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2299 y Ff(j)2617 2308 y Fd(1)2678 2340 y Fg(\012)f Fj(:)17 b(:)g(:)22 b Fg(\012)h Fj(b)3055 2299 y Ff(j)3084 2307 y Fc(s)3121 2340 y Fj(:)p Fk(\(1\))100 2607 y(\(10\))41 b(Sho)m(w)28 b(that)g(for)f(eac)m(h)i(c)m(hange)g(of)f(bases)h Fj(L)f Fk(:)g Fj(B)k Fg(\000)-59 b(!)27 b Fj(C)35 b Fk(with)28 b Fj(c)2574 2622 y Ff(j)2638 2607 y Fk(=)2741 2533 y Fb(P)2863 2607 y Fj(\025)2920 2571 y Ff(i)2920 2632 y(j)2956 2607 y Fj(b)2997 2622 y Ff(i)3054 2607 y Fk(\(with)g(in)m(v)m(erse)315 2733 y(matrix)j(\()p Fj(\026)729 2697 y Ff(i)729 2758 y(j)765 2733 y Fk(\)\))h(the)h(follo)m(wing)d Fa(tr)-5 b(ansformation)34 b(formula)e Fk(holds)0 2983 y(\(2\))292 b Fj(a)p Fk(\()p Fj(B)5 b Fk(\))623 2936 y Ff(i)647 2945 y Fd(1)682 2936 y Ff(;:::)n(;i)804 2944 y Fc(r)623 3008 y Ff(j)652 3017 y Fd(1)686 3008 y Ff(;:::)o(;j)814 3016 y Fc(s)878 2983 y Fk(=)1041 2858 y Ff(n)990 2888 y Fb(X)981 3100 y Ff(k)1018 3109 y Fd(1)1053 3100 y Fe(=1)1159 2983 y Fj(:)17 b(:)g(:)1350 2858 y Ff(n)1299 2888 y Fb(X)1291 3100 y Ff(k)1328 3108 y Fc(r)1362 3100 y Fe(=1)1520 2858 y Ff(n)1470 2888 y Fb(X)1469 3100 y Ff(l)1490 3109 y Fd(1)1524 3100 y Fe(=1)1631 2983 y Fj(:)g(:)g(:)1813 2858 y Ff(n)1763 2888 y Fb(X)1762 3100 y Ff(l)1783 3108 y Fc(s)1817 3100 y Fe(=1)1923 2983 y Fj(\025)1980 2939 y Ff(i)2004 2948 y Fd(1)1980 3011 y Ff(k)2017 3020 y Fd(1)2072 2983 y Fj(:)g(:)g(:)f(\025)2260 2940 y Ff(i)2284 2948 y Fc(r)2260 3011 y Ff(k)2297 3019 y Fc(r)2336 2983 y Fj(\026)2395 2939 y Ff(l)2416 2948 y Fd(1)2395 3008 y Ff(j)2424 3017 y Fd(1)2479 2983 y Fj(:)h(:)g(:)f(\026)2669 2940 y Ff(l)2690 2948 y Fc(s)2669 3008 y Ff(j)2698 3016 y Fc(s)2734 2983 y Fj(a)p Fk(\()p Fj(C)7 b Fk(\))2938 2936 y Ff(k)2975 2945 y Fd(1)3010 2936 y Ff(;:::)n(;k)3145 2944 y Fc(r)2938 3011 y Ff(l)2959 3020 y Fd(1)2994 3011 y Ff(;:::)n(;l)3113 3019 y Fc(s)100 3243 y Fk(\(11\))41 b(Sho)m(w)33 b(that)f(ev)m(ery)j(family)30 b(of)i(sc)m(hemes)i(of)e(co) s(e\016cien)m(ts)1375 3400 y(\()p Fj(a)p Fk(\()p Fj(B)5 b Fk(\))p Fg(j)p Fj(B)37 b Fk(basis)c(of)f Fj(V)21 b Fk(\))315 3565 y(with)39 b Fj(a)p Fk(\()p Fj(B)5 b Fk(\))41 b(=)f(\()p Fj(a)p Fk(\()p Fj(B)5 b Fk(\))1151 3518 y Ff(i)1175 3527 y Fd(1)1210 3518 y Ff(;:::)n(;i)1332 3526 y Fc(r)1151 3591 y Ff(j)1180 3600 y Fd(1)1214 3591 y Ff(;:::)n(;j)1341 3599 y Fc(s)1378 3565 y Fk(\))40 b(and)g Fj(a)p Fk(\()p Fj(B)5 b Fk(\))1859 3518 y Ff(i)1883 3527 y Fd(1)1918 3518 y Ff(;:::)n(;i)2040 3526 y Fc(r)1859 3591 y Ff(j)1888 3600 y Fd(1)1922 3591 y Ff(;:::)o(;j)2050 3599 y Fc(s)2127 3565 y Fg(2)40 b Fi(K)64 b Fk(satisfying)39 b(the)h(transformation)315 3687 y(form)m(ula)c(\(2\))j(de\014nes)h(a)f (unique)g(tensor)h(\(indep)s(enden)m(t)g(of)e(the)h(c)m(hoice)g(of)g (the)g(basis\))315 3803 y Fj(t)27 b Fg(2)i Fj(V)550 3767 y Fh(\012)p Ff(r)665 3803 y Fg(\012)23 b Fk(\()p Fj(V)881 3767 y Fh(\003)921 3803 y Fk(\))959 3767 y Fh(\012)p Ff(s)1083 3803 y Fk(suc)m(h)34 b(that)e(\(1\))g(holds.)315 3919 y Fa(R)n(ule)44 b(for)h(physicists:)65 b(A)45 b(tensor)g(is)g(a)f (c)-5 b(ol)5 b(le)-5 b(ction)44 b(of)h(schemes)f(of)h(c)-5 b(o)g(e\016cients)44 b(that)315 4035 y(tr)-5 b(ansform)34 b(ac)-5 b(c)g(or)g(ding)34 b(to)h(the)f(tr)-5 b(ansformation)34 b(formula)h(for)g(tensors.)100 4152 y Fk(\(12\))41 b(Sho)m(w)d(that)f (\()p Fj(A;)17 b Fg(r)36 b Fk(:)h Fj(A)25 b Fg(\012)h Fj(A)37 b Fg(\000)-60 b(!)36 b Fj(A;)17 b(\021)40 b Fk(:)c Fi(K)60 b Fg(\000)-60 b(!)36 b Fj(A)p Fk(\))h(is)h(a)f Fi(K)17 b Fk(-algebra)42 b(if)37 b(and)g(only)g(if)g Fj(A)315 4297 y Fk(with)j(the)h(m)m(ultiplication)c Fj(A)28 b Fg(\002)g Fj(A)1708 4240 y Fh(\012)1677 4297 y Fg(\000)-60 b(!)42 b Fj(A)28 b Fg(\012)g Fj(A)2186 4240 y Fh(r)2157 4297 y Fg(\000)-59 b(!)41 b Fj(A)g Fk(and)g(the)g(unit)g Fj(\021)t Fk(\(1\))f(is)g(a)h(ring)315 4413 y(and)d Fj(\021)k Fk(:)c Fi(K)61 b Fg(\000)-59 b(!)37 b Fk(Cen)m(t)r(\()p Fj(A)p Fk(\))h(is)g(a)h(ring)e(homomorphism)f(in)m(to)i(the)h Fa(c)-5 b(enter)38 b Fk(of)g Fj(A)p Fk(,)j(where)315 4529 y(Cen)m(t)q(\()p Fj(A)p Fk(\))27 b(:=)h Fg(f)p Fj(a)g Fg(2)g Fj(A)p Fg(j8)p Fj(b)g Fg(2)g Fj(A)g Fk(:)g Fj(ab)g Fk(=)g Fj(ba)p Fg(g)p Fk(.)0 5020 y(Due)33 b(date:)43 b(T)-8 b(uesda)m(y)g(,)35 b(06.11.2001,)c(16:15)h(in)f(Lecture)j(Hall)d (138)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF