42:
2468:
1383:, there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will
1278:
4350:
712:(whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
3800:
1501:. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics (orbits of 1-parameter subgroups acting by left or right multiplication) so give the same exponential map.
2563:
680:
to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
2929:
2286:
4672:
3012:
2706:
3590:
4569:
4524:
1649:
3420:
1826:
4129:
2861:
2275:
2818:
3535:
841:
3245:
964:
1885:
1141:
2005:
4137:
2646:
1717:
4068:
3313:
3916:
3805:
Globally, the exponential map is not necessarily surjective. Furthermore, the exponential map may not be a local diffeomorphism at all points. For example, the exponential map from
4791:
3057:
881:
3107:
1074:
4449:
4027:
1464:
928:
1575:
3830:
2136:
2057:
1957:
4473:
4403:
3684:
3621:
1534:
1330:
766:
658:
4732:
2188:
1607:
4054:
2748:
1908:
2595:
1484:
1094:
4615:
2091:
3987:
3964:
4589:
4379:
4096:
3641:
3491:
3464:
3444:
3155:
3131:
2159:
2028:
1928:
1350:
1302:
1122:
1016:
992:
798:
738:
706:
678:
3689:
3943:-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix
2485:
459:
2463:{\displaystyle \mathbf {w} :=(it+ju+kv)\mapsto \exp(it+ju+kv)=\cos(|\mathbf {w} |)1+\sin(|\mathbf {w} |){\frac {\mathbf {w} }{|\mathbf {w} |}}.\,}
2870:
1391:
is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of
4623:
2940:
507:
2661:
512:
3548:
502:
497:
4529:
4490:
1612:
3324:
1728:
1505:
317:
4101:
1308:. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra
581:
464:
2823:
5015:
4842:
4075:
1373:
20:
5034:
612:
2204:
4681:. It is called by various names such as logarithmic coordinates, exponential coordinates or normal coordinates. See the
2762:
4988:
1124:. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
3504:
810:
5044:
4807:
3841:
3163:
935:
1273:{\displaystyle \exp(X)=\sum _{k=0}^{\infty }{\frac {X^{k}}{k!}}=I+X+{\frac {1}{2}}X^{2}+{\frac {1}{6}}X^{3}+\cdots }
4866:
4345:{\displaystyle \mathrm {Ad} _{\exp X}(Y)=\exp(\mathrm {ad} _{X})(Y)=Y++{\frac {1}{2!}}]+{\frac {1}{3!}}]]+\cdots }
2610:
be a finite dimensional real vector space and view it as a Lie group under the operation of vector addition. Then
5003:
3837:
1847:
474:
1969:
2613:
1676:
3251:
5064:
5007:
2598:
469:
449:
3874:
414:
4745:
4682:
3032:
856:
5059:
5054:
3922:
For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.
3062:
1029:
19:
For the exponential map from a subset of the tangent space of a
Riemannian manifold to the manifold, see
4408:
4000:
1422:
886:
4802:
4057:
3542:
1539:
454:
3808:
2099:
2033:
1933:
4454:
4384:
3665:
3602:
1515:
1311:
747:
639:
4694:
4484:
605:
89:
2171:
1580:
3595:
It follows from the inverse function theorem that the exponential map, therefore, restricts to a
4735:
4032:
3110:
2715:
971:
409:
372:
340:
327:
4837:. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg.
1893:
5082:
2568:
2201:) whose tangent space at 1 can be identified with the space of purely imaginary quaternions,
1469:
1079:
709:
441:
109:
4594:
2066:
184:
174:
164:
154:
5025:
3833:
2756:
1673:) whose tangent space at 1 can be identified with the imaginary line in the complex plane,
685:
69:
59:
8:
5087:
4061:
3795:{\displaystyle g=\exp(X_{1})\exp(X_{2})\cdots \exp(X_{n}),\quad X_{j}\in {\mathfrak {g}}}
1833:
598:
586:
427:
257:
3969:
3946:
3852:
In these important special cases, the exponential map is known to always be surjective:
4812:
4574:
4364:
4081:
3626:
3476:
3470:
3449:
3429:
3140:
3116:
2144:
2013:
1913:
1416:
1335:
1287:
1132:
1107:
1001:
977:
846:
which can be defined in several different ways. The typical modern definition is this:
783:
723:
691:
663:
358:
348:
5040:
5011:
4984:
4848:
4838:
1408:
1387:
in general agree with the exponential map in the Lie group sense. That is to say, if
422:
385:
537:
275:
2060:
1577:
is the unique Lie group homomorphism corresponding to the Lie algebra homomorphism
1128:
777:
557:
237:
229:
221:
213:
205:
138:
119:
79:
5021:
3966:. (Thus, the image excludes matrices with real, negative eigenvalues, other than
2558:{\displaystyle \{s\in S^{3}\subset \mathbf {H} :\operatorname {Re} (s)=\cos(R)\}}
1960:
1888:
1305:
542:
295:
280:
51:
3596:
3134:
2864:
1097:
995:
562:
380:
285:
1497:
It is the exponential map of a canonical right-invariant affine connection on
547:
5076:
4852:
1842:
1666:
773:
270:
99:
1670:
1101:
567:
552:
353:
335:
265:
4832:
2924:{\displaystyle \lbrace \cosh t+\jmath \ \sinh t:t\in \mathbb {R} \rbrace }
1403:
Other equivalent definitions of the Lie-group exponential are as follows:
4983:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
4981:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
2166:
1662:
769:
634:
393:
309:
33:
4667:{\displaystyle \log :U{\overset {\sim }{\to }}N\subset \mathbb {R} ^{n}}
3925:
The image of the exponential map of the connected but non-compact group
3007:{\displaystyle \jmath t\mapsto \exp(\jmath t)=\cosh t+\jmath \ \sinh t.}
1490:
with the initial point at the identity element and the initial velocity
3538:
2479:
1023:
688:
of mathematical analysis is a special case of the exponential map when
532:
398:
290:
3426:
The preceding identity does not hold in general; the assumption that
741:
626:
29:
4067:
2701:{\displaystyle \operatorname {exp} :\operatorname {Lie} (V)=V\to V}
1487:
3585:{\displaystyle \exp _{*}\colon {\mathfrak {g}}\to {\mathfrak {g}}}
489:
4564:{\displaystyle N\subset {\mathfrak {g}}\simeq \mathbb {R} ^{n}}
4519:{\displaystyle \operatorname {exp} :N{\overset {\sim }{\to }}U}
3926:
2191:
1644:{\displaystyle \operatorname {Lie} (\mathbb {R} )=\mathbb {R} }
1368:
right translations, then the Lie-theoretic exponential map for
4956:
3415:{\displaystyle \exp(X+Y)=\exp(X)\exp(Y),\quad {\text{if }}=0}
2141:
corresponds to the exponential map for the complex Lie group
1821:{\displaystyle it\mapsto \exp(it)=e^{it}=\cos(t)+i\sin(t),\,}
1355:
1364:
is compact, it has a
Riemannian metric invariant under left
4124:{\displaystyle \operatorname {Ad} _{*}=\operatorname {ad} }
3024:
41:
2856:{\displaystyle \lbrace \jmath t:t\in \mathbb {R} \rbrace }
2194:
form a Lie group (isomorphic to the special unitary group
4475:
determines a coordinate system near the identity element
3592:, is the identity map (with the usual identifications).
1407:
It is the exponential map of a canonical left-invariant
5000:
Differential geometry, Lie groups, and symmetric spaces
4748:
4697:
4685:
for an example of how they are used in applications.
4626:
4597:
4577:
4532:
4493:
4457:
4411:
4387:
4367:
4140:
4104:
4084:
4035:
4003:
3972:
3949:
3877:
3811:
3692:
3668:
3629:
3605:
3551:
3507:
3479:
3452:
3432:
3327:
3254:
3166:
3143:
3119:
3065:
3035:
2943:
2873:
2826:
2765:
2718:
2664:
2616:
2571:
2488:
2289:
2207:
2174:
2147:
2102:
2069:
2036:
2016:
1972:
1936:
1916:
1896:
1850:
1731:
1679:
1615:
1583:
1542:
1518:
1472:
1425:
1338:
1314:
1290:
1144:
1110:
1082:
1032:
1004:
980:
938:
889:
859:
813:
786:
750:
726:
694:
666:
642:
3992:
3496:
3469:
The image of the exponential map always lies in the
2652:
with its tangent space at 0, and the exponential map
1127:
We have a more concrete definition in the case of a
2270:{\displaystyle \{it+ju+kv:t,u,v\in \mathbb {R} \}.}
5039:, vol. 1 (New ed.), Wiley-Interscience,
4785:
4726:
4666:
4609:
4583:
4563:
4518:
4467:
4443:
4397:
4373:
4344:
4123:
4090:
4048:
4021:
3981:
3958:
3910:
3847:
3824:
3794:
3678:
3635:
3615:
3584:
3529:
3485:
3458:
3438:
3414:
3307:
3239:
3149:
3125:
3101:
3051:
3006:
2923:
2855:
2813:{\displaystyle z=x+y\jmath ,\quad \jmath ^{2}=+1,}
2812:
2742:
2700:
2640:
2589:
2557:
2462:
2277:The exponential map for this Lie group is given by
2269:
2182:
2153:
2130:
2085:
2051:
2022:
1999:
1951:
1922:
1902:
1879:
1820:
1719:The exponential map for this Lie group is given by
1711:
1643:
1601:
1569:
1528:
1478:
1458:
1344:
1324:
1296:
1272:
1116:
1088:
1068:
1010:
986:
958:
922:
875:
835:
792:
760:
732:
700:
672:
652:
5074:
3530:{\displaystyle \exp \colon {\mathfrak {g}}\to G}
836:{\displaystyle \exp \colon {\mathfrak {g}}\to G}
460:Representation theory of semisimple Lie algebras
5032:
4962:
3240:{\displaystyle \exp((t+s)X)=\exp(tX)\exp(sX)\,}
1135:and is given by the ordinary series expansion:
959:{\displaystyle \gamma \colon \mathbb {R} \to G}
5033:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
3939:is not the whole group. Its image consists of
606:
4721:
4698:
4060:at the identity. Then the following diagram
2918:
2874:
2850:
2827:
2552:
2489:
2261:
2208:
2063:, we can identify it with the tangent space
1703:
1680:
4526:is a diffeomorphism from some neighborhood
2601:). Compare this to the first example above.
1880:{\displaystyle X=\mathbb {C} ^{n}/\Lambda }
4830:
4356:
2000:{\displaystyle \pi :\mathbb {C} ^{n}\to X}
1832:that is, the same formula as the ordinary
1356:Comparison with Riemannian exponential map
613:
599:
498:Particle physics and representation theory
40:
4654:
4551:
3901:
3865:is connected and nilpotent (for example,
3646:It is then not difficult to show that if
3304:
3236:
2914:
2846:
2641:{\displaystyle \operatorname {Lie} (V)=V}
2459:
2257:
2176:
2039:
1981:
1939:
1859:
1817:
1712:{\displaystyle \{it:t\in \mathbb {R} \}.}
1699:
1637:
1626:
1374:exponential map of this Riemannian metric
1131:. The exponential map coincides with the
946:
4997:
3836:is not a local diffeomorphism; see also
3308:{\displaystyle \exp(-X)=\exp(X)^{-1}.\,}
3025:Elementary properties of the exponential
2010:from the quotient by the lattice. Since
1419:is given by left translation. That is,
1100:of either the right- or left-invariant
465:Representations of classical Lie groups
16:Map from a Lie algebra to its Lie group
5075:
3911:{\displaystyle G=GL_{n}(\mathbb {C} )}
2474:This map takes the 2-sphere of radius
2931:since the exponential map is given by
21:Exponential map (Riemannian geometry)
5036:Foundations of Differential Geometry
4978:
4950:
4938:
4926:
4914:
4902:
4890:
4878:
4786:{\displaystyle (g,h)\mapsto gh^{-1}}
4029:be a Lie group homomorphism and let
3052:{\displaystyle X\in {\mathfrak {g}}}
1506:Lie group–Lie algebra correspondence
1398:
876:{\displaystyle X\in {\mathfrak {g}}}
318:Lie group–Lie algebra correspondence
4541:
4460:
4390:
4074:In particular, when applied to the
3817:
3814:
3787:
3671:
3608:
3577:
3567:
3516:
3102:{\displaystyle \gamma (t)=\exp(tX)}
3044:
1521:
1317:
1069:{\displaystyle \exp(tX)=\gamma (t)}
868:
822:
753:
645:
13:
5006:, vol. 34, Providence, R.I.:
4444:{\displaystyle X_{1},\dots ,X_{n}}
4188:
4185:
4146:
4143:
4066:
4022:{\displaystyle \phi \colon G\to H}
1959:) the torus comes equipped with a
1897:
1874:
1459:{\displaystyle \exp(X)=\gamma (1)}
1179:
923:{\displaystyle \exp(X)=\gamma (1)}
14:
5099:
4808:Derivative of the exponential map
3993:Exponential map and homomorphisms
3842:derivative of the exponential map
3497:The exponential near the identity
1570:{\displaystyle t\mapsto \exp(tX)}
1494:(thought of as a tangent vector).
4867:Baker-Campbell-Hausdorff formula
4571:of the origin to a neighborhood
3825:{\displaystyle {\mathfrak {so}}}
2512:
2444:
2433:
2418:
2385:
2291:
2131:{\displaystyle \pi :T_{0}X\to X}
2052:{\displaystyle \mathbb {C} ^{n}}
1952:{\displaystyle \mathbb {Z} ^{n}}
5004:Graduate Studies in Mathematics
4677:is then a coordinate system on
4468:{\displaystyle {\mathfrak {g}}}
4398:{\displaystyle {\mathfrak {g}}}
4131:, we have the useful identity:
3848:Surjectivity of the exponential
3771:
3679:{\displaystyle {\mathfrak {g}}}
3662:of exponentials of elements of
3616:{\displaystyle {\mathfrak {g}}}
3599:from some neighborhood of 0 in
3385:
2787:
1529:{\displaystyle {\mathfrak {g}}}
1508:also gives the definition: for
1325:{\displaystyle {\mathfrak {g}}}
761:{\displaystyle {\mathfrak {g}}}
708:is the multiplicative group of
653:{\displaystyle {\mathfrak {g}}}
4971:
4944:
4932:
4920:
4908:
4896:
4884:
4872:
4859:
4831:Birkenhake, Christina (2004).
4824:
4764:
4761:
4749:
4742:such that the group operation
4708:
4638:
4505:
4333:
4330:
4327:
4315:
4306:
4297:
4276:
4273:
4261:
4252:
4231:
4219:
4207:
4201:
4198:
4180:
4168:
4162:
4013:
3905:
3897:
3765:
3752:
3740:
3727:
3718:
3705:
3572:
3521:
3403:
3391:
3379:
3373:
3364:
3358:
3346:
3334:
3289:
3282:
3270:
3261:
3233:
3224:
3215:
3206:
3194:
3188:
3176:
3173:
3096:
3087:
3075:
3069:
2968:
2959:
2950:
2731:
2725:
2712:is the identity map, that is,
2692:
2683:
2677:
2629:
2623:
2584:
2578:
2549:
2543:
2531:
2525:
2449:
2439:
2427:
2423:
2413:
2409:
2394:
2390:
2380:
2376:
2364:
2337:
2328:
2325:
2298:
2122:
1991:
1811:
1805:
1790:
1784:
1756:
1747:
1738:
1630:
1622:
1587:
1564:
1555:
1546:
1453:
1447:
1438:
1432:
1157:
1151:
1063:
1057:
1048:
1039:
950:
917:
911:
902:
896:
827:
715:
513:Galilean group representations
508:Poincaré group representations
1:
5008:American Mathematical Society
4727:{\displaystyle \{Ug|g\in G\}}
3019:
2863:forms the Lie algebra of the
2599:Exponential of a Pauli vector
503:Lorentz group representations
470:Theorem of the highest weight
4818:
3650:is connected, every element
2478:inside the purely imaginary
2183:{\displaystyle \mathbb {H} }
1602:{\displaystyle t\mapsto tX.}
998:at the identity is equal to
7:
5060:Encyclopedia of Mathematics
4998:Helgason, Sigurdur (2001),
4963:Kobayashi & Nomizu 1996
4796:
1669:is a Lie group (called the
1655:
1022:It follows easily from the
10:
5104:
4803:List of exponential topics
3869:connected and abelian), or
3623:to a neighborhood of 1 in
2648:via the identification of
2192:quaternions of unit length
1096:may be constructed as the
455:Lie algebra representation
18:
4834:Complex Abelian Varieties
4405:, each choice of a basis
4049:{\displaystyle \phi _{*}}
3859:is connected and compact,
2743:{\displaystyle \exp(v)=v}
2030:is locally isomorphic to
4485:inverse function theorem
1903:{\displaystyle \Lambda }
450:Lie group representation
4979:Hall, Brian C. (2015),
4734:gives a structure of a
4683:closed-subgroup theorem
4357:Logarithmic coordinates
2590:{\displaystyle \sin(R)}
2565:, a 2-sphere of radius
1479:{\displaystyle \gamma }
1089:{\displaystyle \gamma }
475:Borel–Weil–Bott theorem
4917:Exercises 2.9 and 2.10
4865:This follows from the
4787:
4736:real-analytic manifold
4728:
4668:
4611:
4610:{\displaystyle e\in G}
4585:
4565:
4520:
4487:, the exponential map
4469:
4445:
4399:
4375:
4346:
4125:
4092:
4071:
4050:
4023:
3983:
3960:
3912:
3844:for more information.
3826:
3796:
3680:
3637:
3617:
3586:
3531:
3487:
3466:commute is important.
3460:
3440:
3416:
3309:
3241:
3151:
3127:
3111:one-parameter subgroup
3103:
3053:
3008:
2925:
2857:
2814:
2744:
2702:
2642:
2591:
2559:
2464:
2271:
2184:
2155:
2139:
2132:
2087:
2086:{\displaystyle T_{0}X}
2053:
2024:
2008:
2001:
1961:universal covering map
1953:
1924:
1904:
1881:
1822:
1713:
1645:
1603:
1571:
1530:
1480:
1460:
1346:
1326:
1298:
1274:
1183:
1118:
1090:
1070:
1012:
988:
972:one-parameter subgroup
960:
924:
877:
837:
794:
762:
734:
702:
674:
654:
373:Semisimple Lie algebra
328:Adjoint representation
5055:"Exponential mapping"
4788:
4729:
4669:
4612:
4586:
4566:
4521:
4483:, as follows. By the
4470:
4446:
4400:
4376:
4347:
4126:
4093:
4070:
4051:
4024:
3984:
3961:
3913:
3840:on this failure. See
3827:
3797:
3681:
3638:
3618:
3587:
3532:
3488:
3461:
3441:
3417:
3310:
3242:
3152:
3128:
3104:
3054:
3009:
2926:
2858:
2815:
2745:
2703:
2643:
2592:
2560:
2465:
2272:
2185:
2156:
2133:
2095:
2088:
2054:
2025:
2002:
1965:
1954:
1925:
1905:
1882:
1823:
1714:
1665:centered at 0 in the
1646:
1604:
1572:
1531:
1481:
1461:
1347:
1327:
1299:
1275:
1163:
1119:
1091:
1071:
1013:
989:
961:
925:
878:
853:: The exponential of
838:
795:
763:
735:
710:positive real numbers
703:
675:
655:
442:Representation theory
4746:
4695:
4624:
4595:
4575:
4530:
4491:
4455:
4409:
4385:
4365:
4138:
4102:
4082:
4033:
4001:
3970:
3947:
3875:
3809:
3690:
3666:
3627:
3603:
3549:
3505:
3501:The exponential map
3477:
3450:
3430:
3325:
3252:
3164:
3141:
3117:
3063:
3033:
2941:
2871:
2824:
2763:
2757:split-complex number
2716:
2662:
2614:
2569:
2486:
2287:
2205:
2172:
2145:
2100:
2067:
2034:
2014:
1970:
1934:
1914:
1894:
1848:
1841:More generally, for
1729:
1677:
1613:
1581:
1540:
1516:
1470:
1423:
1336:
1312:
1288:
1142:
1108:
1080:
1030:
1002:
978:
936:
887:
857:
811:
784:
748:
724:
692:
686:exponential function
664:
640:
3157:. It follows that:
3137:at the identity is
2820:the imaginary line
1834:complex exponential
1372:coincides with the
772:(thought of as the
587:Table of Lie groups
428:Compact Lie algebra
4813:Matrix exponential
4793:is real-analytic.
4783:
4724:
4664:
4607:
4581:
4561:
4516:
4465:
4441:
4395:
4371:
4361:Given a Lie group
4342:
4121:
4088:
4072:
4046:
4019:
3982:{\displaystyle -I}
3979:
3959:{\displaystyle -I}
3956:
3908:
3822:
3792:
3676:
3633:
3613:
3582:
3527:
3483:
3471:identity component
3456:
3436:
3412:
3305:
3237:
3147:
3123:
3099:
3049:
3004:
2921:
2853:
2810:
2740:
2698:
2638:
2587:
2555:
2460:
2267:
2180:
2151:
2128:
2083:
2049:
2020:
1997:
1949:
1930:(so isomorphic to
1920:
1900:
1887:for some integral
1877:
1818:
1709:
1641:
1599:
1567:
1526:
1476:
1456:
1417:parallel transport
1342:
1322:
1294:
1270:
1133:matrix exponential
1114:
1086:
1066:
1008:
984:
956:
920:
873:
833:
790:
758:
730:
698:
670:
650:
633:is a map from the
359:Affine Lie algebra
349:Simple Lie algebra
90:Special orthogonal
5017:978-0-8218-2848-9
4844:978-3-662-06307-1
4691:: The open cover
4644:
4584:{\displaystyle U}
4511:
4381:with Lie algebra
4374:{\displaystyle G}
4295:
4250:
4091:{\displaystyle G}
3636:{\displaystyle G}
3486:{\displaystyle G}
3459:{\displaystyle Y}
3439:{\displaystyle X}
3389:
3150:{\displaystyle X}
3126:{\displaystyle G}
2991:
2894:
2454:
2154:{\displaystyle X}
2061:complex manifolds
2023:{\displaystyle X}
1923:{\displaystyle n}
1409:affine connection
1399:Other definitions
1345:{\displaystyle G}
1297:{\displaystyle I}
1252:
1229:
1204:
1117:{\displaystyle X}
1011:{\displaystyle X}
987:{\displaystyle G}
793:{\displaystyle G}
733:{\displaystyle G}
701:{\displaystyle G}
673:{\displaystyle G}
625:In the theory of
623:
622:
423:Split Lie algebra
386:Cartan subalgebra
248:
247:
139:Simple Lie groups
5095:
5068:
5049:
5028:
4993:
4966:
4960:
4954:
4953:Proposition 3.35
4948:
4942:
4936:
4930:
4924:
4918:
4912:
4906:
4900:
4894:
4888:
4882:
4876:
4870:
4863:
4857:
4856:
4828:
4792:
4790:
4789:
4784:
4782:
4781:
4733:
4731:
4730:
4725:
4711:
4673:
4671:
4670:
4665:
4663:
4662:
4657:
4645:
4637:
4616:
4614:
4613:
4608:
4590:
4588:
4587:
4582:
4570:
4568:
4567:
4562:
4560:
4559:
4554:
4545:
4544:
4525:
4523:
4522:
4517:
4512:
4504:
4474:
4472:
4471:
4466:
4464:
4463:
4450:
4448:
4447:
4442:
4440:
4439:
4421:
4420:
4404:
4402:
4401:
4396:
4394:
4393:
4380:
4378:
4377:
4372:
4351:
4349:
4348:
4343:
4296:
4294:
4283:
4251:
4249:
4238:
4197:
4196:
4191:
4161:
4160:
4149:
4130:
4128:
4127:
4122:
4114:
4113:
4097:
4095:
4094:
4089:
4055:
4053:
4052:
4047:
4045:
4044:
4028:
4026:
4025:
4020:
3988:
3986:
3985:
3980:
3965:
3963:
3962:
3957:
3917:
3915:
3914:
3909:
3904:
3896:
3895:
3831:
3829:
3828:
3823:
3821:
3820:
3801:
3799:
3798:
3793:
3791:
3790:
3781:
3780:
3764:
3763:
3739:
3738:
3717:
3716:
3685:
3683:
3682:
3677:
3675:
3674:
3642:
3640:
3639:
3634:
3622:
3620:
3619:
3614:
3612:
3611:
3591:
3589:
3588:
3583:
3581:
3580:
3571:
3570:
3561:
3560:
3536:
3534:
3533:
3528:
3520:
3519:
3492:
3490:
3489:
3484:
3465:
3463:
3462:
3457:
3445:
3443:
3442:
3437:
3421:
3419:
3418:
3413:
3390:
3387:
3318:More generally:
3314:
3312:
3311:
3306:
3300:
3299:
3246:
3244:
3243:
3238:
3156:
3154:
3153:
3148:
3132:
3130:
3129:
3124:
3108:
3106:
3105:
3100:
3058:
3056:
3055:
3050:
3048:
3047:
3013:
3011:
3010:
3005:
2989:
2930:
2928:
2927:
2922:
2917:
2892:
2862:
2860:
2859:
2854:
2849:
2819:
2817:
2816:
2811:
2797:
2796:
2749:
2747:
2746:
2741:
2707:
2705:
2704:
2699:
2647:
2645:
2644:
2639:
2596:
2594:
2593:
2588:
2564:
2562:
2561:
2556:
2515:
2507:
2506:
2477:
2469:
2467:
2466:
2461:
2455:
2453:
2452:
2447:
2442:
2436:
2431:
2426:
2421:
2416:
2393:
2388:
2383:
2294:
2276:
2274:
2273:
2268:
2260:
2200:
2189:
2187:
2186:
2181:
2179:
2160:
2158:
2157:
2152:
2137:
2135:
2134:
2129:
2118:
2117:
2092:
2090:
2089:
2084:
2079:
2078:
2058:
2056:
2055:
2050:
2048:
2047:
2042:
2029:
2027:
2026:
2021:
2006:
2004:
2003:
1998:
1990:
1989:
1984:
1958:
1956:
1955:
1950:
1948:
1947:
1942:
1929:
1927:
1926:
1921:
1909:
1907:
1906:
1901:
1886:
1884:
1883:
1878:
1873:
1868:
1867:
1862:
1827:
1825:
1824:
1819:
1774:
1773:
1718:
1716:
1715:
1710:
1702:
1650:
1648:
1647:
1642:
1640:
1629:
1608:
1606:
1605:
1600:
1576:
1574:
1573:
1568:
1535:
1533:
1532:
1527:
1525:
1524:
1485:
1483:
1482:
1477:
1465:
1463:
1462:
1457:
1351:
1349:
1348:
1343:
1331:
1329:
1328:
1323:
1321:
1320:
1303:
1301:
1300:
1295:
1279:
1277:
1276:
1271:
1263:
1262:
1253:
1245:
1240:
1239:
1230:
1222:
1205:
1203:
1195:
1194:
1185:
1182:
1177:
1129:matrix Lie group
1123:
1121:
1120:
1115:
1104:associated with
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1017:
1015:
1014:
1009:
993:
991:
990:
985:
965:
963:
962:
957:
949:
929:
927:
926:
921:
882:
880:
879:
874:
872:
871:
842:
840:
839:
834:
826:
825:
799:
797:
796:
791:
778:identity element
767:
765:
764:
759:
757:
756:
739:
737:
736:
731:
707:
705:
704:
699:
679:
677:
676:
671:
659:
657:
656:
651:
649:
648:
615:
608:
601:
558:Claude Chevalley
415:Complexification
258:Other Lie groups
144:
143:
52:Classical groups
44:
26:
25:
5103:
5102:
5098:
5097:
5096:
5094:
5093:
5092:
5073:
5072:
5071:
5053:
5047:
5018:
4991:
4974:
4969:
4961:
4957:
4949:
4945:
4937:
4933:
4925:
4921:
4913:
4909:
4905:Corollary 11.10
4901:
4897:
4889:
4885:
4877:
4873:
4864:
4860:
4845:
4829:
4825:
4821:
4799:
4774:
4770:
4747:
4744:
4743:
4707:
4696:
4693:
4692:
4658:
4653:
4652:
4636:
4625:
4622:
4621:
4617:. Its inverse:
4596:
4593:
4592:
4576:
4573:
4572:
4555:
4550:
4549:
4540:
4539:
4531:
4528:
4527:
4503:
4492:
4489:
4488:
4459:
4458:
4456:
4453:
4452:
4435:
4431:
4416:
4412:
4410:
4407:
4406:
4389:
4388:
4386:
4383:
4382:
4366:
4363:
4362:
4359:
4287:
4282:
4242:
4237:
4192:
4184:
4183:
4150:
4142:
4141:
4139:
4136:
4135:
4109:
4105:
4103:
4100:
4099:
4083:
4080:
4079:
4078:of a Lie group
4040:
4036:
4034:
4031:
4030:
4002:
3999:
3998:
3995:
3971:
3968:
3967:
3948:
3945:
3944:
3932:
3900:
3891:
3887:
3876:
3873:
3872:
3850:
3813:
3812:
3810:
3807:
3806:
3786:
3785:
3776:
3772:
3759:
3755:
3734:
3730:
3712:
3708:
3691:
3688:
3687:
3670:
3669:
3667:
3664:
3663:
3628:
3625:
3624:
3607:
3606:
3604:
3601:
3600:
3576:
3575:
3566:
3565:
3556:
3552:
3550:
3547:
3546:
3515:
3514:
3506:
3503:
3502:
3499:
3478:
3475:
3474:
3451:
3448:
3447:
3431:
3428:
3427:
3386:
3326:
3323:
3322:
3292:
3288:
3253:
3250:
3249:
3165:
3162:
3161:
3142:
3139:
3138:
3118:
3115:
3114:
3064:
3061:
3060:
3043:
3042:
3034:
3031:
3030:
3027:
3022:
2942:
2939:
2938:
2913:
2872:
2869:
2868:
2845:
2825:
2822:
2821:
2792:
2788:
2764:
2761:
2760:
2717:
2714:
2713:
2663:
2660:
2659:
2615:
2612:
2611:
2570:
2567:
2566:
2511:
2502:
2498:
2487:
2484:
2483:
2475:
2448:
2443:
2438:
2437:
2432:
2430:
2422:
2417:
2412:
2389:
2384:
2379:
2290:
2288:
2285:
2284:
2256:
2206:
2203:
2202:
2195:
2175:
2173:
2170:
2169:
2146:
2143:
2142:
2113:
2109:
2101:
2098:
2097:
2074:
2070:
2068:
2065:
2064:
2043:
2038:
2037:
2035:
2032:
2031:
2015:
2012:
2011:
1985:
1980:
1979:
1971:
1968:
1967:
1943:
1938:
1937:
1935:
1932:
1931:
1915:
1912:
1911:
1895:
1892:
1891:
1869:
1863:
1858:
1857:
1849:
1846:
1845:
1766:
1762:
1730:
1727:
1726:
1698:
1678:
1675:
1674:
1658:
1636:
1625:
1614:
1611:
1610:
1582:
1579:
1578:
1541:
1538:
1537:
1520:
1519:
1517:
1514:
1513:
1471:
1468:
1467:
1424:
1421:
1420:
1401:
1358:
1337:
1334:
1333:
1316:
1315:
1313:
1310:
1309:
1306:identity matrix
1289:
1286:
1285:
1258:
1254:
1244:
1235:
1231:
1221:
1196:
1190:
1186:
1184:
1178:
1167:
1143:
1140:
1139:
1109:
1106:
1105:
1081:
1078:
1077:
1031:
1028:
1027:
1003:
1000:
999:
979:
976:
975:
945:
937:
934:
933:
888:
885:
884:
867:
866:
858:
855:
854:
821:
820:
812:
809:
808:
802:exponential map
785:
782:
781:
752:
751:
749:
746:
745:
725:
722:
721:
718:
693:
690:
689:
665:
662:
661:
660:of a Lie group
644:
643:
641:
638:
637:
631:exponential map
619:
574:
573:
572:
543:Wilhelm Killing
527:
519:
518:
517:
492:
481:
480:
479:
444:
434:
433:
432:
419:
403:
381:Dynkin diagrams
375:
365:
364:
363:
345:
323:Exponential map
312:
302:
301:
300:
281:Conformal group
260:
250:
249:
241:
233:
225:
217:
209:
190:
180:
170:
160:
141:
131:
130:
129:
110:Special unitary
54:
24:
17:
12:
11:
5:
5101:
5091:
5090:
5085:
5070:
5069:
5051:
5045:
5030:
5016:
4995:
4990:978-3319134666
4989:
4975:
4973:
4970:
4968:
4967:
4955:
4943:
4931:
4919:
4907:
4895:
4893:Corollary 3.47
4883:
4881:Corollary 3.44
4871:
4858:
4843:
4822:
4820:
4817:
4816:
4815:
4810:
4805:
4798:
4795:
4780:
4777:
4773:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4723:
4720:
4717:
4714:
4710:
4706:
4703:
4700:
4675:
4674:
4661:
4656:
4651:
4648:
4643:
4640:
4635:
4632:
4629:
4606:
4603:
4600:
4580:
4558:
4553:
4548:
4543:
4538:
4535:
4515:
4510:
4507:
4502:
4499:
4496:
4462:
4438:
4434:
4430:
4427:
4424:
4419:
4415:
4392:
4370:
4358:
4355:
4354:
4353:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4293:
4290:
4286:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4248:
4245:
4241:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4195:
4190:
4187:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4159:
4156:
4153:
4148:
4145:
4120:
4117:
4112:
4108:
4087:
4076:adjoint action
4043:
4039:
4018:
4015:
4012:
4009:
4006:
3994:
3991:
3978:
3975:
3955:
3952:
3930:
3920:
3919:
3907:
3903:
3899:
3894:
3890:
3886:
3883:
3880:
3870:
3860:
3849:
3846:
3819:
3816:
3789:
3784:
3779:
3775:
3770:
3767:
3762:
3758:
3754:
3751:
3748:
3745:
3742:
3737:
3733:
3729:
3726:
3723:
3720:
3715:
3711:
3707:
3704:
3701:
3698:
3695:
3673:
3632:
3610:
3597:diffeomorphism
3579:
3574:
3569:
3564:
3559:
3555:
3526:
3523:
3518:
3513:
3510:
3498:
3495:
3482:
3455:
3435:
3424:
3423:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3316:
3315:
3303:
3298:
3295:
3291:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3247:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3146:
3135:tangent vector
3122:
3109:is the unique
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3046:
3041:
3038:
3026:
3023:
3021:
3018:
3017:
3016:
3015:
3014:
3003:
3000:
2997:
2994:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2933:
2932:
2920:
2916:
2912:
2909:
2906:
2903:
2900:
2897:
2891:
2888:
2885:
2882:
2879:
2876:
2865:unit hyperbola
2852:
2848:
2844:
2841:
2838:
2835:
2832:
2829:
2809:
2806:
2803:
2800:
2795:
2791:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2752:
2751:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2710:
2709:
2708:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2654:
2653:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2603:
2602:
2586:
2583:
2580:
2577:
2574:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2514:
2510:
2505:
2501:
2497:
2494:
2491:
2472:
2471:
2470:
2458:
2451:
2446:
2441:
2435:
2429:
2425:
2420:
2415:
2411:
2408:
2405:
2402:
2399:
2396:
2392:
2387:
2382:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2293:
2279:
2278:
2266:
2263:
2259:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2178:
2150:
2127:
2124:
2121:
2116:
2112:
2108:
2105:
2082:
2077:
2073:
2046:
2041:
2019:
1996:
1993:
1988:
1983:
1978:
1975:
1964:
1963:
1946:
1941:
1919:
1899:
1876:
1872:
1866:
1861:
1856:
1853:
1838:
1837:
1830:
1829:
1828:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1772:
1769:
1765:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1721:
1720:
1708:
1705:
1701:
1697:
1694:
1691:
1688:
1685:
1682:
1657:
1654:
1653:
1652:
1639:
1635:
1632:
1628:
1624:
1621:
1618:
1598:
1595:
1592:
1589:
1586:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1523:
1502:
1495:
1486:is the unique
1475:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1400:
1397:
1379:For a general
1357:
1354:
1341:
1319:
1293:
1282:
1281:
1269:
1266:
1261:
1257:
1251:
1248:
1243:
1238:
1234:
1228:
1225:
1220:
1217:
1214:
1211:
1208:
1202:
1199:
1193:
1189:
1181:
1176:
1173:
1170:
1166:
1162:
1159:
1156:
1153:
1150:
1147:
1113:
1098:integral curve
1085:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1020:
1019:
1007:
996:tangent vector
983:
970:is the unique
968:
967:
966:
955:
952:
948:
944:
941:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
870:
865:
862:
844:
843:
832:
829:
824:
819:
816:
789:
755:
729:
717:
714:
697:
669:
647:
621:
620:
618:
617:
610:
603:
595:
592:
591:
590:
589:
584:
576:
575:
571:
570:
565:
563:Harish-Chandra
560:
555:
550:
545:
540:
538:Henri Poincaré
535:
529:
528:
525:
524:
521:
520:
516:
515:
510:
505:
500:
494:
493:
488:Lie groups in
487:
486:
483:
482:
478:
477:
472:
467:
462:
457:
452:
446:
445:
440:
439:
436:
435:
431:
430:
425:
420:
418:
417:
412:
406:
404:
402:
401:
396:
390:
388:
383:
377:
376:
371:
370:
367:
366:
362:
361:
356:
351:
346:
344:
343:
338:
332:
330:
325:
320:
314:
313:
308:
307:
304:
303:
299:
298:
293:
288:
286:Diffeomorphism
283:
278:
273:
268:
262:
261:
256:
255:
252:
251:
246:
245:
244:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
200:
199:
195:
194:
193:
192:
186:
182:
176:
172:
166:
162:
156:
149:
148:
142:
137:
136:
133:
132:
128:
127:
117:
107:
97:
87:
77:
70:Special linear
67:
60:General linear
56:
55:
50:
49:
46:
45:
37:
36:
15:
9:
6:
4:
3:
2:
5100:
5089:
5086:
5084:
5081:
5080:
5078:
5066:
5062:
5061:
5056:
5052:
5048:
5046:0-471-15733-3
5042:
5038:
5037:
5031:
5027:
5023:
5019:
5013:
5009:
5005:
5001:
4996:
4992:
4986:
4982:
4977:
4976:
4965:, p. 43.
4964:
4959:
4952:
4947:
4940:
4935:
4929:Exercise 3.22
4928:
4923:
4916:
4911:
4904:
4899:
4892:
4887:
4880:
4875:
4868:
4862:
4854:
4850:
4846:
4840:
4836:
4835:
4827:
4823:
4814:
4811:
4809:
4806:
4804:
4801:
4800:
4794:
4778:
4775:
4771:
4767:
4758:
4755:
4752:
4741:
4737:
4718:
4715:
4712:
4704:
4701:
4690:
4686:
4684:
4680:
4659:
4649:
4646:
4641:
4633:
4630:
4627:
4620:
4619:
4618:
4604:
4601:
4598:
4578:
4556:
4546:
4536:
4533:
4513:
4508:
4500:
4497:
4494:
4486:
4482:
4478:
4436:
4432:
4428:
4425:
4422:
4417:
4413:
4368:
4339:
4336:
4324:
4321:
4318:
4312:
4309:
4303:
4300:
4291:
4288:
4284:
4279:
4270:
4267:
4264:
4258:
4255:
4246:
4243:
4239:
4234:
4228:
4225:
4222:
4216:
4213:
4210:
4204:
4193:
4177:
4174:
4171:
4165:
4157:
4154:
4151:
4134:
4133:
4132:
4118:
4115:
4110:
4106:
4085:
4077:
4069:
4065:
4063:
4059:
4041:
4037:
4016:
4010:
4007:
4004:
3990:
3976:
3973:
3953:
3950:
3942:
3938:
3936:
3929:
3923:
3892:
3888:
3884:
3881:
3878:
3871:
3868:
3864:
3861:
3858:
3855:
3854:
3853:
3845:
3843:
3839:
3835:
3803:
3782:
3777:
3773:
3768:
3760:
3756:
3749:
3746:
3743:
3735:
3731:
3724:
3721:
3713:
3709:
3702:
3699:
3696:
3693:
3661:
3657:
3653:
3649:
3644:
3630:
3598:
3593:
3562:
3557:
3553:
3544:
3540:
3524:
3511:
3508:
3494:
3480:
3472:
3467:
3453:
3433:
3409:
3406:
3400:
3397:
3394:
3382:
3376:
3370:
3367:
3361:
3355:
3352:
3349:
3343:
3340:
3337:
3331:
3328:
3321:
3320:
3319:
3301:
3296:
3293:
3285:
3279:
3276:
3273:
3267:
3264:
3258:
3255:
3248:
3230:
3227:
3221:
3218:
3212:
3209:
3203:
3200:
3197:
3191:
3185:
3182:
3179:
3170:
3167:
3160:
3159:
3158:
3144:
3136:
3120:
3112:
3093:
3090:
3084:
3081:
3078:
3072:
3066:
3039:
3036:
3001:
2998:
2995:
2992:
2986:
2983:
2980:
2977:
2974:
2971:
2965:
2962:
2956:
2953:
2947:
2944:
2937:
2936:
2935:
2934:
2910:
2907:
2904:
2901:
2898:
2895:
2889:
2886:
2883:
2880:
2877:
2866:
2842:
2839:
2836:
2833:
2830:
2807:
2804:
2801:
2798:
2793:
2789:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2758:
2754:
2753:
2737:
2734:
2728:
2722:
2719:
2711:
2695:
2689:
2686:
2680:
2674:
2671:
2668:
2665:
2658:
2657:
2656:
2655:
2651:
2635:
2632:
2626:
2620:
2617:
2609:
2605:
2604:
2600:
2581:
2575:
2572:
2546:
2540:
2537:
2534:
2528:
2522:
2519:
2516:
2508:
2503:
2499:
2495:
2492:
2481:
2473:
2456:
2406:
2403:
2400:
2397:
2373:
2370:
2367:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2334:
2331:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2295:
2283:
2282:
2281:
2280:
2264:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2198:
2193:
2190:, the set of
2168:
2164:
2163:
2162:
2148:
2138:
2125:
2119:
2114:
2110:
2106:
2103:
2094:
2093:, and the map
2080:
2075:
2071:
2062:
2044:
2017:
2007:
1994:
1986:
1976:
1973:
1962:
1944:
1917:
1890:
1870:
1864:
1854:
1851:
1844:
1843:complex torus
1840:
1839:
1835:
1831:
1814:
1808:
1802:
1799:
1796:
1793:
1787:
1781:
1778:
1775:
1770:
1767:
1763:
1759:
1753:
1750:
1744:
1741:
1735:
1732:
1725:
1724:
1723:
1722:
1706:
1695:
1692:
1689:
1686:
1683:
1672:
1668:
1667:complex plane
1664:
1660:
1659:
1633:
1619:
1616:
1596:
1593:
1590:
1584:
1561:
1558:
1552:
1549:
1543:
1511:
1507:
1503:
1500:
1496:
1493:
1489:
1473:
1450:
1444:
1441:
1435:
1429:
1426:
1418:
1414:
1410:
1406:
1405:
1404:
1396:
1394:
1390:
1386:
1382:
1377:
1375:
1371:
1367:
1363:
1353:
1339:
1307:
1291:
1267:
1264:
1259:
1255:
1249:
1246:
1241:
1236:
1232:
1226:
1223:
1218:
1215:
1212:
1209:
1206:
1200:
1197:
1191:
1187:
1174:
1171:
1168:
1164:
1160:
1154:
1148:
1145:
1138:
1137:
1136:
1134:
1130:
1125:
1111:
1103:
1099:
1083:
1060:
1054:
1051:
1045:
1042:
1036:
1033:
1025:
1005:
997:
981:
973:
969:
953:
942:
939:
932:
931:
914:
908:
905:
899:
893:
890:
863:
860:
852:
849:
848:
847:
830:
817:
814:
807:
806:
805:
803:
787:
779:
775:
774:tangent space
771:
743:
727:
713:
711:
695:
687:
684:The ordinary
682:
667:
636:
632:
628:
616:
611:
609:
604:
602:
597:
596:
594:
593:
588:
585:
583:
580:
579:
578:
577:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
530:
523:
522:
514:
511:
509:
506:
504:
501:
499:
496:
495:
491:
485:
484:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
447:
443:
438:
437:
429:
426:
424:
421:
416:
413:
411:
408:
407:
405:
400:
397:
395:
392:
391:
389:
387:
384:
382:
379:
378:
374:
369:
368:
360:
357:
355:
352:
350:
347:
342:
339:
337:
334:
333:
331:
329:
326:
324:
321:
319:
316:
315:
311:
306:
305:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
272:
269:
267:
264:
263:
259:
254:
253:
242:
236:
234:
228:
226:
220:
218:
212:
210:
204:
203:
202:
201:
197:
196:
191:
189:
183:
181:
179:
173:
171:
169:
163:
161:
159:
153:
152:
151:
150:
146:
145:
140:
135:
134:
125:
121:
118:
115:
111:
108:
105:
101:
98:
95:
91:
88:
85:
81:
78:
75:
71:
68:
65:
61:
58:
57:
53:
48:
47:
43:
39:
38:
35:
31:
28:
27:
22:
5083:Lie algebras
5058:
5035:
4999:
4980:
4958:
4946:
4941:Theorem 3.28
4934:
4922:
4910:
4898:
4886:
4874:
4861:
4833:
4826:
4739:
4688:
4687:
4678:
4676:
4480:
4476:
4360:
4073:
3996:
3940:
3934:
3927:
3924:
3921:
3866:
3862:
3856:
3851:
3804:
3659:
3655:
3651:
3647:
3645:
3594:
3543:differential
3500:
3468:
3425:
3317:
3028:
2649:
2607:
2196:
2140:
2096:
2009:
1966:
1671:circle group
1509:
1498:
1491:
1415:, such that
1412:
1402:
1392:
1388:
1384:
1380:
1378:
1369:
1365:
1361:
1359:
1283:
1126:
1102:vector field
1021:
883:is given by
850:
845:
801:
719:
683:
630:
624:
568:Armand Borel
553:Hermann Weyl
354:Loop algebra
336:Killing form
322:
310:Lie algebras
187:
177:
167:
157:
123:
113:
103:
93:
83:
73:
63:
34:Lie algebras
4972:Works cited
2480:quaternions
2167:quaternions
1663:unit circle
770:Lie algebra
716:Definitions
635:Lie algebra
548:Élie Cartan
394:Root system
198:Exceptional
5088:Lie groups
5077:Categories
4058:derivative
3539:smooth map
3059:, the map
3020:Properties
1076:. The map
1024:chain rule
851:Definition
627:Lie groups
533:Sophus Lie
526:Scientists
399:Weyl group
120:Symplectic
80:Orthogonal
30:Lie groups
5065:EMS Press
4951:Hall 2015
4939:Hall 2015
4927:Hall 2015
4915:Hall 2015
4903:Hall 2015
4891:Hall 2015
4879:Hall 2015
4853:851380558
4819:Citations
4776:−
4765:↦
4716:∈
4650:⊂
4642:∼
4639:→
4602:∈
4547:≃
4537:⊂
4509:∼
4506:→
4426:…
4340:⋯
4178:
4155:
4111:∗
4042:∗
4038:ϕ
4014:→
4008::
4005:ϕ
3974:−
3951:−
3838:cut locus
3783:∈
3750:
3744:⋯
3725:
3703:
3573:→
3563::
3558:∗
3545:at zero,
3522:→
3512::
3371:
3356:
3332:
3294:−
3280:
3265:−
3259:
3222:
3204:
3171:
3085:
3067:γ
3040:∈
2996:
2987:ȷ
2978:
2963:ȷ
2957:
2951:↦
2945:ȷ
2911:∈
2899:
2890:ȷ
2881:
2843:∈
2831:ȷ
2790:ȷ
2782:ȷ
2723:
2693:→
2675:
2621:
2576:
2541:
2523:
2509:⊂
2496:∈
2407:
2374:
2335:
2329:↦
2254:∈
2123:→
2104:π
1992:→
1974:π
1898:Λ
1875:Λ
1803:
1782:
1745:
1739:↦
1696:∈
1620:
1588:↦
1553:
1547:↦
1474:γ
1445:γ
1430:
1268:⋯
1180:∞
1165:∑
1149:
1084:γ
1055:γ
1037:
951:→
943::
940:γ
909:γ
894:
864:∈
828:→
818::
804:is a map
742:Lie group
410:Real form
296:Euclidean
147:Classical
4797:See also
4098:, since
4062:commutes
3388:if
3029:For all
1910:of rank
1656:Examples
1488:geodesic
582:Glossary
276:Poincaré
5067:, 2001
5026:1834454
4056:be its
3832:(3) to
3660:product
2755:In the
2165:In the
1889:lattice
1609:(note:
1304:is the
800:). The
776:to the
768:be its
490:physics
271:Lorentz
100:Unitary
5043:
5024:
5014:
4987:
4851:
4841:
4689:Remark
3541:. Its
3133:whose
2990:
2893:
2867:group
2759:plane
1466:where
1284:where
994:whose
930:where
629:, the
266:Circle
3834:SO(3)
3658:is a
3537:is a
2597:(cf.
1026:that
740:be a
341:Index
5041:ISBN
5012:ISBN
4985:ISBN
4849:OCLC
4839:ISBN
4479:for
3997:Let
3446:and
2993:sinh
2975:cosh
2896:sinh
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2606:Let
1661:The
1504:The
744:and
720:Let
291:Loop
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4738:to
4628:log
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4495:exp
4451:of
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3509:exp
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3113:of
3082:exp
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2720:exp
2672:Lie
2666:exp
2618:Lie
2573:sin
2538:cos
2482:to
2404:sin
2371:cos
2332:exp
2199:(2)
2059:as
1800:sin
1779:cos
1742:exp
1617:Lie
1550:exp
1512:in
1427:exp
1411:on
1385:not
1366:and
1360:If
1332:of
1146:exp
1034:exp
974:of
891:exp
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780:of
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