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4707 y Fr(M)1937 4671 y Fo(A)2032 4707 y Ft(of)g(\()p Fs(A;)17 b(B)5 b Ft(\))p Fm(-Hopf)39 b(mo)-5 b(dules)38 b Ft(for)g(a)g(righ)m(t)-263 4828 y(coideal)32 b(subalgebra)h Fs(B)38 b Ft(of)33 b(a)f(Hopf)h(algebra)f Fs(A)p Ft(.)45 b(Ob)5 b(jects)35 b(in)d Fr(M)2179 4792 y Fo(A)2179 4852 y(B)2272 4828 y Ft(\(resp)s(ectiv)m(ely)2842 4843 y Fo(B)2903 4828 y Fr(M)3023 4792 y Fo(A)3080 4828 y Ft(\))h(are)g(righ)m(t)-263 4948 y(\(resp)s(ectiv)m(ely)46 b(left\))e Fs(B)5 b Ft(-mo)s(dules)44 b Fs(V)66 b Ft(whic)m(h)45 b(are)g(righ)m(t)f Fs(A)p Ft(-como)s(dules)g(suc)m(h)i(that)f(the)g (como)s(dule)1603 5197 y(4)p eop %%Page: 5 5 5 4 bop -263 614 a Ft(structure)44 b(map)d(\001)472 629 y Fo(V)533 614 y Ft(:)17 b Fs(V)65 b Fr(!)43 b Fs(V)50 b Fr(\012)29 b Fs(A)42 b Ft(is)g(righ)m(t)f(\(resp)s(ectiv)m(ely)i (left\))e Fs(B)5 b Ft(-linear,)42 b(where)i Fs(V)50 b Fr(\012)29 b Fs(A)42 b Ft(is)f(a)-263 735 y(righ)m(t)e(\(resp)s(ectiv)m (ely)i(left\))d Fs(B)5 b Ft(-mo)s(dule)38 b(via)h(\()p Fs(v)31 b Fr(\012)c Fs(a)p Ft(\))p Fs(b)41 b Ft(:=)1935 668 y Fl(P)2039 735 y Fs(v)t(b)2131 750 y Fq(1)2198 735 y Fr(\012)27 b Fs(ab)2394 750 y Fq(2)2474 735 y Ft(\(resp)s(ectiv)m (ely)41 b Fs(b)p Ft(\()p Fs(v)31 b Fr(\012)c Fs(a)p Ft(\))40 b(=)-263 789 y Fl(P)-158 855 y Fs(b)-117 870 y Fq(1)-77 855 y Fs(v)27 b Fr(\012)d Fs(b)139 870 y Fq(2)179 855 y Fs(a)p Ft(\))34 b(for)h(all)d Fs(v)j Fr(2)d Fs(V)21 b Ft(,)35 b Fs(a)d Fr(2)f Fs(A)p Ft(,)36 b(and)e Fs(b)e Fr(2)g Fs(B)5 b Ft(.)49 b(Morphisms)35 b(are)f(righ)m(t)g Fs(A)p Ft(-colinear)f(and)i(righ)m(t)-263 976 y(\(resp)s(ectiv)m(ely)26 b(left\))e Fs(B)5 b Ft(-linear)23 b(maps.)41 b(Note)25 b(that)f Fs(B)33 b Fr(2)28 b(M)1918 939 y Fo(A)1918 1000 y(B)2003 976 y Ft(and)d Fs(B)33 b Fr(2)2386 991 y Fo(B)2447 976 y Fr(M)2567 939 y Fo(A)2648 976 y Ft(where)27 b(the)e(restriction) -263 1096 y(of)32 b(\001:)17 b Fs(A)28 b Fr(!)g Fs(A)22 b Fr(\012)g Fs(A)33 b Ft(is)f(the)h(como)s(dule)e(structure.)-263 1324 y Fj(Theorem)37 b(1.2)49 b Fm([MW)q(,)39 b(2.1])f(L)-5 b(et)39 b Fs(A)f Fm(b)-5 b(e)38 b(a)g(Hopf)g(algebr)-5 b(a)38 b(with)g(bije)-5 b(ctive)38 b(antip)-5 b(o)g(de)37 b(and)h Fs(B)h Fr(\032)c Fs(A)j Fm(a)-263 1445 y(right)k(c)-5 b(oide)g(al)41 b(sub)-5 b(algebr)g(a.)66 b(L)-5 b(et)1028 1419 y Ft(\026)1002 1445 y Fs(A)41 b Ft(:=)g Fs(A=)-5 b(AB)1529 1408 y Fq(+)1629 1445 y Fm(with)42 b(quotient)g(map)g Fs(\031)t Ft(:)17 b Fs(A)40 b Fr(!)2836 1419 y Ft(\026)2810 1445 y Fs(A)p Fm(.)66 b(Then)3266 1419 y Ft(\026)3240 1445 y Fs(A)42 b Fm(is)g(a)-263 1565 y(quotient)35 b(c)-5 b(o)g(algebr)g(a)34 b(and)g(a)h(quotient)g(left)g Fs(A)p Fm(-mo)-5 b(dule)34 b(of)h Fs(A)p Fm(,)g(and)f(the)h(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)-197 1768 y(\(1\))48 b Fs(A)35 b Fm(is)g(faithful)5 b(ly)35 b(c)-5 b(o\015at)35 b(as)f(a)h(left)1271 1743 y Ft(\026)1245 1768 y Fs(A)p Fm(-c)-5 b(omo)g(dule)34 b(via)g Fs(\031)t Fm(,)h(and)f Fs(B)f Ft(=)2457 1732 y Fq(co)2542 1715 y(\026)2523 1732 y Fo(A)2580 1768 y Fs(A)p Fm(.)-197 1972 y(\(2\))48 b Fs(A)35 b Fm(is)g(faithful)5 b(ly)35 b(\015at)g(as)g(a)f(left)h Fs(B)5 b Fm(-mo)-5 b(dule.)-197 2175 y(\(3\))48 b Fs(A)35 b Fm(is)g(pr)-5 b(oje)g(ctive)34 b(as)h(a)f(left)h Fs(B)5 b Fm(-mo)-5 b(dule,)34 b(and)h Fs(B)40 b Fm(is)34 b(a)h(left)g Fs(B)5 b Fm(-dir)-5 b(e)g(ct)34 b(summand)g(in)h Fs(A)p Fm(.)-197 2379 y(\(4\))48 b Fs(A)35 b Fm(is)g(\015at)g(as)g(a)f(left)h Fs(B)5 b Fm(-mo)-5 b(dule,)34 b(and)h Fs(B)40 b Fm(is)34 b(a)h(simple)f(obje)-5 b(ct)34 b(in)h Fr(M)2532 2342 y Fo(A)2532 2403 y(B)2592 2379 y Fm(.)-197 2582 y(\(5\))48 b(The)35 b(functor)f Fr(M)640 2546 y Fo(A)640 2607 y(B)728 2582 y Fr(!)28 b(M)994 2529 y Fq(\026)976 2546 y Fo(A)1032 2582 y Fm(,)35 b Fs(M)k Fr(7!)27 b Fs(M)5 b(=)-5 b(M)10 b(B)1683 2546 y Fq(+)1777 2582 y Fm(is)35 b(an)g(e)-5 b(quivalenc)g(e.)-263 2810 y Fi(Pr)n(oof:)50 b Ft(This)33 b(is)f([MW)q(,)g(2.1])h(when)g (\(3\))f(is)h(replaced)f(b)m(y)-263 2931 y(\(3)-176 2895 y Fp(0)-153 2931 y Ft(\))h Fs(A)f Ft(is)h(a)f(pro)5 b(jectiv)m(e)33 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35 b(b)m(y)e([M1,)g(1.11])f(the)h(mapping)-156 3967 y Fl(\032)-93 4088 y Fs(B)-14 3989 y Fl(\014)-14 4038 y(\014)-14 4088 y(\014)13 4028 y Fs(B)j Fr(\032)30 b Fs(A)k Ft(is)g(a)g(righ)m(t)f (coideal)f(subalge-)13 4148 y(bra,)c Fs(A)f Ft(is)g(left)f(faithfully)e (\015at)j(o)m(v)m(er)h Fs(B)1431 3967 y Fl(\033)1521 4088 y Fr(!)1648 3967 y Fl(\032)1710 4088 y Fs(I)1761 3989 y Fl(\014)1761 4038 y(\014)1761 4088 y(\014)1789 4025 y Fs(I)38 b Fr(\032)30 b Fs(A)k Ft(is)g(a)f(coideal)g(and)h(left)f (ideal)g(and)1789 4146 y Fs(A)f Ft(is)g(left)g(faithfully)e(co\015at)j (o)m(v)m(er)g Fs(A=I)3348 3967 y Fl(\033)-263 4348 y Ft(is)f(a)h(bijection)e(b)s(et)m(w)m(een)j(sub-)f(and)g(quotien)m(t)g (ob)5 b(jects)33 b(of)f Fs(A)h Ft(satisfying)e(the)i(corresp)s(onding)g (condi-)-263 4468 y(tions)f(in)g(1.2.)-263 4696 y Fj(Remark)37 b(1.3)169 b Ft(1.)48 b(If)28 b(w)m(e)i(apply)e(1.2)g(to)g(the)g(dual)g (algebras)f Fs(B)2193 4660 y Fq(op)2300 4696 y Fr(\032)h Fs(A)2478 4660 y Fq(op)2585 4696 y Ft(\()p Fs(A)2696 4660 y Fq(op)2803 4696 y Ft(is)g(a)g(Hopf)g(algebra)-19 4817 y(since)48 b(the)h(an)m(tip)s(o)s(de)d(of)i Fs(A)f Ft(is)h(bijectiv)m(e\),)j(w)m(e)e(get)f(the)g(dual)f(theorem)g(where) 3138 4791 y(\026)3112 4817 y Fs(A)h Ft(is)f(no)m(w)-19 4937 y Fs(A=B)182 4901 y Fq(+)241 4937 y Fs(A)p Ft(,)33 b Fr(M)494 4901 y Fo(A)494 4962 y(B)587 4937 y Ft(is)f(replaced)h(b)m (y)1205 4952 y Fo(B)1266 4937 y Fr(M)1386 4901 y Fo(A)1443 4937 y Ft(,)f(and)h Fs(A)f Ft(is)h(considered)g(as)g(a)f(righ)m(t)g Fs(B)5 b Ft(-mo)s(dule.)1603 5197 y(5)p eop %%Page: 6 6 6 5 bop -143 614 a Ft(2.)48 b(Let)f Fs(A)f Ft(b)s(e)g(a)g(Hopf)h (algebra)e(and)h Fs(B)56 b Fr(\032)c Fs(A)46 b Ft(a)g(righ)m(t)f (coideal)g(subalgebra.)84 b(W)-8 b(e)47 b(note)f(the)-19 735 y(follo)m(wing)26 b(simplicit)m(y)f(criterion:)40 b(If)28 b Fs(B)33 b Ft(is)28 b(a)g(left)f Fs(B)5 b Ft(-direct)28 b(summand)f(in)h Fs(A)g Ft(then)h Fs(B)k Ft(is)27 b(simple)-19 855 y(in)32 b Fr(M)215 819 y Fo(A)215 880 y(B)275 855 y Ft(.)-263 1083 y Fi(Pr)n(oof:)51 b Ft(Let)33 b Fs(f)11 b Ft(:)17 b Fs(A)28 b Fr(!)g Fs(B)38 b Ft(b)s(e)c(a)e(left)g Fs(B)5 b Ft(-linear)32 b(map)g(suc)m(h)i(that)f Fs(f)11 b Fr(j)2256 1098 y Fo(B)2345 1083 y Ft(=)28 b(id)o(.)45 b(Let)33 b Fs(X)j Fr(\032)29 b Fs(B)38 b Ft(b)s(e)c(a)e(non-)-263 1204 y(zero)d(sub)s(ob)5 b(ject)29 b(in)e Fr(M)608 1168 y Fo(A)608 1228 y(B)668 1204 y Ft(.)42 b(Then)29 b Fs(X)8 b(A)27 b Ft(is)h(a)f(non-zero)h(Hopf)g(mo)s(dule)e(in)h Fr(M)2539 1168 y Fo(A)2539 1228 y(A)2596 1204 y Ft(.)42 b(Since)27 b Fr(M)3034 1168 y Fo(A)3034 1228 y(A)3119 1176 y Fr(\030)3120 1208 y Ft(=)3224 1219 y Fo(k)3267 1204 y Fr(M)g Ft(b)m(y)-263 1324 y(the)35 b(fundamen)m(tal)e(theorem)h (of)g(Hopf)g(mo)s(dules)f([Sw)q(,)i(4.1.1],)f Fs(A)g Ft(is)g(simple)f(in)g Fr(M)2841 1288 y Fo(A)2841 1349 y(A)2932 1324 y Ft(and)i Fs(X)8 b(A)27 b Ft(=)h Fs(A)p Ft(.)-263 1445 y(Hence)41 b(there)f(exist)g(\014nitely)e(man)m(y)h (elemen)m(ts)h Fs(x)1595 1460 y Fo(i)1663 1445 y Fr(2)f Fs(X)8 b Ft(,)41 b Fs(a)1976 1460 y Fo(i)2043 1445 y Fr(2)f Fs(A)f Ft(suc)m(h)i(that)2706 1378 y Fl(P)2794 1465 y Fo(i)2838 1445 y Fs(x)2893 1460 y Fo(i)2922 1445 y Fs(a)2973 1460 y Fo(i)3040 1445 y Ft(=)e(1.)64 b(Then)-263 1565 y(1)28 b(=)f Fs(f)11 b Ft(\(1\))27 b(=)232 1499 y Fl(P)320 1586 y Fo(i)364 1565 y Fs(x)419 1580 y Fo(i)448 1565 y Fs(f)11 b Ft(\()p Fs(a)596 1580 y Fo(i)624 1565 y Ft(\))28 b Fr(2)g Fs(X)8 b Ft(,)32 b(and)h Fs(X)i Ft(=)28 b Fs(B)5 b Ft(.)1994 b Fk(2)-263 1735 y Ft(F)-8 b(or)33 b(completeness)h(w)m(e)g(giv)m(e)g(the)f(short)h(pro)s(of)e(of)h(the)h (follo)m(wing)d(imp)s(ortan)m(t)g(observ)-5 b(ation)33 b(of)g(Kop-)-263 1856 y(pinen.)-263 2084 y Fj(Lemma)38 b(1.4)48 b Fm([Ko])38 b(L)-5 b(et)39 b Fs(A)f Fm(b)-5 b(e)38 b(a)f(Hopf)i(algebr)-5 b(a)37 b(with)h(bije)-5 b(ctive)37 b(antip)-5 b(o)g(de)37 b Fs(S)44 b Fm(and)38 b Fs(B)h Fr(\032)34 b Fs(A)k Fm(a)g(right)-263 2204 y(\(r)-5 b(esp.)45 b(left\))34 b(c)-5 b(oide)g(al)34 b(with)h Ft(1)27 b Fr(2)h Fs(B)5 b Fm(.)45 b(Then)34 b Fs(S)6 b Ft(\()p Fs(AB)1594 2168 y Fq(+)1653 2204 y Ft(\))28 b(=)f Fs(B)1901 2168 y Fq(+)1961 2204 y Fs(A)35 b Fm(\(r)-5 b(esp.)44 b Fs(S)6 b Ft(\()p Fs(B)2537 2168 y Fq(+)2596 2204 y Fs(A)p Ft(\))27 b(=)h Fs(AB)2990 2168 y Fq(+)3049 2204 y Fm(\).)-263 2433 y Fi(Pr)n(oof:)74 b Ft(W)-8 b(e)44 b(assume)g(that)f Fs(B)49 b Ft(is)43 b(a)h(righ)m(t)e(coideal)h(\(and)g (then)h(apply)g(the)g(result)f(to)h(the)g(dual)-263 2553 y(coalgebra)39 b(of)f Fs(A)h Ft(to)g(get)g(the)h(lemma)d(for)h(left)h (coideals\).)62 b(Let)39 b Fr(f)p Fs(x)2276 2568 y Fo(j)2352 2553 y Fr(j)f Fs(j)45 b Fr(2)40 b Fs(J)9 b Fr(g)39 b Ft(b)s(e)g(a)g(basis)g(of)g Fs(B)3431 2517 y Fq(+)3490 2553 y Ft(.)-263 2673 y(Then)32 b(for)e(all)f Fs(x)f Fr(2)g Fs(B)5 b Ft(,)31 b(\001\()p Fs(x)p Ft(\))d(=)g(1)18 b Fr(\012)g Fs(x)h Ft(+)1259 2607 y Fl(P)1347 2694 y Fo(j)1400 2673 y Fs(x)1455 2688 y Fo(j)1510 2673 y Fr(\012)g Fs(y)1654 2688 y Fo(j)1720 2673 y Ft(for)30 b(some)h Fs(y)2158 2688 y Fo(j)2221 2673 y Fr(2)e Fs(A)p Ft(.)42 b(No)m(w)32 b(assume)f Fs(x)d Fr(2)g Fs(B)3270 2637 y Fq(+)3360 2673 y Ft(and)-263 2794 y(apply)33 b Fs(\026)21 b Fr(\016)h Ft(\()p Fs(S)28 b Fr(\012)23 b Ft(id)o(\))33 b(and)f Fs(\026)22 b Fr(\016)g Ft(\(id)f Fr(\012)i Fs(S)6 b Ft(\),)32 b(where)i Fs(\026)e Ft(denotes)i(m)m(ultiplication)28 b(in)k Fs(A)p Ft(:)422 3014 y(0)27 b(=)h(1)p Fs(")p Ft(\()p Fs(x)p Ft(\))f(=)g Fs(x)c Ft(+)1134 2931 y Fl(X)1178 3113 y Fo(j)1271 3014 y Fs(S)6 b Ft(\()p Fs(x)1430 3029 y Fo(j)1466 3014 y Ft(\))p Fs(y)1552 3029 y Fo(j)1588 3014 y Fs(;)50 b Ft(and)32 b(0)c(=)f Fs(S)6 b Ft(\()p Fs(x)p Ft(\))22 b(+)2351 2931 y Fl(X)2395 3113 y Fo(j)2488 3014 y Fs(x)2543 3029 y Fo(j)2580 3014 y Fs(S)6 b Ft(\()p Fs(y)2732 3029 y Fo(j)2768 3014 y Ft(\))p Fs(:)-263 3318 y Ft(Therefore,)40 b Fs(B)291 3282 y Fq(+)387 3318 y Fr(\032)c Fs(S)6 b Ft(\()p Fs(B)683 3282 y Fq(+)742 3318 y Ft(\))p Fs(A)38 b Ft(and)g Fs(S)6 b Ft(\()p Fs(B)1269 3282 y Fq(+)1328 3318 y Ft(\))36 b Fr(\032)g Fs(B)1594 3282 y Fq(+)1654 3318 y Fs(A)p Ft(.)58 b(Hence)39 b Fs(B)2186 3282 y Fq(+)2245 3318 y Fs(A)e Ft(=)f Fs(S)6 b Ft(\()p Fs(B)2650 3282 y Fq(+)2709 3318 y Ft(\))p Fs(A)37 b Ft(and)h Fs(S)6 b Ft(\()p Fs(AB)3308 3282 y Fq(+)3367 3318 y Ft(\))36 b(=)-263 3439 y Fs(S)6 b Ft(\()p Fs(B)-80 3403 y Fq(+)-21 3439 y Ft(\))p Fs(A)28 b Ft(=)g Fs(B)301 3403 y Fq(+)360 3439 y Fs(A)p Ft(.)2982 b Fk(2)-263 3717 y Fj(Corollary)36 b(1.5)49 b Fm(L)-5 b(et)37 b Fs(A)g Fm(b)-5 b(e)36 b(a)h(Hopf)f(algebr) -5 b(a)36 b(with)h(bije)-5 b(ctive)35 b(antip)-5 b(o)g(de)36 b(and)g Fs(I)j Fr(\032)32 b Fs(A)k Fm(a)h(c)-5 b(oide)g(al)35 b(and)-263 3837 y(left)g(ide)-5 b(al)34 b(with)h(quotient)g(map)f Fs(\031)t Ft(:)17 b Fs(A)28 b Fr(!)f Fs(A=I)8 b Fm(.)45 b(Assume)34 b Fs(A=I)43 b Fm(is)35 b(c)-5 b(osemisimple,)32 b(and)j(let)1351 4057 y Fs(A=I)g Ft(=)1655 3974 y Fl(M)1656 4158 y Fo(j)t Fp(2)p Fo(J)1797 4057 y Fs(C)1867 4072 y Fo(j)-263 4362 y Fm(b)-5 b(e)35 b(the)g(dir)-5 b(e)g(ct)35 b(sum)f(of)h(the)g(simple)f(sub)-5 b(c)g(o)g(algebr)g(as)33 b Fs(C)1730 4377 y Fo(j)1801 4362 y Fm(for)i Fs(j)f Fr(2)28 b Fs(J)9 b Fm(,)35 b(of)f Fs(A=I)8 b Fm(.)45 b(Then)-197 4565 y(\(1\))j Fs(I)37 b Ft(=)30 b Fs(AB)319 4529 y Fq(+)378 4565 y Fm(,)36 b(and)f Fs(A)h Fm(is)g(left)f(and)h(right)f(faithful)5 b(ly)36 b(\015at)h(over)e Fs(B)41 b Fm(and)35 b(left)h(and)f(right)h (faithful)5 b(ly)-19 4686 y(c)-5 b(o\015at)35 b(over)g Fs(A=I)42 b Fm(via)35 b Fs(\031)t Fm(.)1603 5197 y Ft(6)p eop %%Page: 7 7 7 6 bop -197 614 a Fm(\(2\))48 b(F)-7 b(or)34 b(al)5 b(l)35 b Fs(j)e Fr(2)28 b Fs(J)9 b Fm(,)35 b(let)-19 849 y Fo(j)18 834 y Fs(A)28 b Ft(:=)f Fr(f)p Fs(a)h Fr(2)g Fs(A)g Fr(j)628 751 y Fl(X)765 834 y Fs(\031)t Ft(\()p Fs(a)913 849 y Fq(1)952 834 y Ft(\))19 b Fr(\012)h Fs(a)1157 849 y Fq(2)1224 834 y Fr(2)29 b Fs(C)1389 849 y Fo(j)1444 834 y Fr(\012)20 b Fs(A)p Fr(g)p Fs(;)116 b(A)1880 849 y Fo(j)1945 834 y Ft(:=)27 b Fr(f)p Fs(a)h Fr(2)g Fs(A)g Fr(j)2454 751 y Fl(X)2591 834 y Fs(a)2642 849 y Fq(1)2701 834 y Fr(\012)19 b Fs(\031)t Ft(\()p Fs(a)2945 849 y Fq(2)2985 834 y Ft(\))27 b Fr(2)i Fs(A)19 b Fr(\012)h Fs(C)3404 849 y Fo(j)3440 834 y Fr(g)p Fs(:)-19 1054 y Fm(Then)41 b Fs(A)f Ft(=)470 988 y Fl(L)563 1075 y Fo(j)t Fp(2)p Fo(J)708 1069 y(j)744 1054 y Fs(A)g Ft(=)973 988 y Fl(L)1065 1075 y Fo(j)t Fp(2)p Fo(J)1210 1054 y Fs(S)6 b Ft(\()p Fs(A)1387 1069 y Fo(j)1423 1054 y Ft(\))p Fm(,)43 b(and)e(for)g(al)5 b(l)41 b Fs(j)6 b Fm(,)2157 1069 y Fo(j)2193 1054 y Fs(A)42 b Fm(r)-5 b(esp)g(e)g(ctively)41 b Fs(S)6 b Ft(\()p Fs(A)3001 1069 y Fo(j)3037 1054 y Ft(\))42 b Fm(is)f(a)g(right)-19 1175 y(c)-5 b(oide)g(al)45 b(in)g Fs(A)g Fm(and)g(a)h(\014nitely)f(gener)-5 b(ate)g(d)45 b(and)g(pr)-5 b(oje)g(ctive)45 b(right)g(r)-5 b(esp)g(e)g(ctively)45 b(left)h(mo)-5 b(dule)-19 1295 y(over)33 b Fs(B)5 b Fm(.)44 b(If)33 b Ft(1)27 b Fr(2)h Fs(J)43 b Fm(is)33 b(the)g(distinguishe)-5 b(d)32 b(index)g(with)h Fs(C)2092 1310 y Fq(1)2159 1295 y Ft(=)28 b Fs(k)s(\031)t Ft(\(1\))p Fm(,)33 b(then)g Fs(B)f Ft(=)2989 1310 y Fq(1)3028 1295 y Fs(A)c Ft(=)g Fs(S)6 b Ft(\()p Fs(A)3410 1310 y Fq(1)3449 1295 y Ft(\))p Fm(.)-263 1523 y Fi(Pr)n(oof:)-192 1727 y Ft(\(1\))48 b(By)33 b([Sc)m(h)q(,)g(1.3],)f Fs(A)h Ft(is)f(righ)m(t)g(faithfully)e (co\015at)i(o)m(v)m(er)i Fs(A=I)40 b Ft(if)32 b(and)h(only)f(if)22 1930 y(\(a\))49 b Fs(A)33 b Ft(is)f(righ)m(t)f(co\015at)i(o)m(v)m(er)h Fs(A=I)40 b Ft(and)17 2092 y(\(b\))49 b Fs(\031)36 b Ft(splits)c(as)h(a)f(map)g(of)g(righ)m(t)g Fs(A=I)8 b Ft(-como)s(dules.)-19 2296 y(Since)41 b Fs(A=I)49 b Ft(is)40 b(cosemisimple,)h(an)m(y)g(exact)h(sequence)h(of)d(righ)m(t)g Fs(A=I)8 b Ft(-como)s(dules)39 b(splits.)68 b(In)-19 2416 y(particular,)29 b(\()p Fs(a)p Ft(\))i(and)f(\()p Fs(b)p Ft(\))g(hold.)42 b(Th)m(us)32 b(w)m(e)f(see)h(that)e Fs(A)g Ft(is)g(righ)m(t)f(and)h(b)m(y)h(the)g(same)f(argumen)m(t)-19 2536 y(left)40 b(faithfully)e(co\015at)j(o)m(v)m(er)h Fs(A=I)8 b Ft(.)68 b(Since)41 b Fs(A)f Ft(is)h(righ)m(t)f(faithfully)e (co\015at)j(o)m(v)m(er)g Fs(A=I)8 b Ft(,)43 b(w)m(e)f(get)-19 2657 y(from)30 b(1.1)h(that)f Fs(AB)727 2621 y Fq(+)814 2657 y Ft(=)e Fs(I)39 b Ft(and)31 b Fs(A)g Ft(is)f(righ)m(t)g (faithfully)f(\015at)h(o)m(v)m(er)i Fs(B)5 b Ft(.)43 b(Then)33 b Fs(B)f Fr(\032)d Fs(A)i Ft(is)f(a)h(righ)m(t)-19 2777 y(coideal)26 b(subalgebra,)j Fs(B)k Ft(=)1023 2741 y Fq(co)q Fo(A=)l(AB)1283 2717 y Fh(+)1338 2777 y Fs(A)28 b Ft(and)g Fs(A)g Ft(is)f(left)g(faithfully)e(co\015at)i(o)m(v)m(er)i Fs(A=)-5 b(AB)3132 2741 y Fq(+)3191 2777 y Ft(.)42 b(Hence)-19 2897 y(the)33 b(equiv)-5 b(alen)m(t)33 b(conditions)e(in)h(1.2)g(hold,) g(and)h Fs(A)f Ft(is)h(left)e(faithfully)f(\015at)j(o)m(v)m(er)g Fs(B)5 b Ft(,)33 b(to)s(o.)-192 3101 y(\(2\))48 b(By)30 b(\(1\),)f Fs(B)34 b Ft(is)28 b(a)g(righ)m(t)g(coideal)g(subalgebra)g (of)h Fs(A)p Ft(,)g Fs(A)g Ft(is)f(left)g(and)h(righ)m(t)f(faithfully)e (\015at)j(o)m(v)m(er)h Fs(B)-19 3221 y Ft(and)197 3196 y(\026)171 3221 y Fs(A)e Ft(=)f Fs(A=)-5 b(AB)644 3185 y Fq(+)736 3221 y Ft(is)32 b(cosemisimple.)41 b(Hence)34 b(b)m(y)f(1.2,)433 3450 y Fr(M)553 3409 y Fo(A)553 3475 y(B)641 3450 y Fr(!)28 b(M)907 3392 y Fq(\026)889 3409 y Fo(A)945 3450 y Fs(;)50 b(M)38 b Fr(7!)27 b Fs(M)5 b(=)-5 b(M)10 b(B)1607 3409 y Fq(+)1667 3450 y Fs(;)49 b Ft(and)33 b Fr(M)2071 3392 y Fq(\026)2053 3409 y Fo(A)2137 3450 y Fr(!)27 b(M)2384 3409 y Fo(A)2384 3475 y(B)2445 3450 y Fs(;)49 b(V)g Fr(7!)28 b Fs(V)21 b Fk(2)2926 3454 y Fq(\026)2907 3471 y Fo(A)2965 3450 y Fs(A;)-19 3670 y Ft(are)46 b(quasi-in)m(v)m(erse)h(category)g(equiv)-5 b(alences.)84 b(F)-8 b(or)45 b(an)m(y)i(righ)m(t)2445 3645 y(\026)2419 3670 y Fs(A)p Ft(-como)s(dule)e Fs(V)21 b Ft(,)50 b(the)c(Hopf)-19 3791 y(mo)s(dule)37 b(structure)j(on)e Fs(V)22 b Fk(2)1068 3795 y Fq(\026)1049 3811 y Fo(A)1106 3791 y Fs(A)39 b Ft(is)f(giv)m(en)h(b)m(y)g(m)m(ultiplication)34 b(and)k(com)m(ultiplication)c(on)39 b Fs(A)p Ft(.)-19 3911 y(Since)264 3886 y(\026)238 3911 y Fs(A)31 b Ft(=)448 3844 y Fl(L)541 3932 y Fo(j)594 3911 y Fs(C)664 3926 y Fo(j)734 3911 y Ft(is)j(a)g(decomp)s(osition)f(of)h(righ)m(t)1938 3886 y(\026)1913 3911 y Fs(A)p Ft(-como)s(dules,)2548 3886 y(\026)2522 3911 y Fs(A)p Fk(2)2688 3915 y Fq(\026)2669 3931 y Fo(A)2727 3911 y Fs(A)2831 3883 y Fr(\030)2832 3915 y Ft(=)2939 3844 y Fl(L)3031 3932 y Fo(j)3084 3911 y Fs(C)3154 3926 y Fo(j)3191 3911 y Fk(2)3284 3915 y Fq(\026)3265 3931 y Fo(A)3322 3911 y Fs(A)h Ft(as)-19 4031 y(righ)m(t)22 b Fs(A)p Ft(-como)s(dules)f(and)h(righ)m(t)g Fs(B)5 b Ft(-mo)s(dules.)39 b(Moreo)m(v)m(er,)26 b(for)c(all)e Fs(j)28 b Ft(the)23 b(natural)e(isomorphism)-19 4162 y Fs(A)146 4092 y Fp(\030)146 4113 y Fq(=)93 4162 y Fr(\000)-16 b(!)318 4137 y Ft(\026)293 4162 y Fs(A)p Fk(2)459 4166 y Fq(\026)440 4183 y Fo(A)497 4162 y Fs(A)p Ft(,)41 b Fs(A)e Fr(7!)889 4096 y Fl(P)993 4162 y Fs(\031)t Ft(\()p Fs(a)1141 4177 y Fq(1)1181 4162 y Ft(\))26 b Fr(\012)h Fs(a)1400 4177 y Fq(2)1440 4162 y Ft(,)41 b(maps)1770 4177 y Fo(j)1806 4162 y Fs(A)f Ft(on)m(to)f Fs(C)2215 4177 y Fo(j)2251 4162 y Fk(2)2344 4166 y Fq(\026)2325 4183 y Fo(A)2382 4162 y Fs(A)p Ft(.)64 b(Hence)40 b Fs(A)f Ft(=)3069 4096 y Fl(L)3162 4183 y Fo(j)3215 4177 y(j)3251 4162 y Fs(A)g Ft(is)g(a)-19 4295 y(decomp)s(osition)32 b(in)g Fr(M)859 4259 y Fo(A)859 4320 y(B)920 4295 y Ft(.)45 b(By)34 b(construction,)1733 4310 y Fq(1)1772 4295 y Fs(A)29 b Ft(=)1979 4259 y Fq(co)2064 4242 y(\026)2046 4259 y Fo(A)2103 4295 y Fs(A)g Ft(=)g Fs(B)5 b Ft(.)45 b(All)32 b(the)2791 4310 y Fo(j)2827 4295 y Fs(A)i Ft(are)f(pro)5 b(jectiv)m(e)-19 4416 y(righ)m(t)44 b Fs(B)5 b Ft(-mo)s(dules)43 b(since)i Fs(A)g Ft(is)f(pro)5 b(jectiv)m(e)46 b(as)f(a)f(righ)m(t)g Fs(B)5 b Ft(-mo)s(dule)42 b(b)m(y)k(1.3\(1\).)79 b(They)46 b(are)-19 4536 y(\014nitely)41 b(generated)i(o)m(v)m(er)f Fs(B)47 b Ft(since)1358 4551 y Fo(j)1394 4536 y Fs(A)1511 4508 y Fr(\030)1511 4540 y Ft(=)1631 4536 y Fs(C)1701 4551 y Fo(j)1737 4536 y Fk(2)1830 4540 y Fq(\026)1811 4557 y Fo(A)1869 4536 y Fs(A)41 b Ft(and)h Fs(C)2252 4551 y Fo(j)2330 4536 y Ft(is)f(\014nite)g(dimensional.)68 b(More)-19 4656 y(generally)-8 b(,)36 b(let)g Fs(V)57 b Ft(b)s(e)37 b(an)m(y)f(\014nite)g(dimensional)e(righ)m(t)2061 4631 y(\026)2035 4656 y Fs(A)p Ft(-como)s(dule.)53 b(Then)37 b Fs(M)45 b Ft(=)33 b Fs(V)22 b Fk(2)3304 4660 y Fq(\026)3285 4677 y Fo(A)3342 4656 y Fs(A)36 b Ft(is)-19 4777 y(\014nitely)31 b(generated)h(as)g(a)f(righ)m(t)f Fs(B)5 b Ft(-mo)s(dule.)41 b(T)-8 b(o)32 b(see)g(this)f(write)g Fs(M)42 b Ft(as)32 b(the)g(ascending)f(union)-19 4897 y(of)37 b(all)e(Hopf)j(submo)s (dules)f Fs(X)8 b(B)42 b Ft(where)d Fs(X)45 b Ft(is)36 b(a)i(\014nite)f(dimensional)d(righ)m(t)j Fs(A)p Ft(-sub)s(como)s(dule) 1603 5197 y(7)p eop %%Page: 8 8 8 7 bop -19 635 a Ft(of)33 b Fs(M)10 b Ft(.)45 b(Let)33 b Fs(F)14 b Ft(:)j Fr(M)685 599 y Fo(A)685 659 y(D)830 565 y Fp(\030)830 586 y Fq(=)777 635 y Fr(\000)-16 b(!)28 b(M)1104 582 y Fq(\026)1086 599 y Fo(A)1175 635 y Ft(b)s(e)33 b(the)h(category)f(equiv)-5 b(alence)33 b(of)g(1.2.)44 b(Then)34 b Fs(F)14 b Ft(\()p Fs(M)c Ft(\))3234 607 y Fr(\030)3235 639 y Ft(=)3340 635 y Fs(V)54 b Ft(is)-19 755 y(the)35 b(ascending)g(union)e(of)h(all)e Fs(F)14 b Ft(\()p Fs(X)8 b(B)d Ft(\).)48 b(Since)34 b Fs(V)56 b Ft(is)34 b(\014nite)g(dimensional,)e Fs(F)14 b Ft(\()p Fs(M)c Ft(\))31 b(=)f Fs(F)14 b Ft(\()p Fs(X)8 b(B)d Ft(\))-19 875 y(for)32 b(some)h Fs(X)8 b Ft(,)32 b(hence)i Fs(M)39 b Ft(=)27 b Fs(X)8 b(B)38 b Ft(is)32 b Fs(B)5 b Ft(-\014nitely)31 b(generated.)-19 1036 y(T)-8 b(o)41 b(get)g(the)g(decomp)s(osition)e(of)h(left)g Fs(B)5 b Ft(-mo)s(dules)39 b(w)m(e)j(apply)e(the)h(previous)g(result)g(to)f Fs(A)3411 1000 y Fq(op)3490 1036 y Ft(.)-19 1157 y(Then)30 b Fs(B)311 1120 y Fq(op)418 1157 y Fr(\032)e Fs(A)596 1120 y Fq(op)703 1157 y Ft(is)h(a)f(righ)m(t)h(coideal)e(subalgebra)i (and)g Fs(A)2173 1120 y Fq(op)2280 1157 y Ft(is)g(left)f(and)h(righ)m (t)f(faithfully)e(\015at)-19 1277 y(o)m(v)m(er)34 b Fs(B)269 1241 y Fq(op)348 1277 y Ft(.)43 b(By)33 b(Koppinen's)g(lemma)d(1.4,)j Fs(S)6 b Ft(\()p Fs(AB)1836 1241 y Fq(+)1895 1277 y Ft(\))27 b(=)h Fs(B)2143 1241 y Fq(+)2202 1277 y Fs(A)33 b Ft(and)1008 1505 y Fs(\033)t Ft(:)17 b Fs(A=)-5 b(AB)1380 1464 y Fq(+)1519 1435 y Fp(\030)1519 1456 y Fq(=)1466 1505 y Fr(\000)-16 b(!)27 b Fs(A=B)1855 1464 y Fq(+)1915 1505 y Fs(A;)50 b Ft(\026)-50 b Fs(a)28 b Fr(7!)p 2270 1420 193 4 v 27 w Fs(S)6 b Ft(\()p Fs(a)p Ft(\))p Fs(;)-19 1717 y Ft(is)38 b(a)f(coalgebra)g(an)m(tiisomorphism.)57 b(Therefore)39 b Fs(A=B)2048 1681 y Fq(+)2107 1717 y Fs(A)e Ft(=)2330 1651 y Fl(L)2422 1738 y Fo(j)2475 1717 y Fs(\033)t Ft(\()p Fs(C)2642 1732 y Fo(j)2679 1717 y Ft(\))g(is)h(a)g(direct)f(sum)h(of)-19 1837 y(simple)24 b(sub)s(coalgebras)i(and)f Fs(A)1148 1801 y Fq(op)1227 1837 y Fs(=)-5 b(A)1344 1801 y Fq(op)1422 1837 y Ft(\()p Fs(B)1539 1801 y Fq(op)1618 1837 y Ft(\))1656 1801 y Fq(+)1743 1837 y Ft(=)27 b Fs(A=B)2047 1801 y Fq(+)2106 1837 y Fs(A)f Ft(is)f(cosemisimple.)39 b(Th)m(us)27 b(w)m(e)g(kno)m(w) -19 1958 y(from)32 b(the)h(previous)g(pro)s(of)e(that)i Fs(A)27 b Ft(=)1438 1891 y Fl(L)1530 1979 y Fo(j)1583 1973 y(j)1646 1933 y Ft(~)1620 1958 y Fs(A)33 b Ft(where)g(for)f(all)f Fs(j)6 b Ft(,)835 2198 y Fo(j)897 2157 y Ft(~)871 2183 y Fs(A)28 b Ft(:=)g Fr(f)p Fs(a)f Fr(2)h Fs(A)g Fr(j)1482 2100 y Fl(X)1624 2176 y(e)1618 2183 y Fs(\031)t Ft(\()p Fs(a)1766 2198 y Fq(1)1806 2183 y Ft(\))22 b Fr(\012)g Fs(a)2016 2198 y Fq(2)2084 2183 y Fr(2)28 b Fs(\033)t Ft(\()p Fs(C)2345 2198 y Fo(j)2381 2183 y Ft(\))22 b Fr(\012)h Fs(A)p Fr(g)-19 2415 y Ft(is)39 b(a)g(\014nitely)g(generated) h(pro)5 b(jectiv)m(e)40 b(left)f Fs(B)5 b Ft(-mo)s(dule)37 b(and)j(a)f(righ)m(t)g(coideal,)g(and)3144 2430 y Fq(1)3209 2390 y Ft(~)3183 2415 y Fs(A)h Ft(=)f Fs(B)5 b Ft(.)-19 2536 y(Here)220 2529 y Fl(e)213 2536 y Fs(\031)t Ft(:)17 b Fs(A)31 b Fr(!)f Fs(A=B)751 2500 y Fq(+)810 2536 y Fs(A)35 b Ft(is)f(the)h(canonical)e(map.)48 b(Finally)-8 b(,)32 b(for)i(all)e Fs(j)6 b Ft(,)2627 2551 y Fo(j)2689 2511 y Ft(~)2663 2536 y Fs(A)31 b Ft(=)g Fs(S)6 b Ft(\()p Fs(A)3051 2551 y Fo(j)3087 2536 y Ft(\))35 b(since)f(for)-19 2656 y(all)d Fs(a)d Fr(2)g Fs(A)p Ft(,)699 2868 y Fs(a)g Fr(2)872 2883 y Fo(j)934 2843 y Ft(~)908 2868 y Fs(A)83 b Fr(\()-17 b(\))1330 2785 y Fl(X)1473 2861 y(e)1467 2868 y Fs(\031)t Ft(\()p Fs(a)1615 2883 y Fq(1)1654 2868 y Ft(\))22 b Fr(\012)h Fs(a)1865 2883 y Fq(2)1932 2868 y Fr(2)28 b Fs(\033)t Ft(\()p Fs(C)2193 2883 y Fo(j)2229 2868 y Ft(\))23 b Fr(\012)f Fs(A)1064 3034 y Fr(\()-17 b(\))1330 2951 y Fl(X)1467 3034 y Fs(\031)t Ft(\()p Fs(S)1630 2993 y Fp(\000)p Fq(1)1724 3034 y Ft(\()p Fs(a)1813 3049 y Fq(1)1852 3034 y Ft(\)\))22 b Fr(\012)h Fs(S)2116 2993 y Fp(\000)p Fq(1)2210 3034 y Ft(\()p Fs(a)2299 3049 y Fq(2)2338 3034 y Ft(\))28 b Fr(2)g Fs(C)2568 3049 y Fo(j)2627 3034 y Fr(\012)22 b Fs(A)1064 3193 y Fr(\()-17 b(\))83 b Fs(S)1396 3152 y Fp(\000)p Fq(1)1490 3193 y Ft(\()p Fs(a)p Ft(\))28 b Fr(2)g Fs(A)1812 3208 y Fo(j)1849 3193 y Fs(:)3442 3445 y Fk(2)-263 3824 y Fu(2)166 b(A)36 b(class)i(of)g (homogeneous)e(spaces)h(de\014ned)d(b)-5 b(y)37 b(in\014nites-)-15 4007 y(imal)56 b(in)-5 b(v)c(arian)k(ts)-263 4226 y Ft(W)d(e)32 b(\014rst)g(collect)e(some)i(w)m(ell-kno)m(wn)f(results)h(and)f (notations)g(on)g(dualit)m(y)g(\(cf.)43 b([M,)32 b(Chapter)g(9],)g([J,) -263 4346 y(I.1.4],)k([T2,)g(section)f(1]\).)51 b(Let)35 b Fs(U)45 b Ft(b)s(e)36 b(an)f(algebra.)49 b(The)36 b Fm(dual)h(c)-5 b(o)g(algebr)g(a)34 b Fs(U)2599 4310 y Fq(0)2671 4346 y Fr(\032)e Fs(U)2856 4310 y Fp(\003)2931 4346 y Ft(is)j(spanned)h(b)m(y)-263 4467 y(the)h(matrix)e(co)s (e\016cien)m(ts)j(of)e(all)f(\014nite)h(dimensional)e(left)i Fs(U)10 b Ft(-mo)s(dules)72 b Fs(V)22 b Ft(.)55 b(If)36 b Fs(\032)p Ft(:)17 b Fs(U)46 b Fr(!)34 b Ft(End\()p Fs(V)22 b Ft(\))36 b(is)-263 4587 y(the)30 b(represen)m(tation)g(of)f Fs(U)10 b Ft(,)30 b Fs(C)854 4551 y Fo(V)945 4587 y Ft(denotes)g(the)g (image)d(of)i(the)h(dual)e(coalgebra)g(\(End)q(\()p Fs(V)21 b Ft(\)\))3058 4551 y Fp(\003)3127 4587 y Ft(under)30 b Fs(\032)3450 4551 y Fp(\003)3490 4587 y Ft(.)-263 4707 y(Th)m(us)39 b Fs(C)66 4671 y Fo(V)163 4707 y Ft(is)d(the)i Fs(k)s 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Fs(f)38 b Fr(2)28 b Fs(V)2474 4912 y Fp(\003)2513 4948 y Ft(,)i Fs(v)i Fr(2)c Fs(V)21 b Ft(.)43 b(Then)30 b Fs(U)3218 4912 y Fq(0)3287 4948 y Ft(is)e(the)1603 5197 y(8)p eop %%Page: 9 9 9 8 bop -263 614 a Ft(sum)36 b(of)f(all)e(the)j(sub)s(coalgebras)g Fs(C)1072 578 y Fo(V)1133 614 y Ft(,)g(and)f Fs(C)1465 578 y Fo(V)1506 587 y Fh(1)1541 578 y Fp(\010)p Fo(V)1637 587 y Fh(2)1708 614 y Ft(=)e Fs(C)1894 578 y Fo(V)1935 587 y Fh(1)1998 614 y Ft(+)24 b Fs(C)2175 578 y Fo(V)2216 587 y Fh(2)2290 614 y Ft(for)35 b(\014nite)g(dimensional)e(left)i Fs(U)10 b Ft(-)-263 735 y(mo)s(dules)32 b Fs(V)177 750 y Fq(1)216 735 y Fs(;)17 b(V)317 750 y Fq(2)356 735 y Ft(.)43 b(The)33 b(natural)f(\()p Fs(U;)17 b(U)10 b Ft(\)-bimo)s(dule) 29 b(structure)34 b(on)e Fs(U)2316 699 y Fp(\003)2388 735 y Ft(and)g(on)g(all)f Fs(C)2925 699 y Fo(V)2985 735 y Ft('s)i(is)f(denoted)-263 855 y(b)m(y)h Fs(x)22 b Fr(\001)f Fs(a)33 b Ft(and)f Fs(a)22 b Fr(\001)f Fs(x)p Ft(,)32 b(for)g(all)e Fs(x)e Fr(2)g Fs(U)10 b Ft(,)33 b Fs(a)28 b Fr(2)g Fs(U)1353 819 y Fp(\003)1393 855 y Ft(,)33 b(where)g(\()p Fs(x)22 b Fr(\001)f Fs(a)p Ft(\)\()p Fs(u)p Ft(\))27 b(:=)h Fs(a)p Ft(\()p Fs(ux)p Ft(\))k(and)g(\()p Fs(a)22 b Fr(\001)f Fs(x)p Ft(\)\()p Fs(u)p Ft(\))27 b(:=)h Fs(a)p Ft(\()p Fs(xu)p Ft(\))-263 976 y(for)37 b(all)d Fs(u)g Fr(2)i Fs(U)10 b Ft(.)56 b(Note)37 b(that)g(the)g(dual)f(algebra)f Fs(U)1651 939 y Fp(\003)1728 976 y Ft(is)h(a)h(left)e(and)i(righ)m(t)f Fs(U)10 b Ft(-mo)s(dule)36 b(algebra)f(with)-263 1096 y(resp)s(ect)f(to)e(these)i(actions,)f(since)g(for)f(all)e Fs(a;)17 b(b)28 b Fr(2)g Fs(U)1656 1060 y Fp(\003)1729 1096 y Ft(and)k Fs(x)d Fr(2)f Fs(U)10 b Ft(,)450 1316 y Fs(x)23 b Fr(\001)f Ft(\()p Fs(ab)p Ft(\))28 b(=)877 1233 y Fl(X)997 1316 y Ft(\()p Fs(x)1090 1331 y Fq(1)1152 1316 y Fr(\001)22 b Fs(a)p Ft(\)\()p Fs(x)1384 1331 y Fq(2)1446 1316 y Fr(\001)g Fs(b)p Ft(\))p Fs(;)50 b Ft(\()p Fs(ab)p Ft(\))22 b Fr(\001)g Fs(x)28 b Ft(=)2079 1233 y Fl(X)2199 1316 y Ft(\()p Fs(a)22 b Fr(\001)g Fs(x)2415 1331 y Fq(1)2455 1316 y Ft(\)\()p Fs(b)g Fr(\001)g Fs(x)2699 1331 y Fq(2)2739 1316 y Ft(\))p Fs(:)-263 1536 y Ft(No)m(w)32 b(let)f Fs(U)41 b Ft(b)s(e)32 b(a)f(Hopf)g(algebra.)41 b(A)32 b Fm(tensor)h(c)-5 b(ate)g(gory)31 b Fr(C)37 b Ft(of)31 b(\014nite)g(dimensional)e(left)h Fs(U)10 b Ft(-mo)s(dules)30 b(is)-263 1656 y(a)j(class)f Fr(C)39 b Ft(of)32 b(\014nite)h(dimensional)d(left)h Fs(U)10 b Ft(-mo)s(dules)32 b(suc)m(h)i(that)-19 1860 y Fs(k)d Fr(2)d(C)39 b Ft(\()p Fs(k)c Ft(as)e(the)g(trivial)d Fs(U)10 b Ft(-mo)s(dule)31 b(via)h Fs(")p Ft(\),)-19 2063 y(if)j Fs(X)r(;)17 b(Y)55 b Fr(2)34 b(C)6 b Ft(,)37 b(then)f Fs(X)d Fr(\010)25 b Fs(Y)54 b Fr(2)34 b(C)42 b Ft(and)36 b Fs(X)d Fr(\012)24 b Fs(Y)55 b Fr(2)34 b(C)42 b Ft(\(with)36 b(diagonal)d Fs(U)10 b Ft(-action)36 b(on)f Fs(X)e Fr(\012)25 b Fs(Y)c Ft(,)-19 2183 y Fs(u)p Ft(\()p Fs(x)27 b Fr(\012)h Fs(y)t Ft(\))39 b(:=)534 2117 y Fl(P)638 2183 y Fs(u)694 2198 y Fq(1)733 2183 y Fs(x)27 b Fr(\012)h Fs(u)976 2198 y Fq(2)1015 2183 y Fs(y)42 b Ft(for)e(all)d Fs(u)j Fr(2)g Fs(U)10 b Ft(,)42 b Fs(x)f Fr(2)f Fs(X)47 b Ft(and)40 b Fs(y)j Fr(2)e Fs(Y)21 b Ft(\),)41 b(and)f Fs(X)2946 2147 y Fp(\003)3025 2183 y Fr(2)h(C)46 b Ft(\(where)-19 2304 y(\()p Fs(uf)11 b Ft(\)\()p Fs(v)t Ft(\))48 b(:=)g Fs(f)11 b Ft(\()p Fs(S)6 b Ft(\()p Fs(u)p Ft(\))p Fs(v)t Ft(\))44 b(for)g(all)f Fs(u)48 b Fr(2)h Fs(U)10 b Ft(,)49 b Fs(f)59 b Fr(2)49 b Fs(X)1917 2268 y Fp(\003)1956 2304 y Ft(,)f(and)d Fs(v)53 b Fr(2)c Fs(X)8 b Ft(,)48 b(where)e Fs(S)k Ft(denotes)c(the)-19 2424 y(an)m(tip)s(o)s(de)32 b(of)g Fs(U)10 b Ft(\).)-263 2628 y(By)34 b(de\014nition,)f(the)h(dual) e(Hopf)i(algebra)e(with)h(resp)s(ect)i(to)e Fr(C)6 b Ft(,)34 b Fs(U)2194 2591 y Fq(0)2184 2652 y Fp(C)2263 2628 y Fr(\032)29 b Fs(U)2445 2591 y Fq(0)2515 2628 y Fr(\032)g Fs(U)2697 2591 y Fp(\003)2737 2628 y Ft(,)34 b(is)f(spanned)i(b)m(y)f(all)-263 2748 y(matrix)d(co)s(e\016cien)m(ts)j (of)e(all)f Fs(V)49 b Fr(2)28 b(C)6 b Ft(.)44 b(Th)m(us)1327 2968 y Fs(U)1403 2927 y Fq(0)1393 2993 y Fp(C)1471 2968 y Ft(:=)1614 2885 y Fl(X)1601 3069 y Fo(V)16 b Fp(2C)1763 2968 y Fs(C)1840 2927 y Fo(V)1900 2968 y Fs(:)-263 3258 y Ft(A)40 b(tensor)h(category)f Fr(C)46 b Ft(is)40 b(called)e Fm(semisimple)h Ft(if)f(all)g Fs(V)62 b Fr(2)41 b(C)46 b Ft(are)40 b(isomorphic)e(to)h(direct)h(sums)g(of)-263 3379 y(simple)34 b(mo)s(dules)f(in)i Fr(C)6 b Ft(.)50 b(If)35 b Fr(C)41 b Ft(is)34 b(semisimple,)f(then)j Fs(U)1798 3343 y Fq(0)1788 3403 y Fp(C)1872 3379 y Ft(is)f(cosemisimple,)e(hence) j(the)f(an)m(tip)s(o)s(de)f(of)-263 3499 y Fs(U)-187 3463 y Fq(0)-197 3524 y Fp(C)-118 3499 y Ft(is)28 b(bijectiv)m(e.)42 b(In)30 b(general,)f(if)e(the)i(an)m(tip)s(o)s(de)f(of)g Fs(U)40 b Ft(is)28 b(bijectiv)m(e)h(and)f(for)g(all)f Fs(X)36 b Fr(2)28 b(C)35 b Ft(also)27 b Fs(X)3259 3463 y Fp(\003)3327 3499 y Ft(with)-263 3620 y Fs(U)10 b Ft(-action)32 b(giv)m(en)g(b)m(y)h Fs(S)594 3583 y Fp(\000)p Fq(1)720 3620 y Ft(\(that)f(is)f Fr(\003)p Fs(u)21 b Fr(\001)f Fs(f)11 b Ft(\)\()p Fs(v)t Ft(\))27 b(:=)h Fs(f)11 b Ft(\()p Fs(S)1786 3583 y Fp(\000)p Fq(1)1880 3620 y Ft(\()p Fs(u)p Ft(\))p Fs(v)t Ft(\)\))31 b(is)g(in)g Fr(C)6 b Ft(,)33 b(then)g(the)f(an)m(tip)s(o)s(de)f(of)h Fs(U)3477 3583 y Fq(0)3467 3644 y Fp(C)-263 3740 y Ft(is)g(bijectiv)m(e.)-263 3968 y Fj(Remark)37 b(2.1)49 b Ft(Let)41 b Fs(V)62 b Ft(b)s(e)41 b(a)g(\014nite)f(dimensional)e(left)i Fs(U)10 b Ft(-mo)s(dule)39 b(with)i(represen)m(tation)g Fs(\032)p Ft(:)17 b Fs(U)52 b Fr(!)-263 4089 y Ft(End)q(\()p Fs(V)22 b Ft(\).)-143 4292 y(1.)48 b Fs(V)55 b Ft(is)33 b(semisimple)e(as)i(a)g Fs(U)10 b Ft(-mo)s(dule)32 b(if)g(and)h(only)g(if)f Fs(C)2005 4256 y Fo(V)2099 4292 y Ft(is)h(a)g(cosemisimple)e(coalgebra.)44 b(If)33 b Fs(V)-19 4412 y Ft(is)f(simple,)f(then)j Fs(C)710 4376 y Fo(V)803 4412 y Ft(is)e(a)g(simple)f(coalgebra.)-143 4616 y(2.)48 b Fs(V)36 b Fr(\012)15 b Fs(V)245 4580 y Fp(\003)312 4616 y Fr(!)27 b Fs(C)516 4580 y Fo(V)577 4616 y Ft(,)j Fs(v)18 b Fr(\012)d Fs(f)39 b Fr(7!)27 b Fs(c)1047 4631 y Fo(f)t(;v)1176 4616 y Ft(is)i(a)f(map)g(of)h(\()p Fs(U;)17 b(U)10 b Ft(\)-bimo)s(dules.)40 b(Here,)30 b Fs(V)36 b Fr(\012)15 b Fs(V)2982 4580 y Fp(\003)3050 4616 y Ft(is)29 b(a)f(\()p Fs(U;)17 b(U)10 b Ft(\)-)-19 4736 y(bimo)s(dule)32 b(via)h Fs(u)p Ft(\()p Fs(v)26 b Fr(\012)e Fs(f)11 b Ft(\))29 b(=)h Fs(uv)c Fr(\012)d Fs(f)45 b Ft(and)34 b(\()p Fs(v)26 b Fr(\012)e Fs(f)11 b Ft(\))p Fs(u)29 b Ft(:=)g Fs(v)e Fr(\012)c Fs(f)11 b(u)p Ft(,)34 b(where)h(\()p Fs(f)11 b(u)p Ft(\)\()p Fs(u)3026 4700 y Fp(0)3048 4736 y Ft(\))29 b(:=)h Fs(f)11 b Ft(\()p Fs(uu)3457 4700 y Fp(0)3479 4736 y Ft(\))-19 4857 y(for)32 b(all)f Fs(u;)17 b(u)422 4820 y Fp(0)471 4857 y Fr(2)29 b Fs(U)10 b Ft(,)33 b Fs(v)e Fr(2)d Fs(V)22 b Ft(,)33 b(and)f Fs(f)39 b Fr(2)28 b Fs(V)1461 4820 y Fp(\003)1501 4857 y Ft(.)1603 5197 y(9)p eop %%Page: 10 10 10 9 bop -143 614 a Ft(3.)48 b(If)40 b Fs(V)60 b Ft(is)39 b(simple)f(and)i Fs(k)i Ft(is)d(algebraically)d(closed,)41 b(then)f Fs(V)49 b Fr(\012)27 b Fs(V)2428 578 y Fp(\003)2506 614 y Fr(!)39 b Fs(C)2722 578 y Fo(V)2783 614 y Ft(,)i Fs(v)31 b Fr(\012)c Fs(f)50 b Fr(7!)39 b Fs(c)3312 629 y Fo(f)t(;v)3451 614 y Ft(is)-19 735 y(bijectiv)m(e.)-143 933 y(4.)48 b(Assume)39 b Fr(C)k Ft(is)37 b(semisimple.)57 b(Let)37 b Fr(E)46 b Ft(b)s(e)38 b(a)f(complete)g(set)h(of)f(represen)m (tativ)m(es)j(of)d(the)h(simple)-19 1054 y(mo)s(dules)32 b(in)g Fr(C)6 b Ft(.)43 b(Then)34 b(\(P)m(eter-W)-8 b(eyl)33 b(decomp)s(osition\))1461 1257 y Fs(U)1537 1216 y Fq(0)1527 1282 y Fp(C)1605 1257 y Ft(=)1719 1174 y Fl(M)1709 1358 y Fo(V)15 b Fp(2E)1872 1257 y Fs(C)1949 1216 y Fo(V)2010 1257 y Fs(:)-263 1569 y Fi(Pr)n(oof:)-143 1758 y Ft(1.)48 b(If)37 b Fs(V)59 b Ft(is)37 b(a)g(semisimple)d Fs(U)10 b Ft(-mo)s(dule,)38 b(then)f Fs(V)59 b Ft(is)37 b(a)g(faithful)e(and)i (semisimple)e Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))q(-mo)s(dule,)-19 1878 y(hence)37 b Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))36 b(is)e(semisimple)f(\(since)i(the)h(radical)d(of)i Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))36 b(annihilates)d Fs(V)21 b Ft(\).)51 b(If)35 b Fs(V)57 b Ft(is)34 b(simple,)-19 1998 y(the)39 b(\014nite)f(dimensional)e(semisimple)g(algebra)h Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))39 b(is)f(simple)f(\(this)h(follo)m (ws)f(for)h(instance)-19 2119 y(from)31 b(the)h(theorem)g(of)f(Artin-W) -8 b(edderburn\).)44 b(Con)m(v)m(ersely)-8 b(,)34 b(if)c Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))33 b(is)e(semisimple,)f(then)i Fs(V)-19 2239 y Ft(is)e(semisimple)d(o)m(v)m(er)k Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))30 b(and)g Fs(U)10 b Ft(.)43 b(This)30 b(pro)m(v)m(es)i(the)e(claim)d(b)m(y)k(dualit)m(y)e(since)h Fs(C)3054 2203 y Fo(V)3142 2212 y Fr(\030)3143 2243 y Ft(=)3248 2239 y Fs(\032)p Ft(\()p Fs(U)10 b Ft(\))3450 2191 y Fp(\003)3490 2239 y Ft(.)-143 2438 y(2.)48 b(is)32 b(clear.)-143 2636 y(3.)48 b(By)33 b(the)g(densit)m(y)h(theorem,)e Fs(\032)h Ft(is)f(on)m(to.)-143 2835 y(4.)48 b([T2)q(,)32 b(1.4].)3442 3024 y Fk(2)-263 3190 y Ft(Let)j Fr(C)40 b Ft(b)s(e)35 b(a)f(tensor)h(category.)49 b(A)34 b(subalgebra)g Fs(K)k Fr(\032)31 b Fs(U)45 b Ft(is)33 b(called)g Fr(C)6 b Fm(-semisimple)33 b Ft(if)g(all)f Fs(V)52 b Fr(2)31 b(C)41 b Ft(are)-263 3310 y(semisimple)31 b(as)h(left)g Fs(K)7 b Ft(-mo)s(dules)31 b(\(b)m(y)j(restriction\).)-263 3519 y Fj(Theorem)j(2.2)49 b Fm(L)-5 b(et)38 b Fs(U)47 b Fm(b)-5 b(e)37 b(a)f(Hopf)h(algebr)-5 b(a,)36 b Fs(K)j Fr(\032)32 b Fs(U)47 b Fm(a)37 b(left)f(c)-5 b(oide)g(al)36 b(sub)-5 b(algebr)g(a)36 b(and)g Fr(C)44 b Fm(a)36 b(tensor)-263 3640 y(c)-5 b(ate)g(gory)36 b(of)f(\014nite)h(dimensional)d(left)j Fs(U)10 b Fm(-mo)-5 b(dules.)47 b(L)-5 b(et)36 b Fs(A)29 b Ft(:=)g Fs(U)2201 3603 y Fq(0)2191 3664 y Fp(C)2276 3640 y Fm(b)-5 b(e)36 b(the)f(dual)h(Hopf)f(algebr)-5 b(a)35 b(with)-263 3760 y(r)-5 b(esp)g(e)g(ct)31 b(to)h Fr(C)6 b Fm(,)32 b Fs(B)h Ft(:=)27 b Fr(f)p Fs(a)h Fr(2)g Fs(A)g Fr(j)f Fs(a)14 b Fr(\001)g Fs(K)1098 3724 y Fq(+)1185 3760 y Ft(=)27 b(0)p Fr(g)k Fm(and)1630 3735 y Ft(\026)1604 3760 y Fs(A)d Ft(:=)f Fs(A=)-5 b(AB)2104 3724 y Fq(+)2163 3760 y Fm(.)44 b(Assume)31 b(that)g(the)g(antip)-5 b(o)g(de)31 b(of)g Fs(A)-263 3880 y Fm(is)k(bije)-5 b(ctive.)44 b(Then)-197 4069 y(\(1\))k Fs(B)33 b Fr(\032)28 b Fs(A)35 b Fm(is)g(a)g(right)f(c) -5 b(oide)g(al)34 b(sub)-5 b(algebr)g(a)34 b(with)h Fs(B)e Ft(=)1926 4033 y Fq(co)2012 4016 y(\026)1993 4033 y Fo(A)2050 4069 y Fs(A)p Fm(.)-197 4268 y(\(2\))48 b(If)38 b Fs(K)44 b Fm(is)38 b Fr(C)6 b Fm(-semisimple,)37 b(then)1192 4243 y Ft(\026)1166 4268 y Fs(A)h Fm(is)g(c)-5 b(osemisimple)36 b(and)h Fs(A)h Fm(is)f(faithful)5 b(ly)38 b(\015at)h(as)e(a)h(left)f (and)-19 4388 y(right)43 b Fs(B)5 b Fm(-mo)-5 b(dule.)67 b(Mor)-5 b(e)42 b(pr)-5 b(e)g(cisely,)44 b(ac)-5 b(c)g(or)g(ding)41 b(to)i(1.5,)g Fs(A)f Ft(=)2420 4322 y Fl(L)2512 4409 y Fo(j)t Fp(2)p Fo(J)2657 4403 y(j)2694 4388 y Fs(A)g Fm(r)-5 b(esp)g(e)g(ctively)42 b Fs(A)g Ft(=)-19 4442 y Fl(L)74 4529 y Fo(j)t Fp(2)p Fo(J)218 4509 y Fs(S)6 b Ft(\()p Fs(A)395 4524 y Fo(j)432 4509 y Ft(\))33 b Fm(is)f(a)h(dir)-5 b(e)g(ct)33 b(sum)g(of)g(\014nitely)g(gener)-5 b(ate)g(d)32 b(and)g(pr)-5 b(oje)g(ctive)32 b(right)h(r)-5 b(esp)g(e)g(ctively)33 b(left)-19 4629 y Fs(B)5 b Fm(-mo)-5 b(dules)34 b(and)h(of)f(right)h(c)-5 b(oide)g(als)34 b(with)g Fs(B)f Ft(=)1784 4644 y Fq(1)1823 4629 y Fs(A)28 b Ft(=)g Fs(S)6 b Ft(\()p Fs(A)2205 4644 y Fq(1)2244 4629 y Ft(\))p Fm(.)-197 4828 y(\(3\))48 b(If)31 b Fs(K)38 b Fm(is)31 b(a)g(Hopf)g(sub)-5 b(algebr)g(a)30 b(of)h Fs(U)42 b Fm(then)30 b(in)h(\(2\))g(for)g(al)5 b(l)30 b Fs(j)6 b Fm(,)2177 4843 y Fo(j)2214 4828 y Fs(A)31 b Fm(is)g(also)f(\014nitely)h(gener)-5 b(ate)g(d)30 b(and)-19 4948 y(pr)-5 b(oje)g(ctive)34 b(as)h(a)g(left)f Fs(B)5 b Fm(-mo)-5 b(dule.)1578 5197 y Ft(10)p eop %%Page: 11 11 11 10 bop -197 614 a Fm(\(4\))48 b(If)38 b Fs(K)45 b Fm(is)38 b(c)-5 b(ommutative)38 b(and)g Fs(k)j Fm(is)d(algebr)-5 b(aic)g(al)5 b(ly)37 b(close)-5 b(d,)38 b(then)2409 589 y Ft(\026)2383 614 y Fs(A)g Fm(is)g(sp)-5 b(anne)g(d)37 b(by)i(gr)-5 b(oup-like)-19 735 y(elements)34 b(if)h(and)f(only)h(if)g Fs(K)42 b Fm(is)34 b Fr(C)6 b Fm(-semisimple.)-263 963 y Fi(Pr)n(oof:)-192 1166 y Ft(\(1\))48 b(Let)178 1141 y(~)153 1166 y Fs(A)29 b Ft(b)s(e)g(the)h(image)d(of)i Fs(A)g Ft(under)h(the)g(restriction)e(map)g Fs(U)2227 1130 y Fq(0)2295 1166 y Fr(!)f Fs(K)2512 1130 y Fq(0)2581 1166 y Ft(whic)m(h)j(is)e(the)i(coalgebra)-19 1287 y(map)j(dual)g(to)g (the)h(inclusion)e(of)h(algebras)f Fs(K)37 b Fr(\032)29 b Fs(U)10 b Ft(.)47 b(Then)35 b Fs(\031)t Ft(:)17 b Fs(A)29 b Fr(!)2597 1262 y Ft(~)2572 1287 y Fs(A)p Ft(,)34 b Fs(\031)t Ft(\()p Fs(a)p Ft(\)\()p Fs(u)p Ft(\))28 b(:=)h Fs(a)p Ft(\()p Fs(u)p Ft(\))k(for)-19 1407 y(all)23 b Fs(a)28 b Fr(2)g Fs(A)d Ft(and)g Fs(u)i Fr(2)h Fs(U)10 b Ft(,)27 b(is)d(a)h(surjectiv)m(e)h(coalgebra)d(map.)41 b(Moreo)m(v)m(er,)27 b(the)f(k)m(ernel)f(of)f Fs(\031)29 b Ft(is)24 b(a)h(left)-19 1528 y(ideal)34 b(in)g Fs(A)i Ft(since)f(for)g(all)e Fs(a)f Fr(2)h Ft(Ker\()p Fs(\031)t Ft(\),)i Fs(c)d Fr(2)h Fs(A)i Ft(and)g Fs(u)d Fr(2)g Fs(K)7 b Ft(,)36 b(\()p Fs(ca)p Ft(\)\()p Fs(u)p Ft(\))c(=)2762 1461 y Fl(P)2866 1528 y Fs(c)p Ft(\()p Fs(u)3002 1543 y Fq(1)3041 1528 y Ft(\))p Fs(a)p Ft(\()p Fs(u)3224 1543 y Fq(2)3263 1528 y Ft(\))g(=)g(0.)-19 1648 y(Here,)42 b(the)d(last)f(equalit)m(y)g(holds)h(b)s(ecause)h Fs(K)46 b Ft(is)39 b(a)f(left)g(coideal)g(in)g Fs(U)10 b Ft(.)63 b(Th)m(us)40 b Fs(\031)t Ft(:)17 b Fs(A)38 b Fr(!)3365 1623 y Ft(~)3339 1648 y Fs(A)h Ft(is)-19 1768 y(a)f(surjectiv)m(e)i (map)d(of)h(coalgebras)g(and)g(left)g Fs(A)p Ft(-mo)s(dules.)59 b(W)-8 b(e)39 b(\014rst)g(note)f(that)g Fs(B)43 b Ft(=)3293 1732 y Fq(co)3378 1715 y(~)3360 1732 y Fo(A)3417 1768 y Fs(A)p Ft(.)-19 1889 y(F)-8 b(or)41 b(if)f Fs(a)k Fr(2)f Fs(A)p Ft(,)h(then)e Fs(a)g Ft(is)f(left)1247 1864 y(~)1222 1889 y Fs(A)p Ft(-coin)m(v)-5 b(arian)m(t)40 b(if)g(and)i(only)f(if) 2456 1822 y Fl(P)2560 1889 y Fs(a)2611 1904 y Fq(1)2651 1889 y Ft(\()p Fs(x)p Ft(\))p Fs(a)2833 1904 y Fq(2)2873 1889 y Ft(\()p Fs(y)t Ft(\))h(=)h Fs(")p Ft(\()p Fs(x)p Ft(\))p Fs(a)p Ft(\()p Fs(y)t Ft(\))-19 2009 y(for)d(all)f Fs(x)i Fr(2)h Fs(K)7 b Ft(,)43 b Fs(y)h Fr(2)e Fs(U)10 b Ft(.)68 b(By)41 b(de\014nition)e(of)i(the)g(righ)m(t)e Fs(U)10 b Ft(-mo)s(dule)40 b(structure)h(on)g Fs(A)f Ft(\(as)h(a)-19 2130 y(submo)s(dule)29 b(of)h Fs(U)654 2093 y Fp(\003)694 2130 y Ft(\),)g(the)g(last)f(equation)g(is)g(equiv) -5 b(alen)m(t)30 b(to)f Fs(a)16 b Fr(\001)g Fs(K)2404 2093 y Fq(+)2491 2130 y Ft(=)28 b(0)h(or)g Fs(a)f Fr(2)g Fs(B)5 b Ft(.)43 b(Since)30 b Fs(\031)j Ft(is)-19 2260 y(a)d(map)f(of)g(left)g Fs(A)p Ft(-mo)s(dules,)g Fs(B)k Ft(=)1273 2224 y Fq(co)1359 2207 y(~)1340 2224 y Fo(A)1397 2260 y Fs(A)d Ft(is)f(a)h(righ)m(t)f(coideal)f(subalgebra)i(of)f Fs(A)p Ft(,)h(and)g(therefore)-19 2381 y Fs(B)36 b Fr(\032)198 2344 y Fq(co)284 2328 y(\026)265 2344 y Fo(A)322 2381 y Fs(A)p Ft(.)48 b(T)-8 b(o)34 b(see)h(that)f Fs(B)i Ft(=)1203 2344 y Fq(co)1288 2328 y(\026)1270 2344 y Fo(A)1327 2381 y Fs(A)p Ft(,)e(note)g(that)g Fs(AB)2044 2344 y Fq(+)2134 2381 y Fr(\032)d Ft(Ker\()p Fs(\031)t Ft(\))j(since)g Fs(\031)k Ft(is)c(left)f Fs(A)p Ft(-linear)-19 2501 y(and)f(for)e(all)g Fs(b)e Fr(2)g Fs(B)694 2465 y Fq(+)753 2501 y Ft(,)812 2435 y Fl(P)916 2501 y Fs(\031)t Ft(\()p Fs(b)1054 2516 y Fq(1)1094 2501 y Ft(\))19 b Fr(\012)h Fs(b)1289 2516 y Fq(2)1357 2501 y Ft(=)27 b Fs(\031)t Ft(\(1\))20 b Fr(\012)g Fs(b)p Ft(,)32 b(hence)g Fs(\031)t Ft(\()p Fs(b)p Ft(\))c(=)g Fs(\031)t Ft(\(1\))p Fs(")p Ft(\()p Fs(b)p Ft(\))f(=)h(0.)43 b(Th)m(us)32 b Fs(\031)k Ft(can)-19 2632 y(b)s(e)d(factorized)f(as)h Fs(A)28 b Fr(!)f Fs(A=)-5 b(AB)1178 2596 y Fq(+)1264 2632 y Ft(=)1394 2606 y(\026)1368 2632 y Fs(A)28 b Fr(!)1622 2606 y Ft(~)1596 2632 y Fs(A)p Ft(,)33 b(and)1919 2596 y Fq(co)2004 2579 y(\026)1985 2596 y Fo(A)2042 2632 y Fs(A)28 b Fr(\032)2248 2596 y Fq(co)2333 2579 y(~)2315 2596 y Fo(A)2372 2632 y Fs(A)g Ft(=)f Fs(B)5 b Ft(.)-192 2835 y(\(2\))48 b(W)-8 b(e)29 b(no)m(w)g(assume)g(that)f Fs(K)36 b Ft(is)28 b Fr(C)6 b Ft(-semisimple.)39 b(Then)1994 2810 y(~)1969 2835 y Fs(A)28 b Ft(is)g(cosemisimple.)40 b(F)-8 b(or)27 b(b)m(y)j (de\014nition,)7 2930 y(~)-19 2955 y Fs(A)38 b Ft(is)g(the)g(sum)g(of)g (all)e Fs(C)916 2919 y Fo(V)1015 2955 y Ft(restricted)i(to)g Fs(K)7 b Ft(,)40 b(for)d Fs(V)59 b Fr(2)37 b(C)6 b Ft(.)60 b(Since)38 b Fs(K)46 b Ft(is)37 b Fr(C)6 b Ft(-semisimple,)37 b(an)m(y)-19 3076 y Fs(V)66 b Fr(2)44 b(C)49 b Ft(is)41 b Fs(K)7 b Ft(-isomorphic)40 b(to)i(some)g(direct)g(sum)g Fs(X)2014 3091 y Fq(1)2082 3076 y Fr(\010)30 b(\001)17 b(\001)g(\001)27 b(\010)i Fs(X)2521 3091 y Fo(n)2610 3076 y Ft(of)42 b(simple)e Fs(K)7 b Ft(-mo)s(dules)-19 3196 y Fs(X)62 3211 y Fo(i)90 3196 y Ft(.)89 b(Hence)49 b Fs(\031)t Ft(\()p Fs(C)685 3160 y Fo(V)746 3196 y Ft(\))f(is)f (cosemisimple)e(as)j(the)g(sum)g(of)f(the)h(simple)f(sub)s(coalgebras)g Fs(C)3428 3160 y Fo(X)3486 3170 y Fg(i)-19 3317 y Ft(of)33 b Fs(K)183 3280 y Fq(0)223 3317 y Ft(.)47 b(Th)m(us)571 3291 y(~)545 3317 y Fs(A)29 b Fr(\032)h Fs(K)844 3280 y Fq(0)918 3317 y Ft(is)j(a)g(cosemisimple)e(sub)s(coalgebra.)47 b(Since)2592 3291 y(~)2566 3317 y Fs(A)34 b Ft(is)f(cosemisimple)f(and) -19 3437 y Fs(\031)t Ft(:)17 b Fs(A)35 b Fr(!)352 3412 y Ft(~)326 3437 y Fs(A)i Ft(is)f(a)h(surjectiv)m(e)h(map)d(of)i (coalgebras)f(and)h(left)f Fs(A)p Ft(-mo)s(dules,)g(w)m(e)i(conclude)f (from)-19 3557 y(1.5)30 b(that)371 3532 y(~)346 3557 y Fs(A)d Ft(=)h Fs(A=)-5 b(AB)819 3521 y Fq(+)905 3557 y Ft(=)1035 3532 y(\026)1009 3557 y Fs(A)31 b Ft(and)f Fs(A)h Ft(is)f(left)g(and)g(righ)m(t)g(faithfully)e(\015at)i(o)m(v)m (er)i Fs(B)j Ft(and)c(w)m(e)h(ha)m(v)m(e)-19 3678 y(the)h(decomp)s (ositions)e(of)i(1.5.)-192 3881 y(\(3\))48 b(Assume)37 b Fs(K)43 b Ft(is)35 b(a)h(Hopf)f(subalgebra)h(of)f Fs(U)10 b Ft(.)54 b(Then)37 b Fs(U)1996 3845 y Fq(0)2069 3881 y Fr(!)c Fs(K)2292 3845 y Fq(0)2368 3881 y Ft(is)i(a)g(Hopf)h(algebra)f (map)g(and)7 3976 y(\026)-19 4002 y Fs(A)28 b Ft(=)g Fs(A=)-5 b(AB)455 3965 y Fq(+)540 4002 y Ft(is)26 b(a)g(quotien)m(t)h (Hopf)f(algebra)f(of)h Fs(A)p Ft(.)42 b(The)27 b(an)m(tip)s(o)s(de)f (de\014nes)i(a)e(bijection)g(in)f(the)-19 4122 y(set)35 b(of)f(all)e(simple)g(sub)s(coalgebras)i Fs(C)1383 4137 y Fo(j)1454 4122 y Ft(of)1592 4097 y(\026)1566 4122 y Fs(A)p Ft(.)48 b(By)35 b(\(2\))f(it)f(su\016ces)j(to)e(sho)m(w)h(that)f Fs(S)6 b Ft(\()p Fs(C)3211 4137 y Fo(j)3247 4122 y Ft(\))30 b(=)g Fs(C)3491 4137 y Fo(l)-19 4242 y Ft(for)i Fs(j;)17 b(l)30 b Fr(2)f Fs(J)9 b Ft(,)32 b(implies)e Fs(S)6 b Ft(\()925 4257 y Fo(j)962 4242 y Fs(A)p Ft(\))27 b(=)h Fs(A)1277 4257 y Fo(l)1303 4242 y Ft(.)44 b(Indeed,)34 b(for)e(an)m(y)h Fs(a)28 b Fr(2)g Fs(A)p Ft(,)847 4446 y Fs(a)g Fr(2)1020 4461 y Fo(j)1057 4446 y Fs(A)g Fr(\()-17 b(\))1396 4380 y Fl(P)1501 4446 y Ft(\026)-50 b Fs(a)1551 4461 y Fq(1)1613 4446 y Fr(\012)22 b Fs(a)1763 4461 y Fq(2)1831 4446 y Fr(2)28 b Fs(C)1995 4461 y Fo(j)2053 4446 y Fr(\012)23 b Fs(A)1158 4567 y Fr(\()-17 b(\))1396 4500 y Fl(P)1500 4567 y Fs(S)6 b Ft(\()q(\026)-50 b Fs(a)1655 4582 y Fq(1)1694 4567 y Ft(\))22 b Fr(\012)h Fs(S)6 b Ft(\()p Fs(a)2009 4582 y Fq(2)2048 4567 y Ft(\))28 b Fr(2)g Fs(S)6 b Ft(\()p Fs(C)2382 4582 y Fo(j)2418 4567 y Ft(\))22 b Fr(\012)h Fs(A)1158 4687 y Fr(\()-17 b(\))1396 4621 y Fl(P)1500 4687 y Fs(S)6 b Ft(\()p Fs(a)p Ft(\))1693 4702 y Fq(1)1754 4687 y Fr(\012)p 1854 4602 193 4 v 23 w Fs(S)g Ft(\()p Fs(a)p Ft(\))2047 4716 y Fq(2)2114 4687 y Fr(2)28 b Fs(A)22 b Fr(\012)h Fs(C)2473 4702 y Fo(l)1158 4807 y Fr(\()-17 b(\))55 b Fs(S)6 b Ft(\()p Fs(a)p Ft(\))27 b Fr(2)h Fs(A)1783 4822 y Fo(l)1809 4807 y Fs(:)1578 5197 y Ft(11)p eop %%Page: 12 12 12 11 bop -192 623 a Ft(\(4\))48 b(Assume)53 b Fs(K)60 b Ft(is)51 b(comm)m(utativ)m(e.)102 b(Then)53 b Fs(K)1666 586 y Fq(0)1758 623 y Ft(and)1993 597 y(~)1967 623 y Fs(A)f Ft(are)h(co)s(comm)m(utativ)m(e.)101 b(If)52 b Fs(K)59 b Ft(is)52 b Fr(C)6 b Ft(-)-19 743 y(semisimple,)39 b(b)m(y)h(\(2\),)871 718 y(\026)846 743 y Fs(A)f Ft(=)1098 718 y(~)1072 743 y Fs(A)h Ft(is)e(cosemisimple)f(hence)k(spanned)g(b)m (y)f(group-lik)m(e)e(elemen)m(ts)-19 863 y(b)s(ecause)43 b Fs(k)h Ft(is)d(algebraically)d(closed.)71 b(Con)m(v)m(ersely)43 b(assume)f(that)f Fs(A=)-5 b(AB)2818 827 y Fq(+)2918 863 y Ft(is)41 b(spanned)i(b)m(y)-19 984 y(group-lik)m(e)34 b(elemen)m(ts.)52 b(Since)1167 959 y(~)1141 984 y Fs(A)36 b Ft(is)e(a)h(coalgebra)f(quotien)m(t)i(of)f Fs(A=)-5 b(AB)2641 948 y Fq(+)2699 984 y Ft(,)36 b(also)2986 959 y(~)2961 984 y Fs(A)f Ft(is)g(spanned)-19 1104 y(b)m(y)25 b(group-lik)m(e)e(elemen)m(ts)h(and)g(hence)i(cosemisimple.)38 b(By)24 b(de\014nition,)2634 1079 y(~)2608 1104 y Fs(A)g Ft(is)g(the)g(sum)g(of)g(all)d(the)-19 1225 y(dual)30 b(coalgebras)g Fs(\032)716 1240 y Fo(V)777 1225 y Ft(\()p Fs(K)7 b Ft(\),)31 b(where)g Fs(\032)1330 1240 y Fo(V)1391 1225 y Ft(:)17 b Fs(U)39 b Fr(!)27 b Ft(End)q(\()p Fs(V)21 b Ft(\))30 b(is)g(the)h(represen)m(tation)g(corresp)s(onding)-19 1345 y(to)j Fs(V)53 b Fr(2)31 b(C)6 b Ft(.)50 b(Hence)36 b(for)e(all)e Fs(V)53 b Fr(2)31 b(C)6 b Ft(,)36 b Fs(\032)1402 1360 y Fo(V)1463 1345 y Ft(\()p Fs(K)7 b Ft(\))1629 1309 y Fp(\003)1703 1345 y Ft(is)34 b(cosemisimple)e(or)i(equiv)-5 b(alen)m(tly)34 b Fs(\032)3106 1360 y Fo(V)3168 1345 y Ft(\()p Fs(K)7 b Ft(\))34 b(is)g(a)-19 1465 y(semisimple)d(algebra)g (or)h Fs(V)54 b Ft(is)32 b(a)h(semisimple)d(mo)s(dule)h(o)m(v)m(er)j Fs(K)7 b Ft(.)43 b(Th)m(us)34 b Fs(K)40 b Ft(is)32 b Fr(C)6 b Ft(-semisimple.)3442 1644 y Fk(2)-263 1806 y Ft(In)36 b(the)g(situation)d(of)i(2.2\(4\))g(w)m(e)h(no)m(w)g(assume)g (that)f(also)f Fr(C)42 b Ft(is)34 b(semisimple.)50 b(Then)36 b(the)g(decomp)s(o-)-263 1927 y(sitions)h(in)g(2.2)g(can)h(b)s(e)g (describ)s(ed)g(more)f(concretely)-8 b(.)60 b(In)38 b(this)f(case)i(w)m (e)f(will)e(giv)m(e)h(an)h(alternativ)m(e)-263 2047 y(pro)s(of)44 b(not)g(using)h(2.2.)79 b(W)-8 b(e)45 b(will)d(explicitly)g(determine)j (dual)e(bases)j(of)e(the)h(\014nitely)f(generated)-263 2168 y(and)33 b(pro)5 b(jectiv)m(e)34 b Fs(B)5 b Ft(-mo)s(dule)31 b(summands)h(of)g Fs(A)h Ft(and)g(the)g(set)g Fs(G)p Ft(\()2187 2142 y(\026)2161 2168 y Fs(A)p Ft(\))g(of)f(group-lik)m(e)f (elemen)m(ts)i(of)f(the)-263 2288 y(quotien)m(t)h(coalgebra)581 2263 y(\026)556 2288 y Fs(A)p Ft(.)-263 2408 y(F)-8 b(or)33 b(an)m(y)i(c)m(haracter)f Fs(\037)c Fr(2)g Ft(Alg\()p Fs(K)r(;)17 b(k)s Ft(\))33 b(of)h Fs(K)7 b Ft(,)34 b(a)f(left)g Fs(K)7 b Ft(-mo)s(dule)32 b Fs(M)10 b Ft(,)35 b(and)e(a)h(righ)m(t)f Fs(K)7 b Ft(-mo)s(dule)32 b Fs(N)44 b Ft(w)m(e)-263 2529 y(will)31 b(denote)i(the)g(eigenspaces)h(of)e Fs(\037)h Ft(b)m(y)766 2735 y Fo(\037)814 2720 y Fs(M)39 b Ft(:=)27 b Fr(f)p Fs(m)h Fr(2)g Fs(M)39 b Fr(j)27 b(8)p Fs(x)i Fr(2)f Fs(K)7 b Ft(:)17 b Fs(xm)28 b Ft(=)f Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(m)p Fr(g)-263 2912 y Ft(and)742 3032 y Fs(N)830 3047 y Fo(\037)906 3032 y Ft(:)h Fs(te)g Ft(=)f Fr(f)p Fs(n)h Fr(2)g Fs(N)38 b Fr(j)28 b(8)p Fs(x)g Fr(2)g Fs(K)7 b Ft(:)17 b Fs(nx)28 b Ft(=)g Fs(n\037)p Ft(\()p Fs(x)p Ft(\))p Fr(g)p Fs(:)-263 3211 y Fj(Corollary)36 b(2.3)49 b Fm(L)-5 b(et)28 b Fs(U)38 b Fm(b)-5 b(e)27 b(a)g(Hopf)g(algebr)-5 b(a)27 b(with)g(bije)-5 b(ctive)26 b(antip)-5 b(o)g(de)27 b(over)g(an)g(algebr)-5 b(aic)g(al)5 b(ly)26 b(close)-5 b(d)-263 3332 y(\014eld)25 b Fs(k)s Fm(,)j Fs(K)34 b Fr(\032)29 b Fs(U)36 b Fm(a)25 b(left)h(c)-5 b(oide)g(al)24 b(sub)-5 b(algebr)g(a)25 b(and)g Fr(C)32 b Fm(a)25 b(tensor)g(c)-5 b(ate)g(gory)26 b(of)f(\014nite)g (dimensional)f(left)h Fs(U)10 b Fm(-)-263 3452 y(mo)-5 b(dules.)44 b(De\014ne)31 b Fs(A;)17 b(B)38 b Fm(and)895 3427 y Ft(\026)870 3452 y Fs(A)33 b Fm(as)f(in)h(2.2.)44 b(Assume)33 b(that)g Fs(K)40 b Fm(is)33 b(c)-5 b(ommutative)32 b(and)h Fr(C)6 b Fm(-semisimple)-263 3572 y(and)32 b Fr(C)39 b Fm(is)32 b(semisimple.)42 b(L)-5 b(et)33 b Fs(X)8 b Ft(\()p Fs(K)r(;)17 b Fr(C)6 b Ft(\))31 b Fm(b)-5 b(e)32 b(the)h(set)f(of)g(al)5 b(l)32 b Fs(\037)c Fr(2)g Ft(Alg\()p Fs(K)r(;)17 b(k)s Ft(\))32 b Fm(with)2711 3587 y Fo(\037)2759 3572 y Fs(V)49 b Fr(6)p Ft(=)28 b Fr(f)p Ft(0)p Fr(g)k Fm(for)g(some)-263 3693 y(simple)i Fs(U)10 b Fm(-mo)-5 b(dule)35 b Fs(V)49 b Fr(2)28 b(C)6 b Fm(.)45 b(Then)-197 3872 y(\(1\))j Fs(A)35 b Ft(=)199 3806 y Fl(L)291 3893 y Fo(\037)p Fp(2)p Fo(X)5 b Fq(\()p Fo(K)q(;)p Fp(C)t Fq(\))641 3872 y Fs(A)714 3887 y Fo(\037)801 3872 y Fm(and)38 b Fs(A)c Ft(=)1211 3806 y Fl(L)1304 3893 y Fo(\037)p Fp(2)p Fo(X)5 b Fq(\()p Fo(K)q(;)p Fp(C)t Fq(\))1654 3872 y Fs(S)h Ft(\()1758 3887 y Fo(\037)1806 3872 y Fs(A)p Ft(\))38 b Fm(ar)-5 b(e)38 b(dir)-5 b(e)g(ct)39 b(sums)f(of)g(right)g(c)-5 b(oide)g(als)37 b(and)-19 3992 y(\014nitely)f(gener)-5 b(ate)g(d)35 b(and)g(pr)-5 b(oje)g(ctive)35 b(right)h(\(left,)g(r)-5 b(esp)g(e)g(ctively\))35 b Fs(B)5 b Fm(-mo)-5 b(dules)35 b(with)h Fs(B)e Ft(=)c Fs(A)3375 4007 y Fo(")3441 3992 y Ft(=)-19 4113 y Fs(S)6 b Ft(\()85 4128 y Fo(")122 4113 y Fs(A)p Ft(\))p Fm(.)-197 4308 y(\(2\))48 b(The)35 b(natur)-5 b(al)35 b(c)-5 b(o)g(algebr)g(a)33 b(map)1170 4283 y Ft(\026)1144 4308 y Fs(A)28 b Fr(!)g Fs(K)1463 4272 y Fq(0)1537 4308 y Fm(given)34 b(by)h(r)-5 b(estriction)35 b(de\014nes)f(a)g(bije)-5 b(ction)1146 4516 y Fs(G)p Ft(\()1287 4491 y(\026)1261 4516 y Fs(A)p Ft(\))1453 4446 y Fp(\030)1453 4467 y Fq(=)1400 4516 y Fr(\000)-16 b(!)27 b Fs(X)8 b Ft(\()p Fs(K)r(;)17 b Fr(C)6 b Ft(\))p Fs(;)51 b(g)31 b Fr(7!)c Fs(\037)2284 4531 y Fo(g)2325 4516 y Fs(;)-19 4707 y Fm(and)33 b(for)g(any)g(gr)-5 b(oup-like)32 b(element)h Fs(g)e Fr(2)1510 4682 y Ft(\026)1485 4707 y Fs(A)p Fm(,)i(the)g(right)h(eigensp)-5 b(ac)g(e)31 b Fs(A)2557 4722 y Fo(\037)2601 4730 y Fg(g)2675 4707 y Fm(of)h Fs(\037)2848 4722 y Fo(g)2922 4707 y Fm(is)h(the)g(sp)-5 b(ac)g(e)32 b(of)-19 4828 y(left)i Fs(g)t Fm(-invariant)f(elements)1054 4843 y Fo(g)1094 4828 y Fs(A)28 b Ft(:=)g Fr(f)p Fs(a)f Fr(2)h Fs(A)g Fr(j)1705 4761 y Fl(P)1810 4828 y Ft(\026)-50 b Fs(a)1860 4843 y Fq(1)1920 4828 y Fr(\012)21 b Fs(a)2069 4843 y Fq(2)2136 4828 y Ft(=)28 b Fs(g)23 b Fr(\012)e Fs(a)p Fr(g)p Fm(,)34 b(and)2762 4843 y Fo(\037)2806 4851 y Fg(g)2846 4828 y Fs(A)28 b Ft(=)f Fs(A)3123 4843 y Fo(g)3191 4828 y Ft(:=)h Fr(f)p Fs(a)f Fr(2)-19 4948 y Fs(A)h Fr(j)137 4882 y Fl(P)242 4948 y Fs(a)293 4963 y Fq(1)354 4948 y Fr(\012)c Ft(\026)-50 b Fs(a)505 4963 y Fq(2)572 4948 y Ft(=)28 b Fs(a)22 b Fr(\012)h Fs(g)t Fr(g)p Fm(.)1578 5197 y Ft(12)p eop %%Page: 13 13 13 12 bop -263 614 a Fi(Pr)n(oof:)-192 791 y Ft(\(1\))48 b(W)-8 b(e)36 b(\014rst)f(consider)h(the)f(decomp)s(osition)f Fs(A)e Ft(=)1767 724 y Fl(L)1859 811 y Fo(\037)1924 791 y Fs(A)1997 806 y Fo(\037)2045 791 y Ft(.)51 b(Let)35 b Fr(E)44 b Ft(b)s(e)35 b(a)g(set)h(of)e(represen)m(tativ)m(es)-19 911 y(of)45 b(the)h(isomorphism)d(classes)k(of)e(the)h(simple)e(mo)s (dules)g(in)h Fr(C)6 b Ft(.)82 b(By)46 b(2.1,)j Fs(A)h Ft(=)3119 845 y Fl(L)3211 932 y Fo(V)16 b Fp(2E)3379 911 y Fs(C)3456 875 y Fo(V)-19 1032 y Ft(is)38 b(a)h(decomp)s(osition)e (in)m(to)g(\014nite)i(dimensional)d(righ)m(t)i(\(and)g(left\))g Fs(U)10 b Ft(-mo)s(dules.)61 b(Since)39 b Fs(K)45 b Ft(is)-19 1152 y(comm)m(utativ)m(e)32 b(and)g Fs(k)k Ft(is)c(algebraically)d (closed,)k(all)d(\014nite)i(dimensional)e(simple)h Fs(K)7 b Ft(-mo)s(dules)-19 1272 y(are)37 b(1-dimensional)c(and)j(giv)m(en)h (b)m(y)g(c)m(haracters)h(of)e Fs(K)7 b Ft(.)55 b(By)37 b(assumption)f(an)m(y)h Fs(V)56 b Fr(2)34 b(E)45 b Ft(is)36 b Fs(K)7 b Ft(-)-19 1393 y(semisimple)27 b(and)i(has)g(a)g(basis)g(\()p Fs(v)1227 1408 y Fo(i)1255 1393 y Ft(\))g(of)f(eigen)m(v)m(ectors)j Fs(v)2018 1408 y Fo(i)2074 1393 y Fr(2)d Fs(V)2225 1408 y Fo(\037)2269 1418 y Fg(i)2328 1393 y Ft(for)g(some)h(c)m(haracters)h Fs(\037)3235 1408 y Fo(i)3292 1393 y Ft(of)f Fs(K)7 b Ft(.)-19 1513 y(Hence)32 b Fs(xv)371 1528 y Fo(i)428 1513 y Ft(=)27 b Fs(\037)592 1528 y Fo(i)621 1513 y Ft(\()p Fs(x)p Ft(\))p Fs(v)799 1528 y Fo(i)858 1513 y Ft(for)j(all)f Fs(i)i Ft(and)g(all)e Fs(x)f Fr(2)g Fs(K)7 b Ft(.)43 b(Let)31 b(\()p Fs(f)2121 1528 y Fo(i)2149 1513 y Ft(\))g(b)s(e)g(the)h (dual)e(basis)g(of)h(\()p Fs(v)3163 1528 y Fo(i)3191 1513 y Ft(\))g(in)f Fs(V)3450 1477 y Fp(\003)3490 1513 y Ft(.)-19 1633 y(Then)36 b Fs(f)286 1648 y Fo(i)338 1633 y Fr(\001)23 b Fs(x)32 b Ft(=)f Fs(\037)644 1648 y Fo(i)672 1633 y Ft(\()p Fs(x)p Ft(\))p Fs(f)851 1648 y Fo(i)914 1633 y Ft(for)j(all)f Fs(i)i Ft(and)f(all)f Fs(x)f Fr(2)f Fs(K)7 b Ft(.)50 b(Th)m(us)36 b(it)e(follo)m(ws)f(from)g (2.1,)i(\(2\))g(and)f(\(3\))-19 1754 y(that)c(for)g(all)f Fs(V)49 b Fr(2)28 b(E)9 b Ft(,)31 b Fs(C)868 1718 y Fo(V)956 1726 y Fr(\030)957 1758 y Ft(=)1061 1754 y Fs(V)40 b Fr(\012)18 b Fs(V)1331 1718 y Fp(\003)1401 1754 y Ft(is)30 b(a)g(semisimple)f(righ)m(t)g Fs(K)7 b Ft(-mo)s(dule,)29 b(and)i(w)m(e)g(can)g(write)1145 1942 y Fs(A)c Ft(=)1349 1859 y Fl(M)1390 2034 y Fo(\037)1491 1942 y Fs(A)1564 1957 y Fo(\037)1640 1942 y Ft(=)1754 1859 y Fl(M)1743 2043 y Fo(V)16 b Fp(2E)1907 1859 y Fl(M)1948 2034 y Fo(\037)2032 1942 y Ft(\()p Fs(C)2147 1901 y Fo(V)2208 1942 y Ft(\))2246 1957 y Fo(\037)2326 1942 y Fs(:)-19 2201 y Ft(By)42 b(de\014nition,)g Fs(A)687 2216 y Fo(")765 2201 y Ft(=)g Fs(B)5 b Ft(.)68 b(F)-8 b(or)40 b(an)m(y)i Fs(\037)p Ft(,)h Fs(A)1637 2216 y Fo(\037)1725 2201 y Ft(is)e(a)f(righ)m(t)g Fs(B)5 b Ft(-submo)s(dule)40 b(of)h Fs(A)f Ft(since)i(for)e(all)-19 2321 y Fs(a)28 b Fr(2)g Fs(A)227 2336 y Fo(\037)275 2321 y Ft(,)33 b Fs(b)28 b Fr(2)g Fs(B)38 b Ft(and)33 b Fs(x)28 b Fr(2)g Fs(K)7 b Ft(,)833 2510 y(\()p Fs(ab)p Ft(\))23 b Fr(\001)f Fs(x)83 b Ft(=)1371 2427 y Fl(X)1491 2510 y Ft(\()p Fs(a)22 b Fr(\001)g Fs(x)1707 2525 y Fq(1)1747 2510 y Ft(\)\()p Fs(b)h Fr(\001)e Fs(x)1991 2525 y Fq(2)2031 2510 y Ft(\))1212 2676 y(=)1371 2593 y Fl(X)1491 2676 y Ft(\()p Fs(a)h Fr(\001)g Fs(x)1707 2691 y Fq(1)1747 2676 y Ft(\))p Fs(b")p Ft(\()p Fs(x)1965 2691 y Fq(2)2005 2676 y Ft(\))p Fs(;)49 b Ft(since)33 b Fs(x)2413 2691 y Fq(2)2481 2676 y Fr(2)28 b Fs(K)1212 2821 y Ft(=)83 b(\()p Fs(a)22 b Fr(\001)g Fs(x)p Ft(\))p Fs(b)1212 2967 y Ft(=)83 b Fs(ab\037)p Ft(\()p Fs(x)p Ft(\))p Fs(:)-19 3155 y Ft(It)33 b(is)f(easy)i(to)e(c)m(hec)m(k)j(that)d(all)f (eigenspaces)i Fs(A)1717 3170 y Fo(\037)1798 3155 y Ft(are)g(righ)m(t)e (coideals)h(in)g Fs(A)p Ft(.)-19 3312 y(Let)40 b Fs(\037)g Ft(b)s(e)h(a)e(c)m(haracter)i(of)f Fs(K)7 b Ft(.)66 b(It)40 b(remains)f(to)g(sho)m(w)i(that)f Fs(A)2379 3327 y Fo(\037)2467 3312 y Ft(is)g(\014nitely)f(generated)i(and)-19 3433 y(pro)5 b(jectiv)m(e)46 b(as)e(a)h(righ)m(t)e Fs(B)5 b Ft(-mo)s(dule.)78 b(W)-8 b(e)45 b(can)f(assume)h(that)g Fs(A)2464 3448 y Fo(\037)2560 3433 y Fr(6)p Ft(=)j Fr(f)p Ft(0)p Fr(g)43 b Ft(or)h(equiv)-5 b(alen)m(tly)-19 3553 y Fs(\037)37 b Fr(2)f Fs(X)8 b Ft(\()p Fs(K)r(;)17 b Fr(C)6 b Ft(\).)58 b(Hence)39 b(there)g(exists)f(a)f(simple)f(mo)s (dule)h Fs(V)57 b Fr(2)37 b(E)46 b Ft(with)38 b(\()p Fs(C)2846 3517 y Fo(V)2906 3553 y Ft(\))2944 3568 y Fo(\037)3028 3553 y Fr(6)p Ft(=)f Fr(f)p Ft(0)p Fr(g)p Ft(.)58 b(Let)-19 3674 y(\()p Fs(v)66 3689 y Fo(i)95 3674 y Ft(\),)32 b(\()p Fs(f)278 3689 y Fo(i)306 3674 y Ft(\))h(b)s(e)f(dual)g(bases)i(of)e (eigen)m(v)m(ectors)i(of)e Fs(V)54 b Ft(and)32 b Fs(V)2128 3637 y Fp(\003)2200 3674 y Ft(as)g(b)s(efore.)44 b(Th)m(us)34 b Fs(f)2945 3689 y Fo(j)3004 3674 y Fr(\001)21 b Fs(x)28 b Ft(=)g Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(f)3480 3689 y Fo(j)-19 3794 y Ft(and)37 b Fs(xv)277 3809 y Fo(j)348 3794 y Ft(=)d Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(v)697 3809 y Fo(j)771 3794 y Ft(for)h(some)i Fs(j)42 b Ft(and)36 b(all)f Fs(x)f Fr(2)h Fs(K)7 b Ft(.)55 b(De\014ne)37 b Fs(v)h Ft(:=)c Fs(v)2524 3809 y Fo(j)2561 3794 y Ft(,)j Fs(f)45 b Ft(:=)34 b Fs(f)2903 3809 y Fo(j)2976 3794 y Ft(and)j(consider)-19 3914 y(the)c(matrix)e(co)s(e\016cien)m(t)j Fs(c)964 3929 y Fo(f)t(;v)1064 3914 y Ft(.)43 b(Since)33 b(\()p Fs(v)1474 3929 y Fo(i)1502 3914 y Ft(\),)g(\()p Fs(f)1686 3929 y Fo(i)1714 3914 y Ft(\))f(are)h(dual)f(bases)i(and)1227 4103 y(\001\()p Fs(c)1388 4118 y Fo(f)t(;v)1488 4103 y Ft(\))28 b(=)1657 4020 y Fl(X)1705 4202 y Fo(i)1794 4103 y Fs(c)1836 4118 y Fo(f)t(;v)1928 4128 y Fg(i)1980 4103 y Fr(\012)23 b Fs(c)2122 4118 y Fo(f)2156 4128 y Fg(i)2182 4118 y Fo(;v)2245 4103 y Fs(;)-19 4361 y Ft(w)m(e)38 b(get)f(1)f(=)f Fs(f)11 b Ft(\()p Fs(v)t Ft(\))34 b(=)i Fs(")p Ft(\()p Fs(c)950 4376 y Fo(f)t(;v)1049 4361 y Ft(\))g(=)1234 4295 y Fl(P)1322 4382 y Fo(i)1366 4361 y Fs(c)1408 4376 y Fo(f)t(;v)1500 4386 y Fg(i)1531 4361 y Fs(S)6 b Ft(\()p Fs(c)1677 4376 y Fo(f)1711 4386 y Fg(i)1737 4376 y Fo(;v)1799 4361 y Ft(\),)38 b(where)h Fs(S)j Ft(is)37 b(the)g(an)m(tip)s(o)s(de)g(of)f Fs(A)p Ft(.)57 b(De\014ne)-19 4482 y Fs(a)32 4497 y Fo(i)88 4482 y Ft(:=)28 b Fs(c)261 4497 y Fo(f)t(;v)353 4507 y Fg(i)383 4482 y Ft(,)k Fs(b)483 4497 y Fo(i)539 4482 y Ft(:=)c Fs(S)6 b Ft(\()p Fs(c)816 4497 y Fo(f)850 4507 y Fg(i)876 4497 y Fo(;v)938 4482 y Ft(\))32 b(and)f Fs(\036)1254 4497 y Fo(i)1282 4482 y Ft(\()p Fs(a)p Ft(\))d(:=)f Fs(b)1608 4497 y Fo(i)1637 4482 y Fs(a)32 b Ft(for)f(all)e Fs(a)f Fr(2)g Fs(A)2248 4497 y Fo(\037)2296 4482 y Ft(.)43 b(By)32 b(the)g(dual)e(basis)i(lemma)d(it)-19 4602 y(su\016ces)35 b(to)d(pro)m(v)m(e)i(that)e Fs(a)957 4617 y Fo(i)1013 4602 y Fr(2)d Fs(A)1181 4617 y Fo(\037)1261 4602 y Ft(and)k Fs(\036)1509 4617 y Fo(i)1537 4602 y Ft(\()p Fs(a)p Ft(\))27 b(=)h Fs(b)1836 4617 y Fo(i)1865 4602 y Fs(a)g Fr(2)g Fs(B)37 b Ft(for)32 b(all)f Fs(i)i Ft(and)f Fs(a)c Fr(2)g Fs(A)2935 4617 y Fo(\037)2983 4602 y Ft(.)-19 4760 y(F)-8 b(or)32 b(all)f Fs(u)c Fr(2)h Fs(U)43 b Ft(and)33 b Fs(x)28 b Fr(2)g Fs(K)7 b Ft(,)479 4948 y(\()p Fs(a)568 4963 y Fo(i)618 4948 y Fr(\001)22 b Fs(x)p Ft(\))p Fs(u)28 b Ft(=)f Fs(f)11 b Ft(\()p Fs(xuv)1203 4963 y Fo(i)1231 4948 y Ft(\))28 b(=)f(\()p Fs(f)33 b Fr(\001)22 b Fs(x)p Ft(\)\()p Fs(uv)1803 4963 y Fo(i)1831 4948 y Ft(\))28 b(=)f Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(f)11 b Ft(\()p Fs(uv)2392 4963 y Fo(i)2420 4948 y Ft(\))28 b(=)f Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(a)2832 4963 y Fo(i)2861 4948 y Ft(\()p Fs(u)p Ft(\))p Fs(;)1578 5197 y Ft(13)p eop %%Page: 14 14 14 13 bop -19 614 a Ft(hence)34 b Fs(a)303 629 y Fo(i)359 614 y Fr(2)28 b Fs(A)526 629 y Fo(\037)574 614 y Ft(.)-19 776 y(Finally)-8 b(,)30 b(for)i(all)f Fs(a)d Fr(2)g Fs(A)863 791 y Fo(\037)943 776 y Ft(and)33 b Fs(x)28 b Fr(2)g Fs(K)7 b Ft(,)787 996 y(\()p Fs(b)866 1011 y Fo(i)894 996 y Fs(a)p Ft(\))23 b Fr(\001)e Fs(x)84 b Ft(=)1353 913 y Fl(X)1473 996 y Ft(\()p Fs(b)1552 1011 y Fo(i)1602 996 y Fr(\001)22 b Fs(x)1707 1011 y Fq(1)1747 996 y Ft(\)\()p Fs(a)g Fr(\001)g Fs(x)2001 1011 y Fq(2)2041 996 y Ft(\))1194 1162 y(=)1353 1079 y Fl(X)1473 1162 y Ft(\()p Fs(b)1552 1177 y Fo(i)1602 1162 y Fr(\001)g Fs(x)1707 1177 y Fq(1)1747 1162 y Ft(\))p Fs(a\037)p Ft(\()p Fs(x)1990 1177 y Fq(2)2030 1162 y Ft(\))98 b(since)33 b Fs(x)2460 1177 y Fq(2)2527 1162 y Fr(2)28 b Fs(K)1194 1328 y Ft(=)83 b(\()p Fs(b)1432 1343 y Fo(i)1483 1328 y Fr(\001)1532 1245 y Fl(X)1669 1328 y Fs(x)1724 1343 y Fq(1)1764 1328 y Fs(\037)p Ft(\()p Fs(x)1918 1343 y Fq(2)1958 1328 y Ft(\)\))p Fs(a)1194 1474 y Ft(=)g(\()p Fs(b)1432 1489 y Fo(i)1460 1474 y Fs(a)p Ft(\))p Fs(")p Ft(\()p Fs(x)p Ft(\))p Fs(;)-19 1694 y Ft(since)32 b(\()p Fs(b)298 1709 y Fo(i)347 1694 y Fr(\001)396 1627 y Fl(P)500 1694 y Fs(x)555 1709 y Fq(1)595 1694 y Fs(\037)p Ft(\()p Fs(x)749 1709 y Fq(2)789 1694 y Ft(\)\))27 b(=)h Fs(b)1037 1709 y Fo(i)1065 1694 y Fs(")p Ft(\()p Fs(x)p Ft(\).)43 b(Th)m(us)34 b Fs(b)1600 1709 y Fo(i)1649 1694 y Fr(\001)20 b Fs(a)28 b Fr(2)g Fs(B)5 b Ft(.)43 b(The)33 b(last)e(equalit)m(y)g(holds)h(since)g(for)f (all)-19 1814 y Fs(u)c Fr(2)i Fs(U)10 b Ft(,)111 2034 y(\()p Fs(b)190 2049 y Fo(i)241 2034 y Fr(\001)290 1951 y Fl(X)427 2034 y Fs(x)482 2049 y Fq(1)522 2034 y Fs(\037)p Ft(\()p Fs(x)676 2049 y Fq(2)716 2034 y Ft(\)\)\()p Fs(u)p Ft(\))82 b(=)h Fs(S)6 b Ft(\()p Fs(c)1311 2049 y Fo(f)1345 2059 y Fg(i)1371 2049 y Fo(;v)1433 2034 y Ft(\)\()1509 1951 y Fl(X)1646 2034 y Fs(x)1701 2049 y Fq(1)1740 2034 y Fs(\037)p Ft(\()p Fs(x)1894 2049 y Fq(2)1934 2034 y Ft(\))p Fs(u)p Ft(\))27 b(=)h Fs(c)2239 2049 y Fo(f)2273 2059 y Fg(i)2299 2049 y Fo(;v)2361 2034 y Ft(\()p Fs(S)6 b Ft(\()p Fs(u)p Ft(\))2614 1951 y Fl(X)2750 2034 y Fs(S)g Ft(\()p Fs(x)2909 2049 y Fq(1)2948 2034 y Ft(\))p Fs(\037)p 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2794 y Fo(\037)3065 2788 y(\037)3113 2773 y Fs(A)p Ft(.)47 b(Hence)-19 2893 y Fs(A)d Ft(=)217 2827 y Fl(L)310 2914 y Fo(\037)374 2893 y Fs(S)6 b Ft(\()478 2908 y Fo(\037)526 2893 y Fs(A)p Ft(\))42 b(since)g Fs(S)48 b Ft(is)41 b(bijectiv)m(e.)71 b(One)43 b(easily)e(sees)i(that)f(all)e (the)i Fs(S)6 b Ft(\()2941 2908 y Fo(\037)2989 2893 y Fs(A)p Ft(\))42 b(are)g(righ)m(t)-19 3014 y(coideals.)g(F)-8 b(rom)28 b(Koppinen's)h(lemma)f(for)g(left)h(coideals)f(\(apply)h(1.4)g (to)h(the)f(dual)g(coalgebra\))-19 3134 y(w)m(e)34 b(get)205 3354 y Fs(K)295 3313 y Fq(+)354 3354 y Fs(U)k Ft(=)28 b Fs(S)628 3313 y Fp(\000)p Fq(1)722 3354 y Ft(\()p Fs(U)10 b(K)926 3313 y Fq(+)986 3354 y Ft(\))28 b(=)f Fs(S)1221 3313 y Fp(\000)p Fq(1)1315 3354 y Ft(\()p Fs(K)1443 3313 y Fq(+)1502 3354 y Ft(\))p Fs(U)43 b Ft(and)33 b Fs(B)g Ft(=)27 b Fr(f)p Fs(a)h Fr(2)g Fs(A)g Fr(j)f Fs(a)c Fr(\001)f Fs(S)2618 3313 y Fp(\000)p Fq(1)2712 3354 y Ft(\()p Fs(K)2840 3313 y Fq(+)2899 3354 y Ft(\))27 b(=)h Fr(f)p Ft(0)p Fr(gg)p 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Ft(Let)34 b Fs(\037)29 b Fr(2)h Fs(X)8 b Ft(\()p Fs(K)r(;)17 b Fr(C)6 b Ft(\).)45 b(W)-8 b(e)34 b(ha)m(v)m(e)h(to)e(sho)m(w)i(that)e Fs(S)6 b Ft(\()1841 629 y Fo(\037)1889 614 y Fs(A)p Ft(\))33 b(is)g(\014nitely)g(generated) h(and)g(pro)5 b(jectiv)m(e)-19 735 y(as)37 b(a)f(left)f Fs(B)5 b Ft(-mo)s(dule.)53 b(Using)36 b(the)g(notations)f(ab)s(o)m(v)m (e)j(it)d(su\016ces)j(to)e(sho)m(w)h(that)f Fs(b)3091 750 y Fo(i)3154 735 y Fr(2)e Fs(S)6 b Ft(\()3358 750 y Fo(\037)3406 735 y Fs(A)p Ft(\))-19 855 y(and)33 b Fs(S)6 b Ft(\()p Fs(a)p Ft(\))p Fs(a)415 870 y Fo(i)471 855 y Fr(2)28 b Fs(B)38 b Ft(for)32 b(all)e Fs(i)j Ft(and)g Fs(a)27 b Fr(2)1390 870 y Fo(\037)1438 855 y Fs(A)p Ft(.)43 b(F)-8 b(or)32 b(all)e Fs(u)e Fr(2)g Fs(U)43 b Ft(and)33 b Fs(x)28 b Fr(2)g Fs(K)7 b Ft(,)663 1075 y(\()p Fs(x)22 b Fr(\001)g Fs(c)870 1090 y Fo(f)904 1100 y Fg(i)930 1090 y Fo(;v)992 1075 y Ft(\)\()p Fs(u)p Ft(\))27 b(=)h Fs(f)1341 1090 y Fo(i)1369 1075 y Ft(\()p Fs(uxv)t Ft(\))g(=)f Fs(\037)p Ft(\()p Fs(x)p 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Ft(\()p Fs(c)2244 2128 y Fo(f)t(;v)2336 2138 y Fg(i)2367 2113 y Ft(\)\))p Fs(a)27 b Ft(=)1196 2272 y(=)83 b Fs(")p Ft(\()p Fs(x)p Ft(\))p Fs(S)1598 2230 y Fp(\000)p Fq(1)1692 2272 y Ft(\()p Fs(c)1772 2287 y Fo(f)t(;v)1864 2297 y Fg(i)1895 2272 y Ft(\))p Fs(a)27 b Ft(=)1196 2417 y(=)83 b Fs(")p Ft(\()p Fs(x)p Ft(\))p Fs(S)1598 2376 y Fp(\000)p Fq(1)1692 2417 y Ft(\()p Fs(S)6 b Ft(\()p Fs(a)p Ft(\))p Fs(a)1974 2432 y Fo(i)2002 2417 y Ft(\))p Fs(:)-19 2637 y Ft(In)33 b(the)g(pro)s(of)f(w)m(e)h (used)h(the)f(equalit)m(y)927 2774 y Fl(X)1063 2857 y Fs(\037)p Ft(\()p Fs(x)1217 2872 y Fq(2)1257 2857 y Ft(\))p Fs(x)1350 2872 y Fq(1)1412 2857 y Fr(\001)22 b Fs(S)1528 2816 y Fp(\000)p Fq(1)1622 2857 y Ft(\()p Fs(c)1702 2872 y Fo(f)t(;v)1794 2882 y Fg(i)1825 2857 y Ft(\))27 b(=)h Fs(")p Ft(\()p Fs(x)p Ft(\))p Fs(S)2237 2816 y Fp(\000)p Fq(1)2331 2857 y Ft(\()p Fs(c)2411 2872 y Fo(f)t(;v)2503 2882 y Fg(i)2533 2857 y Ft(\))-19 3077 y(whic)m(h)33 b(holds)g(since)g(for)f(all)e Fs(u)d Fr(2)h Fs(U)129 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Ft(suc)m(h)h(that)f(for)f(all)f Fs(x)i Fr(2)f Fs(K)7 b Ft(,)39 b Fs(c)2916 4828 y Fo(f)t(;v)3017 4813 y Ft(\()p Fs(x)p Ft(\))e(=)f Fs(\037)p Ft(\()p Fs(x)p Ft(\).)-19 4933 y(Therefore)e Fs(\037)f Ft(is)f(a)g(group-lik)m(e)f (elemen)m(t)i(of)1648 4908 y(~)1623 4933 y Fs(A)p Ft(.)1578 5197 y(15)p eop %%Page: 16 16 16 15 bop -19 614 a Ft(By)30 b(de\014nition,)e(the)i(space)f(of)g(left) f Fs(\037)p Ft(-in)m(v)-5 b(arian)m(ts)27 b(is)i(the)g(eigenspace)h Fs(A)2627 629 y Fo(\037)2675 614 y Ft(,)f(since)h(for)e(all)f Fs(a)h Fr(2)g Fs(A)p Ft(,)-19 668 y Fl(P)85 735 y Fs(\031)t Ft(\()p Fs(a)233 750 y Fq(1)273 735 y Ft(\))11 b Fr(\012)g Fs(a)461 750 y Fq(2)529 735 y Ft(=)28 b Fs(\037)11 b Fr(\012)g Fs(a)28 b Ft(if)f(and)g(only)g(if)f(for)g(all)g Fs(x)i Fr(2)g Fs(K)34 b Ft(and)28 b Fs(u)f Fr(2)h Fs(U)10 b Ft(,)29 b Fs(a)p Ft(\()p Fs(xu)p Ft(\))f(=)2865 668 y Fl(P)2969 735 y Fs(a)3020 750 y Fq(1)3060 735 y Ft(\()p Fs(x)p Ft(\))p Fs(a)3242 750 y Fq(2)3282 735 y Ft(\()p Fs(u)p Ft(\))f(=)-19 855 y 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1243 y Fr(\030)2232 1275 y Ft(=)2362 1246 y(~)2336 1271 y Fs(A)p Ft(.)3442 1453 y Fk(2)-263 1615 y Ft(In)38 b(the)g(next)g(theorem)f(w)m(e)i (consider)e(the)h(sp)s(ecial)e(case)j(of)d(2.1)h(when)i Fs(K)k Ft(=)35 b Fs(k)s Ft([)p Fs(x)p Ft(])j(for)f(some)g(\()p Fs(g)t(;)17 b Ft(1\))p Fm(-)-263 1736 y(primitive)37 b(element)e Fs(x)f Fr(2)f Fs(U)10 b Ft(,)37 b(that)e(is)g(\001\()p Fs(x)p Ft(\))f(=)f Fs(g)27 b Fr(\012)e Fs(x)g Ft(+)f Fs(x)g Fr(\012)h Ft(1,)36 b Fs(g)j Ft(group-lik)m(e)34 b(in)h Fs(U)10 b Ft(.)54 b(Then)36 b Fs(K)43 b Ft(is)35 b(a)-263 1856 y(comm)m(utativ)m(e)f(left)g(coideal)g(subalgebra)g(of)g Fs(U)10 b Ft(.)51 b(If)35 b Fr(C)41 b Ft(is)34 b(a)h(tensor)g(category) g(of)g(\014nite)f(dimensional)-263 1977 y(left)i Fs(U)10 b Ft(-mo)s(dules,)37 b(w)m(e)g(call)e(an)h(elemen)m(t)g Fs(x)f Fr(2)f Fs(U)47 b Fr(C)6 b Fm(-semisimple)35 b Ft(if)g(for)h(all)e Fs(V)56 b Fr(2)34 b(C)6 b Ft(,)38 b(the)f(linear)e(map)-263 2097 y Fs(V)50 b Fr(!)27 b Fs(V)21 b Ft(,)27 b Fs(v)32 b Fr(7!)27 b 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Fs(K)53 b Fm(is)44 b(a)h(left)g(c)-5 b(oide)g(al)45 b(sub)-5 b(algebr)g(a)44 b(of)h Fs(U)10 b Fm(,)48 b(and)c(if)h Fs(U)56 b Fm(is)45 b(p)-5 b(ointe)g(d)44 b(\(or)h(mor)-5 b(e)45 b(gener)-5 b(al)5 b(ly)44 b Fs(U)56 b Fm(is)-19 4948 y(faithful)5 b(ly)35 b(left)g(c)-5 b(o\015at)35 b(over)g Fs(U)5 b(=I)j(U)i Fm(\),)35 b(then)f Fs(I)8 b(U)39 b Ft(=)27 b Fs(K)1956 4912 y Fq(+)2015 4948 y Fs(U)10 b Fm(.)1578 5197 y Ft(17)p eop %%Page: 18 18 18 17 bop -197 614 a Fm(\(2\))48 b(If)35 b Fs(S)6 b Ft(\()p Fs(I)i Ft(\))277 578 y Fp(\003)343 614 y Fr(\032)28 b Fs(I)8 b Fm(,)35 b(then)g Fs(K)871 578 y Fp(\003)938 614 y Ft(=)27 b Fs(K)7 b Fm(.)-263 803 y Fi(Pr)n(oof:)-192 977 y Ft(\(1\))48 b(Since)32 b Fs(I)8 b(U)43 b Ft(is)32 b(a)f(righ)m(t)h(ideal)e(and)i(coideal,)f(the)i(set)g(of)e(righ)m(t)h Fs(U)5 b(=I)j(U)i Ft(-coin)m(v)-5 b(arian)m(t)30 b(elemen)m(ts)j Fs(K)-19 1098 y Ft(is)h(a)g(left)g(coideal)f(subalgebra.)49 b(If)34 b Fs(U)45 b Ft(is)34 b(p)s(oin)m(ted,)h(then)g(b)m(y)g([M2)q(,) g(1.3])f Fs(U)45 b Ft(is)34 b(left)f(\(and)i(righ)m(t\))-19 1218 y(faithfully)d(co\015at)j(o)m(v)m(er)h Fs(U)5 b(=I)j(U)i Ft(.)50 b(Hence)36 b(w)m(e)g(kno)m(w)g(from)d(Theorem)i(1.1)g (\(applied)e(to)i Fs(A)3283 1182 y Fq(op)12 b(cop)3479 1218 y Ft(\))-19 1338 y(that)33 b Fs(I)8 b(U)38 b Ft(=)27 b Fs(K)541 1302 y Fq(+)601 1338 y Fs(U)10 b Ft(.)-192 1532 y(\(2\))48 b(First)32 b(note)i(that)e Fs(K)738 1496 y Fq(+)826 1532 y Fr(\032)d Fs(I)8 b(U)44 b Ft(since)33 b(for)g(an)m(y)g Fs(x)c Fr(2)g Fs(K)7 b Ft(,)1995 1466 y Fl(P)2100 1532 y Fs(x)2155 1547 y Fq(1)2217 1532 y Fr(\012)29 b Ft(\026)-55 b Fs(x)2372 1547 y Fq(2)2440 1532 y Ft(=)29 b Fs(x)23 b Fr(\012)2722 1511 y Ft(\026)2722 1532 y(1,)33 b(hence)41 b(\026)-55 b Fs(x)29 b Ft(=)f Fs(")p Ft(\()p Fs(x)p Ft(\))3468 1511 y(\026)3468 1532 y(1)-19 1652 y(in)k Fs(U)5 b(=I)j(U)i Ft(.)44 b(By)33 b(Koppinen's)g(lemma)d(1.4,)j Fs(S)6 b Ft(\()p Fs(K)1769 1616 y Fq(+)1828 1652 y Fs(U)k Ft(\))28 b(=)f Fs(U)10 b(K)2239 1616 y Fq(+)2299 1652 y Ft(.)44 b(Then)502 1838 y(\()p Fs(K)630 1797 y Fq(+)689 1838 y Ft(\))727 1797 y Fp(\003)794 1838 y Fr(\032)29 b Ft(\()p Fs(U)10 b(K)1104 1797 y Fq(+)1163 1838 y Ft(\))1201 1797 y Fp(\003)1325 1838 y Ft(=)83 b Fs(S)6 b Ft(\()p Fs(K)1678 1797 y Fq(+)1737 1838 y Fs(U)k Ft(\))1851 1797 y Fp(\003)1891 1838 y Fs(;)114 b Ft(since)33 b Fs(U)10 b(K)2437 1802 y Fq(+)2525 1838 y Ft(=)27 b Fs(S)6 b Ft(\()p Fs(K)2822 1802 y Fq(+)2881 1838 y Fs(U)k Ft(\))1324 1983 y Fr(\032)83 b Fs(S)6 b Ft(\()p Fs(I)i(U)i Ft(\))1753 1942 y Fp(\003)1793 1983 y Fs(;)114 b Ft(since)33 b Fs(K)2263 1947 y Fq(+)2350 1983 y Fr(\032)28 b Fs(I)8 b(U)1325 2129 y Ft(=)83 b Fs(S)6 b Ft(\()p Fs(I)i Ft(\))1677 2087 y Fp(\003)1716 2129 y Fs(U)1324 2274 y Fr(\032)83 b Fs(I)8 b(U;)114 b Ft(since)33 b Fs(S)6 b Ft(\()p Fs(I)i Ft(\))2174 2238 y Fp(\003)2241 2274 y Fr(\032)28 b Fs(I)8 b(:)-19 2459 y Ft(Clearly)41 b Fs(K)419 2423 y Fp(\003)500 2459 y Ft(is)h(again)e(a)h(left)g(coideal)g(subalgebra)g(with)g(\()p Fs(K)2338 2423 y Fp(\003)2378 2459 y Ft(\))2416 2423 y Fq(+)2518 2459 y Ft(=)i(\()p Fs(K)2765 2423 y Fq(+)2824 2459 y Ft(\))2862 2423 y Fp(\003)2902 2459 y Ft(,)h(and)e(w)m(e)h(ha)m (v)m(e)-19 2580 y(sho)m(wn)34 b(that)f(\()p Fs(K)617 2544 y Fp(\003)656 2580 y Ft(\))694 2544 y Fq(+)753 2580 y Fs(U)38 b Fr(\032)29 b Fs(I)8 b(U)i Ft(.)43 b(Therefore)1009 2775 y Fs(K)1099 2733 y Fp(\003)1166 2775 y Fr(\032)28 b Fs(U)1347 2733 y Fq(co)13 b Fo(U)s(=)p Fq(\()p Fo(K)1604 2710 y Fe(\003)1640 2733 y Fq(\))1667 2710 y Fh(+)1718 2733 y Fo(U)1805 2775 y Fr(\032)28 b Fs(U)1986 2733 y Fq(co)13 b Fo(U)s(=I)5 b(U)2274 2775 y Ft(=)27 b Fs(K)r(:)-19 2960 y Ft(Since)33 b Fr(\003)f Ft(is)g(an)h(in)m(v)m(olution,)e(w)m(e)j (get)e Fs(K)1433 2924 y Fp(\003)1500 2960 y Ft(=)c Fs(K)7 b Ft(.)3442 3134 y Fk(2)-263 3294 y Ft(As)35 b(an)f(illustration)d(of)j (3.1\(1\),)g(let)g Fs(g)t(;)17 b(h)33 b Ft(b)s(e)i(group-lik)m(e)e (elemen)m(ts)i(of)e Fs(U)45 b Ft(and)35 b Fs(x)c Fr(2)g Fs(U)45 b Ft(with)34 b(\001\()p Fs(x)p Ft(\))d(=)-263 3415 y Fs(g)24 b Fr(\012)d Fs(x)f Ft(+)h Fs(x)f Fr(\012)h Fs(h)p Ft(.)43 b(Then)33 b Fs(k)s(x)f Ft(is)g(a)f(one-dimensional)e (coideal,)i Fs(k)s Ft([)p Fs(xh)2209 3379 y Fp(\000)p Fq(1)2304 3415 y Ft(])g(is)h(a)f(left)g(coideal)f(subalgebra,)-263 3535 y(and)j Fs(xU)39 b Ft(=)27 b Fs(k)s Ft([)p Fs(xh)382 3499 y Fp(\000)p Fq(1)477 3535 y Ft(])504 3499 y Fq(+)564 3535 y Fs(U)10 b Ft(.)-263 3724 y Fj(Remark)37 b(3.2)49 b Ft(Let)27 b Fs(U)38 b Ft(b)s(e)27 b(a)f(Hopf)h Fr(\003)p Ft(-algebra)e(and)i Fr(C)33 b Ft(a)26 b(tensor)i(category)f(of)f (\014nite)g(dimensional)f(left)-263 3845 y Fs(U)10 b Ft(-mo)s(dules.)43 b(F)-8 b(or)31 b(an)m(y)h(left)f Fs(U)10 b Ft(-mo)s(dule)30 b Fs(V)53 b Ft(let)1511 3819 y(\026)1496 3845 y Fs(V)g Ft(b)s(e)32 b Fs(V)53 b Ft(as)32 b(an)f(additiv)m(e)g (group)g(with)h(the)g(follo)m(wing)-263 3965 y(left)g Fs(U)10 b Ft(-mo)s(dule)31 b(structure,)j(denoted)g(b)m(y)f Fs(?)p Ft(:)44 b(F)-8 b(or)31 b(an)m(y)j Fs(v)d Fr(2)d Fs(V)54 b Ft(and)33 b Fs(u)27 b Fr(2)h Fs(U)10 b Ft(,)1291 4151 y Fs(u)22 b(?)g(v)31 b Ft(:=)d Fs(S)6 b Ft(\()p Fs(u)p Ft(\))1847 4109 y Fp(\003)1885 4151 y Fs(v)t(:)-263 4336 y Ft(In)33 b(particular,)350 4311 y(\026)335 4336 y Fs(V)53 b Ft(is)32 b(a)f(complex)h(v)m(ector)h(space)g(with)e Fs(\013)22 b(?)e(v)32 b Ft(:=)37 b(\026)-59 b Fs(\013)q(v)36 b Ft(for)31 b(complex)g(n)m(um)m(b)s(ers)i Fs(\013)q Ft(.)3356 4311 y(\026)3341 4336 y Fs(V)53 b Ft(is)-263 4457 y(the)33 b(restriction)f(of)g Fs(V)54 b Ft(to)32 b Fs(U)44 b Ft(via)31 b(the)i(ring)f(isomorphism)1203 4642 y Fs(U)39 b Fr(!)27 b Fs(U;)49 b(u)27 b Fr(7!)g Fs(S)6 b Ft(\()p Fs(u)p Ft(\))1985 4601 y Fp(\003)2024 4642 y Fs(:)-263 4828 y Ft(A)35 b(tensor)f(category)h Fr(C)40 b Ft(will)32 b(b)s(e)i(called)f(a)h Fm(tensor)i Fr(\003)p Fm(-c)-5 b(ate)g(gory)34 b Ft(if)f(for)g(all)f Fs(V)52 b Fr(2)31 b(C)40 b Ft(also)2958 4803 y(\026)2943 4828 y Fs(V)52 b Fr(2)31 b(C)6 b Ft(.)48 b(Let)35 b Fr(C)-263 4948 y Ft(b)s(e)e(a)f(tensor)i Fr(\003)p Ft(-category.)43 b(Then)1578 5197 y(18)p eop %%Page: 19 19 19 18 bop -192 623 a Ft(\(1\))48 b Fs(A)37 b Ft(:=)f Fs(U)306 587 y Fq(0)296 647 y Fp(C)384 623 y Ft(is)h(a)h(Hopf)f Fr(\003)p Ft(-algebra)f(with)h Fr(\003)p Ft(-structure)i(de\014ned)g(b) m(y)g Fs(a)2512 587 y Fp(\003)2551 623 y Ft(\()p Fs(u)p Ft(\))d(:=)p 2858 538 364 4 v 36 w Fs(a)p Ft(\()p Fs(S)6 b Ft(\()p Fs(u)p Ft(\))3145 594 y Fp(\003)3184 623 y Ft(\))38 b(for)f(all)-19 743 y Fs(a)28 b Fr(2)g Fs(A)33 b Ft(and)g Fs(u)27 b Fr(2)h Fs(U)10 b Ft(.)-192 946 y(\(2\))48 b(If)34 b Fs(I)39 b Fr(\032)31 b Fs(U)44 b Ft(is)34 b(a)g(coideal)f (with)g Fs(S)6 b Ft(\()p Fs(I)i Ft(\))1306 910 y Fp(\003)1376 946 y Fr(\032)31 b Fs(I)8 b Ft(,)34 b(then)h Fs(B)g Ft(:=)c Fr(f)p Fs(a)f Fr(2)h Fs(A)g Fr(j)f Fs(a)23 b Fr(\001)g Fs(I)38 b Ft(=)31 b(0)p Fr(g)p Ft(,)j(the)g(algebra)f(of)-19 1067 y(in\014nitesimal)d(in)m(v)-5 b(arian)m(ts)31 b(de\014ned)j(b)m(y) g Fs(I)8 b Ft(,)32 b(is)g(a)h Fr(\003)p Ft(-subalgebra)e(of)h Fs(A)p Ft(.)-263 1295 y Fi(Pr)n(oof:)-192 1499 y Ft(\(1\))48 b(The)37 b(full)e(Hopf)h(dual)f Fs(U)898 1462 y Fq(0)974 1499 y Ft(is)h(a)g(Hopf)g Fr(\003)p Ft(-algebra)f(with)g Fr(\003)p Ft(-structure)i(as)g(describ)s(ed)g(ab)s(o)m(v)m(e.)55 b Fs(A)-19 1619 y Ft(is)31 b(closed)g(under)h(the)g Fr(\003)p Ft(-structure)f(since)h(for)f(all)e Fs(V)49 b Fr(2)28 b(C)6 b Ft(,)32 b Fs(v)f Fr(2)d Fs(V)22 b Ft(,)31 b(linear)f (functionals)g Fs(f)41 b Ft(on)31 b Fs(V)-19 1739 y Ft(and)i Fs(u)27 b Fr(2)h Fs(U)10 b Ft(,)1119 1860 y Fs(c)1161 1819 y Fp(\003)1161 1884 y Fo(f)t(;v)1261 1860 y Ft(\()p Fs(u)p Ft(\))27 b(=)p 1524 1775 422 4 v 28 w Fs(f)11 b Ft(\()p Fs(S)6 b Ft(\()p Fs(u)p Ft(\))1819 1831 y Fp(\003)1857 1860 y Fs(v)t Ft(\))27 b(=)h Fs(c)2134 1864 y Fq(\026)2119 1881 y Fo(f)7 b(;v)2221 1860 y Ft(\()p Fs(u)p Ft(\))p Fs(;)-19 2034 y Ft(where)34 b Fs(c)320 2038 y Fq(\026)305 2055 y Fo(f)7 b(;v)438 2034 y Ft(is)32 b(the)g(matrix)f(co)s(e\016cien) m(t)h(of)1600 2009 y(\026)1585 2034 y Fs(V)53 b Ft(and)33 b(the)f(linear)e(functional)2801 2008 y(\026)2780 2034 y Fs(f)42 b Ft(on)3020 2009 y(\026)3005 2034 y Fs(V)54 b Ft(is)31 b(de\014ned)-19 2168 y(b)m(y)138 2142 y(\026)117 2168 y Fs(f)10 b Ft(\()p Fs(w)s Ft(\))27 b(:=)p 482 2083 208 4 v 28 w Fs(f)11 b Ft(\()p Fs(w)s Ft(\))32 b(for)g(all)e Fs(w)g Fr(2)1215 2143 y Ft(\026)1200 2168 y Fs(V)50 b Ft(=)27 b Fs(V)22 b Ft(.)-192 2371 y(\(2\))48 b(is)32 b(easy)i(to)e(c)m(hec)m(k)j([KD)o(,)e(1.9].)3442 2575 y Fk(2)-263 2745 y Ft(Let)h Fs(U)45 b Ft(b)s(e)33 b(a)h(Hopf)f Fr(\003)p Ft(-algebra)f(and)i Fr(C)40 b Ft(a)33 b(tensor)i(category)f (of)f(\014nite)h(dimensional)d(left)i Fs(U)10 b Ft(-mo)s(dules.)-263 2865 y(W)-8 b(e)32 b(call)e Fr(C)37 b Fm(unitarizable)31 b Ft(if)f(for)g(all)g Fs(V)49 b Fr(2)28 b(C)37 b Ft(there)32 b(is)f(a)g(hermitian)e(inner)i(pro)s(duct)h Fr(h)p Fs(;)17 b Fr(i)p Ft(:)g Fs(V)39 b Fr(\002)20 b Fs(V)50 b Fr(!)27 b Ff(C)15 b Ft(,)-263 2986 y(conjugate)33 b(linear)e(in)h(the)h (\014rst)g(and)g(linear)e(in)h(the)h(second)h(v)-5 b(ariable)31 b(suc)m(h)j(that)e(for)g(all)f Fs(x)d Fr(2)h Fs(U)43 b Ft(and)-263 3106 y Fs(v)t(;)17 b(w)30 b Fr(2)e Fs(V)22 b Ft(,)32 b Fr(h)p Fs(xv)t(;)17 b(w)s Fr(i)27 b Ft(=)g Fr(h)p Fs(v)t(;)17 b(x)784 3070 y Fp(\003)823 3106 y Fs(w)s Fr(i)p Ft(.)-263 3334 y Fj(Corollary)36 b(3.3)49 b Fm(L)-5 b(et)46 b Fs(U)55 b Fm(b)-5 b(e)45 b(a)g(p)-5 b(ointe)g(d)44 b(Hopf)h Fr(\003)p Fm(-algebr)-5 b(a,)46 b Fr(C)51 b Fm(a)45 b(unitarizable)g(tensor)f(c)-5 b(ate)g(gory)45 b(of)-263 3455 y(\014nite)37 b(dimensional)f(left)h Fs(U)10 b Fm(-mo)-5 b(dules,)38 b(and)f Fs(I)j Fr(\032)32 b Fs(U)48 b Fm(a)38 b(c)-5 b(oide)g(al)36 b(with)h Fs(S)6 b Ft(\()p Fs(I)i Ft(\))2536 3419 y Fp(\003)2607 3455 y Fr(\032)33 b Fs(I)8 b Fm(.)52 b(De\014ne)36 b Fs(A)d Ft(:=)f Fs(U)3477 3419 y Fq(0)3467 3479 y Fp(C)-263 3575 y Fm(and)j Fs(B)d Ft(:=)c Fr(f)p Fs(a)g Fr(2)g Fs(A)g Fr(j)f Fs(a)22 b Fr(\001)g Fs(I)36 b Ft(=)27 b(0)p Fr(g)p Fm(.)45 b(Then)-197 3779 y(\(1\))j Fs(K)35 b Ft(:=)28 b Fs(U)306 3742 y Fq(co)13 b Fo(U)s(=I)5 b(U)597 3779 y Fm(is)30 b(a)h(left)g(c)-5 b(oide)g(al)30 b(sub)-5 b(algebr)g(a)30 b(of)h Fs(U)10 b Fm(,)32 b Fs(B)h Ft(:=)28 b Fr(f)p Fs(a)g Fr(2)g Fs(A)g Fr(j)f(8)p Fs(x)h Fr(2)g Fs(K)7 b Ft(:)17 b Fs(a)d Fr(\001)g Fs(x)27 b Ft(=)h Fs(a")p Ft(\()p Fs(x)p Ft(\))p Fr(g)-19 3899 y Fm(is)35 b(a)g(right)f(c)-5 b(oide)g(al)34 b(sub)-5 b(algebr)g(a)34 b(of)h Fs(A)g Fm(and)f Fs(K)42 b Fm(is)35 b Fr(C)6 b Fm(-semisimple.)-197 4102 y(\(2\))48 b Fs(A=)-5 b(AB)250 4066 y Fq(+)350 4102 y Fm(is)40 b(c)-5 b(osemisimple,)39 b Fs(A)i Fm(is)f(faithful)5 b(ly)41 b(\015at)g(as)f(a)g(left)h(and)f(a) g(right)g Fs(B)5 b Fm(-mo)-5 b(dule)40 b(and)g(a)-19 4223 y(dir)-5 b(e)g(ct)39 b(sum)f(of)g(\014nitely)h(gener)-5 b(ate)g(d)37 b(and)h(pr)-5 b(oje)g(ctive)38 b(left)h Fs(B)5 b Fm(-mo)-5 b(dules)37 b(and)h(right)h Fs(B)5 b Fm(-mo)-5 b(dules)-19 4343 y(as)35 b(in)f(2.2.)-263 4571 y Fi(Pr)n(oof:)1578 5197 y Ft(19)p eop %%Page: 20 20 20 19 bop -192 614 a Ft(\(1\))48 b(Since)39 b Fs(U)49 b Ft(is)39 b(p)s(oin)m(ted,)h(w)m(e)f(get)g(from)f(3.1\(1\))f(that)i 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Ff(C)15 b(P)1105 822 y Fo(n)p Fp(\000)p Fq(1)1242 858 y Ft(.)41 b(Again)24 b(the)i(extension)g Fs(B)33 b Fr(\032)28 b Fs(A)d Ft(satis\014es)h(the)g(assumptions)-19 978 y(in)32 b(3.3.)-143 1181 y(4.)48 b([D])24 b(con)m(tains)h(a)f(surv) m(ey)i(of)e(similar)d(constructions)k(of)f(quan)m(tum)h(homogeneous)f (spaces)i(whic)m(h)-19 1301 y(are)33 b(analogs)e(of)h(compact)h (symmetric)e(spaces.)-263 1634 y Fu(4)166 b(Semisimple)55 b(sk)-5 b(ew-primitiv)g(e)57 b(elemen)-5 b(ts)55 b(in)g Fd(U)2933 1655 y Fs(q)2984 1634 y Fc(\()p Fd(sl)s Fc(\(2\)\))-263 1853 y Ft(Let)37 b Fs(k)h Ft(b)s(e)f(an)e(algebraically)e(closed)j (\014eld)g(and)g(let)f Fs(q)40 b Ft(b)s(e)c(a)g(non-zero)g(elemen)m(t)g (of)f Fs(k)s Ft(,)i(whic)m(h)g(is)e(not)-263 1973 y(a)41 b(ro)s(ot)e(of)h(unit)m(y)-8 b(.)68 b(F)-8 b(or)39 b(all)g(p)s(ositiv)m (e)h(in)m(tegers)h Fs(n)f Ft(let)g Fs(E)1854 1988 y Fo(n)1942 1973 y Ft(b)s(e)h(the)g(unit)e(matrix)g(with)i Fs(n)f Ft(ro)m(ws)i(and)-263 2094 y(columns.)-263 2214 y(First)32 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Ft(\))27 b(=)h Fs(e)495 4369 y Fl(0)495 4515 y(B)495 4565 y(B)495 4615 y(B)495 4664 y(B)495 4714 y(B)495 4764 y(B)495 4817 y(@)584 4422 y Ft(0)98 b([)p Fs(n)p Ft(])216 b(0)f Fr(\001)17 b(\001)g(\001)96 b Ft(0)584 4543 y(0)130 b(0)g([)p Fs(n)23 b Fr(\000)f Ft(1])98 b Fr(\001)17 b(\001)g(\001)96 b Ft(0)595 4635 y(.)595 4669 y(.)595 4702 y(.)736 4644 y(.)774 4669 y(.)812 4694 y(.)1032 4644 y(.)1070 4669 y(.)1108 4694 y(.)1329 4644 y(.)1367 4669 y(.)1406 4694 y(.)1548 4635 y(.)1548 4669 y(.)1548 4702 y(.)584 4836 y(0)130 b(0)1032 4778 y(.)1070 4803 y(.)1108 4828 y(.)1329 4778 y(.)1367 4803 y(.)1406 4828 y(.)1537 4836 y(1)584 4956 y(0)g(0)214 b Fr(\001)17 b(\001)g(\001)213 b Ft(0)131 b(0)1602 4369 y Fl(1)1602 4515 y(C)1602 4565 y(C)1602 4615 y(C)1602 4664 y(C)1602 4714 y(C)1602 4764 y(C)1602 4817 y(A)1691 4689 y Fs(;)50 b(\032)1818 4704 y Fo(e;n)1917 4689 y Ft(\()p Fs(F)14 b Ft(\))27 b(=)2201 4369 y Fl(0)2201 4515 y(B)2201 4565 y(B)2201 4615 y(B)2201 4664 y(B)2201 4714 y(B)2201 4764 y(B)2201 4817 y(@)2290 4422 y Ft(0)130 b(0)h Fr(\001)17 b(\001)g(\001)128 b Ft(0)i(0)2290 4543 y(1)g(0)h Fr(\001)17 b(\001)g(\001)128 b Ft(0)i(0)2290 4677 y(0)103 b([2])2655 4619 y(.)2693 4644 y(.)2731 4669 y(.)2895 4677 y(0)130 b(0)2301 4770 y(.)2301 4803 y(.)2301 4836 y(.)2442 4778 y(.)2480 4803 y(.)2518 4828 y(.)2655 4778 y(.)2693 4803 y(.)2731 4828 y(.)2868 4778 y(.)2906 4803 y(.)2944 4828 y(.)3085 4770 y(.)3085 4803 y(.)3085 4836 y(.)2290 4956 y(0)g(0)h Fr(\001)17 b(\001)g(\001)96 b Ft([)p Fs(n)p Ft(])j(0)3140 4369 y Fl(1)3140 4515 y(C)3140 4565 y(C)3140 4615 y(C)3140 4664 y(C)3140 4714 y(C)3140 4764 y(C)3140 4817 y(A)3229 4689 y Fs(;)1578 5197 y Ft(21)p eop %%Page: 22 22 22 21 bop -263 614 a Ft(and)676 955 y Fs(\032)726 970 y Fo(e;n)825 955 y Ft(\()p Fs(K)7 b Ft(\))28 b(=)g Fs(e)1185 634 y Fl(0)1185 780 y(B)1185 830 y(B)1185 880 y(B)1185 930 y(B)1185 980 y(B)1185 1029 y(B)1185 1083 y(@)1274 697 y Fs(q)1321 661 y Fo(n)1533 697 y Ft(0)165 b Fr(\001)17 b(\001)g(\001)190 b Ft(0)243 b(0)1296 818 y(0)120 b Fs(q)1512 782 y Fo(n)p Fp(\000)p Fq(2)1747 818 y Fr(\001)17 b(\001)g(\001)190 b Ft(0)243 b(0)1307 905 y(.)1307 938 y(.)1307 971 y(.)1505 913 y(.)1544 938 y(.)1582 963 y(.)1753 913 y(.)1791 938 y(.)1829 963 y(.)2066 905 y(.)2066 938 y(.)2066 971 y(.)2358 905 y(.)2358 938 y(.)2358 971 y(.)1296 1092 y(0)188 b(0)165 b Fr(\001)17 b(\001)g(\001)95 b Fs(q)2007 1056 y Fp(\000)p Fo(n)p Fq(+2)2347 1092 y Ft(0)1296 1212 y(0)188 b(0)165 b Fr(\001)17 b(\001)g(\001)190 b Ft(0)j Fs(q)2344 1176 y Fp(\000)p Fo(n)2462 634 y Fl(1)2462 780 y(C)2462 830 y(C)2462 880 y(C)2462 930 y(C)2462 980 y(C)2462 1029 y(C)2462 1083 y(A)2551 955 y Fs(:)-263 1368 y Ft(Here)30 b(w)m(e)g(need)f(the)g(righ)m(t)f(action)g(and)h(use)g(the)h(transp)s (osed)f(matrices.)42 b(W)-8 b(e)29 b(\014rst)g(consider)g(the)g(case) -263 1488 y Fs(e)f Ft(=)g(1.)43 b(Let)939 1609 y Fs(x)28 b Ft(=)g Fs(\013)q Ft(\()p Fs(K)1317 1567 y Fp(\000)p Fq(1)1433 1609 y Fr(\000)22 b Ft(1\))g(+)g Fs(\014)6 b(E)g(K)1968 1567 y Fp(\000)p Fq(1)2085 1609 y Ft(+)22 b Fs(\015)5 b(F)-263 1782 y Ft(b)s(e)33 b(a)g(\()p Fs(K)80 1746 y Fp(\000)p Fq(1)174 1782 y Fs(;)17 b Ft(1\)-primitiv)m(e)30 b(elemen)m(t)i(of)h Fs(U)1293 1797 y Fo(q)1331 1782 y Ft(\()p Fs(sl)r Ft(\(2\)\),)g(where)h Fs(\013)q Ft(,)e Fs(\014)6 b Ft(,)33 b(and)g Fs(\015)k Ft(are)c(\014xed)h(elemen)m(ts)f (of)f(the)-263 1903 y(ground)k(\014eld)f Fs(k)s Ft(,)h(whic)m(h)f(do)g (not)h(v)-5 b(anish)34 b(sim)m(ultaneously)-8 b(.)50 b(Then)37 b(the)e(\(righ)m(t\))g(action)f(of)h Fs(x)g Ft(on)g(the)-263 2023 y(c)m(hosen)g(basis)d(of)g Fs(V)457 2038 y Fq(1)p Fo(;n)591 2023 y Ft(can)h(b)s(e)g(represen)m(ted)i(b)m(y) f(the)f(matrix)57 2531 y Fs(M)151 2546 y Fo(n)226 2531 y Ft(:=)356 2161 y Fl(0)356 2307 y(B)356 2357 y(B)356 2406 y(B)356 2456 y(B)356 2506 y(B)356 2556 y(B)356 2606 y(B)356 2655 y(B)356 2709 y(@)446 2232 y Ft(\()p Fs(q)531 2196 y Fp(\000)p Fo(n)654 2232 y Fr(\000)23 b Ft(1\))p Fs(\013)334 b(\015)560 b Ft(0)435 b Fr(\001)17 b(\001)g(\001)389 b Ft(0)514 2391 y Fs(q)561 2355 y Fp(\000)p Fo(n)662 2391 y Ft([)p Fs(n)p Ft(])p Fs(\014)189 b Ft(\()p Fs(q)1103 2355 y Fq(2)p Fp(\000)p Fo(n)1262 2391 y Fr(\000)23 b Ft(1\))p Fs(\013)282 b Ft([2])p Fs(\015)2301 2333 y Ft(.)2339 2346 y(.)2377 2358 y(.)2415 2370 y(.)2453 2383 y(.)2850 2325 y(.)2850 2358 y(.)2850 2391 y(.)650 2550 y(0)302 b Fs(q)1048 2514 y Fq(2)p Fp(\000)p Fo(n)1185 2550 y Ft([)p Fs(n)22 b Fr(\000)h Ft(1])p Fs(\014)103 b Ft(\()p Fs(q)1711 2514 y Fq(4)p Fp(\000)p Fo(n)1870 2550 y Fr(\000)23 b Ft(1\))p Fs(\013)2301 2492 y Ft(.)2339 2505 y(.)2377 2517 y(.)2415 2530 y(.)2453 2542 y(.)2839 2550 y(0)661 2643 y(.)661 2676 y(.)661 2709 y(.)1175 2651 y(.)1213 2664 y(.)1251 2676 y(.)1289 2689 y(.)1328 2701 y(.)1783 2651 y(.)1821 2664 y(.)1859 2676 y(.)1897 2689 y(.)1935 2701 y(.)2301 2651 y(.)2339 2664 y(.)2377 2676 y(.)2415 2689 y(.)2453 2701 y(.)2779 2709 y([)p Fs(n)p Ft(])p Fs(\015)650 2830 y Ft(0)508 b Fr(\001)17 b(\001)g(\001)523 b Ft(0)320 b Fs(q)2264 2794 y Fo(n)p Fp(\000)p Fq(2)2401 2830 y Ft([1])p Fs(\014)103 b Ft(\()p Fs(q)2747 2794 y Fo(n)2816 2830 y Fr(\000)22 b Ft(1\))p Fs(\013)3081 2161 y Fl(1)3081 2307 y(C)3081 2357 y(C)3081 2406 y(C)3081 2456 y(C)3081 2506 y(C)3081 2556 y(C)3081 2606 y(C)3081 2655 y(C)3081 2709 y(A)3170 2531 y Fs(:)-263 3036 y Ft(The)37 b Fb(quan)m(tum)f(plane)f Fs(k)670 3051 y Fo(q)708 3036 y Ft([)p Fs(a;)17 b(b)p Ft(])40 b(is)35 b(the)h Fs(k)s Ft(-algebra)e(generated)i(b)m(y)h(the)f(elemen)m(ts)g Fs(a)g Ft(and)f Fs(b)h Ft(with)f(the)-263 3156 y(relation)c Fs(ba)d Ft(=)g Fs(q)t(ab)k Ft([K,)h(Chapter)g(IV].)g(It)f(is)g(a)g Fs(U)1577 3171 y Fo(q)1615 3156 y Ft(\()p Fs(sl)r Ft(\(2\)\))g(left)f (mo)s(dule)g(algebra,)h(where)h(the)g(action)-263 3277 y(of)i Fs(U)-83 3292 y Fo(q)-45 3277 y Ft(\()p Fs(sl)r Ft(\(2\)\))f(is)g(giv)m(en)h(in)f([K,)h(Chapter)h(VI)s(I.3].)50 b(Here)35 b(w)m(e)h(need)f(the)g(corresp)s(onding)g(righ)m(t)f(action) -263 3397 y(giv)m(en)f(b)m(y)-263 3616 y Fs(b)r Fr(\001)r Fs(E)i Ft(=)27 b(0)p Fs(;)39 b(a)r Fr(\001)r Fs(E)34 b Ft(=)28 b Fs(b;)40 b(b)r Fr(\001)r Fs(F)i Ft(=)27 b Fs(a;)40 b(a)r Fr(\001)r Fs(F)i Ft(=)27 b(0)p Fs(;)39 b(b)r Fr(\001)r Fs(K)d Ft(=)27 b Fs(q)t(b;)40 b(a)r Fr(\001)r Fs(K)35 b Ft(=)28 b Fs(q)2143 3574 y Fp(\000)p Fq(1)2237 3616 y Fs(a;)40 b(b)r Fr(\001)r Fs(K)2518 3574 y Fp(\000)p Fq(1)2640 3616 y Ft(=)28 b Fs(q)2791 3574 y Fp(\000)p Fq(1)2885 3616 y Fs(b;)40 b(a)r Fr(\001)r Fs(K)3166 3574 y Fp(\000)p Fq(1)3288 3616 y Ft(=)28 b Fs(q)t(a:)-263 3834 y Ft(The)48 b(quan)m(tum)e(plane)g(has)h(a)f(natural)f(gradation,) j(giv)m(en)f(b)m(y)g(the)g(degrees)g(of)f(the)h(monomials,)-263 3954 y(therefore)36 b(there)g(are)f(no)g(zero)h(divisors.)50 b(Let)36 b Fs(k)1543 3969 y Fo(q)1580 3954 y Ft([)p Fs(a;)17 b(b)p Ft(])1770 3969 y Fo(n)1853 3954 y Ft(denote)36 b(the)f(v)m(ector)i(subspace)g(of)d(homoge-)-263 4075 y(neous)29 b(p)s(olynomials)24 b(of)k(degree)g Fs(n)g Ft(in)f Fs(k)1192 4090 y Fo(q)1229 4075 y Ft([)p Fs(a;)17 b(b)p Ft(])t(.)43 b(Then)29 b Fs(k)1794 4090 y Fo(q)1831 4075 y Ft([)p Fs(a;)17 b(b)p Ft(])2021 4090 y Fo(n)2097 4075 y Ft(is)27 b(a)g(simple)f Fs(U)2631 4090 y Fo(q)2669 4075 y Ft(\()p Fs(sl)r Ft(\(2\)\))i(righ)m(t)e(mo)s(dule)-263 4195 y(isomorphic)31 b(to)h Fs(V)407 4210 y Fq(1)p Fo(;n)542 4195 y Ft([K,)g(Theorem)h(VI)s(I.3.3].)-263 4422 y Fj(Lemma)38 b(4.1)114 b Fm(\(1\))48 b(Ther)-5 b(e)35 b(is)f(a)h(non)f(zer)-5 b(o)35 b(element)f Fs(\030)e Fr(2)c Fs(k)2021 4437 y Fo(q)2059 4422 y Ft([)p Fs(a;)17 b(b)p Ft(])2249 4437 y Fq(2)2324 4422 y Fm(such)34 b(that)i Fs(\030)26 b Fr(\001)c Fs(x)28 b Ft(=)g(0)p Fm(.)-197 4625 y(\(2\))48 b(F)-7 b(or)35 b(al)5 b(l)35 b Fs(z)t(;)17 b(\030)33 b Fr(2)c Fs(k)619 4640 y Fo(q)657 4625 y Ft([)p Fs(a;)17 b(b)p Ft(])t Fm(,)36 b(wher)-5 b(e)35 b Fs(\030)27 b Fr(\001)22 b Fs(x)29 b Ft(=)f(0)p Fm(,)36 b(the)f(e)-5 b(quation)35 b Ft(\()p Fs(z)t(\030)5 b Ft(\))23 b Fr(\001)f Fs(x)29 b Ft(=)f(\()p Fs(z)g Fr(\001)22 b Fs(x)p Ft(\))p Fs(\030)40 b Fm(holds,)35 b(that)g(is,)-19 4745 y(the)g(action)g(of)f Fs(x)h Fm(is)g Fs(B)823 4760 y Fo(X)891 4745 y Fm(-right)f(line)-5 b(ar,)34 b(wher)-5 b(e)35 b Fs(B)1813 4760 y Fo(X)1908 4745 y Ft(=)27 b Fr(f)p Fs(\030)32 b Fr(2)c Fs(k)2281 4760 y Fo(q)2319 4745 y Ft([)p Fs(a;)17 b(b)p Ft(])t(:)g Fs(\030)27 b Fr(\001)22 b Fs(x)28 b Ft(=)g(0)p Fr(g)p Fm(.)-197 4948 y(\(3\))48 b(The)35 b(eigensp)-5 b(ac)g(es)33 b(of)h(the)h(action)f(of)h Fs(x)g Fm(on)g Fs(k)1662 4963 y Fo(q)1700 4948 y Ft([)p Fs(a;)17 b(b)p Ft(])1890 4963 y Fo(n)1972 4948 y Fm(ar)-5 b(e)35 b(one-dimensional.)1578 5197 y Ft(22)p eop %%Page: 23 23 23 22 bop -263 614 a Fi(Pr)n(oof:)-192 818 y Ft(\(1\))48 b(The)34 b(determinan)m(t)e(of)g Fs(M)937 833 y Fq(2)1009 818 y Ft(v)-5 b(anishes.)-192 1021 y(\(2\))48 b(The)34 b(quan)m(tum)f(plane)f(is)g(a)g(mo)s(dule)f(algebra)h(of)g Fs(U)1896 1036 y Fo(q)1934 1021 y Ft(\()p Fs(sl)r Ft(\(2\)\):)679 1241 y(\()p Fs(z)t(\030)5 b Ft(\))22 b Fr(\001)g Fs(x)28 b Ft(=)g(\()p Fs(z)e Fr(\001)c Fs(K)1360 1200 y Fp(\000)p Fq(1)1455 1241 y Ft(\)\()p Fs(\030)k Fr(\001)c Fs(x)p Ft(\))g(+)g(\()p Fs(z)27 b Fr(\001)22 b Fs(x)p Ft(\)\()p Fs(\030)27 b Fr(\001)22 b Ft(1\))27 b(=)h(\()p Fs(z)f Fr(\001)21 b Fs(x)p Ft(\))p Fs(\030)5 b(:)-192 1503 y Ft(\(3\))48 b(Let)37 b Fs(\025)d Fr(2)h Fs(k)s Ft(.)56 b(and)36 b(let)g Fs(E)899 1518 y Fo(n)p Fq(+1)1073 1503 y Ft(b)s(e)h(the)g(unit)f(matrix)f(with)h Fs(n)22 b Ft(+)g(1)37 b(ro)m(ws)g(and)g(columns.)55 b(In)36 b(the)-19 1623 y(cases)41 b Fs(\014)46 b Fr(6)p Ft(=)40 b(0)f(or)g Fs(\015)45 b Fr(6)p Ft(=)40 b(0)f(the)i(\014rst)f(or)f(last)g Fs(n)h Ft(columns)f(resp)s(ectiv)m(ely)i(of)e Fs(M)2902 1638 y Fo(n)2976 1623 y Fr(\000)28 b Fs(\025E)3210 1638 y Fo(n)p Fq(+1)3387 1623 y Ft(are)-19 1743 y(linearly)42 b(indep)s(enden)m(t.)79 b(In)45 b(the)f(case)h Fs(\014)53 b Ft(=)47 b Fs(\015)52 b Ft(=)47 b(0)d(there)h(m)m(ust)f(b)s(e)g Fs(u)j Fr(6)p Ft(=)g(0,)g(and)d(in)f(the)-19 1864 y(diagonal)28 b(matrix)g Fs(M)777 1879 y Fo(n)842 1864 y Fr(\000)17 b Fs(\025E)1065 1879 y Fo(n)p Fq(+1)1233 1864 y Ft(at)29 b(most)h(one)g(en)m(try)h(in)f(the)g(diagonal)e(v)-5 b(anishes,)31 b(for)f(since)g Fs(q)-19 1984 y Ft(is)i(not)h(a)f(ro)s (ot)g(of)g(unit)m(y)-8 b(,)32 b(all)f(en)m(tries)i(in)f(the)h(diagonal) d(are)j(pairwise)f(distinct.)3442 2188 y Fk(2)-263 2358 y Ft(F)-8 b(or)38 b(con)m(v)m(enience,)k(w)m(e)e(\014x)f(a)f(solution) 1242 2296 y Fr(p)p 1325 2296 47 4 v 62 x Fs(q)k Ft(of)c(the)h(equation) f Fs(x)2160 2322 y Fq(2)2238 2358 y Ft(=)f Fs(q)43 b Ft(and)38 b(de\014ne)i Fs(q)2967 2322 y Fo(l)3030 2358 y Ft(:=)3171 2296 y Fr(p)p 3254 2296 V 62 x Fs(q)3301 2314 y Fq(2)p Fo(l)3400 2358 y Ft(for)-263 2478 y Fs(l)30 b Fr(2)-100 2439 y Fq(1)p -100 2455 36 4 v -100 2513 a(2)-54 2478 y Ff(Z)o Ft(.)37 b(F)-8 b(or)32 b(eac)m(h)i(nonnegativ)m (e)f(in)m(teger)f Fs(n)h Ft(set)g Fs(I)1616 2493 y Fo(n)1691 2478 y Ft(:=)27 b Fr(f\000)1958 2439 y Fo(n)p 1958 2455 43 4 v 1962 2513 a Fq(2)2011 2478 y Fs(;)17 b Ft(1)22 b Fr(\000)2236 2439 y Fo(n)p 2236 2455 V 2240 2513 a Fq(2)2288 2478 y Fs(;)17 b(:)g(:)g(:)f(;)2517 2439 y Fo(n)p 2517 2455 V 2521 2513 a Fq(2)2592 2478 y Fr(\000)22 b Ft(1)p Fs(;)2794 2439 y Fo(n)p 2794 2455 V 2798 2513 a Fq(2)2847 2478 y Fr(g)p Ft(.)-263 2707 y Fj(Prop)s(osition)36 b(4.2)114 b Fm(\(1\))49 b(L)-5 b(et)242 2927 y Fs(P)305 2942 y Fo(n)352 2927 y Ft(\()p Fs(Y)21 b Ft(\))27 b(=)h(det\()p Fs(M)904 2942 y Fo(n)974 2927 y Fr(\000)22 b Fs(Y)g(E)1224 2942 y Fo(n)p Fq(+1)1361 2927 y Ft(\))28 b(=)f(\()p Fr(\000)p Ft(1\))1732 2885 y Fo(n)p Fq(+1)1869 2927 y Ft(\()p Fs(Y)1986 2885 y Fo(n)p Fq(+1)2145 2927 y Ft(+)22 b Fs(z)2288 2942 y Fo(n)2335 2927 y Fs(Y)2414 2885 y Fo(n)2483 2927 y Ft(+)g Fs(d)2632 2942 y Fo(n)p Fp(\000)p Fq(1)2768 2927 y Fs(Y)2847 2885 y Fo(n)p Fp(\000)p Fq(1)3006 2927 y Ft(+)g Fs(:)17 b(:)g(:)p Ft(\))-19 3147 y Fm(b)-5 b(e)48 b(the)g(char)-5 b(acteristic)g(al)47 b(p)-5 b(olynomial)47 b(of)h Fs(M)1703 3162 y Fo(n)1750 3147 y Fm(.)85 b(Then)47 b(for)h Fs(n)k Fr(\025)h Ft(2)48 b Fm(the)g(p)-5 b(olynomial)47 b Fs(P)3380 3162 y Fo(n)p Fp(\000)p Fq(2)-19 3267 y Fm(divides)34 b Fs(P)368 3282 y Fo(n)415 3267 y Fm(.)-197 3470 y(\(2\))48 b(Fix)34 b Fs(R)29 b Fr(2)f Fs(k)38 b Fm(such)d(that)1279 3650 y Fs(R)1354 3609 y Fq(2)1421 3650 y Ft(=)28 b Fs(\013)1588 3609 y Fq(2)1649 3650 y Ft(+)1816 3582 y(4)p Fs(\014)6 b(\015)f(q)2029 3546 y Fp(\000)p Fq(1)p 1757 3627 426 4 v 1757 3718 a Ft(\()p Fs(q)26 b Fr(\000)d Fs(q)2011 3689 y Fp(\000)p Fq(1)2105 3718 y Ft(\))2143 3689 y Fq(2)2192 3650 y Fs(:)-19 3880 y Fm(Then)34 b Fs(M)329 3895 y Fo(n)412 3880 y Fm(has)g(the)h Fs(n)22 b Ft(+)g(1)35 b Fm(\(not)f(ne)-5 b(c)g(essarily)34 b(distinct\))h(eigenvalues)959 4145 y Fs(\025)1016 4160 y Fo(r)1081 4145 y Ft(:=)1222 4077 y Fs(\013)p 1222 4121 63 4 v 1229 4213 a Ft(2)1294 4145 y(\()p Fs(q)1379 4103 y Fo(r)1439 4145 y Fr(\000)23 b Fs(q)1586 4103 y Fp(\000)p Fo(r)1679 4145 y Ft(\))1717 4103 y Fq(2)1778 4145 y Ft(+)1886 4077 y(1)p 1886 4121 49 4 v 1886 4213 a(2)1945 4145 y(\()p Fs(q)2030 4103 y Fq(2)p Fo(r)2125 4145 y Fr(\000)g Fs(q)2272 4103 y Fp(\000)p Fq(2)p Fo(r)2399 4145 y Ft(\))p Fs(R)q(;)-19 4390 y Fm(for)35 b Fs(r)30 b Fr(2)e Fs(I)348 4405 y Fo(n)395 4390 y Fm(.)-263 4618 y Fi(Pr)n(oof:)1578 5197 y Ft(23)p eop %%Page: 24 24 24 23 bop -192 614 a Ft(\(1\))48 b(Let)30 b Fs(\026)e Ft(b)s(e)i(an)f Fs(f)11 b Ft(-fold)27 b(zero)i(of)g Fs(P)1155 629 y Fo(n)p Fp(\000)p Fq(2)1292 614 y Ft(,)h(i.)e(e.)h(the)h (dimension)e(of)g(the)i(generalized)e(eigenspace)i(for)-19 735 y(the)k(eigen)m(v)-5 b(alue)33 b Fs(\026)f Ft(in)h Fs(k)874 750 y Fo(q)912 735 y Ft([)p Fs(a;)17 b(b)p Ft(])1102 750 y Fo(n)p Fp(\000)p Fq(2)1272 735 y Ft(is)33 b Fs(f)11 b Ft(,)33 b(and)g(let)g Fs(v)1869 750 y Fq(1)1909 735 y Fs(;)17 b(:)g(:)g(:)e(;)i(v)2174 750 y Fo(f)2253 735 y Ft(b)s(e)33 b(a)g(basis)g(of)g Fs(V)2898 699 y Fo(n)p Fp(\000)p Fq(2)2877 759 y Fo(\026)3035 735 y Ft(.)46 b(Therefore)-19 855 y(there)31 b(exists)g(a)f(p)s(ositiv)m(e)g(in)m (teger)g Fs(t)g Ft(suc)m(h)i(that)e Fs(v)1792 870 y Fo(\027)1852 855 y Fr(\001)17 b Ft(\()p Fs(x)h Fr(\000)f Fs(\026)p Ft(\))2199 819 y Fo(t)2256 855 y Ft(=)28 b(0)i(for)f Fs(\027)35 b Ft(=)27 b(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(f)11 b Ft(.)42 b(Let)30 b Fs(\030)35 b Ft(b)s(e)-19 976 y(as)i(in)f(Lemma)f (4.1.)55 b(Then)37 b Fs(v)1087 991 y Fq(1)1127 976 y Fs(\030)5 b(;)17 b(:)g(:)g(:)e(;)i(v)1440 991 y Fo(f)1485 976 y Fs(\030)41 b Ft(are)c(linearly)d(indep)s(enden)m(t)k(elemen)m(ts) e(of)g Fs(k)3214 991 y Fo(q)3252 976 y Ft([)p Fs(a;)17 b(b)p Ft(])3442 991 y Fo(n)3490 976 y Ft(,)-19 1096 y(since)33 b Fs(k)271 1111 y Fo(q)309 1096 y Ft([)p Fs(a;)17 b(b)p Ft(])37 b(is)32 b(an)h(in)m(tegral)e(domain.)42 b(Moreo)m(v)m(er)34 b(b)m(y)f(part)g(\(2\))f(of)g(the)h(lemma,)933 1323 y(\()p Fs(v)1018 1338 y Fo(\027)1061 1323 y Fs(\030)5 b Ft(\))22 b Fr(\001)g Ft(\()p Fs(x)g Fr(\000)h Fs(\026)p Ft(\))1531 1282 y Fo(t)1588 1323 y Ft(=)1691 1227 y Fl(\020)1741 1323 y Fs(v)1788 1338 y Fo(\027)1853 1323 y Fr(\001)f Ft(\()p Fs(x)h Fr(\000)f Fs(\026)p Ft(\))2215 1282 y Fo(t)2244 1227 y Fl(\021)2311 1323 y Fs(\030)32 b Ft(=)27 b(0)p Fs(;)-19 1535 y Ft(whence)d Fs(v)359 1550 y Fo(\027)402 1535 y Fs(\030)i Ft(for)c Fs(\027)34 b Ft(=)27 b(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(f)32 b Ft(b)s(elongs)21 b(to)h(the)g (generalized)f(eigenspace)i(for)e(the)h(eigen)m(v)-5 b(alue)21 b Fs(\026)-19 1656 y Ft(in)32 b Fs(k)146 1671 y Fo(q)184 1656 y Ft([)p Fs(a;)17 b(b)p Ft(])374 1671 y Fo(n)421 1656 y Ft(.)44 b(Therefore)34 b Fs(\026)e Ft(is)g(at)g(least)g(an)h Fs(f)11 b Ft(-fold)30 b(zero)j(of)f Fs(P)2269 1671 y Fo(n)2316 1656 y Ft(.)-192 1857 y(\(2\))48 b(F)-8 b(or)32 b Fs(n)c Ft(=)f(0)33 b(and)f Fs(n)c Ft(=)g(1)k(one)h (easily)f(computes)-19 2069 y Fs(P)44 2084 y Fq(0)84 2069 y Ft(\()p Fs(Y)21 b Ft(\))27 b(=)h(0)6 b Fr(\000)g Fs(Y)50 b Ft(=)27 b Fr(\000)p Fs(Y)54 b Ft(and)33 b Fs(P)1158 2084 y Fq(1)1197 2069 y Ft(\()p Fs(Y)22 b Ft(\))27 b(=)h Fs(Y)1561 2027 y Fq(2)1607 2069 y Fr(\000)6 b Fs(\013)q Ft(\()1791 2002 y Fr(p)p 1874 2002 47 4 v 67 x Fs(q)k Fr(\000)2010 2002 y(p)p 2094 2002 V 2094 2069 a Fs(q)2141 2020 y Fp(\000)p Fq(1)2235 2069 y Ft(\))2273 2027 y Fq(2)2312 2069 y Fs(Y)28 b Fr(\000)6 b Fs(q)2527 2027 y Fp(\000)p Fq(1)2622 2069 y Fs(\014)g(\015)12 b Fr(\000)6 b Fs(\013)2892 2027 y Fq(2)2931 2069 y Ft(\()2969 2002 y Fr(p)p 3051 2002 V 3051 2069 a Fs(q)11 b Fr(\000)3188 2002 y(p)p 3271 2002 V 67 x Fs(q)3318 2020 y Fp(\000)p Fq(1)3412 2069 y Ft(\))3450 2027 y Fq(2)3490 2069 y Fs(:)-19 2281 y Ft(No)m(w)31 b(assume)g Fs(n)d Fr(\025)h Ft(2.)42 b(Due)31 b(to)f(deg)18 b Fs(P)1393 2296 y Fo(n)1440 2281 y Ft(\()p Fs(Y)j Ft(\))28 b(=)f(2)18 b(+)g(deg)g Fs(P)2113 2296 y Fo(n)p Fp(\000)p Fq(2)2250 2281 y Ft(\()p Fs(Y)k Ft(\))30 b(and)h(part)f(\(a\))g(only)g(the)h(t)m(w)m(o)-19 2401 y(extra)i(zeros)h Fs(a)529 2416 y Fo(n)608 2401 y Ft(and)f Fs(b)839 2416 y Fo(n)919 2401 y Ft(of)f Fs(P)1093 2416 y Fo(n)1172 2401 y Ft(ha)m(v)m(e)i(to)f(b)s(e)f(determined.)44 b(Vieta's)32 b(Theorem)h(sa)m(ys)492 2613 y Fs(z)537 2628 y Fo(n)p Fp(\000)p Fq(2)696 2613 y Fr(\000)23 b Fs(a)847 2628 y Fo(n)916 2613 y 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b Fs(a)474 2985 y Fo(n)554 2970 y Ft(and)e Fs(b)784 2985 y Fo(n)864 2970 y Ft(m)m(ust)h(satisfy)g(the)g(quadratic)f(equation)g (in)g Fs(T)607 3182 y(T)678 3141 y Fq(2)739 3182 y Fr(\000)23 b Ft(\()p Fs(z)922 3197 y Fo(n)p Fp(\000)p Fq(2)1081 3182 y Fr(\000)g Fs(z)1226 3197 y Fo(n)1273 3182 y Ft(\))p Fs(T)36 b Ft(+)22 b Fs(d)1553 3197 y Fo(n)1622 3182 y Fr(\000)g Fs(d)1772 3197 y Fo(n)p Fp(\000)p Fq(2)1931 3182 y Ft(+)g Fs(z)2074 3197 y Fo(n)p Fp(\000)p Fq(2)2212 3182 y Ft(\()p Fs(z)2295 3197 y Fo(n)p Fp(\000)p Fq(2)2454 3182 y Fr(\000)h Fs(z)2599 3197 y Fo(n)2646 3182 y Ft(\))28 b(=)f(0)p Fs(:)-19 3394 y Ft(Let)33 b Fs(l)d Ft(=)d Fs(n=)p Ft(2.)44 b(W)-8 b(e)33 b(compute)f(the)h(co)s(e\016cien)m(ts)h(of)e (this)g(equation:)22 3595 y(\(a\))49 b(The)33 b(co)s(e\016cien)m(t)h Fs(z)897 3610 y Fo(n)976 3595 y Ft(is)f(the)g(negativ)m(e)f(trace)h(of) f Fs(M)2076 3610 y Fo(n)2124 3595 y Ft(.)43 b(Therefore)772 3807 y Fs(z)817 3822 y Fo(n)886 3807 y Fr(\000)23 b Fs(z)1031 3822 y Fo(n)p Fp(\000)p 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b(are)g(no)f(limits)e(of)i(summation.) 1012 4612 y Fs(D)1093 4627 y Fo(n)1223 4612 y Ft(:=)99 b Fs(\013)1488 4571 y Fq(2)1545 4529 y Fl(X)1544 4704 y Fo(r)r()g Ft(1)d(cannot)i(b)s (e)f(diagonalized\).)-263 3159 y(In)33 b(order)f(to)g(pro)m(v)m(e)i (\(2\))60 b Fr(\))f Ft(\(1\),)32 b(assume)h(there)g(is)f(a)g(p)s (olynomial)d Fs(P)2355 3174 y Fo(n)2434 3159 y Ft(with)j(a)g(double)g (zero.)44 b(Then)-263 3279 y(b)m(y)g(4.2)e(there)h(are)g(distinct)f (half)f(in)m(tegers)i Fs(l)i Ft(and)e Fs(m)g Ft(suc)m(h)h(that)e Fs(l)32 b Ft(+)c Fs(m)43 b Ft(is)g(an)f(in)m(teger)h(\(b)s(ecause)-263 3400 y Fs(l)r(;)17 b(m)28 b Fr(2)g Fs(I)62 3415 y Fo(n)109 3400 y Ft(\))33 b(and)g Fs(\025)427 3415 y Fo(l)480 3400 y Ft(=)28 b Fs(\025)641 3415 y Fo(m)707 3400 y Ft(.)44 b(Hence)141 3597 y Fs(\013)p 141 3641 63 4 v 148 3732 a Ft(2)214 3664 y(\()p Fs(q)299 3623 y Fo(l)347 3664 y Fr(\000)22 b Fs(q)493 3623 y Fp(\000)p Fo(l)574 3664 y Ft(\))612 3623 y Fq(2)674 3664 y Ft(+)782 3597 y(1)p 782 3641 49 4 v 782 3732 a(2)840 3664 y(\()p Fs(q)925 3623 y Fq(2)p Fo(l)1009 3664 y Fr(\000)g Fs(q)1155 3623 y Fp(\000)p Fq(2)p Fo(l)1271 3664 y Ft(\))p Fs(R)29 b Ft(=)1525 3597 y Fs(\013)p 1525 3641 63 4 v 1532 3732 a Ft(2)1598 3664 y(\()p Fs(q)1683 3623 y Fo(m)1772 3664 y Fr(\000)22 b Fs(q)1918 3623 y Fp(\000)p Fo(m)2039 3664 y Ft(\))2077 3623 y Fq(2)2139 3664 y Ft(+)2247 3597 y(1)p 2247 3641 49 4 v 2247 3732 a(2)2306 3664 y(\()p Fs(q)2391 3623 y Fq(2)p Fo(m)2514 3664 y Fr(\000)h Fs(q)2661 3623 y Fp(\000)p Fq(2)p Fo(m)2818 3664 y Ft(\))p Fs(R)61 b Fr(\))262 3901 y(\))431 3834 y Fs(\013)p 431 3878 63 4 v 438 3969 a Ft(2)504 3901 y(\()p Fs(q)589 3860 y Fq(2)p Fo(l)672 3901 y Ft(+)22 b Fs(q)817 3860 y Fp(\000)p Fq(2)p Fo(l)955 3901 y Fr(\000)h Fs(q)1102 3860 y Fq(2)p Fo(m)1226 3901 y Fr(\000)g Fs(q)1373 3860 y Fp(\000)p Fq(2)p Fo(m)1529 3901 y Ft(\))f(+)1697 3834 y Fs(R)p 1697 3878 76 4 v 1710 3969 a Ft(2)1782 3901 y(\()p Fs(q)1867 3860 y Fq(2)p Fo(l)1950 3901 y Fr(\000)h Fs(q)2097 3860 y Fp(\000)p Fq(2)p Fo(l)2235 3901 y Fr(\000)g Fs(q)2382 3860 y Fq(2)p Fo(m)2506 3901 y Ft(+)f Fs(q)2651 3860 y Fp(\000)p Fq(2)p Fo(m)2807 3901 y Ft(\))28 b(=)f(0)-231 4138 y Fr(\))60 b Ft(\()p Fs(q)14 4097 y Fo(l)q Fp(\000)p Fo(m)178 4138 y Fr(\000)21 b Fs(q)323 4097 y Fo(m)p Fp(\000)p Fo(l)466 4138 y Ft(\)\()552 4071 y Fs(\013)p 552 4115 63 4 v 559 4206 a Ft(2)625 4138 y(\()p Fs(q)710 4097 y Fo(l)q Fq(+)p Fo(m)874 4138 y Fr(\000)g Fs(q)1019 4097 y Fp(\000)p Fo(l)q Fp(\000)p Fo(m)1217 4138 y Ft(\))g(+)1383 4071 y Fs(R)p 1383 4115 76 4 v 1396 4206 a Ft(2)1468 4138 y(\()p Fs(q)1553 4097 y Fo(l)q Fq(+)p Fo(m)1717 4138 y Ft(+)g Fs(q)1861 4097 y Fp(\000)p Fo(l)q Fp(\000)p Fo(m)2059 4138 y Ft(\)\))28 b(=)f(0)60 b Fr(\))f Ft(\()p Fs(q)2619 4097 y Fo(l)q Fp(\000)p Fo(m)2783 4138 y Fr(\000)22 b Fs(q)2929 4097 y Fo(m)p Fp(\000)p Fo(l)3072 4138 y Ft(\))p Fs(\025)3167 4153 y Fo(l)q Fq(+)p Fo(m)3337 4138 y Ft(=)28 b(0)p Fs(:)-263 4384 y Ft(Since)22 b Fs(q)k Ft(is)21 b(not)h(a)g(ro)s(ot)f(of)g(unit)m(y)-8 b(,)24 b(this)e(yields)f Fs(\025)1434 4399 y Fo(l)q Fq(+)p Fo(m)1605 4384 y Ft(=)28 b(0.)39 b(The)23 b(assumption)e Fs(R)29 b Fr(6)p Ft(=)f(0)21 b(implies)f Fs(l)r Ft(+)p Fs(m)28 b Fr(6)p Ft(=)g(0,)-263 4504 y(whence)37 b(0)c(is)h(a)g(double)g(zero)h (of)f Fs(P)1046 4519 y Fq(2)p Fp(j)p Fo(l)q Fq(+)p Fo(m)p Fp(j)1263 4504 y Ft(,)h(and)g(b)m(y)g(the)f(Lemma,)g(the)g(generalized) g(eigenspace)h(of)f(0)-263 4624 y(do)s(es)39 b(not)e(equal)h(the)g (eigenspace)h(in)e Fs(V)1236 4640 y Fq(1)p Fo(;)p Fq(2)p Fp(j)p Fo(l)q Fq(+)p Fo(m)p Fp(j)1508 4624 y Ft(.)60 b(Finally)-8 b(,)36 b(condition)g(\(3\))i(is)f(equiv)-5 b(alen)m(t)38 b(to)f Fs(R)h Fr(6)p Ft(=)e(0)-263 4745 y(and)d Fs(\025)-16 4760 y Fo(n)31 4745 y Fs(\025)88 4760 y Fp(\000)p Fo(n)217 4745 y Fr(6)p Ft(=)28 b(0)k(for)g(all)f(p)s (ositiv)m(e)h(in)m(tegers)h Fs(n)p Ft(,)f(i.)g(e.)h(to)f(condition)f (\(2\).)1027 b Fk(2)1578 5197 y Ft(27)p eop %%Page: 28 28 28 27 bop -263 614 a Ft(There)34 b(is)e(a)h(similar)c(statemen)m(t)k (ab)s(out)f(\(1)p Fs(;)17 b(K)7 b Ft(\)-primitiv)m(e)29 b(elemen)m(ts)k(of)f Fs(U)2535 629 y Fo(q)2574 614 y Ft(\()p Fs(sl)r Ft(\(2\)\).)-263 807 y Fj(Remark)37 b(4.5)169 b Ft(1.)48 b(W)-8 b(e)33 b(consider)g Fs(y)e Ft(:=)c Fs(xK)7 b Ft(.)45 b(Then)33 b Fs(y)j Ft(is)c(\(1)p Fs(;)17 b(K)7 b Ft(\)-primitiv)m(e)29 b(and)219 995 y Fs(M)323 954 y Fp(0)313 1020 y Fo(n)443 995 y Ft(:=)83 b Fs(\032)679 1010 y Fo(e;n)779 995 y Ft(\()p Fs(y)t Ft(\))27 b(=)g Fs(\032)1087 1010 y Fo(e;n)1187 995 y Ft(\()p Fs(K)7 b Ft(\))p Fs(M)1447 1010 y Fo(n)1522 995 y Ft(=)457 1430 y(=)629 1060 y Fl(0)629 1206 y(B)629 1256 y(B)629 1305 y(B)629 1355 y(B)629 1405 y(B)629 1455 y(B)629 1505 y(B)629 1554 y(B)629 1608 y(@)719 1131 y Ft(\(1)21 b Fr(\000)i Fs(q)974 1095 y Fp(\000)p Fo(n)1076 1131 y Ft(\))p Fs(\013)224 b(q)1447 1095 y Fo(n)p Fp(\000)p Fq(2)1584 1131 y Fs(\015)452 b Ft(0)356 b Fr(\001)17 b(\001)g(\001)310 b Ft(0)861 1290 y([)p Fs(n)p Ft(])p Fs(\014)246 b Ft(\(1)22 b Fr(\000)g Fs(q)1529 1254 y Fq(2)p Fp(\000)p Fo(n)1666 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y(C)3163 1355 y(C)3163 1405 y(C)3163 1455 y(C)3163 1505 y(C)3163 1554 y(C)3163 1608 y(A)3252 1430 y Fs(:)-19 1903 y Ft(Let)32 b Fs(M)259 1867 y Fp(00)249 1928 y Fo(n)333 1903 y Ft(the)g(matrix)e(obtained)h(from)f Fs(M)1549 1867 y Fp(0)1539 1928 y Fo(n)1617 1903 y Ft(after)h (replacing)f Fs(\013)i Ft(b)m(y)h Fr(\000)p Fs(\013)f Ft(and)g(in)m(terc)m(hanging)e Fs(\014)-19 2024 y Ft(and)c Fs(\015)5 b Ft(.)41 b(Comparison)24 b(with)i Fs(M)1133 2039 y Fo(n)1205 2024 y Ft(yields)g(that)f(for)g(all)e Fs(i;)17 b(j)34 b Fr(\024)28 b Fs(n)8 b Ft(+)g(1,)26 b(the)g Fs(i;)17 b(j)6 b Ft(-en)m(try)27 b(of)e Fs(M)3218 2039 y Fo(n)3290 2024 y Ft(is)g(the)-19 2144 y(\()p Fs(n)s Ft(+)s(2)s Fr(\000)s Fs(i;)17 b(n)s Ft(+)s(2)s Fr(\000)s Fs(j)6 b Ft(\)-en)m(try)28 b(of)23 b Fs(M)1212 2108 y Fp(00)1202 2169 y Fo(n)1256 2144 y Ft(,)i(whence)g(the)f(c)m (haracteristic)f(p)s(olynomials)e(of)h Fs(M)3118 2159 y Fo(n)3189 2144 y Ft(and)i Fs(M)3474 2108 y Fp(00)3464 2169 y Fo(n)-19 2264 y Ft(are)j(iden)m(tical.)40 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y(eigenspaces)c(are)f(di\013eren)m(t)g(from)e(the)i(generalized)f (eigenspaces.)-143 4605 y(3.)48 b(If)33 b(and)f(only)h(if)e(there)i(is) f(an)h(elemen)m(t)f Fs(n)c Fr(2)1605 4566 y Fq(1)p 1605 4582 36 4 v 1605 4639 a(2)1650 4605 y Ff(N)33 b Fr(n)21 b Ff(N)43 b Ft(satisfying)1166 4882 y Fs(q)1213 4841 y Fp(\000)p Fq(1)1307 4882 y Fs(\014)6 b(\015)1441 4736 y Fl( )1516 4814 y Fs(q)1563 4778 y Fo(n)1632 4814 y Ft(+)22 b Fs(q)1777 4778 y Fp(\000)p Fo(n)p 1516 4858 363 4 v 1543 4950 a Fs(q)k Fr(\000)c Fs(q)1758 4921 y Fp(\000)p Fq(1)1889 4736 y Fl(!)1955 4759 y Fq(2)2022 4882 y Ft(=)27 b Fr(\000)p Fs(\013)2265 4841 y Fq(2)2305 4882 y Fs(;)1578 5197 y Ft(28)p eop %%Page: 29 29 29 28 bop -19 614 a Ft(then)32 b(there)g(is)e(a)h(non)g(zero)g(elemen)m (t)g Fs(\030)1425 578 y Fp(0)1475 614 y Fr(2)e Fs(k)1621 629 y Fo(q)1658 614 y Ft([)p Fs(a;)17 b(b)p Ft(])1848 629 y Fq(2)p Fo(n)1962 614 y Ft(suc)m(h)32 b(that)f Fs(\030)2438 578 y Fp(0)2480 614 y Fr(\001)18 b Fs(x)29 b Ft(=)e(0)p Fs(:)k 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Fr(!)27 b Fs(k)j Ft(suc)m(h)d(that)f Fs(u)9 b Fr(\001)g Fs(w)30 b Ft(=)e Fs(\037)p Ft(\()p Fs(u)p Ft(\))p Fs(w)-263 4122 y Ft(for)33 b(all)e Fs(u)e Fr(2)g Fs(U)10 b Ft(.)46 b(Let)34 b Fs(v)575 4137 y Fq(1)614 4122 y Ft(,)p Fs(:)17 b(:)g(:)f Ft(,)34 b Fs(v)880 4137 y Fo(n)960 4122 y Ft(b)s(e)g(a)f(basis)g(of)g Fs(V)54 b Ft(consisting)33 b(of)g(eigen)m(v)m(ectors)i(of)e(the)g(action)g(of)f Fs(x)p Ft(,)-263 4242 y(i.)g(e.)h Fs(x)23 b Fr(\001)e Fs(v)101 4257 y Fo(j)166 4242 y Ft(=)27 b Fs(\025)326 4257 y Fo(j)363 4242 y Fs(v)410 4257 y Fo(j)479 4242 y Ft(for)32 b Fs(j)i Ft(=)27 b(1,)p Fs(:)17 b(:)g(:)f Ft(,)33 b Fs(n)g Ft(and)f Fs(\025)1409 4257 y Fo(j)1473 4242 y Fr(2)c Fs(k)s Ft(.)44 b(Then)-263 4462 y Fs(x)s Fr(\001)s Ft(\()p Fs(w)6 b Fr(\012)s Fs(v)67 4477 y Fo(j)104 4462 y Ft(\))27 b(=)h(\()p Fs(g)7 b Fr(\001)s Fs(w)s Ft(\))s Fr(\012)s Ft(\()p Fs(x)s Fr(\001)s Fs(v)764 4477 y Fo(j)799 4462 y Ft(\))s(+)s(\()p Fs(x)s Fr(\001)s Fs(w)s Ft(\))s Fr(\012)s Fs(v)1287 4477 y Fo(j)1351 4462 y Ft(=)27 b Fs(\037)p Ft(\()p Fs(g)t Ft(\))p Fs(w)6 b Fr(\012)s Fs(\025)1855 4477 y Fo(j)1891 4462 y Fs(v)1938 4477 y Fo(j)1978 4462 y Ft(+)s Fs(\037)p Ft(\()p Fs(x)p Ft(\))p Fs(w)g Fr(\012)s Fs(v)2452 4477 y Fo(j)2516 4462 y Ft(=)28 b(\()p Fs(\037)p Ft(\()p Fs(g)t Ft(\))p Fs(\025)2903 4477 y Fo(j)2941 4462 y Ft(+)s Fs(\037)p Ft(\()p Fs(x)p Ft(\)\))p Fs(w)6 b Fr(\012)s Fs(v)3453 4477 y Fo(j)3490 4462 y Fs(;)-263 4682 y Ft(whence)35 b(the)f(set)g Fr(f)p Fs(w)24 b Fr(\012)f Fs(v)692 4697 y Fq(1)732 4682 y Ft(,)p Fs(:)17 b(:)g(:)f Ft(,)33 b Fs(w)25 b Fr(\012)e Fs(v)1192 4697 y Fo(n)1239 4682 y Fr(g)33 b Ft(is)f(a)h(basis)g(of)g Fs(k)s(w)25 b Fr(\012)e Fs(V)54 b Ft(consisting)33 b(of)f(eigen)m(v)m (ectors)j(of)d Fs(x)p Ft(.)-263 4802 y Fk(2)1578 5197 y Ft(29)p eop %%Page: 30 30 30 29 bop -263 639 a Fu(5)166 b(Application)57 b(to)f Fd(U)1177 660 y Fs(q)1228 639 y Fc(\()p Fa(g)o Fc(\))-263 858 y Ft(As)30 b(in)f(the)h(last)f(section,)h(w)m(e)h(assume)f(that)f Fs(k)k Ft(is)c(an)g(algebraically)e(closed)i(\014eld)h(of)f(c)m (haracteristic)g(0.)-263 978 y(Let)36 b(\()p Fs(a)4 993 y Fo(ij)65 978 y Ft(\))g(b)s(e)g(an)f Fs(n)25 b Fr(\002)f Fs(n)36 b Ft(matrix)f(with)g(in)m(teger)g(co)s(e\016cien)m(ts)i(suc)m (h)g(that)f Fs(a)2548 993 y Fo(ii)2633 978 y Ft(=)d(2)j(for)f(all)f Fs(i)p Ft(,)i Fs(a)3265 993 y Fo(ij)3359 978 y Fr(6)p Ft(=)d(0)-263 1099 y(for)d(all)e Fs(i)g Fr(6)p Ft(=)f Fs(j)6 b Ft(,)31 b(and)f(there)g(are)g(relativ)m(ely)f(prime)g(in)m (tegers)h Fs(d)1984 1114 y Fq(1)2023 1099 y Fs(;)17 b(:)g(:)g(:)f(;)h (d)2293 1114 y Fo(n)2367 1099 y Fr(2)28 b(f)p Ft(1)p Fs(;)17 b Ft(2)p Fs(;)g Ft(3)p Fr(g)28 b Ft(suc)m(h)k(that)d(\()p Fs(d)3339 1114 y Fo(i)3367 1099 y Fs(a)3418 1114 y Fo(ij)3479 1099 y Ft(\))-263 1219 y(is)j(a)h(symmetric)f(p)s(ositiv)m(e)f (de\014nite)i(matrix.)-263 1339 y(Th)m(us)i(\()p Fs(a)74 1354 y Fo(ij)134 1339 y Ft(\))e(is)f(the)h(Cartan)f(matrix)f(of)h(a)h (\014nite)f(dimensional)e(semisimple)g(Lie)i(algebra)g Fn(g)p Ft(.)-263 1460 y(Let)g Fs(q)f Fr(2)d Fs(k)22 b Fr(n)d(f)p Ft(0)p Fr(g)30 b Ft(b)s(e)h(not)g(a)f(ro)s(ot)g(of)h(1)f (and)i(de\014ne)g Fs(q)1690 1475 y Fo(i)1746 1460 y Ft(:=)27 b Fs(q)1923 1424 y Fo(d)1959 1434 y Fg(i)1990 1460 y Ft(.)43 b(The)32 b(standard)f Fs(q)t Ft(-deformation)e Fs(U)3353 1475 y Fo(q)3391 1460 y Ft(\()p Fn(g)p Ft(\))-263 1580 y(is)j(the)h(algebra)f(generated)h(b)m(y)h Fs(E)997 1595 y Fo(i)1025 1580 y Fs(;)17 b(F)1132 1595 y Fo(i)1160 1580 y Fs(;)g(K)1287 1595 y Fo(i)1315 1580 y Fs(;)g(K)1449 1539 y Fp(\000)p Fq(1)1442 1603 y Fo(i)1576 1580 y Ft(sub)5 b(ject)34 b(to)e(the)h(follo)m(wing)d(relations)828 1791 y Fs(K)911 1806 y Fo(i)939 1791 y Fs(K)1029 1750 y Fp(\000)p Fq(1)1022 1816 y Fo(i)1151 1791 y Ft(=)e(1)f(=)h Fs(K)1525 1750 y Fp(\000)p Fq(1)1518 1816 y Fo(i)1619 1791 y Fs(K)1702 1806 y Fo(i)1730 1791 y Fs(;)50 b(K)1890 1806 y Fo(i)1918 1791 y Fs(K)2001 1806 y Fo(j)2065 1791 y Ft(=)28 b Fs(K)2252 1806 y Fo(j)2288 1791 y Fs(K)2371 1806 y Fo(i)2399 1791 y Fs(;)770 1947 y(K)853 1962 y Fo(i)881 1947 y Fs(E)953 1962 y Fo(j)990 1947 y Fs(K)1080 1905 y Fp(\000)p Fq(1)1073 1971 y Fo(i)1202 1947 y Ft(=)f Fs(q)1352 1892 y Fo(a)1389 1902 y Fg(ij)1348 1969 y Fo(i)1448 1947 y Fs(E)1520 1962 y Fo(j)1557 1947 y Fs(;)49 b(K)1716 1962 y Fo(i)1744 1947 y Fs(F)1807 1962 y Fo(j)1844 1947 y Fs(K)1934 1905 y Fp(\000)p Fq(1)1927 1971 y Fo(i)2056 1947 y Ft(=)28 b Fs(q)2207 1892 y Fp(\000)p Fo(a)2299 1902 y Fg(ij)2203 1969 y Fo(i)2358 1947 y Fs(F)2421 1962 y Fo(j)2458 1947 y Fs(;)1030 2159 y(E)1102 2174 y Fo(i)1130 2159 y Fs(F)1193 2174 y Fo(j)1252 2159 y Fr(\000)23 b Fs(F)1415 2174 y Fo(j)1452 2159 y Fs(E)1524 2174 y Fo(i)1580 2159 y Ft(=)k Fs(\016)1726 2174 y Fo(ij)1797 2092 y Fs(K)1880 2107 y Fo(i)1930 2092 y Fr(\000)c Fs(K)2120 2050 y Fp(\000)p Fq(1)2113 2115 y Fo(i)p 1797 2136 418 4 v 1838 2230 a Fs(q)1881 2245 y Fo(i)1932 2230 y Fr(\000)g Fs(q)2079 2189 y Fp(\000)p Fq(1)2075 2253 y Fo(i)-263 2427 y Ft(for)31 b(all)f Fs(i;)17 b(j)6 b Ft(,)31 b(and)h(the)g Fs(q)t Ft(-deformed)e(Serre)j(relation)c(\(see)k([J,)f(5.1.1\(vi\)])e(b)s(et)m (w)m(een)k(the)e Fs(E)3024 2442 y Fo(i)3052 2427 y Ft('s)g(resp.)g(the) -263 2547 y Fs(F)-200 2562 y Fo(i)-171 2547 y Ft('s)k(whic)m(h)h(w)m(e) g(do)f(not)f(need)i(explicitly)-8 b(.)52 b Fs(U)1435 2562 y Fo(q)1473 2547 y Ft(\()p Fn(g)p Ft(\))36 b(is)g(a)f(Hopf)h (algebra)f(where)i(all)d(the)i(elemen)m(ts)h Fs(K)3489 2562 y Fo(i)-263 2667 y Ft(are)c(group-lik)m(e)e(and)453 2878 y(\001\()p Fs(E)644 2893 y Fo(i)672 2878 y Ft(\))d(=)g(1)21 b Fr(\012)i Fs(E)1084 2893 y Fo(i)1135 2878 y Ft(+)f Fs(E)1305 2893 y Fo(i)1355 2878 y Fr(\012)h Fs(K)1538 2893 y Fo(i)1566 2878 y Fs(;)49 b Ft(\001\()p Fs(F)1824 2893 y Fo(i)1853 2878 y Ft(\))27 b(=)h Fs(K)2112 2837 y Fp(\000)p Fq(1)2105 2903 y Fo(i)2229 2878 y Fr(\012)22 b Fs(F)2391 2893 y Fo(i)2442 2878 y Ft(+)g Fs(F)2603 2893 y Fo(i)2653 2878 y Fr(\012)h Ft(1)-263 3090 y(for)38 b(all)e Fs(i)p Ft(.)59 b(F)-8 b(or)37 b(1)g Fr(\024)g Fs(i)g Fr(\024)g Fs(n)i Ft(let)e Fs(U)1025 3105 y Fo(i)1091 3090 y Ft(b)s(e)h(the)h(subalgebra)e(of)h Fs(U)2079 3105 y Fo(q)2117 3090 y Ft(\()p Fn(g)p Ft(\))g(generated)h(b)m(y)f Fs(K)2954 3105 y Fo(i)2983 3090 y Fs(;)17 b(K)3117 3048 y Fp(\000)p Fq(1)3110 3112 y Fo(i)3211 3090 y Fs(;)g(E)3327 3105 y Fo(i)3355 3090 y Fs(;)g(F)3462 3105 y Fo(i)3490 3090 y Ft(.)-263 3210 y(Then)34 b Fs(U)58 3225 y Fo(i)114 3182 y Fr(\030)115 3214 y Ft(=)219 3210 y Fs(U)285 3225 y Fo(q)317 3235 y Fg(i)348 3210 y Ft(\()p Fs(sl)r Ft(\(2\)\))e(as)h (Hopf)f(algebras.)-263 3330 y(Let)41 b Fr(C)47 b Ft(b)s(e)41 b(the)g(semisimple)d(tensor)k(category)f(of)f(all)e(\014nite)j (dimensional)d(left)i Fs(U)2851 3345 y Fo(q)2889 3330 y Ft(\()p Fn(g)p Ft(\)-mo)s(dules)f(of)-263 3451 y(t)m(yp)s(e)27 b(1,)g(that)f(is)f(all)f(eigen)m(v)-5 b(alues)25 b(of)h(the)g(left)f(m) m(ultiplication)c(with)k Fs(K)2322 3466 y Fo(i)2376 3451 y Ft(for)g(all)50 b Fs(i)26 b Ft(are)f(of)h(the)g(form)e Fs(q)3423 3415 y Fo(m)3490 3451 y Ft(,)-263 3571 y Fs(m)k Fr(2)g Ff(Z)d Ft(\(see)31 b(for)g(example)f([J,)h(4.3]\).)43 b(The)31 b(dual)f(Hopf)h(algebra)f Fs(U)2203 3586 y Fo(q)2241 3571 y Ft(\()p Fn(g)p Ft(\))2367 3523 y Fq(0)2367 3603 y Fp(C)2443 3571 y Ft(is)g(the)h Fs(q)t Ft(-deformed)f(algebra)-263 3691 y(of)i(regular)f(functions)g(on)h(the)g(simply)f(connected,)i (connected)h(semisimple)29 b(algebraic)i(group)g(with)-263 3812 y(Lie)j(algebra)g Fn(g)p Ft(.)49 b(Let)35 b Fs(x)c Fr(2)g Fs(U)799 3827 y Fo(q)838 3812 y Ft(\()p Fn(g)p Ft(\))j(b)s(e)h(a)f(\()p Fs(g)t(;)17 b Ft(1\)-primitiv)m(e)31 b(elemen)m(t)j(whic)m(h)h(is)f(not)g(a)g(scalar)g(m)m(ultiple)-263 3932 y(of)d Fs(g)22 b Fr(\000)e Ft(1.)42 b(Then)33 b(there)e(is)g(some) g Fs(i)g Ft(suc)m(h)i(that)d Fs(g)h Ft(=)d Fs(K)1736 3891 y Fp(\000)p Fq(1)1729 3955 y Fo(i)1861 3932 y Ft(and)j Fs(x)e Fr(2)f Fs(U)2293 3947 y Fo(i)2352 3932 y Ft(is)j(a)f Fs(k)s Ft(-linear)f(com)m(bination)g(of)-263 4053 y Fs(K)-173 4011 y Fp(\000)p Fq(1)-180 4075 y Fo(i)-57 4053 y Fr(\000)22 b Ft(1,)32 b Fs(E)222 4068 y Fo(i)250 4053 y Fs(K)340 4011 y Fp(\000)p Fq(1)333 4075 y Fo(i)467 4053 y Ft(and)g Fs(F)719 4068 y Fo(i)779 4053 y Ft([CM)q(,)h(Theorem)f(A].)g(W)-8 b(e)33 b(call)d Fs(x)j Fm(semisimple)d Ft(if)h(m)m(ultiplication)d (with)j Fs(x)-263 4173 y Ft(is)h(a)h(diagonalizable)c(op)s(erator)j(on) g(all)f(\014nite)h(dimensional)e(left)i Fs(U)2238 4188 y Fo(q)2276 4173 y Ft(\()p Fn(g)p Ft(\)-mo)s(dules.)-263 4391 y Fj(Lemma)38 b(5.1)48 b Fm(F)-7 b(or)31 b(any)g Ft(1)c Fr(\024)h Fs(i)g Fr(\024)g Fs(n)k Fm(let)f Fs(V)52 b Fm(b)-5 b(e)31 b(a)g(\014nite)g(dimensional)e(left)i Fs(U)2580 4406 y Fo(i)2608 4391 y Fm(-mo)-5 b(dule)30 b(of)h(typ)-5 b(e)32 b(1)e(and)-263 4512 y Fs(x)e Fr(2)h Fs(U)-19 4527 y Fo(i)37 4512 y Fr(\032)f Fs(U)208 4527 y Fo(q)246 4512 y Ft(\()p Fn(g)p Ft(\))35 b Fm(a)g Ft(\()p Fs(K)620 4470 y Fp(\000)p Fq(1)613 4534 y Fo(i)714 4512 y Fs(;)17 b Ft(1\))p Fm(-primitive)34 b(element.)-197 4707 y(\(1\))48 b(Ther)-5 b(e)24 b(exists)h(a)g(\014nite)f(dimensional) f(left)i Fs(U)1580 4722 y Fo(q)1619 4707 y Ft(\()p Fn(g)p Ft(\))p Fm(-mo)-5 b(dule)24 b Fs(W)38 b Fm(of)25 b(typ)-5 b(e)25 b(1)g(and)f(a)h(one-dimensional)-19 4828 y(left)35 b Fs(U)214 4843 y Fo(q)253 4828 y Ft(\()p Fn(g)p Ft(\))p Fm(-mo)-5 b(dule)34 b Fs(k)s(a)h Fm(with)g Fs(E)1169 4843 y Fo(j)1206 4828 y Fs(a)28 b Ft(=)g(0)p Fm(,)35 b Fs(F)1566 4843 y Fo(j)1602 4828 y Fs(a)29 b Ft(=)f(0)34 b Fm(and)h Fs(K)2142 4843 y Fo(j)2178 4828 y Fs(a)29 b Ft(=)e Fs(\013)2423 4843 y Fo(j)2460 4828 y Fs(a)35 b Fm(wher)-5 b(e)35 b Fs(\013)2884 4843 y Fo(j)2948 4828 y Fr(2)29 b(f)p Ft(1)p Fs(;)17 b Fr(\000)p Ft(1)p Fr(g)34 b Fm(for)-19 4948 y(al)5 b(l)35 b Ft(1)27 b Fr(\024)h Fs(j)34 b Fr(\024)28 b Fs(n)35 b Fm(such)g(that)g Fs(V)57 b Fm(is)34 b(isomorphic)g(to)h(a)f Fs(U)1980 4963 y Fo(i)2009 4948 y Fm(-submo)-5 b(dule)34 b(of)g Fs(k)s(a)23 b Fr(\012)f Fs(W)49 b Fm(and)34 b Fs(\013)3245 4963 y Fo(i)3301 4948 y Ft(=)28 b(1)p Fm(.)1578 5197 y Ft(30)p eop %%Page: 31 31 31 30 bop -197 614 a Fm(\(2\))48 b Fs(x)36 b Fm(is)e(semisimple)f(in)i Fs(U)860 629 y Fo(i)923 614 y Fm(if)g(and)f(only)h(if)f Fs(x)i Fm(is)e(semisimple)g(in)g Fs(U)2392 629 y Fo(q)2430 614 y Ft(\()p Fn(g)p Ft(\))p Fm(.)-263 832 y Fi(Pr)n(oof:)-192 1028 y Ft(\(1\))48 b(By)40 b([J)q(,)h(10.1.14])e(or)g(the)h (classi\014cation)e(of)i(highest)f(w)m(eigh)m(t)h(mo)s(dules)f(of)g Fs(U)2882 1043 y Fo(q)2920 1028 y Ft(\()p Fn(g)p Ft(\),)j Fs(V)61 b Ft(is)39 b(con-)-19 1148 y(tained)f(in)f(a)h(\014nite)g (dimensional)e(left)h Fs(U)1531 1163 y Fo(q)1569 1148 y Ft(\()p Fn(g)p Ft(\)-mo)s(dule)2093 1123 y(~)2078 1148 y Fs(V)21 b Ft(.)60 b(F)-8 b(rom)37 b(the)i(description)e(of)h (\014nite)-19 1269 y(dimensional)g(left)h Fs(U)770 1284 y Fo(q)808 1269 y Ft(\()p Fn(g)p Ft(\)-mo)s(dules)g(it)g(is)g(kno)m(wn) i(that)2117 1244 y(~)2102 1269 y Fs(V)2221 1241 y Fr(\030)2222 1273 y Ft(=)2338 1269 y Fs(k)s(a)28 b Fr(\012)f Fs(W)14 b Ft(,)42 b(where)f Fs(W)54 b Ft(is)39 b(a)h(left)-19 1389 y Fs(U)47 1404 y Fo(q)86 1389 y Ft(\()p Fn(g)p Ft(\)-mo)s(dule)24 b(of)i(t)m(yp)s(e)h(1)f(and)g Fs(E)1230 1404 y Fo(j)1267 1389 y Fs(a)i Ft(=)g(0,)f Fs(F)1616 1404 y Fo(j)1653 1389 y Fs(a)g Ft(=)h(0,)f Fs(K)2021 1404 y Fo(j)2058 1389 y Fs(a)h Ft(=)f Fs(\013)2302 1404 y Fo(j)2339 1389 y Fs(a)p Ft(,)h Fs(\013)2507 1404 y Fo(j)2571 1389 y Fr(2)g(f)p Ft(1)p Fs(;)17 b Fr(\000)p Ft(1)p Fr(g)26 b Ft(for)f(all)g Fs(j)32 b Ft(\(see)-19 1509 y([J,)j(4.3]\).)49 b(In)35 b(remains)e(to)h(sho)m(w)i(that)e Fs(\013)1515 1524 y Fo(i)1574 1509 y Ft(=)d(1.)48 b(Since)35 b Fs(V)56 b Ft(is)34 b(of)g(t)m(yp)s(e)h(1,)g(there)g(is)f(a)g(non-zero)-19 1630 y Fs(v)e Fr(2)c Fs(V)49 b Ft(suc)m(h)30 b(that)e Fs(K)766 1645 y Fo(i)794 1630 y Fs(v)k Ft(=)27 b Fs(q)1023 1594 y Fo(m)1090 1630 y Fs(v)k Ft(for)d(some)g Fs(m)g Fr(2)g Ff(Z)p Ft(.)36 b(Let)28 b Fs(a)13 b Fr(\012)g Fs(w)32 b Ft(b)s(e)c(the)h(image)d(of)i Fs(v)k Ft(in)27 b Fs(k)s(a)13 b Fr(\012)g Fs(W)h Ft(.)-19 1750 y(Then)648 1871 y Fs(q)695 1829 y Fo(m)761 1871 y Ft(\()p Fs(a)23 b Fr(\012)f Fs(w)s Ft(\))27 b(=)h Fs(K)1297 1886 y Fo(i)1325 1871 y Ft(\()p Fs(a)22 b Fr(\012)h Fs(w)s Ft(\))k(=)g Fs(K)1860 1886 y Fo(i)1889 1871 y Fs(a)22 b Fr(\012)h Fs(K)2145 1886 y Fo(i)2173 1871 y Fs(w)30 b Ft(=)d Fs(\013)2438 1886 y Fo(i)2467 1871 y Fs(a)22 b Fr(\012)h Fs(K)2723 1886 y Fo(i)2751 1871 y Fs(w)s(;)-19 2041 y Ft(hence)34 b Fs(K)335 2056 y Fo(i)363 2041 y Fs(w)c Ft(=)e Fs(\013)630 2000 y Fp(\000)p Fq(1)629 2064 y Fo(i)724 2041 y Fs(q)771 2005 y Fo(m)837 2041 y Fs(w)s Ft(.)43 b(Since)33 b Fs(W)46 b Ft(is)32 b(of)g(t)m(yp)s(e)i(1,)e(w)m(e)i(conclude)f Fs(\013)2517 2056 y Fo(i)2573 2041 y Ft(=)27 b(1.)-192 2242 y(\(2\))48 b(If)g Fs(x)g Ft(is)f(semisimple)f(in)h Fs(U)1015 2257 y Fo(i)1043 2242 y Ft(,)52 b(then)c(trivially)d Fs(x)j Ft(is)g(semisimple)d(as)j(an)g(elemen)m(t)f(in)g Fs(U)3325 2257 y Fo(q)3364 2242 y Ft(\()p Fn(g)p Ft(\).)-19 2362 y(Con)m(v)m(ersely)40 b(assume)f Fs(x)g Ft(is)e(semisimple)f(in)h Fs(U)1705 2377 y Fo(q)1743 2362 y Ft(\()p Fn(g)p Ft(\))q(.)59 b(Let)39 b Fs(V)59 b Ft(b)s(e)39 b(an)m(y)f(\014nite)g(dimensional)e (left)-19 2483 y Fs(U)47 2498 y Fo(i)76 2483 y Ft(-mo)s(dule.)44 b(By)34 b([J,)g(10.1.14])f(or)g(the)h(classi\014cation)e(of)h(highest)h (w)m(eigh)m(t)g(mo)s(dules)e(of)h Fs(U)3325 2498 y Fo(q)3364 2483 y Ft(\()p Fn(g)p Ft(\),)-19 2603 y Fs(V)62 b Ft(is)39 b(con)m(tained)h(in)f(a)g(\014nite)h(dimensional)d(left)i Fs(U)1908 2618 y Fo(q)1946 2603 y Ft(\()p Fn(g)p Ft(\)-mo)s(dule)2471 2578 y(~)2456 2603 y Fs(V)21 b Ft(.)65 b(Hence)41 b(m)m(ultiplication) -19 2723 y(with)32 b Fs(x)h Ft(is)f(diagonalizable)e(on)1174 2698 y(~)1159 2723 y Fs(V)54 b Ft(and)33 b(then)g(on)f Fs(V)22 b Ft(,)33 b(to)s(o.)3442 2919 y Fk(2)-263 3184 y Fj(Theorem)k(5.2)49 b Fm(L)-5 b(et)46 b Ft(1)h Fr(\024)h Fs(i)f Fr(\024)h Fs(n)p Fm(,)g Fs(\013)q(;)17 b(\014)6 b(;)17 b(\015)51 b Fr(2)c Fs(k)s Fm(,)h(and)d Fs(x)j Ft(=)f Fs(\013)q Ft(\()p Fs(K)2321 3143 y Fp(\000)p Fq(1)2314 3207 y Fo(i)2445 3184 y Fr(\000)30 b Ft(1\))g(+)g Fs(\014)6 b(K)2926 3143 y Fp(\000)p Fq(1)2919 3207 y Fo(i)3020 3184 y Fs(E)3092 3199 y Fo(i)3150 3184 y Ft(+)30 b Fs(\015)5 b(F)3375 3199 y Fo(i)3450 3184 y Fr(2)-263 3305 y Fs(U)-197 3320 y Fo(q)-158 3305 y Ft(\()p Fn(g)p Ft(\))22 b Fr(n)g(f)p Ft(0)p Fr(g)34 b Fm(a)h Ft(\()p Fs(K)458 3263 y Fp(\000)p Fq(1)451 3327 y Fo(i)552 3305 y Fs(;)17 b Ft(1\))p Fm(-primitive)34 b(element.)44 b(Assume)34 b Fs(\013)1962 3268 y Fq(2)2024 3305 y Ft(+)22 b(4)p Fs(q)2218 3268 y Fp(\000)p Fq(1)2312 3305 y Fs(\014)6 b(\015)f(=)p Ft(\()p Fs(q)25 b Fr(\000)e Fs(q)2731 3268 y Fp(\000)p Fq(1)2825 3305 y Ft(\))2863 3268 y Fq(2)2930 3305 y Fr(6)p Ft(=)k(0)p Fm(.)-263 3425 y(L)-5 b(et)44 b Fr(C)49 b Fm(b)-5 b(e)43 b(the)g(tensor)g(c)-5 b(ate)g(gory)43 b(of)f(\014nite)h(dimensional)e(left)i Fs(U)2181 3440 y Fo(q)2220 3425 y Ft(\()p Fn(g)p Ft(\))p Fm(-mo)-5 b(dules)42 b(of)h(typ)-5 b(e)43 b(1.)69 b(De\014ne)-263 3545 y Fs(A)28 b Ft(:=)g Fs(U)35 3560 y Fo(q)73 3545 y Ft(\()p Fn(g)p Ft(\))199 3497 y Fq(0)199 3577 y Fp(C)279 3545 y Fm(and)34 b Fs(B)f Ft(:=)28 b Fr(f)p Fs(a)g Fr(2)g Fs(A)f Fr(j)h Fs(a)22 b Fr(\001)g Fs(x)28 b Ft(=)g(0)p Fr(g)p Fm(.)44 b(Then)34 b(the)h(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)-197 3741 y(\(1\))48 b Fs(x)36 b Fm(is)e(semisimple.)-197 3942 y(\(2\))48 b(Ther)-5 b(e)34 b(is)h(no)g(nonne)-5 b(gative)33 b(inte)-5 b(ger)35 b Fs(n)g Fm(satisfying)1121 4232 y Fs(\013)1184 4190 y Fq(2)1245 4232 y Ft(+)22 b Fs(\014)6 b(\015)f(q)1507 4190 y Fp(\000)p Fq(1)1617 4085 y Fl( )1693 4164 y Fs(q)1740 4128 y Fo(n)1809 4164 y Ft(+)22 b Fs(q)1954 4128 y Fp(\000)p Fo(n)p 1693 4208 363 4 v 1719 4300 a Fs(q)k Fr(\000)d Fs(q)1935 4271 y Fp(\000)p Fq(1)2065 4085 y Fl(!)2131 4108 y Fq(2)2198 4232 y Ft(=)28 b(0)p Fs(:)-197 4546 y Fm(\(3\))48 b Fs(A=)-5 b(AB)250 4510 y Fq(+)344 4546 y Fm(is)35 b(sp)-5 b(anne)g(d)33 b(by)i(gr)-5 b(oup-like)34 b(elements)-220 4747 y(\(3\))-90 4711 y Fp(0)-19 4747 y Fs(A=B)182 4711 y Fq(+)241 4747 y Fs(A)h Fm(is)g(sp)-5 b(anne)g(d)34 b(by)h(gr)-5 b(oup-like)34 b(elements)-197 4948 y(\(4\))48 b Fs(A)35 b Fm(is)g(faithful)5 b(ly)35 b(\015at)g(as)g(a)f(left)h Fs(B)5 b Fm(-mo)-5 b(dule)1578 5197 y Ft(31)p eop %%Page: 32 32 32 31 bop -220 614 a Fm(\(4\))-90 578 y Fp(0)-19 614 y Fs(A)35 b Fm(is)g(faithful)5 b(ly)35 b(\015at)g(as)g(a)f(right)h Fs(B)5 b Fm(-mo)-5 b(dule)-197 813 y(\(5\))48 b Fs(B)40 b Fm(is)35 b(a)g Fs(B)5 b Fm(-dir)-5 b(e)g(ct)34 b(summand)g(in)h Fs(A)g Fm(as)f(a)h(left)g Fs(B)5 b Fm(-mo)-5 b(dule)-220 1011 y(\(5\))-90 975 y Fp(0)-19 1011 y Fs(B)40 b Fm(is)35 b(a)g Fs(B)5 b Fm(-dir)-5 b(e)g(ct)34 b(summand)g(in)h Fs(A)g Fm(as)f(a)h(right)g Fs(B)5 b Fm(-mo)-5 b(dule)-197 1209 y(\(6\))48 b Fs(B)40 b Fm(is)35 b(simple)f(in)g Fr(M)743 1173 y Fo(A)743 1234 y(B)-220 1408 y Fm(\(6\))-90 1372 y Fp(0)-19 1408 y Fs(B)40 b Fm(is)35 b(simple)f(in)623 1423 y Fo(B)684 1408 y Fr(M)804 1372 y Fo(A)861 1408 y Fm(.)-263 1616 y Fi(Pr)n(oof:)50 b Ft(\(1\))55 b Fr(\()-17 b(\))55 b Ft(\(2\):)43 b(By)33 b(5.1\(2\))f(and)h(section)g(3.)-263 1736 y(\(1\))60 b Fr(\))g Ft(\(3\))32 b(and)h(\(3\))60 b Fr(\))g Ft(\(4\))32 b(follo)m(w)f(from)g(2.4.)-263 1857 y(\(4\))60 b Fr(\))g Ft(\(5\))32 b(follo)m(ws)f(from)h(1.2.)-263 1977 y(\(5\))60 b Fr(\))g Ft(\(6\))32 b(is)g(1.3\(2\).)-263 2098 y(\(6\))63 b Fr(\))g Ft(\(1\):)45 b(By)35 b(2.4\(2\),)e (assumption)g(\(6\))g(implies)e Fs(B)k Ft(=)30 b Fr(f)p Fs(a)f Fr(2)h Fs(A)g Fr(j)f(9)p Fs(n)h Fr(\025)g Ft(1:)17 b Fs(a)23 b Fr(\001)g Fs(x)2881 2061 y Fo(n)2958 2098 y Ft(=)29 b(0)p Fr(g)p Ft(.)46 b(Let)34 b Fs(W)-263 2218 y Ft(b)s(e)39 b(an)m(y)g(\014nite)f(dimensional)d(simple)i(left)g Fs(U)1415 2233 y Fo(q)1454 2218 y Ft(\()p Fn(g)p Ft(\)-mo)s(dule)f(of)i (t)m(yp)s(e)h(1.)60 b(By)39 b(2.1,)g(part)f(\(2\))g(and)g(\(3\),)-263 2338 y Fs(W)32 b Fr(\012)19 b Fs(W)63 2302 y Fp(\003)129 2311 y Fr(\030)130 2342 y Ft(=)235 2338 y Fs(C)312 2302 y Fo(W)423 2338 y Ft(as)30 b(righ)m(t)g(\(and)h(left\))e Fs(U)1272 2353 y Fo(q)1311 2338 y Ft(\()p Fn(g)p Ft(\)-mo)s(dules.)41 b(Hence)32 b(if)d Fs(f)g Fr(\001)18 b Fs(x)2442 2302 y Fo(n)2517 2338 y Ft(=)28 b(0)i(for)g(some)g Fs(n)e Fr(\025)g Ft(1)j(and)-263 2459 y Fs(f)45 b Fr(2)35 b Fs(W)37 2423 y Fp(\003)76 2459 y Ft(,)i(then)g Fs(f)f Fr(\001)24 b Fs(x)35 b Ft(=)f(0.)54 b(Or)36 b(equiv)-5 b(alen)m(tly)d(,)37 b(if)e Fs(\036)p Ft(:)17 b Fs(W)1849 2423 y Fp(\003)1922 2459 y Fr(!)34 b Fs(W)2162 2423 y Fp(\003)2237 2459 y Ft(is)i(righ)m(t)g(m)m(ultiplication)31 b(with)36 b Fs(x)p Ft(,)-263 2579 y(then)j(Ker)q(\()p Fs(\036)p Ft(\))e(=)g(Ker\()p Fs(\036)660 2543 y Fo(n)707 2579 y Ft(\))h(for)g(all)e Fs(n)i Fr(\025)g Ft(1.)61 b(If)38 b Fs( )t Ft(:)17 b Fs(W)51 b Fr(!)37 b Fs(W)52 b Ft(is)38 b(left)f(m)m(ultiplication)d(with)k Fs(x)p Ft(,)i(then)-263 2699 y Fs( )-196 2663 y Fp(\003)-118 2699 y Ft(=)d Fs(\036)p Ft(,)j(and)f(w)m(e)h(get)e(im)o(\()p Fs( )t Ft(\))f(=)h(im)n(\()p Fs( )1250 2663 y Fo(n)1297 2699 y Ft(\),)i(or)f(equiv)-5 b(alen)m(tly)d(,)39 b(Ker\()p Fs( )t Ft(\))f(=)g(Ker\()p Fs( )2810 2663 y Fo(n)2857 2699 y Ft(\))g(for)g(all)f Fs(n)h Fr(\025)g Ft(1.)-263 2820 y(Th)m(us)31 b(for)d(all)e(simple)h(mo)s(dules)h(in)g Fr(C)6 b Ft(,)30 b(hence)g(for)e(all)e(mo)s(dules)i Fs(W)42 b Ft(in)28 b Fr(C)35 b Ft(\(b)s(ecause)30 b Fr(C)35 b Ft(is)28 b(semisimple\),)-263 2940 y(w)m(e)34 b(ha)m(v)m(e)g(sho)m(wn) 493 3143 y Fr(f)p Fs(w)c Fr(2)e Fs(W)41 b Fr(j)27 b Fs(x)c Fr(\001)f Fs(w)30 b Ft(=)d(0)p Fr(g)h Ft(=)f Fr(f)p Fs(w)j Fr(2)e Fs(W)42 b Fr(j)27 b(9)p Fs(n)h Fr(\025)g Ft(1:)17 b Fs(x)2313 3101 y Fo(n)2382 3143 y Fr(\001)22 b Fs(w)30 b Ft(=)e(0)p Fr(g)p Fs(:)-263 3345 y Ft(W)-8 b(e)25 b(w)m(an)m(t)h(to)e (sho)m(w)i(the)f(same)f(statemen)m(t)h(o)m(v)m(er)h Fs(U)1581 3360 y Fo(i)1609 3345 y Ft(.)41 b(Let)25 b Fs(V)46 b Ft(b)s(e)25 b(a)f(\014nite)h(dimensional)d(left)h Fs(U)3144 3360 y Fo(i)3173 3345 y Ft(-mo)s(dule)-263 3465 y(of)38 b(t)m(yp)s(e)g(1.)58 b(By)39 b(5.1\(1\),)f Fs(V)59 b Ft(is)37 b(isomorphic)f(to)h(a)g Fs(U)1680 3480 y Fo(i)1709 3465 y Ft(-submo)s(dule)f(of)i Fs(k)s(a)26 b Fr(\012)g Fs(W)14 b Ft(,)38 b(where)h Fs(W)51 b Ft(is)37 b(a)h(left)-263 3586 y Fs(U)-197 3601 y Fo(q)-158 3586 y Ft(\()p Fn(g)p Ft(\)-mo)s(dule)30 b(of)i(t)m(yp)s(e)i(1)e(and)h Fs(E)1018 3601 y Fo(i)1046 3586 y Fs(a)28 b Ft(=)g Fs(F)1292 3601 y Fo(i)1320 3586 y Fs(a)g Ft(=)g(0,)k Fs(K)1694 3601 y Fo(i)1722 3586 y Fs(a)c Ft(=)f Fs(a)p Ft(.)44 b(Hence)715 3788 y Fs(xa)28 b Ft(=)f Fs(\013)q Ft(\()p Fs(K)1143 3747 y Fp(\000)p Fq(1)1136 3813 y Fo(i)1260 3788 y Fr(\000)22 b Ft(1\))p Fs(a)g Ft(+)g Fs(\014)6 b(K)1768 3747 y Fp(\000)p Fq(1)1761 3813 y Fo(i)1862 3788 y Fs(E)1934 3803 y Fo(i)1963 3788 y Fs(a)22 b Ft(+)g Fs(\015)5 b(F)2253 3803 y Fo(i)2281 3788 y Fs(a)28 b Ft(=)g(0)p Fs(;)-263 3990 y Ft(and)33 b(the)g(action)f(of)g Fs(x)h Ft(on)f(an)m(y)i(elemen)m(t)e Fs(a)22 b Fr(\012)h Fs(w)s Ft(,)32 b Fs(w)e Fr(2)e Fs(W)14 b Ft(,)32 b(is)g(giv)m(en)h(b)m(y)149 4193 y Fs(x)p Ft(\()p Fs(a)23 b Fr(\012)f Fs(w)s Ft(\))82 b(=)h Fs(K)857 4152 y Fp(\000)p Fq(1)850 4218 y Fo(i)952 4193 y Fs(a)22 b Fr(\012)h Fs(xw)i Ft(+)d Fs(xa)h Fr(\012)f Fs(w)100 b Ft(since)33 b(\001\()p Fs(x)p Ft(\))28 b(=)g Fs(K)2444 4152 y Fp(\000)p Fq(1)2437 4218 y Fo(i)2561 4193 y Fr(\012)22 b Fs(x)h Ft(+)f Fs(x)g Fr(\012)h Ft(1)p Fs(;)608 4338 y Ft(=)83 b Fs(a)23 b Fr(\012)f Fs(xw)s(:)-263 4541 y Ft(In)33 b(particular,)563 4661 y Fr(f)p Fs(v)e Fr(2)e Fs(V)49 b Fr(j)27 b Fs(x)c Fr(\001)f Fs(v)31 b Ft(=)c(0)p Fr(g)h Ft(=)f Fr(f)p Fs(v)32 b Fr(2)c Fs(V)49 b Fr(j)27 b(9)p Fs(n)h Fr(\025)h Ft(1:)17 b Fs(x)2265 4620 y Fo(n)2334 4661 y Fr(\001)k Fs(v)32 b Ft(=)27 b(0)p Fr(g)p Fs(;)-263 4828 y Ft(since)k(this)f(equalit)m(y)g(holds)g(for)g Fs(W)14 b Ft(.)42 b(Th)m(us)32 b(w)m(e)f(see)h(that)e(condition)f (\(2\))h(of)g(theorem)g(4.4)g(is)f(satis\014ed)-263 4948 y(\(here)f(w)m(e)h(use)f(the)g(assumption)f Fs(\013)1020 4912 y Fq(2)1070 4948 y Ft(+)12 b(4)p Fs(q)1254 4912 y Fp(\000)p Fq(1)1348 4948 y Fs(\014)6 b(\015)f(=)p Ft(\()p Fs(q)15 b Fr(\000)d Fs(q)1746 4912 y Fp(\000)p Fq(1)1840 4948 y Ft(\))1878 4912 y Fq(2)1945 4948 y Fr(6)p Ft(=)27 b(0\).)42 b(Hence)28 b(b)m(y)h(4.4\(1\),)e(m)m(ultiplication)1578 5197 y(32)p eop %%Page: 33 33 33 32 bop -263 614 a Ft(with)31 b Fs(x)g Ft(is)f(diagonalizable)e(on)i (all)f Fs(U)1106 629 y Fo(q)1138 639 y Fg(i)1168 614 y Ft(\()p Fs(sl)r Ft(\(2\)\)-mo)s(dules)g(of)i(t)m(yp)s(e)g(1,)g(and)g Fs(x)g Ft(is)g(diagonalizable)c(as)k(an)-263 735 y(elemen)m(t)i(in)f Fs(U)278 750 y Fo(q)310 760 y Fg(i)340 735 y Ft(\()p Fs(sl)r Ft(\(2\)\))g(b)m(y)i(4.6.)-263 855 y(De\014ne)d Fs(y)g Ft(:=)d Fs(xK)385 870 y Fo(i)413 855 y Ft(.)43 b(Then)32 b Fs(y)h Ft(is)d(\(1)p Fs(;)17 b(K)1127 870 y Fo(i)1155 855 y Ft(\)-primitiv)m(e,)28 b(and)j Fs(B)i Ft(=)27 b Fr(f)p Fs(a)h Fr(j)f Fs(a)18 b Fr(\001)g Fs(y)31 b Ft(=)c(0)p Fr(g)p Ft(.)43 b(W)-8 b(e)31 b(no)m(w)g(rep)s(eat)f(the) -263 976 y(previous)j(argumen)m(ts)g(with)f Fs(x)h Ft(replaced)g(b)m(y) h Fs(y)t Ft(.)42 b(Consider)33 b(the)g(statemen)m(t)-263 1096 y(\(1\))-138 1060 y Fp(0)-82 1096 y Fs(y)i Ft(is)e(semisimple.) -263 1216 y(W)-8 b(e)33 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3554 y([J])146 b(A.)31 b(Joseph,)h(Quan)m(tum)f(groups)g(and)g(theis)g(primitiv)m(e)d(ideals,) i(Springer,)h(Berlin{New)g(Y)-8 b(ork,)-13 3674 y(1995)-263 3878 y([K])120 b(C.)33 b(Kassel,)f(In)m(tro)s(duction)h(to)f(Quan)m (tum)g(Groups,)h(GTM)g(155,)f(Springer)g(New)h(Y)-8 b(ork)33 b(1995)-263 4081 y([Ko])71 b(M.)47 b(Koppinen,)j(Coideals)c (subalgebras)h(in)f(Hopf)g(algebras:)71 b(freeness,)52 b(in)m(tegrals,)e(smash)-13 4202 y(pro)s(ducts,)33 b(Comm.)e(Algebra)h (21)g(\(1993\),)g(427{444)-263 4405 y([KD])49 b(T.)32 b(K.)f(Ko)s(orn)m(winder,)h(M.)g(Dijkh)m(uizen,)f(Quan)m(tum)h (Homogeneous)f(Spaces,)i(Quan)m(tum)f(Du-)-13 4525 y(alit)m(y)f(and)i (Quan)m(tum)f(2-Spheres,)h(Geom.)f(Dedicata)g(52)g(\(1994\),)f(291{315) -263 4729 y([KS])66 b(A.)27 b(U.)h(Klim)m(yk,)f(K.)g(Sc)m(hm)s(\177)-51 b(udgen,)29 b(Quan)m(tum)e(groups)h(and)f(their)g(represen)m(tations,)j (Springer,)-13 4849 y(Berlin)h(1997)1578 5197 y(34)p eop %%Page: 35 35 35 34 bop -263 614 a Ft([KV])49 b(L.)38 b(I.)f(Korogo)s(dsky)-8 b(,)39 b(L.)f(L.)f(V)-8 b(aksman,)39 b(Quan)m(tum)e Fs(G)p Ft(-spaces)h(and)g(Heisen)m(b)s(erg)g(algebra,)g(in:)-13 735 y(Quan)m(tum)29 b(Groups,)i(Pro)s(ceedings)f(of)f(W)-8 b(orkshops)31 b(held)f(in)f(the)h(Euler)g(In)m(ternational)e(Math-)-13 855 y(ematical)g(Institute)j(1990,)f(P)-8 b(.)31 b(P)-8 b(.)30 b(Kulish)g(\(ed.\),)h(Lecture)h(notes)f(in)f(Math.)h(1510,)f (pp.)h(56{66,)-13 976 y(Springer,)h(Berlin,)f(1992)-263 1179 y([M1])58 b(A.)37 b(Masuok)-5 b(a,)40 b(Quotien)m(t)d(theory)h(of) f(Hopf)h(algebras,)g(in:)53 b(Adv)-5 b(ances)39 b(in)e(Hopf)g (algebras,)i(J.)-13 1299 y(Bergen,)33 b(S.)g(Mon)m(tgomery)g(\(eds.\),) g(Dekk)m(er,)h(1994)-263 1503 y([M2])58 b(A.)50 b(Masuok)-5 b(a,)55 b(On)50 b(Hopf)g(algebras)f(with)h(co)s(comm)m(utativ)m(e)f (coradicals,)k(J.)d(Algebra)g(144)-13 1623 y(\(1991\),)31 b(451{466)-263 1826 y([MW])50 b(A.)32 b(Masuok)-5 b(a,)32 b(D.)g(Wigner,)f(F)-8 b(aithful)30 b(\015atness)j(of)e(Hopf)h (algebras,)f(J.)h(Algebra)f(170)g(\(1994\),)-13 1947 y(156{164)-263 2150 y([M])107 b(S.)34 b(Mon)m(tgomery)-8 b(,)34 b(Hopf)g(algebras)g(and)g(their)f(actions)h(on)g(rings,)g(CBMS)h (Regional)d(Confer-)-13 2271 y(ence)h(Series)g(in)f(Math.)h(82,)f (Amer.)h(Math.)f(So)s(c.)h(1993)-263 2474 y([M)s(\177)-51 b(u])52 b(E.)35 b(M)s(\177)-51 b(uller,)34 b(Konstruktion)g(v)m(on)i (Rec)m(h)m(tscoidealun)m(teralgebren,)g(Diploma)c(Thesis,)k(Munic)m(h) -13 2594 y(1995)-263 2798 y([NM])50 b(M.)58 b(Noumi,)63 b(K.)58 b(Mimac)m(hi,)63 b(Ask)m(ey-Wilson)c(P)m(olynomials)c(as)k (spherical)e(functions)h(on)-13 2918 y Fs(S)6 b(U)119 2933 y Fo(q)157 2918 y Ft(\(2\),)51 b(in:)74 b(Quan)m(tum)47 b(Groups,)52 b(Pro)s(ceedings)d(of)e(W)-8 b(orkshops)50 b(held)e(in)f(the)h(Euler)g(In-)-13 3039 y(ternational)28 b(Mathematical)g(Institute)i(1990,)f(P)-8 b(.)31 b(P)-8 b(.)30 b(Kulish)f(\(ed.\),)i(Lecture)g(notes)f(in)g(Math.)-13 3159 y(1510,)h(pp.)i(98{103,)f(Springer,)g(Berlin,)f(1992)-263 3362 y([P])130 b(P)-8 b(.)33 b(P)m(o)s(dle)-5 b(\023)-44 b(s,)32 b(Quan)m(tum)h(Spheres,)h(Lett.)f(Math.)g(Ph)m(ys.)h(14)e (\(1987\),)g(193{202)-263 3566 y([R])124 b(L.)32 b(Ro)m(w)m(en,)i(Ring) d(Theory)-8 b(,)34 b(V)-8 b(ol.)32 b(I,)g(Academic)g(Press,)j(Boston,)e (1988)-263 3769 y([Sc)m(h])50 b(H.-J.)39 b(Sc)m(hneider,)j(Principal)37 b(homogeneous)j(spaces)g(for)f(arbitrary)f(Hopf)h(algebras,)h(Israel) -13 3890 y(J.)32 b(Math.)h(72)f(\(1990\),)g(167{195)-263 4093 y([Sw])72 b(Sw)m(eedler,)34 b(Hopf)e(algebras,)g(Benjamin)f(New)j (Y)-8 b(ork)32 b(1969)-263 4296 y([T1])77 b(M.)29 b(T)-8 b(ak)m(euc)m(hi,)30 b(Relativ)m(e)e(Hopf)g(mo)s(dules|equiv)-5 b(alences)28 b(and)h(freeness)h(criteria,)e(J.)h(Algebra)-13 4417 y(60)j(\(1979\),)f(452{471)-263 4620 y([T2])77 b(M.)48 b(T)-8 b(ak)m(euc)m(hi,)54 b(Hopf)47 b(algebra)g(tec)m(hniques)j (applied)d(to)h(the)h(quan)m(tum)f(group)g Fs(U)3174 4635 y Fo(q)3212 4620 y Ft(\()p Fs(sl)r Ft(\(2\)\),)-13 4741 y(Con)m(temp.)33 b(Math.)g(134)f(\(1992\),)f(309{323)1578 5197 y(35)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF