20:
3526:
52:
828:
2778:
2341:
2527:
1808:
while that of the hermitian matrix is real and positive and since in the case of a matrix from the special linear group the product of these two determinants must be 1, then each of them must be 1. Therefore, a special linear matrix can be written as the product of a
2122:
1915:) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of
3164:
2920:
2773:{\displaystyle {\begin{aligned}\left&=T_{ik}&&{\text{for }}i\neq k\\\left&=\mathbf {1} &&{\text{for }}i\neq \ell ,j\neq k\\\left(T_{12}T_{21}^{-1}T_{12}\right)^{4}&=\mathbf {1} \end{aligned}}}
1542:
2532:
1848:) and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues. A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the
2336:{\displaystyle \operatorname {Alt} (3)\cong <\operatorname {E} (2,\mathbf {F} _{2})=\operatorname {SL} (2,\mathbf {F} _{2})=\operatorname {GL} (2,\mathbf {F} _{2})\cong \operatorname {Sym} (3),}
1740:
2977:
497:
472:
435:
3067:
2829:
3012:
3193:
3294:
3216:
3035:
2030:. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so ≤ SL), but in general do not coincide with it.
1245:
799:
1293:
2819:
form another subgroup of GL, with SL as an index 2 subgroup (necessarily normal); in characteristic 2 this is the same as SL. This forms a
1487:
1298:
1288:
1283:
3590:
1103:
3562:
3467:
1367:
1250:
357:
3543:
3569:
2022:
Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the
1398:
307:
3627:; Robertson, Edmund; Williams, Peter (1992), "Presentations for 3-dimensional special linear groups over integer rings",
3269:
and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant
1574:
of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
792:
302:
3576:
3609:
3691:
3558:
1260:
3547:
3254:
718:
1255:
1235:
785:
1703:
1636:; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.
1200:
1108:
2464:
402:
216:
2460:
1645:
1240:
134:
2932:
1908:
1814:
1627:
3159:{\displaystyle \operatorname {SL} ^{\pm }(2k+1,F)\cong \operatorname {SL} (2k+1,F)\times \{\pm I\}}
2915:{\displaystyle 1\to \operatorname {SL} (n,F)\to \operatorname {SL} ^{\pm }(n,F)\to \{\pm 1\}\to 1.}
2812:
1818:
1391:
875:
600:
334:
211:
99:
3583:
480:
455:
418:
3681:
3536:
3457:
3062:
2452:
2397:
2428:
2396:
is generated by transvections. For more general rings the stable difference is measured by the
1810:
1195:
1158:
1126:
1113:
750:
540:
1459:
1227:
895:
624:
2982:
970:
960:
950:
940:
3656:
2820:
2017:
1552:
1471:
1451:
845:
552:
170:
104:
3169:
8:
3686:
2456:
2444:
2062:
2046:
2023:
1785:
1384:
1372:
1213:
1043:
139:
34:
3276:
3198:
3017:
3644:
3364:
1849:
1144:
1134:
124:
96:
2347:
1940:
1571:
1208:
1171:
529:
372:
266:
1323:
1061:
695:
3636:
2448:
2385:
1873:
1841:
1822:
1793:
1475:
1463:
1435:
1343:
1023:
1015:
1007:
999:
991:
924:
905:
865:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
3652:
3235:
2393:
2384:
In some circumstances these coincide: the special linear group over a field or a
2351:
1865:
1467:
1328:
1081:
1066:
837:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
3624:
1789:
1765:
1348:
1166:
1071:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
1333:
3675:
3462:
3447:
2440:
2027:
1056:
885:
690:
612:
446:
319:
185:
3344:
1675:
1582:
1353:
1338:
1139:
1121:
1051:
545:
244:
233:
180:
155:
150:
109:
80:
43:
23:
3665:
Lie groups, Lie algebras, and representations: An elementary introduction
2455:
of SL using transvections with some relations. Transvections satisfy the
1805:
1801:
1769:
1698:
1479:
1455:
1412:
1179:
1095:
819:
19:
3667:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
3648:
1797:
1773:
1318:
1184:
1076:
712:
440:
1853:
815:
533:
3640:
3525:
1537:{\displaystyle \det \colon \operatorname {GL} (n,R)\to R^{\times }.}
2489:
is given by two of the
Steinberg relations, plus a third relation (
1784:
Any invertible matrix can be uniquely represented according to the
70:
2509:(1) be the elementary matrix with 1's on the diagonal and in the
1275:
412:
326:
3296:, but in even dimension there is no one natural identification.
1856:
hermitian matrix, and therefore the topology of this is that of
3438:
3429:
1623:
51:
2925:
This sequence splits by taking any matrix with determinant
2459:, but these are not sufficient: the resulting group is the
2003:
827:
2463:, which is not the special linear group, but rather the
3623:
2490:
3273:). In odd dimension these are naturally identified by
3279:
3201:
3172:
3070:
3020:
2985:
2935:
2832:
2530:
2431:
of the special linear group and elementary matrices.
2125:
1706:
1490:
483:
458:
421:
3550:. Unsourced material may be challenged and removed.
2815:other than 2, the set of matrices with determinant
3288:
3210:
3187:
3158:
3029:
3006:
2971:
2914:
2772:
2335:
1734:
1570:These elements are "special" in that they form an
1536:
491:
466:
429:
3673:
3629:Proceedings of the American Mathematical Society
3234:does not split, and in general is a non-trivial
2439:If working over a ring where SL is generated by
2033:The group generated by transvections is denoted
1491:
1246:Representation theory of semisimple Lie algebras
1927:-dimensional Euclidean space. Thus, the group
1392:
793:
3153:
3144:
2903:
2894:
2101:, transvections need not be commutators (of
3635:(1), American Mathematical Society: 19–26,
2434:
2361:is a field with more than 2 elements, then
1604:
3379:can be written as a semidirect product of
1399:
1385:
1284:Particle physics and representation theory
826:
800:
786:
3610:Learn how and when to remove this message
1458:1, with the group operations of ordinary
485:
460:
423:
2072:, transvections are commutators, so for
1825:in the real case) having determinant 1.
18:
3468:Representations of classical Lie groups
3299:
2783:are a complete set of relations for SL(
2388:is generated by transvections, and the
1251:Representations of classical Lie groups
3674:
2373:is a field with more than 3 elements,
358:Classification of finite simple groups
3014:is odd, the negative identity matrix
2491:Conder, Robertson & Williams 1992
1735:{\displaystyle {\mathfrak {sl}}(n,F)}
1621:can be characterized as the group of
3662:
3548:adding citations to reliable sources
3519:
3510:
3498:
3486:
3324:splits over its determinant (we use
2105:matrices), as seen for example when
1104:Lie group–Lie algebra correspondence
16:Group of matrices with determinant 1
2346:where Alt(3) and Sym(3) denote the
2116:, the field of two elements, then
2004:Relations to other subgroups of GL(
1712:
1709:
13:
2929:, for example the diagonal matrix
2470:A sufficient set of relations for
2467:of the commutator subgroup of GL.
2216:
2026:of GL, and the group generated by
1907:is the product of the topology of
1888:is also simply connected, for all
14:
3703:
2513:position, and 0's elsewhere (and
3524:
3061:and thus the group splits as an
2972:{\displaystyle (-1,1,\dots ,1).}
2762:
2662:
2299:
2266:
2233:
2197:
2164:
1972:. In particular this means that
1804:of the unitary matrix is on the
50:
3535:needs additional citations for
1996:, is not simply connected, for
1828:Thus the topology of the group
1639:
3504:
3492:
3480:
3138:
3117:
3105:
3084:
2963:
2936:
2906:
2891:
2888:
2876:
2860:
2857:
2845:
2836:
2327:
2321:
2309:
2288:
2276:
2255:
2243:
2222:
2210:
2207:
2186:
2174:
2153:
2144:
2138:
2132:
1729:
1717:
1518:
1515:
1503:
1299:Galilean group representations
1294:Poincaré group representations
719:Infinite dimensional Lie group
1:
3473:
1289:Lorentz group representations
1256:Theorem of the highest weight
2392:special linear group over a
1892:greater than or equal to 2.
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
3423:
2798:
2465:universal central extension
1779:
217:List of group theory topics
10:
3708:
2015:
1646:Special linear Lie algebra
1643:
1632:linear transformations of
1241:Lie algebra representation
1815:special orthogonal matrix
1609:The special linear group
2435:Generators and relations
1817:in the real case) and a
1605:Geometric interpretation
1236:Lie group representation
335:Elementary abelian group
212:Glossary of group theory
3692:Linear algebraic groups
3663:Hall, Brian C. (2015),
3458:Projective linear group
3241:Over the real numbers,
3063:internal direct product
2398:special Whitehead group
1261:Borel–Weil–Bott theorem
3559:"Special linear group"
3513:Sections 13.2 and 13.3
3290:
3212:
3189:
3160:
3031:
3008:
3007:{\displaystyle n=2k+1}
2973:
2916:
2774:
2337:
1844:of the topology of SU(
1811:special unitary matrix
1736:
1538:
1159:Semisimple Lie algebra
1114:Adjoint representation
751:Linear algebraic group
493:
468:
431:
27:
3291:
3213:
3190:
3161:
3032:
3009:
2974:
2917:
2775:
2338:
1821:hermitian matrix (or
1737:
1539:
1460:matrix multiplication
1228:Representation theory
494:
469:
432:
22:
3544:improve this article
3277:
3255:connected components
3199:
3188:{\displaystyle n=2k}
3170:
3068:
3018:
2983:
2933:
2830:
2821:short exact sequence
2528:
2123:
1788:as the product of a
1704:
1572:algebraic subvariety
1553:multiplicative group
1488:
1472:general linear group
1417:special linear group
481:
456:
419:
3257:, corresponding to
2732:
2493:, p. 19). Let
2457:Steinberg relations
2047:elementary matrices
2024:commutator subgroup
1876:, we conclude that
1786:polar decomposition
1601:is sometimes used.
1373:Table of Lie groups
1214:Compact Lie algebra
125:Group homomorphisms
35:Algebraic structure
3365:semidirect product
3289:{\displaystyle -I}
3286:
3211:{\displaystyle -I}
3208:
3185:
3156:
3030:{\displaystyle -I}
3027:
3004:
2969:
2912:
2770:
2768:
2715:
2451:), one can give a
2333:
2063:Steinberg relation
1750:) consists of all
1732:
1534:
1145:Affine Lie algebra
1135:Simple Lie algebra
876:Special orthogonal
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
28:
3620:
3619:
3612:
3594:
3501:Proposition 13.11
3367:), and therefore
2672:
2600:
2018:Whitehead's lemma
1941:fundamental group
1819:positive definite
1563:excluding 0 when
1409:
1408:
1209:Split Lie algebra
1172:Cartan subalgebra
1034:
1033:
925:Simple Lie groups
810:
809:
385:
384:
267:Alternating group
224:
223:
3699:
3668:
3659:
3615:
3608:
3604:
3601:
3595:
3593:
3552:
3528:
3520:
3514:
3508:
3502:
3496:
3490:
3484:
3390:
3378:
3362:
3342:
3323:
3300:Structure of GL(
3295:
3293:
3292:
3287:
3272:
3268:
3252:
3233:
3229:
3217:
3215:
3214:
3209:
3194:
3192:
3191:
3186:
3165:
3163:
3162:
3157:
3080:
3079:
3060:
3048:
3036:
3034:
3033:
3028:
3013:
3011:
3010:
3005:
2978:
2976:
2975:
2970:
2928:
2921:
2919:
2918:
2913:
2872:
2871:
2818:
2779:
2777:
2776:
2771:
2769:
2765:
2753:
2752:
2747:
2743:
2742:
2741:
2731:
2723:
2714:
2713:
2673:
2670:
2667:
2665:
2653:
2649:
2648:
2647:
2632:
2631:
2601:
2598:
2595:
2593:
2592:
2573:
2569:
2568:
2567:
2552:
2551:
2508:
2488:
2481:
2449:Euclidean domain
2418:
2386:Euclidean domain
2380:
2368:
2342:
2340:
2339:
2334:
2308:
2307:
2302:
2275:
2274:
2269:
2242:
2241:
2236:
2206:
2205:
2200:
2173:
2172:
2167:
2104:
2100:
2090:
2078:
2071:
2061:. By the second
2060:
2044:
2000:greater than 1.
1995:
1983:
1971:
1957:
1938:
1926:
1906:
1895:The topology of
1887:
1874:simply connected
1863:
1839:
1823:symmetric matrix
1794:hermitian matrix
1772:is given by the
1759:
1741:
1739:
1738:
1733:
1716:
1715:
1696:
1689:
1673:
1620:
1600:
1543:
1541:
1540:
1535:
1530:
1529:
1464:matrix inversion
1450:
1436:commutative ring
1429:
1401:
1394:
1387:
1344:Claude Chevalley
1201:Complexification
1044:Other Lie groups
930:
929:
838:Classical groups
830:
812:
811:
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
3707:
3706:
3702:
3701:
3700:
3698:
3697:
3696:
3672:
3671:
3641:10.2307/2159559
3625:Conder, Marston
3616:
3605:
3599:
3596:
3553:
3551:
3541:
3529:
3518:
3517:
3509:
3505:
3497:
3493:
3485:
3481:
3476:
3426:
3380:
3368:
3352:
3325:
3313:
3310:
3278:
3275:
3274:
3270:
3258:
3242:
3236:group extension
3231:
3219:
3200:
3197:
3196:
3171:
3168:
3167:
3075:
3071:
3069:
3066:
3065:
3050:
3038:
3019:
3016:
3015:
2984:
2981:
2980:
2934:
2931:
2930:
2926:
2867:
2863:
2831:
2828:
2827:
2816:
2809:
2767:
2766:
2761:
2754:
2748:
2737:
2733:
2724:
2719:
2709:
2705:
2704:
2700:
2699:
2696:
2695:
2669:
2666:
2661:
2654:
2640:
2636:
2624:
2620:
2619:
2615:
2612:
2611:
2597:
2594:
2585:
2581:
2574:
2560:
2556:
2544:
2540:
2539:
2535:
2531:
2529:
2526:
2525:
2506:
2499:
2494:
2483:
2471:
2461:Steinberg group
2437:
2404:
2400:
2394:Dedekind domain
2374:
2362:
2352:symmetric group
2303:
2298:
2297:
2270:
2265:
2264:
2237:
2232:
2231:
2201:
2196:
2195:
2168:
2163:
2162:
2124:
2121:
2120:
2115:
2102:
2095:
2080:
2073:
2066:
2050:
2034:
2020:
2014:
1985:
1973:
1966:
1964:
1952:
1928:
1916:
1896:
1877:
1866:Euclidean space
1857:
1829:
1782:
1764:with vanishing
1751:
1708:
1707:
1705:
1702:
1701:
1691:
1679:
1663:
1648:
1642:
1610:
1607:
1590:
1589:, the notation
1525:
1521:
1489:
1486:
1485:
1468:normal subgroup
1442:
1419:
1405:
1360:
1359:
1358:
1329:Wilhelm Killing
1313:
1305:
1304:
1303:
1278:
1267:
1266:
1265:
1230:
1220:
1219:
1218:
1205:
1189:
1167:Dynkin diagrams
1161:
1151:
1150:
1149:
1131:
1109:Exponential map
1098:
1088:
1087:
1086:
1067:Conformal group
1046:
1036:
1035:
1027:
1019:
1011:
1003:
995:
976:
966:
956:
946:
927:
917:
916:
915:
896:Special unitary
840:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
17:
12:
11:
5:
3705:
3695:
3694:
3689:
3684:
3682:Linear algebra
3670:
3669:
3660:
3618:
3617:
3532:
3530:
3523:
3516:
3515:
3503:
3491:
3478:
3477:
3475:
3472:
3471:
3470:
3465:
3460:
3455:
3445:
3436:
3425:
3422:
3421:
3420:
3309:
3298:
3285:
3282:
3218:is already in
3207:
3204:
3184:
3181:
3178:
3175:
3166:. However, if
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3078:
3074:
3026:
3023:
3003:
3000:
2997:
2994:
2991:
2988:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2923:
2922:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2870:
2866:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2813:characteristic
2808:
2797:
2781:
2780:
2764:
2760:
2757:
2755:
2751:
2746:
2740:
2736:
2730:
2727:
2722:
2718:
2712:
2708:
2703:
2698:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2668:
2664:
2660:
2657:
2655:
2652:
2646:
2643:
2639:
2635:
2630:
2627:
2623:
2618:
2614:
2613:
2610:
2607:
2604:
2596:
2591:
2588:
2584:
2580:
2577:
2575:
2572:
2566:
2563:
2559:
2555:
2550:
2547:
2543:
2538:
2534:
2533:
2504:
2497:
2436:
2433:
2402:
2354:on 3 letters.
2344:
2343:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2306:
2301:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2273:
2268:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2240:
2235:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2204:
2199:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2171:
2166:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2113:
2013:
2002:
1962:
1796:with positive
1790:unitary matrix
1781:
1778:
1760:matrices over
1731:
1728:
1725:
1722:
1719:
1714:
1711:
1644:Main article:
1641:
1638:
1606:
1603:
1545:
1544:
1533:
1528:
1524:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1466:. This is the
1441:is the set of
1407:
1406:
1404:
1403:
1396:
1389:
1381:
1378:
1377:
1376:
1375:
1370:
1362:
1361:
1357:
1356:
1351:
1349:Harish-Chandra
1346:
1341:
1336:
1331:
1326:
1324:Henri Poincaré
1321:
1315:
1314:
1311:
1310:
1307:
1306:
1302:
1301:
1296:
1291:
1286:
1280:
1279:
1274:Lie groups in
1273:
1272:
1269:
1268:
1264:
1263:
1258:
1253:
1248:
1243:
1238:
1232:
1231:
1226:
1225:
1222:
1221:
1217:
1216:
1211:
1206:
1204:
1203:
1198:
1192:
1190:
1188:
1187:
1182:
1176:
1174:
1169:
1163:
1162:
1157:
1156:
1153:
1152:
1148:
1147:
1142:
1137:
1132:
1130:
1129:
1124:
1118:
1116:
1111:
1106:
1100:
1099:
1094:
1093:
1090:
1089:
1085:
1084:
1079:
1074:
1072:Diffeomorphism
1069:
1064:
1059:
1054:
1048:
1047:
1042:
1041:
1038:
1037:
1032:
1031:
1030:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
986:
985:
981:
980:
979:
978:
972:
968:
962:
958:
952:
948:
942:
935:
934:
928:
923:
922:
919:
918:
914:
913:
903:
893:
883:
873:
863:
856:Special linear
853:
846:General linear
842:
841:
836:
835:
832:
831:
823:
822:
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
15:
9:
6:
4:
3:
2:
3704:
3693:
3690:
3688:
3685:
3683:
3680:
3679:
3677:
3666:
3661:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3626:
3622:
3621:
3614:
3611:
3603:
3592:
3589:
3585:
3582:
3578:
3575:
3571:
3568:
3564:
3561: –
3560:
3556:
3555:Find sources:
3549:
3545:
3539:
3538:
3533:This article
3531:
3527:
3522:
3521:
3512:
3507:
3500:
3495:
3488:
3483:
3479:
3469:
3466:
3464:
3463:Conformal map
3461:
3459:
3456:
3453:
3449:
3448:Modular group
3446:
3444:
3442:
3437:
3435:
3433:
3428:
3427:
3418:
3414:
3410:
3406:
3402:
3398:
3397:
3396:
3394:
3388:
3384:
3376:
3372:
3366:
3360:
3356:
3350:
3346:
3340:
3336:
3332:
3328:
3321:
3317:
3307:
3303:
3297:
3283:
3280:
3266:
3262:
3256:
3250:
3246:
3239:
3237:
3227:
3223:
3205:
3202:
3182:
3179:
3176:
3173:
3150:
3147:
3141:
3135:
3132:
3129:
3126:
3123:
3120:
3114:
3111:
3108:
3102:
3099:
3096:
3093:
3090:
3087:
3081:
3076:
3072:
3064:
3058:
3054:
3046:
3042:
3024:
3021:
3001:
2998:
2995:
2992:
2989:
2986:
2966:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2909:
2900:
2897:
2885:
2882:
2879:
2873:
2868:
2864:
2854:
2851:
2848:
2842:
2839:
2833:
2826:
2825:
2824:
2822:
2814:
2806:
2802:
2796:
2794:
2790:
2786:
2758:
2756:
2749:
2744:
2738:
2734:
2728:
2725:
2720:
2716:
2710:
2706:
2701:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2658:
2656:
2650:
2644:
2641:
2637:
2633:
2628:
2625:
2621:
2616:
2608:
2605:
2602:
2589:
2586:
2582:
2578:
2576:
2570:
2564:
2561:
2557:
2553:
2548:
2545:
2541:
2536:
2524:
2523:
2522:
2520:
2516:
2512:
2507:
2500:
2492:
2486:
2479:
2475:
2468:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2441:transvections
2432:
2430:
2429:stable groups
2426:
2422:
2416:
2412:
2409:) := SL(
2408:
2399:
2395:
2391:
2387:
2382:
2378:
2372:
2366:
2360:
2355:
2353:
2349:
2330:
2324:
2318:
2315:
2312:
2304:
2294:
2291:
2285:
2282:
2279:
2271:
2261:
2258:
2252:
2249:
2246:
2238:
2228:
2225:
2219:
2213:
2202:
2192:
2189:
2183:
2180:
2177:
2169:
2159:
2156:
2150:
2147:
2141:
2135:
2129:
2126:
2119:
2118:
2117:
2112:
2108:
2098:
2092:
2088:
2084:
2076:
2069:
2064:
2058:
2054:
2048:
2042:
2038:
2031:
2029:
2028:transvections
2025:
2019:
2011:
2007:
2001:
1999:
1993:
1989:
1981:
1977:
1969:
1961:
1955:
1950:
1946:
1942:
1939:has the same
1936:
1932:
1924:
1920:
1914:
1910:
1904:
1900:
1893:
1891:
1885:
1881:
1875:
1871:
1867:
1864:-dimensional
1861:
1855:
1851:
1847:
1843:
1837:
1833:
1826:
1824:
1820:
1816:
1812:
1807:
1803:
1799:
1795:
1791:
1787:
1777:
1775:
1771:
1767:
1763:
1758:
1754:
1749:
1745:
1726:
1723:
1720:
1700:
1694:
1690:of dimension
1687:
1683:
1677:
1671:
1667:
1661:
1657:
1653:
1647:
1637:
1635:
1631:
1629:
1625:
1618:
1614:
1602:
1598:
1594:
1588:
1584:
1580:
1575:
1573:
1568:
1567:is a field).
1566:
1562:
1558:
1554:
1550:
1531:
1526:
1522:
1512:
1509:
1506:
1500:
1497:
1494:
1484:
1483:
1482:
1481:
1477:
1474:given by the
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1440:
1437:
1433:
1427:
1423:
1418:
1414:
1402:
1397:
1395:
1390:
1388:
1383:
1382:
1380:
1379:
1374:
1371:
1369:
1366:
1365:
1364:
1363:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1316:
1309:
1308:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1281:
1277:
1271:
1270:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1233:
1229:
1224:
1223:
1215:
1212:
1210:
1207:
1202:
1199:
1197:
1194:
1193:
1191:
1186:
1183:
1181:
1178:
1177:
1175:
1173:
1170:
1168:
1165:
1164:
1160:
1155:
1154:
1146:
1143:
1141:
1138:
1136:
1133:
1128:
1125:
1123:
1120:
1119:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1101:
1097:
1092:
1091:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1049:
1045:
1040:
1039:
1028:
1022:
1020:
1014:
1012:
1006:
1004:
998:
996:
990:
989:
988:
987:
983:
982:
977:
975:
969:
967:
965:
959:
957:
955:
949:
947:
945:
939:
938:
937:
936:
932:
931:
926:
921:
920:
911:
907:
904:
901:
897:
894:
891:
887:
884:
881:
877:
874:
871:
867:
864:
861:
857:
854:
851:
847:
844:
843:
839:
834:
833:
829:
825:
824:
821:
817:
814:
813:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
25:
21:
3664:
3632:
3628:
3606:
3600:January 2008
3597:
3587:
3580:
3573:
3566:
3554:
3542:Please help
3537:verification
3534:
3506:
3494:
3482:
3451:
3440:
3431:
3416:
3412:
3408:
3404:
3400:
3392:
3386:
3382:
3374:
3370:
3358:
3354:
3348:
3345:monomorphism
3338:
3334:
3330:
3326:
3319:
3315:
3311:
3305:
3301:
3264:
3260:
3248:
3244:
3240:
3225:
3221:
3056:
3052:
3044:
3040:
2924:
2810:
2804:
2800:
2792:
2788:
2784:
2782:
2518:
2514:
2510:
2502:
2495:
2484:
2477:
2473:
2469:
2453:presentation
2438:
2424:
2420:
2414:
2410:
2406:
2389:
2383:
2376:
2370:
2364:
2358:
2357:However, if
2356:
2345:
2110:
2106:
2096:
2093:
2086:
2082:
2074:
2067:
2056:
2052:
2040:
2036:
2032:
2021:
2009:
2005:
1997:
1991:
1987:
1979:
1975:
1967:
1959:
1953:
1948:
1947:), that is,
1944:
1934:
1930:
1922:
1918:
1912:
1902:
1898:
1894:
1889:
1883:
1879:
1869:
1859:
1845:
1835:
1831:
1827:
1783:
1761:
1756:
1752:
1747:
1743:
1692:
1685:
1681:
1676:Lie subgroup
1669:
1665:
1659:
1655:
1651:
1649:
1640:Lie subgroup
1633:
1622:
1616:
1612:
1608:
1596:
1592:
1586:
1583:finite field
1578:
1576:
1569:
1564:
1560:
1556:
1548:
1546:
1447:
1443:
1438:
1431:
1425:
1421:
1416:
1410:
1354:Armand Borel
1339:Hermann Weyl
1140:Loop algebra
1122:Killing form
1096:Lie algebras
973:
963:
953:
943:
909:
899:
889:
879:
869:
859:
855:
849:
820:Lie algebras
640:
628:
616:
604:
592:
580:
568:
564:
556:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
24:Cayley table
3489:Section 2.5
3049:but not in
2823:of groups:
2443:(such as a
2419:, where SL(
2348:alternating
1868:. Since SU(
1850:exponential
1806:unit circle
1802:determinant
1798:eigenvalues
1770:Lie bracket
1699:Lie algebra
1628:orientation
1480:determinant
1456:determinant
1413:mathematics
1334:Élie Cartan
1180:Root system
984:Exceptional
530:Topological
369:alternating
26:of SL(2,3).
3687:Lie groups
3676:Categories
3570:newspapers
3474:References
3312:The group
2427:) are the
2016:See also:
1774:commutator
1630:preserving
1559:(that is,
1430:of degree
1319:Sophus Lie
1312:Scientists
1185:Weyl group
906:Symplectic
866:Orthogonal
816:Lie groups
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
3511:Hall 2015
3499:Hall 2015
3487:Hall 2015
3281:−
3203:−
3195:is even,
3148:±
3142:×
3115:
3109:≅
3082:
3077:±
3022:−
2955:…
2940:−
2907:→
2898:±
2892:→
2874:
2869:±
2861:→
2843:
2837:→
2726:−
2690:≠
2681:ℓ
2678:≠
2671:for
2645:ℓ
2606:≠
2599:for
2501: :=
2369:, and if
2319:
2313:≅
2286:
2253:
2220:
2184:
2151:
2142:≅
2130:
1984:, unlike
1854:traceless
1585:of order
1527:×
1519:→
1501:
1495::
1196:Real form
1082:Euclidean
933:Classical
701:Conformal
589:Euclidean
196:nilpotent
3450:(PSL(2,
3424:See also
3329:≅ GL(1,
3253:has two
2521:). Then
2423:) and E(
1780:Topology
1452:matrices
1368:Glossary
1062:Poincaré
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
3657:1079696
3649:2159559
3584:scholar
3407:) = SL(
3343:as the
3333:) → GL(
1842:product
1840:is the
1581:is the
1551:is the
1478:of the
1470:of the
1434:over a
1276:physics
1057:Lorentz
886:Unitary
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
3655:
3647:
3586:
3579:
3572:
3565:
3557:
3439:SL(2,
3430:SL(2,
3363:, see
3037:is in
2390:stable
2350:resp.
2065:, for
1970:> 2
1943:as SO(
1925:− 1)/2
1800:. The
1792:and a
1768:. The
1742:of SL(
1697:. The
1624:volume
1547:where
1476:kernel
1415:, the
1052:Circle
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
3645:JSTOR
3591:JSTOR
3577:books
3347:from
2795:≥ 3.
2445:field
2375:E(2,
2363:E(2,
2103:2 × 2
2049:) or
2045:(for
1921:+ 2)(
1872:) is
1852:of a
1766:trace
1674:is a
1650:When
1577:When
1454:with
1127:Index
730:Sp(∞)
727:SU(∞)
140:image
3563:news
3415:) ⋊
2482:for
2413:)/E(
2379:) =
2367:) =
2214:<
2094:For
2089:) ≤
1965:for
1958:and
1951:for
1862:− 1)
1813:(or
1626:and
1462:and
1077:Loop
818:and
724:O(∞)
713:Loop
532:and
3637:doi
3633:115
3546:by
3399:GL(
3391:by
3381:SL(
3369:GL(
3353:GL(
3351:to
3314:GL(
3259:SL(
3243:SL(
3220:SL(
3051:SL(
3039:SL(
2979:If
2811:In
2799:SL(
2791:),
2487:≥ 3
2472:SL(
2447:or
2381:.
2316:Sym
2127:Alt
2109:is
2099:= 2
2077:≥ 3
2070:≥ 3
2051:TV(
1986:SL(
1974:SL(
1956:= 2
1929:SL(
1897:SL(
1878:SL(
1830:SL(
1695:− 1
1680:GL(
1678:of
1664:SL(
1658:or
1654:is
1611:SL(
1591:SL(
1555:of
1492:det
1420:SL(
1411:In
908:Sp(
898:SU(
878:SO(
858:SL(
848:GL(
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
3678::
3653:MR
3651:,
3643:,
3631:,
3454:))
3411:,
3403:,
3395::
3385:,
3373:,
3357:,
3337:,
3318:,
3271:−1
3263:,
3247:,
3238:.
3232:SL
3230:,
3112:SL
3073:SL
2927:−1
2910:1.
2865:SL
2840:SL
2817:±1
2787:,
2739:12
2721:21
2711:12
2517:≠
2511:ij
2505:ij
2498:ij
2476:,
2401:SK
2283:GL
2250:SL
2181:GL
2148:GL
2091:.
2085:,
2081:E(
2079:,
2055:,
2039:,
2035:E(
2008:,
1990:,
1978:,
1933:,
1909:SO
1901:,
1882:,
1834:,
1776:.
1755:×
1746:,
1684:,
1668:,
1662:,
1615:,
1595:,
1498:GL
1446:×
1424:,
888:U(
868:O(
615:U(
591:E(
579:O(
37:→
3639::
3613:)
3607:(
3602:)
3598:(
3588:·
3581:·
3574:·
3567:·
3540:.
3452:Z
3443:)
3441:C
3434:)
3432:R
3419:.
3417:F
3413:F
3409:n
3405:F
3401:n
3393:F
3389:)
3387:F
3383:n
3377:)
3375:F
3371:n
3361:)
3359:F
3355:n
3349:F
3341:)
3339:F
3335:n
3331:F
3327:F
3322:)
3320:F
3316:n
3308:)
3306:F
3304:,
3302:n
3284:I
3267:)
3265:R
3261:n
3251:)
3249:R
3245:n
3228:)
3226:F
3224:,
3222:n
3206:I
3183:k
3180:2
3177:=
3174:n
3154:}
3151:I
3145:{
3139:)
3136:F
3133:,
3130:1
3127:+
3124:k
3121:2
3118:(
3106:)
3103:F
3100:,
3097:1
3094:+
3091:k
3088:2
3085:(
3059:)
3057:F
3055:,
3053:n
3047:)
3045:F
3043:,
3041:n
3025:I
3002:1
2999:+
2996:k
2993:2
2990:=
2987:n
2967:.
2964:)
2961:1
2958:,
2952:,
2949:1
2946:,
2943:1
2937:(
2904:}
2901:1
2895:{
2889:)
2886:F
2883:,
2880:n
2877:(
2858:)
2855:F
2852:,
2849:n
2846:(
2834:1
2807:)
2805:F
2803:,
2801:n
2793:n
2789:Z
2785:n
2763:1
2759:=
2750:4
2745:)
2735:T
2729:1
2717:T
2707:T
2702:(
2693:k
2687:j
2684:,
2675:i
2663:1
2659:=
2651:]
2642:k
2638:T
2634:,
2629:j
2626:i
2622:T
2617:[
2609:k
2603:i
2590:k
2587:i
2583:T
2579:=
2571:]
2565:k
2562:j
2558:T
2554:,
2549:j
2546:i
2542:T
2537:[
2519:j
2515:i
2503:e
2496:T
2485:n
2480:)
2478:Z
2474:n
2425:A
2421:A
2417:)
2415:A
2411:A
2407:A
2405:(
2403:1
2377:A
2371:A
2365:A
2359:A
2331:,
2328:)
2325:3
2322:(
2310:)
2305:2
2300:F
2295:,
2292:2
2289:(
2280:=
2277:)
2272:2
2267:F
2262:,
2259:2
2256:(
2247:=
2244:)
2239:2
2234:F
2229:,
2226:2
2223:(
2217:E
2211:]
2208:)
2203:2
2198:F
2193:,
2190:2
2187:(
2178:,
2175:)
2170:2
2165:F
2160:,
2157:2
2154:(
2145:[
2139:)
2136:3
2133:(
2114:2
2111:F
2107:A
2097:n
2087:A
2083:n
2075:n
2068:n
2059:)
2057:A
2053:n
2043:)
2041:A
2037:n
2012:)
2010:A
2006:n
1998:n
1994:)
1992:C
1988:n
1982:)
1980:R
1976:n
1968:n
1963:2
1960:Z
1954:n
1949:Z
1945:n
1937:)
1935:R
1931:n
1923:n
1919:n
1917:(
1913:n
1911:(
1905:)
1903:R
1899:n
1890:n
1886:)
1884:C
1880:n
1870:n
1860:n
1858:(
1846:n
1838:)
1836:C
1832:n
1762:F
1757:n
1753:n
1748:F
1744:n
1730:)
1727:F
1724:,
1721:n
1718:(
1713:l
1710:s
1693:n
1688:)
1686:F
1682:n
1672:)
1670:F
1666:n
1660:C
1656:R
1652:F
1634:R
1619:)
1617:R
1613:n
1599:)
1597:q
1593:n
1587:q
1579:R
1565:R
1561:R
1557:R
1549:R
1532:.
1523:R
1516:)
1513:R
1510:,
1507:n
1504:(
1448:n
1444:n
1439:R
1432:n
1428:)
1426:R
1422:n
1400:e
1393:t
1386:v
1026:8
1024:E
1018:7
1016:E
1010:6
1008:E
1002:4
1000:F
994:2
992:G
974:n
971:D
964:n
961:C
954:n
951:B
944:n
941:A
912:)
910:n
902:)
900:n
892:)
890:n
882:)
880:n
872:)
870:n
862:)
860:n
852:)
850:n
801:e
794:t
787:v
683:8
681:E
675:7
673:E
667:6
665:E
659:4
657:F
651:2
649:G
643:)
641:n
631:)
629:n
619:)
617:n
607:)
605:n
595:)
593:n
583:)
581:n
571:)
569:n
559:)
557:n
499:)
486:Z
474:)
461:Z
437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
;_Cayley_table.svg)
