4919:
3000:
1571:
1851:
1363:
1134:
1358:
1566:{\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\},}
3045:
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
2469:. 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
1220:
298:
4288:
3474:
893:
811:
771:
987:
945:
219:
4144:
5266:
1084:
1054:
859:
835:
668:
567:
519:
492:
468:
426:
396:
372:
348:
324:
208:
142:
2607:
The closed subgroup theorem now simplifies the hypotheses considerably, a priori widening the class of homogeneous spaces. Every closed subgroup yields a homogeneous space.
1353:{\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\},}
593:
2268:
4500:. The topology generated by these bases is the relative topology. The conclusion is that the relative topology is the same as the group topology.
3366:
186:
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"
2316:
2321:
2311:
2306:
4661:. By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in
2126:
2390:
2273:
17:
5333:
864:
782:
742:
2421:
91:
5378:
1817:
for which one can find points in an arbitrarily small neighborhood (in the relative topology) of the identity that are
4928:, Theorem 1, Section 2.7 Rossmann states the theorem for linear groups. The statement is that there is an open subset
5489:
5465:
5414:
5396:
3164:
4706:
and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations
527:
that are sufficiently close to the identity. That is to say, it is necessary to prove the following critical lemma:
4907:
2957:
2283:
4979:
5426:(1929), "Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen",
2278:
2258:
1214:
950:
908:
2223:
2131:
1605:
is composed of infinitely many almost parallel line segments on the surface of the torus. This means that
3477:
3019:
2263:
5428:
1065:
1035:
840:
816:
649:
548:
500:
473:
449:
407:
377:
353:
329:
305:
189:
123:
1104:
coming from these coordinates is not the subset topology. That it so say, the identity component of
4793:
3560:
2544:
2414:
1898:
2610:
In a similar way, the closed subgroup theorem simplifies the hypothesis in the following theorem.
4458:. This is, by the way it is constructed, a neighborhood basis both in the group topology and the
2621:
990:
675:
2985:
is closed so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently,
2865:
2522:
2218:
2181:
2149:
2136:
1612:
293:{\displaystyle {\mathfrak {h}}=\left\{X\mid e^{tX}\in H,\,\,\forall t\in \mathbf {R} \right\}.}
5346:
4283:{\displaystyle (e^{S_{i}})^{m_{i}}=e^{m_{i}S_{i}}=e^{m_{i}\|S_{i}\|Y_{i}}\rightarrow e^{tY}.}
3252:
2250:
1918:
5072:
2991:
is an embedded Lie subgroup if and only if its group topology equals its relative topology.
1993:
1983:
1973:
1963:
5108:
1878:
1868:
8:
5484:
4468:
is analytic, the left and right translates of this neighborhood basis by a group element
2407:
2395:
2236:
2066:
5445:
4798:
3682:
2167:
2157:
1804:
578:
5461:
5449:
5410:
5392:
5374:
4459:
3011:
2594:
2231:
2194:
1827:. For closed subgroups this is not the case as the proof below of the theorem shows.
1596:
87:
2346:
2084:
5437:
5423:
5205:
3003:
2904:
2366:
2046:
2038:
2030:
2022:
2014:
1947:
1928:
1888:
99:
83:
47:
5329:
2672:
2351:
2104:
2089:
1860:
57:
1209:
For an example of a subgroup that is not an embedded Lie subgroup, consider the
3150:
We begin by establishing the key lemma stated in the "overview" section above.
2625:
2371:
2189:
2094:
612:
be the inverse of the exponential map. Then there is some smaller neighborhood
5353:
2356:
95:
5478:
5279:
is sequentially compact, meaning every sequence has a convergent subsequence.
3990:
3679:
3624:
3160:
2722:
is closed. A subgroup is locally closed if every point has a neighborhood in
2079:
1908:
1808:
3056:
3027:
3007:
2999:
2376:
2361:
2162:
2144:
2074:
168:. The first step is to identify something that could be the Lie algebra of
5241:
5193:
5181:
5169:
5157:
4894:. Here the metric obtained from the Hilbert–Schmidt inner product is used.
5373:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
5371:
Lie groups, Lie algebras, and representations: An elementary introduction
4658:
2202:
2118:
1842:
1098:
into a Lie group. It is important to note, however, that the topology on
31:
5217:
5441:
4539:. This is an analytic bijection with analytic inverse. Furthermore, if
3268:
3015:
2712:
2341:
2207:
2099:
103:
4916:, Theorem 20.10. Lee states and proves this theorem in all generality.
2648:
has a unique smooth manifold structure such that the action is smooth.
1658:, yet they cannot be connected to the identity with a path staying in
5409:, Oxford Graduate Texts in Mathematics, Oxford Science Publications,
1838:
1698:
is an analytic bijection, its inverse is not continuous. That is, if
1586:
79:
76:
61:
4988:, For linear groups, Hall proves a similar result in Corollary 3.45.
3713:. Suppose for the purpose of obtaining a contradiction that for all
3052:
is formally identical, except that elements of the Lie algebra are
5282:
2298:
428:
must be big enough to capture some interesting information about
43:
2434:
Because of the conclusion of the theorem, some authors chose to
773:. That is to say, in exponential coordinates near the identity,
5294:
5084:
90:) agreeing with the embedding. One of several results known as
5229:
5145:
5050:
5048:
5306:
4009:
henceforth refers to this subsequence. It will be shown that
1210:
1144:
2667:
being closed, hence an embedded Lie group, are given below.
1811:, hence not an embedding. There are also examples of groups
5045:
5033:
1850:
1493:
1379:
1251:
5096:
2890:
to a diagonal matrix with two entries of irrational ratio.
5391:, Springer Graduate Texts in Mathematics, vol. 218,
5008:
5006:
3469:{\displaystyle \Phi (S,T)=e^{tS}e^{tT}=I+tS+tT+O(t^{2}),}
3010:
as given here. He was prominent in many areas, including
1637:
one can find points in an arbitrarily small neighborhood
5341:
Biographical
Memoirs of the National Academy of Sciences
1133:
5356:(1930), "La théorie des groupes finis et continus et l'
5060:
1119:
In particular, the lemma stated above does not hold if
5003:
4964:
is an analytic bijection onto an open neighborhood of
1500:
1386:
1258:
1173:
in lowest terms, the helix will close up on itself at
1013:, which is the condition for an embedded submanifold.
888:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
806:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
766:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
5023:
5021:
4991:
4147:
3369:
1652:
of the identity that are exponentials of elements of
1366:
1223:
1068:
1038:
953:
911:
867:
843:
819:
785:
745:
652:
581:
551:
503:
476:
452:
410:
380:
356:
332:
308:
222:
192:
126:
5120:
4700:
are analytic since these operations are analytic in
1086:. Rossmann then goes on to introduce coordinates on
2960:, then the connected Lie subgroup with Lie algebra
1128:
5407:Lie Groups – An Introduction Through Linear Groups
5018:
4282:
3993:, there is a convergent subsequence converging to
3468:
1565:
1352:
1078:
1048:
981:
939:
887:
853:
829:
805:
765:
662:
587:
561:
513:
486:
462:
420:
398:, which one might hope to be the tangent space of
390:
366:
342:
318:
292:
202:
136:
1782:is a Lie group. With this topology the injection
1615:. In the group topology, the small open sets are
404:at the identity. For this idea to work, however,
5476:
2269:Representation theory of semisimple Lie algebras
1016:It is worth noting that Rossmann shows that for
4385:and the exponential restricted to the open set
3675:such that the conclusion of the theorem holds.
3645:with analytic inverse. It remains to show that
2652:
1619:line segments on the surface of the torus and
102:'s 1929 proof of a special case for groups of
3697:, linearly ordered by reverse inclusion with
2415:
4246:
4233:
3983:. It takes its values in the unit sphere in
3065:and the exponential mapping is the time one
1205:is irrational, the helix winds indefinitely.
721:. In these coordinates, the lemma says that
674:Once this has been established, one can use
5422:
5081:, Chapter 2, Proposition 1 and Corollary 7.
5012:
4400:is in analytic bijection with the open set
4340:. This is a contradiction. Hence, for some
5117:, See comment to Corollary 5, Section 2.2.
3006:in 1929 proved the theorem in the case of
2422:
2408:
2307:Particle physics and representation theory
1849:
1032:(not necessarily closed), the Lie algebra
3145:
1232:
575:diffeomorphically onto some neighborhood
267:
266:
5404:
5247:
5199:
5187:
5175:
5163:
5126:
5114:
5090:
5078:
4925:
2998:
1132:
5455:
5328:
5288:
5272:
5027:
4418:
3567:analytic, and thus there are open sets
2274:Representations of classical Lie groups
1631:The example shows that for some groups
982:{\displaystyle n=\dim({\mathfrak {g}})}
940:{\displaystyle k=\dim({\mathfrak {h}})}
180:at the identity. The challenge is that
14:
5477:
5352:
5214:, The result follows from Theorem 5.6.
4997:
3559:, the identity. The hypothesis of the
2491:homogeneous space construction theorem
521:actually captures all the elements of
4503:Next, construct coordinate charts on
4298:is a group, the left hand side is in
2813:is a Lie subalgebra such that for no
2525:structure such that the quotient map
2521:, the left coset space, has a unique
1708:corresponds to a small open interval
5368:
5223:
5211:
5151:
5139:
5054:
5039:
4985:
4817:For this one can choose open balls,
2127:Lie group–Lie algebra correspondence
1092:that make the identity component of
569:such that the exponential map sends
497:The key step, then, is to show that
162:is a smooth embedded submanifold of
94:, it was first published in 1930 by
5386:
5312:
5300:
5260:
5235:
5102:
5066:
4913:
2903:be a Lie subalgebra. If there is a
1758:. However, with the group topology
1479:
1071:
1041:
971:
929:
880:
870:
846:
822:
798:
788:
758:
748:
655:
554:
506:
479:
455:
413:
383:
359:
335:
311:
225:
195:
150:be an arbitrary closed subgroup of
129:
24:
3370:
1752:due to the appearance of the sets
1679:is not a Lie group. While the map
1147:laid out on the surface picturing
268:
25:
5501:
1116:but not an embedded submanifold.
302:It is not difficult to show that
5389:Introduction to Smooth manifolds
4490:gives neighborhood bases at all
2657:A few sufficient conditions for
1551:
1463:
1338:
1129:Example of a non-closed subgroup
278:
174:, that is, the tangent space of
120:be a Lie group with Lie algebra
5364:, vol. XLII, pp. 1–61
5253:
5132:
4811:
2642:is a closed Lie subgroup, then
1830:
1215:irrational winding of the torus
1079:{\displaystyle {\mathfrak {g}}}
1049:{\displaystyle {\mathfrak {h}}}
854:{\displaystyle {\mathfrak {g}}}
830:{\displaystyle {\mathfrak {h}}}
663:{\displaystyle {\mathfrak {h}}}
562:{\displaystyle {\mathfrak {g}}}
514:{\displaystyle {\mathfrak {h}}}
487:{\displaystyle {\mathfrak {h}}}
463:{\displaystyle {\mathfrak {h}}}
421:{\displaystyle {\mathfrak {h}}}
391:{\displaystyle {\mathfrak {g}}}
367:{\displaystyle {\mathfrak {h}}}
343:{\displaystyle {\mathfrak {g}}}
319:{\displaystyle {\mathfrak {h}}}
203:{\displaystyle {\mathfrak {h}}}
156:. It is necessary to show that
137:{\displaystyle {\mathfrak {g}}}
5142:. See definition in Chapter 1.
4673:is an embedded submanifold of
4478:gives a neighborhood basis at
4261:
4169:
4148:
3881:is a neighborhood basis, with
3460:
3447:
3385:
3373:
2322:Galilean group representations
2317:Poincaré group representations
1110:is an immersed submanifold of
976:
966:
934:
924:
13:
1:
5321:
4454:form a neighborhood basis at
3793:, there is a unique sequence
3165:Hilbert–Schmidt inner product
2312:Lorentz group representations
2279:Theorem of the highest weight
1595:and hence not closed. In the
989:. Thus, we have exhibited a "
5334:"John von Neumann 1903–1957"
4901:
4592:, then in these coordinates
4484:. These bases restricted to
1821:exponentials of elements of
440:were some large subgroup of
7:
5226:, Exercise 14 in Chapter 5.
4787:
3363:. Expand the exponentials,
3115:, so the specialization to
2970:
2653:Conditions for being closed
2547:. The left action given by
109:
10:
5506:
4462:. Since multiplication in
3934:Normalize the sequence in
3020:foundations of mathematics
2264:Lie algebra representation
727:corresponds to a point in
539:Take a small neighborhood
38:(sometimes referred to as
5456:Willard, Stephen (1970),
5429:Mathematische Zeitschrift
5250:, Problem 2. Section 2.7.
5202:, Problem 5. Section 2.7.
5190:, Problem 4. Section 2.7.
5178:, Problem 3. Section 2.7.
5166:, Problem 1. Section 2.7.
4682:Moreover, multiplication
4415:. This proves the lemma.
3294:is uniquely expressed as
2975:An embedded Lie subgroup
2839:, the group generated by
1643:in the relative topology
1599:, a small open subset of
5490:Theorems in group theory
4804:
4794:Inverse function theorem
4566:. By fixing a basis for
3561:inverse function theorem
3069:of the vector field. If
2994:
2946:is simply connected and
2259:Lie group representation
1628:locally path connected.
684:, that is, writing each
5405:Rossmann, Wulf (2002),
5369:Hall, Brian C. (2015),
4872:for some large enough
3026:The proof is given for
2788:is a compact group and
2622:transitive group action
2445:as closed subgroups of
2284:Borel–Weil–Bott theorem
1062:is a Lie subalgebra of
991:slice coordinate system
676:exponential coordinates
470:turned out to be zero,
326:is a Lie subalgebra of
36:closed-subgroup theorem
18:Closed subgroup theorem
5460:, Dover Publications,
4784:are analytic as well.
4284:
4044:of integers such that
4033:and choose a sequence
3782:is a bijection on the
3470:
3173:be the Lie algebra of
3146:Proof of the key lemma
3023:
2866:one-parameter subgroup
2794:is a closed set, then
2523:real-analytic manifold
2182:Semisimple Lie algebra
2137:Adjoint representation
1613:locally path connected
1567:
1474:with Lie algebra
1354:
1206:
1080:
1050:
983:
941:
889:
855:
837:is just a subspace of
831:
807:
767:
664:
589:
563:
515:
494:would not be helpful.
488:
464:
422:
392:
368:
344:
320:
294:
204:
138:
104:linear transformations
98:, who was inspired by
4285:
3471:
3253:orthogonal complement
3127:instead of arbitrary
3002:
2251:Representation theory
1807:immersion, but not a
1568:
1355:
1136:
1081:
1051:
984:
942:
890:
856:
832:
808:
768:
665:
590:
564:
516:
489:
465:
423:
393:
369:
345:
321:
295:
205:
139:
5345:. See in particular
4636:is the dimension of
4419:Proof of the theorem
4145:
3735:contains an element
3367:
1364:
1221:
1066:
1036:
951:
909:
865:
841:
817:
783:
743:
694:(not necessarily in
650:
579:
549:
501:
474:
450:
408:
378:
354:
330:
306:
220:
190:
124:
50:. It states that if
27:Group theory theorem
5387:Lee, J. M. (2003),
5303:, Proposition 8.22.
2718:A subgroup that is
2509:closed Lie subgroup
2495: —
2396:Table of Lie groups
2237:Compact Lie algebra
1723:, there is no open
1003:looks locally like
595:of the identity in
537: —
434:. If, for example,
5442:10.1007/BF01187749
5362:Mémorial Sc. Math.
5275:, By problem 17G,
5238:, Corollary 15.30.
4799:Lie correspondence
4667:. This shows that
4642:. This shows that
4280:
3683:neighborhood basis
3663:contain open sets
3563:is satisfied with
3466:
3267:decomposes as the
3024:
2493:
2168:Affine Lie algebra
2158:Simple Lie algebra
1899:Special orthogonal
1563:
1534:
1446:
1350:
1315:
1207:
1076:
1046:
979:
937:
885:
861:, this means that
851:
827:
803:
763:
660:
585:
559:
535:
511:
484:
460:
418:
388:
364:
340:
316:
290:
200:
134:
5424:von Neumann, John
5315:, Corollary 8.25.
5291:, Corollary 10.5.
5259:See for instance
5154:, Corollary 3.45.
4460:relative topology
3012:quantum mechanics
2488:
2443:matrix Lie groups
2439:linear Lie groups
2432:
2431:
2232:Split Lie algebra
2195:Cartan subalgebra
2057:
2056:
1948:Simple Lie groups
1597:relative topology
1579:irrational. Then
1475:
1360:and its subgroup
1143:. Imagine a bent
588:{\displaystyle V}
545:of the origin in
531:
374:is a subspace of
350:. In particular,
46:in the theory of
16:(Redirected from
5497:
5470:
5458:General Topology
5452:
5419:
5401:
5383:
5365:
5344:
5338:
5316:
5310:
5304:
5298:
5292:
5286:
5280:
5270:
5264:
5257:
5251:
5245:
5239:
5233:
5227:
5221:
5215:
5209:
5203:
5197:
5191:
5185:
5179:
5173:
5167:
5161:
5155:
5149:
5143:
5136:
5130:
5124:
5118:
5112:
5106:
5100:
5094:
5088:
5082:
5076:
5070:
5064:
5058:
5052:
5043:
5037:
5031:
5025:
5016:
5013:von Neumann 1929
5010:
5001:
4995:
4989:
4983:
4977:
4975:
4969:
4963:
4937:
4923:
4917:
4911:
4895:
4893:
4877:
4871:
4861:
4859:
4858:
4849:
4846:
4815:
4783:
4765:
4747:
4741:
4723:
4705:
4699:
4693:
4688:, and inversion
4687:
4678:
4672:
4666:
4656:
4641:
4635:
4629:
4591:
4585:
4580:and identifying
4579:
4565:
4548:
4538:
4508:
4499:
4489:
4483:
4477:
4467:
4457:
4453:
4449:
4438:
4432:
4414:
4399:
4384:
4370:
4357:
4345:
4339:
4329:
4315:
4309:
4303:
4297:
4289:
4287:
4286:
4281:
4276:
4275:
4260:
4259:
4258:
4257:
4245:
4244:
4232:
4231:
4214:
4213:
4212:
4211:
4202:
4201:
4184:
4183:
4182:
4181:
4167:
4166:
4165:
4164:
4140:
4128:
4085:
4074:
4067:
4043:
4032:
4026:
4008:
4002:
3989:and since it is
3988:
3982:
3981:
3979:
3978:
3966:
3963:
3939:
3930:
3920:
3905:
3895:
3880:
3876:
3872:
3852:
3837:
3821:
3792:
3781:
3777:
3745:
3734:
3718:
3712:
3696:
3689:
3674:
3668:
3662:
3653:
3644:
3635:
3622:
3618:
3605:
3595:
3566:
3558:
3550:
3540:
3533:
3521:
3519:
3518:
3513:
3510:
3487:
3475:
3473:
3472:
3467:
3459:
3458:
3416:
3415:
3403:
3402:
3362:
3344:
3325:
3307:
3293:
3283:
3266:
3260:
3250:
3219:
3178:
3172:
3158:
3142:matters little.
3141:
3126:
3114:
3102:
3096:
3084:
3078:
3064:
3051:
3044:
3004:John von Neumann
2990:
2984:
2965:
2955:
2938:
2932:
2924:
2918:
2912:
2905:simply connected
2902:
2889:
2884:is similar over
2883:
2873:
2863:
2850:
2844:
2838:
2830:
2812:
2799:
2793:
2787:
2781:
2747:
2741:
2731:
2710:
2704:
2698:
2692:
2686:
2673:classical groups
2666:
2647:
2641:
2619:
2600:
2592:
2582:
2542:
2520:
2506:
2496:
2484:
2468:
2456:
2424:
2417:
2410:
2367:Claude Chevalley
2224:Complexification
2067:Other Lie groups
1953:
1952:
1861:Classical groups
1853:
1835:
1834:
1826:
1816:
1802:
1781:
1766:
1757:
1751:
1740:
1722:
1707:
1697:
1678:
1663:
1657:
1651:
1642:
1636:
1624:
1610:
1604:
1594:
1584:
1578:
1572:
1570:
1569:
1564:
1559:
1555:
1554:
1543:
1539:
1538:
1483:
1482:
1476:
1473:
1471:
1467:
1466:
1455:
1451:
1450:
1443:
1442:
1407:
1406:
1359:
1357:
1356:
1351:
1346:
1342:
1341:
1324:
1320:
1319:
1312:
1311:
1279:
1278:
1241:
1240:
1235:
1204:
1198:
1188:
1182:
1176:
1172:
1171:
1170:
1164:
1152:
1142:
1124:
1115:
1109:
1103:
1097:
1091:
1085:
1083:
1082:
1077:
1075:
1074:
1061:
1055:
1053:
1052:
1047:
1045:
1044:
1031:
1025:
1012:
1002:
988:
986:
985:
980:
975:
974:
946:
944:
943:
938:
933:
932:
904:
894:
892:
891:
886:
884:
883:
874:
873:
860:
858:
857:
852:
850:
849:
836:
834:
833:
828:
826:
825:
812:
810:
809:
804:
802:
801:
792:
791:
778:
772:
770:
769:
764:
762:
761:
752:
751:
738:
732:
726:
720:
709:
699:
693:
683:
669:
667:
666:
661:
659:
658:
645:
637:
627:
621:
611:
600:
594:
592:
591:
586:
574:
568:
566:
565:
560:
558:
557:
544:
538:
526:
520:
518:
517:
512:
510:
509:
493:
491:
490:
485:
483:
482:
469:
467:
466:
461:
459:
458:
445:
439:
433:
427:
425:
424:
419:
417:
416:
403:
397:
395:
394:
389:
387:
386:
373:
371:
370:
365:
363:
362:
349:
347:
346:
341:
339:
338:
325:
323:
322:
317:
315:
314:
299:
297:
296:
291:
286:
282:
281:
256:
255:
229:
228:
215:
209:
207:
206:
201:
199:
198:
185:
179:
173:
167:
161:
155:
149:
143:
141:
140:
135:
133:
132:
119:
100:John von Neumann
92:Cartan's theorem
84:smooth structure
74:
68:
55:
40:Cartan's theorem
21:
5505:
5504:
5500:
5499:
5498:
5496:
5495:
5494:
5475:
5474:
5473:
5468:
5417:
5399:
5381:
5336:
5324:
5319:
5311:
5307:
5299:
5295:
5287:
5283:
5271:
5267:
5258:
5254:
5246:
5242:
5234:
5230:
5222:
5218:
5210:
5206:
5198:
5194:
5186:
5182:
5174:
5170:
5162:
5158:
5150:
5146:
5137:
5133:
5125:
5121:
5113:
5109:
5101:
5097:
5089:
5085:
5077:
5073:
5065:
5061:
5057:, Theorem 3.42.
5053:
5046:
5042:, Theorem 3.20.
5038:
5034:
5026:
5019:
5011:
5004:
4996:
4992:
4984:
4980:
4971:
4965:
4939:
4929:
4924:
4920:
4912:
4908:
4904:
4899:
4898:
4892:
4885:
4879:
4873:
4850:
4847:
4844:
4843:
4841:
4839:
4830:
4818:
4816:
4812:
4807:
4790:
4767:
4749:
4743:
4725:
4707:
4701:
4695:
4689:
4683:
4674:
4668:
4662:
4654:
4643:
4637:
4631:
4623:
4610:
4599:
4593:
4587:
4581:
4567:
4556:
4550:
4540:
4516:
4510:
4509:. First define
4504:
4491:
4485:
4479:
4469:
4463:
4455:
4451:
4448:
4440:
4434:
4433:, the image in
4424:
4421:
4401:
4386:
4372:
4368:
4359:
4356:
4347:
4341:
4331:
4317:
4311:
4305:
4299:
4293:
4268:
4264:
4253:
4249:
4240:
4236:
4227:
4223:
4222:
4218:
4207:
4203:
4197:
4193:
4192:
4188:
4177:
4173:
4172:
4168:
4160:
4156:
4155:
4151:
4146:
4143:
4142:
4138:
4130:
4126:
4117:
4104:
4095:
4087:
4084:
4076:
4075:. For example,
4069:
4062:
4053:
4045:
4042:
4034:
4028:
4010:
4004:
3994:
3984:
3976:
3967:
3964:
3962:
3954:
3953:
3951:
3949:
3941:
3935:
3922:
3915:
3907:
3897:
3894:
3882:
3878:
3874:
3871:
3862:
3854:
3847:
3839:
3832:
3823:
3820:
3811:
3802:
3794:
3791:
3783:
3779:
3772:
3759:
3751:
3744:
3736:
3729:
3720:
3714:
3711:
3704:
3698:
3691:
3685:
3670:
3664:
3661:
3655:
3652:
3646:
3643:
3637:
3634:
3628:
3627:bijection from
3620:
3617:
3607:
3604:
3597:
3585:
3574:
3568:
3564:
3556:
3552:
3542:
3539:
3531:
3514:
3511:
3506:
3505:
3503:
3493:
3489:
3485:
3454:
3450:
3408:
3404:
3395:
3391:
3368:
3365:
3364:
3346:
3330:
3309:
3295:
3285:
3271:
3262:
3256:
3221:
3193:
3180:
3174:
3168:
3154:
3148:
3128:
3116:
3104:
3098:
3086:
3080:
3070:
3060:
3047:
3031:
2997:
2986:
2976:
2973:
2961:
2947:
2934:
2926:
2920:
2914:
2908:
2894:
2885:
2879:
2878:if and only if
2869:
2855:
2846:
2845:, is closed in
2840:
2832:
2814:
2804:
2795:
2789:
2783:
2752:
2743:
2733:
2723:
2706:
2700:
2694:
2688:
2676:
2658:
2655:
2643:
2633:
2615:
2605:
2596:
2584:
2577:
2571:
2561:
2554:
2548:
2543:is an analytic
2526:
2512:
2498:
2494:
2474:
2473:submanifold of
2458:
2446:
2428:
2383:
2382:
2381:
2352:Wilhelm Killing
2336:
2328:
2327:
2326:
2301:
2290:
2289:
2288:
2253:
2243:
2242:
2241:
2228:
2212:
2190:Dynkin diagrams
2184:
2174:
2173:
2172:
2154:
2132:Exponential map
2121:
2111:
2110:
2109:
2090:Conformal group
2069:
2059:
2058:
2050:
2042:
2034:
2026:
2018:
1999:
1989:
1979:
1969:
1950:
1940:
1939:
1938:
1919:Special unitary
1863:
1833:
1822:
1812:
1803:is an analytic
1797:
1783:
1779:
1768:
1765:
1759:
1753:
1742:
1738:
1724:
1709:
1699:
1695:
1680:
1676:
1665:
1659:
1653:
1650:
1644:
1638:
1632:
1620:
1606:
1600:
1590:
1580:
1574:
1550:
1533:
1532:
1521:
1515:
1514:
1509:
1496:
1495:
1492:
1491:
1487:
1478:
1477:
1472:
1462:
1445:
1444:
1426:
1422:
1420:
1414:
1413:
1408:
1393:
1389:
1382:
1381:
1378:
1377:
1373:
1365:
1362:
1361:
1337:
1314:
1313:
1298:
1294:
1292:
1286:
1285:
1280:
1265:
1261:
1254:
1253:
1250:
1249:
1245:
1236:
1231:
1230:
1222:
1219:
1218:
1200:
1194:
1184:
1178:
1174:
1166:
1160:
1159:
1154:
1148:
1138:
1131:
1125:is not closed.
1120:
1111:
1105:
1099:
1093:
1087:
1070:
1069:
1067:
1064:
1063:
1057:
1040:
1039:
1037:
1034:
1033:
1027:
1021:
1004:
994:
970:
969:
952:
949:
948:
928:
927:
910:
907:
906:
896:
879:
878:
869:
868:
866:
863:
862:
845:
844:
842:
839:
838:
821:
820:
818:
815:
814:
797:
796:
787:
786:
784:
781:
780:
774:
757:
756:
747:
746:
744:
741:
740:
734:
728:
722:
711:
701:
695:
685:
679:
672:
654:
653:
651:
648:
647:
639:
629:
623:
613:
602:
596:
580:
577:
576:
570:
553:
552:
550:
547:
546:
540:
536:
522:
505:
504:
502:
499:
498:
478:
477:
475:
472:
471:
454:
453:
451:
448:
447:
441:
435:
429:
412:
411:
409:
406:
405:
399:
382:
381:
379:
376:
375:
358:
357:
355:
352:
351:
334:
333:
331:
328:
327:
310:
309:
307:
304:
303:
277:
248:
244:
237:
233:
224:
223:
221:
218:
217:
216:by the formula
211:
194:
193:
191:
188:
187:
181:
175:
169:
163:
157:
151:
145:
128:
127:
125:
122:
121:
115:
112:
86:(and hence the
70:
64:
58:closed subgroup
51:
28:
23:
22:
15:
12:
11:
5:
5503:
5493:
5492:
5487:
5472:
5471:
5466:
5453:
5420:
5415:
5402:
5397:
5384:
5380:978-3319134666
5379:
5366:
5358:Analysis Situs
5350:
5325:
5323:
5320:
5318:
5317:
5305:
5293:
5281:
5265:
5252:
5240:
5228:
5216:
5204:
5192:
5180:
5168:
5156:
5144:
5131:
5119:
5107:
5105:, Example 7.3.
5095:
5093:, Section 2.3.
5083:
5071:
5059:
5044:
5032:
5017:
5002:
4990:
4978:
4918:
4905:
4903:
4900:
4897:
4896:
4890:
4883:
4835:
4826:
4809:
4808:
4806:
4803:
4802:
4801:
4796:
4789:
4786:
4652:
4619:
4608:
4597:
4554:
4514:
4444:
4420:
4417:
4364:
4352:
4279:
4274:
4271:
4267:
4263:
4256:
4252:
4248:
4243:
4239:
4235:
4230:
4226:
4221:
4217:
4210:
4206:
4200:
4196:
4191:
4187:
4180:
4176:
4171:
4163:
4159:
4154:
4150:
4134:
4122:
4113:
4100:
4091:
4080:
4058:
4049:
4038:
3972:
3958:
3945:
3911:
3890:
3873:converging to
3867:
3858:
3843:
3828:
3816:
3807:
3798:
3787:
3778:. Then, since
3768:
3755:
3740:
3725:
3709:
3702:
3659:
3650:
3641:
3632:
3615:
3602:
3583:
3572:
3554:
3541:is seen to be
3534:
3491:
3465:
3462:
3457:
3453:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3414:
3411:
3407:
3401:
3398:
3394:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3189:
3147:
3144:
3054:left invariant
2996:
2993:
2972:
2969:
2968:
2967:
2940:
2919:isomorphic to
2907:compact group
2891:
2852:
2801:
2749:
2720:locally closed
2716:
2675:are closed in
2654:
2651:
2650:
2649:
2626:isotropy group
2620:is a set with
2575:
2569:
2559:
2552:
2486:
2430:
2429:
2427:
2426:
2419:
2412:
2404:
2401:
2400:
2399:
2398:
2393:
2385:
2384:
2380:
2379:
2374:
2372:Harish-Chandra
2369:
2364:
2359:
2354:
2349:
2347:Henri Poincaré
2344:
2338:
2337:
2334:
2333:
2330:
2329:
2325:
2324:
2319:
2314:
2309:
2303:
2302:
2297:Lie groups in
2296:
2295:
2292:
2291:
2287:
2286:
2281:
2276:
2271:
2266:
2261:
2255:
2254:
2249:
2248:
2245:
2244:
2240:
2239:
2234:
2229:
2227:
2226:
2221:
2215:
2213:
2211:
2210:
2205:
2199:
2197:
2192:
2186:
2185:
2180:
2179:
2176:
2175:
2171:
2170:
2165:
2160:
2155:
2153:
2152:
2147:
2141:
2139:
2134:
2129:
2123:
2122:
2117:
2116:
2113:
2112:
2108:
2107:
2102:
2097:
2095:Diffeomorphism
2092:
2087:
2082:
2077:
2071:
2070:
2065:
2064:
2061:
2060:
2055:
2054:
2053:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2009:
2008:
2004:
2003:
2002:
2001:
1995:
1991:
1985:
1981:
1975:
1971:
1965:
1958:
1957:
1951:
1946:
1945:
1942:
1941:
1937:
1936:
1926:
1916:
1906:
1896:
1886:
1879:Special linear
1876:
1869:General linear
1865:
1864:
1859:
1858:
1855:
1854:
1846:
1845:
1832:
1829:
1795:
1777:
1763:
1736:
1693:
1674:
1648:
1562:
1558:
1553:
1549:
1546:
1542:
1537:
1531:
1528:
1525:
1522:
1520:
1517:
1516:
1513:
1510:
1508:
1505:
1502:
1501:
1499:
1494:
1490:
1486:
1481:
1470:
1465:
1461:
1458:
1454:
1449:
1441:
1438:
1435:
1432:
1429:
1425:
1421:
1419:
1416:
1415:
1412:
1409:
1405:
1402:
1399:
1396:
1392:
1388:
1387:
1385:
1380:
1376:
1372:
1369:
1349:
1345:
1340:
1336:
1333:
1330:
1327:
1323:
1318:
1310:
1307:
1304:
1301:
1297:
1293:
1291:
1288:
1287:
1284:
1281:
1277:
1274:
1271:
1268:
1264:
1260:
1259:
1257:
1252:
1248:
1244:
1239:
1234:
1229:
1226:
1130:
1127:
1073:
1043:
978:
973:
968:
965:
962:
959:
956:
936:
931:
926:
923:
920:
917:
914:
882:
877:
872:
848:
824:
800:
795:
790:
760:
755:
750:
657:
584:
556:
529:
508:
481:
457:
415:
385:
361:
337:
313:
289:
285:
280:
276:
273:
270:
265:
262:
259:
254:
251:
247:
243:
240:
236:
232:
227:
197:
131:
111:
108:
88:group topology
26:
9:
6:
4:
3:
2:
5502:
5491:
5488:
5486:
5483:
5482:
5480:
5469:
5467:0-486-43479-6
5463:
5459:
5454:
5451:
5447:
5443:
5439:
5435:
5432:(in German),
5431:
5430:
5425:
5421:
5418:
5416:0-19-859683-9
5412:
5408:
5403:
5400:
5398:0-387-95448-1
5394:
5390:
5385:
5382:
5376:
5372:
5367:
5363:
5359:
5355:
5351:
5348:
5342:
5335:
5331:
5327:
5326:
5314:
5309:
5302:
5297:
5290:
5285:
5278:
5274:
5269:
5262:
5256:
5249:
5248:Rossmann 2002
5244:
5237:
5232:
5225:
5220:
5213:
5208:
5201:
5200:Rossmann 2002
5196:
5189:
5188:Rossmann 2002
5184:
5177:
5176:Rossmann 2002
5172:
5165:
5164:Rossmann 2002
5160:
5153:
5148:
5141:
5135:
5128:
5127:Rossmann 2002
5123:
5116:
5115:Rossmann 2002
5111:
5104:
5099:
5092:
5091:Rossmann 2002
5087:
5080:
5079:Rossmann 2002
5075:
5068:
5063:
5056:
5051:
5049:
5041:
5036:
5029:
5024:
5022:
5014:
5009:
5007:
4999:
4994:
4987:
4982:
4974:
4968:
4962:
4958:
4954:
4950:
4946:
4942:
4936:
4932:
4927:
4926:Rossmann 2002
4922:
4915:
4910:
4906:
4889:
4882:
4876:
4869:
4865:
4857:
4853:
4838:
4834:
4829:
4825:
4821:
4814:
4810:
4800:
4797:
4795:
4792:
4791:
4785:
4782:
4778:
4774:
4770:
4764:
4760:
4756:
4752:
4748:is embedded,
4746:
4740:
4736:
4732:
4728:
4722:
4718:
4714:
4710:
4704:
4698:
4692:
4686:
4680:
4677:
4671:
4665:
4660:
4651:
4647:
4640:
4634:
4628:), 0, ..., 0)
4627:
4622:
4618:
4614:
4607:
4603:
4596:
4590:
4584:
4578:
4574:
4570:
4564:
4560:
4553:
4547:
4543:
4536:
4532:
4528:
4524:
4520:
4513:
4507:
4501:
4498:
4494:
4488:
4482:
4476:
4472:
4466:
4461:
4447:
4443:
4437:
4431:
4427:
4416:
4413:
4409:
4405:
4398:
4394:
4390:
4383:
4379:
4375:
4367:
4362:
4355:
4350:
4344:
4338:
4334:
4328:
4324:
4320:
4314:
4308:
4302:
4296:
4290:
4277:
4272:
4269:
4265:
4254:
4250:
4241:
4237:
4228:
4224:
4219:
4215:
4208:
4204:
4198:
4194:
4189:
4185:
4178:
4174:
4161:
4157:
4152:
4137:
4133:
4125:
4121:
4116:
4112:
4108:
4103:
4099:
4094:
4090:
4083:
4079:
4072:
4066:
4061:
4057:
4052:
4048:
4041:
4037:
4031:
4025:
4021:
4017:
4013:
4007:
4001:
3997:
3992:
3987:
3975:
3971:
3961:
3957:
3948:
3944:
3938:
3932:
3929:
3925:
3919:
3914:
3910:
3904:
3900:
3893:
3889:
3885:
3870:
3866:
3861:
3857:
3851:
3846:
3842:
3836:
3831:
3827:
3819:
3815:
3810:
3806:
3801:
3797:
3790:
3786:
3776:
3771:
3767:
3763:
3758:
3754:
3749:
3743:
3739:
3733:
3728:
3724:
3717:
3708:
3701:
3695:
3688:
3684:
3681:
3676:
3673:
3667:
3658:
3649:
3640:
3631:
3626:
3625:real-analytic
3614:
3610:
3601:
3593:
3589:
3582:
3578:
3571:
3562:
3549:
3545:
3537:
3529:
3525:
3517:
3509:
3501:
3497:
3483:
3479:
3463:
3455:
3451:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3412:
3409:
3405:
3399:
3396:
3392:
3388:
3382:
3379:
3376:
3361:
3358:
3354:
3350:
3342:
3338:
3334:
3329:Define a map
3327:
3324:
3320:
3316:
3312:
3306:
3302:
3298:
3292:
3288:
3282:
3278:
3274:
3270:
3265:
3259:
3254:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3217:
3213:
3209:
3205:
3201:
3197:
3192:
3187:
3183:
3177:
3171:
3166:
3162:
3161:inner product
3157:
3151:
3143:
3139:
3135:
3131:
3124:
3120:
3112:
3108:
3103:is closed in
3101:
3094:
3090:
3083:
3077:
3073:
3068:
3063:
3058:
3057:vector fields
3055:
3050:
3042:
3038:
3034:
3029:
3028:matrix groups
3021:
3017:
3013:
3009:
3008:matrix groups
3005:
3001:
2992:
2989:
2983:
2979:
2964:
2959:
2954:
2950:
2945:
2941:
2937:
2933:is closed in
2930:
2923:
2917:
2911:
2906:
2901:
2897:
2892:
2888:
2882:
2877:
2872:
2868:generated by
2867:
2862:
2858:
2853:
2849:
2843:
2836:
2829:
2825:
2821:
2817:
2811:
2807:
2802:
2798:
2792:
2786:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2750:
2746:
2742:is closed in
2740:
2736:
2730:
2726:
2721:
2717:
2714:
2709:
2703:
2697:
2691:
2684:
2680:
2674:
2670:
2669:
2668:
2665:
2661:
2646:
2640:
2636:
2631:
2627:
2623:
2618:
2613:
2612:
2611:
2608:
2604:
2602:
2599:
2591:
2587:
2581:
2574:
2568:
2564:
2558:
2551:
2546:
2541:
2537:
2533:
2529:
2524:
2519:
2515:
2510:
2505:
2501:
2492:
2485:
2482:
2478:
2472:
2466:
2462:
2454:
2450:
2444:
2440:
2437:
2425:
2420:
2418:
2413:
2411:
2406:
2405:
2403:
2402:
2397:
2394:
2392:
2389:
2388:
2387:
2386:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2339:
2332:
2331:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2304:
2300:
2294:
2293:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2256:
2252:
2247:
2246:
2238:
2235:
2233:
2230:
2225:
2222:
2220:
2217:
2216:
2214:
2209:
2206:
2204:
2201:
2200:
2198:
2196:
2193:
2191:
2188:
2187:
2183:
2178:
2177:
2169:
2166:
2164:
2161:
2159:
2156:
2151:
2148:
2146:
2143:
2142:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2124:
2120:
2115:
2114:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2072:
2068:
2063:
2062:
2051:
2045:
2043:
2037:
2035:
2029:
2027:
2021:
2019:
2013:
2012:
2011:
2010:
2006:
2005:
2000:
1998:
1992:
1990:
1988:
1982:
1980:
1978:
1972:
1970:
1968:
1962:
1961:
1960:
1959:
1955:
1954:
1949:
1944:
1943:
1934:
1930:
1927:
1924:
1920:
1917:
1914:
1910:
1907:
1904:
1900:
1897:
1894:
1890:
1887:
1884:
1880:
1877:
1874:
1870:
1867:
1866:
1862:
1857:
1856:
1852:
1848:
1847:
1844:
1840:
1837:
1836:
1828:
1825:
1820:
1815:
1810:
1809:homeomorphism
1806:
1801:
1794:
1790:
1786:
1776:
1772:
1762:
1756:
1750:
1746:
1735:
1731:
1727:
1721:
1717:
1713:
1706:
1702:
1692:
1688:
1684:
1673:
1669:
1662:
1656:
1647:
1641:
1635:
1629:
1627:
1623:
1618:
1614:
1609:
1603:
1598:
1593:
1588:
1583:
1577:
1560:
1556:
1547:
1544:
1540:
1535:
1529:
1526:
1523:
1518:
1511:
1506:
1503:
1497:
1488:
1484:
1468:
1459:
1456:
1452:
1447:
1439:
1436:
1433:
1430:
1427:
1423:
1417:
1410:
1403:
1400:
1397:
1394:
1390:
1383:
1374:
1370:
1367:
1347:
1343:
1334:
1331:
1328:
1325:
1321:
1316:
1308:
1305:
1302:
1299:
1295:
1289:
1282:
1275:
1272:
1269:
1266:
1262:
1255:
1246:
1242:
1237:
1227:
1224:
1216:
1212:
1203:
1197:
1193:rotations in
1192:
1187:
1183:rotations in
1181:
1169:
1163:
1157:
1151:
1146:
1141:
1135:
1126:
1123:
1117:
1114:
1108:
1102:
1096:
1090:
1060:
1030:
1024:
1019:
1014:
1011:
1007:
1001:
997:
992:
963:
960:
957:
954:
921:
918:
915:
912:
903:
899:
895:is just like
875:
793:
777:
753:
737:
733:precisely if
731:
725:
718:
714:
708:
704:
698:
692:
688:
682:
677:
671:
643:
636:
632:
626:
622:such that if
620:
616:
610:
606:
599:
582:
573:
543:
534:
528:
525:
495:
444:
438:
432:
402:
300:
287:
283:
274:
271:
263:
260:
257:
252:
249:
245:
241:
238:
234:
230:
214:
184:
178:
172:
166:
160:
154:
148:
118:
107:
105:
101:
97:
93:
89:
85:
81:
78:
73:
67:
63:
59:
54:
49:
45:
41:
37:
33:
19:
5457:
5433:
5427:
5406:
5388:
5370:
5361:
5357:
5354:Cartan, Élie
5340:
5308:
5296:
5289:Willard 1970
5284:
5276:
5273:Willard 1970
5268:
5255:
5243:
5231:
5219:
5207:
5195:
5183:
5171:
5159:
5147:
5134:
5122:
5110:
5098:
5086:
5074:
5069:, Chapter 5.
5062:
5035:
5028:Bochner 1958
4993:
4981:
4972:
4966:
4960:
4956:
4952:
4948:
4944:
4940:
4934:
4930:
4921:
4909:
4887:
4880:
4874:
4867:
4863:
4855:
4851:
4836:
4832:
4827:
4823:
4819:
4813:
4780:
4776:
4772:
4768:
4762:
4758:
4754:
4750:
4744:
4742:. But since
4738:
4734:
4730:
4726:
4720:
4716:
4712:
4708:
4702:
4696:
4690:
4684:
4681:
4675:
4669:
4663:
4649:
4645:
4638:
4632:
4625:
4620:
4616:
4612:
4605:
4601:
4594:
4588:
4582:
4576:
4572:
4568:
4562:
4558:
4551:
4545:
4541:
4534:
4530:
4526:
4522:
4518:
4511:
4505:
4502:
4496:
4492:
4486:
4480:
4474:
4470:
4464:
4445:
4441:
4435:
4429:
4425:
4422:
4411:
4407:
4403:
4396:
4392:
4388:
4381:
4377:
4373:
4365:
4360:
4353:
4348:
4342:
4336:
4332:
4326:
4322:
4318:
4312:
4306:
4300:
4294:
4291:
4135:
4131:
4129:will do, as
4123:
4119:
4114:
4110:
4106:
4101:
4097:
4092:
4088:
4081:
4077:
4070:
4064:
4059:
4055:
4050:
4046:
4039:
4035:
4029:
4023:
4019:
4015:
4011:
4005:
4003:. The index
3999:
3995:
3985:
3973:
3969:
3959:
3955:
3946:
3942:
3936:
3933:
3927:
3923:
3917:
3912:
3908:
3902:
3898:
3891:
3887:
3883:
3868:
3864:
3859:
3855:
3849:
3844:
3840:
3834:
3829:
3825:
3817:
3813:
3808:
3804:
3799:
3795:
3788:
3784:
3774:
3769:
3765:
3761:
3756:
3752:
3750:on the form
3747:
3741:
3737:
3731:
3726:
3722:
3715:
3706:
3699:
3693:
3686:
3677:
3671:
3665:
3656:
3647:
3638:
3629:
3612:
3608:
3599:
3591:
3587:
3580:
3576:
3569:
3547:
3543:
3535:
3527:
3523:
3515:
3507:
3499:
3495:
3482:differential
3481:
3359:
3356:
3352:
3348:
3340:
3336:
3332:
3328:
3322:
3318:
3314:
3310:
3304:
3300:
3296:
3290:
3286:
3280:
3276:
3272:
3263:
3257:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3215:
3211:
3207:
3203:
3199:
3195:
3190:
3185:
3181:
3175:
3169:
3155:
3152:
3149:
3137:
3133:
3129:
3122:
3118:
3110:
3106:
3099:
3092:
3088:
3081:
3075:
3071:
3066:
3061:
3053:
3048:
3040:
3036:
3032:
3025:
2987:
2981:
2977:
2974:
2962:
2952:
2948:
2943:
2935:
2928:
2921:
2915:
2909:
2899:
2895:
2886:
2880:
2875:
2870:
2860:
2856:
2847:
2841:
2834:
2827:
2823:
2819:
2815:
2809:
2805:
2796:
2790:
2784:
2777:
2773:
2769:
2765:
2761:
2757:
2753:
2744:
2738:
2734:
2728:
2724:
2719:
2707:
2701:
2695:
2689:
2682:
2678:
2663:
2659:
2656:
2644:
2638:
2634:
2629:
2616:
2609:
2606:
2597:
2595:homogeneous
2589:
2585:
2579:
2572:
2566:
2562:
2556:
2549:
2539:
2535:
2531:
2527:
2517:
2513:
2508:
2503:
2499:
2490:
2487:
2480:
2476:
2470:
2464:
2460:
2452:
2448:
2442:
2438:
2435:
2433:
2377:Armand Borel
2362:Hermann Weyl
2163:Loop algebra
2145:Killing form
2119:Lie algebras
1996:
1986:
1976:
1966:
1932:
1922:
1912:
1902:
1892:
1882:
1872:
1843:Lie algebras
1831:Applications
1823:
1818:
1813:
1799:
1792:
1788:
1784:
1774:
1770:
1760:
1754:
1748:
1744:
1733:
1729:
1725:
1719:
1715:
1711:
1704:
1700:
1690:
1686:
1682:
1671:
1667:
1664:. The group
1660:
1654:
1645:
1639:
1633:
1630:
1625:
1621:
1616:
1607:
1601:
1591:
1581:
1575:
1208:
1201:
1195:
1190:
1185:
1179:
1167:
1161:
1155:
1149:
1139:
1121:
1118:
1112:
1106:
1100:
1094:
1088:
1058:
1028:
1022:
1017:
1015:
1009:
1005:
999:
995:
901:
897:
775:
735:
729:
723:
716:
712:
706:
702:
696:
690:
686:
680:
673:
641:
634:
630:
624:
618:
614:
608:
604:
597:
571:
541:
532:
530:
523:
496:
442:
436:
430:
400:
301:
212:
182:
176:
170:
164:
158:
152:
146:
116:
113:
71:
65:
52:
39:
35:
29:
5436:(1): 3–42,
5347:p. 441
5330:Bochner, S.
4998:Cartan 1930
4659:slice chart
4316:is closed,
3678:Consider a
3478:pushforward
3179:defined as
3167:), and let
3163:(e.g., the
2966:is closed.
2864:, then the
2713:quaternions
2632:of a point
2357:Élie Cartan
2203:Root system
2007:Exceptional
1681:exp :
993:" in which
779:looks like
739:belongs to
646:belongs to
628:belongs to
96:Élie Cartan
32:mathematics
5485:Lie groups
5479:Categories
5322:References
5263:Chapter 21
4938:such that
4878:such that
4086:such that
3853:such that
3619:such that
3284:, so each
3269:direct sum
3085:closed in
3016:set theory
2876:not closed
2800:is closed.
2732:such that
2630:stabilizer
2545:submersion
2342:Sophus Lie
2335:Scientists
2208:Weyl group
1929:Symplectic
1889:Orthogonal
1839:Lie groups
1137:The torus
601:, and let
144:. Now let
48:Lie groups
5450:122565679
5343:: 438–456
5224:Hall 2015
5212:Hall 2015
5152:Hall 2015
5140:Hall 2015
5055:Hall 2015
5040:Hall 2015
4986:Hall 2015
4902:Citations
4346:the sets
4262:→
4247:‖
4234:‖
3931:as well.
3680:countable
3371:Φ
3331:Φ :
2219:Real form
2105:Euclidean
1956:Classical
1805:injective
1787: : (
1548:∈
1545:θ
1530:θ
1507:θ
1460:∈
1457:θ
1440:θ
1431:π
1404:θ
1398:π
1335:∈
1332:ϕ
1326:θ
1309:ϕ
1303:π
1276:θ
1270:π
1020:subgroup
964:
922:
876:⊂
794:⊂
754:⊂
275:∈
269:∀
258:∈
242:∣
82:with the
80:Lie group
62:Lie group
5332:(1958),
5313:Lee 2003
5301:Lee 2003
5261:Lee 2003
5236:Lee 2003
5103:Lee 2003
5067:Lee 2003
4914:Lee 2003
4788:See also
4771: :
4753: :
4729: :
4711: :
4630:, where
4615:), ...,
4517: :
4371:satisfy
4330:, hence
4310:. Since
4304:for all
3896:. Since
3877:because
3746:that is
3476:and the
3159:with an
3018:and the
2971:Converse
2822:∖
2782:, where
2687:, where
2624:and the
2471:embedded
2391:Glossary
2085:Poincaré
1213:and an "
813:. Since
110:Overview
77:embedded
5000:, § 26.
4860:
4842:
4831:| diam(
4549:, then
4141:. Then
4118:+ 1) ||
3991:compact
3980:
3952:
3822:, with
3551:, i.e.
3520:
3504:
3261:. Then
3241:) = 0 ∀
3097:, then
2925:, then
2831:, then
2593:into a
2511:, then
2299:physics
2080:Lorentz
1909:Unitary
1611:is not
1165:⁄
905:, with
638:, then
69:, then
44:theorem
42:) is a
5464:
5448:
5413:
5395:
5377:
4533:↦ log(
4450:under
4292:Since
4027:. Fix
3251:, the
3220:. Let
3153:Endow
2956:is an
2711:, the
2601:-space
2583:turns
2436:define
2075:Circle
1617:single
1177:after
1175:(1, 1)
715:= log(
75:is an
34:, the
5446:S2CID
5337:(PDF)
5138:E.g.
4959:) → e
4805:Notes
4657:is a
4604:) = (
4586:with
4363:= Φ(Β
4105:|| ≤
4063:|| →
3623:is a
3596:with
3586:⊂ GL(
3335:→ GL(
3308:with
3132:⊂ GL(
3079:with
3035:= GL(
3030:with
2995:Proof
2958:ideal
2913:with
2826:, ∈
2705:, or
2565:) = (
2507:is a
2150:Index
1741:with
1718:<
1714:<
1587:dense
1573:with
1211:torus
1199:. If
1153:. If
1145:helix
700:) as
603:log:
533:Lemma
60:of a
56:is a
5462:ISBN
5411:ISBN
5393:ISBN
5375:ISBN
4840:) =
4766:and
4724:and
4561:) ∈
4423:For
4406:) ∩
4395:) ⊂
4358:and
3906:and
3886:e =
3838:and
3824:0 ≠
3730:) ∩
3692:0 ∈
3669:and
3654:and
3606:and
3598:0 ∈
3557:= Id
3502:) =
3355:) ↦
3198:) =
3067:flow
2893:Let
2671:All
2489:The
2100:Loop
1841:and
1798:) →
1747:) ⊂
1743:log(
1189:and
947:and
710:for
640:log(
446:but
114:Let
5438:doi
5360:",
4970:in
4951:, (
4822:= {
4694:in
4439:of
4351:= Β
4325:, ∀
4139:→ 0
4109:≤ (
4073:→ ∞
4068:as
4018:, ∀
3748:not
3690:at
3636:to
3538:= 0
3484:at
3480:or
3345:by
3255:of
3233:| (
3225:= {
3188:∈ M
3184:= {
3117:GL(
3105:GL(
3087:GL(
3059:on
2942:If
2874:is
2854:If
2803:If
2760:= {
2751:If
2693:is
2677:GL(
2628:or
2614:If
2555:⋅ (
2497:If
2475:GL(
2459:GL(
2457:or
2447:GL(
2441:or
1931:Sp(
1921:SU(
1901:SO(
1881:SL(
1871:GL(
1819:not
1728:⊂ (
1685:→ (
1589:in
1585:is
1217:".
1056:of
1026:of
1018:any
961:dim
919:dim
678:on
210:of
30:In
5481::
5444:,
5434:30
5339:,
5047:^
5020:^
5005:^
4955:,
4947:→
4943:×
4933:⊂
4886:⊂
4866:∈
4862:,
4854:+
4779:→
4775:×
4761:→
4757:×
4737:→
4733:×
4719:→
4715:×
4679:.
4648:,
4575:⊕
4571:=
4544:∈
4529:,
4525:→
4521:⊂
4495:∈
4473:∈
4428:≥
4410:⊂
4402:Φ(
4391:∩
4380:∩
4376:=
4335:∈
4321:∈
4127:||
4096:||
4054:||
4022:∈
4014:∈
3998:∈
3977:||
3968:||
3950:=
3940:,
3926:∈
3921:,
3916:∈
3901:∈
3863:∈
3848:∈
3833:∈
3812:+
3803:=
3773:∈
3764:,
3760:=
3721:Φ(
3719:,
3705:⊂
3611:∈
3590:,
3579:,
3575:⊂
3546:+
3528:tT
3526:,
3524:tS
3522:Φ(
3516:dt
3498:,
3488:,
3351:,
3339:,
3326:.
3321:∈
3317:,
3313:∈
3303:+
3299:=
3289:∈
3279:⊕
3275:=
3245:∈
3237:,
3229:∈
3214:∈
3206:∈
3202:|
3136:,
3121:,
3109:,
3091:,
3074:⊂
3039:,
3014:,
2980:⊂
2951:⊂
2939:.
2927:Γ(
2898:⊂
2859:∈
2833:Γ(
2818:∈
2808:⊂
2776:∈
2772:,
2768:∈
2764:|
2762:ab
2758:AB
2756:=
2737:∩
2699:,
2681:,
2662:⊂
2637:∈
2603:.
2534:→
2502:⊂
2479:,
2463:,
2451:,
1911:U(
1891:O(
1791:,
1773:,
1767:,
1732:,
1703:⊂
1689:,
1670:,
1626:is
1158:=
1008:⊂
998:⊂
900:⊂
705:=
689:∈
670:.
633:∩
617:⊂
607:→
106:.
5440::
5349:.
5277:s
5129:.
5030:.
5015:.
4976:.
4973:G
4967:H
4961:H
4957:H
4953:X
4949:G
4945:H
4941:U
4935:g
4931:U
4891:1
4888:U
4884:1
4881:B
4875:m
4870:}
4868:N
4864:k
4856:m
4852:k
4848:/
4845:1
4837:k
4833:B
4828:k
4824:B
4820:Β
4781:H
4777:H
4773:H
4769:i
4763:H
4759:H
4755:H
4751:m
4745:H
4739:G
4735:H
4731:H
4727:i
4721:G
4717:H
4713:H
4709:m
4703:G
4697:H
4691:i
4685:m
4676:G
4670:H
4664:H
4655:)
4653:1
4650:φ
4646:e
4644:(
4639:h
4633:m
4626:h
4624:(
4621:m
4617:x
4613:h
4611:(
4609:1
4606:x
4602:h
4600:(
4598:1
4595:φ
4589:R
4583:g
4577:s
4573:h
4569:g
4563:h
4559:h
4557:(
4555:1
4552:φ
4546:H
4542:h
4537:)
4535:g
4531:g
4527:g
4523:G
4519:e
4515:1
4512:φ
4506:H
4497:H
4493:h
4487:H
4481:g
4475:G
4471:g
4465:G
4456:I
4452:Φ
4446:j
4442:B
4436:H
4430:i
4426:j
4412:H
4408:H
4404:U
4397:h
4393:h
4389:U
4387:(
4382:V
4378:H
4374:e
4369:)
4366:i
4361:V
4354:i
4349:U
4343:i
4337:h
4333:Y
4327:t
4323:H
4319:e
4313:H
4307:i
4301:H
4295:H
4278:.
4273:Y
4270:t
4266:e
4255:i
4251:Y
4242:i
4238:S
4229:i
4225:m
4220:e
4216:=
4209:i
4205:S
4199:i
4195:m
4190:e
4186:=
4179:i
4175:m
4170:)
4162:i
4158:S
4153:e
4149:(
4136:i
4132:S
4124:i
4120:S
4115:i
4111:m
4107:t
4102:i
4098:S
4093:i
4089:m
4082:i
4078:m
4071:i
4065:t
4060:i
4056:S
4051:i
4047:m
4040:i
4036:m
4030:t
4024:R
4020:t
4016:H
4012:e
4006:i
4000:s
3996:Y
3986:s
3974:i
3970:S
3965:/
3960:i
3956:S
3947:i
3943:Y
3937:s
3928:H
3924:e
3918:H
3913:i
3909:h
3903:H
3899:e
3892:i
3888:h
3884:e
3879:Β
3875:0
3869:i
3865:B
3860:i
3856:X
3850:h
3845:i
3841:T
3835:s
3830:i
3826:S
3818:i
3814:T
3809:i
3805:S
3800:i
3796:X
3789:i
3785:B
3780:Φ
3775:h
3770:i
3766:T
3762:e
3757:i
3753:h
3742:i
3738:h
3732:H
3727:i
3723:B
3716:i
3710:1
3707:U
3703:1
3700:B
3694:g
3687:Β
3672:V
3666:U
3660:1
3657:V
3651:1
3648:U
3642:1
3639:V
3633:1
3630:U
3621:Φ
3616:1
3613:V
3609:I
3603:1
3600:U
3594:)
3592:R
3588:n
3584:1
3581:V
3577:g
3573:1
3570:U
3565:Φ
3555:∗
3553:Φ
3548:T
3544:S
3536:t
3532:|
3530:)
3512:/
3508:d
3500:T
3496:S
3494:(
3492:∗
3490:Φ
3486:0
3464:,
3461:)
3456:2
3452:t
3448:(
3445:O
3442:+
3439:T
3436:t
3433:+
3430:S
3427:t
3424:+
3421:I
3418:=
3413:T
3410:t
3406:e
3400:S
3397:t
3393:e
3389:=
3386:)
3383:T
3380:,
3377:S
3374:(
3360:e
3357:e
3353:T
3349:S
3347:(
3343:)
3341:R
3337:n
3333:g
3323:h
3319:T
3315:s
3311:S
3305:T
3301:S
3297:X
3291:g
3287:X
3281:h
3277:s
3273:g
3264:g
3258:h
3249:}
3247:h
3243:T
3239:T
3235:S
3231:g
3227:S
3223:s
3218:}
3216:R
3212:t
3210:∀
3208:H
3204:e
3200:g
3196:R
3194:(
3191:n
3186:X
3182:h
3176:H
3170:h
3156:g
3140:)
3138:R
3134:n
3130:G
3125:)
3123:R
3119:n
3113:)
3111:R
3107:n
3100:H
3095:)
3093:R
3089:n
3082:G
3076:G
3072:H
3062:G
3049:G
3043:)
3041:R
3037:n
3033:G
3022:.
2988:H
2982:G
2978:H
2963:h
2953:g
2949:h
2944:G
2936:G
2931:)
2929:h
2922:h
2916:k
2910:k
2900:g
2896:h
2887:C
2881:X
2871:X
2861:g
2857:X
2851:.
2848:G
2842:e
2837:)
2835:h
2828:h
2824:h
2820:g
2816:X
2810:g
2806:h
2797:H
2791:B
2785:A
2780:}
2778:B
2774:b
2770:A
2766:a
2754:H
2748:.
2745:U
2739:U
2735:H
2729:G
2727:⊂
2725:U
2715:.
2708:H
2702:C
2696:R
2690:F
2685:)
2683:n
2679:F
2664:G
2660:H
2645:X
2639:X
2635:x
2617:X
2598:G
2590:H
2588:/
2586:G
2580:H
2578:)
2576:2
2573:g
2570:1
2567:g
2563:H
2560:2
2557:g
2553:1
2550:g
2540:H
2538:/
2536:G
2532:G
2530::
2528:π
2518:H
2516:/
2514:G
2504:G
2500:H
2483:)
2481:C
2477:n
2467:)
2465:C
2461:n
2455:)
2453:R
2449:n
2423:e
2416:t
2409:v
2049:8
2047:E
2041:7
2039:E
2033:6
2031:E
2025:4
2023:F
2017:2
2015:G
1997:n
1994:D
1987:n
1984:C
1977:n
1974:B
1967:n
1964:A
1935:)
1933:n
1925:)
1923:n
1915:)
1913:n
1905:)
1903:n
1895:)
1893:n
1885:)
1883:n
1875:)
1873:n
1824:h
1814:H
1800:G
1796:g
1793:τ
1789:H
1785:ι
1780:)
1778:g
1775:τ
1771:H
1769:(
1764:g
1761:τ
1755:V
1749:U
1745:V
1739:)
1737:r
1734:τ
1730:H
1726:V
1720:ε
1716:θ
1712:ε
1710:−
1705:h
1701:U
1696:)
1694:r
1691:τ
1687:H
1683:h
1677:)
1675:r
1672:τ
1668:H
1666:(
1661:U
1655:h
1649:r
1646:τ
1640:U
1634:H
1622:H
1608:H
1602:H
1592:G
1582:H
1576:a
1561:,
1557:}
1552:R
1541:|
1536:)
1527:a
1524:i
1519:0
1512:0
1504:i
1498:(
1489:{
1485:=
1480:h
1469:}
1464:R
1453:|
1448:)
1437:a
1434:i
1428:2
1424:e
1418:0
1411:0
1401:i
1395:2
1391:e
1384:(
1375:{
1371:=
1368:H
1348:,
1344:}
1339:R
1329:,
1322:|
1317:)
1306:i
1300:2
1296:e
1290:0
1283:0
1273:i
1267:2
1263:e
1256:(
1247:{
1243:=
1238:2
1233:T
1228:=
1225:G
1202:a
1196:θ
1191:q
1186:φ
1180:p
1168:q
1162:p
1156:a
1150:H
1140:G
1122:H
1113:G
1107:H
1101:H
1095:H
1089:H
1072:g
1059:H
1042:h
1029:G
1023:H
1010:R
1006:R
1000:G
996:H
977:)
972:g
967:(
958:=
955:n
935:)
930:h
925:(
916:=
913:k
902:R
898:R
881:g
871:h
847:g
823:h
799:g
789:h
776:H
759:g
749:h
736:X
730:H
724:X
719:)
717:g
713:X
707:e
703:g
697:H
691:W
687:g
681:W
656:h
644:)
642:h
635:H
631:W
625:h
619:V
615:W
609:U
605:V
598:G
583:V
572:U
555:g
542:U
524:H
507:h
480:h
456:h
443:G
437:H
431:H
414:h
401:H
384:g
360:h
336:g
312:h
288:.
284:}
279:R
272:t
264:,
261:H
253:X
250:t
246:e
239:X
235:{
231:=
226:h
213:H
196:h
183:H
177:H
171:H
165:G
159:H
153:G
147:H
130:g
117:G
72:H
66:G
53:H
20:)
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