2964:
4346:
2843:
67:
878:. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the
3749:
1654:
3591:
2501:
4341:{\displaystyle {\begin{aligned}\left&=0\\\left&=0\\\left&=i\hbar cp_{i}\\\left&=0\\\left&=i\hbar \epsilon _{ijk}p_{k}\\\left&={\frac {i\hbar }{c}}{\mathcal {H}}\delta _{ij}\\\left&=i\hbar \epsilon _{ijk}L_{k}\\\left&=i\hbar \epsilon _{ijk}K_{k}\\\left&=-i\hbar \epsilon _{ijk}L_{k}\end{aligned}}}
3089:
38:
2838:{\displaystyle {\begin{aligned}&=0\,\\{\frac {1}{i}}~&=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho }P_{\mu }\,\\{\frac {1}{i}}~&=\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho }\,,\end{aligned}}}
2489:
4763:
3586:{\displaystyle {\begin{aligned}&=i\epsilon _{mnk}P_{k}~,\\&=0~,\\&=i\eta _{ik}P_{0}~,\\&=-iP_{i}~,\\&=i\epsilon _{mnk}J_{k}~,\\&=i\epsilon _{mnk}K_{k}~,\\&=-i\epsilon _{mnk}J_{k}~,\end{aligned}}}
4856:
1408:
1264:
1476:
889:
for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a
1171:
4647:, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted.
5045:
3737:
3038:
3754:
3094:
2506:
1561:
2378:
1628:
1517:
2322:
2386:
4427:
4390:
2276:
512:
487:
450:
3081:
2954:
1587:
4667:
4454:
2910:
4510:
4626:
4606:
4578:
4554:
4530:
2890:
2866:
1024:
of the spacetime translations group and the
Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess
3679:
4774:
1349:
1182:
2071:
814:
4918:
2119:
5034:
1052:
3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time
1419:
1113:
2124:
4938:
3740:
2114:
2109:
2967:
A diagram of the commutation structure of the
Poincaré algebra. The edges of the diagram connect generators with nonzero commutators.
894:" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with
5019:
1929:
2193:
2076:
5196:
5322:
5226:
3684:
928:, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article.
372:
1092:
5311:
5236:
2974:
2224:
322:
17:
1526:
807:
317:
844:
5288:
5206:
5179:
2331:
5332:
5022:). The group defined in this paper would now be described as the homogeneous Lorentz group with scalar multipliers.
4933:
4457:
3083:. In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as
2869:
2086:
1595:
3596:
where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification
2246:
733:
2484:{\textstyle \exp \left(ia_{\mu }P^{\mu }\right)\exp \left({\frac {i}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}
1484:
5366:
5361:
4928:
2081:
2061:
1302:
800:
2381:
2026:
1934:
5033:
2285:
417:
231:
4923:
4472:
2066:
1332:, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a
1322:
971:
867:
149:
5356:
4464:
4948:
5111:
5040:
Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
4943:
2217:
1701:
615:
349:
226:
114:
5228:
Relativistic
Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory
4655:
The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The
4395:
4358:
2279:
495:
470:
433:
4758:{\displaystyle \operatorname {IO} (1,d-1):=\mathbf {R} ^{1,d-1}\rtimes \operatorname {O} (1,d-1)}
2252:
910:
5083:
4969:
2971:
The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations,
2915:
2021:
1984:
1952:
1939:
1566:
1100:
1080:
984:
895:
891:
765:
555:
3043:
2053:
1721:
948:
639:
1796:
1786:
1776:
1766:
1590:
5123:
4982:
4585:
2895:
1681:
1671:
1340:
1279:
1036:
579:
567:
185:
119:
5252:
4432:
8:
5351:
4633:
4468:
2210:
2198:
2039:
1869:
1021:
154:
49:
5135:
5127:
4986:
4485:
871:
854:
that is of importance as a model in our understanding of the most basic fundamentals of
5147:
5006:
4611:
4591:
4581:
4563:
4539:
4515:
2963:
2875:
2851:
2325:
1970:
1960:
1104:
1035:
10 generators (in four spacetime dimensions) associated with the
Poincaré symmetry, by
1016:
940:
921:
886:
875:
139:
111:
5371:
5328:
5307:
5300:
5284:
5232:
5202:
5175:
5139:
5035:"Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern"
5010:
4644:
4352:
2034:
1997:
1333:
1318:
1287:
953:
902:
848:
836:
544:
387:
281:
31:
5151:
5002:
3599:
2149:
832:
5131:
4998:
4990:
4557:
2169:
1849:
1841:
1833:
1825:
1817:
1750:
1731:
1691:
1329:
695:
687:
679:
671:
663:
651:
591:
531:
521:
363:
305:
180:
2495:. In component form, the Poincaré algebra is given by the commutation relations:
5169:
4908:
2957:
2154:
1907:
1892:
1663:
1295:
1271:
1084:
1049:
3 for the angular momentum – associated with rotations between spatial dimensions
779:
772:
758:
715:
603:
526:
356:
270:
210:
90:
4584:
quantum number. In practice, charge conjugation and parity are violated by many
4913:
4533:
4480:
2174:
1992:
1897:
1310:
906:
840:
786:
722:
412:
392:
329:
294:
215:
205:
190:
175:
129:
106:
4851:{\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)}
4661:-dimensional Poincaré group is analogously defined by the semi-direct product
2159:
1403:{\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {SL} (2,\mathbf {C} ),}
1259:{\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)}
5345:
5222:
5143:
1882:
1711:
1275:
1088:
1073:
1066:
1011:
914:
879:
705:
627:
461:
334:
200:
1046:
3 for the momentum – associated with translations through spatial dimensions
4629:
2179:
2164:
1965:
1947:
1877:
1096:
560:
259:
248:
195:
170:
165:
124:
95:
58:
5060:
4968:
2492:
2242:
2005:
1921:
1645:
5046:
The
Fundamental Equations for Electromagnetic Processes in Moving Bodies
2249:
of the Lie algebra of the
Lorentz group. More specifically, the proper (
1095:
of the origin. The
Poincaré group itself is the minimal subgroup of the
5015:
4994:
2144:
2010:
1902:
727:
455:
30:
For the
Poincaré group (fundamental group) of a topological space, see
1641:
1471:{\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {Spin} (1,3),}
1077:
1069:
851:
548:
1166:{\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)\,,}
898:
being produced as the composition of an even number of reflections.
4637:
4479:
squared of each particle (i.e. its mass squared) and the intrinsic
1520:
1062:
966:
85:
2101:
1314:
925:
855:
427:
341:
37:
66:
4460:; they serve as labels for the representations of the group.
1270:
Another way of putting this is that the
Poincaré group is a
1107:
of the spacetime translations group and the Lorentz group,
1043:
1 for the energy – associated with translations through time
2380:, is connected to the identity and is thus provided by the
1653:
1306:
3732:{\textstyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}
988:, transformations connecting two uniformly moving bodies (
5084:"Survey of Symmetry and Conservation Laws: More Poincare"
5327:(2nd ed.). Cambridge University Press. p. 62.
1061:
The Poincaré group is the group of Minkowski spacetime
5306:. Vol. 1. Cambridge: Cambridge University press.
5059:
4488:
4435:
4398:
4361:
3687:
3602:
3046:
3033:{\textstyle J_{i}={\frac {1}{2}}\epsilon _{imn}M^{mn}}
2977:
2389:
2334:
2288:
2255:
1317:
or half integer) and are associated with particles in
4777:
4670:
4614:
4594:
4566:
4542:
4518:
4463:
The Poincaré group is the full symmetry group of any
3752:
3092:
2918:
2898:
2878:
2854:
2504:
1598:
1569:
1529:
1487:
1422:
1352:
1185:
1116:
498:
473:
436:
1519:
are not able to describe fields with spin 1/2; i.e.
5112:"On the six components of optical angular momentum"
1556:{\displaystyle \operatorname {SL} (2,\mathbf {C} )}
5299:
4850:
4757:
4620:
4600:
4572:
4548:
4524:
4504:
4448:
4421:
4384:
4340:
3731:
3673:
3585:
3075:
3032:
2948:
2904:
2884:
2860:
2837:
2483:
2372:
2316:
2270:
1622:
1589:matrices with unit determinant, isomorphic to the
1581:
1555:
1511:
1470:
1402:
1282:of it; it is sometimes dubbed, informally, as the
1258:
1165:
506:
481:
444:
2892:is the generator of Lorentz transformations, and
909:is a comparable ten-parameter group that acts on
882:of an object is also unaffected by such a shift.
5343:
2256:
2072:Representation theory of semisimple Lie algebras
4966:
4862:The Lie algebra retains its form, with indices
3743:. In terms of the physical parameters, we have
3681:permits reduction of the Lorentz subalgebra to
2373:{\textstyle \mathrm {SO} (1,3)_{+}^{\uparrow }}
970:in space, forming the non-abelian Lie group of
952:(displacements) in time and space, forming the
5194:
5057:
5031:
1481:is more important, because representations of
1413:which may be identified with the double cover
4881:. The alternative representation in terms of
2218:
808:
4975:Rendiconti del Circolo Matematico di Palermo
1343:, the universal cover of the Poincaré group
4919:Representation theory of the Poincaré group
5320:
5278:
5231:. Cambridge University Press. p. 10.
5221:
5198:General Principles of Quantum Field Theory
4939:Particle physics and representation theory
4835:
3739:and efficient treatment of its associated
2225:
2211:
2110:Particle physics and representation theory
1652:
1623:{\displaystyle \operatorname {Spin} (1,3)}
1243:
870:of Minkowski space that do not change the
815:
801:
3635:
2827:
2654:
2548:
1159:
917:to relate co-moving frames of reference.
500:
475:
438:
5297:
2962:
1512:{\displaystyle \operatorname {SO} (1,3)}
1301:Its positive energy unitary irreducible
1286:. In turn, it can also be obtained as a
36:
5201:(2nd ed.). Springer. p. 272.
5109:
2077:Representations of classical Lie groups
14:
5344:
4970:"Sur la dynamique de l'électron"
4899:has no analogue in higher dimensions.
373:Classification of finite simple groups
5167:
4640:may be constructed from those given.
4475:. These are usually specified by the
2317:{\textstyle {\Lambda ^{0}}_{0}\geq 1}
5163:
5161:
2324:) part of the Lorentz subgroup (its
1930:Lie group–Lie algebra correspondence
1099:which includes all translations and
931:
4650:
3715:
3712:
3693:
3690:
1633:
866:The Poincaré group consists of all
24:
4768:with the analogous multiplication
4728:
4085:
3852:
3808:
3764:
2339:
2336:
2292:
2259:
1138:
27:Group of flat spacetime symmetries
25:
5383:
5171:BMS Particles in Three Dimensions
5158:
5110:Barnett, Stephen M (2011-06-01).
4305:
4226:
4150:
4074:
3995:
3885:
1056:
913:. Instead of boosts, it features
4703:
1546:
1425:
1390:
1355:
1119:
65:
5283:. World Scientific Publishing.
5020:On the Dynamics of the Electron
2245:of the Poincaré group. It is a
5245:
5215:
5188:
5103:
5076:
5051:
5025:
4967:Poincaré, Henri (1905-12-14),
4960:
4845:
4814:
4808:
4796:
4790:
4778:
4752:
4734:
4695:
4677:
3726:
3720:
3704:
3698:
3662:
3603:
3531:
3505:
3456:
3430:
3381:
3355:
3319:
3293:
3247:
3221:
3198:
3172:
3123:
3097:
2943:
2919:
2704:
2672:
2595:
2566:
2535:
2509:
2365:
2356:
2343:
2125:Galilean group representations
2120:Poincaré group representations
1617:
1605:
1550:
1536:
1506:
1494:
1462:
1450:
1394:
1380:
1253:
1222:
1216:
1204:
1198:
1186:
1156:
1144:
1039:, imply 10 conservation laws:
734:Infinite dimensional Lie group
13:
1:
5272:
5168:Oblak, Blagoje (2017-08-01).
5136:10.1088/2040-8978/13/6/064010
4929:Symmetry in quantum mechanics
4473:representations of this group
4422:{\textstyle W_{\mu }W^{\mu }}
4385:{\textstyle P_{\mu }P^{\mu }}
2115:Lorentz group representations
2082:Theorem of the highest weight
835:(1905), was first defined by
5302:The Quantum Theory of Fields
4638:time-reversal quantum number
2271:{\textstyle \det \Lambda =1}
1591:Lorentz-signature spin group
1087:, while the six-dimensional
924:, i.e. under the effects of
507:{\displaystyle \mathbb {Z} }
482:{\displaystyle \mathbb {Z} }
445:{\displaystyle \mathbb {Z} }
7:
4934:Pauli–Lubanski pseudovector
4902:
4636:in quantum field theory, a
4458:Pauli–Lubanski pseudovector
1284:inhomogeneous Lorentz group
972:three-dimensional rotations
956:of spacetime translations (
861:
847:. It is a ten-dimensional
232:List of group theory topics
10:
5388:
4870:now taking values between
2067:Lie algebra representation
1176:with group multiplication
1103:. More precisely, it is a
1065:. It is a ten-dimensional
868:coordinate transformations
29:
5298:Weinberg, Steven (1995).
5066:Physikalische Zeitschrift
5044:(Wikisource translation:
4465:relativistic field theory
3076:{\textstyle K_{i}=M_{i0}}
2949:{\displaystyle (+,-,-,-)}
1582:{\displaystyle 2\times 2}
1563:is the group of complex
1309:(nonnegative number) and
998:The last two symmetries,
5253:"Topics: Poincaré Group"
5174:. Springer. p. 80.
5003:2027/uiug.30112063899089
4954:
4944:Continuous spin particle
2062:Lie group representation
1091:is also a subgroup, the
939:is the full symmetry of
885:In total, there are ten
350:Elementary abelian group
227:Glossary of group theory
5281:Group Theory in Physics
5195:N.N. Bogolubov (1989).
4924:Wigner's classification
2087:Borel–Weil–Bott theorem
1328:In accordance with the
1323:Wigner's classification
1290:of the de Sitter group
1101:Lorentz transformations
1072:. The four-dimensional
1030:relativistic invariance
911:absolute time and space
4949:super-Poincaré algebra
4852:
4759:
4622:
4602:
4586:quantum field theories
4574:
4550:
4526:
4506:
4450:
4423:
4386:
4342:
3733:
3675:
3587:
3077:
3034:
2968:
2956:Minkowski metric (see
2950:
2906:
2886:
2862:
2839:
2485:
2374:
2318:
2272:
1985:Semisimple Lie algebra
1940:Adjoint representation
1624:
1583:
1557:
1513:
1472:
1404:
1260:
1167:
766:Linear algebraic group
508:
483:
446:
42:
5061:"Raum und Zeit"
4853:
4760:
4628:are forfeited. Since
4623:
4603:
4588:; where this occurs,
4575:
4551:
4527:
4507:
4451:
4449:{\textstyle W_{\mu }}
4424:
4387:
4343:
3734:
3676:
3588:
3078:
3035:
2966:
2951:
2907:
2905:{\displaystyle \eta }
2887:
2863:
2840:
2486:
2375:
2319:
2273:
2247:Lie algebra extension
2054:Representation theory
1625:
1584:
1558:
1514:
1473:
1405:
1261:
1168:
509:
484:
447:
40:
5367:Theory of relativity
5362:Quantum field theory
5324:Quantum Field Theory
5058:Minkowski, Hermann,
5032:Minkowski, Hermann,
4775:
4668:
4612:
4592:
4564:
4540:
4516:
4486:
4469:elementary particles
4433:
4396:
4359:
4355:of this algebra are
3750:
3685:
3600:
3090:
3044:
2975:
2916:
2896:
2876:
2852:
2502:
2387:
2332:
2286:
2253:
1596:
1567:
1527:
1485:
1420:
1350:
1341:quantum field theory
1183:
1114:
1010:, together make the
496:
471:
434:
5321:L.H. Ryder (1996).
5279:Wu-Ki Tung (1985).
5257:www.phy.olemiss.edu
5128:2011JOpt...13f4010B
4987:1906RCMP...21..129P
4505:{\textstyle J^{PC}}
4467:. As a result, all
2369:
2199:Table of Lie groups
2040:Compact Lie algebra
1292:SO(4, 1) ~ Sp(2, 2)
1026:Poincaré invariance
1022:semi-direct product
845:Minkowski spacetime
140:Group homomorphisms
50:Algebraic structure
4995:10.1007/bf03013466
4848:
4755:
4618:
4598:
4582:charge-conjugation
4570:
4546:
4522:
4502:
4446:
4419:
4382:
4353:Casimir invariants
4338:
4336:
3729:
3671:
3583:
3581:
3073:
3030:
2969:
2946:
2902:
2882:
2858:
2835:
2833:
2481:
2370:
2355:
2326:identity component
2314:
2268:
1971:Affine Lie algebra
1961:Simple Lie algebra
1702:Special orthogonal
1620:
1579:
1553:
1509:
1468:
1400:
1298:goes to infinity.
1256:
1163:
1105:semidirect product
1017:Lorentz invariance
941:special relativity
922:general relativity
887:degrees of freedom
872:spacetime interval
616:Special orthogonal
504:
479:
442:
323:Lagrange's theorem
43:
5313:978-0-521-55001-7
5116:Journal of Optics
4645:topological space
4621:{\displaystyle C}
4601:{\displaystyle P}
4573:{\displaystyle C}
4549:{\displaystyle P}
4525:{\displaystyle J}
4081:
3575:
3497:
3422:
3347:
3285:
3213:
3164:
2999:
2885:{\displaystyle M}
2872:of translations,
2861:{\displaystyle P}
2671:
2667:
2565:
2561:
2448:
2235:
2234:
2035:Split Lie algebra
1998:Cartan subalgebra
1860:
1859:
1751:Simple Lie groups
1334:homogeneous space
1319:quantum mechanics
1288:group contraction
1037:Noether's theorem
954:abelian Lie group
937:Poincaré symmetry
932:Poincaré symmetry
903:classical physics
837:Hermann Minkowski
825:
824:
400:
399:
282:Alternating group
239:
238:
32:Fundamental group
18:Poincaré symmetry
16:(Redirected from
5379:
5338:
5317:
5305:
5294:
5267:
5266:
5264:
5263:
5249:
5243:
5242:
5238:978-1-13950-4324
5219:
5213:
5212:
5192:
5186:
5185:
5165:
5156:
5155:
5107:
5101:
5100:
5098:
5097:
5091:frankwilczek.com
5088:
5080:
5074:
5073:
5063:
5055:
5049:
5043:
5037:
5029:
5023:
5013:
4972:
4964:
4898:
4889:
4880:
4873:
4869:
4865:
4857:
4855:
4854:
4849:
4764:
4762:
4761:
4756:
4724:
4723:
4706:
4660:
4651:Other dimensions
4627:
4625:
4624:
4619:
4607:
4605:
4604:
4599:
4579:
4577:
4576:
4571:
4555:
4553:
4552:
4547:
4536:quantum number,
4531:
4529:
4528:
4523:
4511:
4509:
4508:
4503:
4501:
4500:
4455:
4453:
4452:
4447:
4445:
4444:
4428:
4426:
4425:
4420:
4418:
4417:
4408:
4407:
4391:
4389:
4388:
4383:
4381:
4380:
4371:
4370:
4347:
4345:
4344:
4339:
4337:
4333:
4332:
4323:
4322:
4291:
4287:
4286:
4285:
4273:
4272:
4254:
4253:
4244:
4243:
4215:
4211:
4210:
4209:
4197:
4196:
4178:
4177:
4168:
4167:
4139:
4135:
4134:
4133:
4121:
4120:
4102:
4101:
4089:
4088:
4082:
4077:
4069:
4060:
4056:
4055:
4054:
4042:
4041:
4023:
4022:
4013:
4012:
3984:
3980:
3979:
3978:
3966:
3965:
3937:
3933:
3932:
3931:
3919:
3918:
3900:
3899:
3874:
3870:
3869:
3868:
3856:
3855:
3830:
3826:
3825:
3824:
3812:
3811:
3786:
3782:
3781:
3780:
3768:
3767:
3738:
3736:
3735:
3730:
3719:
3718:
3697:
3696:
3680:
3678:
3677:
3672:
3661:
3660:
3645:
3644:
3631:
3630:
3615:
3614:
3592:
3590:
3589:
3584:
3582:
3573:
3572:
3571:
3562:
3561:
3530:
3529:
3517:
3516:
3495:
3494:
3493:
3484:
3483:
3455:
3454:
3442:
3441:
3420:
3419:
3418:
3409:
3408:
3380:
3379:
3367:
3366:
3345:
3344:
3343:
3318:
3317:
3305:
3304:
3283:
3282:
3281:
3272:
3271:
3246:
3245:
3233:
3232:
3211:
3197:
3196:
3184:
3183:
3162:
3161:
3160:
3151:
3150:
3122:
3121:
3109:
3108:
3082:
3080:
3079:
3074:
3072:
3071:
3056:
3055:
3039:
3037:
3036:
3031:
3029:
3028:
3016:
3015:
3000:
2992:
2987:
2986:
2955:
2953:
2952:
2947:
2911:
2909:
2908:
2903:
2891:
2889:
2888:
2883:
2867:
2865:
2864:
2859:
2844:
2842:
2841:
2836:
2834:
2826:
2825:
2813:
2812:
2797:
2796:
2784:
2783:
2768:
2767:
2755:
2754:
2739:
2738:
2726:
2725:
2703:
2702:
2687:
2686:
2669:
2668:
2660:
2653:
2652:
2643:
2642:
2627:
2626:
2617:
2616:
2594:
2593:
2581:
2580:
2563:
2562:
2554:
2534:
2533:
2521:
2520:
2490:
2488:
2487:
2482:
2480:
2476:
2475:
2474:
2462:
2461:
2449:
2441:
2428:
2424:
2423:
2422:
2413:
2412:
2379:
2377:
2376:
2371:
2368:
2363:
2342:
2323:
2321:
2320:
2315:
2307:
2306:
2301:
2300:
2299:
2277:
2275:
2274:
2269:
2239:Poincaré algebra
2227:
2220:
2213:
2170:Claude Chevalley
2027:Complexification
1870:Other Lie groups
1756:
1755:
1664:Classical groups
1656:
1638:
1637:
1634:Poincaré algebra
1629:
1627:
1626:
1621:
1588:
1586:
1585:
1580:
1562:
1560:
1559:
1554:
1549:
1518:
1516:
1515:
1510:
1477:
1475:
1474:
1469:
1440:
1439:
1428:
1409:
1407:
1406:
1401:
1393:
1370:
1369:
1358:
1330:Erlangen program
1296:de Sitter radius
1293:
1265:
1263:
1262:
1257:
1172:
1170:
1169:
1164:
1134:
1133:
1122:
817:
810:
803:
759:Algebraic groups
532:Hyperbolic group
522:Arithmetic group
513:
511:
510:
505:
503:
488:
486:
485:
480:
478:
451:
449:
448:
443:
441:
364:Schur multiplier
318:Cauchy's theorem
306:Quaternion group
254:
253:
80:
79:
69:
56:
45:
44:
21:
5387:
5386:
5382:
5381:
5380:
5378:
5377:
5376:
5342:
5341:
5335:
5314:
5291:
5275:
5270:
5261:
5259:
5251:
5250:
5246:
5239:
5220:
5216:
5209:
5193:
5189:
5182:
5166:
5159:
5108:
5104:
5095:
5093:
5086:
5082:
5081:
5077:
5056:
5052:
5030:
5026:
4965:
4961:
4957:
4909:Euclidean group
4905:
4896:
4891:
4887:
4882:
4875:
4871:
4867:
4863:
4776:
4773:
4772:
4707:
4702:
4701:
4669:
4666:
4665:
4656:
4653:
4613:
4610:
4609:
4593:
4590:
4589:
4565:
4562:
4561:
4541:
4538:
4537:
4517:
4514:
4513:
4493:
4489:
4487:
4484:
4483:
4481:quantum numbers
4440:
4436:
4434:
4431:
4430:
4413:
4409:
4403:
4399:
4397:
4394:
4393:
4376:
4372:
4366:
4362:
4360:
4357:
4356:
4335:
4334:
4328:
4324:
4312:
4308:
4292:
4281:
4277:
4268:
4264:
4263:
4259:
4256:
4255:
4249:
4245:
4233:
4229:
4216:
4205:
4201:
4192:
4188:
4187:
4183:
4180:
4179:
4173:
4169:
4157:
4153:
4140:
4129:
4125:
4116:
4112:
4111:
4107:
4104:
4103:
4094:
4090:
4084:
4083:
4070:
4068:
4061:
4050:
4046:
4037:
4033:
4032:
4028:
4025:
4024:
4018:
4014:
4002:
3998:
3985:
3974:
3970:
3961:
3957:
3956:
3952:
3949:
3948:
3938:
3927:
3923:
3914:
3910:
3909:
3905:
3902:
3901:
3895:
3891:
3875:
3864:
3860:
3851:
3850:
3849:
3845:
3842:
3841:
3831:
3820:
3816:
3807:
3806:
3805:
3801:
3798:
3797:
3787:
3776:
3772:
3763:
3762:
3761:
3757:
3753:
3751:
3748:
3747:
3741:representations
3711:
3710:
3689:
3688:
3686:
3683:
3682:
3674:{\textstyle =0}
3656:
3652:
3640:
3636:
3626:
3622:
3610:
3606:
3601:
3598:
3597:
3580:
3579:
3567:
3563:
3551:
3547:
3534:
3525:
3521:
3512:
3508:
3502:
3501:
3489:
3485:
3473:
3469:
3459:
3450:
3446:
3437:
3433:
3427:
3426:
3414:
3410:
3398:
3394:
3384:
3375:
3371:
3362:
3358:
3352:
3351:
3339:
3335:
3322:
3313:
3309:
3300:
3296:
3290:
3289:
3277:
3273:
3264:
3260:
3250:
3241:
3237:
3228:
3224:
3218:
3217:
3201:
3192:
3188:
3179:
3175:
3169:
3168:
3156:
3152:
3140:
3136:
3126:
3117:
3113:
3104:
3100:
3093:
3091:
3088:
3087:
3064:
3060:
3051:
3047:
3045:
3042:
3041:
3021:
3017:
3005:
3001:
2991:
2982:
2978:
2976:
2973:
2972:
2958:Sign convention
2917:
2914:
2913:
2897:
2894:
2893:
2877:
2874:
2873:
2853:
2850:
2849:
2846:
2832:
2831:
2818:
2814:
2805:
2801:
2789:
2785:
2776:
2772:
2760:
2756:
2747:
2743:
2731:
2727:
2718:
2714:
2707:
2695:
2691:
2679:
2675:
2659:
2656:
2655:
2648:
2644:
2635:
2631:
2622:
2618:
2609:
2605:
2598:
2589:
2585:
2573:
2569:
2553:
2550:
2549:
2538:
2529:
2525:
2516:
2512:
2505:
2503:
2500:
2499:
2467:
2463:
2454:
2450:
2440:
2439:
2435:
2418:
2414:
2408:
2404:
2400:
2396:
2388:
2385:
2384:
2364:
2359:
2335:
2333:
2330:
2329:
2302:
2295:
2291:
2290:
2289:
2287:
2284:
2283:
2254:
2251:
2250:
2231:
2186:
2185:
2184:
2155:Wilhelm Killing
2139:
2131:
2130:
2129:
2104:
2093:
2092:
2091:
2056:
2046:
2045:
2044:
2031:
2015:
1993:Dynkin diagrams
1987:
1977:
1976:
1975:
1957:
1935:Exponential map
1924:
1914:
1913:
1912:
1893:Conformal group
1872:
1862:
1861:
1853:
1845:
1837:
1829:
1821:
1802:
1792:
1782:
1772:
1753:
1743:
1742:
1741:
1722:Special unitary
1666:
1636:
1597:
1594:
1593:
1568:
1565:
1564:
1545:
1528:
1525:
1524:
1486:
1483:
1482:
1429:
1424:
1423:
1421:
1418:
1417:
1389:
1359:
1354:
1353:
1351:
1348:
1347:
1336:for the group.
1305:are indexed by
1303:representations
1291:
1272:group extension
1184:
1181:
1180:
1123:
1118:
1117:
1115:
1112:
1111:
1085:normal subgroup
1059:
943:. It includes:
934:
864:
821:
792:
791:
780:Abelian variety
773:Reductive group
761:
751:
750:
749:
748:
699:
691:
683:
675:
667:
640:Special unitary
551:
537:
536:
518:
517:
499:
497:
494:
493:
474:
472:
469:
468:
437:
435:
432:
431:
423:
422:
413:Discrete groups
402:
401:
357:Frobenius group
302:
289:
278:
271:Symmetric group
267:
251:
241:
240:
91:Normal subgroup
77:
57:
48:
35:
28:
23:
22:
15:
12:
11:
5:
5385:
5375:
5374:
5369:
5364:
5359:
5357:Henri Poincaré
5354:
5340:
5339:
5333:
5318:
5312:
5295:
5289:
5274:
5271:
5269:
5268:
5244:
5237:
5214:
5207:
5187:
5180:
5157:
5102:
5075:
5050:
5024:
4958:
4956:
4953:
4952:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4914:Galilean group
4911:
4904:
4901:
4894:
4885:
4860:
4859:
4847:
4844:
4841:
4838:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4766:
4765:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4722:
4719:
4716:
4713:
4710:
4705:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4652:
4649:
4617:
4597:
4569:
4545:
4521:
4499:
4496:
4492:
4443:
4439:
4416:
4412:
4406:
4402:
4379:
4375:
4369:
4365:
4349:
4348:
4331:
4327:
4321:
4318:
4315:
4311:
4307:
4304:
4301:
4298:
4295:
4293:
4290:
4284:
4280:
4276:
4271:
4267:
4262:
4258:
4257:
4252:
4248:
4242:
4239:
4236:
4232:
4228:
4225:
4222:
4219:
4217:
4214:
4208:
4204:
4200:
4195:
4191:
4186:
4182:
4181:
4176:
4172:
4166:
4163:
4160:
4156:
4152:
4149:
4146:
4143:
4141:
4138:
4132:
4128:
4124:
4119:
4115:
4110:
4106:
4105:
4100:
4097:
4093:
4087:
4080:
4076:
4073:
4067:
4064:
4062:
4059:
4053:
4049:
4045:
4040:
4036:
4031:
4027:
4026:
4021:
4017:
4011:
4008:
4005:
4001:
3997:
3994:
3991:
3988:
3986:
3983:
3977:
3973:
3969:
3964:
3960:
3955:
3951:
3950:
3947:
3944:
3941:
3939:
3936:
3930:
3926:
3922:
3917:
3913:
3908:
3904:
3903:
3898:
3894:
3890:
3887:
3884:
3881:
3878:
3876:
3873:
3867:
3863:
3859:
3854:
3848:
3844:
3843:
3840:
3837:
3834:
3832:
3829:
3823:
3819:
3815:
3810:
3804:
3800:
3799:
3796:
3793:
3790:
3788:
3785:
3779:
3775:
3771:
3766:
3760:
3756:
3755:
3728:
3725:
3722:
3717:
3714:
3709:
3706:
3703:
3700:
3695:
3692:
3670:
3667:
3664:
3659:
3655:
3651:
3648:
3643:
3639:
3634:
3629:
3625:
3621:
3618:
3613:
3609:
3605:
3594:
3593:
3578:
3570:
3566:
3560:
3557:
3554:
3550:
3546:
3543:
3540:
3537:
3535:
3533:
3528:
3524:
3520:
3515:
3511:
3507:
3504:
3503:
3500:
3492:
3488:
3482:
3479:
3476:
3472:
3468:
3465:
3462:
3460:
3458:
3453:
3449:
3445:
3440:
3436:
3432:
3429:
3428:
3425:
3417:
3413:
3407:
3404:
3401:
3397:
3393:
3390:
3387:
3385:
3383:
3378:
3374:
3370:
3365:
3361:
3357:
3354:
3353:
3350:
3342:
3338:
3334:
3331:
3328:
3325:
3323:
3321:
3316:
3312:
3308:
3303:
3299:
3295:
3292:
3291:
3288:
3280:
3276:
3270:
3267:
3263:
3259:
3256:
3253:
3251:
3249:
3244:
3240:
3236:
3231:
3227:
3223:
3220:
3219:
3216:
3210:
3207:
3204:
3202:
3200:
3195:
3191:
3187:
3182:
3178:
3174:
3171:
3170:
3167:
3159:
3155:
3149:
3146:
3143:
3139:
3135:
3132:
3129:
3127:
3125:
3120:
3116:
3112:
3107:
3103:
3099:
3096:
3095:
3070:
3067:
3063:
3059:
3054:
3050:
3040:, and boosts,
3027:
3024:
3020:
3014:
3011:
3008:
3004:
2998:
2995:
2990:
2985:
2981:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2901:
2881:
2857:
2830:
2824:
2821:
2817:
2811:
2808:
2804:
2800:
2795:
2792:
2788:
2782:
2779:
2775:
2771:
2766:
2763:
2759:
2753:
2750:
2746:
2742:
2737:
2734:
2730:
2724:
2721:
2717:
2713:
2710:
2708:
2706:
2701:
2698:
2694:
2690:
2685:
2682:
2678:
2674:
2666:
2663:
2658:
2657:
2651:
2647:
2641:
2638:
2634:
2630:
2625:
2621:
2615:
2612:
2608:
2604:
2601:
2599:
2597:
2592:
2588:
2584:
2579:
2576:
2572:
2568:
2560:
2557:
2552:
2551:
2547:
2544:
2541:
2539:
2537:
2532:
2528:
2524:
2519:
2515:
2511:
2508:
2507:
2497:
2479:
2473:
2470:
2466:
2460:
2457:
2453:
2447:
2444:
2438:
2434:
2431:
2427:
2421:
2417:
2411:
2407:
2403:
2399:
2395:
2392:
2382:exponentiation
2367:
2362:
2358:
2354:
2351:
2348:
2345:
2341:
2338:
2313:
2310:
2305:
2298:
2294:
2267:
2264:
2261:
2258:
2233:
2232:
2230:
2229:
2222:
2215:
2207:
2204:
2203:
2202:
2201:
2196:
2188:
2187:
2183:
2182:
2177:
2175:Harish-Chandra
2172:
2167:
2162:
2157:
2152:
2150:Henri Poincaré
2147:
2141:
2140:
2137:
2136:
2133:
2132:
2128:
2127:
2122:
2117:
2112:
2106:
2105:
2100:Lie groups in
2099:
2098:
2095:
2094:
2090:
2089:
2084:
2079:
2074:
2069:
2064:
2058:
2057:
2052:
2051:
2048:
2047:
2043:
2042:
2037:
2032:
2030:
2029:
2024:
2018:
2016:
2014:
2013:
2008:
2002:
2000:
1995:
1989:
1988:
1983:
1982:
1979:
1978:
1974:
1973:
1968:
1963:
1958:
1956:
1955:
1950:
1944:
1942:
1937:
1932:
1926:
1925:
1920:
1919:
1916:
1915:
1911:
1910:
1905:
1900:
1898:Diffeomorphism
1895:
1890:
1885:
1880:
1874:
1873:
1868:
1867:
1864:
1863:
1858:
1857:
1856:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1812:
1811:
1807:
1806:
1805:
1804:
1798:
1794:
1788:
1784:
1778:
1774:
1768:
1761:
1760:
1754:
1749:
1748:
1745:
1744:
1740:
1739:
1729:
1719:
1709:
1699:
1689:
1682:Special linear
1679:
1672:General linear
1668:
1667:
1662:
1661:
1658:
1657:
1649:
1648:
1635:
1632:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1578:
1575:
1572:
1552:
1548:
1544:
1541:
1538:
1535:
1532:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1479:
1478:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1438:
1435:
1432:
1427:
1411:
1410:
1399:
1396:
1392:
1388:
1385:
1382:
1379:
1376:
1373:
1368:
1365:
1362:
1357:
1280:representation
1268:
1267:
1255:
1252:
1249:
1246:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1174:
1173:
1162:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1132:
1129:
1126:
1121:
1058:
1057:Poincaré group
1055:
1054:
1053:
1050:
1047:
1044:
996:
995:
981:
963:
933:
930:
915:shear mappings
907:Galilean group
863:
860:
841:isometry group
839:(1908) as the
833:Henri Poincaré
831:, named after
829:Poincaré group
823:
822:
820:
819:
812:
805:
797:
794:
793:
790:
789:
787:Elliptic curve
783:
782:
776:
775:
769:
768:
762:
757:
756:
753:
752:
747:
746:
743:
740:
736:
732:
731:
730:
725:
723:Diffeomorphism
719:
718:
713:
708:
702:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
660:
659:
648:
647:
636:
635:
624:
623:
612:
611:
600:
599:
588:
587:
580:Special linear
576:
575:
568:General linear
564:
563:
558:
552:
543:
542:
539:
538:
535:
534:
529:
524:
516:
515:
502:
490:
477:
464:
462:Modular groups
460:
459:
458:
453:
440:
424:
421:
420:
415:
409:
408:
407:
404:
403:
398:
397:
396:
395:
390:
385:
382:
376:
375:
369:
368:
367:
366:
360:
359:
353:
352:
347:
338:
337:
335:Hall's theorem
332:
330:Sylow theorems
326:
325:
320:
312:
311:
310:
309:
303:
298:
295:Dihedral group
291:
290:
285:
279:
274:
268:
263:
252:
247:
246:
243:
242:
237:
236:
235:
234:
229:
221:
220:
219:
218:
213:
208:
203:
198:
193:
188:
186:multiplicative
183:
178:
173:
168:
160:
159:
158:
157:
152:
144:
143:
135:
134:
133:
132:
130:Wreath product
127:
122:
117:
115:direct product
109:
107:Quotient group
101:
100:
99:
98:
93:
88:
78:
75:
74:
71:
70:
62:
61:
41:Henri Poincaré
26:
9:
6:
4:
3:
2:
5384:
5373:
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5349:
5347:
5336:
5330:
5326:
5325:
5319:
5315:
5309:
5304:
5303:
5296:
5292:
5290:9971-966-57-3
5286:
5282:
5277:
5276:
5258:
5254:
5248:
5240:
5234:
5230:
5229:
5224:
5218:
5210:
5208:0-7923-0540-X
5204:
5200:
5199:
5191:
5183:
5181:9783319618784
5177:
5173:
5172:
5164:
5162:
5153:
5149:
5145:
5141:
5137:
5133:
5129:
5125:
5122:(6): 064010.
5121:
5117:
5113:
5106:
5092:
5085:
5079:
5071:
5067:
5062:
5054:
5047:
5041:
5036:
5028:
5021:
5018:translation:
5017:
5012:
5008:
5004:
5000:
4996:
4992:
4988:
4984:
4980:
4976:
4971:
4963:
4959:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4906:
4900:
4897:
4888:
4878:
4842:
4839:
4836:
4832:
4829:
4826:
4823:
4820:
4817:
4811:
4805:
4802:
4799:
4793:
4787:
4784:
4781:
4771:
4770:
4769:
4749:
4746:
4743:
4740:
4737:
4731:
4725:
4720:
4717:
4714:
4711:
4708:
4698:
4692:
4689:
4686:
4683:
4680:
4674:
4671:
4664:
4663:
4662:
4659:
4648:
4646:
4641:
4639:
4635:
4631:
4615:
4595:
4587:
4583:
4567:
4559:
4543:
4535:
4519:
4497:
4494:
4490:
4482:
4478:
4477:four-momentum
4474:
4470:
4466:
4461:
4459:
4441:
4437:
4414:
4410:
4404:
4400:
4377:
4373:
4367:
4363:
4354:
4329:
4325:
4319:
4316:
4313:
4309:
4302:
4299:
4296:
4294:
4288:
4282:
4278:
4274:
4269:
4265:
4260:
4250:
4246:
4240:
4237:
4234:
4230:
4223:
4220:
4218:
4212:
4206:
4202:
4198:
4193:
4189:
4184:
4174:
4170:
4164:
4161:
4158:
4154:
4147:
4144:
4142:
4136:
4130:
4126:
4122:
4117:
4113:
4108:
4098:
4095:
4091:
4078:
4071:
4065:
4063:
4057:
4051:
4047:
4043:
4038:
4034:
4029:
4019:
4015:
4009:
4006:
4003:
3999:
3992:
3989:
3987:
3981:
3975:
3971:
3967:
3962:
3958:
3953:
3945:
3942:
3940:
3934:
3928:
3924:
3920:
3915:
3911:
3906:
3896:
3892:
3888:
3882:
3879:
3877:
3871:
3865:
3861:
3857:
3846:
3838:
3835:
3833:
3827:
3821:
3817:
3813:
3802:
3794:
3791:
3789:
3783:
3777:
3773:
3769:
3758:
3746:
3745:
3744:
3742:
3723:
3707:
3701:
3668:
3665:
3657:
3653:
3649:
3646:
3641:
3637:
3632:
3627:
3623:
3619:
3616:
3611:
3607:
3576:
3568:
3564:
3558:
3555:
3552:
3548:
3544:
3541:
3538:
3536:
3526:
3522:
3518:
3513:
3509:
3498:
3490:
3486:
3480:
3477:
3474:
3470:
3466:
3463:
3461:
3451:
3447:
3443:
3438:
3434:
3423:
3415:
3411:
3405:
3402:
3399:
3395:
3391:
3388:
3386:
3376:
3372:
3368:
3363:
3359:
3348:
3340:
3336:
3332:
3329:
3326:
3324:
3314:
3310:
3306:
3301:
3297:
3286:
3278:
3274:
3268:
3265:
3261:
3257:
3254:
3252:
3242:
3238:
3234:
3229:
3225:
3214:
3208:
3205:
3203:
3193:
3189:
3185:
3180:
3176:
3165:
3157:
3153:
3147:
3144:
3141:
3137:
3133:
3130:
3128:
3118:
3114:
3110:
3105:
3101:
3086:
3085:
3084:
3068:
3065:
3061:
3057:
3052:
3048:
3025:
3022:
3018:
3012:
3009:
3006:
3002:
2996:
2993:
2988:
2983:
2979:
2965:
2961:
2959:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2899:
2879:
2871:
2855:
2845:
2828:
2822:
2819:
2815:
2809:
2806:
2802:
2798:
2793:
2790:
2786:
2780:
2777:
2773:
2769:
2764:
2761:
2757:
2751:
2748:
2744:
2740:
2735:
2732:
2728:
2722:
2719:
2715:
2711:
2709:
2699:
2696:
2692:
2688:
2683:
2680:
2676:
2664:
2661:
2649:
2645:
2639:
2636:
2632:
2628:
2623:
2619:
2613:
2610:
2606:
2602:
2600:
2590:
2586:
2582:
2577:
2574:
2570:
2558:
2555:
2545:
2542:
2540:
2530:
2526:
2522:
2517:
2513:
2496:
2494:
2477:
2471:
2468:
2464:
2458:
2455:
2451:
2445:
2442:
2436:
2432:
2429:
2425:
2419:
2415:
2409:
2405:
2401:
2397:
2393:
2390:
2383:
2360:
2352:
2349:
2346:
2327:
2311:
2308:
2303:
2296:
2281:
2280:orthochronous
2265:
2262:
2248:
2244:
2240:
2228:
2223:
2221:
2216:
2214:
2209:
2208:
2206:
2205:
2200:
2197:
2195:
2192:
2191:
2190:
2189:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2142:
2135:
2134:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2107:
2103:
2097:
2096:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2059:
2055:
2050:
2049:
2041:
2038:
2036:
2033:
2028:
2025:
2023:
2020:
2019:
2017:
2012:
2009:
2007:
2004:
2003:
2001:
1999:
1996:
1994:
1991:
1990:
1986:
1981:
1980:
1972:
1969:
1967:
1964:
1962:
1959:
1954:
1951:
1949:
1946:
1945:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1927:
1923:
1918:
1917:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1875:
1871:
1866:
1865:
1854:
1848:
1846:
1840:
1838:
1832:
1830:
1824:
1822:
1816:
1815:
1814:
1813:
1809:
1808:
1803:
1801:
1795:
1793:
1791:
1785:
1783:
1781:
1775:
1773:
1771:
1765:
1764:
1763:
1762:
1758:
1757:
1752:
1747:
1746:
1737:
1733:
1730:
1727:
1723:
1720:
1717:
1713:
1710:
1707:
1703:
1700:
1697:
1693:
1690:
1687:
1683:
1680:
1677:
1673:
1670:
1669:
1665:
1660:
1659:
1655:
1651:
1650:
1647:
1643:
1640:
1639:
1631:
1614:
1611:
1608:
1602:
1599:
1592:
1576:
1573:
1570:
1542:
1539:
1533:
1530:
1522:
1503:
1500:
1497:
1491:
1488:
1465:
1459:
1456:
1453:
1447:
1444:
1441:
1436:
1433:
1430:
1416:
1415:
1414:
1397:
1386:
1383:
1377:
1374:
1371:
1366:
1363:
1360:
1346:
1345:
1344:
1342:
1337:
1335:
1331:
1326:
1324:
1320:
1316:
1312:
1308:
1304:
1299:
1297:
1289:
1285:
1281:
1277:
1276:Lorentz group
1273:
1250:
1247:
1244:
1240:
1237:
1234:
1231:
1228:
1225:
1219:
1213:
1210:
1207:
1201:
1195:
1192:
1189:
1179:
1178:
1177:
1160:
1153:
1150:
1147:
1141:
1135:
1130:
1127:
1124:
1110:
1109:
1108:
1106:
1102:
1098:
1094:
1090:
1089:Lorentz group
1086:
1082:
1079:
1075:
1074:abelian group
1071:
1068:
1064:
1051:
1048:
1045:
1042:
1041:
1040:
1038:
1033:
1031:
1027:
1023:
1019:
1018:
1013:
1012:Lorentz group
1009:
1008:
1003:
1002:
993:
992:
987:
986:
982:
979:
978:
973:
969:
968:
964:
961:
960:
955:
951:
950:
946:
945:
944:
942:
938:
929:
927:
923:
918:
916:
912:
908:
904:
899:
897:
893:
888:
883:
881:
880:proper length
877:
873:
869:
859:
857:
853:
850:
846:
842:
838:
834:
830:
818:
813:
811:
806:
804:
799:
798:
796:
795:
788:
785:
784:
781:
778:
777:
774:
771:
770:
767:
764:
763:
760:
755:
754:
744:
741:
738:
737:
735:
729:
726:
724:
721:
720:
717:
714:
712:
709:
707:
704:
703:
700:
694:
692:
686:
684:
678:
676:
670:
668:
662:
661:
657:
653:
650:
649:
645:
641:
638:
637:
633:
629:
626:
625:
621:
617:
614:
613:
609:
605:
602:
601:
597:
593:
590:
589:
585:
581:
578:
577:
573:
569:
566:
565:
562:
559:
557:
554:
553:
550:
546:
541:
540:
533:
530:
528:
525:
523:
520:
519:
491:
466:
465:
463:
457:
454:
429:
426:
425:
419:
416:
414:
411:
410:
406:
405:
394:
391:
389:
386:
383:
380:
379:
378:
377:
374:
371:
370:
365:
362:
361:
358:
355:
354:
351:
348:
346:
344:
340:
339:
336:
333:
331:
328:
327:
324:
321:
319:
316:
315:
314:
313:
307:
304:
301:
296:
293:
292:
288:
283:
280:
277:
272:
269:
266:
261:
258:
257:
256:
255:
250:
249:Finite groups
245:
244:
233:
230:
228:
225:
224:
223:
222:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
163:
162:
161:
156:
153:
151:
148:
147:
146:
145:
142:
141:
137:
136:
131:
128:
126:
123:
121:
118:
116:
113:
110:
108:
105:
104:
103:
102:
97:
94:
92:
89:
87:
84:
83:
82:
81:
76:Basic notions
73:
72:
68:
64:
63:
60:
55:
51:
47:
46:
39:
33:
19:
5334:0-52147-8146
5323:
5301:
5280:
5260:. Retrieved
5256:
5247:
5227:
5217:
5197:
5190:
5170:
5119:
5115:
5105:
5094:. Retrieved
5090:
5078:
5069:
5065:
5053:
5039:
5027:
4978:
4974:
4962:
4892:
4883:
4876:
4861:
4767:
4657:
4654:
4642:
4630:CPT symmetry
4476:
4462:
4350:
3595:
2970:
2847:
2498:
2238:
2236:
2180:Armand Borel
2165:Hermann Weyl
1966:Loop algebra
1948:Killing form
1922:Lie algebras
1887:
1799:
1789:
1779:
1769:
1735:
1725:
1715:
1705:
1695:
1685:
1675:
1646:Lie algebras
1480:
1412:
1338:
1327:
1300:
1283:
1278:by a vector
1269:
1175:
1097:affine group
1081:translations
1060:
1034:
1029:
1025:
1015:
1006:
1005:
1000:
999:
997:
990:
989:
983:
976:
975:
965:
958:
957:
949:translations
947:
936:
935:
919:
900:
884:
865:
828:
826:
710:
655:
643:
631:
619:
607:
595:
583:
571:
342:
299:
286:
275:
264:
260:Cyclic group
138:
125:Free product
96:Group action
59:Group theory
54:Group theory
53:
4981:: 129–176,
2493:Lie algebra
2243:Lie algebra
2160:Élie Cartan
2006:Root system
1810:Exceptional
849:non-abelian
545:Topological
384:alternating
5352:Lie groups
5346:Categories
5273:References
5262:2021-07-18
5223:T. Ohlsson
5096:2021-02-14
5016:Wikisource
2145:Sophus Lie
2138:Scientists
2011:Weyl group
1732:Symplectic
1692:Orthogonal
1642:Lie groups
1093:stabilizer
1067:noncompact
1063:isometries
1014:(see also
652:Symplectic
592:Orthogonal
549:Lie groups
456:Free group
181:continuous
120:Direct sum
5144:2040-8978
5011:120211823
4840:⋅
4830:β
4827:⋅
4818:α
4800:β
4794:⋅
4782:α
4747:−
4732:
4726:⋊
4718:−
4690:−
4675:
4634:invariant
4442:μ
4415:μ
4405:μ
4378:μ
4368:μ
4310:ϵ
4306:ℏ
4300:−
4231:ϵ
4227:ℏ
4155:ϵ
4151:ℏ
4092:δ
4075:ℏ
4000:ϵ
3996:ℏ
3886:ℏ
3708:⊕
3647:−
3549:ϵ
3542:−
3471:ϵ
3396:ϵ
3330:−
3262:η
3138:ϵ
3003:ϵ
2941:−
2935:−
2929:−
2900:η
2870:generator
2823:ρ
2820:μ
2810:σ
2807:ν
2803:η
2794:σ
2791:μ
2781:ρ
2778:ν
2774:η
2770:−
2765:ρ
2762:ν
2752:σ
2749:μ
2745:η
2741:−
2736:σ
2733:ν
2723:ρ
2720:μ
2716:η
2700:σ
2697:ρ
2684:ν
2681:μ
2650:μ
2640:ρ
2637:ν
2633:η
2629:−
2624:ν
2614:ρ
2611:μ
2607:η
2591:ρ
2578:ν
2575:μ
2531:ν
2518:μ
2472:ν
2469:μ
2459:ν
2456:μ
2452:ω
2433:
2420:μ
2410:μ
2394:
2366:↑
2309:≥
2293:Λ
2260:Λ
2022:Real form
1908:Euclidean
1759:Classical
1603:
1574:×
1534:
1492:
1448:
1442:⋊
1378:
1372:⋊
1294:, as the
1248:⋅
1238:β
1235:⋅
1226:α
1208:β
1202:⋅
1190:α
1142:
1136:⋊
1078:spacetime
1070:Lie group
967:rotations
896:rotations
852:Lie group
716:Conformal
604:Euclidean
211:nilpotent
5372:Symmetry
5225:(2011).
5152:55243365
5042:: 53–111
4903:See also
4512:, where
4471:fall in
2491:of this
2194:Glossary
1888:Poincaré
1523:. Here
1521:fermions
874:between
862:Overview
711:Poincaré
556:Solenoid
428:Integers
418:Lattices
393:sporadic
388:Lie type
216:solvable
206:dihedral
191:additive
176:infinite
86:Subgroup
5124:Bibcode
5072:: 75–88
4983:Bibcode
4580:is the
4556:is the
4532:is the
4456:is the
2912:is the
2868:is the
2241:is the
2102:physics
1883:Lorentz
1712:Unitary
1315:integer
1274:of the
1020:); the
926:gravity
856:physics
706:Lorentz
628:Unitary
527:Lattice
467:PSL(2,
201:abelian
112:(Semi-)
5331:
5310:
5287:
5235:
5205:
5178:
5150:
5142:
5009:
4558:parity
4429:where
3574:
3496:
3421:
3346:
3284:
3212:
3163:
2848:where
2670:
2564:
1878:Circle
985:boosts
905:, the
876:events
561:Circle
492:SL(2,
381:cyclic
345:-group
196:cyclic
171:finite
166:simple
150:kernel
5148:S2CID
5087:(PDF)
5007:S2CID
4955:Notes
4643:As a
1953:Index
1321:(see
1083:is a
892:boost
745:Sp(∞)
742:SU(∞)
155:image
5329:ISBN
5308:ISBN
5285:ISBN
5233:ISBN
5203:ISBN
5176:ISBN
5140:ISSN
4890:and
4874:and
4866:and
4608:and
4560:and
4534:spin
4392:and
4351:The
2237:The
1903:Loop
1644:and
1600:Spin
1445:Spin
1311:spin
1307:mass
1004:and
827:The
739:O(∞)
728:Loop
547:and
5132:doi
4999:hdl
4991:doi
4879:− 1
4632:is
2960:).
2430:exp
2391:exp
2328:),
2278:),
2257:det
1734:Sp(
1724:SU(
1704:SO(
1684:SL(
1674:GL(
1339:In
1325:).
1076:of
1028:or
920:In
901:In
843:of
654:Sp(
642:SU(
618:SO(
582:SL(
570:GL(
5348::
5255:.
5160:^
5146:.
5138:.
5130:.
5120:13
5118:.
5114:.
5089:.
5070:10
5068:,
5064:,
5048:).
5038:,
5005:,
4997:,
4989:,
4979:21
4977:,
4973:,
4699::=
4672:IO
1714:U(
1694:O(
1630:.
1531:SL
1489:SO
1375:SL
1032:.
994:).
980:);
962:);
858:.
630:U(
606:E(
594:O(
52:→
5337:.
5316:.
5293:.
5265:.
5241:.
5211:.
5184:.
5154:.
5134::
5126::
5099:.
5014:(
5001::
4993::
4985::
4895:i
4893:K
4886:i
4884:J
4877:d
4872:0
4868:ν
4864:µ
4858:.
4846:)
4843:g
4837:f
4833:,
4824:f
4821:+
4815:(
4812:=
4809:)
4806:g
4803:,
4797:(
4791:)
4788:f
4785:,
4779:(
4753:)
4750:1
4744:d
4741:,
4738:1
4735:(
4729:O
4721:1
4715:d
4712:,
4709:1
4704:R
4696:)
4693:1
4687:d
4684:,
4681:1
4678:(
4658:d
4616:C
4596:P
4568:C
4544:P
4520:J
4498:C
4495:P
4491:J
4438:W
4411:W
4401:W
4374:P
4364:P
4330:k
4326:L
4320:k
4317:j
4314:i
4303:i
4297:=
4289:]
4283:j
4279:K
4275:,
4270:i
4266:K
4261:[
4251:k
4247:K
4241:k
4238:j
4235:i
4224:i
4221:=
4213:]
4207:j
4203:K
4199:,
4194:i
4190:L
4185:[
4175:k
4171:L
4165:k
4162:j
4159:i
4148:i
4145:=
4137:]
4131:j
4127:L
4123:,
4118:i
4114:L
4109:[
4099:j
4096:i
4086:H
4079:c
4072:i
4066:=
4058:]
4052:j
4048:K
4044:,
4039:i
4035:p
4030:[
4020:k
4016:p
4010:k
4007:j
4004:i
3993:i
3990:=
3982:]
3976:j
3972:L
3968:,
3963:i
3959:p
3954:[
3946:0
3943:=
3935:]
3929:j
3925:p
3921:,
3916:i
3912:p
3907:[
3897:i
3893:p
3889:c
3883:i
3880:=
3872:]
3866:i
3862:K
3858:,
3853:H
3847:[
3839:0
3836:=
3828:]
3822:i
3818:L
3814:,
3809:H
3803:[
3795:0
3792:=
3784:]
3778:i
3774:p
3770:,
3765:H
3759:[
3727:)
3724:2
3721:(
3716:u
3713:s
3705:)
3702:2
3699:(
3694:u
3691:s
3669:0
3666:=
3663:]
3658:n
3654:K
3650:i
3642:n
3638:J
3633:,
3628:m
3624:K
3620:i
3617:+
3612:m
3608:J
3604:[
3577:,
3569:k
3565:J
3559:k
3556:n
3553:m
3545:i
3539:=
3532:]
3527:n
3523:K
3519:,
3514:m
3510:K
3506:[
3499:,
3491:k
3487:K
3481:k
3478:n
3475:m
3467:i
3464:=
3457:]
3452:n
3448:K
3444:,
3439:m
3435:J
3431:[
3424:,
3416:k
3412:J
3406:k
3403:n
3400:m
3392:i
3389:=
3382:]
3377:n
3373:J
3369:,
3364:m
3360:J
3356:[
3349:,
3341:i
3337:P
3333:i
3327:=
3320:]
3315:0
3311:P
3307:,
3302:i
3298:K
3294:[
3287:,
3279:0
3275:P
3269:k
3266:i
3258:i
3255:=
3248:]
3243:k
3239:P
3235:,
3230:i
3226:K
3222:[
3215:,
3209:0
3206:=
3199:]
3194:0
3190:P
3186:,
3181:i
3177:J
3173:[
3166:,
3158:k
3154:P
3148:k
3145:n
3142:m
3134:i
3131:=
3124:]
3119:n
3115:P
3111:,
3106:m
3102:J
3098:[
3069:0
3066:i
3062:M
3058:=
3053:i
3049:K
3026:n
3023:m
3019:M
3013:n
3010:m
3007:i
2997:2
2994:1
2989:=
2984:i
2980:J
2944:)
2938:,
2932:,
2926:,
2923:+
2920:(
2880:M
2856:P
2829:,
2816:M
2799:+
2787:M
2758:M
2729:M
2712:=
2705:]
2693:M
2689:,
2677:M
2673:[
2665:i
2662:1
2646:P
2620:P
2603:=
2596:]
2587:P
2583:,
2571:M
2567:[
2559:i
2556:1
2546:0
2543:=
2536:]
2527:P
2523:,
2514:P
2510:[
2478:)
2465:M
2446:2
2443:i
2437:(
2426:)
2416:P
2406:a
2402:i
2398:(
2361:+
2357:)
2353:3
2350:,
2347:1
2344:(
2340:O
2337:S
2312:1
2304:0
2297:0
2282:(
2266:1
2263:=
2226:e
2219:t
2212:v
1852:8
1850:E
1844:7
1842:E
1836:6
1834:E
1828:4
1826:F
1820:2
1818:G
1800:n
1797:D
1790:n
1787:C
1780:n
1777:B
1770:n
1767:A
1738:)
1736:n
1728:)
1726:n
1718:)
1716:n
1708:)
1706:n
1698:)
1696:n
1688:)
1686:n
1678:)
1676:n
1618:)
1615:3
1612:,
1609:1
1606:(
1577:2
1571:2
1551:)
1547:C
1543:,
1540:2
1537:(
1507:)
1504:3
1501:,
1498:1
1495:(
1466:,
1463:)
1460:3
1457:,
1454:1
1451:(
1437:3
1434:,
1431:1
1426:R
1398:,
1395:)
1391:C
1387:,
1384:2
1381:(
1367:3
1364:,
1361:1
1356:R
1313:(
1266:.
1254:)
1251:g
1245:f
1241:,
1232:f
1229:+
1223:(
1220:=
1217:)
1214:g
1211:,
1205:(
1199:)
1196:f
1193:,
1187:(
1161:,
1157:)
1154:3
1151:,
1148:1
1145:(
1139:O
1131:3
1128:,
1125:1
1120:R
1007:K
1001:J
991:K
977:J
974:(
959:P
890:"
816:e
809:t
802:v
698:8
696:E
690:7
688:E
682:6
680:E
674:4
672:F
666:2
664:G
658:)
656:n
646:)
644:n
634:)
632:n
622:)
620:n
610:)
608:n
598:)
596:n
586:)
584:n
574:)
572:n
514:)
501:Z
489:)
476:Z
452:)
439:Z
430:(
343:p
308:Q
300:n
297:D
287:n
284:A
276:n
273:S
265:n
262:Z
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
