2532:
1949:
38:
2527:{\displaystyle {\begin{aligned}&=-iK_{\mu }\,,\\&=iP_{\mu }\,,\\&=2i(\eta _{\mu \nu }D-M_{\mu \nu })\,,\\&=i(\eta _{\mu \nu }K_{\rho }-\eta _{\mu \rho }K_{\nu })\,,\\&=i(\eta _{\rho \mu }P_{\nu }-\eta _{\rho \nu }P_{\mu })\,,\\&=i(\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\nu \sigma }M_{\mu \rho })\,,\end{aligned}}}
2886:
3112:
For two-dimensional
Euclidean space or one-plus-one dimensional spacetime, the space of conformal symmetries is much larger. In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional,
2662:
3102:
1544:
2657:
2881:{\displaystyle {\begin{aligned}&J_{\mu \nu }=M_{\mu \nu }\,,\\&J_{-1,\mu }={\frac {1}{2}}(P_{\mu }-K_{\mu })\,,\\&J_{0,\mu }={\frac {1}{2}}(P_{\mu }+K_{\mu })\,,\\&J_{-1,0}=D.\end{aligned}}}
1755:
1944:
1094:, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by
2667:
1954:
3263:
3321:
and showed it had the conformal property (proportional to a form preserver). Bateman and
Cunningham showed that this conformal group is "the largest group of transformations leaving
2981:
1655:
1480:
928:
1872:
1829:
1794:
1389:
1354:
1315:
1604:
1433:
2562:
1194:
483:
458:
421:
1688:
3201:
3140:
2916:
2990:
3166:
1564:
1241:
1218:
1131:
1267:
1160:
3388:
wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the
3538:
1485:
785:
2574:
2571:
In fact, this Lie algebra is isomorphic to the Lie algebra of the
Lorentz group with one more space and one more time dimension, that is,
3452:"Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie", 2
814:
of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the
3479:
3839:
3537:
3206:
1693:
343:
293:
1881:
3607:
3492:
778:
288:
1064:
3838:
Martin
Schottenloher, The conformal group, chapter 2 of A mathematical introduction to conformal field theory, 2008 (
3815:
3788:
3683:
3659:
3517:
3435:
3732:
3370:
1095:
3539:"The conformal transformations of a space of four dimensions and their applications to geometrical optics"
704:
3825:
3374:
771:
2921:
1613:
1438:
876:
3864:
3647:
3351:
388:
202:
3203:
local conformal symmetries are described by the infinite dimensional space of vector fields of the form
1842:
1799:
1764:
1758:
1359:
1324:
1283:
1569:
1398:
1318:
1107:
120:
2537:
1168:
3571:
3298:
945:
3606:
3758:
3302:
1051:
the various frames of reference, for varying velocity with respect to a rest frame, are related by
586:
320:
197:
85:
466:
441:
404:
3322:
3286:
2659:. It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define
937:
3651:
3297:
groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to
1277:
1040:
990:
736:
526:
3701:
3097:{\displaystyle {\tilde {\eta }}_{ab}=\operatorname {diag} (-1,+1,-1,\cdots ,-1,+1,\cdots ,+1)}
3373:
on the projective line over that ring. Elements of the spacetime conformal group were called
3306:
1660:
610:
3171:
3119:
2891:
3689:
3335:
1079:
550:
538:
156:
90:
3739:
3377:
by
Bateman. The particulars of the spacetime quadratic form study have been absorbed into
3313:
a transformation preserving the quadratic form. Harry
Bateman's paper in 1910 studied the
8:
3602:
3466:
3378:
3347:
3282:
3145:
2984:
1111:
1060:
811:
807:
125:
20:
1078:. Pseudo-Euclidean geometry is supported by alternative complex planes where points are
3640:
3406:
3358:
3343:
3265:
Hence the local conformal symmetries of 2d
Euclidean space is the infinite-dimensional
1549:
1226:
1203:
1116:
1048:
958:
815:
110:
82:
1246:
1139:
3811:
3794:
3784:
3755:
Conformal groups and
Related Symmetries: Physical Results and Mathematical Background
3728:
3679:
3655:
3513:
3488:
3431:
1392:
515:
358:
252:
3389:
1566:
is considered as a single spacetime vector). The conformal compactification is then
681:
3829:
3620:
3584:
3551:
3113:
these do not necessarily extend to a Lie group of well-defined global symmetries.
2565:
1607:
1044:
941:
826:
666:
658:
650:
642:
634:
622:
562:
502:
492:
334:
276:
151:
3309:
are not confined to kinematic motions, but rather are required only to be locally
3736:
3686:
3570:
3463:
3451:
3314:
1134:
1070:
A method to generate an appropriate conformal group is to mimic the steps of the
969:
750:
743:
729:
574:
497:
327:
241:
181:
61:
3845:
1539:{\displaystyle (\mathbf {x} ,\mathbf {t} )\mapsto X=(\mathbf {x} ,\mathbf {t} )}
1071:
962:
3762:
3705:
3608:"The principle of Relativity in Electrodynamics and an Extension Thereof"
3459:
1087:
834:
757:
693:
383:
363:
300:
265:
186:
176:
161:
146:
100:
77:
3588:
1831:
to itself as it maps the origin to infinity, and maps infinity to the origin.
3858:
3624:
3555:
3533:
3455:
3401:
3366:
3334:
has contributed to the mathematics of spacetime conformal transformations in
3278:
1197:
1091:
1075:
1056:
676:
598:
432:
305:
171:
3798:
3720:
3709:
3362:
3331:
3325:
structurally invariant." The conformal group of spacetime has been denoted
3266:
531:
230:
219:
166:
141:
136:
95:
66:
29:
3808:
Differential
Geometry: Cartan's Generalization of Klein's Erlangen Program
1063:
which preserves the differential angle between rapidities. Thus, they are
3750:
3385:
3339:
1875:
1083:
983:
799:
1223:
More concretely, this is the group of angle-preserving smooth maps from
3318:
3294:
698:
426:
3142:, the local conformal symmetries all extend to global symmetries. For
2652:{\displaystyle {\mathfrak {conf}}(p,q)\cong {\mathfrak {so}}(p+1,q+1)}
1024:
519:
3833:
1052:
56:
3425:
3369:. For the spacetime conformal group, it is sufficient to consider
3642:
Masters of theory: Cambridge and the rise of mathematical physics
3508:
Di
Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997).
398:
312:
3384:
Commenting on the continued interest shown in physical science,
954:
37:
3107:
821:
Several specific conformal groups are particularly important:
1036:
3702:
Introduction to the Classical Theory of Particles and Fields
3572:"The Transformation of the Electrodynamical Equations"
3849:
3824:
Peter Scherk (1960) "Some Concepts of Conformal Geometry",
3507:
1035:
In Euclidean geometry one can expect the standard circular
1750:{\displaystyle {\text{Conf}}(p,q):={\text{Conf}}(N^{p,q})}
3168:
Euclidean space, after changing to a complex coordinate
1395:). This conformal compactification can be defined using
3676:
Lie Groups, Lie Algebras and some of their Applications
3725:
A Simple Non-Euclidean Geometry and its Physical Basis
3487:. Springer Science & Business Media. p. 23.
3481:
A Mathematical Introduction to Conformal Field Theory
3209:
3174:
3148:
3122:
2993:
2924:
2894:
2665:
2577:
2540:
1952:
1884:
1845:
1802:
1767:
1696:
1663:
1616:
1572:
1552:
1488:
1441:
1401:
1362:
1327:
1286:
1249:
1229:
1206:
1171:
1142:
1119:
879:
469:
444:
407:
3601:
3342:. Since split-complex numbers and dual numbers form
1939:{\displaystyle \{M_{\mu \nu },P_{\mu },K_{\mu },D\}}
1606:
with 'antipodal points' identified. This happens by
1317:
is the conformal group of the manifold arising from
982:, the conformal group is generated by inversions in
3639:
3350:, the linear fractional transformations require a
3257:
3195:
3160:
3134:
3096:
2975:
2910:
2880:
2651:
2556:
2526:
1938:
1866:
1823:
1788:
1749:
1682:
1649:
1598:
1558:
1538:
1474:
1427:
1383:
1348:
1309:
1261:
1235:
1212:
1188:
1154:
1125:
922:
477:
452:
415:
940:, the conformal orthogonal group is equal to the
3856:
3428:An Introduction to Clifford Algebras and Spinors
1435:, considered as a submanifold of null points in
3637:
3272:
3781:Transformation Groups in Differential Geometry
3613:Proceedings of the London Mathematical Society
3577:Proceedings of the London Mathematical Society
3568:
3544:Proceedings of the London Mathematical Society
3532:
3477:
3426:Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016).
1878:of the conformal group is given by the basis
779:
3258:{\displaystyle l_{n}=-z^{n+1}\partial _{z}.}
2534:and with all other brackets vanishing. Here
1933:
1885:
1055:, a hyperbolic angle. One way to describe a
3108:Conformal group in two spacetime dimensions
3357:It has been traditional since the work of
1946:with the following commutation relations:
1243:to itself. However, when the signature of
1101:
786:
772:
3778:
3678:, page 349, Robert E. Krieger Publishing
2888:It can then be shown that the generators
2837:
2770:
2700:
2516:
2345:
2241:
2137:
2050:
2001:
1848:
1805:
1770:
1619:
1444:
1365:
471:
446:
409:
3430:. Oxford University Press. p. 140.
1690:is the conformal compactification, then
1280:, the definition is slightly different.
1086:. Just as the Möbius group requires the
3317:of a transformation that preserves the
1074:as the conformal group of the ordinary
852:is the group of linear transformations
3857:
3805:
2976:{\displaystyle a,b=-1,0,\cdots ,n=p+q}
1650:{\displaystyle \mathbb {R} ^{p+1,q+1}}
1475:{\displaystyle \mathbb {R} ^{p+1,q+1}}
1067:with respect to the hyperbolic angle.
923:{\displaystyle Q(Tx)=\lambda ^{2}Q(x)}
840:, then the conformal orthogonal group
344:Classification of finite simple groups
3783:. Classics in Mathematics. Springer.
1757:. In particular, this group includes
3456:Proceedings of the Imperial Academy
2617:
2614:
2589:
2586:
2583:
2580:
961:. This group is also known as the
13:
3772:
3501:
3243:
1867:{\displaystyle \mathbb {R} ^{p,q}}
1824:{\displaystyle \mathbb {R} ^{p,q}}
1789:{\displaystyle \mathbb {R} ^{p,q}}
1384:{\displaystyle \mathbb {R} ^{p,q}}
1349:{\displaystyle \mathbf {E} ^{p,q}}
1310:{\displaystyle {\text{Conf}}(p,q)}
1269:is not definite, the 'angle' is a
14:
3876:
3371:linear fractional transformations
1599:{\displaystyle S^{p}\times S^{q}}
1428:{\displaystyle S^{p}\times S^{q}}
1096:linear fractional transformations
1030:
2557:{\displaystyle \eta _{\mu \nu }}
1529:
1521:
1501:
1493:
1330:
1189:{\displaystyle {\text{Conf}}(M)}
860:for which there exists a scalar
36:
3744:
3714:
3694:
1273:which is potentially infinite.
3668:
3631:
3595:
3562:
3526:
3478:Schottenloher, Martin (2008).
3471:
3444:
3419:
3375:spherical wave transformations
3091:
3025:
3001:
2834:
2808:
2767:
2741:
2646:
2622:
2606:
2594:
2513:
2397:
2388:
2356:
2342:
2290:
2281:
2252:
2238:
2186:
2177:
2148:
2134:
2099:
2087:
2061:
2031:
2012:
1979:
1960:
1744:
1725:
1714:
1702:
1533:
1517:
1508:
1505:
1489:
1321:of the pseudo-Euclidean space
1304:
1292:
1256:
1250:
1183:
1177:
1149:
1143:
917:
911:
892:
883:
705:Infinite dimensional Lie group
1:
3826:American Mathematical Monthly
3810:, Springer-Verlag, New York,
3412:
1039:to be characteristic, but in
3301:, though with respect to an
3291:conformal group of spacetime
3273:Conformal group of spacetime
1834:
478:{\displaystyle \mathbb {Z} }
453:{\displaystyle \mathbb {Z} }
416:{\displaystyle \mathbb {Z} }
7:
3765:, see preface for quotation
3753:& H.-D. Doebner (1985)
3648:University of Chicago Press
3395:
3361:in 1914 to use the ring of
3352:projective line over a ring
3285:, two young researchers at
1839:For Pseudo-Euclidean space
1356:(sometimes identified with
953:The conformal group of the
203:List of group theory topics
10:
3881:
3450:Tsurusaburo Takasu (1941)
3354:to be bijective mappings.
3299:orthogonal transformations
1796:, which is not a map from
1319:conformal compactification
1098:on the appropriate plane.
3289:, broached the idea of a
1065:conformal transformations
1023:All conformal groups are
996:, the conformal group is
833:is a vector space with a
3759:Lecture Notes in Physics
3638:Warwick, Andrew (2003).
3458:17(8): 330–8, link from
3303:isotropic quadratic form
3116:For spacetime dimension
321:Elementary abelian group
198:Glossary of group theory
3674:Robert Gilmore (1994)
3589:10.1112/plms/s2-8.1.223
3569:Bateman, Harry (1910).
3287:University of Liverpool
1683:{\displaystyle N^{p,q}}
1102:Mathematical definition
938:definite quadratic form
3779:Kobayashi, S. (1972).
3700:Boris Kosyakov (2007)
3625:10.1112/plms/s2-8.1.77
3556:10.1112/plms/s2-7.1.70
3512:. New York: Springer.
3510:Conformal field theory
3305:. The liberties of an
3259:
3197:
3196:{\displaystyle z=x+iy}
3162:
3136:
3135:{\displaystyle n>2}
3098:
2987:relations with metric
2977:
2912:
2911:{\displaystyle J_{ab}}
2882:
2653:
2558:
2528:
1940:
1868:
1825:
1790:
1751:
1684:
1651:
1600:
1560:
1540:
1476:
1429:
1385:
1350:
1311:
1278:Pseudo-Euclidean space
1263:
1237:
1214:
1190:
1156:
1127:
1041:pseudo-Euclidean space
991:pseudo-Euclidean space
924:
737:Linear algebraic group
479:
454:
417:
3806:Sharpe, R.W. (1997),
3307:electromagnetic field
3293:They argued that the
3260:
3198:
3163:
3137:
3099:
2978:
2913:
2883:
2654:
2559:
2529:
1941:
1869:
1826:
1791:
1752:
1685:
1652:
1601:
1561:
1541:
1477:
1430:
1386:
1351:
1312:
1264:
1238:
1215:
1191:
1157:
1128:
1080:split-complex numbers
959:inversions in circles
925:
480:
455:
418:
3603:Cunningham, Ebenezer
3207:
3172:
3146:
3120:
2991:
2922:
2892:
2663:
2575:
2538:
1950:
1882:
1843:
1800:
1765:
1694:
1661:
1614:
1570:
1550:
1486:
1439:
1399:
1360:
1325:
1284:
1247:
1227:
1204:
1169:
1140:
1117:
957:is generated by the
877:
467:
442:
405:
3379:Lie sphere geometry
3323:Maxwell’s equations
3283:Ebenezer Cunningham
3161:{\displaystyle n=2}
1112:Riemannian manifold
1061:hyperbolic rotation
1047:. In the study of
944:times the group of
808:inner product space
111:Group homomorphisms
21:Algebraic structure
3865:Conformal geometry
3407:Conformal symmetry
3359:Ludwik Silberstein
3255:
3193:
3158:
3132:
3094:
2973:
2908:
2878:
2876:
2649:
2554:
2524:
2522:
1936:
1864:
1821:
1786:
1747:
1680:
1647:
1596:
1556:
1536:
1472:
1425:
1391:after a choice of
1381:
1346:
1307:
1259:
1233:
1210:
1186:
1152:
1123:
1049:special relativity
1043:there is also the
920:
864:such that for all
816:conformal geometry
587:Special orthogonal
475:
450:
413:
294:Lagrange's theorem
3365:to represent the
3004:
2806:
2739:
1723:
1700:
1559:{\displaystyle X}
1482:by the inclusion
1393:orthonormal basis
1290:
1236:{\displaystyle M}
1213:{\displaystyle M}
1175:
1126:{\displaystyle M}
796:
795:
371:
370:
253:Alternating group
210:
209:
3872:
3820:
3802:
3766:
3748:
3742:
3718:
3712:
3698:
3692:
3672:
3666:
3665:
3645:
3635:
3629:
3628:
3610:
3599:
3593:
3592:
3574:
3566:
3560:
3559:
3541:
3530:
3524:
3523:
3505:
3499:
3498:
3486:
3475:
3469:
3448:
3442:
3441:
3423:
3328:
3264:
3262:
3261:
3256:
3251:
3250:
3241:
3240:
3219:
3218:
3202:
3200:
3199:
3194:
3167:
3165:
3164:
3159:
3141:
3139:
3138:
3133:
3103:
3101:
3100:
3095:
3015:
3014:
3006:
3005:
2997:
2982:
2980:
2979:
2974:
2917:
2915:
2914:
2909:
2907:
2906:
2887:
2885:
2884:
2879:
2877:
2864:
2863:
2844:
2833:
2832:
2820:
2819:
2807:
2799:
2794:
2793:
2777:
2766:
2765:
2753:
2752:
2740:
2732:
2727:
2726:
2707:
2699:
2698:
2683:
2682:
2669:
2658:
2656:
2655:
2650:
2621:
2620:
2593:
2592:
2566:Minkowski metric
2563:
2561:
2560:
2555:
2553:
2552:
2533:
2531:
2530:
2525:
2523:
2512:
2511:
2499:
2498:
2483:
2482:
2470:
2469:
2454:
2453:
2441:
2440:
2425:
2424:
2412:
2411:
2387:
2386:
2371:
2370:
2352:
2341:
2340:
2331:
2330:
2315:
2314:
2305:
2304:
2280:
2279:
2264:
2263:
2248:
2237:
2236:
2227:
2226:
2211:
2210:
2201:
2200:
2176:
2175:
2160:
2159:
2144:
2133:
2132:
2114:
2113:
2086:
2085:
2073:
2072:
2057:
2049:
2048:
2030:
2029:
2008:
2000:
1999:
1978:
1977:
1956:
1945:
1943:
1942:
1937:
1926:
1925:
1913:
1912:
1900:
1899:
1873:
1871:
1870:
1865:
1863:
1862:
1851:
1830:
1828:
1827:
1822:
1820:
1819:
1808:
1795:
1793:
1792:
1787:
1785:
1784:
1773:
1756:
1754:
1753:
1748:
1743:
1742:
1724:
1721:
1701:
1698:
1689:
1687:
1686:
1681:
1679:
1678:
1656:
1654:
1653:
1648:
1646:
1645:
1622:
1605:
1603:
1602:
1597:
1595:
1594:
1582:
1581:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1532:
1524:
1504:
1496:
1481:
1479:
1478:
1473:
1471:
1470:
1447:
1434:
1432:
1431:
1426:
1424:
1423:
1411:
1410:
1390:
1388:
1387:
1382:
1380:
1379:
1368:
1355:
1353:
1352:
1347:
1345:
1344:
1333:
1316:
1314:
1313:
1308:
1291:
1288:
1268:
1266:
1265:
1262:{\displaystyle }
1260:
1242:
1240:
1239:
1234:
1219:
1217:
1216:
1211:
1196:is the group of
1195:
1193:
1192:
1187:
1176:
1173:
1161:
1159:
1158:
1155:{\displaystyle }
1153:
1132:
1130:
1129:
1124:
1045:hyperbolic angle
1018:
981:
942:orthogonal group
929:
927:
926:
921:
907:
906:
851:
827:orthogonal group
788:
781:
774:
730:Algebraic groups
503:Hyperbolic group
493:Arithmetic group
484:
482:
481:
476:
474:
459:
457:
456:
451:
449:
422:
420:
419:
414:
412:
335:Schur multiplier
289:Cauchy's theorem
277:Quaternion group
225:
224:
51:
50:
40:
27:
16:
15:
3880:
3879:
3875:
3874:
3873:
3871:
3870:
3869:
3855:
3854:
3846:Conformal Group
3834:10.2307/2308920
3818:
3791:
3775:
3773:Further reading
3770:
3769:
3749:
3745:
3719:
3715:
3699:
3695:
3673:
3669:
3662:
3636:
3632:
3600:
3596:
3567:
3563:
3531:
3527:
3520:
3506:
3502:
3495:
3484:
3476:
3472:
3449:
3445:
3438:
3424:
3420:
3415:
3398:
3326:
3315:Jacobian matrix
3311:proportional to
3275:
3246:
3242:
3230:
3226:
3214:
3210:
3208:
3205:
3204:
3173:
3170:
3169:
3147:
3144:
3143:
3121:
3118:
3117:
3110:
3007:
2996:
2995:
2994:
2992:
2989:
2988:
2985:Lorentz algebra
2923:
2920:
2919:
2899:
2895:
2893:
2890:
2889:
2875:
2874:
2850:
2846:
2842:
2841:
2828:
2824:
2815:
2811:
2798:
2783:
2779:
2775:
2774:
2761:
2757:
2748:
2744:
2731:
2713:
2709:
2705:
2704:
2691:
2687:
2675:
2671:
2666:
2664:
2661:
2660:
2613:
2612:
2579:
2578:
2576:
2573:
2572:
2545:
2541:
2539:
2536:
2535:
2521:
2520:
2504:
2500:
2491:
2487:
2475:
2471:
2462:
2458:
2446:
2442:
2433:
2429:
2417:
2413:
2404:
2400:
2379:
2375:
2363:
2359:
2350:
2349:
2336:
2332:
2323:
2319:
2310:
2306:
2297:
2293:
2272:
2268:
2259:
2255:
2246:
2245:
2232:
2228:
2219:
2215:
2206:
2202:
2193:
2189:
2168:
2164:
2155:
2151:
2142:
2141:
2125:
2121:
2106:
2102:
2081:
2077:
2068:
2064:
2055:
2054:
2044:
2040:
2025:
2021:
2006:
2005:
1995:
1991:
1973:
1969:
1953:
1951:
1948:
1947:
1921:
1917:
1908:
1904:
1892:
1888:
1883:
1880:
1879:
1852:
1847:
1846:
1844:
1841:
1840:
1837:
1809:
1804:
1803:
1801:
1798:
1797:
1774:
1769:
1768:
1766:
1763:
1762:
1732:
1728:
1720:
1697:
1695:
1692:
1691:
1668:
1664:
1662:
1659:
1658:
1623:
1618:
1617:
1615:
1612:
1611:
1590:
1586:
1577:
1573:
1571:
1568:
1567:
1551:
1548:
1547:
1528:
1520:
1500:
1492:
1487:
1484:
1483:
1448:
1443:
1442:
1440:
1437:
1436:
1419:
1415:
1406:
1402:
1400:
1397:
1396:
1369:
1364:
1363:
1361:
1358:
1357:
1334:
1329:
1328:
1326:
1323:
1322:
1287:
1285:
1282:
1281:
1248:
1245:
1244:
1228:
1225:
1224:
1205:
1202:
1201:
1172:
1170:
1167:
1166:
1164:conformal group
1141:
1138:
1137:
1135:conformal class
1118:
1115:
1114:
1104:
1033:
1017:
997:
976:
970:Euclidean space
902:
898:
878:
875:
874:
841:
804:conformal group
792:
763:
762:
751:Abelian variety
744:Reductive group
732:
722:
721:
720:
719:
670:
662:
654:
646:
638:
611:Special unitary
522:
508:
507:
489:
488:
470:
468:
465:
464:
445:
443:
440:
439:
408:
406:
403:
402:
394:
393:
384:Discrete groups
373:
372:
328:Frobenius group
273:
260:
249:
242:Symmetric group
238:
222:
212:
211:
62:Normal subgroup
48:
28:
19:
12:
11:
5:
3878:
3868:
3867:
3853:
3852:
3843:
3836:
3822:
3816:
3803:
3789:
3774:
3771:
3768:
3767:
3763:Springer books
3743:
3713:
3706:Springer books
3693:
3667:
3660:
3630:
3594:
3561:
3534:Bateman, Harry
3525:
3518:
3500:
3494:978-3540686255
3493:
3470:
3460:Project Euclid
3443:
3436:
3417:
3416:
3414:
3411:
3410:
3409:
3404:
3397:
3394:
3390:Poincaré group
3274:
3271:
3254:
3249:
3245:
3239:
3236:
3233:
3229:
3225:
3222:
3217:
3213:
3192:
3189:
3186:
3183:
3180:
3177:
3157:
3154:
3151:
3131:
3128:
3125:
3109:
3106:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3013:
3010:
3003:
3000:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2905:
2902:
2898:
2873:
2870:
2867:
2862:
2859:
2856:
2853:
2849:
2845:
2843:
2840:
2836:
2831:
2827:
2823:
2818:
2814:
2810:
2805:
2802:
2797:
2792:
2789:
2786:
2782:
2778:
2776:
2773:
2769:
2764:
2760:
2756:
2751:
2747:
2743:
2738:
2735:
2730:
2725:
2722:
2719:
2716:
2712:
2708:
2706:
2703:
2697:
2694:
2690:
2686:
2681:
2678:
2674:
2670:
2668:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2619:
2616:
2611:
2608:
2605:
2602:
2599:
2596:
2591:
2588:
2585:
2582:
2551:
2548:
2544:
2519:
2515:
2510:
2507:
2503:
2497:
2494:
2490:
2486:
2481:
2478:
2474:
2468:
2465:
2461:
2457:
2452:
2449:
2445:
2439:
2436:
2432:
2428:
2423:
2420:
2416:
2410:
2407:
2403:
2399:
2396:
2393:
2390:
2385:
2382:
2378:
2374:
2369:
2366:
2362:
2358:
2355:
2353:
2351:
2348:
2344:
2339:
2335:
2329:
2326:
2322:
2318:
2313:
2309:
2303:
2300:
2296:
2292:
2289:
2286:
2283:
2278:
2275:
2271:
2267:
2262:
2258:
2254:
2251:
2249:
2247:
2244:
2240:
2235:
2231:
2225:
2222:
2218:
2214:
2209:
2205:
2199:
2196:
2192:
2188:
2185:
2182:
2179:
2174:
2171:
2167:
2163:
2158:
2154:
2150:
2147:
2145:
2143:
2140:
2136:
2131:
2128:
2124:
2120:
2117:
2112:
2109:
2105:
2101:
2098:
2095:
2092:
2089:
2084:
2080:
2076:
2071:
2067:
2063:
2060:
2058:
2056:
2053:
2047:
2043:
2039:
2036:
2033:
2028:
2024:
2020:
2017:
2014:
2011:
2009:
2007:
2004:
1998:
1994:
1990:
1987:
1984:
1981:
1976:
1972:
1968:
1965:
1962:
1959:
1957:
1955:
1935:
1932:
1929:
1924:
1920:
1916:
1911:
1907:
1903:
1898:
1895:
1891:
1887:
1861:
1858:
1855:
1850:
1836:
1833:
1818:
1815:
1812:
1807:
1783:
1780:
1777:
1772:
1746:
1741:
1738:
1735:
1731:
1727:
1719:
1716:
1713:
1710:
1707:
1704:
1677:
1674:
1671:
1667:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1621:
1608:projectivising
1593:
1589:
1585:
1580:
1576:
1555:
1535:
1531:
1527:
1523:
1519:
1516:
1513:
1510:
1507:
1503:
1499:
1495:
1491:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1446:
1422:
1418:
1414:
1409:
1405:
1378:
1375:
1372:
1367:
1343:
1340:
1337:
1332:
1306:
1303:
1300:
1297:
1294:
1258:
1255:
1252:
1232:
1209:
1198:conformal maps
1185:
1182:
1179:
1151:
1148:
1145:
1122:
1103:
1100:
1088:Riemann sphere
1032:
1031:Angle analysis
1029:
1021:
1020:
1015:
987:
966:
950:
949:
933:
932:
931:
930:
919:
916:
913:
910:
905:
901:
897:
894:
891:
888:
885:
882:
835:quadratic form
825:The conformal
818:of the space.
794:
793:
791:
790:
783:
776:
768:
765:
764:
761:
760:
758:Elliptic curve
754:
753:
747:
746:
740:
739:
733:
728:
727:
724:
723:
718:
717:
714:
711:
707:
703:
702:
701:
696:
694:Diffeomorphism
690:
689:
684:
679:
673:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
631:
630:
619:
618:
607:
606:
595:
594:
583:
582:
571:
570:
559:
558:
551:Special linear
547:
546:
539:General linear
535:
534:
529:
523:
514:
513:
510:
509:
506:
505:
500:
495:
487:
486:
473:
461:
448:
435:
433:Modular groups
431:
430:
429:
424:
411:
395:
392:
391:
386:
380:
379:
378:
375:
374:
369:
368:
367:
366:
361:
356:
353:
347:
346:
340:
339:
338:
337:
331:
330:
324:
323:
318:
309:
308:
306:Hall's theorem
303:
301:Sylow theorems
297:
296:
291:
283:
282:
281:
280:
274:
269:
266:Dihedral group
262:
261:
256:
250:
245:
239:
234:
223:
218:
217:
214:
213:
208:
207:
206:
205:
200:
192:
191:
190:
189:
184:
179:
174:
169:
164:
159:
157:multiplicative
154:
149:
144:
139:
131:
130:
129:
128:
123:
115:
114:
106:
105:
104:
103:
101:Wreath product
98:
93:
88:
86:direct product
80:
78:Quotient group
72:
71:
70:
69:
64:
59:
49:
46:
45:
42:
41:
33:
32:
9:
6:
4:
3:
2:
3877:
3866:
3863:
3862:
3860:
3851:
3847:
3844:
3841:
3837:
3835:
3831:
3827:
3823:
3819:
3817:0-387-94732-9
3813:
3809:
3804:
3800:
3796:
3792:
3790:3-540-58659-8
3786:
3782:
3777:
3776:
3764:
3760:
3756:
3752:
3747:
3741:
3738:
3734:
3730:
3726:
3722:
3717:
3711:
3707:
3703:
3697:
3691:
3688:
3685:
3684:0-89464-759-8
3681:
3677:
3671:
3663:
3661:0-226-87375-7
3657:
3653:
3649:
3644:
3643:
3634:
3626:
3622:
3618:
3614:
3609:
3604:
3598:
3590:
3586:
3582:
3578:
3573:
3565:
3557:
3553:
3549:
3545:
3540:
3535:
3529:
3521:
3519:9780387947853
3515:
3511:
3504:
3496:
3490:
3483:
3482:
3474:
3468:
3465:
3461:
3457:
3453:
3447:
3439:
3437:9780191085789
3433:
3429:
3422:
3418:
3408:
3405:
3403:
3402:Conformal map
3400:
3399:
3393:
3391:
3387:
3382:
3380:
3376:
3372:
3368:
3367:Lorentz group
3364:
3363:biquaternions
3360:
3355:
3353:
3349:
3345:
3341:
3337:
3336:split-complex
3333:
3329:
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3279:Harry Bateman
3270:
3268:
3252:
3247:
3237:
3234:
3231:
3227:
3223:
3220:
3215:
3211:
3190:
3187:
3184:
3181:
3178:
3175:
3155:
3152:
3149:
3129:
3126:
3123:
3114:
3105:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3022:
3019:
3016:
3011:
3008:
2998:
2986:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2903:
2900:
2896:
2871:
2868:
2865:
2860:
2857:
2854:
2851:
2847:
2838:
2829:
2825:
2821:
2816:
2812:
2803:
2800:
2795:
2790:
2787:
2784:
2780:
2771:
2762:
2758:
2754:
2749:
2745:
2736:
2733:
2728:
2723:
2720:
2717:
2714:
2710:
2701:
2695:
2692:
2688:
2684:
2679:
2676:
2672:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2609:
2603:
2600:
2597:
2569:
2567:
2549:
2546:
2542:
2517:
2508:
2505:
2501:
2495:
2492:
2488:
2484:
2479:
2476:
2472:
2466:
2463:
2459:
2455:
2450:
2447:
2443:
2437:
2434:
2430:
2426:
2421:
2418:
2414:
2408:
2405:
2401:
2394:
2391:
2383:
2380:
2376:
2372:
2367:
2364:
2360:
2354:
2346:
2337:
2333:
2327:
2324:
2320:
2316:
2311:
2307:
2301:
2298:
2294:
2287:
2284:
2276:
2273:
2269:
2265:
2260:
2256:
2250:
2242:
2233:
2229:
2223:
2220:
2216:
2212:
2207:
2203:
2197:
2194:
2190:
2183:
2180:
2172:
2169:
2165:
2161:
2156:
2152:
2146:
2138:
2129:
2126:
2122:
2118:
2115:
2110:
2107:
2103:
2096:
2093:
2090:
2082:
2078:
2074:
2069:
2065:
2059:
2051:
2045:
2041:
2037:
2034:
2026:
2022:
2018:
2015:
2010:
2002:
1996:
1992:
1988:
1985:
1982:
1974:
1970:
1966:
1963:
1958:
1930:
1927:
1922:
1918:
1914:
1909:
1905:
1901:
1896:
1893:
1889:
1877:
1859:
1856:
1853:
1832:
1816:
1813:
1810:
1781:
1778:
1775:
1760:
1739:
1736:
1733:
1729:
1717:
1711:
1708:
1705:
1675:
1672:
1669:
1665:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1609:
1591:
1587:
1583:
1578:
1574:
1553:
1525:
1514:
1511:
1497:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1420:
1416:
1412:
1407:
1403:
1394:
1376:
1373:
1370:
1341:
1338:
1335:
1320:
1301:
1298:
1295:
1279:
1274:
1272:
1253:
1230:
1221:
1207:
1199:
1180:
1165:
1146:
1136:
1120:
1113:
1109:
1099:
1097:
1093:
1092:compact space
1089:
1085:
1081:
1077:
1076:complex plane
1073:
1068:
1066:
1062:
1058:
1057:Lorentz boost
1054:
1050:
1046:
1042:
1038:
1028:
1026:
1013:
1009:
1005:
1001:
995:
992:
988:
985:
979:
974:
971:
967:
964:
960:
956:
952:
951:
947:
943:
939:
935:
934:
914:
908:
903:
899:
895:
889:
886:
880:
873:
872:
871:
867:
863:
859:
855:
849:
845:
839:
836:
832:
828:
824:
823:
822:
819:
817:
813:
809:
805:
801:
789:
784:
782:
777:
775:
770:
769:
767:
766:
759:
756:
755:
752:
749:
748:
745:
742:
741:
738:
735:
734:
731:
726:
725:
715:
712:
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688:
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628:
624:
621:
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612:
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584:
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568:
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462:
437:
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428:
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377:
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365:
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220:Finite groups
216:
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68:
65:
63:
60:
58:
55:
54:
53:
52:
47:Basic notions
44:
43:
39:
35:
34:
31:
26:
22:
18:
17:
3828:67(1): 1−30
3807:
3780:
3754:
3746:
3733:0387-90332-1
3727:, Springer,
3724:
3721:Isaak Yaglom
3716:
3710:Google Books
3704:, page 216,
3696:
3675:
3670:
3641:
3633:
3616:
3612:
3597:
3580:
3576:
3564:
3547:
3543:
3528:
3509:
3503:
3480:
3473:
3446:
3427:
3421:
3383:
3356:
3340:dual numbers
3332:Isaak Yaglom
3330:
3310:
3290:
3276:
3267:Witt algebra
3115:
3111:
2570:
1838:
1275:
1270:
1222:
1220:to itself.
1163:
1105:
1084:dual numbers
1072:Möbius group
1069:
1034:
1022:
1011:
1007:
1003:
999:
993:
984:hyperspheres
977:
972:
963:Möbius group
869:
865:
861:
857:
853:
847:
843:
837:
830:
820:
803:
797:
686:
626:
614:
602:
590:
578:
566:
554:
542:
313:
270:
257:
246:
235:
231:Cyclic group
109:
96:Free product
67:Group action
30:Group theory
25:Group theory
24:
3751:A. O. Barut
3650:. pp.
3646:. Chicago:
3583:: 223–264.
3386:A. O. Barut
1876:Lie algebra
1271:hyper-angle
800:mathematics
516:Topological
355:alternating
3413:References
3319:light cone
3295:kinematics
1610:the space
1025:Lie groups
623:Symplectic
563:Orthogonal
520:Lie groups
427:Free group
152:continuous
91:Direct sum
3619:: 77–98.
3550:: 70–89.
3277:In 1908,
3244:∂
3224:−
3080:⋯
3062:−
3056:⋯
3047:−
3029:−
3023:
3002:~
2999:η
2983:obey the
2953:⋯
2938:−
2852:−
2830:μ
2817:μ
2791:μ
2763:μ
2755:−
2750:μ
2724:μ
2715:−
2696:ν
2693:μ
2680:ν
2677:μ
2610:≅
2550:ν
2547:μ
2543:η
2509:ρ
2506:μ
2496:σ
2493:ν
2489:η
2485:−
2480:σ
2477:ν
2467:ρ
2464:μ
2460:η
2456:−
2451:ρ
2448:ν
2438:σ
2435:μ
2431:η
2422:σ
2419:μ
2409:ρ
2406:ν
2402:η
2384:σ
2381:ρ
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2338:μ
2328:ν
2325:ρ
2321:η
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2302:μ
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2130:ν
2127:μ
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2108:μ
2104:η
2083:ν
2070:μ
2046:μ
2027:μ
1997:μ
1986:−
1975:μ
1923:μ
1910:μ
1897:ν
1894:μ
1835:Conf(p,q)
1759:inversion
1584:×
1509:↦
1413:×
1106:Given a (
946:dilations
900:λ
687:Conformal
575:Euclidean
182:nilpotent
3859:Category
3799:31374337
3605:(1910).
3536:(1908).
3396:See also
1059:is as a
1053:rapidity
1014:+ 1) / Z
682:Poincaré
527:Solenoid
399:Integers
389:Lattices
364:sporadic
359:Lie type
187:solvable
177:dihedral
162:additive
147:infinite
57:Subgroup
3723:(1979)
3690:1275599
2564:is the
1546:(where
810:is the
677:Lorentz
599:Unitary
498:Lattice
438:PSL(2,
172:abelian
83:(Semi-)
3814:
3797:
3787:
3740:520230
3731:
3682:
3658:
3652:416–24
3516:
3491:
3434:
3348:fields
3346:, not
3327:C(1,3)
1874:, the
1162:, the
1108:Pseudo
1006:) ≃ O(
980:> 2
955:sphere
936:For a
829:. If
806:of an
802:, the
532:Circle
463:SL(2,
352:cyclic
316:-group
167:cyclic
142:finite
137:simple
121:kernel
3761:#261
3485:(PDF)
3467:14282
3344:rings
2918:with
1657:. If
1200:from
1133:with
1037:angle
1010:+ 1,
998:Conf(
989:In a
812:group
716:Sp(∞)
713:SU(∞)
126:image
3850:nLab
3812:ISBN
3795:OCLC
3785:ISBN
3729:ISBN
3708:via
3680:ISBN
3656:ISBN
3514:ISBN
3489:ISBN
3432:ISBN
3338:and
3281:and
3127:>
3020:diag
1722:Conf
1699:Conf
1289:Conf
1276:For
1174:Conf
1090:, a
710:O(∞)
699:Loop
518:and
3848:in
3840:pdf
3830:doi
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