033 V 1187 1032 V 1187 1031 V 1187 1031 V 1187 1030 V 1187 1029 V 228 1470 a(implies)498 1550 y Fs(\034)525 1529 y Fj(\000)p Fl(1)519 1562 y(1)580 1550 y Fs(:)8 b(:)g(:)g(\034)673 1529 y Fj(\000)p Fl(1)667 1562 y Fq(i)p Fj(\000)p Fl(1)726 1550 y Fs(\034)753 1529 y Fl(2)747 1562 y Fq(i)773 1550 y Fs(\034)794 1557 y Fq(i)p Fj(\000)p Fl(1)862 1550 y Fs(:)g(:)g(:)f(\034)948 1557 y Fl(1)982 1550 y Fu(=)14 b Fs(\034)1055 1557 y Fq(i)1077 1550 y Fs(:)8 b(:)g(:)g(\034)1164 1557 y Fl(2)1184 1550 y Fs(\034)1211 1529 y Fl(2)1205 1562 y(1)1230 1550 y Fs(\034)1257 1529 y Fj(\000)p Fl(1)1251 1562 y(2)1312 1550 y Fs(:)g(:)g(:)g(\034)1405 1529 y Fj(\000)p Fl(1)1399 1562 y Fq(i)1635 1550 y Fu(\(16\))228 1637 y(and)19 b(similarly)c Fs(\034)547 1644 y Fl(1)575 1637 y Fs(:)8 b(:)g(:)g(\034)662 1644 y Fq(i)p Fj(\000)p Fl(1)721 1637 y Fs(\034)748 1619 y Fl(2)742 1649 y Fq(i)768 1637 y Fs(\034)795 1617 y Fj(\000)p Fl(1)789 1649 y Fq(i)p Fj(\000)p Fl(1)856 1637 y Fs(:)g(:)g(:)g(\034)949 1617 y Fj(\000)p Fl(1)943 1648 y(1)1013 1637 y Fu(=)17 b Fs(\034)1095 1617 y Fj(\000)p Fl(1)1089 1649 y Fq(i)1150 1637 y Fs(:)8 b(:)g(:)g(\034)1243 1617 y Fj(\000)p Fl(1)1237 1648 y(2)1290 1637 y Fs(\034)1317 1619 y Fl(2)1311 1649 y(1)1336 1637 y Fs(\034)1357 1644 y Fl(2)1385 1637 y Fs(:)g(:)g(:)g(\034)1472 1644 y Fq(i)1504 1637 y Fu(for)19 b(all)e Fs(i)g Fu(=)228 1695 y(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)p Fu(.)278 1753 y(Let)16 b Fs(B)402 1760 y Fq(n)439 1753 y Fr(3)e Fs(')g Fr(7!)22 b Fu(~)-32 b Fs(')14 b Fr(2)g Fs(S)719 1760 y Fq(n)758 1753 y Fu(denote)j(the)f(canonical)g(epimorphism.)278 1811 y(F)l(or)e(eac)o(h)h(braid)g Fs(')e Fr(2)h Fs(B)727 1818 y Fq(n)766 1811 y Fu(there)g(exists)g(a)h(braid)g Fs(')1219 1819 y Fl(\()p Fq(i)p Fl(\))1274 1811 y Fr(2)f Fs(B)1358 1818 y Fq(n)p Fl(+1)1442 1811 y Fu(suc)o(h)g(that)h(the)228 1870 y(diagram)276 1976 y Fs(P)j Fr(\012)11 b Fs(:)d(:)g(:)j Fr(\012)f Fs(P)531 1958 y Fl(2)562 1976 y Fr(\012)h Fs(:)d(:)g(:)j Fr(\012)f Fs(P)657 b(P)19 b Fr(\012)10 b Fs(:)e(:)g(:)j Fr(\012)g Fs(P)p 783 1959 621 2 v 1362 1958 a Fk(-)875 1939 y Fu(1)h Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)g Fs(f)16 b Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fu(1)503 2022 y Fs(i)276 2269 y(P)18 b Fr(\012)11 b Fs(:)d(:)g(:)j Fr(\012)f Fs(P)531 2251 y Fl(2)562 2269 y Fr(\012)h Fs(:)d(:)g(:)j Fr(\012)f Fs(P)657 b(P)19 b Fr(\012)10 b Fs(:)e(:)g(:)j Fr(\012)g Fs(P)p 783 2252 V 1362 2251 a Fk(-)875 2232 y Fu(1)h Fr(\012)e Fs(:)e(:)g(:)j Fr(\012)g Fs(f)16 b Fr(\012)11 b Fs(:)d(:)g(:)i Fr(\012)h Fu(1)500 2312 y Fs(j)p 522 2217 2 225 v 522 2217 a Fk(?)542 2108 y Fs(')574 2115 y Fl(\()p Fq(i)p Fl(\))p 1545 2217 V 1546 2217 a Fk(?)1566 2111 y Fs(')228 2387 y Fu(comm)o(utes)f(for)k(all)f Fs(f)19 b Fu(:)14 b Fs(P)700 2368 y Fl(2)733 2387 y Fr(\000)-30 b(!)14 b Fs(P)21 b Fu(in)13 b Fr(C)j Fu(\(where)d Fs(j)k Fu(=)22 b(~)-32 b Fs(')p Fu(\()p Fs(i)p Fu(\)\).)20 b(The)13 b(braid)h Fs(')1594 2394 y Fl(\()p Fq(i)p Fl(\))1649 2387 y Fu(can)228 2445 y(b)q(e)k(giv)o(en)f(explicitly)l(,)d(but)k(w)o (e)g(are)f(only)h(in)o(terested)e(in)i(the)f(follo)o(wing)h(sp)q(ecial) 228 2503 y(forms)499 2549 y Fs(\034)520 2557 y Fq(j)r Fl(\()p Fq(i)p Fl(\))594 2549 y Fu(=)c Fs(\034)667 2556 y Fq(j)r Fl(+1)788 2549 y Fu(if)i Fs(j)g(>)e(i)p Fu(;)182 b Fs(\034)1155 2557 y Fq(i)p Fj(\000)p Fl(1\()p Fq(i)p Fl(\))1270 2549 y Fu(=)14 b Fs(\034)1343 2556 y Fq(i)1357 2549 y Fs(\034)1378 2556 y Fq(i)p Fj(\000)p Fl(1)1438 2549 y Fu(;)499 2607 y Fs(\034)520 2615 y Fq(j)r Fl(\()p Fq(i)p Fl(\))594 2607 y Fu(=)g Fs(\034)667 2614 y Fq(j)788 2607 y Fu(if)i Fs(j)g(<)e(i)d Fr(\000)g Fu(1;)142 b Fs(\034)1200 2615 y Fq(i)p Fl(\()p Fq(i)p Fl(\))1270 2607 y Fu(=)14 b Fs(\034)1343 2614 y Fq(i)1357 2607 y Fs(\034)1378 2614 y Fq(i)p Fl(+1)p eop %%Page: 21 21 21 20 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(21)228 225 y Fu(whic)o(h)15 b(can)i(b)q(e)f(easily)g(v)o(eri\014ed.)278 283 y(By)f(\(16\))i(w)o(e)f(ha)o(v)o(e)f(for)i(all)f Fs(z)f Fr(2)f Fs(P)907 265 y Fq(n)p Fl(+1)976 283 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))654 456 y Fs(\034)681 436 y Fl(2)675 469 y Fq(i)701 456 y Fs(\034)722 463 y Fq(i)p Fj(\000)p Fl(1)789 456 y Fs(:)g(:)g(:)g(\034)876 463 y Fl(1)896 456 y Fu(\()p Fs(z)r Fu(\))13 b(=)h Fs(\034)1045 463 y Fq(i)p Fj(\000)p Fl(1)1113 456 y Fs(:)8 b(:)g(:)f(\034)1199 463 y Fl(1)1219 456 y Fu(\()p Fs(z)r Fu(\))p Fs(:)339 b Fu(\(17\))228 634 y FB(Lemm)o(a)16 b(8.1.)k Fo(F)l(or)d Fs(z)f Fr(2)e Fs(P)749 616 y Fq(n)p Fl(+1)818 634 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))p Fo(,)17 b Fs(')d Fr(2)g Fs(B)1128 641 y Fq(n)1169 634 y Fo(and)k Fs(j)e Fu(:=)22 b(~)-32 b Fs(')p Fu(\()p Fs(i)p Fu(\))17 b Fo(we)h(have)559 806 y Fs(')591 813 y Fl(\()p Fq(i)p Fl(\))633 806 y Fs(\034)654 813 y Fq(i)p Fj(\000)p Fl(1)721 806 y Fs(:)8 b(:)g(:)g(\034)808 813 y Fl(1)828 806 y Fu(\()p Fs(z)r Fu(\))17 b(=)d Fs(\034)981 813 y Fq(j)r Fj(\000)p Fl(1)1053 806 y Fs(:)8 b(:)g(:)f(\034)1139 813 y Fl(1)1159 806 y Fu(\(1)12 b Fr(\012)e Fs(')p Fu(\)\()p Fs(z)r Fu(\);)604 864 y Fs(')636 871 y Fl(\()p Fq(i)p Fl(\))678 864 y Fs(\034)699 871 y Fq(i)721 864 y Fs(:)e(:)g(:)g(\034) 808 871 y Fl(1)828 864 y Fu(\()p Fs(z)r Fu(\))17 b(=)d Fs(\034)981 871 y Fq(j)1008 864 y Fs(:)8 b(:)g(:)f(\034)1094 871 y Fl(1)1114 864 y Fu(\(1)12 b Fr(\012)e Fs(')p Fu(\)\()p Fs(z)r Fu(\))p Fs(:)228 1060 y Fo(Pr)n(o)n(of.)19 b Fu(T)l(o)d(pro)o(v) o(e)f(this)h(w)o(e)g(\014rst)g(observ)o(e)f(that)h(these)g(t)o(w)o(o)g (relations)f(are)h(com-)228 1120 y(patible)21 b(with)i(the)f(group)h (structure)f(of)h Fs(B)1062 1127 y Fq(n)1085 1120 y Fu(.)39 b(F)l(or)31 b(~)-32 b Fs(')1274 1107 y Fu(~)1264 1120 y Fs( )q Fu(\()p Fs(i)p Fu(\))24 b(=)32 b(~)-32 b Fs(')p Fu(\()p Fs(j)s Fu(\))24 b(=)g Fs(k)h Fu(w)o(e)228 1178 y(ha)o(v)o(e)249 1337 y Fs(')281 1345 y Fl(\()p Fq(j)r Fl(\))326 1337 y Fs( )358 1345 y Fl(\()p Fq(i)p Fl(\))399 1337 y Fs(\034)420 1344 y Fq(i)p Fj(\000)p Fl(1)488 1337 y Fs(:)8 b(:)g(:)g(\034)575 1344 y Fl(1)595 1337 y Fu(\()p Fs(z)r Fu(\))16 b(=)d Fs(')757 1345 y Fl(\()p Fq(j)r Fl(\))803 1337 y Fs(\034)824 1344 y Fq(j)r Fj(\000)p Fl(1)896 1337 y Fs(:)8 b(:)g(:)f(\034)982 1344 y Fl(1)1002 1337 y Fu(\(1)12 b Fr(\012)f Fs( )r Fu(\)\()p Fs(z)r Fu(\))i(=)g Fs(\034)1308 1344 y Fq(k)q Fj(\000)p Fl(1)1383 1337 y Fs(:)8 b(:)g(:)g(\034)1470 1344 y Fl(1)1490 1337 y Fu(\(1)j Fr(\012)g Fs(' )r Fu(\)\()p Fs(z)r Fu(\);)294 1396 y Fs(')326 1403 y Fl(\()p Fq(j)r Fl(\))371 1396 y Fs( )403 1403 y Fl(\()p Fq(i)p Fl(\))445 1396 y Fs(\034)466 1403 y Fq(i)488 1396 y Fs(:)d(:)g(:)g(\034)575 1403 y Fl(1)595 1396 y Fu(\()p Fs(z)r Fu(\))16 b(=)d Fs(')757 1403 y Fl(\()p Fq(j)r Fl(\))803 1396 y Fs(\034)824 1403 y Fq(j)851 1396 y Fs(:)8 b(:)g(:)f(\034)937 1403 y Fl(1)957 1396 y Fu(\(1)12 b Fr(\012)e Fs( )r Fu(\)\()p Fs(z)r Fu(\))k(=)f Fs(\034)1263 1403 y Fq(k)1293 1396 y Fs(:)8 b(:)g(:)g(\034)1380 1403 y Fl(1)1400 1396 y Fu(\(1)j Fr(\012)g Fs(' )r Fu(\)\()p Fs(z)r Fu(\))228 1566 y(so)23 b(w)o(e)g(only)f(ha)o(v)o(e)g(to)h(sho)o(w)h(these)e(relations)h(for)g (the)g(generators)g Fs(')i Fu(=)g Fs(\034)1690 1573 y Fq(j)1709 1566 y Fu(,)228 1624 y Fs(j)17 b Fu(=)c(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)k Fr(\000)e Fu(1.)22 b(In)16 b(these)g(cases)g(w)o(e)g(ha)o(v)o(e)355 1793 y Fs(\034)376 1801 y Fq(j)r Fl(\()p Fq(i)p Fl(\))434 1793 y Fs(\034)455 1800 y Fq(i)p Fj(\000)p Fl(1)522 1793 y Fs(:)8 b(:)g(:)g(\034)609 1800 y Fl(1)629 1793 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)781 1800 y Fq(j)r Fl(+1)844 1793 y Fs(\034)865 1800 y Fq(i)p Fj(\000)p Fl(1)933 1793 y Fs(:)8 b(:)g(:)f(\034)1019 1800 y Fl(1)1039 1793 y Fu(\()p Fs(z)r Fu(\))708 1851 y(=)14 b Fs(\034)781 1858 y Fq(i)p Fj(\000)p Fl(1)848 1851 y Fs(:)8 b(:)g(:)g(\034)935 1858 y Fl(1)955 1851 y Fs(\034)976 1858 y Fq(j)r Fl(+1)1039 1851 y Fu(\()p Fs(z)r Fu(\))708 1910 y(=)14 b Fs(\034)781 1917 y Fq(i)p Fj(\000)p Fl(1)848 1910 y Fs(:)8 b(:)g(:)g(\034)935 1917 y Fl(1)955 1910 y Fu(\(1)j Fr(\012)g Fs(\034)1080 1917 y Fq(j)1098 1910 y Fu(\)\()p Fs(z)r Fu(\))178 b(for)16 b Fs(j)h(>)d(i)p Fu(;)355 1968 y Fs(\034)376 1975 y Fq(j)r Fl(\()p Fq(i)p Fl(\))434 1968 y Fs(\034)455 1975 y Fq(i)p Fj(\000)p Fl(1)522 1968 y Fs(:)8 b(:)g(:)g(\034)609 1975 y Fl(1)629 1968 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)781 1975 y Fq(j)799 1968 y Fs(\034)820 1975 y Fq(i)p Fj(\000)p Fl(1)888 1968 y Fs(:)8 b(:)g(:)f(\034)974 1975 y Fl(1)994 1968 y Fu(\()p Fs(z)r Fu(\))708 2026 y(=)14 b Fs(\034)781 2033 y Fq(i)p Fj(\000)p Fl(1)848 2026 y Fs(:)8 b(:)g(:)g(\034)935 2033 y Fq(j)953 2026 y Fs(\034)974 2033 y Fq(j)r Fl(+1)1038 2026 y Fs(\034)1059 2033 y Fq(j)1086 2026 y Fs(:)g(:)g(:)f(\034)1172 2033 y Fl(1)1192 2026 y Fu(\()p Fs(z)r Fu(\))708 2084 y(=)14 b Fs(\034)781 2091 y Fq(i)p Fj(\000)p Fl(1)848 2084 y Fs(:)8 b(:)g(:)g(\034)935 2091 y Fq(j)r Fl(+1)998 2084 y Fs(\034)1019 2091 y Fq(j)1038 2084 y Fs(\034)1059 2091 y Fq(j)r Fl(+1)1131 2084 y Fs(:)g(:)g(:)f(\034)1217 2091 y Fl(1)1237 2084 y Fu(\()p Fs(z)r Fu(\))708 2142 y(=)14 b Fs(\034)781 2149 y Fq(i)p Fj(\000)p Fl(1)848 2142 y Fs(:)8 b(:)g(:)g(\034)935 2149 y Fl(1)955 2142 y Fs(\034)976 2149 y Fq(j)r Fl(+1)1039 2142 y Fu(\()p Fs(z)r Fu(\))708 2200 y(=)14 b Fs(\034)781 2207 y Fq(i)p Fj(\000)p Fl(1)848 2200 y Fs(:)8 b(:)g(:)g(\034)935 2207 y Fl(1)955 2200 y Fu(\(1)j Fr(\012)g Fs(\034)1080 2207 y Fq(j)1098 2200 y Fu(\)\()p Fs(z)r Fu(\))178 b(for)16 b Fs(j)h(<)d(i)c Fr(\000)h Fu(1;)314 2258 y Fs(\034)335 2266 y Fq(i)p Fj(\000)p Fl(1\()p Fq(i)p Fl(\))434 2258 y Fs(\034)455 2265 y Fq(i)p Fj(\000)p Fl(1)522 2258 y Fs(:)d(:)g(:)g(\034)609 2265 y Fl(1)629 2258 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)781 2265 y Fq(i)795 2258 y Fs(\034)816 2265 y Fq(i)p Fj(\000)p Fl(1)875 2258 y Fs(\034)896 2265 y Fq(i)p Fj(\000)p Fl(1)964 2258 y Fs(:)8 b(:)g(:)f(\034)1050 2265 y Fl(1)1070 2258 y Fu(\()p Fs(z)r Fu(\))708 2316 y(=)14 b Fs(\034)781 2323 y Fq(i)795 2316 y Fs(\034)816 2323 y Fq(i)p Fj(\000)p Fl(2)884 2316 y Fs(:)8 b(:)g(:)f(\034)970 2323 y Fl(1)990 2316 y Fu(\()p Fs(z)r Fu(\))708 2374 y(=)14 b Fs(\034)781 2381 y Fq(i)p Fj(\000)p Fl(2)848 2374 y Fs(:)8 b(:)g(:)g(\034)935 2381 y Fl(1)955 2374 y Fs(\034)976 2381 y Fq(i)990 2374 y Fu(\()p Fs(z)r Fu(\))708 2433 y(=)14 b Fs(\034)781 2440 y Fq(i)p Fj(\000)p Fl(2)848 2433 y Fs(:)8 b(:)g(:)g(\034)935 2440 y Fl(1)955 2433 y Fu(\(1)j Fr(\012)g Fs(\034)1080 2440 y Fq(i)p Fj(\000)p Fl(1)1139 2433 y Fu(\)\()p Fs(z)r Fu(\);)359 2491 y Fs(\034)380 2498 y Fq(i)p Fl(\()p Fq(i)p Fl(\))434 2491 y Fs(\034)455 2498 y Fq(i)p Fj(\000)p Fl(1)522 2491 y Fs(:)d(:)g(:)g(\034)609 2498 y Fl(1)629 2491 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)781 2498 y Fq(i)795 2491 y Fs(\034)816 2498 y Fq(i)p Fl(+1)875 2491 y Fs(\034)896 2498 y Fq(i)p Fj(\000)p Fl(1)964 2491 y Fs(:)8 b(:)g(:)f(\034)1050 2498 y Fl(1)1070 2491 y Fu(\()p Fs(z)r Fu(\))708 2549 y(=)14 b Fs(\034)781 2556 y Fq(i)803 2549 y Fs(:)8 b(:)g(:)g(\034)890 2556 y Fl(1)910 2549 y Fs(\034)931 2556 y Fq(i)p Fl(+1)990 2549 y Fu(\()p Fs(z)r Fu(\))708 2607 y(=)14 b Fs(\034)781 2614 y Fq(i)803 2607 y Fs(:)8 b(:)g(:)g(\034)890 2614 y Fl(1)910 2607 y Fu(\(1)j Fr(\012)g Fs(\034)1035 2614 y Fq(i)1049 2607 y Fu(\)\()p Fs(z)r Fu(\);)p eop %%Page: 22 22 22 21 bop 228 125 a Fp(22)552 b(BODO)13 b(P)m(AREIGIS)422 224 y Fs(\034)443 232 y Fq(j)r Fl(\()p Fq(i)p Fl(\))501 224 y Fs(\034)522 231 y Fq(i)545 224 y Fs(:)8 b(:)g(:)f(\034)631 231 y Fl(1)651 224 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)803 231 y Fq(j)r Fl(+1)867 224 y Fs(\034)888 231 y Fq(i)910 224 y Fs(:)8 b(:)g(:)g(\034)997 231 y Fl(1)1017 224 y Fu(\()p Fs(z)r Fu(\))730 282 y(=)14 b Fs(\034)803 289 y Fq(i)826 282 y Fs(:)8 b(:)g(:)f(\034)912 289 y Fl(1)932 282 y Fs(\034)953 289 y Fq(j)r Fl(+1)1017 282 y Fu(\()p Fs(z)r Fu(\))730 340 y(=)14 b Fs(\034)803 347 y Fq(i)826 340 y Fs(:)8 b(:)g(:)f(\034)912 347 y Fl(1)932 340 y Fu(\(1)12 b Fr(\012)e Fs(\034)1057 347 y Fq(j)1076 340 y Fu(\)\()p Fs(z)r Fu(\))177 b(for)17 b Fs(j)f(>)e(i)p Fu(;)422 399 y Fs(\034)443 406 y Fq(j)r Fl(\()p Fq(i)p Fl(\))501 399 y Fs(\034)522 406 y Fq(i)545 399 y Fs(:)8 b(:)g(:)f(\034)631 406 y Fl(1)651 399 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)803 406 y Fq(j)822 399 y Fs(\034)843 406 y Fq(i)865 399 y Fs(:)8 b(:)g(:)g(\034)952 406 y Fl(1)972 399 y Fu(\()p Fs(z)r Fu(\))730 457 y(=)14 b Fs(\034)803 464 y Fq(i)826 457 y Fs(:)8 b(:)g(:)f(\034)912 464 y Fq(j)931 457 y Fs(\034)952 464 y Fq(j)r Fl(+1)1015 457 y Fs(\034)1036 464 y Fq(j)1063 457 y Fs(:)h(:)g(:)g(\034)1150 464 y Fl(1)1169 457 y Fu(\()p Fs(z)r Fu(\))730 515 y(=)14 b Fs(\034)803 522 y Fq(i)826 515 y Fs(:)8 b(:)g(:)f(\034)912 522 y Fq(j)r Fl(+1)976 515 y Fs(\034)997 522 y Fq(j)1015 515 y Fs(\034)1036 522 y Fq(j)r Fl(+1)1108 515 y Fs(:)h(:)g(:)g(\034) 1195 522 y Fl(1)1214 515 y Fu(\()p Fs(z)r Fu(\))730 573 y(=)14 b Fs(\034)803 580 y Fq(i)826 573 y Fs(:)8 b(:)g(:)f(\034)912 580 y Fl(1)932 573 y Fs(\034)953 580 y Fq(j)r Fl(+1)1017 573 y Fu(\()p Fs(z)r Fu(\))730 631 y(=)14 b Fs(\034)803 638 y Fq(i)826 631 y Fs(:)8 b(:)g(:)f(\034)912 638 y Fl(1)932 631 y Fu(\(1)12 b Fr(\012)e Fs(\034)1057 638 y Fq(j)1076 631 y Fu(\)\()p Fs(z)r Fu(\))177 b(for)17 b Fs(j)f(<)e(i)d Fr(\000)g Fu(1;)336 689 y Fs(\034)357 697 y Fq(i)p Fj(\000)p Fl(1\()p Fq(i)p Fl(\))456 689 y Fs(\034)477 696 y Fq(i)p Fj(\000)p Fl(1)545 689 y Fs(:)d(:)g(:)f (\034)631 696 y Fl(1)651 689 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)803 696 y Fq(i)817 689 y Fs(\034)838 696 y Fq(i)p Fj(\000)p Fl(1)898 689 y Fs(\034)919 696 y Fq(i)933 689 y Fs(\034)954 696 y Fq(i)p Fj(\000)p Fl(1)1013 689 y Fs(\034)1034 696 y Fq(i)p Fj(\000)p Fl(2)1102 689 y Fs(:)8 b(:)g(:)g(\034)1189 696 y Fl(1)1208 689 y Fu(\()p Fs(z)r Fu(\))730 747 y(=)14 b Fs(\034)803 754 y Fq(i)p Fj(\000)p Fl(1)862 747 y Fs(\034)883 754 y Fq(i)898 747 y Fs(\034)925 729 y Fl(2)919 760 y Fq(i)p Fj(\000)p Fl(1)978 747 y Fs(\034)999 754 y Fq(i)p Fj(\000)p Fl(2)1067 747 y Fs(:)8 b(:)g(:)f(\034)1153 754 y Fl(1)1173 747 y Fu(\()p Fs(z)r Fu(\))730 805 y(=)14 b Fs(\034)803 812 y Fq(i)p Fj(\000)p Fl(1)862 805 y Fs(\034)883 812 y Fq(i)898 805 y Fs(\034)919 812 y Fq(i)p Fj(\000)p Fl(2)986 805 y Fs(:)8 b(:)g(:)g(\034)1073 812 y Fl(1)1093 805 y Fu(\()p Fs(z)r Fu(\))730 863 y(=)14 b Fs(\034)803 870 y Fq(i)p Fj(\000)p Fl(1)862 863 y Fs(\034)883 870 y Fq(i)p Fj(\000)p Fl(2)951 863 y Fs(:)8 b(:)g(:)g(\034)1038 870 y Fl(1)1058 863 y Fs(\034)1079 870 y Fq(i)1093 863 y Fu(\()p Fs(z)r Fu(\))730 922 y(=)14 b Fs(\034)803 929 y Fq(i)p Fj(\000)p Fl(1)871 922 y Fs(:)8 b(:)g(:)f(\034)957 929 y Fl(1)977 922 y Fu(\(1)12 b Fr(\012)e Fs(\034)1102 929 y Fq(i)p Fj(\000)p Fl(1)1162 922 y Fu(\)\()p Fs(z)r Fu(\);)427 980 y Fs(\034)448 987 y Fq(i)p Fl(\()p Fq(i)p Fl(\))501 980 y Fs(\034)522 987 y Fq(i)545 980 y Fs(:)e(:)g(:)f(\034)631 987 y Fl(1)651 980 y Fu(\()p Fs(z)r Fu(\))16 b(=)e Fs(\034)803 987 y Fq(i)817 980 y Fs(\034)838 987 y Fq(i)p Fl(+1)898 980 y Fs(\034)919 987 y Fq(i)941 980 y Fs(:)8 b(:)g(:)g(\034)1028 987 y Fl(1)1048 980 y Fu(\()p Fs(z)r Fu(\))730 1038 y(=)14 b Fs(\034)803 1045 y Fq(i)p Fl(+1)862 1038 y Fs(\034)883 1045 y Fq(i)898 1038 y Fs(\034)919 1045 y Fq(i)p Fl(+1)986 1038 y Fs(:)8 b(:)g(:)g(\034)1073 1045 y Fl(1)1093 1038 y Fu(\()p Fs(z)r Fu(\))730 1096 y(=)14 b Fs(\034)803 1103 y Fq(i)p Fl(+1)862 1096 y Fs(\034)883 1103 y Fq(i)906 1096 y Fs(:)8 b(:)g(:)g(\034)993 1103 y Fl(1)1013 1096 y Fs(\034)1034 1103 y Fq(i)p Fl(+1)1093 1096 y Fu(\()p Fs(z)r Fu(\))730 1154 y(=)14 b Fs(\034)803 1161 y Fq(i)p Fl(+1)871 1154 y Fs(:)8 b(:)g(:)f(\034)957 1161 y Fl(1)977 1154 y Fu(\(1)12 b Fr(\012)e Fs(\034)1102 1161 y Fq(i)1117 1154 y Fu(\)\()p Fs(z)r Fu(\))228 1229 y(where)16 b(w)o(e)g(used)g (\(17\))h(in)f(the)g(3.)g(and)h(7.)f(equations.)468 b Fi(\003)228 1320 y FB(Lemm)o(a)16 b(8.2.)k Fo(F)l(or)d(al)r(l)i Fs(z)d Fr(2)e Fs(P)819 1302 y Fq(n)p Fl(+1)888 1320 y Fu(\()p Fr(\000)p Fu(1)p Fs(;)8 b(\020)t Fu(\))17 b Fo(and)h(al)r(l)g Fs(f)i Fu(:)13 b Fs(P)1326 1302 y Fl(2)1360 1320 y Fr(\000)-29 b(!)13 b Fs(P)25 b Fo(we)18 b(have)553 1405 y Fu(\()p Fs(P)610 1385 y Fq(i)p Fj(\000)p Fl(1)680 1405 y Fr(\012)11 b Fs(f)16 b Fr(\012)11 b Fs(P)858 1385 y Fq(n)p Fj(\000)p Fq(i)921 1405 y Fu(\))p Fs(\034)961 1412 y Fq(i)p Fj(\000)p Fl(1)1029 1405 y Fs(:)d(:)g(:)f(\034)1115 1412 y Fl(1)1135 1405 y Fu(\()p Fs(z)r Fu(\))14 b Fr(2)g Fs(P)1297 1385 y Fq(n)1321 1405 y Fu(\()p Fs(\020)t Fu(\))p Fs(:)228 1496 y Fo(Pr)n(o)n(of.)19 b Fu(F)l(or)d(all)g Fs(')e Fr(2)g Fs(B)662 1503 y Fq(n)702 1496 y Fu(and)i(all)g Fs(k)g Fu(=)e(1)p Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(n)16 b Fu(w)o(e)g(ha)o(v)o(e)452 1577 y Fs(\034)479 1559 y Fl(2)473 1589 y Fq(k)499 1577 y Fs(')g Fu(\()p Fs(P)604 1559 y Fq(i)p Fj(\000)p Fl(1)674 1577 y Fr(\012)11 b Fs(f)16 b Fr(\012)11 b Fs(P)852 1559 y Fq(n)p Fj(\000)p Fq(i)915 1577 y Fu(\))p Fs(\034)955 1584 y Fq(i)p Fj(\000)p Fl(1)1023 1577 y Fs(:)d(:)g(:)g(\034)1110 1584 y Fl(1)1129 1577 y Fu(\()p Fs(z)r Fu(\))14 b(=)547 1635 y(=)g Fs(\034)626 1617 y Fl(2)620 1647 y Fq(k)645 1635 y Fu(\()p Fs(P)702 1617 y Fq(j)r Fj(\000)p Fl(1)777 1635 y Fr(\012)c Fs(f)17 b Fr(\012)11 b Fs(P)955 1617 y Fq(n)p Fj(\000)p Fq(j)1022 1635 y Fu(\))p Fs(')1073 1643 y Fl(\()p Fq(i)p Fl(\))1114 1635 y Fs(\034)1135 1642 y Fq(j)r Fj(\000)p Fl(1)1207 1635 y Fs(:)d(:)g(:)g(\034)1294 1642 y Fl(1)1314 1635 y Fu(\()p Fs(z)r Fu(\))547 1693 y(=)14 b(\()p Fs(P)656 1675 y Fq(j)r Fj(\000)p Fl(1)730 1693 y Fr(\012)d Fs(f)16 b Fr(\012)11 b Fs(P)908 1675 y Fq(n)p Fj(\000)p Fq(j)976 1693 y Fu(\))p Fs(\034)1022 1675 y Fl(2)1016 1707 y Fq(k)q Fl(\()p Fq(j)r Fl(\))1081 1693 y Fs(\034)1102 1700 y Fq(j)r Fj(\000)p Fl(1)1173 1693 y Fs(:)d(:)g(:)g(\034)1260 1700 y Fl(1)1280 1693 y Fu(\(1)j Fr(\012)g Fs(')p Fu(\)\()p Fs(z)r Fu(\))547 1756 y(=)j(\()p Fs(P)656 1738 y Fq(j)r Fj(\000)p Fl(1)730 1756 y Fr(\012)d Fs(f)16 b Fr(\012)11 b Fs(P)908 1738 y Fq(n)p Fj(\000)p Fq(j)976 1756 y Fu(\))p Fs(\034)1016 1763 y Fq(j)r Fj(\000)p Fl(1)1087 1756 y Fs(:)d(:)g(:)g(\034)1174 1763 y Fl(1)1194 1756 y Fu(\(1)j Fr(\012)g Fs(\034)1325 1738 y Fl(2)1319 1768 y Fq(k)1344 1756 y Fs(')p Fu(\)\()p Fs(z)r Fu(\))547 1814 y(=)j Fs(')p Fu(\()p Fs(P)688 1796 y Fq(i)p Fj(\000)p Fl(1)758 1814 y Fr(\012)d Fs(f)16 b Fr(\012)11 b Fs(P)936 1796 y Fq(n)p Fj(\000)p Fq(i)999 1814 y Fu(\))p Fs(\034)1039 1821 y Fq(i)p Fj(\000)p Fl(1)1107 1814 y Fs(:)d(:)g(:)f(\034)1193 1821 y Fl(1)1213 1814 y Fu(\()p Fs(z)r Fu(\))228 1902 y(hence)15 b(\()p Fs(P)420 1884 y Fq(i)p Fj(\000)p Fl(1)491 1902 y Fr(\012)c Fs(f)16 b Fr(\012)11 b Fs(P)669 1884 y Fq(n)p Fj(\000)p Fq(i)732 1902 y Fu(\))p Fs(\034)772 1909 y Fq(i)p Fj(\000)p Fl(1)840 1902 y Fs(:)d(:)g(:)f(\034)926 1909 y Fl(1)946 1902 y Fu(\()p Fs(z)r Fu(\))14 b Fr(2)g Fs(P)1108 1884 y Fq(n)1131 1902 y Fu(\()p Fs(\020)t Fu(\))p Fs(:)475 b Fi(\003)278 1992 y Fu(No)o(w)16 b(w)o(e)g(can)g(giv)o(e)f (the)228 2083 y Fo(Pr)n(o)n(of.)k Fu(of)d(Prop)q(osition)i(3.1:)278 2141 y(W)l(e)12 b(\014rst)g(sho)o(w)h(that)g Fs(P)711 2123 y Fq(n)p Fl(+1)780 2141 y Fu(\()p Fs(\020)t Fu(\))g Fr(\022)h Fs(P)c Fr(\012)s Fu(\()p Fs(P)1049 2123 y Fq(n)1073 2141 y Fu(\()p Fs(\020)t Fu(\)\))j Fr(\022)h Fs(P)1259 2123 y Fq(n)p Fl(+1)1328 2141 y Fu(.)20 b(Let)12 b Fs(z)k Fu(=)1536 2108 y Fm(P)1579 2151 y Fq(k)1609 2141 y Fs(z)1632 2148 y Fq(k)q(;)p Fl(1)1683 2141 y Fr(\012)228 2199 y Fs(:)8 b(:)g(:)t Fr(\012)d Fs(z)357 2206 y Fq(k)q(;n)p Fl(+1)468 2199 y Fu(b)q(e)13 b(in)g Fs(P)623 2181 y Fq(n)p Fl(+1)692 2199 y Fu(\()p Fs(\020)t Fu(\))g(with)g(linearly)f(indep)q (enden)o(t)h Fs(z)1345 2206 y Fq(k)q(;)p Fl(1)1393 2199 y Fu(.)20 b(Let)14 b Fs(';)8 b(\034)1587 2206 y Fq(i)1615 2199 y Fr(2)14 b Fs(B)1699 2206 y Fq(n)228 2257 y Fu(b)q(e)i(giv)o(en.) k(De\014ne)c(1)11 b Fr(\012)f Fs(')k Fr(2)g Fs(B)805 2264 y Fq(n)p Fl(+1)890 2257 y Fu(resp.)21 b(1)11 b Fr(\012)f Fs(\034)1117 2264 y Fq(i)1145 2257 y Fr(2)k Fs(B)1229 2264 y Fq(n)p Fl(+1)1314 2257 y Fu(b)o(y)h(the)h(op)q(eration)h(of)228 2323 y Fs(')d Fu(resp.)21 b Fs(\034)417 2330 y Fq(i)446 2323 y Fu(on)15 b(the)f(factors)i Fs(z)775 2330 y Fq(k)q(;)p Fl(2)831 2323 y Fr(\012)8 b Fs(:)g(:)g(:)e Fr(\012)i Fs(z)1012 2330 y Fq(k)q(;n)p Fl(+1)1109 2323 y Fu(,)14 b(e.g.)21 b(1)8 b Fr(\012)g Fs(\034)1338 2297 y Fl(\()p Fq(n)p Fl(\))1332 2334 y Fq(i)1402 2323 y Fu(=)14 b Fs(\034)1481 2297 y Fl(\()p Fq(n)p Fl(+1\))1475 2334 y Fq(i)p Fl(+1)1576 2323 y Fu(.)21 b(Then)249 2376 y Fm(P)301 2409 y Fs(z)324 2416 y Fq(k)q(;)p Fl(1)383 2409 y Fr(\012)11 b Fs(')465 2391 y Fj(\000)p Fl(1)512 2409 y Fs(\034)539 2391 y Fl(2)533 2421 y Fq(i)559 2409 y Fs(')p Fu(\()610 2376 y Fm(P)662 2409 y Fs(z)685 2416 y Fq(k)q(;)p Fl(2)744 2409 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(z)935 2416 y Fq(k)q(;n)p Fl(+1)1032 2409 y Fu(\))17 b(=)c(\(1)f Fr(\012)f Fs(')1256 2391 y Fj(\000)p Fl(1)1303 2409 y Fs(\034)1330 2391 y Fl(2)1324 2421 y Fq(i)1349 2409 y Fs(')p Fu(\)\()p Fs(z)r Fu(\))1068 2467 y(=)1119 2434 y Fm(P)1163 2477 y Fq(k)1193 2467 y Fs(z)1216 2474 y Fq(k)q(;)p Fl(1)1275 2467 y Fr(\012)g Fs(\020)1350 2449 y Fl(2)1378 2434 y Fm(P)1430 2467 y Fs(z)1453 2474 y Fq(k)q(;)p Fl(2)1513 2467 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(z)1704 2474 y Fq(k)q(;n)p Fl(+1)1801 2467 y Fs(:)228 2550 y Fu(Since)16 b(the)g Fs(z)463 2557 y Fq(k)q(;)p Fl(1)529 2550 y Fu(are)h(linearly)e(indep)q (enden)o(t,)h(the)g(terms)1298 2516 y Fm(P)1350 2550 y Fs(z)1373 2557 y Fq(k)q(;)p Fl(2)1433 2550 y Fr(\012)11 b Fs(:)d(:)g(:)j Fr(\012)g Fs(z)1625 2557 y Fq(k)q(;n)p Fl(+1)228 2608 y Fu(are)16 b(in)g Fs(P)404 2590 y Fq(n)428 2608 y Fu(\()p Fs(\020)t Fu(\))g(hence)f Fs(z)h Fr(2)e Fs(P)k Fr(\012)11 b Fs(P)865 2590 y Fq(n)889 2608 y Fu(\()p Fs(\020)t Fu(\).)p eop %%Page: 23 23 23 22 bop 249 125 a Fp(ON)16 b(LIE)h(ALGEBRAS)g(IN)f(THE)g(CA)m(TEGOR)m (Y)g(OF)21 b(YETTER-DRINFELD)c(MODULES)-18 b(23)278 225 y Fu(No)o(w)16 b(w)o(e)g(sho)o(w)h(that)g(a)g(factorization)g(as)g(giv) o(en)f(in)g(the)g(follo)o(wing)h(diagram)228 283 y(exists)651 335 y Fs(P)689 317 y Fq(n)p Fl(+1)758 335 y Fu(\()p Fs(\020)t Fu(\))193 b Fs(P)19 b Fr(\012)11 b Fs(P)1152 317 y Fq(n)1175 335 y Fu(\()p Fs(\020)t Fu(\))p 836 323 165 2 v 959 322 a Fk(-)909 313 y Fs(\023)657 579 y(P)695 561 y Fl(2)715 579 y Fu(\()p Fr(\000)p Fu(1\))238 b Fs(P)18 b Fr(\012)11 b Fs(P)q(:)p 830 567 209 2 v 997 566 a Fk(-)926 556 y Fs(\023)p 736 532 2 176 v 736 532 a Fk(?)756 456 y Fu(1)g Fr(\012)g Fu([.,.])p 1126 532 V 1126 532 a Fk(?)1146 456 y Fu(1)g Fr(\012)g Fu([.,.])228 648 y(The)h(morphism)d(1)r Fr(\012)r Fu([)p Fs(:;)f(:)p Fu(])13 b(:)g Fs(P)c Fr(\012)r Fs(P)851 629 y Fq(n)875 648 y Fu(\()p Fs(\020)t Fu(\))14 b Fr(\000)-30 b(!)13 b Fs(P)d Fr(\012)r Fs(P)19 b Fu(is)12 b(in)f Fr(C)s Fu(.)20 b(Consider)12 b(the)g(braid-)228 706 y(ing)17 b Fs(\034)22 b Fu(:)16 b Fs(P)j Fr(\012)12 b Fs(P)522 688 y Fq(n)545 706 y Fu(\()p Fs(\020)t Fu(\))k Fr(\000)-30 b(!)16 b Fs(P)737 688 y Fq(n)760 706 y Fu(\()p Fs(\020)t Fu(\))c Fr(\012)g Fs(P)7 b Fu(.)25 b(Since)17 b(it)g(is)g(a)h(natural)g(transformation)228 764 y(the)e(diagram)571 845 y Fs(P)i Fr(\012)11 b Fs(P)708 827 y Fq(n)731 845 y Fu(\()p Fs(\020)t Fu(\))362 b Fs(P)1194 827 y Fq(n)1217 845 y Fu(\()p Fs(\020)t Fu(\))11 b Fr(\012)g Fs(P)p 809 833 333 2 v 1100 832 a Fk(-)962 823 y Fs(\034)486 1089 y(P)19 b Fr(\012)10 b Fu(\()p Fs(P)19 b Fr(\012)10 b Fs(:)e(:)g(:)j Fr(\012)g Fs(P)c Fu(\))192 b(\()p Fs(P)18 b Fr(\012)11 b Fs(:)d(:)g(:)j Fr(\012)f Fs(P)d Fu(\))12 b Fr(\012)f Fs(P)p 893 1077 164 2 v 1015 1076 a Fk(-)959 1057 y Fs(')p 682 1042 2 176 v 683 1042 a Fk(?)p 1267 1042 V 542 w(?)228 1167 y Fu(comm)o(utes)16 b(with)k Fs(')f Fu(=)g Fs(\034)704 1174 y Fq(n)736 1167 y Fs(:)8 b(:)g(:)g(\034)823 1174 y Fl(1)862 1167 y Fu(=)20 b Fs(\034)941 1175 y Fl(\()p Fq(P)q(;P)1015 1166 y Fh(n)1036 1175 y Fl(\))1052 1167 y Fu(,)g(so)g Fs(\034)6 b Fu(\()1195 1134 y Fm(P)1239 1178 y Fq(k)1268 1167 y Fs(z)1291 1174 y Fq(k)q(;)p Fl(1)1353 1167 y Fr(\012)13 b Fu(\()1424 1134 y Fm(P)1476 1167 y Fs(z)1499 1174 y Fq(k)q(;)p Fl(2)1561 1167 y Fr(\012)g Fs(:)8 b(:)g(:)k Fr(\012)228 1225 y Fs(z)251 1232 y Fq(k)q(;n)p Fl(+1)348 1225 y Fu(\)\))i(=)g Fs(\034)473 1232 y Fq(n)505 1225 y Fs(:)8 b(:)g(:)f(\034)591 1232 y Fl(1)611 1225 y Fu(\()630 1192 y Fm(P)674 1236 y Fq(k)703 1225 y Fs(z)726 1232 y Fq(k)q(;)p Fl(1)786 1225 y Fr(\012)k Fs(z)859 1232 y Fq(k)q(;)p Fl(2)918 1225 y Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fs(z)1109 1232 y Fq(k)q(;n)p Fl(+1)1207 1225 y Fu(\).)21 b(Hence)15 b(w)o(e)g(get)587 1308 y Fs(\034)6 b Fu(\(1)11 b Fr(\012)g Fu([)p Fs(:;)d(:)p Fu(]\)\()p Fs(z)r Fu(\))k(=)h(\([)p Fs(:;)8 b(:)p Fu(])i Fr(\012)g Fu(1\))p Fs(\034)1161 1315 y Fq(n)1194 1308 y Fs(:)e(:)g(:)f(\034)1280 1315 y Fl(1)1300 1308 y Fu(\()p Fs(z)r Fu(\))272 b(\(18\))228 1391 y(and)17 b(similarly)587 1474 y Fs(\034)6 b Fu(\([)p Fs(:;)i(:)p Fu(])h Fr(\012)i Fu(1\)\()p Fs(z)r Fu(\))j(=)f(\(1)f Fr(\012)f Fu([)p Fs(:;)d(:)p Fu(]\))p Fs(\034)1164 1481 y Fl(1)1190 1474 y Fs(:)g(:)g(:)f(\034)1276 1481 y Fq(n)1300 1474 y Fu(\()p 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Fs(b)1657 466 y Fq(i)1684 459 y Fu(=)228 517 y Fs(\034)255 499 y Fl(2)249 529 y(1)274 517 y Fu(\()293 483 y Fm(P)345 517 y Fs(a)371 524 y Fq(i)390 517 y Fr(\012)5 b Fs(b)455 524 y Fq(i)469 517 y Fu(\))13 b(=)553 483 y Fm(P)605 517 y Fs(\034)632 499 y Fl(2)626 529 y(1)652 517 y Fu(\()p Fs(a)697 524 y Fq(i)710 517 y Fu(\))5 b Fr(\012)g Fs(b)799 524 y Fq(i)813 517 y Fu(,)14 b(hence)e Fs(\034)1000 499 y Fl(2)994 529 y(1)1020 517 y Fu(\()p Fs(a)1065 524 y Fq(i)1078 517 y Fu(\))i(=)g Fs(a)1189 524 y Fq(i)1216 517 y Fu(and)g Fs(y)h Fr(2)f Fs(P)1432 499 y Fl(2)1452 517 y Fu(\()p Fr(\000)p Fu(1\))5 b Fr(\012)g Fs(P)1640 499 y Fq(n)p Fj(\000)p Fl(1)1709 517 y Fu(.)228 575 y(So)17 b(w)o(e)f(get)g Fs(\034)470 582 y Fq(i)p Fj(\000)p Fl(1)538 575 y Fs(:)8 b(:)g(:)f(\034)624 582 y Fl(1)644 575 y Fs(\034)665 582 y Fq(i)688 575 y Fs(:)h(:)g(:)f(\034)774 582 y Fl(2)794 575 y Fu(\()p Fs(y)r Fu(\))14 b(=)g Fs(\034)945 582 y Fq(i)p Fj(\000)p Fl(1)1012 575 y Fs(:)8 b(:)g(:)g(\034)1099 582 y Fl(1)1119 575 y Fu(\()p Fs(z)r Fu(\))13 b Fr(2)i Fs(P)1281 557 y Fq(i)p Fj(\000)p Fl(1)1351 575 y Fr(\012)c Fs(P)1439 557 y Fl(2)1459 575 y Fu(\()p Fr(\000)p Fu(1\))g Fr(\012)g Fs(P)1659 557 y Fq(n)p Fj(\000)p Fq(i)228 633 y Fu(and)17 b(\(1)11 b Fr(\012)g Fs(:)d(:)g(:)i Fr(\012)h Fu([)p Fs(:;)d(:)p Fu(])h Fr(\012)i Fs(:)d(:)g(:)i Fr(\012)h Fu(1\))p Fs(\034)864 640 y Fq(i)p Fj(\000)p Fl(1)932 633 y Fs(:)d(:)g(:)g(\034)1019 640 y Fl(1)1039 633 y Fu(\()p Fs(z)r Fu(\))13 b Fr(2)h Fs(P)1200 615 y Fq(n)1240 633 y Fu(is)i(de\014ned.)278 691 y(By)e(Lemma)e(8.2)j(w)o(e)f(ha)o(v)o (e)g(\(1)8 b Fr(\012)g Fs(:)g(:)g(:)f Fr(\012)h Fu([)p Fs(:;)g(:)p Fu(])1071 698 y Fl(2)1096 691 y Fr(\012)g Fs(:)g(:)g(:)f Fr(\012)h Fu(1\)\()p Fs(z)r Fu(\))13 b Fr(2)h Fs(P)1459 673 y Fq(n)1483 691 y Fu(\()p Fs(\020)t Fu(\),)g(so)i(that)228 749 y([)p Fs(:;)8 b Fu([)p Fs(:;)g(:)p Fu(])356 756 y Fl(2)373 749 y Fs(;)g(:)p Fu(])423 756 y Fq(n)445 749 y Fs(\034)466 756 y Fq(i)p Fj(\000)p Fl(1)534 749 y Fs(:)g(:)g(:)f(\034)620 756 y Fl(1)640 749 y Fu(\()p Fs(z)r Fu(\))16 b(is)g(w)o(ell)f(de\014ned.)650 b Fi(\003)830 848 y Ft(References)249 927 y FA([1])19 b(George)c(E.)f(Andrews:)21 b Fv(The)15 b(The)n(ory)h(of)f(Partitions.)f FA(Encyclop)q(edia)h(of)f (Mathematics)313 977 y(and)g(its)g(Applications.)e(V)m(ol.2.)g (Addison-W)m(esley)m(,)h(1976.)249 1026 y([2])19 b(Lusztig,)14 b(George:)k Fv(On)d(Quantum)h(Gr)n(oups.)d FA(J.)h(Alg.)f(131)g (\(1990\),)g(466-475)249 1076 y([3])19 b(Ma)r(jid,)12 b(Shahn:)18 b Fv(Br)n(aide)n(d)c(gr)n(oups.)f FA(J.)g(Pure)i(and)e (Appl.)f(Algebra)h(86)g(\(1993\),)f(187-221.)249 1126 y([4])19 b(Susan)f(Mon)o(tgomery:)25 b Fv(Hopf)18 b(A)o(lgebr)n(as)g (and)h(Their)f(A)n(ctions)g(on)h(R)o(ings.)f FA(CBMS)g(82,)313 1176 y(AMS-NSF,)c(1993.)249 1226 y([5])19 b(Bo)q(do)d(P)o(areigis:)21 b Fv(On)16 b(Lie)g(A)o(lgebr)n(as)g(in)g(Br)n(aide)n(d)h(Cate)n (gories.)d FA(T)m(o)h(app)q(ear)h(in)f(Banac)o(h)313 1275 y(Cen)o(ter)g(Publ.)249 1325 y([6])k(Bo)q(do)14 b(P)o(areigis:)k Fv(Symmetric)c(Y)m(etter-Drinfeld)f(Cate)n(gories)i (ar)n(e)f(T)m(rivial.)313 1375 y FA(www.mathematik.)o(uni-m)m(uenc)o (hen.de)e(/p)q(ersonen/professoren.h)o(tml.)249 1425 y([7])19 b(Da)o(vid)14 b(E.)h(Radford:)20 b Fv(The)c(Structur)n(e)f(of) h(Hopf)g(A)o(lgebr)n(as)f(with)h(a)g(Pr)n(oje)n(ction.)e FA(J.)h(Alg.)313 1475 y(92)e(\(1985\),)g(322-)g(347.)249 1525 y([8])19 b(Y)m(orc)o(k)f(Sommerh\177)-21 b(auser:)26 b Fv(Deforme)n(d)18 b(Enveloping)i(A)o(lgebr)n(as.)d FA(New)i(Y)m(ork)f(Journal)g(of)313 1574 y(Mathematics)13 b(2)h(\(1996\),)f(1-23.)249 1624 y([9])19 b(Y)m(etter,)g(Da)o(vid)e (N.:)25 b Fv(Quantum)19 b(Gr)n(oups)f(and)h(R)n(epr)n(esentations)g(of) f(Monoidal)i(Cate-)313 1674 y(gories.)13 b FA(Math.)h(Pro)q(c.)g(Cam)o (b.)e(Phil.)g(So)q(c.)i(108)f(No)h(2)f(\(1990\),)g(261-290.)278 1767 y Fz(Ma)m(thema)m(tisches)25 b(Institut)f(der)h(Universit)1141 1764 y(\177)1140 1767 y(at,)i(Theresienstr.39,)g(80333)228 1817 y(M)270 1814 y(\177)269 1817 y(unchen,)17 b(Germany)278 1867 y Fv(E-mail)d(addr)n(ess)s FA(:)k Fa(pareigis@rz.mathemat)o(ik.u)o (ni-mu)o(enche)o(n.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF