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(his)g(sta)n(y)g(at)g(the)h(Institute)g(b)n(y)f(the)h(Graduiertenk)n (olleg.)328 4765 y(He)37 b(v)n(ery)e(gratefully)h(ac)n(kno)n(wledges)e (the)i(hospitalit)n(y)g(he)h(receiv)n(ed)e(from)h(the)h(Institute)g (and)f(the)328 4864 y(supp)r(ort)27 b(he)h(receiv)n(ed)f(from)g(the)h (Graduiertenk)n(olleg.)443 4935 y Fr(y)477 4965 y Fq(Researc)n(h)e (supp)r(orted)i(in)g(part)f(b)n(y)g(NSF)h(Gran)n(t)f(DMS)i(9802178.) 1922 5214 y Fp(1)p eop %%Page: 2 2 2 1 bop 328 631 a Fo(In)l(tro)t(duction)328 850 y Fp(Let)27 b Fn(A)g Fp(b)s(e)h(a)e(\014nite-dimensional)e(Hopf)j(algebra)f(with)h (an)m(tip)s(o)s(de)f Fn(s)h Fp(o)m(v)m(er)h(a)f(\014eld)g Fn(k)s Fp(.)41 b(By)328 970 y(an)35 b(earlier)f(result)h([14,)g (Theorem)h(1])f(of)g(the)g(\014rst)h(author)f(the)h(order)f(of)g Fn(s)3147 934 y Fu(2)3221 970 y Fp(is)g(\014nite.)328 1091 y(In)30 b(this)f(pap)s(er)g(w)m(e)i(study)f(some)g(of)f(the)h (\014ner)g(prop)s(erties)f(of)g Fn(s)2653 1055 y Fu(2)2722 1091 y Fp(whic)m(h)h(are)f(related)g(to)328 1211 y(the)k(traces)g Fm(tr)r Fp(\()p Fn(s)932 1175 y Fu(2)p Ft(m)1034 1211 y Fp(\))f(of)g(p)s(o)m(w)m(ers)i(of)e Fn(s)1695 1175 y Fu(2)1735 1211 y Fp(.)474 1332 y(Supp)s(ose)25 b(that)f Fn(t)k Fp(:)f Fn(V)50 b Fl(\000)-16 b(!)27 b Fn(V)45 b Fp(is)24 b(a)f(linear)f(endomorphism)h(of)g(a)h(\014nite-dimensional) 328 1452 y(v)m(ector)34 b(space)f Fn(V)54 b Fp(o)m(v)m(er)34 b Fn(k)s Fp(.)44 b(W)-8 b(e)32 b(sa)m(y)i(that)e Fn(t)h Fp(satis\014es)g(the)g Fk(vanishing)h(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition)918 1672 y Fp(if)31 b(whenev)m(er)k Fn(m)28 b Fl(\025)g Fp(0)33 b(and)f Fn(t)1959 1636 y Ft(m)2054 1672 y Fl(6)p Fp(=)27 b(1)2206 1687 y Ft(V)2299 1672 y Fp(then)34 b Fm(tr)q Fp(\()p Fn(t)2664 1636 y Ft(m)2731 1672 y Fp(\))28 b(=)f(0.)328 1892 y(No)m(w)34 b(supp)s(ose)h(that)e Fn(V)56 b Fp(is)33 b(an)g(algebra)g(o)m(v)m(er)h Fn(k)s Fp(.)47 b(W)-8 b(e)34 b(sa)m(y)g(that)g Fn(t)g Fp(satis\014es)g(the)g Fk(str)-5 b(ong)328 2012 y(vanishing)33 b(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition)521 2232 y Fp(if)32 b(whenev)m(er)j Fn(m)28 b Fl(\025)g Fp(0)33 b(and)f Fn(t)1563 2196 y Ft(m)1658 2232 y Fl(6)p Fp(=)27 b(1)1810 2247 y Ft(V)1903 2232 y Fp(then)34 b Fm(tr)q Fp(\()p Fn(t)2268 2196 y Ft(m)2335 2232 y Fl(\016)p Fn(r)s Fp(\()p Fn(a)p Fp(\)\))27 b(=)h(0)k(for)g(all)f Fn(a)d Fl(2)g Fn(V)21 b(;)328 2452 y Fp(where)j Fn(r)s Fp(\()p Fn(a)p Fp(\))j(:)h Fn(V)49 b Fl(\000)-16 b(!)27 b Fn(V)44 b Fp(is)23 b(righ)m(t)e(m)m (ultiplication)d(b)m(y)24 b Fn(a)e Fp(whic)m(h)i(is)e(therefore)h (de\014ned)h(b)m(y)328 2573 y Fn(r)s Fp(\()p Fn(a)p Fp(\)\()p Fn(x)p Fp(\))k(=)f Fn(xa)32 b Fp(for)f(all)f Fn(x)e Fl(2)g Fn(V)22 b Fp(.)43 b(If)31 b Fn(t)h Fp(satis\014es)g(the)g(strong)g(v)-5 b(anishing)30 b(trace)i(condition)328 2693 y(then)k Fn(t)g Fp(satis\014es)h(the)f(v)-5 b(anishing)35 b(trace)h(condition.)52 b(W)-8 b(e)36 b(sho)m(w)h(that)f Fn(s)2993 2657 y Fu(2)3068 2693 y Fp(satis\014es)g(the)328 2813 y(strong)d(v)-5 b(anishing)31 b(trace)i(condition)e(in)h(a)g(n)m(um)m(b)s(er)h(of)f(in) m(teresting)g(cases.)474 2934 y(Researc)m(h)h(for)d(this)h(pap)s(er)g (b)s(egan)f(with)h(a)g(question)g(raised)g(b)m(y)g(P)m(a)m(v)m(el)h (Etingof)e(to)328 3054 y(a)37 b(n)m(um)m(b)s(er)h(of)g(Hopf)f (algebraists)g(at)g(a)g(conference)j(held)d(in)g(Durham,)h(England,)h (in)328 3175 y(the)32 b(summer)g(of)g(1999.)42 b(In)32 b(our)g(terminology)e(the)j(question)f(w)m(as)h(whether)h(or)d(not)h Fn(s)3526 3138 y Fu(2)328 3295 y Fp(satis\014es)37 b(the)h(v)-5 b(anishing)35 b(trace)i(condition.)55 b(In)m(v)m(estigation)36 b(of)g(this)h(question)g(led)f(us)328 3415 y(to)30 b(form)m(ulate)f (the)i(strong)g(v)-5 b(anishing)30 b(trace)h(condition)e(and)i(to)f (study)i(it)e(connection)328 3536 y(with)43 b(Hopf)h(algebra)e (automorphisms)g(of)h Fn(A)p Fp(,)k(in)c(particular)f(with)h Fn(s)2992 3500 y Fu(2)3031 3536 y Fp(.)77 b(When)44 b(the)328 3656 y(c)m(haracteristic)24 b(of)h Fn(k)i Fp(is)e(zero)g(and)f Fn(s)1617 3620 y Fu(2)1681 3656 y Fp(satis\014es)i(the)f(strong)g(v)-5 b(anishing)23 b(trace)i(condition,)328 3777 y(m)m(uc)m(h)33 b(can)g(b)s(e)g(said)f(ab)s(out)g Fn(s)1426 3740 y Fu(2)1465 3777 y Fp(,)h(the)g(eigen)m(v)-5 b(alues)33 b(and)f(eigenspaces)i(of)e Fn(s)3060 3740 y Fu(2)3100 3777 y Fp(,)g(and)h(there)328 3897 y(are)g(implications)28 b(for)k(the)h(algebra)f(structure)i(of)e Fn(A)p Fp(.)474 4017 y(The)e(strong)g(v)-5 b(anishing)28 b(trace)h(condition)f(has)h(implications)d(for)i(classi\014cation)g(of) 328 4138 y(\014nite-dimensional)36 b(Hopf)k(algebras.)64 b(F)-8 b(or)39 b(example,)i(under)f(the)g(assumption)f(that)328 4258 y Fn(s)374 4222 y Fu(2)443 4258 y Fp(satis\014es)31 b(the)g(strong)f(v)-5 b(anishing)29 b(trace)h(condition,)f(the)i (classi\014cation)d(of)i(all)e(Hopf)328 4378 y(algebras)45 b(of)h(dimension)e Fn(p)1369 4342 y Fu(2)1409 4378 y Fp(,)49 b(where)e Fn(p)f Fp(is)f(an)h(o)s(dd)g(prime,)i(o)m(v)m(er)f (an)f(algebraically)328 4499 y(closed)41 b(\014eld)f(of)g(c)m (haracteristic)h(zero)g(is)f(complete.)67 b(See)41 b(the)h(discussion)e (follo)m(wing)328 4619 y(Prop)s(osition)31 b(6.)474 4740 y(The)41 b(strong)f(v)-5 b(anishing)38 b(trace)j(condition)d(for)h(a)h (linear)e(endomorphism)g Fn(t)i Fp(of)g Fn(A)328 4860 y Fp(can)d(b)s(e)g(expressed)j(in)c(a)h(w)m(a)m(y)h(whic)m(h)g(is)e(v)m (ery)j(useful)e(for)f(us)i(in)e(terms)h(of)g(in)m(tegrals.)1922 5214 y(2)p eop %%Page: 3 3 3 2 bop 328 631 a Fp(Let)35 b(\003)c Fl(2)h Fn(A)j Fp(and)g Fn(\025)c Fl(2)h Fn(A)1261 595 y Fj(\003)1335 631 y Fp(b)s(e)j(left)f (and)h(righ)m(t)f(in)m(tegrals)f(for)i Fn(A)g Fp(and)f Fn(A)2992 595 y Fj(\003)3067 631 y Fp(resp)s(ectiv)m(ely)328 751 y(whic)m(h)f(satisfy)g Fn(\025)p Fp(\(\003\))27 b(=)g(1)33 b(and)f(set)1481 969 y Fn(\027)1529 984 y Ft(A)1586 969 y Fp(\()p Fn(t)p Fp(\))c(=)g Fn(s)p Fp(\(\003)1981 984 y Fu(\(2\))2075 969 y Fp(\))p Fn(t)p Fp(\(\003)2254 984 y Fu(\(1\))2348 969 y Fp(\))p Fn(:)328 1186 y Fp(Then)33 b(w)m(e)g(sho)m(w)g(in)e(Lemma)f(3)i(b\))f(that)h Fm(tr)r Fp(\()p Fn(t)p Fl(\016)p Fn(r)s Fp(\()p Fn(a)p Fp(\)\))27 b(=)h(0)j(for)g(all)f Fn(a)e Fl(2)g Fn(A)k Fp(if)f(and)h(only)f(if)328 1306 y Fn(\027)376 1321 y Ft(A)433 1306 y Fp(\()p Fn(t)p Fp(\))d(=)f(0.)43 b(Th)m(us)32 b Fn(t)f Fp(satis\014es)g(the)g(strong)g (v)-5 b(anishing)29 b(trace)i(condition)e(if)g(and)i(only)f(if)328 1427 y(whenev)m(er)35 b Fn(m)28 b Fl(\025)g Fp(0)33 b(and)f Fn(t)1280 1391 y Ft(m)1375 1427 y Fl(6)p Fp(=)27 b(1)1527 1442 y Ft(A)1617 1427 y Fp(then)33 b Fn(\027)1887 1442 y Ft(A)1944 1427 y Fp(\()p Fn(t)2017 1391 y Ft(m)2084 1427 y Fp(\))27 b(=)h(0.)474 1547 y(Our)35 b(study)i(of)d(the)i(strong) f(v)-5 b(anishing)34 b(trace)i(condition)d(for)i Fn(s)2843 1511 y Fu(2)2917 1547 y Fp(splits)g(v)m(ery)h(nat-)328 1667 y(urally)47 b(in)m(to)g(t)m(w)m(o)i(cases:)77 b Fn(s)1397 1631 y Fu(2)1484 1667 y Fp(semisimple,)50 b(and)f Fn(s)2277 1631 y Fu(2)2365 1667 y Fp(not)f(semisimple,)i(as)e(a)g (linear)328 1788 y(endomorphism)29 b(of)h Fn(A)p Fp(.)43 b(Supp)s(ose)31 b(that)f Fn(s)1869 1752 y Fu(2)1939 1788 y Fp(is)f(not)h(semisimple)e(and)j(that)f(the)h(c)m(harac-)328 1908 y(teristic)g(of)g Fn(k)k Fp(is)d(not)f(2.)43 b(W)-8 b(e)33 b(sho)m(w)g(in)e(Prop)s(osition)f(3)h(that)h(either)g Fn(s)2897 1872 y Fu(2)2968 1908 y Fp(or)g Fn(s)3133 1872 y Fj(\003)p Fu(2)3239 1908 y Fp(satis\014es)328 2029 y(the)44 b(strong)f(v)-5 b(anishing)42 b(trace)i(condition;)j(hence)e Fn(s)2312 1992 y Fu(2)2395 2029 y Fp(\(and)e(th)m(us)h Fn(s)2904 1992 y Fj(\003)p Fu(2)2979 2029 y Fp(\))f(satis\014es)h(the) 328 2149 y(v)-5 b(anishing)31 b(trace)i(condition.)474 2269 y(The)46 b(case)f Fn(s)951 2233 y Fu(2)1035 2269 y Fp(semisimple)d(is)h(far)h(more)g(in)m(teresting.)78 b(If)45 b Fn(s)2776 2233 y Fu(2)2863 2269 y Fp(=)i(1)3035 2284 y Ft(A)3137 2269 y Fp(then)e Fn(s)3417 2233 y Fu(2)3500 2269 y Fp(is)328 2390 y(semisimple)31 b(and)j(satis\014es)h(the)f (strong)g(v)-5 b(anishing)33 b(trace)h(condition)f(v)-5 b(acuously)d(.)47 b(W)-8 b(e)328 2510 y(note)39 b(that)f Fn(s)814 2474 y Fu(2)891 2510 y Fp(=)g(1)1054 2525 y Ft(A)1149 2510 y Fp(if)f Fn(A)i Fp(is)f(either)g(comm)m(utativ)m(e)g (or)g(co)s(comm)m(utativ)m(e,)h(or)f(if)g Fn(A)g Fp(is)328 2631 y(semisimple)i(or)i(cosemisimple)f(and)h(the)h(c)m(haracteristic)g (of)f Fn(k)j Fp(is)d(zero.)74 b(Our)42 b(main)328 2751 y(results)33 b(when)h Fn(s)939 2715 y Fu(2)1011 2751 y Fp(is)e(semisimple)f(are)i(that)f Fn(s)2024 2715 y Fu(2)2096 2751 y Fp(satis\014es)i(the)f(strong)g(v)-5 b(anishing)31 b(trace)328 2871 y(condition)h(if)h Fn(A)g Fp(is)h(p)s(oin)m(ted)f(or)g(if)g Fn(A)1691 2835 y Fj(\003)1764 2871 y Fp(is)g(p)s(oin)m(ted.)46 b(These)36 b(results)e(follo)m(w)e (from)g(The-)328 2992 y(orem)g(1)g(and)h(Theorem)g(4)f(resp)s(ectiv)m (ely)-8 b(.)474 3112 y(The)39 b(ideas)f(dev)m(elop)s(ed)h(in)e(this)h (pap)s(er)g(enable)g(us)h(to)f(pro)m(v)m(e)h(Theorem)f(6)g(whic)m(h)328 3232 y(together)29 b(with)g(Theorem)g(1)f(implies)f(when)j Fn(A)f Fp(is)f(p)s(oin)m(ted)h(and)g Fn(k)j Fp(is)c(an)h(algebraically) 328 3353 y(closed)k(\014eld)f(of)g(c)m(haracterictic)g(zero)h(that:)436 3554 y(a\))49 b(The)30 b(order)f Fn(r)i Fp(of)d Fn(s)1248 3518 y Fu(2)1317 3554 y Fp(divides)g(Dim)15 b Fn(A=)p Fp(Dim)f Fn(B)5 b Fp(,)30 b(where)g Fn(B)j Fp(=)27 b Fn(A)2859 3569 y Fu(0)2928 3554 y Fp(is)h(the)h(coradical)572 3674 y(of)j Fn(A)p Fp(,)h(and)f(the)h(minimal)c(p)s(olynomial)g(of)j Fn(s)2214 3638 y Fu(2)2286 3674 y Fp(is)g Fn(X)2473 3638 y Ft(r)2533 3674 y Fl(\000)22 b Fp(1;)431 3877 y(b\))49 b(The)33 b(eigen)m(v)-5 b(alues)33 b(of)f Fn(s)1434 3841 y Fu(2)1506 3877 y Fp(are)g(the)h Fn(r)1883 3841 y Ft(th)1986 3877 y Fp(ro)s(ots)f(of)g(unit)m(y)h(in)f Fn(k)s Fp(;)442 4079 y(c\))49 b(The)33 b(eigenspaces)h(for)e Fn(s)1483 4043 y Fu(2)1555 4079 y Fp(all)f(ha)m(v)m(e)j(the)f(same)f(dimension)f (whic)m(h)i(is)f(Dim)15 b Fn(A=r)s Fp(;)431 4282 y(d\))49 b(If)c Fn(e)h Fp(is)g(an)f(idemp)s(oten)m(t)g(of)g Fn(A)h Fp(whic)m(h)g(satis\014es)h Fn(s)2515 4246 y Fu(2)2554 4282 y Fp(\()p Fn(e)p Fp(\))j(=)g Fn(e)c Fp(then)h Fn(r)h Fp(divides)572 4403 y(Dim)14 b Fn(Ae)p Fp(.)474 4604 y(The)38 b(b)s(ound)g(describ)s(ed)f(in)g(a\))g(is)f(sharp.)58 b(When)38 b Fn(A)f Fp(is)f(p)s(oin)m(ted)h(it)f(follo)m(ws)g(from)328 4724 y(a\))c(that)h Fn(r)d Fp(=)d(Dim)15 b Fn(A)33 b Fp(if)e(and)i(only)f(if)f(Dim)15 b Fn(A)27 b Fp(=)h(1.)474 4844 y(More)h(generally)-8 b(,)28 b(Theorem)h(6)f(holds)g(for)g(a)g (Hopf)g(algebra)f(automorphism)f Fn(t)j Fp(of)f Fn(A)328 4965 y Fp(whic)m(h)e(satis\014es)g(the)g(strong)f(v)-5 b(anishing)24 b(trace)i(condition)e(when)i Fn(k)j Fp(is)c(an)g (algebraically)1922 5214 y(3)p eop %%Page: 4 4 4 3 bop 328 631 a Fp(closed)43 b(\014eld)f(of)g(c)m(haracteristic)g (zero.)73 b(Here)43 b Fn(t)g Fp(replaces)g Fn(s)2593 595 y Fu(2)2675 631 y Fp(and)f Fn(B)48 b Fp(is)41 b(the)i(unique)328 751 y(maximal)20 b(semisimple)i(\(and)i(hence)h(the)f(maximal)c (cosemisimple\))i(sub-Hopf)i(algebra)328 872 y(of)32 b Fn(A)p Fp(.)474 992 y(The)43 b(pap)s(er)f(is)f(organized)g(as)h (follo)m(ws.)69 b(In)42 b(Section)g(1)f(results)h(are)g(deriv)m(ed)g (for)328 1112 y(certain)35 b(linear)e(automorphisms)g(of)i Fn(V)56 b Fp(when)36 b Fn(k)i Fp(is)d(algebraically)c(closed,)36 b(has)g(c)m(har-)328 1233 y(acteristic)41 b(zero)h(and)g Fn(t)g Fp(satis\014es)h(certain)f(trace)g(conditions.)70 b(In)42 b(the)h(next)g(section)328 1353 y(w)m(e)35 b(study)g(the)f (elemen)m(t)g Fn(\027)1322 1368 y Ft(A)1379 1353 y Fp(\()p Fn(t)p Fp(\))g(for)g(linear)e(endomorphisms)h Fn(t)h Fp(of)g Fn(A)g Fp(and)g(pro)m(v)m(e)h(that)328 1474 y(either)25 b Fn(\027)645 1489 y Ft(A)703 1474 y Fp(\()p Fn(s)787 1437 y Fu(2)p Ft(m)888 1474 y Fp(\))j(=)f(0)f(for)f(all)e Fn(m)28 b Fl(\025)g Fp(0)e(or)f Fn(\027)1855 1489 y Ft(A)1908 1470 y Fi(\003)1948 1474 y Fp(\()p Fn(s)2032 1437 y Fj(\003)p Fu(2)p Ft(m)2169 1474 y Fp(\))j(=)f(0)e(for)g(all)f Fn(m)k Fl(\025)g Fp(0)d(when)i Fn(s)3269 1437 y Fu(2)3334 1474 y Fp(is)e(not)328 1594 y(semisimple)h(and)j(the)g(c)m(haracteristic)g (of)f Fn(k)j Fp(is)e(not)f(2.)42 b(A)29 b(k)m(ey)h(step)f(in)f(the)h (pro)s(of)f(is)g(the)328 1714 y(observ)-5 b(ation)35 b(that)g(if)f Fn(t)h Fp(is)g(an)g(algebra)f(endomorphism)g(of)h Fn(A)g Fp(and)h Fn(B)k Fp(is)35 b(a)g(sub-Hopf)328 1835 y(algebra)h(of)g Fn(A)h Fp(whic)m(h)g(satis\014es)h Fn(t)p Fp(\()p Fn(B)5 b Fp(\))35 b Fl(\022)g Fn(B)5 b Fp(,)38 b(then)f Fn(\027)2301 1850 y Ft(B)2362 1835 y Fp(\()p Fn(t)p Fl(j)2463 1850 y Ft(B)2524 1835 y Fp(\))e(=)f(0)j(implies)d Fn(\027)3176 1850 y Ft(A)3233 1835 y Fp(\()p Fn(t)p Fp(\))h(=)g(0.)328 1955 y(Thereb)m(y)g(the)f(pro)s(of)e(reduces)j(to)e(the)h(case)g Fn(A)f Fp(or)g Fn(A)2280 1919 y Fj(\003)2353 1955 y Fp(is)g(the)g (group)g(algebra)f Fn(k)s Fp([)p Fn(Z)3387 1970 y Ft(p)3427 1955 y Fp(])h(of)328 2076 y(the)g(cyclic)f(group)h(of)f(order)g Fn(p)h Fp(and)g(the)g(\014eld)f Fn(k)k Fp(has)c(c)m(haracteristic)h Fn(p)28 b(>)f Fp(0.)474 2196 y(In)h(Section)f(3)g(w)m(e)h(sho)m(w)g (that)f Fn(s)1629 2160 y Fu(2)1696 2196 y Fp(satis\014es)h(the)f (strong)h(v)-5 b(anishing)26 b(trace)h(condition)328 2316 y(when)47 b Fn(s)642 2280 y Fu(2)728 2316 y Fp(is)f(semisimple)e (and)i(the)h(coradical)e Fn(A)2232 2331 y Fu(0)2318 2316 y Fp(of)g Fn(A)i Fp(is)f(co)s(comm)m(utativ)m(e.)83 b(W)-8 b(e)328 2437 y(reduce)37 b(to)e(the)g(case)i(where)f Fn(A)g Fp(is)f(generated)h(b)m(y)g Fn(a;)17 b(x)36 b Fp(as)f(an)h(algebra)e(whic)m(h)i(satisfy)328 2557 y(the)d(relation)f Fn(xa)d Fp(=)f Fn(q)t(ax)p Fp(,)34 b(where)g Fn(q)f Fl(2)c Fn(k)36 b Fp(is)c(a)h(ro)s(ot)f(of)h(unit)m(y)-8 b(,)33 b(and)h(whose)g(copro)s(ducts)328 2677 y(are)f(giv)m(en)f(b)m(y)i (\001\()p Fn(a)p Fp(\))28 b(=)f Fn(a)p Fl(\012)q Fn(a)33 b Fp(and)f(\001\()p Fn(x)p Fp(\))d(=)e Fn(x)p Fl(\012)q Fn(a)22 b Fp(+)h(1)p Fl(\012)p Fn(x)33 b Fp(resp)s(ectiv)m(ely)-8 b(.)474 2798 y(W)g(e)31 b(con)m(tin)m(ue)g(the)f(study)i(of)d Fn(s)1614 2762 y Fu(2)1684 2798 y Fp(when)i(it)e(is)h(semisimple)e(in)h (Section)h(4)g(and)g(pro)m(v)m(e)328 2918 y(that)38 b Fn(s)591 2882 y Fu(2)669 2918 y Fp(satis\014es)i(the)f(strong)f(v)-5 b(anishing)38 b(trace)g(condition)g(when)h(the)g(coradical)e(of)328 3039 y Fn(A)401 3002 y Fj(\003)480 3039 y Fp(is)j(co)s(comm)m(utativ)m (e.)65 b(Our)40 b(pro)s(of)f(is)h(based)g(on)g(the)h(stucture)g(theory) g(of)f(\014nite-)328 3159 y(dimensional)34 b(graded)j(Hopf)f(algebras.) 54 b(A)37 b(k)m(ey)h(step)f(in)f(the)h(pro)s(of)f(is)g(the)g(follo)m (wing)328 3279 y(observ)-5 b(ation.)47 b(Supp)s(ose)35 b(that)f Fn(t)g Fp(is)f(a)h(Hopf)g(algebra)f(endomorphism)f(of)i Fn(A)g Fp(and)g(that)328 3400 y Fn(I)45 b Fp(is)36 b(a)h(nilp)s(oten)m (t)f(Hopf)h(ideal)e(of)i Fn(A)g Fp(whic)m(h)h(satis\014es)f Fn(t)p Fp(\()p Fn(I)8 b Fp(\))36 b Fl(\022)f Fn(I)8 b Fp(.)57 b(Then)38 b Fn(\027)3131 3415 y Ft(A=I)3259 3400 y Fp(\()3298 3381 y(\026)3297 3400 y Fn(t)q Fp(\))d(=)g(0)328 3520 y(implies)27 b Fn(\027)704 3535 y Ft(A)761 3520 y Fp(\()p Fn(t)p Fp(\))h(=)f(0,)j(where)1389 3502 y(\026)1387 3520 y Fn(t)f Fp(is)g(the)h(Hopf)f(algebra)f(automorphism)f(of)h Fn(A=I)37 b Fp(induced)328 3641 y(b)m(y)c Fn(t)p Fp(.)474 3761 y(In)j(Section)f(5)f(w)m(e)j(discuss)f(the)g(c)m(haracters)g Fn(\037)2200 3776 y Ft(A)2289 3761 y Fl(2)d Fn(A)2461 3725 y Fj(\003)2535 3761 y Fp(and)i Fn(x)2782 3776 y Ft(A)2872 3761 y Fl(2)d Fn(A)j Fp(of)g(the)h(righ)m(t)328 3881 y(regular)28 b(represen)m(tations)j(of)e Fn(A)h Fp(and)f Fn(A)1798 3845 y Fj(\003)1867 3881 y Fp(resp)s(ectiv)m(ely)i (using)e(the)h(prop)s(erties)f(of)g Fn(A)h Fp(as)328 4002 y(a)i(F)-8 b(rob)s(enius)32 b(algebra.)43 b(W)-8 b(e)33 b(deduce)h(the)f(trace)g(form)m(ula)1103 4192 y(\(Dim)14 b Fn(A)p Fp(\))p Fm(tr)r Fp(\()p Fn(t)1594 4151 y Fj(\003)1633 4192 y Fl(j)1661 4207 y Ft(\037)1705 4218 y Fh(A)1755 4207 y Ft(A)1808 4188 y Fi(\003)1848 4192 y Fp(\))28 b(=)f Fn(<\027)2141 4207 y Ft(A)2194 4188 y Fi(\003)2234 4192 y Fp(\()p Fn(t)2307 4151 y Fj(\003)2347 4192 y Fp(\))p Fn(;)17 b(\027)2477 4207 y Ft(A)2534 4192 y Fp(\(1)2621 4207 y Ft(A)2677 4192 y Fp(\))p Fn(>)328 4383 y Fp(for)23 b(Hopf)h(algebra)e(automorphisms)g Fn(t)i Fp(of)f Fn(A)p Fp(.)41 b(This)23 b(form)m(ula)f(generalizes)i([6)o(,)i (Equation)328 4503 y(\(6\)].)43 b(Using)32 b(the)h(results)g(of)f (Section)h(4)f(w)m(e)i(sho)m(w)f(that)1535 4742 y Fn(x)1590 4757 y Ft(A)1675 4742 y Fp(=)1778 4621 y Fg(\022)1852 4675 y Fp(Dim)15 b Fn(A)p 1849 4719 279 4 v 1849 4811 a Fp(Dim)g Fn(B)2138 4621 y Fg(\023)2216 4742 y Fn(x)2271 4757 y Ft(B)2332 4742 y Fn(;)328 4965 y Fp(where)34 b Fn(B)j Fp(is)32 b(the)h(sub-Hopf)g(algebra)f(of)g Fn(A)g Fp(generated)i(b)m(y)f(the)g(coradical)e(of)h Fn(A)p Fp(.)1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 474 631 a Fp(When)34 b(the)f(c)m(haracteristic)g(of)f Fn(k)k Fp(do)s(es)d(not)g(divide)f(Dim)14 b Fn(A)33 b Fp(then)g Fn(\037)2997 646 y Ft(A)3055 631 y Fn(=)p Fp(Dim)14 b Fn(A)33 b Fp(and)328 751 y Fn(x)383 766 y Ft(A)440 751 y Fn(=)p Fp(Dim)15 b Fn(A)25 b Fp(are)g(imp)s(ortan)m(t)f(non-zero) h(idemp)s(oten)m(ts.)41 b(F)-8 b(or)24 b(example,)j(when)f(the)g(c)m (har-)328 872 y(acteristic)i(of)g Fn(k)k Fp(is)c(zero)h(and)g Fn(A)g Fp(is)f(semisimple)f(then)i Fn(\037)2380 887 y Ft(A)2437 872 y Fn(=)p Fp(Dim)15 b Fn(A)28 b Fp(and)h Fn(x)3028 887 y Ft(A)3086 872 y Fn(=)p Fp(Dim)14 b Fn(A)29 b Fp(are)328 992 y(non-zero)k(t)m(w)m(o-sided)g(in)m(tegrals)e(of)h Fn(A)1735 956 y Fj(\003)1807 992 y Fp(and)h Fn(A)g Fp(resp)s(ectiv)m (ely)-8 b(.)474 1112 y(In)29 b(Section)g(6)f(w)m(e)i(examine)e (implications)d(of)j(the)h(ideas)f(dev)m(elop)s(ed)i(in)d(this)i(pap)s (er)328 1233 y(for)e(\014nite-dimensional)d(p)s(oin)m(ted)k(Hopf)f (algebras)g(o)m(v)m(er)h(an)g(algebraically)c(closed)k(\014eld)328 1353 y(of)39 b(c)m(harcteristic)h(zero)h(and)f(w)m(e)g(examine)g (implications)c(for)j(the)i(Drinfel'd)d(double.)328 1474 y(Our)32 b(discussion)h(of)f(p)s(oin)m(ted)h(Hopf)f(algebras)g(is)g (based)i(on)e(certain)g(idemp)s(oten)m(ts.)474 1594 y(W)-8 b(e)29 b(w)m(an)m(t)g(to)g(p)s(oin)m(t)e(out)h(that)h(it)e(ma)m(y)h(v)m (ery)i(w)m(ell)e(b)s(e)h(true)g(that)f(the)h(square)g(of)f(the)328 1714 y(an)m(tip)s(o)s(de)40 b(of)h(an)m(y)g(\014nite-dimensional)d (Hopf)j(algebra)f(o)m(v)m(er)i Fn(k)i Fp(satis\014es)d(the)h(strong)328 1835 y(v)-5 b(anishing)31 b(trace)i(condition.)42 b(Needless)34 b(to)e(sa)m(y)i(w)m(e)g(kno)m(w)f(of)f(no)h(coun)m(terexample.)474 1955 y(W)-8 b(e)31 b(shall)f(assume)h(that)g(the)g(reader)g(has)g(some) g(familiarit)m(y)26 b(with)31 b(Hopf)f(algebras)328 2076 y([9,)i(18].)44 b(Throughout)32 b(this)h(pap)s(er)f Fn(k)k Fp(is)c(a)g(\014eld.)474 2196 y(Finally)e(the)j(authors)g(w)m(ould)g (lik)m(e)e(to)i(thank)g(the)g(referee)g(for)f(useful)h(commen)m(ts.)328 2529 y Fo(1)161 b(Preliminaries)328 2748 y Fp(W)-8 b(e)25 b(b)s(egin)g(with)g(sev)m(eral)h(observ)-5 b(ations)25 b(ab)s(out)f(a)h(linear)f(endomorphism)g Fn(t)h Fp(of)g(a)g(\014nite-) 328 2868 y(dimensional)36 b(v)m(ector)k(space)g Fn(V)61 b Fp(o)m(v)m(er)40 b Fn(k)s Fp(.)62 b(The)40 b(endomorphism)d Fn(t)i Fp(is)g Fk(semisimple)e Fp([3,)328 2989 y(A)i(VI)s(I,)g(8])f(if) g(eac)m(h)i Fn(t)p Fp(-stable)e(subspace)j(of)d Fn(V)61 b Fp(has)39 b(a)f Fn(t)p Fp(-stable)h(direct)f(summand,)i(or)328 3109 y(equiv)-5 b(alen)m(tly)29 b(if)g(the)i(minimal)26 b(p)s(olynomial)g(of)k Fn(t)g Fp(o)m(v)m(er)h Fn(k)i Fp(has)e(no)f(m)m(ultiple)d(factors.)43 b(If)328 3229 y Fn(k)29 b Fp(is)d(algebraically)d(closed,)28 b(then)f Fn(t)f Fp(is)g(semisimple)e(if)h(and)h(only)g(if)f Fn(t)h Fp(is)g(diagonalizable.)474 3350 y(If)36 b Fn(t)610 3314 y Ft(r)682 3350 y Fp(=)d(1)840 3365 y Ft(V)937 3350 y Fp(for)j(some)g(p)s(ositiv)m(e)f(in)m(teger)h Fn(r)s Fp(,)h(then)f(the)h(smallest)d(suc)m(h)k Fn(r)g Fp(is)e(called)328 3470 y(the)f Fk(or)-5 b(der)36 b(of)e Fn(t)p Fp(;)h(otherwise)g(the)g (order)g(of)f Fn(t)g Fp(is)g(said)g(to)g(b)s(e)g(in\014nite.)48 b(W)-8 b(e)35 b(denote)g(the)328 3590 y(order)g(of)e Fn(t)i Fp(b)m(y)g(order)q(\()p Fn(t)p Fp(\))f(when)i(it)d(is)h (\014nite.)49 b(W)-8 b(e)35 b(ha)m(v)m(e)h(remark)m(ed)f(that)f(the)h (order)f(of)328 3711 y(the)41 b(an)m(tip)s(o)s(de)f(of)g(a)g (\014nite-dimensional)e(Hopf)i(algebra)f(o)m(v)m(er)j Fn(k)i Fp(is)c(\014nite.)67 b(If)41 b Fn(t)f Fp(has)328 3831 y(\014nite)28 b(order)h(then)h Fn(t)f Fp(is)f(semisimple)e(if)i (and)h(only)f(if)g(the)h(c)m(haracteristic)g(of)f Fn(k)k Fp(do)s(es)d(not)328 3952 y(divide)j(the)h(order)g(of)f Fn(t)p Fp(.)328 4180 y Ff(Lemma)37 b(1)49 b Fk(Supp)-5 b(ose)33 b(that)h(the)g(\014eld)f Fn(k)k Fk(is)d(algebr)-5 b(aic)g(al)5 b(ly)32 b(close)-5 b(d,)33 b(has)g(char)-5 b(acteristic)328 4300 y Fp(0)29 b Fk(and)f(that)i Fn(V)50 b Fk(is)29 b(a)g(\014nite-dimensional)d(ve)-5 b(ctor)29 b(sp)-5 b(ac)g(e)29 b(over)f Fn(k)s Fk(.)43 b(Assume)29 b(further)g(that)328 4421 y Fn(t)41 b Fk(is)f(a)g(line)-5 b(ar)40 b(automorphism)f(of)h Fn(V)62 b Fk(with)41 b(\014nite)f(or)-5 b(der)40 b Fn(r)j Fk(and)d(that)g Fn(t)h Fk(satis\014es)f(the)328 4541 y(vanishing)33 b(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition.)44 b(Then:)436 4744 y Fp(a\))49 b Fk(The)34 b(minimal)g(p)-5 b(olynomial)33 b(for)i Fn(t)g Fk(is)g Fn(X)2065 4708 y Ft(r)2125 4744 y Fl(\000)22 b Fp(1)p Fk(.)431 4948 y Fp(b\))49 b Fn(r)37 b Fk(divides)d Fp(Dim)15 b Fn(V)21 b Fk(.)1922 5214 y Fp(5)p eop %%Page: 6 6 6 5 bop 442 631 a Fp(c\))49 b Fk(The)31 b(eigenvalues)g(of)h Fn(t)g Fk(ar)-5 b(e)32 b(the)g Fn(r)1824 595 y Ft(th)1926 631 y Fk(r)-5 b(o)g(ots)32 b(of)g(unity)h(in)e Fn(k)36 b Fk(and)31 b(e)-5 b(ach)31 b(eigensp)-5 b(ac)g(e)572 751 y(has)34 b(dimension)f Fp(\(Dim)15 b Fn(V)21 b Fp(\))p Fn(=r)s Fk(.)328 1029 y Fe(Pr)n(oof:)63 b Fp(W)-8 b(e)38 b(can)h(assume)g(that)g Fn(r)h(>)e Fp(1.)61 b(Let)39 b Fn(p)p Fp(\()p Fn(X)8 b Fp(\))37 b(=)h Fn(a)2585 1044 y Fu(0)2651 1029 y Fp(+)26 b Fn(a)2804 1044 y Fu(1)2844 1029 y Fn(X)34 b Fp(+)26 b Fl(\001)17 b(\001)g(\001)24 b Fp(+)i Fn(X)3394 993 y Ft(`)3466 1029 y Fp(b)s(e)328 1150 y(the)40 b(minimal)35 b(p)s(olynomial)g(of)k Fn(t)p Fp(.)63 b(Since)40 b Fn(t)f Fp(is)g(an)g(automorphism)e(of)i Fn(V)60 b Fp(necessarily)328 1270 y Fn(a)379 1285 y Fu(0)456 1270 y Fl(6)p Fp(=)38 b(0.)61 b(Since)39 b Fn(t)1003 1234 y Ft(r)1067 1270 y Fl(\000)27 b Fp(1)1220 1285 y Ft(V)1319 1270 y Fp(=)37 b(0)i(it)e(follo)m(ws)g(that)i Fn(`)f Fl(\024)g Fn(r)s Fp(.)61 b(T)-8 b(aking)38 b(the)h(trace)g(of)f (b)s(oth)328 1391 y(sides)29 b(of)f(the)i(equation)e(0)f(=)h Fn(a)1455 1406 y Fu(0)1495 1391 y Fp(1)1544 1406 y Ft(V)1619 1391 y Fp(+)14 b Fn(a)1760 1406 y Fu(1)1799 1391 y Fn(t)g Fp(+)g Fl(\001)j(\001)g(\001)d Fp(+)g Fn(t)2195 1354 y Ft(`)2256 1391 y Fp(giv)m(es)30 b Fm(tr)q Fp(\()p Fn(t)2634 1354 y Ft(`)2668 1391 y Fp(\))d(=)h Fl(\000)p Fn(a)2965 1406 y Fu(0)3005 1391 y Fp(\(Dim)14 b Fn(V)22 b Fp(\))27 b Fl(6)p Fp(=)h(0.)328 1511 y(Therefore)34 b Fn(`)29 b Fl(\025)g Fn(r)s Fp(.)45 b(Consequen)m(tly)36 b Fn(`)29 b Fp(=)f Fn(r)36 b Fp(and)d Fn(X)2204 1475 y Ft(r)2265 1511 y Fl(\000)23 b Fp(1)33 b(is)g(the)g(minimal)c(p)s(olynomial)328 1631 y(of)j Fn(t)p Fp(.)474 1752 y(T)-8 b(o)36 b(complete)g(the)g(pro)s 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2447 y(.)1642 2481 y(.)1917 2414 y(.)1917 2447 y(.)1917 2481 y(.)2392 2414 y(.)2392 2447 y(.)2392 2481 y(.)1559 2601 y Fn(!)1624 2560 y Ft(r)r Fj(\000)p Fu(1)1620 2623 y(1)1835 2601 y Fn(!)1900 2560 y Ft(r)r Fj(\000)p Fu(1)1896 2623 y(2)2110 2601 y Fl(\001)17 b(\001)g(\001)81 b Fn(!)2374 2565 y Ft(r)r Fj(\000)p Fu(1)2370 2626 y Ft(r)2543 2130 y Fg(1)2543 2276 y(C)2543 2326 y(C)2543 2376 y(C)2543 2425 y(C)2543 2479 y(A)2632 2400 y Fn(:)328 2814 y Fp(Then)829 3109 y Fl(A)926 2838 y Fg(0)925 2985 y(B)925 3034 y(B)925 3084 y(B)925 3134 y(B)925 3187 y(@)1040 2907 y Fn(n)1098 2922 y Fu(1)1040 3027 y Fn(n)1098 3042 y Fu(2)1075 3123 y Fp(.)1075 3156 y(.)1075 3189 y(.)1040 3310 y Fn(n)1098 3325 y Ft(r)1179 2838 y Fg(1)1179 2985 y(C)1179 3034 y(C)1179 3084 y(C)1179 3134 y(C)1179 3187 y(A)1279 3109 y Fp(=)1383 2838 y Fg(0)1383 2985 y(B)1383 3034 y(B)1383 3084 y(B)1383 3134 y(B)1383 3187 y(@)1512 2907 y Fp(Dim)15 b Fn(V)1561 3027 y Fm(tr)r Fp(\()p Fn(t)p Fp(\))1638 3123 y(.)1638 3156 y(.)1638 3189 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b(ctor)43 b(sp)-5 b(ac)g(e)41 b(over)h(the)h(\014eld)f Fn(k)j Fk(and)328 4573 y(supp)-5 b(ose)31 b(that)g Fn(k)k Fk(has)c(char)-5 b(acteristic)30 b Fp(0)p Fk(.)44 b(L)-5 b(et)31 b Fn(t)h Fk(b)-5 b(e)31 b(a)g(diagonalizable)e(line)-5 b(ar)31 b(automor-)328 4693 y(phism)c(of)h Fn(V)50 b Fk(which)27 b(has)h(\014nite)g(or)-5 b(der)28 b Fn(r)j Fk(and)c(supp)-5 b(ose)28 b(that)g Fm(tr)r Fp(\()p Fn(t)7 b Fp(+)g Fn(t)2817 4657 y Fu(2)2865 4693 y Fp(+)g Fl(\001)17 b(\001)g(\001)5 b Fp(+)i Fn(t)3189 4657 y Ft(r)r Fj(\000)p Fu(1)3318 4693 y Fp(\))28 b(=)f(0)p Fk(.)328 4814 y(Then)34 b Fp(Dim)14 b Fn(V)50 b Fp(=)27 b Fn(r)s Fp(\(Dim)14 b Fn(U)c Fp(\))p Fk(,)36 b(wher)-5 b(e)34 b Fn(U)45 b Fk(is)35 b(the)g(subsp)-5 b(ac)g(e)34 b(of)h(\014xe)-5 b(d)34 b(p)-5 b(oints)34 b(of)h Fn(t)p Fk(.)1922 5214 y Fp(6)p eop %%Page: 7 7 7 6 bop 328 631 a Fe(Pr)n(oof:)68 b Fp(Since)41 b Fn(t)h Fp(has)f(\014nite)g(order)g Fn(r)j Fp(and)d(is)g(diagonalizable,)f Fn(V)63 b Fp(has)41 b(a)g(basis)g(of)328 751 y(eigen)m(v)m(ectors)25 b(for)f Fn(t)g Fp(b)s(elonging)e(to)i Fn(r)1655 715 y Ft(th)1748 751 y Fp(ro)s(ots)g(of)f(unit)m(y)i Fn(!)2397 766 y Fu(1)2436 751 y Fn(;)17 b(:)g(:)g(:)f(;)h(!)2716 766 y Ft(n)2762 751 y Fp(,)26 b(where)f Fn(n)j Fp(=)f(Dim)n Fn(V)22 b Fp(.)328 872 y(W)-8 b(e)33 b(ma)m(y)f(assume)h(that)g Fn(!)1317 887 y Fu(1)1356 872 y Fn(;)17 b(:)g(:)g(:)f(;)h(!)1636 887 y Ft(s)1700 872 y Fl(6)p Fp(=)27 b(1)32 b(and)h(that)f Fn(!)2346 887 y Ft(s)p Fu(+1)2501 872 y Fp(=)27 b Fl(\001)17 b(\001)g(\001)26 b Fp(=)i Fn(!)2913 887 y Ft(n)2987 872 y Fp(=)g(1.)43 b(Since)1286 1055 y Fm(tr)r Fp(\()p Fn(t)1429 1014 y Ft(`)1462 1055 y Fp(\))28 b(=)f Fn(!)1696 1014 y Ft(`)1692 1080 y Fu(1)1754 1055 y Fp(+)22 b Fl(\001)17 b(\001)g(\001)j Fp(+)i Fn(!)2153 1014 y Ft(`)2149 1080 y(s)2208 1055 y Fp(+)g(\()p Fn(n)g Fl(\000)h Fn(s)p Fp(\))328 1239 y(for)35 b(all)e(1)f Fl(\024)h Fn(`)f Fl(\024)h Fn(r)26 b Fl(\000)f Fp(1,)36 b(and)f Fn(!)1529 1254 y Ft({)1580 1239 y Fp(+)24 b Fl(\001)17 b(\001)g(\001)22 b Fp(+)i Fn(!)1985 1203 y Ft(r)r Fj(\000)p Fu(1)1981 1264 y Ft({)2145 1239 y Fp(=)32 b Fl(\000)p Fp(1)k(for)e(all)g(1)e Fl(\024)h Fn({)f Fl(\024)h Fn(s)p Fp(,)j(it)e(follo)m(ws)328 1360 y(that)911 1480 y Fl(\000)p Fn(s)22 b Fp(+)g(\()p Fn(r)j Fl(\000)e Fp(1\)\()p Fn(n)f Fl(\000)g Fn(s)p Fp(\))28 b(=)f Fm(tr)r Fp(\()p Fn(t)22 b Fp(+)g Fn(t)2178 1439 y Fu(2)2240 1480 y Fp(+)g Fl(\001)17 b(\001)g(\001)k Fp(+)h Fn(t)2610 1439 y Ft(r)r Fj(\000)p Fu(1)2738 1480 y Fp(\))28 b(=)f(0)p Fn(:)328 1639 y Fp(Th)m(us)34 b Fn(n)28 b Fp(=)g Fn(r)s Fp(\()p Fn(n)21 b Fl(\000)i Fn(s)p Fp(\))33 b(whic)m(h)g(is)f(the)h(conclusion)f(of)g(the)h(lemma.)41 b Fd(2)474 1799 y Fp(No)m(w)30 b(let)e Fn(A)h Fp(b)s(e)g(a)f (\014nite-dimensional)e(Hopf)j(algebra)f(o)m(v)m(er)h Fn(k)s Fp(.)43 b(Recall)27 b(that)i Fn(a)f Fl(2)g Fn(A)328 1919 y Fp(is)h Fk(gr)-5 b(ouplike)29 b Fp(if)g(\001\()p Fn(a)p Fp(\))f(=)f Fn(a)p Fl(\012)q Fn(a)j Fp(and)g Fn(\017)p Fp(\()p Fn(a)p Fp(\))e(=)f(1.)43 b(The)30 b(set)h(of)e(grouplik)m(e)g (elemen)m(ts)h Fn(G)p Fp(\()p Fn(A)p Fp(\))328 2039 y(of)g Fn(A)h Fp(is)f(linearly)f(indep)s(enden)m(t)i(and)g(is)f(a)h(m)m (ultiplicativ)m(e)c(subgroup)k(of)f Fn(A)p Fp(.)43 b(Th)m(us)32 b(the)328 2160 y(span)40 b Fn(k)s Fp([)p Fn(G)p Fp(\()p Fn(A)p Fp(\)])g(of)g Fn(G)p Fp(\()p Fn(A)p Fp(\))g(is)f(the)h(group)g (algebra)f(of)g Fn(G)h Fp(o)m(v)m(er)h Fn(k)s Fp(.)66 b(By)40 b([10,)h(Theorem)328 2280 y(7])d(the)g(order)g(of)g Fn(G)f Fp(divides)h(Dim)15 b Fn(A)p Fp(.)59 b(In)39 b(particular)d(the) i(order)g(of)g(an)m(y)g Fn(a)g Fl(2)f Fn(G)p Fp(\()p Fn(A)p Fp(\),)328 2401 y(denoted)c(b)m(y)h(order\()p Fn(a)p Fp(\),)f(divides)f(Dim)15 b Fn(A)p Fp(.)474 2521 y(Let)34 b Fn(g)g Fl(2)c Fn(G)p Fp(\()p Fn(A)p Fp(\))k(and)g Fn(\013)d Fl(2)f Fn(G)p Fp(\()p Fn(A)1655 2485 y Fj(\003)1694 2521 y Fp(\))k(b)s(e)g(the)h(distinguished)e(grouplik)m(e)g(elemen)m (ts)h(of)328 2641 y Fn(A)f Fp(and)f Fn(A)696 2605 y Fj(\003)768 2641 y Fp(resp)s(ectiv)m(ely)-8 b(.)45 b(Then)1639 2825 y Fn(s)1685 2784 y Fu(4)1752 2825 y Fp(=)27 b Fn({)1886 2840 y Ft(g)1927 2825 y Fl(\016)o Fp(\()p Fn({)2045 2842 y Ft(\013)2090 2823 y Fi(\000)p Fc(1)2178 2825 y Fp(\))2216 2784 y Fj(\003)3441 2825 y Fp(\(1\))328 3009 y(is)d(the)i(comp)s (osition)c(of)j(t)m(w)m(o)g(comm)m(uting)e(automorphisms)h(of)g Fn(A)p Fp(,)j(where)f Fn({)3107 3024 y Ft(g)3172 3009 y Fp(is)e(de\014ned)328 3129 y(b)m(y)37 b Fn({)498 3144 y Ft(g)538 3129 y Fp(\()p Fn(x)p Fp(\))d(=)f Fn(g)t(xg)969 3093 y Fj(\000)p Fu(1)1098 3129 y Fp(for)i(all)f Fn(x)g Fl(2)f Fn(A)j Fp(and)g(the)g(automorphism)e Fn({)2730 3146 y Ft(\013)2775 3127 y Fi(\000)p Fc(1)2898 3129 y Fp(of)i Fn(A)3086 3093 y Fj(\003)3161 3129 y Fp(is)f(de\014ned)328 3250 y(b)m(y)k Fn({)500 3266 y Ft(\013)545 3247 y Fi(\000)p Fc(1)632 3250 y Fp(\()p Fn(p)p Fp(\))e(=)g Fn(\013)970 3213 y Fj(\000)p Fu(1)1064 3250 y Fn(p\013)i Fp(for)e(all)f Fn(p)h Fl(2)h Fn(A)1772 3213 y Fj(\003)1811 3250 y Fp(.)60 b(See)39 b([14,)f(16])g(for)f(example.)60 b(In)38 b(particular)328 3370 y(the)33 b(order)g(of)f Fn(s)908 3334 y Fu(2)980 3370 y Fp(is)g(\014nite)g(and)995 3554 y(order\()p Fn(s)1301 3513 y Fu(2)1341 3554 y Fp(\))55 b(divides)g(2)1817 3457 y Fg(\020)1868 3554 y Fp(lcm)o(\(order\()p Fn(g)t Fp(\))p Fn(;)17 b Fp(order)o(\()p Fn(\013)q Fp(\)\))2849 3457 y Fg(\021)3441 3554 y Fp(\(2\))328 3754 y(b)m(y)31 b(\(1\).)42 b(By)31 b(\(2\))f(it)f(follo)m(ws)g(that)h Fn(s)1629 3718 y Fu(2)1699 3754 y Fp(is)f(semisimple)f(if)h(the)i(c)m (haracteristic)f(of)g Fn(k)j Fp(is)d(not)328 3874 y(2)i(and)h(do)s(es)g (not)f(divide)g(order\()p Fn(g)t Fp(\),)g(order\()p Fn(\013)q Fp(\).)328 4202 y Fo(2)161 b(General)51 b(Results)g(for)g Fb(\027)2060 4223 y Fn(A)2138 4202 y Fa(\()p Fb(t)p Fa(\))f Fo(and)h Fb(\027)2719 4223 y Fn(A)2797 4202 y Fa(\()p Fb(s)2914 4150 y Fp(2)p Fn(m)3052 4202 y Fa(\))g Fo(in)h(the)570 4385 y(Sp)t(ecial)j(Case)e(when)g Fb(s)1997 4332 y Fp(2)2103 4385 y Fo(Is)h(not)f(Semisimple)328 4604 y Fp(Let)35 b Fn(A)g Fp(b)s(e)g(a)g(\014nite-dimensional)d(Hopf)i(algebra)g(with)h (an)m(tip)s(o)s(de)f Fn(s)h Fp(o)m(v)m(er)h(the)f(\014eld)g Fn(k)328 4724 y Fp(and)d(let)f(\003)h(b)s(e)g(a)g(\014xed)h(non-zero)f (left)f(in)m(tegral)g(for)g Fn(A)p Fp(.)44 b(F)-8 b(or)31 b(a)h(linear)e(endomorphism)328 4844 y Fn(t)j Fp(of)f Fn(A)g Fp(w)m(e)i(de\014ne)1481 4965 y Fn(\027)1529 4980 y Ft(A)1586 4965 y Fp(\()p Fn(t)p Fp(\))28 b(=)g Fn(s)p Fp(\(\003)1981 4980 y Fu(\(2\))2075 4965 y Fp(\))p Fn(t)p Fp(\(\003)2254 4980 y Fu(\(1\))2348 4965 y Fp(\))p Fn(:)1028 b Fp(\(3\))1922 5214 y(7)p eop %%Page: 8 8 8 7 bop 328 631 a Fp(Since)25 b(the)g(subspace)i(of)d(left)g(in)m (tegrals)f(for)h Fn(A)h Fp(is)f(one-dimensional,)g(c)m(ho)s(osing)g(a)h (di\013er-)328 751 y(en)m(t)31 b(non-zero)g(left)e(in)m(tegral)g(to)h (de\014ne)i Fn(\027)1851 766 y Ft(A)1908 751 y Fp(\()p Fn(t)p Fp(\))e(results)h(in)f(a)g(non-zero)g(scalar)g(m)m(ultiple)328 872 y(of)i(the)h(original.)474 992 y(If)24 b Fn(\027)611 1007 y Ft(A)668 992 y Fp(\()p Fn(t)p Fp(\))k(=)f(0)c(then)h Fm(tr)r Fp(\()p Fn(t)p Fp(\))k(=)f(0)d(b)m(y)g(part)f(a\))g(of)g(the)h (next)g(lemma.)38 b(W)-8 b(e)24 b(are)g(in)m(terested)328 1112 y(in)f(determining)g(when)i Fm(tr)r Fp(\()p Fn(t)p Fp(\))j(=)f(0)d(for)g(certain)f Fn(t)p Fp(,)j(sp)s(eci\014cally)e(when) h Fn(t)j Fp(=)f Fn(s)3087 1076 y Fu(2)p Ft(m)3213 1112 y Fp(for)d(some)328 1233 y(in)m(teger)32 b Fn(m)p Fp(.)44 b(F)-8 b(or)32 b Fn(a)c Fl(2)g Fn(A)33 b Fp(let)e Fn(`)p Fp(\()p Fn(a)p Fp(\))p Fn(;)17 b(r)s Fp(\()p Fn(a)p Fp(\))32 b(b)s(e)h(the)g(endomorphisms)f(of)g Fn(A)h Fp(de\014ned)h(b)m(y)1231 1439 y Fn(`)p Fp(\()p Fn(a)p Fp(\)\()p Fn(x)p Fp(\))28 b(=)f Fn(ax)98 b Fp(and)g Fn(r)s Fp(\()p Fn(a)p Fp(\)\()p Fn(x)p Fp(\))28 b(=)f Fn(xa)328 1645 y Fp(for)32 b(all)e Fn(x)f Fl(2)f Fn(A)p Fp(.)474 1766 y(First)40 b(w)m(e)i(will)d(see)j (that)f Fn(\027)1499 1781 y Ft(A)1556 1766 y Fp(\()p Fn(t)p Fp(\))h(=)f(0)g(if)f(and)h(only)f(if)g(the)h(traces)h(of)e (certain)h(re-)328 1886 y(lated)h(endomorphisms)h(v)-5 b(anish.)74 b(This)44 b(result)f(has)g(a)g(symmetric)f(form)g(if)g Fn(t)h Fp(is)g(an)328 2006 y(algebra)34 b(automorphism)f(whic)m(h)j (comm)m(utes)f(with)g Fn(s)p Fp(.)51 b(W)-8 b(e)35 b(note)h(that)f (Hopf)g(algebra)328 2127 y(endomorphisms)d(comm)m(ute)g(with)g(the)h (an)m(tip)s(o)s(de.)328 2339 y Ff(Lemma)k(3)49 b Fk(L)-5 b(et)39 b Fn(A)f Fk(b)-5 b(e)38 b(a)g(\014nite-dimensional)e(Hopf)j (algebr)-5 b(a)37 b(with)h(antip)-5 b(o)g(de)38 b Fn(s)g Fk(over)328 2460 y(the)d(\014eld)f Fn(k)k Fk(and)c(supp)-5 b(ose)34 b(that)i Fn(t)f Fk(is)f(a)h(line)-5 b(ar)34 b(endomorphism)f(of)i Fn(A)p Fk(.)45 b(Then:)436 2651 y Fp(a\))k Fm(tr)r Fp(\()p Fn(t)p Fp(\))44 b(=)f Fn(\025)p Fp(\()p Fn(\027)1059 2666 y Ft(A)1116 2651 y Fp(\()p Fn(t)p Fp(\)\))h(=)g Fn(\025)p Fp(\()p Fn(s)p Fp(\(\003)1676 2667 y Fu(\(2\))1770 2651 y Fp(\))p Fn(t)p Fp(\(\003)1949 2667 y Fu(\(1\))2043 2651 y Fp(\)\))p Fk(,)h(wher)-5 b(e)43 b Fp(\003)h Fk(is)f(a)g(left)h(inte)-5 b(gr)g(al)43 b(for)h Fn(A)572 2772 y Fk(and)34 b Fn(\025)h Fk(is)f(a)h(right)g(inte) -5 b(gr)g(al)34 b(for)h Fn(A)1851 2735 y Fj(\003)1926 2772 y Fk(which)f(satisfy)g Fn(\025)p Fp(\(\003\))28 b(=)f(1)p Fk(.)431 2971 y Fp(b\))49 b Fn(\027)620 2986 y Ft(A)677 2971 y Fp(\()p Fn(t)p Fp(\))28 b(=)f(0)35 b Fk(if)g(and)f(only)h(if)f Fm(tr)r Fp(\()p Fn(r)s Fp(\()p Fn(a)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))27 b(=)h(0)35 b Fk(for)f(al)5 b(l)35 b Fn(a)28 b Fl(2)g Fn(A)p Fk(.)442 3170 y Fp(c\))49 b Fk(If)39 b Fn(t)i Fk(is)f(an)f(algebr)-5 b(a)40 b(automorphism)f(of)h Fn(A)g Fk(then)g Fn(\027)2492 3185 y Ft(A)2549 3170 y Fp(\()p Fn(t)2622 3134 y Fj(\000)p Fu(1)2717 3170 y Fp(\))d(=)h(0)h Fk(if)h(and)g(only)g(if)572 3291 y Fm(tr)r Fp(\()p Fn(l)r Fp(\()p Fn(a)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))27 b(=)h(0)35 b Fk(for)f(al)5 b(l)35 b Fn(a)28 b Fl(2)g Fn(A)p Fk(.)328 3549 y Fe(Pr)n(oof:)51 b Fp(Let)33 b(\003)g(b)s(e)g(a)g(left)f(in)m(tegral)g(for)h Fn(A)g Fp(and)g Fn(\025)g Fp(b)s(e)g(a)g(righ)m(t)g(in)m(tegral)e(for)i Fn(A)3306 3513 y Fj(\003)3378 3549 y Fp(suc)m(h)328 3670 y(that)f Fn(\025)p Fp(\(\003\))27 b(=)h(1.)43 b(Then)625 3876 y Fm(tr)r Fp(\()p Fn(t)p Fp(\))27 b(=)h Fn(\025)p Fp(\()p Fn(s)p Fp(\(\003)1184 3891 y Fu(\(2\))1278 3876 y Fp(\))p Fn(t)p Fp(\(\003)1457 3891 y Fu(\(1\))1551 3876 y Fp(\)\))f(=)h Fn(\025)p Fp(\()p Fn(s)p Fp(\()p Fn(t)p Fp(\(\003)2078 3891 y Fu(\(2\))2172 3876 y Fp(\)\)\003)2316 3891 y Fu(\(1\))2410 3876 y Fp(\))f(=)h Fn(\025)p Fp(\()p Fn(t)p Fp(\()p Fn(s)p Fp(\(\003)2899 3891 y Fu(\(2\))2993 3876 y Fp(\)\)\003)3137 3891 y Fu(\(1\))3231 3876 y Fp(\))328 4082 y(for)f(all)e(linear)h(endomorphisms)h Fn(t)g Fp(of)g Fn(A)h Fp(b)m(y)g([16,)g(Theorem)g(2].)42 b(Th)m(us)29 b(part)e(a\))g(follo)m(ws.)474 4202 y(T)-8 b(o)31 b(sho)m(w)i(part)e (b\))g(w)m(e)h(recall)d(that)i Fn(A)1873 4166 y Fj(\003)1944 4202 y Fp(is)g(a)f(free)i(left)e Fn(A)p Fp(-mo)s(dule)f(with)i(basis)g Fl(f)p Fn(\025)p Fl(g)p Fp(,)328 4323 y(where)j(\()p Fn(a)p Fl(\001)p Fn(\025)p Fp(\)\()p Fn(x)p Fp(\))27 b(=)h Fn(\025)p Fp(\()p Fn(xa)p Fp(\))33 b(for)f(all)e Fn(a;)17 b(x)28 b Fl(2)g Fn(A)p Fp(.)44 b(Therefore)1079 4529 y(\()p Fn(a)p Fl(\001)p Fn(\025)p Fp(\)\()p Fn(\027)1377 4544 y Ft(A)1434 4529 y Fp(\()p Fn(t)p Fp(\)\))83 b(=)g Fn(\025)p Fp(\()p Fn(\027)1968 4544 y Ft(A)2025 4529 y Fp(\()p Fn(t)p Fp(\))p Fn(a)p Fp(\))1666 4674 y(=)g Fn(\025)p Fp(\()p Fn(s)p Fp(\(\003)2072 4690 y Fu(\(2\))2166 4674 y Fp(\))p Fn(t)p Fp(\(\003)2345 4690 y Fu(\(1\))2439 4674 y Fp(\))p Fn(a)p Fp(\))1666 4819 y(=)g Fn(\025)p Fp(\()p Fn(s)p Fp(\(\003)2072 4835 y Fu(\(2\))2166 4819 y Fp(\)\()p Fn(r)s Fp(\()p Fn(a)p Fp(\))p Fl(\016)o Fn(t)p Fp(\)\(\003)2644 4835 y Fu(\(1\))2738 4819 y Fp(\)\))1666 4965 y(=)g Fm(tr)r Fp(\()p Fn(r)s Fp(\()p Fn(a)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))1922 5214 y(8)p eop %%Page: 9 9 9 8 bop 328 631 a Fp(for)46 b(all)f Fn(a)51 b Fl(2)h Fn(A)p Fp(.)86 b(Since)47 b Fn(A)1389 595 y Fj(\003)1480 631 y Fp(=)k Fn(A)p Fl(\001)p Fn(\025)c Fp(it)e(follo)m(ws)h(that)g Fn(\027)2531 646 y Ft(A)2588 631 y Fp(\()p Fn(t)p Fp(\))52 b(=)f(0)c(if)e(and)i(only)f(if)328 751 y(\()p Fn(a)p Fl(\001)p Fn(\025)p Fp(\)\()p Fn(\027)626 766 y Ft(A)683 751 y Fp(\()p Fn(t)p Fp(\)\))36 b(=)g(0)h(for)h(all)d Fn(a)i Fl(2)f Fn(A)p Fp(,)j(or)f(equiv)-5 b(alen)m(tly)37 b(if)f(and)i(only)f(if)f Fm(tr)r Fp(\()p Fn(r)s Fp(\()p Fn(a)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))h(=)f(0)328 872 y(for)c(all)e Fn(a)e Fl(2)g Fn(A)p Fp(.)474 992 y(The)49 b(pro)s(of)e(of)g(part)h(c\))f(is)h(similar)c(to)j(that)h(of)f(part)h (b\).)89 b(W)-8 b(e)48 b(note)g(that)f Fn(A)3526 956 y Fj(\003)328 1112 y Fp(is)c(also)h(a)f(free)i(righ)m(t)e Fn(A)p Fp(-mo)s(dule)f(with)i(basis)g Fl(f)p Fn(\025)p Fl(g)p Fp(,)i(where)g(\()p Fn(\025)p Fl(\001)o Fn(a)p Fp(\)\()p Fn(x)p Fp(\))i(=)f Fn(\025)p Fp(\()p Fn(ax)p Fp(\))d(for)328 1233 y(all)e Fn(a;)17 b(x)48 b Fl(2)g Fn(A)p Fp(.)79 b(Th)m(us)45 b(using)f([16,)j(Theorem)e(2])f(again)f(w)m (e)i(see)g(that)f Fm(tr)r Fp(\()p Fn(l)r Fp(\()p Fn(a)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))k(=)328 1353 y Fn(\025)p Fp(\()p Fn(at)p Fp(\()p 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Ff(Corollary)j(1)49 b Fk(L)-5 b(et)31 b Fn(A)h Fk(b)-5 b(e)31 b(a)h(\014nite-dimensional)d(Hopf)i(algebr)-5 b(a)31 b(over)g(the)h(\014eld)f Fn(k)j Fk(and)328 2184 y(supp)-5 b(ose)30 b(that)h Fn(t)g Fk(is)f(a)h(line)-5 b(ar)30 b(automorphism)g(of)g Fn(A)p Fk(.)44 b(If)30 b Fn(\027)2431 2199 y Ft(A)2488 2184 y Fp(\()p Fn(t)p Fp(\))e(=)f(0)k Fk(then)f Fn(\027)3070 2199 y Ft(A)3128 2184 y Fp(\()p Fn(r)s Fp(\()p Fn(a)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))e(=)328 2304 y(0)35 b Fk(for)f(al)5 b(l)35 b Fn(a)28 b Fl(2)g Fn(A)p Fk(.)45 b Fd(2)474 2532 y Fp(Supp)s(ose)f Fn(t)f Fp(is)f(an)h(algebra)f(automorphism)e(whic)m(h)k(comm)m(utes)e (with)h Fn(s)p Fp(.)74 b(Since)328 2653 y Fn(s)374 2617 y Fj(\000)p Fu(1)500 2653 y Fp(is)32 b(the)g(an)m(tip)s(o)s(de)f(of)h Fn(A)1352 2617 y Ft(cop)1456 2653 y Fp(,)g(taking)g(\003)f(as)h(the)h (non-zero)f(left)f(in)m(tegral)f(for)i Fn(A)3343 2617 y Ft(cop)3479 2653 y Fp(to)328 2773 y(de\014ne)i Fn(\027)658 2788 y Ft(A)711 2769 y Fh(cop)809 2773 y Fp(\()p Fn(t)p 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Fn(a)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))28 b(=)f(0)35 b Fk(for)g(al)5 b(l)34 b Fn(a)28 b Fl(2)g Fn(A)p Fk(.)431 4192 y Fp(d\))49 b Fm(tr)r Fp(\()p Fn(l)r Fp(\()p Fn(a)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))27 b(=)h(0)35 b Fk(for)f(al)5 b(l)35 b Fn(a)28 b Fl(2)g Fn(A)p Fk(.)328 4440 y Fe(Pr)n(oof:)49 b Fp(By)33 b(Lemma)d(3)h(it)g (follo)m(ws)g(that)g(a\))h(and)g(c\))g(are)f(equiv)-5 b(alen)m(t,)32 b(b\))g(and)g(d\))g(are)328 4560 y(equiv)-5 b(alen)m(t.)47 b(T)-8 b(o)34 b(sho)m(w)h(that)f(a\))f(and)h(b\))g(are)g (equiv)-5 b(alen)m(t)34 b(w)m(e)h(apply)e(the)h(equiv)-5 b(alence)328 4680 y(of)43 b(a\))g(and)h(c\))f(to)h Fn(A)1109 4644 y Ft(cop)1213 4680 y Fp(.)76 b(Hence)45 b Fn(\027)1665 4695 y Ft(A)1722 4680 y Fp(\()p Fn(t)p Fp(\))i(=)f(0)d(if)f(and)i(only) f(if)f Fn(\027)2768 4695 y Ft(A)2821 4676 y Fh(cop)2920 4680 y Fp(\()p Fn(t)p Fp(\))k(=)g(0.)76 b(Th)m(us)328 4801 y Fn(\027)376 4816 y Ft(A)429 4797 y Fh(cop)528 4801 y Fp(\()p Fn(t)p Fp(\))27 b(=)h(0)k(if)g(and)g(only)g(if)g Fn(\027)1482 4816 y Ft(A)1539 4801 y Fp(\()p Fn(t)1612 4765 y Fj(\000)p Fu(1)1707 4801 y Fp(\))27 b(=)h(0)k(b)m(y)h(\(4\).)44 b Fd(2)474 4965 y Fp(When)34 b Fn(t)28 b Fp(=)f Fn(s)971 4929 y Fu(2)1043 4965 y Fp(m)m(uc)m(h)33 b(more)f(can)h(b)s(e)g(said)f (ab)s(out)g Fn(\027)2387 4980 y Ft(A)2444 4965 y Fp(\()p Fn(s)2528 4929 y Fu(2)2567 4965 y Fp(\).)1922 5214 y(9)p eop %%Page: 10 10 10 9 bop 328 631 a Ff(Prop)s(osition)35 b(2)49 b Fk(L)-5 b(et)40 b Fn(A)g Fk(b)-5 b(e)39 b(a)g(\014nite-dimensional)f(Hopf)h (algebr)-5 b(a)39 b(with)g(antip)-5 b(o)g(de)39 b Fn(s)328 751 y Fk(over)34 b(the)h(\014eld)f Fn(k)s Fk(.)45 b(Then:)436 955 y Fp(a\))k Fn(\027)620 970 y Ft(A)677 955 y Fp(\()p Fn(s)761 919 y Fu(2)800 955 y Fp(\))28 b Fl(6)p Fp(=)f(0)35 b Fk(if)g(and)f(only)h(if)f Fn(A)h Fk(is)g(semisimple.)431 1158 y Fp(b\))49 b Fk(If)40 b(the)g(char)-5 b(acteristic)40 b(of)h Fn(k)i Fk(is)d Fp(0)h Fk(and)f Fn(m)h Fk(is)f(an)g(inte)-5 b(ger)40 b(r)-5 b(elatively)41 b(prime)f(to)572 1279 y(the)35 b(or)-5 b(der)34 b(of)h Fn(s)1151 1242 y Fu(2)1225 1279 y Fk(then)g Fm(tr)q Fp(\()p Fn(s)1595 1242 y Fu(2)p Ft(m)1697 1279 y Fp(\))28 b(=)f(0)p Fk(.)442 1482 y Fp(c\))49 b Fk(Supp)-5 b(ose)27 b Fn(B)32 b Fk(is)c(a)f(sub-Hopf)h(algebr)-5 b(a)26 b(of)i Fn(A)g Fk(and)f Fn(t)g Fk(is)h(an)f(algebr)-5 b(a)27 b(endomorphism)572 1602 y(of)34 b Fn(A)h Fk(which)f(satis\014es) g Fn(t)p Fp(\()p Fn(B)5 b Fp(\))28 b Fl(\022)h Fn(B)5 b Fk(.)44 b(Then)34 b Fn(\027)2213 1617 y Ft(B)2274 1602 y Fp(\()p Fn(t)p Fl(j)2375 1617 y Ft(B)2436 1602 y Fp(\))27 b(=)h(0)35 b Fk(implies)e Fn(\027)3070 1617 y Ft(A)3128 1602 y Fp(\()p Fn(t)p Fp(\))27 b(=)h(0)p Fk(.)328 1856 y Fe(Pr)n(oof:)63 b Fp(Since)39 b Fn(\027)1043 1871 y Ft(A)1100 1856 y Fp(\()p Fn(s)1184 1819 y Fu(2)1223 1856 y Fp(\))g(=)f Fn(s)p Fp(\(\003)1566 1871 y Fu(\(2\))1660 1856 y Fp(\))p Fn(s)1744 1819 y Fu(2)1783 1856 y Fp(\(\003)1889 1871 y Fu(\(1\))1983 1856 y Fp(\))g(=)g Fn(s)p Fp(\()p Fn(s)p Fp(\(\003)2409 1871 y Fu(\(1\))2503 1856 y Fp(\)\003)2609 1871 y Fu(\(2\))2703 1856 y Fp(\))g(=)g Fn(\017)p Fp(\(\003\)1,)i(and)f Fn(A)g Fp(is)328 1976 y(semisimple)30 b(if)i(and)g(only)g(if)g Fn(\017)p Fp(\(\003\)1)27 b Fl(6)p Fp(=)h(0,)k(part)g(a\))h(follo)m (ws.)474 2096 y(T)-8 b(o)25 b(pro)m(v)m(e)g(b\))f(w)m(e)i(\014rst)e (consider)h(the)g(case)g(when)g Fm(tr)r Fp(\()p Fn(s)2438 2060 y Fu(2)2477 2096 y Fp(\))j Fl(6)p Fp(=)f(0.)41 b(Then)25 b Fn(A)f Fp(is)g(semisim-)328 2217 y(ple)37 b(and)h(cosemisimple)d(b)m (y)j([5,)h(Theorem)f(2.5],)h(and)e Fn(s)2433 2181 y Fu(2)2509 2217 y Fp(=)f(1)2670 2232 y Ft(A)2764 2217 y Fp(b)m(y)i([6,)h(Theorem)f (3].)328 2337 y(Hence)f(b\))f(is)g(v)-5 b(acuous)36 b(in)f(this)h (case.)54 b(No)m(w)37 b(w)m(e)g(assume)f(that)g Fm(tr)r Fp(\()p Fn(s)2867 2301 y Fu(2)2906 2337 y Fp(\))d(=)g(0.)54 b(W)-8 b(e)36 b(ma)m(y)328 2457 y(also)c(assume)h(that)f Fn(k)k Fp(is)c(algebraically)d(closed.)44 b(Th)m(us)34 b Fn(s)2442 2421 y Fu(2)2514 2457 y Fp(is)e(diagonalizable.)474 2578 y(Let)40 b Fn(r)i Fp(b)s(e)d(the)h(order)g(of)f Fn(s)1482 2542 y Fu(2)1521 2578 y Fp(.)64 b(Let)40 b Fl(f)p Fn(x)1899 2593 y Ft({)1926 2578 y Fl(g)1976 2593 y Fu(1)p Fj(\024)p Ft({)p Fj(\024)p Ft(n)2231 2578 y Fp(b)s(e)f(a)g(basis)h(of)f Fn(A)g Fp(of)g(eigen)m(v)m(ectors)328 2698 y(of)f Fn(s)491 2662 y Fu(2)530 2698 y Fp(.)62 b(W)-8 b(rite)38 b Fn(s)942 2662 y Fu(2)981 2698 y Fp(\()p Fn(x)1074 2713 y Ft({)1102 2698 y Fp(\))g(=)f Fn(!)1356 2662 y Ft(a)1393 2670 y Fh({)1424 2698 y Fn(x)1479 2713 y Ft({)1545 2698 y Fp(for)h(all)f Fn({)p Fp(,)j(where)g Fn(!)h Fp(is)e(a)f(ro)s(ot) f(of)i(unit)m(y)f(of)g(order)h Fn(r)s Fp(,)328 2819 y(and)e(0)f Fl(\024)g Fn(a)771 2834 y Ft({)835 2819 y Fn(<)f(r)40 b Fp(for)d(all)f Fn(i)p Fp(.)58 b(Since)37 b Fm(tr)r Fp(\()p Fn(s)1856 2782 y Fu(2)1895 2819 y Fp(\))f(=)g(0)h(it)f(follo)m (ws)g(that)h Fn(!)k Fp(is)c(a)g(ro)s(ot)f(of)h(the)328 2939 y(p)s(olynomial)h Fn(X)930 2903 y Ft(a)967 2912 y Fc(1)1034 2939 y Fp(+)29 b Fl(\001)17 b(\001)g(\001)26 b Fp(+)i Fn(X)1476 2903 y Ft(a)1513 2911 y Fh(n)1560 2939 y Fp(.)71 b(Let)42 b(\010)1912 2954 y Ft(r)1992 2939 y Fp(b)s(e)g(the)g(minimal)37 b(p)s(olynomial)i(o)m(v)m(er)j(the) 328 3059 y(rational)29 b(n)m(um)m(b)s(ers)k(of)e(a)g(ro)s(ot)g(of)g (unit)m(y)h(of)g(order)f Fn(r)s Fp(.)43 b(Then)33 b Fn(X)2669 3023 y Ft(a)2706 3032 y Fc(1)2765 3059 y Fp(+)21 b Fl(\001)c(\001)g (\001)h Fp(+)j Fn(X)3184 3023 y Ft(a)3221 3031 y Fh(n)3295 3059 y Fp(=)28 b Fn(f)11 b Fp(\010)3528 3074 y Ft(r)328 3180 y Fp(for)35 b(some)g(p)s(olynomial)d Fn(f)11 b Fp(.)51 b(Since)36 b Fn(m)f Fp(is)g(relativ)m(ely)f(prime)g(to)h Fn(r)j Fp(necessarily)e Fn(!)3315 3144 y Ft(m)3416 3180 y Fp(is)f(a)328 3300 y(ro)s(ot)d(of)g(\010)718 3315 y Ft(r)756 3300 y Fp(.)44 b(Th)m(us)34 b(0)27 b(=)h Fn(!)1319 3264 y Ft(ma)1418 3273 y Fc(1)1479 3300 y Fp(+)22 b Fl(\001)17 b(\001)g(\001)j Fp(+)i Fn(!)1878 3264 y Ft(ma)1977 3272 y Fh(n)2052 3300 y Fp(=)27 b Fm(tr)r Fp(\()p Fn(s)2309 3264 y Fu(2)p Ft(m)2411 3300 y Fp(\).)474 3421 y(T)-8 b(o)40 b(sho)m(w)h(part)e(c\))h(w)m(e)g(use)h(the)f(fact)f(that)g Fn(A)h Fp(is)f(a)g(free)h(left)f Fn(B)5 b Fp(-mo)s(dule)38 b(b)m(y)i([10,)328 3541 y(Theorem)27 b(7].)42 b(Cho)s(ose)28 b(a)e(basis)h Fl(f)p Fn(m)1657 3556 y Fu(1)1697 3541 y Fn(;)17 b(:)g(:)g(:)f(;)h(m)2001 3556 y Ft(r)2039 3541 y Fl(g)27 b Fp(for)f Fn(A)h Fp(as)h(a)e(left)h Fn(B)5 b Fp(-mo)s(dule)25 b(and)i(write)328 3661 y(\003)37 b(=)g Fn(b)587 3676 y Fu(1)627 3661 y Fn(m)712 3676 y Fu(1)778 3661 y Fp(+)26 b Fl(\001)17 b(\001)g(\001)24 b Fp(+)i Fn(b)1165 3676 y Ft(s)1202 3661 y Fn(m)1287 3676 y Ft(s)1362 3661 y Fp(for)38 b(some)g Fn(b)1808 3676 y Fu(1)1848 3661 y Fn(;)17 b(:)g(:)g(:)f(;)h(b)2108 3676 y Ft(r)2183 3661 y Fl(2)38 b Fn(B)5 b Fp(.)61 b(Let)38 b(\003)2702 3676 y Ft(B)2801 3661 y Fp(b)s(e)g(a)g(non-zero)h(left)328 3782 y(in)m(tegral)27 b(for)i Fn(B)5 b Fp(.)42 b(F)-8 b(rom)28 b(the)h(equation)g Fn(b)p Fp(\003)f(=)f Fn(\017)p Fp(\()p Fn(b)p Fp(\)\003)j(for)e(all)f Fn(b)h Fl(2)g Fn(B)34 b Fp(w)m(e)d(deduce)f(that)f Fn(b)3538 3797 y Ft({)328 3902 y Fp(is)g(a)g(left)f(in)m(tegral)g(for)g(all)f(1)h Fl(\024)g Fn({)g Fl(\024)g Fn(r)s Fp(.)42 b(Th)m(us)31 b(for)e(eac)m(h)h(1)d Fl(\024)h Fn({)g Fl(\024)g Fn(r)k Fp(w)m(e)e(ha)m(v)m(e)h Fn(b)3189 3917 y Ft({)3244 3902 y Fp(=)d Fn(\013)3410 3917 y Ft({)3437 3902 y Fp(\003)3505 3917 y Ft(B)328 4022 y Fp(for)k(some)g Fn(\013)783 4037 y Ft({)838 4022 y Fl(2)d Fn(k)s Fp(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g(\003)27 b(=)h(\003)2013 4037 y Ft(B)2073 4022 y Fn(a)p Fp(,)33 b(where)h Fn(a)28 b Fp(=)f Fn(\013)2710 4037 y Fu(1)2750 4022 y Fn(m)2835 4037 y Fu(1)2897 4022 y Fp(+)22 b Fl(\001)17 b(\001)g(\001)j Fp(+)i Fn(\013)3293 4037 y Ft(r)3331 4022 y Fn(m)3416 4037 y Ft(r)3454 4022 y Fp(.)474 4143 y(Supp)s(ose)43 b Fn(t)f Fp(is)g(an)g(algebra)e(endomorphism)h(of)h Fn(A)g Fp(whic)m(h)g(satis\014es)h Fn(t)p Fp(\()p Fn(B)5 b Fp(\))43 b Fl(\022)i Fn(B)5 b Fp(.)328 4263 y(The)33 b(calculation)436 4483 y Fn(s)p Fp(\(\003)588 4499 y Fu(\(2\))682 4483 y Fp(\))p Fn(t)p Fp(\(\003)861 4499 y Fu(\(1\))955 4483 y Fp(\))28 b(=)f Fn(s)p Fp(\(\003)1276 4499 y Ft(B)15 b Fu(\(2\))1439 4483 y Fn(a)1490 4499 y Fu(\(2\))1584 4483 y Fp(\))p Fn(t)p Fp(\(\003)1763 4499 y Ft(B)g Fu(\(1\))1925 4483 y Fn(a)1976 4499 y Fu(\(1\))2071 4483 y Fp(\))28 b(=)f Fn(s)p Fp(\()p Fn(a)2375 4499 y Fu(\(2\))2470 4483 y Fp(\))p Fn(s)p Fp(\(\003)2660 4499 y Ft(B)14 b Fu(\(2\))2822 4483 y Fp(\))p Fn(t)p Fp(\(\003)3001 4499 y Ft(B)h Fu(\(1\))3163 4483 y Fp(\))p Fn(t)p Fp(\()p Fn(a)3325 4499 y Fu(\(1\))3420 4483 y Fp(\))328 4703 y(sho)m(ws)36 b(that)f Fn(\027)872 4718 y Ft(A)929 4703 y Fp(\()p Fn(t)p Fp(\))c(=)g Fn(s)p Fp(\()p Fn(a)1313 4719 y Fu(\(2\))1408 4703 y Fp(\))p Fn(\027)1494 4718 y Ft(B)1554 4703 y Fp(\()p Fn(t)p Fl(j)1655 4718 y Ft(B)1716 4703 y Fp(\))p Fn(t)p Fp(\()p Fn(a)1878 4719 y Fu(\(1\))1972 4703 y Fp(\).)50 b(P)m(art)35 b(c\))g(follo)m(ws)e (from)h(this)g(equation.)328 4824 y Fd(2)1898 5214 y Fp(10)p eop %%Page: 11 11 11 10 bop 474 631 a Fp(P)m(art)46 b(b\))f(of)g(the)h(preceding)g(prop)s (osition)d(implies)g(that)i(if)f(the)i(order)g(of)f Fn(s)3416 595 y Fu(2)3500 631 y Fp(is)328 751 y(prime)f(and)i Fn(k)i Fp(has)e(c)m(haracteristic)f(zero)h(then)g Fn(s)2218 715 y Fu(2)2303 751 y Fp(satis\014es)g(the)f(strong)h(v)-5 b(anishing)328 872 y(trace)33 b(condition.)474 992 y(By)g(part)f(c\))g (of)g(the)g(preceding)h(prop)s(osition)d(in)h(man)m(y)h(cases)i Fn(\027)2851 1007 y Ft(A)2908 992 y Fp(\()p Fn(s)2992 956 y Fu(2)p Ft(m)3094 992 y Fp(\))27 b(=)h(0)k(when)328 1112 y(the)h(c)m(haracteristic)f(of)g Fn(k)k Fp(is)c(p)s(ositiv)m(e.) 328 1316 y Ff(Corollary)k(2)49 b Fk(L)-5 b(et)28 b Fn(A)h Fk(b)-5 b(e)28 b(a)g(\014nite-dimensional)e(Hopf)i(algebr)-5 b(a)27 b(with)h(antip)-5 b(o)g(de)28 b Fn(s)g Fk(over)328 1436 y(the)41 b(\014eld)g Fn(k)k Fk(which)40 b(c)-5 b(ontains)41 b(non-semisimple)e(sub-Hopf)i(algebr)-5 b(a)40 b(which)h(is)g(either) 328 1557 y(c)-5 b(ommutative)34 b(or)h(c)-5 b(o)g(c)g(ommutative.)44 b(Then)34 b Fn(\027)2021 1572 y Ft(A)2078 1557 y Fp(\()p Fn(s)2162 1521 y Fu(2)p Ft(m)2264 1557 y Fp(\))28 b(=)f(0)35 b Fk(for)f(al)5 b(l)35 b(inte)-5 b(gers)34 b Fn(m)p Fk(.)328 1810 y Fe(Pr)n(oof:)68 b Fp(Let)42 b Fn(B)k Fp(b)s(e)c(a)e(sub-Hopf)i (algebra)e(of)h Fn(A)g Fp(whic)m(h)h(satis\014es)g(the)g(h)m(yp)s (othesis)328 1930 y(of)35 b(the)h(lemma.)51 b(Since)36 b Fn(B)41 b Fp(is)35 b(comm)m(utativ)m(e)g(or)h(co)s(comm)m(utativ)m(e) e Fn(s)2884 1894 y Fu(2)2924 1930 y Fl(j)2952 1945 y Ft(B)3045 1930 y Fp(=)f(1)3203 1945 y Ft(B)3264 1930 y Fp(.)53 b(Since)328 2051 y Fn(B)e Fp(is)46 b(not)h(semisimple)d Fn(\027)1309 2066 y Ft(B)1370 2051 y Fp(\()p Fn(s)1454 2014 y Fu(2)1493 2051 y Fl(j)1521 2066 y Ft(B)1581 2051 y Fp(\))52 b(=)f(0)46 b(b)m(y)h(part)g(a\))f(of)g(Prop)s(osition)f(2.) 84 b(F)-8 b(or)46 b(an)m(y)328 2171 y(in)m(teger)i Fn(m)g Fp(note)g(that)g Fn(B)53 b Fp(is)47 b(in)m(v)-5 b(arian)m(t)46 b(under)j(the)f(algebra)f(map)g Fn(s)3032 2135 y Fu(2)p Ft(m)3182 2171 y Fp(and)h(that)328 2291 y Fn(\027)376 2306 y Ft(B)437 2291 y Fp(\()p Fn(s)521 2255 y Fu(2)p Ft(m)623 2291 y Fl(j)651 2306 y Ft(B)711 2291 y Fp(\))28 b(=)g Fn(\027)929 2306 y Ft(B)990 2291 y Fp(\(1)p Fl(j)1105 2306 y Ft(B)1165 2291 y Fp(\))g(=)g Fn(\027)1383 2306 y Ft(B)1444 2291 y Fp(\()p Fn(s)1528 2255 y Fu(2)1567 2291 y Fl(j)1595 2306 y Ft(B)1656 2291 y Fp(\))g(=)g(0.)44 b(Hence)34 b(the)g(Corollary)d(follo)m(ws)g(from)h(part)328 2412 y(c\))h(of)f(Prop)s(osition)f(2.)43 b Fd(2)474 2582 y Fp(When)35 b Fn(s)806 2546 y Fu(2)880 2582 y Fp(is)e(not)h(a)g (semisimple)e(endomorphism)h(of)g Fn(A)p Fp(,)i(and)f(the)h(c)m (haracteristic)328 2702 y(of)d Fn(k)k Fp(is)c(not)g(2,)h(it)e(is)h(alw) m(a)m(ys)h(the)g(case)h(that)e Fm(tr)r Fp(\()p Fn(s)2153 2666 y Fu(2)p Ft(m)2255 2702 y Fp(\))27 b(=)h(0.)328 2906 y Ff(Prop)s(osition)35 b(3)49 b Fk(L)-5 b(et)40 b Fn(A)g Fk(b)-5 b(e)39 b(a)g(\014nite-dimensional)f(Hopf)h(algebr)-5 b(a)39 b(with)g(antip)-5 b(o)g(de)39 b Fn(s)328 3026 y Fk(over)30 b(the)h(\014eld)e Fn(k)s Fk(.)44 b(Supp)-5 b(ose)29 b(that)i Fn(s)1636 2990 y Fu(2)1706 3026 y Fk(is)g(not)f(a)g (semisimple)f(endomorphism)g(of)h Fn(A)g Fk(and)328 3147 y(that)35 b(the)g(char)-5 b(acteristic)34 b(of)h Fn(k)j Fk(is)c(not)h Fp(2)p Fk(.)45 b(Then:)436 3350 y Fp(a\))k Fk(Either)35 b Fn(\027)919 3365 y Ft(A)976 3350 y Fp(\()p Fn(s)1060 3314 y Fu(2)p Ft(m)1162 3350 y Fp(\))27 b(=)h(0)34 b Fk(for)h(al)5 b(l)34 b(inte)-5 b(gers)34 b Fn(m)h Fk(or)g Fn(\027)2365 3365 y Ft(A)2418 3346 y Fi(\003)2458 3350 y Fp(\()p Fn(s)2542 3314 y Fj(\003)12 b Fu(2)p Ft(m)2691 3350 y Fp(\))27 b(=)h(0)34 b Fk(for)h(al)5 b(l)34 b(inte)-5 b(gers)572 3470 y Fn(m)p Fk(.)431 3674 y Fp(b\))49 b Fm(tr)r Fp(\()p Fn(s)726 3638 y Fu(2)p Ft(m)827 3674 y Fp(\))28 b(=)f(0)35 b Fk(for)g(al)5 b(l)34 b(inte)-5 b(gers)35 b Fn(m)p Fk(.)328 3927 y Fe(Pr)n(oof:)49 b Fp(W)-8 b(e)32 b(ma)m(y)g(assume)g(that)g Fn(k)j Fp(is)c(algebraically) e(closed.)43 b(Since)32 b(the)g(c)m(haracter-)328 4047 y(istic)25 b(of)i Fn(k)i Fp(is)d(not)h(2)f(and)h Fn(s)1279 4011 y Fu(2)1345 4047 y Fp(is)f(not)g(semisimple,)g(it)f(follo)m(ws)g (that)i(the)g(c)m(haracteristic)f(of)328 4168 y Fn(k)j Fp(is)c Fn(p)i(>)h Fp(2)d(and)h Fn(p)f Fp(divides)h(order\()p Fn(g)t Fp(\))e(or)i Fn(p)f Fp(divides)g(order)q(\()p Fn(\013)q Fp(\))g(b)m(y)h(\(2\).)41 b(Th)m(us)27 b(the)f(group)328 4288 y(algebra)34 b Fn(k)s Fp([)p Fn(Z)820 4303 y Ft(p)860 4288 y Fp(])h(of)f(the)i(cyclic)f(group)f Fn(Z)1822 4303 y Ft(p)1897 4288 y Fp(of)h(order)g Fn(p)g Fp(o)m(v)m(er)h Fn(k)i Fp(is)d(a)g(sub-Hopf)g(algebra)328 4408 y(of)c Fn(A)h Fp(or)f Fn(A)734 4372 y Fj(\003)774 4408 y Fp(.)43 b(Consequen)m(tly)34 b(part)d(a\))h(follo)m(ws)e(b)m(y)j(Corollary)d (2.)43 b(Since)31 b Fm(tr)r Fp(\()p Fn(t)3176 4372 y Fj(\003)3216 4408 y Fp(\))c(=)h Fm(tr)r Fp(\()p Fn(t)p Fp(\))328 4529 y(for)h(all)f(linear)g(endomorphisms)h Fn(t)h Fp(of)f Fn(A)p Fp(,)h(part)g(b\))g(follo)m(ws)e(b)m(y)j(part)e (a\))h(and)f(part)h(a\))f(of)328 4649 y(Lemma)i(3.)43 b Fd(2)474 4819 y Fp(W)-8 b(e)33 b(end)h(this)e(section)h(with)g(a)f (result)h(on)f(the)i(order)f(of)f(a)g(semisimple)f(coalgebra)328 4940 y(automorphism.)1898 5214 y(11)p eop %%Page: 12 12 12 11 bop 328 631 a Ff(Lemma)37 b(4)49 b Fk(L)-5 b(et)35 b Fn(C)42 b Fk(b)-5 b(e)35 b(a)f(\014nite-dimensional)f(c)-5 b(o)g(algebr)g(a)34 b(over)g(the)h(\014eld)g Fn(k)j Fk(and)c(sup-)328 751 y(p)-5 b(ose)34 b(that)h Fn(t)f Fk(is)g(a)h(semisimple)d(c)-5 b(o)g(algebr)g(a)34 b(automorphism)f(of)h Fn(C)7 b Fk(.)45 b(Then)33 b(the)i(or)-5 b(der)34 b(of)328 872 y Fn(t)h Fk(is)g(the)g(or)-5 b(der)34 b(of)h(the)g(c)-5 b(o)g(algebr)g(a)33 b(automorphism)h Fn(t)p Fl(j)2314 887 y Ft(C)2364 896 y Fc(1)2438 872 y Fk(of)g Fn(C)2622 887 y Fu(1)2662 872 y Fk(.)328 1125 y Fe(Pr)n(oof:)50 b Fp(Since)32 b Fn(t)h Fp(is)f(a)g(coalgebra)f(automorphism)f(of)i(the)h(\014nite-dimensional) c(coal-)328 1245 y(gebra)f Fn(C)7 b Fp(,)28 b(the)h(restriction)d Fn(t)p Fl(j)1409 1260 y Ft(C)1459 1268 y Fh(n)1534 1245 y Fp(is)h(a)h(coalgebra)f(automorphism)e(of)i Fn(C)2949 1260 y Ft(n)3024 1245 y Fp(for)g(all)f Fn(n)i Fl(\025)g Fp(0.)328 1366 y(W)-8 b(e)34 b(ma)m(y)f(as)g(w)m(ell)g(assume)h(that)f Fn(t)p Fl(j)1641 1381 y Ft(C)1691 1390 y Fc(1)1763 1366 y Fp(has)h(\014nite)f(order)g Fn(r)s Fp(.)46 b(T)-8 b(o)33 b(pro)m(v)m(e)i(the)f(lemma)d(w)m(e)328 1486 y(need)e(only)f(sho)m(w)h (that)f Fn(t)1240 1450 y Ft(r)1279 1486 y Fl(j)1307 1501 y Ft(C)1393 1486 y Fp(=)g(1)1546 1501 y Ft(C)1604 1486 y Fp(,)i(or)d(equiv)-5 b(alen)m(tly)28 b(that)g Fn(t)2555 1450 y Ft(r)2593 1486 y Fl(j)2621 1501 y Ft(C)2671 1509 y Fh(n)2746 1486 y Fp(=)f(1)2898 1501 y Ft(C)2948 1509 y Fh(n)3023 1486 y Fp(for)h(all)e Fn(n)i Fl(\025)g Fp(1.)328 1606 y(The)i(latter)f(w)m(e)h(do)f(b)m(y)h(induction)f(on)g Fn(n)p Fp(.)43 b(The)30 b(case)g Fn(n)e Fp(=)g(1)h(follo)m(ws)f(b)m(y)i (the)g(de\014nition)328 1727 y(of)i Fn(r)s Fp(.)474 1847 y(Supp)s(ose)h(that)f Fn(n)c(>)f Fp(1)32 b(and)g Fn(t)1561 1811 y Ft(r)1599 1847 y Fl(j)1627 1862 y Ft(C)1677 1871 y Fh(n)p Fi(\000)p Fc(1)1830 1847 y Fp(=)c(1)1983 1862 y Ft(C)2033 1871 y Fh(n)p Fi(\000)p Fc(1)2158 1847 y Fp(.)43 b(Then)33 b Fn(t)2517 1811 y Ft(r)2576 1847 y Fl(\000)22 b Fp(1)2724 1862 y Ft(C)2814 1847 y Fp(v)-5 b(anishes)33 b(on)f Fn(C)3402 1862 y Ft(n)p Fj(\000)p Fu(1)3539 1847 y Fp(.)328 1968 y(Therefore)467 2188 y(\001\()p Fn(t)621 2146 y Ft(r)681 2188 y Fl(\000)23 b Fp(1)830 2203 y Ft(C)889 2188 y Fp(\)\()p Fn(C)1035 2203 y Ft(n)1082 2188 y Fp(\))83 b(=)f(\()p Fn(t)1434 2146 y Ft(r)1473 2188 y Fl(\012)p Fn(t)1585 2146 y Ft(r)1645 2188 y Fl(\000)23 b Fp(1)1794 2203 y Ft(C)1853 2188 y Fl(\012)p Fp(1)1979 2203 y Ft(C)2038 2188 y Fp(\)\(\001\()p Fn(C)2303 2203 y Ft(n)2350 2188 y Fp(\)\))1361 2395 y Fl(\022)29 b Fp(\(\()p Fn(t)1578 2354 y Ft(r)1638 2395 y Fl(\000)22 b Fp(1)1786 2410 y Ft(C)1845 2395 y Fp(\))p Fl(\012)q Fn(t)1996 2354 y Ft(r)2056 2395 y Fp(+)g(1)2203 2410 y Ft(C)2262 2395 y Fl(\012)p Fp(\()p Fn(t)2412 2354 y Ft(r)2473 2395 y Fl(\000)g Fp(1)2621 2410 y Ft(C)2680 2395 y Fp(\)\))2773 2249 y Fg( )2877 2287 y Ft(n)2838 2312 y Fg(X)2842 2493 y Ft({)p Fu(=0)2975 2395 y Fn(C)3045 2410 y Ft(n)p Fj(\000)p Ft({)3170 2395 y Fl(\012)p Fn(C)3317 2410 y Ft({)3345 2249 y Fg(!)1361 2603 y Fl(\022)29 b Fn(C)1537 2618 y Ft(n)1583 2603 y Fl(\012)q Fn(C)1731 2618 y Fu(0)1792 2603 y Fp(+)22 b Fn(C)1960 2618 y Fu(0)1999 2603 y Fl(\012)q Fn(C)2147 2618 y Ft(n)2194 2603 y Fn(:)328 2823 y Fp(Th)m(us)37 b(\()p Fn(t)651 2787 y Ft(r)714 2823 y Fl(\000)25 b Fp(1)865 2838 y Ft(C)924 2823 y Fp(\)\()p Fn(C)1070 2838 y Ft(n)1117 2823 y Fp(\))33 b Fl(\022)h Fn(C)1369 2838 y Fu(1)1408 2823 y Fp(,)j(and)f(consequen)m(tly)i(\()p Fn(t)2319 2787 y Ft(r)2382 2823 y Fl(\000)25 b Fp(1)2533 2838 y Ft(C)2592 2823 y Fp(\))2630 2787 y Fu(2)2669 2823 y Fp(\()p Fn(C)2777 2838 y Ft(n)2824 2823 y Fp(\))33 b(=)g(\(0\).)54 b(The)37 b(last)328 2943 y(equation)25 b(implies)d(that)j(the)g (minimal)c(p)s(olynomial)g Fn(m)p Fp(\()p Fn(X)8 b Fp(\))25 b(of)f Fn(t)p Fl(j)2715 2958 y Ft(C)2765 2966 y Fh(n)2837 2943 y Fp(divides)h(\()p Fn(X)3285 2907 y Ft(r)3329 2943 y Fl(\000)6 b Fp(1\))3499 2907 y Fu(2)3539 2943 y Fp(.)328 3064 y(Since)47 b Fn(t)g Fp(is)g(semisimple)e(the)i(restriction)f Fn(t)p Fl(j)2027 3079 y Ft(C)2077 3087 y Fh(n)2171 3064 y Fp(is)h(semisimple.)85 b(Therefore)48 b Fn(m)p Fp(\()p Fn(X)8 b Fp(\))328 3184 y(divides)32 b Fn(X)745 3148 y Ft(r)805 3184 y Fl(\000)23 b Fp(1;)32 b(consequen)m(tly)j Fn(t)1626 3148 y Ft(r)1664 3184 y Fl(j)1692 3199 y Ft(C)1742 3207 y Fh(n)1811 3184 y Fl(\000)23 b Fp(1)1960 3199 y Ft(C)2010 3207 y Fh(n)2084 3184 y Fp(=)k(0)33 b(as)g(required.)44 b Fd(2)328 3517 y Fo(3)161 b(The)46 b(Elemen)l(t)g Fb(\027)1589 3538 y Fn(A)1666 3517 y Fa(\()p Fb(s)1783 3465 y Fp(2)p Fn(m)1922 3517 y Fa(\))g Fo(when)g Fb(s)2502 3465 y Fp(2)2601 3517 y Fo(Is)h(Semisimple)570 3700 y(and)54 b Fb(A)g Fo(is)g(P)l(oin)l(ted)328 3919 y Fp(Let)49 b Fn(A)g Fp(b)s(e)f(a)h (\014nite-dimensional)c(p)s(oin)m(ted)k(Hopf)f(algebra)g(o)m(v)m(er)h (the)g(\014eld)g Fn(k)j Fp(and)328 4039 y(supp)s(ose)43 b(that)e Fn(s)970 4003 y Fu(2)1050 4039 y Fp(is)g(a)g(semisimple)e (endomorphism)h(of)h Fn(A)p Fp(.)69 b(The)42 b(purp)s(ose)g(of)f(this) 328 4159 y(section)36 b(is)f(to)h(sho)m(w)h(that)e Fn(s)1386 4123 y Fu(2)1461 4159 y Fp(satis\014es)i(the)f(strong)g(v)-5 b(anishing)35 b(trace)h(condition.)52 b(T)-8 b(o)328 4280 y(pro)m(v)m(e)34 b(this)e(result)h(w)m(e)g(will)d(use)k(the)f (follo)m(wing)d(t)m(w)m(o)j(tec)m(hnical)f(lemmas.)474 4400 y(The)48 b(\014rst)f(lemma)d(in)m(v)m(olv)m(es)j Fn(q)t Fp(-binomial)42 b(co)s(e\016cien)m(ts.)86 b(Let)47 b Fn(q)j Fp(b)s(e)d(a)f(non-zero)328 4520 y(elemen)m(t)35 b(of)h Fn(k)s Fp(.)53 b(T)-8 b(o)35 b(describ)s(e)i Fn(q)t Fp(-binomial)31 b(co)s(e\016cien)m(ts)37 b(w)m(e)g(consider)f(the)g (algebra)f Fn(A)328 4641 y Fp(o)m(v)m(er)45 b Fn(k)j Fp(generated)d(b)m(y)g(sym)m(b)s(ols)f Fn(a;)17 b(x)45 b Fp(sub)5 b(ject)46 b(to)e(the)h(relation)d Fn(xa)49 b Fp(=)e Fn(q)t(ax)p Fp(.)80 b(The)328 4761 y Fn(a)379 4725 y Ft({)407 4761 y Fn(x)462 4725 y Ft(|)493 4761 y Fp('s)34 b(where)h(0)29 b Fl(\024)h Fn({;)17 b(|)33 b Fp(form)f(a)h(linear)f(basis)i(for)f Fn(A)p Fp(.)47 b(F)-8 b(or)33 b(0)c Fl(\024)h Fn({)f Fl(\024)h Fn(n)k Fp(the)g Fn(q)t Fp(-binomial)1898 5214 y(12)p eop %%Page: 13 13 13 12 bop 328 681 a Fp(co)s(e\016cien)m(t)798 535 y Fg( )872 620 y Fn(n)886 740 y({)939 535 y Fg(!)981 785 y Ft(q)1066 681 y Fp(is)47 b(the)h(co)s(e\016cien)m(t)g(of)f Fn(a)2009 645 y Ft(n)p Fj(\000)p Ft({)2134 681 y Fn(x)2189 645 y Ft({)2264 681 y Fp(in)f(the)i(expansion)g(of)f(\()p Fn(a)32 b Fp(+)g Fn(x)p Fp(\))3491 645 y Ft(n)3539 681 y Fp(;)328 955 y(otherwise)762 809 y Fg( )836 894 y Fn(n)850 1015 y({)903 809 y Fg(!)945 1059 y Ft(q)1011 955 y Fp(=)27 b(0)32 b(b)m(y)i(de\014nition.)328 1246 y Ff(Lemma)j(5)49 b Fk(L)-5 b(et)36 b Fn(N)k(>)29 b Fp(1)35 b Fk(and)h(supp)-5 b(ose)35 b(that)h(the)g(\014eld)f Fn(k)j Fk(c)-5 b(ontains)35 b(a)h(primitive)f Fn(N)3495 1209 y Ft(th)328 1366 y Fk(r)-5 b(o)g(ot)35 b(of)g(unity)g Fn(!)t Fk(.)44 b(Then:)436 1626 y Fp(a\))572 1480 y Fg( )646 1565 y Fn(N)33 b Fl(\000)22 b Fp(1)755 1685 y Fn(`)913 1480 y Fg(!)955 1730 y Ft(!)1033 1626 y Fp(=)28 b(\()p Fl(\000)p Fp(1\))1339 1590 y Ft(`)1372 1626 y Fn(!)1437 1590 y Fj(\000)p Ft(`)p Fu(\()p Ft(`)p Fu(+1\))p Ft(=)p Fu(2)1804 1626 y Fk(for)34 b(al)5 b(l)35 b Fp(0)27 b Fl(\024)h Fn(`)g Fl(\024)g Fn(N)33 b Fl(\000)22 b Fp(1)p Fk(.)431 1971 y Fp(b\))49 b Fk(If)38 b(the)g(char)-5 b(acteristic)38 b(of)g Fn(k)j Fk(is)d Fn(p)c(>)g Fp(0)k Fk(then)2270 1824 y Fg( )2344 1910 y Fn(N)10 b(p)2481 1873 y Ft(s)2541 1910 y Fl(\000)22 b Fp(1)2496 2030 y Fn(`)2697 1824 y Fg(!)2740 2074 y Ft(!)2824 1971 y Fp(=)34 b(\()p Fl(\000)p Fp(1\))3136 1934 y Ft(`)3169 1971 y Fn(!)3234 1934 y Fj(\000)p Ft(`)p Fu(\()p Ft(`)p Fu(+1\))p Ft(=)p Fu(2)572 2152 y Fk(for)g(al)5 b(l)35 b Fn(s)28 b Fl(\025)g Fp(0)34 b Fk(and)h Fp(0)27 b Fl(\024)h Fn(`)g Fl(\024)g Fn(N)10 b(p)1811 2116 y Ft(s)1870 2152 y Fl(\000)23 b Fp(1)p Fk(.)328 2405 y Fe(Pr)n(oof:)50 b Fp(P)m(art)33 b(a\))f(follo)m(ws)f(from)g(the)i(equations)1108 2687 y(0)28 b(=)1288 2541 y Fg( )1363 2626 y Fn(N)1386 2747 y(`)1459 2541 y Fg(!)1502 2791 y Ft(!)1580 2687 y Fp(=)f Fn(!)1748 2646 y Ft(`)1797 2541 y Fg( )1871 2626 y Fn(N)33 b Fl(\000)22 b Fp(1)1980 2747 y Fn(`)2138 2541 y Fg(!)2181 2791 y Ft(!)2253 2687 y Fp(+)2351 2541 y Fg( )2425 2626 y Fn(N)33 b Fl(\000)23 b Fp(1)2449 2747 y Fn(`)f Fl(\000)h Fp(1)2693 2541 y Fg(!)2735 2791 y Ft(!)328 2974 y Fp(for)37 b(all)e(1)g Fl(\024)i Fn(`)e Fl(\024)h Fn(N)g Fl(\000)26 b Fp(1.)57 b(See)39 b([11)o(,)g(pages)f(691{692])d(for)i(example.)57 b(T)-8 b(o)38 b(sho)m(w)g(part)328 3094 y(b\))29 b(w)m(e)h(\014rst)g (note)f(that)g(an)m(y)h(non-negativ)m(e)f(in)m(teger)g Fn(m)h Fp(has)f(a)g(unique)h(decomp)s(osition)328 3215 y Fn(m)g Fp(=)f Fn(m)633 3230 y Ft(D)698 3215 y Fn(N)k Fp(+)23 b Fn(m)993 3230 y Ft(R)1051 3215 y Fp(,)34 b(where)h Fn(m)1480 3230 y Ft(D)1544 3215 y Fn(;)17 b(m)1673 3230 y Ft(R)1765 3215 y Fp(are)33 b(in)m(tegers)h(and)g(0)29 b Fl(\024)i Fn(m)2752 3230 y Ft(R)2839 3215 y Fn(<)f(N)10 b Fp(.)47 b(By)34 b(part)g(b\))328 3335 y(of)e([11,)g(Prop)s(osition)f (1],)i(for)f(example,)1445 3471 y Fg( )1519 3556 y Fn(m)1541 3677 y(`)1612 3471 y Fg(!)1655 3721 y Ft(!)1733 3617 y Fp(=)1836 3471 y Fg( )1910 3556 y Fn(m)1995 3571 y Ft(R)1933 3677 y Fn(`)1974 3692 y Ft(R)2062 3471 y Fg(!)2104 3721 y Ft(!)2171 3471 y Fg( )2245 3556 y Fn(m)2330 3571 y Ft(D)2267 3677 y Fn(`)2308 3692 y Ft(D)2403 3471 y Fg(!)328 3978 y Fp(for)k(all)f(0)g Fl(\024)g Fn(`)g(<)f(m)p Fp(,)39 b(where)1440 3832 y Fg( )1514 3917 y Fn(m)1599 3932 y Ft(D)1536 4038 y Fn(`)1577 4053 y Ft(D)1671 3832 y Fg(!)1754 3978 y Fp(is)e(the)g(ordinary)f(binomial)d(sym)m(b)s(ol.)56 b(Supp)s(ose)328 4160 y(that)36 b(the)h(c)m(haracteristic)f(of)g Fn(k)k Fp(is)35 b Fn(p)g(>)f Fp(0,)j Fn(s)d Fl(\025)h Fp(0)h(and)g(0)e Fl(\024)h Fn(`)f(<)g(N)10 b(p)2919 4123 y Ft(s)2957 4160 y Fp(.)54 b(When)38 b Fn(s)c Fp(=)g(0)328 4280 y(part)e(b\))h(is)f(part)g(a\).)44 b(Th)m(us)34 b(w)m(e)f(ma)m(y)g(assume)g Fn(s)27 b(>)h Fp(0.)43 b(In)33 b(this)f(case)546 4416 y Fg( )620 4501 y Fn(N)10 b(p)757 4465 y Ft(s)816 4501 y Fl(\000)23 b Fp(1)772 4622 y Fn(`)973 4416 y Fg(!)1015 4666 y Ft(!)1093 4562 y Fp(=)1197 4416 y Fg( )1271 4501 y Fn(N)33 b Fl(\000)22 b Fp(1)1351 4622 y Fn(`)1392 4637 y Ft(R)1538 4416 y Fg(!)1580 4666 y Ft(!)1647 4416 y Fg( )1722 4501 y Fn(p)1771 4465 y Ft(s)1829 4501 y Fl(\000)h Fp(1)1797 4622 y Fn(`)1838 4637 y Ft(D)1986 4416 y Fg(!)2060 4562 y Fp(=)28 b(\()p Fl(\000)p Fp(1\))2366 4521 y Ft(`)2395 4532 y Fh(R)2449 4562 y Fn(!)2514 4521 y Fj(\000)p Ft(`)2598 4532 y Fh(R)2647 4521 y Fu(\()p Ft(`)2703 4532 y Fh(R)2754 4521 y Fu(+1\))p Ft(=)p Fu(2)2963 4416 y Fg( )3037 4501 y Fn(p)3086 4465 y Ft(s)3145 4501 y Fl(\000)22 b Fp(1)3113 4622 y Fn(`)3154 4637 y Ft(D)3301 4416 y Fg(!)1898 5214 y Fp(13)p eop %%Page: 14 14 14 13 bop 328 681 a Fp(b)m(y)39 b(part)f(a\).)59 b(Since)38 b(0)f(=)1318 535 y Fg( )1392 620 y Fn(p)1441 584 y Ft(s)1392 740 y Fn(m)1486 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Fu(+1\))p Ft(=)p Fu(2)2505 1269 y Fp(\()p Fl(\000)p Fp(1\))2707 1227 y Ft(`)2736 1238 y Fh(D)2795 1269 y Fn(:)328 1514 y Fp(By)d(considering)f(the)i (parities)d(of)i Fn(N)34 b Fp(and)24 b Fn(`)1908 1529 y Ft(D)1995 1514 y Fp(one)g(can)g(sho)m(w)h(that)f(this)f(last)g (expression)328 1635 y(is)32 b(\()p Fl(\000)p Fp(1\))628 1599 y Ft(`)661 1635 y Fn(!)726 1599 y Fj(\000)p Ft(`)p Fu(\()p Ft(`)p Fu(+1\))p Ft(=)p Fu(2)1058 1635 y Fp(.)43 b Fd(2)474 1795 y Fp(The)25 b(pro)s(of)e(of)g(the)h(main)e(result)h(of) g(this)g(section)h(reduces)i(to)d(the)h(follo)m(wing)d(sp)s(ecial)328 1915 y(case.)328 2089 y Ff(Lemma)37 b(6)49 b Fk(L)-5 b(et)39 b Fn(A)f Fk(b)-5 b(e)38 b(a)g(\014nite-dimensional)e(Hopf)j (algebr)-5 b(a)37 b(with)h(antip)-5 b(o)g(de)38 b Fn(s)g Fk(over)328 2209 y(the)d(\014eld)f Fn(k)k Fk(gener)-5 b(ate)g(d)34 b(by)h(non-zer)-5 b(o)34 b(elements)g Fn(a;)17 b(x)35 b Fk(which)f(satisfy)1223 2394 y Fp(\001\()p Fn(a)p Fp(\))28 b(=)f Fn(a)p Fl(\012)q Fn(a;)72 b Fp(\001\()p Fn(x)p 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Fj(\000)p Fu(1)p Fj(\000)p Ft(`)1898 5214 y Fp(15)p eop %%Page: 16 16 16 15 bop 563 693 a Fp(=)722 585 y Ft(N)7 b Fj(\000)p Fu(1)739 610 y Fg(X)739 794 y Ft(`)p Fu(=0)892 693 y Fn(!)957 652 y Fj(\000)p Ft(`)1044 693 y Fn(!)1109 652 y Ft(`)p Fu(\()p Fj(\000)p Ft(N)g Fu(+1\))1404 693 y Fn(\014)1465 652 y Ft(N)g Fj(\000)p Fu(1)p Fj(\000)p Ft(`)1706 693 y Fn(a)1757 652 y Fj(\000)p Ft(N)g Fu(+1)1970 693 y Fn(x)2025 652 y Ft(N)g Fj(\000)p Fu(1)563 998 y Fp(=)722 851 y Fg( )788 890 y Ft(N)g Fj(\000)p Fu(1)804 915 y Fg(X)805 1099 y Ft(`)p Fu(=0)958 998 y Fn(\014)1019 956 y Ft(N)g Fj(\000)p Fu(1)p Fj(\000)p Ft(`)1259 851 y Fg(!)1342 998 y Fn(a)1393 956 y Fj(\000)p Ft(N)g Fu(+1)1605 998 y Fn(x)1660 956 y Ft(N)g Fj(\000)p Fu(1)1818 998 y Fn(:)328 1260 y Fp(Th)m(us)34 b Fn(\027)623 1275 y Ft(A)680 1260 y Fp(\()p Fn(t)p Fp(\))28 b(=)g Fm(tr)q Fp(\()p Fn(t)p Fp(\))p Fn(a)1154 1224 y Fu(1)p Fj(\000)p Ft(N)1312 1260 y Fn(x)1367 1224 y Ft(N)7 b Fj(\000)p Fu(1)1525 1260 y Fp(.)474 1381 y(W)-8 b(e)46 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y Fh(r)1766 2486 y Fp(=)28 b Fn(\013)q Fp(\()p Fn(a)2022 2445 y Ft(N)7 b(p)2121 2421 y Fh(r)2181 2486 y Fl(\000)23 b Fp(1\))f(+)g Fn(\013)2550 2501 y Fu(0)2589 2486 y Fn(x)2644 2445 y Ft(N)2734 2486 y Fp(+)g Fl(\001)17 b(\001)g(\001)j Fp(+)i Fn(\013)3130 2501 y Ft(r)r Fj(\000)p Fu(1)3259 2486 y Fn(x)3314 2445 y Ft(N)7 b(p)3413 2421 y Fh(r)r Fi(\000)p Fc(1)328 2669 y Fp(for)40 b(some)h Fn(\013)q(;)17 b(\013)907 2684 y Fu(0)946 2669 y Fn(;)g(:)g(:)g(:)f(;)h(\013)1227 2684 y Ft(r)r Fj(\000)p Fu(1)1397 2669 y Fl(2)43 b Fn(k)h Fp(hold)c(with)h(the)g(restriction)f(that)h Fn(\013)2990 2684 y Ft({)3059 2669 y Fl(6)p Fp(=)h(0)f(implies)328 2790 y(that)32 b Fn(a)590 2754 y Ft(N)7 b(p)689 2730 y Fh(r)756 2790 y Fp(=)27 b Fn(a)910 2754 y Ft(N)7 b(p)1009 2730 y Fh({)1039 2790 y Fp(.)474 2910 y(W)-8 b(e)31 b(ha)m(v)m(e)h (noted)g(that)e Fn(\025)e Fp(=)p 1530 2826 562 4 v 27 w Fn(a)1581 2881 y Fu(1)p Fj(\000)p Ft(N)7 b(p)1770 2863 y Fh(r)1809 2910 y Fn(x)1864 2881 y Ft(N)g(p)1963 2863 y Fh(r)1998 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631 y Fn(a)3511 595 y Ft({)3539 631 y Fp(.)474 751 y(Let)46 b Fn(t)h Fp(b)s(e)f(an)g(algebra)f(endomorphism)f(of)i Fn(A)g Fp(whic)m(h)g(satis\014es)h Fn(t)p Fp(\()p Fn(a)p Fp(\))k(=)f Fn(a)d Fp(and)328 872 y Fn(t)p Fp(\()p Fn(x)p Fp(\))28 b(=)g Fn(\014)6 b(x)32 b Fp(for)g(some)h Fn(\014)g Fl(2)28 b Fn(k)s Fp(.)44 b(F)-8 b(or)31 b Fn({)d Fl(\025)g Fp(0)33 b(let)e Fn(\014)2090 887 y Ft({)2146 872 y Fp(=)2249 805 y Fg(P)2337 825 y Ft(N)7 b(p)2436 801 y Fh({)2462 825 y Fj(\000)p Fu(1)2337 894 y Ft(|)p Fu(=0)2573 872 y Fn(\014)2634 836 y Ft(|)2664 872 y Fp(.)43 b(W)-8 b(e)33 b(will)e(sho)m(w)i(that)944 1166 y Fn(\027)992 1181 y Ft(A)1049 1166 y Fp(\()p Fn(t)p Fp(\))28 b(=)g Fn(a)1343 1124 y Fu(1)p Fj(\000)p Ft(N)7 b(p)1532 1101 y Fh(r)1587 1019 y Fg( )1653 1166 y Fm(tr)r Fp(\()p Fn(t)p Fp(\))p Fn(x)1889 1124 y Ft(N)g(p)1988 1101 y Fh(r)2022 1124 y Fj(\000)p Fu(1)2139 1166 y Fl(\000)2238 1058 y Ft(r)r Fj(\000)p Fu(1)2240 1083 y Fg(X)2244 1264 y Ft({)p Fu(=0)2379 1166 y Fn(n\013)2499 1181 y Ft({)2527 1166 y Fn(\014)2582 1181 y Ft({)2609 1166 y Fn(x)2664 1124 y Ft(N)g(p)2763 1101 y Fh({)2789 1124 y Fj(\000)p Fu(1)2884 1019 y Fg(!)3441 1166 y Fp(\(8\))328 1451 y(whic)m(h)37 b(implies)d Fn(\027)994 1466 y Ft(A)1051 1451 y Fp(\()p Fn(s)1135 1415 y Fu(2)p Ft(m)1237 1451 y Fp(\))h(=)f(0.)56 b(F)-8 b(or)35 b(let)h Fn(t)f Fp(=)g Fn(s)2102 1415 y Fu(2)p Ft(m)2203 1451 y Fp(.)56 b(Since)37 b Fn(!)2610 1415 y Fj(\000)p Ft(m)2765 1451 y Fl(6)p Fp(=)e(1)h(it)g(follo)m(ws)f(that)328 1571 y Fn(\014)383 1586 y Ft({)438 1571 y Fp(=)28 b(\(1)14 b Fl(\000)g Fn(!)799 1535 y Fj(\000)p Ft(mN)7 b(p)1015 1512 y Fh({)1044 1571 y Fp(\))p Fn(=)p Fp(\(1)14 b Fl(\000)g Fn(!)1388 1535 y Fj(\000)p Ft(m)1507 1571 y Fp(\))28 b(=)g(0)g(for)f(all)g(0)g Fl(\024)h Fn({)h Fp(and)f(consequen)m(tly)j Fn(\027)3078 1586 y Ft(A)3135 1571 y Fp(\()p Fn(s)3219 1535 y Fu(2)p Ft(m)3321 1571 y Fp(\))c(=)h(0.)474 1692 y(Let)34 b Fn(g)f Fl(2)c Fn(G)p Fp(\()p Fn(A)p Fp(\),)34 b(0)29 b Fl(\024)h Fn({)g(<)f(r)36 b Fp(and)e(let)f Fn(c)c Fp(=)g Fn(g)t(x)2158 1656 y Ft(N)7 b(p)2257 1632 y Fh({)2283 1656 y Fj(\000)p Fu(1)2377 1692 y Fp(.)47 b(T)-8 b(o)33 b(calculate)g Fn(\027)3051 1707 y Ft(A)3108 1692 y Fp(\()p Fn(t)p Fp(\))h(w)m(e)g(\014rst)328 1812 y(determine)c Fn(s)p Fp(\()p Fn(c)907 1828 y Fu(\(2\))1001 1812 y Fp(\))p Fn(t)p Fp(\()p Fn(c)1154 1828 y Fu(\(1\))1248 1812 y Fp(\).)43 b(F)-8 b(ollo)m(wing)27 b(the)k(steps)g(of)f(the)g (calculation)e(of)i Fn(\027)3121 1827 y Ft(A)3178 1812 y Fp(\()p Fn(t)p Fp(\))g(in)f(the)328 1933 y(c)m(haracteristic)36 b(0)h(case,)i(and)d(using)h(part)f(b\))h(of)f(Lemma)g(5,)h(one)g(can)g (easily)f(deduce)328 2053 y(that)41 b Fn(s)p Fp(\()p Fn(c)674 2068 y Fu(\(2\))768 2053 y Fp(\))p Fn(t)p Fp(\()p Fn(c)921 2068 y Fu(\(1\))1016 2053 y Fp(\))h(=)h Fn(\014)1270 2068 y Ft({)1297 2053 y Fn(a)1348 2017 y Fu(1)p Fj(\000)p Ft(N)7 b(p)1537 1993 y Fh({)1568 2053 y Fn(x)1623 2017 y Ft(N)g(p)1722 1993 y Fh({)1748 2017 y Fj(\000)p Fu(1)1842 2053 y Fp(.)70 b(Since)42 b Fn(\013)2265 2068 y Ft({)2335 2053 y Fl(6)p Fp(=)g(0)f(implies)e Fn(a)2934 2017 y Ft(N)7 b(p)3033 1993 y Fh({)3106 2053 y Fp(=)43 b Fn(a)3276 2017 y Ft(N)7 b(p)3375 1993 y Fh(r)3455 2053 y Fp(w)m(e)328 2173 y(ha)m(v)m(e)34 b(established)f(\(8\).)43 b(Our)32 b(pro)s(of)g(of)g(the)h(lemma)d(is)i(complete.)43 b Fd(2)328 2370 y Ff(Lemma)37 b(7)49 b Fk(L)-5 b(et)28 b Fn(k)i Fk(b)-5 b(e)27 b(algebr)-5 b(aic)g(al)5 b(ly)27 b(close)-5 b(d)26 b(and)h Fn(A)h Fk(b)-5 b(e)27 b(a)g(\014nite-dimensional)e(p)-5 b(ointe)g(d)328 2490 y(Hopf)50 b(algebr)-5 b(a)48 b(with)i(antip)-5 b(o)g(de)49 b Fn(s)h Fk(over)f Fn(k)s Fk(.)89 b(Assume)50 b(that)g Fn(s)2690 2454 y Fu(2)2779 2490 y Fk(is)f(semisimple)f(and)328 2611 y Fn(s)374 2575 y Fu(2)p Ft(m)525 2611 y Fl(6)p Fp(=)i(1)700 2626 y Ft(A)804 2611 y Fk(for)c(some)g Fn(m)p Fk(.)81 b(Then)46 b(ther)-5 b(e)46 b(is)h(a)f(gr)-5 b(oup-like)46 b(element)g Fn(a)k Fl(2)g Fn(G)p Fp(\()p Fn(A)p Fp(\))p Fk(,)g(a)328 2731 y(non-zer)-5 b(o)34 b(element)g Fn(x)28 b Fl(2)g Fn(A)35 b Fk(and)f(a)h(r)-5 b(o)g(ot)35 b(of)g(unity)g Fn(\032)28 b Fl(2)g Fn(k)38 b Fk(such)d(that)873 2943 y Fp(\001\()p Fn(x)p Fp(\))28 b(=)f Fn(x)c Fl(\012)g Fn(a)f Fp(+)g(1)g Fl(\012)g Fn(x;)73 b(s)1936 2902 y Fu(2)1975 2943 y Fp(\()p Fn(x)p Fp(\))28 b(=)g Fn(\032x)100 b Fk(and)f Fn(\032)2747 2902 y Ft(m)2842 2943 y Fl(6)p Fp(=)27 b(1)p Fn(:)328 3203 y Fe(Pr)n(oof:)71 b Fp(By)44 b(Lemma)d(4)h(w)m(e)i(ha)m(v)m(e)g(that)f(order\()p Fn(s)2272 3167 y Fu(2)2311 3203 y Fp(\))g(=)f(order)h(\()p Fn(s)2859 3167 y Fu(2)2899 3203 y Fl(j)2927 3218 y Ft(A)2980 3227 y Fc(1)3018 3203 y Fp(\).)74 b(Therefore)328 3323 y Fn(s)374 3287 y Fu(2)p Ft(m)476 3323 y Fl(j)504 3338 y Ft(A)557 3347 y Fc(1)623 3323 y Fl(6)p Fp(=)27 b(1)775 3338 y Ft(A)828 3347 y Fc(1)867 3323 y Fp(.)474 3444 y(F)-8 b(or)37 b Fn(a)f Fl(2)h Fn(G)p Fp(\()p Fn(A)p Fp(\))h(let)e Fn(V)1310 3459 y Ft(a)1388 3444 y Fp(=)g Fl(f)p Fn(x)h Fl(2)f Fn(A)17 b Fl(j)g Fp(\001\()p Fn(x)p Fp(\))36 b(=)g Fn(x)p Fl(\012)q Fn(a)26 b Fp(+)f(1)p Fl(\012)q Fn(x)p Fl(g)36 b(\022)h Fn(A)3005 3459 y Fu(1)3044 3444 y Fp(.)59 b(Since)38 b Fn(A)f Fp(is)328 3564 y(p)s(oin)m(ted,)32 b Fn(A)780 3579 y Fu(1)852 3564 y Fp(is)g(generated)i(as)f(a)f(left)g Fn(A)1840 3579 y Fu(0)1879 3564 y Fp(-mo)s(dule)f(b)m(y)i(1)g(and)2662 3498 y Fg(P)2750 3585 y Ft(a)p Fj(2)p Ft(G)p Fu(\()p Ft(A)p Fu(\))3018 3564 y Fn(V)3075 3579 y Ft(a)3116 3564 y Fp(.)474 3685 y(By)41 b(assumption)e Fn(s)1204 3648 y Fu(2)1283 3685 y Fp(is)g(a)h(semisimple)e(endomorphism)g(of)i Fn(A)p Fp(.)65 b(Therefore)41 b Fn(s)3421 3648 y Fu(2)3500 3685 y Fp(is)328 3805 y(diagonalizable)29 b(since)k Fn(k)j Fp(is)c(algebraically)e(closed.)44 b(Supp)s(ose)34 b(that)e Fn(a)c Fl(2)h Fn(G)p Fp(\()p Fn(A)p Fp(\).)44 b(Since)328 3925 y Fn(s)374 3889 y Fu(2)440 3925 y Fp(is)25 b(a)h(coalgebra)g (endomorphism)f(of)g Fn(A)i Fp(and)f Fn(s)2117 3889 y Fu(2)2157 3925 y Fp(\()p Fn(g)t Fp(\))g(=)i Fn(g)i Fp(for)25 b(all)g Fn(g)30 b Fl(2)f Fn(G)p Fp(\()p Fn(A)p Fp(\))d(it)f(follo)m(ws) 328 4046 y(that)32 b Fn(s)585 4010 y Fu(2)625 4046 y Fp(\()p Fn(V)720 4061 y Ft(a)761 4046 y Fp(\))c Fl(\022)g Fn(V)989 4061 y Ft(a)1030 4046 y Fp(.)474 4166 y(No)m(w)40 b Fn(s)749 4130 y Fu(2)p Ft(m)851 4166 y Fl(j)879 4181 y Ft(A)932 4190 y Fc(1)1008 4166 y Fl(6)p Fp(=)f(1)1172 4181 y Ft(A)1225 4190 y Fc(1)1263 4166 y Fp(.)62 b(Since)39 b Fn(s)1659 4130 y Fu(2)1738 4166 y Fp(is)f(an)h(algebra)f (endomorphism,)h Fn(s)3069 4130 y Fu(2)p Ft(m)3171 4166 y Fl(j)3199 4181 y Ft(V)3240 4189 y Fh(a)3320 4166 y Fl(6)p Fp(=)f(1)3483 4181 y Ft(V)3524 4189 y Fh(a)328 4287 y Fp(for)d(some)h Fn(a)e Fl(2)g Fn(G)p Fp(\()p Fn(A)p Fp(\).)54 b(Since)36 b Fn(s)1524 4250 y Fu(2)1563 4287 y Fl(j)1591 4302 y Ft(V)1632 4310 y Fh(a)1710 4287 y Fp(is)f(diagonalizable)e(it)i(follo)m(ws)g(that)g Fn(s)3134 4250 y Fu(2)p Ft(m)3236 4287 y Fp(\()p Fn(x)p Fp(\))f Fl(6)p Fp(=)f Fn(x)328 4407 y Fp(where)43 b Fn(x)i Fl(2)f Fn(V)886 4422 y Ft(a)970 4407 y Fp(is)d(an)h(eigen)m(v)m(ector)h(for)f Fn(s)1943 4371 y Fu(2)2024 4407 y Fp(b)s(elonging)f(to)g(some)h Fn(\032)j Fl(2)f Fn(k)s Fp(.)72 b(Since)42 b Fn(s)3526 4371 y Fu(2)328 4527 y Fp(has)e(\014nite)f(order,)i Fn(\032)e Fp(is)g(a)g(primitiv)m(e)e Fn(N)1849 4491 y Ft(th)1959 4527 y Fp(ro)s(ot)h(of)h(unit)m(y)g(for)g(some)g Fn(N)50 b Fl(\025)39 b Fp(1.)64 b(Since)328 4648 y Fn(s)374 4611 y Fu(2)p Ft(m)476 4648 y Fp(\()p Fn(x)p Fp(\))28 b Fl(6)p Fp(=)f Fn(x)33 b Fp(it)f(follo)m(ws)f(that)h Fn(N)39 b(>)27 b Fp(1)32 b(and)h Fn(\032)1996 4611 y Ft(m)2091 4648 y Fl(6)p Fp(=)27 b(1.)43 b Fd(2)328 4844 y Ff(Theorem)37 b(1)49 b Fk(L)-5 b(et)31 b Fn(A)f Fk(b)-5 b(e)31 b(a)f (\014nite-dimensional)e(Hopf)j(algebr)-5 b(a)30 b(with)g(antip)-5 b(o)g(de)30 b Fn(s)g Fk(over)328 4965 y(the)i(\014eld)g Fn(k)s Fk(.)44 b(Supp)-5 b(ose)32 b(that)h Fn(s)1434 4929 y Fu(2)1506 4965 y Fk(is)f(a)g(semisimple)f(endomorphism)f(of)i Fn(A)h Fk(and)f(that)h(the)1898 5214 y Fp(17)p eop %%Page: 18 18 18 17 bop 328 631 a Fk(c)-5 b(or)g(adic)g(al)39 b Fn(A)811 646 y Fu(0)891 631 y Fk(of)h Fn(A)g Fk(is)g(c)-5 b(o)g(c)g(ommutative.) 60 b(Then)40 b Fn(s)2251 595 y Fu(2)2330 631 y Fk(satis\014es)g(the)g (str)-5 b(ong)40 b(vanishing)328 751 y(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition.)328 1005 y Fe(Pr)n(oof:)72 b Fp(Let)44 b Fn(K)50 b Fp(b)s(e)43 b(an)g(algebraic)f(closure)h(of)f Fn(k)s Fp(.)76 b(Since)43 b(\()p Fn(A)p Fl(\012)p Fn(K)7 b Fp(\))2969 1020 y Fu(0)3054 1005 y Fl(\022)47 b Fn(A)3251 1020 y Fu(0)3290 1005 y Fl(\012)q Fn(K)j Fp(it)328 1125 y(follo)m(ws)33 b(that)h(the)g(coradical)e(of)i Fn(A)p Fl(\012)q Fn(K)41 b Fp(is)33 b(co)s(comm)m(utativ)m(e.)47 b(Th)m(us)36 b(since)f Fn(K)41 b Fp(is)33 b(alge-)328 1245 y(braically)26 b(closed)i Fn(A)p Fl(\012)q Fn(K)35 b Fp(is)28 b(p)s(oin)m(ted.)42 b(T)-8 b(o)28 b(pro)m(v)m(e)i(the)e (theorem,)h(it)f(is)f(easy)j(to)e(see)h(that)328 1366 y(w)m(e)34 b(ma)m(y)e(replace)g Fn(A)h Fp(with)f Fn(A)p Fl(\012)q Fn(K)7 b Fp(;)33 b(th)m(us)g(w)m(e)h(will)c(assume)j(that)g Fn(A)f Fp(is)g(p)s(oin)m(ted.)474 1486 y(Supp)s(ose)23 b(that)f Fn(s)1093 1450 y Fu(2)p Ft(m)1223 1486 y Fl(6)p Fp(=)27 b(1)1375 1501 y Ft(A)1432 1486 y Fp(.)40 b(Pic)m(k)22 b(elemen)m(ts)h Fn(a;)17 b(x)28 b Fl(2)g Fn(A)22 b Fp(and)g Fn(\032)g Fp(according)f(to)h(Lemma)328 1606 y(7)36 b(and)g(let)g Fn(B)j Fp(=)34 b Fn(k)s Fl(f)p Fn(a;)17 b(x)p Fl(g)36 b Fp(b)s(e)h(the)f(subalgebra)g(of)g Fn(A)g Fp(generated)i(b)m(y)f Fn(a)f Fp(and)h Fn(x)p Fp(.)55 b(Then)328 1727 y Fn(B)43 b Fp(is)37 b(a)h(sub-bialgebra)e(of)h Fn(A)p Fp(.)60 b(The)39 b(calculations)d Fn(s)p Fp(\()p Fn(a)p Fp(\))g(=)h Fn(a)2634 1691 y Fj(\000)p Fu(1)2766 1727 y Fp(and)h Fn(s)p Fp(\()p Fn(x)p Fp(\))f(=)f Fl(\000)p Fn(xa)3470 1691 y Fj(\000)p Fu(1)328 1847 y Fp(sho)m(w)29 b(that)f Fn(B)k Fp(is)c(a)f(sub-Hopf)h(algebra)f(of)h Fn(A)p Fp(.)42 b(Using)27 b(the)h(last)f(equation)h(w)m(e)h(calculate)328 1968 y Fn(s)374 1931 y Fu(2)413 1968 y Fp(\()p Fn(x)p Fp(\))55 b(=)f Fn(axa)886 1931 y Fj(\000)p Fu(1)1036 1968 y Fp(=)g Fn(\032x)p Fp(.)92 b(In)48 b(particular)f Fn(xa)55 b Fp(=)f Fn(!)t(ax)p Fp(,)f(where)c Fn(!)58 b Fp(=)c Fn(\032)3131 1931 y Fj(\000)p Fu(1)3226 1968 y Fp(.)91 b(Since)328 2088 y Fn(s)374 2052 y Fu(2)p Ft(m)476 2088 y Fp(\()p Fn(x)p Fp(\))31 b Fl(6)p Fp(=)f Fn(x)35 b Fp(it)f(follo)m(ws)f(that)h Fn(s)1515 2052 y Fu(2)p Ft(m)1617 2088 y Fl(j)1645 2103 y Ft(B)1736 2088 y Fl(6)p Fp(=)d(1)1892 2103 y Ft(B)1952 2088 y Fp(.)49 b(Therefore)35 b Fn(\027)2519 2103 y Ft(B)2580 2088 y Fp(\()p Fn(s)2664 2052 y Fu(2)p Ft(m)2766 2088 y Fl(j)2794 2103 y Ft(B)2854 2088 y Fp(\))c(=)g(0)j(b)m(y)h(Lemma)328 2208 y(6,)d(and)h(consequen)m (tly)i Fn(\027)1252 2223 y Ft(A)1309 2208 y Fp(\()p Fn(s)1393 2172 y Fu(2)p Ft(m)1495 2208 y Fp(\))27 b(=)h(0)k(b)m(y)i(part)e(c\))h (of)f(Prop)s(osition)f(2.)43 b Fd(2)474 2379 y Fp(When)35 b Fn(s)806 2342 y Fu(2)880 2379 y Fp(is)f(conjugation)f(b)m(y)i(a)f (grouplik)m(e)f(elemen)m(t,)i(part)f(a\))g(of)g(Prop)s(osition)e(3)328 2499 y(can)h(b)s(e)g(strengthened)h(a)e(bit)g(in)g(the)h(p)s(oin)m(ted) f(case.)328 2702 y Ff(Corollary)k(3)49 b Fk(L)-5 b(et)28 b Fn(A)h Fk(b)-5 b(e)28 b(a)g(\014nite-dimensional)e(Hopf)i(algebr)-5 b(a)27 b(with)h(antip)-5 b(o)g(de)28 b Fn(s)g Fk(over)328 2823 y(the)43 b(\014eld)f Fn(k)s Fk(.)68 b(Supp)-5 b(ose)42 b(that)h(the)g(c)-5 b(or)g(adic)g(al)41 b Fn(A)2108 2838 y Fu(0)2191 2823 y Fk(of)h Fn(A)h Fk(is)f(c)-5 b(o)g(c)g(ommutative)42 b(and)g(that)328 2943 y Fn(s)374 2907 y Fu(2)413 2943 y Fp(\()p Fn(x)p Fp(\))35 b(=)f Fn(axa)846 2907 y Fj(\000)p Fu(1)980 2943 y Fk(for)k(al)5 b(l)38 b Fn(x)d Fl(2)g Fn(A)p Fk(,)k(wher)-5 b(e)38 b Fn(a)c Fl(2)h Fn(G)p Fp(\()p Fn(A)p Fp(\))p Fk(.)55 b(Then)38 b Fn(s)2695 2907 y Fu(2)2773 2943 y Fk(satis\014es)g(the)g(str)-5 b(ong)328 3064 y(vanishing)33 b(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition.)328 3317 y Fe(Pr)n(oof:)51 b Fp(W)-8 b(e)33 b(ma)m(y)g(assume)h(that)f Fn(k)j Fp(is)d (algebraically)d(closed.)45 b(If)33 b(the)h(c)m(haracteristic)328 3437 y(of)h Fn(k)k Fp(do)s(es)d(not)f(divide)g(order\()p Fn(a)p Fp(\))h(then)g Fn(s)1880 3401 y Fu(2)1955 3437 y Fp(is)f(semisimple)f(and)h(Theorem)h(1)g(applies.)328 3557 y(Supp)s(ose)29 b(that)f(the)h(c)m(haracteristic)f(of)g Fn(k)j Fp(divides)d(order)q(\()p Fn(s)2492 3521 y Fu(2)2531 3557 y Fp(\).)42 b(Then)29 b Fn(k)j Fp(has)c(c)m(haracter-)328 3678 y(istic)k Fn(p)e Fl(\025)g Fp(2)j(and)h(the)g(group)f(algebra)g Fn(k)s Fp([)p Fn(Z)1931 3693 y Ft(p)1970 3678 y Fp(])h(of)f(the)h (cyclic)g(group)f Fn(Z)2926 3693 y Ft(p)2999 3678 y Fp(is)g(a)h (sub-Hopf)328 3798 y(algebra)d(of)h Fn(A)p Fp(;)h(th)m(us)h(Corollary)d (2)h(applies.)43 b Fd(2)474 3919 y Fp(A)i(v)m(ery)h(basic)e(example)g (to)g(whic)m(h)h(Corollary)e(3)h(applies)g(is)g(the)h(T)-8 b(aft)44 b(\(Hopf)7 b(\))328 4039 y(algebra)39 b(T)747 4054 y Ft(n;q)848 4039 y Fp(\()p Fn(k)s Fp(\),)j(where)g Fn(n)f Fp(is)f(a)g(p)s(ositiv)m(e)g(in)m(teger)g(and)g Fn(q)45 b Fl(2)d Fn(k)h Fp(is)d(a)g(primitiv)m(e)e Fn(n)3495 4003 y Ft(th)328 4159 y Fp(ro)s(ot)c(of)g(unit)m(y)g([19].)50 b(As)35 b(an)f(algebra)g(T)1834 4174 y Ft(n;q)1935 4159 y Fp(\()p Fn(k)s Fp(\))g(is)g(generated)h(b)m(y)h Fn(a;)17 b(x)35 b Fp(sub)5 b(ject)36 b(to)e(the)328 4280 y(relations)44 b Fn(a)788 4244 y Ft(n)885 4280 y Fp(=)50 b(1,)e Fn(x)1190 4244 y Ft(n)1287 4280 y Fp(=)i(0)45 b(and)h Fn(xa)k Fp(=)g Fn(q)t(ax)p Fp(.)83 b(Th)m(us)47 b(Dim)14 b(T)2784 4295 y Ft(n;q)2885 4280 y Fp(\()p Fn(k)s Fp(\))50 b(=)g Fn(n)3249 4244 y Fu(2)3288 4280 y Fp(.)83 b(The)328 4400 y(coalgebra)41 b(structure)j(of)e(T)1390 4415 y Ft(n;q)1491 4400 y Fp(\()p Fn(k)s Fp(\))h(is)f(determined)g(b)m(y)i(\001\()p Fn(a)p Fp(\))h(=)g Fn(a)p Fl(\012)p Fn(a)e Fp(and)g(\001\()p Fn(x)p Fp(\))i(=)328 4521 y Fn(x)p Fl(\012)q Fn(a)22 b Fp(+)g(1)p Fl(\012)p Fn(x)p Fp(.)1898 5214 y(18)p eop %%Page: 19 19 19 18 bop 328 648 a Fo(4)161 b(The)49 b(Elemen)l(t)f Fb(\027)1594 669 y Fn(A)1671 648 y Fa(\()p Fb(s)1788 596 y Fp(2)p Fn(m)1927 648 y Fa(\))h Fo(when)f Fb(s)2512 596 y Fp(2)2614 648 y Fo(is)i(Semisimple)570 830 y(and)k Fb(A)987 778 y Fl(\003)1095 830 y Fo(is)g(P)l(oin)l(ted)328 1049 y Fp(Let)38 b Fn(A)g Fp(b)s(e)g(a)g(\014nite-dimensional)c(Hopf)k (algebra)f(o)m(v)m(er)i(the)f(\014eld)g Fn(k)i Fp(and)e(let)g Fn(B)j Fl(\022)d Fn(A)328 1170 y Fp(b)s(e)c(a)g(sub-Hopf)g(algebra)f (of)g Fn(A)h Fp(con)m(taining)f(the)h(coradical)f Fn(A)2658 1185 y Fu(0)2731 1170 y Fp(of)h Fn(A)p Fp(.)47 b(W)-8 b(e)35 b(b)s(egin)e(our)328 1290 y(discussion)g(with)f(an)g(analysis)g (of)g(the)h(asso)s(ciated)g(graded)g(Hopf)f(algebra)g(gr)3239 1314 y Ft(B)3300 1290 y Fp(\()p Fn(A)p Fp(\).)474 1410 y(Recall)26 b(that)g(if)g Fn(U;)17 b(V)49 b Fl(\022)28 b Fn(A)f Fp(are)g(subspaces,)j(the)e(w)m(edge)g Fn(U)21 b Fl(^)11 b Fn(V)49 b Fl(\022)29 b Fn(A)e Fp(is)f(the)h(in)m(v)m(erse) 328 1531 y(image)k(under)i(the)g(com)m(ultiplication)28 b(of)k Fn(U)h Fl(\012)23 b Fn(A)f Fp(+)g Fn(A)g Fl(\012)h Fn(V)e Fp(.)44 b(Note)33 b(that)713 1743 y Fn(F)776 1758 y Fu(0)815 1743 y Fp(\()p Fn(A)p Fp(\))28 b(=)g Fn(B)k Fl(\022)c Fn(F)1370 1758 y Fu(1)1410 1743 y Fp(\()p Fn(A)p Fp(\))g(=)f Fn(B)h Fl(^)22 b Fn(B)33 b Fl(\022)28 b Fn(F)2155 1758 y Fu(2)2195 1743 y Fp(\()p Fn(A)p Fp(\))f(=)h Fn(B)f Fl(^)c Fn(B)k Fl(^)c Fn(B)32 b Fl(\022)d Fn(:)17 b(:)g(:)328 1955 y Fp(is)32 b(a)g(Hopf)h(algebra)e(\014ltration)g(of)h(A)g(with) 1933 1888 y Fg(S)2002 1975 y Ft(n)p Fj(\025)p Fu(0)2155 1955 y Fn(F)2218 1970 y Ft(n)2266 1955 y Fp(\()p Fn(A)p Fp(\))27 b(=)h Fn(A)p Fp(.)474 2075 y(De\014ne)33 b Fn(F)839 2090 y Fj(\000)p Fu(1)934 2075 y Fp(\()p Fn(A)p Fp(\))28 b(=)f(0)32 b(and)h(let)363 2287 y(gr)449 2310 y Ft(B)510 2287 y Fp(\()p Fn(A)p Fp(\))28 b(=)794 2204 y Fg(M)790 2387 y Ft(n)p Fj(\025)p Fu(0)940 2287 y Fp(gr)1026 2310 y Ft(B)1087 2287 y Fp(\()p Fn(A)p Fp(\)\()p Fn(n)p Fp(\))p Fn(;)49 b Fp(where)34 b(gr)1815 2310 y Ft(B)1876 2287 y Fp(\()p Fn(A)p Fp(\)\()p Fn(n)p Fp(\))27 b(=)h Fn(F)2353 2302 y Ft(n)2400 2287 y Fp(\()p Fn(A)p Fp(\))p Fn(=F)2661 2302 y Ft(n)p Fj(\000)p Fu(1)2798 2287 y Fp(\()p Fn(A)p Fp(\))33 b(for)f(all)e Fn(n)e Fl(\025)g Fp(0)p Fn(;)328 2575 y Fp(b)s(e)33 b(the)g(asso)s(ciated)f(graded)h(Hopf)f(algebra.)474 2696 y(The)45 b(pro)5 b(jection)44 b Fn(\031)52 b Fp(:)47 b(gr)1429 2719 y Ft(B)1490 2696 y Fp(\()p Fn(A)p Fp(\))g Fl(!)g Fp(gr)1920 2719 y Ft(B)1981 2696 y Fp(\()p Fn(A)p Fp(\)\(0\))g(=)g Fn(B)j Fp(is)43 b(a)h(Hopf)g(algebra)f(map,)328 2816 y(and)34 b(the)h(inclusion)e(map)h Fn(B)i Fl(\022)31 b Fp(gr)1622 2839 y Ft(B)1682 2816 y Fp(\()p Fn(A)p Fp(\))k(is)f(a)g (Hopf)g(algebra)f(section)i(of)f Fn(\031)t Fp(.)48 b(Let)35 b Fn(R)d Fp(=)328 2936 y(gr)415 2960 y Ft(B)475 2936 y Fp(\()p Fn(A)p Fp(\))624 2900 y Ft(coB)778 2936 y Fp(b)s(e)d(the)f (righ)m(t)g(coin)m(v)-5 b(arian)m(t)27 b(elemen)m(ts)i(with)f(resp)s (ect)i(to)e(the)h(pro)5 b(jection)28 b Fn(\031)t Fp(.)328 3057 y(By)35 b(de\014nition)f Fn(R)e Fp(=)f Fl(f)p Fn(x)h Fl(2)g Fp(gr)1453 3080 y Ft(B)1514 3057 y Fp(\()p Fn(A)p Fp(\))f Fl(j)g Fp(\(1)24 b Fl(\012)g Fn(\031)t Fp(\)\(\001\()p Fn(x)p Fp(\)\))31 b(=)h Fn(x)24 b Fl(\012)g Fp(1)p Fl(g)p Fp(.)49 b(By)35 b([15,)g(Theorem)328 3177 y(3])d(there)i(is)e(an)g (isomorphism)e(of)i(Hopf)h(algebras)1601 3389 y(gr)1688 3413 y Ft(B)1748 3389 y Fp(\()p Fn(A)p Fp(\))1925 3361 y Fl(\030)1926 3393 y Fp(=)2030 3389 y Fn(R)q Fp(#)p Fn(B)5 b(;)328 3601 y Fp(de\014ned)35 b(b)m(y)f(m)m(ultiplication)29 b(in)j(gr)1629 3624 y Ft(B)1690 3601 y Fp(\()p Fn(A)p Fp(\),)h(where)i Fn(R)f Fp(has)g(the)g(structure)g(of)f(a)g(braided)328 3721 y(Hopf)g(algebra)f(in)g(the)h(category)g(of)g(left)f(Y)-8 b(etter-Drinfeld)31 b(mo)s(dules)h(o)m(v)m(er)i Fn(B)5 b Fp(.)45 b(See)34 b([9])328 3842 y(and)f(the)g(discussion)f(in)g([2,)h (Section)f(2].)328 4061 y Ff(Lemma)37 b(8)49 b Fk(L)-5 b(et)41 b Fn(A)g Fk(b)-5 b(e)40 b(a)h(\014nite-dimensional)d(Hopf)j (algebr)-5 b(a)40 b(over)g(the)h(\014eld)f Fn(k)j Fk(and)328 4181 y(supp)-5 b(ose)32 b(that)g Fn(B)38 b Fk(is)32 b(a)g(sub-Hopf)g (algebr)-5 b(a)31 b(of)h Fn(A)h Fk(which)e(c)-5 b(ontains)32 b(the)g(c)-5 b(or)g(adic)g(al)31 b Fn(A)3414 4196 y Fu(0)3486 4181 y Fk(of)328 4301 y Fn(A)p Fk(.)45 b(Then:)436 4498 y Fp(a\))k Fn(R)29 b Fp(=)778 4432 y Fg(L)870 4519 y Ft(n)934 4498 y Fn(R)q Fp(\()p Fn(n)p Fp(\))g Fk(with)g Fn(R)q Fp(\()p Fn(n)p Fp(\))f(=)f Fn(R)10 b Fl(\\)f Fk(gr)1963 4521 y Ft(B)2024 4498 y Fp(\()p Fn(A)p Fp(\)\()p Fn(n)p Fp(\))29 b Fk(for)f(al)5 b(l)29 b Fn(n)f Fl(\025)g Fp(0)g Fk(is)h(a)g(gr)-5 b(ade)g(d)28 b(Hopf)572 4618 y(algebr)-5 b(a)34 b(in)g(the)h(br)-5 b(aide)g(d)34 b(sense)g(and)h(gr)2053 4642 y Ft(B)2113 4618 y Fp(\()p Fn(A)p Fp(\)\()p Fn(n)p Fp(\))2424 4591 y Fl(\030)2425 4622 y Fp(=)2529 4618 y Fn(R)q Fp(\()p Fn(n)p Fp(\)#)p Fn(B)41 b Fk(for)34 b(al)5 b(l)35 b Fn(n)28 b Fl(\025)g Fp(0)p Fk(.)431 4819 y Fp(b\))49 b Fn(R)q Fp(\()p Fn(n)p Fp(\))35 b(=)h Fl(f)q Fp(\026)-50 b Fn(a)35 b Fl(2)h Fn(F)1229 4834 y Ft(n)1276 4819 y Fp(\()p Fn(A)p Fp(\))p Fn(=F)1537 4834 y Ft(n)p Fj(\000)p Fu(1)1674 4819 y Fp(\()p Fn(A)p Fp(\))f Fl(j)g Fn(a)h Fl(2)g Fn(F)2173 4834 y Ft(n)2220 4819 y Fp(\()p Fn(A)p Fp(\))p Fn(;)17 b Fp(\001\()p Fn(a)p Fp(\))25 b Fl(\000)h Fn(a)f Fl(\012)h Fp(1)35 b Fl(2)h Fn(F)3177 4834 y Ft(n)p Fj(\000)p Fu(1)3314 4819 y Fp(\()p Fn(A)p Fp(\))25 b Fl(\012)572 4940 y Fn(F)635 4955 y Ft(n)682 4940 y Fp(\()p Fn(A)p Fp(\))p Fl(g)35 b Fk(for)f(al)5 b(l)35 b Fn(n)28 b Fl(\025)g Fp(0)p Fk(.)1898 5214 y Fp(19)p eop %%Page: 20 20 20 19 bop 328 631 a Fe(Pr)n(oof:)46 b Fp(P)m(art)26 b(a\))f(is)h (explained)f(in)g([2,)i(Lemma)e(2.1].)40 b(T)-8 b(o)26 b(see)h(part)f(b\))g(let)f Fn(a)i Fl(2)i Fn(F)3343 646 y Ft(n)3390 631 y Fp(\()p Fn(A)p Fp(\).)328 751 y(The)50 b(residue)f(class)h(\026)-50 b Fn(a)55 b Fl(2)h Fn(F)1428 766 y Ft(n)1475 751 y Fp(\()p Fn(A)p Fp(\))p Fn(=F)1736 766 y Ft(n)p Fj(\000)p Fu(1)1873 751 y Fp(\()p Fn(A)p Fp(\))49 b(is)f(in)g Fn(R)i Fp(if)e(and)h(only)f(if)h(\026)-50 b Fn(a)3138 767 y Fu(\(1\))3232 751 y Fl(\012)r Fp(\026)g Fn(a)3361 767 y Fu(\(2\))3488 751 y Fl(\000)329 891 y Fp(\026)g Fn(a)26 b Fl(\012)h Fp(1)37 b Fl(2)h Fp(gr)786 915 y Ft(B)847 891 y Fp(\()p Fn(A)p Fp(\))p Fl(\012)1090 795 y Fg(\020)1139 825 y(L)1232 912 y Ft(i)p Fj(\025)p Fu(1)1367 891 y Fp(gr)1453 915 y Ft(B)1514 891 y Fp(\()p Fn(A)p Fp(\)\()p Fn(i)p Fp(\))1772 795 y Fg(\021)1822 891 y Fp(.)61 b(The)39 b(last)f(condition)f(means)h(exactly)h(that)328 1024 y(\001\()p Fn(a)p Fp(\))22 b Fl(\000)h Fn(a)f Fl(\012)h Fp(1)32 b(is)g(an)h(elemen)m(t)f(in)g Fn(F)1683 1039 y Ft(n)p Fj(\000)p Fu(1)1820 1024 y Fp(\()p Fn(A)p Fp(\))23 b Fl(\012)f Fn(F)2154 1039 y Ft(n)2201 1024 y Fp(\()p Fn(A)p Fp(\))p Fn(:)33 b Fd(2)474 1188 y Fp(W)-8 b(e)43 b(will)e(need)i(the)g(dual)f(of)g(Prop)s(osition)f(1.)74 b(F)-8 b(or)42 b(all)e Fn(p)45 b Fl(2)g Fn(A)2903 1152 y Fj(\003)2943 1188 y Fp(,)g(let)d Fn(R)q Fp(\()p Fn(p)p Fp(\))h(and)328 1309 y Fn(L)p Fp(\()p Fn(p)p Fp(\))32 b(b)s(e)h(the)f(linear)f(endomorphisms)g(of)h Fn(A)g Fp(de\014ned)h(b)m(y)g Fn(R)q Fp(\()p Fn(p)p Fp(\)\()p Fn(a)p Fp(\))28 b(=)g Fn(p)p Fp(\()p Fn(a)3099 1324 y Fu(\(1\))3193 1309 y Fp(\))p Fn(a)3282 1324 y Fu(\(2\))3409 1309 y Fp(and)328 1429 y Fn(L)p Fp(\()p Fn(p)p Fp(\)\()p Fn(a)p Fp(\))42 b(=)g Fn(a)857 1445 y Fu(\(1\))951 1429 y Fn(p)p Fp(\()p Fn(a)1089 1445 y Fu(\(2\))1183 1429 y Fp(\))f(resp)s(ectiv)m(ely)h(for)e(all)f Fn(a)j Fl(2)g Fn(A)p Fp(.)68 b(Note)41 b(that)g Fn(R)q Fp(\()p Fn(p)p Fp(\))g(and)g Fn(L)p Fp(\()p Fn(p)p Fp(\))328 1550 y(are)e(the)g (transp)s(oses)h(of)e(left)g(and)h(righ)m(t)f(m)m(ultiplication)c(resp) s(ectiv)m(ely)40 b(b)m(y)g Fn(p)e Fp(on)h Fn(A)3499 1513 y Fj(\003)3539 1550 y Fp(.)328 1670 y(Since)e Fm(tr)q Fp(\()p Fn(t)p Fp(\))e(=)f Fm(tr)r Fp(\()p Fn(t)1055 1634 y Fj(\003)1094 1670 y Fp(\))j(for)f(all)e(linear)h(endomorphisms)h Fn(t)h Fp(of)f Fn(A)p Fp(,)h(the)g(dual)f(v)m(ersion)h(of)328 1790 y(Prop)s(osition)31 b(1)h(can)h(b)s(e)g(form)m(ulated:)328 1976 y Ff(Prop)s(osition)i(4)49 b Fk(L)-5 b(et)40 b Fn(A)g Fk(b)-5 b(e)39 b(a)g(\014nite-dimensional)f(Hopf)h(algebr)-5 b(a)39 b(with)g(antip)-5 b(o)g(de)39 b Fn(s)328 2096 y Fk(and)f Fn(t)h Fk(b)-5 b(e)38 b(a)h(c)-5 b(o)g(algebr)g(a)37 b(automorphism)h(of)g Fn(A)h Fk(which)f(c)-5 b(ommutes)38 b(with)h Fn(s)p Fk(.)56 b(Then)38 b(the)328 2217 y(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)436 2402 y Fp(a\))49 b Fn(\027)620 2417 y Ft(A)673 2398 y Fi(\003)713 2402 y Fp(\()p Fn(t)786 2366 y Fj(\003)826 2402 y Fp(\))27 b(=)h(0)p Fn(:)431 2600 y Fp(b\))49 b Fn(\027)620 2615 y Ft(A)673 2596 y Fi(\003)713 2600 y Fp(\()p Fn(t)786 2564 y Fj(\003\000)p Fu(1)916 2600 y Fp(\))27 b(=)h(0)p Fn(:)442 2797 y Fp(c\))49 b Fm(tr)r Fp(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))28 b(=)g(0)35 b Fk(for)f(al)5 b(l)35 b Fn(p)27 b Fl(2)h Fn(A)1747 2761 y Fj(\003)1787 2797 y Fk(.)431 2995 y Fp(d\))49 b Fm(tr)r Fp(\()p Fn(R)q Fp(\()p Fn(p)p Fp(\))p Fl(\016)o Fn(t)p Fp(\))28 b(=)g(0)34 b Fk(for)h(al)5 b(l)34 b Fn(p)28 b Fl(2)g Fn(A)1756 2959 y Fj(\003)1796 2995 y Fk(.)328 3180 y Fd(2)328 3385 y Ff(Theorem)37 b(2)49 b Fk(L)-5 b(et)34 b Fn(A)g Fk(b)-5 b(e)33 b(a)g(\014nite-dimensional)f(Hopf)h (algebr)-5 b(a)33 b(over)g(the)h(\014eld)f Fn(k)j Fk(and)328 3505 y(supp)-5 b(ose)33 b(that)g Fn(B)39 b Fk(is)33 b(a)g(sub-Hopf)g (algebr)-5 b(a)33 b(of)g Fn(A)g Fk(c)-5 b(ontaining)32 b(the)i(c)-5 b(or)g(adic)g(al)32 b(of)h Fn(A)p Fk(.)44 b(L)-5 b(et)328 3626 y Fn(t)33 b Fk(a)g(Hopf)g(algebr)-5 b(a)32 b(automorphism)g(of)g Fn(A)p Fk(.)45 b(Assume)33 b Fn(t)p Fp(\()p Fn(B)5 b Fp(\))27 b(=)h Fn(B)5 b Fk(,)33 b(let)g(gr)p Fp(\()p Fn(t)p Fp(\))g Fk(denote)g(the)328 3746 y(induc)-5 b(e)g(d)30 b(Hopf)g(algebr)-5 b(a)30 b(automorphism)f(on)h(gr)2088 3770 y Ft(B)2149 3746 y Fp(\()p Fn(A)p Fp(\))h Fk(and)f(let)g Fn(R)f Fp(=)e Fk(gr)2938 3770 y Ft(B)2999 3746 y Fp(\()p Fn(A)p Fp(\))3148 3710 y Ft(coB)3273 3746 y Fk(.)44 b(Then)1127 3945 y Fm(tr)r Fp(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))28 b(=)f Fm(tr)r Fp(\()p Fk(gr)p Fp(\()p Fn(t)p Fp(\))p Fl(j)2013 3960 y Ft(R)2070 3945 y Fp(\))c Fl(\001)e Fm(tr)r Fp(\(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))p Fl(j)2668 3960 y Ft(B)2728 3945 y Fp(\))328 4145 y Fk(for)35 b(al)5 b(l)34 b Fn(p)28 b Fl(2)g Fn(A)867 4108 y Fj(\003)906 4145 y Fk(.)328 4393 y Fe(Pr)n(oof:)93 b Fp(Let)54 b Fn(n)62 b Fl(\025)h Fp(1.)104 b(By)54 b(part)f(a\))f(of)h (Lemma)e(8)i(the)g(m)m(ultiplication)c(map)328 4513 y Fn(R)q Fp(\()p Fn(n)p Fp(\)#)p Fn(B)33 b Fl(!)28 b Fp(gr)939 4537 y Ft(B)1000 4513 y Fp(\()p Fn(A)p Fp(\)\()p Fn(n)p Fp(\))k(is)g(bijectiv)m(e.)43 b(No)m(w)33 b(let)g(\026)-50 b Fn(a)28 b Fl(2)g Fn(R)q Fp(\()p Fn(n)p Fp(\),)33 b Fn(a)27 b Fl(2)i Fn(F)2879 4528 y Ft(n)2926 4513 y Fp(\()p Fn(A)p Fp(\))j(and)g Fn(b)c Fl(2)h Fn(B)5 b Fp(.)328 4634 y(Then)39 b(\001\()p Fn(a)p Fp(\))26 b Fl(\000)g Fn(a)f Fl(\012)h Fp(1)36 b Fl(2)h Fn(F)1355 4649 y Ft(n)p Fj(\000)p Fu(1)1492 4634 y Fp(\()p Fn(A)p Fp(\))26 b Fl(\012)g Fn(F)1833 4649 y Ft(n)1880 4634 y Fp(\()p Fn(A)p Fp(\))38 b(b)m(y)g(part)f(b\))h(of)f(Lemma)f(8.)59 b(Therefore)328 4754 y Fn(a)379 4770 y Fu(\(1\))473 4754 y Fn(b)514 4770 y Fu(\(1\))631 4754 y Fl(\012)23 b Fn(a)782 4770 y Fu(\(2\))877 4754 y Fn(b)918 4770 y Fu(\(2\))1035 4754 y Fl(\000)f Fn(ab)1226 4770 y Fu(\(1\))1343 4754 y Fl(\012)h Fn(b)1484 4770 y Fu(\(2\))1606 4754 y Fl(2)28 b Fn(F)1763 4769 y Ft(n)p Fj(\000)p Fu(1)1901 4754 y Fp(\()p Fn(A)p Fp(\))22 b Fl(\012)h Fn(F)2235 4769 y Ft(n)2282 4754 y Fp(\()p Fn(A)p Fp(\).)43 b(Hence)34 b(w)m(e)g(compute)987 4965 y(gr\()p Fn(L)p Fp(\()p Fn(p)p Fp(\)\)\()q(\026)-50 b Fn(ab)p Fp(\))28 b(=)p 1640 4880 688 4 v 27 w Fn(a)1691 4980 y Fu(\(1\))1786 4965 y Fn(b)1827 4980 y Fu(\(1\))1922 4965 y Fn(p)p Fp(\()p Fn(a)2060 4980 y Fu(\(2\))2154 4965 y Fn(b)2195 4980 y Fu(\(2\))2290 4965 y Fp(\))g(=)g(\026)-50 b Fn(ab)2551 4980 y Fu(\(1\))2646 4965 y Fn(p)p Fp(\()p Fn(b)2774 4980 y Fu(\(2\))2869 4965 y Fp(\))1898 5214 y(20)p eop %%Page: 21 21 21 20 bop 328 631 a Fp(for)32 b(all)e Fn(p)e Fl(2)g Fn(A)856 595 y Fj(\003)896 631 y Fp(,)k(where)i(gr\()p Fn(L)p Fp(\()p Fn(p)p Fp(\)\))e(denotes)i(the)f(induced)g(map)f(on)h(gr)2946 654 y Ft(B)3007 631 y Fp(\()p Fn(A)p Fp(\).)474 751 y(W)-8 b(e)37 b(ha)m(v)m(e)h(sho)m(wn)g(that)f(the)g(induced)g(map)f(gr\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))h(can)g(b)s(e)g (iden)m(ti\014ed)f(with)328 872 y(the)46 b(map)g(\(gr)o(\()p Fn(t)p Fp(\))p Fl(j)1003 887 y Ft(R)1061 872 y Fp(\))p Fl(\012)p Fp(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))p Fl(j)1556 887 y Ft(B)1617 872 y Fp(\).)83 b(This)47 b(pro)m(v)m(es)g Fm(tr)r Fp(\(gr\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\)\))k(=)f Fm(tr)r Fp(\(gr)o(\()p Fn(t)p Fp(\))p Fl(j)3411 887 y Ft(R)3469 872 y Fp(\))31 b Fl(\001)328 992 y Fm(tr)r Fp(\(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))p Fl(j)816 1007 y Ft(B)876 992 y Fp(\),)i(and)f(the)h(theorem)g (follo)m(ws)e(since)i Fm(tr)r Fp(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\))27 b(=)h Fm(tr)r Fp(\(gr)o(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\))p Fl(\016)p Fn(t)p Fp(\)\).)44 b Fd(2)474 1160 y Fp(Theorem)36 b(2)g(implies)d(an)j (analog)e(of)h(part)g(c\))h(of)f(Prop)s(osition)f(2)i(under)g(an)g (addi-)328 1280 y(tional)30 b(assumption.)328 1477 y Ff(Corollary)36 b(4)49 b Fk(L)-5 b(et)47 b Fn(A)g Fk(b)-5 b(e)47 b(a)g(\014nite-dimensional)e(Hopf)i(algebr)-5 b(a)46 b(over)h(the)g(\014eld)g Fn(k)s Fk(.)328 1597 y(Supp)-5 b(ose)42 b(that)i Fn(I)50 b Fl(\022)44 b Fn(A)f Fk(a)g(nilp)-5 b(otent)43 b(Hopf)f(ide)-5 b(al.)69 b(L)-5 b(et)43 b Fn(t)h Fk(b)-5 b(e)43 b(a)f(Hopf)h(algebr)-5 b(a)43 b(auto-)328 1718 y(morphism)34 b(of)h Fn(A)g Fk(such)g(that)h Fn(t)p Fp(\()p Fn(I)8 b Fp(\))28 b Fl(\022)h Fn(I)43 b Fk(and)35 b(let)2139 1699 y Fp(\026)2137 1718 y Fn(t)h Fk(denote)f(the)g(induc)-5 b(e)g(d)34 b(Hopf)i(algebr)-5 b(a)328 1838 y(automorphism)34 b(on)g Fn(A=I)8 b Fk(.)45 b(If)34 b Fn(\027)1504 1854 y Ft(A=I)1632 1838 y Fp(\()1671 1820 y(\026)1670 1838 y Fn(t)p Fp(\))28 b(=)g(0)34 b Fk(then)h Fn(\027)2223 1853 y Ft(A)2280 1838 y Fp(\()p Fn(t)p Fp(\))28 b(=)g(0)p Fk(.)328 2082 y Fe(Pr)n(oof:)68 b Fp(Let)41 b Fn(\031)47 b Fp(:)42 b Fn(A)g Fl(\000)-16 b(!)42 b Fn(A=I)50 b Fp(=)42 b Fn(B)47 b Fp(b)s(e)41 b(the)h(pro)5 b(jection)41 b(and)g(iden)m(tify)f Fn(B)3295 2046 y Fj(\003)3376 2082 y Fp(with)328 2203 y Fn(\031)387 2166 y Fj(\003)426 2203 y Fp(\()p Fn(B)543 2166 y Fj(\003)583 2203 y Fp(\).)81 b(Since)46 b Fn(I)57 b Fl(\022)49 b Fp(Rad)17 b Fn(A)45 b Fp(it)f(follo)m(ws)g(that)h(the)h(sub-Hopf)g (algebra)e Fn(B)3245 2166 y Fj(\003)3329 2203 y Fp(of)h Fn(A)3526 2166 y Fj(\003)328 2323 y Fp(con)m(tains)31 b(the)h(coradical)e(of)h Fn(A)1470 2287 y Fj(\003)1510 2323 y Fp(.)43 b(Observ)m(e)33 b(that)2165 2305 y(\026)2164 2323 y Fn(t)2199 2287 y Fj(\003)2266 2323 y Fp(=)28 b Fn(t)2405 2287 y Fj(\003)2444 2323 y Fl(j)2472 2338 y Ft(B)2528 2319 y Fi(\003)2600 2323 y Fp(and)k Fn(L)p Fp(\()p Fn(\031)t Fp(\()p Fn(a)p Fp(\)\))27 b(=)h Fn(L)p Fp(\()p Fn(a)p Fp(\))p Fl(j)3469 2338 y Ft(B)3525 2319 y Fi(\003)328 2443 y Fp(for)k(all)e Fn(a)e Fl(2)g Fn(A)p Fp(.)474 2564 y(Supp)s(ose)34 b(that)e Fn(\027)1116 2579 y Ft(A=I)1245 2564 y Fp(\()1284 2545 y(\026)1283 2564 y Fn(t)p Fp(\))27 b(=)h(0.)43 b(Then)825 2776 y Fm(tr)r Fp(\(\()p Fn(L)p Fp(\()p Fn(a)p Fp(\))p Fl(\016)p Fn(t)1249 2735 y Fj(\003)1289 2776 y Fp(\))p Fl(j)1355 2791 y Ft(B)1411 2772 y Fi(\003)1451 2776 y Fp(\))27 b(=)h Fm(tr)r Fp(\()p Fn(L)p Fp(\()p Fn(\031)t Fp(\()p Fn(a)p Fp(\)\))p Fl(\016)2107 2757 y Fp(\026)2106 2776 y Fn(t)2141 2735 y Fj(\003)2180 2776 y Fp(\))g(=)g Fm(tr)q Fp(\()2458 2757 y(\026)2457 2776 y Fn(t)p Fl(\016)p Fn(r)s Fp(\()p Fn(\031)t Fp(\()p Fn(a)p Fp(\)\)\))f(=)h(0)328 2988 y(for)41 b(all)f Fn(a)j Fl(2)h Fn(A)e Fp(b)m(y)g(Prop)s(osition)e(4.)71 b(No)m(w)42 b(w)m(e)h(use)g(Theorem)f(2)f(to)h(conclude)g(that)328 3108 y Fm(tr)r Fp(\()p Fn(t)p Fl(\016)p Fn(r)s Fp(\()p Fn(a)p Fp(\)\))30 b(=)h Fm(tr)q Fp(\()p Fn(L)p Fp(\()p Fn(a)p Fp(\))p Fl(\016)q Fn(t)1256 3072 y Fj(\003)1295 3108 y Fp(\))g(=)g(0)j(for)g(all)e Fn(a)f Fl(2)g Fn(A)p Fp(,)k(whence)h Fn(\027)2547 3123 y Ft(A)2605 3108 y Fp(\()p Fn(t)p Fp(\))30 b(=)h(0)j(b)m(y)h(Prop)s(osition)328 3229 y(4)d(again.)42 b Fd(2)474 3396 y Fp(No)m(w)31 b(w)m(e)g(assume)f (that)g Fn(A)g Fp(is)g(p)s(oin)m(ted)f(and)h Fn(B)j Fp(=)28 b Fn(A)2397 3411 y Fu(0)2436 3396 y Fp(.)43 b(The)31 b(graded)f(Hopf)g(algebra)328 3517 y(asso)s(ciated)37 b(to)g(the)h(coradical)e(\014ltration)f(will)h(b)s(e)h(denoted)i(b)m(y) f(gr\()p Fn(A)p Fp(\).)58 b(Our)37 b(goal)f(is)328 3637 y(to)43 b(sho)m(w)h(in)e(Theorem)i(2)f(that)g Fm(tr)q Fp(\(gr\()p Fn(t)p Fp(\))p Fl(j)1904 3652 y Ft(R)1962 3637 y Fp(\))i(=)h(0)d(under)h(additional)c(assumptions.)328 3758 y(W)-8 b(e)33 b(will)d(need)k(the)f(next)g(lemma)d(to)i(reduce)i (to)e(a)h(subalgebra)f(where)i(the)f(trace)f(can)328 3878 y(easily)g(b)s(e)h(computed.)328 4097 y Ff(Lemma)k(9)49 b Fk(L)-5 b(et)30 b Fn(A)g Fk(b)-5 b(e)30 b(a)f(\014nite-dimensional)f (algebr)-5 b(a)29 b(over)g(the)h(\014eld)f Fn(k)k Fk(and)c(supp)-5 b(ose)328 4218 y(that)39 b Fn(B)45 b Fk(a)38 b(lo)-5 b(c)g(al)39 b(sub)-5 b(algebr)g(a)38 b(of)h Fn(A)g Fk(with)g(maximal)f (ide)-5 b(al)38 b Fn(B)2585 4181 y Fu(+)2683 4218 y Fk(and)h Fn(B)5 b(=B)3084 4181 y Fu(+)3178 4190 y Fl(\030)3179 4222 y Fp(=)3291 4218 y Fn(k)s Fk(.)57 b(L)-5 b(et)328 4338 y Fn(\033)42 b Fp(:)c Fn(A)g Fl(!)g Fn(A)j Fk(b)-5 b(e)40 b(an)g(algebr)-5 b(a)40 b(homomorphism)e(such)i(that)h Fn(\033)t Fp(\()p Fn(B)2756 4302 y Fu(+)2815 4338 y Fp(\))d Fl(\022)h Fn(B)3086 4302 y Fu(+)3145 4338 y Fk(.)62 b(Assume)328 4458 y(that)48 b Fn(A)g Fk(is)g(fr)-5 b(e)g(e)47 b(as)h(a)g(left)g Fn(B)5 b Fk(-mo)-5 b(dule)47 b(by)h(r)-5 b(estriction.)83 b(If)48 b Fn(\033)k Fk(is)47 b(a)h(diagonalizable)328 4579 y(endomorphism)f(of)h Fn(A)p Fk(,)k(then)d(ther)-5 b(e)49 b(exists)f(a)h Fn(\033)t Fk(-stable)f(line)-5 b(ar)48 b(subsp)-5 b(ac)g(e)48 b Fn(X)62 b Fl(\022)54 b Fn(A)328 4699 y Fk(such)40 b(that)g(the)g(multiplic)-5 b(ation)40 b(in)g Fn(A)g Fk(de\014nes)f(an)h(isomorphism)e Fn(B)31 b Fl(\012)c Fn(X)45 b Fl(!)37 b Fn(A)p Fk(.)60 b(In)328 4819 y(p)-5 b(articular)1469 4940 y Fm(tr)r Fp(\()p Fn(\033)t Fl(j)1664 4955 y Ft(B)1724 4940 y Fp(\))p Fm(tr)r Fp(\()p Fn(\033)t Fl(j)1957 4955 y Ft(X)2024 4940 y Fp(\))27 b(=)h Fm(tr)r Fp(\()p Fn(\033)t Fp(\))p Fn(:)1898 5214 y Fp(21)p eop %%Page: 22 22 22 21 bop 328 631 a Fe(Pr)n(oof:)48 b Fp(Let)31 b Fl(f)p Fn(a)993 646 y Ft({)1020 631 y Fl(g)1070 646 y Fu(1)p Fj(\024)p Ft({)p Fj(\024)p Ft(n)1315 631 y Fp(b)s(e)f(a)g(left)f Fn(B)5 b Fp(-basis)30 b(of)f Fn(A)p Fp(,)i(and)f(let)f Fn(V)52 b Fp(b)s(e)30 b(the)h Fn(k)s Fp(-span)f(of)f(the)328 751 y Fn(a)379 766 y Ft(i)407 751 y Fp('s.)50 b(Then)35 b Fn(A)c Fp(=)g Fn(B)1095 715 y Fu(+)1154 751 y Fn(V)45 b Fl(\010)24 b Fn(V)d Fp(,)35 b(and)g Fn(B)1768 715 y Fu(+)1827 751 y Fn(A)c Fp(=)g(\()p Fn(B)2155 715 y Fu(+)2214 751 y Fp(\))2252 715 y Fu(2)2291 751 y Fn(V)45 b Fp(+)23 b Fn(B)2571 715 y Fu(+)2630 751 y Fn(V)53 b Fp(=)30 b Fn(B)2925 715 y Fu(+)2985 751 y Fn(V)55 b Fp(is)34 b Fn(\033)t Fp(-stable.)328 872 y(Since)g Fn(\033)39 b Fp(is)33 b(diagonalizable,)f(there)j(exists)g(a)f Fn(\033)t Fp(-stable)f(subspace)j Fn(X)j Fl(\022)31 b Fn(A)j Fp(suc)m(h)i(that) 328 992 y Fn(A)31 b Fp(=)f Fn(B)617 956 y Fu(+)676 992 y Fn(V)45 b Fl(\010)24 b Fn(X)8 b Fp(.)48 b(Since)35 b Fn(B)1379 956 y Fu(+)1438 992 y Fn(A)c Fp(=)f Fn(B)1727 956 y Fu(+)1786 992 y Fn(V)22 b Fp(,)34 b(w)m(e)i(deduce)g(form)d(the)h (last)g(equation)g(that)328 1112 y Fn(B)407 1076 y Fu(+)466 1112 y Fn(V)50 b Fp(=)29 b(\()p Fn(B)795 1076 y Fu(+)854 1112 y Fp(\))892 1076 y Fu(2)931 1112 y Fn(V)44 b Fp(+)23 b Fn(B)1210 1076 y Fu(+)1269 1112 y Fn(X)41 b Fp(from)31 b(whic)m(h)j Fn(B)1980 1076 y Fu(+)2039 1112 y Fn(V)50 b Fl(\022)29 b Fp(\()p Fn(B)2369 1076 y Fu(+)2429 1112 y Fp(\))2467 1076 y Ft(m)2533 1112 y Fn(V)44 b Fp(+)22 b Fn(B)2811 1076 y Fu(+)2871 1112 y Fn(X)40 b Fp(follo)m(ws)32 b(for)h(all)328 1233 y(natural)27 b(n)m(um)m(b)s(ers)h Fn(m)g Fl(\025)h Fp(1.)41 b(Since)28 b Fn(B)1718 1197 y Fu(+)1805 1233 y Fp(is)g(a)f(nilp)s(oten)m(t)g(ideal)f(of)i Fn(A)p Fp(,)h(w)m(e)g(conclude)f(that)328 1353 y Fn(B)407 1317 y Fu(+)466 1353 y Fn(A)42 b Fp(=)g Fn(B)778 1317 y Fu(+)837 1353 y Fn(V)63 b Fp(=)42 b Fn(B)1154 1317 y Fu(+)1213 1353 y Fn(X)8 b Fp(.)68 b(Therefore)42 b Fn(A)g Fp(=)f Fn(B)2158 1317 y Fu(+)2218 1353 y Fn(X)35 b Fl(\010)28 b Fn(X)8 b Fp(,)43 b(and)e(the)g(m)m(ultiplication)328 1474 y(map)c Fn(B)30 b Fl(\012)c Fn(X)44 b Fl(!)35 b Fn(A)j Fp(is)f(surjectiv)m(e,)j(hence)f(bijectiv)m(e)e(since)h Fn(B)5 b Fl(\012)p Fn(X)46 b Fp(and)37 b Fn(A)h Fp(ha)m(v)m(e)g(the)328 1594 y(same)32 b(dimension.)474 1714 y(The)i(claim)c(ab)s(out)i(the)h (traces)g(follo)m(ws)f(easily)-8 b(.)42 b Fd(2)328 1879 y Ff(Theorem)37 b(3)49 b Fk(L)-5 b(et)26 b Fn(A)g Fk(b)-5 b(e)25 b(a)g(\014nite-dimensional)f(p)-5 b(ointe)g(d)25 b(Hopf)g(algebr)-5 b(a)25 b(with)h(antip)-5 b(o)g(de)328 1999 y Fn(s)32 b Fk(over)g(the)h(\014eld)e Fn(k)s Fk(.)44 b(Assume)33 b(that)g Fn(s)1718 1963 y Fu(2)1789 1999 y Fk(is)g(semisimple)d(and)i(set)h Fn(R)28 b Fp(=)g Fk(gr)p Fp(\()p Fn(A)p Fp(\))3165 1963 y Ft(coA)3283 1972 y Fc(0)3321 1999 y Fn(:)33 b Fk(If)e Fn(m)328 2120 y Fk(is)k(an)f(inte)-5 b(ger)35 b(such)f(that)h Fn(s)1361 2084 y Fu(2)p Ft(m)1491 2120 y Fl(6)p Fp(=)28 b(1)1644 2135 y Ft(A)1735 2120 y Fk(then)35 b Fm(tr)r Fp(\()p Fk(gr)o Fp(\()p Fn(s)2229 2084 y Fu(2)p Ft(m)2331 2120 y Fp(\))p Fl(j)2397 2135 y Ft(R)2455 2120 y Fp(\))27 b(=)h(0)p Fk(.)328 2321 y Fe(Pr)n(oof:)52 b Fp(W)-8 b(e)34 b(ma)m(y)f(assume)h(that)f Fn(k)k Fp(is)c(algebraically)d(closed.)46 b(By)34 b(Lemma)e(7,)i(there) 328 2442 y(are)i(a)g(group-lik)m(e)f(elemen)m(t)h Fn(a)e Fl(2)h Fn(G)p Fp(\()p Fn(A)p Fp(\))p Fn(;)17 b Fp(0)33 b Fl(6)p Fp(=)h Fn(x)g Fl(2)g Fn(A)j Fp(and)f(a)g(ro)s(ot)f(of)h(unit)m (y)g Fn(\032)f Fl(2)f Fn(k)39 b Fp(of)328 2562 y(order)29 b Fn(N)38 b(>)28 b Fp(1)g(suc)m(h)j(that)d(\001\()p Fn(x)p Fp(\))h(=)e Fn(x)15 b Fl(\012)g Fn(a)g Fp(+)g(1)g Fl(\012)g Fn(x;)i(s)2264 2526 y Fu(2)2303 2562 y Fp(\()p Fn(x)p Fp(\))28 b(=)f Fn(\032x;)17 b(\032)2764 2526 y Ft(m)2859 2562 y Fl(6)p Fp(=)28 b(1)p Fn(:)h Fp(Let)g Fn(g)i Fp(=)c Fn(a)3471 2526 y Fj(\000)p Fu(1)328 2682 y Fp(and)33 b Fn(y)d Fp(=)e Fn(xa)806 2646 y Fj(\000)p Fu(1)901 2682 y Fp(.)43 b(Then)34 b(\001\()p Fn(y)t Fp(\))27 b(=)h Fn(y)d Fl(\012)e Fp(1)e(+)h Fn(g)k Fl(\012)d Fn(y)35 b Fp(and)e Fn(s)2399 2646 y Fu(2)2438 2682 y Fp(\()p Fn(y)t Fp(\))27 b(=)g Fn(\032y)t Fp(.)474 2803 y(Let)35 b Fn(z)k Fp(b)s(e)c(the)g(residue)g(class)g(of)f Fn(y)k Fp(in)33 b(gr\()p Fn(A)p Fp(\)\(1\))e(=)g Fn(A)2490 2818 y Fu(1)2529 2803 y Fn(=)-5 b(A)2646 2818 y Fu(0)2685 2803 y Fp(.)50 b(Then)35 b Fn(z)40 b Fp(is)34 b(non-zero)328 2923 y(and)25 b(\001)591 2939 y(gr)p Fu(\()p Ft(A)p Fu(\))790 2923 y Fp(\()p Fn(z)t Fp(\))k(=)e Fn(g)11 b Fl(\012)d Fn(z)k Fp(+)c Fn(z)k Fl(\012)c Fp(1.)41 b(Hence)27 b Fn(z)32 b Fl(2)c Fn(R)h Fp(=)f(gr)o(\()p Fn(A)p Fp(\))2486 2887 y Ft(coA)2604 2896 y Fc(0)2643 2923 y Fp(,)f(\001)2778 2938 y Ft(R)2836 2923 y Fp(\()p Fn(z)t Fp(\))h(=)f(1)8 b Fl(\012)g Fn(z)k Fp(+)c Fn(z)k Fl(\012)c Fp(1)328 3044 y(and)32 b Fn(\016)t Fp(\()p Fn(z)t Fp(\))d(=)e Fn(g)e Fl(\012)d Fn(z)t Fp(.)44 b(Here,)33 b(\001)1450 3059 y Ft(R)1540 3044 y Fp(and)g Fn(\016)j Fp(are)c(the)h(com)m (ultiplication)27 b(and)33 b(the)g(coaction)328 3164 y(of)g(the)h(Y)-8 b(etter-Drinfeld)32 b(Hopf)h(algebra)g Fn(R)h Fp(o)m(v)m(er)h Fn(k)s(G)p Fp(\()p Fn(A)p Fp(\).)46 b(Since)34 b(gr\()p Fn(s)p Fp(\)\()p Fn(z)t Fp(\))c(=)f Fl(\000)p Fn(g)3395 3128 y Fj(\000)p Fu(1)3489 3164 y Fn(z)t Fp(,)328 3284 y(w)m(e)34 b(compute)e(\(gr\()p Fn(s)p Fp(\)\))1152 3248 y Fu(2)1191 3284 y Fp(\()p Fn(z)t Fp(\))c(=)g Fn(g)1499 3248 y Fj(\000)p Fu(1)1592 3284 y Fn(z)t(g)t Fp(.)474 3405 y(Th)m(us)42 b Fn(z)i Fl(2)c Fn(P)14 b Fp(\()p Fn(R)q Fp(\))40 b(is)f Fn(g)t Fp(-homogeneous)f(and)i (the)h(action)d(of)i Fn(g)j Fp(on)c Fn(z)45 b Fp(is)39 b(giv)m(en)h(b)m(y)328 3525 y Fn(g)25 b Fl(\001)d Fn(z)33 b Fp(=)27 b Fn(g)t(z)t(g)782 3489 y Fj(\000)p Fu(1)903 3525 y Fp(=)h Fn(q)t(z)37 b Fp(with)32 b Fn(q)g Fp(=)27 b Fn(\032)1586 3489 y Fj(\000)p Fu(1)1714 3525 y Fp(a)32 b(ro)s(ot)f(of)i(unit)m(y)f(of)g(order)h Fn(N)10 b Fp(.)474 3646 y(Since)36 b(gr\()p Fn(A)p Fp(\))g(is)f(\014nite-dimensional,)e Fn(z)38 b Fl(2)c Fp(gr)o(\()p Fn(A)p Fp(\)\(1\))i(is)f(nilp)s(oten)m (t.)52 b(Hence)37 b Fn(z)3409 3609 y Ft(n)3490 3646 y Fp(=)328 3766 y(0)p Fn(;)17 b(z)470 3730 y Ft(n)p Fj(\000)p Fu(1)635 3766 y Fl(6)p Fp(=)28 b(0)f(for)g(some)g Fn(n)h Fl(\025)g Fp(2.)41 b(Then)29 b(1)p Fn(;)17 b(z)t(;)g(:)g(:)g(:)f(;)h(z) 2166 3730 y Ft(n)p Fj(\000)p Fu(1)2331 3766 y Fp(are)27 b(linearly)f(indep)s(enden)m(t,)k(and)328 3886 y(b)m(y)j(the)g Fn(q)t Fp(-binomial)c(form)m(ula,)834 4127 y(0)f(=)f(\001)1095 4142 y Ft(R)1153 4127 y Fp(\()p Fn(z)1240 4086 y Ft(n)1288 4127 y Fp(\))g(=)h(\()p Fn(z)f Fl(\012)22 b Fp(1)g(+)g(1)g Fl(\012)h Fn(z)t Fp(\))2093 4086 y Ft(n)2168 4127 y Fp(=)2272 4019 y Ft(n)p Fj(\000)p Fu(1)2278 4044 y Fg(X)2282 4226 y Ft({)p Fu(=1)2421 3981 y Fg( )2495 4066 y Fn(n)2509 4187 y({)2562 3981 y Fg(!)2604 4231 y Ft(q)2659 4127 y Fn(z)2708 4086 y Ft({)2758 4127 y Fl(\012)g Fn(z)2907 4086 y Ft(n)p Fj(\000)p Ft({)3032 4127 y Fn(:)328 4456 y Fp(Hence)618 4310 y Fg( )692 4395 y Fn(n)706 4516 y({)759 4310 y Fg(!)801 4560 y Ft(q)868 4456 y Fp(=)28 b(0)33 b(for)f(all)f(1)d Fl(\024)h Fn({)g Fl(\024)g Fn(n)22 b Fl(\000)h Fp(1.)45 b(Then)34 b(w)m(e)g(kno)m(w)g(from)e([11)o(,)i (Corollary)d(2])328 4650 y(that)h Fn(N)43 b Fp(divides)33 b(n,)g(and)f(w)m(e)i(obtain)804 4891 y Fm(tr)q Fp(\(gr\()p Fn(s)1082 4850 y Fu(2)p Ft(m)1184 4891 y Fp(\))p Fl(j)1250 4907 y Ft(k)r Fu([)p Ft(z)s Fu(])1367 4891 y Fp(\))27 b(=)1536 4784 y Ft(n)p Fj(\000)p Fu(1)1542 4808 y Fg(X)1546 4990 y Ft({)p Fu(=0)1686 4891 y Fn(q)1733 4850 y Ft(m{)1850 4891 y Fp(=)g(0)33 b(since)g Fn(q)2321 4850 y Ft(m)2415 4891 y Fl(6)p Fp(=)27 b(1)33 b(and)f Fn(q)2836 4850 y Ft(n)2911 4891 y Fp(=)27 b(1)p Fn(:)1898 5214 y Fp(22)p eop %%Page: 23 23 23 22 bop 328 631 a Fp(Since)33 b Fn(k)s Fp([)p Fn(z)t Fp(])g(is)f(a)h(lo)s(cal)d(subalgebra)i(of)g Fn(R)q Fp(,)h Fm(tr)r Fp(\(gr)o(\()p Fn(s)2195 595 y Fu(2)p Ft(m)2297 631 y Fp(\))p Fl(j)2363 646 y Ft(R)2420 631 y Fp(\))28 b(=)g(0)k(b)m(y)h(Lemma)f(9.)43 b Fd(2)328 905 y Ff(Theorem)37 b(4)49 b Fk(L)-5 b(et)31 b Fn(A)f Fk(b)-5 b(e)31 b(a)f (\014nite-dimensional)e(Hopf)j(algebr)-5 b(a)30 b(with)g(antip)-5 b(o)g(de)30 b Fn(s)g Fk(over)328 1026 y(the)37 b(\014eld)g Fn(k)s Fk(.)52 b(Assume)38 b(that)g Fn(s)1455 989 y Fu(2)1531 1026 y Fk(is)g(semisimple)d(and)i(the)g(c)-5 b(or)g(adic)g(al)37 b(of)g Fn(A)3093 989 y Fj(\003)3170 1026 y Fk(is)g(c)-5 b(o)g(c)g(om-)328 1146 y(mutative.)45 b(Then)34 b Fn(s)1069 1110 y Fu(2)1143 1146 y Fk(satis\014es)g(the)h(str)-5 b(ong)35 b(vanishing)e(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition.)328 1420 y Fe(Pr)n(oof:)60 b Fp(W)-8 b(e)38 b(ma)m(y)g(assume)g(that)f Fn(k)k Fp(is)c(algebraically)d(closed.)59 b(Then)39 b(the)f(theorem)328 1541 y(follo)m(ws)31 b(b)m(y)j(dualit)m(y)d(from)h(Theorem)h(3,)f (Theorem)h(2)f(and)h(Prop)s(osition)e(4.)43 b Fd(2)328 1873 y Fo(5)161 b(A)50 b(Generalization)h(of)f(a)g(Basic)h(T)-13 b(race)49 b(F)-13 b(orm)l(ula)328 2092 y Fp(W)-8 b(e)46 b(w)m(an)m(t)h(to)f(generalize)f(an)h(imp)s(ortan)m(t)e(trace)j(form)m (ula)d(concerning)i(the)h(square)328 2212 y(of)38 b(the)g(an)m(tip)s(o) s(de)f(in)h([6,)h(Equation)f(\(6\)])g(to)g(higher)f(ev)m(en)j(p)s(o)m (w)m(ers;)j(indeed)38 b(to)g(Hopf)328 2333 y(algebra)j(automorphisms)f (of)h Fn(A)p Fp(.)72 b(It)42 b(turns)h(out)e(that)h(the)h(elemen)m(ts)f Fn(\027)3081 2348 y Ft(A)3138 2333 y Fp(\()p Fn(t)p Fp(\))g(are)g(an) 328 2453 y(ingredien)m(t)32 b(of)g(this)g(generalization.)41 b(Our)32 b(pro)s(of)g(follo)m(ws)f(the)i(metho)s(d)f(giv)m(en)g(in)g ([17,)328 2574 y(Theorem)h(3.13],)f(and)h(w)m(e)g(form)m(ulate)e(the)i (result)g(dually)e(to)h([5,)h(6].)474 2694 y(Recall)40 b(that)g(a)h(\014nite-dimensional)d(algebra)i Fn(A)h Fp(o)m(v)m(er)h(a)e(\014eld)h Fn(k)j Fp(is)d(a)f Fk(F)-7 b(r)i(ob)g(enius)328 2814 y(algebr)g(a)28 b Fp(if)f(there)i(is)f(a)g (linear)f(function)h Fn(')g Fp(:)g Fn(A)g Fl(!)f Fn(k)s Fp(,)i(a)g(natural)e(n)m(um)m(b)s(er)i Fn(n)p Fp(,)g(and)g(there)328 2935 y(are)k(elemen)m(ts)f Fn(a)940 2950 y Ft({)968 2935 y Fn(;)17 b(b)1053 2950 y Ft({)1108 2935 y Fl(2)28 b Fn(A)33 b Fp(for)f(all)e(1)e Fl(\024)g Fn({)g Fl(\024)g Fn(n)33 b Fp(suc)m(h)h(that)1317 3204 y Fn(x)28 b Fp(=)1542 3096 y Ft(n)1503 3121 y Fg(X)1507 3302 y Ft({)p Fu(=1)1640 3204 y Fn(')p Fp(\()p Fn(xa)1848 3219 y Ft({)1876 3204 y Fp(\))p Fn(b)1955 3219 y Ft({)2015 3204 y Fp(for)k(all)f Fn(x)d Fl(2)g Fn(A:)864 b Fp(\(9\))328 3485 y(W)-8 b(e)39 b(sa)m(y)g(that)f Fn(')h Fp(is)f(a)g Fk(F)-7 b(r)i(ob)g(enius)39 b(homomorphism)d Fp(and,)k(with)e(an)g(abuse)i(of)e(termi-)328 3605 y(nology)-8 b(,)32 b(that)g Fl(f)p Fn(a)971 3620 y Ft({)998 3605 y Fl(g)p Fn(;)17 b Fl(f)p Fn(b)1183 3620 y Ft({)1210 3605 y Fl(g)33 b Fp(are)f Fk(dual)j(b)-5 b(ases)34 b(of)e Fn(A)p Fp(.)474 3726 y(W)-8 b(e)49 b(note)g(that)f(if) f(\(9\))h(holds)g(then)h(the)f(linear)f(map)h Fn(A)54 b Fl(\000)-16 b(!)54 b Fn(A)3023 3690 y Fj(\003)3111 3726 y Fp(de\014ned)c(b)m(y)328 3846 y Fn(a)h Fl(7!)g Fn(')p Fl(\001)p Fn(a)46 b Fp(is)g(an)g(isomorphism)e(of)i(righ)m(t)g Fn(A)p Fp(-mo)s(dules.)83 b(In)47 b(this)f(case)h Fn(A)52 b Fl(\000)-17 b(!)51 b Fn(A)3526 3810 y Fj(\003)328 3967 y Fp(de\014ned)46 b(b)m(y)g Fn(a)j Fl(7!)f Fn(a)p Fl(\001)p Fn(')d Fp(is)f(an)h(isomorphism)d(of)j(left)f Fn(A)p Fp(-mo)s(dules)f(whic)m(h)j(is)e(also)g(a)328 4087 y(consequence)36 b(of)1317 4259 y Fn(x)28 b Fp(=)1542 4151 y Ft(n)1503 4176 y Fg(X)1507 4358 y Ft({)p Fu(=1)1640 4259 y Fn(a)1691 4274 y Ft({)1718 4259 y Fn(')p Fp(\()p Fn(b)1861 4274 y Ft({)1889 4259 y Fn(x)p Fp(\))33 b(for)f(all)f Fn(x)d Fl(2)g Fn(A:)815 b Fp(\(10\))328 4502 y(Observ)m(e)35 b(that)e(\(9\))g(and)g(\(10\))g(are)g(logically)c(equiv)-5 b(alen)m(t.)45 b(T)-8 b(o)34 b(see)g(this)f(note)g(that)g(\(9\))328 4622 y(holds)f(if)g(and)g(only)g(if)1306 4891 y(\()p Fn(y)t Fl(\001)o Fn(')p Fp(\)\()p Fn(x)p Fp(\))c(=)f(\()p Fn(y)t Fl(\001)o Fn(')p Fp(\)\()2082 4784 y Ft(n)2044 4808 y Fg(X)2047 4990 y Ft({)p Fu(=1)2180 4891 y Fn(')p Fp(\()p Fn(xa)2388 4906 y Ft({)2416 4891 y Fp(\))p Fn(b)2495 4906 y Ft({)2523 4891 y Fp(\))p Fn(;)1898 5214 y Fp(23)p eop %%Page: 24 24 24 23 bop 328 631 a Fp(or)1306 784 y(\()p Fn(')p Fl(\001)o Fn(x)p Fp(\)\()p Fn(y)t Fp(\))28 b(=)f(\()p Fn(')p Fl(\001)p Fn(x)p Fp(\)\()2086 676 y Ft(n)2048 701 y Fg(X)2051 882 y Ft({)p Fu(=1)2184 784 y Fn(a)2235 799 y Ft({)2263 784 y Fn(')p Fp(\()p Fn(b)2406 799 y Ft({)2434 784 y Fn(y)t Fp(\)\))p Fn(;)328 1025 y Fp(for)32 b(all)e Fn(x;)17 b(y)31 b Fl(2)d Fn(A)p Fp(.)44 b(The)34 b(latter)d(holds)h(if)g(and)h (only)f(if)f(\(10\))h(holds.)328 1248 y Ff(Lemma)37 b(10)49 b Fk(L)-5 b(et)48 b Fn(A)g Fk(b)-5 b(e)47 b(a)g(F)-7 b(r)i(ob)g(enius)47 b(algebr)-5 b(a)46 b(over)h(the)h(\014eld)f Fn(k)k Fk(with)c(F)-7 b(r)i(ob)g(enius)328 1369 y(homomorphism)35 b Fn(')i Fk(and)f(dual)h(b)-5 b(ases)36 b Fl(f)p Fn(a)1872 1384 y Ft({)1899 1369 y Fl(g)p Fn(;)17 b Fl(f)p Fn(b)2084 1384 y Ft({)2112 1369 y Fl(g)p Fk(.)50 b(L)-5 b(et)38 b Fn(\013)32 b Fl(2)g Fn(k)40 b Fk(and)c Fn(v)g Fl(2)c Fn(A)37 b Fk(such)f(that)328 1489 y Fn(v)379 1453 y Fu(2)446 1489 y Fp(=)27 b Fn(\013)q(v)t Fk(,)35 b(and)f(let)h Fn(t)28 b Fp(:)f Fn(v)t(A)h Fl(!)f Fn(v)t(A)35 b Fk(b)-5 b(e)35 b(a)f(line)-5 b(ar)34 b(map.)45 b(Then)1095 1756 y Fn(\013)q Fm(tr)q Fp(\()p Fn(t)p Fp(\))28 b(=)1508 1648 y Ft(n)1470 1673 y Fg(X)1473 1855 y Ft({)p Fu(=1)1606 1756 y Fn(')p Fp(\()p Fn(t)p Fp(\()p Fn(v)t(b)1873 1771 y Ft({)1901 1756 y Fp(\))p Fn(a)1990 1771 y Ft({)2017 1756 y Fp(\))g(=)2225 1648 y Ft(n)2186 1673 y Fg(X)2190 1855 y Ft({)p Fu(=1)2323 1756 y Fn(')p Fp(\()p Fn(b)2466 1771 y Ft({)2493 1756 y Fn(t)p Fp(\()p Fn(v)t(a)2668 1771 y Ft({)2696 1756 y Fp(\)\))p Fn(:)328 2091 y Fe(Pr)n(oof:)62 b Fp(F)-8 b(or)38 b(a)g(\014nite-dimensional)d(v)m(ector)k(space)h Fn(V)60 b Fp(o)m(v)m(er)39 b Fn(k)j Fp(w)m(e)d(iden)m(tify)f Fn(V)3370 2055 y Fj(\003)3410 2091 y Fl(\012)p Fn(V)328 2212 y Fp(with)32 b(End)17 b(\()p Fn(V)22 b Fp(\))32 b(b)m(y)h(\()p Fn(v)1152 2175 y Fj(\003)1191 2212 y Fl(\012)q Fn(v)t Fp(\)\()p Fn(u)p Fp(\))26 b(=)i Fn(v)1671 2175 y Fj(\003)1710 2212 y Fp(\()p Fn(u)p Fp(\))p Fn(v)35 b Fp(for)d(all)e Fn(v)2259 2175 y Fj(\003)2326 2212 y Fl(2)e Fn(V)2499 2175 y Fj(\003)2571 2212 y Fp(and)k Fn(v)t(;)17 b(u)27 b Fl(2)h Fn(V)21 b Fp(.)44 b(It)32 b(is)g(easy)328 2332 y(to)g(see)i(that)e Fm(tr)r Fp(\()p Fn(v)975 2296 y Fj(\003)1014 2332 y Fl(\012)q Fn(v)t Fp(\))27 b(=)g Fn(v)1362 2296 y Fj(\003)1402 2332 y Fp(\()p Fn(v)t Fp(\).)474 2452 y(F)-8 b(or)32 b(all)f Fn(x)d Fl(2)g Fn(A)k Fp(w)m(e)i(ha)m(v)m(e)1012 2704 y Fn(\013)q(v)t(x)28 b Fp(=)f Fn(v)1363 2663 y Fu(2)1402 2704 y Fn(x)i Fp(=)1628 2596 y Ft(n)1589 2621 y Fg(X)1592 2802 y Ft({)p Fu(=1)1726 2704 y Fn(')p Fp(\()p Fn(v)t(xa)1985 2719 y Ft({)2012 2704 y Fp(\))p Fn(v)t(b)2142 2719 y Ft({)2197 2704 y Fp(=)2339 2596 y Ft(n)2301 2621 y Fg(X)2304 2802 y Ft({)p Fu(=1)2437 2704 y Fn(v)t(a)2539 2719 y Ft({)2567 2704 y Fn(')p Fp(\()p Fn(b)2710 2719 y Ft({)2737 2704 y Fn(v)t(x)p Fp(\))328 2989 y(b)m(y)k(\(9\))g(and)f(\(10\))g(resp)s(ectiv)m(ely)-8 b(.)45 b(Th)m(us)1152 3261 y Fn(\013)q(t)27 b Fp(=)1420 3153 y Ft(n)1381 3178 y Fg(X)1384 3360 y Ft({)p Fu(=1)1518 3261 y Fn(a)1569 3276 y Ft({)1596 3261 y Fl(\001)p Fn(')p Fl(\012)p Fn(t)p Fp(\()p Fn(v)t(b)1930 3276 y Ft({)1958 3261 y Fp(\))g(=)2166 3153 y Ft(n)2127 3178 y Fg(X)2130 3360 y Ft({)p Fu(=1)2264 3261 y Fn(')p Fl(\001)o Fn(b)2396 3276 y Ft({)2424 3261 y Fl(\012)p Fn(t)p Fp(\()p Fn(v)t(a)2676 3276 y Ft({)2704 3261 y Fp(\))328 3540 y(from)k(whic)m(h)i(the)g(trace) g(form)m(ulas)e(follo)m(w.)42 b Fd(2)474 3709 y Fp(Let)e Fn(A)f Fp(b)s(e)g(a)g(\014nite-dimensional)d(Hopf)j(algebra)f(with)g (an)m(tip)s(o)s(de)h Fn(s)p Fp(,)h(let)f(\003)f Fl(2)i Fn(A)328 3829 y Fp(b)s(e)h(a)g(left)g(in)m(tegral)e(of)i Fn(A)g Fp(and)g(let)g Fn(\025)h Fl(2)h Fn(A)1965 3793 y Fj(\003)2046 3829 y Fp(b)s(e)e(a)g(righ)m(t)f(in)m(tegral)g(of)h Fn(A)3078 3793 y Fj(\003)3158 3829 y Fp(suc)m(h)i(that)328 3950 y Fn(\025)p Fp(\(\003\))27 b(=)h(1.)43 b(Then)1523 4070 y Fn(x)28 b Fp(=)g Fn(\025)p Fp(\()p Fn(x)p Fp(\003)1928 4086 y Fu(\(1\))2022 4070 y Fp(\))p Fn(s)p Fp(\(\003)2212 4086 y Fu(\(2\))2306 4070 y Fp(\))p Fn(:)1021 b Fp(\(11\))328 4242 y(W)-8 b(e)37 b(recall)f(the)h(standard)h(pro)s(of:)52 b(F)-8 b(or)36 b(all)f Fn(x)g Fl(2)h Fn(A)h Fp(w)m(e)h(compute)g Fn(\025)p Fp(\()p Fn(x)p Fp(\003)3039 4258 y Fu(\(1\))3133 4242 y Fp(\))p Fn(s)p Fp(\(\003)3323 4258 y Fu(\(2\))3417 4242 y Fp(\))d(=)328 4363 y Fn(\025)p Fp(\()p Fn(x)478 4378 y Fu(\(1\))573 4363 y Fp(\003)641 4378 y Fu(\(1\))735 4363 y Fp(\))p Fn(x)828 4378 y Fu(\(2\))922 4363 y Fp(\003)990 4378 y Fu(\(2\))1084 4363 y Fn(s)p Fp(\(\003)1236 4378 y Fu(\(3\))1330 4363 y Fp(\))k(=)f Fn(\025)p Fp(\()p Fn(x)1671 4378 y Fu(\(1\))1766 4363 y Fp(\003\))p Fn(x)1927 4378 y Fu(\(2\))2060 4363 y Fp(=)g Fn(x;)h Fp(where)i(the)e(\014rst)g (equalit)m(y)g(holds)328 4483 y(since)26 b Fn(\025)g Fp(is)f(a)h(righ)m(t)f(in)m(tegral)g(and)h(the)g(last)f(equalit)m(y)h (follo)m(ws)e(since)j(\003)e(is)h(a)f(left)g(in)m(tegral)328 4604 y(and)i Fn(\025)p Fp(\(\003\))g(=)h(1.)41 b(Th)m(us)28 b Fn(A)f Fp(is)g(a)g(F)-8 b(rob)s(enius)26 b(algebra)g(with)g(F)-8 b(rob)s(enius)27 b(homomorphism)328 4724 y Fn(\025)32 b Fp(and)h(dual)f(bases)i Fl(f)p Fp(\003)1198 4739 y Fu(\(1\))1291 4724 y Fl(g)p Fn(;)17 b Fl(f)p Fn(s)p Fp(\(\003)1587 4739 y Fu(\(2\))1681 4724 y Fp(\))p Fl(g)p Fn(:)474 4844 y Fp(Let)52 b Fn(\037)729 4859 y Ft(A)845 4844 y Fl(2)60 b Fn(A)1044 4808 y Fj(\003)1135 4844 y Fp(b)s(e)51 b(the)h(c)m (haracter)g(of)f(the)g(regular)f(represen)m(tation,)57 b(that)51 b(is)328 4965 y Fn(\037)389 4980 y Ft(A)446 4965 y Fp(\()p Fn(x)p Fp(\))h(=)g Fm(tr)q Fp(\()p Fn(l)r Fp(\()p Fn(x)p Fp(\)\))47 b(for)g(all)d Fn(x)52 b Fl(2)g Fn(A)p Fp(.)86 b(Since)47 b Fn(A)g Fp(is)f(a)g(F)-8 b(rob)s(enius)47 b(algebra)e(Lemma)1898 5214 y(24)p eop %%Page: 25 25 25 24 bop 328 631 a Fp(10)43 b(implies)d(that)j Fn(\037)1093 646 y Ft(A)1150 631 y Fp(\()p Fn(x)p Fp(\))j(=)f Fm(tr)r Fp(\()p Fn(r)s Fp(\()p Fn(x)p Fp(\)\))e(for)g(all)e Fn(x)46 b Fl(2)g Fn(A)d Fp(as)g(w)m(ell.)74 b(Using)43 b(this)g(latter)328 751 y(form)m(ulation)30 b(of)i Fn(\037)1026 766 y Ft(A)1115 751 y Fp(w)m(e)i(see)g(that)1235 971 y Fn(\037)1296 986 y Ft(A)1353 971 y Fp(\()p Fn(x)p Fp(\))28 b(=)g Fn(\025)p Fp(\()p Fn(\027)1759 986 y Ft(A)1816 971 y Fp(\(1)1903 986 y Ft(A)1960 971 y Fp(\))p Fn(x)p Fp(\))33 b(for)f(all)e Fn(x)e Fl(2)g Fn(A)734 b Fp(\(12\))328 1191 y(b)m(y)33 b(part)g(a\))f(of)g(Lemma)f(3,)i(where)g(\003)g(is)f(used)i(to)e (de\014ne)i Fn(\027)2514 1206 y Ft(A)2571 1191 y Fp(\(1)2658 1206 y Ft(A)2715 1191 y Fp(\).)474 1312 y(The)46 b(isomorphism)d Fn(A)31 b Fl(\012)g Fn(A)49 b Fl(!)g Fn(A)31 b Fl(\012)g Fn(A;)17 b(x)31 b Fl(\012)g Fn(y)52 b Fl(7!)d Fn(x)2580 1327 y Fu(\(1\))2705 1312 y Fl(\012)32 b Fn(x)2869 1327 y Fu(\(2\))2963 1312 y Fn(y)t(;)45 b Fp(implies)d(the)328 1432 y(follo)m(wing)30 b(w)m(ell-kno)m(wn)j(iden)m(tit)m(y)f(in)g Fn(A)1794 1396 y Fj(\003)1575 1652 y Fn(\037)1636 1611 y Fu(2)1636 1677 y Ft(A)1721 1652 y Fp(=)c(\(Dim)14 b Fn(A)p Fp(\))p Fn(\037)2234 1667 y Ft(A)2291 1652 y Fn(:)1074 b Fp(\(13\))474 1872 y(No)m(w)33 b Fn(\037)757 1887 y Ft(A)837 1872 y Fl(\016)22 b Fn(t)28 b Fp(=)f Fn(\037)1136 1887 y Ft(A)1226 1872 y Fp(for)32 b(an)m(y)h(algebra)e(automorphism)g Fn(t)i Fp(of)f Fn(A)p Fp(,)g(or)h(equiv)-5 b(alen)m(tly)1676 2092 y Fn(T)14 b Fp(\()p Fn(\037)1846 2107 y Ft(A)1903 2092 y Fp(\))28 b(=)f Fn(\037)2133 2107 y Ft(A)2190 2092 y Fn(;)1175 b Fp(\(14\))328 2312 y(where)34 b Fn(T)41 b Fp(=)28 b Fn(t)847 2276 y Fj(\003)886 2312 y Fp(.)44 b(In)33 b(particular)e Fn(S)1595 2276 y Fu(2)1634 2312 y Fp(\()p Fn(\037)1733 2327 y Ft(A)1790 2312 y Fp(\))c(=)h Fn(\037)2020 2327 y Ft(A)2110 2312 y Fp(where)33 b Fn(S)39 b Fp(is)32 b(the)h(an)m(tip)s(o)s(de)f(of)g Fn(A)3344 2276 y Fj(\003)3383 2312 y Fp(.)474 2433 y(By)h(the)g(same)g(argumen)m (t,)f(for)g(an)m(y)h(grouplik)m(e)f(elemen)m(t)h Fn(\036)27 b Fl(2)h Fn(A)2872 2396 y Fj(\003)2912 2433 y Fp(,)1721 2653 y Fn(\037)1782 2668 y Ft(A)1867 2653 y Fp(=)f Fn(\037)2031 2668 y Ft(A)2088 2653 y Fn(\036;)1219 b Fp(\(15\))328 2873 y(since)38 b(for)g(all)e Fn(a)h Fl(2)h Fn(A)p Fp(,)i(\()p Fn(\037)1299 2888 y Ft(A)1356 2873 y Fn(\036)p Fp(\)\()p Fn(a)p Fp(\))d(=)g Fn(\037)1790 2888 y Ft(A)1847 2873 y Fp(\()p Fn(a)1936 2888 y Fu(\(1\))2030 2873 y Fp(\))p Fn(\036)p Fp(\()p Fn(a)2215 2888 y Fu(\(2\))2309 2873 y Fp(\))g(=)g Fn(\037)2558 2888 y Ft(A)2616 2873 y Fp(\()p Fn(a)2705 2888 y Fu(\(1\))2799 2873 y Fn(\036)p Fp(\()p Fn(a)2946 2888 y Fu(\(2\))3040 2873 y Fp(\)\),)j(and)e Fn(a)f Fl(7!)328 2993 y Fn(a)379 3008 y Fu(\(1\))473 2993 y Fn(\036)p Fp(\()p Fn(a)620 3008 y Fu(\(2\))715 2993 y Fp(\))32 b(is)g(an)h(algebra)e(automorphism)g(of)h Fn(A)p Fp(.)474 3113 y(F)-8 b(or)25 b(clarit)m(y)g(w)m(e)j(will)23 b(use)k(the)g(notation)d Fl(h)p Fn(p;)17 b(x)p Fl(i)28 b Fp(=)f Fn(p)p Fp(\()p 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Fp(\()p Fn(T)14 b Fp(\))p Fn(;)j(\027)2468 3793 y Ft(A)2524 3778 y Fp(\(1)2611 3793 y Ft(A)2668 3778 y Fp(\))p Fl(i)p Fn(;)328 3998 y Fk(wher)-5 b(e)34 b Fp(\003)h Fk(is)f(use)-5 b(d)35 b(to)g(de\014ne)f Fn(\027)1476 4013 y Ft(A)1533 3998 y Fp(\(1)1620 4013 y Ft(A)1677 3998 y Fp(\))h Fk(and)f Fn(S)2005 3961 y Fj(\000)p Fu(1)2099 3998 y Fp(\()p Fn(\025)p Fp(\))h Fk(to)g(de\014ne)f Fn(\027)2716 4013 y Ft(A)2769 3994 y Fi(\003)2809 3998 y Fp(\()p Fn(T)14 b Fp(\))p Fn(:)328 4251 y Fe(Pr)n(oof:)50 b Fp(W)-8 b(e)33 b(\014rst)g(note)f (that)1371 4471 y Fn(S)6 b Fp(\()p Fn(\027)1523 4486 y Ft(A)1576 4467 y Fi(\003)1616 4471 y Fp(\()p Fn(T)14 b Fp(\)\))27 b(=)h Fn(T)14 b Fp(\()p Fn(\025)2098 4486 y Fu(\(2\))2192 4471 y Fp(\))p Fn(S)6 b Fp(\()p Fn(\025)2391 4486 y Fu(\(1\))2485 4471 y Fp(\))869 b(\(16\))328 4703 y(since)44 b Fn(S)50 b Fp(and)43 b Fn(T)58 b Fp(comm)m(ute.)76 b(It)44 b(follo)m(ws)f(that)2176 4671 y Fg(e)2165 4703 y Fp(\003)k(:)f Fn(A)2426 4667 y Fj(\003)2513 4703 y Fl(!)g Fn(k)h Fp(de\014ned)e(b)m(y)3261 4671 y Fg(e)3251 4703 y Fp(\003)o(\()p Fn(p)p Fp(\))i(=)328 4823 y Fn(p)p Fp(\(\003\))c(for)h(all)e Fn(p)47 b Fl(2)h Fn(A)1155 4787 y Fj(\003)1238 4823 y Fp(is)c(a)g(F)-8 b(rob)s(enius)43 b(homomorphism)e(of)j Fn(A)2797 4787 y Fj(\003)2881 4823 y Fp(with)f(dual)h(bases)328 4944 y Fl(f)p Fn(S)6 b Fp(\()p Fn(\025)539 4959 y Fu(\(1\))633 4944 y Fp(\))p Fl(g)p Fn(;)17 b Fl(f)p Fn(\025)872 4959 y Fu(\(2\))965 4944 y Fl(g)p Fp(;)42 b(v)m(erify)d(\(10\))g(using)f(the)h(fact)g(that)2437 4911 y Fg(e)2426 4944 y Fp(\003)f(is)h(a)f(left)g(in)m(tegral)g(for)g Fn(A)3491 4908 y Fj(\003\003)1898 5214 y Fp(25)p eop %%Page: 26 26 26 25 bop 328 631 a Fp(and)28 b(that)h Fn(\025)f Fp(is)g(a)g(righ)m(t)g (in)m(tegral)e(for)i Fn(A)1777 595 y Fj(\003)1817 631 y Fp(.)42 b(Using)28 b(\(14\))g(w)m(e)h(observ)m(e)h(that)f Fn(T)42 b Fp(de\014nes)30 b(an)328 751 y(endomorphism)d(of)g Fn(\037)1147 766 y Ft(A)1204 751 y Fn(A)1277 715 y Fj(\003)1317 751 y Fp(.)42 b(Hence)29 b(w)m(e)g(get)f(from)e(Lemma)h(10)g(and)h (\(13\))f(with)h Fn(v)j Fp(=)d Fn(\037)3509 766 y Ft(A)590 919 y Fp(\(Dim)14 b Fn(A)p Fp(\))j Fm(tr)r Fp(\()p Fn(T)d Fl(j)1162 934 y Ft(\037)1206 945 y Fh(A)1255 934 y Ft(A)1308 915 y Fi(\003)1348 919 y Fp(\)\))28 b(=)1621 887 y Fg(e)1610 919 y Fp(\003\()p Fn(T)14 b Fp(\()p Fn(\037)1886 934 y Ft(A)1943 919 y Fn(\025)2000 935 y Fu(\(2\))2094 919 y Fp(\))p Fn(S)6 b Fp(\()p Fn(\025)2293 935 y Fu(\(1\))2387 919 y Fp(\)\))1452 1040 y(=)82 b Fl(h)p Fn(\037)1710 1055 y Ft(A)1767 1040 y Fn(T)14 b Fp(\()p Fn(\025)1933 1055 y Fu(\(2\))2027 1040 y Fp(\))p Fn(S)6 b Fp(\()p Fn(\025)2226 1055 y Fu(\(1\))2320 1040 y Fp(\))p Fn(;)17 b Fp(\003)p Fl(i)1452 1160 y Fp(=)82 b Fl(h)p Fn(\037)1710 1175 y Ft(A)1767 1160 y Fn(S)6 b Fp(\()p Fn(\027)1919 1175 y Ft(A)1972 1156 y Fi(\003)2012 1160 y Fp(\()p Fn(T)14 b Fp(\)\))p Fn(;)j Fp(\003)p Fl(i)647 b Fp(b)m(y)33 b(\(16\))1452 1280 y(=)82 b Fn(\037)1671 1295 y Ft(A)1729 1280 y Fp(\(\003)1835 1296 y Fu(\(1\))1928 1280 y 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b(\(11\))p Fn(:)328 1772 y Fd(2)474 1926 y Fp(If)e Fn(T)41 b Fp(=)28 b Fn(S)838 1890 y Fu(2)908 1926 y Fp(then)j(the)g(trace)g(form)m(ula)e(of)h (Theorem)h(5)g(is)f(the)h(trace)g(form)m(ula)e(in)h([6,)328 2047 y(Equation)i(\(6\)])h(since)844 2209 y Fl(h)p Fn(\027)931 2224 y Ft(A)984 2205 y Fi(\003)1024 2209 y Fp(\()p Fn(S)1128 2168 y Fu(2)1168 2209 y Fp(\))p Fn(;)17 b(\027)1298 2224 y Ft(A)1354 2209 y Fp(\(1)1441 2224 y Ft(A)1498 2209 y Fp(\))p Fl(i)27 b Fp(=)h Fl(h)p Fn(\025)p Fp(\(1\))p Fn(";)17 b(S)6 b Fp(\(\003)2189 2225 y Fu(\(2\))2281 2209 y Fp(\)\003)2387 2225 y Fu(\(1\))2481 2209 y Fl(i)28 b Fp(=)f Fn(\025)p Fp(\(1\))p Fn(")p Fp(\(\003\))p Fn(:)474 2372 y Fp(F)-8 b(or)34 b(a)f(\014nite-dimensional)e(Hopf)j(algebra)f Fn(A)h Fp(o)m(v)m(er)h(the)g(\014eld)f Fn(k)j Fp(let)c Fn(x)3089 2387 y Ft(A)3177 2372 y Fp(=)d Fn(\037)3344 2387 y Ft(A)3397 2368 y Fi(\003\003)3499 2372 y Fl(2)328 2492 y Fn(A)401 2456 y Fj(\003\003)510 2492 y Fp(=)j Fn(A)p Fp(.)55 b(Th)m(us)37 b Fn(x)1079 2507 y Ft(A)1173 2492 y Fp(is)e(determined)h(b)m(y)h Fn(p)p Fp(\()p Fn(x)2068 2507 y Ft(A)2126 2492 y Fp(\))d(=)f Fm(tr)r Fp(\()p Fn(R)q Fp(\()p Fn(p)p Fp(\)\))j(for)f(all)f Fn(p)g Fl(2)g Fn(A)3236 2456 y Fj(\003)3276 2492 y Fp(,)j(or)f(b)m(y)328 2613 y Fn(p)p Fp(\()p Fn(x)470 2628 y Ft(A)527 2613 y Fp(\))28 b(=)g Fm(tr)r Fp(\()p Fn(L)p Fp(\()p Fn(p)p Fp(\)\))33 b(for)f(all)f Fn(p)d Fl(2)h Fn(A)1597 2577 y Fj(\003)1636 2613 y Fp(.)45 b(Supp)s(ose)34 b(that)e Fn(B)38 b Fp(is)33 b(a)f(sub-Hopf)h(algebra)f(of)h Fn(A)p Fp(.)328 2733 y(Then)k Fn(A)e Fp(is)h(a)f(free)h(left)f(\(or)g(righ)m(t\))g Fn(B)5 b Fp(-mo)s(dule)34 b(b)m(y)i([10,)g(Theorem)g(7].)53 b(Th)m(us)37 b(Dim)15 b Fn(B)328 2854 y Fp(divides)32 b(Dim)15 b Fn(A)32 b Fp(whic)m(h)i(means)e(that)h(Dim)14 b Fn(A=)p Fp(Dim)g Fn(B)38 b Fp(is)32 b(an)h(in)m(teger.)474 2974 y(W)-8 b(e)40 b(end)g(this)f(section)g(b)m(y)h(describing)f(the)g (relationship)f(b)s(et)m(w)m(een)j Fn(x)3130 2989 y Ft(A)3226 2974 y Fp(and)f Fn(x)3478 2989 y Ft(B)3539 2974 y Fp(,)328 3094 y(where)34 b Fn(B)j Fp(is)32 b(the)h(sub-Hopf)g(algebra)f(of)g Fn(A)g Fp(generated)i(b)m(y)f(the)g(coradical)e(of)h Fn(A)p Fp(.)328 3249 y Ff(Prop)s(osition)j(5)49 b Fk(L)-5 b(et)40 b Fn(A)g Fk(b)-5 b(e)39 b(a)g(\014nite-dimensional)e(Hopf)i (algebr)-5 b(a)39 b(over)g(the)g(\014eld)g Fn(k)328 3369 y Fk(and)g(supp)-5 b(ose)39 b(that)h Fn(B)k Fk(is)c(the)f(sub-Hopf)g (algebr)-5 b(a)39 b(of)g Fn(A)h Fk(gener)-5 b(ate)g(d)39 b(by)g(the)h(c)-5 b(or)g(adic)g(al)328 3489 y(of)35 b Fn(A)p Fk(.)44 b(Then)1535 3652 y Fn(x)1590 3667 y Ft(A)1675 3652 y Fp(=)1778 3531 y Fg(\022)1852 3585 y Fp(Dim)15 b Fn(A)p 1849 3629 279 4 v 1849 3721 a Fp(Dim)g Fn(B)2138 3531 y Fg(\023)2216 3652 y Fn(x)2271 3667 y Ft(B)2332 3652 y Fn(:)328 3878 y Fe(Pr)n(oof:)62 b Fp(By)38 b(Theorem)h(2)f(w)m (e)h(ha)m(v)m(e)g Fn(p)p Fp(\()p Fn(x)1917 3893 y Ft(A)1974 3878 y Fp(\))f(=)f Fm(tr)q Fp(\(gr\(1)2444 3893 y Ft(A)2501 3878 y Fp(\))p Fl(j)2567 3893 y Ft(R)2624 3878 y Fp(\))p Fn(p)p Fp(\()p Fn(x)2804 3893 y Ft(B)2865 3878 y Fp(\))h(for)f(all)g Fn(p)g Fl(2)g Fn(A)3499 3841 y Fj(\003)3539 3878 y Fp(.)328 3998 y(Since)32 b Fn(R)q Fl(\012)p Fn(B)h Fl(')28 b Fn(A)k Fp(as)g(v)m(ector)h(spaces)g(and)f(gr\(1)2123 4013 y Ft(A)2179 3998 y Fl(j)2207 4013 y Ft(R)2265 3998 y Fp(\))27 b(=)h(1)2483 4013 y Ft(R)2572 3998 y Fp(the)k(prop)s(osition)e(follo)m (ws.)328 4118 y Fd(2)328 4443 y Fo(6)161 b(Applications)37 b(to)e(Hopf)h(Algebra)g(Automorphisms)570 4625 y Fb(t)54 b Fo(and)f(F)-13 b(urther)53 b(Results)g(on)g Fb(\027)2441 4647 y Fn(A)2519 4625 y Fa(\()p Fb(t)p Fa(\))328 4844 y Fp(Supp)s(ose)36 b(that)f Fn(t)g Fp(is)f(a)h(Hopf)f(algebra)g (automorphism)f(of)h Fn(A)h Fp(whic)m(h)h(has)f(\014nite)g(order)328 4965 y(and)47 b(assume)g(that)g Fn(t)g Fp(satis\014es)g(the)h(strong)e (v)-5 b(anishing)46 b(trace)h(condition.)85 b(In)47 b(this)1898 5214 y(26)p eop %%Page: 27 27 27 26 bop 328 631 a Fp(section)22 b(w)m(e)i(study)f Fn(t)g Fp(with)f(particular)f(emphasis)h(on)g(the)h(case)g Fn(t)28 b Fp(=)g Fn(s)2845 595 y Fu(2)2884 631 y Fp(.)40 b(Our)22 b(discussion)328 751 y(is)32 b(based)i(on)e(the)h(follo)m(wing)d (general)i(lemma.)328 919 y Ff(Lemma)37 b(11)49 b Fk(L)-5 b(et)27 b Fn(t)f Fk(b)-5 b(e)27 b(an)f(algebr)-5 b(a)25 b(automorphism)h(of)g(a)g(\014nite-dimensional)f(algebr)-5 b(a)328 1039 y Fn(A)35 b Fk(over)g(a)g(\014eld)g Fn(k)j Fk(of)d(char)-5 b(acteristic)35 b(zer)-5 b(o.)46 b(Supp)-5 b(ose)34 b(that)i Fn(t)g Fk(has)e(\014nite)h(or)-5 b(der)35 b Fn(r)j Fk(and)328 1159 y(satis\014es)32 b(the)h(str)-5 b(ong)33 b(vanishing)f(tr)-5 b(ac)g(e)33 b(c)-5 b(ondition.)43 b(If)33 b Fn(e)g Fk(is)g(an)g(idemp)-5 b(otent)32 b(of)h Fn(A)g Fk(and)328 1280 y Fn(t)p Fp(\()p Fn(e)p Fp(\))28 b(=)f Fn(e)36 b Fk(then)e Fn(r)k Fk(divides)c Fp(Dim)14 b Fn(Ae)p Fk(.)328 1485 y Fe(Pr)n(oof:)45 b Fp(W)-8 b(e)24 b(ma)m(y)f(assume)h(that)f Fn(k)k Fp(is)c(algebraically)d(closed)k(and) f(that)g Fn(e)28 b Fl(6)p Fp(=)g(0.)40 b(Since)24 b Fn(t)328 1605 y Fp(is)e(an)g(algebra)g(endomorphism)f(of)h Fn(A)g Fp(and)h Fn(t)p Fp(\()p Fn(e)p Fp(\))28 b(=)g Fn(e)p Fp(,)c(the)f(idemp)s(oten)m(t)f(endomorphism)328 1726 y Fn(r)s Fp(\()p Fn(e)p Fp(\))32 b(of)g Fn(A)h Fp(and)g Fn(t)f Fp(comm)m(ute.)43 b(In)33 b(particular)e Fn(t)p Fp(\()p Fn(Ae)p Fp(\))d Fl(\022)g Fn(Ae)p Fp(.)474 1846 y(No)m(w)41 b Fm(tr)r Fp(\()p Fn(t)847 1810 y Ft(m)914 1846 y Fl(j)942 1861 y Ft(Ae)1031 1846 y Fp(\))g(=)g Fm(tr)q Fp(\()p Fn(t)1369 1810 y Ft(m)1436 1846 y Fl(\016)p Fn(r)s Fp(\()p Fn(e)p Fp(\)\))f(for)g(all)e Fn(m)j Fl(\025)g Fp(0.)67 b(Since)40 b Fn(t)h Fp(satis\014es)g(the)g(strong)328 1967 y(v)-5 b(anishing)25 b(trace)i(condition)f(it)f(th)m(us)j(follo)m (ws)e(that)g Fn(t)p Fl(j)2297 1982 y Ft(Ae)2413 1967 y Fp(satis\014es)i(the)f(v)-5 b(anishing)25 b(trace)328 2087 y(condition.)42 b(Since)32 b Fn(Ae)c Fl(6)p Fp(=)g(\(0\))k(it)f (follo)m(ws)g(that)h Fn(t)p Fl(j)2146 2102 y Ft(Ae)2268 2087 y Fp(has)h(\014nite)f(order)g Fn(r)s Fp(.)43 b(Our)32 b(conclu-)328 2207 y(sion)g(that)g Fn(r)k Fp(divides)c(Dim)14 b Fn(Ae)33 b Fp(no)m(w)h(follo)m(ws)d(b)m(y)i(Lemma)e(1.)44 b Fd(2)474 2365 y Fp(Let)37 b Fn(A)g Fp(b)s(e)g(a)f (\014nite-dimensional)e(Hopf)j(algebra)e(o)m(v)m(er)j Fn(k)i Fp(and)d(let)f Fn(n)f Fl(\025)g Fp(0.)56 b(Then)328 2486 y(the)30 b(sum)g(of)f(all)f(sub)s(coalgebras)i Fn(C)37 b Fp(of)29 b Fn(A)h Fp(whic)m(h)h(are)e(in)m(v)-5 b(arian)m(t)29 b(under)i Fn(s)3038 2450 y Fu(2)3107 2486 y Fp(and)f(satisfy)328 2606 y Fn(s)374 2570 y Fu(2)p Ft(n)456 2606 y Fl(j)484 2621 y Ft(C)571 2606 y Fp(=)d(1)723 2621 y Ft(C)810 2606 y Fp(is)h(in)f(fact)h(a)g(sub-Hopf)h(algebra)e(of)g Fn(A)p Fp(.)42 b(Th)m(us)30 b Fn(A)e Fp(has)h(a)f(unique)h(sub-Hopf)328 2727 y(algebra)i Fn(B)38 b Fp(maximal)29 b(with)k(resp)s(ect)g(to)g (the)g(prop)s(ert)m(y)g(that)f Fn(s)2678 2690 y Fu(2)p Ft(n)2760 2727 y Fl(j)2788 2742 y Ft(B)2876 2727 y Fp(=)c(1)3029 2742 y Ft(B)3089 2727 y Fp(.)474 2847 y(No)m(w)38 b(supp)s(ose)g(that)f (the)h(c)m(haracteristic)f(of)g Fn(k)j Fp(is)c(zero.)58 b(Since)37 b Fn(A)g Fp(is)g(semisimple)328 2967 y(if)c(and)i(only)f(if) f Fn(s)964 2931 y Fu(2)1004 2967 y Fl(j)1032 2982 y Ft(A)1119 2967 y Fp(=)e(1)1275 2982 y Ft(A)1366 2967 y Fp(b)m(y)36 b([6],)f(w)m(e)g(conclude)g(that)f Fn(A)h Fp(con)m(tains)f(a)h(unique)g (maxi-)328 3088 y(mal)d(semisimple)g(sub-Hopf)i(algebra)f(whic)m(h)h(w) m(e)h(denote)g(b)m(y)g Fn(A)2728 3103 y Ft(s)2764 3088 y Fp(.)48 b(As)35 b(an)e(immediate)328 3208 y(consequence)j(of)c(Lemma) f(11:)328 3375 y Ff(Corollary)36 b(5)49 b Fk(L)-5 b(et)28 b Fn(A)h Fk(b)-5 b(e)28 b(a)g(\014nite-dimensional)e(Hopf)i(algebr)-5 b(a)27 b(with)h(antip)-5 b(o)g(de)28 b Fn(s)g Fk(over)328 3496 y(a)j(\014eld)g Fn(k)k Fk(of)d(char)-5 b(acteristic)31 b(zer)-5 b(o,)32 b(let)f Fn(A)1854 3511 y Ft(s)1923 3496 y Fk(b)-5 b(e)31 b(the)h(unique)g(maximal)e(semisimple)g(sub-)328 3616 y(Hopf)g(algebr)-5 b(a)30 b(of)g Fn(A)p Fk(,)i(supp)-5 b(ose)30 b(that)g Fn(s)1723 3580 y Fu(2)1793 3616 y Fk(has)g(or)-5 b(der)31 b Fn(r)i Fk(and)d(that)h Fn(s)2719 3580 y Fu(2)2789 3616 y Fk(satis\014es)e(the)i(str)-5 b(ong)328 3737 y(vanishing)28 b(tr)-5 b(ac)g(e)29 b(c)-5 b(ondition.)42 b(Then)29 b Fn(r)j Fk(divides)c Fp(Dim)15 b Fn(Ae)29 b Fk(for)h(al)5 b(l)29 b(idemp)-5 b(otents)28 b Fn(e)g Fl(2)g Fn(A)3499 3752 y Ft(s)3536 3737 y Fk(.)328 3857 y Fd(2)474 4062 y Fp(The)40 b(dimension)d(of)h(the)h(unique)g(maximal)d(semisimple)g (Hopf-subalgebra)i(of)g(a)328 4183 y(\014nite-dimensional)25 b(Hopf)k(algebra)f Fn(A)g Fp(o)m(v)m(er)i(a)f(\014eld)f(of)g(c)m (haracteristic)h(zero)g(is)f(related)328 4303 y(to)j(the)h(order)g(of)g (a)f(Hopf)h(algebra)e(automorphism)g(of)h Fn(A)h Fp(whic)m(h)g (satis\014es)h(the)f(strong)328 4423 y(v)-5 b(anishing)31 b(trace)i(condition.)328 4604 y Ff(Theorem)k(6)49 b Fk(L)-5 b(et)32 b Fn(A)h Fk(b)-5 b(e)32 b(a)g(\014nite-dimensional)e(Hopf)i (algebr)-5 b(a)32 b(over)g(a)g(\014eld)f(of)h(char-)328 4724 y(acteristic)i(zer)-5 b(o,)33 b(let)h Fn(B)f Fp(=)28 b Fn(A)1406 4739 y Ft(s)1477 4724 y Fk(b)-5 b(e)33 b(the)h(unique)g (maximal)f(semisimple)f(sub-Hopf)i(alge-)328 4844 y(br)-5 b(a)34 b(of)g Fn(A)p Fk(,)g(supp)-5 b(ose)33 b(that)i Fn(t)f Fk(is)g(a)g(Hopf)g(algebr)-5 b(a)33 b(automorphism)g(of)h Fn(A)g Fk(of)g(\014nite)g(or)-5 b(der)328 4965 y Fn(r)37 b Fk(and)e Fn(t)g Fk(also)f(satis\014es)g(the)h(str)-5 b(ong)35 b(vanishing)e(tr)-5 b(ac)g(e)35 b(c)-5 b(ondition.)44 b(Then)1898 5214 y Fp(27)p eop %%Page: 28 28 28 27 bop 436 631 a Fp(a\))49 b Fn(r)37 b Fk(divides)d Fp(Dim)15 b Fn(A=)p Fp(Dim)f Fn(B)5 b Fk(,)35 b(and)f(the)h(minimal)f (p)-5 b(olynomial)33 b(of)i Fn(t)g Fk(is)g Fn(X)3246 595 y Ft(r)3306 631 y Fl(\000)22 b Fp(1)p Fk(.)431 834 y Fp(b\))49 b Fk(The)34 b(eigenvalues)g(of)g Fn(t)h Fk(ar)-5 b(e)35 b(the)g Fn(r)1841 798 y Ft(th)1946 834 y Fk(r)-5 b(o)g(ots)35 b(of)f(unity)i(in)e Fn(k)s Fk(.)442 1038 y Fp(c\))49 b Fk(The)30 b(eigensp)-5 b(ac)g(es)30 b(for)h Fn(t)h Fk(al)5 b(l)30 b(have)h(the)g(same)g(dimension)f(which)g(is)h Fp(\(Dim)14 b Fn(A)p Fp(\))p Fn(=r)s Fk(.)431 1241 y Fp(d\))49 b Fk(If)h Fn(e)h Fk(is)g(an)f(idemp)-5 b(otent)51 b(of)f Fn(A)h Fk(which)f(satis\014es)g Fn(t)p Fp(\()p Fn(e)p Fp(\))58 b(=)f Fn(e)52 b Fk(then)e Fn(r)k Fk(divides)572 1362 y Fp(Dim)14 b Fn(Ae)p Fk(.)328 1640 y Fe(Pr)n(oof:)44 b Fp(Let)23 b Fn(e)g Fp(b)s(e)g(a)f(left)g(in)m(tegral)f(for)h Fn(B)28 b Fp(whic)m(h)23 b(satis\014es)g Fn(\017)p Fp(\()p Fn(e)p Fp(\))28 b(=)g(1.)40 b(The)23 b(calculation)328 1760 y Fn(e)373 1724 y Fu(2)440 1760 y Fp(=)28 b Fn(\017)p Fp(\()p Fn(e)p Fp(\))p Fn(e)g Fp(=)g Fn(e)e Fp(sho)m(ws)i(that)d Fn(e)i Fp(is)e(an)h(idemp)s(oten)m(t)f(of)h Fn(A)p Fp(.)41 b(By)27 b(assumption)e Fn(t)i Fp(is)e(a)h(Hopf)328 1880 y(algebra)j(automorphisms)g(of)h Fn(A)p Fp(.)42 b(Th)m(us)32 b Fn(t)p Fp(\()p Fn(B)5 b Fp(\))31 b(is)f(a)g(semisimple)e(sub-Hopf)i (algebra)f(of)328 2001 y Fn(A)k Fp(whic)m(h)h(means)f(that)g Fn(t)p Fp(\()p Fn(B)5 b Fp(\))30 b Fl(\022)f Fn(B)38 b Fp(b)m(y)c(the)g(maximalilt)m(y)29 b(and)k(uniqueness)j(of)d Fn(B)5 b Fp(.)45 b(As)328 2121 y(a)31 b(consequence)j Fn(t)p Fl(j)1020 2136 y Ft(B)1111 2121 y Fp(is)c(a)h(Hopf)g(algebra)f (automorphism)e(of)j Fn(B)36 b Fp(from)30 b(whic)m(h)h Fn(t)p Fp(\()p Fn(e)p Fp(\))d(=)g Fn(e)328 2242 y Fp(follo)m(ws.)42 b(A)m(t)33 b(this)f(p)s(oin)m(t)g(w)m(e)i(conclude)f(that)f Fn(r)j Fp(divides)e(Dim)14 b Fn(Ae)33 b Fp(b)m(y)g(Lemma)f(11.)474 2362 y(T)-8 b(o)30 b(calculate)f(Dim)14 b Fn(Ae)30 b Fp(w)m(e)h(use)g(the)f(fact)g(that)g Fn(A)g Fp(is)f(a)h(free)g(righ)m (t)f Fn(B)5 b Fp(-mo)s(dule.)41 b(Let)328 2482 y Fl(f)p Fn(m)463 2497 y Fu(1)503 2482 y Fn(;)17 b(:)g(:)g(:)e(;)i(m)806 2497 y Ft(s)843 2482 y Fl(g)32 b Fp(b)s(e)h(a)g(basis)f(for)g Fn(A)h Fp(as)g(a)f(free)h(righ)m(t)f Fn(B)5 b Fp(-mo)s(dule.)41 b(The)34 b(calculation)546 2702 y Fn(Ae)28 b Fp(=)f(\()p Fn(m)918 2717 y Fu(1)958 2702 y Fn(B)5 b Fl(\010)17 b(\001)g(\001)g (\001)e(\010)q Fn(m)1427 2717 y Ft(s)1464 2702 y Fn(B)5 b Fp(\))p Fn(e)28 b Fp(=)f Fn(m)1842 2717 y Fu(1)1882 2702 y Fn(B)5 b(e)p Fl(\010)17 b(\001)g(\001)g(\001)e(\010)q Fn(m)2396 2717 y Ft(s)2433 2702 y Fn(B)5 b(e)28 b Fp(=)f Fn(k)s(m)2827 2717 y Fu(1)2867 2702 y Fl(\010)17 b(\001)g(\001)g(\001)e (\010)q Fn(k)s(m)3311 2717 y Ft(s)328 2922 y Fp(sho)m(ws)34 b(that)e(Dim)15 b Fn(Ae)28 b Fp(=)g Fn(s)f Fp(=)h(Dim)14 b Fn(A=)p Fp(Dim)h Fn(B)5 b Fp(.)474 3043 y(Th)m(us)37 b(w)m(e)f(ha)m(v)m(e)h(sho)m(wn)f(the)g(\014rst)f(half)f(of)h(part)g (a\).)51 b(The)36 b(second)g(half)e(of)h(a\))g(and)328 3163 y(parts)29 b(b\))h(and)f(c\))h(are)f(consequences)k(of)28 b(Lemma)g(1.)42 b(P)m(art)30 b(d\))f(follo)m(ws)f(b)m(y)i(Lemma)e(11.) 328 3284 y Fd(2)328 3512 y Ff(Corollary)36 b(6)49 b Fk(L)-5 b(et)33 b Fn(A)g Fk(b)-5 b(e)33 b(a)g(\014nite-dimensional)e(p)-5 b(ointe)g(d)32 b(Hopf)h(algebr)-5 b(a)33 b(over)f(a)h(\014eld)328 3632 y(of)38 b(char)-5 b(acteristic)37 b(zer)-5 b(o)38 b(with)g(c)-5 b(or)g(adic)g(al)37 b Fn(B)h Fp(=)c Fn(A)2181 3647 y Fu(0)2220 3632 y Fk(,)39 b(and)e(supp)-5 b(ose)38 b(that)g Fn(s)3089 3596 y Fu(2)3167 3632 y Fk(has)f(or)-5 b(der)328 3753 y Fn(r)s Fk(.)44 b(Then)34 b(a\))h({)g(d\))f(in)h(The)-5 b(or)g(em)33 b(6)i(hold)f(for)h Fn(t)28 b Fp(=)f Fn(s)2223 3716 y Fu(2)2263 3753 y Fk(.)45 b(In)34 b(p)-5 b(articular,)1414 3973 y Fn(r)37 b Fk(divides)d Fp(Dim)15 b Fn(A=)p Fp(Dim)f Fn(A)2413 3988 y Fu(0)2453 3973 y Fn(;)328 4193 y Fk(and)34 b Fn(r)d Fp(=)c(Dim)15 b Fn(A)35 b Fk(if)f(and)h(only)f(if)h Fn(A)28 b Fp(=)f Fn(k)s Fp(1)p Fk(.)328 4471 y Fe(Pr)n(oof:)49 b Fp(Since)32 b Fn(A)g Fp(is)f(p)s(oin)m(ted,)g Fn(A)1627 4486 y Ft(s)1692 4471 y Fp(=)d Fn(A)1869 4486 y Fu(0)1940 4471 y Fp(and)j(Theorem)h(6)g(applies)e(b)m(y)j(Theorem)f(1.)474 4591 y(T)-8 b(o)37 b(see)h(that)f Fn(r)g Fp(=)e(Dim)14 b Fn(A)37 b Fp(if)f(and)h(only)f(if)g Fn(A)f Fp(=)f Fn(k)s Fp(1,)k(supp)s(ose)g(that)f Fn(r)g Fp(=)e(Dim)15 b Fn(A)p Fp(.)328 4711 y(Then)39 b(eac)m(h)g(eigenspace)g(for)e Fn(s)1496 4675 y Fu(2)1573 4711 y Fp(is)h(one-dimensional)d(b)m(y)k (part)f(b\).)60 b(In)38 b(particular)e(1)328 4832 y(is)f(the)i(only)e (grouplik)m(e)g(elemen)m(t)h(in)f Fn(A)p Fp(.)54 b(Let)36 b(\003)g(b)s(e)g(a)g(non-zero)g(left)f(in)m(tegral)f(for)i Fn(A)p Fp(.)328 4952 y(Then)e Fn(s)629 4916 y Fu(2)668 4952 y Fp(\(\003\))27 b(=)h Fn(\013)q Fp(\()p Fn(g)t Fp(\)\003,)j(where)j Fn(\013)29 b Fl(2)f Fn(G)p Fp(\()p Fn(A)1914 4916 y Fj(\003)1953 4952 y Fp(\))33 b(and)f Fn(g)g Fl(2)c Fn(G)p Fp(\()p Fn(A)p Fp(\).)43 b(See)34 b([16)o(])f(for)f(example.)1898 5214 y(28)p eop %%Page: 29 29 29 28 bop 328 631 a Fp(Therefore)39 b Fn(g)g Fp(=)d(1)i(and)g (consequen)m(tly)h Fn(\013)q Fp(\()p Fn(g)t Fp(\))c(=)i(1.)58 b(This)38 b(means)g(\003)e Fl(2)g Fn(k)s Fp(1,)j(whic)m(h)f(is)328 751 y(equiv)-5 b(alen)m(t)32 b(to)h Fn(A)27 b Fp(=)h Fn(k)s Fp(1.)474 872 y(Another)35 b(w)m(a)m(y)g(to)f(see)h(this)f(is)f (to)h(note)g(that)g(if)f(Dim)15 b Fn(A)2544 887 y Fu(0)2614 872 y Fp(=)30 b(1)k(then)g Fn(A)3099 887 y Fu(1)3169 872 y Fp(=)c Fn(A)3348 887 y Fu(0)3388 872 y Fp(,)k(for)328 992 y(otherwise)f Fn(A)f Fp(w)m(ould)g(con)m(tain)f(a)h(non-zero)g (primitiv)m(e)e(elemen)m(t)i(whic)m(h)g(is)g(imp)s(ossible)328 1112 y(in)k(c)m(haracteristic)h(zero)g(since)g Fn(A)g Fp(is)f(\014nite-dimensional.)54 b(In)37 b(this)f(case)i Fn(A)3157 1127 y Fu(0)3232 1112 y Fp(=)c Fn(A)3415 1127 y Fu(1)3490 1112 y Fp(=)328 1233 y Fl(\001)17 b(\001)g(\001)26 b Fp(=)h Fn(A)p Fp(.)44 b Fd(2)474 1403 y Fp(F)-8 b(or)27 b(the)i(T)-8 b(aft)27 b(algebras)g Fn(r)k Fp(=)c(Dim)15 b Fn(A=)p Fp(Dim)f Fn(A)2161 1418 y Fu(0)2201 1403 y Fp(.)41 b(Th)m(us)30 b(the)e(estimate)f(of)g(Corollary)328 1523 y(6)32 b(on)h(the)g(order)f(of)h Fn(s)1125 1487 y Fu(2)1196 1523 y Fp(is)g(sharp)g(in)e(the)i(case)h(of)e(p)s(oin)m (ted)g(Hopf)h(algebras.)474 1644 y(In)g(the)g(general)f(case)i(m)m(uc)m (h)f(less)g(is)f(kno)m(wn.)45 b(By)33 b(\(2\))f(it)f(follo)m(ws)h(that) 1362 1864 y(order)q(\()p Fn(s)1669 1823 y Fu(2)1708 1864 y Fp(\))g(divides)h(2\(Dim)14 b Fn(A)p Fp(\))p Fn(:)474 2084 y Fp(W)-8 b(e)32 b(assume)g(that)f(the)h(c)m(haracteristic)g(of)f Fn(k)j Fp(is)d(zero)h(and)g(in)m(tro)s(duce)f(the)h(elemen)m(t)328 2204 y Fn(x)383 2219 y Ft(A)483 2204 y Fp(of)42 b(Section)h(5)f(in)m (to)g(our)h(discussion.)74 b(Let)43 b Fn(B)48 b Fp(b)s(e)43 b(the)g(sub-Hopf)g(algebra)e(of)i Fn(A)328 2325 y Fp(generated)i(b)m(y) h(the)f(coradical)e(of)h Fn(A)p Fp(.)80 b(Since)45 b Fn(\017)p Fp(\()p Fn(x)2238 2340 y Ft(A)2295 2325 y Fp(\))j(=)h(\(Dim) 14 b Fn(A)p Fp(\)1)48 b Fl(6)p Fp(=)g(0)c(it)g(follo)m(ws)328 2445 y(that)g Fn(x)606 2460 y Ft(A)710 2445 y Fl(6)p Fp(=)i(0.)77 b(No)m(w)45 b Fn(x)1274 2460 y Ft(A)1378 2445 y Fp(=)i(\(Dim)14 b Fn(A=)p Fp(Dim)g Fn(B)5 b Fp(\))p Fn(x)2231 2460 y Ft(B)2336 2445 y Fp(b)m(y)45 b(Prop)s(osition)d(5.)77 b(Th)m(us)45 b(the)328 2565 y(idemp)s(oten)m(t)35 b Fn(x)901 2580 y Ft(A)959 2565 y Fn(=)p Fp(Dim)14 b Fn(A)33 b Fp(=)h Fn(x)1478 2580 y Ft(B)1539 2565 y Fn(=)p Fp(Dim)14 b Fn(B)39 b Fl(2)34 b Fn(B)5 b Fp(.)53 b(Mimic)m(king)34 b(the)j(pro)s(of)e(of)g(Theorem)328 2686 y(6)d(one)h(can)g(easily)f (sho)m(w)h(that:)328 2889 y Ff(Prop)s(osition)i(6)49 b Fk(L)-5 b(et)36 b Fn(A)g Fk(b)-5 b(e)35 b(a)h(\014nite-dimensional)d (Hopf)i(algebr)-5 b(a)35 b(over)g(a)h(\014eld)e Fn(k)39 b Fk(of)328 3010 y(char)-5 b(acteristic)38 b(zer)-5 b(o.)55 b(Supp)-5 b(ose)37 b(that)i Fn(t)g Fk(is)f(a)g(Hopf)g(algebr)-5 b(a)38 b(automorphism)f(of)i Fn(A)f Fk(of)328 3130 y(\014nite)33 b(or)-5 b(der)34 b Fn(r)i Fk(and)d(that)h Fn(t)g Fk(satis\014es)f(the)g (str)-5 b(ong)34 b(vanishing)e(tr)-5 b(ac)g(e)33 b(c)-5 b(ondition.)44 b(L)-5 b(et)34 b Fn(B)328 3250 y Fk(b)-5 b(e)33 b(the)h(sub-Hopf)f(algebr)-5 b(a)33 b(of)g Fn(A)h Fk(gener)-5 b(ate)g(d)33 b(by)g(the)h(c)-5 b(or)g(adic)g(al)32 b(of)i Fn(A)p Fk(.)44 b(Then)33 b Fn(r)j Fk(divides)328 3371 y Fp(\(Dim)14 b Fn(A)j Fp(Dim)e Fn(B)5 b(x)989 3386 y Ft(B)1050 3371 y Fp(\))p Fn(=)p Fp(Dim)14 b Fn(B)5 b Fk(.)45 b Fd(2)474 3624 y Fp(Let)35 b Fn(A)724 3639 y Ft(s)796 3624 y Fp(b)s(e)g(the)g(unique)g(maximal)c(semisimple)i (sub-Hopf)i(algebra)e(of)i Fn(A)p Fp(.)50 b(Since)328 3744 y(semisimple)23 b(Hopf)h(algebras)h(in)f(c)m(haracteristic)h(zero) g(are)g(also)f(cosemisimple)f([5,)j(The-)328 3865 y(orem)33 b(3.3],)g(w)m(e)h(conclude)g(that)f Fn(A)1616 3880 y Ft(s)1682 3865 y Fl(\022)c Fn(A)1861 3880 y Fu(0)1930 3865 y Fl(\022)g Fn(B)5 b Fp(,)34 b(where)g Fn(B)39 b Fp(is)33 b(the)g(sub-Hopf)h(algebra)328 3985 y(of)k Fn(A)h Fp(generated)h(b)m(y)f(the)g(coradical)f(of)g Fn(A)p Fp(.)62 b(Th)m(us)40 b Fn(A)2347 4000 y Ft(s)2422 3985 y Fp(=)e Fn(B)44 b Fp(if)38 b(and)h(only)f(if)f Fn(A)3334 4000 y Fu(0)3413 3985 y Fp(is)h(a)328 4105 y(sub-Hopf)h(algebra)e(of)i Fn(A)p Fp(.)62 b(In)39 b(this)f(case)i Fn(x)1971 4120 y Ft(A)2066 4105 y Fp(=)e(\(Dim)15 b Fn(A=)p Fp(Dim)f Fn(B)5 b Fp(\))p Fn(x)2911 4120 y Ft(B)3011 4105 y Fp(is)38 b(a)h(non-zero)328 4226 y(left)31 b(in)m(tegral)f(of)h Fn(B)36 b Fp(b)m(y)d([7,)f(Prop)s(osition)e(9])h(since)h Fn(s)2257 4190 y Fu(2)2324 4226 y Fp(=)c(1)2477 4241 y Ft(A)2533 4226 y Fp(;)k(th)m(us)h(Dim)15 b Fn(B)5 b(x)3140 4241 y Ft(A)3225 4226 y Fp(=)27 b(1)32 b(and)328 4346 y(the)h(conclusions)f(of)h(Theorem)f(6)h(and)f(Prop)s(osition)f(6)i (are)f(the)h(same.)474 4467 y(There)28 b(are)e(in)m(teresting)g (connections)h(b)s(et)m(w)m(een)h(the)f(strong)f(v)-5 b(anishing)25 b(trace)i(con-)328 4587 y(dition)33 b(and)i(the)h (classi\014cation)d(problem)h(for)h(\014nite-dimensional)c(Hopf)k (algebras)g Fn(A)328 4707 y Fp(of)h(dimension)e Fn(p)956 4671 y Fu(2)996 4707 y Fp(,)j(where)g Fn(p)f Fp(is)g(an)g(o)s(dd)g (prime,)g(o)m(v)m(er)h(an)f(algebraically)d(closed)j(\014eld)328 4828 y(of)f(c)m(haracteristic)h(zero.)53 b(Let)35 b Fn(s)h Fp(b)s(e)g(the)g(an)m(tip)s(o)s(de)f(of)g Fn(A)p Fp(.)52 b(By)37 b(the)f(results)g(of)f([1])g(the)328 4948 y(Hopf)41 b(algebra)f Fn(A)i Fp(is)e(the)i(group)f(algebra)f(of)h Fn(Z)2145 4965 y Ft(p)2181 4946 y Fc(2)2261 4948 y Fp(or)g Fn(Z)2456 4963 y Ft(p)2495 4948 y Fl(\002)p Fn(Z)2639 4963 y Ft(p)2720 4948 y Fp(o)m(v)m(er)i Fn(k)s Fp(,)g(or)e(is)g(a)g(T) -8 b(aft)1898 5214 y(29)p eop %%Page: 30 30 30 29 bop 328 631 a Fp(algebra)33 b(T)741 646 y Ft(n;k)846 631 y Fp(\()p Fn(q)t Fp(\),)h(or)g Fn(s)1197 595 y Fu(2)1270 631 y Fp(has)g(order)g(2)p Fn(p)p Fp(.)48 b(If)34 b Fn(s)2019 595 y Fu(2)2092 631 y Fp(satis\014es)g(the)h(strong)f(v)-5 b(anishing)32 b(trace)328 751 y(condition)f(then)i(the)g(last)f(p)s (ossibilit)m(y)f(is)h(ruled)g(out.)474 872 y(Let)j Fn(A)g Fp(b)s(e)g(an)m(y)h(Hopf)e(algebra)g(o)m(v)m(er)i Fn(k)s Fp(.)50 b(The)36 b(subspace)h(of)d(\014xed)i(p)s(oin)m(ts)f(of)f Fn(s)3426 836 y Fu(2)3500 872 y Fp(is)328 992 y(a)i(subalgebra)f(of)h Fn(B)k Fp(of)c Fn(A)p Fp(.)53 b(If)36 b Fn(A)g Fp(is)f (\014nite-dimensional)e(and)j(p)s(oin)m(ted,)h(and)f Fn(k)i Fp(is)e(an)328 1112 y(algebraically)h(closed)k(\014eld)g(of)f(c) m(haracteristic)h(0,)h(then)g(Dim)14 b Fn(B)46 b Fp(divides)41 b(Dim)14 b Fn(A)41 b Fp(b)m(y)328 1233 y(part)32 b(b\))h(of)f(Theorem)h (6.)43 b(Generally)31 b Fn(A)i Fp(is)f(not)h(a)f(free)h(\(left\))f Fn(B)5 b Fp(-mo)s(dule.)328 1436 y Ff(Example)36 b(1)49 b Fk(L)-5 b(et)31 b Fn(A)g Fk(b)-5 b(e)31 b(a)f(\014nite-dimensional)f (unimo)-5 b(dular)30 b(Hopf)h(algebr)-5 b(a)30 b(with)g(an-)328 1557 y(tip)-5 b(o)g(de)36 b Fn(s)g Fk(over)g Fn(k)s Fk(,)g(and)g(supp) -5 b(ose)35 b(that)i Fn(s)1825 1521 y Fu(2)1894 1557 y Fl(6)p Fp(=)30 b(1)2049 1572 y Ft(A)2106 1557 y Fk(.)49 b(Then)35 b Fn(A)h Fk(is)g(not)g(a)g(fr)-5 b(e)g(e)36 b(left)g(mo)-5 b(dule)328 1677 y(over)34 b(the)h(sub)-5 b(algebr)g(a)34 b Fn(B)40 b Fk(of)35 b(\014xe)-5 b(d)34 b(p)-5 b(oints)34 b(of)h Fn(s)2067 1641 y Fu(2)2106 1677 y Fk(.)474 1880 y Fp(T)-8 b(o)30 b(see)i(this,)e(let)f(\003)h(b)s(e)h (a)e(non-zero)i(t)m(w)m(o-sided)f(in)m(tegral)f(for)g Fn(A)p Fp(.)43 b(Then)31 b Fn(s)p Fp(\(\003\))c(=)h(\003.)328 2001 y(Therefore)35 b(\003)30 b Fl(2)h Fn(B)5 b Fp(.)48 b(No)m(w)35 b Fn(s)1390 1965 y Fu(2)1460 2001 y Fl(6)p Fp(=)30 b(1)1615 2016 y Ft(A)1706 2001 y Fp(implies)i(that)h Fn(B)j Fl(6)p Fp(=)30 b Fn(A)p Fp(.)48 b(Supp)s(ose)36 b Fl(f)p Fn(m)3135 2016 y Fu(1)3174 2001 y Fn(;)17 b(:)g(:)g(:)f(;)h(m) 3478 2016 y Ft(r)3516 2001 y Fl(g)328 2121 y Fp(is)41 b(a)f(basis)h(for)g Fn(A)g Fp(as)g(a)g(left)f Fn(B)5 b Fp(-mo)s(dule.)67 b(Then)43 b Fn(r)i(>)d Fp(1)e(and)h Fn(\017)p Fp(\()p Fn(m)2880 2136 y Ft({)2908 2121 y Fp(\))i Fl(6)p Fp(=)f(0)e(for)h(some)328 2242 y(1)27 b Fl(\024)i Fn({)e Fl(\024)i Fn(r)s Fp(.)40 b(W)-8 b(e)27 b(ma)m(y)f(assume)g(that) g Fn(\017)p Fp(\()p Fn(m)1852 2257 y Fu(1)1892 2242 y Fp(\))h Fl(6)p Fp(=)h(0.)41 b(Since)26 b Fn(\017)p Fp(\()p Fn(m)2588 2257 y Fu(2)2628 2242 y Fp(\)\003)p Fn(m)2819 2257 y Fu(1)2867 2242 y Fl(\000)9 b Fn(\017)p Fp(\()p Fn(m)3115 2257 y Fu(1)3155 2242 y Fp(\)\003)p Fn(m)3346 2257 y Fu(2)3413 2242 y Fp(=)28 b(0)328 2362 y(determines)41 b(a)g(non-trivial)d(dep)s(endency)43 b(relation)c(among)h Fn(m)2721 2377 y Fu(1)2760 2362 y Fn(;)17 b(:)g(:)g(:)f(;)h(m)3064 2377 y Ft(r)3102 2362 y Fp(,)43 b(it)d(follo)m(ws)328 2482 y(that)32 b Fn(A)h Fp(can)g(not)f(b)s(e)h(a)f(free)h(left)f Fn(B)5 b Fp(-mo)s(dule.)474 2603 y(There)34 b(are)e(an)h(abundance)g (of)f(examples)h(whic)m(h)g(satisfy)f(the)h(h)m(yp)s(othesis)g(of)f (Ex-)328 2723 y(ample)24 b(1.)40 b(Let)25 b Fn(A)g Fp(b)s(e)h(an)m(y)f (\014nite-dimensional)d(Hopf)j(algebra)e(o)m(v)m(er)j Fn(k)i Fp(whose)f(an)m(tip)s(o)s(de)328 2844 y Fn(s)39 b Fp(satis\014es)h Fn(s)825 2807 y Fu(2)903 2844 y Fl(6)p Fp(=)f(1)1067 2859 y Ft(A)1123 2844 y Fp(.)63 b(Then)41 b(the)e(quan)m(tum)h(double)e Fn(D)s Fp(\()p Fn(A)p Fp(\))h(of)g Fn(A)g Fp(satis\014es)h(the)f(h)m(y-)328 2964 y(p)s(othesis)33 b(of)f(Example)g(1.)474 3084 y(Let)42 b Fn(A)f Fp(b)s(e)g (\014nite-dimensional)d(Hopf)j(algebra)f(o)m(v)m(er)i(an)m(y)g(\014eld) e Fn(k)s Fp(.)69 b(Recall)40 b(that)328 3205 y Fn(D)s Fp(\()p Fn(A)p Fp(\))27 b(=)h Fn(A)765 3168 y Fj(\003)12 b Ft(cop)916 3205 y Fl(\012)q Fn(A)31 b Fp(as)g(coalgebras.)42 b(One)31 b(w)m(a)m(y)h(of)f(describing)f(the)h(m)m(ultiplication)c(in) 328 3325 y Fn(D)s Fp(\()p Fn(A)p Fp(\))32 b(is)g(b)m(y)i(the)f(form)m (ula)1064 3545 y(\()p Fn(p)p Fl(\012)q Fn(a)p Fp(\)\()p Fn(q)t Fl(\012)p Fn(b)p Fp(\))28 b(=)g Fn(p)p Fp(\()p Fn(a)1829 3561 y Fu(\(1\))1923 3545 y Fn(*q)t(\()o(s)2211 3504 y Fj(\000)p Fu(1)2305 3545 y Fp(\()p Fn(a)2394 3561 y Fu(\(3\))2489 3545 y Fp(\)\))p Fl(\012)p Fn(a)2693 3561 y Fu(\(2\))2788 3545 y Fn(b)328 3765 y Fp(for)k(all)e Fn(p;)17 b(q)32 b Fl(2)c Fn(A)947 3729 y Fj(\003)1019 3765 y Fp(and)k Fn(a;)17 b(b)29 b Fl(2)f Fn(A)p Fp(.)43 b(The)34 b(an)m(tip)s(o)s(de)d Fm(s)i Fp(of)f Fn(D)s Fp(\()p Fn(A)p Fp(\))h(is)f(giv)m(en)h(b)m(y)926 3985 y Fm(s)p Fp(\()p Fn(p)p Fl(\012)q Fn(a)p Fp(\))28 b(=)f Fm(s)p Fp(\(\()p Fn(p)p Fl(\012)q Fp(1\)\()p Fn(\017)p Fl(\012)p Fn(a)p Fp(\)\))h(=)g(\()p Fn(\017)p Fl(\012)q Fn(s)p Fp(\()p Fn(a)p Fp(\)\)\()p Fn(S)2558 3944 y Fj(\000)p Fu(1)2652 3985 y Fp(\()p Fn(p)p Fp(\))p Fl(\012)p Fp(1\))p Fn(;)328 4205 y Fp(where)49 b Fn(s)f Fp(is)f(the)i(an)m(tip)s(o)s(de)e (of)g Fn(A)h Fp(and)g Fn(S)60 b Fp(=)54 b Fn(s)2183 4169 y Fj(\003)2222 4205 y Fp(.)90 b(Observ)m(e)50 b(that)d Fm(s)2992 4169 y Fu(2)3086 4205 y Fp(=)54 b Fn(S)3282 4169 y Fj(\000)p Fu(2)3376 4205 y Fl(\012)p Fn(s)3499 4169 y Fu(2)3539 4205 y Fp(.)328 4325 y(Th)m(us)36 b Fn(s)623 4289 y Fu(2)697 4325 y Fp(is)f(a)f(semisimple)e(endomorphism)i (of)g Fn(A)h Fp(if)f(and)g(only)h(if)e Fm(s)i Fp(is)g(a)f(semisimple) 328 4446 y(endomorphism)d(of)h Fn(D)s Fp(\()p Fn(A)p Fp(\).)474 4566 y(Let)44 b Fn(\025)f Fp(b)s(e)h(a)f Fk(left)g Fp(in)m(tegral)f(for)h Fn(A)1770 4530 y Fj(\003)1853 4566 y Fp(and)g(\003)h(b)s(e)f(a)g Fk(right)h Fp(in)m(tegral)e(for)g Fn(A)p Fp(.)77 b(Then)328 4687 y Fn(\025)p Fl(\012)p Fp(\003)38 b(is)f(a)h(t)m(w)m(o-sided)g(in)m(tegral)e(for)i Fn(D)s Fp(\()p Fn(A)p Fp(\))f(b)m(y)i(part)f(a\))f(of)g([12,)i(Theorem) f(4].)60 b(Let)38 b Fn(t)328 4807 y Fp(b)s(e)c(Hopf)g(algebra)e (automorphism)g(of)i Fn(A)f Fp(and)h(set)h Fn(T)44 b Fp(=)29 b Fn(t)2496 4771 y Fj(\003)2536 4807 y Fp(.)47 b(Then)35 b Fn(T)2937 4771 y Fj(\000)p Fu(1)3031 4807 y Fl(\012)q Fn(t)f Fp(is)f(a)h(Hopf)1898 5214 y(30)p eop %%Page: 31 31 31 30 bop 328 631 a Fp(algebra)31 b(automorphism)e(of)j Fn(D)s Fp(\()p Fn(A)p Fp(\).)43 b(W)-8 b(e)32 b(lea)m(v)m(e)h(the)f (reader)g(with)g(the)g(short)g(exercise)328 751 y(of)g(establishing)391 934 y Fn(\027)439 950 y Ft(D)r Fu(\()p Ft(A)p Fu(\))611 934 y Fp(\()p Fn(T)720 893 y Fj(\000)p Fu(1)814 934 y Fl(\012)q Fn(t)p Fp(\))c(=)1096 838 y Fg(\020)1146 934 y Fn(s)p Fp(\(\003)1298 950 y Fu(\(4\))1392 934 y Fp(\))p Fn(*)o Fp(\()p Fn(S)1631 893 y Fj(\000)p Fu(1)1725 934 y Fl(\016)p Fn(T)1846 893 y Fj(\000)p Fu(1)1940 934 y Fp(\()p Fn(\027)2026 949 y Ft(A)2079 930 y Fi(\003)2119 934 y Fp(\()p Fn(T)14 b Fp(\)\)\))p Fn(\()o Fp(\003)2507 950 y Fu(\(2\))2601 838 y Fg(\021)2667 934 y Fl(\012)q Fn(s)p Fp(\(\003)2897 950 y Fu(\(3\))2991 934 y Fp(\))p Fn(t)p Fp(\(\003)3170 950 y Fu(\(1\))3264 934 y Fp(\))p Fn(;)63 b Fp(\(17\))328 1130 y(where)34 b Fn(\027)658 1145 y Ft(A)711 1126 y Fi(\003)751 1130 y Fp(\()p Fn(T)14 b Fp(\))32 b(is)g(de\014ned)i(with)e(resp)s(ect)i(to)e(the)h(left)f(in) m(tegral)f Fn(\025)i Fp(for)f Fn(A)3045 1094 y Fj(\003)3084 1130 y Fp(.)474 1250 y(Regard)i Fn(A)f Fp(and)h Fn(A)1183 1214 y Fj(\003)12 b Ft(cop)1368 1250 y Fp(as)34 b(sub-Hopf)f(algebras)g (of)g Fn(D)s Fp(\()p Fn(A)p Fp(\))g(b)m(y)i(the)f(iden)m(ti\014cations) 328 1371 y Fn(a)k Fp(=)f Fn(\017)p Fl(\012)q Fn(a)i Fp(and)f Fn(p)g Fp(=)f Fn(p)p Fl(\012)q Fp(1)h(for)g(all)e Fn(a)i Fl(2)g Fn(A)g Fp(and)h Fn(p)e Fl(2)h Fn(A)2405 1334 y Fj(\003)2445 1371 y Fp(.)61 b(Let)38 b Fn(t)h Fp(b)s(e)g(a)f(Hopf)g (algebra)328 1491 y(automorphism)33 b(of)h Fn(A)h Fp(and)g(consider)h (the)f(Hopf)g(algebra)f(automorphism)f Fn(T)3246 1455 y Fj(\000)p Fu(1)3340 1491 y Fl(\012)p Fn(t)i Fp(of)328 1611 y Fn(D)s Fp(\()p Fn(A)p Fp(\),)e(where)i Fn(T)43 b Fp(=)29 b Fn(t)1144 1575 y Fj(\003)1183 1611 y Fp(.)46 b(Observ)m(e)35 b(that)f(\()p Fn(T)1953 1575 y Fj(\000)p Fu(1)2047 1611 y Fl(\012)p Fn(t)p Fp(\))p Fl(j)2225 1626 y Ft(A)2311 1611 y Fp(=)29 b Fn(t)34 b Fp(and)f(\()p Fn(T)2784 1575 y Fj(\000)p Fu(1)2878 1611 y Fl(\012)p Fn(t)p Fp(\))p Fl(j)3056 1626 y Ft(A)3109 1607 y Fi(\003)p Fh(cop)3269 1611 y Fp(=)c Fn(T)3445 1575 y Fj(\000)p Fu(1)3539 1611 y Fp(.)328 1732 y(No)m(w)j Fn(\027)597 1747 y Ft(A)650 1728 y Fi(\003)690 1732 y Fp(\()p Fn(T)14 b Fp(\))27 b(=)h(0)j(if)f(and)h(only)g(if)f Fn(\027)1673 1747 y Ft(A)1726 1728 y Fi(\003)11 b Fh(cop)1867 1732 y Fp(\()p Fn(T)1976 1696 y Fj(\000)p Fu(1)2070 1732 y Fp(\))28 b(=)g(0)j(b)m(y)h(\(4\).)43 b(Th)m(us)32 b(either)f Fn(\027)3217 1747 y Ft(A)3275 1732 y Fp(\()p Fn(t)p Fp(\))c(=)h(0)328 1852 y(or)k Fn(\027)495 1867 y Ft(A)548 1848 y Fi(\003)588 1852 y Fp(\()p Fn(T)14 b Fp(\))28 b(=)f(0)32 b(implies)f(that)h Fn(\027)1538 1868 y Ft(D)r Fu(\()p Ft(A)p Fu(\))1710 1852 y Fp(\()p Fn(T)1819 1816 y Fj(\000)p Fu(1)1913 1852 y Fl(\012)p Fn(t)p Fp(\))c(=)g(0)k(b)m(y)i(part)e(c\))h(of)f(Prop)s (osition)f(2.)328 2024 y Ff(Prop)s(osition)k(7)49 b Fk(L)-5 b(et)35 b Fn(A)f Fk(b)-5 b(e)33 b(a)h(\014nite-dimensional)d(Hopf)j (algebr)-5 b(a)33 b(over)h(any)g(\014eld)f Fn(k)s Fk(,)328 2144 y(supp)-5 b(ose)39 b(that)i Fn(t)f Fk(is)g(a)g(Hopf)g(algebr)-5 b(a)39 b(automorphism)g(of)h Fn(A)g Fk(and)g(let)g Fn(T)51 b Fp(=)38 b Fn(t)3217 2108 y Fj(\003)3256 2144 y Fk(.)61 b(Then)328 2265 y(the)35 b(Hopf)g(algebr)-5 b(a)34 b(automorphism)g Fn(T)1767 2228 y Fj(\000)p Fu(1)1861 2265 y Fl(\012)p Fn(t)h Fk(of)g Fn(D)s Fp(\()p Fn(A)p Fp(\))f Fk(satis\014es)g(the)h (fol)5 b(lowing:)436 2436 y Fp(a\))49 b Fk(If)34 b Fn(t)h Fk(satis\014es)f(the)h(vanishing)f(tr)-5 b(ac)g(e)34 b(c)-5 b(ondition)34 b(then)h(so)g(do)-5 b(es)34 b Fn(T)2996 2400 y Fj(\000)p Fu(1)3090 2436 y Fl(\012)p Fn(t)p Fk(.)431 2629 y Fp(b\))49 b Fk(If)41 b(either)i Fn(t)f Fk(or)g Fn(T)56 b Fk(satis\014es)42 b(the)g(str)-5 b(ong)42 b(vanishing)f(tr)-5 b(ac)g(e)42 b(c)-5 b(ondition)41 b(then)h(so)572 2750 y(do)-5 b(es)34 b Fn(T)857 2713 y Fj(\000)p Fu(1)951 2750 y Fl(\012)q Fn(t)p Fk(.)328 2960 y Fe(Pr)n(oof:)44 b Fp(In)22 b(ligh)m(t)e(of)h(the)h(commen)m(ts)g(ab)s(o)m(v)m(e)h(w)m (e)f(need)h(only)e(note)h(that)g Fm(tr)r Fp(\(\()p Fn(T)3187 2924 y Fj(\000)p Fu(1)3281 2960 y Fl(\012)p Fn(t)p Fp(\))3431 2924 y Ft(m)3498 2960 y Fp(\))27 b(=)328 3081 y Fm(tr)r Fp(\()p Fn(T)507 3045 y Fj(\000)p Ft(m)628 3081 y Fp(\))p Fm(tr)q Fp(\()p Fn(t)808 3045 y Ft(m)875 3081 y Fp(\))33 b(for)f(all)e Fn(m)e Fl(\025)g Fp(0.)44 b Fd(2)474 3240 y Fp(In)m(tro)s(ducing)35 b(\(17\))f(in)m(to)h(the)g(discussion)g(at)g (this)g(p)s(oin)m(t)f(w)m(e)i(deduce)h(some)d(infor-)328 3361 y(mation)d(ab)s(out)h Fn(\027)986 3376 y Ft(A)1039 3357 y Fi(\003)1079 3361 y Fp(\()p Fn(T)14 b Fp(\))32 b(when)i Fn(\027)1561 3376 y Ft(A)1618 3361 y Fp(\()p Fn(t)p Fp(\))28 b(=)f(0.)328 3533 y Ff(Prop)s(osition)35 b(8)49 b Fk(L)-5 b(et)40 b Fn(A)g Fk(b)-5 b(e)39 b(a)g (\014nite-dimensional)f(Hopf)h(algebr)-5 b(a)39 b(with)g(antip)-5 b(o)g(de)39 b Fn(s)328 3653 y Fk(over)d(the)g(\014eld)g Fn(k)j Fk(and)c(supp)-5 b(ose)36 b(that)h Fn(t)f Fk(is)g(a)g(Hopf)g (algebr)-5 b(a)35 b(automorphism)h(of)g Fn(A)p Fk(.)48 b(If)328 3773 y Fn(\027)376 3788 y Ft(A)433 3773 y Fp(\()p Fn(t)p Fp(\))28 b(=)f(0)35 b Fk(then)740 3873 y Fg(\020)790 3969 y Fn(s)p Fp(\(\003)942 3984 y Fu(\(4\))1036 3969 y Fp(\))p Fn(*)o Fp(\()p Fn(S)1275 3928 y Fj(\000)p Fu(1)1369 3969 y Fl(\016)p Fn(T)1490 3928 y Fj(\000)p Fu(1)1584 3969 y Fp(\()p Fn(\027)1670 3984 y Ft(A)1723 3965 y Fi(\003)1763 3969 y Fp(\()p Fn(T)14 b Fp(\)\)\))p Fn(\()o Fp(\003)2151 3984 y Fu(\(2\))2246 3873 y Fg(\021)2312 3969 y Fl(\012)p Fn(s)p Fp(\(\003)2541 3984 y Fu(\(3\))2635 3969 y Fp(\))p Fn(t)p Fp(\(\003)2814 3984 y Fu(\(1\))2908 3969 y Fp(\))28 b(=)f(0)p Fn(;)328 4152 y Fk(wher)-5 b(e)34 b Fp(\003)h Fk(is)f(a)h(right)g(inte)-5 b(gr)g(al)34 b(for)h Fn(A)p Fk(,)g Fn(S)e Fp(=)28 b Fn(s)2012 4116 y Fj(\003)2086 4152 y Fk(and)34 b Fn(T)42 b Fp(=)27 b Fn(t)2512 4116 y Fj(\003)2552 4152 y Fk(.)45 b Fd(2)474 4363 y Fp(F)-8 b(or)41 b(a)h(Hopf)g(algebra)e(automorphism)g Fn(t)i Fp(of)g Fn(A)f Fp(it)g(ma)m(y)h(v)m(ery)h(w)m(ell)e(b)s(e)i(the)f(case) 328 4483 y(that)f Fn(\027)596 4498 y Ft(A)653 4483 y Fp(\()p Fn(t)p Fp(\))h(=)g(0)f(and)g Fn(\027)1260 4498 y Ft(A)1313 4479 y Fi(\003)1353 4483 y Fp(\()p Fn(t)1426 4447 y Fj(\003)1466 4483 y Fp(\))h Fl(6)p Fp(=)g(0.)68 b(F)-8 b(or)40 b(example,)j(supp)s(ose)f(that)f Fn(A)g Fp(is)g(a)g(\014nite-)328 4604 y(dimensional)26 b(Hopf)j(algebra)f (whic)m(h)h(is)f(not)h(semisimple)d(whose)k(dual)f Fn(A)3034 4567 y Fj(\003)3102 4604 y Fp(is)f(semisim-)328 4724 y(ple.)41 b(Case)29 b(in)e(p)s(oin)m(t,)h(the)h(group)e(algebra)g Fn(k)s Fp([)p Fn(Z)2057 4739 y Ft(p)2097 4724 y Fp(])h(of)f(the)h (cyclic)g(group)g(of)f(order)h Fn(p)g Fp(o)m(v)m(er)328 4844 y Fn(k)39 b Fp(when)e(the)g(c)m(haracteristic)e(of)h Fn(k)j Fp(is)c Fn(p)f Fl(\025)g Fp(2.)53 b(Then)37 b Fn(\027)2385 4859 y Ft(A)2443 4844 y Fp(\()p Fn(s)2527 4808 y Fu(2)2566 4844 y Fp(\))c(=)h(0)i(and)g Fn(\027)3073 4859 y Ft(A)3126 4840 y Fi(\003)3166 4844 y Fp(\()p Fn(s)3250 4808 y Fu(2)12 b Fj(\003)3336 4844 y Fp(\))34 b Fl(6)p Fp(=)f(0)328 4965 y(b)m(y)g(part)g(a\))f(of)g(Prop)s(osition)f(2.)1898 5214 y(31)p eop %%Page: 32 32 32 31 bop 328 631 a Fo(References)377 850 y Fp([1])48 b(N.)34 b(Andruskiewitsc)m(h)h(and)e(H.-J.)g(Sc)m(hneider,)i(Hopf)e (algebras)f(of)h(order)g Fn(p)3336 814 y Fu(2)3409 850 y Fp(and)528 970 y(braided)g(Hopf)f(algebras)g(of)g(order)h Fn(p)p Fp(,)f Fk(J.)j(of)g(A)n(lgebr)-5 b(a)32 b Ff(199)g Fp(\(1998\),)g(430{454.)377 1174 y([2])48 b(N.)56 b(Andruskiewitsc)m(h) h(and)f(H.-J.)f(Sc)m(hneider,)62 b(Lifting)54 b(of)h(quan)m(tum)g (linear)528 1294 y(spaces)44 b(and)e(p)s(oin)m(ted)f(Hopf)g(algebras)g (of)h(order)f Fn(p)2466 1258 y Fu(3)2506 1294 y Fp(,)j Fk(J.)f(A)n(lgebr)-5 b(a)41 b Ff(209)h Fp(\(1998\),)528 1415 y(658{691.)377 1618 y([3])48 b(N.)33 b(Bourbaki,)f(Alg)m(\022)-46 b(ebre,)33 b(Chapitres)g(4)f(\022)-49 b(a)32 b(7,)h(Masson,)g(1981.)377 1821 y([4])48 b(P)-8 b(.)74 b(Etingof)e(and)i(S.)f(Gelaki,)82 b(On)74 b(\014nite-dimensional)c(semisimple)h(and)528 1942 y(cosemisimple)51 b(Hopf)i(algebras)f(in)g(p)s(ositiv)m(e)g(c)m (haracteristic,)110 b(In)m(ternational)528 2062 y(Mathematics)33 b(Researc)m(h)h(Notices)e Ff(16)h Fp(\(1998\),)e(851{864.)377 2266 y([5])48 b(R.)30 b(G.)g(Larson)g(and)g(D.)g(E.)g(Radford,)h (Finite-dimensional)25 b(cosemisimple)j(Hopf)528 2386 y(algebras)j(in)g(c)m(haracteristic)g(0)g(are)h(semisimple,)d Fk(J.)34 b(A)n(lgebr)-5 b(a)31 b Ff(117)g Fp(\(1988\),)g(267{)528 2506 y(289.)377 2710 y([6])48 b(R.)39 b(G.)e(Larson)i(and)f(D.)g(E.)g (Radford,)i(Semisimple)35 b(cosemisimple)h(Hopf)i(alge-)528 2830 y(bras,)33 b Fk(A)n(mer.)i(J.)g(Math.)d Ff(110)h Fp(\(1988\),)f(187{195.)377 3033 y([7])48 b(R.)43 b(G.)e(Larson)i(and)f (M.)h(E.)f(Sw)m(eedler,)k(An)d(asso)s(ciativ)m(e)f(orthogonal)e (bilinear)528 3154 y(form)32 b(for)g(Hopf)g(algebrtas,)g Fk(A)n(mer.)i(J.)h(Math.)e Ff(91)g Fp(\(1969\),)e(75{94.)377 3357 y([8])48 b(A.)31 b(Masuok)-5 b(a,)30 b(On)g(Hopf)g(algebras)f (with)h(co)s(comm)m(utativ)m(e)f(coradicals,)g Fk(J.)j(A)n(lge-)528 3478 y(br)-5 b(a)33 b Ff(144)f Fp(\(1991\),)g(451{466.)377 3681 y([9])48 b(S.)25 b(Mon)m(tgomery)-8 b(,)26 b(Hopf)f(algebras)f (and)g(their)g(actions)g(on)h(rings,)g(CBMS)h(Lecture)528 3801 y(Notes)34 b(82,)e(Amer.)g(Math.)h(So)s(c.,)g(1993.)328 4005 y([10])48 b(W.)24 b(D.Nic)m(hols)e(and)h(M.)g(B.)g(Zo)s(eller,)g (A)g(Hopf)g(algebra)f(freeness)j(theorem,)f Fk(A)n(mer.)528 4125 y(J.)35 b(of)g(Math.)e Ff(111)f Fp(\(1989\),)g(381{385.)328 4329 y([11])48 b(D.)33 b(E.)g(Radford,)f(Finite-dimensional)d(simple-p) s(oin)m(ted)h(Hopf)j(algebras,)f Fk(J.)j(A)n(l-)528 4449 y(gebr)-5 b(a)32 b Ff(211)h Fp(\(1999\),)f(686{710.)328 4652 y([12])48 b(D.)36 b(E.)g(Radford,)h(Minimal)c(quasitriangular)g (Hopf)j(algebras,)g Fk(J.)i(A)n(lgebr)-5 b(a)36 b Ff(157)528 4773 y Fp(\(1993\),)c(285{315.)1898 5214 y(32)p eop %%Page: 33 33 33 32 bop 328 631 a Fp([13])48 b(D.)39 b(E.)f(Radford,)i(The)f(group)g (of)f(automorphisms)f(of)h(a)g(semisimple)e(Hopf)i(al-)528 751 y(gebra)43 b(o)m(v)m(er)g(a)e(\014eld)h(of)g(c)m(haracteristic)g(0) g(is)f(\014nite,)j Fk(A)n(mer.)f(J.)h(of)f(Math.)g Ff(112)528 872 y Fp(\(1990\),)32 b(331{357.)328 1075 y([14])48 b(D.)37 b(E.)g(Radford,)h(The)g(order)f(of)f(the)i(an)m(tip)s(o)s(de)e(of)g(a)h (\014nite-dimensional)d(Hopf)528 1196 y(algebra)e(is)g(\014nite,)g Fk(A)n(mer.)j(J.)g(of)f(Math.)f Ff(98)g Fp(\(1976\),)e(333{355.)328 1399 y([15])48 b(D.)e(E.)g(Radford,)i(The)e(structure)h(of)e(Hopf)h (algebras)f(with)g(a)g(pro)5 b(jection,)49 b Fk(J.)528 1519 y(A)n(lgebr)-5 b(a)32 b Ff(92)p Fp(\(1985\),)g(322{347.)328 1723 y([16])48 b(D.)40 b(E.)g(Radford,)h(The)f(trace)g(function)f(and)h (Hopf)f(algebras.)g Fk(J.)j(A)n(lgebr)-5 b(a)39 b Ff(163)528 1843 y Fp(\(1994\),)32 b(583{622.)328 2046 y([17])48 b(H.-J.)27 b(Sc)m(hneider,)i(Lectures)g(on)e(Hopf)f(algebras,)i (\(1995\),)f(T)-8 b(raba)5 b(jos)27 b(de)h(Matem-)528 2167 y(atica)k(31/95)f(\(F)-8 b(aMAF\).)328 2370 y([18])48 b(Moss)34 b(Sw)m(eedler,)g("Hopf)e(Algebras",)g(W.)h(A.)g(Benjamin,)e (New)j(Y)-8 b(ork,)33 b(1969.)328 2574 y([19])48 b(E.)23 b(J.)g(T)-8 b(aft,)25 b(The)e(order)g(of)f(the)h(an)m(tip)s(o)s(de)f (of)g(a)g(\014nite)g(dimensional)e(Hopf)j(algebra,)528 2694 y Fk(Pr)-5 b(o)g(c.)35 b(Natl.)g(A)-5 b(c)g(ad.)35 b(Sci.)f(USA)f Ff(68)g Fp(\(1971\),)f(2631{2633.)1898 5214 y(33)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF