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b(=)f Fm(\023)1375 2394 y Ff(1)1415 2379 y Fr(\()p Fm(x)p Fr(\))c(+)f Fm(\023)1701 2394 y Ff(2)1741 2379 y Fr(\()p Fm(x)p Fr(\))28 b(=)f Fm(x)c Fl(\012)f Fr(1)g(+)g(1)g Fl(\012)h Fm(x:)149 2540 y Fr(The)34 b(counit)e(is)g Fm(")27 b Fr(=)h Fm(e)968 2555 y Fb(K)1052 2540 y Fr(=)f(0)h Fl(2)g Fn(K)17 b Fr(-)6 b Fo(cAlg)n Fr(\()p Fn(K)18 b Fr([)p Fm(x)p Fr(])p Fm(;)g Fn(K)23 b Fr(\))2069 2513 y Fl(\030)2070 2544 y Fr(=)2174 2540 y Fn(G)2246 2555 y Fh(a)2294 2540 y Fr(\()p Fn(K)17 b Fr(\))38 b(hence)1608 2702 y Fm(")p Fr(\()p Fm(x)p Fr(\))28 b(=)f(0)p Fm(:)149 2866 y Fr(The)34 b(an)m(tip)s(o)s(de)e(is)g Fm(S)h Fr(=)28 b(id)1130 2823 y Fd(\000)p Ff(1)1130 2896 y Fb(K)5 b Ff([)p Fh(x)p Ff(])1293 2866 y Fl(2)28 b Fn(K)17 b Fr(-)6 b Fo(cAlg)n Fr(\()p Fn(K)18 b Fr([)p Fm(x)p Fr(])p Fm(;)f Fn(K)23 b Fr([)q Fm(x)p Fr(]\))2240 2838 y Fl(\030)2241 2870 y Fr(=)2345 2866 y Fn(G)2416 2881 y Fh(a)2464 2866 y Fr(\()p Fn(K)17 b Fr([)q Fm(x)p Fr(]\))38 b(hence)1556 3040 y Fm(S)6 b Fr(\()p Fm(x)p Fr(\))28 b(=)f Fl(\000)p Fm(x:)149 3202 y Fr(2.)44 b(The)33 b(a\016ne)g (algebraic)e(group)i(called)e Fk(multiplic)-5 b(ative)34 b(gr)-5 b(oup)1361 3363 y Fn(G)1433 3378 y Fh(m)1533 3363 y Fr(:)28 b Fn(K)17 b Fr(-)6 b Fo(cAlg)26 b Fl(\000)-60 b(!)28 b Fo(Ab)0 3525 y Fr(with)41 b Fn(G)302 3540 y Fh(m)375 3525 y Fr(\()p Fm(A)p Fr(\))h(:=)h Fm(A)785 3489 y Fd(\003)867 3525 y Fr(=)f Fm(U)10 b Fr(\()p Fm(A)p Fr(\))42 b(from)e(Lemma)g(2.3.7)h(is)f(represen)m(ted)k(b)m(y)e(the)g (Hopf)f(algebra)0 3641 y Fn(K)17 b Fr([)q Fm(x;)g(x)253 3605 y Fd(\000)p Ff(1)353 3641 y Fr(])28 b(=)g Fn(K)17 b Fr([)p Fm(x)q(;)g(y)t Fr(])p Fm(=)p Fr(\()p Fm(xy)30 b Fl(\000)23 b Fr(1\).)43 b(W)-8 b(e)33 b(determine)f(com)m (ultiplication,)c(counit,)33 b(and)f(an)m(tip)s(o)s(de.)149 3757 y(By)i(Corollary)c(2.2.7)i(the)h(com)m(ultiplication)28 b(is)23 3919 y(\001)g(=)g Fm(\023)270 3934 y Ff(1)332 3919 y Fl(\003)22 b Fm(\023)438 3934 y Ff(2)505 3919 y Fl(2)28 b Fn(K)18 b Fr(-)5 b Fo(cAlg)o Fr(\()p Fn(K)17 b Fr([)q 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Fh(m)2512 4342 y Fr(\()p Fn(K)g Fr(\))39 b(hence)1608 4503 y Fm(")p Fr(\()p Fm(x)p Fr(\))28 b(=)f(1)p Fm(:)149 4667 y Fr(The)45 b(an)m(tip)s(o)s(de)f(is)f Fm(S)53 b Fr(=)47 b(id)1203 4624 y Fd(\000)p Ff(1)1203 4698 y Fb(K)5 b Ff([)p Fh(x;x)1369 4679 y Fc(\000)p Fe(1)1456 4698 y Ff(])1527 4667 y Fl(2)47 b Fn(K)18 b Fr(-)5 b Fo(cAlg)o Fr(\()p Fn(K)17 b Fr([)q Fm(x;)g(x)2263 4631 y Fd(\000)p Ff(1)2363 4667 y Fr(])p Fm(;)g Fn(K)h Fr([)p Fm(x;)f(x)2687 4631 y Fd(\000)p Ff(1)2788 4667 y Fr(]\))2900 4640 y Fl(\030)2901 4671 y Fr(=)3024 4667 y Fn(G)3096 4682 y Fh(a)3144 4667 y Fr(\()p Fn(K)g Fr([)p Fm(x)q(;)g(x)3435 4631 y Fd(\000)p Ff(1)3535 4667 y Fr(]\))0 4792 y(hence)1547 4913 y Fm(S)6 b Fr(\()p Fm(x)p Fr(\))28 b(=)g Fm(x)1931 4872 y Fd(\000)p Ff(1)2025 4913 y Fm(:)149 5054 y Fr(3.)44 b(The)33 b(a\016ne)g (algebraic)e(group)i(called)e Fk(additive)j(matrix)h(gr)-5 b(oup)1343 5216 y Fn(M)1431 5174 y Ff(+)1431 5240 y Fh(n)1524 5216 y Fr(:)28 b Fn(K)17 b Fr(-)6 b Fo(cAlg)26 b Fl(\000)-60 b(!)28 b Fo(Ab)p Fm(;)p eop %%Page: 48 6 48 5 bop 0 236 a Fp(48)333 b(2.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fr(with)c Fn(M)306 389 y Ff(+)306 450 y Fh(n)371 425 y Fr(\()p Fm(A)p Fr(\))h(the)f(additiv)m(e)g(group)g(of)g Fm(n)14 b Fl(\002)g Fm(n)p Fr(-matrices)28 b(with)g(co)s(e\016cien)m(ts)h(in)f Fm(A)g Fr(is)g(represen)m(ted)0 541 y(b)m(y)36 b(the)f(comm)m(utativ)m (e)f(algebra)g Fm(M)1341 505 y Ff(+)1331 566 y Fh(n)1432 541 y Fr(=)e Fn(K)17 b Fr([)p Fm(x)1694 556 y Fh(ij)1760 541 y Fl(j)p Fr(1)32 b Fl(\024)g Fm(i;)17 b(j)37 b Fl(\024)32 b Fm(n)p Fr(])k(\(Lemma)d(2.3.9\).)50 b(This)35 b(algebra)0 658 y(m)m(ust)e(b)s(e)f(a)h(Hopf)f(algebra.)149 774 y(The)f(com)m (ultiplication)25 b(is)k(\001)f(=)g Fm(\023)1404 789 y Ff(1)1460 774 y Fl(\003)16 b Fm(\023)1560 789 y Ff(2)1627 774 y Fl(2)28 b Fn(K)18 b Fr(-)5 b Fo(cAlg)o Fr(\()p Fm(M)2195 738 y Ff(+)2185 799 y Fh(n)2255 774 y Fm(;)17 b(M)2403 738 y Ff(+)2393 799 y Fh(n)2478 774 y Fl(\012)f Fm(M)2675 738 y Ff(+)2665 799 y Fh(n)2736 774 y Fr(\))2801 746 y Fl(\030)2802 778 y Fr(=)2907 774 y Fn(M)2995 738 y Ff(+)2995 799 y Fh(n)3060 774 y Fr(\()p Fm(M)3202 738 y Ff(+)3192 799 y Fh(n)3278 774 y Fl(\012)g Fm(M)3475 738 y Ff(+)3465 799 y Fh(n)3535 774 y Fr(\).)0 890 y(Hence)846 1023 y(\001\()p Fm(x)1020 1038 y Fh(ij)1081 1023 y Fr(\))28 b(=)g Fm(\023)1285 1038 y Ff(1)1325 1023 y Fr(\()p Fm(x)1418 1038 y Fh(ij)1478 1023 y Fr(\))22 b(+)h Fm(\023)1671 1038 y Ff(2)1710 1023 y Fr(\()p Fm(x)1803 1038 y Fh(ij)1864 1023 y Fr(\))28 b(=)f Fm(x)2088 1038 y Fh(ij)2172 1023 y Fl(\012)22 b Fr(1)g(+)g(1)g Fl(\012)h Fm(x)2666 1038 y Fh(ij)2727 1023 y Fm(:)149 1176 y Fr(The)34 b(counit)e(is)g Fm(")27 b Fr(=)h Fm(e)968 1191 y Fb(K)1052 1176 y Fr(=)f(\(0\))h Fl(2)g Fn(K)17 b Fr(-)6 b Fo(cAlg)n Fr(\()p Fm(M)1875 1140 y Ff(+)1865 1201 y Fh(n)1935 1176 y Fm(;)17 b Fn(K)g Fr(\))2122 1148 y Fl(\030)2123 1180 y Fr(=)2227 1176 y Fn(M)2315 1140 y Ff(+)2315 1201 y Fh(n)2380 1176 y Fr(\()p Fn(K)h Fr(\))38 b(hence)1578 1349 y Fm(")p Fr(\()p Fm(x)1717 1364 y Fh(ij)1777 1349 y Fr(\))28 b(=)f(0)p Fm(:)149 1529 y Fr(The)34 b(an)m(tip)s(o)s(de)e(is)g Fm(S)h Fr(=)28 b(id)1130 1486 y Fd(\000)p Ff(1)1130 1570 y Fh(M)1205 1540 y Fe(+)1198 1581 y Fg(n)1288 1529 y Fl(2)g Fn(K)17 b Fr(-)6 b Fo(cAlg)n Fr(\()p Fm(M)1855 1492 y Ff(+)1845 1553 y Fh(n)1915 1529 y Fm(;)17 b(M)2063 1492 y Ff(+)2053 1553 y Fh(n)2123 1529 y Fr(\))2188 1501 y Fl(\030)2189 1533 y Fr(=)2293 1529 y Fn(M)2381 1492 y Ff(+)2381 1553 y Fh(n)2446 1529 y Fr(\()p Fm(M)2588 1492 y Ff(+)2578 1553 y Fh(n)2648 1529 y Fr(\))33 b(hence)1495 1721 y Fm(S)6 b Fr(\()p Fm(x)1654 1736 y Fh(ij)1715 1721 y Fr(\))28 b(=)f Fl(\000)p Fm(x)2016 1736 y Fh(ij)2078 1721 y Fm(:)149 1894 y Fr(4.)43 b(The)31 b(matrix)e(algebra)g Fn(M)1208 1909 y Fh(n)1261 1894 y Fr(\()p Fm(A)p Fr(\))h(also)f(has)i (a)e(noncomm)m(utativ)m(e)h(m)m(ultiplication,)c(the)k(ma-)0 2011 y(trix)j(m)m(ultiplication,)c(de\014ning)k(a)h(monoid)d(structure) k Fn(M)2148 1974 y Fd(\002)2148 2035 y Fh(n)2213 2011 y Fr(\()p Fm(A)p Fr(\).)46 b(Th)m(us)35 b Fn(K)18 b Fr([)p Fm(x)2837 2026 y Fh(ij)2904 2011 y Fr(])33 b(carries)h(another)0 2127 y(coalgebra)40 b(structure)h(whic)m(h)g(de\014nes)i(a)d(bialgebra) f Fm(M)2107 2091 y Fd(\002)2097 2151 y Fh(n)2208 2127 y Fr(=)i Fn(K)18 b Fr([)p Fm(x)2479 2142 y Fh(ij)2546 2127 y Fr(].)67 b(Ob)m(viously)41 b(there)g(is)f(no)0 2243 y(an)m(tip)s(o)s(de.)149 2359 y(The)31 b(com)m(ultiplication)25 b(is)k(\001)f(=)g Fm(\023)1404 2374 y Ff(1)1460 2359 y Fl(\003)16 b Fm(\023)1560 2374 y Ff(2)1627 2359 y Fl(2)28 b Fn(K)18 b Fr(-)5 b Fo(cAlg)o Fr(\()p Fm(M)2195 2323 y Fd(\002)2185 2384 y Fh(n)2255 2359 y Fm(;)17 b(M)2403 2323 y Fd(\002)2393 2384 y Fh(n)2478 2359 y Fl(\012)f Fm(M)2675 2323 y Fd(\002)2665 2384 y Fh(n)2736 2359 y Fr(\))2801 2332 y Fl(\030)2802 2363 y Fr(=)2907 2359 y Fn(M)2995 2323 y Fd(\002)2995 2384 y Fh(n)3060 2359 y Fr(\()p Fm(M)3202 2323 y Fd(\002)3192 2384 y Fh(n)3278 2359 y Fl(\012)g Fm(M)3475 2323 y Fd(\002)3465 2384 y Fh(n)3535 2359 y Fr(\).)0 2476 y(Hence)34 b(\001\(\()p Fm(x)502 2491 y Fh(ij)563 2476 y Fr(\)\))28 b(=)f Fm(\023)804 2491 y Ff(1)844 2476 y Fr(\(\()p Fm(x)975 2491 y Fh(ij)1036 2476 y Fr(\)\))22 b Fl(\001)g Fm(\023)1218 2491 y Ff(2)1258 2476 y Fr(\(\()p Fm(x)1389 2491 y Fh(ij)1449 2476 y Fr(\)\))28 b(=)g(\()p Fm(x)1750 2491 y Fh(ij)1810 2476 y Fr(\))23 b Fl(\012)f Fr(\()p Fm(x)2063 2491 y Fh(ij)2124 2476 y Fr(\))33 b(or)1317 2674 y(\001\()p Fm(x)1491 2689 y Fh(ik)1558 2674 y Fr(\))28 b(=)1727 2579 y Fa(X)1783 2789 y Fh(j)1888 2674 y Fm(x)1943 2689 y Fh(ij)2026 2674 y Fl(\012)23 b Fm(x)2181 2689 y Fh(j)t(k)2256 2674 y Fm(:)149 2954 y Fr(The)34 b(counit)e(is)g Fm(")27 b Fr(=)h Fm(e)968 2969 y Fb(K)1052 2954 y Fr(=)f Fm(E)34 b Fl(2)28 b Fn(K)18 b Fr(-)5 b Fo(cAlg)o Fr(\()p Fm(M)1829 2917 y Fd(\002)1819 2978 y Fh(n)1888 2954 y Fm(;)17 b Fn(K)h Fr(\))2075 2926 y Fl(\030)2076 2958 y Fr(=)2180 2954 y Fn(M)2268 2917 y Fd(\002)2268 2978 y Fh(n)2334 2954 y Fr(\()p Fn(K)f 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