360:
1991:
1759:
2200:
There, the bracket on the left side is the operation of the original algebra, the bracket on the right is the commutator of the composition of operators, and the identity states that the
499:
2104:
2195:
2672:
401:
2445:
2223:
905:
42:
that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the
1506:
1263:
727:
693:
80:
1930:
1017:
2557:
1549:
783:
655:
613:
575:
537:
191:
979:
1087:
2583:
1427:
1146:
2635:
1369:
461:
441:
421:
255:
239:
215:
171:
2615:
1809:
1459:
1298:
1055:
115:
46:, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician
1769:
2882:
2120:
Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:
1940:
2778:
2877:
2872:
2793:"Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium quemcunque propositas integrandi"
2110:
1765:
1945:
1557:
466:
365:
Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form
1999:
2788:
2226:
47:
2126:
2255:
The following identity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:
138:
2643:
368:
2867:
2808:
2694:
2689:
2109:
Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a
118:
2262:
2684:
2247:
2203:
50:. He derived the Jacobi identity for Poisson brackets in his 1862 paper on differential equations.
2750:
794:
1464:
1158:
706:
672:
59:
2840:
2774:
1936:
1825:
1093:
990:
126:
2458:
1522:
735:
622:
580:
542:
504:
176:
2820:
1513:
934:
194:
39:
1060:
2114:
355:{\displaystyle x\times (y\times z)\ +\ y\times (z\times x)\ +\ z\times (x\times y)\ =\ 0.}
218:
122:
2562:
1382:
1098:
2638:
2620:
1517:
1327:
446:
426:
406:
224:
200:
156:
2843:
2588:
1782:
1432:
1271:
1028:
88:
2861:
1149:
1057:
satisfies the Jacobi identity, it may be said that it behaves as if it were given by
142:
134:
54:
43:
2452:
703:, which measures the failure of commutativity in matrix multiplication. Instead of
1779:
Most common examples of the Jacobi identity come from the bracket multiplication
17:
1812:
666:
616:
83:
31:
2749:
Alekseev, Ilya; Ivanov, Sergei O. (18 April 2016). "Higher Jacobi
Identities".
2243:
2239:
700:
130:
2848:
2773:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
2771:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
1939:, the Jacobi identity admits two equivalent reformulations. Defining the
1816:
1089:
in some associative algebra even if it is not actually defined that way.
2113:. That form of the Jacobi identity is also used to define the notion of
2824:
2755:
2455:, with the Lie bracket acting as both a product and a derivative:
695:
matrices, which may be thought of as infinitesimal motions of an
1148:, the Jacobi identity may be rewritten as a modification of the
2792:
2650:
2811:(1991). "Jacobi and the Birth of Lie's Theory of Groups".
501:. Alternatively, we may observe that the ordered triples
2646:
2623:
2591:
2565:
2461:
2265:
2206:
2129:
2002:
1948:
1828:
1785:
1560:
1525:
1467:
1435:
1385:
1330:
1274:
1161:
1101:
1063:
1031:
993:
937:
797:
738:
709:
675:
625:
583:
545:
507:
469:
449:
429:
409:
371:
258:
227:
203:
179:
159:
91:
62:
2225:
map sending each element to its adjoint action is a
2666:
2629:
2609:
2577:
2551:
2439:
2217:
2189:
2098:
1985:
1924:
1803:
1753:
1543:
1500:
1453:
1421:
1363:
1292:
1257:
1140:
1081:
1049:
1011:
973:
899:
777:
721:
699:-dimensional vector space. The Ă— operation is the
687:
649:
607:
569:
531:
493:
455:
435:
415:
395:
354:
233:
209:
185:
165:
109:
74:
1986:{\displaystyle \operatorname {ad} _{x}:y\mapsto }
2797:Journal fĂĽr die reine und angewandte Mathematik
1314:
669:is constructed from the (associative) ring of
2617:is literally a derivative operator acting on
1754:{\displaystyle ++=0,\qquad +\{,X\}+\{,Y\}=0.}
8:
1742:
1718:
1712:
1688:
1673:
1661:
1636:
1624:
1606:
1594:
1576:
1564:
1538:
1526:
931:that is closed under the bracket operation:
494:{\displaystyle x\mapsto y\mapsto z\mapsto x}
1019:, the Jacobi identity continues to hold on
2099:{\displaystyle \operatorname {ad} _{x}=+.}
1300:is the action of the infinitesimal motion
788:In that notation, the Jacobi identity is:
121:, the Jacobi identity is satisfied by the
2754:
2655:
2649:
2648:
2645:
2622:
2590:
2564:
2460:
2451:The Jacobi identity is equivalent to the
2264:
2207:
2205:
2175:
2162:
2134:
2128:
2078:
2041:
2007:
2001:
1953:
1947:
1827:
1784:
1559:
1524:
1466:
1434:
1384:
1329:
1273:
1160:
1100:
1062:
1030:
992:
936:
796:
737:
708:
674:
624:
582:
544:
506:
468:
448:
428:
408:
370:
257:
226:
202:
178:
158:
90:
61:
2190:{\displaystyle \operatorname {ad} _{}=.}
2715:C. G. J. Jacobi (1862), §26, Theorem V.
2707:
910:That is easily checked by computation.
1935:Because the bracket multiplication is
665:The simplest informative example of a
1819:. The Jacobi identity is written as:
117:both satisfy the Jacobi identity. In
7:
2736:
729:, the Lie bracket notation is used:
463:are permuted according to the cycle
2667:{\displaystyle {\mathcal {L}}_{X}Y}
396:{\displaystyle a\times (b\times c)}
2242:is the analogous identity for the
2211:
2208:
27:Property of some binary operations
25:
2725:
1770:Baker–Campbell–Hausdorff formula
2883:Properties of binary operations
2714:
1657:
2604:
2592:
2546:
2543:
2531:
2522:
2516:
2507:
2495:
2492:
2486:
2483:
2471:
2462:
2440:{\displaystyle ]]+]]+]]+]]=0.}
2428:
2425:
2422:
2410:
2401:
2392:
2386:
2383:
2380:
2368:
2359:
2350:
2344:
2341:
2338:
2326:
2317:
2308:
2302:
2299:
2296:
2284:
2275:
2266:
2181:
2155:
2147:
2135:
2090:
2065:
2059:
2034:
2028:
2016:
1980:
1968:
1965:
1913:
1910:
1898:
1889:
1883:
1880:
1868:
1859:
1853:
1850:
1838:
1829:
1798:
1786:
1733:
1721:
1703:
1691:
1682:
1658:
1645:
1621:
1615:
1591:
1585:
1561:
1495:
1483:
1471:
1468:
1448:
1436:
1416:
1413:
1398:
1389:
1386:
1358:
1355:
1340:
1331:
1287:
1275:
1246:
1243:
1231:
1222:
1216:
1213:
1201:
1192:
1186:
1177:
1165:
1162:
1135:
1123:
1114:
1102:
1044:
1032:
1025:. Thus, if a binary operation
950:
938:
919:is an associative algebra and
882:
879:
867:
858:
852:
849:
837:
828:
822:
819:
807:
798:
751:
739:
644:
626:
602:
584:
564:
546:
526:
508:
485:
479:
473:
390:
378:
337:
325:
307:
295:
277:
265:
129:, it is satisfied by operator
104:
92:
1:
2218:{\displaystyle \mathrm {ad} }
1429:), is equal to the action of
1766:Lie bracket of vector fields
1512:There is also a plethora of
141:of quantum mechanics by the
900:{\displaystyle ]+]+]\ =\ 0}
2899:
1763:
1501:{\displaystyle ,\cdot \ ]}
1312:, that can be stated as:
1258:{\displaystyle ,Z]=]-]~.}
722:{\displaystyle X\times Y}
688:{\displaystyle n\times n}
75:{\displaystyle a\times b}
2585:are vector fields, then
2227:Lie algebra homomorphism
1993:, the identity becomes:
1925:{\displaystyle ]+]+]=0.}
1514:graded Jacobi identities
1012:{\displaystyle X,Y\in V}
137:and equivalently in the
48:Carl Gustav Jacob Jacobi
2878:Non-associative algebra
2873:Mathematical identities
2769:Hall, Brian C. (2015),
2552:{\displaystyle ]=,Z]+]}
1544:{\displaystyle \{X,Y\}}
1371:), minus the action of
778:{\displaystyle =XY-YX.}
661:Commutator bracket form
650:{\displaystyle (x,y,z)}
608:{\displaystyle (z,x,y)}
570:{\displaystyle (y,z,x)}
532:{\displaystyle (x,y,z)}
186:{\displaystyle \times }
139:phase space formulation
2668:
2631:
2611:
2579:
2553:
2441:
2219:
2191:
2100:
1987:
1926:
1805:
1755:
1545:
1510:
1502:
1455:
1423:
1365:
1294:
1259:
1142:
1083:
1051:
1013:
975:
974:{\displaystyle =XY-YX}
901:
779:
723:
689:
651:
619:of the ordered triple
609:
571:
533:
495:
457:
437:
417:
397:
356:
235:
211:
187:
167:
111:
76:
2813:Arch. Hist. Exact Sci
2695:Three subgroups lemma
2690:Super Jacobi identity
2669:
2632:
2612:
2580:
2554:
2442:
2220:
2192:
2101:
1988:
1927:
1806:
1756:
1546:
1503:
1456:
1424:
1366:
1295:
1260:
1143:
1094:antisymmetry property
1084:
1082:{\displaystyle XY-YX}
1052:
1014:
976:
902:
780:
724:
690:
652:
610:
572:
534:
496:
458:
438:
418:
398:
357:
236:
212:
188:
168:
112:
84:Lie bracket operation
77:
2697:(Hall–Witt identity)
2644:
2621:
2589:
2563:
2459:
2263:
2204:
2127:
2000:
1946:
1826:
1783:
1558:
1523:
1465:
1433:
1383:
1328:
1272:
1159:
1150:associative property
1099:
1061:
1029:
991:
935:
795:
736:
707:
673:
623:
581:
543:
505:
467:
447:
427:
407:
369:
256:
225:
201:
177:
157:
119:analytical mechanics
89:
60:
44:associative property
2844:"Jacobi Identities"
2685:Structure constants
2578:{\displaystyle X,Y}
913:More generally, if
38:is a property of a
2841:Weisstein, Eric W.
2825:10.1007/BF00375135
2664:
2627:
2607:
2575:
2549:
2437:
2240:Hall–Witt identity
2233:Related identities
2215:
2187:
2096:
1983:
1922:
1801:
1751:
1541:
1498:
1451:
1422:{\displaystyle (]}
1419:
1361:
1290:
1255:
1141:{\displaystyle =-}
1138:
1079:
1047:
1009:
971:
897:
775:
719:
685:
647:
605:
567:
529:
491:
453:
433:
413:
393:
352:
231:
207:
183:
163:
107:
72:
2726:T. Hawkins (1991)
2630:{\displaystyle Y}
1494:
1412:
1364:{\displaystyle ]}
1354:
1251:
925:is a subspace of
893:
887:
617:even permutations
456:{\displaystyle z}
436:{\displaystyle y}
416:{\displaystyle x}
348:
342:
318:
312:
288:
282:
234:{\displaystyle +}
210:{\displaystyle 0}
195:binary operations
166:{\displaystyle +}
127:quantum mechanics
18:Jacobi identities
16:(Redirected from
2890:
2854:
2853:
2828:
2804:
2789:Jacobi, C. G. J.
2783:
2761:
2760:
2758:
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2665:
2660:
2659:
2654:
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2634:
2633:
2628:
2616:
2614:
2613:
2610:{\displaystyle }
2608:
2584:
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2012:
2011:
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1989:
1984:
1958:
1957:
1941:adjoint operator
1931:
1929:
1928:
1923:
1810:
1808:
1807:
1804:{\displaystyle }
1802:
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1752:
1550:
1548:
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1507:
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1504:
1499:
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1458:
1457:
1454:{\displaystyle }
1452:
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1410:
1370:
1368:
1367:
1362:
1352:
1310:
1304:
1299:
1297:
1296:
1293:{\displaystyle }
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1262:
1261:
1256:
1249:
1147:
1145:
1144:
1139:
1088:
1086:
1085:
1080:
1056:
1054:
1053:
1050:{\displaystyle }
1048:
1023:
1018:
1016:
1015:
1010:
985:
980:
978:
977:
972:
929:
923:
917:
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904:
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403:, the variables
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123:Poisson brackets
116:
114:
113:
110:{\displaystyle }
108:
81:
79:
78:
73:
40:binary operation
21:
2898:
2897:
2893:
2892:
2891:
2889:
2888:
2887:
2858:
2857:
2839:
2838:
2835:
2809:Hawkins, Thomas
2807:
2787:
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2764:
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2260:
2246:operation in a
2235:
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2125:
2124:
2115:Leibniz algebra
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2037:
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1997:
1949:
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1943:
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1823:
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1780:
1777:
1772:
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1521:
1520:
1518:anticommutators
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1380:
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1302:
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1096:
1059:
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1026:
1021:
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988:
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921:
915:
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792:
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245:Jacobi identity
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223:
222:
219:neutral element
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174:
155:
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151:
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58:
57:
36:Jacobi identity
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2833:External links
2831:
2830:
2829:
2819:(3): 187-278.
2805:
2785:
2780:978-3319134666
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2639:Lie derivative
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1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1540:
1537:
1534:
1531:
1528:
1497:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1450:
1447:
1444:
1441:
1438:
1418:
1415:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1360:
1357:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1316:The action of
1289:
1286:
1283:
1280:
1277:
1266:
1265:
1254:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1078:
1075:
1072:
1069:
1066:
1046:
1043:
1040:
1037:
1034:
1008:
1005:
1002:
999:
996:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
908:
907:
896:
890:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
786:
785:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
718:
715:
712:
684:
681:
678:
662:
659:
646:
643:
640:
637:
634:
631:
628:
604:
601:
598:
595:
592:
589:
586:
566:
563:
560:
557:
554:
551:
548:
528:
525:
522:
519:
516:
513:
510:
490:
487:
484:
481:
478:
475:
472:
452:
432:
412:
392:
389:
386:
383:
380:
377:
374:
363:
362:
351:
345:
339:
336:
333:
330:
327:
324:
321:
315:
309:
306:
303:
300:
297:
294:
291:
285:
279:
276:
273:
270:
267:
264:
261:
230:
206:
182:
162:
150:
147:
106:
103:
100:
97:
94:
71:
68:
65:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2895:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2865:
2863:
2851:
2850:
2845:
2842:
2837:
2836:
2832:
2826:
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2776:
2772:
2767:
2766:
2757:
2752:
2745:
2742:
2738:
2733:
2730:
2727:
2722:
2719:
2716:
2711:
2708:
2701:
2696:
2693:
2691:
2688:
2686:
2683:
2682:
2678:
2661:
2656:
2640:
2637:, namely the
2624:
2601:
2598:
2595:
2572:
2569:
2566:
2540:
2537:
2534:
2528:
2525:
2519:
2513:
2510:
2504:
2501:
2498:
2489:
2480:
2477:
2474:
2468:
2465:
2454:
2450:
2449:
2434:
2431:
2419:
2416:
2413:
2407:
2404:
2398:
2395:
2389:
2377:
2374:
2371:
2365:
2362:
2356:
2353:
2347:
2335:
2332:
2329:
2323:
2320:
2314:
2311:
2305:
2293:
2290:
2287:
2281:
2278:
2272:
2269:
2259:
2258:
2254:
2253:
2249:
2245:
2241:
2237:
2236:
2232:
2230:
2228:
2184:
2176:
2172:
2168:
2163:
2159:
2152:
2144:
2141:
2138:
2131:
2123:
2122:
2121:
2118:
2116:
2112:
2093:
2087:
2084:
2079:
2075:
2071:
2068:
2062:
2056:
2053:
2050:
2047:
2042:
2038:
2031:
2025:
2022:
2019:
2013:
2008:
2004:
1996:
1995:
1994:
1977:
1974:
1971:
1962:
1959:
1954:
1950:
1942:
1938:
1937:antisymmetric
1919:
1916:
1907:
1904:
1901:
1895:
1892:
1886:
1877:
1874:
1871:
1865:
1862:
1856:
1847:
1844:
1841:
1835:
1832:
1822:
1821:
1820:
1818:
1814:
1795:
1792:
1789:
1774:
1771:
1767:
1748:
1745:
1739:
1736:
1730:
1727:
1724:
1715:
1709:
1706:
1700:
1697:
1694:
1685:
1679:
1676:
1670:
1667:
1664:
1654:
1651:
1648:
1642:
1639:
1633:
1630:
1627:
1618:
1612:
1609:
1603:
1600:
1597:
1588:
1582:
1579:
1573:
1570:
1567:
1554:
1553:
1552:
1535:
1532:
1529:
1519:
1515:
1509:
1489:
1486:
1480:
1477:
1474:
1445:
1442:
1439:
1407:
1404:
1401:
1395:
1392:
1378:
1374:
1349:
1346:
1343:
1337:
1334:
1323:
1319:
1313:
1311:
1305:
1284:
1281:
1278:
1252:
1240:
1237:
1234:
1228:
1225:
1219:
1210:
1207:
1204:
1198:
1195:
1189:
1183:
1180:
1174:
1171:
1168:
1155:
1154:
1153:
1151:
1132:
1129:
1126:
1120:
1117:
1111:
1108:
1105:
1095:
1090:
1076:
1073:
1070:
1067:
1064:
1041:
1038:
1035:
1024:
1006:
1003:
1000:
997:
994:
986:
968:
965:
962:
959:
956:
953:
947:
944:
941:
930:
924:
918:
911:
894:
888:
876:
873:
870:
864:
861:
855:
846:
843:
840:
834:
831:
825:
816:
813:
810:
804:
801:
791:
790:
789:
772:
769:
766:
763:
760:
757:
754:
748:
745:
742:
732:
731:
730:
716:
713:
710:
702:
698:
682:
679:
676:
668:
660:
658:
641:
638:
635:
632:
629:
618:
599:
596:
593:
590:
587:
561:
558:
555:
552:
549:
523:
520:
517:
514:
511:
488:
482:
476:
470:
450:
430:
410:
387:
384:
381:
375:
372:
349:
343:
334:
331:
328:
322:
319:
313:
304:
301:
298:
292:
289:
283:
274:
271:
268:
262:
259:
252:
251:
250:
248:
228:
220:
204:
196:
180:
160:
148:
146:
144:
143:Moyal bracket
140:
136:
135:Hilbert space
132:
128:
124:
120:
101:
98:
95:
85:
69:
66:
63:
56:
55:cross product
51:
49:
45:
41:
37:
33:
19:
2868:Lie algebras
2847:
2816:
2812:
2800:
2796:
2770:
2744:
2732:
2721:
2710:
2453:Product Rule
2199:
2119:
2108:
1934:
1813:Lie algebras
1778:
1775:Adjoint form
1511:
1461:, (operator
1376:
1375:followed by
1372:
1321:
1320:followed by
1317:
1315:
1307:
1301:
1267:
1091:
1020:
982:
926:
920:
914:
912:
909:
787:
696:
664:
364:
242:
152:
52:
35:
29:
2739:Example 3.3
1551:, such as:
981:belongs to
667:Lie algebra
131:commutators
32:mathematics
2862:Categories
2756:1604.05281
2702:References
2244:commutator
2111:derivation
1764:See also:
1516:involving
1379:(operator
1324:(operator
1092:Using the
701:commutator
615:, are the
197:, and let
149:Definition
2849:MathWorld
2737:Hall 2015
2085:
2048:
2014:
1966:↦
1817:Lie rings
1490:⋅
1408:⋅
1350:⋅
1220:−
1121:−
1071:−
1004:∈
963:−
764:−
714:×
680:×
486:↦
480:↦
474:↦
385:×
376:×
332:×
323:×
302:×
293:×
272:×
263:×
181:×
67:×
2803:: 1-181.
2791:(1862).
2679:See also
987:for all
82:and the
2559:. If
217:be the
193:be two
2777:
1493:
1411:
1353:
1250:
892:
886:
347:
341:
317:
311:
287:
281:
241:. The
34:, the
2751:arXiv
2248:group
133:on a
125:. In
2775:ISBN
2238:The
1815:and
1768:and
577:and
443:and
221:for
173:and
153:Let
53:The
2821:doi
1811:on
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1306:on
1268:If
249:is
30:In
2864::
2846:.
2817:42
2815:.
2801:60
2799:.
2795:.
2435:0.
2229:.
2173:ad
2160:ad
2132:ad
2117:.
2076:ad
2039:ad
2005:ad
1951:ad
1920:0.
1749:0.
1152::
657:.
539:,
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145:.
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2759:.
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2605:]
2602:Y
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2532:[
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2526:Y
2523:[
2520:+
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2496:[
2493:[
2490:=
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2466:X
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2267:[
2250:.
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2153:=
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2091:]
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2023:,
2020:y
2017:[
2009:x
1981:]
1978:y
1975:,
1972:x
1969:[
1963:y
1960::
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1205:Y
1202:[
1199:,
1196:X
1193:[
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1178:]
1175:Y
1172:,
1169:X
1166:[
1163:[
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1133:X
1130:,
1127:Y
1124:[
1118:=
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1112:Y
1109:,
1106:X
1103:[
1077:X
1074:Y
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1045:]
1042:Y
1039:,
1036:X
1033:[
1022:V
1007:V
1001:Y
998:,
995:X
984:V
969:X
966:Y
960:Y
957:X
954:=
951:]
948:Y
945:,
942:X
939:[
928:A
922:V
916:A
895:0
889:=
883:]
880:]
877:Y
874:,
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868:[
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859:[
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847:X
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838:[
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814:,
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808:[
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799:[
773:.
770:X
767:Y
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755:=
752:]
749:Y
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743:X
740:[
717:Y
711:X
697:n
683:n
677:n
645:)
642:z
639:,
636:y
633:,
630:x
627:(
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600:y
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588:z
585:(
565:)
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547:(
527:)
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512:x
509:(
489:x
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388:c
382:b
379:(
373:a
344:=
338:)
335:y
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326:(
320:z
314:+
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296:(
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266:(
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102:b
99:,
96:a
93:[
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20:)
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