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b(group)e(algebra)f(can)i(b)s(e)g(constructed)h(as)f (the)g(free)g(v)m(ector)g(space)h Fd(K)17 b Fj(G)42 b Fo(with)35 b(basis)g Fj(G)0 541 y Fo(and)f(the)f(algebra)g(structure)h (of)f Fd(K)18 b Fj(G)39 b Fo(is)33 b(giv)m(en)h(b)m(y)g Fd(K)18 b Fj(G)28 b Fl(\012)23 b Fd(K)18 b Fj(G)35 b Fl(3)30 b Fj(g)c Fl(\012)d Fj(h)29 b Fl(7!)g Fj(g)t(h)g Fl(2)g Fd(K)18 b Fj(G)39 b Fo(and)34 b(the)0 658 y(unit)e Fj(\021)f Fo(:)d Fd(K)51 b Fl(3)28 b Fj(\013)h Fl(7!)e Fj(\013)q(e)h Fl(2)g Fd(K)17 b Fj(G)p Fo(.)149 774 y(The)41 b(group)e(algebra)g Fd(K)17 b Fj(G)46 b Fo(is)39 b(a)g(Hopf)h(algebra.) 63 b(The)41 b(com)m(ultiplication)35 b(is)k(giv)m(en)h(b)m(y)h(the)0 890 y(diagram)1430 999 y Fj(G)371 b Fd(K)18 b Fj(G)p 1535 966 314 4 v 1766 964 a Fg(-)1680 946 y Ff(\023)1610 1225 y(f)1536 1116 y Fg(@)1619 1199 y(@)1702 1282 y(@)1785 1365 y(@)1804 1384 y(@)-83 b(R)1740 1482 y Fd(K)18 b Fj(G)28 b Fl(\012)22 b Fd(K)c Fj(G)p 1954 1384 4 351 v 1956 1384 a Fg(?)1995 1231 y Fh(\001)0 1593 y Fo(with)39 b Fj(f)11 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y(4.)42 b(\(Nic)m(hols\).)h(Let)32 b Fj(x)h Fo(b)s(e)g(an)f(elemen)m(t)g(in)g (a)g(Hopf)h(algebra)e Fj(H)40 b Fo(with)32 b(an)m(tip)s(o)s(de)g Fj(S)6 b Fo(.)43 b(Assume)267 4748 y(that)32 b(for)g(an)m(y)h Fj(a)g Fo(in)f Fj(H)40 b Fo(w)m(e)34 b(ha)m(v)m(e)1372 4826 y Fc(X)1532 4921 y Fj(b)1573 4936 y Ff(i)1602 4921 y Fj(xS)6 b Fo(\()p Fj(c)1803 4936 y Ff(i)1831 4921 y Fo(\))28 b(=)f Fj(")p Fo(\()p Fj(a)p Fo(\))p Fj(x)267 5099 y Fo(where)34 b(\001)p Fj(a)28 b Fo(=)812 5025 y Fc(P)934 5099 y Fj(b)975 5114 y Ff(i)1026 5099 y Fl(\012)23 b Fj(c)1168 5114 y Ff(i)1196 5099 y Fo(.)43 b(Conjecture:)i Fj(x)33 b Fo(is)f(in)g(the)h(cen)m(ter)h(of)e Fj(H)8 b Fo(.)416 5216 y(\()p Fj(ax)29 b Fo(=)692 5141 y Fc(P)814 5216 y Fj(a)865 5231 y Fh(\(1\))959 5216 y Fj(x")p Fo(\()p Fj(a)1149 5231 y Fh(\(2\))1244 5216 y Fo(\))f(=)1413 5141 y Fc(P)1535 5216 y Fj(a)1586 5231 y Fh(\(1\))1680 5216 y Fj(xS)6 b Fo(\()p Fj(a)1890 5231 y Fh(\(2\))1985 5216 y Fo(\))p Fj(a)2074 5231 y Fh(\(3\))2169 5216 y Fo(\))27 b(=)2338 5141 y Fc(P)2459 5216 y Fj(")p Fo(\()p Fj(a)2594 5231 y Fh(\(1\))2689 5216 y Fo(\))p Fj(xa)2833 5231 y Fh(\(2\))2955 5216 y Fo(=)h Fj(xa:)p Fo(\))p eop %%Page: 39 9 39 8 bop 1425 236 a Fk(1.)35 b(HOPF)e(ALGEBRAS)1321 b(39)416 425 y Fo(In)28 b(the)f(remaining)e(six)i(conjectures)i Fj(H)34 b Fo(is)27 b(a)f(\014nite-dimensional)e(Hopf)j(algebra)f(o)m(v) m(er)267 541 y(an)32 b(algebraically)e(closed)i(\014eld.)149 658 y(5.)42 b(If)30 b Fj(H)38 b Fo(is)30 b(semisimple)e(on)j(either)f (side)g(\(i.e.)43 b(either)30 b Fj(H)38 b Fo(or)30 b(the)h(dual)e Fj(H)2855 622 y Fi(\003)2925 658 y Fo(is)h(semisimple)e(as)267 774 y(an)k(algebra\))g(the)h(square)g(of)g(the)g(an)m(tip)s(o)s(de)e (is)h(the)h(iden)m(tit)m(y)-8 b(.)416 890 y(\(Y)g(es)35 b(if)e(c)m(har\()p Fd(K)18 b Fo(\))40 b(=)34 b(0)g([Larson-Radford],)g (y)m(es)i(if)d(c)m(har\()p Fd(K)18 b Fo(\))40 b(is)34 b(large)f([Sommerh\177)-49 b(au-)267 1006 y(ser]\))149 1123 y(6.)42 b(The)35 b(size)g(of)f(the)h(matrices)f(o)s(ccurring)g(in) g(an)m(y)h(full)e(matrix)g(constituen)m(t)i(of)f Fj(H)42 b Fo(divides)267 1239 y(the)33 b(dimension)e(of)h Fj(H)8 b Fo(.)416 1355 y(\(Y)-8 b(es)32 b(is)e(Hopf)h(algebra)f(is)g (de\014ned)i(o)m(v)m(er)g Fd(Z)c Fo([Larson];)k(in)e(general)h(not)f (kno)m(wn;)j(w)m(ork)267 1471 y(b)m(y)g([Mon)m(tgomery-Withersp)s(o)s (on],)f([Zh)m(u],)i([Gelaki]\))149 1588 y(7.)42 b(If)33 b Fj(H)41 b Fo(is)33 b(semisimple)e(on)i(b)s(oth)h(sides)g(the)f(c)m (haracteristic)h(do)s(es)f(not)h(divide)f(the)g(dimen-)267 1704 y(sion.)416 1820 y(\(Larson-Radford\))149 1936 y(8.)42 b(If)32 b(the)h(dimension)f(of)g Fj(H)40 b Fo(is)32 b(prime)f(then)i Fj(H)40 b Fo(is)32 b(comm)m(utativ)m(e)g(and)h(co)s(comm)m(utativ)m(e.) 416 2052 y(\(Y)-8 b(es)34 b(in)d(c)m(haracteristic)i(0)f([Zh)m(u:)44 b(1994]\))416 2169 y(Remark.)f(Kac,)33 b(Larson,)g(and)f(Sw)m(eedler)i (ha)m(v)m(e)g(partial)c(results)j(on)g(5)f({)g(8.)416 2285 y(\(W)-8 b(as)33 b(also)f(pro)m(v)m(ed)i(b)m(y)f([Kac]\))416 2401 y(In)i(the)g(t)m(w)m(o)g(\014nal)e(conjectures)j(assume)f(that)f (the)h(c)m(haracteristic)f(do)s(es)g(not)h(divide)267 2517 y(the)e(dimension)e(of)h Fj(H)8 b Fo(.)149 2634 y(9.)42 b(The)33 b(dimension)f(of)g(the)h(radical)d(is)i(the)h(same)g (on)f(b)s(oth)h(sides.)416 2750 y(\(Coun)m(terexample)c(b)m(y)f([Nic)m (hols];)h(coun)m(terexample)g(in)e(F)-8 b(rob)s(enius-)27 b(Lusztig)g(k)m(ernel)267 2866 y(of)32 b Fj(U)444 2881 y Ff(q)482 2866 y Fo(\()p Fj(sl)r Fo(\(2\)\))g([Sc)m(hneider]\))101 2982 y(10.)41 b(There)29 b(are)g(only)e(a)h(\014nite)g(n)m(um)m(b)s(er) h(\(up)f(to)g(isomorphism\))e(of)i(Hopf)g(algebras)f(of)h(a)g(giv)m(en) 267 3099 y(dimension.)416 3215 y(\(Y)-8 b(es)34 b(for)e(semisimple,)e (cosemisimple)g(Hopf)j(algebras:)43 b(Stefan)32 b(1997\))416 3331 y(\(Coun)m(terexamples:)45 b([Andruskiewitsc)m(h,)34 b(Sc)m(hneider],)g([Beattie,)f(others])g(1997\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF