796:
32:
860:
are used to classify semisimple Lie groups or algebras (or algebraic groups) over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows. Satake diagrams, however, can be generalized to any field (see above) and fall
1167:
The unadorned Dynkin diagram (i.e., that with only white nodes and no arrows), when interpreted as a Satake diagram, represents the split real form of the Lie algebra, whereas it represents the compact form when interpreted as a Vogan diagram.
865:, whereas Vogan diagrams are defined specifically over the reals. Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.
941:
1128:
1019:
1162:
1053:
1077:
976:
890:
1457:
449:
497:
704:
A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules.
502:
492:
487:
307:
571:
454:
1352:
1286:
1258:
1216:
902:
602:
783:
is represented by drawing conjugate points of the Dynkin diagram near each other, and the Satake–Tits diagram is called a Satake diagram.
464:
1182:
1238:
1090:
981:
1436:
1242:
459:
439:
1133:
1024:
404:
312:
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444:
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79:
683:
656:
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399:
362:
330:
317:
1510:
1297:
431:
99:
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1177:
1058:
957:
893:
871:
59:
49:
827:
648:
588:
576:
417:
247:
1295:
Satake, Ichirô (1960), "On representations and compactifications of symmetric
Riemannian spaces",
1490:
1382:
1322:
348:
338:
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1420:
1348:
1314:
1282:
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1212:
952:
862:
844:
833:
527:
412:
375:
265:
1452:
1433:
Algebraic Groups and
Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)
836:
correspond to the Satake diagram with only white (i.e., non blackened) and unpaired vertices.
1466:
1410:
1400:
1347:, Lecture Notes in Pure and Applied Mathematics, vol. 3, New York: Marcel Dekker Inc.,
1306:
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1204:
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674:
over a field is a generalization of the Satake diagram to arbitrary fields, introduced by
671:
532:
285:
270:
41:
1396:
1415:
779:
is the field of real numbers, the absolute Galois group has order 2, and its action on
632:
552:
537:
370:
275:
1453:"Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque"
795:
1504:
1494:
868:
The essential difference is that the Satake diagram of a real semisimple Lie algebra
857:
693:
260:
89:
771:, together with the action of the Galois group, with the simple roots vanishing on
675:
557:
542:
343:
325:
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1342:
1276:
652:
620:
616:
383:
299:
23:
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1208:
522:
388:
280:
1478:
1318:
682:), that reduces the classification of reductive algebraic groups to that of
624:
19:
1424:
1278:
Lie groups and Lie algebras III: structure of Lie groups and Lie algebras
1405:
1083:), whereas Vogan diagrams are defined starting from a maximally compact
747:) has a Dynkin diagram with respect to some choice of positive roots of
1431:
Tits, Jacques (1966), "Classification of algebraic semisimple groups",
1326:
479:
1250:
936:{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}
1310:
31:
1387:
1199:, Graduate Texts in Mathematics, vol. 225, Berlin, New York:
751:. This Dynkin diagram has a natural action of the Galois group of
659:
of the complex Lie algebra corresponding to the Dynkin diagram.
790:
655:. The Satake diagrams associated to a Dynkin diagram classify
830:
correspond to the Satake diagram with all vertices blackened.
1369:
Spindel, Philippe; Persson, Daniel; Henneaux, Marc (2008),
1371:"Spacelike Singularities and Hidden Symmetries of Gravity"
1123:{\displaystyle \theta ({\mathfrak {h}})={\mathfrak {h}}}
1014:{\displaystyle \theta ({\mathfrak {h}})={\mathfrak {h}}}
1235:
Differential geometry, Lie groups, and symmetric spaces
807:
1079:
appears as the Lie algebra of the maximal split torus
1344:
Classification theory of semi-simple algebraic groups
1275:
Onishchik, A. L.; Vinberg, Ėrnest
Borisovich (1994),
1136:
1093:
1061:
1027:
984:
960:
947:) is defined by starting from a maximally noncompact
905:
874:
1055:
is as small as possible (in the presentation above,
1157:{\displaystyle {\mathfrak {h}}\cap {\mathfrak {k}}}
1048:{\displaystyle {\mathfrak {h}}\cap {\mathfrak {k}}}
1156:
1122:
1087:-stable Cartan subalgebra, that is, one for which
1071:
1047:
1013:
970:
935:
884:
840:
450:Representation theory of semisimple Lie algebras
1458:Journal für die reine und angewandte Mathematik
852:Differences between Satake and Vogan diagrams
731:defined over the separable algebraic closure
643:, p.109) whose configurations classify
596:
8:
696:of a Lie group, although they look similar.
711:is an algebraic group defined over a field
759:. Also some of the simple roots vanish on
603:
589:
488:Particle physics and representation theory
133:
30:
15:
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907:
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455:Representations of classical Lie groups
187:
136:
18:
640:
7:
692:Satake diagrams are not the same as
679:
308:Lie group–Lie algebra correspondence
1149:
1139:
1115:
1102:
1064:
1040:
1030:
1006:
993:
963:
928:
918:
908:
877:
1241:, vol. 34, Providence, R.I.:
14:
727:to be a maximal torus containing
1183:List of irreducible Tits indices
794:
775:colored black. In the case when
1239:Graduate Studies in Mathematics
1072:{\displaystyle {\mathfrak {h}}}
971:{\displaystyle {\mathfrak {h}}}
885:{\displaystyle {\mathfrak {g}}}
767:is given by the Dynkin diagram
1107:
1097:
998:
988:
943:(the +1 and −1 eigenspaces of
861:under the general paradigm of
503:Galilean group representations
498:Poincaré group representations
1:
1437:American Mathematical Society
1243:American Mathematical Society
493:Lorentz group representations
460:Theorem of the highest weight
1375:Living Reviews in Relativity
841:Onishchik & Vinberg 1994
719:be a maximal split torus in
715:, such as the reals. We let
1233:Helgason, Sigurdur (2001),
899:and associated Cartan pair
1527:
445:Lie algebra representation
1471:10.1515/crll.1971.247.196
1209:10.1007/978-1-4757-4094-3
1164:is as large as possible.
978:, that is, one for which
839:A table can be found at (
631:is a generalization of a
440:Lie group representation
1341:Satake, Ichiro (1971),
465:Borel–Weil–Bott theorem
1451:Tits, Jacques (1971),
1158:
1124:
1073:
1049:
1015:
972:
937:
886:
647:Lie algebras over the
363:Semisimple Lie algebra
318:Adjoint representation
1298:Annals of Mathematics
1195:Bump, Daniel (2004),
1159:
1125:
1074:
1050:
1016:
973:
938:
887:
432:Representation theory
1435:, Providence, R.I.:
1178:Relative root system
1134:
1091:
1059:
1025:
982:
958:
903:
872:
845:Table 4, pp. 229–230
828:Compact Lie algebras
662:More generally, the
1406:10.12942/lrr-2008-1
1397:2008LRR....11....1H
765:Satake–Tits diagram
668:Satake–Tits diagram
577:Table of Lie groups
418:Compact Lie algebra
1439:, pp. 33–62,
1154:
1120:
1069:
1045:
1011:
968:
933:
882:
834:Split Lie algebras
806:. You can help by
689:algebraic groups.
349:Affine Lie algebra
339:Simple Lie algebra
80:Special orthogonal
1354:978-0-8247-1607-3
1301:, Second Series,
1288:978-3-540-54683-2
1260:978-0-8218-2848-9
1218:978-0-387-21154-1
953:Cartan subalgebra
894:Cartan involution
863:Galois cohomology
824:
823:
613:
612:
413:Split Lie algebra
376:Cartan subalgebra
238:
237:
129:Simple Lie groups
1518:
1497:
1465:(247): 196–220,
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1390:
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856:Both Satake and
819:
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605:
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591:
548:Claude Chevalley
405:Complexification
248:Other Lie groups
134:
42:Classical groups
34:
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1311:10.2307/1969880
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1251:10.1090/gsm/034
1232:
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1201:Springer-Verlag
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811:
804:needs expansion
789:
702:
672:algebraic group
670:of a reductive
609:
564:
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533:Wilhelm Killing
517:
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371:Dynkin diagrams
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313:Exponential map
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271:Conformal group
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100:Special unitary
44:
12:
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858:Vogan diagrams
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837:
831:
822:
821:
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799:
788:
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701:
698:
694:Vogan diagrams
635:introduced by
633:Dynkin diagram
629:Satake diagram
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528:Henri Poincaré
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478:Lie groups in
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276:Diffeomorphism
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60:Special linear
57:
50:General linear
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1312:
1308:
1305:(1): 77–110,
1304:
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1035:
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859:
851:
846:
842:
838:
835:
832:
829:
826:
825:
818:
815:December 2009
809:
805:
802:This section
800:
797:
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792:
786:
784:
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766:
762:
758:
754:
750:
746:
742:
738:
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722:
718:
714:
710:
707:Suppose that
705:
699:
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690:
688:
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673:
669:
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587:
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583:
578:
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569:
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567:
559:
556:
554:
551:
549:
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544:
541:
539:
536:
534:
531:
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521:
520:
513:
512:
504:
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491:
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486:
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475:
474:
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208:
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194:
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95:
91:
88:
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81:
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71:
68:
65:
61:
58:
55:
51:
48:
47:
43:
38:
37:
33:
29:
28:
25:
21:
17:
1511:Lie algebras
1462:
1456:
1432:
1378:
1374:
1343:
1302:
1296:
1281:, Springer,
1277:
1234:
1196:
1166:
1084:
1080:
948:
944:
896:
867:
855:
812:
808:adding to it
803:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
708:
706:
703:
691:
667:
663:
661:
653:real numbers
628:
621:Lie algebras
617:mathematical
614:
558:Armand Borel
543:Hermann Weyl
344:Loop algebra
326:Killing form
300:Lie algebras
177:
167:
157:
147:
113:
103:
93:
83:
73:
63:
53:
24:Lie algebras
723:, and take
684:anisotropic
538:Élie Cartan
384:Root system
188:Exceptional
1197:Lie groups
1189:References
700:Definition
664:Tits index
657:real forms
625:Lie groups
523:Sophus Lie
516:Scientists
389:Weyl group
110:Symplectic
70:Orthogonal
20:Lie groups
1495:116999784
1479:0075-4102
1388:0710.1818
1319:0003-486X
1145:∩
1095:θ
1036:∩
986:θ
924:⊕
687:reductive
619:study of
400:Real form
286:Euclidean
137:Classical
1505:Category
1425:28179821
1381:(1): 1,
1172:See also
951:-stable
787:Examples
572:Glossary
266:Poincaré
1487:0277536
1445:0224710
1416:5255974
1393:Bibcode
1363:0316588
1335:0118775
1327:1969880
1269:1834454
1227:2062813
739:. Then
678: (
639: (
615:In the
480:physics
261:Lorentz
90:Unitary
1493:
1485:
1477:
1443:
1423:
1413:
1361:
1351:
1333:
1325:
1317:
1285:
1267:
1257:
1225:
1215:
763:. The
645:simple
637:Satake
256:Circle
1491:S2CID
1383:arXiv
1323:JSTOR
892:with
649:field
331:Index
1475:ISSN
1463:1971
1421:PMID
1349:ISBN
1315:ISSN
1283:ISBN
1255:ISBN
1213:ISBN
1130:and
1021:and
680:1966
676:Tits
641:1960
627:, a
623:and
281:Loop
22:and
1467:doi
1411:PMC
1401:doi
1307:doi
1247:doi
1205:doi
810:.
735:of
666:or
651:of
112:Sp(
102:SU(
82:SO(
62:SL(
52:GL(
1507::
1489:,
1483:MR
1481:,
1473:,
1461:,
1455:,
1441:MR
1419:,
1409:,
1399:,
1391:,
1379:11
1377:,
1373:,
1359:MR
1357:,
1331:MR
1329:,
1321:,
1313:,
1303:71
1265:MR
1263:,
1253:,
1245:,
1237:,
1223:MR
1221:,
1211:,
1203:,
847:).
843:,
92:U(
72:O(
1469::
1403::
1395::
1385::
1309::
1249::
1207::
1150:k
1140:h
1116:h
1111:=
1108:)
1103:h
1098:(
1085:θ
1081:S
1065:h
1041:k
1031:h
1007:h
1002:=
999:)
994:h
989:(
964:h
949:θ
945:θ
929:p
919:k
914:=
909:g
897:θ
878:g
817:)
813:(
781:D
777:k
773:S
769:D
761:S
757:k
755:/
753:K
749:T
745:K
743:(
741:G
737:k
733:K
729:S
725:T
721:G
717:S
713:k
709:G
604:e
597:t
590:v
230:8
228:E
222:7
220:E
214:6
212:E
206:4
204:F
198:2
196:G
178:n
175:D
168:n
165:C
158:n
155:B
148:n
145:A
116:)
114:n
106:)
104:n
96:)
94:n
86:)
84:n
76:)
74:n
66:)
64:n
56:)
54:n
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