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Fm(\037)p Fr(\()p Fm(g)841 1471 y Fh(\033)q Fg(\()p Fh(l)p Fg(\))902 1464 y Fm(;)g(h)p Fr(\))971 1415 y Ff(\021\020)1110 1422 y(Y)1029 1516 y Fh(l\033)q Fg(\()p Fh(j)r Fg(\))1252 1464 y Fm(\037)p Fr(\()p Fm(h;)g(g)1375 1471 y Fh(\033)q Fg(\()p Fh(l)p Fg(\))1437 1464 y Fr(\))1456 1415 y Ff(\021)1494 1464 y Fr(=)350 1616 y(=)402 1568 y Ff(\020)435 1558 y Fh(\033)q Fg(\()p Fh(j)r Fk(\000)p Fg(1\))p Fk(\000)p Fg(1)486 1575 y Ff(Y)481 1665 y Fh(r)q Fg(=1)598 1616 y Fm(\037)p Fr(\()p Fm(h;)g(g)721 1623 y Fh(r)740 1616 y Fr(\))759 1568 y Ff(\021)784 1616 y Fm(\037)p Fr(\()p Fm(h;)g(g)907 1624 y Fh(\033)q Fg(\()p Fh(j)r Fk(\000)p Fg(1\))1019 1616 y Fr(\))1038 1568 y Ff(\020)1198 1575 y(Y)1071 1669 y Fh(j)r Fk(\000)p Fg(1)p Fh(\033)q Fg(\()p Fh(l)p Fg(\))1386 1616 y Fm(\037)p Fr(\()p Fm(g)1459 1624 y Fh(\033)q Fg(\()p Fh(l)p Fg(\))1520 1616 y Fm(;)g(h)p Fr(\))1589 1568 y Ff(\021)1614 1616 y Fl(\001)399 1731 y(\001)413 1683 y Ff(\020)572 1690 y(Y)446 1784 y Fh(l\033)q Fg(\()p Fh(j)r Fk(\000)p Fg(1\))761 1731 y Fm(\037)p Fr(\()p Fm(h;)g(g)884 1739 y Fh(\033)q Fg(\()p Fh(l)p Fg(\))945 1731 y Fr(\))964 1683 y Ff(\021)0 1881 y Fr(W)l(e)16 b(c)o(hange)g(parameters)f(b)o(y)h Fm(\034)k Fr(=)14 b Fm(\033)688 1863 y Fk(\000)p Fg(1)751 1881 y Fr(with)i Fm(\034)6 b Fr(\()p Fm(p)p Fr(\))11 b(+)g(1)j(=)g Fm(\034)6 b Fr(\()p Fm(q)r Fr(\))15 b(and)i(ha)o(v)o(e)e(to)i(sho)o(w)214 1973 y Ff(\020)247 1964 y Fh(q)q Fk(\000)p Fg(1)252 1979 y Ff(Y)247 2070 y Fh(r)q Fg(=1)325 2021 y Fm(\037)p Fr(\()p Fm(h;)8 b(g)448 2028 y Fh(r)467 2021 y Fr(\))486 1973 y Ff(\021\020)624 1979 y(Y)544 2073 y Fh(q)q(>l;\034)t Fg(\()p Fh(q)q Fg(\))p Fh(<\034)t Fg(\()p Fh(l)p Fg(\))766 2021 y Fm(\037)p Fr(\()p Fm(g)839 2028 y Fh(l)852 2021 y Fm(;)g(h)p Fr(\))921 1973 y Ff(\021\020)1060 1979 y(Y)979 2073 y Fh(l>q)q(;\034)t Fg(\()p Fh(l)p Fg(\))p Fh(<\034)t Fg(\()p Fh(q)q Fg(\))1202 2021 y Fm(\037)p Fr(\()p Fm(h;)g(g)1325 2028 y Fh(l)1338 2021 y Fr(\))1357 1973 y Ff(\021)1395 2021 y Fr(=)351 2161 y(=)403 2113 y 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Fn(Lie)19 b(Algebras)d(and)i(Universal)f(Enveloping)f(Algebras)0 1383 y Ft(De\014nition)i(4.1.)h Fr(An)d(ob)s(ject)g Fm(P)21 b Fr(=)701 1350 y Ff(L)755 1383 y Fm(P)786 1390 y Fh(g)821 1383 y Fl(2)14 b(M)928 1365 y Fh(k)q(G)993 1383 y Fr(together)i(with)g (op)q(erations)522 1485 y([)p Fm(:)8 b(:)g(:)f Fr(])13 b(:)h Fm(P)687 1492 y Fh(g)704 1497 y Fd(1)734 1485 y Fl(\012)d Fm(:)d(:)g(:)i Fl(\012)h Fm(P)933 1492 y Fh(g)950 1496 y Fe(n)988 1485 y Fl(\000)-9 b(!)14 b Fm(P)1113 1492 y Fh(g)1130 1497 y Fd(1)1147 1492 y Fg(+)p Fh(:::)p Fg(+)p Fh(g)1248 1496 y Fe(n)0 1586 y Fr(for)20 b(all)f Fm(n)g Fl(2)h Fi(N)p Fr(,)h(all)e(primitiv)o(e)d Fm(n)p Fr(-th)k(ro)q(ots)h(of)f(unit)o(y)f Fm(\020)t Fr(,)h(and)g(all)f Fm(\020)t Fr(-families)e(\()p Fm(g)1523 1593 y Fg(1)1543 1586 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(g)1684 1593 y Fh(n)1708 1586 y Fr(\),)19 b(is)0 1644 y(called)c(a)i(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Lie)17 b(algebr)n(a)g Fr(if)f(the)g(follo)o(wing)g (iden)o(tities)e(hold:)65 1721 y(\(1\))22 b(for)c(all)h(primitiv)n(e)d Fm(n)p Fr(-th)j(ro)q(ots)h(of)e(unit)o(y)g Fm(\020)t Fr(,)h(all)f Fm(\020)t Fr(-families)f(\()p Fm(g)1322 1728 y Fg(1)1342 1721 y Fm(;)8 b(:)g(:)g(:)15 b(;)8 b(g)1482 1728 y Fh(n)1506 1721 y Fr(\),)19 b(all)f Fm(\033)h Fl(2)f Fm(S)1756 1728 y Fh(n)1780 1721 y Fr(,)149 1779 y(and)e(all)g Fm(x)339 1786 y Fh(i)367 1779 y Fl(2)e Fm(P)445 1786 y Fh(g)462 1791 y Fe(i)461 1881 y Fr([)p Fm(x)503 1888 y Fg(1)522 1881 y Fm(;)8 b(:)g(:)g(:)15 b(;)8 b(x)667 1888 y Fh(n)690 1881 y Fr(])14 b(=)g Fm(\032)p Fr(\()p Fm(\033)o(;)8 b Fr(\()p Fm(g)905 1888 y Fg(1)924 1881 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(g)1065 1888 y Fh(n)1089 1881 y Fr(\)\)[)p Fm(x)1169 1889 y Fh(\033)q Fg(\(1\))1236 1881 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(x)1382 1889 y Fh(\033)q Fg(\()p Fh(n)p Fg(\))1454 1881 y Fr(])p Fm(;)65 1982 y Fr(\(2\))22 b(for)16 b(all)f(primitiv)o(e)e Fm(n)p Fr(-th)j(ro)q(ots)h(of)g(unit)o(y)e Fm(\020)t Fr(,)h(all)f Fm(\020)t Fr(-families)f(\()p Fm(g)1295 1989 y Fg(1)1315 1982 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(g)1456 1989 y Fh(n)p Fg(+1)1524 1982 y Fr(\),)16 b(and)h(for)f(all)149 2040 y Fm(x)177 2047 y Fh(i)204 2040 y Fl(2)e Fm(P)282 2047 y Fh(g)299 2052 y Fe(i)315 2040 y Fr(\()p Fm(H)t Fr(\))348 2127 y Fh(n)p Fg(+1)352 2140 y Ff(X)353 2231 y Fh(i)p Fg(=1)423 2181 y Fm(\032)p Fr(\(\()p Fm(i)8 b(:)g(:)g(:)g Fr(1\))p Fm(;)g Fr(\()p Fm(g)684 2188 y Fg(1)704 2181 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(g)845 2188 y Fh(n)p Fg(+1)913 2181 y Fr(\)\)[)p Fm(x)993 2188 y Fh(i)1007 2181 y Fm(;)g Fr([)p Fm(x)1071 2188 y Fg(1)1090 2181 y Fm(;)g(:)g(:)g(:)15 b(;)c Fr(^)-27 b Fm(x)1235 2188 y Fh(i)1249 2181 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(x)1395 2188 y Fh(n)p Fg(+1)1463 2181 y Fr(]])13 b(=)h(0)p Fm(;)65 2320 y Fr(\(3\))22 b(for)e(all)g(primitiv)o(e)d Fm(n)p Fr(-th)k(ro)q(ots)h(of)f(unit)o(y)e Fm(\020)t Fr(,)j(all)e Fm(\020)t Fr(-families)e(\()p Fm(g)1341 2327 y Fg(1)1361 2320 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(g)1502 2327 y Fh(n)1525 2320 y Fr(\),)21 b(all)f Fm(h)i Fl(2)f Fm(G)149 2378 y Fr(suc)o(h)16 b(that)g(all)g(\()p Fm(h;)8 b(g)524 2385 y Fh(i)538 2378 y Fr(\))17 b(are)f(\()p Fl(\000)p Fr(1\)-families,)d(all)j Fm(y)1055 2385 y Fh(i)1083 2378 y Fl(2)e Fm(P)1161 2385 y Fh(g)1178 2390 y Fe(i)1193 2378 y Fr(,)i(and)h(all)f Fm(x)d Fl(2)h Fm(P)1505 2385 y Fh(h)331 2520 y Fr([)p Fm(x;)8 b Fr([)p Fm(y)433 2527 y Fg(1)451 2520 y Fm(;)g(y)497 2527 y Fg(2)516 2520 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(y)658 2527 y Fh(n)681 2520 y Fr(]])13 b(=)793 2466 y Fh(n)774 2478 y Ff(X)775 2569 y Fh(i)p Fg(=1)842 2472 y Ff(\020)877 2466 y Fh(i)p Fk(\000)p Fg(1)879 2478 y Ff(Y)875 2569 y Fh(j)r Fg(=1)945 2520 y Fm(\037)p Fr(\()p Fm(g)1018 2527 y Fh(j)1036 2520 y Fm(;)8 b(h)p Fr(\))1105 2472 y Ff(\021)1130 2520 y Fr([)p Fm(y)1168 2527 y Fg(1)1187 2520 y Fm(;)g(:)g(:)g(:)16 b(;)8 b Fr([)p Fm(x;)g(y)1393 2527 y Fh(i)1405 2520 y Fr(])p Fm(;)g(:)g(:)g(:)16 b(;)8 b(y)1561 2527 y Fh(n)1584 2520 y Fr(])p Fm(:)p eop %%Page: 14 14 14 13 bop 0 92 a Fs(14)683 b(BODO)18 b(P)m(AREIGIS)50 200 y Fr(The)13 b(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)k(algebras)i(form)e (a)h(category)g(in)g(a)g(straigh)o(tforw)o(ard)h(w)o(a)o(y)f(and)g(the) g(construc-)0 258 y(tion)20 b(giv)o(en)f(in)h(section)g(3)h(de\014nes)f (a)g(functor)h Fm(P)27 b Fr(:)20 b(\()p Fm(G;)8 b(\037)p Fr(\)-Hopf)24 b Fl(\000)-8 b(!)20 b Fr(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)20 b(as)h(can)f(b)q(e)0 316 y(easily)15 b(v)o(eri\014ed.)50 375 y(The)k(theorems)f(3.2,)i(2.5,)g(and)f(2.6)h(sho)o(w)g(that)f(this) g(notion)h(of)g(generalized)e(Lie)h(algebras)0 433 y(passes)e(its)f (test)g(on)h(the)f(set)g(of)h(primitiv)n(e)c(elemen)o(ts)h(of)i(a)h (Hopf)f(algebra:)0 522 y Ft(Corollary)j(4.2.)h Fj(L)n(et)e Fm(H)23 b Fj(b)n(e)c(a)g Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Hopf)19 b(algebr)n(a)h(in)f Fl(M)1165 504 y Fh(k)q(G)1214 522 y Fj(.)26 b(Then)20 b(the)f(set)g(of)g(primitive)0 580 y(elements)h Fm(P)7 b Fr(\()p Fm(H)t Fr(\))17 b Fj(forms)g(a)g Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Lie)18 b(algebr)n(a.)0 669 y Ft(Lemma)e(4.3.)j Fj(L)n(et)g Fm(A)f Fj(b)n(e)h(a)g Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-algebr)n(a)19 b(in)g Fl(M)985 650 y Fh(k)q(G)1034 669 y Fj(.)26 b(Then)20 b Fm(A)e Fj(c)n(arries)g(the)h(structur)n(e)g(of)f(a)0 727 y Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Lie)18 b(algebr)n(a)g Fm(A)434 709 y Fh(L)477 727 y Fj(with)g(the)f(op)n(er)n(ations)346 817 y Fr([)p Fm(x)388 824 y Fg(1)407 817 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(x)553 824 y Fh(n)576 817 y Fr(])13 b(:=)682 776 y Ff(X)669 868 y Fh(\033)q Fk(2)p Fh(S)735 872 y Fe(n)765 817 y Fm(\032)p Fr(\()p Fm(\033)o(;)8 b Fr(\()p Fm(g)900 824 y Fg(1)919 817 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(g)1060 824 y Fh(n)1084 817 y Fr(\)\))p Fm(x)1150 825 y Fh(\033)q Fg(\(1\))1229 817 y Fl(\001)j Fm(:)d(:)g(:)i Fl(\001)h Fm(x)1375 825 y Fh(\033)q Fg(\()p Fh(n)p Fg(\))0 946 y Fj(for)17 b(al)r(l)i(\(r)n(o)n(ots)c(of)j(unity\))f Fm(\020)h Fl(2)c Fm(k)602 928 y Fk(\003)622 946 y Fj(,)k(al)r(l)g Fm(\020)t Fj(-families)h Fr(\()p Fm(g)991 953 y Fg(1)1011 946 y Fm(;)8 b(:)g(:)g(:)15 b(;)8 b(g)1151 953 y Fh(n)1175 946 y Fr(\))p Fj(,)17 b(and)h(al)r(l)h Fm(x)1419 953 y Fh(i)1446 946 y Fl(2)14 b Fm(A)1530 953 y Fh(g)1547 958 y Fe(i)1562 946 y Fj(.)0 1034 y(Pr)n(o)n(of.)19 b Fr(This)d(is)g(a)h(rephrasing)g(of)f(theorems)f(2.5)i(and)g(2.6.)p 1151 1007 33 2 v 1151 1036 2 30 v 1182 1036 V 1151 1038 33 2 v 50 1121 a(This)f(lemma)d(de\014nes)j(a)h(functor)f(\()p Fm(G;)8 b(\037)p Fr(\)-)q(Alg)14 b Fl(3)g Fm(A)f Fl(7!)h Fm(A)1121 1103 y Fh(L)1160 1121 y Fl(2)g Fr(\()p Fm(G;)8 b(\037)p Fr(\)-Lie.)0 1210 y Ft(Theorem)17 b(4.4.)i Fj(L)n(et)g Fm(P)25 b Fj(b)n(e)19 b(a)f Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Lie)19 b(algebr)n(a.)25 b(Then)19 b(ther)n(e)g(is)f(a)g (universal)i(asso)n(ciative)0 1268 y(enveloping)g(algebr)n(a)e Fm(U)5 b Fr(\()p Fm(P)i Fr(\))18 b Fj(in)g Fl(M)656 1250 y Fh(k)677 1268 y Fm(G)p Fj(.)0 1357 y(Pr)n(o)n(of.)h Fr(W)l(e)26 b(de\014ne)f Fm(U)5 b Fr(\()p Fm(P)i Fr(\))30 b(:=)g Fm(T)7 b Fr(\()p Fm(P)g Fr(\))p Fm(=I)29 b Fr(where)c Fm(T)7 b Fr(\()p Fm(P)g Fr(\))26 b(is)f(the)h(tensor)g(algebra)g(and)g Fm(I)34 b Fr(:=)0 1415 y(\([)p Fm(x)61 1422 y Fg(1)80 1415 y Fm(;)8 b(:)g(:)g(:)16 b(;)8 b(x)226 1422 y Fh(n)249 1415 y Fr(])k Fl(\000)326 1381 y Ff(P)370 1425 y Fh(\033)q Fk(2)p Fh(S)436 1429 y Fe(n)468 1415 y Fm(\032)p Fr(\()p Fm(\033)o(;)c Fr(\()p Fm(g)603 1422 y Fg(1)622 1415 y Fm(;)g(:)g(:)g(:)16 b(;)8 b(g)763 1422 y Fh(n)787 1415 y Fr(\)\))p Fm(x)853 1422 y Fh(\033)q Fg(\(1\))933 1415 y Fl(\001)13 b Fm(:)8 b(:)g(:)j Fl(\001)h Fm(x)1083 1422 y Fh(\033)q Fg(\()p Fh(n)p Fg(\))1155 1415 y Fr(\))19 b(is)f(the)g(ideal)f(generated)h(b)o(y)g(all)0 1473 y(terms)10 b(formed)g(for)i(all)f(\(ro)q(ots)i(of)e(unit)o(y\))g Fm(\020)18 b Fl(2)c Fm(k)866 1455 y Fk(\003)886 1473 y Fr(,)e(all)f Fm(\020)t Fr(-families)e(\()p Fm(g)1231 1480 y Fg(1)1251 1473 y Fm(;)f(:)g(:)g(:)16 b(;)8 b(g)1392 1480 y Fh(n)1416 1473 y Fr(\),)j(and)i(all)e Fm(x)1642 1480 y Fh(i)1669 1473 y Fl(2)j Fm(P)1747 1480 y Fh(g)1764 1485 y Fe(i)1780 1473 y Fr(.)50 1531 y(The)21 b(tensor)i(algebra)f(is)f (constructed)h(in)f Fl(M)929 1513 y Fh(k)q(G)999 1531 y Fr(with)h(the)f(natural)h(grading.)39 b(Since)21 b(the)0 1590 y(relations)16 b(are)g(homogeneous,)g(the)g(algebra)h Fm(U)5 b Fr(\()p Fm(P)i Fr(\))17 b(is)f(also)h(in)e Fl(M)1240 1572 y Fh(k)q(G)1289 1590 y Fr(.)50 1648 y(By)23 b(the)g(univ)o(ersal)g (prop)q(ert)o(y)g(of)h(the)f(tensor)h(algebra)g(an)o(y)g(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)23 b(homomorphism)0 1706 y Fm(f)e Fr(:)15 b Fm(P)23 b Fl(\000)-8 b(!)15 b Fm(A)261 1688 y Fh(L)304 1706 y Fr(extends)i(uniquely)f(to)i(a)f(\()p Fm(G;)8 b(\037)p Fr(\)-algebra)18 b(homomorphism)d Fm(g)j Fr(:)d Fm(U)5 b Fr(\()p Fm(P)i Fr(\))16 b Fl(\000)-8 b(!)15 b Fm(A)p Fr(,)0 1764 y(so)i(that)g Fm(U)5 b Fr(\()p Fm(P)i Fr(\))16 b(is)g(the)g(univ)o(ersal)g(asso)q(ciativ)o(e)g (algebra)g(generated)h(b)o(y)e Fm(P)7 b Fr(.)p 1440 1737 V 1440 1767 2 30 v 1471 1767 V 1440 1769 33 2 v 0 1851 a Ft(Corollary)19 b(4.5.)h Fj(The)d(functor)g(-)644 1833 y Fh(L)684 1851 y Fr(:)c(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-)p Fr(Alg)14 b Fl(\000)-8 b(!)13 b Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-)q Fr(Lie)16 b Fj(has)h(the)h(left)g(adjoint)f (func-)0 1910 y(tor)g Fm(U)i Fr(:)14 b(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-)p Fr(Lie)13 b Fl(\000)-8 b(!)14 b Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-)p Fr(Alg)p Fj(.)50 1998 y Fr(Observ)o(e)16 b(that)h(the)f(iden)o(tities)f(for)i(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)17 b(algebras)g(pla)o(y)g(no)g(sp)q(ecial)f (role)h(in)f(the)h(con-)0 2056 y(struction)h(of)h(the)g(univ)o(ersal)e (en)o(v)o(eloping)g(algebra.)28 b(Its)19 b(construction)f(dep)q(ends)h (only)g(on)g(the)0 2114 y(giv)o(en)h(op)q(erations,)j(whic)o(h)d(m)o (ust)f(b)q(e)i(homogeneous)g(so)h(that)f(the)g(residue)f(class)h (algebra)g(is)0 2173 y Fm(G)p Fr(-graded)d(again.)50 2231 y(If)23 b(w)o(e)h(consider)g(the)g(set)g Fm(P)7 b Fr(\()p Fm(H)t Fr(\))24 b(of)g(primitiv)o(e)d(elemen)o(ts)g(of)k(an)f (arbitrary)g(\()p Fm(G;)8 b(\037)p Fr(\)-Hopf)0 2289 y(algebra)20 b Fm(H)k Fr(then)19 b(there)g(are)h(man)o(y)e(homogeneous) i(partially)e(de\014ned)i(common)d(op)q(erations)0 2347 y(in)g(all)f Fm(P)7 b Fr(\()p Fm(H)t Fr(\).)24 b(W)l(e)17 b(used)g(the)f(most)h(basic)g(ones)g(in)g(our)g(generalization)f(of)i (Lie)e(algebras.)25 b(W)l(e)0 2406 y(w)o(ould)14 b(lik)o(e)e(to)i(kno)o (w)g(if)f(the)h(brac)o(k)o(et-op)q(erations)g(and)g(iden)o(tities)e (giv)o(en)h(in)h(the)f(de\014nition)h(of)g(a)0 2464 y(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)13 b(algebra)h(are)f(a)h(generating)f(set)g (for)h(all)e(p)q(ossible)i(op)q(erations)g(and)g(relations)f(for)h(the) 0 2522 y Fm(G)p Fr(-graded)h(mo)q(dule)d Fm(P)7 b Fr(\()p Fm(H)t Fr(\))14 b(for)g(ev)o(ery)d(Hopf)j(algebra)g(in)f Fl(M)1111 2504 y Fh(k)q(G)1159 2522 y Fr(,)h(ev)o(ery)e(ab)q(elian)h (group)i Fm(G)p Fr(,)f(ev)o(ery)0 2580 y(bic)o(haracter)e Fm(\037)h Fr(and)h(ev)o(ery)d(in)o(tegral)i(domain)f Fm(k)r Fr(.)20 b(This)14 b(should)f(b)q(e)h(true)f(for)g(the)g(iden)o (tities)e(if)i(all)p eop %%Page: 15 15 15 14 bop 409 92 a Fs(ON)18 b(LIE)g(ALGEBRAS)g(IN)g(BRAIDED)g(CA)m (TEGORIES)367 b(15)0 200 y Fr(maps)17 b(from)f(\()p Fm(G;)8 b(\037)p Fr(\)-Lie)17 b(algebras)h(in)o(to)f(their)f(univ)o(ersal)g(en) o(v)o(eloping)g(algebras)i(are)f(injectiv)o(e.)0 258 y(T)l(o)h(pro)o(v)o(e)f(this,)g(some)f(kind)h(of)h(generalization)f(of) h(the)f(P)o(oincar)o(\023)-23 b(e-Birkho\013-Witt)17 b(theorem)e(is)0 316 y(needed.)50 378 y(No)o(w)g(w)o(e)g(sho)o(w)h(ho)o (w)f(to)h(construct)f(Hopf)h(algebras)g(in)f(the)g(braided)g(category)g (of)h Fm(G)p Fr(-graded)0 436 y(v)o(ector)f(spaces)i(using)g(univ)o (ersal)e(en)o(v)o(eloping)g(algebras.)0 551 y Ft(Theorem)i(4.6.)i Fj(L)n(et)d Fm(P)23 b Fj(b)n(e)17 b(a)e Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Lie)17 b(algebr)n(a.)22 b(Then)17 b(the)f(universal)h(enveloping)i(algebr)n(a)0 609 y Fm(U)5 b Fr(\()p Fm(P)i Fr(\))18 b Fj(is)f(a)h Fr(\()p Fm(G;)8 b(\037)p Fr(\))p Fj(-Hopf)17 b(algebr)n(a.)0 725 y(Pr)n(o)n(of.)i Fr(W)l(e)g(de\014ne)f(a)i(homomorphism)c Fm(\016)k Fr(:)e Fm(P)26 b Fl(\000)-9 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