4908:
2989:
1560:
1840:
1352:
1123:
1347:
1555:{\displaystyle H=\left\{\left.{\begin{pmatrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{pmatrix}}\right|\theta \in \mathbf {R} \right\}{\text{with Lie algebra }}{\mathfrak {h}}=\left\{\left.{\begin{pmatrix}i\theta &0\\0&ia\theta \end{pmatrix}}\right|\theta \in \mathbf {R} \right\},}
3034:
for concreteness and relative simplicity, since matrices and their exponential mapping are easier concepts than in the general case. Historically, this case was proven first, by John von
Neumann in 1929, and inspired Cartan to prove the full closed subgroup theorem in 1930. The proof for general
2458:. In this setting, one proves that every element of the group sufficiently close to the identity is the exponential of an element of the Lie algebra. (The proof is practically identical to the proof of the closed subgroup theorem presented below.) It follows every closed subgroup is an
1209:
287:
4277:
3463:
882:
800:
760:
976:
934:
208:
4133:
5255:
1073:
1043:
848:
824:
657:
556:
508:
481:
457:
415:
385:
361:
337:
313:
197:
131:
2596:
The closed subgroup theorem now simplifies the hypotheses considerably, a priori widening the class of homogeneous spaces. Every closed subgroup yields a homogeneous space.
1342:{\displaystyle G=\mathbb {T} ^{2}=\left\{\left.{\begin{pmatrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{pmatrix}}\right|\theta ,\phi \in \mathbf {R} \right\},}
582:
2257:
4489:. The topology generated by these bases is the relative topology. The conclusion is that the relative topology is the same as the group topology.
3355:
175:
is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra"
2305:
2310:
2300:
2295:
4650:. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in
2115:
2379:
2262:
5322:
853:
771:
731:
2410:
80:
5367:
1806:
for which one can find points in an arbitrarily small neighborhood (in the relative topology) of the identity that are
4917:, Theorem 1, Section 2.7 Rossmann states the theorem for linear groups. The statement is that there is an open subset
5478:
5454:
5403:
5385:
3153:
4695:
and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations
516:
that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma:
4896:
2946:
2272:
4968:
5415:(1929), "Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen",
2267:
2247:
1203:
939:
897:
2212:
2120:
1594:
is composed of infinitely many almost parallel line segments on the surface of the torus. This means that
3466:
3008:
2252:
5417:
1054:
1024:
829:
805:
638:
537:
489:
462:
438:
396:
366:
342:
318:
294:
178:
112:
1093:
coming from these coordinates is not the subset topology. That it so say, the identity component of
4782:
3549:
2533:
2403:
1887:
2599:
In a similar way, the closed subgroup theorem simplifies the hypothesis in the following theorem.
4447:. This is, by the way it is constructed, a neighborhood basis both in the group topology and the
2610:
979:
664:
2974:
is closed so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently,
2854:
2511:
2207:
2170:
2138:
2125:
1601:
282:{\displaystyle {\mathfrak {h}}=\left\{X\mid e^{tX}\in H,\,\,\forall t\in \mathbf {R} \right\}.}
5335:
4272:{\displaystyle (e^{S_{i}})^{m_{i}}=e^{m_{i}S_{i}}=e^{m_{i}\|S_{i}\|Y_{i}}\rightarrow e^{tY}.}
3241:
2239:
1907:
5061:
2980:
is an embedded Lie subgroup if and only if its group topology equals its relative topology.
1982:
1972:
1962:
1952:
5097:
1867:
1857:
8:
5473:
4457:
is analytic, the left and right translates of this neighborhood basis by a group element
2396:
2384:
2225:
2055:
5434:
4787:
3671:
2156:
2146:
1793:
567:
5450:
5438:
5399:
5381:
5363:
4448:
3000:
2583:
2220:
2183:
1816:. For closed subgroups this is not the case as the proof below of the theorem shows.
1585:
76:
2335:
2073:
5426:
5412:
5194:
2992:
2893:
2355:
2035:
2027:
2019:
2011:
2003:
1936:
1917:
1877:
88:
72:
36:
5318:
2661:
2340:
2093:
2078:
1849:
46:
1198:
For an example of a subgroup that is not an embedded Lie subgroup, consider the
3139:
We begin by establishing the key lemma stated in the "overview" section above.
2614:
2360:
2178:
2083:
601:
be the inverse of the exponential map. Then there is some smaller neighborhood
5342:
2345:
84:
5467:
5268:
is sequentially compact, meaning every sequence has a convergent subsequence.
3979:
3668:
3613:
3149:
2711:
is closed. A subgroup is locally closed if every point has a neighborhood in
2068:
1897:
1797:
3045:
3016:
2996:
2988:
2365:
2350:
2151:
2133:
2063:
157:. The first step is to identify something that could be the Lie algebra of
5230:
5182:
5170:
5158:
5146:
4883:. Here the metric obtained from the Hilbert–Schmidt inner product is used.
5362:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
5360:
Lie groups, Lie algebras, and representations: An elementary introduction
4647:
2191:
2107:
1831:
1087:
into a Lie group. It is important to note, however, that the topology on
20:
5206:
5430:
4528:. This is an analytic bijection with analytic inverse. Furthermore, if
3257:
3004:
2701:
2330:
2196:
2088:
92:
4905:, Theorem 20.10. Lee states and proves this theorem in all generality.
2637:
has a unique smooth manifold structure such that the action is smooth.
1647:, yet they cannot be connected to the identity with a path staying in
5398:, Oxford Graduate Texts in Mathematics, Oxford Science Publications,
1827:
1687:
is an analytic bijection, its inverse is not continuous. That is, if
1575:
68:
65:
50:
4977:, For linear groups, Hall proves a similar result in Corollary 3.45.
3702:. Suppose for the purpose of obtaining a contradiction that for all
3041:
is formally identical, except that elements of the Lie algebra are
5271:
2287:
417:
must be big enough to capture some interesting information about
32:
2423:
Because of the conclusion of the theorem, some authors chose to
762:. That is to say, in exponential coordinates near the identity,
5283:
5073:
79:) agreeing with the embedding. One of several results known as
5218:
5134:
5039:
5037:
5295:
3998:
henceforth refers to this subsequence. It will be shown that
1199:
1133:
2656:
being closed, hence an embedded Lie group, are given below.
1800:, hence not an embedding. There are also examples of groups
5034:
5022:
1839:
1482:
1368:
1240:
5085:
2879:
to a diagonal matrix with two entries of irrational ratio.
5380:, Springer Graduate Texts in Mathematics, vol. 218,
4997:
4995:
3458:{\displaystyle \Phi (S,T)=e^{tS}e^{tT}=I+tS+tT+O(t^{2}),}
2999:
as given here. He was prominent in many areas, including
1626:
one can find points in an arbitrarily small neighborhood
5330:
Biographical
Memoirs of the National Academy of Sciences
1122:
5345:(1930), "La théorie des groupes finis et continus et l'
5049:
1108:
In particular, the lemma stated above does not hold if
4992:
4953:
is an analytic bijection onto an open neighborhood of
1489:
1375:
1247:
1162:
in lowest terms, the helix will close up on itself at
1002:, which is the condition for an embedded submanifold.
877:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
795:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
755:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
5012:
5010:
4980:
4136:
3358:
1641:
of the identity that are exponentials of elements of
1355:
1212:
1057:
1027:
942:
900:
856:
832:
808:
774:
734:
641:
570:
540:
492:
465:
441:
399:
369:
345:
321:
297:
211:
181:
115:
5109:
4689:
are analytic since these operations are analytic in
1075:. Rossmann then goes on to introduce coordinates on
2949:, then the connected Lie subgroup with Lie algebra
1117:
5396:Lie Groups – An Introduction Through Linear Groups
5007:
4271:
3982:, there is a convergent subsequence converging to
3457:
1554:
1341:
1067:
1037:
970:
928:
876:
842:
818:
794:
754:
651:
576:
550:
502:
475:
451:
409:
387:, which one might hope to be the tangent space of
379:
355:
331:
307:
281:
191:
125:
1771:is a Lie group. With this topology the injection
1604:. In the group topology, the small open sets are
393:at the identity. For this idea to work, however,
5465:
2258:Representation theory of semisimple Lie algebras
1005:It is worth noting that Rossmann shows that for
4374:and the exponential restricted to the open set
3664:such that the conclusion of the theorem holds.
3634:with analytic inverse. It remains to show that
2641:
1608:line segments on the surface of the torus and
91:'s 1929 proof of a special case for groups of
3686:, linearly ordered by reverse inclusion with
2404:
4235:
4222:
3972:. It takes its values in the unit sphere in
3054:and the exponential mapping is the time one
1194:is irrational, the helix winds indefinitely.
710:. In these coordinates, the lemma says that
663:Once this has been established, one can use
5411:
5070:, Chapter 2, Proposition 1 and Corollary 7.
5001:
4389:is in analytic bijection with the open set
4329:. This is a contradiction. Hence, for some
5106:, See comment to Corollary 5, Section 2.2.
2995:in 1929 proved the theorem in the case of
2411:
2397:
2296:Particle physics and representation theory
1838:
1021:(not necessarily closed), the Lie algebra
3134:
1221:
564:diffeomorphically onto some neighborhood
256:
255:
5393:
5236:
5188:
5176:
5164:
5152:
5115:
5103:
5079:
5067:
4914:
2987:
1121:
5444:
5317:
5277:
5261:
5016:
4407:
3556:analytic, and thus there are open sets
2263:Representations of classical Lie groups
1620:The example shows that for some groups
971:{\displaystyle n=\dim({\mathfrak {g}})}
929:{\displaystyle k=\dim({\mathfrak {h}})}
169:at the identity. The challenge is that
5466:
5341:
5203:, The result follows from Theorem 5.6.
4986:
3548:, the identity. The hypothesis of the
2480:homogeneous space construction theorem
510:actually captures all the elements of
4492:Next, construct coordinate charts on
4287:is a group, the left hand side is in
2802:is a Lie subalgebra such that for no
2514:structure such that the quotient map
2510:, the left coset space, has a unique
1697:corresponds to a small open interval
5357:
5212:
5200:
5140:
5128:
5043:
5028:
4974:
4806:For this one can choose open balls,
2116:Lie group–Lie algebra correspondence
1081:that make the identity component of
558:such that the exponential map sends
486:The key step, then, is to show that
151:is a smooth embedded submanifold of
83:, it was first published in 1930 by
5375:
5301:
5289:
5249:
5224:
5091:
5055:
4902:
2892:be a Lie subalgebra. If there is a
1747:. However, with the group topology
1468:
1060:
1030:
960:
918:
869:
859:
835:
811:
787:
777:
747:
737:
644:
543:
495:
468:
444:
402:
372:
348:
324:
300:
214:
184:
139:be an arbitrary closed subgroup of
118:
13:
3359:
1741:due to the appearance of the sets
1668:is not a Lie group. While the map
1136:laid out on the surface picturing
257:
14:
5490:
1105:but not an embedded submanifold.
291:It is not difficult to show that
5378:Introduction to Smooth manifolds
4479:gives neighborhood bases at all
2646:A few sufficient conditions for
1540:
1452:
1327:
1118:Example of a non-closed subgroup
267:
163:, that is, the tangent space of
109:be a Lie group with Lie algebra
5353:, vol. XLII, pp. 1–61
5242:
5121:
4800:
2631:is a closed Lie subgroup, then
1819:
1204:irrational winding of the torus
1068:{\displaystyle {\mathfrak {g}}}
1038:{\displaystyle {\mathfrak {h}}}
843:{\displaystyle {\mathfrak {g}}}
819:{\displaystyle {\mathfrak {h}}}
652:{\displaystyle {\mathfrak {h}}}
551:{\displaystyle {\mathfrak {g}}}
503:{\displaystyle {\mathfrak {h}}}
476:{\displaystyle {\mathfrak {h}}}
452:{\displaystyle {\mathfrak {h}}}
410:{\displaystyle {\mathfrak {h}}}
380:{\displaystyle {\mathfrak {g}}}
356:{\displaystyle {\mathfrak {h}}}
332:{\displaystyle {\mathfrak {g}}}
308:{\displaystyle {\mathfrak {h}}}
192:{\displaystyle {\mathfrak {h}}}
145:. It is necessary to show that
126:{\displaystyle {\mathfrak {g}}}
5131:. See definition in Chapter 1.
4662:is an embedded submanifold of
4467:gives a neighborhood basis at
4250:
4158:
4137:
3870:is a neighborhood basis, with
3449:
3436:
3374:
3362:
2311:Galilean group representations
2306:Poincaré group representations
1099:is an immersed submanifold of
965:
955:
923:
913:
1:
5310:
4443:form a neighborhood basis at
3782:, there is a unique sequence
3154:Hilbert–Schmidt inner product
2301:Lorentz group representations
2268:Theorem of the highest weight
1584:and hence not closed. In the
978:. Thus, we have exhibited a "
5323:"John von Neumann 1903–1957"
4890:
4581:, then in these coordinates
4473:. These bases restricted to
1810:exponentials of elements of
429:were some large subgroup of
7:
5215:, Exercise 14 in Chapter 5.
4776:
3352:. Expand the exponentials,
3104:, so the specialization to
2959:
2642:Conditions for being closed
2536:. The left action given by
98:
10:
5495:
4451:. Since multiplication in
3923:Normalize the sequence in
3009:foundations of mathematics
2253:Lie algebra representation
716:corresponds to a point in
528:Take a small neighborhood
27:(sometimes referred to as
5445:Willard, Stephen (1970),
5418:Mathematische Zeitschrift
5239:, Problem 2. Section 2.7.
5191:, Problem 5. Section 2.7.
5179:, Problem 4. Section 2.7.
5167:, Problem 3. Section 2.7.
5155:, Problem 1. Section 2.7.
4671:Moreover, multiplication
4404:. This proves the lemma.
3283:is uniquely expressed as
2964:An embedded Lie subgroup
2828:, the group generated by
1632:in the relative topology
1588:, a small open subset of
5479:Theorems in group theory
4793:
4783:Inverse function theorem
4555:. By fixing a basis for
3550:inverse function theorem
3058:of the vector field. If
2983:
2935:is simply connected and
2248:Lie group representation
1617:locally path connected.
673:, that is, writing each
5394:Rossmann, Wulf (2002),
5358:Hall, Brian C. (2015),
4861:for some large enough
3015:The proof is given for
2777:is a compact group and
2611:transitive group action
2434:as closed subgroups of
2273:Borel–Weil–Bott theorem
1051:is a Lie subalgebra of
980:slice coordinate system
665:exponential coordinates
459:turned out to be zero,
315:is a Lie subalgebra of
25:closed-subgroup theorem
5449:, Dover Publications,
4773:are analytic as well.
4273:
4033:of integers such that
4022:and choose a sequence
3771:is a bijection on the
3459:
3162:be the Lie algebra of
3135:Proof of the key lemma
3012:
2855:one-parameter subgroup
2783:is a closed set, then
2512:real-analytic manifold
2171:Semisimple Lie algebra
2126:Adjoint representation
1602:locally path connected
1556:
1463:with Lie algebra
1343:
1195:
1069:
1039:
972:
930:
878:
844:
826:is just a subspace of
820:
796:
756:
653:
578:
552:
504:
483:would not be helpful.
477:
453:
411:
381:
357:
333:
309:
283:
193:
127:
93:linear transformations
87:, who was inspired by
4274:
3460:
3242:orthogonal complement
3116:instead of arbitrary
2991:
2240:Representation theory
1796:immersion, but not a
1557:
1344:
1125:
1070:
1040:
973:
931:
879:
845:
821:
797:
757:
654:
579:
553:
505:
478:
454:
412:
382:
358:
334:
310:
284:
194:
128:
5334:. See in particular
4625:is the dimension of
4408:Proof of the theorem
4134:
3724:contains an element
3356:
1353:
1210:
1055:
1025:
940:
898:
854:
830:
806:
772:
732:
683:(not necessarily in
639:
568:
538:
490:
463:
439:
397:
367:
343:
319:
295:
209:
179:
113:
39:. It states that if
16:Group theory theorem
5376:Lee, J. M. (2003),
5292:, Proposition 8.22.
2707:A subgroup that is
2498:closed Lie subgroup
2484: —
2385:Table of Lie groups
2226:Compact Lie algebra
1712:, there is no open
992:looks locally like
584:of the identity in
526: —
423:. If, for example,
5431:10.1007/BF01187749
5351:Mémorial Sc. Math.
5264:, By problem 17G,
5227:, Corollary 15.30.
4788:Lie correspondence
4656:. This shows that
4631:. This shows that
4269:
3672:neighborhood basis
3652:contain open sets
3552:is satisfied with
3455:
3256:decomposes as the
3013:
2482:
2157:Affine Lie algebra
2147:Simple Lie algebra
1888:Special orthogonal
1552:
1523:
1435:
1339:
1304:
1196:
1065:
1035:
968:
926:
874:
850:, this means that
840:
816:
792:
752:
649:
574:
548:
524:
500:
473:
449:
407:
377:
353:
329:
305:
279:
189:
123:
5413:von Neumann, John
5304:, Corollary 8.25.
5280:, Corollary 10.5.
5248:See for instance
5143:, Corollary 3.45.
4449:relative topology
3001:quantum mechanics
2477:
2432:matrix Lie groups
2428:linear Lie groups
2421:
2420:
2221:Split Lie algebra
2184:Cartan subalgebra
2046:
2045:
1937:Simple Lie groups
1586:relative topology
1568:irrational. Then
1464:
1349:and its subgroup
1132:. Imagine a bent
577:{\displaystyle V}
534:of the origin in
520:
363:is a subspace of
339:. In particular,
35:in the theory of
5486:
5459:
5447:General Topology
5441:
5408:
5390:
5372:
5354:
5333:
5327:
5305:
5299:
5293:
5287:
5281:
5275:
5269:
5259:
5253:
5246:
5240:
5234:
5228:
5222:
5216:
5210:
5204:
5198:
5192:
5186:
5180:
5174:
5168:
5162:
5156:
5150:
5144:
5138:
5132:
5125:
5119:
5113:
5107:
5101:
5095:
5089:
5083:
5077:
5071:
5065:
5059:
5053:
5047:
5041:
5032:
5026:
5020:
5014:
5005:
5002:von Neumann 1929
4999:
4990:
4984:
4978:
4972:
4966:
4964:
4958:
4952:
4926:
4912:
4906:
4900:
4884:
4882:
4866:
4860:
4850:
4848:
4847:
4838:
4835:
4804:
4772:
4754:
4736:
4730:
4712:
4694:
4688:
4682:
4677:, and inversion
4676:
4667:
4661:
4655:
4645:
4630:
4624:
4618:
4580:
4574:
4569:and identifying
4568:
4554:
4537:
4527:
4497:
4488:
4478:
4472:
4466:
4456:
4446:
4442:
4438:
4427:
4421:
4403:
4388:
4373:
4359:
4346:
4334:
4328:
4318:
4304:
4298:
4292:
4286:
4278:
4276:
4275:
4270:
4265:
4264:
4249:
4248:
4247:
4246:
4234:
4233:
4221:
4220:
4203:
4202:
4201:
4200:
4191:
4190:
4173:
4172:
4171:
4170:
4156:
4155:
4154:
4153:
4129:
4117:
4074:
4063:
4056:
4032:
4021:
4015:
3997:
3991:
3978:and since it is
3977:
3971:
3970:
3968:
3967:
3955:
3952:
3928:
3919:
3909:
3894:
3884:
3869:
3865:
3861:
3841:
3826:
3810:
3781:
3770:
3766:
3734:
3723:
3707:
3701:
3685:
3678:
3663:
3657:
3651:
3642:
3633:
3624:
3611:
3607:
3594:
3584:
3555:
3547:
3539:
3529:
3522:
3510:
3508:
3507:
3502:
3499:
3476:
3464:
3462:
3461:
3456:
3448:
3447:
3405:
3404:
3392:
3391:
3351:
3333:
3314:
3296:
3282:
3272:
3255:
3249:
3239:
3208:
3167:
3161:
3147:
3131:matters little.
3130:
3115:
3103:
3091:
3085:
3073:
3067:
3053:
3040:
3033:
2993:John von Neumann
2979:
2973:
2954:
2944:
2927:
2921:
2913:
2907:
2901:
2894:simply connected
2891:
2878:
2873:is similar over
2872:
2862:
2852:
2839:
2833:
2827:
2819:
2801:
2788:
2782:
2776:
2770:
2736:
2730:
2720:
2699:
2693:
2687:
2681:
2675:
2662:classical groups
2655:
2636:
2630:
2608:
2589:
2581:
2571:
2531:
2509:
2495:
2485:
2473:
2457:
2445:
2413:
2406:
2399:
2356:Claude Chevalley
2213:Complexification
2056:Other Lie groups
1942:
1941:
1850:Classical groups
1842:
1824:
1823:
1815:
1805:
1791:
1770:
1755:
1746:
1740:
1729:
1711:
1696:
1686:
1667:
1652:
1646:
1640:
1631:
1625:
1613:
1599:
1593:
1583:
1573:
1567:
1561:
1559:
1558:
1553:
1548:
1544:
1543:
1532:
1528:
1527:
1472:
1471:
1465:
1462:
1460:
1456:
1455:
1444:
1440:
1439:
1432:
1431:
1396:
1395:
1348:
1346:
1345:
1340:
1335:
1331:
1330:
1313:
1309:
1308:
1301:
1300:
1268:
1267:
1230:
1229:
1224:
1193:
1187:
1177:
1171:
1165:
1161:
1160:
1159:
1153:
1141:
1131:
1113:
1104:
1098:
1092:
1086:
1080:
1074:
1072:
1071:
1066:
1064:
1063:
1050:
1044:
1042:
1041:
1036:
1034:
1033:
1020:
1014:
1001:
991:
977:
975:
974:
969:
964:
963:
935:
933:
932:
927:
922:
921:
893:
883:
881:
880:
875:
873:
872:
863:
862:
849:
847:
846:
841:
839:
838:
825:
823:
822:
817:
815:
814:
801:
799:
798:
793:
791:
790:
781:
780:
767:
761:
759:
758:
753:
751:
750:
741:
740:
727:
721:
715:
709:
698:
688:
682:
672:
658:
656:
655:
650:
648:
647:
634:
626:
616:
610:
600:
589:
583:
581:
580:
575:
563:
557:
555:
554:
549:
547:
546:
533:
527:
515:
509:
507:
506:
501:
499:
498:
482:
480:
479:
474:
472:
471:
458:
456:
455:
450:
448:
447:
434:
428:
422:
416:
414:
413:
408:
406:
405:
392:
386:
384:
383:
378:
376:
375:
362:
360:
359:
354:
352:
351:
338:
336:
335:
330:
328:
327:
314:
312:
311:
306:
304:
303:
288:
286:
285:
280:
275:
271:
270:
245:
244:
218:
217:
204:
198:
196:
195:
190:
188:
187:
174:
168:
162:
156:
150:
144:
138:
132:
130:
129:
124:
122:
121:
108:
89:John von Neumann
81:Cartan's theorem
73:smooth structure
63:
57:
44:
29:Cartan's theorem
5494:
5493:
5489:
5488:
5487:
5485:
5484:
5483:
5464:
5463:
5462:
5457:
5406:
5388:
5370:
5325:
5313:
5308:
5300:
5296:
5288:
5284:
5276:
5272:
5260:
5256:
5247:
5243:
5235:
5231:
5223:
5219:
5211:
5207:
5199:
5195:
5187:
5183:
5175:
5171:
5163:
5159:
5151:
5147:
5139:
5135:
5126:
5122:
5114:
5110:
5102:
5098:
5090:
5086:
5078:
5074:
5066:
5062:
5054:
5050:
5046:, Theorem 3.42.
5042:
5035:
5031:, Theorem 3.20.
5027:
5023:
5015:
5008:
5000:
4993:
4985:
4981:
4973:
4969:
4960:
4954:
4928:
4918:
4913:
4909:
4901:
4897:
4893:
4888:
4887:
4881:
4874:
4868:
4862:
4839:
4836:
4833:
4832:
4830:
4828:
4819:
4807:
4805:
4801:
4796:
4779:
4756:
4738:
4732:
4714:
4696:
4690:
4684:
4678:
4672:
4663:
4657:
4651:
4643:
4632:
4626:
4620:
4612:
4599:
4588:
4582:
4576:
4570:
4556:
4545:
4539:
4529:
4505:
4499:
4498:. First define
4493:
4480:
4474:
4468:
4458:
4452:
4444:
4440:
4437:
4429:
4423:
4422:, the image in
4413:
4410:
4390:
4375:
4361:
4357:
4348:
4345:
4336:
4330:
4320:
4306:
4300:
4294:
4288:
4282:
4257:
4253:
4242:
4238:
4229:
4225:
4216:
4212:
4211:
4207:
4196:
4192:
4186:
4182:
4181:
4177:
4166:
4162:
4161:
4157:
4149:
4145:
4144:
4140:
4135:
4132:
4131:
4127:
4119:
4115:
4106:
4093:
4084:
4076:
4073:
4065:
4064:. For example,
4058:
4051:
4042:
4034:
4031:
4023:
4017:
3999:
3993:
3983:
3973:
3965:
3956:
3953:
3951:
3943:
3942:
3940:
3938:
3930:
3924:
3911:
3904:
3896:
3886:
3883:
3871:
3867:
3863:
3860:
3851:
3843:
3836:
3828:
3821:
3812:
3809:
3800:
3791:
3783:
3780:
3772:
3768:
3761:
3748:
3740:
3733:
3725:
3718:
3709:
3703:
3700:
3693:
3687:
3680:
3674:
3659:
3653:
3650:
3644:
3641:
3635:
3632:
3626:
3623:
3617:
3616:bijection from
3609:
3606:
3596:
3593:
3586:
3574:
3563:
3557:
3553:
3545:
3541:
3531:
3528:
3520:
3503:
3500:
3495:
3494:
3492:
3482:
3478:
3474:
3443:
3439:
3397:
3393:
3384:
3380:
3357:
3354:
3353:
3335:
3319:
3298:
3284:
3274:
3260:
3251:
3245:
3210:
3182:
3169:
3163:
3157:
3143:
3137:
3117:
3105:
3093:
3087:
3075:
3069:
3059:
3049:
3036:
3020:
2986:
2975:
2965:
2962:
2950:
2936:
2923:
2915:
2909:
2903:
2897:
2883:
2874:
2868:
2867:if and only if
2858:
2844:
2835:
2834:, is closed in
2829:
2821:
2803:
2793:
2784:
2778:
2772:
2741:
2732:
2722:
2712:
2695:
2689:
2683:
2677:
2665:
2647:
2644:
2632:
2622:
2604:
2594:
2585:
2573:
2566:
2560:
2550:
2543:
2537:
2532:is an analytic
2515:
2501:
2487:
2483:
2463:
2462:submanifold of
2447:
2435:
2417:
2372:
2371:
2370:
2341:Wilhelm Killing
2325:
2317:
2316:
2315:
2290:
2279:
2278:
2277:
2242:
2232:
2231:
2230:
2217:
2201:
2179:Dynkin diagrams
2173:
2163:
2162:
2161:
2143:
2121:Exponential map
2110:
2100:
2099:
2098:
2079:Conformal group
2058:
2048:
2047:
2039:
2031:
2023:
2015:
2007:
1988:
1978:
1968:
1958:
1939:
1929:
1928:
1927:
1908:Special unitary
1852:
1822:
1811:
1801:
1792:is an analytic
1786:
1772:
1768:
1757:
1754:
1748:
1742:
1731:
1727:
1713:
1698:
1688:
1684:
1669:
1665:
1654:
1648:
1642:
1639:
1633:
1627:
1621:
1609:
1595:
1589:
1579:
1569:
1563:
1539:
1522:
1521:
1510:
1504:
1503:
1498:
1485:
1484:
1481:
1480:
1476:
1467:
1466:
1461:
1451:
1434:
1433:
1415:
1411:
1409:
1403:
1402:
1397:
1382:
1378:
1371:
1370:
1367:
1366:
1362:
1354:
1351:
1350:
1326:
1303:
1302:
1287:
1283:
1281:
1275:
1274:
1269:
1254:
1250:
1243:
1242:
1239:
1238:
1234:
1225:
1220:
1219:
1211:
1208:
1207:
1189:
1183:
1173:
1167:
1163:
1155:
1149:
1148:
1143:
1137:
1127:
1120:
1114:is not closed.
1109:
1100:
1094:
1088:
1082:
1076:
1059:
1058:
1056:
1053:
1052:
1046:
1029:
1028:
1026:
1023:
1022:
1016:
1010:
993:
983:
959:
958:
941:
938:
937:
917:
916:
899:
896:
895:
885:
868:
867:
858:
857:
855:
852:
851:
834:
833:
831:
828:
827:
810:
809:
807:
804:
803:
786:
785:
776:
775:
773:
770:
769:
763:
746:
745:
736:
735:
733:
730:
729:
723:
717:
711:
700:
690:
684:
674:
668:
661:
643:
642:
640:
637:
636:
628:
618:
612:
602:
591:
585:
569:
566:
565:
559:
542:
541:
539:
536:
535:
529:
525:
511:
494:
493:
491:
488:
487:
467:
466:
464:
461:
460:
443:
442:
440:
437:
436:
430:
424:
418:
401:
400:
398:
395:
394:
388:
371:
370:
368:
365:
364:
347:
346:
344:
341:
340:
323:
322:
320:
317:
316:
299:
298:
296:
293:
292:
266:
237:
233:
226:
222:
213:
212:
210:
207:
206:
205:by the formula
200:
183:
182:
180:
177:
176:
170:
164:
158:
152:
146:
140:
134:
117:
116:
114:
111:
110:
104:
101:
75:(and hence the
59:
53:
47:closed subgroup
40:
17:
12:
11:
5:
5492:
5482:
5481:
5476:
5461:
5460:
5455:
5442:
5409:
5404:
5391:
5386:
5373:
5369:978-3319134666
5368:
5355:
5347:Analysis Situs
5339:
5314:
5312:
5309:
5307:
5306:
5294:
5282:
5270:
5254:
5241:
5229:
5217:
5205:
5193:
5181:
5169:
5157:
5145:
5133:
5120:
5108:
5096:
5094:, Example 7.3.
5084:
5082:, Section 2.3.
5072:
5060:
5048:
5033:
5021:
5006:
4991:
4979:
4967:
4907:
4894:
4892:
4889:
4886:
4885:
4879:
4872:
4824:
4815:
4798:
4797:
4795:
4792:
4791:
4790:
4785:
4778:
4775:
4641:
4608:
4597:
4586:
4543:
4503:
4433:
4409:
4406:
4353:
4341:
4268:
4263:
4260:
4256:
4252:
4245:
4241:
4237:
4232:
4228:
4224:
4219:
4215:
4210:
4206:
4199:
4195:
4189:
4185:
4180:
4176:
4169:
4165:
4160:
4152:
4148:
4143:
4139:
4123:
4111:
4102:
4089:
4080:
4069:
4047:
4038:
4027:
3961:
3947:
3934:
3900:
3879:
3862:converging to
3856:
3847:
3832:
3817:
3805:
3796:
3787:
3776:
3767:. Then, since
3757:
3744:
3729:
3714:
3698:
3691:
3648:
3639:
3630:
3621:
3604:
3591:
3572:
3561:
3543:
3530:is seen to be
3523:
3480:
3454:
3451:
3446:
3442:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3403:
3400:
3396:
3390:
3387:
3383:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3178:
3136:
3133:
3043:left invariant
2985:
2982:
2961:
2958:
2957:
2956:
2929:
2908:isomorphic to
2896:compact group
2880:
2841:
2790:
2738:
2709:locally closed
2705:
2664:are closed in
2643:
2640:
2639:
2638:
2615:isotropy group
2609:is a set with
2564:
2558:
2548:
2541:
2475:
2419:
2418:
2416:
2415:
2408:
2401:
2393:
2390:
2389:
2388:
2387:
2382:
2374:
2373:
2369:
2368:
2363:
2361:Harish-Chandra
2358:
2353:
2348:
2343:
2338:
2336:Henri Poincaré
2333:
2327:
2326:
2323:
2322:
2319:
2318:
2314:
2313:
2308:
2303:
2298:
2292:
2291:
2286:Lie groups in
2285:
2284:
2281:
2280:
2276:
2275:
2270:
2265:
2260:
2255:
2250:
2244:
2243:
2238:
2237:
2234:
2233:
2229:
2228:
2223:
2218:
2216:
2215:
2210:
2204:
2202:
2200:
2199:
2194:
2188:
2186:
2181:
2175:
2174:
2169:
2168:
2165:
2164:
2160:
2159:
2154:
2149:
2144:
2142:
2141:
2136:
2130:
2128:
2123:
2118:
2112:
2111:
2106:
2105:
2102:
2101:
2097:
2096:
2091:
2086:
2084:Diffeomorphism
2081:
2076:
2071:
2066:
2060:
2059:
2054:
2053:
2050:
2049:
2044:
2043:
2042:
2041:
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
1998:
1997:
1993:
1992:
1991:
1990:
1984:
1980:
1974:
1970:
1964:
1960:
1954:
1947:
1946:
1940:
1935:
1934:
1931:
1930:
1926:
1925:
1915:
1905:
1895:
1885:
1875:
1868:Special linear
1865:
1858:General linear
1854:
1853:
1848:
1847:
1844:
1843:
1835:
1834:
1821:
1818:
1784:
1766:
1752:
1725:
1682:
1663:
1637:
1551:
1547:
1542:
1538:
1535:
1531:
1526:
1520:
1517:
1514:
1511:
1509:
1506:
1505:
1502:
1499:
1497:
1494:
1491:
1490:
1488:
1483:
1479:
1475:
1470:
1459:
1454:
1450:
1447:
1443:
1438:
1430:
1427:
1424:
1421:
1418:
1414:
1410:
1408:
1405:
1404:
1401:
1398:
1394:
1391:
1388:
1385:
1381:
1377:
1376:
1374:
1369:
1365:
1361:
1358:
1338:
1334:
1329:
1325:
1322:
1319:
1316:
1312:
1307:
1299:
1296:
1293:
1290:
1286:
1282:
1280:
1277:
1276:
1273:
1270:
1266:
1263:
1260:
1257:
1253:
1249:
1248:
1246:
1241:
1237:
1233:
1228:
1223:
1218:
1215:
1119:
1116:
1062:
1032:
967:
962:
957:
954:
951:
948:
945:
925:
920:
915:
912:
909:
906:
903:
871:
866:
861:
837:
813:
789:
784:
779:
749:
744:
739:
646:
573:
545:
518:
497:
470:
446:
404:
374:
350:
326:
302:
278:
274:
269:
265:
262:
259:
254:
251:
248:
243:
240:
236:
232:
229:
225:
221:
216:
186:
120:
100:
97:
77:group topology
15:
9:
6:
4:
3:
2:
5491:
5480:
5477:
5475:
5472:
5471:
5469:
5458:
5456:0-486-43479-6
5452:
5448:
5443:
5440:
5436:
5432:
5428:
5424:
5421:(in German),
5420:
5419:
5414:
5410:
5407:
5405:0-19-859683-9
5401:
5397:
5392:
5389:
5387:0-387-95448-1
5383:
5379:
5374:
5371:
5365:
5361:
5356:
5352:
5348:
5344:
5340:
5337:
5331:
5324:
5320:
5316:
5315:
5303:
5298:
5291:
5286:
5279:
5274:
5267:
5263:
5258:
5251:
5245:
5238:
5237:Rossmann 2002
5233:
5226:
5221:
5214:
5209:
5202:
5197:
5190:
5189:Rossmann 2002
5185:
5178:
5177:Rossmann 2002
5173:
5166:
5165:Rossmann 2002
5161:
5154:
5153:Rossmann 2002
5149:
5142:
5137:
5130:
5124:
5117:
5116:Rossmann 2002
5112:
5105:
5104:Rossmann 2002
5100:
5093:
5088:
5081:
5080:Rossmann 2002
5076:
5069:
5068:Rossmann 2002
5064:
5057:
5052:
5045:
5040:
5038:
5030:
5025:
5018:
5013:
5011:
5003:
4998:
4996:
4988:
4983:
4976:
4971:
4963:
4957:
4951:
4947:
4943:
4939:
4935:
4931:
4925:
4921:
4916:
4915:Rossmann 2002
4911:
4904:
4899:
4895:
4878:
4871:
4865:
4858:
4854:
4846:
4842:
4827:
4823:
4818:
4814:
4810:
4803:
4799:
4789:
4786:
4784:
4781:
4780:
4774:
4771:
4767:
4763:
4759:
4753:
4749:
4745:
4741:
4737:is embedded,
4735:
4729:
4725:
4721:
4717:
4711:
4707:
4703:
4699:
4693:
4687:
4681:
4675:
4669:
4666:
4660:
4654:
4649:
4640:
4636:
4629:
4623:
4617:), 0, ..., 0)
4616:
4611:
4607:
4603:
4596:
4592:
4585:
4579:
4573:
4567:
4563:
4559:
4553:
4549:
4542:
4536:
4532:
4525:
4521:
4517:
4513:
4509:
4502:
4496:
4490:
4487:
4483:
4477:
4471:
4465:
4461:
4455:
4450:
4436:
4432:
4426:
4420:
4416:
4405:
4402:
4398:
4394:
4387:
4383:
4379:
4372:
4368:
4364:
4356:
4351:
4344:
4339:
4333:
4327:
4323:
4317:
4313:
4309:
4303:
4297:
4291:
4285:
4279:
4266:
4261:
4258:
4254:
4243:
4239:
4230:
4226:
4217:
4213:
4208:
4204:
4197:
4193:
4187:
4183:
4178:
4174:
4167:
4163:
4150:
4146:
4141:
4126:
4122:
4114:
4110:
4105:
4101:
4097:
4092:
4088:
4083:
4079:
4072:
4068:
4061:
4055:
4050:
4046:
4041:
4037:
4030:
4026:
4020:
4014:
4010:
4006:
4002:
3996:
3990:
3986:
3981:
3976:
3964:
3960:
3950:
3946:
3937:
3933:
3927:
3921:
3918:
3914:
3908:
3903:
3899:
3893:
3889:
3882:
3878:
3874:
3859:
3855:
3850:
3846:
3840:
3835:
3831:
3825:
3820:
3816:
3808:
3804:
3799:
3795:
3790:
3786:
3779:
3775:
3765:
3760:
3756:
3752:
3747:
3743:
3738:
3732:
3728:
3722:
3717:
3713:
3706:
3697:
3690:
3684:
3677:
3673:
3670:
3665:
3662:
3656:
3647:
3638:
3629:
3620:
3615:
3614:real-analytic
3603:
3599:
3590:
3582:
3578:
3571:
3567:
3560:
3551:
3538:
3534:
3526:
3518:
3514:
3506:
3498:
3490:
3486:
3472:
3468:
3452:
3444:
3440:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3401:
3398:
3394:
3388:
3385:
3381:
3377:
3371:
3368:
3365:
3350:
3347:
3343:
3339:
3331:
3327:
3323:
3318:Define a map
3316:
3313:
3309:
3305:
3301:
3295:
3291:
3287:
3281:
3277:
3271:
3267:
3263:
3259:
3254:
3248:
3243:
3237:
3233:
3229:
3225:
3221:
3217:
3213:
3206:
3202:
3198:
3194:
3190:
3186:
3181:
3176:
3172:
3166:
3160:
3155:
3151:
3150:inner product
3146:
3140:
3132:
3128:
3124:
3120:
3113:
3109:
3101:
3097:
3092:is closed in
3090:
3083:
3079:
3072:
3066:
3062:
3057:
3052:
3047:
3046:vector fields
3044:
3039:
3031:
3027:
3023:
3018:
3017:matrix groups
3010:
3006:
3002:
2998:
2997:matrix groups
2994:
2990:
2981:
2978:
2972:
2968:
2953:
2948:
2943:
2939:
2934:
2930:
2926:
2922:is closed in
2919:
2912:
2906:
2900:
2895:
2890:
2886:
2881:
2877:
2871:
2866:
2861:
2857:generated by
2856:
2851:
2847:
2842:
2838:
2832:
2825:
2818:
2814:
2810:
2806:
2800:
2796:
2791:
2787:
2781:
2775:
2768:
2764:
2760:
2756:
2752:
2748:
2744:
2739:
2735:
2731:is closed in
2729:
2725:
2719:
2715:
2710:
2706:
2703:
2698:
2692:
2686:
2680:
2673:
2669:
2663:
2659:
2658:
2657:
2654:
2650:
2635:
2629:
2625:
2620:
2616:
2612:
2607:
2602:
2601:
2600:
2597:
2593:
2591:
2588:
2580:
2576:
2570:
2563:
2557:
2553:
2547:
2540:
2535:
2530:
2526:
2522:
2518:
2513:
2508:
2504:
2499:
2494:
2490:
2481:
2474:
2471:
2467:
2461:
2455:
2451:
2443:
2439:
2433:
2429:
2426:
2414:
2409:
2407:
2402:
2400:
2395:
2394:
2392:
2391:
2386:
2383:
2381:
2378:
2377:
2376:
2375:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2328:
2321:
2320:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2293:
2289:
2283:
2282:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2245:
2241:
2236:
2235:
2227:
2224:
2222:
2219:
2214:
2211:
2209:
2206:
2205:
2203:
2198:
2195:
2193:
2190:
2189:
2187:
2185:
2182:
2180:
2177:
2176:
2172:
2167:
2166:
2158:
2155:
2153:
2150:
2148:
2145:
2140:
2137:
2135:
2132:
2131:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2109:
2104:
2103:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2057:
2052:
2051:
2040:
2034:
2032:
2026:
2024:
2018:
2016:
2010:
2008:
2002:
2001:
2000:
1999:
1995:
1994:
1989:
1987:
1981:
1979:
1977:
1971:
1969:
1967:
1961:
1959:
1957:
1951:
1950:
1949:
1948:
1944:
1943:
1938:
1933:
1932:
1923:
1919:
1916:
1913:
1909:
1906:
1903:
1899:
1896:
1893:
1889:
1886:
1883:
1879:
1876:
1873:
1869:
1866:
1863:
1859:
1856:
1855:
1851:
1846:
1845:
1841:
1837:
1836:
1833:
1829:
1826:
1825:
1817:
1814:
1809:
1804:
1799:
1798:homeomorphism
1795:
1790:
1783:
1779:
1775:
1765:
1761:
1751:
1745:
1739:
1735:
1724:
1720:
1716:
1710:
1706:
1702:
1695:
1691:
1681:
1677:
1673:
1662:
1658:
1651:
1645:
1636:
1630:
1624:
1618:
1616:
1612:
1607:
1603:
1598:
1592:
1587:
1582:
1577:
1572:
1566:
1549:
1545:
1536:
1533:
1529:
1524:
1518:
1515:
1512:
1507:
1500:
1495:
1492:
1486:
1477:
1473:
1457:
1448:
1445:
1441:
1436:
1428:
1425:
1422:
1419:
1416:
1412:
1406:
1399:
1392:
1389:
1386:
1383:
1379:
1372:
1363:
1359:
1356:
1336:
1332:
1323:
1320:
1317:
1314:
1310:
1305:
1297:
1294:
1291:
1288:
1284:
1278:
1271:
1264:
1261:
1258:
1255:
1251:
1244:
1235:
1231:
1226:
1216:
1213:
1205:
1201:
1192:
1186:
1182:rotations in
1181:
1176:
1172:rotations in
1170:
1158:
1152:
1146:
1140:
1135:
1130:
1124:
1115:
1112:
1106:
1103:
1097:
1091:
1085:
1079:
1049:
1019:
1013:
1008:
1003:
1000:
996:
990:
986:
981:
952:
949:
946:
943:
910:
907:
904:
901:
892:
888:
884:is just like
864:
782:
766:
742:
726:
722:precisely if
720:
714:
707:
703:
697:
693:
687:
681:
677:
671:
666:
660:
632:
625:
621:
615:
611:such that if
609:
605:
599:
595:
588:
571:
562:
532:
523:
517:
514:
484:
433:
427:
421:
391:
289:
276:
272:
263:
260:
252:
249:
246:
241:
238:
234:
230:
227:
223:
219:
203:
173:
167:
161:
155:
149:
143:
137:
107:
96:
94:
90:
86:
82:
78:
74:
70:
67:
62:
56:
52:
48:
43:
38:
34:
30:
26:
22:
5446:
5422:
5416:
5395:
5377:
5359:
5350:
5346:
5343:Cartan, Élie
5329:
5297:
5285:
5278:Willard 1970
5273:
5265:
5262:Willard 1970
5257:
5244:
5232:
5220:
5208:
5196:
5184:
5172:
5160:
5148:
5136:
5123:
5111:
5099:
5087:
5075:
5063:
5058:, Chapter 5.
5051:
5024:
5017:Bochner 1958
4982:
4970:
4961:
4955:
4949:
4945:
4941:
4937:
4933:
4929:
4923:
4919:
4910:
4898:
4876:
4869:
4863:
4856:
4852:
4844:
4840:
4825:
4821:
4816:
4812:
4808:
4802:
4769:
4765:
4761:
4757:
4751:
4747:
4743:
4739:
4733:
4731:. But since
4727:
4723:
4719:
4715:
4709:
4705:
4701:
4697:
4691:
4685:
4679:
4673:
4670:
4664:
4658:
4652:
4638:
4634:
4627:
4621:
4614:
4609:
4605:
4601:
4594:
4590:
4583:
4577:
4571:
4565:
4561:
4557:
4551:
4547:
4540:
4534:
4530:
4523:
4519:
4515:
4511:
4507:
4500:
4494:
4491:
4485:
4481:
4475:
4469:
4463:
4459:
4453:
4434:
4430:
4424:
4418:
4414:
4411:
4400:
4396:
4392:
4385:
4381:
4377:
4370:
4366:
4362:
4354:
4349:
4342:
4337:
4331:
4325:
4321:
4315:
4311:
4307:
4301:
4295:
4289:
4283:
4280:
4124:
4120:
4118:will do, as
4112:
4108:
4103:
4099:
4095:
4090:
4086:
4081:
4077:
4070:
4066:
4059:
4053:
4048:
4044:
4039:
4035:
4028:
4024:
4018:
4012:
4008:
4004:
4000:
3994:
3992:. The index
3988:
3984:
3974:
3962:
3958:
3948:
3944:
3935:
3931:
3925:
3922:
3916:
3912:
3906:
3901:
3897:
3891:
3887:
3880:
3876:
3872:
3857:
3853:
3848:
3844:
3838:
3833:
3829:
3823:
3818:
3814:
3806:
3802:
3797:
3793:
3788:
3784:
3777:
3773:
3763:
3758:
3754:
3750:
3745:
3741:
3739:on the form
3736:
3730:
3726:
3720:
3715:
3711:
3704:
3695:
3688:
3682:
3675:
3666:
3660:
3654:
3645:
3636:
3627:
3618:
3601:
3597:
3588:
3580:
3576:
3569:
3565:
3558:
3536:
3532:
3524:
3516:
3512:
3504:
3496:
3488:
3484:
3471:differential
3470:
3348:
3345:
3341:
3337:
3329:
3325:
3321:
3317:
3311:
3307:
3303:
3299:
3293:
3289:
3285:
3279:
3275:
3269:
3265:
3261:
3252:
3246:
3235:
3231:
3227:
3223:
3219:
3215:
3211:
3204:
3200:
3196:
3192:
3188:
3184:
3179:
3174:
3170:
3164:
3158:
3144:
3141:
3138:
3126:
3122:
3118:
3111:
3107:
3099:
3095:
3088:
3081:
3077:
3070:
3064:
3060:
3055:
3050:
3042:
3037:
3029:
3025:
3021:
3014:
2976:
2970:
2966:
2963:
2951:
2941:
2937:
2932:
2924:
2917:
2910:
2904:
2898:
2888:
2884:
2875:
2869:
2864:
2859:
2849:
2845:
2836:
2830:
2823:
2816:
2812:
2808:
2804:
2798:
2794:
2785:
2779:
2773:
2766:
2762:
2758:
2754:
2750:
2746:
2742:
2733:
2727:
2723:
2717:
2713:
2708:
2696:
2690:
2684:
2678:
2671:
2667:
2652:
2648:
2645:
2633:
2627:
2623:
2618:
2605:
2598:
2595:
2586:
2584:homogeneous
2578:
2574:
2568:
2561:
2555:
2551:
2545:
2538:
2528:
2524:
2520:
2516:
2506:
2502:
2497:
2492:
2488:
2479:
2476:
2469:
2465:
2459:
2453:
2449:
2441:
2437:
2431:
2427:
2424:
2422:
2366:Armand Borel
2351:Hermann Weyl
2152:Loop algebra
2134:Killing form
2108:Lie algebras
1985:
1975:
1965:
1955:
1921:
1911:
1901:
1891:
1881:
1871:
1861:
1832:Lie algebras
1820:Applications
1812:
1807:
1802:
1788:
1781:
1777:
1773:
1763:
1759:
1749:
1743:
1737:
1733:
1722:
1718:
1714:
1708:
1704:
1700:
1693:
1689:
1679:
1675:
1671:
1660:
1656:
1653:. The group
1649:
1643:
1634:
1628:
1622:
1619:
1614:
1610:
1605:
1596:
1590:
1580:
1570:
1564:
1197:
1190:
1184:
1179:
1174:
1168:
1156:
1150:
1144:
1138:
1128:
1110:
1107:
1101:
1095:
1089:
1083:
1077:
1047:
1017:
1011:
1006:
1004:
998:
994:
988:
984:
890:
886:
764:
724:
718:
712:
705:
701:
695:
691:
685:
679:
675:
669:
662:
630:
623:
619:
613:
607:
603:
597:
593:
586:
560:
530:
521:
519:
512:
485:
431:
425:
419:
389:
290:
201:
171:
165:
159:
153:
147:
141:
135:
105:
102:
60:
54:
41:
28:
24:
18:
5425:(1): 3–42,
5336:p. 441
5319:Bochner, S.
4987:Cartan 1930
4648:slice chart
4305:is closed,
3667:Consider a
3467:pushforward
3168:defined as
3156:), and let
3152:(e.g., the
2955:is closed.
2853:, then the
2702:quaternions
2621:of a point
2346:Élie Cartan
2192:Root system
1996:Exceptional
1670:exp :
982:" in which
768:looks like
728:belongs to
635:belongs to
617:belongs to
85:Élie Cartan
21:mathematics
5474:Lie groups
5468:Categories
5311:References
5252:Chapter 21
4927:such that
4867:such that
4075:such that
3842:such that
3608:such that
3273:, so each
3258:direct sum
3074:closed in
3005:set theory
2865:not closed
2789:is closed.
2721:such that
2619:stabilizer
2534:submersion
2331:Sophus Lie
2324:Scientists
2197:Weyl group
1918:Symplectic
1878:Orthogonal
1828:Lie groups
1126:The torus
590:, and let
133:. Now let
37:Lie groups
5439:122565679
5332:: 438–456
5213:Hall 2015
5201:Hall 2015
5141:Hall 2015
5129:Hall 2015
5044:Hall 2015
5029:Hall 2015
4975:Hall 2015
4891:Citations
4335:the sets
4251:→
4236:‖
4223:‖
3920:as well.
3669:countable
3360:Φ
3320:Φ :
2208:Real form
2094:Euclidean
1945:Classical
1794:injective
1776: : (
1537:∈
1534:θ
1519:θ
1496:θ
1449:∈
1446:θ
1429:θ
1420:π
1393:θ
1387:π
1324:∈
1321:ϕ
1315:θ
1298:ϕ
1292:π
1265:θ
1259:π
1009:subgroup
953:
911:
865:⊂
783:⊂
743:⊂
264:∈
258:∀
247:∈
231:∣
71:with the
69:Lie group
51:Lie group
5321:(1958),
5302:Lee 2003
5290:Lee 2003
5250:Lee 2003
5225:Lee 2003
5092:Lee 2003
5056:Lee 2003
4903:Lee 2003
4777:See also
4760: :
4742: :
4718: :
4700: :
4619:, where
4604:), ...,
4506: :
4360:satisfy
4319:, hence
4299:. Since
4293:for all
3885:. Since
3866:because
3735:that is
3465:and the
3148:with an
3007:and the
2960:Converse
2811:∖
2771:, where
2676:, where
2613:and the
2460:embedded
2380:Glossary
2074:Poincaré
1202:and an "
802:. Since
99:Overview
66:embedded
4989:, § 26.
4849:
4831:
4820:| diam(
4538:, then
4130:. Then
4107:+ 1) ||
3980:compact
3969:
3941:
3811:, with
3540:, i.e.
3509:
3493:
3250:. Then
3230:) = 0 ∀
3086:, then
2914:, then
2820:, then
2582:into a
2500:, then
2288:physics
2069:Lorentz
1898:Unitary
1600:is not
1154:⁄
894:, with
627:, then
58:, then
33:theorem
31:) is a
5453:
5437:
5402:
5384:
5366:
4522:↦ log(
4439:under
4281:Since
4016:. Fix
3240:, the
3209:. Let
3142:Endow
2945:is an
2700:, the
2590:-space
2572:turns
2425:define
2064:Circle
1606:single
1166:after
1164:(1, 1)
704:= log(
64:is an
23:, the
5435:S2CID
5326:(PDF)
5127:E.g.
4948:) → e
4794:Notes
4646:is a
4593:) = (
4575:with
4352:= Φ(Β
4094:|| ≤
4052:|| →
3612:is a
3585:with
3575:⊂ GL(
3324:→ GL(
3297:with
3121:⊂ GL(
3068:with
3024:= GL(
3019:with
2984:Proof
2947:ideal
2902:with
2815:, ∈
2694:, or
2554:) = (
2496:is a
2139:Index
1730:with
1707:<
1703:<
1576:dense
1562:with
1200:torus
1188:. If
1142:. If
1134:helix
689:) as
592:log:
522:Lemma
49:of a
45:is a
5451:ISBN
5400:ISBN
5382:ISBN
5364:ISBN
4829:) =
4755:and
4713:and
4550:) ∈
4412:For
4395:) ∩
4384:) ⊂
4347:and
3895:and
3875:e =
3827:and
3813:0 ≠
3719:) ∩
3681:0 ∈
3658:and
3643:and
3595:and
3587:0 ∈
3546:= Id
3491:) =
3344:) ↦
3187:) =
3056:flow
2882:Let
2660:All
2478:The
2089:Loop
1830:and
1787:) →
1736:) ⊂
1732:log(
1178:and
936:and
699:for
629:log(
435:but
103:Let
5427:doi
5349:",
4959:in
4940:, (
4811:= {
4683:in
4428:of
4340:= Β
4314:, ∀
4128:→ 0
4098:≤ (
4062:→ ∞
4057:as
4007:, ∀
3737:not
3679:at
3625:to
3527:= 0
3473:at
3469:or
3334:by
3244:of
3222:| (
3214:= {
3177:∈ M
3173:= {
3106:GL(
3094:GL(
3076:GL(
3048:on
2931:If
2863:is
2843:If
2792:If
2749:= {
2740:If
2682:is
2666:GL(
2617:or
2603:If
2544:⋅ (
2486:If
2464:GL(
2448:GL(
2446:or
2436:GL(
2430:or
1920:Sp(
1910:SU(
1890:SO(
1870:SL(
1860:GL(
1808:not
1717:⊂ (
1674:→ (
1578:in
1574:is
1206:".
1045:of
1015:of
1007:any
950:dim
908:dim
667:on
199:of
19:In
5470::
5433:,
5423:30
5328:,
5036:^
5009:^
4994:^
4944:,
4936:→
4932:×
4922:⊂
4875:⊂
4855:∈
4851:,
4843:+
4768:→
4764:×
4750:→
4746:×
4726:→
4722:×
4708:→
4704:×
4668:.
4637:,
4564:⊕
4560:=
4533:∈
4518:,
4514:→
4510:⊂
4484:∈
4462:∈
4417:≥
4399:⊂
4391:Φ(
4380:∩
4369:∩
4365:=
4324:∈
4310:∈
4116:||
4085:||
4043:||
4011:∈
4003:∈
3987:∈
3966:||
3957:||
3939:=
3929:,
3915:∈
3910:,
3905:∈
3890:∈
3852:∈
3837:∈
3822:∈
3801:+
3792:=
3762:∈
3753:,
3749:=
3710:Φ(
3708:,
3694:⊂
3600:∈
3579:,
3568:,
3564:⊂
3535:+
3517:tT
3515:,
3513:tS
3511:Φ(
3505:dt
3487:,
3477:,
3340:,
3328:,
3315:.
3310:∈
3306:,
3302:∈
3292:+
3288:=
3278:∈
3268:⊕
3264:=
3234:∈
3226:,
3218:∈
3203:∈
3195:∈
3191:|
3125:,
3110:,
3098:,
3080:,
3063:⊂
3028:,
3003:,
2969:⊂
2940:⊂
2928:.
2916:Γ(
2887:⊂
2848:∈
2822:Γ(
2807:∈
2797:⊂
2765:∈
2761:,
2757:∈
2753:|
2751:ab
2747:AB
2745:=
2726:∩
2688:,
2670:,
2651:⊂
2626:∈
2592:.
2523:→
2491:⊂
2468:,
2452:,
2440:,
1900:U(
1880:O(
1780:,
1762:,
1756:,
1721:,
1692:⊂
1678:,
1659:,
1615:is
1147:=
997:⊂
987:⊂
889:⊂
694:=
678:∈
659:.
622:∩
606:⊂
596:→
95:.
5429::
5338:.
5266:s
5118:.
5019:.
5004:.
4965:.
4962:G
4956:H
4950:H
4946:H
4942:X
4938:G
4934:H
4930:U
4924:g
4920:U
4880:1
4877:U
4873:1
4870:B
4864:m
4859:}
4857:N
4853:k
4845:m
4841:k
4837:/
4834:1
4826:k
4822:B
4817:k
4813:B
4809:Β
4770:H
4766:H
4762:H
4758:i
4752:H
4748:H
4744:H
4740:m
4734:H
4728:G
4724:H
4720:H
4716:i
4710:G
4706:H
4702:H
4698:m
4692:G
4686:H
4680:i
4674:m
4665:G
4659:H
4653:H
4644:)
4642:1
4639:φ
4635:e
4633:(
4628:h
4622:m
4615:h
4613:(
4610:m
4606:x
4602:h
4600:(
4598:1
4595:x
4591:h
4589:(
4587:1
4584:φ
4578:R
4572:g
4566:s
4562:h
4558:g
4552:h
4548:h
4546:(
4544:1
4541:φ
4535:H
4531:h
4526:)
4524:g
4520:g
4516:g
4512:G
4508:e
4504:1
4501:φ
4495:H
4486:H
4482:h
4476:H
4470:g
4464:G
4460:g
4454:G
4445:I
4441:Φ
4435:j
4431:B
4425:H
4419:i
4415:j
4401:H
4397:H
4393:U
4386:h
4382:h
4378:U
4376:(
4371:V
4367:H
4363:e
4358:)
4355:i
4350:V
4343:i
4338:U
4332:i
4326:h
4322:Y
4316:t
4312:H
4308:e
4302:H
4296:i
4290:H
4284:H
4267:.
4262:Y
4259:t
4255:e
4244:i
4240:Y
4231:i
4227:S
4218:i
4214:m
4209:e
4205:=
4198:i
4194:S
4188:i
4184:m
4179:e
4175:=
4168:i
4164:m
4159:)
4151:i
4147:S
4142:e
4138:(
4125:i
4121:S
4113:i
4109:S
4104:i
4100:m
4096:t
4091:i
4087:S
4082:i
4078:m
4071:i
4067:m
4060:i
4054:t
4049:i
4045:S
4040:i
4036:m
4029:i
4025:m
4019:t
4013:R
4009:t
4005:H
4001:e
3995:i
3989:s
3985:Y
3975:s
3963:i
3959:S
3954:/
3949:i
3945:S
3936:i
3932:Y
3926:s
3917:H
3913:e
3907:H
3902:i
3898:h
3892:H
3888:e
3881:i
3877:h
3873:e
3868:Β
3864:0
3858:i
3854:B
3849:i
3845:X
3839:h
3834:i
3830:T
3824:s
3819:i
3815:S
3807:i
3803:T
3798:i
3794:S
3789:i
3785:X
3778:i
3774:B
3769:Φ
3764:h
3759:i
3755:T
3751:e
3746:i
3742:h
3731:i
3727:h
3721:H
3716:i
3712:B
3705:i
3699:1
3696:U
3692:1
3689:B
3683:g
3676:Β
3661:V
3655:U
3649:1
3646:V
3640:1
3637:U
3631:1
3628:V
3622:1
3619:U
3610:Φ
3605:1
3602:V
3598:I
3592:1
3589:U
3583:)
3581:R
3577:n
3573:1
3570:V
3566:g
3562:1
3559:U
3554:Φ
3544:∗
3542:Φ
3537:T
3533:S
3525:t
3521:|
3519:)
3501:/
3497:d
3489:T
3485:S
3483:(
3481:∗
3479:Φ
3475:0
3453:,
3450:)
3445:2
3441:t
3437:(
3434:O
3431:+
3428:T
3425:t
3422:+
3419:S
3416:t
3413:+
3410:I
3407:=
3402:T
3399:t
3395:e
3389:S
3386:t
3382:e
3378:=
3375:)
3372:T
3369:,
3366:S
3363:(
3349:e
3346:e
3342:T
3338:S
3336:(
3332:)
3330:R
3326:n
3322:g
3312:h
3308:T
3304:s
3300:S
3294:T
3290:S
3286:X
3280:g
3276:X
3270:h
3266:s
3262:g
3253:g
3247:h
3238:}
3236:h
3232:T
3228:T
3224:S
3220:g
3216:S
3212:s
3207:}
3205:R
3201:t
3199:∀
3197:H
3193:e
3189:g
3185:R
3183:(
3180:n
3175:X
3171:h
3165:H
3159:h
3145:g
3129:)
3127:R
3123:n
3119:G
3114:)
3112:R
3108:n
3102:)
3100:R
3096:n
3089:H
3084:)
3082:R
3078:n
3071:G
3065:G
3061:H
3051:G
3038:G
3032:)
3030:R
3026:n
3022:G
3011:.
2977:H
2971:G
2967:H
2952:h
2942:g
2938:h
2933:G
2925:G
2920:)
2918:h
2911:h
2905:k
2899:k
2889:g
2885:h
2876:C
2870:X
2860:X
2850:g
2846:X
2840:.
2837:G
2831:e
2826:)
2824:h
2817:h
2813:h
2809:g
2805:X
2799:g
2795:h
2786:H
2780:B
2774:A
2769:}
2767:B
2763:b
2759:A
2755:a
2743:H
2737:.
2734:U
2728:U
2724:H
2718:G
2716:⊂
2714:U
2704:.
2697:H
2691:C
2685:R
2679:F
2674:)
2672:n
2668:F
2653:G
2649:H
2634:X
2628:X
2624:x
2606:X
2587:G
2579:H
2577:/
2575:G
2569:H
2567:)
2565:2
2562:g
2559:1
2556:g
2552:H
2549:2
2546:g
2542:1
2539:g
2529:H
2527:/
2525:G
2521:G
2519::
2517:π
2507:H
2505:/
2503:G
2493:G
2489:H
2472:)
2470:C
2466:n
2456:)
2454:C
2450:n
2444:)
2442:R
2438:n
2412:e
2405:t
2398:v
2038:8
2036:E
2030:7
2028:E
2022:6
2020:E
2014:4
2012:F
2006:2
2004:G
1986:n
1983:D
1976:n
1973:C
1966:n
1963:B
1956:n
1953:A
1924:)
1922:n
1914:)
1912:n
1904:)
1902:n
1894:)
1892:n
1884:)
1882:n
1874:)
1872:n
1864:)
1862:n
1813:h
1803:H
1789:G
1785:g
1782:τ
1778:H
1774:ι
1769:)
1767:g
1764:τ
1760:H
1758:(
1753:g
1750:τ
1744:V
1738:U
1734:V
1728:)
1726:r
1723:τ
1719:H
1715:V
1709:ε
1705:θ
1701:ε
1699:−
1694:h
1690:U
1685:)
1683:r
1680:τ
1676:H
1672:h
1666:)
1664:r
1661:τ
1657:H
1655:(
1650:U
1644:h
1638:r
1635:τ
1629:U
1623:H
1611:H
1597:H
1591:H
1581:G
1571:H
1565:a
1550:,
1546:}
1541:R
1530:|
1525:)
1516:a
1513:i
1508:0
1501:0
1493:i
1487:(
1478:{
1474:=
1469:h
1458:}
1453:R
1442:|
1437:)
1426:a
1423:i
1417:2
1413:e
1407:0
1400:0
1390:i
1384:2
1380:e
1373:(
1364:{
1360:=
1357:H
1337:,
1333:}
1328:R
1318:,
1311:|
1306:)
1295:i
1289:2
1285:e
1279:0
1272:0
1262:i
1256:2
1252:e
1245:(
1236:{
1232:=
1227:2
1222:T
1217:=
1214:G
1191:a
1185:θ
1180:q
1175:φ
1169:p
1157:q
1151:p
1145:a
1139:H
1129:G
1111:H
1102:G
1096:H
1090:H
1084:H
1078:H
1061:g
1048:H
1031:h
1018:G
1012:H
999:R
995:R
989:G
985:H
966:)
961:g
956:(
947:=
944:n
924:)
919:h
914:(
905:=
902:k
891:R
887:R
870:g
860:h
836:g
812:h
788:g
778:h
765:H
748:g
738:h
725:X
719:H
713:X
708:)
706:g
702:X
696:e
692:g
686:H
680:W
676:g
670:W
645:h
633:)
631:h
624:H
620:W
614:h
608:V
604:W
598:U
594:V
587:G
572:V
561:U
544:g
531:U
513:H
496:h
469:h
445:h
432:G
426:H
420:H
403:h
390:H
373:g
349:h
325:g
301:h
277:.
273:}
268:R
261:t
253:,
250:H
242:X
239:t
235:e
228:X
224:{
220:=
215:h
202:H
185:h
172:H
166:H
160:H
154:G
148:H
142:G
136:H
119:g
106:G
61:H
55:G
42:H
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