%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Twisted.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -O 0cm,2cm Twisted %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.06.04:1801 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 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Fi(i)1531 1966 y Fn(\))6 b Fj(\012)0 2026 y Fk(\031)30 2008 y Fg(\003)74 2026 y Fn(=)20 b(1)14 b Fj(\012)g Fk(e)248 2008 y Fg(\003)271 2026 y Fn(.)34 b(So)21 b(b)o(y)f(de\014nition)f(this)i(elemen)o(t)f(satis\014es)g Fk(h)p Fn(\()1217 1989 y Ff(P)1269 2041 y Fi(i)1294 2026 y Fk(x)1322 2033 y Fi(i)1353 2026 y Fj(\012)14 b Fk(y)1430 2033 y Fi(i)1447 2026 y Fn(\)\()p Fk(\033)r Fn(\))22 b(=)0 2049 y Ff(P)53 2101 y Fi(i)77 2086 y Fk(x)105 2093 y Fi(i)123 2086 y Fk(\033)r Fn(\()p Fk(y)196 2093 y Fi(i)213 2086 y Fn(\))14 b(=)g Fk(\016)321 2093 y Fi(e;\033)379 2086 y Fn(,)i(hence)g(for)h(all)f Fk(s)e Fj(2)g Fk(L)290 2212 y(s)g Fn(=)380 2164 y Ff(X)403 2269 y Fi(\033)460 2212 y Fk(\033)r Fn(\()p Fk(s)p Fn(\))p Fk(\016)573 2219 y Fi(e;\033)646 2212 y Fn(=)698 2164 y Ff(X)709 2270 y Fi(i;\033)779 2212 y Fk(x)807 2219 y Fi(i)824 2212 y Fk(\033)r Fn(\()p Fk(y)897 2219 y Fi(i)914 2212 y Fk(s)p Fn(\))h(=)1023 2164 y Ff(X)1052 2270 y Fi(i)1104 2212 y Fk(x)1132 2219 y Fi(i)1149 2212 y Fn(tr\()p 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2526 y Fd(\000)p Fc(1)870 2536 y Fi(\031)913 2526 y Fn(=)h Fk(\016)990 2533 y Fi(\033)o(;\031)1051 2526 y Fn(.)26 b(F)l(urthermore)15 b(if)j(w)o(e)g(sub-)0 2586 y(stitute)24 b Fk(s)j Fn(=)e Fk(x)307 2593 y Fi(j)352 2586 y Fn(in)f(\(4\))g(w)o(e)g(get)g Fk(x)704 2593 y Fi(j)752 2586 y Fn(=)816 2549 y Ff(P)869 2601 y Fi(i)894 2586 y Fk(x)922 2593 y Fi(i)947 2549 y Ff(P)1000 2601 y Fi(\031)1035 2586 y Fk(\031)r Fn(\()p Fk(y)1108 2593 y Fi(i)1126 2586 y Fn(\))p Fk(\031)r Fn(\()p Fk(x)1222 2593 y Fi(j)1244 2586 y Fn(\).)44 b(This)23 b(implies)0 2608 y Ff(P)53 2661 y Fi(\031)88 2646 y Fk(\031)r Fn(\()p Fk(y)161 2653 y Fi(i)178 2646 y Fn(\))p Fk(\031)r Fn(\()p Fk(x)274 2653 y Fi(j)297 2646 y Fn(\))14 b(=)g Fk(\016)405 2653 y Fi(ij)440 2646 y Fn(.)p eop %%Page: 5 5 5 4 bop 100 215 a Fn(If)12 b(w)o(e)g(substitute)g Fk(s)i Fn(=)g(1)e(in)g(\(4\),)i(w)o(e)e(get)h(the)f(co)q(e\016cien)o(ts)g(of)h (the)f(unit)g(elemen)o(t)0 275 y(of)17 b Fk(L)f Fn(from)g(1)e(=)316 238 y Ff(P)369 290 y Fi(i)394 275 y Fn(tr\()p Fk(y)476 282 y Fi(i)493 275 y Fn(\))p Fk(x)540 282 y Fi(i)574 275 y Fn(as)559 404 y Fk(")582 411 y Fi(i)613 404 y Fn(=)f(tr\()p Fk(y)747 411 y Fi(i)764 404 y Fn(\))i(=)855 356 y Ff(X)850 463 y Fi(\031)q Fg(2)p Fi(F)941 404 y Fk(\014)969 411 y Fi(\031)q(;i)1022 404 y Fk(:)100 599 y Fn(If)g(w)o(e)g(substitute)g Fk(s)f Fn(=)g Fk(x)570 606 y Fi(i)587 599 y Fk(x)615 606 y Fi(j)652 599 y Fn(in)h(\(4\),)h(w)o(e)f(get)h(the)g(m)o (ultiplication)d(co)q(e\016cien)o(ts)0 659 y(of)k Fk(L)f Fn(from)g Fk(x)253 666 y Fi(i)270 659 y Fk(x)298 666 y Fi(j)333 659 y Fn(=)386 622 y Ff(P)439 674 y Fi(k)471 659 y Fk(x)499 666 y Fi(k)524 659 y Fn(tr\()p Fk(y)606 666 y Fi(k)631 659 y Fk(x)659 666 y Fi(i)677 659 y Fk(x)705 666 y Fi(j)726 659 y Fn(\))e(=)812 622 y Ff(P)864 674 y Fi(k)897 659 y Fk(x)925 666 y Fi(k)958 622 y Ff(P)1011 674 y Fi(\031)1046 659 y Fk(\031)r Fn(\()p Fk(x)1123 666 y Fi(i)1141 659 y Fn(\))p Fk(\031)r Fn(\()p Fk(x)1237 666 y Fi(j)1259 659 y Fn(\))p Fk(\031)r Fn(\()p Fk(y)1351 666 y Fi(k)1377 659 y Fn(\))j(as)554 788 y Fk(\013)586 767 y Fi(k)586 800 y(ij)635 788 y Fn(=)692 740 y Ff(X)687 847 y Fi(\031)q Fg(2)p Fi(F)778 788 y Fk(\015)804 795 y Fi(i;\031)857 788 y Fk(\015)883 795 y Fi(j;\031)938 788 y Fk(\014)966 795 y Fi(\031)q(;k)1027 788 y Fk(:)100 983 y Fn(Finally)h(if)i(w)o(e)g(substitute)f Fk(s)h Fn(=)e Fk(\033)r Fn(\()p Fk(x)811 990 y Fi(i)829 983 y Fn(\))i(in)g(\(4\),)h (w)o(e)e(get)i(the)f(co)q(e\016cien)o(ts)f(for)0 1043 y(the)e Fk(F)7 b Fn(-action)16 b(from)g Fk(\033)r Fn(\()p Fk(x)486 1050 y Fi(i)503 1043 y Fn(\))e(=)589 1005 y Ff(P)642 1058 y Fi(k)674 1043 y Fk(x)702 1050 y Fi(k)736 1005 y Ff(P)788 1058 y Fi(\031)824 1043 y Fk(\031)r Fn(\()p Fk(y)897 1050 y Fi(k)922 1043 y Fn(\))p Fk(\031)r(\033)r Fn(\()p Fk(x)1048 1050 y Fi(i)1066 1043 y Fn(\))j(as)513 1172 y Fk(\030)537 1151 y Fi(k)535 1184 y(i;\033)601 1172 y Fn(=)659 1124 y Ff(X)654 1231 y Fi(\031)q Fg(2)p Fi(F)744 1172 y Fk(\015)770 1179 y Fi(i;\031)q(\033)847 1172 y Fk(\014)875 1179 y Fi(\031)q(;k)936 1172 y Fk(:)100 b Fj(t)-35 b(u)100 1379 y Fn(So)21 b(w)o(e)g(ha)o(v)o(e)g(describ)q(ed) g(the)h(algebra)f(structure)f(of)i Fk(L)g Fn(solely)f(in)g(terms)g(of)0 1439 y(the)f(matrix)g Fk(C)j Fn(\(and)d(its)g(in)o(v)o(erse)f Fk(B)r Fn(\).)33 b(The)20 b(description)f(of)h(the)h Fk(F)7 b Fn(-action)19 b(uses)0 1498 y(the)i(righ)o(t)f(regular)f (action)i(of)g Fk(F)28 b Fn(on)20 b(itself.)35 b(The)21 b(action)f(is)h(th)o(us)f(giv)o(en)g(b)o(y)g(the)0 1558 y(induced)d(p)q(erm)o(utations)f(of)i(the)g(columns)e(of)i Fk(C)t Fn(.)25 b(Observ)o(e)17 b(that)h(the)g(co)q(e\016cien)o(ts)0 1618 y(of)h Fk(C)k Fn(and)18 b Fk(B)k Fn(are)c(in)h Fk(L)p Fn(,)h(the)f(co)q(e\016cien)o(ts)g(of)g(the)g(unit,)h(the)f(m)o (ultiplication)e(and)0 1678 y(the)d Fk(F)7 b Fn(-action,)13 b(ho)o(w)o(ev)o(er,)g(are)g(all)g(elemen)o(ts)g(in)g Fk(K)18 b Fn(\(since)13 b(they)h(are)g(obtained)e(from)0 1738 y(the)17 b(trace\).)100 1826 y(W)l(e)f(consider)g(the)g (isomorphism)191 1955 y Fk(L)225 1935 y Fi(n)266 1942 y Fj(\030)266 1958 y Fn(=)318 1955 y Fk(L)11 b Fj(\012)g Fk(K)459 1935 y Fi(n)500 1942 y Fj(\030)500 1958 y Fn(=)552 1955 y Fk(L)g Fj(\012)g Fk(L)723 1927 y Fi(h)695 1955 y Fj(\000)-9 b(!)14 b Fk(L)d Fj(\012)g Fn(\()p Fk(K)t(F)c Fn(\))1007 1935 y Fg(\003)1044 1942 y Fj(\030)1044 1958 y Fn(=)1097 1955 y Fk(L)k Fj(\012)f Fk(K)1237 1935 y Fi(F)1284 1942 y Fj(\030)1284 1958 y Fn(=)1337 1955 y Fk(L)1371 1935 y Fi(F)0 2072 y Fn(where)15 b Fk(L)177 2054 y Fi(F)226 2072 y Fn(is)h(the)g(set)g(of)g Fk(n)p Fn(-tuples)f(of)h(elemen)o(ts)f(from)h Fk(L)p Fn(,)f(indexed)h(b)o(y)f (the)i(set)f Fk(F)7 b Fn(,)0 2131 y(or)12 b(the)g(set)h(of)f(maps)f (from)g Fk(F)20 b Fn(to)12 b Fk(L)p Fn(.)20 b(This)12 b(isomorphism)d(sends)i(\()p Fk(l)1239 2138 y Fi(i)1256 2131 y Fj(j)p Fk(i)i Fn(=)h(1)p Fk(;)8 b(:)g(:)g(:)h(;)f(n)p Fn(\))k(to)0 2191 y(\()19 2154 y Ff(P)72 2206 y Fi(i)97 2191 y Fk(l)112 2198 y Fi(i)128 2191 y Fk(\031)r Fn(\()p Fk(x)205 2198 y Fi(i)223 2191 y Fn(\))p Fj(j)p Fk(\031)k Fj(2)e Fk(F)7 b Fn(\))14 b(=)g(\()491 2154 y Ff(P)544 2206 y Fi(i)569 2191 y Fk(l)584 2198 y Fi(i)600 2191 y Fk(\015)626 2198 y Fi(i;\031)679 2191 y Fn(\),)f(i.e.)20 b(it)13 b(is)e(m)o(ultiplication)f(b)o(y)i Fk(C)j Fn(on)d(the)g(righ)o (t.)0 2251 y(The)18 b(in)o(v)o(erse)e(map)h(is)h(m)o(ultiplication)e(b) o(y)h Fk(B)r Fn(.)26 b(F)l(or)17 b Fk(a;)8 b(b)17 b Fj(2)f Fk(L)1165 2233 y Fi(F)1216 2251 y Fn(w)o(e)i(get)g(\()p Fk(a)13 b Fj(\001)f Fk(b)p Fn(\))p Fk(B)19 b Fn(=)0 2311 y Fk(aB)e(?)d(bB)r Fn(,)24 b(where)d Fk(?)h Fn(denotes)f(the)h(m)o (ultiplication)e(on)h Fk(L)1124 2293 y Fi(n)1173 2311 y Fn(with)g(m)o(ultiplication)0 2371 y(co)q(e\016cien)o(ts)14 b(\()p Fk(\013)301 2352 y Fi(k)301 2383 y(ij)337 2371 y Fn(\))h(and)f Fk(L)500 2352 y Fi(F)547 2371 y Fn(has)g(comp)q(onen)o (t)o(wise)f(m)o(ultiplication.)19 b(Similarly)13 b(the)0 2437 y(map)k(is)g(also)g(compatible)g(with)h(the)g Fk(F)7 b Fn(-action,)17 b(on)h Fk(L)1047 2419 y Fi(F)1098 2437 y Fn(comp)q(onen)o(t)o(wise)e(and)h(on)0 2497 y Fk(L)34 2479 y Fi(n)78 2497 y Fn(with)f(co)q(e\016cien)o(ts)g Fk(\030)467 2479 y Fi(k)465 2510 y(i;\033)518 2497 y Fn(.)100 2586 y(The)c(co)q(e\016cien)o(ts)h(as)f(calculated)h(in)g (Lemma)f(1)h(can)f(also)h(b)q(e)g(used)f(to)h(describ)q(e)0 2646 y(a)i Fk(K)t Fn(-algebra)f(structure)g(on)g Fk(S)589 2628 y Fi(n)631 2646 y Fn(for)h(an)o(y)g(\(non-comm)o(utativ)o(e\))e Fk(K)t Fn(-algebra)h Fk(S)j Fn(and)p eop %%Page: 6 6 6 5 bop 0 215 a Fn(an)15 b(action)g(of)g Fk(F)22 b Fn(on)15 b Fk(S)427 197 y Fi(n)468 215 y Fn(just)g(b)o(y)g(applying)f(the)h (isomorphism)d Fk(L)c Fj(\012)1290 222 y Fi(K)1336 215 y Fk(S)1384 202 y Fj(\030)1384 218 y Fn(=)1436 215 y Fk(K)1482 197 y Fi(n)1517 215 y Fj(\012)1556 222 y Fi(K)0 275 y Fk(S)49 262 y Fj(\030)49 277 y Fn(=)103 275 y Fk(S)137 257 y Fi(n)163 275 y Fn(.)25 b(If)18 b Fk(S)i Fn(is)d(an)g(algebra)f (of)i(the)g(form)e Fk(S)912 262 y Fj(\030)912 277 y Fn(=)966 275 y Fk(L)c Fj(\012)f Fk(T)c Fn(,)18 b(then)f(w)o(e)g(can)h(ev)o(en)f (use)0 335 y(the)g(isomorphism)c(of)k Fk(F)7 b Fn(-algebras)564 442 y(\()p Fk(L)12 b Fj(\012)f Fk(T)c Fn(\))734 422 y Fi(n)775 429 y Fj(\030)775 445 y Fn(=)827 442 y(\()p Fk(L)12 b Fj(\012)f Fk(T)c Fn(\))997 422 y Fi(F)0 550 y Fn(where)12 b(\()p Fk(L)r Fj(\012)r Fk(T)7 b Fn(\))291 532 y Fi(F)337 550 y Fn(has)12 b(comp)q(onen)o(t)o(wise)e(op)q (erations)h(and)h(\()p Fk(L)r Fj(\012)r Fk(T)7 b Fn(\))1244 532 y Fi(n)1284 550 y Fn(has)12 b(op)q(erations)0 610 y(with)21 b(co)q(e\016cien)o(ts)g(from)g(Lemma)g(1.)36 b(The)21 b(isomorphism)d(is)j(still)g(giv)o(en)g(b)o(y)g(the)0 670 y(matrices)16 b Fk(B)j Fn(and)c Fk(C)t Fn(.)196 784 y Fl(3.)24 b(The)19 b(Constr)o(uction)f(of)g(a)g(Twisted)h(Gr)o(oup)f (Ring)100 898 y Fn(W)l(e)h(w)o(an)o(t)f(to)h(construct)g(the)g Fk(K)t Fn(-algebra)e Fk(K)966 905 y Fm(\000)993 898 y Fk(G)j Fn(for)e(an)h Fk(F)7 b Fn(-group)17 b Fk(G)i Fn(b)o(y)g(us-)0 958 y(ing)g(de\014ning)e(generators)h(and)h(relations)f(of)h Fk(G)p Fn(.)30 b(W)l(e)19 b(\014rst)g(ha)o(v)o(e)f(to)i(lo)q(ok)f(at)g (free)0 1018 y(algebras.)100 1105 y(Let)f Fk(X)23 b Fn(b)q(e)18 b(a)g(set)g(and)g Fk(G)583 1112 y Fi(f)627 1105 y Fn(b)q(e)g(the)g (free)g(monoid)f(on)h Fk(X)t Fn(.)26 b(Then)18 b(the)g(monoid)0 1164 y(ring)d Fk(LG)175 1171 y Fi(f)218 1164 y Fn(and)g(the)i(free)f (\(non-comm)o(utativ)o(e\))f Fk(L)p Fn(-algebra)g Fk(L)p Fj(h)p Fk(X)t Fj(i)i Fn(coincide.)k(This)0 1224 y(is)f(due)g(to)h(the)g (fact)h(that)f(the)g(underlying)e(functor)h(from)g(the)h(category)g (Alg)f(of)0 1284 y(non-comm)o(utativ)o(e)9 b Fk(L)p Fn(-algebras)h(to)h (the)h(category)g(Set)f(of)h(sets)f(factors)g(through)f(the)0 1344 y(category)20 b(Mon)e(of)i(monoids.)28 b(The)19 b(free)h(constructions)d(are)i(the)h(corresp)q(onding)0 1404 y(left)d(adjoin)o(t)f(functors.)100 1491 y(A)c(similar)d(argumen)o (t)h(can)i(b)q(e)f(used)g(in)g(a)h(sligh)o(tly)e(more)h(complicated)g (setting.)0 1550 y(Let)k Fk(F)21 b Fn(b)q(e)14 b(a)g(\014nite)g(group.) 20 b(An)o(y)14 b(of)g(the)g(categories)g(Set,)g(Mon,)g(Alg)g(named)f (ab)q(o)o(v)o(e)0 1610 y(can)j(b)q(e)h(used)f(to)g(construct)g(a)h(new) f(category)h Fk(F)7 b Fn(-Set,)16 b Fk(F)7 b Fn(-Mon,)15 b Fk(F)7 b Fn(-Alg)16 b(of)h(ob)s(jects)0 1670 y(on)f(whic)o(h)e Fk(F)23 b Fn(acts)16 b(as)g(automorphisms)c(\(actually)17 b(pairs)d(consisting)h(of)h(an)f(ob)s(ject)0 1730 y(plus)i(an)i (action\))f(and)g(morphisms)e(whic)o(h)h(are)h(compatible)g(with)g(the) h Fk(F)7 b Fn(-action.)0 1789 y(It)19 b(is)g(easy)g(to)g(see)g(that)g (the)g(left)h(adjoin)o(t)e(functors)g(of)h(the)g(underlying)e(functors) 0 1849 y(Alg)h Fj(\000)-9 b(!)18 b Fn(Set,)g(Alg)g Fj(\000)-8 b(!)17 b Fn(Mon,)h(and)f(Mon)g Fj(\000)-9 b(!)18 b Fn(Set)g(can)f(b)q (e)h(used)f(as)h(left)g(adjoin)o(t)0 1909 y(functors)c(for)h(the)h (underlying)d(functors)h Fk(F)7 b Fn(-Alg)15 b Fj(\000)-8 b(!)15 b Fk(F)7 b Fn(-Set,)15 b Fk(F)7 b Fn(-Alg)14 b Fj(\000)-8 b(!)15 b Fk(F)7 b Fn(-Mon,)0 1969 y(and)16 b Fk(F)7 b Fn(-Mon)15 b Fj(\000)-8 b(!)16 b Fk(F)7 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Fk(F)7 b Fj(i)20 b Fn(is)e(the)i(monoid)e(algebra)g Fk(LG)807 2593 y Fi(f)852 2586 y Fn(o)o(v)o(er)h(the)g(free)h Fk(F)7 b Fn(-monoid)17 b(generated)0 2646 y(b)o(y)f Fk(Y)11 b Fn(.)p eop %%Page: 7 7 7 6 bop 100 215 a Fn(F)l(or)21 b(a)g(set)i Fk(Y)34 b Fn(=)22 b Fj(f)p Fk(g)498 222 y Fi(j)519 215 y Fj(j)p Fk(j)j Fj(2)e Fk(I)t Fj(g)f Fn(consider)f Fk(Y)26 b Fj(\002)14 b Fk(F)29 b Fn(with)22 b(elemen)o(ts)f Fk(\031)r Fn(\()p Fk(g)1478 222 y Fi(j)1500 215 y Fn(\))i(:=)0 275 y(\()p Fk(g)43 282 y Fi(j)64 275 y Fk(;)8 b(\031)r Fn(\))20 b(and)e(a)g(set)h(of)g("v)m(ariables")e Fj(f)p Fk(S)750 282 y Fi(ij)785 275 y Fj(j)p Fk(i)g Fn(=)g(1)p Fk(;)8 b(:)g(:)g(:)h(;)f(n)p Fn(;)g Fk(j)20 b Fj(2)d Fk(I)t Fj(g)i Fn(.)28 b(Then)18 b(the)h(map)0 335 y(\010)14 b(:)f Fk(LG)150 342 y Fi(f)190 335 y Fj(\000)-8 b(!)13 b Fk(L)p Fj(h)p Fk(S)368 342 y Fi(ij)403 335 y Fj(i)k Fn(de\014ned)f(b)o(y)551 494 y(\010\()p Fk(\031)r Fn(\()p Fk(g)679 501 y Fi(j)700 494 y Fn(\)\))f(:=)843 432 y Fi(n)819 446 y Ff(X)823 553 y Fi(i)p Fm(=1)900 494 y Fk(\015)926 501 y Fi(i;\031)978 494 y Fk(S)1009 501 y Fi(ij)0 655 y Fn(is)h(an)g Fk(L)p Fn(-algebra)f(isomorphism)f(with)i (in)o(v)o(erse)f(map)536 779 y(\011\()p Fk(S)625 786 y Fi(ij)660 779 y Fn(\))f(:=)764 732 y Ff(X)759 839 y Fi(\031)q Fg(2)p Fi(F)850 779 y Fk(\014)878 786 y Fi(\031)q(;i)931 779 y Fk(\031)r Fn(\()p Fk(g)1004 786 y Fi(j)1025 779 y Fn(\))p Fk(:)0 944 y Fn(This)g(follo)o(ws)g(immediately)h(from)f(the) h(fact)h(that)g Fk(B)h Fn(and)e Fk(C)j Fn(are)d(in)o(v)o(erse)e (matrices)0 1003 y(of)g(eac)o(h)f(other)g(and)g(that)i Fk(L)p Fn(-algebra)d(homomorphis)o(ms)e(are)k(uniquely)f(determined)0 1063 y(b)o(y)22 b(the)h(action)g(on)f(the)h(free)g(generating)f(set.)41 b(The)23 b(algebra)e(isomorphism)f(\010,)0 1123 y(ho)o(w)o(ev)o(er,)15 b(is)h(not)g(an)g Fk(F)7 b Fn(-isomorphism,)13 b(since)j(no)g Fk(F)7 b Fn(-structure)15 b(has)h(b)q(een)h(de\014ned)0 1183 y(on)f Fk(L)p Fj(h)p Fk(S)153 1190 y Fi(ij)188 1183 y Fj(i)p Fn(,)h(but)f(it)h(could)f(b)q(e)h(used)e(to)i(de\014ne)f(suc)o (h)g(a)g(structure.)100 1272 y(W)l(e)g(w)o(an)o(t)g(to)h(use)f(the)h (isomorphism)c(\010)k(to)g(giv)o(e)f(a)h(di\013eren)o(t)e(represen)o (tation)0 1332 y(of)g(a)f(group)g(ring)f Fk(LG)i Fn(of)g(an)f Fk(F)7 b Fn(-group)13 b Fk(G)i Fn(b)o(y)f(generators)g(and)g (relations.)20 b(F)l(or)13 b(that)0 1391 y(purp)q(ose)20 b(w)o(e)h(\014rst)f(construct)g(the)i(monoid)d(algebra)h Fk(LG)1124 1398 y Fi(f)1171 1391 y Fn(of)h(a)g(free)g Fk(F)7 b Fn(-monoid)0 1451 y Fk(G)39 1458 y Fi(f)80 1451 y Fn(whic)o(h)13 b(has)h Fk(G)h Fn(as)f(a)h(quotien)o(t)f(group.)20 b(This)14 b(algebra)g(is)g(isomorphic)e(to)j Fk(L)p Fj(h)p Fk(S)1540 1458 y Fi(ij)1575 1451 y Fj(i)0 1511 y Fn(b)o(y)22 b(the)i(isomorphism)19 b(\010.)41 b(Then)22 b(w)o(e)h(imp)q(ort)f(the)h (relations)f(of)h Fk(G)g Fn(written)g(as)0 1571 y(pairs)d(of)h(elemen)o (ts)g(of)g Fk(G)497 1578 y Fi(f)523 1571 y Fn(.)37 b(Their)20 b(di\013erences)h(generate)g(the)g(ideal)g Fk(J)26 b Fn(of)c Fk(LG)1569 1578 y Fi(f)0 1630 y Fn(to)c(b)q(e)g(factored)f(out) h(to)g(obtain)f(\(a)h(represen)o(tation)d(of)t(\))j Fk(LG)p Fn(.)25 b(This)17 b(ideal)g(will)g(b)q(e)0 1690 y(transferred)h(to)i (the)h(ideal)e Fk(M)26 b Fn(of)20 b Fk(L)p Fj(h)p Fk(S)751 1697 y Fi(ij)785 1690 y Fj(i)h Fn(via)f(the)g(giv)o(en)f(isomorphism)e (\010.)32 b(The)0 1750 y(cen)o(tral)11 b(p)q(oin)o(t)g(of)h(the)h (calculation)e(will)g(b)q(e)h(that)g(the)g(generators)f(of)h Fk(M)18 b Fn(are)11 b(already)0 1810 y(de\014ned)22 b(o)o(v)o(er)g Fk(K)k Fn(\(instead)d(of)g Fk(L)p Fn(\),)h(i.e.)40 b(they)23 b(are)g(elemen)o(ts)f(of)h Fk(K)t Fj(h)p Fk(S)1392 1817 y Fi(ij)1426 1810 y Fj(i)h Fn(whic)o(h)0 1870 y(generate)d(an)g(ideal)g Fk(M)455 1851 y Fg(0)492 1870 y Fj(\022)h Fk(K)t Fj(h)p Fk(S)649 1877 y Fi(ij)684 1870 y Fj(i)p Fn(.)37 b(Th)o(us)20 b(w)o(e)h(obtain)g(a)g Fk(K)t Fn(-algebra)f Fk(K)1453 1877 y Fm(\000)1481 1870 y Fk(G)i Fn(:=)0 1929 y Fk(K)t Fj(h)p Fk(S)96 1936 y Fi(ij)131 1929 y Fj(i)p Fk(=)m(M)225 1911 y Fg(0)257 1929 y Fn(with)16 b(the)h(prop)q(ert)o(y)f Fk(L)11 b Fj(\012)744 1936 y Fi(K)793 1929 y Fk(K)835 1936 y Fm(\000)862 1929 y Fk(G)915 1916 y Fj(\030)915 1932 y Fn(=)968 1929 y Fk(LG)p Fn(.)100 2019 y(More)21 b(precisely)g(let)h Fk(G)g Fn(b)q(e)g(an)g Fk(F)7 b Fn(-group)20 b(generated)h(as)h(an)f Fk(F)7 b Fn(-)p Fb(monoid)23 b Fn(b)o(y)0 2078 y(the)g(set)f Fk(Y)35 b Fn(=)24 b Fj(f)p Fk(g)351 2085 y Fi(i)367 2078 y Fj(j)p Fk(i)g Fj(2)g Fk(I)501 2085 y Fm(1)523 2078 y Fj(g)p Fn(.)40 b(Then)22 b(there)g(is)g(a)h(surjectiv)o(e)f Fk(F)7 b Fn(-homomorph)o(ism)0 2138 y Fk(\036)p Fn(:)h Fk(G)91 2145 y Fi(f)136 2138 y Fj(\000)-8 b(!)19 b Fk(G)p Fn(,)i(where)e Fk(G)496 2145 y Fi(f)542 2138 y Fn(is)h(the)g(free)g Fk(F)7 b Fn(-monoid)18 b(on)i Fk(Y)11 b Fn(.)32 b(A)20 b(relation)f(for)h Fk(G)g Fn(is)0 2198 y(a)e(pair)g(of)h(elemen)o(ts)e(\()p Fk(r)453 2205 y Fm(1)477 2198 y Fk(;)8 b(r)521 2205 y Fm(2)543 2198 y Fn(\))18 b Fj(2)f Fk(G)669 2205 y Fi(f)708 2198 y Fj(\002)12 b Fk(G)798 2205 y Fi(f)842 2198 y Fn(with)19 b Fk(\036)p Fn(\()p Fk(r)1029 2205 y Fm(1)1052 2198 y Fn(\))e(=)g Fk(\036)p Fn(\()p Fk(r)1215 2205 y Fm(2)1238 2198 y Fn(\).)28 b(The)18 b(set)h(of)g(all)0 2258 y(relations)13 b(for)h Fk(G)g Fn(is)g(an)g Fk(F)7 b Fn(-submonoid)k(of)k Fk(G)841 2265 y Fi(f)873 2258 y Fj(\002)6 b Fk(G)957 2265 y Fi(f)983 2258 y Fn(.)21 b(Let)15 b Fj(f)p Fk(r)1152 2265 y Fi(j)1187 2258 y Fn(=)f(\()p Fk(r)1281 2265 y Fi(j)r Fm(1)1322 2258 y Fk(;)8 b(r)1366 2265 y Fi(j)r Fm(2)1408 2258 y Fn(\))p Fj(j)p Fk(j)16 b Fj(2)e Fk(I)1547 2265 y Fm(2)1570 2258 y Fj(g)0 2317 y Fn(b)q(e)24 b(a)h(generating)e (set)h(of)h(the)f Fk(F)7 b Fn(-submonoid)21 b(of)k(relations.)43 b(Then)24 b Fk(G)g Fn(can)g(b)q(e)0 2377 y(considered)15 b(as)h(a)g(quotien)o(t)g(of)h Fk(G)638 2384 y Fi(f)680 2377 y Fn(mo)q(dulo)f(the)g(congruence)f(relation)h(generated)0 2437 y(b)o(y)g Fj(f)p Fk(r)116 2444 y Fi(j)138 2437 y Fj(g)p Fn(.)100 2526 y(W)l(e)i(consider)f(t)o(w)o(o)g(elemen)o(ts)h Fk(r)i Fn(and)d Fk(s)i Fn(of)f Fk(G)963 2533 y Fi(f)1007 2526 y Fn(\(written)h(with)f(non-negativ)o(e)0 2586 y(p)q(o)o(w)o(ers)f (and)g Fk(F)7 b Fn(-m)o(ultiples)16 b(of)i(the)g(generators)f(in)h Fk(Y)11 b Fn(\).)27 b(Let)18 b Fk(J)23 b Fn(b)q(e)18 b(the)h(ideal)e(gen-)0 2646 y(erated)g(b)o(y)f Fj(f)p Fk(\031)r Fn(\()p Fk(r)q Fn(\))d Fj(\000)e Fk(\031)r Fn(\()p Fk(s)p Fn(\))p Fj(j)p Fk(\031)18 b Fj(2)c Fk(F)7 b Fj(g)17 b Fn(in)g Fk(LG)812 2653 y Fi(f)838 2646 y Fn(.)23 b(Then)16 b Fk(M)k Fn(:=)14 b(\010\()p Fk(J)5 b Fn(\))17 b(is)g(generated)f(b)o(y)p eop %%Page: 8 8 8 7 bop 0 215 a Fj(f)p Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)q Fn(\))14 b Fj(\000)e Fk(\031)r Fn(\()p Fk(s)p Fn(\)\))p Fj(j)p Fk(\031)21 b Fj(2)c Fk(F)7 b Fj(g)p Fn(.)27 b(W)l(e)18 b(w)o(an)o(t)g(to)g(c)o(hange)g(this)g(generating)f (set)h(to)h(a)f(set)0 275 y(of)f Fk(K)t Fn(-linear)e(com)o(binations)f (of)j(the)g Fk(S)735 282 y Fi(ij)769 275 y Fn(.)22 b(F)l(or)16 b(this)g(purp)q(ose)f(w)o(e)h(form)388 385 y Fj(f)418 337 y Ff(X)413 444 y Fi(\031)q Fg(2)p Fi(F)503 385 y Fk(\014)531 392 y Fi(\031)q(;i)584 385 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)q Fn(\))d Fj(\000)e Fk(\031)r Fn(\()p Fk(s)p Fn(\)\))p Fj(j)p Fk(i)16 b Fn(=)d(1)p Fk(;)8 b(:)g(:)g(:)h(;)f(n)p Fj(g)p Fk(:)0 583 y Fl(Lemma)20 b(2.)27 b Fe(The)19 b(ideal)e Fk(M)25 b Fe(in)17 b Fk(L)p Fj(h)p Fk(S)699 590 y Fi(ij)734 583 y Fj(i)i Fe(generated)f(b)o(y)g Fj(f)p Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)q Fn(\))c Fj(\000)e Fk(\031)r Fn(\()p Fk(s)p Fn(\)\))p Fj(j)p Fk(\031)21 b Fj(2)c Fk(F)7 b Fj(g)0 643 y Fe(is)22 b(also)g(generated)g(b)o(y)g (the)g(set)h Fj(f)670 605 y Ff(P)722 658 y Fi(\031)q Fg(2)p Fi(F)815 643 y Fk(\014)843 650 y Fi(\031)q(;i)896 643 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)q Fn(\))17 b Fj(\000)e Fk(\031)r Fn(\()p Fk(s)p Fn(\)\))p Fj(j)p Fk(i)25 b Fn(=)f(1)p Fk(;)8 b(:)g(:)g(:)g(;)g(n)p Fj(g)24 b(\022)0 702 y Fk(K)t Fj(h)p Fk(S)96 709 y Fi(ij)131 702 y Fj(i)p Fe(.)0 812 y Fl(Pr)o(oof:)i Fn(Since)17 b(the)h(matrix)f(\()p Fk(\014)627 819 y Fi(\031)q(;i)680 812 y Fn(\))h(is)f(in)o(v)o (ertible,)f(this)h(set)h(generates)f(the)g(same)0 872 y(ideal)i Fk(M)25 b Fn(in)19 b Fk(L)p Fj(h)p Fk(S)340 879 y Fi(ij)375 872 y Fj(i)p Fn(.)31 b(W)l(e)19 b(sho)o(w)g(that)g(the) h(co)q(e\016cien)o(ts)f(of)h(the)f(pro)q(ducts)g(of)g(the)0 932 y Fk(S)31 939 y Fi(ij)66 932 y Fn('s)d(are)g(all)g(in)g Fk(K)t Fn(.)22 b(Let)17 b Fk(r)f Fn(=)e Fk(\034)610 939 y Fm(1)632 932 y Fn(\()p Fk(g)675 939 y Fm(1)697 932 y Fn(\))e Fj(\001)e Fk(:)e(:)g(:)k Fj(\001)f Fk(\034)869 939 y Fi(p)891 932 y Fn(\()p Fk(g)934 939 y Fi(p)958 932 y Fn(\).)22 b(Then)16 b(w)o(e)g(obtain)88 995 y Ff(X)83 1101 y Fi(\031)q Fg(2)p Fi(F)173 1042 y Fk(\014)201 1049 y Fi(\031)q(;i)255 1042 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(\034)381 1049 y Fm(1)404 1042 y Fn(\()p Fk(g)447 1049 y Fm(1)469 1042 y Fn(\))c Fj(\001)e Fk(:)e(:)g(:)k Fj(\001)f Fk(\034)641 1049 y Fi(p)663 1042 y Fn(\()p Fk(g)706 1049 y Fi(p)729 1042 y Fn(\)\))373 1181 y(=)431 1134 y Ff(X)426 1241 y Fi(\031)q Fg(2)p Fi(F)516 1181 y Fk(\014)544 1188 y Fi(\031)q(;i)597 1181 y Fn(\010\()p Fk(\031)r(\034)704 1188 y Fm(1)727 1181 y Fn(\()p Fk(g)770 1188 y Fm(1)792 1181 y Fn(\)\))h Fj(\001)f Fk(:)d(:)g(:)j Fj(\001)g Fn(\010\()p Fk(\031)r(\034)1068 1188 y Fi(p)1092 1181 y Fn(\()p Fk(g)1135 1188 y Fi(p)1158 1181 y Fn(\)\))373 1321 y(=)431 1273 y Ff(X)426 1380 y Fi(\031)q Fg(2)p Fi(F)544 1273 y Ff(X)516 1379 y Fi(i)530 1384 y Fc(1)550 1379 y Fi(;:::)o(;i)623 1384 y Fa(p)652 1321 y Fk(\014)680 1328 y Fi(\031)q(;i)733 1321 y Fk(\015)759 1328 y Fi(i)773 1333 y Fc(1)793 1328 y Fi(;\031)q(\034)847 1333 y Fc(1)869 1321 y Fk(S)900 1328 y Fi(i)914 1333 y Fc(1)933 1328 y Fi(;)p Fm(1)979 1321 y Fj(\001)g Fk(:)d(:)g(:)j Fj(\001)g Fk(\015)1124 1328 y Fi(i)1138 1333 y Fa(p)1158 1328 y Fi(;\031)q(\034)1212 1333 y Fa(p)1237 1321 y Fk(S)1268 1328 y Fi(i)1282 1333 y Fa(p)1302 1328 y Fi(;p)373 1466 y Fn(=)454 1419 y Ff(X)426 1525 y Fi(i)440 1530 y Fc(1)459 1525 y Fi(;:::;i)533 1530 y Fa(p)554 1466 y Fn(\()578 1419 y Ff(X)573 1526 y Fi(\031)q Fg(2)p Fi(F)663 1466 y Fk(\014)691 1473 y Fi(\031)q(;i)744 1466 y Fk(\015)770 1473 y Fi(i)784 1478 y Fc(1)804 1473 y Fi(;\031)q(\034)858 1478 y Fc(1)891 1466 y Fj(\001)g Fk(:)d(:)g(:)k Fj(\001)e Fk(\015)1036 1473 y Fi(i)1050 1478 y Fa(p)1071 1473 y Fi(;\031)q(\034)1125 1478 y Fa(p)1149 1466 y Fn(\))p Fk(S)1199 1473 y Fi(i)1213 1478 y Fc(1)1233 1473 y Fi(;)p Fm(1)1278 1466 y Fj(\001)h Fk(:)d(:)g(:)j Fj(\001)g Fk(S)1428 1473 y Fi(i)1442 1478 y Fa(p)1463 1473 y Fi(;p)1498 1466 y Fk(:)0 1618 y Fn(Since)16 b(the)h(co)q(e\016cien)o(ts)165 1669 y Ff(X)160 1775 y Fi(\031)q Fg(2)p Fi(F)250 1716 y Fk(\014)278 1723 y Fi(\031)q(;i)331 1716 y Fk(\015)357 1723 y Fi(i)371 1728 y Fc(1)390 1723 y Fi(;\031)q(\034)444 1728 y Fc(1)478 1716 y Fj(\001)11 b Fk(:)d(:)g(:)j Fj(\001)g Fk(\015)623 1723 y Fi(i)637 1728 y Fa(p)658 1723 y Fi(;\031)q(\034)712 1728 y Fa(p)750 1716 y Fn(=)808 1669 y Ff(X)803 1775 y Fi(\031)q Fg(2)p Fi(F)893 1716 y Fk(\031)r Fn(\()p Fk(y)966 1723 y Fi(i)983 1716 y Fn(\))p Fk(\031)r(\034)1054 1723 y Fm(1)1077 1716 y Fn(\()p Fk(x)1124 1723 y Fi(i)1138 1728 y Fc(1)1161 1716 y Fn(\))d Fk(:)g(:)g(:)h(\031)r(\034)1307 1723 y Fi(p)1330 1716 y Fn(\()p Fk(x)1377 1723 y Fi(i)1391 1728 y Fa(p)1415 1716 y Fn(\))0 1861 y(are)16 b(the)h(trace)g(of)f (certain)h(elemen)o(ts)e(they)i(are)f(in)h Fk(K)t Fn(.)k Fj(t)-35 b(u)0 1984 y Fl(Theorem)17 b(3.)27 b Fe(Let)15 b Fk(G)h Fe(b)q(e)f(an)g Fk(F)7 b Fe(-group)14 b(with)h Fk(F)7 b Fe(-monoid)13 b(generators)h Fj(f)p Fk(g)1417 1991 y Fi(i)1433 1984 y Fj(j)p Fk(i)g Fj(2)g Fk(I)1547 1991 y Fm(1)1570 1984 y Fj(g)0 2043 y Fe(and)e Fk(F)7 b Fe(-monoid)j(relations)i Fj(f)p Fk(r)567 2050 y Fi(i)584 2043 y Fj(j)p Fk(i)i Fj(2)g Fk(I)698 2050 y Fm(2)720 2043 y Fj(g)e Fe(generating)g(all)g Fk(F)7 b Fe(-monoid)j(relations)i (of)g Fk(G)p Fe(.)0 2103 y(Let)j Fk(M)140 2085 y Fg(0)169 2103 y Fe(b)q(e)f(the)g(ideal)g(of)g(the)h Fk(K)t Fe(-algebra)d Fk(K)t Fj(h)p Fk(S)902 2110 y Fi(ij)937 2103 y Fj(j)p Fk(i)i Fn(=)f(1)p Fk(;)8 b(:)g(:)g(:)h(;)f(n)p Fn(;)g Fk(j)16 b Fj(2)e Fk(I)1328 2110 y Fm(1)1350 2103 y Fj(i)h Fe(generated)0 2163 y(b)o(y)h(the)h(set)280 2268 y Fj(f)310 2221 y Ff(X)305 2328 y Fi(\031)q Fg(2)p Fi(F)395 2268 y Fk(\014)423 2275 y Fi(\031)q(;i)476 2268 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)602 2275 y Fi(j)r Fm(1)644 2268 y Fn(\))12 b Fj(\000)f Fk(\031)r Fn(\()p Fk(r)796 2275 y Fi(j)r Fm(2)837 2268 y Fn(\)\))p Fj(j)p Fk(i)k Fn(=)f(1)p Fk(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(;)g Fk(j)16 b Fj(2)f Fk(I)1268 2275 y Fm(2)1290 2268 y Fj(g)0 2416 y Fe(where)d Fn(\010\()p Fk(\031)r Fn(\()p Fk(g)268 2423 y Fi(j)290 2416 y Fn(\)\))j(:=)409 2379 y Ff(P)462 2431 y Fi(i)487 2416 y Fk(\015)513 2423 y Fi(i;\031)565 2416 y Fk(S)596 2423 y Fi(ij)631 2416 y Fe(.)21 b(Then)12 b Fk(K)834 2423 y Fm(\000)862 2416 y Fk(G)i Fn(:=)f Fk(K)t Fj(h)p Fk(S)1077 2423 y Fi(ij)1112 2416 y Fj(i)p Fk(=)m(M)1206 2398 y Fg(0)1234 2416 y Fe(is)g(a)f(Hopf)i(algebra)0 2476 y(and)i(an)g Fk(L)p Fe(-form)f(of)i(the)g(Hopf)g(algebra)e Fk(K)t(G)p Fe(,)i(i.e.)k Fn(\010:)8 b Fk(L)j Fj(\012)1116 2483 y Fi(K)1166 2476 y Fk(K)t(G)1265 2463 y Fj(\030)1265 2478 y Fn(=)1317 2476 y Fk(L)g Fj(\012)1401 2483 y Fi(K)1450 2476 y Fk(K)1492 2483 y Fm(\000)1520 2476 y Fk(G)p Fe(.)0 2586 y Fl(Pr)o(oof:)26 b Fn(By)21 b(h)o(yp)q(othesis)e(the)h Fk(F)7 b Fn(-group)18 b Fk(G)i Fn(is)g(represen)o(ted)e(as)i(a)g (quotien)o(t)f(of)i(a)0 2646 y(free)d Fk(F)7 b Fn(-monoid)15 b(b)o(y)i(certain)h(relations.)23 b(If)18 b Fk(r)f Fn(=)e(\()p Fk(r)977 2653 y Fm(1)1001 2646 y Fk(;)8 b(r)1045 2653 y Fm(2)1068 2646 y Fn(\))18 b(is)f(a)g(relation)g(in)g Fk(G)h Fn(then)p eop %%Page: 9 9 9 8 bop 0 215 a Fk(r)22 222 y Fm(1)65 215 y Fn(=)21 b Fk(r)147 222 y Fm(2)190 215 y Fn(in)f Fk(G)h Fn(hence)g Fk(\031)r Fn(\()p Fk(r)526 222 y Fm(1)549 215 y Fn(\))g(=)f Fk(\031)r Fn(\()p Fk(r)719 222 y Fm(2)743 215 y Fn(\))h(for)f(all)g Fk(\031)j Fj(2)e Fk(F)7 b Fn(.)34 b(It)21 b(is)f(easy)h(to)g(see)f (that)0 275 y(the)g(group)f(algebra)h Fk(K)t(G)g Fn(is)g(the)g(quotien) o(t)g Fk(K)t(G)946 282 y Fi(f)972 275 y Fk(=J)1030 257 y Fg(0)1064 275 y Fn(where)f Fk(G)1250 282 y Fi(f)1296 275 y Fn(is)h(the)h(free)f Fk(F)7 b Fn(-)0 335 y(monoid)20 b(generated)h(b)o(y)g Fj(f)p Fk(g)537 342 y Fi(i)553 335 y Fj(j)p Fk(i)h Fj(2)g Fk(I)683 342 y Fm(1)705 335 y Fj(g)f Fn(and)g Fk(J)886 317 y Fg(0)921 335 y Fn(is)g(the)g(ideal)g (of)h Fk(K)t(G)1337 342 y Fi(f)1384 335 y Fn(generated)0 394 y(b)o(y)g Fj(f)p Fk(\031)r Fn(\()p Fk(r)171 401 y Fi(i)p Fm(1)209 394 y Fn(\))15 b Fj(\000)g Fk(\031)r Fn(\()p Fk(r)368 401 y Fi(i)p Fm(2)405 394 y Fn(\))p Fj(j)p Fk(i)25 b Fj(2)f Fk(I)559 401 y Fm(2)581 394 y Fk(;)8 b(\031)27 b Fj(2)d Fk(F)7 b Fj(g)p Fn(.)40 b(If)23 b(w)o(e)f(iden)o(tify)h Fk(L)15 b Fj(\012)1244 401 y Fi(K)1297 394 y Fk(K)t(G)23 b Fn(and)f Fk(LG)p Fn(,)0 454 y Fk(L)q Fj(\012)74 461 y Fi(K)112 454 y Fk(K)t Fj(h)p Fk(S)208 461 y Fi(ij)243 454 y Fj(i)12 b Fn(and)f Fk(L)p Fj(h)p Fk(S)450 461 y Fi(ij)485 454 y Fj(i)p Fn(,)h(more)f(generally)f (the)i(tensor)f(pro)q(duct)f(with)i Fk(L)f Fn(with)g(the)0 514 y(m)o(ultiplication)k(with)h Fk(L)p Fn(,)h(then)g(w)o(e)f(get)h(an) f(algebra)g(isomorphism)e(\010:)8 b Fk(L)j Fj(\012)g Fk(K)t(G)1556 501 y Fj(\030)1556 517 y Fn(=)0 574 y Fk(L)g Fj(\012)g Fk(K)t Fj(h)p Fk(S)191 581 y Fi(ij)226 574 y Fj(i)p Fk(=)m(M)320 556 y Fg(0)335 574 y Fn(,)16 b(since)g Fk(L)h Fn(is)f(faithfully)g(\015at.)100 636 y(The)d(diagonal)e(on)i Fk(L)p Fj(h)p Fk(S)544 643 y Fi(ij)579 636 y Fj(i)p Fk(=L)t Fj(\001)t Fk(M)732 618 y Fg(0)760 636 y Fn(is)g(induced)f(b)o(y)h (\010:)8 b Fk(LG)14 b Fj(\000)-9 b(!)14 b Fk(L)p Fj(h)p Fk(S)1378 643 y Fi(ij)1413 636 y Fj(i)p Fk(=L)t Fj(\001)t Fk(M)1566 618 y Fg(0)1581 636 y Fn(.)0 696 y(W)l(e)j(ha)o(v)o(e)104 820 y(\001\()p Fk(s)188 827 y Fi(k)q(l)226 820 y Fn(\))d(=)g(\001\()378 773 y Ff(X)373 879 y Fi(\031)q Fg(2)p Fi(F)463 820 y Fk(\014)491 827 y Fi(\031)q(;k)552 820 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(g)680 827 y Fi(l)696 820 y Fn(\)\)\))259 959 y(=)317 912 y Ff(X)312 1019 y Fi(\031)q Fg(2)p Fi(F)402 959 y Fk(\014)430 966 y Fi(\031)q(;k)491 959 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(g)619 966 y Fi(l)635 959 y Fn(\)\))e Fj(\012)f Fn(\010\()p Fk(\031)r Fn(\()p Fk(g)863 966 y Fi(l)879 959 y Fn(\)\))j(=)987 912 y Ff(X)984 1018 y Fi(i;j;\031)1071 959 y Fk(\015)1097 966 y Fi(i;\031)1149 959 y Fk(\015)1175 966 y Fi(j;\031)1230 959 y Fk(\014)1258 966 y Fi(\031)q(;k)1319 959 y Fk(s)1342 966 y Fi(il)1383 959 y Fj(\012)d Fk(s)1456 966 y Fi(j)r(l)259 1102 y Fn(=)312 1055 y Ff(X)325 1161 y Fi(i;j)392 1102 y Fk(\013)424 1081 y Fi(k)424 1114 y(ij)459 1102 y Fk(s)482 1109 y Fi(il)523 1102 y Fj(\012)f Fk(s)595 1109 y Fi(j)r(l)0 1287 y Fn(where)k Fk(s)165 1294 y Fi(ij)215 1287 y Fn(is)g(the)h (residue)e(class)h(of)g Fk(S)714 1294 y Fi(ij)749 1287 y Fn(.)21 b(Since)14 b(the)h(co)q(e\016cien)o(ts)f Fk(\013)1278 1269 y Fi(k)1278 1300 y(ij)1328 1287 y Fn(are)g(in)g Fk(K)k Fn(the)0 1346 y(diagonal)f(is)h(already)g(de\014ned)f(in)h Fk(K)t Fj(h)p Fk(s)752 1353 y Fi(ij)788 1346 y Fj(i)f Fn(=)g Fk(K)t Fj(h)p Fk(S)976 1353 y Fi(ij)1011 1346 y Fj(i)p Fk(=)m(M)1105 1328 y Fg(0)1120 1346 y Fn(.)28 b(In)18 b(a)g(similar)f(w)o(a)o(y)h(w)o(e)0 1406 y(obtain)g Fk(")p Fn(\()p Fk(s)222 1413 y Fi(ij)258 1406 y Fn(\))e(=)349 1369 y Ff(P)401 1421 y Fi(\031)437 1406 y Fk(\014)465 1413 y Fi(\031)q(;i)534 1406 y Fn(=)g Fk(")612 1413 y Fi(i)647 1406 y Fn(\(indep)q(enden)o(t)i(of)g Fk(j)s Fn(!\))26 b(also)18 b(de\014ned)f(on)h Fk(K)t Fj(h)p Fk(s)1526 1413 y Fi(ij)1561 1406 y Fj(i)p Fn(.)0 1466 y(Since)e Fk(L)p Fj(h)p Fk(s)206 1473 y Fi(ij)242 1466 y Fj(i)h Fn(is)f(a)g(bialgebra)f(\(isomorphic)g(to)h Fk(LG)p Fn(\))h(and)f Fk(L=K)k Fn(is)c(faithfully)h(\015at,)0 1526 y(w)o(e)h(get)g(that)g Fk(K)t Fj(h)p Fk(s)356 1533 y Fi(ij)392 1526 y Fj(i)g Fn(is)g(also)f(a)h(bialgebra.)24 b(It)18 b(is)g(a)g(bialgebra)e(form)h(of)h Fk(K)t(G)p Fn(.)26 b(By)0 1585 y([2)15 b(Remark)g(follo)o(wing)f(Thm.1])h(a)g (bialgebra)f(form)h(of)g(a)h(Hopf)g(algebra)e(is)h(already)0 1645 y(a)i(Hopf)f(algebra.)21 b Fj(t)-35 b(u)100 1759 y Fn(The)13 b(Hopf)h(algebra)e Fk(K)531 1766 y Fm(\000)558 1759 y Fk(G)i Fn(:=)g Fk(K)t Fj(h)p Fk(s)766 1766 y Fi(ij)801 1759 y Fj(i)g Fn(is)f(called)g(a)h(\000)p Fb(-twiste)m(d)h(gr)m(oup)h (ring)p Fn(.)21 b(W)l(e)0 1819 y(ha)o(v)o(e)15 b(constructed)g(the)h (\000-t)o(wisted)f(group)g(ring)f(b)o(y)i(generators)e Fk(s)1261 1826 y Fi(ij)1313 1819 y Fn(and)h(relations)0 1879 y(using)g(the)i(form)o(ula)578 1953 y Fk(s)601 1960 y Fi(ij)650 1953 y Fn(=)708 1905 y Ff(X)703 2012 y Fi(\031)q Fg(2)p Fi(F)793 1953 y Fk(\031)r Fn(\()p Fk(y)866 1960 y Fi(i)884 1953 y Fn(\))p Fk(\031)r Fn(\()p Fk(g)976 1960 y Fi(j)997 1953 y Fn(\))0 2101 y(in)f Fk(LG)p Fn(.)22 b(Eac)o(h)16 b(relation)g Fk(r)i Fn(for)e Fk(G)h Fn(induces)f Fk(n)g Fn(relations)g(in)g Fk(K)1173 2108 y Fm(\000)1200 2101 y Fk(G)482 2186 y Ff(X)477 2293 y Fi(\031)q Fg(2)p Fi(F)568 2233 y Fk(\014)596 2240 y Fi(\031)q(;i)649 2233 y Fn(\010\()p Fk(\031)r Fn(\()p Fk(r)775 2240 y Fm(1)799 2233 y Fn(\))11 b Fj(\000)g Fk(\031)r Fn(\()p Fk(r)950 2240 y Fm(2)973 2233 y Fn(\)\))k(=)e(0)p Fk(:)0 2407 y Fn(These)18 b(generators)f(and)h(relations)f(are)h(obtained)g(from)f (the)i(de\014ning)e Fk(F)7 b Fn(-genera-)0 2466 y(tors)13 b(and)f(relations)g(for)g Fk(G)p Fn(.)21 b(Eac)o(h)13 b(generator)f Fk(g)898 2473 y Fi(j)932 2466 y Fn(\(in)h(fact)g(eac)o(h) g(elemen)o(t\))g(of)g Fk(G)g Fn(and)0 2526 y(its)k Fk(F)7 b Fn(-orbit)16 b(in)h Fk(G)e Fj(\022)g Fk(K)t(G)i Fn(corresp)q(onds)e (to)j(a)f(family)g(\()p Fk(s)1090 2533 y Fi(ij)1125 2526 y Fj(j)p Fk(i)e Fn(=)g(1)p Fk(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(\))18 b(in)f Fk(K)1528 2533 y Fm(\000)1555 2526 y Fk(G)0 2586 y Fn(b)o(y)e(the)g(giv)o(en)g(form)o(ula.)20 b(The)15 b(diagonal)e(of)j Fk(K)879 2593 y Fm(\000)906 2586 y Fk(G)g Fn(applied)d(to)j(an)f(elemen)o(t)f(of)i(suc)o(h)0 2646 y(a)h(family)f(can)h(b)q(e)g(expressed)f(solely)g(b)o(y)g(the)h (elemen)o(ts)f(of)h(the)g(same)f(family)h(\(and)p eop %%Page: 10 10 10 9 bop 0 215 a Fn(co)q(e\016cien)o(ts)18 b(in)h Fk(K)t Fn(\))g(so)f(w)o(e)g(de\014ne)h(a)f(\000)p Fb(-gr)m(oup-like)j(se)m (quenc)m(e)e Fn(of)g(a)g(Hopf)g(algebra)0 275 y Fk(H)i Fn(to)c(b)q(e)g(an)f Fk(n)p Fn(-tuple)g(\()p Fk(s)476 282 y Fi(i)493 275 y Fn(\))h(in)f Fk(H)21 b Fn(suc)o(h)16 b(that)343 391 y(\001\()p Fk(s)427 398 y Fi(k)452 391 y Fn(\))e(=)538 343 y Ff(X)551 449 y Fi(i;j)618 391 y Fk(\013)650 370 y Fi(k)650 403 y(ij)685 391 y Fk(s)708 398 y Fi(i)736 391 y Fj(\012)d Fk(s)809 398 y Fi(j)896 391 y Fn(and)66 b Fk(")p Fn(\()p Fk(s)1108 398 y Fi(i)1126 391 y Fn(\))14 b(=)f Fk(")1234 398 y Fi(i)0 547 y Fn(holds.)0 633 y Fl(Cor)o(ollar)m(y)g(4.)27 b Fe(Let)13 b Fk(G)h Fe(b)q(e)f(an)g Fk(F)7 b Fe(-group.)19 b(Let)14 b Fk(L)f Fe(b)q(e)g(a)g(comm)o(utativ)o(e)f(ring)h(whic)o(h)0 693 y(con)o(tains)j(subrings)e Fk(K)19 b Fj(\022)14 b Fk(L)541 675 y Fg(0)569 693 y Fj(\022)h Fk(L)h Fe(suc)o(h)g(that)i Fk(L)929 675 y Fg(0)957 693 y Fn(:)c Fk(K)21 b Fe(is)c(an)f Fk(F)7 b Fe(-Galois)16 b(extension.)0 753 y(Then)j(the)g(group)f (algebra)h Fk(LG)g Fe(has)g(generators)f Fj(f)p Fk(s)1019 760 y Fi(il)1048 753 y Fj(j)p Fk(i)h Fn(=)f(1)p Fk(;)8 b(:)g(:)g(:)h(;)f(n)p Fn(;)g Fk(l)19 b Fj(2)g Fk(I)t Fj(g)g Fe(suc)o(h)0 813 y(that)e(for)f(all)g Fk(l)f Fj(2)f Fk(I)261 918 y Fn(\001\()p Fk(s)345 925 y Fi(k)q(l)382 918 y Fn(\))h(=)468 871 y Ff(X)548 918 y Fk(\013)580 898 y Fi(k)580 931 y(ij)615 918 y Fk(s)638 925 y Fi(il)679 918 y Fj(\012)c Fk(s)752 925 y Fi(j)r(l)902 918 y Fe(and)116 b Fk(")p Fn(\()p Fk(s)1164 925 y Fi(il)1194 918 y Fn(\))14 b(=)g Fk(")1303 925 y Fi(i)1320 918 y Fk(:)0 1034 y Fe(The)i(relations) g(for)g(the)h Fj(f)p Fk(s)515 1041 y Fi(ij)550 1034 y Fj(g)g Fe(are)f(those)g(of)h(Theorem)e(3.)0 1148 y Fl(Pr)o(oof:)26 b Fn(follo)o(ws)20 b(immediately)f(from)h(the)g(isomorphism)d Fk(LG)1254 1134 y Fj(\030)1254 1150 y Fn(=)1313 1148 y Fk(L)c Fj(\012)1399 1155 y Fi(L)1426 1145 y Fd(0)1456 1148 y Fk(L)1490 1130 y Fg(0)1517 1148 y Fj(\012)1556 1155 y Fi(K)0 1207 y Fk(K)t(G)99 1194 y Fj(\030)99 1210 y Fn(=)151 1207 y Fk(L)e 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Fj(j)p Fk(i)f Fj(2)f Fk(I)1183 222 y Fm(1)1206 215 y Fj(g)i Fn(of)h Fk(G)g Fn(and)f(a)g(gen-)0 275 y(erating)e(set)h(of)g(relations)e Fj(f)p Fk(r)549 282 y Fi(i)566 275 y Fj(j)p Fk(i)g Fj(2)g Fk(I)680 282 y Fm(2)702 275 y Fj(g)i Fn(.)21 b(Construct)15 b Fk(K)1052 282 y Fm(\000)1080 275 y Fk(G)g Fn(as)h(in)f(Theorem)f(3.)22 b(F)l(or)0 335 y(the)f(generating)e(elemen)o(ts)h(of)h Fk(G)f Fn(let)h Fk(f)5 b Fn(\()p Fk(g)813 342 y Fi(j)835 335 y Fn(\))21 b(=)934 297 y Ff(P)986 350 y Fi(i)1011 335 y Fk(x)1039 342 y Fi(i)1070 335 y Fj(\012)13 b Fk(t)1140 342 y Fi(ij)1196 335 y Fj(2)20 b Fk(U)5 b Fn(\()p Fk(L)15 b Fj(\012)e Fk(S)s Fn(\).)34 b(The)0 394 y(co)q(e\016cien)o(ts)16 b Fk(t)270 401 y Fi(ij)319 394 y Fj(2)e Fk(S)19 b Fn(are)d(uniquely)g (determined.)21 b(Observ)o(e)15 b(that)407 557 y Fk(f)5 b Fn(\()p Fk(\031)r Fn(\()p Fk(g)528 564 y Fi(j)550 557 y Fn(\)\))15 b(=)f Fk(\031)r(f)5 b Fn(\()p Fk(g)758 564 y Fi(j)780 557 y Fn(\))14 b(=)889 495 y Fi(n)865 510 y Ff(X)869 616 y Fi(i)p Fm(=1)946 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Fk(\014)935 928 y Fi(\031)q(;i)988 921 y Fn(\()p Fk(\031)r Fn(\()p Fk(r)1078 928 y Fm(1)1102 921 y Fn(\))11 b Fj(\000)f Fk(\031)r Fn(\()p Fk(r)1252 928 y Fm(2)1275 921 y Fn(\)\))17 b(for)f(some)f Fk(i)f Fj(2)0 981 y(f)p Fn(1)p Fk(;)8 b(:)g(:)g(:)g(;)g(n)p Fj(g)18 b Fn(b)q(e)f(one)g(of)g(the)h(de\014ning) e(relations)g(for)g(the)i(ideal)f Fk(M)1265 963 y Fg(0)1279 981 y Fn(.)24 b(W)l(e)18 b(calculate)0 1048 y(the)f(action)f(of)301 1035 y Ff(e)292 1048 y Fk(f)22 b Fn(only)17 b(on)f(the)h(\014rst)e (part)h(of)h(this)f(term)40 1163 y Ff(e)31 1176 y Fk(f)5 b Fn(\()84 1129 y Ff(X)79 1236 y Fi(\031)q Fg(2)p Fi(F)170 1176 y Fk(\014)198 1183 y Fi(\031)q(;i)251 1176 y Fk(\031)r Fn(\()p Fk(r)322 1183 y Fm(1)346 1176 y Fn(\)\))14 b(=)460 1163 y Ff(e)451 1176 y Fk(f)6 b Fn(\()505 1129 y Ff(X)500 1236 y Fi(\031)q Fg(2)p Fi(F)590 1176 y Fk(\014)618 1183 y Fi(\031)q(;i)671 1176 y Fk(\031)r Fn(\()p Fk(\034)742 1183 y Fm(1)765 1176 y Fn(\()p Fk(g)808 1183 y Fm(1)830 1176 y Fn(\))i 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y(i;\033)1179 1278 y Fk(x)1207 1285 y Fi(k)1232 1278 y Fk(s)1255 1285 y Fi(i)1272 1278 y Fn(\()p Fk(g)r Fn(\)\))238 1453 y(=)297 1406 y Ff(X)290 1513 y Fi(i;k)q(;\031)383 1453 y Fk(\015)409 1460 y Fi(i;\031)q(\033)486 1453 y Fk(\014)514 1460 y Fi(\031)q(;k)575 1453 y Fk(\015)601 1460 y Fi(k)q(;\034)660 1453 y Fk(s)683 1460 y Fi(i)700 1453 y Fn(\()p Fk(g)r Fn(\))14 b(=)830 1406 y Ff(X)859 1512 y Fi(i)911 1453 y Fk(\015)937 1460 y Fi(i;\034)t(\033)1012 1453 y Fk(s)1035 1460 y Fi(i)1051 1453 y Fn(\()p Fk(g)r Fn(\))h(=)e(\()p Fk(\034)6 b(\033)r Fn(\)\()p Fk(g)r Fn(\))p Fk(:)0 1596 y Fn(F)l(urthermore)14 b(w)o(e)i(ha)o(v)o(e)80 1706 y Fk(\031)r Fn(\()p Fk(g)r(h)p Fn(\))e(=)g Fk(\031)r Fn(\()319 1658 y Ff(X)333 1764 y Fi(i;j)400 1706 y Fk(x)428 1713 y Fi(i)445 1706 y Fk(x)473 1713 y Fi(j)494 1706 y Fk(s)517 1713 y Fi(i)534 1706 y Fn(\()p Fk(g)r Fn(\))p Fk(s)621 1713 y Fi(j)643 1706 y Fn(\()p Fk(h)p Fn(\)\))g(=)g Fk(\031)r Fn(\()845 1658 y Ff(X)871 1765 y Fi(k)926 1706 y Fk(x)954 1713 y Fi(k)987 1658 y Ff(X)1001 1764 y Fi(i;j)1067 1706 y Fk(\013)1099 1685 y Fi(k)1099 1718 y(ij)1134 1706 y Fk(s)1157 1713 y Fi(i)1174 1706 y Fn(\()p Fk(g)r Fn(\))p Fk(s)1261 1713 y Fi(j)1283 1706 y Fn(\()p Fk(h)p Fn(\)\))217 1848 y(=)270 1801 y Ff(X)295 1908 y Fi(k)350 1848 y Fk(\031)r Fn(\()p Fk(x)427 1855 y Fi(k)453 1848 y Fn(\))480 1801 y Ff(X)494 1907 y Fi(i;j)561 1848 y Fk(\013)593 1828 y Fi(k)593 1861 y(ij)628 1848 y Fk(s)651 1855 y Fi(i)668 1848 y Fn(\()p Fk(g)r Fn(\))p Fk(s)755 1855 y Fi(j)776 1848 y Fn(\()p Fk(h)p Fn(\)\))h(=)929 1801 y Ff(X)943 1907 y Fi(i;j)1010 1848 y Fk(\031)r Fn(\()1059 1801 y Ff(X)1084 1908 y Fi(k)1139 1848 y Fk(\013)1171 1828 y Fi(k)1171 1861 y(ij)1206 1848 y Fk(x)1234 1855 y Fi(k)1259 1848 y Fn(\))p Fk(s)1301 1855 y Fi(i)1319 1848 y Fn(\()p Fk(g)r Fn(\))p Fk(s)1406 1855 y Fi(j)1427 1848 y Fn(\()p Fk(h)p Fn(\)\))217 1991 y(=)270 1944 y Ff(X)284 2050 y Fi(i;j)350 1991 y Fk(\031)r Fn(\()p Fk(x)427 1998 y Fi(i)445 1991 y Fn(\))p Fk(\031)r Fn(\()p Fk(x)541 1998 y Fi(j)563 1991 y Fn(\))p Fk(s)605 1998 y Fi(i)622 1991 y Fn(\()p Fk(g)r Fn(\))p Fk(s)709 1998 y Fi(j)731 1991 y Fn(\()p Fk(h)p Fn(\))f(=)g Fk(\031)r Fn(\()p Fk(g)r Fn(\))p Fk(\031)r Fn(\()p Fk(h)p Fn(\))p Fk(:)0 2157 y Fn(By)j(de\014nition)f(w)o(e)g(also)g(ha)o(v)o(e)g Fk(e)p Fn(\()p Fk(g)r Fn(\))e(=)g Fk(g)k Fn(so)e(that)h Fk(G)g Fn(is)f(an)g Fk(F)7 b Fn(-group.)100 2245 y(The)14 b(c)o(hoice)g(of)g Fk(F)7 b Fn(-monoid)13 b(generators)g Fj(f)p Fk(g)919 2252 y Fi(j)940 2245 y Fj(g)h Fn(for)g Fk(G)h Fn(de\014nes)e Fk(K)t Fn(-algebra)g(gen-)0 2304 y(erators)23 b Fk(s)197 2311 y Fi(i)214 2304 y Fn(\()p Fk(g)257 2311 y Fi(j)278 2304 y Fn(\))h(for)g Fk(H)k Fn(with)c(the)g(form)o(ulas)d Fk(g)921 2311 y Fi(j)968 2304 y Fn(=)1033 2267 y Ff(P)1085 2319 y Fi(i)1110 2304 y Fk(x)1138 2311 y Fi(i)1155 2304 y Fk(s)1178 2311 y Fi(i)1195 2304 y Fn(\()p Fk(g)1238 2311 y Fi(j)1259 2304 y Fn(\))j(and)f Fk(s)1429 2311 y Fi(i)1446 2304 y Fn(\()p Fk(g)1489 2311 y Fi(j)1510 2304 y Fn(\))k(=)0 2327 y Ff(P)53 2379 y Fi(\031)88 2364 y Fk(\014)116 2371 y Fi(\031)q(;i)169 2364 y Fk(\031)r Fn(\()p Fk(g)242 2371 y Fi(j)263 2364 y Fn(\).)c(The)16 b(relations)g(are)g(transformed)e(as)j(in)f(Theorem)f (3.)22 b Fj(t)-35 b(u)100 2466 y Fn(The)18 b(connection)g(with)g(some)f (kno)o(wn)h(results)f(should)g(b)q(e)h(men)o(tioned)f(here.)0 2526 y(In)h([6)f(Thm.6.4.])25 b(the)18 b(category)h(of)f(\014nite)g (etale)g(group)f(sc)o(hemes)f(o)o(v)o(er)h(a)h(\014eld)g Fk(K)0 2586 y Fn(w)o(as)10 b(found)h(to)g(b)q(e)h(equiv)m(alen)o(t)f (to)g(the)h(category)f(of)h(\014nite)e Fj(G)s Fn(-groups,)g(where)h Fj(G)j Fn(is)c(the)0 2646 y(pro\014nite)17 b(Galois)h(group)f(of)h(the) h(separable)d(closure)i(of)g Fk(K)t Fn(,)g(acting)h(con)o(tin)o(uously) p eop %%Page: 14 14 14 13 bop 0 215 a Fn(on)19 b(the)g(\014nite)g(groups.)27 b(These)19 b(group)f(sc)o(hemes,)g(represen)o(ted)f(b)o(y)i(comm)o (utativ)o(e)0 275 y(Hopf)12 b(algebras,)g(b)q(ecome)f(constan)o(t)h (group)f(sc)o(hemes)f(already)h(after)i(a)f(\014nite)f(Galois)0 335 y(\014eld)16 b(extension)f Fk(L)p Fn(,)i(i.e.)k Fk(L)10 b Fj(\012)g Fk(H)615 317 y Fg(\003)653 321 y Fj(\030)653 337 y Fn(=)705 335 y Fk(LG)17 b Fn(for)e(a)i(\014nite)f(group)e Fk(G)p Fn(.)22 b(The)16 b(action)h(of)f Fj(G)0 394 y Fn(on)g Fk(G)h Fn(is)f(giv)o(en)g(b)o(y)g(the)h(action)g(of)f Fk(F)21 b Fn(=)14 b(Aut\()p Fk(L=K)t Fn(\))j(on)f Fk(G)h Fn(via)g(the)f(isomorphism)109 513 y Fk(K)t Fn(-Alg\()p Fk(H)q(;)8 b(L)p Fn(\))399 500 y Fj(\030)399 516 y Fn(=)452 513 y Fk(L)p Fn(-Alg)o(\()p Fk(L)k Fj(\012)682 520 y Fi(K)731 513 y Fk(H)q(;)c(L)p Fn(\))863 500 y Fj(\030)863 516 y Fn(=)915 513 y Fk(L)p Fn(-Coalg)o(\()p Fk(L;)g(L)k Fj(\012)1250 520 y Fi(K)1299 513 y Fk(H)1344 493 y Fg(\003)1367 513 y Fn(\))399 575 y Fj(\030)399 591 y Fn(=)452 588 y(Group-Lik)o(es)n(\()p Fk(L)f Fj(\012)g Fk(H)878 567 y Fg(\003)901 588 y Fn(\))934 575 y Fj(\030)934 591 y Fn(=)987 588 y(Group-Lik)o(es)n(\()p Fk(LG)p Fn(\))1380 575 y Fj(\030)1380 591 y Fn(=)1432 588 y Fk(G:)0 704 y Fn(Theorem)k(7)g(is)g(a)h(generalization)f(of)g(this)h(theorem)f(to)h (in\014nite)f(groups)f(and)h(a)o(w)o(a)o(y)0 764 y(from)21 b(the)i(\014eld)e(requiremen)o(ts)f(for)i Fk(L)g Fn(and)f Fk(K)t Fn(.)39 b(On)21 b(the)i(other)e(hand)g(our)h(the-)0 824 y(orem)f(also)g(generalizes)g(part)g(of)h(the)h(kno)o(wn)e(an)o (tiequiv)m(alence)g(b)q(et)o(w)o(een)h(group)0 883 y(sc)o(hemes)15 b(of)h(m)o(ultiplicativ)o(e)f(t)o(yp)q(e)h(and)g(ab)q(elian)f Fj(G)s Fn(-groups)f([6)i(Thm.7.3.])k(to)d(non-)0 943 y(comm)o(utativ)o(e)j(Hopf)i(algebras.)35 b(In)21 b(fact)h(our)f (Theorem)f(3)i(giv)o(es)f(a)g(description)0 1003 y(of)h(the)g(represen) o(ting)e(algebras)g(of)i(group)f(sc)o(hemes)f(of)i(m)o(ultiplicativ)o (e)f(t)o(yp)q(e)h(b)o(y)0 1063 y(generators)15 b(and)h(relations)g(of)g (the)h(corresp)q(onding)d(c)o(haracter)i(group.)579 1182 y Fl(6.)24 b(Tw)o(o)18 b(Examples)100 1301 y Fn(W)l(e)26 b(w)o(an)o(t)f(to)h(illustrate)f(our)g(results)g(with)g(t)o(w)o(o)h (examples.)49 b(Let)26 b Fk(C)1504 1308 y Fm(2)1556 1301 y Fn(=)0 1361 y Fj(f)p Fk(\031)53 1368 y Fm(1)76 1361 y Fk(;)8 b(\031)126 1368 y Fm(2)149 1361 y Fj(g)22 b Fn(=)h Fj(f)p Fk(\017;)8 b(\033)r Fj(g)22 b Fn(b)q(e)g(the)h(cyclic)f (group)f(with)g(t)o(w)o(o)h(elemen)o(ts.)37 b(Assume)21 b(that)0 1421 y(2)g(is)g(in)o(v)o(ertible)f(in)h Fk(K)t Fn(.)37 b(Consider)20 b(the)i Fk(C)826 1428 y Fm(2)848 1421 y Fn(-Galois)e(extension)h Fk(L)h Fn(=)g Fk(K)t Fn(\()p Fk(i)p Fn(\))g(of)g Fk(K)0 1481 y Fn(with)16 b Fk(i)130 1462 y Fm(2)167 1481 y Fn(=)d Fj(\000)p Fn(1)k(and)f(basis)f Fk(x)547 1488 y Fm(1)583 1481 y Fn(=)f(1)p Fk(;)8 b(x)711 1488 y Fm(2)748 1481 y Fn(=)13 b Fk(i)p Fn(.)22 b(W)l(e)17 b(form)f(the)h(matrix)346 1626 y(\()q Fk(\015)392 1633 y Fi(i\031)433 1626 y Fn(\))d(=)519 1556 y Ff(\022)564 1597 y Fk(\031)592 1604 y Fm(1)614 1597 y Fn(\()p Fk(x)661 1604 y Fm(1)684 1597 y Fn(\))51 b Fk(\031)782 1604 y Fm(2)804 1597 y Fn(\()p Fk(x)851 1604 y Fm(1)875 1597 y Fn(\))564 1656 y Fk(\031)592 1663 y Fm(1)614 1656 y Fn(\()p Fk(x)661 1663 y Fm(2)684 1656 y Fn(\))g Fk(\031)782 1663 y Fm(2)804 1656 y Fn(\()p Fk(x)851 1663 y Fm(2)875 1656 y Fn(\))902 1556 y Ff(\023)953 1626 y Fn(=)1005 1556 y Ff(\022)1050 1597 y Fn(1)66 b(1)1054 1656 y Fk(i)54 b Fj(\000)p Fk(i)1189 1556 y Ff(\023)1234 1626 y Fk(:)0 1772 y Fn(with)16 b(the)h(in)o(v)o(erse)e(matrix)585 1909 y(\()p Fk(\014)632 1916 y Fi(ij)668 1909 y Fn(\))f(=)753 1838 y Ff(\022)804 1859 y Fm(1)p 804 1867 20 2 v 804 1896 a(2)912 1859 y(1)p 905 1867 34 2 v 905 1896 a(2)p Fi(i)804 1919 y Fm(1)p 804 1927 20 2 v 804 1956 a(2)880 1939 y Fj(\000)932 1919 y Fm(1)p 925 1927 34 2 v 925 1956 a(2)p Fi(i)973 1838 y Ff(\023)0 2055 y Fn(Then)i(the)h(m)o (ultiplication)d(co)q(e\016cien)o(ts)j(are)316 2178 y Ff(\000)339 2218 y Fk(\013)371 2197 y Fi(k)371 2230 y(ij)406 2178 y Ff(\001)442 2218 y Fn(=)495 2133 y Ff( )560 2156 y Fm(2)534 2171 y Ff(X)539 2278 y Fi(l)p Fm(=1)615 2218 y Fk(\015)641 2225 y Fi(k)q(l)678 2218 y Fk(\014)706 2225 y Fi(li)735 2218 y Fk(\014)763 2225 y Fi(lj)797 2133 y Ff(!)851 2218 y Fn(=)903 2148 y Ff(\022)948 2188 y Fn(1)69 b(0)948 2248 y(0)50 b Fj(\000)p Fn(1)1095 2146 y Ff(\014)1095 2176 y(\014)1095 2206 y(\014)1095 2235 y(\014)1134 2188 y Fn(0)g(1)1134 2248 y(1)g(0)1242 2148 y Ff(\023)0 2387 y Fn(where)20 b(the)h(t)o(w)o(o)f(adjacen)o(t)g (matices)h(are)f(\()p Fk(\013)862 2369 y Fm(1)862 2400 y Fi(ij)897 2387 y Fn(\))h(and)f(\()p Fk(\013)1089 2369 y Fm(2)1089 2400 y Fi(ij)1124 2387 y Fn(\).)35 b(The)20 b(co)q(e\016cien)o(ts)g(of)0 2447 y(the)d(action)f(of)h Fk(F)24 b Fn(on)16 b Fk(L)g Fn(are)376 2558 y Ff(\000)399 2598 y Fk(\030)423 2578 y Fi(k)421 2611 y(i;\033)473 2558 y Ff(\001)510 2598 y Fn(=)563 2513 y Ff( )607 2551 y(X)602 2658 y Fi(\031)q Fg(2)p Fi(F)692 2598 y Fk(\015)718 2605 y Fi(i;\031)q(\033)795 2598 y Fk(\014)823 2605 y Fi(\031)q(;k)884 2513 y Ff(!)938 2598 y Fn(=)990 2528 y Ff(\022)1035 2569 y Fn(1)69 b(0)1035 2628 y(0)50 b Fj(\000)p Fn(1)1182 2528 y Ff(\023)p eop %%Page: 15 15 15 14 bop 0 215 a Fn(and)16 b(the)h(co)q(e\016cien)o(ts)f(of)g(the)h (unit)f(are)h(\()p Fk(")808 222 y Fm(1)831 215 y Fk(;)8 b(")876 222 y Fm(2)898 215 y Fn(\))14 b(=)g(\(1)p Fk(;)8 b Fn(0\).)100 306 y(F)l(or)19 b(a)h(\000-t)o(wisted)g(group)g(ring)f Fk(K)757 313 y Fm(\000)784 306 y Fk(G)i Fn(w)o(e)f(use)g(the)h (isomorphism)c(\()p Fk(LG)p Fn(\))1507 288 y Fi(n)1556 293 y Fj(\030)1556 309 y Fn(=)0 366 y(\()p Fk(LG)p Fn(\))111 348 y Fi(F)162 366 y Fn(as)f(describ)q(ed)g(at)h(the)g(end)f(of)h (paragraph)e(2)h(giv)o(en)h(b)o(y)f(m)o(ultiplication)f(b)o(y)0 426 y Fk(B)20 b Fn(resp.)k Fk(C)t Fn(.)g(W)l(e)18 b(denote)g(the)g(m)o (ultiplication)d(in)i(the)h(algebra)f(\()p Fk(LG)p Fn(\))1353 408 y Fi(n)1398 426 y Fn(b)o(y)h Fk(?)p Fn(,)f(the)0 485 y(action)23 b(of)f Fk(F)30 b Fn(with)23 b(no)f(sp)q(ecial)g (notation.)40 b(W)l(e)23 b(still)f(consider)f Fk(K)1317 492 y Fm(\000)1345 485 y Fk(G)i Fn(and)e Fk(K)t(G)0 545 y Fn(em)o(b)q(edded)14 b(in)g Fk(LG)p Fn(.)22 b(Eac)o(h)14 b(elemen)o(t)h 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Fn(.)30 b(So)19 b(there)0 935 y(is)f(a)h(relation)f Fk(\033)r Fn(\()p Fk(g)r Fn(\))p Fk(g)h Fn(=)f(1)g(\(if)i(w)o(e)e(consider)f Fk(G)i Fn(as)g(a)f(m)o (ultiplicativ)o(e)g(group\).)27 b(W)l(e)0 995 y(de\014ne)463 1073 y(\()p Fk(c;)8 b(s)p Fn(\))15 b(:=)e(\()p Fk(g)r(;)8 b(\033)r Fn(\()p Fk(g)r Fn(\)\))837 1003 y Ff(\022)897 1024 y Fm(1)p 897 1032 20 2 v 897 1060 a(2)1012 1024 y(1)p 1012 1032 V 1012 1060 a(2)897 1083 y(1)p 890 1092 34 2 v 890 1120 a(2)p Fi(i)979 1103 y Fj(\000)1031 1083 y Fm(1)p 1024 1092 V 1024 1120 a(2)p Fi(i)1072 1003 y Ff(\023)1117 1073 y Fk(;)0 1204 y Fn(i.e.)24 b Fk(c)15 b Fn(=)185 1184 y Fm(1)p 185 1192 20 2 v 185 1221 a(2)211 1204 y Fn(\()p Fk(g)e Fn(+)e Fk(g)343 1186 y Fg(\000)p Fm(1)396 1204 y Fn(\))18 b(and)f Fk(s)e Fn(=)636 1184 y Fm(1)p 629 1192 34 2 v 629 1221 a(2)p Fi(i)669 1204 y Fn(\()p Fk(g)f Fj(\000)d Fk(g)802 1186 y Fg(\000)p Fm(1)855 1204 y Fn(\))18 b(in)f Fk(LG)p Fn(.)24 b(These)17 b(elemen)o(ts)g(generate)0 1263 y(a)k Fk(K)t Fn(-subalgebra)e 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