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3250 y Fq(f)11 b Fu(\()p Fq(x)p Fu(\))p Fq(g)t Fu(\()p Fq(y)t Fu(\).)40 b(In)26 b(order)h(to)f(obtain)f(a)h(unique)h(homomorphism)d Fq(\034)38 b Fu(de\014ned)28 b(on)e(the)h(tensor)g(pro)s(duct)0 3366 y(w)m(e)32 b(ha)m(v)m(e)h(to)e(sho)m(w)i(that)e Fq(\034)988 3330 y Fn(0)1039 3366 y Fu(:)d Fr(K)1166 3330 y Fl(X)1259 3366 y Fp(\002)20 b Fr(K)1427 3330 y Fl(Y)1522 3366 y Fp(\000)-60 b(!)27 b Fr(K)1738 3330 y Fl(X)5 b Fn(\002)p Fl(Y)1954 3366 y Fu(is)31 b(a)g(bilinear)e(map)i (:)43 b Fq(\034)2821 3330 y Fn(0)2844 3366 y Fu(\()p Fq(f)31 b Fu(+)19 b Fq(f)3115 3330 y Fn(0)3138 3366 y Fq(;)e(g)t Fu(\)\()p Fq(x;)g(y)t Fu(\))26 b(=)0 3483 y(\()p Fq(f)15 b Fu(+)t Fq(f)240 3446 y Fn(0)263 3483 y Fu(\)\()p Fq(x)p Fu(\))p Fq(g)t Fu(\()p Fq(y)t Fu(\))26 b(=)h(\()p Fq(f)11 b Fu(\()p Fq(x)p Fu(\))t(+)t Fq(f)1111 3446 y Fn(0)1134 3483 y Fu(\()p Fq(x)p Fu(\)\))p Fq(g)t Fu(\()p Fq(y)t Fu(\))27 b(=)g Fq(f)11 b Fu(\()p Fq(x)p Fu(\))p Fq(g)t Fu(\()p Fq(y)t Fu(\))t(+)t Fq(f)2124 3446 y Fn(0)2146 3483 y 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Fj(\001)3267 3983 y Fp(\000)-16 b(!)27 b Fr(K)3527 3947 y Fl(X)0 4099 y Fu(where)35 b(\001)c(:)f Fq(X)39 b Fp(\000)-60 b(!)30 b Fq(X)h Fp(\002)24 b Fq(X)42 b Fu(in)33 b Fs(Set)h Fu(is)g(the)g(diagonal)e(map)h(\001\()p Fq(x)p Fu(\))e(:=)g(\()p Fq(x;)17 b(x)p Fu(\).)48 b(F)-8 b(urthermore)34 b(w)m(e)0 4245 y(get)26 b(a)g(unit)f Fq(\021)32 b Fu(:)c Fr(K)636 4208 y Fn(f\003)q(g)819 4188 y Fk(K)865 4165 y Fi(\017)780 4245 y Fp(\000)-16 b(!)27 b Fr(K)1039 4208 y Fl(X)1139 4245 y Fu(where)g Fq(\017)h Fu(:)g Fq(X)36 b Fp(\000)-60 b(!)27 b(f\003g)f Fu(is)f(the)i(unique)g(map)e(in)m(to)g(the)i(one)f(elemen)m(t)0 4361 y(set.)55 b(One)36 b(v)m(eri\014es)i(easily)d(that)h(\()p Fr(K)1336 4325 y Fl(X)1410 4361 y Fq(;)17 b(\021)t(;)g Fp(r)p Fu(\))35 b(is)h(a)g Fr(K)17 b Fu(-algebra.)59 b(Tw)m(o)37 b(prop)s(erties)f(are)g(essen)m(tial)0 4477 y(here,)e(the)f(asso)s(ciativit)m(y)f(and)h(the)g(unit)f(of)h Fr(K)56 b Fu(and)33 b(the)g(fact)g(that)g(\()p Fq(X)r(;)17 b Fu(\001)p Fq(;)g(\017)p Fu(\))33 b(is)f(a)h(\\comonoid")0 4593 y(in)f(the)h(category)g Fs(Set)p Fu(:)1208 5208 y Fq(X)d Fp(\002)22 b Fq(X)383 b(X)30 b Fp(\002)23 b Fq(X)30 b Fp(\002)22 b Fq(X)p 1536 5181 318 4 v 1771 5179 a Fh(-)1620 5243 y Fg(\001)p Fn(\002)p Fg(1)1313 4721 y Fq(X)594 b(X)30 b Fp(\002)22 b Fq(X)p 1430 4693 529 4 v 1876 4691 a Fh(-)1665 4672 y Fg(\001)p 1356 5110 4 351 v 1357 5110 a Fh(?)1260 4958 y Fg(\001)p 2136 5110 V 2137 5110 a Fh(?)2176 4954 y Fg(1)p Fn(\002)p Fg(\001)p eop %%Page: 4 4 4 3 bop 0 236 a Fm(4)416 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)970 436 y Fq(X)984 b(X)30 b Fp(\002)22 b Fq(X)p 1088 408 919 4 v 1924 406 a Fh(-)1518 387 y Fg(\001)p 1013 728 4 254 v 1015 728 a Fh(?)917 624 y Fg(\001)1664 617 y(1)1699 628 y Fi(X)1219 502 y Fh(P)1302 530 y(P)1385 558 y(P)1468 585 y(P)1552 613 y(P)1635 641 y(P)1718 668 y(P)1801 696 y(P)1884 724 y(P)1897 728 y(P)-83 b(q)p 2183 728 V 205 w(?)2224 619 y Fg(1)p Fn(\002)p Fl(\017)865 821 y Fq(X)30 b Fp(\002)23 b Fq(X)477 b Fp(f\003g)21 b(\002)i Fq(X)2022 794 y Fp(\030)2022 826 y Fu(=)2127 821 y Fq(X)2243 794 y Fp(\030)2244 826 y Fu(=)2348 821 y Fq(X)30 b Fp(\002)23 b(f\003g)p Fq(:)p 1193 798 413 4 v 1523 796 a Fh(-)1339 858 y Fl(\017)p Fn(\002)p Fg(1)0 997 y Fu(Since)29 b Fr(K)323 961 y Fu(-)395 997 y(is)g(a)g(functor)g (these)i(t)m(w)m(o)f(diagrams)d(carry)j(o)m(v)m(er)g(to)f(the)h (category)f Fs(V)-9 b(ec)29 b Fu(and)g(pro)s(duce)0 1114 y(the)k(required)g(diagrams)e(for)h(a)g Fr(K)18 b Fu(-algebra.)149 1230 y(F)-8 b(or)36 b(a)h(map)e Fq(f)46 b Fu(:)34 b Fq(X)43 b Fp(\000)-60 b(!)34 b Fq(Y)58 b Fu(w)m(e)38 b(obtain)d(a)i (homomorphism)d(of)i(algebras)g Fr(K)2983 1194 y Fl(f)3069 1230 y Fu(:)f Fr(K)3202 1194 y Fl(Y)3304 1230 y Fp(\000)-60 b(!)34 b Fr(K)3527 1194 y Fl(X)0 1346 y Fu(b)s(ecause)g(the)f(diagrams) 908 1552 y Fr(K)979 1516 y Fl(Y)1068 1552 y Fp(\012)23 b Fr(K)1239 1516 y Fl(Y)1713 1552 y Fr(K)1785 1516 y Fl(Y)e Fn(\002)p Fl(Y)p 1335 1517 350 4 v 1601 1515 a Fh(-)1490 1497 y Fl(\034)901 2040 y Fr(K)973 2003 y Fl(X)1068 2040 y Fp(\012)i Fr(K)1239 2003 y Fl(X)1707 2040 y Fr(K)1778 2003 y Fl(X)5 b Fn(\002)p Fl(X)p 1342 2005 337 4 v -359 w Fh(-)1490 1984 y Fl(\034)2500 1556 y Fr(K)2572 1520 y Fl(Y)p 1992 1517 480 4 v 2389 1515 a Fh(-)2179 1496 y Fk(K)2225 1473 y Fj(\001)2497 2044 y Fr(K)2568 2008 y Fl(X)p 1998 2005 470 4 v 2385 2003 a Fh(-)2180 1984 y Fk(K)2226 1960 y Fj(\001)p 1105 1935 4 351 v 1107 1935 a Fh(?)1146 1785 y Fk(K)1191 1761 y Fi(f)1237 1785 y Fn(\012)p Fk(K)1338 1761 y Fi(f)p 1836 1935 V 1838 1935 a Fh(?)1877 1788 y Fk(K)1923 1765 y Fi(f)h Ff(\002)p Fi(f)p 2568 1935 V 2569 1935 a Fh(?)2608 1788 y Fk(K)2654 1765 y Fi(f)0 2191 y Fu(and)1599 2310 y Fr(K)1671 2274 y Fn(f\003g)1814 2282 y Fp(\030)1815 2314 y Fu(=)1919 2310 y Fr(K)1823 2521 y Fl(\021)1832 2420 y Fh(A)1874 2503 y(A)1915 2586 y(A)1957 2669 y(A)1966 2688 y(A)-42 b(U)1736 2521 y Fl(\021)1722 2420 y Fh(\001)1681 2503 y(\001)1639 2586 y(\001)1598 2669 y(\001)1588 2688 y(\001)g(\013)1485 2797 y Fr(K)1557 2761 y Fl(Y)1969 2797 y Fr(K)2041 2761 y Fl(X)p 1652 2759 289 4 v 1857 2757 a Fh(-)1750 2738 y Fk(K)1796 2714 y Fi(f)0 2918 y Fu(comm)m(ute.)149 3034 y(Th)m(us)1371 3176 y Fr(K)1443 3134 y Fu(-)1513 3176 y(:)28 b Fs(Set)f Fp(\000)-60 b(!)28 b Fr(K)18 b Fu(-)5 b Fs(cAlg)0 3337 y Fu(is)32 b(a)g(con)m(tra)m(v)-5 b(arian)m(t)33 b(functor.)149 3453 y(By)45 b(the)f(de\014nition)g(of)f(the)i (set-theoretic)f(\(cartesian\))g(pro)s(duct)g(w)m(e)h(kno)m(w)g(that)f Fr(K)3404 3417 y Fl(X)3524 3453 y Fu(=)0 3495 y Fe(Q)94 3599 y Fl(X)178 3570 y Fr(K)18 b Fu(.)49 b(This)32 b(iden)m(tit)m(y)f (do)s(es)h(not)g(only)f(hold)g(on)h(the)g(set)g(lev)m(el,)g(it)f(holds) g(also)g(for)g(the)h(algebra)0 3688 y(structures)i(on)f Fr(K)663 3652 y Fl(X)769 3688 y Fu(resp.)1014 3614 y Fe(Q)1108 3718 y Fl(X)1192 3688 y Fr(K)17 b Fu(.)149 3805 y(W)-8 b(e)33 b(no)m(w)h(construct)f(an)g(in)m(v)m(erse)h(functor) 1316 3987 y(Sp)s(ec)28 b(:)g Fr(K)18 b Fu(-)5 b Fs(cAlg)26 b Fp(\000)-60 b(!)28 b Fs(Set)p Fq(:)0 4168 y Fu(F)-8 b(or)35 b(eac)m(h)h(p)s(oin)m(t)f Fq(x)f Fp(2)f Fq(X)43 b Fu(there)37 b(is)e(a)g(maximal)e(ideal)h Fq(m)2131 4183 y Fl(x)2211 4168 y Fu(of)2325 4094 y Fe(Q)2419 4198 y Fl(X)2503 4168 y Fr(K)59 b Fu(de\014ned)37 b(b)m(y)g Fq(m)3179 4183 y Fl(x)3256 4168 y Fu(:=)c Fp(f)p Fq(f)44 b Fp(2)0 4285 y Fu(Map\()p Fq(X)r(;)17 b Fr(K)h Fu(\))p Fp(j)p Fq(f)11 b Fu(\()p Fq(x)p Fu(\))39 b(=)34 b(0)p Fp(g)p Fu(.)54 b(If)36 b Fq(X)44 b Fu(is)35 b(a)h(\014nite)g(set)h (then)g(these)g(are)f(exactly)h(all)d(maximal)f(ideals)0 4401 y(of)111 4326 y Fe(Q)205 4430 y Fl(X)289 4401 y Fr(K)18 b Fu(.)50 b(T)-8 b(o)33 b(sho)m(w)h(this)f(w)m(e)g(observ)m(e)i (the)e(follo)m(wing.)42 b(The)34 b(surjectiv)m(e)g(homomorphism)c Fq(p)3501 4416 y Fl(x)3573 4401 y Fu(:)0 4442 y Fe(Q)94 4546 y Fl(X)178 4517 y Fr(K)52 b Fp(\000)-60 b(!)28 b Fr(K)57 b Fu(has)33 b(k)m(ernel)h Fq(m)1087 4532 y Fl(x)1164 4517 y Fu(hence)g Fq(m)1520 4532 y Fl(x)1597 4517 y Fu(is)f(a)f (maximal)e(ideal.)43 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y(No)m(w)c(w)m(e)f(sho)m(w)h(that)e Fp(X)42 b Fu(is)26 b(represen)m(table)i(b)m(y)f Fp(O)s Fu(\()p Fp(X)15 b Fu(\))28 b(=)f Fr(K)18 b Fu([)p Fq(x)2395 3481 y Fg(1)2441 3466 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)2731 3481 y Fl(n)2778 3466 y Fu(])p Fq(=)p Fu(\()p Fq(p)2941 3481 y Fg(1)2980 3466 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(p)3265 3481 y Fl(m)3331 3466 y Fu(\).)41 b(Ob-)0 3583 y(serv)m(e)26 b(that)e(\()p Fq(p)526 3598 y Fg(1)565 3583 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(p)850 3598 y Fl(m)916 3583 y Fu(\))24 b(denotes)h(the)g(\(t)m(w)m(o-sided\))f(ideal)f(in)g Fr(K)18 b Fu([)p Fq(x)2467 3598 y Fg(1)2513 3583 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)2803 3598 y Fl(n)2850 3583 y Fu(])25 b(generated)f(b)m(y)i (the)0 3699 y(p)s(olynomials)21 b Fq(p)583 3714 y Fg(1)623 3699 y Fq(;)c(:)g(:)g(:)32 b(;)17 b(p)907 3714 y Fl(m)973 3699 y Fu(.)41 b(W)-8 b(e)25 b(kno)m(w)h(that)e(eac)m(h)h Fq(n)p Fu(-tup)s(el)f(\()p Fq(a)2286 3714 y Fg(1)2326 3699 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(a)2612 3714 y Fl(n)2659 3699 y Fu(\))28 b Fp(2)g Fq(A)2892 3663 y Fl(n)2963 3699 y Fu(uniquely)d(deter-)0 3815 y(mines)33 b(an)g(algebra)f (homomorphism)e Fq(f)40 b Fu(:)29 b Fr(K)17 b Fu([)p Fq(x)1743 3830 y Fg(1)1788 3815 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(x)2078 3830 y Fl(n)2126 3815 y Fu(])28 b Fp(\000)-60 b(!)29 b Fq(A)34 b Fu(b)m(y)g Fq(f)11 b Fu(\()p Fq(x)2722 3830 y Fg(1)2761 3815 y Fu(\))29 b(=)g Fq(a)2984 3830 y Fg(1)3023 3815 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(f)11 b Fu(\()p Fq(x)3411 3830 y Fl(n)3457 3815 y Fu(\))29 b(=)0 3931 y Fq(a)51 3946 y Fl(n)98 3931 y Fu(.)52 b(\(The)37 b(p)s(olynomial)32 b(ring)i Fr(K)18 b Fu([)p Fq(x)1283 3946 y Fg(1)1329 3931 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)1619 3946 y Fl(n)1666 3931 y Fu(])36 b(in)e Fr(K)18 b Fu(-)5 b Fs(cAlg)34 b Fu(is)h(free)h(o)m(v)m(er)h(the)f(set)g Fp(f)p Fq(x)3146 3946 y Fg(1)3186 3931 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(x)3476 3946 y Fl(n)3523 3931 y Fp(g)p Fu(,)0 4047 y(or)36 b Fr(K)17 b Fu([)p Fq(x)277 4062 y Fg(1)322 4047 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)613 4062 y Fl(n)660 4047 y Fu(])36 b(together)g(with)g(the)g(em)m(b)s(edding)g Fq(\023)e Fu(:)f Fp(f)p Fq(x)2238 4062 y Fg(1)2278 4047 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(x)2569 4062 y Fl(n)2616 4047 y Fp(g)33 b(\000)-60 b(!)34 b Fr(K)17 b Fu([)p Fq(x)3004 4062 y Fg(1)3049 4047 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(x)3339 4062 y Fl(n)3387 4047 y Fu(])36 b(is)f(a)0 4164 y(couniv)m(ersal)h (solution)f(of)h(the)g(problem)f(giv)m(en)i(b)m(y)g(the)g(underlying)e (functor)i Fr(A)61 b Fu(:)34 b Fr(K)18 b Fu(-)5 b Fs(cAlg)33 b Fp(\000)-60 b(!)0 4280 y Fs(Set)33 b Fu(and)g(the)g(set)g Fp(f)p Fq(x)805 4295 y Fg(1)845 4280 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(x)1136 4295 y Fl(n)1183 4280 y Fp(g)28 b(2)g Fs(Set)p Fu(.\))44 b(This)33 b(homomorphism)d(of)j(algebras)f(maps)g(p) s(olyno-)0 4396 y(mials)i Fq(p)p Fu(\()p Fq(x)401 4411 y Fg(1)441 4396 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(x)731 4411 y Fl(n)779 4396 y Fu(\))36 b(in)m(to)g Fq(f)11 b Fu(\()p Fq(p)p Fu(\))34 b(=)g Fq(p)p Fu(\()p Fq(a)1521 4411 y Fg(1)1561 4396 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(a)1847 4411 y Fl(n)1894 4396 y Fu(\).)56 b(Hence)38 b(\()p Fq(a)2398 4411 y Fg(1)2437 4396 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(a)2724 4411 y Fl(n)2770 4396 y Fu(\))37 b(is)f(a)g(common)g (zero)0 4512 y(of)27 b(the)i(p)s(olynomials)c Fq(p)857 4527 y Fg(1)896 4512 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(p)1181 4527 y Fl(m)1275 4512 y Fu(if)26 b(and)i(only)g(if)f Fq(f)11 b Fu(\()p Fq(p)1985 4527 y Fl(i)2013 4512 y Fu(\))27 b(=)h Fq(p)2231 4527 y Fl(i)2259 4512 y Fu(\()p Fq(a)2348 4527 y Fg(1)2387 4512 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(a)2674 4527 y Fl(n)2721 4512 y Fu(\))27 b(=)h(0,)h(i.e.)41 b Fq(p)3210 4527 y Fg(1)3249 4512 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(p)3534 4527 y Fl(m)0 4629 y Fu(are)27 b(in)g(the)h(k)m(ernel)g (of)e Fq(f)11 b Fu(.)42 b(This)27 b(happ)s(ens)i(if)d(and)h(only)g(if)f Fq(f)38 b Fu(v)-5 b(anishes)28 b(on)g(the)f(ideal)f(\()p Fq(p)3172 4644 y Fg(1)3211 4629 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(p)3496 4644 y Fl(m)3562 4629 y Fu(\))0 4745 y(or)32 b(in)g(other)h(w)m(ord)g(can)g(b)s(e)g(factorized)f(through)g (the)h(residue)g(class)g(map)799 4902 y Fq(\027)i Fu(:)27 b Fr(K)18 b Fu([)p Fq(x)1090 4917 y Fg(1)1136 4902 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)1426 4917 y Fl(n)1473 4902 y Fu(])28 b Fp(\000)-60 b(!)28 b Fr(K)17 b Fu([)p Fq(x)1827 4917 y Fg(1)1872 4902 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)2163 4917 y Fl(n)2210 4902 y Fu(])p Fq(=)p Fu(\()p Fq(p)2373 4917 y Fg(1)2412 4902 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(p)2696 4917 y Fl(m)2763 4902 y Fu(\))0 5059 y(This)33 b(induces)g(a)f(bijection)179 5216 y(Mor)355 5231 y Fk(K)5 b Fu(-)g Fc(cAlg)604 5216 y Fu(\()p Fr(K)18 b Fu([)p Fq(x)796 5231 y Fg(1)842 5216 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)1132 5231 y Fl(n)1179 5216 y Fu(])p Fq(=)p Fu(\()p Fq(p)1342 5231 y Fg(1)1381 5216 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(p)1666 5231 y Fl(m)1732 5216 y Fu(\))p Fq(;)g(A)p Fu(\))27 b Fp(3)h Fq(f)39 b Fp(7!)27 b Fu(\()p Fq(f)11 b Fu(\()p Fq(x)2450 5231 y Fg(1)2489 5216 y Fu(\))p Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(f)11 b Fu(\()p Fq(x)2915 5231 y Fl(n)2962 5216 y Fu(\)\))27 b Fp(2)h(X)15 b Fu(\()p Fq(A)p Fu(\))p Fq(:)p eop %%Page: 7 7 7 6 bop 542 236 a Fm(1.)34 b(THE)g(PRINCIPLES)f(OF)g(COMMUT)-6 b(A)g(TIVE)33 b(ALGEBRAIC)g(GEOMETR)-6 b(Y)478 b(7)0 425 y Fu(No)m(w)33 b(it)f(is)g(easy)h(to)g(see)g(that)g(this)f (bijection)f(is)h(a)h(natural)e(isomorphism)f(\(in)i Fq(A)p Fu(\).)p 3529 425 4 66 v 3533 363 59 4 v 3533 425 V 3591 425 4 66 v 149 638 a(If)26 b(no)g(p)s(olynomials)d(are)j (giv)m(en)g(for)g(the)g(ab)s(o)m(v)m(e)h(construction,)g(then)g(the)f (functor)h(under)f(this)0 755 y(construction)i(is)f(the)i(a\016ne)f (space)h Fr(A)1388 719 y Fl(n)1469 755 y Fu(of)e(dimension)g Fq(n)p Fu(.)42 b(By)28 b(giving)f(p)s(olynomials)d(the)29 b(functor)0 871 y Fp(X)43 b Fu(b)s(ecomes)28 b(a)g(subfunctor)h(of)e Fr(A)1227 835 y Fl(n)1280 871 y Fu(,)i(b)s(ecause)h(it)d(de\014nes)i (subsets)h Fp(X)15 b Fu(\()p Fq(A)p Fu(\))28 b Fp(\022)g Fr(A)2868 835 y Fl(n)2921 871 y Fu(\()p Fq(A)p Fu(\))f(=)h Fq(A)3274 835 y Fl(n)3321 871 y Fu(.)42 b(Both)0 987 y(functors)26 b(are)g(represen)m(table)h(functors.)42 b(The)27 b(em)m(b)s(edding)e(is)h(induced)g(b)m(y)h(the)f(homomorphism) 0 1103 y(of)32 b(algebras)g Fq(\027)i Fu(:)28 b Fr(K)17 b Fu([)q Fq(x)782 1118 y Fg(1)827 1103 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)1118 1118 y Fl(n)1165 1103 y Fu(])28 b Fp(\000)-60 b(!)27 b Fr(K)18 b Fu([)p Fq(x)1518 1118 y Fg(1)1564 1103 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)1854 1118 y Fl(n)1901 1103 y Fu(])p Fq(=)p Fu(\()p Fq(p)2064 1118 y Fg(1)2104 1103 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(p)2388 1118 y Fl(m)2454 1103 y Fu(\).)149 1300 y Fs(Problem)36 b(1.1.1.)199 b Fu(1.)41 b(Determine)36 b(the)h(a\016ne)g(algebra)e(of)h(the)h(functor) f(\\unit)g(circle")267 1416 y Fq(S)333 1380 y Fg(1)405 1416 y Fu(in)31 b Fr(A)584 1380 y Fg(2)630 1416 y Fu(.)149 1532 y(2.)42 b(Determine)31 b(the)i(a\016ne)g(algebra)f(of)g(the)h (functor)g(\\unit)e(sphere")j Fq(S)2803 1496 y Fl(n)p Fn(\000)p Fg(1)2973 1532 y Fu(in)e Fr(A)3152 1496 y Fl(n)3205 1532 y Fu(.)149 1648 y(3.)42 b(Let)33 b Fp(X)47 b Fu(denote)33 b(the)g(plane)f(curv)m(e)i Fq(y)d Fu(=)c Fq(x)1799 1612 y Fg(2)1839 1648 y Fu(.)44 b(Then)33 b Fp(X)47 b Fu(is)32 b(isomorphic)f(to)h(the)h(a\016ne)g(line.)149 1765 y(4.)42 b(Let)35 b Fp(Y)43 b Fu(denote)36 b(the)f(plane)g(curv)m(e)i Fq(xy)e Fu(=)d(1.)51 b(Then)36 b Fp(Y)43 b Fu(is)35 b(not)g(isomorphic) e(to)i(the)g(a\016ne)267 1881 y(line.)58 b(\(Hin)m(t:)53 b(An)38 b(isomorphism)d Fr(K)18 b Fu([)p Fq(x;)f(x)1806 1845 y Fn(\000)p Fg(1)1907 1881 y Fu(])36 b Fp(\000)-60 b(!)37 b Fr(K)17 b Fu([)p Fq(y)t Fu(])43 b(sends)d Fq(x)e Fu(to)f(a)h(p)s(olynomial)c Fq(p)p Fu(\()p Fq(y)t Fu(\))267 1997 y(whic)m(h)43 b(m)m(ust)f(b)s(e)g(in)m(v)m(ertible.)72 b(Consider)43 b(the)f(highest)h(co)s(e\016cien)m(t)g(of)e Fq(p)p Fu(\()p Fq(y)t Fu(\))g(and)i(sho)m(w)267 2113 y(that)32 b Fq(p)p Fu(\()p Fq(y)t Fu(\))27 b Fp(2)h Fr(K)17 b Fu(.)50 b(But)33 b(that)f(means)h(that)f(the)h(map)f(cannot)g(b)s(e)h (bijectiv)m(e.\))149 2230 y(5.)42 b(Let)47 b Fr(K)77 b Fu(=)53 b Fr(C)73 b Fu(b)s(e)48 b(the)g(\014eld)f(of)g(complex)g(n)m (um)m(b)s(ers.)89 b(Sho)m(w)49 b(that)e(the)h(unit)f(functor)267 2346 y Fq(U)d Fu(:)34 b Fr(K)17 b Fu(-)6 b Fs(cAlg)32 b Fp(\000)-60 b(!)34 b Fs(Set)i Fu(in)f(Lemma)f(2.3.7)i(is)g(naturally) e(isomorphic)g(to)i(the)g(unit)g(circle)267 2462 y(functor)f Fq(S)674 2426 y Fg(1)713 2462 y Fu(.)51 b(\(Hin)m(t:)d(There)36 b(is)e(an)h(algebra)f(isomorphism)e(b)s(et)m(w)m(een)38 b(the)d(represen)m(ting)267 2578 y(algebras)d Fr(K)17 b Fu([)p Fq(e)q(;)g(e)880 2542 y Fn(\000)p Fg(1)980 2578 y Fu(])32 b(and)h Fr(K)18 b Fu([)p Fq(c;)f(s)p Fu(])p Fq(=)p Fu(\()p Fq(c)1616 2542 y Fg(2)1683 2578 y Fu(+)22 b Fq(s)1827 2542 y Fg(2)1888 2578 y Fp(\000)h Fu(1\).\))149 2695 y(6.)267 2658 y Fn(\003)337 2695 y Fu(Let)30 b Fr(K)54 b Fu(b)s(e)31 b(an)f(algebraically)d(closed)k(\014eld.)43 b(Let)30 b Fq(p)g Fu(b)s(e)h(an)f(irreducible)g(square)h(p)s(olyno-)267 2811 y(mial)d(in)i Fr(K)17 b Fu([)p Fq(x)q(;)g(y)t Fu(].)48 b(Let)31 b Fp(Z)39 b Fu(b)s(e)31 b(the)g(conic)f(section)h(de\014ned)i (b)m(y)e Fq(p)g Fu(with)f(the)i(a\016ne)f(algebra)267 2927 y Fr(K)17 b Fu([)p Fq(x)q(;)g(y)t Fu(])p Fq(=)p Fu(\()p Fq(p)p Fu(\).)79 b(Sho)m(w)44 b(that)f Fp(Z)51 b Fu(is)42 b(naturally)g(isomorphic)f(either)i(to)g Fp(X)57 b Fu(or)43 b(to)f Fp(Y)51 b Fu(from)267 3043 y(parts)33 b(3.)43 b(resp.)h(4.)149 3240 y Fs(Remark)37 b(1.1.8.)50 b Fu(A\016ne)37 b(algebras)f(of)h(a\016ne)g(sc)m(hemes)i(are)e (\014nitely)f(generated)h(comm)m(u-)0 3356 y(tativ)m(e)32 b(algebras)f(and)h(an)m(y)g(suc)m(h)i(algebra)c(is)i(an)g(a\016ne)g (algebra)f(of)g(some)h(a\016ne)g(sc)m(heme,)i(since)0 3472 y Fq(A)101 3444 y Fp(\030)102 3476 y Fu(=)206 3472 y Fr(K)17 b Fu([)q Fq(x)360 3487 y Fg(1)405 3472 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)696 3487 y Fl(n)743 3472 y Fu(])p Fq(=)p Fu(\()p Fq(p)906 3487 y Fg(1)945 3472 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(p)1230 3487 y Fl(m)1296 3472 y Fu(\))32 b(\(Hilb)s(ert)f(basis)i (theorem\).)149 3588 y(The)i(p)s(olynomials)c Fq(p)944 3603 y Fg(1)983 3588 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(p)1267 3603 y Fl(m)1367 3588 y Fu(are)34 b(not)f(uniquely)h (determined)f(b)m(y)i(the)f(a\016ne)g(algebra)e(of)0 3705 y(an)e(a\016ne)h(sc)m(heme.)44 b(Not)31 b(ev)m(en)h(the)e(ideal)f (generated)j(b)m(y)f(the)g(p)s(olynomials)c(in)j(the)h(p)s(olynomial)0 3821 y(ring)e Fr(K)17 b Fu([)p Fq(x)352 3836 y Fg(1)397 3821 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)688 3836 y Fl(n)735 3821 y Fu(])30 b(is)f(uniquely)h(determined.)43 b(Also)29 b(the)i(n)m(um)m(b)s(er)f(of)f(v)-5 b(ariables)29 b Fq(x)3127 3836 y Fg(1)3167 3821 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(x)3457 3836 y Fl(n)3534 3821 y Fu(is)0 3937 y(not)32 b(uniquely)h(determined.)149 4053 y(The)e Fr(K)18 b Fu(-p)s(oin)m(ts)35 b(\()p Fq(\013)848 4068 y Fg(1)887 4053 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(\013)1185 4068 y Fl(n)1231 4053 y Fu(\))28 b Fp(2)g(X)15 b Fu(\()p Fr(K)i Fu(\))36 b(of)29 b(an)h(a\016ne)g(sc)m (heme)i Fp(X)44 b Fu(\(with)29 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y(t)m(w)m(o)f(complex)f (p)s(oin)m(ts)g(\(with)g(co)s(e\016cien)m(ts)h(in)e(the)i(\014eld)f Fr(C)69 b Fu(of)43 b(complex)g(n)m(um)m(b)s(ers\))h(and)g(the)0 4867 y(sc)m(heme)39 b Fp(X)52 b Fu(has)38 b(in\014nitely)e(man)m(y)i (complex)f(p)s(oin)m(ts,)h(hence)h Fp(X)15 b Fu(\()p Fr(C)20 b Fu(\))42 b Fp(6)2585 4839 y(\030)2586 4871 y Fu(=)2698 4867 y Fp(Y)8 b Fu(\()p Fr(C)20 b Fu(\).)64 b(This)38 b(do)s(es)g(not)0 4983 y(result)g(from)f(the)i(em)m(b)s (eddings)f(in)m(to)f(di\013eren)m(t)i(spaces)g Fr(A)2188 4947 y Fg(2)2272 4983 y Fu(resp.)61 b Fr(A)2599 4947 y Fg(1)2644 4983 y Fu(.)g(In)38 b(fact)g(w)m(e)h(also)f(ha)m(v)m(e)0 5099 y Fp(O)s Fu(\()p Fp(Y)8 b Fu(\))27 b(=)h Fr(K)17 b Fu([)q Fq(x)p Fu(])p Fq(=)p Fu(\()p Fq(x)687 5063 y Fg(2)739 5099 y Fu(+)6 b(1\))936 5072 y Fp(\030)937 5103 y Fu(=)1041 5099 y Fr(K)18 b Fu([)p Fq(x;)f(y)t Fu(])p Fq(=)p Fu(\()p Fq(x)1460 5063 y Fg(2)1511 5099 y Fu(+)6 b(1)p Fq(;)17 b(y)t Fu(\),)25 b(so)h Fp(Y)32 b Fu(can)25 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Fu(should)f(b)s(e)h(an)m(ything)f(lik)m(e)g(an)g(algebra)g(of) g(functions)g(on)h(these)g(geometric)f(sets.)44 b(It)0 774 y(is)38 b(not)g(ev)m(en)i(clear)e(where)h(these)h(functions)e (should)h(assume)g(their)e(v)-5 b(alues.)61 b(Only)38 b(if)f(w)m(e)j(can)0 890 y(sho)m(w)29 b(that)f Fq(A)g Fu(can)h(b)s(e)f(view)m(ed)h(as)g(a)e(reasonable)h(algebra)f(of)h (functions,)h(w)m(e)g(should)f(talk)f(ab)s(out)0 1006 y(a)35 b(realization)f(of)h(the)h(dream)f(of)g(geometry)-8 b(.)53 b(But)36 b(this)f(will)f(b)s(e)i(done)g(in)f(the)h(follo)m(wing) d(theo-)0 1123 y(rem.)55 b(W)-8 b(e)37 b(will)e(consider)i(functions)f (as)h(maps)f(\(co)s(ordinate)g(functions\))h(from)e(the)i(geometric)0 1239 y(set)f Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))35 b(to)g(the)h(set)h(of) e(co)s(ordinates)g Fq(B)5 b Fu(,)36 b(maps)f(that)g(are)h(natural)e(in) h Fq(B)5 b Fu(.)52 b(Suc)m(h)37 b(co)s(ordinate)0 1355 y(functions)c(are)f(just)h(natural)e(transformations)g(from)h Fp(X)47 b Fu(to)32 b(the)h(underlying)f(functor)h Fr(A)22 b Fu(.)149 1537 y Fs(Theorem)38 b(1.1.11.)49 b Fo(L)-5 b(et)32 b Fp(X)46 b Fo(b)-5 b(e)32 b(a)f(ge)-5 b(ometric)31 b(sp)-5 b(ac)g(e)31 b(with)h(the)f(a\016ne)g(algebr)-5 b(a)31 b Fq(A)d Fu(=)g Fp(O)s Fu(\()p Fp(X)15 b Fu(\))p Fo(.)0 1654 y(Then)27 b Fq(A)348 1626 y Fp(\030)349 1658 y Fu(=)453 1654 y(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))34 b Fo(as)27 b Fr(K)18 b Fo(-algebr)-5 b(as,)34 b(wher)-5 b(e)27 b Fr(A)56 b Fu(:)28 b Fr(K)17 b Fo(-)6 b Fs(cAlg)26 b Fp(\000)-58 b(!)28 b Fs(Set)g Fo(is)g(the)g(underlying)f(func-)0 1770 y(tor)32 b(or)f(a\016ne)g(line.)43 b(The)31 b(isomorphism)f Fq(A)1640 1742 y Fp(\030)1641 1774 y Fu(=)1745 1770 y(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))38 b Fo(induc)-5 b(es)31 b(a)g(natur)-5 b(al)32 b(tr)-5 b(ansformation)0 1886 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))27 b Fp(\000)-58 b(!)28 b Fq(B)40 b Fo(\(natur)-5 b(al)35 b(in)g Fq(B)5 b Fo(\).)149 2068 y Fd(Pr)n(oof.)42 b Fu(First)52 b(w)m(e)i(de\014ne)g(an)f(isomorphism)d(b)s(et)m(w)m(een)55 b(the)e(sets)h Fq(A)f Fu(and)g(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\).)0 2185 y(Because)48 b(of)d Fp(X)65 b Fu(=)50 b(Mor)950 2200 y Fk(K)t Fu(-)6 b Fc(cAlg)1199 2185 y Fu(\()p Fq(A;)17 b Fu(-)o(\))50 b(=:)h Fr(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fq(A;)17 b Fu(-\))45 b(and)h Fr(A)79 b Fu(=)50 b(Mor)2858 2200 y Fk(K)5 b Fu(-)h Fc(cAlg)3107 2185 y Fu(\()p Fr(K)18 b Fu([)p Fq(x)p Fu(])p Fq(;)g Fu(-)5 b(\))50 b(=:)0 2301 y Fr(K)17 b Fu(-)6 b Fs(cAlg)o Fu(\()p Fr(K)17 b Fu([)p Fq(x)q Fu(])p Fq(;)g Fu(-)5 b(\))32 b(the)h(Y)-8 b(oneda)33 b(Lemma)e(giv)m(es)i(us)61 2472 y(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))34 b(=)27 b(Nat\()p Fr(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fq(A;)17 b Fu(-)o(\))p Fq(;)g Fr(K)g Fu(-)6 b Fs(cAlg)n Fu(\()p Fr(K)18 b Fu([)p Fq(x)p Fu(])p Fq(;)f Fu(-)6 b(\)\))2164 2444 y Fp(\030)2165 2476 y Fu(=)2269 2472 y Fr(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fr(K)17 b Fu([)p Fq(x)q Fu(])p Fq(;)g(A)p Fu(\))33 b(=)28 b Fr(A)22 b Fu(\()p Fq(A)p Fu(\))3361 2444 y Fp(\030)3361 2476 y Fu(=)3466 2472 y Fq(A)0 2642 y Fu(on)49 b(the)g(set)g(lev)m(el.)92 b(Let)49 b Fq(\036)55 b Fu(:)g Fq(A)h Fp(\000)-60 b(!)55 b Fu(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))55 b(denote)49 b(the)g(giv)m(en)g (isomorphism.)90 b Fq(\036)48 b Fu(is)0 2759 y(de\014ned)e(b)m(y)f Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))49 b(:=)f Fq(p)p Fu(\()p Fq(a)p Fu(\).)79 b(By)45 b(the)g(Y)-8 b(oneda)45 b(Lemma)e(its)h(in)m (v)m(erse)i(is)e(giv)m(en)h(b)m(y)0 2875 y Fq(\036)58 2839 y Fn(\000)p Fg(1)152 2875 y Fu(\()p Fq(\013)28 b Fu(:=)g Fq(\013)q Fu(\(\()p Fq(A)p Fu(\)\(1\)\()p Fq(x)p Fu(\).)149 2991 y(Nat\()p Fp(X)15 b Fq(;)i Fr(A)22 b Fu(\))36 b(carries)31 b(an)g(algebra)f(structure)i(giv)m(en)f(b)m(y)h (the)f(algebra)f(structure)i(of)e(the)h(co)s(ef-)0 3107 y(\014cien)m(ts.)50 b(F)-8 b(or)33 b(a)h(co)s(e\016cien)m(t)h(algebra)e Fq(B)5 b Fu(,)35 b(a)f Fq(B)5 b Fu(-p)s(oin)m(t)33 b Fq(p)e Fu(:)g Fq(A)g Fp(\000)-60 b(!)30 b Fq(B)40 b Fu(in)33 b Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))31 b(=)f Fr(K)18 b Fu(-)5 b Fs(Alg)o Fu(\()p Fq(A;)17 b(B)5 b Fu(\),)0 3224 y(and)46 b Fq(\013)q(;)17 b(\014)56 b Fp(2)c Fu(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))52 b(w)m(e)47 b(ha)m(v)m(e)h Fq(\013)q Fu(\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\))51 b Fp(2)g Fr(A)22 b Fu(\()p Fq(B)5 b Fu(\))57 b(=)51 b Fq(B)5 b Fu(.)85 b(Hence)47 b(\()p Fq(\013)32 b Fu(+)g Fq(\014)6 b Fu(\)\()p Fq(B)f Fu(\)\()p Fq(p)p Fu(\))50 b(:=)0 3340 y(\()p Fq(\013)q Fu(\()p Fq(B)5 b Fu(\))29 b(+)g Fq(\014)6 b Fu(\()p Fq(B)f Fu(\)\)\()p Fq(p)p Fu(\))46 b(=)f Fq(\013)q Fu(\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\))29 b(+)h Fq(\014)6 b Fu(\()p Fq(B)f Fu(\)\()p Fq(p)p Fu(\))42 b(and)i(\()p Fq(\013)30 b Fp(\001)f Fq(\014)6 b Fu(\)\()p Fq(B)f Fu(\)\()p Fq(p)p Fu(\))45 b(:=)h(\()p Fq(\013)q Fu(\()p Fq(B)5 b Fu(\))29 b Fp(\001)g Fq(\014)6 b Fu(\()p Fq(B)f Fu(\)\)\()p Fq(p)p Fu(\))45 b(=)0 3456 y Fq(\013)q Fu(\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\))22 b Fp(\001)f Fq(\014)6 b Fu(\()p Fq(B)f Fu(\)\()p Fq(p)p Fu(\))32 b(mak)m(e)h(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))39 b(an)32 b(algebra.)149 3572 y(Let)41 b Fq(a)g Fu(b)s(e)h(an)e (arbitrary)g(elemen)m(t)h(in)f Fq(A)p Fu(.)68 b(By)42 b(the)f(isomorphism)d(giv)m(en)j(ab)s(o)m(v)m(e)h(this)f(ele-)0 3688 y(men)m(t)h(induces)i(an)e(algebra)f(homomorphism)f Fq(g)1856 3703 y Fl(a)1941 3688 y Fu(:)45 b Fr(K)17 b Fu([)p Fq(x)q Fu(])50 b Fp(\000)-60 b(!)45 b Fq(A)d Fu(mapping)f Fq(x)h Fu(on)m(to)h Fq(a)p Fu(.)73 b(This)0 3805 y(algebra)47 b(homomorphism)f(induces)j(the)g(natural)e(transformation)g Fq(\036)p Fu(\()p Fq(a)p Fu(\))54 b(:)h Fp(X)69 b(\000)-60 b(!)55 b Fr(A)22 b Fu(.)97 b(On)0 3937 y(the)41 b Fq(B)5 b Fu(-lev)m(el)39 b(it)h(is)f(just)i(the)g(comp)s(osition)d(with)i Fq(g)1935 3952 y Fl(a)1976 3937 y Fu(,)i(i.e.)66 b Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\))41 b(=)f(\()p Fr(K)18 b Fu([)p Fq(x)p Fu(])3169 3876 y Fl(g)3203 3884 y Fi(a)3124 3937 y Fp(\000)-16 b(!)41 b Fq(A)3502 3876 y Fl(p)3439 3937 y Fp(\000)-16 b(!)0 4053 y Fq(B)5 b Fu(\).)94 b(Since)50 b(suc)m(h)g(a)f(homomorphism)e(is)i(completely)f (describ)s(ed)i(b)m(y)h(the)f(image)d(of)i Fq(x)h Fu(w)m(e)0 4169 y(get)40 b Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))41 b(=)g Fq(p)p Fu(\()p Fq(a)p Fu(\).)67 b(T)-8 b(o)41 b(compare)f(the)h(algebra)e (structures)j(of)e Fq(A)g Fu(and)h(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))0 4286 y(let)45 b Fq(a;)17 b(a)300 4250 y Fn(0)374 4286 y Fp(2)51 b Fq(A)p Fu(.)83 b(W)-8 b(e)46 b(ha)m(v)m(e)h Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))50 b(=)h Fq(p)p Fu(\()p Fq(a)p Fu(\))45 b(and)h Fq(\036)p Fu(\()p Fq(a)2437 4250 y Fn(0)2461 4286 y Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))50 b(=)g Fq(p)p Fu(\()p Fq(a)3224 4250 y Fn(0)3247 4286 y Fu(\),)g(hence)0 4402 y Fq(\036)p Fu(\()p Fq(a)26 b Fu(+)h Fq(a)327 4366 y Fn(0)350 4402 y Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))39 b(=)f Fq(p)p Fu(\()p Fq(a)27 b Fu(+)f Fq(a)1270 4366 y Fn(0)1294 4402 y Fu(\))38 b(=)g Fq(p)p Fu(\()p Fq(a)p Fu(\))27 b(+)f Fq(p)p Fu(\()p Fq(a)1927 4366 y Fn(0)1951 4402 y Fu(\))38 b(=)h Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))26 b(+)g Fq(\036)p Fu(\()p Fq(a)3013 4366 y Fn(0)3037 4402 y Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))38 b(=)0 4518 y(\()p Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\))q(+)q Fq(\036)p Fu(\()p Fq(a)728 4482 y Fn(0)751 4518 y Fu(\)\()p Fq(B)g Fu(\)\()p Fq(p)p Fu(\)\)\()p Fq(x)p Fu(\))28 b(=)g(\()p Fq(\036)p Fu(\()p Fq(a)p Fu(\)\()p Fq(B)5 b Fu(\))q(+)q Fq(\036)p Fu(\()p Fq(a)1973 4482 y Fn(0)1996 4518 y Fu(\)\()p Fq(B)g Fu(\)\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))28 b(=)f(\()p Fq(\036)p Fu(\()p Fq(a)p Fu(\))q(+)q Fq(\036)p Fu(\()p Fq(a)3062 4482 y Fn(0)3086 4518 y Fu(\)\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\).)0 4634 y(Analogously)36 b(w)m(e)j(get)e Fq(\036)p Fu(\()p Fq(aa)1072 4598 y Fn(0)1096 4634 y Fu(\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\))36 b(=)g Fq(p)p Fu(\()p Fq(aa)1882 4598 y Fn(0)1905 4634 y Fu(\))g(=)g Fq(p)p Fu(\()p Fq(a)p Fu(\))p Fq(p)p Fu(\()p Fq(a)2405 4598 y Fn(0)2428 4634 y Fu(\))g(=)g(\()p Fq(\036)p Fu(\()p Fq(a)p Fu(\))26 b Fp(\001)f Fq(\036)p Fu(\()p Fq(a)3063 4598 y Fn(0)3086 4634 y Fu(\)\)\()p Fq(B)5 b Fu(\)\()p Fq(p)p Fu(\)\()p Fq(x)p Fu(\),)0 4751 y(and)42 b(th)m(us)i Fq(\036)p Fu(\()p Fq(a)28 b Fu(+)h Fq(a)755 4714 y Fn(0)778 4751 y Fu(\))44 b(=)g Fq(\036)p Fu(\()p Fq(a)p Fu(\))29 b(+)f Fq(\036)p Fu(\()p Fq(a)1445 4714 y Fn(0)1469 4751 y Fu(\))42 b(and)g Fq(\036)p Fu(\()p Fq(aa)1946 4714 y Fn(0)1969 4751 y Fu(\))i(=)g Fq(\036)p Fu(\()p Fq(a)p Fu(\))29 b Fp(\001)f Fq(\036)p Fu(\()p Fq(a)2588 4714 y Fn(0)2611 4751 y Fu(\).)73 b(Hence)43 b(addition)e(and)0 4867 y(m)m(ultiplication)26 b(in)j(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))37 b(are)30 b(de\014ned)i(b)m(y)f(the)g(addition)e(and)h(the)h(m)m(ultiplication)26 b(of)k(the)0 4983 y(v)-5 b(alues)32 b Fq(p)p Fu(\()p Fq(a)p Fu(\))23 b(+)f Fq(p)p Fu(\()p Fq(a)725 4947 y Fn(0)748 4983 y Fu(\))33 b(resp.)44 b Fq(p)p Fu(\()p Fq(a)p Fu(\))p Fq(p)p Fu(\()p Fq(a)1377 4947 y Fn(0)1400 4983 y Fu(\).)149 5099 y(W)-8 b(e)38 b(describ)s(e)g(the)g(action)e Fq( )t Fu(\()p Fq(B)5 b Fu(\))36 b(:)f Fq(A)26 b Fp(\002)g(X)15 b Fu(\()p Fq(B)5 b Fu(\))35 b Fp(\000)-60 b(!)36 b Fq(B)42 b Fu(of)37 b Fq(A)g Fu(on)g Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\).)58 b(Let)37 b Fq(p)f Fu(:)f Fq(A)h Fp(\000)-60 b(!)36 b Fq(B)0 5216 y Fu(b)s(e)i(a)g Fq(B)5 b Fu(-p)s(oin)m(t)37 b(in)g Fr(K)18 b Fu(-)5 b Fs(cAlg)o Fu(\()p Fq(A;)17 b(B)5 b Fu(\))37 b(=)f Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\).)60 b(F)-8 b(or)37 b(eac)m(h)i Fq(a)e Fp(2)h Fq(A)g Fu(the)g(image)f Fq(\036)p Fu(\()p Fq(a)p Fu(\))f(:)h Fp(X)52 b(\000)-60 b(!)37 b Fr(A)p eop %%Page: 10 10 10 9 bop 0 236 a Fm(10)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fu(is)37 b(a)h(natural)f(transformation)f(hence)j(w)m(e)g (ha)m(v)m(e)g(maps)f Fq( )t Fu(\()p Fq(B)5 b Fu(\))36 b(:)h Fq(A)26 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))36 b Fp(\000)-60 b(!)37 b Fq(B)43 b Fu(suc)m(h)c(that)0 541 y Fq( )t Fu(\()p Fq(B)5 b Fu(\)\()p Fq(a;)17 b(p)p Fu(\))38 b(=)g Fq(p)p Fu(\()p Fq(a)p Fu(\).)62 b(Finally)36 b(eac)m(h)k(homomorphism)c(of)i(algebras)g Fq(f)49 b Fu(:)39 b Fq(B)k Fp(\000)-60 b(!)38 b Fq(B)3135 505 y Fn(0)3198 541 y Fu(induces)h(a)0 658 y(comm)m(utativ)m(e)32 b(diagram)1359 1327 y Fq(A)22 b Fp(\002)g(X)15 b Fu(\()p Fq(B)1756 1291 y Fn(0)1779 1327 y Fu(\))500 b Fq(B)2396 1291 y Fn(0)p 1846 1304 442 4 v 2205 1302 a Fh(-)1944 1396 y Fq( )t Fu(\()p Fq(B)2128 1360 y Fn(0)2151 1396 y Fu(\))1370 839 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))522 b Fq(B)p 1835 816 466 4 v 2218 814 a Fh(-)1956 770 y Fq( )t Fu(\()p Fq(B)5 b Fu(\))p 1586 1234 4 351 v 1588 1234 a Fh(?)1134 1083 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(f)c Fu(\))p 2366 1234 V 2368 1234 a Fh(?)2407 1083 y Fq(f)0 1534 y Fu(Th)m(us)34 b Fq( )t Fu(\()p Fq(B)5 b Fu(\))28 b(:)g Fq(A)22 b Fp(\002)g(X)15 b Fu(\()p Fq(B)5 b Fu(\))28 b Fp(\000)-60 b(!)28 b Fq(B)37 b Fu(is)32 b(a)h(natural)e(transformation.)p 3529 1534 4 66 v 3533 1472 59 4 v 3533 1534 V 3591 1534 4 66 v 149 1714 a Fs(Remark)37 b(1.1.12.)50 b Fu(Observ)m(e)35 b(that)d(the)i(isomorphism)c Fq(A)2367 1687 y Fp(\030)2368 1718 y Fu(=)2473 1714 y(Nat\()p Fp(X)15 b Fq(;)i Fr(A)k Fu(\))39 b(induces)34 b(a)e(natu-)0 1830 y(ral)e(transformation)e Fq(A)19 b Fp(\002)g(X)c Fu(\()p Fq(B)5 b Fu(\))27 b Fp(\000)-60 b(!)28 b Fq(B)36 b Fu(\(natural)29 b(in)h Fq(B)5 b Fu(\).)43 b(In)32 b(particular)d(the) i(a\016ne)g(algebra)f Fq(A)0 1947 y Fu(can)j(b)s(e)h(view)m(ed)g(as)f (the)h(set)g(of)e(functions)i(from)d(the)j(set)g(of)f Fq(B)5 b Fu(-p)s(oin)m(ts)32 b Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))33 b(in)m(to)f(the)i(\\base")0 2063 y(ring)27 b Fq(B)32 b Fu(\(functions)c(whic)m(h)g(are)g(natural)e(in)h Fq(B)5 b Fu(\).)42 b(In)28 b(this)f(sense)j(the)e(algebra)e Fq(A)i Fu(ma)m(y)g(b)s(e)f(consid-)0 2179 y(ered)35 b(as)g(function)e (algebra)g(of)h(the)h(geometric)e(space)i Fp(X)15 b Fu(.)49 b(Th)m(us)35 b(w)m(e)h(will)c(call)g Fq(A)j Fu(the)f Fo(function)0 2295 y(algebr)-5 b(a)32 b Fu(of)g Fp(X)15 b Fu(.)149 2412 y(One)37 b(can)f(sho)m(w)h(that)f(the)g(algebra)f Fq(A)h Fu(is)f(univ)m(ersal)h(with)g(resp)s(ect)h(to)e(the)i(prop)s (ert)m(y)-8 b(,)37 b(that)0 2528 y(for)d(eac)m(h)i(comm)m(utativ)m(e)e (algebra)g Fq(D)j Fu(and)e(eac)m(h)h(natural)e(transformation)e Fq(\032)g Fu(:)g Fq(D)26 b Fp(\002)e(X)15 b Fu(\(-)o(\))32 b Fp(\000)-60 b(!)31 b Fu(-)0 2644 y(there)i(is)f(a)h(unique)g (homomorphism)d(of)i(algebras)g Fq(f)38 b Fu(:)28 b Fq(D)i Fp(\000)-60 b(!)28 b Fq(A)p Fu(,)33 b(suc)m(h)h(that)e(the)h(triangle) 1480 3285 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))230 b Fq(B)p 1944 3262 173 4 v 2034 3260 a Fh(-)1950 3329 y Fl( )r Fg(\()p Fl(B)s Fg(\))2002 3033 y Fl(\032)p Fg(\()p Fl(B)s Fg(\))1766 2924 y 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(algebraic)f(geometry)h(to)g(reco)m(v)m(er)i(the)f(geometric)f(space)0 4752 y(of)39 b(\(rational\))e(p)s(oin)m(ts)i(from)g(the)h(function)f (algebra)g Fq(A)p Fu(.)64 b(W)-8 b(e)40 b(will)e(not)h(follo)m(w)f (this)i(approac)m(h)0 4869 y(since)33 b(it)f(do)s(es)h(not)f(easily)g (extend)i(to)e(noncomm)m(utativ)m(e)g(geometry)-8 b(.)149 5049 y Fs(Problem)36 b(1.1.2.)49 b Fu(Let)33 b Fp(X)47 b Fu(b)s(e)33 b(an)g(a\016ne)g(sc)m(heme)h(with)e(a\016ne)h(algebra) 1120 5216 y Fq(A)28 b Fu(=)g Fr(K)17 b Fu([)p Fq(x)1479 5231 y Fg(1)1524 5216 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)1815 5231 y Fl(n)1862 5216 y Fu(])p Fq(=)p Fu(\()p Fq(p)2025 5231 y Fg(1)2064 5216 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(p)2348 5231 y Fl(m)2415 5216 y Fu(\))p Fq(:)p eop %%Page: 11 11 11 10 bop 542 236 a Fm(1.)34 b(THE)g(PRINCIPLES)f(OF)g(COMMUT)-6 b(A)g(TIVE)33 b(ALGEBRAIC)g(GEOMETR)-6 b(Y)439 b(11)0 425 y Fu(De\014ne)47 b(\\co)s(ordinate)e(functions")g Fq(q)1383 440 y Fl(i)1463 425 y Fu(:)50 b Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))51 b Fp(\000)-60 b(!)51 b Fq(B)g Fu(whic)m(h)46 b(describ)s(e)h(the)g(co)s(ordinates)e(of)0 541 y Fq(B)5 b Fu(-p)s(oin)m(ts)32 b(and)h(iden)m(tify)e(these)j(co)s(ordinate)e (functions)h(with)f(elemen)m(ts)h(of)f Fq(A)p Fu(.)149 716 y(No)m(w)i(w)m(e)f(will)d(study)k(morphisms)e(b)s(et)m(w)m(een)i (geometric)e(spaces.)149 890 y Fs(Theorem)38 b(1.1.13.)49 b Fo(L)-5 b(et)39 b Fp(X)49 b(\022)34 b Fr(A)1460 854 y Fl(r)1543 890 y Fo(and)k Fp(Y)k(\022)35 b Fr(A)2023 854 y Fl(s)2104 890 y Fo(b)-5 b(e)38 b(a\016ne)g(algebr)-5 b(aic)37 b(varieties)h(and)g(let)0 1006 y Fq(\036)27 b Fu(:)h Fp(X)43 b(\000)-58 b(!)28 b(Y)42 b Fo(b)-5 b(e)35 b(a)g(natur)-5 b(al)35 b(tr)-5 b(ansformation.)43 b(Then)35 b(ther)-5 b(e)34 b(ar)-5 b(e)35 b(p)-5 b(olynomials)744 1163 y Fq(p)793 1178 y Fg(1)833 1163 y Fu(\()p Fq(x)926 1178 y Fg(1)965 1163 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(x)1256 1178 y Fl(r)1294 1163 y Fu(\))p Fq(;)g(:)g(:)g(:)32 b(;)17 b(p)1616 1178 y Fl(s)1653 1163 y Fu(\()p Fq(x)1746 1178 y Fg(1)1785 1163 y 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b(i.e.)d(the)i(morphisms)d(b)-5 b(etwe)g(en)42 b(a\016ne)0 1689 y(algebr)-5 b(aic)34 b(varieties)g(ar)-5 b(e)35 b(of)f(p)-5 b(olynomial)34 b(typ)-5 b(e.)149 1864 y Fd(Pr)n(oof.)42 b Fu(Let)47 b Fp(O)s Fu(\()p Fp(X)15 b Fu(\))51 b(=)f Fr(K)18 b Fu([)p Fq(x)1298 1879 y Fg(1)1344 1864 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)1634 1879 y Fl(r)1672 1864 y Fu(])p Fq(=I)54 b Fu(and)47 b Fp(O)s Fu(\()p Fp(Y)8 b Fu(\))50 b(=)h Fr(K)18 b Fu([)p Fq(y)2606 1879 y Fg(1)2651 1864 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(y)2934 1879 y Fl(s)2970 1864 y Fu(])p Fq(=J)9 b Fu(.)85 b(F)-8 b(or)45 b Fq(A)52 b Fp(2)0 1980 y Fr(K)17 b Fu(-)6 b Fs(Alg)42 b Fu(and)h(\()p Fq(a)614 1995 y Fg(1)654 1980 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(a)940 1995 y Fl(r)978 1980 y Fu(\))46 b Fp(2)g(X)15 b Fu(\()p Fq(A)p Fu(\))42 b(let)h Fq(f)56 b Fu(:)46 b Fr(K)17 b Fu([)q Fq(x)1934 1995 y Fg(1)1979 1980 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)2270 1995 y Fl(r)2308 1980 y Fu(])p Fq(=I)53 b Fp(\000)-60 b(!)46 b Fq(A)d Fu(with)g Fq(f)11 b Fu(\()p Fq(x)3144 1995 y Fl(i)3172 1980 y Fu(\))46 b(=)f Fq(a)3428 1995 y Fl(i)3500 1980 y Fu(b)s(e)0 2096 y(the)36 b(homomorphism)d (obtained)i(from)g Fp(X)15 b Fu(\()p Fq(A)p Fu(\))1768 2068 y Fp(\030)1769 2100 y Fu(=)1879 2096 y Fr(K)i Fu(-)6 b Fs(Alg)n Fu(\()p Fr(K)18 b Fu([)p Fq(x)2352 2111 y Fg(1)2398 2096 y Fq(;)f(:)g(:)g(:)32 b(;)17 b(x)2688 2111 y Fl(r)2726 2096 y Fu(])p Fq(=I)8 b(;)17 b(A)p Fu(\).)52 b(The)37 b(natural)0 2212 y(transformation)20 b Fq(\036)h Fu(is)g(giv)m(en)h(b)m(y)h(comp)s(osition)c(with)i(a)h(homomorphism)d Fq(g)31 b Fu(:)d Fr(K)17 b Fu([)p Fq(y)2962 2227 y Fg(1)3007 2212 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(y)3291 2227 y Fl(s)3327 2212 y Fu(])p Fq(=J)36 b Fp(\000)-60 b(!)0 2328 y Fr(K)17 b Fu([)q Fq(x)154 2343 y Fg(1)199 2328 y Fq(;)g(:)g(:)g(:)33 b(;)17 b(x)490 2343 y Fl(r)528 2328 y Fu(])p Fq(=I)40 b Fu(hence)34 b(w)m(e)g(get)187 2485 y Fq(\036)p Fu(\()p Fq(A)p Fu(\))28 b(:)g Fr(K)17 b Fu(-)6 b Fs(cAlg)n Fu(\()p Fr(K)18 b Fu([)p Fq(x)1000 2500 y Fg(1)1046 2485 y 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Fn(\000)p Fg(1)181 3985 y Fu(:)27 b Fr(P)p Fu(\()p Fq(A)p Fu(\))j Fp(3)e Fu(\()p Fq(a;)17 b(b)p Fu(\))28 b Fp(7!)f Fq(a)h Fp(2)g Fr(A)22 b Fu(\()p Fq(A)p Fu(\).)p eop %%Page: 12 12 12 11 bop 0 236 a Fm(12)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)509 425 y Fs(2.)57 b(Quan)m(tum)37 b(Spaces)i(and)f(Noncomm)m (utativ)m(e)d(Geometry)149 600 y Fu(No)m(w)k(w)m(e)g(come)f(to)f (noncomm)m(utativ)m(e)g(geometric)g(spaces)j(and)e(their)f(function)h (algebras.)0 716 y(Man)m(y)33 b(of)d(the)i(basic)g(principles)e(of)h (comm)m(utativ)m(e)f(algebraic)g(geometry)h(as)h(in)m(tro)s(duced)g(in) e(1.1)0 832 y(carry)43 b(o)m(v)m(er)g(to)f(noncomm)m(utativ)m(e)g (geometry)-8 b(.)72 b(Our)42 b(main)f(aim,)i(ho)m(w)m(ev)m(er,)k(is)42 b(to)g(study)i(the)0 948 y(symmetries)34 b(\(automorphisms\))g(of)g (noncomm)m(utativ)m(e)g(spaces)i(whic)m(h)g(lead)e(to)g(the)i(notion)d (of)0 1064 y(a)f(quan)m(tum)h(group.)149 1181 y(Since)c(the)g 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b(\000)-60 b(!)40 b(Y)47 b Fu(is)39 b(a)0 2034 y(natural)31 b(transformation.)149 2306 y(This)i(de\014nition)f(implies)e(immediately)149 2579 y Fs(Corollary)36 b(1.2.2.)50 b Fo(The)40 b(nonc)-5 b(ommutative)40 b(sp)-5 b(ac)g(es)41 b(form)f(a)i(c)-5 b(ate)g(gory)41 b Fs(QS)g Fo(that)h(is)f(dual)0 2695 y(to)35 b(the)g(c)-5 b(ate)g(gory)35 b(of)f Fr(K)18 b Fo(-algebr)-5 b(as.)149 2967 y Fs(Remark)37 b(1.2.3.)50 b Fu(Th)m(us)30 b(one)g(often)f(calls)f(the)i(dual)e(category)h Fr(K)18 b Fu(-)5 b Fs(Alg)2810 2924 y Fl(op)2913 2967 y Fu(category)30 b(of)e(non-)0 3084 y(comm)m(utativ)m(e)k(spaces.)149 3200 y(If)39 b Fq(A)f Fu(is)g(a)g(\014nitely)f(generated)i(algebra)e (then)i(it)e(ma)m(y)h(b)s(e)h(considered)g(as)f(a)g(residue)h(class)0 3316 y(algebra)29 b Fq(A)439 3288 y Fp(\030)440 3320 y Fu(=)545 3316 y Fr(K)17 b Fp(h)p Fq(x)710 3331 y Fg(1)756 3316 y Fq(;)g(:)g(:)g(:)32 b(;)17 b(x)1046 3331 y Fl(n)1093 3316 y Fp(i)p Fq(=I)37 b Fu(of)30 b(a)f(p)s(olynomial)d(algebra)j(in)g 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b(case,)j(but)f(that)f(holds)g(in)g (the)h(comm)m(utativ)m(e)e(as)i(w)m(ell)e(as)i(in)f(the)g(noncomm)m (uta-)0 5216 y(tiv)m(e)j(situation.)p eop %%Page: 13 13 13 12 bop 637 236 a Fm(2.)35 b(QUANTUM)d(SP)-6 b(A)n(CES)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(GEOMETR)-6 b(Y)535 b(13)149 425 y Fs(Lemma)38 b(1.2.4.)49 b Fo(L)-5 b(et)39 b Fq(D)i Fo(b)-5 b(e)38 b(a)g(set)h(and)f Fq(\036)c Fu(:)g Fq(D)28 b Fp(\002)d(X)15 b Fu(\()p Fo(-)p Fu(\))34 b Fp(\000)-58 b(!)35 b Fr(A)22 b Fu(\()p Fo(-)6 b Fu(\))38 b Fo(b)-5 b(e)38 b(a)h(natur)-5 b(al)38 b(tr)-5 b(ansfor-)0 541 y(mation.)44 b(Then)34 b(ther)-5 b(e)35 b(exists)g(a)f(unique)h (map)f Fq(f)39 b Fu(:)27 b Fq(D)k Fp(\000)-58 b(!)28 b Fq(A)35 b Fo(such)g(that)g(the)g(diagr)-5 b(am)1235 1065 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))620 b Fq(B)p 1700 1042 563 4 v 2180 1040 a Fh(-)1900 1109 y Fl( )r Fg(\()p Fl(B)s Fg(\))1230 773 y Fq(D)25 b Fp(\002)d(X)15 b Fu(\()p Fq(B)5 b Fu(\))p 1451 972 4 156 v 1453 972 a Fh(?)1283 911 y Fl(f)i 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Fq(A)p Fu(\)\(1\)\()p Fq(x)p Fu(\))i(but)0 5216 y(w)m(e)e(iden)m(tify)e Fr(A)22 b Fu(\()p Fq(B)5 b Fu(\))38 b(and)33 b Fq(B)k Fu(b)m(y)d Fr(A)22 b Fu(\()p Fq(B)5 b Fu(\))34 b Fp(3)28 b Fq(p)f Fp(7!)h Fq(p)p Fu(\()p Fq(x)p Fu(\))f Fp(2)i Fq(B)5 b Fu(.\))p eop %%Page: 14 14 14 13 bop 0 236 a Fm(14)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)149 425 y Fu(Then)34 b(w)m(e)g(get)77 593 y Fq( )t Fu(\()p Fq(B)5 b Fu(\)\()p Fq(f)32 b Fp(\002)23 b Fu(1\)\()p Fq(d;)17 b(p)p Fu(\))82 b(=)27 b Fq( )t Fu(\()p Fq(B)5 b Fu(\)\()p Fq(f)11 b Fu(\()p Fq(d)p Fu(\))p Fq(;)17 b(p)p Fu(\))906 710 y(=)27 b Fq( )t Fu(\()p Fq(B)5 b Fu(\)\()p Fq(f)1328 673 y Fn(0)1351 710 y Fu(\()p Fq(d)p Fu(\)\()p Fq(A)p Fu(\)\(1\)\()p Fq(x)p Fu(\))p Fq(;)17 b(p)p Fu(\))32 b(\(b)m(y)h(de\014nition)f(of)g Fq(f)11 b Fu(\))906 826 y(=)27 b Fq(p)c Fp(\016)f Fq(f)1212 790 y Fn(0)1235 826 y Fu(\()p Fq(d)p Fu(\)\()p Fq(A)p Fu(\)\(1\))31 b(\(since)i(w)m(e)h(ma)m(y)e(omit)f Fq(x)p Fu(\))906 942 y(=)c Fq(p)c Fp(\016)f Fq(\036)p Fu(\()p Fq(A)p Fu(\)\()p Fq(d;)17 b Fu(1\))31 b(\(b)m(y)i(de\014nition)f(of)g Fq(f)2388 906 y Fn(0)2411 942 y Fu(\))906 1058 y(=)27 b Fq(\036)p Fu(\()p Fq(B)5 b Fu(\)\()p Fq(D)25 b Fp(\002)e(X)15 b Fu(\()p Fq(p)p Fu(\)\)\()p Fq(d;)i Fu(1\))31 b(\(since)i Fq(\036)f Fu(is)g(a)g(natural)g(transformation\))906 1175 y(=)27 b Fq(\036)p Fu(\()p Fq(B)5 b Fu(\)\()p Fq(d;)17 b(p)p Fu(\))p Fq(:)0 1346 y Fu(Hence)34 b(the)f(diagram)d(in)i(the)h (Lemma)e(comm)m(utes.)149 1462 y(T)-8 b(o)47 b(sho)m(w)h(the)f (uniqueness)i(of)d Fq(f)58 b Fu(let)46 b Fq(g)55 b Fu(:)d Fq(D)i Fp(\000)-60 b(!)52 b Fq(A)47 b Fu(b)s(e)g(a)f(homomorphism)e (suc)m(h)k(that)0 1579 y Fq( )t Fu(\()p Fq(B)5 b Fu(\)\()p Fq(g)25 b Fp(\002)e Fu(1\))k(=)h Fq(\036)p Fu(\()p Fq(B)5 b Fu(\).)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)0 1754 y Fq(f)11 b Fu(\()p Fq(d)p Fu(\))27 b(=)h Fq(f)376 1713 y Fn(0)399 1754 y Fu(\()p Fq(d)p Fu(\)\()p Fq(A)p Fu(\)\(1\))e(=)i Fq(\036)p Fu(\()p Fq(A)p Fu(\)\()p Fq(d;)17 b Fu(1\))26 b(=)i Fq( )t Fu(\()p Fq(A)p Fu(\)\()p Fq(g)6 b Fp(\002)s Fu(1\)\()p Fq(d;)17 b Fu(1\))26 b(=)i Fq( )t Fu(\()p Fq(A)p Fu(\)\()p Fq(f)11 b Fu(\()p Fq(d)p Fu(\))p Fq(;)17 b Fu(1\))26 b(=)h(1)s Fp(\016)s Fq(g)t Fu(\()p Fq(d)p Fu(\))f(=)h Fq(g)t Fu(\()p Fq(d)p Fu(\))0 1929 y(hence)34 b Fq(f)k Fu(=)28 b Fq(g)t Fu(.)p 3529 1929 4 66 v 3533 1867 59 4 v 3533 1929 V 3591 1929 4 66 v 149 2114 a Fs(Problem)36 b(1.2.3.)49 b Fa(De\014nition:)61 b Fu(Let)40 b Fq(D)j Fu(b)s(e)e(an)f(algebra.)66 b(A)40 b(natural)f(transformation)0 2230 y Fq(\036)k Fu(:)g Fq(D)31 b Fp(\002)d(X)58 b(\000)-60 b(!)43 b Fr(A)70 b Fu(is)41 b(called)f(an)h Fo(algebr)-5 b(a)43 b(action)e Fu(if)f Fq(\036)p Fu(\()p Fq(B)5 b Fu(\)\(-)o Fq(;)17 b(p)p Fu(\))43 b(:)g Fq(D)j Fp(\000)-60 b(!)43 b Fr(A)22 b Fu(\()p Fq(B)5 b Fu(\))48 b(=)43 b Fq(B)k Fu(is)41 b(an)0 2347 y(algebra)31 b(homomorphism)f(for)i(all)f Fq(B)37 b Fu(and)c(all)e Fq(p)c Fp(2)h(X)15 b Fu(\()p Fq(B)5 b Fu(\).)149 2463 y Fa(L)-6 b(emma:)44 b Fu(The)33 b(natural)f(transformation)e Fq( )i Fu(:)c Fq(A)22 b Fp(\002)h(X)42 b(\000)-60 b(!)28 b Fr(A)60 b Fu(is)32 b(an)h(algebra)e(action.)149 2579 y Fa(The)-6 b(or)g(em:)47 b Fu(Let)34 b Fq(D)i Fu(b)s(e)e(an)g(algebra)f(and)h Fq(\036)29 b Fu(:)h Fq(D)25 b Fp(\002)f(X)15 b Fu(\(-)o(\))30 b Fp(\000)-60 b(!)29 b Fr(A)22 b Fu(\(-)6 b(\))33 b(b)s(e)h(an)g (algebra)f(action.)0 2695 y(Then)c(there)g(exists)f(a)g(unique)g (algebra)f(homomorphism)e Fq(f)39 b Fu(:)28 b Fq(D)i Fp(\000)-60 b(!)28 b Fq(A)g Fu(suc)m(h)h(that)f(the)g(diagram)1430 3344 y Fq(A)22 b Fp(\002)h(X)15 b Fu(\()p Fq(B)5 b Fu(\))230 b Fq(B)p 1895 3321 173 4 v 1985 3319 a Fh(-)1900 3389 y Fl( )r Fg(\()p Fl(B)s Fg(\))1952 3092 y Fl(\036)p Fg(\()p Fl(B)s Fg(\))1716 2983 y Fh(@)1799 3066 y(@)1882 3149 y(@)1965 3232 y(@)1984 3251 y(@)-83 b(R)1425 2857 y Fq(D)25 b Fp(\002)d(X)15 b Fu(\()p Fq(B)5 b Fu(\))p 1646 3251 4 351 v 1648 3251 a Fh(?)1478 3092 y Fl(f)i Fn(\002)p Fg(1)0 3534 y Fu(comm)m(utes.)149 3719 y Fs(De\014nition)37 b(1.2.5.)49 b Fu(The)33 b(noncomm)m(utativ)m(e)f(space)i Fr(A)2250 3668 y Fg(2)p Fn(j)p Fg(0)2250 3729 y Fl(q)2383 3719 y Fu(with)e(the)h(function)f(algebra)1106 3900 y Fp(O)s Fu(\()p Fr(A)1292 3859 y Fg(2)p Fn(j)p Fg(0)1292 3925 y Fl(q)1392 3900 y Fu(\))c(:=)f Fr(K)18 b Fp(h)p Fq(x;)f(y)t Fp(i)p Fq(=)p Fu(\()p Fq(xy)30 b Fp(\000)23 b Fq(q)2256 3859 y Fn(\000)p Fg(1)2350 3900 y Fq(y)t(x)p Fu(\))0 4080 y(with)28 b Fq(q)k Fp(2)c Fr(K)33 b Fp(n)15 b(f)p Fu(0)p Fp(g)33 b Fu(is)28 b(called)g(the)h Fo(\(deforme)-5 b(d\))30 b(quantum)i(plane)p Fu(.)41 b(The)30 b(noncomm)m(utativ)m(e)e (space)0 4214 y Fr(A)66 4163 y Fg(0)p Fn(j)p Fg(2)66 4224 y Fl(q)199 4214 y Fu(with)k(the)h(function)f(algebra)1032 4396 y Fp(O)s Fu(\()p Fr(A)1218 4354 y Fg(0)p Fn(j)p Fg(2)1218 4420 y Fl(q)1318 4396 y Fu(\))c(:=)g Fr(K)17 b Fp(h)p Fq(\030)5 b(;)17 b(\021)t Fp(i)p Fq(=)p Fu(\()p Fq(\030)1943 4354 y Fg(2)1986 4396 y Fq(;)g(\021)2082 4354 y Fg(2)2121 4396 y Fq(;)g(\030)5 b(\021)25 b Fu(+)d 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5178 y Fq(:)p eop %%Page: 15 15 15 14 bop 637 236 a Fm(2.)35 b(QUANTUM)d(SP)-6 b(A)n(CES)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(GEOMETR)-6 b(Y)535 b(15)149 425 y Fs(De\014nition)37 b(1.2.6.)49 b Fu(Let)43 b Fp(X)57 b Fu(b)s(e)43 b(a)f(noncomm)m(utativ)m(e)g(space)i(with)e (function)g(algebra)g Fq(A)0 541 y Fu(and)g(let)e Fp(X)419 556 y Fl(c)495 541 y Fu(b)s(e)i(the)g(restriction)e(of)h(the)h(functor) g Fp(X)57 b Fu(:)43 b Fr(K)17 b Fu(-)6 b Fs(Alg)42 b Fp(\000)-60 b(!)42 b Fs(Set)g Fu(to)f(the)h(category)f(of)0 658 y(comm)m(utativ)m(e)35 b(algebras:)49 b Fp(X)1081 673 y Fl(c)1148 658 y Fu(:)33 b Fr(K)18 b Fu(-)5 b Fs(cAlg)32 b Fp(\000)-60 b(!)33 b Fs(Set)p Fu(.)53 b(Then)37 b(w)m(e)f(call)e Fp(X)2617 673 y Fl(c)2688 658 y Fu(the)i Fo(c)-5 b(ommutative)37 b(p)-5 b(art)0 774 y Fu(of)32 b(the)h(noncomm)m(utativ)m(e)f(space)i Fp(X)15 b Fu(.)149 965 y Fs(Lemma)38 b(1.2.7.)49 b Fo(The)d(c)-5 b(ommutative)46 b(p)-5 b(art)47 b Fp(X)1936 980 y Fl(c)2017 965 y Fo(of)f(a)h(nonc)-5 b(ommutative)45 b(sp)-5 b(ac)g(e)46 b Fp(X)61 b Fo(is)46 b(an)0 1081 y(a\016ne)34 b(variety.)149 1272 y Fd(Pr)n(oof.)42 b Fu(The)34 b(underlying)f(functor)g Fr(A)56 b Fu(:)29 b Fr(K)17 b Fu(-)6 b Fs(cAlg)27 b Fp(\000)-60 b(!)28 b Fr(K)17 b Fu(-)6 b Fs(Alg)32 b Fu(has)h(a)g(left)f(adjoin)m(t) g(functor)0 1388 y Fr(K)17 b Fu(-)6 b Fs(Alg)44 b Fp(3)h Fq(A)g Fp(7!)g Fq(A=)p Fu([)p Fq(A;)17 b(A)p Fu(])45 b Fp(2)h Fr(K)17 b Fu(-)6 b Fs(cAlg)41 b Fu(where)j([)p Fq(A;)17 b(A)p Fu(])43 b(denotes)h(the)f(t)m(w)m(o-sided)g(ideal)f(of)g Fq(A)0 1505 y Fu(generated)d(b)m(y)h(the)e(elemen)m(ts)h Fq(ab)27 b Fp(\000)f Fq(ba)p Fu(.)62 b(In)39 b(fact)f(for)g(eac)m(h)h (homomorphism)d(of)i(algebras)g Fq(f)48 b Fu(:)0 1621 y Fq(A)28 b Fp(\000)-60 b(!)28 b Fq(B)k Fu(with)27 b(a)f(comm)m(utativ) m(e)h(algebra)f Fq(B)32 b Fu(there)c(is)e(a)h(factorization)e(through)i Fq(A=)p Fu([)p Fq(A;)17 b(A)p Fu(])28 b(since)0 1737 y Fq(f)43 b Fu(v)-5 b(anishes)33 b(on)g(the)g(elemen)m(ts)g Fq(ab)23 b Fp(\000)f Fq(ba)p Fu(.)149 1853 y(Hence)32 b(if)d Fq(A)e Fu(=)h Fp(O)s Fu(\()p Fp(X)15 b Fu(\))30 b(is)f(the)i(function)e(algebra)g(of)h Fp(X)44 b Fu(then)31 b Fq(A=)p Fu([)p Fq(A;)17 b(A)p Fu(])30 b(is)f(the)i(represen)m(ting)0 1970 y(algebra)g(for)h Fp(X)561 1985 y Fl(c)596 1970 y Fu(.)p 3529 1970 4 66 v 3533 1907 59 4 v 3533 1970 V 3591 1970 4 66 v 149 2161 a Fs(Remark)37 b(1.2.8.)50 b Fu(F)-8 b(or)41 b(an)m(y)i(comm)m(utativ)m(e)e(algebra)g(\(of)h(co)s (e\016cien)m(ts\))h Fq(B)48 b Fu(the)43 b(spaces)g Fp(X)0 2277 y Fu(and)i Fp(X)273 2292 y Fl(c)352 2277 y Fu(ha)m(v)m(e)h(the)f (same)g Fq(B)5 b Fu(-p)s(oin)m(ts:)67 b Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))48 b(=)g Fp(X)1977 2292 y Fl(c)2011 2277 y Fu(\()p Fq(B)5 b Fu(\).)80 b(The)46 b(t)m(w)m(o)f(spaces)h(di\013er)e (only)g(for)0 2393 y(noncomm)m(utativ)m(e)39 b(algebras)g(of)g(co)s (e\016cien)m(ts.)66 b(In)40 b(particular)e(for)h(comm)m(utativ)m(e)g (\014elds)h Fq(B)45 b Fu(as)0 2521 y(algebras)c(of)g(co)s(e\016cien)m (ts)h(the)g(quan)m(tum)f(plane)g 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b(the)h(p)s(oin)m(ts)f(of)g Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))28 b(=)h Fr(K)18 b Fu(-)5 b Fs(Alg)o Fu(\()p Fq(A;)17 b(B)5 b Fu(\))33 b(as)g(a)g(subspace)j(of)d Fr(A)3372 3025 y Fg(2)3418 3061 y Fu(\()p Fq(B)5 b Fu(\).)0 3177 y(What)33 b(is)f Fp(X)443 3192 y Fl(c)478 3177 y Fu(\()p Fq(B)5 b Fu(\))32 b(and)h(what)g(is)f(the)h(a\016ne)g(algebra)e (of)h Fp(X)2148 3192 y Fl(c)2183 3177 y Fu(?)149 3368 y(T)-8 b(o)35 b(understand)h(ho)m(w)f(Hopf)f(algebras)g(\014t)g(in)m (to)g(the)h(con)m(text)h(of)e(noncomm)m(utativ)m(e)f(spaces)0 3485 y(w)m(e)h(ha)m(v)m(e)g(to)e(b)s(etter)h(understand)h(the)f(tensor) g(pro)s(duct)g(in)e Fr(K)18 b Fu(-)5 b Fs(Alg)o Fu(.)149 3676 y Fs(De\014nition)37 b(1.2.9.)49 b Fu(Let)42 b Fq(A)i Fu(=)f Fp(O)s Fu(\()p Fp(X)15 b Fu(\))41 b(and)h Fq(A)1960 3639 y Fn(0)2027 3676 y Fu(=)i Fp(O)s Fu(\()p Fp(Y)8 b Fu(\))41 b(b)s(e)h(the)h(function)e(algebras)g(of)0 3792 y(the)i(noncomm)m(utativ)m(e)g(spaces)h Fp(X)58 b Fu(resp.)76 b Fp(Y)8 b Fu(.)74 b(Tw)m(o)44 b Fq(B)5 b Fu(-p)s(oin)m(ts)43 b Fq(p)i Fu(:)h Fq(A)f Fp(\000)-60 b(!)46 b Fq(B)i Fu(in)42 b Fp(X)15 b Fu(\()p Fq(B)5 b Fu(\))43 b(and)0 3908 y Fq(p)49 3872 y Fn(0)111 3908 y Fu(:)38 b Fq(A)249 3872 y Fn(0)311 3908 y Fp(\000)-60 b(!)39 b Fq(B)44 b Fu(in)38 b Fp(Y)8 b Fu(\()p Fq(B)d Fu(\))39 b(are)g(called)f Fo(c)-5 b(ommuting)40 b(p)-5 b(oints)38 b Fu(if)g(w)m(e)i(ha)m(v)m(e)g(for)e(all)f Fq(a)i Fp(2)g Fq(A)g Fu(and)g(all)0 4024 y Fq(a)51 3988 y Fn(0)102 4024 y Fp(2)28 b Fq(A)269 3988 y Fn(0)1322 4170 y Fq(p)p Fu(\()p Fq(a)p Fu(\))p Fq(p)1547 4129 y Fn(0)1570 4170 y Fu(\()p Fq(a)1659 4129 y Fn(0)1683 4170 y Fu(\))g(=)f Fq(p)1901 4129 y Fn(0)1924 4170 y Fu(\()p Fq(a)2013 4129 y Fn(0)2037 4170 y Fu(\))p Fq(p)p Fu(\()p Fq(a)p Fu(\))p Fq(;)0 4335 y Fu(i.e.)43 b(if)31 b(the)i(images)f(of)g (the)h(t)m(w)m(o)g(homomorphisms)d Fq(p)j Fu(and)g Fq(p)2257 4299 y Fn(0)2312 4335 y Fu(comm)m(ute.)149 4526 y Fs(Remark)k(1.2.10.) 50 b Fu(T)-8 b(o)39 b(sho)m(w)i(that)f(the)g(p)s(oin)m(ts)f Fq(p)g Fu(and)h Fq(p)2368 4490 y Fn(0)2431 4526 y Fu(comm)m(ute,)h(it)d (is)i(su\016cien)m(t)g(to)0 4643 y(c)m(hec)m(k)e(that)d(the)h(images)f (of)g(the)h(algebra)e(generators)i Fq(p)p Fu(\()p Fq(x)2221 4658 y Fg(1)2261 4643 y Fu(\))p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(p)p Fu(\()p Fq(x)2676 4658 y Fl(m)2743 4643 y Fu(\))35 b(comm)m(ute)g(with)h(the)0 4759 y(images)44 b(of)h(the)h(algebra)f(generators)h Fq(p)1527 4723 y Fn(0)1550 4759 y Fu(\()p Fq(y)1636 4774 y Fg(1)1675 4759 y Fu(\))p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(p)1997 4723 y Fn(0)2020 4759 y Fu(\()p Fq(y)2106 4774 y Fl(n)2153 4759 y Fu(\))45 b(under)i(the)f(m)m(ultiplication.)78 b(This)0 4875 y(means)33 b(that)f(w)m(e)i(ha)m(v)m(e)1587 5021 y Fq(b)1628 5036 y Fl(i)1656 5021 y Fq(b)1697 4979 y Fn(0)1697 5045 y Fl(j)1762 5021 y Fu(=)28 b Fq(b)1907 4979 y Fn(0)1907 5045 y Fl(j)1944 5021 y Fq(b)1985 5036 y Fl(i)0 5191 y Fu(for)k(the)h Fq(B)5 b Fu(-p)s(oin)m(ts)32 b(\()p Fq(b)801 5206 y Fg(1)841 5191 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(b)1117 5206 y Fl(m)1184 5191 y Fu(\))27 b Fp(2)i(X)15 b Fu(\()p Fq(B)5 b Fu(\))32 b(and)g(\()p Fq(b)1885 5155 y Fn(0)1885 5215 y Fg(1)1925 5191 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(b)2202 5155 y Fn(0)2202 5215 y Fl(n)2249 5191 y Fu(\))27 b Fp(2)i(Y)8 b Fu(\()p Fq(B)d Fu(\).)p eop %%Page: 16 16 16 15 bop 0 236 a Fm(16)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)149 425 y Fs(De\014nition)37 b(1.2.11.)49 b Fu(The)33 b(functor)625 587 y(\()p Fp(X)43 b(?)28 b(Y)8 b Fu(\)\()p Fq(B)d Fu(\))27 b(:=)h Fp(f)p Fu(\()p Fq(p;)17 b(p)1538 546 y Fn(0)1560 587 y Fu(\))28 b Fp(2)g(X)15 b Fu(\()p Fq(B)5 b Fu(\))22 b Fp(\002)h(Y)8 b Fu(\()p Fq(B)d Fu(\))p Fp(j)p Fq(p;)17 b(p)2483 546 y Fn(0)2537 587 y Fu(comm)m(ute)p Fp(g)0 749 y Fu(is)32 b(called)f(the)i Fo(ortho)-5 b(gonal)34 b(pr)-5 b(o)g(duct)33 b Fu(of)f(the)h(noncomm)m(utativ)m(e)f(spaces)i Fp(X)47 b Fu(and)33 b Fp(Y)8 b Fu(.)149 926 y Fs(Remark)37 b(1.2.12.)50 b Fu(T)-8 b(ogether)27 b(with)f Fp(X)41 b Fu(and)27 b Fp(Y)34 b Fu(the)27 b(orthogonal)e(pro)s(duct)i Fp(X)42 b(?)28 b(Y)34 b Fu(is)26 b(again)0 1043 y(a)33 b(functor,)h(since)g(homomorphisms)d Fq(f)40 b Fu(:)29 b Fq(B)35 b Fp(\000)-60 b(!)29 b Fq(B)1895 1006 y Fn(0)1952 1043 y Fu(are)k(compatible)f(with)h(the)h(m)m(ultiplication)0 1159 y(and)f(th)m(us)g(preserv)m(e)i(comm)m(uting)c(p)s(oin)m(ts.)43 b(Hence)34 b Fp(X)42 b(?)28 b(Y)41 b Fu(is)32 b(a)g(subfunctor)i(of)e Fp(X)37 b(\002)22 b(Y)8 b Fu(.)149 1336 y Fs(Lemma)38 b(1.2.13.)49 b Fo(If)35 b Fp(X)51 b Fo(and)36 b Fp(Y)43 b Fo(ar)-5 b(e)36 b(nonc)-5 b(ommutative)35 b(sp)-5 b(ac)g(es,)35 b(then)h Fp(X)45 b(?)30 b(Y)44 b Fo(is)36 b(a)g(non-)0 1452 y(c)-5 b(ommutative)34 b(sp)-5 b(ac)g(e)34 b(with)h(function)g (algebr)-5 b(a)34 b Fp(O)s Fu(\()p Fp(X)42 b(?)28 b(Y)8 b Fu(\))28 b(=)f Fp(O)s Fu(\()p Fp(X)15 b Fu(\))22 b Fp(\012)h(O)s Fu(\()p Fp(Y)8 b Fu(\))p Fo(.)149 1569 y(If)37 b Fp(X)51 b Fo(and)36 b Fp(Y)44 b Fo(have)36 b(\014nitely)h(gener)-5 b(ate)g(d)36 b(function)g(algebr)-5 b(as)36 b(then)g(the)h(function)f(algebr)-5 b(a)36 b(of)0 1685 y Fp(X)42 b(?)28 b(Y)43 b Fo(is)35 b(also)f(\014nitely)h(gener)-5 b(ate)g(d.)149 1862 y Fd(Pr)n(oof.)42 b Fu(Let)35 b Fq(A)c Fu(:=)g Fp(O)s Fu(\()p Fp(X)15 b Fu(\))34 b(and)h Fq(A)1492 1826 y Fn(0)1547 1862 y Fu(:=)c Fp(O)s Fu(\()p Fp(Y)8 b Fu(\).)49 b(Let)34 b(\()p Fq(p;)17 b(p)2346 1826 y Fn(0)2369 1862 y Fu(\))31 b Fp(2)h Fu(\()p Fp(X)45 b(?)32 b(Y)8 b Fu(\)\()p Fq(B)d Fu(\))34 b(b)s(e)h(a)f(pair)f(of)0 1978 y(comm)m(uting)23 b(p)s(oin)m(ts.)41 b(Then)26 b(there)g(is)f(a)g (unique)g(homomorphism)e(of)h(algebras)h Fq(h)j Fu(:)f Fq(A)7 b Fp(\012)g Fq(A)3324 1942 y Fn(0)3376 1978 y Fp(\000)-60 b(!)28 b Fq(B)0 2094 y Fu(suc)m(h)34 b(that)e(the)h(follo)m (wing)d(diagram)h(comm)m(utes)1270 2348 y Fq(A)305 b(A)23 b Fp(\012)f Fq(A)1916 2312 y Fn(0)p 1372 2317 248 4 v 1537 2315 a Fh(-)1479 2297 y Fq(\023)1441 2571 y(p)1375 2467 y Fh(@)1458 2550 y(@)1541 2633 y(@)1624 2716 y(@)1643 2735 y(@)-83 b(R)1741 2837 y Fq(B)5 b(:)p 1792 2735 4 351 v 1794 2735 a Fh(?)1699 2594 y Fq(h)2233 2348 y(A)2306 2312 y Fn(0)p 1969 2317 236 4 v 1969 2315 a Fh(\033)2058 2297 y Fq(\023)2092 2261 y Fn(0)2098 2586 y Fq(p)2147 2550 y Fn(0)2130 2467 y Fh(\000)2047 2550 y(\000)1964 2633 y(\000)1881 2716 y(\000)1862 2735 y(\000)-83 b(\011)0 2945 y Fu(De\014ne)38 b Fq(h)p Fu(\()p Fq(a)26 b Fp(\012)g Fq(a)632 2909 y Fn(0)655 2945 y Fu(\))36 b(:=)g Fq(p)p Fu(\()p Fq(a)p Fu(\))p Fq(p)1093 2909 y Fn(0)1117 2945 y Fu(\()p Fq(a)1206 2909 y Fn(0)1229 2945 y Fu(\))h(and)h(c)m(hec)m(k)i (the)e(necessary)i(prop)s(erties.)58 b(Observ)m(e)39 b(that)e(for)0 3061 y(an)g(arbitrary)f(homomorphism)e(of)i(algebras)h Fq(h)e Fu(:)g Fq(A)25 b Fp(\012)h Fq(A)2179 3025 y Fn(0)2238 3061 y Fp(\000)-60 b(!)35 b Fq(B)42 b Fu(the)37 b(images)f(of)g(elemen) m(ts)h(of)0 3177 y(the)32 b(form)e Fq(a)21 b Fp(\012)f Fu(1)32 b(and)f(1)20 b Fp(\012)h Fq(a)1052 3141 y Fn(0)1107 3177 y Fu(comm)m(ute)31 b(since)h(these)h(elemen)m(ts)f(already)f(comm) 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Fp(\012)h Fq(a)1093 3753 y Fn(0)1093 3814 y Fl(j)1129 3789 y Fu(.)p 3529 3789 4 66 v 3533 3726 59 4 v 3533 3789 V 3591 3789 4 66 v 149 3966 a Fs(Prop)s(osition)36 b(1.2.14.)49 b Fo(The)35 b(ortho)-5 b(gonal)34 b(pr)-5 b(o)g(duct)35 b(of)f(nonc)-5 b(ommutative)34 b(sp)-5 b(ac)g(es)34 b(is)h(asso-)0 4082 y(ciative,)f(i.e.)45 b(for)34 b(nonc)-5 b(ommutative)34 b(sp)-5 b(ac)g(es)34 b Fp(X)15 b Fo(,)34 b Fp(Y)8 b Fo(,)35 b(and)f Fp(Z)43 b Fo(we)35 b(have)1138 4244 y Fu(\()p Fp(X)42 b(?)28 b(Y)8 b Fu(\))28 b Fp(?)g(Z)1748 4217 y(\030)1749 4249 y Fu(=)1853 4244 y Fp(X)42 b(?)28 b Fu(\()p Fp(Y)36 b(?)28 b(Z)8 b Fu(\))p Fq(:)149 4422 y Fd(Pr)n(oof.)42 b Fu(Let)48 b Fq(B)53 b Fu(b)s(e)48 b(a)g(co)s(e\016cien)m(t)g(algebra)f(and)h(let) f Fq(p)2333 4437 y Fl(x)2431 4422 y Fp(2)54 b(X)15 b Fu(\()p Fq(B)5 b Fu(\),)51 b Fq(p)2919 4437 y Fl(y)3014 4422 y Fp(2)j(Y)8 b Fu(\()p Fq(B)d Fu(\),)52 b(and)0 4538 y Fq(p)49 4553 y Fl(z)121 4538 y Fp(2)33 b(Z)8 b Fu(\()p Fq(B)d Fu(\))35 b(b)s(e)h(p)s(oin)m(ts)f(suc)m(h)h(that)f(\(\() p Fq(p)1483 4553 y Fl(x)1527 4538 y Fq(;)17 b(p)1620 4553 y Fl(y)1661 4538 y Fu(\))p Fq(;)g(p)1792 4553 y Fl(z)1832 4538 y Fu(\))35 b(is)g(a)g(pair)f(of)h(comm)m(uting)e(p)s (oin)m(ts)i(in)g(\(\()p Fp(X)47 b(?)0 4654 y(Y)8 b Fu(\))27 b Fp(?)i(Z)8 b Fu(\)\()p Fq(B)d Fu(\).)43 b(In)33 b(particular)d(\()p Fq(p)1247 4669 y Fl(x)1291 4654 y Fq(;)17 b(p)1384 4669 y Fl(y)1425 4654 y Fu(\))32 b(is)g(also)f(a)h(pair)f(of)h(comm)m(uting) e(p)s(oin)m(ts.)44 b(Th)m(us)33 b(w)m(e)h(ha)m(v)m(e)0 4770 y(for)e(all)f Fq(a)c Fp(2)h Fq(A)g Fu(:=)g Fp(O)s Fu(\()p Fp(X)15 b Fu(\),)32 b Fq(a)1043 4734 y Fn(0)1094 4770 y Fp(2)c Fq(A)1261 4734 y Fn(0)1313 4770 y Fu(:=)f Fp(O)s Fu(\()p Fp(Y)8 b Fu(\),)32 b(and)h Fq(a)1976 4734 y Fn(0)q(0)2046 4770 y Fp(2)c Fq(A)2214 4734 y Fn(00)2284 4770 y Fu(:=)e Fp(O)s Fu(\()p Fp(Z)8 b Fu(\))0 4932 y Fq(p)49 4947 y Fl(x)93 4932 y Fu(\()p Fq(a)p Fu(\))p Fq(p)269 4947 y Fl(y)310 4932 y Fu(\()p Fq(a)399 4891 y Fn(0)423 4932 y Fu(\))p Fq(p)510 4947 y Fl(z)549 4932 y Fu(\()p Fq(a)638 4891 y Fn(0)q(0)681 4932 y Fu(\))28 b(=)f(\()p Fq(p)937 4947 y Fl(x)981 4932 y Fq(;)17 b(p)1074 4947 y Fl(y)1115 4932 y Fu(\)\()p Fq(a)22 b Fp(\012)h Fq(a)1415 4891 y Fn(0)1438 4932 y Fu(\))p Fq(p)1525 4947 y Fl(z)1565 4932 y Fu(\()p Fq(a)1654 4891 y Fn(00)1696 4932 y Fu(\))28 b(=)g Fq(p)1915 4947 y Fl(z)1954 4932 y Fu(\()p Fq(a)2043 4891 y Fn(0)q(0)2086 4932 y Fu(\)\()p Fq(p)2211 4947 y Fl(x)2255 4932 y Fq(;)17 b(p)2348 4947 y Fl(y)2389 4932 y Fu(\)\()p Fq(a)22 b Fp(\012)g Fq(a)2688 4891 y Fn(0)2712 4932 y Fu(\))28 b(=)f Fq(p)2930 4947 y Fl(z)2970 4932 y Fu(\()p Fq(a)3059 4891 y Fn(00)3101 4932 y Fu(\))p Fq(p)3188 4947 y Fl(x)3232 4932 y Fu(\()p Fq(a)p Fu(\))p Fq(p)3408 4947 y Fl(y)3450 4932 y Fu(\()p Fq(a)3539 4891 y Fn(0)3562 4932 y Fu(\))0 5094 y(and)1260 5216 y Fq(p)1309 5231 y Fl(x)1353 5216 y Fu(\()p Fq(a)p Fu(\))p Fq(p)1529 5231 y Fl(y)1570 5216 y Fu(\()p Fq(a)1659 5174 y Fn(0)1683 5216 y Fu(\))h(=)f Fq(p)1901 5231 y Fl(y)1942 5216 y Fu(\()p Fq(a)2031 5174 y Fn(0)2055 5216 y Fu(\))p Fq(p)2142 5231 y Fl(x)2186 5216 y Fu(\()p Fq(a)p Fu(\))p Fq(:)p eop %%Page: 17 17 17 16 bop 637 236 a Fm(2.)35 b(QUANTUM)d(SP)-6 b(A)n(CES)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(GEOMETR)-6 b(Y)535 b(17)0 425 y Fu(If)33 b(w)m(e)i(c)m(ho)s(ose)f Fq(a)c Fu(=)f(1)k(then)h(w)m(e)h(get)e Fq(p)1401 440 y Fl(y)1442 425 y Fu(\()p Fq(a)1531 389 y Fn(0)1555 425 y Fu(\))p Fq(p)1642 440 y Fl(z)1682 425 y Fu(\()p Fq(a)1771 389 y Fn(00)1813 425 y Fu(\))c(=)g Fq(p)2034 440 y Fl(z)2074 425 y Fu(\()p Fq(a)2163 389 y Fn(00)2206 425 y Fu(\))p Fq(p)2293 440 y Fl(y)2334 425 y Fu(\()p Fq(a)2423 389 y Fn(0)2446 425 y Fu(\).)47 b(F)-8 b(or)32 b(arbitrary)h Fq(a;)17 b(a)3295 389 y Fn(0)3318 425 y Fq(;)g(a)3413 389 y Fn(00)3489 425 y Fu(w)m(e)0 541 y(then)33 b(get)165 698 y Fq(p)214 713 y Fl(x)258 698 y Fu(\()p Fq(a)p Fu(\))p Fq(p)434 713 y Fl(y)476 698 y Fu(\()p Fq(a)565 657 y Fn(0)588 698 y Fu(\))p Fq(p)675 713 y Fl(z)715 698 y Fu(\()p 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b(are)h(also)f(pairs)g(of)g(comm)m(uting)f(p)s(oin)m(ts.)p 3529 855 4 66 v 3533 793 59 4 v 3533 855 V 3591 855 4 66 v 149 1030 a Fs(Problem)36 b(1.2.5.)49 b Fu(Sho)m(w)32 b(that)e(the)h(orthogonal)e(pro)s(duct)i(of)f(quan)m(tum)h(spaces)i Fp(X)42 b(?)28 b(Y)38 b Fu(is)0 1146 y(a)h(tensor)g(pro)s(duct)g(for)f (the)i(category)f Fs(QS)g Fu(\(in)f(the)h(sense)i(of)d(monoidal)e (categories)j({)g(if)e(y)m(ou)0 1262 y(kno)m(w)d(already)e(what)h(that) f(is\).)p eop %%Page: 18 18 18 17 bop 0 236 a Fm(18)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)316 425 y Fs(3.)56 b(Quan)m(tum)38 b(Monoids)f(and)i(their)d (Actions)g(on)i(Quan)m(tum)f(Spaces)149 600 y Fu(W)-8 b(e)39 b(use)g(the)f(orthogonal)e(pro)s(duct)i(in)m(tro)s(duced)g(in)f (the)i(previous)f(section)g(as)g(\\pro)s(duct")0 716 y(to)d(de\014ne)i(the)e(notion)g(of)f(a)i(monoid)d(\(some)i(ma)m(y)g (call)f(it)g(an)i(algebra)e(w.r.t.)52 b(the)36 b(orthogonal)0 832 y(pro)s(duct\).)53 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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 Fq(I)41 b Fu(hence)619 1457 y Fq(a)670 1421 y Fn(0)694 1457 y Fq(b)735 1421 y Fn(0)786 1457 y Fu(=)28 b Fq(q)937 1421 y Fn(\000)p Fg(1)1031 1457 y Fq(b)1072 1421 y Fn(0)1096 1457 y Fq(a)1147 1421 y Fn(0)1170 1457 y Fq(;)17 b(a)1265 1421 y Fn(0)1288 1457 y Fq(c)1330 1421 y Fn(0)1381 1457 y Fu(=)28 b Fq(q)1532 1421 y Fn(\000)p Fg(1)1626 1457 y Fq(c)1668 1421 y Fn(0)1691 1457 y Fq(a)1742 1421 y Fn(0)1766 1457 y Fq(;)17 b(b)1851 1421 y Fn(0)1874 1457 y Fq(d)1925 1421 y Fn(0)1976 1457 y Fu(=)27 b Fq(q)2126 1421 y Fn(\000)p Fg(1)2220 1457 y Fq(d)2271 1421 y Fn(0)2294 1457 y Fq(b)2335 1421 y Fn(0)2359 1457 y Fq(;)17 b(c)2445 1421 y Fn(0)2468 1457 y Fq(d)2519 1421 y Fn(0)2570 1457 y Fu(=)27 b Fq(q)2720 1421 y Fn(\000)p Fg(1)2815 1457 y Fq(d)2866 1421 y Fn(0)2889 1457 y Fq(c)2931 1421 y Fn(0)2954 1457 y Fq(;)1009 1573 y(b)1050 1537 y Fn(0)1074 1573 y Fq(c)1116 1537 y Fn(0)1167 1573 y Fu(=)g Fq(c)1312 1537 y Fn(0)1336 1573 y Fq(b)1377 1537 y Fn(0)1400 1573 y Fq(;)17 b(a)1495 1537 y Fn(0)1519 1573 y Fq(d)1570 1537 y Fn(0)1615 1573 y Fp(\000)22 b Fq(q)1761 1537 y Fn(\000)p Fg(1)1855 1573 y Fq(b)1896 1537 y Fn(0)1920 1573 y Fq(c)1962 1537 y Fn(0)2013 1573 y Fu(=)28 b Fq(d)2168 1537 y Fn(0)2191 1573 y Fq(a)2242 1537 y Fn(0)2287 1573 y Fp(\000)23 b Fq(q)t(c)2476 1537 y Fn(0)2499 1573 y Fq(b)2540 1537 y Fn(0)2564 1573 y Fq(:)0 1757 y Fu(W)-8 b(e)37 b(write)g(these)h(quadruples)f(as)g(2)25 b Fp(\002)g Fu(2-matrices)36 b(and)g(call)f(them)i Fo(quantum)i(matric) -5 b(es)p Fu(.)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 Fo(The)h(algebr)-5 b(aic)34 b(appr)-5 b(o)g(ach:)149 2455 y Fu(The)34 b(algebra)d Fq(M)785 2470 y Fl(q)824 2455 y Fu(\(2\))h(is)g(a)g(bialgebra)f(with)h (the)h(diagonal)1105 2707 y(\001)1203 2567 y Fe(\022)1277 2647 y Fq(a)87 b(b)1281 2764 y(c)h(d)1461 2567 y Fe(\023)1562 2707 y Fu(=)1666 2567 y Fe(\022)1739 2647 y Fq(a)g(b)1744 2764 y(c)g(d)1924 2567 y Fe(\023)2020 2707 y Fp(\012)2119 2567 y Fe(\022)2193 2647 y Fq(a)g(b)2197 2764 y(c)g(d)2378 2567 y Fe(\023)2468 2707 y Fq(;)0 2957 y Fu(i.e.)77 b(b)m(y)45 b(\001\()p Fq(a)p Fu(\))i(=)g Fq(a)30 b Fp(\012)h Fq(a)f Fu(+)f Fq(b)i Fp(\012)f Fq(c)p Fu(,)47 b(\001\()p Fq(b)p Fu(\))g(=)g Fq(a)30 b Fp(\012)h Fq(b)f Fu(+)g Fq(b)g Fp(\012)h Fq(d)p Fu(,)46 b(\001\()p Fq(c)p Fu(\))h(=)g Fq(c)30 b Fp(\012)g Fq(a)g Fu(+)g Fq(d)g Fp(\012)g Fq(c)44 b Fu(and)0 3073 y(\001\()p Fq(d)p Fu(\))28 b(=)f Fq(c)22 b Fp(\012)h Fq(b)f Fu(+)g Fq(d)g Fp(\012)h Fq(d)p Fu(,)32 b(and)h(with)f(the)h(counit)1352 3322 y Fq(")1415 3182 y Fe(\022)1488 3263 y Fq(a)87 b(b)1492 3379 y(c)h(d)1673 3182 y Fe(\023)1774 3322 y Fu(=)1877 3182 y Fe(\022)1950 3263 y Fu(1)83 b(0)1950 3379 y(0)g(1)2131 3182 y Fe(\023)2221 3322 y Fq(;)0 3572 y Fu(i.e.)53 b Fq(")p Fu(\()p Fq(a)p Fu(\))32 b(=)h(1,)k Fq(")p Fu(\()p Fq(b)p Fu(\))c(=)g(0,)j Fq(")p Fu(\()p Fq(c)p Fu(\))d(=)g(0,)j(and)g Fq(")p Fu(\()p Fq(d)p Fu(\))d(=)g(1.)52 b(W)-8 b(e)37 b(ha)m(v)m(e)g(to)e(pro)m(v)m(e) i(that)f(\001)g(and)g Fq(")f Fu(are)0 3688 y(homomorphisms)i(of)i (algebras)f(and)i(that)f(the)g(coalgebra)f(la)m(ws)i(are)f (satis\014ed.)64 b(T)-8 b(o)39 b(obtain)f(a)0 3804 y(homomorphism)25 b(of)j(algebras)f(\001)h(:)g Fq(M)1424 3819 y Fl(q)1462 3804 y Fu(\(2\))f Fp(\000)-60 b(!)28 b Fq(M)1853 3819 y Fl(q)1892 3804 y Fu(\(2\))13 b Fp(\012)g Fq(M)2214 3819 y Fl(q)2251 3804 y Fu(\(2\))28 b(w)m(e)h(de\014ne)g(\001)f(:)g Fr(K)17 b Fp(h)p Fq(a;)g(b;)h(c;)f(d)p Fp(i)32 b(\000)-60 b(!)0 3921 y Fq(M)94 3936 y Fl(q)132 3921 y Fu(\(2\))22 b Fp(\012)h Fq(M)473 3936 y Fl(q)511 3921 y Fu(\(2\))33 b(on)f(the)i(free)f(algebra)e(\(the)i(p)s(olynomial)d(ring)h(in)h (noncomm)m(uting)f(v)-5 b(ariables\))0 4037 y Fr(K)17 b Fp(h)p Fq(a;)h(b;)f(c;)g(d)p Fp(i)46 b Fu(generated)d(b)m(y)f(the)h (set)f Fp(f)p Fq(a;)17 b(b;)g(c;)g(d)p Fp(g)41 b Fu(and)h(sho)m(w)h (that)e(it)g(v)-5 b(anishes)43 b(on)e(the)h(ideal)0 4153 y Fq(I)47 b Fu(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 Fq(M)1435 4284 y Fl(q)1474 4269 y Fu(\(2\))f Fp(\000)-60 b(!)28 b Fq(M)1865 4284 y Fl(q)1903 4269 y Fu(\(2\))20 b Fp(\012)h Fq(M)2240 4284 y Fl(q)2279 4269 y Fu(\(2\).)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 Fq(I)8 b Fu(:)42 4566 y(\001\()p Fq(ab)23 b Fp(\000)f Fq(q)422 4530 y Fn(\000)p Fg(1)516 4566 y Fq(ba)p Fu(\))29 b(=)e(\001\()p Fq(a)p Fu(\)\001\()p Fq(b)p Fu(\))c Fp(\000)g Fq(q)1354 4530 y Fn(\000)p Fg(1)1448 4566 y Fu(\001\()p Fq(b)p Fu(\)\001\()p Fq(a)p Fu(\))29 b(=)42 4683 y(=)e(\()p Fq(a)22 b Fp(\012)h Fq(a)f Fu(+)g Fq(b)h Fp(\012)g Fq(c)p Fu(\)\()p Fq(a)f Fp(\012)g Fq(b)h Fu(+)f Fq(b)h Fp(\012)f Fq(d)p Fu(\))g Fp(\000)h Fq(q)1564 4647 y Fn(\000)p Fg(1)1658 4683 y Fu(\()p Fq(a)f Fp(\012)h Fq(b)f Fu(+)g Fq(b)h Fp(\012)g Fq(d)p Fu(\)\()p Fq(a)e Fp(\012)i Fq(a)f Fu(+)g Fq(b)h Fp(\012)g Fq(c)p Fu(\))42 4799 y(=)k Fq(aa)c Fp(\012)f Fq(ab)h Fu(+)f Fq(ab)h Fp(\012)g Fq(ad)f Fu(+)g Fq(ba)g Fp(\012)h Fq(cb)g Fu(+)f Fq(bb)h Fp(\012)f Fq(cd)g Fp(\000)h Fq(q)1903 4763 y Fn(\000)p Fg(1)1997 4799 y Fu(\()p Fq(aa)f Fp(\012)h Fq(ba)g Fu(+)f Fq(ab)h Fp(\012)f Fq(bc)h Fu(+)f Fq(ba)h Fp(\012)f Fq(da)g Fu(+)g Fq(bb)i Fp(\012)e Fq(dc)p Fu(\))42 4915 y(=)27 b Fq(aa)c Fp(\012)f Fu(\()p Fq(ab)h Fp(\000)g Fq(q)669 4879 y Fn(\000)p Fg(1)763 4915 y Fq(ba)p Fu(\))g(+)f Fq(ab)g Fp(\012)h Fu(\()p Fq(ad)f Fp(\000)h Fq(q)1537 4879 y Fn(\000)p Fg(1)1631 4915 y Fq(bc)p Fu(\))f(+)g Fq(ba)h Fp(\012)g Fu(\()p Fq(cb)f Fp(\000)h Fq(q)2377 4879 y Fn(\000)p Fg(1)2471 4915 y Fq(da)p Fu(\))f(+)g Fq(bb)h Fp(\012)g Fu(\()p Fq(cd)e Fp(\000)i Fq(q)3235 4879 y Fn(\000)p Fg(1)3329 4915 y Fq(dc)p Fu(\))42 5031 y(=)k Fq(ba)c Fp(\012)g Fu(\()p Fq(q)445 4995 y Fn(\000)p Fg(1)539 5031 y Fq(ad)f Fp(\000)g Fq(q)809 4995 y Fn(\000)p Fg(2)903 5031 y Fq(bc)h Fu(+)f Fq(cb)h Fp(\000)f Fq(q)1359 4995 y Fn(\000)p Fg(1)1453 5031 y Fq(da)p Fu(\))28 b Fp(\021)g Fu(0)99 b(mo)s(d)33 b(\()p Fq(I)8 b Fu(\))p Fq(:)0 5216 y Fu(The)34 b(reader)f(should)f(c)m(hec)m(k)j(the)e(other)g (iden)m(tities.)p eop %%Page: 21 21 21 20 bop 451 236 a Fm(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 Fu(The)34 b(coasso)s(ciativit)m(y)d(follo)m(ws)h(from)140 657 y(\(\001)22 b Fp(\012)h Fu(1\)\001)566 516 y Fe(\022)639 597 y Fq(a)88 b(b)643 713 y(c)g(d)824 516 y Fe(\023)925 657 y Fu(=)27 b(\001)1126 516 y Fe(\022)1200 597 y Fq(a)87 b(b)1204 713 y(c)h(d)1384 516 y Fe(\023)1480 657 y Fp(\012)1580 516 y Fe(\022)1653 597 y Fq(a)g(b)1658 713 y(c)f(d)1838 516 y Fe(\023)1939 657 y Fu(=)27 b(\()2080 516 y Fe(\022)2154 597 y Fq(a)87 b(b)2158 713 y(c)h(d)2339 516 y Fe(\023)2434 657 y Fp(\012)2534 516 y Fe(\022)2607 597 y Fq(a)g(b)2612 713 y(c)f(d)2792 516 y Fe(\023)2865 657 y Fu(\))22 b Fp(\012)3025 516 y Fe(\022)3098 597 y Fq(a)88 b(b)3103 713 y(c)g(d)3283 516 y Fe(\023)3384 657 y Fu(=)118 896 y(=)222 755 y Fe(\022)295 836 y Fq(a)g(b)300 952 y(c)f(d)480 755 y Fe(\023)575 896 y Fp(\012)23 b Fu(\()713 755 y Fe(\022)786 836 y Fq(a)88 b(b)791 952 y(c)f(d)971 755 y Fe(\023)1067 896 y Fp(\012)1166 755 y Fe(\022)1240 836 y Fq(a)g(b)1244 952 y(c)h(d)1425 755 y Fe(\023)1498 896 y Fu(\))27 b(=)1667 755 y Fe(\022)1740 836 y Fq(a)88 b(b)1745 952 y(c)g(d)1925 755 y Fe(\023)2021 896 y Fp(\012)22 b Fu(\001)2218 755 y Fe(\022)2292 836 y Fq(a)87 b(b)2296 952 y(c)h(d)2477 755 y Fe(\023)2578 896 y Fu(=)27 b(\(1)22 b Fp(\012)h Fu(\001\)\001)3107 755 y Fe(\022)3180 836 y Fq(a)88 b(b)3185 952 y(c)f(d)3365 755 y Fe(\023)3455 896 y Fq(:)0 1127 y Fu(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 Fo(The)h(ge)-5 b(ometric)34 b(appr)-5 b(o)g(ach:)149 1360 y Fq(M)243 1375 y Fl(q)282 1360 y Fu(\(2\))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 Fe(\022)1550 1588 y Fq(a)1601 1603 y Fg(1)1728 1588 y Fq(b)1769 1603 y Fg(1)1554 1704 y Fq(c)1596 1719 y Fg(1)1723 1704 y Fq(d)1774 1719 y Fg(1)1814 1508 y Fe(\023)1915 1648 y Fu(and)2101 1508 y Fe(\022)2174 1588 y Fq(a)2225 1603 y Fg(2)2352 1588 y Fq(b)2393 1603 y Fg(2)2178 1704 y Fq(c)2220 1719 y Fg(2)2348 1704 y Fq(d)2399 1719 y Fg(2)2438 1508 y Fe(\023)2539 1648 y Fu(in)f Fp(M)2769 1663 y Fl(q)2806 1648 y Fu(\(2\)\()p Fq(A)p Fu(\).)42 b(Then)29 b(their)0 1820 y(matrix)i(pro)s(duct)738 1911 y Fe(\022)811 1992 y Fq(a)862 2007 y Fg(1)989 1992 y Fq(b)1030 2007 y Fg(1)815 2108 y Fq(c)857 2123 y Fg(1)985 2108 y Fq(d)1036 2123 y Fg(1)1075 1911 y Fe(\023)17 b(\022)1238 1992 y Fq(a)1289 2007 y Fg(2)1416 1992 y Fq(b)1457 2007 y Fg(2)1243 2108 y Fq(c)1285 2123 y Fg(2)1412 2108 y Fq(d)1463 2123 y Fg(2)1502 1911 y Fe(\023)1603 2052 y Fu(=)1706 1911 y Fe(\022)1780 1992 y Fq(a)1831 2007 y Fg(1)1870 1992 y Fq(a)1921 2007 y Fg(2)1983 1992 y Fu(+)22 b Fq(b)2122 2007 y Fg(1)2162 1992 y Fq(c)2204 2007 y Fg(2)2326 1992 y Fq(a)2377 2007 y Fg(1)2417 1992 y Fq(b)2458 2007 y Fg(2)2520 1992 y Fu(+)g Fq(b)2659 2007 y Fg(1)2699 1992 y Fq(d)2750 2007 y Fg(2)1780 2108 y Fq(c)1822 2123 y Fg(1)1861 2108 y Fq(a)1912 2123 y Fg(2)1974 2108 y Fu(+)g Fq(d)2123 2123 y Fg(1)2162 2108 y Fq(c)2204 2123 y Fg(2)2326 2108 y Fq(c)2368 2123 y Fg(1)2408 2108 y Fq(b)2449 2123 y Fg(2)2511 2108 y Fu(+)g Fq(d)2660 2123 y Fg(1)2699 2108 y Fq(d)2750 2123 y Fg(2)2789 1911 y Fe(\023)0 2283 y Fu(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(\()p Fq(a)131 2466 y Fg(1)170 2451 y Fq(a)221 2466 y Fg(2)283 2451 y Fu(+)22 b Fq(b)422 2466 y Fg(1)462 2451 y Fq(c)504 2466 y Fg(2)543 2451 y Fu(\)\()p Fq(a)670 2466 y Fg(1)710 2451 y Fq(b)751 2466 y Fg(2)813 2451 y Fu(+)g Fq(b)952 2466 y Fg(1)992 2451 y Fq(d)1043 2466 y Fg(2)1082 2451 y Fu(\))27 b(=)h Fq(a)1302 2466 y Fg(1)1342 2451 y Fq(a)1393 2466 y Fg(2)1432 2451 y Fq(a)1483 2466 y Fg(1)1523 2451 y Fq(b)1564 2466 y Fg(2)1626 2451 y Fu(+)22 b Fq(a)1775 2466 y Fg(1)1815 2451 y Fq(a)1866 2466 y Fg(2)1905 2451 y Fq(b)1946 2466 y Fg(1)1986 2451 y Fq(d)2037 2466 y Fg(2)2098 2451 y Fu(+)g Fq(b)2237 2466 y Fg(1)2277 2451 y Fq(c)2319 2466 y Fg(2)2359 2451 y Fq(a)2410 2466 y Fg(1)2449 2451 y Fq(b)2490 2466 y Fg(2)2552 2451 y Fu(+)g Fq(b)2691 2466 y Fg(1)2731 2451 y Fq(c)2773 2466 y Fg(2)2813 2451 y Fq(b)2854 2466 y Fg(1)2894 2451 y Fq(d)2945 2466 y Fg(2)581 2568 y Fu(=)28 b Fq(a)736 2583 y Fg(1)775 2568 y Fq(a)826 2583 y Fg(1)866 2568 y Fq(a)917 2583 y Fg(2)957 2568 y Fq(b)998 2583 y Fg(2)1060 2568 y Fu(+)22 b Fq(a)1209 2583 y Fg(1)1248 2568 y Fq(b)1289 2583 y Fg(1)1329 2568 y Fq(a)1380 2583 y Fg(2)1420 2568 y Fq(d)1471 2583 y Fg(2)1532 2568 y Fu(+)g Fq(b)1671 2583 y Fg(1)1711 2568 y Fq(a)1762 2583 y Fg(1)1802 2568 y Fq(c)1844 2583 y Fg(2)1883 2568 y Fq(b)1924 2583 y Fg(2)1986 2568 y Fu(+)g Fq(b)2125 2583 y Fg(1)2165 2568 y Fq(b)2206 2583 y Fg(1)2246 2568 y Fq(c)2288 2583 y Fg(2)2327 2568 y Fq(d)2378 2583 y Fg(2)581 2684 y Fu(=)28 b Fq(q)732 2648 y Fn(\000)p Fg(1)826 2684 y Fq(a)877 2699 y Fg(1)916 2684 y Fq(a)967 2699 y Fg(1)1007 2684 y Fq(b)1048 2699 y Fg(2)1088 2684 y Fq(a)1139 2699 y Fg(2)1201 2684 y Fu(+)22 b Fq(q)1346 2648 y Fn(\000)p Fg(1)1440 2684 y Fq(b)1481 2699 y Fg(1)1521 2684 y Fq(a)1572 2699 y Fg(1)1612 2684 y Fu(\()p Fq(d)1701 2699 y Fg(2)1740 2684 y Fq(a)1791 2699 y Fg(2)1852 2684 y Fu(+)g(\()p Fq(q)2035 2648 y Fn(\000)p Fg(1)2152 2684 y Fp(\000)g Fq(q)t Fu(\))p Fq(b)2377 2699 y Fg(2)2417 2684 y Fq(c)2459 2699 y Fg(2)2498 2684 y Fu(\))g(+)g Fq(b)2697 2699 y Fg(1)2737 2684 y Fq(a)2788 2699 y Fg(1)2828 2684 y Fq(b)2869 2699 y Fg(2)2909 2684 y Fq(c)2951 2699 y Fg(2)3012 2684 y Fu(+)g Fq(q)3157 2648 y Fn(\000)p Fg(1)3252 2684 y Fq(b)3293 2699 y Fg(1)3333 2684 y Fq(b)3374 2699 y Fg(1)3413 2684 y Fq(d)3464 2699 y Fg(2)3504 2684 y Fq(c)3546 2699 y Fg(2)581 2800 y Fu(=)28 b Fq(q)732 2764 y Fn(\000)p Fg(1)826 2800 y Fu(\()p Fq(a)915 2815 y Fg(1)954 2800 y Fq(a)1005 2815 y Fg(1)1045 2800 y Fq(b)1086 2815 y Fg(2)1126 2800 y Fq(a)1177 2815 y Fg(2)1239 2800 y Fu(+)22 b Fq(a)1388 2815 y Fg(1)1427 2800 y Fq(b)1468 2815 y Fg(1)1508 2800 y Fq(b)1549 2815 y Fg(2)1589 2800 y Fq(c)1631 2815 y Fg(2)1693 2800 y Fu(+)g Fq(b)1832 2815 y Fg(1)1872 2800 y Fq(a)1923 2815 y Fg(1)1962 2800 y Fq(d)2013 2815 y Fg(2)2052 2800 y Fq(a)2103 2815 y Fg(2)2165 2800 y Fu(+)g Fq(b)2304 2815 y Fg(1)2344 2800 y Fq(b)2385 2815 y Fg(1)2425 2800 y Fq(d)2476 2815 y Fg(2)2515 2800 y Fq(c)2557 2815 y Fg(2)2597 2800 y Fu(\))581 2916 y(=)28 b Fq(q)732 2880 y Fn(\000)p Fg(1)826 2916 y Fu(\()p Fq(a)915 2931 y Fg(1)954 2916 y Fq(b)995 2931 y Fg(2)1035 2916 y Fq(a)1086 2931 y Fg(1)1126 2916 y Fq(a)1177 2931 y Fg(2)1239 2916 y Fu(+)22 b Fq(a)1388 2931 y Fg(1)1427 2916 y Fq(b)1468 2931 y Fg(2)1508 2916 y Fq(b)1549 2931 y Fg(1)1589 2916 y Fq(c)1631 2931 y Fg(2)1693 2916 y Fu(+)g Fq(b)1832 2931 y Fg(1)1872 2916 y Fq(d)1923 2931 y Fg(2)1962 2916 y Fq(a)2013 2931 y Fg(1)2052 2916 y Fq(a)2103 2931 y Fg(2)2165 2916 y Fu(+)g Fq(b)2304 2931 y Fg(1)2344 2916 y Fq(d)2395 2931 y Fg(2)2434 2916 y Fq(b)2475 2931 y Fg(1)2515 2916 y Fq(c)2557 2931 y Fg(2)2597 2916 y Fu(\))581 3033 y(=)28 b Fq(q)732 2996 y Fn(\000)p Fg(1)826 3033 y Fu(\()p Fq(a)915 3048 y Fg(1)954 3033 y Fq(b)995 3048 y Fg(2)1057 3033 y Fu(+)22 b Fq(b)1196 3048 y Fg(1)1236 3033 y Fq(d)1287 3048 y Fg(2)1327 3033 y Fu(\)\()p Fq(a)1454 3048 y Fg(1)1493 3033 y Fq(a)1544 3048 y Fg(2)1606 3033 y Fu(+)g Fq(b)1745 3048 y Fg(1)1785 3033 y Fq(c)1827 3048 y Fg(2)1866 3033 y Fu(\))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 Fp(M)1325 3335 y Fl(q)1362 3320 y Fu(\(2\))g Fp(?)g(M)1756 3335 y Fl(q)1794 3320 y Fu(\(2\)\()p Fq(A)p Fu(\))f Fp(\000)-60 b(!)36 b(M)2376 3335 y Fl(q)2414 3320 y Fu(\(2\)\()p Fq(A)p Fu(\))h(\(natural)f(in)h Fq(A)p Fu(\).)58 b(It)37 b(is)0 3501 y(asso)s(ciativ)m(e)32 b(and)h(has)g(unit)1056 3361 y Fe(\022)1129 3441 y Fu(1)83 b(0)1129 3558 y(0)g(1)1310 3361 y Fe(\023)1383 3501 y Fu(.)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(\(\()1132 3667 y Fe(\022)1206 3748 y Fq(a)1257 3763 y Fg(1)1384 3748 y Fq(b)1425 3763 y Fg(1)1210 3864 y Fq(c)1252 3879 y Fg(1)1379 3864 y Fq(d)1430 3879 y Fg(1)1469 3667 y Fe(\023)1559 3808 y Fq(;)1603 3667 y Fe(\022)1676 3748 y Fq(a)1727 3763 y Fg(2)1855 3748 y Fq(b)1896 3763 y Fg(2)1681 3864 y Fq(c)1723 3879 y Fg(2)1850 3864 y Fq(d)1901 3879 y Fg(2)1940 3667 y Fe(\023)2014 3808 y Fu(\))p Fq(;)2096 3667 y Fe(\022)2169 3748 y Fq(a)2220 3763 y Fg(3)2347 3748 y Fq(b)2388 3763 y Fg(3)2173 3864 y Fq(c)2215 3879 y Fg(3)2342 3864 y Fq(d)2393 3879 y Fg(3)2432 3667 y Fe(\023)2506 3808 y Fu(\))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 Fe(\022)1159 4211 y Fq(a)1210 4226 y Fg(1)1338 4211 y Fq(b)1379 4226 y Fg(1)1164 4327 y Fq(c)1206 4342 y Fg(1)1333 4327 y Fq(d)1384 4342 y Fg(1)1423 4130 y Fe(\023)1513 4271 y Fq(;)17 b Fu(\()1595 4130 y Fe(\022)1668 4211 y Fq(a)1719 4226 y Fg(2)1846 4211 y Fq(b)1887 4226 y Fg(2)1673 4327 y Fq(c)1715 4342 y Fg(2)1842 4327 y Fq(d)1893 4342 y Fg(2)1932 4130 y Fe(\023)2022 4271 y Fq(;)2066 4130 y Fe(\022)2139 4211 y Fq(a)2190 4226 y Fg(3)2317 4211 y Fq(b)2358 4226 y Fg(3)2143 4327 y Fq(c)2185 4342 y Fg(3)2313 4327 y Fq(d)2364 4342 y Fg(3)2403 4130 y Fe(\023)2476 4271 y Fu(\)\))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 Fe(\022)491 4614 y Fu(1)83 b(0)491 4731 y(0)g(1)672 4534 y Fe(\023)17 b(\022)835 4614 y Fq(a)88 b(b)840 4731 y(c)f(d)1020 4534 y Fe(\023)1145 4674 y Fu(=)1272 4534 y Fe(\022)1345 4614 y Fq(a)h(b)1350 4731 y(c)f(d)1530 4534 y Fe(\023)1655 4674 y Fu(=)1782 4534 y Fe(\022)1855 4614 y Fq(a)h(b)1860 4731 y(c)g(d)2040 4534 y Fe(\023)17 b(\022)2203 4614 y Fu(1)83 b(0)2203 4731 y(0)g(1)2384 4534 y Fe(\023)2504 4674 y Fu(for)46 b(all)e(quan)m(tum)j(matrices)0 4781 y Fe(\022)73 4862 y Fq(a)88 b(b)78 4978 y(c)g(d)258 4781 y Fe(\023)359 4922 y Fp(2)28 b(M)573 4937 y Fl(q)611 4922 y Fu(\(2\)\()p Fq(B)5 b Fu(\))32 b(w)m(e)i(see)g(that)e Fp(M)1556 4937 y Fl(q)1594 4922 y Fu(\(2\))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 Fp(M)2239 5114 y Fl(q)2277 5099 y Fu(\(2\))g(and)h(the)f(com)m (ultiplication)c(of)0 5216 y Fq(M)94 5231 y Fl(q)132 5216 y Fu(\(2\))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 22 22 21 bop 0 236 a Fm(22)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fq(M)94 440 y Fl(q)132 425 y Fu(\(2\))22 b Fp(\012)h Fq(M)473 440 y Fl(q)511 425 y Fu(\(2\))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(\()p Fq(\023)72 603 y Fg(1)112 588 y Fq(;)17 b(\023)190 603 y Fg(2)229 588 y Fu(\))28 b Fp(2)g(M)509 603 y Fl(q)547 588 y Fu(\(2\))f Fp(?)h(M)924 603 y Fl(q)962 588 y Fu(\(2\)\()p Fq(M)1219 603 y Fl(q)1257 588 y Fu(\(2\))17 b Fp(\012)g Fq(M)1587 603 y Fl(q)1626 588 y Fu(\(2\)\))27 b(=)h Fr(K)17 b Fu(-)6 b Fs(Alg)n Fu(\()p Fq(M)2333 603 y Fl(q)2372 588 y Fu(\(2\))17 b Fp(\012)g Fq(M)2702 603 y Fl(q)2741 588 y Fu(\(2\))p Fq(;)g(M)3004 603 y Fl(q)3041 588 y Fu(\(2\))g Fp(\012)h Fq(M)3372 603 y Fl(q)3410 588 y Fu(\(2\)\))p Fq(:)0 816 y Fu(Since)38 b Fq(\023)294 831 y Fg(1)370 816 y Fu(=)483 675 y Fe(\022)556 756 y Fq(a)88 b(b)561 872 y(c)f(d)741 675 y Fe(\023)840 816 y Fp(\012)26 b Fu(1)37 b(=)1141 675 y Fe(\022)1214 756 y Fq(a)23 b Fp(\012)f Fu(1)88 b Fq(b)22 b Fp(\012)h Fu(1)1219 872 y Fq(c)f Fp(\012)h Fu(1)87 b Fq(d)22 b Fp(\012)h Fu(1)1740 675 y Fe(\023)1851 816 y Fu(and)38 b Fq(\023)2080 831 y Fg(2)2157 816 y Fu(=)e(1)26 b Fp(\012)2447 675 y Fe(\022)2520 756 y Fq(a)88 b(b)2525 872 y(c)f(d)2705 675 y Fe(\023)2815 816 y Fu(=)2927 675 y Fe(\022)3001 756 y Fu(1)22 b Fp(\012)g Fq(a)88 b Fu(1)22 b Fp(\012)h Fq(b)3005 872 y Fu(1)f Fp(\012)h Fq(c)87 b Fu(1)22 b Fp(\012)h Fq(d)3527 675 y Fe(\023)0 1063 y Fu(w)m(e)38 b(ha)m(v)m(e)g(id)33 b(=)i(\()p Fq(\023)675 1078 y Fg(1)715 1063 y Fq(;)17 b(\023)793 1078 y Fg(2)832 1063 y Fu(\))35 b(=)f(\()1053 922 y Fe(\022)1126 1003 y Fq(a)88 b(b)1131 1119 y(c)g(d)1311 922 y Fe(\023)1410 1063 y Fp(\012)25 b Fu(1)p Fq(;)17 b Fu(1)24 b Fp(\012)1780 922 y Fe(\022)1854 1003 y Fq(a)88 b(b)1858 1119 y(c)g(d)2039 922 y Fe(\023)2112 1063 y Fu(\).)56 b(The)37 b(Y)-8 b(oneda)37 b(Lemma)e(de\014nes)k(the)0 1241 y(diagonal)24 b(as)j(the)h(image)d(of)h(the)h(iden)m(tit)m(y)g (under)g Fr(K)18 b Fu(-)5 b Fs(Alg)o Fu(\()p Fq(M)2243 1256 y Fl(q)2282 1241 y Fu(\(2\))10 b Fp(\012)g Fq(M)2598 1256 y Fl(q)2637 1241 y Fu(\(2\))p Fq(;)17 b(M)2900 1256 y Fl(q)2937 1241 y Fu(\(2\))10 b Fp(\012)g Fq(M)3253 1256 y Fl(q)3293 1241 y Fu(\(2\)\))27 b Fp(\000)-60 b(!)0 1421 y Fr(K)17 b Fu(-)6 b Fs(Alg)o Fu(\()p Fq(M)414 1436 y Fl(q)452 1421 y Fu(\(2\))p Fq(;)17 b(M)715 1436 y Fl(q)753 1421 y Fu(\(2\))j Fp(\012)i Fq(M)1091 1436 y Fl(q)1129 1421 y Fu(\(2\)\))32 b(b)m(y)h(the)f(m)m(ultiplication.)39 b(So)32 b(\001\()2544 1281 y Fe(\022)2617 1361 y Fq(a)88 b(b)2622 1478 y(c)f(d)2802 1281 y Fe(\023)2875 1421 y Fu(\))28 b(=)g(\001)f(=)h Fq(\023)3291 1436 y Fg(1)3352 1421 y Fp(\003)21 b Fq(\023)3457 1436 y Fg(2)3524 1421 y Fu(=)0 1669 y(\()38 1528 y Fe(\022)111 1609 y Fq(a)88 b(b)116 1725 y(c)g(d)296 1528 y Fe(\023)392 1669 y Fp(\012)22 b Fu(1\))g Fp(\003)g Fu(\(1)g Fp(\012)881 1528 y Fe(\022)954 1609 y Fq(a)88 b(b)959 1725 y(c)f(d)1139 1528 y Fe(\023)1212 1669 y Fu(\))28 b(=)1381 1528 y Fe(\022)1455 1609 y Fq(a)87 b(b)1459 1725 y(c)h(d)1640 1528 y Fe(\023)1735 1669 y Fp(\012)1835 1528 y Fe(\022)1908 1609 y Fq(a)g(b)1913 1725 y(c)f(d)2093 1528 y Fe(\023)2166 1669 y Fu(.)149 1846 y(Th)m(us)35 b Fq(M)491 1861 y Fl(q)529 1846 y Fu(\(2\))d (de\014nes)i(a)f(quan)m(tum)g(monoid)d Fp(M)1966 1861 y Fl(q)2004 1846 y Fu(\(2\))i(with)293 2074 y Fp(M)413 2089 y Fl(q)450 2074 y Fu(\(2\)\()p Fq(B)5 b Fu(\))28 b(=)861 1933 y Fe(\032\022)1009 2014 y Fq(a)1060 1978 y Fn(0)1171 2014 y Fq(b)1212 1978 y Fn(0)1014 2130 y Fq(c)1056 2094 y Fn(0)1167 2130 y Fq(d)1218 2094 y Fn(0)1241 1933 y Fe(\023)1314 1989 y(\014)1314 2049 y(\014)1347 2074 y Fq(a)1398 2033 y Fn(0)1422 2074 y Fq(;)17 b(b)1507 2033 y Fn(0)1530 2074 y Fq(;)g(c)1616 2033 y Fn(0)1639 2074 y Fq(;)g(d)1734 2033 y Fn(0)1785 2074 y Fp(2)28 b Fq(B)5 b Fu(;)17 b Fq(a)2053 2033 y Fn(0)2076 2074 y Fq(b)2117 2033 y Fn(0)2169 2074 y Fu(=)27 b Fq(q)2319 2033 y Fn(\000)p Fg(1)2413 2074 y Fq(b)2454 2033 y Fn(0)2478 2074 y Fq(a)2529 2033 y Fn(0)2552 2074 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(b)2829 2033 y Fn(0)2852 2074 y Fq(c)2894 2033 y Fn(0)2945 2074 y Fu(=)28 b Fq(c)3091 2033 y Fn(0)3114 2074 y Fq(b)3155 2033 y Fn(0)3189 1933 y Fe(\033)3280 2074 y Fq(:)0 2305 y Fu(This)34 b(is)f(the)h(deformed)f(v)m(ersion)h (of)f Fp(M)1482 2263 y Fn(\002)1482 2329 y Fg(2)1575 2305 y Fu(the)h(m)m(ultiplicativ)m(e)c(monoid)i(of)h(the)h(2)22 b Fp(\002)h Fu(2-matrices)0 2421 y(of)32 b(comm)m(utativ)m(e)g (algebras.)149 2549 y(2.)43 b(Let)31 b Fq(A)514 2498 y Fg(2)p Fn(j)p Fg(0)514 2560 y Fl(q)636 2549 y Fu(=)d Fr(K)17 b Fp(h)p Fq(x;)h(y)t Fp(i)p Fq(=)p Fu(\()p Fq(xy)26 b Fp(\000)19 b Fq(q)1400 2513 y Fn(\000)p Fg(1)1494 2549 y Fq(y)t(x)p Fu(\))30 b(b)s(e)h(the)h(function)e(algebra)g(of)g(the)h (quan)m(tum)g(plane)0 2683 y Fr(A)66 2632 y Fg(2)p Fn(j)p Fg(0)66 2694 y Fl(q)166 2683 y Fu(.)44 b(By)33 b(the)g(de\014nition)e (1.2.5)h(w)m(e)i(ha)m(v)m(e)897 2906 y Fr(A)963 2865 y Fg(2)p Fn(j)p Fg(0)963 2931 y Fl(q)1063 2906 y Fu(\()p Fq(A)1174 2865 y Fn(0)1198 2906 y Fu(\))27 b(=)1367 2765 y Fe(\032\022)1515 2846 y Fq(x)1517 2962 y(y)1570 2765 y Fe(\023)1644 2821 y(\014)1644 2881 y(\014)1677 2906 y Fq(x;)17 b(y)31 b Fp(2)d Fq(A)2022 2865 y Fn(0)2045 2906 y Fu(;)17 b Fq(xy)31 b Fu(=)d Fq(q)2374 2865 y Fn(\000)p Fg(1)2468 2906 y Fq(y)t(x)2585 2765 y Fe(\033)2676 2906 y Fq(:)0 3125 y Fu(The)34 b(set)401 3324 y Fp(M)521 3339 y Fl(q)559 3324 y Fu(\(2\)\()p Fq(A)795 3283 y Fn(0)818 3324 y Fu(\))27 b(=)987 3184 y Fe(\032\022)1135 3265 y Fq(u)83 b(x)1137 3381 y(y)91 b(z)1329 3184 y Fe(\023)1402 3240 y(\014)1402 3299 y(\014)1436 3324 y Fq(u;)17 b(x;)g(y)t(;)g(z)30 b Fp(2)e Fq(A)1973 3283 y Fn(0)1997 3324 y Fu(;)17 b Fq(ux)27 b Fu(=)h Fq(q)2330 3283 y Fn(\000)p Fg(1)2424 3324 y Fq(xu;)17 b(:)g(:)g(:)32 b(;)17 b(xy)31 b Fu(=)d Fq(y)t(x)3124 3184 y Fe(\033)0 3543 y Fu(op)s(erates)33 b(on)g(this)f(quan)m(tum)h(plane)f(b)m(y)h(matrix)e(m)m(ultiplication) 317 3762 y Fp(M)437 3777 y Fl(q)475 3762 y Fu(\(2\)\()p Fq(A)711 3721 y Fn(0)734 3762 y Fu(\))c Fp(?)i Fr(A)971 3721 y Fg(2)p Fn(j)p Fg(0)971 3787 y Fl(q)1071 3762 y Fu(\()p Fq(A)1182 3721 y Fn(0)1205 3762 y Fu(\))f Fp(3)g Fu(\()1403 3622 y Fe(\022)1476 3702 y Fq(a)88 b(b)1481 3819 y(c)f(d)1661 3622 y Fe(\023)1751 3762 y Fq(;)1795 3622 y Fe(\022)1868 3702 y Fq(x)1870 3819 y(y)1924 3622 y Fe(\023)1997 3762 y Fu(\))27 b Fp(7!)2190 3622 y Fe(\022)2263 3702 y Fq(a)88 b(b)2268 3819 y(c)f(d)2448 3622 y Fe(\023)2544 3762 y Fp(\001)2593 3622 y Fe(\022)2667 3702 y Fq(x)2669 3819 y(y)2722 3622 y Fe(\023)2823 3762 y Fp(2)28 b Fr(A)2983 3721 y Fg(2)p Fn(j)p Fg(0)2983 3787 y Fl(q)3083 3762 y Fu(\()p Fq(A)3194 3721 y Fn(0)3218 3762 y Fu(\))p Fq(:)0 3981 y Fu(Again)42 b(one)g(should)h(c)m(hec)m(k)i(that)d(the)h (required)g(equations)g(are)g(preserv)m(ed.)76 b(Since)43 b(w)m(e)h(ha)m(v)m(e)0 4097 y(a)k(matrix)e(m)m(ultiplication)e(w)m(e)49 b(get)f(an)g(op)s(eration)f(as)h(in)f(the)i(preceding)f(prop)s (osition.)89 b(In)0 4226 y(particular)31 b Fq(A)523 4175 y Fg(2)p Fn(j)p Fg(0)523 4236 y Fl(q)650 4226 y Fu(is)h(a)g Fq(M)923 4241 y Fl(q)962 4226 y Fu(\(2\)-como)s(dule)e(algebra.)149 4406 y(As)38 b(in)f(example)g(1.)57 b(w)m(e)39 b(get)e(the)h(com)m (ultiplication)33 b(as)k Fq(\016)t Fu(\()2358 4266 y Fe(\022)2432 4346 y Fq(x)2434 4463 y(y)2487 4266 y Fe(\023)2560 4406 y Fu(\))f(=)g Fq(\016)k Fu(=)35 b(\()2978 4266 y Fe(\022)3052 4346 y Fq(a)87 b(b)3056 4463 y(c)h(d)3236 4266 y Fe(\023)3335 4406 y Fp(\012)26 b Fu(1\))f Fp(\003)0 4654 y Fu(\(1)d Fp(\012)208 4513 y Fe(\022)282 4594 y Fq(x)284 4710 y(y)337 4513 y Fe(\023)411 4654 y Fu(\))27 b(=)580 4513 y Fe(\022)653 4594 y Fq(a)88 b(b)658 4710 y(c)f(d)838 4513 y Fe(\023)933 4654 y Fp(\012)1033 4513 y Fe(\022)1106 4594 y Fq(x)1109 4710 y(y)1162 4513 y Fe(\023)1252 4654 y Fq(:)149 4857 y Fu(3.)41 b(Let)23 b Fq(A)504 4806 y Fg(0)p Fn(j)p Fg(2)504 4868 y Fl(q)626 4857 y Fu(=)k Fr(K)18 b Fp(h)p Fq(\030)5 b(;)17 b(\021)t Fp(i)p Fq(=)p Fu(\()p Fq(\030)1158 4821 y Fg(2)1201 4857 y Fq(;)g(\021)1297 4821 y Fg(2)1336 4857 y Fq(;)g(\030)5 b(\021)h Fu(+)s Fq(q)t(\021)t(\030)f Fu(\))21 b(b)s(e)i(the)h(function) e(algebra)g(of)h(the)g(dual)g(quan)m(tum)0 4991 y(plane)32 b Fr(A)326 4940 y Fg(0)p Fn(j)p Fg(2)326 5001 y Fl(q)426 4991 y Fu(.)44 b(By)33 b(the)g(de\014nition)f(1.2.5)g(w)m(e)h(ha)m(v)m (e)476 5184 y Fr(A)542 5143 y Fg(0)p Fn(j)p Fg(2)542 5209 y Fl(q)642 5184 y Fu(\()p Fq(A)753 5143 y Fn(0)777 5184 y Fu(\))27 b(=)946 5073 y Fe(n)1012 5103 y(\000)1058 5182 y Fq(a)1109 5146 y Fn(0)1216 5182 y Fq(b)1257 5146 y Fn(0)1280 5103 y Fe(\001)1326 5069 y(\014)1326 5129 y(\014)1326 5189 y(\014)1359 5184 y Fq(a)1410 5143 y Fn(0)1434 5184 y Fq(;)17 b(b)1519 5143 y Fn(0)1570 5184 y Fp(2)28 b Fq(A)1737 5143 y Fn(0)1760 5184 y Fu(;)17 b Fq(a)1855 5143 y Fn(0)1879 5132 y Fg(2)1946 5184 y Fu(=)27 b(0)p Fq(;)17 b(b)2183 5143 y Fn(0)2207 5132 y Fg(2)2274 5184 y Fu(=)27 b(0)p Fq(;)17 b(a)2521 5143 y Fn(0)2544 5184 y Fq(b)2585 5143 y Fn(0)2637 5184 y Fu(=)27 b Fp(\000)p Fq(q)t(b)2905 5143 y Fn(0)2929 5184 y Fq(a)2980 5143 y Fn(0)3014 5073 y Fe(o)3097 5184 y Fq(:)p eop %%Page: 23 23 23 22 bop 451 236 a Fm(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 Fu(The)46 b(quan)m(tum)g(monoid)e Fq(M)1092 440 y Fl(q)1130 425 y Fu(\(2\))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 Fr(A)312 713 y Fg(0)p Fn(j)p Fg(2)312 779 y Fl(q)412 754 y Fu(\()p Fq(A)523 713 y Fn(0)546 754 y Fu(\))28 b Fp(?)g(M)837 769 y Fl(q)875 754 y Fu(\(2\)\()p Fq(A)1111 713 y Fn(0)1134 754 y Fu(\))f Fp(3)i Fu(\()1332 674 y Fe(\000)1377 752 y Fq(\030)88 b(\021)1559 674 y Fe(\001)1622 754 y Fq(;)1666 614 y Fe(\022)1739 694 y Fq(a)f(b)1743 811 y(c)h(d)1924 614 y Fe(\023)1997 754 y Fu(\))27 b Fp(7!)2190 674 y Fe(\000)2235 752 y Fq(\030)88 b(\021)2418 674 y Fe(\001)2485 754 y Fp(\001)2535 614 y Fe(\022)2609 694 y Fq(a)f(b)2613 811 y(c)h(d)2793 614 y Fe(\023)2894 754 y Fp(2)29 b Fr(A)3055 713 y Fg(0)p Fn(j)p Fg(2)3055 779 y Fl(q)3155 754 y Fu(\()p Fq(A)3266 713 y Fn(0)3289 754 y Fu(\))p Fq(:)0 999 y Fu(This)35 b(giv)m(es)g(another)g(example)f (of)h(a)f Fq(M)1501 1014 y Fl(q)1540 999 y Fu(\(2\)-como)s(dule)e (algebra)i Fq(A)2552 948 y Fg(0)p Fn(j)p Fg(2)2552 1009 y Fl(q)2678 999 y Fp(\000)-60 b(!)31 b Fq(A)2899 948 y Fg(0)p Fn(j)p Fg(2)2899 1009 y Fl(q)3017 999 y Fp(\012)25 b Fq(M)3213 1014 y Fl(q)3251 999 y Fu(\(2\))34 b(with)0 1179 y Fq(\016)t Fu(\()85 1099 y Fe(\000)131 1177 y Fq(\030)87 b(\021)313 1099 y Fe(\001)358 1179 y Fu(\))28 b(=)g Fq(\016)j Fu(=)d(\()744 1099 y Fe(\000)789 1177 y Fq(\030)88 b(\021)971 1099 y Fe(\001)1039 1179 y Fp(\012)23 b Fu(1\))f Fp(\003)g Fu(\(1)f Fp(\012)1528 1039 y Fe(\022)1601 1119 y Fq(a)88 b(b)1606 1236 y(c)g(d)1786 1039 y Fe(\023)1860 1179 y Fu(\))27 b(=)2029 1099 y Fe(\000)2075 1177 y Fq(\030)87 b(\021)2257 1099 y Fe(\001)2324 1179 y Fp(\012)2424 1039 y Fe(\022)2497 1119 y Fq(a)h(b)2502 1236 y(c)g(d)2682 1039 y Fe(\023)2772 1179 y Fq(:)149 1409 y Fu(What)40 b(is)e(no)m(w)i(the)f(reason)h(for)e(the)i(remark)-5 b(able)38 b(relations)f(on)i Fp(M)2731 1424 y Fl(q)2769 1409 y Fu(\(2\)?)62 b(It)39 b(is)g(based)h(on)0 1525 y(a)e(fact)f(that)h(w)m(e)h(will)c(sho)m(w)k(later)e(namely)g(that)g Fq(M)1973 1540 y Fl(q)2012 1525 y Fu(\(2\))g(is)h(the)g(univ)m(ersal)g (quan)m(tum)g(monoid)0 1662 y(acting)33 b(on)g(the)h(quan)m(tum)g (plane)f Fr(A)1339 1611 y Fg(2)q Fn(j)p Fg(0)1339 1672 y Fl(q)1473 1662 y Fu(from)g(the)h(left)e(and)i(on)g(the)g(dual)f(quan) m(tum)h(plane)f Fr(A)3500 1611 y Fg(0)p Fn(j)p Fg(2)3500 1672 y Fl(q)0 1778 y Fu(from)28 b(the)i(righ)m(t.)42 b(This)30 b(ho)m(w)m(ev)m(er)i(happ)s(ens)f(in)e(the)h(category)g(of)f (quan)m(tum)h(planes)f(represen)m(ted)0 1894 y(b)m(y)41 b(quadratic)f(algebras.)65 b(Here)41 b(w)m(e)g(will)d(sho)m(w)j(a)f (simpler)e(theorem)i(for)g(\014nite)f(dimensional)0 2011 y(algebras.)149 2191 y Fs(Problem)d(1.3.6.)49 b Fu(Determine)35 b(the)h Fr(H)21 b Fu(-p)s(oin)m(ts)41 b(of)35 b(the)i(quan)m(tum)f (plane)f Fr(A)3000 2140 y Fg(2)p Fn(j)p Fg(0)3000 2201 y Fl(q)3136 2191 y Fu(where)i Fr(H)62 b Fu(is)0 2307 y(the)33 b Fr(R)5 b Fu(-algebra)37 b(of)32 b(the)h(quaternions.)149 2476 y Fs(De\014nition)k(1.3.7.)198 b Fu(1.)42 b(Let)d Fp(X)54 b Fu(b)s(e)39 b(a)g(quan)m(tum)h(space.)65 b(A)39 b(quan)m(tum)g(space)i Fp(M)p Fu(\()p Fp(X)15 b Fu(\))267 2592 y(together)33 b(with)g(a)g(morphism)e(of)i(quan)m(tum)h(spaces)g Fq(\026)29 b Fu(:)g Fp(M)p Fu(\()p Fp(X)15 b Fu(\))27 b Fp(?)j(X)43 b(\000)-60 b(!)29 b(X)48 b Fu(is)32 b(called)g(a)267 2708 y Fo(quantum)39 b(sp)-5 b(ac)g(e)39 b(acting)g(universal)5 b(ly)37 b Fu(on)h Fp(X)52 b Fu(\(or)37 b(simply)f(a)h Fo(universal)i(quantum)h(sp)-5 b(ac)g(e)267 2824 y(for)36 b Fp(X)15 b Fu(\))37 b(if)e(for)i(ev)m(ery)h(quan)m(tum)g(space)g Fp(Y)44 b Fu(and)37 b(ev)m(ery)i(morphism)c(of)h(quan)m(tum)h(spaces) 267 2940 y Fq(f)h Fu(:)28 b Fp(Y)35 b(?)29 b(X)42 b(\000)-60 b(!)28 b(X)38 b Fu(there)25 b(is)f(a)f(unique)i(morphism)d(of)i(quan)m (tum)g(spaces)i Fq(g)31 b Fu(:)d Fp(Y)35 b(\000)-60 b(!)28 b(M)p Fu(\()p Fp(X)15 b Fu(\))267 3057 y(suc)m(h)34 b(that)e(the)h (follo)m(wing)d(diagram)g(comm)m(utes)1403 3676 y Fp(M)p Fu(\()p Fp(X)15 b Fu(\))27 b Fp(?)h(X)196 b(X)15 b Fq(:)p 1932 3654 124 4 v 1973 3652 a Fh(-)1972 3699 y Fl(\026)1957 3424 y(f)1722 3315 y Fh(@)1805 3398 y(@)1888 3481 y(@)1971 3564 y(@)1989 3583 y(@)-83 b(R)1507 3194 y Fp(Y)35 b(?)28 b(X)p 1652 3583 4 351 v 1653 3583 a Fh(?)1433 3424 y Fl(g)r Fn(?)p Fg(1)1560 3435 y Ff(X)149 3836 y Fu(2.)42 b(Let)36 b Fq(A)g Fu(b)s(e)h(a)f Fr(K)17 b Fu(-algebra.)59 b(A)36 b Fr(K)18 b Fu(-algebra)40 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))37 b(together)f(with)g(a)g(homomorphism)d(of)267 3952 y(algebras)g Fq(\016)g Fu(:)d Fq(A)f Fp(\000)-60 b(!)30 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))24 b Fp(\012)f Fq(A)34 b Fu(is)f(called)f(an)i Fo(algebr)-5 b(a)35 b(c)-5 b(o)g(acting)35 b(universal)5 b(ly)35 b(on)f Fq(A)f Fu(\(or)267 4068 y(simply)28 b(a)g Fo(universal)j(algebr)-5 b(a)31 b(for)e Fq(A)p Fu(\))g(if)f(for)h(ev)m(ery)i Fr(K)17 b Fu(-algebra)34 b Fq(B)g Fu(and)29 b(ev)m(ery)i(homomor-)267 4184 y(phism)k(of)h Fr(K)18 b Fu(-algebras)41 b Fq(f)k Fu(:)35 b Fq(A)f Fp(\000)-60 b(!)34 b Fq(B)c Fp(\012)c Fq(A)36 b Fu(there)h(exists)h(a)e(unique)h(homomorphism)c(of)267 4301 y(algebras)f Fq(g)f Fu(:)c Fq(M)10 b Fu(\()p Fq(A)p Fu(\))29 b Fp(\000)-60 b(!)28 b Fq(B)37 b Fu(suc)m(h)d(that)f(the)g (follo)m(wing)d(diagram)g(comm)m(utes)1426 4464 y Fq(A)226 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1528 4440 169 4 v 1614 4438 a Fh(-)1594 4420 y Fl(\016)1604 4699 y(f)1530 4590 y Fh(@)1613 4673 y(@)1696 4756 y(@)1779 4839 y(@)1798 4858 y(@)-83 b(R)1813 4956 y Fq(B)27 b Fp(\012)c Fq(A)p 1948 4858 4 351 v 1950 4858 a Fh(?)1989 4698 y Fl(g)r Fn(\012)p Fg(1)2115 4709 y Fi(A)149 5099 y Fu(By)38 b(the)g(univ)m(ersal)f(prop)s(erties)h(the)g(univ)m(ersal)f (algebra)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\))38 b(for)f Fq(A)h Fu(and)f(the)h(univ)m(ersal)0 5216 y(quan)m(tum)33 b(space)h Fp(M)p Fu(\()p Fp(X)15 b Fu(\))31 b(for)h Fp(X)47 b Fu(are)33 b(unique)g(up)g(to)f(isomorphism.)p eop %%Page: 24 24 24 23 bop 0 236 a Fm(24)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)149 425 y Fs(Prop)s(osition)36 b(1.3.8.)199 b Fu(1.)41 b Fo(L)-5 b(et)29 b Fq(A)g Fo(b)-5 b(e)28 b(a)g Fr(K)18 b Fo(-algebr)-5 b(a)33 b(with)c(universal)f(algebr)-5 b(a)27 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))30 b Fo(and)267 541 y Fq(\016)38 b Fu(:)d Fq(A)g Fp(\000)-58 b(!)35 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))26 b Fp(\012)g Fq(A)p Fo(.)56 b(Then)38 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))39 b Fo(is)g(a)f(bialgebr)-5 b(a)38 b(and)g Fq(A)h Fo(is)f(an)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))p Fo(-c)-5 b(omo)g(dule)267 658 y(algebr)g(a)34 b(by)h Fq(\016)t Fo(.)149 774 y Fu(2.)42 b Fo(If)31 b Fq(B)37 b Fo(is)31 b(a)h(bialgebr)-5 b(a)30 b(and)h(if)h Fq(f)38 b Fu(:)28 b Fq(A)g Fp(\000)-58 b(!)28 b Fq(B)20 b Fp(\012)15 b Fq(A)33 b Fo(de\014nes)d(the)i(structur)-5 b(e)33 b(of)e(a)h Fq(B)5 b Fo(-c)-5 b(omo)g(dule)267 890 y(algebr)g(a)26 b(on)g Fq(A)h Fo(then)f(ther)-5 b(e)27 b(is)f(a)h(unique)f(homomorphism)f Fq(g)31 b Fu(:)c Fq(M)10 b Fu(\()p Fq(A)p Fu(\))29 b Fp(\000)-58 b(!)28 b Fq(B)k Fo(of)26 b(bialgebr)-5 b(as)267 1006 y(such)34 b(that)i(the)f(fol)5 b(lowing)33 b(diagr)-5 b(am)34 b(c)-5 b(ommutes)1380 1188 y Fq(A)226 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1482 1165 169 4 v 1568 1163 a Fh(-)1542 1144 y Fq(\016)1541 1432 y(f)1485 1315 y Fh(@)1568 1398 y(@)1651 1481 y(@)1734 1564 y(@)1753 1583 y(@)-83 b(R)1767 1681 y Fq(B)27 b Fp(\012)c Fq(A)p 1902 1583 4 351 v 1904 1583 a Fh(?)1943 1429 y Fq(g)i Fp(\012)e Fu(1)2164 1444 y Fl(A)149 1826 y Fu(The)43 b(corresp)s(onding)e(statemen)m(t)i(for)e (quan)m(tum)g(spaces)j(and)d(quan)m(tum)h(monoids)e(is)i(the)0 1942 y(follo)m(wing.)149 2117 y Fs(Prop)s(osition)36 b(1.3.9.)199 b Fu(1.)41 b Fo(L)-5 b(et)51 b Fp(X)65 b Fo(b)-5 b(e)50 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 Fp(M)p Fu(\()p Fp(X)15 b Fu(\))29 b Fo(and)h Fq(\026)d Fu(:)h Fp(M)p Fu(\()p Fp(X)15 b Fu(\))27 b Fp(?)h Fq(A)g Fp(\000)-58 b(!)28 b Fq(A)p Fo(.)44 b(Then)29 b Fp(M)p Fu(\()p Fp(X)15 b Fu(\))30 b Fo(is)g(a)h(quantum)f(monoid)g(and)267 2349 y Fp(X)49 b Fo(is)35 b(an)f Fp(M)p Fu(\()p Fp(X)15 b Fu(\))p Fo(-sp)-5 b(ac)g(e)34 b(by)h Fq(\026)p Fo(.)149 2465 y Fu(2.)42 b Fo(If)36 b Fp(Y)44 b Fo(is)36 b(another)g(quantum)g (monoid)g(and)g(if)g Fq(f)41 b Fu(:)31 b Fp(Y)38 b(?)31 b(X)45 b(\000)-58 b(!)31 b(X)51 b Fo(de\014nes)35 b(the)i(structur)-5 b(e)267 2582 y(of)40 b(a)g Fp(Y)8 b Fo(-sp)-5 b(ac)g(e)39 b(on)g Fp(X)55 b Fo(then)40 b(ther)-5 b(e)40 b(is)g(a)g(unique)g (morphism)f(of)h(quantum)g(monoids)f Fq(g)i Fu(:)267 2698 y Fp(Y)35 b(\000)-58 b(!)28 b(M)p Fu(\()p Fp(X)15 b Fu(\))35 b Fo(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 Fp(M)p Fu(\()p Fp(X)15 b Fu(\))27 b Fp(?)h(X)196 b(X)15 b Fq(:)p 1932 3301 124 4 v 1973 3299 a Fh(-)1972 3347 y Fl(\026)1957 3072 y(f)1722 2963 y Fh(@)1805 3046 y(@)1888 3129 y(@)1971 3212 y(@)1989 3231 y(@)-83 b(R)1507 2841 y Fp(Y)35 b(?)28 b(X)p 1652 3231 4 351 v 1653 3231 a Fh(?)1433 3072 y Fl(g)r Fn(?)p Fg(1)1560 3083 y Ff(X)149 3496 y Fd(Pr)n(oof.)42 b Fu(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 Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)533 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(A)p 1375 4137 476 4 v 1768 4135 a Fh(-)1470 4201 y Fg(1)1505 4215 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1667 4201 y Fn(\012)p Fl(\016)1085 3770 y Fq(A)909 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1187 3747 852 4 v 1956 3745 a Fh(-)1595 3727 y Fl(\016)p 1120 4067 4 254 v 1121 4067 a Fh(?)1049 3964 y Fl(\016)p 2290 4067 V 2291 4067 a Fh(?)2330 3958 y Fg(\001)p Fn(\012)p Fg(1)2479 3969 y Fi(A)0 4352 y Fu(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 Fq(M)10 b Fu(\()p Fq(A)p Fu(\))0 4468 y(with)38 b(resp)s(ect)i(to)e (the)h(algebra)e(morphism)g(\(1)1763 4483 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))1975 4468 y Fp(\012)27 b Fq(\016)t Fu(\))p Fq(\016)t Fu(.)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 Fq(\017)c Fu(:)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))28 b Fp(\000)-60 b(!)28 b Fr(K)56 b Fu(suc)m(h)34 b(that)1418 4748 y Fq(A)227 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1520 4725 169 4 v 1606 4723 a Fh(-)1587 4704 y Fl(\016)1553 4983 y Fg(1)1588 4994 y Fi(A)1523 4875 y Fh(@)1606 4958 y(@)1689 5041 y(@)1772 5124 y(@)1791 5142 y(@)-83 b(R)1703 5241 y Fq(A)1804 5213 y Fp(\030)1805 5245 y Fu(=)1909 5241 y Fr(K)46 b Fp(\012)22 b Fq(A)p 1941 5143 4 351 v 1942 5143 a Fh(?)1981 4983 y Fl(\017)p Fn(\012)p Fg(1)2100 4994 y Fi(A)p eop %%Page: 25 25 25 24 bop 451 236 a Fm(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 Fu(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 Fq(A)1670 b(M)10 b Fu(\()p Fq(A)p Fu(\))22 b Fp(\012)h Fq(A)p 816 737 1612 4 v 2345 735 a Fh(-)1604 716 y Fl(\016)p 749 1057 4 254 v 750 1057 a Fh(?)789 953 y Fl(\016)p 2679 1057 V 2681 1057 a Fh(?)2720 947 y Fg(\001)p Fn(\012)p Fg(1)2869 958 y Fi(A)526 1150 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)1294 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1004 1127 1237 4 v 2158 1125 a Fh(-)1479 1079 y Fg(1)1514 1093 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1676 1079 y Fn(\012)p Fl(\016)p 739 1447 4 254 v 741 1447 a Fh(?)503 1337 y Fg(\001)p Fn(\012)p Fg(1)652 1348 y Fi(A)p 768 1447 V 770 1447 a Fh(?)809 1330 y Fg(1)844 1344 y Fi(M)g Fj(\()p Fi(A)p Fj(\))1006 1330 y Fn(\012)p Fl(\016)p 2660 1447 V 2661 1447 a Fh(?)2172 1329 y Fg(\001)p Fn(\012)p Fg(1)2321 1343 y Fi(M)g Fj(\()p Fi(A)p Fj(\))2482 1329 y Fn(\012)p Fg(1)2572 1340 y Fi(A)p 2689 1447 V 2691 1447 a Fh(?)2730 1329 y Fg(1)2765 1343 y Fi(M)g Fj(\()p Fi(A)p Fj(\))2926 1329 y Fn(\012)p Fg(\001)p Fn(\012)p Fg(1)3130 1340 y Fi(A)329 1540 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)918 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1181 1517 861 4 v 1959 1515 a Fh(-)1343 1469 y Fg(1)1378 1483 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1540 1469 y Fn(\012)p Fg(1)1630 1483 y Fi(M)g Fj(\()p Fi(A)p Fj(\))1791 1469 y Fn(\012)p Fl(\016)0 1744 y Fu(and)859 1903 y Fq(A)1251 b(M)10 b Fu(\()p Fq(A)p Fu(\))22 b Fp(\012)h Fq(A)p 961 1880 1193 4 v 2071 1878 a Fh(-)1540 1859 y Fl(\016)p 894 2200 4 254 v 896 2200 a Fh(?)935 2096 y Fl(\016)p 2259 2200 V 2261 2200 a Fh(?)2022 2090 y Fg(\001)p Fn(\012)p Fg(1)2171 2101 y Fi(A)p 2551 2590 4 644 v 2553 2590 a Fh(?)2592 2276 y Fg(1)2627 2290 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))2789 2276 y Fn(\012)p Fg(1)2879 2287 y Fi(A)671 2293 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)534 b(M)10 b Fu(\()p Fq(A)p Fu(\))22 b Fp(\012)h Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1149 2270 476 4 v 1542 2268 a Fh(-)1244 2222 y Fg(1)1279 2236 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1441 2222 y Fn(\012)p Fl(\016)1079 2471 y Fg(1)1114 2485 y Fi(M)g Fj(\()p Fi(A)p Fj(\))1276 2471 y Fn(\012)p Fg(1)1366 2482 y Fi(A)1100 2364 y Fh(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 Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(A)2122 2655 y Fp(\030)2123 2687 y Fu(=)2227 2683 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fr(K)46 b Fp(\012)22 b Fq(A)p 2259 2590 4 254 v 2261 2590 a Fh(?)1801 2471 y Fg(1)1836 2485 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1998 2471 y Fn(\012)p Fl(\017)p Fn(\012)p Fg(1)2172 2482 y Fi(A)0 2867 y Fu(and)951 3025 y Fq(A)909 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1053 3002 852 4 v 1822 3000 a Fh(-)1461 2982 y Fl(\016)p 937 3712 4 644 v 939 3712 a Fh(?)814 3406 y Fg(1)849 3417 y Fi(A)p 1034 3322 4 254 v 1036 3322 a Fh(?)1075 3218 y Fl(\016)p 2107 3322 V 2109 3322 a Fh(?)2148 3212 y Fg(\001)p Fn(\012)p Fg(1)2297 3223 y Fi(A)1007 3415 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(A)436 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1485 3392 379 4 v 1781 3390 a Fh(-)1531 3344 y Fg(1)1566 3358 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1728 3344 y Fn(\012)p Fl(\016)p 1034 3712 4 254 v 1036 3712 a Fh(?)1075 3601 y Fl(\017)p Fn(\012)p Fg(1)1194 3612 y Fi(A)p 2107 3712 V 2109 3712 a Fh(?)2148 3593 y Fl(\017)p Fn(\012)p Fg(1)2267 3607 y Fi(M)g Fj(\()p Fi(A)p Fj(\))2428 3593 y Fn(\012)p Fg(1)2518 3604 y Fi(A)951 3805 y Fq(A)505 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)2005 3778 y Fp(\030)2006 3809 y Fu(=)2111 3805 y Fr(K)45 b Fp(\012)23 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(A:)p 1053 3782 448 4 v 1418 3780 a Fh(-)1259 3761 y Fl(\016)149 3989 y Fu(In)29 b(fact)g(these)h(diagrams)c(imply)h(b)m(y)i(the)g (uniqueness)i(of)d(the)h(induced)g(homomorphisms)d(of)0 4105 y(algebras)34 b(\(\001)24 b Fp(\012)g Fu(1)675 4121 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))861 4105 y Fu(\)\001)32 b(=)f(\(1)1206 4121 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))1416 4105 y Fp(\012)24 b Fu(\001\)\001,)36 b(\(1)1867 4121 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))2077 4105 y Fp(\012)24 b Fq(\017)p Fu(\)\001)31 b(=)g(1)2523 4121 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))2745 4105 y Fu(and)34 b Fq(\017)24 b Fp(\012)g Fu(\(1)3187 4121 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))3374 4105 y Fu(\)\001)31 b(=)0 4221 y(1)49 4237 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))236 4221 y Fu(.)43 b(Finally)30 b Fq(A)j Fu(is)f(an)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\)-como)s(dule)32 b(algebra)f(b)m(y)j(the)f(de\014nition)e(of)h(\001)h(and)g Fq(\017)p Fu(.)149 4338 y(No)m(w)27 b(assume)f(that)g(a)f(structure)i (of)f(a)f Fq(B)5 b Fu(-como)s(dule)24 b(algebra)h(on)h Fq(A)f Fu(is)h(giv)m(en)g(b)m(y)g(a)g(bialgebra)0 4454 y Fq(B)i Fu(and)22 b Fq(f)39 b Fu(:)28 b Fq(A)f Fp(\000)-60 b(!)28 b Fq(B)7 b Fp(\012)r Fq(A)p Fu(.)40 b(Then)24 b(there)f(is)g(a)f(unique)h(homomorphism)d(of)i(algebras)g Fq(g)31 b Fu(:)c Fq(M)10 b Fu(\()p Fq(A)p Fu(\))29 b Fp(\000)-60 b(!)0 4570 y Fq(B)38 b Fu(suc)m(h)c(that)e(the)h(diagram) 1426 4748 y Fq(A)226 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1528 4725 169 4 v 1614 4723 a Fh(-)1594 4704 y Fl(\016)1604 4984 y(f)1530 4875 y Fh(@)1613 4958 y(@)1696 5041 y(@)1779 5124 y(@)1798 5142 y(@)-83 b(R)1813 5241 y Fq(B)27 b Fp(\012)c Fq(A)p 1948 5143 4 351 v 1950 5143 a Fh(?)1989 4982 y Fl(g)r Fn(\012)p Fg(1)2115 4993 y Fi(A)p eop %%Page: 26 26 26 25 bop 0 236 a Fm(26)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fu(comm)m(utes.)43 b(Then)34 b(the)f(follo)m(wing)d (diagram)698 653 y Fq(A)325 b(M)10 b Fu(\()p Fq(A)p Fu(\))22 b Fp(\012)h Fq(A)p 800 629 267 4 v 984 627 a Fh(-)916 609 y Fl(\016)1096 653 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))22 b Fp(\012)h Fq(A)534 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1573 610 476 4 v 1966 608 a Fh(-)1712 577 y Fg(\001)p Fn(\012)p Fg(1)1861 588 y Fi(A)p 1573 649 V 1966 647 a Fh(-)1669 712 y Fg(1)1704 726 y Fi(M)5 b Fj(\()p Fi(A)p Fj(\))1865 712 y Fn(\012)p Fl(\016)925 839 y(f)837 751 y Fh(Q)920 807 y(Q)1003 862 y(Q)1086 917 y(Q)1134 949 y(Q)-83 b(s)p 1318 949 4 254 v 103 w(?)1359 838 y Fl(g)r Fn(\012)p Fg(1)1485 849 y Fi(A)p 2293 949 V 2295 949 a Fh(?)2334 838 y Fl(g)r Fn(\012)p Fl(g)r Fn(\012)p Fg(1)2551 849 y Fi(A)1183 1047 y Fq(B)27 b Fp(\012)c Fq(A)795 b(B)28 b Fp(\012)22 b Fq(B)27 b Fp(\012)c Fq(A)p 1486 1000 739 4 v 2142 998 a Fh(-)1728 967 y Fg(\001)1787 978 y Fi(B)1840 967 y Fn(\012)p Fg(1)1930 978 y Fi(A)p 1486 1039 V 2141 1037 a Fh(-)1762 1102 y Fg(1)1797 1113 y Fi(B)1850 1102 y Fn(\012)p Fl(f)0 1277 y Fu(implies)38 b(\(\()p Fq(g)31 b Fp(\012)d Fq(g)t Fu(\)\001)f Fp(\012)h Fu(1)949 1292 y Fl(A)1006 1277 y Fu(\))p Fq(\016)45 b Fu(=)c(\()p Fq(g)31 b Fp(\012)d Fq(g)j Fp(\012)d Fu(1)1702 1292 y Fl(A)1759 1277 y Fu(\)\(\001)f Fp(\012)h Fu(1)2097 1292 y Fl(A)2154 1277 y Fu(\))p Fq(\016)45 b Fu(=)c(\()p Fq(g)31 b Fp(\012)d Fq(g)j Fp(\012)d Fu(1)2850 1292 y Fl(A)2907 1277 y Fu(\)\(1)3032 1293 y Fl(M)7 b Fg(\()p Fl(A)p Fg(\))3246 1277 y Fp(\012)28 b Fq(\016)t Fu(\))p Fq(\016)45 b Fu(=)0 1394 y(\()p Fq(g)27 b Fp(\012)e Fu(\()p Fq(g)i Fp(\012)d Fu(1)476 1409 y Fl(A)533 1394 y Fu(\))p Fq(\016)t Fu(\))p Fq(\016)36 b Fu(=)c(\(1)930 1409 y Fl(B)1014 1394 y Fp(\012)25 b Fu(\()p Fq(g)i Fp(\012)d Fu(1)1378 1409 y Fl(A)1435 1394 y Fu(\))p Fq(\016)t Fu(\)\()p Fq(g)j Fp(\012)d Fu(1)1820 1409 y Fl(A)1877 1394 y Fu(\))p Fq(\016)36 b Fu(=)c(\(1)2189 1409 y Fl(B)2274 1394 y Fp(\012)24 b Fq(f)11 b Fu(\))p Fq(f)43 b Fu(=)32 b(\(\001)2790 1409 y Fl(B)2875 1394 y Fp(\012)24 b Fu(1)3025 1409 y Fl(A)3082 1394 y Fu(\))p Fq(f)43 b Fu(=)32 b(\(\001)3438 1409 y Fl(B)3523 1394 y Fp(\012)0 1510 y Fu(1)49 1525 y Fl(A)106 1510 y Fu(\)\()p Fq(g)25 b Fp(\012)e Fu(1)403 1525 y Fl(A)459 1510 y Fu(\))p Fq(\016)32 b Fu(=)c(\(\001)795 1525 y Fl(B)855 1510 y Fq(g)e Fp(\012)d Fu(1)1077 1525 y Fl(A)1133 1510 y Fu(\))p Fq(\016)37 b Fu(hence)d(\()p Fq(g)25 b Fp(\012)e Fq(g)t Fu(\)\001)k(=)g(\001)2113 1525 y Fl(B)2174 1510 y Fq(g)t Fu(.)43 b(F)-8 b(urthermore)32 b(the)h(diagram)1182 1711 y Fq(A)714 b(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p 1284 1688 657 4 v 1858 1686 a Fh(-)1594 1668 y Fl(\016)1861 2106 y Fq(B)28 b Fp(\012)22 b Fq(A)1505 1898 y Fl(f)1355 1796 y Fh(H)1438 1838 y(H)1521 1879 y(H)1604 1921 y(H)1687 1962 y(H)1770 2004 y(H)1779 2008 y(H)-83 b(j)p 1997 2008 4 254 v 136 w(?)1783 1896 y Fl(g)r Fn(\012)p Fg(1)1909 1907 y Fi(A)p 2192 2398 4 644 v 2193 2398 a Fh(?)2232 2092 y Fl(\017)p Fn(\012)p Fg(1)2351 2103 y Fi(A)1759 2497 y Fq(A)1860 2469 y Fp(\030)1861 2501 y Fu(=)1965 2497 y Fr(K)46 b Fp(\012)22 b Fq(A)1463 2092 y Fg(1)1498 2103 y Fi(A)1287 1838 y Fh(@)1370 1921 y(@)1453 2004 y(@)1536 2087 y(@)1619 2170 y(@)1702 2253 y(@)1785 2336 y(@)1847 2398 y(@)-83 b(R)p 1997 2398 4 254 v 68 w(?)1738 2287 y Fl(\017)1767 2298 y Fi(B)1819 2287 y Fn(\012)p Fg(1)1909 2298 y Fi(A)0 2662 y Fu(implies)30 b Fq(\017)370 2677 y Fl(B)431 2662 y Fq(g)h Fu(=)d Fq(\017)p Fu(.)44 b(Th)m(us)34 b Fq(g)h Fu(is)e(a)f(homomorphism)e(of)i (bialgebras.)p 3529 2662 4 66 v 3533 2599 59 4 v 3533 2662 V 3591 2662 4 66 v 149 2875 a(Since)f(univ)m(ersal)g(algebras)f (for)g(algebras)g Fq(A)h Fu(tend)h(to)e(b)s(ecome)h(v)m(ery)h(big)e (they)h(do)g(not)g(exist)0 2991 y(in)38 b(general.)63 b(But)39 b(a)g(theorem)g(of)f(T)-8 b(am)m(bara's)39 b(sa)m(ys)i(that)e (they)h(exist)f(for)g(\014nite)f(dimensional)0 3108 y(algebras)32 b(\(o)m(v)m(er)h(a)g(\014eld)f Fr(K)18 b Fu(\).)149 3304 y Fs(De\014nition)37 b(1.3.10.)49 b Fu(If)e Fp(X)63 b Fu(is)47 b(a)g(quan)m(tum)h(space)h(with)e(\014nite)g(dimensional)e (function)0 3420 y(algebra)31 b(then)j(w)m(e)f(call)e Fp(X)47 b Fu(a)33 b Fo(\014nite)h(quantum)h(sp)-5 b(ac)g(e)p Fu(.)149 3617 y(The)33 b(follo)m(wing)c(theorem)i(is)g(the)i(quan)m (tum)e(space)i(v)m(ersion)f(and)g(equiv)-5 b(alen)m(t)32 b(to)f(a)g(theorem)0 3733 y(of)h(T)-8 b(am)m(bara.)149 3929 y Fs(Theorem)38 b(1.3.11.)49 b Fo(L)-5 b(et)33 b Fp(X)48 b Fo(b)-5 b(e)32 b(a)h(\014nite)f(quantum)h(sp)-5 b(ac)g(e.)44 b(Then)32 b(ther)-5 b(e)32 b(exists)h(a)g(\(univer-)0 4045 y(sal\))h(quantum)h(sp)-5 b(ac)g(e)34 b Fp(M)p Fu(\()p Fp(X)15 b Fu(\))34 b Fo(with)h(morphism)f(of)g(quantum)h(sp)-5 b(ac)g(es)34 b Fq(\026)27 b Fu(:)h Fp(M)p Fu(\()p Fp(X)15 b Fu(\))27 b Fp(?)h(X)43 b(\000)-58 b(!)28 b(X)15 b Fo(.)149 4242 y Fu(The)34 b(algebra)d(v)m(ersion)j(of)e(this)g(theorem)g(is)149 4438 y Fs(Theorem)38 b(1.3.12.)49 b Fu(\(T)-8 b(am)m(bara\))48 b Fo(L)-5 b(et)50 b Fq(A)f Fo(b)-5 b(e)49 b(a)g(\014nite)g(dimensional) e Fr(K)17 b Fo(-algebr)-5 b(a.)93 b(Then)0 4554 y(ther)-5 b(e)38 b(exists)g(a)f(\(universal\))g Fr(K)18 b Fo(-algebr)-5 b(a)43 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))39 b Fo(with)e(homomorphism)f (of)i(algebr)-5 b(as)37 b Fq(\016)g Fu(:)d Fq(A)f Fp(\000)-58 b(!)0 4671 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))23 b Fp(\012)g Fq(A)p Fo(.)149 4867 y Fd(Pr)n(oof.)42 b Fu(W)-8 b(e)43 b(are)f(going)f(to)h(construct)i(the)f Fr(K)17 b Fu(-algebra)47 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))43 b(quite)f(explicitly)-8 b(.)71 b(First)0 4983 y(w)m(e)50 b(observ)m(e)g(that)f Fq(A)824 4947 y Fn(\003)919 4983 y Fu(=)55 b(Hom)1253 4998 y Fk(K)1309 4983 y Fu(\()p Fq(A;)17 b Fr(K)g Fu(\))55 b(is)48 b(a)g(coalgebra)g(\(cf.)92 b(problem)48 b(A.6.8\))g(with)h(the) 0 5099 y(structural)44 b(morphism)e(\001)47 b(:)g Fq(A)1204 5063 y Fn(\003)1291 5099 y Fp(\000)-60 b(!)47 b Fu(\()p Fq(A)30 b Fp(\012)g Fq(A)p Fu(\))1814 5063 y Fn(\003)1901 5072 y Fp(\030)1902 5103 y Fu(=)2026 5099 y Fq(A)2099 5063 y Fn(\003)2168 5099 y Fp(\012)g Fq(A)2348 5063 y Fn(\003)2388 5099 y Fu(.)78 b(Denote)44 b(the)g(dual)f(basis)h(b)m(y)0 5141 y Fe(P)105 5167 y Fl(n)105 5245 y(i)p Fg(=1)240 5216 y Fq(a)291 5231 y Fl(i)341 5216 y Fp(\012)24 b Fu(\026)-51 b Fq(a)491 5179 y Fl(i)547 5216 y Fp(2)28 b Fq(A)22 b Fp(\012)g Fq(A)908 5179 y Fn(\003)948 5216 y Fu(.)43 b(No)m(w)33 b(let)f Fq(T)14 b Fu(\()p Fq(A)21 b Fp(\012)i Fq(A)1757 5179 y Fn(\003)1796 5216 y Fu(\))32 b(b)s(e)h(the)g(tensor)g (algebra)e(of)h(the)g(v)m(ector)i(space)p eop %%Page: 27 27 27 26 bop 451 236 a Fm(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(27)0 425 y Fq(A)22 b Fp(\012)h Fq(A)268 389 y Fn(\003)307 425 y Fu(.)44 b(Consider)33 b(elemen)m(ts)g(of)f(the)h(tensor)g(algebra)644 579 y Fq(xy)25 b Fp(\012)e Fq(\020)35 b Fp(2)28 b Fq(A)22 b Fp(\012)h Fq(A)1312 543 y Fn(\003)1351 579 y Fq(;)644 695 y(x)f Fp(\012)h Fq(y)i Fp(\012)e Fu(\001\()p Fq(\020)8 b Fu(\))27 b Fp(2)h Fq(A)22 b Fp(\012)h Fq(A)f Fp(\012)h Fq(A)1786 659 y Fn(\003)1847 695 y Fp(\012)g Fq(A)2020 659 y Fn(\003)2087 667 y Fp(\030)2088 699 y Fu(=)2192 695 y Fq(A)g Fp(\012)f Fq(A)2460 659 y Fn(\003)2522 695 y Fp(\012)h Fq(A)f Fp(\012)h Fq(A)2890 659 y Fn(\003)2929 695 y Fq(;)644 811 y Fu(1)f Fp(\012)g Fq(\020)35 b Fp(2)28 b Fq(A)22 b Fp(\012)h Fq(A)1254 775 y Fn(\003)1294 811 y Fq(;)644 928 y(\020)8 b Fu(\(1\))26 b Fp(2)i Fr(K)18 b Fq(:)0 1085 y Fu(The)34 b(follo)m(wing)29 b(elemen)m(ts)1321 1245 y Fq(xy)d Fp(\012)d Fq(\020)29 b Fp(\000)22 b Fq(x)h Fp(\012)g Fq(y)i Fp(\012)e Fu(\001\()p Fq(\020)8 b Fu(\))-2280 b(\(1\))0 1407 y(and)1541 1568 y(1)22 b Fp(\012)h Fq(\020)29 b Fp(\000)23 b Fq(\020)8 b Fu(\(1\))-2060 b(\(2\))0 1734 y(generate)33 b(a)f(t)m(w)m(o-sided)i(ideal)d Fq(I)k Fp(\022)28 b Fq(T)14 b Fu(\()p Fq(A)22 b Fp(\012)h Fq(A)1699 1698 y Fn(\003)1738 1734 y Fu(\).)44 b(No)m(w)33 b(w)m(e)h(de\014ne) 1317 1895 y Fq(M)10 b Fu(\()p Fq(A)p Fu(\))29 b(:=)e Fq(T)14 b Fu(\()p Fq(A)22 b Fp(\012)h Fq(A)2106 1854 y Fn(\003)2145 1895 y Fu(\))p Fq(=I)0 2056 y Fu(and)36 b(the)h(co)s(op)s(eration)e Fq(\016)j Fu(:)c Fq(A)g Fp(3)g Fq(a)g Fp(\000)-60 b(!)1486 1981 y Fe(P)1591 2008 y Fl(n)1591 2085 y(i)p Fg(=1)1709 2056 y Fu(\()p Fq(a)25 b Fp(\012)h Fu(\026)-50 b Fq(a)1976 2020 y Fl(i)2004 2056 y Fu(\))25 b Fp(\012)g Fq(a)2220 2071 y Fl(i)2282 2056 y Fp(2)35 b Fq(T)14 b Fu(\()p Fq(A)24 b Fp(\012)h Fq(A)2764 2020 y Fn(\003)2804 2056 y Fu(\))p Fq(=I)32 b Fp(\012)25 b Fq(A)p Fu(.)55 b(This)36 b(is)g(a)0 2172 y(w)m(ell-de\014ned)d(linear)e (map.)149 2289 y(T)-8 b(o)33 b(sho)m(w)i(that)d(this)h(map)f(is)h(a)f (homomorphism)e(of)j(algebras)f(w)m(e)i(\014rst)g(describ)s(e)f(the)h (m)m(ul-)0 2405 y(tiplication)39 b(of)k Fq(A)g Fu(b)m(y)h Fq(a)911 2420 y Fl(i)939 2405 y Fq(a)990 2420 y Fl(j)1072 2405 y Fu(=)1193 2330 y Fe(P)1298 2434 y Fl(k)1357 2405 y Fq(\013)1420 2369 y Fl(k)1419 2429 y(ij)1480 2405 y Fq(a)1531 2420 y Fl(k)1574 2405 y Fu(.)74 b(Then)44 b(the)f(com)m (ultiplication)c(of)j Fq(A)3041 2369 y Fn(\003)3124 2405 y Fu(is)g(giv)m(en)h(b)m(y)0 2533 y(\001\()q(\026)-50 b Fq(a)170 2497 y Fl(k)213 2533 y Fu(\))56 b(=)439 2458 y Fe(P)544 2562 y Fl(ij)621 2533 y Fq(\013)684 2497 y Fl(k)683 2557 y(ij)745 2533 y Fu(\026)-50 b Fq(a)795 2497 y Fl(i)857 2533 y Fp(\012)35 b Fu(\026)-50 b Fq(a)1019 2497 y Fl(j)1105 2533 y Fu(since)50 b(\(\001\()q(\026)-50 b Fq(a)1569 2497 y Fl(k)1612 2533 y Fu(\))p Fq(;)17 b(a)1745 2548 y Fl(l)1804 2533 y Fp(\012)34 b Fq(a)1966 2548 y Fl(m)2033 2533 y Fu(\))56 b(=)g(\()q(\026)-50 b Fq(a)2348 2497 y Fl(k)2391 2533 y Fq(;)17 b(a)2486 2548 y Fl(l)2511 2533 y Fq(a)2562 2548 y Fl(m)2629 2533 y Fu(\))56 b(=)2855 2458 y Fe(P)2960 2562 y Fl(r)3015 2533 y Fq(\013)3078 2497 y Fl(r)3077 2559 y(l)q(m)3165 2533 y Fu(\()q(\026)-50 b Fq(a)3254 2497 y Fl(k)3297 2533 y Fq(;)17 b(a)3392 2548 y Fl(r)3430 2533 y Fu(\))56 b(=)0 2665 y Fq(\013)63 2629 y Fl(k)62 2691 y(l)q(m)196 2665 y Fu(=)316 2591 y Fe(P)422 2694 y Fl(ij)499 2665 y Fq(\013)562 2629 y Fl(k)561 2690 y(ij)621 2665 y Fu(\()q(\026)-50 b Fq(a)710 2629 y Fl(i)739 2665 y Fq(;)17 b(a)834 2680 y Fl(l)860 2665 y Fu(\)\()q(\026)-50 b Fq(a)987 2629 y Fl(j)1023 2665 y Fq(;)17 b(a)1118 2680 y Fl(m)1185 2665 y Fu(\))45 b(=)g(\()1427 2591 y Fe(P)1532 2694 y Fl(ij)1609 2665 y Fq(\013)1672 2629 y Fl(k)1671 2690 y(ij)1733 2665 y Fu(\026)-51 b Fq(a)1782 2629 y Fl(i)1840 2665 y Fp(\012)31 b Fu(\026)-51 b Fq(a)1997 2629 y Fl(j)2034 2665 y Fq(;)17 b(a)2129 2680 y Fl(l)2184 2665 y Fp(\012)30 b Fq(a)2342 2680 y Fl(m)2408 2665 y Fu(\).)74 b(No)m(w)44 b(write)e(1)j(=)3254 2591 y Fe(P)3376 2665 y Fq(\014)3437 2629 y Fl(k)3479 2665 y Fq(a)3530 2680 y Fl(k)3573 2665 y Fu(.)0 2796 y(Then)j(w)m(e)g(get)g Fq(\017)p Fu(\()q(\026)-50 b Fq(a)733 2760 y Fl(i)761 2796 y Fu(\))53 b(=)f Fq(\014)1041 2760 y Fl(i)1116 2796 y Fu(since)c Fq(\017)p Fu(\()q(\026)-50 b Fq(a)1498 2760 y Fl(i)1526 2796 y Fu(\))53 b(=)f(\()q(\026)-50 b Fq(a)1834 2760 y Fl(i)1862 2796 y Fq(;)17 b Fu(1\))52 b(=)2174 2721 y Fe(P)2279 2825 y Fl(j)2332 2796 y Fq(\014)2393 2760 y Fl(j)2429 2796 y Fu(\()q(\026)-50 b Fq(a)2518 2760 y Fl(i)2546 2796 y Fq(;)17 b(a)2641 2811 y Fl(j)2678 2796 y Fu(\))52 b(=)g Fq(\014)2957 2760 y Fl(i)2985 2796 y Fu(.)87 b(So)47 b(w)m(e)i(ha)m(v)m(e)0 2926 y Fq(\016)t Fu(\()p Fq(a)p Fu(\))p Fq(\016)t Fu(\()p Fq(b)p Fu(\))28 b(=)g(\()508 2851 y Fe(P)613 2878 y Fl(n)613 2955 y(i)p Fg(=1)731 2926 y Fu(\()p Fq(a)22 b Fp(\012)i Fu(\026)-51 b Fq(a)992 2890 y Fl(i)1021 2926 y Fu(\))22 b Fp(\012)g Fq(a)1231 2941 y Fl(i)1259 2926 y Fu(\))g Fp(\001)g Fu(\()1407 2851 y Fe(P)1512 2878 y 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Fq(f)11 b Fu(.)40 b(First)22 b(of)i(all)d Fq(f)38 b Fu(:)28 b Fq(A)g Fp(\000)-60 b(!)28 b Fq(B)9 b Fp(\012)t Fq(A)0 3538 y Fu(induces)31 b(uniquely)g(determined)f(linear)e(maps)i Fq(f)1815 3553 y Fl(i)1871 3538 y Fu(:)e Fq(A)g Fp(\000)-60 b(!)28 b Fq(B)35 b Fu(with)30 b Fq(f)11 b Fu(\()p Fq(a)p Fu(\))27 b(=)2818 3463 y Fe(P)2923 3567 y Fl(i)2968 3538 y Fq(f)3016 3553 y Fl(i)3044 3538 y Fu(\()p Fq(a)p Fu(\))18 b Fp(\012)g Fq(a)3335 3553 y Fl(i)3394 3538 y Fu(since)0 3654 y(the)33 b Fq(a)219 3669 y Fl(i)281 3654 y Fu(form)e(a)i(basis.)45 b(Since)33 b Fq(f)44 b Fu(is)32 b(a)h(homomorphism)d(of)j(algebras)f(w) m(e)i(get)f(from)3118 3579 y Fe(P)3223 3683 y Fl(k)3282 3654 y Fq(f)3330 3669 y Fl(k)3373 3654 y Fu(\()p Fq(a)p Fu(\))23 b Fp(\012)0 3770 y Fq(a)51 3785 y Fl(k)146 3770 y Fu(=)52 b Fq(f)11 b Fu(\()p Fq(ab)p Fu(\))52 b(=)g Fq(f)11 b Fu(\()p Fq(a)p Fu(\))p Fq(f)g Fu(\()p Fq(b)p Fu(\))52 b(=)1223 3695 y Fe(P)1329 3799 y Fl(ij)1389 3770 y Fu(\()p Fq(f)1475 3785 y Fl(i)1503 3770 y Fu(\()p Fq(a)p Fu(\))32 b Fp(\012)h Fq(a)1823 3785 y Fl(i)1851 3770 y Fu(\)\()p Fq(f)1975 3785 y Fl(j)2012 3770 y Fu(\()p Fq(b)p Fu(\))f Fp(\012)g Fq(a)2321 3785 y Fl(j)2358 3770 y Fu(\))52 b(=)2576 3695 y Fe(P)2681 3799 y Fl(ij)2758 3770 y Fq(f)2806 3785 y Fl(i)2835 3770 y Fu(\()p Fq(a)p Fu(\))p Fq(f)3010 3785 y Fl(j)3046 3770 y Fu(\()p Fq(b)p Fu(\))32 b Fp(\012)h Fq(a)3356 3785 y Fl(i)3384 3770 y Fq(a)3435 3785 y Fl(j)3524 3770 y Fu(=)0 3828 y Fe(P)105 3932 y Fl(ij)t(k)221 3903 y Fq(\013)284 3867 y Fl(k)283 3927 y(ij)343 3903 y Fq(f)391 3918 y Fl(i)420 3903 y Fu(\()p Fq(a)p Fu(\))p Fq(f)595 3918 y Fl(j)631 3903 y Fu(\()p Fq(b)p Fu(\))23 b Fp(\012)f Fq(a)921 3918 y Fl(k)997 3903 y Fu(b)m(y)33 b(comparison)f(of)g(co)s(e\016cien)m(ts) 1247 4102 y Fq(f)1295 4117 y Fl(k)1338 4102 y Fu(\()p Fq(ab)p Fu(\))c(=)1638 4008 y Fe(X)1681 4218 y Fl(ij)1798 4102 y Fq(\013)1861 4061 y Fl(k)1860 4127 y(ij)1921 4102 y Fq(f)1969 4117 y Fl(i)1997 4102 y Fu(\()p Fq(a)p Fu(\))p Fq(f)2172 4117 y Fl(j)2209 4102 y 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(the)i(ideal)e(w)m(e)i(get)0 4726 y Fq(g)t Fu(\()p Fq(ab)32 b Fp(\012)h Fu(\026)-50 b Fq(a)373 4689 y Fl(k)447 4726 y Fp(\000)32 b Fq(a)g Fp(\012)g Fq(b)h Fp(\012)f Fu(\001\()q(\026)-50 b Fq(a)1101 4689 y Fl(k)1144 4726 y Fu(\)\))51 b(=)g(\(1)32 b Fp(\012)h Fu(\026)-50 b Fq(a)1677 4689 y Fl(k)1720 4726 y Fu(\))1775 4651 y Fe(P)1880 4755 y Fl(l)1922 4726 y Fq(f)1970 4741 y Fl(l)1996 4726 y Fu(\()p Fq(ab)p Fu(\))32 b Fp(\012)h Fq(a)2357 4741 y Fl(l)2414 4726 y Fp(\000)2524 4651 y Fe(P)2629 4755 y Fl(r)r(sij)2772 4726 y Fq(\013)2835 4689 y Fl(k)2834 4750 y(ij)2895 4726 y Fu(\(1)e Fp(\012)j Fu(\026)-51 b Fq(a)3173 4689 y Fl(i)3202 4726 y Fu(\)\()p Fq(f)3326 4741 y Fl(r)3364 4726 y Fu(\()p Fq(a)p Fu(\))32 b Fp(\012)0 4858 y Fq(a)51 4873 y Fl(r)89 4858 y Fu(\))e Fp(\001)g Fu(\(1)g Fp(\012)i Fu(\026)-50 b Fq(a)491 4822 y Fl(j)528 4858 y Fu(\)\()p Fq(f)652 4873 y Fl(s)689 4858 y Fu(\()p Fq(b)p Fu(\))30 b Fp(\012)h Fq(a)995 4873 y Fl(s)1032 4858 y Fu(\))48 b(=)g Fq(f)1290 4873 y Fl(k)1333 4858 y Fu(\()p Fq(ab)p Fu(\))30 b Fp(\000)1639 4783 y Fe(P)1744 4887 y Fl(ij)1822 4858 y Fq(\013)1885 4822 y Fl(k)1884 4883 y(ij)1944 4858 y Fq(f)1992 4873 y Fl(i)2020 4858 y Fu(\()p Fq(a)p Fu(\))p Fq(f)2195 4873 y Fl(j)2232 4858 y Fu(\()p Fq(b)p Fu(\))48 b(=)g(0.)79 b(W)-8 b(e)45 b(ha)m(v)m(e)h(furthermore)0 4983 y Fq(g)t Fu(\(1)26 b Fp(\012)h Fq(\020)34 b Fp(\000)28 b Fq(\020)8 b Fu(\(1\)\))38 b(=)h(\(1)27 b Fp(\012)g Fq(\020)8 b Fu(\))p Fq(f)j Fu(\(1\))26 b Fp(\000)h Fq(\020)8 b Fu(\(1\))38 b(=)i(\(1)26 b Fp(\012)i Fq(\020)8 b Fu(\)\(1)25 b Fp(\012)j Fu(1\))e Fp(\000)i Fq(\020)8 b Fu(\(1\))38 b(=)i(1)p Fq(\020)8 b Fu(\(1\))25 b Fp(\000)i Fq(\020)8 b Fu(\(1\))39 b(=)g(0.)0 5099 y(Th)m(us)c(the)f (homomorphism)d(of)i(algebras)f Fq(g)37 b Fu(v)-5 b(anishes)34 b(on)g(the)g(ideal)d Fq(I)42 b Fu(so)33 b(it)g(ma)m(y)g(b)s(e)h (factored)0 5216 y(through)41 b Fq(M)10 b Fu(\()p Fq(A)p Fu(\))42 b(=)g Fq(T)14 b Fu(\()p Fq(A)p Fu(\))p Fq(=I)8 b Fu(.)68 b(Denote)41 b(this)f(factorization)f(also)h(b)m(y)h Fq(g)t Fu(.)68 b(Then)42 b(the)f(diagram)p eop %%Page: 28 28 28 27 bop 0 236 a Fm(28)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fu(comm)m(utes)36 b(since)h(\()p Fq(g)28 b Fp(\012)d Fu(1)969 440 y Fl(A)1026 425 y Fu(\))p Fq(\016)t Fu(\()p Fq(a)p Fu(\))34 b(=)g(\()p Fq(g)28 b Fp(\012)d Fu(1)1646 440 y Fl(A)1703 425 y Fu(\)\()1779 350 y Fe(P)1884 454 y Fl(i)1912 425 y Fu(\()p Fq(a)g Fp(\012)h Fu(\026)-50 b Fq(a)2179 389 y Fl(i)2207 425 y Fu(\))25 b Fp(\012)g Fq(a)2423 440 y Fl(i)2452 425 y Fu(\))34 b(=)2634 350 y Fe(P)2739 454 y Fl(i)2767 425 y Fu(\(1)24 b Fp(\012)j Fu(\026)-50 b Fq(a)3032 389 y Fl(i)3060 425 y Fu(\))p Fq(f)11 b Fu(\()p Fq(a)p Fu(\))24 b Fp(\012)i Fq(a)3462 440 y Fl(i)3524 425 y Fu(=)0 468 y Fe(P)105 572 y Fl(ij)182 543 y Fq(f)230 558 y Fl(j)267 543 y Fu(\()p Fq(a)p Fu(\)\()q(\026)-50 b Fq(a)483 506 y Fl(i)511 543 y Fq(;)17 b(a)606 558 y Fl(j)643 543 y Fu(\))22 b Fp(\012)g Fq(a)853 558 y Fl(i)909 543 y Fu(=)1013 468 y Fe(P)1118 572 y Fl(i)1163 543 y Fq(f)1211 558 y Fl(i)1239 543 y Fu(\()p Fq(a)p Fu(\))g Fp(\012)h Fq(a)1539 558 y Fl(i)1595 543 y Fu(=)28 b Fq(f)11 b Fu(\()p Fq(a)p Fu(\).)149 662 y(W)-8 b(e)38 b(still)c(ha)m(v)m(e)k (to)e(sho)m(w)i(that)f Fq(g)j Fu(is)c(uniquely)h(determined.)56 b(Assume)37 b(that)g(w)m(e)h(also)e(ha)m(v)m(e)0 778 y(\()p Fq(h)24 b Fp(\012)h Fu(1)269 793 y Fl(A)326 778 y Fu(\))p Fq(\016)37 b Fu(=)c Fq(f)46 b Fu(then)873 704 y Fe(P)978 807 y Fl(i)1023 778 y Fq(h)p Fu(\()p Fq(a)24 b Fp(\012)i Fu(\026)-50 b Fq(a)1345 742 y Fl(i)1374 778 y Fu(\))24 b Fp(\012)h Fq(a)1589 793 y Fl(i)1650 778 y Fu(=)33 b(\()p Fq(h)24 b Fp(\012)h Fu(1)2028 793 y Fl(A)2085 778 y Fu(\))p Fq(\016)t Fu(\()p Fq(a)p Fu(\))33 b(=)g Fq(f)11 b Fu(\()p Fq(a)p Fu(\))33 b(=)2767 704 y Fe(P)2872 807 y Fl(i)2917 778 y Fq(f)2965 793 y Fl(i)2993 778 y Fu(\()p Fq(a)p Fu(\))25 b Fp(\012)f Fq(a)3297 793 y Fl(i)3362 778 y Fu(hence)0 896 y Fq(h)p Fu(\()p Fq(a)e Fp(\012)i Fu(\026)-50 b Fq(a)318 860 y Fl(i)346 896 y Fu(\))28 b(=)g Fq(f)564 911 y Fl(i)592 896 y Fu(\()p Fq(a)p Fu(\))g(=)f Fq(g)t Fu(\()p Fq(a)22 b Fp(\012)i Fu(\026)-51 b Fq(a)1162 860 y Fl(i)1191 896 y Fu(\),)32 b(i.e.)43 b Fq(g)31 b Fu(=)d Fq(h)p Fu(.)p 3529 896 4 66 v 3533 833 59 4 v 3533 896 V 3591 896 4 66 v 149 1070 a Fs(De\014nition)37 b(1.3.13.)49 b Fu(Let)42 b Fq(A)g Fu(b)s(e)h(a)e Fr(K)18 b Fu(-algebra.)76 b(The)44 b(univ)m(ersal)d (algebra)g Fq(M)10 b Fu(\()p Fq(A)p Fu(\))43 b(for)f Fq(A)0 1186 y Fu(that)32 b(is)g(a)h(bialgebra)d(is)j(also)e(called)h (the)h Fo(c)-5 b(o)g(endomorphism)32 b(bialgebr)-5 b(a)31 b Fu(of)i Fq(A)p Fu(.)149 1361 y Fs(Problem)j(1.3.7.)199 b Fu(1.)41 b(Determine)28 b(explicitly)f(the)i(dual)f(coalgebra)g Fq(A)2951 1325 y Fn(\003)3019 1361 y Fu(of)h(the)g(algebra)267 1477 y Fq(A)f Fu(:=)f Fr(K)18 b Fp(h)p Fq(x)p Fp(i)p Fq(=)p Fu(\()p Fq(x)845 1441 y Fg(2)890 1477 y Fu(\).)44 b(\(Hin)m(t:)f(Find)31 b(a)i(basis)f(for)g Fq(A)p Fu(.\))149 1593 y(2.)42 b(Determine)28 b(and)i(describ)s(e)g(the)h(co)s (endomorphism)d(bialgebra)g(of)h Fq(A)g Fu(from)g(problem)f(1.1.)267 1709 y(\(Hin)m(t:)43 b(Determine)31 b(\014rst)i(a)f(set)h(of)f(algebra) f(generators)i(of)f Fq(M)10 b Fu(\()p Fq(A)p Fu(\).)44 b(Then)34 b(describ)s(e)f(the)267 1826 y(relations.\))149 1942 y(3.)42 b(Determine)31 b(explicitly)g(the)i(dual)f(coalgebra)f Fq(A)2056 1906 y Fn(\003)2128 1942 y Fu(of)i Fq(A)27 b Fu(:=)h Fr(K)18 b Fp(h)p Fq(x)p Fp(i)p Fq(=)p Fu(\()p Fq(x)2818 1906 y Fg(3)2863 1942 y Fu(\).)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 Fq(A)g Fu(from)g(problem)f(1.3.)149 2174 y(5.)42 b(\(*\))f(Determine)f(explicitly)f(the)j(dual)f(coalgebra) f Fq(A)2266 2138 y Fn(\003)2347 2174 y Fu(of)h Fq(A)h Fu(:=)h Fr(K)17 b Fp(h)p Fq(x;)h(y)t Fp(i)p Fq(=I)53 b Fu(where)43 b(the)267 2291 y(ideal)31 b Fq(I)40 b Fu(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 Fq(xy)26 b Fp(\000)c Fq(q)1687 2406 y Fn(\000)p Fg(1)1781 2447 y Fq(y)t(x;)17 b(x)1987 2406 y Fg(2)2027 2447 y Fq(;)g(y)2123 2406 y Fg(2)2161 2447 y Fq(:)149 2604 y Fu(6.)42 b(\(*\))32 b(Determine)g(the)h(co)s (endomorphism)e(bialgebra)f(of)i Fq(A)h Fu(from)e(problem)h(1.5.)149 2721 y(7.)42 b(Let)32 b Fq(A)h Fu(b)s(e)f(a)g(\014nite)g(dimensional)e Fr(K)18 b Fu(-algebra)37 b(with)32 b(univ)m(ersal)g(bialgebra)e Fq(A)e Fp(\000)-60 b(!)28 b Fq(B)f Fp(\012)22 b Fq(A)p Fu(.)267 2837 y(Sho)m(w)402 2953 y(i\))41 b(that)k Fq(A)806 2917 y Fl(op)929 2953 y Fp(\000)-60 b(!)49 b Fq(B)1174 2917 y Fl(op)1279 2953 y Fp(\012)31 b Fq(A)1460 2917 y Fl(op)1579 2953 y Fu(is)45 b(univ)m(ersal)g(\(where)h Fq(A)2520 2917 y Fl(op)2639 2953 y Fu(has)g(the)g(m)m(ultiplication)509 3069 y Fp(r)p Fq(\034)39 b Fu(:)28 b Fq(A)22 b Fp(\012)h Fq(A)k Fp(\000)-60 b(!)28 b Fq(A)22 b Fp(\012)h Fq(A)28 b Fp(\000)-60 b(!)28 b Fq(A)p Fu(\);)375 3185 y(ii\))40 b(that)32 b Fq(A)821 3158 y Fp(\030)822 3190 y Fu(=)926 3185 y Fq(A)999 3149 y Fl(op)1105 3185 y Fu(implies)f Fq(B)1543 3158 y Fp(\030)1544 3190 y Fu(=)1649 3185 y Fq(B)1728 3149 y Fl(op)1834 3185 y Fu(\(as)i(bialgebras\);)348 3302 y(iii\))39 b(that)25 b(for)h(comm)m(utativ)m(e)f(algebras)g Fq(A)h Fu(the)h(algebra)d Fq(B)31 b Fu(satis\014es)c Fq(B)2965 3274 y Fp(\030)2966 3306 y Fu(=)3070 3302 y Fq(B)3149 3266 y Fl(op)3249 3302 y Fu(but)f(that)509 3418 y Fq(B)37 b Fu(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 Fq(B)1532 3507 y Fp(\030)1533 3538 y Fu(=)1637 3534 y Fq(B)1716 3498 y Fl(op)1816 3534 y Fu(for)i(the)h(bialgebra)e Fq(B)33 b Fu(=)27 b Fr(K)18 b Fp(h)p Fq(a;)f(b)p Fp(i)p Fq(=)p Fu(\()p Fq(a)3169 3498 y Fg(2)3214 3534 y Fq(;)g(ab)8 b Fu(+)g Fq(ba)p Fu(\).)509 3650 y(\(compare)32 b(problem)f(1.7)h(2\).) p eop %%Page: 29 29 29 28 bop 451 236 a Fm(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(29)149 425 y Fs(Sk)m(etc)m(h)38 b(of)f(solution:)149 541 y Fu(1/2.)j(The)25 b(bialgebra)d(has)i(the)h(form)d Fq(B)33 b Fu(=)28 b Fr(K)17 b Fp(h)p Fq(a;)g(b)p Fp(i)p Fq(=)p Fu(\()p Fq(a)2149 505 y Fg(2)2194 541 y Fq(;)g(ab)t Fu(+)t Fq(ba)p Fu(\))26 b(with)d(\001\()p Fq(a)p Fu(\))29 b(=)e Fq(a)t Fp(\012)t Fu(1)t(+)t Fq(b)t Fp(\012)t Fq(a)p Fu(,)0 658 y(\001\()p Fq(b)p Fu(\))h(=)g Fq(b)22 b Fp(\012)h Fq(b)33 b Fu(and)g Fq(")p Fu(\()p Fq(a)p Fu(\))27 b(=)h(0,)k Fq(")p Fu(\()p Fq(b)p Fu(\))c(=)f(1.)44 b(The)33 b(coaction)f(is)g Fq(\016)t Fu(\()p Fq(x)p Fu(\))c(=)f Fq(a)c Fp(\012)f Fu(1)g(+)g Fq(b)h Fp(\012)g Fq(x)p Fu(.)149 774 y(3/4.)52 b Fq(A)36 b Fu(has)g(the)g(basis)f(1)p Fq(;)17 b(x;)g(x)1321 738 y Fg(2)1361 774 y Fu(.)52 b(The)37 b(dual)d(coalgebra)h(has)h(the)g (dual)f(basis)g Fq(e;)17 b(\020)8 b(;)17 b(\020)3337 789 y Fg(2)3410 774 y Fu(with)0 890 y(\001\()p Fq(e)p Fu(\))28 b(=)g Fq(e)22 b Fp(\012)h Fq(e)p Fu(,)32 b(\001\()p Fq(\020)8 b Fu(\))27 b(=)h Fq(\020)h Fp(\012)23 b Fq(e)f Fu(+)g Fq(e)h Fp(\012)f Fq(\020)40 b Fu(and)32 b(\001\()p 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Fg(2)1055 1239 y Fu(.)43 b(The)34 b(m)m(ultiplication)28 b(table)j(and)i(the)g(relations)e(arise)h(from) 515 1433 y(1)22 b Fp(\012)g Fq(e)28 b Fu(=)g(1)p Fq(;)515 1549 y Fu(1)22 b Fp(\012)g Fq(\020)35 b Fu(=)28 b(1)22 b Fp(\012)g Fq(\020)1080 1564 y Fg(2)1147 1549 y Fu(=)27 b(0)p Fq(;)515 1665 y(x)570 1629 y Fg(2)632 1665 y Fp(\012)22 b Fq(e)28 b Fu(=)g(\()p Fq(x)22 b Fp(\012)h Fq(e)p Fu(\)\()p Fq(x)g Fp(\012)f Fq(e)p Fu(\))p Fq(;)515 1781 y(x)570 1745 y Fg(2)632 1781 y Fp(\012)g Fq(\020)35 b Fu(=)28 b(\()p Fq(x)22 b Fp(\012)h Fq(\020)8 b Fu(\)\()p Fq(x)21 b Fp(\012)i Fq(e)p Fu(\))f(+)g(\()p Fq(x)h Fp(\012)g Fq(e)p Fu(\)\()p Fq(x)f Fp(\012)h Fq(\020)8 b Fu(\))p Fq(;)515 1898 y(x)570 1861 y Fg(2)632 1898 y Fp(\012)22 b Fq(\020)774 1913 y Fg(2)841 1898 y Fu(=)28 b(\()p Fq(x)22 b Fp(\012)h Fq(\020)1203 1913 y Fg(2)1242 1898 y Fu(\)\()p Fq(x)f Fp(\012)h Fq(e)p Fu(\))f(+)g(\()p Fq(x)h Fp(\012)f Fq(\020)8 b Fu(\)\()p Fq(x)22 b Fp(\012)h Fq(\020)8 b Fu(\))21 b(+)h(\()p Fq(x)g Fp(\012)h Fq(e)p 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Fu(\)\()p Fq(x)f Fp(\012)h Fq(\020)3144 2399 y Fg(2)3183 2384 y Fu(\))0 2570 y(W)-8 b(e)33 b(use)h(the)f (abbreviation)e Fp(f)p Fq(u;)17 b(v)t Fp(g)26 b Fu(:=)i Fq(u)1535 2534 y Fg(2)1574 2570 y Fq(v)e Fu(+)c Fq(uv)t(u)f Fu(+)h Fq(v)t(u)2134 2534 y Fg(2)2205 2570 y Fu(and)32 b(ha)m(v)m(e)1400 2770 y Fq(a)1451 2733 y Fg(3)1518 2770 y Fu(=)c(0)p Fq(;)1400 2886 y Fp(f)p Fq(a;)17 b(b)p Fp(g)28 b Fu(=)f(0)p Fq(;)1400 3002 y Fp(f)p Fq(a;)17 b(c)p Fp(g)22 b Fu(+)g Fp(f)p Fq(b;)17 b(a)p Fp(g)27 b Fu(=)h(0)p Fq(:)0 3205 y Fu(The)34 b(condition)d(\(1)22 b Fp(\012)g Fq(\016)t Fu(\))p Fq(\016)32 b Fu(=)27 b(\(\001)c Fp(\012)f Fu(1\))p Fq(\016)36 b Fu(implies)1003 3404 y(\001\()p Fq(a)p Fu(\))28 b(=)g Fq(a)22 b Fp(\012)h Fu(1)f(+)g Fq(b)g Fp(\012)h Fq(a)f Fu(+)g Fq(c)g Fp(\012)h Fq(a)2234 3368 y Fg(2)2274 3404 y Fq(;)1003 3520 y Fu(\001\()p Fq(b)p Fu(\))28 b(=)g Fq(b)23 b Fp(\012)f Fq(b)h Fu(+)f Fq(c)g Fp(\012)h Fu(\()p Fq(ba)f Fu(+)g Fq(ab)p Fu(\))p Fq(;)1003 3637 y Fu(\001\()p Fq(c)p Fu(\))28 b(=)f Fq(b)c Fp(\012)g Fq(c)f Fu(+)g Fq(c)g Fp(\012)h Fq(b)1864 3601 y Fg(2)1926 3637 y Fu(+)f Fq(c)g Fp(\012)g Fu(\()p Fq(ca)h Fu(+)f Fq(ac)p Fu(\))p Fq(;)1003 3753 y(\017)p Fu(\()p Fq(a)p Fu(\))28 b(=)g(0)p Fq(;)1003 3869 y(\017)p Fu(\()p Fq(b)p Fu(\))h(=)e(1)p Fq(;)1003 3985 y(\017)p Fu(\()p Fq(c)p Fu(\))h(=)g(0)p Fq(:)149 4188 y Fu(5/6.)44 b Fq(A)33 b Fu(has)g(the)h(basis)e(1)p Fq(;)17 b(x;)g(y)t(;)g(xy)t Fu(.)43 b(The)33 b(dual)g(basis)f(of)h Fq(A)2362 4152 y Fn(\003)2434 4188 y Fu(is)g(denoted)g(b)m(y)h Fq(e;)17 b(\030)5 b(;)17 b(\021)t(;)g(\022)s Fu(.)43 b(The)0 4304 y(diagonal)30 b(is)979 4458 y(\001\()p Fq(e)p Fu(\))e(=)f Fq(e)22 b Fp(\012)h Fq(e;)979 4574 y Fu(\001\()p Fq(\030)5 b Fu(\))27 b(=)g Fq(\030)g Fp(\012)22 b Fq(e)h Fu(+)f Fq(e)g Fp(\012)h Fq(\030)5 b(;)979 4690 y Fu(\001\()p Fq(\021)t Fu(\))27 b(=)h Fq(\021)d Fp(\012)e Fq(e)f Fu(+)g Fq(e)h Fp(\012)f Fq(\021)t(;)979 4806 y Fu(\001\()p Fq(\022)s Fu(\))28 b(=)f Fq(\022)e Fp(\012)e Fq(e)f Fu(+)g Fq(e)h Fp(\012)f Fq(\022)k Fu(+)c Fq(\030)k Fp(\012)d Fq(\021)j Fu(+)c Fq(q)t(\021)j Fp(\012)e Fq(\030)5 b(:)0 4983 y Fu(Th)m(us)52 b(the)f(co)s(endomorphism)e(bialgebra)f(has)j(the)g (algebra)e(generators)i Fq(a)34 b Fp(\012)h Fq(\020)57 b Fu(with)50 b Fq(a)59 b Fp(2)0 5099 y(f)p Fu(1)p Fq(;)17 b(x;)g(y)t(;)g(xy)t Fp(g)43 b Fu(and)j Fq(\020)58 b Fp(2)50 b(f)p Fq(e;)17 b(\030)5 b(;)17 b(\021)t(;)g(\022)s Fp(g)p Fu(.)81 b(The)47 b(generators)f(of)g(the)g(relations)e(\(of)h Fq(I)8 b Fu(\))46 b(are)g(giv)m(en)0 5216 y(b)m(y)k(the)g(equations)f (1.1)g(and)h(1.2.)93 b(They)51 b(imply)c(that)i(1)33 b Fp(\012)h Fq(e)50 b Fu(is)e(the)i(unit)f(elemen)m(t,)k(that)p eop %%Page: 30 30 30 29 bop 0 236 a Fm(30)378 b(1.)35 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)f(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)0 425 y Fu(1)22 b Fp(\012)h Fq(\030)32 b Fu(=)27 b(1)22 b Fp(\012)h Fq(\021)31 b Fu(=)d(1)22 b Fp(\012)g Fq(\022)31 b Fu(=)d(0)k(and)h(that)199 575 y Fq(ab)23 b Fp(\012)g Fq(e)28 b Fu(=)f(\()p Fq(a)c Fp(\012)f Fq(e)p Fu(\)\()p Fq(b)h Fp(\012)g Fq(e)p Fu(\))p Fq(;)199 691 y(ab)g Fp(\012)g Fq(\030)32 b Fu(=)c(\()p Fq(a)22 b Fp(\012)g Fq(\030)5 b Fu(\)\()p Fq(b)22 b Fp(\012)h Fq(e)p Fu(\))f(+)g(\()p Fq(a)h Fp(\012)f Fu(1\)\()p Fq(b)g Fp(\012)h Fq(\030)5 b Fu(\))p Fq(;)199 807 y(ab)23 b Fp(\012)g Fq(\021)31 b Fu(=)d(\()p Fq(a)22 b Fp(\012)h Fq(\021)t Fu(\)\()p Fq(b)f Fp(\012)h Fq(e)p Fu(\))f(+)g(\()p Fq(a)g Fp(\012)h Fu(1\)\()p Fq(b)f Fp(\012)h Fq(\021)t Fu(\))p Fq(;)199 924 y(ab)g Fp(\012)g Fq(\022)31 b Fu(=)c(\()p Fq(a)c Fp(\012)f Fq(\022)s Fu(\)\()p Fq(b)h Fp(\012)f Fq(e)p Fu(\))h(+)f(\()p Fq(a)g Fp(\012)h Fu(1\)\()p Fq(b)f Fp(\012)h Fq(\022)s Fu(\))f(+)g(\()p Fq(a)g Fp(\012)h Fq(\030)5 b Fu(\)\()p Fq(b)22 b Fp(\012)g Fq(\021)t Fu(\))g(+)g Fq(q)t Fu(\()p Fq(a)g Fp(\012)h Fq(\021)t Fu(\)\()p Fq(b)f Fp(\012)h Fq(\030)5 b Fu(\))p Fq(:)0 1077 y Fu(F)-8 b(urthermore)32 b(for)g Fq(ab)h Fu(w)m(e)h(ha)m(v)m(e)g(to)e(tak)m(e)h(in)m(to)f (accoun)m(t)i(the)f(relations)e(in)g Fq(A)p Fu(.)149 1193 y(W)-8 b(e)33 b(de\014ne)750 1323 y Fq(a)28 b Fu(:=)g Fq(x)22 b Fp(\012)h Fq(e;)96 b(b)28 b Fu(:=)g Fq(x)22 b Fp(\012)h Fq(\030)5 b(;)87 b(c)27 b Fu(:=)h Fq(x)23 b Fp(\012)f Fq(\021)t(;)88 b(d)28 b Fu(:=)f Fq(x)c Fp(\012)g Fq(\022)s(;)760 1439 y(e)28 b Fu(:=)g Fq(y)d Fp(\012)e Fq(e;)83 b(f)39 b Fu(:=)27 b Fq(y)e Fp(\012)e Fq(\030)5 b(;)83 b(g)31 b Fu(:=)c Fq(y)f Fp(\012)c Fq(\021)t(;)83 b(h)28 b Fu(:=)f Fq(x)c Fp(\012)g Fq(\022)s(;)0 1598 y Fu(and)29 b(get)g Fq(\016)t Fu(\()p Fq(x)p Fu(\))f(=)f Fq(a)14 b Fp(\012)g Fu(1)g(+)g Fq(b)g Fp(\012)g Fq(x)g Fu(+)g Fq(c)g Fp(\012)h Fq(y)23 b Fu(+)14 b Fq(d)g Fp(\012)g Fq(xy)34 b Fu(and)29 b Fq(\016)t Fu(\()p Fq(y)t Fu(\))d(=)i Fq(e)14 b Fp(\012)g Fu(1)g(+)g Fq(f)27 b Fp(\012)14 b Fq(x)g Fu(+)g Fq(g)20 b Fp(\012)14 b Fq(y)19 b Fu(+)14 b Fq(h)g Fp(\012)g Fq(xy)t Fu(.)0 1714 y(Hence)34 b Fq(B)k Fu(is)32 b(generated)h(b)m(y)h Fq(a;)17 b(:)g(:)g(:)32 b(;)17 b(h)33 b Fu(as)f(an)h(algebra.)42 b(The)34 b(relations)d(are)740 1864 y Fq(a)791 1828 y Fg(2)859 1864 y Fu(=)c Fq(e)1007 1828 y Fg(2)1074 1864 y Fu(=)h(0)p Fq(;)740 1980 y(ab)23 b Fu(+)f Fq(ba)28 b Fu(=)g Fq(ac)22 b Fu(+)g Fq(ca)28 b Fu(=)g Fq(ef)33 b Fu(+)22 b Fq(f)11 b(e)27 b Fu(=)h Fq(eg)e Fu(+)c Fq(g)t(e)27 b Fu(=)g(0)p Fq(;)740 2096 y(ad)22 b Fu(+)g Fq(da)g Fu(+)g Fq(bc)h Fu(+)f Fq(q)t(cb)28 b Fu(=)f Fq(eh)c Fu(+)f Fq(he)g Fu(+)g Fq(f)11 b(g)26 b Fu(+)c Fq(q)t(g)t(f)37 b Fu(=)28 b(0)p Fq(;)740 2213 y(ae)g Fu(=)g Fq(q)t(ea;)740 2329 y(af)33 b Fu(+)22 b Fq(be)29 b Fu(=)e Fq(q)t Fu(\()p Fq(f)11 b(a)22 b Fu(+)g Fq(eb)p Fu(\))p Fq(;)740 2445 y(ag)k Fu(+)c Fq(ce)28 b Fu(=)f Fq(q)t Fu(\()p Fq(g)t(a)22 b Fu(+)g Fq(ec)p Fu(\))p Fq(;)740 2561 y(ah)h Fp(\000)f Fq(q)t(ha)h Fu(+)f Fq(de)g Fp(\000)g Fq(q)t(ed)g Fu(+)g Fq(bg)k Fp(\000)d Fq(q)1985 2525 y Fg(2)2024 2561 y Fq(g)t(b)f Fu(+)g Fq(q)t(cf)33 b Fp(\000)22 b Fq(q)t(f)11 b(c)27 b Fu(=)h(0)p Fq(:)0 2714 y Fu(The)34 b(diagonal)c(is)580 2864 y(\001\()p Fq(a)p Fu(\))e(=)f Fq(a)c Fp(\012)f Fu(1)g(+)g Fq(b)h Fp(\012)f Fq(a)h Fu(+)f Fq(c)g Fp(\012)h Fq(e)f Fu(+)g Fq(d)g Fp(\012)g Fq(ae;)580 2980 y Fu(\001\()p Fq(b)p Fu(\))28 b(=)g Fq(b)22 b Fp(\012)h Fq(b)f Fu(+)g Fq(c)h Fp(\012)f Fq(f)33 b Fu(+)22 b Fq(d)g Fp(\012)g Fu(\()p Fq(af)33 b Fu(+)22 b Fq(be)p Fu(\))p Fq(;)580 3097 y Fu(\001\()p Fq(c)p Fu(\))27 b(=)h Fq(b)23 b Fp(\012)f Fq(c)g Fu(+)g Fq(c)h Fp(\012)f Fq(g)k Fu(+)c Fq(d)f Fp(\012)i Fu(\()p Fq(ag)j Fu(+)c Fq(ce)p Fu(\))p Fq(;)580 3213 y Fu(\001\()p Fq(d)p Fu(\))27 b(=)h Fq(b)22 b Fp(\012)h Fq(d)f Fu(+)g Fq(c)g Fp(\012)h Fq(h)f Fu(+)g Fq(d)g Fp(\012)g Fu(\()p Fq(ah)h Fu(+)f Fq(de)g Fu(+)g Fq(bg)k Fu(+)c Fq(q)2506 3177 y Fn(\000)p Fg(1)2600 3213 y Fq(cf)11 b Fu(\))130 b(etc.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF