1716:
40:
152:
1162:
A symmetric space with a compatible complex structure is called
Hermitian. The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric
1721:
For the infinite (A, B, C, D) series of Dynkin diagrams, a connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices. For those of type A
2162:
is not 'simple' as an abstract group, and according to most (but not all) definitions this is not a simple Lie group. Further, most authors do not count its Lie algebra as a simple Lie algebra. It is listed here so that the list of "irreducible simply connected symmetric spaces" is complete. Note
1104:
of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of
1956:; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to the infinite families, largely because their descriptions make use of
1497:
in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such a real group is the
1238:
for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup.
1242:
The fundamental group listed in the table below is the fundamental group of the simple group with trivial center. Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group).
6851:
only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. The A, D and E series groups are all simply laced, but no group of type B, C, F, or G is simply laced.
1126:
Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces).
920:
In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a
5642:
The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
1122:
First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)
1600:
1015:, because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counter-example are the
1542:
1415:
4434:
1218:
6705:
6473:
6336:
6120:
6055:
5934:
5903:
5845:
5775:
2092:
1680:
1475:
1352:
1308:
2183:
2160:
2118:
2058:
1653:
915:
886:
825:
787:
1699:. It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of the
929:
of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
1871:
1040:
1620:
1495:
1435:
1328:
866:
as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether
6896:
1154:. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.
1077:
center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)
569:
862:
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is
793:(the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of
617:
1007:. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the corresponding
622:
612:
607:
895:
normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but
1070:
953:
normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is
427:
691:
574:
3166:
compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space.
1114:
104:
1703:. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by
1173:, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.
6955:
1130:
The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each
76:
722:
4586:
1046:, and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are
1622:, and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and
83:
7007:
6993:
6979:
6921:
123:
57:
1722:
and C we can find explicit matrix representations of the corresponding simply connected Lie group as matrix groups.
2291:
1547:
90:
3712:
2896:
840:
797:. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "
584:
6931:
992:
account for many special examples and configurations in other branches of mathematics, as well as contemporary
61:
1255:
579:
559:
72:
19:
This article is about the
Killing-Cartan classification. For a smaller list of groups that commonly occur in
891:
The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed
524:
432:
1505:
1378:
4706:
3286:
3002:
1866:, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by
2185:
is the only such non-compact symmetric space without a compact dual (although it has a compact quotient
1183:
6935:
1251:
Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:
761:
564:
6681:
6449:
6312:
6096:
6031:
5910:
5879:
5821:
5751:
2068:
1810:)/{I, −I} of projective unitary symplectic matrices. The symplectic groups have a double-cover by the
1658:
1444:
1073:
is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with
4562:
1361:
1333:
1289:
981:
1695:
Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is
4445:
3810:
2466:
2367:
1828:
1824:
1770:
1766:
1749:
1074:
1016:
715:
199:
2166:
2143:
2101:
2041:
1629:
1100:. This reduces the problem of classifying the real simple Lie algebras to that of finding all the
898:
869:
808:
770:
5415:
1715:
50:
922:
1700:
1623:
1269:
519:
482:
450:
437:
28:
5421:
Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers.
1502:, which appears in infinite-dimensional representation theory and physics. When one takes for
97:
7027:
6881:
3579:
Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler.
1741:
1737:
1355:
1043:
551:
219:
1835:
and as its associated centerless compact group the projective special orthogonal group PSO(2
1088:
is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless
1004:
1000:
798:
179:
169:
8:
7022:
6891:
993:
969:
961:
708:
696:
537:
367:
24:
20:
1092:
is already the complexification of a Lie algebra, in which case the complexification of
1022:
6886:
1957:
1605:
1480:
1420:
1313:
1064:
985:
965:
946:
938:
757:
468:
458:
7003:
6989:
6975:
6951:
6917:
3593:
2763:
1811:
1499:
1368:
1138:, one compact and one non-compact. The non-compact one is a cover of the quotient of
926:
532:
495:
647:
385:
6943:
3902:
3475:
3102:
2562:
1950:
1943:
1936:
1929:
1922:
1795:
1791:
977:
743:
667:
347:
339:
331:
323:
315:
229:
189:
1777:. This group is not simply connected however: its universal (double) cover is the
1277:
854:. The final classification is often referred to as Killing-Cartan classification.
6909:
3527:
Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.
1704:
1259:
1047:
847:
794:
753:
740:
652:
405:
390:
161:
7000:
Symmetries, Lie algebras, and representations: a graduate course for physicists.
6866:
6848:
1973:
1438:
1273:
1265:
1163:
space; this gives the complete list of irreducible
Hermitian symmetric spaces.
1081:
672:
490:
395:
6947:
1708:
851:
657:
7016:
6876:
6861:
2199:
2012:
1696:
1655:
with that Lie algebra, called the "simply connected Lie group" associated to
1372:
1080:
Over the complex numbers the semisimple Lie algebras are classified by their
380:
209:
16:
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
5372:
Rosenfeld hyperbolic projective plane over the complexified Cayley numbers.
1748:
and as its associated centerless compact group the projective unitary group
1012:
954:
863:
790:
677:
662:
463:
445:
375:
5367:
Rosenfeld elliptic projective plane over the complexified Cayley numbers.
4633:
1690:
1221:
1008:
973:
942:
503:
419:
143:
1146:, and the compact one is a cover of the quotient of the compact form of
6871:
3159:. Copies of complex projective space in quaternionic projective space.
2494:. Copies of complex projective space in quaternionic projective space.
1867:
1848:
1778:
642:
508:
400:
1258:
The classification of simple Lie algebras over the complex numbers by
1166:
The four families are the types A III, B I and D I for
949:. An important technical point is that a simple Lie group may contain
6844:
1101:
746:
139:
1882:, corresponding to a covering map homomorphism from SU(4) to SO(6).
1272:, classified by additional decorations of its Dynkin diagram called
1115:
Symmetric space § Classification of
Riemannian symmetric spaces
756:. The list of simple Lie groups can be used to read off the list of
39:
3576:
Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler.
1965:
1225:
5261:
Complex structures on R compatible with the
Euclidean structure.
599:
6170:
Non-division quaternionic subalgebras of non-division octonions
988:
possibilities not corresponding to any familiar geometry. These
937:
An equivalent definition of a simple Lie group follows from the
3999:, or the product of a group of order 2 and the symmetric group
1980:
5418:
in the projective plane over the complexified Cayley numbers.
767:
Together with the commutative Lie group of the real numbers,
1921:
above, there are five so-called exceptional Dynkin diagrams
1847:) is not simply connected; its universal cover is again the
3524:
Quaternionic projective planes in Cayley projective plane.
835:
matrices with determinant equal to 1 is simple for all odd
151:
1790:
has as its associated simply connected group the group of
1802:
and as its associated centerless group the Lie group PSp(
1736:
has as its associated simply connected compact group the
2946:, when it is the upper half plane or unit complex disc.
1765:
has as its associated centerless compact groups the odd
6986:
Differential geometry, Lie groups, and symmetric spaces
5637:
5629:
Hyperbolic Cayley projective plane. Quaternion-Kähler.
1874:. Thus SO(4) is not a simple group. Also, the diagram D
1851:, but the latter again has a center (cf. its article).
2751:
This is the automorphism group of the Cayley algebra.
6684:
6452:
6315:
6099:
6034:
5913:
5882:
5824:
5754:
2169:
2146:
2104:
2071:
2044:
1661:
1632:
1608:
1550:
1508:
1483:
1447:
1423:
1381:
1336:
1316:
1292:
1186:
1025:
901:
872:
846:
The first classification of simple Lie groups was by
811:
773:
1544:
the full fundamental group, the resulting Lie group
1220: stand for the real numbers, complex numbers,
6897:
Classification of low-dimensional real Lie algebras
1602:is the universal cover of the centerless Lie group
984:of simple Lie groups that there exist also several
964:, which provide a group-theoretic underpinning for
64:. Unsourced material may be challenged and removed.
6699:
6467:
6330:
6114:
6049:
5928:
5897:
5839:
5769:
2177:
2154:
2112:
2086:
2052:
1674:
1647:
1614:
1594:
1536:
1489:
1469:
1429:
1409:
1346:
1322:
1310:, there is a unique "centerless" simple Lie group
1302:
1212:
1034:
909:
880:
819:
789:, and that of the unit-magnitude complex numbers,
781:
1725:
7014:
570:Representation theory of semisimple Lie algebras
1286:For every (real or complex) simple Lie algebra
1284:Classification of centerless simple Lie groups
1823:has as its associated compact group the even
1256:Classification of simple complex Lie algebras
1157:
716:
6930:
5626:Cayley projective plane. Quaternion-Kähler.
1595:{\displaystyle {\tilde {G}}^{K=\pi _{1}(G)}}
1268:Each simple complex Lie algebra has several
1119:Symmetric spaces are classified as follows.
1105:order at most 2 of the complex Lie algebra.
4017:projective complex special orthogonal group
1417:of the fundamental group of some Lie group
839: > 1, when it is isomorphic to the
1266:Classification of simple real Lie algebras
723:
709:
608:Particle physics and representation theory
150:
6687:
6455:
6318:
6102:
6037:
5916:
5885:
5827:
5757:
4593:Grassmannian of maximal positive definite
2171:
2148:
2106:
2074:
2046:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1019:in even dimension. These have the matrix
941:: A connected Lie group is simple if its
903:
874:
813:
775:
752:which does not have nontrivial connected
124:Learn how and when to remove this message
27:. For groups of dimension at most 3, see
6908:
4430:compatible with the Hermitian structure
1960:. For example, the group associated to G
4983:in which case it is quaternion-Kähler.
4961:in which case it is quaternion-Kähler.
3713:projective complex special linear group
1972:is the automorphism group of a certain
972:and related geometries in the sense of
850:, and this work was later perfected by
575:Representations of classical Lie groups
7015:
6834:
6173:Quaternionic subalgebras of octonions
1246:
1058:
6940:Representation Theory: A First Course
6271:Hermitian, quasi-split, quaternionic
5971:Hermitian, quasi-split, quaternionic
1843:)/{I, −I}. As with the B series, SO(2
1684:
5638:Simple Lie groups of small dimension
2932:, in which case it is the 2-sphere.
1858:is two isolated nodes, the same as A
1537:{\displaystyle K\subset \pi _{1}(G)}
1410:{\displaystyle K\subset \pi _{1}(G)}
428:Lie group–Lie algebra correspondence
62:adding citations to reliable sources
33:
7002:Cambridge University Press, 2003.
5266:Quaternionic quadratic forms on R.
4981:is 1, quaternionic hyperbolic space
4959:is 1, quaternionic projective space
1664:
1362:Classification of simple Lie groups
1339:
1295:
1108:
13:
6965:
5121:Grassmannian of positive definite
4765:Grassmannian of positive definite
2133:
1213:{\displaystyle \mathbb {R,C,H,O} }
14:
7039:
3162:Hermitian. Complex structures on
3155:Hermitian. Complex structures of
2925:or set of RP in CP. Hermitian if
2490:Hermitian. Complex structures of
1964:is the automorphism group of the
1885:In addition to the four families
1071:Lie algebra of a simple Lie group
6700:{\displaystyle \mathbb {H} ^{7}}
6468:{\displaystyle \mathbb {H} ^{6}}
6331:{\displaystyle \mathbb {H} ^{5}}
6115:{\displaystyle \mathbb {H} ^{2}}
6050:{\displaystyle \mathbb {H} ^{4}}
5929:{\displaystyle \mathbb {C} ^{3}}
5898:{\displaystyle \mathbb {R} ^{3}}
5840:{\displaystyle \mathbb {H} ^{3}}
5770:{\displaystyle \mathbb {H} ^{2}}
2292:projective special unitary group
2087:{\displaystyle \mathbb {R} ^{*}}
1872:quaternions and spatial rotation
1714:
1675:{\displaystyle {\mathfrak {g}}.}
1470:{\displaystyle {\tilde {G}}^{K}}
1375:. Given a (nontrivial) subgroup
1053:
38:
2897:projective special linear group
1968:, and the group associated to F
1371:of any Lie group is a discrete
1347:{\displaystyle {\mathfrak {g}}}
1303:{\displaystyle {\mathfrak {g}}}
960:Simple Lie groups include many
932:
841:projective special linear group
49:needs additional citations for
4271:Order 2 (complex conjugation)
4177:Order 2 (complex conjugation)
4130:Order 2 (complex conjugation)
3897:Order 2 (complex conjugation)
3805:Order 2 (complex conjugation)
2135:
1726:Overview of the classification
1639:
1587:
1581:
1558:
1531:
1525:
1455:
1404:
1398:
1142:by a maximal compact subgroup
1096:is a product of two copies of
980:. It emerged in the course of
623:Galilean group representations
618:Poincaré group representations
1:
6902:
5745:Split, Hermitian, hyperbolic
4435:quaternionic hyperbolic space
2028:Dimension of symmetric space
857:
613:Lorentz group representations
580:Theorem of the highest weight
294:
284:
274:
264:
2178:{\displaystyle \mathbb {R} }
2155:{\displaystyle \mathbb {R} }
2113:{\displaystyle \mathbb {R} }
2053:{\displaystyle \mathbb {R} }
1648:{\displaystyle {\tilde {G}}}
1437:, one can use the theory of
910:{\displaystyle \mathbb {R} }
881:{\displaystyle \mathbb {R} }
820:{\displaystyle \mathbb {R} }
782:{\displaystyle \mathbb {R} }
7:
6855:
5707:= Sp(1) = SU(2)=Spin(3), SO
5322:Quasi-split but not split.
4426:Quaternionic structures on
1792:unitary symplectic matrices
1176:
955:simple as an abstract group
762:Riemannian symmetric spaces
10:
7044:
3591:
3587:
2761:
2197:
2193:
2010:
2006:
1688:
1158:Hermitian symmetric spaces
1112:
1062:
565:Lie algebra representation
18:
6948:10.1007/978-1-4612-0979-9
6250:Complex projective space
6247:Complex hyperbolic space
6025:Hyperbolic, quaternionic
5977:Complex projective plane
5974:Complex hyperbolic plane
4300:
4293:
3992:Noncyclic of order 4 for
3600:
2770:
2206:
2025:Outer automorphism group
2019:
1825:special orthogonal groups
1767:special orthogonal groups
1441:to construct a new group
1084:, of types "ABCDEFG". If
1017:special orthogonal groups
999:As a counterexample, the
6914:Exceptional Lie Algebras
6637:Hermitian, quaternionic
6090:Siegel upper half space
5876:Euclidean structures on
5416:Cayley projective planes
5184:| ≤ 2, quasi-split
5159:is 4, quaternion-Kähler
5118:is 4, quaternion-Kähler
4803:is 4, quaternion-Kähler
4762:is 4, quaternion-Kähler
4609:is 2, quaternion-Kähler
4446:complex hyperbolic space
3811:special orthogonal group
3287:special orthogonal group
3003:special orthogonal group
2935:Euclidean structures on
2757:
2467:compact symplectic group
2368:special orthogonal group
2127:
560:Lie group representation
4783:is 1, Hyperbolic space
2001:
1231:In the symbols such as
1003:is neither simple, nor
888:is a simple Lie group.
585:Borel–Weil–Bott theorem
6998:Fuchs and Schweigert,
6701:
6469:
6332:
6116:
6093:Complex structures on
6051:
5930:
5899:
5841:
5771:
5139:is 1, Hyperbolic Space
5098:is 1, Projective space
4742:is 1, Projective space
3283:identity component of
3000:identity component of
2179:
2156:
2114:
2088:
2054:
1676:
1649:
1616:
1596:
1538:
1491:
1471:
1431:
1411:
1367:One can show that the
1354:and which has trivial
1348:
1324:
1304:
1214:
1036:
925:, whose center is the
911:
882:
821:
783:
483:Semisimple Lie algebra
438:Adjoint representation
29:Bianchi classification
6702:
6470:
6333:
6117:
6052:
5931:
5900:
5842:
5772:
5108:is 2 ; Hermitian
4083:Order 4 (non-cyclic)
3985:Order 4 (cyclic when
2530:Order 4 (cyclic when
2180:
2157:
2115:
2089:
2055:
1738:special unitary group
1677:
1650:
1617:
1597:
1539:
1492:
1472:
1432:
1412:
1349:
1330:whose Lie algebra is
1325:
1305:
1215:
1150:by the same subgroup
1037:
912:
883:
822:
784:
552:Representation theory
6765:Split, quaternionic
6682:
6514:Split, quaternionic
6450:
6313:
6167:Split, quaternionic
6097:
6032:
5911:
5880:
5822:
5752:
5457:Non-cyclic, order 4
2994:Non-cyclic, order 4
2167:
2144:
2102:
2069:
2042:
1870:multiplication; see
1659:
1630:
1606:
1548:
1506:
1481:
1445:
1421:
1379:
1334:
1314:
1290:
1184:
1023:
1001:general linear group
962:classical Lie groups
899:
870:
809:
799:special linear group
771:
58:improve this article
6892:Table of Lie groups
6835:Simply laced groups
6187:SU(4) = Spin(6), SO
5991:Sp(2) = Spin(5), SO
5907:Real structures on
5575:Quaternion-Kähler.
5572:Quaternion-Kähler.
5475:Quaternion-Kähler.
5472:Quaternion-Kähler.
5319:Quaternion-Kähler.
5316:Quaternion-Kähler.
4816:| ≤ 1, split.
3901:projective complex
3757:with fixed volume.
3752:Hermitian forms on
3701:, 4 (noncyclic) if
2921:Real structures on
2869:Infinite cyclic if
2561:projective special
1958:exceptional objects
1247:Full classification
1059:Simple Lie algebras
994:theoretical physics
970:projective geometry
758:simple Lie algebras
697:Table of Lie groups
538:Compact Lie algebra
25:Table of Lie groups
21:theoretical physics
6973:Einstein manifolds
6887:Catastrophe theory
6841:simply laced group
6697:
6465:
6328:
6112:
6047:
5926:
5895:
5837:
5767:
4319:Outer automorphism
3619:Outer automorphism
2789:Outer automorphism
2220:Outer automorphism
2175:
2152:
2110:
2084:
2050:
1701:Peter–Weyl theorem
1685:Compact Lie groups
1672:
1645:
1612:
1592:
1534:
1487:
1467:
1427:
1407:
1344:
1320:
1300:
1210:
1065:simple Lie algebra
1035:{\displaystyle -I}
1032:
990:exceptional groups
966:spherical geometry
939:Lie correspondence
907:
878:
817:
779:
735:In mathematics, a
469:Affine Lie algebra
459:Simple Lie algebra
200:Special orthogonal
73:"Simple Lie group"
6957:978-1-4612-0979-9
6882:Kac–Moody algebra
6832:
6831:
6678:Hyperbolic space
6548:Split, hermitian
6446:Hyperbolic space
6309:Hyperbolic space
6087:Split, Hermitian
6028:Hyperbolic space
5818:Hyperbolic space
5748:Hyperbolic plane
5635:
5634:
4587:quaternion-Kähler
4291:
4290:
3594:Complex Lie group
3585:
3584:
2764:Split Lie algebra
2755:
2754:
2125:
2124:
1812:metaplectic group
1642:
1615:{\displaystyle G}
1561:
1500:metaplectic group
1490:{\displaystyle K}
1458:
1430:{\displaystyle G}
1373:commutative group
1369:fundamental group
1323:{\displaystyle G}
1134:simple Lie group
1011:has a degenerate
927:fundamental group
733:
732:
533:Split Lie algebra
496:Cartan subalgebra
358:
357:
249:Simple Lie groups
134:
133:
126:
108:
7035:
6961:
6927:
6910:Jacobson, Nathan
6706:
6704:
6703:
6698:
6696:
6695:
6690:
6474:
6472:
6471:
6466:
6464:
6463:
6458:
6337:
6335:
6334:
6329:
6327:
6326:
6321:
6121:
6119:
6118:
6113:
6111:
6110:
6105:
6056:
6054:
6053:
6048:
6046:
6045:
6040:
5935:
5933:
5932:
5927:
5925:
5924:
5919:
5904:
5902:
5901:
5896:
5894:
5893:
5888:
5846:
5844:
5843:
5838:
5836:
5835:
5830:
5776:
5774:
5773:
5768:
5766:
5765:
5760:
5657:Symmetric space
5646:
5645:
5508:Infinite cyclic
5353:Infinite cyclic
5304:Cyclic, order 6
5237:Infinite cyclic
5183:
5171:
5080:Grassmannian of
4941:Grassmannian of
4815:
4724:Grassmannian of
4631:
4567:Grassmannian of
4455:
4443:
4339:symmetric space
4334:symmetric space
4329:symmetric space
4298:
4297:
4012:
3998:
3903:symplectic group
3707:
3700:
3691:
3639:symmetric space
3634:symmetric space
3629:symmetric space
3598:
3597:
3416:Cyclic, order 4
3280:
3266:
3103:symplectic group
3094:Infinite cyclic
2945:
2931:
2809:symmetric space
2804:symmetric space
2799:symmetric space
2768:
2767:
2563:orthogonal group
2557:
2543:
2513:
2441:
2343:
2302:
2286:
2279:
2270:
2247:
2204:
2203:
2184:
2182:
2181:
2176:
2174:
2161:
2159:
2158:
2153:
2151:
2139:
2119:
2117:
2116:
2111:
2109:
2093:
2091:
2090:
2085:
2083:
2082:
2077:
2059:
2057:
2056:
2051:
2049:
2031:Symmetric space
2017:
2016:
1995:
1994:
1990:
1987:
1878:is the same as A
1718:
1681:
1679:
1678:
1673:
1668:
1667:
1654:
1652:
1651:
1646:
1644:
1643:
1635:
1624:simply connected
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1591:
1590:
1580:
1579:
1563:
1562:
1554:
1543:
1541:
1540:
1535:
1524:
1523:
1496:
1494:
1493:
1488:
1476:
1474:
1473:
1468:
1466:
1465:
1460:
1459:
1451:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1397:
1396:
1353:
1351:
1350:
1345:
1343:
1342:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1299:
1298:
1219:
1217:
1216:
1211:
1209:
1172:
1109:Symmetric spaces
1048:reductive groups
1041:
1039:
1038:
1033:
978:Erlangen program
916:
914:
913:
908:
906:
887:
885:
884:
879:
877:
826:
824:
823:
818:
816:
788:
786:
785:
780:
778:
754:normal subgroups
737:simple Lie group
725:
718:
711:
668:Claude Chevalley
525:Complexification
368:Other Lie groups
254:
253:
162:Classical groups
154:
136:
135:
129:
122:
118:
115:
109:
107:
66:
42:
34:
7043:
7042:
7038:
7037:
7036:
7034:
7033:
7032:
7013:
7012:
6968:
6966:Further reading
6958:
6932:Fulton, William
6924:
6905:
6858:
6837:
6820:
6801:
6780:
6761:
6742:
6723:
6691:
6686:
6685:
6683:
6680:
6679:
6671:
6652:
6634:
6616:
6597:
6563:
6544:
6510:
6491:
6459:
6454:
6453:
6451:
6448:
6447:
6439:
6420:
6402:
6390:
6386:
6360:
6322:
6317:
6316:
6314:
6311:
6310:
6302:
6298:
6267:
6213:
6209:
6190:
6164:
6140:
6106:
6101:
6100:
6098:
6095:
6094:
6083:
6079:
6041:
6036:
6035:
6033:
6030:
6029:
6021:
6017:
5994:
5920:
5915:
5914:
5912:
5909:
5908:
5889:
5884:
5883:
5881:
5878:
5877:
5869:
5831:
5826:
5825:
5823:
5820:
5819:
5811:
5807:
5803:
5799:
5761:
5756:
5755:
5753:
5750:
5749:
5741:
5737:
5733:
5710:
5681:
5640:
5606:
5602:
5587:
5557:
5550:
5535:
5502:
5487:
5454:
5448:
5433:
5399:
5384:
5371:
5366:
5347:
5332:
5301:
5295:
5280:
5278:
5265:
5260:
5231:
5198:
5196:
5175:
5173:
5163:
5150:
5149:is 2, Hermitian
5140:
5130:
5109:
5099:
5089:
5005:
4999:
4997:
4982:
4972:
4960:
4950:
4931:
4896:
4888:
4848:
4836:
4830:
4828:
4807:
4794:
4793:is 2, Hermitian
4784:
4774:
4753:
4752:is 2; Hermitian
4743:
4733:
4655:
4649:
4647:
4623:
4621:
4600:
4594:
4592:
4576:
4566:
4535:
4526:
4491:
4478:
4472:
4470:
4449:
4438:
4403:
4390:
4358:
4356:
4338:
4333:
4328:
4320:
4315:
4310:
4309:Maximal compact
4296:
4285:
4265:
4250:
4238:
4218:
4203:
4191:
4171:
4156:
4144:
4124:
4109:
4097:
4077:
4062:
4050:
4026:
4019:
4007:
4005:
3993:
3982:
3951:
3937:
3913:
3906:
3891:
3860:
3846:
3822:
3814:
3799:
3768:
3749:
3723:
3716:
3702:
3695:
3686:
3682:
3651:
3638:
3633:
3628:
3620:
3615:
3610:
3609:Maximal compact
3603:Real dimension
3596:
3590:
3561:
3554:
3539:
3509:
3502:
3487:
3455:
3440:
3413:
3398:
3370:
3355:
3344:
3337:
3331:is the same as
3330:
3323:
3317:is the same as
3316:
3290:
3275:
3273:
3261:
3215:
3202:
3195:
3189:is the same as
3188:
3181:
3175:is the same as
3174:
3113:
3106:
3088:
3057:
3044:
3038:is the same as
3037:
3006:
2960:
2940:
2939:. Hermitian if
2926:
2907:
2900:
2887:
2874:
2866:
2855:
2824:
2808:
2803:
2798:
2790:
2785:
2780:
2779:Maximal compact
2766:
2760:
2733:
2706:
2679:
2652:
2625:
2614:
2607:
2601:is the same as
2600:
2593:
2587:is the same as
2586:
2573:
2566:
2552:
2550:
2538:
2508:
2506:
2470:
2436:
2434:
2421:
2415:is the same as
2414:
2408:
2406:
2399:
2393:is the same as
2392:
2379:
2371:
2338:
2336:
2324:
2317:
2311:is the same as
2310:
2296:
2295:
2281:
2274:
2265:
2242:
2240:
2221:
2216:
2202:
2196:
2170:
2168:
2165:
2164:
2147:
2145:
2142:
2141:
2130:
2105:
2103:
2100:
2099:
2078:
2073:
2072:
2070:
2067:
2066:
2045:
2043:
2040:
2039:
2015:
2009:
2004:
1996:
1992:
1988:
1985:
1983:
1971:
1963:
1954:
1947:
1940:
1933:
1926:
1920:
1911:
1902:
1893:
1881:
1877:
1865:
1861:
1857:
1822:
1789:
1764:
1735:
1728:
1705:Wilhelm Killing
1693:
1687:
1663:
1662:
1660:
1657:
1656:
1634:
1633:
1631:
1628:
1627:
1607:
1604:
1603:
1575:
1571:
1564:
1553:
1552:
1551:
1549:
1546:
1545:
1519:
1515:
1507:
1504:
1503:
1482:
1479:
1478:
1461:
1450:
1449:
1448:
1446:
1443:
1442:
1439:covering spaces
1422:
1419:
1418:
1392:
1388:
1380:
1377:
1376:
1338:
1337:
1335:
1332:
1331:
1315:
1312:
1311:
1294:
1293:
1291:
1288:
1287:
1274:Satake diagrams
1260:Dynkin diagrams
1249:
1237:
1187:
1185:
1182:
1181:
1179:
1167:
1160:
1117:
1111:
1082:Dynkin diagrams
1067:
1061:
1056:
1024:
1021:
1020:
935:
923:universal cover
917:is not simple.
902:
900:
897:
896:
873:
871:
868:
867:
860:
848:Wilhelm Killing
812:
810:
807:
806:
795:group extension
774:
772:
769:
768:
729:
684:
683:
682:
653:Wilhelm Killing
637:
629:
628:
627:
602:
591:
590:
589:
554:
544:
543:
542:
529:
513:
491:Dynkin diagrams
485:
475:
474:
473:
455:
433:Exponential map
422:
412:
411:
410:
391:Conformal group
370:
360:
359:
351:
343:
335:
327:
319:
300:
290:
280:
270:
251:
241:
240:
239:
220:Special unitary
164:
130:
119:
113:
110:
67:
65:
55:
43:
32:
17:
12:
11:
5:
7041:
7031:
7030:
7025:
7011:
7010:
6996:
6982:
6967:
6964:
6963:
6962:
6956:
6928:
6922:
6904:
6901:
6900:
6899:
6894:
6889:
6884:
6879:
6874:
6869:
6867:Coxeter matrix
6864:
6857:
6854:
6849:Dynkin diagram
6836:
6833:
6830:
6829:
6827:
6825:
6822:
6818:
6815:
6811:
6810:
6808:
6806:
6803:
6799:
6794:
6790:
6789:
6787:
6785:
6782:
6778:
6775:
6771:
6770:
6768:
6766:
6763:
6759:
6756:
6752:
6751:
6749:
6747:
6744:
6740:
6737:
6733:
6732:
6730:
6728:
6725:
6721:
6718:
6714:
6713:
6707:
6694:
6689:
6676:
6673:
6669:
6666:
6662:
6661:
6659:
6657:
6654:
6650:
6647:
6643:
6642:
6640:
6638:
6635:
6632:
6629:
6625:
6624:
6622:
6620:
6617:
6614:
6611:
6607:
6606:
6604:
6602:
6599:
6595:
6592:
6588:
6587:
6585:
6583:
6580:
6577:
6573:
6572:
6570:
6568:
6565:
6561:
6558:
6554:
6553:
6551:
6549:
6546:
6542:
6539:
6535:
6534:
6532:
6530:
6527:
6524:
6520:
6519:
6517:
6515:
6512:
6508:
6505:
6501:
6500:
6498:
6496:
6493:
6489:
6486:
6482:
6481:
6475:
6462:
6457:
6444:
6441:
6437:
6434:
6430:
6429:
6427:
6425:
6422:
6418:
6415:
6411:
6410:
6407:
6404:
6400:
6397:
6395:
6392:
6388:
6384:
6381:
6377:
6376:
6373:
6370:
6367:
6365:
6362:
6358:
6355:
6351:
6350:
6347:
6344:
6338:
6325:
6320:
6307:
6304:
6300:
6296:
6293:
6289:
6288:
6285:
6282:
6275:
6272:
6269:
6265:
6262:
6258:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6235:
6234:
6231:
6228:
6221:
6218:
6215:
6211:
6207:
6204:
6200:
6199:
6197:
6195:
6192:
6188:
6185:
6181:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6157:
6153:
6152:
6150:
6148:
6146:
6144:
6141:
6138:
6133:
6129:
6128:
6125:
6122:
6109:
6104:
6091:
6088:
6085:
6081:
6077:
6074:
6070:
6069:
6066:
6063:
6057:
6044:
6039:
6026:
6023:
6019:
6015:
6012:
6008:
6007:
6005:
6003:
6001:
5999:
5996:
5992:
5989:
5985:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5962:
5961:
5959:
5957:
5955:
5953:
5950:
5947:
5943:
5942:
5939:
5936:
5923:
5918:
5905:
5892:
5887:
5874:
5871:
5867:
5864:
5860:
5859:
5856:
5853:
5847:
5834:
5829:
5816:
5813:
5809:
5805:
5801:
5797:
5794:
5790:
5789:
5786:
5783:
5777:
5764:
5759:
5746:
5743:
5739:
5735:
5731:
5728:
5724:
5723:
5721:
5719:
5717:
5715:
5712:
5708:
5702:
5698:
5697:
5694:
5691:
5689:
5686:
5683:
5682:(ℝ) = Spin(2)
5679:
5672:
5668:
5667:
5664:
5661:
5658:
5655:
5653:
5650:
5639:
5636:
5633:
5632:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5604:
5600:
5595:
5592:
5589:
5585:
5579:
5578:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5548:
5543:
5540:
5537:
5533:
5527:
5526:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5500:
5495:
5492:
5489:
5485:
5479:
5478:
5476:
5473:
5470:
5467:
5461:
5458:
5455:
5452:
5446:
5441:
5438:
5435:
5431:
5425:
5424:
5422:
5419:
5412:
5409:
5406:
5403:
5400:
5397:
5392:
5389:
5386:
5382:
5376:
5375:
5373:
5368:
5363:
5360:
5357:
5354:
5351:
5345:
5340:
5337:
5334:
5330:
5324:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5293:
5288:
5285:
5282:
5281:(quasi-split)
5276:
5270:
5269:
5267:
5262:
5257:
5247:
5241:
5238:
5235:
5226:
5221:
5214:
5204:
5192:
5186:
5185:
5160:
5119:
5078:
5073:
5062:
5061:≥ 3, order 8.
5051:
5049:
5038:
5027:
5017:
4993:
4987:
4986:
4984:
4962:
4939:
4933:
4922:
4919:
4900:
4897:
4892:
4884:
4879:
4868:
4858:
4824:
4818:
4817:
4804:
4763:
4722:
4717:
4704:
4702:
4700:
4689:
4678:
4668:
4643:
4637:
4636:
4610:
4589:
4560:
4554:
4543:
4541:
4539:
4530:
4521:
4516:
4511:
4501:
4466:
4460:
4459:
4457:
4448:(of dimension
4437:(of dimension
4431:
4424:
4413:
4399:
4396:
4394:
4391:
4386:
4381:
4375:
4364:
4350:
4344:
4343:
4340:
4335:
4330:
4325:
4322:
4317:
4312:
4307:
4304:
4301:
4295:
4292:
4289:
4288:
4286:
4283:
4279:Compact group
4277:
4274:
4272:
4269:
4266:
4263:
4258:
4255:
4252:
4248:
4242:
4241:
4239:
4236:
4232:Compact group
4230:
4227:
4225:
4222:
4219:
4216:
4211:
4208:
4205:
4201:
4195:
4194:
4192:
4189:
4185:Compact group
4183:
4180:
4178:
4175:
4172:
4169:
4164:
4161:
4158:
4154:
4148:
4147:
4145:
4142:
4138:Compact group
4136:
4133:
4131:
4128:
4125:
4122:
4117:
4114:
4111:
4107:
4101:
4100:
4098:
4095:
4091:Compact group
4089:
4086:
4084:
4081:
4078:
4075:
4070:
4067:
4064:
4060:
4054:
4053:
4051:
4048:
4044:Compact group
4042:
4032:
4021:
4014:
4003:
3990:
3983:
3978:
3973:
3968:
3957:
3947:
3941:
3940:
3938:
3935:
3931:Compact group
3929:
3919:
3908:
3898:
3895:
3892:
3887:
3882:
3877:
3866:
3856:
3850:
3849:
3847:
3844:
3840:Compact group
3838:
3828:
3816:
3806:
3803:
3800:
3795:
3790:
3785:
3774:
3766:
3760:
3759:
3750:
3745:
3741:Compact group
3739:
3729:
3718:
3709:
3692:
3685:Cyclic, order
3683:
3678:
3673:
3668:
3657:
3647:
3641:
3640:
3635:
3630:
3625:
3622:
3617:
3612:
3607:
3604:
3601:
3589:
3586:
3583:
3582:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3552:
3547:
3544:
3541:
3537:
3531:
3530:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3500:
3495:
3492:
3489:
3485:
3479:
3478:
3472:
3470:
3468:
3465:
3462:
3459:
3456:
3453:
3448:
3445:
3442:
3438:
3432:
3431:
3429:
3427:
3425:
3422:
3420:
3417:
3414:
3411:
3406:
3403:
3400:
3396:
3390:
3389:
3387:
3385:
3383:
3380:
3377:
3374:
3371:
3368:
3363:
3360:
3357:
3353:
3347:
3346:
3342:
3335:
3328:
3321:
3314:
3309:
3307:
3305:
3300:
3281:
3271:
3258:
3247:
3236:
3231:
3221:
3211:
3205:
3204:
3200:
3193:
3186:
3179:
3172:
3167:
3160:
3153:
3143:
3108:
3098:
3095:
3092:
3083:
3078:
3073:
3063:
3053:
3047:
3046:
3042:
3035:
3030:
3028:
3026:
3016:
2998:
2995:
2992:
2981:
2976:
2966:
2956:
2950:
2949:
2947:
2933:
2919:
2909:
2902:
2893:
2880:
2867:
2860:
2850:
2845:
2840:
2830:
2820:
2814:
2813:
2810:
2805:
2800:
2795:
2792:
2787:
2782:
2777:
2774:
2771:
2759:
2756:
2753:
2752:
2749:
2747:
2744:
2741:
2738:
2735:
2731:
2725:
2724:
2722:
2720:
2717:
2714:
2711:
2708:
2704:
2698:
2697:
2695:
2693:
2690:
2687:
2684:
2681:
2677:
2671:
2670:
2668:
2666:
2663:
2660:
2657:
2654:
2650:
2644:
2643:
2641:
2639:
2636:
2633:
2630:
2627:
2623:
2617:
2616:
2612:
2605:
2598:
2591:
2584:
2579:
2568:
2558:
2548:
2535:
2528:
2525:
2515:
2502:
2496:
2495:
2488:
2462:
2459:
2456:
2453:
2443:
2430:
2424:
2423:
2419:
2412:
2404:
2397:
2390:
2385:
2373:
2364:
2361:
2358:
2355:
2345:
2332:
2326:
2325:
2322:
2315:
2308:
2303:
2288:
2271:
2264:Cyclic, order
2262:
2259:
2249:
2236:
2230:
2229:
2226:
2223:
2218:
2213:
2210:
2207:
2195:
2192:
2191:
2190:
2173:
2150:
2129:
2126:
2123:
2122:
2120:
2108:
2097:
2094:
2081:
2076:
2064:
2061:
2048:
2036:
2035:
2032:
2029:
2026:
2023:
2020:
2008:
2005:
2003:
2000:
1982:
1974:Albert algebra
1969:
1961:
1952:
1945:
1938:
1931:
1924:
1916:
1907:
1898:
1889:
1879:
1875:
1863:
1859:
1855:
1818:
1785:
1760:
1731:
1727:
1724:
1689:Main article:
1686:
1683:
1671:
1666:
1641:
1638:
1611:
1589:
1586:
1583:
1578:
1574:
1570:
1567:
1560:
1557:
1533:
1530:
1527:
1522:
1518:
1514:
1511:
1486:
1464:
1457:
1454:
1426:
1406:
1403:
1400:
1395:
1391:
1387:
1384:
1365:
1364:
1359:
1341:
1319:
1297:
1281:
1263:
1248:
1245:
1235:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1178:
1175:
1159:
1156:
1113:Main article:
1110:
1107:
1063:Main article:
1060:
1057:
1055:
1052:
1031:
1028:
982:classification
934:
931:
905:
876:
859:
856:
815:
777:
731:
730:
728:
727:
720:
713:
705:
702:
701:
700:
699:
694:
686:
685:
681:
680:
675:
673:Harish-Chandra
670:
665:
660:
655:
650:
648:Henri Poincaré
645:
639:
638:
635:
634:
631:
630:
626:
625:
620:
615:
610:
604:
603:
598:Lie groups in
597:
596:
593:
592:
588:
587:
582:
577:
572:
567:
562:
556:
555:
550:
549:
546:
545:
541:
540:
535:
530:
528:
527:
522:
516:
514:
512:
511:
506:
500:
498:
493:
487:
486:
481:
480:
477:
476:
472:
471:
466:
461:
456:
454:
453:
448:
442:
440:
435:
430:
424:
423:
418:
417:
414:
413:
409:
408:
403:
398:
396:Diffeomorphism
393:
388:
383:
378:
372:
371:
366:
365:
362:
361:
356:
355:
354:
353:
349:
345:
341:
337:
333:
329:
325:
321:
317:
310:
309:
305:
304:
303:
302:
296:
292:
286:
282:
276:
272:
266:
259:
258:
252:
247:
246:
243:
242:
238:
237:
227:
217:
207:
197:
187:
180:Special linear
177:
170:General linear
166:
165:
160:
159:
156:
155:
147:
146:
132:
131:
46:
44:
37:
15:
9:
6:
4:
3:
2:
7040:
7029:
7026:
7024:
7021:
7020:
7018:
7009:
7008:0-521-54119-0
7005:
7001:
6997:
6995:
6994:0-8218-2848-7
6991:
6987:
6983:
6981:
6980:0-387-15279-2
6977:
6974:
6970:
6969:
6959:
6953:
6949:
6945:
6941:
6937:
6933:
6929:
6925:
6923:0-8247-1326-5
6919:
6916:. CRC Press.
6915:
6911:
6907:
6906:
6898:
6895:
6893:
6890:
6888:
6885:
6883:
6880:
6878:
6877:Coxeter group
6875:
6873:
6870:
6868:
6865:
6863:
6862:Cartan matrix
6860:
6859:
6853:
6850:
6846:
6842:
6828:
6826:
6823:
6816:
6813:
6812:
6809:
6807:
6804:
6798:
6795:
6792:
6791:
6788:
6786:
6783:
6776:
6773:
6772:
6769:
6767:
6764:
6757:
6754:
6753:
6750:
6748:
6745:
6738:
6735:
6734:
6731:
6729:
6726:
6719:
6716:
6715:
6712:
6708:
6692:
6677:
6674:
6667:
6664:
6663:
6660:
6658:
6655:
6648:
6645:
6644:
6641:
6639:
6636:
6630:
6627:
6626:
6623:
6621:
6618:
6612:
6609:
6608:
6605:
6603:
6600:
6593:
6590:
6589:
6586:
6584:
6581:
6578:
6575:
6574:
6571:
6569:
6567:Quaternionic
6566:
6559:
6556:
6555:
6552:
6550:
6547:
6540:
6537:
6536:
6533:
6531:
6528:
6525:
6522:
6521:
6518:
6516:
6513:
6506:
6503:
6502:
6499:
6497:
6494:
6487:
6484:
6483:
6480:
6476:
6460:
6445:
6442:
6435:
6432:
6431:
6428:
6426:
6423:
6416:
6413:
6412:
6408:
6405:
6398:
6396:
6393:
6382:
6379:
6378:
6374:
6371:
6368:
6366:
6363:
6356:
6353:
6352:
6348:
6345:
6343:
6339:
6323:
6308:
6305:
6294:
6291:
6290:
6286:
6283:
6280:
6277:Grassmannian
6276:
6273:
6270:
6263:
6260:
6259:
6255:
6252:
6249:
6246:
6243:
6240:
6237:
6236:
6232:
6229:
6226:
6223:Grassmannian
6222:
6219:
6216:
6205:
6202:
6201:
6198:
6196:
6193:
6186:
6183:
6182:
6178:
6175:
6172:
6169:
6166:
6161:
6158:
6155:
6154:
6151:
6149:
6147:
6145:
6142:
6137:
6134:
6131:
6130:
6126:
6123:
6107:
6092:
6089:
6086:
6075:
6072:
6071:
6067:
6064:
6062:
6058:
6042:
6027:
6024:
6013:
6010:
6009:
6006:
6004:
6002:
6000:
5997:
5990:
5987:
5986:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5963:
5960:
5958:
5956:
5954:
5951:
5948:
5945:
5944:
5940:
5937:
5921:
5906:
5890:
5875:
5872:
5865:
5862:
5861:
5857:
5854:
5852:
5848:
5832:
5817:
5814:
5795:
5792:
5791:
5787:
5784:
5782:
5778:
5762:
5747:
5744:
5729:
5726:
5725:
5722:
5720:
5718:
5716:
5713:
5711:(ℝ) = PSU(2)
5706:
5703:
5700:
5699:
5695:
5692:
5690:
5687:
5684:
5677:
5673:
5670:
5669:
5665:
5662:
5660:Compact dual
5659:
5656:
5654:
5651:
5648:
5647:
5644:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5599:
5596:
5593:
5590:
5584:
5581:
5580:
5577:
5574:
5571:
5568:
5565:
5562:
5559:
5554:
5547:
5544:
5541:
5538:
5532:
5529:
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5505:
5499:
5496:
5493:
5490:
5484:
5481:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5451:
5445:
5442:
5439:
5436:
5430:
5427:
5426:
5423:
5420:
5417:
5413:
5410:
5407:
5404:
5401:
5396:
5393:
5390:
5387:
5381:
5378:
5377:
5374:
5369:
5364:
5361:
5358:
5355:
5352:
5350:
5344:
5341:
5338:
5335:
5329:
5326:
5325:
5321:
5318:
5315:
5312:
5309:
5306:
5303:
5298:
5292:
5289:
5286:
5283:
5275:
5272:
5271:
5268:
5263:
5258:
5255:
5251:
5248:
5245:
5242:
5239:
5236:
5234:
5229:
5225:
5222:
5219:
5215:
5212:
5208:
5205:
5202:
5195:
5191:
5188:
5187:
5182:
5178:
5170:
5166:
5161:
5158:
5154:
5148:
5144:
5138:
5134:
5128:
5124:
5120:
5117:
5113:
5107:
5103:
5097:
5093:
5087:
5083:
5079:
5077:
5074:
5071:
5067:
5063:
5060:
5056:
5052:
5050:
5047:
5043:
5039:
5036:
5032:
5028:
5025:
5021:
5018:
5016:
5012:
5008:
5003:
4996:
4992:
4989:
4988:
4985:
4980:
4976:
4970:
4966:
4963:
4958:
4954:
4948:
4944:
4940:
4938:
4934:
4930:
4926:
4920:
4917:
4913:
4909:
4905:
4901:
4898:
4895:
4891:
4887:
4883:
4880:
4877:
4873:
4869:
4866:
4862:
4859:
4856:
4852:
4847:
4843:
4839:
4834:
4827:
4823:
4820:
4819:
4814:
4810:
4805:
4802:
4798:
4792:
4788:
4782:
4778:
4772:
4768:
4764:
4761:
4757:
4751:
4747:
4741:
4737:
4731:
4727:
4723:
4721:
4718:
4716:
4714:
4710:
4705:
4703:
4701:
4698:
4694:
4690:
4687:
4683:
4679:
4676:
4672:
4669:
4666:
4662:
4658:
4653:
4646:
4642:
4639:
4638:
4635:
4630:
4626:
4619:
4615:
4611:
4608:
4604:
4598:
4595:subspaces of
4590:
4588:
4584:
4580:
4574:
4571:subspaces of
4570:
4564:
4561:
4559:
4555:
4552:
4548:
4544:
4542:
4540:
4538:
4533:
4529:
4524:
4520:
4517:
4515:
4512:
4509:
4505:
4502:
4499:
4495:
4489:
4485:
4481:
4476:
4469:
4465:
4462:
4461:
4458:
4453:
4447:
4441:
4436:
4432:
4429:
4425:
4422:
4418:
4414:
4411:
4407:
4402:
4397:
4395:
4392:
4389:
4385:
4382:
4379:
4376:
4373:
4369:
4365:
4362:
4354:
4349:
4346:
4345:
4341:
4336:
4331:
4326:
4323:
4318:
4313:
4308:
4305:
4302:
4299:
4287:
4282:
4278:
4275:
4273:
4270:
4267:
4262:
4259:
4256:
4253:
4247:
4244:
4243:
4240:
4235:
4231:
4228:
4226:
4223:
4220:
4215:
4212:
4209:
4206:
4200:
4197:
4196:
4193:
4188:
4184:
4181:
4179:
4176:
4173:
4168:
4165:
4162:
4159:
4153:
4150:
4149:
4146:
4141:
4137:
4134:
4132:
4129:
4126:
4121:
4118:
4115:
4112:
4106:
4103:
4102:
4099:
4094:
4090:
4087:
4085:
4082:
4079:
4074:
4071:
4068:
4065:
4059:
4056:
4055:
4052:
4047:
4043:
4040:
4036:
4033:
4030:
4025:
4018:
4015:
4010:
4002:
3996:
3991:
3988:
3984:
3981:
3977:
3974:
3972:
3969:
3966:
3962:
3958:
3956:≥ 4) complex
3955:
3950:
3946:
3943:
3942:
3939:
3934:
3930:
3927:
3923:
3920:
3917:
3912:
3905:
3904:
3899:
3896:
3893:
3890:
3886:
3883:
3881:
3878:
3875:
3871:
3867:
3865:≥ 3) complex
3864:
3859:
3855:
3852:
3851:
3848:
3843:
3839:
3836:
3832:
3829:
3826:
3820:
3813:
3812:
3807:
3804:
3801:
3798:
3794:
3791:
3789:
3786:
3783:
3779:
3775:
3773:≥ 2) complex
3772:
3765:
3762:
3761:
3758:
3755:
3751:
3748:
3744:
3740:
3737:
3733:
3730:
3727:
3721:
3715:
3714:
3710:
3705:
3698:
3693:
3689:
3684:
3681:
3677:
3674:
3672:
3669:
3666:
3662:
3658:
3656:≥ 1) complex
3655:
3650:
3646:
3643:
3642:
3636:
3631:
3626:
3623:
3618:
3613:
3608:
3605:
3602:
3599:
3595:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3558:
3551:
3548:
3545:
3542:
3536:
3533:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3506:
3499:
3496:
3493:
3490:
3484:
3481:
3480:
3477:
3473:
3471:
3469:
3466:
3463:
3460:
3457:
3452:
3449:
3446:
3443:
3437:
3434:
3433:
3430:
3428:
3426:
3423:
3421:
3418:
3415:
3410:
3407:
3404:
3401:
3395:
3392:
3391:
3388:
3386:
3384:
3381:
3378:
3375:
3372:
3367:
3364:
3361:
3358:
3352:
3349:
3348:
3341:
3334:
3327:
3320:
3313:
3310:
3308:
3306:
3304:
3301:
3298:
3294:
3289:
3288:
3282:
3278:
3270:
3264:
3259:
3256:
3252:
3248:
3245:
3241:
3237:
3235:
3232:
3229:
3225:
3222:
3219:
3214:
3210:
3207:
3206:
3199:
3192:
3185:
3178:
3171:
3168:
3165:
3161:
3158:
3154:
3151:
3147:
3144:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3112:
3105:
3104:
3099:
3096:
3093:
3091:
3086:
3082:
3079:
3077:
3074:
3071:
3067:
3064:
3061:
3056:
3052:
3049:
3048:
3041:
3034:
3031:
3029:
3027:
3024:
3020:
3017:
3014:
3010:
3005:
3004:
2999:
2996:
2993:
2990:
2986:
2982:
2980:
2977:
2974:
2970:
2967:
2964:
2959:
2955:
2952:
2951:
2948:
2943:
2938:
2934:
2929:
2924:
2920:
2917:
2913:
2910:
2905:
2899:
2898:
2894:
2891:
2885:
2881:
2878:
2872:
2868:
2864:
2859:
2853:
2849:
2846:
2844:
2841:
2838:
2834:
2831:
2828:
2823:
2819:
2816:
2815:
2811:
2806:
2801:
2796:
2793:
2788:
2783:
2778:
2775:
2772:
2769:
2765:
2750:
2748:
2745:
2742:
2739:
2736:
2730:
2727:
2726:
2723:
2721:
2718:
2715:
2712:
2709:
2703:
2700:
2699:
2696:
2694:
2691:
2688:
2685:
2682:
2676:
2673:
2672:
2669:
2667:
2664:
2661:
2658:
2655:
2649:
2646:
2645:
2642:
2640:
2637:
2634:
2631:
2628:
2622:
2619:
2618:
2611:
2604:
2597:
2590:
2583:
2580:
2577:
2572:
2565:
2564:
2559:
2555:
2547:
2541:
2536:
2533:
2529:
2526:
2523:
2519:
2516:
2511:
2505:
2501:
2498:
2497:
2493:
2489:
2486:
2482:
2478:
2474:
2469:
2468:
2463:
2460:
2457:
2454:
2451:
2447:
2444:
2439:
2433:
2429:
2426:
2425:
2418:
2411:
2403:
2396:
2389:
2386:
2383:
2377:
2370:
2369:
2365:
2362:
2359:
2356:
2353:
2349:
2346:
2341:
2335:
2331:
2328:
2327:
2321:
2314:
2307:
2304:
2300:
2294:
2293:
2289:
2284:
2277:
2272:
2268:
2263:
2260:
2257:
2253:
2250:
2245:
2239:
2235:
2232:
2231:
2227:
2224:
2219:
2214:
2211:
2208:
2205:
2201:
2200:Compact group
2188:
2138:
2137:
2132:
2131:
2121:
2098:
2095:
2079:
2065:
2062:
2038:
2037:
2033:
2030:
2027:
2024:
2021:
2018:
2014:
2013:Abelian group
1999:
1997:
1977:
1975:
1967:
1959:
1955:
1948:
1941:
1934:
1927:
1919:
1915:
1910:
1906:
1901:
1897:
1892:
1888:
1883:
1873:
1869:
1854:The diagram D
1852:
1850:
1846:
1842:
1838:
1834:
1832:
1826:
1821:
1815:
1813:
1809:
1805:
1801:
1799:
1793:
1788:
1782:
1780:
1776:
1774:
1768:
1763:
1757:
1755:
1753:
1747:
1745:
1739:
1734:
1723:
1719:
1717:
1712:
1710:
1706:
1702:
1698:
1692:
1682:
1669:
1636:
1625:
1609:
1584:
1576:
1572:
1568:
1565:
1555:
1528:
1520:
1516:
1512:
1509:
1501:
1484:
1462:
1452:
1440:
1424:
1401:
1393:
1389:
1385:
1382:
1374:
1370:
1363:
1360:
1357:
1317:
1285:
1282:
1279:
1278:Ichirô Satake
1275:
1271:
1267:
1264:
1261:
1257:
1254:
1253:
1252:
1244:
1240:
1234:
1229:
1227:
1223:
1174:
1170:
1164:
1155:
1153:
1149:
1145:
1141:
1137:
1133:
1128:
1124:
1120:
1116:
1106:
1103:
1099:
1095:
1091:
1087:
1083:
1078:
1076:
1072:
1066:
1054:Related ideas
1051:
1049:
1045:
1029:
1026:
1018:
1014:
1010:
1006:
1002:
997:
995:
991:
987:
983:
979:
975:
971:
967:
963:
958:
956:
952:
948:
944:
940:
930:
928:
924:
918:
894:
889:
865:
855:
853:
849:
844:
842:
838:
834:
830:
804:
800:
796:
792:
765:
763:
759:
755:
751:
748:
745:
742:
738:
726:
721:
719:
714:
712:
707:
706:
704:
703:
698:
695:
693:
690:
689:
688:
687:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
640:
633:
632:
624:
621:
619:
616:
614:
611:
609:
606:
605:
601:
595:
594:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
561:
558:
557:
553:
548:
547:
539:
536:
534:
531:
526:
523:
521:
518:
517:
515:
510:
507:
505:
502:
501:
499:
497:
494:
492:
489:
488:
484:
479:
478:
470:
467:
465:
462:
460:
457:
452:
449:
447:
444:
443:
441:
439:
436:
434:
431:
429:
426:
425:
421:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
373:
369:
364:
363:
352:
346:
344:
338:
336:
330:
328:
322:
320:
314:
313:
312:
311:
307:
306:
301:
299:
293:
291:
289:
283:
281:
279:
273:
271:
269:
263:
262:
261:
260:
256:
255:
250:
245:
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235:
231:
228:
225:
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218:
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211:
208:
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198:
195:
191:
188:
185:
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167:
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149:
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128:
125:
117:
106:
103:
99:
96:
92:
89:
85:
82:
78:
75: –
74:
70:
69:Find sources:
63:
59:
53:
52:
47:This article
45:
41:
36:
35:
30:
26:
22:
7028:Lie algebras
6999:
6985:
6972:
6942:. Springer.
6939:
6913:
6840:
6838:
6796:
6746:Quasi-split
6710:
6478:
6341:
6278:
6224:
6159:
6135:
6060:
5850:
5780:
5704:
5675:
5641:
5608:
5597:
5582:
5552:
5545:
5530:
5503:
5497:
5482:
5463:
5449:
5443:
5428:
5394:
5379:
5348:
5342:
5327:
5296:
5290:
5273:
5253:
5249:
5243:
5232:
5227:
5223:
5217:
5210:
5206:
5200:
5193:
5189:
5180:
5176:
5168:
5164:
5156:
5152:
5146:
5142:
5136:
5132:
5126:
5122:
5115:
5111:
5105:
5101:
5095:
5091:
5085:
5081:
5075:
5069:
5065:
5058:
5054:
5045:
5041:
5034:
5030:
5023:
5019:
5014:
5010:
5006:
5001:
4994:
4990:
4978:
4974:
4968:
4964:
4956:
4952:
4946:
4942:
4936:
4928:
4924:
4915:
4911:
4907:
4903:
4893:
4889:
4885:
4881:
4875:
4871:
4864:
4860:
4854:
4850:
4845:
4841:
4837:
4832:
4825:
4821:
4812:
4808:
4800:
4796:
4790:
4786:
4780:
4776:
4770:
4766:
4759:
4755:
4749:
4745:
4739:
4735:
4729:
4725:
4719:
4712:
4708:
4696:
4692:
4685:
4681:
4674:
4670:
4664:
4660:
4656:
4651:
4644:
4640:
4632:| ≤ 1,
4628:
4624:
4617:
4613:
4606:
4602:
4596:
4582:
4578:
4572:
4568:
4557:
4550:
4546:
4536:
4531:
4527:
4522:
4518:
4513:
4507:
4503:
4497:
4493:
4487:
4483:
4479:
4474:
4467:
4463:
4451:
4439:
4427:
4420:
4419:− 1)(2
4416:
4409:
4405:
4400:
4387:
4383:
4377:
4371:
4370:− 1)(2
4367:
4360:
4352:
4347:
4327:Dimension of
4324:Other names
4280:
4260:
4245:
4233:
4213:
4198:
4186:
4166:
4151:
4139:
4119:
4104:
4092:
4072:
4057:
4045:
4038:
4034:
4028:
4023:
4016:
4008:
4000:
3994:
3986:
3979:
3975:
3970:
3964:
3960:
3953:
3948:
3944:
3932:
3925:
3921:
3915:
3910:
3900:
3888:
3884:
3879:
3873:
3869:
3862:
3857:
3853:
3841:
3834:
3830:
3824:
3818:
3808:
3796:
3792:
3787:
3781:
3777:
3770:
3763:
3756:
3753:
3746:
3742:
3735:
3731:
3725:
3719:
3711:
3703:
3696:
3687:
3679:
3675:
3670:
3664:
3660:
3653:
3648:
3644:
3627:Dimension of
3624:Other names
3556:
3549:
3534:
3504:
3497:
3482:
3450:
3435:
3408:
3393:
3365:
3350:
3345:is abelian.
3339:
3332:
3325:
3318:
3311:
3302:
3296:
3292:
3284:
3276:
3268:
3262:
3254:
3250:
3243:
3239:
3233:
3227:
3223:
3217:
3212:
3208:
3197:
3190:
3183:
3176:
3169:
3163:
3156:
3149:
3145:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3110:
3100:
3089:
3084:
3080:
3075:
3069:
3065:
3059:
3054:
3050:
3039:
3032:
3022:
3018:
3012:
3008:
3001:
2988:
2984:
2978:
2972:
2968:
2962:
2957:
2953:
2941:
2936:
2927:
2922:
2915:
2911:
2903:
2895:
2889:
2883:
2876:
2870:
2862:
2857:
2851:
2847:
2842:
2836:
2832:
2826:
2821:
2817:
2797:Dimension of
2794:Other names
2728:
2701:
2674:
2647:
2620:
2615:is abelian.
2609:
2602:
2595:
2588:
2581:
2575:
2570:
2560:
2553:
2545:
2539:
2531:
2521:
2517:
2509:
2503:
2499:
2491:
2484:
2480:
2476:
2472:
2464:
2449:
2445:
2437:
2431:
2427:
2416:
2409:
2401:
2394:
2387:
2381:
2375:
2366:
2351:
2347:
2339:
2333:
2329:
2319:
2312:
2305:
2298:
2290:
2282:
2275:
2266:
2255:
2251:
2243:
2237:
2233:
2225:Other names
2186:
2134:
1978:
1917:
1913:
1908:
1904:
1899:
1895:
1890:
1886:
1884:
1853:
1844:
1840:
1836:
1830:
1819:
1816:
1807:
1803:
1797:
1786:
1783:
1772:
1761:
1758:
1751:
1743:
1732:
1729:
1720:
1713:
1694:
1366:
1283:
1250:
1241:
1232:
1230:
1180:
1168:
1165:
1161:
1151:
1147:
1143:
1139:
1135:
1131:
1129:
1125:
1121:
1118:
1097:
1093:
1089:
1085:
1079:
1068:
1013:Killing form
998:
989:
959:
950:
936:
933:Alternatives
919:
892:
890:
861:
845:
836:
832:
828:
802:
766:
749:
736:
734:
678:Armand Borel
663:Hermann Weyl
464:Loop algebra
446:Killing form
420:Lie algebras
297:
287:
277:
267:
248:
233:
223:
213:
203:
193:
183:
173:
144:Lie algebras
120:
111:
101:
94:
87:
80:
68:
56:Please help
51:verification
48:
6936:Harris, Joe
6675:Hyperbolic
6443:Hyperbolic
6306:Hyperbolic
6264:SU(2,2), SO
5678:= U(1) = SO
5523:Hermitian.
5520:Hermitian.
5256:− 1)
5213:− 1)
5026:− 1)
4634:quasi-split
4337:Non-Compact
4314:Fundamental
4041:− 1)
3967:− 1)
3637:Non-Compact
3614:Fundamental
3441:VIII split
3285:projective
3249:Order 4 if
3220:≥ 4) split
3101:projective
3062:≥ 3) split
2965:≥ 2) split
2865:−1)/2
2829:≥ 1) split
2807:Non-Compact
2784:Fundamental
2524:− 1)
2465:projective
2215:Fundamental
1709:Élie Cartan
1691:root system
1222:quaternions
1132:non-compact
1009:Lie algebra
986:exceptional
974:Felix Klein
943:Lie algebra
852:Élie Cartan
744:non-abelian
658:Élie Cartan
504:Root system
308:Exceptional
7023:Lie groups
7017:Categories
6984:Helgason,
6903:References
6872:Weyl group
6784:Hermitian
6727:Hermitian
6619:Hermitian
6495:Hermitian
6244:Hermitian
5688:Real line
5370:Hermitian.
5365:Hermitian.
5264:Hermitian.
5259:Hermitian.
4591:Hermitian.
4433:Copies of
4380:− 1
4306:Real rank
4303:Dimension
3606:Real rank
3592:See also:
3253:odd, 8 if
2776:Real rank
2773:Dimension
2762:See also:
2514:) compact
2442:) compact
2344:) compact
2248:) compact
2212:Real rank
2209:Dimension
2198:See also:
2140:The group
2060:(Abelian)
2022:Dimension
2011:See also:
1868:quaternion
1849:spin group
1779:spin group
1626:Lie group
1270:real forms
1102:real forms
1005:semisimple
858:Definition
643:Sophus Lie
636:Scientists
509:Weyl group
230:Symplectic
190:Orthogonal
140:Lie groups
114:April 2010
84:newspapers
6845:Lie group
5174:If |
4806:If |
4622:If |
4620:=1, split
4563:Hermitian
4553:), A III
4454:− 1
4442:− 1
4311:subgroup
3611:subgroup
2781:subgroup
2534:is odd).
2483:), PUSp(2
2080:∗
1979:See also
1966:octonions
1862:∪ A
1640:~
1573:π
1559:~
1517:π
1513:⊂
1456:~
1390:π
1386:⊂
1226:octonions
1027:−
893:connected
747:Lie group
741:connected
520:Real form
406:Euclidean
257:Classical
6938:(2004).
6912:(1971).
6856:See also
6824:Complex
6805:Complex
6656:Compact
6582:Compact
6529:Compact
6424:Compact
6394:Complex
6364:Complex
6241:SU(3,1)
6194:Compact
6143:Compact
5998:Compact
5968:SU(1,2)
5952:Compact
5815:Complex
5800:(ℂ) = Sp
5734:(ℝ) = Sp
5714:Compact
5685:Abelian
5614:Order 2
5560:Order 2
5511:Order 2
5460:Trivial
5405:Order 2
5402:Trivial
5356:Trivial
5307:Order 2
5240:Order 2
5230:−1
4899:Order 2
4534:−1
4525:−1
4393:Order 2
4355:−1
4342:Remarks
4251:complex
4204:complex
4157:complex
4110:complex
4063:complex
3989:is odd)
3809:complex
3564:Order 2
3540:I split
3512:Order 2
3488:I split
3419:Order 2
3399:V split
3376:Order 2
3373:Order 2
3356:I split
3126:), PSp(2
3118:), PSp(2
3087:−1
2812:Remarks
2734:compact
2707:compact
2680:compact
2653:compact
2626:compact
2479:), PUSp(
2475:), PSp(2
2228:Remarks
2034:Remarks
1839:) = SO(2
1276:, after
1177:Notation
951:discrete
692:Glossary
386:Poincaré
6971:Besse,
6709:Sphere
6477:Sphere
6387:(ℂ), Sp
6340:Sphere
6299:(ℍ), SO
6274:ℝ in ℝ
6220:ℝ in ℝ
6210:(ℝ), SO
6080:(ℝ), Sp
6059:Sphere
6018:(ℝ), Sp
5849:Sphere
5808:(ℝ), SO
5804:(ℂ), SO
5779:Sphere
5738:(ℝ), SO
5652:Groups
5414:Set of
5172:, split
4910:, 2 if
4835:> 2)
4654:> 1)
4408:), SU(2
4332:Compact
3632:Compact
3588:Complex
3464:E VIII
3138:), PSp(
3130:), PSp(
2918:+ 3)/2
2802:Compact
2280:, 2 if
2194:Compact
2007:Abelian
1991:⁄
1806:) = Sp(
1697:compact
1075:trivial
1042:in the
600:physics
381:Lorentz
210:Unitary
98:scholar
7006:
6992:
6978:
6954:
6920:
6847:whose
6601:Split
6579:SU(5)
6526:Sp(3)
6369:SU(3)
6281:(2,4)
6227:(3,3)
6217:Split
5949:SU(3)
5873:Split
5514:E VII
5359:E III
4585:is 2;
4321:group
4316:group
4294:Others
3997:> 4
3621:group
3616:group
3338:, and
3265:> 4
3182:, and
2791:group
2786:group
2608:, and
2542:> 4
2285:> 1
2222:group
2217:group
1949:, and
1912:, and
1356:center
1224:, and
1044:center
947:simple
864:simple
376:Circle
100:
93:
86:
79:
71:
23:, see
6843:is a
5663:Rank
5620:F II
5603:(Spin
5566:E IX
5408:E IV
5310:E II
5246:(2n)
5203:≥ 4)
5125:s in
5084:s in
4967:s in
4945:s in
4902:1 if
4867:+ 1)
4849:(1 ≤
4769:s in
4728:s in
4677:+ 1)
4510:+ 2)
4492:(1 ≤
4444:) in
4423:+ 1)
4374:+ 1)
4363:≥ 2)
4006:when
3928:+ 1)
3876:+ 1)
3837:+ 1)
3784:+ 1)
3738:+ 2)
3694:2 if
3667:+ 2)
3260:2 if
3257:even
3230:- 1)
3152:+ 1)
3072:+ 1)
3025:+ 1)
2975:+ 1)
2892:≥ 2.
2888:2 if
2882:1 if
2875:2 if
2839:+ 2)
2758:Split
2537:2 if
2452:+ 1)
2354:+ 1)
2273:1 if
2258:+ 2)
2163:that
2128:Notes
1477:with
827:) of
801:" SL(
739:is a
451:Index
105:JSTOR
91:books
7004:ISBN
6990:ISBN
6976:ISBN
6952:ISBN
6918:ISBN
6821:(ℂ)
6802:(ℂ)
6781:(ℝ)
6762:(ℝ)
6743:(ℝ)
6724:(ℝ)
6672:(ℝ)
6653:(ℝ)
6598:(ℝ)
6564:(ℝ)
6545:(ℝ)
6511:(ℝ)
6492:(ℝ)
6440:(ℝ)
6421:(ℝ)
6403:(ℝ)
6399:Spin
6391:(ℂ)
6361:(ℂ)
6303:(ℝ)
6268:(ℝ)
6214:(ℝ)
6191:(ℝ)
6084:(ℝ)
6022:(ℝ)
5995:(ℝ)
5870:(ℝ)
5812:(ℂ)
5742:(ℝ)
5666:Dim
5649:Dim
5569:112
5539:248
5491:133
5488:VII
5437:133
5333:III
5220:/2⌋
5057:and
5044:)SO(
5029:min(
5004:≥ 4)
4932:(R)
4870:min(
4695:)SO(
4680:min(
4477:≥ 1)
4207:104
4182:248
4160:496
4135:133
4113:266
4066:156
3570:G I
3518:F I
3467:128
3444:248
3402:133
3379:E I
3291:PSO(
3242:)SO(
3196:and
3015:+1)
2991:+1)
2987:)SO(
2908:(R)
2879:≥ 2
2683:248
2656:133
2471:PSp(
2400:and
2318:and
2301:+ 1)
2297:PSU(
2002:List
1829:SO(2
1775:+ 1)
1771:SO(2
1754:+ 1)
1746:+ 1)
1707:and
1069:The
791:U(1)
760:and
401:Loop
142:and
77:news
6944:doi
6814:30
6793:28
6774:28
6760:4,4
6755:28
6741:5,3
6736:28
6722:6,2
6717:28
6670:7,1
6665:28
6646:28
6633:3,2
6628:24
6615:4,1
6610:24
6591:24
6576:24
6562:4,2
6557:21
6538:21
6523:21
6509:4,3
6504:21
6490:5,2
6485:21
6438:6,1
6433:21
6414:21
6409:10
6380:20
6354:16
6301:5,1
6292:15
6266:4,2
6261:15
6238:15
6212:3,3
6203:15
6184:15
6156:14
6132:14
6078:3,2
6073:10
6020:2,2
6016:4,1
6011:10
5988:10
5806:3,1
5740:2,1
5674:ℝ,
5623:16
5611:))
5591:52
5588:II
5536:IX
5517:54
5469:64
5466:VI
5434:VI
5411:26
5388:78
5385:IV
5362:32
5336:78
5313:40
5284:78
5197:III
5162:If
5155:or
5151:If
5145:or
5141:If
5135:or
5131:If
5114:or
5110:If
5104:or
5100:If
5094:or
5090:If
5064:SO(
5053:If
5040:SO(
5013:= 2
4977:or
4973:If
4955:or
4951:If
4799:or
4795:If
4789:or
4785:If
4779:or
4775:If
4758:or
4754:If
4748:or
4744:If
4738:or
4734:If
4707:SO(
4691:SO(
4667:+1
4663:= 2
4612:If
4605:or
4601:If
4581:or
4577:If
4545:SU(
4490:+ 1
4471:III
4456:).
4276:14
4254:28
4229:52
4088:78
4020:PSO
4011:= 4
3907:PSp
3717:PSL
3706:≥ 2
3699:= 1
3690:+ 1
3543:14
3521:28
3491:52
3424:70
3382:42
3359:78
3279:= 4
3274:if
3238:SO(
3216:I (
3107:PSp
3058:I (
3007:SO(
2983:SO(
2961:I (
2944:= 1
2930:= 1
2901:PSL
2886:= 1
2873:= 1
2856:or
2825:I (
2737:14
2710:52
2629:78
2567:PSO
2556:= 4
2551:if
2512:≥ 4
2440:≥ 3
2342:≥ 2
2278:= 1
2269:+ 1
2246:≥ 1
1796:Sp(
1750:PU(
1742:SU(
1711:).
1171:= 2
976:'s
945:is
831:by
232:Sp(
222:SU(
202:SO(
182:SL(
172:GL(
60:by
7019::
6988:.
6950:.
6934:;
6839:A
6817:SL
6777:SO
6758:SO
6739:SO
6720:SO
6668:SO
6649:SO
6631:SU
6613:SU
6594:SL
6560:Sp
6541:Sp
6507:SO
6488:SO
6436:SO
6417:SO
6406:2
6383:SO
6375:8
6372:2
6357:SL
6349:5
6346:1
6295:SL
6287:8
6284:2
6256:6
6253:1
6233:9
6230:3
6206:SL
6179:8
6176:2
6127:6
6124:2
6076:SO
6068:4
6065:1
6014:SO
5983:4
5980:1
5965:8
5946:8
5941:5
5938:2
5866:SL
5863:8
5858:3
5855:1
5796:SL
5793:6
5788:2
5785:1
5730:SL
5727:3
5701:3
5696:1
5693:0
5671:1
5617:1
5594:1
5563:1
5551:×
5542:4
5494:3
5440:4
5391:2
5339:2
5287:4
5279:II
5244:SO
5209:(2
5167:=
5076:pq
5072:)
5048:)
5037:)
5022:(2
4937:pq
4927:,2
4921:Sp
4918:.
4914:=
4906:≠
4878:)
4863:(2
4857:)
4853:≤
4840:=
4829:II
4720:pq
4699:)
4688:)
4673:(2
4558:pq
4500:)
4496:≤
4486:=
4482:+
4412:)
4398:SL
4366:(2
4357:II
4268:1
4257:2
4224:2
4221:1
4210:4
4174:1
4163:8
4127:2
4116:7
4080:3
4069:6
4037:(2
4031:)
4013:.
3963:(2
3924:(2
3918:)
3894:2
3872:(2
3833:(2
3827:)
3821:+1
3815:SO
3802:2
3780:(2
3728:)
3722:+1
3708:.
3573:8
3567:1
3555:×
3546:2
3515:1
3503:×
3494:4
3476:E8
3474:@
3461:1
3458:2
3447:8
3405:7
3362:6
3324:,
3299:)
3267:,
3246:)
3226:(2
3203:.
3142:)
3097:1
3068:(2
3045:.
2997:1
2971:(2
2906:+1
2854:/2
2746:1
2743:1
2740:0
2719:1
2716:1
2713:0
2692:1
2689:1
2686:0
2665:1
2662:2
2659:0
2638:2
2635:3
2632:0
2594:,
2578:)
2544:,
2527:0
2520:(2
2487:)
2461:1
2458:2
2455:0
2448:(2
2422:.
2384:)
2378:+1
2372:SO
2363:1
2360:2
2357:0
2350:(2
2287:.
2261:0
2189:).
2136:^†
2096:1
2063:1
1998:.
1976:.
1942:,
1935:,
1928:,
1903:,
1894:,
1827:,
1814:.
1794:,
1781:.
1769:,
1756:.
1740:,
1228:.
1050:.
996:.
968:,
957:.
843:.
805:,
764:.
212:U(
192:O(
6960:.
6946::
6926:.
6819:4
6800:2
6797:G
6779:8
6711:S
6693:7
6688:H
6651:8
6596:5
6543:6
6479:S
6461:6
6456:H
6419:7
6401:5
6389:4
6385:5
6359:3
6342:S
6324:5
6319:H
6297:2
6279:G
6225:G
6208:4
6189:6
6163:2
6160:G
6139:2
6136:G
6108:2
6103:H
6082:4
6061:S
6043:4
6038:H
5993:5
5922:3
5917:C
5891:3
5886:R
5868:3
5851:S
5833:3
5828:H
5810:3
5802:2
5798:2
5781:S
5763:2
5758:H
5736:2
5732:2
5709:3
5705:S
5680:2
5676:S
5609:R
5607:(
5605:9
5601:4
5598:B
5586:4
5583:F
5556:1
5553:A
5549:7
5546:E
5534:8
5531:E
5504:S
5501:6
5498:E
5486:7
5483:E
5464:E
5453:1
5450:A
5447:6
5444:D
5432:7
5429:E
5398:4
5395:F
5383:6
5380:E
5349:S
5346:5
5343:D
5331:6
5328:E
5300:1
5297:A
5294:5
5291:A
5277:6
5274:E
5254:n
5252:(
5250:n
5233:R
5228:n
5224:A
5218:n
5216:⌊
5211:n
5207:n
5201:n
5199:(
5194:n
5190:D
5181:q
5179:−
5177:p
5169:q
5165:p
5157:q
5153:p
5147:q
5143:p
5137:q
5133:p
5129:.
5127:R
5123:R
5116:q
5112:p
5106:q
5102:p
5096:q
5092:p
5088:.
5086:R
5082:R
5070:q
5068:,
5066:p
5059:q
5055:p
5046:q
5042:p
5035:q
5033:,
5031:p
5024:n
5020:n
5015:n
5011:q
5009:+
5007:p
5002:n
5000:(
4998:I
4995:n
4991:D
4979:q
4975:p
4971:.
4969:H
4965:H
4957:q
4953:p
4949:.
4947:H
4943:H
4935:4
4929:q
4925:p
4923:2
4916:q
4912:p
4908:q
4904:p
4894:q
4890:C
4886:p
4882:C
4876:q
4874:,
4872:p
4865:n
4861:n
4855:q
4851:p
4846:q
4844:+
4842:p
4838:n
4833:n
4831:(
4826:n
4822:C
4813:q
4811:−
4809:p
4801:q
4797:p
4791:q
4787:p
4781:q
4777:p
4773:.
4771:R
4767:R
4760:q
4756:p
4750:q
4746:p
4740:q
4736:p
4732:.
4730:R
4726:R
4715:)
4713:q
4711:,
4709:p
4697:q
4693:p
4686:q
4684:,
4682:p
4675:n
4671:n
4665:n
4661:q
4659:+
4657:p
4652:n
4650:(
4648:I
4645:n
4641:B
4629:q
4627:−
4625:p
4618:q
4616:=
4614:p
4607:q
4603:p
4599:.
4597:C
4583:q
4579:p
4575:.
4573:C
4569:p
4565:.
4556:2
4551:q
4549:,
4547:p
4537:S
4532:q
4528:A
4523:p
4519:A
4514:p
4508:n
4506:(
4504:n
4498:q
4494:p
4488:n
4484:q
4480:p
4475:n
4473:(
4468:n
4464:A
4452:n
4450:2
4440:n
4428:C
4421:n
4417:n
4415:(
4410:n
4406:H
4404:(
4401:n
4388:n
4384:C
4378:n
4372:n
4368:n
4361:n
4359:(
4353:n
4351:2
4348:A
4284:2
4281:G
4264:2
4261:G
4249:2
4246:G
4237:4
4234:F
4217:4
4214:F
4202:4
4199:F
4190:8
4187:E
4170:8
4167:E
4155:8
4152:E
4143:7
4140:E
4123:7
4120:E
4108:7
4105:E
4096:6
4093:E
4076:6
4073:E
4061:6
4058:E
4049:n
4046:D
4039:n
4035:n
4029:C
4027:(
4024:n
4022:2
4009:n
4004:3
4001:S
3995:n
3987:n
3980:n
3976:D
3971:n
3965:n
3961:n
3959:2
3954:n
3952:(
3949:n
3945:D
3936:n
3933:C
3926:n
3922:n
3916:C
3914:(
3911:n
3909:2
3889:n
3885:C
3880:n
3874:n
3870:n
3868:2
3863:n
3861:(
3858:n
3854:C
3845:n
3842:B
3835:n
3831:n
3825:C
3823:(
3819:n
3817:2
3797:n
3793:B
3788:n
3782:n
3778:n
3776:2
3771:n
3769:(
3767:n
3764:B
3754:C
3747:n
3743:A
3736:n
3734:(
3732:n
3726:C
3724:(
3720:n
3704:n
3697:n
3688:n
3680:n
3676:A
3671:n
3665:n
3663:(
3661:n
3659:2
3654:n
3652:(
3649:n
3645:A
3560:1
3557:A
3553:1
3550:A
3538:2
3535:G
3508:1
3505:A
3501:3
3498:C
3486:4
3483:F
3454:8
3451:D
3439:8
3436:E
3412:7
3409:A
3397:7
3394:E
3369:4
3366:C
3354:6
3351:E
3343:1
3340:D
3336:1
3333:A
3329:2
3326:D
3322:3
3319:A
3315:3
3312:D
3303:n
3297:n
3295:,
3293:n
3277:n
3272:3
3269:S
3263:n
3255:n
3251:n
3244:n
3240:n
3234:n
3228:n
3224:n
3218:n
3213:n
3209:D
3201:1
3198:A
3194:1
3191:B
3187:1
3184:C
3180:2
3177:B
3173:2
3170:C
3164:R
3157:H
3150:n
3148:(
3146:n
3140:n
3136:R
3134:,
3132:n
3128:n
3124:R
3122:,
3120:n
3116:R
3114:(
3111:n
3109:2
3090:S
3085:n
3081:A
3076:n
3070:n
3066:n
3060:n
3055:n
3051:C
3043:1
3040:A
3036:1
3033:B
3023:n
3021:(
3019:n
3013:n
3011:,
3009:n
2989:n
2985:n
2979:n
2973:n
2969:n
2963:n
2958:n
2954:B
2942:n
2937:R
2928:n
2923:C
2916:n
2914:(
2912:n
2904:n
2890:n
2884:n
2877:n
2871:n
2863:n
2861:(
2858:B
2852:n
2848:D
2843:n
2837:n
2835:(
2833:n
2827:n
2822:n
2818:A
2732:2
2729:G
2705:4
2702:F
2678:8
2675:E
2651:7
2648:E
2624:6
2621:E
2613:1
2610:D
2606:1
2603:A
2599:2
2596:D
2592:3
2589:A
2585:3
2582:D
2576:R
2574:(
2571:n
2569:2
2554:n
2549:3
2546:S
2540:n
2532:n
2522:n
2518:n
2510:n
2507:(
2504:n
2500:D
2492:H
2485:n
2481:n
2477:n
2473:n
2450:n
2446:n
2438:n
2435:(
2432:n
2428:C
2420:2
2417:C
2413:2
2410:B
2407:.
2405:1
2402:C
2398:1
2395:A
2391:1
2388:B
2382:R
2380:(
2376:n
2374:2
2352:n
2348:n
2340:n
2337:(
2334:n
2330:B
2323:1
2320:C
2316:1
2313:B
2309:1
2306:A
2299:n
2283:n
2276:n
2267:n
2256:n
2254:(
2252:n
2244:n
2241:(
2238:n
2234:A
2187:S
2172:R
2149:R
2107:R
2075:R
2047:R
1993:2
1989:1
1986:+
1984:7
1981:E
1970:4
1962:2
1953:8
1951:E
1946:7
1944:E
1939:6
1937:E
1932:4
1930:F
1925:2
1923:G
1918:i
1914:D
1909:i
1905:C
1900:i
1896:B
1891:i
1887:A
1880:3
1876:3
1864:1
1860:1
1856:2
1845:r
1841:r
1837:r
1833:)
1831:r
1820:r
1817:D
1808:r
1804:r
1800:)
1798:r
1787:r
1784:C
1773:r
1762:r
1759:B
1752:r
1744:r
1733:r
1730:A
1670:.
1665:g
1637:G
1610:G
1588:)
1585:G
1582:(
1577:1
1569:=
1566:K
1556:G
1532:)
1529:G
1526:(
1521:1
1510:K
1485:K
1463:K
1453:G
1425:G
1405:)
1402:G
1399:(
1394:1
1383:K
1358:.
1340:g
1318:G
1296:g
1280:.
1262:.
1236:6
1233:E
1207:O
1204:,
1201:H
1198:,
1195:C
1192:,
1189:R
1169:p
1152:H
1148:G
1144:H
1140:G
1136:G
1098:L
1094:L
1090:L
1086:L
1030:I
904:R
875:R
837:n
833:n
829:n
814:R
803:n
776:R
750:G
724:e
717:t
710:v
350:8
348:E
342:7
340:E
334:6
332:E
326:4
324:F
318:2
316:G
298:n
295:D
288:n
285:C
278:n
275:B
268:n
265:A
236:)
234:n
226:)
224:n
216:)
214:n
206:)
204:n
196:)
194:n
186:)
184:n
176:)
174:n
127:)
121:(
116:)
112:(
102:·
95:·
88:·
81:·
54:.
31:.
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