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1976 y(Pr)n(o)n(of.)19 b Fq(The)e(monomorphism)d Fo(H)19 b Fk(3)14 b Fo(f)20 b Fk(7!)15 b Fq(\()829 1936 y Fj(R)877 1976 y Fo(\))f(f)5 b Fq(\))15 b Fk(2)g Fo(H)1095 1958 y Fm(\003)1132 1976 y Fq(is)h(comp)q(osed)g(of)h Fo(S)h Fq(:)c Fo(H)19 b Fk(!)14 b Fo(H)21 b Fq(and)0 2035 y Fo(H)d Fk(3)c Fo(f)19 b Fk(7!)14 b Fq(\()p Fo(f)19 b(*)337 1995 y Fj(R)370 2035 y Fq(\))14 b Fk(2)g Fo(H)494 2017 y Fm(\003)514 2035 y Fq(.)p 1765 2035 V 1767 2004 30 2 v 1767 2035 V 1796 2035 2 33 v 50 2118 a(W)l(e)20 b(call)f(a)i (generalized)d(in)o(tegral)i Fo(a)g Fk(2)h Fo(H)857 2100 y Fm(\003)897 2118 y Fq(a)f Ff(r)n(ational)h(inte)n(gr)n(al)f Fq(if)g Fo(a)g Fq(is)g(of)g(the)g(form)f Fo(a)h Fq(=)0 2139 y Fj(P)61 2136 y(R)84 2194 y Fl(i)112 2177 y Fo(\))14 b(f)199 2184 y Fl(i)213 2177 y Fq(.)0 2260 y Ft(Corollary)k(4.)24 b Ff(F)l(or)17 b(every)h(r)n(ational)f(inte)n(gr)n(al)h Fo(a)13 b Fk(2)h Fo(H)1022 2242 y Fm(\003)1060 2260 y Ff(ther)n(e)j(is)g(a)h(unique)h Fo(g)d Fk(2)e Fo(H)21 b Ff(such)d(that)626 2362 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y Fq(One)h(of)h(the)g(\014rst)f(to)h(study)g(this)g(prop)q(ert)o(y)f (of)h(the)g(in)o(tegral)1224 172 y Fj(R)1276 213 y Fq(to)g(represen)o (t)f(other)h(linear)0 271 y(functionals)d(w)o(as)h(J.)f(v.)f(Neumann)g (in)h([4].)50 329 y(If)k Fo(H)k Fq(is)c(\014nite)g(dimensional)f(then)h (the)g(isomorphism)e(In)o(t)1177 336 y Fl(l)1190 329 y Fq(\()p Fo(H)1253 311 y Fm(\003)1273 329 y Fq(\))13 b Fk(\012)h Fo(H)25 b Fk(3)1477 289 y Fj(R)1519 329 y Fk(\012)p Fo(f)h Fk(7!)20 b Fq(\()1697 289 y Fj(R)1751 329 y Fo(\))0 390 y(f)5 b Fq(\))14 b Fk(2)g Fo(H)153 371 y Fm(\003)173 369 y Fl(r)q(at)224 390 y Fq(\()p Fk(\022)f Fo(H)339 371 y Fm(\003)359 390 y Fq(\))j(sho)o(ws)h(In)o(t)597 397 y Fl(l)610 390 y Fq(\()p Fo(H)673 371 y Fm(\003)693 390 y Fq(\))f(has)h(dimension)e(1.)50 448 y(W)l(e)i(c)o(ho)q(ose)h(for) f(the)h(rest)f(of)h(this)f(pap)q(er)h(a)g(non)g(zero)f(left)f(in)o(v)m (arian)o(t)h(in)o(tegral)1552 408 y Fj(R)1602 448 y Fq(whenev)o(er)0 506 y(w)o(e)f(are)g(in)g(the)g(situation)h(of)f Fo(H)21 b Fq(\014nite)15 b(dimensional.)50 564 y(Let)e Fo(H)18 b Fq(b)q(e)13 b(\014nite)g(dimensional.)19 b(Since)788 524 y Fj(R)830 564 y Fo(a)13 b Fq(is)g(a)h(left)e(in)o(v)m(arian)o(t)h (in)o(tegral)g(and)h(dim)n(\(In)o(t)1666 571 y Fl(l)1679 564 y Fq(\()p Fo(H)1742 546 y Fm(\003)1762 564 y Fq(\)\))0 622 y(=)g(1)i(there)g(is)g(a)h(unique)e(mo)q(d\()p Fo(a)p Fq(\))e Fk(2)h Fn(K)23 b Fq(with)736 666 y Fj(R)769 706 y Fo(a)14 b Fq(=)f(mo)q(d\()p Fo(a)p Fq(\))1018 666 y Fj(R)1051 706 y Fo(:)0 791 y Fq(One)k(c)o(hec)o(ks)g(that)h(mo)q(d)e(:) g Fo(H)548 773 y Fm(\003)584 791 y Fk(!)g Fn(K)25 b Fq(is)17 b(an)h(algebra)g(homomorphism)d(called)i(the)g Ff(mo)n(dulus)i(of)0 849 y Fo(H)44 831 y Fm(\003)64 849 y Fq(.)31 b(If)19 b(mo)q(d)f(=)i Fo(")f Fq(=)g(1)454 856 y Fl(H)486 847 y Fe(\003)525 849 y Fq(then)h Fo(H)684 831 y Fm(\003)723 849 y Fq(is)f(called)g Ff(unimo)n(dular)p Fq(.)30 b(This)20 b(is)f(equiv)m(alen)o(t)f(to)1666 809 y Fj(R)1718 849 y Fq(also)0 907 y(b)q(eing)e(righ)o(t)g(in)o(v)m(arian)o(t)g(or)g(In)o (t)576 914 y Fl(l)589 907 y Fq(\()p Fo(H)652 889 y Fm(\003)672 907 y Fq(\))e(=)f(In)o(t)819 914 y Fl(r)838 907 y Fq(\()p Fo(H)901 889 y Fm(\003)921 907 y Fq(\).)0 998 y Ft(Corollary)18 b(5.)24 b Ff(If)e Fo(H)27 b Ff(is)c(\014nite)h(dimensional)f(then)h (for)e(every)h Fo(a)g Fk(2)h Fo(H)1386 980 y Fm(\003)1429 998 y Ff(ther)n(e)e(is)h(a)g(unique)0 1056 y Fo(g)16 b Fk(2)e Fo(H)22 b Ff(such)c(that)f Fk(h)p Fo(a;)8 b(f)d Fk(i)15 b Fq(=)539 1016 y Fj(R)581 1056 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(S)s Fq(\()p Fo(g)r Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)17 b Ff(for)g(al)r(l)i Fo(f)g Fk(2)14 b Fo(H)t Ff(.)0 1148 y Ft(Corollary)k(6.)24 b Ff(If)18 b Fo(H)k Ff(is)d(\014nite)g(dimensional)h(then)f Fo(S)f Fq(:)d Fo(H)20 b Fk(!)15 b Fo(H)22 b Ff(and)d Fo(H)h Fk(3)15 b Fo(f)21 b Fk(7!)15 b Fq(\()p Fo(f)21 b(*)1699 1108 y Fj(R)1732 1148 y Fq(\))16 b Fk(2)0 1206 y Fo(H)44 1188 y Fm(\003)81 1206 y Ff(ar)n(e)h(bije)n(ctive.)50 1296 y Fq(If)e Fo(G)h Fq(is)g(a)g(\014nite)f(group)i(then)e(ev)o(ery)f (generalized)h(in)o(tegral)g Fo(a)e Fk(2)h Fn(K)1292 1278 y Fl(G)1340 1296 y Fq(can)i(b)q(e)g(written)f(with)h(a)0 1355 y(uniquely)f(determined)f Fo(g)i Fk(2)e Fo(H)20 b Fq(as)436 1462 y Fk(h)p Fo(a;)8 b(f)d Fk(i)14 b Fq(=)617 1395 y Fj(Z)675 1462 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(S)s Fq(\()p Fo(g)r Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)14 b Fq(=)1051 1415 y Fj(X)1051 1521 y Fl(x)p Fm(2)p Fl(G)1131 1462 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(g)r Fq(\()p Fo(x)1298 1442 y Fm(\000)p Fi(1)1345 1462 y Fq(\))-1364 b(\(11\))0 1593 y(for)17 b(all)e Fo(f)k Fk(2)c Fo(H)t Fq(.)50 1651 y(If)g Fo(G)i Fq(is)f(a)h(\014nite)e(Ab)q(elian)h(group)h (then)f(eac)o(h)f(group)j(elemen)o(t)13 b(\(rational)j(in)o(tegral\))f Fo(y)h Fk(2)e Fo(G)g Fk(\022)0 1709 y Fn(K)t Fo(G)19 b Fq(can)e(b)q(e)f(written)g(as)651 1773 y Fo(y)f Fq(=)742 1726 y Fj(X)743 1832 y Fl(x)p Fm(2)p Fl(G)823 1726 y Fj(X)822 1840 y Fl(\037)p Fm(2)876 1832 y Fi(^)868 1840 y Fl(G)904 1773 y Fo(\014)932 1780 y Fl(\037)955 1773 y Fk(h)p Fo(x)1002 1753 y Fm(\000)p Fi(1)1049 1773 y Fo(;)8 b(\037)p Fk(i)p Fo(x)0 1912 y Fq(since)20 b Fk(h)p Fo(y)r(;)8 b(f)d Fk(i)21 b Fq(=)g Fk(h)p Fq(\()357 1872 y Fj(R)412 1912 y Fo(\))482 1875 y Fj(P)534 1927 y Fl(\037)p Fm(2)588 1918 y Fi(^)580 1927 y Fl(G)618 1912 y Fo(\014)646 1919 y Fl(\037)669 1912 y Fo(\037)p Fq(\))p Fo(;)8 b(f)d Fk(i)21 b Fq(=)g Fk(h)888 1872 y Fj(R)922 1912 y Fo(;)8 b(f)d(S)s Fq(\()1025 1875 y Fj(P)1077 1927 y Fl(\037)p Fm(2)1131 1918 y Fi(^)1123 1927 y Fl(G)1161 1912 y Fo(\014)1189 1919 y Fl(\037)1212 1912 y Fo(\037)p Fq(\))p Fk(i)21 b Fq(=)1361 1875 y Fj(P)1414 1927 y Fl(x)p Fm(2)p Fl(G)1487 1912 y Fk(h)p Fo(x;)8 b(f)d Fk(i)1612 1875 y Fj(P)1665 1927 y Fl(\037)p Fm(2)1718 1918 y Fi(^)1711 1927 y Fl(G)1748 1912 y Fo(\014)1776 1919 y Fl(\037)0 1977 y Fk(h)p Fo(x;)j(S)s Fq(\()p Fo(\037)p Fq(\))p Fk(i)23 b Fq(=)f Fk(h)292 1940 y Fj(P)345 1991 y Fl(x)p Fm(2)p Fl(G)427 1940 y Fj(P)479 1991 y Fl(\037)p Fm(2)533 1983 y Fi(^)525 1991 y Fl(G)563 1977 y Fo(\014)591 1984 y Fl(\037)614 1977 y Fk(h)p Fo(x)661 1959 y Fm(\000)p Fi(1)708 1977 y Fo(;)8 b(\037)p Fk(i)p Fo(x;)g(f)d Fk(i)p Fo(:)22 b Fq(In)f(particular)g(the)g(matrix)f(\()p Fk(h)p Fo(x)1529 1959 y Fm(\000)p Fi(1)1576 1977 y Fo(;)8 b(\037)p Fk(i)p Fq(\))22 b(is)f(in-)0 2037 y(v)o(ertible.)490 2152 y(4.)28 b Fp(The)18 b(Naka)m(y)m(ama)e(A)o(utomorphism)50 2240 y Fq(Let)f Fo(H)20 b Fq(b)q(e)c(\014nite)f(dimensional.)k(Since)c Fk(h)819 2200 y Fj(R)852 2240 y Fo(;)8 b(f)d(g)r Fk(i)15 b Fq(=)f Fk(h)p Fq(\()1052 2200 y Fj(R)1099 2240 y Fo(\()g(f)5 b Fq(\))p Fo(;)j(g)r Fk(i)16 b Fq(as)g(a)g(functional)g(on)g Fo(g)h Fq(is)f(a)0 2298 y(generalized)f(in)o(tegral,)g(there)h(is)g(a)g (unique)g Fo(\027)s Fq(\()p Fo(f)5 b Fq(\))14 b Fk(2)g Fo(H)21 b Fq(suc)o(h)16 b(that)686 2382 y Fk(h)705 2342 y Fj(R)739 2382 y Fo(;)8 b(f)d(g)r Fk(i)14 b Fq(=)g Fk(h)919 2342 y Fj(R)953 2382 y Fo(;)8 b(g)r(\027)s Fq(\()p Fo(f)d Fq(\))p Fk(i)-1113 b Fq(\(12\))0 2467 y(or)531 2497 y Fj(Z)589 2565 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(g)r Fq(\()p Fo(x)p Fq(\))p Fo(dx)14 b Fq(=)893 2497 y Fj(Z)952 2565 y Fo(g)r Fq(\()p Fo(x)p Fq(\))p Fo(\027)s Fq(\()p Fo(f)5 b Fq(\)\()p Fo(x)p Fq(\))p Fo(dx:)-1270 b Fq(\(13\))p eop %%Page: 5 5 5 4 bop 323 118 a Fg(F)o(OURIER)16 b(TRANSF)o(ORMS)i(O)o(VER)e(FINITE)g (QUANTUM)g(GR)o(OUPS)304 b(5)0 213 y Fq(Although)17 b(the)g(functions)g Fo(f)s(;)8 b(g)17 b Fk(2)f Fo(H)21 b Fq(of)c(the)g(quan)o(tum)f(group)i (do)f(not)h(comm)o(ute)c(under)j(m)o(ul-)0 271 y(tiplication,)e(there)g (is)h(a)h(simple)d(comm)o(utation)g(rule)h(if)h(the)g(pro)q(duct)h(is)f (in)o(tegrated.)0 362 y Ft(Prop)r(osition)i(7.)24 b Ff(The)j(map)g Fo(\027)35 b Fq(:)c Fo(H)36 b Fk(!)c Fo(H)f Ff(is)c(an)g(algebr)n(a)h (automorphism,)g(c)n(al)r(le)n(d)g(the)0 420 y Fq(Nak)m(a)o(y)o(ama)15 b(automorphism)p Ff(.)0 512 y(Pr)n(o)n(of.)k Fq(It)f(is)g(clear)g(that) h Fo(\027)j Fq(is)c(a)h(linear)f(map.)27 b(W)l(e)18 b(ha)o(v)o(e)1095 471 y Fj(R)1137 512 y Fo(f)5 b(\027)s Fq(\()p Fo(g)r(h)p Fq(\))18 b(=)1358 471 y Fj(R)1399 512 y Fo(g)r(hf)23 b Fq(=)1555 471 y Fj(R)1597 512 y Fo(hf)5 b(\027)s Fq(\()p Fo(g)r Fq(\))18 b(=)0 531 y Fj(R)42 571 y Fo(f)5 b(\027)s Fq(\()p Fo(g)r Fq(\))p Fo(\027)s Fq(\()p Fo(h)p Fq(\))14 b(hence)g Fo(\027)s Fq(\()p Fo(g)r(h)p Fq(\))g(=)g Fo(\027)s Fq(\()p Fo(g)r Fq(\))p Fo(\027)s Fq(\()p Fo(h)p Fq(\))g(and)876 531 y Fj(R)917 571 y Fo(f)5 b(\027)s Fq(\(1\))15 b(=)1102 531 y Fj(R)1143 571 y Fo(f)20 b Fq(hence)13 b Fo(\027)s Fq(\(1\))h(=)g(1.)21 b(F)l(urthermore)0 631 y(if)h Fo(\027)s Fq(\()p Fo(g)r Fq(\))j(=)f(0)f(then)g(0)i(=)f Fk(h)523 590 y Fj(R)557 631 y Fo(;)8 b(f)d(\027)s Fq(\()p Fo(g)r Fq(\))p Fk(i)25 b Fq(=)g Fk(h)824 590 y Fj(R)858 631 y Fo(;)8 b(g)r(f)d Fk(i)25 b Fq(=)f Fk(h)p Fq(\()p Fo(f)31 b(*)1206 590 y Fj(R)1239 631 y Fq(\))p Fo(;)8 b(g)r Fk(i)23 b Fq(for)g(all)f Fo(f)30 b Fk(2)25 b Fo(H)i Fq(hence)0 689 y Fk(h)p Fo(a;)8 b(g)r Fk(i)14 b Fq(=)g(0)j(for)f(all)g Fo(a)d Fk(2)h Fo(H)490 671 y Fm(\003)527 689 y Fq(hence)h Fo(g)h Fq(=)e(0.)21 b(So)c Fo(\027)j Fq(is)c(injectiv)o(e)d(hence)j (bijectiv)o(e.)p 1765 689 2 33 v 1767 657 30 2 v 1767 689 V 1796 689 2 33 v 0 780 a Ft(Corollary)i(8.)24 b Ff(The)18 b(map)f Fo(H)h Fk(3)c Fo(f)19 b Fk(7!)14 b Fq(\()755 740 y Fj(R)802 780 y Fo(\()f(f)5 b Fq(\))14 b Fk(2)g Fo(H)1017 762 y Fm(\003)1055 780 y Ff(is)j(an)h(isomorphism.)0 872 y(Pr)n(o)n(of.)h Fq(W)l(e)d(ha)o(v)o(e)658 937 y(\()677 897 y Fj(R)724 937 y Fo(\()d(f)5 b Fq(\))14 b(=)g(\()p Fo(\027)s Fq(\()p Fo(f)5 b Fq(\))15 b Fo(*)1090 897 y Fj(R)1123 937 y Fq(\))0 1014 y(since)22 b Fk(h)p Fq(\()164 974 y Fj(R)223 1014 y Fo(\()i(f)5 b Fq(\))p Fo(;)j(g)r Fk(i)26 b Fq(=)f Fk(h)518 974 y Fj(R)552 1014 y Fo(;)8 b(f)d(g)r Fk(i)25 b Fq(=)g Fk(h)754 974 y Fj(R)788 1014 y Fo(;)8 b(g)r(\027)s Fq(\()p Fo(f)d Fq(\))p Fk(i)26 b Fq(=)f Fk(h)p Fq(\()p Fo(\027)s Fq(\()p Fo(f)5 b Fq(\))26 b Fo(*)1269 974 y Fj(R)1302 1014 y Fq(\))p Fo(;)8 b(g)r Fk(i)p Fq(.)41 b(This)23 b(implies)d(the)0 1072 y(corollary)l(.)p 1765 1072 V 1767 1041 30 2 v 1767 1072 V 1796 1072 2 33 v 50 1164 a(If)c Fo(G)g Fq(is)g(a)h(\014nite)f(group)h(and)g Fo(H)h Fq(=)c Fn(K)746 1146 y Fl(G)795 1164 y Fq(then)i Fo(H)21 b Fq(is)16 b(comm)o(utativ)n(e)d(hence)j Fo(\027)h Fq(=)c(id.)527 1284 y(5.)28 b Fp(The)18 b(Dira)o(c)h(Del)m(t)m(a)f (Function)50 1371 y Fq(An)e(elemen)o(t)e Fo(\016)i Fk(2)f Fo(H)21 b Fq(is)16 b(called)g(a)h Ff(Dir)n(ac)g Fo(\016)r Ff(-function)h Fq(if)e Fo(\016)i Fq(is)f(a)g(left)e(in)o(v)m(arian)o(t) h(in)o(tegral)g(in)h Fo(H)0 1429 y Fq(with)f Fk(h)130 1389 y Fj(R)164 1429 y Fo(;)8 b(\016)r Fk(i)13 b Fq(=)h(1,)i(i.e.)k(if) c Fo(\016)i Fq(satis\014es)482 1544 y Fo(f)5 b(\016)15 b Fq(=)f Fo(")p Fq(\()p Fo(f)5 b Fq(\))p Fo(\016)115 b Fq(and)1028 1477 y Fj(Z)1086 1544 y Fo(\016)r Fq(\()p Fo(x)p Fq(\))p Fo(dx)13 b Fq(=)h(1)0 1654 y(for)j(all)e Fo(f)k Fk(2)c Fo(H)t Fq(.)21 b(If)16 b Fo(H)k Fq(has)d(a)g(Dirac)f Fo(\016)r Fq(-function)g(then)g(w)o(e)g(write)616 1698 y Fj(Z)665 1711 y Fm(\003)693 1765 y Fo(a)p Fq(\()p Fo(x)p Fq(\))p Fo(dx)e Fq(=)903 1725 y Fj(R)937 1738 y Fm(\003)956 1765 y Fo(a)g Fq(:=)f Fk(h)p Fo(a;)8 b(\016)r Fk(i)p Fo(:)-1185 b Fq(\(14\))0 1880 y Ft(Prop)r(osition)18 b(9.)-17 b Ff(.)46 1952 y Fq(1.)25 b Ff(If)17 b Fo(H)k Ff(is)d(\014nite)g(dimensional)h(then)f(ther)n(e)g(exists)g(a)f(unique) i(Dir)n(ac)e Fo(\016)r Ff(-function)h Fo(\016)r Ff(.)46 2010 y Fq(2.)25 b Ff(If)17 b Fo(H)k Ff(is)d(in\014nite)h(dimensional)f (then)h(ther)n(e)e(exists)h(no)g(Dir)n(ac)f Fo(\016)r Ff(-function.)0 2102 y(Pr)n(o)n(of.)i Fq(1.)i(Since)13 b Fo(H)18 b Fk(3)c Fo(f)19 b Fk(7!)14 b Fq(\()p Fo(f)19 b(*)669 2062 y Fj(R)703 2102 y Fq(\))13 b Fk(2)h Fo(H)826 2084 y Fm(\003)860 2102 y Fq(is)g(an)g(isomorphism)d(there)i(is)h(a)g Fo(\016)h Fk(2)f Fo(H)k Fq(suc)o(h)c(that)0 2161 y(\()p Fo(\016)k(*)124 2121 y Fj(R)158 2161 y Fq(\))e(=)h Fo(":)g Fq(Then)h(\()p Fo(f)5 b(\016)19 b(*)585 2121 y Fj(R)618 2161 y Fq(\))e(=)f(\()p Fo(f)22 b(*)17 b Fq(\()p Fo(\016)h(*)963 2121 y Fj(R)997 2161 y Fq(\)\))e(=)h(\()p Fo(f)22 b(*)16 b(")p Fq(\))g(=)h Fo(")p Fq(\()p Fo(f)5 b Fq(\))p Fo(")17 b Fq(=)f Fo(")p Fq(\()p Fo(f)5 b Fq(\)\()p Fo(\016)18 b(*)1748 2121 y Fj(R)1781 2161 y Fq(\))0 2221 y(whic)o(h)d(implies)d Fo(f)5 b(\016)16 b Fq(=)e Fo(")p Fq(\()p Fo(f)5 b Fq(\))p Fo(\016)r Fq(.)20 b(F)l(urthermore)14 b(w)o(e)h(ha)o(v)o(e)g Fk(h)1054 2181 y Fj(R)1087 2221 y Fo(;)8 b(\016)r Fk(i)14 b Fq(=)g Fk(h)1237 2181 y Fj(R)1270 2221 y Fo(;)8 b Fq(1)1316 2228 y Fl(H)1350 2221 y Fo(\016)r Fk(i)14 b Fq(=)g Fk(h)p Fq(\()p Fo(\016)h(*)1597 2181 y Fj(R)1630 2221 y Fq(\))p Fo(;)8 b Fq(1)1695 2228 y Fl(H)1729 2221 y Fk(i)14 b Fq(=)0 2279 y Fo(")p Fq(\(1)66 2286 y Fl(H)100 2279 y Fq(\))g(=)f(1)208 2286 y Fd(K)237 2279 y Fq(.)50 2337 y(2.)21 b(is)16 b([6])g(exercise)f(V.4.)p 1765 2337 V 1767 2306 30 2 v 1767 2337 V 1796 2337 2 33 v 0 2428 a Ft(Lemma)h(10.)24 b Ff(L)n(et)e Fo(H)27 b Ff(b)n(e)c(a)g(\014nite)h (dimensional)g(Hopf)e(algebr)n(a.)39 b(Then)1428 2388 y Fj(R)1485 2428 y Fk(2)24 b Fo(H)1586 2410 y Fm(\003)1629 2428 y Ff(is)e(a)h(left)0 2487 y(inte)n(gr)n(al)18 b(i\013)431 2580 y Fo(a)p Fq(\()476 2533 y Fj(X)556 2540 y(R)589 2598 y Fi(\(1\))647 2580 y Fk(\012)11 b Fo(S)s Fq(\()749 2540 y Fj(R)782 2598 y Fi(\(2\))829 2580 y Fq(\)\))j(=)g(\()952 2533 y Fj(X)1032 2540 y(R)1065 2598 y Fi(\(1\))1123 2580 y Fk(\012)d Fo(S)s Fq(\()1225 2540 y Fj(R)1258 2598 y Fi(\(2\))1306 2580 y Fq(\)\))p Fo(a)-1370 b Fq(\(15\))p eop %%Page: 6 6 6 5 bop 0 118 a Fg(6)722 b(BODO)17 b(P)m(AREIGIS)0 213 y Ff(i\013)504 266 y Fj(X)585 313 y Fo(S)s Fq(\()p Fo(a)p Fq(\))682 273 y Fj(R)714 331 y Fi(\(1\))772 313 y Fk(\012)822 273 y Fj(R)855 331 y Fi(\(2\))916 313 y Fq(=)968 266 y Fj(X)1048 273 y(R)1082 331 y Fi(\(1\))1140 313 y Fk(\012)11 b Fo(a)1216 273 y Fj(R)1249 331 y Fi(\(2\))0 313 y Fq(\(16\))0 414 y Ff(i\013)612 467 y Fj(X)692 515 y Fo(f)716 522 y Fi(\(1\))763 515 y Fk(h)782 475 y Fj(R)816 515 y Fo(;)d(f)862 522 y Fi(\(2\))909 515 y Fk(i)14 b Fq(=)g Fk(h)1013 475 y Fj(R)1046 515 y Fo(;)8 b(f)d Fk(i)p Fq(1)1140 522 y Fl(H)1175 515 y Fo(:)-1189 b Fq(\(17\))0 624 y Ff(Pr)n(o)n(of.)19 b Fq(Let)236 583 y Fj(R)286 624 y Fq(b)q(e)d(a)h(left)e(in)o(tegral.)21 b(Then)46 683 y Fj(X)126 730 y Fo(a)152 738 y Fi(\(1\))199 690 y Fj(R)232 747 y Fi(\(1\))290 730 y Fk(\012)11 b Fo(S)s Fq(\()392 690 y Fj(R)425 747 y Fi(\(2\))472 730 y Fq(\))p Fo(S)s Fq(\()p Fo(a)569 738 y Fi(\(2\))616 730 y Fq(\))j(=)700 683 y Fj(X)772 730 y Fq(\()p Fo(a)817 690 y Fj(R)850 730 y Fq(\))869 738 y Fi(\(1\))927 730 y Fk(\012)d Fo(S)s Fq(\(\()p Fo(a)1074 690 y Fj(R)1107 730 y Fq(\))1126 738 y Fi(\(2\))1173 730 y Fq(\))j(=)f Fo(")p Fq(\()p Fo(a)p Fq(\)\()1363 683 y Fj(X)1443 690 y(R)1476 747 y Fi(\(1\))1534 730 y Fk(\012)e Fo(S)s Fq(\()1636 690 y Fj(R)1669 747 y Fi(\(2\))1716 730 y Fq(\)\))0 831 y(for)17 b(all)e Fo(a)f Fk(2)g Fo(H)t Fq(.)21 b(Hence)130 912 y(\()149 875 y Fj(P)210 872 y(R)243 929 y Fi(\(1\))301 912 y Fk(\012)11 b Fo(S)s Fq(\()403 872 y Fj(R)436 929 y Fi(\(2\))483 912 y Fq(\)\))p Fo(a)16 b Fq(=)615 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Fq(=)e Fk(h)p Fo(a;)8 b Fq(1)1584 1603 y Fl(H)1618 1596 y Fk(ih)1656 1555 y Fj(R)1690 1596 y Fo(;)g(f)d Fk(i)p Fq(.)50 1654 y(Con)o(v)o(ersely)24 b(if)h Fo(\025)31 b Fk(2)f Fo(H)526 1636 y Fm(\003)571 1654 y Fq(with)c(\(17\))g(is)g(giv)o(en)f(then)g Fk(h)p Fo(a\025;)8 b(f)d Fk(i)31 b Fq(=)1362 1616 y Fj(P)1415 1654 y Fk(h)p Fo(a;)8 b(f)1506 1661 y Fi(\(1\))1553 1654 y Fk(ih)p Fo(\025;)g(f)1665 1661 y Fi(\(2\))1713 1654 y Fk(i)30 b Fq(=)0 1712 y Fk(h)p Fo(a;)8 b Fq(1)91 1719 y Fl(H)125 1712 y Fk(ih)p Fo(\025;)g(f)d Fk(i)18 b Fq(hence)d Fo(a\025)f Fq(=)g Fo(")p Fq(\()p Fo(a)p Fq(\))p Fo(\025)p Fq(.)p 1765 1712 2 33 v 1767 1681 30 2 v 1767 1712 V 1796 1712 2 33 v 50 1812 a(If)i Fo(G)g Fq(is)g(a)h(\014nite)f(group)h (then)664 1948 y Fo(\016)r Fq(\()p Fo(x)p Fq(\))c(=)819 1863 y Fj(\()859 1914 y Fq(0)k(if)e Fo(x)f Fk(6)p Fq(=)g Fo(e)p Fq(;)859 1984 y(1)j(if)e Fo(x)f Fq(=)g Fo(e:)0 1948 y Fq(\(18\))0 2090 y(In)i(fact)f(since)g Fo(\016)j Fq(is)d(left)g(in)o(v)m(arian)o(t)g(w)o(e)h(get)g 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Fi(^)743 2613 y Fl(G)770 2546 y Fk(h)p Fo(\037;)8 b(x)p Fk(i)13 b Fq(=)h Fk(j)p Fo(G)p Fk(j)p Fo(\016)1042 2553 y Fl(e;x)1090 2546 y Fo(:)p eop %%Page: 7 7 7 6 bop 323 118 a Fg(F)o(OURIER)16 b(TRANSF)o(ORMS)i(O)o(VER)e(FINITE)g (QUANTUM)g(GR)o(OUPS)304 b(7)0 213 y Fq(Hence)145 175 y Fj(P)198 227 y Fl(x)p Fm(2)p Fl(G;\037)p Fm(2)332 219 y Fi(^)325 227 y Fl(G)353 213 y Fk(h)p Fo(\037;)8 b(x)p Fk(i)14 b Fq(=)g Fk(j)p Fo(G)p Fk(j)g Fq(=)f Fo(\013)700 195 y Fm(\000)p Fi(1)764 213 y Fq(and)732 319 y Fo(\016)i Fq(=)f Fk(j)p Fo(G)p Fk(j)887 299 y Fm(\000)p Fi(1)943 272 y Fj(X)943 386 y Fl(\037)p Fm(2)996 378 y Fi(^)989 386 y Fl(G)1024 319 y Fo(\037:)-1069 b Fq(\(19\))608 502 y(6.)28 b Fp(F)o(ourier)18 b(Transf)o(orms)50 589 y Fq(Let)e Fo(H)k Fq(b)q(e)15 b(\014nite)h(dimensional)e(for)i(the)f (rest)h(of)g(this)f(pap)q(er.)22 b(In)15 b(Corollary)h(8)g(w)o(e)f(ha)o (v)o(e)g(seen)0 647 y(that)k(the)e(map)h Fo(H)j Fk(3)c Fo(f)23 b Fk(7!)17 b Fq(\()548 607 y Fj(R)598 647 y Fo(\()g(f)5 b Fq(\))17 b Fk(2)h Fo(H)824 629 y Fm(\003)862 647 y Fq(is)g(an)g(isomorphism.)25 b(This)18 b(map)g(will)f(b)q(e)h(called)0 705 y(the)e Ff(F)l(ourier)h(tr)n(ansform)p Fq(.)0 802 y Ft(Theorem)g(11.)24 b Ff(The)17 b(F)l(ourier)g(tr)n(ansform)f Fo(H)i Fk(3)c Fo(f)20 b Fk(7!)1053 789 y Fj(e)1044 802 y Fo(f)g Fk(2)14 b Fo(H)1179 784 y Fm(\003)1216 802 y Ff(is)j(bije)n(ctive)i(with)579 884 y Fj(e)570 897 y Fo(f)g Fq(=)14 b(\()684 857 y Fj(R)731 897 y Fo(\()g(f)5 b Fq(\))14 b(=)907 850 y Fj(X)979 897 y Fk(h)998 857 y Fj(R)1032 915 y Fi(\(1\))1079 897 y Fo(;)8 b(f)d Fk(i)1149 857 y Fj(R)1183 915 y Fi(\(2\))0 897 y Fq(\(20\))0 993 y Ff(The)18 b(inverse)g(F)l(ourier)f(tr)n(ansform)f(is)i(de\014ne)n(d)g (by)636 1088 y Fj(e)-27 b Fo(a)13 b Fq(=)728 1040 y Fj(X)808 1088 y Fo(S)841 1067 y Fm(\000)p Fi(1)888 1088 y Fq(\()p Fo(\016)929 1095 y Fi(\(1\))976 1088 y Fq(\))p Fk(h)p Fo(a;)8 b(\016)1084 1095 y Fi(\(2\))1130 1088 y Fk(i)p Fo(:)-1163 b Fq(\(21\))0 1183 y Ff(Sinc)n(e)19 b(these)f(maps)f(ar)n(e) 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1765 2370 2 33 v 1767 2339 30 2 v 1767 2370 V 1796 2370 2 33 v 50 2464 a(If)g Fo(G)g Fq(is)g(a)h(\014nite)f(group)h(and)g Fo(H)h Fq(=)c Fn(K)746 2446 y Fl(G)795 2464 y Fq(then)753 2546 y Fj(e)744 2559 y Fo(f)19 b Fq(=)839 2512 y Fj(X)840 2617 y Fl(x)p Fm(2)p Fl(G)919 2559 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(x:)p eop %%Page: 8 8 8 7 bop 0 118 a Fg(8)722 b(BODO)17 b(P)m(AREIGIS)0 213 y Fq(Since)d(\001\()p Fo(\016)r Fq(\))f(=)294 176 y Fj(P)347 227 y Fl(x)p Fm(2)p Fl(G)428 213 y Fo(x)456 195 y Fm(\000)p Fi(1)503 187 y Fm(\003)532 213 y Fk(\012)8 b Fo(x)607 195 y Fm(\003)642 213 y Fq(where)15 b(the)g Fo(x)893 195 y Fm(\003)926 213 y Fk(2)f Fn(K)1009 195 y Fl(G)1057 213 y Fq(are)h(the)g(dual)h(basis)g(to)f(the)g Fo(x)f Fk(2)g Fo(G)p Fq(,)h(w)o(e)0 271 y(get)720 349 y Fj(e)-27 b Fo(a)14 b Fq(=)812 301 y Fj(X)813 407 y Fl(x)p Fm(2)p Fl(G)884 349 y Fk(h)p Fo(a;)8 b(x)979 328 y Fm(\003)999 349 y Fk(i)p Fo(x)1046 328 y Fm(\003)1065 349 y Fo(:)50 488 y Fq(If)k Fo(G)g 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Fi(^)1240 963 y Fl(G)1277 949 y Fo(a)p Fq(\()p Fo(\037)p Fq(\))p Fo(\037)1403 931 y Fm(\000)p Fi(1)1449 949 y Fo(;)338 1019 y(f)19 b Fq(=)14 b Fk(j)p Fo(G)p Fk(j)499 1001 y Fm(\000)p Fi(1)554 982 y Fj(P)607 1033 y Fl(\037)p Fm(2)660 1025 y Fi(^)653 1033 y Fl(G)699 1006 y Fj(e)690 1019 y Fo(f)6 b Fq(\()p Fo(\037)p Fq(\))p Fo(\037)820 1001 y Fm(\000)p Fi(1)866 1019 y Fo(;)48 b(a)14 b Fq(=)1019 982 y Fj(P)1072 1033 y Fl(x)p Fm(2)p Fl(G)1152 1019 y Fj(e)-27 b Fo(a)p Fq(\()p Fo(x)p Fq(\))p Fo(x:)0 982 y Fq(\(25\))0 1110 y(This)16 b(implies)507 1211 y Fj(e)499 1224 y Fo(f)5 b Fq(\()p Fo(\037)p Fq(\))13 b(=)662 1176 y Fj(X)663 1282 y Fl(x)p Fm(2)p Fl(G)742 1224 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(\037)p Fq(\()p Fo(x)p Fq(\))13 b(=)999 1156 y Fj(Z)1057 1224 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(\037)p Fq(\()p Fo(x)p Fq(\))p Fo(dx)-1302 b Fq(\(26\))0 1360 y(with)16 b(in)o(v)o(erse)f(transform)594 1449 y Fj(e)-27 b Fo(a)o Fq(\()p Fo(x)p Fq(\))14 b(=)g Fk(j)p Fo(G)p Fk(j)818 1429 y Fm(\000)p 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Fm(\003)1322 2053 y Ff(by)500 2151 y Fk(h)p Fo(a)11 b Fk(\003)g Fo(b;)d(f)d Fk(i)14 b Fq(:=)762 2104 y Fj(X)834 2151 y Fk(h)p Fo(a;)8 b(S)934 2131 y Fm(\000)p Fi(1)981 2151 y Fq(\()p Fo(\016)1022 2159 y Fi(\(1\))1069 2151 y Fq(\))p Fo(f)d Fk(ih)p Fo(b;)j(\016)1220 2159 y Fi(\(2\))1267 2151 y Fk(i)p Fo(:)0 2249 y Ff(Then)18 b(the)g(fol)r(lowing)i(tr)n(ansformation)c(rule)i(holds)g(for)f Fo(f)s(;)8 b(g)15 b Fk(2)f Fo(H)t Ff(:)785 2332 y Fj(f)782 2345 y Fo(f)5 b(g)17 b Fq(=)911 2332 y Fj(e)903 2345 y Fo(f)f Fk(\003)11 b Fj(e)-28 b Fo(g)r(:)-1018 b Fq(\(29\))0 2435 y Ff(In)14 b(p)n(articular)g Fo(H)324 2417 y Fm(\003)357 2435 y Ff(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 2486 y Fj(f)0 2497 y Fq(1)24 2504 y Fl(H)72 2497 y Fq(=)124 2457 y Fj(R)157 2497 y Ff(,)j(i.e.)709 2548 y Fj(R)753 2588 y Fk(\003)11 b Fo(a)j Fq(=)f Fo(a)e Fk(\003)953 2548 y Fj(R)1000 2588 y Fq(=)j Fo(a:)-1092 b Fq(\(30\))p eop %%Page: 9 9 9 8 bop 323 118 a 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Fq(\(32\))0 576 y Ff(Pr)n(o)n(of.)19 b Fq(W)l(e)24 b(ha)o(v)o(e)g(the)h(rules)f(for)h(the)f(F)l(ourier)g(transform.)47 b(F)l(rom)23 b(\(24\))i(w)o(e)g(get)f Fk(h)p Fo(a;)8 b(f)d Fk(i)29 b Fq(=)0 634 y Fk(h)19 594 y Fj(R)53 634 y Fo(;)7 b Fj(e)-27 b Fo(a)o(f)5 b Fk(i)15 b Fq(=)214 594 y Fj(R)255 634 y(e)-27 b Fo(a)o Fq(\()p Fo(x)p Fq(\))p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(dx)16 b Fq(and)h(from)e(\(23\))i(w)o (e)f(get)424 744 y Fk(h)p Fo(a;)8 b(f)d Fk(i)14 b Fq(=)g Fk(h)p Fo(S)657 723 y Fm(\000)p Fi(1)704 744 y Fq(\()p Fo(a)p Fq(\))777 730 y Fj(e)768 744 y Fo(f)5 b(;)j(\016)r Fk(i)14 b Fq(=)928 676 y Fj(Z)977 689 y Fm(\003)1005 744 y Fo(S)1038 723 y Fm(\000)p Fi(1)1085 744 y Fq(\()p Fo(a)p Fq(\)\()p Fo(x)p Fq(\))1224 730 y Fj(e)1215 744 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(dx:)p 1765 844 2 33 v 1767 813 30 2 v 1767 844 V 1796 844 2 33 v 50 927 a Fq(The)19 b(F)l(ourier)f(transform)g(leads)h(to)g(an)g(in)o (teresting)e(in)o(tegral)h(transform)h(on)g Fo(H)k Fq(b)o(y)18 b(double)0 985 y(application.)0 1076 y Ft(Prop)r(osition)g(17.)24 b Ff(The)d(double)h(tr)n(ansform)894 1063 y Fq(\025)884 1076 y Fo(f)j Fq(:=)20 b(\()p Fo(\016)i(\()e Fq(\()1156 1036 y Fj(R)1209 1076 y Fo(\()g(f)5 b Fq(\)\))22 b Ff(de\014nes)g(an)f (automor-)0 1134 y(phism)c Fo(H)h Fk(!)c Fo(H)21 b Ff(with)664 1202 y Fq(\025)654 1215 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))13 b(=)812 1148 y Fj(Z)870 1215 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(\016)r Fq(\()p Fo(xy)r Fq(\))p Fo(dx:)0 1326 y Ff(Pr)n(o)n(of.)19 b Fq(W)l(e)d(ha)o(v)o(e)290 1408 y Fk(h)p Fo(y)r(;)368 1395 y Fq(\025)357 1408 y Fo(f)6 b Fk(i)16 b Fq(=)e Fk(h)p Fo(y)r(;)8 b Fq(\()p Fo(\016)15 b(\()f Fq(\()679 1368 y Fj(R)726 1408 y Fo(\()f(f)5 b Fq(\)\))p Fk(i)15 b Fq(=)f Fk(h)p Fq(\()979 1368 y Fj(R)1026 1408 y Fo(\()g(f)5 b Fq(\))p Fo(y)r(;)j(\016)r Fk(i)422 1466 y Fq(=)474 1429 y Fj(P)527 1466 y Fk(h)p Fq(\()565 1426 y Fj(R)612 1466 y Fo(\()14 b(f)5 b Fq(\))p Fo(;)j(\016)767 1474 y Fi(\(1\))813 1466 y Fk(ih)p Fo(y)r(;)g(\016)921 1474 y Fi(\(2\))968 1466 y Fk(i)15 b Fq(=)1053 1429 y Fj(P)1106 1466 y Fk(h)1125 1426 y Fj(R)1159 1466 y Fo(;)8 b(f)d(\016)1232 1474 y Fi(\(1\))1279 1466 y Fk(ih)p Fo(y)r(;)j(\016) 1387 1474 y Fi(\(2\))1433 1466 y Fk(i)422 1524 y Fq(=)474 1487 y Fj(P)527 1524 y Fk(h)546 1484 y Fj(R)569 1542 y Fi(\(1\))617 1524 y Fo(;)g(f)d Fk(ih)706 1484 y Fj(R)730 1542 y Fi(\(2\))777 1524 y Fo(;)j(\016)821 1532 y Fi(\(1\))868 1524 y Fk(ih)p Fo(y)r(;)g(\016)976 1532 y Fi(\(2\))1023 1524 y Fk(i)14 b Fq(=)1108 1487 y Fj(P)1160 1524 y Fk(h)1179 1484 y Fj(R)1203 1542 y Fi(\(1\))1250 1524 y Fo(;)8 b(f)d Fk(ih)1339 1484 y Fj(R)1364 1542 y Fi(\(2\))1419 1524 y Fo(y)r(;)j(\016)r Fk(i)422 1591 y Fq(=)474 1553 y Fj(P)527 1591 y Fk(h)546 1550 y Fj(R)569 1608 y Fi(\(1\))617 1591 y Fo(;)g(f)d Fk(ih)706 1550 y Fj(R)730 1608 y Fi(\(2\))777 1591 y Fo(;)j Fq(\()p Fo(y)15 b(*)f(\016)r Fq(\))p Fk(i)g Fq(=)1047 1553 y Fj(P)1100 1591 y Fk(h)1119 1550 y Fj(R)1153 1591 y Fo(;)8 b(f)d Fq(\()p Fo(y)15 b(*)f(\016)r Fq(\))p Fk(i)422 1657 y Fq(=)474 1617 y Fj(R)515 1657 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(\016)r Fq(\()p Fo(xy)r Fq(\))p Fo(dx)0 1736 y Fq(since)15 b Fk(h)p Fo(x;)8 b Fq(\()p Fo(y)16 b(*)d(\016)r Fq(\))p Fk(i)h Fq(=)g Fk(h)p Fo(xy)r(;)8 b(\016)r Fk(i)p Fq(.)p 1765 1736 V 1767 1705 30 2 v 1767 1736 V 1796 1736 2 33 v 755 1835 a Fp(References)21 1914 y Fs([1])19 b(A.)f(Haar:)25 b Fh(Der)18 b(Ma\031b)n(e)n(gri\013)h(in)g (der)f(The)n(orie)g(der)g(kontinuierlichen)g(Grupp)n(en.)g Fs(Annals)g(of)f(Math.,)g(34)85 1964 y(\(147-169\),)12 b(1933.)21 2013 y([2])19 b(R.)d(G.)g(Larson,)h(M.)g(E.)f(Sw)o(eedler:) 25 b Fc(An)17 b(Asso)q(ciativ)o(e)h(Orthogonal)e(Bilinear)g(F)m(orm)f (for)i(Hopf)f(Algebras.)85 2063 y Fs(Am.)c(J.)i(Math.)g(91)f (\(75-93\),)f(1969.)21 2113 y([3])19 b(S.)14 b(Mon)o(tgomery:)i Fh(Hopf)f(A)o(lgebr)n(as)f(and)i(Their)e(A)n(ctions)h(on)g(R)o(ings.)f Fs(CBMS)g(82,)f(AMS)h(1993.)21 2163 y([4])19 b(J.)14 b(v.)f(Neumann:)k Fh(On)e(rings)g(of)g(op)n(er)n(ators.)f(III.)f Fs(Annals)h(of)f(Math.,)g(41)h(\(94-161\),)e(1940.)21 2213 y([5])19 b(F.)d(Riesz,)g(B.)g(Sz.-Nagy:)22 b Fh(V)m(orlesungen)17 b(\177)-22 b(ub)n(er)18 b(F)m(unktionalanalysis.)e Fs(VEB)h(Deutsc)o (her)g(V)m(erlag)f(der)h(Wis-)85 2262 y(sensc)o(haften,)e(Berlin,)f (1956.)21 2312 y([6])19 b(M.)14 b(E.)f(Sw)o(eedler:)19 b Fh(Hopf)c(A)o(lgebr)n(as.)e Fs(Benjamin,)f(New)i(Y)m(ork,)f(1969.)50 2406 y Fb(Ma)m(thema)m(tisches)j(Institut)f(der)h(Universit)886 2403 y(\177)885 2406 y(at)f(M)992 2403 y(\177)991 2406 y(unchen,)i(Germany)50 2455 y Fh(E-mail)d(addr)n(ess)s Fs(:)19 b Fa(pareigis@rz.math)o(emat)o(ik.un)o(i-mue)o(nchen)o(.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF