1586:
35:
692:
It is important to note that the converse of the first result above is false: Even if the
Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group. For example, the Killing form on the Lie algebra of the Heisenberg group
2063:
938:). It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; the existence of an Ad-invariant inner product on the compact real form greatly simplifies the development.
2289:
2168:
688:
In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand (for which the corresponding subgroup is a torus) and a summand on which the
Killing form is negative definite.
2656:
2475:
2560:
2379:
1974:
892:
859:
1979:
936:
1575:
2221:
1207:
1254:
1122:
1441:
1370:
1482:
1326:
2603:
2422:
1003:
973:
815:
791:
767:
731:
452:
2226:
1873:
1847:
1791:
1735:
1680:
1625:
2769:
1821:
1765:
1710:
1655:
1402:
1290:
1162:
1080:
2739:
2712:
2685:
2506:
2325:
2094:
1909:
500:
2099:
505:
495:
490:
310:
574:
457:
2914:
2608:
2427:
605:
2974:
1489:
2936:
2511:
2330:
467:
1129:
817:
is semisimple, this inner product can be taken to be the negative of the
Killing form. Thus relative to this inner product, Ad(
1913:
2058:{\displaystyle \operatorname {SU} (1)\cong \operatorname {SO} (1)\cong \operatorname {Sp} (0)\cong \operatorname {SO} (0).}
864:
828:
462:
442:
407:
315:
901:
1498:
945:
on the representability of Lie algebras: just as every finite-dimensional Lie algebra in characteristic 0 embeds in
693:
is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.
641:; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose
447:
2829:
2809:
2173:
822:
680:
If the
Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact
1444:
1329:
1210:
598:
82:
1166:
1216:
1084:
1876:
1590:
1406:
1335:
895:
649:; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest
1450:
1294:
1046:
665:, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:
402:
365:
333:
320:
1125:
703:
670:
434:
102:
2565:
2384:
978:
948:
796:
772:
748:
712:
62:
52:
591:
579:
250:
2866:
2846:
351:
341:
2284:{\displaystyle \operatorname {SU} (2)\cong \operatorname {Spin} (3)\cong \operatorname {Sp} (1)}
2932:
2910:
2785:
1585:
1042:
1025:
646:
638:
530:
415:
378:
268:
1852:
1826:
1770:
1714:
1659:
1604:
2924:
2744:
1796:
1740:
1685:
1630:
1485:
1377:
1265:
1257:
1137:
1055:
942:
550:
230:
222:
214:
206:
198:
177:
167:
157:
147:
131:
112:
72:
2717:
2690:
2663:
2482:
2301:
2292:
2070:
1885:
535:
288:
273:
44:
2163:{\displaystyle {\mathfrak {su}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}}
1594:
1014:
1010:
555:
540:
373:
278:
661:
Formally, one may define a compact Lie algebra either as the Lie algebra of a compact
2968:
742:
263:
92:
2954:
1597:
yield exceptional isomorphisms of compact Lie algebras and corresponding Lie groups.
642:
619:
560:
545:
346:
328:
258:
637:. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a
634:
386:
302:
26:
623:
525:
391:
283:
2931:, Progress in Mathematics, vol. 140 (2nd ed.), Boston: Birkhäuser,
2780:
1029:
662:
650:
22:
2909:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
2907:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
706:; note that the analogous result is true for compact groups in general.
482:
2651:{\displaystyle \operatorname {SU} (4)\cong \operatorname {Spin} (6).}
2470:{\displaystyle \operatorname {Sp} (2)\cong \operatorname {Spin} (5).}
653:
of a corresponding complex Lie algebra, namely the complexification.
34:
1041:
The compact Lie algebras are classified and named according to the
1032:, split Lie algebras being "as far as possible" from being compact.
1584:
2771:
respectively, with corresponding isomorphisms of Lie algebras.
2555:{\displaystyle {\mathfrak {su}}_{4}\cong {\mathfrak {so}}_{6}}
2374:{\displaystyle {\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}}
669:
The
Killing form on the Lie algebra of a compact Lie group is
1976:
is the trivial diagram, corresponding to the trivial group
1969:{\displaystyle A_{0}\cong B_{0}\cong C_{0}\cong D_{0}}
2747:
2720:
2693:
2666:
2611:
2568:
2514:
2485:
2430:
2387:
2333:
2304:
2229:
2176:
2102:
2073:
1982:
1916:
1888:
1855:
1829:
1799:
1773:
1743:
1717:
1688:
1662:
1633:
1607:
1501:
1453:
1409:
1380:
1338:
1297:
1268:
1219:
1169:
1140:
1087:
1058:
981:
951:
904:
867:
831:
799:
775:
751:
715:
887:{\displaystyle \operatorname {ad} \ {\mathfrak {g}}}
854:{\displaystyle \operatorname {SO} ({\mathfrak {g}})}
1495:Compact real forms of the exceptional Lie algebras
2763:
2733:
2706:
2679:
2650:
2597:
2554:
2500:
2469:
2416:
2373:
2319:
2283:
2215:
2162:
2088:
2057:
1968:
1903:
1867:
1841:
1815:
1785:
1759:
1729:
1704:
1674:
1649:
1619:
1569:
1476:
1435:
1396:
1364:
1320:
1284:
1248:
1201:
1156:
1116:
1074:
997:
967:
930:
886:
853:
809:
785:
761:
725:
2223:and the corresponding isomorphisms of Lie groups
1601:The classification is non-redundant if one takes
931:{\displaystyle {\mathfrak {so}}({\mathfrak {g}})}
2605:and the corresponding isomorphism of Lie groups
2424:and the corresponding isomorphism of Lie groups
453:Representation theory of semisimple Lie algebras
1570:{\displaystyle E_{6},E_{7},E_{8},F_{4},G_{2}.}
793:is the Lie algebra of some compact group. If
599:
8:
2562:corresponds to the isomorphisms of diagrams
2381:corresponds to the isomorphisms of diagrams
2170:corresponds to the isomorphisms of diagrams
2216:{\displaystyle A_{1}\cong B_{1}\cong C_{1}}
769:admits an Ad-invariant inner product, then
2857:
2855:
2820:
2818:
1488:) (properly, the compact form is PSO, the
606:
592:
491:Particle physics and representation theory
136:
33:
18:
2752:
2746:
2725:
2719:
2698:
2692:
2671:
2665:
2610:
2586:
2573:
2567:
2546:
2537:
2536:
2526:
2517:
2516:
2513:
2484:
2429:
2405:
2392:
2386:
2365:
2356:
2355:
2345:
2336:
2335:
2332:
2303:
2228:
2207:
2194:
2181:
2175:
2154:
2145:
2144:
2134:
2125:
2124:
2114:
2105:
2104:
2101:
2072:
1981:
1960:
1947:
1934:
1921:
1915:
1887:
1854:
1828:
1804:
1798:
1772:
1748:
1742:
1716:
1693:
1687:
1661:
1638:
1632:
1606:
1558:
1545:
1532:
1519:
1506:
1500:
1462:
1456:
1455:
1452:
1421:
1412:
1411:
1408:
1385:
1379:
1353:
1341:
1340:
1337:
1309:
1300:
1299:
1296:
1273:
1267:
1228:
1222:
1221:
1218:
1181:
1172:
1171:
1168:
1145:
1139:
1099:
1090:
1089:
1086:
1063:
1057:
983:
982:
980:
953:
952:
950:
919:
918:
906:
905:
903:
878:
877:
866:
842:
841:
830:
801:
800:
798:
777:
776:
774:
753:
752:
750:
717:
716:
714:
1202:{\displaystyle {\mathfrak {so}}_{2n+1},}
1128:(properly, the compact form is PSU, the
941:This can be seen as a compact analog of
2797:
1249:{\displaystyle {\mathfrak {o}}_{2n+1},}
1117:{\displaystyle {\mathfrak {su}}_{n+1},}
458:Representations of classical Lie groups
190:
139:
21:
1436:{\displaystyle {\mathfrak {so}}_{2n},}
1365:{\displaystyle {\mathfrak {usp}}_{n},}
2862:
2842:
2825:
2805:
2714:as diagrams, these are isomorphic to
1477:{\displaystyle {\mathfrak {o}}_{2n},}
1321:{\displaystyle {\mathfrak {sp}}_{n},}
1024:Compact Lie algebras are opposite to
7:
2887:
975:every compact Lie algebra embeds in
311:Lie group–Lie algebra correspondence
2541:
2538:
2521:
2518:
2360:
2357:
2340:
2337:
2149:
2146:
2129:
2126:
2109:
2106:
1490:projective special orthogonal group
1457:
1416:
1413:
1348:
1345:
1342:
1304:
1301:
1223:
1176:
1173:
1094:
1091:
987:
984:
957:
954:
920:
910:
907:
879:
843:
802:
778:
754:
718:
677:, not negative definite in general.
14:
2929:Lie Groups Beyond an Introduction
2598:{\displaystyle A_{3}\cong D_{3},}
2417:{\displaystyle B_{2}\cong C_{2},}
998:{\displaystyle {\mathfrak {so}}.}
968:{\displaystyle {\mathfrak {gl}},}
1130:projective special unitary group
1017:of the complex Lie algebra with
1013:of a compact Lie algebra is the
810:{\displaystyle {\mathfrak {g}}}
786:{\displaystyle {\mathfrak {g}}}
762:{\displaystyle {\mathfrak {g}}}
726:{\displaystyle {\mathfrak {g}}}
627:
2642:
2636:
2624:
2618:
2461:
2455:
2443:
2437:
2278:
2272:
2260:
2254:
2242:
2236:
2049:
2043:
2031:
2025:
2013:
2007:
1995:
1989:
925:
915:
848:
838:
506:Galilean group representations
501:Poincaré group representations
1:
496:Lorentz group representations
463:Theorem of the highest weight
2960:Encyclopaedia of Mathematics
2828:, Propositions 4.26, 4.27,
2991:
2975:Properties of Lie algebras
823:orthogonal transformations
733:for the compact Lie group
448:Lie algebra representation
702:Compact Lie algebras are
1877:exceptional isomorphisms
1591:exceptional isomorphisms
1445:special orthogonal group
1330:compact symplectic group
1211:special orthogonal group
443:Lie group representation
2905:Hall, Brian C. (2015),
1868:{\displaystyle n\geq 1}
1842:{\displaystyle n\geq 0}
1786:{\displaystyle n\geq 4}
1730:{\displaystyle n\geq 3}
1675:{\displaystyle n\geq 2}
1620:{\displaystyle n\geq 1}
1047:semisimple Lie algebras
1026:split real Lie algebras
896:skew-symmetric matrices
468:Borel–Weil–Bott theorem
2765:
2764:{\displaystyle D_{5},}
2735:
2708:
2681:
2652:
2599:
2556:
2502:
2471:
2418:
2375:
2321:
2285:
2217:
2164:
2090:
2059:
1970:
1905:
1869:
1843:
1817:
1816:{\displaystyle D_{n}.}
1787:
1761:
1760:{\displaystyle C_{n},}
1731:
1706:
1705:{\displaystyle B_{n},}
1676:
1651:
1650:{\displaystyle A_{n},}
1621:
1598:
1571:
1478:
1437:
1398:
1397:{\displaystyle D_{n}:}
1366:
1322:
1286:
1285:{\displaystyle C_{n}:}
1250:
1203:
1158:
1157:{\displaystyle B_{n}:}
1118:
1076:
1075:{\displaystyle A_{n}:}
999:
969:
932:
888:
855:
811:
787:
763:
727:
366:Semisimple Lie algebra
321:Adjoint representation
2766:
2736:
2734:{\displaystyle A_{4}}
2709:
2707:{\displaystyle E_{5}}
2682:
2680:{\displaystyle E_{4}}
2653:
2600:
2557:
2503:
2472:
2419:
2376:
2322:
2286:
2218:
2165:
2091:
2060:
1971:
1906:
1870:
1844:
1823:If one instead takes
1818:
1788:
1762:
1732:
1707:
1677:
1652:
1622:
1588:
1572:
1484:corresponding to the
1479:
1443:corresponding to the
1438:
1399:
1367:
1328:corresponding to the
1323:
1287:
1256:corresponding to the
1251:
1209:corresponding to the
1204:
1159:
1126:special unitary group
1124:corresponding to the
1119:
1077:
1000:
970:
933:
889:
856:
812:
788:
764:
728:
435:Representation theory
2865:, Proposition 4.24,
2845:, Proposition 4.25,
2745:
2718:
2691:
2664:
2609:
2566:
2512:
2501:{\displaystyle n=3,}
2483:
2428:
2385:
2331:
2320:{\displaystyle n=2,}
2302:
2227:
2174:
2100:
2089:{\displaystyle n=1,}
2071:
1980:
1914:
1904:{\displaystyle n=0,}
1886:
1875:one obtains certain
1853:
1827:
1797:
1771:
1741:
1715:
1686:
1660:
1631:
1605:
1499:
1451:
1407:
1378:
1336:
1332:; sometimes written
1295:
1266:
1217:
1167:
1138:
1085:
1056:
979:
949:
902:
865:
829:
797:
773:
749:
713:
1021:vertices blackened.
580:Table of Lie groups
421:Compact Lie algebra
16:Mathematical theory
2955:Lie group, compact
2761:
2731:
2704:
2677:
2648:
2595:
2552:
2498:
2467:
2414:
2371:
2317:
2281:
2213:
2160:
2086:
2055:
1966:
1901:
1865:
1839:
1813:
1783:
1757:
1727:
1702:
1672:
1647:
1617:
1599:
1567:
1474:
1433:
1394:
1362:
1318:
1282:
1246:
1199:
1154:
1114:
1072:
1043:compact real forms
995:
965:
928:
884:
851:
807:
783:
759:
745:,. Conversely, if
723:
352:Affine Lie algebra
342:Simple Lie algebra
83:Special orthogonal
2925:Knapp, Anthony W.
2916:978-0-387-40122-5
2786:Split Lie algebra
2660:If one considers
2291:(the 3-sphere or
876:
647:negative definite
639:compact Lie group
616:
615:
416:Split Lie algebra
379:Cartan subalgebra
241:
240:
132:Simple Lie groups
2982:
2941:
2919:
2891:
2885:
2879:
2876:
2870:
2859:
2850:
2839:
2833:
2822:
2813:
2802:
2770:
2768:
2767:
2762:
2757:
2756:
2740:
2738:
2737:
2732:
2730:
2729:
2713:
2711:
2710:
2705:
2703:
2702:
2686:
2684:
2683:
2678:
2676:
2675:
2657:
2655:
2654:
2649:
2604:
2602:
2601:
2596:
2591:
2590:
2578:
2577:
2561:
2559:
2558:
2553:
2551:
2550:
2545:
2544:
2531:
2530:
2525:
2524:
2508:the isomorphism
2507:
2505:
2504:
2499:
2476:
2474:
2473:
2468:
2423:
2421:
2420:
2415:
2410:
2409:
2397:
2396:
2380:
2378:
2377:
2372:
2370:
2369:
2364:
2363:
2350:
2349:
2344:
2343:
2327:the isomorphism
2326:
2324:
2323:
2318:
2293:unit quaternions
2290:
2288:
2287:
2282:
2222:
2220:
2219:
2214:
2212:
2211:
2199:
2198:
2186:
2185:
2169:
2167:
2166:
2161:
2159:
2158:
2153:
2152:
2139:
2138:
2133:
2132:
2119:
2118:
2113:
2112:
2096:the isomorphism
2095:
2093:
2092:
2087:
2064:
2062:
2061:
2056:
1975:
1973:
1972:
1967:
1965:
1964:
1952:
1951:
1939:
1938:
1926:
1925:
1910:
1908:
1907:
1902:
1874:
1872:
1871:
1866:
1848:
1846:
1845:
1840:
1822:
1820:
1819:
1814:
1809:
1808:
1792:
1790:
1789:
1784:
1766:
1764:
1763:
1758:
1753:
1752:
1736:
1734:
1733:
1728:
1711:
1709:
1708:
1703:
1698:
1697:
1681:
1679:
1678:
1673:
1656:
1654:
1653:
1648:
1643:
1642:
1626:
1624:
1623:
1618:
1576:
1574:
1573:
1568:
1563:
1562:
1550:
1549:
1537:
1536:
1524:
1523:
1511:
1510:
1486:orthogonal group
1483:
1481:
1480:
1475:
1470:
1469:
1461:
1460:
1442:
1440:
1439:
1434:
1429:
1428:
1420:
1419:
1403:
1401:
1400:
1395:
1390:
1389:
1371:
1369:
1368:
1363:
1358:
1357:
1352:
1351:
1327:
1325:
1324:
1319:
1314:
1313:
1308:
1307:
1291:
1289:
1288:
1283:
1278:
1277:
1258:orthogonal group
1255:
1253:
1252:
1247:
1242:
1241:
1227:
1226:
1208:
1206:
1205:
1200:
1195:
1194:
1180:
1179:
1163:
1161:
1160:
1155:
1150:
1149:
1123:
1121:
1120:
1115:
1110:
1109:
1098:
1097:
1081:
1079:
1078:
1073:
1068:
1067:
1004:
1002:
1001:
996:
991:
990:
974:
972:
971:
966:
961:
960:
937:
935:
934:
929:
924:
923:
914:
913:
893:
891:
890:
885:
883:
882:
874:
860:
858:
857:
852:
847:
846:
816:
814:
813:
808:
806:
805:
792:
790:
789:
784:
782:
781:
768:
766:
765:
760:
758:
757:
732:
730:
729:
724:
722:
721:
709:The Lie algebra
608:
601:
594:
551:Claude Chevalley
408:Complexification
251:Other Lie groups
137:
45:Classical groups
37:
19:
2990:
2989:
2985:
2984:
2983:
2981:
2980:
2979:
2965:
2964:
2950:
2945:
2939:
2923:
2917:
2904:
2900:
2895:
2894:
2886:
2882:
2877:
2873:
2860:
2853:
2840:
2836:
2823:
2816:
2803:
2799:
2794:
2777:
2748:
2743:
2742:
2721:
2716:
2715:
2694:
2689:
2688:
2667:
2662:
2661:
2607:
2606:
2582:
2569:
2564:
2563:
2535:
2515:
2510:
2509:
2481:
2480:
2426:
2425:
2401:
2388:
2383:
2382:
2354:
2334:
2329:
2328:
2300:
2299:
2225:
2224:
2203:
2190:
2177:
2172:
2171:
2143:
2123:
2103:
2098:
2097:
2069:
2068:
1978:
1977:
1956:
1943:
1930:
1917:
1912:
1911:
1884:
1883:
1851:
1850:
1825:
1824:
1800:
1795:
1794:
1769:
1768:
1744:
1739:
1738:
1713:
1712:
1689:
1684:
1683:
1658:
1657:
1634:
1629:
1628:
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2878:SpringerLink
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1581:Isomorphisms
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738:
734:
691:
687:
681:
672:
660:
643:Killing form
631:
626:, there are
620:mathematical
617:
561:Armand Borel
546:Hermann Weyl
420:
347:Loop algebra
329:Killing form
303:Lie algebras
180:
170:
160:
150:
116:
106:
96:
86:
76:
66:
56:
27:Lie algebras
2830:pp. 249–250
2810:pp. 248–251
635:Lie algebra
541:Élie Cartan
387:Root system
191:Exceptional
2898:References
2863:Knapp 2002
2843:Knapp 2002
2826:Knapp 2002
2806:Knapp 2002
1030:real forms
821:) acts by
697:Properties
684:Lie group.
682:semisimple
657:Definition
624:Lie theory
526:Sophus Lie
519:Scientists
392:Weyl group
113:Symplectic
73:Orthogonal
23:Lie groups
2890:Chapter 7
2888:Hall 2015
2781:Real form
2634:
2628:≅
2616:
2580:≅
2533:≅
2453:
2447:≅
2435:
2399:≅
2352:≅
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2264:≅
2252:
2246:≅
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2201:≅
2188:≅
2141:≅
2121:≅
2041:
2035:≅
2023:
2017:≅
2005:
1999:≅
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1954:≅
1941:≅
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1860:≥
1834:≥
1778:≥
1722:≥
1667:≥
1612:≥
872:
836:
704:reductive
671:negative
663:Lie group
651:real form
622:field of
403:Real form
289:Euclidean
140:Classical
2969:Category
2927:(2002),
2775:See also
894:acts by
675:definite
575:Glossary
269:Poincaré
2867:pp. 249
2847:pp. 249
632:compact
618:In the
483:physics
264:Lorentz
93:Unitary
2935:
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1028:among
875:
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259:Circle
2958:, in
2792:Notes
630:of a
334:Index
2933:ISBN
2911:ISBN
2741:and
2687:and
2631:Spin
2479:For
2450:Spin
2298:For
2249:Spin
2067:For
1882:For
1793:for
1767:and
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1627:for
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284:Loop
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