%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: ln2_5.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips ln2_5 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.11.03:0757 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 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Fn(is)g(a)g(noncomm)m(utativ)m(e)f (nonco)s(com-)0 716 y(m)m(utativ)m(e)j(Hopf)h(algebra.)149 887 y Fk(Remark)k(2.5.2.)50 b Fn(W)-8 b(e)45 b(shall)f(consider)h(all)f (Hopf)h(algebras)f(as)h(quan)m(tum)h(groups.)81 b(Ob-)0 1003 y(serv)m(e,)54 b(ho)m(w)m(ev)m(er,)g(that)48 b(the)h(comm)m (utativ)m(e)e(Hopf)h(algebras)f(ma)m(y)h(b)s(e)g(considered)h(as)f (a\016ne)0 1119 y(algebraic)40 b(groups)i(and)g(that)f(the)h(co)s(comm) m(utativ)m(e)f(Hopf)g(algebras)g(ma)m(y)g(b)s(e)h(considered)h(as)0 1236 y(formal)27 b(groups.)43 b(Their)30 b(prop)s(ert)m(y)g(as)g(a)g (quan)m(tum)f(space)i(or)f(as)f(a)h(quan)m(tum)g(monoid)e(will)f(pla)m (y)0 1352 y(some)33 b(role.)43 b(But)33 b(often)g(the)g(\(p)s(ossibly)f (nonexisting\))g(dual)g(Hopf)h(algebra)f(will)e(ha)m(v)m(e)k(the)g (geo-)0 1468 y(metrical)24 b(meaning.)40 b(The)27 b(follo)m(wing)d (examples)i Fi(SL)1935 1483 y Fh(q)1973 1468 y Fn(\(2\))g(and)h Fi(G)21 b(L)2452 1483 y Fh(q)2490 1468 y Fn(\(2\))k(will)g(ha)m(v)m(e)i (a)f(geometrical)0 1584 y(meaning.)149 1755 y Fk(Example)37 b(2.5.3.)49 b Fn(The)34 b(smallest)d(prop)s(er)i(quan)m(tum)g(group,)f (i.e.)44 b(the)33 b(smallest)e(noncom-)0 1872 y(m)m(utativ)m(e)h(nonco) s(comm)m(utativ)m(e)g(Hopf)g(algebra,)g(is)g(the)h(4-dimensional)c (algebra)1059 2029 y Fg(H)1140 2044 y Ff(4)1207 2029 y Fn(:=)f Fi(K)17 b Fe(h)p Fg(g)t(;)g(x)p Fe(i)p Fg(=)p Fn(\()p Fg(g)1775 1987 y Ff(2)1841 2029 y Fe(\000)22 b Fn(1)p Fg(;)17 b(x)2088 1987 y Ff(2)2128 2029 y Fg(;)g(xg)25 b Fn(+)d Fg(g)t(x)p Fn(\))0 2185 y(whic)m(h)33 b(w)m(as)h(\014rst)f (describ)s(ed)g(b)m(y)h(M.)f(Sw)m(eedler.)44 b(The)34 b(coalgebra)d(structure)j(is)e(giv)m(en)h(b)m(y)936 2335 y(\001\()p Fg(g)t Fn(\))27 b(=)h Fg(g)d Fe(\012)e Fg(g)t(;)263 b Fn(\001\()p Fg(x)p Fn(\))28 b(=)f Fg(x)c Fe(\012)g Fn(1)f(+)g Fg(g)j Fe(\012)e Fg(x;)1041 2451 y(")p Fn(\()p Fg(g)t Fn(\))j(=)i(1)p Fg(;)648 b(")p Fn(\()p Fg(x)p Fn(\))27 b(=)h(0)p Fg(;)868 2568 y(S)6 b Fn(\()p Fg(g)t Fn(\))27 b(=)g Fg(g)1242 2531 y Fd(\000)p Ff(1)1336 2568 y Fn(\(=)g 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Fg(q)2889 789 y Fd(\000)p Ff(1)2983 825 y Fg(bc)p Fn(\))p Fg(;)468 941 y Fn(\()p Fg(ad)f Fe(\000)h Fg(q)777 905 y Fd(\000)p Ff(1)871 941 y Fg(bc)p Fn(\))p Fg(c)28 b Fn(=)g Fg(c)p Fn(\()p Fg(ad)22 b Fe(\000)g Fg(q)1516 905 y Fd(\000)p Ff(1)1611 941 y Fg(bc)p Fn(\))p Fg(;)121 b Fn(\()p Fg(da)22 b Fe(\000)h Fg(q)t(bc)p Fn(\))p Fg(d)k Fn(=)h Fg(d)p Fn(\()p Fg(ad)21 b Fe(\000)i Fg(q)2851 905 y Fd(\000)p Ff(1)2945 941 y Fg(bc)p Fn(\))p Fg(:)0 1111 y Fn(W)-8 b(e)33 b(can)g(form)e(the)i(quan) m(tum)g(determinan)m(t)f(of)g(an)h(arbitrary)e(quan)m(tum)i(matrix)e (in)h Fg(A)h Fn(b)m(y)655 1341 y(det)790 1356 y Fh(q)845 1201 y Fc(\022)918 1282 y Fg(a)969 1245 y Fd(0)1080 1282 y Fg(b)1121 1245 y Fd(0)923 1398 y Fg(c)965 1362 y Fd(0)1076 1398 y Fg(d)1127 1362 y Fd(0)1150 1201 y Fc(\023)1251 1341 y Fn(:=)27 b Fg(a)1432 1300 y Fd(0)1456 1341 y Fg(d)1507 1300 y Fd(0)1552 1341 y Fe(\000)22 b Fg(q)1698 1300 y Fd(\000)p Ff(1)1793 1341 y Fg(b)1834 1300 y Fd(0)1857 1341 y Fg(c)1899 1300 y Fd(0)1950 1341 y Fn(=)28 b Fg(d)2105 1300 y Fd(0)2128 1341 y Fg(a)2179 1300 y Fd(0)2224 1341 y Fe(\000)23 b Fg(q)t(c)2413 1300 y Fd(0)2436 1341 y Fg(b)2477 1300 y Fd(0)2529 1341 y Fn(=)k Fg(')p Fn(\(det)2869 1356 y Fh(q)2908 1341 y Fn(\))0 1577 y(if)j Fg(')d Fn(:)h Fg(M)328 1592 y Fh(q)366 1577 y Fn(\(2\))g Fe(\000)-60 b(!)27 b Fg(A)k Fn(is)f(the)i(algebra)d(homomorphism)f(asso)s(ciated)j (with)f(the)h(quan)m(tum)g(matrix)0 1617 y Fc(\022)73 1698 y Fg(a)124 1662 y Fd(0)235 1698 y Fg(b)276 1662 y Fd(0)78 1814 y Fg(c)120 1778 y Fd(0)231 1814 y Fg(d)282 1778 y Fd(0)305 1617 y Fc(\023)378 1758 y Fn(.)149 2005 y(Giv)m(en)25 b(t)m(w)m(o)g(comm)m(uting)e(quan)m(tum)i(2)6 b Fe(\002)g Fn(2-matrices)2105 1865 y Fc(\022)2179 1945 y Fg(a)2230 1909 y Fd(0)2341 1945 y Fg(b)2382 1909 y Fd(0)2183 2061 y Fg(c)2225 2025 y Fd(0)2336 2061 y Fg(d)2387 2025 y Fd(0)2410 1865 y Fc(\023)2500 2005 y Fg(;)2544 1865 y Fc(\022)2617 1945 y Fg(a)2668 1909 y Fd(00)2798 1945 y Fg(b)2839 1909 y Fd(0)q(0)2622 2061 y Fg(c)2664 2025 y Fd(00)2794 2061 y Fg(d)2845 2025 y Fd(00)2887 1865 y Fc(\023)2960 2005 y Fn(.)41 b(The)26 b(quan)m(tum)0 2177 y(determinan)m(t)32 b(preserv)m(es)j(the)e(pro)s(duct,)g(since)504 2407 y(det)640 2422 y Fh(q)678 2407 y Fn(\()716 2267 y Fc(\022)789 2347 y Fg(a)840 2311 y Fd(0)951 2347 y Fg(b)992 2311 y Fd(0)794 2464 y Fg(c)836 2428 y Fd(0)947 2464 y Fg(d)998 2428 y Fd(0)1021 2267 y Fc(\023)17 b(\022)1184 2347 y Fg(a)1235 2311 y Fd(00)1365 2347 y Fg(b)1406 2311 y Fd(0)q(0)1189 2464 y Fg(c)1231 2428 y Fd(00)1361 2464 y Fg(d)1412 2428 y Fd(0)o(0)1454 2267 y Fc(\023)1527 2407 y Fn(\))28 b(=)f(det)1832 2422 y Fh(q)1886 2267 y Fc(\022)1960 2347 y Fg(a)2011 2311 y Fd(0)2034 2347 y Fg(a)2085 2311 y Fd(0)q(0)2150 2347 y Fn(+)22 b Fg(b)2289 2311 y Fd(0)2313 2347 y Fg(c)2355 2311 y Fd(00)2480 2347 y Fg(a)2531 2311 y Fd(0)2555 2347 y Fg(b)2596 2311 y Fd(00)2661 2347 y Fn(+)g Fg(b)2800 2311 y Fd(0)2823 2347 y Fg(d)2874 2311 y Fd(00)1960 2464 y Fg(c)2002 2428 y Fd(0)2025 2464 y Fg(a)2076 2428 y Fd(00)2141 2464 y Fn(+)g Fg(d)2290 2428 y Fd(0)2313 2464 y Fg(c)2355 2428 y Fd(00)2480 2464 y Fg(c)2522 2428 y Fd(0)2545 2464 y Fg(b)2586 2428 y Fd(0)q(0)2651 2464 y Fn(+)h Fg(d)2801 2428 y Fd(0)2823 2464 y Fg(d)2874 2428 y Fd(00)2917 2267 y Fc(\023)678 2583 y Fn(=)k(\()p Fg(a)870 2547 y Fd(0)894 2583 y Fg(a)945 2547 y Fd(00)1010 2583 y Fn(+)22 b Fg(b)1149 2547 y Fd(0)1172 2583 y Fg(c)1214 2547 y Fd(0)q(0)1257 2583 y Fn(\)\()p Fg(c)1375 2547 y Fd(0)1398 2583 y Fg(b)1439 2547 y Fd(0)q(0)1504 2583 y Fn(+)g Fg(d)1653 2547 y Fd(0)1676 2583 y Fg(d)1727 2547 y Fd(00)1769 2583 y Fn(\))g Fe(\000)h Fg(q)1976 2547 y Fd(\000)p Ff(1)2070 2583 y Fn(\()p Fg(a)2159 2547 y Fd(0)2182 2583 y Fg(b)2223 2547 y Fd(0)q(0)2288 2583 y Fn(+)f Fg(b)2427 2547 y Fd(0)2451 2583 y Fg(d)2502 2547 y Fd(00)2544 2583 y Fn(\)\()p Fg(c)2662 2547 y Fd(0)2686 2583 y Fg(a)2737 2547 y Fd(00)2801 2583 y Fn(+)g Fg(d)2950 2547 y Fd(0)2973 2583 y Fg(c)3015 2547 y Fd(00)3058 2583 y Fn(\))678 2699 y(=)27 b Fg(a)832 2663 y Fd(0)856 2699 y Fg(c)898 2663 y Fd(0)921 2699 y Fg(a)972 2663 y Fd(0)q(0)1015 2699 y Fg(b)1056 2663 y Fd(00)1121 2699 y Fn(+)22 b Fg(b)1260 2663 y Fd(0)1284 2699 y Fg(c)1326 2663 y Fd(0)1349 2699 y Fg(c)1391 2663 y Fd(00)1433 2699 y Fg(b)1474 2663 y Fd(0)q(0)1539 2699 y Fn(+)g Fg(a)1688 2663 y Fd(0)1712 2699 y Fg(d)1763 2663 y Fd(0)1786 2699 y Fg(a)1837 2663 y Fd(00)1880 2699 y Fg(d)1931 2663 y Fd(0)o(0)1995 2699 y Fn(+)g Fg(b)2134 2663 y Fd(0)2158 2699 y Fg(d)2209 2663 y Fd(0)2232 2699 y Fg(c)2274 2663 y Fd(00)2316 2699 y Fg(d)2367 2663 y Fd(00)775 2816 y Fe(\000)p Fg(q)899 2780 y Fd(\000)p Ff(1)994 2816 y Fn(\()p Fg(a)1083 2780 y Fd(0)1106 2816 y Fg(c)1148 2780 y Fd(0)1172 2816 y Fg(b)1213 2780 y Fd(00)1256 2816 y Fg(a)1307 2780 y Fd(00)1371 2816 y Fn(+)g Fg(b)1510 2780 y Fd(0)1534 2816 y Fg(c)1576 2780 y Fd(0)1600 2816 y Fg(d)1651 2780 y Fd(0)o(0)1693 2816 y Fg(a)1744 2780 y Fd(00)1808 2816 y Fn(+)g Fg(a)1957 2780 y Fd(0)1981 2816 y Fg(d)2032 2780 y Fd(0)2055 2816 y Fg(b)2096 2780 y Fd(0)q(0)2139 2816 y Fg(c)2181 2780 y Fd(00)2245 2816 y Fn(+)g Fg(b)2384 2780 y Fd(0)2408 2816 y Fg(d)2459 2780 y Fd(0)2482 2816 y Fg(d)2533 2780 y Fd(00)2575 2816 y Fg(c)2617 2780 y Fd(00)2660 2816 y Fn(\))678 2932 y(=)27 b Fg(b)822 2896 y Fd(0)846 2932 y Fg(c)888 2896 y Fd(0)911 2932 y Fg(c)953 2896 y Fd(0)q(0)996 2932 y Fg(b)1037 2896 y Fd(00)1102 2932 y Fn(+)22 b Fg(a)1251 2896 y Fd(0)1274 2932 y Fg(d)1325 2896 y Fd(0)1348 2932 y Fg(a)1399 2896 y Fd(0)q(0)1442 2932 y Fg(d)1493 2896 y Fd(00)1557 2932 y Fe(\000)h Fg(q)1704 2896 y Fd(\000)p Ff(1)1798 2932 y Fg(b)1839 2896 y Fd(0)1863 2932 y Fg(c)1905 2896 y Fd(0)1928 2932 y Fg(d)1979 2896 y Fd(00)2021 2932 y Fg(a)2072 2896 y Fd(0)q(0)2137 2932 y Fe(\000)g Fg(q)2284 2896 y Fd(\000)p Ff(1)2378 2932 y Fg(a)2429 2896 y Fd(0)2452 2932 y Fg(d)2503 2896 y Fd(0)2526 2932 y Fg(b)2567 2896 y Fd(0)q(0)2610 2932 y Fg(c)2652 2896 y Fd(00)678 3048 y Fn(=)k Fg(b)822 3012 y Fd(0)846 3048 y Fg(c)888 3012 y Fd(0)911 3048 y Fg(c)953 3012 y Fd(0)q(0)996 3048 y Fg(b)1037 3012 y Fd(00)1102 3048 y Fn(+)22 b Fg(a)1251 3012 y Fd(0)1274 3048 y Fg(d)1325 3012 y Fd(0)1348 3048 y Fg(a)1399 3012 y Fd(0)q(0)1442 3048 y Fg(d)1493 3012 y Fd(00)1557 3048 y Fe(\000)h Fg(q)1704 3012 y Fd(\000)p Ff(1)1798 3048 y Fg(b)1839 3012 y Fd(0)1863 3048 y Fg(c)1905 3012 y Fd(0)1928 3048 y Fg(d)1979 3012 y Fd(00)2021 3048 y Fg(a)2072 3012 y Fd(0)q(0)2137 3048 y Fe(\000)g Fg(q)2284 3012 y Fd(\000)p Ff(1)2378 3048 y Fg(a)2429 3012 y Fd(0)2452 3048 y Fg(d)2503 3012 y Fd(0)2526 3048 y Fg(b)2567 3012 y Fd(0)q(0)2610 3048 y Fg(c)2652 3012 y Fd(00)775 3164 y Fe(\000)p Fg(q)899 3128 y Fd(\000)p Ff(1)994 3164 y Fg(b)1035 3128 y Fd(0)1059 3164 y Fg(c)1101 3128 y Fd(0)1124 3164 y Fn(\()p Fg(a)1213 3128 y Fd(00)1256 3164 y Fg(d)1307 3128 y Fd(00)1371 3164 y Fe(\000)g Fg(d)1522 3128 y Fd(0)o(0)1564 3164 y Fg(a)1615 3128 y Fd(00)1679 3164 y Fe(\000)g Fg(q)1826 3128 y Fd(\000)p Ff(1)1920 3164 y Fg(b)1961 3128 y Fd(0)q(0)2004 3164 y Fg(c)2046 3128 y Fd(00)2111 3164 y Fn(+)f Fg(q)t(c)2298 3128 y Fd(00)2340 3164 y Fg(b)2381 3128 y Fd(0)q(0)2424 3164 y Fn(\))678 3281 y(=)27 b Fg(a)832 3244 y Fd(0)856 3281 y Fg(d)907 3244 y Fd(0)930 3281 y Fg(a)981 3244 y Fd(00)1024 3281 y Fg(d)1075 3244 y Fd(0)o(0)1139 3281 y Fe(\000)22 b Fg(q)1285 3244 y Fd(\000)p Ff(1)1380 3281 y Fg(a)1431 3244 y Fd(0)1454 3281 y Fg(d)1505 3244 y Fd(0)1528 3281 y Fg(b)1569 3244 y Fd(0)q(0)1612 3281 y Fg(c)1654 3244 y Fd(00)1719 3281 y Fe(\000)g Fg(q)1865 3244 y Fd(\000)p Ff(1)1959 3281 y Fg(b)2000 3244 y Fd(0)2024 3281 y Fg(c)2066 3244 y Fd(0)2089 3281 y Fn(\()p Fg(a)2178 3244 y Fd(0)q(0)2221 3281 y Fg(d)2272 3244 y Fd(00)2336 3281 y Fe(\000)h Fg(q)2483 3244 y Fd(\000)p Ff(1)2577 3281 y Fg(b)2618 3244 y Fd(0)q(0)2661 3281 y Fg(c)2703 3244 y Fd(00)2745 3281 y Fn(\))678 3397 y(=)k(\()p Fg(a)870 3361 y Fd(0)894 3397 y Fg(d)945 3361 y Fd(0)990 3397 y Fe(\000)c Fg(q)1137 3361 y Fd(\000)p Ff(1)1231 3397 y Fg(b)1272 3361 y Fd(0)1296 3397 y Fg(c)1338 3361 y Fd(0)1361 3397 y Fn(\)\()p Fg(a)1488 3361 y Fd(00)1530 3397 y Fg(d)1581 3361 y Fd(00)1646 3397 y Fe(\000)f Fg(q)1792 3361 y Fd(\000)p Ff(1)1886 3397 y Fg(b)1927 3361 y Fd(0)q(0)1970 3397 y Fg(c)2012 3361 y Fd(00)2055 3397 y Fn(\))678 3576 y(=)27 b(det)917 3591 y Fh(q)972 3436 y Fc(\022)1045 3516 y Fg(a)1096 3480 y Fd(0)1207 3516 y Fg(b)1248 3480 y Fd(0)1049 3633 y Fg(c)1091 3596 y Fd(0)1202 3633 y Fg(d)1253 3596 y Fd(0)1276 3436 y Fc(\023)1366 3576 y Fn(det)1502 3591 y Fh(q)1556 3436 y Fc(\022)1630 3516 y Fg(a)1681 3480 y Fd(00)1811 3516 y Fg(b)1852 3480 y Fd(0)q(0)1634 3633 y Fg(c)1676 3596 y Fd(0)q(0)1806 3633 y Fg(d)1857 3596 y Fd(00)1900 3436 y Fc(\023)1990 3576 y Fg(:)0 2992 y Fn(\(1\))0 3812 y(In)33 b(particular)e(w)m(e)j(ha)m(v)m (e)g(\001\(det)1195 3827 y Fh(q)1234 3812 y Fn(\))27 b(=)h(det)1539 3827 y Fh(q)1593 3812 y Fe(\012)17 b Fn(det)1823 3827 y Fh(q)1894 3812 y Fn(and)32 b Fg(")p Fn(\(det)2302 3827 y Fh(q)2340 3812 y Fn(\))c(=)g(1.)44 b(The)33 b(quan)m(tum)g (determi-)0 3928 y(nan)m(t)g(is)f(a)g(group)h(lik)m(e)f(elemen)m(t)g (\(see)i(2.1.6\).)149 4044 y(No)m(w)g(w)m(e)f(de\014ne)h(an)f(algebra) 1197 4218 y Fg(S)6 b(L)1329 4233 y Fh(q)1367 4218 y Fn(\(2\))28 b(:=)f Fg(M)1744 4233 y Fh(q)1783 4218 y Fn(\(2\))p Fg(=)p Fn(\(det)2129 4233 y Fh(q)2190 4218 y Fe(\000)22 b Fn(1\))p Fg(:)0 4393 y Fn(The)34 b(algebra)d Fg(S)6 b(L)674 4408 y Fh(q)712 4393 y Fn(\(2\))32 b(represen)m(ts)j(the)e(functor)601 4631 y Fi(S)o(L)722 4646 y Fh(q)760 4631 y Fn(\(2\)\()p Fg(A)p Fn(\))28 b(=)f Fe(f)1215 4491 y Fc(\022)1289 4571 y Fg(a)1340 4535 y Fd(0)1451 4571 y Fg(b)1492 4535 y Fd(0)1293 4688 y Fg(c)1335 4652 y Fd(0)1446 4688 y Fg(d)1497 4652 y Fd(0)1520 4491 y Fc(\023)1621 4631 y Fe(2)h(M)1835 4646 y Fh(q)1873 4631 y Fn(\(2\)\()p Fg(A)p Fn(\))p Fe(j)p Fn(det)2310 4646 y Fh(q)2364 4491 y Fc(\022)2438 4571 y Fg(a)2489 4535 y Fd(0)2600 4571 y Fg(b)2641 4535 y Fd(0)2442 4688 y Fg(c)2484 4652 y Fd(0)2595 4688 y Fg(d)2646 4652 y Fd(0)2669 4491 y Fc(\023)2770 4631 y Fn(=)g(1)p Fe(g)p Fg(:)0 4867 y Fn(There)42 b(is)d(a)h(surjectiv)m(e)i (homomorphism)c(of)h(algebras)h Fg(M)2226 4882 y Fh(q)2265 4867 y Fn(\(2\))g Fe(\000)-60 b(!)41 b Fg(S)6 b(L)2720 4882 y Fh(q)2758 4867 y Fn(\(2\))40 b(and)h Fi(S)o(L)3242 4882 y Fh(q)3280 4867 y Fn(\(2\))f(is)g(a)0 4983 y(subfunctor)34 b(of)e Fe(M)717 4998 y Fh(q)754 4983 y Fn(\(2\).)149 5099 y(Let)45 b Fg(X)r(;)17 b(Y)65 b Fn(b)s(e)44 b(comm)m(uting)e(quan) m(tum)i(matrices)f(satisfying)g(det)2654 5114 y Fh(q)2692 5099 y Fn(\()p Fg(X)8 b Fn(\))47 b(=)g(1)f(=)h(det)3381 5114 y Fh(q)3419 5099 y Fn(\()p Fg(Y)21 b Fn(\).)0 5216 y(Since)66 b(det)424 5231 y Fh(q)462 5216 y Fn(\()p Fg(X)8 b Fn(\))17 b(det)779 5231 y Fh(q)817 5216 y Fn(\()p Fg(Y)k Fn(\))85 b(=)g(det)1352 5231 y Fh(q)1390 5216 y Fn(\()p Fg(X)8 b(Y)21 b Fn(\))66 b(for)g(comm)m(uting)f(quan)m(tum)h(matrices)g (w)m(e)h(get)p eop %%Page: 55 4 55 3 bop 1378 236 a Fl(5.)35 b(QUANTUM)d(GR)n(OUPS)1274 b(55)0 425 y Fn(det)135 440 y Fh(q)174 425 y Fn(\()p Fg(X)8 b(Y)21 b Fn(\))47 b(=)g(1,)g(hence)e Fi(SL)1114 440 y Fh(q)1152 425 y Fn(\(2\))f(is)g(a)g(quan)m(tum)g(submonoid)f(of)h Fe(M)2695 440 y Fh(q)2733 425 y Fn(\(2\))f(and)i Fg(S)6 b(L)3235 440 y Fh(q)3273 425 y Fn(\(2\))44 b(is)f(a)0 541 y(bialgebra)31 b(with)h(diagonal)1105 769 y(\001)1203 629 y Fc(\022)1277 710 y Fg(a)87 b(b)1281 826 y(c)h(d)1461 629 y Fc(\023)1562 769 y Fn(=)1666 629 y Fc(\022)1739 710 y Fg(a)g(b)1744 826 y(c)g(d)1924 629 y Fc(\023)2020 769 y Fe(\012)2119 629 y Fc(\022)2193 710 y Fg(a)g(b)2197 826 y(c)g(d)2378 629 y Fc(\023)2468 769 y Fg(;)0 997 y Fn(and)1352 1165 y Fg(")1415 1025 y Fc(\022)1488 1105 y Fg(a)f(b)1492 1222 y(c)h(d)1673 1025 y Fc(\023)1774 1165 y Fn(=)1877 1025 y Fc(\022)1950 1105 y Fn(1)83 b(0)1950 1222 y(0)g(1)2131 1025 y Fc(\023)2221 1165 y Fg(:)149 1378 y Fn(T)-8 b(o)30 b(sho)m(w)g(that)f Fg(S)6 b(L)868 1393 y Fh(q)907 1378 y Fn(\(2\))29 b(has)g(an)h(an)m(tip)s(o)s(de)e(w)m (e)j(\014rst)e(de\014ne)i(a)e(homomorphism)e(of)i(algebras)0 1494 y Fg(T)41 b Fn(:)28 b Fg(M)247 1509 y Fh(q)286 1494 y Fn(\(2\))f Fe(\000)-60 b(!)28 b Fg(M)677 1509 y Fh(q)715 1494 y Fn(\(2\))840 1458 y Fh(op)946 1494 y Fn(b)m(y)1161 1731 y Fg(T)1249 1590 y Fc(\022)1322 1671 y Fg(a)88 b(b)1327 1787 y(c)f(d)1507 1590 y Fc(\023)1608 1731 y Fn(:=)1739 1590 y Fc(\022)1917 1671 y Fg(d)188 b Fe(\000)p Fg(q)t(b)1812 1787 y Fe(\000)p Fg(q)1936 1751 y Fd(\000)p Ff(1)2031 1787 y Fg(c)140 b(a)2321 1590 y Fc(\023)2411 1731 y Fg(:)0 2034 y Fn(W)-8 b(e)33 b(c)m(hec)m(k)i(that)d Fg(T)42 b Fn(:)27 b Fi(K)884 1894 y Fc(\034)5 b(\022)1037 1974 y Fg(a)88 b(b)1042 2091 y(c)g(d)1222 1894 y Fc(\023)q(\035)1398 2034 y Fe(\000)-60 b(!)28 b Fg(M)1637 2049 y Fh(q)1675 2034 y Fn(\(2\))1800 1998 y Fh(op)1906 2034 y Fn(v)-5 b(anishes)33 b(on)g(the)g(ideal)e Fg(I)8 b Fn(.)442 2277 y Fg(T)14 b Fn(\()p Fg(ab)23 b Fe(\000)f Fg(q)812 2236 y Fd(\000)p Ff(1)906 2277 y Fg(ba)p Fn(\))29 b(=)e Fg(T)14 b Fn(\()p Fg(b)p Fn(\))p Fg(T)g Fn(\()p Fg(a)p Fn(\))22 b Fe(\000)h Fg(q)1723 2236 y Fd(\000)p Ff(1)1817 2277 y Fg(T)14 b Fn(\()p Fg(a)p Fn(\))p Fg(T)g Fn(\()p Fg(b)p Fn(\))28 b(=)f Fe(\000)p Fg(q)t(bd)c Fn(+)f Fg(q)2718 2236 y Fd(\000)p Ff(1)2812 2277 y Fg(q)t(db)28 b Fn(=)f(0)p Fg(:)0 2449 y Fn(W)-8 b(e)46 b(lea)m(v)m(e)h(the)g(c)m(hec)m(k)h(of)e (the)g(other)g(de\014ning)g(relations)f(to)h(the)g(reader.)85 b(F)-8 b(urthermore)45 b Fg(T)0 2565 y Fn(restricts)e(to)f(a)h (homomorphism)c(of)k(algebras)e Fg(S)51 b Fn(:)45 b Fg(S)6 b(L)2130 2580 y Fh(q)2168 2565 y Fn(\(2\))44 b Fe(\000)-60 b(!)45 b Fg(S)6 b(L)2631 2580 y Fh(q)2669 2565 y Fn(\(2\))2794 2529 y Fh(op)2910 2565 y Fn(since)43 b Fg(T)14 b Fn(\(det)3403 2580 y Fh(q)3441 2565 y Fn(\))45 b(=)0 2681 y Fg(T)14 b Fn(\()p Fg(ad)30 b Fe(\000)h Fg(q)396 2645 y Fd(\000)p Ff(1)490 2681 y Fg(bc)p Fn(\))49 b(=)g Fg(T)14 b Fn(\()p Fg(d)p Fn(\))p Fg(T)g Fn(\()p Fg(a)p Fn(\))30 b Fe(\000)h Fg(q)1366 2645 y Fd(\000)p Ff(1)1460 2681 y Fg(T)14 b Fn(\()p Fg(c)p Fn(\))p Fg(T)g Fn(\()p Fg(b)p Fn(\))48 b(=)h Fg(ad)30 b Fe(\000)h Fg(q)2297 2645 y Fd(\000)p 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5155 y Fd(\000)p Ff(1)2264 5191 y Fn(.)p eop %%Page: 56 5 56 4 bop 0 236 a Fl(56)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)149 425 y Fk(Example)37 b(2.5.7.)49 b Fn(Let)33 b Fg(M)1191 440 y Fh(q)1229 425 y Fn(\(2\))g(b)s(e)f(as)h(in)f(the)h(example)f(2.5.5.)43 b(W)-8 b(e)33 b(de\014ne)1339 589 y Fg(GL)1482 604 y Fh(q)1520 589 y Fn(\(2\))27 b(:=)h Fg(M)1897 604 y Fh(q)1935 589 y Fn(\(2\)[)p Fg(t)p Fn(])p Fg(=J)0 752 y Fn(with)j Fg(J)40 b Fn(generated)32 b(b)m(y)g(the)f(elemen)m(ts)h Fg(t)19 b Fe(\001)g Fn(\()p Fg(ad)g Fe(\000)g Fg(q)1859 716 y Fd(\000)p Ff(1)1953 752 y Fg(bc)p Fn(\))h Fe(\000)g Fn(1)p Fg(:)30 b Fn(The)j(algebra)c Fg(GL)2979 767 y Fh(q)3018 752 y Fn(\(2\))h(represen)m(ts)0 868 y(the)j(functor)346 1068 y Fi(G)22 b(L)490 1083 y Fh(q)528 1068 y Fn(\(2\)\()p Fg(A)p Fn(\))27 b(=)h Fe(f)983 928 y Fc(\022)1056 1009 y Fg(a)1107 972 y Fd(0)1218 1009 y Fg(b)1259 972 y Fd(0)1061 1125 y Fg(c)1103 1089 y Fd(0)1214 1125 y Fg(d)1265 1089 y 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Fg(M)1282 1829 y Fh(q)1321 1814 y Fn(\(2\))185 b Fg(M)1725 1829 y Fh(q)1764 1814 y Fn(\(2\)[)p Fg(t)p Fn(])p 1474 1793 129 4 v 1520 1791 a Fa(-)2172 1814 y Fg(G)2249 1829 y Fh(q)2287 1814 y Fn(\(2\))p 2006 1793 138 4 v 2061 1791 a Fa(-)1768 2312 y Fg(A)1385 1942 y Fa(@)1468 2025 y(@)1551 2108 y(@)1634 2191 y(@)1653 2210 y(@)-83 b(R)p 1803 2210 4 351 v 68 w(?)2141 1942 y(\000)2058 2025 y(\000)1975 2108 y(\000)1892 2191 y(\000)1873 2210 y(\000)g(\011)0 2528 y Fn(with)37 b Fg(t)e Fe(7!)g Fn(det)568 2543 y Fh(q)622 2388 y Fc(\022)696 2468 y Fg(a)747 2432 y Fd(0)858 2468 y Fg(b)899 2432 y Fd(0)700 2585 y Fg(c)742 2548 y Fd(0)853 2585 y Fg(d)904 2548 y Fd(0)927 2388 y Fc(\023)1000 2410 y Fd(\000)p Ff(1)1111 2528 y Fg(:)i Fn(Th)m(us)i Fi(G)21 b(L)1571 2543 y Fh(q)1609 2528 y Fn(\(2\)\()p Fg(A)p Fn(\))37 b(is)f(a)h(subset)h(of)f Fe(M)2647 2543 y Fh(q)2685 2528 y Fn(\(2\)\()p Fg(A)p Fn(\).)56 b(Observ)m(e)39 b(that)0 2706 y Fg(M)94 2721 y Fh(q)132 2706 y Fn(\(2\))28 b Fe(\000)-60 b(!)27 b Fg(GL)572 2721 y Fh(q)611 2706 y Fn(\(2\))32 b(is)g Fj(not)h Fn(surjectiv)m(e.)149 2822 y(Since)h(the)g(quan)m(tum)g(determinan)m(t) f(preserv)m(es)k(pro)s(ducts)d(and)g(the)g(pro)s(duct)g(of)f(in)m(v)m (ertible)0 2938 y(elemen)m(ts)f(is)f(again)g(in)m(v)m(ertible)g(w)m(e)i (get)f Fi(G)21 b(L)1633 2953 y Fh(q)1672 2938 y Fn(\(2\))31 b(is)g(a)h(quan)m(tum)g(submonoid)f(of)g Fe(M)3140 2953 y Fh(q)3178 2938 y Fn(\(2\),)h(hence)0 3119 y(\001)c(:)g Fg(GL)307 3134 y Fh(q)345 3119 y Fn(\(2\))f Fe(\000)-60 b(!)28 b Fg(GL)785 3134 y Fh(q)824 3119 y Fn(\(2\))t Fe(\012)t Fg(GL)1177 3134 y Fh(q)1216 3119 y Fn(\(2\))23 b(with)h(\001)1676 2978 y Fc(\022)1749 3059 y Fg(a)88 b(b)1754 3175 y(c)f(d)1934 2978 y Fc(\023)2035 3119 y Fn(=)2138 2978 y Fc(\022)2212 3059 y Fg(a)h(b)2216 3175 y(c)g(d)2397 2978 y Fc(\023)2474 3119 y Fe(\012)2555 2978 y Fc(\022)2630 3059 y Fg(a)f(b)2634 3175 y(c)h(d)2814 2978 y Fc(\023)2912 3119 y Fn(and)24 b(\001\()p Fg(t)p Fn(\))k(=)f Fg(t)t Fe(\012)t Fg(t)p Fn(.)149 3297 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