2097:
While any transitive Lie groupoid is isomorphic to some gauge groupoid, not all transitive Lie algebroids are Atiyah algebroids of some principal bundle. Integrability is the crucial property to distinguish the two concepts: a transitive Lie algebroid is integrable if and only if it is isomorphic to
2396:
269:
897:
483:
2264:
1832:
687:
2059:
1383:
1338:
802:
2502:
1911:
1600:
2440:
2142:
713:
1280:
1794:
405:
932:
1070:
1689:
1239:
2196:
2169:
1996:
1866:
1766:
1658:
1568:
1210:
1169:
1013:
1630:
555:
373:
2269:
650:
532:
2083:
2021:
1969:
1933:
1738:
1716:
1540:
1505:
1483:
1457:
1435:
1409:
1300:
1139:
1114:
1092:
1042:
986:
957:
822:
761:
739:
623:
601:
579:
506:
347:
327:
190:
160:
126:
104:
79:
56:
2090:
Note that these two properties are independent. Integrable Lie algebroids does not need to be transitive; conversely, transitive Lie algebroids (often called
827:
17:
1508:-invariant vertical vector fields. In algebraic or analytic contexts, it is often convenient to view the Atiyah sequence as an exact sequence of
2647:
2628:
199:
2584:
Janusz
Grabowski; Alexei Kotov & Norbert Poncin (2011), "Geometric structures encoded in the lie structure of an Atiyah algebroid",
413:
2201:
626:
350:
1801:
656:
2027:
1348:
1306:
770:
282:
285:. It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in
1873:
2445:
1575:
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2676:
2112:
694:
164:
1244:
2408:
1509:
290:
2522:
1773:
2666:
2593:
378:
2657:
Tom
Mestdag & Bavo Langerock (2005), "A Lie algebroid framework for non-holonomic systems",
903:
1049:
2643:
2624:
2544:
1695:
1665:
1215:
2684:
2603:
2570:
2534:
2175:
2148:
1975:
1845:
1745:
1637:
1547:
1178:
1148:
992:
279:
36:
1609:
537:
355:
2107:
509:
16:"Atiyah sequence" redirects here. For the spectral sequence of Atiyah and Hirzebruch, see
2680:
632:
514:
2068:
2006:
1954:
1918:
1723:
1701:
1525:
1490:
1468:
1442:
1420:
1394:
1285:
1124:
1099:
1077:
1027:
971:
942:
807:
765:
746:
724:
608:
586:
564:
491:
332:
312:
275:
175:
168:
145:
111:
89:
64:
41:
2575:
2539:
2699:
2688:
1142:
134:
1838:
1016:
307:
286:
138:
1461:
1413:
717:
24:
1390:
of the vector bundles. More precisely, the sections of the Atiyah algebroid of
2607:
2548:
2198:. Similarly, the curvatures of such connections correspond to the two forms
629:
on the vector bundles in this sequence. Moreover, since the vertical bundle
130:
2561:
Michael F. Atiyah (1957), "Complex analytic connections in fibre bundles",
2391:{\displaystyle \Omega _{\sigma }(X,Y):=_{A}-\sigma (_{{\mathfrak {X}}(M)})}
83:
1302:-invariant. Last, the kernel of the anchor is isomorphic precisely to
2671:
443:
264:{\displaystyle 0\to P\times _{G}{\mathfrak {g}}\to TP/G\to TM\to 0.}
892:{\displaystyle 0\to P\times _{G}{\mathfrak {g}}\to TP/G\to TM\to 0}
278:, who introduced the construction to study the existence theory of
2612:
2598:
478:{\displaystyle 0\to VP\to TP{\xrightarrow {d\pi }}\pi ^{*}TM\to 0}
2642:, London Mathematical Society lecture notes, vol. 213, CUP,
2623:, London Mathematical Society lecture notes, vol. 124, CUP,
1464:, which is an extension of the Lie algebra of vector fields on
2172:
are in bijective correspondence with principal connections on
2024:
and its isotropy Lie algebra bundle is the associated bundle
1797:
of the action at an arbitrary point, is the action algebroid
1355:
824:
of the exact sequence above yields a short exact sequence:
2621:
Lie groupoids and Lie algebroids in differential geometry
1095:, with source and target given by the two projections of
2442:
of principal bundles induces a Lie algebroid morphism
1045:, and whose morphisms are elements of the quotient of
2448:
2411:
2272:
2259:{\displaystyle \Omega _{\sigma }\in \Omega ^{2}(M,P)}
2204:
2178:
2151:
2115:
2071:
2030:
2009:
1978:
1957:
1921:
1876:
1848:
1804:
1776:
1748:
1726:
1704:
1668:
1640:
1612:
1578:
1550:
1528:
1493:
1471:
1445:
1423:
1397:
1351:
1344:
The Atiyah sequence yields a short exact sequence of
1309:
1288:
1247:
1218:
1181:
1151:
1127:
1102:
1080:
1052:
1030:
995:
974:
945:
906:
830:
810:
773:
749:
727:
697:
659:
635:
611:
589:
567:
540:
517:
494:
416:
381:
358:
335:
315:
202:
178:
148:
114:
92:
67:
44:
2640:General theory of lie groupoids and lie algebroids
2496:
2434:
2390:
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2190:
2163:
2136:
2077:
2053:
2015:
1990:
1963:
1927:
1905:
1860:
1826:
1788:
1760:
1732:
1710:
1683:
1652:
1624:
1594:
1562:
1534:
1499:
1477:
1451:
1429:
1403:
1377:
1332:
1294:
1274:
1233:
1204:
1163:
1133:
1108:
1086:
1064:
1036:
1007:
980:
951:
926:
891:
816:
796:
755:
733:
707:
681:
644:
617:
595:
573:
549:
526:
500:
477:
399:
367:
341:
321:
263:
184:
154:
120:
98:
73:
50:
2527:Transactions of the American Mathematical Society
2523:"Complex analytic connections in fibre bundles"
2098:the Atiyah algebroid of some principal bundle.
1741:, the Atiyah algebroid of the principal bundle
2002:Transitive (so its unique orbit is the entire
2144:of the Atiyah sequence of a principal bundle
8:
1827:{\displaystyle {\mathfrak {h}}\times M\to M}
1619:
1613:
682:{\displaystyle P\times {\mathfrak {g}}\to P}
2054:{\displaystyle P\times _{G}{\mathfrak {g}}}
1914:(sometimes also called Atiyah algebroid of
1378:{\displaystyle {\mathcal {C}}^{\infty }(M)}
1333:{\displaystyle P\times _{G}{\mathfrak {g}}}
797:{\displaystyle P\times _{G}{\mathfrak {g}}}
652:is isomorphic to the trivial vector bundle
163:. Explicitly, it is given by the following
2504:between the respective Atiyah algebroids.
1769:, with structure group the isotropy group
2670:
2597:
2574:
2538:
2481:
2464:
2447:
2410:
2368:
2367:
2366:
2335:
2277:
2271:
2244:
2243:
2222:
2209:
2203:
2177:
2150:
2114:
2070:
2045:
2044:
2038:
2029:
2008:
1977:
1956:
1920:
1877:
1875:
1847:
1806:
1805:
1803:
1775:
1747:
1725:
1703:
1667:
1639:
1611:
1580:
1579:
1577:
1549:
1527:
1492:
1470:
1444:
1422:
1396:
1360:
1354:
1353:
1350:
1324:
1323:
1317:
1308:
1287:
1246:
1217:
1194:
1180:
1150:
1126:
1101:
1079:
1051:
1029:
994:
973:
944:
910:
905:
866:
851:
850:
844:
829:
809:
788:
787:
781:
772:
748:
726:
699:
698:
696:
667:
666:
658:
634:
610:
588:
566:
539:
516:
493:
457:
438:
415:
380:
357:
334:
314:
238:
223:
222:
216:
201:
177:
147:
113:
91:
66:
43:
18:Atiyah–Hirzebruch spectral sequence
2513:
1906:{\displaystyle \mathrm {Der} (E)\to M}
1605:The Atiyah algebroid of the principal
1521:The Atiyah algebroid of the principal
2064:Integrable (to the gauge groupoid of
1512:of local sections of vector bundles.
7:
2497:{\displaystyle d\phi :TP/G\to TP/G'}
1950:The Atiyah algebroid of a principal
1595:{\displaystyle {\mathfrak {g}}\to *}
2369:
2245:
2046:
1807:
1581:
1325:
852:
789:
700:
668:
224:
2274:
2219:
2206:
2094:) are not necessarily integrable.
1884:
1881:
1878:
1361:
14:
2576:10.1090/s0002-9947-1957-0086359-5
2540:10.1090/S0002-9947-1957-0086359-5
804:. In conclusion, the quotient by
1869:is the general linear algebroid
1386:-modules by taking the space of
407:defines a short exact sequence:
2137:{\displaystyle \sigma :TM\to A}
742:, its quotient by the diagonal
708:{\displaystyle {\mathfrak {g}}}
2472:
2421:
2385:
2380:
2374:
2363:
2350:
2347:
2332:
2328:
2322:
2313:
2307:
2301:
2295:
2283:
2253:
2250:
2240:
2228:
2182:
2155:
2128:
1982:
1946:Transitivity and integrability
1897:
1894:
1888:
1852:
1818:
1752:
1675:
1644:
1586:
1554:
1372:
1366:
1275:{\displaystyle d\pi :TP\to TM}
1263:
1222:
1155:
1023:, whose objects are points of
999:
883:
874:
857:
834:
673:
469:
429:
420:
391:
255:
246:
229:
206:
1:
2435:{\displaystyle \phi :P\to P'}
1241:is given by the differential
1837:The Atiyah algebroid of the
1789:{\displaystyle H\subseteq G}
1438:-invariant vector fields on
400:{\displaystyle \pi :P\to M}
2722:
2689:10.1088/0305-4470/38/5/011
2102:Relations with connections
1073:by the diagonal action of
967:Recall that any principal
927:{\displaystyle P/G\cong M}
15:
2638:Kirill Mackenzie (2005),
2619:Kirill Mackenzie (1987),
2608:10.1007/s00031-011-9126-9
2092:abstract Atiyah sequences
1661:is the tangent algebroid
1065:{\displaystyle P\times P}
604:-bundle, then the group
2563:Trans. Amer. Math. Soc.
1684:{\displaystyle TM\to M}
1234:{\displaystyle A\to TM}
1212:, while the anchor map
1172:of its gauge groupoid.
899:of vector bundles over
488:of vector bundles over
2659:J. Phys. A: Math. Gen.
2521:Atiyah, M. F. (1957).
2498:
2436:
2392:
2260:
2192:
2191:{\displaystyle P\to M}
2165:
2164:{\displaystyle P\to M}
2138:
2079:
2055:
2017:
1992:
1991:{\displaystyle P\to M}
1965:
1929:
1907:
1862:
1861:{\displaystyle E\to M}
1828:
1790:
1762:
1761:{\displaystyle G\to M}
1734:
1712:
1685:
1654:
1653:{\displaystyle M\to M}
1626:
1596:
1564:
1563:{\displaystyle G\to *}
1536:
1501:
1479:
1453:
1431:
1405:
1379:
1334:
1296:
1276:
1235:
1206:
1205:{\displaystyle A=TP/G}
1165:
1164:{\displaystyle A\to M}
1135:
1110:
1088:
1066:
1038:
1009:
1008:{\displaystyle P\to M}
982:
953:
935:, which is called the
928:
893:
818:
798:
757:
735:
709:
683:
646:
619:
597:
575:
551:
528:
502:
479:
401:
369:
343:
323:
265:
186:
156:
122:
100:
75:
52:
2586:Transformation Groups
2499:
2437:
2393:
2261:
2193:
2166:
2139:
2080:
2056:
2018:
1993:
1966:
1930:
1908:
1863:
1829:
1791:
1763:
1735:
1713:
1686:
1655:
1627:
1625:{\displaystyle \{e\}}
1597:
1565:
1537:
1502:
1480:
1454:
1432:
1406:
1380:
1335:
1297:
1277:
1236:
1207:
1166:
1136:
1117:. By definition, the
1111:
1089:
1067:
1039:
1010:
983:
954:
929:
894:
819:
799:
758:
736:
710:
684:
647:
620:
598:
576:
552:
550:{\displaystyle d\pi }
529:
503:
480:
402:
370:
368:{\displaystyle d\pi }
344:
324:
266:
187:
157:
123:
101:
76:
53:
2446:
2409:
2270:
2202:
2176:
2149:
2113:
2069:
2028:
2007:
1976:
1955:
1919:
1874:
1846:
1802:
1774:
1746:
1724:
1702:
1666:
1638:
1610:
1576:
1548:
1526:
1491:
1469:
1443:
1421:
1395:
1349:
1307:
1286:
1245:
1216:
1179:
1149:
1125:
1100:
1078:
1050:
1028:
993:
972:
943:
904:
828:
808:
771:
747:
725:
695:
657:
633:
609:
587:
565:
538:
515:
492:
414:
379:
356:
333:
313:
200:
176:
165:short exact sequence
146:
112:
90:
65:
42:
2681:2005JPhA...38.1097M
1841:of a vector bundle
1571:is the Lie algebra
450:
291:geometric mechanics
2494:
2432:
2388:
2256:
2188:
2161:
2134:
2075:
2051:
2013:
1988:
1961:
1925:
1903:
1858:
1824:
1786:
1758:
1730:
1708:
1681:
1650:
1622:
1592:
1560:
1532:
1497:
1475:
1449:
1427:
1401:
1375:
1330:
1292:
1272:
1231:
1202:
1161:
1131:
1106:
1084:
1062:
1034:
1015:has an associated
1005:
978:
963:As a Lie algebroid
949:
924:
889:
814:
794:
753:
731:
705:
679:
645:{\displaystyle VP}
642:
615:
593:
571:
547:
527:{\displaystyle VP}
524:
498:
475:
397:
375:of the projection
365:
339:
319:
274:It is named after
261:
182:
152:
118:
96:
71:
48:
2649:978-0-521-49928-6
2630:978-0-521-34882-9
2078:{\displaystyle P}
2016:{\displaystyle M}
1964:{\displaystyle G}
1928:{\displaystyle E}
1733:{\displaystyle M}
1711:{\displaystyle G}
1535:{\displaystyle G}
1500:{\displaystyle G}
1478:{\displaystyle M}
1452:{\displaystyle P}
1430:{\displaystyle G}
1404:{\displaystyle P}
1295:{\displaystyle G}
1134:{\displaystyle P}
1109:{\displaystyle M}
1087:{\displaystyle G}
1037:{\displaystyle M}
981:{\displaystyle G}
952:{\displaystyle P}
817:{\displaystyle G}
756:{\displaystyle G}
734:{\displaystyle G}
618:{\displaystyle G}
596:{\displaystyle G}
574:{\displaystyle P}
534:is the kernel of
501:{\displaystyle P}
451:
342:{\displaystyle M}
322:{\displaystyle P}
185:{\displaystyle M}
155:{\displaystyle P}
121:{\displaystyle G}
99:{\displaystyle M}
74:{\displaystyle P}
51:{\displaystyle G}
2713:
2691:
2674:
2652:
2633:
2610:
2601:
2579:
2578:
2553:
2552:
2542:
2518:
2503:
2501:
2500:
2495:
2493:
2485:
2468:
2441:
2439:
2438:
2433:
2431:
2397:
2395:
2394:
2389:
2384:
2383:
2373:
2372:
2340:
2339:
2282:
2281:
2265:
2263:
2262:
2257:
2249:
2248:
2227:
2226:
2214:
2213:
2197:
2195:
2194:
2189:
2170:
2168:
2167:
2162:
2143:
2141:
2140:
2135:
2084:
2082:
2081:
2076:
2060:
2058:
2057:
2052:
2050:
2049:
2043:
2042:
2022:
2020:
2019:
2014:
1997:
1995:
1994:
1989:
1970:
1968:
1967:
1962:
1934:
1932:
1931:
1926:
1912:
1910:
1909:
1904:
1887:
1867:
1865:
1864:
1859:
1833:
1831:
1830:
1825:
1811:
1810:
1795:
1793:
1792:
1787:
1767:
1765:
1764:
1759:
1739:
1737:
1736:
1731:
1717:
1715:
1714:
1709:
1690:
1688:
1687:
1682:
1659:
1657:
1656:
1651:
1631:
1629:
1628:
1623:
1601:
1599:
1598:
1593:
1585:
1584:
1569:
1567:
1566:
1561:
1541:
1539:
1538:
1533:
1506:
1504:
1503:
1498:
1484:
1482:
1481:
1476:
1458:
1456:
1455:
1450:
1436:
1434:
1433:
1428:
1410:
1408:
1407:
1402:
1384:
1382:
1381:
1376:
1365:
1364:
1359:
1358:
1339:
1337:
1336:
1331:
1329:
1328:
1322:
1321:
1301:
1299:
1298:
1293:
1281:
1279:
1278:
1273:
1240:
1238:
1237:
1232:
1211:
1209:
1208:
1203:
1198:
1175:It follows that
1170:
1168:
1167:
1162:
1140:
1138:
1137:
1132:
1119:Atiyah algebroid
1115:
1113:
1112:
1107:
1093:
1091:
1090:
1085:
1071:
1069:
1068:
1063:
1043:
1041:
1040:
1035:
1014:
1012:
1011:
1006:
987:
985:
984:
979:
958:
956:
955:
950:
933:
931:
930:
925:
914:
898:
896:
895:
890:
870:
856:
855:
849:
848:
823:
821:
820:
815:
803:
801:
800:
795:
793:
792:
786:
785:
762:
760:
759:
754:
740:
738:
737:
732:
714:
712:
711:
706:
704:
703:
688:
686:
685:
680:
672:
671:
651:
649:
648:
643:
624:
622:
621:
616:
602:
600:
599:
594:
580:
578:
577:
572:
556:
554:
553:
548:
533:
531:
530:
525:
507:
505:
504:
499:
484:
482:
481:
476:
462:
461:
452:
439:
406:
404:
403:
398:
374:
372:
371:
366:
348:
346:
345:
340:
329:over a manifold
328:
326:
325:
320:
280:complex analytic
270:
268:
267:
262:
242:
228:
227:
221:
220:
191:
189:
188:
183:
161:
159:
158:
153:
127:
125:
124:
119:
105:
103:
102:
97:
80:
78:
77:
72:
57:
55:
54:
49:
29:Atiyah algebroid
2721:
2720:
2716:
2715:
2714:
2712:
2711:
2710:
2696:
2695:
2656:
2650:
2637:
2631:
2618:
2613:arXiv:0905.1226
2611:, available as
2583:
2560:
2557:
2556:
2520:
2519:
2515:
2510:
2486:
2444:
2443:
2424:
2407:
2406:
2403:
2362:
2331:
2273:
2268:
2267:
2218:
2205:
2200:
2199:
2174:
2173:
2147:
2146:
2111:
2110:
2104:
2067:
2066:
2034:
2026:
2025:
2005:
2004:
1974:
1973:
1953:
1952:
1948:
1943:
1917:
1916:
1872:
1871:
1844:
1843:
1800:
1799:
1772:
1771:
1744:
1743:
1722:
1721:
1700:
1699:
1664:
1663:
1636:
1635:
1608:
1607:
1574:
1573:
1546:
1545:
1524:
1523:
1518:
1489:
1488:
1467:
1466:
1441:
1440:
1419:
1418:
1393:
1392:
1352:
1347:
1346:
1313:
1305:
1304:
1284:
1283:
1243:
1242:
1214:
1213:
1177:
1176:
1147:
1146:
1123:
1122:
1098:
1097:
1076:
1075:
1048:
1047:
1026:
1025:
991:
990:
970:
969:
965:
941:
940:
937:Atiyah sequence
902:
901:
840:
826:
825:
806:
805:
777:
769:
768:
745:
744:
723:
722:
693:
692:
655:
654:
631:
630:
607:
606:
585:
584:
582:is a principal
563:
562:
536:
535:
513:
512:
510:vertical bundle
490:
489:
453:
412:
411:
377:
376:
354:
353:
331:
330:
311:
310:
304:
299:
212:
198:
197:
174:
173:
144:
143:
110:
109:
88:
87:
63:
62:
40:
39:
33:Atiyah sequence
21:
12:
11:
5:
2719:
2717:
2709:
2708:
2698:
2697:
2694:
2693:
2654:
2648:
2635:
2629:
2616:
2581:
2555:
2554:
2533:(1): 181–207.
2512:
2511:
2509:
2506:
2492:
2489:
2484:
2480:
2477:
2474:
2471:
2467:
2463:
2460:
2457:
2454:
2451:
2430:
2427:
2423:
2420:
2417:
2414:
2402:
2399:
2387:
2382:
2379:
2376:
2371:
2365:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2338:
2334:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2280:
2276:
2255:
2252:
2247:
2242:
2239:
2236:
2233:
2230:
2225:
2221:
2217:
2212:
2208:
2187:
2184:
2181:
2160:
2157:
2154:
2133:
2130:
2127:
2124:
2121:
2118:
2103:
2100:
2088:
2087:
2074:
2062:
2048:
2041:
2037:
2033:
2012:
1987:
1984:
1981:
1960:
1947:
1944:
1942:
1939:
1938:
1937:
1924:
1902:
1899:
1896:
1893:
1890:
1886:
1883:
1880:
1857:
1854:
1851:
1835:
1823:
1820:
1817:
1814:
1809:
1785:
1782:
1779:
1757:
1754:
1751:
1729:
1707:
1692:
1680:
1677:
1674:
1671:
1649:
1646:
1643:
1621:
1618:
1615:
1603:
1591:
1588:
1583:
1559:
1556:
1553:
1531:
1517:
1514:
1496:
1474:
1448:
1426:
1400:
1374:
1371:
1368:
1363:
1357:
1327:
1320:
1316:
1312:
1291:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1230:
1227:
1224:
1221:
1201:
1197:
1193:
1190:
1187:
1184:
1160:
1157:
1154:
1130:
1105:
1083:
1061:
1058:
1055:
1033:
1021:gauge groupoid
1004:
1001:
998:
977:
964:
961:
948:
923:
920:
917:
913:
909:
888:
885:
882:
879:
876:
873:
869:
865:
862:
859:
854:
847:
843:
839:
836:
833:
813:
791:
784:
780:
776:
766:adjoint bundle
764:action is the
752:
730:
702:
678:
675:
670:
665:
662:
641:
638:
614:
592:
570:
546:
543:
523:
520:
497:
486:
485:
474:
471:
468:
465:
460:
456:
449:
446:
442:
437:
434:
431:
428:
425:
422:
419:
396:
393:
390:
387:
384:
364:
361:
338:
318:
303:
300:
298:
295:
276:Michael Atiyah
272:
271:
260:
257:
254:
251:
248:
245:
241:
237:
234:
231:
226:
219:
215:
211:
208:
205:
181:
169:vector bundles
151:
117:
95:
70:
47:
13:
10:
9:
6:
4:
3:
2:
2718:
2707:
2704:
2703:
2701:
2690:
2686:
2682:
2678:
2673:
2668:
2665:: 1097–1111,
2664:
2660:
2655:
2651:
2645:
2641:
2636:
2632:
2626:
2622:
2617:
2614:
2609:
2605:
2600:
2595:
2591:
2587:
2582:
2577:
2572:
2568:
2564:
2559:
2558:
2550:
2546:
2541:
2536:
2532:
2528:
2524:
2517:
2514:
2507:
2505:
2490:
2487:
2482:
2478:
2475:
2469:
2465:
2461:
2458:
2455:
2452:
2449:
2428:
2425:
2418:
2415:
2412:
2405:Any morphism
2400:
2398:
2377:
2359:
2356:
2353:
2344:
2341:
2336:
2325:
2319:
2316:
2310:
2304:
2298:
2292:
2289:
2286:
2278:
2237:
2234:
2231:
2223:
2215:
2210:
2185:
2179:
2171:
2158:
2152:
2131:
2125:
2122:
2119:
2116:
2109:
2101:
2099:
2095:
2093:
2086:
2072:
2063:
2039:
2035:
2031:
2023:
2010:
2001:
2000:
1999:
1985:
1979:
1971:
1958:
1945:
1940:
1935:
1922:
1913:
1900:
1891:
1868:
1855:
1849:
1840:
1836:
1834:
1821:
1815:
1812:
1796:
1783:
1780:
1777:
1768:
1755:
1749:
1740:
1727:
1718:
1705:
1697:
1693:
1691:
1678:
1672:
1669:
1660:
1647:
1641:
1632:
1616:
1604:
1602:
1589:
1570:
1557:
1551:
1542:
1529:
1520:
1519:
1515:
1513:
1511:
1507:
1494:
1485:
1472:
1463:
1459:
1446:
1437:
1424:
1415:
1411:
1398:
1389:
1385:
1369:
1342:
1340:
1318:
1314:
1310:
1289:
1269:
1266:
1260:
1257:
1254:
1251:
1248:
1228:
1225:
1219:
1199:
1195:
1191:
1188:
1185:
1182:
1173:
1171:
1158:
1152:
1144:
1143:Lie algebroid
1128:
1120:
1116:
1103:
1094:
1081:
1072:
1059:
1056:
1053:
1044:
1031:
1022:
1019:, called its
1018:
1002:
996:
988:
975:
962:
960:
946:
938:
934:
921:
918:
915:
911:
907:
886:
880:
877:
871:
867:
863:
860:
845:
841:
837:
831:
811:
782:
778:
774:
767:
763:
750:
741:
728:
719:
715:
689:
676:
663:
660:
639:
636:
628:
625:
612:
603:
590:
581:
568:
558:
544:
541:
521:
518:
511:
495:
472:
466:
463:
458:
454:
447:
444:
440:
435:
432:
426:
423:
417:
410:
409:
408:
394:
388:
385:
382:
362:
359:
352:
336:
316:
309:
302:As a sequence
301:
296:
294:
292:
288:
284:
281:
277:
258:
252:
249:
243:
239:
235:
232:
217:
213:
209:
203:
196:
195:
194:
192:
179:
170:
166:
162:
149:
140:
137:of the gauge
136:
135:Lie algebroid
132:
128:
115:
106:
93:
85:
81:
68:
60:
58:
45:
34:
30:
26:
19:
2706:Lie algebras
2672:math/0410460
2662:
2658:
2639:
2620:
2589:
2585:
2566:
2562:
2530:
2526:
2516:
2404:
2145:
2105:
2096:
2091:
2089:
2065:
2003:
1951:
1949:
1915:
1870:
1842:
1839:frame bundle
1798:
1770:
1742:
1720:
1698:
1662:
1634:
1606:
1572:
1544:
1522:
1487:
1465:
1439:
1417:
1391:
1345:
1343:
1303:
1174:
1145:
1118:
1096:
1074:
1046:
1024:
1020:
1017:Lie groupoid
968:
966:
936:
900:
743:
721:
691:
653:
605:
583:
561:
559:
508:, where the
487:
351:differential
308:fiber bundle
305:
287:gauge theory
273:
172:
142:
108:
86:
61:
38:
32:
28:
22:
2592:: 137–160,
2569:: 181–207,
2266:defined by:
1998:is always:
1719:-action on
1462:Lie bracket
1414:Lie algebra
1282:, which is
718:Lie algebra
297:Definitions
283:connections
25:mathematics
2508:References
2108:splittings
1941:Properties
1696:transitive
37:principal
2599:0905.1226
2549:0002-9947
2473:→
2453:ϕ
2422:→
2413:ϕ
2401:Morphisms
2345:σ
2342:−
2320:σ
2305:σ
2279:σ
2275:Ω
2220:Ω
2216:∈
2211:σ
2207:Ω
2183:→
2156:→
2129:→
2117:σ
2036:×
1983:→
1898:→
1853:→
1819:→
1813:×
1781:⊆
1753:→
1676:→
1645:→
1590:∗
1587:→
1558:∗
1555:→
1362:∞
1315:×
1264:→
1252:π
1223:→
1156:→
1057:×
1000:→
919:≅
884:→
875:→
858:→
842:×
835:→
779:×
674:→
664:×
545:π
470:→
459:∗
455:π
448:π
430:→
421:→
392:→
383:π
363:π
256:→
247:→
230:→
214:×
207:→
133:, is the
131:Lie group
2700:Category
2491:′
2429:′
1972:-bundle
1694:Given a
1633:-bundle
1543:-bundle
1516:Examples
1388:sections
989:-bundle
690:, where
441:→
306:For any
139:groupoid
107:, where
84:manifold
2677:Bibcode
1510:sheaves
1486:by the
1412:is the
1141:is the
716:is the
82:over a
59:-bundle
35:, of a
2646:
2627:
2547:
2106:Right
1460:under
349:, the
27:, the
2667:arXiv
2594:arXiv
171:over
129:is a
31:, or
2644:ISBN
2625:ISBN
2545:ISSN
627:acts
289:and
2685:doi
2604:doi
2571:doi
2535:doi
1416:of
1121:of
939:of
720:of
560:If
167:of
141:of
23:In
2702::
2683:,
2675:,
2663:38
2661:,
2602:,
2590:16
2588:,
2567:85
2565:,
2543:.
2531:85
2529:.
2525:.
2299::=
1341:.
959:.
557:.
293:.
259:0.
193::
2692:.
2687::
2679::
2669::
2653:.
2634:.
2615:.
2606::
2596::
2580:.
2573::
2551:.
2537::
2488:G
2483:/
2479:P
2476:T
2470:G
2466:/
2462:P
2459:T
2456::
2450:d
2426:P
2419:P
2416::
2386:)
2381:)
2378:M
2375:(
2370:X
2364:]
2360:Y
2357:,
2354:X
2351:[
2348:(
2337:A
2333:]
2329:)
2326:Y
2323:(
2317:,
2314:)
2311:X
2308:(
2302:[
2296:)
2293:Y
2290:,
2287:X
2284:(
2254:)
2251:]
2246:g
2241:[
2238:P
2235:,
2232:M
2229:(
2224:2
2186:M
2180:P
2159:M
2153:P
2132:A
2126:M
2123:T
2120::
2085:)
2073:P
2061:)
2047:g
2040:G
2032:P
2011:M
1986:M
1980:P
1959:G
1936:)
1923:E
1901:M
1895:)
1892:E
1889:(
1885:r
1882:e
1879:D
1856:M
1850:E
1822:M
1816:M
1808:h
1784:G
1778:H
1756:M
1750:G
1728:M
1706:G
1679:M
1673:M
1670:T
1648:M
1642:M
1620:}
1617:e
1614:{
1582:g
1552:G
1530:G
1495:G
1473:M
1447:P
1425:G
1399:P
1373:)
1370:M
1367:(
1356:C
1326:g
1319:G
1311:P
1290:G
1270:M
1267:T
1261:P
1258:T
1255::
1249:d
1229:M
1226:T
1220:A
1200:G
1196:/
1192:P
1189:T
1186:=
1183:A
1159:M
1153:A
1129:P
1104:M
1082:G
1060:P
1054:P
1032:M
1003:M
997:P
976:G
947:P
922:M
916:G
912:/
908:P
887:0
881:M
878:T
872:G
868:/
864:P
861:T
853:g
846:G
838:P
832:0
812:G
790:g
783:G
775:P
751:G
729:G
701:g
677:P
669:g
661:P
640:P
637:V
613:G
591:G
569:P
542:d
522:P
519:V
496:P
473:0
467:M
464:T
445:d
436:P
433:T
427:P
424:V
418:0
395:M
389:P
386::
360:d
337:M
317:P
253:M
250:T
244:G
240:/
236:P
233:T
225:g
218:G
210:P
204:0
180:M
150:P
116:G
94:M
69:P
46:G
20:.
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