233:
752:
886:
793:
696:
615:
544:
503:
574:
471:
387:
351:
327:
297:
273:
171:
147:
119:
92:
912:
845:
819:
932:
635:
176:
655:
235:
is a toral subalgebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional
1004:
970:
1022:
701:
850:
757:
660:
579:
508:
59:
476:
549:
446:
95:
354:
236:
47:
368:
332:
308:
278:
254:
152:
128:
122:
100:
73:
329:
over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if
50:
over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero
55:
43:
891:
824:
798:
358:
302:
For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra.
1000:
966:
917:
620:
240:
228:{\displaystyle \operatorname {ad} ({\mathfrak {h}})\subset {\mathfrak {gl}}({\mathfrak {g}})}
244:
980:
996:
976:
962:
640:
39:
1016:
989:
406:
954:
362:
248:
961:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York:
243:
and vice versa. In particular, a maximal toral Lie subalgebra in this setting is
17:
31:
51:
54:
elements. Over an algebraically closed field, every toral Lie algebra is
747:{\displaystyle -\lambda y=\operatorname {ad} _{\mathfrak {h}}(y)x}
239:, over an algebraically closed field of characteristic 0 is a
505:
is diagonalizable, it is enough to show the eigenvalues of
365:
is identically zero, contradicting semisimplicity. Hence,
42:
of a general linear Lie algebra all of whose elements are
991:
Introduction to Lie
Algebras and Representation Theory
881:{\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)}
788:{\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)}
691:{\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)}
610:{\displaystyle \operatorname {ad} _{\mathfrak {h}}(x)}
539:{\displaystyle \operatorname {ad} _{\mathfrak {h}}(x)}
920:
894:
853:
827:
801:
760:
704:
663:
643:
623:
582:
552:
511:
479:
449:
371:
335:
311:
281:
257:
179:
155:
131:
103:
76:
988:
926:
906:
880:
839:
813:
787:
746:
690:
649:
629:
609:
568:
538:
497:
465:
381:
345:
321:
291:
267:
227:
165:
141:
113:
86:
27:Lie algebra all of which elements are semisimple
305:In a finite-dimensional semisimple Lie algebra
888:with eigenvalue zero, a contradiction. Thus,
8:
389:must have a nonzero semisimple element, say
754:is a linear combination of eigenvectors of
247:, coincides with its centralizer, and the
942:
919:
893:
859:
858:
852:
826:
800:
766:
765:
759:
722:
721:
703:
669:
668:
662:
642:
622:
588:
587:
581:
560:
559:
551:
517:
516:
510:
478:
457:
456:
448:
431:
373:
372:
370:
337:
336:
334:
313:
312:
310:
283:
282:
280:
259:
258:
256:
216:
215:
203:
202:
190:
189:
178:
157:
156:
154:
133:
132:
130:
105:
104:
102:
78:
77:
75:
353:has only nilpotent elements, then it is
66:In semisimple and reductive Lie algebras
419:
795:with nonzero eigenvalues. But, unless
498:{\displaystyle \operatorname {ad} (x)}
427:
425:
423:
7:
569:{\displaystyle y\in {\mathfrak {h}}}
466:{\displaystyle x\in {\mathfrak {h}}}
860:
767:
723:
670:
589:
561:
518:
458:
374:
338:
314:
284:
260:
217:
207:
204:
191:
158:
134:
106:
79:
25:
382:{\displaystyle {\mathfrak {g}}}
346:{\displaystyle {\mathfrak {g}}}
322:{\displaystyle {\mathfrak {g}}}
292:{\displaystyle {\mathfrak {h}}}
268:{\displaystyle {\mathfrak {g}}}
166:{\displaystyle {\mathfrak {g}}}
142:{\displaystyle {\mathfrak {h}}}
114:{\displaystyle {\mathfrak {g}}}
87:{\displaystyle {\mathfrak {h}}}
875:
869:
782:
776:
738:
732:
685:
679:
604:
598:
533:
527:
492:
486:
222:
212:
196:
186:
1:
409:, in the theory of Lie groups
60:simultaneously diagonalizable
987:Humphreys, James E. (1972),
657:is a sum of eigenvectors of
443:Proof (from Humphreys): Let
397:is then a toral subalgebra.
945:, Ch. IV, § 15.3. Corollary
1039:
1023:Properties of Lie algebras
907:{\displaystyle \lambda =0}
840:{\displaystyle -\lambda y}
814:{\displaystyle \lambda =0}
58:; thus, its elements are
927:{\displaystyle \square }
630:{\displaystyle \lambda }
959:Linear algebraic groups
121:is called toral if the
928:
908:
882:
841:
815:
789:
748:
692:
651:
631:
611:
570:
540:
499:
467:
383:
347:
323:
293:
269:
229:
167:
143:
123:adjoint representation
115:
96:semisimple Lie algebra
88:
929:
909:
883:
847:is an eigenvector of
842:
816:
790:
749:
693:
652:
632:
612:
576:be an eigenvector of
571:
541:
500:
468:
393:; the linear span of
384:
348:
324:
294:
270:
237:reductive Lie algebra
230:
168:
144:
116:
89:
995:, Berlin, New York:
918:
892:
851:
825:
799:
758:
702:
661:
641:
621:
580:
550:
509:
477:
447:
369:
333:
309:
279:
255:
177:
153:
129:
101:
74:
924:
904:
878:
837:
811:
785:
744:
688:
647:
627:
607:
566:
546:are all zero. Let
536:
495:
463:
379:
343:
319:
299:is nondegenerate.
289:
265:
225:
163:
139:
111:
84:
1006:978-0-387-90053-7
972:978-0-387-97370-8
650:{\displaystyle x}
241:Cartan subalgebra
18:Toral Lie algebra
16:(Redirected from
1030:
1009:
994:
983:
946:
940:
934:
933:
931:
930:
925:
913:
911:
910:
905:
887:
885:
884:
879:
865:
864:
863:
846:
844:
843:
838:
820:
818:
817:
812:
794:
792:
791:
786:
772:
771:
770:
753:
751:
750:
745:
728:
727:
726:
697:
695:
694:
689:
675:
674:
673:
656:
654:
653:
648:
636:
634:
633:
628:
617:with eigenvalue
616:
614:
613:
608:
594:
593:
592:
575:
573:
572:
567:
565:
564:
545:
543:
542:
537:
523:
522:
521:
504:
502:
501:
496:
472:
470:
469:
464:
462:
461:
441:
435:
434:, Ch. II, § 8.1.
429:
388:
386:
385:
380:
378:
377:
361:), but then its
352:
350:
349:
344:
342:
341:
328:
326:
325:
320:
318:
317:
298:
296:
295:
290:
288:
287:
274:
272:
271:
266:
264:
263:
245:self-normalizing
234:
232:
231:
226:
221:
220:
211:
210:
195:
194:
172:
170:
169:
164:
162:
161:
148:
146:
145:
140:
138:
137:
120:
118:
117:
112:
110:
109:
93:
91:
90:
85:
83:
82:
36:toral subalgebra
21:
1038:
1037:
1033:
1032:
1031:
1029:
1028:
1027:
1013:
1012:
1007:
997:Springer-Verlag
986:
973:
963:Springer-Verlag
953:
950:
949:
941:
937:
916:
915:
890:
889:
854:
849:
848:
823:
822:
821:, we have that
797:
796:
761:
756:
755:
717:
700:
699:
664:
659:
658:
639:
638:
619:
618:
583:
578:
577:
548:
547:
512:
507:
506:
475:
474:
445:
444:
442:
438:
430:
421:
416:
403:
367:
366:
359:Engel's theorem
331:
330:
307:
306:
277:
276:
253:
252:
175:
174:
151:
150:
127:
126:
99:
98:
72:
71:
68:
28:
23:
22:
15:
12:
11:
5:
1036:
1034:
1026:
1025:
1015:
1014:
1011:
1010:
1005:
984:
971:
948:
947:
943:Humphreys 1972
935:
923:
903:
900:
897:
877:
874:
871:
868:
862:
857:
836:
833:
830:
810:
807:
804:
784:
781:
778:
775:
769:
764:
743:
740:
737:
734:
731:
725:
720:
716:
713:
710:
707:
687:
684:
681:
678:
672:
667:
646:
626:
606:
603:
600:
597:
591:
586:
563:
558:
555:
535:
532:
529:
526:
520:
515:
494:
491:
488:
485:
482:
460:
455:
452:
436:
432:Humphreys 1972
418:
417:
415:
412:
411:
410:
402:
399:
376:
340:
316:
286:
275:restricted to
262:
224:
219:
214:
209:
206:
201:
198:
193:
188:
185:
182:
160:
136:
108:
81:
67:
64:
48:diagonalizable
40:Lie subalgebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1035:
1024:
1021:
1020:
1018:
1008:
1002:
998:
993:
992:
985:
982:
978:
974:
968:
964:
960:
956:
955:Borel, Armand
952:
951:
944:
939:
936:
921:
901:
898:
895:
872:
866:
855:
834:
831:
828:
808:
805:
802:
779:
773:
762:
741:
735:
729:
718:
714:
711:
708:
705:
682:
676:
665:
644:
624:
601:
595:
584:
556:
553:
530:
524:
513:
489:
483:
480:
453:
450:
440:
437:
433:
428:
426:
424:
420:
413:
408:
407:Maximal torus
405:
404:
400:
398:
396:
392:
364:
360:
356:
303:
300:
250:
246:
242:
238:
199:
183:
180:
124:
97:
70:A subalgebra
65:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
990:
958:
938:
439:
394:
390:
363:Killing form
304:
301:
249:Killing form
69:
35:
29:
32:mathematics
414:References
44:semisimple
922:◻
896:λ
867:
832:λ
829:−
803:λ
774:
730:
709:λ
706:−
698:and then
677:
625:λ
596:
557:∈
525:
484:
454:∈
355:nilpotent
200:⊂
184:
52:nilpotent
1017:Category
957:(1991),
473:. Since
401:See also
981:1102012
637:. Then
56:abelian
1003:
979:
969:
94:of a
38:is a
1001:ISBN
967:ISBN
46:(or
34:, a
251:of
149:on
125:of
30:In
1019::
999:,
977:MR
975:,
965:,
914:.
856:ad
763:ad
719:ad
666:ad
585:ad
514:ad
481:ad
422:^
181:ad
173:,
62:.
902:0
899:=
876:)
873:y
870:(
861:h
835:y
809:0
806:=
783:)
780:y
777:(
768:h
742:x
739:)
736:y
733:(
724:h
715:=
712:y
686:)
683:y
680:(
671:h
645:x
605:)
602:x
599:(
590:h
562:h
554:y
534:)
531:x
528:(
519:h
493:)
490:x
487:(
459:h
451:x
395:x
391:x
375:g
357:(
339:g
315:g
285:h
261:g
223:)
218:g
213:(
208:l
205:g
197:)
192:h
187:(
159:g
135:h
107:g
80:h
20:)
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