2578:, which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it is of noncompact type. The spaces of Euclidean type have rank equal to their dimension and are isometric to a Euclidean space of that dimension. Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type. In both cases there are two classes.
9268:
55:
10437:
10087:
2523:
if it is not the product of two or more
Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian
9091:
1043:
Locally
Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.
10308:
9885:
10521:
on the tangent space at the identity coset. Thus the
Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with
9411:
1593:
10181:
6102:
The classification of
Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric space
10574:. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with
9748:, as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis
10502:. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
2337:
6093:
is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the
Riemannian and pseudo-Riemannian case.
6280:
is semisimple. This is the analogue of the
Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with
9263:{\displaystyle \langle X,Y\rangle _{\mathfrak {g}}={\begin{cases}\langle X,Y\rangle _{p}\quad &X,Y\in T_{p}M\cong {\mathfrak {m}}\\-B(X,Y)\quad &X,Y\in {\mathfrak {h}}\\0&{\mbox{otherwise}}\end{cases}}}
6156:
3262:
1460:
2063:
9486:
5267:
697:
In geometric terms, a complete, simply connected
Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold
4897:
4506:
4024:
3472:
2170:. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a
5596:
4316:
5986:
5506:
5460:
5323:
5166:
5120:
4953:
4680:
4563:
4416:
4370:
1267:
9552:
9310:
2729:
Specializing to the
Riemannian symmetric spaces of class A and compact type, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces
4983:
9648:
10707:, in Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland, American Mathematical Society,
9683:
5406:
5066:
9874:
4799:
4218:
933:
973:
if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor. A locally symmetric space is said to be a
2721:
In both class A and class B there is thus a correspondence between symmetric spaces of compact type and non-compact type. This is known as duality for
Riemannian symmetric spaces.
10529:, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.
5744:
3968:
3416:
3201:
2149:
9792:
6679:
6621:
6586:
6523:
6433:
3620:
3005:
10432:{\displaystyle \langle \cdot ,\cdot \rangle ={\frac {1}{\lambda _{1}}}\left.B\right|_{{\mathfrak {m}}_{1}}+\cdots +{\frac {1}{\lambda _{d}}}\left.B\right|_{{\mathfrak {m}}_{d}}}
5856:
4131:
3799:
3576:
3369:
3041:
2846:
2810:
10082:{\displaystyle \langle Y_{i}^{\#},Y_{j}\rangle =\lambda _{i}\langle Y_{i},Y_{j}\rangle =B(Y_{i},Y_{j})=\langle Y_{j}^{\#},Y_{i}\rangle =\lambda _{j}\langle Y_{j},Y_{i}\rangle }
9445:
5786:
5668:
5636:
4712:
3832:
3758:
3653:
3153:
9742:
3920:
2960:
2151:
see the definition and following proposition on page 209, chapter IV, section 3 in
Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information.
10300:
10114:
9816:
9707:
9585:
9518:
6547:
6457:
6402:
6374:
6327:
6303:
6278:
6235:
6211:
6183:
1871:
1847:
1819:
1792:
1768:
1740:
1716:
1692:
1668:
1644:
1620:
1409:
1381:
1349:
1313:
2422:, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (
5914:
5889:
2902:
874:
3092:
2105:
4087:
3532:
3325:
472:
10276:
5820:
5702:
5540:
5357:
5200:
5017:
4841:
4746:
4626:
4597:
4450:
4260:
4165:
9340:
3868:
3689:
2698:
is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of
1119:
750:
9345:
4051:
3289:
3119:
2929:
1471:
797:
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by
520:
10247:
10122:
3496:
10480:
of a Riemannian manifold at a point acts irreducibly on the tangent space, then either the manifold is a locally Riemannian symmetric space, or it is in one of
3889:
6709:
The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.
3724:
525:
756:, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of
9045:
An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex
515:
510:
10446:
the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital (
330:
2690:
The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type,
1746:, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that
594:
477:
5934:
construction. The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a
6638:
The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite
10498:
A Riemannian symmetric space that is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called a
2592:
is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type).
11035:
10849:
10712:
8964:
Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every point
990:
10928:
6117:
3207:
1421:
625:
10548:
10538:
2491:
of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group
2706:. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups
2249:
1079:
of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space.
2504:
1998:
10938:
10890:
10867:
10784:
10762:
11004:(1956), "Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces, with applications to Dirichlet series",
9450:
6305:
semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if
5206:
487:
10633:
33:
4847:
4456:
3974:
3422:
5546:
4266:
11053:
5945:
5465:
5419:
5282:
5125:
5079:
4912:
4639:
4522:
4375:
4329:
2745:, together with a geometric interpretation, if readily available. The labelling of these spaces is the one given by Cartan.
1203:
801:. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
482:
462:
9520:
can be further factored into eigenspaces classified by the Killing form. This is accomplished by defining an adjoint map
1291:
fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra
11058:
9523:
427:
335:
9276:
11068:
4958:
1958:. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its
1743:
1121:, which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points.
9593:
10900:
5939:
9656:
6186:
5363:
5277:
5023:
4907:
1412:
757:
651:
467:
9824:
4752:
4171:
10571:
10499:
10493:
2451:
7711:
to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case
879:
10613:
10607:
6352:. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that
6082:
6009:
5931:
2069:
1165:
783:
618:
102:
10547:) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called
5935:
5754:
4517:
10881:
Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
9046:
8946:
8860:
5708:
3932:
3380:
3165:
2110:
422:
385:
353:
340:
9751:
6653:
6595:
6560:
6497:
6407:
3587:
2972:
6238:
5826:
4101:
3769:
3546:
3339:
3011:
2816:
2780:
2650:
687:
683:
454:
122:
10442:
In certain practical applications, this factorization can be interpreted as the spectrum of operators,
9420:
5759:
5641:
5609:
4685:
3805:
3731:
3626:
3126:
2495:
of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality that
9712:
3896:
2936:
1598:
The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer
10638:
10281:
10095:
9797:
9688:
9557:
9499:
6528:
6438:
6383:
6355:
6308:
6284:
6259:
6216:
6192:
6164:
6037:
1852:
1828:
1800:
1773:
1749:
1721:
1697:
1673:
1649:
1625:
1601:
1390:
1362:
1330:
1294:
1125:
1002:
82:
72:
9127:
11063:
6650:
determines a real form. From this it is easy to construct tables of symmetric spaces for any given
2628:
2557:
667:
655:
647:
611:
599:
440:
270:
10457:
Classification of symmetric spaces proceeds based on whether or not the Killing form is definite.
5897:
5872:
2852:
1068:, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple
844:
10990:
7707:
For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing
6086:
6051:
Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If
6025:
3047:
2083:
663:
371:
361:
2637:
The compact simply connected Lie groups are the universal covers of the classical Lie groups SO(
6376:
is simple. It remains to describe the latter case. For this, one needs to classify involutions
4057:
3502:
3295:
11031:
10934:
10886:
10863:
10845:
10780:
10758:
10708:
10648:
6698:
6070:
1021:
753:
691:
550:
435:
398:
288:
10543:
A Riemannian symmetric space that is additionally equipped with a parallel subbundle of End(T
10255:
9406:{\displaystyle B(X,Y)=\operatorname {trace} (\operatorname {ad} X\circ \operatorname {ad} Y)}
5797:
5679:
5517:
5334:
5177:
4994:
4818:
4723:
4603:
4574:
4427:
4237:
4142:
10982:
10915:
10827:
10805:
10733:
10692:
10621:
10451:
9315:
5930:) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a
3838:
3659:
2596:
1588:{\displaystyle \subset {\mathfrak {h}},\;\subset {\mathfrak {m}},\;\subset {\mathfrak {h}}.}
1089:
1065:
1061:
1006:
725:
570:
250:
242:
234:
226:
218:
197:
187:
177:
167:
151:
132:
92:
10901:"A uniform description of compact symmetric spaces as Grassmannians using the magic square"
10454:
that can classify the different representations under which different orbitals transform.)
10176:{\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{1}\oplus \cdots \oplus {\mathfrak {m}}_{d}}
5996:. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.
2527:
The next step is to show that any irreducible, simply connected Riemannian symmetric space
10667:
Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis", Third edition, Springer
10567:
6090:
6033:
6029:
2543:
2488:
1076:
1053:
555:
308:
293:
64:
4030:
3268:
3098:
2908:
17:
10302:
semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes:
10192:
6040:(with zero, positive and negative curvature respectively). De Sitter space of dimension
6004:
An important class of symmetric spaces generalizing the Riemannian symmetric spaces are
3478:
10966:
10879:
10793:
10751:
10680:
10643:
10481:
10477:
10471:
8938:
6245:
6074:
3874:
2710:(up to conjugation). Such involutions extend to involutions of the complexification of
2483:) be the algebraic data associated to it. To classify the possible isometry classes of
2457:
1926:
1010:
998:
679:
575:
560:
393:
298:
6689:. This extends the compact/non-compact duality from the Riemannian case, where either
6459:
is a complex simple Lie algebra, and the corresponding symmetric spaces have the form
3695:
1032:). In fact, already the identity component of the isometry group acts transitively on
11047:
10721:
9070:
944:
798:
790:
This definition includes more than the Riemannian definition, and reduces to it when
283:
112:
11001:
10970:
10772:
10746:
10517:
contains a central circle. A quarter turn by this circle acts as multiplication by
9414:
9082:
8942:
8866:
6044:
may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension
6012:(nondegenerate instead of positive definite on each tangent space). In particular,
4091:
3536:
3329:
2073:
955:
need not be isometric, nor can it be extended, in general, from a neighbourhood of
580:
565:
366:
348:
278:
10705:
Harmonic analysis on semisimple symmetric spaces: A survey of some general results
1821:
with a direct sum decomposition satisfying these three conditions, the linear map
10617:
639:
406:
322:
46:
6089:. Conversely a manifold with such a connection is locally symmetric (i.e., its
977:
if in addition its geodesic symmetries can be extended to isometries on all of
10919:
6550:
5892:
1959:
1083:
1069:
675:
545:
411:
303:
10818:
Cartan, Élie (1927), "Sur une classe remarquable d'espaces de Riemann, II",
1894:
1141:
772:
42:
9417:. The minus sign appears because the Killing form is negative-definite on
2595:
The examples in class B are completely described by the classification of
6244:
However, the irreducible symmetric spaces can be classified. As shown by
5867:
671:
659:
10738:
763:
From the point of view of Lie theory, a symmetric space is the quotient
10994:
10944:
Chapter XI contains a good introduction to Riemannian symmetric spaces.
10832:
10810:
10696:
1877:
Riemannian symmetric spaces satisfy the Lie-theoretic characterization
502:
682:
to give a complete classification. Symmetric spaces commonly occur in
6340:
As in the Riemannian case there are semisimple symmetric spaces with
6151:{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}
3257:{\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (q))\,}
1455:{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}
1057:
1013:
locally Riemannian symmetric space is actually Riemannian symmetric.
10986:
1164:
of a typical point is an open subgroup of the fixed point set of an
54:
10703:
van den Ban, E. P.; Flensted-Jensen, M.; Schlichtkrull, H. (1997),
6681:, and furthermore, there is an obvious duality given by exchanging
1323:, whose square is the identity. It follows that the eigenvalues of
2456:
The algebraic description of Riemannian symmetric spaces enabled
2332:{\displaystyle s_{p}:M\to M,\quad h'K\mapsto h\sigma (h^{-1}h')K}
2627:
is its maximal compact subgroup. In both cases, the rank is the
10562:
is quaternion-Kähler if and only if isotropy representation of
2570:
has nonpositive (but not identically zero) sectional curvature.
2058:{\displaystyle \sigma :G\to G,h\mapsto s_{p}\circ h\circ s_{p}}
10796:(1926), "Sur une classe remarquable d'espaces de Riemann, I",
6237:
is not semisimple (or even reductive) in general, it can have
1086:
is locally symmetric but not symmetric, with the exception of
9481:{\displaystyle \langle \cdot ,\cdot \rangle _{\mathfrak {g}}}
6701:, i.e., its fixed point set is a maximal compact subalgebra.
6475:: these are the analogues of the Riemannian symmetric spaces
6020:
dimensional pseudo-Riemannian symmetric spaces of signature (
782:
that is (a connected component of) the invariant group of an
9061:
Some properties and forms of symmetric spaces can be noted.
8941:.) Selberg proved that weakly symmetric spaces give rise to
5262:{\displaystyle \mathrm {Spin} (16)/\{\pm \mathrm {vol} \}\,}
2519:
A simply connected Riemannian symmetric space is said to be
1905:
is Riemannian homogeneous). Therefore, if we fix some point
10844:, CBMS Regional Conference, American Mathematical Society,
10683:(1999), "Weakly symmetric spaces and spherical varieties",
10403:
10349:
9256:
8869:
extended Cartan's definition of symmetric space to that of
29:(pseudo-)Riemannian manifold whose geodesics are reversible
2186:: such an inner product always exists by averaging, since
9879:
These are orthogonal with respect to the metric, in that
6065:
is a symmetric space, then Nomizu showed that there is a
4892:{\displaystyle \mathrm {SO} (12)\cdot \mathrm {SU} (2)\,}
4501:{\displaystyle \mathrm {SO} (10)\cdot \mathrm {SO} (2)\,}
4019:{\displaystyle \mathrm {Sp} (p)\times \mathrm {Sp} (q)\,}
3467:{\displaystyle \mathrm {SO} (p)\times \mathrm {SO} (q)\,}
2542:
has vanishing curvature, and is therefore isometric to a
706:) is said to be symmetric if and only if, for each point
666:
about every point. This can be studied with the tools of
10777:
Essays in the History of Lie Groups and Algebraic Groups
5591:{\displaystyle \mathrm {Sp} (3)\cdot \mathrm {SU} (2)\,}
4311:{\displaystyle \mathrm {SU} (6)\cdot \mathrm {SU} (2)\,}
1885:
is a Riemannian symmetric space, the identity component
6248:, there is a dichotomy: an irreducible symmetric space
5981:{\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{n},}
5501:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
5455:{\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}
5318:{\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}
5161:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
5115:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
4948:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
4675:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
4558:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
4411:{\displaystyle (\mathbb {C} \otimes \mathbb {H} )P^{2}}
4365:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
2714:, and these in turn classify non-compact real forms of
10959:
Symmetric spaces II: Compact Spaces and Classification
10860:
Differential geometry, Lie groups and symmetric spaces
10092:
since the Killing form is symmetric. This factorizes
9744:
is the Killing form. This map is sometimes called the
9247:
10311:
10284:
10258:
10195:
10125:
10098:
9888:
9827:
9800:
9754:
9715:
9691:
9659:
9596:
9560:
9526:
9502:
9453:
9423:
9348:
9318:
9279:
9094:
6656:
6598:
6563:
6531:
6500:
6441:
6410:
6386:
6358:
6311:
6287:
6262:
6219:
6195:
6167:
6120:
5948:
5900:
5875:
5829:
5800:
5762:
5711:
5682:
5644:
5612:
5549:
5520:
5468:
5422:
5366:
5337:
5285:
5209:
5180:
5128:
5082:
5026:
4997:
4961:
4915:
4850:
4821:
4755:
4726:
4688:
4642:
4606:
4577:
4525:
4459:
4430:
4378:
4332:
4269:
4240:
4174:
4145:
4104:
4060:
4033:
3977:
3935:
3899:
3877:
3841:
3808:
3772:
3734:
3698:
3662:
3629:
3590:
3549:
3505:
3481:
3425:
3383:
3342:
3298:
3271:
3210:
3168:
3129:
3101:
3050:
3014:
2975:
2939:
2911:
2855:
2819:
2783:
2460:
to obtain a complete classification of them in 1926.
2252:
2113:
2086:
2001:
1855:
1831:
1803:
1776:
1752:
1724:
1700:
1676:
1652:
1628:
1604:
1474:
1424:
1393:
1365:
1333:
1297:
1206:
1092:
882:
847:
728:
10971:"Invariant affine connections on homogeneous spaces"
8881:
with a transitive connected Lie group of isometries
2107:
and its identity component (hence an open subgroup)
1262:{\displaystyle G^{\sigma }=\{g\in G:\sigma (g)=g\}.}
10726:
Annales Scientifiques de l'École Normale Supérieure
2623:is a simply connected complex simple Lie group and
837:and reverses geodesics through that point, i.e. if
10878:
10750:
10624:can be interpreted as reductive symmetric spaces.
10431:
10294:
10270:
10241:
10175:
10108:
10081:
9868:
9810:
9786:
9736:
9701:
9677:
9642:
9579:
9547:{\displaystyle {\mathfrak {m}}\to {\mathfrak {m}}}
9546:
9512:
9480:
9439:
9405:
9334:
9304:
9262:
6673:
6615:
6580:
6549:may be viewed as the fixed point set of a complex
6541:
6517:
6451:
6427:
6396:
6368:
6321:
6297:
6272:
6229:
6205:
6177:
6150:
6008:, in which the Riemannian metric is replaced by a
5980:
5908:
5883:
5850:
5814:
5780:
5738:
5696:
5662:
5630:
5590:
5534:
5500:
5454:
5400:
5351:
5317:
5261:
5194:
5160:
5114:
5060:
5011:
4977:
4947:
4891:
4835:
4793:
4740:
4706:
4674:
4620:
4591:
4557:
4500:
4444:
4410:
4364:
4310:
4254:
4212:
4159:
4125:
4081:
4045:
4018:
3962:
3914:
3883:
3862:
3826:
3793:
3752:
3718:
3683:
3647:
3614:
3570:
3526:
3490:
3466:
3410:
3363:
3319:
3283:
3256:
3195:
3147:
3113:
3086:
3035:
2999:
2954:
2923:
2896:
2840:
2804:
2331:
2143:
2099:
2057:
1865:
1841:
1813:
1786:
1762:
1734:
1710:
1686:
1662:
1638:
1614:
1587:
1454:
1403:
1375:
1343:
1307:
1261:
1124:An example of a non-Riemannian symmetric space is
1113:
1052:Basic examples of Riemannian symmetric spaces are
927:
868:
744:
10873:The standard book on Riemannian symmetric spaces.
10582: = 2 (these are isomorphic), BDI with
9305:{\displaystyle \langle \cdot ,\cdot \rangle _{p}}
2174:-invariant inner product on the tangent space to
10768:Contains a compact introduction and many tables.
4978:{\displaystyle \mathbb {H} \otimes \mathbb {O} }
4061:
3506:
3299:
2619:is the diagonal subgroup. For non-compact type,
2603:is a compact simply connected simple Lie group,
2418:If one starts with a Riemannian symmetric space
1280:(including, of course, the identity component).
1072:equipped with a bi-invariant Riemannian metric.
752:as minus the identity (every symmetric space is
473:Representation theory of semisimple Lie algebras
10930:Foundations of Differential Geometry, Volume II
10899:Huang, Yongdong; Leung, Naichung Conan (2010).
10724:(1957), "Les espaces symétriques noncompacts",
9643:{\displaystyle \langle X,Y^{\#}\rangle =B(X,Y)}
10927:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
10598: = 1, EII, EVI, EIX, FI and G.
6592:extends to a complex antilinear involution of
10820:Bulletin de la Société Mathématique de France
10798:Bulletin de la Société Mathématique de France
9678:{\displaystyle \langle \cdot ,\cdot \rangle }
5401:{\displaystyle E_{7}\cdot \mathrm {SU} (2)\,}
5061:{\displaystyle E_{6}\cdot \mathrm {SO} (2)\,}
2962:that leave the complex determinant invariant
2446:Classification of Riemannian symmetric spaces
1327:are ±1. The +1 eigenspace is the Lie algebra
619:
8:
10324:
10312:
10076:
10050:
10034:
10003:
9962:
9936:
9920:
9889:
9869:{\displaystyle Y_{i}^{\#}=\lambda _{i}Y_{i}}
9672:
9660:
9616:
9597:
9467:
9454:
9293:
9280:
9143:
9130:
9108:
9095:
8877:. These are defined as Riemannian manifolds
6646:determines a complex symmetric space, while
5255:
5238:
4794:{\displaystyle \mathrm {SU} (8)/\{\pm I\}\,}
4787:
4778:
4213:{\displaystyle \mathrm {Sp} (4)/\{\pm I\}\,}
4206:
4197:
2221:is Riemannian symmetric, consider any point
1253:
1220:
10842:Analysis on Non-Riemannian Symmetric Spaces
6627:and hence also a complex linear involution
6241:representations which are not irreducible.
2556:has nonnegative (but not identically zero)
2080:is contained between the fixed point group
1975:, hence compact. Moreover, if we denote by
670:, leading to consequences in the theory of
10566:contains an Sp(1) summand acting like the
6256:is either flat (i.e., an affine space) or
6024: − 1,1), are important in
5927:
3728:Space of orthogonal complex structures on
1966:is a subgroup of the orthogonal group of T
1548:
1511:
935:It follows that the derivative of the map
928:{\displaystyle f(\gamma (t))=\gamma (-t).}
626:
612:
511:Particle physics and representation theory
156:
53:
38:
10831:
10809:
10737:
10421:
10415:
10414:
10412:
10393:
10384:
10367:
10361:
10360:
10358:
10339:
10330:
10310:
10286:
10285:
10283:
10257:
10224:
10218:
10217:
10207:
10201:
10200:
10194:
10167:
10161:
10160:
10144:
10138:
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9991:
9978:
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9896:
9887:
9860:
9850:
9837:
9832:
9826:
9802:
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9799:
9778:
9759:
9753:
9714:
9693:
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9690:
9658:
9610:
9595:
9571:
9559:
9538:
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9528:
9527:
9525:
9504:
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9470:
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9425:
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9422:
9347:
9323:
9317:
9296:
9278:
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9231:
9230:
9184:
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9171:
9146:
9122:
9112:
9111:
9093:
6665:
6659:
6658:
6655:
6607:
6601:
6600:
6597:
6572:
6566:
6565:
6562:
6533:
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6509:
6503:
6502:
6499:
6443:
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6419:
6413:
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6409:
6388:
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6385:
6360:
6359:
6357:
6313:
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6310:
6289:
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6286:
6264:
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6221:
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6218:
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6194:
6169:
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6142:
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6122:
6121:
6119:
5969:
5960:
5952:
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5902:
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5899:
5877:
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5874:
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5805:
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5735:
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5087:
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4851:
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4773:
4756:
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4725:
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4641:
4617:
4611:
4605:
4588:
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4497:
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4331:
4307:
4290:
4270:
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4175:
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4059:
4032:
4015:
3998:
3978:
3976:
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3898:
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3790:
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3697:
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3551:
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3463:
3446:
3426:
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3407:
3384:
3382:
3349:
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3344:
3341:
3297:
3270:
3253:
3236:
3219:
3211:
3209:
3192:
3169:
3167:
3136:
3132:
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3128:
3100:
3049:
3032:
3015:
3013:
2996:
2976:
2974:
2946:
2942:
2941:
2938:
2910:
2886:
2854:
2837:
2820:
2818:
2801:
2784:
2782:
2306:
2257:
2251:
2137:
2131:
2121:
2112:
2091:
2085:
2049:
2030:
2000:
1857:
1856:
1854:
1833:
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1802:
1778:
1777:
1775:
1754:
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1723:
1702:
1701:
1699:
1678:
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1675:
1654:
1653:
1651:
1630:
1629:
1627:
1606:
1605:
1603:
1576:
1575:
1563:
1562:
1553:
1552:
1539:
1538:
1526:
1525:
1516:
1515:
1502:
1501:
1489:
1488:
1479:
1478:
1473:
1446:
1445:
1436:
1435:
1426:
1425:
1423:
1395:
1394:
1392:
1367:
1366:
1364:
1359:), and the −1 eigenspace will be denoted
1335:
1334:
1332:
1299:
1298:
1296:
1211:
1205:
1197:is an open subgroup of the invariant set
1091:
881:
846:
733:
727:
8673:
8349:
7976:
7812:
7720:
7431:
7152:
6711:
2747:
2531:is of one of the following three types:
1276:is open, it is a union of components of
991:Cartan–Ambrose–Hicks theorem
11028:Harmonic Analysis on Commutative Spaces
10660:
2463:For a given Riemannian symmetric space
2438:) completely describe the structure of
813:be a connected Riemannian manifold and
478:Representations of classical Lie groups
210:
159:
41:
9077:can be lifted to a scalar product on
8937:was later shown to be unnecessary by
3155:compatible with the Hermitian metric
7:
9312:is the Riemannian metric defined on
9050:
8334: / SO(9,1)×SO(1,1)
8143: / SO(5,5)×SO(1,1)
5739:{\displaystyle \mathrm {Spin} (9)\,}
3963:{\displaystyle \mathrm {Sp} (p+q)\,}
3411:{\displaystyle \mathrm {SO} (p+q)\,}
3196:{\displaystyle \mathrm {SU} (p+q)\,}
3123:Space of quaternionic structures on
2144:{\displaystyle (G^{\sigma })_{o}\,,}
1024:(meaning that the isometry group of
951:. On a general Riemannian manifold,
331:Lie group–Lie algebra correspondence
10416:
10362:
10287:
10219:
10202:
10162:
10139:
10128:
10101:
9803:
9787:{\displaystyle Y_{1},\ldots ,Y_{n}}
9694:
9539:
9529:
9505:
9472:
9426:
9232:
9185:
9113:
6674:{\displaystyle {\mathfrak {g}}^{c}}
6660:
6616:{\displaystyle {\mathfrak {g}}^{c}}
6602:
6581:{\displaystyle {\mathfrak {g}}^{c}}
6567:
6534:
6518:{\displaystyle {\mathfrak {g}}^{c}}
6504:
6444:
6428:{\displaystyle {\mathfrak {g}}^{c}}
6414:
6389:
6361:
6314:
6290:
6265:
6222:
6198:
6170:
6143:
6133:
6123:
3615:{\displaystyle \mathrm {SO} (2n)\,}
3000:{\displaystyle \mathrm {SU} (2n)\,}
2499:is simply connected. (This implies
1941:. By differentiating the action at
1858:
1834:
1806:
1779:
1755:
1727:
1703:
1679:
1655:
1631:
1607:
1577:
1564:
1554:
1540:
1527:
1517:
1503:
1490:
1480:
1447:
1437:
1427:
1396:
1368:
1336:
1300:
10950:Symmetric spaces I: General Theory
10933:, Wiley Classics Library edition,
10533:Quaternion-Kähler symmetric spaces
10016:
9902:
9838:
9611:
9572:
8855:Weakly symmetric Riemannian spaces
8633: / SO(10,2)×SL(2,
8555: / SO(8,4)×SU(2)
8535: / SO(12)× Sp(1)
8425: / SO(12)× Sp(1)
8275: / SO(8,2)×SO(2)
8258: / SU(4,2)×SU(2)
8209: / SO(6,4)×SO(2)
8192: / SU(4,2)×SU(2)
7962: / Sp(2,1)×Sp(1)
7925: / Sp(2,1)×Sp(1)
6028:, the most notable examples being
6006:pseudo-Riemannian symmetric spaces
5851:{\displaystyle \mathrm {SO} (4)\,}
5834:
5831:
5722:
5719:
5716:
5713:
5574:
5571:
5554:
5551:
5384:
5381:
5251:
5248:
5245:
5220:
5217:
5214:
5211:
5044:
5041:
4875:
4872:
4855:
4852:
4760:
4757:
4484:
4481:
4464:
4461:
4294:
4291:
4274:
4271:
4179:
4176:
4126:{\displaystyle \mathbb {H} ^{p+q}}
4002:
3999:
3982:
3979:
3940:
3937:
3922:compatible with the inner product
3810:
3794:{\displaystyle \mathrm {Sp} (n)\,}
3777:
3774:
3631:
3595:
3592:
3571:{\displaystyle \mathbb {R} ^{p+q}}
3450:
3447:
3430:
3427:
3388:
3385:
3364:{\displaystyle \mathbb {C} ^{p+q}}
3237:
3220:
3212:
3173:
3170:
3036:{\displaystyle \mathrm {Sp} (n)\,}
3019:
3016:
2980:
2977:
2841:{\displaystyle \mathrm {SO} (n)\,}
2824:
2821:
2805:{\displaystyle \mathrm {SU} (n)\,}
2788:
2785:
2737:. They are here given in terms of
2505:long exact sequence of a fibration
2407:is a geodesic symmetry and, since
1797:Conversely, given any Lie algebra
1646:. The second condition means that
1355:(since this is the Lie algebra of
943:is minus the identity map on the
25:
11030:, American Mathematical Society,
10779:, American Mathematical Society,
10549:quaternion-Kähler symmetric space
10539:Quaternion-Kähler symmetric space
10476:If the identity component of the
9440:{\displaystyle {\mathfrak {h}}~;}
8871:weakly symmetric Riemannian space
8485: / SO(6,6)×SL(2,
8264: / SU(5,1)×SL(2,
8244: / SO(10)×SO(2)
8198: / SU(3,3)×SL(2,
8069: / SO(10)×SO(2)
6073:(i.e. an affine connection whose
5781:{\displaystyle \mathbb {O} P^{2}}
5663:{\displaystyle \mathbb {H} P^{2}}
5631:{\displaystyle \mathbb {O} P^{2}}
4707:{\displaystyle \mathbb {OP} ^{2}}
3827:{\displaystyle \mathrm {U} (n)\,}
3753:{\displaystyle \mathbb {R} ^{2n}}
3648:{\displaystyle \mathrm {U} (n)\,}
3148:{\displaystyle \mathbb {C} ^{2n}}
2415:is a Riemannian symmetric space.
1984:: M → M the geodesic symmetry of
1945:we obtain an isometric action of
1917:is diffeomorphic to the quotient
1873:, is an involutive automorphism.
1016:Every Riemannian symmetric space
10840:Flensted-Jensen, Mogens (1986),
10634:Orthogonal symmetric Lie algebra
9737:{\displaystyle B(\cdot ,\cdot )}
8168: / SU(6)×SU(2)
8062: / SU(6)×SU(2)
7902: / Sp(3)×Sp(1)
7874: / Sp(3)×Sp(1)
7790: / SU(2)×SU(2)
7771: / SU(2)×SU(2)
6487:a complex simple Lie group, and
5961:
5953:
5606:Space of symmetric subspaces of
5416:Space of symmetric subspaces of
5076:Space of symmetric subspaces of
4636:Space of symmetric subspaces of
4326:Space of symmetric subspaces of
3915:{\displaystyle \mathbb {H} ^{n}}
2955:{\displaystyle \mathbb {C} ^{n}}
2574:A more refined invariant is the
2389:equal to minus the identity on T
1742:. Thus any symmetric space is a
997:is locally Riemannian symmetric
722:and acting on the tangent space
34:Symmetric space (disambiguation)
10554:An irreducible symmetric space
10505:An irreducible symmetric space
10295:{\displaystyle {\mathfrak {g}}}
10109:{\displaystyle {\mathfrak {m}}}
9811:{\displaystyle {\mathfrak {m}}}
9702:{\displaystyle {\mathfrak {m}}}
9580:{\displaystyle Y\mapsto Y^{\#}}
9513:{\displaystyle {\mathfrak {m}}}
9215:
9152:
6542:{\displaystyle {\mathfrak {g}}}
6525:is simple. The real subalgebra
6452:{\displaystyle {\mathfrak {g}}}
6397:{\displaystyle {\mathfrak {g}}}
6380:of a (real) simple Lie algebra
6369:{\displaystyle {\mathfrak {g}}}
6322:{\displaystyle {\mathfrak {g}}}
6298:{\displaystyle {\mathfrak {g}}}
6273:{\displaystyle {\mathfrak {g}}}
6230:{\displaystyle {\mathfrak {h}}}
6206:{\displaystyle {\mathfrak {h}}}
6178:{\displaystyle {\mathfrak {m}}}
3893:Space of complex structures on
2278:
2190:is compact, and by acting with
1866:{\displaystyle {\mathfrak {m}}}
1842:{\displaystyle {\mathfrak {h}}}
1814:{\displaystyle {\mathfrak {g}}}
1787:{\displaystyle {\mathfrak {h}}}
1763:{\displaystyle {\mathfrak {m}}}
1735:{\displaystyle {\mathfrak {g}}}
1711:{\displaystyle {\mathfrak {h}}}
1687:{\displaystyle {\mathfrak {h}}}
1663:{\displaystyle {\mathfrak {m}}}
1639:{\displaystyle {\mathfrak {g}}}
1615:{\displaystyle {\mathfrak {h}}}
1404:{\displaystyle {\mathfrak {g}}}
1376:{\displaystyle {\mathfrak {m}}}
1344:{\displaystyle {\mathfrak {h}}}
1308:{\displaystyle {\mathfrak {g}}}
10461:Applications and special cases
10230:
10196:
9997:
9971:
9731:
9719:
9637:
9625:
9564:
9534:
9400:
9376:
9364:
9352:
9212:
9200:
5966:
5949:
5940:double Lagrangian Grassmannian
5926:A more modern classification (
5844:
5838:
5732:
5726:
5584:
5578:
5564:
5558:
5485:
5469:
5439:
5423:
5394:
5388:
5302:
5286:
5230:
5224:
5145:
5129:
5099:
5083:
5054:
5048:
4932:
4916:
4885:
4879:
4865:
4859:
4770:
4764:
4659:
4643:
4542:
4526:
4494:
4488:
4474:
4468:
4395:
4379:
4349:
4333:
4304:
4298:
4284:
4278:
4189:
4183:
4076:
4064:
4012:
4006:
3992:
3986:
3956:
3944:
3857:
3845:
3820:
3814:
3787:
3781:
3713:
3699:
3678:
3666:
3641:
3635:
3608:
3599:
3521:
3509:
3460:
3454:
3440:
3434:
3404:
3392:
3314:
3302:
3250:
3247:
3241:
3230:
3224:
3216:
3189:
3177:
3081:
3066:
3063:
3051:
3029:
3023:
2993:
2984:
2883:
2871:
2868:
2856:
2834:
2828:
2798:
2792:
2585:is a (real) simple Lie group;
2363:is an isometry with (clearly)
2323:
2299:
2290:
2269:
2166:with a compact isotropy group
2128:
2114:
2023:
2011:
1569:
1549:
1532:
1512:
1495:
1475:
1244:
1238:
1108:
1096:
919:
910:
901:
898:
892:
886:
857:
851:
714:, there exists an isometry of
526:Galilean group representations
521:Poincaré group representations
1:
10466:Symmetric spaces and holonomy
8929:. (Selberg's assumption that
6161:is said to be irreducible if
5988:for normed division algebras
2511:is connected by assumption.)
2198:-invariant Riemannian metric
2076:such that the isotropy group
1005:, and furthermore that every
516:Lorentz group representations
483:Theorem of the highest weight
11019:Spaces of constant curvature
10513:is Hermitian if and only if
9685:is the Riemannian metric on
8945:, so that in particular the
8873:, or in current terminology
8817: / SO(12,4) or E
6491:a maximal compact subgroup.
5909:{\displaystyle \mathbb {H} }
5891:which are isomorphic to the
5884:{\displaystyle \mathbb {O} }
5866:Space of subalgebras of the
2933:Space of real structures on
2897:{\displaystyle (n-1)(n+2)/2}
971:locally Riemannian symmetric
869:{\displaystyle \gamma (0)=p}
11021:(5th ed.), McGraw–Hill
10877:Helgason, Sigurdur (1984),
10858:Helgason, Sigurdur (1978),
10669:(See section 5.3, page 256)
9085:. This is done by defining
9073:on the Riemannian manifold
8760: / SO(8,8) or E
6014:Lorentzian symmetric spaces
3087:{\displaystyle (n-1)(2n+1)}
2849:
2100:{\displaystyle G^{\sigma }}
1825:, equal to the identity on
1744:reductive homogeneous space
1020:is complete and Riemannian
758:pseudo-Riemannian manifolds
674:; or algebraically through
11085:
10605:
10536:
10491:
10488:Hermitian symmetric spaces
10469:
8858:
6337:might not be irreducible.
6187:irreducible representation
5278:Rosenfeld projective plane
4908:Rosenfeld projective plane
4098:-dimensional subspaces of
3543:-dimensional subspaces of
3336:-dimensional subspaces of
2449:
2380:and (by differentiating) d
2354:. Then one can check that
1849:and minus the identity on
975:(globally) symmetric space
652:pseudo-Riemannian manifold
468:Lie algebra representation
31:
10920:10.1007/s00208-010-0549-8
10590: = 4, CII with
10572:quaternionic vector space
10500:Hermitian symmetric space
10494:Hermitian symmetric space
10450:the Killing form being a
9081:by combining it with the
9065:Lifting the metric tensor
4082:{\displaystyle \min(p,q)}
3527:{\displaystyle \min(p,q)}
3320:{\displaystyle \min(p,q)}
2770:Geometric interpretation
2452:List of simple Lie groups
1889:of the isometry group of
1694:-invariant complement to
18:Locally symmetric variety
11026:Wolf, Joseph A. (2007),
11017:Wolf, Joseph A. (1999),
10614:Bott periodicity theorem
10608:Bott periodicity theorem
10602:Bott periodicity theorem
8961:) is multiplicity free.
8933:should be an element of
6069:-invariant torsion-free
6010:pseudo-Riemannian metric
6000:General symmetric spaces
5932:Freudenthal magic square
1001:its curvature tensor is
463:Lie group representation
11006:J. Indian Math. Society
10271:{\displaystyle i\neq j}
9047:semisimple Lie algebras
9042:is a symmetric space.
8980:, there is an isometry
8626: / SU(6,2)
8548: / SU(6,2)
8542: / SU(4,4)
8458: / SU(4,4)
8316: / Sp(3,1)
8251: / Sp(2,2)
8175: / Sp(3,1)
8101: / Sp(2,2)
7971: / SO(8,1)
7934: / SO(5,4)
5936:Lagrangian Grassmannian
5815:{\displaystyle G_{2}\,}
5755:Cayley projective plane
5697:{\displaystyle F_{4}\,}
5535:{\displaystyle F_{4}\,}
5352:{\displaystyle E_{8}\,}
5195:{\displaystyle E_{8}\,}
5012:{\displaystyle E_{7}\,}
4836:{\displaystyle E_{7}\,}
4741:{\displaystyle E_{7}\,}
4621:{\displaystyle F_{4}\,}
4592:{\displaystyle E_{6}\,}
4518:Cayley projective plane
4445:{\displaystyle E_{6}\,}
4255:{\displaystyle E_{6}\,}
4160:{\displaystyle E_{6}\,}
1897:acting transitively on
1622:is a Lie subalgebra of
1152:is a homogeneous space
488:Borel–Weil–Bott theorem
10747:Besse, Arthur Lancelot
10433:
10296:
10272:
10243:
10177:
10110:
10083:
9870:
9812:
9788:
9738:
9703:
9679:
9644:
9581:
9548:
9514:
9482:
9441:
9407:
9336:
9335:{\displaystyle T_{p}M}
9306:
9264:
8947:unitary representation
8875:weakly symmetric space
8861:Weakly symmetric space
8753: / SO(16)
8727: / SO(16)
6675:
6617:
6582:
6543:
6519:
6453:
6429:
6398:
6370:
6323:
6299:
6274:
6231:
6207:
6179:
6152:
6098:Classification results
5982:
5928:Huang & Leung 2010
5910:
5885:
5852:
5816:
5782:
5740:
5698:
5664:
5632:
5592:
5536:
5502:
5456:
5402:
5353:
5319:
5263:
5196:
5162:
5116:
5062:
5013:
4979:
4949:
4893:
4837:
4795:
4742:
4708:
4676:
4622:
4593:
4559:
4502:
4446:
4412:
4366:
4312:
4256:
4214:
4161:
4127:
4083:
4047:
4020:
3964:
3916:
3885:
3864:
3863:{\displaystyle n(n+1)}
3828:
3795:
3754:
3720:
3685:
3684:{\displaystyle n(n-1)}
3649:
3616:
3572:
3528:
3492:
3468:
3412:
3365:
3321:
3285:
3258:
3197:
3149:
3115:
3088:
3037:
3001:
2956:
2925:
2898:
2842:
2806:
2651:exceptional Lie groups
2487:, first note that the
2333:
2182:at the identity coset
2145:
2101:
2059:
1867:
1843:
1815:
1788:
1764:
1736:
1712:
1688:
1664:
1640:
1616:
1589:
1456:
1405:
1387:is an automorphism of
1377:
1345:
1309:
1283:As an automorphism of
1263:
1179:is an automorphism of
1115:
1114:{\displaystyle L(2,1)}
929:
870:
833:if it fixes the point
746:
745:{\displaystyle T_{p}M}
650:(or more generally, a
386:Semisimple Lie algebra
341:Adjoint representation
11054:Differential geometry
10957:Loos, Ottmar (1969),
10948:Loos, Ottmar (1969),
10908:Mathematische Annalen
10434:
10297:
10273:
10244:
10178:
10111:
10084:
9871:
9813:
9789:
9746:generalized transpose
9739:
9704:
9680:
9645:
9582:
9549:
9515:
9483:
9442:
9408:
9337:
9307:
9265:
8905:there is an isometry
8690: / SO(16,
8451: / SU(8)
8418: / SU(8)
8377: / SO(12,
8094: / Sp(4)
8055: / Sp(4)
8019: / SO(10,
7953: / SO(9)
7883: / SO(9)
6676:
6618:
6583:
6544:
6520:
6454:
6430:
6399:
6371:
6324:
6300:
6275:
6232:
6208:
6180:
6153:
5983:
5911:
5886:
5853:
5817:
5783:
5741:
5699:
5665:
5633:
5593:
5537:
5503:
5457:
5403:
5354:
5320:
5264:
5197:
5163:
5117:
5063:
5014:
4980:
4950:
4894:
4838:
4796:
4743:
4709:
4677:
4623:
4594:
4560:
4503:
4447:
4413:
4367:
4313:
4257:
4215:
4162:
4128:
4084:
4048:
4021:
3965:
3917:
3886:
3865:
3829:
3796:
3755:
3721:
3686:
3650:
3617:
3573:
3529:
3493:
3469:
3413:
3366:
3322:
3286:
3259:
3198:
3150:
3116:
3089:
3038:
3002:
2957:
2926:
2899:
2843:
2807:
2725:Classification result
2515:Classification scheme
2346:is the involution of
2334:
2158:is a symmetric space
2146:
2102:
2060:
1962:at any point) and so
1868:
1844:
1816:
1789:
1765:
1737:
1713:
1689:
1665:
1641:
1617:
1590:
1457:
1406:
1378:
1346:
1310:
1264:
1160:where the stabilizer
1116:
1028:acts transitively on
930:
871:
825:of a neighborhood of
747:
688:representation theory
684:differential geometry
455:Representation theory
10639:Relative root system
10309:
10282:
10256:
10193:
10123:
10096:
9886:
9825:
9798:
9752:
9713:
9689:
9657:
9594:
9558:
9524:
9500:
9451:
9421:
9346:
9316:
9277:
9092:
8821: / Sk(8,
8764: / Sk(8,
8643: / Sk(6,
8616: / SL(4,
8561: / Sk(6,
8495: / Sk(6,
8474: / SL(4,
8464: / SL(8,
8366: / SL(8,
8323: / SL(3,
8215: / Sk(5,
8181: / Sp(8,
8132: / SL(3,
8118: / SL(6,
8107: / Sp(8,
8004: / SL(6,
7993: / Sp(8,
7911: / Sp(6,
7848: / SO(9,
7831: / Sp(6,
7799: / SL(2,
7741: / SL(2,
7076: / S(GL(
7028:) / Sp(2
6827: / S(GL(
6746: / S(GL(
6654:
6596:
6561:
6529:
6498:
6439:
6435:is not simple, then
6408:
6384:
6356:
6309:
6285:
6260:
6217:
6193:
6165:
6118:
6038:anti-de Sitter space
5946:
5898:
5873:
5827:
5798:
5760:
5709:
5680:
5642:
5610:
5547:
5518:
5466:
5420:
5364:
5335:
5283:
5207:
5178:
5126:
5080:
5024:
4995:
4959:
4913:
4848:
4819:
4753:
4724:
4686:
4640:
4604:
4575:
4523:
4457:
4428:
4376:
4330:
4267:
4238:
4172:
4143:
4102:
4058:
4031:
3975:
3933:
3897:
3875:
3839:
3806:
3770:
3732:
3696:
3660:
3627:
3588:
3547:
3503:
3479:
3423:
3381:
3340:
3296:
3269:
3208:
3166:
3127:
3099:
3048:
3012:
2973:
2937:
2909:
2853:
2817:
2781:
2694:is such a group and
2599:. For compact type,
2503:is connected by the
2250:
2111:
2084:
1999:
1853:
1829:
1801:
1774:
1750:
1722:
1698:
1674:
1650:
1626:
1602:
1472:
1422:
1391:
1363:
1331:
1295:
1204:
1132:Algebraic definition
1126:anti-de Sitter space
1090:
1003:covariantly constant
880:
845:
805:Geometric definition
726:
32:For other uses, see
11059:Riemannian geometry
10757:, Springer-Verlag,
10739:10.24033/asens.1054
10020:
9906:
9842:
9488:positive-definite.
8972:and tangent vector
7632: / Sp(2
7602:) / GL(
7570:) / Sp(
7452: / Sp(2
7326:) / GL(
7286:) / SO(
6988:) / GL(
6944: / S(U(
6930:) / Sk(
6494:Thus we may assume
4046:{\displaystyle 4pq}
3284:{\displaystyle 2pq}
3114:{\displaystyle n-1}
2924:{\displaystyle n-1}
2558:sectional curvature
841:is a geodesic with
821:. A diffeomorphism
668:Riemannian geometry
648:Riemannian manifold
600:Table of Lie groups
441:Compact Lie algebra
11069:Homogeneous spaces
10885:, Academic Press,
10862:, Academic Press,
10833:10.24033/bsmf.1113
10811:10.24033/bsmf.1105
10753:Einstein Manifolds
10697:10.1007/BF01236659
10594: = 1 or
10586: = 4 or
10578: = 2 or
10429:
10292:
10278:. For the case of
10268:
10242:{\displaystyle =0}
10239:
10173:
10106:
10079:
10006:
9892:
9866:
9828:
9808:
9784:
9734:
9699:
9675:
9640:
9577:
9544:
9510:
9496:The tangent space
9478:
9437:
9403:
9332:
9302:
9260:
9255:
9251:
9034:is independent of
9010:the derivative of
7693: / GL(
7654: / Sp(
7526: / Sp(
7485: / GL(
7418: / SL(
7382: / SO(
7360: / Sk(
7240: / SO(
7206: / GL(
7173: / SO(
7120: / Sp(
7106: / GL(
7062: / Sk(
7002: / Sp(
6912: / SO(
6867: / Sp(
6849: / GL(
6813: / SO(
6779: / Sp(
6732: / SO(
6671:
6613:
6578:
6539:
6515:
6471:is a real form of
6449:
6425:
6394:
6366:
6319:
6295:
6270:
6227:
6203:
6175:
6148:
6026:general relativity
5978:
5906:
5893:quaternion algebra
5881:
5848:
5812:
5778:
5736:
5694:
5660:
5628:
5588:
5532:
5498:
5452:
5398:
5349:
5315:
5259:
5192:
5158:
5112:
5058:
5009:
4975:
4945:
4889:
4833:
4791:
4738:
4704:
4672:
4618:
4589:
4555:
4498:
4442:
4408:
4362:
4308:
4252:
4210:
4157:
4123:
4079:
4043:
4016:
3960:
3912:
3881:
3860:
3824:
3791:
3750:
3716:
3681:
3645:
3612:
3568:
3524:
3491:{\displaystyle pq}
3488:
3464:
3408:
3361:
3317:
3281:
3254:
3193:
3145:
3111:
3084:
3033:
2997:
2952:
2921:
2894:
2838:
2802:
2524:symmetric spaces.
2329:
2141:
2097:
2055:
1863:
1839:
1811:
1784:
1760:
1732:
1708:
1684:
1660:
1636:
1612:
1585:
1452:
1401:
1373:
1341:
1319:, also denoted by
1305:
1259:
1111:
925:
866:
778:by a Lie subgroup
742:
664:inversion symmetry
372:Affine Lie algebra
362:Simple Lie algebra
103:Special orthogonal
11037:978-0-8218-4289-8
10851:978-0-8218-0711-8
10714:978-0-8218-0609-8
10679:Akhiezer, D. N.;
10649:Cartan involution
10399:
10345:
10116:into eigenspaces
9433:
9250:
8852:
8851:
8672:
8671:
8348:
8347:
7975:
7974:
7811:
7810:
7705:
7704:
7668: / U(
7618: = Sp(2
7584: / U(
7438: = Sp(2
7430:
7429:
7396: / U(
7300: / U(
7151:
7150:
6699:Cartan involution
6111:with Lie algebra
6071:affine connection
5919:
5918:
3884:{\displaystyle n}
3539:of oriented real
2597:simple Lie groups
1929:of the action of
1066:hyperbolic spaces
1062:projective spaces
831:geodesic symmetry
692:harmonic analysis
636:
635:
436:Split Lie algebra
399:Cartan subalgebra
261:
260:
152:Simple Lie groups
16:(Redirected from
11076:
11040:
11022:
11013:
10997:
10962:
10953:
10943:
10923:
10905:
10895:
10884:
10872:
10854:
10836:
10835:
10814:
10813:
10789:
10767:
10756:
10742:
10741:
10717:
10699:
10671:
10665:
10622:orthogonal group
10568:unit quaternions
10528:
10452:Casimir operator
10438:
10436:
10435:
10430:
10428:
10427:
10426:
10425:
10420:
10419:
10411:
10400:
10398:
10397:
10385:
10374:
10373:
10372:
10371:
10366:
10365:
10357:
10346:
10344:
10343:
10331:
10301:
10299:
10298:
10293:
10291:
10290:
10277:
10275:
10274:
10269:
10248:
10246:
10245:
10240:
10229:
10228:
10223:
10222:
10212:
10211:
10206:
10205:
10182:
10180:
10179:
10174:
10172:
10171:
10166:
10165:
10149:
10148:
10143:
10142:
10132:
10131:
10115:
10113:
10112:
10107:
10105:
10104:
10088:
10086:
10085:
10080:
10075:
10074:
10062:
10061:
10049:
10048:
10033:
10032:
10019:
10014:
9996:
9995:
9983:
9982:
9961:
9960:
9948:
9947:
9935:
9934:
9919:
9918:
9905:
9900:
9875:
9873:
9872:
9867:
9865:
9864:
9855:
9854:
9841:
9836:
9817:
9815:
9814:
9809:
9807:
9806:
9793:
9791:
9790:
9785:
9783:
9782:
9764:
9763:
9743:
9741:
9740:
9735:
9708:
9706:
9705:
9700:
9698:
9697:
9684:
9682:
9681:
9676:
9649:
9647:
9646:
9641:
9615:
9614:
9586:
9584:
9583:
9578:
9576:
9575:
9553:
9551:
9550:
9545:
9543:
9542:
9533:
9532:
9519:
9517:
9516:
9511:
9509:
9508:
9487:
9485:
9484:
9479:
9477:
9476:
9475:
9446:
9444:
9443:
9438:
9431:
9430:
9429:
9412:
9410:
9409:
9404:
9341:
9339:
9338:
9333:
9328:
9327:
9311:
9309:
9308:
9303:
9301:
9300:
9269:
9267:
9266:
9261:
9259:
9258:
9252:
9248:
9236:
9235:
9189:
9188:
9176:
9175:
9151:
9150:
9118:
9117:
9116:
8893:such that given
8885:and an isometry
8840: / E
8836:×SU(2) or E
8832: / E
8806: / E
8787: / E
8775: / E
8734: / E
8701: / E
8674:
8664: / E
8654: / E
8605: / E
8586: / E
8576: / E
8516: / E
8506: / E
8432: / E
8392: / E
8350:
8341: / F
8306: / F
8288: / F
8226: / F
8150: / F
8076: / F
8034: / F
7977:
7813:
7721:
7717:
7501: = Sp(
7432:
7342: = Sk(
7153:
6712:
6680:
6678:
6677:
6672:
6670:
6669:
6664:
6663:
6622:
6620:
6619:
6614:
6612:
6611:
6606:
6605:
6587:
6585:
6584:
6579:
6577:
6576:
6571:
6570:
6548:
6546:
6545:
6540:
6538:
6537:
6524:
6522:
6521:
6516:
6514:
6513:
6508:
6507:
6458:
6456:
6455:
6450:
6448:
6447:
6434:
6432:
6431:
6426:
6424:
6423:
6418:
6417:
6403:
6401:
6400:
6395:
6393:
6392:
6375:
6373:
6372:
6367:
6365:
6364:
6328:
6326:
6325:
6320:
6318:
6317:
6304:
6302:
6301:
6296:
6294:
6293:
6279:
6277:
6276:
6271:
6269:
6268:
6236:
6234:
6233:
6228:
6226:
6225:
6212:
6210:
6209:
6204:
6202:
6201:
6184:
6182:
6181:
6176:
6174:
6173:
6157:
6155:
6154:
6149:
6147:
6146:
6137:
6136:
6127:
6126:
6064:
6048: + 1.
5987:
5985:
5984:
5979:
5974:
5973:
5964:
5956:
5942:of subspaces of
5922:As Grassmannians
5915:
5913:
5912:
5907:
5905:
5890:
5888:
5887:
5882:
5880:
5868:octonion algebra
5857:
5855:
5854:
5849:
5837:
5821:
5819:
5818:
5813:
5810:
5809:
5787:
5785:
5784:
5779:
5777:
5776:
5767:
5745:
5743:
5742:
5737:
5725:
5703:
5701:
5700:
5695:
5692:
5691:
5669:
5667:
5666:
5661:
5659:
5658:
5649:
5637:
5635:
5634:
5629:
5627:
5626:
5617:
5597:
5595:
5594:
5589:
5577:
5557:
5541:
5539:
5538:
5533:
5530:
5529:
5507:
5505:
5504:
5499:
5497:
5496:
5484:
5476:
5461:
5459:
5458:
5453:
5451:
5450:
5438:
5430:
5407:
5405:
5404:
5399:
5387:
5376:
5375:
5358:
5356:
5355:
5350:
5347:
5346:
5324:
5322:
5321:
5316:
5314:
5313:
5301:
5293:
5268:
5266:
5265:
5260:
5254:
5237:
5223:
5201:
5199:
5198:
5193:
5190:
5189:
5167:
5165:
5164:
5159:
5157:
5156:
5144:
5136:
5121:
5119:
5118:
5113:
5111:
5110:
5098:
5090:
5067:
5065:
5064:
5059:
5047:
5036:
5035:
5018:
5016:
5015:
5010:
5007:
5006:
4984:
4982:
4981:
4976:
4974:
4966:
4954:
4952:
4951:
4946:
4944:
4943:
4931:
4923:
4898:
4896:
4895:
4890:
4878:
4858:
4842:
4840:
4839:
4834:
4831:
4830:
4800:
4798:
4797:
4792:
4777:
4763:
4747:
4745:
4744:
4739:
4736:
4735:
4713:
4711:
4710:
4705:
4703:
4702:
4697:
4681:
4679:
4678:
4673:
4671:
4670:
4658:
4650:
4627:
4625:
4624:
4619:
4616:
4615:
4598:
4596:
4595:
4590:
4587:
4586:
4564:
4562:
4561:
4556:
4554:
4553:
4541:
4533:
4507:
4505:
4504:
4499:
4487:
4467:
4451:
4449:
4448:
4443:
4440:
4439:
4417:
4415:
4414:
4409:
4407:
4406:
4394:
4386:
4371:
4369:
4368:
4363:
4361:
4360:
4348:
4340:
4317:
4315:
4314:
4309:
4297:
4277:
4261:
4259:
4258:
4253:
4250:
4249:
4219:
4217:
4216:
4211:
4196:
4182:
4166:
4164:
4163:
4158:
4155:
4154:
4132:
4130:
4129:
4124:
4122:
4121:
4110:
4094:of quaternionic
4088:
4086:
4085:
4080:
4052:
4050:
4049:
4044:
4025:
4023:
4022:
4017:
4005:
3985:
3969:
3967:
3966:
3961:
3943:
3921:
3919:
3918:
3913:
3911:
3910:
3905:
3890:
3888:
3887:
3882:
3869:
3867:
3866:
3861:
3833:
3831:
3830:
3825:
3813:
3800:
3798:
3797:
3792:
3780:
3759:
3757:
3756:
3751:
3749:
3748:
3740:
3725:
3723:
3722:
3719:{\displaystyle }
3717:
3709:
3690:
3688:
3687:
3682:
3654:
3652:
3651:
3646:
3634:
3621:
3619:
3618:
3613:
3598:
3577:
3575:
3574:
3569:
3567:
3566:
3555:
3533:
3531:
3530:
3525:
3497:
3495:
3494:
3489:
3473:
3471:
3470:
3465:
3453:
3433:
3417:
3415:
3414:
3409:
3391:
3370:
3368:
3367:
3362:
3360:
3359:
3348:
3326:
3324:
3323:
3318:
3290:
3288:
3287:
3282:
3263:
3261:
3260:
3255:
3240:
3223:
3215:
3202:
3200:
3199:
3194:
3176:
3154:
3152:
3151:
3146:
3144:
3143:
3135:
3120:
3118:
3117:
3112:
3093:
3091:
3090:
3085:
3042:
3040:
3039:
3034:
3022:
3006:
3004:
3003:
2998:
2983:
2961:
2959:
2958:
2953:
2951:
2950:
2945:
2930:
2928:
2927:
2922:
2903:
2901:
2900:
2895:
2890:
2847:
2845:
2844:
2839:
2827:
2811:
2809:
2808:
2803:
2791:
2748:
2564:Non-compact type
2338:
2336:
2335:
2330:
2322:
2314:
2313:
2286:
2262:
2261:
2230:
2150:
2148:
2147:
2142:
2136:
2135:
2126:
2125:
2106:
2104:
2103:
2098:
2096:
2095:
2064:
2062:
2061:
2056:
2054:
2053:
2035:
2034:
1872:
1870:
1869:
1864:
1862:
1861:
1848:
1846:
1845:
1840:
1838:
1837:
1820:
1818:
1817:
1812:
1810:
1809:
1793:
1791:
1790:
1785:
1783:
1782:
1769:
1767:
1766:
1761:
1759:
1758:
1741:
1739:
1738:
1733:
1731:
1730:
1717:
1715:
1714:
1709:
1707:
1706:
1693:
1691:
1690:
1685:
1683:
1682:
1669:
1667:
1666:
1661:
1659:
1658:
1645:
1643:
1642:
1637:
1635:
1634:
1621:
1619:
1618:
1613:
1611:
1610:
1594:
1592:
1591:
1586:
1581:
1580:
1568:
1567:
1558:
1557:
1544:
1543:
1531:
1530:
1521:
1520:
1507:
1506:
1494:
1493:
1484:
1483:
1461:
1459:
1458:
1453:
1451:
1450:
1441:
1440:
1431:
1430:
1410:
1408:
1407:
1402:
1400:
1399:
1382:
1380:
1379:
1374:
1372:
1371:
1350:
1348:
1347:
1342:
1340:
1339:
1314:
1312:
1311:
1306:
1304:
1303:
1268:
1266:
1265:
1260:
1216:
1215:
1120:
1118:
1117:
1112:
1007:simply connected
985:Basic properties
934:
932:
931:
926:
875:
873:
872:
867:
829:is said to be a
751:
749:
748:
743:
738:
737:
678:, which allowed
628:
621:
614:
571:Claude Chevalley
428:Complexification
271:Other Lie groups
157:
65:Classical groups
57:
39:
21:
11084:
11083:
11079:
11078:
11077:
11075:
11074:
11073:
11044:
11043:
11038:
11025:
11016:
11000:
10987:10.2307/2372398
10965:
10956:
10947:
10941:
10926:
10903:
10898:
10893:
10876:
10870:
10857:
10852:
10839:
10817:
10792:
10787:
10771:
10765:
10745:
10720:
10715:
10702:
10678:
10675:
10674:
10666:
10662:
10657:
10630:
10610:
10604:
10558: /
10541:
10535:
10523:
10509: /
10496:
10490:
10474:
10468:
10463:
10413:
10402:
10401:
10389:
10359:
10348:
10347:
10335:
10307:
10306:
10280:
10279:
10254:
10253:
10216:
10199:
10191:
10190:
10159:
10136:
10121:
10120:
10094:
10093:
10066:
10053:
10040:
10024:
9987:
9974:
9952:
9939:
9926:
9910:
9884:
9883:
9856:
9846:
9823:
9822:
9796:
9795:
9774:
9755:
9750:
9749:
9711:
9710:
9687:
9686:
9655:
9654:
9606:
9592:
9591:
9567:
9556:
9555:
9522:
9521:
9498:
9497:
9494:
9466:
9449:
9448:
9419:
9418:
9344:
9343:
9319:
9314:
9313:
9292:
9275:
9274:
9254:
9253:
9244:
9238:
9237:
9216:
9191:
9190:
9167:
9153:
9142:
9123:
9107:
9090:
9089:
9067:
9059:
8988:, depending on
8863:
8857:
8843:
8839:
8835:
8831:
8820:
8816:
8809:
8805:
8799:
8790:
8786:
8778:
8774:
8763:
8759:
8752:
8746:
8737:
8733:
8726:
8717:
8704:
8700:
8689:
8680:
8667:
8663:
8659:
8657:
8653:
8642:
8638:
8632:
8625:
8621:
8615:
8608:
8604:
8598:
8589:
8585:
8581:
8579:
8575:
8560:
8556:
8554:
8547:
8543:
8541:
8534:
8528:
8519:
8515:
8511:
8509:
8505:
8494:
8490:
8484:
8473:
8469:
8463:
8459:
8457:
8450:
8444:
8435:
8431:
8424:
8417:
8408:
8395:
8391:
8376:
8365:
8356:
8344:
8340:
8333:
8322:
8315:
8309:
8305:
8299:
8291:
8287:
8276:
8274:
8263:
8259:
8257:
8250:
8243:
8237:
8229:
8225:
8214:
8210:
8208:
8197:
8193:
8191:
8180:
8176:
8174:
8167:
8161:
8153:
8149:
8142:
8131:
8127:
8117:
8106:
8102:
8100:
8093:
8087:
8079:
8075:
8068:
8061:
8054:
8045:
8037:
8033:
8018:
8003:
7992:
7983:
7970:
7961:
7952:
7944:
7933:
7924:
7920:
7910:
7901:
7893:
7882:
7873:
7862:
7847:
7830:
7819:
7798:
7789:
7781:
7770:
7759:
7740:
7729:
7712:
7688:
7649:
7593:
7561:
7559:
7550:
7541:
7532:
7413:
7377:
7317:
7277:
7275:
7266:
7257:
7248:
7101:
7019:
6979:
6977:
6968:
6959:
6950:
6921:
6844:
6707:
6657:
6652:
6651:
6623:commuting with
6599:
6594:
6593:
6564:
6559:
6558:
6527:
6526:
6501:
6496:
6495:
6479: /
6463: /
6437:
6436:
6411:
6406:
6405:
6382:
6381:
6354:
6353:
6333: /
6307:
6306:
6283:
6282:
6258:
6257:
6252: /
6215:
6214:
6191:
6190:
6163:
6162:
6116:
6115:
6107: /
6100:
6091:universal cover
6060: /
6052:
6034:De Sitter space
6030:Minkowski space
6002:
5965:
5944:
5943:
5924:
5896:
5895:
5871:
5870:
5825:
5824:
5801:
5796:
5795:
5768:
5758:
5757:
5707:
5706:
5683:
5678:
5677:
5650:
5640:
5639:
5618:
5608:
5607:
5545:
5544:
5521:
5516:
5515:
5488:
5464:
5463:
5442:
5418:
5417:
5367:
5362:
5361:
5338:
5333:
5332:
5305:
5281:
5280:
5205:
5204:
5181:
5176:
5175:
5148:
5124:
5123:
5102:
5078:
5077:
5027:
5022:
5021:
4998:
4993:
4992:
4957:
4956:
4935:
4911:
4910:
4846:
4845:
4822:
4817:
4816:
4751:
4750:
4727:
4722:
4721:
4689:
4684:
4683:
4662:
4638:
4637:
4607:
4602:
4601:
4578:
4573:
4572:
4545:
4521:
4520:
4455:
4454:
4431:
4426:
4425:
4398:
4374:
4373:
4352:
4328:
4327:
4265:
4264:
4241:
4236:
4235:
4170:
4169:
4146:
4141:
4140:
4105:
4100:
4099:
4056:
4055:
4029:
4028:
3973:
3972:
3931:
3930:
3900:
3895:
3894:
3873:
3872:
3837:
3836:
3804:
3803:
3768:
3767:
3735:
3730:
3729:
3694:
3693:
3658:
3657:
3625:
3624:
3586:
3585:
3550:
3545:
3544:
3501:
3500:
3477:
3476:
3421:
3420:
3379:
3378:
3343:
3338:
3337:
3294:
3293:
3267:
3266:
3206:
3205:
3164:
3163:
3130:
3125:
3124:
3097:
3096:
3046:
3045:
3010:
3009:
2971:
2970:
2940:
2935:
2934:
2907:
2906:
2851:
2850:
2815:
2814:
2779:
2778:
2733: /
2727:
2686:
2679:
2672:
2665:
2658:
2649:) and the five
2544:Euclidean space
2517:
2489:universal cover
2454:
2448:
2411:was arbitrary,
2406:
2394:
2388:
2371:
2362:
2315:
2302:
2279:
2253:
2248:
2247:
2222:
2217: /
2206: /
2178: /
2162: /
2127:
2117:
2109:
2108:
2087:
2082:
2081:
2045:
2026:
1997:
1996:
1983:
1971:
1954:
1879:
1851:
1850:
1827:
1826:
1799:
1798:
1772:
1771:
1748:
1747:
1720:
1719:
1696:
1695:
1672:
1671:
1648:
1647:
1624:
1623:
1600:
1599:
1470:
1469:
1420:
1419:
1411:, this gives a
1389:
1388:
1361:
1360:
1329:
1328:
1293:
1292:
1207:
1202:
1201:
1192:
1156: /
1146:symmetric space
1140:be a connected
1134:
1088:
1087:
1077:Riemann surface
1054:Euclidean space
1050:
1040:is connected).
987:
878:
877:
843:
842:
807:
771:of a connected
767: /
729:
724:
723:
644:symmetric space
632:
587:
586:
585:
556:Wilhelm Killing
540:
532:
531:
530:
505:
494:
493:
492:
457:
447:
446:
445:
432:
416:
394:Dynkin diagrams
388:
378:
377:
376:
358:
336:Exponential map
325:
315:
314:
313:
294:Conformal group
273:
263:
262:
254:
246:
238:
230:
222:
203:
193:
183:
173:
154:
144:
143:
142:
123:Special unitary
67:
37:
30:
23:
22:
15:
12:
11:
5:
11082:
11080:
11072:
11071:
11066:
11061:
11056:
11046:
11045:
11042:
11041:
11036:
11023:
11014:
10998:
10975:Amer. J. Math.
10963:
10954:
10945:
10939:
10924:
10896:
10891:
10874:
10868:
10855:
10850:
10837:
10815:
10790:
10785:
10769:
10763:
10743:
10722:Berger, Marcel
10718:
10713:
10700:
10685:Transf. Groups
10681:Vinberg, E. B.
10673:
10672:
10659:
10658:
10656:
10653:
10652:
10651:
10646:
10644:Satake diagram
10641:
10636:
10629:
10626:
10620:of the stable
10606:Main article:
10603:
10600:
10537:Main article:
10534:
10531:
10492:Main article:
10489:
10486:
10478:holonomy group
10472:Holonomy group
10470:Main article:
10467:
10464:
10462:
10459:
10440:
10439:
10424:
10418:
10410:
10407:
10404:
10396:
10392:
10388:
10383:
10380:
10377:
10370:
10364:
10356:
10353:
10350:
10342:
10338:
10334:
10329:
10326:
10323:
10320:
10317:
10314:
10289:
10267:
10264:
10261:
10250:
10249:
10238:
10235:
10232:
10227:
10221:
10215:
10210:
10204:
10198:
10184:
10183:
10170:
10164:
10158:
10155:
10152:
10147:
10141:
10135:
10130:
10103:
10090:
10089:
10078:
10073:
10069:
10065:
10060:
10056:
10052:
10047:
10043:
10039:
10036:
10031:
10027:
10023:
10018:
10013:
10009:
10005:
10002:
9999:
9994:
9990:
9986:
9981:
9977:
9973:
9970:
9967:
9964:
9959:
9955:
9951:
9946:
9942:
9938:
9933:
9929:
9925:
9922:
9917:
9913:
9909:
9904:
9899:
9895:
9891:
9877:
9876:
9863:
9859:
9853:
9849:
9845:
9840:
9835:
9831:
9805:
9781:
9777:
9773:
9770:
9767:
9762:
9758:
9733:
9730:
9727:
9724:
9721:
9718:
9696:
9674:
9671:
9668:
9665:
9662:
9651:
9650:
9639:
9636:
9633:
9630:
9627:
9624:
9621:
9618:
9613:
9609:
9605:
9602:
9599:
9574:
9570:
9566:
9563:
9541:
9536:
9531:
9507:
9493:
9490:
9474:
9469:
9465:
9462:
9459:
9456:
9436:
9428:
9402:
9399:
9396:
9393:
9390:
9387:
9384:
9381:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9351:
9331:
9326:
9322:
9299:
9295:
9291:
9288:
9285:
9282:
9271:
9270:
9257:
9245:
9243:
9240:
9239:
9234:
9229:
9226:
9223:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9199:
9196:
9193:
9192:
9187:
9182:
9179:
9174:
9170:
9166:
9163:
9160:
9157:
9154:
9149:
9145:
9141:
9138:
9135:
9132:
9129:
9128:
9126:
9121:
9115:
9110:
9106:
9103:
9100:
9097:
9066:
9063:
9058:
9055:
9049:, is given in
9028:
9027:
9008:
8939:Ernest Vinberg
8859:Main article:
8856:
8853:
8850:
8849:
8841:
8837:
8833:
8829:
8826:
8818:
8814:
8811:
8807:
8803:
8800:
8797:
8793:
8792:
8788:
8784:
8776:
8772:
8769:
8761:
8757:
8754:
8750:
8747:
8744:
8740:
8739:
8735:
8731:
8728:
8724:
8721:
8718:
8715:
8711:
8710:
8702:
8698:
8695:
8687:
8684:
8681:
8678:
8670:
8669:
8668:×SO(1,1)
8665:
8661:
8655:
8651:
8648:
8640:
8630:
8627:
8623:
8613:
8610:
8606:
8602:
8599:
8596:
8592:
8591:
8587:
8583:
8577:
8573:
8570:
8558:
8552:
8549:
8545:
8539:
8536:
8532:
8529:
8526:
8522:
8521:
8517:
8513:
8510:×SO(1,1)
8507:
8503:
8500:
8492:
8482:
8479:
8471:
8461:
8455:
8452:
8448:
8445:
8442:
8438:
8437:
8433:
8429:
8426:
8422:
8419:
8415:
8412:
8409:
8406:
8402:
8401:
8393:
8389:
8386:
8374:
8371:
8363:
8360:
8357:
8354:
8346:
8345:
8342:
8338:
8335:
8331:
8328:
8320:
8317:
8313:
8310:
8307:
8303:
8300:
8297:
8293:
8292:
8289:
8285:
8282:
8272:
8269:
8261:
8255:
8252:
8248:
8245:
8241:
8238:
8235:
8231:
8230:
8227:
8223:
8220:
8212:
8206:
8203:
8195:
8189:
8186:
8178:
8172:
8169:
8165:
8162:
8159:
8155:
8154:
8151:
8147:
8144:
8140:
8137:
8129:
8115:
8112:
8104:
8098:
8095:
8091:
8088:
8085:
8081:
8080:
8077:
8073:
8070:
8066:
8063:
8059:
8056:
8052:
8049:
8046:
8043:
8039:
8038:
8035:
8031:
8028:
8016:
8013:
8001:
7998:
7990:
7987:
7984:
7981:
7973:
7972:
7968:
7963:
7959:
7954:
7950:
7945:
7942:
7936:
7935:
7931:
7926:
7922:
7908:
7903:
7899:
7894:
7891:
7885:
7884:
7880:
7875:
7871:
7866:
7863:
7860:
7854:
7853:
7845:
7840:
7828:
7823:
7820:
7817:
7809:
7808:
7796:
7791:
7787:
7782:
7779:
7773:
7772:
7768:
7763:
7760:
7757:
7751:
7750:
7738:
7733:
7730:
7727:
7703:
7702:
7663:
7627:
7612:
7611:
7579:
7555:
7546:
7537:
7530:
7521:
7495:
7494:
7480:
7447:
7428:
7427:
7391:
7355:
7336:
7335:
7295:
7271:
7262:
7253:
7244:
7235:
7220:
7219:
7201:
7168:
7149:
7148:
7115:
7071:
7057:
7038:
7037:
6997:
6973:
6964:
6955:
6948:
6939:
6907:
6881:
6880:
6862:
6822:
6808:
6793:
6792:
6774:
6741:
6727:
6706:
6703:
6668:
6662:
6610:
6604:
6575:
6569:
6536:
6512:
6506:
6446:
6422:
6416:
6391:
6363:
6316:
6292:
6267:
6246:Katsumi Nomizu
6239:indecomposable
6224:
6200:
6172:
6159:
6158:
6145:
6140:
6135:
6130:
6125:
6099:
6096:
6075:torsion tensor
6001:
5998:
5977:
5972:
5968:
5963:
5959:
5955:
5951:
5923:
5920:
5917:
5916:
5904:
5879:
5864:
5861:
5858:
5846:
5843:
5840:
5836:
5833:
5822:
5808:
5804:
5793:
5789:
5788:
5775:
5771:
5766:
5752:
5749:
5746:
5734:
5731:
5728:
5724:
5721:
5718:
5715:
5704:
5690:
5686:
5675:
5671:
5670:
5657:
5653:
5648:
5638:isomorphic to
5625:
5621:
5616:
5604:
5601:
5598:
5586:
5583:
5580:
5576:
5573:
5569:
5566:
5563:
5560:
5556:
5553:
5542:
5528:
5524:
5513:
5509:
5508:
5495:
5491:
5487:
5483:
5479:
5475:
5471:
5462:isomorphic to
5449:
5445:
5441:
5437:
5433:
5429:
5425:
5414:
5411:
5408:
5396:
5393:
5390:
5386:
5383:
5379:
5374:
5370:
5359:
5345:
5341:
5330:
5326:
5325:
5312:
5308:
5304:
5300:
5296:
5292:
5288:
5275:
5272:
5269:
5257:
5253:
5250:
5247:
5243:
5240:
5236:
5232:
5229:
5226:
5222:
5219:
5216:
5213:
5202:
5188:
5184:
5173:
5169:
5168:
5155:
5151:
5147:
5143:
5139:
5135:
5131:
5122:isomorphic to
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5074:
5071:
5068:
5056:
5053:
5050:
5046:
5043:
5039:
5034:
5030:
5019:
5005:
5001:
4990:
4986:
4985:
4973:
4969:
4965:
4942:
4938:
4934:
4930:
4926:
4922:
4918:
4905:
4902:
4899:
4887:
4884:
4881:
4877:
4874:
4870:
4867:
4864:
4861:
4857:
4854:
4843:
4829:
4825:
4814:
4810:
4809:
4807:
4804:
4801:
4789:
4786:
4783:
4780:
4776:
4772:
4769:
4766:
4762:
4759:
4748:
4734:
4730:
4719:
4715:
4714:
4701:
4696:
4693:
4669:
4665:
4661:
4657:
4653:
4649:
4645:
4634:
4631:
4628:
4614:
4610:
4599:
4585:
4581:
4570:
4566:
4565:
4552:
4548:
4544:
4540:
4536:
4532:
4528:
4514:
4511:
4508:
4496:
4493:
4490:
4486:
4483:
4479:
4476:
4473:
4470:
4466:
4463:
4452:
4438:
4434:
4423:
4419:
4418:
4405:
4401:
4397:
4393:
4389:
4385:
4381:
4359:
4355:
4351:
4347:
4343:
4339:
4335:
4324:
4321:
4318:
4306:
4303:
4300:
4296:
4293:
4289:
4286:
4283:
4280:
4276:
4273:
4262:
4248:
4244:
4233:
4229:
4228:
4226:
4223:
4220:
4208:
4205:
4202:
4199:
4195:
4191:
4188:
4185:
4181:
4178:
4167:
4153:
4149:
4138:
4134:
4133:
4120:
4117:
4114:
4109:
4089:
4078:
4075:
4072:
4069:
4066:
4063:
4053:
4042:
4039:
4036:
4026:
4014:
4011:
4008:
4004:
4001:
3997:
3994:
3991:
3988:
3984:
3981:
3970:
3958:
3955:
3952:
3949:
3946:
3942:
3939:
3928:
3924:
3923:
3909:
3904:
3891:
3880:
3870:
3859:
3856:
3853:
3850:
3847:
3844:
3834:
3822:
3819:
3816:
3812:
3801:
3789:
3786:
3783:
3779:
3776:
3765:
3761:
3760:
3747:
3744:
3739:
3726:
3715:
3712:
3708:
3704:
3701:
3691:
3680:
3677:
3674:
3671:
3668:
3665:
3655:
3643:
3640:
3637:
3633:
3622:
3610:
3607:
3604:
3601:
3597:
3594:
3583:
3579:
3578:
3565:
3562:
3559:
3554:
3534:
3523:
3520:
3517:
3514:
3511:
3508:
3498:
3487:
3484:
3474:
3462:
3459:
3456:
3452:
3449:
3445:
3442:
3439:
3436:
3432:
3429:
3418:
3406:
3403:
3400:
3397:
3394:
3390:
3387:
3376:
3372:
3371:
3358:
3355:
3352:
3347:
3327:
3316:
3313:
3310:
3307:
3304:
3301:
3291:
3280:
3277:
3274:
3264:
3252:
3249:
3246:
3243:
3239:
3235:
3232:
3229:
3226:
3222:
3218:
3214:
3203:
3191:
3188:
3185:
3182:
3179:
3175:
3172:
3161:
3157:
3156:
3142:
3139:
3134:
3121:
3110:
3107:
3104:
3094:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3043:
3031:
3028:
3025:
3021:
3018:
3007:
2995:
2992:
2989:
2986:
2982:
2979:
2968:
2964:
2963:
2949:
2944:
2931:
2920:
2917:
2914:
2904:
2893:
2889:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2848:
2836:
2833:
2830:
2826:
2823:
2812:
2800:
2797:
2794:
2790:
2787:
2776:
2772:
2771:
2768:
2765:
2762:
2757:
2752:
2726:
2723:
2702:that contains
2684:
2677:
2670:
2663:
2656:
2572:
2571:
2561:
2547:
2536:Euclidean type
2516:
2513:
2450:Main article:
2447:
2444:
2402:
2390:
2384:
2367:
2358:
2340:
2339:
2328:
2325:
2321:
2318:
2312:
2309:
2305:
2301:
2298:
2295:
2292:
2289:
2285:
2282:
2277:
2274:
2271:
2268:
2265:
2260:
2256:
2194:, we obtain a
2154:To summarize,
2140:
2134:
2130:
2124:
2120:
2116:
2094:
2090:
2066:
2065:
2052:
2048:
2044:
2041:
2038:
2033:
2029:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
1979:
1967:
1950:
1927:isotropy group
1878:
1875:
1860:
1836:
1808:
1781:
1770:brackets into
1757:
1729:
1705:
1681:
1657:
1633:
1609:
1596:
1595:
1584:
1579:
1574:
1571:
1566:
1561:
1556:
1551:
1547:
1542:
1537:
1534:
1529:
1524:
1519:
1514:
1510:
1505:
1500:
1497:
1492:
1487:
1482:
1477:
1463:
1462:
1449:
1444:
1439:
1434:
1429:
1415:decomposition
1398:
1370:
1338:
1302:
1270:
1269:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1214:
1210:
1188:
1133:
1130:
1110:
1107:
1104:
1101:
1098:
1095:
1075:Every compact
1049:
1046:
999:if and only if
986:
983:
969:is said to be
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
865:
862:
859:
856:
853:
850:
806:
803:
741:
736:
732:
634:
633:
631:
630:
623:
616:
608:
605:
604:
603:
602:
597:
589:
588:
584:
583:
578:
576:Harish-Chandra
573:
568:
563:
558:
553:
551:Henri Poincaré
548:
542:
541:
538:
537:
534:
533:
529:
528:
523:
518:
513:
507:
506:
501:Lie groups in
500:
499:
496:
495:
491:
490:
485:
480:
475:
470:
465:
459:
458:
453:
452:
449:
448:
444:
443:
438:
433:
431:
430:
425:
419:
417:
415:
414:
409:
403:
401:
396:
390:
389:
384:
383:
380:
379:
375:
374:
369:
364:
359:
357:
356:
351:
345:
343:
338:
333:
327:
326:
321:
320:
317:
316:
312:
311:
306:
301:
299:Diffeomorphism
296:
291:
286:
281:
275:
274:
269:
268:
265:
264:
259:
258:
257:
256:
252:
248:
244:
240:
236:
232:
228:
224:
220:
213:
212:
208:
207:
206:
205:
199:
195:
189:
185:
179:
175:
169:
162:
161:
155:
150:
149:
146:
145:
141:
140:
130:
120:
110:
100:
90:
83:Special linear
80:
73:General linear
69:
68:
63:
62:
59:
58:
50:
49:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
11081:
11070:
11067:
11065:
11062:
11060:
11057:
11055:
11052:
11051:
11049:
11039:
11033:
11029:
11024:
11020:
11015:
11011:
11007:
11003:
11002:Selberg, Atle
10999:
10996:
10992:
10988:
10984:
10980:
10976:
10972:
10968:
10964:
10960:
10955:
10951:
10946:
10942:
10940:0-471-15732-5
10936:
10932:
10931:
10925:
10921:
10917:
10914:(1): 79–106.
10913:
10909:
10902:
10897:
10894:
10892:0-12-338301-3
10888:
10883:
10882:
10875:
10871:
10869:0-12-338460-5
10865:
10861:
10856:
10853:
10847:
10843:
10838:
10834:
10829:
10825:
10821:
10816:
10812:
10807:
10803:
10799:
10795:
10791:
10788:
10786:0-8218-0288-7
10782:
10778:
10774:
10773:Borel, Armand
10770:
10766:
10764:0-387-15279-2
10760:
10755:
10754:
10748:
10744:
10740:
10735:
10732:(2): 85–177,
10731:
10727:
10723:
10719:
10716:
10710:
10706:
10701:
10698:
10694:
10690:
10686:
10682:
10677:
10676:
10670:
10664:
10661:
10654:
10650:
10647:
10645:
10642:
10640:
10637:
10635:
10632:
10631:
10627:
10625:
10623:
10619:
10615:
10609:
10601:
10599:
10597:
10593:
10589:
10585:
10581:
10577:
10573:
10569:
10565:
10561:
10557:
10552:
10550:
10546:
10540:
10532:
10530:
10526:
10520:
10516:
10512:
10508:
10503:
10501:
10495:
10487:
10485:
10483:
10479:
10473:
10465:
10460:
10458:
10455:
10453:
10449:
10445:
10422:
10408:
10405:
10394:
10390:
10386:
10381:
10378:
10375:
10368:
10354:
10351:
10340:
10336:
10332:
10327:
10321:
10318:
10315:
10305:
10304:
10303:
10265:
10262:
10259:
10236:
10233:
10225:
10213:
10208:
10189:
10188:
10187:
10168:
10156:
10153:
10150:
10145:
10133:
10119:
10118:
10117:
10071:
10067:
10063:
10058:
10054:
10045:
10041:
10037:
10029:
10025:
10021:
10011:
10007:
10000:
9992:
9988:
9984:
9979:
9975:
9968:
9965:
9957:
9953:
9949:
9944:
9940:
9931:
9927:
9923:
9915:
9911:
9907:
9897:
9893:
9882:
9881:
9880:
9861:
9857:
9851:
9847:
9843:
9833:
9829:
9821:
9820:
9819:
9779:
9775:
9771:
9768:
9765:
9760:
9756:
9747:
9728:
9725:
9722:
9716:
9669:
9666:
9663:
9634:
9631:
9628:
9622:
9619:
9607:
9603:
9600:
9590:
9589:
9588:
9568:
9561:
9492:Factorization
9491:
9489:
9463:
9460:
9457:
9434:
9416:
9397:
9394:
9391:
9388:
9385:
9382:
9379:
9373:
9370:
9367:
9361:
9358:
9355:
9349:
9329:
9324:
9320:
9297:
9289:
9286:
9283:
9241:
9227:
9224:
9221:
9218:
9209:
9206:
9203:
9197:
9194:
9180:
9177:
9172:
9168:
9164:
9161:
9158:
9155:
9147:
9139:
9136:
9133:
9124:
9119:
9104:
9101:
9098:
9088:
9087:
9086:
9084:
9080:
9076:
9072:
9071:metric tensor
9064:
9062:
9056:
9054:
9052:
9048:
9043:
9041:
9037:
9033:
9025:
9021:
9017:
9013:
9009:
9006:
9002:
8999:
8998:
8997:
8995:
8991:
8987:
8983:
8979:
8975:
8971:
8967:
8962:
8960:
8956:
8952:
8948:
8944:
8943:Gelfand pairs
8940:
8936:
8932:
8928:
8924:
8920:
8916:
8912:
8908:
8904:
8900:
8896:
8892:
8888:
8884:
8880:
8876:
8872:
8868:
8865:In the 1950s
8862:
8854:
8847:
8827:
8824:
8812:
8801:
8795:
8794:
8782:
8770:
8767:
8755:
8748:
8742:
8741:
8729:
8722:
8719:
8713:
8712:
8708:
8696:
8693:
8685:
8682:
8676:
8675:
8649:
8647:)×Sp(1)
8646:
8636:
8628:
8619:
8611:
8609:× SO(2)
8600:
8594:
8593:
8571:
8568:
8564:
8550:
8537:
8530:
8524:
8523:
8501:
8499:)×Sp(1)
8498:
8488:
8480:
8477:
8467:
8453:
8446:
8440:
8439:
8436:× SO(2)
8427:
8420:
8413:
8410:
8404:
8403:
8399:
8387:
8384:
8380:
8372:
8369:
8361:
8358:
8352:
8351:
8336:
8329:
8327:)×Sp(1)
8326:
8318:
8311:
8301:
8295:
8294:
8283:
8281:)×SO(2)
8280:
8270:
8267:
8253:
8246:
8239:
8233:
8232:
8221:
8219:)×SO(2)
8218:
8204:
8201:
8187:
8184:
8170:
8163:
8157:
8156:
8145:
8138:
8136:)×SU(2)
8135:
8125:
8121:
8113:
8110:
8096:
8089:
8083:
8082:
8071:
8064:
8057:
8050:
8047:
8041:
8040:
8029:
8026:
8022:
8014:
8011:
8007:
7999:
7996:
7988:
7985:
7979:
7978:
7967:
7964:
7958:
7955:
7949:
7946:
7941:
7938:
7937:
7930:
7927:
7918:
7914:
7907:
7904:
7898:
7895:
7890:
7887:
7886:
7879:
7876:
7870:
7867:
7864:
7859:
7856:
7855:
7851:
7844:
7841:
7838:
7834:
7827:
7824:
7821:
7815:
7814:
7806:
7803:)× SL(2,
7802:
7795:
7792:
7786:
7783:
7778:
7775:
7774:
7767:
7764:
7761:
7756:
7753:
7752:
7748:
7745:)× SL(2,
7744:
7737:
7734:
7731:
7726:
7723:
7722:
7719:
7715:
7710:
7700:
7696:
7692:
7687:
7684: =
7683:
7680: +
7679:
7675:
7671:
7667:
7664:
7661:
7657:
7653:
7647:
7643:
7639:
7635:
7631:
7628:
7625:
7621:
7617:
7614:
7613:
7609:
7605:
7601:
7597:
7591:
7587:
7583:
7580:
7577:
7573:
7569:
7565:
7558:
7554:
7549:
7545:
7540:
7536:
7529:
7525:
7522:
7520:
7517: =
7516:
7513: +
7512:
7508:
7504:
7500:
7497:
7496:
7492:
7488:
7484:
7481:
7479:
7476: =
7475:
7472: +
7471:
7467:
7463:
7459:
7455:
7451:
7448:
7445:
7441:
7437:
7434:
7433:
7425:
7421:
7417:
7411:
7407:
7403:
7399:
7395:
7392:
7389:
7385:
7381:
7375:
7371:
7367:
7363:
7359:
7356:
7353:
7349:
7345:
7341:
7338:
7337:
7333:
7329:
7325:
7321:
7315:
7311:
7307:
7303:
7299:
7296:
7293:
7289:
7285:
7281:
7274:
7270:
7265:
7261:
7256:
7252:
7247:
7243:
7239:
7236:
7233:
7229:
7225:
7222:
7221:
7217:
7213:
7209:
7205:
7202:
7200:
7196:
7192:
7188:
7184:
7180:
7176:
7172:
7169:
7166:
7162:
7158:
7155:
7154:
7147:
7143:
7139:
7135:
7131:
7127:
7123:
7119:
7116:
7113:
7109:
7105:
7099:
7095:
7091:
7087:
7083:
7079:
7075:
7072:
7069:
7065:
7061:
7058:
7055:
7051:
7047:
7043:
7040:
7039:
7035:
7031:
7027:
7023:
7017:
7013:
7009:
7005:
7001:
6998:
6995:
6991:
6987:
6983:
6976:
6972:
6967:
6963:
6958:
6954:
6947:
6943:
6940:
6937:
6933:
6929:
6925:
6919:
6915:
6911:
6908:
6906:
6902:
6898:
6894:
6890:
6886:
6883:
6882:
6878:
6874:
6870:
6866:
6863:
6860:
6856:
6852:
6848:
6842:
6838:
6834:
6830:
6826:
6823:
6820:
6816:
6812:
6809:
6806:
6802:
6798:
6795:
6794:
6790:
6786:
6782:
6778:
6775:
6773:
6769:
6765:
6761:
6757:
6753:
6749:
6745:
6742:
6739:
6735:
6731:
6728:
6725:
6721:
6717:
6714:
6713:
6710:
6704:
6702:
6700:
6696:
6692:
6688:
6684:
6666:
6649:
6645:
6641:
6636:
6634:
6630:
6626:
6608:
6591:
6573:
6556:
6552:
6510:
6492:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6420:
6379:
6351:
6347:
6343:
6338:
6336:
6332:
6255:
6251:
6247:
6242:
6240:
6188:
6138:
6128:
6114:
6113:
6112:
6110:
6106:
6097:
6095:
6092:
6088:
6084:
6080:
6077:vanishes) on
6076:
6072:
6068:
6063:
6059:
6055:
6049:
6047:
6043:
6039:
6035:
6031:
6027:
6023:
6019:
6015:
6011:
6007:
5999:
5997:
5995:
5991:
5975:
5970:
5957:
5941:
5937:
5933:
5929:
5921:
5894:
5869:
5865:
5862:
5859:
5841:
5823:
5806:
5802:
5794:
5791:
5790:
5773:
5769:
5756:
5753:
5750:
5747:
5729:
5705:
5688:
5684:
5676:
5673:
5672:
5655:
5651:
5623:
5619:
5605:
5602:
5599:
5581:
5567:
5561:
5543:
5526:
5522:
5514:
5511:
5510:
5493:
5489:
5477:
5447:
5443:
5431:
5415:
5412:
5409:
5391:
5377:
5372:
5368:
5360:
5343:
5339:
5331:
5328:
5327:
5310:
5306:
5294:
5279:
5276:
5273:
5270:
5241:
5234:
5227:
5203:
5186:
5182:
5174:
5171:
5170:
5153:
5149:
5137:
5107:
5103:
5091:
5075:
5072:
5069:
5051:
5037:
5032:
5028:
5020:
5003:
4999:
4991:
4988:
4987:
4967:
4940:
4936:
4924:
4909:
4906:
4903:
4900:
4882:
4868:
4862:
4844:
4827:
4823:
4815:
4812:
4811:
4808:
4805:
4802:
4784:
4781:
4774:
4767:
4749:
4732:
4728:
4720:
4717:
4716:
4699:
4682:isometric to
4667:
4663:
4651:
4635:
4632:
4629:
4612:
4608:
4600:
4583:
4579:
4571:
4568:
4567:
4550:
4546:
4534:
4519:
4516:Complexified
4515:
4512:
4509:
4491:
4477:
4471:
4453:
4436:
4432:
4424:
4421:
4420:
4403:
4399:
4387:
4372:isometric to
4357:
4353:
4341:
4325:
4322:
4319:
4301:
4287:
4281:
4263:
4246:
4242:
4234:
4231:
4230:
4227:
4224:
4221:
4203:
4200:
4193:
4186:
4168:
4151:
4147:
4139:
4136:
4135:
4118:
4115:
4112:
4097:
4093:
4090:
4073:
4070:
4067:
4054:
4040:
4037:
4034:
4027:
4009:
3995:
3989:
3971:
3953:
3950:
3947:
3929:
3926:
3925:
3907:
3892:
3878:
3871:
3854:
3851:
3848:
3842:
3835:
3817:
3802:
3784:
3766:
3763:
3762:
3745:
3742:
3727:
3710:
3706:
3702:
3692:
3675:
3672:
3669:
3663:
3656:
3638:
3623:
3605:
3602:
3584:
3581:
3580:
3563:
3560:
3557:
3542:
3538:
3535:
3518:
3515:
3512:
3499:
3485:
3482:
3475:
3457:
3443:
3437:
3419:
3401:
3398:
3395:
3377:
3374:
3373:
3356:
3353:
3350:
3335:
3331:
3328:
3311:
3308:
3305:
3292:
3278:
3275:
3272:
3265:
3244:
3233:
3227:
3204:
3186:
3183:
3180:
3162:
3159:
3158:
3140:
3137:
3122:
3108:
3105:
3102:
3095:
3078:
3075:
3072:
3069:
3060:
3057:
3054:
3044:
3026:
3008:
2990:
2987:
2969:
2966:
2965:
2947:
2932:
2918:
2915:
2912:
2905:
2891:
2887:
2880:
2877:
2874:
2865:
2862:
2859:
2831:
2813:
2795:
2777:
2774:
2773:
2769:
2766:
2763:
2761:
2758:
2756:
2753:
2750:
2749:
2746:
2744:
2740:
2736:
2732:
2724:
2722:
2719:
2717:
2713:
2709:
2705:
2701:
2697:
2693:
2688:
2683:
2676:
2669:
2662:
2655:
2652:
2648:
2644:
2640:
2635:
2633:
2632:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2598:
2593:
2591:
2586:
2584:
2579:
2577:
2569:
2565:
2562:
2559:
2555:
2551:
2548:
2545:
2541:
2537:
2534:
2533:
2532:
2530:
2525:
2522:
2514:
2512:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2461:
2459:
2453:
2445:
2443:
2441:
2437:
2433:
2429:
2425:
2421:
2416:
2414:
2410:
2405:
2401:
2397:
2393:
2387:
2383:
2379:
2375:
2370:
2366:
2361:
2357:
2353:
2349:
2345:
2326:
2319:
2316:
2310:
2307:
2303:
2296:
2293:
2287:
2283:
2280:
2275:
2272:
2266:
2263:
2258:
2254:
2246:
2245:
2244:
2243:) and define
2242:
2238:
2234:
2229:
2225:
2220:
2216:
2213:To show that
2211:
2209:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2152:
2138:
2132:
2122:
2118:
2092:
2088:
2079:
2075:
2071:
2050:
2046:
2042:
2039:
2036:
2031:
2027:
2020:
2017:
2014:
2008:
2005:
2002:
1995:
1994:
1993:
1991:
1987:
1982:
1978:
1974:
1970:
1965:
1961:
1957:
1953:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1876:
1874:
1824:
1795:
1745:
1582:
1572:
1559:
1545:
1535:
1522:
1508:
1498:
1485:
1468:
1467:
1466:
1442:
1432:
1418:
1417:
1416:
1414:
1386:
1358:
1354:
1326:
1322:
1318:
1290:
1286:
1281:
1279:
1275:
1256:
1250:
1247:
1241:
1235:
1232:
1229:
1226:
1223:
1217:
1212:
1208:
1200:
1199:
1198:
1196:
1191:
1186:
1182:
1178:
1174:
1170:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1131:
1129:
1127:
1122:
1105:
1102:
1099:
1093:
1085:
1080:
1078:
1073:
1071:
1067:
1063:
1059:
1055:
1047:
1045:
1041:
1039:
1035:
1031:
1027:
1023:
1019:
1014:
1012:
1008:
1004:
1000:
996:
993:implies that
992:
984:
982:
980:
976:
972:
968:
964:
962:
958:
954:
950:
946:
945:tangent space
942:
938:
922:
916:
913:
907:
904:
895:
889:
883:
863:
860:
854:
848:
840:
836:
832:
828:
824:
820:
816:
812:
804:
802:
800:
799:Marcel Berger
795:
793:
789:
785:
781:
777:
774:
770:
766:
761:
759:
755:
739:
734:
730:
721:
717:
713:
709:
705:
701:
695:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
629:
624:
622:
617:
615:
610:
609:
607:
606:
601:
598:
596:
593:
592:
591:
590:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
543:
536:
535:
527:
524:
522:
519:
517:
514:
512:
509:
508:
504:
498:
497:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
460:
456:
451:
450:
442:
439:
437:
434:
429:
426:
424:
421:
420:
418:
413:
410:
408:
405:
404:
402:
400:
397:
395:
392:
391:
387:
382:
381:
373:
370:
368:
365:
363:
360:
355:
352:
350:
347:
346:
344:
342:
339:
337:
334:
332:
329:
328:
324:
319:
318:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
276:
272:
267:
266:
255:
249:
247:
241:
239:
233:
231:
225:
223:
217:
216:
215:
214:
209:
204:
202:
196:
194:
192:
186:
184:
182:
176:
174:
172:
166:
165:
164:
163:
158:
153:
148:
147:
138:
134:
131:
128:
124:
121:
118:
114:
111:
108:
104:
101:
98:
94:
91:
88:
84:
81:
78:
74:
71:
70:
66:
61:
60:
56:
52:
51:
48:
44:
40:
35:
27:
19:
11027:
11018:
11009:
11005:
10981:(1): 33–65,
10978:
10974:
10958:
10949:
10929:
10911:
10907:
10880:
10859:
10841:
10823:
10819:
10801:
10797:
10794:Cartan, Élie
10776:
10752:
10729:
10725:
10704:
10688:
10684:
10668:
10663:
10611:
10595:
10591:
10587:
10583:
10579:
10575:
10563:
10559:
10555:
10553:
10544:
10542:
10524:
10518:
10514:
10510:
10506:
10504:
10497:
10475:
10456:
10447:
10443:
10441:
10251:
10185:
10091:
9878:
9745:
9652:
9495:
9415:Killing form
9272:
9083:Killing form
9078:
9074:
9068:
9060:
9044:
9039:
9035:
9031:
9029:
9023:
9019:
9015:
9011:
9004:
9000:
8996:, such that
8993:
8989:
8985:
8981:
8977:
8973:
8969:
8965:
8963:
8958:
8954:
8950:
8934:
8930:
8926:
8922:
8918:
8914:
8910:
8906:
8902:
8898:
8894:
8890:
8889:normalising
8886:
8882:
8878:
8874:
8870:
8867:Atle Selberg
8864:
8845:
8822:
8810:×Sp(1)
8791:×SU(2)
8780:
8765:
8738:×Sp(1)
8706:
8691:
8658:×SO(2)
8644:
8634:
8617:
8590:×SO(2)
8580:×SO(2)
8566:
8565:)×SL(2,
8562:
8520:×SO(2)
8496:
8486:
8475:
8465:
8397:
8382:
8381:)×Sp(2,
8378:
8367:
8343:4(−20)
8324:
8278:
8265:
8216:
8199:
8182:
8133:
8123:
8122:)×SL(2,
8119:
8108:
8024:
8023:)×SO(2,
8020:
8009:
8008:)×SL(2,
8005:
7994:
7965:
7956:
7947:
7939:
7928:
7916:
7915:)×Sp(2,
7912:
7905:
7896:
7888:
7877:
7868:
7857:
7849:
7842:
7836:
7835:)×Sp(2,
7832:
7825:
7804:
7800:
7793:
7784:
7776:
7765:
7754:
7746:
7742:
7735:
7724:
7713:
7708:
7706:
7698:
7694:
7690:
7685:
7681:
7677:
7673:
7669:
7665:
7659:
7655:
7651:
7645:
7641:
7637:
7633:
7629:
7623:
7619:
7615:
7607:
7603:
7599:
7595:
7589:
7585:
7581:
7575:
7571:
7567:
7563:
7556:
7552:
7547:
7543:
7538:
7534:
7527:
7523:
7518:
7514:
7510:
7506:
7502:
7498:
7490:
7486:
7482:
7477:
7473:
7469:
7465:
7461:
7457:
7453:
7449:
7443:
7439:
7435:
7423:
7419:
7415:
7409:
7405:
7401:
7397:
7393:
7387:
7383:
7379:
7373:
7369:
7365:
7361:
7357:
7351:
7347:
7343:
7339:
7331:
7327:
7323:
7319:
7313:
7309:
7305:
7301:
7297:
7291:
7287:
7283:
7279:
7272:
7268:
7263:
7259:
7254:
7250:
7245:
7241:
7237:
7231:
7227:
7223:
7215:
7211:
7207:
7203:
7198:
7194:
7190:
7186:
7182:
7178:
7174:
7170:
7164:
7160:
7156:
7145:
7141:
7137:
7133:
7129:
7125:
7121:
7117:
7111:
7107:
7103:
7097:
7093:
7089:
7085:
7081:
7077:
7073:
7067:
7063:
7059:
7053:
7049:
7045:
7041:
7033:
7029:
7025:
7021:
7015:
7011:
7007:
7003:
6999:
6993:
6989:
6985:
6981:
6974:
6970:
6965:
6961:
6956:
6952:
6945:
6941:
6935:
6931:
6927:
6923:
6917:
6913:
6909:
6904:
6900:
6896:
6892:
6888:
6884:
6876:
6872:
6868:
6864:
6858:
6854:
6850:
6846:
6840:
6836:
6832:
6828:
6824:
6818:
6814:
6810:
6804:
6800:
6796:
6788:
6784:
6780:
6776:
6771:
6767:
6763:
6759:
6755:
6751:
6747:
6743:
6737:
6733:
6729:
6723:
6719:
6715:
6708:
6694:
6690:
6686:
6682:
6647:
6643:
6639:
6637:
6632:
6628:
6624:
6589:
6554:
6493:
6488:
6484:
6480:
6476:
6472:
6468:
6464:
6460:
6377:
6349:
6345:
6341:
6339:
6334:
6330:
6253:
6249:
6243:
6160:
6108:
6104:
6101:
6078:
6066:
6061:
6057:
6053:
6050:
6045:
6041:
6021:
6017:
6013:
6005:
6003:
5993:
5989:
5925:
4095:
4092:Grassmannian
3540:
3537:Grassmannian
3333:
3330:Grassmannian
2759:
2754:
2742:
2738:
2734:
2730:
2728:
2720:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2689:
2681:
2674:
2667:
2660:
2653:
2646:
2642:
2638:
2636:
2630:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2594:
2589:
2587:
2582:
2580:
2575:
2573:
2567:
2563:
2553:
2550:Compact type
2549:
2539:
2535:
2528:
2526:
2520:
2518:
2508:
2500:
2496:
2492:
2484:
2480:
2476:
2472:
2468:
2464:
2462:
2455:
2439:
2435:
2431:
2427:
2423:
2419:
2417:
2412:
2408:
2403:
2399:
2395:
2391:
2385:
2381:
2377:
2373:
2368:
2364:
2359:
2355:
2351:
2347:
2343:
2341:
2240:
2236:
2232:
2231:(a coset of
2227:
2223:
2218:
2214:
2212:
2207:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2153:
2077:
2074:automorphism
2067:
1989:
1985:
1980:
1976:
1972:
1968:
1963:
1955:
1951:
1946:
1942:
1938:
1934:
1930:
1925:denotes the
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1890:
1886:
1882:
1880:
1822:
1796:
1597:
1464:
1384:
1356:
1352:
1324:
1320:
1316:
1288:
1284:
1282:
1277:
1273:
1271:
1194:
1189:
1184:
1180:
1176:
1172:
1168:
1161:
1157:
1153:
1149:
1145:
1137:
1135:
1123:
1081:
1074:
1051:
1042:
1037:
1033:
1029:
1025:
1017:
1015:
994:
988:
978:
974:
970:
966:
965:
960:
956:
952:
948:
940:
936:
838:
834:
830:
826:
822:
818:
814:
810:
808:
796:
794:is compact.
791:
787:
779:
775:
768:
764:
762:
719:
715:
711:
707:
703:
699:
696:
662:contains an
643:
637:
581:Armand Borel
566:Hermann Weyl
367:Loop algebra
349:Killing form
323:Lie algebras
200:
190:
180:
170:
136:
126:
116:
106:
96:
86:
76:
47:Lie algebras
26:
10826:: 114–134,
10804:: 214–216,
10618:loop spaces
9447:this makes
9051:Wolf (2007)
8844:×SL(2,
8779:×SL(2,
8705:×Sp(2,
8396:×SO(2,
7640:)×Sp(2
7460:)×Sp(2
6553:involution
6329:is simple,
3332:of complex
2521:irreducible
2458:Élie Cartan
1022:homogeneous
817:a point of
640:mathematics
561:Élie Cartan
407:Root system
211:Exceptional
11064:Lie groups
11048:Categories
10967:Nomizu, K.
10961:, Benjamin
10952:, Benjamin
10655:References
10482:7 families
9057:Properties
8913:such that
7542:)×Sp(
7258:)×SO(
7181:)×SO(
7084:)×GL(
6835:)×GL(
6754:)×GL(
6551:antilinear
2764:Dimension
2507:, because
2072:Lie group
2070:involutive
1992:, the map
1901:(that is,
1413:direct sum
1166:involution
1084:lens space
1070:Lie groups
959:to all of
784:involution
676:Lie theory
660:isometries
546:Sophus Lie
539:Scientists
412:Weyl group
133:Symplectic
93:Orthogonal
43:Lie groups
10391:λ
10379:⋯
10337:λ
10325:⟩
10322:⋅
10316:⋅
10313:⟨
10263:≠
10157:⊕
10154:⋯
10151:⊕
10077:⟩
10051:⟨
10042:λ
10035:⟩
10017:#
10004:⟨
9963:⟩
9937:⟨
9928:λ
9921:⟩
9903:#
9890:⟨
9848:λ
9839:#
9769:…
9729:⋅
9723:⋅
9673:⟩
9670:⋅
9664:⋅
9661:⟨
9617:⟩
9612:#
9598:⟨
9573:#
9565:↦
9535:→
9468:⟩
9464:⋅
9458:⋅
9455:⟨
9395:
9389:∘
9383:
9374:
9294:⟩
9290:⋅
9284:⋅
9281:⟨
9249:otherwise
9228:∈
9195:−
9181:≅
9165:∈
9144:⟩
9131:⟨
9109:⟩
9096:⟨
6960:)×U(
6213:. Since
6139:⊕
6083:curvature
5958:⊗
5568:⋅
5478:⊗
5432:⊗
5378:⋅
5295:⊗
5242:±
5138:⊗
5092:⊗
5038:⋅
4968:⊗
4925:⊗
4869:⋅
4782:±
4652:⊗
4535:⊗
4478:⋅
4388:⊗
4342:⊗
4288:⋅
4201:±
3996:×
3673:−
3444:×
3234:×
3106:−
3058:−
2916:−
2863:−
2308:−
2297:σ
2291:↦
2270:→
2123:σ
2093:σ
2043:∘
2037:∘
2024:↦
2012:→
2003:σ
1895:Lie group
1573:⊂
1536:⊂
1499:⊂
1443:⊕
1236:σ
1227:∈
1213:σ
1144:. Then a
1142:Lie group
1036:(because
914:−
908:γ
890:γ
849:γ
773:Lie group
423:Real form
309:Euclidean
160:Classical
10969:(1954),
10775:(2001),
10749:(1987),
10691:: 3–24,
10628:See also
8720:–
8683:–
8411:–
8359:–
8277:or Sk(5,
8048:–
7986:–
7865:–
7822:–
7762:–
7732:–
6588:, while
6467:, where
6087:parallel
6016:, i.e.,
2629:rank of
2320:′
2284:′
2235:, where
1921:, where
1383:. Since
1272:Because
1175:). Thus
1048:Examples
1011:complete
754:complete
672:holonomy
654:) whose
595:Glossary
289:Poincaré
11012:: 47–87
10995:2372398
10612:In the
9554:taking
9413:is the
6348:×
5938:, or a
2398:. Thus
2350:fixing
1171:in Aut(
1058:spheres
718:fixing
503:physics
284:Lorentz
113:Unitary
11034:
10993:
10937:
10889:
10866:
10848:
10783:
10761:
10711:
10616:, the
9653:where
9432:
9342:, and
9273:Here,
9018:sends
9003:fixes
8842:7(−25)
8838:8(−24)
8830:8(−24)
8819:8(−24)
8815:8(−24)
8804:8(−24)
8798:8(−24)
8783:) or E
8666:6(−26)
8662:7(−25)
8656:6(−14)
8652:7(−25)
8641:7(−25)
8631:7(−25)
8624:7(−25)
8614:7(−25)
8603:7(−25)
8597:7(−25)
8588:6(−14)
8339:6(−26)
8332:6(−26)
8321:6(−26)
8314:6(−26)
8304:6(−26)
8298:6(−26)
8290:4(−20)
8286:6(−14)
8273:6(−14)
8262:6(−14)
8256:6(−14)
8249:6(−14)
8242:6(−14)
8236:6(−14)
7969:4(−20)
7960:4(−20)
7951:4(−20)
7943:4(−20)
7594:or Sp(
7562:or Sp(
7318:or SO(
7278:or SO(
7136:even,
7020:or SU(
6980:or SU(
6922:or SU(
6705:Tables
6185:is an
6081:whose
5172:EVIII
2751:Label
2645:), Sp(
2641:), SU(
2611:×
2342:where
2068:is an
1670:is an
1082:Every
1064:, and
680:Cartan
279:Circle
10991:JSTOR
10904:(PDF)
10570:on a
10186:with
9818:with
9371:trace
9030:When
8834:7(−5)
8789:7(−5)
8584:7(−5)
8574:7(−5)
8559:7(−5)
8553:7(−5)
8546:7(−5)
8540:7(−5)
8533:7(−5)
8527:7(−5)
7412:even
7404:/2),
7376:even
7368:/2),
7354:even
7316:even
7308:/2),
7218:even
7128:/2),
7100:even
7056:even
7044:= SL(
7018:even
7010:/2),
6887:= SU(
6879:even
6861:even
6799:= SL(
6791:even
6718:= SL(
6697:is a
6483:with
6404:. If
4989:EVII
4955:over
4422:EIII
3582:DIII
3160:AIII
2767:Rank
2467:let (
1960:1-jet
1893:is a
1465:with
1183:with
876:then
656:group
646:is a
354:Index
11032:ISBN
10935:ISBN
10887:ISBN
10864:ISBN
10846:ISBN
10781:ISBN
10759:ISBN
10709:ISBN
10448:i.e.
10444:e.g.
10252:for
9709:and
9069:The
9022:to −
8992:and
8921:and
8785:8(8)
8777:7(7)
8773:8(8)
8762:8(8)
8758:8(8)
8751:8(8)
8745:8(8)
8660:or E
8639:or E
8622:or E
8582:or E
8578:6(2)
8557:or E
8544:or E
8518:6(2)
8514:7(7)
8512:or E
8508:6(6)
8504:7(7)
8493:7(7)
8491:or E
8483:7(7)
8472:7(7)
8470:or E
8462:7(7)
8460:or E
8456:7(7)
8449:7(7)
8443:7(7)
8260:or E
8228:4(4)
8224:6(2)
8213:6(2)
8211:or E
8207:6(2)
8196:6(2)
8194:or E
8190:6(2)
8179:6(2)
8177:or E
8173:6(2)
8166:6(2)
8160:6(2)
8152:4(4)
8148:6(6)
8141:6(6)
8130:6(6)
8128:or E
8116:6(6)
8105:6(6)
8103:or E
8099:6(6)
8092:6(6)
8086:6(6)
7932:4(4)
7923:4(4)
7921:or F
7909:4(4)
7900:4(4)
7892:4(4)
7797:2(2)
7788:2(2)
7780:2(2)
7226:=SO(
7159:=SO(
7092:)),
6762:)),
6685:and
6036:and
5992:and
5674:FII
5410:112
5329:EIX
5271:128
4813:EVI
4569:EIV
4232:EII
3927:CII
3375:BDI
2967:AII
2741:and
2615:and
2576:rank
2376:) =
1949:on T
1193:and
1187:= id
1148:for
1136:Let
989:The
809:Let
690:and
642:, a
304:Loop
45:and
10983:doi
10916:doi
10912:350
10828:doi
10806:doi
10734:doi
10693:doi
10527:= 2
9794:of
9587:as
9014:at
8984:of
8976:at
8968:in
8953:on
8949:of
8909:in
8901:in
7716:= 0
7689:or
7676:),
7650:or
7509:),
7468:),
7422:/4,
7414:or
7400:/2,
7386:/2,
7378:or
7364:/2,
7350:),
7346:/2,
7304:/2,
7214:),
7210:/2,
7189:),
7124:/2,
7110:/2,
7102:or
7088:/2,
7080:/2,
7066:/2,
7052:),
7048:/2,
7006:/2,
6978:))
6895:),
6875:),
6857:),
6853:/2,
6845:or
6843:))
6787:),
6693:or
6557:of
6189:of
6085:is
5748:16
5600:28
5512:FI
5070:54
4901:64
4803:70
4718:EV
4630:26
4510:32
4320:40
4222:42
4137:EI
4062:min
3764:CI
3507:min
3300:min
2775:AI
2607:is
2588:B.
2581:A.
2202:on
1988:at
1937:at
1933:on
1919:G/K
1909:of
1881:If
1718:in
1351:of
1315:of
947:of
939:at
786:of
710:of
658:of
638:In
135:Sp(
125:SU(
105:SO(
85:SL(
75:GL(
11050::
11010:20
11008:,
10989:,
10979:76
10977:,
10973:,
10910:.
10906:.
10824:55
10822:,
10802:54
10800:,
10730:74
10728:,
10687:,
10551:.
10484:.
9392:ad
9380:ad
9053:.
9038:,
8927:σx
8925:=
8923:sy
8919:σy
8917:=
8915:sx
8897:,
8848:)
8825:)
8768:)
8709:)
8694:)
8637:)
8620:)
8569:)
8489:)
8478:)
8400:)
8385:)
8370:)
8268:)
8202:)
8185:)
8126:)
8111:)
8027:)
8012:)
7997:)
7852:)
7839:)
7807:)
7749:)
7718:.
7714:kl
7701:)
7662:)
7648:)
7626:)
7610:)
7592:)
7578:)
7560:)
7493:)
7446:)
7426:)
7390:)
7334:)
7294:)
7276:)
7234:)
7197:=
7193:+
7167:)
7144:=
7140:+
7114:)
7070:)
7036:)
6996:)
6938:)
6920:)
6903:=
6899:+
6821:)
6807:)
6770:=
6766:+
6740:)
6726:)
6635:.
6344:=
6056:=
6032:,
5863:2
5860:8
5792:G
5751:1
5603:4
5413:4
5274:8
5228:16
5073:3
4904:4
4863:12
4806:7
4633:2
4513:2
4472:10
4323:4
4225:6
2718:.
2687:.
2680:,
2673:,
2666:,
2659:,
2634:.
2566::
2552::
2538::
2479:,
2475:,
2471:,
2442:.
2434:,
2430:,
2426:,
2239:∈
2228:hK
2226:=
2210:.
2184:eK
1913:,
1794:.
1287:,
1128:.
1060:,
1056:,
1009:,
981:.
963:.
788:G.
760:.
702:,
694:.
686:,
115:U(
95:O(
10985::
10922:.
10918::
10830::
10808::
10736::
10695::
10689:4
10596:q
10592:p
10588:q
10584:p
10580:q
10576:p
10564:K
10560:K
10556:G
10545:M
10525:p
10519:i
10515:K
10511:K
10507:G
10423:d
10417:m
10409:|
10406:B
10395:d
10387:1
10382:+
10376:+
10369:1
10363:m
10355:|
10352:B
10341:1
10333:1
10328:=
10319:,
10288:g
10266:j
10260:i
10237:0
10234:=
10231:]
10226:j
10220:m
10214:,
10209:i
10203:m
10197:[
10169:d
10163:m
10146:1
10140:m
10134:=
10129:m
10102:m
10072:i
10068:Y
10064:,
10059:j
10055:Y
10046:j
10038:=
10030:i
10026:Y
10022:,
10012:j
10008:Y
10001:=
9998:)
9993:j
9989:Y
9985:,
9980:i
9976:Y
9972:(
9969:B
9966:=
9958:j
9954:Y
9950:,
9945:i
9941:Y
9932:i
9924:=
9916:j
9912:Y
9908:,
9898:i
9894:Y
9862:i
9858:Y
9852:i
9844:=
9834:i
9830:Y
9804:m
9780:n
9776:Y
9772:,
9766:,
9761:1
9757:Y
9732:)
9726:,
9720:(
9717:B
9695:m
9667:,
9638:)
9635:Y
9632:,
9629:X
9626:(
9623:B
9620:=
9608:Y
9604:,
9601:X
9569:Y
9562:Y
9540:m
9530:m
9506:m
9473:g
9461:,
9435:;
9427:h
9401:)
9398:Y
9386:X
9377:(
9368:=
9365:)
9362:Y
9359:,
9356:X
9353:(
9350:B
9330:M
9325:p
9321:T
9298:p
9287:,
9242:0
9233:h
9225:Y
9222:,
9219:X
9213:)
9210:Y
9207:,
9204:X
9201:(
9198:B
9186:m
9178:M
9173:p
9169:T
9162:Y
9159:,
9156:X
9148:p
9140:Y
9137:,
9134:X
9125:{
9120:=
9114:g
9105:Y
9102:,
9099:X
9079:G
9075:M
9040:M
9036:X
9032:s
9026:.
9024:X
9020:X
9016:x
9012:s
9007:;
9005:x
9001:s
8994:X
8990:x
8986:M
8982:s
8978:x
8974:X
8970:M
8966:x
8959:M
8957:(
8955:L
8951:G
8935:G
8931:σ
8911:G
8907:s
8903:M
8899:y
8895:x
8891:G
8887:σ
8883:G
8879:M
8846:R
8828:E
8823:H
8813:E
8808:7
8802:E
8796:E
8781:R
8771:E
8766:H
8756:E
8749:E
8743:E
8736:7
8732:8
8730:E
8725:8
8723:E
8716:8
8714:E
8707:C
8703:7
8699:8
8697:E
8692:C
8688:8
8686:E
8679:8
8677:E
8650:E
8645:H
8635:R
8629:E
8618:H
8612:E
8607:6
8601:E
8595:E
8572:E
8567:R
8563:H
8551:E
8538:E
8531:E
8525:E
8502:E
8497:H
8487:R
8481:E
8476:H
8468:)
8466:R
8454:E
8447:E
8441:E
8434:6
8430:7
8428:E
8423:7
8421:E
8416:7
8414:E
8407:7
8405:E
8398:C
8394:6
8390:7
8388:E
8383:C
8379:C
8375:7
8373:E
8368:C
8364:7
8362:E
8355:7
8353:E
8337:E
8330:E
8325:H
8319:E
8312:E
8308:4
8302:E
8296:E
8284:E
8279:H
8271:E
8266:R
8254:E
8247:E
8240:E
8234:E
8222:E
8217:H
8205:E
8200:R
8188:E
8183:R
8171:E
8164:E
8158:E
8146:E
8139:E
8134:H
8124:R
8120:R
8114:E
8109:R
8097:E
8090:E
8084:E
8078:4
8074:6
8072:E
8067:6
8065:E
8060:6
8058:E
8053:6
8051:E
8044:6
8042:E
8036:4
8032:6
8030:E
8025:C
8021:C
8017:6
8015:E
8010:C
8006:C
8002:6
8000:E
7995:C
7991:6
7989:E
7982:6
7980:E
7966:F
7957:F
7948:F
7940:F
7929:F
7919:)
7917:R
7913:R
7906:F
7897:F
7889:F
7881:4
7878:F
7872:4
7869:F
7861:4
7858:F
7850:C
7846:4
7843:F
7837:C
7833:C
7829:4
7826:F
7818:4
7816:F
7805:R
7801:R
7794:G
7785:G
7777:G
7769:2
7766:G
7758:2
7755:G
7747:C
7743:C
7739:2
7736:G
7728:2
7725:G
7709:σ
7699:R
7697:,
7695:n
7691:G
7686:n
7682:ℓ
7678:k
7674:ℓ
7672:,
7670:k
7666:G
7660:C
7658:,
7656:n
7652:G
7646:R
7644:,
7642:l
7638:R
7636:,
7634:k
7630:G
7624:R
7622:,
7620:n
7616:G
7608:H
7606:,
7604:p
7600:p
7598:,
7596:p
7590:q
7588:,
7586:p
7582:G
7576:C
7574:,
7572:n
7568:n
7566:,
7564:n
7557:q
7553:ℓ
7551:,
7548:p
7544:ℓ
7539:q
7535:k
7533:,
7531:p
7528:k
7524:G
7519:n
7515:q
7511:p
7507:q
7505:,
7503:p
7499:G
7491:C
7489:,
7487:n
7483:G
7478:n
7474:ℓ
7470:k
7466:C
7464:,
7462:ℓ
7458:C
7456:,
7454:k
7450:G
7444:C
7442:,
7440:n
7436:G
7424:H
7420:n
7416:G
7410:ℓ
7408:,
7406:k
7402:ℓ
7398:k
7394:G
7388:C
7384:n
7380:G
7374:ℓ
7372:,
7370:k
7366:ℓ
7362:k
7358:G
7352:n
7348:H
7344:n
7340:G
7332:R
7330:,
7328:n
7324:n
7322:,
7320:n
7314:q
7312:,
7310:p
7306:q
7302:p
7298:G
7292:C
7290:,
7288:n
7284:n
7282:,
7280:n
7273:q
7269:l
7267:,
7264:p
7260:ℓ
7255:q
7251:k
7249:,
7246:p
7242:k
7238:G
7232:q
7230:,
7228:p
7224:G
7216:n
7212:C
7208:n
7204:G
7199:n
7195:ℓ
7191:k
7187:C
7185:,
7183:ℓ
7179:C
7177:,
7175:k
7171:G
7165:C
7163:,
7161:n
7157:G
7146:n
7142:ℓ
7138:k
7134:ℓ
7132:,
7130:k
7126:ℓ
7122:k
7118:G
7112:C
7108:n
7104:G
7098:ℓ
7096:,
7094:k
7090:H
7086:ℓ
7082:H
7078:k
7074:G
7068:H
7064:n
7060:G
7054:n
7050:H
7046:n
7042:G
7034:R
7032:,
7030:p
7026:p
7024:,
7022:p
7016:q
7014:,
7012:p
7008:q
7004:p
7000:G
6994:C
6992:,
6990:p
6986:p
6984:,
6982:p
6975:q
6971:l
6969:,
6966:p
6962:l
6957:q
6953:k
6951:,
6949:p
6946:k
6942:G
6936:H
6934:,
6932:p
6928:p
6926:,
6924:p
6918:q
6916:,
6914:p
6910:G
6905:n
6901:q
6897:p
6893:q
6891:,
6889:p
6885:G
6877:n
6873:R
6871:,
6869:n
6865:G
6859:n
6855:C
6851:n
6847:G
6841:R
6839:,
6837:l
6833:R
6831:,
6829:k
6825:G
6819:l
6817:,
6815:k
6811:G
6805:R
6803:,
6801:n
6797:G
6789:n
6785:C
6783:,
6781:n
6777:G
6772:n
6768:ℓ
6764:k
6760:C
6758:,
6756:ℓ
6752:C
6750:,
6748:k
6744:G
6738:C
6736:,
6734:n
6730:G
6724:C
6722:,
6720:n
6716:G
6695:τ
6691:σ
6687:τ
6683:σ
6667:c
6661:g
6648:τ
6644:τ
6642:∘
6640:σ
6633:τ
6631:∘
6629:σ
6625:τ
6609:c
6603:g
6590:σ
6574:c
6568:g
6555:τ
6535:g
6511:c
6505:g
6489:K
6485:G
6481:K
6477:G
6473:G
6469:H
6465:H
6461:G
6445:g
6421:c
6415:g
6390:g
6378:σ
6362:g
6350:H
6346:H
6342:G
6335:H
6331:G
6315:g
6291:g
6266:g
6254:H
6250:G
6223:h
6199:h
6171:m
6144:m
6134:h
6129:=
6124:g
6109:H
6105:G
6079:M
6067:G
6062:H
6058:G
6054:M
6046:n
6042:n
6022:n
6018:n
5994:B
5990:A
5976:,
5971:n
5967:)
5962:B
5954:A
5950:(
5903:H
5878:O
5845:)
5842:4
5839:(
5835:O
5832:S
5807:2
5803:G
5774:2
5770:P
5765:O
5733:)
5730:9
5727:(
5723:n
5720:i
5717:p
5714:S
5689:4
5685:F
5656:2
5652:P
5647:H
5624:2
5620:P
5615:O
5585:)
5582:2
5579:(
5575:U
5572:S
5565:)
5562:3
5559:(
5555:p
5552:S
5527:4
5523:F
5494:2
5490:P
5486:)
5482:O
5474:H
5470:(
5448:2
5444:P
5440:)
5436:O
5428:O
5424:(
5395:)
5392:2
5389:(
5385:U
5382:S
5373:7
5369:E
5344:8
5340:E
5311:2
5307:P
5303:)
5299:O
5291:O
5287:(
5256:}
5252:l
5249:o
5246:v
5239:{
5235:/
5231:)
5225:(
5221:n
5218:i
5215:p
5212:S
5187:8
5183:E
5154:2
5150:P
5146:)
5142:O
5134:C
5130:(
5108:2
5104:P
5100:)
5096:O
5088:H
5084:(
5055:)
5052:2
5049:(
5045:O
5042:S
5033:6
5029:E
5004:7
5000:E
4972:O
4964:H
4941:2
4937:P
4933:)
4929:O
4921:H
4917:(
4886:)
4883:2
4880:(
4876:U
4873:S
4866:)
4860:(
4856:O
4853:S
4828:7
4824:E
4788:}
4785:I
4779:{
4775:/
4771:)
4768:8
4765:(
4761:U
4758:S
4733:7
4729:E
4700:2
4695:P
4692:O
4668:2
4664:P
4660:)
4656:O
4648:C
4644:(
4613:4
4609:F
4584:6
4580:E
4551:2
4547:P
4543:)
4539:O
4531:C
4527:(
4495:)
4492:2
4489:(
4485:O
4482:S
4475:)
4469:(
4465:O
4462:S
4437:6
4433:E
4404:2
4400:P
4396:)
4392:H
4384:C
4380:(
4358:2
4354:P
4350:)
4346:O
4338:C
4334:(
4305:)
4302:2
4299:(
4295:U
4292:S
4285:)
4282:6
4279:(
4275:U
4272:S
4247:6
4243:E
4207:}
4204:I
4198:{
4194:/
4190:)
4187:4
4184:(
4180:p
4177:S
4152:6
4148:E
4119:q
4116:+
4113:p
4108:H
4096:p
4077:)
4074:q
4071:,
4068:p
4065:(
4041:q
4038:p
4035:4
4013:)
4010:q
4007:(
4003:p
4000:S
3993:)
3990:p
3987:(
3983:p
3980:S
3957:)
3954:q
3951:+
3948:p
3945:(
3941:p
3938:S
3908:n
3903:H
3879:n
3858:)
3855:1
3852:+
3849:n
3846:(
3843:n
3821:)
3818:n
3815:(
3811:U
3788:)
3785:n
3782:(
3778:p
3775:S
3746:n
3743:2
3738:R
3714:]
3711:2
3707:/
3703:n
3700:[
3679:)
3676:1
3670:n
3667:(
3664:n
3642:)
3639:n
3636:(
3632:U
3609:)
3606:n
3603:2
3600:(
3596:O
3593:S
3564:q
3561:+
3558:p
3553:R
3541:p
3522:)
3519:q
3516:,
3513:p
3510:(
3486:q
3483:p
3461:)
3458:q
3455:(
3451:O
3448:S
3441:)
3438:p
3435:(
3431:O
3428:S
3405:)
3402:q
3399:+
3396:p
3393:(
3389:O
3386:S
3357:q
3354:+
3351:p
3346:C
3334:p
3315:)
3312:q
3309:,
3306:p
3303:(
3279:q
3276:p
3273:2
3251:)
3248:)
3245:q
3242:(
3238:U
3231:)
3228:p
3225:(
3221:U
3217:(
3213:S
3190:)
3187:q
3184:+
3181:p
3178:(
3174:U
3171:S
3141:n
3138:2
3133:C
3109:1
3103:n
3082:)
3079:1
3076:+
3073:n
3070:2
3067:(
3064:)
3061:1
3055:n
3052:(
3030:)
3027:n
3024:(
3020:p
3017:S
2994:)
2991:n
2988:2
2985:(
2981:U
2978:S
2948:n
2943:C
2919:1
2913:n
2892:2
2888:/
2884:)
2881:2
2878:+
2875:n
2872:(
2869:)
2866:1
2860:n
2857:(
2835:)
2832:n
2829:(
2825:O
2822:S
2799:)
2796:n
2793:(
2789:U
2786:S
2760:K
2755:G
2743:K
2739:G
2735:K
2731:G
2716:G
2712:G
2708:G
2704:K
2700:G
2696:K
2692:G
2685:2
2682:G
2678:4
2675:F
2671:8
2668:E
2664:7
2661:E
2657:6
2654:E
2647:n
2643:n
2639:n
2631:G
2625:K
2621:G
2617:K
2613:M
2609:M
2605:G
2601:M
2590:G
2583:G
2568:M
2560:.
2554:M
2546:.
2540:M
2529:M
2509:G
2501:K
2497:M
2493:G
2485:M
2481:g
2477:σ
2473:K
2469:G
2465:M
2440:M
2436:g
2432:σ
2428:K
2424:G
2420:M
2413:M
2409:p
2404:p
2400:s
2396:M
2392:p
2386:p
2382:s
2378:p
2374:p
2372:(
2369:p
2365:s
2360:p
2356:s
2352:K
2348:G
2344:σ
2327:K
2324:)
2317:h
2311:1
2304:h
2300:(
2294:h
2288:K
2281:h
2276:,
2273:M
2267:M
2264::
2259:p
2255:s
2241:G
2237:h
2233:K
2224:p
2219:K
2215:G
2208:K
2204:G
2200:g
2196:G
2192:G
2188:K
2180:K
2176:G
2172:K
2168:K
2164:K
2160:G
2156:M
2139:,
2133:o
2129:)
2119:G
2115:(
2089:G
2078:K
2051:p
2047:s
2040:h
2032:p
2028:s
2021:h
2018:,
2015:G
2009:G
2006::
1990:p
1986:M
1981:p
1977:s
1973:M
1969:p
1964:K
1956:M
1952:p
1947:K
1943:p
1939:p
1935:M
1931:G
1923:K
1915:M
1911:M
1907:p
1903:M
1899:M
1891:M
1887:G
1883:M
1859:m
1835:h
1823:σ
1807:g
1780:h
1756:m
1728:g
1704:h
1680:h
1656:m
1632:g
1608:h
1583:.
1578:h
1570:]
1565:m
1560:,
1555:m
1550:[
1546:,
1541:m
1533:]
1528:m
1523:,
1518:h
1513:[
1509:,
1504:h
1496:]
1491:h
1486:,
1481:h
1476:[
1448:m
1438:h
1433:=
1428:g
1397:g
1385:σ
1369:m
1357:G
1353:H
1337:h
1325:σ
1321:σ
1317:G
1301:g
1289:σ
1285:G
1278:G
1274:H
1257:.
1254:}
1251:g
1248:=
1245:)
1242:g
1239:(
1233::
1230:G
1224:g
1221:{
1218:=
1209:G
1195:H
1190:G
1185:σ
1181:G
1177:σ
1173:G
1169:σ
1162:H
1158:H
1154:G
1150:G
1138:G
1109:)
1106:1
1103:,
1100:2
1097:(
1094:L
1038:M
1034:M
1030:M
1026:M
1018:M
995:M
979:M
967:M
961:M
957:p
953:f
949:p
941:p
937:f
923:.
920:)
917:t
911:(
905:=
902:)
899:)
896:t
893:(
887:(
884:f
864:p
861:=
858:)
855:0
852:(
839:γ
835:p
827:p
823:f
819:M
815:p
811:M
792:H
780:H
776:G
769:H
765:G
740:M
735:p
731:T
720:p
716:M
712:M
708:p
704:g
700:M
698:(
627:e
620:t
613:v
253:8
251:E
245:7
243:E
237:6
235:E
229:4
227:F
221:2
219:G
201:n
198:D
191:n
188:C
181:n
178:B
171:n
168:A
139:)
137:n
129:)
127:n
119:)
117:n
109:)
107:n
99:)
97:n
89:)
87:n
79:)
77:n
36:.
20:)
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