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y Fl(A)766 1640 y Fi(\012)p Fj(r)881 1676 y Fk(\012)23 b Fl(A)1054 1640 y Fi(\012)p Fj(s)2052 1676 y Fl(A)2125 1640 y Fi(\012)p Fj(r)2240 1676 y Fk(\012)g Fl(A)2413 1640 y Fi(\012)p Fj(s)2527 1676 y Fk(\012)g Fl(B)k Fk(\012)c Fl(B)p 1175 1644 849 4 v 1941 1642 a Fb(-)1326 1607 y Fj(')p Fg(\()p Fj(X)1462 1583 y Fd(\012)p Fc(r)1545 1607 y Fg(\))p Fi(\012)p Fj(')p Fg(\()p Fj(X)1763 1583 y Fd(\012)p Fc(s)1845 1607 y Fg(\))1180 1871 y Fl(A)1253 1835 y Fi(\012)p Fj(r)1368 1871 y Fk(\012)g Fl(A)1541 1835 y Fi(\012)p Fj(s)1655 1871 y Fk(\012)f Fl(B)1833 1835 y Fi(\012)p Fj(r)1949 1871 y Fk(\012)g Fl(B)2127 1835 y Fi(\012)p Fj(s)1193 1731 y Fb(X)1276 1752 y(X)1344 1769 y(X)-83 b(z)546 b(\030)2056 1748 y(\030)2124 1731 y(\030)-83 b(:)p 918 2354 4 644 v 920 2354 a(?)p 1698 2110 4 205 v 1700 2110 a(?)p 2478 2354 4 644 v 2480 2354 a(?)1327 2266 y Fl(A)1400 2230 y Fi(\012)p Fg(\()p Fj(r)r Fg(+)p Fj(s)p Fg(\))1658 2266 y Fk(\012)22 b Fl(B)1836 2230 y Fi(\012)p Fg(\()p Fj(r)r Fg(+)p Fj(s)p Fg(\))1193 2354 y Fb(\030)1276 2333 y(\030)1344 2316 y(\030)-83 b(:)546 b(X)2056 2337 y(X)2124 2354 y(X)-83 b(z)765 2461 y Fl(A)838 2425 y Fi(\012)p Fg(\()p Fj(r)r Fg(+)p Fj(s)p Fg(\))2225 2461 y Fl(A)2298 2425 y Fi(\012)p Fg(\()p Fj(r)r Fg(+)p Fj(s)p Fg(\))2555 2461 y Fk(\012)23 b Fl(B)p 1103 2424 1093 4 v 2113 2422 a Fb(-)1464 2501 y Fj(')p Fg(\()p Fj(X)1600 2478 y Fd(\012)p Fa(\()p Fc(r)r Fa(+)p Fc(s)p Fa(\))1807 2501 y Fg(\))0 2636 y Fp(so)33 b(that)f Fl(')c Fp(:)f Fl(!)538 2651 y Fj(A)623 2636 y Fk(\000)-60 b(!)28 b Fl(!)829 2651 y Fj(A)908 2636 y Fk(\012)22 b Fl(B)38 b Fp(is)32 b(a)g(monoidal)e(natural)h (transformation.)149 2752 y(Con)m(v)m(ersely)k(let)c Fl(')d Fp(:)f Fl(!)990 2767 y Fj(A)1075 2752 y Fk(\000)-60 b(!)28 b Fl(!)1281 2767 y Fj(A)1358 2752 y Fk(\012)22 b Fl(B)37 b Fp(b)s(e)c(a)e(natural)g(transformation.)42 b(Let)32 b Fl(f)38 b Fp(:=)28 b Fl(')p Fp(\()p Fl(X)8 b Fp(\))27 b(:)h Fl(A)0 2868 y Fk(\000)-60 b(!)28 b Fl(A)22 b Fk(\012)h Fl(B)5 b Fp(.)43 b(Then)34 b(the)f(follo)m(wing)d(comm)m (ute)1078 3045 y Fl(A)22 b Fk(\012)h Fl(A)506 b(A)22 b Fk(\012)h Fl(A)f Fk(\012)h Fl(B)k Fk(\012)c Fl(B)p 1375 3017 449 4 v 1741 3015 a Fb(-)1530 2983 y Fj(f)7 b Fi(\012)p Fj(f)p 1210 3337 4 254 v 1212 3337 a Fb(?)1118 3223 y Fg(=)p 2185 3337 V 2187 3337 a Fb(?)2226 3228 y Fg(1)p Fi(\012)p Fj(m)1078 3435 y Fl(A)22 b Fk(\012)h Fl(A)607 b(A)22 b Fk(\012)g Fl(A)h Fk(\012)f Fl(B)p 1375 3407 549 4 v 1841 3405 a Fb(-)1508 3369 y Fj(')p Fg(\()p Fj(X)5 b Fi(\012)p Fj(X)g Fg(\))p 1210 3727 4 254 v 1212 3727 a Fb(?)1111 3614 y Fj(m)p 2185 3727 V 2187 3727 a Fb(?)2226 3618 y Fj(m)p Fi(\012)p Fg(1)1176 3825 y Fl(A)801 b(A)22 b Fk(\012)h Fl(B)p 1277 3797 744 4 v 1938 3795 a Fb(-)1628 3763 y Fj(f)0 3952 y Fp(and)1333 4055 y Fh(K)627 b(K)45 b Fk(\012)23 b Fh(K)p 1439 4027 546 4 v 1902 4025 a Fb(-)1685 3985 y Fi(\030)1685 4006 y Fg(=)p 1370 4347 4 254 v 1372 4347 a Fb(?)1278 4233 y Fg(=)p 2150 4347 V 2152 4347 a Fb(?)1333 4445 y Fh(K)626 b(K)45 b Fk(\012)23 b Fl(B)p 1439 4417 545 4 v 1901 4415 a Fb(-)p 1370 4737 4 254 v 1372 4737 a(?)1292 4624 y Fj(u)p 2150 4737 V 2152 4737 a Fb(?)2191 4628 y Fj(u)p Fi(\012)p Fg(1)1335 4835 y Fl(A)593 b(A)23 b Fk(\012)f Fl(B)5 b(:)p 1437 4807 536 4 v 1890 4805 a Fb(-)1684 4773 y Fj(f)0 4942 y Fp(Hence)34 b Fl(f)k Fp(:)28 b Fl(A)g Fk(\000)-60 b(!)28 b Fl(A)22 b Fk(\012)g Fl(B)38 b Fp(is)32 b(a)h(homomorphism)c(of)j(algebras.)149 5058 y(Th)m(us)j(w)m(e)e(ha)m (v)m(e)h(de\014ned)g(an)f(isomorphism)988 5216 y Fh(K)18 b Fm(Alg)t Fp(\()p Fl(A;)f(A)22 b Fk(\012)h Fl(B)5 b Fp(\))1732 5188 y Fk(\030)1733 5220 y Fp(=)1837 5216 y(Nat)1997 5174 y Fi(\012)2056 5216 y Fp(\()p Fl(!)2155 5231 y Fj(A)2212 5216 y Fl(;)17 b(!)2317 5231 y Fj(A)2395 5216 y Fk(\012)23 b Fl(B)5 b Fp(\))p eop %%Page: 105 4 105 3 bop 791 236 a Fn(7.)35 b(THE)e(QUANTUM)g(MONOID)f(OF)h(A)g (QUANTUM)f(SP)-6 b(A)n(CE)650 b(105)0 425 y Fp(that)33 b(is)g(natural)f(in)h Fl(B)5 b Fp(.)46 b(If)33 b Fl(A)g Fp(is)g(\014nite)g(dimensional)e(then)j(the)g(left)e(hand)i(side)g(is)e (represen)m(ted)0 541 y(b)m(y)27 b(the)g(T)-8 b(am)m(bara)26 b(bialgebra)f Fl(M)1208 556 y Fj(r)1246 541 y Fp(\()p Fl(A)p Fp(\))i(and)f(the)h(righ)m(t)f(hand)h(side)g(b)m(y)g(the)g (bialgebra)e(co)s(end\()p Fl(!)3478 556 y Fj(A)3535 541 y Fp(\).)0 658 y(Th)m(us)34 b(b)s(oth)e(bialgebras)g(m)m(ust)g(b)s(e)h (isomorphic.)p 3529 658 4 66 v 3533 595 59 4 v 3533 658 V 3591 658 4 66 v 149 832 a Fm(Corollary)j(3.7.2.)50 b Ff(Ther)-5 b(e)72 b(is)h(a)g(unique)g(isomorphism)f(of)h(bialgebr)-5 b(as)72 b Fl(M)3237 847 y Fj(r)3275 832 y Fp(\()p Fl(A)p Fp(\))3523 804 y Fk(\030)3523 836 y Fp(=)0 948 y(co)s(end)q(\()p Fl(!)346 963 y Fj(A)402 948 y Fp(\))35 b Ff(such)g(that)g(the)g(diagr) -5 b(am)1379 1097 y Fl(A)213 b(A)22 b Fk(\012)h Fl(M)1954 1112 y Fj(r)1992 1097 y Fp(\()p Fl(A)p Fp(\))p 1481 1074 156 4 v 1554 1072 a Fb(-)1484 1223 y(@)1567 1306 y(@)1650 1389 y(@)1733 1472 y(@)1752 1491 y(@)-83 b(R)1586 1584 y Fl(A)22 b Fk(\012)h Fp(co)s(end\()p Fl(!)2126 1599 y Fj(A)2183 1584 y Fp(\))p 1902 1491 4 351 v 1903 1491 a Fb(?)0 1719 y Ff(c)-5 b(ommutes)149 1894 y Fe(Pr)n(oof.)42 b Fp(This)33 b(is)f(a)h(direct)f(consequence)k(of)c(the)h(univ)m(ersal) f(prop)s(ert)m(y)-8 b(.)p 3529 1894 4 66 v 3533 1831 59 4 v 3533 1894 V 3591 1894 4 66 v 149 2060 a(Th)m(us)28 b(the)e(T)-8 b(am)m(bara)25 b(bialgebra)f(that)h(represen)m(ts)j(the)e (univ)m(ersal)g(quan)m(tum)g(monoid)e(acting)0 2176 y(on)i(a)g (\014nite)h(quan)m(tum)f(space)i(ma)m(y)e(b)s(e)h(reconstructed)h(b)m (y)f(the)g(T)-8 b(annak)j(a-Krein)25 b(reconstruction)0 2292 y(from)d(represen)m(tation)i(theory)-8 b(.)41 b(Similar)20 b(reconstructions)k(can)f(b)s(e)h(giv)m(en)f(for)g(more)g(complicated)0 2408 y(quan)m(tum)33 b(spaces)h(suc)m(h)g(as)f(so)g(called)e(quadratic) h(quan)m(tum)h(spaces.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF