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3177 y(What)33 b(is)f Fj(X)443 3192 y Fh(c)478 3177 y Fp(\()p Fl(B)5 b Fp(\))32 b(and)h(what)g(is)f(the)h(a\016ne)g(algebra)e (of)h Fj(X)2148 3192 y Fh(c)2183 3177 y Fp(?)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 Fk(K)18 b Fp(-)5 b Fm(Alg)o Fp(.)149 3676 y Fm(De\014nition)37 b(1.2.9.)49 b Fp(Let)42 b Fl(A)i Fp(=)f Fj(O)s Fp(\()p Fj(X)15 b Fp(\))41 b(and)h Fl(A)1960 3639 y Fe(0)2027 3676 y Fp(=)i Fj(O)s Fp(\()p Fj(Y)8 b Fp(\))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 Fj(X)58 b Fp(resp.)76 b Fj(Y)8 b Fp(.)74 b(Tw)m(o)44 b Fl(B)5 b Fp(-p)s(oin)m(ts)43 b Fl(p)i Fp(:)h Fl(A)f Fj(\000)-60 b(!)46 b Fl(B)i Fp(in)42 b Fj(X)15 b Fp(\()p Fl(B)5 b Fp(\))43 b(and)0 3908 y Fl(p)49 3872 y Fe(0)111 3908 y Fp(:)38 b Fl(A)249 3872 y Fe(0)311 3908 y Fj(\000)-60 b(!)39 b 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b Fp(\()p Fl(B)d Fp(\).)p eop %%Page: 16 6 16 5 bop 0 236 a Fn(16)341 b(1.)76 b(COMMUT)-6 b(A)g(TIVE)33 b(AND)g(NONCOMMUT)-6 b(A)g(TIVE)32 b(ALGEBRAIC)h(GEOMETR)-6 b(Y)149 425 y Fm(De\014nition)37 b(1.2.11.)49 b Fp(The)33 b(functor)625 587 y(\()p Fj(X)43 b(?)28 b(Y)8 b Fp(\)\()p Fl(B)d Fp(\))27 b(:=)h Fj(f)p Fp(\()p Fl(p;)17 b(p)1538 546 y Fe(0)1560 587 y Fp(\))28 b Fj(2)g(X)15 b Fp(\()p Fl(B)5 b Fp(\))22 b Fj(\002)h(Y)8 b Fp(\()p Fl(B)d Fp(\))p Fj(j)p Fl(p;)17 b(p)2483 546 y Fe(0)2537 587 y Fp(comm)m(ute)p Fj(g)0 749 y Fp(is)32 b(called)f(the)i Fi(ortho)-5 b(gonal)34 b(pr)-5 b(o)g(duct)33 b Fp(of)f(the)h(noncomm)m(utativ)m(e)f(spaces)i Fj(X)47 b Fp(and)33 b Fj(Y)8 b Fp(.)149 926 y Fm(Remark)37 b(1.2.12.)50 b Fp(T)-8 b(ogether)27 b(with)f Fj(X)41 b Fp(and)27 b Fj(Y)34 b Fp(the)27 b(orthogonal)e(pro)s(duct)i Fj(X)42 b(?)28 b(Y)34 b Fp(is)26 b(again)0 1043 y(a)33 b(functor,)h(since)g(homomorphisms)d Fl(f)40 b Fp(:)29 b Fl(B)35 b Fj(\000)-60 b(!)29 b Fl(B)1895 1006 y Fe(0)1952 1043 y 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Fp(\))27 b Fj(?)i(Z)8 b Fp(\)\()p Fl(B)d Fp(\).)43 b(In)33 b(particular)d(\()p Fl(p)1247 4669 y Fh(x)1291 4654 y Fl(;)17 b(p)1384 4669 y Fh(y)1425 4654 y Fp(\))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 Fl(a)c Fj(2)h Fl(A)g Fp(:=)g Fj(O)s Fp(\()p Fj(X)15 b Fp(\),)32 b Fl(a)1043 4734 y Fe(0)1094 4770 y Fj(2)c Fl(A)1261 4734 y Fe(0)1313 4770 y Fp(:=)f Fj(O)s Fp(\()p Fj(Y)8 b Fp(\),)32 b(and)h Fl(a)1976 4734 y Fe(0)q(0)2046 4770 y Fj(2)c Fl(A)2214 4734 y Fe(00)2284 4770 y Fp(:=)e Fj(O)s Fp(\()p Fj(Z)8 b Fp(\))0 4932 y Fl(p)49 4947 y Fh(x)93 4932 y Fp(\()p Fl(a)p Fp(\))p Fl(p)269 4947 y Fh(y)310 4932 y Fp(\()p Fl(a)399 4891 y Fe(0)423 4932 y Fp(\))p Fl(p)510 4947 y Fh(z)549 4932 y Fp(\()p Fl(a)638 4891 y Fe(0)q(0)681 4932 y Fp(\))28 b(=)f(\()p Fl(p)937 4947 y Fh(x)981 4932 y Fl(;)17 b(p)1074 4947 y Fh(y)1115 4932 y Fp(\)\()p Fl(a)22 b Fj(\012)h Fl(a)1415 4891 y Fe(0)1438 4932 y Fp(\))p Fl(p)1525 4947 y Fh(z)1565 4932 y Fp(\()p Fl(a)1654 4891 y Fe(00)1696 4932 y Fp(\))28 b(=)g Fl(p)1915 4947 y Fh(z)1954 4932 y Fp(\()p Fl(a)2043 4891 y Fe(0)q(0)2086 4932 y Fp(\)\()p Fl(p)2211 4947 y Fh(x)2255 4932 y Fl(;)17 b(p)2348 4947 y Fh(y)2389 4932 y Fp(\)\()p Fl(a)22 b Fj(\012)g Fl(a)2688 4891 y Fe(0)2712 4932 y Fp(\))28 b(=)f Fl(p)2930 4947 y Fh(z)2970 4932 y Fp(\()p Fl(a)3059 4891 y Fe(00)3101 4932 y Fp(\))p Fl(p)3188 4947 y Fh(x)3232 4932 y Fp(\()p Fl(a)p Fp(\))p Fl(p)3408 4947 y Fh(y)3450 4932 y Fp(\()p Fl(a)3539 4891 y Fe(0)3562 4932 y Fp(\))0 5094 y(and)1260 5216 y Fl(p)1309 5231 y Fh(x)1353 5216 y Fp(\()p Fl(a)p Fp(\))p Fl(p)1529 5231 y Fh(y)1570 5216 y Fp(\()p Fl(a)1659 5174 y Fe(0)1683 5216 y Fp(\))h(=)f Fl(p)1901 5231 y Fh(y)1942 5216 y Fp(\()p Fl(a)2031 5174 y Fe(0)2055 5216 y Fp(\))p Fl(p)2142 5231 y Fh(x)2186 5216 y Fp(\()p Fl(a)p Fp(\))p Fl(:)p eop %%Page: 17 7 17 6 bop 637 236 a Fn(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 Fp(If)33 b(w)m(e)i(c)m(ho)s(ose)f Fl(a)c Fp(=)f(1)k(then)h(w)m(e)h(get)e Fl(p)1401 440 y Fh(y)1442 425 y Fp(\()p Fl(a)1531 389 y Fe(0)1555 425 y Fp(\))p Fl(p)1642 440 y Fh(z)1682 425 y Fp(\()p Fl(a)1771 389 y Fe(00)1813 425 y Fp(\))c(=)g Fl(p)2034 440 y Fh(z)2074 425 y Fp(\()p Fl(a)2163 389 y Fe(00)2206 425 y Fp(\))p Fl(p)2293 440 y Fh(y)2334 425 y Fp(\()p Fl(a)2423 389 y Fe(0)2446 425 y Fp(\).)47 b(F)-8 b(or)32 b(arbitrary)h Fl(a;)17 b(a)3295 389 y Fe(0)3318 425 y Fl(;)g(a)3413 389 y Fe(00)3489 425 y Fp(w)m(e)0 541 y(then)33 b(get)165 698 y Fl(p)214 713 y Fh(x)258 698 y Fp(\()p Fl(a)p Fp(\))p Fl(p)434 713 y Fh(y)476 698 y Fp(\()p Fl(a)565 657 y Fe(0)588 698 y Fp(\))p Fl(p)675 713 y Fh(z)715 698 y Fp(\()p Fl(a)804 657 y Fe(00)846 698 y Fp(\))28 b(=)f Fl(p)1064 713 y Fh(z)1104 698 y Fp(\()p Fl(a)1193 657 y Fe(0)q(0)1236 698 y Fp(\))p Fl(p)1323 713 y Fh(x)1367 698 y Fp(\()p Fl(a)p Fp(\))p Fl(p)1543 713 y Fh(y)1584 698 y Fp(\()p Fl(a)1673 657 y Fe(0)1696 698 y Fp(\))h(=)g Fl(p)1915 713 y Fh(z)1954 698 y Fp(\()p Fl(a)2043 657 y Fe(0)q(0)2086 698 y Fp(\))p Fl(p)2173 713 y Fh(y)2214 698 y Fp(\()p Fl(a)2303 657 y Fe(0)2327 698 y Fp(\))p Fl(p)2414 713 y Fh(x)2457 698 y Fp(\()p Fl(a)p Fp(\))g(=)g Fl(p)2765 713 y Fh(y)2806 698 y Fp(\()p Fl(a)2895 657 y Fe(0)2919 698 y Fp(\))p Fl(p)3006 713 y Fh(z)3045 698 y Fp(\()p Fl(a)3134 657 y Fe(0)q(0)3177 698 y Fp(\))p Fl(p)3264 713 y Fh(x)3308 698 y Fp(\()p Fl(a)p Fp(\))0 855 y(hence)34 b(\()p Fl(p)358 870 y Fh(y)399 855 y Fl(;)17 b(p)492 870 y Fh(z)532 855 y Fp(\))32 b(and)h(\()p Fl(p)879 870 y Fh(x)923 855 y Fl(;)17 b Fp(\()p Fl(p)1054 870 y Fh(y)1095 855 y Fl(;)g(p)1188 870 y Fh(z)1227 855 y Fp(\)\))32 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 Fm(Problem)36 b(1.2.3.)49 b Fp(Sho)m(w)32 b(that)e(the)h(orthogonal)e(pro)s(duct)i(of)f(quan)m(tum)h(spaces)i Fj(X)42 b(?)28 b(Y)38 b Fp(is)0 1146 y(a)h(tensor)g(pro)s(duct)g(for)f (the)i(category)f Fm(QS)g Fp(\(in)f(the)h(sense)i(of)d(monoidal)e (categories)j({)g(if)e(y)m(ou)0 1262 y(kno)m(w)d(already)e(what)h(that) f(is\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF