2719:
3044:
1306:
2834:
3834:
1549:
1184:
2450:
2874:
3268:
3710:
1687:
651:
267:
2283:
821:
346:
941:
3433:
4050:
1871:
736:
2407:
3560:
2048:
1190:
1041:
1957:
556:
3993:
1468:
2732:
3727:
1495:
1117:
3923:
is the pullback of the Maurer–Cartan form along some section of the tautological bundle. This is a consequence of the existence of primitives of the
2714:{\displaystyle d\theta ^{i}(E_{j},E_{k})=-\theta ^{i}()=-\sum _{r}{c_{jk}}^{r}\theta ^{i}(E_{r})=-{c_{jk}}^{i}=-{\frac {1}{2}}({c_{jk}}^{i}-{c_{kj}}^{i}),}
3039:{\displaystyle d\omega =\sum _{i}E_{i}(e)\otimes d\theta ^{i}\,=\,-{\frac {1}{2}}\sum _{ijk}{c_{jk}}^{i}E_{i}(e)\otimes \theta ^{j}\wedge \theta ^{k}.}
3055:
1338:
3611:
1576:
2054:
577:
178:
4104:
2220:
3470:
1326:
4157:
1074:
over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique
1075:
761:
1474:
464:
289:
868:
3397:
352:
3998:
1805:
663:
430:
157:
2318:
3508:
1996:
1301:{\displaystyle \forall g\in G\quad \omega _{g}=\mathrm {Ad} (h)(R_{h}^{*}\omega _{e}),{\text{ where }}h=g^{-1},}
3886:, satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifold
993:
463:
known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the
3282:
460:
68:
1047:
3441:
In the method of moving frames, one sometimes considers a local section of the tautological bundle, say
1882:
523:
389:
4147:
1555:
3955:
1962:
but the left-invariant fields span the tangent space at any point (the push-forward of a basis in
4152:
3924:
2424:
2079:
381:
2829:{\displaystyle d\theta ^{i}=-{\frac {1}{2}}\sum _{jk}{c_{jk}}^{i}\theta ^{j}\wedge \theta ^{k}.}
1974:
under a diffeomorphism is still a basis), so the equation is true for any pair of vector fields
1396:
4066:
3829:{\displaystyle \theta _{V}=\operatorname {Ad} (h_{UV}^{-1})\theta _{U}+(h_{UV})^{*}\omega _{H}}
4126:
4100:
3374:
3297:
393:
48:
2856:. To relate it to the previous definition, which only involved the Maurer–Cartan form
4114:
4081:
3293:
1071:
278:
2169:
is left-invariant, applying the Maurer–Cartan form to it simply returns the value of
1544:{\displaystyle {\mathfrak {g}}=T_{e}G\cong \{{\hbox{left-invariant vector fields on G}}\}.}
78:
As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the
3388:
3307:
2131:
1179:{\displaystyle \omega _{e}=\mathrm {id} :T_{e}G\rightarrow {\mathfrak {g}},{\text{ and}}}
3273:
which establishes the equivalence of the two forms of the Maurer–Cartan equation.
4119:
3903:
1737:
657:
64:
4141:
89:
72:
739:
376:, but without a fixed choice of unit element. This motivation came, in part, from
4117:(1964). "Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.".
453:
characterization of the principal homogeneous space. It is a one-form defined on
17:
3721:
gives a compatibility condition relating the two sections on the overlap region:
3460:
508:
377:
351:
A question of importance to Cartan and his contemporaries was how to identify a
79:
31:
2096:
39:
2065:
One can also view the Maurer–Cartan form as being constructed from a
1477:, and the bracket on the right-hand side is the bracket on the Lie algebra
4097:
Differential
Geometry: Cartan's Generalization of Klein's Erlangen Program
2438:). A simple calculation, using the definition of the exterior derivative
3263:{\displaystyle d\omega (E_{j},E_{k})=-\sum _{i}{c_{jk}}^{i}E_{i}(e)=-=-,}
385:
362:
1489:.) These facts may be used to establish an isomorphism of Lie algebras
57:
that carries the basic infinitesimal information about the structure of
4086:
3705:{\displaystyle h_{UV}(x)=s_{V}\circ s_{U}^{-1}(x),\quad x\in U\cap V.}
467:
of the Lie algebra and in this way obtain, locally, a group action on
1682:{\displaystyle d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ().}
405:, but usually without a fixed choice of origin corresponding to the
3587:
are a pair of local sections defined, respectively, over open sets
3285:. In this context, one may view the Maurer–Cartan form as a
646:{\displaystyle L_{g}:G\to G\quad {\mbox{where}}\quad L_{g}(h)=gh,}
406:
4130:
1709:-valued function obtained by duality from pairing the one-form
1046:
In this sense, the Maurer–Cartan form is always the left
283:
A Lie group acts on itself by multiplication under the mapping
262:{\displaystyle \omega (v)=(L_{g^{-1}})_{*}v,\quad v\in T_{g}G.}
3502:
over the base. The Maurer–Cartan equation implies that
3281:
Maurer–Cartan forms play an important role in Cartan's
2278:{\displaystyle \omega =\sum _{i}E_{i}(e)\otimes \theta ^{i}.}
3438:
which is the condition for the vanishing of the curvature.
4067:"Sur la structure des groupes infinis de transformations"
3469:
need only be a local section over the submanifold.) The
1483:. (This may be used as the definition of the bracket on
1529:
1329:
of forms along the right-translation in the group and
605:
392:
forming a Lie group. The geometries of interest were
4001:
3958:
3730:
3614:
3511:
3400:
3058:
2877:
2735:
2453:
2321:
2223:
2212:. Thus, the Maurer–Cartan form can be written
1999:
1885:
1808:
1579:
1498:
1399:
1193:
1120:
996:
871:
764:
666:
580:
526:
292:
181:
3892:
locally with the structure of the homogeneous space
4074:
2301:Suppose that the Lie brackets of the vector fields
816:{\displaystyle (L_{g})_{*}X=X\quad \forall g\in G.}
388:on a space, where the symmetries of the space were
4118:
4044:
3987:
3828:
3704:
3554:
3427:
3262:
3038:
2828:
2713:
2401:
2277:
2042:
1951:
1865:
1681:
1543:
1462:
1300:
1178:
1035:
935:
815:
730:
645:
550:
341:{\displaystyle G\times G\ni (g,h)\mapsto gh\in G.}
340:
261:
3367:. The Maurer–Cartan form on the Lie group
936:{\displaystyle \omega _{g}(v)=(L_{g^{-1}})_{*}v.}
3428:{\displaystyle d\omega +\omega \wedge \omega =0}
2089:consisting of left-invariant vector fields, and
4045:{\displaystyle T_{gh}G{\text{ if }}X\in T_{h}G}
1473:where the bracket on the left-hand side is the
122:which is a linear mapping of the tangent space
3387:}, then this Cartan connection is an ordinary
1866:{\displaystyle X(\omega (Y))=Y(\omega (X))=0,}
731:{\displaystyle (L_{g})_{*}:T_{h}G\to T_{gh}G.}
3850:is the Maurer–Cartan form on the group
3377:for this principal bundle. In particular, if
8:
2402:{\displaystyle =\sum _{k}{c_{ij}}^{k}E_{k}.}
1535:
1525:
71:, and bears his name together with that of
3555:{\displaystyle d\theta +{\frac {1}{2}}=0.}
2427:of the Lie algebra (relative to the basis
2043:{\displaystyle d\omega +{\frac {1}{2}}=0.}
88:. The Lie algebra is identified with the
4085:
4033:
4018:
4006:
4000:
3976:
3966:
3957:
3820:
3810:
3797:
3781:
3765:
3757:
3735:
3729:
3662:
3657:
3644:
3619:
3613:
3599:, then they are related by an element of
3521:
3510:
3399:
3245:
3223:
3186:
3164:
3136:
3126:
3116:
3111:
3104:
3085:
3072:
3057:
3027:
3014:
2992:
2982:
2972:
2967:
2954:
2940:
2936:
2932:
2926:
2901:
2891:
2876:
2817:
2804:
2794:
2784:
2779:
2769:
2755:
2743:
2734:
2699:
2689:
2684:
2674:
2664:
2659:
2645:
2633:
2623:
2618:
2602:
2589:
2579:
2569:
2564:
2557:
2535:
2522:
2506:
2487:
2474:
2461:
2452:
2390:
2380:
2370:
2365:
2358:
2342:
2329:
2320:
2266:
2244:
2234:
2222:
2009:
1998:
1884:
1807:
1578:
1528:
1513:
1500:
1499:
1497:
1398:
1286:
1271:
1259:
1249:
1244:
1220:
1211:
1192:
1171:
1162:
1161:
1149:
1134:
1125:
1119:
1023:
1014:
1001:
995:
921:
906:
901:
876:
870:
782:
772:
763:
713:
697:
684:
674:
665:
616:
604:
585:
579:
525:
291:
247:
224:
209:
204:
180:
172:along the left-translation in the group:
384:where one was interested in a notion of
3936:
2852:This equation is also often called the
1036:{\displaystyle \omega _{g}=g^{-1}\,dg.}
116:is thus a one-form defined globally on
3912:into the homogeneous space, such that
3902:. In other words, there is locally a
3473:of the Maurer–Cartan form along
429:abstractly characterized by having a
7:
2726:
2214:
2145:is a Maurer–Cartan frame, and
1355:is a left-invariant vector field on
501:be the tangent space of a Lie group
3880:defined on open sets in a manifold
3326:is a smooth manifold of dimension
2055:bracket of Lie algebra-valued forms
1501:
1163:
517:acts on itself by left translation
3049:The frame components are given by
2862:, take the exterior derivative of
1224:
1221:
1194:
1138:
1135:
798:
25:
4121:Lectures on differential geometry
1570:are arbitrary vector fields then
1531:left-invariant vector fields on G
417:A principal homogeneous space of
1952:{\displaystyle d\omega (X,Y)+=0}
1060:Characterization as a connection
551:{\displaystyle L:G\times G\to G}
110:. The Maurer–Cartan form
3683:
3463:of the homogeneous space, then
1206:
797:
611:
603:
236:
3973:
3959:
3807:
3790:
3774:
3750:
3677:
3671:
3634:
3628:
3543:
3531:
3254:
3251:
3238:
3229:
3216:
3210:
3201:
3198:
3192:
3176:
3170:
3157:
3148:
3142:
3091:
3065:
3004:
2998:
2913:
2907:
2705:
2655:
2608:
2595:
2544:
2541:
2515:
2512:
2493:
2467:
2348:
2322:
2256:
2250:
2031:
2019:
1940:
1937:
1931:
1922:
1916:
1910:
1904:
1892:
1851:
1848:
1842:
1836:
1827:
1824:
1818:
1812:
1673:
1670:
1658:
1655:
1646:
1643:
1637:
1631:
1622:
1619:
1613:
1607:
1598:
1586:
1457:
1454:
1448:
1439:
1433:
1427:
1421:
1418:
1406:
1403:
1390:are both left-invariant, then
1265:
1237:
1234:
1228:
1158:
918:
894:
888:
882:
779:
765:
706:
681:
667:
656:and this induces a map of the
628:
622:
597:
542:
320:
317:
305:
221:
197:
191:
185:
1:
3605:in each fibre of the bundle:
3351:induces the structure of an
2864:
273:Motivation and interpretation
67:as a basic ingredient of his
3988:{\displaystyle (L_{g})_{*}X}
3943:Introduced by Cartan (1904).
3292:defined on the tautological
2854:Maurer–Cartan equation
1988:Maurer–Cartan equation
1761:is the Lie derivative along
1475:Lie bracket of vector fields
82:associated to the Lie group
4099:. Springer-Verlag, Berlin.
3859:A system of non-degenerate
2153:Maurer–Cartan coframe
1084:. Indeed, it is the unique
1064:If we regard the Lie group
965:by a matrix valued mapping
353:principal homogeneous space
4174:
1990:. It is often written as
1463:{\displaystyle \omega ()=}
431:free and transitive action
276:
3479:defines a non-degenerate
2067:Maurer–Cartan frame
2061:Maurer–Cartan frame
1799:are left-invariant, then
1554:By the definition of the
98:at the identity, denoted
1986:. This is known as the
1078:on the principal bundle
828:Maurer–Cartan form
447:Maurer–Cartan form
36:Maurer–Cartan form
3357:-principal bundle over
3283:method of moving frames
1740:of this function along
1050:of the identity map of
461:integrability condition
370:identical to the group
69:method of moving frames
4046:
3989:
3830:
3706:
3556:
3429:
3277:On a homogeneous space
3264:
3040:
2830:
2715:
2403:
2279:
2180:at the identity. Thus
2044:
1953:
1867:
1715:with the vector field
1683:
1545:
1464:
1302:
1180:
1048:logarithmic derivative
1037:
947:Extrinsic construction
937:
817:
732:
647:
561:such that for a given
552:
482:Intrinsic construction
342:
263:
156:. It is given as the
63:. It was much used by
4158:Differential geometry
4095:R. W. Sharpe (1996).
4065:Cartan, Élie (1904).
4047:
3990:
3831:
3707:
3557:
3430:
3265:
3041:
2831:
2716:
2404:
2280:
2045:
1954:
1868:
1684:
1546:
1465:
1303:
1181:
1038:
981:, then one can write
938:
818:
733:
648:
553:
507:at the identity (its
449:gives an appropriate
343:
264:
49:differential one-form
3999:
3956:
3728:
3715:The differential of
3612:
3509:
3459:. (If working on a
3398:
3337:. The quotient map
3056:
2875:
2733:
2451:
2319:
2221:
1997:
1883:
1806:
1577:
1496:
1397:
1341:on the Lie algebra.
1191:
1118:
1076:principal connection
994:
869:
841:-valued one-form on
762:
664:
578:
524:
290:
179:
27:Mathematical concept
3773:
3670:
2724:so that by duality
2425:structure constants
1556:exterior derivative
1254:
847:defined on vectors
47:is a distinguished
4087:10.24033/asens.538
4042:
3995:gives a vector in
3985:
3925:Darboux derivative
3826:
3753:
3702:
3653:
3552:
3425:
3296:associated with a
3260:
3109:
3036:
2965:
2896:
2826:
2777:
2711:
2562:
2399:
2363:
2275:
2239:
2053:Here denotes the
2040:
1949:
1863:
1787:In particular, if
1679:
1541:
1533:
1460:
1378:. Furthermore, if
1298:
1240:
1176:
1033:
933:
813:
728:
643:
609:
548:
394:homogeneous spaces
382:Erlangen programme
338:
259:
18:Maurer-Cartan form
4125:. Prentice-Hall.
4021:
3529:
3375:Cartan connection
3298:homogeneous space
3100:
2950:
2948:
2887:
2850:
2849:
2765:
2763:
2653:
2553:
2354:
2299:
2298:
2230:
2017:
1773:-valued function
1532:
1274:
1273: where
1174:
738:A left-invariant
608:
16:(Redirected from
4165:
4134:
4124:
4115:Shlomo Sternberg
4110:
4091:
4089:
4071:
4052:
4051:
4049:
4048:
4043:
4038:
4037:
4022:
4019:
4014:
4013:
3994:
3992:
3991:
3986:
3981:
3980:
3971:
3970:
3950:
3944:
3941:
3922:
3911:
3901:
3891:
3885:
3879:
3868:
3864:
3855:
3849:
3835:
3833:
3832:
3827:
3825:
3824:
3815:
3814:
3805:
3804:
3786:
3785:
3772:
3764:
3740:
3739:
3720:
3711:
3709:
3708:
3703:
3669:
3661:
3649:
3648:
3627:
3626:
3604:
3598:
3592:
3586:
3575:
3561:
3559:
3558:
3553:
3530:
3522:
3501:
3488:
3484:
3478:
3468:
3458:
3434:
3432:
3431:
3426:
3386:
3372:
3366:
3356:
3350:
3336:
3325:
3315:
3305:
3294:principal bundle
3291:
3289:
3269:
3267:
3266:
3261:
3250:
3249:
3228:
3227:
3191:
3190:
3169:
3168:
3141:
3140:
3131:
3130:
3125:
3124:
3123:
3108:
3090:
3089:
3077:
3076:
3045:
3043:
3042:
3037:
3032:
3031:
3019:
3018:
2997:
2996:
2987:
2986:
2981:
2980:
2979:
2964:
2949:
2941:
2931:
2930:
2906:
2905:
2895:
2861:
2844:
2835:
2833:
2832:
2827:
2822:
2821:
2809:
2808:
2799:
2798:
2793:
2792:
2791:
2776:
2764:
2756:
2748:
2747:
2727:
2720:
2718:
2717:
2712:
2704:
2703:
2698:
2697:
2696:
2679:
2678:
2673:
2672:
2671:
2654:
2646:
2638:
2637:
2632:
2631:
2630:
2607:
2606:
2594:
2593:
2584:
2583:
2578:
2577:
2576:
2561:
2540:
2539:
2527:
2526:
2511:
2510:
2492:
2491:
2479:
2478:
2466:
2465:
2443:
2437:
2422:
2408:
2406:
2405:
2400:
2395:
2394:
2385:
2384:
2379:
2378:
2377:
2362:
2347:
2346:
2334:
2333:
2311:
2293:
2284:
2282:
2281:
2276:
2271:
2270:
2249:
2248:
2238:
2215:
2211:
2179:
2168:
2150:
2144:
2129:
2105:
2094:
2088:
2077:
2049:
2047:
2046:
2041:
2018:
2010:
1985:
1979:
1973:
1958:
1956:
1955:
1950:
1872:
1870:
1869:
1864:
1798:
1792:
1783:
1772:
1766:
1760:
1745:
1735:
1720:
1714:
1708:
1702:
1688:
1686:
1685:
1680:
1569:
1563:
1550:
1548:
1547:
1542:
1534:
1530:
1518:
1517:
1505:
1504:
1488:
1482:
1469:
1467:
1466:
1461:
1389:
1383:
1377:
1371:
1360:
1354:
1336:
1324:
1307:
1305:
1304:
1299:
1294:
1293:
1275:
1272:
1264:
1263:
1253:
1248:
1227:
1216:
1215:
1185:
1183:
1182:
1177:
1175:
1172:
1167:
1166:
1154:
1153:
1141:
1130:
1129:
1108:
1102:
1098:
1083:
1072:principal bundle
1069:
1055:
1042:
1040:
1039:
1034:
1022:
1021:
1006:
1005:
986:
980:
964:
956:
942:
940:
939:
934:
926:
925:
916:
915:
914:
913:
881:
880:
861:
846:
840:
834:
822:
820:
819:
814:
787:
786:
777:
776:
754:
747:
737:
735:
734:
729:
721:
720:
702:
701:
689:
688:
679:
678:
652:
650:
649:
644:
621:
620:
610:
606:
590:
589:
570:
557:
555:
554:
549:
516:
506:
500:
472:
458:
444:
438:
428:
422:
413:
404:
375:
369:
360:
347:
345:
344:
339:
279:Lie group action
268:
266:
265:
260:
252:
251:
229:
228:
219:
218:
217:
216:
171:
155:
143:
133:
121:
115:
109:
97:
87:
62:
56:
46:
21:
4173:
4172:
4168:
4167:
4166:
4164:
4163:
4162:
4138:
4137:
4113:
4107:
4094:
4069:
4064:
4061:
4056:
4055:
4029:
4002:
3997:
3996:
3972:
3962:
3954:
3953:
3951:
3947:
3942:
3938:
3933:
3921:
3913:
3907:
3893:
3887:
3881:
3878:
3870:
3866:
3860:
3851:
3848:
3840:
3816:
3806:
3793:
3777:
3731:
3726:
3725:
3716:
3640:
3615:
3610:
3609:
3600:
3594:
3588:
3585:
3577:
3574:
3566:
3507:
3506:
3490:
3486:
3480:
3474:
3464:
3442:
3396:
3395:
3389:connection form
3378:
3368:
3358:
3352:
3338:
3327:
3317:
3311:
3308:closed subgroup
3301:
3287:
3286:
3279:
3241:
3219:
3182:
3160:
3132:
3112:
3110:
3081:
3068:
3054:
3053:
3023:
3010:
2988:
2968:
2966:
2922:
2897:
2873:
2872:
2857:
2842:
2813:
2800:
2780:
2778:
2739:
2731:
2730:
2685:
2683:
2660:
2658:
2619:
2617:
2598:
2585:
2565:
2563:
2531:
2518:
2502:
2483:
2470:
2457:
2449:
2448:
2439:
2436:
2428:
2421:
2413:
2412:The quantities
2386:
2366:
2364:
2338:
2325:
2317:
2316:
2310:
2302:
2291:
2262:
2240:
2219:
2218:
2202:
2193:
2181:
2178:
2170:
2167:
2159:
2146:
2143:
2135:
2132:Kronecker delta
2128:
2119:
2107:
2100:
2099:of sections of
2090:
2083:
2082:of sections of
2076:
2070:
2063:
1995:
1994:
1981:
1975:
1969:
1963:
1881:
1880:
1804:
1803:
1794:
1788:
1774:
1768:
1762:
1747:
1741:
1722:
1716:
1710:
1704:
1693:
1575:
1574:
1565:
1559:
1509:
1494:
1493:
1484:
1478:
1395:
1394:
1385:
1379:
1373:
1372:is constant on
1362:
1356:
1350:
1347:
1330:
1322:
1314:
1282:
1255:
1207:
1189:
1188:
1145:
1121:
1116:
1115:
1104:
1100:
1094:
1085:
1079:
1065:
1062:
1051:
1010:
997:
992:
991:
982:
978:
966:
958:
957:is embedded in
952:
949:
917:
902:
897:
872:
867:
866:
862:by the formula
857:
848:
842:
836:
830:
778:
768:
760:
759:
749:
743:
709:
693:
680:
670:
662:
661:
612:
581:
576:
575:
562:
522:
521:
512:
502:
496:
487:
484:
479:
468:
465:exponential map
454:
440:
434:
424:
418:
409:
396:
390:transformations
371:
365:
356:
288:
287:
281:
275:
243:
220:
205:
200:
177:
176:
167:
161:
160:of a vector in
151:
145:
135:
129:
123:
117:
111:
105:
99:
93:
83:
58:
52:
42:
28:
23:
22:
15:
12:
11:
5:
4171:
4169:
4161:
4160:
4155:
4150:
4140:
4139:
4136:
4135:
4111:
4105:
4092:
4060:
4057:
4054:
4053:
4041:
4036:
4032:
4028:
4025:
4020: if
4017:
4012:
4009:
4005:
3984:
3979:
3975:
3969:
3965:
3961:
3945:
3935:
3934:
3932:
3929:
3917:
3904:diffeomorphism
3874:
3844:
3837:
3836:
3823:
3819:
3813:
3809:
3803:
3800:
3796:
3792:
3789:
3784:
3780:
3776:
3771:
3768:
3763:
3760:
3756:
3752:
3749:
3746:
3743:
3738:
3734:
3713:
3712:
3701:
3698:
3695:
3692:
3689:
3686:
3682:
3679:
3676:
3673:
3668:
3665:
3660:
3656:
3652:
3647:
3643:
3639:
3636:
3633:
3630:
3625:
3622:
3618:
3581:
3570:
3563:
3562:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3528:
3525:
3520:
3517:
3514:
3436:
3435:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3391:, and we have
3373:yields a flat
3278:
3275:
3271:
3270:
3259:
3256:
3253:
3248:
3244:
3240:
3237:
3234:
3231:
3226:
3222:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3189:
3185:
3181:
3178:
3175:
3172:
3167:
3163:
3159:
3156:
3153:
3150:
3147:
3144:
3139:
3135:
3129:
3122:
3119:
3115:
3107:
3103:
3099:
3096:
3093:
3088:
3084:
3080:
3075:
3071:
3067:
3064:
3061:
3047:
3046:
3035:
3030:
3026:
3022:
3017:
3013:
3009:
3006:
3003:
3000:
2995:
2991:
2985:
2978:
2975:
2971:
2963:
2960:
2957:
2953:
2947:
2944:
2939:
2935:
2929:
2925:
2921:
2918:
2915:
2912:
2909:
2904:
2900:
2894:
2890:
2886:
2883:
2880:
2848:
2847:
2838:
2836:
2825:
2820:
2816:
2812:
2807:
2803:
2797:
2790:
2787:
2783:
2775:
2772:
2768:
2762:
2759:
2754:
2751:
2746:
2742:
2738:
2722:
2721:
2710:
2707:
2702:
2695:
2692:
2688:
2682:
2677:
2670:
2667:
2663:
2657:
2652:
2649:
2644:
2641:
2636:
2629:
2626:
2622:
2616:
2613:
2610:
2605:
2601:
2597:
2592:
2588:
2582:
2575:
2572:
2568:
2560:
2556:
2552:
2549:
2546:
2543:
2538:
2534:
2530:
2525:
2521:
2517:
2514:
2509:
2505:
2501:
2498:
2495:
2490:
2486:
2482:
2477:
2473:
2469:
2464:
2460:
2456:
2432:
2417:
2410:
2409:
2398:
2393:
2389:
2383:
2376:
2373:
2369:
2361:
2357:
2353:
2350:
2345:
2341:
2337:
2332:
2328:
2324:
2306:
2297:
2296:
2287:
2285:
2274:
2269:
2265:
2261:
2258:
2255:
2252:
2247:
2243:
2237:
2233:
2229:
2226:
2198:
2189:
2174:
2163:
2139:
2124:
2115:
2074:
2062:
2059:
2051:
2050:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2016:
2013:
2008:
2005:
2002:
1965:
1960:
1959:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1874:
1873:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1738:Lie derivative
1690:
1689:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1552:
1551:
1540:
1537:
1527:
1524:
1521:
1516:
1512:
1508:
1503:
1471:
1470:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1346:
1343:
1339:adjoint action
1318:
1311:
1310:
1309:
1308:
1297:
1292:
1289:
1285:
1281:
1278:
1270:
1267:
1262:
1258:
1252:
1247:
1243:
1239:
1236:
1233:
1230:
1226:
1223:
1219:
1214:
1210:
1205:
1202:
1199:
1196:
1186:
1170:
1165:
1160:
1157:
1152:
1148:
1144:
1140:
1137:
1133:
1128:
1124:
1090:
1061:
1058:
1044:
1043:
1032:
1029:
1026:
1020:
1017:
1013:
1009:
1004:
1000:
987:explicitly as
974:
948:
945:
944:
943:
932:
929:
924:
920:
912:
909:
905:
900:
896:
893:
890:
887:
884:
879:
875:
853:
824:
823:
812:
809:
806:
803:
800:
796:
793:
790:
785:
781:
775:
771:
767:
727:
724:
719:
716:
712:
708:
705:
700:
696:
692:
687:
683:
677:
673:
669:
658:tangent bundle
654:
653:
642:
639:
636:
633:
630:
627:
624:
619:
615:
602:
599:
596:
593:
588:
584:
559:
558:
547:
544:
541:
538:
535:
532:
529:
492:
483:
480:
478:
475:
459:satisfying an
423:is a manifold
349:
348:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
274:
271:
270:
269:
258:
255:
250:
246:
242:
239:
235:
232:
227:
223:
215:
212:
208:
203:
199:
196:
193:
190:
187:
184:
163:
147:
125:
101:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4170:
4159:
4156:
4154:
4151:
4149:
4146:
4145:
4143:
4132:
4128:
4123:
4122:
4116:
4112:
4108:
4106:0-387-94732-9
4102:
4098:
4093:
4088:
4083:
4079:
4075:
4068:
4063:
4062:
4058:
4039:
4034:
4030:
4026:
4023:
4015:
4010:
4007:
4003:
3982:
3977:
3967:
3963:
3949:
3946:
3940:
3937:
3930:
3928:
3926:
3920:
3916:
3910:
3905:
3900:
3896:
3890:
3884:
3877:
3873:
3863:
3857:
3854:
3847:
3843:
3821:
3817:
3811:
3801:
3798:
3794:
3787:
3782:
3778:
3769:
3766:
3761:
3758:
3754:
3747:
3744:
3741:
3736:
3732:
3724:
3723:
3722:
3719:
3699:
3696:
3693:
3690:
3687:
3684:
3680:
3674:
3666:
3663:
3658:
3654:
3650:
3645:
3641:
3637:
3631:
3623:
3620:
3616:
3608:
3607:
3606:
3603:
3597:
3591:
3584:
3580:
3573:
3569:
3565:Moreover, if
3549:
3546:
3540:
3537:
3534:
3526:
3523:
3518:
3515:
3512:
3505:
3504:
3503:
3500:
3497:
3493:
3483:
3477:
3472:
3467:
3462:
3457:
3453:
3449:
3445:
3439:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3394:
3393:
3392:
3390:
3385:
3381:
3376:
3371:
3365:
3361:
3355:
3349:
3345:
3341:
3335:
3331:
3324:
3320:
3314:
3309:
3304:
3299:
3295:
3284:
3276:
3274:
3257:
3246:
3242:
3235:
3232:
3224:
3220:
3213:
3207:
3204:
3195:
3187:
3183:
3179:
3173:
3165:
3161:
3154:
3151:
3145:
3137:
3133:
3127:
3120:
3117:
3113:
3105:
3101:
3097:
3094:
3086:
3082:
3078:
3073:
3069:
3062:
3059:
3052:
3051:
3050:
3033:
3028:
3024:
3020:
3015:
3011:
3007:
3001:
2993:
2989:
2983:
2976:
2973:
2969:
2961:
2958:
2955:
2951:
2945:
2942:
2937:
2933:
2927:
2923:
2919:
2916:
2910:
2902:
2898:
2892:
2888:
2884:
2881:
2878:
2871:
2870:
2869:
2867:
2866:
2860:
2855:
2846:
2839:
2837:
2823:
2818:
2814:
2810:
2805:
2801:
2795:
2788:
2785:
2781:
2773:
2770:
2766:
2760:
2757:
2752:
2749:
2744:
2740:
2736:
2729:
2728:
2725:
2708:
2700:
2693:
2690:
2686:
2680:
2675:
2668:
2665:
2661:
2650:
2647:
2642:
2639:
2634:
2627:
2624:
2620:
2614:
2611:
2603:
2599:
2590:
2586:
2580:
2573:
2570:
2566:
2558:
2554:
2550:
2547:
2536:
2532:
2528:
2523:
2519:
2507:
2503:
2499:
2496:
2488:
2484:
2480:
2475:
2471:
2462:
2458:
2454:
2447:
2446:
2445:
2442:
2435:
2431:
2426:
2420:
2416:
2396:
2391:
2387:
2381:
2374:
2371:
2367:
2359:
2355:
2351:
2343:
2339:
2335:
2330:
2326:
2315:
2314:
2313:
2312:are given by
2309:
2305:
2295:
2288:
2286:
2272:
2267:
2263:
2259:
2253:
2245:
2241:
2235:
2231:
2227:
2224:
2217:
2216:
2213:
2210:
2206:
2201:
2197:
2192:
2188:
2184:
2177:
2173:
2166:
2162:
2156:
2154:
2149:
2142:
2138:
2133:
2127:
2123:
2118:
2114:
2110:
2104:
2098:
2093:
2087:
2081:
2073:
2068:
2060:
2058:
2056:
2037:
2034:
2028:
2025:
2022:
2014:
2011:
2006:
2003:
2000:
1993:
1992:
1991:
1989:
1984:
1978:
1972:
1968:
1946:
1943:
1934:
1928:
1925:
1919:
1913:
1907:
1901:
1898:
1895:
1889:
1886:
1879:
1878:
1877:
1860:
1857:
1854:
1845:
1839:
1833:
1830:
1821:
1815:
1809:
1802:
1801:
1800:
1797:
1791:
1785:
1781:
1777:
1771:
1765:
1758:
1754:
1750:
1744:
1739:
1733:
1729:
1725:
1719:
1713:
1707:
1700:
1696:
1676:
1667:
1664:
1661:
1652:
1649:
1640:
1634:
1628:
1625:
1616:
1610:
1604:
1601:
1595:
1592:
1589:
1583:
1580:
1573:
1572:
1571:
1568:
1562:
1557:
1538:
1522:
1519:
1514:
1510:
1506:
1492:
1491:
1490:
1487:
1481:
1476:
1451:
1445:
1442:
1436:
1430:
1424:
1415:
1412:
1409:
1400:
1393:
1392:
1391:
1388:
1382:
1376:
1369:
1365:
1359:
1353:
1344:
1342:
1340:
1334:
1328:
1321:
1317:
1295:
1290:
1287:
1283:
1279:
1276:
1268:
1260:
1256:
1250:
1245:
1241:
1231:
1217:
1212:
1208:
1203:
1200:
1197:
1187:
1168:
1155:
1150:
1146:
1142:
1131:
1126:
1122:
1114:
1113:
1112:
1111:
1110:
1107:
1097:
1093:
1088:
1082:
1077:
1073:
1068:
1059:
1057:
1054:
1049:
1030:
1027:
1024:
1018:
1015:
1011:
1007:
1002:
998:
990:
989:
988:
985:
977:
973:
969:
962:
955:
946:
930:
927:
922:
910:
907:
903:
898:
891:
885:
877:
873:
865:
864:
863:
860:
856:
851:
845:
839:
833:
829:
810:
807:
804:
801:
794:
791:
788:
783:
773:
769:
758:
757:
756:
753:
746:
742:is a section
741:
725:
722:
717:
714:
710:
703:
698:
694:
690:
685:
675:
671:
659:
640:
637:
634:
631:
625:
617:
613:
600:
594:
591:
586:
582:
574:
573:
572:
569:
565:
545:
539:
536:
533:
530:
527:
520:
519:
518:
515:
510:
505:
499:
495:
490:
481:
476:
474:
471:
466:
462:
457:
452:
451:infinitesimal
448:
443:
437:
432:
427:
421:
415:
412:
408:
403:
399:
395:
391:
387:
383:
379:
374:
368:
364:
361:. That is, a
359:
354:
335:
332:
329:
326:
323:
314:
311:
308:
302:
299:
296:
293:
286:
285:
284:
280:
272:
256:
253:
248:
244:
240:
237:
233:
230:
225:
213:
210:
206:
201:
194:
188:
182:
175:
174:
173:
170:
166:
159:
154:
150:
142:
138:
132:
128:
120:
114:
108:
104:
96:
91:
90:tangent space
86:
81:
76:
74:
73:Ludwig Maurer
70:
66:
61:
55:
50:
45:
41:
37:
33:
19:
4120:
4096:
4077:
4073:
3948:
3939:
3918:
3914:
3908:
3898:
3894:
3888:
3882:
3875:
3871:
3861:
3858:
3852:
3845:
3841:
3838:
3717:
3714:
3601:
3595:
3589:
3582:
3578:
3571:
3567:
3564:
3498:
3495:
3491:
3481:
3475:
3465:
3455:
3451:
3447:
3443:
3440:
3437:
3383:
3379:
3369:
3363:
3359:
3353:
3347:
3343:
3339:
3333:
3329:
3322:
3318:
3312:
3302:
3280:
3272:
3048:
2863:
2858:
2853:
2851:
2840:
2723:
2440:
2433:
2429:
2418:
2414:
2411:
2307:
2303:
2300:
2289:
2208:
2204:
2199:
2195:
2190:
2186:
2182:
2175:
2171:
2164:
2160:
2157:
2152:
2147:
2140:
2136:
2125:
2121:
2116:
2112:
2108:
2102:
2091:
2085:
2071:
2066:
2064:
2052:
1987:
1982:
1976:
1970:
1966:
1961:
1875:
1795:
1789:
1786:
1779:
1775:
1769:
1763:
1756:
1752:
1748:
1746:. Similarly
1742:
1731:
1727:
1723:
1717:
1711:
1705:
1698:
1694:
1691:
1566:
1560:
1553:
1485:
1479:
1472:
1386:
1380:
1374:
1367:
1363:
1357:
1351:
1348:
1332:
1319:
1315:
1312:
1105:
1095:
1091:
1086:
1080:
1066:
1063:
1052:
1045:
983:
975:
971:
967:
960:
953:
950:
858:
854:
849:
843:
837:
831:
827:
825:
751:
744:
740:vector field
655:
567:
563:
560:
513:
503:
497:
493:
488:
485:
477:Construction
469:
455:
450:
446:
441:
435:
425:
419:
416:
410:
401:
397:
372:
366:
357:
350:
282:
168:
164:
152:
148:
140:
136:
130:
126:
118:
112:
106:
102:
94:
84:
77:
59:
53:
43:
35:
29:
4080:: 153–206.
3461:submanifold
1109:satisfying
755:such that
660:to itself:
509:Lie algebra
378:Felix Klein
158:pushforward
80:Lie algebra
65:Élie Cartan
32:mathematics
4148:Lie groups
4142:Categories
4059:References
3952:Subtlety:
2106:such that
2097:dual basis
1345:Properties
277:See also:
4153:Equations
4027:∈
3978:∗
3818:ω
3812:∗
3779:θ
3767:−
3748:
3733:θ
3694:∩
3688:∈
3664:−
3651:∘
3541:θ
3535:θ
3516:θ
3417:ω
3414:∧
3411:ω
3405:ω
3236:ω
3214:ω
3208:−
3155:−
3102:∑
3098:−
3063:ω
3025:θ
3021:∧
3012:θ
3008:⊗
2952:∑
2938:−
2924:θ
2917:⊗
2889:∑
2882:ω
2815:θ
2811:∧
2802:θ
2767:∑
2753:−
2741:θ
2681:−
2643:−
2615:−
2587:θ
2555:∑
2551:−
2504:θ
2500:−
2459:θ
2444:, yields
2356:∑
2264:θ
2260:⊗
2232:∑
2225:ω
2029:ω
2023:ω
2004:ω
1929:ω
1914:ω
1890:ω
1840:ω
1816:ω
1653:ω
1650:−
1635:ω
1626:−
1611:ω
1584:ω
1523:≅
1446:ω
1431:ω
1401:ω
1288:−
1257:ω
1251:∗
1209:ω
1201:∈
1195:∀
1173: and
1159:→
1123:ω
1103:-form on
1016:−
999:ω
923:∗
908:−
874:ω
805:∈
799:∀
784:∗
707:→
686:∗
598:→
543:→
537:×
330:∈
321:↦
303:∋
297:×
241:∈
226:∗
211:−
183:ω
40:Lie group
3865:-valued
3485:-valued
3471:pullback
3446: :
2423:are the
2134:. Then
1327:pullback
571:we have
386:symmetry
363:manifold
134:at each
4131:64-7993
3869:-forms
3316:, then
2095:be the
2069:. Let
1767:of the
1736:is the
1703:is the
1361:, then
1337:is the
1325:is the
1099:valued
445:. The
4129:
4103:
3839:where
3489:-form
3332:− dim
3300:. If
2158:Since
2130:, the
1721:, and
1313:where
38:for a
34:, the
4070:(PDF)
3931:Notes
3306:is a
3290:-form
2151:is a
2080:basis
2078:be a
1692:Here
1558:, if
1070:as a
835:is a
607:where
407:coset
144:into
4127:LCCN
4101:ISBN
3593:and
3576:and
3328:dim
2207:) ∈
2194:) =
2120:) =
1980:and
1793:and
1564:and
1384:and
826:The
486:Let
4082:doi
3906:of
3382:= {
3310:of
2865:(1)
1876:so
1349:If
1331:Ad(
1089:= T
959:GL(
951:If
852:∈ T
748:of
511:).
491:≅ T
439:on
433:of
380:'s
355:of
92:of
51:on
30:In
4144::
4078:21
4076:.
4072:.
3927:.
3856:.
3745:Ad
3550:0.
3494:=
3454:→
3342:→
2868::
2419:ij
2155:.
2057:.
2038:0.
1784:.
1759:))
1734:))
1056:.
976:ij
970:=(
566:∈
473:.
414:.
411:eH
139:∈
75:.
4133:.
4109:.
4090:.
4084::
4040:G
4035:h
4031:T
4024:X
4016:G
4011:h
4008:g
4004:T
3983:X
3974:)
3968:g
3964:L
3960:(
3919:U
3915:θ
3909:M
3899:H
3897:/
3895:G
3889:M
3883:M
3876:U
3872:θ
3867:1
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3853:H
3846:H
3842:ω
3822:H
3808:)
3802:V
3799:U
3795:h
3791:(
3788:+
3783:U
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3755:h
3751:(
3742:=
3737:V
3718:h
3700:.
3697:V
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3685:x
3681:,
3678:)
3675:x
3672:(
3667:1
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3655:s
3646:V
3642:s
3638:=
3635:)
3632:x
3629:(
3624:V
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3617:h
3602:H
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3579:s
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3568:s
3547:=
3544:]
3538:,
3532:[
3527:2
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3450:/
3448:G
3444:s
3423:0
3420:=
3408:+
3402:d
3384:e
3380:H
3370:G
3364:H
3362:/
3360:G
3354:H
3348:H
3346:/
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3340:G
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3330:G
3323:H
3321:/
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3313:G
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3255:]
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3247:k
3243:E
3239:(
3233:,
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3217:(
3211:[
3205:=
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3196:e
3193:(
3188:k
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3180:,
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3174:e
3171:(
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3162:E
3158:[
3152:=
3149:)
3146:e
3143:(
3138:i
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3128:i
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3114:c
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3087:k
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3066:(
3060:d
3034:.
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2999:(
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2908:(
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2824:.
2819:k
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2750:=
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2706:)
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2542:]
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2513:(
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2497:=
2494:)
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2468:(
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2273:.
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2254:e
2251:(
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2203:(
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2196:E
2191:i
2187:E
2185:(
2183:ω
2176:i
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2122:δ
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2113:E
2111:(
2109:θ
2103:G
2101:T
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2035:=
2032:]
2026:,
2020:[
2015:2
2012:1
2007:+
2001:d
1983:Y
1977:X
1971:G
1967:e
1964:T
1947:0
1944:=
1941:]
1938:)
1935:Y
1932:(
1926:,
1923:)
1920:X
1917:(
1911:[
1908:+
1905:)
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1843:(
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1831:=
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1813:(
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1796:Y
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1782:)
1780:X
1778:(
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1770:g
1764:Y
1757:X
1755:(
1753:ω
1751:(
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1732:Y
1730:(
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1726:(
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1712:ω
1706:g
1701:)
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1697:(
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1677:.
1674:)
1671:]
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1665:,
1662:X
1659:[
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1638:(
1632:(
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1587:(
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1567:Y
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1539:.
1536:}
1526:{
1520:G
1515:e
1511:T
1507:=
1502:g
1486:g
1480:g
1458:]
1455:)
1452:Y
1449:(
1443:,
1440:)
1437:X
1434:(
1428:[
1425:=
1422:)
1419:]
1416:Y
1413:,
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1407:[
1404:(
1387:Y
1381:X
1375:G
1370:)
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1366:(
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1335:)
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1198:g
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1143::
1139:d
1136:i
1132:=
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1101:1
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668:(
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