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b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fr(The)34 b(quan)m(tum)e(space)i Fm(M)993 440 y Fg(q)1031 425 y Fr(\(2\))e(asso)s(ciated)g(with)h(the)g(algebra)e Fn(M)2478 440 y Fg(q)2516 425 y Fr(\(2\))i(is)f(giv)m(en)g(b)m(y)334 615 y Fm(M)454 630 y Fg(q)492 615 y Fr(\(2\)\()p Fn(A)p Fr(\))o(=)c Fk(K)17 b Fr(-)6 b Fo(Alg)o Fr(\()p Fn(M)1283 630 y Fg(q)1321 615 y Fr(\(2\))p Fn(;)17 b(A)p Fr(\))765 794 y(=)869 653 y Fd(\032)o(\022)1017 734 y Fn(a)1068 698 y Fh(0)1179 734 y Fn(b)1220 698 y Fh(0)1021 850 y Fn(c)1063 814 y Fh(0)1174 850 y Fn(d)1225 814 y Fh(0)1248 653 y Fd(\023)1338 794 y Fm(j)p Fn(a)1417 758 y Fh(0)1440 794 y Fn(;)g(b)1525 758 y Fh(0)1549 794 y Fn(;)g(c)1635 758 y Fh(0)1658 794 y Fn(;)g(d)1753 758 y Fh(0)1803 794 y Fm(2)28 b Fn(A)p Fr(;)17 b Fn(a)2065 758 y Fh(0)2089 794 y Fn(b)2130 758 y Fh(0)2181 794 y Fr(=)28 b Fn(q)2332 758 y Fh(\000)p Fi(1)2426 794 y Fn(b)2467 758 y Fh(0)2491 794 y Fn(a)2542 758 y Fh(0)2565 794 y Fn(;)17 b(:)g(:)g(:)32 b(;)17 b(b)2841 758 y Fh(0)2865 794 y Fn(c)2907 758 y Fh(0)2958 794 y Fr(=)27 b Fn(c)3103 758 y Fh(0)3127 794 y Fn(b)3168 758 y Fh(0)3191 653 y Fd(\033)0 1043 y Fr(where)32 b(eac)m(h)g(homomorphism)c(of)j(algebras)f Fn(f)39 b Fr(:)27 b Fn(M)1908 1058 y Fg(q)1947 1043 y Fr(\(2\))g Fm(\000)-60 b(!)28 b Fn(A)j Fr(is)f(describ)s(ed)i(b)m(y)g(the)g (quadruple)0 1160 y(\()p Fn(a)89 1124 y Fh(0)112 1160 y Fn(;)17 b(b)197 1124 y Fh(0)221 1160 y Fn(;)g(c)307 1124 y Fh(0)330 1160 y Fn(;)g(d)425 1124 y Fh(0)448 1160 y Fr(\))29 b(of)h(images)e(of)h(the)i(algebra)d(generators)j Fn(a;)17 b(b;)g(c;)g(d)p Fr(.)42 b(The)30 b(images)f(m)m(ust)h(satisfy) g(the)0 1276 y(same)i(relations)g(that)g(generate)h(the)g(t)m(w)m (o-sided)g(ideal)e Fn(I)41 b Fr(hence)619 1457 y Fn(a)670 1421 y Fh(0)694 1457 y Fn(b)735 1421 y Fh(0)786 1457 y Fr(=)28 b Fn(q)937 1421 y Fh(\000)p Fi(1)1031 1457 y Fn(b)1072 1421 y Fh(0)1096 1457 y Fn(a)1147 1421 y Fh(0)1170 1457 y Fn(;)17 b(a)1265 1421 y Fh(0)1288 1457 y Fn(c)1330 1421 y Fh(0)1381 1457 y Fr(=)28 b Fn(q)1532 1421 y Fh(\000)p Fi(1)1626 1457 y Fn(c)1668 1421 y Fh(0)1691 1457 y Fn(a)1742 1421 y Fh(0)1766 1457 y Fn(;)17 b(b)1851 1421 y Fh(0)1874 1457 y Fn(d)1925 1421 y Fh(0)1976 1457 y Fr(=)27 b Fn(q)2126 1421 y Fh(\000)p Fi(1)2220 1457 y Fn(d)2271 1421 y Fh(0)2294 1457 y Fn(b)2335 1421 y Fh(0)2359 1457 y Fn(;)17 b(c)2445 1421 y Fh(0)2468 1457 y Fn(d)2519 1421 y Fh(0)2570 1457 y Fr(=)27 b Fn(q)2720 1421 y Fh(\000)p Fi(1)2815 1457 y Fn(d)2866 1421 y Fh(0)2889 1457 y Fn(c)2931 1421 y Fh(0)2954 1457 y Fn(;)1009 1573 y(b)1050 1537 y Fh(0)1074 1573 y Fn(c)1116 1537 y Fh(0)1167 1573 y Fr(=)g Fn(c)1312 1537 y Fh(0)1336 1573 y Fn(b)1377 1537 y Fh(0)1400 1573 y Fn(;)17 b(a)1495 1537 y Fh(0)1519 1573 y Fn(d)1570 1537 y Fh(0)1615 1573 y Fm(\000)22 b Fn(q)1761 1537 y Fh(\000)p Fi(1)1855 1573 y Fn(b)1896 1537 y Fh(0)1920 1573 y Fn(c)1962 1537 y Fh(0)2013 1573 y Fr(=)28 b Fn(d)2168 1537 y Fh(0)2191 1573 y Fn(a)2242 1537 y Fh(0)2287 1573 y Fm(\000)23 b Fn(q)t(c)2476 1537 y Fh(0)2499 1573 y Fn(b)2540 1537 y Fh(0)2564 1573 y Fn(:)0 1757 y Fr(W)-8 b(e)37 b(write)g(these)h(quadruples)f(as)g(2)25 b Fm(\002)g Fr(2-matrices)36 b(and)g(call)f(them)i Fl(quantum)i(matric) -5 b(es)p Fr(.)55 b(The)0 1873 y(un)m(usual)33 b(comm)m(utation)d (relations)i(are)g(c)m(hosen)i(so)f(that)f(the)h(follo)m(wing)d (examples)j(w)m(ork.)149 1990 y(The)f(quan)m(tum)e(space)h(of)f(quan)m (tum)g(matrices)f(turns)i(out)f(to)g(b)s(e)g(a)g(quan)m(tum)h(monoid.) 41 b(W)-8 b(e)0 2106 y(giv)m(e)40 b(b)s(oth)h(the)g(algebraic)e(\(with) h(function)g(algebras\))g(and)g(the)h(geometric)f(\(with)g(quan)m(tum)0 2222 y(spaces\))34 b(approac)m(h)f(to)f(de\014ne)i(the)f(m)m (ultiplication.)149 2338 y(a\))g Fl(The)h(algebr)-5 b(aic)34 b(appr)-5 b(o)g(ach:)149 2455 y Fr(The)34 b(algebra)d Fn(M)785 2470 y Fg(q)824 2455 y Fr(\(2\))h(is)g(a)g(bialgebra)f(with)h (the)h(diagonal)1105 2707 y(\001)1203 2567 y Fd(\022)1277 2647 y Fn(a)87 b(b)1281 2764 y(c)h(d)1461 2567 y Fd(\023)1562 2707 y Fr(=)1666 2567 y Fd(\022)1739 2647 y Fn(a)g(b)1744 2764 y(c)g(d)1924 2567 y Fd(\023)2020 2707 y Fm(\012)2119 2567 y Fd(\022)2193 2647 y Fn(a)g(b)2197 2764 y(c)g(d)2378 2567 y Fd(\023)2468 2707 y Fn(;)0 2957 y Fr(i.e.)77 b(b)m(y)45 b(\001\()p Fn(a)p Fr(\))i(=)g Fn(a)30 b Fm(\012)h Fn(a)f Fr(+)f Fn(b)i Fm(\012)f Fn(c)p Fr(,)47 b(\001\()p Fn(b)p Fr(\))g(=)g Fn(a)30 b Fm(\012)h Fn(b)f Fr(+)g Fn(b)g Fm(\012)h Fn(d)p Fr(,)46 b(\001\()p Fn(c)p Fr(\))h(=)g Fn(c)30 b Fm(\012)g Fn(a)g Fr(+)g Fn(d)g Fm(\012)g Fn(c)44 b Fr(and)0 3073 y(\001\()p Fn(d)p Fr(\))28 b(=)f Fn(c)22 b Fm(\012)h Fn(b)f Fr(+)g Fn(d)g Fm(\012)h Fn(d)p Fr(,)32 b(and)h(with)f(the)h(counit)1352 3322 y Fn(")1415 3182 y Fd(\022)1488 3263 y Fn(a)87 b(b)1492 3379 y(c)h(d)1673 3182 y Fd(\023)1774 3322 y Fr(=)1877 3182 y Fd(\022)1950 3263 y Fr(1)83 b(0)1950 3379 y(0)g(1)2131 3182 y Fd(\023)2221 3322 y Fn(;)0 3572 y Fr(i.e.)53 b Fn(")p Fr(\()p Fn(a)p Fr(\))32 b(=)h(1,)k Fn(")p Fr(\()p Fn(b)p Fr(\))c(=)g(0,)j Fn(")p Fr(\()p Fn(c)p Fr(\))d(=)g(0,)j(and)g Fn(")p 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b Fr(or)39 b(more)f(simply)g(on)h(the)h(generators)f(of)g (the)g(ideal.)62 b(Then)40 b(it)e(factors)i(through)f(a)f(unique)0 4269 y(homomorphism)29 b(of)i(algebras)g(\001)d(:)g Fn(M)1435 4284 y Fg(q)1474 4269 y Fr(\(2\))f Fm(\000)-60 b(!)28 b Fn(M)1865 4284 y Fg(q)1903 4269 y Fr(\(2\))20 b Fm(\012)h Fn(M)2240 4284 y Fg(q)2279 4269 y Fr(\(2\).)42 b(W)-8 b(e)33 b(c)m(hec)m(k)h(this)d(only)g(for)h(one)0 4386 y(generator)h(of)f(the)h(ideal)e Fn(I)8 b Fr(:)42 4566 y(\001\()p Fn(ab)23 b Fm(\000)f Fn(q)422 4530 y Fh(\000)p Fi(1)516 4566 y Fn(ba)p Fr(\))29 b(=)e(\001\()p Fn(a)p Fr(\)\001\()p Fn(b)p Fr(\))c Fm(\000)g Fn(q)1354 4530 y Fh(\000)p Fi(1)1448 4566 y Fr(\001\()p Fn(b)p Fr(\)\001\()p Fn(a)p Fr(\))29 b(=)42 4683 y(=)e(\()p Fn(a)22 b Fm(\012)h Fn(a)f Fr(+)g Fn(b)h Fm(\012)g Fn(c)p Fr(\)\()p Fn(a)f Fm(\012)g Fn(b)h Fr(+)f Fn(b)h Fm(\012)f Fn(d)p Fr(\))g Fm(\000)h Fn(q)1564 4647 y Fh(\000)p Fi(1)1658 4683 y Fr(\()p Fn(a)f Fm(\012)h Fn(b)f Fr(+)g Fn(b)h Fm(\012)g Fn(d)p Fr(\)\()p Fn(a)e Fm(\012)i Fn(a)f Fr(+)g Fn(b)h Fm(\012)g Fn(c)p Fr(\))42 4799 y(=)k Fn(aa)c Fm(\012)f Fn(ab)h Fr(+)f Fn(ab)h Fm(\012)g Fn(ad)f Fr(+)g Fn(ba)g Fm(\012)h Fn(cb)g Fr(+)f Fn(bb)h Fm(\012)f Fn(cd)g Fm(\000)h Fn(q)1903 4763 y Fh(\000)p Fi(1)1997 4799 y Fr(\()p Fn(aa)f Fm(\012)h Fn(ba)g Fr(+)f Fn(ab)h Fm(\012)f Fn(bc)h Fr(+)f Fn(ba)h Fm(\012)f Fn(da)g Fr(+)g Fn(bb)i Fm(\012)e Fn(dc)p Fr(\))42 4915 y(=)27 b Fn(aa)c Fm(\012)f Fr(\()p Fn(ab)h Fm(\000)g Fn(q)669 4879 y Fh(\000)p Fi(1)763 4915 y Fn(ba)p Fr(\))g(+)f Fn(ab)g Fm(\012)h Fr(\()p Fn(ad)f Fm(\000)h Fn(q)1537 4879 y Fh(\000)p Fi(1)1631 4915 y Fn(bc)p Fr(\))f(+)g Fn(ba)h Fm(\012)g Fr(\()p Fn(cb)f Fm(\000)h Fn(q)2377 4879 y Fh(\000)p Fi(1)2471 4915 y Fn(da)p Fr(\))f(+)g Fn(bb)h Fm(\012)g Fr(\()p Fn(cd)e Fm(\000)i Fn(q)3235 4879 y Fh(\000)p Fi(1)3329 4915 y Fn(dc)p Fr(\))42 5031 y(=)k Fn(ba)c Fm(\012)g Fr(\()p Fn(q)445 4995 y Fh(\000)p Fi(1)539 5031 y Fn(ad)f Fm(\000)g Fn(q)809 4995 y Fh(\000)p Fi(2)903 5031 y Fn(bc)h Fr(+)f Fn(cb)h Fm(\000)f Fn(q)1359 4995 y Fh(\000)p Fi(1)1453 5031 y Fn(da)p Fr(\))28 b Fm(\021)g Fr(0)99 b(mo)s(d)33 b(\()p Fn(I)8 b Fr(\))p Fn(:)0 5216 y Fr(The)34 b(reader)f(should)f(c)m(hec)m(k)j(the)e(other)g (iden)m(tities.)p eop %%Page: 21 5 21 4 bop 451 236 a Fp(3.)35 b(QUANTUM)d(MONOIDS)f(AND)i(THEIR)f(A)n (CTIONS)h(ON)f(QUANTUM)h(SP)-6 b(A)n(CES)347 b(21)149 425 y Fr(The)34 b(coasso)s(ciativit)m(y)d(follo)m(ws)h(from)140 657 y(\(\001)22 b Fm(\012)h Fr(1\)\001)566 516 y Fd(\022)639 597 y Fn(a)88 b(b)643 713 y(c)g(d)824 516 y Fd(\023)925 657 y Fr(=)27 b(\001)1126 516 y Fd(\022)1200 597 y Fn(a)87 b(b)1204 713 y(c)h(d)1384 516 y Fd(\023)1480 657 y Fm(\012)1580 516 y Fd(\022)1653 597 y Fn(a)g(b)1658 713 y(c)f(d)1838 516 y Fd(\023)1939 657 y Fr(=)27 b(\()2080 516 y Fd(\022)2154 597 y Fn(a)87 b(b)2158 713 y(c)h(d)2339 516 y Fd(\023)2434 657 y Fm(\012)2534 516 y Fd(\022)2607 597 y Fn(a)g(b)2612 713 y(c)f(d)2792 516 y Fd(\023)2865 657 y Fr(\))22 b Fm(\012)3025 516 y Fd(\022)3098 597 y Fn(a)88 b(b)3103 713 y(c)g(d)3283 516 y Fd(\023)3384 657 y Fr(=)118 896 y(=)222 755 y Fd(\022)295 836 y Fn(a)g(b)300 952 y(c)f(d)480 755 y Fd(\023)575 896 y Fm(\012)23 b Fr(\()713 755 y Fd(\022)786 836 y Fn(a)88 b(b)791 952 y(c)f(d)971 755 y Fd(\023)1067 896 y Fm(\012)1166 755 y Fd(\022)1240 836 y Fn(a)g(b)1244 952 y(c)h(d)1425 755 y Fd(\023)1498 896 y Fr(\))27 b(=)1667 755 y Fd(\022)1740 836 y Fn(a)88 b(b)1745 952 y(c)g(d)1925 755 y Fd(\023)2021 896 y Fm(\012)22 b Fr(\001)2218 755 y Fd(\022)2292 836 y Fn(a)87 b(b)2296 952 y(c)h(d)2477 755 y Fd(\023)2578 896 y Fr(=)27 b(\(1)22 b Fm(\012)h Fr(\001\)\001)3107 755 y Fd(\022)3180 836 y Fn(a)88 b(b)3185 952 y(c)f(d)3365 755 y Fd(\023)3455 896 y Fn(:)0 1127 y Fr(The)34 b(reader)f(should)f(c)m(hec)m(k)j(the)e (prop)s(erties)f(of)g(the)h(counit.)149 1243 y(b\))g Fl(The)h(ge)-5 b(ometric)34 b(appr)-5 b(o)g(ach:)149 1360 y Fn(M)243 1375 y Fg(q)282 1360 y Fr(\(2\))30 b(has)g(a)g(rather)g (remark)-5 b(able)29 b(\(and)h(actually)f(w)m(ell)g(kno)m(wn\))j(com)m (ultiplication)25 b(that)30 b(is)0 1476 y(b)s(etter)37 b(understo)s(o)s(d)h(b)m(y)g(using)e(the)i(induced)f(m)m(ultiplication) c(of)j(comm)m(uting)g(p)s(oin)m(ts.)56 b(Giv)m(en)0 1648 y(t)m(w)m(o)29 b(comm)m(uting)d(quan)m(tum)j(matrices)1476 1508 y Fd(\022)1550 1588 y Fn(a)1601 1603 y Fi(1)1728 1588 y Fn(b)1769 1603 y Fi(1)1554 1704 y Fn(c)1596 1719 y Fi(1)1723 1704 y Fn(d)1774 1719 y Fi(1)1814 1508 y Fd(\023)1915 1648 y Fr(and)2101 1508 y Fd(\022)2174 1588 y Fn(a)2225 1603 y Fi(2)2352 1588 y Fn(b)2393 1603 y Fi(2)2178 1704 y Fn(c)2220 1719 y Fi(2)2348 1704 y Fn(d)2399 1719 y Fi(2)2438 1508 y Fd(\023)2539 1648 y Fr(in)f Fm(M)2769 1663 y Fg(q)2806 1648 y Fr(\(2\)\()p Fn(A)p Fr(\).)42 b(Then)29 b(their)0 1820 y(matrix)i(pro)s(duct)738 1911 y Fd(\022)811 1992 y Fn(a)862 2007 y Fi(1)989 1992 y Fn(b)1030 2007 y Fi(1)815 2108 y Fn(c)857 2123 y Fi(1)985 2108 y Fn(d)1036 2123 y Fi(1)1075 1911 y Fd(\023)17 b(\022)1238 1992 y Fn(a)1289 2007 y Fi(2)1416 1992 y Fn(b)1457 2007 y Fi(2)1243 2108 y Fn(c)1285 2123 y Fi(2)1412 2108 y Fn(d)1463 2123 y Fi(2)1502 1911 y Fd(\023)1603 2052 y Fr(=)1706 1911 y Fd(\022)1780 1992 y Fn(a)1831 2007 y Fi(1)1870 1992 y Fn(a)1921 2007 y Fi(2)1983 1992 y Fr(+)22 b Fn(b)2122 2007 y Fi(1)2162 1992 y Fn(c)2204 2007 y Fi(2)2326 1992 y Fn(a)2377 2007 y Fi(1)2417 1992 y Fn(b)2458 2007 y Fi(2)2520 1992 y Fr(+)g Fn(b)2659 2007 y Fi(1)2699 1992 y Fn(d)2750 2007 y Fi(2)1780 2108 y Fn(c)1822 2123 y Fi(1)1861 2108 y Fn(a)1912 2123 y Fi(2)1974 2108 y Fr(+)g Fn(d)2123 2123 y Fi(1)2162 2108 y Fn(c)2204 2123 y Fi(2)2326 2108 y Fn(c)2368 2123 y Fi(1)2408 2108 y Fn(b)2449 2123 y Fi(2)2511 2108 y Fr(+)g Fn(d)2660 2123 y Fi(1)2699 2108 y Fn(d)2750 2123 y Fi(2)2789 1911 y Fd(\023)0 2283 y Fr(is)32 b(again)f(a)i(quan)m(tum)f(matrix.)42 b(T)-8 b(o)33 b(pro)m(v)m(e)h(this)e(w)m(e)i(only)e(c)m(hec)m(k)j(one)d (of)h(the)g(relations)42 2451 y(\()p Fn(a)131 2466 y Fi(1)170 2451 y Fn(a)221 2466 y Fi(2)283 2451 y Fr(+)22 b Fn(b)422 2466 y Fi(1)462 2451 y Fn(c)504 2466 y Fi(2)543 2451 y Fr(\)\()p Fn(a)670 2466 y Fi(1)710 2451 y Fn(b)751 2466 y Fi(2)813 2451 y Fr(+)g Fn(b)952 2466 y Fi(1)992 2451 y Fn(d)1043 2466 y Fi(2)1082 2451 y Fr(\))27 b(=)h Fn(a)1302 2466 y Fi(1)1342 2451 y Fn(a)1393 2466 y Fi(2)1432 2451 y Fn(a)1483 2466 y Fi(1)1523 2451 y Fn(b)1564 2466 y Fi(2)1626 2451 y Fr(+)22 b Fn(a)1775 2466 y Fi(1)1815 2451 y Fn(a)1866 2466 y Fi(2)1905 2451 y Fn(b)1946 2466 y Fi(1)1986 2451 y Fn(d)2037 2466 y Fi(2)2098 2451 y Fr(+)g Fn(b)2237 2466 y Fi(1)2277 2451 y Fn(c)2319 2466 y Fi(2)2359 2451 y Fn(a)2410 2466 y Fi(1)2449 2451 y Fn(b)2490 2466 y Fi(2)2552 2451 y Fr(+)g Fn(b)2691 2466 y Fi(1)2731 2451 y Fn(c)2773 2466 y Fi(2)2813 2451 y Fn(b)2854 2466 y Fi(1)2894 2451 y Fn(d)2945 2466 y Fi(2)581 2568 y Fr(=)28 b Fn(a)736 2583 y Fi(1)775 2568 y Fn(a)826 2583 y Fi(1)866 2568 y Fn(a)917 2583 y Fi(2)957 2568 y Fn(b)998 2583 y Fi(2)1060 2568 y Fr(+)22 b Fn(a)1209 2583 y Fi(1)1248 2568 y Fn(b)1289 2583 y Fi(1)1329 2568 y Fn(a)1380 2583 y Fi(2)1420 2568 y Fn(d)1471 2583 y Fi(2)1532 2568 y Fr(+)g Fn(b)1671 2583 y Fi(1)1711 2568 y Fn(a)1762 2583 y Fi(1)1802 2568 y Fn(c)1844 2583 y Fi(2)1883 2568 y Fn(b)1924 2583 y Fi(2)1986 2568 y Fr(+)g Fn(b)2125 2583 y Fi(1)2165 2568 y Fn(b)2206 2583 y Fi(1)2246 2568 y Fn(c)2288 2583 y Fi(2)2327 2568 y Fn(d)2378 2583 y Fi(2)581 2684 y Fr(=)28 b Fn(q)732 2648 y Fh(\000)p Fi(1)826 2684 y Fn(a)877 2699 y Fi(1)916 2684 y Fn(a)967 2699 y Fi(1)1007 2684 y Fn(b)1048 2699 y Fi(2)1088 2684 y Fn(a)1139 2699 y Fi(2)1201 2684 y Fr(+)22 b Fn(q)1346 2648 y Fh(\000)p Fi(1)1440 2684 y Fn(b)1481 2699 y Fi(1)1521 2684 y Fn(a)1572 2699 y Fi(1)1612 2684 y Fr(\()p Fn(d)1701 2699 y Fi(2)1740 2684 y Fn(a)1791 2699 y Fi(2)1852 2684 y Fr(+)g(\()p Fn(q)2035 2648 y Fh(\000)p Fi(1)2152 2684 y Fm(\000)g Fn(q)t Fr(\))p Fn(b)2377 2699 y Fi(2)2417 2684 y Fn(c)2459 2699 y Fi(2)2498 2684 y Fr(\))g(+)g Fn(b)2697 2699 y Fi(1)2737 2684 y Fn(a)2788 2699 y Fi(1)2828 2684 y Fn(b)2869 2699 y Fi(2)2909 2684 y Fn(c)2951 2699 y Fi(2)3012 2684 y Fr(+)g Fn(q)3157 2648 y Fh(\000)p Fi(1)3252 2684 y Fn(b)3293 2699 y Fi(1)3333 2684 y Fn(b)3374 2699 y Fi(1)3413 2684 y Fn(d)3464 2699 y Fi(2)3504 2684 y Fn(c)3546 2699 y Fi(2)581 2800 y Fr(=)28 b Fn(q)732 2764 y Fh(\000)p Fi(1)826 2800 y Fr(\()p Fn(a)915 2815 y Fi(1)954 2800 y Fn(a)1005 2815 y Fi(1)1045 2800 y Fn(b)1086 2815 y Fi(2)1126 2800 y Fn(a)1177 2815 y Fi(2)1239 2800 y Fr(+)22 b Fn(a)1388 2815 y Fi(1)1427 2800 y Fn(b)1468 2815 y Fi(1)1508 2800 y Fn(b)1549 2815 y Fi(2)1589 2800 y Fn(c)1631 2815 y Fi(2)1693 2800 y Fr(+)g Fn(b)1832 2815 y Fi(1)1872 2800 y Fn(a)1923 2815 y Fi(1)1962 2800 y Fn(d)2013 2815 y Fi(2)2052 2800 y Fn(a)2103 2815 y Fi(2)2165 2800 y Fr(+)g Fn(b)2304 2815 y Fi(1)2344 2800 y Fn(b)2385 2815 y Fi(1)2425 2800 y Fn(d)2476 2815 y Fi(2)2515 2800 y Fn(c)2557 2815 y Fi(2)2597 2800 y Fr(\))581 2916 y(=)28 b Fn(q)732 2880 y Fh(\000)p Fi(1)826 2916 y Fr(\()p Fn(a)915 2931 y Fi(1)954 2916 y Fn(b)995 2931 y Fi(2)1035 2916 y Fn(a)1086 2931 y Fi(1)1126 2916 y Fn(a)1177 2931 y Fi(2)1239 2916 y Fr(+)22 b Fn(a)1388 2931 y Fi(1)1427 2916 y Fn(b)1468 2931 y Fi(2)1508 2916 y Fn(b)1549 2931 y Fi(1)1589 2916 y Fn(c)1631 2931 y Fi(2)1693 2916 y Fr(+)g Fn(b)1832 2931 y Fi(1)1872 2916 y Fn(d)1923 2931 y Fi(2)1962 2916 y Fn(a)2013 2931 y Fi(1)2052 2916 y Fn(a)2103 2931 y Fi(2)2165 2916 y Fr(+)g Fn(b)2304 2931 y Fi(1)2344 2916 y Fn(d)2395 2931 y Fi(2)2434 2916 y Fn(b)2475 2931 y Fi(1)2515 2916 y Fn(c)2557 2931 y Fi(2)2597 2916 y Fr(\))581 3033 y(=)28 b Fn(q)732 2996 y Fh(\000)p Fi(1)826 3033 y Fr(\()p Fn(a)915 3048 y Fi(1)954 3033 y Fn(b)995 3048 y Fi(2)1057 3033 y Fr(+)22 b Fn(b)1196 3048 y Fi(1)1236 3033 y Fn(d)1287 3048 y Fi(2)1327 3033 y Fr(\)\()p Fn(a)1454 3048 y Fi(1)1493 3033 y Fn(a)1544 3048 y Fi(2)1606 3033 y Fr(+)g Fn(b)1745 3048 y Fi(1)1785 3033 y Fn(c)1827 3048 y Fi(2)1866 3033 y Fr(\))0 3204 y(W)-8 b(e)22 b(ha)m(v)m(e)i(used)f(that)e(the)i(t)m(w)m(o)f(p)s(oin)m (ts)g(are)g(comm)m(uting)e(p)s(oin)m(ts.)40 b(This)22 b(m)m(ultiplication)17 b(ob)m(viously)0 3320 y(is)37 b(a)g(natural)f(transformation)g Fm(M)1325 3335 y Fg(q)1362 3320 y Fr(\(2\))g Fm(?)g(M)1756 3335 y Fg(q)1794 3320 y Fr(\(2\)\()p Fn(A)p Fr(\))f Fm(\000)-60 b(!)36 b(M)2376 3335 y Fg(q)2414 3320 y Fr(\(2\)\()p Fn(A)p Fr(\))h(\(natural)f(in)h Fn(A)p Fr(\).)58 b(It)37 b(is)0 3501 y(asso)s(ciativ)m(e)32 b(and)h(has)g(unit)1056 3361 y Fd(\022)1129 3441 y Fr(1)83 b(0)1129 3558 y(0)g(1)1310 3361 y Fd(\023)1383 3501 y Fr(.)43 b(F)-8 b(or)32 b(the)h(asso)s(ciativit)m(y)e(observ)m(e)k(that) d(b)m(y)i(1.2.14)1056 3808 y(\(\()1132 3667 y Fd(\022)1206 3748 y Fn(a)1257 3763 y Fi(1)1384 3748 y Fn(b)1425 3763 y Fi(1)1210 3864 y Fn(c)1252 3879 y Fi(1)1379 3864 y Fn(d)1430 3879 y Fi(1)1469 3667 y Fd(\023)1559 3808 y Fn(;)1603 3667 y Fd(\022)1676 3748 y Fn(a)1727 3763 y Fi(2)1855 3748 y Fn(b)1896 3763 y Fi(2)1681 3864 y Fn(c)1723 3879 y Fi(2)1850 3864 y Fn(d)1901 3879 y Fi(2)1940 3667 y Fd(\023)2014 3808 y Fr(\))p Fn(;)2096 3667 y Fd(\022)2169 3748 y Fn(a)2220 3763 y Fi(3)2347 3748 y Fn(b)2388 3763 y Fi(3)2173 3864 y Fn(c)2215 3879 y Fi(3)2342 3864 y Fn(d)2393 3879 y Fi(3)2432 3667 y Fd(\023)2506 3808 y Fr(\))0 4039 y(is)e(a)g(pair)g(of)g(comm)m(uting)f(p)s(oin)m(ts)h(if)f (and)i(only)f(if)1048 4271 y(\()1086 4130 y Fd(\022)1159 4211 y Fn(a)1210 4226 y Fi(1)1338 4211 y Fn(b)1379 4226 y Fi(1)1164 4327 y Fn(c)1206 4342 y Fi(1)1333 4327 y Fn(d)1384 4342 y Fi(1)1423 4130 y Fd(\023)1513 4271 y Fn(;)17 b Fr(\()1595 4130 y Fd(\022)1668 4211 y Fn(a)1719 4226 y Fi(2)1846 4211 y Fn(b)1887 4226 y Fi(2)1673 4327 y Fn(c)1715 4342 y Fi(2)1842 4327 y Fn(d)1893 4342 y Fi(2)1932 4130 y Fd(\023)2022 4271 y Fn(;)2066 4130 y Fd(\022)2139 4211 y Fn(a)2190 4226 y Fi(3)2317 4211 y Fn(b)2358 4226 y Fi(3)2143 4327 y Fn(c)2185 4342 y Fi(3)2313 4327 y Fn(d)2364 4342 y Fi(3)2403 4130 y Fd(\023)2476 4271 y Fr(\)\))0 4502 y(is)32 b(a)g(pair)g(of)g(comm)m(uting)f(p)s(oin) m(ts.)149 4674 y(Since)418 4534 y Fd(\022)491 4614 y Fr(1)83 b(0)491 4731 y(0)g(1)672 4534 y Fd(\023)17 b(\022)835 4614 y Fn(a)88 b(b)840 4731 y(c)f(d)1020 4534 y Fd(\023)1145 4674 y Fr(=)1272 4534 y Fd(\022)1345 4614 y Fn(a)h(b)1350 4731 y(c)f(d)1530 4534 y Fd(\023)1655 4674 y Fr(=)1782 4534 y Fd(\022)1855 4614 y Fn(a)h(b)1860 4731 y(c)g(d)2040 4534 y Fd(\023)17 b(\022)2203 4614 y Fr(1)83 b(0)2203 4731 y(0)g(1)2384 4534 y Fd(\023)2504 4674 y Fr(for)46 b(all)e(quan)m(tum)j(matrices)0 4781 y Fd(\022)73 4862 y Fn(a)88 b(b)78 4978 y(c)g(d)258 4781 y Fd(\023)359 4922 y Fm(2)28 b(M)573 4937 y Fg(q)611 4922 y Fr(\(2\)\()p Fn(B)5 b Fr(\))32 b(w)m(e)i(see)g(that)e Fm(M)1556 4937 y Fg(q)1594 4922 y Fr(\(2\))g(is)g(a)g(quan)m(tum)h(monoid.)149 5099 y(It)j(remains)f(to)g(sho)m(w)h(that)f(the)h(m)m(ultiplication)31 b(of)k Fm(M)2239 5114 y Fg(q)2277 5099 y Fr(\(2\))g(and)h(the)f(com)m (ultiplication)c(of)0 5216 y Fn(M)94 5231 y Fg(q)132 5216 y Fr(\(2\))39 b(corresp)s(ond)h(to)e(eac)m(h)i(other)f(b)m(y)h (the)f(Y)-8 b(oneda)40 b(Lemma.)61 b(The)40 b(iden)m(tit)m(y)e (morphism)f(of)p eop %%Page: 22 6 22 5 bop 0 236 a Fp(22)341 b(1.)76 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)g(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fn(M)94 440 y Fg(q)132 425 y Fr(\(2\))22 b Fm(\012)h Fn(M)473 440 y Fg(q)511 425 y Fr(\(2\))32 b(is)g(giv)m(en)h(b)m(y)h(the)f(pair)e(of)h(comm)m(uting)f(p)s(oin)m (ts)0 588 y(\()p Fn(\023)72 603 y Fi(1)112 588 y Fn(;)17 b(\023)190 603 y Fi(2)229 588 y Fr(\))28 b Fm(2)g(M)509 603 y Fg(q)547 588 y Fr(\(2\))f Fm(?)h(M)924 603 y Fg(q)962 588 y Fr(\(2\)\()p Fn(M)1219 603 y Fg(q)1257 588 y Fr(\(2\))17 b Fm(\012)g Fn(M)1587 603 y Fg(q)1626 588 y Fr(\(2\)\))27 b(=)h Fk(K)17 b Fr(-)6 b Fo(Alg)n Fr(\()p Fn(M)2333 603 y Fg(q)2372 588 y Fr(\(2\))17 b Fm(\012)g Fn(M)2702 603 y Fg(q)2741 588 y Fr(\(2\))p Fn(;)g(M)3004 603 y Fg(q)3041 588 y Fr(\(2\))g Fm(\012)h Fn(M)3372 603 y Fg(q)3410 588 y Fr(\(2\)\))p Fn(:)0 816 y Fr(Since)38 b Fn(\023)294 831 y Fi(1)370 816 y Fr(=)483 675 y Fd(\022)556 756 y Fn(a)88 b(b)561 872 y(c)f(d)741 675 y Fd(\023)840 816 y Fm(\012)26 b Fr(1)37 b(=)1141 675 y Fd(\022)1214 756 y Fn(a)23 b Fm(\012)f Fr(1)88 b Fn(b)22 b Fm(\012)h Fr(1)1219 872 y Fn(c)f Fm(\012)h Fr(1)87 b Fn(d)22 b Fm(\012)h Fr(1)1740 675 y Fd(\023)1851 816 y Fr(and)38 b Fn(\023)2080 831 y Fi(2)2157 816 y Fr(=)e(1)26 b Fm(\012)2447 675 y Fd(\022)2520 756 y Fn(a)88 b(b)2525 872 y(c)f(d)2705 675 y Fd(\023)2815 816 y Fr(=)2927 675 y Fd(\022)3001 756 y Fr(1)22 b Fm(\012)g Fn(a)88 b Fr(1)22 b Fm(\012)h Fn(b)3005 872 y Fr(1)f Fm(\012)h Fn(c)87 b Fr(1)22 b Fm(\012)h Fn(d)3527 675 y Fd(\023)0 1063 y Fr(w)m(e)38 b(ha)m(v)m(e)g(id)33 b(=)i(\()p Fn(\023)675 1078 y Fi(1)715 1063 y Fn(;)17 b(\023)793 1078 y Fi(2)832 1063 y Fr(\))35 b(=)f(\()1053 922 y Fd(\022)1126 1003 y Fn(a)88 b(b)1131 1119 y(c)g(d)1311 922 y Fd(\023)1410 1063 y Fm(\012)25 b Fr(1)p Fn(;)17 b Fr(1)24 b Fm(\012)1780 922 y Fd(\022)1854 1003 y Fn(a)88 b(b)1858 1119 y(c)g(d)2039 922 y Fd(\023)2112 1063 y Fr(\).)56 b(The)37 b(Y)-8 b(oneda)37 b(Lemma)e(de\014nes)k(the)0 1241 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1725 y(c)f(d)1139 1528 y Fd(\023)1212 1669 y Fr(\))28 b(=)1381 1528 y Fd(\022)1455 1609 y Fn(a)87 b(b)1459 1725 y(c)h(d)1640 1528 y Fd(\023)1735 1669 y Fm(\012)1835 1528 y Fd(\022)1908 1609 y Fn(a)g(b)1913 1725 y(c)f(d)2093 1528 y Fd(\023)2166 1669 y Fr(.)149 1846 y(Th)m(us)35 b Fn(M)491 1861 y Fg(q)529 1846 y Fr(\(2\))d (de\014nes)i(a)f(quan)m(tum)g(monoid)d Fm(M)1966 1861 y Fg(q)2004 1846 y Fr(\(2\))i(with)293 2074 y Fm(M)413 2089 y Fg(q)450 2074 y Fr(\(2\)\()p Fn(B)5 b Fr(\))28 b(=)861 1933 y Fd(\032\022)1009 2014 y Fn(a)1060 1978 y Fh(0)1171 2014 y Fn(b)1212 1978 y Fh(0)1014 2130 y Fn(c)1056 2094 y Fh(0)1167 2130 y Fn(d)1218 2094 y Fh(0)1241 1933 y Fd(\023)1314 1989 y(\014)1314 2049 y(\014)1347 2074 y Fn(a)1398 2033 y Fh(0)1422 2074 y Fn(;)17 b(b)1507 2033 y Fh(0)1530 2074 y Fn(;)g(c)1616 2033 y Fh(0)1639 2074 y Fn(;)g(d)1734 2033 y Fh(0)1785 2074 y Fm(2)28 b Fn(B)5 b Fr(;)17 b Fn(a)2053 2033 y Fh(0)2076 2074 y Fn(b)2117 2033 y Fh(0)2169 2074 y Fr(=)27 b Fn(q)2319 2033 y Fh(\000)p 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Fm(\000)p Fn(q)t(b)2905 5143 y Fh(0)2929 5184 y Fn(a)2980 5143 y Fh(0)3014 5073 y Fd(o)3097 5184 y Fn(:)p eop %%Page: 23 7 23 6 bop 451 236 a Fp(3.)35 b(QUANTUM)d(MONOIDS)f(AND)i(THEIR)f(A)n (CTIONS)h(ON)f(QUANTUM)h(SP)-6 b(A)n(CES)347 b(23)0 425 y Fr(The)46 b(quan)m(tum)g(monoid)e Fn(M)1092 440 y Fg(q)1130 425 y Fr(\(2\))i(also)e(op)s(erates)i(on)g(the)f(dual)g(quan)m(tum)h (plane)f(b)m(y)h(matrix)0 541 y(m)m(ultiplication)246 754 y Fk(A)312 713 y Fi(0)p Fh(j)p Fi(2)312 779 y Fg(q)412 754 y Fr(\()p Fn(A)523 713 y Fh(0)546 754 y Fr(\))28 b Fm(?)g(M)837 769 y Fg(q)875 754 y Fr(\(2\)\()p Fn(A)1111 713 y Fh(0)1134 754 y Fr(\))f Fm(3)i Fr(\()1332 674 y Fd(\000)1377 752 y Fn(\030)88 b(\021)1559 674 y Fd(\001)1622 754 y Fn(;)1666 614 y Fd(\022)1739 694 y Fn(a)f(b)1743 811 y(c)h(d)1924 614 y Fd(\023)1997 754 y Fr(\))27 b Fm(7!)2190 674 y Fd(\000)2235 752 y Fn(\030)88 b(\021)2418 674 y Fd(\001)2485 754 y Fm(\001)2535 614 y Fd(\022)2609 694 y Fn(a)f(b)2613 811 y(c)h(d)2793 614 y Fd(\023)2894 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b(a)g(quantum)h(sp)-5 b(ac)g(e)50 b(with)g(universal)g (quantum)267 2233 y(sp)-5 b(ac)g(e)30 b Fm(M)p Fr(\()p Fm(X)15 b Fr(\))29 b Fl(and)h Fn(\026)d Fr(:)h Fm(M)p Fr(\()p Fm(X)15 b Fr(\))27 b Fm(?)h Fn(A)g Fm(\000)-58 b(!)28 b Fn(A)p Fl(.)44 b(Then)29 b Fm(M)p Fr(\()p Fm(X)15 b Fr(\))30 b Fl(is)g(a)h(quantum)f(monoid)g(and)267 2349 y Fm(X)49 b Fl(is)35 b(an)f Fm(M)p Fr(\()p Fm(X)15 b Fr(\))p Fl(-sp)-5 b(ac)g(e)34 b(by)h Fn(\026)p Fl(.)149 2465 y Fr(2.)42 b Fl(If)36 b Fm(Y)44 b Fl(is)36 b(another)g(quantum)g (monoid)g(and)g(if)g Fn(f)41 b Fr(:)31 b Fm(Y)38 b(?)31 b(X)45 b(\000)-58 b(!)31 b(X)51 b Fl(de\014nes)35 b(the)i(structur)-5 b(e)267 2582 y(of)40 b(a)g Fm(Y)8 b Fl(-sp)-5 b(ac)g(e)39 b(on)g Fm(X)55 b Fl(then)40 b(ther)-5 b(e)40 b(is)g(a)g(unique)g (morphism)f(of)h(quantum)g(monoids)f Fn(g)i Fr(:)267 2698 y Fm(Y)35 b(\000)-58 b(!)28 b(M)p Fr(\()p Fm(X)15 b Fr(\))35 b Fl(such)f(that)h(the)g(fol)5 b(lowing)34 b(diagr)-5 b(am)34 b(c)-5 b(ommutes)1403 3324 y Fm(M)p Fr(\()p Fm(X)15 b Fr(\))27 b Fm(?)h(X)196 b(X)15 b Fn(:)p 1932 3301 124 4 v 1973 3299 a Fj(-)1972 3347 y Fg(\026)1957 3072 y(f)1722 2963 y Fj(@)1805 3046 y(@)1888 3129 y(@)1971 3212 y(@)1989 3231 y(@)-83 b(R)1507 2841 y Fm(Y)35 b(?)28 b(X)p 1652 3231 4 351 v 1653 3231 a Fj(?)1433 3072 y Fg(g)r Fh(?)p Fi(1)1560 3083 y Ff(X)149 3496 y Fe(Pr)n(oof.)42 b Fr(W)-8 b(e)40 b(giv)m(e)e(the)i(pro)s(of)d(for)i(the)g(algebra)e(v)m (ersion)j(of)e(the)h(prop)s(osition.)61 b(Consider)0 3612 y(the)33 b(follo)m(wing)d(comm)m(utativ)m(e)h(diagram)897 4160 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)533 b(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)f Fn(A)p 1375 4137 476 4 v 1768 4135 a Fj(-)1470 4201 y Fi(1)1505 4215 y Fc(M)5 b Fb(\()p Fc(A)p Fb(\))1667 4201 y Fh(\012)p Fg(\016)1085 3770 y Fn(A)909 b(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)p 1187 3747 852 4 v 1956 3745 a Fj(-)1595 3727 y Fg(\016)p 1120 4067 4 254 v 1121 4067 a Fj(?)1049 3964 y Fg(\016)p 2290 4067 V 2291 4067 a Fj(?)2330 3958 y Fi(\001)p Fh(\012)p Fi(1)2479 3969 y Fc(A)0 4352 y Fr(where)47 b(the)g(morphism)d(of)i(algebras)f(\001)i(is) e(de\014ned)j(b)m(y)f(the)g(univ)m(ersal)f(prop)s(ert)m(y)h(of)e Fn(M)10 b Fr(\()p Fn(A)p Fr(\))0 4468 y(with)38 b(resp)s(ect)i(to)e (the)h(algebra)e(morphism)g(\(1)1763 4483 y Fg(M)7 b Fi(\()p Fg(A)p Fi(\))1975 4468 y Fm(\012)27 b Fn(\016)t Fr(\))p Fn(\016)t Fr(.)60 b(F)-8 b(urthermore)38 b(there)h(is)f(a)g (unique)0 4584 y(morphism)31 b(of)h(algebras)g Fn(\017)c Fr(:)g Fn(M)10 b Fr(\()p Fn(A)p Fr(\))28 b Fm(\000)-60 b(!)28 b Fk(K)56 b Fr(suc)m(h)34 b(that)1418 4748 y Fn(A)227 b(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)p 1520 4725 169 4 v 1606 4723 a Fj(-)1587 4704 y Fg(\016)1553 4983 y Fi(1)1588 4994 y Fc(A)1523 4875 y Fj(@)1606 4958 y(@)1689 5041 y(@)1772 5124 y(@)1791 5142 y(@)-83 b(R)1703 5241 y Fn(A)1804 5213 y Fm(\030)1805 5245 y Fr(=)1909 5241 y Fk(K)46 b Fm(\012)22 b Fn(A)p 1941 5143 4 351 v 1942 5143 a Fj(?)1981 4983 y Fg(\017)p Fh(\012)p Fi(1)2100 4994 y Fc(A)p eop %%Page: 25 9 25 8 bop 451 236 a Fp(3.)35 b(QUANTUM)d(MONOIDS)f(AND)i(THEIR)f(A)n (CTIONS)h(ON)f(QUANTUM)h(SP)-6 b(A)n(CES)347 b(25)0 425 y Fr(comm)m(utes.)149 541 y(The)34 b(coalgebra)e(axioms)f(arise)h(from) f(the)i(follo)m(wing)d(comm)m(utativ)m(e)i(diagrams)714 760 y Fn(A)1670 b(M)10 b Fr(\()p Fn(A)p Fr(\))22 b Fm(\012)h Fn(A)p 816 737 1612 4 v 2345 735 a Fj(-)1604 716 y Fg(\016)p 749 1057 4 254 v 750 1057 a Fj(?)789 953 y Fg(\016)p 2679 1057 V 2681 1057 a Fj(?)2720 947 y Fi(\001)p Fh(\012)p Fi(1)2869 958 y Fc(A)526 1150 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)1294 b(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)f Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)p 1004 1127 1237 4 v 2158 1125 a Fj(-)1479 1079 y Fi(1)1514 1093 y Fc(M)5 b Fb(\()p Fc(A)p Fb(\))1676 1079 y Fh(\012)p Fg(\016)p 739 1447 4 254 v 741 1447 a Fj(?)503 1337 y Fi(\001)p Fh(\012)p Fi(1)652 1348 y Fc(A)p 768 1447 V 770 1447 a Fj(?)809 1330 y Fi(1)844 1344 y Fc(M)g Fb(\()p Fc(A)p Fb(\))1006 1330 y Fh(\012)p Fg(\016)p 2660 1447 V 2661 1447 a Fj(?)2172 1329 y Fi(\001)p Fh(\012)p Fi(1)2321 1343 y Fc(M)g Fb(\()p Fc(A)p Fb(\))2482 1329 y Fh(\012)p Fi(1)2572 1340 y Fc(A)p 2689 1447 V 2691 1447 a Fj(?)2730 1329 y Fi(1)2765 1343 y Fc(M)g Fb(\()p Fc(A)p Fb(\))2926 1329 y Fh(\012)p Fi(\001)p Fh(\012)p Fi(1)3130 1340 y Fc(A)329 1540 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)f Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)918 b(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)f Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)p 1181 1517 861 4 v 1959 1515 a Fj(-)1343 1469 y Fi(1)1378 1483 y Fc(M)5 b Fb(\()p Fc(A)p Fb(\))1540 1469 y Fh(\012)p Fi(1)1630 1483 y Fc(M)g Fb(\()p Fc(A)p Fb(\))1791 1469 y Fh(\012)p Fg(\016)0 1744 y Fr(and)859 1903 y Fn(A)1251 b(M)10 b Fr(\()p Fn(A)p Fr(\))22 b Fm(\012)h Fn(A)p 961 1880 1193 4 v 2071 1878 a Fj(-)1540 1859 y Fg(\016)p 894 2200 4 254 v 896 2200 a Fj(?)935 2096 y Fg(\016)p 2259 2200 V 2261 2200 a Fj(?)2022 2090 y Fi(\001)p Fh(\012)p Fi(1)2171 2101 y Fc(A)p 2551 2590 4 644 v 2553 2590 a Fj(?)2592 2276 y Fi(1)2627 2290 y Fc(M)5 b Fb(\()p Fc(A)p Fb(\))2789 2276 y Fh(\012)p Fi(1)2879 2287 y Fc(A)671 2293 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)534 b(M)10 b Fr(\()p Fn(A)p Fr(\))22 b Fm(\012)h Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fn(A)p 1149 2270 476 4 v 1542 2268 a Fj(-)1244 2222 y Fi(1)1279 2236 y Fc(M)5 b Fb(\()p Fc(A)p Fb(\))1441 2222 y Fh(\012)p Fg(\016)1079 2471 y Fi(1)1114 2485 y Fc(M)g Fb(\()p Fc(A)p Fb(\))1276 2471 y Fh(\012)p Fi(1)1366 2482 y Fc(A)1100 2364 y Fj(P)1184 2392 y(P)1267 2419 y(P)1350 2447 y(P)1433 2475 y(P)1516 2502 y(P)1599 2530 y(P)1682 2558 y(P)1765 2585 y(P)1778 2590 y(P)-83 b(q)1646 2683 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)f Fn(A)2122 2655 y Fm(\030)2123 2687 y Fr(=)2227 2683 y Fn(M)10 b Fr(\()p Fn(A)p Fr(\))23 b Fm(\012)g Fk(K)46 b Fm(\012)22 b Fn(A)p 2259 2590 4 254 v 2261 2590 a Fj(?)1801 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Fh(\003)2128 1942 y Fr(of)i Fn(A)27 b Fr(:=)h Fk(K)18 b Fm(h)p Fn(x)p Fm(i)p Fn(=)p Fr(\()p Fn(x)2818 1906 y Fi(3)2863 1942 y Fr(\).)149 2058 y(4.)42 b(Determine)28 b(and)i(describ)s(e)g(the)h(co)s(endomorphism)d (bialgebra)g(of)h Fn(A)g Fr(from)g(problem)f(1.3.)149 2174 y(5.)42 b(\(*\))f(Determine)f(explicitly)f(the)j(dual)f(coalgebra) f Fn(A)2266 2138 y Fh(\003)2347 2174 y Fr(of)h Fn(A)h Fr(:=)h Fk(K)17 b Fm(h)p Fn(x;)h(y)t Fm(i)p Fn(=I)53 b Fr(where)43 b(the)267 2291 y(ideal)31 b Fn(I)40 b Fr(is)32 b(generated)i(as)e(a)h(t)m(w)m(o-sided)g(ideal)e(b)m(y)i(the)g(p)s (olynomials)1412 2447 y Fn(xy)26 b Fm(\000)c Fn(q)1687 2406 y Fh(\000)p Fi(1)1781 2447 y Fn(y)t(x;)17 b(x)1987 2406 y Fi(2)2027 2447 y Fn(;)g(y)2123 2406 y Fi(2)2161 2447 y Fn(:)149 2604 y Fr(6.)42 b(\(*\))32 b(Determine)g(the)h(co)s (endomorphism)e(bialgebra)f(of)i Fn(A)h Fr(from)e(problem)h(1.5.)149 2721 y(7.)42 b(Let)32 b Fn(A)h Fr(b)s(e)f(a)g(\014nite)g(dimensional)e Fk(K)18 b Fr(-algebra)37 b(with)32 b(univ)m(ersal)g(bialgebra)e Fn(A)e Fm(\000)-60 b(!)28 b Fn(B)f Fm(\012)22 b Fn(A)p Fr(.)267 2837 y(Sho)m(w)402 2953 y(i\))41 b(that)k Fn(A)806 2917 y Fg(op)929 2953 y Fm(\000)-60 b(!)49 b Fn(B)1174 2917 y Fg(op)1279 2953 y Fm(\012)31 b Fn(A)1460 2917 y Fg(op)1579 2953 y Fr(is)45 b(univ)m(ersal)g(\(where)h Fn(A)2520 2917 y Fg(op)2639 2953 y Fr(has)g(the)g(m)m(ultiplication)509 3069 y Fm(r)p Fn(\034)39 b Fr(:)28 b Fn(A)22 b Fm(\012)h Fn(A)k Fm(\000)-60 b(!)28 b Fn(A)22 b Fm(\012)h Fn(A)28 b Fm(\000)-60 b(!)28 b Fn(A)p Fr(\);)375 3185 y(ii\))40 b(that)32 b Fn(A)821 3158 y Fm(\030)822 3190 y Fr(=)926 3185 y Fn(A)999 3149 y Fg(op)1105 3185 y Fr(implies)f Fn(B)1543 3158 y Fm(\030)1544 3190 y Fr(=)1649 3185 y Fn(B)1728 3149 y Fg(op)1834 3185 y Fr(\(as)i(bialgebras\);)348 3302 y(iii\))39 b(that)25 b(for)h(comm)m(utativ)m(e)f(algebras)g Fn(A)h Fr(the)h(algebra)d Fn(B)31 b Fr(satis\014es)c Fn(B)2965 3274 y Fm(\030)2966 3306 y Fr(=)3070 3302 y Fn(B)3149 3266 y Fg(op)3249 3302 y Fr(but)f(that)509 3418 y Fn(B)37 b Fr(need)d(not)e(b)s(e)h(comm)m(utativ)m(e.)351 3534 y(iv\))41 b(Find)25 b(an)g(isomorphism)e Fn(B)1532 3507 y Fm(\030)1533 3538 y Fr(=)1637 3534 y Fn(B)1716 3498 y Fg(op)1816 3534 y Fr(for)i(the)h(bialgebra)e Fn(B)33 b Fr(=)27 b Fk(K)18 b Fm(h)p Fn(a;)f(b)p Fm(i)p Fn(=)p Fr(\()p Fn(a)3169 3498 y Fi(2)3214 3534 y Fn(;)g(ab)8 b Fr(+)g Fn(ba)p Fr(\).)509 3650 y(\(compare)32 b(problem)f(1.2)h(2\).) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF