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3380 a Fy(,)889 3380 y SDict begin H.S end 889 3380 a Black Fy(20)p Black 979 3321 a SDict begin H.R end 979 3321 a 979 3380 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.st) cvn H.B /ANN pdfmark end 979 3380 a Fy(,)1035 3380 y SDict begin H.S end 1035 3380 a Black Fy(23)p Black 1126 3321 a SDict begin H.R end 1126 3321 a 1126 3380 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.v) cvn H.B /ANN pdfmark end 1126 3380 a Fy(].)41 b(A)30 b(more)h(detailed)e(review)h (can)h(b)s(e)e(seen)i(in)e([)2791 3380 y SDict begin H.S end 2791 3380 a Black Fy(1)p Black 2837 3321 a SDict begin H.R end 2837 3321 a 2837 3380 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ai1) cvn H.B /ANN pdfmark end 2837 3380 a Fy(].)225 3545 y(On)43 b(the)h(other)g(hand,)i(from)e(a)g (Lie)g(algebra)g Fw(g)f Fy(one)h(can)h(obtain)e(a)h(Lie)g(conformal)f (algebra)h FI(C)7 b(ur)s Fw(g)p Fy(,)125 3678 y(and)31 b(from)h(a)h(Lie)e(conformal)h(algebra)g FI(R)h Fy(one)g(can)f(get)i (an)e(asso)s(ciated)g(in\014nite-dimensional)c(Lie)j(algebra)125 3810 y(\()p FI(Lie)g(R)q Fy(\))p 435 3824 22 4 v 30 w(,)f(whic)m(h)f (is)g(called)h(the)g(annihilation)d(algebra)j(of)h FI(R)f Fy([)2332 3810 y SDict begin H.S end 2332 3810 a Black Fy(3)p Black 2378 3752 a SDict begin H.R end 2378 3752 a 2378 3810 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ak) cvn H.B /ANN pdfmark end 2378 3810 a Fy(].)41 b(Conformal)29 b(algebras)h(w)m(ere)h(in)m(tro)s(duced)125 3943 y(b)m(y)23 b(Kac)h([)449 3943 y SDict begin H.S end 449 3943 a Black Fy(13)p Black 540 3885 a SDict begin H.R end 540 3885 a 540 3943 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.k) cvn H.B /ANN pdfmark end 540 3943 a 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b(a)1076 3013 y Fu(i)1104 3064 y Ft(\()p Fu(k)1168 3074 y FB(i)1195 3064 y Ft(\))1226 3046 y Fv(\001)15 b(\001)g(\001)1332 3064 y Ft(\()p Fu(k)1396 3073 y FB(n)p FC(\000)p FJ(1)1517 3064 y Ft(\))1549 3046 y FI(a)1597 3013 y Fu(n)1669 3046 y Fy(=)25 b Fv(\000)p FI(k)1883 3060 y Fu(i)1911 3046 y FI(a)1959 3013 y Ft(1)1999 3064 y(\()p Fu(k)2063 3073 y FJ(1)2098 3064 y Ft(\))2129 3046 y Fv(\001)15 b(\001)g(\001)2235 3064 y Ft(\()p Fu(k)2299 3074 y FB(i)p FC(\000)p FJ(1)2404 3064 y Ft(\))2436 3046 y FI(a)2484 3013 y Fu(i)2512 3064 y Ft(\()p Fu(k)2576 3074 y FB(i)2603 3064 y Fs(\000)p Ft(1\))2724 3046 y Fv(\001)g(\001)g(\001)2831 3064 y Ft(\()p Fu(k)2895 3073 y FB(n)p FC(\000)p FJ(1)3016 3064 y Ft(\))3047 3046 y FI(a)3095 3013 y Fu(n)3163 3046 y Fy(for)20 b FI(i)26 b Fy(=)f(1)p FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(n)q Fv(\000)q Fy(1,)437 3211 y FI(a)485 3178 y Ft(1)524 3230 y(\()p Fu(k)588 3239 y FJ(1)623 3230 y Ft(\))655 3211 y Fv(\001)g(\001)g(\001)761 3230 y Ft(\()p Fu(k)825 3239 y FB(n)p FC(\000)p FJ(1)946 3230 y Ft(\))977 3211 y FI(@)5 b(a)1078 3178 y Fu(n)1151 3211 y Fy(=)25 b FI(@)5 b Fy(\()p FI(a)1383 3178 y Ft(1)1423 3230 y(\()p Fu(k)1487 3239 y FJ(1)1522 3230 y Ft(\))1553 3211 y FI(a)1601 3178 y Ft(2)1641 3230 y(\()p Fu(k)1705 3239 y FJ(2)1740 3230 y Ft(\))1771 3211 y Fv(\001)15 b(\001)g(\001)1877 3230 y Ft(\()p Fu(k)1941 3239 y FB(n)p FC(\000)p FJ(1)2062 3230 y Ft(\))2094 3211 y FI(a)2142 3178 y Fu(n)2189 3211 y Fy(\))g(+)2325 3143 y Fm(P)2421 3169 y Fu(n)p Fs(\000)p Ft(1)2421 3238 y Fu(i)p Ft(=1)2573 3211 y FI(k)2620 3225 y Fu(i)2648 3211 y FI(a)2696 3178 y Ft(1)2736 3230 y(\()p Fu(k)2800 3239 y FJ(1)2835 3230 y Ft(\))2866 3211 y Fv(\001)g(\001)g (\001)2972 3230 y Ft(\()p Fu(k)3036 3240 y FB(i)p FC(\000)p FJ(1)3141 3230 y Ft(\))3173 3211 y FI(a)3221 3178 y Fu(i)3249 3230 y Ft(\()p Fu(k)3313 3240 y FB(i)3340 3230 y Fs(\000)p Ft(1\))3461 3211 y Fv(\001)g(\001)g(\001)3567 3230 y Ft(\()p Fu(k)3631 3239 y FB(n)p FC(\000)p FJ(1)3753 3230 y Ft(\))3784 3211 y FI(a)3832 3178 y Fu(n)3879 3211 y Fy(,)225 3376 y(\(C3)371 3343 y Fs(0)394 3376 y Fy(\))31 b FI(a)508 3343 y Ft(1)547 3395 y(\()p Fu(k)611 3404 y FJ(1)646 3395 y Ft(\))678 3376 y FI(a)726 3343 y Ft(2)765 3395 y(\()p Fu(k)829 3404 y FJ(2)864 3395 y Ft(\))896 3376 y Fv(\001)15 b(\001)g(\001)1002 3395 y Ft(\()p Fu(k)1066 3404 y FB(n)p FC(\000)p FJ(1)1187 3395 y Ft(\))1218 3376 y FI(a)1266 3343 y Fu(n)1339 3376 y Fy(=)25 b Fv(\000)p FI(a)1554 3343 y Ft(1)1593 3395 y(\()p Fu(k)1657 3404 y FJ(1)1692 3395 y Ft(\))1738 3376 y Fv(\001)15 b(\001)g(\001)i FI(a)1908 3343 y Fu(i)p Ft(+1)2026 3395 y(\()p Fu(k)2090 3405 y FB(i)p FJ(+1)2194 3395 y Ft(\))2225 3376 y FI(a)2273 3343 y Fu(i)2301 3395 y Ft(\()p Fu(k)2365 3405 y FB(i)2392 3395 y Ft(\))2424 3376 y Fv(\001)e(\001)g(\001)2530 3395 y Ft(\()p Fu(k)2594 3404 y FB(n)p FC(\000)p FJ(1)2715 3395 y Ft(\))2746 3376 y FI(a)2794 3343 y Fu(n)2872 3376 y Fy(for)30 b FI(i)25 b Fy(=)g(1)p FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(n)21 b Fv(\000)f Fy(1,)406 3542 y(=)25 b Fv(\000)685 3473 y Fm(P)588 3623 y Fu(j;j)666 3632 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)877 3542 y Fy(\()p Fv(\000)p Fy(1\))1063 3509 y Fu(j)t Ft(+)p Fu(k)1188 3518 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1400 3509 y Fs(\000)p Fu(j)1484 3518 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1700 3468 y Fm(\000)1742 3501 y Fu(k)1779 3510 y FJ(1)1746 3573 y Fu(j)1775 3582 y FJ(1)1813 3468 y Fm(\001)1870 3542 y Fv(\001)15 b(\001)g(\001)1991 3468 y Fm(\000)2033 3501 y Fu(k)2070 3510 y FB(n)p FC(\000)p FJ(2)2037 3573 y Fu(j)2066 3582 y FB(n)p FC(\000)p FJ(2)2190 3468 y Fm(\001)2232 3542 y FI(@)2285 3509 y Ft(\()p Fu(j)t Ft(\))2377 3542 y FI(a)2425 3509 y Ft(1)2464 3560 y(\()p Fu(j)2520 3569 y FJ(1)2555 3560 y Ft(\))2586 3542 y FI(a)2634 3509 y Ft(2)2674 3560 y(\()p Fu(j)2730 3569 y FJ(2)2764 3560 y Ft(\))2811 3542 y Fv(\001)g(\001)g(\001)h FI(a)2980 3509 y Fu(n)3027 3560 y Ft(\()p Fu(j)t Ft(+)p Fu(k)3179 3569 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)3391 3560 y Fs(\000)p Fu(j)3475 3569 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)3687 3560 y Ft(\))3718 3542 y FI(a)3766 3509 y Fu(n)p Fs(\000)p Ft(1)3903 3542 y Fy(,)225 3813 y(\(C4)371 3780 y Fs(0)394 3813 y Fy(\))496 3727 y Fu(n)469 3745 y Fm(P)460 3895 y Fu(i)p Ft(=1)574 3813 y Fy(\()p Fv(\000)p Fy(1\))760 3780 y Fu(n)p Fs(\000)p Fu(i)887 3813 y FI(a)935 3780 y Ft(1)974 3832 y(\()p Fu(k)1038 3841 y FJ(1)1073 3832 y Ft(\))1120 3813 y Fv(\001)f(\001)g(\001)h FI(a)1289 3780 y Fu(i)p Fs(\000)p Ft(1)1408 3832 y(\()p Fu(k)1472 3842 y FB(i)p FC(\000)p FJ(1)1576 3832 y Ft(\))1608 3813 y FI(a)1656 3780 y Fu(i)p Ft(+1)1774 3832 y(\()p Fu(k)1838 3842 y FB(i)p FJ(+1)1942 3832 y Ft(\))1974 3813 y Fv(\001)f(\001)g (\001)2080 3832 y Ft(\()p Fu(k)2144 3841 y FB(n)p FC(\000)p FJ(1)2265 3832 y Ft(\))2296 3813 y FI(a)2344 3780 y Fu(n)2391 3832 y Ft(\()p Fu(k)2455 3840 y FB(n)2498 3832 y Ft(\))2530 3813 y Fy(\()p FI(a)2613 3780 y Fu(i)2641 3832 y Ft(\()p Fu(k)2705 3842 y FB(i)2732 3832 y Ft(\))2763 3813 y FI(b)2802 3780 y Ft(2)2842 3832 y(\()p Fu(k)2906 3841 y FB(n)p FJ(+1)3026 3832 y Ft(\))3057 3813 y Fv(\001)g(\001)g(\001)3163 3832 y Ft(\()p Fu(k)3227 3841 y FJ(2)p FB(n)p FC(\000)p FJ(2)3379 3832 y Ft(\))3410 3813 y FI(b)3449 3780 y Fu(n)3496 3813 y Fy(\))406 4031 y(=)575 3963 y Fm(P)502 4113 y Fu(j)531 4122 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)758 3958 y Fm(\000)800 3991 y Fu(k)837 4000 y FJ(1)804 4063 y Fu(j)833 4072 y FJ(1)871 3958 y Fm(\001)928 4031 y Fv(\001)g(\001)g(\001)1049 3958 y Fm(\000)1091 3991 y Fu(k)1128 4000 y FB(n)p FC(\000)p FJ(1)1095 4063 y Fu(j)1124 4072 y FB(n)p FC(\000)p FJ(1)1249 3958 y Fm(\001)1290 4031 y Fy(\()p FI(a)1373 3998 y Ft(1)1413 4050 y(\()p Fu(j)1469 4059 y FJ(1)1504 4050 y Ft(\))1535 4031 y Fv(\001)g(\001)g (\001)1641 4050 y Ft(\()p Fu(j)1697 4059 y FB(n)p FC(\000)p FJ(1)1818 4050 y Ft(\))1850 4031 y FI(a)1898 3998 y Fu(n)1945 4031 y Fy(\))1980 4050 y Ft(\(+)p Fu(k)2099 4059 y FJ(1)p FC(!)p FB(n)2233 4050 y Fs(\000)p Fu(j)2317 4059 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2529 4050 y Ft(\))2560 4031 y FI(b)2599 3998 y Ft(2)2639 4050 y(\()p Fu(k)2703 4059 y FB(n)p FJ(+1)2822 4050 y Ft(\))2854 4031 y Fv(\001)g(\001)g(\001)2960 4050 y Ft(\()p Fu(k)3024 4059 y FJ(2)p FB(n)p FC(\000)p FJ(2)3176 4050 y Ft(\))3207 4031 y FI(b)3246 3998 y Fu(n)3293 4031 y Fy(.)125 4291 y SDict begin H.S end 125 4291 a 125 4291 a SDict begin 16 H.A end 125 4291 a 125 4291 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.3) cvn /DEST pdfmark end 125 4291 a Black FH(Example)36 b(2.3.)p Black 43 w Fo(L)-5 b(et)35 b Fy(\()p Fw(g)p FI(;)15 b Fy([)p Fv(\001)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b Fv(\001)p Fy(]\))36 b Fo(b)-5 b(e)35 b(an)g FI(n)p Fo(-Lie)e(algebr)-5 b(a.)49 b(Then)35 b FI(C)7 b(ur)37 b Fw(g)28 b Fy(=)h Fp(C)18 b Fy([)p FI(@)5 b Fy(])28 b Fv(\012)21 b Fw(g)35 b Fo(c)-5 b(arries)35 b(an)h FI(n)p Fo(-Lie)125 4424 y(c)-5 b(onformal)35 b(algebr)-5 b(a)34 b(structur)-5 b(e)33 b(given)f(by)601 4608 y FI(g)647 4571 y Ft(1)687 4630 y(\(0\))781 4608 y Fv(\001)15 b(\001)g(\001)887 4627 y Ft(\(0\))982 4608 y FI(g)1028 4571 y Fu(n)1101 4608 y Fy(=)24 b([)p FI(g)1267 4571 y Ft(1)1308 4608 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(g)1571 4571 y Fu(n)1619 4608 y Fy(])p FI(;)48 b(g)1763 4571 y Ft(1)1803 4630 y(\()p Fu(k)1867 4639 y FJ(1)1902 4630 y Ft(\))1933 4608 y Fv(\001)15 b(\001)g(\001)2039 4627 y Ft(\()p Fu(k)2103 4636 y FB(n)p FC(\000)p FJ(1)2224 4627 y Ft(\))2256 4608 y FI(g)2302 4571 y Fu(n)2375 4608 y Fy(=)25 b(0)33 b Fo(if)f FI(k)2684 4622 y Ft(1)2744 4608 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(k)3099 4622 y Fu(n)p Fs(\000)p Ft(1)3262 4608 y FI(>)25 b Fy(0)p FI(;)125 4792 y Fo(for)33 b FI(g)316 4759 y Ft(1)356 4792 y FI(;)15 b(g)442 4759 y Ft(2)483 4792 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(g)745 4759 y Fu(n)819 4792 y Fv(2)24 b Fw(g)p Fo(.)42 b(The)33 b(c)-5 b(orr)g(esp)g(onding)36 b Fv(f)p FI(\025)1878 4806 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2121 4792 y Fv(g)p Fo(-br)-5 b(acket)33 b(is)1427 4977 y Fy([)p FI(g)1498 4939 y Ft(1)1538 4999 y Fu(\025)1579 5008 y FJ(1)1618 4977 y Fv(\001)15 b(\001)g(\001)1724 4992 y Fu(\025)1765 5001 y FB(n)p FC(\000)p FJ(1)1890 4977 y FI(g)1936 4939 y Fu(n)1984 4977 y Fy(])25 b(=)g([)p FI(g)2201 4939 y Ft(1)2241 4977 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(g)2504 4939 y Fu(n)2552 4977 y Fy(])p FI(:)125 5161 y(C)7 b(ur)40 b Fw(g)d Fo(is)h(c)-5 b(al)5 b(le)-5 b(d)39 b(the)f Fl(curr)-5 b(ent)44 b FI(n)p Fl(-Lie)g(c)-5 b(onformal)43 b(algebr)-5 b(a)37 b Fo(asso)-5 b(ciate)g(d)41 b(to)d Fw(g)p Fo(.)57 b(It)38 b(is)g(not)h(di\016cult)f(to)125 5294 y(che)-5 b(ck)41 b(that)i FI(C)7 b(ur)43 b Fw(g)e Fo(is)g(simple)h(if)f(and)h(only)g(if)f Fw(g)g Fo(is)g(simple)h(and)g (the)g(isomorphism)i(of)d FI(n)p Fo(-Lie)f(algebr)-5 b(as)125 5427 y(c)g(orr)g(esp)g(onds)48 b(to)d(the)g(isomorphism)j(of)c (asso)-5 b(ciate)g(d)47 b(curr)-5 b(ent)46 b FI(n)p Fo(-Lie)d(c)-5 b(onformal)47 b(algebr)-5 b(as.)79 b(Sinc)-5 b(e)44 b(ther)-5 b(e)p Black Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(4)893 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 125 77 a Fo(exists)43 b(only)g(one)g (\014nite-dimensional)h(simple)g FI(n)p Fo(-Lie)d(algebr)-5 b(a)44 b(up)f(to)g(isomorphism)j Fy([)3220 77 y SDict begin H.S end 3220 77 a Black Fy(16)p Black 3311 18 a SDict begin H.R end 3311 18 a 3311 77 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Li) cvn H.B /ANN pdfmark end 3311 77 a Fy(])p Fo(,)f(\014nite)e(simple)125 210 y(curr)-5 b(ent)33 b FI(n)p Fo(-Lie)f(c)-5 b(onformal)34 b(algebr)-5 b(a)34 b(is)f(unique)f(up)g(to)i(isomorphism.)225 445 y Fy(There)25 b(is)g(an)i FI(n)p Fy(-Lie)e(algebra)h(structure)g(con)m (tained)g(in)f(the)i(distribution)22 b(notion)k(of)g(an)g FI(n)p Fy(-Lie)g(conformal)125 578 y(algebra.)125 781 y SDict begin H.S end 125 781 a 125 781 a SDict begin 16 H.A end 125 781 a 125 781 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.4) cvn /DEST pdfmark end 125 781 a Black FH(Lemma)33 b(2.4.)p Black 42 w Fo(L)-5 b(et)34 b Fy(\()p FI(R)q(;)15 b Fv(f\017)1094 799 y Ft(\()p Fu(k)1158 808 y FJ(1)1193 799 y Ft(\))1224 781 y Fv(\017)1270 799 y Ft(\()p Fu(k)1334 808 y FJ(2)1369 799 y Ft(\))1400 781 y Fv(\001)g(\001)g(\001)1506 799 y Ft(\()p Fu(k)1570 808 y FB(n)p FC(\000)p FJ(1)1691 799 y Ft(\))1723 781 y Fv(\017g)p Fy(\))34 b Fo(b)-5 b(e)33 b(an)h FI(n)p Fo(-Lie)d(c)-5 b(onformal)36 b(algebr)-5 b(a.)44 b(Then)33 b Fv(\017)3424 799 y Ft(\(0\))3519 781 y Fv(\017)3564 799 y Ft(\(0\))3659 781 y Fv(\001)15 b(\001)g(\001)3765 799 y Ft(\(0\))3859 781 y Fv(\017)125 914 y Fo(is)31 b(an)h FI(n)p Fo(-Lie)e(br)-5 b(acket)32 b(of)f FI(R)q Fo(,)g(and)i(with)f(r)-5 b(esp)g(e)g(ct)33 b(to)f(the)f Fy(0)p Fo(-th)h(pr)-5 b(o)g(duct)34 b FI(@)5 b(R)32 b Fo(is)f(an)h(ide)-5 b(al)32 b(of)g FI(R)g Fo(so)g(that)g FI(R)q(=@)5 b(R)125 1046 y Fo(is)32 b(an)i FI(n)p Fo(-Lie)d(algebr)-5 b(a.)p Black 125 1268 a(Pr)g(o)g(of.)p Black 43 w Fy(Denote)22 b([)p FI(a)781 1235 y Ft(1)820 1268 y FI(;)15 b(a)908 1235 y Ft(2)948 1268 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1213 1235 y Fu(n)1260 1268 y Fy(])26 b(:=)f FI(a)1480 1235 y Ft(1)1520 1287 y(\(0\))1614 1268 y FI(a)1662 1235 y Ft(2)1701 1287 y(\(0\))1796 1268 y Fv(\001)15 b(\001)g(\001)1902 1287 y Ft(\(0\))1996 1268 y FI(a)2044 1235 y Fu(n)2111 1268 y Fy(for)20 b(all)g FI(a)2405 1235 y Ft(1)2444 1268 y FI(;)15 b(a)2532 1235 y Ft(2)2572 1268 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2837 1235 y Fu(n)2909 1268 y Fv(2)25 b FI(R)q Fy(.)37 b(The)20 b(sk)m(ew-symmetry)125 1401 y(and)30 b(Filipp)s(o)m(v)e(iden)m(tit)m(y) h(can)i(b)s(e)f(easily)f(obtained)h(from)g(\(C3)2273 1368 y Fs(0)2297 1401 y Fy(\))g(and)g(\(C4)2685 1368 y Fs(0)2709 1401 y Fy(\).)41 b(\(C2)2956 1368 y Fs(0)2980 1401 y Fy(\))30 b(giv)m(es)533 1577 y([)p FI(a)606 1540 y Ft(1)646 1577 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(@)5 b(a)963 1540 y Fu(i)992 1577 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1257 1540 y Fu(n)1304 1577 y Fy(])26 b(=)f(0)31 b FI(if)40 b(i)25 b Fy(=)g(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b(n)20 b Fv(\000)g Fy(1)p FI(;)46 b(and)31 b Fy([)p FI(a)2679 1540 y Ft(1)2718 1577 y FI(;)15 b(a)2806 1540 y Ft(2)2846 1577 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(@)5 b(a)3164 1540 y Fu(n)3212 1577 y Fy(])25 b Fv(2)g FI(@)5 b(R)q(:)125 1753 y Fy(Therefore,)30 b FI(@)5 b(R)32 b Fy(is)d(an)h(ideal)f(of)i FI(R)g Fy(and)f FI(R)q(=@)5 b(R)31 b Fy(is)f(an)g FI(n)p Fy(-Lie)g(algebra.)1266 b Fk(\003)225 1975 y Fy(Kac)33 b(et.)49 b(al.)f(classi\014ed)31 b(Lie)i(conformal)f(algebras)h(of)g(rank)f(one)i(in)d([)2647 1975 y SDict begin H.S end 2647 1975 a Black Fy(3)p Black 2693 1916 a SDict begin H.R end 2693 1916 a 2693 1975 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ak) cvn H.B /ANN pdfmark end 2693 1975 a Fy(])i(and)f(sho)m(w)m(ed)i(Virasoro)e (algebra)125 2108 y(is)j(the)h(only)g(one)g(non-comm)m(utativ)m(e)i (Lie)d(conformal)h(algebras)g(of)g(rank)g(one.)58 b(In)m(terestingly)-8 b(,)38 b(w)m(e)e(\014nd)f(a)125 2241 y(di\013eren)m(t)30 b(conclusion)f(in)g FI(n)p Fy(-Lie)g(conformal)h(algebra)h(when)e FI(n)c Fv(\025)g Fy(3.)125 2444 y SDict begin H.S end 125 2444 a 125 2444 a SDict begin 16 H.A end 125 2444 a 125 2444 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.5) cvn /DEST pdfmark end 125 2444 a Black FH(Prop)s(osition)39 b(2.5.)p Black 43 w Fo(L)-5 b(et)35 b FI(R)30 b Fy(:=)f Fp(C)17 b Fy([)q FI(@)5 b Fy(])p FI(e)41 b Fo(b)-5 b(e)35 b(an)g FI(n)p Fo(-Lie)e(c)-5 b(onformal)37 b(algebr)-5 b(a)36 b(of)f(r)-5 b(ank)36 b(one)f(for)g FI(n)29 b Fv(\025)g Fy(3)p Fo(.)48 b(Then)125 2576 y FI(R)33 b Fo(is)g(c)-5 b(ommutative.)p Black 125 2798 a(Pr)g(o)g(of.)p Black 43 w Fy(Set)30 b([)p FI(e)629 2813 y Fu(\025)670 2822 y FJ(1)710 2798 y FI(e)752 2813 y Fu(\025)793 2822 y FJ(2)832 2798 y Fv(\001)15 b(\001)g(\001)938 2813 y Fu(\025)979 2822 y FB(n)p FC(\000)p FJ(1)1104 2798 y FI(e)p Fy(])26 b(=)f FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)1529 2812 y Ft(1)1569 2798 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1839 2812 y Fu(n)p Fs(\000)p Ft(1)1976 2798 y Fy(\))p FI(e)p Fy(.)42 b(F)-8 b(rom)31 b(\(C2\))g(and)e(\(C4\),) j(w)m(e)f(ha)m(v)m(e)684 2974 y FI(f)10 b Fy(\()p Fv(\000)p FI(\025)898 2988 y Ft(1)957 2974 y Fv(\000)20 b(\001)15 b(\001)g(\001)21 b(\000)f FI(\025)1318 2988 y Fu(n)1365 2974 y FI(;)15 b(\025)1458 2988 y Ft(1)1498 2974 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1768 2988 y Fu(n)p Fs(\000)p Ft(1)1906 2974 y Fy(\))p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2177 2988 y Ft(1)2237 2974 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2598 2988 y Fu(n)2645 2974 y FI(;)15 b(\025)2738 2988 y Fu(n)p Ft(+1)2876 2974 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)3145 2988 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)3318 2974 y Fy(\))614 3197 y(=)744 3084 y Fu(n)699 3111 y Fm(X)708 3307 y Fu(i)p Ft(=1)831 3197 y Fy(\()p Fv(\000)p Fy(1\))1017 3160 y Fu(n)p Fs(\000)p Fu(i)1144 3197 y FI(f)10 b Fy(\()p FI(@)25 b Fy(+)1442 3084 y Fu(n)1398 3111 y Fm(X)1402 3307 y Fu(j)t Ft(=1)1408 3375 y Fu(j)t Fs(6)p Ft(=)p Fu(i)1544 3197 y FI(\025)1597 3211 y Fu(j)1634 3197 y FI(;)15 b(\025)1727 3211 y Fu(i)1755 3197 y FI(;)g(\025)1848 3211 y Fu(n)p Ft(+1)1986 3197 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2256 3211 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2428 3197 y Fy(\))p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2699 3211 y Ft(1)2740 3197 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)2975 3173 y Fy(^)2956 3197 y FI(\025)3009 3211 y Fu(i)3038 3197 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)3308 3211 y Fu(n)3355 3197 y Fy(\))p FI(:)3718 3197 y SDict begin H.S end 3718 3197 a 3718 3197 a SDict begin 19.25 H.A end 3718 3197 a 3718 3197 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.2) cvn /DEST pdfmark end 3718 3197 a Fy(\(2.2\))125 3516 y(Besides,)30 b(\(C3\))h(implies)924 3692 y FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)1160 3706 y Ft(1)1200 3692 y FI(;)g Fv(\001)g(\001)g (\001)32 b FI(;)15 b(\025)1470 3706 y Fu(n)p Fs(\000)p Ft(1)1607 3692 y Fy(\))26 b(=)f Fv(\000)p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2071 3706 y Ft(1)2110 3692 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2380 3706 y Fu(j)t Ft(+1)2507 3692 y FI(;)g(\025)2600 3706 y Fu(j)2637 3692 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2907 3706 y Fu(n)p Fs(\000)p Ft(1)3045 3692 y Fy(\))p FI(;)125 3868 y Fy(for)35 b FI(j)40 b Fy(=)33 b(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)34 b FI(;)15 b(n)23 b Fv(\000)h Fy(2.)56 b(Hence)37 b(if)d FI(\025)1515 3882 y Fu(j)1586 3868 y Fy(=)f FI(\025)1743 3883 y Fu(k)1822 3868 y Fy(for)i(some)h FI(j)k Fv(6)p Fy(=)33 b FI(k)39 b Fy(or)d FI(\025)2636 3882 y Fu(j)2706 3868 y Fy(=)e(0)i(for)f(some)h FI(j)5 b Fy(,)38 b(then)d(w)m(e)h(ha)m(v)m(e)125 4001 y FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)361 4015 y Ft(1)401 4001 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)670 4015 y Fu(n)p Fs(\000)p Ft(1)808 4001 y Fy(\))26 b(=)f(0.)225 4166 y(Plugging)k FI(\025)658 4180 y Ft(2)722 4166 y Fy(=)c FI(\025)871 4180 y Fu(n)p Ft(+1)1008 4166 y FI(;)15 b(\025)1101 4180 y Ft(3)1167 4166 y Fy(=)25 b FI(\025)1316 4180 y Fu(n)p Ft(+2)1453 4166 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)1722 4180 y Fu(n)p Fs(\000)p Ft(1)1885 4166 y Fy(=)25 b FI(\025)2034 4180 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2237 4166 y Fy(in)m(to)30 b(\()2456 4166 y SDict begin H.S end 2456 4166 a Black Fy(2.2)p Black 2573 4107 a SDict begin H.R end 2573 4107 a 2573 4166 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 2573 4166 a Fy(\),)h(w)m(e)g(can)f(obtain)g(that)715 4342 y FI(f)10 b Fy(\()p Fv(\000)p FI(\025)929 4356 y Ft(1)988 4342 y Fv(\000)20 b(\001)15 b(\001)g(\001)22 b(\000)d FI(\025)1349 4356 y Fu(n)1396 4342 y FI(;)c(\025)1489 4356 y Ft(1)1529 4342 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1799 4356 y Fu(n)p Fs(\000)p Ft(1)1937 4342 y Fy(\))p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2208 4356 y Ft(1)2268 4342 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2629 4356 y Fu(n)2676 4342 y FI(;)15 b(\025)2769 4356 y Fu(n)p Ft(+1)2907 4342 y FI(;)g Fv(\001)g(\001)g (\001)31 b FI(;)15 b(\025)3176 4356 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)3349 4342 y Fy(\))645 4565 y(=)o(\()p Fv(\000)p Fy(1\))901 4527 y Fu(n)p Fs(\000)p Ft(1)1039 4565 y FI(f)10 b Fy(\()p FI(@)26 b Fy(+)1338 4451 y Fu(n)1293 4479 y Fm(X)1298 4674 y Fu(j)t Ft(=2)1440 4565 y FI(\025)1493 4579 y Fu(j)1529 4565 y FI(;)15 b(\025)1622 4579 y Ft(1)1662 4565 y FI(;)g(\025)1755 4579 y Ft(2)1795 4565 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2065 4579 y Fu(n)p Fs(\000)p Ft(1)2203 4565 y Fy(\))p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2474 4579 y Ft(2)2514 4565 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2783 4579 y Fu(n)2831 4565 y Fy(\))p FI(:)3718 4565 y SDict begin H.S end 3718 4565 a 3718 4565 a SDict begin 19.25 H.A end 3718 4565 a 3718 4565 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.3) cvn /DEST pdfmark end 3718 4565 a Fy(\(2.3\))125 4816 y(Substituting)28 b FI(\025)699 4830 y Ft(1)758 4816 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)1119 4830 y Fu(n)1192 4816 y Fy(=)25 b(0)30 b(in)m(to)h(\()1583 4816 y SDict begin H.S end 1583 4816 a Black Fy(2.3)p Black 1699 4757 a SDict begin H.R end 1699 4757 a 1699 4816 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 1699 4816 a Fy(\),)g(w)m(e)g(obtain)891 5041 y FI(f)10 b Fy(\()p FI(@)25 b Fy(+)1189 4927 y Fu(n)1145 4955 y Fm(X)1149 5150 y Fu(j)t Ft(=2)1291 5041 y FI(\025)1344 5055 y Fu(j)1381 5041 y FI(;)15 b Fv(\000)1551 4927 y Fu(n)1507 4955 y Fm(X)1511 5150 y Fu(j)t Ft(=2)1653 5041 y FI(\025)1706 5055 y Fu(j)1743 5041 y FI(;)g(\025)1836 5055 y Ft(2)1876 5041 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2146 5055 y Fu(n)p Fs(\000)p Ft(1)2283 5041 y Fy(\))p FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)2554 5055 y Ft(2)2594 5041 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2864 5055 y Fu(n)2911 5041 y Fy(\))26 b(=)f(0)p FI(;)125 5294 y Fy(whic)m(h)35 b(implies)e(either)i FI(f)10 b Fy(\()p FI(@)29 b Fy(+)1230 5226 y Fm(P)1326 5252 y Fu(n)1326 5321 y(j)t Ft(=2)1468 5294 y FI(\025)1521 5308 y Fu(j)1557 5294 y FI(;)15 b Fv(\000)1683 5226 y Fm(P)1779 5252 y Fu(n)1779 5321 y(j)t Ft(=2)1921 5294 y FI(\025)1974 5308 y Fu(j)2011 5294 y FI(;)g(\025)2104 5308 y Ft(2)2144 5294 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2413 5308 y Fu(n)p Fs(\000)p Ft(1)2551 5294 y Fy(\))35 b(=)f(0)j(or)f FI(f)10 b Fy(\()p FI(@)5 b(;)15 b(\025)3161 5308 y Ft(2)3201 5294 y FI(;)g Fv(\001)g(\001)g (\001)31 b FI(;)15 b(\025)3470 5308 y Fu(n)3518 5294 y Fy(\))35 b(=)f(0.)58 b(In)125 5427 y(b)s(oth)30 b(cases,)h(it)f(can)h (b)s(e)e(inferred)g(that)i FI(f)10 b Fy(\()p FI(x)1650 5441 y Ft(1)1689 5427 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(x)1958 5441 y Fu(n)2005 5427 y Fy(\))25 b(=)g(0.)p Black Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)118 b(5)p Black 225 77 a Fy(Therefore,)30 b(w)m(e)h(ha)m(v)m(e)g([)p FI(e)1072 92 y Fu(\025)1113 101 y FJ(1)1153 77 y FI(e)1195 92 y Fu(\025)1236 101 y FJ(2)1275 77 y Fv(\001)15 b(\001)g(\001)1381 92 y Fu(\025)1422 101 y FB(n)p FC(\000)p FJ(1)1547 77 y FI(e)p Fy(])26 b(=)f(0.)2028 b Fk(\003)225 312 y Fy(Ho)m(w)m(ev)m (er,)32 b(one)f(can)g(easily)e(\014nd)g(a)i(non-comm)m(utativ)m(e)g FI(n)p Fy(-Lie)f(conformal)g(algebra)g(free)h(of)f(rank)g(t)m(w)m(o.) 125 523 y SDict begin H.S end 125 523 a 125 523 a SDict begin 16 H.A end 125 523 a 125 523 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.6) cvn /DEST pdfmark end 125 523 a Black FH(Prop)s(osition)37 b(2.6.)p Black 43 w Fo(L)-5 b(et)33 b FI(R)28 b Fy(:=)f Fp(C)17 b Fy([)p FI(@)6 b Fy(])p FI(e)1478 537 y Ft(1)1545 523 y Fv(\010)20 b Fp(C)e Fy([)p FI(@)5 b Fy(])q FI(e)1842 537 y Ft(2)1921 523 y Fo(b)-5 b(e)33 b(an)h FI(n)p Fo(-Lie)e(c)-5 b(onformal)36 b(algebr)-5 b(a)34 b(fr)-5 b(e)g(e)34 b(of)g(r)-5 b(ank)34 b(two.)45 b(If)125 655 y FI(I)36 b Fy(:=)29 b Fp(C)17 b Fy([)q FI(@)5 b Fy(])p FI(e)532 669 y Ft(2)612 655 y Fo(is)34 b(a)h(c)-5 b(onformal)37 b(ide)-5 b(al)36 b(of)f FI(R)g Fo(and)h FI(n)28 b Fv(\025)h Fy(3)p Fo(,)35 b FI(R)g Fo(is)g(determine)-5 b(d)36 b(by)f Fy([)p FI(e)2979 669 y Ft(1)3019 673 y Fu(\025)3060 682 y FJ(1)3099 655 y Fv(\001)15 b(\001)g(\001)3205 670 y Fu(\025)3246 679 y FB(n)p FC(\000)p FJ(2)3371 655 y FI(e)3413 669 y Ft(1)3453 673 y Fu(\025)3494 682 y FB(n)p FC(\000)p FJ(1)3619 655 y FI(e)3661 669 y Ft(2)3701 655 y Fy(])34 b Fo(and)125 788 y Fy([)p FI(e)192 802 y Ft(1)232 806 y Fu(\025)273 815 y FJ(1)312 788 y Fv(\001)15 b(\001)g(\001)418 803 y Fu(\025)459 812 y FB(n)p FC(\000)p FJ(2)584 788 y FI(e)626 802 y Ft(1)666 788 y Fy(])p Fo(.)42 b(F)-7 b(urthermor)i(e,)35 b(the)e(br)-5 b(ackets)33 b(of)g FI(R)g Fo(ar)-5 b(e)34 b(given)e(by)225 953 y(\(i\))g(either)728 1197 y Fy([)p FI(e)795 1211 y Fu(i)819 1220 y FJ(1)859 1224 y Fu(\025)900 1233 y FJ(1)939 1197 y FI(e)981 1211 y Fu(i)1005 1220 y FJ(2)1044 1224 y Fu(\025)1085 1233 y FJ(2)1124 1197 y Fv(\001)15 b(\001)g(\001)1230 1212 y Fu(\025)1271 1221 y FB(n)p FC(\000)p FJ(1)1396 1197 y FI(e)1438 1211 y Fu(i)1462 1219 y FB(n)1509 1197 y Fy(])25 b(=)1655 1042 y Fm(\()1729 1135 y Fy(0)p FI(;)926 b Fo(if)33 b FI(i)2845 1149 y Ft(1)2900 1135 y Fv(\001)15 b(\001)g(\001)h FI(i)3052 1149 y Fu(n)3124 1135 y Fv(\025)25 b Fy(2)p FI(;)1729 1196 y Fm(Q)1814 1291 y Ft(1)p Fs(\024)p Fu(i<j)t Fs(\024)p Fu(n)p Fs(\000)p Ft(1)2208 1264 y Fy(\()p FI(\025)2296 1278 y Fu(i)2345 1264 y Fv(\000)20 b FI(\025)2489 1278 y Fu(j)2525 1264 y Fy(\))p FI(g)s(;)94 b Fo(if)33 b FI(i)2845 1278 y Ft(1)2900 1264 y Fv(\001)15 b(\001)g(\001)h FI(i)3052 1278 y Fu(n)3124 1264 y Fy(=)25 b(1)p FI(;)3718 1197 y SDict begin H.S end 3718 1197 a 3718 1197 a SDict begin 19.25 H.A end 3718 1197 a 3718 1197 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.4) cvn /DEST pdfmark end 3718 1197 a Fy(\(2.4\))125 1441 y Fo(wher)-5 b(e)34 b FI(i)413 1455 y Ft(1)453 1441 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(i)701 1455 y Fu(n)775 1441 y Fv(2)27 b(f)p Fy(1)p FI(;)15 b Fy(2)p Fv(g)p Fo(,)35 b FI(g)i Fo(is)d(a)g(p)-5 b(olynomial)36 b(in)e Fv(S)7 b(P)2108 1455 y Fu(n)p Fs(\000)p Ft(1)2245 1441 y Fy([)p FI(@)e Fy(])34 b Fo(and)h Fv(S)7 b(P)2692 1455 y Fu(n)p Fs(\000)p Ft(1)2863 1441 y Fo(is)33 b(the)h(set)g(of)f(al)5 b(l)34 b(symmetric)125 1573 y(p)-5 b(olynomials)36 b(with)d FI(n)20 b Fv(\000)g Fy(1)33 b Fo(variables.)225 1739 y(\(ii\))f(or)1138 2040 y Fy([)p FI(e)1205 2054 y Fu(i)1229 2063 y FJ(1)1268 2067 y Fu(\025)1309 2076 y FJ(1)1348 2040 y FI(e)1390 2054 y Fu(i)1414 2063 y FJ(2)1453 2067 y Fu(\025)1494 2076 y FJ(2)1533 2040 y Fv(\001)15 b(\001)g(\001)1639 2055 y Fu(\025)1680 2064 y FB(n)p FC(\000)p FJ(1)1805 2040 y FI(e)1847 2054 y Fu(i)1871 2062 y FB(n)1918 2040 y Fy(])26 b(=)2065 1826 y Fm(8)2065 1908 y(>)2065 1935 y(<)2065 2099 y(>)2065 2126 y(:)2146 1914 y Fy(0)p FI(;)100 b Fo(if)32 b FI(i)2435 1928 y Ft(1)2490 1914 y Fv(\001)15 b(\001)g(\001)h FI(i)2642 1928 y Fu(n)2715 1914 y Fv(\025)25 b Fy(4)p FI(;)2146 2043 y(h;)93 b Fo(if)32 b FI(i)2435 2057 y Ft(1)2490 2043 y Fv(\001)15 b(\001)g(\001)h FI(i)2642 2057 y Fu(n)2715 2043 y Fy(=)25 b(2)p FI(;)2146 2173 y Fy(0)p FI(;)100 b Fo(if)32 b FI(i)2435 2187 y Ft(1)2490 2173 y Fv(\001)15 b(\001)g(\001)h FI(i)2642 2187 y Fu(n)2715 2173 y Fy(=)25 b(1)p FI(;)3718 2040 y SDict begin H.S end 3718 2040 a 3718 2040 a SDict begin 19.25 H.A end 3718 2040 a 3718 2040 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.5) cvn /DEST pdfmark end 3718 2040 a Fy(\(2.5\))125 2341 y Fo(wher)-5 b(e)34 b FI(i)413 2355 y Ft(1)478 2341 y Fv(\024)25 b FI(i)605 2355 y Ft(2)670 2341 y Fv(\024)g(\001)15 b(\001)g(\001)26 b(\024)f FI(i)1024 2355 y Fu(n)1096 2341 y Fv(2)g(f)p Fy(1)p FI(;)15 b Fy(2)p Fv(g)p Fo(,)35 b(and)e FI(h)26 b Fv(2)f(S)7 b(P)1937 2355 y Fu(n)p Fs(\000)p Ft(1)2107 2341 y Fo(satisfying)887 2464 y Fu(n)842 2491 y Fm(X)851 2686 y Fu(i)p Ft(=1)974 2577 y Fy(\()p Fv(\000)p Fy(1\))1160 2540 y Fu(n)p Fs(\000)p Fu(i)1286 2577 y FI(h)p Fy(\()p FI(\025)1426 2591 y Fu(i)1455 2577 y FI(;)15 b(\025)1548 2591 y Fu(n)p Ft(+1)1686 2577 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1956 2591 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2129 2577 y Fy(\))p FI(h)p Fy(\()p FI(\025)2304 2591 y Ft(1)2344 2577 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2579 2553 y Fy(^)2561 2577 y FI(\025)2614 2591 y Fu(i)2643 2577 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2912 2591 y Fu(n)2960 2577 y Fy(\))25 b(=)g(0)p FI(:)3718 2577 y SDict begin H.S end 3718 2577 a 3718 2577 a SDict begin 19.25 H.A end 3718 2577 a 3718 2577 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 3718 2577 a Fy(\(2.6\))125 2767 y Fo(.)p Black 125 3002 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(F)d(rom)33 b(Prop)s(osition)1138 3002 y SDict begin H.S end 1138 3002 a Black Fy(2.5)p Black 1255 2944 a SDict begin H.R end 1255 2944 a 1255 3002 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1255 3002 a Fy(,)g FI(I)39 b Fy(and)32 b FI(R)q(=I)40 b Fy(are)33 b(comm)m(utativ)m(e)h FI(n)p Fy(-Lie)e(conformal)g (algebras,)h(whic)m(h)e(im-)125 3135 y(plies)1547 3321 y([)p FI(e)1614 3335 y Ft(2)1654 3339 y Fu(\025)1695 3348 y FJ(1)1734 3321 y FI(e)1776 3335 y Ft(2)1816 3339 y Fu(\025)1857 3348 y FJ(2)1896 3321 y Fv(\001)15 b(\001)g(\001)2002 3336 y Fu(\025)2043 3345 y FB(n)p FC(\000)p FJ(1)2168 3321 y FI(e)2210 3335 y Ft(2)2250 3321 y Fy(])25 b(=)g(0)p FI(;)790 3481 y Fy([)p FI(e)857 3495 y Fu(i)881 3504 y FJ(1)921 3508 y Fu(\025)962 3517 y FJ(1)1000 3481 y FI(e)1042 3495 y Fu(i)1066 3504 y FJ(2)1106 3508 y Fu(\025)1147 3517 y FJ(2)1185 3481 y Fv(\001)15 b(\001)g(\001)1292 3496 y Fu(\025)1333 3505 y FB(n)p FC(\000)p FJ(1)1458 3481 y FI(e)1500 3495 y Fu(i)1524 3503 y FB(n)1571 3481 y Fy(])25 b Fv(2)g FI(I)7 b Fy([)p FI(\025)1832 3495 y Ft(1)1872 3481 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2142 3495 y Fu(n)p Fs(\000)p Ft(1)2279 3481 y Fy(])p FI(;)46 b Fy(for)30 b FI(i)2545 3495 y Ft(1)2585 3481 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(i)2833 3495 y Fu(n)2906 3481 y Fv(2)25 b(f)p Fy(1)p FI(;)15 b Fy(2)p Fv(g)p FI(:)225 3700 y FH(Step)34 b(1:)41 b Fy(Denote)1112 3886 y([)p FI(e)1179 3900 y Ft(2)1219 3904 y Fu(\025)1260 3913 y FJ(1)1299 3886 y Fv(\001)15 b(\001)g(\001)1405 3901 y Fu(\025)1446 3910 y FB(n)p FC(\000)p FJ(2)1571 3886 y FI(e)1613 3900 y Ft(2)1653 3904 y Fu(\025)1694 3913 y FB(n)p FC(\000)p FJ(1)1819 3886 y FI(e)1861 3900 y Ft(1)1901 3886 y Fy(])25 b(=)g FI(f)2092 3900 y Ft(1)2131 3886 y Fy(\()p FI(@)5 b(;)15 b(\025)2312 3900 y Ft(1)2353 3886 y FI(;)g Fv(\001)g(\001)g (\001)31 b FI(;)15 b(\025)2622 3900 y Fu(n)p Fs(\000)p Ft(1)2760 3886 y Fy(\))p FI(e)2837 3900 y Ft(2)2877 3886 y FI(;)988 4046 y Fy([)p FI(e)1055 4060 y Ft(2)1095 4064 y Fu(\025)1136 4073 y FJ(1)1175 4046 y Fv(\001)g(\001)g(\001)1281 4061 y Fu(\025)1322 4070 y FB(n)p FC(\000)p FJ(2)1447 4046 y FI(e)1489 4060 y Ft(2)1529 4064 y Fu(\025)1570 4073 y FB(n)p FC(\000)p FJ(2)1695 4046 y FI(e)1737 4060 y Ft(1)1777 4064 y Fu(\025)1818 4073 y FB(n)p FC(\000)p FJ(1)1943 4046 y FI(e)1985 4060 y Ft(1)2025 4046 y Fy(])25 b(=)g FI(f)2216 4060 y Ft(2)2255 4046 y Fy(\()p FI(@)5 b(;)15 b(\025)2436 4060 y Ft(1)2477 4046 y FI(;)g Fv(\001)g(\001)g (\001)31 b FI(;)15 b(\025)2746 4060 y Fu(n)p Fs(\000)p Ft(1)2884 4046 y Fy(\))p FI(e)2961 4060 y Ft(2)3001 4046 y FI(;)2002 4174 y Fy(.)2002 4208 y(.)2002 4241 y(.)1033 4401 y([)p FI(e)1100 4415 y Ft(2)1140 4419 y Fu(\025)1181 4428 y FJ(1)1220 4401 y FI(e)1262 4415 y Ft(2)1302 4419 y Fu(\025)1343 4428 y FJ(2)1381 4401 y FI(e)1423 4415 y Ft(1)1463 4419 y Fu(\025)1504 4428 y FJ(3)1543 4401 y Fv(\001)g(\001)g(\001)1649 4416 y Fu(\025)1690 4425 y FB(n)p FC(\000)p FJ(1)1815 4401 y FI(e)1857 4415 y Ft(1)1897 4401 y Fy(])26 b(=)f FI(f)2089 4415 y Fu(n)p Fs(\000)p Ft(2)2225 4401 y Fy(\()p FI(@)5 b(;)15 b(\025)2406 4415 y Ft(1)2447 4401 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2717 4415 y Fu(n)p Fs(\000)p Ft(1)2854 4401 y Fy(\))p FI(e)2931 4415 y Ft(2)2971 4401 y FI(:)125 4587 y Fy(W)-8 b(e)31 b(claim)f FI(f)569 4601 y Ft(1)633 4587 y Fy(=)25 b FI(f)774 4601 y Ft(2)838 4587 y Fy(=)g Fv(\001)15 b(\001)g(\001)27 b Fy(=)d FI(f)1206 4601 y Fu(n)p Fs(\000)p Ft(2)1368 4587 y Fy(=)h(0.)225 4752 y(W)-8 b(e)26 b(\014rst)e(sho)m(w)h FI(f)823 4766 y Ft(1)888 4752 y Fy(=)g(0.)39 b(It)25 b(is)f(easy)i(to)g(see)g(that)g FI(f)1952 4766 y Ft(1)1991 4752 y Fy(\()p FI(@)5 b(;)15 b(\025)2172 4766 y Ft(1)2212 4752 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2482 4766 y Fu(n)p Fs(\000)p Ft(1)2620 4752 y Fy(\))25 b(is)f(sk)m(ew-symmetric)h(with)f(resp)s(ect)125 4885 y(to)31 b FI(\025)289 4899 y Fu(i)348 4885 y Fy(b)m(y)f(\(C3\).)41 b(F)-8 b(rom)31 b(\(C4\),)722 5071 y([[)p FI(e)814 5085 y Ft(2)855 5089 y Fu(\025)896 5098 y FJ(1)934 5071 y Fv(\001)15 b(\001)g(\001)1041 5086 y Fu(\025)1082 5095 y FB(n)p FC(\000)p FJ(2)1207 5071 y FI(e)1249 5085 y Ft(2)1288 5089 y Fu(\025)1329 5098 y FB(n)p FC(\000)p FJ(1)1455 5071 y FI(e)1497 5085 y Ft(1)1536 5071 y Fy(])1561 5086 y Fu(\025)1602 5095 y FJ(1)1637 5086 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(\025)1847 5094 y FB(n)1894 5071 y FI(e)1936 5085 y Ft(2)1976 5089 y Fu(\025)2017 5098 y FB(n)p FJ(+1)2140 5071 y Fv(\001)g(\001)g(\001)2247 5086 y Fu(\025)2288 5095 y FJ(2)p FB(n)p FC(\000)p FJ(3)2443 5071 y FI(e)2485 5085 y Ft(2)2525 5089 y Fu(\025)2566 5098 y FJ(2)p FB(n)p FC(\000)p FJ(2)2722 5071 y FI(e)2764 5085 y Ft(1)2803 5071 y Fy(])652 5309 y(=)737 5195 y Fu(n)p Fs(\000)p Ft(1)738 5222 y Fm(X)747 5418 y Fu(i)p Ft(=1)870 5309 y Fy(\()p Fv(\000)p Fy(1\))1056 5271 y Fu(n)p Fs(\000)p Fu(i)1183 5309 y Fy([)p FI(e)1250 5323 y Ft(2)1290 5326 y Fu(\025)1331 5335 y FJ(1)1385 5309 y Fv(\001)g(\001)g(\001)35 b Fy(^)-64 b FI(e)1548 5323 y Ft(2)1588 5326 y Fu(\025)1629 5336 y FB(i)1675 5309 y Fv(\001)15 b(\001)g(\001)h FI(e)1838 5323 y Ft(2)1878 5326 y Fu(\025)1919 5335 y FB(n)p FC(\000)p FJ(1)2044 5309 y FI(e)2086 5323 y Ft(1)2126 5326 y Fu(\025)2167 5334 y FB(n)2214 5309 y Fy([)p FI(e)2281 5323 y Ft(2)2321 5326 y Fu(\025)2362 5336 y FB(i)2392 5309 y FI(e)2434 5323 y Ft(2)2474 5326 y Fu(\025)2515 5335 y FB(n)p FJ(+1)2639 5309 y Fv(\001)f(\001)g(\001)2745 5324 y Fu(\025)2786 5333 y FJ(2)p FB(n)p FC(\000)p FJ(3)2942 5309 y FI(e)2984 5323 y Ft(2)3024 5326 y Fu(\025)3065 5335 y FJ(2)p FB(n)p FC(\000)p FJ(2)3220 5309 y FI(e)3262 5323 y Ft(1)3302 5309 y Fy(]])p FI(;)p Black Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(6)893 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 125 77 a Fy(whic)m(h)29 b(is)g(equiv)-5 b(alen)m(t)30 b(to)308 255 y FI(f)353 269 y Ft(1)392 255 y Fy(\()p Fv(\000)p FI(\025)551 269 y Ft(1)611 255 y Fv(\000)19 b(\001)c(\001)g(\001)22 b(\000)e FI(\025)972 269 y Fu(n)1019 255 y FI(;)15 b(\025)1112 269 y Ft(1)1151 255 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1421 269 y Fu(n)p Fs(\000)p Ft(1)1559 255 y Fy(\))p FI(f)1639 269 y Ft(1)1678 255 y Fy(\()p FI(@)5 b(;)15 b(\025)1859 269 y Ft(1)1920 255 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2281 269 y Fu(n)2328 255 y FI(;)15 b(\025)2421 269 y Fu(n)p Ft(+1)2559 255 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2828 269 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)3001 255 y Fy(\))237 492 y(=)323 378 y Fu(n)p Fs(\000)p Ft(1)324 406 y Fm(X)332 601 y Fu(i)p Ft(=1)456 492 y Fy(\()p Fv(\000)p Fy(1\))642 454 y Fu(n)p Ft(+1)p Fs(\000)p Fu(i)859 492 y FI(f)904 506 y Ft(1)943 492 y Fy(\()p FI(@)26 b Fy(+)1187 378 y Fu(n)1143 406 y Fm(X)1147 601 y Fu(j)t Ft(=1)1153 669 y Fu(j)t Fs(6)p Ft(=)p Fu(i)1289 492 y FI(\025)1342 506 y Fu(j)1379 492 y FI(;)15 b(\025)1472 506 y Fu(i)1500 492 y FI(;)g(\025)1593 506 y Fu(n)p Ft(+1)1731 492 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2001 506 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2173 492 y Fy(\))p FI(f)2253 506 y Ft(1)2293 492 y Fy(\()p FI(@)5 b(;)15 b(\025)2474 506 y Ft(1)2514 492 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2749 468 y Fy(^)2731 492 y FI(\025)2784 506 y Fu(i)2813 492 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)3082 506 y Fu(n)p Fs(\000)p Ft(1)3220 492 y FI(;)g Fv(\000)p FI(@)26 b Fv(\000)3540 378 y Fu(n)3496 406 y Fm(X)3500 601 y Fu(j)t Ft(=1)3506 669 y Fu(j)t Fs(6)p Ft(=)p Fu(i)3642 492 y FI(\025)3695 506 y Fu(j)3732 492 y Fy(\))p FI(:)3718 768 y SDict begin H.S end 3718 768 a 3718 768 a SDict begin 19.25 H.A end 3718 768 a 3718 768 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 3718 768 a Fy(\(2.7\))125 921 y(Similar)32 b(to)j(Prop)s(osition)1052 921 y SDict begin H.S end 1052 921 a Black Fy(2.5)p Black 1168 862 a SDict begin H.R end 1168 862 a 1168 921 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1168 921 a Fy(,)h(w)m(e)f(set)p FI(\025)1531 935 y Ft(2)1604 921 y Fy(=)d FI(\025)1760 935 y Fu(n)p Ft(+1)1897 921 y FI(;)15 b(\025)1990 935 y Ft(3)2063 921 y Fy(=)32 b FI(\025)2219 935 y Fu(n)p Ft(+2)2356 921 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2626 935 y Fu(n)p Fs(\000)p Ft(1)2796 921 y Fy(=)33 b FI(\025)2953 935 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)3125 921 y FI(;)15 b(\025)3218 935 y Ft(1)3281 921 y Fy(+)23 b Fv(\001)15 b(\001)g(\001)24 b Fy(+)f FI(\025)3651 935 y Fu(n)3730 921 y Fy(=)33 b(0,)125 1053 y(then)d(w)m(e)h(can)f (deduce)h(that)f FI(f)1177 1067 y Ft(1)1242 1053 y Fy(=)25 b(0)30 b(from)g(the)h(simpli\014ed)26 b(form)m(ula)k(of)h(\()2667 1053 y SDict begin H.S end 2667 1053 a Black Fy(2.7)p Black 2783 995 a SDict begin H.R end 2783 995 a 2783 1053 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.7) cvn H.B /ANN pdfmark end 2783 1053 a Fy(\))497 1281 y FI(f)542 1295 y Ft(1)581 1281 y Fy(\()p FI(@)26 b Fy(+)825 1167 y Fu(n)781 1195 y Fm(X)785 1390 y Fu(j)t Ft(=2)927 1281 y FI(\025)980 1295 y Fu(j)1017 1281 y FI(;)15 b Fv(\000)p Fy(\()1207 1167 y Fu(n)1163 1195 y Fm(X)1167 1390 y Fu(j)t Ft(=2)1310 1281 y FI(\025)1363 1295 y Fu(j)1399 1281 y Fy(\))p FI(;)g(\025)1527 1295 y Ft(2)1567 1281 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1837 1295 y Fu(n)p Fs(\000)p Ft(1)1975 1281 y Fy(\))p FI(f)2055 1295 y Ft(1)2094 1281 y Fy(\()p FI(@)5 b(;)15 b(\025)2275 1295 y Ft(2)2316 1281 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2585 1295 y Fu(n)p Fs(\000)p Ft(1)2723 1281 y FI(;)g Fv(\000)p FI(@)26 b Fv(\000)19 b Fy(\()3078 1167 y Fu(n)3033 1195 y Fm(X)3037 1390 y Fu(j)t Ft(=1)3043 1458 y Fu(j)t Fs(6)p Ft(=)p Fu(i)3180 1281 y FI(\025)3233 1295 y Fu(j)3270 1281 y Fy(\)\))26 b(=)f(0)p FI(:)125 1597 y Fy(W)-8 b(e)31 b(further)f(consider)459 1775 y([[)p FI(e)551 1789 y Ft(2)591 1792 y Fu(\025)632 1801 y FJ(1)671 1775 y Fv(\001)15 b(\001)g(\001)777 1790 y Fu(\025)818 1799 y FB(n)p FC(\000)p FJ(3)944 1775 y FI(e)986 1789 y Ft(2)1025 1792 y Fu(\025)1066 1801 y FB(n)p FC(\000)p FJ(2)1192 1775 y FI(e)1234 1789 y Ft(1)1273 1792 y Fu(\025)1314 1801 y FB(n)p FC(\000)p FJ(1)1439 1775 y FI(e)1481 1789 y Ft(1)1521 1775 y Fy(])1546 1790 y Fu(\025)1587 1799 y FJ(1)1622 1790 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(\025)1832 1798 y FB(n)1879 1775 y FI(e)1921 1789 y Ft(2)1960 1792 y Fu(\025)2001 1801 y FB(n)p FJ(+1)2125 1775 y Fv(\001)g(\001)g(\001) 2231 1790 y Fu(\025)2272 1799 y FJ(2)p FB(n)p FC(\000)p FJ(4)2428 1775 y FI(e)2470 1789 y Ft(2)2510 1792 y Fu(\025)2551 1801 y FJ(2)p FB(n)p FC(\000)p FJ(3)2706 1775 y FI(e)2748 1789 y Ft(1)2788 1792 y Fu(\025)2829 1801 y FJ(2)p FB(n)p FC(\000)p FJ(2)2985 1775 y FI(e)3027 1789 y Ft(1)3067 1775 y Fy(])388 2012 y(=)474 1898 y Fu(n)p Fs(\000)p Ft(2)475 1926 y Fm(X)484 2121 y Fu(i)p Ft(=1)607 2012 y Fy(\()p Fv(\000)p Fy(1\))793 1974 y Fu(n)p Fs(\000)p Fu(i)920 2012 y Fy([)p FI(e)987 2026 y Ft(2)1027 2030 y Fu(\025)1068 2039 y FJ(1)1122 2012 y Fv(\001)g(\001)g(\001)34 b Fy(^)-63 b FI(e)1285 2026 y Ft(2)1325 2030 y Fu(\025)1366 2040 y FB(i)1412 2012 y Fv(\001)15 b(\001)g(\001)h FI(e)1575 2026 y Ft(2)1615 2030 y Fu(\025)1656 2039 y FB(n)p FC(\000)p FJ(2)1781 2012 y FI(e)1823 2026 y Ft(1)1863 2030 y Fu(\025)1904 2039 y FB(n)p 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b(;)15 b(\025)1789 2439 y Ft(1)1850 2425 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2211 2439 y Fu(n)2258 2425 y FI(;)15 b(\025)2351 2439 y Fu(n)p Ft(+1)2488 2425 y FI(;)g Fv(\001)g(\001)g (\001)32 b FI(;)15 b(\025)2758 2439 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2931 2425 y Fy(\))167 2662 y(=)253 2548 y Fu(n)p Fs(\000)p Ft(2)254 2576 y Fm(X)262 2771 y Fu(i)p Ft(=1)386 2662 y Fy(\()p Fv(\000)p Fy(1\))572 2624 y Fu(n)p Fs(\000)p Fu(i)699 2662 y FI(f)744 2676 y Ft(2)782 2662 y Fy(\()p FI(@)26 b Fy(+)1027 2548 y Fu(n)982 2576 y Fm(X)987 2771 y Fu(j)t Ft(=1)992 2839 y Fu(j)t Fs(6)p Ft(=)p Fu(i)1129 2662 y FI(\025)1182 2676 y Fu(j)1218 2662 y FI(;)15 b(\025)1311 2676 y Fu(i)1340 2662 y FI(;)g(\025)1433 2676 y Fu(n)p Ft(+1)1570 2662 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1840 2676 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2013 2662 y Fy(\))p FI(f)2093 2676 y Ft(2)2132 2662 y Fy(\()p FI(@)5 b(;)15 b(\025)2313 2676 y Ft(1)2354 2662 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2589 2638 y Fy(^)2571 2662 y FI(\025)2624 2676 y Fu(i)2652 2662 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2922 2676 y Fu(n)p Fs(\000)p Ft(2)3060 2662 y FI(;)g Fv(\000)p FI(@)25 b Fv(\000)3379 2548 y Fu(n)3335 2576 y Fm(X)3340 2771 y Fu(j)t Ft(=1)3345 2839 y Fu(j)t Fs(6)p Ft(=)p Fu(i)3482 2662 y FI(\025)3535 2676 y Fu(j)3571 2662 y FI(;)15 b(\025)3664 2676 y Fu(n)p Fs(\000)p Ft(1)3802 2662 y Fy(\))p FI(:)3718 2937 y SDict begin H.S end 3718 2937 a 3718 2937 a SDict begin 19.25 H.A end 3718 2937 a 3718 2937 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 3718 2937 a Fy(\(2.8\))125 3091 y(Based)21 b(on)g(the)g(sk)m (ew-symmetry)g(of)g FI(f)1418 3105 y Ft(2)1457 3091 y Fy(,)i(substituting)18 b(in)i FI(\025)2151 3105 y Ft(2)2215 3091 y Fy(=)25 b FI(\025)2364 3105 y Fu(n)p Ft(+1)2502 3091 y FI(;)15 b(\025)2595 3105 y Ft(3)2660 3091 y Fy(=)25 b FI(\025)2809 3105 y Fu(n)p Ft(+2)2946 3091 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)3216 3105 y Fu(n)p 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Fy(yields)e(the)i(sk)m (ew-symmetry)g(of)g FI(f)2149 224 y Fu(n)2222 210 y Fy(with)e(resp)s (ect)i(to)h FI(\025)2894 224 y Fu(i)2922 210 y Fy(.)39 b(So)27 b(there)g(is)e(a)j(one)f(to)g(one)125 342 y(corresp)s(ondence)j (b)s(et)m(w)m(een)h FI(f)1156 356 y Fu(n)1233 342 y Fy(and)e(a)i(p)s (olynomial)d FI(g)34 b Fy(in)29 b Fv(S)7 b(P)2270 356 y Fu(n)p Fs(\000)p Ft(1)2407 342 y Fy([)p FI(@)e Fy(])32 b(b)m(y)1501 548 y FI(f)1546 562 y Fu(n)1618 548 y Fy(=)1851 461 y Fm(Y)1714 657 y Ft(1)p Fs(\024)p Fu(i<j)t Fs(\024)p Fu(n)p Fs(\000)p Ft(1)2103 548 y Fy(\()p FI(\025)2191 562 y Fu(i)2240 548 y Fv(\000)20 b FI(\025)2384 562 y Fu(j)2421 548 y Fy(\))p FI(g)s(:)225 861 y Fy(\(ii\))29 b(If)h FI(f)512 875 y Fu(n)584 861 y Fy(=)25 b(0.)41 b(It)30 b(can)h(b)s(e)f(deduced)f(from)h(\(C4\))h(that)779 1066 y([[)p FI(e)871 1080 y Ft(1)911 1084 y Fu(\025)952 1093 y FJ(1)991 1066 y Fv(\001)15 b(\001)g(\001)1097 1081 y Fu(\025)1138 1090 y FB(n)p FC(\000)p FJ(1)1264 1066 y FI(e)1306 1080 y Ft(1)1345 1066 y Fy(])1370 1081 y Fu(\025)1411 1090 y FJ(1)1446 1081 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(\025)1656 1089 y FB(n)1703 1066 y FI(e)1745 1080 y Ft(1)1785 1084 y Fu(\025)1826 1093 y FB(n)p FJ(+1)1949 1066 y Fv(\001)g(\001)g(\001)2055 1081 y Fu(\025)2096 1090 y FJ(2)p FB(n)p FC(\000)p FJ(3)2252 1066 y FI(e)2294 1080 y Ft(1)2334 1084 y Fu(\025)2375 1093 y FJ(2)p FB(n)p FC(\000)p FJ(2)2531 1066 y FI(e)2573 1080 y Ft(2)2612 1066 y Fy(])708 1289 y(=)839 1176 y Fu(n)794 1203 y Fm(X)803 1399 y Fu(i)p Ft(=1)926 1289 y Fy(\()p Fv(\000)p Fy(1\))1112 1252 y Fu(n)p Fs(\000)p Fu(i)1238 1289 y Fy([)p FI(e)1305 1303 y Ft(1)1345 1307 y Fu(\025)1386 1316 y FJ(1)1440 1289 y Fv(\001)g(\001)g(\001)35 b Fy(^)-63 b FI(e)1604 1303 y Ft(1)1643 1307 y Fu(\025)1684 1317 y FB(i)1715 1289 y Fv(\001)15 b(\001)g(\001)1821 1304 y Fu(\025)1862 1313 y FB(n)p FC(\000)p FJ(1)1987 1289 y FI(e)2029 1303 y Ft(1)2069 1307 y Fu(\025)2110 1315 y FB(n)2157 1289 y Fy([)p FI(e)2224 1303 y Ft(1)2264 1307 y Fu(\025)2305 1317 y FB(i)2336 1289 y FI(e)2378 1303 y Ft(1)2417 1307 y Fu(\025)2458 1316 y FB(n)p FJ(+1)2582 1289 y Fv(\001)g(\001)g(\001) 2688 1304 y Fu(\025)2729 1313 y FJ(2)p FB(n)p FC(\000)p FJ(3)2885 1289 y FI(e)2927 1303 y Ft(1)2967 1307 y Fu(\025)3008 1316 y FJ(2)p FB(n)p FC(\000)p FJ(2)3163 1289 y FI(e)3205 1303 y Ft(2)3245 1289 y Fy(]])p FI(:)125 1552 y Fy(Substituting)28 b(in)h FI(f)797 1566 y Fu(n)p Fs(\000)p Ft(1)933 1552 y Fy(,)i(w)m(e)g(obtain)397 1691 y Fu(n)352 1718 y Fm(X)361 1914 y Fu(i)p Ft(=1)484 1804 y Fy(\()p Fv(\000)p Fy(1\))670 1767 y Fu(n)p Fs(\000)p Fu(i)796 1804 y FI(f)841 1818 y Fu(n)p Fs(\000)p Ft(1)978 1804 y Fy(\()p FI(@)26 b Fy(+)1222 1691 y Fu(n)1178 1718 y Fm(X)1182 1914 y Fu(j)t Ft(=1)1188 1982 y Fu(j)t Fs(6)p Ft(=)p Fu(i)1324 1804 y FI(\025)1377 1818 y Fu(j)1414 1804 y FI(;)15 b(\025)1507 1818 y Fu(i)1536 1804 y FI(;)g(\025)1629 1818 y Fu(n)p Ft(+1)1766 1804 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2036 1818 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2209 1804 y Fy(\))p FI(f)2289 1818 y Fu(n)p Fs(\000)p Ft(1)2426 1804 y Fy(\()p FI(@)5 b(;)15 b(\025)2607 1818 y Ft(1)2647 1804 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2882 1780 y Fy(^)2864 1804 y FI(\025)2917 1818 y Fu(i)2946 1804 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)3215 1818 y Fu(n)3263 1804 y Fy(\))25 b(=)g(0)p FI(:)3718 1804 y SDict begin H.S end 3718 1804 a 3718 1804 a SDict begin 19.25 H.A end 3718 1804 a 3718 1804 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 3718 1804 a Fy(\(2.9\))125 2208 y(Set)31 b FI(f)327 2222 y Fu(n)p Fs(\000)p Ft(1)464 2208 y Fy(\()p FI(@)5 b(;)15 b(\025)645 2222 y Ft(1)685 2208 y FI(;)g Fv(\001)g(\001)g(\001) 32 b FI(;)15 b(\025)955 2222 y Fu(n)p Fs(\000)p Ft(1)1093 2208 y Fy(\))26 b(=)1277 2122 y Fu(m)1260 2140 y Fm(P)1251 2290 y Fu(i)p Ft(=0)1380 2208 y FI(g)1423 2222 y Fu(i)1452 2208 y Fy(\()p FI(\025)1540 2222 y Ft(1)1580 2208 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1850 2222 y Fu(n)p Fs(\000)p Ft(1)1987 2208 y Fy(\))p FI(@)2075 2175 y Fu(i)2135 2208 y Fy(with)30 b FI(g)2386 2222 y Fu(m)2479 2208 y Fv(6)p Fy(=)c(0)31 b(and)f(assume)h FI(m)26 b Fv(\025)g Fy(1.)43 b(Computing)125 2383 y(the)30 b(co)s(e\016cien)m(ts)i(of)e FI(@)898 2350 y Ft(2)p Fu(m)p Fs(\000)p Ft(1)1090 2383 y Fy(,)h(w)m(e)g(ha)m(v)m(e)595 2520 y Fu(n)551 2547 y Fm(X)560 2742 y Fu(i)p Ft(=1)682 2633 y Fy(\()p Fv(\000)p Fy(1\))868 2596 y Fu(n)p Fs(\000)p Fu(i)995 2633 y FI(g)1038 2647 y Fu(m)1105 2633 y Fy(\()p FI(\025)1193 2647 y Fu(i)1222 2633 y FI(;)15 b(\025)1315 2647 y Fu(n)p Ft(+1)1452 2633 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1722 2647 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)1895 2633 y Fy(\))p FI(g)1973 2647 y Fu(m)p Fs(\000)p Ft(1)2130 2633 y Fy(\()p FI(\025)2218 2647 y Ft(1)2258 2633 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2493 2609 y Fy(^)2475 2633 y FI(\025)2528 2647 y Fu(i)2557 2633 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2826 2647 y Fu(n)2874 2633 y Fy(\))556 2938 y(+)691 2825 y Fu(n)647 2852 y Fm(X)656 3047 y Fu(i)p Ft(=1)778 2938 y Fy(\()p Fv(\000)p Fy(1\))964 2901 y Fu(n)p Fs(\000)p Fu(i)1091 2938 y FI(g)1134 2952 y Fu(m)p Fs(\000)p Ft(1)1291 2938 y Fy(\()p FI(\025)1379 2952 y Fu(i)1408 2938 y FI(;)g(\025)1501 2952 y Fu(n)p Ft(+1)1638 2938 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1908 2952 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2081 2938 y Fy(\))p FI(g)2159 2952 y Fu(m)2226 2938 y Fy(\()p FI(\025)2314 2952 y Ft(1)2354 2938 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2589 2914 y Fy(^)2571 2938 y FI(\025)2624 2952 y Fu(i)2653 2938 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)2922 2952 y Fu(n)2970 2938 y Fy(\))556 3243 y(+)691 3129 y Fu(n)647 3157 y Fm(X)656 3352 y Fu(i)p Ft(=1)778 3243 y Fy(\()p Fv(\000)p Fy(1\))964 3205 y Fu(n)p Fs(\000)p Fu(i)1091 3243 y FI(m)p Fy(\()1251 3129 y Fu(n)1206 3157 y Fm(X)1210 3352 y Fu(j)t Ft(=1)1216 3420 y Fu(j)t Fs(6)p Ft(=)p Fu(i)1353 3243 y FI(\025)1406 3257 y Fu(j)1442 3243 y Fy(\))p FI(g)1520 3257 y Fu(m)1588 3243 y Fy(\()p FI(\025)1676 3257 y Fu(i)1704 3243 y FI(;)g(\025)1797 3257 y Fu(n)p Ft(+1)1935 3243 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2205 3257 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2377 3243 y Fy(\))p FI(g)2455 3257 y Fu(m)2523 3243 y Fy(\()p FI(\025)2611 3257 y Ft(1)2651 3243 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)2886 3219 y Fy(^)2867 3243 y FI(\025)2920 3257 y Fu(i)2949 3243 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)3219 3257 y Fu(n)3266 3243 y Fy(\))26 b(=)f(0)p FI(:)125 3586 y Fy(Since)k FI(g)405 3600 y Fu(i)464 3586 y Fy(is)h(sk)m(ew-symmetric,)g(putting)f(in)g FI(\025)1723 3600 y Ft(2)1788 3586 y Fy(=)c FI(\025)1937 3600 y Fu(n)p Ft(+1)2074 3586 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2344 3600 y Fu(n)p Fs(\000)p Ft(1)2507 3586 y Fy(=)25 b FI(\025)2656 3600 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2859 3586 y Fy(yields)1069 3791 y FI(m)p Fy(\()p FI(\025)1237 3805 y Fu(n)1304 3791 y Fv(\000)20 b FI(\025)1448 3805 y Ft(1)1488 3791 y Fy(\))p FI(g)1566 3805 y Fu(m)1633 3791 y Fy(\()p FI(\025)1721 3805 y Ft(1)1761 3791 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2031 3805 y Fu(n)p Fs(\000)p Ft(1)2168 3791 y Fy(\))p FI(g)2246 3805 y Fu(m)2313 3791 y Fy(\()p FI(\025)2401 3805 y Ft(2)2441 3791 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)2711 3805 y Fu(n)2758 3791 y Fy(\))26 b(=)f(0)125 3997 y(so)k(that)g FI(g)473 4011 y Fu(m)566 3997 y Fy(=)c(0.)40 b(This)27 b(is)h(a)h(con)m(tradiction.)40 b(Hence)30 b FI(f)2044 4011 y Fu(n)p Fs(\000)p Ft(1)2209 3997 y Fy(is)e(a)h(constan)m(t)h (with)e(resp)s(ect)h(to)g FI(@)5 b Fy(.)41 b(Then)28 b(\()3753 3997 y SDict begin H.S end 3753 3997 a Black Fy(2.9)p Black 3869 3938 a SDict begin H.R end 3869 3938 a 3869 3997 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 3869 3997 a Fy(\))125 4130 y(turns)h(in)m(to)757 4248 y Fu(n)713 4276 y Fm(X)722 4471 y Fu(i)p Ft(=1)844 4362 y Fy(\()p Fv(\000)p Fy(1\))1030 4324 y Fu(n)p Fs(\000)p Fu(i)1157 4362 y FI(f)1202 4376 y Fu(n)p Fs(\000)p Ft(1)1339 4362 y Fy(\()p FI(\025)1427 4376 y Fu(i)1455 4362 y FI(;)15 b(\025)1548 4376 y Fu(n)p Ft(+1)1686 4362 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)1956 4376 y Ft(2)p Fu(n)p Fs(\000)p Ft(2)2129 4362 y Fy(\))p FI(f)2209 4376 y Fu(n)p Fs(\000)p Ft(1)2346 4362 y Fy(\()p FI(\025)2434 4376 y Ft(1)2473 4362 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)2708 4338 y Fy(^)2690 4362 y FI(\025)2743 4376 y Fu(i)2772 4362 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)3042 4376 y Fu(n)3089 4362 y Fy(\))26 b(=)f(0)p FI(:)3672 4362 y SDict begin H.S end 3672 4362 a 3672 4362 a SDict begin 19.25 H.A end 3672 4362 a 3672 4362 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 3672 4362 a Fy(\(2.10\))125 4624 y(As)30 b(men)m(tioned)f(ab)s(o)m(v)m(e,)i (it)e(can)h(b)s(e)f(easily)g(pro)m(v)m(ed)h(that)g FI(R)h Fy(is)d(an)i FI(n)p Fy(-Lie)f(conformal)g(algebra)h(if)f(and)g(only)f (if)125 4757 y FI(f)170 4771 y Fu(n)p Fs(\000)p Ft(1)337 4757 y Fy(is)h(a)i(sk)m(ew-symmetric)f(p)s(olynomial)e(with)h FI(n)20 b Fv(\000)g Fy(1)31 b(v)-5 b(ariables)29 b(satisfying)g(\()2900 4757 y SDict begin H.S end 2900 4757 a Black Fy(2.10)p Black 3062 4698 a SDict begin H.R end 3062 4698 a 3062 4757 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 3062 4757 a Fy(\).)712 b Fk(\003)125 4981 y SDict begin H.S end 125 4981 a 125 4981 a SDict begin 16 H.A end 125 4981 a 125 4981 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.7) cvn /DEST pdfmark end 125 4981 a Black FH(Remark)35 b(2.7.)p Black 42 w Fo(Supp)-5 b(ose)35 b(that)f FI(n)26 b Fy(=)g(3)33 b Fo(and)h FI(h)p Fy(\()p FI(x;)15 b(y)s Fy(\))28 b(=)2079 4913 y Fm(P)2089 5063 y Fu(i;j)2190 4981 y FI(a)2238 4995 y Fu(ij)2299 4981 y FI(x)2351 4948 y Fu(i)2379 4981 y FI(y)2427 4948 y Fu(j)2490 4981 y Fv(2)d(S)7 b(P)2709 4995 y Ft(2)2748 4981 y Fo(,)33 b(then)h FI(h)p Fy(\()p FI(x;)15 b(y)s Fy(\))34 b Fo(is)f(a)h(solution)g(of)125 5161 y(\()162 5161 y SDict begin H.S end 162 5161 a Black Fo(2.6)p Black 283 5102 a SDict begin H.R end 283 5102 a 283 5161 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 283 5161 a Fo(\))e(if)f(and)i(only)g(if)e(the)h FI(m)g Fo(or)-5 b(der)33 b(antisymmetric)g(matrix)h Fv(f)p FI(a)2381 5175 y Fu(ij)2442 5161 y Fv(g)e Fo(satis\014es)g FI(a)2907 5175 y Fu(ij)2968 5161 y FI(a)3016 5176 y Fu(k)r(l)3099 5161 y Fv(\000)18 b FI(a)3236 5176 y Fu(ik)3303 5161 y FI(a)3351 5176 y Fu(j)t(l)3428 5161 y Fy(+)g FI(a)3565 5176 y Fu(il)3615 5161 y FI(a)3663 5176 y Fu(j)t(k)3763 5161 y Fy(=)25 b(0)125 5294 y Fo(for)36 b(al)5 b(l)37 b FI(i;)15 b(j;)g(k)s(;)g(l)36 b Fy(=)31 b(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b(m)p Fo(.)52 b(It)36 b(c)-5 b(an)37 b(b)-5 b(e)36 b(e)-5 b(asily)37 b(che)-5 b(cke)g(d)36 b(that)h(every)f(3)h(or)-5 b(der)37 b(antisymmetric)h(matrix)125 5427 y(is)32 b(a)h(solution,)h(but)f(not)g (al)5 b(l)33 b(4)g(or)-5 b(der)34 b(antisymmetric)g(matric)-5 b(es)34 b(me)-5 b(et)33 b(the)g(c)-5 b(ondition.)p Black Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(8)893 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 125 -6 a SDict begin H.S end 125 -6 a 125 -6 a SDict begin 16 H.A end 125 -6 a 125 -6 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 125 -6 a 769 77 a Fy(3.)46 b Fx(The)35 b(associa)-6 b(ted)32 b(infinite-dimensional)h FI(n)p Fx(-Lie)i(algebras)225 296 y Fy(In)45 b(this)g(section,)50 b(w)m(e)d(in)m(tro)s(duce)e(the)h (notion)f(of)h(asso)s(ciated)h(in\014nite-dimensional)41 b FI(n)p Fy(-Lie)k(algebras)125 429 y Fq(L)18 b FI(ie)295 443 y Fu(p)365 429 y FI(R)40 b Fy(of)g(an)f FI(n)p Fy(-Lie)g(conformal) f(algebra)i FI(R)q Fy(.)67 b(Based)40 b(on)f(the)h(structure)e(of)i Fq(L)18 b FI(ie)3147 443 y Fu(p)3217 429 y FI(R)q Fy(,)42 b(w)m(e)e(can)f(de\014ne)125 562 y(a)28 b(linearly)d(compact)k(top)s (ology)f(on)g(the)g(subalgebra)e(\()p Fq(L)18 b FI(ie)2188 576 y Fu(p)2259 562 y FI(R)q Fy(\))p 2368 576 22 4 v 30 w(.)39 b(W)-8 b(e)29 b(sho)m(w)f(torsionless)e(\014nite)h FI(n)p Fy(-Lie)g(con-)125 694 y(formal)e(algebra)g FI(R)h Fy(is)f(determined)f(b)m(y)h(the)h(linearly)d(compact)k FI(n)p Fy(-Lie)e(algebra)g(\()p Fq(L)18 b FI(ie)3075 708 y Fu(p)3146 694 y FI(R)q Fy(\))p 3255 708 V 30 w(,)26 b(whic)m(h)f(is)f(useful)125 827 y(for)30 b(further)f(discussion)f(on)i (simple)e(\014nite)i FI(n)p Fy(-Lie)f(conformal)h(algebras.)225 992 y Fq(L)17 b FI(ie)394 1006 y Fu(p)465 992 y FI(R)31 b Fy(are)g(de\014ned)e(in)g(the)i(follo)m(wing)d(w)m(a)m(y)-8 b(.)125 1211 y SDict begin H.S end 125 1211 a 125 1211 a SDict begin 16 H.A end 125 1211 a 125 1211 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.1) cvn /DEST pdfmark end 125 1211 a Black FH(Lemma)30 b(3.1.)p Black 40 w Fo(Supp)-5 b(ose)32 b FI(R)f Fo(is)f(an)h FI(n)p Fo(-Lie)e(c)-5 b(onformal)32 b(algebr)-5 b(a,)2381 1188 y Fy(~)2361 1211 y FI(R)26 b Fy(=)f FI(R)q Fy([)p FI(t)2680 1178 y Fs(\006)p Ft(1)2774 1211 y Fy(])p Fo(.)41 b(Then)31 b Fp(C)18 b Fy([)3200 1187 y(~)3188 1211 y FI(@)11 b Fy(])p Fo(-mo)-5 b(dule)3631 1188 y Fy(~)3612 1211 y FI(R)31 b Fo(is)f(an)125 1344 y FI(n)p Fo(-Lie)h(c)-5 b(onformal)35 b(algebr)-5 b(a)34 b(with)g(the)f(c)-5 b(onformal)35 b(pr)-5 b(o)g(ducts)34 b(de\014ne)-5 b(d)34 b(by)325 1543 y Fy(\()p FI(a)408 1505 y Ft(1)468 1543 y Fv(\012)20 b FI(f)604 1557 y Ft(1)643 1543 y Fy(\))678 1561 y Ft(\()p Fu(k)742 1570 y FJ(1)777 1561 y Ft(\))808 1543 y Fv(\001)15 b(\001)g(\001)914 1561 y Ft(\()p Fu(k)978 1570 y FB(n)p FC(\000)p FJ(1)1099 1561 y Ft(\))1131 1543 y Fy(\()p FI(a)1214 1505 y Fu(n)1282 1543 y Fv(\012)k FI(f)1417 1557 y Fu(n)1464 1543 y Fy(\))738 1712 y(=)951 1626 y Fm(X)823 1823 y Fu(j)852 1832 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1064 1823 y Fs(2)p Fn(N)1154 1832 y FJ(+)1210 1712 y Fy(\()p FI(a)1293 1675 y Ft(1)1333 1731 y(\()p Fu(k)1397 1740 y FJ(1)1432 1731 y Ft(+)p Fu(j)1516 1740 y FJ(1)1550 1731 y Ft(\))1581 1712 y Fv(\001)c(\001)g (\001)1687 1731 y Ft(\()p Fu(k)1751 1740 y FB(n)p FC(\000)p FJ(1)1872 1731 y Ft(+)p Fu(j)1956 1740 y FB(n)p FC(\000)p FJ(1)2077 1731 y Ft(\))2108 1712 y FI(a)2156 1675 y Fu(n)2203 1712 y Fy(\))21 b Fv(\012)f Fy(\(\()p FI(@)2473 1664 y Ft(\()p Fu(j)2529 1673 y FJ(1)2564 1664 y Ft(\))2468 1736 y Fu(t)2596 1712 y FI(f)2641 1726 y Ft(1)2680 1712 y Fy(\))15 b Fv(\001)g(\001)g(\001)i Fy(\()p FI(@)2940 1664 y Ft(\()p Fu(j)2996 1673 y FB(n)p FC(\000)p FJ(1)3117 1664 y Ft(\))2935 1736 y Fu(t)3149 1712 y FI(f)3194 1726 y Fu(n)p Fs(\000)p Ft(1)3330 1712 y Fy(\))p FI(f)3410 1726 y Fu(n)3457 1712 y Fy(\))p FI(;)3718 1712 y SDict begin H.S end 3718 1712 a 3718 1712 a SDict begin 19.25 H.A end 3718 1712 a 3718 1712 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 3718 1712 a Fy(\(3.1\))125 1998 y Fo(for)33 b(al)5 b(l)33 b FI(a)448 1965 y Ft(1)488 1998 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)752 1965 y Fu(n)825 1998 y Fv(2)25 b FI(R)q(;)15 b(f)1066 2012 y Ft(1)1105 1998 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(f)1367 2012 y Fu(n)1439 1998 y Fv(2)25 b Fp(C)17 b Fy([)q FI(t)1643 1965 y Fs(\006)p Ft(1)1743 1998 y Fy(])p FI(;)e(k)1855 2012 y Ft(1)1895 1998 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(k)2159 2012 y Fu(n)p Fs(\000)p Ft(1)2322 1998 y Fv(2)25 b Fp(N)2468 2012 y Ft(+)2533 1998 y Fo(.)p Black 125 2250 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(F)d(or)30 b(con)m(v)m(enience)h(of)g(calculations,)e(w)m (e)h(con)m(v)m(ert)i(\()2178 2250 y SDict begin H.S end 2178 2250 a Black Fy(3.1)p Black 2294 2192 a SDict begin H.R end 2294 2192 a 2294 2250 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 2294 2250 a Fy(\))f(in)m(to)e(the)i(corresp)s(onding)c Fv(f)p FI(\025)3381 2264 y Ft(1)3421 2250 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(\025)3691 2264 y Fu(n)p Fs(\000)p Ft(1)3829 2250 y Fv(g)p Fy(-)125 2383 y(brac)m(k)m(ets.)307 2582 y([)p FI(a)380 2544 y Ft(1)439 2582 y Fv(\012)20 b FI(f)575 2596 y Ft(1)614 2604 y Fu(\025)655 2613 y FJ(1)694 2582 y Fv(\001)15 b(\001)g(\001)800 2597 y Fu(\025)841 2606 y FB(n)p FC(\000)p FJ(1)966 2582 y FI(a)1014 2544 y Fu(n)1082 2582 y Fv(\012)20 b FI(f)1218 2596 y Fu(n)1264 2582 y Fy(])236 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pdfmark end 3718 4344 a Fy(\(3.2\))125 4542 y(Then)29 b(w)m(e)i(only)f(need)g(to)h(v)m(erify)f(\(C1\)-\(C4\))i (are)f(satis\014ed.)39 b(Let)31 b(us)f(c)m(hec)m(k)i(\(C3\))f(as)g(an)f (example.)225 4708 y(a\).)41 b(F)-8 b(or)31 b FI(i)26 b Fy(=)f(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b(n)20 b Fv(\000)g Fy(2,)374 4906 y([)p FI(a)447 4869 y Ft(1)507 4906 y Fv(\012)f FI(f)642 4920 y Ft(1)681 4928 y Fu(\025)722 4937 y FJ(1)776 4906 y Fv(\001)c(\001)g(\001)i FI(a)946 4869 y Fu(i)p Ft(+1)1084 4906 y Fv(\012)j FI(f)1220 4920 y Fu(i)p Ft(+1)1338 4931 y Fu(\025)1379 4941 y FB(i)p FJ(+1)1487 4906 y FI(a)1535 4869 y Fu(i)1583 4906 y Fv(\012)g FI(f)1719 4920 y Fu(i)1747 4928 y(\025)1788 4938 y FB(i)1834 4906 y Fv(\001)15 b(\001)g(\001)h FI(a)2003 4869 y Fu(n)2070 4906 y Fv(\012)k FI(f)2206 4920 y Fu(n)2252 4906 y Fy(])303 5066 y(=[)p FI(a)447 5029 y Ft(1)486 5081 y Fu(\025)527 5090 y FJ(1)562 5081 y Ft(+)p Fu(@)654 5089 y FB(t)678 5104 y FJ(1)736 5066 y Fv(\001)15 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FI(a)1016 5191 y Fu(i)1045 5244 y(\025)1086 5254 y FB(i)1112 5244 y Ft(+)p Fu(@)1204 5252 y FB(t)1228 5268 y(i)1263 5229 y FI(a)1311 5191 y Fu(i)p Ft(+1)1429 5244 y Fu(\025)1470 5254 y FB(i)p FJ(+1)1573 5244 y Ft(+)p Fu(@)1665 5252 y FB(t)1689 5268 y(i)p FJ(+1)1816 5229 y Fv(\001)f(\001)g(\001)h FI(a)1985 5191 y Fu(n)2032 5229 y Fy(])21 b Fv(\012)f Fy(\()p FI(f)2249 5243 y Ft(1)2288 5229 y Fy(\()p FI(t)2356 5243 y Ft(1)2396 5229 y Fy(\))15 b Fv(\001)g(\001)g(\001)h FI(f)2612 5243 y Fu(n)p Fs(\000)p Ft(1)2749 5229 y Fy(\()p FI(t)2817 5243 y Fu(n)p Fs(\000)p Ft(1)2954 5229 y Fy(\))p FI(f)3034 5243 y Fu(n)3081 5229 y Fy(\()p FI(t)p Fy(\)\))p Fv(j)3244 5243 y Fu(t)3269 5252 y FJ(1)3305 5243 y Ft(=)p Fu(t;)p Fs(\001\001\001)11 b Fu(;t)3521 5252 y FB(n)p FC(\000)p FJ(1)3642 5243 y Ft(=)p Fu(t)303 5390 y Fy(=)20 b Fv(\000)g Fy([)p FI(a)558 5353 y Ft(1)618 5390 y Fv(\012)g FI(f)754 5404 y Ft(1)793 5412 y Fu(\025)834 5421 y FJ(1)872 5390 y Fv(\001)15 b(\001)g(\001)978 5405 y Fu(\025)1019 5414 y FB(n)p FC(\000)p FJ(1)1145 5390 y FI(a)1193 5353 y Fu(n)1260 5390 y Fv(\012)20 b FI(f)1396 5404 y Fu(n)1442 5390 y Fy(])p FI(:)p Black Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)118 b(9)p Black 225 77 a Fy(b\).)369 249 y([)p FI(a)442 212 y Ft(1)502 249 y Fv(\012)20 b FI(f)638 263 y Ft(1)677 271 y Fu(\025)718 280 y FJ(1)772 249 y Fv(\001)15 b(\001)g(\001)h FI(a)941 212 y Fu(n)p 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FJ(1)3149 423 y Ft(=)p Fu(t)298 570 y Fy(=)20 b Fv(\000)g Fy([)p FI(a)553 532 y Ft(1)593 585 y Fu(\025)634 594 y FJ(1)669 585 y Ft(+)p Fu(@)761 593 y FB(t)785 608 y FJ(1)842 570 y Fv(\001)15 b(\001)g(\001)i FI(a)1012 532 y Fu(n)1059 585 y Fs(\000)p Fu(@)t Fs(\000)p Fu(\025)1251 594 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1463 585 y Fs(\000)p Fu(@)1555 593 y FB(t)1579 608 y FJ(1)p FC(!)p FB(n)1717 585 y Ft(+)p Fu(@)1809 593 y FB(t)1833 601 y(n)1883 570 y FI(a)1931 532 y Fu(n)p Fs(\000)p Ft(1)2068 570 y Fy(])k Fv(\012)f Fy(\()p FI(f)2285 584 y Ft(1)2324 570 y Fy(\()p FI(t)2392 584 y Ft(1)2432 570 y Fy(\))15 b Fv(\001)g(\001)g(\001)h FI(f)2648 584 y Fu(n)p Fs(\000)p Ft(1)2785 570 y Fy(\()p FI(t)2853 584 y Fu(n)p Fs(\000)p Ft(1)2990 570 y Fy(\))p FI(f)3070 584 y Fu(n)3117 570 y Fy(\()p FI(t)3185 584 y Fu(n)3232 570 y Fy(\)\))p Fv(j)3327 584 y Fu(t)3352 593 y FJ(1)3388 584 y Ft(=)p Fu(t;)p Fs(\001\001\001)11 b Fu(;t)3604 592 y FB(n)3647 584 y Ft(=)p Fu(t)298 729 y Fy(=)20 b Fv(\000)g Fy([)p FI(a)553 692 y Ft(1)593 744 y Fu(\025)634 753 y FJ(1)669 744 y Ft(+)p Fu(@)761 752 y FB(t)785 767 y FJ(1)842 729 y Fv(\001)15 b(\001)g(\001)i FI(a)1012 692 y Fu(n)1059 760 y Fs(\000)1123 743 y Ft(~)1114 760 y Fu(@)t Fs(\000)p Fu(\025)1251 769 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1463 760 y Ft(+)p Fu(@)1555 768 y FB(t)1579 776 y(n)1629 729 y FI(a)1677 692 y Fu(n)p Fs(\000)p Ft(1)1815 729 y Fy(])j Fv(\012)g Fy(\()p FI(f)2031 743 y Ft(1)2070 729 y Fy(\()p FI(t)2138 743 y Ft(1)2178 729 y Fy(\))15 b Fv(\001)g(\001)g(\001)i FI(f)2395 743 y Fu(n)p Fs(\000)p Ft(1)2531 729 y Fy(\()p FI(t)2599 743 y Fu(n)p Fs(\000)p Ft(1)2737 729 y Fy(\))p FI(f)2817 743 y Fu(n)2863 729 y Fy(\()p FI(t)2931 743 y Fu(n)2979 729 y Fy(\)\))p Fv(j)3074 743 y Fu(t)3099 752 y FJ(1)3135 743 y Ft(=)p Fu(t;)p Fs(\001\001\001)10 b Fu(;t)3350 751 y FB(n)3393 743 y Ft(=)p Fu(t)298 892 y Fy(=)20 b Fv(\000)g Fy([)p FI(a)553 855 y Ft(1)613 892 y Fv(\012)g FI(f)749 906 y Ft(1)788 914 y Fu(\025)829 923 y FJ(1)883 892 y Fv(\001)15 b(\001)g(\001)h FI(a)1052 855 y Fu(n)1119 892 y Fv(\012)k FI(f)1255 906 y Fu(n)1302 923 y Fs(\000)1366 906 y Ft(~)1357 923 y Fu(@)t Fs(\000)p Fu(\025)1494 932 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1710 892 y FI(a)1758 855 y Fu(n)p Fs(\000)p Ft(1)1916 892 y Fv(\012)f FI(f)2051 906 y Fu(n)p Fs(\000)p Ft(1)2188 892 y Fy(])3834 1065 y Fk(\003)125 1265 y SDict begin H.S end 125 1265 a 125 1265 a SDict begin 16 H.A end 125 1265 a 125 1265 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.2) cvn /DEST pdfmark end 125 1265 a Black FH(Corollary)31 b(3.2.)p Black 38 w Fo(\(i\))e(F)-7 b(or)29 b(e)-5 b(ach)29 b FI(p)c Fv(\025)g Fy(1)p FI(;)15 b(p)26 b Fv(2)f Fp(N)1751 1279 y Ft(+)1816 1265 y Fo(,)1893 1242 y Fy(~)1873 1265 y FI(R)1942 1279 y Fu(p)2007 1265 y Fy(:=)h FI(R)q Fy([)p FI(t)2257 1227 y Fs(\006)p Ft(1)2257 1291 y(1)2351 1265 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(t)2600 1232 y Fs(\006)p Ft(1)2600 1288 y Fu(p)2695 1265 y Fy(])28 b Fo(is)h(an)f FI(n)p Fo(-Lie)f(c)-5 b(onformal)31 b(algebr)-5 b(a)125 1398 y(with)334 1374 y Fy(~)322 1398 y FI(@)370 1412 y Fu(p)436 1398 y Fy(:=)25 b FI(@)g Fv(\012)20 b Fy(1)15 b Fv(\001)g(\001)g(\001)i Fy(1)k(+)f(1)g Fv(\012)g FI(@)1264 1412 y Fu(t)1289 1421 y FJ(1)1349 1398 y Fv(\012)g Fy(1)15 b Fv(\001)g(\001)g(\001)i Fy(1)j(+)g Fv(\001)15 b(\001)g(\001)21 b Fy(+)f(1)h Fv(\012)f(\001)15 b(\001)g(\001)21 b(\012)f Fy(1)h Fv(\012)f FI(@)2574 1412 y Fu(t)2599 1420 y FB(p)2672 1398 y Fo(and)34 b(the)f(c)-5 b(onformal)35 b(pr)-5 b(o)g(ducts)260 1571 y Fy(\()p FI(a)343 1533 y Ft(1)403 1571 y Fv(\012)20 b FI(f)539 1585 y Ft(1)578 1571 y Fy(\))613 1589 y Ft(\()p Fu(k)677 1598 y FJ(1)712 1589 y Ft(\))744 1571 y Fv(\001)15 b(\001)g(\001)850 1589 y Ft(\()p Fu(k)914 1598 y FB(n)p FC(\000)p FJ(1)1035 1589 y Ft(\))1067 1571 y Fy(\()p FI(a)1150 1533 y Fu(n)1217 1571 y Fv(\012)20 b FI(f)1353 1585 y Fu(n)1400 1571 y Fy(\))190 1757 y(=)332 1671 y Fm(X)276 1884 y Fu(j)309 1854 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)305 1905 y(1)p FC(!)p FB(p)520 1757 y Fy(\()p FI(a)603 1719 y Ft(1)643 1781 y(\()p Fu(k)707 1790 y FJ(1)742 1781 y Ft(+)p Fu(j)830 1758 y FJ(1)826 1802 y(1)p FC(!)p FB(p)953 1781 y Ft(\))984 1757 y Fv(\001)15 b(\001)g(\001)1090 1788 y Ft(\()p Fu(k)1154 1797 y FB(n)p FC(\000)p FJ(1)1275 1788 y Ft(+)p Fu(j)1363 1758 y FB(n)p FC(\000)p FJ(1)1359 1809 y(1)p FC(!)p FB(p)1486 1788 y Ft(\))1518 1757 y FI(a)1566 1719 y Fu(n)1613 1757 y Fy(\))20 b Fv(\012)g Fy(\(\()p FI(@)1882 1705 y Ft(\()p Fu(j)1942 1682 y FJ(1)1938 1726 y(1)1978 1705 y Ft(\))1877 1781 y Fu(t)1902 1790 y FJ(1)2025 1757 y Fv(\001)15 b(\001)g(\001)h FI(@)2199 1699 y Ft(\()p Fu(j)2259 1676 y FJ(1)2255 1716 y FB(p)2293 1699 y Ft(\))2194 1781 y Fu(t)2219 1789 y FB(p)2325 1757 y FI(f)2370 1771 y Ft(1)2409 1757 y Fy(\))f Fv(\001)g(\001)g(\001)i Fy(\()p FI(@)2669 1705 y Ft(\()p Fu(j)2729 1675 y FB(n)p FC(\000)p FJ(1)2725 1726 y(1)2850 1705 y Ft(\))2664 1781 y Fu(t)2689 1790 y FJ(1)2897 1757 y Fv(\001)e(\001)g(\001)h FI(@)3071 1705 y Ft(\()p Fu(j)3131 1676 y FB(n)p FC(\000)p FJ(1)3127 1716 y FB(p)3252 1705 y Ft(\))3066 1781 y Fu(t)3091 1789 y FB(p)3284 1757 y FI(f)3329 1771 y Fu(n)p Fs(\000)p Ft(1)3465 1757 y Fy(\))p FI(f)3545 1771 y Fu(n)3592 1757 y Fy(\))p FI(:)3718 1757 y SDict begin H.S end 3718 1757 a 3718 1757 a SDict begin 19.25 H.A end 3718 1757 a 3718 1757 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 3718 1757 a Fy(\(3.3\))225 2089 y Fo(\(ii\))37 b(Denote)g Fq(L)18 b FI(ie)879 2103 y Fu(p)952 2089 y FI(R)34 b Fy(:=)1205 2066 y(~)1185 2089 y FI(R)1254 2103 y Fu(p)1294 2089 y FI(=)1351 2065 y Fy(~)1339 2089 y FI(@)1387 2103 y Fu(p)1447 2066 y Fy(~)1427 2089 y FI(R)1496 2103 y Fu(p)1573 2089 y Fo(and)k FI(a)1802 2103 y Fu(m)1864 2112 y FJ(1)p FC(!)p FB(p)2030 2089 y Fy(=)33 b FI(a)2182 2103 y Fu(m)2244 2112 y FJ(1)2279 2103 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2451 2111 y FB(p)2525 2089 y Fy(:=)34 b FI(a)24 b Fv(\012)f FI(t)2854 2049 y Fu(m)2916 2058 y FJ(1)2854 2115 y Ft(1)2970 2089 y Fv(\001)15 b(\001)g(\001)h FI(t)3124 2040 y Fu(m)3186 2048 y FB(p)3124 2101 y Fu(p)3261 2089 y Fv(2)33 b Fq(L)18 b FI(ie)3525 2103 y Fu(p)3598 2089 y FI(R)q Fo(.)55 b(The)125 2222 y(br)-5 b(acket)435 2394 y Fy([)p FI(a)508 2357 y Ft(1)508 2426 y Fu(m)570 2403 y FJ(1)570 2448 y(1)p FC(!)p FB(p)703 2394 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)967 2357 y Fu(n)967 2417 y(m)1029 2394 y FB(n)1029 2438 y FJ(1)p FC(!)p FB(p)1162 2394 y Fy(])365 2605 y(=)507 2519 y Fm(X)450 2732 y Fu(j)483 2703 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)479 2754 y(1)p FC(!)p FB(p)710 2532 y Fm(\000)752 2558 y Fu(m)814 2534 y FJ(1)814 2578 y(1)767 2644 y Fu(j)800 2621 y FJ(1)796 2665 y(1)849 2532 y Fm(\001)906 2605 y Fv(\001)g(\001)g(\001)1027 2532 y Fm(\000)1069 2554 y Fu(m)1131 2525 y FB(n)p FC(\000)p FJ(1)1131 2565 y FB(p)1084 2647 y Fu(j)1117 2618 y FB(n)p FC(\000)p FJ(1)1113 2658 y FB(p)1252 2532 y Fm(\001)1294 2605 y Fy(\()p FI(a)1377 2568 y Ft(1)1416 2629 y(\()p Fu(j)1476 2606 y FJ(1)1472 2650 y(1)p FC(!)p FB(p)1600 2629 y Ft(\))1631 2605 y Fv(\001)g(\001)g(\001)1738 2636 y Ft(\()p Fu(j)1798 2606 y FB(n)p FC(\000)p FJ(1)1794 2657 y(1)p FC(!)p FB(p)1921 2636 y Ft(\))1953 2605 y FI(a)2001 2568 y Fu(n)2048 2605 y Fy(\))2083 2636 y Fu(m)2145 2613 y FB(n)2145 2657 y FJ(1)2188 2636 y Ft(+)p Fu(m)2305 2606 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2305 2657 y(1)2517 2636 y Fs(\000)p Fu(j)2605 2606 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2601 2657 y(1)2817 2636 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2989 2617 y FB(n)2989 2653 y(p)3032 2636 y Ft(+)p Fu(m)3149 2606 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)3149 2647 y FB(p)3361 2636 y Fs(\000)p Fu(j)3449 2606 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)3445 2647 y FB(p)3718 2876 y SDict begin H.S end 3718 2876 a 3718 2876 a SDict begin 19.25 H.A end 3718 2876 a 3718 2876 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 3718 2876 a Fy(\(3.4\))125 3048 y Fo(is)32 b(a)h(wel)5 b(l-de\014ne)-5 b(d)34 b FI(n)p Fo(-Lie)d(br)-5 b(acket)33 b(on)h Fq(L)17 b FI(ie)1650 3062 y Fu(p)1723 3048 y FI(R)q Fo(.)42 b(Besides,)32 b Fq(L)18 b FI(ie)2389 3062 y Fu(p)2462 3048 y FI(R)33 b Fo(is)g(a)g Fp(C)18 b Fy([)p FI(@)5 b Fy(])p Fo(-mo)l(d)q(ule)38 b(via)1112 3279 y SDict begin H.S end 1112 3279 a 1112 3279 a SDict begin 16 H.A end 1112 3279 a 1112 3279 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.5) cvn /DEST pdfmark end 1112 3279 a FI(@)5 b Fy(\()p FI(a)1248 3293 y Fu(m)1310 3302 y FJ(1)1346 3293 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1518 3301 y FB(p)1558 3279 y Fy(\))26 b(=)f Fv(\000)p Fy(\()1869 3161 y Fu(p)1821 3193 y Fm(X)1830 3388 y Fu(i)p Ft(=0)1968 3279 y FI(m)2048 3293 y Fu(i)2076 3279 y FI(a)2124 3293 y Fu(m)2186 3302 y FJ(1)2221 3293 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2393 3303 y FB(i)2419 3293 y Fs(\000)p Ft(1)p Fu(;)p Fs(\001\001\001)o Fu(m)2650 3301 y FB(p)2690 3279 y Fy(\))p FI(:)p Black 968 w Fy(\(3.5\))p Black 225 3546 a Fo(\(iii\))32 b(The)h(induc)-5 b(e)g(d)33 b FI(@)976 3560 y Fu(t)1001 3570 y FB(i)1065 3546 y Fo(ar)-5 b(e)34 b(derivations)g(of)f Fq(L)18 b FI(ie)1965 3560 y Fu(p)2037 3546 y FI(R)q Fo(,)33 b(given)f(by)1231 3718 y SDict begin H.S end 1231 3718 a 1231 3718 a SDict begin 16 H.A end 1231 3718 a 1231 3718 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 1231 3718 a FI(@)1279 3732 y Fu(t)1304 3742 y FB(i)1335 3718 y Fy(\()p FI(a)1418 3732 y Fu(m)1480 3741 y FJ(1)1515 3732 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)1688 3740 y FB(p)1728 3718 y Fy(\))25 b(=)g FI(m)1964 3732 y Fu(i)1992 3718 y FI(a)2040 3732 y Fu(m)2102 3741 y FJ(1)2137 3732 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)2310 3742 y FB(i)2336 3732 y Fs(\000)p Ft(1)p Fu(;)p Fs(\001\001\001)o Fu(m)2567 3740 y FB(p)2607 3718 y FI(:)p Black 1086 w Fy(\(3.6\))p Black 125 3891 a Fo(Thus)33 b FI(@)38 b Fo(in)32 b(\(ii\))h(is)f(also)i(a)f(derivation)h(of)f Fq(L)18 b FI(ie)1785 3905 y Fu(p)1858 3891 y FI(R)q Fo(.)p Black 125 4108 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(\(i\))31 b(Based)i(on)e(the)h(fact)h(that)1486 4085 y(~)1466 4108 y FI(R)1535 4122 y Fu(p)p Ft(+1)1693 4108 y Fy(=)1811 4085 y(~)1791 4108 y FI(R)1860 4122 y Fu(p)1900 4108 y Fy([)p FI(t)1958 4070 y Fs(\006)p Ft(1)1958 4134 y Fu(p)p Ft(+1)2088 4108 y Fy(],)f(the)g(result)f(can)h(b)s(e)f(inductiv) m(ely)f(obtained)h(from)125 4241 y(Lemma)449 4241 y SDict begin H.S end 449 4241 a Black Fy(3.1)p Black 566 4182 a SDict begin H.R end 566 4182 a 566 4241 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 566 4241 a Fy(.)225 4406 y(\(ii\))e(The)h(statemen)m(t)i(is)e (ob)m(viously)f(true)h(b)m(y)g(\(i\))g(and)g(Lemma)2425 4406 y SDict begin H.S end 2425 4406 a Black Fy(2.4)p Black 2541 4348 a SDict begin H.R end 2541 4348 a 2541 4406 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.4) cvn H.B /ANN pdfmark end 2541 4406 a Fy(.)225 4571 y(\(iii\))42 b(Since)h FI(@)714 4585 y Fu(t)739 4595 y FB(i)814 4571 y Fy(comm)m(utes)i(with)1487 4547 y(~)1475 4571 y FI(@)1523 4585 y Fu(p)1563 4571 y Fy(,)j(\()1671 4571 y SDict begin H.S end 1671 4571 a Black Fy(3.6)p Black 1787 4513 a SDict begin H.R end 1787 4513 a 1787 4571 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 1787 4571 a Fy(\))d(is)e(w)m(ell-de\014ned.)79 b(The)44 b(statemen)m(t)i(can)e(b)s(e)f(pro)m(v)m(ed)h(b)m(y)125 4704 y(v)m(erifying)755 4883 y FI(@)803 4897 y Fu(t)828 4907 y FB(i)859 4883 y Fy([)p FI(a)932 4846 y Ft(1)932 4915 y Fu(m)994 4893 y FJ(1)994 4937 y(1)p FC(!)p FB(p)1126 4883 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1391 4846 y Fu(n)1391 4906 y(m)1453 4883 y FB(n)1453 4927 y FJ(1)p FC(!)p FB(p)1586 4883 y Fy(])25 b(=)1776 4770 y Fu(n)1732 4797 y Fm(X)1736 4993 y Fu(j)t Ft(=1)1863 4883 y Fy([)p FI(a)1936 4846 y Ft(1)1936 4915 y Fu(m)1998 4893 y FJ(1)1998 4937 y(1)p FC(!)p FB(p)2131 4883 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(@)2395 4897 y Fu(t)2420 4907 y FB(i)2452 4883 y Fy(\()p FI(a)2535 4839 y Fu(j)2535 4932 y(m)2597 4900 y FB(j)2597 4953 y FJ(1)p FC(!)p FB(p)2730 4883 y Fy(\))p FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3030 4846 y Fu(n)3030 4906 y(m)3092 4883 y FB(n)3092 4927 y FJ(1)p FC(!)p FB(p)3224 4883 y Fy(])p FI(:)125 5107 y Fy(T)-8 b(ak)m(e)32 b FI(@)393 5121 y Fu(t)418 5130 y FJ(1)487 5107 y Fy(as)f(an)f(example.)474 5215 y Fu(n)430 5243 y Fm(X)434 5438 y Fu(j)t Ft(=1)561 5329 y Fy([)p FI(a)634 5291 y Ft(1)634 5361 y Fu(m)696 5338 y FJ(1)696 5382 y(1)731 5361 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)904 5342 y FJ(1)904 5377 y FB(p)944 5329 y FI(;)k Fv(\001)g(\001)g(\001)32 b FI(;)15 b(@)1209 5343 y Fu(t)1234 5352 y FJ(1)1274 5329 y Fy(\()p FI(a)1357 5285 y Fu(j)1357 5378 y(m)1419 5345 y FB(j)1419 5399 y FJ(1)1454 5378 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1626 5345 y FB(j)1626 5388 y(p)1666 5329 y Fy(\))p FI(;)15 b Fv(\001)g(\001)g(\001)33 b FI(;)15 b(a)1967 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494 y Fm(P)125 643 y Fu(i)p Ft(=1)254 562 y FI(e)296 576 y Fu(i)345 562 y Fv(\012)21 b FI(f)482 576 y Fu(i)540 562 y Fy(and)31 b(an)g(elemen)m(t)g(of)1305 539 y(~)1285 562 y FI(R)1354 576 y Fu(p)1425 562 y Fy(as)1578 475 y Fu(r)1546 494 y Fm(P)1537 643 y Fu(i)p Ft(=0)1717 475 y Fu(l)1680 494 y Fm(P)1667 643 y Fu(j)t Ft(=1)1804 562 y FI(@)1852 576 y Fu(i)1881 562 y FI(e)1923 576 y Fu(j)1980 562 y Fv(\012)21 b FI(f)2117 576 y Fu(ij)2177 562 y Fy(.)42 b(Th)m(us)30 b(for)h(an)g(elemen)m(t)g(in)f(the)h(in)m(tersection)g(of)125 758 y FI(V)20 b Fy([)p FI(t)256 720 y Fs(\006)p Ft(1)256 785 y(1)351 758 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(t)600 726 y Fs(\006)p Ft(1)600 781 y Fu(p)695 758 y Fy(])30 b(and)939 735 y(~)927 758 y FI(@)975 772 y Fu(p)1035 736 y Fy(~)1015 758 y FI(R)1084 772 y Fu(p)1124 758 y Fy(,)g(w)m(e)h(ha)m(v)m(e)1341 1006 y(~)1330 1030 y FI(@)1378 1044 y Fu(p)1417 1030 y Fy(\()1502 916 y Fu(r)1452 943 y Fm(X)1461 1139 y Fu(i)p Ft(=0)1654 916 y Fu(l)1599 943 y 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Fu(;j)3346 2973 y FB(p)3397 2998 y Fv(\012)h FI(t)3513 2954 y Fu(j)3542 2963 y FJ(1)3513 3024 y Ft(1)3596 2998 y Fv(\001)j(\001)g(\001)h FI(t)3750 2949 y Fu(j)3779 2957 y FB(p)3750 3010 y Fu(p)3844 2998 y Fv(2)125 3183 y FI(R)q Fy([)p FI(t)253 3197 y Ft(1)292 3183 y FI(;)f Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)542 3197 y Fu(p)582 3183 y Fy(])30 b(suc)m(h)g(that)900 3351 y(~)889 3375 y FI(@)937 3389 y Fu(p)977 3375 y Fy(\()1065 3289 y Fm(X)1012 3484 y Fu(j)1041 3493 y FJ(1)1075 3484 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)1214 3492 y FB(p)1265 3375 y FI(s)1308 3337 y Fu(j)1337 3346 y FJ(1)1371 3337 y Fu(;)p Fs(\001\001\001)g Fu(;j)1510 3345 y FB(p)1569 3375 y Fv(\012)20 b FI(t)1693 3331 y Fu(j)1722 3340 y FJ(1)1693 3401 y Ft(1)1776 3375 y Fv(\001)15 b(\001)g(\001)h FI(t)1930 3326 y Fu(j)1959 3334 y FB(p)1930 3387 y Fu(p)1998 3375 y Fy(\))26 b(=)2204 3289 y Fm(X)2155 3484 y Fu(i)2179 3493 y FJ(1)2214 3484 y Fu(;)p Fs(\001\001\001)10 b Fu(;i)2348 3492 y FB(p)2399 3375 y FI(r)2443 3337 y Fu(i)2467 3346 y FJ(1)2501 3337 y Fu(;)p Fs(\001\001\001)g Fu(;i)2635 3345 y FB(p)2695 3375 y Fv(\012)20 b FI(t)2819 3334 y Fu(i)2843 3343 y FJ(1)2819 3401 y Ft(1)2897 3375 y Fv(\001)15 b(\001)g(\001)h FI(t)3051 3326 y Fu(i)3075 3334 y FB(p)3051 3387 y Fu(p)3115 3375 y FI(:)125 3641 y Fy(It)30 b(can)h(b)s(e)f(pro)m (v)m(en)g(b)m(y)h(induction)d(on)i FI(p)p Fy(.)225 3807 y(Let)g FI(p)25 b Fy(=)g(1.)41 b(F)-8 b(or)32 b FI(r)22 b Fv(\012)e FI(t)1016 3774 y Fu(m)1016 3831 y Ft(1)1108 3807 y Fv(2)25 b FI(R)q Fy([)p FI(t)1322 3768 y Fs(\006)p Ft(1)1322 3833 y(1)1416 3807 y Fy(])30 b(where)g(\()p FI(@)c Fv(\000)20 b FI(c)p Fy(\))2008 3774 y Fu(j)2045 3807 y FI(r)28 b Fy(=)d(0)31 b(for)f(a)h(\014xed)e FI(j)5 b Fy(,)32 b(it)e(can)g(b)s(e)g(computed)g(that)180 4262 y FI(r)22 b Fv(\012)e FI(t)367 4224 y Fu(m)367 4284 y Ft(1)459 4262 y Fy(=)555 3912 y Fm(8)555 3994 y(>)555 4021 y(>)555 4048 y(>)555 4076 y(>)555 4103 y(>)555 4130 y(>)555 4157 y(<)555 4321 y(>)555 4348 y(>)555 4376 y(>)555 4403 y(>)555 4430 y(>)555 4457 y(>)555 4485 y(:)647 4020 y Fy(~)636 4044 y FI(@)684 4058 y Ft(1)723 4044 y Fy(\()800 3953 y Fu(j)767 3976 y Fm(P)758 4126 y Fu(i)p Ft(=1)965 4008 y Fu(m)p Ft(!)p 898 4023 216 4 v 898 4076 a(\()p Fu(m)p Ft(+)p Fu(i)p Ft(\)!)1124 4044 y Fy(\()p Fv(\000)p FI(@)5 b Fy(\))1318 4011 y Fu(i)p Fs(\000)p Ft(1)1437 4044 y FI(r)23 b Fv(\012)d FI(t)1625 4006 y Fu(m)p Ft(+)p Fu(i)1625 4070 y Ft(1)1770 4044 y Fy(\))p FI(;)1408 b Fy(if)30 b FI(m)25 b Fv(\025)g Fy(0)p FI(;)15 b(c)26 b Fy(=)f(0)p FI(;)636 4259 y Fv(\000)719 4235 y Fy(~)707 4259 y FI(@)755 4273 y Ft(1)794 4259 y Fy(\()855 4172 y Fu(m)838 4191 y Fm(P)829 4341 y Fu(i)p Ft(=0)969 4223 y Fu(m)p Ft(!)p 969 4238 83 4 v 988 4290 a Fu(i)p Ft(!)1061 4259 y Fy(\()p Fv(\000)p FI(f)10 b Fy(\()p FI(@)5 b Fy(\)\))1380 4226 y Fu(m)p Ft(+1)p Fs(\000)p Fu(i)1616 4259 y FI(r)23 b Fv(\012)d FI(t)1804 4226 y Fu(i)1804 4283 y Ft(1)1843 4259 y Fy(\))p FI(;)1335 b Fy(if)30 b FI(m)25 b Fv(\025)g Fy(0)p FI(;)15 b(c)26 b Fv(6)p Fy(=)f(0)p FI(;)636 4488 y Fv(\000)719 4464 y Fy(~)707 4488 y FI(@)755 4502 y Ft(1)794 4488 y Fy(\()829 4402 y Fs(\000)p Fu(m)p Fs(\000)p Ft(1)885 4420 y Fm(P)876 4570 y Fu(i)p Ft(=0)1062 4443 y(\()p Fs(\000)p Fu(m)p Fs(\000)p Ft(1)p Fs(\000)p Fu(i)p Ft(\)!)p 1062 4467 361 4 v 1101 4521 a(\()p Fs(\000)p Fu(m)p Fs(\000)p Ft(1\)!)1433 4488 y Fy(\()p Fv(\000)p FI(@)5 b Fy(\))1627 4455 y Fu(i)p Fs(\000)p Ft(1)1746 4488 y FI(r)23 b Fv(\012)d FI(t)1934 4450 y Fu(m)p Ft(+)p Fu(i)1934 4514 y Ft(1)2079 4488 y Fy(\))g Fv(\000)2359 4452 y Ft(1)p 2235 4467 282 4 v 2235 4521 a(\()p Fs(\000)p Fu(m)p Fs(\000)p Ft(1\)!)2527 4488 y Fy(\()p Fv(\000)p FI(@)5 b Fy(\))2721 4455 y Fu(i)p Fs(\000)p Ft(1)2840 4488 y FI(r)23 b Fv(\012)d FI(t)3028 4450 y Fs(\000)p Ft(1)3028 4514 y(1)3122 4488 y FI(;)91 b Fy(if)30 b FI(m)25 b Fv(\024)g(\000)p Fy(1)p FI(;)3718 4655 y SDict begin H.S end 3718 4655 a 3718 4655 a SDict begin 19.25 H.A end 3718 4655 a 3718 4655 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.7) cvn /DEST pdfmark end 3718 4655 a Fy(\(3.7\))125 4841 y(where)32 b FI(f)10 b Fy(\()p FI(@)5 b Fy(\))33 b(is)f(a)h(p)s(olynomial)d(in)h Fp(C)18 b Fy([)p FI(@)5 b Fy(])39 b(suc)m(h)33 b(that)g FI(@)5 b(f)10 b Fy(\()p FI(@)5 b Fy(\))p FI(r)32 b Fy(=)d FI(r)s Fy(.)47 b(By)33 b(\()2672 4805 y Fu(@)p 2672 4820 42 4 v 2677 4872 a(c)2745 4841 y Fv(\000)22 b Fy(1\))2918 4808 y Fu(j)2955 4841 y FI(r)32 b Fy(=)c(0,)34 b FI(f)42 b Fy(is)32 b(w)m(ell-de\014ned.)125 5009 y(Besides,)e(that)h FI(r)23 b Fv(\012)d FI(t)858 4971 y Fs(\000)p Ft(1)858 5035 y(1)977 5009 y Fv(2)1075 4985 y Fy(~)1063 5009 y FI(@)1111 5023 y Ft(1)1170 4986 y Fy(~)1151 5009 y FI(R)1220 5023 y Ft(1)1289 5009 y Fy(connotes)32 b(the)e(existence)h(of)2372 4918 y Fu(q)2341 4941 y Fm(P)2311 5093 y Fu(i)p Ft(=)p Fs(\000)p Fu(l)2481 5009 y FI(r)2522 5023 y Fu(i)2571 5009 y Fv(\012)20 b FI(t)2695 4976 y Fu(i)2695 5034 y Ft(1)2764 5009 y Fy(suc)m(h)30 b(that)682 5333 y FI(r)23 b Fv(\012)d FI(t)870 5294 y Fs(\000)p Ft(1)870 5359 y(1)989 5333 y Fy(=)1071 5309 y(~)1060 5333 y FI(@)1108 5347 y Ft(1)1148 5333 y Fy(\()1244 5214 y Fu(q)1195 5246 y Fm(X)1183 5444 y Fu(i)p Ft(=)p Fs(\000)p Fu(l)1354 5333 y FI(r)1395 5347 y Fu(i)1443 5333 y Fv(\012)g FI(t)1567 5295 y Fu(i)1567 5355 y Ft(1)1606 5333 y Fy(\))p Black Black eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(12)858 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 989 159 a Fy(=)c Fv(\000)g FI(l)r(r)1241 174 y Fs(\000)p Fu(l)1342 159 y Fv(\012)g FI(t)1466 121 y Fs(\000)p Fu(l)q Fs(\000)p Ft(1)1466 185 y(1)1657 159 y Fy(+)1781 46 y Fs(\000)p Ft(2)1760 73 y Fm(X)1748 270 y Fu(i)p Ft(=)p Fs(\000)p Fu(l)1904 159 y Fy(\()p FI(@)5 b(r)2033 173 y Fu(i)2082 159 y Fy(+)20 b(\()p FI(i)g Fy(+)g(1\))p FI(r)2471 173 y Fu(i)p Ft(+1)2590 159 y Fy(\))h Fv(\012)f FI(t)2770 122 y Fu(i)2770 182 y Ft(1)2829 159 y Fy(+)g FI(@)5 b(r)3014 173 y Fs(\000)p Ft(1)3129 159 y Fv(\012)20 b FI(t)3253 121 y Fs(\000)p Ft(1)3253 185 y(1)1444 492 y Fy(+)1538 374 y Fu(q)r Fs(\000)p Ft(1)1535 406 y Fm(X)1543 601 y Fu(i)p Ft(=0)1666 492 y Fy(\()p FI(@)5 b(r)1795 506 y Fu(i)1844 492 y Fy(+)20 b(\()p FI(i)h Fy(+)f(1\))p FI(r)2234 506 y Fu(i)p Ft(+1)2353 492 y Fy(\))g Fv(\012)g FI(t)2532 455 y Fu(i)2532 515 y Ft(1)2592 492 y Fy(+)g FI(@)5 b(r)2777 506 y Fu(q)2835 492 y Fv(\012)20 b FI(t)2959 448 y Fu(q)2959 518 y Ft(1)2998 492 y FI(:)125 783 y Fy(Con)m(trasting)36 b(the)h(co)s(e\016cien)m(ts,) i(w)m(e)e(ha)m(v)m(e)h FI(r)1682 798 y Fs(\000)p Fu(l)1798 783 y Fy(=)e(0)h(when)e FI(l)j Fy(is)e(large)h(enough.)59 b(Then)35 b FI(r)3273 798 y Fs(\000)p Fu(l)q Ft(+1)3480 783 y Fy(=)g FI(r)3627 798 y Fs(\000)p Fu(l)q Ft(+2)3834 783 y Fy(=)125 916 y Fv(\001)15 b(\001)g(\001)29 b Fy(=)e FI(r)398 930 y Fs(\000)p Ft(1)521 916 y Fy(=)g(0)32 b(are)h(deduced)e (in)f(succession.)45 b(So)32 b FI(r)e Fy(=)e FI(@)5 b(r)2175 930 y Fs(\000)p Ft(1)2297 916 y Fy(=)28 b(0)k(and)g(th)m(us)2873 893 y(~)2853 916 y FI(R)2922 930 y Ft(1)2989 916 y Fy(=)c FI(R)q(t)3191 877 y Fs(\000)p Ft(1)3191 942 y(1)3306 916 y Fv(\010)3410 892 y Fy(~)3398 916 y FI(@)3446 930 y Ft(1)3505 893 y Fy(~)3486 916 y FI(R)3555 930 y Ft(1)3594 916 y Fy(.)46 b(Hence)125 1048 y Fq(L)18 b FI(ie)295 1062 y Ft(1)365 1048 y FI(R)26 b Fv(')f FI(R)q(t)659 1010 y Fs(\000)p Ft(1)659 1075 y(1)753 1048 y Fy(.)225 1214 y(No)m(w)38 b(assume)f(that)h(the)g(statemen)m(t)h(is)e(true)g (for)h FI(p)f Fy(=)g FI(k)s Fy(.)62 b(Let)38 b(us)f(consider)f FI(r)28 b Fv(\012)c FI(t)3182 1173 y Fu(m)3244 1182 y FJ(1)3182 1240 y Ft(1)3299 1214 y Fv(\001)15 b(\001)g(\001)h FI(t)3453 1164 y Fu(m)3515 1176 y FB(k)q FJ(+1)3453 1243 y Fu(k)r Ft(+1)3672 1214 y Fy(where)125 1347 y FI(m)205 1361 y Ft(1)244 1347 y FI(;)f Fv(\001)g(\001)g(\001)32 b FI(;)15 b(m)541 1362 y Fu(k)r Ft(+1)721 1347 y Fv(\025)47 b Fy(0.)80 b(Based)44 b(on)f(the)h(assumption,)h(there)f(exists)f FI(f)2655 1361 y Fu(m)2717 1373 y FB(k)q FJ(+1)2883 1347 y Fv(2)k FI(R)q Fy([)p FI(t)3119 1361 y Ft(1)3158 1347 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)3408 1362 y Fu(k)3451 1347 y Fy(])44 b(suc)m(h)f(that)136 1455 y(~)125 1479 y FI(@)173 1494 y Fu(k)216 1479 y FI(f)261 1493 y Fu(m)323 1505 y FB(k)q FJ(+1)469 1479 y Fy(=)27 b FI(r)c Fv(\012)e FI(t)756 1439 y Fu(m)818 1448 y FJ(1)756 1505 y Ft(1)872 1479 y Fv(\001)15 b(\001)g(\001)h FI(t)1026 1436 y Fu(m)1088 1448 y FB(k)1026 1509 y Fu(k)1131 1479 y Fy(.)43 b(Similarly)-8 b(,)29 b FI(f)1650 1493 y Fu(m)1712 1505 y FB(k)q FJ(+1)1827 1493 y Fs(\000)p Ft(1)1921 1479 y FI(;)15 b(f)2006 1493 y Fu(m)2068 1505 y FB(k)q FJ(+1)2184 1493 y Fs(\000)p Ft(2)2278 1479 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(f)2540 1493 y Ft(0)2610 1479 y Fy(can)32 b(b)s(e)f(found)f(b)m(y)h(recurrence)g(rela-)125 1612 y(tions)359 1588 y(~)348 1612 y FI(@)396 1627 y Fu(k)439 1612 y FI(f)484 1626 y Fu(i)536 1612 y Fy(=)25 b Fv(\000)p Fy(\()p FI(i)c Fy(+)f(1\))p FI(f)1006 1626 y Fu(i)p Ft(+1)1125 1612 y Fy(.)40 b(Then)30 b(w)m(e)g(ha)m(v)m(e)1225 1906 y FI(r)22 b Fv(\012)e FI(t)1412 1866 y Fu(m)1474 1875 y FJ(1)1412 1932 y Ft(1)1528 1906 y Fv(\001)15 b(\001)g(\001)i FI(t)1683 1856 y Fu(m)1745 1868 y FB(k)q FJ(+1)1683 1936 y Fu(k)r Ft(+1)1889 1906 y Fy(=)1997 1882 y(~)1985 1906 y FI(@)2033 1921 y Fu(k)r Ft(+1)2167 1906 y Fy(\()2202 1782 y Fu(m)2264 1794 y FB(k)q FJ(+1)2225 1820 y Fm(X)2234 2015 y Fu(i)p Ft(=0)2395 1906 y FI(f)2440 1920 y Fu(i)2487 1906 y Fv(\012)j FI(t)2611 1869 y Fu(i)2611 1929 y(k)r Ft(+1)2744 1906 y Fy(\))p FI(:)3834 2195 y Fk(\003)225 2494 y Fy(F)-8 b(rom)29 b(no)m(w)g(on,)h(let)f FI(R)h Fy(b)s(e)e(a)i(\014nite)e FI(n)p Fy(-Lie)g(conformal)h(algebra.)40 b(By)30 b([)2635 2494 y SDict begin H.S end 2635 2494 a Black Fy(18)p Black 2726 2435 a SDict begin H.R end 2726 2435 a 2726 2494 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.n) cvn H.B /ANN pdfmark end 2726 2494 a Fy(],)g(there)f(is)f(a) i(\014nite-dimensional)125 2627 y(space)h FI(V)50 b Fy(suc)m(h)30 b(that)1512 2823 y FI(R)c Fy(=)f(\()p Fp(C)18 b Fy([)p FI(@)6 b Fy(])26 b Fv(\012)20 b FI(V)g Fy(\))h Fv(\010)f FI(T)13 b(or)32 b(R)q(;)125 3038 y Fy(where)i FI(T)13 b(or)33 b(R)g Fy(:=)g Fv(f)p FI(r)j Fv(2)c FI(R)q Fv(j)p FI(f)10 b Fy(\()p FI(@)5 b Fy(\))p FI(r)36 b Fy(=)c(0)f(for)f(some)h (non-zero)g FI(f)10 b Fy(\()p FI(@)5 b Fy(\))33 b Fv(2)f Fp(C)18 b Fy([)p FI(@)5 b Fy(])q Fv(g)p Fy(.)60 b(Cho)s(ose)35 b Fv(f)p FI(v)3298 3005 y Fu(i)3327 3038 y Fv(g)g Fy(as)g(a)g(basis)f (of)125 3171 y FI(V)20 b Fy(,)31 b(and)e(w)m(e)i(can)g(de\014ne)f(a)g (series)g(of)h(subspaces)e(of)i Fq(L)18 b FI(ie)2106 3185 y Fu(p)2176 3171 y FI(R)658 3405 y SDict begin H.S end 658 3405 a 658 3405 a SDict begin 16 H.A end 658 3405 a 658 3405 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.8) cvn /DEST pdfmark end 658 3405 a Fv(f)p Fq(L)800 3367 y Ft(\()p Fu(m)p Ft(\))782 3427 y Fu(p)947 3405 y Fy(=)25 b FI(span)p Fv(f)p FI(v)1327 3367 y Fu(i)1324 3427 y(m)1386 3436 y FJ(1)1420 3427 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)1593 3435 y FB(p)1633 3405 y Fv(j)p FI(m)1738 3419 y Ft(1)1777 3405 y FI(;)k Fv(\001)g(\001)g(\001)32 b FI(;)15 b(m)2074 3419 y Fu(p)2139 3405 y Fv(\025)25 b Fy(0)p FI(;)15 b(m)2400 3419 y Ft(1)2461 3405 y Fy(+)20 b Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(m)2849 3419 y Fu(p)2913 3405 y Fv(\025)25 b FI(m)p Fv(gg)p FI(:)p Black 514 w Fy(\(3.8\))p Black 225 3671 a(Recall)34 b(that)i(a)g(top)s (ological)f(space)h(is)e(called)g(a)i(linearly)d(compact)j(space)g(if)e (it)h(is)f(a)i(pro)s(duct)e(of)h(\014nite)125 3804 y(dimensional)27 b(discrete)i(spaces)i([)1262 3804 y SDict begin H.S end 1262 3804 a Black Fy(14)p Black 1353 3745 a SDict begin H.R end 1353 3745 a 1353 3804 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.l) cvn H.B /ANN pdfmark end 1353 3804 a Fy(,)1408 3804 y SDict begin H.S end 1408 3804 a Black Fy(11)p Black 1499 3745 a SDict begin H.R end 1499 3745 a 1499 3804 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.g) cvn H.B /ANN pdfmark end 1499 3804 a Fy(],)f(and)f(a)i (linearly)c(compact)k(algebra)f(is)f(a)h(linearly)e(compact)j(space)125 3936 y(equipp)s(ed)36 b(with)h(a)h(con)m(tin)m(uous)g(structure)g(map)g ([)1916 3936 y SDict begin H.S end 1916 3936 a Black Fy(11)p Black 2007 3878 a SDict begin H.R end 2007 3878 a 2007 3936 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.g) cvn H.B /ANN pdfmark end 2007 3936 a Fy(,)2070 3936 y SDict begin H.S end 2070 3936 a Black Fy(8)p Black 2116 3878 a SDict begin H.R end 2116 3878 a 2116 3936 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ck) cvn H.B /ANN pdfmark end 2116 3936 a Fy(].)64 b(F)-8 b(rom)39 b(the)f(ab)s(o)m(v)m (e)i(de\014nition,)e(w)m(e)g(obtain)g(the)125 4069 y(follo)m(wing)29 b(four)g(results)h(immediately)-8 b(.)225 4235 y(\(1\))33 b(F)-8 b(rom)33 b(\()649 4235 y SDict begin H.S end 649 4235 a Black Fy(3.4)p Black 765 4176 a SDict begin H.R end 765 4176 a 765 4235 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 765 4235 a Fy(\))g(and)f(\()1047 4235 y SDict begin H.S end 1047 4235 a Black Fy(3.5)p Black 1163 4176 a SDict begin H.R end 1163 4176 a 1163 4235 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.5) cvn H.B /ANN pdfmark end 1163 4235 a Fy(\),)h Fq(L)1353 4187 y Ft(\(0\))1335 4246 y Fu(p)1480 4235 y Fy(is)e(a)i FI(@)5 b Fy(-stable)33 b(subalgebra)e(of)h Fq(L)18 b FI(ie)2732 4249 y Fu(p)2803 4235 y FI(R)q Fy(.)46 b(F)-8 b(urthermore,)33 b(if)e Fv(f)p FI(v)3676 4202 y Fu(i)3705 4235 y Fv(g)i Fy(are)125 4367 y Fp(C)18 b Fy([)p FI(@)5 b Fy(]-)q(indep)s(enden)m(t) 34 b(in)29 b FI(R)q Fy(,)h Fq(L)1168 4319 y Ft(\(0\))1150 4379 y Fu(p)1288 4367 y Fv(')25 b FI(V)20 b Fy([)p FI(t)1515 4381 y Ft(1)1554 4367 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)1804 4381 y Fu(p)1844 4367 y Fy(])31 b(b)m(y)f(Prop)s (osition)2515 4367 y SDict begin H.S end 2515 4367 a Black Fy(3.3)p Black 2631 4309 a SDict begin H.R end 2631 4309 a 2631 4367 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 2631 4367 a Fy(.)225 4533 y(\(2\))h(If)f FI(i)25 b(<)g(j)5 b Fy(,)32 b Fq(L)809 4485 y Ft(\()p Fu(i)p Ft(\))792 4544 y Fu(p)918 4533 y Fv(\033)25 b Fq(L)1111 4485 y Ft(\()p Fu(j)t Ft(\))1093 4544 y Fu(p)1232 4533 y Fy(and)30 b Fq(L)1506 4485 y Ft(\()p Fu(i)p Ft(\))1488 4544 y Fu(p)1589 4533 y FI(=)p Fq(L)1731 4485 y Ft(\()p Fu(j)t Ft(\))1713 4544 y Fu(p)1853 4533 y Fy(is)g(of)g(\014nite)f(dimension.)225 4698 y(\(3\))i(F)-8 b(or)31 b(an)m(y)g FI(i)737 4712 y Ft(1)777 4698 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(i)1024 4712 y Fu(p)1090 4698 y Fv(\025)25 b Fy(0,)31 b([)p Fq(L)1409 4650 y Ft(\()p Fu(i)1460 4659 y FJ(1)1495 4650 y Ft(\))1391 4710 y Fu(p)1527 4698 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b Fq(L)1841 4649 y Ft(\()p Fu(i)1892 4657 y FB(p)1928 4649 y Ft(\))1823 4710 y Fu(p)1960 4698 y Fy(])26 b Fv(\032)e Fq(L)2203 4649 y Ft(\()p Fu(i)2254 4658 y FJ(1)2289 4649 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(i)2482 4657 y FB(p)2518 4649 y Fs(\000)p Fu(s)p Ft(\))2185 4710 y Fu(p)2668 4698 y Fy(for)30 b(some)h FI(s)e Fy(b)m(y)i(\()3269 4698 y SDict begin H.S end 3269 4698 a Black Fy(3.4)p Black 3385 4639 a SDict begin H.R end 3385 4639 a 3385 4698 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 3385 4698 a Fy(\).)225 4863 y(\(4\))46 b Fq(L)482 4815 y Ft(\(0\))465 4875 y Fu(p)622 4863 y Fy(can)f(b)s(e)f(seen)h(as)h(a)f(linearly)e(compact)j FI(n)p Fy(-Lie)e(algebra)h(with)f(the)h(top)s(ology)g(induced)e(b)m(y) 125 4996 y Fv(f)p Fq(L)267 4948 y Ft(\()p Fu(m)p Ft(\))249 5008 y Fu(p)389 4996 y Fv(g)p Fy(,)32 b(as)g(it)f(is)g(the)h(top)s (ological)f(pro)s(duct)g(of)g Fq(L)1954 4948 y Ft(\()p Fu(i)p Ft(\))1936 5008 y Fu(p)2037 4996 y FI(=)p Fq(L)2180 4948 y Ft(\()p Fu(i)p Ft(+1\))2161 5008 y Fu(p)2353 4996 y FI(;)15 b(i)28 b Fv(\025)f Fy(0)32 b(as)g(a)g(space)g(and)f(has)h(a)g (con)m(tin)m(uous)125 5129 y FI(n)p Fy(-Lie)e(brac)m(k)m(et)i(as)e(an)g (algebra.)225 5294 y(It)g(should)e(b)s(e)i(noted)h(that)f(although)g Fq(L)1660 5246 y Ft(\()p Fu(m)p Ft(\))1642 5306 y Fu(p)1812 5294 y Fy(dep)s(ends)e(on)i(the)h(c)m(hoice)g(of)g Fv(f)p FI(v)2912 5261 y Fu(i)2941 5294 y Fv(g)p Fy(,)g Fq(L)3138 5246 y Ft(\(0\))3121 5306 y Fu(p)3263 5294 y Fy(and)f(the)g(induced)125 5427 y(top)s(ology)h(on)f Fq(L)722 5379 y Ft(\(0\))704 5439 y Fu(p)846 5427 y Fy(is)g(indep)s(enden)m(t)e(of)j(that)f(c)m (hoice.)42 b(So)30 b(w)m(e)h(can)g(ha)m(v)m(e)g(the)g(follo)m(wing)e (de\014nition.)p Black Black eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)83 b(13)p Black 125 91 a SDict begin H.S end 125 91 a 125 91 a SDict begin 16 H.A end 125 91 a 125 91 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.4) cvn /DEST pdfmark end 125 91 a Black FH(De\014nition)43 b(3.4.)p Black 46 w Fy(Denote)38 b(\()p Fq(L)18 b FI(ie)1352 105 y Fu(p)1423 91 y FI(R)q Fy(\))p 1532 105 22 4 v 66 w(:=)37 b Fq(L)1823 43 y Ft(\(0\))1806 103 y Fu(p)1954 91 y Fy(=)f Fv(f)p FI(a)2154 105 y Fu(m)2216 114 y FJ(1)2252 105 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2424 113 y FB(p)2464 91 y Fv(j)p FI(a)37 b Fv(2)f FI(R)q(;)15 b(m)2861 105 y Ft(1)2901 91 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(m)3197 105 y Fu(p)3274 91 y Fv(\025)36 b Fy(0)p Fv(g)p FI(:)i Fy(Then)e(the)125 224 y(top)s(ological)29 b FI(n)p Fy(-Lie)f(algebra)h (\()p Fq(L)18 b FI(ie)1344 238 y Fu(p)1415 224 y FI(R)q Fy(\))p 1524 238 V 59 w(is)28 b(called)g(the)h Fo(annihilation)34 b(algebr)-5 b(a)33 b(of)e(level)g FI(p)e Fy(asso)s(ciated)h(to)g FI(n)p Fy(-)125 360 y(Lie)h(conformal)g(algebra)g FI(R)q Fy(.)43 b(The)31 b(completion)1912 334 y Fp(\\)1813 360 y Fy(\()p Fq(L)18 b FI(ie)2018 374 y Fu(p)2089 360 y FI(R)q Fy(\))p 2198 374 V 61 w(of)31 b(\()p Fq(L)18 b FI(ie)2564 374 y Fu(p)2635 360 y FI(R)q Fy(\))p 2744 374 V 61 w(with)30 b(resp)s(ect)h(to)h(its)f(top)s(ology)125 493 y(is)e(called)h(the)g Fo(c)-5 b(omplete)g(d)35 b(annihilation)g (algebr)-5 b(a)34 b(of)f(level)f FI(p)e Fy(of)h FI(R)q Fy(.)125 693 y SDict begin H.S end 125 693 a 125 693 a SDict begin 16 H.A end 125 693 a 125 693 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.5) cvn /DEST pdfmark end 125 693 a Black FH(Example)f(3.5.)p Black 40 w Fo(F)-7 b(or)30 b(the)g(curr)-5 b(ent)31 b FI(n)p Fo(-Lie)d(c)-5 b(onformal)32 b(algebr)-5 b(a)31 b FI(C)7 b(ur)32 b Fw(g)e Fo(asso)-5 b(ciate)g(d)32 b(to)e(an)g FI(n)p Fo(-Lie)f(algebr)-5 b(a)31 b Fw(g)p Fo(,)125 826 y Fy(\()p Fq(L)18 b FI(ie)330 840 y Fu(p)403 826 y FI(C)7 b(ur)37 b Fw(g)p Fy(\))p 690 840 22 4 v 65 w Fo(is)e(nothing)h(but)e(the)h FI(n)p Fo(-Lie)f(algebr)-5 b(a)36 b Fw(g)p Fy([)p FI(t)2152 840 y Ft(1)2191 826 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)2441 840 y Fu(p)2481 826 y Fy(])35 b Fo(endowe)-5 b(d)37 b(with)e(the)h(top)-5 b(olo)g(gy)37 b(induc)-5 b(e)g(d)125 959 y(by)32 b Fv(f)p Fq(L)386 911 y Ft(\()p Fu(m)p Ft(\))367 971 y Fu(p)532 959 y Fy(=)821 891 y Fm(P)628 1037 y Fu(m)690 1046 y FJ(1)725 1037 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)956 1045 y FB(p)993 1037 y Ft(=)p Fu(m)1125 959 y FI(t)1158 918 y Fu(m)1220 927 y FJ(1)1158 985 y Ft(1)1274 959 y Fv(\001)15 b(\001)g(\001)h FI(t)1428 910 y Fu(m)1490 918 y FB(p)1428 971 y Fu(p)1531 959 y Fw(g)p Fy([)p FI(t)1635 973 y Ft(1)1674 959 y FI(;)f Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)1924 973 y Fu(p)1964 959 y Fy(])p Fv(g)p Fo(.)225 1235 y Fy(A)25 b(\014nite)g FI(n)p Fy(-Lie)g(conformal)h(algebra)g FI(R)g Fy(is)f(called)g Fo(torsionless)36 b Fy(if)24 b FI(T)13 b(or)29 b(R)d Fy(=)e(0.)40 b(T)-8 b(o)26 b(sho)m(w)g(the)g(imp)s(ortance)125 1368 y(of)k(\()p Fq(L)19 b FI(ie)434 1382 y Fu(p)504 1368 y FI(R)q Fy(\))p 613 1382 V 60 w(to)31 b(torsionless)f(\014nite)f FI(R)q Fy(,)h(the)h(follo)m(wing)e(lemma)g(is)h(necessary)-8 b(.)125 1568 y SDict begin H.S end 125 1568 a 125 1568 a SDict begin 16 H.A end 125 1568 a 125 1568 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.6) cvn /DEST pdfmark end 125 1568 a Black FH(Lemma)33 b(3.6.)p Black 42 w Fo(F)-7 b(or)33 b FI(a)917 1535 y Ft(1)957 1568 y FI(;)15 b Fv(\001)g(\001)g (\001)32 b FI(;)15 b(a)1222 1535 y Fu(n)1294 1568 y Fv(2)25 b FI(R)q(;)15 b(m)1570 1535 y Ft(1)1570 1593 y(1)1610 1568 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(m)1906 1535 y Fu(n)1906 1591 y(p)1979 1568 y Fv(2)25 b Fp(N)6 b Fo(,)267 1764 y Fy(\()p FI(a)350 1726 y Ft(1)390 1788 y(\()417 1738 y Fj(P)503 1788 y Fu(m)565 1765 y FJ(1)565 1809 y(1)p FC(!)p FB(p)693 1788 y Ft(\))725 1764 y Fv(\001)15 b(\001)g(\001)831 1795 y Ft(\()858 1745 y Fj(P)945 1795 y Fu(m)1007 1765 y FB(n)p FC(\000)p FJ(1)1007 1816 y(1)p FC(!)p FB(p)1134 1795 y Ft(\))1166 1764 y FI(a)1214 1726 y Fu(n)1261 1764 y Fy(\))1296 1778 y Fu(m)1358 1755 y FB(n)1358 1799 y FJ(1)p FC(!)p FB(p)1516 1764 y Fy(=)1668 1678 y Fm(X)1612 1891 y Fu(j)1645 1861 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1641 1912 y(1)p FC(!)p FB(p)1856 1764 y Fy(\()p Fv(\000)p Fy(1\))2042 1673 y Fj(P)2129 1723 y Fu(m)2191 1693 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2191 1744 y(1)p FC(!)p FB(p)2404 1723 y Fs(\000)2459 1673 y Fj(P)2545 1723 y Fu(j)2578 1693 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2574 1744 y(1)p FC(!)p FB(p)2794 1690 y Fm(\000)2836 1716 y Fu(m)2898 1693 y FJ(1)2898 1737 y(1)2851 1803 y Fu(j)2884 1780 y FJ(1)2880 1824 y(1)2933 1690 y Fm(\001)2989 1764 y Fv(\001)g(\001)g(\001)3111 1690 y Fm(\000)3152 1713 y Fu(m)3214 1683 y FB(n)p FC(\000)p FJ(1)3214 1723 y FB(p)3167 1806 y Fu(j)3200 1776 y FB(n)p FC(\000)p FJ(1)3196 1816 y FB(p)3336 1690 y Fm(\001)1196 2047 y Fy([)p FI(a)1269 2010 y Ft(1)1269 2079 y Fu(j)1302 2056 y FJ(1)1298 2100 y(1)p FC(!)p FB(p)1430 2047 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)1694 2009 y Fu(n)p Fs(\000)p Ft(1)1694 2092 y Fu(j)1727 2063 y FB(n)p FC(\000)p FJ(1)1723 2114 y(1)p FC(!)p FB(p)1855 2047 y FI(;)g(a)1943 2010 y Fu(n)1943 2086 y(m)2005 2063 y FB(n)2005 2107 y FJ(1)2048 2086 y Ft(+)p Fu(m)2165 2056 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2165 2107 y(1)2378 2086 y Fs(\000)p Fu(j)2466 2056 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2462 2107 y(1)2677 2086 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2849 2067 y FB(n)2849 2103 y(p)2892 2086 y Ft(+)p Fu(m)3009 2056 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)3009 2097 y FB(p)3222 2086 y Fs(\000)p Fu(j)3310 2056 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)3306 2097 y FB(p)3525 2047 y Fy(])p FI(:)3718 2047 y SDict begin H.S end 3718 2047 a 3718 2047 a SDict begin 19.25 H.A end 3718 2047 a 3718 2047 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.9) cvn /DEST pdfmark end 3718 2047 a Fy(\(3.9\))p Black 125 2302 a Fo(Pr)-5 b(o)g(of.)p Black 43 w Fy(Let)31 b FI(a)617 2317 y Fu(\025)658 2326 y FJ(1)692 2317 y Fu(;)p Fs(\001\001\001)11 b Fu(;\025)844 2325 y FB(p)909 2302 y Fy(:=)1208 2234 y Fm(P)1030 2386 y Fu(m)1092 2395 y FJ(1)1127 2386 y Fu(;)p Fs(\001\001\001)f Fu(;m)1299 2394 y FB(p)1336 2386 y Fs(2)p Fn(N)1426 2395 y FJ(+)1496 2302 y FI(\025)1549 2316 y Ft(1)1589 2265 y(\()p Fu(m)1678 2274 y FJ(1)1713 2265 y Ft(\))1760 2302 y Fv(\001)15 b(\001)g(\001)h FI(\025)1934 2316 y Fu(p)1974 2265 y Ft(\()p Fu(m)2063 2273 y FB(p)2100 2265 y Ft(\))2132 2302 y FI(a)2180 2316 y Fu(m)2242 2325 y FJ(1)2277 2316 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2449 2324 y FB(p)2489 2302 y Fy(,)31 b(and)f(w)m(e)g(ha)m(v)m(e)453 2541 y([)p FI(a)526 2503 y Ft(1)526 2573 y Fu(\025)567 2550 y FJ(1)567 2594 y(1)602 2573 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)753 2554 y FJ(1)753 2589 y FB(p)793 2541 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1058 2503 y Fu(n)1058 2564 y(\025)1099 2541 y FB(n)1099 2585 y FJ(1)1142 2564 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)1293 2545 y FB(n)1293 2580 y(p)1340 2541 y Fy(])382 2744 y(=)558 2657 y Fm(X)468 2864 y Fu(m)530 2841 y FJ(1)530 2885 y(1)565 2864 y Fu(;)p Fs(\001\001\001)g Fu(;m)737 2845 y FB(n)737 2881 y(p)795 2744 y FI(\025)848 2706 y Ft(1)848 2766 y(1)888 2690 y(\()p Fu(m)977 2666 y FJ(1)977 2710 y(1)1012 2690 y Ft(\))1059 2744 y Fv(\001)15 b(\001)g(\001)h FI(\025)1233 2706 y Fu(n)1233 2766 y(p)1280 2704 y Ft(\()p Fu(m)1369 2681 y FB(n)1369 2721 y(p)1412 2704 y Ft(\))1444 2744 y Fy([)p FI(a)1517 2706 y Ft(1)1517 2776 y Fu(m)1579 2753 y FJ(1)1579 2797 y(1)1614 2776 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1786 2757 y FJ(1)1786 2792 y FB(p)1827 2744 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)2091 2706 y Fu(n)2091 2766 y(m)2153 2743 y FB(n)2153 2787 y FJ(1)2197 2766 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2369 2747 y FB(n)2369 2783 y(p)2416 2744 y Fy(])382 3048 y(=)733 2961 y Fm(X)468 3175 y Fu(m)530 3152 y FJ(1)530 3196 y(1)565 3175 y Fu(;)p Fs(\001\001\001)g Fu(;m)737 3156 y FB(n)737 3191 y(p)780 3175 y Fu(;j)833 3152 y FJ(1)829 3196 y(1)866 3175 y Fu(;)p Fs(\001\001\001)h Fu(;j)1010 3145 y FB(n)p FC(\000)p FJ(1)1006 3185 y FB(p)1145 2974 y Fm(\000)1186 3000 y Fu(m)1248 2976 y FJ(1)1248 3020 y(1)1201 3086 y Fu(j)1234 3063 y FJ(1)1230 3107 y(1)1283 2974 y Fm(\001)1340 3048 y Fv(\001)k(\001)g(\001)1461 2974 y Fm(\000)1503 2997 y Fu(m)1565 2967 y FB(n)p FC(\000)p FJ(1)1565 3007 y FB(p)1518 3090 y Fu(j)1551 3060 y FB(n)p FC(\000)p FJ(1)1547 3100 y FB(p)1686 2974 y Fm(\001)1728 3048 y FI(\025)1781 3010 y Ft(1)1781 3070 y(1)1820 2993 y(\()p Fu(m)1909 2970 y FJ(1)1909 3014 y(1)1945 2993 y Ft(\))1992 3048 y Fv(\001)g(\001)g(\001)h FI(\025)2166 3010 y Fu(n)2166 3070 y(p)2213 3008 y Ft(\()p Fu(m)2302 2984 y FB(n)2302 3024 y(p)2345 3008 y Ft(\))635 3326 y Fy(\()p FI(a)718 3288 y Ft(1)758 3349 y(\()p Fu(j)818 3326 y FJ(1)814 3370 y(1)852 3349 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(j)1054 3330 y FJ(1)1050 3366 y FB(p)1088 3349 y Ft(\))1119 3326 y Fv(\001)f(\001)g(\001)1225 3356 y Ft(\()p Fu(j)1285 3326 y FB(n)p FC(\000)p FJ(1)1281 3377 y(1)1406 3356 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(j)1608 3326 y FB(n)p FC(\000)p FJ(1)1604 3367 y FB(p)1728 3356 y Ft(\))1759 3326 y FI(a)1807 3288 y Fu(n)1855 3326 y Fy(\))1890 3356 y Fu(m)1952 3333 y FJ(1)1952 3377 y(1)1987 3356 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)2218 3333 y FB(n)2218 3377 y FJ(1)2260 3356 y Fs(\000)p Fu(j)2348 3333 y FJ(1)2344 3377 y(1)2382 3356 y Fs(\000\001\001\001)o(\000)p Fu(j)2584 3326 y FB(n)p FC(\000)p FJ(1)2580 3377 y(1)2704 3356 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2876 3337 y FJ(1)2876 3373 y FB(p)2912 3356 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)3143 3337 y FB(n)3143 3373 y(p)3186 3356 y Fs(\000)p Fu(j)3274 3337 y FJ(1)3270 3373 y FB(p)3308 3356 y Fs(\000\001\001\001)n(\000)p Fu(j)3509 3326 y FB(n)p FC(\000)p FJ(1)3505 3367 y FB(p)382 3513 y Fy(=)1458 3427 y Fm(X)979 3633 y Fi(j)1009 3608 y 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Ft(\))580 564 y Fy([)p FI(a)653 527 y Ft(1)653 596 y Fu(k)692 573 y FJ(1)690 617 y(1)726 596 y Fu(;)p Fs(\001\001\001)10 b Fu(;k)875 577 y FJ(1)873 613 y FB(p)913 564 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1178 527 y Fu(n)1178 587 y(k)1217 565 y FB(n)1215 609 y FJ(1)1260 587 y Fu(;)p Fs(\001\001\001)10 b Fu(;k)1409 568 y FB(n)1407 604 y(p)1455 564 y Fy(])327 771 y(=)678 684 y Fm(X)413 898 y Fu(m)475 875 y FJ(1)475 919 y(1)510 898 y Fu(;)p Fs(\001\001\001)g Fu(;m)682 879 y FB(n)682 914 y(p)725 898 y Fu(;j)778 875 y FJ(1)774 919 y(1)812 898 y Fu(;)p Fs(\001\001\001)g Fu(;j)955 868 y FB(n)p FC(\000)p FJ(1)951 908 y FB(p)1090 771 y FI(\025)1143 733 y Ft(1)1143 793 y(1)1182 717 y(\()p Fu(m)1271 693 y FJ(1)1271 737 y(1)1307 717 y Ft(\))1353 771 y Fv(\001)15 b(\001)g(\001)i FI(\025)1528 733 y Fu(n)1528 793 y(p)1575 731 y Ft(\()p Fu(m)1664 707 y FB(n)1664 747 y(p)1707 731 y Ft(\))1739 771 y Fy(\()p Fv(\000)p Fy(1\))1925 733 y Fu(m)1987 710 y FJ(1)1987 754 y(1)2022 733 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)2253 703 y FB(n)p FC(\000)p FJ(1)2253 744 y FB(p)2374 733 y Fs(\000)p Fu(j)2462 710 y FJ(1)2458 754 y(1)2496 733 y Fs(\000\001\001\001)o (\000)p Fu(j)2698 703 y FB(n)p FC(\000)p FJ(1)2694 744 y FB(p)2822 697 y Fm(\000)2864 723 y Fu(m)2926 700 y FJ(1)2926 744 y(1)2879 809 y Fu(j)2912 787 y FJ(1)2908 831 y(1)2961 697 y Fm(\001)3018 771 y Fv(\001)e(\001)g(\001)3139 697 y Fm(\000)3181 720 y Fu(m)3243 690 y FB(n)p FC(\000)p FJ(1)3243 730 y FB(p)3196 813 y Fu(j)3229 783 y FB(n)p FC(\000)p FJ(1)3225 823 y FB(p)3364 697 y Fm(\001)580 1049 y Fy([)p FI(a)653 1012 y Ft(1)653 1081 y Fu(j)686 1059 y FJ(1)682 1103 y(1)720 1081 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)863 1063 y FJ(1)859 1098 y FB(p)901 1049 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1166 1011 y Fu(n)p Fs(\000)p Ft(1)1166 1095 y Fu(j)1199 1065 y FB(n)p FC(\000)p FJ(1)1195 1116 y(1)1320 1095 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)1463 1065 y FB(n)p FC(\000)p FJ(1)1459 1105 y FB(p)1587 1049 y FI(;)15 b(a)1675 1012 y Fu(n)1675 1088 y(m)1737 1065 y FJ(1)1737 1110 y(1)1772 1088 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)2003 1065 y FB(n)2003 1110 y FJ(1)2046 1088 y Fs(\000)p Fu(j)2134 1065 y FJ(1)2130 1110 y(1)2168 1088 y Fs(\000\001\001\001)n(\000)p Fu(j)2369 1059 y FB(n)p FC(\000)p FJ(1)2365 1110 y(1)2489 1088 y Fu(;)p Fs(\001\001\001)c Fu(;m)2662 1069 y FJ(1)2662 1105 y FB(p)2698 1088 y Ft(+)p Fs(\001\001\001)n Ft(+)p Fu(m)2928 1069 y FB(n)2928 1105 y(p)2971 1088 y Fs(\000)p Fu(j)3059 1069 y FJ(1)3055 1105 y FB(p)3093 1088 y Fs(\000\001\001\001) o(\000)p Fu(j)3295 1059 y FB(n)p FC(\000)p FJ(1)3291 1099 y FB(p)3419 1049 y Fy(])p FI(:)3672 1049 y SDict begin H.S end 3672 1049 a 3672 1049 a SDict begin 19.25 H.A end 3672 1049 a 3672 1049 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.11) cvn /DEST pdfmark end 3672 1049 a Fy(\(3.11\))3834 1264 y Fk(\003)125 1469 y SDict begin H.S end 125 1469 a 125 1469 a SDict begin 16 H.A end 125 1469 a 125 1469 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.7) cvn /DEST pdfmark end 125 1469 a Black FH(Corollary)36 b(3.7.)p Black 43 w Fo(\(i\))c FI(a)978 1484 y Fu(\025)1019 1493 y FJ(1)1054 1484 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)1205 1492 y FB(p)1271 1469 y Fy(=)25 b(0)33 b Fo(if)f(and)h(only)h(if)e FI(a)25 b Fv(2)g FI(T)13 b(or)35 b(R)q Fo(.)225 1634 y(\(ii\))e(L)-5 b(et)35 b FI(R)f Fo(b)-5 b(e)34 b(an)g FI(n)p Fo(-Lie)f(c)-5 b(onformal)36 b(algebr)-5 b(a)35 b FI(R)g Fo(and)g(a)f(fr)-5 b(e)g(e)34 b Fp(C)18 b Fy([)p FI(@)6 b Fy(])p Fo(-mo)-5 b(d)q(ule.)51 b(Then)35 b FI(R)f Fo(is)g(c)-5 b(ommutative)125 1767 y(if)32 b(and)i(only)f(if)f Fy(\()p Fq(L)19 b FI(ie)881 1781 y Fu(p)954 1767 y FI(R)q Fy(\))p 1063 1781 22 4 v 63 w Fo(is)32 b(c)-5 b(ommutative.)p Black 125 1993 a(Pr)g(o)g(of.)p Black 43 w Fy(\(i\))35 b(F)-8 b(rom)37 b(Prop)s(osition)1276 1993 y SDict begin H.S end 1276 1993 a Black Fy(3.3)p Black 1393 1934 a SDict begin H.R end 1393 1934 a 1393 1993 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 1393 1993 a Fy(,)g FI(a)1503 2008 y Fu(\025)1544 2017 y FJ(1)1579 2008 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)1730 2016 y FB(p)1804 1993 y Fy(=)34 b(0)i(if)f FI(a)f Fv(2)f FI(T)13 b(or)33 b(R)q Fy(.)57 b(If)35 b FI(a)44 b(=)-55 b Fv(2)34 b FI(T)13 b(or)32 b(R)q Fy(,)37 b FI(a)d Fy(=)3366 1925 y Fm(P)3376 2074 y Fu(i;j)3477 1993 y FI(c)3516 2007 y Fu(ij)3577 1993 y FI(@)3630 1960 y Fu(i)3659 1993 y FI(v)3706 1960 y Fu(j)3766 1993 y Fy(+)24 b FI(r)125 2182 y Fy(where)30 b(non-zero)h(elemen)m(t)1093 2113 y Fm(P)1103 2263 y Fu(i;j)1204 2182 y FI(c)1243 2196 y Fu(ij)1304 2182 y FI(@)1357 2149 y Fu(i)1385 2182 y FI(v)1432 2149 y Fu(j)1494 2182 y Fv(2)25 b Fp(C)18 b Fy([)p FI(@)5 b Fy(])27 b Fv(\012)20 b FI(V)50 b Fy(and)30 b FI(r)e Fv(2)d FI(T)13 b(or)32 b(R)q Fy(.)41 b(Since)1125 2496 y SDict begin H.S end 1125 2496 a 1125 2496 a SDict begin 16 H.A end 1125 2496 a 1125 2496 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.12) cvn /DEST pdfmark end 1125 2496 a Fy(\()p FI(@)5 b(a)p Fy(\))1296 2510 y Fu(m)1358 2519 y FJ(1)1394 2510 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1566 2518 y FB(p)1632 2496 y Fy(=)25 b Fv(\000)p Fy(\()1882 2377 y Fu(p)1834 2410 y Fm(X)1843 2605 y Fu(i)p Ft(=0)1980 2496 y FI(m)2060 2510 y Fu(i)2088 2496 y FI(a)2136 2510 y Fu(m)2198 2519 y FJ(1)2233 2510 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2405 2520 y FB(i)2432 2510 y Fs(\000)p Ft(1)p Fu(;)p Fs(\001\001\001)n Fu(m)2662 2518 y FB(p)2703 2496 y Fy(\))p Black 934 w(\(3.12\))p Black 125 2732 a(holds)29 b(in)g Fq(L)18 b FI(ie)639 2746 y Fu(p)709 2732 y FI(R)q Fy(,)31 b FI(a)883 2747 y Fu(\025)924 2756 y FJ(1)958 2747 y Fu(;)p Fs(\001\001\001)11 b Fu(;\025)1110 2755 y FB(p)1175 2732 y Fv(6)p Fy(=)25 b(0)31 b(for)f FI(m)1566 2746 y Fu(i)1624 2732 y Fy(su\016cien)m(tly)f(large.)225 2897 y(\(ii\))g(It)i(can)f(b)s(e)g(immediately)f(deduced)g(from)h(\() 1886 2897 y SDict begin H.S end 1886 2897 a Black Fy(3.4)p Black 2003 2839 a SDict begin H.R end 2003 2839 a 2003 2897 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 2003 2897 a Fy(\))g(and)g(\()2280 2897 y SDict begin H.S end 2280 2897 a Black Fy(3.9)p Black 2396 2839 a SDict begin H.R end 2396 2839 a 2396 2897 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 2396 2897 a Fy(\).)1378 b Fk(\003)225 3123 y Fy(Based)30 b(on)h(the)f(ab)s(o)m(v)m(e)i(discussion,)c(w)m(e)j (can)f(giv)m(e)h(the)g(follo)m(wing)e(theorem.)125 3328 y SDict begin H.S end 125 3328 a 125 3328 a SDict begin 16 H.A end 125 3328 a 125 3328 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.8) cvn /DEST pdfmark end 125 3328 a Black FH(Theorem)48 b(3.8.)p Black 49 w Fo(Supp)-5 b(ose)46 b FI(R)f Fo(and)g FI(S)k Fo(ar)-5 b(e)45 b(torsionless)h (\014nite)f FI(n)p Fo(-Lie)e(c)-5 b(onformal)46 b(algebr)-5 b(as.)78 b(Then)45 b(for)125 3461 y(every)33 b(top)-5 b(olo)g(gic)g(al)36 b FI(n)p Fo(-Lie)31 b(algebr)-5 b(a)34 b(homomorphism)j FI(')26 b Fy(:)g(\()p Fq(L)18 b FI(ie)2338 3475 y Fu(p)2411 3461 y FI(R)q Fy(\))p 2520 3475 22 4 v 56 w Fv(!)26 b Fy(\()p Fq(L)18 b FI(ie)2894 3475 y Fu(p)2967 3461 y FI(S)5 b Fy(\))p 3067 3475 V 63 w Fo(c)-5 b(omp)g(atible)35 b(with)f FI(@)3816 3475 y Fu(t)3841 3485 y FB(i)3872 3461 y Fo(-)125 3594 y(actions,)j(ther)-5 b(e)37 b(exists)f(a)h(unique)d(homomorphism)40 b(of)c FI(n)p Fo(-Lie)f(c)-5 b(onformal)38 b(algebr)-5 b(as)37 b FI(\036)32 b Fy(:)f FI(R)h Fv(!)f FI(S)41 b Fo(inducing)125 3727 y FI(')p Fo(.)p Black 125 3952 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(Cho)s(ose)34 b FI(r)h Fv(2)d FI(R)q Fy(.)54 b(As)35 b FI(S)k Fy(is)34 b(torsionless,)h(b)m(y)f(\()2011 3952 y SDict begin H.S end 2011 3952 a Black Fy(3.6)p Black 2128 3894 a SDict begin H.R end 2128 3894 a 2128 3952 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 2128 3952 a Fy(\))h(and)f(Corollary)2785 3952 y SDict begin H.S end 2785 3952 a Black Fy(3.7)p Black 2902 3894 a SDict begin H.R end 2902 3894 a 2902 3952 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.7) cvn H.B /ANN pdfmark end 2902 3952 a Fy(,)i(there)f(is)e(a)i(unique)e FI(s)f Fv(2)g FI(S)125 4085 y Fy(suc)m(h)f(that)g FI(')p Fy(\()p FI(r)663 4099 y Fu(m)725 4108 y FJ(1)761 4099 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)933 4107 y FB(p)974 4085 y Fy(\))26 b(=)g FI(s)1175 4099 y Fu(m)1237 4108 y FJ(1)1272 4099 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1444 4107 y FB(p)1515 4085 y Fy(for)31 b(all)f FI(m)1862 4099 y Ft(1)1901 4085 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(m)2198 4099 y Fu(p)2264 4085 y Fv(2)26 b Fp(N)7 b Fy(.)48 b(Let)32 b Fv(f)p FI(s)2736 4052 y Fu(i)2764 4085 y Fv(g)f Fy(b)s(e)g(a)g Fp(C)18 b Fy([)p FI(@)5 b Fy(])q(-basis)36 b(of)31 b FI(S)5 b Fy(.)42 b(Then)125 4218 y(there)28 b(exist)g FI(c)607 4185 y Fu(i)607 4243 y(j)636 4252 y FJ(1)671 4243 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)810 4251 y FB(p)874 4218 y Fv(2)25 b Fp(C)52 b Fy(suc)m(h)28 b(that)g FI(')p Fy(\()p FI(r)1586 4232 y Fu(m)1648 4241 y FJ(1)1684 4232 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1856 4240 y FB(p)1897 4218 y Fy(\))25 b(=)2146 4150 y Fm(P)2053 4300 y Fu(i;j)2126 4309 y FJ(1)2160 4300 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)2299 4308 y FB(p)2350 4218 y FI(c)2389 4185 y Fu(i)2389 4243 y(j)2418 4252 y FJ(1)2452 4243 y Fu(;)p Fs(\001\001\001)g Fu(;j)2591 4251 y FB(p)2631 4218 y FI(s)2674 4185 y Fu(i)2674 4243 y(j)2703 4252 y FJ(1)2737 4243 y Fu(;)p Fs(\001\001\001)g Fu(;j)2876 4251 y FB(p)2915 4218 y Fy(.)40 b(Since)27 b FI(')i Fy(comm)m(utes)g(with)125 4398 y FI(@)173 4412 y Fu(t)198 4422 y FB(i)229 4398 y Fy(,)i FI(j)322 4413 y Fu(l)373 4398 y Fv(\024)25 b FI(m)549 4413 y Fu(l)605 4398 y Fy(for)31 b FI(l)c Fy(=)e(1)p FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(p)p Fy(.)41 b(Th)m(us)29 b(the)i(relation)e(can)i(b)s(e)f (rephrased)f(as)424 4638 y FI(')p Fy(\()p FI(r)559 4652 y Fu(m)621 4661 y FJ(1)657 4652 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)829 4660 y FB(p)869 4638 y Fy(\))26 b(=)1016 4551 y Fm(X)1069 4747 y Fu(i)1190 4523 y(m)1252 4532 y FJ(1)1173 4551 y Fm(X)1162 4747 y Fu(j)1191 4756 y FJ(1)1225 4747 y Ft(=0)1331 4638 y Fv(\001)15 b(\001)g(\001)1480 4514 y Fu(m)1542 4522 y FB(p)1464 4551 y Fm(X)1452 4747 y Fu(j)1481 4755 y FB(p)1517 4747 y Ft(=0)1632 4600 y Fu(m)1694 4609 y FJ(1)1729 4600 y Ft(!)p 1632 4617 117 4 v 1649 4669 a Fu(j)1678 4678 y FJ(1)1712 4669 y Ft(!)1774 4638 y Fv(\001)g(\001)g(\001)1905 4591 y Fu(m)1967 4599 y FB(p)2003 4591 y Ft(!)p 1905 4617 119 4 v 1922 4669 a Fu(j)1951 4677 y FB(p)1987 4669 y Ft(!)2033 4638 y FI(d)2080 4600 y Fu(i)2080 4660 y(m)2142 4669 y FJ(1)2177 4660 y Fs(\000)p Fu(j)2261 4669 y FJ(1)2295 4660 y Fu(;)p Fs(\001\001\001)c Fu(;m)2468 4668 y FB(p)2504 4660 y Fs(\000)p Fu(j)2588 4668 y FB(p)2627 4638 y FI(s)2670 4600 y Fu(i)2670 4660 y(j)2699 4669 y FJ(1)2733 4660 y Fu(;)p Fs(\001\001\001)f Fu(;j)2872 4668 y FB(p)930 4970 y Fy(=)1016 4884 y Fm(X)1069 5079 y Fu(i)1190 4855 y(m)1252 4864 y FJ(1)1173 4884 y Fm(X)1162 5079 y Fu(j)1191 5088 y FJ(1)1225 5079 y Ft(=0)1331 4970 y Fv(\001)15 b(\001)g(\001)1480 4847 y Fu(m)1542 4855 y FB(p)1464 4884 y Fm(X)1452 5079 y Fu(j)1481 5087 y FB(p)1517 5079 y Ft(=0)1622 4970 y FI(d)1669 4933 y Fu(i)1669 4993 y(m)1731 5002 y FJ(1)1766 4993 y Fs(\000)p Fu(j)1850 5002 y FJ(1)1884 4993 y Fu(;)p Fs(\001\001\001)c Fu(;m)2057 5001 y FB(p)2093 4993 y Fs(\000)p Fu(j)2177 5001 y FB(p)2216 4970 y FI(@)2264 4984 y Fu(t)2289 4993 y FJ(1)2329 4933 y Fu(m)2391 4942 y FJ(1)2426 4933 y Fs(\000)p Fu(j)2510 4942 y FJ(1)2563 4970 y Fv(\001)k(\001)g(\001)h FI(@)2732 4984 y Fu(t)2757 4992 y FB(p)2799 4933 y Fu(m)2861 4941 y FB(p)2897 4933 y Fs(\000)p Fu(j)2981 4941 y FB(p)3021 4970 y FI(s)3064 4933 y Fu(i)3064 4993 y(m)3126 5002 y FJ(1)3160 4993 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)3333 5001 y FB(p)3672 4970 y SDict begin H.S end 3672 4970 a 3672 4970 a SDict begin 19.25 H.A end 3672 4970 a 3672 4970 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.13) cvn /DEST pdfmark end 3672 4970 a Fy(\(3.13\))125 5241 y(with)39 b(constan)m(ts)j FI(d)802 5208 y Fu(i)802 5267 y(j)831 5276 y FJ(1)866 5267 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)1005 5275 y FB(p)1044 5241 y Fy(.)72 b(Then)40 b FI(\036)p Fy(\()p FI(r)s Fy(\))j(=)1806 5173 y Fm(P)1713 5323 y Fu(i;j)1786 5332 y FJ(1)1820 5323 y Fu(;)p Fs(\001\001\001)10 b Fu(;j)1959 5331 y FB(p)2009 5241 y FI(d)2056 5208 y Fu(i)2056 5267 y(j)2085 5276 y FJ(1)2120 5267 y Fu(;)p Fs(\001\001\001)g Fu(;j)2259 5275 y FB(p)2298 5241 y FI(@)2346 5255 y Fu(t)2371 5264 y FJ(1)2411 5205 y Fu(j)2440 5214 y FJ(1)2493 5241 y Fv(\001)15 b(\001)g(\001)i FI(@)2663 5255 y Fu(t)2688 5263 y FB(p)2729 5205 y Fu(j)2758 5213 y FB(p)2797 5241 y FI(s)2840 5208 y Fu(i)2909 5241 y Fy(is)40 b(the)h(only)e(map)i (satis\014es)125 5427 y(\()p FI(\036)p Fy(\()p FI(r)s Fy(\)\))363 5441 y Fu(m)425 5450 y FJ(1)461 5441 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)633 5449 y FB(p)699 5427 y Fy(=)25 b FI(')p Fy(\()p FI(r)930 5441 y Fu(m)992 5450 y FJ(1)1028 5441 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1200 5449 y FB(p)1240 5427 y Fy(\))31 b(for)f(all)f FI(m)1651 5441 y Ft(1)1690 5427 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(m)1987 5441 y Fu(p)2052 5427 y Fv(2)25 b Fp(N)7 b Fy(,)36 b(if)30 b FI(\036)p Fy(\()p FI(r)s Fy(\))25 b Fv(2)g FI(S)5 b Fy(.)p Black Black eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)83 b(15)p Black 225 77 a Fy(Denote)30 b(the)f(origin)f(of)h(\()p Fq(L)18 b FI(ie)1254 91 y Fu(p)1324 77 y FI(R)q Fy(\))p 1433 91 22 4 v 59 w(resp.)40 b(\()p Fq(L)18 b FI(ie)1921 91 y Fu(p)1991 77 y FI(S)5 b Fy(\))p 2091 91 V 59 w(b)m(y)29 b FH(0)2323 91 y Fu(R)2410 77 y Fy(resp.)39 b FH(0)2690 91 y Fu(S)2769 77 y Fy(and)29 b(the)g(n)m(uclear)f(base)h(\()3650 77 y SDict begin H.S end 3650 77 a Black Fy(3.8)p Black 3767 18 a SDict begin H.R end 3767 18 a 3767 77 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.8) cvn H.B /ANN pdfmark end 3767 77 a Fy(\))g(of)125 210 y(\()p Fq(L)18 b FI(ie)330 224 y Fu(p)401 210 y FI(R)q Fy(\))p 510 224 22 4 v 58 w(resp.)39 b(\()p Fq(L)18 b FI(ie)996 224 y Fu(p)1067 210 y FI(S)5 b Fy(\))p 1167 224 V 58 w(b)m(y)28 b Fq(L)1442 162 y Ft(\()p Fu(m)p Ft(\))1424 221 y Fu(p)1564 210 y Fy(\()p FI(R)q Fy(\))g(resp.)40 b Fq(L)2056 162 y Ft(\()p Fu(m)p Ft(\))2039 221 y Fu(p)2178 210 y Fy(\()p FI(S)5 b Fy(\).)41 b(Since)27 b FI(')p Fy(\()p FH(0)2757 224 y Fu(R)2815 210 y Fy(\))e(=)g FH(0)3024 224 y Fu(S)3103 210 y Fy(and)i FI(')i Fy(is)e(con)m(tin)m(uous,)125 342 y(for)37 b Fq(L)367 294 y Ft(\(1\))350 354 y Fu(p)462 342 y Fy(\()p FI(S)5 b Fy(\))37 b Fv(3)e FH(0)779 356 y Fu(S)830 342 y Fy(,)j(there)f(exists)g(a)g(neigh)m(b)s(orho)s(o)s(d)e FI(U)47 b Fy(of)37 b FH(0)2321 356 y Fu(R)2416 342 y Fy(suc)m(h)f(that)i FI(')p Fy(\()p FI(U)10 b Fy(\))37 b Fv(\032)f Fq(L)3273 294 y Ft(\(1\))3255 354 y Fu(p)3367 342 y Fy(\()p FI(S)5 b Fy(\).)61 b(That)37 b(is,)125 475 y FI(')p Fy(\()p Fq(L)317 427 y Ft(\()p Fu(m)p Ft(\))298 487 y Fu(p)438 475 y Fy(\()p FI(R)q Fy(\)\))26 b Fv(\032)f Fq(L)832 427 y Ft(\(1\))814 487 y Fu(p)926 475 y Fy(\()p FI(S)5 b Fy(\))23 b(for)e FI(m)g Fy(su\016cien)m(tly)g(large,)i(whic)m (h)e(means)g FI(d)2566 442 y Fu(i)2566 498 y(m)2628 507 y FJ(1)2664 498 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2836 506 y FB(p)2902 475 y Fy(=)24 b(0)f(for)e FI(m)3275 489 y Ft(1)3317 475 y Fy(+)s Fv(\001)15 b(\001)g(\001)s Fy(+)s FI(m)3653 489 y Fu(p)3718 475 y Fv(\035)25 b Fy(0.)125 608 y(So)38 b FI(\036)g Fy(is)g(a)g(w)m(ell-de\014ned)e(map)i(from)g FI(R)h Fy(to)g FI(S)5 b Fy(.)64 b(The)38 b(comm)m(utativit)m(y)h(of)f FI(')h Fy(with)e FI(@)3135 622 y Fu(t)3160 632 y FB(i)3229 608 y Fy(brings)f(ab)s(out)i(the)125 741 y(compatibilit)m(y)33 b(of)i FI(\036)g Fy(and)f FI(@)5 b Fy(.)55 b(By)35 b(the)g (de\014nition)e(of)i Fq(L)17 b FI(ie)2190 755 y Fu(p)2261 741 y FI(R)q Fy(,)36 b(\()p FI(@)5 b(r)s Fy(\))2559 755 y Fu(m)2621 764 y FJ(1)2657 755 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2829 763 y FB(p)2902 741 y Fy(=)32 b FI(@)5 b Fy(\()p FI(r)3134 755 y Fu(m)3196 764 y FJ(1)3232 755 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)3404 763 y FB(p)3444 741 y Fy(\).)55 b(Then)34 b(w)m(e)125 874 y(ha)m(v)m(e)953 1044 y(\()p FI(\036)p Fy(\()p FI(@)5 b(r)s Fy(\)\))1244 1058 y Fu(m)1306 1067 y FJ(1)1342 1058 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1514 1066 y FB(p)1580 1044 y Fy(=)25 b FI(')p Fy(\(\()p FI(@)5 b(r)s Fy(\))1937 1058 y Fu(m)1999 1067 y FJ(1)2035 1058 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2207 1066 y FB(p)2248 1044 y Fy(\))25 b(=)p FI(')p Fy(\()p FI(@)5 b Fy(\()p FI(r)2602 1058 y Fu(m)2664 1067 y FJ(1)2700 1058 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)2873 1066 y FB(p)2913 1044 y Fy(\)\))2308 1204 y(=)p FI(@)5 b(')p Fy(\()p FI(r)2567 1218 y Fu(m)2629 1227 y FJ(1)2665 1218 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2837 1226 y FB(p)2878 1204 y Fy(\))2308 1364 y(=)p FI(@)5 b Fy(\(\()p FI(\036)p Fy(\()p FI(r)s Fy(\)\))2705 1378 y Fu(m)2767 1387 y FJ(1)2804 1378 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)2976 1386 y FB(p)3016 1364 y Fy(\))2308 1524 y(=\()p FI(@)5 b Fy(\()p FI(\036)p Fy(\()p FI(r)s Fy(\)\)\))2740 1538 y Fu(m)2802 1547 y FJ(1)2839 1538 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)3011 1546 y FB(p)3052 1524 y FI(:)125 1694 y Fy(Th)m(us)43 b FI(\036)p Fy(\()p FI(@)5 b(r)s Fy(\))50 b(=)e FI(@)5 b Fy(\()p FI(\036)p Fy(\()p FI(r)s Fy(\)\).)84 b(Similarly)-8 b(,)45 b(b)m(y)f(\()1756 1694 y SDict begin H.S end 1756 1694 a Black Fy(3.9)p Black 1873 1636 a SDict begin H.R end 1873 1636 a 1873 1694 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 1873 1694 a Fy(\),)k FI(\036)d Fy(comm)m(utes)g(with)e(the)i(conformal)e(brac)m(k)m (ets)j(as)f FI(')125 1827 y Fy(comm)m(utes)g(with)e(the)i FI(n)p Fy(-Lie)f(brac)m(k)m(ets.)84 b(Hence)45 b FI(\036)g Fy(is)e(an)h FI(n)p Fy(-Lie)g(conformal)g(algebra)h(homomorphism)125 1960 y(b)s(et)m(w)m(een)31 b FI(R)g Fy(and)f FI(S)5 b Fy(.)3834 2125 y Fk(\003)125 2201 y SDict begin H.S end 125 2201 a 125 2201 a SDict begin 16 H.A end 125 2201 a 125 2201 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 125 2201 a 991 2366 a Fy(4.)46 b Fx(Cohomology)33 b(of)h FI(n)p Fx(-Lie)h(conf)n(ormal)d(algebra)225 2585 y Fy(In)c(this)g(section,)h(w)m(e)h(establish)d(the)i(represen)m (tation)g(and)f(cohomology)i(theory)g(of)f(an)g FI(n)p Fy(-Lie)f(conformal)125 2718 y(algebra)i FI(R)h Fy(and)f(explain)f (their)g(relation)h(with)f(that)i(of)f FI(n)p Fy(-Lie)g(conformal)g (algebras)g(\()p Fq(L)19 b FI(ie)3314 2732 y Fu(p)3384 2718 y FI(R)q Fy(\))p 3493 2732 22 4 v 30 w(.)125 2913 y SDict begin H.S end 125 2913 a 125 2913 a SDict begin 16 H.A end 125 2913 a 125 2913 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.1) cvn /DEST pdfmark end 125 2913 a Black FH(De\014nition)35 b(4.1.)p Black 43 w Fy(Let)c FI(V)51 b Fy(and)30 b FI(W)43 b Fy(b)s(e)30 b Fp(C)18 b Fy([)p FI(@)5 b Fy(]-)q(mo)s(dules.)46 b(A)30 b Fp(C)18 b Fy(-linear)35 b(map)c FI(f)2813 2880 y Fu(\025)2854 2889 y FJ(1)2888 2880 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)3039 2889 y FB(n)p FC(\000)p FJ(2)3190 2913 y Fy(:)26 b FI(V)46 b Fv(!)25 b FI(W)43 b Fy(is)30 b(called)125 3046 y(a)h Fo(c)-5 b(onformal)35 b(line)-5 b(ar)33 b(map)38 b Fy(if)888 3216 y FI(f)943 3179 y Fu(\025)984 3188 y FJ(1)1018 3179 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)1169 3188 y FB(n)p FC(\000)p FJ(2)1294 3238 y Fu(\025)1340 3216 y FI(@)5 b(v)29 b Fy(=)24 b(\()p FI(@)i Fy(+)20 b FI(\025)g Fy(+)g FI(\025)1978 3230 y Ft(1)2038 3216 y Fy(+)g Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2399 3230 y Fu(n)p Fs(\000)p Ft(2)2536 3216 y Fy(\)\()p FI(f)2661 3179 y Fu(\025)2702 3188 y FJ(1)2737 3179 y Fu(;)p Fs(\001\001\001)10 b Fu(;\025)2888 3188 y FB(n)p FC(\000)p FJ(2)3013 3238 y Fu(\025)3058 3216 y FI(v)s Fy(\))125 3387 y(for)30 b FI(v)f Fv(2)24 b FI(V)d Fy(.)225 3614 y(Similar)28 b(to)33 b([)677 3614 y SDict begin H.S end 677 3614 a Black Fy(3)p Black 722 3556 a SDict begin H.R end 722 3556 a 722 3614 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ak) cvn H.B /ANN pdfmark end 722 3614 a Fy(],)g(w)m(e)f(use)f FI(C)7 b(hom)p Fy(\()p FI(V)e(;)15 b(W)e Fy(\))33 b(to)f(denote)g(the)g (set)g(of)g(all)f(conformal)g(linear)f(maps)i(from)f FI(V)52 b Fy(to)125 3747 y FI(W)39 b Fy(and)27 b FI(C)7 b(end)27 b(V)45 b Fy(:=)25 b FI(C)7 b(hom)p Fy(\()p FI(V)e(;)15 b(V)21 b Fy(\).)40 b(Then)26 b(the)h(generalized)g FI(C)7 b(end)27 b(V)47 b Fy(also)27 b(has)g(a)g(Lie)g(conformal)f(algebra)125 3880 y(structure)k(de\014ned)f(b)m(y)463 4051 y SDict begin H.S end 463 4051 a 463 4051 a SDict begin 16 H.A end 463 4051 a 463 4051 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.1) cvn /DEST pdfmark end 463 4051 a 930 w Fy(\()p FI(@)5 b(f)1536 4018 y Fu(\025)1577 4027 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1793 4051 y Fy(\))1829 4078 y Fu(\025)1874 4051 y FI(v)29 b Fy(:=)c Fv(\000)p FI(\025f)2247 4018 y Fu(\025)2288 4027 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2503 4073 y Fu(\025)2549 4051 y FI(v)s(;)546 4211 y Fy([)p FI(f)626 4178 y Fu(\025)667 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)884 4232 y Fu(\025)929 4211 y FI(g)975 4178 y Fu(\026)1017 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1234 4211 y Fy(])1259 4225 y Fu(\026)1306 4211 y FI(v)k Fy(:=)c FI(f)1555 4178 y Fu(\025)1596 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1812 4232 y Fu(\025)1857 4211 y Fy(\()p FI(g)1938 4178 y Fu(\026)1980 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2198 4232 y Fu(\026)p Fs(\000)p Fu(\025)2341 4211 y FI(v)s Fy(\))20 b Fv(\000)g FI(g)2580 4178 y Fu(\026)2622 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2840 4232 y Fu(\026)p Fs(\000)p Fu(\025)2982 4211 y Fy(\()p FI(f)3072 4178 y Fu(\025)3113 4187 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)3330 4232 y Fu(\025)3375 4211 y FI(v)s Fy(\))p FI(:)p Black 236 w Fy(\(4.1\))p Black 125 4381 a(Besides,)43 b(it)d(is)f(ob)m(vious)h(for)g(an)h FI(n)p Fy(-Lie)e(conformal)h(algebra)h FI(R)g Fy(that)g FI(L)p Fy(\()2735 4342 y Fv(\000)-71 b(!)2757 4381 y FI(a)21 b Fy(\))43 b(:=)f FI(a)3090 4348 y Ft(1)3129 4396 y Fu(\025)3170 4405 y FJ(1)3209 4381 y Fv(\001)15 b(\001)g(\001)3315 4396 y Fu(\025)3356 4405 y FB(n)p FC(\000)p FJ(2)3481 4381 y FI(a)3529 4348 y Fu(n)p Fs(\000)p Ft(1)3707 4381 y Fy(is)39 b(an)125 4514 y(elemen)m(t)31 b(in)e FI(C)7 b(end)30 b(R)h Fy(b)m(y)1271 4647 y FI(L)p Fy(\()1368 4608 y Fv(\000)-71 b(!)1389 4647 y FI(a)22 b Fy(\))1494 4662 y Fu(\025)1540 4647 y FI(r)27 b Fy(=)e([)p FI(a)1777 4609 y Ft(1)1817 4662 y Fu(\025)1858 4671 y FJ(1)1897 4647 y Fv(\001)15 b(\001)g(\001)2003 4662 y Fu(\025)2044 4671 y FB(n)p FC(\000)p FJ(2)2169 4647 y FI(a)2217 4609 y Fu(n)p Fs(\000)p Ft(1)2354 4662 y Fu(\025)2400 4647 y FI(r)s Fy(])p FI(;)g(r)28 b Fv(2)d FI(R)q(:)125 4842 y SDict begin H.S end 125 4842 a 125 4842 a SDict begin 16 H.A end 125 4842 a 125 4842 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.2) cvn /DEST pdfmark end 125 4842 a Black FH(De\014nition)34 b(4.2.)p Black 42 w Fy(F)-8 b(or)31 b(an)e FI(n)p Fy(-Lie)g(conformal)h(algebra)g FI(R)q Fy(,)f FI(f)2260 4809 y Fu(\025)2301 4818 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2542 4842 y Fv(2)c FI(C)7 b(end)30 b(R)g Fy(is)f(called)g(a)h Fo(derivation)38 b Fy(of)125 4975 y FI(R)31 b Fy(if)161 5186 y FI(f)216 5149 y Fu(\025)257 5158 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)473 5208 y Fu(\025)518 5186 y Fy([)p FI(a)591 5149 y Ft(1)631 5200 y Fu(\026)673 5209 y FJ(1)712 5186 y FI(a)760 5149 y Ft(2)800 5200 y Fu(\026)842 5209 y FJ(2)881 5186 y Fv(\001)15 b(\001)g(\001)987 5200 y Fu(\026)1029 5209 y FB(n)p FC(\000)p FJ(1)1154 5186 y FI(a)1202 5149 y Fu(n)1249 5186 y Fy(])26 b(=)1396 5073 y Fu(n)p Fs(\000)p Ft(1)1397 5100 y Fm(X)1405 5295 y Fu(i)p Ft(=1)1529 5186 y Fy([)p FI(a)1602 5149 y Ft(1)1641 5200 y Fu(\026)1683 5209 y FJ(1)1738 5186 y Fv(\001)15 b(\001)g(\001)h FI(a)1907 5149 y Fu(i)p Fs(\000)p Ft(1)2025 5200 y Fu(\026)2067 5210 y FB(i)p FC(\000)p FJ(1)2177 5186 y Fy(\()p FI(f)2267 5149 y Fu(\025)2308 5158 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2524 5208 y Fu(\025)2569 5186 y FI(a)2617 5149 y Fu(i)2646 5186 y Fy(\))2681 5201 y Fu(\025)p Ft(+)p Fu(\025)2818 5210 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)3030 5201 y Ft(+)p Fu(\026)3127 5211 y FB(i)3158 5186 y FI(a)3206 5149 y Fu(i)p Ft(+1)3325 5200 y Fu(\026)3367 5210 y FB(i)p FJ(+1)3475 5186 y Fv(\001)f(\001)g(\001)3581 5200 y Fu(\026)3623 5209 y FB(n)p FC(\000)p FJ(1)3748 5186 y FI(a)3796 5149 y Fu(n)3843 5186 y Fy(])1538 5423 y(+)20 b([)p FI(a)1702 5386 y Ft(1)1742 5437 y Fu(\026)1784 5446 y FJ(1)1823 5423 y Fv(\001)15 b(\001)g(\001)1929 5437 y Fu(\026)1971 5446 y FB(n)p FC(\000)p FJ(2)2097 5423 y FI(a)2145 5386 y Fu(n)p Fs(\000)p Ft(1)2282 5437 y Fu(\026)2324 5446 y FB(n)p FC(\000)p FJ(1)2449 5423 y Fy(\()p FI(f)2539 5386 y Fu(\025)2580 5395 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2796 5445 y Fu(\025)2842 5423 y FI(a)2890 5386 y Fu(n)2937 5423 y Fy(\)])p FI(:)3718 5423 y SDict begin H.S end 3718 5423 a 3718 5423 a SDict begin 19.25 H.A end 3718 5423 a 3718 5423 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.2) cvn /DEST pdfmark end 3718 5423 a Fy(\(4.2\))p Black Black eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop -331 0 a SDict begin /product where{pop 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b FI(D)s(er)k(R)d Fo(is)f(a)h(sub)-5 b(algebr)g(a)34 b(of)f FI(C)7 b(end)33 b(R)q Fo(.)225 375 y(\(ii\))42 b(Every)h FI(L)p Fy(\()765 336 y Fv(\000)-71 b(!)787 375 y FI(a)21 b Fy(\))43 b Fo(is)g(a)g(derivation)i(of)e FI(R)q Fo(.)72 b(A)n(nd)43 b(elements)g(in)g(the)g(line)-5 b(ar)44 b(sp)-5 b(ac)g(e)44 b(sp)-5 b(anne)g(d)45 b(by)e(such)125 508 y(derivations)34 b(ar)-5 b(e)34 b(c)-5 b(al)5 b(le)-5 b(d)33 b Fl(inner)g Fo(derivations.)p Black 125 728 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(\(i\))49 b(W)-8 b(e)50 b(only)e(need)g(to)i (sho)m(w)f([)p FI(f)1628 695 y Fu(\025)1669 704 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1885 749 y Fu(\025)1930 728 y FI(g)1976 695 y Fu(\026)2018 704 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2235 728 y Fy(])g(de\014ned)f(in)g(\()2801 728 y SDict begin H.S end 2801 728 a Black Fy(4.1)p Black 2917 669 a SDict begin H.R end 2917 669 a 2917 728 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.1) cvn H.B /ANN 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Fu(\026)706 5446 y FJ(1)745 5423 y FI(a)793 5386 y Ft(2)832 5437 y Fu(\026)874 5446 y FJ(2)914 5423 y Fv(\001)15 b(\001)g(\001)1020 5437 y Fu(\026)1062 5446 y FB(n)p FC(\000)p FJ(1)1187 5423 y FI(a)1235 5386 y Fu(n)1282 5423 y Fy(])p Black Black eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)83 b(17)p Black 211 159 a Fy(=)297 46 y Fu(n)p Fs(\000)p Ft(1)298 73 y Fm(X)307 268 y Fu(i)p Ft(=1)430 159 y Fy(\()p Fv(\000)p Fy(1\))616 122 y Fu(n)p Fs(\000)p Fu(i)743 159 y Fy([)p FI(a)816 122 y Ft(1)856 173 y Fu(\026)898 182 y FJ(1)952 159 y Fv(\001)15 b(\001)g(\001)h FI(a)1121 122 y Fu(i)p Fs(\000)p Ft(1)1240 173 y Fu(\026)1282 183 y FB(i)p FC(\000)p FJ(1)1391 159 y FI(a)1439 122 y Fu(i)p Ft(+1)1558 173 y Fu(\026)1600 183 y FB(i)p FJ(+1)1707 159 y Fv(\001)f(\001)g(\001)1814 173 y Fu(\026)1856 182 y FB(n)p FC(\000)p FJ(1)1981 159 y FI(a)2029 122 y Fu(n)2076 174 y Fs(\000)p Fu(@)t Fs(\000)p Fu(\025)p Fs(\000)p Fu(\025)2364 183 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2576 174 y Fs(\000)p Fu(\026)2673 183 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2890 159 y Fy(\()p FI(L)p Fy(\()3022 96 y Fv(\000)-71 b(!)3049 159 y FI(b)26 b Fy(\))3149 186 y Fs(\000)p Fu(@)t Fs(\000)p Fu(\026)3342 196 y FB(i)3369 186 y Fs(\000)p Fu(\025)3465 195 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)3681 159 y FI(a)3729 122 y Fu(i)3757 159 y Fy(\)])302 409 y(+)20 b([)p FI(a)466 372 y Ft(1)506 423 y Fu(\026)548 432 y FJ(1)587 409 y Fv(\001)15 b(\001)g(\001)693 423 y Fu(\026)735 432 y FB(n)p FC(\000)p FJ(2)861 409 y FI(a)909 372 y Fu(n)p Fs(\000)p Ft(1)1046 423 y Fu(\026)1088 432 y FB(n)p FC(\000)p FJ(1)1213 409 y Fy(\()p FI(L)p Fy(\()1345 346 y Fv(\000)-71 b(!)1372 409 y FI(b)26 b Fy(\))1472 436 y Fu(\025)p Ft(+)p Fu(\026)1610 445 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1827 409 y FI(a)1875 372 y Fu(n)1922 409 y Fy(\)])211 653 y(=)297 539 y Fu(n)p Fs(\000)p Ft(1)298 567 y Fm(X)307 762 y Fu(i)p Ft(=1)430 653 y Fy(\()p Fv(\000)p Fy(1\))616 615 y Fu(n)p Fs(\000)p Fu(i)743 653 y Fy([)p FI(a)816 615 y Ft(1)856 667 y Fu(\026)898 676 y FJ(1)952 653 y Fv(\001)15 b(\001)g(\001)h FI(a)1121 615 y Fu(i)p Fs(\000)p Ft(1)1240 667 y Fu(\026)1282 677 y FB(i)p FC(\000)p FJ(1)1391 653 y FI(a)1439 615 y Fu(i)p Ft(+1)1558 667 y Fu(\026)1600 677 y FB(i)p FJ(+1)1707 653 y Fv(\001)f(\001)g(\001)1814 667 y Fu(\026)1856 676 y FB(n)p FC(\000)p FJ(1)1981 653 y FI(a)2029 615 y Fu(n)2076 668 y Fs(\000)p Fu(@)t Fs(\000)p Fu(\025)p Fs(\000)p Fu(\025)2364 677 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2576 668 y Fs(\000)p Fu(\026)2673 677 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2890 653 y Fy(\()p FI(L)p Fy(\()3022 590 y Fv(\000)-71 b(!)3049 653 y FI(b)26 b Fy(\))3149 680 y Fu(\025)3194 653 y FI(a)3242 615 y Fu(i)3270 653 y Fy(\)])302 903 y(+)20 b([)p FI(a)466 865 y Ft(1)506 917 y Fu(\026)548 926 y FJ(1)587 903 y Fv(\001)15 b(\001)g(\001)693 917 y Fu(\026)735 926 y FB(n)p FC(\000)p FJ(2)861 903 y FI(a)909 865 y Fu(n)p Fs(\000)p Ft(1)1046 917 y Fu(\026)1088 926 y FB(n)p FC(\000)p FJ(1)1213 903 y Fy(\()p FI(L)p Fy(\()1345 840 y Fv(\000)-71 b(!)1372 903 y FI(b)26 b Fy(\))1472 930 y Fu(\025)1518 903 y FI(a)1566 865 y Fu(n)1613 903 y Fy(\)])211 1140 y(=)297 1027 y Fu(n)p Fs(\000)p Ft(1)298 1054 y Fm(X)307 1249 y Fu(i)p Ft(=1)430 1140 y Fy([)p FI(a)503 1103 y Ft(1)543 1154 y Fu(\026)585 1163 y FJ(1)639 1140 y Fv(\001)15 b(\001)g(\001)i FI(a)809 1103 y Fu(i)p Fs(\000)p Ft(1)927 1154 y Fu(\026)969 1164 y FB(i)p FC(\000)p FJ(1)1078 1140 y Fy(\()p FI(L)p Fy(\()1210 1077 y Fv(\000)-71 b(!)1237 1140 y FI(b)26 b Fy(\))1337 1167 y Fu(\025)1383 1140 y FI(a)1431 1103 y Fu(i)1459 1140 y Fy(\))1494 1155 y Fu(\025)p Ft(+)p Fu(\025)1631 1164 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1844 1155 y Ft(+)p Fu(\026)1941 1165 y FB(i)1971 1140 y FI(a)2019 1103 y Fu(i)p Ft(+1)2138 1154 y Fu(\026)2180 1164 y FB(i)p FJ(+1)2288 1140 y Fv(\001)15 b(\001)g(\001)2394 1154 y Fu(\026)2436 1163 y FB(n)p FC(\000)p FJ(1)2561 1140 y FI(a)2609 1103 y Fu(n)2656 1140 y Fy(])302 1390 y(+)20 b([)p FI(a)466 1353 y Ft(1)506 1404 y Fu(\026)548 1413 y FJ(1)587 1390 y Fv(\001)15 b(\001)g(\001)693 1404 y Fu(\026)735 1413 y FB(n)p FC(\000)p FJ(2)861 1390 y FI(a)909 1353 y Fu(n)p Fs(\000)p Ft(1)1046 1404 y Fu(\026)1088 1413 y FB(n)p FC(\000)p FJ(1)1213 1390 y Fy(\()p FI(L)p Fy(\()1345 1327 y Fv(\000)-71 b(!)1372 1390 y FI(b)26 b Fy(\))1472 1417 y Fu(\025)1518 1390 y FI(a)1566 1353 y Fu(n)1613 1390 y Fy(\)])p FI(:)3834 1618 y Fk(\003)125 1858 y SDict begin H.S end 125 1858 a 125 1858 a SDict begin 16 H.A end 125 1858 a 125 1858 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.4) cvn /DEST pdfmark end 125 1858 a Black FH(Remark)34 b(4.4.)p Black 42 w Fo(It)f(c)-5 b(an)33 b(b)-5 b(e)33 b(e)-5 b(asily)34 b(c)-5 b(ompute)g(d)34 b(that)393 2145 y Fy([)p FI(L)p Fy(\()515 2106 y Fv(\000)-71 b(!)537 2145 y FI(a)22 b Fy(\))642 2160 y Fu(\025)687 2145 y FI(L)p Fy(\()784 2082 y Fv(\000)-71 b(!)811 2145 y FI(b)26 b Fy(\)])936 2159 y Fu(\026)983 2145 y FI(r)i Fy(=)1137 2031 y Fu(n)p Fs(\000)p Ft(2)1138 2059 y Fm(X)1147 2254 y Fu(i)p Ft(=1)1270 2145 y Fy(\()p FI(b)1344 2108 y Ft(1)1384 2159 y Fu(\026)1426 2168 y FJ(1)1481 2145 y Fv(\001)15 b(\001)g(\001)h FI(b)1641 2108 y Fu(i)p Fs(\000)p Ft(1)1759 2159 y Fu(\026)1801 2169 y FB(i)p FC(\000)p FJ(1)1911 2145 y Fy(\()p FI(L)p Fy(\()2043 2106 y Fv(\000)-71 b(!)2065 2145 y FI(a)21 b Fy(\))2169 2172 y Fu(\025)2215 2145 y FI(b)2254 2108 y Fu(i)2282 2145 y Fy(\))2317 2160 y Fu(\025)p Ft(+)p Fu(\025)2454 2169 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2667 2160 y Ft(+)p Fu(\026)2764 2170 y FB(i)2795 2145 y FI(b)2834 2108 y Fu(i)p Ft(+1)2952 2159 y Fu(\026)2994 2169 y FB(i)p FJ(+1)3117 2145 y Fv(\001)15 b(\001)g(\001)h FI(b)3277 2108 y Fu(n)p Fs(\000)p Ft(1)3414 2145 y Fy(\))3449 2160 y Fu(\026)p Fs(\000)p Fu(\025)3592 2145 y FI(r)1142 2379 y Fy(+)k(\()p FI(b)1307 2341 y Ft(1)1347 2393 y Fu(\026)1389 2402 y FJ(1)1444 2379 y Fv(\001)15 b(\001)g(\001)h FI(b)1604 2341 y Fu(n)p Fs(\000)p Ft(2)1741 2393 y Fu(\026)1783 2402 y FB(n)p FC(\000)p FJ(2)1908 2379 y Fy(\()p FI(L)p Fy(\()2040 2340 y Fv(\000)-71 b(!)2062 2379 y FI(a)22 b Fy(\))2167 2406 y Fu(\025)2213 2379 y FI(b)2252 2341 y Fu(n)p Fs(\000)p Ft(1)2389 2379 y Fy(\)\))2459 2394 y Fu(\025)2500 2403 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)2713 2394 y Ft(+)p Fu(\026)2814 2379 y FI(r)3718 2379 y SDict begin H.S end 3718 2379 a 3718 2379 a SDict begin 19.25 H.A end 3718 2379 a 3718 2379 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 3718 2379 a Fy(\(4.5\))125 2616 y Fo(for)33 b FI(L)p Fy(\()367 2576 y Fv(\000)-71 b(!)389 2616 y FI(a)21 b Fy(\))26 b(=)f FI(a)663 2583 y Ft(1)702 2631 y Fu(\025)743 2640 y FJ(1)782 2616 y Fv(\001)15 b(\001)g(\001)888 2631 y Fu(\025)929 2640 y FB(n)p FC(\000)p FJ(2)1054 2616 y FI(a)1102 2583 y Fu(n)p Fs(\000)p Ft(1)1240 2616 y FI(;)g(L)p Fy(\()1377 2552 y Fv(\000)-71 b(!)1403 2616 y FI(b)26 b Fy(\))g(=)f FI(b)1664 2583 y Ft(1)1703 2630 y Fu(\026)1745 2639 y FJ(1)1784 2616 y Fv(\001)15 b(\001)g(\001)1890 2630 y Fu(\026)1932 2639 y FB(n)p FC(\000)p FJ(2)2058 2616 y FI(b)2097 2583 y Fu(n)p Fs(\000)p Ft(1)2234 2616 y FI(;)g(r)28 b Fv(2)d FI(R)q Fo(.)125 2855 y SDict begin H.S end 125 2855 a 125 2855 a SDict begin 16 H.A end 125 2855 a 125 2855 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.5) cvn /DEST pdfmark end 125 2855 a Black FH(De\014nition)35 b(4.5.)p Black 42 w Fy(Let)c FI(R)f Fy(b)s(e)g(an)g FI(n)p Fy(-Lie)f(conformal)h(algebra.)41 b(A)30 b Fp(C)18 b Fy([)p FI(@)5 b Fy(])q(-mo)s(dule)35 b FI(M)40 b Fy(is)29 b(called)g(a)i Fo(c)-5 b(onformal)125 2988 y(r)g(epr)g(esentation)40 b Fy(of)31 b FI(R)g Fy(if)e(it)h(is)f(endo)m(w)m(ed)i(with)e(the)i Fv(f)p FI(\025)2022 3002 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2265 2988 y Fv(g)p Fy(-action)h FI(a)2662 2955 y Ft(1)2701 3003 y Fu(\025)2742 3012 y FJ(1)2781 2988 y Fv(\001)15 b(\001)g(\001)2887 3003 y Fu(\025)2928 3012 y FB(n)p FC(\000)p FJ(2)3053 2988 y FI(a)3101 2955 y Fu(n)p Fs(\000)p Ft(1)3238 3003 y Fu(\025)3279 3012 y FB(n)p FC(\000)p FJ(1)3405 2988 y FI(v)33 b Fy(satisfying)225 3153 y(\(M1\))e FI(a)502 3120 y Ft(1)542 3168 y Fu(\025)583 3177 y FJ(1)622 3153 y Fv(\001)15 b(\001)g(\001)728 3168 y Fu(\025)769 3177 y FB(n)p FC(\000)p FJ(2)894 3153 y FI(a)942 3120 y Fu(n)p Fs(\000)p Ft(1)1104 3153 y Fv(2)25 b FI(C)7 b(end)30 b(M)10 b Fy(,)31 b(i.e.)40 b FI(a)1794 3120 y Ft(1)1834 3168 y Fu(\025)1875 3177 y FJ(1)1913 3153 y Fv(\001)15 b(\001)g(\001)2020 3168 y Fu(\025)2061 3177 y FB(n)p FC(\000)p FJ(2)2186 3153 y FI(a)2234 3120 y Fu(n)p Fs(\000)p Ft(1)2371 3168 y Fu(\025)2412 3177 y FB(n)p FC(\000)p FJ(1)2537 3153 y FI(v)29 b Fv(2)c Fp(C)17 b Fy([)p FI(@)6 b(;)15 b(\025)2927 3167 y Ft(1)2973 3153 y FI(;)g(\025)3066 3167 y Ft(2)3106 3153 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(\025)3375 3167 y Fu(n)p Fs(\000)p Ft(1)3513 3153 y Fy(])20 b Fv(\012)3629 3172 y Fn(C)10 b Ft([)p Fu(@)t Ft(])3781 3153 y FI(M)g Fy(,)225 3319 y(\(M2\))31 b FI(a)502 3286 y Ft(1)542 3334 y Fu(\025)583 3343 y FJ(1)622 3319 y Fv(\001)15 b(\001)g(\001)728 3334 y Fu(\025)769 3344 y FB(i)p FC(\000)p FJ(1)878 3319 y FI(@)5 b(a)979 3286 y Fu(i)1007 3334 y(\025)1048 3344 y FB(i)1079 3319 y Fv(\001)15 b(\001)g(\001)1185 3334 y Fu(\025)1226 3343 y FB(n)p FC(\000)p FJ(2)1351 3319 y FI(a)1399 3286 y Fu(n)p Fs(\000)p Ft(1)1537 3334 y Fu(\025)1578 3343 y FB(n)p FC(\000)p FJ(1)1703 3319 y FI(v)28 b Fy(=)d Fv(\000)p FI(\025)1995 3333 y Fu(i)2023 3319 y FI(a)2071 3286 y Ft(1)2111 3334 y Fu(\025)2152 3343 y FJ(1)2191 3319 y Fv(\001)15 b(\001)g(\001)2297 3334 y Fu(\025)2338 3343 y FB(n)p FC(\000)p FJ(2)2463 3319 y FI(a)2511 3286 y Fu(n)p Fs(\000)p Ft(1)2648 3334 y Fu(\025)2689 3343 y FB(n)p FC(\000)p FJ(1)2814 3319 y FI(v)34 b Fy(for)c FI(i)c Fy(=)f(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b(n)20 b Fv(\000)g Fy(1,)437 3484 y([)p FI(a)510 3451 y Ft(1)549 3499 y Fu(\025)590 3508 y FJ(1)629 3484 y Fv(\001)15 b(\001)g(\001)735 3499 y Fu(\025)776 3508 y FB(n)p FC(\000)p FJ(2)902 3484 y FI(a)950 3451 y Fu(n)p Fs(\000)p Ft(1)1087 3499 y Fu(\025)1128 3508 y FB(n)p FC(\000)p FJ(1)1253 3484 y FI(@)5 b(v)s Fy(])26 b(=)f(\()p FI(@)h Fy(+)20 b FI(\025)1753 3498 y Ft(1)1813 3484 y Fy(+)f FI(\025)1956 3498 y Ft(2)2016 3484 y Fy(+)h Fv(\001)15 b(\001)g(\001)21 b Fy(+)f FI(\025)2377 3498 y Fu(n)p Fs(\000)p Ft(1)2514 3484 y Fy(\))p FI(a)2597 3451 y Ft(1)2637 3499 y Fu(\025)2678 3508 y FJ(1)2717 3484 y Fv(\001)15 b(\001)g(\001)2823 3499 y Fu(\025)2864 3508 y FB(n)p FC(\000)p FJ(2)2989 3484 y FI(a)3037 3451 y Fu(n)p Fs(\000)p Ft(1)3175 3499 y Fu(\025)3216 3508 y FB(n)p FC(\000)p FJ(1)3341 3484 y FI(v)s Fy(,)225 3649 y(\(M3\))31 b FI(a)502 3616 y Ft(1)542 3664 y Fu(\025)583 3673 y FJ(1)637 3649 y Fv(\001)15 b(\001)g(\001)h FI(a)806 3616 y Fu(i)834 3664 y(\025)875 3674 y FB(i)906 3649 y FI(a)954 3616 y Fu(i)p Ft(+1)1072 3664 y Fu(\025)1113 3674 y FB(i)p FJ(+1)1221 3649 y Fv(\001)f(\001)g(\001)1327 3664 y Fu(\025)1368 3673 y FB(n)p FC(\000)p FJ(1)1493 3649 y FI(v)29 b Fy(=)c Fv(\000)p FI(a)1781 3616 y Ft(1)1820 3664 y Fu(\025)1861 3673 y FJ(1)1915 3649 y Fv(\001)15 b(\001)g(\001)h FI(a)2084 3616 y Fu(i)p Ft(+1)2203 3664 y Fu(\025)2244 3674 y FB(i)p FJ(+1)2351 3649 y FI(a)2399 3616 y Fu(i)2428 3664 y(\025)2469 3674 y FB(i)2499 3649 y Fv(\001)f(\001)g(\001)2605 3664 y Fu(\025)2646 3673 y FB(n)p FC(\000)p FJ(1)2772 3649 y FI(v)33 b Fy(for)d FI(i)c Fy(=)f(1)p FI(;)15 b Fy(2)p FI(;)g Fv(\001)g(\001)g(\001)33 b FI(;)15 b(n)20 b Fv(\000)g Fy(2,)225 3846 y(\(M4\))490 3759 y Fu(n)463 3777 y Fm(P)454 3927 y Fu(i)p Ft(=1)569 3846 y Fy(\()p Fv(\000)p Fy(1\))755 3813 y Fu(n)p Fs(\000)p Fu(i)881 3846 y FI(a)929 3813 y Ft(1)969 3861 y Fu(\025)1010 3870 y FJ(1)1064 3846 y Fv(\001)15 b(\001)g(\001)h FI(a)1233 3813 y Fu(i)p Fs(\000)p Ft(1)1351 3861 y Fu(\025)1392 3871 y FB(i)p FC(\000)p FJ(1)1502 3846 y FI(a)1550 3813 y Fu(i)p Ft(+1)1668 3861 y Fu(\025)1709 3871 y FB(i)p FJ(+1)1817 3846 y Fv(\001)f(\001)g(\001)1923 3861 y Fu(\025)1964 3870 y FB(n)p FC(\000)p FJ(1)2089 3846 y FI(a)2137 3813 y Fu(n)2184 3861 y(\025)2225 3869 y FB(n)2272 3846 y Fy(\()p FI(a)2355 3813 y Fu(i)2383 3861 y(\025)2424 3871 y FB(i)2455 3846 y FI(b)2494 3813 y Ft(1)2533 3861 y Fu(\025)2574 3870 y FB(n)p FJ(+1)2698 3846 y Fv(\001)g(\001)g(\001)2804 3861 y Fu(\025)2845 3870 y FJ(2)p FB(n)p FC(\000)p FJ(3)3001 3846 y FI(b)3040 3813 y Fu(n)p Fs(\000)p Ft(2)3177 3861 y Fu(\025)3218 3870 y FJ(2)p FB(n)p FC(\000)p FJ(2)3374 3846 y FI(v)s Fy(\))406 4053 y(=)25 b([)p FI(a)575 4020 y Ft(1)615 4068 y Fu(\025)656 4077 y FJ(1)695 4053 y FI(a)743 4020 y Ft(2)782 4068 y Fu(\025)823 4077 y FJ(2)862 4053 y Fv(\001)15 b(\001)g(\001)968 4068 y Fu(\025)1009 4077 y FB(n)p FC(\000)p FJ(1)1135 4053 y FI(a)1183 4020 y Fu(n)1230 4053 y Fy(])1255 4068 y Fu(\025)1296 4077 y FJ(1)1331 4068 y Ft(+)p Fs(\001\001\001)n Ft(+)p Fu(\025)1540 4076 y FB(n)1587 4053 y FI(b)1626 4020 y Ft(1)1666 4068 y Fu(\025)1707 4077 y FB(n)p FJ(+1)1830 4053 y Fv(\001)g(\001)g(\001) 1936 4068 y Fu(\025)1977 4077 y FJ(2)p FB(n)p FC(\000)p FJ(3)2133 4053 y FI(b)2172 4020 y Fu(n)p Fs(\000)p Ft(2)2309 4068 y Fu(\025)2350 4077 y FJ(2)p FB(n)p FC(\000)p FJ(2)2506 4053 y FI(v)s Fy(,)406 4218 y FI(a)454 4185 y Ft(1)494 4233 y Fu(\025)535 4242 y FJ(1)574 4218 y Fv(\001)g(\001)g(\001)680 4233 y Fu(\025)721 4242 y FB(n)p FC(\000)p FJ(2)846 4218 y FI(a)894 4185 y Fu(n)p Fs(\000)p Ft(1)1031 4233 y Fu(\025)1077 4218 y Fy(\()p FI(b)1151 4185 y Ft(1)1190 4232 y Fu(\026)1232 4241 y FJ(1)1272 4218 y Fv(\001)g(\001)g(\001)1378 4232 y Fu(\026)1420 4241 y FB(n)p FC(\000)p FJ(2)1545 4218 y FI(b)1584 4185 y Fu(n)p Fs(\000)p Ft(1)1721 4232 y Fu(\026)1768 4218 y FI(v)s Fy(\))21 b Fv(\000)f FI(b)2001 4185 y Ft(1)2040 4232 y Fu(\026)2082 4241 y FJ(1)2121 4218 y Fv(\001)15 b(\001)g(\001)2227 4232 y Fu(\026)2269 4241 y FB(n)p FC(\000)p FJ(2)2395 4218 y FI(b)2434 4185 y Fu(n)p Fs(\000)p Ft(1)2571 4232 y Fu(\026)2618 4218 y Fy(\()p FI(a)2701 4185 y Ft(1)2740 4233 y Fu(\025)2781 4242 y FJ(1)2820 4218 y Fv(\001)g(\001)g(\001)2926 4233 y Fu(\025)2967 4242 y FB(n)p FC(\000)p FJ(2)3093 4218 y FI(a)3141 4185 y Fu(n)p Fs(\000)p Ft(1)3278 4233 y Fu(\025)3323 4218 y FI(v)s Fy(\))406 4383 y(=)25 b([\()p FI(a)610 4350 y Ft(1)650 4398 y Fu(\025)691 4407 y FJ(1)730 4383 y Fv(\001)15 b(\001)g(\001)836 4398 y Fu(\025)877 4407 y FB(n)p FC(\000)p FJ(2)1003 4383 y FI(a)1051 4350 y Fu(n)p Fs(\000)p Ft(1)1188 4383 y Fy(\))1223 4398 y Fu(\025)1268 4383 y Fy(\()p FI(b)1342 4350 y Ft(1)1382 4397 y Fu(\026)1424 4406 y FJ(1)1463 4383 y Fv(\001)g(\001)g(\001)1570 4397 y Fu(\026)1612 4406 y FB(n)p FC(\000)p FJ(2)1737 4383 y FI(b)1776 4350 y Fu(n)p Fs(\000)p Ft(1)1913 4383 y Fy(\)])1973 4398 y Fu(\025)p Ft(+)p Fu(\026)2116 4383 y FI(v)s(;)125 4516 y Fy(for)30 b(all)f FI(a)438 4483 y Ft(1)478 4516 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)743 4483 y Fu(n)790 4516 y FI(;)g(b)869 4483 y Ft(1)909 4516 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(b)1165 4483 y Fu(n)p Fs(\000)p Ft(1)1327 4516 y Fv(2)25 b FI(R)q(;)15 b(v)29 b Fv(2)c FI(M)10 b Fy(.)125 4756 y SDict begin H.S end 125 4756 a 125 4756 a SDict begin 16 H.A end 125 4756 a 125 4756 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.6) cvn /DEST pdfmark end 125 4756 a Black FH(Remark)41 b(4.6.)p Black 45 w Fo(In)d(essenc)-5 b(e,)39 b(the)f(notion)h(of)f(r)-5 b(epr)g(esentations)41 b(is)c(given)h(by)f(r)-5 b(eplacing)39 b(the)f(last)h(element)125 4889 y(in)33 b Fv(f)p FI(\025)335 4903 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)578 4889 y Fv(g)p Fo(-br)-5 b(acket)34 b(of)g FI(R)f Fo(by)g(an)h(element)f(of)h FI(M)10 b Fo(.)43 b(Sinc)-5 b(e)34 b(it)f(is)g(a)g(left)h(mo)-5 b(dule)34 b(action,)g(we)f(lose)h(p)-5 b(art)35 b(of)125 5022 y(skew-symmetry)h(in)e(\(C3\).)48 b(Due)34 b(to)h(the)g(two)g (di\013er)-5 b(ent)35 b(p)-5 b(ositions)37 b(of)d(elements)h(in)g(the)f (Filipp)-5 b(ov)37 b(identity)125 5155 y(\(C4\),)c(ther)-5 b(e)34 b(ar)-5 b(e)33 b(two)h(e)-5 b(quations)33 b(in)g(\(M4\).)225 5427 y Fy(F)-8 b(rom)30 b(\(C4\))i(and)d(the)i(ab)s(o)m(v)m(e)g (de\014nition,)e(the)h(follo)m(wing)f(lemma)h(can)h(b)s(e)e(deduced.)p Black Black eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(18)858 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 125 77 a SDict begin H.S end 125 77 a 125 77 a SDict begin 16 H.A end 125 77 a 125 77 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.7) cvn /DEST pdfmark end 125 77 a Black FH(Lemma)42 b(4.7.)p Black 46 w Fo(Supp)-5 b(ose)41 b FI(M)50 b Fo(is)40 b(a)g(c)-5 b(onformal)42 b(r)-5 b(epr)g(esentation)43 b(of)d FI(n)p Fo(-Lie)e(c)-5 b(onformal)42 b(algebr)-5 b(a)41 b FI(R)q Fo(.)63 b(Then)125 210 y FI(R)21 b Fv(\010)f FI(M)42 b Fo(is)33 b(an)g FI(n)p Fo(-Lie)e(c)-5 b(onformal)35 b(algebr)-5 b(a)34 b(with)g(the)f(c)-5 b(onformal)35 b(br)-5 b(ackets)33 b(de\014ne)-5 b(d)34 b(as)252 380 y Fy([)p FI(a)325 343 y Ft(1)385 380 y Fy(+)19 b FI(v)522 343 y Ft(1)562 395 y Fu(\025)603 404 y FJ(1)642 380 y FI(a)690 343 y Ft(2)750 380 y Fy(+)h FI(v)888 343 y Ft(2)928 395 y Fu(\025)969 404 y FJ(2)1007 380 y Fv(\001)15 b(\001)g(\001)1114 395 y Fu(\025)1155 404 y FB(n)p FC(\000)p FJ(1)1280 380 y FI(a)1328 343 y Fu(n)1395 380 y Fy(+)20 b FI(v)1533 343 y Fu(n)1580 380 y Fy(])181 617 y(=[)p FI(a)325 580 y Ft(1)364 632 y Fu(\025)405 641 y FJ(1)444 617 y Fv(\001)15 b(\001)g(\001)550 632 y Fu(\025)591 641 y FB(n)p FC(\000)p FJ(1)717 617 y FI(a)765 580 y Fu(n)812 617 y Fy(])20 b(+)948 504 y Fu(n)p Fs(\000)p Ft(1)949 531 y Fm(X)957 727 y Fu(i)p Ft(=1)1081 617 y Fy(\()p Fv(\000)p Fy(1\))1267 580 y Fu(n)p Fs(\000)p Fu(i)1394 617 y FI(a)1442 580 y Ft(1)1481 632 y Fu(\025)1522 641 y FJ(1)1576 617 y Fv(\001)15 b(\001)g(\001)h FI(a)1745 580 y Fu(i)1774 632 y(\025)1815 642 y FB(i)1845 617 y FI(a)1893 580 y Fu(i)p Ft(+1)2012 632 y Fu(\025)2053 642 y FB(i)p FJ(+1)2160 617 y Fv(\001)f(\001)g(\001)2267 632 y Fu(\025)2308 641 y FB(n)p FC(\000)p FJ(1)2433 617 y FI(a)2481 580 y Fu(n)2528 636 y Fs(\000)p Fu(@)t Fs(\000)2679 586 y Fj(P)2765 636 y Fu(\025)2806 646 y FB(i)2837 617 y FI(v)2884 580 y Fu(i)2932 617 y Fy(+)20 b([)p FI(a)3096 580 y Ft(1)3136 632 y Fu(\025)3177 641 y FJ(1)3231 617 y Fv(\001)15 b(\001)g(\001)h FI(a)3400 580 y Fu(n)p Fs(\000)p Ft(1)3537 632 y Fu(\025)3578 641 y FB(n)p FC(\000)p FJ(1)3704 617 y FI(v)3751 580 y Fu(n)3798 617 y Fy(])p FI(:)3718 812 y SDict begin H.S end 3718 812 a 3718 812 a SDict begin 19.25 H.A end 3718 812 a 3718 812 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 3718 812 a Fy(\(4.6\))p Black 125 1007 a Fo(Pr)-5 b(o)g(of.)p Black 43 w Fy(Our)43 b(goal)i(is)e(to)i(sho)m(w)f(\(C1\)-\(C4\))i(hold) d(for)h(\()2124 1007 y SDict begin H.S end 2124 1007 a Black Fy(4.6)p Black 2241 949 a SDict begin H.R end 2241 949 a 2241 1007 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.6) cvn H.B /ANN pdfmark end 2241 1007 a Fy(\).)83 b(In)43 b(fact,)49 b(\(C1\)-\(C3\))d(of)f FI(R)30 b Fv(\010)f FI(M)54 b Fy(can)45 b(b)s(e)125 1140 y(immediately)31 b(deduced)g(from)h(\(C1\)-\(C3\))j(of)e FI(R)g Fy(and\(M1\)-\(M3\).)50 b(A)32 b(sligh)m(tly)f(more)i(complex)f(discussion)125 1273 y(is)d(giv)m(en)i(for)f(the)g(v)m(eri\014cation)g(of)634 1444 y([[)p FI(a)732 1406 y Ft(1)792 1444 y Fy(+)20 b FI(v)930 1406 y Ft(1)970 1459 y Fu(\025)1011 1468 y FJ(1)1050 1444 y Fv(\001)15 b(\001)g(\001)1156 1459 y Fu(\025)1197 1468 y FB(n)p FC(\000)p FJ(1)1322 1444 y FI(a)1370 1406 y Fu(n)1437 1444 y Fy(+)20 b FI(v)1575 1406 y Fu(n)1623 1444 y Fy(])1648 1459 y Fu(\025)1689 1468 y FJ(1)1724 1459 y Ft(+)p Fs(\001\001\001)n Ft(+)p Fu(\025)1933 1467 y FB(n)1980 1444 y FI(b)2019 1406 y Ft(2)2079 1444 y Fy(+)g FI(w)2237 1406 y Ft(2)2277 1459 y Fu(\025)2318 1468 y FB(n)p FJ(+1)2441 1444 y Fv(\001)15 b(\001)g(\001)2547 1459 y Fu(\025)2588 1468 y FJ(2)p FB(n)p FC(\000)p FJ(2)2744 1444 y FI(b)2783 1406 y Fu(n)2850 1444 y Fy(+)20 b FI(w)3008 1406 y Fu(n)3056 1444 y Fy(])563 1667 y(=)694 1553 y Fu(n)649 1580 y Fm(X)658 1776 y Fu(i)p Ft(=1)781 1667 y Fy(\()p Fv(\000)p Fy(1\))967 1629 y Fu(n)p Fs(\000)p Fu(i)1093 1667 y Fy([)p FI(a)1166 1629 y Ft(1)1226 1667 y Fy(+)g FI(v)1364 1629 y Ft(1)1404 1682 y Fu(\025)1445 1691 y FJ(1)1499 1667 y Fv(\001)15 b(\001)g(\001)h FI(a)1668 1629 y Fu(i)p Fs(\000)p Ft(1)1807 1667 y Fy(+)k FI(v)1945 1629 y Fu(i)p Fs(\000)p Ft(1)2063 1682 y Fu(\025)2104 1692 y FB(i)p FC(\000)p FJ(1)2214 1667 y FI(a)2262 1629 y Fu(i)p Ft(+1)2400 1667 y Fy(+)g FI(v)2538 1629 y Fu(i)p Ft(+1)2657 1682 y Fu(\025)2698 1692 y FB(i)p FJ(+1)2805 1667 y Fv(\001)15 b(\001)g(\001)2911 1682 y Fu(\025)2952 1691 y FB(n)p FC(\000)p FJ(1)3078 1667 y FI(a)3126 1629 y Fu(n)3193 1667 y Fy(+)20 b FI(v)3331 1629 y Fu(n)3378 1682 y(\025)3419 1690 y FB(n)816 1902 y Fy([)p FI(a)889 1864 y Fu(i)938 1902 y Fy(+)g FI(v)1076 1864 y Fu(i)1104 1917 y(\025)1145 1927 y FB(i)1176 1902 y FI(b)1215 1864 y Ft(2)1274 1902 y Fy(+)g FI(w)1432 1864 y Ft(2)1472 1917 y Fu(\025)1513 1926 y FB(n)p FJ(+1)1637 1902 y Fv(\001)15 b(\001)g(\001)1743 1917 y Fu(\025)1784 1926 y FJ(2)p FB(n)p FC(\000)p FJ(2)1940 1902 y FI(b)1979 1864 y Fu(n)2046 1902 y Fy(+)20 b FI(w)2204 1864 y Fu(n)2251 1902 y Fy(]])p FI(;)3718 1902 y SDict begin H.S end 3718 1902 a 3718 1902 a SDict begin 19.25 H.A end 3718 1902 a 3718 1902 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.7) cvn /DEST pdfmark end 3718 1902 a Fy(\(4.7\))125 2072 y(where)34 b FI(a)440 2039 y Ft(1)480 2072 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)745 2039 y Fu(n)792 2072 y FI(;)g(b)871 2039 y Ft(2)911 2072 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(b)1167 2039 y Fu(n)1247 2072 y Fv(2)33 b FI(R)q Fy(,)j FI(v)1519 2039 y Ft(1)1559 2072 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(v)1823 2039 y Fu(n)1870 2072 y FI(;)g(w)1977 2039 y Ft(2)2018 2072 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(w)2301 2039 y Fu(n)2382 2072 y Fv(2)33 b FI(M)10 b Fy(.)55 b(Putting)34 b(in)g(\()3139 2072 y SDict begin H.S end 3139 2072 a Black Fy(4.6)p Black 3255 2014 a SDict begin H.R end 3255 2014 a 3255 2072 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.6) cvn H.B /ANN pdfmark end 3255 2072 a Fy(\),)j(w)m(e)f (obtain)e(the)125 2205 y(expansion)f(of)i(t)m(w)m(o)h(sides)d(of)i(\() 1200 2205 y SDict begin H.S end 1200 2205 a Black Fy(4.7)p Black 1316 2146 a SDict begin H.R end 1316 2146 a 1316 2205 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.7) cvn H.B /ANN pdfmark end 1316 2205 a Fy(\).)53 b(The)34 b(summands)f(are)i(in)e(the)i(follo)m(wing)e(three)h(forms:)49 b(\(a\))35 b(terms)g(in)125 2338 y FI(R)q Fy(,)29 b(\(b\))h(terms)f (con)m(tained)g FI(v)1109 2305 y Fu(i)1137 2338 y Fy(,)h(\(c\))g(terms) f(con)m(tained)h FI(w)2062 2305 y Fu(i)2090 2338 y Fy(.)41 b(T)-8 b(erms)29 b(of)g(t)m(yp)s(e)g(\(a\))h(in)e(t)m(w)m(o)j(sides)d (are)h(equal)g(due)125 2471 y(to)h(\(C4\))h(of)f FI(R)q Fy(.)40 b(The)29 b(equiv)-5 b(alences)30 b(of)g(terms)f(of)h(t)m(yp)s (e)g(\(a\))h(in)e(t)m(w)m(o)i(sides)d(can)i(b)s(e)g(inferred)d(from)j (\(M3\))h(and)125 2604 y(the)36 b(last)f(equation)g(in)f(\(M4\),)39 b(while)33 b(that)j(of)g(terms)f(of)h(t)m(yp)s(e)g(\(b\))f(can)h(b)s(e) f(inferred)e(from)i(\(M3\))i(and)e(the)125 2736 y(\014rst)30 b(equation)g(in)f(\(M4\).)2821 b Fk(\003)125 2912 y SDict begin H.S end 125 2912 a 125 2912 a SDict begin 16 H.A end 125 2912 a 125 2912 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.8) cvn /DEST pdfmark end 125 2912 a Black FH(Lemma)47 b(4.8.)p Black 49 w Fo(L)-5 b(et)45 b FI(R)f Fo(b)-5 b(e)45 b(an)f FI(n)p Fo(-Lie)f(c)-5 b(onformal)47 b(algebr)-5 b(a.)78 b(If)44 b FI(M)54 b Fo(is)44 b(a)h(c)-5 b(onformal)46 b(r)-5 b(epr)g(esentation)47 b(of)125 3045 y(c)-5 b(onformal)39 b(algebr)-5 b(a)39 b FI(R)q Fo(,)f(then)f Fq(V)1265 3059 y Fu(p)1305 3045 y Fy(\()p FI(M)10 b Fy(\))34 b(:=)g FI(M)10 b Fy([)p FI(t)1793 3007 y Fs(\006)p Ft(1)1793 3071 y(1)1887 3045 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)2137 3012 y Fs(\006)p Ft(1)2137 3068 y Fu(p)2231 3045 y Fy(])p FI(=)2313 3021 y Fy(~)2301 3045 y FI(@)2349 3059 y Fu(p)2390 3045 y FI(M)10 b Fy([)p FI(t)2546 3007 y Fs(\006)p Ft(1)2546 3071 y(1)2641 3045 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(t)2890 3012 y Fs(\006)p Ft(1)2890 3068 y Fu(p)2985 3045 y Fy(])37 b Fo(is)g(a)h(r)-5 b(epr)g(esentation)40 b(of)125 3178 y FI(n)p Fo(-Lie)31 b(algebr)-5 b(a)34 b Fq(L)18 b FI(ie)849 3192 y Fu(p)922 3178 y FI(R)q Fo(.)41 b(F)-7 b(urthermor)i(e,)35 b FI(M)43 b Fo(c)-5 b(arries)34 b(a)f Fy(\()p Fq(L)18 b FI(ie)2329 3192 y Fu(p)2402 3178 y FI(R)q Fy(\))p 2511 3192 22 4 v 30 w Fo(-r)-5 b(epr)g(esentation)35 b(structur)-5 b(e.)p Black 125 3393 a(Pr)g(o)g(of.)p Black 43 w Fy(A)21 b(discussion)e(similar)g(to)j(Lemma)1620 3393 y SDict begin H.S end 1620 3393 a Black Fy(3.1)p Black 1736 3334 a SDict begin H.R end 1736 3334 a 1736 3393 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 1736 3393 a 21 w Fy(will)d(sho)m(w)i FI(M)10 b Fy([)p FI(t)2292 3355 y Fs(\006)p Ft(1)2292 3419 y(1)2387 3393 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(t)2637 3360 y Fs(\006)p Ft(1)2637 3415 y Fu(p)2731 3393 y Fy(])22 b(is)e(a)i(conformal)f(represen)m(tation)125 3526 y(of)248 3503 y(~)229 3526 y FI(R)298 3540 y Fu(p)337 3526 y Fy(.)42 b(Then)30 b(b)m(y)g(\(M2\),)j(a)e(similar)d(pro)s(of)i(of)h(Corollary) 2143 3526 y SDict begin H.S end 2143 3526 a Black Fy(3.2)p Black 2259 3467 a SDict begin H.R end 2259 3467 a 2259 3526 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 2259 3526 a 31 w Fy(can)g(explain)e Fq(V)2830 3540 y Fu(p)2869 3526 y Fy(\()p FI(M)10 b Fy(\))32 b(is)d(a)i(represen)m(tation)g(of)125 3659 y FI(n)p Fy(-Lie)f(algebra)g Fq(L)18 b FI(ie)851 3673 y Fu(p)921 3659 y FI(R)q Fy(.)41 b(Explicitly)-8 b(,)28 b(the)j Fq(L)17 b FI(ie)1816 3673 y Fu(p)1887 3659 y FI(R)q Fy({represen)m(tation)31 b(structure)f(on)g Fq(V)3170 3673 y Fu(p)3209 3659 y Fy(\()p FI(M)10 b Fy(\))31 b(is)f(giv)m(en)g(b)m(y)210 3844 y(\()p FI(a)293 3807 y Ft(1)293 3876 y Fu(m)355 3853 y FJ(1)355 3897 y(1)390 3876 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)562 3857 y FJ(1)562 3893 y FB(p)603 3844 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)867 3806 y Fu(n)p Fs(\000)p Ft(1)867 3889 y Fu(m)929 3859 y FB(n)p FC(\000)p FJ(1)929 3910 y(1)1051 3889 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1223 3859 y FB(n)p FC(\000)p FJ(1)1223 3900 y FB(p)1348 3844 y Fy(\))p FI(v)1427 3858 y Fu(m)1489 3835 y FB(n)1489 3879 y FJ(1)1533 3858 y Fu(;)p Fs(\001\001\001)g Fu(;m)1705 3839 y FB(n)1705 3874 y(p)1777 3844 y Fy(=)1929 3758 y Fm(X)1873 3971 y Fu(j)1906 3941 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1902 3992 y(1)p FC(!)p FB(p)2132 3770 y Fm(\000)2174 3796 y Fu(m)2236 3773 y FJ(1)2236 3817 y(1)2189 3883 y Fu(j)2222 3860 y FJ(1)2218 3904 y(1)2271 3770 y Fm(\001)2328 3844 y Fv(\001)15 b(\001)g(\001)2449 3770 y Fm(\000)2491 3793 y Fu(m)2553 3763 y FB(n)p FC(\000)p FJ(1)2553 3804 y FB(p)2506 3886 y Fu(j)2539 3856 y FB(n)p FC(\000)p FJ(1)2535 3897 y FB(p)2674 3770 y Fm(\001)628 4127 y Fy(\()p FI(a)711 4089 y Ft(1)751 4150 y(\()p Fu(j)811 4128 y FJ(1)807 4172 y(1)845 4150 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(j)1047 4131 y FJ(1)1043 4167 y FB(p)1081 4150 y Ft(\))1127 4127 y Fv(\001)g(\001)g(\001)i FI(a)1297 4089 y Fu(n)p Fs(\000)p Ft(1)1434 4157 y(\()p Fu(j)1494 4128 y FB(n)p FC(\000)p FJ(1)1490 4178 y(1)1614 4157 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(j)1816 4128 y FB(n)p FC(\000)p FJ(1)1812 4168 y FB(p)1936 4157 y Ft(\))1968 4127 y FI(v)s Fy(\))2050 4157 y Fu(m)2112 4134 y FJ(1)2112 4178 y(1)2148 4157 y Ft(+)p Fs(\001\001\001)n Ft(+)p Fu(m)2378 4134 y FB(n)2378 4178 y FJ(1)2421 4157 y Fs(\000)p Fu(j)2509 4134 y FJ(1)2505 4178 y(1)2543 4157 y Fs(\000\001\001\001)o(\000)p Fu(j)2745 4128 y FB(n)p FC(\000)p FJ(1)2741 4178 y(1)2865 4157 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)3037 4138 y FJ(1)3037 4174 y FB(p)3073 4157 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)3304 4138 y FB(n)3304 4174 y(p)3347 4157 y Fs(\000)p Fu(j)3435 4138 y FJ(1)3431 4174 y FB(p)3468 4157 y Fs(\000\001\001\001) o(\000)p Fu(j)3670 4128 y FB(n)p FC(\000)p FJ(1)3666 4168 y FB(p)3794 4127 y FI(;)3718 4260 y SDict begin H.S end 3718 4260 a 3718 4260 a SDict begin 19.25 H.A end 3718 4260 a 3718 4260 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.8) cvn /DEST pdfmark end 3718 4260 a Fy(\(4.8\))125 4421 y(where)21 b FI(v)423 4435 y Fu(m)485 4444 y FJ(1)520 4435 y Fu(;)p Fs(\001\001\001)11 b Fu(;m)693 4443 y FB(p)758 4421 y Fy(:=)25 b FI(v)7 b Fv(\012)s FI(t)1037 4380 y Fu(m)1099 4389 y FJ(1)1037 4447 y Ft(1)1153 4421 y Fv(\001)15 b(\001)g(\001)h FI(t)1307 4371 y Fu(m)1369 4379 y FB(p)1307 4432 y Fu(p)1435 4421 y Fv(2)25 b Fq(V)1578 4435 y Fu(p)1617 4421 y Fy(\()p FI(M)10 b Fy(\).)39 b(And)21 b(the)h(induced)e(\()p Fq(L)e FI(ie)2723 4435 y Fu(p)2794 4421 y FI(R)q Fy(\))p 2903 4435 22 4 v 30 w(-represen)m(tation)k(structure)125 4553 y(on)30 b FI(M)41 b Fy(is)29 b(de\014ned)g(as)558 4724 y(\()p FI(a)641 4687 y Ft(1)641 4756 y Fu(m)703 4733 y FJ(1)703 4777 y(1)739 4756 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)911 4737 y FJ(1)911 4773 y FB(p)951 4724 y FI(;)16 b Fv(\001)f(\001)g(\001)31 b FI(;)15 b(a)1216 4686 y Fu(n)p Fs(\000)p Ft(1)1216 4769 y Fu(m)1278 4739 y FB(n)p FC(\000)p FJ(1)1278 4790 y(1)1400 4769 y Fu(;)p Fs(\001\001\001)10 b Fu(;m)1572 4739 y FB(n)p FC(\000)p FJ(1)1572 4780 y FB(p)1697 4724 y Fy(\))p FI(v)29 b Fy(=)c(\()p FI(a)1984 4687 y Ft(1)2024 4748 y(\()p Fu(m)2113 4725 y FJ(1)2113 4769 y(1)2148 4748 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)2379 4729 y FJ(1)2379 4764 y FB(p)2416 4748 y Ft(\))2462 4724 y Fv(\001)15 b(\001)g(\001)h FI(a)2631 4687 y Fu(n)p Fs(\000)p Ft(1)2769 4755 y(\()p Fu(m)2858 4725 y FB(n)p FC(\000)p FJ(1)2858 4776 y(1)2979 4755 y Ft(+)p Fs(\001\001\001)o Ft(+)p Fu(m)3210 4725 y FB(n)p FC(\000)p FJ(1)3210 4765 y FB(p)3331 4755 y Ft(\))3363 4724 y FI(v)s Fy(\))p FI(:)3718 4724 y SDict begin H.S end 3718 4724 a 3718 4724 a SDict begin 19.25 H.A end 3718 4724 a 3718 4724 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.9) cvn /DEST pdfmark end 3718 4724 a Fy(\(4.9\))3834 4898 y Fk(\003)225 5113 y Fy(No)m(w)28 b(it)f(is)g(time)h(to)g(construct)h(the)f(basic)f (complex)g(of)h FI(R)q Fy(.)40 b(Let)28 b FI(R)h Fy(b)s(e)e(an)h FI(n)p Fy(-Lie)f(conformal)g(algebra)h(and)125 5246 y FI(M)40 b Fy(b)s(e)30 b(a)h(conformal)f(represen)m(tation)g(of)h FI(R)q Fy(.)40 b(De\014ne)2008 5223 y Fm(e)1990 5246 y FI(C)2062 5213 y Fu(q)2099 5246 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))31 b(as)g(the)g(set)g(of)f Fp(C)18 b Fy(-linear)35 b(maps)995 5416 y FI(\015)30 b Fy(:)c FI(R)1193 5379 y Fs(\012)1248 5355 y FB(n)p FC(\000)p FJ(1)1393 5416 y Fv(\012)1484 5374 y Fg(\()p Fi(q)17 b Ff(\000)e Fg(1\))1543 5440 y Ff(\001)c(\001)g(\001)1704 5416 y Fv(\012)19 b FI(R)1864 5379 y Fs(\012)1919 5355 y FB(n)p FC(\000)p FJ(1)2064 5416 y Fv(\012)h FI(R)26 b Fv(!)f FI(M)10 b Fy([)p FI(\025)2542 5372 y Ft(1)p Fs(!)p Fu(p)p Fs(\000)p Ft(1)2542 5442 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2786 5416 y FI(;)15 b(\025)2879 5379 y Fu(p)p Fs(\000)p Ft(1)2879 5439 y Fu(n)3009 5416 y Fy(])p Black Black eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST pdfmark end -214 -197 a Black 278 -123 a Fu(n)p Ft(-LIE)30 b(CONF)n(ORMAL)g(ALGEBRAS)h(AND)f(ITS)h(ASSOCIA)-6 b(TED)31 b(INFINITE-DIMENSIONAL)g Fu(n)p Ft(-LIE)f(ALGEBRAS)83 b(19)p Black 563 81 a FI(a)611 43 y Ft(1)611 103 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)874 81 y Fv(\012)20 b(\001)15 b(\001)g(\001)21 b(\012)f FI(a)1230 36 y Fu(q)r Fs(\000)p Ft(1)1230 107 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1493 81 y Fv(\012)g FI(a)1632 43 y Fu(q)r Fs(\000)p Ft(1)1632 103 y Fu(n)1786 81 y Fv(7!)25 b FI(\015)1949 114 y Fu(\025)1990 81 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)1990 135 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2202 114 y Fu(;\025)2263 81 y FB(q)r FC(\000)p FJ(1)2263 125 y FB(n)2380 81 y Fy(\()p FI(a)2463 43 y Ft(1)2463 103 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2706 81 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2971 36 y Fu(q)r Fs(\000)p Ft(1)2971 107 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3215 81 y FI(;)g(a)3303 43 y Fu(q)r Fs(\000)p Ft(1)3303 103 y Fu(n)3431 81 y Fy(\))125 247 y(satisfying)29 b(the)i(follo)m(wing)d(conditions.)225 412 y(\(B1\))j FH(Conformal)j(an)m(tilinearit)m(y)-9 b(.)538 596 y FI(\015)585 630 y Fu(\025)626 597 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)626 651 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)838 630 y Fu(;\025)899 597 y FB(q)r FC(\000)p FJ(1)899 640 y FB(n)1016 596 y Fy(\()p FI(a)1099 559 y Ft(1)1099 619 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1342 596 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1607 559 y Fu(i)1607 619 y Ft(1)1647 596 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)15 b(@)5 b(a)1964 559 y Fu(i)1964 619 y(j)2002 596 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2267 559 y Fu(i)2267 619 y(n)p Fs(\000)p Ft(1)2404 596 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2669 552 y Fu(q)r Fs(\000)p Ft(1)2669 623 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2912 596 y FI(;)g(a)3000 559 y Fu(q)r Fs(\000)p Ft(1)3000 619 y Fu(n)3129 596 y Fy(\))1654 780 y(=)25 b Fv(\000)p FI(\025)1874 743 y Fu(i)1874 803 y(j)1910 780 y FI(\015)1957 814 y Fu(\025)1998 781 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)1998 835 y(1)p FC(!)p FB(n)2202 814 y Fu(;\025)2263 781 y FB(q)r FC(\000)p FJ(1)2263 824 y FB(n)2380 780 y Fy(\()p FI(a)2463 743 y Ft(1)2463 803 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2706 780 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2971 736 y Fu(q)r Fs(\000)p Ft(1)2971 806 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3215 780 y FI(;)g(a)3303 743 y Fu(q)r Fs(\000)p Ft(1)3303 803 y Fu(n)3431 780 y Fy(\))p FI(:)225 997 y Fy(\(B2\))34 b FH(Sk)m(ew-symmetry)-9 b(.)45 b Fy(F)-8 b(or)33 b(eac)m(h)h(\014xed)e FI(i)p Fy(,)h FI(\015)38 b Fy(is)31 b(sk)m(ew-symmetric)i(with)e(resp)s(ect)i(to)g(sim)m (ultaneous)125 1130 y(p)s(erm)m(utations)c(of)i FI(a)833 1097 y Fu(i)833 1156 y(j)900 1130 y Fy(and)e FI(\025)1129 1097 y Fu(i)1129 1156 y(j)1166 1130 y Fy(.)125 1340 y SDict begin H.S end 125 1340 a 125 1340 a SDict begin 16 H.A end 125 1340 a 125 1340 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.9) cvn /DEST pdfmark end 125 1340 a Black FH(Remark)23 b(4.9.)p Black 33 w Fo(The)h(image)g(b)-5 b(e)23 b(interpr)-5 b(ete)g(d)25 b(as)f(the)g(r)-5 b(esult)24 b(of)f FI(\015)5 b Fo(-action)24 b(on)g FI(a)2794 1307 y Ft(1)2794 1365 y(1)2834 1369 y Fu(\025)2875 1346 y FJ(1)2875 1390 y(1)2929 1340 y Fv(\001)15 b(\001)g(\001)h FI(a)3098 1307 y Fu(i)3098 1366 y(j)3134 1383 y(\025)3175 1360 y FB(i)3175 1405 y(j)3227 1340 y Fv(\001)f(\001)g(\001)i FI(a)3397 1296 y Fu(q)r Fs(\000)p Ft(1)3397 1366 y Fu(n)p Fs(\000)p Ft(1)3534 1377 y Fu(\025)3575 1345 y FB(q)r FC(\000)p FJ(1)3575 1399 y FB(n)p FC(\000)p FJ(1)3700 1340 y FI(a)3748 1296 y Fu(q)r Fs(\000)p Ft(1)3748 1352 y Fu(n)3876 1374 y(\025)3917 1341 y FB(q)r FC(\000)p FJ(1)3917 1384 y FB(n)4034 1340 y Fo(.)125 1588 y SDict begin H.S end 125 1588 a 125 1588 a SDict begin 16 H.A end 125 1588 a 125 1588 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.10) cvn /DEST pdfmark end 125 1588 a Black FH(Prop)s(osition)36 b(4.10.)p Black 43 w Fo(F)-7 b(or)33 b FI(\015)e Fv(2)1298 1565 y Fm(e)1279 1588 y FI(C)1351 1555 y Fu(q)1389 1588 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))p Fo(,)286 1773 y Fy(\()p FI(D)s(\015)5 b Fy(\))486 1806 y Fu(\025)527 1773 y FJ(1)p FC(!)p FB(q)527 1827 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)740 1806 y Fu(;\025)801 1773 y FB(q)801 1816 y(n)847 1773 y Fy(\()p FI(a)930 1735 y Ft(1)930 1795 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1173 1773 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1438 1728 y Fu(q)1438 1799 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1682 1773 y FI(;)g(a)1770 1735 y Fu(q)1770 1795 y(n)1817 1773 y Fy(\))215 1965 y(=)380 1878 y Fm(X)301 2074 y Ft(1)p Fs(\024)p Fu(i<j)t Fs(\024)p Fu(q)591 1965 y Fy(\()p Fv(\000)p Fy(1\))777 1927 y Fu(i)806 1965 y FI(\015)853 2012 y Fu(\025)894 1980 y FJ(1)p FC(!)981 1968 y FJ(^)985 1980 y FB(i)q FC(!)p FB(j)s FC(\000)p FJ(1)894 2033 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1180 2012 y Fu(;)p Ft([)p Fu(\025)1261 1989 y FB(i)1261 2033 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1472 2012 y Fu(;\025)1533 1980 y FB(j)1533 2033 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1745 2012 y Ft(])p Fu(;\025)1826 1980 y FB(j)s FJ(+1)p FC(!)p FB(q)1826 2033 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2038 2012 y Fu(;\025)2099 1980 y FB(q)2099 2023 y(n)2145 1965 y Fy(\()p FI(a)2228 1927 y Ft(1)2228 1987 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2472 1965 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)17 b Fy(^)-47 b FI(a)2736 1927 y Fu(i)2736 1987 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2980 1965 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3245 1920 y Fu(j)t Fs(\000)p Ft(1)3245 1991 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3488 1965 y FI(;)2330 2220 y Fy([)p FI(a)2403 2182 y Fu(i)2403 2242 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2646 2220 y FI(;)g(a)2734 2175 y Fu(j)2734 2246 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2978 2220 y Fy(])p FI(;)g(a)3091 2175 y Fu(j)t Ft(+1)3091 2246 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3334 2220 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3599 2182 y Fu(q)3599 2242 y(n)3647 2220 y Fy(\))306 2448 y(+)445 2329 y Fu(q)397 2361 y Fm(X)405 2557 y Fu(i)p Ft(=1)528 2448 y Fy(\()p Fv(\000)p Fy(1\))714 2410 y Fu(i)743 2448 y FI(\015)790 2495 y Fu(\025)831 2462 y FJ(1)p FC(!)918 2450 y FJ(^)922 2462 y FB(i)q FC(!)p FB(q)831 2516 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1044 2495 y Fu(;\025)1105 2462 y FB(q)1105 2506 y(n)1147 2495 y Ft(+)p Fu(\025)1243 2472 y FB(i)1243 2516 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1459 2448 y Fy(\()p FI(a)1542 2410 y Ft(1)1542 2470 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1786 2448 y FI(;)g Fv(\001)g(\001)g(\001)31 b FI(;)17 b Fy(^)-47 b FI(a)2050 2410 y Fu(i)2050 2470 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2294 2448 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2559 2403 y Fu(q)2559 2474 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2802 2448 y FI(;)g Fy([)p FI(a)2915 2410 y Fu(i)2915 2470 y Ft(1)2955 2474 y Fu(\025)2996 2451 y FB(i)2996 2495 y FJ(1)3035 2448 y Fv(\001)g(\001)g(\001)3141 2472 y Fu(\025)3182 2449 y FB(i)3182 2493 y(n)p FC(\000)p FJ(2)3307 2448 y FI(a)3355 2410 y Fu(i)3355 2470 y(n)p Fs(\000)p Ft(1)3493 2481 y Fu(\025)3534 2458 y FB(i)3534 2502 y(n)p FC(\000)p FJ(1)3659 2448 y FI(a)3707 2410 y Fu(q)3707 2470 y(n)3754 2448 y Fy(]\))306 2757 y(+)445 2639 y Fu(q)397 2671 y Fm(X)405 2866 y Fu(i)p Ft(=1)528 2757 y Fy(\()p Fv(\000)p 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Ft(1)3108 3102 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3352 3076 y FI(;)g(a)3440 3032 y Fu(q)3440 3103 y(i)3478 3076 y Fy(\)\))p FI(;)3672 3076 y SDict begin H.S end 3672 3076 a 3672 3076 a SDict begin 19.25 H.A end 3672 3076 a 3672 3076 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.10) cvn /DEST pdfmark end 3672 3076 a Fy(\(4.10\))125 3343 y Fo(wher)-5 b(e)29 b Fy([)p FI(a)450 3310 y Fu(i)450 3367 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)693 3343 y FI(;)15 b(a)781 3298 y Fu(j)781 3369 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1025 3343 y Fy(])28 b Fo(and)h Fy([)p FI(\025)1328 3310 y Fu(i)1328 3367 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1571 3343 y FI(;)15 b(\025)1664 3298 y Fu(j)1664 3369 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1908 3343 y Fy(])28 b Fo(ar)-5 b(e)28 b(the)h(c)-5 b(orr)g(esp)g(onding)31 b(elements)d(and)h(indexes)g(of)f(the)125 3476 y(exp)-5 b(ansion)25 b(of)f Fy([\()p FI(a)740 3443 y Fu(i)740 3500 y Ft(1)780 3504 y Fu(\025)821 3481 y FB(i)821 3525 y FJ(1)860 3476 y Fv(\001)15 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Fy(,)40 b(the)e(v)-5 b(alue)37 b(of)h(the)g(\014rst)e (summand)g(of)i(\()3707 4166 y SDict begin H.S end 3707 4166 a Black Fy(4.10)p Black 3869 4108 a SDict begin H.R end 3869 4108 a 3869 4166 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.10) cvn H.B /ANN pdfmark end 3869 4166 a Fy(\))125 4299 y(equals)h(the)g(original)f(v)-5 b(alue)39 b(m)m(ultiplying)c Fv(\000)p FI(\025)1774 4266 y Fu(i)1774 4325 y(j)1811 4299 y Fy(.)67 b(Similar)37 b(c)m(hanges)j(are)g(seen)f (in)f(the)i(second)f(and)g(third)125 4432 y(summands)c(b)m(y)i(\(C2\))i (of)e FI(R)h Fy(and)f(\(M2\))h(of)g FI(M)10 b Fy(.)61 b(F)-8 b(or)38 b(the)g(fourth)e(summand,)i(if)e FI(i)h Fv(6)p Fy(=)f FI(q)s Fy(,)j(w)m(e)f(can)g(get)g(the)125 4565 y(m)m(ultiple)28 b Fv(\000)p FI(\025)605 4532 y Fu(i)605 4590 y(j)672 4565 y Fy(b)m(y)i(\(B1\))i(of)e FI(\015)36 b Fy(and)29 b(if)h FI(i)25 b Fy(=)g FI(q)s Fy(,)31 b(the)f(m)m(ultiple)e Fv(\000)p FI(\025)2343 4521 y 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471 y Fv(\001)15 b(\001)g(\001)2366 505 y Fu(\025)2407 473 y FB(j)2407 526 y(n)p FC(\000)p FJ(2)2533 471 y FI(a)2581 427 y Fu(j)2581 497 y(n)p Fs(\000)p Ft(1)2718 508 y Fu(\025)2759 476 y FB(j)2759 529 y(n)p FC(\000)p FJ(1)2884 471 y Fy(\()p FI(a)2967 433 y Fu(i)2967 493 y Ft(1)3007 498 y Fu(\025)3048 475 y FB(i)3048 519 y FJ(1)3087 471 y Fv(\001)g(\001)g(\001)3193 496 y Fu(\025)3234 473 y FB(i)3234 517 y(n)p FC(\000)p FJ(2)3359 471 y FI(a)3407 433 y Fu(i)3407 493 y(n)p Fs(\000)p Ft(1)3544 505 y Fu(\025)3585 482 y FB(i)3585 526 y(n)p FC(\000)p FJ(1)3710 471 y FI(v)s Fy(\))p FI(;)125 697 y Fy(the)29 b(transp)s(osition)f(of)h FI(a)979 664 y Fu(i)979 723 y(j)1045 697 y Fy(and)f FI(a)1268 664 y Fu(i)1268 723 y(j)t Ft(+1)1424 697 y Fy(c)m(hanges)j(the)e(sign)f (of)i(ev)m(ery)g(term)f(in)f(the)i(\014rst)e(summand.)39 b(By)29 b(\(C3\),)125 830 y(\(M3\))j(and)f(\(B2\),)i(it)e(is)f(easy)i (to)f(see)h(that)g(the)f(signs)f(of)h(the)h(second)f(and)f(third)g (summands)f(are)i(c)m(hanged.)125 963 y(F)-8 b(or)32 b(the)f(fourth)f(summand,)g(if)g FI(i)d Fv(6)p Fy(=)f FI(q)s Fy(,)31 b(the)g(sign)f(is)g(c)m(hanged)i(b)m(y)f(\(B2\))i(and)d (if)g FI(i)d Fy(=)f FI(q)s Fy(,)31 b(the)h(sign)e(is)g(c)m(hanged)125 1096 y(b)m(y)g(\()p Fv(\000)p Fy(1\))437 1063 y Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(i)p Ft(+1)743 1096 y Fy(.)225 1261 y(Then)f(w)m(e)i(consider)762 1469 y(\()p FI(D)s(\015)5 b Fy(\))962 1503 y Fu(\025)1003 1470 y FJ(1)p FC(!)p FB(q)1003 1524 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1216 1503 y Fu(;\025)1277 1470 y FB(q)1277 1513 y(n)1323 1469 y Fy(\()p FI(a)1406 1432 y Ft(1)1406 1492 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1649 1469 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1914 1425 y Fu(q)r Fs(\000)p Ft(1)1914 1495 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2158 1469 y FI(;)g(a)2246 1425 y Fu(q)2246 1495 y Ft(1)2285 1469 y FI(;)g Fv(\001)g(\001)g(\001) 32 b FI(;)15 b(a)2550 1425 y Fu(q)2550 1495 y(n)p Fs(\000)p Ft(1)2688 1469 y FI(;)g(a)2776 1432 y Fu(q)2776 1492 y(n)2823 1469 y Fy(\))691 1653 y(+\()p FI(D)s(\015)5 b Fy(\))962 1687 y Fu(\025)1003 1654 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)1003 1708 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1216 1687 y Fu(;\025)1277 1654 y FB(q)1277 1708 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(2)1489 1687 y Fu(;\025)1550 1654 y FB(q)1550 1697 y(n)1592 1687 y Fu(;\025)1653 1654 y FB(q)1653 1708 y(n)p FC(\000)p FJ(1)1778 1653 y Fy(\()p FI(a)1861 1616 y Ft(1)1861 1676 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2104 1653 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2369 1609 y Fu(q)r Fs(\000)p Ft(1)2369 1679 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2612 1653 y FI(;)g(a)2700 1609 y Fu(q)2700 1679 y Ft(1)2740 1653 y FI(;)g Fv(\001)g(\001)g(\001) 32 b FI(;)15 b(a)3005 1616 y Fu(q)3005 1676 y(n)3052 1653 y FI(;)g(a)3140 1609 y Fu(q)3140 1679 y(n)p Fs(\000)p Ft(1)3278 1653 y Fy(\))p FI(:)125 1871 y Fy(As)27 b(discussed)e(ab)s(o) m(v)m(e,)k(w)m(e)f(only)e(need)h(to)h(consider)e(the)h(case)i(when)d FI(a)2548 1827 y Fu(p)2548 1897 y(n)p Fs(\000)p Ft(1)2712 1871 y Fy(and)g FI(a)2933 1827 y Fu(p)2933 1883 y(n)3007 1871 y Fy(are)i(separated.)40 b(Putting)125 2004 y(in)30 b(\()267 2004 y SDict begin H.S end 267 2004 a Black Fy(4.5)p Black 383 1946 a SDict begin H.R end 383 1946 a 383 2004 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 383 2004 a Fy(\),)i(w)m(e)g(\014nd)e(terms)h (in)f(the)h(\014rst)f(and)h(second)g(summand)f(o)s(ccur)h(in)f(pairs)g (with)f(opp)s(osite)i(signs)f(and)125 2137 y(th)m(us)g(cancel)h(eac)m (h)g(other)g(out.)41 b(And)29 b(this)h(also)g(happ)s(ens)e(in)h(the)i (third)e(and)g(fourth)h(summand.)225 2302 y FH(Step)k(3:)41 b Fy(No)m(w)31 b(w)m(e)g(are)f(going)h(to)g(sho)m(w)f FI(D)1735 2269 y Ft(2)1800 2302 y Fy(=)25 b(0.)41 b(Substituting)28 b(in)h(\()2669 2302 y SDict begin H.S end 2669 2302 a Black Fy(4.10)p Black 2831 2244 a SDict begin H.R end 2831 2244 a 2831 2302 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.10) cvn H.B /ANN pdfmark end 2831 2302 a Fy(\),)h(w)m(e)h(ha)m (v)m(e)209 2511 y(\()p FI(D)322 2473 y Ft(2)361 2511 y FI(\015)5 b Fy(\))448 2544 y Fu(\025)489 2512 y FJ(1)p FC(!)p FB(q)489 2565 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)702 2544 y Fu(;\025)763 2512 y FB(q)763 2555 y(n)810 2511 y Fy(\()p FI(a)893 2473 y Ft(1)893 2533 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1136 2511 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1401 2467 y Fu(q)1401 2537 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1644 2511 y FI(;)g(a)1732 2473 y Fu(q)1732 2533 y(n)1780 2511 y Fy(\))138 2703 y(=)303 2617 y Fm(X)224 2813 y Ft(1)p Fs(\024)p Fu(i<j)t Fs(\024)p Fu(q)514 2703 y Fy(\()p Fv(\000)p Fy(1\))700 2666 y Fu(i)729 2703 y Fy(\()p FI(D)s(\015)5 b Fy(\))929 2750 y Fu(\025)970 2718 y FJ(1)p FC(!)1057 2706 y FJ(^)1061 2718 y FB(i)r FC(!)p FB(j)s FC(\000)p FJ(1)970 2772 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1256 2750 y Fu(;)p Ft([)p Fu(\025)1337 2728 y FB(i)1337 2772 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1549 2750 y Fu(;\025)1610 2718 y FB(j)1610 2772 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1822 2750 y Ft(])p Fu(;\025)1903 2718 y FB(j)s FJ(+1)p FC(!)p FB(q)1903 2772 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2115 2750 y Fu(;\025)2176 2718 y FB(q)2176 2761 y(n)2222 2703 y Fy(\()p FI(a)2305 2666 y Ft(1)2305 2726 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2548 2703 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)2813 2666 y Fu(i)2813 2726 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3057 2703 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)3321 2659 y Fu(j)t Fs(\000)p Ft(1)3321 2729 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3565 2703 y FI(;)2209 2958 y Fy([)p FI(a)2282 2921 y Fu(i)2282 2981 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2525 2958 y FI(;)g(a)2613 2914 y Fu(j)2613 2984 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2857 2958 y Fy(])p FI(;)g(a)2970 2914 y Fu(j)t Ft(+1)2970 2984 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3213 2958 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3478 2921 y Fu(q)3478 2981 y(n)3525 2958 y Fy(\))229 3186 y(+)369 3068 y Fu(q)320 3100 y Fm(X)328 3295 y Fu(i)p Ft(=1)451 3186 y Fy(\()p Fv(\000)p Fy(1\))637 3148 y Fu(i)666 3186 y Fy(\()p FI(D)s(\015)5 b Fy(\))866 3233 y Fu(\025)907 3201 y FJ(1)p FC(!)994 3189 y FJ(^)998 3201 y FB(i)r FC(!)p FB(q)907 3255 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1120 3233 y Fu(;\025)1181 3201 y FB(q)1181 3244 y(n)1224 3233 y Ft(+)p Fu(\025)1320 3211 y FB(i)1320 3255 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1536 3186 y Fy(\()p FI(a)1619 3148 y Ft(1)1619 3208 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1862 3186 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)2127 3148 y Fu(i)2127 3208 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2371 3186 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)2635 3142 y Fu(q)2635 3212 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2879 3186 y FI(;)g Fy([)p FI(a)2992 3148 y Fu(i)2992 3208 y Ft(1)3032 3213 y Fu(\025)3073 3190 y FB(i)3073 3234 y FJ(1)3112 3186 y Fv(\001)g(\001)g(\001)3218 3211 y Fu(\025)3259 3188 y FB(i)3259 3232 y(n)p FC(\000)p FJ(2)3384 3186 y FI(a)3432 3148 y Fu(i)3432 3208 y(n)p Fs(\000)p Ft(1)3569 3220 y Fu(\025)3610 3197 y FB(i)3610 3241 y(n)p FC(\000)p FJ(1)3736 3186 y FI(a)3784 3148 y Fu(q)3784 3208 y(n)3831 3186 y Fy(]\))229 3496 y(+)369 3377 y Fu(q)320 3409 y Fm(X)328 3605 y Fu(i)p Ft(=1)451 3496 y Fy(\()p Fv(\000)p Fy(1\))637 3458 y Fu(i)p Ft(+1)756 3496 y FI(a)804 3458 y Fu(i)804 3518 y Ft(1)844 3522 y Fu(\025)885 3499 y FB(i)885 3543 y FJ(1)924 3496 y Fv(\001)g(\001)g(\001)1030 3520 y Fu(\025)1071 3497 y FB(i)1071 3541 y(n)p FC(\000)p FJ(2)1196 3496 y FI(a)1244 3458 y Fu(i)1244 3518 y(n)p Fs(\000)p Ft(1)1381 3529 y Fu(\025)1422 3506 y FB(i)1422 3550 y(n)p FC(\000)p FJ(1)1548 3496 y Fy(\(\()p FI(D)s(\015)5 b Fy(\))1783 3543 y Fu(\025)1824 3510 y FJ(1)p FC(!)1911 3498 y FJ(^)1915 3510 y FB(i)q FC(!)p FB(q)1824 3564 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2037 3543 y Fu(;\025)2098 3510 y FB(q)2098 3554 y(n)2144 3496 y Fy(\()p FI(a)2227 3458 y Ft(1)2227 3518 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2471 3496 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)17 b Fy(^)-47 b FI(a)2735 3458 y Fu(i)2735 3518 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2979 3496 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3244 3451 y Fu(q)3244 3522 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3487 3496 y FI(;)g(a)3575 3458 y Fu(q)3575 3518 y(n)3623 3496 y Fy(\)\))229 3815 y(+)320 3701 y Fu(n)p Fs(\000)p Ft(1)321 3728 y Fm(X)329 3924 y Fu(i)p Ft(=1)453 3815 y Fy(\()p Fv(\000)p Fy(1\))639 3777 y Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(i)p Ft(+1)945 3815 y FI(a)993 3770 y Fu(q)993 3841 y Ft(1)1032 3845 y Fu(\025)1073 3812 y FB(q)1073 3866 y FJ(1)1127 3815 y Fv(\001)g(\001)g(\001)j Fy(^)-47 b FI(a)1296 3770 y Fu(q)1296 3842 y(i)1350 3815 y Fv(\001)15 b(\001)g(\001)h FI(a)1519 3770 y Fu(q)1519 3841 y(n)p Fs(\000)p Ft(1)1656 3852 y Fu(\025)1697 3819 y FB(q)1697 3873 y(n)p FC(\000)p FJ(1)1822 3815 y FI(a)1870 3777 y Fu(q)1870 3837 y(n)1917 3841 y(\025)1958 3809 y FB(q)1958 3852 y(n)2005 3815 y Fy(\(\()p FI(D)s(\015)5 b Fy(\))2240 3848 y Fu(\025)2281 3815 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)2281 3869 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2494 3848 y Fu(;\025)2555 3815 y FB(q)2555 3870 y(i)2594 3815 y Fy(\()p FI(a)2677 3777 y Ft(1)2677 3837 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2920 3815 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3185 3770 y Fu(q)r Fs(\000)p Ft(1)3185 3841 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3428 3815 y FI(;)g(a)3516 3770 y Fu(q)3516 3842 y(i)3555 3815 y Fy(\)\))3672 4009 y SDict begin H.S end 3672 4009 a 3672 4009 a SDict begin 19.25 H.A end 3672 4009 a 3672 4009 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.11) cvn /DEST pdfmark end 3672 4009 a Fy(\(4.11\))125 4198 y(Then)33 b(w)m(e)i(discuss)d(eac)m(h)j(summand)e(in)f(the)j (expansion)e(of)h(\()2277 4198 y SDict begin H.S end 2277 4198 a Black Fy(4.11)p Black 2439 4139 a SDict begin H.R end 2439 4139 a 2439 4198 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.11) cvn H.B /ANN pdfmark end 2439 4198 a Fy(\))g(in)f(turn.)51 b(F)-8 b(or)35 b(the)f(sak)m(e)h(of)f(simplicit)m (y)-8 b(,)125 4331 y(w)m(e)35 b(use)f FI(S)480 4345 y Fu(ij)541 4331 y Fy(,)i FI(i;)15 b(j)38 b Fy(=)32 b(1)p FI(;)15 b Fy(2)p FI(;)g Fy(3)p FI(;)g Fy(4)38 b(to)d(mean)g(the)g FI(j)5 b Fy(th)35 b(summand)e(of)i(the)g(expansion)e(of)i(the)g FI(i)p Fy(th)g(summand)e(in)125 4464 y(\()160 4464 y SDict begin H.S end 160 4464 a Black Fy(4.11)p Black 322 4405 a SDict begin H.R end 322 4405 a 322 4464 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.11) cvn H.B /ANN pdfmark end 322 4464 a Fy(\).)225 4629 y FH(\()p FI(S)322 4643 y Ft(11)396 4629 y FH(\).)40 b Fy(T)-8 b(erms)30 b(in)f FI(S)944 4643 y Ft(11)1049 4629 y Fy(can)h(b)s(e)g(divided)e(in) m(to)i(the)h(follo)m(wing)d(three)j(t)m(yp)s(es.)487 4837 y(\()p FI(a)p Fy(\))g FI(\015)683 4872 y Fs(\001\001\001)10 b Fu(;)p Ft([)p Fu(\025)834 4849 y FB(i)834 4893 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1046 4872 y Fu(;\025)1107 4839 y FB(j)1107 4893 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1319 4872 y Ft(])p Fu(;)p Fs(\001\001\001)g Fu(;)p Ft([)p Fu(\025)1510 4849 y FB(k)1510 4893 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1721 4872 y Fu(;\025)1782 4849 y FB(l)1782 4893 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1994 4872 y Ft(])p Fu(;)p Fs(\001\001\001)g Fu(;\025)2165 4839 y FB(q)2165 4882 y(n)2212 4837 y Fy(\()p Fv(\001)15 b(\001)g(\001)31 b FI(;)15 b Fy([)p FI(a)2496 4800 y Fu(i)2496 4860 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2740 4837 y FI(;)g(a)2828 4793 y Fu(j)2828 4863 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3071 4837 y Fy(])p FI(;)2320 5019 y Fv(\001)g(\001)g(\001)31 b FI(;)15 b Fy([)p FI(a)2569 4981 y Fu(k)2569 5041 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2813 5019 y FI(;)g(a)2901 4981 y Fu(l)2901 5041 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3145 5019 y Fy(])p FI(;)g 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FH(\()p FI(S)322 1775 y Ft(14)396 1761 y FH(,)25 b FI(S)506 1775 y Ft(41)581 1761 y FH(,)g FI(S)691 1775 y Ft(42)766 1761 y FH(,)g(part)e(of)h FI(S)1200 1775 y Ft(24)1274 1761 y FH(,)i FI(S)1385 1775 y Ft(43)1482 1761 y FH(and)e(part)e(of)i FI(S)2052 1775 y Ft(34)2126 1761 y FH(\).)37 b Fy(Since)19 b(the)i(expansion)e(of)h([)p FI(L)p Fy(\()3234 1684 y Fv(\000)-71 b(!)3241 1761 y FI(a)3289 1728 y Fu(i)3326 1761 y Fy(\))3361 1788 y Fu(\025)3402 1765 y FB(i)3402 1809 y(n)p FC(\000)p FJ(1)3527 1761 y FI(L)p Fy(\()3624 1684 y Fv(\000)g(!)3627 1761 y FI(a)3675 1728 y Fu(j)3716 1761 y Fy(\)])3776 1796 y Fu(\025)3817 1773 y FB(i)3817 1817 y(n)p FC(\000)p FJ(1)3938 1796 y Ft(+)p Fu(\025)4034 1763 y FB(j)4034 1817 y(n)p FC(\000)p FJ(1)125 1894 y Fy(is)727 1964 y Fu(n)p Fs(\000)p Ft(1)728 1991 y Fm(X)738 2189 y Fu(l)q Ft(=1)860 2078 y Fy(\()p FI(a)943 2033 y Fu(j)943 2104 y Ft(1)983 2112 y Fu(\025)1024 2079 y FB(j)1024 2133 y FJ(1)1078 2078 y Fv(\001)15 b(\001)g(\001)i FI(a)1248 2033 y Fu(j)1248 2107 y(l)q Fs(\000)p Ft(1)1364 2118 y Fu(\025)1405 2085 y FB(j)1405 2142 y(l)p FC(\000)p FJ(1)1512 2078 y Fy(\()p FI(L)p Fy(\()1644 1996 y Fv(\000)-71 b(!)1651 2078 y FI(a)1699 2040 y Fu(i)1735 2078 y Fy(\))1771 2104 y Fu(\025)1812 2082 y FB(i)1812 2126 y(n)p FC(\000)p FJ(1)1937 2078 y FI(a)1985 2033 y Fu(j)1985 2107 y(l)2021 2078 y Fy(\))2056 2112 y Fu(\025)2097 2079 y FB(j)2097 2135 y(l)2130 2112 y Ft(+)p Fu(\025)2226 2089 y FB(i)2226 2133 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2443 2078 y FI(a)2491 2033 y Fu(j)2491 2107 y(l)q Ft(+1)2607 2118 y Fu(\025)2648 2085 y FB(j)2648 2142 y(l)p FJ(+1)2769 2078 y Fv(\001)15 b(\001)g(\001)h FI(a)2938 2033 y Fu(j)2938 2104 y(n)p Fs(\000)p Ft(1)3075 2078 y Fy(\))3110 2112 y Fu(\025)3151 2079 y FB(j)3151 2133 y(n)p FC(\000)p FJ(1)3277 2078 y FI(;)125 2298 y Fy(terms)30 b(in)f FI(S)540 2312 y Ft(14)645 2298 y Fy(ha)m(v)m(e)j(the)e(follo)m(wing)f(forms.)267 2478 y(\()p FI(a)p Fy(\))i FI(a)464 2434 y Fu(q)464 2504 y Ft(1)504 2508 y Fu(\025)545 2476 y FB(q)545 2529 y FJ(1)599 2478 y Fv(\001)15 b(\001)g(\001)j Fy(^)-47 b FI(a)768 2434 y Fu(q)768 2507 y(k)826 2478 y Fv(\001)15 b(\001)g(\001)h FI(a)995 2434 y Fu(q)995 2504 y(n)p Fs(\000)p Ft(1)1133 2515 y Fu(\025)1174 2482 y FB(q)1174 2536 y(n)p FC(\000)p FJ(1)1299 2478 y FI(a)1347 2440 y Fu(q)1347 2500 y(n)1394 2505 y(\025)1435 2472 y FB(q)1435 2515 y(n)1482 2478 y Fy(\()p FI(\015)1564 2512 y Fs(\001\001\001)11 b Fu(;)p Ft([)p Fu(\025)1716 2489 y FB(i)1716 2533 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1927 2512 y Fu(;\025)1988 2480 y FB(j)1988 2533 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2200 2512 y Ft(])p Fu(;)p Fs(\001\001\001)f Fu(;\025)2371 2480 y FB(q)2371 2536 y(k)2413 2478 y Fy(\()p Fv(\001)15 b(\001)g(\001)32 b FI(;)15 b Fy([)p FI(a)2698 2440 y Fu(i)2698 2500 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2941 2478 y FI(;)g(a)3029 2434 y Fu(j)3029 2504 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3273 2478 y Fy(])p FI(;)g Fv(\001)g(\001)g(\001) 32 b FI(;)15 b(a)3563 2434 y Fu(q)3563 2507 y(k)3606 2478 y Fy(\)\))p FI(;)267 2693 y Fy(\()p FI(b)p Fy(\))31 b FI(a)455 2649 y Fu(q)455 2719 y Ft(1)495 2723 y Fu(\025)536 2691 y FB(q)536 2745 y FJ(1)590 2693 y Fv(\001)15 b(\001)g(\001)i Fy(^)-46 b FI(a)759 2649 y Fu(q)759 2720 y(j)812 2693 y Fv(\001)15 b(\001)g(\001)i FI(a)982 2649 y Fu(q)982 2719 y(n)p Fs(\000)p Ft(1)1119 2730 y Fu(\025)1160 2698 y FB(q)1160 2751 y(n)p FC(\000)p FJ(1)1285 2693 y FI(a)1333 2656 y Fu(q)1333 2716 y(n)1380 2720 y(\025)1421 2687 y FB(q)1421 2730 y(n)1468 2693 y Fy(\()p FI(\015)1550 2741 y Fu(\025)1591 2708 y FJ(1)p FC(!)1678 2696 y FJ(^)1682 2708 y FB(i)q FC(!)p FB(q)r FC(\000)p FJ(1)1591 2762 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1879 2741 y Fu(;\025)1940 2708 y FB(q)1940 2763 y(j)1974 2741 y Ft(+)p Fu(\025)2070 2718 y FB(i)2070 2762 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2286 2693 y Fy(\()p Fv(\001)e(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)2546 2656 y Fu(i)2546 2716 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2789 2693 y FI(;)15 b Fv(\001)g(\001)g (\001)32 b FI(;)15 b Fy(\()p FI(L)p Fy(\()3138 2612 y Fv(\000)-71 b(!)3145 2693 y FI(a)3193 2656 y Fu(i)3230 2693 y Fy(\))3265 2720 y Fu(\025)3306 2697 y FB(i)3306 2741 y(n)p FC(\000)p FJ(1)3432 2693 y FI(a)3480 2649 y Fu(q)3480 2720 y(j)3518 2693 y Fy(\)\)\))p FI(;)267 2922 y Fy(\()p FI(c)p Fy(\))47 b Fv(\001)15 b(\001)g(\001)i Fy(^)-46 b FI(a)592 2878 y Fu(q)592 2952 y(k)650 2922 y Fv(\001)15 b(\001)g(\001)h Fy(\()p FI(L)p Fy(\()903 2841 y Fv(\000)-71 b(!)910 2922 y FI(a)958 2885 y Fu(i)995 2922 y Fy(\))1030 2949 y Fu(\025)1071 2926 y FB(i)1071 2970 y(n)p FC(\000)p FJ(1)1196 2922 y FI(a)1244 2878 y Fu(q)1244 2950 y(j)1282 2922 y Fy(\))1317 2947 y Fu(\025)1358 2914 y FB(q)1358 2969 y(j)1393 2947 y Ft(+)p Fu(\025)1489 2924 y FB(i)1489 2968 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1721 2922 y Fv(\001)15 b(\001)g(\001)h FI(a)1890 2885 y Fu(q)1890 2945 y(n)1937 2949 y(\025)1978 2917 y FB(q)1978 2960 y(n)2025 2922 y Fy(\()p FI(\015)2107 2970 y Fu(\025)2148 2937 y FJ(1)p FC(!)2235 2925 y FJ(^)2239 2937 y FB(i)q FC(!)p FB(q)r FC(\000)p FJ(1)2148 2991 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2436 2970 y Fu(;\025)2497 2937 y FB(q)2497 2994 y(k)2534 2970 y Ft(+)p Fu(\025)2630 2947 y FB(i)2630 2991 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2847 2922 y Fy(\()p Fv(\001)f(\001)g(\001)32 b FI(;)16 b Fy(^)-46 b FI(a)3107 2885 y Fu(i)3107 2945 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3350 2922 y FI(;)15 b Fv(\001)g(\001)g (\001)32 b FI(;)15 b(a)3615 2878 y Fu(q)3615 2952 y(k)3658 2922 y Fy(\)\))p FI(;)267 3148 y Fy(\()p FI(d)p Fy(\))47 b Fv(\001)15 b(\001)g(\001)h Fy(\()p FI(L)p Fy(\()684 3067 y Fv(\000)-71 b(!)691 3148 y FI(a)739 3110 y Fu(i)776 3148 y Fy(\))811 3175 y Fu(\025)852 3152 y FB(i)852 3196 y(n)p FC(\000)p FJ(1)977 3148 y FI(a)1025 3104 y Fu(q)1025 3175 y(j)1063 3148 y Fy(\))1098 3173 y Fu(\025)1139 3140 y FB(q)1139 3195 y(j)1174 3173 y Ft(+)p Fu(\025)1270 3150 y FB(i)1270 3194 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1502 3148 y Fv(\001)15 b(\001)g(\001)i Fy(^)-46 b FI(a)1671 3104 y Fu(q)1671 3177 y(k)1729 3148 y Fv(\001)15 b(\001)g(\001)h FI(a)1898 3110 y Fu(q)1898 3170 y(n)1945 3175 y(\025)1986 3142 y FB(q)1986 3185 y(n)2033 3148 y Fy(\()p FI(\015)2115 3195 y Fu(\025)2156 3163 y FJ(1)p FC(!)2243 3151 y FJ(^)2247 3163 y FB(i)q FC(!)p FB(q)r FC(\000)p FJ(1)2156 3217 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2444 3195 y Fu(;\025)2505 3163 y FB(q)2505 3219 y(k)2542 3195 y Ft(+)p Fu(\025)2638 3173 y FB(i)2638 3217 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2855 3148 y Fy(\()p Fv(\001)f(\001)g(\001)32 b FI(;)16 b Fy(^)-46 b FI(a)3115 3110 y Fu(i)3115 3170 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3358 3148 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3623 3104 y Fu(q)3623 3177 y(k)3666 3148 y Fy(\)\))p FI(:)225 3380 y Fy(T)-8 b(erms)27 b(of)g(t)m(yp)s(e)h (\(a\))g(in)f FI(S)1103 3394 y Ft(14)1204 3380 y Fy(and)g(terms)h(of)f FI(S)1785 3394 y Ft(41)1887 3380 y Fy(are)h(in)e(the)i(same)g(form.)39 b(But)27 b(they)h(ha)m(v)m(e)h(opp)s(osite)d(signs)125 3513 y(in)g(di\013eren)m(t)g(summands)g(for)h(\014xed)f FI(i;)15 b(j;)g(k)s Fy(.)42 b(So)27 b(they)g(are)h(o\013set.)41 b(Similarly)-8 b(,)24 b(terms)j(of)h(t)m(yp)s(e)f(\(b\))g(in)f FI(S)3656 3527 y Ft(14)3758 3513 y Fy(and)125 3645 y(part)k(terms)g(of) h FI(S)735 3659 y Ft(42)840 3645 y Fy(cancel)f(eac)m(h)i(other)f(out)f (for)g(\014xed)g FI(i)c(<)f(j)30 b Fy(=)25 b FI(p)p Fy(.)225 3811 y(Fix)k FI(i;)15 b(k)s Fy(,)32 b(terms)e(of)h(t)m(yp)s(e)f(\(c\))i (and)d(\(d\))i(plus)d(corresp)s(onding)h(terms)h(in)f FI(S)2788 3825 y Ft(24)258 4004 y Fy(\()p Fv(\000)p Fy(1\))444 3966 y Fu(i)p Ft(+)p Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(k)768 4004 y Fv(\001)15 b(\001)g(\001)i Fy(^)-46 b FI(a)937 3959 y Fu(q)937 4033 y(k)995 4004 y Fv(\001)15 b(\001)g(\001)h Fy(\()p FI(L)p Fy(\()1248 3922 y Fv(\000)-71 b(!)1255 4004 y FI(a)1303 3966 y Fu(i)1340 4004 y Fy(\))1375 4031 y Fu(\025)1416 4008 y FB(i)1416 4052 y(n)p FC(\000)p FJ(1)1541 4004 y FI(a)1589 3966 y Fu(q)1589 4026 y(n)1636 4004 y Fy(\))1671 4028 y Fu(\025)1712 3996 y FB(q)1712 4039 y(n)1755 4028 y Ft(+)p Fu(\025)1851 4005 y FB(i)1851 4049 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2068 4004 y Fy(\()p FI(\015)2150 4051 y Fu(\025)2191 4019 y FJ(1)p FC(!)2278 4007 y FJ(^)2282 4019 y FB(i)q FC(!)p FB(q)r FC(\000)p FJ(1)2191 4072 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2478 4051 y Fu(;\025)2539 4019 y FB(q)2539 4075 y(k)2577 4051 y Ft(+)p Fu(\025)2673 4028 y FB(i)2673 4072 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2890 4004 y Fy(\()p Fv(\001)15 b(\001)g(\001)31 b FI(;)17 b Fy(^)-47 b FI(a)3149 3966 y Fu(i)3149 4026 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3393 4004 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3658 3959 y Fu(q)3658 4033 y(k)3701 4004 y Fy(\)\))125 4198 y(equal)418 4386 y(\()p Fv(\000)p Fy(1\))604 4349 y Fu(i)p Ft(+)p Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(k)913 4386 y Fy([)p FI(L)p Fy(\()1035 4305 y Fv(\000)-71 b(!)1042 4386 y FI(a)1090 4349 y Fu(i)1127 4386 y Fy(\))1162 4413 y Fu(\025)1203 4390 y FB(i)1203 4434 y(n)p FC(\000)p FJ(1)1328 4386 y FI(L)p Fy(\()1425 4313 y Fv(\000)g(!)1425 4386 y FI(a)1473 4342 y Fu(q)1473 4415 y(k)1516 4386 y Fy(\)])1576 4411 y Fu(\025)1617 4388 y FB(i)1617 4432 y(n)p FC(\000)p FJ(1)1739 4411 y Ft(+)p Fu(\025)1835 4378 y FB(q)1835 4421 y(n)1882 4386 y Fy(\()p FI(\015)1964 4433 y Fu(\025)2005 4401 y FJ(1)p FC(!)2092 4389 y FJ(^)2096 4401 y FB(i)q FC(!)p FB(q)r FC(\000)p FJ(1)2005 4455 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2293 4433 y Fu(;\025)2354 4401 y FB(q)2354 4457 y(k)2391 4433 y Ft(+)p Fu(\025)2487 4411 y FB(i)2487 4455 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2704 4386 y Fy(\()p Fv(\001)15 b(\001)g(\001)32 b FI(;)16 b Fy(^)-46 b FI(a)2964 4349 y Fu(i)2964 4409 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3207 4386 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3472 4342 y Fu(q)3472 4415 y(k)3515 4386 y Fy(\)\))p FI(;)125 4623 y Fy(where)30 b FI(L)p Fy(\()485 4551 y Fv(\000)-71 b(!)485 4623 y FI(a)533 4579 y Fu(q)533 4653 y(k)576 4623 y Fy(\))26 b(:=)f FI(a)806 4579 y Fu(q)806 4649 y Ft(1)845 4654 y Fu(\025)886 4621 y FB(q)886 4675 y FJ(1)940 4623 y Fv(\001)15 b(\001)g(\001)j Fy(^)-47 b FI(a)1109 4579 y Fu(q)1109 4653 y(k)1152 4623 y Fv(\001)15 b(\001)g(\001)1258 4646 y Fu(\025)1299 4613 y FB(q)1299 4667 y(n)p FC(\000)p FJ(1)1424 4623 y FI(a)1472 4579 y Fu(q)1472 4635 y(n)1520 4623 y Fy(.)40 b(It)31 b(can)f(b)s(e)g(matc)m (hed)h(with)e(terms)i(of)f FI(S)2960 4637 y Ft(43)484 4845 y Fy(\()p Fv(\000)p Fy(1\))670 4808 y Fu(i)p Ft(+)p Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(k)r Ft(+2)1069 4845 y FI(L)p Fy(\()1166 4772 y Fv(\000)-71 b(!)1166 4845 y FI(a)1214 4801 y Fu(q)1214 4874 y(k)1258 4845 y Fy(\))1293 4879 y Fu(\025)1334 4846 y FB(q)1334 4889 y(n)1381 4845 y FI(L)p Fy(\()1478 4764 y Fv(\000)g(!)1485 4845 y FI(a)1533 4808 y Fu(i)1569 4845 y Fy(\))1604 4872 y Fu(\025)1645 4849 y FB(i)1645 4893 y(n)p FC(\000)p FJ(1)1770 4845 y Fy(\(\()p FI(\015)1887 4893 y Fu(\025)1928 4860 y FJ(1)p FC(!)2015 4848 y FJ(^)2019 4860 y FB(i)r FC(!)p FB(q)r FC(\000)p FJ(1)1928 4914 y(1)p 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y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1710 3589 y FI(;)g(a)1798 3551 y Fu(q)r Fs(\000)p Ft(1)1798 3611 y Fu(n)1926 3589 y Fy(\))1252 3772 y(=)25 b(\()p FI(@)h Fy(+)20 b FI(\025)1601 3728 y Ft(1)p Fs(!)p Fu(q)r Fs(\000)p Ft(1)1601 3798 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1864 3772 y Fy(+)g FI(\025)2008 3735 y Fu(q)r Fs(\000)p Ft(1)2008 3795 y Fu(n)2136 3772 y Fy(\))p FI(\015)2218 3806 y Fu(\025)2259 3773 y FJ(1)p FC(!)p FB(q)r FC(\000)p FJ(1)2259 3827 y(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2472 3806 y Fu(;\025)2533 3773 y FB(q)r FC(\000)p FJ(1)2533 3816 y FB(n)2650 3772 y Fy(\()p FI(a)2733 3735 y Ft(1)2733 3795 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2976 3772 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3241 3728 y Fu(q)r Fs(\000)p Ft(1)3241 3798 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3484 3772 y FI(;)g(a)3572 3735 y Fu(q)r Fs(\000)p Ft(1)3572 3795 y Fu(n)3701 3772 y Fy(\))p FI(:)3672 3910 y SDict begin H.S end 3672 3910 a 3672 3910 a SDict begin 19.25 H.A end 3672 3910 a 3672 3910 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.12) cvn /DEST pdfmark end 3672 3910 a Fy(\(4.12\))125 4133 y SDict begin H.S end 125 4133 a 125 4133 a SDict begin 16 H.A end 125 4133 a 125 4133 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.11) cvn /DEST pdfmark end 125 4133 a Black FH(Corollary)36 b(4.11.)p Black 43 w FI(D)s(@)30 b Fy(=)25 b FI(@)5 b(D)s Fo(.)42 b(Thus)33 b FI(@)1602 4110 y Fm(e)1584 4133 y FI(C)1656 4100 y Fs(\003)1695 4133 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))34 b Fo(is)e(a)h(sub)-5 b(c)g(omplex)35 b(of)2775 4110 y Fm(e)2757 4133 y FI(C)2829 4100 y Fs(\003)2868 4133 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))p Fo(.)p Black 125 4391 a(Pr)-5 b(o)g(of.)p Black 43 w Fy(Consider)28 b(the)i(expansion)f(of)i(\()p FI(D)s Fy(\()p FI(@)5 b(\015)g Fy(\)\))1790 4425 y Fu(\025)1831 4392 y FJ(1)p FC(!)p FB(q)1831 4446 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2045 4425 y Fu(;\025)2106 4392 y FB(q)2106 4435 y(n)2152 4391 y Fy(\()p FI(a)2235 4358 y Ft(1)2235 4416 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2479 4391 y FI(;)15 b Fv(\001)g(\001)g(\001)31 b FI(;)15 b(a)2743 4347 y Fu(q)2743 4417 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2987 4391 y FI(;)g(a)3075 4347 y Fu(q)3075 4403 y(n)3122 4391 y Fy(\))31 b(b)m(y)f(\()3349 4391 y SDict begin H.S end 3349 4391 a Black Fy(4.10)p Black 3511 4333 a SDict begin H.R end 3511 4333 a 3511 4391 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.10) cvn H.B /ANN pdfmark end 3511 4391 a Fy(\).)41 b(W)-8 b(e)31 b(can)125 4548 y(easily)e(extract)j FI(@)25 b Fy(+)20 b FI(\025)903 4504 y Ft(1)p Fs(!)p Fu(q)903 4574 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1166 4548 y Fy(+)f FI(\025)1309 4504 y Fu(q)1309 4560 y(n)1387 4548 y Fy(from)29 b(the)i(\014rst)e(t)m(w)m(o)j(summands.)39 b(And)29 b FI(@)c Fy(+)20 b FI(\025)3019 4504 y Ft(1)p Fs(!)p Fu(q)3019 4574 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3282 4548 y Fy(+)g FI(\025)3426 4504 y Fu(q)3426 4560 y(n)3503 4548 y Fy(of)30 b(the)h(last)125 4681 y(t)m(w)m(o)h (summands)c(can)j(b)s(e)f(deduced)f(b)m(y)h(applying)f(\(M2\).)1722 b Fk(\003)225 4939 y Fy(Then)29 b(w)m(e)i(giv)m(e)g(the)f(follo)m(wing) f(de\014nitions.)125 5161 y SDict begin H.S end 125 5161 a 125 5161 a SDict begin 16 H.A end 125 5161 a 125 5161 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.12) cvn /DEST pdfmark end 125 5161 a Black FH(De\014nition)d(4.12.)p Black 35 w Fy(The)c(complex)1398 5138 y Fm(e)1379 5161 y FI(C)1451 5128 y Fs(\003)1490 5161 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))23 b(endo)m(w)m(ed)g(with)d(di\013eren)m(tial)h FI(D)k Fy(is)c(called)g(the)h Fo(b)-5 b(asic)26 b(c)-5 b(omplex)125 5294 y(of)38 b FI(R)g Fo(with)g(c)-5 b(o)g(e\016cients)38 b(in)f FI(M)10 b Fy(.)57 b(And)34 b(the)i(quotien)m(t)g(complex)f FI(C)2473 5261 y Fs(\003)2512 5294 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))35 b(=)2947 5271 y Fm(e)2929 5294 y FI(C)3001 5261 y Fs(\003)3040 5294 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))p FI(=@)3436 5271 y Fm(e)3416 5294 y FI(C)3488 5261 y Fs(\003)3529 5294 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))36 b(is)125 5427 y(called)29 b(the)i Fo(r)-5 b(e)g(duc)g(e)g(d)34 b(c)-5 b(omplex)35 b(of)e FI(R)g Fo(with)g(c)-5 b(o)g(e\016cients)34 b(in)e FI(M)10 b Fy(.)p Black Black eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop -331 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -331 0 a -214 -197 a SDict begin H.S end -214 -197 a -214 -197 a SDict begin H.R end -214 -197 a -214 -197 a SDict begin [/View [/XYZ H.V]/Dest (page.24) cvn /DEST pdfmark end -214 -197 a Black 125 -123 a Ft(24)858 b(MENGJUN)23 b(W)-8 b(ANG)1664 -146 y FJ(1)1699 -123 y Ft(,)23 b(LIPENG)i(LUO)2192 -146 y FJ(2)2249 -123 y Ft(AND)f(ZHIXIANG)g(WU)2942 -146 y FJ(3)p Black 225 77 a Fy(Recall)35 b(the)h(complex)f(of)h(an)f FI(n)p Fy(-Lie)g(algebra)h Fw(g)f Fy(with)f(co)s(e\016cien)m(ts)j(in)d (a)i Fw(g)p Fy(-represen)m(tation)g(\()p FI(V)5 b(;)15 b(\032)p Fy(\))37 b(\(See,)125 210 y(e.g.,[)310 210 y SDict begin H.S end 310 210 a Black Fy(17)p Black 403 151 a SDict begin H.R end 403 151 a 403 210 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.lszc) cvn H.B /ANN pdfmark end 403 210 a Fy(]\))31 b(is)e(comp)s(osed)h(b)m(y)225 375 y(\(i\))g FI(q)s Fy(-co)s(c)m(hains)g FI(C)861 342 y Fu(q)899 375 y Fy(\()p Fw(g)p FI(;)15 b(V)20 b Fy(\))26 b(:=)f Fv(f)p FI(\015)31 b Fy(:)26 b Fw(g)1495 342 y Fs(^)1542 318 y FB(n)p FC(\000)p FJ(1)1687 375 y Fv(\012)1778 332 y Fg(\()p Fi(q)17 b Ff(\000)e Fg(1\))1836 399 y Ff(\001)c(\001)h (\001)1997 375 y Fv(\012)20 b Fw(g)2134 342 y Fs(^)2181 318 y FB(n)p FC(\000)p FJ(1)2326 375 y Fv(^)g Fw(g)25 b Fv(!)g FI(V)c Fv(g)p Fy(,)225 540 y(\(ii\))29 b(di\013eren)m(tial)g FI(D)f Fy(:)d FI(C)1056 507 y Fu(q)1094 540 y Fy(\()p Fw(g)p FI(;)15 b(V)21 b Fy(\))k Fv(!)g FI(C)1537 507 y Fu(q)r Ft(+1)1665 540 y Fy(\()p Fw(g)p FI(;)15 b(V)21 b Fy(\))31 b(giv)m(en)f(b)m(y)396 728 y(\()p FI(D)s(\015)5 b Fy(\)\()p FI(a)679 690 y Ft(1)679 750 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)923 728 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)1188 684 y Fu(q)1188 754 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1431 728 y FI(;)g(a)1519 690 y Fu(q)1519 750 y(n)1567 728 y Fy(\))325 895 y(=)491 809 y Fm(X)411 1005 y Ft(1)p Fs(\024)p Fu(i<j)t Fs(\024)p Fu(q)701 895 y Fy(\()p Fv(\000)p Fy(1\))887 858 y Fu(i)917 895 y FI(\015)5 b Fy(\()p FI(a)1052 858 y Ft(1)1052 918 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1295 895 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)1560 858 y Fu(i)1560 918 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1803 895 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2068 851 y Fu(j)t Fs(\000)p Ft(1)2068 921 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2312 895 y FI(;)g Fy([)p FI(a)2425 858 y Fu(i)2425 918 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2668 895 y FI(;)g(a)2756 851 y Fu(j)2756 921 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3000 895 y Fy(])p FI(;)g(a)3113 851 y Fu(j)t Ft(+1)3113 921 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3356 895 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)3621 858 y Fu(q)3621 918 y(n)3669 895 y Fy(\))416 1218 y(+)556 1100 y Fu(q)507 1132 y Fm(X)516 1327 y Fu(i)p Ft(=1)639 1218 y Fy(\()p Fv(\000)p Fy(1\))825 1181 y Fu(i)854 1218 y FI(\015)5 b Fy(\()p FI(a)989 1181 y Ft(1)989 1241 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1232 1218 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)1497 1181 y Fu(i)1497 1241 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1740 1218 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2005 1174 y Fu(q)2005 1244 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2249 1218 y FI(;)g Fy([)p FI(a)2362 1181 y Fu(i)2362 1241 y Ft(1)2402 1218 y FI(;)g Fv(\001)g(\001)g(\001)q FI(;)g(a)2636 1181 y Fu(i)2636 1241 y(n)p Fs(\000)p Ft(1)2774 1218 y FI(;)g(a)2862 1181 y Fu(q)2862 1241 y(n)2909 1218 y Fy(]\))416 1528 y(+)556 1409 y Fu(q)507 1442 y Fm(X)516 1637 y Fu(i)p Ft(=1)639 1528 y Fy(\()p Fv(\000)p Fy(1\))825 1490 y Fu(i)p Ft(+1)944 1528 y FI(\032)p Fy(\()p FI(a)1074 1490 y Fu(i)1074 1550 y Ft(1)1114 1528 y FI(;)g Fv(\001)g(\001)g(\001)q FI(;)g(a)1348 1490 y Fu(i)1348 1550 y(n)p Fs(\000)p Ft(1)1486 1528 y Fy(\)\()p FI(\015)5 b Fy(\()p FI(a)1691 1490 y Ft(1)1691 1550 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1935 1528 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)16 b Fy(^)-46 b FI(a)2200 1490 y Fu(i)2200 1550 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2443 1528 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2708 1484 y Fu(q)2708 1554 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2951 1528 y FI(;)g(a)3039 1490 y Fu(q)3039 1550 y(n)3087 1528 y Fy(\)\))416 1847 y(+)507 1733 y Fu(n)p Fs(\000)p Ft(1)508 1760 y Fm(X)517 1956 y Fu(i)p Ft(=1)640 1847 y Fy(\()p Fv(\000)p Fy(1\))826 1809 y Fu(n)p Ft(+)p Fu(q)r Fs(\000)p Fu(i)p Ft(+1)1132 1847 y FI(\032)p Fy(\()p FI(a)1262 1803 y Fu(q)1262 1873 y Ft(1)1302 1847 y FI(;)g 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Fm(\001)2260 590 y Fv(\001)15 b(\001)g(\001)2381 516 y Fm(\000)2422 537 y Fu(m)2484 514 y FB(i)2484 558 y FJ(1)32 b FB(p)2435 634 y Fu(k)2474 611 y FB(i)2472 655 y FJ(1)f FB(p)2582 516 y Fm(\001)2639 590 y Fv(\001)15 b(\001)g(\001)2760 516 y Fm(\000)2801 538 y Fu(m)2863 515 y FB(i)2863 559 y(n)p FC(\000)p FJ(1)32 b(1)2814 633 y Fu(k)2853 610 y FB(i)2851 654 y(n)p FC(\000)p FJ(1)f(1)3046 516 y Fm(\001)3102 590 y Fv(\001)15 b(\001)g(\001)3224 516 y Fm(\000)3265 537 y Fu(m)3327 514 y FB(i)3327 558 y(n)p FC(\000)p FJ(1)32 b FB(p)3278 634 y Fu(k)3317 611 y FB(i)3315 655 y(n)p FC(\000)p FJ(1)f FB(p)3511 516 y Fm(\001)247 868 y FI(\015)294 915 y Ft(\()p Fu(m)383 883 y FJ(1)p FC(!)470 871 y FJ(^)474 883 y FB(i)q FC(!)p FB(q)383 936 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)596 915 y Fu(;m)678 883 y FB(q)678 926 y(n)720 915 y Ft(+)p Fu(m)837 892 y FB(i)837 936 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1050 915 y Fs(\000)p Fu(k)1144 892 y FB(i)1142 936 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1354 915 y Ft(\))1385 868 y Fy(\()p FI(a)1468 830 y Ft(1)1468 890 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)1712 868 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)16 b Fy(^)-46 b FI(a)1977 830 y Fu(i)1977 890 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2220 868 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)15 b(a)2485 824 y Fu(q)2485 894 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2728 868 y FI(;)g(a)2816 830 y Fu(i)2816 890 y Ft(1)2856 894 y(\()p Fu(k)2922 872 y FB(i)2920 916 y FJ(1)2955 894 y Ft(\))2987 868 y Fv(\001)g(\001)g(\001)3093 892 y Ft(\()p Fu(k)3159 869 y FB(i)3157 914 y(n)p FC(\000)p FJ(2)3278 892 y Ft(\))3309 868 y FI(a)3357 830 y Fu(i)3357 890 y(n)p Fs(\000)p Ft(1)3494 901 y(\()p Fu(k)3560 878 y FB(i)3558 922 y(n)p FC(\000)p FJ(1)3680 901 y Ft(\))3711 868 y FI(a)3759 830 y Fu(q)3759 890 y(n)3806 868 y Fy(\))176 1087 y(=)366 1000 y Fm(X)262 1208 y Fu(k)301 1185 y FB(i)299 1229 y FJ(1)333 1208 y Fu(;)p Fs(\001\001\001)10 b Fu(;k)482 1185 y FB(i)480 1229 y(n)p FC(\000)p FJ(1)601 1087 y Fy(\()p Fv(\000)p Fy(1\))787 1049 y Fu(i)816 1013 y Fm(\000)857 1038 y Fu(m)919 1015 y FB(i)919 1059 y FJ(1)870 1126 y Fu(k)909 1103 y FB(i)907 1147 y FJ(1)954 1013 y Fm(\001)1011 1087 y Fv(\001)15 b(\001)g(\001)1132 1013 y Fm(\000)1174 1035 y Fu(m)1236 1012 y FB(i)1236 1056 y(n)p FC(\000)p FJ(1)1187 1129 y Fu(k)1226 1106 y FB(i)1224 1150 y(n)p FC(\000)p FJ(1)1357 1013 y Fm(\001)1399 1087 y FI(\015)1446 1134 y Ft(\()p Fu(m)1535 1101 y FJ(1)p FC(!)1622 1089 y FJ(^)1626 1101 y FB(i)r FC(!)p FB(q)1535 1155 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)1748 1134 y Fu(;m)1830 1101 y FB(q)1830 1145 y(n)1873 1134 y Ft(+)p Fu(m)1990 1111 y FB(i)1990 1155 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2202 1134 y Fs(\000)p Fu(k)2296 1111 y FB(i)2294 1155 y FJ(1)p FC(!)p FB(n)p FC(\000)p FJ(1)2506 1134 y Ft(\))2538 1087 y Fy(\()p FI(a)2621 1049 y Ft(1)2621 1109 y(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)2864 1087 y FI(;)g Fv(\001)g(\001)g(\001)32 b FI(;)17 b Fy(^)-47 b FI(a)3129 1049 y Fu(i)3129 1109 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)3372 1087 y FI(;)15 b Fv(\001)g(\001)g(\001)32 b FI(;)610 1362 y(a)658 1317 y Fu(q)658 1388 y Ft(1)p Fs(!)p Fu(n)p Fs(\000)p Ft(1)901 1362 y FI(;)15 b(a)989 1324 y Fu(i)989 1384 y Ft(1)1029 1388 y(\()p Fu(k)1095 1365 y FB(i)1093 1409 y FJ(1)1128 1388 y Ft(\))1160 1362 y Fv(\001)g(\001)g(\001)1266 1386 y Ft(\()p Fu(k)1332 1363 y FB(i)1330 1407 y(n)p FC(\000)p FJ(2)1451 1386 y Ft(\))1482 1362 y FI(a)1530 1324 y Fu(i)1530 1384 y(n)p Fs(\000)p Ft(1)1668 1395 y(\()p Fu(k)1734 1372 y FB(i)1732 1416 y(n)p FC(\000)p FJ(1)1853 1395 y Ft(\))1884 1362 y FI(a)1932 1324 y Fu(q)1932 1384 y(n)1979 1362 y Fy(\))p FI(:)3834 1576 y Fk(\003)225 1805 y Fy(As)30 b(a)h(result,)e(w)m(e)i(obtain)f(an)g(isomorphism)e(on)i(cohomology) 2433 1782 y Fm(e)2411 1805 y FI(H)2494 1772 y Fs(\003)2534 1805 y Fy(\()p FI(R)q(;)15 b(M)10 b Fy(\))26 b Fv(')f FI(H)3017 1772 y Fs(\003)3056 1805 y Fy(\(\()p Fq(L)19 b FI(ie)3297 1819 y Ft(1)3367 1805 y FI(R)q Fy(\))p 3476 1819 22 4 v 30 w FI(;)c(M)10 b Fy(\).)125 2012 y SDict begin H.S end 125 2012 a 125 2012 a SDict begin 16 H.A end 125 2012 a 125 2012 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.14) cvn /DEST pdfmark end 125 2012 a Black FH(Remark)32 b(4.14.)p Black 41 w Fo(F)-7 b(r)i(om)32 b(the)f(pr)-5 b(o)g(of)33 b(of)e(Pr)-5 b(op)g(osition)1975 2030 y SDict begin H.S end 1975 2030 a Black -18 x Fo(4.10)p Black 2142 1954 a SDict begin H.R end 2142 1954 a 2142 2012 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.10) cvn H.B /ANN pdfmark end 2142 2012 a Fo(,)31 b(one)g(c)-5 b(an)32 b(\014nd)f(ther)-5 b(e)32 b(ar)-5 b(e)31 b(mor)-5 b(e)32 b(than)g(one)f(c)-5 b(oho-)125 2145 y(molo)g(gy)34 b(c)-5 b(omplex)35 b(with)e(the)g(same)h FI(q)s Fo(-c)-5 b(o)g(chains)33 b(in)g(the)g(b)-5 b(asic)33 b(c)-5 b(omplex.)44 b(F)-7 b(or)33 b(example,)h(if)f(we)f(only)i(r)-5 b(eserve)125 2278 y(the)43 b(\014rst)f(summand)i(of)f(\()1059 2295 y SDict begin H.S end 1059 2295 a Black -17 x Fo(4.10)p Black 1226 2219 a SDict begin H.R end 1226 2219 a 1226 2278 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.10) cvn H.B /ANN pdfmark end 1226 2278 a Fo(\),)i FI(D)g Fo(is)d(stil)5 b(l)42 b(a)h(wel)5 b(l-de\014ne)-5 b(d)43 b(di\013er)-5 b(ential.)72 b(These)42 b(c)-5 b(omplexes)44 b(of)f FI(n)p Fo(-Lie)125 2411 y(c)-5 b(onformal)37 b(algebr)-5 b(a)35 b FI(R)h Fo(ar)-5 b(e)35 b(matche)-5 b(d)36 b(with)g(c)-5 b(orr)g(esp)g(onding)37 b(c)-5 b(omplexes)36 b(of)f FI(n)p Fo(-Lie)e(algebr)-5 b(a)36 b Fy(\()p Fq(L)18 b FI(ie)3550 2425 y Fu(p)3623 2411 y FI(R)q Fy(\))p 3732 2425 22 4 v 65 w Fo(in-)125 2543 y(tr)-5 b(o)g(duc)g(e)g(d)35 b(in)e Fy([)616 2543 y SDict begin H.S end 616 2543 a Black Fy(1)p Black 661 2485 a SDict begin H.R end 661 2485 a 661 2543 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ai1) cvn H.B /ANN pdfmark end 661 2543 a Fy(])g Fo(in)g(the)g(same)g(way)h (to)f(The)-5 b(or)g(em)1884 2561 y SDict begin H.S end 1884 2561 a Black -18 x Fo(4.13)p Black 2052 2485 a SDict begin H.R end 2052 2485 a 2052 2543 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.13) cvn H.B /ANN pdfmark end 2052 2543 a Fo(.)1750 2816 y SDict begin H.S end 1750 2816 a 1750 2816 a SDict begin 16 H.A end 1750 2816 a 1750 2816 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1750 2816 a Fx(References)p Black 125 2961 a SDict begin H.S end 125 2961 a FG([1])205 2961 y SDict begin 11 H.A end 205 2961 a 205 2961 a SDict begin [/View [/XYZ H.V]/Dest (cite.ai1) cvn /DEST pdfmark end 205 2961 a Black 224 w FG(J.)28 b(A.)g(de)f(Azc\023)-38 b(arraga)29 b(and)f(J.)g(M.)g(Izquierdo,)g FD(n)p Fd(-ary)g(algebras:) 40 b(a)28 b(review)g(with)g(applications)p FG(,)i(J.)f(Ph)n(ys.)e(A:)h (Math.)429 3052 y(Theor.)f(43)f(\(2010\).)p Black 125 3144 a SDict begin H.S end 125 3144 a FG([2])205 3144 y SDict begin 11 H.A end 205 3144 a 205 3144 a SDict begin [/View [/XYZ H.V]/Dest (cite.ai2) cvn /DEST pdfmark end 205 3144 a Black 224 w FG(J.)19 b(A.)g(de)g(Azc\023)-38 b(arraga)20 b(and)f(J.)h(M.)f(Izquierdo,)h Fd(Cohomology)g(of)g(Filipp) r(o)n(v)f(algebras)i(and)d(an)h(analogue)h(of)g(Whitehead's)429 3235 y(lemma)p FG(,)25 b(J.)h(Ph)n(ys.)g(Conf.)h(Ser.)e(175)i (\(2009\).)p Black 125 3326 a SDict begin H.S end 125 3326 a FG([3])205 3326 y SDict begin 11 H.A end 205 3326 a 205 3326 a SDict begin [/View [/XYZ H.V]/Dest (cite.ak) cvn /DEST pdfmark end 205 3326 a Black 224 w FG(A.)20 b(D')h(Andrea)e(and)h(V.)h(Kac,)h Fd(Structure)d(theory)h(of)h (\014nite)g(conformal)g(algebras)p FG(,)i(Selecta)e(Math.,New)h(ser.,)g (\(4\)1998,)429 3418 y(377-418.)p Black 125 3509 a SDict begin H.S end 125 3509 a FG([4])205 3509 y SDict begin 11 H.A end 205 3509 a 205 3509 a SDict begin [/View [/XYZ H.V]/Dest (cite.bdk) cvn /DEST pdfmark end 205 3509 a Black 224 w FG(B.)h(Bak)l(alo)n(v,)g(A.)f(D'Andrea)g(and)g(V.)g(G.)g (Kac,)i Fd(Theory)e(of)h(\014nite)f(pseudoalgebras)p FG(,)i(Adv.)e(Math.,)h(162\(2001\),)j(1-140.)p Black 125 3600 a SDict begin H.S end 125 3600 a FG([5])205 3600 y SDict begin 11 H.A end 205 3600 a 205 3600 a SDict begin [/View [/XYZ H.V]/Dest (cite.bgs) cvn /DEST pdfmark end 205 3600 a Black 224 w FG(C.)d(Bai,)i(L.)d(Guo)h(and)f(Y.)h (Sheng,)g Fd(Bialgebras,)j(the)c(classical)j(Y)-6 b(ang-Baxter)22 b(equation)g(and)h(Manin)f(triples)i(for)f(3-Lie)429 3692 y(algebras)p FG(,)k(Adv.)e(Theor.)i(Math.)f(Ph)n(ys.,)g (23\(1\):27-74,)i(2019.)p Black 125 3783 a SDict begin H.S end 125 3783 a FG([6])205 3783 y SDict begin 11 H.A end 205 3783 a 205 3783 a SDict begin [/View [/XYZ H.V]/Dest (cite.bk) cvn /DEST pdfmark end 205 3783 a Black 224 w FG(B.)e(Bak)l(alo)n(v)g(and)g(V.)f(G.)h(Kac,)h Fd(Field)f(algebras)p FG(,)h(In)n(t.)e(Math.)i(Res.)f(Not.)f (2003\(3\),)j(123-159.)p Black 125 3874 a SDict begin H.S end 125 3874 a FG([7])205 3874 y SDict begin 11 H.A end 205 3874 a 205 3874 a SDict begin [/View [/XYZ H.V]/Dest (cite.bkv) cvn /DEST pdfmark end 205 3874 a Black 224 w FG(B.)33 b(Bak)l(alo)n(v,)h(V.)e(G.)g(Kac)h(and)e(A.)h(A.)g(V)-6 b(orono)n(v,)33 b Fd(Cohomology)g(of)g(conformal)g(algebras)p FG(,)i(Comm.)d(Math.)g(Ph)n(ys.,)429 3966 y(200\(1999\),)c(561-598.)p Black 125 4057 a SDict begin H.S end 125 4057 a FG([8])205 4057 y SDict begin 11 H.A end 205 4057 a 205 4057 a SDict begin [/View [/XYZ H.V]/Dest (cite.ck) cvn /DEST pdfmark end 205 4057 a Black 224 w FG(N.)j(Can)n(tarini)i(and)e(V.)g (G.)h(Kac,)h Fd(Classi\014cation)h(of)e(simple)f(linearly)i(compact)e FD(n)p Fd(-Lie)g(sup)r(eralgebras)p FG(,)k(Comm)n(un.)429 4148 y(Math.)26 b(Ph)n(ys.)g(298\(2010\),)i(833-853.)p Black 125 4240 a SDict begin H.S end 125 4240 a FG([9])205 4240 y SDict begin 11 H.A end 205 4240 a 205 4240 a SDict begin [/View [/XYZ H.V]/Dest (cite.ct) cvn /DEST pdfmark end 205 4240 a Black 224 w FG(R.)22 b(Chatterjee)i(and)e(L.)h (T)-6 b(akh)n(ta)t(jan,)24 b Fd(Asp)r(ects)e(of)i(classical)h(and)d (quan)n(tum)e(Nam)n(bu)h(mec)n(hanics)p FG(,)i(Lett.)f(Math.)h(Ph)n (ys.)429 4331 y(37)j(\(1996\))h(475-482.)p Black 125 4422 a SDict begin H.S end 125 4422 a FG([10])243 4422 y SDict begin 11 H.A end 243 4422 a 243 4422 a SDict begin [/View [/XYZ H.V]/Dest (cite.f) cvn /DEST pdfmark end 243 4422 a Black 186 w FG(V.)e(T.)i(Filipp)r(o)n(v,)f FD(n)p Fd(-Lie)g(algebras)p FG(,)i(Sib.)d(Mat.)i(Zh.)e(26)i(\(1985\),)g (126-140.)p Black 125 4514 a SDict begin H.S end 125 4514 a FG([11])243 4514 y SDict begin 11 H.A end 243 4514 a 243 4514 a SDict begin [/View [/XYZ H.V]/Dest (cite.g) cvn /DEST pdfmark end 243 4514 a Black 186 w FG(V.Guillemin,)h Fd(A)f(Jordan-H\177)-38 b(older)28 b(decomp)r(osition)g(for)g(a)g (certain)g(class)h(of)g(in\014nite)e(dimensional)h(Lie)g(algebras)p FG(,)i(J.)429 4605 y(Di\013.)c(Geom.)f(2\(1968\),)j(313-345.)p Black 125 4696 a SDict begin H.S end 125 4696 a FG([12])243 4696 y SDict begin 11 H.A end 243 4696 a 243 4696 a SDict begin [/View [/XYZ H.V]/Dest (cite.h) cvn /DEST pdfmark end 243 4696 a Black 186 w FG(J.)37 b(Hopp)r(e,)h Fd(On)e(M-Algebras,)k(the)35 b(quan)n(tisation)i(of)g(Nam)n(bu-mec)n (hanics,)f(and)g(v)n(olume)f(preserving)h(di\013eomor-)429 4787 y(phisms)p FG(,)25 b(Helv.)h(Ph)n(ys.)f(Acta)h(70)g(\(1997\),)h (302-317.)p Black 125 4879 a SDict begin H.S end 125 4879 a FG([13])243 4879 y SDict begin 11 H.A end 243 4879 a 243 4879 a SDict begin [/View [/XYZ H.V]/Dest (cite.k) cvn /DEST pdfmark end 243 4879 a Black 186 w FG(V.)e(G.)i(Kac,)f Fd(V)-6 b(ertex)24 b(algebras)k(for)e(b)r(eginners)p FG(,)g(Univ.)g(Lect.)g(Series)g(10,)h(AMS)e(\(1996\).)i(Second)e (edition)h(1998.)p Black 125 4970 a SDict begin H.S end 125 4970 a FG([14])243 4970 y SDict begin 11 H.A end 243 4970 a 243 4970 a SDict begin [/View [/XYZ H.V]/Dest (cite.l) cvn /DEST pdfmark end 243 4970 a Black 186 w FG(S.)f(Lefsc)n (hetz,)i Fd(Algebraic)g(top)r(ology)p FG(,)g(Amer.)d(Math.)j(So)r(c.)f 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