%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ln4.dvi %%CreationDate: Mon Jun 26 16:17:40 2000 %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: DVIPS16 -a ln4 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.06.26:1541 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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80041A1A7F991D>I<137F3801C1C038070070000E7F487F003C131E0038130E0078130F 00707F00F01480A80078EB0F00A20038130E003C131E001C131C6C5B6C5B3801C1C0D800 7FC7FC191A7E991E>II82 DI<00 7FB5FC38701C0700401301A200C0148000801300A300001400B13803FFE0191A7F991C> I<39FFE07FC0390E000E001404B200065B12076C5B6C6C5A3800E0C0013FC7FC1A1A7F99 1D>I<39FF801FC0391C00070014066C1304A36C5BA26C6C5AA36C6C5AA26C6C5AA3EB70 80A213790139C7FCA2131EA3130CA21A1A7F991D>I<39FF801FE0391E00070014066C13 046C130CEB800800035BEA01C06D5A00001360EB7040EB78801338011DC7FC131F130EAA EBFFC01B1A7F991D>89 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin letter %%EndSetup %%Page: 116 1 116 0 bop 871 2683 a Fr(116)p eop %%Page: 117 2 117 1 bop 753 395 a Fq(CHAPTER)24 b(4)532 545 y Fp(The)e (In\014nitesimal)j(Theory)459 686 y Fo(1.)j(In)n(tegrals)18 b(and)h(F)-5 b(ourier)18 b(T)-5 b(ransforms)75 773 y Fq(Assume)14 b(for)j(this)f(c)o(hapter)g(that)72 b(is)16 b(a)g(\014eld.)75 858 y Fo(Lemm)o(a)g(4.1.1.)23 b Fm(L)n(et)18 b Fl(C)i Fm(b)n(e)e(a)g(\014nite)g(dimensional)h(c)n(o)n(algebr)n(a.)j (Every)c(right)f Fl(C)t Fm(-c)n(omo)n(dule)0 916 y Fl(M)i Fm(is)13 b(a)g(left)h Fl(C)270 898 y Fk(\003)290 916 y Fm(-mo)n(dule)g(by)f Fl(c)549 898 y Fk(\003)569 916 y Fl(m)g Fq(=)677 878 y Fj(P)738 916 y Fl(m)781 924 y Fi(\()p Fh(M)t Fi(\))848 916 y Fg(h)p Fl(c)888 898 y Fk(\003)908 916 y Fl(;)8 b(m)973 924 y Fi(\(1\))1019 916 y Fg(i)14 b Fm(and)f(c)n(onversely)i(by)e Fl(\016)r Fq(\()p Fl(m)p Fq(\))g(=)1601 878 y Fj(P)1654 930 y Fh(i)1676 916 y Fl(c)1697 898 y Fk(\003)1697 928 y Fh(i)1717 916 y Fl(m)r Fg(\012)0 974 y Fl(c)21 981 y Fh(i)53 974 y Fm(wher)n(e)190 937 y Fj(P)251 974 y Fl(c)272 956 y Fk(\003)272 986 y Fh(i)303 974 y Fg(\012)e Fl(c)374 981 y Fh(i)405 974 y Fm(is)17 b(the)h(dual)g(b)n(asis.)75 1059 y Ff(Pr)o(oof.)h Fq(W)l(e)d(c)o(hec)o(k)f(that)h Fl(M)22 b Fq(b)q(ecomes)15 b(a)i(left)e Fl(C)1016 1041 y Fk(\003)1035 1059 y Fq(-mo)q(dule)257 1124 y(\()p Fl(c)297 1106 y Fk(\003)317 1124 y Fl(c)338 1106 y Fk(0)349 1100 y(\003)369 1124 y Fq(\))p Fl(m)h Fq(=)499 1086 y Fj(P)560 1124 y Fl(m)603 1132 y Fi(\()p Fh(M)t Fi(\))669 1124 y Fg(h)p Fl(c)709 1106 y Fk(\003)729 1124 y Fl(c)750 1106 y Fk(0)762 1100 y(\003)782 1124 y Fl(;)8 b(m)847 1132 y Fi(\(1\))893 1124 y Fg(i)15 b Fq(=)978 1086 y Fj(P)1039 1124 y Fl(m)1082 1132 y Fi(\()p Fh(M)t Fi(\))1149 1124 y Fg(h)p Fl(c)1189 1106 y Fk(\003)1209 1124 y Fl(;)8 b(m)1274 1132 y Fi(\(1\))1321 1124 y Fg(ih)p Fl(c)1380 1106 y Fk(0)1392 1100 y(\003)1412 1124 y Fl(;)g(m)1477 1132 y Fi(\(2\))1523 1124 y Fg(i)447 1182 y Fq(=)14 b Fl(c)520 1164 y Fk(\003)548 1145 y Fj(P)609 1182 y Fl(m)652 1190 y Fi(\()p Fh(M)t Fi(\))718 1182 y Fg(h)p Fl(c)758 1164 y Fk(0)770 1158 y(\003)790 1182 y Fl(;)8 b(m)855 1190 y Fi(\(1\))902 1182 y Fg(i)14 b Fq(=)g Fl(c)1008 1164 y Fk(\003)1027 1182 y Fq(\()p Fl(c)1067 1164 y Fk(0)1079 1158 y(\003)1099 1182 y Fl(m)p Fq(\))p Fl(:)0 1258 y Fq(It)g(is)h(easy)f(to)h(c)o(hec)o(k)e(that)i(the)f(t)o(w)o(o)h 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Fi(\(2\))1267 2597 y Fg(i)p Fl(:)p eop %%Page: 129 14 129 13 bop 474 118 a Fr(1.)17 b(INTEGRALS)f(AND)h(F)o(OURIER)g(TRANSF)o (ORMS)403 b(129)0 213 y Fm(Then)18 b(the)g(fol)r(lowing)i(tr)n (ansformation)c(rule)i(holds)g(for)f Fl(f)s(;)8 b(g)15 b Fg(2)f Fl(H)t Fm(:)785 296 y Fj(f)782 309 y Fl(f)5 b(g)17 b Fq(=)911 296 y Fj(e)903 309 y Fl(f)f Fg(\003)11 b Fj(e)-28 b Fl(g)r(:)-1018 b Fq(\(23\))0 399 y Fm(In)14 b(p)n(articular)g Fl(H)324 381 y Fk(\003)357 399 y Fm(with)h(the)f(c)n (onvolution)i(multiplic)n(ation)f(is)f(an)g(asso)n(ciative)g(algebr)n (a)h(with)f(unit)4 450 y Fj(f)0 461 y Fq(1)24 468 y Fh(H)72 461 y Fq(=)124 421 y Fj(R)157 461 y Fm(,)j(i.e.)709 512 y Fj(R)753 552 y Fg(\003)11 b Fl(a)j Fq(=)f Fl(a)e Fg(\003)953 512 y Fj(R)1000 552 y Fq(=)j Fl(a:)-1092 b Fq(\(24\))75 646 y Ff(Pr)o(oof.)19 b Fq(Giv)o(en)c Fl(f)s(;)8 b(g)r(;)g(h)14 b Fg(2)g Fl(H)637 628 y Fk(\003)657 646 y Fq(.)21 b(Then)183 742 y Fg(h)204 729 y Fj(f)202 742 y Fl(f)5 b(g)s(;)j(h)p Fg(i)17 b Fq(=)d Fg(h)414 702 y 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Fq(One)j(of)h(the)f(most)g(imp)q (ortan)o(t)f(form)o(ulas)h(for)h(F)l(ourier)f(transforms)g(is)g(the)g (Planc)o(herel)f(for-)0 1611 y(m)o(ula)g(on)h(the)g(in)o(v)m(ariance)g (of)g(the)g(inner)g(pro)q(duct)h(under)f(F)l(ourier)g(transforms.)21 b(W)l(e)16 b(ha)o(v)o(e)75 1705 y Fo(Theorem)g(4.1.29.)23 b Fm(\(The)18 b(Plancher)n(el)h(formula\))712 1802 y Fg(h)p Fl(a;)8 b(f)d Fg(i)15 b Fq(=)e Fg(h)922 1789 y Fj(e)912 1802 y Fl(f)6 b(;)i(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)p Fl(:)-1088 b Fq(\(25\))75 1899 y Ff(Pr)o(oof.)19 b Fq(First)d(w)o(e)g(ha)o(v)o(e)f(from)g(\(16\))96 1997 y Fg(h)p Fl(a;)8 b(f)d Fg(i)17 b Fq(=)280 1959 y Fj(P)333 1997 y Fg(h)352 1956 y Fj(R)385 2014 y Fi(\(1\))432 1997 y Fl(;)7 b Fj(e)-27 b Fl(a)p Fg(ih)518 1956 y Fj(R)552 2014 y Fi(\(2\))599 1997 y Fl(;)8 b(S)654 1978 y Fk(\000)p Fi(1)701 1997 y Fq(\()p Fl(\016)742 2004 y Fi(\(1\))789 1997 y Fq(\))p Fg(ih)855 1983 y Fj(e)846 1997 y Fl(f)e(;)i(\016)920 2004 y Fi(\(2\))966 1997 y Fg(i)14 b Fq(=)1051 1959 y 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b(de\014ne)g Fl(d)p Fq(\()p Fl(X)295 2295 y Fi(1)323 2288 y Fl(:)8 b(:)g(:)g(X)429 2295 y Fh(n)453 2288 y Fq(\))14 b(:=)551 2251 y Fj(P)604 2264 y Fh(n)604 2302 y(i)p Fi(=1)671 2288 y Fl(X)711 2295 y Fi(1)740 2288 y Fl(:)8 b(:)g(:)f(X)845 2295 y Fh(i)p Fk(\000)p Fi(1)914 2288 y Fg(\012)h Fl(dX)1026 2295 y Fh(i)1050 2288 y Fg(\012)h Fl(X)1138 2295 y Fh(i)p Fi(+1)1206 2288 y Fl(:)f(:)g(:)f(X)1311 2295 y Fh(n)1351 2288 y Fq(in)15 b(particular)f Fl(d)p Fq(\()p Fl(X)1714 2295 y Fh(i)1729 2288 y Fq(\))g(=)0 2346 y(1)6 b Fg(\012)g Fl(dX)140 2353 y Fh(i)160 2346 y Fg(\012)g Fq(1.)21 b(T)l(o)14 b(see)f(that)h Fl(d)g Fq(is)g(a)g(deriv)m(ation)f(it)g(su\016ces)h(to)g (sho)o(w)g(this)g(on)g(the)f(basis)h(elemen)o(ts:)199 2419 y Fl(d)p Fq(\()p Fl(X)283 2426 y Fi(1)312 2419 y Fl(:)8 b(:)g(:)g(X)418 2426 y Fh(k)439 2419 y Fl(X)479 2426 y Fh(k)q Fi(+1)555 2419 y Fl(:)g(:)g(:)f(X)660 2426 y Fh(n)684 2419 y Fq(\))439 2484 y(=)491 2446 y Fj(P)544 2459 y Fh(k)544 2498 y(j)r Fi(=1)615 2484 y Fl(X)655 2491 y Fi(1)684 2484 y Fl(:)h(:)g(:)f(X)789 2491 y Fh(j)r Fk(\000)p Fi(1)864 2484 y Fg(\012)k Fl(dX)979 2491 y Fh(j)1009 2484 y Fg(\012)g Fl(X)1099 2491 y Fh(j)r Fi(+1)1171 2484 y Fl(:)d(:)g(:)g(X)1277 2491 y Fh(k)1299 2484 y Fl(X)1339 2491 y Fh(k)q Fi(+1)1414 2484 y Fl(:)g(:)g(:)f(X)1519 2491 y Fh(n)456 2545 y Fq(+)502 2508 y Fj(P)554 2521 y Fh(n)554 2560 y(j)r Fi(=)p Fh(k)q Fi(+1)673 2545 y Fl(X)713 2552 y Fi(1)741 2545 y Fl(:)h(:)g(:)g(X)847 2552 y Fh(k)869 2545 y Fl(X)909 2552 y Fh(k)q Fi(+1)984 2545 y Fl(:)g(:)g(:)f(X)1089 2552 y Fh(j)r Fk(\000)p Fi(1)1164 2545 y Fg(\012)k Fl(dX)1279 2552 y Fh(j)1309 2545 y Fg(\012)g Fl(X)1399 2552 y Fh(j)r Fi(+1)1471 2545 y Fl(:)d(:)g(:)g(X)1577 2552 y Fh(n)439 2607 y Fq(=)14 b Fl(d)p Fq(\()p Fl(X)575 2614 y Fi(1)604 2607 y Fl(:)8 b(:)g(:)g(X)710 2614 y Fh(k)731 2607 y Fq(\))p Fl(X)790 2614 y Fh(k)q Fi(+1)866 2607 y Fl(:)g(:)g(:)f(X)971 2614 y Fh(n)1006 2607 y Fq(+)k Fl(X)1095 2614 y Fi(1)1124 2607 y Fl(:)d(:)g(:)f(X)1229 2614 y Fh(k)1251 2607 y Fl(d)p Fq(\()p Fl(X)1335 2614 y Fh(k)q Fi(+1)1410 2607 y Fl(:)h(:)g(:)g(X)1516 2614 y Fh(n)1540 2607 y Fq(\))p eop %%Page: 132 17 132 16 bop 0 118 a Fr(132)534 b(4.)17 b(THE)f(INFINITESIMAL)g(THEOR)m (Y)0 213 y Fq(No)o(w)i(let)g Fl(D)h Fq(:)e Fl(B)i Fg(\000)-30 b(!)17 b Fl(M)24 b Fq(b)q(e)18 b(a)h(deriv)m(ation.)27 b(De\014ne)18 b Fl(')g Fq(b)o(y)g Fl(')p Fq(\(1)13 b Fg(\012)f Fl(dX)1321 220 y Fh(i)1349 213 y Fg(\012)g Fq(1\))17 b(:=)g Fl(D)q Fq(\()p Fl(X)1629 220 y Fh(i)1645 213 y Fq(\).)27 b(This)0 271 y(map)14 b(ob)o(viously)g(extends)g(to)h (a)g(homomorphism)d(of)j Fl(B)s Fq(-)p Fl(B)s Fq(-bimo)q(dules.)k(F)l (urthermore)13 b(w)o(e)h(ha)o(v)o(e)106 352 y Fl('d)p Fq(\()p Fl(X)222 359 y Fi(1)251 352 y Fl(:)8 b(:)g(:)f(X)356 359 y Fh(n)380 352 y Fq(\))17 b(=)c Fl(')p Fq(\()518 315 y Fj(P)571 367 y Fh(j)597 352 y Fl(X)637 359 y Fi(1)666 352 y Fl(:)8 b(:)g(:)f(X)771 359 y Fh(j)r Fk(\000)p Fi(1)846 352 y Fg(\012)k Fl(dX)961 359 y Fh(j)991 352 y Fg(\012)g Fl(X)1081 359 y Fh(j)r Fi(+1)1153 352 y Fl(:)d(:)g(:)g(X)1259 359 y Fh(n)1283 352 y Fq(\))416 414 y(=)467 377 y Fj(P)520 429 y Fh(j)546 414 y Fl(X)586 421 y Fi(1)615 414 y Fl(:)g(:)g(:)g(X)721 421 y Fh(j)r Fk(\000)p Fi(1)784 414 y Fl(')p Fq(\(1)k Fg(\012)f Fl(dX)986 421 y Fh(j)1016 414 y Fg(\012)g Fq(1\))p Fl(X)1149 421 y Fh(j)r Fi(+1)1221 414 y Fl(:)d(:)g(:)g(X)1327 421 y Fh(n)1364 414 y Fq(=)14 b Fl(D)q Fq(\()p Fl(X)1516 421 y Fi(1)1545 414 y Fl(:)8 b(:)g(:)g(X)1651 421 y Fh(n)1675 414 y Fq(\))0 501 y(hence)15 b Fl('d)g Fq(=)e Fl(D)q Fq(.)75 559 y(T)l(o)h(sho)o(w)h(the)f(uniqueness)g(of)h Fl(')f Fq(let)f Fl( )i Fq(:)f(\012)867 566 y Fh(B)911 559 y Fg(\000)-30 b(!)14 b Fl(M)19 b Fq(b)q(e)c(a)f(bimo)q(dule)f (homomorphism)e(suc)o(h)0 617 y(that)18 b Fl( )r(d)f Fq(=)f Fl(D)q Fq(.)27 b(Then)18 b Fl( )r Fq(\(1)12 b Fg(\012)g Fl(dX)653 624 y Fh(i)680 617 y Fg(\012)g Fq(1\))17 b(=)f Fl( )r(d)p Fq(\()p Fl(X)963 624 y Fh(i)978 617 y Fq(\))g(=)h Fl(D)q Fq(\()p Fl(X)1168 624 y Fh(i)1183 617 y Fq(\))f(=)h Fl(')p Fq(\(1)12 b Fg(\012)g Fl(dX)1476 624 y Fh(i)1503 617 y Fg(\012)g Fq(1\).)27 b(Since)17 b Fl( )0 675 y Fq(and)g Fl(')f Fq(are)g Fl(B)s Fq(-)p Fl(B)s Fq(-bimo)q(dules)f(homomorphisms)f(this)i(extends)f(to)i Fl( )e Fq(=)f Fl(')p Fq(.)75 733 y(b\))19 b(No)o(w)g(let)f Fl(A)g Fq(:=)57 b Fg(h)p Fl(X)550 740 y Fh(i)564 733 y Fg(j)p Fl(i)18 b Fg(2)h Fl(J)5 b Fg(i)p Fl(=I)23 b Fq(b)q(e)c(an)g(arbitrary)g(algebra)h(with)e Fl(B)j Fq(=)58 b Fg(h)p Fl(X)1634 740 y Fh(i)1648 733 y Fg(j)p Fl(i)18 b Fg(2)h Fl(J)5 b Fg(i)0 792 y Fq(free.)20 b(De\014ne)418 857 y(\012)453 864 y Fh(A)496 857 y Fq(:=)13 b(\012)596 864 y Fh(B)627 857 y Fl(=)p Fq(\()p Fl(I)t Fq(\012)731 864 y Fh(B)772 857 y Fq(+)e(\012)856 864 y Fh(B)887 857 y Fl(I)j Fq(+)d Fl(B)s(d)1037 864 y Fh(B)1067 857 y Fq(\()p Fl(I)t Fq(\))g(+)g Fl(d)1216 864 y Fh(B)1246 857 y Fq(\()p Fl(I)t Fq(\))p Fl(B)s Fq(\))p Fl(:)0 932 y Fq(W)l(e)g(\014rst)h(sho)o (w)g(that)g Fl(I)t Fq(\012)453 939 y Fh(B)485 932 y Fq(+)r(\012)560 939 y Fh(B)590 932 y Fl(I)5 b Fq(+)r Fl(B)s(d)722 939 y Fh(B)752 932 y Fq(\()p Fl(I)t Fq(\))r(+)r Fl(d)883 939 y Fh(B)912 932 y Fq(\()p Fl(I)t Fq(\))p Fl(B)13 b Fq(is)e(a)h Fl(B)s Fq(-)p Fl(B)s Fq(-subbimo)q(dule.)19 b(Since)10 b(\012)1679 939 y Fh(B)1721 932 y Fq(and)0 990 y Fl(I)21 b Fq(are)d Fl(B)s Fq(-)p Fl(B)s Fq(-bimo)q(dules)e(the)i (terms)e Fl(I)t Fq(\012)755 997 y Fh(B)803 990 y Fq(and)i(\012)934 997 y Fh(B)965 990 y Fl(I)j Fq(are)d(bimo)q(dules.)24 b(F)l(urthermore)16 b(w)o(e)i(ha)o(v)o(e)0 1048 y Fl(bd)46 1055 y Fh(B)76 1048 y Fq(\()p Fl(i)p Fq(\))p Fl(b)152 1030 y Fk(0)177 1048 y Fq(=)c Fl(bd)275 1055 y Fh(B)305 1048 y Fq(\()p Fl(ib)362 1030 y Fk(0)373 1048 y Fq(\))c Fg(\000)f Fl(bid)513 1055 y Fh(B)543 1048 y Fq(\()p Fl(b)583 1030 y Fk(0)594 1048 y Fq(\))14 b Fg(2)g Fl(B)s(d)739 1055 y Fh(B)769 1048 y Fq(\()p Fl(I)t Fq(\))9 b(+)g Fl(I)t Fq(\012)950 1055 y Fh(B)996 1048 y Fq(hence)15 b Fl(I)t Fq(\012)1192 1055 y Fh(B)1231 1048 y Fq(+)9 b(\012)1313 1055 y Fh(B)1344 1048 y Fl(I)k Fq(+)c Fl(B)s(d)1491 1055 y Fh(B)1521 1048 y Fq(\()p Fl(I)t Fq(\))g(+)g Fl(d)1666 1055 y Fh(B)1697 1048 y Fq(\()p Fl(I)t Fq(\))p Fl(B)0 1106 y Fq(is)16 b(a)h(bimo)q(dule.)75 1164 y(No)o(w)g Fl(I)t Fq(\012)248 1171 y Fh(B)296 1164 y Fq(and)h(\012)427 1171 y Fh(B)458 1164 y Fl(I)j Fq(are)c(subbimo)q(dules)g(of)h Fl(I)t Fq(\012)1008 1171 y Fh(B)1050 1164 y Fq(+)12 b(\012)1135 1171 y Fh(B)1165 1164 y Fl(I)k Fq(+)c Fl(B)s(d)1318 1171 y Fh(B)1348 1164 y Fq(\()p Fl(I)t Fq(\))f(+)h Fl(d)1498 1171 y Fh(B)1529 1164 y Fq(\()p Fl(I)t Fq(\))p Fl(B)s Fq(.)24 b(Hence)0 1222 y Fl(A)13 b Fq(=)h Fl(B)s(=I)20 b Fq(acts)c(on)h(b)q(oth)g(sides)f(on)h(\012)711 1229 y Fh(A)756 1222 y Fq(so)g(that)g(\012)957 1229 y Fh(A)1002 1222 y Fq(b)q(ecomes)e(an)h Fl(A)p Fq(-)p Fl(A)p Fq(-bimo)q(dule.)75 1280 y(Let)k Fl(\027)j Fq(:)c(\012)281 1287 y Fh(B)332 1280 y Fg(\000)-30 b(!)19 b Fq(\012)445 1287 y Fh(A)494 1280 y Fq(and)h(also)h Fl(\027)i Fq(:)c Fl(B)k Fg(\000)-30 b(!)19 b Fl(A)h Fq(b)q(e)g(the)f(residue)h(homomorphisms.)29 b(Since)0 1338 y Fl(\027)s(d)52 1345 y Fh(B)83 1338 y Fq(\()p Fl(i)p Fq(\))13 b Fg(2)h Fl(\027)s(d)250 1345 y Fh(B)281 1338 y Fq(\()p Fl(I)t Fq(\))f(=)h(0)g Fg(\022)g Fq(\012)536 1345 y Fh(A)579 1338 y Fq(w)o(e)f(get)h(a)g(unique)f (factorization)h(map)f Fl(d)1333 1345 y Fh(A)1375 1338 y Fq(:)h Fl(A)f Fg(\000)-30 b(!)13 b Fq(\012)1560 1345 y Fh(A)1603 1338 y Fq(suc)o(h)h(that)784 1629 y Fl(A)145 b Fq(\012)1001 1636 y Fh(A)p 835 1616 117 2 v 910 1615 a Fa(-)872 1648 y Fh(d)890 1654 y Fc(A)783 1434 y Fl(B)g Fq(\012)1000 1441 y Fh(B)p 837 1421 114 2 v 909 1420 a Fa(-)871 1405 y Fh(d)889 1411 y Fc(B)p 802 1581 2 127 v 803 1581 a Fa(?)763 1525 y Fh(\027)p 997 1581 V 997 1581 a Fa(?)1017 1525 y Fh(\027)0 1732 y Fq(comm)o(utes.)18 b(Since)e Fl(d)401 1739 y Fh(A)429 1732 y Fq(\()p 448 1691 21 2 v Fl(b)p Fq(\))e(=)p 554 1689 115 2 v 14 w Fl(d)579 1739 y Fh(B)609 1732 y Fq(\()p Fl(b)p Fq(\))i(it)g(is)g(clear) g(that)g Fl(d)1029 1739 y Fh(A)1074 1732 y Fq(is)g(a)h(deriv)m(ation.) 75 1790 y(Let)12 b Fl(D)k Fq(:)d Fl(A)h Fg(\000)-30 b(!)13 b Fl(M)18 b Fq(b)q(e)13 b(a)g(deriv)m(ation.)19 b(The)13 b Fl(A)p Fq(-)p Fl(A)p Fq(-bimo)q(dule)e Fl(M)18 b Fq(is)12 b(also)h(a)g Fl(B)s Fq(-)p Fl(B)s Fq(-bimo)q(dule)e(b)o(y)0 1848 y Fl(bm)i Fq(=)p 129 1808 21 2 v 14 w Fl(bm)p Fq(.)21 b(F)l(urthermore)14 b Fl(D)q(\027)k Fq(:)13 b Fl(B)k Fg(\000)-31 b(!)14 b Fl(A)g Fg(\000)-31 b(!)14 b Fl(M)22 b Fq(is)15 b(again)j(a)e(deriv)m(ation.)21 b(Let)16 b Fl(')1524 1855 y Fh(B)1568 1848 y Fq(:)e(\012)1631 1855 y Fh(B)1675 1848 y Fg(\000)-30 b(!)14 b Fl(M)0 1907 y Fq(b)q(e)22 b(the)g(unique)f(factorization)h(map)f(for)h(the)g Fl(B)s Fq(-deriv)m(ation)g Fl(D)q(\027)s Fq(.)39 b(Consider)23 b(the)e(follo)o(wing)0 1965 y(diagram)747 2037 y Fl(B)169 b Fq(\012)988 2044 y Fh(B)p 801 2024 138 2 v 897 2023 a Fa(-)847 2008 y Fh(d)865 2014 y Fc(B)748 2232 y Fl(A)145 b Fq(\012)965 2239 y Fh(A)p 799 2219 117 2 v 874 2218 a Fa(-)836 2203 y Fh(d)854 2209 y Fc(A)p 766 2184 2 127 v 767 2184 a Fa(?)p 961 2184 V 153 w(?)p 961 2379 V 962 2379 a(?)918 2324 y Fh( )p 1010 2379 2 322 v 1010 2379 a Fa(?)1030 2222 y Fh(')963 2430 y Fl(M)804 2327 y Fh(D)801 2294 y Fa(@)842 2336 y(@)884 2377 y(@)886 2379 y(@)-42 b(R)0 2489 y Fq(W)l(e)21 b(w)o(an)o(t)h(to)f(construct)h Fl( )h Fq(suc)o(h)e(that)h(the)f(diagram)g(comm)o(utes.)34 b(Let)22 b Fl(i!)i Fg(2)f Fl(I)t Fq(\012)1608 2496 y Fh(B)1638 2489 y Fq(.)37 b(Then)0 2547 y Fl(')p Fq(\()p Fl(i!)r Fq(\))16 b(=)p 189 2508 17 2 v 16 w Fl(i')p Fq(\()p Fl(!)r Fq(\))g(=)g(0)i(and)h(similarly)c Fl(')p Fq(\()p Fl(!)r(i)p Fq(\))g(=)i(0.)26 b(Let)17 b Fl(bd)1105 2554 y Fh(B)1136 2547 y Fq(\()p Fl(i)p Fq(\))f Fg(2)g Fl(B)s(d)1321 2554 y Fh(B)1351 2547 y Fq(\()p Fl(I)t Fq(\))h(then)h Fl(')p Fq(\()p Fl(bd)1642 2554 y Fh(B)1672 2547 y Fq(\()p Fl(i)p Fq(\)\))e(=)p 0 2567 21 2 v 0 2608 a Fl(b'd)78 2615 y Fh(B)108 2608 y Fq(\()p Fl(i)p Fq(\))e(=)p 229 2567 V 14 w Fl(b)o(D)q Fq(\()p 309 2569 17 2 v Fl(i)q Fq(\))g(=)f(0)j(and)f(similarly)e Fl(')p Fq(\()p Fl(d)819 2615 y Fh(B)849 2608 y Fq(\()p Fl(i)p 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Fh(H)1137 211 y Fc(o)1170 213 y Fg(\012)14 b Fq(id)1289 220 y Fi(\()p Fh(\016)q Fi(\))1335 213 y Fq(\)\()p Fl(f)5 b Fq(\))22 b(=)g Fl(f)d Fg(\012)1611 220 y Fi(\()p Fh(\016)q Fi(\))1671 213 y Fl(f)26 b Fq(and)0 271 y(\()p Fl(")42 278 y Fh(H)74 269 y Fc(o)104 271 y Fg(\012)11 b Fq(id)220 278 y Fi(\()p Fh(\016)q Fi(\))266 271 y Fq(\)\()p Fl(f)5 b Fq(\))15 b(=)e(1)k(i\013)f Fl(f)j Fg(2)14 b Fl(G)671 278 y Fi(\()p Fh(\016)q Fi(\))718 271 y Fq(\()p Fl(H)781 253 y Fh(o)812 271 y Fg(\012)49 b Fq(\()p Fl(\016)r Fq(\)\).)75 329 y(Hence)16 b(w)o(e)h(ha)o(v)o(e)g(a)h(bijectiv)o(e)d(map)i Fl(!)h Fq(:)54 b(-)p Fo(cAlg)p Fq(\()p Fl(H)q(;)47 b Fq(\()p Fl(\016)r Fq(\)\))16 b Fg(3)g Fl(f)21 b Fq(=)16 b Fl(f)1388 336 y Fi(0)1420 329 y Fq(+)c Fl(f)1494 336 y Fi(1)1514 329 y Fl(\016)17 b Fg(7!)f Fl(f)1643 336 y Fi(0)1675 329 y Fg(\012)c Fq(1)g(+)0 387 y Fl(f)24 394 y Fi(1)58 387 y Fg(\012)h Fl(\016)22 b Fg(2)f Fl(G)272 395 y Fi(\()p Fh(\016)q Fi(\))319 387 y Fq(\()p Fl(H)382 369 y Fh(o)415 387 y 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33 v 75 2351 a Fo(Corollary)18 b(4.4.5.)23 b Fm(L)n(et)13 b Fl(H)18 b Fm(b)n(e)13 b(a)h(Hopf)f(algebr)n(a)h(that)g (is)f(\014nitely)i(gener)n(ate)n(d)f(a)f(s)h(an)f(algebr)n(a.)0 2409 y(Then)18 b Fo(Lie)o Fq(\()p Fl(H)265 2391 y Fh(o)284 2409 y Fq(\))g Fm(is)f(\014nite)i(dimensional.)75 2492 y Ff(Pr)o(oof.)g Fq(Let)i Fl(H)j Fq(=)60 b Fg(h)p Fl(a)566 2499 y Fi(1)586 2492 y Fl(;)8 b(:)g(:)g(:)15 b(;)8 b(a)729 2499 y Fh(n)752 2492 y Fg(i)p Fq(.)34 b(Since)19 b Fl(H)25 b Fq(=)73 b Fg(\010)14 b Fl(I)23 b Fq(w)o(e)d(can)h(c)o(ho)q(ose)f Fl(a)1578 2499 y Fi(1)1618 2492 y Fq(=)h(1)f(and)0 2550 y Fl(a)26 2557 y Fi(2)45 2550 y Fl(;)8 b(:)g(:)g(:)16 b(;)8 b(a)189 2557 y Fh(n)228 2550 y Fg(2)17 b Fl(I)t Fq(.)25 b(Th)o(us)18 b(an)o(y)g(elemen)o(t)d(in)i Fl(i)f Fg(2)h Fl(I)k Fq(can)d(b)q(e)g(written)f(as)1321 2512 y Fj(P)1382 2550 y Fl(\013)1413 2557 y Fh(J)1437 2550 y Fl(a)1463 2557 y Fh(j)1477 2562 y Fb(1)1505 2550 y Fl(:)8 b(:)g(:)f(a)1596 2557 y Fh(j)1610 2563 y Fc(k)1649 2550 y Fq(so)19 b(that)0 2608 y Fl(I)t(=I)76 2590 y Fi(2)109 2608 y Fq(=)p 200 2580 46 2 v 53 w Fl(a)226 2615 y Fi(2)256 2608 y Fq(+)11 b Fl(:)d(:)g(:)i Fq(+)p 422 2580 50 2 v 11 w Fl(a)448 2615 y Fh(n)471 2608 y Fq(.)21 b(This)c(giv)o(es)e(the) h(result.)p 1765 2608 2 33 v 1767 2577 30 2 v 1767 2608 V 1796 2608 2 33 v eop %%Page: 143 28 143 27 bop 207 118 a Fr(4.)17 b(DERIV)l(A)m(TIONS)f(AND)g(LIE)h (ALGEBRAS)g(OF)g(AFFINE)f(ALGEBRAIC)h(GR)o(OUPS)136 b(143)75 213 y Fo(Prop)r(osition)17 b(4.4.6.)24 b Fm(L)n(et)e Fl(H)28 b Fm(b)n(e)23 b(a)g(c)n(ommutative)h(Hopf)f(algebr)n(a)g(and) 1489 220 y Fh(H)1522 213 y Fl(M)29 b Fm(b)n(e)23 b(an)g Fl(H)t Fm(-)0 271 y(mo)n(dule.)29 b(Then)20 b(we)g(have)g Fq(\012)544 278 y Fh(H)595 257 y Fg(\030)596 273 y Fq(=)652 271 y Fl(H)d Fg(\012)12 b Fl(I)t(=I)836 253 y Fi(2)874 271 y Fm(and)20 b Fl(d)e Fq(:)f Fl(H)22 b Fg(\000)-29 b(!)17 b Fl(H)g Fg(\012)12 b Fl(I)t(=I)1368 253 y Fi(2)1407 271 y Fm(is)19 b(given)i(by)f Fl(d)p Fq(\()p Fl(a)p Fq(\))d(=)0 297 y Fj(P)61 335 y Fl(a)87 342 y Fi(\(1\))145 335 y Fg(\012)p 195 291 260 2 v 11 w Fq(\(id)7 b Fg(\000)p Fl(")p Fq(\)\()p Fl(a)388 342 y Fi(\(2\))435 335 y Fq(\))p Fm(.)75 422 y Ff(Pr)o(oof.)19 b Fq(Consider)k(the)g(algebra)h Fl(B)k Fq(:=)c Fl(H)c Fg(\010)c Fl(M)28 b Fq(with)23 b(\()p Fl(a;)8 b(m)p Fq(\)\()p Fl(a)1369 404 y Fk(0)1379 422 y Fl(;)g(m)1444 404 y Fk(0)1455 422 y Fq(\))26 b(=)f(\()p Fl(aa)1634 404 y Fk(0)1645 422 y Fl(;)8 b(am)1736 404 y Fk(0)1762 422 y Fq(+)0 480 y Fl(a)26 462 y Fk(0)37 480 y Fl(m)p Fq(\).)31 b(Let)19 b Fg(G)j Fq(=)59 b(-)p Fo(cAlg)o Fq(\()p Fl(H)q(;)8 b Fq(-)q(\).)31 b(Then)20 b(w)o(e)f(ha)o(v)o(e)f Fg(G)s Fq(\()p Fl(B)s Fq(\))h Fg(\022)g Fq(Hom)n(\()p Fl(H)q(;)8 b(B)s Fq(\))1443 466 y Fg(\030)1443 482 y Fq(=)1501 480 y(Hom)n(\()p Fl(H)q(;)g(H)t Fq(\))14 b Fg(\010)0 538 y Fq(Hom)o(\()p Fl(H)q(;)8 b(M)d Fq(\).)21 b(An)14 b(elemen)o(t)d(\()p Fl(';)d(D)q Fq(\))15 b Fg(2)f Fq(Hom)n(\()p Fl(H)q(;)8 b(B)s Fq(\))14 b(is)h(in)f Fg(G)s Fq(\()p Fl(B)s Fq(\))f(i\013)h(\()p Fl(';)8 b(D)q Fq(\)\(1\))15 b(=)f(\()p Fl(')p Fq(\(1\))p Fl(;)8 b(D)q Fq(\(1\)\))0 596 y(=)33 b(\(1)p Fl(;)8 b Fq(0\),)30 b(hence)c Fl(')p Fq(\(1\))33 b(=)g(1)28 b(and)g Fl(D)q Fq(\(1\))33 b(=)g(0,)d(and)e(\()p Fl(')p Fq(\()p Fl(ab)p Fq(\))p Fl(;)8 b(D)q Fq(\()p Fl(ab)p Fq(\)\))32 b(=)h(\()p Fl(';)8 b(D)q Fq(\)\()p Fl(ab)p Fq(\))32 b(=)0 654 y(\()p Fl(';)8 b(D)q Fq(\)\()p Fl(a)p Fq(\)\()p Fl(';)g(D)q Fq(\)\()p Fl(b)p Fq(\))30 b(=)f(\()p Fl(')p Fq(\()p Fl(a)p Fq(\))p Fl(;)8 b(D)q Fq(\()p Fl(a)p Fq(\)\)\()p Fl(')p Fq(\()p Fl(b)p Fq(\))p Fl(;)g(D)q Fq(\()p Fl(b)p Fq(\)\))29 b(=)h(\()p Fl(')p Fq(\()p Fl(a)p Fq(\))p Fl(')p Fq(\()p Fl(b)p Fq(\))p Fl(;)8 b(')p Fq(\()p Fl(a)p Fq(\))p Fl(D)q Fq(\()p Fl(b)p Fq(\))16 b(+)h Fl(D)q Fq(\()p Fl(a)p Fq(\))p Fl(')p Fq(\()p Fl(b)p Fq(\),)0 712 y(hence)f Fl(')p Fq(\()p Fl(ab)p Fq(\))f(=)g Fl(')p Fq(\()p Fl(a)p Fq(\))p Fl(')p Fq(\()p Fl(b)p Fq(\))h(and)h Fl(D)q Fq(\()p Fl(ab)p Fq(\))f(=)f Fl(')p Fq(\()p Fl(a)p Fq(\))p Fl(D)q Fq(\()p Fl(b)p Fq(\))c(+)h Fl(D)q Fq(\()p Fl(a)p Fq(\))p Fl(')p Fq(\()p Fl(b)p Fq(\).)23 b(So)18 b(\()p Fl(';)8 b(D)q Fq(\))17 b(is)g(in)g Fg(G)s Fq(\()p Fl(B)s Fq(\))f(i\013)0 771 y Fl(')i Fg(2)g(G)s Fq(\()p Fl(H)t Fq(\))g(and)h Fl(D)h Fq(is)f(a)g Fl(')p Fq(-deriv)m(ation.)27 b(The)19 b Fg(\003)p Fq(-m)o(ultiplic)o(ation)d (in)j(Hom)n(\()p Fl(H)q(;)8 b(B)s Fq(\))19 b(is)f(giv)o(en)g(b)o(y)0 829 y(\()p Fl(';)8 b(D)q Fq(\))14 b Fg(\003)f Fq(\()p Fl(')236 811 y Fk(0)247 829 y Fl(;)8 b(D)310 811 y Fk(0)322 829 y Fq(\))19 b(=)g(\()p Fl(')13 b Fg(\003)g Fl(')551 811 y Fk(0)562 829 y Fl(;)8 b(')13 b Fg(\003)g Fl(D)708 811 y Fk(0)734 829 y Fq(+)g Fl(D)h Fg(\003)f Fl(')909 811 y Fk(0)921 829 y Fq(\))19 b(b)o(y)g(applying)g(this)g(to)g(an)h (elemen)o(t)c Fl(a)i Fg(2)h Fl(H)t Fq(.)0 887 y(Since)j(\()p Fl(';)8 b Fq(0\))25 b Fg(2)h(G)s Fq(\()p Fl(B)s Fq(\))c(and)h(\()p Fl(u";)8 b(D)q Fq(\))26 b Fg(2)f(G)s Fq(\()p Fl(B)s Fq(\))d(for)i(ev)o (ery)d Fl(")p Fq(-deriv)m(ation)i Fl(D)q Fq(,)i(there)d(is)h(a)g(bi-)0 945 y(jection)f(Der)243 952 y Fh(")261 945 y Fq(\()p Fl(H)q(;)8 b(M)d Fq(\))439 931 y Fg(\030)440 947 y Fq(=)502 945 y Fg(f)p Fq(\()p Fl(u";)j(D)659 952 y Fh(")677 945 y Fq(\))25 b Fg(2)f(G)808 952 y Fh(")826 945 y Fq(\()p Fl(B)s Fq(\))p Fg(g)953 931 y(\030)954 947 y Fq(=)1016 945 y Fg(f)p Fq(\(1)1084 952 y Fh(H)1118 945 y Fl(;)8 b(D)1180 952 y Fi(1)1200 945 y Fq(\))25 b Fg(2)f(G)1331 952 y Fi(1)1351 945 y Fq(\()p Fl(B)s Fq(\))p Fg(g)1477 931 y(\030)1478 947 y Fq(=)1540 945 y(Der)k(\()p Fl(H)q(;)8 b(M)d Fq(\))0 1003 y(b)o(y)20 b(\()p Fl(u";)8 b(D)204 1010 y Fh(")222 1003 y Fq(\))22 b Fg(7!)f Fq(\(1)p Fl(;)8 b Fq(0\))15 b Fg(\003)f Fq(\()p Fl(u";)8 b(D)628 1010 y Fh(")646 1003 y Fq(\))21 b(=)h(\(1)p Fl(;)8 b Fq(1)15 b Fg(\003)f Fl(D)929 1010 y Fh(")947 1003 y Fq(\))22 b Fg(2)f(G)1072 1010 y Fi(1)1092 1003 y Fq(\()p Fl(B)s Fq(\))f(with)h(in)o(v)o(erse)e(map)g(\(1)p Fl(;)8 b(D)1689 1010 y Fi(1)1710 1003 y Fq(\))21 b Fg(7!)0 1061 y Fq(\()p Fl(S;)8 b Fq(0\))k Fg(\003)f Fq(\(1)p Fl(;)d(D)267 1068 y Fi(1)288 1061 y Fq(\))14 b(=)h(\()p Fl(u";)8 b(S)14 b Fg(\003)d Fl(D)586 1068 y Fi(1)607 1061 y Fq(\))j Fg(2)h(G)718 1068 y Fh(")736 1061 y Fq(\()p Fl(B)s Fq(\).)23 b(Hence)16 b(w)o(e)g(ha)o(v)o(e)g(isomorphisms)f(Der)28 b(\()p Fl(H)q(;)8 b(M)d Fq(\))1761 1047 y Fg(\030)1762 1063 y Fq(=)0 1119 y(Der)78 1126 y Fh(")96 1119 y Fq(\()p Fl(H)q(;)j(M)d Fq(\))264 1105 y Fg(\030)264 1121 y Fq(=)316 1119 y(Hom)o(\()p Fl(I)t(=I)513 1101 y Fi(2)532 1119 y Fl(;)j(M)d Fq(\))639 1105 y Fg(\030)639 1121 y Fq(=)692 1119 y(Hom)793 1126 y Fh(H)827 1119 y Fq(\()p Fl(H)15 b Fg(\012)c Fl(I)t(=I)1027 1101 y Fi(2)1046 1119 y Fl(;)d(M)d Fq(\).)75 1180 y(The)18 b(univ)o(ersal)f Fl(")p Fq(-deriv)m(ation)h(for)h(v)o(ector)e(spaces)i (is)p 1083 1140 111 2 v 18 w(id)8 b Fg(\000)p Fl(")17 b Fq(:)g Fl(A)g Fg(\000)-31 b(!)17 b Fl(I)t(=I)1447 1162 y Fi(2)1466 1180 y Fq(.)28 b(The)18 b(univ)o(ersal)0 1244 y Fl(")p Fq(-deriv)m(ation)j(for)h Fl(H)t Fq(-mo)q(dules)f(is)h Fl(D)705 1251 y Fh(")723 1244 y Fq(\()p Fl(a)p Fq(\))h(=)g(1)15 b Fg(\012)p 963 1201 212 2 v 14 w Fq(\(id)8 b Fg(\000)p Fl(")p Fq(\)\()p Fl(a)p Fq(\))22 b Fg(2)h Fl(A)14 b Fg(\012)g Fl(I)t(=I)1434 1226 y Fi(2)1453 1244 y Fq(.)38 b(The)21 b(univ)o(ersal)0 1308 y(1-deriv)m(ation)i(for)g Fl(H)t Fq(-mo)q(dules)g(is)f(1)16 b Fg(\003)f Fl(D)791 1315 y Fh(")833 1308 y Fq(with)23 b(\(1)16 b Fg(\003)f Fl(D)1090 1315 y Fh(")1109 1308 y Fq(\)\()p Fl(a)p Fq(\))24 b(=)1279 1271 y Fj(P)1340 1308 y Fl(a)1366 1316 y Fi(\(1\))1428 1308 y Fg(\012)p 1483 1265 260 2 v 16 w Fq(\(id)8 b Fg(\000)p Fl(")p Fq(\)\()p Fl(a)1677 1316 y Fi(\(2\))1723 1308 y Fq(\))25 b Fg(2)0 1368 y Fl(A)11 b Fg(\012)f Fl(I)t(=I)173 1350 y Fi(2)192 1368 y Fq(.)p 1765 1368 2 33 v 1767 1337 30 2 v 1767 1368 V 1796 1368 2 33 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF